Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines [1 ed.] 0323886639, 9780323886635

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Structural Mechanics and Design of Metal Pipes A Systematic Approach for Onshore and Offshore Pipelines

Spyros A. Karamanos Professor of Structural Mechanics, University of Thessaly, Volos, Greece

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2023 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-323-88663-5 For Information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Matthew Deans Acquisitions Editor: Dennis McGonagle Editorial Project Manager: Aera F. Gariguez Production Project Manager: Kamesh R Cover Designer: Miles Hitchen Typeset by Aptara, New Delhi, India

Dedication

To my father, Anthony S. Karamanos

Foreword

Since the development of the first industrial facilities, metal pipes have been employed as basic components to facilitate the operation of machinery, mainly used to contain steam under pressure and as means of transporting water and other liquids depending on the application. In more modern years, metal pipes are used for transportation and storage of traditional energy resources, mainly in the Oil & Gas and Nuclear sectors, as well as for the transportation of water and basic chemical and petrochemical products. As we are already going through the era of “Energy Transition,” pipes remain key components for transporting energy resources. Further to traditional onshore and offshore Oil & Gas applications where pipes are dominant elements, to achieve the “Net Zero” targets in the following years’ pipes will be used in even more demanding applications, for example, in high-pressure/high-temperature applications, for the transportation of hydrogen and ammonia, and in carbon capture & storage applications. This book is a complete guide that describes the basic principles used for the analysis of onshore and offshore pipe response under basic and more elaborate loading conditions. Starting from the first principles, the theoretical formulation of the mechanical problem and the basic equations for each case are presented in detail. Insightful discussion on the structural response under various conditions is also provided, allowing for the reader to understand in depth the key features of pipe response and design. The traditional methods as well as the “state-of-the-art” in the analysis and design of pipes and pipelines are discussed while the limitations and benefits that each analysis approach offers are explained. The book is built on the knowledge and experience of Professor Karamanos gained over many years of scientific and professional work, R&D projects, and numerous publications which resulted in significant contributions to this field, a combination that is hard to find in any other textbook. Its structure can serve engineers of all levels. It can be a useful reference document of theoretical and applied knowledge for students in the field who would like to gain a good understanding of pipe analysis and design. It can be also used as a “go-by” for early and midcareer professionals who want a complete guidebook with reference to pipeline design codes and discussion of their provisions. Finally, it can also serve mature engineering professionals who are looking for a complete source of information and description of the “state-of-the-art.” It is very

xii

Foreword

fortunate that Professor Karamanos has accomplished to deliver such a complete book on this topic. Dr. George E. Varelis Principal Subsea Systems Engineer Innovation and Technology Lead Worley, UK

Preface

The origins of this book go back to 1989 when I started my graduate studies at The University of Texas at Austin (UT Austin) under the supervision of John Tassoulas. The first research project I worked at UT Austin was on the structural stability of deepwater pipelines under combined loading, sponsored by the newly established Offshore Technology Research Center. These were exciting times for the offshore industry, extending its deep-water activities in the Gulf of Mexico and elsewhere, and planning the construction of pipelines in water depths that exceeded 7000 feet (2000 m). John’s supervision and guidance of my research were unique toward developing state-of-theart computational methodologies and exploring the mechanical behavior of offshore pipelines. During my post-doc year at Delft University of Technology, in 1996, I continued my research on tubulars and tubular structures with Jaap Wardenier. Among other projects, I started my cooperation with Nol Gresnigt on onshore pipeline mechanics. Nol’s experience in pipeline mechanics and especially in large-scale laboratory testing has been exceptional. Our cooperation became very close when I started teaching at the University of Thessaly in 1999, and since then, we have cooperated in numerous projects. All those years, our research team had a strong interaction with Nol through common projects on issues related to pipeline mechanics and design, and this has been a great asset for us. Nol’s friendship, cooperation, and support continue until today, and I really appreciate his instructive advice and comments on several parts of the book. The book may be considered as a personal journey in the “world of pipelines,” which are the spearhead of metal tubulars. My initial intention was to include also topics related to piping components and systems, very common in industrial facilities, power plants, and terminals. However, this would increase the size of the book by a significant amount, and therefore, these topics will be part of a future publication. In the present book, the mechanics and design of pipes are tackled from a structural engineering point-of-view. Emphasis is given on pipe stress–strain analysis considered as a long cylindrical shell, and on buckling and structural instability under combined action, whereas topics related to material behavior, pipe fracture, or assessment of aged pipes are treated briefly and will be the subject of a future book. The contents of this book are divided into four parts. Part I is introductory and offers an overview of pipeline engineering in terms of pipe manufacturing, design, and construction (Chapters 1 and 2). The structural mechanics of pipes is treated extensively in Part II; Chapters 3 and 4 refer to elastic pipes, whereas in Chapters 5 and 6, the mechanics of metallic pipes is presented. Part III focuses on the structural design of onshore and offshore pipelines (Chapters 7 and 8), with reference to major pipeline specifications. It also comprises “strain-based design” against geohazards (Chapter 9).

xiv

Preface

Finally, special topics are presented in Part IV: large-diameter steel water transmission pipelines (Chapter 10), upheaval and lateral buckling (Chapter 11), and mechanicallylined pipes (Chapter 12). The book is addressed to both researchers and practicing engineers that wish to deepen their knowledge and understanding of pipe mechanics. It may be used as a reference book for researchers and graduate students working in the field of pipes and tubular structures. It may be also useful to practicing engineers in this field, as a complete guidebook for pipeline mechanical design, which offers extensive background to pipeline design standards, and discusses their relevant provisions. There are several ways to read and make use of this book. Below are some hints: r

r

r

r r

Part II (Chapters 3–6) contains the basics of pipe mechanics and the necessary background for understanding current design standards and specifications. Apart from their usefulness to practicing engineers, parts of those chapters can be incorporated in an elective or graduate course on structural mechanics or advanced mechanics of materials. Chapters 1, 2, 7, and/or 8 constitute a set of chapters suitable for an introductory course on pipeline mechanical design (onshore and/or offshore) addressed to practicing engineers. This set of chapters may also be addressed to researchers in this field for an overview of pipeline engineering. Chapter 9 on pipeline design against geohazards is a standalone chapter that introduces strainbased design to researchers and engineering professionals. It may complement the above introductory course on pipeline mechanical design. It may also be part of a graduate course on geohazard (or seismic) design of critical infrastructure systems. Chapter 10 is another standalone chapter, offering an overview of large-diameter steel pipes for water transmission. Chapters 3 and 4 explain in simple terms, the development of stress and strain in deforming elastic pipes, presenting elegant analytical solutions, and numerical simulations. They can be used as part of a structural mechanics course, or as an introduction to the mechanics of elastic tubes from soft/biological materials.

During my career, I had the chance to cooperate and interact with numerous individuals and groups that influenced my research on pipes and pipelines. While a graduate student at UT Austin, I met Stelios Kyriakides, a worldwide expert in the field. Stelios’ work and in particular his high-quality experiments have been inspiring for me and for my students, and his legacy is apparent in Chapters 5 and 6 of this book. In Thessaly, I had the opportunity to cooperate with Philip Perdikaris and Charis Papatheocharis in performing numerous laboratory experiments, which gave added value to the numerical models of our research team and improved our understanding on the structural behavior of pipe and tubular components. In addition, my longtime cooperation with Panos Dakoulas was essential for developing a strong and unique expertise in “pipelines and geohazards”. I would also like to thank all our partners at Centro Sviluppo Materiali, and, particularly, Giuseppe Demofonti and Elisabetta Mecozzi, for their cooperation in numerous European research projects, many of which refer to steel pipes and tubular structures. Many sincere thanks to Brent Keil and Rich Mielke, Northwest Pipe Company, and Bob Card, LAN, for our longtime cooperation in steel water pipelines, which is reflected in Chapter 10 of the present book. I am grateful to Corinth Pipeworks, and particularly to Thanasis Tazedakis,

Preface

xv

Chris Palagkas, Tim Dourdounis, and John Voudouris, for our longtime collaboration in many issues related to pipe manufacturing. In all those collaborations with research and industrial partners, the systematic and continuous administrative support of Ioanna Charalambous-Moisidou has been tremendous and indispensable. Finally, I would like to thank my colleagues in Thessaly and in Edinburgh for providing a fertile academic environment, necessary for writing this book. As a professor, I had the honor to supervise top-quality PhD students. I would like to thank all my PhD students; their excellent research has formed the foundation of the present book. I am proud that most of them are already well-established in the field of pipelines and energy infrastructure, in Greece or abroad. Special thanks go to George E. Varelis for writing the Foreword of the book and for his constructive input in upheaval and lateral buckling (Chapter 11). Many thanks also go to Aris G. Stamou and Apostolos Nasikas for their great help in the sections of collapse and local buckling (Chapters 3, 5, and 6); Ilias Gavriilidis for his input in lined pipe mechanics (Chapter 12); Daniel Vasilikis for providing information on confined cylinder buckling (Chapters 3 and 5); Polynikis Vazouras and Gregory Sarvanis for their strong input in the “strain-demand” section (Chapter 9); Aglaia Pournara for her work on buckled pipes (Chapter 9); Patricia Pappa for her assistance in issues related to pipeline construction (Chapter 1); Giannoula Chatzopoulou and Kostis Chatziioannou for their input in cyclic plasticity (Appendix D). I am indebted to AVAX S.A., George Tasakos and Foteini Marnari for providing several photos on onshore pipeline construction, including the left photo of the cover page. Many thanks go to Duane DeGeer, Intecsea, and Chris Timms, C-FER Technologies, for their valuable input and support. I am also grateful to Saipem S.p.A. and to Riccardo Castellani and Luigino Vitali, for providing photos from offshore pipeline installation, including the right photo on the cover page. It would be impossible to accomplish writing this book without the precious, continuous, and meticulous support of Kelly Georgiadi-Stefanidi. Throughout the writing process, Kelly has been my direct and closest assistant and has devoted tremendous efforts in managing and reviewing the manuscript, in organizing the figures and the references and in scrutinizing the proofs. Many thanks also go to Dennis McGonangle, Kamesh Ramajogi and Aera Gariguez, who coordinated this project on behalf of Elsevier. Finally, I would like to express my sincere gratitude to the Karamanos family: my wife Peny, my kids Ioanna and Tony, and my parents Anthony and Lily, for their continuous support and love. The endless hours I spent in the preparation of this book would have otherwise been spent with them. Spyros A. Karamanos Fall 2022 Volos, Greece

Contents

Foreword Preface

Part I

xi xiii

Introduction to Pipelines

1

1

Introduction to pipeline engineering 1.1 Historical note 1.2 Hydrocarbon pipeline projects 1.3 Introduction to hydrocarbon pipeline design and construction 1.4 Pipeline design considerations 1.5 Onshore pipeline construction 1.6 Offshore pipeline installation References

3 3 4 11 13 17 23 40

2

Line pipe manufacturing 2.1 General considerations and steel material properties 2.2 Pipes manufactured from steel coil 2.3 Pipes manufactured from steel plates 2.4 Seamless pipes References

43 43 47 51 55 57

Part II 3

4

Pipe Mechanics

Structural mechanics of elastic rings 3.1 Ring stresses under internal pressure 3.2 Ring buckling under external pressure 3.3 Confined ring deformation and buckling under uniform external pressure 3.4 Ring under transverse loading References Structural mechanics of elastic cylinders 4.1 Stresses in circular cylinders under internal or external pressure 4.2 Structural stability equations of cylindrical shells 4.3 Buckling of elastic cylindrical shells under axial compression

59 61 61 63 80 88 92 95 95 100 106

viii

Contents

4.4

A note on post-buckling behavior of axially-compressed elastic cylinders 4.5 Buckling of elastic cylindrical shells under uniform external pressure 4.6 Uniform bending of an elastic tube 4.7 Uniform bending of an elastic tube in the presence of pressure 4.8 Buckling of an elastic tube under bending References

5

6

8

110 112 122 124 130

Mechanical behavior of metal pipes under internal and external pressure 5.1 A brief note on pipe response under internal pressure 5.2 External pressure collapse and post-buckling response 5.3 Factors influencing pipe collapse 5.4 Buckle propagation and arrest in long metal cylinders 5.5 Effect of tension on collapse and buckling propagation 5.6 Externally-pressurized cylinders under lateral confinement References

133 133 135 152 159 173 177 184

Metal pipes and tubes under structural loading 6.1 Metal pipe subjected to transverse loading 6.2 Uniform axial compression of a metal pipe 6.3 A note on constitutive modeling for buckling calculations 6.4 Bending of long metal pipes 6.5 Effect of internal pressure on bending response 6.6 Bending of externally pressurized pipes References

187 187 200 210 211 214 219 228

Part III 7

109

Pipeline Design

Basic onshore pipeline mechanical design 7.1 Brief introduction to pipeline standards, pipe sizes and pressure design 7.2 ASME B31.8 Gas transmission & distribution piping systems 7.3 ASME B31.4 Pipeline transportation systems for liquids and slurries 7.4 EN 1594 Gas supply systems – Pipelines for maximum operating pressure over 16 bar – Functional requirements References Offshore pipeline mechanical design 8.1 Offshore pipeline mechanical design framework 8.2 Mechanical design of offshore pipelines according to API 1111

231 233 233 236 245 248 250 251 251 253

Contents

ix

8.3

DNV-ST-F101 provisions for the mechanical design of offshore pipelines 8.4 A short note on buckle propagation and arrestor design 8.5 Discussion of API and DNV collapse formulae 8.6 Other formulae for predicting the collapse pressure of pipes and tubes 8.7 Discussion of collapse formulae and the slenderness approach 8.8 Effect of pipe manufacturing on the collapse pressure References

9

Pipeline analysis and design in geohazard areas 9.1 Aspects of strain-based design 9.2 Strain demand calculation (pipeline strain analysis) 9.3 Strain resistance verification of the pipeline 9.4 Outline of mitigation measures References

Part IV

Special Topics

257 265 266 271 277 279 283 287 287 290 307 316 317

321

10

Buried steel water pipeline design 10.1 Manufacturing of large diameter pipes and in-situ construction 10.2 Wall thickness determination 10.3 Ring deflection 10.4 External pressure collapse design 10.5 Steel pipeline joints 10.6 Longitudinal stress 10.7 Lap-welded joints under severe longitudinal action 10.8 Improving the structural performance of lap-welded joints References

323 323 330 332 334 337 338 340 348 353

11

Global buckling of pipelines 11.1 Driving force for global buckling 11.2 Upheaval buckling 11.3 Lateral buckling References

355 355 358 367 372

12

Mechanically lined pipes 12.1 Fabrication of lined pipes 12.2 A brief presentation of the TU Delft experiments 12.3 Structural response of lined pipes under monotonic bending 12.4 Effect of fabrication process on bending response of lined pipes 12.5 Structural response of lined pipes under cyclic loading References

375 375 379 382 392 400 411

x

Contents

A - Constitutive equations for linear elastic cylinders B - Column buckling C - Inelastic bending and the plastic hinge concept D - Metal plasticity fundamentals E - Geometry and equilibrium of plastic collapse mechanism F - End effects on internally pressurized thin-walled cylinders G - Stresses in thick-walled pressurized cylinders H - Glossary Instead of an Epilogue

413 417 427 441 467 471 475 481 489

Index

491

Part I Introduction to Pipelines 1. Introduction to pipeline engineering 3 2. Line pipe manufacturing 43

Introduction to pipeline engineering 1.1

1

Historical note

The history of hydrocarbon pipelines starts in early 19th century in London, UK, where the Westminster gas light company constructed gas pipes below public streets1,2 . The gas was burnt to light the streets of London using lamp posts (Miesner and Leffler, 2006). This concept was soon adopted by gas light companies in major US cities, using pipes made of lead or hollowed wooden logs. In mid-nineteenth century, hollowed logs were also used for transporting oil or gas from the production well to the nearest refinery, in a continuous manner. The transition from hollowed wooden logs and lead pipes to cast iron pipes provided more opportunities to the oil and gas industry. At the same time, the first steel tubes emerged, manufactured from steel sheets, rolled to a circular shape and lap or butt welded. In late 19th century, the invention of the roll piercing process by the Mannesmann brothers has been a milestone in steel pipe fabrication and its industrial production. In mid-twentieth century, the advancement of welding technology enabled the fabrication of welded pipes, opening new opportunities to the pipeline and piping industrial sector. The demands and requirements imposed by the offshore industrial sector have motivated significant developments in pipeline engineering. The discovery of large offshore fields in the 70’s, both in the Gulf of Mexico and in the North Sea, signaled the beginning of a new era in pipeline technology (Veldman and Lagers, 1997). The need to transport gas from the enormous gas reserves located in North Africa to European markets, motivated the construction of several offshore pipelines across the Mediterranean Sea in deep water. Furthermore, the exploitation of new offshore hydrocarbon reserves located in the North Sea, the Gulf of Mexico, the Persian Gulf, Brazil, West Africa, South-East Asia, West Australia, and recently in East Mediterranean, in increasingly deeper waters, combined with the stricter environmental requirements, have imposed new challenges for pipeline design and construction. Finally, the exploitation of huge oil and gas reserves in the Caspian Sea and the need to transport them in European markets, lead to the design and construction of large-diameter pipelines that cross the Caucasus mountains, Anatolia and South-East Europe, and are subjected to severe geohazard threats, imposing significant challenges for their structural integrity. 1 The

present chapter refers to hydrocarbon pipelines, which have been developed rather independently of water transmission pipelines. Steel water pipelines will be presented in Chapter 10 of this book. 2 It is also said that Chinese, several thousand years ago, used bamboo sealed with mud to transport natural gas. However, there exists very limited, if any, information on this issue.

Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00018-9 c 2023 Elsevier Inc. All rights reserved. Copyright 

4

1.2

Structural Mechanics and Design of Metal Pipes

Hydrocarbon pipeline projects

Major hydrocarbon pipeline projects require an investment of billions of Euros (or dollars) and long-term planning until their construction starts. Usually, these major pipelines are quite long, crossing different countries and continental borders. Therefore, in addition to financial aspects, geopolitical issues arising from tension and conflicts between neighboring countries, may be decisive for the design and completion of a major pipeline project. Furthermore, environmental and safety issues may affect the planning and the final decision for such a project. The following examples describe some major pipeline projects presenting their key technical features, as well as some important non-technical information. They refer to onshore pipeline projects in geohazard areas and to offshore pipeline projects, because they constitute two topics associated with unique design aspects, which are discussed extensively in the present book.

1.2.1 Onshore pipeline projects A large number of onshore pipelines are in operation and their design process has been well established. However, the construction of large-diameter hydrocarbon pipelines in earthquake-prone or geohazard areas, has imposed a number of challenges for their structural integrity. Two landmark pipeline projects are presented below. The first is the Baku-Tbilisi-Ceyhan (BTC) crude oil pipeline, and the second is the “Southern Gas Corridor” connecting Baku, Azerbaijan, with Lecce, Italy, which is composed by three consecutive pipelines: (a) the South Caucasus Pipeline expansion (SCPX); (b) the Trans-Anatolian Pipeline (TANAP); (c) the Trans-Adriatic Pipeline (TAP). Herein, TANAP and TAP pipelines are described in more detail (see also Table 1.1). Finally, a short mention to the Interconnector Greece-Bulgaria (IGB) pipeline as branch of the TAP pipeline is made.

1.2.1.1

Baku-Tbilisi-Ceyhan (BTC) pipeline

The Baku-Tbilisi-Ceyhan (BTC) pipeline transports crude oil from the Caspian Sea (Baku, Azerbaijan) to the Mediterranean Sea (Ceyhan, Turkey). It has been proposed as an alternative to crude oil transportation with tankers through the Black Sea and the strait of Bosporus. The highly congested Bosporus strait and the ensuing environmental issues have imposed a major drawback in the tanker transportation solution, and this was a decisive factor for the final decision for constructing the BTC pipeline (Güney and Gudmestad, 1999). The pipeline was commissioned in late 2005 and is designed to deliver up to one million barrels of crude oil per day from the Sangachal terminal near Baku, Azerbaijan, to Ceyhan, Turkey, in the Mediterranean through Tbilisi, Georgia. The total pipeline length is 1,760 km, of which 442 km are in Azerbaijan, 248 km in Georgia and 1060 km in Turkey. The first part of the BTC pipeline in Azerbaijan has a diameter of 42 inches (1,070 mm). The diameter size increases to 46 inches (1,170 mm) in its second part in the Caucasus mountains and in Georgia. Then, it reverts to 1,070 mm in Turkey, and reduces to 34 inches (865 mm) near its final destination in Ceyhan. The line pipe is

Introduction to pipeline engineering

5

Table 1.1 Three important onshore pipeline projects (BTC, TANAP, TAP) and a summary for their technical and operational details. Trans-Anatolian Pipeline (TANAP) Natural gas 1,841 km 16 × 109 m3 /year 2018 Turkey

Trans-Adriatic Pipeline (TAP) Natural gas 878 km 10 × 109 m3 /year 2020 Greece, Albania, Italy

Internal pressure

Baku-TbilisiCeyhan (BTC) Crude oil 1,768 km 160 × 103 m3 /day 2006 Azerbaijan, Georgia, Turkey 125 bar

95.5 bar

Pipe size

42 in, 46 in, 34 in

95 bar 145 bar (offshore) 48 in, 36 in (offshore)

56 in, 48 in and 2 × 36 in (offshore) X70 X70 70 m (Marmara Sea) 810 m (Adriatic Sea)

Type of content Length Capacity Commission Year Countries involved

Line pipe Steel Grade X65 Water depth (offshore n/a part) Special issues Seismic, Landslides Seismic, Anatolia fault crossing

Seismic, Landslides

API 5L steel grade X65, with thickness up to 25.8 mm depending on the diameter size and the location along its alignment. The main technical challenges of the BTC pipeline project are the highly seismic and geohazard areas crossed, and have motivated a significant amount of research. There exist numerous active seismic faults along the alignment in Azerbaijan, Georgia and Turkey (Hengesh et al., 2004). Furthermore, in several mountainous areas there is a high risk of landslide action due to slope instability. The design of BTC pipeline in those areas required the design and implementation of innovative technical solutions for mitigating those threats and has been a milestone in pipeline design practice against geohazards (Shilston et al., 2004).

1.2.1.2

Southern Gas Corridor

The Southern Gas Corridor is a European initiative for developing a natural gas supply route from the Caspian Sea and the Middle East to Europe, in an attempt to establish diverse sources of energy supply. The route from Azerbaijan to Europe starts from the Shah Deniz 2 Gas Field and consists of the South Caucasus Pipeline (SCPX), the Trans-Anatolian Pipeline (TANAP), and the Trans-Adriatic Pipeline (TAP), reaching its final destination in San Foca, near Lecce, Italy. In the following, the TANAP and TAP pipelines are briefly described.

1.2.1.2.1

TANAP pipeline

The Trans-Anatolian Natural Gas Pipeline (TANAP) Project is the second segment of the Southern Gas Corridor. It starts from the Turkish-Georgian border at Türkgözü/Posof/Ardahan where it connects to SCPX and ends at the Greek-Turkish border in ˙Ipsala/Edirne, where it connects to TAP pipeline. It includes a short offshore

6

Structural Mechanics and Design of Metal Pipes

part that crosses the Sea of Marmara. There are two off-take stations at Eski¸sehir and at East Thrace, which connect the pipeline with the local Turkish gas distribution system. Its maximum discharge is 16 × 109 m3 (570 × 109 ft3 ) of gas per year and was commissioned in 2018. The TANAP pipeline has total length 1,841 km and nominal capacity 31 × 109 m3 per year, in high-flow conditions. Its diameter size is 56 inches up to the Eski¸sehir Compressor Station and 48 inches from Eski¸sehir Compressor Station to the GreekTurkish border. The onshore length is 1,832 km, made of API 5L X70 line pipe, with design pressure equal to 95.5 bar. The 56-inch-diameter part of TANAP has three different wall thicknesses of 19.45 mm, 23.34 mm and 28.01 mm, depending on the location, whereas the part with the 48-inch-diameter has three thicknesses of 16.67 mm, 20.01 mm and 24.01 mm. The 18-km-long offshore section of TANAP crosses the Canakkale Strait (Dardanelles) in the Sea of Marmara. It consists of two 36-inch-diameter pipelines, made of API 5L X65 line pipe, with 22.9 mm wall thickness, installed at a maximum water depth of approximately 70 m. Seismically-induced geohazards exist along the entire alignment of TANAP pipeline, given the fact that Turkey represents one of the most active seismic countries in the planet. Seismic threats consist of strong ground shaking action, active tectonic faults, soil liquefaction and lateral spreading, and landslides, constituting severe threats for the structural integrity of the pipeline (Robl et al., 2020). In particular, TANAP crosses nine active faults including the North Anatolian Fault Zone (NAFZ) which is crossed twice. This is a notorious seismic fault: its surface rupture is associated with a maximum horizontal displacement of more than 7 m.

1.2.1.2.2

TAP pipeline

The Trans-Adriatic Pipeline (TAP) starts at the Greek-Turkish border at Kipoi, Evros, where it connects with TANAP gas pipeline. It passes through Greece and Albania, and after crossing the Adriatic Sea, it comes ashore in South Italy, at San Foca, near Lecce. The total length of TAP pipeline is 878 km, of which 550 km are in Greece, 215 km in Albania, 105 km are offshore in the Adriatic Sea, and the final 8 km are located in Italy. At its highest point the TAP pipeline rises up to 1,800 m in the Albanian mountains, and the offshore part is installed at a maximum water depth of 810 m. The pipeline was commissioned in 2020. The current capacity of TAP pipeline is 10 × 109 m3 (350 × 109 ft3 ) of natural gas per year, of which 8 × 109 m3 (280 × 109 ft3 ) are delivered to Italy, 1 × 109 m3 (35 × 109 ft3 ) to Greece, and 1 × 109 m3 (35 × 109 ft3 ) to Bulgaria, through the IGB (Gas Interconnector Greece-Bulgaria) pipeline. It is made of API 5L X70 48-inchdiameter line pipes, designed for internal pressure of 95 bar in the onshore section and API 5L X65 36-inch-diameter line pipes, designed for internal pressure of 145 bar on the offshore section. Geohazards exist along the TAP pipeline route, including several tectonic fault crossings and numerous soil liquefaction areas, associated with lateral spreading and buoyancy (Slejko et al., 2021). In addition, especially in the Albanian section of the

Introduction to pipeline engineering

7

Table 1.2 A list of landmark offshore pipeline projects.

Pipeline Franpipe Nord Stream Langeled Maghreb

Max. depth (m) 70 210 360 400

Diameter (in) 42 2 × 48 44 2 × 22

Length of offshore pipeline (km) 840 1,224 1,166 47

Time of commission 1998 2012 2006 1996

Trans-Med 610 pipeline Green Stream 1,150

3 × 20/2 × 26 155

1983

32

520

2004

Blue Stream Medgaz

2,150 2,160

2 × 24 24

396 210

2005 2010

Turk Stream 2,200 Perdido Norte 2,530

2 × 32 18

930 312

2020 2010

Location North Sea Baltic Sea North Sea Mediterranean Sea Mediterranean Sea Mediterranean Sea Black Sea Mediterranean Sea Black Sea Gulf of Mexico

pipeline, there exist several areas of potential landslide action, which constitute severe threats for TAP pipeline integrity (Marinos et al., 2019).

1.2.1.3

Interconnector Greece-Bulgaria (IGB) pipeline

The IGB pipeline is a branch of TAP pipeline interconnecting Komotini, Greece, with Stara Zagora, Bulgaria. It is a 32-inch-diameter, 182-km long pipeline (of which 31 km are in Greece), operating at 55 bar internal pressure. The IGB pipeline is designed for 3 × 109 m3 (105 × 109 ft3 ) annual capacity, which may be expanded up to 5 × 109 m3 per year. It acts as a strategic gas infrastructure providing diversification of gas supply to Bulgaria and to Southeast Europe gas market. Because of its reverse flow capability, it also improves Greece’s energy security.

1.2.2 Offshore pipeline projects The construction of offshore pipelines is a fascinating engineering process. Offshore technology has allowed the construction of deep-water pipelines in water depths that exceed 2,000 meters. Because of such large depths, the design of those pipelines is required to confront several challenges. Table 1.2 lists a few landmark offshore pipeline projects sorted by the corresponding water depth. Four of those pipeline projects are described in more detail below, together with the famous Oman-India Pipeline, and their technical characteristics are summarized in Table 1.3. A short note on the EastMed pipeline project is also made.

1.2.2.1 Blue Stream pipeline Blue Stream is a gas pipeline that transmits natural gas from Russia to Turkey crossing the Black Sea, bypassing several countries. It was commissioned in 2005, and at

8

Table 1.3 Landmark offshore gas pipeline projects. Blue Stream Natural gas 396 km 16 × 109 m3 /year 2005 Russia, Turkey 250 bar 609.6 mm × 31.8 mm

Medgaz Natural gas 210 km 10.5 × 109 m3 /year 2010 Algeria, Spain 220 bar 624 mm × 29.9 mm

Nord Stream Natural gas 1,224 km 55 × 109 m3 /year 2011 Russia, Germany 220 bar 1,220 mm × 41 mm

South (Turk) Stream Natural gas 910 km 31.5 × 109 m3 /year 2020 Russia, Turkey 300 bar 812.8 mm × 39 mm

Oman-India Natural gas 1140 km

X65 2,150 m

X70 2,160 m

SAWL 485 213 m

SAWL 450 2,200 m

X60 3,500 m

n/a Oman, India 660.4 mm × 41.3 mm

Structural Mechanics and Design of Metal Pipes

Type of content Length Capacity Commission Year Countries involved Internal pressure Diameter × thickness (max) Steel Grade Maximum water depth

Introduction to pipeline engineering

9

full capacity it is capable of conveying 16 × 109 m3 of natural gas from Russia to Turkey. The offshore part of Blue Stream pipeline connects Dzhubga, Russia with Samsun, Turkey. It is 396-km long, consisting of a pair of 24-inch outside diameter steel pipelines. At the time of its construction, Blue Stream was the deepest offshore pipeline project in the world, and it is still considered a landmark submarine pipeline project (maximum water depth of 2,150 m). The two pipelines are made of API 5L grade X65 line pipes, with maximum wall thickness of 31.8 mm, installed in deep waters using the J-lay method, and extensive testing was performed to qualify the collapse capacity of the line pipes (DeGeer, 2005). Furthermore, buckle arrestors with outside diameter equal to 652 mm and thickness equal to 52.7 mm are employed.

1.2.2.2

Medgaz pipeline

The Medgaz pipeline is a 210 km subsea pipeline between Beni Saf, Algeria and Almería, Spain, and was commissioned in 2010 at a maximum depth of 2,160 m. It is a 24-inch-diameter pipeline laid across the Mediterranean Sea with the capacity to carry 8 billion cubic meters per year of natural gas, but this capacity was extended to 10.5 billion cubic meters per year. This capacity is expected to double in a subsequent planned upgrade. The pipeline is made of API 5L grade X70 line pipe, with outer diameter 624 mm and thickness 29.9 mm. In its deepest part, the pipeline was installed with the J-lay method. Each buckle arrestor is 4 m long with outer diameter equal to 675 mm and wall thickness 55.6 mm. An extensive experimental program was conducted for determining the collapse strength of line pipes, including a series of full-scale collapse tests (DeGeer et al., 2007).

1.2.2.3 Nord Stream pipeline The Nord Stream Gas Pipeline (NSGP) project supplies Europe with natural gas from Russia through the Baltic Sea and Germany, and consists of a twin-pipeline system with a combined capacity of 55 billion cubic meters per year. The first pipeline was commissioned in 2011 and the second in 2012. The offshore part of Nord Stream is 1,224-km-long and connects Vyborg, near Leningrad, Russia to Lubmin, near Greifswald, Germany. The diameter of the pipe is 1,220 mm (48 in), made of SAWL 485 grade carbon steel line pipe (equivalent to API 5L X70), with wall thickness ranging from 26.8 mm to 41 mm. The pipeline is installed in maximum water depth of 213 m and the working pressure is 220 bar. Buckle arrestors are required at the deepest sections of the pipeline to avoid propagation buckling. The buckle arrestors are 12.2 m long pipe segments with 41 mm thickness. Due to uneven seabed, the formation of free spans is associated with significant local pipeline bending moments in the pipeline (Bruschi, 2012; Pettinelli et al., 2012), which have been mitigated by means of specialpurpose seabed intervention works (e.g., rock dumping).

1.2.2.4 South Stream (Turk Stream) pipeline The South Stream project was aimed at constructing a long, deep-sea pipeline, to transport natural gas from the Black Sea to Bulgaria and through Serbia, Hungary, Slovenia and further to Austria. However, the project was cancelled in 2014, seven

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years after its start. The Turk Stream pipeline project was announced in late 2014, replacing South Stream pipeline project. Turk Stream starts from the Russkaya, near Anapa, Russia, crosses the Black Sea, and terminates at Kıyıköy, in the European part of Turkey. In most of its part, the Turk Stream pipeline follows the South Stream alignment but deviates from it in the west part of the Black Sea, going southwest to Turkey instead of continuing west towards Bulgaria. The offshore section of Turk Stream is a 910-km-long natural gas pipeline, which crosses the Black Sea at depths of 2,200 m. It consists of two parallel pipelines running across the Black Sea, each having a diameter of 32 inches. The pipeline operates at 300 bar internal pressure, and it is made of SAWL 450 steel grade line pipe with 39 mm wall thickness to withstand the high external pressure at those depths. A substantial amount of collapse testing has been performed in support of Turk Stream pipeline construction (Timms et al., 2018). It is the first large-diameter offshore pipeline (with diameter larger than 30 inches) installed in water depth that exceeds 2,000 meters.

1.2.2.5 Oman-India pipeline The famous Oman-India Pipeline project (OIP) has been a landmark in offshore pipeline engineering. Its design was conducted in the early 90’s and refers to a 1140-km-long pipeline, with diameter sizes of 20 to 26 inches, designed for 3,500 meters of water depth. This extreme water depth imposed a number of significant technical challenges, including collapse resistance in ultra-deep-water conditions. This project is still considered as the state-of-the-art of offshore pipeline design. The OIP design had to confront numerous technical matters, such as the development of a qualified deep-water pipeline repair system, pipe mill upgrades necessary to manufacture the thick-walled line pipe, the upgrade of lay vessels with adequate tension capacity to enable the installation of pipes in 3,500 m water depth, and the mitigation of deep offshore geohazards, such as mudflows, seismic faults and slope failures (McKeehan, 1995). The OIP pipeline was not constructed, for geopolitical reasons, but it is still considered as a milestone in offshore pipeline engineering, with immense contributions to the “state-of-the-art” of deep-water pipeline design.

1.2.2.6

EastMed pipeline

The last offshore pipeline project mentioned in this brief introduction is the EastMed pipeline project. It refers to a 1,900-kilometer subsea pipeline aimed at delivering natural gas from the recently discovered gas fields of East Mediterranean Sea to European markets. The pre-FEED stage of the EastMed project has been completed in 2018, and the project is currently at the FEED stage. The proposed alignment crosses the East Mediterranean Sea from east to west. The first part of EastMed pipeline connects the gas fields in East Mediterranean, south of Cyprus, with the island of Crete, Greece. Subsequently it crosses the south part of the Aegean Sea, it becomes onshore in Greece and then, crossing the Adriatic Sea, it connects to Italy. Its construction has not started yet, and it is expected to be commissioned by 2025. Upon its construction, it will be the longest and deepest underwater pipeline in the

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world, to be installed in water depths that reach or even exceed 3,000 meters. Apart from its design against collapse in those water depths, the construction of EastMed is associated with numerous technical challenges from deep-water geohazards, including seismic actions, underwater landslides and mud flows. The pipeline, in its first stage of operation, is expected to deliver 10 × 109 m3 of natural gas per year.

1.3

Introduction to hydrocarbon pipeline design and construction

1.3.1 Initial steps of a pipeline project To start a hydrocarbon pipeline project the need for such a pipeline has to be established. At that stage, the following parties are necessary to agree on this project: (a) the hydrocarbon producer, (b) the hydrocarbon consumer (client) and (c) the investor of the project. Upon agreement, the interested parties form the “pipeline consortium”, which is the owner of the pipeline, and appoint the contractor and the project manager. At the initial stage, several alternatives for hydrocarbon transportation are considered, e.g., road transportation, railway, or tankers. In this process, the following features have to be taken into account, and make the pipeline an attractive solution for hydrocarbon transportation: r r r r

The pipeline constitutes the safest way for transporting energy resources, and exhibits the lowest rate of incidents, casualties etc. compared with other transportation means. It requires an important initial investment, but it is a cost-effective investment, with highest return of the invested capital. It has a lifespan of at least 40 years and requires relatively low maintenance cost. It is much less aggressive to the environment than other transportation means (road, rail, tanker).

Overall, the pipeline project should be economically feasible, and the final decision for such a project should be based on the expected rate of return of the invested capital.

1.3.2 Introduction to pipeline mechanical design The mechanical design of a pipeline, also called structural design, constitutes a major part of the pipeline engineering project, which aims at determining the pipe diameter and wall thickness, the steel grade, the method of pipe manufacturing, and the method of installation (mainly for offshore pipelines). The magnitude of hydrocarbon supply is the major parameter for determining the size of the pipeline. In addition, fluid containment properties (for either liquid or gas) determine the operating pressure and temperature and should be considered as the initial input to mechanical design. Pipeline design is performed in accordance with the codes and standards and other specifications imposed by the owner. In major onshore hydrocarbon pipeline projects,

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Structural Mechanics and Design of Metal Pipes

ASME standards are usually followed in many parts of the world, unless a complete set of national standards is available. In offshore pipelines, the DNV standards are mostly used, but API and ASME standards are also employed, especially in North America. The optimum pipeline alignment should be chosen in terms of topography, easy access, and geological issues (including seismic and other geohazards). In onshore pipelines, a survey of the installation site is usually commissioned, and the pipeline alignment is selected. The main loading condition is internal pressure, which is associated primarily with the development of hoop stresses in the pipe wall. However, additional sources of pipe wall stresses exist, such as temperature, soil settlements, geohazard and seismic actions, and pipeline crossings with railways and highways. The design of offshore pipelines is more challenging and depends on the water depth. Selecting the pipeline alignment is a critical task, which requires thorough underwater investigation and depends, among other issues, on the relief and the geological parameters of the sea floor. Those parameters influence the selection of the basic pipeline parameters, as well as the installation method to be used. It is important to underline that, in an offshore pipeline, the most severe loading conditions occur during the installation phase, rather than in its operation.

1.3.3 Pipe fabrication Once all the basic parameters of the pipe are specified, they are submitted to the pipe mill for the production of the line pipes. “Line pipe” is the pipe segment produced in the pipe mill and constitutes the basic component for pipeline construction. The pipe mill receives the raw material from the steel producer (steel mill) in the form of steel plates, coils or billets, depending on the type of line pipe to be fabricated (see Chapter 2). Line pipe fabrication is a very systematic process, which follows strict specifications. Those include: r r r r r

dimensional tolerances (e.g., cross-sectional ovality, wall eccentricity, out-of-straightness); minimum specified yield stress and maximum yield-to-tensile (Y/T) stress ratio for ductility; toughness and related mechanical characteristics of the pipe material, such as the non-ductile transition temperature; weld characteristics (for seam-welded pipes); special corrosion resistance requirements.

During the line pipe manufacturing process, continuous communication between the designer, the contractor and the fabricator is essential. Direct communication with the steel mill producer that supplies the raw material to the pipe mill (plates, coils, or billets) is also necessary. This allows for efficient control of the fabrication process, reduces the cost, resolves any problems that may arise and results in cost optimization of the pipeline project.

1.3.4 Pipeline construction Upon manufacturing, the line pipes are shipped to the construction site for building the pipeline. In offshore projects, they are shipped to a yard close to the offshore project and transferred in smaller quantities to the lay barge for marine installation. The overall

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Figure 1.1 Lowering of a straight pipeline section in the trench (photo by S. A. Karamanos).

installation is administered by a construction project engineer, who should be in close communication with the pipeline design engineer and the pipe mill. A description of pipeline construction process for onshore and offshore pipelines is offered in Sections 1.5 and 1.6 respectively.

1.4

Pipeline design considerations

Similar to any other engineering design project, the design of a pipeline consists of a systematic set of tasks. On the other hand, it may not be considered as a simple sequence of tasks but requires several “iterations” because of numerous interactions among different factors. The pipeline has the simplest geometry in structural engineering: an elongated cylinder (Fig. 1.1). Therefore, one may underestimate the importance and difficulty of pipeline design. Pipeline design is a topic involving important technological implications and very strict requirements that require high-level design experience in order to converge to an optimal design. It also involves a series of design calculations. In most cases, those calculations are not very complicated, and nowadays they can be performed by the use of computer methods using special-purpose software in an efficient and economical manner. This allows the pipeline designer to concentrate on the non-quantitative aspects of the design process and optimize the pipeline design. It is also important to underline that pipeline design follows a different philosophy than traditional structural design. In many structural systems, e.g., steel buildings, optimization of structural design refers mainly to simplifying the construction and the corresponding structural details, rather than saving quantities of steel. On the other hand, optimizing pipeline design in terms of saving steel material, mainly by reducing pipe wall thickness, may lead to substantial savings. Those savings are due to material cost reduction, but also to transportation and installation cost, as well as to welding cost. As an example, the use of a more elaborate design approach for a 48-inch-diameter onshore pipeline that reduces its wall thickness from 22 mm to 19 mm, results in significant material cost savings, and furthermore, transporting and handling lighter line pipes become easier and more economical.

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Structural Mechanics and Design of Metal Pipes

1.4.1 Route selection The selection of pipeline route (alignment) is a critical part of the design process. A poorly chosen alignment may result in very costly surprises and delays at a later stage of the project, with serious consequences for the progress of pipeline construction. Problems may arise if the alignment has conflicts with public authorities and other operators, or violates environmental requirements. In addition, good understanding of geomorphological factors should be acquired at the initial stage of design, especially in areas with severe geohazards (seismic or landslides). In cases where geohazards are expected to occur, appointing an experienced team of geologists to visit the site, to inspect the proposed pipeline alignment and report any alignment conflicts or potential geohazards can be of significant benefit for the pipeline project and its timely execution.

1.4.2 Pipe materials Materials need to be specified at an early stage of pipeline design. In hydrocarbon pipelines, steel material dominates the market. Currently, steel grades X60, X65 and X70 according to API 5L (American Petroleum Institute, 2018) are mainly used in pipeline applications. Grade X70 steel is quite common in onshore pipeline projects. In offshore pipelines, X70 is less frequent, but its use is steadily increasing. In special projects, especially offshore, besides carbon steel, various kinds of stainless steel can be used, mainly in the form of clad or lined bi-material pipes (see also Chapter 12). In offshore pipeline projects, the designer may need to consider materials for anticorrosion coatings, concrete coatings, or materials for thermal insulation, depending on the requirements of the project.

1.4.3 Pipeline design for optimum thickness An onshore pipe needs to be strong enough not to burst. It should be also capable of resisting actions from hydraulics and from ground-induced actions, if any. An offshore pipe needs to be strong enough not to burst, and not to deform excessively under external pressure (buckling), especially during its installation phase, when the pipeline is empty with no internal pressure. Furthermore, an offshore pipeline should be heavy enough to be hydrodynamically stable on the seabed, and safe against upheaval buckling and vortex-induced vibrations (VIV) in spans on the seabed. On the other hand, a heavy pipe is always more expensive. Apart from the amount of material, a heavy pipe is more difficult to transport, bend, weld and install. Therefore, it is the designer’s responsibility to determine an optimum thickness for the pipeline to be constructed, considering all relevant parameters throughout the pipeline construction project.

1.4.4 Pipeline constructability The designed pipeline must be constructable. More specifically, the primary task of the designer is to ensure that the pipeline can be constructed easily and economically, without technical difficulties. Towards this purpose, the designer must ensure that the pipeline can be constructed by a large number of contractors as possible, so that the owner is in a strong negotiating position for the award of the construction contract. This

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Figure 1.2 Fractured steel pipe after burst due to excessive internal pressure (photo by S. A. Karamanos).

is possible in onshore pipeline projects. On the other hand, the choice of contractors in offshore pipelines is more limited, because offshore construction requires more specialized equipment and personnel. The choice of contractor becomes quite limited in the case of deep offshore pipeline projects, where few contractors have the necessary equipment and expertise for performing the installation task.

1.4.5 Pipeline protection The pipeline must be secured against internal and external corrosion, and this requires coating of the pipeline, as well as cathodic protection on site. Protecting the pipeline against internal corrosion may also require the use of special chemicals injected into the pipeline to control the corrosion (corrosion inhibitors), or the use of pipes lined with a thin-walled pipe made of a Corrosion Resistance Alloy (see also Chapter 12). The pipeline must also be protected from various sources of external damage such as dropped objects, excessive settlement, geohazards and seismic actions. For the case of offshore pipelines, trawl gear and ship anchors may impose significant threats. Finally, changes of seabed level should be accounted for.

1.4.6 Principal structural failure modes of pipelines Under excessive internal pressure, pipelines burst. This happens because the pipeline wall may not be able to resist tensile stresses higher than a certain limit, and this causes pipe wall rupture with catastrophic consequences (Fig. 1.2). Burst is accompanied with explosion, which is a serious threat for human lives, especially in onshore gas pipelines. Furthermore, it may destroy nearby properties, facilities, or infrastructure. Burst in offshore oil pipeline is also dangerous, it may also be accompanied with explosion, and constitutes a serious environmental threat due to oil spill. Burst and the associated spill can be a very serious matter in offshore pipelines or flowlines conveying liquid hydrocarbons, especially in deep water, where limited access to the damaged area exists and the spread of oil into the marine environment is sometimes very difficult to stop. In offshore pipelines, apart from burst, structural instability due to external pressure is also very important, primarily during their installation process. In most cases, pipelines are installed empty, and are subjected to external pressure, which may cause buckling and collapse (Fig. 1.3). Collapse leads to flattening of the cross-section and,

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Structural Mechanics and Design of Metal Pipes

Figure 1.3 Collapse of a steel pipe due to external pressure (photo courtesy: Corinth Pipeworks S.A.).

subsequently, propagation of this collapsed pattern along the pipeline, destroying a significant length. Buckle propagation constitutes a significant issue in the course of pipeline design and a major threat for their structural integrity.

1.4.7 Pipeline specifications In onshore pipeline design, in many parts of the world, the American Standards are used. The basic design documents are issued by the American Society of Mechanical Engineers (ASME): r r

B31.4 Pipeline Transportation Systems for Liquids and Slurries (American Society of Mechanical Engineers, 2019); B31.8 Gas Transmission & Distribution Piping Systems (American Society of Mechanical Engineers, 2018).

For pipe welding, either in the pipe mill during pipe fabrication or on site, the API Standard 1104 (American Petroleum Institute, 2021) issued by the American Petroleum Institute is used. In addition, API 5L specification is used for specifying and qualifying line pipe material. Despite the wide acceptance of American Standards, some other notable specifications are employed, for example the Canadian standard CSA Z662 (Canadian Standard Association, 2019), the European Norm for gas pipelines EN 1594 (European Committee for Standardization, 2013) or the Dutch standard for pipelines NEN 3650 Part 1 and Part 2 (Nederlands Normalisatie-Instituut, 2020). It is also noted that pipeline owners often have their own detailed design and construction specifications, in addition to the standard specifications. Offshore pipelines have been designed traditionally with the American Standards issued by the American Petroleum Institute (API). The major standard for offshore pipeline design is API RP 1111 (American Petroleum Institute, 2015), which is also applicable for construction, operation, and maintenance issues. Line pipe specification and qualification is made according to API 5L, and API 1104 is used for welding. However, in recent years, the European DNV standards (of Norwegian origin) are becoming increasingly popular. The main standard for offshore pipeline and construction is DNV-ST-F101 (Det Norske Veritas, 2021). Offshore pipeline design is also covered by the CSA Z662 standard.

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Figure 1.4 Marked and beveled line pipe at the construction site (photo by S. A. Karamanos).

1.5

Onshore pipeline construction

Upon manufacturing, the line pipes are transported from the pipe mill to the construction site, for constructing the pipeline. Before leaving the pipe mill, each line pipe is marked and labeled, so that its exact position in the pipeline alignment is well defined (Fig. 1.4). The construction process consists of the following three stages: r r r

transportation and field bending of line pipes; welding, inspection, and joint coating; trenching, lowering-in, tie-in, backfilling and right-of-way.

Pipeline construction takes place within the right-of-way (ROW). This is a strip of land, usually 15 – 20 meters wide, where permission is given to a pipeline company for constructing and operating the pipeline.

1.5.1 Transportation and bending Some line pipes will require bending to accommodate changes in direction in the pipeline alignment and elevation of the trench. Bending can be done before or after trenching is completed. It is a cold bending process, which uses a hydraulic bending machine and an internal mandrel, which is necessary to prevent local buckling of the bent pipe (Fig. 1.5). During the bending process, inspection is required to ensure that: (a) the line pipe is being handled properly without coating damage; (b) the maximum allowable limits of curvature and cross-sectional ovalization are met as specified in the relevant design standard; (c) the pipe surface after bending is smooth and free from wrinkles, dents and flat spots. Furthermore, it is suggested that bending is not performed close to a circumferential (girth) weld and that the longitudinal seam of the pipe is on the neutral axis to minimize the influence of bending.

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Structural Mechanics and Design of Metal Pipes

Figure 1.5 Bending of a line pipe with a bending machine (photo courtesy: Avax S.A.).

After bending, the line pipes are lined up at the correct location (specified by the pipeline design) in the right-of-way, next to the pipeline alignment, ready to be welded and construct the pipeline (Fig. 1.6).

1.5.2 Line pipe welding, inspection and joint coating 1.5.2.1

Welding process

Welding of the line pipes to construct the pipeline is a key stage in pipeline construction, and it is performed at several stations simultaneously (Fig. 1.7A). The welding processes that are commonly used for pipeline applications are Shielded Metal Arc Welding (SMAW), which is a manual welding method (Fig. 1.7B), and Gas Metal Arc Welding (GMAW), which is a mechanized welding technique. Shielded Metal Arc Welding (SMAW) uses consumable stick electrodes that melt when an electric arc is struck and maintained between the tip of the electrode and steel material being welded. These electrodes have two main parts: the core wire and the external flux coating. The core wire provides the necessary metal to fill the weld joint. The flux coating shields the arc and the molten metal, protects them from the atmosphere, adds alloying elements to the weld metal and forms the protective layer during and after solidification of the weld metal. SMAW still remains a widely used process for pipeline in-situ welding. With the appropriate choice of

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Figure 1.6 Line pipes lined up on the right-of-way ready for welding (IGB pipeline project, photo courtesy: Avax S.A.).

consumables and welding technique, SMAW can be applied to all welding positions and meet a wide range of requirements in the mechanical properties. However, the quality of SMAW depends on the welder’s manual skills, and therefore, all welders should be properly qualified. Alternatively, the Gas Metal Arc Welding (GMAW) is a mechanized welding process that employs a consumable wire that melts when an electric arc develops between the wire and the steel material, shielded by an externally supplied gas. The welding wire is fed continuously during the GMAW process. Multiple welding passes are required to complete each weld. The number of passes depends on the pipe wall thickness and the welding process used. The first pass is called the “stringer bed” or “root pass”, which is followed by the second pass, which is also called the “hot pass”. Subsequently, the required number of fill passes are then conducted, and finally, a “cap pass” is performed to complete the girth welding process. Fig. 1.8 depicts the macrographic images of a SMAW weld (Fig. 1.8A) and a GMAW weld (Fig. 1.8B), both performed in X65 steel pipes with 24 in outer diameter and 12.5 mm wall thickness. The macro images show that the mechanized process (GMAW) results in a more uniform weld profile with smoother weld cap surface, and smaller size of heat-affected zone. The usual length of line pipe is 12 meters. Nevertheless, the contractor may often “double join” the pipe by welding two line pipes together and form a single

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(A)

(B)

Figure 1.7 (A) Pipeline welding stations (IGB pipeline project, photo courtesy: Avax S.A.); (B) SMAW welding (photo by Visser & Smit Hanab, courtesy: Carel Kramer Photography).

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(A)

(B)

Figure 1.8 Macrographic pictures of girth welds from a 24-inch diameter 12.5-mm-thick X65 steel pipe: (A) SMAW weld, (B) GMAW weld. (photo courtesy: Rina Consulting – Centro Sviluppo Materiali S.p.A.).

24-meter-long section, referred to as the “double joint”. This “double joining” process is usually performed at the pipe stockpile site to reduce handling and transportation costs. The advantage of double joining the pipe is that half as many welds are needed to be performed on the construction site, reducing the rate of weld defects and increasing the daily production rate for the pipeline construction. However, the extra difficulty in transporting and handling the longer “double joints” on the road, sometimes over rough terrain, may render double joining impractical for some locations.

1.5.2.2

Nondestructive testing and marking

All on-site pipeline welds are inspected with non-destructive testing, using radiographic or automatic ultrasonic testing (AUT) inspection, and the corresponding report of weld quality is provided. Each weld is marked with a tag that contains a unique number, which also appears in the inspection report. The presence and the nature of any defects in the weld that failed to meet the acceptance requirements of API 1104 and any additional requirements stated by the construction specifications, should be marked to indicate a need for repair or replacement. On large diameter pipelines jointed using mechanical welding, AUT inspection is usually preferable than radiography to identify the size and the location of weld defects, such as porosity, slag, inclusion, lack of fusion, and lack of penetration. The results of ultrasonic inspection are obtained more quickly than radiography, enabling corrective actions to be taken faster and in a more efficient manner. Ultrasonic inspection is also used to detect laminations in the parent material of the pipe, or to confirm minimum wall thickness after grinding of the weld. Weld inspection is also performed with the magnetic particle technique (MPI), which is used primarily to identify surface and near-surface discontinuities, as well as with liquid dye penetrant inspection (LPI), used primarily to identify discontinuities (e.g., cracks after the surface is cleaned, porosity and luck of fusion) on surface of the weld.

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Figure 1.9 Pipeline joint coating equipment (IGB pipeline project, photo courtesy: Avax S.A.).

1.5.2.3 Coating of welds Upon completion of welding and inspection, all weld areas after nondestructive testing should be cleaned and coated (Fig. 1.9). Usually, heat shrink sleeves using fusion bond epoxy are used. These are shrinkable polyethylene sleeves that provide anti corrosion protection for butt wells. Prior to coating, it is necessary to clean the weld properly and preheat the area at the appropriate temperature.

1.5.3 Trenching, lowering-in, tie-in and right-of-way 1.5.3.1

Trenching and lowering-in

Trenches are usually excavated using wheel ditchers, assisted by backhoes to excavate trenches in areas that require extra depth, as well as in pipeline bends, and in wet or rocky areas. Minimum trench dimensions should be determined to ensure that minimum cover requirements set by the pipeline specifications are satisfied, and that the backfill material is able to flow around the pipe and fill the trench properly, with the required degree of soil compaction. Properly graded material should be used underneath the pipeline (bedding) to support the pipeline installed in the trench, like a “mattress”, and control its interaction with soil and other external loads (Fig. 1.10). The pipeline is installed (“lowered”) into the excavated trench using side-boom tractors (Fig. 1.11). In rocky ground conditions, the excavated material is often run

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Figure 1.10 Bedding in the pipeline trench (IGB pipeline project, photo courtesy: Avax S.A.).

through a padding machine (Fig. 1.12), so that potentially damaging material (e.g., rock) is washed out, and fine grain soil is used to fill-in the trench around the pipeline. This material provides good lateral support and longitudinal restraint to the buried pipe, preventing pipe ovalization due to overburden loads and local overstressing of the pipeline by restricting its movement.

1.5.3.2 Tie-in welding Tie-in welds are the welds conducted within the trench to complete pipeline construction (Fig. 1.13). In many instances, they are performed in parallel with backfilling activity, if the required space is left by the backfill crew. A minimum of 1 meter on each side of a tie-in should be left to allow for the pipe parts to be welded. To perform the “stringer bead” pass, an external line-up clamp is used, which holds the two parts of the pipeline together. After welding, the tie-in weld is inspected using non-destructive testing, subsequently it is coated, and finally the open trench is backfilled.

1.5.3.3 Backfilling and right-of-way The backfill material should be compacted using heavy equipment, to the compaction level prescribed by the construction specifications (Fig. 1.14). The ditch line should be approximately 1 meter higher than prescribed to mitigate erosion effects, and openings should be left to allow for water drainage. After backfilling, the right-of-way is restored and marked.

1.6

Offshore pipeline installation

The installation of pipelines on the seabed constitutes one of the most challenging procedures in offshore pipeline engineering. Compared with onshore pipeline construction, offshore pipeline construction and installation is a much more sophisticated and demanding procedure. A high-level of engineering expertise is required, and the size and cost of the various types of installation vessels used have reached such a technical

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Structural Mechanics and Design of Metal Pipes

(A)

(B)

Figure 1.11 Pipeline lowering in the trench (IGB pipeline project, photos courtesy: Avax S.A.).

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Figure 1.12 Padding in the pipeline trench (IGB pipeline project, photo courtesy: Avax S.A.).

Figure 1.13 Tie-in welding of gas pipeline in the trench (photo by S. A. Karamanos).

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Structural Mechanics and Design of Metal Pipes

Figure 1.14 Backfilling of pipeline trench (IGB pipeline project, photo courtesy: Avax S.A.).

level of sophistication that offshore pipelaying has developed into an engineering discipline of its own. The installation methods most commonly employed are: r r r r

S-lay (shallow to deep water) J-lay (intermediate to deep water) Reel lay (intermediate to deep water) Tow-in (shallow to intermediate water depths)

and are briefly discussed in following.

1.6.1 S-lay installation method The S-lay is a traditional pipeline installation method, very popular for shallow and moderately deep water, shown schematically in Fig. 1.15. The pipeline segments (line pipes) are transported to the pipelay vessel, here referred to as “S-lay vessel”, they are welded with girth welds on the vessel, and subsequently, the welded pipeline is eased off the stinger, which is located at the stern of the vessel, as the vessel moves forward. The pipeline moves downward from the stinger through the water until it reaches its final position on the seafloor. In this installation process, the pipeline forms an “S” shape in the water.

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Figure 1.15 Schematic representation of S-lay offshore pipeline installation method.

There are several special features related to this installation process. The main feature is the shape of the pipeline from the vessel to the sea bottom. It is a doublecurvature “S-shape” configuration with two bending areas, shown in Fig. 1.15. The first is called “overbend area”, characterized by high tension and bending. The second bending area is the “sagbend”, which is near the seabed. At that location, the pipeline is subjected to a combination of external pressure and bending, together with a small amount of tension, and it is the most critical configuration during the installation process. In between these two bending areas, the pipeline is nearly straight, and is subjected to a combination of external pressure and tension, with rather small bending deformation. During its installation, the pipeline is also subjected to hydrodynamic forces from waves and currents. Another important feature of this installation method, related to the shape of the deformed pipeline, is the stinger, which affects the “overbend” area of the pipeline configuration. This is a large structural component, usually a truss, that supports the pipe when leaving the vessel (Fig. 1.16). The stinger extends from the stern of the vessel to support the pipe, controlling its initial curvature during installation. In some large S-lay vessels the length of the stinger may exceed 120 meters. Furthermore, some S-lay vessels have adjustable or articulated stingers, which can be shortened, lengthened and adjust their curvature according to the installation depth, minimizing the bending action on the pipeline at the overbend area and allowing for S-lay pipeline installation in water depths that may exceed 2,000 meters (Heerema, 1995; Faldini et al., 2014). It is also important to apply tension on the pipeline during the S-lay process to prevent excessive bending in the pipeline, and maintain the curvature of the sagbend region within acceptable limits. Tension is applied via tension rollers with rubber pads that press and hold the pipe without damaging its surface (Fig. 1.17). In such

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Figure 1.16 Offshore pipeline over the stinger of Saipem’s Castorosei S-lay vessel (photo courtesy: Saipem S.p.A.).

Figure 1.17 Tensioner device in Saipem’s Castoro 8 vessel for S-lay pipeline installation (photo courtesy: Saipem S.p.A.).

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a way, the pipeline shape is controlled against excessive bending in the presence of external pressure, and local buckling at the sagbend region is prevented (Section 6.6). Such a buckle would cause collapse of the pipeline cross-section, triggering the buckle propagation phenomenon (Section 5.4). Sometimes, buckling propagation occurs under high pressure and propagates with high speed, inducing high strains to the pipeline and leading to possible fracture of the pipe wall, referred to as “wet buckle”. Welds are performed on board to connect the pipe segments, using the SMAW method or the GMAW method. Welding is a critical task and requires special and qualified crew. A series of stations are located along the pipe-assembly line for welding, inspecting and field coating. It is important to minimize any delays in the welding process, mainly due to repair of weldments, which have significant economic consequences. One should note that the largest percentage of the cost of an offshore pipeline project is related to the installation process, because of the high cost of hiring the pipelay vessel and its personnel. Therefore, the faster the pipe is welded on the vessel, the faster it is installed, and this minimizes the cost of use of the pipelay vessel. Fig. 1.18 shows the Castorone vessel by Saipem S.p.A., one of the largest S-lay pipelay vessels currently in operation, with length of stinger equal to 120 meters (Faldini et al., 2014). Fig. 1.19, shows the first part of the pipelaying process, where three pipes are connected together to form a “triple-joint”. Subsequently, the “triple joints” are transported to the “fire line” for welding each triple joint to the next one (Fig. 1.20), followed by inspection and coating of the welds, before launched into the seawater through the stinger (Fig. 1.21). S-lay vessels are capable of installing 3-6 kilometers of pipeline per day, depending mainly on their diameter, thickness, welding specifications, and coating requirements, as well as on weather conditions.

1.6.2 J-lay installation method The J-lay method has been developed to overcome some of the deficiencies of S-lay installation in deep water (Jo, 1993; Wilkins, 1994). The J-lay installation method is shown schematically in Fig. 1.22, and induces less stress in the pipeline near the sea surface by inserting the pipeline into the sea in a vertical or nearly-vertical position. The pipe is lifted and put in a tall tower on the boat. At the bottom of the tower, each pipe segment is welded to the previous segment, and it is inserted into the sea. Unlike the double-curvature configuration obtained during S-lay, the pipeline during J-lay installation has a single curvature, and its shape under the water is reminiscent of a “J”. Again, the sagbend region is associated with external pressure and bending, and this is the most critical loading condition for the structural integrity of the pipeline. Fig. 1.23 shows Saipem’s S7000 J-lay vessel (Faldini, 1999), which installed the deepoffshore part of the Blue Stream pipeline in the Black Sea at water depths that exceeded 2,000 meters (see Table 1.3), and Fig. 1.24 depicts J-lay vessel FDS 2 (Faldini et al., 2014), also owned by Saipem S.p.A. In J-lay process, tension should be also applied on top of the launched pipeline, so that pipe bending at sagbed is controlled. Top tension in J-lay is smaller than in S-lay, and the reduced stress at the top section of the pipeline allows J-lay to be more efficient in deep water applications than S-lay. Additionally, a J-lay pipeline can withstand more

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Structural Mechanics and Design of Metal Pipes

Figure 1.18 Saipem’s Castorone S-lay vessel (photo courtesy: Saipem S.p.A.).

Introduction to pipeline engineering

31

Figure 1.19 Production of “triple-joints” in Saipem’s Castorone S-lay vessel (photo courtesy: Saipem S.p.A.).

efficiently the vessel motions and the effects of underwater currents than a pipeline installed with S-lay. However, these advantages come with an important price: there is only one station for welding, inspection, and coating in the tower, and at this station the pipes to be welded are in vertical position. This constitutes the main disadvantage of the J-lay method in comparison with S-lay. Fig. 1.25 shows the welding station in the S7000 vessel and Fig. 1.26 depicts the fit-up stage of two consecutive pipes, before welding. Double-joining or triple-joining of line pipes, forming longer pipe sections before inserting them in the tower can be extremely helpful in such a process, minimizing welding in the tower. Some vessels may also use “quadruple-joints”, welding four pipes in the prefabrication stage, as shown in Fig. 1.27 for vessel FDS 2.

1.6.3 Reeling installation method The main feature of the S-lay and the J-lay installation methods is that the pipe segments, after their transportation on site, are connected to each other on the vessel, and this requires welding on the vessel. On the other hand, the reeling method is developed for the main purpose of avoiding welding on the vessel during pipeline installation. Reel barges (reelships) are capable of installing relatively small diameter pipes and are equipped with a reel, which is either vertical (Friman et al., 1978) or horizontal (Malahy, 1995), where the pipe is wrapped around. In horizontal reel vessels

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Structural Mechanics and Design of Metal Pipes

Figure 1.20 Girth welding of two triple-joints in Castorone vessel (photo courtesy: Saipem S.p.A.).

Figure 1.21 Pipeline launched in water through the stinger in the “splash area” (Castorone vessel photo courtesy: Saipem S.p.A.).

Introduction to pipeline engineering

Figure 1.22 Schematic representation of J-lay pipeline installation method.

Figure 1.23 Saipem’s S7000 J-lay vessel (photo courtesy: Saipem S.p.A.).

33

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Structural Mechanics and Design of Metal Pipes

Figure 1.24 Saipem’s FDS 2 J-lay vessel (photo courtesy: Saipem S.p.A.).

Figure 1.25 Welding station in Saipem’s S7000 J-lay vessel (photo courtesy: Saipem S.p.A.).

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Figure 1.26 Pipe fit-up before welding in Saipem’s FDS 2 J-lay vessel (photo courtesy: Saipem S.p.A.).

Figure 1.27 Quadruple-joint pipe in vertical position during welding in the tower of Saipem’s FDS 2 J-lay vessel (photo courtesy: Saipem S.p.A.).

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Structural Mechanics and Design of Metal Pipes

(A)

(B)

Figure 1.28 (A) Schematic representation of reeling pipeline installation method. (B) Adjustable tower of reel vessel.

the pipeline starts its configuration horizontally and requires a stinger similar to the S-lay installation, while in vertical reel vessels the initial pipeline angle is usually adjustable (as shown in Fig. 1.28) and may be either vertical or with a certain angle. Fig. 1.29 shows vertical-reel ship Apache II of TechnipFMC. Using the reeling method, welding of pipe segments is performed onshore. This allows for performing welding in a much more controlled environment, minimizing the defect rate of girth welds. Furthermore, the operation time of the laying vessel is reduced, leading to substantial savings in installation cost. Reeled pipe is lifted from the dock to the vessel, spooled around the reel, and simply unspooled and straightened as installation is performed. When all of the pipe on the reel is installed, the vessel either returns to shore for loading the reel with the new pipe, or a new reel is installed, lifted from a transport vessel, which also returns the spent reel to shore, saving time and cost. From the structural design point-of-view, during the reeling process, the pipe undergoes two cycles of severe bending loading, prior to entering the water (Fig. 1.30). If repair of a weld is needed, then three additional bending cycles are applied. Therefore, the pipe and, primarily, its welds should be designed for resisting five severe bending cycles, in addition to the combined loading conditions at the sagbend region near the seabed. The five bending cycles induce high strains at the pipe wall, with maximum strains that may exceed 2% and, because of alternating plastic deformation, the material properties of the parent steel material and the weld are affected. This may lead to girth weld fracture due to ultra-low cycle fatigue, or pipe wall buckling, especially at the

Introduction to pipeline engineering

(A)

(B)

Figure 1.29 TechnipFMC Apache II reel vessel, in Leith Harbor, Edinburgh, Scotland, UK (photos by S. A. Karamanos).

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Structural Mechanics and Design of Metal Pipes

Figure 1.30 Schematic representation of the two complete loading cycles, in terms of the corresponding bending moment – curvature diagram, during the reeling process.

vicinity of the weld. Modern specifications contain strict rules for performing girth welds in pipes for reeling, capable of sustaining those large values of cyclic strain.

1.6.4 Towing (tow-in) methods Towing methods can be regarded as alternative to the more conventional installation techniques described above (Brown, 2006). In towing, a pipeline section is constructed onshore and is then towed to the installation site, as shown in Fig. 1.31. This has the clear advantage of welding and inspecting the pipeline section onshore, in a controlled environment. Towing a pipeline section can be beneficial for relatively short pipeline sections, in shore approaches, and in pulling a pipeline from onshore to offshore. Traditionally, towing methods have been used for relatively shallow water applications. In recent years, the interest of towing techniques for installing pipelines and flowlines has increased, in particular for deep-water applications (Alliot et al., 2006). Furthermore, towing does not require the sophisticated equipment of the other installation methods described above. There is also an increasing number of subsea fields with short flowlines (less than 20 km), and this makes towing installation quite attractive. There exist four variations of the towing method shown in Fig. 1.31. In all those approaches, the ends of the pipelines are configured with termination structures called sleds, so that one end can be connected to subsea equipment and components. In surface towing (Fig. 1.31A), the pipeline is towed by the tugboat, and is equipped with buoyancy devices so that it floats at the sea surface. Because of the length involved, two tugboats might be required to control the towing process, one in front and one in the tail of the pipeline. Once on the offshore site, the buoyancy devices are removed or flooded, and the pipeline is lowered on the seabed. Mid-depth towing (Fig. 1.31B) requires a second “hold-back” tugboat and fewer buoyancy modules. Both the depth of submerge and the shape of the pipeline are controlled by the speed of the tugboat. In this case, the pipeline settles to the sea bottom on its own, when the forward progression

Introduction to pipeline engineering

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(A)

(B)

(C)

(D)

Figure 1.31 Pipeline towing methods: (A) surface tow; (B) mid-surface tow; (C) off-bottom tow; (D) bottom tow.

seizes. The third case is off-bottom tow (Fig. 1.31C), which uses buoyancy devices and steel chains for extra weight. The chains are used to sink the pipeline to a depth near the sea bottom, but as the chain links reach the sea floor, their suspended length is reduced, their effective weight reduces, and the buoyancy keeps the towed pipeline at

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Structural Mechanics and Design of Metal Pipes

a certain distance above the sea bottom. As in the other versions of towing, once on the final location, the buoyancy is removed, and the pipeline settles to the sea floor. The last variation of pipeline towing is bottom tow, shown in Fig. 1.31D. In this case, the pipeline is sunk to the sea bottom and then towed along the sea floor. Bottom tow is an attractive and economical solution in soft and flat seabed, and in relatively shallow water. For more technical details on towing methods, the reader is referred to Ley and Reynolds (2006).

References Alliot, V., Zhang, H., Perinet, D., & Sinha, S. (2006). Development of towing techniques for deepwater flowlines and risers. In Offshore Technology Conference, OTC 17826. American Petroleum Institute. (2015). Design, Construction, Operation, and Maintenance of Offshore Hydrocarbon Pipelines (Limit State Design). API RP 1111. Washington, DC. American Petroleum Institute. (2018). Specification for Line Pipe. API SPEC 5L. Washington, DC. American Petroleum Institute. (2021). Welding of Pipelines and Related Facilities. API STD 1104. Washington, DC. American Society of Mechanical Engineers. (2018). Gas transmission and distribution piping systems. ASME B31.8 Standard, New York, NY. American Society of Mechanical Engineers. (2019). Pipeline transportation systems for liquid hydrocarbons and Slurries. ASME B31.4 Standard, New York, NY. Brown, R. J. (2006). Past present and future towing of pipelines and risers. In Offshore Technology Conference, OTC 18047. Bruschi, R. (2012). From the Longest to the Deepest Pipelines. International Offshore and Polar Engineering Conference, ISOPE-I-12-223. Canadian Standard Association. (2019). Oil and gas pipeline systems. CSA-Z662 Standard. Mississauga, ON, Canada. DeGeer, D., Timms, C., & Lobanov, V. (2005). Blue Stream Collapse Test Program. ASME International Conference on Offshore Mechanics and Arctic Engineering, OMAE 200567260. DeGeer, D., Timms, C., Wolodko, J., Yarmuch, M., Preston, R., & MacKinnon, D. (2007). Local Buckling Assessments for the Medgaz Pipeline. In ASME International Conference on Offshore Mechanics and Arctic Engineering, OMAE 2007-29493. Det Norske Veritas. (2021). Submarine Pipeline Systems. STANDARD DNV-ST-F101. Høvik, Norway. European Committee for Standardization. (2013). Gas supply systems-Pipelines for maximum operating pressure over 16 bar. EN 1594 Standard. Brussels, Belgium. Faldini, R. (1999). S7000: A New Horizon. Offshore Technology Conference, OTC 10712. Faldini, R., Marchini, S., Oldani, A., & Pellegrini, R. (2014). CastorOne and FDS 2: Getting Stronger for Laying Deeper. In Offshore Technology Conference, OTC 25110. Friman, K. R., Uyeda, S. T., & Bidstrup, H. (1978). First reel pipelay ship under construction – Applications up to 16-inch diameter pipe 3000 feet of water. Offshore Technology Conference 1, OTC 3069 (pp. 193–198). Güney, D., & Gudmestad, O. T. (1999). Oil transport alternatives from the Caspian Sea. International Offshore and Polar Engineering Conference.

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Heerema, E. P. (1995). DP pipelay vessel’solitaire’: plunging into the deep. In ASME International Conference on Offshore Mechanics and Arctic Engineering, OMAE 5, 525–537. Hengesh, J. V., Angell, M., & Lettis, W. R. (2004). Characterization of Surface Rupture Hazards for BTC Pipeline Fault Crossings, Turkey. In Terrain and geohazard challenges facing onshore oil and gas pipelines, Conference Proceedings. Jo, C. H. (1993). Limitation and comparison of S-lay and J-lay methods. In 3rd International Offshore and Polar Engineering Conference, 201–206. Ley, T., & Reynolds, D. (2006). Pulling and towing of pipelines and bundles. In Offshore Technology Conference, OTC 18233. Malahy, R. C. (1995). Installation of DP system and adaptation of the reel barge Chickasaw for deepwater pipelay. Offshore Technology Conference, 3, OTC 7815, 113–121. Marinos, V., Stoumpos, G., & Papazachos, C. (2019). Landslide Hazard and Risk Assessment for a Natural Gas Pipeline Project: The Case of the Trans Adriatic Pipeline, Albania Section. Geosciences, 9(2), 61. McKeehan, D. S. (1995). Oman-India Gas Pipeline Technology Development. Offshore Pipeline Technology Conference, OPT 95. Miesner, T. O., & Leffler, W. L. (2006). Oil & gas pipelines in nontechnical language. Tulsa, Oklahoma: PennWell Corporation. Nederlands Normalisatie-Instituut. (2020). Requirements for pipeline systems. NEN 3650. Part1: General, and Part-2: Steel pipelines. Delft, Netherlands. Pettinelli, D., Bergomi, S. P., Bruschi, R., Zenobi, D., Rott, W., & Gjedrem, T. (2012). Nord Stream Project - Segmented Pipeline System: Sizing vs Design for Operation. International Offshore and Polar Engineering Conference, ISOPE-I-12-238. Robl, K., Ta¸sdemir, A., & Sa¸ ¸ smaz, A. (2020). The Impact of Geohazards on the Trans Anatolian Natural Gas Pipeline Project – TANAP. Pipeline Technology Journal, 1, 116–120. Shilston, D. T., Lee, E. M., Pollos-Pirallo, S., Morgan, D., Clarke, J., Fookes, P. G., & Brunsden, D. (2004). No Access-Terrain evaluation and site investigations for design of the TransCaucasus Oil and Gas Pipelines in Georgia. In Terrain and geohazard challenges facing onshore oil and gas pipelines, Conference Proceedings. Slejko, D., Rebez, A., Santulin, M., Garcia–Pelaez, J., Sandron, D., Tamaro, A., Civile, D., Volpi, V., Caputo, R., Ceramicola, S., Chatzipetros, A., Daja, S., Fabris, P., Geletti, R., Karvelis, P., Moratto, L., Papazachos, C., Pavlides, S., Rapti, D., Rossi, G., Saraò, A., Sboras, S., Vuan, A., Zecchin, M., Zgur, F., & Zuliani, D. (2021). Seismic hazard for the Trans Adriatic Pipeline (TAP). Part 1: probabilistic seismic hazard analysis along the pipeline. Bulletin of Earthquake Engineering, 19, 3349–3388. Timms, C., Swanek, D., DeGeer, D., Meijer, A., Liu, P., Jurdik, E., & Chaudhuri, L. (2018). Turkstream Collapse Test Program. In ASME International Conference on Ocean, Offshore and Arctic Engineering, OMAE 2018-78454. Veldman, H. E., & Lagers, G. H. G. (1997). 50 Years Offshore. The Netherlands: Foundation of Offshore Studies, Delft. Wilkins, J. R. (1994). From S-lay to J-lay, ASME International Conference on Offshore Mechanics and Arctic Engineering, OMAE, 5, 319-331.

Line pipe manufacturing

2

The principal component of a pipeline is the “line pipe”, which is fabricated in the pipe mill. Line pipe fabrication may appear a rather trivial manufacturing process to the non-expert, mainly because of its very simple geometry, i.e., an elongated circular cylinder. However, pipe manufacturing requires high skills and good expertise in metallurgy, welding and metal forming technology to provide a product that meets the specified material properties and is suitable for pipeline applications. The main features of line pipe fabrication are presented in this chapter, and an overview of different manufacturing methods is offered.

2.1

General considerations and steel material properties

The “line pipe”, also called “pipe joint”, constitutes the principal component of a pipeline. It is fabricated in the pipe mill, which is either part of an integrated steel industry company or an independent industrial unit. In the latter case, pipe fabricators order the steel material from a steel producer. The line pipe is usually a 40-feet long (12 m) section of steel tube, but this length may vary depending on specific project requirements. After their fabrication, the “line pipes” are hydrotested, coated, and transported to the construction site, where they are connected by on-site welding to form the “pipeline”. Carbon steel line pipe can be manufactured using several different techniques, each of which produces a pipe with certain characteristics. These characteristics include strength, wall thickness, corrosion resistance, and temperature and pressure limitations. For example, steel pipes having the same wall thickness and the same steel material but manufactured by different methods may have significant differences in their pressure and structural strength capacity.

2.1.1 Line pipe steels In most hydrocarbon pipeline projects, the steel material of line pipes should conform to the provisions of the American Petroleum Institute (API) specification 5L (American Petroleum Institute, 2018). In some cases, pipe specification level PSL2 is considered, which imposes strict requirements for the use of line pipe in demanding applications (e.g., offshore, geohazards). About two decades ago, the API 5L specification was converted into an international standard ISO 3183 (International Organization for Standardization, 2019), issued by the International Organization for Standardization. Both API 5L and ISO 3183 cover the selection and use of steel for both seamless and welded line pipes manufactured in the pipe mill. Line pipe steels are characterized primarily by their yield strength. In ISO 3183, SI units are adopted for the yield strength, and line pipe steel materials are designated by Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00007-4 c 2023 Elsevier Inc. All rights reserved. Copyright 

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Table 2.1 API 5L Seamless Steel Pipe Yield & Tensile Strength. API 5L Grades API 5L X42 API 5L X52 API 5L X60 API 5L X65 API 5L X70

Yield Strength minimum ksi (MPa) 42 (290) 52 (359) 60 (414) 65 (449) 70 (485)

Tensile Strength minimum ksi (MPa) 60 (414) 66 (455) 75 (518) 77 (513) 82 (566)

capital letter L followed by the Specified Minimum Yield Strength (SMYS) in MPa. In API 5L specification, they are designated by capital letter X followed by a number indicating the yield strength in thousands of pounds (kilo-pounds) per square inch, denoted as ksi. The various types of steel, referred to as “steel grades”, used in line pipes are shown in Table 2.1, which states the API 5L designations in ksi with those of ISO 3183, which are based on yield strength in MPa. As an example, steel grade X60 has a yield strength of 60 ksi or 60,000 pounds per square inch (psi) or, in SI units, 448.5 MPa (1 MPa = 0.145 ksi). The equivalent grade in ISO 3183 is L450, i.e., steel with yield stress equal to 450 MPa. It should be underlined that the value of SMYS is a minimum value, and that the real yield stress of purchased steel is always higher.

2.1.2 Steel material requirements Line pipe steel material must have adequate strength while retaining its ductility, toughness and weldability, which are outlined below. One should note though that there exists some conflict between these properties. Strength is the ability of the pipe steel and the associated weldments to resist the longitudinal and circumferential tensile stresses imposed on the pipe during its installation and operation. Steel material strength is characterized mainly by its yield strength which constitutes a primary design parameter. Increasing the yield strength, pipe wall thickness requirement decreases, and this can be an essential economic factor for the pipeline project. One may consider that the use of thinner pipes in a pipeline project reduces material and welding costs, but also transportation costs and the cost of equipment for pipe installation. The latter is important in both onshore and offshore pipeline projects, but becomes very significant in offshore hydrocarbon pipeline construction, where thinner pipes may reduce the loads on the pipelay vessel and, in particular, the loads to be sustained by the vessel stinger and the tensioner (see also Section 1.6). Currently, hydrocarbon pipelines use high strength steels. For onshore pipeline applications, grade X70 steels are widely used, whereas in offshore applications the use of X70 steel is more limited. On the other hand, large-diemeter steel pipelines for water transmission (Chapter 10) typically use steel grade up to X42. The strength of a line pipe steel is measured in a tensile testing machine using strip specimens extracted from the pipe (Fig. 2.1). As mentioned above, the measured tensile strength (yield or ultimate) is different than the nominal value of the steel grade. In several cases, mainly in offshore pipeline design against collapse, the compression

Line pipe manufacturing

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45

(B)

(C)

Figure 2.1 (A) Tensile testing of a steel strip specimen; (B) strip specimens from heavy wall plates tested under tension to failure; (C) tensile specimens: 1a and 2a are undeformed specimens, 1b and 2b are the corresponding fractured specimens; 1a and 1b are mild steel specimens grade 275, 2a and 2b are high-strength steel specimens grade 590 (photos by S.A. Karamanos).

Figure 2.2 (A) Anti-buckling devices for testing steel strip specimens in compression (photo by S.A. Karamanos); (B) Charpy test set-up (photo courtesy: EΒETAM/MIRTEC S.A.).

properties of the pipe wall material are more important than the tensile properties and need to be determined experimentally. However, compression of strip specimens will result in bucking, and therefore, anti-buckling devices are required (Fig. 2.2A). In welded pipes, one should further notice that the mechanical properties of steel material of the line pipe as a final product of the fabrication process, is different than those of the steel plate or the steel coil received from the steel producer. The main reason for this difference is cold forming and welding during the manufacturing process. In addition, material properties may be affected by the pipe coating process. Furthermore, because of the manufacturing process, the pipe material may be anisotropic, which means different yield/strength properties in the longitudinal and the transverse directions, and this has to be addressed during material characterization of the pipe material. Ductility is the ability of the pipe to absorb overstressing under significant deformation and is an important property of steel pipe material. An important feature of steel material is that its ductility reduces as the yield stress of the steel material increases. This imposes a challenge to the development of the relevant steel material specification,

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in prescribing the appropriate combination of strengthening mechanisms to ensure that steel material strength is achieved without sacrificing its ductility. Ductility is measured in a tensile testing machine, using a strip specimen, as a percent of the measured elongation with respect to the original length of the strip specimen. Simple bend tests on strip specimens or flattening tests on rings extracted from pipes and examination of the steel surface for cracks are also used for validating steel material ductility. To ensure that the steel material has adequate ductility, an adequate margin between yield and tensile strength should exist. In several specifications, this is expressed as a maximum value of the ratio between the yield strength (Y) and the tensile strength (Τ) of the material, referred to as Y/T ratio. For example, a value of the Y/T ratio equal to 0.94 means that the yield strength of the steel material is 94% of the tensile strength or alternatively the tensile strength is 6% above the yield strength. Typical values of this limit ratio used in pipe specifications are between 0.90 and 0.92, but this value may differ in special cases, e.g., in sour-service pipelines. Toughness is the ability of a material to absorb energy and plastically deform without fracturing. Sometimes it is defined as the resistance of the material to impact loads. Tough materials yield under impact loading and fail in a ductile and progressive manner. Non-tough materials exhibit brittle fracture, which is sudden and may have catastrophic consequences. Among several parameters, most of which are metallurgical, toughness also depends on the level of temperature. The temperature at which the material behavior changes from ductile to brittle is called “non-ductile transition temperature”, denoted as NDTT, or simply “transition temperature”. The required toughness of the steel material and the acceptable level of NDTT depend on the service of the pipeline and the environment to which the pipe will be exposed during its service. As an example, gas pipelines require significant toughness. There is a very high energy stored in those pipelines because of gas compression, and therefore if a pipe wall rupture occurs, gas expansion results in a rapid decrease in temperature, possibly leading to a shift from ductile to brittle behavior of the pipeline steel material. This initial pipe rupture may propagate at high speed until the “driving force” of the crack falls below a critical level, or the crack hits either a sufficiently tough section of the pipe or a crack arrestor device. A second example refers to hydrocarbon pipelines in arctic regions installed and operating at low temperature. In such cases, a low value of NDTT is required, without sacrificing pipe material strength and weldability. Several mechanical tests are available to examine whether the steel material has adequate toughness. Most of those tests are associated with the sudden application of forces on appropriate specimens. Drop weight testing, and Charpy testing (Fig. 2.2B) are typical test methods widely used by pipeline industry. The purpose of these tests is to ensure that the pipe material meets the specified requirements for the NDTT and the absorbed energy. Welding constitutes a significant part of the total cost of pipeline construction. In the pipe mill, for line pipe fabrication, it is usually performed according to the relevant provisions of API 5L standard. Good weldability of pipeline metals has to be ensured and this is an essential factor for a successful pipeline project.

Line pipe manufacturing

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47

(B)

(C)

Figure 2.3 (A) Seamless pipe; (B) pipe with longitudinal weld; (C) pipe with spiral weld [schematic].

2.1.3 Categorization of pipe fabrication methods Fig. 2.3 shows schematically three typical line pipes in their final form. The first pipe is seamless, which means that it does not contain any internal weld. The second and the third are welded pipes and are fabricated by appropriate cold forming of a steel plate or a coil in a circular configuration, followed by welding. Historically, in the nineteenth century, rolled strips of metal sheet were hot formed into a circular shape and butt of lap welded in the same heat. At the end of the nineteenth century, processes for fabricating seamless pipes become available, dominating the market until WWII. During the last decades, the developments in manufacturing process and particularly in welding technology led to a rapid increase of the use of welded pipes. Currently, around 70% of the steel tube production market is covered by welded pipes. Each of these methods for producing pipe has several advantages and disadvantages. As an example, butt-welded pipes with longitudinal seam weld formed from rolled plates, have more uniform wall thickness and can be more easily inspected for defects prior to forming and welding. This manufacturing method is particularly useful when thin wall pipes and long pipe lengths are needed. On the other hand, the welded seam might present defects that violate the specified quality control requirements. The following sections of the present Chapter outline several fabrication processes, used in hydrocarbon and water pipeline industry. The two main categories are “welded pipes” and “seamless pipes”. In the first category, pipes made of steel coil are distinguished from pipes manufactured from steel plate.

2.2

Pipes manufactured from steel coil

Steel coil is a finished steel product such as sheet or strip which has been wound or coiled after hot or cold rolling. For pipeline applications, hot-rolled coils are used (Fig. 2.4). To fabricate pipes, the coiled steel plate is uncoiled and flattened. Subsequently the flattened sheet passes through a series of rolls to form the pipe.

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(A)

Structural Mechanics and Design of Metal Pipes

(B)

Figure 2.4 Steel coils for the fabrication of large diameter spiral-welded pipes for water transmission projects (photos by S.A. Karamanos).

Figure 2.5 Schematic representation of the ERW pipe manufacturing process.

Two types of line pipe are fabricated from coil plate: (a) ERW pipes and (b) spiralwelded pipes.

2.2.1 ERW pipes The manufacturing process for ERW pipes is shown schematically in Fig. 2.5. The fabrication process starts with receiving the steel coils from the steel producer (Fig. 2.6A). After decoiling (Fig. 2.6B), the flattened metal sheet undergoes edge milling and passes through a series of rolls that gradually bend the sheet into a circular shape ready for welding in the longitudinal direction. The weld seam is made by electric resistance welding (ERW), a method that does not use welding consumables. An AC electric current is used to heat the adjacent faces over the pipe. Subsequently, the two faces of the sheet are pressed together to produce the longitudinal weld. In the traditional form of ERW fabrication process, the

Line pipe manufacturing

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(A)

(B)

(C)

(D)

Figure 2.6 ERW-HFI pipe manufacturing: (A) steel coils for HFI pipe manufacturing process; (B) decoiling process; (C) HFI welding process; (D) heat treatment of HFI weld (courtesy of photos: Corinth Pipeworks SA).

Figure 2.7 Macrographic picture of a HFI pipe weld (photo courtesy: Corinth Pipeworks SA).

frequency of the AC current has been rather low. More recently, the AC current is induced through induction coils at high frequency that may exceed 100,000 Hz (Fig. 2.6C). The latter has overcome several drawbacks of the traditional ERW process, and it is widely used today by numerous pipe fabricators. For this reason, the method is also referred to as HFI-ERW pipe manufacturing process or simply HFI (High Frequency Induction) process (Brauer et al., 2010). It is also referred to as HFW process. The ERW/HFI weld is very fine and sometimes non-detectable with naked eye (see Fig. 2.7). Subsequently, the weld is heat treated (Fig. 2.6D) and is inspected by ultrasonic testing. Using this fabrication process, pipes with diameter size up to 24 inches and thickness up to about 0.5 inch can be produced. In particular, ERW pipes with sizes ranging between 10 inches and 16 inches are widely used in reeling pipeline applications, and are strong competitors to seamless pipes, because of their lower fabrication cost.

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Structural Mechanics and Design of Metal Pipes

Figure 2.8 Spiral-welded pipe manufacturing process; adjustment of the relative location of the three rollers controls the diameter of the pipe.

The ERW fabrication process is continuous, and the new coil is connected by welding to the end of the previous coil. After welding and inspection, the pipe weld undergoes annealing, cooling, sizing and cutting of the continuous pipe to segments of appropriate length with the use of cutting saw. The line pipes are hydrotested, coated, and ready for being shipped to the construction site.

2.2.2 Spiral welded pipes The concept of spiral-welded pipe fabrication is the twisting of a metal strip into a spiral shape, followed by welding where the edges join one another to form a seam (Fig. 2.3C). This fabrication method starts from a steel coil (Fig. 2.4). The flat and uncoiled steel sheet, after edge trimming for welding preparation, passes through a triplet of rollers positioned in a “pyramid” configuration, and oriented at an appropriate angle with respect to the de-coiling direction. In such a way, the steel coil is bent into a spiral shape (Fig. 2.8). There are two welding stations just after passing through the pyramid rollers, one internal and one external, which perform pipe welding with the submerged arc welding method (SAW). In SAW, the metal is joined by fusing with an electric arc that develops between a metal electrode and the edge of the pipe. A layer of granular, fusible material covers the weld area and protects the arc and the molten metal. Internal welding is performed first (Fig. 2.9A), followed by external welding.

Line pipe manufacturing

51

(A)

(B)

Figure 2.9 (A) Internal welding in spiral-welded (HSAW) pipes; (B) macrographic picture of a spiral weld (courtesy of photos: Northwest Pipe Company).

Fig. 2.9B shows the macrographic picture of a weld from a large-diameter spiralwelded pipe. The shape of the weld seam in those pipes has a spiral (helical) configuration and therefore these pipes are referred to as HSAW or SAWH pipes. After the welding process, the weld is inspected either visually or by ultrasonic testing (UT) to ensure that the weld is acceptable, and finally, the pipe is cut in line pipe segments, usually with the use of a plasma torch. Adjusting the angle of the three rollers and considering the width of the coil, the desired pipe diameter can be achieved. Using this method, very large diameter pipes of relatively small pipe wall thickness (high values of diameter-to-thickness ratio) can be fabricated. As an example, pipes with diameter size that exceed 120 inches (3 m) can be manufactured, to be used in water transmission pipeline projects in North America (Fig. 2.10). On the other hand, up to a few years ago, the use of spiral welded pipes in hydrocarbon pipelines has been quite limited. The oil and gas sector has been reluctant in using spiral-welded pipes, especially in demanding pipeline applications, e.g., in offshore applications, or in projects involving geohazards. However, this perception is gradually changing. The results of the recent European project SBD-SPIPE (Iob et al., 2018; Chatzopoulou et al., 2019) have shown that spiral-welded pipes can be used in those applications and that they may be suitable for strain-based design.

2.3

Pipes manufactured from steel plates

An efficient method for fabricating relatively large diameter butt-welded pipes is by cold bending a steel plate through forming dies or rollers into a hollow circular shape.

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Structural Mechanics and Design of Metal Pipes

(A)

(B)

(C)

(D)

Figure 2.10 Manufacturing of large-diameter spiral-welded pipe at the pipe mill: (A) arrangement of spiral-welding process; (B) spiral bending of the steel sheet; (C) cutting of spiral-welded line pipe with plasma flame; (D) spiral-welded pipe after welding and cutting, ready for inspection and hydrotesting (courtesy of photos: Northwest Pipe Company).

Subsequently, a longitudinal seam is produced welding together the two ends of the plate, as shown in Fig. 2.3B. The most popular methods of this category are UOE and JCO-E processes, which are both press-bending processes. Alternatively, the plate can be bent to a circular shape passing through a set of rollers, and this is referred to as “roller-bending” process. Those methods are outlined below.

2.3.1 UOE pipes This process starts from a flat plate of appropriate thickness and width, and consists of four sequential mechanical steps, as shown in Fig. 2.11, (a) crimping of the plate edges, (b) U-ing of the plate, (c) O-ing of the plate and welding of the two edges of the plate, and finally, (d) expansion (E), applying either internal pressure or, most usually, a mechanical expander. Welding is performed using the submerged arc welding (SAW) technique and at least two welding passes are performed; one internal, which is performed first, and one external, performed after rotating the pipe by 180°. Subsequently, the weld is inspected by ultrasonic testing and X-ray radiography, and finally, the pipe is subjected to expansion.

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53

Figure 2.11 Schematic representation of the UOE pipe manufacturing process.

The expansion stage is necessary for controlling the final shape of the pipe crosssection. Furthermore, during this fabrication process, the steel plate is subjected to a substantial amount of cold bending and therefore, residual stresses develop. A significant part of those residual stresses can be alleviated during pipe expansion, but in this case, the applied tensile deformation results in a decrease of yield stress in compression, because of the Bauschinger effect, and this can be very important for offshore pipes, because it affects pipe resistance against high external pressure. After expansion, the line pipe is hydrotested, coated, and finally shipped on site for pipeline construction (onshore or offshore). Pipes with size between 16 and 64 inches can be efficiently manufactured using this method. UOE pipes have been used in numerous pipeline applications, both onshore and offshore. In particular, their use in deep offshore applications has triggered significant experimental and numerical research in an attempt to understand the effects of the manufacturing process on the collapse resistance of the pipe (e.g., Herynk et al., 2007; Chatzopoulou et al., 2016). A significant part of this research has been directed towards (a) optimizing the expansion process in terms of the pipe resistance to external pressure and (b) improving the value of the so-called “fabrication factor” in the DNV-ST-F101 standard for UOE pipes (Det Norske Veritas, 2021), which accounts for the effects of manufacturing process on the collapse pressure of offshore pipes. This issue is discussed more extensively in Chapter 8.

2.3.2 JCO-E pipes The JCO-E manufacturing process is an efficient method for manufacturing thickwalled steel line pipes. It follows a concept similar to UOE process, gradually bending a steel plate into a circular cylindrical shape, and consists of five sequential mechanical

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Structural Mechanics and Design of Metal Pipes

Figure 2.12 Schematic representation of the JCO-E pipe manufacturing process.

(A)

(B)

Figure 2.13 JCO-E pipe fabrication in the pipe mill; (A) steel plate ready for the forming process; (B) plate crimping (courtesy of photos: Corinth Pipeworks SA).

steps, shown schematically in Fig. 2.12, (a) crimping of the plate edges, (b) the J phase, where the plate is formed into a J-shape, (c) the C phase, where the deformed plate is pressed into a quasi-round shape, (d) the O phase, where the deformed plate obtains a round shape, and subsequently both ends of the plate are welded and (e) the expansion (E) phase where the pipe geometry is improved through a mechanical expander. Fig. 2.13 shows the first stage of this process, consisting of steel plate preparation and its edge crimping. Subsequently, the J-ing, C-ing and O-ing stages (simply referred to as “JCO”) are performed using a series of punches with the use of an elongated forming die. The die moves vertically and causes local bending of the plate, as shown in Fig. 2.14. The JCO forming is a progressive press-bending process of the steel plate in multiple steps and constitutes the main difference from the two-step UO forming process. During the JCO forming process, the steel plate deforms more uniformly, the

Line pipe manufacturing

(A)

55

(B)

Figure 2.14 JCO forming with the vertical forming press; (courtesy of photos: Corinth Pipeworks SA).

residual stresses induced by cold bending are small and the surface does not produce scratches. After bending to a circular shape, the weld is performed using submerged arc welding (SAW) with at least two passes, one internal and one external (Fig. 2.15AC). Following weld inspection and verification, the pipe is expanded (Fig. 2.15D), hydrotested and coated, and it is ready for transportation to the construction site. Two important parameters that refer to the JCO-E process are: (a) the number of punches during the JCO stages, and (b) the level of final expansion. The interested reader is referred to the recent publications of Chatzopoulou et al. (2017) and Antoniou et al. (2019) on the effect of those parameters in the mechanical response of JCO-E pipes, candidates for offshore pipeline applications, in terms of their resistance to external pressure collapse.

2.3.3 Roll bending process The Roll Bending (RB) process, also referred to as “plate rolling”, as opposed to the JCO and UO press-bending processes, is performed with a pipe rolling machine. This consists of a set of three or four rollers, positioned in a “pyramid” configuration to bend the plate into a pipe (Fig. 2.16). The plate passes several times through the rollers, which can move and gradually increase the curvature imposed to the plate until the desired curvature is achieved. Subsequently, internal and external submerged arc welding (SAW) is performed, producing a longitudinal weld. If the pipe is expanded after welding, this method is called RBE pipe.

2.4

Seamless pipes

The invention of the cross-roll piercing process by the Mannesmann brothers in the end of the nineteenth century has been a milestone in the development and establishment of seamless pipe manufacturing. Seamless pipe is formed by piercing a solid steel rod,

56

(A)

Structural Mechanics and Design of Metal Pipes

(B)

(C) (D)

Figure 2.15 JCO-E pipe fabrication: (A) internal welding; (B) external welding; (C) macrographic picture of a longitudinal weld from a JCO-E pipe; (D) expansion with the use of mechanical mandrels (courtesy of photos: Corinth Pipeworks SA).

in high temperature, with a mandrel to produce a pipe that has no seam weld. Fig. 2.17 depicts the basic concept of seamless pipe manufacturing process (Bellmann and Kümmerling, 1993). The process starts with a solid steel bar or rod, called “billet”, which is heated at high temperature. The solid, near-molten billet is inserted in a set of rotating barrel-shaped rolls, as shown in Fig. 2.17. The hot billet also rotates, and a piercing device penetrates the billet, forming the hollow cylinder. There exist several variations of this manufacturing process. After the pipe is formed, it is normalized or quenched and tempered (Q&T) for homogenization of its mechanical properties and for improving its toughness and strength. Finally, the pipe is hydrotested, coated (if necessary) and shipped to the construction site. Pipes of various lengths and diameter size up to 28 inches can be produced with the seamless fabrication process. Practically there is no restriction on the size of pipe wall thickness in seamless pipes, and this is an important advantage. The main competitor of seamless pipe is the ERW (HFI) pipe because of its lower fabrication cost. The main advantages of seamless pipes with respect to ERW pipes are: (a) the lack of weld seem, (b) the quasi-uniform distribution of material properties and (c) the very low residual stresses. On the other hand, seamless pipes are more expensive than ERW pipes, they

Line pipe manufacturing

57

(B)

(A)

Figure 2.16 Roll Bending for pipe manufacturing. (A) Three-roller plate bending machine. (B) Steel plate being rolled (courtesy of photos: Northwest Pipe Company).

Figure 2.17 Seamless pipe manufacturing process.

are likely to have non-uniform thickness around their cross-section and they usually have significant roughness on their inside and outside surfaces.

References American Petroleum Institute. (2018). Specification for Line Pipe. API SPEC 5L. Washington, DC. Antoniou, K., Chatzopoulou, G., Karamanos, S. A., Tazedakis, A., Palagas, C., & Dourdounis, E. (2019). Numerical Simulation of JCO-E Pipe Manufacturing Process and its Effect

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on the External Pressure Capacity of the Pipe. Journal of Offshore Mechanics and Arctic Engineering, ASME, 141(1) Article Number: 011704. Bellmann, M., & Kümmerling, R. (1993). Kriterien für die umformtechnische Bewertung von Lochschrägwalzwerken für die Herstellung nahtloser Rohre. Stahl und Eisen, 113(8), 47– 53. Brauer, H., Löbbe, H., & Bick, M. (2010). HFI-welded pipes: where are the limits? In Proceedings of the 8th International Pipeline Conference, IPC 2010-31233. Chatzopoulou, G., Karamanos, S. A., & Varelis, G. E. (2016). Finite element analysis of UOE manufacturing process and its effect on mechanical behavior of offshore pipes. International Journal of Solids and Structures, 83, 13–27. Chatzopoulou, G., Antoniou, K., & Karamanos, S. A. (2017). Numerical Simulation of JCO Pipe Forming Process and its Effect on the External Pressure Capacity of the Pipe. In Proceedings of the 36th International Conference on Ocean, Offshore and Arctic Engineering, OMAE 2017. Chatzopoulou, G., Sarvanis, G. C., Karamanos, S. A., Mecozzi, E., & Hilgert, O. (2019). The effect of spiral cold-bending manufacturing process on pipeline mechanical behavior. International Journal of Solids and Structures, 166, 167–182. Det Norske Veritas. (2021). Submarine Pipeline Systems. STANDARD DNV-ST-F101. Høvik, Norway. Herynk, M. D., Kyriakides, S., Onoufriou, A., & Yun, H. D. (2007). Effects of the UOE/UOC pipe manufacturing processes on pipe collapse pressure. International Journal of Mechanical Sciences, 49(5), 533–553. Iob et al. (2018). Strain-based design of spiral-welded pipes for demanding pipeline applications. Research Fund for Coal and Steel, European Commission, project No. RFSR-CT2013-00025. Brussels, Belgium. International Organization for Standardization. (2019). Petroleum and natural gas industries – Steel pipe for pipeline transportation systems. ISO standard 3183. Geneva, Switzerland.

Part II Pipe Mechanics 3. 4. 5. 6.

Structural mechanics of elastic rings 61 Structural mechanics of elastic cylinders 95 Mechanical behavior of metal pipes under internal and external pressure Metal pipes and tubes under structural loading 187

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Structural mechanics of elastic rings

3

The present chapter offers an introduction to structural behavior of pipes focusing on the mechanical response of rings made of linear elastic material, subjected to uniform pressure (internal or external) and transverse loading. The elastic material is assumed isotropic and homogeneous. Emphasis is given on structural instability (buckling) of rings, caused by external pressure under unconfined and confined conditions. The analysis and the results presented in the present chapter are essential for understanding the mechanical behavior and design of metal pipes, discussed in subsequent chapters. The case of a circular ring with rectangular cross-section is of particular interest for the present analysis. It represents a longitudinal “slice” of a long cylinder and has direct application in pipeline mechanics, as shown in Fig. 3.1. The cylinder diameter, radius and thickness are denoted as D, r and t respectively, and the length of this slice is b, also shown in Fig. 3.1. In many instances, the value of b is taken equal to 1. Under uniform pressure, the response of a long cylinder may not vary in the longitudinal direction, and therefore, it can be considered as a two-dimensional problem under plane-strain conditions. In the following analysis, we consider rings with rectangular cross-section, as “slices” of a long cylinder, because they are directly related to pipes and tubes. However, the formulation may also be applicable to the case of standalone rings of arbitrary cross-section, with minor adjustments of the relevant equations. In the analytical formulation that follows, the ring thickness is considered very small with respect to ring diameter. This refers to the so-called thin-walled ring theory.

3.1

Ring stresses under internal pressure

Consider a circular ring of constant thickness, subjected to uniform internal pressure p. The ring is thin walled, in the sense that the thickness of its cross-section t is significantly smaller than its diameter D. Furthermore, for convenience it is assumed that b = 1. Equilibrium of the free-body diagram of the half-ring (see Fig. 3.2) results in the following equation for the tensile stress σ in the pipe wall, in terms of pressure p, the ring diameter D and the ring thickness t: D (3.1) 2t Eq. (3.1) is the well-known Barlow formula (Barlow, 1837), after the English mathematician Peter Barlow (1776–1862), and is the basic equation for the pressure design of pipelines and pressure vessels. Note that Eq. (3.1) is based only on equilibrium and it is valid for any material (elastic or inelastic). Under internal pressure, the response is axisymmetric with no dependence on the angle θ (see Fig. 3.2). Due to ring expansion, the strain in its circumferential direction (hoop strain) can be readily computed from σθ = p

Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00010-4 c 2023 Elsevier Inc. All rights reserved. Copyright 

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Structural Mechanics and Design of Metal Pipes

Figure 3.1 Ring as a “slice” of a long cylinder.

Figure 3.2 Stresses developed in a ring under internal pressure.

the corresponding change of length: εθ =

w r

(3.2)

Considering linear elastic material that obeys Hooke’s law (σ θ = Eεθ , where E is the Young’s modulus of the elastic material), the tensile strain in the pipe wall in the circumferential (hoop) direction is equal to: εθ =

pr Et

(3.3)

Structural mechanics of elastic rings

(A)

63

(B)

Figure 3.3 (A) Axisymmetric deformation of ring under external pressure; (B) stresses developed in the externally-pressurized ring.

Finally, using Eq. (3.3) the corresponding outward radial displacement w can be expressed in terms of internal pressure p as follows: w=

pr2 Et

(3.4)

The above equations are valid for a standalone ring of rectangular cross-section t × b (b=1). If plane strain conditions are considered for the ring, then E should be replaced by E/(1-ve 2 ), where ve is Poisson’s ratio (see Appendix A).

3.2

Ring buckling under external pressure

If the ring is subjected to uniform external pressure p, one may consider that the above equations may still be valid with an opposite sign (see Fig. 3.3) and that the response of the ring is also axisymmetric. However, in externally-pressurized rings, the ring wall is under compressive stress and may become structurally unstable, leading to buckling and inducing significant deformation to the ring. Upon buckling, its geometry becomes totally different than the initial circular configuration. We will prove that it takes an oval shape. To understand the mechanical behavior of an elastic ring, it is necessary to present the governing equations asssociated with in-plane ring deformation.

3.2.1 Ring kinematics The local Cartesian coordinate system x, ¯ y, ¯ z¯, shown in Fig. 3.1, is used for the case of a ring “extracted” from a long cylinder. The ring is treated as two-dimensional on the x¯ − y¯ plane (see Fig. 3.4) and is considered as a closed curved beam of circular shape. The ring configuration is characterized by its axis (also called “reference line” of the ring), which is assumed to coincide with the centroid of the ring cross-section. The diameter and radius of the initially circular reference line is denoted by r.

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Structural Mechanics and Design of Metal Pipes

Figure 3.4 Deformation parameters of a circular ring.

At this stage, an important clarification is necessary to avoid confusion. Two “crosssections” or “sections” are considered in the present analysis (Fig. 3.1): (a) The ring cross-section, with area Ar = bt and moment of inertia about the reference line Ir = bt 3 /12 (see also Fig. 3.4). (b) The cylinder (or pipe, or tube) cross-section, with area A = 2πrt and moment of inertia about the cylinder centroidal axis I = π r3 t.

The two-dimensional problem of ring deformation can be tackled assuming that ring sections, initially plane and normal to the ring axis, remain – after ring deformation – plane and normal to the deformed ring axis. This is a direct extension of the EulerBernoulli assumption for straight beams, widely used in Mechanics of Materials. Under this assumption, instead of monitoring the motion of all material points of the ring as a two-dimensional continuum, one may monitor the displacement of the reference line only. The accuracy of this assumption has been demonstrated for thin rings, i.e., for ring with large diameter-to-thickness (D/t) ratio. Furthermore, the displacement of reference line is expressed in terms of its components in a polar coordinate system (ρ, θ), whereas z¯ is the third coordinate in the direction normal to the x, ¯ y¯ plane, and runs along the cylinder (see also Fig. 3.1). Sometimes, instead of θ , we employ coordinates, which is the length along the circumference (s = rθ and ds = rdθ ). In this polar coordinate system, the corresponding components of in-plane displacement of the reference line are: 1. the radial displacement w(θ ) 2. the hoop (or circumferential or tangential) displacement v(θ)

Both displacements w and v are functions of angle θ. Each point C on the reference line, initially with coordinates x, ¯ y¯ is displaced at C∗ with coordinates x¯∗ , y¯∗ . From Fig. 3.4, one may write: x¯ = r cos θ

(3.5)

Structural mechanics of elastic rings

65

y¯ = r sin θ

(3.6)

x¯ ∗ = (r + w) cos θ − v sin θ

(3.7)

y¯ ∗ = (r + w) sin θ + v cos θ

(3.8)

and

The circumferential (or hoop) strain at a specific point H of the ring cross-section (see Fig. 3.4), not necessarily located on the reference line, is expressed as follows: εθ (ζ ) = εθm + ζ kθ

(3.9)

where ζ is a local coordinate on the ring cross-section, with −t/2 ≤ ζ ≤ t/2, expressing the radial distance of this point from the reference line. Clearly, ρ = r + ζ . The first term on the right-hand side of Eq. (3.9) is the membrane strain of the ring:  εθm = where (·) = βθ =

v + w r



1 + βθ2 2

(3.10)

d(·) and the rotation β θ of the ring cross-section is expressed as follows: dθ

v − w r

(3.11)

Eq. (3.10) indicates that the membrane strain consists of a linear and a quadratic part. The latter represents membrane stretching due to transverse deformation. The second term on the right-hand side of Eq. (3.9) is the bending strain of the ring, which is linear across the ring thickness, and the bending curvature kθ is defined as: kθ =

v − w r2

(3.12)

Note that the linear part of the membrane strain consists of two parts: 1 dv dv = , 1. a part due to differential displacement in the tangential direction s (or θ) ds r dθ w 2. a second part due to radial displacement . r

Furthermore, the total rotation of the ring cross-section β θ consists of two parts: 1 dw dw = , 1. a counterclockwise part due to the differential radial displacement ds r dθ v 2. a clockwise part because of tangential displacement . r

Finally, a direct comparison of the above expressions with the corresponding expressions from beam-column theory is offered in Table 3.1.

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Table 3.1 Comparison of expressions for beam-column kinematics with the corresponding expressions for circular rings (Brush and Almroth, 1975).1

membrane strain cross-sectional rotation bending curvature 1

Beam-column   du 1 dw 2 + dx 2 dx dw − dx d2w − 2 dx

Ring     1 v dw 2 dv +w + − ds 2 r ds v dw − r ds 1 dv d 2 w − 2 r ds ds

In this Table, x denotes the coordinate along the beam-column.

3.2.2 Stresses, stress resultants and constitutive equation Linear elastic material is assumed for the ring, which follows Hooke’s law. For the standalone ring case, the state of stress is uniaxial, only the circumferential stress is nonzero and the corresponding stress-strain relationship is expressed in the following simple form: σθ = E εθ

(3.13)

For a ring which is the slice of a long cylinder (see Fig. 3.1), the strain in the longitudinal direction ε z is zero, the state of stress in the pipe wall is bi-axial (σ θ and σ z ), and the corresponding stress-strain relationship is expressed as follows (Appendix A): E εθ 1 − νe 2

σθ =

(3.14)

where ν e is Poisson’s ratio, equal to about 0.30 for all metals in the elastic range. In the following, we focus our discussion on the latter case, i.e., the ring as a slice of a long cylinder, because of its direct application to pipes and tubes, and Eq. (3.14) will be used. On the other hand, with minor adjustments, the equations are also applicable to standalone rings of any shape of their cross-section. Appropriate integration of stresses over the ring cross-section results in the axial force Nθ and the bending moment Mθ of the ring cross-section, which are often referred to as “stress resultants”: t/2 Nθ =

σθ dζ

(3.15)

σθ ζ dζ

(3.16)

−t/2

t/2 Mθ = −t/2

Structural mechanics of elastic rings

67

3.2.3 Equilibrium equations The equilibrium equations of the ring refer to both tangential and radial direction, and can be expressed as follows in terms of the stress resultants Nθ and Mθ (Brush and Almroth, 1975): r Nθ + Mθ − r Nθ βθ − p r2 βθ = 0

(3.17)

  Mθ − r Nθ − r (Nθ βθ ) − p r v + w = p r2

(3.18)

3.2.4 Final equations The governing equations of the ring are obtained combining the constitutive Eqs. (3.14)–(3.16) with ring kinematics (3.9)–(3.12) and subsequently inserting into the above equilibrium Eqs. (3.17) and (3.18). They are expressed in terms of the unknown displacements w(θ) and v(θ) in the following form:       Ir 1 v − w 2 v − w  v + w + + r 2 r Ar r2 r         v + w 1 v − w 2 v − w pr v − w − + − = 0 (3.19) r 2 r r EAr r        Ir v + w 1 v − w 2 v − w  − + Ar r2 r r 2 r           pr v + w v + w 1 v − w 2 v − w pr − − + = r 2 r r EAr r EAr



(3.20) One should underline that the ring is a closed system, and therefore, Eqs. (3.19) and (3.20) do not require specific boundary conditions. Instead, periodic solutions are sought.

3.2.5 Prebuckling state and bifurcation The axisymmetric solution presented in Section 3.1 for internal pressure, can be also used to express the prebuckling solution of our problem, if a change of sign is considered in the pressure p. The prebuckling displacements are: w0 = − v0 = 0

pr2 Et

(3.21) (3.22)

where subscript ( r)0 refers to the prebuckling state (Fig. 3.5), and the corresponding

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Structural Mechanics and Design of Metal Pipes

Figure 3.5 Prebuckling configuration and buckling shape (n = 2) of a ring under external pressure.

axial force per unit length is equal to: Nθ0 = −pr

(3.23)

Eqs. (3.21)–(3.23) constitute a rather trivial solution, obtained using elementary Statics and Mechanics of Materials. One can readily verify that it satisfies trivially the final Eqs. (3.19) and (3.20) for any value of pressure p. However, this solution may not be the only solution of our problem. Our purpose is to determine under which conditions an additional (non-trivial) solution of this problem may be possible. From a physical perspective, when uniform external pressure is applied on the ring, hoop compression develops in the ring wall. At the stage where this compression exceeds a critical level, structural instability occurs associated with ring buckling. It is important to underline that the external pressure p is the “destabilizing” parameter of our system, and the problem can be stated as follows: determine the level of external pressure, which allows for a solution additional to the above trivial prebuckling solution. The presence of such an additional (non-trivial) solution at a specific level of external pressure indicates that, at this pressure level, a “bifurcation” occurs, and from the mathematical point-of-view two possibilities exist: the ring either (a) continues to follow the prebuckling state in Eqs. (3.21)–(3.23) and remains circular under uniform compression, or (b) bifurcates to the additional (non-trivial) solution. The latter is associated with buckling, and the value of pressure at which the solution bifurcates is called “critical pressure” or “buckling pressure”. Denoting this additional solution by w1 and v1 , one may write: w → w0 + w1

(3.24)

v → v0 + v1

(3.25)

Structural mechanics of elastic rings

69

We limit the present analysis at the vicinity of the bifurcation point, i.e., just after bifurcation, with the purpose of determining the critical pressure and the corresponding deformation mode. Therefore, we assume that displacements w1 and v1 are small. Under the assumption of small displacements w1 and v1 , inserting Eqs. (3.24) and (3.25) into the final Eqs. (3.19) and (3.20), and keeping only first-order terms, one readily obtains the following system of homogenous equations:     EAr r2 v1 + w1 + EIr v1 + w1 = 0

(3.26)

      EAr r2 v1 + w1 − EIr v1 − w1 + pr3 w1 + w1 = 0

(3.27)

The above equations are linear in terms of w1 and v1 , and are often referred to as “linearized equations of ring buckling”. In mathematical terms, the above set of Eqs. (3.26) and (3.27) constitutes a Sturm-Liouville eigenvalue problem. Given the geometry of the problem, we seek periodic non-trivial solutions for w1 and v1 . Towards this purpose, we assume that: v 1 = B1 sin n θ

(3.28)

w 1 = B2 cos n θ

(3.29)

and

where B1 and B2 are constants to be determined and n is a wave number. Inserting Eqs. (3.28) and (3.29) into the linearized Eqs. (3.26) and (3.27), one obtains the following set of equations:     (3.30) EAr r 2 −n 2 B 1 − nB 2 + EIr −n 2 B 1 − n 3 B 2 sin nθ = 0   3    EIr −n B 1 − n 4 B 2 − EAr r 2 (nB 1 + B 2 ) − pr3 B 2 − n2 B 2 cos nθ = 0 (3.31)



or equivalently,     B1 n 1 + c n 2 n2 (1 + c)   =0 B2 1 + c n4 − n2 − 1 γ n 1 + c n2

(3.32)

pr Ir and γ = , are non-dimensional parameters of ring geometry 2 Ar r EAr and loading, respectively. For a non-trivial solution to exist, the determinant of the 2 × 2 matrix should be zero: where c =

det

   n2 (1 + c)   n 1+ cn 2  =0 1 + c n4 − n2 − 1 γ n 1 + c n2

(3.33)

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Structural Mechanics and Design of Metal Pipes

Figure 3.6 Rigid body translation of a circular ring (n = 1).

and this leads to the computation of the eigenvalues of the homogenous problem from Eqs. (3.26) and (3.27), and the corresponding eigenmodes. We examine first the special case of n = 1. In that case, one readily obtains from Eq. (3.33): (1 + c)(B 1 + B 2 ) = 0

⇔

B 1 = −B 2

(3.34)

and the corresponding displacements are: w 1 (1) = B cos θ

(3.35)

v 1 (1) = −B sin θ

(3.36)

where subscript ( r)(1) refers to n = 1. The above displacement field represents a rigid body translation of the ring (see Fig. 3.6), and therefore, it is not of interest in our discussion. Consequently, to obtain solutions associated with buckling, values of n greater than 1 should be considered. For n ≥ 2, Eq. (3.33) leads to the following values of external pressure (eigenvalues): n2 − 1 p = pn = 1+c



EIr r3

 (3.37)

3.2.6 Buckling pressure and mode The “critical pressure” pcr is the lower eigenvalue computed from Eq. (3.37), and this corresponds to n = 2. Therefore, pcr = p2 =

 t 3 3 EIr E 2E  t  3 = = 1 + c r3 4 (1 + c) r 1+c D

(3.38)

Structural mechanics of elastic rings

71

and the corresponding buckling mode is computed from the following condition: 4 (1 + c)B 1 + 2 (1 + 4 c) B 2 = 0

(3.39)

which stems from Eq. (3.33) if n ≥ 2 is assumed. Equivalently,  B 2 = −2

 1+c B1 1 + 4c

(3.40)

Considering that the ring is thin walled and that the value of diameter-to-thickness ratio is large, c=

  t3 1 1  t 2 Ir = 1 = Ar r 2 12 t r 2 3 D

(3.41)

and 1 + c ࣃ 1. Therefore, the critical pressure can be expressed as: pcr =

3EIr r3

(3.42)

an equation presented first by Bresse (1866). Equivalently,  t 3 p c r = 2E D

(3.43)

Additionally, from Eq. (3.40): B 2 = −2 B 1

(3.44)

and the corresponding buckling mode is: w 1 (2) = B cos 2θ v 1 (2) = −

(3.45)

B sin 2θ 2

(3.46)

where B is an arbitrary constant. The mode is shown schematically in Fig. 3.5. For the case of a ring under plane strain conditions, Eq. (3.43) is modified as follows (Bryan, 1888): pcr =

2E  t  3 1 − νe 2 D

(3.47)

From expressions (3.45) and (3.46) that define the radial and tangential displacements of the buckling mode, one may compute from Eqs. (3.10) and (3.11) the corresponding membrane circumferential strain of the cylinder wall:  εm1(2) =

w1(2) + v 1(2) r

 +

1 2



v1(2) − w 1(2) r

2 =

  9 B 2 8 r

(3.48)

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Structural Mechanics and Design of Metal Pipes

Figure 3.7 Buckling of an axially-compressed simply supported bar (beam-column).

This expression is quadratic in terms of the buckling mode amplitude B because the linear part vanishes. Therefore, if only first-order terms are considered, the membrane strain is negligible, and the ring exhibits no extension or contraction of its periphery. This type of deformation is often referred to as “inextensional” deformation and is experessed by v1 + w1 = 0. Therefore, it is reasonable to consider that the membrane strain of the initial post-buckling configuration is zero, and this is “first-order accurate” for thin-walled rings.

3.2.7 A heuristic formulation and solution of elastic ring buckling It is possible to obtain the governing buckling equation for the ring under external pressure and the corresponding buckling pressure, using a heuristic formulation. This starts from the well-known buckling equation of a straight elastic bar of length L and cross-sectional moment of inertia Ib (Fig. 3.7) compressed by a force F, also referred to as “beam-column”: EIb

d4w d2w + F =0 dx4 dx2

(3.49)

Eq. (3.49) is written equivalently, Lb (EIb Lb (w)) − FLb (w) = 0

(3.50)

where Lb is the curvature operator of the beam-column model: Lb (.) = −

d 2 (.) dx2

(3.51)

For the case of an inextensional ring (v’ + w = 0), the ring hoop curvature kθ from Eq. (3.12) can be written: kθ = −

 1 w + w 2 r

(3.52)

Structural mechanics of elastic rings

73

and this leads to the following curvature operator for the ring model: Lr (.) = −

1   (.) + (.) r2

(3.53)

Therefore, adapting Eq. (3.50) for the case of the “inextensional ring” and considering the width b of the ring, one obtains: Lr (EIr Lr (w)) − (prb)Lr (w) = 0

(3.54)

Eq. (3.54) results in the following homogenous equation: 

   w(4) + 2w + w + p∗ w + w = 0

(3.55)

where p∗ =

pr3 b EIr

(3.56)

Solutions of Eq. (3.55) are sought in the following form: w(θ ) = B cos nθ

(3.57)

and inserting into Eq. (3.55), one obtains the eigenvalues of the problem:   n4 − 2n2 + 1 = p∗ n2 − 1

(3.58)

Excluding the rigid body motion n = 1 for the reasons explained earlier, p∗ = n2 − 1

(3.59)

and the lowest eigenvalue is p∗ = 3, or equivalently, pcr = 2E

 t 3 D

(3.60)

which is Eq. (3.43). For the case of a ring under plane strain conditions (Fig. 3.1), E should be replaced with E/(1 – ve 2 ) and the critical pressure is equal to Eq. (3.47).

3.2.8 Effects of initial ovality and first yield In practice, no ring or pipe is perfectly circular. On the contrary, they are always “nonperfect” or “imperfect”, in the sense that their configuration deviates from the circular shape, and this may affect their structural response under external pressure. From the mathematical perspective, an “imperfect” ring exhibits in-plane bending deformation as soon as external pressure is applied. Without loss of generality, consider a ring with

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Structural Mechanics and Design of Metal Pipes

Figure 3.8 Ring with oval initial imperfection.

an initial non-circular shape in the following form, which is similar to the buckling mode (see Fig. 3.8): w(θ ˆ ) = αˆ cos 2θ v(θ ˆ )=−

αˆ sin 2θ 2

(3.61) (3.62)

where αˆ is the amplitude of initial imperfection. Note that the initial oval shape is “inextensional”, because vˆ + wˆ = 0. It can be shown (Brush and Almroth, 1975) that the initial oval shape, upon application of external pressure remains oval, but with increased amplitude:   1 cos 2θ (3.63) w(θ ) = α cos 2θ = αˆ 1 − p/p c r   α αˆ 1 v(θ ) = − sin 2θ = − sin 2θ (3.64) 2 2 1 − p/p c r where pcr is the critical buckling pressure of the perfect ring and 1/(1 − p/pcr ) is an amplification factor. It is interesting to note that the displacements expressed in Eqs. (3.63) and (3.64) are also inextensional (v + w = 0) and similar to the initial imperfection. Eqs. (3.63) and (3.64) may be also written:  

1 w(θ ˆ ) w(θ ) (3.65) = ˆ ) v(θ ) 1 − p/pcr v(θ and α = αˆ

1 1 − p/pcr

(3.66)

Eq. (3.66) is plotted schematically in Fig. 3.9. The bending moment Mθ around the

Structural mechanics of elastic rings

75

Figure 3.9 Pressure-deformation response of externally pressurized elastic rings.

ring is:

 M(θ ) = EIr kθ = −EIr

w + w r2



 −

wˆ  + wˆ r2

 =

3EIr (α − α) ˆ cos 2θ r2 (3.67)

and the corresponding hoop stress at the inner and outer part of the ring is: pr σθ (θ ) = − ± t



Mθ (θ ) t 3 /12



t 2

(3.68)

Because of the trigonometric function cos 2θ in Eq. (3.67), the maximum bending deformation and stress occur at four equally-spaced cross-sections around the circumference of the ring, namely at θ equal to 0, π /2, π and 3π /2. At those locations, the bending moment is: Mmax

  3EIr p = 2 αˆ r pcr − p

(3.69)

and the corresponding maximum stress is the sum of the absolute values of the membrane and bending stress: σ max

   r  1  prαˆ    = p  +  ·  2   (1 − p/p c r ) t t /6 

(3.70)

First yielding occurs when this stress reaches the yield stress σ Y of the material. Setting σ max = σ Y in Eq. (3.70) and rearranging, the corresponding value of pressure

76

Structural Mechanics and Design of Metal Pipes

pF can be computed from the solution of the following quadratic equation: 

p 2F

   σY t σY t 6r αˆ − + 1+ pcr pcr = 0 pF + r t r r

(3.71)

Eq. (3.71) is attributed to Timoshenko (1933), and it is analogous to the well-known Perry’s formula for straight bars (columns), as presented in the stability textbook by Timoshenko and Gere (1961).

3.2.9 Measures of ring in-plane deformation Previous analysis demonstrated that the in-plane deformation of a ring under external pressure is characterized by ovalization. In pipeline design and analysis, the following parameter has been widely employed to quantify ring ovalization (e.g., Kyriakides and Corona, 2007): =

Dmax − Dmin Dmax + Dmin

(3.72)

where Dmax and Dmin are the maximum and minimum diameter of the ring (Fig. 3.8B). The value of ovalization at the initial stage before the application of pressure, is called initial ovalization or initial ovality. As an example, for the case of an initially imperfect ring described by Eqs. (3.61) and (3.62), according to Eq. (3.72), the value of initial ovalization of the ring is: ˆ = αˆ r

(3.73)

Sometimes, cross-sectional ovalization is expressed in the following form: =

Dmax − Dmin 2D

(3.74)

which provides a value very similar to the one provided by Eq. (3.72) considering that Dmax +Dmin ∼ = 2D. Upon application of external pressure, the ovalization becomes: =

α r

(3.75)

It is also noted that in several pipeline design documents, such as DNV-ST-F101 (Det Norske Veritas, 2021), the following measure of initial ovalization is employed: Dmax − Dmin Oˆ 0 = D ˆ It is readily concluded that Oˆ 0  2 .

(3.76)

Structural mechanics of elastic rings

77

Figure 3.10 Area enclosed by the reference line of the ring in the undeformed (pre-buckling) and deformed (post-buckling) configuration.

3.2.10 Energy formulation Energy principles constitute powerful tools for solving complex problems in Structural Mechanics in a simple and efficient manner. They also constitute the basis for modern numerical methods, such as the finite element method. Herein, we adopt the principle of stationary value of potential energy , which is defined as the sum of deformation (strain) energy U and the potential of external work V:

= U + V

(3.77)

To simplify the analysis, the elastic ring is considered inextensional (v + w = 0), so that only bending deformation is accounted for in the strain energy expression. Therefore, 1 U = 2

2πR 2π 1 M kθ ds = EIr kθ2 rdθ 2 0

(3.78)

0

The potential of external work V is defined as the negative of the work W produced by the external loading V = −W. For a ring under uniform external pressure, this work is the product of the external pressure p times the change of area enclosed by the ring AE (Fig. 3.10). The area enclosed by the reference line of the deformed ring can be written trivially as follows: A∗E

 = A∗E

dA∗E

1 = 2

  A∗E

∂ y¯∗ ∂ x¯∗ + ∂ x¯∗ ∂ y¯∗



dA∗E

1 = 2



div x¯ ∗ dA∗E

(3.79)

A∗E

where x¯∗ and y¯∗ are the coordinates of a point in the deformed reference line of the ring. Applying divergence theorem and using Eqs. (3.7) and (3.8), Eq. (3.79)

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Structural Mechanics and Design of Metal Pipes

Figure 3.11 Second-order work by the compressive force in the ring.

leads to: A∗E

1 = 2

2π



 r2 + r v + 2 r w + v2 − v w + v w + w2 dθ

(3.80)

o

Considering also that: 1 2

2π r2 dθ = π r2

(3.81)

o

and due to periodicity: 2π

v dθ = v (2π ) − v (0) = 0

(3.82)

o

one obtains: A∗E

1 = 2

2π



 2rw + v2 − v w + v w + w2 dθ + π r2

(3.83)

o

so that the work due to external pressure Wp1 is:

Wp1

1 = −p A = − p 2

2π



 2rw + v2 − vw + v w + w2 dθ

(3.84)

o

Furthermore, one must consider to the external work term the additional secondorder work Wp2 of the compressive force −pr of the ring. Referring to Fig. 3.11, the shortening u of the elementary length s along the ring circumference is: u = s (1 − cos βθ )  s

βθ 2 2

(3.85)

Structural mechanics of elastic rings

79

where β θ is the rotation of s. Therefore,

WP2

1 = pr 2

2πr βθ 2 ds

(3.86)

0

and

WP2

1 = pr2 2

2π 

v − w r

2 dθ

(3.87)

0

Combining Eqs. (3.77), (3.78), (3.84) and (3.87): 1 EI

= 2 r3

2π



  2

w+w

1 dθ − p 2

0

2π



  2

v−w

0

1 dθ + p 2

2π



 2rw + v2 − vw dθ

0

(3.88) For equilibrium, the value of should be stationary. The following assumed function is assumed for the radial displacement: w(θ ) = α cos 2θ

(3.89)

where α is the radial displacement amplitude, and because of inextensionality: v(θ ) = −

α sin 2θ 2

(3.90)

Inserting Eqs. (3.89) and (3.90) into Eq. (3.88), one obtains the following discretized form of the potential energy:

(α) =

9πEIr 2 9π p 2 3π p 2 2 α − α α − 2r3 8 8

(3.91)

and enforcing its stationary value: d =0 dα

(3.92)

the critical pressure of Eq. (3.42) is obtained.

3.2.11 A note on the post-buckling response of elastic rings The above analysis is capable of predicting the buckling pressure and the corresponding buckling mode but does not provide any information on post-buckling response. To

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Structural Mechanics and Design of Metal Pipes

determine whether this bifurcation is stable or unstable, and investigate ring deformation under external pressure beyond bifurcation, a higher-order ring theory is necessary. A detailed presentation of this theory is outside the scope of the present book. Nevertheless, a few interesting aspects are stated in this note. Budiansky, in his classical publication for elastic buckling (Budiansky, 1974) developed a nonlinear formulation for the post-bifurcation analysis of a ring under uniform external pressure based on Koiter’s general post-buckling theory (Koiter, 1963), resulting in the following asymptotic expressions for the initial post-buckling equilibrium state in terms of the change of area enclosed by the ring AE and the crosssectional ovalization of the ring :   p 9 AE =1+ (3.93) pcr 16 AE0 27 p = 1 + 2 (3.94) pcr 32 In Eq. (3.93), AE denotes the change of the area enclosed by the ring, and AE0 = πr2 is the initial area enclosed by the ring. Both Eqs. (3.93) and (3.94) show an increase of pressure in the post-buckling path with respect to the critical pressure pcr , which implies that the post-buckling response is stable, with small sensitivity to the presence of initial imperfections. This result has been verified in subsequent works by El Naschie (1975) and Kyriakides and Babcock (1981). Eqs. (3.93) and (3.94) are plotted in Fig. 3.12, together with finite element results considering several values of initial imperfection amplitude for an isotropic elastic ring (E = 210 GPa, ν e = 0.3) with outer diameter (OD) 600 mm and thickness (t) 15 mm. The finite element results verify the stable nature of bifurcation and the imperfection insensitivity of this problem. Furthermore, the results show that for very small values of initial imperfection amplitude, the finite element results are in very close agreement with the analytical expressions (3.93) and (3.94).

3.3

Confined ring deformation and buckling under uniform external pressure

The structural response of externally-pressurized rings under lateral confinement is an issue related to thin-walled underground pipelines for water transmission, with typical diameter-to-thickness ratio ranging between 100 and 300, which are buried or concreteencased under hydrostatic conditions due to high water table or vacuum conditions. In this section, we discuss the elastic solution of the problem of an elastic ring within a rigid cavity, subjected to uniform external pressure, first presented by Glock (1977). The problem is shown schematically in Fig. 3.13; an elastic ring is placed within a rigid cavity, which fits perfectly the ring geometry. The ring is externally pressurized, but the rigid cavity prevents any outward deflection. Under those confined conditions, it is interesting to examine whether the ring may buckle at certain level of pressure. Initially, the ring is perfectly fitted within the cavity. Upon external pressure application, the ring contracts uniformly, so that a small gap develops between its

Structural mechanics of elastic rings

(A)

(B)

Figure 3.12 Nonlinear response of elastic rings under external pressure for different values of initial imperfection and comparison with Budiansky’s analytical solution as expressed by Eqs. (3.93) and (3.94); (A) diagrams of pressure versus change of enclosed area and (B) pressure-ovality diagrams.

81

82

Structural Mechanics and Design of Metal Pipes

(A)

(B)

Figure 3.13 Elastic ring under uniform external pressure and confined conditions.

outer surface and the cavity wall. When the value of external pressure p reaches the ring buckling pressure under unconfined conditions pcr , the ring buckles and starts deforming as an oval, as examined in the previous section. However, because of the confinement imposed by the cavity, the oval shape may not develop. Instead, the ring accommodates itself within the cavity and behaves as an arch under transverse distributed loading. With further increase of the external pressure, the arch becomes structurally unstable, exhibiting an inward buckling configuration (Fig. 3.13). Glock’s formulation and solution quantifies this structural behavior (Glock, 1977). The original solution has also been reported in detail by Omara et al. (1997), and more briefly by Vasilikis and Karamanos (2009, 2014). It is summarized below for the sake of completeness.

3.3.1 Buckling of an elastic ring in a rigid cavity under external pressure Glock’s solution (Glock, 1977) refers to elastic rings confined within a rigid medium, with kinematics based on Donnell approximations (Donnell, 1933). More specifically, the total hoop axial strain is given by Eq. (3.9), the membrane strain is given by Eq. (3.10), whereas the ring section rotation and the hoop curvature are expressed as follows: w r w kθ = − 2 r

βθ = −

(3.95) (3.96)

which are simpler expressions than those in ring kinematics expressed in Eqs. (3.11) and (3.12). Ring deformation consists of two parts: part I, the “buckled” region, and part II, the “unbuckled” region. A shape function w(θ) for the buckled

Structural mechanics of elastic rings

83

region is assumed as follows:  w(θ ) = δcos

2

πθ 2φ

 (3.97)

where φ is the angle that defines the border between the “buckled” and the “unbuckled” ring portions, so that − φ ≤ θ ≤ φ. Considering the total potential energy of the ring, assuming a constant hoop axial force around the ring, and requiring minimization of the potential energy with respect to both δ and φ, closed-form expressions for the pressure p, the amplitude of buckled shape δ and the axial force Nθ are obtained in terms of angle φ:  2   δ φ 6pr3 φ 4 + 10 =− r EIr π π ⎡ ⎤   2   5 1 pr3 π ⎣ 80 EIr π ⎦ 1± = 16 − EIr φ 6 3 EAr r2 φ Nθ =

  5 EIr π 2 3 r2 φ

(3.98)

(3.99)

(3.100)

Minimization of pressure in terms of angle φ, results in the following closed-form expressions for the maximum or buckling pressure under confined conditions pGL , the corresponding angle φ cr and the corresponding amplitude of the buckling shape [subscript ( r)GL refers to the critical stage where pressure predicted by Glock (1977) is obtained]:  2/5 EAr r2 pGL r3 = 0.969 EIr EIr

(3.101)

 1/5   EAr r2 π = 0.856 φ GL EIr

(3.102)

    EIr 2/5 δ = 2.819 r GL EAr r2

(3.103)

For the case of a ring under plane-strain conditions, Eq. (3.101) can be written in the following form: pGL =

E  t 2.2 1 − νe 2 D

(3.104)

The most important observation is the striking difference in the values of buckling pressure under confined and unconfined conditions. Combining Eqs. (3.47) and

84

Structural Mechanics and Design of Metal Pipes

(A)

(B)

Figure 3.14 Finite element model for the analysis of externally-pressurized rings within a cavity; (A) model general view; (B) detail at the location of the inward buckle pattern.

(3.104), one obtains:   1 D 0.8 pGL = pcr 2 t

(3.105)

Considering D/t values from 100 to 300, the above ratio ranges from 20 to 48. This means that the value of pcr is of little interest in the present case and refers to a very early stage of loading.

3.3.2 Further results and discussion A finite element model has been developed (Vasilikis and Karamanos 2009, 2014; Vasilikis 2012), to simulate the response of an externally-pressurized ring within the cavity (Fig. 3.14) and the complete pressure-deformation equilibrium path was obtained, as shown in Fig. 3.15A for different values of D/t ratio. Furthermore, the finite element results in terms of the maximum (buckling) pressure are in excellent agreement with Glock’s predictions as shown in Fig. 3.15B. Nevertheless, the numerical results indicate that the ring response is highly unstable beyond the maximum pressure, associated with an “inversion” of the ring wall, resulting in a local curvature of opposite sign.

Structural mechanics of elastic rings

85

Figure 3.15 (A) Load-deflection path from finite element simulation. (B) Buckling pressure of elastic ring under uniform external pressure within a rigid cavity; finite element results versus Glock’s predictions (Glock, 1977).

86

Structural Mechanics and Design of Metal Pipes

The finite element results show that the maximum pressure corresponds to a ring configuration where the local curvature at θ = 0 changes sign, denoted as configuration (2) in Fig. 3.16, and it is consistent with Glock’s solution. Using Eqs. (3.95) and (3.97), the local curvature at θ = 0, denoted as kθ0 , is calculated as follows: kθ0 =

  w (0) δ π 2 = − r2 2r2 φ

(3.106)

The local curvature θ = 0 at maximum pressure stage can be obtained analytically substituting Eqs. (3.102) and (3.103) into Eq. (3.106) to get: kθ0,GL = −1.033

1 1 − r r

(3.107)

In conclusion, the analytical solution (Glock, 1977) and the finite element results (Vasilikis and Karamanos 2009, 2014; Vasilikis 2012), indicate that, at buckling, i.e. where p reaches its critical value pGL ), the total hoop kθ curvature at the θ = 0 location is instantaneously zero, i.e., inversion of the cylinder wall occurs. The unstable post-buckling behavior in Fig. 3.15A also implies that the structural response of the elastic ring is sensitive to initial imperfections. Two types of imperfections are considered, shown in Fig. 3.17; a “gap” imperfection due to different size of the cavity and the ring (Fig. 3.17A) and a local “out-of-roundness” imperfection in the form of a small inward deflection at the critical location (Fig. 3.17B). Numerical results, summarized in Fig. 3.18, indicate that the presence of those imperfections may reduce significantly the maximum pressure that the ring can sustain. Vasilikis and Karamanos (2009) have proposed the following knock-down factor α kd to quantify the reduction of maximum pressure due to initial imperfections: αkd =

0.15 ¯ 0.7

(3.108)

¯ is an initial imperfection parameter, defined as follows: where  ¯ =

δ0 + 3g r

  D t

(3.109)

The accuracy of this equation is shown in Fig. 3.18, plotted against finite element results.

3.4

Ring under transverse loading

The response of a ring subjected to opposite radial forces constitutes a problem with several applications in pipeline mechanics (see Fig. 3.19), as well as in crashworthiness analysis. Those radial forces cause ovalization of the ring cross-section, which may

Structural mechanics of elastic rings

Figure 3.16 Consecutive deformation shapes of a tightly-fitted elastic cylinder; configuration (2) corresponds to the maximum pressure stage.

87

88

Structural Mechanics and Design of Metal Pipes

(A)

(B)

Figure 3.17 Ring imperfections within a cavity: (A) “gap” imperfection due to size mismatch between the cavity and the ring and (B) local “out-of-roundness” imperfection.

Figure 3.18 Knock-down factor for the influence of imperfections on the buckling pressure of ¯ (Vasilikis and Karamanos, confined elastic rings in terms of initial imperfection parameter 2009).

induce significant stresses and strains in the ring. The radial forces are often applied through rigid plates (see Fig. 3.19A), crushing the ring, or through a more concentrated manner, i.e. with the use of wedge indenters (see Fig. 3.19B and Fig. 3.19C). For all cases, at initial stages of loading, ring deformation can be represented by the model in Fig. 3.19D, and it is examined in the following.

Structural mechanics of elastic rings

(A)

89

(B)

(C)

(D)

Figure 3.19 Ring under transverse loading. (A) Crushing between two rigid flat plates. (B) Radial concentrated transverse loads. (C) Concentrated radial loading of a ring resting on a flat plane. (D) Ring deformation.

3.4.1 Energy formulation for non-pressurized ring Assuming elastic material and inextensional ring kinematics, ring flattening can be computed using the energy formation, described in Section 3.2. The potential energy of the problem under consideration can be expressed as follows: 1 EIr

= 2 r3

2π



w + w

2

dθ − 2(−F )w(π /2)

(3.110)

0

Assume a doubly-symmetric trigonometric solution: w(θ ) = α cos 2θ

(3.111)

where α is the amplitude of deformation. Inserting into Eq. (3.110), the discretized form of the potential energy is obtained:

(α) =

9π EIr 2 α − 2Fα 2r3

(3.112)

For equilibrium, the stationary value of function (α) is sought: d =0 dα

(3.113)

and the resulting expression is written as follows: F=

9πEIr α 2r3

(3.114)

90

Structural Mechanics and Design of Metal Pipes

or equivalently, F = 0.589

E  t 3 δV 1 − νe 2 r

(3.115)

so that the shortening of diameter AA in Fig. 3.19D along the opposite loads is expressed as follows:   1 − νe 2 D 3 F δV = 0.212 E t

(3.116)

Because of the assumed shape function (3.111), the lengthening of diameter BB is the same with the shortening of diameter AA (δ H = δ V ).

3.4.2 A more accurate solution Timoshenko and Gere (1961) presented a more elaborate solution of this problem. Considering linear kinematics and the symmetry of the problem, the following equation for the radial displacement has been obtained: w(θ ) =

  4 Fr3 cos θ + θ sin θ − 4EIr π

(3.117)

which is valid for 0 ≤ θ ≤ π /2. Τhe shortening of diameter AA along the opposite loads is obtained from Eq. (3.117), considering the radial displacement at θ = π /2. One readily obtains: E  t 3 δV 1 − νe 2 r

(3.118)

  1 − νe 2 D 3 F E t

(3.119)

F = 0.5601 or δV = 0.223

The lengthening of diameter BB can be computed considering the lateral displacement in Eq. (3.117) at θ = 0. One readily obtains: δH = 0.918 δV

(3.120)

Therefore, in the solution of Timoshenko and Gere (1961) [Eq. (3.117)], δ V = δ H .

Structural mechanics of elastic rings

91

(A)

(B)

Figure 3.20 (A) Externally pressurized elastic ring under two opposite radial loads; (B) ring under two opposite radial loads and lateral confinement.

3.4.3 Effects of pressure or lateral confinement on elastic ring response under transverse loading The model shown in Fig. 3.20A represents the case of a transversely loaded ring (loads F) subjected to opposite loads in the presence of uniform external pressure p. An approximate solution of the problem can be obtained using an enhanced expression of the ring potential energy, to account for the effects of pressure. Combining Eqs. (3.110) and (3.88) the potential energy of the ring can be written as follows: 1 EIr

= 2 r3

2π



  2

w+w

1 dθ − 2(−F )w(π /2) − p 2

0

1 + p 2

2π

2π



v − w

2

dθ

0



 2rw + v2 − vw dθ

(3.121)

0

and assuming a doubly symmetric trigonometric solution: w(θ ) = α cos 2θ

(3.122)

one obtains the discretized form of potential energy:

(α) =

9π EIr 2 9π p 2 3π p 2 α − 2Fα − α − α 2r3 8 8

(3.123)

The stationary value of function (α) reduces in the following expression of the load-deflection relationship: F=

  9π EIr p 1 − α 2r3 3EIr /r3

(3.124)

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Structural Mechanics and Design of Metal Pipes

or δV = 0.212

   1 − νe 2 D 3 1 F E t 1 − p/pcr

(3.125)

The same expressions may also be used for the case of internal pressure considering opposite sign of pressure. In this case, the internal pressure has an opposite contribution to the externally applied loads, reducing the deflection imposed by transverse loading, and increasing the resistance against transverse loading. The model shown in Fig. 3.20B represents the case of a transversely loaded ring (loads F) in the presence of lateral confinement, a case quite common in buried flexible thin-walled pipes (Watkins, 2004). In this model, lateral confinement is represented by the two horizontal springs kH at θ = 0 and θ = π /2. Similar to the previous case, an approximate solution can be obtained enhancing the expression of the ring potential energy in Eq. (3.110), to account for the effects of the lateral spring as follows: 1 EIr

= 2 r3

2π



w + w

2

1 dθ + kH w2 (0) − 2(−F )w(π /2) 2

(3.126)

0

Assuming the doubly-symmetric solution of Eq. (3.111), the solution is obtained in a straightforward manner minimizing the potential energy expression, and can be written in the following form: α=

0.071Fr3 EIr + 0.071kH r3

(3.127)

References Barlow, P. (1837). A Treatise on the Strength of Timber, Cast Iron, Malleable Iron and Other Materials With Rules For Application in architecture, Construction of Suspension bridges, railways, etc.; and an Appendix On the Power of Locomotive engines, and the Effect of Inclined Planes and Gradients. London: John Weale. Bresse, M. (1866). Cours De Méchanique Appliquée (2nd Edition). Paris, France. Brush, D. O., & Almroth, B. O. (1975). Buckling of Bars, Plates, and Shells. New York, NY: McGraw-Hill. Bryan, G. H. (1888). Application of the energy test to the collapse of a long pipe under external pressure. In Proceedings of the Cambridge Philosophical Society, Proc UK: 6 (pp. 287– 292). Budiansky, B. (1974). Theory of Buckling and Post-Buckling Behavior of Elastic Structures. Advances in Applied Mechanics, 14, 1–65. Det Norske Veritas. (2021). Submarine Pipeline Systems. STANDARD DNV-ST-F101. Høvik, Norway. Donnell, L.H. (1933). Stability of Thin-Walled Tubes under Torsion. NACA report 479.

Structural mechanics of elastic rings

93

El Naschie, M. S. (1975). The initial postbuckling of an extensional ring under external pressure. International Journal of Mechanical Sciences, 17, 387–388. Glock, D. (1977). Überkritisches Verhalteneines Starr Ummantelten Kreisrohresbei Wasserdrunck von Aussen und Temperaturdehnung (Post-Critical Behavior of a Rigidly Encased Circular Pipe Subject to External Water Pressure and Thermal Extension). Der Stahlbau, 7, 212–217. Koiter, W. T. (1963). On the concept of stability of equilibrium for continuous bodies. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, B66, 173–177. Kyriakides, S., & Babcock, C. D. (1981). Large Deflection Collapse Analysis of an Inelastic Inextensional Ring Under External Pressure. International Journal of Solid Structures, 17(10), 981–993. Kyriakides, S., & Corona, E. (2007). Mechanics of Offshore Pipelines. Buckling and Collapse: Vol. 1. Elsevier. Omara, A. M., Guice, L. K., Straughan, W. T., & Akl, F. A. (1997). Buckling Models of Thin Circular Pipes Encased in Rigid Cavity. Journal of Engineering Mechanics, ASCE, 123(12), 1294–1301. Timoshenko, S. P. (1933). Working stresses for columns and thin-walled structures. Transaction ASME Applied Mechanics Division I, 173–185. Timoshenko, S. P., & Gere, J. M. (1961). Theory of Elastic Stability (2nd Edition). McGraw-Hill Book Company, International Student Edition. Vasilikis, D., & Karamanos, S. A. (2009). Stability of Confined Thin-Walled Steel Cylinders under External Pressure. International Journal of Mechanical Sciences, 51(1), 21–32. Vasilikis, D. (2012). Structural Stability of Confined Cylindrical Steel Shells. PhD Dissertation. Department of Mechanical Engineering, University of Thessaly. Vasilikis, D., & Karamanos, S. A. (2014). On the Mechanics of Confined Steel Cylinders Under External Pressure. Applied Mechanics Reviews, ASME, 66 Article Number 010801. Watkins, R. K. (2004). Buried Pipe Encased in Concrete. In International Conference on Pipeline Engineering and Construction. ASCE San Diego, CA.

Stresses in thick-walled pressurized cylinders

G

For cylindrical members and vessels subjected to hydrostatic pressure conditions, the application of thin-walled cylindrical shell theory results in the well-known Barlow’s equation for hoop stress, and the corresponding equation for axial stress, whereas the radial stress is zero (see Section 4.1). The equations for thin-walled capped cylindrical shells are summarized below and, despite some small variations, they have been the basis for determining stress in pressurized pipes in most pipeline codes and standards (e.g., American Society of Mechanical Engineers, 2018; American Society of Mechanical Engineers, 2019; European Committee for Standardization, 2013; American Petroleum Institute, 2015; Det Norske Veritas, 2021). (a) Circumferential (hoop) stress σθ = (pint − pext )

D 2t

(G.1)

(b) Radial stress σρ = 0

(G.2)

(c) Longitudinal (axial) stress σx = (pint − pext )

D 4t

(G.3)

where pint − pext is the net pressure on the pipe wall (internal pint minus external pext ), and D is the pipe diameter. In the present book D represents the outer diameter. However, in thin-walled cylinder theory, a distinction between the outer diameter D, the inner diameter Di , and the mean diameter Dm is inconsequential. For pressurized cylindrical shells with values of the D/t ratio less than 15, the cylindrical shell is no longer considered as thin-walled, in the sense that Eqs. (G.1)– (G.3) are not accurate. Instead, thick-shell equations should be used.

G.1

Stresses in capped thick-walled cylinders under internal and external pressure

Consider a thick-walled capped cylindrical shell of inside radius ri and outside radius ro subjected to internal pressure pint and external pressure pext , as shown in Fig. G.1. Determining the stresses in the pipe wall requires the solution of an axisymmetric Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00015-3 c 2023 Elsevier Inc. All rights reserved. Copyright 

476

Structural Mechanics and Design of Metal Pipes

Figure G.1 Thick-walled capped cylinder under internal and external hydrostatic pressure; circumferential (hoop), longitudinal and radial stresses.

two-dimensional elasticity problem, which accounts for the variation of stress through the cylinder thickness. Describing the steps for solving this problem is out of the scope of the present discussion, and the reader is referred to the classical textbook by Timoshenko and Gere (1961) or any other elasticity textbook. The solution can be summarized in the following: (a) Circumferential (hoop) stress σ θ : σθ = r

(pext − pint )ri2 ro2 pint ri2 − pext ro2   − 2 2 ro − ri ρ 2 ro2 − ri2

Internal pressure pint only: σθ,i

r

(G.4)

  ro2 ri2 1+ 2 = pint 2 ro − ri2 ρ

(G.5)

External pressure pext only: σθ,o = −pext

  ri2 ro2 1 + ro2 − ri2 ρ2

(G.6)

(b) Longitudinal (axial) stress σ x : σx =

pint ri2 − pext ro2 ro2 − ri2

(G.7)

Stresses in thick-walled pressurized cylinders

477

Figure G.2 Distribution of hoop and radial stresses across the thickness of a thick-walled cylinder. (A) Internal pressure. (B) External pressure. r

Internal pressure pint only: σx,i = pint

r

ri2 ro2 − ri2

(G.8)

External pressure pext only: σx,o = −pext

ro2

ro2 − ri2

(G.9)

(c) Radial stress σ ρ : σρ = r

(pext − pint )ri2 ro2 pint ri2 − pext ro2   + 2 2 ro − ri ρ 2 ro2 − ri2

Internal pressure pint only: σρ,i = pint

r

(G.10)

  ro2 ri2 1 − ro2 − ri2 ρ2

(G.11)

External pressure pext only: σρ,o = −pext

  ri2 ro2 1 − ro2 − ri2 ρ2

(G.12)

The above equations show that the hoop and the radial stresses vary across the cylinder thickness, and their variation is shown in Fig. G.2A and B for internal pressure pint and external pressure pext , respectively, whereas axial stress is constant through the cylinder thickness. One can readily show that Eqs. (G.6)–(G.12) result in the thinwalled expressions (G.1)–(G.3) when cylinders of very small thickness are considered.

478

G.2

Structural Mechanics and Design of Metal Pipes

Approximate expressions for stresses in thick-walled cylinders

For moderately thick cylindrical shells, it is possible to account for the effect of the “finite” cylinder thickness employing thin-shell theory and using the mean radius rm or the mean diameter Dm . Therefore, for the case of internally pressurized cylinders one may write for the hoop stress: σθ = pint

Dm 2t

(G.13)

and considering that Dm = D − t: σθ = pint

D−t 2

(G.14)

Starting from Eq. (G.14) in the course of pipe design for internal pressure pi , the minimum required thickness is equal to: tmin =

pint D 2(σW + 0.5pint )

(G.15)

where σ W is the allowable stress of the pipe material. A similar expression can be obtained if one considers in Eq. (G.13) that Dm = Di + t. In this case, σθ = pint

Di + t 2

(G.16)

and the minimum required thickness is equal to: tmin =

pDi 2(σW − 0.5pint )

(G.17)

Eqs. (G.15) and (G.17) are similar to the equations for minimum thickness stated in ASME Boiler and Pressure Vessel Code, Section VIII Division 1 (American Society of Mechanical Engineers, 2021b), widely used for pressure vessel design in nuclear facilities (EW is the weld efficiency factor introduced in Section 7.2): tmin =

pint ro σW EW + 0.5pint

(G.18)

tmin =

pint ri σW EW − 0.5pint

(G.19)

Stresses in thick-walled pressurized cylinders

479

and in ASME Boiler and Pressure Vessel Code Section III, Division 1 (American Society of Mechanical Engineers, 2021a) for the design of nuclear piping: pint ro σW EW + 0.4pint pint ri = σW EW − 0.6pint

tmin =

(G.20)

tmin

(G.21)

Equations for pressure design of straight pipes which are analogous to Eqs. (G.18)– (G.21) can be also found in ASME B31.1 (American Society of Mechanical Engineers, 2020) and ASME B31.3 (American Society of Mechanical Engineers, 2021c) standards.

References American Petroleum Institute (2015). Design, construction, operation, and maintenance of offshore hydrocarbon pipelines (Limit State Design). API RP 1111 (reaffirmed 2021). Washington, DC. American Society of Mechanical Engineers. (2018). Gas transmission and distribution piping systems. ASME B31.8 Standard. New York, NY. American Society of Mechanical Engineers. (2019). Pipeline transportation systems for liquid hydrocarbons and other liquids. ASME B31.4 Standard. New York, NY. American Society of Mechanical Engineers. (2020). Power Piping. ASME B31.1 Code for Pressure Piping Standard. New York, NY. American Society of Mechanical Engineers. (2021a). Boiler and Pressure Vessel Code Section III-Rules for Construction of Nuclear Facility Components-Division 1. New York, NY. American Society of Mechanical Engineers. (2021b). Boiler and Pressure Vessel Code Section VIII-Rules for Construction of Pressure Vessels Division 1. New York, NY. American Society of Mechanical Engineers. (2021c). Process Piping. ASME B31.3 Standard. New York, NY. Det Norske Veritas. (2021). Submarine Pipeline Systems. STANDARD DNV-ST-F101. Høvik, Norway. European Committee for Standardization (2013). Gas supply systems-Pipelines for maximum operating pressure over 16 bar. EN 1594 Standard. Brussels, Belgium. Timoshenko, S. P., & Gere, J. M. (1961). Theory of Elastic Stability (Second Edition). McGrawHill Book Company, International Student Edition.

Mechanical behavior of metal pipes under internal and external pressure

5

The present chapter refers to metal rings and cylinders under primarily external pressure. The formulae and results from elastic cylinders and pipes (Chapters 3 and 4) constitute the basis of the present analysis but are adjusted and extended to account for elastic-plastic material behavior, which characterizes metal cylinders and pipes. Following a brief note on metal pipe response under internal pressure, the structural behavior and collapse of metal pipes under external pressure is presented in detail. The special issues of propagating buckles in long cylinders, and metal ring instability under confined conditions are also discussed. In this chapter, the value of the diameterto-thickness ratio of the pipes analyzed is relatively low and therefore, it is necessary to distinguish between the outer diameter D and the mean diameter Dm (Dm = D−t), which corresponds to the reference line of the pipe cross-section.

5.1

A brief note on pipe response under internal pressure

Internal pressure is the primary parameter for pipeline mechanical design. Pipe resistance to internal pressure is the main requirement that the pipe should satisfy to fulfill its fluid transportation function. In pipeline design procedure, this requirement defines the size of pipe thickness. Consider a long metal cylinder subjected to gradually increased internal pressure. At low levels of internal pressure, the pipe response is elastic, and stresses and strains can be calculated using the axisymmetric analysis in Section 4.1. Referring to the biaxial state of stress shown in Fig. 5.1A, from Barlow’s formula and using the mean diameter Dm , the circumferential stress σ θ is equal to: σθ =

pDm 2t

(5.1)

and is always larger than the longitudinal stress σ x regardless of the pipe end conditions (see Section 4.1). Upon increase of internal pressure, the pipe material yields, and the pipe wall develops plastic deformation. Subsequently, with further increase of internal pressure, the pipe material hardens and, eventually, the pipe bursts because of pipe wall fracture, due to excessive tensile strain. Considering that the hoop stress σ θ is always larger than the corresponding longitudinal stress σ x , fracture occurs along a pipe generator (Fig. 5.1B), which is perpendicular to the circumferential tensile stress (σ θ ). Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00003-7 c 2023 Elsevier Inc. All rights reserved. Copyright 

134

Structural Mechanics and Design of Metal Pipes

(A)

(B)

Figure 5.1 (A) Biaxial state of stress in internally pressurized pipes; (B) burst of steel pipe under internal pressure [with permission from Arnold (Nol) M. Gresnigt].

In a long pipe subjected to internal pressure, and in the absence of a serious local defect, predicting the exact location where fracture would initiate is a non-trivial, if not impossible, task. The location depends on the presence of initial imperfections, mainly in terms of pipe wall thickness variations, which are usually due to pipe material corrosion. The presence of gauges in the pipe wall, usually caused by external interference during pipeline construction or operation, may also affect the burst pressure and determine the location of fracture. One may argue that the presence of seam welds (longitudinal or spiral) would be prone to pipe wall fracture. However, those welds are performed in a controlled environment in the pipe mill. Furthermore, the weld material is overmatched, i.e., stronger than the base metal material. Therefore, spiral or longitudinal seam welds, if properly constructed, may not be critical locations for pipe wall fracture under internal pressure. In addition, the internal pressure capacity of a pipeline may not be affected by the strength of the girth (circumferential) welds because the orientation of those welds is not directly affected by the circumferential stress developed in the pipe wall. Therefore, if the girth welds are properly constructed and do not contain significant defects, they do not constitute weak locations in terms of pipeline fracture under internal pressure. Experimental evidence has shown that, most likely, fracture of an internally-pressurized pipe would occur away from the welds, as shown in Fig. 5.1B. There exist several analytical models for predicting the burst pressure of an internally pressurized pipe (Stewart et al., 1994; Zhu and Leis, 2007; Oh et al., 2020) and the interested reader is referred to the review paper by Zhu and Leis (2012) for a concise presentation of pipe burst models. Those models and the available experimental results have demonstrated that an intact pipe, i.e. a pipe without significant defects, is capable of sustaining an internal pressure level equal to or above the yield pressure pY , where pY = 2σY t/Dm and σ Y is the yield stress of the pipe material. Therefore, the value of pY can be used as a lower bound estimate of the burst pressure of the pipe. If the modified yield stress σY∗ in the circumferential direction is considered, which accounts for the presence of axial stress σ x in the pipe through the use the Von Mises yield criterion, as

Mechanical behavior of metal pipes under internal and external pressure

135

described in Section 4.1, the value of the corresponding yield pressure pY∗ = 2σY∗t/Dm offers a more realistic yet lower bound prediction of the burst pressure (Stewart et al., 1994; Bruschi et al., 2013), and it is adopted by several design specifications (see Section 8.3).

5.2

External pressure collapse and post-buckling response

The failure mode of internally pressurized pipes is material-related and relates to pipe material ductility. On the other hand, the behavior of metal pipes under external pressure is characterized by structural instability. In Section 3.2, buckling of an elastic ring under plane strain conditions subjected to external pressure was analyzed and the critical pressure pcr for imperfection-free rings was determined as follows: pcr =

  2E t 3 1 − νe2 Dm

(5.2)

The analysis is representative of the response of an externally pressurized long elastic pipe. The corresponding buckling mode is expressed in terms of the radial displacement as follows: w(2) (θ ) = B cos 2θ

(5.3)

where subscript ()(2) denotes that the buckling mode has two halfwaves in the circumferential direction and B is an arbitrary constant. Using Barlow’s formula (p = 2σ t/Dm ) at the onset of buckling, the critical pressure pcr corresponds to a circumferential stress σ cr expressed as:   E t 2 σcr = 1 − νe2 Dm

(5.4)

5.2.1 The concept of ring slenderness The above elastic solution is valid if the ring material remains elastic until the onset of buckling. This is expressed by the following inequality: σcr < σY

(5.5)

or Dm > t



E  σY 1 − νe2 

(5.6)

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Structural Mechanics and Design of Metal Pipes

Figure 5.2 Schematic diagram showing the dependence of maximum (ultimate) stress of metal pipes on the slenderness parameter, in the absence of initial imperfections.

Using the above notation, and neglecting the presence of initial imperfections, a simple approach for determining the maximum pressure sustained by a ring made of elastic-plastic material would be: (a) to compute pcr and pY (pY = 2σY t/Dm ), (b) to consider the minimum of those two values as the ultimate pressure pmax (pmax = min[pY ,pcr ]). Using Barlow’s formula, and introducing the following slenderness parameter:  ¯ = σY (5.7) λ σcr or ¯ = Dm λ t



 1 − νe2 σY E

(5.8)

this approach can be expressed in terms of the maximum hoop stress σ max (σmax = 0.5 pmax Dm /t) as follows: σmax = σY σmax = σY

1 ¯λ2 if

if

¯ ≥1 λ

¯ ≤1 λ

(5.9) (5.10)

and it is shown graphically in Fig. 5.2. The corresponding ultimate pressure pmax is also referred to as “collapse pressure”. This approach does not account for initial geometric imperfections or strain hardening effects. Those issues will be discussed in subsequent sections of the present chapter. The finite element results in Fig. 5.3A depict the pressure response of a very thick¯ = 0.441, where D is the outer walled X65 steel ring (D/t = 10, σ Y = 448.5 MPa, λ

Mechanical behavior of metal pipes under internal and external pressure

137

Figure 5.3 (Α) Response of a thick-wall X65 steel pipe (D/t = 10) under external pressure; hydrostatic pressure versus lateral pressure conditions; (B) stress-strain diagram for the X65 steel material.

diameter of the pipe) and are obtained from cross-sectional (ring) analysis, considering the ring as a slice of a long pipeline. The stress-strain curve of the pipe material is shown in Fig. 5.3B. In the graph of Fig. 5.3A, the value of pressure p is plotted in terms of the change of area enclosed by the pipe AE normalized by the initial area enclosed by the pipe AE0 . The analysis assumes uniform external pressure and plane strain conditions of the ring, where out-of-plane displacements are zero, representing

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Structural Mechanics and Design of Metal Pipes

ˆ = the real conditions of a long deep-water pipeline. A small value of initial ovality ( 0.2%) is also employed in this analysis to enable smooth transition from pre-buckling to post-buckling. The pressure-area diagram reaches a maximum pressure value, denoted as pmax , which is the collapse pressure of the pipe. Beyond that stage, a rapid drop of pressure occurs, and this indicates a highly unstable post-buckling behavior. Because of the large thickness of this pipe (low value of D/t ratio), the collapse (maximum) pressure pmax depends highly on the yield stress of the pipe material and is equal to 108.7 MPa. From Barlow’s formula (σθ = 0.5pDm /t), this pressure corresponds to a hoop stress σ θ equal to 489.2 MPa, which is significantly higher than the yield stress of the pipe material. The above numerical result is an indication that thick-walled steel pipes with small initial imperfections can sustain high levels of external pressure, corresponding to hoop stress that exceeds the yield stress of the material.

5.2.2 Bifurcation solution in the plastic range It is possible to conduct a bifurcation analysis of externally-pressurized rings in the inelastic range using the analytical formulation outlined below (Ju and Kyriakides, 1988; Kyriakides and Corona, 2007), and obtain closed-form expressions for the corresponding bifurcation pressure p¯ cr . The analysis employs: (a) The rate form of equilibrium equations of the ring: 1 d M˙ θ d N˙ θ + =0 dθ rm dθ     2 d 2 M˙ θ ˙ θ + prm d v˙ − d w˙ − prm d v˙ + w˙ = 0 − r N m dθ 2 dθ dθ 2 dθ

(5.11) (5.12)

where Nθ and Mθ are the stress resultants (axial force and bending moment) in the hoop direction of the ring and the dot on top denotes the rate of a quantity. (b) The incremental form of ring kinematics: ε˙θ = ε˙θm + ζ k˙ θ

(5.13)

where ε˙θm =

  1 d v˙ + w˙ rm dθ

and

  d 2 w˙ 1 d v˙ k˙ θ = − rm dθ dθ 2

(5.14)

(c) The incremental elastic-plastic constitutive Eqs.:   σ˙ x C = xx σ˙ θ Cxθ

Cxθ Cθθ

 ε˙x ε˙θ

(5.15)

also written in more compact form: ˙ = [C]ε˙ σ

(5.16)

Mechanical behavior of metal pipes under internal and external pressure

139

The constitutive matrix [C] in Eq. (5.16) depends on the plasticity model adopted in the analysis. Using J2 -flow theory of plasticity:

   1 ε˙x 1 + q f (2σx − σθ )2 −ve + q f (2σx − σθ )(2σθ − σx ) σ˙ x = ε˙θ σ˙ θ 1 + q f (2σθ − σx )2 E −ve + q f (2σx − σθ )(2σθ − σx ) (5.17) where:   1 E qf = −1 2 4σeq ET

(5.18)

On the other hand, adopting the rate form of J2 -deformation theory of plasticity (see also Appendix D), the corresponding constitutive expressions are:  

 1 1 + qd (2σx − σθ )2 −vS + qd (2σx − σθ )(2σθ − σx ) σ˙ x ε˙x = ε˙θ σ˙ θ 1 + qd (2σθ − σx )2 ES −vS + qd (2σx − σθ )(2σθ − σx ) (5.19) where: qd =

  1 ES − 1 , 2 4σeq ET

vS =

  1 ES 1 + ve − 2 E 2

(5.20)

In Eqs. (5.17)–(5.20), σ eq is the von Mises equivalent stress, and ET and ES are the tangent and secant modulus of the material, respectively (see Appendix D). In each case, matrix [C] is the inverse of the above 2 × 2 constitutive matrices in Eqs. (5.17) and (5.19). Combining the equilibrium, the constitutive and the kinematic equations as expressed in Eqs. (5.11)–(5.16), and assuming a doubly symmetric displacement function for the buckling mode: w1 = cw cos 2θ

(5.21)

v1 = cv sin 2θ

(5.22)

where cw , cv are arbitrary constants, one results in the following expression for the buckling pressure in the inelastic range p¯cr :   Cθθ t 3

1 + (1/4 fL )(t/Dm )2 Dm

p¯ cr = 2

(5.23)

where Cθθ is the appropriate element of the constitutive matrix [C] and fL depends on the type of pressure loading, i.e., plane strain conditions with zero axial strain,

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Structural Mechanics and Design of Metal Pipes

hydrostatic pressure or lateral-only pressure. For the case of hydrostatic pressure: σθ = −

pDm , 2t

σx = −

pDm 4t

(5.24)

and one obtains: Cθθ =

ES + 3(ES /ET )] [1 L

(5.25)

4νS2 1 + 3(ES /ET )

(5.26)

1/4 f

and fL = 1 −

For the cases of zero axial strain (εx = 0) and lateral pressure only (σ x = 0) the reader is referred to the paper by Ju and Kyriakides (1988). If the stress-strain curve of the pipe material is expressed analytically, closed-form expressions for ES , ET , and ν S can be obtained. The Ramberg-Osgood equation below is widely used for expressing the uniaxial stress-strain curve of metals:  nˆ σ σ ε = + αR (5.27) E σ0 where σ 0 is a reference stress, α R is a material constant, often taken equal to 3/7, and nˆ is an exponent that represents material strain hardening. Perhaps the most interesting and useful result of the above inelastic buckling analysis is that the corresponding buckling mode is identical to the one obtained for the elastic case, with two waves around the ring circumference, expressed in Eqs. (5.21) and (5.22). This result is used in the analysis of imperfect rings below.

5.2.3 Analysis of initially oval elastic-plastic rings The bifurcation solutions for the elastic case (Chapter 3) and for the elastic-plastic case presented above, have been obtained under the assumption of zero initial imperfections. Nevertheless, initial imperfections are always present in practical applications and play a significant role on the value of the ultimate (collapse) pressure pmax sustained by the pipe. For the purposes of the present analysis, an initially imperfect ring is considered, as part (slice) of a long cylinder, with an oval initial imperfection in the doubly-symmetric inextensional form expressed in terms of the radial and tangential displacements as follows: w(θ ˆ ) = αˆ cos 2θ

(5.28)

and v(θ ˆ )=−

αˆ sin 2θ 2

(5.29)

Mechanical behavior of metal pipes under internal and external pressure

141

Figure 5.4 Response of an initially oval pipe subjected to uniform external pressure.

The displacement pattern in Eqs. (5.28) and (5.29) is similar to the buckling modes obtained for elastic and elastic-plastic bifurcation of externally-pressurized rings. In Section 3.2, the response of an imperfect ring subjected to uniform external pressure was analyzed, assuming elastic material behavior. It was found that with increasing pressure, the ring deforms retaining its oval configuration, and that the value of initial ovalization αˆ grows to α according to the following formula:  α = αˆ

1 1 − p/p c r

 (5.30)

where pcr is the elastic buckling pressure obtained in Section 3.2. Eq. (5.30) is shown schematically with the red curve in Fig. 5.4. The amplification of ovalization expressed by Eq. (5.30) results eventually in the development of plastic deformation at θ = 0, π/2, π, 3π /2 . First yielding occurs at pressure level pF given by the following expression (see also Section 3.2):

p 2F

      σY t 6r αˆ σY t − + 1+ pcr pF + pcr = 0 rm t rm rm

(5.31)

which has been first reported by Timoshenko (1933). Upon first yielding of the ring, the response becomes more compliant. Eq. (5.30) is no longer valid, and for this reason it is continued with the dotted red line in Fig. 5.4, whereas the true response is lower than the dotted line. Beyond that point, the ovalization increases more rapidly,

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Structural Mechanics and Design of Metal Pipes

Figure 5.5 Development of ring plastic mechanism with four equally-spaced plastic hinges.

Figure 5.6 Geometric features of four-plastic-hinge mechanism.

plastic deformation spreads across the thickness of the ring, forming a plastic hinge at each of four equally-spaced locations (θ = 0, π/2, π, 3π /2), and this results in the plastic-hinge mechanism shown in Fig. 5.5. The formation of the four plastic hinges corresponds to a plastic mechanism and the ring becomes structurally unstable, leading to ring (cylinder) collapse, also shown schematically in Fig. 5.5. Considering the geometric features shown in Fig. 5.6, the equilibrium of forces in the horizontal direction and the balance of moments with respect to B’ (Fig. 5.7), one results in Eq. (5.32) (see Appendix F) that relates the external pressure p with the deformation of the plastic-hinge mechanism. The latter is expressed by the ring deflection α and normalized by the ring radius rm (x = α/rm ): p = pY



 t 1 Dm 2x − x2

(5.32)

Eq. (5.32) represents an unstable structural response characterized by significant reduction of pressure with increasing ring deformation and it is shown schematically in Fig. 5.4 with the green dotted line. For the detailed analysis of the plastic mechanism that leads to the derivation of Eq. (5.32), the reader is referred to Appendix E.

Mechanical behavior of metal pipes under internal and external pressure

143

Figure 5.7 Forces and moments in the four-plastic-hinge mechanism.

The connection of solutions expressed by Eqs. (5.30) and (5.32) with the real behavior of the ring is discussed in the following. Fig. 5.8 shows schematically the elastic solution expressed by Eq. (5.30) plotted with the red curve, the plastic collapse solution expressed by Eq. (5.32) plotted with the green curve, and the true response, plotted with the continuous purple curve. There exists a transition from the elastic behavior to the fully-plastic response. The maximum pressure pmax sustained by the ring is the highest value of the purple curve, somewhat higher than pF , and occurs within this transition part. The accurate calculation of maximum pressure pmax requires a numerical analysis of the externally pressurized ring (e.g., finite element analysis), which accounts for structural instability phenomena and describes the spread of plasticity in the ring and the gradual formation of plastic hinges. However, it is possible to obtain reasonable estimates of pmax using simpler tools. The value of first yielding pressure (pF ) may offer a conservative (lower-bound) estimate of the collapse pressure pmax of the ring. Another estimate of pmax can be obtained considering the intersection between the elastic ovalization path and the plastic mechanism path, as shown in Fig. 5.8. In this approach, the structural response is approximated by two intersecting lines, the red line and the green line, representing the two parts of ring deformation: (a) the purely elastic part, defined by Eq. (5.30) and (b) the fully-plastic part, expressed by the plastic mechanism in Eq. (5.32). The maximum pressure obtained by this approach, denoted as pPL in Fig. 5.8, is the intersection of the two curves and offers an upper bound of the actual collapse pressure pmax . Therefore, the value of collapse pressure is bounded by the value of pF and pPL (pF < pmax < pPL ).

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Structural Mechanics and Design of Metal Pipes

Figure 5.8 Simplified approach for defining the ultimate pressure capacity of an externally pressurized metal pipe.

Three X65 steel pipes with D/t = 20.5, 30.5 and 40 (D is the outer pipe diameter) are analyzed with two-dimensional finite element models under plane strain conditions. The finite element model is shown in Fig. 5.9 for the thick-walled pipe (D/t = 20.5) and consists of eight-node quadratic plane-strain elements. Analogous meshes are used for the other two pipes. The stress-strain diagram of the steel material used in those analyses is the one shown in Fig. 5.3B. One quarter of the pipe cross-section is analyzed due to double-symmetry of the problem. Fig. 5.10–Fig. 5.12 depict the external pressure response of those three pipes and show the equilibrium path of pressure with respect to the ovalization of pipe cross-section, for different values of initial ovality. Cross-sectional ovalization is expressed through the ovalization parameter (or “ovality”): =

Dmax − Dmin Dmax + Dmin

(5.33)

introduced in Chapter 3. The initial value of this ovalization parameter , prior to the ˆ For each curve in Fig. 5.10–Fig. 5.12, application of external pressure, is denoted as . ˆ the value of  corresponds to the starting point of the curve on the horizontal axis of the diagram.

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Figure 5.9 Finite element model and mesh for the collapse analysis of a X65 steel pipe (D/t = 20.5); initial (undeformed) and deformed configuration of the model.

Figure 5.10 Response of an X65 steel pipe subjected to external pressure, for different values of initial imperfection (D/t = 20.5).

The results shown in Fig. 5.10–Fig. 5.12 follow the trends of the purple curve in the qualitative diagram of Fig. 5.8. For each case, a maximum value of pressure (collapse pressure) is reached, beyond which the response is unstable, characterized by an abrupt reduction of pressure and sudden collapse, also reported in relevant experimental data (e.g., Yeh and Kyriakides, 1988). The results also show that the

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Figure 5.11 Response of an X65 steel pipe subjected to external pressure, for different values of initial imperfection (D/t = 30.5).

Figure 5.12 Response of an X65 steel pipe subjected to external pressure, for different values of initial imperfection (D/t = 40).

Mechanical behavior of metal pipes under internal and external pressure

Table 5.1 Properties of X65 steel pipes analyzed in Fig. 5.10–Fig. 5.16. ¯ D/t slenderness λ p0max (MPa) pY (MPa) pcr (MPa) pF (MPa) 20.5 0.86 47.0 46.0 62.2 45.1 30.5 1.30 17.5 30.4 18.0 17.8 40 1.72 7.63 23.0 7.78 7.73

147

p0max /pY 1.02 0.58 0.33

Notes: (1) The material curve in Fig. 5.3B is used for all pipes (yield stress σ Y = 448.5 MPa). ˆ = 0.01% (nearly perfect pipe). (2) The collapse pressure p0max is computed numerically with  (3) The yield pressure pY and the elastic buckling pressure pcr are calculated with the mean pipe diameter. ˆ = 0.01%. (4) The first yield pressure pF is calculated from Eq. (5.31) with 

diameter-to-thickness ratio (D/t) or (Dm /t) is a very important parameter for the value ˆ equal to 0.01%, the of the collapse pressure. Considering a very small value of  collapse pressure is equal to 47 MPa, 17.5 MPa and 7.63 MPa for D/t = 20.5, 30.5 and 40 respectively and it is denoted as p0max in Table 5.1. Because of this very small ˆ those pipes may be considered as “nearly perfect”. The p0max value for the value of , thick-walled pipe (D/t = 20.5) is very close to pY , whereas for the other two values of D/t, the p0max value is comparable with the elastic buckling pressure pcr . This is due to the cross-sectional (ring) slenderness of the pipe. Using Eq. (5.8), the ring slenderness of the thick-walled pipe (D/t = 20.5) is readily calculated less than 1, which means that inelastic behavior governs the response. On the other hand, the ring slenderness of the other two pipes is greater than 1, and therefore, elastic buckling governs the response. It is also interesting to note that regardless of the initial ovality value, in Fig. 5.10-Fig. 5.12 all curves in the post-buckling range converge to one curve, because of the development of the plastic-hinge mechanism. Fig. 5.13 summarizes the results in Fig. 5.10–Fig. 5.12 and depicts the variation of ˆ The pmax values maximum pressure pmax with respect to the value of initial ovality . are normalized by the yield pressure of the pipe pY , calculated using the mean diameter Dm (Dm = D − t) of the pipe cross-section (pY = 2σY t/Dm ). The results demonstrate the significant effect of initial ovality on collapse. For the pipe with D/t = 40, an ˆ = 1% corresponds to 25% reduction of the collapse pressure. For initial ovality  ˆ equal to 1% reduces the the thick-walled pipe (D/t = 20.5), an initial ovality  collapse pressure by 40%. It is also noted that the collapse pressure of the thick-walled ˆ pipe (D/t = 20.5) for a very small value of initial ovality (=0.01%) slightly exceeds the value of pY . This is due to plane strain conditions that increase the compressive yield stress in the circumferential direction due to Von Mises yield criterion (see also Section 4.1 and Appendix D). Several consecutive stages of cross-sectional deformation from the stage of maximum pressure to the stage of complete collapse and the corresponding location of those stages on the pressure-ovalization diagram are shown in Fig. 5.14A,B and Fig. 5.15A,B for the X65 pipes with D/t = 40 and D/t = 20.5 respectively, and for a very small ˆ = 0.01%). It is interesting to note that the configuration value of initial ovality ( of the pipe cross-section at maximum (collapse) pressure is nearly circular, and that the ovalized shape develops gradually into the post-buckling range. The distribution of plastic deformation in both Fig. 5.14C and Fig. 5.15C clearly shows the formation of

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Figure 5.13 Collapse pressure dependence on value of initial cross-sectional ovalization for X65 steel pipes (D/t = 20.5, 30.5 and 40).

plastic mechanism with four equally-spaced plastic hinges, which leads to the collapsed shapes shown in Fig. 5.14D and Fig. 5.15D. The elastic and elastic-plastic ring response for two pipes with D/t = 40 and 20.5 are compared in Fig. 5.16A and Fig. 5.16B, respectively. Both elastic and elastic-plastic solutions were obtained by two-dimensional plane strain finite element simulations, for different levels of initial ovality. The dotted curves refer to the elastic results, whereas the continuous curves correspond to elastic-plastic (X65 steel) response. The nonlinear elastic post-buckling solution for the perfect case, expressed by Eq. (3.94), and the plastic mechanism equation are also depicted in Fig. 5.16. The latter is expressed by Eq. (5.34), which is equivalent to Eq. (5.32) and is obtained using the definition of ovality in Eq. (5.33), as described in Appendix E. p = pY



 t 1 + 2 Dm 2

(5.34)

In the diagrams shown in Fig. 5.16A and Fig. 5.16B, first yielding can be identified as the point where the elastic-plastic curve starts deviating from the corresponding elastic curve. This value of pressure can be calculated analytically with good accuracy from Eq. (5.31). Furthermore, the results in Fig. 5.16A and Fig. 5.16B show that it offers a reasonable, yet somewhat conservative, estimate of the collapse pressure. On the other hand, the value of pressure at the intersection of the elastic path Eq. (5.30) with the plastic path Eq. (5.34) may be quite higher than the collapse strength of the pipe, especially for the thick-walled pipe, as shown in Fig. 5.16B. A significant improvement of this prediction can be achieved if the plastic moment of the ring cross-section MP

(B)

(C)

(D)

Mechanical behavior of metal pipes under internal and external pressure

(A)

Figure 5.14 Post-buckling response and subsequent post-buckling configurations of a thin-walled X65 steel pipe with D/t = 40 subjected to uniform external pressure. (Α) Consecutive configurations from maximum pressure (1) to collapse (6). (B) Pressure-ovality diagram. (C) Distribution of equivalent plastic strain at stage (4). (D) Distribution of equivalent plastic strain at collapsed stage. 149

(C)

(D)

Figure 5.15 Post-buckling response and subsequent post-buckling configurations of a thick-walled X65 steel pipe D/t = 20.5 subjected to uniform external pressure. (Α) Consecutive configurations from maximum pressure (1) to collapse (6). (B) Pressure-ovality diagram. (C) Distribution of equivalent plastic strain at stage (4). (D) Distribution of equivalent plastic strain at collapsed stage.

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(B)

150

(A)

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151

Figure 5.16 Elastic response versus elastic-plastic response of two X65 pipes, subjected to uniform external pressure, for different levels of initial ovality; comparison with analytical expressions. (A) D/t = 40. (B) D/t = 20.5.

is reduced to account for the presence of axial compression due to external pressure. Following the analysis presented in Appendix C, the reduced plastic moment MP  of a rectangular section is:  MP = MP 1 −



p pY

2  (5.35)

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and using Eq. (5.35), the equation of the plastic collapse mechanism becomes: p = pY



  2   t 1 + 2 p 1− Dm 2 pY

(5.36)

Eq. (5.36) is also plotted in Fig. 5.16 and its intersection with the elastic solution constitutes a significantly improved prediction of the collapse pressure, especially for the pipe with D/t = 20.5. Summarizing, the results shown in Fig. 5.16 clearly show that the combination of elastic ovalization solution with the simplified collapse mechanism solution provides accurate predictions for the collapse pressure for the D/t range of interest only if the presence of axial compressive force in the ring and its influence on the maximum bending moment is taken into account. In that case, the intersection between the two equilibrium paths is a reliable upper-bound estimate of pipe collapse pressure.

5.3

Factors influencing pipe collapse

In the previous section, we examined the influence of initial cross-sectional ovality on the collapse pressure of a metal pipe, modeled as a ring. Initial ovality is the primary form of initial imperfection and is a key factor for the value of collapse pressure. However, there exist several additional factors, which are mainly related to the fabrication process of the line pipe, as described in Chapter 2. A distinction should be made between the geometric factors (initial ovality, variation of thickness around the cylinder cross-section) and the factors related to material behavior, namely material anisotropy, residual stresses, and the Bauschinger effect. Those factors are briefly discussed below.

5.3.1 Initial ovality Initial ovality of the cylinder cross-section in the form of a doubly-symmetric pattern that deviates from the ideal circular configuration is a major parameter affecting pipe collapse, as noted in the previous section. This out-of-roundness shape is associated directly with the buckling mode obtained from bifurcation analysis considering elastic and elastic-plastic material behavior. The finite element results in Fig. 5.10–Fig. 5.13 depict the sensitivity of collapse pressure pmax on the amplitude of initial imperfection for three values of D/t, namely 20.5, 30.5 and 40, and demonstrate the significant effect of initial ovality on the collapse strength.

5.3.2 Thickness variation This type of imperfection is quite common in seamless pipes, because of the manufacturing process, and is expressed through the so-called “thickness parameter”: ˆ = tmax − tmin  tmax + tmin

(5.37)

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153

Figure 5.17 Buckling response of metal rings with non-uniform thickness: (A) D/t = 20.5, (B) D/t = 30.5.

where tmax and tmin are the maximum and minimum values of pipe wall thickness around the cross-section (Yeh and Kyriakides, 1986). Analytical expressions for the collapse pressure of a ring with non-uniform thickness under external pressure may not be available, and the bifurcation solution requires either a semi-numerical method or a numerical simulation. Finite element results are depicted in Fig. 5.17 and Fig. 5.18 for two X65 pipes (D/t = 20.5 and 30.5), considering a two-dimensional model with

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Figure 5.18 Variation of maximum pressure with respect to the value of thickness parameter ˆ in X65 pipes for two values of initial ovality. (A) D/t = 20.5, (B) D/t = 30.5. 

the cross-sectional shape shown in the sketch of Fig. 5.17. In this model, the outer and the inner surfaces of the pipe cross-section have an eccentricity e, resulting in a non-uniform thickness around the cross-section (see the sketches in Fig. 5.17). The results show the effect of thickness variation on the collapse pressure. This effect is less pronounced than the effect of initial ovality. Fig. 5.19 shows the collapsed shapes of two pipes (D/t = 20.5) with different combinations of initial ovality and thickness ˆ = 0.05%,  ˆ = 20% and  ˆ = 1.6%,  ˆ = 5%). Because of the nonvariation ( uniform thickness around the pipe, the buckled shape is no longer doubly-symmetric, and it is characterized by a local inward displacement pattern of the pipe wall at the region of minimum thickness.

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Figure 5.19 Collapse configuration of a X65 steel pipe with non-uniform thickness ˆ = 0.05%,  ˆ = 20%; (B)  ˆ = 1.6%,  ˆ = 5%. (D/t = 20.5). (A) 

5.3.3 Material-type imperfections In addition to the geometric imperfections discussed above, the line pipe manufacturing process also introduces inhomogeneities and imperfections on pipe material properties, which influence the value of collapse pressure. These are: (a) material anisotropy in terms of different yielding point in the two principal directions (hoop and longitudinal), (b) residual stresses and (c) the Bauschinger effect. These factors are interrelated and refer primarily to cold-formed bending during line pipe manufacturing. In the present section, those effects are discussed separately, for the sake of simplicity. A more detailed treatment of the effect of the manufacturing

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(A)

(B)

Figure 5.20 Effect of anisotropy on the collapse pressure of X65 steel pipes for different ˆ = 0.2%). values of D/t ratio (initial ovality 

process on the collapse pressure will be presented in Chapter 8, using a finite element simulation. The anisotropy of steel properties refers to the different yield point of the steel material in the hoop and the axial direction of the pipe. Fig. 5.20 shows the dependence of collapse pressure with respect to the level of yielding anisotropy, for different D/t values, obtained with a finite element model. The level of anisotropy is defined as follows: σY,θ (5.38) S= σY,x where σ Y,θ is the yield stress in the hoop direction of the pipe and σ Y,x is the yield stress in the axial direction of the pipe. In Fig. 5.20, pmax (S) is the maximum (collapse) pressure for a value of anisotropy S, and pmax (1) is the maximum pressure for isotropic material behavior (S = 1). To account for the anisotropy on the yield stress, Hill’s yield criterion is employed, as described in Appendix D. The results in Fig. 5.20 show that

Mechanical behavior of metal pipes under internal and external pressure

157

Figure 5.21 (A) Gap-opening method for estimating the amplitude of residual stress in metal pipes. (B) Assumed distribution of residual stresses across the pipe thickness.

a 15% variation of yield stress in hoop direction with respect to the yield stress in the longitudinal direction (S = 0.85), results in a 2%–10% change in collapse pressure, compared with the isotropic case, depending on the D/t ratio. Fig. 5.21 shows a popular and convenient method of measuring cold bendinginduced residual stress. A ring slice is extracted from the pipe and, subsequently, it is cut longitudinally. Due to the presence of residual stresses the ring springs back and opens as shown in Fig. 5.21A, causing a gap at that location. The size of the gap g depends on the magnitude of residual stresses. Assuming elastic unloading and considering the release of bending deformation, the gap size g can be related to the amplitude of residual stresses σ r . From simple Mechanics of Materials, one obtains the following expression for the value of the amplitude of residual stress: σr =

Et g π Dm

(5.39)

Fig. 5.22 shows the effect of residual stresses on the collapse pressure for a X65 pipe with D/t = 30.5 and initial ovality 0.2%, obtained with finite element simulations, assuming a linear distribution of residual (initial) stresses through the ring thickness, as shown in Fig. 5.21B. In Fig. 5.22, pmax (Rr ) is the maximum (collapse) pressure for a specific value of residual stress ratio Rr = σr /σY and pmax (0) is the maximum pressure for the residual stress-free case (Rr = 0). The numerical results show that residual stresses reduce the collapse pressure. As an example, for a maximum value of residual stress σ r equal to 50% of yield stress, the reduction of collapse pressure is 16%, and for σ r equal to 90% of yield stress, the reduction of collapse pressure is 34%. Yielding anisotropy and residual stresses may not be the only effects of cold bending during the pipe manufacturing process. Cold bending induces work hardening on the pipe material, so that yield strength in the direction of loading increases. On the other hand, this increase occurs reducing the yield strength in the opposite direction because of the “Bauschinger effect”, described in more detail in Appendix D.

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Figure 5.22 Effect of residual stresses on the collapse pressure of an X65 pipe (D/t = 30.5, ˆ = 0.2%). 

(A)

(B)

Figure 5.23 Schematic representation of Bauschinger effect on pipe material behavior due to cold bending manufacturing process: (A) consecutive stages of pipe deformation; (B) stress-strain diagram at point C; stages (a) – (d) are consistent with the four stages (a) – (d) in Fig. 5.23A.

Fig. 5.23A depicts in a simplistic manner the main stages of deformation during the manufacturing process of a UOE or JCO-E line pipe (see also Chapter 2). Starting from a flat plate [stage (a)], the plate is rounded, and the two ends are welded together [stage (b)]. Subsequently, expansion is applied, followed by unloading [stage (c)]. Finally, the application of external pressure results in collapse in an oval form [stage (d)], as

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159

Figure 5.24 The shape of a propagating buckle in an offshore pipeline; finite element analysis of a X65 pipe with D/t = 30.5.

described earlier in the present chapter. The stress-strain diagram in Fig. 5.23B refers to the evolution of stress at point C, which is on the outer surface of the pipe, in a circumferential location opposite to the weld (see Fig. 5.23A). During the application of external pressure, point C undergoes compression and because of the Bauschinger effect, yielding occurs at a stress σ Y(−) , which is significantly lower than both σ Y(+) and the initial yield stress σ Y (σ Y(−) < σ Y < σ Y(+) ). Considering that collapse is highly dependent on first yielding, it is readily concluded that the Bauschinger effect due to cold bending may affect significantly the value of collapse pressure. This effect will be discussed in more detail in Chapter 8.

5.4

Buckle propagation and arrest in long metal cylinders

A propagating buckle is a consequence of pipe collapse under external pressure. It is shown in Fig. 5.24, and may be described as a state of deformation that, upon initiation, will travel along the pipeline, causing progressive collapse of the pipeline in its passage.1 The minimum pressure required for a buckle to propagate is typically 1 If

you ever used drinking straws made of soft paper, you may note that the phenomenon of buckling propagation is similar to the collapse of such straws under excessive suction.

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significantly lower than the collapse pressure pmax , for which the pipeline would be designed under normal circumstances. Therefore, the propagating buckle will advance, collapsing a very long part of the pipeline, unless an adequately designed arresting device is encountered during this propagation, or the ambient pressure does not fall below the minimum propagation pressure level. Buckle propagation occurs after pipeline collapse in deep water, and constitutes an impressive buckling problem, which is of essential importance for the structural integrity of offshore pipelines. The publication of Mesloh et al. (1973) was the first to report propagating buckles in offshore pipelines. Subsequent research provided extensive experimental data on the propagation pressure, whereas numerical simulations using finite element models contributed to better understanding the phenomenon. In the following, reference is made to several key publications on this subject. The present section describes this impressive instability problem in offshore pipelines through the answer to a series of questions, in an attempt to identify and clarify the main aspects of buckling propagation in a simple and efficient manner. For an extensive presentation of propagating buckles in offshore pipelines, the reader is referred to the recent book by Kyriakides and Lee (2021).

5.4.1 What exactly is a propagating buckle? Fig. 5.25 shows the deformation stages of a long pipe segment subjected to external pressure, obtained through a rigorous finite element model, which aims at describing the physical problem in detail. The corresponding p − V diagram, where V is the change of volume enclosed by the pipe, is shown in Fig. 5.26. The pipe has diameterto-thickness ratio D/t = 30.5 and is made of X65 steel material (Fig. 5.3B). The model is 20 diameters long and it is made of X65 steel material, with the stress-strain curve shown in Fig. 5.3B. Symmetry conditions are imposed at the front section of the pipe, whereas the far end section is fixed. Initially, a pair of concentrated opposite forces in the radial direction is applied at cross-section A, located at the front end of the pipe segment. Upon removal of this pair of forces, the end section is slightly ovalized, due to plastic deformation and, subsequently, uniform external pressure is applied gradually on the entire pipe segment. The following sequence of events occurs: (a) A maximum pressure pmax is reached at stage 1 in the p − V equilibrium diagram of Fig. 5.26. The value of maximum pressure is 17.1 MPa and, beyond that stage, the pipe becomes unstable, characterized by a rapid drop of pressure. At the stage where maximum pressure is reached, the ovalization of the pipe is insignificant. (b) With continuing deformation, the cross-section at A develops maximum ovalization (stage 2 in Fig. 5.25). Ovalization increases rapidly, while the value of pressure drops. (c) Cross-section A is completely collapsed (flattened), so that the top and bottom generators at the inner surface of the cylinder establish contact (stage 3 in Fig. 5.25) at a pressure equal to 2.46 MPa (“touchdown” point). (d) The pressure increases slightly, and the flattened profile starts propagating along the pipe at constant pressure (stage 4 in Fig. 5.25) equal to 2.9 MPa, as shown in Fig. 5.26. (e) The buckle continues to propagate and the entire pipe segment collapses progressively under constant pressure equal to 2.9 MPa (stage 5 in Fig. 5.25).

Mechanical behavior of metal pipes under internal and external pressure

Figure 5.25 Successive configurations of a X65 steel pipe (D/t = 30.5) during buckling propagation; the pipe is initially dented at its end and subsequently subjected to uniform external pressure.

161

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Figure 5.26 Pressure variation in a collapsible X65 steel pipe (D/t = 30.5) subjected to uniform external pressure, with respect to the change of volume enclosed by the pipe.

Understanding the physics behind the p − V diagram shown in Fig. 5.26 and relating this diagram with the corresponding deformed shapes in Fig. 5.25 is essential for a thorough comprehension of the buckle propagation phenomenon. The main conclusion from the above analysis is that the buckle, upon formation, propagates along the pipe under a constant level of external pressure, which is much lower than the collapse pressure of the pipe (2.9 MPa versus 17.1 MPa). The following paragraphs are aimed at elucidating buckling propagation in a simple and efficient manner.

5.4.2 How can this phenomenon be initiated? There are several reasons for buckling propagation to initiate. A small local imperfection in the form of a small “bump” or dent on the pipeline surface (Fig. 5.27), usually caused accidentally during pipe transportation or pipeline construction, is a typical example of triggering mechanism; at that location, the strength of the pipeline under external pressure is reduced, and upon application of external pressure, local collapse may occur at a pressure level lower than the collapse pressure of the intact pipe, followed by buckling propagation. Buckling propagation may be also triggered during the installation process of an offshore pipeline (Fig. 5.28) where excessive loading due to bending, external pressure and tension in the sagbend area may cause local collapse, which may propagate due to the presence of high external pressure.

5.4.3 Why does the buckle propagate? One should remember that the external pressure is always perpendicular to the deformed surface of the pipeline. Because of the inclined shape of the propagation profile, a component of the total pressure force Fp on the pipeline is always directed in the longitudinal direction. This horizontal component (FpH , shown in Fig. 5.29) is the “driving force” that makes the buckle to propagate.

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163

Figure 5.27 A small “bump” on the pipe wall that may be responsible for triggering buckling propagation.

Figure 5.28 Buckle initiation in the sagbend area during pipeline installation, due to excessive combination of bending, external pressure, and tension.

5.4.4 How much pressure is required for the buckle to propagate? The minimum pressure required for a buckle to propagate is called “propagation pressure” pp . In the diagram of Fig. 5.26, it is represented by the long horizontal section of the p − V curve that follows the “touchdown” point. It is a characteristic “strength” of the cylindrical shell of the pipeline and depends on its material (yield stress) and geometry (D/t ratio). There exist three possibilities:

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Figure 5.29 Schematic representation of propagation profile and horizontal component of pressure force on the pipe surface. r

If the external pressure p around the pipe is lower that the propagation pressure (p < pp ), then the pipe is safe against buckle propagation and this phenomenon does not occur.

r

If the external pressure p reaches the propagation pressure level (p = pp ), then buckling propagation occurs in a quasi-static manner, i.e., under zero velocity. The value of pp is a “threshold value” for this phenomenon, and an important parameter for pipeline design.

r

If the external pressure p exceeds the propagation pressure (p > pp ), then buckling propagation occurs dynamically, with a propagation speed that depends on the level of pressure. Experimental measurements and finite element simulations have demonstrated that the propagation speed may reach several hundred meters per second.

In a practical pipeline application, the pipeline design engineer should compute the propagation pressure pp for a specific steel pipeline, i.e., determine the minimum pressure required for this phenomenon to occur, given the material and the geometry of the pipe.

5.4.5 Can we estimate the propagation pressure? The shapes in Fig. 5.25 show that each cross-section of the pipe deforms from a circular to a collapsed configuration as the buckle passes through it. Therefore, relating this deformation to the collapse problem of a metal ring under external pressure (as examined in Sections 5.2 and 5.3), a rough estimate of the propagation pressure can be obtained by a simple two-dimensional model. The model is based on the four-plastichinge collapse mechanism shown in Fig. 5.30, and was first employed by Palmer and Martin (1975). It is the same plastic-hinge model used for collapse calculations in Section 5.3, described in Appendix E. Considering rigid-plastic material behavior, the internal work Wint dissipated at the four plastic hinges, during deformation of the ring from its circular shape to final collapse, is: Wint = 4M p

π 2

(5.40)

Mechanical behavior of metal pipes under internal and external pressure

165

Figure 5.30 Four-plastic-hinge model for the calculation of propagation pressure (Palmer and Martin, 1975).

where Mp is the plastic moment developed at one plastic hinge, equal to M p = σY t 2 /4 per unit length of the pipe. Furthermore, the corresponding external pressure work is equal to the product of pressure with the change of area enclosed by the collapsing ring, which is equal to the change of area enclosed by the quadrilateral ABCD (Fig. 5.30). Therefore,  √ 2 Wext = p rm 2

(5.41)

Enforcing balance of energy, Wint = Wext , using Eqs. (5.40) and (5.41), and solving for the pressure, one readily obtains a closed-form expression for the propagation pressure:  p p = π σY

t Dm

2 (5.42)

This constitutes an interesting analytical result, that shows that the propagation pressure depends on the yield stress of the pipeline material and its geometry, expressed in terms of the non-dimensional diameter-to-thickness ratio. An enhancement of this simple model, using plastic material behavior with hardening was reported in Kyriakides (1993). Using Eq. (5.42), pp is computed equal to 1.62 MPa, which is a value significantly lower than the value of 2.9 MPa calculated with the finite element analysis in Fig. 5.26. The prediction of Eq. (5.42) can be somewhat improved if the yield stress under plane strain conditions σY∗ is used instead of σY . This leads to a pp value equal to 1.83 MPa, which is still well below the finite element value. To obtain more reliable predictions of pp , the above concept of energy balance can be improved, using an advanced model for calculating the internal work Wint . This model can be either semi-numerical (Kyriakides et al., 1984) or entirely numerical (Katsounas and Tassoulas, 1990). Using such a model, a more accurate diagram that relates the variation of pressure p with the change of area AE enclosed by the pipe cross-section (denoted as AE ), can be obtained. Fig. 5.31A shows a generic p − AE diagram of a collapsing ring, representing the

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Figure 5.31 The Maxwell line concept for calculating the buckle propagation pressure (Chater and Hutchinson, 1984).

pipe cross-section. The initial elastic part is very steep (practically vertical), because of the negligible change of area enclosed by the pipe cross-section in the pre-buckling stage, also shown in the results of Section 5.2. The area below this diagram is equal to the internal work (Wint ) associated with ring deformation, and is expressed as follows: A  ET

Wint =

p dAE

(5.43)

0

Balance of energy requires that the internal work Wint is equal to the external work Wext . The latter is the product of the propagation pressure pp times the total change of area AET from the circular shape of the ring to the final collapsed configuration. Therefore, balance of energy Wint = Wext , results in: 1 pp = AET

A  ET

p dAE

(5.44)

0

Graphically, this concept is expressed by a horizontal line that equates areas (1) and (2) in Fig. 5.31B, above and below the p − AE diagram respectively. The line is referred to as “Maxwell line” and was introduced in this problem by Chater and Hutchinson (1984). In Fig. 5.32 the “Maxwell line” is applied in the response of an X65 pipe with D/t = 30.5 and provides a prediction for the propagation pressure pp equal to 2.18 MPa, which is a better prediction than the one offered by Eq. (5.42).

5.4.6 How accurate are two-dimensional models? The above two-dimensional approaches are quite straightforward; they predict the propagation pressure pp by relating directly the propagating buckle phenomenon to the collapse of a ring under plane strain conditions. However, the values of propagation pressure pp obtained by either Eq. (5.42) or by Maxwell’s line may not be satisfactory when compared with experimental data and this has been noticed in early works

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167

Figure 5.32 Prediction of the propagation pressure with the Maxwell line for a X65-steel with D/t = 30.5; two-dimensional finite element calculation.

Figure 5.33 Stretching of pipe generators (cylinder meridians) during buckling propagation.

(Kyriakides et al., 1984; Katsounas and Tassoulas, 1990). The two-dimensional calculations performed on ring models do not recognize that the propagating buckle profile is actually a three-dimensional state of deformation. More specifically, a two-dimensional analysis neglects longitudinal stretching of the pipe generators. Stretching of pipe generators can be readily verified comparing the initial length of the top generator AB with its deformed length A’B (clearly, A’B>AB) as shown in Fig. 5.33. Furthermore, apart from longitudinal stretching, the two-dimensional models do not account for the exact history of deformation of the pipe cross section from the front to the rear of the buckle propagation profile, which may affect the response on account of the path-dependence of elastic-plastic pipe material behavior (Katsounas and Tassoulas, 1990).

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(B)

Figure 5.34 (A) Progressive collapse of a cross-section during buckle propagation as the buckle pass through (X65 pipe, D/t = 30.5). (B) Comparison of a collapsed ring under external pressure (plane strain conditions) and the collapsed shape of the propagating buckle.

The above deficiencies motivated the development of rigorous numerical simulation tools based on three-dimensional finite element models for the analysis of propagating buckles. These numerical models are quite demanding because they need to handle the large inelastic deformations of the pipe wall and the progressive contact between segments of the pipe interior wall during buckling propagation. The publications by Jensen (1988) and by Katsounas and Tassoulas (1990) were the first to report threedimensional simulations of propagating buckles. In particular, the report by Katsounas and Tassoulas (1990) is an excellent document for understanding the physics of propagating buckles and their simulation with finite elements, and provides numerical results that are in very good agreement with available experimental data. In Fig. 5.34A the initial and the deformed configurations are depicted for a steel pipeline during buckling propagation, obtained from the finite element analysis. The pipeline is made of X65 steel material, has outer diameter 610 mm and thickness 20 mm, and the finite element model is 20-diameters long. The pressure-volume diagram (p − V) for the buckle propagation of this pipe is shown in Fig. 5.26. In the first part of the analysis, the pipeline is slightly dented at its end section, and subsequently, it is subjected to uniform pressure in a quasi-static manner, leading to buckling initiation and propagation. Fig. 5.34B compares the shape of a collapsed ring under plane strain conditions subjected to uniform pressure with the collapsed shape of the propagating buckle, showing a small difference, which is attributed to longitudinal stretching that increases the flatness of the collapsed ring. Fig. 5.35 shows the quasistatic propagating buckle profile from the finite element analysis, traveling along the externally-pressurized steel pipeline.

5.4.7 What happens if the pressure level is higher than the propagation pressure? Upon buckle initiation, if the level of external pressure p is higher than the propagation pressure pp (p > pp ), the buckle will propagate at high speed, possibly destroying a very

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Figure 5.35 Quasi-static buckling propagation profile obtained with three-dimensional finite element model (X65 pipe, D/t = 30.5). (A) Longitudinal view. (B) Three-dimensional view.

long part of the pipeline. This constitutes a dynamic problem, where inertial forces and pipe-fluid interaction effects should be considered. The pioneering experimental work by Kyriakides and Babcock (1979), contributed significantly to understanding the dynamic propagation of buckles in offshore pipelines. More recent experiments on dynamic propagation have been reported by Kyriakides and Netto (2000). Furthermore, experiments on buckle propagation arrest have been reported by Netto and Kyriakides (2000a), accompanied by finite element calculations (Netto and Kyriakides, 2000b). During dynamic buckle propagation the propagation profile is steeper and larger deformations occur than quasi-static propagation, especially at the two edges of the flattened shape. This may lead to potential rupture of pipe wall at either of these two edges, and it is referred to as “wet buckle” in offshore pipeline terminology (Kyriakides and Babcock, 1981). Furthermore, the propagation velocity was found to increase rapidly when the level of pressure increases, and can be very high; for pressure levels approximately twice the propagation pressure (p  2pp ), velocities of several hundred meters per second were recorded, a remarkable result. An excellent numerical contribution on this subject has been presented by Song and Tassoulas (1990, 1993), using an in-house three-dimensional finite element formulation (Fig. 5.36), which enhanced the formulation of Katsounas and Tassoulas (1990) and provided very good results in terms of propagation velocities in comparison with experimental data reported by Kyriakides and Babcock (1979). The above experimental and numerical publications have also examined buckle propagation in air and in water, and indicated that fluid-structure interaction plays an important role on dynamic propagation. In particular, the velocity of propagating buckles in air was calculated much higher than the velocity in water.

5.4.8 Is it possible to prevent buckling propagation? Pipelines are usually designed against collapse, and their thickness is determined according to the maximum pressure pmax (collapse pressure) sustained by the pipe. However, the value of propagation pressure pp is significantly lower than pmax , as

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Figure 5.36 Profiles of dynamic propagating buckles in externally-pressurized pipes [finite element results by Song and Tassoulas (1990) published with permission from OTRC] for an aluminum pipe with D/t = 35.7 and yield stress 289.8 MPa (42 ksi); the quasi-static propagation pressure pp is calculated equal to 1.08 MPa.

shown very clearly in Fig. 5.26. Therefore, if buckle propagation is triggered during the installation process, the buckle, if not somehow arrested, will propagate along the pipeline destroying kilometers of the pipeline with catastrophic consequences. Arresting propagating buckles is essential for safeguarding offshore pipeline integrity, and it is achieved using “buckle arrestors”. These are special-purpose stiff devices, placed at regular intervals along the pipeline, that increase locally pipe stiffness against cross-sectional deformation. When a propagating buckle reaches an arrestor, the arrestor because of its large stiffness stops the propagation of the buckle and prevents further collapse of the pipe. Therefore, if a propagating buckle is triggered in a subsea pipeline in the presence of buckle arrestors, pipeline collapse will be confined within two successive arrestors, allowing for the collapsed pipeline segment to be removed and replaced with a new intact pipe segment. In practice this is a very expensive repair operation, but it is much preferable than the flattening of a very large length of the pipeline. The spacing between arrestors denoted as LD in Fig. 5.37 and in Fig. 5.38 is a decision of the pipeline designer and can be equal to several hundred meters. The “integral buckle arrestor” is the most common type of arrestor used in deepwater pipeline applications. It is a relatively short pipe component, significantly thicker than the pipeline, with the same inside diameter as the pipeline and is butt-welded between two adjacent pipes (Fig. 5.38). The thickness ha and the length La of the arrestor should be determined by the designer, so that the arrestor provides adequate stiffness and fulfills its buckle arresting purpose. Other types of buckle arrestors are: (a) the slip-on arrestor, (b) the clamped arrestor and (c) the spiral arrestor. For more information on those arrestors, the reader is referred to Kyriakides and Lee (2021).

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Figure 5.37 J-lay installation of a deep-water pipeline with arrestors.

Figure 5.38 Sketch of an integral buckle arrestor.

5.4.9 How do we define arrestor resistance against propagation? The cross-over pressure of a buckle arrestor pX is the level of external pressure necessary for a traveling buckle in a pipeline to cross the arrestor and to continue its

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Figure 5.39 Consecutive profiles of buckle propagation, arrest and cross-over (X65 pipe, D/t = 30.5).

propagation beyond the arrestor. Fig. 5.39 and Fig. 5.40 show the performance of an integral arrestor in a X65 steel pipeline with outer diameter 610 mm and thickness 20 mm (D/t = 30.5). The arrestor length La is equal to 1.2D and its total thickness ha is 2.5 times the pipe thickness t. The total length of the model is equal to 20D and the arrestor is located near the middle of the model. Before the application of external pressure, section A is forced to ovalize slightly with the use of two radial forces, as shown in the first stage of Fig. 5.25. Subsequently, uniform external pressure is applied reaching a maximum value pmax equal to 17.1 MPa, corresponding to pipe

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Figure 5.40 Pressure-volume diagram of a propagating buckle in a pipe with an integral arrestor (X65 pipe, D/t = 30.5) and comparison with the corresponding diagram of the same steel pipe without arrestors; stages (1) – (4) denoted in the diagram refer to the pipeline profiles shown in Fig. 5.39.

collapse and buckle initiation. Upon collapse of section A, buckle propagation occurs, associated with flattening of the pipe, and this is represented by the horizontal part of the pressure-volume diagram, denoted as stage (2) in Fig. 5.40 (pp = 2.9 MPa). When the front of the buckle reaches the arrestor, the pressure increases and eventually crosses the arrestor at pressure pX = 14.04 MPa. In this diagram, the corresponding p − V curve for the same pipe without arrestor is also shown for comparison. Fig. 5.39 also shows that the flattened shape in the downstream part of the pipeline is rotated by 90° with respect to the flattened shape in the upstream part. This is called “flipping mode of cross-over”, or simply “flip-flop”, and depends on the dimensions of the arrestor relative to the dimensions of the pipe cross-section (Park and Kyriakides, 1997). For a more detailed analysis of integral arrestor resistance to buckle propagation, including the effects of dynamic propagation, the publications of Park and Kyriakides (1997), Mansour and Tassoulas (1997) and Netto and Kyriakides (2000a, 2000b) are suggested.

5.5

Effect of tension on collapse and buckling propagation

The presence of tension introduces additional stress and strain in the longitudinal direction of the pipeline and influences its external pressure response. The results shown in Sections 5.2 and 5.3 have clearly indicated that the value of compressive yield stress σ Y in the hoop direction is a major parameter for calculating the external pressure capacity of relatively thin-walled pipes. Considering either Von Mises or Tresca yield criterion, it can be readily concluded that the presence of tension in the longitudinal direction of the pipe reduces the value of compressive yield stress in the hoop direction, and therefore, it decreases the collapse pressure pmax of the pipe.

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In relatively thin-walled pipes, elastic buckling governs pressure response, as expressed by Eq. (5.2), which is independent of the yield stress σ Y of the pipe material. Consequently, the presence of tension does not influence the buckling pressure of thin-walled pipes. On the other hand, in thick-walled pipes the collapse pressure depends on the value of yield stress. Therefore, the combined state of hoop and axial stress reduces the material yield stress and decreases the collapse pressure of the pipe. The degrading effect of tension on the collapse pressure may be approximated considering a reduced yield stress σ  Y in the circumferential direction. Using the Von Mises yield criterion, written in the following form for the biaxial state of stress under consideration (σ x ,σ θ ):  2  2    σθ σx σθ σx + − =1 (5.45) σY σY σY σY the reduction of yield stress in compression in the circumferential direction due to axial tensile stress σ x = σ is obtained setting σ θ = σ  Y equal to: ⎡ ⎤   2 σ 3 σ ⎦ (5.46) + 1− σY = σY ⎣− 2σY 4 σY where σ = T/A, T is the tension force, TY = σ Y A is the yield force and A is the crosssectional area of the metal pipe. Equivalently: σY = fτ σY

(5.47)

where  T fτ = − + 2TY

1−

  3 T 2 4 TY

(5.48)

Using Barlow’s formula in Eq. (5.47) and assuming that the collapse pressure is analogous to σ  Y , one may write the following expression for the reduced collapse pressure p max in the presence of tension as follows: pmax = fτ pmax

(5.49)

where pmax in Eq. (5.49) is the value of collapse pressure in the absence of axial force. Kyriakides and Chang (1992) have proposed another equation that expresses the reduction of yield stress in the hoop direction in the presence of axial tension: 1 − (T/TY )2 fσ =  1 − 0.75(T/TY )2

(5.50)

Using Eq. (5.50) one may write: pmax = fσ pmax

(5.51)

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An alternative method to approximate the effects of axial tension on the collapse pressure is to use Eq. (5.47) or σY = fσ σY to compute the reduction of yield stress in the hoop direction σ  Y , and subsequently, using σ  Y , to apply Eq. (5.31) and obtain the pressure that causes first yielding p F :  p2 F −

    ˆ σ Y t 6r  σ Y t + 1+ pcr = 0 pcr pF + rm t rm rm

(5.52)

The value of p F in Eq. (5.52) can be taken as the reduced collapse pressure p max in the presence of tension T. Fig. 5.41A and Fig. 5.41B depict the pressure-ovality diagrams for two X65 steel pipes with D/t = 10 and 30.5 for different levels of tension T, expressed as fractions of the yield tension TY , showing the influence of axial tension on the collapse pressure. The results have been obtained using a finite element model under generalized plane strain conditions, which allows the pipe cross-section to deform in the out-of-plane direction and account for the application of tensile force T. In addition, a T → p loading path is followed in the analysis, where tension T is applied first, and subsequently, keeping the value of tension T constant, external pressure p is applied until collapse. Fig. 5.41C summarizes the numerical results and shows the reduced value of collapse pressure p max with respect to the level of applied tension, normalized by the yield tension TY . Each curve in Fig. 5.41C may be regarded as a “pressure-tension interaction envelope”, which expresses the collapse pressure for different values of tension. Pairs of p and T values that are below the envelop, correspond to safe combinations of external pressure and axial tension. The simplified interaction curve, expressed by Eq. (5.51), is also plotted in Fig. 5.41C, and offers a fairly good approximation of pressure-tension interaction. In a long pipe, the combined loading of pressure and tension can initiate a propagating buckle. In this case, the value of propagation pressure is affected by the presence of tension T. Experimental and numerical studies (Kyriakides and Chang, 1992; Nogueira and Tassoulas, 1995) have indicated a reduction of the propagation pressure if a tensile load is applied. For design purposes, the approach described by either Eq. (5.49) or Eq. (5.51) for the collapse pressure can be adopted for the propagation pressure. In this approach, the reduced propagation pressure p p in the presence of axial tensile force can be approximated as follows: pp = fτ p p

(5.53)

pp = fσ p p

(5.54)

or

where pp is the propagation pressure of the pipe in the absence of tension.

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(A)

(B)

(C)

Figure 5.41 Effect of tension load on the collapse pressure of X65 steel pipes; (A) pressure-ovality response for with D/t = 30.5; (B) pressure-ovality response for with D/t = 10; (C) pressure-tension interaction envelopes (D/t = 10 and 30.5); the numerical results from (A) and (B) are compared with the predictions of Eqs. (5.50) and (5.51).

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Figure 5.42 Finite element model for simulating buckling of a steel ring within a stiff cavity (D/t = 200, σ Y = 313 MPa).

5.6

Externally-pressurized cylinders under lateral confinement

The structural instability of confined cylinders made of elastic material has been examined in Section 3.3. It was shown that Glock’s analytical expression is in excellent agreement with numerical results from finite element models. It was also found that the response and the value of maximum pressure pmax is sensitive to the presence of initial imperfections. The present section examines the response of externallypressurized metal cylinders under confined conditions, accounting for elastic-plastic material behavior. Fig. 5.42 depicts a plane-strain finite element model for the simulation of a thin-walled steel cylinder with D/t = 200 under confined conditions, subjected to uniform external pressure, assuming a frictionless interface between the cylinder and the confinement medium. The steel material has yield stress σ Y = 313 MPa, ultimate stress σ U = 492 MPa and a plastic plateau (zero post-yield hardening) up to nominal strain 1.5%. The confinement medium modulus E’ is assumed equal to 10% of steel Young’s modulus (E’=21,000 MPa), corresponding to practically rigid confinement (e.g., concrete encasement). Fig. 5.43A depicts the pressure-displacement curve of the above steel cylinder in the absence of initial imperfections (zero out-ofroundness and no gap between the cylinder and the cavity). The depicted displacement is the vertical deflection δ of point A at θ =0, as shown in Fig. 5.43B. Fig. 5.43A also depicts the pressure-displacement curve of an elastic cylinder with the same D/t ratio. The comparison of the two pressure-displacement diagrams shows that the ultimate pressure pmax of the steel cylinder (1.44 MPa) is significantly lower than the

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(B)

Figure 5.43 (A) Response of an elastic cylinder and a steel cylinder subjected to uniform external pressure under confined conditions (D/t = 200, rigid cavity, no imperfections); (B) schematic representation of deformed ring.

Figure 5.44 Response of a thin-walled steel cylinder (D/t = 200) confined by a stiff surrounding medium subjected to uniform external pressure for different values of initial imperfection amplitude; comparison with plastic collapse mechanism.

corresponding pressure of the elastic cylinder (1.95 MPa), and this reflects the effect of inelastic material behavior on the ultimate pressure capacity of the cylinder. The pressure-displacement curves in Fig. 5.44 represent the response of the steel cylinder under consideration (D/t = 200) for different values of initial out-ofroundness, defined in Fig. 3.17B. The response is presented in a dimensionless form, where the values of pressure p are normalized by the yield pressure pY∗ under plane strain conditions pY∗ = 1.13(2σY t/D), and the deflection δ of point A is normalized

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Figure 5.45 Response of an initially perfect steel cylinder (D/t = 200), confined by a stiff surrounding medium. (A) Consecutive buckling and post-buckling shapes. (B) Comparison with the corresponding shape of an elastic cylinder with the same geometric properties.

by the cylinder radius r. The amplitude of initial out-of-roundness (δ0 /r) corresponds to the initial value of each pressure-displacement curve on the horizontal axis of the diagram. The results demonstrate that the ultimate pressure pmax is affected by the value of initial imperfection by a substantial amount and is substantially smaller than the yield pressure, even for negligible initial imperfection. The finite element analysis of the steel cylinder also indicates that the maximum pressure pmax , in the absence of initial imperfections, occurs very soon after first

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Figure 5.46 Pressure-displacement diagrams for different values of initial gap between the steel cylinder and a rigid cavity (D/t = 200). (A) gap amplitude g/r = 0.0027; (B) gap amplitude g/r = 0.008.

yielding and corresponds to a deformation stage before “inversion” of the cylinder wall occurs. Fig. 5.45A depicts the successive deformed configurations of the steel cylinder, and stage (a) at deflection 1% of the cylinder radius, corresponds to the maximum pressure pmax . A comparison between the deformed shapes from elastic and steel (elastic-plastic) cylinders, corresponding to the same deflection (δ/rm = 0.068), shows that the shape of steel cylinders is characterized by more abrupt changes of local curvature at the symmetry point A and the “touchdown” point B (Fig. 5.45B), and this is attributed to the concentration of plastic deformation at those points. Upon reaching the maximum pressure pmax , the cylinder becomes structurally unstable, with rapid drop of its pressure capacity, and this is responsible for the severe sensitivity of the pmax value to the presence of initial imperfections. The numerical results in Fig. 5.44A show that the presence of initial out-of-roundness of amplitude

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Figure 5.47 Effects of initial out-of-roundness and initial gap (g/r) on the maximum pressure sustained by a confined steel cylinder embedded in a rigid confinement medium (D/t = 200).

less than 1% of the cylinder diameter (δ0 /rm < 0.02) results in a 60% reduction of the ultimate pressure pmax with respect to the zero-imperfection case (δ 0 = 0). In steel cylinders, the unstable post-buckling behavior is characterized by the development of a plastic collapse mechanism with one stationary plastic hinge at symmetry point A, and two moving hinges at the two “touchdown” points B and B’, shown in Fig. 5.43B. It is possible to develop an approximate closed-form expression for this mechanism, written in the following dimensionless form (Vasilikis and Karamanos, 2009):   t 4 p = (5.55)    2 ∗ pY Dm δ δ 6 − rm rm In Fig. 5.44, Eq. (5.55) is plotted together with the numerical results, verifying that the pressure beyond its maximum value is a rapidly decreasing function of cylinder deformation. However, when compared with finite element results, Eq. (5.55) underestimates the pressure by about 30–35%, which is attributed to the fact that this simplified model does not account for the additional internal work required for hinges B and B to travel along the cylinder perimeter during the progressive collapse of the cylinder. The presence of a small gap between the circular cylinder and the cavity of the confinement medium may also have significant effect on the maximum pressure, as shown in Fig. 5.46. This constitutes another type of imperfection, discussed in more detail in Section 3.3 (Fig. 3.17). The cylinder has a D/t ratio equal to 200 and is made of a steel material with yield stress σ Y equal to 313 MPa. The gap size, denoted as g, is the maximum distance between the cylinder and the surrounding medium at θ = 0, and it is

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Figure 5.48 Comparison between numerical results and analytical predictions from Montel’s simplified equation.

normalized by the cylinder radius r. Comparison of the numerical results in Fig. 5.44, compared with the corresponding results of Fig. 5.46, indicates that the presence of a small gap reduces significantly the ultimate pressure capacity pmax of the cylinder. For δ 0 = 0, the presence of an initial gap size equal to 0.27% of the cylinder radius r reduces the ultimate capacity pmax by 40% with respect to the perfect cylinder. The above analysis demonstrates that the structural response of confined cylinders under external pressure is sensitive to the presence of initial imperfections. The effects

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of initial gap (g/r) and out-of-roundness (δ0 /r) imperfections on the value of maximum pressure (pmax /pY∗ ) are summarized in Fig. 5.47. It is possible to estimate the ultimate external pressure capacity of confined cylinders and the pressure that causes first yielding of the cylinder wall, in a way analogous to the one used for the first yielding formula of laterally unconfined cylinders, expressed by Eq. (5.31). Adopting this concept and using experimental results, Montel (1960) proposed a semi-empirical formula for the buckling pressure of cylinders embedded in a stiff cavity pM , in terms of the material yield stress σ Y , the cylinder geometry D/t, the initial out-of-roundness with amplitude δ 0 and the initial gap between the cylinder and the rigid cavity with maximum value g: pM =

14.1 σY (D/t ) [1 + 1.2(δ0 + 2g)/t] 1.5

(5.56)

Eq. (5.56) is valid for the following ranges of geometric and material parameters: 60≤ D/t ≤ 340, 250 MPa≤ σ Y ≤ 500 MPa, 0.1≤ δ 0 /t ≤ 0.5, g/t ≤ 0.25 and g/r ≤ 0.0025. Clearly, the ultimate pressure pM predicted by Eq. (5.56) is a decreasing function of both δ 0 and g. The predictions of Eq. (5.56), are compared with finite element results in Fig. 5.48. For D/t values between 100 and 200, the value of pM is well below the yield pressure pY ∗ of the cylinder, i.e. the pressure that causes plastic yielding of the cylinder wall. Fig. 5.48 shows that the empirical Eq. (5.56), despite its simplicity, predicts the maximum pressure sustained by a cylinder encased in a stiff boundary within a good level of accuracy. The response of a metal ring embedded within a flexible and deformable medium, i.e., lower values of E , is also of interest to buried pipeline engineering applications. The flexibility of the surrounding medium has a strong effect on the value of maximum external pressure sustained by the steel cylinder, and the interested reader is referred to the relevant publications of Vasilikis and Karamanos (2009, 2014), as well as to the thesis of Vasilikis (2012).

References Bruschi, R., Battistini, A., Gjedrem, T., & Zimmermann, S. (2013). Burst Tests on SAWL 485 Line Pipes for Nord Stream. International Offshore and Polar Engineering Conference, ISOPE-I-13-575. Chater, E., & Hutchinson, J. W. (1984). On the Propagation of Bulges and Buckles. Journal of Applied Mechanics, 51(2), 269–277. Jensen, H. M. (1988). Collapse of hydrostatically loaded cylindrical shells. International Journal of Solids and Structures, 24(1), 51–64. Ju, G. T., & Kyriakides, S. (1988). Thermal Buckling of Offshore Pipelines. Journal of Offshore Mechanics and Arctic Engineering, 110(4), 355–364. Katsounas, A.T., Tassoulas, J.L. (1990). Finite Element Analysis of Propagating Buckles in Deep Water Pipelines. Offshore Technology Research Center, OTRC report B1, College Station, TX.

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Kyriakides, S. (1993). Propagating Instabilities in Structures. Advances in Applied Mechanics, 30, 67–189. Kyriakides, S., & Babcock, C. D. (1979). On the Dynamics and the Arrest of the Propagating Buckle in Offshore Pipelines. In Offshore Technology Conference, OTC-3479. Kyriakides, S., & Babcock, C. D. (1981). Experimental Determination of the Propagation Pressure in Circular Pipes. Journal of Pressure Vessel Technology, 103, 328–336. Kyriakides, S., & Chang, Y. C. (1992). On the Effect of Axial Tension on the Propagation Pressure of Long Cylindrical Shells. International Journal of Mechanical Sciences, 34(1), 3–15. Kyriakides, S., & Corona, E. (2007). Mechanics of Offshore Pipelines: Volume 1 Buckling and Collapse (1st ed., p. 401). Elsevier. Kyriakides, S., & Lee, L.-H. (2021). Mechanics of Offshore Pipelines: Volume 2 Buckle Propagation and Arrest (1st ed., p. 440). Gulf Professional Publishing, Elsevier. Kyriakides, S., & Netto, T. A. (2000). On the dynamics of propagating buckles in pipelines. International Journal of Solids and Structures, 37, 6843–6867. Kyriakides, S., Yeh, M.-K., & Roach, D. (1984). On the Determination of the Propagation Pressure of Long Circular Tubes. Journal of Pressure Vessel Technology, 106(2), 150–159. Mansour, G. N., & Tassoulas, J. L. (1997). Crossover of integral-ring buckle arrestor: computational results. ASCE Journal of Engineering Mechanics, 123(4), 359–366. Mesloh, R. E., Sorenson, J. E., & Atterbury, T. J. (1973). Buckling and Offshore Pipelines (pp. 40–43). Gas Magazine. Montel, R. (1960). Formule semi-empirique pour la détermination de la pression extérieure limite d’instabilité des conduites métalliques lisses noyées dans du béton. La Houille Blanche, (5), 560–568. Netto, T. A., & Kyriakides, S. (2000a). Dynamic performance of integral buckle arrestors for offshore pipelines. Part I: experiments. International Journal of Mechanical Sciences, 42(7), 1405–1423. Netto, T. A., & Kyriakides, S. (2000b). Dynamic performance of integral buckle arrestors for offshore pipelines. Part II: analysis. International Journal of Mechanical Sciences, 42(7), 1425–1452. Nogueira, A. C., & Tassoulas, J. L. (1995). Finite element analysis of buckle propagation in pipelines under tension. International Journal of Mechanical Sciences, 37(3), 249–259. Oh, D. H., Race, J., Oterkus, S., & Chang, E. (2020). A new methodology for the prediction of burst pressure for API 5L X grade flawless pipelines. Ocean Engineering, 212, 107602. Palmer, A. C., & Martin, J. H. (1975). Buckle propagation in submarine pipelines. Nature, 254(5495), 46–48. Park, T.-D., & Kyriakides, S. (1997). On the performance of integral buckle arrestors for offshore pipelines. International Journal of Mechanical Sciences, 39(6), 643–669. Song, H.-W., Tassoulas, J.L. (1990). Dynamics of Propagating Buckles in Deep-Water Pipelines. OTRC report B4, College Station, TX. Song, H.-W., & Tassoulas, J. L. (1993). Finite element analysis of propagating buckles. International Journal for Numerical Methods in Engineering, 36(20), 3529–3552. Stewart, G., Klever, F. J., & Ritchie, D. (1994). An analytical model to predict the burst capacity of pipelines. In International conference on Offshore Mechanics and Arctic Engineering. OMAE-13. Timoshenko, S. P. (1933). Working stresses for columns and thin-walled structures. Trans. ASME, Appl. Mech. Div. I, 173–185. Vasilikis, D. (2012). Structural behavior and stability of cylindrical steel shells with lateral confinement. PhD dissertation, University of Thessaly.

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Vasilikis, D., & Karamanos, S. A. (2009). Stability of confined thin-walled steel cylinders under external pressure. International Journal of Mechanical Sciences, 51(1), 21–32. Vasilikis, D., & Karamanos, S. A. (2014). Mechanics of Confined Thin-Walled Cylinders Subjected to External Pressure. Applied Mechanics Reviews, 66(1) AMR-12-1043. Yeh, M. K., & Kyriakides, S. (1986). On the Collapse of Inelastic Thick-Walled Tubes Under External Pressure. Journal of Energy Resources Technology, 108(1), 35–47. Yeh, M. K., & Kyriakides, S. (1988). Collapse of Deepwater Pipelines. Journal of Energy Resources Technology, 110(1), 1–11. Zhu, X.-K., & Leis, B. N. (2007). Theoretical and Numerical Predictions of Burst Pressure of Pipelines. Journal of Pressure Vessel Technology, 129(4), 644–652. Zhu, X.-K., & Leis, B. N. (2012). Evaluation of burst pressure prediction models for line pipes. International Journal of Pressure Vessels and Piping, 89, 85–97.

Metal pipes and tubes under structural loading

6

The structural response and buckling of long metal cylindrical shells under structural loading is examined in the present chapter. This includes transverse loading, axial compression and longitudinal bending, which are of particular importance for safeguarding the structural integrity of pipelines and pipe components under severe external actions. Their structural response is significantly affected by the level of pressure and whether the pressure is internal or external. In the first part of this chapter, the response to transverse loading is examined. Subsequently, the structural behavior of metal pipes under uniform compression and bending loading is described.

6.1

Metal pipe subjected to transverse loading

The response of an elastic ring of width b and thickness t subjected to two opposite compressive radial loads Q, shown in Fig. 6.1A has been examined in Section 3.4, and the following solution has been obtained, which relates force Q with the lateral displacement wL :    Eb t Qel = 1.12 wL = Kel wL 2 1 − νe rm

(6.1)

In the above equation, subscript ( r)el refers to elastic material behavior, Kel is the ring stiffness for this type of loading, considering elastic material behavior, and is equal to the expression in the brackets. Furthermore, the mean radius rm is used. Eq. (6.1) is represented by the first (linear) part of the load-displacement sketch in Fig. 6.1C. The radial loading may represent either two concentrated loads at diametrically opposite points of the ring (Fig. 6.1A) or crushing of the ring between two rigid plates (Fig. 6.1B). Quite often, the ring problems in Fig. 6.1A and Fig. 6.1B are twodimensional approximations of the long cylinder problems shown in Fig. 6.2 (L → ∞). In that case, one may consider a ring from the cylinder with unit width (b = 1) and the concentrated load Q in Fig. 6.1 should be considered as load per unit length along the cylinder. The linear elastic solution of Eq. (6.1) may also be applicable in a metal ring, provided that the stresses are below the yield stress of the ring material. Therefore, this solution is valid for the initial stages of ring deformation. With further increase of lateral load, because of the “oval” shape of the deformed ring, plastic deformation occurs at four equally-spaced locations of the ring θ = 0, π /2, π , 3π /2. Eventually a plastic hinge is formed at each of those locations, and a four-plastic-hinge mechanism develops, similar to the one presented in Appendix E and used for external pressure Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00019-0 c 2023 Elsevier Inc. All rights reserved. Copyright 

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(A)

(B)

(C)

Figure 6.1 (A) Ring under two opposite radially-directed concentrated loads. (B) Ring crushed between two rigid plates. (C) Idealized elastic-plastic load-deflection diagram for a metal ring subjected to transverse loading.

(A)

(B)

Figure 6.2 (A) Long pipe subjected to uniform line loads along its length acting in the radial direction. (B) Crushing of a long pipe between two rigid parallel plates.

collapse in Chapter 5. The plastic load Qpl corresponding to the formation of the four plastic hinges (Fig. 6.1C), can be obtained from equilibrium of moments in a quarter-ring. As a first approximation, equilibrium at the undeformed configuration is considered, as shown in Fig. 6.3: 2MP = Q pl rm /2

(6.2)

Considering double symmetry of the problem andtakinginto account that the fullyplastic moment of the cylinder wall MP is MP = σY bt 2 /4 , one readily obtains from Eq. (6.2): Q pl = σY

bt 2 rm

(6.3)

If the ring model represents a “slice” of a long cylinder, then a unit width can be considered (b = 1) and Qpl in Eq. (6.3) is the plastic load per unit length of the cylinder,

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189

Figure 6.3 Equilibrium of plastic-hinge mechanism in a metal ring loaded with two opposite radially-directly loads, in the undeformed configuration of the ring.

Figure 6.4 (A) Plastic-hinge mechanism and (B) geometry and equilibrium at the deformed configuration of a quarter of the pipe cross-section.

equal to: Q pl = σY

t2 rm

(6.4)

In this case, in order to represent more accurately the plane strain conditions of the ring, the yield stress σ Y in the above equations should be replaced by the yield stress under plane strain conditions σY∗ (see Section 4.1). According to the above approach, the maximum load sustained by the ring is equal to Qpl , and the bilinear response of Fig. 6.1C is obtained. This solution does not account for the change of ring geometry during the deformation of plastic mechanism. To account for this nonlinear geometric effect, an approach similar to the one followed in Section 5.2 should be adopted. In this analysis, the two cases in Fig. 6.1A and Fig. 6.1B are treated separately, because of the different way that the load is applied. The kinematics and the equilibrium of the concentrated load case in Fig. 6.1A is shown in Fig. 6.4. Equilibrium in the horizontal direction leads directly to: H=0

(6.5)

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Figure 6.5 Ring model for crushing between two parallel rigid plates.

and the balance of moments with respect to point B results into the following expression: Q β = 2MP + Hγ 2

(6.6)

From Eqs. (6.5) and (6.6) one obtains the following equation for the lateral load Qpl in the plastic range, where subscript ( r)pl indicates the response in the plastic regime:  2 σY t 1 (6.7) Q pl = √ rm 1 + 2x − x2 where x = wL /rm . Therefore, the resistance of an elastic-plastic ring under radial loading can be approximated as a combination of Eqs. (6.1) and (6.7). In this approach, the response is considered initially elastic up to the intersection Q max of the elastic solution in Eq. (6.1) with the plastic collapse in Eq. (6.7). This concept of elasticplastic response has been presented in Chapter 5 for the case of external pressure and is also employed herein for transverse loading. The value of load at the intersection of the two curves is an estimate of the maximum load that the ring can carry. One may also consider that the actual maximum force Qmax sustained by the pipe is slightly lower than the intersection load Q max . Using the four-hinge plastic mechanism, the response of an inelastic ring crushed between two rigid plates in Fig. 6.1B can also be described in a simple and efficient manner, based on the deformed configuration shown in Fig. 6.5. The main difference with the previous analysis is that the point of application of each transverse load moves horizontally because of the rigid plates that remain always parallel to each other. The analysis leads to the following expression for the load Qpl in the plastic range:  Q pl =

 σY t 2 1 √ rm 1 − x2

(6.8)

The maximum load that the metal ring can sustain may be approximated by the intersection of the curves defined by Eqs. (6.1) and (6.8).

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191

Figure 6.6 Effects of external pressure on the equilibrium of the collapse mechanism.

The above expressions in Eqs. (6.1), (6.7) and (6.8) need to be enhanced to account for the presence of external or internal pressure. For the elastic branch of the solution, one may use the amplification factor used in Eq. (5.30): Qel =

1 Kel wL 1 − p/pcr

(6.9)

where p is the pressure, which is considered positive if it is external and pcr is the  buckling pressure of the ring under uniform external pressure pcr = critical  elastic 2E/ 1 − νe2 (t/Dm )3 . For the second branch of the solution and for the case of concentrated loads (Fig. 6.1A), the equilibrium equations of the plastic mechanism expressed by Eqs. (6.5) and (6.8), in the presence of pressure, become (see also Fig. 6.6): H = SP sin ϕ

(6.10)

 √ Q β = 2MP − SP rm / 2 − Hγ 2

(6.11)

and

Combining Eqs. (6.10) and (6.11), the plastic collapse solution of the pressurized ring under opposite concentrated radial loads is:   2     σY t 1 p Dm  (6.12) 2x − x2 1− Q pl = √ rm pY t 1 + 2x − x2 Repeating the above procedure for a ring crushed between two rigid plates (Fig. 6.1B), the plastic load of the pressurized ring is expressed as:  2     σY t 1 p Dm 2 1 − x 2x (6.13) 1 − Q pl = √ rm pY t 1 − x2 Karamanos and Eleftheriadis (2004) compared the above model with results from finite element simulations (Fig. 6.7) on steel pipes with yield stress σ Y = 60 ksi, and

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Figure 6.7 Steel pipes loaded radially in the presence of external pressure; two-dimensional (plane strain) finite element simulations versus predictions from analytical models: (A) two opposite line loads, (B) ring crushing between two rigid plates.

D/t = 48, subjected to lateral loading under zero pressure and for two levels of external pressure (16.3% and 32.6% of the collapse pressure pco ). The finite element models consider a ring under plane strain conditions, therefore, for a fair comparison with the analytical solution, the yield stress in the hoop direction under plane strain conditions σY∗ is considered, instead of the yield stress under uniaxial loading conditions. Herein, the value of σY∗ is taken equal to 1.15σ Y . The load per unit length Q is normalized by Q pc = σY∗t 2 /rm , (Q/Q pc ), where rm is the mean pipe radius, and σY∗ is the yield stress in the hoop direction under plane strain conditions, considered equal to 1.15σ Y . The values of pressure are normalized by the collapse pressure pco of the pipe. For the pipes under consideration (D/t = 48), the value of pco is equal to the elastic buckling pressure pcr (pcr = 2Et 3 / (1 − νe 2 )D2m ). The results in Fig. 6.7A refer to concentrated

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(A)

193

(B)

Figure 6.8 Steel pipes loaded with a wedge-shape denting tool; experiments performed at TNO, The Netherlands, for N.V. Nederlandse Gasunie in early 80’s (published with permission).

radial loads (case A in Fig. 6.2), and those in Fig. 6.7B to crushing between two rigid plates (case B in Fig. 6.2). The results show that external pressure has a strong destabilizing effect on the structural response of the pipe. In the presence of relatively small levels of external pressure (less than 1/3 of the collapse pressure of the pipe), the load and the deformation capacity of the pipe under transverse loading decreases by a substantial amount. Comparison between the results of Fig. 6.7A and those of Fig. 6.7B shows that, in the absence of pressure, pipe resistance to concentrated loads is less than pipe resistance to rigid plates. However, increasing the level of external pressure, this difference is alleviated. Finally, all results in Fig. 6.7 demonstrate that the simplified four-hinge solution can provide a fairly good approximation of the response of rings made of elastic-plastic material subjected to transverse loads. The above models are two-dimensional and are representative of either an isolated ring or, most frequently, one of the three-dimensional cases shown in Fig. 6.2, where no variation of loading or support conditions are assumed along the cylinder (pipe) length. However, in numerous cases, a pipe is subjected to a concentrated denting load, which is applied locally in a small part of its length, which is a three-dimensional problem. Fig. 6.8 shows two cases, where the denting tool is a transversely oriented wedge (Wildschut et al., 1984). The above two-dimensional models may not be adequate for simulating the denting response of the pipe under such loading conditions, and more elaborate analytical models should be considered. Several models have been developed to describe the denting response of pressurized tubular members under lateral concentrated loads and to define the force-displacement (F − δ) relationship (see Fig. 6.9). The model presented below is a simplified model, based on a ring-type approximation of the pipe, enhanced to account for threedimensional effects through appropriate “effective lengths” or “effective widths”. The model was introduced by Gresnigt (1984) and was revisited more recently by Gresnigt et al. (2007). The model formulation results in closed-form expressions for the denting response of cylinders, which represent the general features of pipe behavior under

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Figure 6.9 Schematic representation of cylindrical shell denting.

wedge loading in a simple and elegant manner. The total response is assumed to consist of three major parts: (1) the initial elastic solution, (2) the plastic mechanism and, finally, (3) the membrane or stretching part. In all parts, the effects of pressure are taken into account, with appropriate amendment of the corresponding force-displacement equations. Both longitudinal and transverse orientations of the wedge denting tool are considered in the final expressions.

6.1.1 Elastic solution The elastic part of the response is based on the idealized problem of a cylinder subjected to a pair of radial opposite loads F, which are denoted as Fel for this part of the solution. A linear expression relating Fel and the relevant lateral displacement δ can be written in the following form (Gresnigt et al., 2007): Fel =

EIr Bel δ 0.149rm3

(6.14)

where Ir is the moment of inertia of the pipe wall and Bel is the so-called “equivalent pipe length” for elastic response under transverse loading:

(6.15) Bel = 1.33r rm /t + b The effects of pressure are taken into account, considering the well-known pressure amplification factor: αrr = 1/(1 − p/pcr )

(6.16)

where p is the pressure around the pipe, which is positive when it is external and external pressure of a negative when it is internal, and pcr is the critical (buckling)   uniformly pressurized ring pcr = 3EI/rm3 [or pcr = 2E/ 1 − νe2 (t/Dm )3 ], so that: Fel =

EIr √ Bel δ 0.149rm3 αrr

(6.17)

For p = 0 and Bel = 1, Eqs. (6.14) and (6.17) are identical to the formula for ring deformation under opposite radial loads and plane strain conditions expressed

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195

by Eq. (6.1). Furthermore, for the case of internal pressure, which is quite frequent in practical applications, the presence of internal pressure increases pipe resistance. The dimensionless form of Eq. (6.17) is:  3/2 λ0 βel Fel t = 1.582 x¯ (6.18) fel = √ Fpc rm αrr  √ where Fpc = σY t 2 /4 Dm /t is a force-like parameter, x¯ is the normalized denting displacement of the applied force   (x¯ = δ/rm ), βel = Bel /rm is the dimensionless effective length, and λ0 = E/ σY 1 − νe2 is a dimensionless material parameter.

6.1.2 Plastic mechanism solution The fully-plastic solution is based on a plastic mechanism with four equally-spaced plastic hinges (Fig. 6.3). This model assumes rigid-plastic behavior and plane strain conditions. Applying static equilibrium on the circular (non-deformed) configuration of the plastic mechanism in the absence of internal or external pressure, as shown in Fig. 6.3, one readily obtains: 2M p = Q pl rm /2

(6.19)

where Mp is the plastic moment of the tube wall under hoop bending, which is equal to M p = σY B pl t 2 /4, and Bpl is the corresponding “equivalent length” of the pipe for plastic response. Two corrections of the above expression are necessary, to improve the corresponding predictions. First, the yield stress under plane strain conditions should be used (σ Y ∗ ). Considering that the Poisson’s ratio in the full plastic range under plane strain conditions is equal to 0.5 and, adopting the von Mises yield criterion, the yield stress in the hoop direction should be increased by 15 percent (σ Y ∗ = 1.15σ Y ). In addition, the presence of a uniform hoop stress σ θ due to internal pressure (σθ = prm /t) reduces the plastic bending moment capacity of the pipe wall (see also Appendix C). Under those two enhancements, the limit plastic load Fpl is written as follows:   2  σY t 2 σθ (6.20) Fpl = 1.15 1 − 0.75 B pl SB rm σY where the term in the brackets accounts for the reduction of yield stress due to hoop stress σθ = prm /t, through the von Mises yield criterion, and SB is a factor that accounts for the local effects of a transversely oriented denting device due to nonuniform contact. The effective length of the plastic mechanism solution is smaller than the corresponding effective length for the elastic solution, because inelastic deformations are more local, which is verified by experimental results. Based on these observations (Gresnigt, 1984; Gresnigt et al., 2007), length Bpl and factor SB are chosen equal to:

(6.21) B pl = 0.80 rm rm /t + b SB =

rm rm − 0.35 b

(b < 2rm )

(6.22)

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Figure 6.10 Stretching of cylinder generators during the denting process and their resistance.

and SB is equal to unity when a longitudinally oriented wedge is applied. Note that the above plastic load expressed in Eq. (6.20) is constant with no dependence on the denting displacement, and this is due to the consideration of equilibrium in the circular configuration. The dimensionless form of Eq. (6.20) is:   2  Fpl t σθ = 3.25 1 − 0.75 β pl SB f pl = Fpc σY rm

(6.23)

  where, Fpc = σY t 2 /4 Dtm and β pl = B pl /rm is the dimensionless equivalent length factor for plastic response.

6.1.3 Membrane solution The membrane response represents the stretching of pipe meridians that resist the denting process. Such a mechanism is activated when the denting depth δ has a finite value, and can be approximated with the following expression, which comes from static equilibrium of forces shown in Fig. 6.10: Fm = 2FST cos θ

(6.24)

In Eq. (6.24), Fm is the required lateral force for membrane deformation, and FST is the resultant denting force resisted by the stretching of cylinder generators. Considering an “effective length” l of the plastic membrane force in Fig. 6.10, the slope of the deformed generator is computed as follows: δ cos θ  √ 2 δ + l2

(6.25)

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197

Assuming rigid-plastic conditions, the resultant stretching force FST is written as follows: FST = W tσY

(6.26)

where W is an “effective width”, taken equal to 0.8l (Gresnigt, 1984). Furthermore, length l is taken as a portion of the effective length Bp :

l = (rm /8) rm /t

(6.27)

Therefore, using Eqs. (6.25)–(6.27) and Eq. (6.22), Eq. (6.24) can be written as follows:  rm δ

(6.28) SB Fm = 0.2σY rmt 2 t δ + rm3 /64t and in dimensionless form: fm =

r  Fm x¯ m

= 0.566 SB 2 Fpc t x¯ + rm /64t

(6.29)

where the SB factor has been introduced in Eq. (6.22), and x¯ = δ/rm . Eqs. (6.28) or (6.29) may be enhanced to account for plastic anisotropy, where a different yield stress σ Yx may exist in the longitudinal direction than the yield stress in the circumferential direction (σ Yx = σ Y ), and should replace σ Y in Eq. (6.26). To account for plastic anisotropy effects, the plastic anisotropy factor S = σY x /σY is introduced (σ Yx and σ Y are the yield stresses in the axial and hoop direction respectively), and Eq. (6.26) is re-written as follows: δ SB Fm = 1.6 l SσY t 2 δ + rm3 /64t

(6.30)

6.1.4 Pressure effects on inelastic response The effects of pressure on the plastic ring mechanism, as well as on the membrane part of the solution can be accounted for, considering the work done by the pressure due to the change of the cross-sectional area enclosed by the deforming pipe. The kinematics of the four-hinge plastic mechanism are used in the principle of virtual displacements, which equates the virtual work done by the external forces Fpi deforming the ring and the external work of pressure p times the change of area AE enclosed by the deforming ring. Details of this analysis can be found in Gresnigt (1984) and Gresnigt et al. (2007). The analysis results in a linear relationship between the lateral force corresponding to pressure effects Fpi and the total denting displacement δ: Fpi = 2pB pi δ

(6.31)

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Structural Mechanics and Design of Metal Pipes

where Bpi is the corresponding “equivalent length”, which is given by the following expression (Gresnigt, 1984):

(6.32) B pi = 0.4αrr rm rm /t + b Note that in Eq. (6.32) the “equivalent length” Bpi decreases with increasing internal pressure. The dimensionless form of Eq. (6.31) is: Fpi p rm 3/2 = 5.65 β pi x¯ (6.33) f pi = Fpc σY t where β pi = Bpi /R. It is important to note that the last term in the right-hand side of Eqs. (6.15), (6.21) and (6.32), i.e., the denting tool size b, should be included only in the case of longitudinally-oriented wedges, and excluded in transversely-oriented wedges.

6.1.5 Model summary and comparison with experimental data The above analytical model is summarized as follows: r

First part: elastic response, expressed by Eq. (6.17), which is a linear relationship between the denting force and the corresponding displacement δ: F = Fel

r

Second part: plastic ring mechanism Fpl expressed by Eq. (6.20), which is independent of δ; in the presence of pressure, add Fpi from Eq. (6.31), which is linear with respect to δ: F = Fpl + Fpi

r

(6.34)

(6.35)

Third part: membrane response, expressed by Eq. (6.28), which is a nonlinear relationship between the denting force and the corresponding displacement δ; if pressure is present, Fpi from Eq. (6.31) should be added: F = Fm + Fpi

(6.36)

The above formulation, shows clearly the increase of pipe resistance in the presence of internal pressure, expressed by the pressure parameter in Eq. (6.16), as well as by Eq. (6.31). On the other hand, external pressure has a destabilizing effect, assisting the denting process and reducing the denting resistance of the pipe. In the present formulation, pressure effects on the denting resistance can be used for both internal and external pressure, by simply changing the sign of pressure. Fig. 6.11 shows the very good comparison between available experimental results and the model predictions. In this comparison, three test specimens are considered, tested under lateral loading, applied through a wedge denting tool, with rounded edge, as described in Gresnigt (1984) and in Gresnigt et al. (2007). The properties of the tubular specimens are shown in Table 6.1. Despite the lack of clear distinction between the three parts of the model solution (elastic, rigid-plastic and membrane), especially for the case of transverselyoriented wedge, the overall comparison between experimental results and analytical

Metal pipes and tubes under structural loading

(A)

(B)

(C)

Figure 6.11 Comparison of experimental results with the simplified analytical model: (A) specimen 1, (B) specimen 2, (C) specimen 3.

199

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Table 6.1 Properties of laterally loaded specimens. Outer Diameter Specimen (mm) 1 165 2 165 3 133

Thickness (mm) 4.82 4.82 2.72

Length (mm) 2,000 2,000 900

Yield stress (MPa) 290 290 265 (axial) 335 (hoop)

Pressure (MPa) 0 4.0 6.62

Wedge orientation longitudinal (L) longitudinal (L) transverse (T)

predictions is very satisfactory. For more details on this topic, the reader is referred to the papers by Karamanos and Andreadakis (2006) and Gresnigt et al. (2007).

6.2

Uniform axial compression of a metal pipe

When a pipeline is compressed and is adequately restrained against lateral movement so that beam-type buckling is prevented, it may exhibit shell-type buckling referred to as “local buckling”. Typical examples are the compression of high-pressure/hightemperature buried pipes, or ground-induced actions on buried pipelines because of axial soil displacement that induce axial compression to the pipe. Shell-type buckling of axially loaded cylindrical shells with elastic material behavior has been analyzed in detail in Chapter 4. It was noticed that elastic cylinders buckle suddenly and catastrophically, and that the maximum axial load sustained by an elastic cylinder is highly sensitive to the presence of initial geometric imperfections. This behavior also characterizes thin-walled metal cylinders that buckle at stress level below the yield limit of the metal material. On the other hand, the structural instability of relatively thick-walled metal cylinders subjected to uniform axial compression occurs in the inelastic range of the pipe material and is associated with a more gradual sequence of events, where first bifurcation occurs well before the stage of structural failure. Fig. 6.12 illustrates schematically the response of an axially-compressed relatively thick-walled hollow cylinder (tube or pipe) in the inelastic range of the material and offers a good starting point for the present discussion. Initially, the tube exhibits uniform axial shortening and the axial load-displacement (F − u) diagram of Fig. 6.12 follows practically the uniaxial stress-strain curve of the tube material under compression. This load-displacement curve constitutes the fundamental (primary) path of the compressed cylinder and expresses the pre-buckling state of the present problem. At some level of axial load, denoted as (b1) in the F − u diagram, beyond the onset of yielding (Y), first bifurcation occurs and axisymmetric wrinkles develop, which are periodic along the cylinder. With increasing compressive deformation, the wrinkles grow in amplitude, the axial stiffness of the tube reduces, and the F − u diagram deviates from the material compression curve. However, this first bifurcation does not imply structural failure of the tube. On the contrary, the tube can sustain additional loading and deformation before it fails.

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Figure 6.12 Generic diagram showing the response of an axially loaded metal hollow cylinder.

As the tube is further compressed, one of the wrinkles becomes dominant, leading to localization of deformation and to a maximum load, Fmax , corresponding to displacement um , beyond which the cylinder becomes structurally unstable. In some cases, the maximum load Fmax can be significantly higher than the value of Fb1 as shown schematically in Fig. 6.12. The interval between ub1 and um , denoted as u = um − ub1 in Fig. 6.12, can be quite large in thick-walled cylindrical shells (i.e., in tubes with low values of the D/t ratio), but decreases when the D/t ratio increases. The value of u also depends on the strain hardening of the material after initial yielding. At some stage, a second bifurcation occurs, where the wrinkling pattern transforms into a nonaxisymmetric mode, followed by structural failure of the tube. The exact location of this non-axisymmetric bifurcation, and the number m of circumferential waves (lobes) in this non-axisymmetric pattern, both depend on the D/t ratio and the size of initial imperfections. The reader is also referred to the publication of Bardi and Kyriakides (2006), where the main features of this response are extensively described, based on experimental observations on small-scale metal specimens with D/t ranging from 22 to 52. In the following, two of those experiments are simulated with finite element models and are described below to elucidate the structural response of metal tubes under uniform axial compression. The metal tubes under consideration are seamless with diameter-to-thickness (D/t) ratio equal to 28.5 and 43.3, and are made of stainless steel with a uniaxial stress-strain curve that follows the Ramberg-Osgood law: ε=

 nˆ σ σ + αR E σ0

(6.37)

with hardening exponent nˆ = 13, coefficient α R = 3/7 and stress parameter σ 0 = 572 MPa. The yield stress of the material σ Y corresponding to a 0.2% offset strain is 592.6 MPa (85.9 ksi) and this value of stress is used for normalizing the results in a nondimensional form. The tubes have been tested experimentally by Bardi and Kyriakides (2006), and are simulated with finite element models that employ four-node shell

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Figure 6.13 Load-displacement (shortening) diagram for a thick-walled tube (D/t = 28.5); experimental data (Bardi and Kyriakides, 2006) versus finite element simulation.

elements, and a special-purpose elastic-plastic constitutive model suitable for shell buckling calculations (see Section 6.3 for a brief presentation of this model). Fig. 6.13 shows the load-shortening response for the thicker tube (D/t = 28.5). The axial force F and the shortening displacement u are normalized by the plastic (yield) force FY = σ Y πDm t and the gauge length of the specimen L, so that f¯ = F/FY and ε¯ = u/L. The value of ε¯ is referred to as “normalized shortening” and may be considered as a macroscopic compression strain of the tube. An initial geometric imperfection is considered as a linear combination of the axisymmetric mode and the non-axisymmetric mode with two circumferential waves (m = 2), with coefficients 0.05% and 0.8% of pipe thickness respectively. Both modes are obtained from an eigenvalue analysis of the compressed tube. The numerical results show that first bifurcation to a uniform wrinkling pattern occurs at ε¯b1 = 1.56% and the maximum load occurs at ε¯m = 3.43%, well beyond the onset of wrinkles. Fig. 6.14 depicts three configurations of the deformed tube representing the evolution of wrinkles until buckle localization and structural failure. Stage (a) corresponds to ε¯ = 3.64% slightly after the maximum load and shows a uniform wrinkling pattern of the tube. Even though this stage of deformation is well beyond first bifurcation, the wrinkles are hardly visible. Subsequently, in stage (b) one wrinkle grows faster than the others and tube deformation localizes in a non-axisymmetric pattern with two lobes (m = 2) called “mode 2” (ε¯ = 5.82%). With increased axial compression, the localized nonaxisymmetric pattern becomes more pronounced, as shown in stage (c), leading to structural failure of the tube. The load-shortening response of the thinner tube (D/t = 43.3) is shown in Fig. 6.15. An initial geometric imperfection is considered as a linear combination of the axisymmetric and a non-axisymmetric mode with m = 3, with coefficients 0.05% and 6% of pipe thickness respectively. First bifurcation occurs at ε¯b1 = 1.08% and the

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(a)

203

(b)

(c)

Figure 6.14 Three stages of axially compressed thick-walled tube (D/t = 28.5); from uniform wrinkling to a non-axisymmetric localized mode (m = 2); stages (a), (b) and (c) refer to the corresponding points in the diagram of Fig. 6.13.

Figure 6.15 Load-displacement (shortening) diagram for a thick-walled tube (D/t = 43.3), experimental data (Bardi and Kyriakides, 2006) versus finite element simulation.

maximum load is reached at ε¯m = 1.68%. In this tube, due to higher value of the D/t ratio, ε¯m is less distant than ε¯b1 . Fig. 6.16 shows the deformed configuration of the tube at two stages of deformation. Stage (a) is at ε¯ = 2.05%, which is beyond the stage of maximum load. At that stage, wrinkles on the tube wall are only slightly visible. However, with continuing compression, the deformation localizes gradually and reverts to a non-axisymmetric mode, with three lobes (m = 3), shown in Fig. 6.16B, which is reminiscent of a diamond-shape pattern. These observations were also reported in the corresponding experiments. The above numerical results are consistent with the buckled tube shapes shown in Fig. 6.17A, from laboratory experiments performed at the University of Thessaly (Papadimitriou, 2008). They refer to 4-inch-diameter mild steel tubular specimens,

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(a)

(b)

Figure 6.16 Two stages of axially compressed tube (D/t = 43.3); from uniform wrinkling to a non-axisymmetric localized diamond-shape mode (m = 3); stages (a) and (b) refer to the corresponding points in the diagram of Fig. 6.15.

(A)

(B)

Figure 6.17 (A) Buckled mild steel tubular specimens subjected to uniform axial compression (D/t = 44.4), (photo by S. A. Karamanos). (B) Schematic representation of variable-thickness tube suitable for axial compression test.

with yield stress 275 MPa and diameter-to-thickness ratio D/t = 44.4, buckled under uniform axial compression. The specimen on the left corresponds to a stage shortly after the maximum load, whereas the specimen on the right was compressed well beyond the maximum load so that the buckle localized into a highly folded configuration. Both specimens exhibited “mode 3” buckling (m = 3), shown in Fig. 6.18. This shape is reminiscent of the shape of the stainless steel tube in Fig. 6.16, which has a similar value of D/t ratio. To investigate the axial compression performance of long tubular specimens, it is necessary to eliminate the influence of clamping or other constraints at the connection of the specimen with the testing device. Otherwise, this would result in local buckling (“elephant’s foot”) near the tube ends and may affect the buckling response of the tubular specimen. Towards this purpose, the tubular specimens of Fig. 6.17A were properly machined in their central part prior to testing so that the variable-thickness

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205

Figure 6.18 Longitudinal view of axially compressed thick-walled tube (D/t = 44.4) with mode 3 buckling (m = 3), (photo by S. A. Karamanos).

geometry of Fig. 6.17B is obtained. The total length of the specimens was 312 mm, the length of each tapered part was 40 mm and the central part with constant thickness was 170-mm-long. A similar approach for eliminating end effects was followed in the axial compression experiments conducted by Bardi and Kyriakides (2006). A major observation from the above experiments and numerical simulations refers to the shape of the buckling mode at first bifurcation point (onset of wrinkling), which is always axisymmetric. It is instructive to analyze inelastic cylinders and obtain analytical buckling expressions to demonstrate in a rigorous manner that the mode at first bifurcation is always axisymmetric. The analysis presented below follows the methodology described by Gellin (1979), which is analogous to the one presented in Chapter 4 for elastic cylinders but considers elastic-plastic material response. The formulation employs J2 -deformation theory of plasticity (see Appendix D), and the Donnell-Mushtari-Vlasov shell kinematic equations (Brush and Almroth, 1975). First, the fundamental pre-buckling state is obtained, expressed in terms of the stress function f(x, θ) and the transverse (radial) displacement w(x, θ ) of the cylindrical shell: 1 f0 = − σ t rm2 θ 2 2 w0 =

νS σ rm ES

(6.38)

(6.39)

In Eqs. (6.38) and (6.39), subscript ( r)0 refers to pre-buckling state, σ is the average compressive stress, so that the total compressive force is F = σ A, and A is the crosssectional area of the metal cylinder (A = 2πrm t). Furthermore, ES is the secant modulus of the material and νS =

  1 ES 1 + νe − 2 E 2

(6.40)

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The pre-buckling stress state σ x0 , σ θ0 in the cylinder is expressed as follows: σx0 = −

1 f0,θθ = −σ rm2 t

σθ0 = σxθ0 = 0

(6.41) (6.42)

Subsequently, Gellin (1979) examined the possibility of bifurcation from the fundamental state, using the linearized equations of the elastic-plastic cylinder under uniform axial compression and assuming a perturbation of the fundamental (prebuckling) state (w1 ,f1 ). The resulting equations in terms of f1 and w1 are:  4  1 ∂ 4 f1 ∂ 4 f1 ∂ f1 1 ∂ 2 w1 =0 (6.43) A 4 + 2B1 2 2 + − 4 ET t ∂x ∂x ∂s ∂s rm ∂x2  4  ET t 3 ∂ 4 w1 ∂ 4 w1 ∂ 2 w1 ∂ w1 1 ∂ 2 f1 + 2B + +σ =0 A + 2 2 4 2 2 4 2 ∂x ∂x ∂s ∂s rm ∂x ∂x2 12(1 − νd )

(6.44)

where s = rm θ , f1 and w1 are the perturbations of the fundamental solution f0 and w0 , ET is the tangent modulus of the metal material (ET = dσ /dε), and:   3ET 1 1+ A= 4 ES B1 =

ET (1 + νS ) − νT ES

 1 − νd2 + νT 1 + νS   3 ET 1− νd2 = νT2 + 4 ES   1 ES 1 νe − νS = + 2 E 2     1 1 1 E 1 ET = + νe − νS − νT = + 2 ES 2 2 E 2 B2 =

ES ET

(6.45)

(6.46)



(6.47)

(6.48)

(6.49)

(6.50)

Setting ET = ES = E, one can readily show that Eqs. (6.43) and (6.44) reduce to the corresponding linearized equations for elastic cylindrical shells. Harmonic functions are considered for the perturbation functions w1 and f1 : w1 = α1 cos ms ¯ cos nθ

(6.51)

f1 = β1 cos ms ¯ cos nθ

(6.52)

Metal pipes and tubes under structural loading

207

where α 1 and β 1 are constants, m¯ =

2m − 1 π, L

m = 1, 2, 3, ...

(6.53)

and L is a characteristic length. Inserting the assumed solution (6.51) and (6.52) into the linearized equations of the cylindrical shell (6.43) and (6.44), one obtains the following buckling condition:  σ (m, ¯ n) = ET

1 t2  ϕ2 (m,  ¯ n) + 2 2 rm ϕ1 (m, ¯ n) 12 1 − νd

 (6.54)

where σ (m, ¯ n) is the bifurcation stress  ϕ1 = Am¯ + 2B1 2

n rm



2 +

n rm

4

1 m¯ 2

(6.55)

1 m¯ 2

(6.56)

and  ϕ2 = Am¯ + 2B2 2

n rm



2 +

n rm

4

To obtain the values of m¯ and n that minimize the value of bifurcation stress σ (m, ¯ n), it is required that: ∂σ =0 ∂ m¯ leading to the following expression:    12 1 − νd2 ϕ1 = rm t

(6.57)

(6.58)

Subsequently, it is further required that: ∂σ =0 ∂n

(6.59)

and the following bifurcation condition is obtained: (B1 − B2 )n = 0

(6.60)

One can show that for elastic-plastic material, B1 is always less than B2 (B1 < B2 ), and that the equality B1 = B2 is valid only if the material is elastic. Therefore, for the bifurcation into the elastic-plastic domain of the material, the wave number n must be equal to zero, which leads to the conclusion that the buckling mode associated with first bifurcation in the inelastic range is always axisymmetric. This is a great result, which verifies the experimental observations (e.g., Bardi and Kyriakides, 2006).

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Setting n = 0 in Eq. (6.54), the critical wave number in the longitudinal direction of the cylinder is: m¯ cr =

   4 12 1 − νd2 



1

rm t 3ET 1+ 4 ES

(6.61)



which corresponds to the following buckling half-wavelength Lhw and critical stress σ cr :    rm t π 3ET Lhw =   1 + (6.62)  4 ES 4 12 1 − ν 2 d



ET

t σcr =    3 1 − νd2 rm

 (6.63)

and the total axial compressive load is: Fcr = σcr (2π rmt )

(6.64)

Assuming an elastic material (ET = E and ν d = ν e ), Eqs. (6.62) and (6.63) reduce to the corresponding equations of axisymmetric elastic buckling presented in Section 4. Gellin (1979) also analyzed an initially-imperfect cylindrical shell with axisymmetric wrinkling imperfection w(x) ˆ in the form of the buckling mode obtained above (see also Fig. 6.19): w(x) ˆ = αˆ cos m¯ cr x

(6.65)

where αˆ is the imperfection amplitude and m¯ cr is the wave number given by Eq. (6.61). Upon application of axial load F, the tube configuration remains axisymmetric, the wrinkle amplitude increases from αˆ to α, and using a numerical formulation, a nonlinear response is obtained in the form of a nonlinear function F = F(α) that expresses the relation between the imposed axial load F and the corresponding wrinkle amplitude α. Subsequently, the possibility of a second non-symmetric bifurcation on the nonlinear axisymmetric path F = F(α) is examined. Considering the F = F(α) relation as the primary path of the structural system, assuming the following displacement pattern for the non-axisymmetric bifurcation mode: w(x) ˜ = cos nθ

∞  i=1



m¯ cr x ai cos (2i − 1) 2

 (6.66)

Metal pipes and tubes under structural loading

209

Figure 6.19 Axisymmetric wrinkling imperfection in cylinders.

and linearizing the equilibrium equations the possibility of bifurcation from the F = F(α) path to a non-axisymmetric configuration is examined. This procedure constitutes an extension of Koiter’s elastic post-buckling formulation (Koiter, 1963) for inelastic material behavior. In Eq. (6.66) the axial buckling wavelength is assumed to be twice that of the axisymmetric prebuckling solution in Eq. (6.62). This stems from Koiter’s argument that buckling in an axisymmetric configuration will be stimulated in areas with compressive hoop stress and will be restrained in areas with tensile hoop stress. Solution of this bifurcation problem (Gellin, 1979) demonstrates that this non-axisymmetric bifurcation occurs at a load F cr significantly lower than the axisymmetric buckling load Fcr of an imperfection-free cylinder, which is expressed in Eqs. (6.63) and (6.64). Furthermore, Gellin’s results have shown that in metal cylinders with elastic-plastic behavior, the reduction of F cr value with increasing initial imperfection amplitude αˆ is less pronounced than the corresponding reduction in elastic cylindrical shells (Koiter, 1963). This is an indication that elastic-plastic cylindrical shells are less sensitive to initial imperfections than the corresponding elastic ones. The effect of internal pressure on the response of axially compressed cylinders is important for pipelines and pipe components, which are subjected to compressive loads during their operation. Experimental results on small-scale stainless-steel specimens with D/t ratio equal to 28 and 40 (Parquette and Kyriakides, 2006) have shown that internally pressurized specimens remain axisymmetric well beyond their ultimate load. In addition, the value of maximum load Fmax is reduced in the presence of internal pressure, due to early yielding of the material, and the corresponding buckling wavelength increases with respect to its value in the non-pressurized case. In those

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tests, regardless of the level of pressure, the “distance” u between first wrinkling and ultimate load formation was quite significant.

6.3

A note on constitutive modeling for buckling calculations

In the above bifurcation analysis, the use of appropriate material stiffness moduli is essential for estimating accurately the buckling load. One may be tempted to use the moduli of classical J2 -flow theory of plasticity, given that this model is extensively employed in simulating the inelastic response of metals in numerous engineering applications. However, bifurcation predictions based on J2 -flow theory are quite higher than those observed experimentally. This has been attributed to the “corner” or “vertex” developed on the yield surface at the loading point during continuing plastic loading, which was more recently observed experimentally by Kuwabara et al. (2000). As the yield surface becomes non-smooth at the loading point, plastic strain increments may be directed in any direction within the vertex. In such a case, the flow rule becomes non-associative, the material stiffness reduces, and the response becomes more compliant (see also Appendix D). This effect may be negligible for loading paths that are proportional or quasi-proportional in the stress space. However, it may become significant when an abrupt change occurs on the stress path, as in the case of shell bifurcation (buckling) into a wavy pattern. Several attempts have been made to account for this phenomenon and simulate accurately buckling, in accordance with available experimental data. It has been shown that bifurcation calculations that use the moduli of J2 -deformation theory provide better predictions with respect to experimental data in comparison with J2 -flow theory moduli. However, the classical deformation theory is a total strain theory and does not account for elastic unloading. In this context, the combined use of J2 -flow theory for tracing the equilibrium path of a cylindrical shell, and J2 -deformation theory moduli for identifying bifurcation on the equilibrium path, may be a computational compromise and has been employed by Bushnell (1974), Ju and Kyriakides (1992) and more recently by Bardi et al. (2006). Nevertheless, such an approach uses two different material models within the same analysis. A number of constitutive models for metal plasticity with yield surface vertices have been proposed, which account for more compliant behavior and higher plastic deformation in non-proportional loading, when compared to J2 -flow theory. A key model of this category is the J2 -corner theory developed by Christoffersen and Hutchinson (1979), which has been employed in investigating shear band formation (Hutchinson and Tvergaard, 1981) and in modeling structural instability of compressed cylinders (Tvergaard, 1983). Nonetheless, the calibration and implementation of such models in a finite element environment is quite cumbersome, making their use rather unattractive for structural computations. A few special-purpose constitutive laws have also been proposed, mimicking the increased plastic flow caused by yield surface corners, while maintaining the smooth shape of Von Mises yield surfaces (e.g., Simo, 1987;

Metal pipes and tubes under structural loading

(A)

211

(B)

Figure 6.20 Bending response of a non-pressurized thin-walled steel pipe with D/t = 100 (Van Foeken, 1994); (A) moment-curvature diagram; (B) buckled shape of the pipe (photo courtesy: TNO, The Netherlands).

Peek, 2000; Pappa and Karamanos, 2016). In this framework, the recent work by Nasikas et al. (2022), proposes a non-associative model, and presents its rigorous numerical implementation in a finite element environment [see also Nasikas (2022)]. The finite element results in Fig. 6.13–Fig. 6.16 have been obtained using this nonassociative plasticity model.

6.4

Bending of long metal pipes

Bending of metal tubes or pipes has numerous applications in offshore and onshore pipeline engineering. The corresponding elastic problem was presented in detail in Section 4.8. It was demonstrated that the bending response of elastic tubes differs from the response predicted by simple beam theory, mainly because of cross-sectional ovalization and the ensuing limit-point instability (Brazier effect). Furthermore, wrinkling of the cylinder wall may occur on the compression side of the tube, a bifurcation type of instability. For initially straight elastic tubes, it was shown that wrinkling occurs prior to the limit point. The above elastic behavior may also describe the bending response of thin-walled metal tubes or pipes, with D/t value equal to 100 or larger. In this case, inelastic deformation prior to buckling is rather limited, if any, and wrinkling occurs suddenly. The moment-curvature for a pipe with D/t = 100 is shown in Fig. 6.20A. The pipe has diameter 160.4 mm, thickness 1.59 mm, yield stress 327 MPa and was subjected to pure bending, as part of a series of experiments performed by TNO, in Rijswijk, The Netherlands (Van Foeken, 1994). In this graph, the values of bending moment and curvature are normalized by the plastic bending moment of the pipe section MP

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Structural Mechanics and Design of Metal Pipes

Figure 6.21 Bending moment-curvature diagram of an aluminum tube with D/t = 60.5; experimental data (Kyriakides and Ju, 1992) versus finite element simulation.

(MP = σY D2m t) and by the curvature-like parameter kI (kI = t/D2m ), so that the nondimensional values of moment and curvature are obtained (m¯ = M/MP and κ¯ = k/kI ). The maximum bending moment recorded in the experiment is 90% of MP and occurs at normalized curvature κ¯ m = 0.68. The shape of the buckle shown in Fig. 6.20B is a typical diamond-shape buckle at the compression side with wrinkles (“waves”) that develop in both the hoop and the longitudinal direction. It consists of a main buckle, which is symmetric with respect to the plane of bending, and two or more side buckles. Fig. 6.21 shows the moment-curvature diagram of a long aluminum pipe with D/t ratio equal to 60.5 and yield stress 299 MPa, subjected to global bending. The stressstrain curve of the material follows the Ramberg-Osgood law in Eq. (6.37), with nˆ =28, α R = 3/7 and σ 0 = 298.6 MPa. The pipe was experimentally tested by Kyriakides and Ju (1992) and the experiment was simulated recently by Nasikas (2022) using a finite element model, which employs four-node shell elements and the special-purpose constitutive model described in Section 6.3. The model has a small initial imperfection, in the form of wrinkling, with amplitude equal to 0.1% of the pipe radius, which is consistent with the imperfection amplitude reported by Ju and Kyriakides (1992). The combined effects of plasticity and ovalization result in progressive reduction of bending stiffness and flattening of the compressed side of the cylinder. Wrinkles develop on the compression side and the maximum bending moment is reached at κ¯ m = 0.90, with m¯ max = 0.89, which are very similar to the experimental values. The value of maximum moment Mmax is above the yield moment but below the fully-plastic moment MP , which is typical for pipes with similar D/t value. Upon reaching the maximum moment, the pipe exhibits a sudden drop in bending resistance, represented by the “snap-back” in the moment-curvature diagram of Fig. 6.21. Fig. 6.22 shows four consecutive stages of pipe deformation, also denoted in Fig. 6.21. Stage (a) is just before the maximum moment (κ¯ = 0.88), indicating that a slight local buckle has already formed in the middle

Metal pipes and tubes under structural loading

213

(a) (b) (c) (d)

Figure 6.22 Four consecutive configurations of an aluminum pipe (D/t = 60.5) from finite element simulation, showing the growth of local buckling; stages (a)–(d) correspond to the points depicted in the diagram of Fig. 6.21.

section. Stage (b) is at κ¯ = 0.93, which is the end of the “snap-back” of the momentcurvature diagram. Continuation of bending deformation results in rapid development of local buckling [stage (c)], which obtains finally the form of a “diamond-shape” wrinkling pattern shown in stage (d). Thicker pipes, which are common in offshore pipeline applications, exhibit a smoother bending response, and the above phenomena occur more gradually. In particular, pipes with D/t values less than 25, which are suitable for deep offshore applications, exhibit a smooth moment-curvature response, with substantial amount of inelastic deformation before structural failure. Their response is rather smooth, characterized by a maximum bending moment Mmax due to cross-sectional ovalization, which is called “ovalization limit point” or simply “limit point”, and the corresponding normalized value of curvature is denoted as κ¯ m . The value of Mmax is very close to the fully-plastic moment Mp of the pipe cross-section. If the pipe material possesses sufficient strain hardening, the value of Mmax may exceed Mp . Beyond this limit point, ovalization localizes in a several-diameters-long zone and continues to grow, while the bending moment decreases very smoothly. In those thick-walled pipes, ovalization instability is dominant because it is the mechanism associated with the maximum bending moment. Fig. 6.23 depicts the response of a thick-walled aluminum pipe (D/t = 19.5), subjected to pure bending. The stress-strain curve of the material follows the Ramberg-Osgood law in Eq. (6.37), with hardening exponent nˆ = 37, α R = 3/7 and σ 0 = 308.9 MPa. The yield stress of the pipe corresponding to 2% plastic strain is 309 MPa. The pipe was tested experimentally by Kyriakides and Ju (1992) and has also been simulated numerically by Nasikas (2022), with an initial imperfection amplitude 0.1% of the pipe radius. The response in Fig. 6.23 is different from the response shown in Fig. 6.21. The moment-curvature diagram is characterized by a

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Structural Mechanics and Design of Metal Pipes

Figure 6.23 Bending moment-curvature diagram of a moderately thick-walled aluminum tube (D/t = 19.5); experimental data (Kyriakides and Ju, 1992) versus finite element simulation.

smooth limit point instability at κ¯ m = 0.95, with maximum bending moment 7% higher than Mp (m¯ max = 1.07). Beyond that point, the pipe is capable of sustaining further bending deformation without significant loss of bending moment resistance until a value of κ¯ equal to 1.24, as shown in Fig. 6.23. At that stage, the bending resistance starts decreasing abruptly, represented by the rapid descent of the momentcurvature diagram, and the pipe becomes structurally failed. The corresponding value of curvature, denoted as κ¯C , is called “ultimate” or “buckling” curvature. In the present case, κ¯C = 1.24. Fig. 6.24 shows four consecutive stages of pipe deformation, also denoted in the moment-curvature diagram of Fig. 6.23. Stages (a) and (b) correspond to maximum moment (κ¯ m = 0.95) and ultimate curvature (κ¯C = 1.24) respectively. Up to that stage, no local buckle is detected. Further continuation of bending results in the onset of local buckling in stage (c), leading to an “inward kink” [stage (d)] and to complete structural collapse of the pipe.

6.5

Effect of internal pressure on bending response

Bending in the presence of internal pressure is a loading condition associated with externally-imposed actions on a pipeline during its operation, e.g., ground-induced actions. The effect of internal pressure on the bending response of pipes made of elastic material has been discussed in detail in Section 4.9 and it was found that it is a stabilizing factor for the bent pipe. When compared with the non-pressurized bending case, internally pressurized elastic pipes under bending undergo smaller cross-sectional ovalization, with less flattening in the compression zone, leading to bifurcation and pipe wall wrinkling at larger values of curvature. Furthermore, in elastic tubes the buckling wavelength under pressurized bending conditions is smaller than the one of the

Metal pipes and tubes under structural loading

215

(a) (b)

(c) (d) Figure 6.24 Consecutive configurations of pipe deformation under bending (D/t = 19.5); local buckling development (finite element simulation); stages (a)–(d) correspond to the points depicted in the diagram of Fig. 6.23.

non-pressurized case, because of the smaller local hoop curvature at the compression zone (see also Houliara and Karamanos, 2006). Fig. 6.25A shows the moment-curvature diagram of a thin-walled pipe internally pressurized at 10 bar (1 MPa) with D/t = 102, with geometric and material properties very similar to the non-pressurized pipe shown in Fig. 6.20 and discussed in Section 6.4 (Van Foeken, 1994). Despite the similar value of maximum bending moment recorded in the two experiments, the bending-curvature response in Fig. 6.25A is different than the one shown in Fig. 6.20A because of the presence of internal pressure. The corresponding normalized value of curvature κ¯ m is 0.84, which is 23.5% higher than the non-pressurized case (κ¯ m = 0.68). Furthermore, the buckled shape of the pressurized pipe in Fig. 6.25B is characterized by an outward buckle, or “bulge”, as opposed to the diamond-type shape of the non-pressurized case in Fig. 6.20B. The moment-curvature diagrams in Fig. 6.26 show the effects of internal pressure on three metal pipes with D/t = 52, made of stainless steel material, subjected to pressurized bending. Experimental results for those tubes were reported by Limam et al. (2010). The stress-strain curve of the pipe material follows the RambergOsgood law in Eq. (6.37), with nˆ = 9.3, α R = 3/7 and σ 0 = 227 MPa, and the average measured yield stress σ Y of the tubes was measured 258.8 MPa (37.5 ksi). Three pressure levels are considered in the present analysis, namely 0%, 13.8% and 46.7% of nominal yield pressure pY . The corresponding buckling shapes are depicted in Fig. 6.27. Comparison of the moment-curvature diagrams for different levels of pressure indicates a substantial influence of the internal pressure level on

216

(A)

Structural Mechanics and Design of Metal Pipes

(B)

Figure 6.25 Bending of an internally pressurized thin-walled steel pipe with D/t = 102 (Van Foeken, 1994); (A) moment-curvature diagram; (B) buckled shape of the pipe (photo courtesy: TNO, The Netherlands).

the structural response. The results from numerical simulations are in very good agreement with the experimental results (Nasikas, 2022). The buckling process of the pipe begins with the appearance of small amplitude axial wrinkles on its compressed side, which are periodic along its length. At some stage, one wrinkle starts growing rapidly and eventually the deformation localizes and takes the shape of an outward bulge. The maximum bending moment of the pipe m¯ max is reached at κ¯ m followed by structural failure. For the non-pressurized case (Fig. 6.26A) κ¯ m = 1.27, but in the presence of internal pressure the value of κ¯ m is significantly higher and equal to 2.46 and 5.19 for the two pressure levels under consideration (Fig. 6.26B and Fig. 6.26C). The numerical results verify the stabilizing effect of internal pressure, which decreases ovalization and delays local buckling, so that structural instability occurs at a larger value of curvature. For the pipe under consideration, the value of κ¯ m for the 46.7% level is more than 4 times larger than the κ¯ m value for the nonpressurized pipe. It is also shown that for these pipes, the presence of internal pressure increases the maximum bending moment and the buckling wavelength in comparison with pure bending conditions. This is a different result than the one reported in Section 4.9 for elastic pipes. More specifically, the value of buckling wavelength is 6.4 mm in pure bending and increases to 8.2 mm for internal pressure equal to 46.7% of pY . The increase of the buckling wavelength in pressurized metal pipes is attributed to inelastic deformation and the bi-axial state of stress, which influence material rigidity at the local buckling area (Nasikas, 2022). Finally, it is underlined that in internally pressurized pipes the pipe material should be ductile enough to sustain the large bending deformations developed in the postbuckling configuration. Upon local buckling, the deformation concentrates at the buckled area of the pipe, and large tensile strains develop at the tension side of the pipe. Therefore, if pipe material ductility is not sufficient, the pipe wall may fracture,

Metal pipes and tubes under structural loading

(A)

(B)

(C)

Figure 6.26 Bending response of stainless-steel pipes (D/t = 52); (A) zero pressure; (B) internal pressure 13.8% of yield pressure; (C) internal pressure 46.7% of yield pressure; experimental data (Limam et al., 2010) versus finite element simulation.

217

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Structural Mechanics and Design of Metal Pipes

(A) (B)

(C)

Figure 6.27 Effect of internal pressure on the buckled shape of bent pipes (D/t = 52); (A) zero pressure; (B) internal pressure 13.8% of yield pressure; (C) internal pressure 46.7% of yield pressure.

Figure 6.28 Fracture of an internally-pressurized pipe subjected to bending (photo courtesy: Rina Consulting – Centro Sviluppo Materiali SpA).

soon after the maximum bending moment is reached. Such a case is shown in Fig. 6.28 for an X80 48-inch-diameter 22.9-mm-thick UOE pipe tested in bending under 50% of yield pressure (Fonzo et al., 2011). The pipe was intentionally tested well beyond the onset of local buckling and the maximum bending moment, and this resulted in pipe wall rupture at the tension side of the buckled cross-section. This situation is most critical when local buckling occurs at the vicinity of a girth weld, a common case because of the discontinuity induced by the weld. Upon local buckling, the tension side of pipe wall at this location is prone to rupture, given that girth welds are the “weak spots” of the pipeline under severe longitudinal strain, as discussed in more detail in Chapter 9.

Metal pipes and tubes under structural loading

219

Figure 6.29 Collapsed steel specimen (D/t = 23.7) subjected to externally pressurized bending (with permission from Stress Engineering Inc.).

6.6

Bending of externally pressurized pipes

During offshore pipeline installation, the combined loading of bending and external pressure at the sagbend area constitutes a critical loading condition. Opposite to internal pressure, external pressure has a destabilizing effect on pipeline bending response. External pressure assists the growth of cross-sectional ovalization induced by bending, leading to rapid collapse of the pipe due to excessive ovalization. This is represented by a limit point on the moment-curvature diagram. Beyond the limit point, the behavior becomes unstable and this might have serious consequences in practical applications depending on the level of pressure. During deep-water pipeline installation, if the pipe collapses at the sagbend area, the external pressure is usually higher than the propagation pressure, and therefore, a propagating buckle is triggered, which flattens the pipe between two arrestors and requires repair (see Section 5.4). The main features of steel pipes employed in offshore pipeline applications and subjected to pressurized bending are cross-sectional ovalization and collapse, which usually occur before the formation of local buckling. Fig. 6.29 shows the collapsed shape of a pipe specimen with D/t = 23.7, and yield stress equal to 49.7 ksi (342.9 MPa). The experiment is part of a testing program on collapse of deep-water pipelines, conducted in late 80’s by Stress Engineering Inc., Houston, Texas (Fowler, 1991). The specimen was subjected to bending inside a pressure chamber, under constant external pressure equal to about 60% of the pipe collapse pressure. In this test, external pressure was applied first and, keeping the pressure constant, bending was gradually increased until collapse. This sequence is referred to as “p → k” loading sequence and is most representative of the loading conditions in the sagbend region during offshore pipeline installation. The configuration in Fig. 6.29 indicates clearly that the specimen flattened and collapsed due to ovalization instability, without any sign of local wrinkling. The small-scale experimental results reported by Corona and Kyriakides (1988a, 1988b) on 1.25-inch-diameter stainless steel (SS-304) pipes with D/t equal to 35.7 and 19.2 subjected to externally pressurized bending (p → k loading sequence) contributed to better understanding of pipe behavior under combined loading conditions. The experiments demonstrated that ovalization instability, rather than local buckling, governs

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Structural Mechanics and Design of Metal Pipes

Figure 6.30 Pressure-curvature interaction diagram. (A) D/t = 35.7, (B) D/t = 19.2; experimental data compared with finite element results (Karamanos & Tassoulas, 1991a).

the response of those pipes. Fig. 6.30 shows the comparison of these experimental data with finite element results, obtained with a special-purpose cross-sectional analysis of the pipe under generalized plane strain conditions (Karamanos & Tassoulas, 1991a). The diagrams in Fig. 6.31 show the bending response of an X65 steel pipe with D/t =30.5 for different levels of external pressure (p → k). The level of pressure is expressed as a fraction of the collapse pressure of the pipe. The stress-strain curve of

Metal pipes and tubes under structural loading

221

(A)

(B)

(C)

(D)

Figure 6.31 Bending response of an X65 steel pipe (D/t = 30.5) for different levels of external pressure, obtained by a cross-sectional finite element analysis. (A) Moment-curvature diagrams. (B) Ovalization-curvature diagrams. (C) Pressure-curvature interaction diagram. (D) Pressure-moment interaction diagram.

pipe material is depicted in Fig. 5.3B and was used extensively in the calculations of Sections 5.2, 5.3 and 5.4. The results in Fig. 6.31 are obtained from two-dimensional (ovalization) analysis, which assumes uniform cross-sectional deformation along the pipe, and does not account for pipe wall wrinkling. The moment-curvature (m¯ − κ) ¯ diagrams in Fig. 6.31A indicate a strong influence of external pressure on bending deformation capacity. The latter is expressed by the normalized value of curvature κ¯ m corresponding to the point of maximum bending moment (m¯ max ), called “limit point”. Under low level of pressure (p/pco ≤ 0.10), the moment-curvature diagram is very smooth with an extended plateau. Increasing the level of external pressure, the plateau disappears, the value of κ¯ m decreases by a significant amount, and beyond the limit point, the bending moment drops rapidly. The strong influence of external pressure on bending deformation capacity shown in Fig. 6.31A can be explained by the corresponding ovalization diagrams in Fig. 6.31B. The combined action of bending and external pressure results in rapid growth of crosssectional ovalization, leading to structural instability in the form of pipe collapse. The effect of pressure p on bending deformation capacity κ¯ m can be expressed by the pressure-curvature (p − κ¯ m ) diagram in Fig. 6.31C, which is often called “interaction diagram” or “collapse envelope”, in the sense that combinations of external pressure

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Structural Mechanics and Design of Metal Pipes

(A)

(B)

(C)

(D)

Figure 6.32 Bending response of a thick-walled aluminum pipe (D/t = 19.5) for different levels of external pressure, obtained by a cross-sectional finite element analysis. (A) Moment-curvature diagrams. (B) Ovalization-curvature diagrams. (C) Pressure-curvature interaction diagram. (D) Pressure-moment interaction diagram.

that are above this curve may not be safe for the pipe in terms of structural stability. Fig. 6.31D shows the corresponding p − m¯ max interaction diagram for this pipe, which has a clear “convex” shape. The pressurized bending response of a thick-walled aluminum pipe with D/t =19.5 is shown in Fig. 6.32. A pipe with the same geometric and material properties was tested by Kyriakides and Ju (1992) and analyzed in Section 6.4 under bending in the absence of pressure (see Fig. 6.23 and Fig. 6.24). The diagrams in Fig. 6.32 are analogous to the ones presented in Fig. 6.31 for the X65 pipe. The value of collapse pressure pco is equal to 23.5 MPa and was calculated using an analysis similar to the one followed in Sections 5.2 and 5.3. In comparison with the results in Fig. 6.31, the values of κ¯ m and m¯ max of the aluminum pipe are larger, and the moment-curvature diagrams are more “rounded”. Both features are attributed to the relatively low value of D/t ratio of the aluminum pipes and the rounded shape of the stress-strain curve, without plastic plateau. Furthermore, the shape of the p − κ¯ m interaction diagram shown in Fig. 6.32C is “concave”, as opposed to the quasi-linear interaction diagram in Fig. 6.31C for the steel pipe. The thick-walled aluminum pipe D/t =19.5 is also analyzed under pressurized bending conditions using a three-dimensional finite element model, similar to the

Metal pipes and tubes under structural loading

223

Figure 6.33 Bending response of a thick-walled aluminum pipe (D/t = 19.5) for different levels of external pressure, obtained by a three-dimensional finite element analysis. (A) Moment-curvature diagrams. (B) Ovalization-curvature diagrams.

model employed in Section 6.4 for pure bending analysis. This model can simulate the formation of local buckling and is more representative of the real bending response of the pipe than the two-dimensional model employed in Fig. 6.32. For the purposes of the present analysis the pipe is assumed imperfection-free, without initial wrinkling. The bending results for different levels of pressure are shown in Fig. 6.33A and Fig. 6.33B in terms of moment-curvature and ovalization-curvature diagrams respectively. The maximum moment m¯ max and the corresponding curvature κ¯ m are depicted in Fig. 6.32C and Fig. 6.32D with triangular markers and compare very well with the corresponding results from cross-sectional (2D) analysis. This good comparison indicates that cross-sectional models, despite their simplicity, provide reasonable predictions of the

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(a)

(b)

(c)

Figure 6.34 Consecutive buckled shapes of thick-walled aluminum pipe (D/t = 19.5) under pressurized bending (pressure at 16.8% of collapse pressure); stages (a), (b), (c) correspond to the points depicted in the diagram of Fig. 6.33A.

maximum moment m¯ max and the corresponding curvature κ¯ m , for pipes with D/t ratio within the range of interest. The three-dimensional results in Fig. 6.33A imply that reaching a limit point, i.e., a maximum bending moment, does not necessarily mean that the pipe has structurally failed, as noted in Section 6.4 for pure bending. There exists a margin between the limit point (maximum moment) and the “failure” or “ultimate” stage, where abrupt decrease of bending moment begins in the moment-curvature diagram (Fig. 6.33A) and the pipe becomes highly unstable. At that stage, the value of ovalization starts increasing very rapidly as shown in Fig. 6.33B. The value of curvature at this ultimate stage (denoted as κ¯C ) is plotted in Fig. 6.32C with circular markers. The values of κ¯ m and κ¯C are quite distant for zero pressure, but this distance decreases when the level of external pressure increases, and the two values coincide for pressure levels above 50% of the collapse pressure (pco ), as shown in Fig. 6.32C. Fig. 6.34 and Fig. 6.35 depict the bent shapes of the aluminum pipe (D/t = 19.5) for two levels of internal pressure, namely 16.8% and 33.7% of collapse pressure pco . Fig. 6.34 shows three deformed shapes of pipe for the 16.8% level. Stage (a) is just prior to ultimate curvature (κ¯C ) and is characterized by a smooth local buckle in the form of a small inward kink. Stages (b) and (c), correspond to κ¯ = 1.37 and κ¯ = 1.62 respectively, and show that the buckle is further developed, but is smoother and more extended than the one obtained for non-pressurized bending, shown in Fig. 6.24. Fig. 6.35 shows three consecutive shapes of pipe deformation for the 33.7% pressure level. Stage (a ) refers to ultimate curvature (κ¯C ) and is characterized by a smooth inward pattern, which starts spreading along the pipe [stage (b )] and gradually flattens

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225

(ac)

(bc) (cc)

Figure 6.35 Consecutive buckled shapes of thick-walled aluminum pipe (D/t = 19.5) under pressurized bending (pressure at 33.7% of collapse pressure); stages (a ), (b ), (c ) correspond to the points depicted in the diagram of Fig. 6.33A. A propagating buckle pattern develops from stage (b ) to stage (c ).

the entire pipe segment [stage (c )], while the curvature remains unchanged. This behavior can be explained considering the value of propagation pressure pp of the pipe. Using a similar finite element model, the value of pp is calculated equal to 5.1 MPa, which is 21.8% of pco . By consequence, the 33.7% pressure level is high enough to trigger buckle propagation and cause progressive flattening of the entire pipe segment. From a computational point-of-view, this pressure-bending calculation is very demanding, because it corresponds to the abruptly descending branch of the m¯ − κ¯ diagram in Fig. 6.33A and, therefore, a numerical stabilization technique has been necessary to advance the numerical solution and obtain the evolution of flattening in the post-buckling range. In any case, this numerical result and the flattened shapes of Fig. 6.35 demonstrate in a very clear manner what happens in the sagbend region during pipeline installation under an unfavorable combination of bending and external pressure. Fig. 6.36A shows the moment-curvature diagrams for relatively thin-walled aluminum pipes with D/t = 49, for three different levels of external pressure (zero, 21% and 62% of the pipe collapse pressure pco ). The analysis is three-dimensional and accounts for the possible formation of local buckling. Similar pipes have been tested experimentally by Ju and Kyriakides (1991), and the present numerical results are in good agreement with the corresponding experimental data. The yield stress σ Y of these pipes is 307 MPa (44.5 ksi), and the stress-strain curve of the material follows the Ramberg-Osgood law in Eq. (6.37), with nˆ = 29, α R = 3/7 and σ 0 = 307 MPa. The pipe wall is relatively thin, and for this reason the values of κ¯ m and κ¯C practically coincide. The numerical results verify the significant reduction of the κ¯ m

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Figure 6.36 Bending response of a thin-walled pipe (D/t = 49) for different levels of external pressure, obtained by a three-dimensional finite element analysis. (A) Moment-curvature diagrams. (B) Ovalization-curvature diagrams.

value with increasing external pressure. For external pressure equal to 62% of collapse pressure, the deformation capacity is reduced by approximately 40% with respect to the non-pressurized case. The corresponding ovalization-curvature diagram is shown in Fig. 6.36B, showing the significant influence of pressure on the rate of growth of cross-sectional ovalization. The deformed configurations of pipes for zero pressure and for the 21% pressure level are shown in Fig. 6.37, and indicate structural failure under “diamond-shape” local buckling. The buckled shape of the pressurized pipe in Fig. 6.37A is less sharp and more rounded than the non-pressurized pipe, in Fig. 6.37B and this is

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(A)

(B)

Figure 6.37 Deformed shapes of a pipe with D/t = 49 subjected to bending. (A) zero pressure; (B) pressure equal to 21% of the collapse pressure.

(a)

(b)

(c)

Figure 6.38 Consecutive deformed shapes of a pipe with D/t = 49 subjected to bending in the presence of external pressure equal to 62% of the collapse pressure.

attributed to the presence of external pressure, which has a smoothening effect on the buckle configuration because of ovalization. Three consecutive stages of the highly pressurized case (62% level) are shown in Fig. 6.38. Because of the high level of external pressure, despite the formation of an inward local buckle in stages (a) and (b), the response is governed by cross-sectional flattening, which spreads along the pipe, as depicted in the shape of stage (c).

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A final note on this combined loading analysis refers to the loading sequence followed in the above combined loading analysis. In all results presented in this section, external pressure is applied first, followed by bending (p → k loading sequence) and this is representative of the loading conditions in the sagbend region during offshore pipeline installation. Experiments and numerical simulations have shown that the p → k sequence is more critical, compared with the k → p sequence, where external pressure is applied on a pre-bent pipe keeping the bending curvature constant and with the “radial sequence”, where both p and k are increased simultaneously in a proportional manner. The interaction diagram for the p → k sequence is lower than the interaction diagram of the other two cases. A comparison between those three different loading sequences and a relevant discussion can be found in the publications by Corona and Kyriakides (1988a), and by Karamanos and Tassoulas (1991a, 1991b).

References Bardi, F. C., & Kyriakides, S. (2006). Plastic buckling of circular tubes under axial compressionpart I: experiments. International Journal of Mechanical Sciences, 48(8), 830–841. Bardi, F. C., Kyriakides, S., & Yun, H. D. (2006). Plastic buckling of circular tubes under axial compression-part II. Analysis. International Journal of Mechanical Sciences, 48(8), 842– 854. Brush, D. O., & Almroth, B. O. (1975). Buckling of Bars, Plates, and Shells. New York, NY: McGraw-Hill. Bushnell, D. (1974). Bifurcation buckling of shells of revolution including large deflections, plasticity and creep. International Journal of Solids and Structures, 10(11), 1287–1305. Christoffersen, J., & Hutchinson, J. W. (1979). A class of phenomenological corner theories of plasticity. Journal of the Mechanics and Physics of Solids, 27(5–6), 465–487. Corona, E., & Kyriakides, S. (1988a). On the collapse of inelastic tubes under combined bending and pressure. International Journal of Solids and Structures, 24(5), 505–535. Corona, E., & Kyriakides, S. (1988b). Collapse of pipelines under combined bending and external pressure. Proceedings, International Conference on the Behavior of Offshore Structures, 3, 953–964. Fonzo, A., Lucci, A., Ferino, J., Di Biagio, M., Spinelli, C. M., & Flaxa, V. (2011). Full scale investigation on strain capacity of high-grade large diameter pipes. In 18th Joint Technical Meeting. PRCI-AFIA-EPRG. Fowler, J. R., & Langner, C. G. (1991). Limits for Deepwater Pipelines Performance. In Offshore Technology Conference, OTC 6757. Gellin, S. (1979). Effect of an axisymmetric imperfection on the plastic buckling of an axially compressed cylindrical shell. Journal of Applied Mechanics, 46(1), 125–131 ASME. Gresnigt, A.M. (1984). Handrekenmodel en Proeven Indeukdiepte/Rekenmetingen Gastransportbuizen, (Analytical model and indentation tests/strain measurements in gas pipelines), Research Report B-84-425/63.6.0900, IBBC-TNO (now TNO – Built Environment and Geosciences), Delft, The Netherlands [in Dutch]. Gresnigt, A. M., Karamanos, S. A., & Andreadakis, K. P. (2007). Lateral loading of internally pressurized steel pipes. Journal of Pressure Vessel Technology, 129(4), 630–638 ASME. Houliara, S., & Karamanos, S. A. (2006). Buckling and post-buckling of long pressurized elastic thin-walled tubes under in-plane bending. International Journal of Non-Linear Mechanics, 41(4), 491–511.

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Hutchinson, J. W., & Tvergaard, V. (1981). Shear band formation in plane strain. International Journal of Solids and Structures, 17, 451–470. Ju, G. T., & Kyriakides, S. (1991). Bifurcation buckling versus limit load instabilities of elasticplastic tubes under bending and external pressure. Journal of Offshore Mechanics and Arctic Engineering, 113(1), 43–52 ASME. Ju, G. T., & Kyriakides, S. (1992). Bifurcation and localization instabilities in cylindrical shells under bending—II. Predictions. International Journal of Solids and Structures, 29(9), 1143–1171. Karamanos, S. A., & Andreadakis, K. P. (2006). Denting of internally pressurized tubes under lateral loads. International Journal of Mechanical Sciences, 48(10), 1080–1094. Karamanos, S. A., & Eleftheriadis, C. (2004). Collapse of pressurized elastoplastic tubular members under lateral loads. International Journal of Mechanical Sciences, 46(1), 35–56. Karamanos, S. A., & Tassoulas, J. L. (1991a). Stability of inelastic tubes under external pressure and bending, OTRC report B18. College Station, TX. Karamanos, S. A., & Tassoulas, J. L. (1991b). Stability of inelastic tubes under external pressure and bending. Journal of Engineering Mechanics, 117(12), 2845–2861. Koiter, W. (1963). The effect of axisymmetric imperfections on the buckling of cylindrical shells under axial compression. Koninklijke Nederlandse Akademie van Wetenschappen, B66, 265–279. Kuwabara, T., Kuroda, M., Tvergaard, V., & Nomura, K. (2000). Use of abrupt strain path change for determining subsequent yield surface: experimental study with metal sheets. Acta Materialia, 48, 2071–2079. Kyriakides, S., & Ju, G. (1992). Bifurcation and localization instabilities in cylindrical shells under bending—I. Experiments. International Journal of Solids and Structures, 29(9), 1117–1142. Limam, A., Lee, L. H., Corona, E., & Kyriakides, S. (2010). Inelastic wrinkling and collapse of tubes under combined bending and internal pressure. International Journal of Mechanical Sciences, 52(5), 637–647. Nasikas, A. (2022). Non-associative plasticity for structural instability of cylindrical shells in the inelastic range. PhD Thesis, School of Engineering, The University of Edinburgh, Scotland, UK. Nasikas, A., Karamanos, S. A., & Papanicolopulos, S. A. (2022). A framework for formulating and implementing non-associative plasticity models for shell buckling computations. International Journal of Solids and Structures, 111508. Papadimitriou, A. (2008). Elastic-plastic buckling of metal cylindrical shells under axial compression, Graduate Diploma Thesis, Department of Civil Engineering, University of Thessaly, Volos, Greece. Pappa, P., & Karamanos, S. A. (2016). Non-associative J2 plasticity model for finite element buckling analysis of shells in the inelastic range. Computer Methods in Applied Mechanics and Engineering, 300, 689–715. Paquette, J. A., & Kyriakides, S. (2006). Plastic buckling of tubes under axial compression and internal pressure. International Journal of Mechanical Sciences, 48(8), 855–867. Peek, R. (2000). An incrementally continuous deformation theory of plasticity with unloading. International Journal of Solids and Structures, 37(36), 5009–5032. Simo, J. C. (1987). A J2-flow theory exhibiting a corner-like effect and suitable for large-scale computation. Computer Methods in Applied Mechanics and Engineering, 62(2), 169–194. Tvergaard, V. (1983). Plastic buckling of axially compressed circular cylindrical shells. ThinWalled Structures, 1(2), 139–163.

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Van Foeken, R.J. (1994). Effect of buckling deformation on the burst pressure of pipes. Report 94CON-R1008/FNR. TNO Building and Construction Research, Rijswijk, The Netherlands. Wildschut, H., ter Avest, F. J., van Foeken, R. J., Gresnigt, A. M., Koning, C., & Spiekhout, J. (1984). The influence of the pipe wall thickness on the resistance to external damage of steel gas transmission pipelines. Report for N.V. Nederlandse Gasunie, TNO metaalinstituut. The Netherlands.

Part III Pipeline Design 7. Basic onshore pipeline mechanical design 233 8. Offshore pipeline mechanical design 251 9. Pipeline analysis and design in geohazard areas 287

Basic onshore pipeline mechanical design

7

The present chapter describes the basics of onshore pipeline mechanical design (Fig. 7.1). Emphasis is given to ASME B31.8 (American Society of Mechanical Engineers, 2018a) and ASME B31.4 (American Society of Mechanical Engineers, 2019) standards, which are the most popular design standards for onshore hydrocarbon pipelines. These standards are used not only in the US, but also in numerous onshore pipeline projects around the world. The relevant provisions of European standard EN 1594 (European Committee for Standardization, 2013) for high-pressure gas pipelines, issued by the Technical Committee TC67 of the European Committee of Standardization, are also presented. The discussion focuses on pipeline mechanical design under “operation loading” conditions, which is also referred to as “pressure design” because internal pressure is the primary loading condition. This also includes the effect of longitudinal stresses developed from pressure and temperature, or from small amount of pipeline subsidence. However, it may not include extreme loading conditions, such as severe groundinduced actions in geohazard areas, e.g., seismic or landslide actions. In this case, a different approach is necessary, which is presented in detail in Chapter 9. The design folmulae in this chapter are presented in accordance with notation of the relevant document, with minor modifications for consistency with the general notation followed in the present book.

7.1

Brief introduction to pipeline standards, pipe sizes and pressure design

The ASME B31.4 standards are widely used for designing onshore pipelines and piping components and systems. The ASME B31 project started in 1926, and the first document was issued in 1935, entitled “American Tentative Standard Code for Pressure Piping”. Since that time, many documents have been issued, and several revisions of each document have been made. Currently the relevant standards within ASME B31 are the following, published as separate documents: r r r r r r r r

B31.1, Power Piping B31.3, Process Piping B31.4, Pipeline Transportation Systems for Liquid Hydrocarbons and Other Liquids B31.5, Refrigeration Piping and Heat Transfer Components B31.8, Gas Transmission and Distribution Piping Systems B31.8S, Managing System Integrity of Gas Pipelines B31.9, Building Services Piping B31.11, Slurry Transportation Piping Systems

Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00011-6 c 2023 Elsevier Inc. All rights reserved. Copyright 

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Figure 7.1 Onshore natural gas pipeline construction (IGB project; photo courtesy: Avax SA). r r

B31.12, Hydrogen Piping and Pipelines B31G, Manual for Determining the Remaining Strength of Corroded Pipelines

For a concise description of the above documents and their relationship with other ASME or API standards, the reader is referred to the piping handbooks by Nayyar (2000) and Ellenberger (2005).

7.1.1 Pipe sizes The ASME B36.10M specification (American Society of Mechanical Engineers, 2018b) covers the dimensions of seamless and welded steel pipes for high and low temperature service; it provides the values of pipe outside (outer) and inside (inner) diameter (OD, ID), pipe thickness (WT), and pipe weight (in pounds per foot and

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235

kilogram per meter). The nominal weight per unit length Wpipe in SI units (kilograms per meter), is calculated using the following formula: Wpipe = 0.0246615(D − t ) t

(7.1)

where D and t are the outside diameter (OD) and the thickness (WT) of the pipe respectively, both in mm. The size of all pipes is identified by their nominal pipe size. In North America, the size is referred to as “nominal pipe size” (NPS xx) expressed in inches. Pipe size up to NPS 12 (DN 300), inclusive, is based on an outside diameter (OD) that is somewhat larger that the nominal size. As an example, NPS 12 means an outside diameter equal to 12.75 in. The outside diameter of pipes with OD NPS 14 and larger is equal to the nominal value. Using SI units, pipe size is expressed by “diamètre nominale” (nominal diameter, DN xx), and this is used primarily in Europe. Table 7.1 shows the correspondence of pipe sizes NPS and DN. The details on the dimensions and mechanical properties of those pipes can be found in Table 1 of ASME standard 36.10M.

7.1.2 Basic considerations in pipeline design under internal pressure Burst of the steel pipeline is the major mode of failure associated with internal pressure. It occurs in the form of pipe wall rupture, because of excessive hoop stress developed by internal pressure (Barlow’s formula), it is a catastrophic failure, associated with loss of containment. Preventing burst is the principal target of “pressure design” of pipelines and it is based on the so-called “allowable stress design” approach. In this design approach, the maximum stress developed within the pipe wall should be less than the allowable stress, which is a percent of the yield stress of the material. The following generic formula can be used for introducing the reader to the philosophy of “pressure design” of pipelines. It is based on Barlow’s formula and describes a generic method for pipe wall thickness determination based on the hoop stress: t=

1 pD fs1 fs2 2σY

(7.2)

where p is the design pressure, D is the outer diameter of the pipe, σ Y is the material yield stress (see steel grade), fs1 is a design factor (or “usage factor”) multiplying the yield stress, usually equal to 0.72 for normal conditions and fs2 is a manufacturing tolerance factor (e.g. if a wall thickness 12.5 percent below nominal is allowed, it is 0.875). As an example, for a 30-inch Χ60 pipeline operating at 150 bars of internal pressure, considering a design factor fs1 = 0.72, and a manufacturing factor fs2 = 0.875, the thickness is readily calculated from Eq. (7.2) equal to 21.9 mm. The formula in Eq. (7.2) has been introduced by Palmer (1996) for education purposes, and represents a generic form of the internal pressure equations that appear in almost all pipeline design standards and specifications.

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Table 7.1 Nominal pipe size (NPS), nominal diameter (DN) and real diameter according to ASME B36.10M up to 92-inch-diameter size. Nominal Pipe Size (NPS) 4 4 12 5 6 7 8 9 10 12 14 16 18 20 22 24 26 28 30 32 34 36 40 42 44 46 48 52 56 60 64 68 72 76 80 88 92

7.2

Nominal Diameter (DN) 100 115 125 150 – 200 – 250 300 350 400 450 500 550 600 650 700 750 800 850 900 1,000 1,050 1,100 1,150 1,200 1,300 1,400 1,500 1,600 1,700 1,800 1,900 2,000 2,200 2,300

Real outside diameter (in) 4.500 5.000 5.563 6.625 7.625 8.625 9.625 10.75 12.75 14.00 16.00 18.00 20.00 22.00 24.00 26.00 28.00 30.00 32.00 34.00 36.00 40.00 42.00 44.00 46.00 48.00 52.00 56.00 60.00 64.00 68.00 72.00 76.00 80.00 88.00 92.00

Real outside diameter (mm) 114.30 127.00 141.30 168.28 193.68 219.08 244.48 273.05 323.85 355.60 406.40 457.20 508.00 558.80 609.60 660.40 711.20 762.00 812.80 863.60 914.40 1,016.00 1,066.80 1,117.60 1,168.40 1,219.20 1,320.80 1,422.40 1,524.00 1,625.60 1,727.20 1,828.80 1,930.40 2,032,00 2,235.20 2,336.80

ASME B31.8 Gas transmission & distribution piping systems

This code refers mainly to gas transportation pipelines between sources and terminals. It includes gas metering, regulating, and gathering pipelines. It also contains rules for corrosion protection and, in combination with its supplement ASME B31.8S, covers

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Table 7.2 Basic design factor FD . Location class 1 (division 1) 2 (division 2) 3 4 5

Design factor 0.80 0.72 0.60 0.50 0.40

the management of the integrity of such pipelines. The main issues of “pressure design” are presented below.

7.2.1 Internal pressure design The internal pressure design in ASME B31.8 (American Society of Mechanical Engineers, 2018a) is based on the following formula (paragraph 841.1.1): p≤

2St FD EW T D

(7.3)

where p is the design pressure, D is the outer diameter of the pipe, S is the specified minimum yield strength (SMYS) of the pipe material, t is the pipe thickness, FD is the design factor, EW is the longitudinal joint factor, and T is the temperature derating factor. For thick-walled pipes with D/t 15, and this can be readily shown by expanding the logarithm in Eq. (8.1). For low values of D/t, i.e. when D/t < 15, Eq. (8.1) is recommended for calculating the burst pressure. The main feature of those equations is the consideration of ultimate tensile stress σ U , in addition to yield stress σ Y , for calculating the burst pressure which is in accordance with experimental observations. Having established the value of burst pressure, the hydrostatic test pressure pt , the pipeline design pressure pd , and the incidental overpressure pa may not exceed the

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following limits: pt ≤ fd fe ft pb

(8.3)

pd ≤ 0.80pt

(8.4)

pa ≤ 0.90pt

(8.5)

where fd is the internal pressure (burst) design factor equal to 0.90 for pipelines and 0.75 for risers, fe is the line pipe weld joint factor according to ASME B31.8 (American Society of Mechanical Engineers, 2018) (denoted as EW in Chapter 7 of the present book), which is equal to 1.0 in almost all cases considered in offshore pipes, and ft is the temperature derating factor, as specified in the pressure design provisions of ASME B31.8 (denoted as T in Chapter 7 of the present book, and equal to 1.0 for temperatures less than 121°C). Finally, it is underlined that in the above calculations, the net internal pressure, i.e., the difference between internal and external pressure should be considered.

8.2.2 Longitudinal tension force design The effective tension Teff on the pipeline should not exceed 60% of yield tension load: Te f f ≤ 0.60 TY

(8.6)

In Eq. (8.6), the effective tension Teff is computed as follows: Te f f = Ta − pint Ai + pext Ao

(8.7)

where pint is the internal pressure, pext is the external pressure, Ai is the internal crosssectional area enclosed by the pipe, A0 is the cross-sectional area enclosed by the outer surface of the pipe, Ta is the axial tension in pipe: Ta = σa A

(8.8)

where σ a is the axial stress in the pipe wall. In addition, in Eq. (8.6), the yield tension TY is: TY = σY A

(8.9)

where A is the cross-sectional area of the steel pipe: A = Ao − Ai =

 π 2 D − D2i 4

(8.10)

or in a simpler form A = π Dmt, where Dm = D − t is the mean diameter of the pipe.

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8.2.3 Combined pressure and tension The combination of primary longitudinal tension and differential (internal) pressure load shall not exceed the value of g1 as follows: 



pint − pext pb



2 +

Te f f TY

2 ≤ g1

(8.11)

where g1 is equal to 0.90 for operational loads and 0.96 for extreme loads and hydrotest loads.

8.2.4 Collapse under external pressure Collapse is the major limit state for the design of offshore pipelines. Determining the collapse pressure of the pipe is essential for safeguarding pipeline integrity during the installation process. If the collapse pressure is exceeded, then the pipeline may collapse into a flattened shape. This may have catastrophic consequences, especially when related to the buckle propagation phenomenon. According to API 1111, the net external pressure (pint − pext ) on the pipeline should not exceed a fraction of the collapse pressure pco of the pipe as follows: (pint − pext ) ≤ f0 pco

(8.12)

where f0 is the collapse factor and is equal to 0.7 for seamless or electric resistance welded (ERW) pipes, and equal to 0.6 for cold expanded pipes (e.g., UOE or JCO-E pipes). The standard allows for f0 = 0.7 in cold expanded pipes, if partial recovery of compressive yield strength has occurred by heat treatment to temperature levels of at least 233 °C (450 °F) for several minutes, during the coating process. However, this increased value of f0 should be validated through an appropriate testing program. The collapse pressure pco in API 1111 is computed using Rankine’s formula for n = 2 (see also Section 8.5 of the present book): pco = 

pY pcr pY2 + p2cr

(8.13)

where the yield pressure pY and the elastic collapse pressure pcr are given by the following equations: pY = 2σY

t D

2E  t 3  1 − v2e D

pcr = 

(8.14) (8.15)

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8.2.5 Buckling due to combined bending and external pressure The reduced bending deformation (strain) ε b in the presence of external pressure is expressed through the following interaction equation:   (pext − pint ) ε b ˆ + =g εb fc pco

(8.16)

or equivalently: εb



  (pext − pint ) ˆ − = εb g  fc pco

 (8.17)

which is valid for D/t values up to 50. In Eq. (8.16), ε b is the maximum bending strain that the pipe can sustain in the presence of net external pressure (pext − pint ), εb is the buckling strain under pure bending (zero pressure) conditions, calculated as follows: εb =

t 2D

(8.18)

  ˆ is the fc is the collapse factor for combined pressure and bending loading, and g  collapse reduction factor because of initial imperfections:   ˆ = g

1 ˆ 1 + 20

(8.19)

ˆ In Eq. (8.19),  ˆ is the initial ovality It is recommended that fc is taken equal to f0 /g(). of the pipe (see Section 3.2) defined as follows: ˆ = Dmax − Dmin  Dmax + Dmin

(8.20)

where Dmax and Dmin are the maximum and minimum diameter of the pipe cross-section respectively. To avoid buckling, maximum bending strains during the installation process (ε1 ) and under in-place conditions (ε2 ) should be limited as follows: f1 ε1 ≤ εb

(8.21)

f2 ε2 ≤ εb

(8.22)

where ε b is the maximum bending strain that the pipe can sustain in the presence of external pressure, calculated from Eqs. (8.16) or (8.17). In Eqs. (8.21) and (8.22), f1 is the bending safety factor for installation bending in the presence of external pressure and f2 is the bending safety factor for in-place bending in the presence of external

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pressure. The bending strains ε1 and ε2 are not simply nominal (global) bending strains and shall account for possible strain raisers. In the case of reeling installation, ε1 = (SAF )

r r + Rreel

(8.23)

where r is radius of pipe cross-section, Rreel is the reel or alignment radius, and SAF is an appropriate strain amplification factor, always greater than 1.

8.2.6 Propagating buckles and buckle arrestors Buckle arrestors may be used when the net external pressure (pext − pint ) is greater than 80% of propagation pressure pp : pext − pint ≥ f p p p

(8.24)

where the propagation pressure pp is given by the following equation: p p = 24σY

 t 2.4 D

(8.25)

and fp is the propagating buckle design factor, equal to 0.80. For the design of buckle arrestors, the API 1111 standard refers to the publications of Park and Kyriakides (1997) and Langner (1999). Furthermore, API 1111 does not provide any specific guidance for buckle arrestor spacing.

8.2.7 Fatigue Several pipeline components such as risers or unsupported free spans, are subjected to repeated loading from the seawater environment, and should be assessed for fatigue. The fatigue life of the component is defined as the time necessary to develop a throughwall-thickness crack. To perform this calculation, Miner’s rule and an appropriate S-N curve are used. The predicted fatigue life of the component should be at least 10 times the service life of the component under consideration. For more details on the fatigue assessment of pipeline components, reference is made to the API 2RD document for pipeline riser design (American Petroleum Institute, 2013).

8.3

DNV-ST-F101 provisions for the mechanical design of offshore pipelines

The first pipeline code was issued by DNV (Det Norske Veritas) in 1976, and since then, significant revisions and amendments have been made. For many years, it was known as DNV-OS-F101 (Det Norske Veritas, 2013). Following the merger between DNV and

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Structural Mechanics and Design of Metal Pipes

Table 8.1 Characteristic wall thickness values. Characteristic thickness t1 t2

Prior to operation t − tfab t

Operation t − tfab − tcorr − tero t − tcorr − tero

GL (Germanischer Lloyd) in 2013, the name of this standard became DNVGL-STF101 and an updated version was issued in 2017 (Det Norske Veritas, 2017). In 2021, DNVGL changed its name to DNV, but retained its post-merger structure. The latest version of the standard, named DNV-ST-F101 (Det Norske Veritas, 2021a), was issued in August 2021 and amended in December 2021. Gradually, this standard achieved global recognition within the oil & gas sector, and currently the majority of new pipeline projects internationally are designed with this standard. It provides acceptance criteria and procedures for pipeline design, fabrication and installation, and applies modern limit-state-design principles with ‘safety classes’ linked to consequences of failure. The main parts of the DNV-ST-F101 standard related to mechanical design of offshore pipelines are summarized in this section.

8.3.1 Design framework for pipe mechanical resistance In DNV-ST-F101, the design provisions for the mechanical resistance of pipelines are offered in Sections 5.3 and 5.4 of the standard. The mechanical resistance RRD can be expressed in the following general form: RRD =

  Rc tc , fc , Oˆ 0 γm γSC

(8.26)

where Rc is the characteristic resistance, tc is the characteristic thickness, Oˆ 0 is the initial out-of-roundness of the pipe, γ m is the material factor, γ SC is the safety class resistance factor, and fc is the characteristic material strength (yield or ultimate). More specifically: r

r

The characteristic thickness is equal to t1 or t2 from Table 8.1, where tfab is the fabrication tolerance, tcorr is the corrosion allowance and tero is the erosion allowance. The condition “prior to operation” is associated with no corrosion (e.g., mill pressure test, installation conditions, or system pressure test conditions). If corrosion exists, the “operation” conditions should be considered. The initial out-of-roundness Oˆ 0 of the pipe is equal to: Dmax − Dmin Oˆ 0 = D

r

(8.27)

ˆ defined in Note that the value of Oˆ 0 is approximately twice the value of ovality parameter , Eq. (8.20). The characteristic strength of the material is related to either yield σ Y or ultimate tensile strength σ U . They are based on the specified minimum yield stress (SMYS) or the specified

Offshore pipeline mechanical design

259

Table 8.2 Safety class resistance factors, γ SC . Limit state Pressure containment (PC) Local buckling, collapse and load controlled conditions (LB) Local buckling under displacement controlled conditions (DC)

Safety class Safety class resistance factor Low Medium High γ SC,PC 1.046 1.138 1.308 1.04 1.14 1.26 γ SC,LB γ SC,DC

2.0

2.5

3.3

minimum tensile stress (SMTS) of the material respectively, and account for material derating because of temperature. r The material resistance factor γ depends on the limit state (LS) under consideration. It is m equal to 1.15 for the serviceability LS, the ultimate LS and the accidental LS, and equal to 1.0 for the fatigue LS. r The safety class resistance factors γ depend on the limit state under consideration and the SC “safety class” of the pipeline (low, medium and high), according to Table 8.2.

8.3.2 Pressure containment resistance (burst) The basic equation to compute pipeline resistance against burst pb (t) is:     t 2 2σcb pb (t ) = √ D−t 3

(8.28)

where stress σ cb is the minimum of σ Y and σU /1.15. DNV-ST-F101 accounts for the ultimate tensile stress σ U in addition to the √ yield stress σ Y in the burst resistance calculation. Furthermore, the coefficient 2/ 3 refers to hydrostatic pressure conditions, as described in Section 4.1 of this book. Therefore, the net internal pressure pint − pext should be less than the factored burst pressure: pb (t1 ) (8.29) pint − pext ≤ γm γSC,PC where pint is the maximum internal pressure that the offshore pipeline is designed to sustain, pext is the external pressure. The burst pressure should be calculated using thickness t1 , which accounts for fabrication tolerance and for thickness reduction due to corrosion. To complete the pressure containment requirements, the design criterion in Eq. (8.29) in DNV-ST-F101 is enhanced with criteria that refer to the pressure that the offshore pipeline system is tested to prior to commissioning and to the hydrostatic test pressure at the pipe mill.

8.3.3 Local buckling: pipe collapse (external pressure only) The maximum net external pressure at any point along the pipeline should be less than the factored collapse pressure pco . This is expressed as follows: pext − pmin ≤

pco (t1 ) γm γSC,LB

(8.30)

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Structural Mechanics and Design of Metal Pipes

Table 8.3 Maximum fabrication factor, α fab . Seamless pipe 1.00

α fab

UO, RB, ERW and HFW pipe 0.93

UOE and JCO-E pipe 0.85

where pext is the external pressure and pmin is the minimum internal pressure in the pipeline (normally taken as zero during installation). The characteristic collapse pressure pco , which expresses pipeline resistance against collapse, is calculated from the following third-order algebraic equation: 

D 2 2 ˆ (t ) (t )] (t ) (t ) (t )p (t )p (t ) O = p − p − p p [pco el co pl co el pl 0 t

(8.31)

A similar equation for the collapse pressure is also adopted by BS 8010 (British Standards Institution, 2009). In Eq. (8.31) the calculation of collapse pressure pco considers thickness t1 . Furthermore, in Eq. (8.31), pressure pel is the well-known elastic buckling (critical) pressure: pel (t ) =

2E  t 3 1 − v2e D

(8.32)

pressure ppl is equal to the yield pressure pY = 2σY t/D multiplied by the so-called “fabrication factor” α fab (obtained from Table 8.3): p pl (t ) = 2σY α f ab

t D

(8.33)

and Oˆ 0 is the initial ovality of the pipe cross-section from Eq. (8.27), including the ovalization caused during the construction phase. In Eq. (8.31) the value of Oˆ 0 should be taken always greater than 0.5%. It is reminded that the value of Oˆ 0 is approximately ˆ defined in Eq. (8.20). twice the value of , The value of α fab represents the reduction of collapse strength in cold-formed pipes, especially in “expanded” pipes (UOE or JCO-E), because of the Bauschinger effect. The α fab values in Table 8.3 are suggested by the DNV-ST-F101 standard. The standard also allows the use of higher values of α fab when the pipes are either heat treated or externally compressed (instead of expanded), if this is documented and supported by experimental data (see also Section 8.8). The solution of Eq. (8.31) in terms of pco can be calculated as follows: 1 pco = y − b 3

(8.34)

where b = −pel

(8.35)

Offshore pipeline mechanical design

and the value of y is computed using the following consecutive steps:

 D c = − p pl (t )2 + p pl (t )pel (t )Oˆ 0 t

261

(8.36)

d = pel (t )p pl (t )2

(8.37)

  1 1 2 u= − b +c 3 3

(8.38)

  1 2 3 1 v= b − bc + d 2 27 3

(8.39)

 −v  = cos √ −u3   √  60π y = −2 −u cos + 3 180 −1



(8.40)

(8.41)

8.3.4 Buckle propagation and arrest Buckle arrestors should be used along the pipeline when the net external pressure does not satisfy the propagating buckle criterion: pext − pmin ≤

p p (t2 ) γm γSC,LB

(8.42)

where pmin is the minimum internal pressure in the pipe, normally equal to zero during the installation of the pipeline. The propagation pressure in Eq. (8.42) can be calculated as follows: p p = 35σY α f ab

 t 2.5 2

D

(8.43)

The calculation in Eq. (8.43) should be performed with thickness t2 , and the formula is applicable for pipelines with 15 < D/t2 < 45. The dimensions of an integral buckle arrestor may be determined by the requirement that the level of external pressure pe does not exceed the cross-over pressure pX of the arrestor: pe ≤

pX 1.1γm γSC,LB

(8.44)

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Structural Mechanics and Design of Metal Pipes

The crossover pressure pX of an integral buckle arrestor can be computed as follows:

    t2 La pX = p p + p p,BA − p p 1 − EXP −20 2 (8.45) D where pp,BA is the propagation pressure of a fictitious infinitely-long pipe with the cross-section of the arrestor, and is calculated by Eq. (8.43) using the buckle arrestor properties. Furthermore, La is the buckle arrestor length.

8.3.5 Criteria for local buckling under structural loading DNV-ST-F101 contains design rules for offshore pipelines subjected to a combination of pressure and structural loading, during both installation and operation. Two main categories of loading patterns are distinguished: (a) load-controlled situations and (b) displacement-controlled situations. In most cases, it is quite difficult to realize whether a load-controlled or a displacement-controlled loading scheme is applicable. However, during the critical stage of pipeline installation, it has been recognized that the loading situation is more similar to a displacement-controlled scheme, and this case is analyzed below.

8.3.5.1

Structural loading displacement-controlled conditions under internal pressure

In a displacement-controlled loading scheme, where the pipe is subjected to longitudinal compressive strain εSd due to applied displacement from bending and axial compressive force, in the presence of internal pressure, the following criterion is adopted by the DNV-ST-F101 standard for the compressive strain εSd that causes local buckling of the pipe wall: εSd ≤

εc (t2 , pmin − pext ) γSC,DC

(8.46)

In Eq. (8.46), calculations should be made with thickness t2 , pmin is the minimum internal pressure associated with the bending strain, εc is the compressive limit strain that causes local buckling, computed as follows: εc (t, pmin − pext ) =

t D

− 0.01

 0.85 1.5 αh



   20 pmin − pext 2 1+ αgw αmat 3 pb (t ) (8.47)

where α mat is a material factor that depends on the shape of the pipe material stressstrain curve (Lüder’s plateau), α h is equal to the pipe material yield-to-tensile (Y/T) ratio, and α gw is a girth weld factor. Eq. (8.46) is valid for 15 ≤ D/t2 ≤ 45 and for positive values of the net internal pressure (pint ≥ pext ).

Offshore pipeline mechanical design

8.3.5.2

263

Structural loading displacement-controlled conditions under external pressure

In a displacement-controlled scheme, the longitudinal compressive strain εSd from bending and axial compression that causes local buckling of the pipe wall in the presence of external pressure pext − pmin should satisfy the following criterion:

εSd εc (t2 , 0)/γSC,DC

0.8 +

pext − pmin  ≤1  pc (t2 )/ γm γSC,LB

(8.48)

which is also applicable for pipes with 15 ≤ D/t2 ≤ 45 and for a positive value of net external pressure (pmin < pext ). From Eq. (8.47):  0.85 1.5 αgw αmat − 0.01 εc (t, 0) = D αh t

(8.49)

The exponent value of 0.8 in Eq. (8.48) accounts for the “concave” shape of the bending strain-pressure interaction diagram, noticed in Chapter 6 of the present book.

8.3.6 Ovalization due to bending The ovalization of a pipe O0 subjected to pressurized bending can be estimated using Eq. (8.50) below, where ε is the bending strain and Oˆ 0 is the initial ovality, before the application of bending load: 



  2  D D 1 2ε O0 = Oˆ 0 + 0.030 1 + 120t t 1 − p/pco

(8.50)

This equation is given in Chapter 13 (“Commentary”) of the DNV-ST-F101 standard, and can be re-written in the following form, considering the ovality parameter :  2   D 1 D ˆ + 0.015 1 + 2ε =  120t t 1 − p/pco 



(8.51)

Considering that the nominal value of bending strain ε is related to bending curvature k through the following equation: ε=

kD 2

(8.52)

and Eq. (8.51) can be written as follows:

    1 ˆ + 0.015 1 + D κ¯ 2 =  120t 1 − p/pco

(8.53)

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Structural Mechanics and Design of Metal Pipes

where the normalized value of curvature κ¯ is equal to k/kI , and kI = t/D2 . Eq. (8.53) is based on the ovalization equation proposed by Murphey and Langner (1985):   D κ¯ 2  = 0.015 1 + 120t

(8.54)

Eq. (8.53) enhances Eq. (8.54) to account for initial ovality and the presence of external pressure. It is interesting to compare the ovalization expression of Eq. (8.53) with the elastic ovalization solution presented by Houliara and Karamanos (2006):   ˆ + κ2 = 

1 1 − p/pcr

(8.55)

where κ=

kr2  1 − νe2 t

(8.56)

and pcr is the elastic buckling pressure. Equivalently, Eq. (8.55) is written as follows:   ˆ + 0.569κ¯ 2 = 

1 1 − p/pcr

(8.57)

8.3.7 Fatigue assessment A fatigue assessment should be conducted to account for stress fluctuations. When repeated loading of variable amplitude occurs, the linear damage hypothesis expressed by Miner’s rule should be employed. The application of Miner’s rule implies that the long-term distribution of stress range is replaced by a stress histogram (stress spectrum), consisting of a number of constant amplitude stress range blocks σ ri (or strain range blocks εri ), and the corresponding number of repetitions, ni . Thus, the fatigue criterion is given by: DF · DFF ≤ 1.0

(8.58)

Ns  ni DF = N i i=1

(8.59)

where DF is Miner’s sum, NS is number of stress blocks in the stress histogram, ni is the number of stress cycles (repetitions) in stress block i, Ni is the number of cycles to failure at constant stress range of magnitude σ ri (or strain range εri ) obtained from an appropriate S-N fatigue curve and DFF is a fatigue design factor, from Table 8.4. Recommend practice for S-N fatigue strength analysis is given in Section 5 of DNVRP-C203 (Det Norske Veritas, 2021b).

Offshore pipeline mechanical design

265

Table 8.4 Allowable design fatigue factor. Safety class DFF

8.4

Low 3

Medium 6

High 10

A short note on buckle propagation and arrestor design

According to current design practice, the external pressure design of offshore pipelines is based on (a) pipe wall thickness calculation from pipe collapse consideration and (b) use of buckle arrestors at appropriate intervals to prevent buckling propagation and limit the collapsed pipe between two consecutive arrestors, at the event of local buckling in deep water. An alternative design philosophy may also be used, based on propagation rather than collapse. In this design framework, buckle propagation is considered as the primary limit state and pipe wall thickness is determined based on the propagation pressure. It is reminded though that for a pipe with the geometric and material properties of interest, the propagation pressure is significantly lower than the collapse pressure and therefore this design philosophy leads to a substantially thicker pipe. The values of collapse and propagation pressure, calculated according to API 1111, are plotted in Fig. 8.3 in terms of the D/t ratio, normalized by the value of nominal yield pressure (pY = 2σY t/D). Comparing the two design alternatives, one has to choose between a thick pipe without buckle arrestors and a thinner pipe with arrestors. The former approach might be suitable for relatively shallow water applications, where other design criteria are more critical (e.g., internal pressure design or on-bottom stability), but in offshore projects beyond a certain depth, using a thick pipe without arrestors increases significantly the

Figure 8.3 Collapse pressure versus propagation pressure (normalized by the nominal yield pressure pY = 2σY t/D), according to the corresponding design formulae in API 1111.

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Structural Mechanics and Design of Metal Pipes

construction cost. Therefore, the latter approach is followed in offshore pipeline design and construction. As an example, consider a 26-inch-diameter UOE pipe, made of X65 steel, (σ Y = 450 MPa) to be used in 1,000 meters of water depth. At that depth, the external pressure acting on the pipeline is 10 MPa (100 bar). Using the API 1111 provisions, and considering an empty pipe during installation, the collapse pressure is the principal limit state. Therefore, from Eqs. (8.12), (8.13) and assuming a design factor equal to 0.6 (welded pipe), a thickness equal to 23.2 mm (0.91 in) is calculated. In this case, the factored propagation pressure (with a design factor of 0.80) is equal to 2.79 MPa, which implies that buckle arrestors are necessary at the depth under consideration. On the other hand, if buckling propagation is considered as the primary limit state, then from Eqs. (8.24), (8.25) and a design factor equal to 0.80, a 39.5-mm (1.56-in) pipe wall thickness is required to sustain 10 MPa, which is a very large thickness resulting to a pipe with D/t = 16.7. The thickness increase compared with the pipe collapse approach is 70%. As a conclusion, significant financial consequences may be implied for the pipeline project if the alternative approach is adopted, even though the use of buckle arrestors would be avoided.

8.5

Discussion of API and DNV collapse formulae

In this section, an attempt is made to present the collapse formulae in API 1111 and in DNV-ST-F101 from a different perspective. The discussion is aimed at better understanding the two collapse methodologies and provide some background for their development. The discussion starts with the general form of Rankine’s formula for calculating ring collapse pressure pco : pco =  n

pcr pY pYn + pncr

(8.60)

Eq. (8.60) is a more general form of Eq. (8.13) and can be written equivalently: 1 pco = pY √ n ¯ 2n 1+λ

(8.61)

¯ is the normalized ring slenderness: where λ  ¯ = λ

pY pcr

(8.62)

or equivalently,  ¯ = Dm λ t

  σY 1 − v2e E

(8.63)

Offshore pipeline mechanical design

267

Figure 8.4 Rankine’s formula in terms of cross-sectional pipe slenderness (n = 1, 2 and 8); comparison with some representative collapse test data reported by Fowler and Langner (1991).

The API 1111 specification uses generalized Rankine’s formula assuming n = 2, and therefore, from Eq. (8.61): pco = pY √

1 ¯4 1+λ

(8.64)

Eq. (8.64) is equivalent to Eq. (8.13). On the other hand, assuming n = 1, Eq. (8.61) becomes: pco = pY

1 ¯2 1+λ

(8.65)

and in this case (n = 1), the collapse pressure is the harmonic mean of pcr and pY . Rankine’s formula offers a very straightforward tool for describing the real collapse resistance of pipes under external pressure, using a single curve for thick-walled and for slender pipes. In Fig. 8.4 this formula is plotted for different values of n, together with some representative experimental results reported by Fowler and Langner (1991). Using this approach, it is possible to account indirectly for the presence of initial imperfection and residual stresses. Increasing the value of n in Eq. (8.61), the behavior of the perfect ring is better approximated, as shown in Fig. 8.4 for n = 8. The case of an initially oval ring under uniform external pressure is considered next, which has been analyzed in Section 5.2. The shape of initially ovality is defined as follows in terms of the radial displacement, wˆ = αˆ cos 2θ

(8.66)

It was shown that, upon application of external pressure p, the deformed shape remains oval but its amplitude increases from αˆ to α, so that the total radial deflection of the ring, including the initial imperfection displacement of Eq. (8.66), is expressed

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Structural Mechanics and Design of Metal Pipes

as follows: w = α cos 2θ

(8.67)

where α = αˆ

pcr pcr − p

(8.68)

Eq. (8.68) is valid provided that the response is elastic. Eventually, inelastic deformation occurs, leading to the formation of a collapse mechanism with four equallyspaced plastic hinges, expressed by the following equation (see Appendix E):  p = pY

t Dm

 

α 2 rm



1 −



α rm

2

(8.69)

Eq. (8.69) can be improved if the effect of membrane hoop force Nθ on the bending moment capacity of the pipe wall is considered (see Appendix C). The membrane hoop force developed in the pipe wall due to pressure is: Nθ = −p rm

(8.70)

Therefore, the reduced bending moment capacity MP of the pipe wall (per unit length of pipe) is:  M p



= MP 1 −

Nθ NY

2  (8.71)

where NY = −σY t

(8.72)

is the compressive plastic force in the circumferential direction. One can readily show that: (−prm ) p Nθ = = (−σY t ) NY pY

(8.73)

where pY = σY t/rm . Combining Eq. (8.71) and Eq. (8.73) one obtains:  M p = MP 1 −



p pY

2  (8.74)

Offshore pipeline mechanical design

269

and Eq. (8.69) becomes:  p = pY

Dm t

 

α 2 rm



1

 

−

α rm



2 1 −

p pY

2  (8.75)

In Section 5.2, it was suggested that the collapse pressure pco (i.e., the maximum pressure pmax that the pipe can sustain) can be estimated at the intersection of the elastic solution and the plastic solution expressed by Eqs. (8.68) and (8.75) respectively. It was also shown that the use of Eq. (8.69) instead of Eq. (8.75), provides less accurate results because it does not account for the influence of axial hoop stress on the plastic moment of the pipe wall. For the purpose of the present analysis, Eq. (8.75) can be somewhat simplified, considering that α/rm is a small quantity at the onset of collapse, so that only the first term in the denominator is used:  p = pY

Dm t



  2  1 p   1− α pY 2 rm

(8.76)

or equivalently:  α=

pY p



   2  Dm p (0.5rm ) 1 − t pY

(8.77)

Combining the elastic solution Eq. (8.68) and the plastic solution Eq. (8.77) one obtains:   2  pY − p2 pY Dm pcr = (0.5rm ) (8.78) αˆ pcr − p p t pY2 The solution of Eq. (8.78) in terms of pressure p, is the intersection of the elastic solution and the plastic solution, and offers an estimate of collapse pressure pco using this elastic-plastic analysis approach. Simple rearrangement of Eq. (8.78) results in the following expression: 

pY2

−

p2co



   D αˆ (pcr − pco ) = 2 pcr pY pco rm t 

(8.79)

The above 3rd degree polynomial (algebraic) equation should be solved to determine ˆ can be the collapse pressure pco . Note that the initial imperfection parameter  introduced in Eq. (8.79), considering that: ˆ = αˆ  rm

(8.80)

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Structural Mechanics and Design of Metal Pipes

ˆ Eq. (8.79) is very similar to the collapse formula and given the fact that Oˆ 0  2, in Eq. (8.31) adopted by DNV-ST-F101 and BS 8010. Eq. (8.79) was first reported by de Winter (1981) through the development of a collapsed ring model, composed by a “rocker element” and a “bending element”. The solution of Eq. (8.79) is presented in Section 8.3 [see Eqs. (8.34)–(8.41)]. Another interesting result is also obtained from the above elastic-plastic analysis, if Eq. (8.76) is considered in its simplest form, neglecting the contribution of membrane hoop force Nθ = −prm on the plastic bending moment MP . In that case, the plastic mechanism equation becomes:  p = pY

Dm t



1  α 2 rm 

(8.81)

or, rearranging,  α=

pY p



 Dm (0.5rm ) t

(8.82)

Considering the intersection of the elastic solution Eq. (8.68) and the elastic-plastic solution Eq. (8.82), an estimate of the collapse pressure pco can be obtained by the solution of the following equation: 

αˆ (pcr − pco )pY = 2 rm



 Dm pcr pco t

(8.83)

or pco = pY

1 ¯2

+λ

(8.84)

where ˆ Dm

= 2 t

(8.85)

If one assumes that: ˆ = 

t 2Dm

(8.86)

then = 1 and Eq. (8.84): pco = pY

1 ¯2 1+λ

(8.87)

Offshore pipeline mechanical design

271

which is identical to Eq. (8.65), i.e., Rankine’s formula for n = 1. The above analysis shows that, under the condition expressed by Eq. (8.86), the modified de Winter’s formula, expressed by Eq. (8.83) can be written in the form of Eq. (8.87), which is a special form of Rankine’s formula.

8.6

Other formulae for predicting the collapse pressure of pipes and tubes

Several methodologies for predicting the collapse pressure of externally pressurized cylinders have been proposed in the literature or in other design standards, and three of them are presented below. The first set of formulae is proposed by the API Bulletin 5C3, the second set has been proposed by API RP 2A provisions for tubular members, and the third methodology was introduced by DeGeer and Cheng (2000).

8.6.1 API Bulletin 5C3 The API Bulletin 5C3 (American Petroleum Institute, 1994) presents the formulae for the collapse pressure calculation of various pipe properties given in API standards, including background information regarding their development and use. It has been widely used in the design of tubular casing and liners for drilling. In that document, the prediction of collapse pressure pco depends on the yield strength of the material σ Y and the diameter-to-thickness ratio D/t, and has four branches. The first branch is yield pressure (for low values of D/t), the fourth branch is elastic buckling, whereas the other two describe “intermediate” or “transition” stages of tube collapse. (a) Yield strength collapse

p co = pˆY = 2σY

D/t − 1 (D/t )2



D ≤ t

if

  D t YP

(8.88)

(b) Plastic collapse pressure   t p co = pˆ Pl = σY A − B − σˆC D

if

    D D D ≤ < t YP t t PT

(8.89)

(c) Transition collapse pressure p co = pˆ Tr

 t = σY F − G if D 

    D D D ≤ < t PT t t TE

(8.90)

  D t TE

(8.91)

(d) Elastic collapse pressure p co

46.95 × 106 = pˆ E = (D/t )(D/t − 1)2

if

D > t

272

Structural Mechanics and Design of Metal Pipes

Table 8.5 Coefficients and limits for API Bulletin 5C3. Steel grade 60 70

A 3.005 3.037

B 0.0566 0.0617

σˆ C (psi) 1,356 1,984

F 1.983 1.984

G 0.0373 0.0403

D

D

t

t

YP

14.4 13.85

PT

24.42 23.38

D t

TE

35.73 33.17

Figure 8.5 Equilibrium of forces in an externally-pressurized moderately thick cylinder.

Coefficients A, B, F and G, the stress parameter σˆC and the above D/t limits in Eqs. (8.88)–(8.91) depend on the steel grade and the D/t ratio. They are summarized in Table 8.5 for steel grades 60 and 70, which are very common for the pipeline applications under consideration. The yield strength collapse in Eq. (8.88) is obtained from the yielding condition of the inner surface of an externally pressurized thick-walled cylinder, according to Eq. (G.6) in Appendix G. In this equation, setting r = ri , σ θ , o = σ Y and pext = pˆY , Eq. (8.88) is readily obtained. Eq. (8.91) is a modified version of Bryan’s formula for elastic buckling of rings under uniform external pressure, with a reduction coefficient. Starting from the critical buckling stress:   E t 2 (8.92) σcr = 1 − νe 2 Dm obtained in Section 5.2 of this book and considering that in an externally-pressurized cylinder one may write the following equilibrium equation (Fig. 8.5): 2σ t = pD the following expression for the critical pressure is obtained:  3  2E t pcr = 1 − νe2 D2m D Furthermore, introduction of a reduction factor equal to 0.75 results in:

 3  2E t pˆ E = 0.75 2 1 − νe D2m D

(8.93)

(8.94)

(8.95)

For E = 30 × 106 psi, ν e = 0.3 and Dm = D − t, one readily obtains Eq. (8.91).

Offshore pipeline mechanical design

273

The plastic collapse formula in Eq. (8.89) stems from statistical regression analysis of a large number of experimental data (Clinedinst, 1963), whereas the transition formula in Eq. (8.90) simply connects Eqs. (8.89) and (8.91) in a smooth manner. When axial tension T is present, API 5C3 suggests that Eqs. (8.88)–(8.91) are used for the calculation of the reduced collapse pressure, but with a modified yield stress σ  Y , according to the following equation that stems directly from the Von Mises yield criterion: ⎡ T + σY = σY ⎣− 2TY



⎤   3 T 2⎦ 1− 4 TY

(8.96)

where TY = σ Y A is the yield force of the pipe cross-section.

8.6.2 API RP 2A The API RP 2A document has been for many years the “flagship” of API standards for offshore structures. It was developed for fixed offshore platforms, but several parts of this document are used by other API standards as well. In this section, the API RP 2A equations for cylindrical member design under hydrostatic pressure are presented in a form suitable for pipelines. Those equations have been developed for ring stiffened tubular members, but can be easily adjusted for the case of long unstiffened tubulars, e.g., pipelines, if the distance between stiffeners is assumed very large. Two versions of those equations are presented below from two versions of this specification, API RP 2A 1993, LRFD, Load and Resistance Factors Design (American Petroleum Institute, 1993), and API RP 2A 2002, WSD, Working Stress Design (American Petroleum Institute, 2002).

8.6.2.1

API RP 2A 1993 (LRFD)

The elastic buckling stress under external pressure σ e constitutes the basis of the API RP 2A methodology. For long unstiffened tubular members, it is defined as follows: σe = 0.88E(t/D)2

(8.97)

The elastic buckling stress in Eq. (8.97) corresponds to a pressure level pe = 2σet/D, equal to

 2E  t 3 p e = 0.8 1 − νe2 D

(8.98)

which is equal to 80% of the theoretical elastic ring collapse strength pcr , most likely to account for initial imperfections or residual stresses in the tubular member. The stress that causes collapse of the tubular member σ co is given by the following

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expressions: σco = σe

if 

σco = 0.7σe σco = σY

σe ≤ 0.55 σY σe σY

if

(8.99)

0.4 if

0.55 σY < σe < 2.44 σY

σe ≥ 2.44 σY

(8.100) (8.101)

The above methodology can be written in an alternative form in terms of normalized ¯ defined as: ring slenderness λ,  ¯ = σY /σe (8.102) λ Using Eq. (8.102), Eqs. (8.99)–(8.101) can be written as follows: 1 ¯ ≥ 1.348 if λ ¯λ2   1 ¯ < 1.348 pco = pY 0.7 0.8 if 0.640 < λ ¯ λ

pco = pY

pco = pY

if

¯ ≤ 0.640 λ

(8.103) (8.104) (8.105)

where pco = 2σcot/D. Upon calculation of the collapse stress σ co , the factored stress σ θ in the tubular member should be: σθ ≤ φh σco

(8.106)

where φ h is the resistance factor for hoop buckling strength (φ h = 0.8). It is reminded that the document under consideration is a Load & Resistance Factor Design (LRFD) specification, where design verification is based on the comparison of factored loads and resistances. Eq. (8.106) can be written equivalently in terms of factored external pressure p on the tubular member (p = 2σθ t/D): p ≤ φh pco

(8.107)

In the presence of axial tension T, the maximum allowed factored pressure p on the tubular member is calculated by the following interaction formula: A2 + B2 − 2νe A B = 1

(8.108)

where A = σx /(φT σY )

(8.109)

B = p /(φh pco )

(8.110)

where φ T is the resistance factor for axial tension resistance (φ T = 0.95) and ν e = 0.3 for metals.

Offshore pipeline mechanical design

8.6.2.2

275

API RP 2A 2002 (WSD)

The API RP 2A 2002 (WSD) document follows a very similar approach for the collapse pressure. The elastic buckling stress σ e is calculated from Eq. (8.97), and the stress that causes collapse of the tubular member σ co is given by the following set of equations: σco = σe

if

σe ≤ 0.55 σY

σco = 0.45 σY + 0.18 σe σco =

1.31 σY 1.15 + (σY /σe )

σco = σY

if

if

if

(8.111) 0.55 σY ≤ σe < 1.6 σY 1.6 σY < σe < 6.2 σY

σe ≥ 6.2 σY

(8.112)

(8.113) (8.114)

In comparison with its LRFD version, the WSD version of API RP 2A uses the same equation for the elastic buckling pressure, and for the yield pressure. However, instead of (Eq. 8.115) it uses Eqs. (8.112) and (8.113). Using the definition of normalized ring slenderness in Eq. (8.102), Eqs. (8.111)–(8.114) can be written in the following equivalent form: 1 ¯ ≥ 1.348 if λ ¯λ2   0.18 ¯ < 1.348 pco = pY 0.45 + 2 if 0.791 ≤ λ ¯ λ   1.31 ¯ < 0.791 if 0.401 < λ pco = pY ¯2 1.15 + λ pco = pY

pco = pY

if

¯ ≤ 0.401 λ

(8.116)

(8.117)

(8.118) (8.119)

For structural safety against collapse, the stress σ θ in the tubular member should satisfy: σθ ≤ σco/SFh

(8.120)

where SFh is the safety factor against hydrostatic collapse, equal to 2 for installation and normal operating conditions. API RP 2A 2002 (WSD) adopts a Working Stress Design (WSD) approach, which is different than LRFD and therefore the values of 1/SFh and φ h are not comparable. Equivalently, the (non-factored) pressure p on the tubular member should be: p ≤ pco/SFh

(8.121)

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Structural Mechanics and Design of Metal Pipes

In the presence of axial tension T, the maximum allowed (non-factored) pressure p on the tubular member is calculated by the interaction Eq. (8.108), where: A = (SFh )σx /σY

(8.122)

B = (SFT )p /pco

(8.123)

where SFT is the resistance factor for axial tension resistance (SFT = 1.67).

8.6.3 DeGeer and Cheng (2000) This methodology was proposed by DeGeer and Cheng (2000) and follows the reduced-modulus approach, a concept introduced initially by Gerard (1962) for buckling of plates and shells. The collapse pressure pco of the pipe is given in terms of the elastic buckling pressure pcr expressed by the following equation: pco = αPL ·η·pcr where α PL is a coefficient given by the following expression: ⎛ ⎞  ˆ   E  T ˆ ⎠ 1− αPL = ⎝1 − 0.2 E 0.005

(8.124)

(8.125)

ET is the tangent modulus, η is the plastic reduction factor defined as follows (Gerard, 1962):    ES + 3ET 1 − νe2 (8.126) η= 1 − νˆ 2 4E νˆ is the inelastic Poisson’s ratio (Gerard, 1962): νˆ = 0.5 −

ES (0.5 − νe ) E

(8.127)

ˆ and  ˆ are parameters that represent the initial ovality and the thickness variation  (see also Section 5.3): ˆ = Dmax − Dmin  Dmax + Dmin t ˆ = max − tmin  tmax + tmin and pcr is the well-known elastic ring buckling pressure: 2E  t 3 pcr = 1 − νe 2 D

(8.128) (8.129)

(8.130)

In order to apply this methodology, the stress-strain curve of the pipe material should be known or assumed εˆ = ε( ˆ σˆ ) or σˆ = σˆ (ε), ˆ which relates the equivalent Von Mises

Offshore pipeline mechanical design

stress: σˆ =



σx2 + σθ2 − σx σθ

277

(8.131)

with the corresponding strain and is calibrated from uniaxial tension. In most cases, the Ramberg-Osgood law is adopted for the stress-strain curve (see Eq. 6.37). In any case, the σˆ − εˆ stress-strain curve allows for determining the secant modulus ES = σˆ /εˆ and ˆ the tangent modulus ET = d σˆ /d ε. The above methodology can be also applied in the presence of an axial force F, which reduces the collapse capacity. The effect of axial force F is taken into account in calculating the axial stress σ x in Eq. (8.131). Despite the elegant form of Eq. (8.124), this methodology requires an iterative procedure for calculating the collapse pressure. The reason is that the values of η and α PL depend on the level of stress on the stress-strain diagram, which depends on the value of external pressure. On the other hand, the main advantages of this methodology are: (a) it accounts for the exact stress-strain curve of the pipe material, which can be determined by laboratory testing and (b) it considers the effects of initial ovality and thickness variation. Therefore, it can be used for calculating the collapse pressure of specific pipes, when material properties and initial imperfections are known. DeGeer and Cheng (2000) have applied their methodology for predicting the collapse pressure of pipes from the Oman-India and the Blue Stream pipelines, and the predictions were very satisfactory.

8.7

Discussion of collapse formulae and the slenderness approach

The collapse equations proposed by API 1111 and by API RP 2A (LRFD and WSD), as presented in Sections 8.5 and 8.6, can be written in the following unified and generic form:   ¯ (8.132) pco = pY f λ   ¯ is a function of normalized ring slenderness defined by Eq. (8.62) or where f λ Eq. (8.102), which satisfies the following requirements:   ¯ = 1 for small values of slenderness (8.133) f λ   ¯ ∼ 1 for large values of slenderness f λ (8.134) ¯2 λ It can be readily shown that the API 1111 formula [Eq. (8.64)], and the API RP 2A equations [Eqs. (8.103)–(8.105) and Eqs. (8.116)–(8.119)] are special forms of the generic equation (8.132). Moreover, the general form of Rankine’s formula, expressed by Eq. (8.61), is also a special case of Eq. (8.132). The approach expressed by Eq. (8.132) is often called “slenderness approach for buckling” and has been adopted, directly or indirectly, by numerous specifications for design of structural members and systems against structural instability. Eq. (8.132) is

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Structural Mechanics and Design of Metal Pipes

Figure 8.6 The “slenderness” concept for the stability (buckling) of structural systems.

shown schematically in Fig. 8.6. As an example, European standard EN 1993 for steel structural design adopts this approach for calculating the buckling resistance of columns, plates and shells (EN 1993, Parts 1-1, 1-5 and 1-6) (European Committee for Standardization, 2005; European Committee for Standardization, 2006; European Committee for Standardization, 2007). The slenderness approach for buckling design of a structural system is based on the ¯ defined as the square root of the ratio of yield calculation of member slenderness λ, √ ¯ = RY /Rcr ). Then buckling load (or stress) Rcr (λ load (or stress) RY over the  elastic  ¯ is defined, which satisfies the requirements (8.133) and an appropriate function f λ (8.134), and takes into account the special features of the instability problem under consideration. In several specifications, Eq. (8.134) is multiplied by an imperfection factor α imp (α imp < 1) to account for the presence of initial imperfections, but this does not change the slenderness concept described herein. Finally, the buckling load of the elastic-plastic system, i.e., the maximum load Rmax that the structural system can sustain, is given by the following equation:   ¯ Rmax = RY f λ

(8.135)

For the case of an externally pressurized metal pipe the applied load is external pressure, and therefore, Rmax is pco , Rcr is pcr and RY is pY . In some cases, a single   ¯ is used, e.g., in API 1111 (Rankine’s formula), but alternatively, a expression for f λ   ¯ function (e.g., in API RP 2A). branched function can be also used for the f λ As a final note, the DNV-ST-F101 collapse equation can be also written in a form compatible with the above slenderness approach. Starting from Eq. (8.31), or its equivalent form Eq. (8.79), one may write:  1−



pco pY

2 

1 − ¯λ2



pco pY

where is given by Eq. (8.85).



= 2 ¯λ



pco pY

 (8.136)

Offshore pipeline mechanical design

Equivalently,       2 

pco pco ¯ 2 pco = 1−λ 1− pY pY pY

279

(8.137)

Eq. (8.136) or Eq. (8.137) for a given value of define an implicit function of ¯ For simplicity, consider a perfect pipe ( = 0). Then, from pco /pY with respect to λ. ¯ → ∞, one readily obtains: Eq. (8.136) and λ 1 pco = 2 ¯ pY λ

(8.138)

¯ = 0, Eq. (8.137) leads to: and for λ pco =1 pY

(8.139)

This means that the DNV-ST-F101 equation for perfect pipes satisfies Eqs. (8.138) and (8.139) and therefore, it is compatible with the slenderness approach expressed by Eqs. (8.132)–(8.134). In conclusion, the slenderness approach constitutes a framework that comprises the collapse formulae proposed in several design standards, and offers an efficient tool for developing buckling strength equations in elastic-plastic mechanical systems.

8.8

Effect of pipe manufacturing on the collapse pressure

In Chapter 2, the UOE and JCO-E manufacturing processes have been presented. In both processes, a steel plate is cold-formed to a circular shape and subsequently expanded circumferentially to its final configuration. This allows for the production of pipes with very low initial ovality (i.e., values of Oˆ 0 significantly lower than 0.5%). In addition, because of expansion, residual stresses in the pipe wall are also very low. However, despite the low values of cross-sectional ovality and residual stresses in those pipes, experimental results have demonstrated that the collapse pressure of UOE pipes can be significantly lower than that of seamless pipes (hot-formed, non-expanded pipes) with the same D/t value and the same grade steel. This reduction of collapse strength can be as high as 25–30% (Stark and McKeehan, 1995). The main reason for the reduction of collapse pressure is the large tensile strain induced by the expansion process, during the final stage of the manufacturing process, so that the pipe wall is subjected to significant hoop tension, well into the inelastic range. As a result, because of the Bauschinger effect, the compressive strength of the material is reduced, resulting in early yielding of the pipe in compression, and by consequence in a lower value of collapse pressure. This issue has been discussed in Section 5.3 of this book, and it is widely recognized as an important factor affecting collapse of expanded pipes. The first attempt to address this issue was reported by Kyriakides et al. (1991), through a semi-analytical model aimed at quantifying the reduction of collapse in a UOE pipe due to cold-forming and expansion. Subsequent numerical studies on this subject have been published by

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Structural Mechanics and Design of Metal Pipes

Figure 8.7 Comparison of compression stress–strain diagrams from transverse JCO-E pipe specimens with the stress-stress diagram obtained from the original plate of the pipe.

(A)

(B)

Figure 8.8 Finite element results on the variation of collapse pressure and ovality after fabrication with respect to expansion strain parameter εE (X65 steel material, D = 26 in, t = 19 mm, D/t = 34.75).

Herynk et al. (2007), Toscano et al. (2008), Chatzopoulou et al. (2016) on UOE pipes and more recently by Antoniou et al. (2019) and Antoniou (2021) on JCO-E pipes. Fig. 8.7 shows the comparison of the stress–strain responses in compression from specimens extracted from a X65 steel JCO-E pipe (D = 26 in, t = 19 mm, D/t = 34.75) with the stress-strain curve of the original plate before pipe fabrication. The plate material exhibits a clear plastic plateau after reaching the yield stress, which extends beyond 1% strain. The compression stress–strain diagrams of the pipe material are “rounded” and significantly lower that the stress-strain diagram of the plate up to 0.6% and this is attributed to cold bending and expansion of the plate. Considering that in

Offshore pipeline mechanical design

281

(A)

(B)

(C)

Figure 8.9 Finite element model for the JCO-E forming of an X65 steel pipe showing the distribution of equivalent plastic strain (D = 26 in, t = 19 mm, D/t = 34.75). (A) Crimping of edges, (B) J-ing step, (C) O-ing.

externally pressurized pipes, the compression strain at collapse is usually less than 0.5%, it is readily concluded that the collapse strength of the JCO-E pipe would be significantly affected by cold forming and expansion. The effects of expansion on pipe ovality after fabrication 0 and on the collapse pressure pco for the JCO-E pipe under consideration are shown in Fig. 8.8. The results have been obtained with a finite element model that simulates the manufacturing process and the subsequent application of external pressure on the pipe (Fig. 8.9 and Fig. 8.10). The level of expansion during the JCO-E process is expressed in terms of

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Structural Mechanics and Design of Metal Pipes

(B)

(A)

(C)

Figure 8.10 Finite element model for the JCO-E forming of an X65 steel pipe showing the distribution of equivalent plastic strain (D = 26 in, t = 19 mm, D/t = 34.75). (A) Prior to expansion, (B) Maximum expansion, (C) Unloading and final configuration.

the so-called “expansion strain” εE , defined as follows: εE =

CE − CW CW

(8.140)

In the above expression, CE and CW are the mid-surface lengths of the pipe circumference immediately after the expansion phase (and removal of the expander segments) and immediately after welding, respectively. For εE values less than about 0.5%, the increase of expansion increases the collapse pressure (Fig. 8.8A), and this is attributed to the decrease of cross-sectional ovality 0 (Fig. 8.8B) and the reduction of residual stresses. However, for εE values beyond 1.2%, the collapse pressure decreases with increasing expansion, despite the small value of ovality 0 , and this is due to the Bauschinger effect (Antoniou et al., 2019; Antoniou, 2021). Therefore, for the pipe under consideration, values of ε E between 0.45% and 1.15% correspond to maximum collapse pressure. The effect of cold-forming on collapse pressure has been introduced in the DNV-STF101 formula Eq. (8.31) through the α fab factor, defined in Eq. (8.33). For “expanded” pipes (UOE and JCO-E) the value of 0.85 is suggested for the α fab parameter, which

Offshore pipeline mechanical design

283

reduces the value ppl in Eq. (8.33) and, by consequence, the value of collapse pressure pco in Eq. (8.31). Previous works have indicated that the collapse strength of UOE pipes can be improved if the pipe is exposed to heat treatment slightly above 200°C for a few minutes (Al-Sharif and Preston, 1996; DeGeer et al., 2004). This mild heat treatment occurs during fusion-bonded epoxy coating of the pipe for external corrosion protection, and increases the yield stress of the material. This increase is the result of strain-aging of the steel material, where the pipe material recovers a significant part of its compressive yield stress and, therefore, it increases its collapse pressure capacity. Because of this heat treatment effect, considering the value for α fab equal to 0.85, the DNV-ST-F101 collapse formula may result in unreasonably low prediction of collapse pressure. This has been noticed in the full-scale collapse tests on UOE pipes reported by Arroyo et al. (2014) and De Lucca et al. (2015). In those publications, based on the test results, a value of α fab equal to 1.0 is suggested for heat-treated pipes. Recent results from fullscale collapse tests and numerical simulations on thick-walled JCO-E pipes (Gavriilidis et al., 2022) are in favor of using α fab = 1.0 in heat-treated pipes. The research on this topic continues, towards more reliable prediction of collapse pressure in deep offshore pipelines.

References Al-Sharif, A. M., & Preston, R. (1996). Simulation of Thick-walled Submarine Pipeline Collapse Under Bending and Hydrostatic Pressure. In Offshore Technology Conference. Houston, Texas OTC-8212-MS. American Petroleum Institute. (1993). Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms - Load and Resistance Factor Design. API RP 2A (LRFD). Washington, DC. American Petroleum Institute. (1994). Bulletin on Formulas and Calculations for Casing, Tubing, Drill Pipe, and Line Pipe Properties. API Bulletin 5C3. Washington, DC. American Petroleum Institute. (2002). Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms-Working Stress Design. API RP 2A (WSD). Washington, DC. American Petroleum Institute. (2013). Dynamic Risers for Floating Production Systems. API STD 2RD (reaffirmed 2020). Washington, DC. American Petroleum Institute. (2015). Design, Construction, Operation, and Maintenance of Offshore Hydrocarbon Pipelines (Limit State Design). API RP 1111 (reaffirmed 2021). Washington, DC. Antoniou, K., Chatzopoulou, G., Karamanos, S. A., Tazedakis, A., Palagas, C., & Dourdounis, E. (2019). Numerical Simulation of JCO-E Pipe Manufacturing Process and Its Effect on the External Pressure Capacity of the Pipe. J. Offshore Mechanics & Arctic Engineering, 141(1) Article Number: 011704. American Society of Mechanical Engineers. (2018). Gas transmission and distribution piping systems. ASME B31.8 Standard. New York, NY. Antoniou, K. (2021). Numerical Simulation of JCO-E Line Pipe Manufacturing and its Influence on The Mechanical Behavior and Strength of Offshore Pipelines. Volos, Greece: Department of Mechanical Engineering.

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Arroyo, F., León, H. R., Silva, R., Mantovano, L., Solano, R. F., Fabio, B., & Azevedo, F. B. (2014). Collapse Resistance Enhancement in UOE SAWL Line Pipes During Coating Heat Treatment for Ultra Deepwater Applications. In ASME, International Conference of Offshore Mechanics and Arctic Engineering, San Francisco, California, USA OMAE2014-24134. British Standards Institution. (2009). Code of Practice for Pipelines Part 3. Pipelines Subsea: Design, Construction and Installation. BS 8010-3, British Standards Institution. London, UK. Chatzopoulou, G., Karamanos, S. A., & Varelis, G. E. (2016). Finite element analysis of UOE manufacturing process and its effect on mechanical behavior of offshore pipes. Int. J. Solid Structures, 83, 13–27. Clinedinst, W.O., 1963. Development of API collapse pressure formulae. Report for the American Petroleum Institute, Dallas, Texas. De Lucca, R., Solano, R. F., Swanek, D., de Azevedo, F. B., Arroyo, F., Alves, H., & Silva, R. (2015). Enhanced Collapse Resistance for Different D/t Ratios of UOE Pipes for Ultra Deepwater Application. In ASME International Conference of Offshore Mechanics and Arctic Engineering, OMAE2015-42092, St. John’s, Newfoundland, Canada. de Winter, P. E. (1981). Deformation Capacity of Steel Tubes in Deep Water. In Offshore Technology Conference, OTC 4035, Houston, Texas. DeGeer, D., & Cheng, J. J. R. (2000). Predicting Pipeline Collapse Resistance. In ASME International Pipeline Conference, IPC2000-235, Calgary, Alberta, Canada. DeGeer, D., Marewski, U., Hillenbrand, H. G., Weber, B., & Crawford, M. (2004). Collapse Testing of Thermally Treated Line Pipe for Ultra-Deepwater Applications. In ASME, International Conference of Offshore Mechanics and Arctic Engineering, Vancouver, British Columbia, Canada, OMAE2004-51569 (pp. 265–273). DeGeer, D., Timms, C., & Lobanov, V. (2005). Blue Stream Collapse Test Program. In ASME International Conference on Offshore Mechanics and Arctic Engineering, OMAE200567260. Halkidiki, Greece. DeGeer, D., Timms, C., Wolodko, J., Yarmuch, M., Preston, R., & MacKinnon, D. (2007). Local Buckling Assessments for the Megdaz Pipeline. ASME International Conference on Offshore Mechanics and Arctic Engineering, OMAE2007-29493. San Diego, California, USA. Det Norske Veritas. (2013). Submarine Pipeline Systems. STANDARD DNV-OS-F101. Høvik, Norway. Det Norske Veritas. (2017). Submarine Pipeline Systems. STANDARD DNVGL-ST-F101. Høvik, Norway. Det Norske Veritas. (2021a). Submarine Pipeline Systems. STANDARD DNV-ST-F101. Høvik, Norway. Det Norske Veritas. (2021b). Fatigue Design of Offshore Steel Structures. RECOMMENDED PRACTICE DNV-RP-C203. Høvik, Norway. Diniz, F. L. B., Raposo, C. V., Neto, M. C., Pitt, D., Venas, A., Stahl, S., & Borsheim, L. (2012). X-Stream – An Innovative Pipeline Solution to the Challenges of Extreme Water Depths. ASME International Conference on Ocean, Offshore and Arctic Engineering, OMAE201283910, Rio de Janeiro, Brazil. European Committee for Standardization. (2005). Eurocode 3: Design of steel structures – Part 1–1: General rules and rules for buildings. EN 1993-1-1. Brussels, Belgium. European Committee for Standardization. (2006). Eurocode 3 – Design of steel structures – Part 1–5: Plated structural elements. EN 1993-1-5. Brussels, Belgium.

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European Committee for Standardization. (2007). Eurocode 3 – Design of steel structures – Part 1–6: Strength and Stability of Shell Structures. EN 1993-1-6. Brussels, Belgium. Fowler, J.R., Langner, C.G. (1991). Limits for Deepwater Pipelines Performance, Offshore Technology Conference, OTC 6757, Houston, Texas. Gavriilidis, I., Stamou, A. G., Karamanos, S. A., Palagas, C., Dourdounis, E., Voudouris, N., & Tazedakis, A. (2022). Collapse of JCO-E pipes; experiments and numerical predictions. In ASME, International Conference of Offshore Mechanics and Arctic Engineering, OMAE2022-79129 Hamburg, Germany. Gerard, G. (1962). Introduction to Structural Stability Theory. New-York, NY: McGraw-Hill Book Co. Herynk, M. D., Kyriakides, S., Onoufriou, A., & Yun, H. D. (2007). Effects of the UOE/UOC pipe manufacturing processes on pipe collapse pressure. International Journal of Mechanical Sciences, 49(5), 533–553. Houliara, S., & Karamanos, S. A. (2006). Buckling and post-buckling of long pressurized elastic thin-walled tubes under in-plane bending. International Journal of Non-Linear Mechanics, 41(4), 491–511. Kyriakides, S., Corona, E., & Fischer, F. J. (1991). On the effect of the UOE manufacturing process on the collapse pressure of long tubes. In Proceedings of Offshore Technology Conference, OTC 6758, Houston, Texas. Langner, C. G. (1999). Buckle Arrestors for Deepwater Pipelines. In Offshore Technology Conference, OTC 10711, Houston, Texas. Murphey, C., & Langner, C. G. (1985). Ultimate pipe strength under bending, collapse and fatigue. In International Conference of Offshore Mechanics and Arctic Engineering (OMAE). ASME, Dallas, USA. Palmer, A. C. (1998). A Radical Alternative Approach to Design and Construction of Pipelines in Deep Water. In Offshore Technology Conference, OTC-8670, Houston, Texas. Park, T. D., & Kyriakides, S. (1997). On the performance of integral buckle arrestors for offshore pipelines. International Journal of Mechanical Sciences, 39(6), 643–669. Stark, P. R., & McKeehan, D. S. (1995). Hydrostatic collapse research in support of the OmanIndia gas pipeline, Houston, Texas. In Offshore Technology Conference 2, OTC 7705 (pp. 105–120). Timms, C., Swanek, D., DeGeer, D., Meijer, A., Liu, P., Jurdik, E., & Chaudhuri, L. (2018). Turkstream Collapse Test Program. In ASME International Conference on Ocean, Offshore and Arctic Engineering, OMAE2018-78454, Madrid, Spain. Toscano, R. G., Raffo, J. L., Fritz, M., Silva, R. C., Hines, J., & Timms, C. (2008). Modeling the UOE Pipe Manufacturing Process. In ASME, International Conference of Offshore Mechanics and Arctic Engineering, Estoril, Portugal (pp. 521–528). OMAE2008-57605.

Pipeline analysis and design in geohazard areas

9

Quite often, pipelines transporting energy and water resources, are constructed in regions with extreme terrains associated with geohazards, including ground shaking, seismic fault action, landslides, soil subsidence or soil liquefaction. In those regions, significant ground movements may occur during the service life of these pipelines and, therefore, safeguarding pipeline integrity towards unhindered delivery of energy and water is an essential requirement of pipeline design. In addition, the increasing stringency of environmental restrictions impose new challenges to pipeline engineers. Nowadays, “geohazards and pipelines” constitutes a hot topic in pipeline engineering research and practice, with significant importance for pipeline structural safety. It is a rather new topic that motivates a substantial amount of research in the engineering community. In the present chapter, methods for the analysis and design of buried steel pipelines in geohazard areas are presented, based on recent developments on this engineering subject. The methods can be used by pipeline engineers to safeguard the structural safety of steel hydrocarbon pipelines against ground-induced actions. For detailed technical presentation of these methodologies, the interested reader is referred to the PRCI guidelines (PRCI, 2017) and to the recent book by Karamanos et al. (2021). The latter is the summary of European project GIPIPE (Vazouras et al., 2015a), a landmark project on geohazards and pipelines, focusing on soil-pipe interaction. In geohazard areas, the pipeline is designed following a “strain-based” design approach. The main aspects of strain-based design are presented, pinpointing its main differences with traditional “stress-based design”, which is also called “pressure design”, and expanding on its two main components, namely strain demand and strain resistance. Special emphasis is given on a simple and efficient methodology for calculating strains from groundinduced actions in the pipeline. The development of advanced numerical simulation methodologies, such as finite elements, enable the investigation of ground-induced actions on pipelines in a more rigorous manner. Strain resistance is also presented briefly in terms of pipeline tensile and compressive strength and deformation capacity. Finally, hints for mitigating the effects of geohazards on pipelines are outlined.

9.1

Aspects of strain-based design

Pipeline design codes and standards, such as ASME B31.4 (American Society of Mechanical Engineers, 2019), ASME B31.8 (American Society of Mechanical Engineers, 2018) or EN 1594 (European Committee for Standardization, 2013), provide a complete set of rules for pipeline design under normal operating conditions. Under those conditions, internal pressure is the primary pipeline design parameter, this Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00016-5 c 2023 Elsevier Inc. All rights reserved. Copyright 

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procedure is called “pressure design”, i.e., pipeline design under normal operation conditions, and has been presented in Chapter 7. However, the above codes and standards do not contain design rules or guidance on how to produce safe designs for pipelines in areas with large ground movements, where large plastic strains are expected to develop. Furthermore, those codes and standards do not provide explicit design criteria or performance limits on how to achieve this important objective. Therefore, it is necessary to supplement ASME B31.4 or B31.8 with design documents that provide direct guidance to engineers on how to design pipelines in geohazard areas. The modern design practice for hydrocarbon pipelines against ground-induced actions follows the so-called “strain-based design” approach. This is a relatively new approach, based on the strain that develops in the pipe wall due to ground movement. Procedures that are “stress-based” are suitable for “pressure design” of pipelines but are inadequate for addressing potential pipeline failure modes that may occur after severe ground-induced actions. Industry acceptance of strain-based design methods has increased tremendously over the last ten years and, nowadays, these methods are widely used by pipeline designers. Earthquakes constitute a major source of ground-induced actions (seismic actions). However, geohazards are not limited to seismic actions. As an example, landslides or differential settlements are not necessarily earthquake driven. Furthermore, ground shaking may not always stem from tectonic phenomena but can be induced by human activity, such as extraction of natural gas from deep soil strata or abandonment of mines, which both cause collapse of soil cavities. Overall, ground-induced actions on pipelines may be categorized as follows: r r

transient actions: strain and curvature on the ground due to traveling wave effects during an earthquake or another event, e.g., soil cavity collapse; permanent actions: ground deformation due to seismic-induced faults, landslides in unstable faults, soil subsidence or liquefaction-induced lateral spreading.

The transient effects are usually substantially less severe for pipeline structural integrity than permanent ground deformations and are not discussed herein. The interested reader is referred to the book by Karamanos et al. (2021). On the other hand, permanent ground deformations may constitute significant threats for pipeline safety and require special attention (O’Rourke and Liu, 2012). Among permanent ground deformations, fault-induced ground deformation and landslide action have been recognized as the most severe actions on steel buried pipelines. Nevertheless, lateral spreading has also caused damage to buried pipelines constructed in liquefaction-prone ground, and soil subsidence may also be a major issue for pipeline integrity in cases where differential settlements may occur. The problem of pipeline deformation under permanent ground action can be represented in a generic form by the sketch in Fig. 9.1, where one part of the ground moves with respect to the other parts. Because of this motion, the buried pipeline that crosses this area tries to adjust itself within the imposed deformation pattern. This pattern may represent several types of ground-induced actions, such as the edge of a landslide, a tectonic fault, the edge of a liquefied zone, or an area of differential

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(A)

(B)

Figure 9.1 Schematic representation of permanent ground-induced deformation in buried pipelines; (A) pipeline in tension, (B) pipeline in compression.

settlement (subsidence). The differential ground movement causes bending and, depending on the orientation of the pipeline, stretching (Fig. 9.1A) or compression (Fig. 9.1B) of the pipeline at this critical location, associated with the development of significant stresses and strains in the pipe wall. Geohazard identification is the first step in this design process. It is conducted by a specialized geologist or a geotechnical engineer and may be performed visually through field or aerial surveys, remote sensing information, LiDAR or photogrammetric surveys, or analysis of in-line inspection data. Upon geohazard identification, a site-specific analysis is necessary to quantify the ground motion in terms of type, size, amount and orientation. As an example, constructing a pipeline in an area of potential landslide, slope stability and deformation analysis is performed to estimate potential slip surfaces, the characteristics of soil deformation in terms of size, the amount and the direction of soil movement. Subsequently, pipeline analysis is conducted by

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the pipeline engineer using the soil deformation estimated in the slope deformation analysis. This analysis consists of two main parts: r r

strain demand calculation, strain resistance verification,

which are outlined in Sections 9.2 and 9.3 respectively. If necessary, appropriate mitigation measures should be applied, outlined in Section 9.4.

9.2

Strain demand calculation (pipeline strain analysis)

This part of the design process refers to the determination of stresses and strains in the pipeline wall induced by the ground movement. It is a procedure that requires a soil-pipe interaction analysis that accounts for properties of soil around the pipeline (soil stiffness and strength) and the mechanical properties of steel pipe, which depend primarily on pipe diameter, thickness and steel grade. The amount of ground displacement is an essential parameter for performing the strain-demand analysis and is determined mainly by geological considerations. However, this value, in order to be used in the strain-demand analysis of the pipeline, should be adjusted according to the pipeline function and, in the case of faults, the effective fault activity (Davis, 2008). There exist several methodologies for computing ground-induced stresses and strains in the pipeline, which consider soil-pipe interaction in a direct or indirect manner. Some methodologies are more simplified, can be applied with handmade calculations, and are more suitable for preliminary design, whereas more rigorous methodologies are based on numerical models, which use computer methods and are used in detailed analysis or assessment of pipelines. During the last two decades, the significant advancement in computer technology and software has allowed the use of these computer methods in pipeline design and analysis. The state-of-the-art on this subject is described in the book by Karamanos et al. (2021) and is briefly presented below for the case of continuous pipelines.

9.2.1 Finite element modeling Finite element modeling is a powerful methodology for simulating the effects of ground-induced actions on a buried pipeline. The finite element analysis of buried pipelines requires some computational effort and expertise but offers an advanced tool for determining stresses and strains within the pipeline wall with significant accuracy with respect to the real situation. There exist two levels of finite element modeling, namely level 1 and 2, briefly described below. Level 1 is adequate for regular design purposes, whereas level 2 is used only in special cases, where increased accuracy is necessary. Level 2 modeling is also suitable for research purposes. Level 1: Beam-type finite element analysis In this type of finite element analysis, the pipe is modeled with beam-type finite elements, which are attached to soil springs (Fig. 9.2). These models have been used

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Figure 9.2 Beam-type finite element model; soil is modeled with nonlinear springs.

mainly for simulating permanent ground-induced actions on pipelines, but they can be used for modeling wave effects on the pipeline as well. The finite element mesh near the soil discontinuity (e.g., fault plane) should be fine enough, so that gradients of stress and strains are accurately described. Some features of these models are stated below. Type of finite elements: The use of regular beam elements for the pipeline model is not recommended, because they cannot account for pressure. Instead, special-purpose “pipe elements” are preferable for pipeline seismic analysis. These are enhanced beamtype elements that account for the effect of hoop stress due to pressure and are available in several commercial general-purpose finite element programs. However, “pipe elements” usually have a circular cross section and do not describe cross-sectional ovalization, which can be quite significant in pipe bends (elbows). Therefore, the use of more elaborate “pipe elements”, capable of describing cross-sectional ovalization, sometimes referred to as “elbow elements”, can further improve the accuracy of the finite element model, especially at pipeline bends (Bathe and Almeida, 1982; Karamanos and Tassoulas, 1996). Alternatively, it is possible to employ regular pipe elements, which are essentially beam elements with circular cross-section, accounting for the ovalization effects at pipe bends using appropriate flexibility factors, and stress intensity factors. Pipe and soil modeling: Pipe material should be modeled as elastic-plastic, considering strain-hardening. The ground surrounding the pipeline should be modeled by nonlinear springs (Fig. 9.3), attached to the pipe nodes and directed in the transverse directions (with stiffness kV and kH in the vertical and lateral direction respectively) and axially (kax ). The springs should account for possible slip between the pipe and the soil. In addition, the vertical spring should be non-symmetric, accounting for the different resistance to the downward and to the upward motion of the pipe (Fig. 9.3C). Expressions for this soil stiffness are offered in ALA guidelines

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(B)

(D) (C)

Figure 9.3 Soil springs for beam-type finite element models.

(American Lifelines Alliance, 2001), based on the type of soil. Alternative expressions for those springs may be found in the NEN 3650 standard (Nederlands Normalisatie Instituut, 2020). The reader is also referred to the works of Xie et al. (2013) and Saiyar et al. (2016) regarding the limitations of soil spring reaction models, especially for the case of flexible pipes. In addition, comparison of “pipe” and “elbow” element methodologies with more rigorous finite element methodologies and experimental data have been reported recently by Sarvanis et al. (2016) and Sarvanis and Karamanos (2016). Analysis procedure and output: The pipeline analysis under permanent groundinduced actions follows three steps: (a) gravity, (b) operational loading (pressure and temperature) and (c) application of the permanent ground displacement. To perform this analysis, the imposed soil displacements should be applied at the ends of the soil springs. The analysis output consists of stress resultants in pipeline cross-sections, as well as the stresses and strains in the longitudinal direction. The user should be cautioned that if the finite elements are not capable of describing cross-sectional distortion, the stresses and strains obtained may be quite different than the real stresses and strains in the pipeline wall, especially when the pipe cross-section is expected to exhibit significant distortion with respect to its initial circular shape. Furthermore, those models may not be suitable for simulating the formation of wrinkles due to local buckling. Consideration of local stresses due to pipe wall wrinkling locations requires a more detailed analysis, with the use of shell elements for modeling the pipe. Level 2: Three-dimensional finite element analysis Three-dimensional finite element models constitute a rigorous numerical tool to simulate buried pipeline behavior under permanent ground deformation. Such a model can

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(B)

Figure 9.4 Three-dimensional continuum-type finite element model; (A) undeformed configuration, (B) deformed configuration.

describe quite accurately the nonlinear geometry of the deforming soil-pipe system (including pressure effects and the distortion of the pipeline cross-section), the inelastic material behavior for both the pipe and the soil, as well as the interaction between the pipe and the soil. However, it requires computational expertise by the analysist. This modeling technique has been developed primarily in the course of the GIPIPE project (Vazouras et al., 2010; 2012; 2015a; 2015b) and opened new opportunities for investigating soil-pipe interaction problems in a detailed and rigorous manner. Discretization: An elongated prismatic model is considered, where the steel pipeline is embedded in the soil, as shown in Fig. 9.4 for the case of a strike-slip fault. Shell elements are employed for modeling the steel pipeline segment, whereas threedimensional “brick” elements are used to simulate the surrounding soil. The discontinuity plane (e.g., fault plane, edge of landslide or lateral spreading) divides the soil in two blocks, where one block moves with respect to the other. The model accounts for gravity, operation loads (pressure and thermal) and the ground-induced movement. The latter is imposed holding one soil block fixed and imposing a displacement pattern in the external nodes of the second block. A fine mesh should be employed at the part of the pipeline where maximum stresses and strains are expected. Similarly, the finite element mesh for the soil should be more refined in the region near the fault and coarser in the region away from the fault. The relative movement of the two blocks is considered to occur within a narrow transition zone of width w to avoid numerical problems. Material models: The constitutive models account for the elastic-plastic behavior of both the pipeline and soil. Von Mises (J2 ) plasticity with isotropic hardening can be employed for describing pipe steel material, calibrated through a uniaxial stressstrain curve from a standard tensile test. There exist several models for describing

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the mechanical behavior of soils. In many instances, an elastic-perfectly plastic MohrCoulomb model can be considered for modeling soil behavior. This model is characterized by the soil cohesion c, the friction angle φ (see Fig. 9.7), the elastic modulus of soil Esoil , and the Poisson’s ratio of soil vsoil . Furthermore, a contact algorithm should be employed to simulate the interface between the outer surface of the steel pipe and the surrounding soil, which takes into account interface friction, and allowing separation of the pipe and the surrounding soil. Pipeline continuity: Special-purpose springs oriented in the longitudinal direction should be considered, attached to the ends of the pipeline, to account for the infinite length of the pipeline and its continuity at the end of the model (see Fig. 9.4), as described in detail by Vazouras et al. (2015b). Alternatively, the pipeline outside the soil block may be continued with level 1 modeling, i.e., using “pipe” or “elbow” elements, with soil springs attached to the nodes. Analysis procedure and output: The analysis is performed in three consecutive steps: (a) gravity of soil and pipe, (b) operational loads (pressure and temperature) and (c) application of the permanent ground displacement. The analysis follows a displacementcontrolled incremental scheme, which increases gradually the ground displacement. At each increment of the nonlinear analysis, stresses and strains at the pipeline wall should be recorded. Furthermore, using a dense mesh of finite elements at the critical pipeline portions, local buckling (wrinkling) formation and post-buckling deformation at the compression side of the pipeline wall can be simulated in an explicit manner. Level 2 models can represent very accurately soil-pipe interaction and the response of buried pipelines under permanent ground deformations. Furthermore, crosssectional distortion and, in particular, local buckling at the pipe wall can be simulated in a rigorous manner. Fig. 9.5 shows the deformed configuration of a 36-inch outer diameter steel pipe (914 mm) with 1/4-inch thickness (D/t = 144), internally pressurized at 90 psi (6.21 bar), crossing a horizontal fault at angle β = 25°. The yield stress of pipe material is equal to 43.9 ksi (303 MPa). The configuration shown in Fig. 9.5 corresponds to a fault displacement of 1.41 m. For visualization purposes, only the bottom half of the soil block is shown. Because of its orientation with respect to the fault, the steel pipeline undergoes longitudinal bending with significant stretching, and does not exhibit local buckling despite a wavy strain pattern that appears in the compressive side of the pipe wall. Fig. 9.6 shows the response of a large-diameter water pipe crossing a horizontal fault at angle β = 0° (normal crossing). The pipe has outer diameter 86.25 in (2,190 mm), wall thickness 0.625 in (15.88 mm), and is internally pressurized at 90 psi (6.21 bar). The yield stress of the pipe steel material is 43.9 ksi (303 MPa). The deformed shape of the pipe depicted in Fig. 9.6 corresponds to a fault displacement equal to 1 m. The pipeline exhibits local buckling at two locations on either side of the fault due to bending deformation.

9.2.2 Experiments on soil-pipe interaction Within the GIPIPE project (Vazouras et al., 2015a; Karamanos et al., 2021), a series of physical experiments on soil-pipe interaction have been performed, representing buried

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Figure 9.5 Deformation of a 36-inch-diamter steel pipeline (thickness 1/4 in, D/t = 144, pressure 6.21 bar) crossing an active horizontal fault at β = 25°; three-dimensional continuum-type finite element model.

pipelines subjected to permanent ground deformation, with the aim of examining experimentally soil-pipeline interaction in the presence of ground-induced actions. A very brief presentation of those experiments is offered below. Four (4) large-scale soil-pipe interaction tests have been performed by Rina Consulting - Centro Sviluppo Materiali S.p.A., at their facilities in Pedrasdefogu, Sardinia, Italy, using a special “landslide/fault” device. The setup has been composed by three adjacent concrete boxes containing the soil and the buried pipe: two fixed boxes and one sliding box in-between, schematically represented in Fig. 9.8A. The setup was 25-meter long in total, in which 219-mm-diameter 5.56-mm-thick L450 (X65) steel pipe specimens were buried. During testing, the central sliding soil box was pulled by two hydraulic actuators, while the other two boxes remained fixed. The specimen was instrumented with strain gauges to measure the local strains on the pipe and to evaluate global pipe deflections by integrating strain measure over the pipe length, as well as with laser LVDTs to measure the pipe displacement at the two ends. The force applied by the hydraulic pulling system was also measured. Fig. 9.8B shows the central sliding soil box and the connectors to the actuators during testing. The four large-scale tests were also accompanied by a series of indoor laboratory tests on axial soil-pipe interaction (3 tests) and transverse soil-pipe interaction (3 tests). More information on those tests is provided in the paper by Sarvanis et al. (2018) and in the book by Karamanos et al. (2021). Ten (10) experiments representing the response of a pipeline subjected to a lateral ground-induced deformation (i.e., a strike-slip fault), have been performed at Stevin Laboratory, Delft University of Technology (TU Delft), The Netherlands, using “virtual” soil conditions in the laboratory (Van Es, 2016). The test setup is shown in

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Figure 9.6 Deformation of a 86.25-inch-diameter steel water pipeline (thickness 0.625 in, D/t = 138, pressure 6.21 bar) crossing a horizontal fault at β = 0°; three-dimensional continuum-type finite element model.

Figure 9.7 Mohr-Coulomb material model for soils; (A) in the σ -τ stress plane, (B) in the deviatoric stress plane.

Fig. 9.9 and Fig. 9.10 and consisted of two rectangular frames, one of which was moveable, while the other remained fixed. The soil was simulated with appropriate nonlinear springs, referred to as “ring-springs”. In each test, the 20-meter-long pipeline specimen was connected to the two frames by means of the ring-springs, shown schematically in Fig. 9.9. Each ring-spring consisted of a collapsing steel ring, shown in Fig. 9.10B, with appropriate diameter, thickness and yield strength, so that the desired soil properties are represented. Six ring-springs were installed on each frame and the loads of each ring-spring were transferred to the specimen through two steel rods and two flexible steel straps allowing ovalization of the pipeline. Axial soil friction

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Figure 9.8 Large-scale experiments performed at CSM facilities, Perdasdefogu, Sardinia, Italy; (A) 3D configuration of the experimental set-up, (B) photo of the central (moving) soil box and the actuators (photo courtesy: Rina Consulting – Centro Sviluppo Materiali S.p.A.).

(A)

(B)

Figure 9.9 Basic setup of the large-scale experiments at TU Delft, The Netherlands (schematic); (A) initial position of frame and undeformed pipe configuration, (B) displaced position of frame and deformed pipe configuration.

has been simulated by applying an axial force at the two ends of the pipeline using hydraulic actuators. Sixty-six (66) strain gauges were attached along the pipe, mainly in the area of expected maximum deformation, and cross-sectional ovalization was monitored at ten locations along the pipe. The experiments examined the effect of

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(B)

(C)

Figure 9.10 Large-scale experiments at TU Delft, The Netherlands; (A) general view of deformed specimen under differential “soil” movement, (B) collapsible tubes simulating soil resistance, (C) Local buckling of pipe specimen (photos by Sjors H.J. van Es, published with permission).

different parameters, e.g., pipeline geometry, internal pressure, types of soil and the presence or absence of a girth weld in a critical segment on either side of the fault. A detailed description of the experiments is presented by Van Es (2016), Van Es and Gresnigt (2016) and the recent book by Karamanos et al. (2021). Pipeline response under normal and reverse fault action was examined in a series of small-scale experiments in the Laboratory of Soil Mechanics of National Technical University (NTU), Athens, Greece. The experimental program included 26 tests and examined both the fault rupture propagation without the presence of the pipe and soil–pipe interaction. A custom-built apparatus was used to simulate fault rupture propagation and fault rupture–soil–structure interaction (Fig. 9.11). It comprises a stationary part and a movable part that electronically translates upwards or downwards, applying fault displacement (reverse or normal) in a quasi-static manner. In this apparatus, maintaining a constant dip angle equal to 45° for both normal and reverse faults, bedrock dislocations up to 0.15 m can be achieved. The inner dimensions of the box are 2.65m × 0.9m × 0.9m. The pipe specimen was placed in the middle of the container and embedded into a 0.65-meter-deep stratum of dense sand. Despite the limited dimensions of the box, soil-pipe interaction was well measured and enabled the calibration of finite element models. The specimens were 35-mm-diameter welded tubes made of stainless-steel grade AISI Type 444. The deformation of two specimens

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(B)

(C)

(D)

Figure 9.11 Small-scale soil-pipe interaction experiments at NTU Athens, Greece; (A) schematic representation of test apparatus, (B) positioning the pipe and filling the box, (C) simulation of normal fault, (D) simulation of reverse fault (photos by Angelos Tsatsis, published with permission).

subjected to normal and reverse fault movement is shown in Fig. 9.12. Both specimens exhibited local buckling, which is more pronounced for the reverse fault. The reader is referred to Tsatsis (2017), Tsatsis et al. (2019) and Karamanos et al. (2021) for more information on those experiments.

9.2.3 Analytical models for fault crossing analysis In addition to numerical models, there exist analytical methodologies for strain analysis of pipelines under geohazard actions. Following the pioneering work of Newmark and Hall (1975), analytical methodologies were proposed in the works of Kennedy et al. (1977), Wang and Yeh (1985) and, more recently, by Takada et al. (2001), Karamitros et al. (2007) and Trifonov and Cherniy (2010). Most of these analytical methodologies refer to formulations that require iterative solution methods, which may not be easy to implement and use in everyday engineering practice. In the following, the main aspects of buried pipes subjected to permanent ground deformations are presented in a simple and efficient manner. The ground-induced displacement of the pipeline is denoted by dF and is representative of the fault crossing case (Fig. 9.13). Nevertheless, depending on the geohazard action under consideration,

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(A)

(B)

(C)

(D)

Figure 9.12 Soil-pipe interaction experiments at NTU Athens; (A) deformation of a 35-mm-diameter specimen under normal fault, (B) local buckle under normal fault, (C) deformation of a 35-mm-diameter specimen under reverse fault, (D) local buckle under reverse fault (photos by Angelos Tsatsis, published with permission).

Figure 9.13 Schematic permanent ground-induced horizontal deformation in buried pipelines (three-dimensional sketch).

this displacement may also represent any other type of soil displacement, e.g., at the edge of a landslide, or the edge of a liquefaction spreading pattern or a differential settlement of the pipeline. Referring to Fig. 9.13 and to Fig. 9.14, the total longitudinal strain in the pipeline consists of the following components: 1. bending strain εb , because of transverse displacement dF cos β; 2. membrane strain ε m1 , because of pipe elongation due to bending; 3. membrane strain εm2 , because of the direct elongation of the pipeline dF sin β, as shown in Fig. 9.14.

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Figure 9.14 Schematic permanent ground-induced horizontal deformation in buried pipelines; (A) transverse and longitudinal deformation of pipeline, (B) strain analysis model.

To understand pipeline mechanics and behavior under this ground-induced displacement, the total displacement of a point on the pipeline axis is considered as the superposition of a transverse component u(x), shown with the dotted brown line in Fig. 9.14A, and a longitudinal component ul (x). The following pattern is assumed for the transverse displacement of the pipeline: u(x) = (dF cos β )ϕ(x)

(9.1)

where ϕ(x) is an assumed shape function. Furthermore, ul (x) is assumed to be a linear function of coordinate x, so that: ul (x) = (dF sin β )(x/LAB )

(9.2)

At point B, x = LAB , and from Eq. (9.2) one obtains ul (LAB ) = dF sin β. The maximum bending strain εb developed in the pipe is: εb =

kmax D D = u (xm ) 2 2

(9.3)

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where xm is the location along the pipeline where curvature, i.e., the second derivative of the pipeline displacement denoted as u”, is maximum. Using Eqs. (9.1) and (9.3):   D (dF cos β ) εb = ϕ  (xm ) 2

(9.4)

which implies a linear dependence of the bending strain εb in terms of the groundinduced displacement. The membrane strain εm1 is due to bending (i.e., due to transverse deformation y(x) only), it depends on the ground-induced displacement dF and it is present even in cases where the pipeline is perpendicular to the fault. Referring to Fig. 9.14A, and considering the transverse deformation of the pipe (represented by the brown dotted curve), the elongation AB of segment AB from AB to AB1 can be calculated as follows: 

LAB

AB =



1 + u 2 dx − LAB

(9.5)

0

so that the corresponding axial strain (also referred to as “membrane” or “stretching” strain) εm is: εm1 =

AB 1 = LAB LAB



LAB

 1 + u 2 dx − 1

(9.6)

0

and is assumed to be uniformly distributed along the pipeline. Using the following series expansion:  1 1 + u 2 = 1 + u2 + · · · 2

(9.7)

keeping only the first two terms (under the assumption of small displacements and rotations), and inserting into the axial membrane strain in Eq. (9.6), one obtains: εm1 =

(dF cos β )2 AB  ϕ0 2 LAB 2LAB

(9.8)

where 

LAB

ϕ0 = LAB

ϕ  dx 2

(9.9)

0

An interesting conclusion from Eq. (9.8) is that the membrane strain εm1 is a quadratic function of ground displacement dF . Finally, the direct elongation component

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εm2 , if assumed uniformly distributed within the length AB, is equal to: εm2 =

dF sin β LAB

(9.10)

and indicates a linear dependence of εm2 with respect to the ground-induced displacement dF . The total strain ε is the superposition of strains εb , εm1 , and εm2 :     dF sin β 1 dF cos β 2 D  (dF cos β ) ε= + ϕ0 + ϕ (xm ) LAB 2 LAB 2

(9.11)

Eq. (9.11) offers a straightforward closed-form expression for the strain in the pipeline due to ground-imposed deformation dF . However, apart from assuming an appropriate function ϕ(x), it is also necessary to know or to estimate the value of LAB to compute this strain, and this is not a trivial issue. The calculation of LAB requires the solution of the coupled soil-pipe interaction problem, considering soil-pipe interaction, and constitutes a challenge in developing an efficient analytical methodology. All existing analytical methodologies attempt to calculate the value of LAB in a direct or indirect manner. In the following, we refer to two straightforward methodologies, which are easy to implement in engineering practice.

9.2.3.1

Newmark and Hall (1975)

In this simplified methodology, the pipeline is assumed to be a cable, neglecting its bending resistance, so that the bending strain is zero (ε b = 0). The strain in the pipeline is computed from the fault displacement dF as follows:  2 dF 1 dF ε= sin β + cos β L 2 L

(9.12)

where L = LAB . In Eq. (9.12), only membrane strain components εm1 and εm2 are considered. Eq. (9.12) is a special form of Eq. (9.11) that neglects bending strain and assumes ϕ 0 = 1. In Eq. (9.12), the length L of the deformed S-shape of the pipeline is the distance between “inflection points” and the following formula is proposed by Newmark and Hall (1975) for its calculation: FP L + = 2 tu



F − FP tu

 (9.13)

where F is the axial tension in the pipeline, FP is the yield axial tension of the pipe crosssection (Fp = πDt σ Y ) and tu is the maximum soil resistance in the axial direction. For typical steel material, the value of F after yielding is practically equal to the yield force

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of the pipe cross-section FP , therefore, L2

FP tu

(9.14)

It is questionable whether Eq. (9.13) or Eq. (9.14) provide accurate predictions for the length of the deformed pipeline L. This issue will be discussed in a subsequent paragraph.

9.2.3.2

American Lifelines Alliance - ALA (2001)

The equation proposed by Newmark and Hall (1975), expressed in Eq. (9.12), is adopted by ALA (2001). However, a factor of 2 is used to multiply the strain, most likely as a “design factor” or a “safety factor”, accounting for possible unconservativeness of Eq. (9.12). The resulting expression is:  2 dF dF (9.15) cos β ε = 2 sin β + L L The length L of the deformed shape is also calculated from Eqs. (9.13) or (9.14) above. Again, the calculation of an accurate value of the “deformed length” L is of primary importance, but it is not addressed in ALA (2001) and will be discussed in the following paragraph.

9.2.4 A simple and efficient analytical model for soil-pipe interaction in permanent ground-induced actions This methodology involves quite some analytical effort for its development, but results in a convenient set of closed-form equations and is very straightforward to use for design purposes. It can be considered as an amendment because this methodology: r r r r r

considers a realistic shape function of the pipeline; accounts for different soil conditions on either side of the discontinuity plane; simulates both horizontal and vertical fault action; accounts for flexible-end conditions of the pipe segment, due to pipeline continuity; offers a reliable and straightforward calculation of the “deformed length”.

The problem under consideration can be stated schematically in Fig. 9.15, which is similar to, yet more enhanced than, the one shown in Fig. 9.14. The buried pipeline crosses a discontinuity plane in the ground at angle β (or θ ), as shown in Fig. 9.15A. In the following analysis, the discontinuity plane is the tectonic fault plane. However, it may also represent any other type of soil discontinuity, such as the edge of a landslide or the interface between liquefied and non-liquefied soil. In Fig. 9.15, the soil block on the right side of the fault moves parallel to the fault direction, by an amount of displacement dF with respect to the soil block on the left side. Due to this differential ground motion, the pipeline is subjected to both bending and stretching, obtaining a stretched S-shape configuration, also shown in Fig. 9.15. The lengths L1 and L2 in Fig. 9.15 correspond to the curved pipe parts of the deformed

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Figure 9.15 Sketches for the analytical methodology: (A) schematic representation of ground-induced deformation of pipeline, (B) deformation of pipeline considering only the transverse component of pipeline deformation.

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Figure 9.16 Length ratio α = L1 /L2 in terms of soil resistance ratio b = qu1 /qu2 ; (A) diagram for b values between 0 and 20, and (B) diagram for b values between 20 and 150.

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(A)

(B)

Figure 9.17 (A) Function F(α) in terms of length ratio α = L1 /L2 . (B) Length ratio αi = Li /L2 in terms of length ratio α = L1 /L2 .

S-shaped pipeline on each side of the fault, while Li is the distance of the inflection point (i.e. the point where the curvature changes sign) from the fault plane. The proposed methodology is based on decomposing the ground displacement dF in a transverse component dF cosβ and a longitudinal component dF sinβ (see Fig. 9.14A). Furthermore, it assumes the following function for the transverse displacement u(x): ⎧       πx x ⎪ ⎪ ˆ 1 + Li ) cos β 1+ 1 sin − 0 ≤ x ≤ L1 +Li d(L ⎪ ⎪ ⎨ L1 + L i   4  L1 + Li  1 π (−x + L1 + Li ) x + L2 − L1 − 2Li u(x) = ˆ d(L − L ) cos β 1+ sin − ⎪ 2 i ⎪ ⎪ 4 L2 − Li L2 − Li ⎪ ⎩ L1 + Li ≤ x ≤ L (9.16) where, dˆ = dF /(L1 + L2 ) is the normalized ground displacement. This assumed shape function expressed by Eq. (9.16) looks rather sophisticated, but has the advantage of satisfying the following three conditions at the two ends: (a) zero displacement, (b) zero slope, and (c) zero curvature, so that the decay of bending deformation is described properly from the physical point-of-view. It should be noticed that assuming a typical double-curvature trigonometric function for u(x), e.g., u(x) =

πx dF 1 − cos 2 L

(9.17)

may not be appropriate, because the corresponding curvature does not decay to zero in a proper manner at the two ends of this segment. At x = 0 and x = L the curvature calculated by double differentiation of Eq. (9.17) is non zero (u (0) = 0 and u (L) = 0), and therefore, a discontinuity of bending curvature and strain exists between this deformed part and the adjacent straight part of the pipeline.

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In the present formulation, it is always assumed that L2 is larger than L1 (L2 ≥ L1 ), which means that the soil resistance qu1 is larger than soil resistance qu2 . Furthermore, the final stage of transverse deformation in Fig. 9.15B can be considered as the superposition of two bending deformation patterns: (a) the configuration due to differential movement of the supports in the transverse direction, which represents the permanent soil movement of the fault in the absence of soil resistance, and (b) the configuration due to distributed loading, representing soil resistance (Sarvanis and Karamanos, 2017). A key assumption is also made: upon first yielding of pipeline material, the values of lengths L1 and L2 remain constant. The main argument in support of this assumption is that, upon first yielding at a specific location, deformation will localize around this point, so that the general shape of the pipeline in terms of lengths L1 , L2 and Li may not change significantly. The accuracy of this assumption has been verified numerically, and more details on this formulation can be found in the article of Sarvanis and Karamanos (2017). The design steps that outline this formulation are summarized in Table 9.1. For the specific case of a horizontal (strike-slip) fault, it is reasonable to assume the same soil conditions on either side of the fault. Therefore, for qu1 = qu2 (both equal to a single value qu ), the methodology becomes more straightforward, and it is summarized for the sake of completeness in Table 9.2 below. In this case, L1 = L2 = L/2, Li = 0, and therefore, the total length L is equal to 2L1 (or 2L2 ).

9.3

Strain resistance verification of the pipeline

Upon calculation of ground-induced strains, as presented in Section 9.2, the pipeline designer should determine whether the pipe wall is capable of sustaining those strains, together with the strains from normal operating conditions, so that pipeline failure is prevented. The term “pipeline failure” may be interpreted in many ways. In the current design philosophy against geohazards, it is widely accepted that pipeline failure is defined as “loss of pressure containment” and therefore, it refers primarily to pipe wall rupture. Preventing pipe wall rupture is the main performance criterion that the pipeline must meet, so that it continues to fulfill its transportation function. In structural engineering terminology, pipe wall rupture is an “ultimate limit state”. Other performance criteria may also be considered in pipeline design, such as local buckling (wrinkling) of pipe wall, or cross-sectional distortion, but they are less critical than pipe wall rupture, because they are not associated directly with loss of pressure containment. Therefore, they are often referred to as “serviceability limit states”. Nevertheless, local buckling under certain conditions may be a threat for pipeline integrity under operating conditions, an issue to be discussed later in this section. In geohazard (seismic) design of a structural system, verifying the integrity of the system against ultimate or serviceability limit states requires the calculation of the so-called “engineering demand parameters”. In case of pipelines under geohazard action, the calculation of strain is the “engineering demand parameter” of interest, and several methods for its calculation (numerical and analytical) have been described in

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Table 9.1 Strain calculation for fault crossing; general case of crossing angle. r r

Input

Geometric and material properties of the pipe (D, t, E, σ Y ) maximum resistance of the surrounding soil (qu1 , qu2 , tu ) with qu1 ≥ qu2 (Fig. 9.3, Fig. 9.15B) r axial soil displacement x corresponding to axial soil resistance t (Fig. 9.3B) u u r imposed fault (ground) displacement d F r crossing angle β

1

calculate the soil resistance ratio b = qu1 /qu2

2

calculate the length ratio α = L1 /L2 from Fig. 9.16 using b = qu1 /qu2

3

find parameter F(α) from Fig. 9.17A using the length ratio α = L1 /L2

4

calculate the length ratio αi = Li /L2 from Fig. 9.17B using α = L1 /L2

5

calculate the ground displacement dY for first yielding of the pipe cross section: σ t  D σ  dY Y Y = F (α) D E D qu2

6

compute the characteristic lengths of the deformed pipeline L1 , L2 and Li (EI is the bending stiffness of the pipe considered as a beam):   √ L1 24 dY EI 1/4 and Li = α i L2 , L2 = L1 = α qu1 + qu2 α

7

calculate the maximum bending strain εb and the membrane strain εm (EA is the axial stiffness of the pipe considered as a beam):    π 2D (32 + π 2 ) ˆ2 2 ω εb = dˆ cos β εm = d cos β + dˆ sin β 8(L1 + Li ) 64 ω+1  L tu where dˆ = dF /(L1 + L2 ) and ω = 2 xu EA

8

calculate the maximum tensile strain εT and the maximum compressive strain εC ; compare with the corresponding strain limits εTu and ε Cu (see Section 9.3): ε T = εb + εm ≤ εTu and ε C = εb − εm ≤ εCu

Section 9.2. It should be always remembered that the pipeline may undergo significant deformation during a severe seismic event, and this is associated with some degree of plasticity in the pipe material. Therefore, the value of stress may not be appropriate for assessing the structural integrity of the pipeline. On the other hand, the value of strain offers a more reliable measure of engineering demand and is used almost exclusively in assessing geohazard actions on pipelines.

9.3.1 Tension strain limit Tensile strain capacity is directly related to pipe wall fracture and loss of pressure containment. The level of tensile strain εTu that the pipeline is capable of resisting depends primarily on the ability of the pipeline girth welds to sustain the corresponding

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Table 9.2 Strain calculation for “perpendicular” fault crossing [angle θ = 90ο (β = 0ο )].

Input

r r r r r

geometric and material properties of the pipe (D, t, E, σ Y ) maximum resistance of the surrounding soil (qu , tu ) axial soil displacement xu corresponding to axial soil resistance tu imposed ground displacement dF crossing angle β

1

calculate the ground displacement dY corresponding to first yielding of the pipe:   42 σY t DσY dY = D 5 E D qu

2

compute the characteristic lengths of the deformed S-shape of the pipeline L:   12dY EI 1/4 L=2 qu

3

calculate the maximum bending strain εb and the membrane strain εm :    π 2D ˆ (32 + π 2 ) ˆ2 2 ω εb = d cos β εm = d cos β + dˆ sin β 4L ω+1  64 L tu ˆ where d = dF /L and ω = 2 xu EA

4

calculate the maximum tensile strain εT and the maximum compressive strain εC and compare with the corresponding strain limits εTu and ε Cu (see Section 9.3): εT = εb + εm ≤ εTu and ε C = εb − εm ≤ εCu

deformation. In the present paragraph, the main parameters that determine the level of tension strain limit are briefly discussed. It is noted that the limit values for the maximum tensile strain εTu refer to the “macroscopic” strain calculated from a strain analysis methodology, described in Section 9.2. The value of this macroscopic strain can be very different than the real strain that exists locally at the girth weld toe, which is significantly higher because of local geometry of the weld and the weld notch effects.

9.3.1.1 Girth weld and HAZ In the absence of serious defects or damage in the base material of the line pipes, the tensile strain capacity εTu is controlled mainly by the strain capacity of the pipeline girth welds, including the heat effected zone (HAZ), which are usually the weakest locations along the pipeline against tensile loading. One should always remember that the pipeline under severe longitudinal action is similar to a chain, so that its strength is determined by the strength of the weakest link, which are the girth welds. The weld material should always be “overmatched”, in the sense that it should have higher yield and ultimate strength than those of the adjacent pipe material, in order to avoid strain concentration (localization) at the girth weld when the pipeline is subjected to tension. In those locations though, the strength and the strain capacity of the weld depends not only on the strength and the ductility of the weld material, but also on:

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Figure 9.18 Macrographic picture of girth weld of a 15-mm-thick L485 steel pipe with “high-low” (“hi-lo”) misalignment (photo by A. M. Gresnigt, published with permission).

(a) the type, size and location of defects in the weld, (b) the strength of the heat affected zone (HAZ) of the pipe material, (c) the presence of geometric imperfections, such as misalignment of the adjacent pipes, referred to as “high-low” (or “hi-lo”) imperfection (Fig. 9.18), and (d) the difference in material strength of the adjacent pipes. Weld defects (imperfections) are inevitable in pipeline girth welds but must be limited by defining allowable values (tolerances). One should always bear in mind that girth welds are field welds, constructed on site, in a less controlled environment than the welds produced in the pipe mill. Therefore, constructing on site girth welds of good quality constitutes a challenge, which is essential for pipeline integrity (Fairchild et al., 2008). Furthermore, softening of the HAZ due to heat input during welding may reduce the strength of the weld and should also be taken into account. The designer must ensure that the girth welds of the pipeline have adequate strength and ductility under the high-strain conditions imposed by the strain demand analysis. Towards this purpose, laboratory testing must be performed to ensure sufficient strength and ductility in the weld and the HAZ, and to verify experimentally that the pipeline is able to deform without fracture and to reach the required strain capacity with acceptable reliability. A typical test is the curved wide plate (CWP) tension test (Fig. 9.19). It is an intermediatescale tension test on a non-flattened specimen extracted from a pipeline containing a girth weld. The specimen may contain a circumferential defect (notch) in the weld, introduced in-purpose. The CWP test constitutes an effective alternative to full-scale testing of the entire pipe cross-section and is widely used for weld qualification in the course of pipeline strain-based design (Fairchild et al., 2008; Denys et al., 2013). For more information on weld defect tolerance and its influence on strain-based pipeline design reference is made to standards, such as DNV-ST-F101 (Det Norske Veritas, 2021) and CSA Z662 (Canadian Standard Association, 2019), and to relevant articles published by experts in this topic, e.g., Fairchild et al. (2008; 2016), Verstraete et al. (2012), Hertelé et al. (2015), or Panico et al. (2017). Quite often there exists a notable difference in real strength between the two pipes joined at the girth weld, even if they have the same specified minimum yield strength (SMYS) and the same nominal thickness. This difference in strength is mainly due to the different yield stress in the two pipes and may result in localization of strain on one side of the girth weld (in the weaker pipe), reducing the joint strength, especially in the case of bending, and has been observed in large-scale laboratory experiments on

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Figure 9.19 Curved wide plate (CWP) experiment (schematic).

tubular members (e.g., Van Es, 2016). Therefore, when ordering line pipes, in addition to specifying their minimum yield strength (SMYS), it is strongly suggested to limit the yield strength value of each pipe to a maximum value. Finally, the presence of “hilo” imperfection in the girth weld may further reduce joint strength and deformation capacity (Fig. 9.18).

9.3.1.2

Tension strain limit commonly used in pipeline standards

It should be underlined that the use of regular codes and standards for pipeline fabrication, design and construction (e.g., ASME B31.4, ASME B31.8, EN 1594), does not guarantee that pipelines subjected to large ground-induced deformations can reach a tensile strain capacity of 2% or 3%, often required by strain demand analysis. The primary focus of these standards is pipeline design for pressure containment, under normal operating conditions (“pressure design”) and this was presented in detail in Chapter 7 of the present book. In “pressure design”, the longitudinal stresses and strains developed should be within the elastic regime of pipe material. Using the provisions of the above standards for performing girth welds, the maximum longitudinal strain allowed is equal to 0.5%, as noted in API 1104 (American Petroleum Institute, 2021), which is often considered as the “elastic” (or better, “quasi-elastic”) strain limit for geohazard design. However, limiting the value of acting longitudinal strain below 0.5% in pipelines subjected to severe ground-induced actions may be a difficult, if not impossible, target to achieve. Values of strain capacity higher than 0.5% can be reached in welded pipelines, but further requirements are necessary to qualify the pipeline and its girth welds for such a high-strain demand. In other words, the above mentioned codes and standards for pressure design establish minimum safety conditions, whereas in areas where significant ground-induced deformations are expected, it is necessary to specify

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special requirements, additional to the regular standards and specifications, in order to qualify the pipeline in terms of high-strain conditions. This imposes several additional requirements for the resistance of girth welds. Adopting a 2% or 3% strain limit for the strength verification of the pipeline in geohazard areas constitutes a real challenge. These are high values of strain, more than 10 times higher than the nominal yield strain εY of the pipe material under uniaxial conditions (εY = σY /E). As an example, for a 12-meter-long pipe, a 3% tensile strain implies elongation of 36 cm (14.2 in)! Useful information on the tensile limit of girth welds subjected to high-strain conditions exists in offshore pipeline standards, in conjunction with the reeling laying method (see Chapter 2), which is often associated with longitudinal strains above 2%. Offshore pipeline standards, such as the DNV-ST-F101 standard for submarine pipeline systems specify rules for determining allowable weld defects in relation to desired strain capacity. In all cases, an Engineering Criticality Assessment (ECA) procedure should be followed, using appropriate laboratory testing. As a final comment, the strain capacity value of 3% stated in several standards and guidelines is a very high value, and should be used only after proper qualification of the girth welds.

9.3.1.3

Requirements and recommendations for high-strain tension limit

To enable the pipeline to sustain high levels of tensile strain, the following requirements and recommendations should be considered for the line pipe material and for the girth welds: 1. The yield-to-tensile strength ratio (Y/T) of the line pipe material should be less than a specific value. In most design documents available, the value of this limit is no greater than about 0.92. 2. The weld metal must be sufficiently “overmatched”, for both yield and tensile strength, so that the weld metal is stronger than the base metal. “Overmatching” aims at preventing strain localization in the weld zone under severe tensile action. 3. To achieve weld overmatching, the real properties of the pipe material and the weld, including HAZ softening, should be considered, as determined by appropriate laboratory testing. Using the nominal values of the material properties may not guarantee overmatching of the constructed welded joint. 4. There is a statistic distribution for both base metal and weld metal strength, and this should be taken into account when selecting the welding consumables for the pipes under consideration. 5. The acceptance criteria for weld imperfections should be established considering the detection capacity of the non-destructive evaluation (NDE) technique to be applied. In high strain applications, such as pipelines in geohazard areas, the use of NDE techniques capable of detecting surface defects (e.g., Automated Ultrasonic Testing) is necessary. 6. The tensile strain capacity of girth welds and the corresponding HAZ should be validated by a suitable Engineering Criticality Assessment (ECA) procedure, which determines the critical weld defect size, in terms of its length and depth. These procedures involve appropriate laboratory testing, e.g., CTOD testing, Curved Wide Plate (CWP) testing.

To achieve weld “overmatching” and adequate deformation capacity, the above requirements should be addressed in the Welding Procedure Specifications (WPS) and

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the Weld Procedure Qualification Records (WPQR) of the pipeline project, as well as in the qualification procedure of the welders.

9.3.2 Compression strain limit Local buckling of the pipe wall is a “structural” limit state of the pipeline. It is characterized by the development of short-wave wrinkles in the pipe wall due to excessive compression, stemming either from uniform axial compression or from longitudinal bending. The reader is referred to Sections 6.2, 6.4 and 6.5 for the necessary background on this topic. Local buckling does not necessarily lead to direct loss of pressure containment, and therefore, there is a debate on whether it constitutes an ultimate limit state or a serviceability limit state. In any case, it has been widely accepted that the development of local buckling constitutes an important limit state, influencing both the operability and the structural integrity of the pipeline, and should be given appropriate attention in the course of pipeline structural design subjected to geohazard action. The shape and the amplitude of the wrinkles in the pipe wall are strongly influenced by the diameter-to-thickness ratio (D/t) and the level of internal pressure in the pipeline. A large number of experimental and numerical results are available on this subject and have contributed towards establishing strain limit values for the initiation (onset) of local buckling, often referred to as critical strains. It has been widely recognized that the available experimental results show a large scatter in terms of the observed critical compressive strain. Recently, in the COMBITUBE project (Bijlaard et al., 2014; Van Es et al., 2016; Vasilikis et al., 2016) various factors that are responsible for this scatter have been investigated. Based on a large number of experimental results, the formula expressed by Eqs. (9.18) and (9.19) below predicts the axial compressive strain εCu needed for the onset of local buckling in the presence of internal pressure:   2 t r − 0.0025 + 3000 p |p|, r Et   2 t r = 0.10  + 3000 p |p| r Et

εCu = 0.25

if

r ≤ 60 t

(9.18)

εCu

if

r > 60 t

(9.19)

where r is the radius of the deformed cross section e.g., due to combination of bending and earth loads in a buried pipe, and can be estimated by the following formula (see also Fig. 9.20): r = r

1 1 − 3w/r ¯

(9.20)

For simplicity in the calculations, r may be taken as r (i.e., the initial radius of the pipe cross-section). Eq. (9.20) was introduced by Gresnigt (1986) and has been adopted by NEN 3650 standard and, in a slightly different form, by the CSA Z662 standard.

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Figure 9.20 Ovalization and local hoop radius of pipe cross-section.

(A)

(B)

Figure 9.21 Local buckle configuration of a non-pressurized 24-inch-diameter pipe subjected to bending (D/t =72, yield stress 313 MPa): (A) buckled shape from experimental testing (photo by A. M. Gresnigt, published with permission), (B) finite element simulation of local buckling in this pipe.

Analytical expressions for the calculation of the ovalization in case of bending with other loads, such as earth loads, may be found in the NEN 3651 standard (Nederlands Normalisatie Instituut, 2019) and in the paper by Gresnigt et al. (2017). Local buckling is associated with the formation of pipe wall wrinkles, as shown in Fig. 9.21. The wrinkles may grow in major or minor folds, associated with significant

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(A)

315

(B)

Figure 9.22 Fracture developed at locally buckled X52 steel 6-inch-diameter pipe, subjected to cyclic bending: (A) side view of buckled area, (B) fatigue crack at the ridge of the buckle (photos by S. A. Karamanos).

plastic deformation, and may render the pipeline unfit for use. In addition, the occurrence of local buckling causes a reduction of the bending moment capacity of the pipe cross-section, leading to localization of bending curvature. In internally-pressurized pipelines, upon local buckling formation, further increase of bending loading results in an increase of local tensile strains at the tension side of the buckled cross section of the pipeline, which may lead to pipe wall fracture, as noticed in Section 6.5.

9.3.2.1

On the severity of local buckles

The presence of local buckles does not necessarily lead to direct loss of pressure containment. In many instances, local buckles do not impose an immediate threat for pipeline structural integrity, and the pipeline may be operational for a certain period. Burst tests on buckled pipes have indicated that, unless the buckled shape is excessively folded and plastically deformed, the burst pressure of the buckled pipe is quite high, comparable with the burst of an intact pipe as reported in the relevant tests performed by Van Foeken (1994). In some of these burst tests, the location of rupture occurred away from the buckled area. Overall, those tests have shown a good performance of buckled pipes under internal pressure, indicating that the presence of buckles may not be an immediate threat for the structural integrity of the pipe against burst. Therefore, upon detection of local buckling in a pipeline, the repair of the buckled pipeline may be scheduled for a later time, considering the priorities and the resources of the pipeline operator. However, operating a buckled pipeline requires the analysis of pressure or temperature fluctuations to assess the possibility of fracture at the buckled area (Dama et al., 2007; Pournara, 2015; Pournara et al., 2015), because of low-cycle fatigue. Fig. 9.22A shows the buckled shape of a 6-inch nominal diameter X52 steel pipe with pipe wall thickness 2.68 mm, which was subjected to cyclic bending after local buckling occurred (Pournara et al., 2015). Fracture occurred after approximately 600

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loading cycles at the ridge of the buckle (Fig. 9.22B). Towards assessing buckled pipeline integrity and estimating its design life, a fitness-for-service analysis (also called fitness-for-purpose) should be carried out, analyzing the strain variations because of load variations (mainly from internal pressure and temperature), and evaluating them with low-cycle fatigue design concepts. If necessary, the operating pressure of a buckled pipeline may be lowered in order to meet an appropriate low-cycle fatigue criterion and a suitable time interval is allowed until repair. A detailed presentation of fitness-for-service analyses in buckled pipes is beyond the scope of the present discussion, and the interested reader is referred to the papers by Dama et al. (2007) and Pournara et al. (2015).

9.3.3 Ovalization of the cross section The pipeline should not exhibit excessive cross-sectional ovalization or distortion, so that inspection and other measuring instruments travelling within the pipeline can pass through for maintenance purposes. This type of cross-sectional deformation is often associated with plastic deformation of the pipe wall. However, excessive elastic deformations may also violate a serviceability limit state. The ultimate limit value for the smallest diameter of the ovalized cross-section Dmin should satisfy the requirement Dmin ≥ 0.85D (NEN 3650), which means that the minimum diameter Dmin should be greater than 85% of the original pipe diameter (Fig. 9.20). In relatively thin-walled pipes, used mainly for water transportation, exceeding an ovalization limit may impose a risk of collapse due to pipe wall inversion, and relevant limits of cross-sectional deformation are discussed in Chapter 10. It is noted though that for the serviceability limit state, most likely not associated with severe ground-induced actions, a much lower permissible value for the ovalization should be taken, which depends on the requirements set by the passage of measuring and (intelligent) in-line inspection tools, referred to as PIGs (Pipeline Inspection Gauges). This is usually agreed between the operator of the pipeline system and the company performing the in-line inspection. A rule of thumb, used in several pipeline standards (e.g., NEN 3650), specifies an allowable ovalization under normal operational conditions of approximately 6%. However, this value should be the decision of the pipeline operator.

9.4

Outline of mitigation measures

If the verification analysis of pipeline resistance shows that the pipeline is not capable of sustaining the ground-induced strains, then mitigation measures should be applied. There are several options, and the selection of a particular approach depends on pipeline location, expected failure mode, potential for collateral damage, risk acceptance philosophy, and the estimated mitigation costs. The first and most obvious measure that should be examined by the designer is the re-alignment of pipeline routing in order to avoid a specific geohazard area. However,

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in many instances, this option may not be feasible and therefore, other mitigation measures should be employed to improve pipeline performance and minimize the risk of pipeline failure because of the expected ground-induced actions. These measures are categorized into the following three categories: (1) reduction of ground-induced action on the pipeline, (2) increase of pipeline ability to resist ground-induced strains, and (3) modification and improvement of post-earthquake assessment procedures for addressing the consequences of pipeline damage. A more detailed treatment of those issues is offered in the GIPIPE guidelines (Karamanos et al., 2021), as well as in the PRCI (2017) guidelines. On the other hand, one should keep in mind that the list of possible mitigation measures in those recommendations may not constitute a cookbook. Mitigation measures should be considered as part of pipeline risk analysis and be evaluated for each particular situation, depending primarily on pipeline location, the expected failure mode and its consequences, as well as the estimated mitigation costs.

References American Lifelines Alliance (2001). ALA Guidelines for the Design of Buried Steel Pipe (with addenda through February 2005). ASCE-FEMA. American Petroleum Institute (2021). Welding of Pipelines and Related Facilities. API 1104 Standard. Washington, DC: American Society of Mechanical Engineers (2018). Gas Transmission and Distribution Piping Systems. ASME B31.8 Standard. American Society of Mechanical Engineers (2019). Pipeline Transportation Systems for Liquid Hydrocarbons and Other Liquids. ASME B31.4 Standard. Bathe, K. J., & Almeida, C. A. (1982). A Simple and Effective Pipe Elbow Element—Interaction Effects. Journal of Applied Mechanics, 49(1), 165–171. Bijlaard, F. S. K., Gresnigt, A. M., Van Es, S. H. J., et al. (2014). Bending Resistance of Steel Tubes in CombiWalls (COMBITUBE), Final Report. RFSR-CT-2011-00034. Brussels, Belgium: Research Fund for Coal and Steel (RFCS), European Commission. Canadian Standard Association (2019). Oil and Gas Pipeline Systems. CSA-Z662 Standard. Mississauga, ON, Canada. Dama, E., Karamanos, S. A., & Gresnigt, A. M. (2007). Failure of Locally Buckled Pipelines. J Press Vess-T ASME, 129(2), 272–279. Davis, C.A. (2008). Assessing geotechnical earthquake hazards for water lifeline systems with uniform confidence. In Proceedings of ASCE Geotechnical Earthquake Engineering and Soil Dynamics IV, Sacramento, CA, paper 4291. Denys, R. M., Hertele, S., & Lefevre, A. A. (2013). Use of curved-wide-plate (CWP) data for the prediction of girth-weld integrity. The Journal of Pipeline Engineering, 12(3), 245–257. Det Norske Veritas (2021). Submarine Pipeline Systems. STANDARD DNV-ST-F101. Høvik, Norway. European Committee for Standardization (2013). Gas Supply Systems Pipelines for Maximum Operating Pressure Over 16 bar. EN 1594 Standard. Brussels, Belgium. Fairchild, D. P., Cheng, W., Ford, S. J., Minnaar, K., Biery, N. E., Kumar, A., et al. (2008). Recent advances in curved wide plate testing and implications for strain-based design. International Journal of Offshore and Polar Engineering, 18(3), 161–170.

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Fairchild, D. P., Panico, M., Crapps, J. M., Cheng, W., Tang, H., & Shafrova, S. (2016). FullScale Pipe Strain Test Quality and Safety Factor Determination for Strain-Based Engineering Critical Assessment. In Proceedings of the 11th International Pipeline Conference IPC2016. Calgary, Alberta, Canada. September 26-30. Gresnigt, A. M. (1986). Plastic Design of Buried Steel Pipes in Settlement Areas. HERON, 31(4), 1–113. Gresnigt, A. M., Van Es, S. H. J., Karamanos, S. A., & Vasilikis, D. (2017). New design rules for tubes in combined walls in EN 1993-5. In Proceedings of EUROSTEEL. Copenhagen, Denmark. September 13–15. Hertelé, S., Denys, R., Horn, A., Van Minnebruggen, K., & De Waele, W. (2015). Framework for Key Influences on Tensile Strain Capacity of Flawed Girth Welds. Journal of Pressure Vessel Technology, Transactions of the ASME, 137(4) Art. no. 041409. Karamanos, S. A., Gresnigt, A. M., & Dijkstra, G. J. (2021). Geohazards and Pipelines. Springer. Karamanos, S. A., & Tassoulas, J. L. (1996). Tubular Members I: stability Analysis and Preliminary Results. Journal of Engineering Mechanics, ASCE, 122(1), 64–71. Karamitros, D. K., Bouckovalas, G. D., & Kouretzis, G. P. (2007). Stress analysis of buried steel pipelines at strike-slip fault crossings. Soil Dynamics and Earthquake Engineering, 27, 200–211. Kennedy, R. P., Chow, A. W., & Williamson, R. A. (1977). Fault movement effects on buried oil pipeline. Journal of Transportation Engineering, ASCE, 103, 617–633. Nederlands Normalisatie Instituut (2020). Requirements For Pipeline Systems. NEN 3650, Part-1: General and Part-2: Steel Pipelines. Delft, The Netherlands. Nederlands Normalisatie Instituut (2019). Requirements For Pipeline Systems. NEN 3651: Additional requirements For Pipelines in Or Nearby Important Public Works. Delft, The Netherlands. Newmark, N. M., & Hall, W. J. (1975). Pipeline design to resist large fault displacement. In Proceedings of U.S. National Conference on Earthquake Engineering (pp. 416–425). O’Rourke, M. J., & Liu, X. (2012). Seismic Design of Buried and Offshore Pipelines. MCEER Monograph, MCEER-12-MN04, Buffalo, NY. Panico, M., Macia, M. L., Fairchild, D. P., & Wentao, C. (2017). A Case Study of Design and Integrity Management Framework for Strain-Based Pipelines. In Proceedings of the 27th International Ocean and Polar Engineering Conference. San Francisco, CA, USA. June 25-30. Pipeline Research Council International (PRCI) (2017). Pipeline Seismic Design and Assessment Guideline. In D. G. Honegger (Ed.), Contract No. PR-268-134501-R01, Technical Committee of Design, Materials and Construction. Houston, TX. Pournara, A. E. (2015). Structural Integrity of Steel Oil & Gas Pipelines With Local Wall distortions, PhD Dissertation. Department of Mechanical Engineering, University of Thessaly. Volos, Greece. Pournara, A. E., Papatheocharis, T., Karamanos, S. A., & Perdikaris, P. C. (2015). Structural integrity of buckled steel pipes. In Proceedings of ASME 34th International Conference on Ocean, Offshore and Arctic Engineering. OMAE 2015 May 31–June 5. Saiyar, M., Ni, P., Take, W. A., & Moore, I. D. (2016). Response of pipelines of differing flexural stiffness to normal faulting. Géotechnique, 66(4), 275–286. Sarvanis, G. C., & Karamanos, S. A. (2016). Analytical Methodologies for Buried Pipeline Design in Geohazard Areas. In Pressure Vessels and Piping Conference. ASME 2016 July 17–21.

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Sarvanis, G. C., Ferino, J., Karamanos, S. A., Vazouras, P., Dakoulas, P., Mecozzi, E., et al. (2016). Soil-pipe interaction models for simulating the mechanical response of buried steel pipelines crossing active faults. In Proceedings of the International Society of Offshore and Polar Engineers, ISOPE2016. Rhodes, Greece. Sarvanis, G. C., & Karamanos, S. A. (2017). Analytical Model for the Strain Analysis of Continuous Buried Pipelines in Geohazard Areas. Engineering Structures, 152, 57–69. Sarvanis, G. C., Karamanos, S. A., Vazouras, P., Mecozzi, E., Lucci, A., & Dakoulas, P. (2018). Permanent earthquake-induced actions in buried pipelines: numerical modeling and experimental verification. Earthquake Engineering and Structural Dynamics, 47(4), 966– 987. Takada, S., Hassani, N., & Fukuda, K. (2001). A new proposal for simplified design of buried steel pipes crossing active faults. Earthquake Engineering and Structural Dynamics, 30, 1243–1257. Trifonov, O. V., & Cherniy, V. P. (2010). A semi-analytical approach to a nonlinear stress–strain analysis of buried steel pipelines crossing active faults. Soil Dynamics and Earthquake Engineering, 30, 1298–1308. Tsatsis, A. (2017). Experimental and Numerical Simulation of Buried Pipelines Subjected to Large Permanent Ground displacements, Fault Rupture and landslides. PhD dissertation. School of Civil Engineering. National Technical University of Athens, Greece. Tsatsis, A., Loli, M., & Gazetas, G. (2019). Pipeline in dense sand subjected to tectonic deformation from normal or reverse faulting. Soil Dynamics and Earthquake Engineering, 127, 105780. Van Es, S. H. J. (2016). Inelastic Local Buckling of Tubes For Combined Walls and pipelines. PhD Dissertation. Delft University of Technology. Van Es, S. H. J., & Gresnigt, A. M. (2016). Experimental and numerical investigation into the behavior of buried steel pipelines under strike-slip fault movement. In Proceedings of the 11th International Pipeline Conference, IPC2016. Calgary, Alberta, Canada. Van Es, S. H. J., Gresnigt, A. M., Vasilikis, D., & Karamanos, S. A. (2016). Ultimate Bending Capacity of Spiral-Welded Steel Tubes - Part I: experiments. Thin Walled Structures, 102, 286–304. Van Foeken, R. J. (1994). Effect of buckling deformation on the burst pressure of pipes. In Report 94-CON-R1008/FNR, TNO Building and Construction Research. Rijswijk, The Netherlands. Vasilikis, D., Karamanos, S. A., van Es, S. H. J., & Gresnigt, A. M. (2016). Ultimate Bending Capacity of Spiral-Welded Steel Tubes - Part II: predictions. Thin Walled Structures, 102, 305–319. Vazouras, P., Karamanos, S. A., & Dakoulas, P. (2010). Finite element analysis of buried steel pipelines under strike-slip fault displacements. Soil Dynamics and Earthquake Engineering, 30(11), 1361–1376. Vazouras, P., Karamanos, S. A., & Dakoulas, P. (2012). Mechanical behavior of buried steel pipes crossing active strike-slip faults. Soil Dynamics and Earthquake Engineering, 41, 164– 180. Vazouras, P., Sarvanis, G., Karamanos, S. A., et al. (2015a). Safety of Buried Steel Pipelines Under Ground Induced Deformations (GIPIPE). Final Report, RFSR-CT-2011-00027. Brussels, Belgium: Research Fund for Coal and Steel (RFCS), European Commission. Vazouras, P., Dakoulas, P., & Karamanos, S. A. (2015b). Pipe–soil interaction and pipeline performance under strike–slip fault movements. Soil Dynamics and Earthquake Engineering, 72, 48–65.

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Verstraete, M., De Waele, W., Denys, R., & Hertelé, S. (2012). Comparison of pipeline girth weld defect acceptance at the onset of yielding according to CSA Z662 and EPRG guidelines. In International Pipeline Conference IPC2012-90593, Calgary, AL, Canada. Wang, L. R. L., & Yeh, Y. A. (1985). A refined seismic analysis and design of buried pipeline for fault movement. Earthquake Engineering and Structural Dynamics, 13, 75–96. Xie, X., Symans, M. D., O’Rourke, M. J., Abdoun, T. H., O’Rourke, T. D., Palmer, M. C., et al. (2013). Numerical modeling of buried HDPE pipelines subjected to normal faulting: a case study. Earthquake Spectra, 29(2), 609–632.

Part IV Special Topics 10. Buried steel water pipeline design 323 11. Global buckling of pipelines 355 12. Mechanically lined pipes 375

Buried steel water pipeline design

10

In North America, large diameter steel water pipelines are used for transmitting water resources. Major transmission pipelines have a diameter that may range between 48 and 144 inches, with a value of diameter-to-thickness (D/t) ratio, which very often exceeds 200 (Fig. 10.1). Steel pipes with D/t values ranging between 200 and 240 are very common in water transmission pipeline applications, whereas pipes with D/t ratio less than 130 are considered as “thick-walled” by the relevant steel pipe industry. In comparison with hydrocarbon pipelines, these pipes are thin-walled, with very flexible cross-section in terms of ring-type deformation. The mechanical interaction between the buried pipe and the soil, is very important in determining the state of stress in the pipeline and ensuring its structural integrity. Furthermore, because of the high value of D/t ratio, the pipe wall is susceptible to local buckling under compressive action. Field joining of those large-diameter thin-walled steel pipes is also an issue with some unique features and requires special attention. In the first part of this Chapter, some aspects of large-diameter steel pipe fabrication and construction are also outlined, pinpointing differences with hydrocarbon pipelines. Subsequently, the structural design of large diameter steel water pipes is presented and analyzed, according to AWWA Manual M11 (American Water Works Association, 2017a). Finally, the structural performance of field lap-welded joints under normal and extreme loading conditions is discussed, and a new concept for improving their resilience is presented.

10.1

Manufacturing of large diameter pipes and in-situ construction

The production of large-diameter steel water pipes is performed according to the AWWA C200 standard (American Water Works Association, 2017b). They are manufactured primarily with the spiral-welded pipe method, which has been presented in detail in Chapter 2 of the present book. The spiral-weld process is continuous, and the continuous cylinder produced by spiral welding is cut with plasma cutting at specific intervals, depending on the required length. This pipe segment is the primary component of pipe production and is called “pipe joint” or “stick” (Fig. 10.2). Normally, the length of a pipe stick is 50 ft, but quite often, depending on the project requirements, 1/2-joints or 1/4-joints are produced, which are pipe segments with length equal to one-half or one quarter of the original length respectively. After cutting, each pipe segment is hydrotested in the pipe mill up to a pressure level of 75% of nominal yield pressure, as required by AWWA C200, or at the level specified by the purchaser (Fig. 10.3). If lap welded joints are required for the pipeline, Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00017-7 c 2023 Elsevier Inc. All rights reserved. Copyright 

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(A)

(B)

Figure 10.1 Large-diameter steel pipes for water transmission: (A) transportation, (B) at the construction site; the pipes are equipped with stulls, which prevent cross-sectional distortion of the flexible pipe (courtesy of photos: Northwest Pipe Company).

each pipe is expanded in one end to form a bell (Fig. 10.4), so that the end part of the adjacent pipe is inserted and welded. In the case of rolled-groove rubber gasket joints, one end of the pipe is expanded (Fig. 10.5A) and the other end is roll-shaped, as shown in Fig. 10.5B and Fig. 10.5C. At the final stage of production in the pipe mill, the pipe is coated, internally and externally. In most cases, polyurethane is used for external coating. It is a spray-applied bonded dielectric coating (Fig. 10.6), constructed according to AWWA C222 standard (American Water Works Association, 2018b), and provides resistance properties against water and chemical action. Polyurethane coating may also be used

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(A)

325

(B)

Figure 10.2 Large-diameter steel pipe “joint” (“stick”): (A) in the pipe mill just after cutting, (B) after hydrotesting (courtesy of photos: Northwest Pipe Company).

Figure 10.3 Hydrotesting of spiral welded pipe (photo courtesy: Northwest Pipe Company).

Figure 10.4 Expanded end of pipe for a bell and spigot welded joint (photo by S. A. Karamanos).

internally, but most often cement mortal lining is used for internal coating in buried pipelines, which passivates the steel material preventing corrosion. Cement mortar lining is applied in the pipe mill according to AWWA C205 standard (American Water Works Association, 2018a), by spinning the pipe and centrifugally casting the cement

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(A)

(C)

(B)

Figure 10.5 Pipe ends for rolled-groove gasket joint: (A) bell end; (B), (C) spigot end (courtesy of photos: Northwest Pipe Company).

Figure 10.6 Spraying of pipe for external coating (photo courtesy: Northwest Pipe Company).

to the pipe interior wall (Fig. 10.7). The thickness of cement mortar lining is usually 1/2 inch but it depends on the requirements of the project. In pipes of large diameter and at joint locations, this lining process is field applied using special equipment. Fig. 10.8A shows a pipe bend (elbow) used in steel water pipelines. It is a mitered bend, composed by a series of butt-welded straight segments (Fig. 10.8B), fabricated in the pipe mill in accordance with the AWWA C208 standard (American Water Works Association, 2017c). The bend angle can vary depending on the pipeline alignment. Induction bends, or cold bends are not used in thin-walled large-diameter pipes. Elbows are fittings fabricated in the pipe mill. Another common fitting is the pipe outlet shown in Fig. 10.9. Outlets can be either single or double and, should be reinforced with ring plates that increase the main pipe thickness around the opening. The amount of

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327

(A)

(B)

Figure 10.7 Cement mortar lining: (A) pipe spinning equipment, (B) centrifuge casting of cement mortar lining (courtesy of photos: Northwest Pipe Company).

(A)

(B)

Figure 10.8 (A) Mitered pipe bend for large diameter water pipelines (photo by S. A. Karamanos); (B) welding of mitered pipe bend parts (photo courtesy: Northwest Pipe Company).

reinforcement depends on the magnitude of the so-called “pressure-diameter value” (PDV), described in detailed in Chapter 7 of AWWA M11. When the value of PDV exceeds a certain limit, crotch-plate reinforcement should be used to stiffen the pipe junction and prevent its excessive ovalization. Wye branches, simply called “wyes”, are also fabricated in the pipe mill (Fig. 10.10). They are used very often in water pipeline systems and are also reinforced with crotch plates (Fig. 10.11). Upon fabrication, hydrotesting and coating in the pipe mill, the pipes and the fittings are transported to the construction site (Fig. 10.12) and are joined to construct the pipeline (Fig. 10.13 and Fig. 10.14). In large-diameter water pipelines, as opposed to hydrocarbon pipelines, the pipes are lowered and joined (welded) within the trench (Fig. 10.15). Steel pipes can be joined by many different rigid or flexible configurations, leading to segmental or welded pipelines respectively. More information on welded joints is provided in Section 10.5. In some cases, special devices, referred to as

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Figure 10.9 Single pipe outlet without crotch plates (photo by S. A. Karamanos).

(A)

(B)

Figure 10.10 (A) Large-diameter wye branch reinforced with crotch plates at the pipe mill (photo by S. A. Karamanos); (B) wye branch at the construction site (photo courtesy: Northwest Pipe Company).

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329

Figure 10.11 Wye branch for large diameter pipe (photo courtesy: Northwest Pipe Company).

(A)

(B)

Figure 10.12 (A) Coated pipes ready for transportation to the construction site. (B) Pipe delivery on site (courtesy of photos: Northwest Pipe Company).

(A)

(B)

Figure 10.13 Lowering into the trench: (A) straight pipe, (B) pipe bend (courtesy of photos: Northwest Pipe Company).

“couplings” are used, which are aimed at allowing the rotation between two adjacent pipes and accommodate differential ground motion (e.g., soil subsidence). There exist several parameters that influence the height of soil cover on top of a buried pipe (Fig. 10.16). From the point-of-view of structural design, the soil on

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Figure 10.14 Large diameter mitered bend already installed in the trench (photo courtesy: Northwest Pipe Company).

(A)

(B)

Figure 10.15 Pipeline welding in the trench: (A) alignment of pipes, (B) lap-welding construction (courtesy of photos: Northwest Pipe Company).

top of a buried pipe induces a compressive stress to the pipe cross-section, and this should be less than the allowable strength of steel. Usually, this requirement may not be difficult to meet. On the other hand, controlling ring deflection as a small percent of pipe diameter is an important design requirement that depends on the type of coating, and this requirement is affected by the height of soil cover and more information is offered in Section 10.3. Furthermore, the minimum height of soil cover also depends on frost conditions, flotation of the pipe (during soil liquefaction), and ring deformation due to surface (live) loads.

10.2

Wall thickness determination

AWWA design manual M11 (American Water Works Association, 2017a) is the main design document for designing buried steel water pipelines and their components. Its

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331

Figure 10.16 Pipeline installation in the trench (photo courtesy: Northwest Pipe Company).

latest edition has been published in 2017. In particular, Chapter 4 of AWWA M11 refers to determining pipe wall thickness, presented briefly in the following. The calculation is based on the classical Barlow’s formula, written as follows: t=

pD 2σW

(10.1)

where: r r r r

t is the pipe wall thickness for the internal pressure, p is the internal pressure, D is the outside diameter of steel pipe cylinder (not including coatings), σ W is the allowable design stress, which is equal to 50% of yield stress σ Y for working pressure conditions, and 75% of yield stress for transient pressure or field test conditions.

The pipe wall thickness should also conform with the following provisions (depending on the pipe size), which stem from pipe handling requirements, r

For pipe diameter less than 54 in (1,372 mm): t=

D 288

(10.2)

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(A)

(B)

Figure 10.17 Schematic representation of ring deflection mechanism: (A) physical problem, (B) ring model with soil springs. r

For pipe diameter equal to or larger than 54 in (1,372 mm): t=

r

D + 20 400

(10.3)

For steel pipe with cement-mortar lining and flexible coating: t=

D 240

(10.4)

In Eqs. (10.2), (10.3) and (10.4), D and t should be expressed in inches.

10.3

Ring deflection

The cross-sectional deflection of buried steel pipes due to earth load and to surface (live) load is an important feature of their mechanical behavior. The problem can be approximated as two-dimensional, under plane strain conditions, and is shown schematically in Fig. 10.17. The vertical force W per unit length of pipe from soil cover and from live load causes ring deflection x and ovalization, which is resisted from soil resistance at both sides of the ring. This problem was analyzed in Section 3.4 of this book, leading to the closed-form expression for ring deflection x, written below with the new notation: x =

0.14 W r3 EIr + 0.071 kH r3

(10.5)

In Eq. (10.5), the vertical load W per unit length of pipe is considered as concentrated at the top of the ring, kH is the lateral stiffness of the surrounding soil, also considered

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333

per unit length, and EIr is the ring-type bending resistance of pipe cross-section (Ir = t 3 /12). To predict ring deflection of unpressurized pipe buried in soil, AWWA M11 adopts the well-known Iowa formula. This formula was first proposed by Spangler (1941), later modified by Watkins and Spangler (1958) and subsequently, has been presented in various forms (Watkins and Anderson, 2000). In AWWA M11, the following form of modified Iowa formula is adopted for vertical deflection x (often referred to as “modified Iowa formula”): x = Dl

Kˆ W rm3 EIr + 0.061E  rm3

(10.6)

In the above equation, W is the total load per unit of pipe length, considering earth and surface (live) loading, Dl is the deflection lag factor usually taken equal to 1, Kˆ is the bedding constant equal to 0.1, rm is the mean radius, EIr is the total bending resistance of the pipe ring including external coating and liner: EIr = (EIr )steel + (EIr )liner + (EIr )coating

(10.7)

and E is the modulus of soil under lateral confinement conditions, discussed below in more detail. Eq. (10.6) provides an estimate of long-term horizontal deflection of unpressurized pipe subjected to earth and live loads. The similarity of Eq. (10.5) with the modified Iowa formula in Eq. (10.6) is very obvious. One must realize that the formula in qualitative terms can be expressed as follows: ring deflection

=

vertical load (ring stiffness) + (soil resistance)

(10.8)

The modulus of soil reaction E is an important variable in the modified Iowa formula. Despite its simplicity, it is an efficient parameter to represent soil stiffness in the pipe-soil system. Values for E in the modified Iowa formula are proposed within AWWA M11 (Table 5-3 of the M11 manual), and have proven to be very reliable over time. Simple calculations for typical buried pipes show that ring deflection is controlled primarily by the soil resistance rather than by pipe stiffness. As an example, consider a steel pipe with D/t = 150, and E = 17 MPa, one readily obtains that the soil resistance term is about 20 times greater than the ring stiffness term. The situation becomes more in favor of soil resistance in the case of thinner pipes with D/t = 200 or greater. In most practical applications, the soil contributes to 90–99% of total resistance to deflection. This means that good compaction of backfill material is the primary means of controlling ring deflection of the pipe rather than the increase pipe wall thickness. The following deflection limits for steel pipes should be considered: r

x ≤ 0.05D for flexible lining and coating such as liquid epoxy linings and coatings, polyurethane linings and coatings, or tape coatings; r x ≤ 0.03D for cement-mortar lining and flexible coatings; r x ≤ 0.02D for cement-mortar coating.

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Structural Mechanics and Design of Metal Pipes

Using those deflection limits, the modified Iowa formula can be written in the following form, in terms of E , and provides the required value of E to achieve the above limits:   CW EI E  = 16.4 − 3 (10.9) rm rm where C is the factor to account for predicted pipe deflection limit, W is the load per unit of pipe length, and rm is the mean radius of the pipe. Therefore, C = 1, for flexible lining and coating; C = 1.67, for cement-mortar lining and flexible coating; C = 2.5, for cement-mortar coating.

The load per unit of pipe length W should include the “soil prism” load above the pipe and the live load (e.g., truck wheel load) on the ground surface. The latter should be calculated at the depth of interest using Boussinesq equations, as presented in Handy and Spangler (2007) or in Chapter 5 of AWWA M11. In internally pressurized pipes, the presence of internal pressure tends to re-round the pipe, has a beneficial effect on pipe resistance, and this effect is extensively described in the book by Watkins and Anderson (2000).

10.4

External pressure collapse design

The pipes under consideration are thin-walled, significantly “thinner” than the ones used for onshore hydrocarbon applications. Therefore, in addition to internal pressure design, large-diameter water pipelines should be also designed against external pressure. Two cases are distinguished: (a) laterally unrestrained (unconfined) conditions; (b) laterally confined conditions. The former case refers to aboveground pipes, and the latter to buried pipes, which are confined by the surrounding soil. (a) Unconfined conditions

In aboveground pipelines, external pressure is due to vacuum conditions, and pipe design is based on Timoshenko Equation which provides an estimate of the ultimate (critical) pressure of the pipe, based on first yielding of the pipe material, denoted as pcr,F . Timoshenko’s formula is written below in a form, compatible with AWWA M11 notation, and requires the solution of a quadratic algebraic equation for calculating pcr,F , expressed as follows: 

pcr,F

2

− [σY /m + (1 + 6mx)pe ]pcr,F + σY pe /m = 0

where: r r r

pcr,F is the critical collapse pressure corresponding to first yielding of pipe material, σ Y is the specified minimum yield strength (SMYS) of pipe material, m is the radius-to-thickness ratio of the pipe (r/ta ),

(10.10)

Buried steel water pipeline design r r

335

r is the outside radius of pipe, ta is the adjusted pipe wall thickness, which accounts for the thickness of lining and coating, defined as follows: ta = t + (tL + tC )/(E/EC )

(10.11)

r r r r r

t is the steel pipe thickness, tL is the cement-mortar lining thickness, tC is the cement-mortar coating thickness, x is the percent deflection of conduit in decimal form (ellipticity), pe is the elastic buckling (collapse) pressure of a perfectly-round elastic tube, defined in Eq. (10.12), r E is the Young’s modulus of steel, r E is the Young’s modulus of cement mortar (E/E =7.5). C C

The elastic collapse pressure pe of a perfect round pipe is computed from the wellknown formula (Section 3.2), adding the contribution of external coating and lining: pe =

 3  3 2E  t 3 2EC 2EC tL tC + + 2 2 2 1 − νe D 1 − νC DI 1 − νC DC

(10.12)

where: r r r r r

D is the outside diameter of steel cylinder, DI is the inside diameter of steel cylinder (cement-mortar lining outside diameter), DC is the outside diameter of cement mortar coating, ν e is Poisson’s ratio for steel, taken as 0.30, ν C is Poisson’s ratio for cement mortar, taken as 0.25.

The first term on the right-hand side of Eq. (10.12) is the classical Bryan (1888) Equation for elastic buckling of externally pressurized pipes. The second and the third term simply add the contribution of cement-mortar coating and lining considering the same buckling expression. Note that vacuum conditions may cause collapse in thinwalled aboveground pipes with the D/t values of interest. Considering a standalone steel pipe with vacuum conditions, the pipe is under uniform external pressure equal to 1 bar (0.1 MPa). According to Bryan’s elastic buckling pressure, in the absence of coating or lining contribution, the pipe will collapse if the D/t value is above 167. Considering imperfection sensitivity, the limit value of D/t that corresponds to collapse can be much lower. Such collapse may occur in several instances in large above-ground diameter pipes, when vacuum conditions occur, possibly due to misfunction of air valves. The collapse of the 94-inch-diameter Bouquet Canyon pipeline in California in 1934, or the recent collapse of a 72-inch-diameter pipe in La Verkin, Utah in 2013, are typical examples of this type of failure. (b) Confined conditions

Buried pipelines may also be subjected to external pressure conditions, which may cause buckling of the thin-walled pipe cross-section in the form of an inward lobe, as discussed in Section 3.3 for elastic material behavior and Section 5.6 for

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elastic-plastic material response. In the latter section, it was shown that Montel’s Equation (Montel, 1960) can be used for predicting the collapse pressure (denoted as pM ) of the laterally confined pipe under uniform external pressure, in terms of the material yield stress σ Y , the cylinder geometry D/t, the initial out-of-roundness with amplitude δ 0 and the initial gap with maximum value g between the cylinder and the rigid cavity: pM =

14.1 σY (D/t )1.5 [1 + 1.2(δ0 + 2g)/t]

(10.13)

In AWWA M11, the following equation is proposed for predicting the ultimate external pressure under confined conditions, denoted as pa :  0.67 (1.2Cn )(EI)0.33 ϕs E  kν RH pa = (FS)r

(10.14)

where: r r r r r

pa is the allowable buckling pressure, FS is the factor of safety, taken equal to 2.0, Cn is a calibration factor to account for nonlinear effects, taken equal to 0.55, ϕ s is a factor to account for variability in stiffness of compacted soil; suggested value is 0.9, kV is the modulus correction factor for Poisson’s ratio, ν S , of the soil, defined as follows: kV = (1 + νS )(1 − 2νS )/(1 − νS )

(10.15)

and in the absence of specific information, it is common in Eq. (10.15) to assume ν s = 0.3, so that kν = 0.74, r

RH is the correction factor for the depth of fill: RH = 11.4/(11 + 2r/Hc )

r r

(10.16)

Hc is the height of soil cover above top of pipe, E is the modulus of soil reaction.

Eq. (10.14) is based on elastic buckling of a ring in a deformable cavity and is independent of the yield stress σ Y (e.g., Duns and Butterfield, 1971; Moore et al., 1989), whereas Montel’s Eq. (10.13) is a semi-empirical equation developed for a rigid cavity on the basis of first yielding of pipe material and available experimental data. For safety against collapse, the value of pa should be larger than the acting external pressure from the combined action of vacuum conditions, hydrostatic conditions (due to high water table) and the backfill weight load on the pipe (prism load).

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337

(A)

(B)

Figure 10.18 Bell-and-spigot joints in water pipelines: (A) rubber gasket joint for segmental pipelines, (B) lap-welded joints: double-welded; single welded ID (inside weld); single welded OD (outside weld).

10.5

Steel pipeline joints

Several types of joints are used in steel water pipelines. The most common types of joints are bell-and-spigot rubber gasket joints used in segmental pipelines, and lapwelded joints for welded pipelines (Fig. 10.18). However, there exist several other types of joints (Fig. 10.19), including the butt-welded joint. Special joints also include bolted flanged connections, couplings, and expansion joints. Fig. 10.18A depicts schematically the rolled–groove rubber gasket joint (see also Fig. 10.5). This joint is widely used in segmental pipelines. Gasket-sealed joints represent the most common non-restrained jointing system not only for steel water pipes, but also for other commonly used pipe materials such as concrete, ductile iron, fiberglass or PVC. This type of joint has been widely employed for more than fifty years in North America, for steel water transmission and distribution pipelines. It has been used in steel water pipes in diameters of up to 78-inch, and working pressures that may exceed 250 psi (maximum transient pressure more than 375 psi). For more details on the performance of gasket joints in steel pressure pipes, the reader is referred to the publication of Kelemen et al. (2011). Fig. 10.20A shows a lap-welded joint, used in numerous large-diameter steel pipelines for water transmission, because of its low construction cost and its proven history of use. This type of joint constitutes a cost-effective alternative to butt-welded

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Figure 10.19 Butt-welded joints and butt-strap joints.

full-penetration joints. It requires the cold forming of a “bell” at one end of each pipe segment, which is manufactured at the pipe mill using a mandrel that expands the end part of the pipe. During on-site construction, the non-expanded end of the adjacent pipe segment (called “spigot”) is inserted and welded to the bell with a single (inside or outside) or double full circumferential fillet weld (Fig. 10.20B). The mechanical performance of field-applied lap-welded joints under longitudinal stresses is of primary importance for safeguarding the structural integrity of water pipelines, especially in geohazard areas (e.g., seismic, landslides, soil subsidence). Section 10.6 below presents the design of lap-welded joints in the context of normal operating conditions. In Section 10.7, the structural performance of lap-welded joints under extreme loading conditions (e.g., seismic geohazard) is examined in detail, based on the results of a recent large-scale industrial project. Finally, in Section 10.8, a new concept for improving the structural performance of lap welded joints is presented.

10.6

Longitudinal stress

Longitudinal stress in the pipeline depends primarily on the boundary (end) conditions of the pipe. Two end conditions exist: (a) thrust (unrestrained) conditions and (b) Poisson/thermal (restrained) conditions, which are analyzed below in terms of their effect on longitudinal stress σ L . Detailed information and the necessary background on longitudinal stress in pressurized pipes under various end conditions is presented in Section 4.1 of the present book. (a) Thrust conditions

In this case, “capped-end” conditions apply, and the longitudinal stress σ L is half the hoop stress σ h : σL =

σh 2

(10.17)

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(C)

Figure 10.20 Lap-welded joint for a 24-inch-diameter pipe: (A) double-welded, (B) single-welded (inside). (C) Sketch of a lap-welded joint with its main geometric features (photos by S. A. Karamanos).

Considering a lap weld with a 45-degree profile shape, the weld throat is 0.707 times the pipe thickness. Therefore, under thrust conditions, the weld should be able to sustain a stress equal to 1.41σ L . If the weld is made of electrodes with yield and strength equal to or above those of the pipe material, i.e., the weld is “overmatched”, one may readily conclude that thrust conditions may not be critical for the lap welded joint. (b) Poisson/thermal conditions

Under restrained conditions, the longitudinal stress is the sum of the thermal stress and the Poisson stress: σL = EαT T + 0.3σh

(10.18)

where E is Young’s modulus of steel, T is the change of temperature, α T is the coefficient of thermal expansion of steel, and Poisson’s ratio is taken equal to 0.3. The following design requirements apply at the lap welded joint: r

for thermal stress only: 1.41(EαT T ) ≤ 0.9σU

(10.19)

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Figure 10.21 Beam action in a pipeline under its own weight. r

for thermal and Poisson conditions: 1.41(EαT T + 0.3σh ) ≤ min [0.9σY , 0.67σU ]

(10.20)

where σ U is the ultimate stress of the pipe material (tensile stress limit). Longitudinal stresses from beam-type deformation of the pipe, developed during pipeline construction, should be added to the pressure and thermal stresses. For proper pipeline alignment in the trench, the ends of each pipe are usually supported on mounds. In the case of a segmental pipeline, if loose soil conditions exist beneath the pipe, beam action occurs (Fig. 10.21). Using simple Statics and Strength of Materials, the maximum longitudinal stress σ b for a simply supported span occurs at mid-span and is equal to:  qw L 2 σb = 2πt D

(10.21)

where L is the length of the pipe section (stick) and qw is the weight of the pipe per unit length and its contents per unit length. The above approach is valid for segmental pipelines. If the joints are welded, the beam should be considered fixed at its both ends. In this case, the maximum longitudinal bending stress σ b occurs at the two end sections, and can be readily computed as follows: σb =

 qw L 2 3πt D

(10.22)

The above stresses assume concentrated reactions at the two end sections. However, because of soil support under the haunches, the maximum longitudinal stresses will be significantly less than the values computed with Eqs. (10.21) and (10.22). In any case, Eqs. (10.21) or (10.22) can be used for design purposes.

10.7

Lap-welded joints under severe longitudinal action

The performance of lap-welded joints under structural loads (axial force and bending) is essential for safeguarding pipeline integrity in seismic or other geohazard

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(B)

(C)

Figure 10.22 Schematic representation and the stress path in a lap-welded joint under: (A) axial tension, (B) axial compression, (C) bending.

areas, where severe ground-induced actions are expected (Karamanos et al., 2017). Because of the bell geometry, the stress path under axial compressive load has an eccentricity, which is slightly greater than pipe thickness, because of a small gap between the bell and the spigot. Therefore, local stresses occur due to bending of the pipe wall at the bell, so that the longitudinal stress and strain is increased with respect to its nominal value at the pipe cylinder, and this may result in early joint failure. One may be tempted to model the problem of axial tension loading of a lap-welded joint as a two-dimensional problem considering a longitudinal slice of the joint shown in Fig. 10.22. In such a case, the maximum stress at cross-section a-a is equal to the sum of stress σ a from the axial force and the bending stress σ b due to eccentricity (Fig. 10.23A): σ =

Fe F + 2 bt bt /6

(10.23)

Considering the minimum eccentricity c = 0, then e = t, and one readily calculates from Eq. (10.23) σ = 7F/bt, which means that the nominal axial stress σa = F/bt is amplified by a factor of 7, because of the bell (Moser and Faulkman, 2008). Considering allowable stress design, this factor leads to the conclusion that the lap-welded joint has a very low efficiency. Repetition of this analysis with elastic-plastic material behavior shows that the axial load causing full plastification of section a-a is slightly below 40% of the nominal plastic force of the cross-section (FY = σ Y bt) as shown in

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(A)

(B)

Figure 10.23 Two-dimensional model for a lap-welded joint under axial tension for determining stresses at the critical part of the bell: (A) elastic analysis, (B) fully-plastic analysis.

Figure 10.24 Load-displacement response of internally pressurized 25.75-inch-diameter lap-welded joints subjected to tension.

Fig. 10.23B (Brockenbrough, 1990; Van Greusen, 2008). This percent is an improvement of the result from elastic analysis, but it is still a very low value compared with experimental results, which show that lap welded joints are capable of sustaining a substantial percent of the strength of the steel cylinder. Fig. 10.24 shows load-displacement curves of lap welded joints obtained experimentally. They refer to a double-welded and a single-welded (outside weld) lapwelded joint subjected to axial tension in the presence of internal pressure (40% of

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(B)

Figure 10.25 (A) Testing device for axial loading of pipe specimens. (B) Lap-welded joint ready for tension test; L0 (equal to 28 in) is the gage length for measuring elongation (photos by S. A. Karamanos).

yield pressure pY ). In both specimens, the pipe outer diameter is 25.75 in (24-inch pipe nominal), with thickness equal to 0.25 in (D/t = 103) and it is made of steel grade 40. The experiments have been part of a large research program, launched in 2015, aimed at determining the structural strength and deformation of lap-welded joints under extreme loading conditions, and were performed in the facilities of Northwest Pipe Company, in Adelanto, CA, using the testing machine shown in Fig. 10.25A. The load capacity of the two joints was measured equal to 40% and 50% higher than the plastic load of the pipe cross-section [FP = σ Y (π Dt)], calculated with the real value of yield stress σ Y , determined from material testing (Fig. 10.24). The elongation of each specimen L was measured with wire transducers that recorded the relative displacement of two points located on either side of the lap joint at 14 in from the end of the bell, so that the total gage length L0 is equal to 711 mm (28 in), as shown Fig. 10.25B. The elongation is expressed by the value of normalized displacement L/L0 and should be regarded as the “average strain” induced in the lapwelded joint. The deformation capacity up to rupture of the joints under consideration has been remarkable, as shown in Fig. 10.24. At rupture, the corresponding normalized displacement L/L0 of the single lap-welded joint was 4.5% (point 1), whereas for the double lap-welded joint the L/L0 value was 22.6% (point 2). These values indicate that these joints are capable of sustaining a significant amount of deformation before rupture and loss of pressure containment. The initial, deformed and ruptured shapes of the double-welded joint specimen are shown in Fig. 10.26, verifying its high deformation capacity. Furthermore, in all tension-loaded specimens, fracture occurred at the base material of the pipe, away from the weld, as shown in Fig. 10.26B and Fig. 10.26C. More information on the tension experiments is offered in the paper by Keil et al. (2020a).

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(A)

(B)

(C)

Figure 10.26 Elongation of a lap-welded joint under tension: (A) configuration just before pipe wall fracture, (B) rupture stage, (C) final fractured configuration (photos by S. A. Karamanos).

The above experimental results demonstrate that the simplified two-dimensional analysis based on a longitudinal slice of the lap welded joint, shown in Fig. 10.23, may not be suitable for predicting the real structural strength of a lap-welded joint. The main reason is that the simplified two-dimensional analysis is a stress-based approach aimed at calculating the maximum local stress, and this is not adequate for the case of extreme loading conditions, where a strain-based approach should be followed (see also Chapter 9). In addition, the simplified two-dimensional analysis does not account for two main features of joint response: (a) the development of significant stress and strain in the circumferential direction due to Poisson-like effects, which is associated with a substantial part of the total strain energy, and (b) the large deformations and the corresponding change of geometry of the joint that occur during the application of loading. Those features can be taken into account only through a finite element model (Chatzopoulou et al., 2018; Sarvanis et al., 2019, 2020). If a lap-welded joint is subjected to uniform axial compression, the stress path has several similarities with the one described in tension-loaded specimens. However, axial compression of lap-welded joints has additional features associated with the occurrence of local buckling, which make this structural response more complex. The key feature is the eccentricity of the stress path in this connection (Fig. 10.22B) that acts as an “initial imperfection”, facilitating the formation of local buckling. A similar yet more elaborate situation occurs when the lap-welded joint is subjected to bending, as shown in Fig. 10.22. In this case, the compression side of the pipe joint is subjected to a loading pattern similar to the one in Fig. 10.22B, whereas the tension side undergoes the loading conditions of Fig. 10.22A.

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(B)

Figure 10.27 Axial compression response of 25.75-inch-diameter lap-welded joint specimens, pressurized at 40% of yield pressure: (A) double-welded grade 36 steel specimen, with D/t = 192, (B) two specimens made of grade 40 steel, with D/t = 103.

A series of axial compression and bending tests have been performed in addition to tension tests on lap-welded joints (Keil et al., 2020a). The axial compression tests were performed with the same apparatus used for the tension tests (Fig. 10.25A). The specimens were internally pressurized to 40% of the yield pressure pY , and the load-shortening diagram of three representative axially-compressed lap-welded joint specimens are depicted in Fig. 10.27. The diagram shows that the compressive strength of the joint is a substantial percent of the plastic load of the pipe section. For the thin-walled pipe specimen in Fig. 10.27A (D/t = 192, steel grade 36) the ultimate load is more than 80% of the plastic load of the pipe cross-section. For the thick-walled specimens in Fig. 10.27B (D/t = 103, steel grade 40) the ultimate load exceeds the plastic load. These values of ultimate load are well above the load capacity predicted by the simplified two-dimensional models of Fig. 10.23. During the experiment, shortening of the specimens was measured in terms of the normalized displacement L/L0 , using a gage length L0 as in the case of tension (Fig. 10.25B). The final configuration of the buckled (crushed) thin-walled pipe specimen in Fig. 10.27A is shown in Fig. 10.28. Buckles developed in a non-axisymmetric pattern. In some parts of the specimen, they occurred at the bell and in some other parts, they developed at the spigot. Overall, the joint exhibited excellent deformation capacity, and values of the shortening parameter L/L0 that exceeded 12% were measured before fracture of the wall and loss of pressure containment. Fig. 10.29 shows the bending response of lap-welded joint pipe specimens, obtained from a series of large-scale physical experiments performed with the four-point bending frame of Fig. 10.30. The value of applied force Q is normalized by the force parameter Qp , which is equal to the ratio of plastic moment of the pipe cross-section (MP = σY D2m t, where Dm = D − t) over the distance e1 , so that QP = MP /e1 . Based on its definition, QP can be considered as the load that causes complete plastification of the pipe cross-section. In Fig. 10.29, the value of Q/Q p is plotted in terms of midpoint displacement u, shown in Fig. 10.30B, normalized by the pipe diameter D (u/D).

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Figure 10.28 Final (buckled) configuration of a thin-walled lap-welded joint pipe specimen (double welded) subjected to axial compression (D/t = 192, steel grade 36) (photo by S. A. Karamanos).

Fig. 10.29A shows the bending response of a thin-walled specimen that contains a lapwelded joint together with the bending diagram of a plain pipe specimen with no weld. Both specimens are pressurized at 40% of yield pressure pY . The bending response shows that lap-welded joints can sustain a bending load equal to 59% of the load that corresponds to the complete plastification of the pipe cross-section. This maximum load is 82% of the bending moment strength of a plain pipe with the same cross-section geometry and material. Most importantly, the lap-welded thin-walled specimen was deformed by a substantial amount without loss of pressure containment, as shown in Fig. 10.31. Fig. 10.29B depicts the load-displacement response of two thick-walled single-welded specimens (D/t = 103, steel grade 40) pressurized at 40% of yield pressure pY . The ultimate load of those two specimens reached approximately 90% of the load that corresponds to the complete plastification of the pipe cross-section. Furthermore, they were deformed by a substantial amount without loss of pressure containment. The buckling shapes of the two specimens are shown in Fig. 10.32. More experimental and numerical results on the bending performance of lap-welded joints are presented in the publications of Keil et al. (2018) and Sarvanis et al. (2020). As a conclusion, in contrast to the results of the simplified two-dimensional models (Fig. 10.23), lap-welded joints constructed properly according to AWWA standards are very resilient under severe structural loading, and this was demonstrated in recent large-scale physical experiments. Numerical simulations with advanced finite element models verified the findings of experimental results (Keil et al., 2020a, 2020b). The

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(A)

(B)

Figure 10.29 Βending response of lap-welded joint specimens: (A) thin-walled pipe double-welded specimen (D/t = 192, steel grade 36) and plain pipe specimen without weld, (B) two thick-walled specimens (D/t = 103, steel grade 40).

ultimate strength of standard lap-welded joints under tension, compression and bending was found equal to a high percent of the strength of the plain pipe and, most importantly, the joints were able to undergo large amount of deformation (axial and bending) without loss of containment. This last feature is of primary importance in geohazard areas, where severe ground-induced actions are expected to occur, and are associated with the development of large longitudinal strains in the pipe wall.

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(A)

(B)

Figure 10.30 Loading frame for large-scale bending of pipe specimens: (A) photo of the frame (photos by S. A. Karamanos), (B) schematic representation.

10.8

Improving the structural performance of lap-welded joints

The experimental results presented in the previous section, and the relevant numerical calculations (Keil et al., 2018, 2020a, 2020b), demonstrated the very good structural performance of standard lap-welded joints under axial loading and bending conditions, and in particular, their ability to undergo substantial deformation, well beyond the maximum load, without loss of water containment. In the above experiments, despite their excellent performance, several specimens exhibited buckling at the bell and through the field-applied fillet weld. Under compressive action, local buckling is likely to occur at the bell due to joint geometry and the eccentric stress path, shown schematically in Fig. 10.22B for the case of an internal weld. It is recalled that the bell is fabricated by cold expansion (Section 10.1), which induces significant work hardening and residual stresses in the steel material. The above arguments have raised some concerns on the use of lap welded joints in geohazard areas, despite the fact that these concerns are not justified by the results of the experiments described in Section 10.7. These concerns motivated the proposal of several welded joint concepts, such as

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(B)

Figure 10.31 Final (buckled) configuration of a thin-walled lap-welded pipe joint specimen subjected to bending (D/t = 192, steel grade 36, double weld): (A) final stage of bending test, (B) after test (photos by S. A. Karamanos).

(A)

(B)

Figure 10.32 Buckled shape of lap-welded joints subjected to bending (D/t = 103, steel grade 40, single welds): (A) inside weld, (B) outside weld (photos by S. A. Karamanos).

the Enduro Bell reinforced lap-welded joint (McPherson et al., 2016) or the JFE joint (Nakazono et al., 2019), both aimed at protecting the bell and the welds from externally-induced actions. However, those solutions are neither simple nor economical to manufacture and install. On the other hand, if the buckles were forced to develop in the pipe cylinder, away from the bell and the weld, pipeline safety would be further increased, and this constitutes the basis of our discussion below. The experiments and the finite element calculations in standard lap-welded joints described in the previous section of this book (Sarvanis et al., 2019, 2020) have shown

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(B)

Figure 10.33 Configuration of a lap-welded joint with projection: (A) pipe specimen with projection (photo by S. A. Karamanos), (B) sketch of the welded joint with projection.

that the location of the buckle, may not always be at the bell, but it depends on the presence of small initial geometric perturbations or distortions at the pipe wall. Note that in Fig. 10.32, the inside-welded specimen buckled at the bell, whereas the outside-welded specimen buckled at the spigot. Using a more mathematical terminology, the buckling response of a standard lap-welded joint is sensitive to the presence of small initial geometric “imperfections” (i.e., small, inevitable, usually undetectable deviations from the theoretical perfect geometry) which may occur during pipe fabrication or field construction. This leads to the following argument (Keil et al., 2020b, 2022): imposing a small initial geometric perturbation in the form of a projection at the spigot near the weld, would enforce the buckle to occur at this specific location, preventing the buckle from occurring at the bell. In such a case, the bell and the weld region would be protected from the excessive deformation that develops in the buckled region. The projection may be placed at the spigot, at a small distance from the weld, and its amplitude (Fig. 10.33) should be simultaneously: (a) large enough to ensure that buckling would occur at the location of projection in the spigot and not in the bell, and (b) small enough, so that the strength of the joint in terms of axial compression and bending is not significantly affected. To satisfy both requirements, the optimum size of the initial geometric projection should be somewhat less than the pipe wall thickness, and this was determined by an extensive finite element study (Keil et al., 2022). From the pipe manufacturing point-of-view, this projection can be introduced within the normal fabrication process of the pipe in the pipe mill before the coating/lining operations. Using the testing devices shown in Fig. 10.25 and Fig. 10.30, a series of lap-welded joints, equipped with this projection at the spigot near the weld, were tested under bending and axial compression. All specimens buckled at the projection. Fig. 10.34 shows the deformed shape of a single (inside)-welded lap joint specimen, equipped with a projection, subjected to bending. The specimen buckled at the projection, whereas the corresponding standard lap-welded joint in Fig. 10.32A exhibited local buckling at the bell. Furthermore, the bending strength of the lap-welded specimens equipped with the projection was measured quite close to the strength of the standard lap-welded specimens. Similar results were obtained for several lap-welded joints equipped with

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Figure 10.34 Final (buckled) configuration of a single (inside)-welded joint specimen with projection subjected to bending (photos by S. A. Karamanos).

Figure 10.35 Final (buckled) configuration of a thin-walled (D/t =192) double-welded joint specimen with projection subjected to axial compression (photo by S. A. Karamanos).

the projection tested under axial compression. Buckling occurred consistently at the location of the projection in an axisymmetric form, without significant reduction on the joint strength with respect to the corresponding standard lap-welded joints (Keil et al., 2022). Fig. 10.35 shows the buckled configuration of a thin-walled (D/t = 192) double-welded joint specimen with projection subjected to axial compression. In this specimen, buckling occurred exactly at the projection, in a controlled manner, in contrast with the corresponding specimen with no projection, shown in Fig. 10.28, where buckles developed over the entire area of the welded joint. In addition to the above monotonic tests, a pipe specimen that contained a projection was tested under cyclic loading until failure. The specimen was subjected first to compression, which caused local buckling at the projection. Subsequently, it was subjected to tension-compression cycles, which induced significant amount of repeated inelastic deformation in the buckled area of the pipe by flattening and folding the pipe wall repeatedly at the buckle (Fig. 10.36), leading to low-cycle fatigue of pipe material. This loading pattern is much more severe than the one expected to occur under groundinduced actions. Nevertheless, the specimen was capable to sustain 26 of those largeamplitude cycles before rupture at the crest of the buckle, as shown in Fig. 10.37B. This

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Figure 10.36 Load-displacement diagram of lap-welded joint with projection subjected to cyclic axial loading.

(A)

(B)

Figure 10.37 Lap-welded joint specimen with projection subjected to cyclic axial loading: (A) specimen configuration under compression, (B) final configuration after fracture (photos by S. A. Karamanos).

was a remarkable result, demonstrating that the projection is capable of sustaining a significant number of loading cycles with repeated excursions into the inelastic range of the material, before developing a through-thickness crack. Based on the above experimental results, and the relevant numerical simulations, the initial projection concept is an efficient, reliable, and economical solution for welded steel pipelines constructed in geohazard areas and constitutes the basis of the InfraShield® system (Northwest Pipe Company, 2022), launched recently (patent

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pending). Current research is underway, for the purpose of applying this concept for improving the structural performance of large-diameter buried steel pipelines subjected to differential settlements (Vazouras et al., 2022).

References American Water Works Association. (2017a). Steel Pipe - A Guide For Design and Installation (5th Edition). AWWA M11. American Water Works Association. (2017b). Steel Water Pipe, 6 in. (150 mm) and Larger. AWWA C200. American Water Works Association. (2017c). Dimensions For Fabricated Steel Water Pipe Fittings. AWWA C208. American Water Works Association. (2018a). Cement–Mortar Protective Lining and Coating for Steel Water Pipe 4 in. (100 mm) and Larger—Shop Applied. AWWA C205. American Water Works Association. (2018b). Polyurethane Coatings and Linings for Steel Water Pipe and Fittings. AWWA C222. Brockenbrough, R. L. (1990). Strength of bell-and-spigot joints. Journal of Structural Engineering, 116(7), 1983–1991 ASCE. Bryan, G. H. (1888). Application of the energy test to the collapse of a long pipe under external pressure. Proceeding of the Cambridge Philosophical Soc, Proc UK, Cambridge, 6, 287– 292. Chatzopoulou, G., Fappas, D., Karamanos, S. A., Keil, B. D., & Mielke, R. D. (2018). Numerical Simulation of Steel Lap Welded Pipe Joint Behavior in Seismic Conditions. In ASCE Pipelines Conference 2018 (pp. 444–455). Toronto, ON, Canada. Duns, C. S., & Butterfield, R. (1971). Flexible buried cylinders Part III: buckling behavior. International Journal of Rock Mechanics and Mining Sciences, 8(6), 613–627. Handy, R. L., & Spangler, M. G. (2007). Geotechnical Engineering: Soil and Foundation Principles and Practice (5th Edition). New York, NY: McGraw Hill. Karamanos, S. A., Sarvanis, G. C., Keil, B. D., & Card, R. J. (2017). Analysis and design of buried steel water pipelines in seismic areas. Journal of Pipeline Systems Engineering and Practice, 8(4), art. no. 04017018. Keil, B. D., Gobler, F., Mielke, R. D., Lucier, G., Sarvanis, G. C., & Karamanos, S. A. (2018). Experimental Results of Steel Lap Welded Pipe Joints in Seismic Conditions. In ASCE Pipelines Conference 2018 (pp. 433–443). Toronto, ON, Canada. Keil, B. D., Lucier, G., Karamanos, S. A., Mielke, R. D., Gobler, F., Fappas, D., et al. (2020a). Experimental investigation of steel lap welded pipe joint performance under severe axial loading conditions in seismic or geohazard areas. In ASCE Pipelines Conference 2020 (pp. 249–258). Keil, B. D., Mielke, R. D., Gobler, F., Lucier, G., Sarvanis, G. C., Chatzopoulou, G., et al. (2020b). Newly developed seismic resilient steel pipe joint safeguards: pipeline structural integrity during severe geohazard events. ASCE Pipelines Conference 2020, 265–276. Keil, B. D., Fappas, D., Gobler, F., Sarvanis, G. C., Chatzopoulou, G., Lucier, G., et al. (2022). A new concept for improving the structural resilience of lap-welded steel pipeline joints. Thin Walled Structures, 171, 108676. Kelemen, N., Keil, B., Mielke, R., Davidenko, G., & Gardner, J. (2011). Performance of Gasket Joints in Steel Pressure Pipes. ASCE Pipelines Conference 2011 (pp. 1278–1287). Seattle, Washington.

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McPherson, D. L., Duffy, M., Koritsa, E., Karamanos, S. A., & Plattsmier, J. R. (2016). Improving the performance of steel pipe welded lap joints in geohazard areas. In ASCE Pipelines Conference 2016, Kansas City, Missouri. Moser, A. P., & Faulkner, S. (2008). Buried Pipe Design (3rd Edition). New York, NY: McGrawHill. Montel, R. (1960). Formule semi-empirique pour la détermination de la pression extérieure limite d’instabilité des conduites métalliques lisses noyées dans du béton. La Houille Blanche, (5), 560–568. Moore, I. D. (1989). Elastic buckling of buried flexible tubes-a review of theory and experiment. Journal of Geotechnical Engineering, ASCE, 115(3), 340–358. Nakazono, H., Hasegava, N., Wham, B. P., O’Rourke, T. D., & Blake, B. (2019). Innovative solution to large ground displacement using steel pipe for crossing fault. In ASCE Pipelines Conference 2019, Nashville, TN. Northwest Pipe Company. Infrashield® Seismic Resilient Joint System. https://www.nwpipe. com/products/engineered-steel-water-pipe/infrashield-seismic-resilient-joint-system/. Sarvanis, G. C., Chatzopoulou, G., Fappas, D., Karamanos, S. A., Keil, B. D., Mielke, R. D., et al. (2019). In Finite Element Analysis of Steel Lap Welded Joint Behavior under Severe Seismic Loading Conditions. ASCE Pipelines Conference 2019 (pp. 325–333). Nashville, TN. Sarvanis, G. C., Chatzopoulou, G., Fappas, D., Karamanos, S. A., Keil, B. D., Lucier, G., et al. (2020). Bending Response of Lap Welded Steel Pipeline Joints. Thin-Walled Structures, 157, 107065. Spangler, M. G. (1941). The Structural Design of Flexible Pipe Culverts. Iowa: Iowa State College of Agriculture and Mechanic Arts. Van Greusen, J. (2008). Seismic design of bell-and-spigot joints for large diameter steel pipe. In ASCE Pipelines Conference 2008, Atlanta, GA. Vazouras, P., Keil, B. D., Dakoulas, P., Mielke, R. D., & Karamanos, S. A. (2022). A novel concept for assuring the performance of steel water pipelines in ground settlement areas. In ASCE Pipelines Conference 2022, Indianapolis, IN. Watkins, R. K., & Anderson, L. R. (2000). Structural Mechanics of Buried Pipes. Boca Raton, Florida: CRC Press LLC. Watkins, R. K., & Spangler, M. G. (1958). Some characteristics of the modulus of passive resistance of soil: a study in similitude. In Proceedings of the 37th Annual Meeting of the Highway Research Board, Washington D.C: 37 (pp. 576–583).

Global buckling of pipelines

11

Global buckling, also called “beam buckling”, is the structural instability of the pipeline considered as a straight (or nearly straight) slender bar, subjected to compression, from internal pressure and temperature. It is a different type of structural instability than local buckling (which is a “shell buckling” phenomenon) and refers to the classical “Euler-type” buckling of compressed bars or columns. The beam-column theory presented in Appendix B, and in particular the beamcolumn response on elastic foundation, constitutes a good starting point for understanding and modeling this phenomenon. However, pipeline global buckling is a more complex phenomenon, and significant research is still performed to elucidate its main features, especially soil-pipe interaction. Motivated by the need for safe design of hightemperature/high-pressure (HT/HP) offshore pipelines, the SAFEBUCK joint industry project (Bruton and Carr, 2011) has been an important step towards understanding this problem, and its findings constituted the basis of the recommended practice document DNV-RP-F110 (Collberg et al., 2011). The present chapter offers an introductory discussion of two main topics of pipeline global buckling: (a) upheaval buckling (Fig. 11.1) and (b) lateral buckling (Fig. 11.2). The two topics have several similarities, and they both refer to pipelines that exhibit significant compression during their operation, because of pressure and temperature. Quite often, global buckling is a sudden event, which can cause overstress at the buckled region, leading to local buckling, rupture and failure. Controlling the shape of the buckle and the sequence of events associated with its formation and growth is essential for safeguarding pipeline integrity. In the first part of this chapter, the driving compressive force that causes global buckling in pipelines under pressure and temperature is presented. Then, the mechanism of upheaval buckling is examined, and a simple design approach is described. The last part of the chapter offers a brief presentation of lateral buckling and possible mitigations measures in offshore pipelines.

11.1

Driving force for global buckling

The first step in global buckling design is the calculation of longitudinal compressive force in the pipe wall and the fluid containment due to the increase of pressure and temperature. This force is the “driving force” for global buckling. A pipeline that operates at specific levels of temperature and internal pressure tends to expand or elongate. However, if the pipeline is not free to expand, it will be subjected to axial compressive load that may cause global buckling. Consider a long straight pipeline subjected to internal pressure p and temperature increase T. The thin-walled cylinder approximation of the pipe allows for writing the Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00014-1 c 2023 Elsevier Inc. All rights reserved. Copyright 

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Figure 11.1 Upheaval buckling of a buried pipeline (schematic).

Figure 11.2 Lateral buckling (snaking) of an offshore pipeline on the seafloor (schematic).

compressive force in a closed-form expression, which can be used for design purposes. The hoop stress σ θ of the pipe wall is calculated from Barlow’s formula: σθ =

pD 2t

(11.1)

and the axial strain εx can be expressed as follows from Hooke’s law (Appendix A): εx =

1 (σx − ve σθ ) + αT T E

(11.2)

If the pipe is axially restrained, the axial strain should be zero (εx = 0), and combining with Eq. (11.2), the axial stress developed in the pipeline is equal to: σx = ve σθ − EαT T

(11.3)

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357

and the corresponding force F1 on the pipe wall is: F1 = πDtσx

(11.4)

In addition, the compressive force on pipe contents F2 should be added to the above force: F2 = −

π D2 p 4

(11.5)

so that the total compressive force F on the pipeline cross-section is: F = −(1 − 2νe )

πD2 p − π D t EαT T 4

(11.6)

In the case of offshore pipelines, the external pressure (pext ) should be subtracted from the internal pressure (pint ), so that in Eq. (11.6) the value of p denotes the net internal pressure of the pipe: p = pint − pext

(11.7)

In Eq. (11.6) the value of residual lay tension TR should be added to the above compressive force. The calculation of this force may not be trivial, but it is possible to estimate it from a detailed analysis and simulation of the pipelaying process. Therefore, the total force is: F = TR − (1 − 2νe )

πD2 p − π D t EαT T 4

(11.8)

Eq. (11.6) [or Eq. (11.8)] provides the resultant compressive force in a pipeline under fully-constrained end conditions, i.e., when the longitudinal movement is entirely prevented. In the presence of expansion loops, some longitudinal expansion movement is allowed to occur, and this reduces the longitudinal compressive force because of the longitudinal tension induced in the pipe wall, which counteracts the compression because of the constrained fluid containment. The first term in the right-hand side of Eq. (11.6), is due to net internal pressure p, whereas the second term depends on the temperature raise T. More specifically, the total force F consists of two components (Fig. 11.3): r

the “pressure” force component: Fpr = −(1 − 2νe )

r

π D2 p 4

(11.9)

the “temperature” or “thermal” force component: Fth = −(π D t )EαT T

(11.10)

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Figure 11.3 Axial forces acting on the pipeline due to axial restraint.

11.2

Upheaval buckling

Upheaval buckling occurs in buried pipelines, both onshore and offshore. In its postbuckling configuration, the buried pipeline “arches upwards” and, under severe compression, it may get out of the ground surface or the seabed, forming a loop (arch) of several tens of meters. The first incidents of global buckling in offshore pipelines were reported in the 80s in the form of upheaval buckling (upward buckles of buried offshore pipelines), mainly due to the construction of several high-pressure/high-temperature pipelines. One of the first incidents reported on upheaval buckling of buried offshore pipelines occurred in 1986 in the Mærsk Olie og Gas A/S Rolf pipeline, in the North Sea (Guijt, 1990), followed by a few other incidents. These incidents triggered a significant amount of research on this topic. The recent advances in computer modeling enabled the simulation of the actual pipeline configuration and its structural behavior, and more detailed design and analysis methodologies for upheaval buckling were applied, with the purpose of either preventing upheaval buckling or mitigating the influence of buckles in pipeline integrity (Palmer et al., 1994).

11.2.1 A simple and efficient analytical formulation for upheaval buckling design The structural stability of the pipeline in its initial position depends on the local profile of the pipe in contact with its foundation, and on whether the downward force applied by the overburden soil is adequate to hold the pipe in its position. More specifically, the governing factors for assessing pipeline structural stability are the pipeline profile configuration, the axial compressive force, the flexural rigidity of the pipe cross-section, and the soil resistance against upward displacement of the pipe. In the following analysis, introduced by Palmer et al. (1990), these factors are combined to provide an elegant and efficient solution for upheaval buckling. Initially, a pipeline profile is defined by considering a horizontal axis x that corresponds to an initially straight pipeline profile, and a height function y(x), which defines

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359

Figure 11.4 Upheaval buckling model parameters.

Figure 11.5 Beam-column model for upheaval buckling.

the real pipeline profile with maximum height δ as shown in Fig. 11.4. The pipeline is idealized as an elastic beam subjected to axial compressive force F and has flexural rigidity EI, where I = πr3 t is the bending moment of inertia of the circular crosssection. An important step in this analysis is to assume a shape function, which represents the pipeline profile. As a first approximation, the following simple trigonometric function is assumed for the imperfection profile with maximum height δ and length L: y(x) = δ cos2

πx L

(11.11)

From elementary beam-column theory (Appendix B), the downward force per unit length q(x) required to maintain the pipeline in equilibrium in this position can be provided through the well-known beam-column equation written below in a form suitable for the purposes of the present analysis (Fig. 11.5): q(x) = −EI

d2y d4y − F dx4 dx2

(11.12)

Inserting the profile function y(x) of Eq. (11.11) into the beam-column Eq. (11.12), the lateral load q(x) required for equilibrium of the pipeline in this position is expressed as follows:   π2 2πx 8δEIπ 4 + 2δF cos q(x) = − L4 L2 L

(11.13)

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and the maximum lateral load from Eq. (11.13) is: qmax = 2δF

π2 8δEIπ 4 − L2 L4

(11.14)

The above result can be written in a more universal form, introducing the dimensionless parameters w and L : qmax EI δF 2  FL2

L = EI

w =

(11.15) (11.16)

Then, Eq. (11.14) can be written as follows:

w = 2π 2

1 1 − 8π 4 4 2

L

L

(11.17)

One should note that Eq. (11.17) is obtained under the assumption of the cosine profile defined in Eq. (11.11). However, the real profile may be quite different than the one defined in Eq. (11.11). Therefore, one may write the following general form of Eq. (11.17):

w = c

1 1 +d 4 2

L

L

(11.18)

or equivalently,

w 2L = c + d −2 L

(11.19)

where c, d are coefficients to be computed. Repeating the above analysis for different assumed shapes of initial imperfection profile, instead of Eq. (11.11), different coefficients c, d are computed. Such a procedure was conducted by Palmer et al. (1990). The analysis found that the relationship between w 2L and −2 L is quasi bilinear, and that two pairs of values (c, d) should be defined; one pair corresponding to small values of L and a second pair to large values of L . Furthermore, special attention is necessary when imperfection profiles of very short length are considered (i.e., small values of L ). During upheaval buckling of very short profiles, the pipeline is only in contact with the crest of the imperfection and therefore, in this case, the value of L is inconsequential, and this is represented by a cut-off for small values of L . Based on the above analysis, the final expression proposed by Palmer et al. (1990) is expressed as follows:

L < 4.49,

w = 0.0646

(11.20)

4.49 ≤ L < 8.06,

w = 5.68/ 2L − 88.35/ 4L

(11.21)

L > 8.06,

w = 9.6/ 2L − 343/ 4L

(11.22)

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Figure 11.6 Design curve for upheaval buckling design, w versus L , and comparison with experimental data from Palmer et al. (1990).

and is plotted in Fig. 11.6, together with experimental results also reported in Palmer et al. (1990). Therefore, given the pipeline bending rigidity EI, the compressive axial force F, and the pipeline profile δ, L, it is possible to calculate L and from Eqs. (11.20)–(11.22) the value of w is obtained. Finally, from Eq. (11.15), the value of qmax required for the equilibrium of the pipeline in this deformed profile is computed. The next question is whether the pipe-soil system is capable of providing the required soil resistance qmax , an issue to be examined in the following paragraph.

11.2.2 Resistance against upheaval buckling The sources of pipeline resistance against upheaval buckling (qmax ) are the pipeline weight qw and the uplift resistance qs of the soil cover. To prevent upheaval buckling, the following condition should be met: qmax ≤

1 (qs + qw ) SF

(11.23)

where SF is an appropriate safety factor. For underwater pipelines, the submerged weight qw should be considered in the above calculation. The soil resistance should be computed differently, depending on soil cohesion (Schaminée et al., 1990). For cohesionless soil (sand), the soil strength is expressed as follows: qs = γ HD(1 + fu H/D)

(11.24)

In Eq. (11.24), the first term represents the weight of the soil prism on top of the soil and the second term refers to the shear resistance of the soil cover against upward

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Figure 11.7 Soil resistance qs for cohesionless soils.

motion of the pipe (Fig. 11.7). In Eq. (11.24), γ is the unit weight of soil, and fu is an uplift coefficient, which is usually determined experimentally and taken as 0.5 for dense materials and 0.1 for loose sands. For cohesive soil (clay) the soil strength is: qs = cD min(3, H/D)

(11.25)

where c is the shear strength of cohesive soil.

11.2.3 Finite element simulation of upheaval buckling There exist a large number of analytical formulations on the mechanical response of buried pipelines that exhibit upheaval buckling, and the interested reader is referred to the early papers by Hobbs (1984) and Ju and Kyriakides (1988). In most analytical formulations the pipeline is considered as a long elastic beam, neglecting crosssectional deformation and local buckling. Furthermore, in those models, soil-pipe interaction is taken into account in an indirect and simplified manner. Soil-pipe interaction is a key feature of upheaval buckling, and in this perspective, the present problem has several similarities with pipeline response under permanent ground deformations presented in Chapter 9. Therefore, similar modeling approaches can be followed. In engineering design practice, level 1 models are used almost exclusively, where the pipeline is modeled with “pipe” elements and the soil is simulated with nonlinear springs attached to the pipe element nodes. In a recent research project, Vazouras et al. (2020) presented an advanced numerical model that simulates upheaval buckling in buried pipelines in a rigorous manner. The model follows the three-dimensional finite element analysis models (level 2) presented in Section 9.2 and has been calibrated and verified with data from physical experiments performed at the Laboratory of Soil Mechanics, in NTU Athens, Greece. Two types of experiments were conducted in the course of that project: (a) uplift resistance (pullout) tests and (b) upheaval buckling tests. In both tests, the pipe was embedded in a soil box, filled with sand of controlled grain size and compaction level. In the uplift experiments, PVC pipes of diameter equal to 100 mm and 200 mm

Global buckling of pipelines

363

(A)

(B)

(C)

Figure 11.8 Uplifting response of 200-mm-diameter buried pipe in moderately dense sand (relative density Dr = 65%) and depth H = 500 mm; pull-out test and distribution of displacements and shear stresses for three values of uplifting displacement; (A) 1.2 mm, (B) 11 mm, (C) 25 mm [results by Vazouras et al. (2020), published with permission].

were employed, and displacement measurements were obtained using particle image velocimetry (PIV), which allowed for accurate computation of shear strains in the soil during pipe uplifting. Fig. 11.8 shows the configuration of the uplifting pipe, the vertical displacement profile and the distribution of shear strains in the soil, for three stages of uplifting deformation. The upheaval buckling tests were performed on Ø20 mm × 1.2 mm aluminum pipes hinged at the two ends using the setup shown in Fig. 11.9. The pipes had an initial imperfection with amplitude, equal to approximately 1 pipe diameter. Fig. 11.10 shows the buckled configuration of the aluminum specimen and its “arching” out of the soil. Upon calibration and validation of their model, Vazouras et al. (2020) simulated the upheaval buckling response of pipelines and analyzed the response of a 36-inchdiameter X65 steel pipeline with thickness 12.7 mm (0.5 in). The total length of the

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Structural Mechanics and Design of Metal Pipes

Figure 11.9 Schematic representation of the upheaval buckling experimental set-up used in NTUA (dimensions in mm).

(A)

(B)

(C)

Figure 11.10 Buckled pipeline specimens in upheaval buckling experiments; (A), (B) “arching” of buckled specimen in the sand box, (C) specimen after test [Vazouras et al. (2020), photos by P. Vazouras, A. Tsatsis, P. Dakoulas, published with permission].

pipeline is 1.07 km, and the soil cover above the pipeline is equal to 1 m. The central part of the model is shown in Fig. 11.11 and consists of a rectangular soil block (70 m × 5 m × 2.4 m) modeled with 8-node reduced-integration solid elements and the embedded pipeline, modeled with reduced-integration 4-node shell elements (S4R). The shell element mesh is quite dense, to enable the simulation of local buckling

Global buckling of pipelines

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Figure 11.11 Finite element model developed for simulating the upheaval buckling of a 36-inch-diameter buried pipeline [results from Vazouras et al. (2020), with permission].

formation. In this part, the configuration of the soil-pipe model is not straight, but has an initial “upward” imperfection profile similar to the profile defined in Eq. (11.11). Soilpipe interaction is considered through a contact algorithm between the outer surface of the steel pipe and the surrounding soil, with coefficient of friction equal to 0.3. Outside the soil block, the pipe is modeled with beam-type “pipe” elements, which are connected to soil springs with stiffness in accordance with the guidance provided by the ALA guidelines (American Lifelines Alliance, 2001). Von Mises (J2 ) plasticity with isotropic hardening is used for modeling the pipe material (Appendix D), with Young’s modulus 70 GPa, Poisson’s ratio 0.3, and yield strength 150 MPa. Finally, a modified Mohr–Coulomb constitutive model was utilized (Anastasopoulos et al., 2007), accounting for the reduction of friction and dilation angle with the magnitude equivalent deviatoric plastic strain. The soil model was calibrated with experimental results of direct shear tests. Fig. 11.12 depicts the post buckling pipeline shape embedded in a clay soil with density 1,500 kg/m3 , cohesion 20 kPa and Young’s modulus 5,000 kPa. The pipeline profile is “imperfect” with respect to a straight alignment, with amplitude δ = 0.6 m, and length L = 20 m. Because of upheaval buckling, the pipeline is bent to a significant amount and exhibits local buckling at points A, B and B . The buckle area at A exhibits the highest strain and pinpoints a critical location of the pipe where pipe wall rupture may occur. For an extended summary of this research effort, the reader is referred to the recent paper by Vazouras et al. (2021).

11.2.4 Design measures for upheaval buckling There exist several methodologies to prevent the formation of upheaval buckling. The most obvious measure would be the reduction of the compression force F, by decreasing the design pressure and the operating pressure. However, this may not be

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Figure 11.12 Local buckling formation on a thermally-buckled 36-inch-diameter buried pipeline; finite element simulation [results from Vazouras et al. (2020), with permission].

possible in practice because the level of pressure and temperature are both dictated from flow assurance requirements. An interesting observation in Eq. (11.6) is that the thermal component of the force (Fth ) depends linearly on pipe wall thickness (t) and therefore one may be tempted to reduce pipe wall thickness. However, pipe wall thickness reduction is not always possible, because it decreases the pipeline resistance to pressure and to externally applied loads and reduces the bending stiffness. Another possibility for reducing the risk of upheaval buckling is the increase of pipeline weight, using concrete coating, which is often used in offshore pipelines. Increase of soil resistance increases pipeline resistance against upheaval buckling. This can be achieved by increasing the soil cover depth or placing heavy rocks above the ground surface at critical locations along the pipeline (Liu and Cross, 2019). This measure is used quite often in submarine pipelines, but in this case, the rock berms should be designed to remain in place, resisting horizontal hydrodynamic forces. Furthermore, reducing the allowable size of initial imperfection (out-of-straightness) δ in the construction specifications of the pipeline reduces the risk of upheaval buckling. The most rational approach in designing a pipeline against upheaval buckling is the use of strain-based design (Kristiansen et al., 2005), which is straightforward to apply and can be very useful for the design and assessment of upheaval buckling. Strain-based design, introduced in Chapter 9 for ground-induced actions, overcomes the main deficiency of traditional stress-based design, where the acting stress should be limited by an allowable stress, which is a fraction of yield stress (e.g., Solano et al., 2012). Stress-based design does not account for the inelastic deformation

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Figure 11.13 “Arching” of buried pipeline following upheaval buckling, exposing its crest.

reserves of the pipeline. On the other hand, strain-based design allows the pipe to deform in the inelastic range, provided that the strain developed in the pipeline wall is below an appropriate deformation (strain) limit. For tensile loading this limit is the strain value beyond which an Engineering Criticality Assessment (ECA) is required for assessing the integrity of the welds. For compressive strains the limit strain corresponding to the onset of buckling should be employed. If these strain limit values are exceeded, appropriate ECA analysis should be conducted. As a conclusion, it is possible to accept the formation of a global buckle provided that an appropriate assessment of those buckles is made, using survey data and advanced numerical tools. One should remember that not all buckles are necessarily dangerous for pipeline integrity. On the other hand, apart from overstressing the buckled region, the occurrence of an upheaval buckle that exposes the pipe above the ground (or seabed) surface (see Fig. 11.10 and Fig. 11.13) may not be acceptable by the pipeline operator. For the case of an underwater pipeline, the exposed “arch” of the pipeline is subjected to hydrodynamic forces, which may induce fatigue damage. Furthermore, the possibility of hooking from trawl gears or anchors is also increased.

11.3

Lateral buckling

Lateral buckling was identified as a potential problem for offshore pipelines resting on the seabed nearly half a century ago (Palmer and Baldry, 1974). It is often referred to as “snaking” of the pipeline. For a long time, this problem did not receive proper attention, but, during the last two decades it became important because of the increasing need for designing offshore pipelines that operate at elevated pressure and temperature, and are constructed in offshore locations where pipeline burial may not be a feasible option (Bruton et al., 2005; Seyfipoor et al., 2019). In many cases, lateral buckles are smooth, and may not threaten the structural integrity of the pipeline. Quite often, they remain undetected, and the associated lateral movement of the pipeline is harmless. However, in a few instances, the associated lateral movements can be large and may lead to localization of displacements, with all pipe movement concentrated in one buckle. At that point, a local buckle (kink) may occur at the bent configuration of the pipe, leading to pipeline rupture due to substantial amount of local bending deformation.

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The driving force for developing lateral buckling is the same as in upheaval buckling, expressed in Eqs. (11.6) or (11.8). Most offshore pipelines are likely to be susceptible to global buckling, because the resistance against lateral displacement is quite limited and might not be able to prevent lateral buckling. As an example, consider a straight 14-inch-diameter X60 steel submarine pipeline laying on the seabed, with pipe wall thickness 18.9 mm, and submerged weight 1.4 kN/m. The pipeline is slightly (partially) embedded in the seabed soil, and the lateral stiffness of soil is ks = 46.7 × 10−3 N/mm2 . The internal pressure p of the pipeline is 200 bar, and the temperature change T is 100°C. Using this data, the pressure component Fpr and the thermal component Fth of the total compression force are computed equal to 0.71 MN and 5.62 MN respectively, summing to a total compressive force F = 6.33 MN. It is possible to estimate the theoretical strength of the pipeline considering a straight elastic beam-column on elastic foundation, without initial geometric imperfection, using the corresponding formulae of Appendix B. Using the values of lateral √ soil stiffness ks and pipe beam bending stiffness EI, the ultimate force (Fu = 2 ks EI) that the pipe can sustain is 3.3 MN, significantly lower than the total force developed from pressure and temperature. One may notice that the above calculation is very simplified and considers an idealized situation, which may not represent the real conditions of the pipeline on the sea bottom, and most importantly, the axial friction between the pipe and the soil, which has a significant influence on the axial resistance and response of the pipeline. Moreover, a real pipeline on the seabed is not straight, the soil lateral stiffness is nonlinear, and inelastic deformation may develop in the pipeline during bending. Because of these reasons, a finite element model would be more appropriate for simulating the mechanical response and lateral buckling of offshore pipelines, considering all the above parameters. In any case, the above simple calculation indicates that, under normal operating conditions, offshore pipelines will most likely exhibit lateral buckling. Simple calculations show that in order to keep the total compressive force of the aforementioned pipeline below Fu , considering an acceptable safety factor, this requires a low value of T, which may not be realistic in many applications. Restraining the lateral deformation of offshore pipelines might be possible by trenching and burying. This would have been a good solution for shallow water sections of the pipeline, especially in shore approaches, but burial and trenching are uncommon in deep water. Similarly, rock dumping of a long section may prevent lateral buckling, but can be very expensive, especially in non-shallow water. Preheating the pipe reduces the axial compression force due to temperature Fth but does not solve the problem. The use of expansion loops at appropriate intervals may offer a solution for shallow water pipelines. Expansion loops are made of pipe bends with radius R = 5D, as shown in Fig. 11.14, are quite flexible devices and are capable of accommodating thermal expansion and reducing the axial force in the pipeline. If used to accommodate the axial expansion of the pipeline, they should be designed properly because significant cross-sectional ovalization develops in the bends, causing high stresses and strains (Guijt, 1999). As a conclusion, the main question is how to mitigate the effects of lateral buckling on pipeline integrity instead of how to prevent it. Towards this purpose the main design strategy is to allow lateral buckling and to control the location and the size of lateral buckles.

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Figure 11.14 Expansion loop used for accommodating thermal expansion effects.

Lateral buckling of offshore pipelines is often associated with the phenomenon of pipeline walking, which is the progressive axial translation of the pipeline under cyclic thermal loading due to: (a) repeated startups and shutdowns and (b) non-uniform distribution of axial force along the pipeline because of axial friction (Carneiro and Carvalhal, 2020). It has also been shown that pipeline walking is accentuated in seabed slopes (Castelo et al., 2019). Pipeline walking increases local deformation and strain in the lateral buckles and may result in loss of tension in a nearby steel catenary riser or, in the case of a flowline, increase of stresses and strains in a jumper. For more information on pipeline walking and on relevant mitigation measures the reader is referred to the results of the SAFEBUCK project (Carr et al., 2006; Bruton and Carr, 2011), and in the more recent papers by Curti et al. (2019) and Carneiro & Franco (2019).

11.3.1 Design strategy and buckle initiation techniques The main concept in offshore pipeline against lateral buckling is to “work with the buckle” and control its consequences, rather than “work against the buckle” and attempt to prevent it. The target of this design concept is to allow the formation of lateral buckling and control the lateral buckles along the pipeline in terms of their location and their size for the operating conditions under consideration. In an offshore pipeline exposed on the seafloor, multiple lateral buckles develop and thermal loading “feeds in” each buckle, expanding the buckle and inducing strain that may exceed the allowable strain limit. Between two consecutive lateral buckles, virtual anchors are formed at points of zero displacement, so that the pipeline consists of a series of pipeline segments “anchored” at their ends. Adopting the terminology of the DNV-RP-F110 document (Det Norske Veritas, 2019), the distance between two consecutive virtual anchor points is called “characteristic virtual anchor spacing”, denoted as (VAS)C . Increasing this distance, the feed-in to each buckle and the corresponding induced strain increase. The first step in lateral buckling design of unburied offshore pipelines is to determine the maximum spacing between the virtual anchors that a single buckle can sustain, i.e., the maximum induced strain is within the allowable limits. This maximum spacing is called “tolerable virtual anchor spacing”, denoted as (VAS)T (Collberg et al., 2011). For a safe design, the characteristic virtual anchor

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Figure 11.15 Snake-lay technique for controlling lateral buckling (plan view).

spacing should be less than the tolerable virtual anchor spacing: (VAS)C < (VAS)T

(11.26)

and this ensures that feed-in is distributed in the buckles along the pipeline in a proper manner, and that the design criteria in each buckle are satisfied. The next design step is to develop a buckle initiation strategy that triggers lateral buckling in a controlled manner, ensuring that (VAS)C is always less than (VAS)T so that the buckles are structurally safe. In support of developing a buckle initiation strategy, a detailed strain analysis of the pipeline is required that accounts for strain localization, inelastic effects, soil-pipe interaction and the correct pipeline alignment, considering the buckle initiation measure followed. There exist several buckle initiation techniques, which are outlined below (Sinclair et al., 2009).

11.3.1.1 Snake-lay The snake-lay method consists of laying the pipeline in a zig-zag alignment shown in Fig. 11.15, instead of following a straight alignment (Preston et al., 1999; Rundsag et al., 2008; Liu et al., 2013). This creates an initially “snaking” shape, which is magnified under axial compression, but in a controlled manner. Typical offset is suggested around 100 m and the pitch of the snaking configuration (LSL ) may vary between 2 and 5 km, so that the local radius of curvature at the buckle initiation locations is kept well below the critical curvature of the pipe (see Chapter 6).

11.3.1.2

Sleepers

Sleepers are pre-installed large-diameter pipe sections, placed in a direction perpendicular to the pipeline alignment (Solano et al., 2012; Liu and Li, 2017). The spacing of sleepers can vary from 1 to 3 km (Fig. 11.16). The pipeline lies on the sleepers, which cause vertical out-of-straightness of the pipeline, and reduce its lateral restraint, thus initiating pipeline lateral buckling and reducing the axial load. This method should be applied in places where fishing equipment and anchors may not cause pipeline “hooking”.

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Figure 11.16 Sleepers used to initiate offshore pipeline lateral buckling.

Figure 11.17 Distributed buoyancy modules used to control lateral buckling of offshore pipelines that lie on the seabed.

11.3.1.3

Distributed buoyancy

The distributed buoyancy method consists of reducing pipe weight over a short pipeline segment, typically up to 100 m, using buoyancy (flotation) modules that embrace the pipe at the desired buckle initiation sites (Peek and Yun, 2007; Solano et al., 2012). Because of reduced weight in this pipeline section, the lateral soil resistance is reduced, and this enables buckle initiation and reduction of the compression loads (Fig. 11.17). This technique should be used with caution to avoid interference of the uplifted pipeline with fishing equipment and anchors. Furthermore, on-bottom stability requirements of the pipeline should be re-examined.

11.3.1.4

Residual curvature

The residual curvature method is a convenient and cost-effective method, which receives increasing attention over the recent years. It is mainly used when the pipeline

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Figure 11.18 Residual curvature method for controlling and mitigating the effects of lateral buckling in reeled pipelines.

is installed with the reel-lay method, as shown in Fig. 11.18 (Roy et al., 2014; Cooper et al., 2017; Gallegillo et al., 2017), but it has also been proposed to be used with the S-lay method (Tewolde, 2017). The method consists of imposing out-of-straightness at several intervals along the pipeline, by introducing residual deformation at selected locations. It is achieved by adjusting the straightener on the reel vessel during laying, such that a residual local curvature is imposed at specific pre-determined locations along the pipeline, as shown in Fig. 11.18. This leads to a periodically repeated out-ofstraightness pattern on the seabed after installation, which acts as a buckle initiator when the pipeline is subjected to high compression force (Zhang and Kyriakides, 2021). The local configuration of the bent pipe is shown in Fig. 11.18 and is associated with a residual bending strain of about 0.25% at approximately every kilometer (Cooper et al., 2017).

References American Lifelines Alliance (2001). ALA guidelines for the design of buried steel pipe (with addenda through February 2005). ASCE-FEMA. Anastasopoulos, I., Gazetas, G., Bransby, M. F., Davies, M. C. R., & El Nahas, A. (2007). Fault rupture propagation through sand: finite-element analysis and validation through centrifuge experiments. Journal of Geotechnical and Geoenvironmental Engineering, 133(8), 943– 958. Bruton, D., & Carr, M. (2011). Overview of the safebuck JIP. In Offshore Technology Conference. OTC 21671. Bruton, D., Carr, M., Crawford, M., & Poiate, E. (2005). The safe design of hot on-bottom pipelines with lateral buckling using the design guideline developed by the safebuck joint industry project. In Deep Offshore Technology Conference. Vitoria, Espirito Santo, Brazil.

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Carneiro, D., & Carvalhal, R. (2020). Pipeline walking due to temperature transient: enhanced analytical calculations. In ASME International Conference on Offshore Mechanics and Arctic Engineering. OMAE2020–18360, 4, art. no. V004T04A035, virtual conference. Carneiro, D., & Franco, L. (2019). Walking anchors: when to fix one or both ways? In ASME International Conference on Offshore Mechanics and Arctic Engineering, OMAE201995359, 5B. Carr, M., Sinclair, F., & Bruton, D. (2006). Pipeline walking—understanding the field layout challenges, and analytical solutions developed for the SAFEBUCK JIP. In Offshore Technology Conference. OTC 17945. Houston, Texas. Castelo, A., White, D., & Tian, Y. (2019). Simple solutions for downslope pipeline walking on elastic-perfectly-plastic soils. Ocean Engineering, 172, 671–683. Collberg, L., Carr, M., & Levond, E. (2011). Safebuck design guideline and DNV RP F110. In Offshore Technology Conference. OTC 21575. Houston, Texas. Cooper, P., Zhao, T., & Kortekaas, F. (2017). Residual curvature method for lateral buckling of deepwater flowlines. In Offshore Pipeline Technology Conference. Amsterdam, The Netherlands. Curti, G., Pavone, D., Marchionni, L., Guyon, V., Perrin, F., & Pirinu, G. (2019). Challenges and lessons learnt from the design, fabrication and installation of pipe walking mitigations. In ASME International Conference on Offshore Mechanics and Arctic Engineering. OMAE2019-95055, 5B. Glasgow, Scotland, UK. Det Norske Veritas (2019). Global buckling of submarine pipelines structural design due to high temperature/high pressure. In Recommended Practice DNV-RP-F110 (amended 2021). Høvik, Norway. Gallegillo, M., Qi, X., Ofoha, G., Bhide, R., Messias, N., & Helland, T. (2017). Post-reeled behaviour of pipelines with global buckling mitigation by the residual curvature method. In ASME International Conference on Ocean, Offshore and Arctic Engineering, OMAE201762481. Trondheim, Norway. Guijt, J. (1990). Upheaval buckling of offshore pipelines: overview and introduction. In Offshore Technology Conference, OTC 6487. Houston, Texas. Guijt, W. (1999). Design considerations of high-temperature pipelines, In International Offshore and Polar Engineering Conference, ISOPE-I-99-223683-689. Brest, France. Hobbs, R. E. (1984). In-service buckling of heated pipelines. Journal of Transportation Engineering, 110, 175–189. Ju, G. T., & Kyriakides, S. (1988). Thermal buckling of offshore pipelines. Journal of Offshore Mechanics & Arctic Engineering, 110, 355–364. Kristiansen, N. Ø., Peek, R., Tørnes, K., & Carr, M. (2005). Designed buckling for HP/HT pipelines. Offshore Magazine, 65(10). Liu, M., & Cross, C. (2019). Pipeline rockberm design principles for UHB mitigation. In ASME International Conference on Ocean, Offshore and Arctic Engineering OMAE2019-95444, Glasgow, Scotland, UK. Liu, M., Sun, J., & Chang, G. (2013). Parametric study on the effectiveness of buckling initiators to control pipeline lateral buckling in deepwater applications. In Offshore Technology Conference, OTC-24485-MS, 1744-1750. Rio de Janeiro, Brazil. Liu, W., & Li, B. (2017). Study of sleeper’s impact on the deep-water pipeline lateral global buckling. In IOP Conference Series: Earth and Environmental Science, 81 MSETEE 2017, paper No. 012119. Palmer, A. C., & Baldry, J. A. S. (1974). Lateral Buckling of Axially-Constrained Pipelines. Journal of Petroleum Technology, JTP forum, 1283–1284.

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Palmer, A. C., Carr,M.,Maltby, T.,McShane, B., & Ingram, J. (1994). Upheaval buckling: What do we know, and what don’t we know? Oslo: Offshore Pipeline Technology Seminar. Palmer, A. C., Ellinas, C. P., Richards, D. M., & Guijt, J. (1990). Design of submarine pipelines against upheaval buckling. In Offshore Technology Conference, OTC 6335. Houston, Texas. Peek, R., & Yun, H. (2007). Floatation to trigger lateral buckles in pipelines on a flat seabed. Journal of Engineering Mechanics, 133(4), 442–451. Preston, R., Drennan, F., & Cameron, C. (1999). Controlled Lateral Buckling of Large Diameter Pipeline by Snaked Lay. International Offshore and Polar Engineering Conference, ISOPEI-99-126, Brest, France. Roy, A., Rao, V., Charnaux, C., Ragupathy, P., & Sriskandarajah, T. (2014). Straightener settings for under-straight residual curvature of reel laid pipeline. In ASME International Conference on Ocean, Offshore and Arctic Engineering, OMAE2014-24513. San Francisco, California. Rundsag, J. O., Tørnes, K., Cumming, G., Rathbone, A. D., & Roberts, C. (2008). Optimised snaked lay geometry. In International Offshore and Polar Engineering Conference, ISOPEI-08-188. Vancouver, Canada. Schaminee, P. E. L., Zorn, N. F., & Schotman, G. J. M. (1990). Soil response for pipeline upheaval buckling analyses: full-scale laboratory tests and modelling. In Offshore Technology Conference, OTC 6486. Houston, Texas. Seyfipour, I., Walker, A., & Kimiaei, M. (2019). Local buckling of subsea pipelines as a walking mitigation technique. Ocean Engineering, 194, 106626. Sinclair, F., Carr, M., Bruton, D., & Farrant, T. (2009). Design challenges and experience with controlled lateral buckle initiation methods. In ASME International Conference on Ocean, Offshore and Arctic Engineering, OMAE2009-79434. Honolulu, Hawaii, USA. Solano, R. F., Antunes, B. R., Hansen, A. S., Bedrossian, A., & Roberts, G. (2012). Comparison of design and operational behaviour of an offshore pipeline with controlled lateral buckling. In ASME International Conference on Ocean, Offshore and Arctic Engineering, OMAE2012-83646. Rio de Janeiro, Brazil. Tewolde, A. (2017). Pipelay With Residual Curvature. MS Thesis. Stavanger, Norway: Faculty of Science and Technology, University of Stavanger. Vazouras, P., Tsatsis, A., & Dakoulas, P. (2020). Numerical Simulation and Experimental Testing of Global Buckling of Buried Pipelines Due to High Pressure and temperature. MIS 5005623 Project. Athens, Greece: Ministry of Education. Vazouras, P., Tsatsis, A., & Dakoulas, P. (2021). Thermal upheaval buckling of buried pipelines: experimental behavior and numerical modeling. Journal of Pipeline Systems Engineering and Practice, ASCE, 12(1), 04020057. Zhang, W., & Kyriakides, S. (2021). Controlled pipeline lateral buckling by reeling induced curvature imperfections. Marine Structures, 77, 102905.

Mechanically lined pipes

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Hydrocarbon mixtures, transported by steel pipelines, may contain corrosive ingredients, such as hydrogen sulphide (H2 S) or hydrogen chloride (HCl), carbon dioxide (CO2 ) and water (H2 O). Those ingredients result in pipeline internal corrosion, which may cause failure with significant environmental and economic impact. To protect the pipeline from internal corrosion, bi-metallic pipes referred to as “clad pipes” are often used, consisting of a thick-walled low-alloy carbon steel (“outer pipe” or “carrier pipe”) which provides strength, and a thin inner layer (“liner pipe”) made of a corrosion resistant alloy (CRA) material for corrosion protection (Fig. 12.1). There exist two categories of bi-metallic pipes, namely the “metallurgically clad” pipes and the “mechanically clad” or “mechanically lined” pipes. In metallurgically clad pipes, the CRA layer is metallurgically bonded (cladded) to the internal surface of the carbon steel pipe, and this can be performed in several ways: weld cladding, centrifugal casting or rolling of laminated plates. Regardless of the method of manufacturing process, the main feature of this bi-metallic pipe is the union of the two metallic materials in one mass, so that a single pipe is produced. On the other hand, mechanically lined pipes, also called simply “lined pipes”, are doublewall bi-metallic pipes, which consist of two separate components, a corrosion-resistant liner inserted into a low-alloy outer cardon steel pipe (Fig. 12.1). The bond between the two pipes is only mechanical and the two pipes are not fused together. This means that the two materials remain two different masses, and this is their main difference from the metallurgically bonded pipes. Lined pipes constitute an efficient and cost-effective solution for internal corrosion protection. The CRA liner concept for corrosion protection is rather old in the pipeline industry but has gained significant attention in recent deep-water pipeline applications, mainly in relatively small diameter pipelines installed by the reeling method. Fig. 12.2 shows schematically a typical girth weld for connecting lined pipes, performed before spooling the lined pipeline around the reel of the installation vessel. The structural behavior of lined pipes under bending loading is of particular interest for safeguarding their structural integrity during the reeling process. Bending loading associated with excessive plastic deformation during reeling may lead to the development of cracks at the girth weld area. Furthermore, repeated loading during reeling may also cause low-cycle fatigue cracks at the girth weld. In addition to weld cracking, bending may also result in the development of wrinkles and buckles at the compression side of the thin-walled liner due to local buckling, while the thick-walled outer pipe remains structurally stable (Fig. 12.3). The wrinkles are an impediment to internal fluid flow, they block pipeline pigging equipment, and may lead to the development of fatigue cracks at the wrinkled area because of repeated operational loads and local stress concentration. Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00013-X c 2023 Elsevier Inc. All rights reserved. Copyright 

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(A)

(B)

Figure 12.1 The lined pipe concept: (A) numerical model, (B) lined pipe components (photo by S. A. Karamanos).

Figure 12.2 Schematic representation of lined pipe weld.

The present chapter focuses on liner wrinkling under predominantly bending loading. Following a brief presentation of the main fabrication methods and the relevant experiments performed at Delft University of Technology (TU Delft), the main aspects of lined pipe bending are examined, including the effects of internal pressure and the influence of the fabrication process on liner wrinkling. Finally, some key features of lined pipe structural behavior under cyclic bending are presented, with reference to the loading conditions during pipeline reeling installation.

12.1

Fabrication of lined pipes

Mechanically lined pipes can be fabricated by either a purely mechanical manufacturing process (Yoshida et al., 1981), or by a thermo-mechanical process, as explained in detail by de Koning et al. (2004). Those two processes are briefly described below. The manufacturing process of hydraulically expanded lined pipes is purely mechanical and starts by inserting the liner into the outer pipe (Fig. 12.4A). The outside diameter of the CRA liner is somewhat smaller than the inside diameter of the outer pipe.

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Figure 12.3 (A), (B) Wrinkled liners in lined pipes tested at TU Delft (photos by S. A. Karamanos). (C) Wrinkles developed in the liner pipe from finite element analysis.

After the liner pipe is inserted, a small gap exists between the two pipes. Subsequently, hydraulic expansion of the pipes is performed by applying internal pressure up to a stage where the two pipes are in contact, and subsequently, depressurization leads to the final stage of the pipe. Depending on the level of pressurization, the outer pipe either remains elastic or expands inelastically, and this influences the degree of mechanical bonding between the two pipes at the final stage. In the first case, the method is called “elastic expansion”, and in the second case “plastic expansion”. Fig. 12.4B shows schematically the steps of fabricating a thermo-mechanically bonded lined pipe, called tight-fit (TFP) pipe. Initialy the outer pipe is heated, and expands, so that the liner pipe is inserted into it. Subsequently, internal pressure is applied expanding the liner. At a certain stage of this expansion the two pipes establish contact and expand together. Finally, depressurization and cooling leads to the final configuration of the pipe, and the mechanical bonding between the two pipes. This fabrication method results in hoop compression of the liner because of its confinement by the outer pipe. The degree of mechanical bonding and the magnitude of residual stress in the liner (Fig. 12.4C) depend, among other factors, on the amount of expansion during the process, the maximum temperature reached by the liner, and the value of initial gap between the two pipes (g0 ). Furthermore, depending on the duration of the extension step, the liner temperature may reach the temperature of the outer pipe (fully-heated liner) or a lower temperature (partially-heated liner).

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(A)

(B)

(C)

Figure 12.4 Lined pipe manufacturing: (A) hydraulically expanded lined pipes, (B) thermomechanical tight-fit pipe (TFP), (C) residual stresses after manufacturing.

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Figure 12.5 TU Delft experiments of lined pipes used by Focke (2007): (A) general view from experiments (photo by Arnoldus (Nol) M. Gresnigt, published with permission), (B) schematic representation of the set-up.

12.2

A brief presentation of the TU Delft experiments

Unique laboratory experiments on lined pipes have been conducted during the period 2003–11 at Delft University of Technology (TU Delft), mainly motivated by the use of lined pipes in reeling applications. The work consisted of two sets of monotonicallyincreasing full-scale bending experiments on 12-inch-diameter TFP lined pipes, described in detail in the Ph.D. theses of Focke (2007) and Hilberink (2011). Those experiments are outlined below for the sake of completeness.

12.2.1 Experiments by Focke (2007) The full-scale experiments performed by Focke (2007) investigated the initiation and development of liner wrinkling and the ovalization of TFP pipes during lined pipe spooling in the reeling process, using a horizontal bending set-up, shown in Fig. 12.5, which simulated the reel geometry. A “lift” force was applied in the rig, to control the initial reaction force of the reel on the pipe at the initial contact point. A series of reel simulators (concrete blocks) were used, with radii ranging from 9 m to 5.5 m.

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The specimens were bent stepwise to smaller bending radii Ri using successively the available reel simulators (Fig. 12.5), except for one specimen, which was bent directly over the smaller radius rig (5m) for comparison purposes. A special-purpose laser wagon was developed to scan the inside of the TFP pipe and measure the size of liner wrinkles prior, during and after the application of bending. In this testing program, seven reeling simulation tests were performed on 12.75-inch outer diameter TFP lined pipes, with the set-up shown schematically in Fig. 12.5B. The 3-mm-thick 316L liner pipe had a longitudinal seam inserted into the ERW X65 steel outer pipe. Two of those specimens contained girth welds in the highly bending region as shown in Fig. 12.5B. Prior to testing the TFP lined pipes, a single-walled 12.75-inch pipe was tested in the bending rig. Furthermore, axial compression tests on TFP pipe segments and smallscale reeling tests on 22-mm outer diameter single-walled pipe were also performed, and an extensive comparison with analytical calculations was made. All those supplemental tests enabled the optimization of the full-scale bending set-up. Measurements on liner wrinkling were obtained throughout the reel bending tests and provided very valuable information on the size of wrinkles (buckling wavelength) and their evolution with the applied curvature. During all tests, the outer pipe remained structurally stable, despite its cross-sectional ovalization and the significant amount of inelastic deformation induced in the pipe by bending. The experimental results also indicated a decrease of liner wrinkling amplitude, when the mechanical bonding between the two pipes increased, and this was attributed to the higher radial stress, which increased friction between the liner and the outer pipe. The presence of a circumferential (girth) weld resulted in more severe wrinkling at lower values of bending curvature, most likely due to the uneven distribution of the contact stress between the reel and the lined pipe specimen at that location. Finally, the experimental results did not show any significant difference in liner wrinkling and ovalization between stepwise bending using successively all reel simulators and single-step bending to the final radius.

12.2.2 Experiments by Hilberink (2011) Full-scale experiments were also performed at TU Delft on 6-meter-long TFP lined pipe specimens, using a four-point bending set-up (Fig. 12.6). Bending was applied by controlling the vertical upward displacement of the two hydraulic cylinders at points A and A’, such that the two upward displacements were always equal, while the vertical displacements of the pipe at B and B’ were restrained. Each cylinder was equipped with a load cell and with an actuator to monitor the force and the corresponding displacement (Hilberink, 2011). In the 1.6-meter-long section BB’ of the bent specimen with constant bending moment, the outer pipe has been instrumented with strain gauges symmetrically placed around the mid span at the compression and the tension side of the pipe. A curvature-measuring device and several ovalization-measuring devices were also placed symmetrically around the mid span of the pipe. Furthermore, a laser trolley, similar to the one used in the experiments of Focke (2007), was used to scan the

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Figure 12.6 TU Delft experiments of lined pipes (Hilberink, 2011): (A) experimental set-up (photo by S. A. Karamanos), (B) schematic representation of the set-up.

inside surface of the liner, including a 360-degree scanning along a defined longitudinal region and longitudinal scanning along the most compressed liner generator. A full scan was performed at each specimen prior to bending to determine the initial geometrical imperfections of the liner. The amount of mechanical bonding between the outer and liner pipe was determined by means of saw cut tests, and measuring the opening gap of the liner. The results from these full-scale tests complemented the experimental results of Focke (2007). Special issues addressed by these tests were the effects of circumferential (girth) weld, outer pipe thickness, pipe coating and internal pressure on the bending response. In particular, it was reported that: r r

the value of curvature at the onset of wrinkling was influenced by the presence of a circumferential weld; the increase of outer pipe thickness resulted in liner pipe wrinkling at larger curvature;

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Figure 12.7 Stress–strain curves of the outer pipe and liner pipe materials (lined pipe LP1). r r

the coating process had negative influence on the specimen during bending, resulting in earlier wrinkling; the presence of internal pressure (equal to 2 MPa) had a positive effect on the bending deformation capacity, resulting in wrinkling initiation at larger values of curvature and in smaller amplitude wrinkles.

12.3

Structural response of lined pipes under monotonic bending

The experimental results from TU Delft described in the previous section have been a very good starting point for the development of numerical models, which elucidated the main aspects of lined pipe wrinkling. Such a model that allowed for an in-depth numerical investigation of liner pipe wrinkling has been presented by Vasilikis and Karamanos (2012), and has been extended more recently by Gavriilidis and Karamanos (2019). The latter work simulated the localization of wrinkling patterns, examining the effects of internal pressure. The results from these two works constitute the basis for the present discussion. In the following, the main features of lined pipe bending are summarized for a 12-inch nominal diameter mechanically lined pipe, denoted as LP1. The outer pipe has outside diameter equal to 325 mm and thickness 14.3 mm, and the liner is 3-mm-thick. The outer pipe is made of carbon steel grade API 5L X65. The nominal stress-strain curve of the X65 material, obtained from a uniaxial tensile test, is shown in Fig. 12.7 with elastic modulus E = 210,000 MPa, Poisson’s ratio ν e = 0.3, yield stress σ Y,o = 566 MPa, plastic plateau up to 2% engineering strain, and strain hardening that corresponds to 614 MPa stress at 9% strain. The thin-walled liner is made of stainless steel (AISI 316L). The stress–strain curve of the liner pipe

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Figure 12.8 Ovalization analysis model and liner pipe detachment (liner pipe LP1): (A) definition of ovalization, (B) front view of liner detachment.

316L material is also shown in Fig. 12.7 with parameters El = 193,000 MPa, ν e = 0.3, proportional limit σ p,l = 250 MPa at 0.13% strain, and yield stress σ Y,l = 298 MPa, corresponding to a 0.2% residual (plastic) strain. Two cases are examined, namely a tight-fit lined pipe (TFP) with compressive prestress (residual stress) in the liner σ res = 166 MPa, and an initially stress-free lined pipe (σ res = 0), called stress-free pipe or “snug-fit” pipe (SFP).

12.3.1 Ovalization and detachment Fig. 12.8 shows a slice model for the ovalization analysis of the bi-metallic lined pipe under consideration. This is practically a cross-sectional analysis, similar to the one described in Section 6.6, also referred to as “generalized plane strain” analysis. The model assumes constant deformation along the pipe and does not take into account the formation of wrinkles in the pipe wall. The main result from this ovalization analysis is that both pipes ovalize, but the thin-walled inner pipe ovalizes more than the outer pipe (ul > uo ), so that the liner detaches from the outer pipe. Liner detachment is depicted in Fig. 12.8 and is an essential feature for understanding lined pipe bending response. Fig. 12.9 shows the evolution of liner detachment with respect to the applied curvature. The value of detachment is normalized by the liner thickness (see also Fig. 12.8A) and the value of bending curvature k is normalized by the curvature-like parameter ko = to/D2m,o (Dm,o is the mean diameter of the outer pipe, with Dm,o = Do − to ), so that κ¯ = k/ko. Because of detachment, the compressed part of the liner cross section forms a thin-walled cylindrical panel, which is subjected to axial compression, and is supported only at its two side edges (a) and (b) that are in contact with the outer pipe, shown schematically in Fig. 12.10. Eventually this compression causes buckling of the panel in the form of wrinkling. Fig. 12.9 also shows that the amount of detachment depends on the level of liner prestress in the circumferential direction because of the manufacturing process. The two diagrams in Fig. 12.9 indicate that the presence of hoop prestressing (TFP pipe)

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Figure 12.9 Detachment between liner and outer pipe, in terms of applied bending curvature (pipe LP1, 2D ovalization analysis).

Figure 12.10 “Curved panel” after detachment, subjected to axial compression (schematic).

delays the growth of detachment, and this has a positive effect on the bending response of lined pipes.

12.3.2 Initial wrinkling When a critical state is reached at the compression zone of the detached liner (i.e., at the “unsupported panel” of Fig. 12.10), the liner becomes structurally unstable and buckles. This is a bifurcation type of structural instability, with transition from a smooth ovalization stage, constant along the pipe, to a periodic wrinkling pattern, shown in Fig. 12.11 from finite element analysis. Determining the stage at which liner wrinkling initiates is not a trivial task. For elastic liners, it is possible to obtain a reliable semianalytical prediction of the onset of wrinkling using the results from the ovalization analysis and considering the Local Buckling Hypothesis (LBH) discussed in Section 4.9. The LBH hypothesis assumes that wrinkling of a bent thin-walled cylinder starts

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Figure 12.11 First bifurcation of liner pipe (lined pipe LP1): (A) uniform wrinkling, (B) half-wavelength of buckling mode.

Figure 12.12 Local Buckling Hypothesis (LBH).

when the local compressive stress at the compression zone σ x0 becomes equal to the buckling stress of an equivalent uniformly compressed cylinder, which has the same elastic material properties, the same thickness and radius equal to the hoop radius of curvature at the compression zone, accounting for the effects of ovalization (Fig. 12.12). Therefore, the buckling condition is expressed as follows:

σx0 = 

E



t

3(1 − νe 2 ) rθ0

 (12.1)

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Figure 12.13 First bifurcation of liner pipe; detachment of points (1) and (2) for SFP and TFP.

Considering that the longitudinal stress σ x0 and hoop curvature 1/rθ0 at the buckling point (i.e., at θ = π /2) are functions of the applied curvature (as obtained by the ovalization analysis described in the previous paragraph), the above equation can be solved in terms of curvature, and this determines the critical curvature of the pipe, corresponding to the onset of liner wrinkling. The corresponding buckling halfwavelength Lhw of the liner pipe is given by the following expression:  Lhw =

π4   12 1 − νe 2

1/4

√ rθ0tl

(12.2)

The LBH predictions for liners with elastic material behavior are in very good agreement with the finite element results reported by Vasilikis (2012) and Vasilikis and Karamanos (2012). However, extending this methodology for metal liners with elastic-plastic material behavior, may not be a straightforward procedure, and therefore, a numerical formulation and solution is much preferable. Fig. 12.13 shows the detachment of points (1) and (2), which are located at the crest and the valley of the wrinkle (see also Fig. 12.11B). The detachment is plotted in terms of the applied bending curvature k, expressed in its normalized form κ¯ = k/ko (ko = to/D2m,o). The value of curvature is k = φ/L, where φ is the relative rotation between the two end sections of the model and L is the length of the model. The numerical results indicate that this wavy pattern does not occur suddenly. On the contrary, there is a gradual deviation of the two curves, implying a “tangential” type of bifurcation with no distinct bifurcation point. The smooth transition to post-buckling in lined steel pipes facilitates its numerical simulation, in the sense that it is not necessary to introduce a negligible imperfection to trigger bifurcation. Because of this “tangential” bifurcation, the definition of buckling (critical) curvature κ¯ cr may be rather ambiguous. To overcome this ambiguity, the buckling curvature κ¯ cr is defined as the

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Figure 12.14 Buckling configuration of liner from a lined pipe tested in the TU Delft experimental program: (A) general view of the buckled area, (B) detail of the buckled configuration (behind the local buckle at the back of the pipe), showing a uniform wrinkling pattern (photos by S. A. Karamanos).

curvature at which the detachment of point (2) in Fig. 12.13 starts decreasing. The values of κ¯ cr for the SFP and TFP pipes under consideration are equal to 0.898 and 1.19 respectively, as shown Fig. 12.13 and correspond to detachment values () equal to 4% and 3.4% of the liner wall thickness tl for the SFP and the TFP pipe respectively. Comparing the bending behavior of SFP and TFP pipes in terms of uniform wrinkling initiation, it is readily concluded that pipe prestressing has a beneficial effect on value of critical curvature κ¯ cr . Fig. 12.14B depicts a uniform pattern developed at the compression part of a lined pipe under bending and is a detail of the buckled shape shown in Fig. 12.14. It is not easy to detect this pattern with naked eye because the corresponding detachment values are less than 5% of liner thickness tl . The numerical results also show that the value of buckling half-wavelength Lhw shown in Fig. 12.11B is not affected by the presence of prestressing. In the present case, the values of Lhw are calculated equal to 43 mm for both the SFP pipe and the TFP pipe.

12.3.3 Secondary bifurcation and localization Upon first wrinkling, the height of the wrinkled pattern increases rapidly, as shown in Fig. 12.13, due to the increase of inward displacement of point (1). On the other hand, point (2), soon after the reversal of displacement direction, establishes contact with the outer pipe. The possibility of this wavy pattern to exhibit a second bifurcation has been examined, considering a lined pipe model of length equal to 2Lhw , where Lhw is the half-wavelength determined above (Fig. 12.11B). The double wavelength of second bifurcation with respect to the wavelength of first bifurcation is consistent with experimental measurements (Focke, 2007), and with Koiter’s argument, discussed in Section 6.2. The buckling shape obtained from this model is shown in Fig. 12.15 and is characterized by a central (main) buckle, which is symmetric with respect to the plane of bending, and four adjacent (minor) buckles, also observed in the experiments (Fig. 12.14A). The liner response in terms of the detachment of points (1), (2) and (3)

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Structural Mechanics and Design of Metal Pipes

(B)

Figure 12.15 Second bifurcation of liner pipe: (A) buckling half-wavelength, (B) entire buckled zone (lined pipe LP1).

is shown in Fig. 12.16 for the 12-inch-diameter pipe under consideration and reflects the entire response of the lined pipe up to this stage. Initially, the liner exhibits constant ovalization along its length and the responses of the three points coincide. Upon uniform wrinkling, the curves of points (1) and (3) continue to coincide, whereas the curve of point (2) deviates from the other two curves and starts decreasing. At some level of bending curvature, the curves for points (1) and (3) deviate, and the shape transforms to the pattern shown in Fig. 12.15. The second bifurcation is also gradual and has a “tangential” form, with no distinct bifurcation point. Following the definition of κ¯ cr (first bifurcation), the buckling  , is defined as the curvature at which curvature at second bifurcation, denoted as κ¯ cr the detachment of point (3) reaches a maximum value before it starts to decrease. The numerical results show the beneficial effects of hoop prestressing on the value   . Second bifurcation (κ¯ cr ) is associated with wrinkle heights of buckling curvature κ¯ cr significantly higher than those corresponding to first bifurcation (κ¯ cr ). Therefore, it is reasonable to consider that liner failure occurs when the second bifurcation takes place. The above numerical results were obtained for imperfection-free (perfect) pipes. Numerical results reported by Vasilikis and Karamanos (2012) indicated that the  is sensitive to initial geometric imperfections, response and the value of curvature κ¯ cr for both SFP and TFP lined pipes. The imperfection sensitivity is more severe when the initial imperfect geometry of the liner has the form of the secondary buckling mode, rather than the first buckling mode. The results show that for imperfection amplitude equal to 10% of the liner thickness (i.e., about 0.3 mm), the value of critical  may be reduced by about 50% with respect to the critical curvature of the curvature κ¯ cr corresponding imperfection-free pipe. Numerical results from the above finite element models were found in very good agreement with the experimental data reported by Focke (2007) and Hilberink (2011) in terms of the value of buckling wavelength and the evolution of wrinkle height with respect to the applied curvature. Soon after second bifurcation and with increasing bending curvature, the buckling shape localizes to the deformation pattern of Fig. 12.17. Upon wrinkling localization,

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Figure 12.16 Finite element analysis of lined pipes; evolution of detachment at points (1), (2) and (3) of pipe LP1.

the deformation is no longer uniform along the pipe and the curvature of the bent/buckled pipe should be considered as an average for the pipe segment under consideration. In such a case, the curvature k may not be uniquely defined. In the present results it is considered equal to k = φ/L, where φ is the relative rotation between the two end sections of the pipe segment and L is the length of the pipe segment under

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(A)

(B)

Figure 12.17 Localized pattern in lined pipes; first and second bifurcation: (A) TU Delft experiment (photo by S. A. Karamanos), (B) finite element simulation.

consideration. The value of curvature corresponding to local buckling formation of the liner is denoted as ku and is a characteristic value of curvature for the pipe segment.

12.3.4 Effect of internal pressure The beneficial effect of internal pressure on the bending deformation capacity has already been noticed in Section 6.5 for single-walled pipes. Full-scale experiments on internally-pressurized lined pipes have shown a significant delay of liner wrinkling with respect to the non-pressurized case (Toguyeni and Banse, 2012; Sriskandarajah et al., 2013a; 2013b). In those studies, the level of pressure was quite high, about 50% of nominal yield pressure of the liner. Furthermore, in a series of patents (Endal et al., 2012; Mair et al., 2013; Howard & Hoss, 2016), the application of high levels of internal pressure (30 bar) is suggested to prevent liner wrinkling during reeling installation of lined pipes. On the other hand, the numerical results by Yuan and Kyriakides (2014; 2015) indicated that modest levels of internal pressure (up to 6.9 bar) result delaying liner wrinkling and buckling by a substantial amount with respect to the non-pressurized case. However, in all the above studies, the optimum level of internal pressure was not determined. The results shown in Fig. 12.18 are part of an extensive study on internallypressurized lined pipe response under bending, using finite element models (Gavriilidis and Karamanos, 2019; Gavriilidis, 2021) and show maximum liner detachment in terms of bending curvature. The diagrams demonstrate that the introduction of a relatively low level of internal pressure, up to 10% of nominal yield pressure of the liner (pY,l = 2σY,l tl /dm,l ) is adequate to stabilize the liner cross-section and prevent liner wrinkling for the bending curvature range of interest. Fig. 12.19 shows that for low internal pressure (up to 5% of pY,l ), liner detachment results in the formation of a local buckling pattern, characterized by a main buckle and four adjacent minor buckles, very similar to the non-pressurized case. However, increasing the internal pressure level above 5% of pY,l , the uniform wrinkling stage vanishes, the liner buckles in the form of a localized inward pattern, and the value of critical curvature ku is significantly increased. At 10% of pY,l , the pressure stabilizes the liner cross-section by a substantial amount and the formation of local buckling is significantly delayed. Furthermore, pressurized bending response is sensitive to initial geometrical imperfections, as shown in

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Figure 12.18 Normalized detachment at the center of the main buckle of lined pipe LP1 for different levels of internal pressure: (A) SFP pipe, (B) TFP pipe.

Fig. 12.20 for SFP and TFP pipes and for internal pressure equal to 10% of pY,l . The SFP lined pipe is more sensitive to initial imperfections. The main conclusion from the above results and all the results reported by Gavriilidis and Karamanos (2019) is that the introduction of relatively low levels of internal pressure, up to 10% of the yield pressure of the liner, delays significantly liner

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Figure 12.19 Liner buckled configuration for a TFP pipe, with the distribution of equivalent plastic strain at normalized curvature κ¯ = 3, for different levels of internal pressure (lined pipe LP1).

wrinkling and increases the deformation capacity of the lined pipe, with beneficial effects on reeling installation process.

12.4

Effect of fabrication process on bending response of lined pipes

The fabrication process of mechanically-lined pipes introduces significant inelastic deformation in the liner pipe, as noticed in Section 12.1, and it is expected to be a significant factor affecting liner wrinkling. For simulating the manufacturing process and its effect on the mechanical behavior of lined pipes under bending, a 12-inch nominal diameter pipe is considered, denoted as LP2. It consists of a thick-walled outer pipe, made of X70 steel grade, and a thin-walled inner pipe, made of stainless steel 316L. The outside diameter (Do ) and wall thickness (to ) of the outer pipe are equal to 12.75 in (323.85 mm) and 15.9 mm respectively. The outside diameter (Dl ) and thickness (tl ) of the liner are 289.25 mm and 2.8 mm, respectively, corresponding to an initial radial gap (g0 ) between the liner and the outer pipe equal to 1.4 mm (50% of tl ). The materials of liner and outer pipe are shown in Fig. 12.21 in terms of their cyclic stress-strain behavior.

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Figure 12.20 Evolution of maximum liner detachment in lined pipes with respect to applied curvature for different imperfection amplitudes (LP1, p/pY = 10%): (A) SFP pipe, (B) TFP pipe.

12.4.1 Simulation of fabrication process Two different procedures of lined pipe manufacturing are presented below: (a) the hydraulic expansion of both pipes up to elastic or plastic deformation in the outer pipe and (b) the thermo-mechanical process of tight-fit-pipes (TFP). The finite element

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Figure 12.21 Stress–strain curves of outer and liner pipe material (lined pipe LP2): (A) outer pipe, (B) liner pipe.

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Figure 12.22 Normalized hoop stress with respect to normalized change in diameter of both pipes (liner and outer), during plastic hydraulic expansion of the outer pipe (lined pipe LP2).

models developed for this analysis assume constant state of stress and deformation along the pipe, and therefore, a slice of the pipe is adequate for modeling this process. The additional assumption of axisymmetric deformation simplifies further the numerical model and allows the consideration of only a small sector of pipe cross-section. Nonlinear kinematic hardening models are used to describe the material behavior of the liner and the outer pipe. The models are outlined in Appendix D and are calibrated with the cyclic stress-strain curves of Fig. 12.21. The most common version of hydraulic expansion consists of applying high internal pressure during the pressurization step of the lined pipe, which exceeds the plastic pressure of the outer pipe (pY,o = 2σY,oto/Dm,o) as shown in Fig. 12.22. It is called “plastic expansion” because of the inelastic deformation induced in the outer pipe. For the 12-inch pipe under consideration, the stress-strain curves of the two materials are shown in Fig. 12.22, for pressurization level 59.9 MPa, equal to 117% of pY,o . Initially, the liner is inserted into the outer pipe, and the two pipes are internally pressurized. The 0 → 1 ) until it establishes contact with liner expands elastically, then inelastically ( 1 . Subsequently, both pipes expand together ( 1 → 2 ). At the the outer pipe at stage  2 , the outer pipe has been subjected to substantial inelastic deformation. end of stage  2 → 3 ). Because of the This step is followed by depressurization of the two pipes ( larger elastic deformation in the outer pipe, compared with the one in the liner, the two pipes continue to remain in contact after depressurization and this results in mechanical bonding. This process has also been simulated with a three-dimensional finite element model, and it was found that for this pipe, the liner pipe wall does not exhibit any wrinkles after depressurization (Gavriilidis and Karamanos, 2020).

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Figure 12.23 Normalized hoop stress with respect to normalized change in the diameter of both pipes (liner and outer), during elastic hydraulic expansion of the outer pipe (lined pipe LP2).

“Elastic expansion” is a special version of hydraulic expansion, described above. First, the liner is inserted into the outer pipe, followed by the application of internal pressure up to a level below the yield pressure of the outer pipe (pY,o ), as shown in 2 , the outer pipe expands linearly and remains elastic. This Fig. 12.23. Up to stage  method is referred to as “elastically expanded” in the sense that the outer pipe remains elastic during the entire fabrication process. In the simulation shown in Fig. 12.23, the pressure level is 80% (41.1 MPa) of pY,o . Depressurization of both pipes leads to the 2 → 3 ). In “elastic expansion”, the final state final configuration of the lined pipe ( depends highly on the initial radial gap (g0 ) values between the liner and the outer pipes. Initial gap values (g0 ) ranging from 35% to 75% of the liner wall thickness (tl ), results in residual radial gap values (gr ) at the end of the process, which are smaller by one order of magnitude, compared to g0 . The tight-fit pipe (TFP) fabrication follows a thermo-mechanical process, and it is shown in Fig. 12.24 for the 12-inch-diameter pipe under consideration. In this analysis, the thermal expansion coefficients of the two materials are α l = 1.62 × 10−5 K−1 and α o = 1.3 × 10−5 K−1 for the liner and the outer pipe respectively, and the outer pipe 0 → 1 ). During heating ( 0 → 1 ), the liner is slightly is heated up to To = 680 K ( deformed due to coupling between the outer and the liner pipe in the numerical model. 1 → 2 ) up to 31.4 MPa, which is Subsequently, the liner is pressurized internally ( nearly 6 times the yield pressure of the liner (pY,l ). At 8.8 MPa of internal pressure, the liner establishes contact with the outer pipe and the two pipes continue to expand together. In this stage, the temperature of the outer pipe is assumed to remain constant.

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Figure 12.24 Tight-fit pipe manufacturing process with full heating of the liner pipe (lined pipe LP2).

Upon contact of the two pipes, the liner is heated, and if the duration of this step is long enough, the liner temperature reaches the temperature of the outer pipe (Tl = To ). In such a case, the process is called “fully-heated” TFP. The thermal hoop strain increases significantly and tends to increase the diameter of the liner (εh,T = (Dl /Dl )T ). However, due to lateral confinement by the outer pipe, the hoop tension in the liner gradually decreases and hoop compression develops, represented by the sharp drop of the hoop stress of the liner shown in the diagram of Fig. 12.24. The analysis also shows that reverse plastic loading (RPL) occurs in the liner material during this pressurization 2 . After depressurization step, denoted by the horizontal arrow before the end of stage  3 , residual hoop compression develops in the liner pipe, due to confinement at stage  by the outer pipe. The “partially-heated” TFP process is shown in Fig. 12.25 for the 12-inch-diameter pipe under consideration. In this case, the liner is partially heated after its contact with the outer pipe up to Tl = 388 K, which is 57% of the outer pipe temperature To . The liner pipe exhibits less thermal hoop expansion, resulting in a smaller decrease of hoop stress, as shown in Fig. 12.25. Reverse plastic loading (RPL) occurs during 3 . At depressurization, denoted with the horizontal arrow before the end of stage  3 ), the liner pipe is under higher hoop compression the end of the process (stage  stress, compared with the fully-heated pipe. Numerical results obtained by Gavriilidis and Karamanos (2020) indicated that the degree of mechanical bonding in TFP lined pipes and the value of residual hoop compression depends highly on the maximum temperature of the liner. An important note should be made at this point, which refers to the material model that should be used for simulating the manufacturing process. The results show that

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Figure 12.25 Tight-fit pipe manufacturing process with partial heating of the liner pipe (lined pipe LP2).

reverse plastic loading may occur during depressurization, and therefore, it is necessary that the material model accounts for the Bauschinger effect. This also implies that the use of a simple plasticity model for describing the liner pipe material (e.g., isotropic hardening), may not be suitable for simulating accurately the manufacturing process.

12.4.2 Effects of fabrication process on bending response Numerical results are presented in Fig. 12.26 for the 12-inch lined pipe LP2 using a two-stage finite element model. In the first stage of the analysis, the manufacturing process is performed and, subsequently, the analysis proceeds to bending of the lined pipe (Gavriilidis and Karamanos, 2020). The model length is 1.32 pipe diameters (11 half-wavelengths) and appropriate symmetry has been imposed. In the z = L plane, a reference node is imposed, appropriately coupled with the corresponding cross-section, so that the lined pipe end may ovalize on the rotated plane during bending. Rotation is applied at the reference node. In all cases, the initial radial gap (g0 ) is equal to 50% of liner thickness. The stress-strain curves of the outer and the liner pipe materials are those depicted in Fig. 12.21 respectively. The maximum normalized detachment () of the liner pipe occurs at the main buckle [point (1), in Fig. 12.15], and it is shown in Fig. 12.26A in terms of bending curvature, for different manufacturing processes. The values of detachment and curvature are normalized by the wall thickness of the liner pipe (tl ), with respect to normalized curvature (κ¯ = k/ko). At some value of curvature, the detachment starts increasing very rapidly and the detachment-curvature diagram becomes nearly vertical,

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(A)

(B)

Figure 12.26 Effect of different manufacturing processes on bending response [lined pipe LP2, initial radial gap (g0 ) 50% of liner thickness]: (Α) detachment of liner pipe at the crest of the main buckle, (Β) liner moment.

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indicating structural instability of the liner. The deformation capacity of the partiallyheated TFP lined pipe is the largest of all cases analyzed, and this is due to the high value of residual stress induced by the manufacturing process. On the other hand, the plastically-expanded lined pipe has the poorest response, and this is attributed to the large inelastic deformation of the two pipes (liner and outer) during the manufacturing process. The abrupt increase of liner detachment shown in Fig. 12.26A corresponds to a sudden drop of the corresponding bending moment curve of the liner plotted in each curve of Fig. 12.26B. In this graph, the value of bending moment is the bending moment carried by the liner only (Mliner ), and is normalized by Mo = σY,oD2m,oto, so that ml = Mliner /Mo. The value of curvature that corresponds to the rapid increase of detachment and its abrupt decrease of bending moment is referred to as the “ultimate curvature” of the liner, denoted as ku , and in dimensionless form as κ¯ u . In the presence of internal pressure, the rapid detachment of liner occurs at higher values of curvature, increasing the deformation capacity of the lined pipe against liner wrinkling, as presented in Fig. 12.27A, for three fabrication methods and for pressure 10% of pY,l . Fig. 12.27B depicts the normalized moment carried by the liner pipe. Each moment-curvature diagram exhibits a sharp drop of moment at the curvature where the value of detachment starts increasing rapidly in Fig. 12.27A. The value of ultimate curvature (κ¯ u ) of the plastically expanded, fully and partially heated TFP pressurized pipes is increased by 27%, 143%, and 115%, respectively in the presence of pressure. The results demonstrate that despite the relatively low level of internal pressure, the ultimate curvature ku increases significantly with respect to the zero-pressure case.

12.5

Structural response of lined pipes under cyclic loading

The numerical results presented in the previous sections focused on monotonically increasing bending of lined pipes. On the other hand, the cyclic response of lined pipes is of particular interest, mainly because of reeling installation, where the pipe is subjected to five bending cycles, i.e. the two bending cycles during regular reeling procedure and three additional cycles in a failure-repair scenario. Cyclic experiments on lined pipes, bent repeatedly over a reel simulator, have shown a gradual increase of liner wrinkling amplitude over the loading cycles, leading to liner pipe failure (Tkaczyk et al., 2011). This is presented in the following, summarizing some representative numerical results from recent publications (Gavriilidis and Karamanos, 2021; 2022). The reader interested in more details on the cyclic response of lined pipes is referred to those two publications for additional numerical results and an extended discussion.

12.5.1 Introduction to cyclic bending behavior Consideration of cyclic bending is of primary importance for the structural performance of lined pipes during reeling installation. Fig. 12.28 shows schematically two bending cycles denoted as (a) and (b), bounded by two values of curvature k1 and k2 . The value of k1 is the maximum curvature applied in cyclic bending, and is smaller than the value of ultimate curvature ku under monotonic bending, discussed in the previous sections. The curvature in cycle (a) ranges between zero and k1 . Cycle (b) involves

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Figure 12.27 Effect of internal pressure for different methods of lined pipe manufacturing (lined pipe LP2): (Α) detachment of liner pipe at the crest of the main buckle, (Β) liner moment.

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Figure 12.28 Bending cycles during reeling (schematic).

negative (opposite sign) values of curvature and represents the effects of straightening. The value of k2 is negative, so that upon unloading the pipe returns to its straight configuration. During reeling installation process, and including the repair scenario, the pipe is subjected to 2 cycles of type (a) and 3 cycles of type (b). Being always below the value of ultimate curvature ku (k1 < ku ), one may argue that the liner is safe against wrinkling, regardless of the number of loading cycles. However, the small, otherwise negligible, amplitude wrinkles that develop in the liner during the first or second cycle may rapidly increase during the subsequent cycles of loading and lead to liner failure. This is due to concentration and accumulation of inelastic deformation in the buckled area over the bending cycles, because of strain ratcheting (see Appendix D). The gradual development of inelastic deformation in the buckled area reduces the local stiffness of liner pipe wall, and increases the wrinkle amplitude over the loading cycles. In conclusion, despite the fact that the curvature range is well below the ultimate curvature value ku (k2 ≤ k ≤ k1 < ku ), the liner may exhibit excessive local buckling deformation after those five loading cycles. The analysis of cyclic response of lined pipes requires the development of suitable finite element models. In the following, lined pipe LP2, introduced in Section 12.4, is analyzed under cyclic bending. The loading sequence and amplitude represent the loading conditions during reeling and correspond to maximum curvature values lower than the ultimate curvature ku from monotonic loading. The analysis has two stages; first the manufacturing process of a mechanically-bonded lined pipe is simulated as described in the previous section and, subsequently, the analysis proceeds to five bending cycles. The limits of cyclic loading are chosen so that they are representative of reel and aligner radii in practical applications. During the initiation of bending loading, the 12 o’clock location of the pipe is under compression and the 6 o’clock location is in tension. The cyclic curves of Fig. 12.21 are considered for the pieces of the model under consideration. In Fig. 12.29, the total bending moment MTotal applied on the entire double-walled pipe and the bending moment corresponding to the liner pipe Mliner are presented with

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Figure 12.29 Cyclic moment-curvature diagram of lined pipe (cycles are numbered in the graph): (A) bending moment of the entire lined pipe, (B) liner bending moment (fully-heated TFP lined pipe LP2).

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Figure 12.30 Evolution of liner detachment with respect to the number of cycles, at 12 and 6 o’clock locations; pure bending of LP2.

respect to the applied curvature of the pipe. In both graphs, the values of moment are normalized by Mo = σY,oD2m,oto, so that mTotal = MTotal /Mo and ml = Mliner /Mo, and the values of applied curvature are normalized by ko = to/D2m,o (κ¯ = k/ko). The diagrams show that the bending moment of the liner pipe exhibits a reduction after the third cycle (Fig. 12.29B), and this is an indication of local buckling and increase of detachment. The detachment () of liner from the outer pipe at 12 o’clock location at the crest of the main buckle is presented in Fig. 12.30 with respect to the number of bending cycles, and its value is normalized by the wall thickness of the liner pipe (tl ). After the third loading cycle, the liner detachment increases very rapidly, and in this perspective, the results in Fig. 12.30 and in Fig. 12.29B are consistent. Fig. 12.31 depicts the consecutive stages of liner deformation, which correspond to the characteristic points “a” – “f” in diagram of Fig. 12.30. The depicted shapes show that during the first two bending cycles a small-amplitude uniform wrinkling pattern develops along the compression side of the liner (stage “b”). During those two cycles, the wrinkle amplitude is less than the wall thickness of the liner pipe (stages “a”, “b”). In the last three cycles, the detachment increases rapidly leading to local buckling of the liner and the formation of a buckling pattern with a main buckle and four minor adjacent buckles (stages “d”, “e”, “f”), which is very similar to the one obtained under monotonic bending. These results imply that the localization of uniform wrinkling occurs between stages “b” and “c”, i.e., during the third loading cycle. The liner at 6 o’clock location also exhibits detachment and local buckling as shown in Fig. 12.30 and in the final configuration of the lined pipe in Fig. 12.32, despite the fact that this point is initially subjected to tension. Referring to Fig. 12.30, at the end of the second cycle, the liner pipe is nearly undeformed at this location, but deformation starts to develop during the third cycle, leading to the shape shown in Fig. 12.32.

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(a)

(b)

(c)

(d)

(e)

(f)

Figure 12.31 Deformed shapes of liner during cyclic loading at 12 o’clock location (pure bending of LP2); stages (a)–(f) correspond to the points depicted in the diagram of Fig. 12.30.

Figure 12.32 Final configuration of lined pipe after five bending cycles; local buckling is present at both 12 and 6 o’clock locations (pure bending of LP2).

12.5.2 Reeling simulation of lined pipes There exist significant differences between bending during reeling, called “reel bending” and the pure bending conditions considered above, where a standalone pipe segment is subjected to end moments. The first difference is that reel bending is applied by spooling (and un-spooling) the pipe around the reel, and this is not a pure bending process (Fig. 12.33). Furthermore, reel bending is accompanied by the application of axial tension, which is always present during the reeling process. To analyze the structural performance of lined pipes accounting for these special features, a specialpurpose finite element model has been developed, shown in Fig. 12.34. The reel is represented as a circular rigid surface of radius Rreel , and bending loading is applied by spooling the pipe around the rigid surface. The total length of the lined pipe model is 150 times the outer diameter of the outer pipe (Do ). One end of the lined pipe is kinematically connected to the reel, as shown in Fig. 12.34, so that it follows the

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Figure 12.33 Lined pipe bending over a reel.

displacements and rotations imposed by the rotation of the reel. The other end of the pipe has a roller support, allowing only for horizontal displacement. The lined pipe is incrementally spooled around the reel by applying an appropriate rotation ω (shown in Fig. 12.34), while a constant tensile force, called “back tension”, is applied at the other end. The model considers half the pipe cross-section, using symmetry with respect to the plane of bending, as shown in Fig. 12.34B. The lined pipe is divided into four segments (SEG-A, SEG-B, SEG-C and SEG-D) with different mesh densities to achieve computational efficiency, as shown in Fig. 12.34B. Segment SEG-B is the densest section of the lined pipe, where the numerical results are extracted. In particular, the central section of SEG-B is the “target” section where measurements are obtained. The final section of the model (SEG-D) does not establish contact with the reel, it is not expected to exhibit local buckling and therefore, it has a coarser mesh. Three circular rigid surfaces, denoted as RB1, RB-2 and RB-3, are considered for performing three-point bending on the SEG-B section after a winding–unwinding cycle. This process represents the straightener that zeroes the curvature before the pipeline leaves the vessel. The numerical model is described in more detail by Gavriilidis and Karamanos (2021). Fig. 12.35 presents the maximum detachment of the liner pipe from the outer pipe at 6 o’clock location with respect to the bending cycles. Considering a reel radius Rreel = 10 m, which is a typical value in practical applications, the spooling process is simulated, subjecting the lined pipe to global bending strain εb equal to 1.59%, where εb = Do/(2Rreel +Do ). Upon completion of the second cycle and the fifth cycle, a threepoint bending of the lined pipe is performed using rigid surfaces RB-1, RB-2 and RB-3, which simulate pipeline straightening. A back tension force is also applied, which is equal to 2% of the yield tension of the outer pipe, expressed by FP = σ Y,o (π Dm,o to ).

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(A)

(B)

Figure 12.34 Finite element model for reel bending analysis: (A) model overview, (B) finite element mesh of liner pipe in segments SEG-A, SEG-B, SEG-D, and in the circumferential direction.

This loading sequence is referred to as Reeling Case I. The detachment of the liner from the outer pipe is calculated in the middle section of the SEG-B segment and normalized by the wall thickness of the liner pipe (tl ). In this model, during pipe spooling, the 6 o’clock location is in contact with the reel and undergoes compression during bending on the real, as shown in Fig. 12.34B and is the critical location in the present case. Fig. 12.35 also compares the evolution of liner detachment for reel bending with the pure bending cyclic case. The two loading cases may be considered as equivalent in the sense that the same curvature range is imposed on the lined pipe. In both analyses, the liner and the outer pipe are free of geometric imperfections, and the manufacturing process follows the fully-heated TFP method, simulated in the first part of the analysis before cyclic bending starts. Under reel bending, the liner detachment in each cycle is lower than the one calculated under pure bending. This is attributed to the residual

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Figure 12.35 Liner normalized detachment () with respect to the number of cycles at critical location of the liner (lined pipe LP2, fully-heated TFP); pure bending versus reel bending (Reeling Case I).

Figure 12.36 Pipeline shape during spooling and unspooling at the end of the first cycle, showing the residual curvature of the pipeline (lined pipe LP2, fully-heated TFP).

curvature observed during unwinding so that the pipeline undergoes bending under a smaller curvature range to the reeled pipe, than in the pure bending case, as shown in Fig. 12.36 for the “target” cross-section of the reeled pipe. Therefore, during reel bending, a smaller strain range is induced in the pipe, so that the evolution of plastic deformation is less rapid, and the formation of local buckling is delayed. Fig. 12.37 shows the consecutive deformation stages of the liner at the 6 o’clock location during cyclic reel bending. The shapes are analogous to the ones in Fig. 12.31, but they

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(a)

(b)

(c)

(d)

(e)

(f)

Figure 12.37 Buckled shape of liner during reel loading (lined pipe LP2, fully-heated TFP); stages (a)–(f) correspond to the points depicted in the diagram of Fig. 12.35.

Figure 12.38 Bending response of lined pipe under reel bending; comparison of results from Reeling Cases I and II (lined pipe LP2, fully-heated TFP).

correspond to less severe local deformation. The numerical results also show that, during unwinding, the liner does not wrinkle at the 12 o’clock location, which is opposite to the contact point with the reel, and this is another major difference of reel bending response with the response under pure bending. The normalized detachment of the liner pipe is presented in Fig. 12.38, showing the difference in liner detachment amplitude between two loading sequences, namely Reeling Case I (defined above) and Reeling Case II. In the latter case, the lined pipe

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(A)

(B)

Figure 12.39 Effect of axial tension on reel bending: (A) tension up to 4% of FP , (B) tension between 4% and 10% of FP (lined pipe LP2, fully-heated TFP, Reeling Case II).

undergoes five reeling cycles up to bending strain εb = 1.59% without straightening. The comparison of numerical results in Fig. 12.38 highlights the effect of reverse bending (effect of straightener) on liner wrinkling. The detachment in Reeling Case I is more pronounced during the last three cycles, after straightening the pipeline at the end of the second cycle. More specifically, liner detachment during spooling of the lined pipe in Reeling Case II is 74%, 36% and 17% lower than in Reeling Case I at the third, fourth and fifth cycle, respectively.

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Finally, the presence of back tension F has a significant effect on the structural response of lined pipes during reeling. The lined pipe is subjected to reel bending without straightening (Reeling Case II). The diagrams of Fig. 12.39 refer to liner detachment for different levels of tension, expressed as yield tension FP , over the loading cycles. The results in Fig. 12.39A correspond to small value of tension (up to 4% of FP ) and show that increasing the back tension level, liner detachment also increases, and as a result, the liner pipe buckles locally at an earlier stage. During the fourth cycle, for 2% FP and 4% FP tension, the liner detachment is 95% and 114% higher than in the case of 1% FP tension. Increasing further the back tension level, liner detachment decreases, as shown in (Fig. 12.39B) for tension levels 4% FP , 7% FP and 10% FP . During the fourth spooling cycle, for the 7% FP and 10% FP levels, the liner detachment is 24% and 173% lower than the one at the 2% FP level.

References de Koning, A. C., Nakasugi, H., & Li, P. (2004). TFP and TFT back in town (Tight Fit CRA lined Pipe and Tubing). Stainless Steel World, 1, 53–61. Endal, G., Levold, E., Ilstad, H., (2012). Method for laying a pipeline having an inner corrosion proof cladding. U.S. Patent 8,226,327. 24 July. Focke, E. S. (2007). Reeling of tight fit pipe. Ph.D. thesis: Faculty of Civil Engineering, Delft University of Technology. Gavriilidis, I. (2021). Mechanical behaviour of steel lined pipes under monotonic and cyclic bending. PhD Thesis: School of Engineering, The University of Edinburgh. Scotland, UK. Gavriilidis, I., & Karamanos, S. A. (2019). Bending and buckling of internally-pressurized steel lined pipes. Ocean Engineering, 171, 540–553. Gavriilidis, I., & Karamanos, S. A. (2020). Effect of manufacturing process on lined pipe bending response. Journal of Offshore Mechanics and Arctic Engineering, 142(5), 051801. Gavriilidis, I., & Karamanos. , S. A. (2021). Liner wrinkling in offshore steel lined pipes during reeling installation. Thin Walled Structures, 166, 108114. Gavriilidis, I., & Karamanos, S. A. (2022). Structural response of steel lined pipes under cyclic bending. International Journal of Solids and Structures, 234–235, 111245. Hilberink, A. (2011). Mechanical Behaviour of Lined pipe. Ph.D. Thesis: Faculty of Civil Engineering, Delft University of Technology. Howard, B., Hoss, J.L. (2016). Method of spooling a bi-metallic pipe. U.S. Patent 15/070,664. Mair, J., Schuller, T., Holler, G., Henneicke, F., Banse, J., (2013). Reeling and unreeling and internally clad pipeline. U.S. Patent 34390, A1. Sriskandarajah, T., Rao, V., & Ragupathy, P. (2013a). Seal weld fatigue assessment for CRA lined pipe for HP/HT applications. In International Offshore and Polar Engineering Conference, ISOPE-I-13-235, Anchorage, Alaska. Sriskandarajah, T., Roberts, G., & Rao, V. (2013b). Fatigue aspects of CRA lined pipe for HP/HT flowlines. In Offshore Technology Conference, OTC-23932-MS, Houston, Texas, USA. Tkaczyk, T., Pepin, A., & Denniel, S. (2011). Integrity of Mechanically Lined Pipes Subjected to Multi-Cycle Plastic Bending. In International Conference on Ocean, Offshore and Arctic Engineering, ASME. Rotterdam, The Netherlands. Toguyeni, G. A., & Banse, J. (2012). Mechanically Lined Pipe: installation by Reel-Lay. In Offshore Technology Conference, OTC-23096-MS. Houston, Texas, USA.

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Vasilikis, D. (2012). Structural Behavior and Stability of Cylindrical Metal Shells with Lateral Confinement. Ph.D. Thesis: Department of Mechanical Engineering, University of Thessaly. Volos, Greece:. Vasilikis, D., & Karamanos, S. A. (2012). Mechanical behavior and wrinkling of lined pipes. International Journal of Solids and Structures, 49(23–24), 3432–3446. Yoshida, T., Mann, T., Matsuda, S., Matsui, S., Atsuta, T., Toma, S., & Itoga, K. (1981). The development of corrosion-resistant tubing. In Offshore Technology Conference, OTC-4153MS. Houston, Texas, USA. Yuan, L., & Kyriakides, S. (2014). Liner wrinkling and collapse of bi-material pipe under bending. International Journal of Solids and Structures, 51(3–4), 599–611. Yuan, L., & Kyriakides, S. (2015). Liner wrinkling and collapse of girth-welded bi-material pipe under bending. Applied Ocean Research, 50, 209–216.

Constitutive equations for linear elastic cylinders A.1

A

Hooke’s law for elastic materials

When stresses and strains are low, metals respond in a linear elastic manner. Furthermore, it is often assumed that elastic properties are the same in all directions and homogenous within the material. Under those assumptions, the material can be described in terms of two material constants. In most engineering applications, those two constants are Young’s modulus E and Poisson’s ratio ve . The corresponding stress– strain relationships are expressed as follows in a polar (cylindrical) coordinate system (ρ, θ , x), which is an orthogonal coordinate system (Chapter 4), suitable for describing quantities in a cylindrical shell: εx =

  1 σx − ve σθ + σρ + αT T E

(A.1)

εθ =

  1 σθ − ve σx + σρ + αT T E

(A.2)

ερ =

 1 σρ − ve (σx + σθ ) + αT T E

(A.3)

γxθ =

1 τxθ G

(A.4)

γxρ =

1 τxρ G

(A.5)

γρθ =

1 τρθ G

(A.6)

where G = E/2(1 + ve ) is the shear modulus of the material, α T is the coefficient of thermal expansion of the material, and T is the temperature differential. Note that thermal effects refer to normal strains only (εx , εθ , ερ ).

A.2

Plane stress conditions for cylindrical shell analysis

In shell analysis, the main assumption is that: σρ = 0 Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00008-6 c 2023 Elsevier Inc. All rights reserved. Copyright 

(A.7)

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Structural Mechanics and Design of Metal Pipes

and therefore, neglecting the thermal terms, the constitutive Eqs. (A.1)–(A.3) for the normal stresses and strains become: εx =

1 (σx − ve σθ ) E

(A.8)

εθ =

1 (σθ − ve σx ) E

(A.9)

ve E

(A.10)

ερ = −(σx + σθ )

Similarly, the shear strain γ xθ is: γxθ =

1 τxθ G

(A.11)

If the shell is analyzed according to Kirchhoff-Love assumptions, then γ ρ θ = γ ρ x = 0 (and therefore, τ ρ θ = τ ρ x = 0). Inverting Eqs. (A.8), (A.9), and (A.11), one obtains the constitutive equations in terms of stresses and strains: σθ =

E (εθ + ve εx ) 1 − v2e

(A.12)

σx =

E (εx + ve εθ ) 1 − v2e

(A.13)

τxθ = Gγxθ

A.3

(A.14)

Bending of cylindrical shells

Consider only bending action in the cylindrical shell (no membrane action). Then, the strain-curvature relationships are expressed as follows: εx = ζ kx

(A.15)

εθ = ζ kθ

(A.16)

γxθ = ζ kxθ

(A.17)

where kx and kθ express the shell curvature in the directions x and θ, respectively, ζ is the coordinate across the shell thickness (ζ = 0 corresponds to mid-surface), and kxθ is the twist of the (x, θ ) plane. The main feature of Eqs. (A.15)–(A.17) is that the strains are distributed linearly across the shell thickness, and their magnitude at a specific point is proportional to the distance of that point from the middle (reference) surface of the shell. Because of the linear relationship between stress and strain, as expressed

Constitutive equations for linear elastic cylinders

415

in Eqs. (A.12)–(A.14), the distribution of normal and shear stresses across the cylinder thickness is also linear and their value is proportional to the distance of the specific point from the reference surface. The corresponding stress resultants are the bending and twisting moments (t is the cylinder thickness) and are defined as follows for sufficiently thin-walled cylindrical shells: t/2 Mx =

σx ζ dζ

(A.18)

σθ ζ dζ

(A.19)

−t/2

t/2 Mθ = −t/2

t/2 Mxθ =

τxθ ζ dζ

(A.20)

−t/2

Substituting Eqs. (A.12)–(A.14) into Eqs. (A.18)–(A.20), and using shell kinematics expressed by Eqs. (A.15)–(A.17), one obtains: ˆ x + ve kθ ) Mx = D(k

(A.21)

ˆ θ + ve kx ) Mθ = D(k

(A.22)

ˆ + ve )kxθ Mxθ = D(1

(A.23)

   where Dˆ = Et 3 / 12 1 − νe2 is the bending rigidity of the shell.

A.4

Uniform cross-sectional deformation of cylindrical shells

In several pipeline applications, we are interested in describing cross-sectional deformation of the cylindrical shell, which is uniform along the length of the cylinder. In this case, one slice of the cylinder with length b can be analyzed, which is a ring, under “plane strain” conditions in the longitudinal direction, and is representative for the entire shell (see Fig. A.1). In mathematical terms, the following condition applies for the slice under consideration: εx = 0

(A.24)

and from Eq. (A.15) one readily obtains: kx = 0

(A.25)

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Structural Mechanics and Design of Metal Pipes

Figure A.1 Ring deformation as part (slice) of a cylinder uniformly deformed along its length; deformation under plane strain conditions.

Consequently, σθ =

E εθ 1 − v2e

(A.26)

σx =

ve E εθ 1 − v2e

(A.27)

The above equations can be also expressed in terms of stress resultants: ˆ θ Mθ = Dk

(A.28)

ˆ θ Mx = ve Dk

(A.29)

One should note the differences between Eqs. (A.26) and (A.28), and the corresponding equations for an isolated ring with a rectangular cross-section b × t, Eqs. (A.30)–(A.32): σθ = Eεθ   Mθ = Ebt 3 /12 kθ

(A.30)

σx = 0, Mx = 0

(A.32)

(A.31)

and

The main reason for the differences between Eqs. (A.30)–(A.32) and Eqs. (A.26)– (A.29) is the plane strain condition expressed in Eq. (A.24) for the cylinder, which is not applicable in the case of an isolated ring.

Column buckling B1

B

Classical beam-column theory

Classical column buckling analysis is based on Bernoulli beam theory. The physical problem is shown in Fig. B.1 and the governing equations are: r

Kinematics: k=−

d2w dx2

(B.1)

where k is the curvature along the beam, and w(x) is the lateral deflection of the beam. r

Constitutive equation: σ = Eε

(B.2)

which relates the longitudinal stress σ and the longitudinal strain ε at a material point of the beam. Integrating Eq. (B.2) over the cross-section, the constitutive equation can be expressed in stress-resultant form that relates the bending moment M with the curvature k: M = (EIb )k

(B.3)

where Ib is the moment of inertia of the beam. r

Equilibrium, which relates the external forces with the stress resultants: d2w d2M − F 2 = −q 2 dx dx

(B.4)

The main difference of the above buckling analysis with respect to the classical beam theory is the consideration of the effect of axial forces in bending. This is often called “second-order” effect or “P-” effect, and this theory is often referred to as “second-order” theory or “P-” theory. This “P-” effect is expressed in the above equations with the second term in the left-hand side of Eq. (B.4) and is also shown schematically in the detail of Fig. B.1. In classical beam theory, this term is missing. Combining Eqs. (B.1), (B.3), and (B.4), one obtains the final equation for the beam, subjected to lateral load q and axial compression F: EIb

d4w d2w + F =q dx4 dx2

Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00002-5 c 2023 Elsevier Inc. All rights reserved. Copyright 

(B.5)

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Structural Mechanics and Design of Metal Pipes

Figure B.1 Beam-column subjected to transverse load and axial compression.

Eq. (B.5) is the classical “beam-column” equation. The term “beam-column” stems from the combined action of lateral load q and axial compression F. If the lateral load q is zero, then Eq. (B.5) becomes: EIb w(4) + F w = 0

(B.6)

Eq. (B.6) is homogeneous and should be solved with the boundary conditions of the problem. For a simply supported beam-column or column, the boundary conditions are: w(0) = 0, w(L) = 0, w (0) = 0, w (L) = 0

(B.7)

The differential Eq. (B.6) with the boundary conditions Eq. (B.7) constitute a continuous eigenvalue problem in terms of F, sometimes referred to as “SturmLouiville” problem. Apart from the trivial solution w(x) = 0, this problem has also a nontrivial solution for the following values of F: Fn =

n2 π 2 EIb , L2

n = 1, 2, 3, ...

(B.8)

and the corresponding solutions (modes) are: w(n) (x) = A sin

nπ x , L

n = 1, 2, 3, ...

(B.9)

where A is an arbitrary constant. The critical (or buckling) load is the lowest value of Fn (the lowest eigenvalue), which is for n = 1: Fcr = F1 =

π 2 EIb L2

(B.10)

Column buckling

419

and the corresponding solution (mode) is: w(1) (x) = A sin

πx L

(B.11)

The critical load Fcr for columns with boundary conditions different than the ones in Eq. (B.7) can be written in the following form: Fcr =

π 2 EIb (KL)2

(B.12)

where K is the effective length factor, so that KL is the distance between two inflection (zero curvature) points in the buckling mode1 . Since the column is straight up to the onset of buckling, one may assume that force F is uniformly distributed in the cross-section, so that the corresponding stress is σ = F/Ab . At the onset of buckling the stress is: σcr =

Fcr Ab

(B.13)

or π 2E σcr =  2

(B.14)

KL rb

√ where rb = Ib /Ab is the radius of inertia of the beam-column cross-section. The ratio KL/rb is called “column slenderness”. For the above analysis to be valid in a metal column, material behavior up to buckling should be elastic. In such a case, the following inequality should apply: σcr ≤ σY

(B.15)

and using Eq. (B.14):  KL ≥ rb

π 2E σY

(B.16)

If σ cr > σ Y , this means that the column will become plastic first and subsequently buckle in the inelastic range. Therefore, the maximum compression stress sustained by 1 Note that Eq. (B.5) is also valid for tensile loading F, simply by changing the sign of F. Nevertheless, in that

case, the solution is different (it is expressed with hyperbolic sine and cosine functions), and a nontrivial solution may not be possible. From the physical point-of-view, when F is tensile, buckling does not occur.

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Structural Mechanics and Design of Metal Pipes

the column is: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨σcr σmax = ⎪ ⎪ ⎪ ⎪ ⎩σY

 π 2E KL ≥ rb  σY π 2E KL < rb σY

if if

¯ is introduced: Next, the “normalized slenderness” parameter λ ¯λ = σY σcr which is equal to: ¯λ = KL σY rb π 2 E

(B.17)

(B.18)

(B.19)

¯ is referred to simply In many instances, the “normalized slenderness” parameter λ as “slenderness”. However, one must be careful in distinguishing this parameter from ¯ also depends on the material of the KL/rb . The latter is purely geometric, whereas λ column. Using Eq. (B.19), Eq. (B.17) can be written: σmax

1 σ 2 ¯ = Yλ σY

if if

¯ ≥1 λ ¯ 0, ∀n, and this shows that the value of x in Eq. (B.41) corresponds to a local minimum. Fig. B.5 shows the graphical representation of Eq. (B.38) for different values of integer n (n = 1, 2, 3...). We are particularly interested in the case of a very long beam. In such a case, the values of L and x are large, but there will always be an integer n so that x ∼ = n and

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Structural Mechanics and Design of Metal Pipes

√ L/n ∼ = π 4 EIb /ks , as stated in Eqs. (B.41) and (B.42). Therefore, the corresponding value of fn is equal to 2 and the corresponding load is the critical load:  Fcr = 2 ks EIb

(B.43)

The L/n ratio is equal to the half-wave length:  Lhw = π

4

EIb ks

(B.44)

From Eq. (B.36), one may write:  Fcr = ks

Lhw π

2

 + EIb

π Lhw

2 (B.45)

and using Eq. (B.44): Fcr = 2π 2

EIb 2 Lhw

(B.46)

Eqs. (B.43) or (B.46) provide the critical load and Eq. (B.44) gives the half-wave length of an axially compressed infinite-length beam-column on elastic foundation. For more details on the structural response of beams and beam-columns on elastic foundation, the reader is referred to the book by Den Hartog (1987).

References American Institute of Steel Construction Standard (2016). Specification for Structural Steel for Buildings. ANSI/AISC 360-16. Chicago, IL, USA. Bjorhovde, R. (1972). Detereministic and Probabilistic Approaches to the Strength of Steel Columns. PhD dissertation. Lehigh University: Bethlehem, PA. Bleich, F. (1952). Buckling Strength of Metal Structures. McGraw-Hill Education. Den Hartog, J. P. (1987). Advanced Strength of Materials. Dover Civil and Mechanical Engineering. European Committee for Standardization (2005). Eurocode 3: Design of steel structures Part 11: General rules and rules for buildings. EN 1993-1-1. Brussels, Belgium. Rankine, W. J. M. (1876). A Manual of Civil Engineering. London, UK: Griffin and Company.

Inelastic bending and the plastic hinge concept

C

The assumption of the elastic material behavior may not be sufficient to describe the actual material behavior in the pipe and pipeline applications considered in the present book. In metals, when the stresses exceed a certain limit, the behavior becomes inelastic, and the metal exhibits irreversible deformation. Fig. C.1 shows the stress– strain curve for a grade 275 steel material. The diagram shows the relationship between the elongation L of the specimen, expressed by the nominal strain ε (defined as ε = L/L) and the corresponding nominal stress (i.e., the force F divided by the initial cross-sectional area A0 of the specimen) σ = F/A0 . The initial part of the diagram is linear corresponding to the elastic behavior of the material (elastic region). The behavior is initially elastic (and very stiff) up to the yield stress σ Y . Upon reaching the yield limit at about 300 MPa, the steel material becomes inelastic or “elastic/plastic”, associated with large increase of strain for a small increase of stress. Immediately after yielding the material exhibits a “plastic plateau,” where stress is nearly constant up to about 1.5% or 2% of strain. Subsequently, strain hardening occurs, and the stress increases up to the maximum stress, referred to as “ultimate tensile stress” σ U . For this steel material, the ultimate tensile stress is 450 MPa and occurs at a strain approximately equal to 19%. Fig. C.2 depicts schematically the main features of this behavior described above. This diagram constitutes the basis of our discussion in this Appendix and will be further expanded in Appendix D for the case of multiaxial stress state.

C1

Inelastic bending of rectangular beams

Consider a beam with a rectangular cross-section, subjected to bending, as shown in Fig. C.3. The following assumptions and conditions are used in the development of the moment–curvature relationship: 1. Strains are proportional to the distance from the neutral axis. Plane sections under bending remain plane after deformation. 2. The stress–strain relationship is idealized and consists of two straight lines, as shown in Fig. C.4. The material is called “elastic – perfectly plastic” and the stress-strain relationship is defined mathematically as follows: σ = Eε

0 < ε ≤ εY

(C.1)

σ = σY

εY < ε < ∞

(C.2)

3. The properties in compression are assumed to be the same as those in tension, and the behavior of fibers in bending is assumed to be the same as in tension or compression. 4. Rotations ϕ and deformations of beam fibers are sufficiently small, so that tan ϕ = ϕ. 5. Equilibrium conditions are expressed by the following equations: Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00006-2 c 2023 Elsevier Inc. All rights reserved. Copyright 

428

Structural Mechanics and Design of Metal Pipes

Figure C.1 Experimental stress–strain curve from a strip specimen made of grade 275 steel material.

Figure C.2 Uniaxial stress–strain diagram of steel material (schematic).

Inelastic bending and the plastic hinge concept

429

Figure C.3 Rectangular beam under bending.

Figure C.4 Idealized stress–strain diagram for steel material (elastic - perfectly plastic).

Normal force F:  F = σ dA

(C.3)

A

Bending moment M:  σ ζ dA

M=

(C.4)

A

where σ is the stress at a specific point on the cross-section, and ζ is the distance of the stress point from the neutral axis.

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Structural Mechanics and Design of Metal Pipes

Figure C.5 Stress distribution across the height of a rectangular beam at various stages of bending; (1) elastic behavior; (2) first yield; (3) elastic–plastic behavior; (4) fully-plastic moment.

For small levels of bending moment, the behavior of the beam is elastic. For convenience, we first consider the case of bending without the presence of axial force. This is called “pure bending.” Axial force will be introduced later in our discussion. The distribution of stress and strain across the height of the cross-section is linear, as shown in stage (1) of Fig. C.5, the value of stress and strain at a certain point is proportional to the distance of this point from the neutral axis, and bending is described by the following equations: r Bending curvature:

k=

ε ζ

(C.5)

r Maximum strain at the extreme fiber:

εm = k

t 2

(C.6)

r Maximum stress at the extreme fiber:

σm = Ek

t 2

(C.7)

r The bending moment is related to curvature as follows:

M = EIb k

(C.8)

Inelastic bending and the plastic hinge concept

431

where Ib = bt 3 /12 is the moment of inertia of the cross-section. Furthermore, the maximum stress at the extreme fiber is: σm =

Mt Ib 2

(C.9)

σm =

6M bt 2

(C.10)

or

First yield occurs when the stress at the extreme fiber reaches the yield stress value σ Y . At that stage, denoted as stage (2) in Fig. C.5, the bending moment MY is: MY = σY Z

(C.11)

where Z = bt 2 /6 is the elastic bending resistance of the rectangular cross-section. The corresponding curvature is: kY =

2σY Et

(C.12)

Upon continuation of bending, the extreme fibers in tension and compression plastify, and the size of the elastic region gradual decreases. The distribution of stress across the beam height is shown in stage (3) of Fig. C.5. Bending resistance is offered only by the elastic part and the total bending moment can be computed from Eq. (C.4) equal to:     σY bt 2 3 1 kY 2 − , M= 6 2 2 k

for

k > kY

(C.13)

shown graphically in Fig. C.6, and the value of curvature is related to the size of the elastic zone (2ζ e ) as follows: k=

2σY 2εY = ζe Eζe

(C.14)

At the limit where ζ e = 0, from Eq. (C.14) k → ∞ and: M = MP =

σY bt 2 4

(C.15)

This value of MP is the limit value of moment in the cross-section, referred to as “fully plastic bending moment” or simply “plastic moment.” Theoretically, the value of MP is reached when the curvature is infinite, but in reality, the value of MP is reached at a finite value of curvature.

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Structural Mechanics and Design of Metal Pipes

Figure C.6 Bending-curvature diagram for a beam with rectangular cross-section; Eq. (C.13) versus idealized “plastic hinge” response. Numbers (1) – (4) are consistent with the numbers in Fig. C.5.

C2

The plastic hinge concept in beams

Consider a beam that is free of structural instability (local buckling) and of material fracture phenomena. The reason this beam is not capable of supporting a load beyond an ultimate load level is that plastic hinges are formed at certain critical sections. We will examine what those plastic hinges are, and how they are formed. The bilinear moment–curvature curve in Fig. C.6 is characteristic of the plastic hinge. The following two features are particularly important: r After the elastic limit M is reached, the M − k curve approaches quite rapidly to the Y

horizontal line corresponding to the plastic moment value MP .

r Upon reaching M , there is an indefinite increase in curvature at constant moment. P

These two features are expressed in the idealized bilinear curve in Fig. C.6, which consists of two straight lines: M = EIb k

(C.16)

M = MP

(C.17)

where: kP =

MP EIb

(C.18)

According to the idealized curve in Fig. C.6, the member remains elastic until the moment reaches MP . Thereafter, rotation occurs at constant moment. This implies that the member acts as if it were hinged, except with a constant restraining moment MP .

Inelastic bending and the plastic hinge concept

433

Comparing the two curves in Fig. C.6, the only effect of the idealization is to wipe out a short piece of curve in the actual moment–curvature relationship. There are several factors that influence the ability of members to form plastic hinges. Some of these are important and are outlined below.

Residual stresses Residual stresses in a steel member are due to uneven cooling, cold bending or welding, and their effect is twofold. First, residual stresses cause initiation of yield at lower load level than predicted by usual stress analysis. Secondly, they may reduce the ultimate capacity of a compression member when its strength is governed by either local or global buckling. In the absence of any structural instability phenomena, they have a negligible effect on the maximum bending strength of a member. For example, in the case of columns supporting primarily axial loads, or in externally pressurized pipes which buckle in the elastic range, the effect of residual stress is of predominant importance.

Strain hardening After structural steel has been strained through the plastic region to about 10 or 15 times the elastic limit strain, further deformation is accompanied by an increase in the stress capacity of the material. This strain hardening is illustrated in Fig. C.2. Strain hardening has a beneficial effect on the bending strength. However, the complications of taking into account strain hardening either in calculating the ultimate load or the deflection are such that it is neglected in simple plastic theory of beams and the idealized curve of Fig. C.4 is used.

Effect of axial force The presence of actual force in a member subjected to bending induces extra stresses and reduces the bending moment capacity of a member. In some cases, these forces are important design factors and therefore, they are treated more extensively in Section C.4 of the present Appendix.

Material fracture The ability of a member to deform in the plastic region and form a plastic hinge might be limited due to material fracture. Steel material is quite ductile and is expected to be able to deform quite well beyond its elastic limit, allowing for the formation of a plastic hinge. On the other hand, the presence of a weldment in the plastic hinge area, may not allow the formation of a plastic hinge. In several cases, upon the development of a certain amount of plastic deformation, rupture of the weld may occur, reducing abruptly the bending moment capacity of the member.

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Structural Mechanics and Design of Metal Pipes

Figure C.7 Distribution of plastic deformation in a beam subjected to lateral point load at mid-span.

C3

Distribution of plastic deformation in a transversely loaded beam

For the idealized moment–curvature curve in Fig. C.6, the plastic hinge forms at discrete points at which all plastic rotations are assumed to occur. Thus, the length of the hinge approaches zero and outside this area, the member is assumed to behave elastically. In actual behavior though, the plastic deformation region extends over a length of member. For example, in the rectangular beam of Fig. C.7, the length of plastic deformation, often referred to as “plastic hinge length,” is equal to one-third of the beam span. In general, the plastic hinge length is the length of the beam over which the bending moment is greater than the yield moment and the size of the hinge length depends on the loading, the boundary conditions, and the geometry of the beam cross-section. In summary: 1. A plastic hinge is the result of yielding due to flexure in the structural member. 2. Although plastic deformation spreads over a length about the location of maximum bending moment, in most of the analytical work, it is assumed that all plastic rotation occurs at this location. 3. Where plastic hinge is located, the steel member acts as if it were hinged, except with a constant restraining moment MP . 4. Plastic hinges form at points of the maximum moment, including points of load application, or fixed supports. 5. In beams with rectangular cross-section, the plastic moment MP is equal to σY bt 2 /4; MP is 50% higher than the yield moment MY , which is the bending moment that corresponds to the onset of plastic deformation (see Fig. C.6).

C4

Influence of axial force on the plastic moment

In addition to causing column instability, if compressive, the presence of axial force N tends to reduce the magnitude of the plastic moment MP . The previous analysis may be modified easily to account for this influence because the important plastic hinge characteristic is still retained in the presence of this axial force, even though the moment

Inelastic bending and the plastic hinge concept

435

Figure C.8 Distribution of stresses in a beam cross-section under the simultaneous action of bending moment and axial force.

Figure C.9 Stresses analysis under the simultaneous action of bending moment and axial force at the fully plastic range.

capacity is reduced. The effect is small in the case of small axial load N and therefore, in several cases, reduction in bending moment capacity maybe ignored. However, in the case of relatively high axial force, the analysis needs to account for the reduction of plastic moment due to axial load. The following analysis refers to compressive axial force, but it can be readily adjusted for the case of tensile axial force. Fig. C.8 shows the stress distribution of the beam at various stages of deformation caused by thrust and moment. Due to the axial force, yielding of the compression side precedes that of the tension side. Eventually, plastification occurs, but since part of the area resists the axial force, the stress block no longer contains equal compression and tension areas, as in the pure bending case. Thus, as shown in Fig. C.9, the total stress distribution represented by (a) may be divided into two parts: a part (b) that is associated with the axial load and a part (c) that corresponds to the bending moment. At ultimate stage, when the entire cross-section is plastic, the axial force and the bending moment are (Fig. C.9): N = σY bζ0 = (σY bt )

ζ0 t

(C.19)

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Structural Mechanics and Design of Metal Pipes

and   t − ζ0 e M = σY b 2

(C.20)

respectively. In Eq. (C.20): e=t−

t + ζ0 t − ζ0 = 2 2

(C.21)

Therefore, the bending moment is: M = σY

 b 2 t − ζ02 4

(C.22)

Equivalently: 

bt 2 M = σY 4



 1−

ζ0 t

2  (C.23)

or M =1− MP



N NY

2 (C.24)

where NY is the force that causes complete yielding of the cross-section. Therefore, for a given level of axial compressive force N and using Eq. (C.24), the reduced plastic moment M P of the rectangular cross-section can be given by the following expression:  M p

C5

= MP 1 −



N NY

2  (C.25)

Extension to inelastic cylindrical bending of plates

From the geometrical point-of-view, a plate may be considered as an assembly of rectangular beams, as shown in Fig. C.10A. However, the bending response of an individual (standalone) beam is different from the bending response of a beam-type strip which is part of the plate. The main difference is the lateral restrain of the beam. Consider a standalone beam with rectangular cross-section. The beam is free to deform laterally, because of Poisson’s effect. Under longitudinal bending, its crosssection does not remain rectangular, but it will deform as shown in Fig. C.10B. The state of stress is uniaxial, classical beam theory is applicable, and the cross-sectional deformation is not considered in the analysis. On the other hand, when the rectangular

Inelastic bending and the plastic hinge concept

437

Figure C.10 Plate bending versus beam bending. (A) Plate as an assembly of strips and standalone beam. (B) Cross-sectional deformation under bending.

beam is a strip of a plate, the shape of the cross-section may not change and this induces stresses in the transverse direction (denoted as σ T ), which are necessary for preserving strain compatibility imposed by the condition: εT = 0

(C.26)

where subscript (·)T denotes the transverse direction. Considering the behavior of an elastic plate subjected to bending curvature k, the moment bending M is (see also Appendix A): M=

Et 3 k  12 1 − νe2

(C.27)

Furthermore, the stresses are: r in the axial direction

 σ =

 E kζ 1 − νe2

(C.28)

r in the transverse direction

 σT =

 νe E kζ 1 − νe2

(C.29)

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Structural Mechanics and Design of Metal Pipes

The latter are responsible for the development of a bending moment equal to ν e M acting in the transverse direction, shown in Fig. C.10B, and it is also described in Appendix A for the case of rings under plane strain conditions. For elastic–plastic material behavior, the main feature of the response described in Section C.1 also applies: first yielding occurs at the extreme (outer) fibers of the crosssection, and with increasing bending deformation, it further develops across the plate thickness, until the entire section becomes plastic. Using Von Mises yield criterion (see Appendix D): σ 2 + σT2 − σT σ = σY2

(C.30)

At first yielding, considering σ T = ν e σ , one readily obtains for the axial stress at the outer fiber: 1 σY σ = 1 + νe − νe2

(C.31)

and this occurs at curvature k = kY∗ : kY∗ =

 2σY∗  1 − νe2 Et

(C.32)

where σY∗ = σY / 1 + νe − νe2 may be considered as the modified yield stress in the axial direction due to plane strain conditions. Further increase of bending curvature creates a plastic zone with increasing width. To determine the relationship between the size of the remaining elastic region 2ζ e , one has to consider the interaction between the axial and the transverse stress through the yield criterion in Eq. (C.30) together with the plane strain condition in Eq. (C.26). The axial stress is equal to:

2 σ = σY / 1 + νep − νep

(C.33)

where ν ep is “Poisson’s ratio” in the elastic-plastic range (σ T = ν ep σ ). At the onset of plastic deformation at the specific point and, considering a value of ν ep equal to 0.5, as in the fully plastic range, one obtains the following upper limit value of stress:  √ σ = 2/ 3 σY

(C.34)

Applying Eq. (C.33) in metals for elastic behavior (ν ep = ν e = 0.3), one obtains σ = 1.125σ Y , which is slightly below the fully plastic value σ = 1.155σ Y computed by Eq. (C.34). Nevertheless, a closed-form analytical solution to this bending problem may not be possible, and a numerical method is required. The problem can be significantly simplified if Tresca criterion is used (see Appendix D). In this case, the yield stress in

Inelastic bending and the plastic hinge concept

439

the axial direction is equal to σ Y and the solution is identical to the uniaxial solution expressed by Eq. (C.13). A “compromise” that allows for the use of Von Mises yield criterion is the use of the uniaxial stress

solution in Eq. (C.13) but with the upper limit modified yield ∗ 2 . This provides a solution that is quite close to the stress σY = σY / 1 + νep − νep “exact” solution obtained with a numerical method. The main reason that this solution approximates quite well the “exact” solution is the fact that the axial stress varies within a rather small range 1.125σ Y ≤ σ < 1.155σ Y . Therefore, considering a constant value equal to the upper limit may not affect the solution significantly.

Metal plasticity fundamentals

D

This Appendix outlines the basics of metal plasticity, necessary to understand the mechanical behavior of metal pipes and tubes. Some features of elastic–plastic material response related to inelastic bending of beams have been presented in Appendix C. The present Appendix extends the discussion on inelastic behavior of metals and presents the basic features of plasticity theory, which are necessary for understanding the physical phenomena described in the present book1 . For a detailed presentation of the theory of plasticity, the reader is referred to a classical textbook on this subject (e.g., Lubliner, 2008; Chen and Han, 1988). Initially, one-dimensional inelastic response of metals is presented, and the key concepts are introduced. Subsequently, multiaxial behavior of metals in the inelastic range is discussed, and the basic plasticity models are outlined. Special emphasis is given to the important issue of reverce or cyclic plasticity. Finally, the deformation theory of plasticity is briefly presented, which is used in structural instability calculations.

D1

One-dimensional inelastic response of metals

Consider a strip specimen from metal material subjected to uniaxial tension. When the stress σ exceeds a certain limit, which is called yield limit and is denoted by σ Y , then the behavior becomes plastic or inelastic. The main feature of this plastic behavior is the small increase of stress σ corresponding to relatively large deformation. This behavior is shown schematically in the diagram of Fig. D.1, for a typical carbon steel material. It depicts the relationship between the elongation L of the specimen, expressed by the nominal strain ε (defined as ε = L/L) and the corresponding nominal stress (i.e., the force F divided by the initial cross-sectional area A0 of the specimen) σ = F/A0 . The initial part of the diagram is linear corresponding to the elastic behavior of the material (elastic region). Upon reaching the yield stress σ Y at point A, the diagram becomes nearly horizontal forming a “plastic plateau.” After a value of strain equal to about 1.5% or 2%, the diagram begins to rise and the stress increases, due to the phenomenon of strain hardening, or simply hardening. Another σ − ε diagram is shown in Fig. D.2 depicting the uniaxial tension behavior characteristic of an austenitic (stainless) steel. Specimens of cold-worked carbon steels exhibit the same behavior. In this case, no plastic plateau exists after the yield point; immediately after yielding the material starts to harden, and a smooth transition from elastic to elastic–plastic response occurs, characterized by a gradually decreasing slope of the σ − ε diagram. In this case, the exact point of yielding is difficult to identify and is usually represented by the material strength at plastic strain 0.2%. 1 The

presentation focuses on basic plasticity of metals. However, this theory of plasticity constitutes the basis for describing the inelastic response of other materials (e.g., geo-materials or concrete).

Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00001-3 c 2023 Elsevier Inc. All rights reserved. Copyright 

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Figure D.1 Inelastic behavior of simple structural steel; loading, unloading from B to C and reloading, again at point B.

Figure D.2 Inelastic behavior of austenitic steel; loading, unloading from B to C and reloading, again at point B.

In both Figs. D.1 and D.2, when the metal specimen is unloaded, uploading is elastic, represented by a straight line parallel to the initial elastic behavior. This means that the strain reduction (strain recovery) from B to C is equal to the total elastic deformation corresponding to stress σ B at point B, that is, equal to: εe = σB /E

(D.1)

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Figure D.3 Tangent and secant moduli in inelastic response.

After unloading at point C (stress σ C is equal to zero) there is a “residual” strain εC , which is equal to the plastic deformation εp developed up to point B. In other words, plastic deformation is “nonreversible” or “permanent”. Let us now consider the elastic–plastic behavior of Fig. D.3. The slope of the tangent at point B of the curve, beyond the yield stress σ Y , is denoted as ET and is called tangent modulus. Usually, the value ET is significantly smaller than Young’s modulus (ET  E). A subsequent increment in strain dε can be written as the sum of an elastic part dεe and a plastic part dεp , so that: dε = dεe + dε p

(D.2)

The corresponding increase in stress dσ is: dσ = E(dε − dε p )

(D.3)

Eq. (D.3) can be written as follows: dσ = ET dε

(D.4)

Defining the hardening modulus: H=

ET E E − ET

(D.5)

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Figure D.4 Loading, unloading, and reloading in the opposite direction (reverse loading).

or H=

ET 1 − ET /E

(D.6)

one can readily write: dσ = H dε p

(D.7)

Suppose now that the tensile specimen has the mechanical behavior, as shown in Fig. D.4, the specimen is loaded into the plastic range at point B, then it is unloaded from point B to point C. Subsequently, we continue loading in the opposite direction, imposing compressive stress to the specimen. The specimen will behave elastically up to some point B . Denoting the stress at this point σ  B , the question is how the value of σ  B compares with σ B . If σB = σB

(D.8)

then our material is considered to obey “isotropic” hardening. On the other hand, if   |σB | + σ  B  = 2σY

(D.9)

then our material exhibits “kinematic” hardening. In other words, the isotropic hardening model increases the size of the elastic region “symmetrically”, that is, both in tension and in compression. On the contrary, the kinematic hardening model “maintains” the initial size of the elastic region, which is equal to 2σ Y . In reality, metals

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445

Figure D.5 Elasto-plastic behavior with constant of elasto-plastic rigidity; loading, unloading and reverse plastic loading. In this case, the tangent modulus ET has a constant value.

exhibit a mechanical behavior that is between the two above hypotheses, and it is further discussed in Section D.6. Note 1 For the sake of simplicity, in several structural and mechanical applications, we consider that the postyielding behavior is represented by a straight line in the stress–strain diagram (Fig. D.5). In this case, the behavior is called “linear hardening”. Sometimes, this linear approximation is quite useful for the understanding of the basic concepts and allows for simple and efficient simulation of elastic–plastic behavior. Linear hardening can be applied to both isotropic and kinematic hardening models. Note 2 In some applications, mainly referring to carbon steel material, it is reasonable to assume that the tangent modulus is zero, that is, the postyield slope of the stress– strain diagram is zero (Fig. D.6). This model is called “elastic/perfectly plastic” and is used in numerous mechanical and structural applications. Apart from its simplicity, it also allows for developing closed-form analytical expressions for the stiffness or the strength of a metal component. This model has also been employed in the inelastic bending analysis presented in Appendix C. Note 3 Unlike elastic behavior of materials, where a one-to-one correspondence exists between the value of stress and the value of strain, the main characteristic of elastic– plastic (inelastic) behavior is that this one-to-one correspondence is no longer valid. One can readily realize that in elastic–plastic behavior, a value of stress may correspond to more than one value of strain (essentially an infinite number of values of strain).

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Figure D.6 Elastic/perfectly plastic material with zero elasto-plastic rigidity; the tangent modulus ET and the hardening modulus H are both zero.

D2

Preliminaries for multiaxial stress inelastic analysis

Consider the 3 × 3 matrix [σ ij ] that contains the components of stress tensor σ with respect to a specific Cartesian coordinate system x, y, z. Then the principal stresses σ i , i = 1,2,3 are the eigenvalues of the following discrete eigenvalue problem: 

 σi j − σ [1] = 0

(D.10)

where [1] is the 3 × 3 identity matrix. Equivalently, the following determinant should vanish: ⎡

σx − σ det ⎣ τxy τxz

τxy σy − σ τyz

⎤ τxz τyz ⎦ = 0 σz − σ

(D.11)

Eq. (D.11) is a third-degree polynomial equation in terms of σ , and its solution provides the three eigenvalues σ 1 , σ 2 , σ 3 , referred to as “principal stresses” of the stress tensor. In the case of a biaxial plane stress problem where σ z = 0 and all shear stress components τ ij are also zero, σ 1 and σ 2 coincide with σ x and σ y , and σ 3 is equal to 0. In the case of a plane stress problem, where all in-plane stress components σ x , σ y , and τ xy are nonzero, then the principal stresses are: σ1,2

σx + σy = ± 2



σx − σy 2

2 2 + τxy

(D.12)

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Figure D.7 Principal stresses and their orientation.

and the orientation of the principal stresses (system x1 ,y1 ,z) with respect to the x, y, z system is given by the following expression (see also Fig. D.7): tan 2θ p =

2τxy σx − σy

(D.13)

The hydrostatic stress σ P is expressed in terms of the normal stress components σ x , σ y , and σ z as follows: σP =

σx + σy + σz 3

(D.14)

One may prove that the value of hydrostatic pressure σ P is independent of the coordinate system chosen to write the stress components σ ij . That means: σx + σy + σz = σx + σy + σz

(D.15)

where σ  x , σ  y , and σ  z are the normal stress components with respect to another coordinate system x , y , z . The sum of diagonal components of matrix [σ ij ] is denoted I1 and is called “first invariant” of the stress tensor. The term “invariant” means that its value does not change with the change of the coordinate system. The deviatoric stress s is of primary importance for modeling inelastic material behavior, and its components sij with respect to a specific Cartesian coordinate system x, y, z are defined as follows: sx = σx − σP

(D.16)

sy = σy − σP

(D.17)

sz = σz − σP

(D.18)

sxy = τxy

(D.19)

syz = τyz

(D.20)

sxz = τxz

(D.21)

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The second invariant of the deviatoric stress tensor, denoted as J2 is of particular importance for inelastic modeling and analysis of metals. It is defined as follows: J2 =

1 s·s 2

(D.22)

where the dot (·) denotes the “scalar product” between two tensors. Eq. (D.22) can be also expressed in terms of deviatoric stress components: 2 − τxz2 − τyz2 J2 = sx sy + sy sz + sx sz − τxy

(D.23)

or in terms of stress components: J2 =

 2 2 1 2 + τxz2 + τyz2 σx − σy + σy − σz + (σx − σz )2 + τxy 6

(D.24)

or, finally, in terms of the principal stresses: J2 =

 1 (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ1 − σ3 )2 6

(D.25)

In plane stress conditions, the second invariant of the deviatoric stress tensor is expressed as: J2 =

 1 2 2 σx + σy 2 − σx σy + 3 τxy 3

(D.26)

and in terms of the two principal stresses: J2 =

D3

 1 2 σ1 + σ2 2 − σ1 σ2 3

(D.27)

Limits of elastic behavior in multiaxial stress conditions (yield criteria)

The limits of elastic behavior in tension and compression are often determined with an appropriate uniaxial test. Exceeding this limit, the material behavior is elastic–plastic and this was discussed in Section D.1. However, this one-dimensional state of stress is rarely encountered in real applications, where multiaxial stress/strain states develop. Therefore, a yield criterion is used to determine when a material point yields under the multiaxial state of stress. The general form of a yield criterion is expressed as follows:

 F σi j = 0

(D.28)

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449

where F(σ ij ) is a function of stresses, referred to as yield function2 . It is noted that Eq. (D.28) expresses a surface in the stress space, often called “yield surface,” which encloses all stress states with elastic material behavior, including the origin of stress space (σ ij = 0). If the stress point σ ij is within the yield surface, then F(σ ij ) < 0 and the behavior is elastic. When the stress σ ij reaches the yield surface, then F(σ ij ) = 0 and inelastic response begins. With increasing loading, strain hardening occurs, as discussed in the following section. In metals, two multiaxial yield criteria are mainly used: the Von Mises yield criterion and the Tresca yield criterion. Both criteria have been used extensively in the analysis and design of metal structural components and compare quite well with existing experimental data. In this comparison, a small superiority of the Von Mises criterion has been observed. Both criteria, in their original form, are “isotropic” criteria, which means that yielding occurs at the same stress level in any direction of loading. Furthermore, experimental evidence has shown that -to a first approximation- yielding of metals may not be affected by hydrostatic stress. Therefore, in both criteria, it is assumed that the metal material develops plastic deformation by distortion (shear) and not by being hydrostatically compressed or expanded. The Von Mises and Tresca criteria are briefly presented next. In addition, the anisotropic yield criterion proposed by Hill is also presented, suitable for describing yielding anisotropy. r

Von Mises yield criterion

It is the most widely used criterion in metals and has been shown to be in very good agreement with experimental data. This criterion assumes that yielding occurs when the distortional energy (or equivalently, the second invariant of the deviatoric stress tensor J2 ) reaches a critical value. According to this criterion, the yield function is expressed as follows: F=

1 1 s · s − σY2 2 3

F=

 3J2 − σY

(D.29)

or (D.30)

where σ Y is the yield stress under uniaxial tension. According to this criterion, yielding occurs when F = 0, which is written equivalently as:  3J2 = σY 2 It

(D.31)

is important to underline the difference between the function F(σ ij ) and the yield criterion, which is expressed by F(σ ij ) = 0. Sometimes, the two terms are used as equivalent, but they are not.

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Figure D.8 Von Mises and Tresca yield criteria in terms of principal stresses in the plane.

or (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 = 2σY 2

(D.32)

Under a two-dimensional stress state (e.g., plane stress), yielding occurs when the principal stresses satisfy the equation: σ1 2 − σ1 σ2 + σ2 2 = σY 2

(D.33)

or equivalently (expressed in general stress components): σx 2 − σx σy + σy 2 + 3τxy 2 = σY 2

(D.34)

The calculated value σ eq below is called the “Von Mises equivalent” stress: σeq =



σ1 2 − σ1 σ2 + σ2 2

(D.35)

σx 2 − σx σy + σy 2 + 3τxy 2

(D.36)

or σeq =



A geometric representation of Von Mises yield criterion in terms of two principal stresses σ 1 , σ 2 is shown in Fig. D.8. r

Tresca yield criterion

Another criterion for determining the first yielding of metals under multiaxial stress conditions was proposed by Tresca. The Tresca criterion assumes that yielding occurs when the shear stress on any plane reaches a critical value. Under the two-dimensional

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451

state of stresses (plane stress) yield occurs when: max (|σ1 |, |σ2 |, |σ1 − σ2 |) = σY

(D.37)

The calculated value σ eq below is called the “Tresca equivalent” stress: σeq = max (|σ1 |, |σ2 |, |σ1 − σ2 |)

(D.38)

Geometrically, this criterion expressed in terms of two principal stresses σ 1 , σ 2 is represented in Fig. D.8, in comparison with the Von Mises criterion. The use of Tresca criterion may have some advantages over the Von Mises criterion: (a) it is sometimes simpler to use in analytical calculations; (b) it is more conservative than Von Mises, providing safer predictions. r

Hill yield criterion

Anisotropy often occurs in rolled plates of metals, where the yield strength may differ somewhat between the rolling, transverse, and thickness directions. It is also quite common in welded pipes because of the cold working. To account for these variations in the modeling of the material behavior, the anisotropic yield criterion proposed by Hill (1998) is used, expressed as: 2 2

2 + 2Lτyz2 + 2Mτxz2 = 1 (D.39) H σx − σy + F σy − σz + G(σz − σx )2 + 2Nτxy where the six constants H, F, G, N, L, M are material constants. Denoting σ Yx , σ Yy , σ Yz , the yield stresses in the three directions x, y, z under uniaxial conditions, and τ Yxy , τ Yyz , τ Yxz the shear yield strengths on the respective orthogonal planes, the empirical constants can be evaluated in terms of the yield strengths as follows: H +G=

1 σY2x

H +F =

1 σY2y

1

2M =

F +G=

1 σY2z

(D.40)

and 2N =

1 τY2xy

2L =

τY2yz

1 τY2xz

(D.41)

For plane stress conditions, and defining the following yield stress ratios: Sθ =

σY θ σY x

Sr =

σY r σY x

Sxθ =

τY xθ σY x

(D.42)

one may write Hill’s yield criterion in the following form (Kyriakides and Yeh, 1988):

 1 1 1 1 σx2 − 1 + 2 − 2 σx σθ + 2 σθ2 + 2 τxθ2 − σY2x = 0 Sr Sθ Sθ Sxθ

(D.43)

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The values of yield stress ratios Sθ , Sr , Sxθ are determined using special-purpose experiments.

D4

Models for multiaxial inelastic behavior of metals

The development of the classical theory of plasticity was motivated by the need for modeling the inelastic behavior of metals. A key assumption of this theory is the decomposition of total strain tensor ε in two parts: an elastic part εe and a plastic (inelastic) part ε p so that: ε = εe + ε p

(D.44)

Therefore, starting from the well-known elasticity equation that relates the stress tensor to the elastic part of strain tensor: σ = D εe

(D.45)

one may write the following expression that relates the rate of stress with the rate of strain: ˙ = D(ε˙ − ε˙ p ) σ

(D.46)

In Eqs. (D.45) and (D.46), D is the fourth-order elastic rigidity tensor, which can be written as follows: D = 3Kv Iv + 2G Id

(D.47)

where G is the shear modulus of the material, Kv is the bulk modulus (Kv = E/[3(1 − 2ve )] ), and Iv , Id are the volumetric and deviatoric fourth-order unit tensors, whose Cartesian components are expressed using the Kronecker delta δ ij as follows: Iidjkl =

 1 1 δik δ jl + δil δ jk − δi j δkl 2 3

(D.48)

Iivjkl =

1 δi j δkl 3

(D.49)

In numerical implementation of metal plasticity, for example, in a finite element environment, the Von Mises yield criterion in Eq. (D.36) is usually adopted, mainly because of the simple and continuous form of the yield function. In such a case, the plasticity model is often called “J2 model”, and we refer to “J2 plasticity”. However, when analytical calculations are performed, Tresca criterion may offer some advantages because of its simplicity. Nevertheless, the following discussion is limited to J2 plasticity.

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The next issue refers to the flow rule, which expresses the evolution of plastic strain ε p . This is expressed as follows: ˙ ∂

ε˙ p = λ ∂σ

(D.50)

˙ is a scalar where the dot on top of a certain quantity denotes the rate of this quantity, λ quantity called plastic multiplier, and = (σ ij ) is a stress function called plastic potential. In classical metal plasticity, function in the flow rule expression is taken equal to the yield function F. In such a case, the flow rule is called associative, and the plastic strain rate is written as: ˙ ε˙ p = λ

∂F ∂σ

(D.51)

where ∂F/∂σ is the outward normal of the yield surface F = 0. From Eq. (D.50), the increment of plastic strain is normal to the yield surface, and this is referred to as the “normality rule.” If the Von Mises function in Eq. (D.36) is chosen, then the plasticity theory is called Von Mises plasticity theory or simply J2 theory. In Von Mises plasticity: ∂F =s ∂σ

(D.52)

˙ ε˙ p = λs

(D.53)

and

˙ is associated with It is important to underline that whether an increment of stress σ an increase of plastic deformation (˙ε p = 0) or not, this depends on: (a) the location of current stress σ in the stress space with respect to the yield surface F = 0, and ˙ with respect to the outward normal ∂F/∂σ. (b) the direction of σ

More specifically, ˙ The stress is 1. If F(σ) < 0, the behavior is elastic for any direction of the stress increment σ. within the yield surface. ˙ ≤ 0, the behavior is elastic. The stress point is on the yield 2. If F(σ) = 0 and (∂F/∂σ) · σ surface and elastic unloading occurs (see Fig. D.9). ˙ > 0, the behavior is elastic–plastic and inelastic deformation 3. If F(σ) = 0 and (∂F/∂σ) · σ develops. The stress point is on the yield surface and plastic loading occurs (Fig. D.9). 4. The case F(σ) > 0 is not an acceptable state of stress.

Item 4 above stems from the essential requirement that upon plastic loading (item 3 above), the new state of stress should be on the new yield surface. In other words, during plastic loading, not only the stress state moves outward with respect to the yield

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Figure D.9 Stress on the yield surface; condition for plastic loading and elastic unloading.

surface, but the yield surface should also move, expand, or distort in a way that the final stress point at the end of the plastic loading increment is on the new yield surface. There are two direct consequences of the requirement in item 4. The first is the consistency condition and it is expressed mathematically as follows: F˙ = 0

(D.54)

imposing that the new stress state is on the yield surface at its new position. The second issue is to define how this new position of yield surface is achieved. Does the yield surface translate, expand, distort, or exhibits a combination of those? The evolution of yield surface in terms of plastic deformation is called hardening. There exist several hardening rules associated with the postyielding evolution of a yield surface. Definition of an appropriate hardening rule in plasticity model is essential for simulating the inelastic behavior of metal materials accurately. Some basic hardening rules are discussed below.

D5

Elementary hardening rules

The following fundamental hardening rules are usually considered in elastic-plastic analysis, presented below in the context of Von Mises (J2 ) plasticity. r

Isotropic hardening

It is the simplest hardening model, and it is suitable for modeling problems associated with loading conditions that do not include reverse loading. In this case, the yield surface expands uniformly in the stress space, without any distortion or translation, and

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455

this is expressed by the yield function: F=

k2 (εq ) 1 s·s− =0 2 3

(D.55)

or equivalently: F = 3J2 − k2 (εq ) = 0

(D.56)

where k(εq ) is a function of the equivalent plastic strain εq , which is a measure of plastic deformation at the specific material point, defined as follows:  ε˙q =

2 p p ε˙ · ε˙ 3

(D.57)

Integrating over time: t εq =

ε˙ q dt

(D.58)

0

In isotropic hardening, the outward normal to the yield surface is: ∂F =s ∂σ

(D.59)

and the flow rule becomes: ˙s ε˙ p = λ

(D.60)

Function k(εq ) should be defined using a uniaxial test from the material under consideration. This is called “calibration” of the model. It is straightforward to show, using Eq. (D.55), that in uniaxial tension, where the only nonzero stress component is σ x , k is equal to σ x . Furthermore, applying Eq. (D.57) in uniaxial tension, one can easily show that εq is equal to εxp (the plastic part of axial strain). Therefore, determining the σx − εxp relationship from a uniaxial material test, function k(εq ) can be defined: k depends on εq the same way σ x depends on εxp in uniaxial tension. Clearly, the value of k for εq = 0 is equal to σ Y , which is the initial yield stress of the material in uniaxial tension. ˙ is calculated, and finally, inverting Using the consistency condition, λ Eq. (D.60) one obtains the following equation that expresses the stress rate with respect to the strain rate: ˙ = Dep ε˙ σ

(D.61)

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where Dep = D −

9 G2 (s  s) k2 (H + 3G)

(D.62)

and H = dk/dεq

(D.63)

is the hardening modulus,  denotes the fourth-order tensor product of two secondorder tensors, and the components of the fourth-order tensor ss with respect to a Cartesian coordinate system are sij skl . r

Kinematic hardening

Kinematic hardening has been proposed for loading histories that involve unloading and reverse plastic loading. In those loading histories, isotropic hardening cannot simulate the Bauschinger effect. Therefore, material hardening is modeled by allowing the yield surface to follow the stress point by translating in the stress space. In its initial version, known as linear kinematic hardening, the size of Von Mises yield surface is kept constant. The analytical expression of the yield surface is: F=

k2 1 (s − a) · (s − a) − 2 3

(D.64)

where a is the so-called “back stress,” which defines the center of the yield surface in the deviatoric plane, and the value of yield stress k is constant. In this hardening rule: ∂F =s−a ∂σ

(D.65)

and the flow rule becomes: ˙ − a) ε˙ p = λ(s

(D.66)

Furthermore, an evolution rule for the back stress a is necessary. The following simple equation was proposed by Prager (1956), which expresses a linear relationship between the rate of back stress with the increment of equivalent plastic strain: a˙ = C ε˙ p

(D.67)

From uniaxial tension conditions, it is possible to show that the value of C is: C=

2 H 3

(D.68)

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457

The incremental stress–strain relationship is expressed by Eq. (D.61), where: Dep = D −

9G2 (ξ  ξ) ξ¯2 (H + 3G)

(D.69)

Also ξ = s − a, and  ξ¯ = r

3 ξ·ξ 2

(D.70)

Perfect plasticity

In this case, the yield criterion can be considered as a special version of the isotropic  hardening rule, where k is constant k˙ = 0 : F=

k2 1 s·s− =0 2 3

(D.71)

In this case, H = 0, and from Eq. (D.62) the elastic–plastic rigidity tensor is: Dep = D −

3G (s  s) σ¯ 2

(D.72)

It can be shown that: Dep s = 0

(D.73)

which implies that the rigidity tensor Dep has a zero eigenvalue and s is the corresponding eigenvector. Therefore: det (Dep ) = 0

(D.74)

and tensor D ep is nonreversible and nonpositive definite.

D6

Basics of cyclic plasticity

Metal components of structural and mechanical systems and, in particular, steel tubes and pipes are often subjected to severe inelastic loading, associated with reverse plastic loading and reloading in the inelastic range. The manufacturing of welded pipes from plates or coils, followed by the application of pressure and/or structural loading results in loading histories characterized by reverse inelastic loading and reloading patterns. Those repeated excursions into the inelastic range of the metal material could lead to excessive accumulation of plastic deformation and, combined with other parameters, for example, geometrical imperfections, this may lead to structural failure.

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Figure D.10 Stress–strain curve of a grade 275 steel specimen loaded between two symmetric strain levels ±1.25% (ε = 2.5%); initial plastic response is followed by reverse plastic loading and the Bauschinger effect.

The accurate description of metal cyclic behavior in the inelastic range is necessary to simulate the performance of structural steel components under strong cyclic loading. For the purposes of our discussion, the main feature that characterizes the stress–strain behavior of the material is yielding under reverse plastic loading. The application of inelastic deformation in one direction causes cold working of the material in this direction and increases the corresponding yield strength. On the other hand, the yield strength in the opposite direction is reduced. The greater the initial inelastic deformation, the lower the yield strength in the opposite direction. This phenomenon is called “Bauschinger effect,” after the German engineer Johann Bauschinger (1834–1893). Cold working of the material in one direction results in accumulation of dislocations at the grain boundaries. Upon reverse loading, the dislocations are assisted by the back stresses at the grain boundaries. Hence the dislocations slide more easily, and the result is that the yield strength for loading in the opposite direction is less than it would be if deformation had continued in the initial direction of loading. In Fig. D.10, the response of a strip specimen made of grade 275 steel (specified minimum yield stress equal to 275 MPa) is shown, subjected first to uniaxial tension up to 1.25% strain (point B), followed by reverse plastic loading. Inelastic deformation occurs at about 250 MPa (point A), followed by strain hardening. Upon reaching 1.25% strain, the specimen is unloaded and reverse loaded (compressed) up to -1.25% (point D). During reverse loading, inelastic deformation occurs at approximately 100 MPa (point C), a value significantly lower than the initial yield stress, due to Bauschinger effect.

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Figure D.11 Stress–strain response of a grade 275 steel specimen; “strain ratcheting” under nonsymmetric stress-controlled loading.

Upon reaching −1.25% compression strain, the specimen is stretched again, exhibiting a response in DEF, similar to the one recorded in BCD. Another feature of cyclic plasticity response is “ratcheting,” which is the progressive increase of strain under nonsymmetric stress-controlled cycles. In this case, each hysteresis loop is translated with respect to the previous one because there exists a residual deformation and the stress–strain loop does not close, as shown in Fig. D.11, for a grade 275 steel specimen. Two main categories of phenomenological plasticity models have been developed to describe the mechanical behavior of metal materials under cyclic loading, classified in terms of the definition of the hardening modulus. In the first category of cyclic plasticity models, the hardening modulus is defined indirectly, by imposing the consistency condition on the yield surface, which couples the hardening modulus with the kinematic hardening rule. Those models are often referred to as “coupled models” (Bari and Hassan, 2000) and may be considered a direct extension of the classical linear kinematic hardening models, described above. The first model of this category was introduced by Armstrong and Frederick (1966), by adding a nonlinear term, referred to as “recovery term,” in the evolution equation of the back stress tensor. This concept was elaborated by Chaboche (1986, 1991) who proposed hardening rules with superimposed back stress equations, for the purpose of representing cyclic material

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behavior more accurately. Further enhancements were proposed by Ohno and Wang (1994) and more recently by Hassan and co-workers (e.g., Rahman et al., 2008). The second category of cyclic plasticity models comprises the so-called “uncoupled” models (Bari and Hassan, 2001). Their main feature is the direct definition of the plastic modulus through an appropriate function, so that the plastic modulus is influenced only indirectly by the kinematic hardening rule. A landmark “uncoupled” cyclic plasticity model was proposed by Dafalias and Popov (1975, 1976) and independently by Krieg (1975), introducing the concept of “bounding-surface” or “twosurface” plasticity. In that model, in addition to the yield surface, an outer surface called “bounding surface” is considered, which obeys kinematic hardening, and the value of the hardening modulus is a function of the distance between two surfaces. In the following, a brief description of the two categories of cyclic plasticity models is offered. r

Non-linear kinematic hardening

The nonlinear kinematic hardening model is an enhanced version of the use of classical kinematic hardening rule [Eqs. (D.64)–(D.70)] for simulating in an accurate manner the features associated with cyclic loading beyond the elastic regime, as described in the previous paragraph, and especially the Bauschinger effect. In particular, the simulation of strain ratcheting constitutes the most challenging issue. Herein, the discussion is limited to J2 plasticity where Von Mises yield surface can translate or change size, and the flow rule is expressed by Eq. (D.66). The evolution of back stress is given by the following equation: a˙ = C ε˙ p − γ ε˙q a

(D.75)

where γ is a dimensionless parameter that controls the maximum value and the rate of saturation of back stress a. The above expression enhances Eq. (D.67), with the addition of the second term in the right-hard side (“recovery term”), and constitutes the simplest form of nonlinear kinematic hardening, initially proposed by Armstrong and Frederick (1966). Chaboche (1986) showed that the use of a single back stress tensor leads to over-prediction of ratcheting and revised the kinematic hardening rule proposing the use of four superimposed nonlinear back stresses as follows:

a˙ =

N  i=1

a˙ i =

N 

 Ci ε˙ p − ε˙q γi ai

(D.76)

i=1

In Eq. (D.76), the value of N (which is the number of back stresses) is usually taken equal to 3 or 4. Enhanced versions of this model were proposed by Chaboche (1991) and Ohno and Wang (1994). More recent versions of the nonlinear kinematic hardening rule were proposed by Bari and Hassan (2002), with the purpose of simulating more accurately the accumulation of plastic strain (ratcheting) during

Metal plasticity fundamentals

461

cyclic plastic loading a˙ =

N  i=1

a˙ i =

N 

Ci ε˙ p −

i=1

N 

γi χi ε˙q [ai δi + (1 − δi )(n · ai )n]

(D.77)

i=1

where δ i is a multiaxial ratcheting parameter, n is the unit outward normal to the yield surface, and χ i is equal to 1 for i = 1, 2, 3 and:   a¯4 χ4 = 1 − √ a·a

(D.78)

In Eq. (D.78),  are the Macaulay brackets and a¯4 is a threshold term that determines whether the recovery term of the fourth back stress tensor is active. The numerical implementation of the above cyclic plasticity models in a finite element program may not be a straightforward process. Despite the proposal of several enhanced versions of nonlinear kinematic hardening, their efficient and robust numerical implementation in a finite element environment may not be a trivial matter. Commercial finite element software programs have implemented nonlinear kinematic hardening in its primitive forms expressed by Eqs. (D.75) and (D.76). However, the most recent versions of this model require an in-house user subroutine, to be used in conjunction with finite element software. The recent works by Chatziioannou et al. (2021a, 2021b) propose a framework for the robust numerical implementation of nonlinear kinematic hardening models and present their capabilities in predicting the cyclic response of piping components, with emphasis on ratcheting. r

Bounding-surface plasticity

The major advantage of uncoupled models over coupled models is the explicit definition of hardening modulus, which allows for simulating very efficiently cyclic metal behavior (i.e., plastic plateau, Bauschinger effect, ratcheting). The distinctive feature of this model is the outer surface, referred to as “Bounding Surface,” which plays the role of a bound for the yield surface (Dafalias and Popov, 1976). Both the yield (inner) surface and the bounding (outer) surface can harden by translating and changing size, obeying a mixed hardening rule under the restriction that the entire yield surface must always remain within the bounding surface. In this context, hardening depends on the distance between the two surfaces, to be defined at a later stage in this section. The expressions describing the yield and the bounding surface are k2 (εq ) 1 (s − a) · (s − a) − =0 2 3 k¯ 2 (εq ) 1 F¯ = (¯s − b) · (¯s − b) − =0 2 3

F=

(D.79) (D.80)

respectively, where s is the deviatoric stress tensor, s¯ is the deviatoric stress on the bounding surface, a and b are the back stress tensors associated with the yield and

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Figure D.12 Schematic representation of bounding-surface plasticity model in the deviatoric stress plane.

the bounding surface respectively, k and k¯ are functions of the equivalent plastic strain ε q , defined in Eqs. (D.57) and (D.58). The√radii R and√R¯ of the yield and the bounding ¯ respectively (Fig. D.12). surface in the deviatoric space are equal to 2/3 k and 2/3 k, During initial plastic loading, the flow rule is controlled by the yield surface. Upon reaching the bounding surface, the two surfaces stay together at the specific stress point, and the flow rule is controlled by the bounding surface. The two surfaces lose contact when reverse plastic loading occurs. The following function for k is adopted, which represents the change of the size of the yield surface, due to Bauschinger effect: k(εq ) = σY + Q(1 − e−bεq )

(D.81)

where σ Y is the initial yield stress of the material, and Q and b are material parameters. Without being restrictive, a similar function can be also assumed for the size of the ¯ outer surface k. The flow rule is expressed by the following equation: ε˙ p =

1 ˙ N (N · σ) H

(D.82)

where H is the hardening modulus, N is a tensor normal to the yield surface, defined as follows:  3 1 (s − a) (D.83) N= 2 |s − a| and || is the Euclidian norm of a tensor. Furthermore, the hardening modulus H is defined directly through an appropriate function of the “distance” δ in stress space

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463

between points A and B, as shown in Fig. D.12. Point A represents the current stress state on the yield surface “loading” point, and point B is referred to as the “congruent point” of A on the bounding surface, which has the same outward normal vector N, as ¯ shown in Fig. D.12 (N = N). The following expression for function H was proposed in the initial version of the model:

¯ H(δ, δin ) = H + h

δ δin − δ

 (D.84)

where H¯ is the modulus of the bounding surface, activated when the two surfaces establish contact, h is a hardening parameter, δ in and δ are the initial and the current distance between the two surfaces: δ=



(¯s − s) · (¯s − s)

(D.85)

For the yield surface motion during a plastic loading increment, the rule proposed by Mroz (1967) is adopted: a˙ = μ˙ v

(D.86)

where v is a unit tensor (v  v = 1) along the direction of segment AB in the deviatoric stress space, as shown in Fig. D.12: v=

1 (¯s − s) |¯s − s|

(D.87)

and μ˙ is a scalar quantity evaluated from the consistency condition at the yield surface. The translational motion of the bounding surface is coupled with the hardening of the yield surface. There exist two possibilities depending on whether two surfaces are in contact or not, which should be examined during plastic loading. More details on the model are presented in the paper by Dafalias and Popov (1976), and in the recent paper by Chatzopoulou and Karamanos (2021). Uncoupled models, despite their advantage in the explicit definition of the hardening modulus, have not been very popular in large-scale computations, mainly because of the challenges involved in their numerical integration and implementation. For a recent work on the implementation of the bounding surface model within a finite element environment, addressing all these numerical challenges, and its application in simulating the mechanical response of piping components under cyclic bending, the interested reader is referred to the recent publication of Chatzopoulou and Karamanos (2021).

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The deformation theory of plasticity

Deformation theory of plasticity is the simplest way to describe inelastic deformation. It employs total stress–strain relationships, which refer to nonlinear elasticity, but in several problems, this is a very reasonable assumption. The main reason for using this type of theory is its mathematical convenience which allows for the derivation of closed-form solutions for numerous practical problems. Deformation theory can be used efficiently in problems associated with nearly proportional stress paths. For this case, it is possible to show that the integration of the J2 incremental relationships provides a final state of deformation which is very similar to the one obtained by J2 deformation theory. For metal pipes and tubular members that exhibit bifurcation buckling, there exists another reason for using J2 deformation theory instead of J2 flow theory, despite the fact that the stress path during bifurcation is far from being considered as proportional. On the contrary, the stress path is highly nonproportional and one would expect that it would be better represented by an incremental theory (e.g., J2 flow theory). However, it has been shown that the use of J2 flow theory provides unreasonably high predictions of the critical load and the corresponding strains and deformations, whereas the use of the J2 deformation theory offers better predictions. An explanation for this paradox may lie on the formation of a corner (vertex) on the yield surface at the loading point, which makes the material behavior more compliant. This phenomenon cannot be represented by classical J2 flow theory, and implies that the normality rule, expressed by Eq. (D.51) is no longer applicable. On the other hand, the use of J2 deformation theory results in a non-associative plastic flow that mimics the above “corner effect” and results in reduced rigidity, which provides better predictions for the bifurcation load and the corresponding deformation, as described in Section 6.3. To clarify those issues, the formulation of J2 deformation theory is briefly presented below. The basic equation relates plastic deformation with stress, as follows: ε p = f (J2 ) s

(D.88)

Applying Eq. (D.88) to uniaxial tension, one readily obtains that function f(J2 ) should be equal to (3/2)(1/ES − 1/E ), where ES is the secant modulus, so that:

 3 1 1 εp = − s (D.89) 2 ES E Therefore, the total strain is expressed in tensor form as:

 3 1 1 ε = Ce σ + − s 2 ES E

(D.90)

where Ce is the fourth-order elastic compliance tensor (Ce = D−1 ). The rate form of deformation theory is obtained by time differentiation of Eq. (D.89)



 3σ˙ eq 1 3 1 1 1 − − s˙ + s (D.91) ε˙ p = 2 ES E 2σeq ET ES

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465

or in terms of total strain rate (see also Neale, 1981):



 3σ˙ eq 1 3 1 1 1 ˙ + s˙ + ε˙ = C σ − − s 2 ES E 2σeq ET ES e

(D.92)

The first term in the right-hand side of Eq. (D.91) is not necessarily normal to the Von Mises yield surface and makes the flow rule non-associative. The instantaneous rigidity tensor Dep d is obtained by inverting Eq. (D.92), so that: ˙ = Dep ˙ σ d ε

(D.93)

where

Dep d =D− h=

ES E E − ES

 

6G2 9G2 1 1 (s  s) Id − 2 − h + 3G k H + 3G h + 3G

(D.94)

(D.95)

and Id is the deviatoric fourth-order identity tensor. It can be shown that the rigidity moduli from J2 deformation theory are more compliant than the ones from the classical J2 flow theory (Nasikas, 2022). Therefore, using the rigidity tensor Dep d in bifurcation calculations results in better predictions of the bifurcation load compared with predictions obtained using the moduli Dep from the classical J2 flow theory in Eq. (D.62).

References Armstrong, P.J., Frederick, C.O., 1966. A mathematical representation of the multiaxial Bauschinger effect. CEGB Report RD/B/N731. Bari, S., & Hassan, T. (2000). Anatomy of coupled constitutive model for ratcheting simulation. International Journal of Plasticity, 16(3-4), 381–409. Bari, S., & Hassan, T. (2001). Kinematic hardening rules in uncoupled modeling for multiaxial ratcheting simulation. International Journal of Plasticity, 17, 885–905. Bari, S., & Hassan, T. (2002). An advancement in cyclic plasticity modeling for multiaxial ratcheting simulation. International Journal of Plasticity, 18(7), 873–894. Chaboche, J. L. (1986). Time-independent constitutive theories for cyclic plasticity. International Journal of Plasticity, 2(2), 149–188. Chaboche, J. L. (1991). On some modifications of kinematic hardening to improve the description of ratcheting effects. International Journal of Plasticity, 7(7), 661–678. Chatziioannou, K., Karamanos, S. A., & Huang, Y. (2021). Simulation of cyclic loading on pipe elbows using advanced plane-stress elastoplasticity models. Journal of Pressure Vessel Technology, 143(2), 021501. Chatziioannou, K., Huang, Y., & Karamanos, S. A. (2021). An implicit numerical scheme for cyclic elastoplasticity and ratcheting under plane stress conditions. Computers & Structures, 249, 106509.

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Chatzopoulou, G., & Karamanos, S. A. (2021). Numerical implementation of bounding-surface model for simulating cyclic inelastic response of metal piping components. Finite Elements in Analysis and Design, 185, 103493. Chen, W. F., & Han, D. J. (1988). Plasticity for Structural Engineers. New York, NY: SpringerVerlag. Dafalias, Y. F., & Popov, E. P. (1975). A model of nonlinearly hardening materials for complex loading. Acta Mechanica, 21, 173–192. Dafalias, Y. F., & Popov, E. P. (1976). Plastic internal variables formalism of cyclic plasticity. Journal of Applied Mechanics, 43(4), 645–651. Hill, R. (1998). The Mathematical Theory of Plasticity. New York, NY: Oxford University Press Inc. Krieg, R. D. (1975). A practical two-surface plasticity theory. Journal of Applied Mechanics, 42, 641–646 ASME. Kyriakides, S., & Yeh, M. K. (1988). Plastic anisotropy in drawn metal tubes. Journal of Engineering for Industry, 110, 303–307 ASME. Lubliner, J. (2008). Plasticity Theory. New York, NY: Dover Publications. Mróz, Z. (1967). On the description of anisotropic workhardening. Journal of the Mechanics and Physics of Solids, 15(3), 163–175. Nasikas, A. (2022). Non-associative plasticity for structural instability of cylindrical shells in the inelastic range. PhD Thesis, School of Engineering, The University of Edinburgh. Scotland, UK. Neale, K. W. (1981). Phenomenological constitutive laws in finite plasticity. Solid Mechanics Archives, 6, 79. Ohno, N., & Wang, J. D. (1994). Kinematic hardening rules for simulation of ratchetting behavior. European Journal of Mechanics – A/Solids, 13(4), 519–531. Prager, W. (1956). A new method of analyzing stresses and strains in work-hardening plastic solids. Journal of Applied Mechanics, 23(4), 493–496. Rahman, S. M., Hassan, T., & Corona, E. (2008). Evaluation of cyclic plasticity models in ratcheting simulation of straight pipes under cyclic bending and steady internal pressure. International Journal of Plasticity, 24(10), 1756–1791.

Geometry and equilibrium of plastic collapse mechanism

E

The plastic mechanism of Fig. E.1 develops in the postbuckling stage of a thin-walled ring with radius r under uniform external pressure. It is also used in modeling pipe deformation under lateral loading conditions. The mechanism is characterized by four equally spaced plastic hinges, which is compatible with the oval shape of buckling and the post-buckling configuration of metal rings. Because of double symmetry, one-quarter of the ring cross-section is considered, as shown in Fig. E.2. A major observation is that in this model the distance between two adjacent plastic hinges should remain constant. Therefore, the length of segments   should be equal to AB √ (undeformed configuration) and A B (deformed configuration) r 2. Because of this, the downward displacement AA of point A should be different than the lateral displacement BB of point B, that is, α = η. Normalizing the value of α with the ring radius r, so that x = α/r, one readily obtains the following geometric expressions for γ and β: γ = r(1 − x)

(E.1)

 β = r 1 + 2x − x2

(E.2)

and

Furthermore, the rotation of A B with respect to its original position AB is equal to θ /2, where θ is the total plastic rotation in each plastic hinge. From Fig. E.2, this rotation can be written as follows: π θ = −ϕ 2 4

(E.3)

π − 2ϕ 2

(E.4)

or θ=

The change of area AE enclosed by the ring is of primary importance in our analysis. It is equal to the change of area of the shaded quadrilateral, as shown in Fig. E.3. Considering the geometry of the quarter-ring shown in Fig. E.2, the area of the shaded quadrilateral in Fig. E.3 is: AE ∗ = 4 ×



1 γβ 2



Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00005-0 c 2023 Elsevier Inc. All rights reserved. Copyright 

(E.5)

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Figure E.1 Plastic collapse mechanism with four equally spaced plastic hinges.

Figure E.2 Geometry of plastic collapse mechanism.

and after some algebra:  AE ∗ = 2r2 (1 − x) 1 + 2x − x2

(E.6)

Therefore, the change of area AE enclosed by the ring, is given by the following expression:   2  1 − (1 − x) 1 + 2x − x2 AE = πr2 − AE ∗ = πr2 π

(E.7)

The equilibrium of the plastic mechanism is shown in Fig. E.4. Each plastic hinge carries a constant bending moment equal to: MP = σY

t2 4

(E.8)

Geometry and equilibrium of plastic collapse mechanism

469

Figure E.3 Change of area enclosed by the ring during its collapse deformation.

Figure E.4 Equilibrium in the plastic collapse mechanism.

The total force from the uniform external pressure on this quarter-ring segment is equal to the product of pressure with the projection of the quarter-ring on the A B line: √ SP = pr 2

(E.9)

Equilibrium of forces in the horizontal direction Fx = 0, requires that: H = SP sin ϕ

(E.10)

and balance of moments with respect to point B , (M)B = 0, leads to: √ SP r

2 = 2MP + Hγ 2

(E.11)

Combining Eqs. (E.10) and (E.11), one obtains the pressure–displacement equilibrium relationship for the collapse mechanism: t 1 p = pY D 2x − x2

(E.12)

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where pY is the plastic pressure: pY = 2σY

t D

(E.13)

Eq. (E.12) can be modified to express the external pressure in terms of ring crosssection ovalization , which is defined as follows: =

Dmax − Dmin Dmax + Dmin

(E.14)

In the plastic mechanism model Dmax = 2β and Dmin = 2γ , and therefore Eq. (E.14) becomes: =

β −γ β +γ

(E.15)

Setting ξ = 2x − x2 , one may readily obtain that:  β =r 1+ξ

(E.16)

 γ =r 1−ξ

(E.17)

and inserting Eqs. (E.16) and (E.17) into Eq. (E.15), one obtains after some algebraic mathematical manipulations: ξ=

2 1 + 2

(E.18)

and therefore, Eq. (E.12) becomes: t 1 + 2 p = pY D 2

(E.19)

Finally, starting from Eq. (E.5) and using Eqs. (E.16)–(E.18), the dependence of pressure on the change of area AE enclosed by the ring cross-section during collapse deformation can be expressed as follows: t

p  = pY D

1 2  1 − 1 − 2 AE /r2

(E.20)

End effects on internally pressurized thin-walled cylinders

F

The outward deflection of an internally pressurized circular cylinder with nondeformable ends is shown in Fig. F.1. The boundary conditions at the two ends prevent both lateral displacement and the corresponding rotation and can be expressed as follows (Reddy, 2007): w(x) =

 pr2  1 − e−bx (cos bx + sin bx) Et

(F.1)

w(x) =

pr2 fw (bx) Et

(F.2)

or

where b is a parameter that depends mainly on the cylindrical shell geometry:   3 1 − νe2 b = r2t 2 4

(F.3)

and fw (z) = 1 − e−z (cos z + sin z)

(F.4)

The displacement in Eq. (F.1) is axisymmetric and is applicable for either fixed or capped ends (provided that the capped plates are thick enough to prevent end rotation). The deflected shape is shown schematically in Fig. F.1 and is associated with bending, and the corresponding bending moment is readily computed as follows: M(x) = −

d2w Et 3  2  12 1 − νe2 dx

(F.5)

which leads to the following expression: M(x) =

p −bx p e (sin bx − cos bx) = 2 fM (bx) 2b2 2b

(F.6)

where: fM (z) = e−z (sin z − cos z) Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00009-8 c 2023 Elsevier Inc. All rights reserved. Copyright 

(F.7)

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Structural Mechanics and Design of Metal Pipes

Figure F.1 Schematic representation of end effects on a cylindrical shell under internal pressure.

Figure F.2 Plot of functions fw (z) and fM (z).

End effects on internally pressurized thin-walled cylinders

473

The maximum bending moment occurs at x = 0 and is equal to: Mmax = −

p 2b2

(F.8)

Functions fw (z) and fM (z) are plotted in Fig. F.2. Defining the “transition length” is of particular importance. One readily observes that the end effects become negligible for z = 4, or equivalently,  t x = 2.19 D D

(F.9)

Using Eq. (F.9) for an internally pressurized cylinder with D/t = 20, the effects are negligible at approximately half pipe diameter, whereas for a cylinder with D/t = 100, the corresponding distance is 22% of the pipe diameter.

Reference Reddy, J. N. (2007). Theory and Analysis of Elastic Plates and Shells. Boca Raton, FL: CRC Press, Taylor & Francis Group.

Structural mechanics of elastic cylinders

4

In this chapter, the structural response of thin-walled elastic cylinders is presented, extending the analysis of elastic rings, discussed in the previous chapter. Our discussion starts with the response under internal pressure, which is a rather straightforward topic, but contains some important features that require special attention. Subsequently, the structural response under external pressure and/or longitudinal action, e.g., axial loading and bending, is examined. These are topics associated with several forms of structural instability and require the adoption of shell or shell-type approach. We focus our discussion on cylindrical shells with circular cross-section, which are directly related to tubes and pipes. Because of its geometry, a cylindrical shell is better described with polar coordinates ρ, θ, x rather than Cartesian coordinates (Fig. 4.1). Direction x is along the cylinder axis and perpendicular to the cross-section of the cylindrical shell. Shell thickness t is assumed constant, and is much smaller than the radius r and the length L of the cylinder, i.e. t  r and t  L. In the applications of interest (tubes and pipes), the length L of the cylinder is also larger than its radius r (r < L).

4.1

Stresses in circular cylinders under internal or external pressure

Consider a thin-walled cylinder of circular cross-section, made of elastic material, subjected to internal pressure. We consider a biaxial state of stress within the cylinder with longitudinal stress σ x and hoop stress σ θ , whereas the radial stress is zero (σ ρ = 0). The stresses are related to strains as follows: εx =

1 (σx − νe σθ ) E

(4.1)

εθ =

1 (σθ − νe σx ) E

(4.2)

The ring equilibrium analysis of the previous section can be applied in a longitudinal slice of the cylinder, so that: σθ =

pD 2t

(4.3)

The uniform expansion of the cylinder cross-section w is calculated from elementary kinematics: εθ =

w r

Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00004-9 c 2023 Elsevier Inc. All rights reserved. Copyright 

(4.4)

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Figure 4.1 Cylinder geometry, coordinate system, displacements of the reference surface and stresses.

For an isolated ring, σ θ = Eεθ (Hooke’s law) and using Eqs. (4.3) and (4.4), one obtains: w=

pr2 Et

(4.5)

However, in a cylinder, in addition to hoop stress σ θ , axial stress σ x and/or axial strain εx may also develop, and their calculation requires careful consideration of the boundary conditions at the end of the cylinder. There exist three types of end conditions: (a) fixed; (b) capped; (c) lateral pressure only, shown in Fig. 4.2 and Fig. 4.3. Each of those cases is examined separately below. (a) Fixed conditions

This case is represented schematically in Fig. 4.2A. The two end sections are encastred (zero displacements) and, therefore, no axial strain develops (εx = 0). From Eq. (4.1): σx = νe σθ

(4.6)

and inserting into Eq. (4.2) one obtains: εθ =

1 − νe 2 σθ E

(4.7)

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97

Figure 4.2 Cylinder under uniform internal pressure. (A) Fixed ends, (B) capped ends and (C) detail at the vicinity of the end section.

(A)

(B)

Figure 4.3 Cylinder under uniform internal pressure on the lateral surface only. (A) Externally applied force to eliminate the “capped-end” force from internal pressure. (B) Pressure is applied only on lateral surface, with zero net axial force.

or σθ =

E εθ 1 − νe 2

(4.8)

The radial expansion of the cylinder is equal to: ερ = −

νe (σθ + σx ) E

(4.9)

This case is also equivalent to a ring analysis under plane strain conditions, a case examined in the previous chapter.

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Structural Mechanics and Design of Metal Pipes

An important note is necessary at this point. The above equations assume a biaxial stress state, which is uniform along the cylinder and are valid sufficiently far from the cylinder ends. In reality, the end sections are fixed and prevent cross-sectional expansion of the cylinder at those ends, so that w = 0. Therefore, there exists a “transition zone” so that the biaxial state of stress is developed, as shown in Fig. 4.2C. In this transition zone, the cylinder wall is subjected to significant bending and the corresponding state of stress is significantly more complex than the biaxial stress state described above. The length of transition zone Ltr depends on the diameter-tothickness ratio and can be estimated using the analysis outlined in Appendix F. As a conclusion, the previous analysis applies outside the transition zone, and this implies that the cylinder is sufficiently long, i.e., L  2Ltr so that this uniform bi-axial state of stress can be developed within the cylinder wall. (b) Capped-end conditions

Capped conditions mean that the end section is attached to a rigid cap but is free to move in the longitudinal direction. Sometimes, they are referred to as “hydrostatic pressure” conditions. Because of this cap, internal pressure exerts an axial force, FP which is the product of pressure p with the end section area π D2 /4 and tensile stress develops at the cylinder wall in the longitudinal direction: σx = FP /A

(4.10)

where A = π Dt . Therefore, σx =

pD 4t

(4.11)

which is equal to σθ /2. From Hooke’s law: εθ =

(1 − νe /2) σθ E

(4.12)

εx =

(1/2 − νe ) σθ E

(4.13)

and

Based on the above expressions for the hoop and axial strains, cross-sectional expansion w and the change of length of the cylinder L can be readily computed. One should underline that this solution is valid sufficiently far from the two capped ends, as noted in the previous case (see also Fig. 4.2B and Fig. 4.2C). (c) Lateral pressure only

For our analysis, this is a straightforward condition, which implies that σ x = 0. Therefore, 1 σθ E νe εx = − σθ E

εθ =

(4.14) (4.15)

Structural mechanics of elastic cylinders

99

Figure 4.4 Stress paths in internally pressurized cylinders, for different types of boundary conditions.

and w=

pr2 Et

(4.16)

On the other hand, this condition can be quite challenging from the practical pointof-view, i.e., on how to implement these conditions in an experiment. One should recognize that some sort of capping is necessary in an experiment, otherwise it would not be possible to apply internal pressure. Therefore, to simulate experimentally this type of end condition, a special arrangement is necessary, so that the capped end force is eliminated (Fig. 4.3A) and lateral conditions are ensured (Fig. 4.3B). The interested reader is referred to the publication of Kyogoku et al. (1981), where a special experimental set-up is described.

4.1.1 First yielding under internal pressure The value of internal pressure pY∗ that causes first yielding of the pipe material can be computed using a suitable yield criterion for bi-axial loading. Extensive experimental data have demonstrated that the von Mises yield criterion, written below for a biaxial state of stress (σ x ,σ θ ), describes yielding of metals in a very accurate manner (Appendix D): σx2 + σθ2 − σx σθ = σY2

(4.17)

This yield criterion is represented in graphical form in Fig. 4.4. Using the results of this section, one obtains: r

for fixed ends [Case (a), Fig. 4.2A]: σθ =

pD 2t

σx = νe σθ

and

 1 t  2σY pY∗ = √ D 1 − νe 2 + νe

(4.18)

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Structural Mechanics and Design of Metal Pipes

Figure 4.5 Deformed shell and Kirchhoff-Love assumption. r

for capped ends [Case (b), Fig. 4.2B]: σθ =

r

pD 2t

σx = 0.5σθ

and

2  t  pY∗ = √ 2σY D 3

(4.19)

for lateral pressure only [Case (c), Fig. 4.3]: σθ =

pD 2t

σx = 0

and

pY∗ = 2σY

t D

(4.20)

The corresponding loading paths for cases (a), (b) and (c) are also depicted in Fig. 4.4. It is interesting to note that, considering a Poisson’s ratio value equal to 0.3 which is typical for metals, the values of yield pressure in cases (a) and (b) are 12.5% and 15% higher than the yield pressure under uniaxial stress conditions.

4.2

Structural stability equations of cylindrical shells

The case of internal pressure examined above has been quite straightforward because it refers to an axisymmetric deformation state of the cylinder, which is uniform along its length, with the exception of the neighborhood of the end sections. On the other hand, other loading conditions of interest, e.g., axial compression, bending or external pressure loading, which are related to structural instabilities, refer to more complex stress and strain states, and the rather simple analysis of Section 4.1 may not be adequate. Instead, a shell-type theory is necessary to describe cylinder deformations and stresses. The presentation of cylindrical shell stability starts with the main assumptions of cylindrical shell behavior, the corresponding kinematics and elasticity in terms of stress resultants. Consideration of equilibrium leads to the governing shell equations and the corresponding linearized buckling equations are obtained using a perturbation method (Brush and Almroth, 1975). We follow the Kirchhoff-Love shell theory, which is the direct extension of Euler-Bernoulli beam theory to plate or shell bending (see Fig. 4.5

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101

Figure 4.6 (A) Cylinder configuration; (B) stresses in the cylinder wall; (C) in-plane membrane forces; (D) bending and twisting moments.

and 4.6). The main hypothesis of this theory is that all through-thickness fibers, initially straight and normal to the reference surface of the shell, remain straight and normal to the deformed reference surface after deformation. In this case, the rotation of the fiber in the x or the θ direction equals the slope of the reference surface along this direction. Therefore, one may concentrate on displacements of the reference surface only. Upon determining the displacements of the reference surface, the displacement and the position of every material point in the cylinder can be computed. There exist more elaborate theories of shells (and plates). One of those theories is based on the assumption that the fibers across the thickness remain straight after deformation, but not perpendicular to the reference surface. This is the Mindlin– Reissner shell theory, which is a direct extension of the Timoshenko beam theory in plates and shells. However, this theory will not be examined in the present analysis.

4.2.1 Shell displacements and rotations We consider the mid-thickness surface as the “reference” surface of our cylindrical shell (see Fig. 4.1). The displacements of the reference surface in the axial, tangential and radial directions are u(x,θ), v(x,θ ) and w(x,θ ) respectively, shown in Fig. 4.1. Each

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fiber, initially normal to the reference surface, according to Kirchhoff-Love theory, remains straight after deformation and its rotations in the x and the θ direction are: βx = −

∂w = −w,x ∂x

βθ = −

1 ∂w w,θ =− r ∂θ r

(4.21)

4.2.2 Cylindrical shell kinematics The strains at the arbitrary point in the cylinder wall (ε¯x , ε¯θ , γ¯xθ ), not necessarily on the reference surface, are: ε¯x = εmx + ζ kx

ε¯θ = εmθ + ζ kθ

where the membrane strains are: εmx = u,x +

1 2 w 2 ,x

εmθ =

γ¯xθ = γmxθ + ζ kxθ

v ,θ + w 1  w,θ 2 + r 2 r

γmx,θ =

(4.22) 1 w,x w,θ u,θ + v,x + r r (4.23)

and the bending curvatures kx , kθ and the twist kxθ are: 1 1 kx = −w,xx kθ = − 2 w,θθ kx θ = − w,xθ (4.24) r 2r The mixed derivative of the radial displacement w, xθ (or kxθ ) is referred to as “twist” of the cylinder surface and expresses the variation of the β x slope along the θ direction, or – equivalently – the variation of the β θ slope along the x direction.

4.2.3 Constitutive equations Material behavior obeys Hooke’s law. Herein, we express the constitutive equations in terms of stress resultants, which are defined by proper integration of stresses in the thickness direction (Fig. 4.6). r

Membrane forces t/2 Nx = −t/2

  ζ dζ σx 1 + r

t/2 Nθ =

t/2 σθ dζ

−t/2

Nxθ = −t/2

  ζ dζ τxθ 1 + r

t/2 Nθx =

τθx dζ

−t/2

(4.25) r

Moments t/2 Mx = −t/2

  ζ ζ dζ σx 1 + r

t/2 Mθ =

t/2 σθ ζ dζ

Mxθ =

−t/2

−t/2

  ζ ζ dζ τx θ 1 + r

t/2 Mθx =

τθx ζ dζ

−t/2

(4.26) r

Shear forces  t/2  ζ dζ Qx = τxz 1 + r −t/2

t/2 Qθ = −t/2

τθz dζ

(4.27)

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103

For sufficiently thin cylindrical shells, ζ /r can be neglected relative to unity in the above equations, and therefore, Nxθ = Nθx and Mxθ = Mθx . Using the above definitions, the membrane strains are related to membrane forces as follows:   1 − νe ˆ mx + ve ε mθ ) ˆ mθ + ve ε mx ) γmx θ N θ = C(ε Nx θ = Cˆ N x = C(ε 2 (4.28) where Cˆ is the membrane (axial) stiffness, defined as Cˆ = Et/(1−νe 2 ). Similarly, the moments of the cylinder wall are expressed in terms of the curvatures and twist: ˆ x + ve k θ ) M x = D(k

ˆ θ + ve k x ) M θ = D(k

ˆ − νe ) kx θ Mx θ = D(1

(4.29)

In the equations, Dˆ is the bending stiffness of cylinder wall, defined as Dˆ =

 above  3 2 Et / 12 1 − νe .

4.2.4 Equilibrium equations The in-plane (membrane) equilibrium equations in terms of the membrane forces are: Nx,x +

Nx θ,θ =0 r

Nxθ,x +

Nθ,θ =0 r

(4.30)

which are typical plane stress equilibrium equations in two-dimensional elasticity. In the transverse direction, there are two equations for the balance of moments: Mx,x +

Mx θ,θ = Qx r

Mxθ,x +

Mθ,θ = Qθ r

(4.31)

and one equation for the equilibrium of forces: Qx,x +

Q θ,θ + q + qN = 0 r

(4.32)

In Eq. (4.32), qN (x,θ ) is the externally applied transverse load on the cylinder surface, and qN (x,θ) is the equivalent “lateral” distributed load due to membrane actions, composed by two parts: r

a first part due to cylinder initial geometry (Fig. 4.7): q¯ = −

r

Nθ r

(4.33)

a second part due to second-order effects because of bending curvature, which is an extension of the second-order effects in beam-columns and flat plates in cylindrical shells: 2 1 q = Nx w,xx + Nxθ w,xθ + 2 Nθ w,θθ r r

(4.34)

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Structural Mechanics and Design of Metal Pipes

Figure 4.7 Lateral load due to initial cylinder curvature.

Therefore,   Nθ 2 1 qN = − + Nx w,xx + Nxθ w,xθ + 2 Nθ w,θθ r r r

(4.35)

Combining Eqs. (4.31), (4.32) and (4.35) it is possible to eliminate the shear forces Qx and Qθ to obtain: 2 1 Mx,xx + Mxθ,xθ + 2 Mθ,θθ = q + qN r r

(4.36)

Using the constitutive Eq. (4.29), the kinematics expressed in Eq. (4.24), and Eq. (4.35), and inserting into Eq. (4.36), one obtains:   ˆ 4 w + 1 Nθ − Nx w,x x + 2 Nx θ w,x θ + 1 Nθ w,θθ = q D∇ (4.37) r r r2

4.2.5 Stress function and final equations Another, very useful equation stems from the strain compatibility equation. Considering the membrane strains in Eq. (4.23), one may write: 1 w,θθ 1 1 1 εxm,θθ + εθm,xx − γxθm ,xθ = w2 ,xθ − w,xx 2 + w,xx 2 r r r r r

(4.38)

Subsequently, a stress function f(x, θ) is introduced as follows: Nx =

1 f,θθ r2

(4.39)

N θ = f,xx

(4.40)

1 N xθ = − f,xθ r

(4.41)

Considering the membrane constitutive Eqs. (4.28) and inserting into Eq. (4.38), one obtains: w,θθ 1 1 1 4 ∇ f = w2 ,xθ − w,xx 2 + w,xx Et r r r

(4.42)

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105

and the equilibrium Eq. (4.37) becomes: ˆ 4 w = q + 1 f,θθ w,θθ − 2 f,x θ w,x θ + f,x x w,x x − f,x x D∇ r2 r r

(4.43)

Eqs. (4.42) and (4.43) constitute the set of nonlinear governing equations for the cylindrical shell under consideration.

4.2.6 Prebuckling solution, bifurcation, and linearized equations The prebuckling solution w0 (x,θ ) and f0 (x,θ ) is a trivial solution of the governing equations. It satisfies the following equations (recall that s = rθ ): f0,ss = Nx0

f0,xx = Nθ0

f0,xs = −Nxθ0

(4.44)

It is also assumed that the derivatives of the prebuckling displacements are zero, i.e., w0,i = 0 and w0,ij = 0. Therefore, the prebuckling solution satisfies the following equations: 1 D∇ 4 w0 + Nθ0 = q(x, θ ) r

(4.45)

∇ 4 f0 = 0

(4.46)

Subsequently, we consider a perturbation w1 , f1 of the prebuckling solution w0 and f0 as follows: w → w0 + w1

(4.47)

f → f0 + f1

(4.48)

Introducing Eqs. (4.47) and (4.48), accounting for the prebuckling solution in Eq. (4.44), and keeping only linear terms with respect to the perturbation, one obtains the following linearized equations:  ˆ 4 w1 = 1 f0,θθ w1,xx + f0,xx w1,θθ − 2 f0,xθ w1,xθ − 1 f1,xx D∇ 2 r r

(4.49)

1 1 4 ∇ f1 = 2 w1,xx Et r

(4.50)

and

Finally, combining Eqs. (4.49) and (4.50), it is possible to eliminate f1 and obtain the following linearized buckling equation in terms of w1 (Bazant and Cedolin, 1991):   ˆ 8 w1 − ∇ 4 N0x w1,x x + 2 N0x θ w1,x θ + 1 N0θ w1,θθ + Et w1,xxxx = 0 (4.51) D∇ r r2 r2

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Structural Mechanics and Design of Metal Pipes

Figure 4.8 Cylindrical shell under uniform axial compression.

where ∇ 8 (.) is the square of the bi-harmonic operator:  ∇ 8 (.) = ∇ 4 ∇ 4 (.) Eq. (4.51) can be applied in different loading cases. Two of those loading cases are examined below.

4.3

Buckling of elastic cylindrical shells under axial compression

In the case of uniform axial compression of elastic cylinders (see Fig. 4.8), the prebuckling solution is f0,xx = f0,xθ = 0 and: f0,θθ = −σ t

(4.52)

N0x = −P

(4.53)

where σ is the applied uniform stress (p = 2πrtσ ). Therefore, the linearized buckling equation becomes:  ˆ 8 w1 − σ t ∇ 4 w1,x x + Et w1,xxxx = 0 D∇ r2

(4.54)

To solve the above problem, we assume harmonic solution for w1 (x,θ ) in both directions, in the following form: w 1 (x, θ ) = C 1 sin

mπ x sin n θ L

(4.55)

or w 1 (x, θ ) = C 1 sin mξ ¯ sin n θ

(4.56)

Structural mechanics of elastic cylinders

(A)

107

(B)

Figure 4.9 (A) Configuration of axially-loaded cylinder under axisymmetric deformation; (B) axisymmetric deformation of a strip along the cylinder.

where mπ r L x ξ= r

m¯ =

(4.57) (4.58)

m, n are wave numbers and L is a length parameter to be discussed later. Inserting Eqs. (4.56), (4.57) and (4.58) into the linearized buckling Eq. (4.54), one obtains: 

σm,n ¯

m¯ 2 + n2 t= m¯ 2

2

 Dˆ m¯ 2 2 +  2 1 − νe Cˆ 2 r m¯ 2 + n2

(4.59)

Eq. (4.59) provides the eigenvalues of the buckling problem. There is an eigenvalue for each pair of m and n. We are interested in the lowest value of σm,n ¯ , which is the buckling load. To obtain the lowest value of axial stress σm,n ¯ , Eq. (4.59) should be minimized in terms of m¯ and n. A simplification of this problem can be made,

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Structural Mechanics and Design of Metal Pipes

setting:  X=

m¯ 2 + n2 m¯

2 (4.60)

so that σ (X ) t = X

1 Dˆ + 1 − νe2 Cˆ 2 r X

(4.61)

Treating X as a continuous variable, and requiring: ∂σ (X ) =0 ∂X

(4.62)

the critical (buckling) stress is: t  E σcr =   r 3 1 − ve 2

(4.63)

t  Et Nx,cr =   r 3 1 − ve 2

(4.64)

or

and the corresponding value of X is:  2 2    r  m¯ + n2 Xcr = = 2 3 1 − ve 2 m¯ t cr

(4.65)

The buckling stress Eq. (4.63) has been reported first for axisymmetric conditions by Lorenz (1908), and later by Timoshenko (1910) and Southwell (1914). The above value of Xcr and the critical value of stress σ cr , correspond to an infinite number of m and n values. This means that there exist an infinite number of eigenmodes (buckling modes), which correspond to the critical load, and therefore, the mechanical behavior of uniformly-compressed elastic cylinders is sensitive to initial imperfections. The case of axisymmetric buckling mode is of particular interest. In that case n = 0, and therefore, 1/2  r    (4.66) Xcr = m¯ 2 = 2 3 1 − ve 2 t The ratio of L/m is the half-wavelength of the axisymmetric buckling mode, also referred to as “buckling half-wavelength” denoted as Lhw . Therefore, √ π rt (4.67) Lhw =   4 12 1 − ve 2

Structural mechanics of elastic cylinders

109

Figure 4.10 Experimental data versus theoretical prediction of buckling stress (see Brush and Almorth, 1975).

4.4

A note on post-buckling behavior of axially-compressed elastic cylinders

Fig. 4.10 shows the comparison of the critical stress expressed in Eq. (4.63) with experimental data. The most important observation is that the analytical expression provides buckling load estimates significantly lower than the buckling loads obtained experimentally and it is inadequate for predicting the real strength of the cylindrical shell under axial compression. For high r/t ratio, the experimental strength is about 20% to 40% of the theoretical value. For low r/t ratio, the buckling strength is closer to the theoretical value, yet still quite lower. During the first half of the 20th century, this has created a significant controversy within the structural mechanics community. It was only after the works of Von Karman, Koiter, Donnell and their co-workers, in the 1940’s that this issue has been resolved (see Kármán, Dunn, & Tsien, 1940; Kármán & Tsien, 1941; Koiter, 1945; Donnell and Wan, 1950). For a historical overview of this topic, the reader is referred to the relevant article by Calladine (1995) and the classical book by Brush and Almroth (1975). The key feature behind this controversy is the sensitivity of cylindrical shell response on the presence of initial geometric imperfections, a feature that stems from the fact that the buckling load corresponds to an infinite number of modes. The above pioneering works between 1941 and 1950 demonstrated that there is a strong interaction of buckling modes, corresponding to the same buckling load and imperfection sensitivity. The interested reader may also refer to “Augusti’s column”, which is a simple bar-spring system that illustrates very clearly the interaction between modes corresponding to the same bifurcation load (Augusti, 1964). The analysis by Donnell and Wan (1950) has further shown that very small imperfections, with amplitude equal to about the shell thickness, may result in significant reduction of the buckling strength of the shell, way below the theoretical bifurcation value of the critical stress obtained in Eq. (4.63). The real response of a cylindrical shell subjected

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Figure 4.11 Schematic representation of load-shortening path for perfect and imperfect cylinders.

Figure 4.12 Cylindrical shell subjected to uniform lateral pressure.

to uniform axial compression is shown schematically in Fig. 4.11. The perfect cylinder has a highly unstable post-buckling response, characterized by a “snap-back” of the post-buckling equilibrium path, whereas the maximum load sustained by the imperfect cylinder is much lower because of the presence of initial imperfections. Few years later, an experimental verification of this argument was provided by (Tennyson, 1969), who performed experiments on liquid photo-elastic plastic material cylindrical shells with nearly zero imperfections and was the first to achieve buckling loads that exceeded 90% of the theoretical buckling load.

4.5

Buckling of elastic cylindrical shells under uniform external pressure

When uniform lateral pressure is applied on the cylinder, as shown in Fig. 4.12, the prebuckling state is expressed as follows: N x0 = N xθ0 = 0 and: Nθ0 = −p r

(4.68)

Structural mechanics of elastic cylinders

111

and the buckling Eq. (4.51) becomes: D∇ 8 w1 +

 Et 1 w 1,xxxx − 2 ∇ 4 Nθ0 w1, θθ = 0 r2 r

(4.69)

We seek solutions of the following form: w1 (x, θ ) = Co sin m¯ ξ sin n θ

(4.70)

where x = Rξ and: m¯ =

mπ r L

(4.71)

Inserting Eq. (4.70) into the differential Eq. (4.69), the following expression is obtained for the eigenvalues of the external pressure:  pmn =

m¯ 2 + n2 n2

2

 Dˆ m¯ 4 2 +  2 1 − νe Cˆ 2 r n2 m¯ 2 + n2

(4.72)

One can readily show that the value of pmn is minimized in terms of m for m = 1, so that: 

2  t 2  (πr/L)2 + n2 (π r/L)4 1 pn r 2  + = 

2 1 − νe Cˆ 2 2 2 2 Et n2 r 12 1 − νe n (π r/L) + n (4.73) For specific values of L/r and r/t, the value of n corresponding to the smallest eigenvalue pn is determined by trial-and-error. We are particularly interested in the case of long cylinders, where L/r → ∞, and the above equation results in: pn = n2

Dˆ r3

(4.74)

In that case, it can be shown that the smallest value of pressure occurs for n = 2 and therefore, pcr = 4

Dˆ r3

(4.75)

or  t 3 8E pcr =  3 1 − νe 2 D

(4.76)

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Structural Mechanics and Design of Metal Pipes

The value of critical pressure in Eq. (4.75) or Eq. (4.76) is 25% higher than the ˆ 3 . This discrepancy is explained by one obtained from ring theory, which is pcr = 3D/r comparing ring and shell kinematics. According to Donnell’s kinematics for cylindrical shells, the fiber rotation in the hoop direction is Eq. (4.21): βθ = −

1 ∂w r ∂θ

(4.77)

which is different than the more exact expression used in ring theory (see Chapter 3): βθ =

v 1 ∂w − r r ∂θ

(4.78)

For a detailed treatment of cylindrical shell buckling under uniform lateral pressure, considering different shell theories and for different values of the L/r ratio, the interested reader is referred to the classical textbook of Brush and Almorth (1975).

4.6

Uniform bending of an elastic tube

Consider a very long circular hollow cylinder, made of linear elastic material, subjected to longitudinal bending (Fig. 4.13). Because of its length, the cylinder is free of boundary conditions and therefore, constant deformation can be assumed along its length. In the following, the long cylinder will be referred to as “pipe” or “tube”. One may be tempted to adopt a classical beam-bending solution for this problem from elementary mechanics of materials. However, this apparently simple problem has several unique features, which make it significantly different than bending of beams with standard cross-section, e.g., rectangular, I-shaped or box. In particular, the response of a circular hollow cylinder under longitudinal bending is characterized by cross-sectional distortion in the form of an oval shape, referred to as “ovalization” (see Fig. 4.13), which reduces the moment of inertia of the cylinder cross-section and, consequently, decreases its bending resistance. We will examine this structural response in detail in the following. The classical solution of beam flexure assumes undeformed cross-section and leads to a linear relation between the bending moment M exerted on the tube and the curvature of the neutral axis k, expressed as M = EIk. For a thin-walled cylindrical hollow section, this expression can be written in a non-dimensional form as follows: m=πκ

(4.79)

where the values of moment M and curvature k are normalized by Me and kN respectively: Me =

Ert 2 1−

νe2

,

kN =

r2



t 1 − νe2

(4.80)

Structural mechanics of elastic cylinders

113

Figure 4.13 Uniform bending of an elongated cylindrical shell with initial bent configuration.

so that: m=

M , Me

κ=

k kN

(4.81)

However, as noticed, the above solution does not account for cross-sectional ovalization and may not be adequate for describing the response of the bent cylinder. Ovalization is due to the inward stress components σ v (see Fig. 4.13) and has a unique effect on the bending response of elongated cylinders, resulting in loss of stiffness and the moment-curvature diagram deviates from the linear solution of Eq. (4.79). With increasing curvature, the slope of the m − κ diagram decreases. When the ovalization mechanism becomes dominant, the moment-curvature path reaches a limit point, beyond which the tube behavior becomes unstable, as expressed by the negative slope of the descending branch of the m − κ diagram (Brazier, 1927). This is a “limit point” instability, also referred to as “ovalization instability” or Brazier effect, shown schematically in Fig. 4.14. In the present analysis, uniform deformation along the tube length is assumed referring to tubes with infinite length (free of boundary conditions) which can be examined through a cross-sectional (two-dimensional) analysis. The analysis aims at providing closed-form expressions for the ovalization phenomenon, and comparison with finite element results obtained elsewhere is also presented (Karamanos, 2002; Houliara and Karamanos, 2006). The analytical expressions will be used as pre-buckling solutions for identifying possible bifurcation instability under bending loading in a subsequent section. The analytical solution adopts a variational formulation, based on the strain energy of the deformed tube segment and considering its longitudinal and circumferential components. The solution is based on a simple Ritz discretization, assuming a

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Structural Mechanics and Design of Metal Pipes

(A)

(B)

Figure 4.14 Schematic representation of limit point (ovalization) instability in elongated cylinders subjected to bending; (A) moment-curvature diagram and limit point and (B) cylinder cross-section kinematics.

trigonometric doubly symmetric shape function. Similar variational approaches have been adopted in previous investigations (Wood, 1958; Boyle, 1981). Before presenting the tube bending analysis, it is interesting to notice that the shape of bending ovalization is reminiscent of the oval shape of a buckled cylinder subjected to external pressure. Therefore, the combined effect of bending and external pressure on a cylinder is expected to make the cylinder highly unstable and this constitutes an important feature for understanding the mechanical behavior of offshore pipelines, to be discussed in a later section. The longitudinal strain energy per unit length UL of a bent tube, including the effects of initial curvature 1/R is: Etr UL = 2

2

2 π

1 k (r + w) sin θ + k v cos θ + (w sin θ + v cos θ ) R

dθ

(4.82)

0

or Etr UL = 2

2 π

2

1 k r sin θ +  (w sin θ + v cos θ ) R

dθ

(4.83)

0

where w and v are the radial and tangential in-plane displacements of the reference line (Fig. 4.13), which are functions of the circumferential coordinate θ . The applied curvature k, associated with bending deformation, is related to the initial radius R and

Structural mechanics of elastic cylinders

115

the current radius R of the bent tube in the longitudinal direction as follows: k=

1 1 −  R R

(4.84)

Inextentionality of the cross-section is also imposed, so that the mean hoop strain εθ0 is zero and therefore, w+

dv =0 dθ

(4.85)

The change of curvature kθ in the circumferential direction is: kθ =

1   v − w r2

(4.86)

and the strain energy due to cross-sectional distortion (hoop strain energy) is: 2π 2π     2 Dˆ E t3  2  v + v dθ = v + v dθ UC = 3 2 3 2r 24 1 − νe r 0

(4.87)

0

The following pattern is assumed for cross-sectional deformation, expressing the cross-sectional displacements w, v in terms of trigonometric doubly symmetric functions: w(θ ) = α cos 2θ v(θ ) = −

(4.88)

α sin 2θ 2

(4.89)

where α is the ovalization amplitude. Those expressions satisfy the inextensionality constraint of Eq. (4.85). The expression for the longitudinal energy becomes: Etr UL = 2

2π 

k r sin θ +

2 α α cos 2θ sin θ − sin 2θ cos θ dθ R 2R

k r sin θ +

2 α α (sin ) (sin ) dθ 3θ − sin θ − 3θ + sin θ 2R 4R (4.91)

(4.90)

0

or equivalently: UL =

Etr 2

 

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Structural Mechanics and Design of Metal Pipes

After performing the integrations in Eq. (4.91):   Etr 2 2 5 α2 3 krα UL = π k r π + 2 π − 2 2 R 8R

(4.92)

Using Eq. (4.89) the hoop strain energy UC is written as follows: UC =

3 π E t 3 α2  8 1 − νe 2 r3

(4.93)

and the total strain energy of the bent cylinder is U = UC + UL .

4.6.1 Solution for the ovalization parameter It is straightforward to obtain the ovalization parameter α in terms of the applied curvature k directly from the minimization of U in terms of α. This is described below for several characteristic cases. (a) Assume an initially curved tube ( R1 = 0). Under the simplifying assumption that 1 ≡ R1 (i.e. small imposed bending deformations), the classical linear von Kármán R (1911) expression is obtained: ∂U =0 ∂α

⇒

α=

6 krR 5 + 6λ2

(4.94)

where t 2 R2 λ = 4 r 2



1 1 − νe 2

 (4.95)

is the geometric parameter of the initially bent tube. (b) Consider an initially straight tube ( R1 = 0), so that R1 = k. Neglecting the quadratic term in the UL expression, the Brazier solution is obtained (Brazier, 1927), resulting in a quadratic expression of the ovalization parameter α in terms of the applied curvature k. More specifically, the expression for the longitudinal and hoop strain energy becomes:   Etπr 2 2 3k2 r α k r − UL = 2 2 3 2 3 πEt α UC =  8 1 − νe 2 r3

(4.96) (4.97)

and minimization of total strain energy in terms of α (∂U/∂α = 0) results in: α=

r 5 k2  1 − νe 2 2 t

(4.98)

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117

(c) As a third case, assume an initially straight tube (i.e., R1 = 0), but include the quadratic ovalization term in the UL expression. Then, it is possible to enhance the Brazier solution for an initially straight tube, and obtain an expression referred to as “modified Brazier”. In this case, upon minimization of the strain energy, a quasiquadratic expression is obtained: α=

k2 r 5  1 − νe 2 2 t

1 5k2 r4 1+  6 1 − νe 2 t 2

(4.99)

(d) If all terms in the expression of UL are included, for the general case of an initially bent tube, the following expression is obtained by minimization of U in terms of α: α=

k r R (1 + k R) + 56 (1 + k R)

λ2

(4.100)

It is interesting to compare the expression in Eq. (4.100) with the corresponding expression derived by Reissner (1959) for slightly initially bent tubes: α=

rR k(1 + k R) λ2

(4.101)

The Reissner (1959) expression in Eq. (4.101) neglects the second term in the denominator of Eq. (4.100), and shows that the solution for the cross-sectional distortion (ovalization) may be considered as the superposition of two parts; a linear part and a quadratic part: α=

r R2 k2 rR k + λ2 λ2

(4.102)

The first term in the right-hand side is linear and refers to the von Kármán (1911) solution for initially curved pipes, and the second term is quadratic and refers to Brazier solution. Based on this formulation and solution, the overall bending response of an initially curved pipe is the superposition of the two solutions.

4.6.2 Expressions for the moment-curvature diagram and the ovalization limit moment Using the above relationship between α and k, it is possible to eliminate α from the expression of the strain energy, and derive the moment-curvature relationship, i.e. determine the function M = M(k), by differentiating the strain energy expression with respect to curvature k: M=

∂U ∂k

(4.103)

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The maximum moment, representing ovalization instability, corresponds to the curvature value that makes the first derivative of function M(k) to vanish. For straight tubes, following the Brazier assumptions (Brazier, 1927), the expression for the strain energy is:   3 E π t 3 α2 Etrπ 2 2 3 2 k r − k rα +  U= 2 2 8 1 − νe2 r3

(4.104)

Using the quadratic ovalization-curvature expression (4.98) obtained by Brazier (1927), the strain energy in Eq. (4.104) becomes:   E t r 3 π 2 3 k4 r 4  2 1 − ν k − e 2 4 t2

U=

(4.105)

The moment-curvature relationship is obtained by differentiating Eq. (4.105): ∂U ∂k

M=

(4.106)

to obtain:   3 k2 r 4  2 M =EIk 1− 1 − ν e 2 t2

(4.107)

For initially bent tubes (1/R = 0), the linear von Kármán formulation results in the following expression for the strain energy: U=

  3 krα 3 π E t 3 α2 E t r π 2 2 5 α2  − k r + + 2 8R2 2 R 8 1 − νe2 r3

(4.108)

Setting: 

1 α = krR 2 λ + 5/6

 (4.109)

the strain energy becomes: Etrπ U= 2



k2 r2 R2 5 3 k2 r 2 R  − k2 r 2 +  8R2 λ2 + 5 2 2R λ2 + 56 6   t 2 k2 R2 Etrπ 3 +   2 4 1 − ν 2 r 2 λ2 + 5 2 e

6



(4.110)

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119

Thus, a linear equation for the moment in terms of curvature is obtained:   ∂U 9 M= = k EI 1 − = k EI fF ∂k 10 + 12 λ2

(4.111)

where fF is the flexibility factor also proposed by Dodge and Moore (1972) for curved pipe components (pipe bends): fF = 1 −

9 10 + 12 λ2

(4.112)

It is also possible to write the moment-curvature equation as derived by Reissner (1959) in the following form:     1 1 3 r4  +k + 2k M = E I k 1 − 2 1 − νe2 4t R R

(4.113)

The above equations of moment and ovalization in terms of the applied curvature can be written in a dimensionless form (m = M/Me , κ = k/kN ). Furthermore, the value of the ovalization is normalized by the radius of the tube r, to obtain the ovalization parameter  = α/r. With the above notation, the expressions in Table 4.1 are obtained. In this Table, Reissner’s equation is also included in a slightly modified form reported 1/2  by Boyle and Spence (1980); the modification refers to the use of Et/ 1 − νe 2 for the membrane stiffness instead of the Et value employed by Reissner (1959). In addition, the m − κ equation obtained by Boyle and Spence (1980), presented by Boyle (1981) is included. In Fig. 4.15, the Brazier and modified-Brazier solutions are plotted and compared with numerical results from finite element analysis (Karamanos, 2002). It is important to notice that, despite its simplicity, Brazier’s solution provides very good predictions in terms of bending moment (see Fig. 4.15A) and ovalization (see Fig. 4.15B) up to a level of curvature κ = 0.5, a feature to be employed in a subsequent section. The successive configurations of tube cross-section subjected to bending, as obtained by the finite element formulation (Karamanos, 2002) is shown in Fig. 4.16. Configurations (1)-(3) indicate significant ovalization prior to the limit point. The finite element values for the limit bending moment and the corresponding ovalization are mov = 0.959 and ov = 0.229, which occur at a normalized curvature value κ ov = 0.476. The corresponding predictions of Brazier’s solution are mov = 0.987, ov = 0.221 and κ ov = 0.470, which are quite close to the numerical values. The bending response of initially curved elastic cylinders (1/R = 0) has several unique features, and for a detailed presentation of this topic the reader is referred to the publications by Karamanos (2002), Houliara and Karamanos (2006) and Houliara (2008). Fig. 4.17 refers to initially bent tubes under opening bending action, i.e., applied bending curvature opposite to the initial curvature, which tends to straighten the tube (see Fig. 4.17B). Ovalization-curvature diagrams are depicted in Fig. 4.17A for

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Table 4.1 Ovalization solutions for elastic tube bending; ovalization-curvature and momentcurvature expressions. Ovalization vs. curvature Brazier (1927)

 = κ2

modified Brazier

 = κ2

von Kármán (1911) nonlinear curved tube

Reissner (1959)

1 5 κ2 1+ 6 κλ = 5 λ2 + 6 = κλ(1 + κλ) 5 λ2 + (1 + κλ) 6 κ  = (1 + κ λ) λ

Moment vs. curvature   3 m = πκ 1 − κ2 2   5 3 m = π κ 1 −  + 2 2 8   9 m = πκ 1 − 10 + 12 λ2 m=

    1 5 1 3  2+ + 2 1 + 4 κλ 4 κλ

πκ 1 − 

3 (1 + κ λ) (1 + 2 κ λ) m=πκ 1− 4 λ2



modified Reissner, Boyle (1981)

m=   3 (1 + κ λ) (1 + 2 κ λ) πκ 1 − 1 − νe 2 4 λ2 1 − νe 2

Boyle & Spence (1980)

m= πκ 1 − νe 2

⎡

⎤

⎢ (1 + κ λ) (1 + 2 κ λ) ⎥ ⎢1 − 3 ⎥ ⎣ 2 5 (1 + κλ)2 ⎦ 4λ  2 1 − νe + 6 λ2

different values of initial curvature κ in . It is quite interesting to note that, regardless of the value of initial curvature κ in , the response is initially characterized by “negative ovalization”, i.e., flattening pattern parallel to the plane of bending, up to the stage where the cylinder becomes straight. Furthermore, when the cylinder becomes straight (zero total curvature), the corresponding ovalization is zero. This is shown in Fig. 4.17A, where all curves pass through the origin. Beyond that stage, the ovalization changes sign.

4.7

Uniform bending of an elastic tube in the presence of pressure

Pressurized bending of elongated cylinders is of particular importance for pipes and pipelines. In this case, the cross-sectional ovalization is expected to be affected by the presence of pressure. From the physical point-of-view, the cases of internal and external pressure have different effects on the bending response. If the tube

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121

Figure 4.15 Response of initially straight elastic tubes predicted by Brazier, modified Brazier and finite elements in the absence of pressure (r/t = 120); (A) moment-curvature diagrams and (B) ovalization-curvature diagrams.

is externally pressurized, the bending-induced ovalization (Brazier effect) is further increased by the presence of external pressure, resulting in rapid growth of ovalization with respect to the applied curvature. On the other hand, the presence of internal pressure reduces pipeline ovalization, and has an opposite effect on the bending response. The following formulation accounts for the presence of pressure, regardless of its sign (internal or external), and the above arguments can be quantified using an enhanced version of the energy formulation described in Sections 4.6, accounting for

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Figure 4.16 Finite element results for cross-sectional ovalization of an initially straight cylinder under uniform bending (zero pressure).

(A)

(B)

Figure 4.17 (A) Evolution of ovalization in initially curved elastic tubes under “opening” bending in the absence of pressure. (B) “Negative” cross-sectional ovalization.

the work of pressure. Τhe pressure potential VP is: VP = p AE = p(AE ∗ − AE0 )

(4.114)

where AE ∗ is the area enclosed by the deformed ring: AE

∗

1 = πr + 2

2π

2

0



2rw + v2 + v w − vw + w2 dθ

(4.115)

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123

and AE0 is the area enclosed initially by the ring, equal to π r2 . In this expression, external pressure p is assumed positive. Furthermore, the second-order work of hoop pressure stress (σ p = pr/t) is: 2π WP =

pr 1 ε˜θm r t dθ = p t 2

0

2π

(v − w ) dθ 2

(4.116)

0

2  where ε˜θm = (1/2) v − w /r is the nonlinear part of the membrane hoop strain. A simple “inextentional” Ritz-type solution for w(θ ) and v(θ) is assumed: α (4.117) w(θ ) = α cos 2θ and v(θ ) = − sin 2θ 2 Neglecting the quadratic terms of α in UL , considering Eqs. (4.115), (4.116) and (4.117), and adding the contribution of initial curvature kin =1/R one obtains:   3 Eπt 3 α 2 3  Eπrt 2 2 3 − pπ 3α 2 + α 2 k r − rαk(k + kin ) + (α, k) = 2 3 2 2 8 (1 − νe )r 8 (4.118) Minimization of  results in the following expressions for the ovalization: = and

κ (κ + κin ) α = r 1 − fp

  3(κ + κin )(2κ + κin ) m = πκ 1 − 4(1 − f p )

(4.119)

(4.120)

that describes the primary path (uniform ovalization). In the above expressions (4.119) fp is the normalized value of pressure, with f p = p/pcr where pcr =

and (4.120),  2E/ 1 − νe2 (t/D)3 . Using Eqs. (4.119) and (4.120), closed-form expressions for the longitudinal stress σ x and for the hoop curvature at the deformed configuration 1/rθ can be obtained (Houliara and Karamanos, 2006):    3(κ + κin )2 (κ + κin )2 σx (θ ) =κ 1− sin θ + sin 3θ (4.121) σe 4(1 − f p ) 4(1 − f p ) 1 1 3κ (κ + κin ) = + cos 2θ (4.122) rθ (θ ) r r(1 − f p ) Expressions (4.119) – (4.122) account for initial curvature of the tube 1/R and for the presence of pressure p, and are similar to those derived by Reissner (1959) for slightly bent tubes, using an asymptotic approximation of the nonlinear ring equations. For the special case of zero pressure (i.e., fp = 0), a moment-curvature equation very similar to Eq. (4.120) was derived by Boyle (1981). Furthermore, in the publications by Karamanos (2002) and Houliara and Karamanos (2006), expressions (4.119) – (4.122) were found to compare quite well with numerical results up to the ovalization

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Figure 4.18 Schematic representation of initial straight elongated cylinder bending ovalization versus bifurcation instability; C is the critical point.

limit point of tubes, with relatively small initial curvature (− 0.4 ≤ κ in ≤ 0.2). This good comparison is a key feature that allows the use of those solutions to describe the nonlinear pre-buckling state of elastic tubes in a simple and efficient manner, as explained in detail in the next paragraph.

4.8

Buckling of an elastic tube under bending

The ovalization solutions derived in Sections 4.6 and 4.7 correspond to a “smooth” bending response, uniform along the tube, which leads to a limit point on the momentcurvature diagram. We referred to this structural response as “ovalization instability” of “Brazier effect”. On the other hand, experimental observations and numerical results have shown that thin-walled cylinders subjected to bending exhibit buckling in the form of bifurcation to a short-wave wrinkling pattern. For initially straight tubes, this bifurcation occurs slightly before the limit point as shown schematically in Fig. 4.18. The prebuckling configuration of a bent tube at the compression zone is characterized by a “saddle point”, i.e., a point of double local curvature of opposite sign in each direction (see Fig. 4.19), which is also affected by cross-sectional ovalization. This is a non-trivial prebuckling state constituting a formidable buckling problem! The bifurcation on the ovalization path of an initially straight non-pressurized cylinder with r/t = 120 is shown in Fig. 4.20, as calculated from finite element models (see Karamanos, 2002). In this graph, (cr) and (ov) denote the bifurcation

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125

Figure 4.19 Saddle point: double curvature of opposite sign at the prebuckling stage of a bent cylinder, in the vicinity of the critical point (e.g., point C in Fig. 4.18).

(A)

(B)

Figure 4.20 Bifurcation on the bending response of an initially straight elastic cylinder (r/t = 120); (A) moment-curvature diagram and (B) buckled shape of the cylinder.

(critical) point and the ovalization limit point respectively. Bifurcation occurs at a critical curvature κ cr = 0.396, which is prior to the limit point instability (κ ov = 0.470), and therefore, it governs the response. The value of κ cr corresponds √ to a bending moment mcr = 0.928 and a buckling half-wavelength Lhw = 1.75 Dt (Karamanos, 2002; Houliara and Karamanos, 2006).

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Detecting bifurcation on the ovalization path may not be a trivial process and requires a numerical formulation as described in detail by Houliara and Karamanos (2006). However, it is possible to use a simplified analytical approach to estimate the location of bifurcation on the primary (ovalization) path, by applying the socalled “local buckling hypothesis” (LBH) on the pre-buckling distorted (ovalized) cross-sectional configuration. LBH has been used in several instances in shell buckling problems, but it has been introduced and described in a systematic manner by Axelrad (1965). LBH assumes that buckling occurs when the axial stress σ x around the cross-section becomes equal to the buckling stress of uniformly compressed cylinder, with the same material and thickness, and radius equal to the local radius r of the bent cylinder at this specific point, including the effects of ovalization. In mathematical terms, LBH is expressed as follows: t  E σx = 3(1 − νe2 ) r

(4.123)

In other words, according to LBH, it is assumed that bifurcation occurs when the axial stress σ x at the “critical” point becomes equal to the right-hand-side of Eq. (4.123), which stems from the expression of buckling stress, Eq. (4.63), substituting the tube radius r with the local radius of tube circumference r , because of ovalization. There are two main assumptions behind LBH: (a) buckling occurs locally at a buckling zone around the critical point and (b) the state of stress within the buckling zone is nearly constant, allowing for the consideration of axisymmetric conditions. Identifying the location of the critical point around the circumference of the tube in the prebuckling state is an important step in this procedure and requires some effort. The critical point around the circumference is at the location where Eq. (4.123) is satisfied first. Fortunately, for initially straight tubes, and for initially bent tubes with relatively small κ in values, the critical point is at the intrados (θ = π /2). Therefore, for those tubes, given the ovalization solution and the corresponding expressions in Eqs. (4.121) and (4.122), the values of stress σx (π /2) and the local curvature 1/rθ (π /2) at the critical point can be determined as follows: 1 1 κ (κ + κin ) 1 = = −3 r rθ (π /2) r r(1 − f p ) σx (π /2) =

 t   (κ + κin )2 κ 1− 1 − fp 1 − νe 2 r E

(4.124)

(4.125)

It is also interesting to notice that using the LBH approach, the bent tube is approximated with a uniformly compressed circular tube of radius r , as shown graphically in Fig. 4.21. The LBH condition in Eq. (4.123), using Eq. (4.124) and (4.125), results in

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Figure 4.21 Concept of “equivalent axially-compressed cylinder” in bent cylinders (initially straight).

the following equation for the critical curvature κ cr :     3κcr (κcr + κin ) (κcr + κin )2 1 κcr 1 − − √ 1− =0 1 − fp 1 − fp 3

(4.126)

Upon solution of Eq. (4.125) in terms of critical curvature κ cr , the value of κ cr is used to estimate the critical bending moment:  mcr = π κcr

3(κcr + κin )(2κcr + κin ) 1− 4(1 − f p )

 (4.127)

Finally, the buckling half-wavelength from the corresponding formula of axially compressed tubes [Eq. (4.67)], using the local radius r instead of the original radius r, is: 

 Lhw

1  =π 12 1 − νe2

1/4

√

r t

(4.128)

The above expression can be written in the following normalized form: L hw s= L0



(1 − f p ) (1 − f p ) − 3κcr (κcr + κin )

(4.129)

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Figure 4.22 Local buckling hypothesis (LBH) predictions for bifurcation of initially straight pressurized elastic tubes compared with finite element results (r/t = 120, 480 and 720); (A) critical bending moment, (B) critical curvature and (C) buckling half-wavelength.

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Figure 4.23 Buckling shape of an initially bent cylinder (κ in =−1.374, r/t = 120) subjected to bending opposite to its initial bend configuration.

where the value of the buckling half-wavelength L hw in Eq. (4.129) is normalized by 1/4  √  L0 = rt π 4 /12 1 − νe 2 , which is the value of half-wavelength of the cylinder subjected to uniform axial compression under axisymmetric conditions. The main conclusion from the analysis and the numerical results of Houliara and Karamanos (2006) is that, provided that the critical location is identified, the predictions obtained from LBH are quite accurate. For an initially straight nonpressurized tube (κ in = 0), the LBH methodology expressed by Eqs. (4.126), (4.127) and (4.128) estimates a buckling moment mcr = 0.9215, which occurs at a critical √ curvature κ cr = 0.386 and corresponds to a buckling half-wavelength Lhw = 1.63 Dt. All of those values are very close to the ones obtained numerically with threedimensional finite element models. For a wide range of pressure (internal and external) the analytical LBH predictions are compared with the finite element predictions in Fig. 4.22, for three elastic tubes with r/t = 120, 480 and 720, and the comparison is very satisfactory. An important observation is that the analytical predictions in their normalized form are independent of the value of the r/t ratio. Furthermore, the analytical predictions are closer to the finite element results for large values of the r/t ratio. In Fig. 4.22, the numerical results closest to the analytical results are those for r/t = 720 in terms of critical bending moment, critical curvature and the corresponding half-wavelength. As a final note on the application of LBH, in initially bent tubes with relatively large κ in values, numerical results have shown that the critical point may not be located at the intrados (θ = π /2). Therefore, to detect possible bifurcation on the ovalization (primary) path, it is necessary to monitor at each loading step the values of stress and local curvature for every point around the cross-section. Fig. 4.23 shows the buckling shape of an initially bent cylinder (κ in = − 1.374) subjected to negative bending, i.e., bending opposite to its initial configuration. Buckling occurs away from the intrados, and this location has been verified using the results of an ovalization analysis by

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Houliara and Karamanos (2006), which monitors the local curvature and longitudinal stress.

References Augusti, G. (1964). Stabilita Di Strutture Elastiche Elementari in Pesenza Di Grandi Spostamenti: Vols. 4-5. Napoli: Academia delle Scienze Fisiche a Matematiche di Napoli. Axelrad, E. L. (1965). Refinement of buckling-load analysis for tube flexure by way of considering precritical deformation [in Russian]. Izvestiya Akademii Nauk SSSR. Otdelenie Tekhnicheskikh Nauk, Mekhanika i Mashinostroenie, 4, 133–139. Bazant, Z. P., & Cedolin, L. (1991). Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories. Oxford University Press. Boyle, J. T., & Spence, J. (1980). A simple analysis for oval pressure pipe bends under external bending. In Proc. 4th Int. Conf. on Pressure Vessel Technology 1980. Boyle, J. T. (1981). The Finite Bending of Curved Pipes. International Journal of Solids and Structures, 17(5), 515–529. Brazier, L. G. (1927). On the Flexure of Thin Cylindrical Shells and Other “Thin” Sections. Proceedings of the Royal Society, series A, 116, 104–114. Brush, D. O., & Almroth, B. O. (1975). Buckling of Bars, Plates, and Shells. New York, NY: McGraw-Hill. Calladine, C. R. (1995). Understanding imperfection-sensitivity in the buckling of thin-walled shells. Thin-Walled Structures, 23(1–4), 215–235. Dodge, W. G., & Moore, S. E. (1972). Stress Indices and Flexible Factors For Moment Loadings On Elbows and Curved Pipe. Oak Ridge, TN: Oak Ridge National Laboratory Technical Report ORNL-TM-3658. Donnell, L. H., & Wan, C. C. (1950). Effect of Imperfections on Buckling of Thin Cylinders and Columns Under Axial Compression. Journal of Applied Mechanics, ASME, 17(1), 73–83. Houliara, S., & Karamanos, S. A. (2006). Buckling and post-buckling of long pressurized elastic thin-walled tubes under in-plane bending. International Journal of Non-Linear Mechanics, 41(4), 491–511. Houliara, S. (2008). Computational Techniques in Structural Stability of Thin-Walled Cylindrical shells. PhD Dissertation. Department of Mechanical Engineering, University of Thessaly. Karamanos, S. A. (2002). Bending Instabilities of Elastic Tubes. International Journal of Solids and Structures, 39(8), 2059–2085. Kármán, T., von (1911). Uber die Formuanderung dunnwandiger Rohre. Zeitschrift Des Vereines deutcher Ingenieure, 55, 1889–1895. Kármán, T., von, Dunn, L., & Tsien, H. S. (1940). The influence of curvature on the buckling characteristics of structures. Journal of the Aeronautical Sciences, 7(7). Kármán, T., von, & Tsien, H. S. (1941). The buckling of thin cylindrical shells under axial compression. Journal of the Aeronautical Sciences, 8(8), 303–312. Koiter, W. T. (1945). On the Stability of Elastic Equilibrium. PhD Thesis, Univ of Delft (in Dutch). Kyogoku, T., Tokimasa, K., Nakanishi, H., & Okazawa, T. (1981). Experimental study of axial tension load collapse strength of oil well casing. Proc. Offshore Technology Conference II, OTC 4108, Vol. II, 387–395. Lorenz, R. (1908). Achsensymmetrische Vetzerrungen in dinnwandigen Hohlzylindern. Zeitschrift des Vereines Deutscher Ingenieure, 52, 1707.

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Reissner, E. (1959). On Finite Bending of Pressurized Tubes. Journal of Applied Mechanics, ASME, 26(3), 386–392. Southwell, R. V. (1914). On the general theory of elastic stability. Philosophical Transactions of the Royal Society A, 213, 497–508. Tennyson, R. C. (1969). Buckling Modes of Circular Cylindrical Shells Under Axial Compression. AIAA Journal, 7(8). Timoshenko, S. P. (1910). Einige Stabilitätsprobleme der Elastizitätstheorie. Zeitschrift für Mathematik und Physik, 58, 337–385. Wood, J. D. (1958). The Flexure of a Uniformly Pressurized, Circular, Cylindrical Shell. Journal of Applied Mechanics, ASME, 25(4), 453–458.

Glossary

H

Anisotropy The condition where a material has different properties (e.g., stiffness, yield, strength) in different directions; see also Plastic Anisotropy. Arrestor see Buckle Arrestor. Barlow Equation The basic equation for computing hoop stress σ θ in a pressurized cylinder, based on thin-walled cylinder theory proposed by Peter Barlow in 1837: σθ = pD/2t. Base Metal The metal material of the line pipe. Bauschinger Effect Early yielding of metal material under reverse loading, and rounding of the stress–strain diagram. Bedding Before installing a pipeline in a trench, proper soil material should be placed in the trench (bedding) to act as a “mattress” that supports the pipeline. Bevel An angle at the edge of pipe end section, necessary for pipe welding. It is formed either in the pipe mill or on site. Bifurcation The point on a load-displacement equilibrium diagram of a structural system where the uniqueness of solution no longer exists. At that point, the response of the system bifurcates into two possible solutions. The concept of bifurcation can be extended to non-structural systems as well. Buckling The transition of a structural system from a smooth configuration to a wavy pattern. Equivalently, the process of switching from a stiff and structurally sound configuration to one that is less stiff and structurally dangerous. Buckle Arrestor Stiff ring-type device placed at certain intervals (usually several hundred meters) along a pipeline for arresting a propagating buckle and limiting the damage (flattening) to the segment between two consecutive arrestors. Butt Welded Joint Welding of two adjacent steel plates, which are placed end-to-end without overlap and welded in the same plane. Charpy Test Typical mechanical test for measuring the fracture toughness of a metal material. Collapse The unstable post-buckling response of a metal pipe under external pressure, which leads to a flattened “dog-bone” shape of the pipe cross-section. Collapse Pressure The maximum value of external pressure that the pipe can sustain before collapse. Cross-over Pressure The value of external pressure that allows a propagating buckle to crossover a buckle arrestor in an offshore pipeline under quasi-static conditions. Diameter In most cases, the size of the pipe is specified by the outside (outer) diameter (D) in inches. In analytical calculations, the mean diameter of the pipe is usually employed (Dm = D − t). In some cases, reference to inside (inner) diameter of the pipe is also made (Di = D − 2t). Double Joint Two line pipes connected with a girth weld before reaching the installation site (for onshore pipelines) or the fire-line of the lay vessel (for offshore pipelines). Ductility Measure of the ability of a metal material to deform well into the inelastic range without rupturing or losing its strength; materials with good ductility are called “ductile”. Materials with low or no ductility are called “brittle”. Elastic buckling Buckling of a structural system while its material is still within the elastic range. Structural Mechanics and Design of Metal Pipes: A Systematic Approach for Onshore and Offshore Pipelines. DOI: https://doi.org/10.1016/B978-0-323-88663-5.00021-9 c 2023 Elsevier Inc. All rights reserved. Copyright 

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Structural Mechanics and Design of Metal Pipes

Electric Resistance Welded Pipe (ERW) Welded pipe fabricated from a steel coil, with the following steps: decoiling, cold-forming into a circular shape, and butt-welding with Electric Resistance Welding (ERW) method, without the addition of a weld (filler) metal. Currently, high-frequency induction (HFI) is used for welding those pipes, and therefore, they are often called “HFI pipes”. Elbow Curved pipe segment; see also Pipe Bend. Field Weld Weld performed on the construction site, not in the pipe mill. Flowline A subsea pipeline that connects the wellhead to an offshore platform (i.e., to the first downstream process component). Fracture Toughness Measure of material ability to resist crack propagation under applied stress. Free Span Section of pipeline that is freely suspended between two projections on the seabed. Girth Weld Circumferential weld performed on site (field weld) connecting two pipe segments. Gas Metal Arc Welding (GMAW) A mechanized welding process. It employs a consumable wire that melts when an electric arc develops between the wire and the steel material, shielded by externally supplied gas. Grade of Steel Pipeline steels are designated with a letter (e.g. X, A or B), which defines mainly manufacturing and chemical specifications, followed by a number that represents the minimum specified yield stress (SMYS) in ksi. For example, grade X60 steel has a yield stress of at least 60 ksi (414 MPa). Heat Affected Zone (HAZ) The pipe metal in the neighborhood of a weld that is affected by heat during the welding process. Integral Buckle Arrestor A thicker section of pipe ring that is welded in a pipeline at chosen intervals for arresting a propagating buckle. Initial Ovality The ovality of a pipe (i.e., its out-of-roundness) as received from the pipe producer, before its installation and service. (a) It is usually expressed through the following ovality parameter (e.g., in API 1111): ˆ = (Dmax − Dmin )/(Dmax + Dmin ).  (b) Alternatively, the following measure of initial ovality can be also used (e.g., in DNV-STF101): Oˆ 0 = (Dmax − Dmin )/D. JCO-E Pipe Welded pipe fabricated from a steel plate using the following steps: Crimping, Jing, C-ing, O-ing, welding (SAW method) and expansion. Often, JCO-E pipes are also called LSAW or SAWL pipes (“L” stands for “longitudinal”); see also UOE Pipe. J-Lay Pipeline installation method where the pipe leaves the vessel in a vertical (or nearly vertical) direction, and the suspended section of the pipeline takes a “J” shape. Jumper A short subsea piping component that connects two subsea components. Lay Barge see Lay Vessel. Lay Vessel Vessel with all the equipment for installing an offshore pipeline. Lap-Welded Joint Welding of two adjacent steel plates placed in an overlapping pattern on top of each other. In lap-welded joints, the weld is fillet-type and can be made on one side (single-welded joint) or on both sides (double-welded joint). Line Pipe The basic pipe component fabricated in the pipe mill. A typical length of line pipe for hydrocarbon pipes is 12 meters, however, this length may vary depending on the pipe producer and the client requirements. Line pipes are welded on site to construct the pipeline. Lined Pipe see Mechanically Lined Pipe. Limit Load The maximum load in a structural response that corresponds to a limit state.

Glossary

483

Limit State A state beyond which a structure or a component cannot fulfill its function requirements in satisfactory manner, in terms of either safety (called “ultimate limit state”) or serviceability (called “serviceability limit state”). Local Buckling In metal shell buckling, wrinkles tend to localize, forming a large-amplitude wrinkle and leading to structural failure of the metal shell. Local Buckling Hypothesis (LBH) A hypothesis proposed in the early 1960s by E. L. Axelrad, stating that local buckling in an elastic pipe under bending occurs when the longitudinal stress at a point on the pipe circumference becomes equal to the buckling stress of a uniformly compressed cylinder made of the same elastic material and having a radius equal to the local radius of bent cylinder at that point. Manifold Subsea structure connecting several flowlines from wells to an outgoing flowline that connects to an offshore platform. Mechanically Lined Pipe Bi-metallic pipe, where a thin-walled pipe (liner pipe), made of corrosion resistant alloy (CRA), is inserted inside a thick-walled carbon steel pipe (outer pipe), in a way that the two pipes are connected with mechanical bonding only, and the two materials remain two distinct masses. Metallurgically Lined Pipe Bi-metallic pipe, where a thin-walled pipe made of corrosion resistant alloy (CRA) is metallurgically bonded (cladded) to the internal surface of a carbon steel pipe, so that the two metallic materials become one mass, and a single pipe is produced. Ovality A measure of pipe cross-section out-of-roundness. It is also referred to as “ovalization”. (a) It is usually defined through the ovality parameter as follows:  = (Dmax − Dmin )/(Dmax + Dmin ). (b) Alternatively, the following measure of ovality may also be used: O0 = (Dmax − Dmin )/D. Ovality of a pipe as received from the pipe mill, is called “initial ovality”; see also Initial Ovality. Ovalization see Ovality. Overbend The section of a pipeline during the installation process that bends over the stinger of the lay vessel. Overmatching It refers to a weld, when the weld (filler) metal is stronger than the base metal of the connected pipes, so that it “overmatches” the base material. Padding In rocky ground conditions, the excavated material is often run through a padding machine, so that potentially damaging material (e.g., rock) is washed out, and fine grain soil is used to fill-in the trench around the pipeline. Pipe Bend Curved pipe segment; see also Elbow. There exist “cold bends”, mainly produced on-site with a cold-bending machine, and “hot bends”, fabricated in an industrial plant, using a thermo-mechanical process (e.g., induction bending). Pipe Mill Industrial unit that fabricates the line pipes. Pipeline A long onshore or offshore pipe that transports fluid (liquid or gas). Pipeline Inspection Gage (PIG) A sophisticated device that travels inside a pipeline, for pipeline inspection and/or maintenance (cleaning). Plastic Anisotropy The condition where the material has different yield stress in different directions. In welded pipes, because of the cold-forming process, the yield stress in the circumferential (hoop) direction is different than the one in the longitudinal direction. Plastic Buckling Buckling of a metal structural component when its material is either partially or fully within the inelastic range. Plastic Force see Yield Force.

484

Structural Mechanics and Design of Metal Pipes

Plastic Hinge If a structural component subjected to bending becomes fully plastic, its bending resistance reaches a maximum (M = MP ), and further rotation occurs at constant moment MP . In this case, the member (or component) acts as if it were hinged, except with a constant restraining moment MP . To differential from classical “hinge”, this is called “plastic hinge”. Plastic Moment The bending moment required to render a cross-section of a bent component fully plastic (excluding any local buckling phenomena). (a) In cross-sectional ring analysis of a pipe, the plastic moment of the pipe wall is MP = σY b t 2 /4 (b is the length of the ring). (b) In longitudinal bending of a pipe, the plastic moment of the pipe cross-section is MP = σ Y ZP , where ZP is the plastic modulus of the circular hollow section. If the pipe is not very thick, one may write ZP = D2m t. Plastic Plateau The transition from elastic to plastic regime of a metal material is often characterized by an abrupt change of slope in the stress–strain diagram, where the stress remains nearly constant up to a strain value of about 1.5% or 2%. This is associated with a material instability, known as Lüders’ banding, and is very common in the response of hotfinished, low-carbon steel material. The region of the stress–strain diagram with the nearly constant stress is called “plastic plateau”. Plastic plateau disappears with strain hardening and with reverse plastic loading (see Bauschinger Effect). Plastic Pressure See Yield Pressure. Pressure Design Pipeline design procedure against normal operating conditions (primarily against pressure and thermal loading conditions). Propagating Buckle If the level of external pressure on an offshore pipeline is high enough, a local collapse pattern will propagate and flatten a large portion of the pipeline. Propagation Pressure The minimum value of external pressure required for a local collapsed pattern to propagate along an externally pressurized pipeline flattening the pipeline under quasi-static conditions. Quadruple Joint Four line pipes connected with girth welds before reaching the installation site forming a long pipe segment. It is used for offshore pipeline construction with the J-Lay method in an efficient manner; in such a case, a “quadruple joint” is usually 48 meters long (see Chapter 1 and Saipem’s FDS 2 J-Lay vessel). Ratcheting Accumulation of plastic strain over the loading cycles, for a material that is loaded repeatedly in the inelastic range, under non-symmetric stress-controlled cycles. Residual Stresses Internal stresses into a pipe wall after the manufacturing process. Residual stresses are self-equilibrated, and exist in the pipe wall in the absence of any external loading. Reeling A pipeline installation method where a pipeline is initially spooled around a large diameter reel (onshore) and subsequently it is installed at an offshore site by unspooling from the reel and straightening. Riser see Steel Catenary Riser (SCR). Ring Slenderness The √ square root of the ratio of yield pressure pY over the elastic buckling ¯ = pY /pcr . It is also referred to as “ring normalized slenderness”; see also pressure pcr : λ Slenderness. Roll Bending Pipe fabrication method, performed with a rolling machine (with three or four rollers in a pyramid location) that bends the plate into a circular shape, followed by seam welding and, sometimes, by expansion. The plate passes several times through the rollers, until the desired curvature is achieved. Sagbend The section of pipe near the seabed that bends as the pipe obtains a natural catenary shape during installation.

Glossary

485

Seamless Pipe A line pipe without a weld (as opposed to welded pipe), produced by a hot forming process. Seamless pipes are typically made up to diameter size of 18 inches. S-Lay Pipeline installation method where the pipe leaves the installation vessel in a horizontal orientation, and the suspended section takes an “S” shape. Sleepers Pre-installed large-diameter pipe sections, placed perpendicular to the pipeline alignment, acting as buckle initiation devices in offshore pipelines exposed on the seafloor. Slenderness It refers to any structural system or component that exhibits structural instability. It is defined √ as the square root of ratio of plastic load RY over the elastic buckling load ¯ = RY /Rcr . Rcr : λ Steel Mill Industrial unit that produces the steel raw material for the pipe mill (steel coils, plates or billets). Steel Catenary Riser (SCR) A steel pipe that connects an offshore pipeline or a well on the sea floor to a fixed or floating hydrocarbon production platform. Shielded Metal Arc Welding (SMAW) A manual welding method, widely common in pipeline field welding. It uses consumable stick electrodes that melt when an electric arc is struck and maintained between the tip of the electrode and steel material being welded. Snaking Global buckling of an offshore pipeline in a wavy-form due to thermal and pressure loading. Specified Minimum Yield Stress (SMYS) The minimum yield strength of pipe material, required by the material specification. Spiral Welded Pipe Welded pipe fabricated from a steel coil, with the following steps: decoiling, cold-bending in an oblique direction (which forms a “spiral” or “helix”), and welding (inside and outside) with the submerged arc welding (SAW) method. It is also called “HSAW pipe” or “SAWH pipe” (“H” stands for “helical”). Stinger A long cantilever structural system that comes out of the stern of the lay vessel and supports the pipeline for a certain distance as it leaves the vessel. Usually, it is a tubular structure (truss), either rigid or articulated to adjust for the optimum shape of the pipe depending on the water depth. Strain Aging Recovery (or partial recovery) of strength of a metal material following plastic deformation due to diffusion of soluble carbon and/or nitrogen atoms to dislocations. Heat treatment accelerates the process of strain aging. Strain-Based Design Pipeline design procedure based on strain analysis and strain limits, as opposed to stress-based design, which refers to stress analysis and stress allowables. It is often used in pipeline design against geohazard (seismic) actions. Strain Demand Pipeline analysis procedure under permanent ground deformation with the purpose of calculating the longitudinal ground-induced strains in the pipeline wall. Strain Hardening Increase of yield stress with plastic deformation in the direction of loading. Strain Resistance The longitudinal strain that the pipeline and its girth welds are capable to sustain under tensile or compressive action. It is an important concept in pipeline design against geohazard action. Stringer Bead The first pass in the process of a girth weld, also called “root pass”. Submerged Arc Welding (SAW) It is a common arc welding process involving the formation of an arc between a continuously fed electrode, which is covered by a granular flux coating, and the workpiece. It is applied as a mechanized process, and it is used in numerous welding applications in the pipe mill. Tensile Strength The maximum engineering stress recorded in a uniaxial tension test. For a ductile material, it corresponds to the onset of necking. It is also called “ultimate stress”. Tensioner Device on a lay-vessel that applies tension on the pipeline and controls its suspended shape during offshore installation.

486

Structural Mechanics and Design of Metal Pipes

Thickness parameter Geometric imperfection parameter, which expresses wall thickness variˆ = (tmax − tmin )/(tmax + tmin ). ation around the pipe cross-section, and defined as follows:  It refers primarily to seamless pipes, and it is also called “pipe wall eccentricity” parameter. Tie-In Welding Welding of pipeline parts in the trench (for hydrocarbon pipelines). Towing Pipeline method for installing prefabricated sections of a pipeline by towing them to the installation site with the use of a tow vessel (tug). Sometimes, two tow vessels are necessary (one leading and one trailing). Transition Temperature The temperature at which the metal material behavior changes from ductile to brittle is called “non-ductile transition temperature” or simply “transition temperature”, denoted as NDTT. Triple Joint Three line pipes connected with girth welds before reaching the fire-line of the lay vessel (for offshore pipelines). It is used in offshore pipeline construction. Tubular Member A structural member of hollow cylindrical shape, which constitutes part of a tubular structure (e.g. an offshore platform). Ultimate Strength or Ultimate Stress see Tensile Strength. UOE Pipe Welded pipe fabricated from a steel plate using the following steps: Crimping, U-ing, O-ing, welding (SAW), and expansion. Sometimes, UOE pipes are also called LSAW or SAWL pipes (“L” stands for “longitudinal”); see also JCO-E Pipe. Upheaval Buckling Global buckling of a section of buried pipeline in the upward direction due to internal pressure and thermal loading. Virtual Anchor Spacing (VAS) In offshore pipelines, which are exposed on the seafloor and exhibit lateral buckling, virtual anchors are formed at points of zero displacement, and the distance between two consecutive virtual anchor points is called “virtual anchor spacing”. Wall Thickness Eccentricity see Thickness Parameter. Weld Metal In a welded connection, it is the metal that has been molten in the welding process and then solidified. Welded Pipe A pipe that is fabricated from a plate or a coil, through a cold-forming process and a longitudinal or spiral seam (ERW or SAW), as opposed to Seamless Pipe. Wet Buckle This is a buckle that leads to localized collapse of a pipeline, which in turn results in fracturing of the pipe wall, allowing water to flood the pipeline. Yield Force The value of axial force that causes the first yielding of a cross-section, in the absence of any other failure mode (e.g., buckling). It is also referred to as “plastic force”. (a) In cross-sectional analysis of pipes (ring analysis), the yield force in the circumferential direction of the pipe is FY = σ Y tb (b is the length of the ring). (b) In an axially loaded pipe (hollow cylinder), the axial yield force is FY = σ Y A (A is the pipe cross-sectional area) and, if the pipe is not very thick, it can be computed as: FY = σ Y (π Dm t). Yield Moment The value of bending moment that causes first yielding of a cross-section (excluding any local buckling phenomena). Sometimes it is called “first yield moment”. (a) In cross-sectional ring analysis of a pipe, the plastic moment of the pipe wall is MY = σY b t 2 /6 (b is the length of the ring). (b) In longitudinal bending of a pipe, the plastic moment of the pipe cross-section is MY = σ Y Z, where Z is the elastic modulus of the circular hollow section. If the pipe is not too thick, one may write Z = π rm2 t. Yield Pressure The value of uniform pressure (internal or external) that causes first yielding of the pipe material, in the absence of any other failure mode (e.g., collapse). Form Barlow’s equation, pY = 2σY t/Dm . In the presence of axial stress in the pipewall the value of yield stress σ Y should be adjusted according to Von Mises yield criterion. In thick-walled pipes,

Glossary

487

the equations of Appendix G should be used for calculating the yield pressure with more accuracy. Yield Stress This is the stress under uniaxial stress conditions that denotes the onset of inelastic deformation, either in tension or in compression. In particular, (a) In hot-finished, low-carbon steel materials that exhibit a plastic plateau, the value of yield stress is well-defined. (b) In metals with rounded stress–strain diagrams, a conventional yield stress definition is used, either r the stress corresponding to total strain 0.5% r the stress corresponding to plastic (irreversible) strain 0.2%. For most materials of interest, the above two values of yield stress are close.

Instead of an epilogue

One year ago, while building my family tree, to my great surprise, a very interesting finding came up, related to pipeline construction: an ancestor of mine, named Spyros Karamanos (1821–1895), uncle of my great-grandfather Anthony, and Mayor of the island of Poros, conceived and realized in 1881, a pioneering infrastructure project for the water supply of Poros from the water fountain of Belesi.1 The fountain was in mainland Greece (Peloponnese) at about 2.4 km total distance from the town of Poros. The project included the construction of several water tanks, and most importantly, a water transmission pipeline with a 250-m offshore part at a maximum depth of 4–5 m, crossing the Poros strait. My ancestor Spyros Karamanos was the Mayor of Poros for two terms (1857– 1862 and 1879–1888) and member of the Greek Parliament during 1865–1869. Apart from his contribution to local society as a Mayor, he was an acclaimed medical doctor and a great benefactor of the island. After his death, he left his entire property to the Municipality of Poros. Most likely, the water supply project with the offshore pipeline has been his greatest achievement, resolving the longtime problem of water shortage in Poros. Based on my investigation, two main technical reports on this project were submitted to the Municipality of Poros before the project construction. The first report is on the chemical analysis of the water from the Belesi fountain and was prepared in 1879 by A. K. Christomanos, Professor of Chemistry at the University of Athens.2 The second report was prepared in 1880 by I. Lazarimos, a prominent engineer of that time in Athens, and it refers to technical issues of water transmission. Unfortunately, despite my efforts, it was not possible to obtain a copy of the second report. It is fortunate though that in the last two pages of the first report, Prof. Christomanos makes his suggestions on the construction of the water supply system (i.e., the pipeline and the water tanks) and refers to waterproof steel pipes with diameter 30–40 cm and capacity of 150 m3 per day (for 7000 inhabitants of the town of Poros), for installation on the seafloor on rigid supports. According to two other sources,3,4 steel pipes capable of resisting 10 bar of internal pressure were used for the onshore part of the pipeline, whereas lead pipes were finally employed for the offshore part. Another source5 describes that the lead pipes were loaded on a barge, tied up with ropes, and welded 1 Many

thanks go to Mr. Ioannis Maniatis, Director of Poros Municipal Library for his valuable help in my investigation. 2 http://jupiter.chem.uoa.gr/pchem/lab/pubs/AX_Mpelesi_1879.pdf. 3 http://www.koutouzis.gr/idreusi.htm. 4 Elias Drosinos, The Mayor Spyros D. Karamanos, Published by the Municipality of Poros, Athens 2001. 5 http://pireorama.blogspot.com/2015/03/blog-post.html.

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Structural Mechanics and Design of Metal Pipes

one to another. Subsequently, using the ropes, the welded pipeline was lowered into the sea until they reached the sea floor. This water supply project was completed in 1881. The total cost of the project was 75,000 drachmas and exceeded the financial resources of the Municipality of Poros by a substantial amount. The contribution of Mayor Karamanos was immense in securing the necessary funding for the project, through a loan from the Harbor Fund that covered approximately half of the total cost. By the time that the present book is published, it has not been possible to obtain more technical details on this landmark and pioneering offshore pipeline project. My investigation will continue until a clear understanding of this project is obtained, as a tribute to my ancestor Spyros Karamanos.

Index

Page numbers followed by “f” and “t” indicate, figures and tables respectively.

Alignment, 5, 6, 9, 10, 12, 14, 17, 18, 257, 316, 326, 330f, 340, 365, 370 Aluminum pipe, 170f, 214, 215f, 224f, 224, 225f, 226, 226f, 362 American Lifelines Alliance (ALA), 304 American Petroleum Institute (API), 14, 16, 43, 252, 253, 257, 271, 273, 311, 475 American Society of Mechanical Engineers (ASME), 16, 233, 234, 237, 245, 253, 287, 475, 478 Arrestor, 7, 9, 46, 169, 171f, 171, 173f, 221, 251, 252, 257, 261, 262, 265, 266 Automatic ultrasonic testing (AUT), 21 Axially-compressed elastic cylinders, 106, 109 simply supported bar, 72f Axisymmetric wrinkling imperfection, 210, 211f

analysis, 114 instability, 113 moment, 38f, 102 stress, 239 Bi-metallic pipe, 375, 483 Blue Stream pipeline, 7, 29, 277 Brazier solution, 116, 117, 119 Bryan’s formula, 272 Buckle arrestor (arrest), 9, 171f, 171, 173f initiation, 163f propagation, 159 response, 79 shape, 68f tubular specimens, 206f Burst pressure, 134, 253, 259, 315 Butt-strap joints, 338f Butt-welded joints, 337, 338f Butt-welded pipes, 47, 51

B

C

Baku-Tbilisi-Ceyhan pipeline (BTC), 4, 5t Barlow’s formula, 61 Bar-spring system, 109 Bauschinger effect, 53, 152, 155, 157, 158f, 260, 279, 282, 397, 456, 458f, 458, 460, 461, 462 Beam-bending solution, 112 Beam-column equation, 359, 418 model, 72, 359f theory, 65, 355, 359 Beam-type finite element models, 292f Bell and spigot joints, 337f Bend radius, 243, 249 Bending bifurcation, 105

Capped-end conditions, 98, 239, 338 Carbon steel, 9, 14, 43, 375, 382, 441, 445, 483, 484, 487 Cement mortar lining, 324, 327f, 332, 333, 334, 335 Charpy testing, 46 Circumferential stress, 65, 133, 134, 135 strain, 71 Coating, 14, 15, 17, 18, 22f, 29, 44, 255, 283, 324, 326f, 327, 329, 331, 332, 333, 334, 335, 350, 366, 381, 382, 485 Collapse design envelope, 223 external pressure formulae, 10, 251, 252f, 252, 255, 256

A

492

mechanism pipe manufacturing, 279 steel pipes, 327 Confined rings, 80 Crotch plate, 326, 328f Curvature-like parameter, 213, 383 Curved wide plate (CWP) experiment, 310 Cyclic plasticity, 441, 457, 459, 460, 461 Cyclic test, 351 Cylinder geometry, 96f, 183, 335 Cylindrical shell buckling, 112 kinematics, 102 stability, 100 structural stability, 100 D Denting process, 198f, 198, 200 Design collapse, 223 framework, 251, 258, 265 mechanical resistance, 258 of pipeline, 71, 235, 249, 288 Det Norske Veritas (DNV), 16, 53, 76, 252, 257, 264, 310, 369, 475 Detachment, 383f, 383, 384f, 386f, 386, 387, 388, 389f, 390, 391f, 393f, 398, 399f, 400, 401f, 402, 404f, 404, 406, 407, 408f, 409, 411 Diameter-to-thickness ratio (D/t), 51, 80, 145, 165, 205, 246, 271, 313 Displacement-controlled loading, 262 DNV collapse formulae, 266 Donnell, 82, 109, 112, 207 Double joint pipe, 19 Ductility, 12, 44, 45, 46, 135, 218, 309, 310, 481 Dynamic Propagation, 169, 171 E EastMed project, 10 Elastic buckling pressure, 141, 145, 193, 264, 275, 276, 335 buckling stress, 273, 275 cylindrical shells, 106, 110, 208, 210 expansion, 376, 396 pipe, 135, 216, 218, 483 -plastic constitutive model, 203

Index

solution, 80, 135, 143, 152, 189, 195, 196, 197, 269, 270 tube, 112, 120t, 120, 121f, 122f, 124, 128f, 129, 213, 216, 335 Electric resistance welded (ERW) pipes, 255, 481 Enduro Bell, 348 Engineering criticality assessment (ECA), 312, 366 Equilibrium equations, 66, 103, 138, 193, 210, 272 Equivalent axially-compressed cylinder, 127f Equivalent pipe length, 196 Euler-Bernoulli beam theory, 100 Externally-pressurized cylinders, 177, 272 External pressure, 133, 135, 138, 141, 142, 144, 145f, 146f, 158, 159, 251, 252f, 255, 256, 257, 259, 272, 274 weld, 323, 325f, 326, 327, 337f F Fabrication factor, 53, 260t Fabrication process, 12, 44, 47, 48, 50, 53, 56, 152, 350, 376, 392, 393, 396, 398 Fatigue, 36, 242, 244, 247, 248, 249, 257, 259, 264, 265t, 315f, 315, 351, 367, 375 Field-applied lap-welded joints, 338 Finite element models, 84f, 84, 124, 129, 144, 145f, 156, 160, 168, 169f, 175, 177, 177f, 193, 203, 214, 224, 226, 281f, 282f, 290, 291f, 292f, 293f, 295f, 296f, 298, 344, 346, 365f, 368, 388, 390, 395, 398, 402, 405, 407f simulation, 85f, 148, 155, 157, 164, 193, 194f, 204f, 205f, 214f, 215f, 216f, 217f, 219f, 314f, 362, 366f, 390f Fixed-end conditions, 96 Fracture, 15f, 27, 36, 45f, 46, 133, 134, 218, 220f, 308, 314, 315f, 343, 344f, 345, 352f, 432, 433, 481, 482 Full-scale collapse test, 9, 252, 283 Full-scale experiments, 379, 380, 390 Fully-plastic solution, 197, 269 G Gap-opening method, 157f

Index

Gas pipeline, 6, 7, 10, 15, 16, 25f, 46, 233, 234f, 243, 248 Gas Metal Arc Welding (GMAW), 18, 482 Gasket (gasketed) joint, 323, 326f, 337f, 337 Geohazard, Geohazard, 3, 4, 287, 288, 289, 299, 307 GIPIPE project, 292, 294 Girth weld, 19, 26, 290, 298, 308, 309, 311, 312 Glock, 80, 82, 83, 84, 85f, 87, 177 H Hardening (85 instances), 136, 140 Heat affected zone (HAZ), 19, 309, 482 High Frequency Induction (HFI), 48, 481 High Frequency Welding (HFW), 48, 260t Hooke’s law, 62, 65, 96, 98, 102, 355, 413 Hoop stress, 12, 74, 95, 96, 133, 136, 138, 197, 210, 235, 238, 241, 242, 269, 291, 338, 355, 395f, 396f, 396, 475, 476, 478, 481 strain, 61, 65, 115, 116, 121, 396 Hydraulic expansion, 376, 393, 395f, 396f, 396 Hydrocarbon pipelines design, 4, 233, 235 projects, 4 Hydrostatic pressure, 98, 137f, 139, 259, 273, 447, 475, 476f I Imperfection (114 instances), 98, 139 Infrashield, 352 Initial imperfection (50 instances), 73, 87 ovality (ovalization), 73, 76, 136, 145, 147, 151f, 152, 157, 256, 259, 263, 264, 276, 277, 279, 482 Integral buckle arrestor, 170, 171f, 261, 262, 482 Interaction diagram, 222f, 223f, 223, 224, 230 Internal pressure, 6, 14 weld, 47, 50, 51f, 56f, 323, 327, 337, 348 Iowa formula, 333, 334 Isotropic hardening, 293, 363, 397, 444, 454, 455, 456, 457

493

J JCO-E pipes, 53, 54f, 55, 56f, 255, 260t, 280f, 280, 281, 283, 482 J2 -deformation theory of plasticity, 139, 207 J-lay installation method, 29, 31 K Kinematic hardening, 393, 444, 445, 456, 459, 460 Kirchhoff-Love shell theory, 100 Koiter, 80, 109, 210, 387 L Lap-welded joints, 323 Lateral buckling, 251, 355, 356f, 367, 368, 369, 370, 371f, 372f Lateral pressure, 96, 98, 100, 110f, 112, 137f, 140 Limit point instability, 113, 124, 213, 215 state, 253, 255 Linearized buckling equation, 100, 105, 106 Line pipe, 4, 6 bending, 376, 382, 383, 406f fabrication, 12 steels, 43 strength, 44 Load & Resistance Factor Design (LRFD), 273, 274, 275, 277 Local buckling, 17, 21, 27, 202, 206, 215, 217f, 218, 221, 224 Local buckling hypothesis (LBH), 126, 128f, 384, 385f, 483 Longitudinal strain energy, 114 Longitudinal stress, 95, 133 Longitudinal tension, 254, 255, 357 M Material-type imperfections, 155 Mechanically lined pipes, 375 Medgaz pipeline, 9 Membrane force, 102, 103 response, 198 solution, 198 Metal beam-type, 202

494

hollow cylinder, 203f pipe, 189, 365 tube, 203, 213 Metallurgically lined pipes, 483 Mindlin-Reissner shell theory, 101 Miner’s rule, 257, 264 Minimum cover, 22, 243 Mohr-Coulomb model, 294 Moment-curvature, 117, 118 diagram, 224f, 224 equation, 226 relationship, 433 Monotonic tests, 351 N Natural gas pipeline, 5, 10 Non-axisymmetric bifurcation, 203, 210 Nondestructive testing and marking, 21 Nonlinear kinematic hardening, 395, 460 Nord Stream Gas Pipeline (NSGP), 9 O Offshore pipeline design, 3, 10, 11, 13, 14, 15, 16, 251, 265 installation, 27f mechanical design, 251 projects, 7 Oman-India Pipeline (OIP), 7, 10 Onshore pipeline construction, 17 design, 3, 10, 11, 13, 14, 15, 16, 235, 236, 241, 242, 244, 245, 248 projects, 4 Ovalization (ovality), 76, 116, 213, 215, 221, 222, 226 Overbend, 27 P Perry’s formula, 75 Pipe fabrication, 3, 12, 16, 47 Pipe outlet, 326 Pipeline alignment, 326 Pipeline constructability, 14 Pipeline construction, 12, 14, 17, 23 Pipeline optimum thickness, 14 Pipeline installation method, 26 Pipeline cover, 243t, 243f

Index

mechanical design, 11, 233 towing methods, 38, 39f welding, 20f, 330f Pipe material, 14 Plastic anisotropy, 199 collapse, 143, 152 hinge, 142f, 427, 432f, 432, 433, 434, 467, 483 mechanism equation, 148 solution, 148 Poisson’s ratio, 63, 66 Poisson stress, 339 Polyurethane coating, 324 Post-buckling, 79 Prebuckling, 67, 105 Pressure amplification factor, 74 containment resistance, 259 cross-over, 171 -curvature interaction, 222f, 223f design, 233 effects on inelastic response, 199 external, 221 internal, 12, 95, 97f, 98, 99, 100, 120 propagation, 166 Pressurized bending, 120, 218, 221 Propagating buckles, 133, 159f, 166, 168 Propagation and arrestor design, 9, 29, 159 Q Quadruple-joint pipe, 35f R Radial displacement, 90 Ramberg-Osgood, 140, 203, 215, 218 Rankine’s formula, 255, 266, 267 Reeling installation method, 31 Residual curvature, 371, 372f stress, 53 tension, 4, 357, 369 Reverse plastic loading (RPL), 397 Rigid-plastic, 197 Ring buckling under external pressure, 63 deflection, 332, 333 equilibrium analysis, 66

Index

plane deformation, 76 plastic mechanism, 148, 192 slenderness, 135 -split test, 252 Ritz discretization, 113 Roll Bending (RB) process, 55 Route selection, 14 S Sagbend, 27 Seamless pipes, 47, 55 Secant modulus, 139 Second-order effects, 103 Secondary bifurcation, 387 Seismic, 5, 287, 288, 291, 307, 308 Shear forces, 102 Shielded Metal Arc Welding (SMAW), 18 S-lay installation method, 26 Snake-lay method, 370 Soil-pipe interaction, 245, 287 Southern Gas Corridor, 5 South Stream project, 9 Spiral weld, 50 Stainless steel pipe, 219f Steel coil, 47 grade, 11 pipeline joints, 337 plate, 51 Strain -based design, 287 compressive, 63, 287, 294, 313 demand, 290 hardening, 291 limit, 308 resistance, 287, 307 tensile, 174, 175 Stress compressive, 152, 207 critical, 135, 193 design, 235 hoop, 12, 96 limit, 249 longitudinal, 476 tensile, 210 Structural failure mode, 15 Submerged arc welding (SAW), 50

495

T Tangent modulus, 208 Tensile strain capacity, 308, 309 strength, 44t Tensioner, 28f Thermal stress, 246 Thickness, 5, 6, 9, 10, 11, 13, 14, 19, 29, 46, 51, 52, 61, 80, 235, 335, 339, 343, 476 Thick-walled, 10, 138 Tie-in welds, 23 Tight-fit pipe (TFP), 393, 396 Timoshenko’s formula, 334 Toughness, 46 Towing installation method, 38 Trans-Adriatic Pipeline (TAP), 5t Trans-Anatolian Pipeline (TANAP), 5t Trigonometric function, 75 Triple-joint pipe, 29 Two-dimensional models, 169 U Uniform external pressure, 63, 68, 80 Ultimate tensile stress, 253, 259 UOE pipes, 52, 53 Upheaval buckling, 245, 358 V Vacuum buckling, 335 Virtual anchor distance, 369 Von Kármán formulation, 118 Von Mises yield criterion, 99, 134, 174, 197, 273 Vortex-induced vibrations (VIV), 14 W Wall thickness determination, 6, 9, 11, 13, 19, 235, 366, 368, 404 variation, 134 Wedge-shape denting tool, 195f Weld defects, 21, 310 Welding process, 18, 19 station, 20f, 50 Weld inspection, 21, 55

496

Wet buckle, 29, 169 Working Stress Design (WSD), 273, 275 Wrinkling, 203, 204 Wye junction, 327

Index

Y Yield stress, 12, 45, 75, 134 Yield-to-tensile stress ratio (Y/T), 12, 262 Young’s modulus, 62, 177, 238, 339