Fundamentals of Tank and Process Equipment Design 3031312252, 9783031312250

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Foundations of Engineering Mechanics

András Nagy

Fundamentals of Tank and Process Equipment Design

Foundations of Engineering Mechanics Series Editors Vladimir I. Babitsky, School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, Leicestershire, UK Jens Wittenburg, Karlsruhe, Germany

The series “Foundations of Engineering Mechanics” includes scientific monographs and graduate-level textbooks on relevant and modern topics of application-oriented mechanics. In particular, the aim of the series is to present selected works of Russian and Eastern European scientists, so far not published in Western countries, drawing from the large pool of experience from major technological research projects. By contributing to the long tradition of enrichment of Western science and teaching by Eastern sources, the volumes of the series address to scientists, institutional and industrial researchers, lecturers and graduate students.

András Nagy

Fundamentals of Tank and Process Equipment Design

András Nagy Department of Building Services and Process Engineering Faculty of Mechanical Engineering Budapest University of Technology and Economics Budapest, Budapest, Hungary

ISSN 1612-1384 ISSN 1860-6237 (electronic) Foundations of Engineering Mechanics ISBN 978-3-031-31225-0 ISBN 978-3-031-31226-7 (eBook) https://doi.org/10.1007/978-3-031-31226-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Chemical industry and energy industry equipment—including large storage tanks— represents structures exposed to heat stress and corrosion load as well, in addition to mechanical load. Their design requires special expertise divergent from general mechanical engineering practice and is assisted by design software programmes ranging from the selection of the appropriate engineering material to strength dimensioning in practice. Standard dimensioning algorithms using material, shape and dimension databases serve as a basis for computer-aided design. In the course of strength calculations for checking after preliminary design, state-of-the-art finite element programmes can be used for determining both the stress state and the elongation state, close to reality, of stress concentrating cross-sections. Thus, in more complicated cases, even crack spread and fatigue simulations can be carried out. The theoretical fundamentals of the algorithms above are described by areas of mechanical engineering science involved in purely elastic and elastic–plastic shell structures, as well as in various engineering materials and corrosion. In addition to the presentation of the theoretical fundamentals of calculation methods and dimensioning methods used in the design of process equipment and storage tanks, the aim of this specialist publication is to interpret and evaluate the calculation results obtained from the elastic and elastic–plastic finite element examination of stress concentrating cross-sections, as well as to demonstrate corrosion-proof design through examples. This specialist publication summarizes knowledge collected and own research results produced in the field of mechanical engineering specialized in the chemical industry over the course of 30 years of research and academic work, serving as a useful aid in both university education and research and in engineering practice. Budapest, Hungary

András Nagy

Acknowledgements First of all, I am highly indebted to my master, Prof. Univ. Dr. László Varga, for his professional guidance and selfless helpfulness. Let me extend my thanks to the companies OPAL Zrt. and OT INDUSTRIES for their assistance in writing this book and to my colleagues Balázs Dudinszky, Ádám Marinkó, Dániel Papp and Dániel Horváth for their assistance in producing the figures. Finally, let me thank here as well to Prof. Univ. Dr. György Fábry for his professional support in writing this book. v

Contents

1 Design Methods, Design Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Equipment and Storage Tank Loads, Classification and Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Impact and Consideration of Loads in Design . . . . . . . . . . . . . . . . . . . 2 Corrosion Loads, Corrosion Resistance, Corrosion-Proof Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Definition of Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Electrochemical Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Manifestations and Types of Electrochemical Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Chemical Corrosion and Its Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Corrosion Prevention, Corrosion Protection . . . . . . . . . . . . . . . . . . . . 2.4.1 Corrosion Prevention by Surface Protection . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning by Taking Membrane Stresses into Account . . . . . . . . . 3.1 Determination of Allowable Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Membrane Stress State of Equipment and Storage Tanks with Axially Symmetrical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Structural Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Membrane Edge Forces, Membrane Elongations . . . . . . . . . . 3.2.3 Membrane Stress State Equilibrium Equations . . . . . . . . . . . 3.3 Dimensioning by Taking Vapour Pressure into Account . . . . . . . . . . 3.3.1 Dimensioning the Cylindrical Part of the Vessel . . . . . . . . . . 3.3.2 Dimensioning Spherical Tanks and Hemispherical Pressure Vessel Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Dimensioning Conical Pressure Vessel Ends . . . . . . . . . . . . . 3.4 Dimensioning by Taking Hydrostatic Pressure, Mass Forces, Environmental Impacts (Snow, Wind), and Centrifugal Force Fields into Account . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.4.1 Dimensioning Cylindrical Fluid Tanks with a Conical Pressure Vessel End . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Dimensioning the Cylindrical Part of Fluid Tanks with Large Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Dimensioning Cylindrical Centrifuges . . . . . . . . . . . . . . . . . . 3.4.4 Dimensioning Tower Structures Exposed to Environmental Impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 Dimensioning Outdoor Spherical Tanks with Large Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Dimensioning Equipment Loaded by External Pressure . . . . . . . . . . . . 4.1 Dimensioning Cylindrical Shells and Conical Pressure Vessel Ends for External Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Dimensioning Doubly Curved Pressure Vessel Ends for External Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Bending Stress State of Axi-symmetrical Shells. Strength Analysis by Taking Bending Stresses into Account . . . . . . . . . . . . . . . . . 5.1 Axially Symmetrical Bending Stress State of Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Bending Stress State of Cylindrical Shells Loaded by Shearing Force at the Rim . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Bending Stress State of Cylindrical Shells Loaded by Moment at the Rim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Bending Stress State of Cylindrical Shells Loaded by Shearing Force at the Main Circle . . . . . . . . . . . . . . . . . . . 5.1.4 Bending Stress State of Cylindrical Shells Loaded by Moment at the Main Circle . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Examination of Shell Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Strength Tests of Cylinder—Cone Connections . . . . . . . . . . . 5.2.2 Strength Tests of Cylinder—Hemisphere Connections . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Pressure Vessel Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Stress States and Dimensioning of Elliptical Pressure Vessel Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Stress States and Dimensioning of Torispheric Pressure Vessel Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Design of Openings on Cylindrical Shells and Pressure Vessel Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Strength Tests of the Support Environment . . . . . . . . . . . . . . . . . . . . .

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6.3.1 Strength Tests of the Support Environment of Spherical Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 7 Strength Test and Dimensioning on Leak Tightness of Flange Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Engineering Design of Pipeline and Equipment Flange Joints . . . . . 7.2 Internal Loads of Flange Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Flange Joint Operation Under Simultaneous Heat and Mechanical Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Tests and Calculations to Verify the Model . . . . . . . . . . . . . . . 7.2.3 Dimensioning on Leak Tightness of Flange Joints Exposed to Simultaneous Heat and Mechanical Load . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Investigation of Stress Concentrating Cross-Sections Using the Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Defining Material Law to Serve as a Basis for Finite Element Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Specimen-Level Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Structural Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Evaluation Method for Elastic–Plastic Finite Element Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Numerical Tests and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165 165 171 176 194 207 211 215 217 217 223 227 229 234

Annex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Chapter 1

Design Methods, Design Guidelines

Abstract Equipment and storage tank loads, classification and requirements. Impact and consideration of loads in design. Mechanical loads, heat (stress) and corrosion loads. Keywords Mechanical loads · Operating pressure · Nominal pressure · Test pressure · Heat and corrosion loads

1.1 Equipment and Storage Tank Loads, Classification and Requirements Process equipment/storage tank is understood as a plate welded, possibly cast or forged, closed tank structure operated under distributed load. Such load may result from: • internal pressure, which can be generated by charge input, storage or heating under pressure, by chemical reactions in the vessel, and by the hydrostatic pressure of the charge, • the mass of the charge and the equipment, • environmental impacts (wind and snow load, seismic effects, vibrations), • additional loads in the course of transport and installation. If the working pressure is higher than atmospheric pressure, the equipment is loaded by overpressure, and if it is lower, the equipment is operated under vacuum. As regards equipment in practice, overpressures and temperatures vary widely (0 MPa ≤ p ≤ 300 − 400 MPa), (−200 ◦ C ≤ T ≤ 800 − 900◦ C). Requirements for vessels: • • • • • •

appropriate strength, proper stiffness, heat or cold resistance, corrosion resistance, manufacturability, repairability, ease of assembly,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Nagy, Fundamentals of Tank and Process Equipment Design, Foundations of Engineering Mechanics, https://doi.org/10.1007/978-3-031-31226-7_1

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Fig. 1.1 Various forms of appearance of equipment and storage tanks (a, external heating chamber mixing autoclave; b, heat exchanger; c, spherical tank; d, column)

• transportability. Equipment and storage tanks can be classified by structural design (see Fig. 1.1), area of application (chemical industry, energy industry, food industry), according to the chemical process taking place in the equipment, and by safety technology aspects as well.

1.2 Impact and Consideration of Loads in Design In case of process equipment/storage tanks, mechanical load, heat (stress) and corrosion load are considered as generic terms. Mechanical loads include all external forces. As a result of mechanical loads, load bearing parts are elongated and internal forces and stresses are generated in crosssections. Elongations can only be elastic, so the corresponding stresses have to stay below the proportionality limit. Exceptions only include certain stress concentrating cross-sections (curvature transitions of pressure vessel ends, flange-shell connections, nozzle environment, and support environment). At such locations, a slight degree of plastic elongation is also permitted in the course of the first test load following manufacturing and stress-relieving heat treatment, as a consequence of which amplitudes of stress are degraded, elastic stress reserve is released in the course of unloading after test load, and the range of elastic elongation is increased during operational load to follow test load. Compressive and shearing stresses may be dangerous even below the proportionality limit as they may cause buckling and loss of stability of the shell. The most typical loads include internal overpressure, loading the concave side of the vessel wall, and external overpressure, loading the convex side of the vessel.

1.2 Impact and Consideration of Loads in Design

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In general, internal overpressure generates tensile stress, and external overpressure generates compressive stress in the wall of the vessel. If internal overpressure is increased in excess of the allowed value, the vessel can burst at membrane stress places, and increased external overpressure may result in buckling of the vessel. Operating pressure means the highest internal or external overpressure produced in the equipment during normal operation in the workflow. Nominal pressure means the highest operating pressure allowed at 20 ◦ C dimensioning temperature at which equipment and storage tanks can be operated permanently. Standards parts characterized by nominal pressure (pipe and equipment flanges, pressure vessel ends, fittings, etc.) are to be selected by taking the dimensioning temperature into account. Dimensioning pressure means the maximum overpressure pertaining to the operating condition, in respect of which the strength calculation is required to be performed. Test pressure means the testing pressure of the equipment/storage tank. In practice in general, pressure tests are carried out using water of environmental temperature, or gas more rarely. In addition to leakproofness and strength control, the purpose of a pressure test is to reduce the amplitude of stress produced in stress concentrating cross-sections by plastic elongation, thereby to extend the elastic load carrying capacity range. Heat stresses Heat stress includes effects accompanied by temperature changes of the structure. A temperature field can be constant or can vary both in space and time. On the one hand, a heat effect can produce internal forces and thermal stresses in the cross-sections of the vessel if elongation caused by heat expansion is inhibited; on the other hand, it can affect engineering material properties (yield stress, tensile strength, specific impact energy, elongation rupture, specific cross-section reduction). The heat resistance of the structure can also be inferred from the behaviour of the engineering material, characterized for instance by the so-called (σ − ε) curve set shown in Fig. 1.2. By the way, the figure shows the tensile diagrams of carbon steel used in practice, in function of temperature. According to the figure, the hot strength of the material can initially be characterized by yield stress at elevated temperature (ReT ). After reaching a certain temperature (T ≥ 380 ÷ 400 ◦ C), the engineering material gets into a state of creeping, where—subject to constant stress σT —it is elongated until rupture. In the state of creeping, structural material behaviour can also be characterized by Fig. 1.3, showing the change in time of elongations at constant temperature in function of load stress. Further material properties σ1/100,000T ; σ B/100,000T can be derived from specimen-level tests similarly to yield stress (ReT ) and tensile strength (RmT ); they are as follows: σ2 = σ1/100,000T is the stress that causes 1% remaining strain on the specimen examined at dimensioning temperature in 100,000 h (more than 11 years);

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Fig. 1.2 Characteristic tensile diagram of carbon steel at various temperatures

Fig. 1.3 Structural material behaviour in the state of creeping

σ B/100,000T is the stress that causes the destruction (rupture) of the specimen at dimensioning temperature. By introducing the duration strengths mentioned, equipment operated at high temperatures (T ≥ 380 ÷ 400 ◦ C) can be dimensioned for lifetime (100,000 h) and elongation (1%). Similarly to heat resistance, the cold resistance of the structure can also be basically inferred from the behaviour of the engineering material (specimen testing, specific impact energy). Figure 1.4 illustrates specific impact energy developments to be measured on a notched specimen in the transition temperature range (TT K V ◦ C). Unfortunately, the results of specimen-level tests are not suitable for assessing the cold resistance of structures as the tendency of brittle-type fracture for manufactured equipment is always higher than for a notched specimen made of the material of the structure. Reasons for this include equipment shape, multiaxis stress state, cold working in production, welding, etc. It is the designer’s task to select the appropriate cold resistant engineering material, as well as to design a structure and to

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prescribe a manufacturing technology to ensure that the effect of factors contributing to embrittlement should prevail to the least possible degree. These are as follows: • Striving for uniform strength. Large leaps of wall thickness, as well as drastic cross-section and curvature changes should be avoided as far as possible. • Stress relieving heat treatment. Vessels are required to be relieved from stresses introduced in the course of cold working and welding. Corrosion loads, corrosion resistance Corrosion loads include any and all chemical and electrochemical processes and the effects thereof, consequent upon which load bearing cross-sections are reduced and the strength characteristics of the engineering material are possibly impaired. The next chapter discusses in sufficient detail the phenomenon of corrosion, diverse corrosion loads, corrosion protection, and corrosion-proof design as well. Fig. 1.4 Boiler plate specific impact energy developments in the transition temperature range

KCU 200 2 [J/cm ] 180 160 140 120 100 80 60 40 20 0 -200

-100

0 100 20°C

200 [C°]

Chapter 2

Corrosion Loads, Corrosion Resistance, Corrosion-Proof Design

Abstract Definition of corrosion. Electrochemical corrosion. Manifestations and types of electrochemical corrosion. Chemical corrosion and its types. Corrosion prevention, corrosion protection. Corrosion prevention by surface protection. Lined equipment design and construction. Design and construction of equipment with coating layer. Keywords Electrochemical corrosion · Chemical corrosion · Corrosion prevention and protecting

2.1 Definition of Corrosion • Corrosion is understood as the destruction of metals by chemical or electrochemical effects, starting on their surfaces in contact with gas, steam, or fluid mediums. • Corrosion loads include any and all chemical and electrochemical processes and the effects thereof, consequent upon which load bearing cross-sections are reduced and the strength characteristics of the engineering material are possibly impaired. The most common type of corrosion is the formation of iron rust. Rust is formed in the presence of water and air. In the course of the process, water is decomposed partly to ions, then iron ions leave the surface of the iron covered by water, and go over to the water by transferring two electrons of negative charge, with molecular hydrogen being generated in the meantime. Finally, ferrous hydroxide is produced and oxidized into ferric hydroxide by oxygen in the air. In most cases, metal corrosion is caused by electrochemical phenomena, but cases of pure chemical corrosion also occur [1–3].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Nagy, Fundamentals of Tank and Process Equipment Design, Foundations of Engineering Mechanics, https://doi.org/10.1007/978-3-031-31226-7_2

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2.2 Electrochemical Corrosion A galvanic cell is produced in the course of chemical corrosion. A galvanic cell produced by two metals of different normal potentials immersed into an electrolyte is termed as a macroelement, and a galvanic cell produced by one metal immersed into an electrolyte is termed as a microelement. Electrolytes are defined as chemicals, the diluted solution or melt of which can conduct electric current by way of mobile charge carriers—anions and cations. Microelements can be produced on metal surfaces of homogeneous structure if metal parts close to each other are in contact with electrolytes of different concentrations. A good example for this is the dissolution of a crystalline substance in a quiescent fluid, as solution concentration increases in the proximity of solid particles along the surface of the equipment in the course of their dissolution. Microelements can also be produced if a metal is in contact with electrolytes of varying absorbed oxygen content (electrolyte dripped on a metal surface). Slack fluids appearing at internal surface disparities and cracks facilitate the production of microelements as a result of differences in absorbed oxygen content. Foreign metal particles stuck on the metal surface can also lead to microelement formation when their potentials are different. On the basis of potential differences, microelements can be produced: • between the boundary and the inside of a crystal particle when one more electronegative constituent precipitates if exposed to heat, and is located on the surface of such particles, and then is dissolved by the electrolyte. This phenomenon called inter-crystal corrosion can be observed in austenite Cr-Ni stainless steels, where chromium carbide precipitating at grain boundaries will be the anode of the microelement. As a result of dissolution, metallic connections between crystal particles are decreased, leading to a drastic reduction of mechanical properties, primarily tensile strength. The corrosion phenomenon mentioned does not damage metal surfaces, so it is invisible to the naked eye. In the presence of inter-crystal corrosion, the metal loses its metallic clinking, so it can be detected by tapping with a hammer, for instance. • in case of alloys, in the presence of tissue elements of different potentials. More electronegative crystals are dissolved by the electrolyte, and a porous spongelike structure is left behind. The phenomenon mentioned is termed as selective corrosion in the literature, a good example for which is brass corrosion, where zinc is released in the presence of an electrolyte and the copper remains. • as a result of different internal mechanical stresses giving rise to potential differences between diversely deformed material parts. The formation of so-called stress corrosion cracks is subject to three criteria, namely damage to the protective layer of the metal surface, tensile stress and special charge (presence of alkaline, slightly acidic saline solutions of chlorine or halogenic content, or reductive alkalis). Tensile stress can be produced by external mechanical load,

2.2 Electrochemical Corrosion

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heat treatment, welding, cold working, milling, and H2 diffusion. Corrosion itself emerges in the form of cracks perpendicular to the direction of load on the surface exposed to tensile stress. Austenite Cr-Ni stainless steels and cold-profiled brass are particularly prone to stress corrosion.

2.2.1 Manifestations and Types of Electrochemical Corrosion As electrochemical corrosion can be traced back to the operation of micro- and macroelements, damage to and corrosive collapse of the metal surface occurs locally, on local anode surfaces. Depending on whether micro- or macroelements are produced and where microelements are formed (on surfaces, between particles), corrosion can cause damage to the surface in the form of spots, holes or punctiform damage; it can also result in the sporadic discontinuation of connections between metal crystals at particle boundaries (inter-crystal), and can make the entire cross-section porous / spongelike (selective). The forms of appearance of electrochemical corrosion can manifest themselves under various circumstances in practice. The ways of contact between the electrolyte and the metal define different forms of corrosion. The types of chemical corrosion can be classified into three main groups, namely types of corrosion occurring as a result of • electrolyte concentration differences; • potential differences between metals in contact with an electrolyte; and • electrochemical reactions combined with strength load. The types of corrosion arising from electrolyte concentration differences can be broken down into further corrosion types resulting from • local electrolyte concentration differences (concentration cell corrosion), and • locally changing dissolved oxygen content in the electrolyte, respectively. Concentration cell corrosion. In an electrolyte, local concentration differences can be caused for instance by the dissolution of a set of crystalline particles in a quiescent fluid as already mentioned. Concentration differences can occur in the solution in case of significant vessel surface temperature changes within a relatively small environment. The warmer surface part is the anode of the microelement, while the colder one acts as a cathode, so the warmer part will be corroded. This type of corrosion phenomenon causes problems particularly in the surroundings of steam inlet nozzles, protection against which can be provided by the design solution shown in Fig. 2.1. In case of tangential steam inlet as shown in the figure, no considerable temperature differences can be produced in the internal shell.

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Fig. 2.1 Protection of steam inlet nozzles against heat shock corrosion

Corrosion types resulting from the locally changing dissolved oxygen content in the electrolyte Split corrosion It is caused by the humidity of the air within closed spaces of engineering structures without aeration, and by slack fluids collected in confined interspaces, since oxygen supply from the absorbed air is scarce at these places. The surface parts covered by humidity at these places will be the anodes of the microelement, so corrosion will start here. Split corrosion almost always results from a design error, so this type of corrosion is also termed as corrosion “starting on the drawing board”. A number of design error examples can be listed for electrochemical corrosion by slack fluids, such as in the case of heat exchangers at pipe-tube sheet connections when pipes are not pressed into the boreholes of the tube sheet (Fig. 2.2). A similar problem occurs in the case of outlet nozzles if such nozzles are welded to the pressure vessel end by being hung in (Fig. 2.3), as well as in the case of lined wall equipment if no aeration is ensured for closed spaces between the lining and the load-bearing carbon steel. As shown in Fig. 2.4, in the case of an acid resistant lining welded onto the internal surface of the carbon steel flange, it is sufficient to have even only one aerate borehole on the outside surface of the flange to prevent split corrosion. Obviously, further design solutions can also be outlined to avoid the risk of split corrosion.

2.2 Electrochemical Corrosion

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Fig. 2.2 Pipe and tube sheet connection design: incorrect a and correct b in terms of split corrosion

Fig. 2.3 Incorrect (nozzle hung in, a and correct (welded in, b flared out, c) designs of outlet nozzle environment

Capillary corrosion is a special type of split corrosion where the electrolyte is located in the pores of a porous material. It primarily occurs in the case of metal parts fixed into the soil, such as iron piles placed into the soil at lakes and rivers. In such a case, the part of the pile in the soil will be the anode, and the part above bottom soil, in pure contact with water will be the cathode, as humid soil contains much less dissolved oxygen than the water layer above it. The same phenomenon can be observed in the case of underground pipes where soil structure is not homogeneous, meaning that sand layers of looser structure and more compact clay layers alternate. The part in contact with the sand is the cathode, and the part in contact with the clay is the anode, so this latter section will be corroded.

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Fig. 2.4 Split corrosion protection of lined carbon steel flange by aerate borehole

Phase separation section corrosion can be observed in equipment containing a boiling fluid. The surface part near the fluid surface is in contact with a medium of much less dissolved oxygen content than the parts below such surface, therefore surface discoloration characteristic of corrosion appears at the boundary of the fluid and vapour phases. A typical type of electrochemical corrosion caused by macroelements is contact corrosion, which occurs when two metals of different normal potentials are immersed into an electrolyte. Always the more electronegative metal will corrode, appearing on the surface in the form of deep grooves (spot corrosion). This type of corrosion is highly frequent in chemical industry equipment where steel and a metal of more negative normal potential than that of steel (e.g. aluminium) are in contact. This phenomenon can be observed in case of a flange joint containing a gasket of a metal or of metal cladding with normal potential different from that of steel (e.g. copper) with a grooved gasket surface as per Fig. 2.5, or in case of the flange joints of the aluminium equipment shown in Fig. 2.6, where prestress is provided by steel screws. The parts to corrode will be the structural part of more negative normal potential (aluminium) in the very first case; the gasket surface in contact with the metal gasket of more positive normal potential than steel (copper) in the second case; and the part of the aluminium flange in contact with the steel screw in the last example. The corrosions above can be eliminated by zinc plating the steel at the point of contact of aluminium and steel components, or by coating the steel parts with insulation lacquer. In case of a flange joint with copper gasket, it may also be sufficient to place the gasket surface of the flange above the surface of the fluid constituting an

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13

Fig. 2.5 Contact corrosion developing in the surroundings of the gasket surface in case of a steel flange with copper gasket

electrolyte as shown in Fig. 2.5, as well as to electrically insulating the contacting surfaces of the steel flange screw and the aluminium flange as shown in Fig. 2.6. The phenomenon of contact corrosion can occur not only in the event of contact between two different metals, but also where parts of identical engineering material are welded together, if the composition of the weld is highly different from that of the basic material (wrongly chosen electrode); therefore, welding wire selection can also be important from the corrosion point of view in respect of welded equipment. Leakage current corrosion Corrosion by direct current occurs where the current leaves the cable due to insulation or conduct errors. Conduits and steel cables laid in parallel with rail tracks are most frequently exposed to it. It may occur by reason of any breakage or any rail joint of high contact resistance that current exits from the conductor and goes through a conduit / cable in the soil before returning to the rail. Current exit points function as anodes and

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2 Corrosion Loads, Corrosion Resistance, Corrosion-Proof Design

Fig. 2.6 Contact corrosion protection by applying an electric insulation layer placed between the two metals

inlet points as cathodes. In this example, a highly intensive corrosion effect can be observed in the exit cross-section from the rail and the conduit, respectively.

2.3 Chemical Corrosion and Its Types Chemical corrosion is the consequence of a chemical reaction between the active agent and the engineering material in contact with it, occurring in a non-electrolytic environment. As opposed to electrochemical corrosion, pure chemical corrosion uniformly attacks the entire metal surface [4–6]. Most frequent types of chemical corrosion Oxygen corrosion In a normal case (at not too high temperatures) oxygen forms a protective layer against corrosion on metal surfaces. At high temperatures, however, scale (metal oxide) is formed on metal surfaces if exposed to oxygen, which is converted into iron rust in case of high relative humidity ( > 60%).

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Carbon dioxide corrosion. It causes highly intensive scale formation and grain growth in case of steels at high temperatures. The effect of carbon dioxide corrosion is the strongest if it works together with oxygen and humidity. Water vapour corrosion It forms a metal hydroxide layer of loose structure on the metal surface, which cannot be considered as a protective layer. Its destructive effect is particularly high if the dissolved oxygen content of the water in contact with water vapour is more than 2 mg/litre. Hydrogen corrosion In general, the hydrogen corrosion of metals occurs with a reductive effect. At high temperatures, it eliminates the protective layer (oxide) of the metal. Methane is generated in the course of the process, as a consequence of which the structure of the steel gets loose and inter-crystal cracks are produced. In parallel therewith, the strength properties of steel are considerably reduced as carbon content is decreased. Before the appearance of hydrogen resistant steels, equipment exposed to the risk of hydrogen corrosion was provided with a corrosion resistant stainless steel lining or a coating layer (metal plating). Corrosion by industrial gases In a damp state, chlorine, ammonia, sulphur dioxide, sulphur trioxide, and hydrogen sulphide are highly corrosive to steels. Corrosion by fluids Various fluids cause chemical corrosion on metals if such fluids do not conduct current, meaning that they are not electrolytes. The corrosion effect of fluids on various metals [7–10] can be demonstrated by so-called corrosion tests. In the course of such tests, the metal plate to be examined is immersed in the given fluid and the change—reduction—in the course of time of metal thickness as a characteristic dimension is determined. Thickness  reduction  mm . The is related to an annual period, yielding the speed of corrosion c = s t year speed of corrosion was used for producing so-called corrosion tables, indispensable for selecting the engineering material of the equipment. The rows of corrosion tables include compounds found in practice, and the columns include a variety of engineering steels. The table makes a distinction between three groups on the basis   of mm , the figure of annual corrosive collapse. If the speed of corrosion is c ≤ 0.1 year steel is resistant to chemical corrosion if the speed of corro   by the active medium; mm mm  sion is within the range of 0.1 year ≤ c ≤ 1.0 year , the engineering metal is   mm acceptable for the given aggressive charge; and in the case of c ≥ 1.0 year it is not resistant, so its application should be avoided. A disadvantage of the corrosion

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2 Corrosion Loads, Corrosion Resistance, Corrosion-Proof Design

table is that it is based on assuming that the speed of corrosion is constant. The corrosion tests mentioned also demonstrated that the degree and speed of corrosion cannot be considered as constant in each case, showing an increasing (accelerating) or decreasing (decelerating) tendency in the course of time.

2.4 Corrosion Prevention, Corrosion Protection The corrosion loads detailed above can be protected against: • in the course of steel production, for instance against the risk of stress corrosion cracks, by the aluminium alloying of boiler plates and by increasing the nickel content of austenite stainless steels. The risk of inter-crystal corrosion can be reduced by applying stabilizers (Ti, Ni, Ta). In the course of steel production, C content can be further decreased by adding the above stabilizers to achieve low carbon content of 0.07 to 0.03%; • by corrosion-proof design, by applying appropriate engineering materials, surface protection, and construction design. • in case of operating equipment, by adding inhibitors to the charge and by cathodic corrosion protection by external power supply. Inhibitors are anti-corrosion substances; adding them in small quantities to equipment content can reduce corrosion. Depending on whether they exert their impact on the anode or the cathode, a distinction can be made between anodic and cathodic inhibitors. Anodic inhibitors hinder the operation of the anode of the microelement by coating the surface of the anode with an insoluble protective layer. Anodic inhibitors are chromate, carbonate or phosphate compounds which exert the desired anticorrosion effect when added in a quantity of up to 1% in a neutral or alkaline electrolyte. Cathodic inhibitors (calcium-hydrocarbonate, zinc sulphate, nickel, copper, and magnesium compounds) reduce the microelement cathode surface, inhibiting hydrogen production. Independent of their quantity, cathodic inhibitors reduce corrosion. In the case of cathodic corrosion protection with an external power source, the required anti-corrosion effect can be achieved by connecting the negative pole of a direct current power source to the surface to be protected, and the positive pole to the so-called external anode placed in the electrolyte. In case of complete cathodic corrosion protection, corrosive local cathode current can be discontinued on the equipment to be protected. The cathodic corrosion elimination above is mainly applied for the protection of underground tanks and conduits. In case of conduits, the external anode wire is laid parallel with the conduit. The system thus established is supplied intermittently in order to eliminate voltage drops caused by resistance.

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Fig. 2.7 Carbon steel plate surface protection

2.4.1 Corrosion Prevention by Surface Protection One of the possible solutions for corrosion prevention is to manufacture the equipment of corrosion resistant stainless engineering steel instead of carbon steel. At the same time, stainless steels and metal alloys are expensive, their availability in terms of quantity is restricted, so designers need to make efforts to find economical solutions. The general rule is that if the required wall thickness of the equipment is s ≤ 6 mm, stainless steel can be applied; while if s ≥ 6 mm, , it is expedient to provide the surface of the cheap, non-corrosion resistant engineering material with surface protection for reasons of economy (Fig. 2.7), that is, to cover it with a material resisting to undesired corrosive effects. Thereby the equipment will be made of double engineering material. The equipment is required to be designed and manufactured by taking into consideration the material and manufacturing technology of the cover. Below is an overview of surface protection procedures as well as design and manufacturing requirements for producing protective covers.

Lined Equipment Design and Construction Lining is defined as a layer of engineering material placed loosely inside the equipment or fixed to its wall only at some places, made of metal, ceramics or plastic in practice (Fig. 2.8). Fig. 2.8 Surface protection by applying metal, ceramic and plastic linings

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Metal lining is provided to equipment of appropriate strength, made of carbon steel, wood and concrete. In respect of strength dimensioning, only the geometric features and strength characteristics of the equipment to be lined may be taken into consideration. In the course of design and manufacturing, additional tasks are represented—as opposed to traditional equipment made of a single material—by fixing the lining metals to each other (independent of the material of the equipment to be lined), and by interconnecting the lining and the equipment wall. Carbon steel equipment is usually provided with stainless steel or lead lining [11]. Lining made of metals that cannot be welded with steel (aluminium) may only be produced if there is no need for welding after inserting the lining. Due to the difference in potential of the basic material and the lining, insulation lining must be placed between the two metals, which may get damaged in the course of welding, so there is a risk of contact corrosion. The heat expansion properties of the basic material and the lining are required to be taken into consideration during the design process since lining rupture (Fig. 2.9a) or lining collapse (Fig. 2.9b) may occur in case of major differences. In the event that the lining can be welded to the steel, then all corrosion resistant iron or metal casting materials can be applied in theory. The lining plate is generally 1.25–4.0 mm thick. Lining must always be provided to finished equipment on which pressure tests have already been completed. Lining pieces are connected to the basic plate by welding. The distance between welds (L, see Fig. 2.10) is influenced by the following aspects: 1. Retaining the corrosion resistance of the lining 2. Heat expansion difference between the basic plate and the lining 3. Manufacturing cost of the equipment 1. In case of stainless steel lining, there can be a risk of corrosion in the surroundings of the welded connections between the load bearing carbon steel basic plate and the lining plate as a consequence of “weld dilution”. A weld gets diluted if there is a considerable carbon concentration difference in the cross-section of the weld between the basic plate and the lining. Lining corrosion resistance is Fig. 2.9 Destruction types caused by heat expansion differences

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Fig. 2.10 Weld distance of lining plates

deteriorated by carbon diffusion due to such concentration difference. The socalled weld dilution above can be protected against by applying electrodes of low carbon content and high alloying substance content, by minimizing the number of welds between the lining and the basic plate, and by applying the design solutions shown in Fig. 2.11. 2. In case of differences in heat expansion capabilities, lining collapse and lining rupture are also affected by weld distance. 3. The number of welds applied also considerably affects manufacturing costs.

Fig. 2.11 Protection against local corrosion caused by diluted welds by applying multilayer welding (a), lining overlapping (b), and cover plating (c)

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To summarize the effects above, it can be established that there is no general rule for the distance between the welds to fix the lining. After the lining is completed, it is essential to test the welds for fluid compactness and gas compactness. The equipment is required to be designed in such a way that the test can be carried out at any time. Lining by metals that cannot be welded with steel Such lining is typically characterized by lead plates. This lining is not permitted to be applied in equipment under high overpressure or vacuum because the lead is locally elongated under such load, it gets porous, and acidity ceases to exist. In vacuum, the lead lining can get buckled. The lead lining must be fixed to the vertical equipment wall. Distances between fixtures should be selected within the range of 500–700 mm depending on the operating pressure of the equipment. Figure 2.12 shows construction solutions for fixing the lead lining.

Fig. 2.12 a Fastening by steel tenons. b Fastening by bolted joints

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Design of equipment with plastic lining Lining by plastic plates is rare: plastic foils are more generally applied for lining. There are two reasons for this: on the one hand, the heat expansion coefficient of plastic is one order of magnitude higher than that of the carbon steel to be lined; and on the other hand, PVC of excellent chemical resistance is easy to glue. Nevertheless, polyethylene and polypropylene plates problematic to glue still justify their application as lining because of their high resistance to chemical and heat effects and good mechanical properties. As a general rule, tanks larger than 3 m3 volume are not economical to be lined by polyethylene or polypropylene. When lining flat-walled tanks by plastic, the plates are fixed to the equipment wall with countersunk bolts, then bolt heads are sealed with pieces of plastic foil, and plate edges are welded together. This method can only be applied in case of flat-walled tanks, and only in the absence of sudden temperature changes. As the heat expansion coefficient of polyethylene and polypropylene is about 14 times higher than that of carbon steels to be lined, already in the case of a temperature rise of t = 60 ◦ C a bolt distribution of less than 250 mm must be applied to ensure that the internal surface is acceptably smooth. Differences between heat expansion coefficients can be considerably reduced by placing a perforated steel plate between to polyethylene or polypropylene plates, and the plastic plates are welded together at the places of perforations. The heat expansion of plates becoming uniform by using this solution of welding practically corresponds to the heat expansion of steel plates. It is expedient to select the thickness of plastic plates between 1.5 and 2.0 mm on the side of the aggressive medium, and between 0.5 and 0.8 mm of the side of the tank wall. Walled equipment design In practice, carbon steel equipment is lined with ceramic plates, bricks or graphite plates. This type of lining is used in boiler plants, autoclaves and tanks exposed to high temperatures and aggressive chemical effects. 5–75 mm thick plates or bricks are embedded in acid resistant putty. By reason of the walling of porous structure, various types of rubber or plastic lining are placed between the load bearing shell and the walling to ensure perfect corrosion protection. In this case, the walling primarily serves for heat insulation, therefore the required thickness is determined by the heat resistance of the corrosion protection layer. In determining the thickness of the walling, it must be taken into consideration, on the one hand, that its heat expansion ability is much lower than that of the load bearing carbon steel, and on the other hand, that the walling can only bear compressive stress without getting damaged. In case of correctly dimensioned equipment, in an operational state the walling is subject to compressive stress—or is unstressed in a boundary case. This can be ensured most easily by pre-tightening the walling in the shell already in the course of manufacturing, thereby generating compressive stresses in the walling and tensile stresses in the shell. This can be carried out by applying acid resistant putties which

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Fig. 2.13 Cylindrical equipment of embossed pressure vessel ends with walling

swell in the course of bonding, on the one hand, and if subjected to an active substance, on the other hand. In order to prevent the walling from buckling, such types of equipment are made to be cylindrical for static reasons, and the internal surfaces of embossed bottoms are brought to plane by bricking or concrete filling (Fig. 2.13).

Design and Construction of Equipment with Coating Layer Corrosion resistant coating layers are solidly connected to the basic material throughout their entire surface. As for their material, they can be metallic, rubber, plastic, and enamel coating layers. A metallic coating layer can be made by rolling or melting on [11, 12]. Plates used as equipment construction material are provided with a rolled coating layer (metal cladding), while the technique of melting on is applied for the corrosion resistant coating layer of finished equipment or equipment parts. Consequently, the design principles to be followed will also be different. Equipment with metal cladding Mostly chemical industry and food industry equipment is made with metal cladding [13], in cases when the requirements below or any of them are desired to be met [14].

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• Production of equipment which is chemically resistant to the internal medium, generally by using ordinary boiler plates. • In case of high temperature and high pressure, considerable wall thickness is required to take up loads, and the quantity of alloyed substance to be used can be reduced by metal cladding. • In addition to corrosion resistance, requirements include good heat transfer capabilities and equipment applicability under vacuum. In case of corrosion and erosion loads, and taking weldability into account as well, in general it is expedient to use metal cladding where coating layer thickness is 5–10% of total wall thickness and ranges between 2.0 and 6.0 mm. From the economy point of view, it is expedient to manufacture equipment with metal cladding where total wall thickness is at least 8 mm. Metal cladding layers are good heat conductors, therefore they are suitable for the corrosion protection of the internal part in case of equipment with external heating chamber apparatus. There is a molecular connection between the two metals along the contact surface, so it behaves as a single material when subjected to load, meaning that the elongation of the layers coincides. The stresses generated are in proportion to the elasticity moduli of the layers. If the melting point of the coating material is much lower than the melting point of the carbon steel to be protected, it must not be taken into consideration in strength calculations. Plates with metal cladding are appropriate for welding, bending, pressing and rimming. Only in the case of welding is it necessary to observe rules different from the working of ordinary boiler plates because the composition of the welded joint on the corrosion resistant side must correspond to that of the coating material. In case of faulty welding, the weld can take up certain components from the basic material, mainly carbon. This can be avoided by using electrodes of higher alloying substance content than that of the coating material when welding the cladding side. If no electrodes of higher alloying substance content are available, the welded joint is required to be made in several layers. Welding design should prescribe a weld shape and a welding technology to ensure that the acid resistant part is in contact with the carbon steel weld and basic material on the smallest possible surface. Figure 2.14 illustrates the technological steps of making butt welds. As shown in the figure, the welding of the two sides must be carried out separately, using carbon steel electrodes on the carbon steel side and electrodes of higher alloying substance content than that of the coating layer on the coating side. First the carbon steel side and then the stainless steel side are to be welded in a way that the stainless steel side should be welded in at least two steps to avoid “weld dilution”, as shown in the figure. Weld dilution is particularly dangerous in case of strong acidic effects, therefore the solutions shown in Fig. 2.15 can be used in such cases. In case of the solution shown in Fig. 2.15a, the diluted root side part of the butt weld on the coating layer side is covered with a lining piece of a material identical with the coating layer after edge finishing. In the event that the coating layer is thick

24 Fig. 2.14 Technological steps of making butt welds when welding metal cladding

Fig. 2.15 Corrosion protection by applying cover lining (a), and lining piece welded in (b)

2 Corrosion Loads, Corrosion Resistance, Corrosion-Proof Design

2.4 Corrosion Prevention, Corrosion Protection

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enough, corrosion protection for the surroundings of the welded joint can be ensured by welding in a lining piece after hollowing the root side (Fig. 2.15b). Figure 2.16 illustrates surface protection for a carbon steel manhole on a cover with metal cladding. The internal surfaces of the carbon steel nozzle welded into the cover, with split weldable extension and blind are respectively lined with stainless steel. Corrosion protection of the surroundings of small curvature radius of the sight glass is ensured by so-called surface layer welding made of stainless steel in several steps. Aeration of the confined spaces below lining plates is provided through the boreholes shown in the figure. Design of equipment coated by rubber and plastic From the construction point of view, equipment to be coated by rubber or plastic requires identical design. The only difference lies in the bond of the coating layer with the basic material [15, 16]. Usually equipment of welded structure is provided with a rubber or plastic coating layer. In the course of manufacturing such steel equipment, it should be ensured that the internal surface to be covered is adequately smooth, welded joints are compact and ground to be smooth on the coating side. Fig. 2.16 Surface protection of carbon steel manhole by corrosion resistant lining

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Welded joints should be made by using properly flowing electrodes, otherwise blisters can occur. The vessel should be of a divided design if possible. When determining wall thickness, loadability without damage of the glued connection and the coating layer should be taken into consideration. In order to reduce deformations and elongations, reinforcements are required to be applied to the equipment. Vessel covers convex towards the inside should be avoided. The vessel is required to be constructed in a way that no air or solvent inclusion should be left between the metal and the coating. Sharp edges and deaeratable fits should be avoided. Pipe bends may be up to 90°. The straight part of a pipe bend may be up to half of its diameter if the external diameter of the pipe is less than 400 mm. A pipe branch may be located from the end of the pipe section by up to 100 mm more than the external diameter of the pipe. No nozzle hung in the inside of the equipment may be applied. The nozzle should be as short as possible. If the external diameter of the nozzle is less than 80 mm, then the length of the nozzle is allowed to be up to twice the external diameter; in case of a pipe diameter exceeding 80 mm, up to three times thereof. It is expedient to design a blank flange rather than a nozzle of small diameter. Only external threaded nozzles may be applied. The thread must be completed before coating. Hollow structures and double walls (reinforcing plate—shell connections) must be provided with aeration boreholes. Rubber coatings Rubber boasts with one of the most versatile application areas. Natural and artificial rubber coatings resist most chemicals, acids and alkalis used in the industry. Hard rubber coatings are applied in case of vibration-proof operation at constant temperatures, subject to increased acid resistance requirements. Soft rubber coatings are applied for the surface protection of tanks exposed to dynamic loads, strong wear or major temperature fluctuations. As the heat expansion of hard rubber is much greater than that of steel, soft rubber coating should be placed, as applicable, between the steel and hard rubber coatings. Plastic coating Vulcanized coatings are not allowed in case of concrete tanks and wooden tubs. Softened PVC, high-pressure polyethylene and polypropylene can be properly applied for coatings [17]. Polypropylene coatings are applied in case of intense corrosion loads at high temperatures. Enamelled equipment. Enamelling has a history of several thousand years. At the beginning, precious metals were enamelled only for decoration purposes. In Hungary, enamelling developed in conjunction with the development of goldsmith’s art in the fifteenth and sixteenth centuries, as enamelling was used for decorating precious metals at that time. It was in the eighteenth century that attempts were made to coat iron vessels with glass for rust protection. Industrial enamelling was commenced in Germany around 1760. Iron plate enamelling was managed to be carried out in the early nineteenth century.

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The spread of chemical industry equipment with acid resistant enamel coating was due to the rapid development of the chemical industry. By the end of the nineteenth century, enamelled iron plates increasingly upstaged enamelled cast iron. Companies manufacturing chemical industry reactors and tanks with enamel coating (De Dietrich, Pfaudler) appeared in more and more countries worldwide. In Hungary, enamelled chemical industry equipment started to be produced in the 1940s (Lampart). Enamel is a not completely melted, inorganic material mainly consisting of oxides and substantially solidified in a vitreous manner, burnt on the surface of the metal to be coated in one or more layers after being ground with or without various additives. Glass composition needs to be selected in a manner that its heat expansion should be in line with the heat expansion of the metal to be coated; it should not soften at too high temperatures and should ensure bonding to the metal. Basic materials of enamels Networking elements used for enamel production include silicon, boron and phosphorus [18, 19]. Silicon dioxide is the most important basic material of glasses and enamels. The silicon dioxide content of enamels is highly varied: in general, they contain 40 to 50%, but certain acid resistant basic enamels and basic enamels for castings may contain even more. In the enamel industry, quartz sand—and quartz powder produced by grinding quartz sand—are used for applying silicon dioxide. Besides quartz, the most important networking element is boron. There is hardly any enamel that would not contain boron trioxide. Boron trioxide increases enamel heat expansion, reduces enamel surface stress, and facilitates the formation of the enamel-metal bond. Phosphorus pentaoxide enhances the dilution of iron oxide in basic enamels. Trisodium phosphate is a material with phosphorus pentaoxide content, containing 18.6% phosphorus pentaoxide, 24.5% sodium oxide and 56.9% water. The substances listed below, containing so-called transitional elements, can stabilize the vitreous state. Aluminium oxide (Al2 O3 )—reduces enamel heat expansion and elasticity, and increases enamel viscosity, heat resistance and wear resistance. Magnesium oxide—increases enamel firing temperature, surface stress and viscosity. Sodium oxide—decreases the enamel’s softening point, increases its heat expansion, enhances enamel shine, and reduces chemical resistance. Calcium oxide—increases chemical resistance and viscosity.

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Binding agents Appropriate bonding between the enamel and the iron plate is ensured by cobalt oxide and nickel oxide added to the enamel. Whitening agents Originally, enamel is colourless and transparent, so it behaves like glass in the optical sense. To make it opaque, various whitening or colouring agents (cryolite, calcium fluorite, lead oxide, antimony compounds, titanium dioxide, zirconium compounds) need to be added. Water Water plays a highly significant role in the enamel industry. Much water is used for soaking steel plates, for grinding enamel sludges and for enamelling as well. Water used in the course of enamelling is subject to strict requirements (pH 7.6, German degree 5.80, etc.). Purpose of enamelling Enamelling has a dual purpose: on the one hand, to protect the coated metal surface, and on the other hand to make the coated surface aesthetic. The greatest advantage of enamelled chemical industry equipment is combining the favourable mechanical properties of steel with the excellent chemical properties of resistance glasses, thus vessels of appropriate strength and chemical resistance can be produced. Vitreous enamel forms a very hard and smooth surface, enabling to perfectly clean the equipment. Practically, a vitreous enamel coating does not contaminate the medium stored. Application areas of enamelled chemical industry equipment Equipment and devices provided with a vitreous enamel coating are suitable for transporting fluids and gases, and for carrying out a number of chemical industry operations including evaporation, crystallization, distillation, extraction, halogenation, sulphonation, oxidation, etc. Vitreous enamel production Vitreous enamel production [20–22] consists of two fundamental series of operations: enamel frit production plus enamel sludge and powder enamel production. Enamel frits can be produced by melting a homogenized basic material mix in a way that basic materials are melted together in a rotating furnace at a temperature around 1500 °C, then the melted enamel is drained from the furnace to make it easy to crush and grind when solidified. This can be achieved by suddenly contacting the high temperature melt with cold water, by precipitating (fritting) it into a tub filled with water. The resulting enamel frit can only be used after grinding and crushing. Enamel sludge of proper consistence can be made from the frit, mill additive and water in a ball mill, which can be applied to a properly prepared metal surface

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by spraying or pouring. A clean to metal surface is required for adequate bonding between the enamel and the metal in the course of firing. Two layers of basic enamel (0.5 mm thick) and further four layers (1.4 mm thick) are applied to the properly prepared metal surface. Each layer is dried, then the basic enamel is fired at 920 °C, and the chemical resistant enamel at 820 °C. The basic enamel ensures bonding between the metal and the enamel. The chemical resistant enamel enables resistance to a variety of mediums. Enamelling design criteria Due to a risk of enamel pitting, the smallest rounding radius on the surface to be enamelled can be rmin = 10–12 mm. It is important to keep wall thickness at a nearly constant value to avoid the development of lines of force and haircrack fractures by more even heating up and cooling down. Enamelability criteria. Enamelable plates must be suitable for cold working and welding; in addition, they must be fault-free, with no pores, cracks, or scale on them. They must be soakable. Non-desoxidized cold-rolled plates are most suitable for enamelling. Maximum accompanying element quantities of enamelable plates [19, 21]: C < 0.06% Mn < 0.4% P < 0.05% S < 0.03% Si < 0.06% Cu < 0.3% Cr < 0.04% Mo < 0.1% N < 0.005% Technological steps of enamelling Preparatory operations include degreasing and soaking the surface to be enamelled. This is followed by enamelling, which comprises several operations different from each other (basic enamelling, cover enamelling, drying and firing operations). Steel plates can be enamelled by dipping, draining or blasting (spraying). The next step is to dry the enamel layer applied in a wet condition. Sludges used for enamelling contain 30–50% water, therefore moisture needs to be removed from the enamel layer by drying before firing. The drying temperature is 150–200 °C, the time is 2–3 min. Plate enamel firing is a highly important operation of enamelling. In the course of firing, a bond develops between the metal and the enamel; the enamel layers melt

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together and the enamel becomes shiny when exposed to heat. Firing is carried out at 850–920 °C. Chemical and physical properties of enamel coatings Acid resistance Vitreous enamel resists to all organic and inorganic oxidizing and reductive acids up to a relatively high temperature (200 °C), except to folic acid and its salts which damage the silicon frame. At higher degrees of concentration, phosphoric acid also attacks the enamel. In case of concentrated phosphoric acid (70% weight), the maximum temperature can be 120 °C [23]. Alkali resistance. Allowable operating temperatures are lower than in the case of acids. Enamelled coating is resistant to alkali solutions up to 60–70 °C. Allowable temperatures also depend on the pH value of the solution, but may not exceed 60 to 150 °C. Temperature is an important factor because the speed of corrosion doubles at every 10 °C. Mechanical properties Similarly to glass, enamel coating breaks and cracks, so small pieces can bounce off, leading to equipment destruction. Therefore due care needs to be taken in the event of transport, cleaning and maintenance. The hardness of enamels ensures proper wear resistance. Heat shock resistance Enamelled vessels are frequently subjected to sudden temperature changes. It is important for the enamel as well as the enamelled equipment to resist heat shocks without damage. Heat shock resistance not only depends on the enamel but on plate properties as well. Thicker plates can dissipate heat better, so the enamel is not exposed to such a strong heat shock as in the case of thinner plates. It is the other way around in case of enamels: the thinner is the layer applied, the more resistant it is. As regards enamel properties, enamel heat shock resistance is improved by increased tensile strength and thermal conductivity, while it is decreased by increased heat expansion coefficient, elasticity modulus, specific weight and specific heat figures. Sudden major temperature changes can lead to enamel rupture and to the development of haircrack fractures. Requirements for welded joints in enamelled equipment manufacturing Welds must be properly enamelable. The composition of filler metals used must be identical with that of the basic material. The heat expansion of welded joints is required to be identical with the heat expansion of the steels to be enamelled. There can be no fault whatsoever down to 4 mm depth below the weld surface to be enamelled.

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Enamelled equipment construction Only engineering steels with the chemical composition detailed earlier are to be used for enamelling. No sharp edges and steep breaks down are allowed to be applied. The smallest curvature radius can be 10–12 mm. Abrupt material thickness and weight changes should be avoided. Nozzles built into the equipment can be drawn out or welded in, but both structural designs are required to meet the minimum corner curvature condition. No welding works are allowed on the outside of the enamelled wall after enamelling. During structural design, care should be taken to the inclusion of appropriate cooling elements (collars and pads), to which other designed elements can be welded on after enamelling. Design work should take into consideration the heat treatment of workpieces without deformation at 800 to 900 °C by proper construction and the design of appropriate firing tools. In terms of enamelled equipment dimensioning, the allowable stress figure should be determined with due care [24]. It is a general rule that under operating loads, steel tensile stress may not exceed the residual compressive stress in the enamel when cooling down. The allowable maximum tensile stress in the enamel is 16–20 MPa. Figures 2.17, 2.18, 2.19 and 2.20 show special structural designs of enamelled process equipment (rimmed nozzles, enamelable flange connections, mixing shaft, connection of heat exchanger pipe and tube sheet). The figures clearly show radial forms for enamelability and unbroken constructions resulting in nearly constant wall thickness.

Fig. 2.17 Flared out nozzles on enamelled autoclave

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Fig. 2.18 Enamelled blind with sight glass

Fig. 2.19 Enamelled armed mixing shaft

References

33

Fig. 2.20 Pipe and tube sheet connection in enamelled heat exchanger

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

Dévay, J.: Fémek korróziója és korrózióvédelme. Budapest, M˝uszaki Könyvkiadó (1979) Erdey-Grúz, T.: Elektródfolyamatok kinetikája. Budapest, Akadémiai Kiadó (1969) Schierhold, P.: Nichtrostende Stähle. Verlag Stahleisen M.B.H, Düsseldorf (1977) Corrosion, Causes and Prevention. New York, McGraw Hill (1951) Fontana, M.G., Green, N.D.: Corrosion Engineering. New York, McGraw Hill. (1967) Kaeschle, H.: Die Korrosion der Metalle. Springer Verlag, Berlin (1979) Donndorf, R.: Werkstoffeinsatz und Korrosionsschutz in der chemischen Industrie. VEB D. Verlag für Grundstoffing, Leipzig (1982) Bünger, J.: Die Korrosion durch Salpetersäure, Werkstoffe u. Korrosion 9, (1938) Bürger, J.: Die korrosion durch Schwefelsäure Wekstoffeu. Korrosion 7, (1956) Engel, L.: Korrosion in Mineralölraffinerien Erdöl u. Kohle 27, H6 (1974) Szántay, B.: Vegyipari készülékek szerkesztése. Budapest, Tankönyvkiadó (1963) Richter, U.: Verfahrenstechnik 3.4. (1963) Titze, H.: Vegyipari készülékek szerkezeti elemei. Budapest, M˝uszaki Könyvkiadó (1966) Dilthey, U., Wanke, R.: Chemie-Ing-Tech. 46.11. (1974) Domininghaus, H.: Plastverarbeiter 30. Nr. 8. (1979) Kovács, L.: M˝uanyag zsebkönyv, 4. kiadás. Budapest, M˝uszaki Könyvkiadó (1979) Erhard, G., Strickle, E.: Maschinenelemente aus thermoplastischen. Kunststoffen , Düsseldorf, VDI-Verlag (1978) Eitel: The Physical Chemistry of Silicates. London, McGrawe Hill Bokk (1967) Wolf, M.B.: Üvegipari táblázatok és számítások. M˝uszaki Könyvkiadó, Budapest (1956) Albert, P.P.: T˝uzzománcozás. Budapest, M˝uszaki Könyvkiadó (1976) Korányi, Gy., Knapp O.: Üvegipari kézikönyv. M˝uszaki Könyvkiadó, Budapest (1964)

34

2 Corrosion Loads, Corrosion Resistance, Corrosion-Proof Design

22. L˝ocsei, B: Vegyipari korrózióálló szerkezeti anyagként alkalmazható üvegek és vitrokerámiák. Budapest (1966) 23. Veres, Gy.: Saválló zománcok. Tankönyvkiadó, Budapest (1966) 24. Reuss, P.: Zománcozott nyomástartó edények szilárdsági méretezése. Kandidátusi értekezés, Budapest (1979)

Chapter 3

Dimensioning of Process Equipment and Storage Tanks. Dimensioning by Taking Membrane Stresses into Account

Abstract Determination of allowable stress. Membrane stress state of equipment and storage tanks with axially symmetrical geometry. Dimensioning by taking vapour pressure into account. Dimensioning the cylindrical, spherical and conical part of the vessel. Dimensioning by taking hydrostatic pressure, mass forces, environmental impacts (snow, wind), and centrifugal force fields into account. Dimensioning cylindrical fluid tanks with a conical pressure vessel end. Dimensioning the cylindrical part of fluid tanks with large diameter. Dimensioning cylindrical centrifuges. Dimensioning tower structures exposed to environmental impacts. Dimensioning outdoor spherical tanks with large diameter. Keywords Membrane edge forces · Membrane stresses · Fluid tanks · Cylindrical centrifuges · Tower structure · Spherical tanks

3.1 Determination of Allowable Stress In addition to heat and corrosion resistance, requirements for process equipment and storage tanks include appropriate strength and proper stiffness. Appropriate strength can be ensured where stresses by load in different equipment cross-sections stay below the limit values allowed. The stiffness requirement (restriction of displacement and elongation by load) can be ensured by the appropriate bending stiffness (I · E) and torsional stiffness (I p · G) of load bearing cross-sections. Strength requirements can be met by strength dimensioning based on membrane stresses (determination of wall thickness (s)), and by the ensuing strength analysis which takes local bending stresses into account. Dimensioning (determination of wall thickness) is usually performed by taking into consideration membrane forces to be calculated by presuming a membrane stress state. The determination of membrane forces equals to solving a statically definite problem. Simple correlations resulting from equilibrium criteria can be directly used to calculate the required wall thicknesses. A pure membrane stress state prevails only on the cylindrical, conical, and spherical parts of equipment and storage tanks free from interfering effects, so the calculated wall thicknesses are valid only for these © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Nagy, Fundamentals of Tank and Process Equipment Design, Foundations of Engineering Mechanics, https://doi.org/10.1007/978-3-031-31226-7_3

35

36

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

parts. The surroundings of nozzles, supports and flanges as well as the curvature transitions of pressure vessel ends require additional strength analysis as a result of the local bending stresses generated there. Membrane stresses are constant along the wall thickness. By limiting membrane stresses as primarily dangerous stresses it can be ensured that the structure will not be destroyed “catastrophically” (cylindrical, conical and spherical parts will not burst) as a result of one-time load. In dimensioning, allowed stress depends on the type of the stress–strain characteristic curve (σ − ∈) of the engineering material (Figs. 3.1 and 3.2) as well. In the case of carbon steels and slightly alloyed steels (Eq. 3.1), and austenite stainless steels (Eq. 3.2), the allowable stress is [1] 

ReT or R p0.2T RmT σ1/100,000T σ B/1,000,00T ; ; ; n1 n2 n3 n4   R p1.0T RmT σ1/100,000T σ B/100,000T = min ; ; ; n1 n2 n3 n4



σallowed = min

(3.1)

σallowed

(3.2)

The numerical values of the safety factors (n i ) in correlations 3.1 and 3.2 can be derived from Table 3.1 depending on load state. In respect of the equipment to be dimensioned, the weakening impact of welded joints on the basic material is taken into consideration by the so-called weld strength factor (v ≤ 1.0), the value of which depends on: weld shape, type, size, and location; welding method; prescribed weld tests, welder’s qualifications, etc. Fig. 3.1 Stress–strain characteristic curve of carbon steels and slightly alloyed steels

3.2 Membrane Stress State of Equipment and Storage Tanks with Axially …

37

Fig. 3.2 Stress–strain characteristic curve of austenite stainless steels

Table 3.1 Numerical values of the safety factors Load

Safety factors n1

n2

n3

n4

Operational state

1.5

2.4–2.6

1.0

1.5

Testing and installation state

1.05

Allowed membrane stress for the structure: , σallowed = σallowed · v

(3.3)

3.2 Membrane Stress State of Equipment and Storage Tanks with Axially Symmetrical Geometry In order to determine the membrane stress state of axially symmetrical equipment and storage tanks, models must be created for structure, load, material and finally for calculations. In the present case: the structural model is → an axially symmetrical thin shell, the load model is → a surface distributed load constant in time and constant or changing in space, the material model is →homogeneous isotropic and linearly elastic (Hooke’s law), the calculation model is → technical shell theory.

38

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

3.2.1 Structural Model The structural model of the process equipment and storage tanks examined (axisymmetrical shell) is shown in Fig. 3.3. Figure 3.4 shows the middle surface of the axi-symmetrical shell bisecting wall thickness by indicating meridian (ϑ) and circumferential (ϕ) angle coordinates and the so-called meridian curve produced as the plane section of the middle section including a rotation axis. In Fig. 3.4, curvature conditions at a discretionary point “P” of the meridian curve are characterized by the radius of circle of curvature r1 (ϑ) and its part up to the rotation axis r2 (ϑ). The figure shows that the middle surface of the axi-symmetrical shell bisecting wall thickness can be generated by revolving the meridian curve around the rotation axis. Correlations of the P point surroundings of the meridian Fig. 3.3 Axi-symmetrical shell as a structural model

Fig. 3.4 Middle surface and meridian curve of axi-symmetrical shell

3.2 Membrane Stress State of Equipment and Storage Tanks with Axially …

39

Fig. 3.5 Environment of the meridian curve at Point P

curve (see Fig. 3.5) are as follows: r2 · sin(ϑ) = r

(3.4a)

r1 · cos(ϑ) · dϑ = dr

(3.4b)

3.2.2 Membrane Edge Forces, Membrane Elongations Figure 3.6 shows the shell element characterized by unit middle surface arc lengths taken out of the surroundings of point “P” of the axi-symmetrical shell as per Fig. 3.4. The figure shows membrane stresses in tangential direction (σ1 ) and in circumferential direction (σ2 ), as well as shear membrane stresses (τ12 = τ21 ) to ensure the static equilibrium of the shell element, produced in connecting cross-sections as a result of surface load (X 1 , X 2 , X 3 ). The resultants of membrane stresses generated in the cross-sections pertaining to unit middle surface arc lengths (see Eqs. 3.5a–3.5c) + 2 s

N1 =

σ1 · (r2 + x3 ) · − 2s

1 · d x3 r2

(3.5a)

40

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

+ 2 s

N2 =

σ2 · (r1 + x3 ) · − 2s

+ 2

+ 2

s

N12 = N21 = − 2s

1 · d x3 r1

(3.5b)

s

1 τ12 · (r2 + x3 ) · · d x3 = r2

τ21 · (r1 + x3 ) · − 2s

1 · d x3 r1

(3.5c)

are termed as tangential N1 , circumferential N2 , and shearing N12 = N21 = N edge forces [2–4]. In case of rs2 < 15 , shells are thin, where 1 + xr23 ≈ 1; 1 + xr13 ≈ 1, so the following will result from Eqs. (3.5a–c): N1 = σ1 · s

(3.6a)

N2 = σ2 · s

(3.6b)

N12 = N21 = N = τ12 · s = τ21 · s

(3.6c)

Fig. 3.6 Interpretation of membrane stresses and edge forces in the axi-symmetrical shell

3.2 Membrane Stress State of Equipment and Storage Tanks with Axially …

41

In the knowledge of the edge forces above, membrane stresses can be calculated by way of division by wall thickness as a result of unit middle surface arc lengths. By introducing edge forces, dimensioning is simplified to the strength analysis of the axi-symmetrical shell (middle surface) in a biaxial stress state. By applying Hooke’s law to elongations, the tangential (ε1 ) and circumferential (ε2 ) elongation and angle distortion of the middle surface [5] will be ε1 =

1 · (N1 − ν · N2 ) E ·s

(3.7a)

ε2 =

1 · (N2 − ν · N1 ) E ·s

(3.7b)

γ12 = γ21 =

1 1 · N12 = · N21 G·s G·s

(3.7c)

In the equations above, E, G, ν indicate the Young modulus, the modulus of torsional shear, and the Poisson ratio of the engineering material, respectively.

3.2.3 Membrane Stress State Equilibrium Equations Figure 3.7 illustrates the equilibrium of an elemental shell piece cut from the middle surface by the two meridian sections at a central angle dϕ from each other and by the section planes perpendicular to the rotation axis. The equations expressing the equilibrium of the edge forces generated in connecting cross-sections as a result of distributed loads (X 1 , X 2 , X 3 ) in the tangential direction x1 , in the circumferential direction x2 , and in the normal direction x3 , respectively—and as a result of removed parts—all acting on the shell piece (removed from the middle surface) with arc lengths r · dϕ and r1 · dϑ as shown in the figure are as follows [6–8]: in the tangential direction x1 , 1 ∂(N1 · r ) ∂(N21 ) + − N2 · cos(ϑ) = −X 1 · r · r1 ∂(ϑ) ∂(ϕ)

(3.8a)

in the circumferential direction x2 , 1 ∂(N12 · r ) ∂(N2 ) + + N21 · cos(ϑ) = −X 2 · r · r1 ∂(ϑ) ∂(ϕ) in the normal direction x3 ,

(3.8b)

42

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

Fig. 3.7 Equilibrium force system impacting shell element cut out of middle surface

N1 N2 + = −X 3 r1 r2

(3.8c)

Pursuant to the duality theorem, N12 = N21 = N .

3.3 Dimensioning by Taking Vapour Pressure into Account In the vast majority of cases, effective load is caused by vapour overpressure in the equipment, which can be taken into consideration for calculations in the form of distributed load along the entire internal surface in a perpendicular direction (x3 ) to a surface of constant value: X 1 = X 2 = 0; X 3 = − p. As both the load and the shell geometry are axially symmetrical, Eqs. (3.8a–c) can be written in the following ≡ 0: simple form by assuming N12 = N21 = 0 and ∂(...) ∂(ϕ) 1 d(N1 · r ) − N2 · cos(ϑ) = 0 · r1 d(ϑ)

(3.9a)

3.3 Dimensioning by Taking Vapour Pressure into Account

N1 N2 + =p r1 r2

43

(3.9b)

Using Eqs. (3.9a–b), the edge forces N1 and N2 sought for can be expressed in function of load ( p) and shell curvature (r1 ; r2 ) conditions as follows. From Eq. 3.9b, N2 = p · r 2 − N1 ·

r2 r1

(3.10a)

which, being substituted into correlation (3.9a) and following completion of the designated operations and reduction, will result in d(N1 · r · sin(ϑ)) = p · r1 · r2 · sin(ϑ) · cos(ϑ) d(ϑ)

(3.10b)

Using correlations (3.4a–b) the solution of (3.10b) is p · N1 = r · sin(ϑ)

r r · d(r )

(3.11)

0

out of which the edge forces sought for can be written in the forms p · r2 2   r2 p · r2 · 2− N2 = 2 r1 N1 =

(3.12a) (3.12b)

respectively. In the knowledge of edge forces, the membrane stresses and membrane elongations in the interference-free parts of the equipment shell will be [5, 9], p · r2 2·s   r2 p · r2 · 2− σ2 = 2·s r1    r2 p · r2 · 1−ν· 2− ε1 = 2· E ·s r1   r2 p · r2 · 2− −ν ε2 = 2· E ·s r1 σ1 =

(3.13a) (3.13b) (3.13c) (3.13d)

44

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

According to Eqs. (3.13a–d), the wall thickness (s) required to safely bear load (p) is determined by vapour pressure and the curvature conditions of the meridian curve.

3.3.1 Dimensioning the Cylindrical Part of the Vessel The structural model of plate welded vessels is a cylindrical shell outlined in Fig. 3.8, the meridian curve of which is a straight line parallel with the rotation axis, so it follows that r1 = ∞; r2 = R. The membrane forces produced: p · r2 p·R = 2 2   r2 p · r2 = p·R · 2− N2 = 2 r1 N1 =

(3.14a) (3.14b)

from which the membrane stresses and membrane elongations in the tangential direction and in the circumferential direction will be: σ1 =

Fig. 3.8 Structural model of cylindrical shells

p·R 2·s

(3.15a)

3.3 Dimensioning by Taking Vapour Pressure into Account

p·R s      p·R r2 1 p · r2 = · 1−ν· 2− · −ν ε1 = 2· E ·s r1 E ·s 2   wH p·R p · r2 r2 ν

ε2 = = · 2− −ν = · 1− R 2· E ·s r1 E ·s 2 σ2 =

45

(3.15b) (3.15c) (3.15d)

Using elongation ε2 in the circumferential direction, the radial displacement of the cylindrical shell caused by internal overpressure will be w H , wH =

ν

p · R2 p · R2 · 1− = · 0.85 E ·s 2 E ·s

(3.16)

where the Poisson ratio is ν = 0.3 . It can be established on the basis of Eqs. (3.15a–b) that the cylindrical shell bears gas pressure load along its entire surface in the form of constant membrane stresses σ1 in the tangential direction and σ2 in the circumferential direction, respectively (equal strength form). Out of these stresses, the one in the circumferential direction is double the one in the tangential direction. This finding coincides with the experience that the destruction of cylindrical shells occurs in the form of burst in the tangential direction, caused by membrane stress in the circumferential direction. Determining the required wall thickness Based on Mohr’s stress circles as shown in Fig. 3.9, the main stresses will be σ I = σ2

(3.17a)

σ I I = σ1

(3.17b)

σI I I ∼ =0

(3.17c)

As in an axially symmetrical case, the main geometrical and load directions (tangential and circumferential) coincide with the main stress directions, so it is expedient to perform dimensioning by applying Tresca Mohr’s equivalent stress based on the principle of the highest shearing stress. σr ed,Mohr = σ I − σ I I I = σ2 =

p·R ≤ σallowed s

(3.18)

from which the required wall thickness will be s≥

p·R σallowed

(3.19)

46

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

Fig. 3.9 Mohr’s stress circles of cylindrical shell under internal pressure

3.3.2 Dimensioning Spherical Tanks and Hemispherical Pressure Vessel Ends The structural model of spherical tanks and hemispherical pressure vessel ends is shown in Fig. 3.10. The meridian curve is a rotation axis centered semicircle, from which it follows that r1 = r2 = R. Membrane forces generated by internal pressure: p · r2 p·R = 2 2   p·R r2 p · r2 = · 2− N2 = 2 r1 2 N1 =

Fig. 3.10 Structural model of spherical tanks and the spherical parts of pressure vessel ends

(3.20a) (3.20b)

3.3 Dimensioning by Taking Vapour Pressure into Account

47

Membrane stresses and membrane elongations in the tangential and circumferential directions, respectively: σ1 = σ2 =

p·R 2·s

  1−ν p·R wG = · ε1 = ε2 = R E ·s 2

(3.20c) (3.20d)

In the knowledge of elongations, the radial displacement of the spherical shell will be   1−ν p · R2 p · R2 · · 0.35 (3.21) wG = = E ·s 2 E ·s Based on Eqs. (3.20a–d), it can be established that a spherical shell bears gas pressure type loads in the form of constant and equal membrane stresses σ1 = σ2 (equal strength form). Contrary to cylindrical shells, the destruction of spherical shells cannot be linked to a favoured direction, so at the limit of load bearing capacity, shell burst can start in an arbitrary direction from a material structure defect location in case of constant wall thickness. Determining wall thickness required for strength Based on Mohr’s stress circle shown in Fig. 3.11. The main stresses will be σ I = σ I I = σ1 = σ2

(3.22a)

σI I I ∼ =0

(3.22b)

Basic equation of dimensioning: Fig. 3.11 Mohr’s stress circle of spherical shell under internal pressure

48

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

p·R ≤ σallowed 2·s

σr ed,Mohr = σ I − σ I I I = σ2 =

(3.23)

from which the required wall thickness will be s≥

p·R 2 · σallowed

(3.24)

In the knowledge of the results, it can be established that in case of identical load (p) and geometry (R), the cylindrical part of the vessel requires double the wall thickness of the hemispheric pressure vessel end. In case of identical wall thickness, 2 ·0.35 is much smaller than the radial displacement of the hemispheric end wG = p·R E·s 2

the expansion of the cylindrical shell w H = p·R · 0.85, meaning that hemispheric E·s ends restrain the expansion of the shell, making it stiffer. This also means that the destruction of a cylindrical vessel with hemispheric ends occurs far from the vessel ends, in the form of burst in the tangential direction starting from a material or manufacturing defect location.

3.3.3 Dimensioning Conical Pressure Vessel Ends The structural model of pre-designed, rolled and welded or forged conical pressure vessel ends is a conical shell shown in Fig. 3.12, the meridian curve of which is a R 2 ·tg(α) = hcos(α) . straight line at an angle α with the rotation axis r1 = ∞; r2 = cos(α) Fig. 3.12 Structural model of conical pressure vessel ends

3.3 Dimensioning by Taking Vapour Pressure into Account

49

Membrane forces resulting from internal pressure: p·R p · h 2 · tg(α) p · r2 = = 2 2 · cos(α) 2 · cos(α)   p·R p · h 2 · tg(α) p · r2 r2 = = N2 = · 2− 2 r1 cos(α) cos(α) N1 =

(3.25a) (3.25b)

Membrane stresses: σ1 =

p · h 2 · tg(α) p·R = 2 · cos(α) · s 2 · cos(α) · s

(3.25c)

p · h 2 · tg(α) p·R = cos(α) · s cos(α) · s

(3.25d)

σ2 =

Based on Eqs. (3.25a–d), it can be established that the membrane stresses generated in the conical pressure vessel end in the tangential direction (σ1 ) and in the circumferential (σ2 ) direction increase linearly starting from the apex of the cone. The dangerous point is to be found in the farthest cross-section from the apex (h 2max ; Rmax ). Similarly to a cylindrical shell, the membrane stress in the circumferential direction is double the one in the tangential direction, so the destruction of the conical shell occurs as a burst in the tangential direction starting from a material or geometric defect in the farthest cross-section from the apex. Determining wall thickness required for strength Main stresses: σ I = σ2

(3.26a)

σ I I = σ1

(3.26b)

σI I I ∼ = 0.

(3.26c)

Basic equation of dimensioning: σr ed,Mohr,max = σ2max =

p · h 2max · tg(α) p · Rmax = ≤ σallowed cos(α) · s cos(α) · s

(3.27)

from which the required wall thickness will be s=

p · h 2max · tg(α) p · Rmax = cos(α) · σallowed cos(α) · σallowed

(3.28)

50

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

3.4 Dimensioning by Taking Hydrostatic Pressure, Mass Forces, Environmental Impacts (Snow, Wind), and Centrifugal Force Fields into Account In addition to vapour or gas pressure loads the wall thickness required for structure strength can be considerably influenced in certain cases by the hydrostatic pressure of the fluid stored in the equipment, in case of outdoor structures by environmental impacts (wind and snow), by equipment tare mass, and in case of industrial centrifuges operated at high rotational speeds by the centrifugal force field, representing external mechanical loads. Consequently, designers must model the effects above, to be followed by determining additional membrane stresses in order to be able to specify the wall thickness required to safely bear the loads listed. Examining the loads above and determining the membrane stress state caused by them require more complex modelling and deeper analysis than those explained previously, the reasons for which include, on the one hand, that the distributed load on the structure is not constant along the surface (X 3 /= const.); on the other hand, that distributed load X 1 in the tangential direction also appears besides load in the normal direction; what is more, there is no axial symmetry under wind pressure. It follows from the above that correlations (3.13a–d) deduced from equilibrium Eqs. (3.8a–c) cannot be used in the course of stress state calculations. The following sections present mechanical models and calculation algorithms taking the loads above into consideration as required for the dimensioning of equipment and storage tanks appearing in practice.

3.4.1 Dimensioning Cylindrical Fluid Tanks with a Conical Pressure Vessel End Figure 3.13 shows an outline of the structure to be analysed, including the main geometric features and their signage. In practice, the support is located at the lowest cross-section of the cylindrical part of the vessel, from which it follows that the entire cylindrical part of the vessel above the support is unloaded in the tangential direction: N1ρt = 0, while the edge force generated in the circumferential direction will be as follows on the basis of equilibrium Eq. 3.8c: N2ρt = ρt · g · R · h 1

(3.29)

N2ρt max = ρt · g · R · H1

(3.30)

the highest value whereof is

3.4 Dimensioning by Taking Hydrostatic Pressure, Mass Forces, …

51

Fig. 3.13 Outline of cylindrical fluid tank with a conical pressure vessel end, including main geometric features and signage

Knowing the curvature conditions of the meridian curve of the conical pressure vessel end, the load on the cone in the circumferential direction will be as follows, based on equilibrium Eq. (3.8c): N2ρt =

ρt · g · tan α (H1 + H2 − h 2 ) · h 2 cos α

(3.31)

2 from the apex of the the maximum of which is produced at a height h 2 = H1 +H 2 cone if H2 > H1 , and its value can be determined from the equation below.

N2ρt max =

ρt · g · tan α (H1 + H2 )2 4 · cos α

(3.32)

If the height of the cone is lower than the height of the cylindrical part H2 < H1 the maximum load in the circumferential direction will be generated at the connection

52

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

of the cone and the cylinder in cross-section h 2 = H2 N2ρt max =

ρt · g · tan α H1 · H2 cos α

(3.33)

Instead of solving equilibrium Eq. (3.8a), the edge force on the cone produced in the tangential direction results from the analysis of the vertical equilibrium condition of the structure defined according to Fig. 3.14 as follows [10, 11]:  2 · r · π · N1ρt · cos α = Vcylinder + Vcone · ρt · g

(3.34)

from which N1ρt

  ρt · g · tan α 2 H1 + H2 − h 2 · h 2 . = 2 · cos α 3

(3.35)

In case of H2 > 3H1 , the maximum value of the edge force in the tangential direction will be generated on the cone at a height h 2 = 43 · (H1 + H2 ) measured from the apex of the cone, the numerical value whereof is provided by the correlation Fig. 3.14 Static equilibrium of conical pressure vessel end

3.4 Dimensioning by Taking Hydrostatic Pressure, Mass Forces, …

53

below: N1ρt max =

3 ρt · g · tan α · (H1 + H2 )2 . 16 cos α

(3.36a)

In case of H2 < 3H1 , the maximum edge force in the tangential direction will be produced at the connecting cross-section of the cone and the cylinder, the measure of which can be calculated from equation N1ρt max

  1 ρt · g · tan α H1 + H2 · H2 = 2 · cos α 3

(3.36b)

The most important statements to be made on the basis of the correlations written: • irrespective of the geometry, the dangerous cross-section is always located on the conical pressure vessel end, and effective load is always produced in the circumferential direction. If H2 < H1 , the dangerous cross-section will be at the juncture of the cone and the cylinder (h 2 = H2 ). Using the basic irregularity of dimensioning, σr ed,Mohr,max =

phid · R , ≤ σallowed s · cos α

(3.37)

Storage tank wall thickness required for strength: s≥

phid , σallowed

·R · cos α

(3.38)

where phid = ρt · g · H1 . If there is vapour pressure in operation besides hydrostatic load, the required wall thickness will be: s≥

( p0 + phid ) · R , σallowed · cos α

In case of H2 > H1 , the dangerous cross-section is located at a height h 2 = on the cone. The basic irregularity of dimensioning: σr ed,Mohr,max =

ρt · g · tan α , (H1 + H2 )2 ≤ σallowed 4 · s · cos α

(3.39) H1 +H2 2

(3.40)

from which s≥

ρt · g · tan α (H1 + H2 )2 , 4 · σallowed · cos α

(3.41)

54

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

Required wall thickness in case of simultaneous vapour pressure:   R (H1 + H2 )2 s ≥ ρt · g + p0 · , 4H2 σallowed · cos α

(3.42)

Furthermore, it can be established that by derogation from the case of pure vapour pressure, effective stresses in the circumferential and tangential directions are not generated in an identical cross-section, the quotient of which varies between 1.33 ≤ σ2ρt max < 2.0 in function of the ratio of cone height and cylindrical height filled with σ1ρt max

 2 . fluid H H1

3.4.2 Dimensioning the Cylindrical Part of Fluid Tanks with Large Diameter In practice, the volume of large-sized storage tanks is V = 20,000–80,000 m3 , the medium stored is crude oil or a processed oil derivative or petrol. Load bearing of the floating or fixed roof structure is usually ensured by a frame structure built of steel beams, while the thin plate shell welded to the frame only serves for space delimitation. The cylindrical part of the storage tank consists of pre-embossed plates welded to each other. The bottom plate end is made on the basis of an individual design, by welding segments together. For security reasons, the bottom plate end is double-walled, so if the space between the two bottom plates is vacuumed, potential leakage can be detected. The structure is built on a reinforced concrete foundation. The plate thickness of the cylindrical part is increased step by step in accordance with the hydrostatic load. Connection between the lowest cylindrical zone and the bottom plate is ensured by a double-fillet welded joint without a reinforcing ring. The material of the entire structure is ordinary carbon steel, in accordance with the medium stored. For the purpose of leakage protection, the cylindrical part in contact with the medium stored is surrounded by an external mantle with larger diameter for loss elimination, which at the same time relieves the internal shell from additional loads by wind pressure. Figure 3.15 shows a schematic outline of a storage tank corresponding to the description above, indicating the cross-section of the cylindrical shell with zones of varying internal wall thickness. This analysis aims to specify the membrane stress state of the cylindrical shell, and then to produce a shell design resulting in favourable material utilisation. Cylindrical shell load By neglecting roof mass load, the cylindrical shell is only loaded by the hydrostatic pressure of the medium stored; furthermore, the bottom plate rests on the concrete foundation, so the cylindrical shell is unloaded in the tangential direction. N1ρt = 0, load in the circumferential direction by using equilibrium Eq. (3.8c) will be

3.4 Dimensioning by Taking Hydrostatic Pressure, Mass Forces, …

55

Fig. 3.15 Geometry, load, and stress state of cylindrical shell with varying wall thickness

N2ρt = R · phid = ρt · g · R · h

(3.43)

Presuming constant wall thickness along the height: σ2ρt max =

ρt · g · R · H R · phid,max = s s

(3.44)

Required wall thickness: s≥

ρt · g · R · H , σallowed

(3.45)

According to Fig. 3.15, if the cylindrical shell is made of zones with varying wall thickness by presuming that membrane stress in the circumferential direction at i = the cross-sections of wall thickness changes reaches its allowable value, σ2ρ t max , σallowed , the correlation between wall thicknesses and zone height will be: Δh i =

, σallowed (si − si−1 ) ρt · g · R

(3.46)

The material utility of the cylindrical shell is characterized by the following integral of membrane stress in the circumferential direction along the height of the shell:

56

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

H σ2ρt (h)dh

(3.47)

0

which in this case is Δh i =

, σallowed ρt ·g·R (si

H σ2ρt (h)dh = 0

− si−1 )

N Σ i=1

,2 2 si2 − si−1 σallowed . 2 · ρt · g · R si

(3.48)

The so-called material utility index K σ results from the comparison of the integral above to the case of constant wall thickness (Δh 1 = H ): H 1.0 ≤ K σ =

0

σ2ρt (h)dh , σallowed ·H 2

< 2.0

(3.49)

which, in the present case, can be written in the simple form below: ΣN Kσ =

i=1

2 si2 −si−1 si

sN

(3.50)

Material quantity of the cylindrical shell by presuming constant wall thickness: M0 = ρsteel · 2R · π · H · s N

(3.51)

which will be M = ρsteel · 2R · π ·

N Σ

·h i · si .

(3.52)

i=1

in case of zones with varying wall thickness. By comparing the cylindrical shell mass in case of varying wall thickness to the case of constant wall thickness, the so-called material savings index can be defined: the smaller it is, the more favourable is the shell structure. ΣN ·h i · si ρsteel · 2R · π · i=1 M = ≤ 1.0 (3.53) KM = M0 ρsteel · 2R · π · H · s N In case of the wall structure with varying wall thickness, if the condition i , σ2ρ = σallowed t max

(3.54)

prevails, the material savings index can be written in the following simple form:

3.4 Dimensioning by Taking Hydrostatic Pressure, Mass Forces, …

ΣN KM =

i=1 (si

− si−1 )si

s N2

57

.

(3.55)

The so-called material utility index K σ and the so-called material savings index K M introduced as above properly characterize the appropriacy of the planned structure. Obviously, the previously outlined stress condition referring to each zone cannot be met as shell design is produced by using the entire width of table sheets. Apart from this, the indices introduced are suitable for qualifying the structure in each case. Table 3.2 shows the numerical values of zone height (Δh i ), wall thickness (si ), and utility indices (K σ , K M ) of a cylindrical shell having a volume of 20,000m3 (R = 18,000 mm, H = 21,000 mm, σallowed = 160 MPa, ρt = 860 mkg3 ), pertaining to three different plate allocations. The data in column (a) refer to an existing storage tank, while columns (b) and (c) include the geometric data and utility indices resulting from correlation (3.54). Figure 3.16 shows the stress distributions corresponding to each case. Based on calculation results, it can be stated that more favourable utility indices are yielded by reducing the wall thickness of the top zone and by refining plate allocation. Table 3.2 Numerical values of zone height (Δhi), wall thickness (si), and utility indices (Kσ, KM) of a cylindrical shell having a volume of 20,000 m3 a

b

c

K σ = 1.5028

K σ = 1.6426

K σ = 1.6843

K M = 0.6129

K M = 0.5625

K M = 0.55

8

7590

6

6321.662

5

5268.052

9

1490

7

1053.61

6

1053.61

11

1490

8

1053.61

7

1053.61

12

1490

9

1053.61

8

1053.61

13

1490

10

1053.61

9

1053.61

15

1490

11

1053.61

10

1053.61

16

1490

12

1053.61

11

1053.61

17

1490

13

1053.61

12

1053.61

19

1490

14

1053.61

13

1053.61

20

1490

15

1053.61

14

1053.61

16

1053.61

15

1053.61

17

1053.61

16

1053.61

18

1053.61

17

1053.61

19

1053.61

18

1053.61

20

1053.61

19

1053.61

20

1053.61

si [mm]

Δh i [mm]

si [mm]

Δh i [mm]

si [mm]

Δh i [mm]

58

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

Fig. 3.16 Stress distribution of the cylindrical shell of a 20,000 m3 tank in case of three different plate allocations

It is worth mentioning that a small degree of local bending load occurs at the juncture of zones with varying thickness as a result of differing plate thickness, which dies away within a relatively short distance, so it does not affect dimensioning. It is also worth mentioning that the circular plate laid on the concrete foundation is capable of a minimum radial deformation in its own plane, so in reality much lower membrane stress is produced in the circumferential direction than the one calculated for the connecting cross-section of the lowest zone and the bottom plate, which is favourable for the weld.

3.4.3 Dimensioning Cylindrical Centrifuges Centrifuges of high rpm are used for the separation of suspensions in a number of technologies within the chemical and pharmaceutical industries. The centrifugal force field in centrifuges can be several thousandfold of the gravity force field, resulting in considerable mechanical load in the wall of the centrifuge drum. Figure 3.17 shows the structural model used for centrifuge dimensioning and the main geometric features.

3.4 Dimensioning by Taking Hydrostatic Pressure, Mass Forces, …

59

Fig. 3.17 Structural and load models of a cylindrical centrifuge drum

Cylindrical shell load is basically composed of two effects, the so-called no-load state resulting from the rotation of the drum, and the pressure distribution from the centrifugal force field developing in the suspension. Load caused by no-load state As in the course of drum rotation the centrifugal force field is always perpendicular to the rotation axis, no load is produced in the tangential direction N10 = 0. The distributed load on the shell in the normal direction comes from the centrifugal force impact on a unit surface area thereof: X 3 = ρshell · s · rtk · ω2

(3.56)

60

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

From equilibrium Eq. (3.8c) in the normal direction, the edge force in the circumferential direction in no-load state will be: N20 = ρshell · s · rtk2 · ω2

(3.57)

from which the so-called no-load stress will be: σ0 =

N20 = ρshell · rtk2 · ω2 s

(3.58)

Charge caused load Pressure distribution within the charge, resulting from the centrifugal force field, can be calculated from the equilibrium d Fc = d F p of forces impacting the fluid charge of mass dm ρt ∼ = ρt · 2rt · π · L · drt , where d Fc = dm ρt · rt · ω2

(3.59a)

d F p = dp · A = dp · 2rt · π · L

(3.59b)

From the equilibrium condition above, the pressure gradient within the fluid will be dp = ρt · ω2 · rt drt

(3.60)

from which pressure distribution within the charge will be p(rt ) =

ρt · ω2  2 2 rt − rtb 2

(3.61)

Distributed load in the normal direction, on the internal surface of the cylindrical centrifuge drum: X 3 = p(rtk ) =

ρt · ω2  2 2 rtk − rtb 2

(3.62a)

which, according to equilibrium Eq. (3.8c), results in an edge force N2t = rtk · p(rtk ) = rtk ·

ρt · ω2  2 2 rtk − rtb 2

(3.62b)

in the circumferential direction. Instead of solving equilibrium Eq. (3.8a), by writing the equilibrium of load in axial direction as defined in Fig. 3.17,

3.4 Dimensioning by Taking Hydrostatic Pressure, Mass Forces, …

61

rtk 2rtk · π · N1t =

2rt · π · p(rt )drt

(3.63)

rtb

after the designated operations are performed, the following correlation is yielded for the edge force in the tangential direction N1t , produced in the centrifuge wall:    2 2 ρt · ω2 · rtk3 rtb . N1t = 1− 8 rtk

(3.64)

By introducing the density relation characteristic of centrifuge operation, defined ρt , as well as a so-called charge coefficient defined by by the correlation λ = ρshell

2 the correlation Ψ = 1 − rrtbtk , the resultant stresses generated in the cylindrical centrifuge shell [11, 13] will be as follows in the tangential direction, σ1e = σ1t =

λ · Ψ 2 rtk N1t = σ0 s 8 s

(3.65a)

in the circumferential direction, σ2e = σ0 + σ2t =

  N20 + N2t λ · Ψ rtk = σ0 1 + s 2 s

(3.65b)

in normal direction, σ3e = − p(rtk ) = −σ0

λ · Ψ 2

(3.65c)

It can be established that the stresses calculated by correlations (3.65a-c) are main stresses at the same time, namely σ I = σ2e σ I I = σ1e σ I I I = σ3e . Using the basic irregularity of dimensioning.  , σr ed,Mohr = σ I − σ I I I ≤ σallowed , the required wall thickness (s) will be. s≥

σ0 · λ · Ψ · rtk , 2σallowed − σ0 (2 + λ · Ψ )

(3.66)

Figure 3.18 shows the required wall thickness as per (3.66) for the cylindrical part of a centrifuge of 110 mm internal radius in function of charge coefficient, while Fig. 3.19 also shows the required wall thickness in function of the internal radius in case of different charge coefficients.

62

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

Fig. 3.18 Required wall thickness of cylindrical centrifuge part in function of charge coefficient

Fig. 3.19 Required wall thickness in case of different charge coefficients, in function of characteristic centrifuge size

3.4.4 Dimensioning Tower Structures Exposed to Environmental Impacts Material transfer processes between mediums of varying composition and physical state are carried out in operational conditions, in vertical cylindrical columns. The material transfer process between mediums flowing opposite each other requires large specific contact surfaces, to be ensured by packing or discrete plate structures. Based on the operational calculations related to the transfer process, large shell heights are yielded as compared to equipment diameter, which results in the fact that

3.4 Dimensioning by Taking Hydrostatic Pressure, Mass Forces, …

63

the columns above are usually installed outdoors. In the design of columns installed outdoors, load investigations must take into account environmental impacts (snow, wind, and seismic as well) in addition to the load caused by mediums flowing opposite each other; besides the impact of steel mass can be considerable due to large shell height. The static dimensioning method to be presented takes the following loads into account: • vapour pressure above the fluid charge ( p0 ); • hydrostatic load ( phid ) pertaining to the flooding condition; • concentrated mass (m 0 ) (end, nozzles on end, other elements fixed to the end, snow, ice, etc. deposited on the end);  • mass of the a cylindrical shell as distributed load mH h ; • static pressure distribution ( ps (ϕ)) around the cylindrical shell, caused by flow resulting from wind impact [12]. Due to their hazards, it is expedient to examine loads not included in the list—such as the aperiodic exciting impact of wind causing dynamic extra loads, and its periodic exciting impact causing harmful oscillations near the own frequency, as well as extra loads caused by seismic impact—according to the standards currently in effect and to take their impact into account in strength calculations. Based on the above, Fig. 3.20 shows the design of the tower structure examined, including the most important geometric features and the loads taken into consideration. Examination of the listed load cases Membrane stresses in the tangential direction σ1 p0 and in the circumferential direction σ2 p0 , caused by vapour pressure above the fluid charge: σ1 p0 =

p0 · R 2s

(3.67a)

σ2 p0 =

p0 · R s

(3.67b)

As the skirt support is located at the bottom cross-section of the cylinder, no stress in the tangential direction is generated from charge mass; at the same time, in the circumferential direction, membrane stress proportionate to hydrostatic pressure is yielded along the height, to be calculated in the known manner. σ1ρt = 0 σ2ρt =

R · ρt · g · h R · phid = s s

(3.68a) (3.68b)

64

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

Fig. 3.20 Tower structure exposed to environmental impacts

Concentrated mass results in force m 0 · g, while the distributed mass of the cylindrical shell results in distributed forces mH · h · g = m · h · g in the tangential direction, respectively. From the static equilibrium condition in the tangential direction, and on the basis of equilibrium Eq. (3.8c): σ1m = −

(m 0 + m · h)g 2R · π · s

σ2m = 0

(3.69a) (3.69.b)

When examining wind load impacts, a mathematically accurate solution is yielded by equilibrium Eqs. (3.8a–c) written for the shell element r1 = ∞, r2 = R, ϑ = π2 , and in case of X 1 = X 2 = 0, X 3 = ps (ϕ) by the following differential Eqs. (3.70a–c) [11, 13, 14]:

3.4 Dimensioning by Taking Hydrostatic Pressure, Mass Forces, …

65

• equation in the tangential direction R

∂ N21 ∂ N1 + =0 ∂h ∂ϕ

(3.70a)

• equation in the circumferential direction R

∂ N12 ∂ N2 + =0 ∂h ∂ϕ

(3.70b)

N2 = R · ps (ϕ)

(3.70c)

• in the normal direction

furthermore, from the duality theorem, N12 = N21 = N ; Solutions of the differential equations above: N1 (h, ϕ) =

1 d 2 ps (ϕ) h 2 · R dϕ 2 2

N2 (ϕ) = R · ps (ϕ) N (h, ϕ) = −h

dps (ϕ) . dϕ

(3.71a) (3.71b) (3.71c)

By analysing the results above, it can be established that only the correlation referring to the edge force N2 (ϕ) in the circumferential direction can be used in practical calculations. In order to specify the edge force in the tangential direction N1 (h, ϕ) and the shearing edge force N (h, ϕ), the solution above uses the derivatives of the approximating function ps (ϕ) fit by the Fourier method for static pressure values measured along the column perimeter. And the solution method based on the derivatives of approximating functions may result in defects of even one order of magnitude as compared to reality. In order to examine loads caused by wind pressure, it is expedient to apply a solution method which eliminates the problem above, meaning that it possibly uses the load function ps (ϕ) in an integral form. Having regard to the geometric conditions of the column (H ≫ R), it is relatively easy to determine the load in the tangential direction if the column is taken as a fixed beam under distributed load of constant intensity [4, 11]. Load in the tangential direction is caused by the bending moment of the beam of circular ring cross-section. Obviously, the approach above cannot be used to examine shearing load, but it is dangerous for bending along tangential directions ϕ = 0 and ϕ = π because of the symmetry N (h, ϕ = 0; ϕ = π ) = 0. By applying the signage in Fig. 3.20, the correlation between the distributed load Vs of the fixed beam model and the pressure distribution function ps (ϕ) will be:

66

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

2π Vs =

ps (ϕ) · R · cos ϕdϕ,

(3.72)

0

bending moment distribution along the longitudinal axis: Ms (h) = Vs

h2 , 2

(3.73)

membrane stress in tangential direction caused by wind pressure in the congestion point (ϕ = 0), σ1s(h) = +

Ms (h) R Iy

(3.74a)

σ1s(h) = −

Ms (h) R Iy

(3.74b)

on the opposite side (ϕ = π ),

where the secondary moment of inertia of the circular ring cross-section is: Iy = R3 · π · s

(3.74c)

The cross-section dangerous for bending is the cross-section of support (fixing) (h = H ), where it is expedient to examine in detail the stress state of the congestion point and of the so-called windshielded point opposite to it. Stress state of the congestion point (ϕ = 0), Stresses in the tangential σ1s and circumferential σ2s directions caused by wind pressure, respectively:  2π σ1s(h=H,ϕ=0) = +

0

σ2s(h=H,ϕ=0) =

ps (ϕ) · cos ϕdϕ H 2 R·π ·s 2

(3.75a)

R · ps (ϕ = 0) s

(3.75b)

ps (ϕ) · cos ϕdϕ H 2 R·π ·s 2

(3.75c)

on the windshielded side (ϕ = π ):  2π σ1s(h=H,ϕ=π ) = −

0

σ2s(h=H,ϕ=π ) =

R · ps (ϕ = π ) . s

(3.75d)

3.4 Dimensioning by Taking Hydrostatic Pressure, Mass Forces, …

67

The following will be yielded by stating the resultant of the membrane stresses caused by the loads detailed above in the dangerous cross-section (h = H ), in the congestion point and in the windshielded point opposite to it. Resultant membrane stresses in the congestion point (ϕ = 0) in the tangential direction σ1e and in the circumferential direction σ2e : σ1e = σ1 p0 + σ1m + σ1s p0 · R (m 0 + m)g − + = 2s 2R · π · s σ2e = σ2 p0 + σ2ρt + σ2s =

 2π 0

ps (ϕ) · cos ϕdϕ H 2 R·π ·s 2

(3.76a)

R · ρt · g · H R · ps (ϕ = 0) p0 · R + + s s s

(3.76b)

where the sign of stresses is σ1e > 0 , σ2e > 0; consequently, the main stresses will be σ I = max(σ1e ; σ2e ) σ I I = min(σ1e ; σ2e ); σ I I I = 0, and the equivalent stress will be σr ed,Mohr,(h=H,ϕ=0) = max(σ1e ; σ2e ). At an opposite point (ϕ = π ), σ1e = σ1 p0 + σ1m + σ1s =

p0 · R (m 0 + m)g − − 2s 2R · π · s

σ2e = σ2 p0 + σ2ρt + σ2s =

 2π 0

ps (ϕ) · cos ϕdϕ H 2 R·π ·s 2

(3.76c)

R · ρt · g · H R · ps (ϕ = π ) p0 · R + + s s s

(3.76d)

Material transfer processes take place under nearly atmospheric  H conditions; on ≫ 1, so the sign the other hand, the height versus diameter ratio of the tower is 2R of stresses will be σ1e < 0, σ2e > 0, from which it follows that σ1 = σ2e , σ I I = 0, σ I I I = σ1e , σr ed,Mohr,(h=H,ϕ=π ) = σ2e + |σ1e | It follows from the above that the effective load to be taken into consideration in the course of dimensioning is generated most probably in the support cross-section, on the so-called windshielded side opposite to the congestion point. Following substitution into the basic irregularity of dimensioning and reductions, the formula below is yielded for column wall thickness required for strength: s≥

R · pr ed , σallowed

(3.77a)

where the resultant of each load case can be defined as the following so-called reduced vapour pressure:  pr ed =

p0 (m 0 + m)g + ρt · g · H + ps (ϕ = π ) + + 2 2R 2 · π

 2π 0

ps (ϕ) · cos ϕdϕ 2π



H R

2 

(3.77b)

68

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

By introducing the so-called reduced vapour pressure defined by correlation (3.77b), the dimensioning of columns exposed to environmental impacts can be traced back to the use of the so-called “boiler plant formula”.

3.4.5 Dimensioning Outdoor Spherical Tanks with Large Diameter Liquified gas is generally stored in outdoor spherical tanks with large diameter. Similarly to cylindrical tower structures, loads result from the pressure of the medium stored, environmental impacts (snow, wind), and the tare weight of the equipment. The following subsections discuss the ways of determining the loads caused by each load case and the membrane stresses generated by such loads. Wall thickness required for strength is calculated by jointly taking into consideration the membrane stresses produced in each case.

Membrane Stresses Caused by Fluid Charge Membrane stresses caused by hydrostatic loads can be determined from the examination of the vertical static equilibrium of the equipment, and on the basis of equilibrium Eq. (3.8c) in the normal direction written for the shell element as follows. Based on Fig. 3.21, load by hydrostatic pressure on an elemental surface piece d A: d F p = phid · d A

(3.78a)

 phid = ρt · g · R 1 − cos ϑ ∗

(3.78b)

dA ∼ = 2 · π · R 2 · sin ϑ ∗ · dϑ ∗

(3.78c)

where

Vertical static equilibrium of the spherical calotte part pertaining to angle range 0 ≤ ϑ ∗ ≤ ϑ: ϑ d F pver tic = 2 · R · π · N1ρt · sin2 ϑ

(3.79a)

0

d F pver tic = d F p · cos ϑ ∗

(3.79b)

3.4 Dimensioning by Taking Hydrostatic Pressure, Mass Forces, …

69

Fig. 3.21 Internal and external forces ensuring the static equilibrium of the spherical calotte parts above and below the support

Following reductions after

solving Eq.  (3.79a) above, the membrane stress in N the tangential direction [15] σ1ρt = s1ρt generated in the cross-sections above the support: ρt · g · R 2 (1 − cos ϑ)(2 cos ϑ + 1) , 0 ≤ ϑ ≤ ϑ0 (3.80a) 6·s 1 + cos ϑ 

N Membrane stress in the circumferential direction σ2ρt = s2ρt based on equilibrium Eq. (3.8c) in the normal direction: σ1ρt =

σ2ρt =

ρt · g · R 2 (1 − cos ϑ)(5 + 4 cos ϑ) , 0 ≤ ϑ ≤ ϑ0 6·s 1 + cos ϑ

(3.80b)

70

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

By writing the equations above for the spherical calotte part below the support, the membrane stresses sought for will be as follows in function of angle coordinate (ϑ) in the meridian direction: σ1ρt =

ρt · g · R 2 5(1 − cos ϑ) + 2cos 2 ϑ 6·s 1 − cos ϑ

(3.81a)

ϑ0 ≤ ϑ ≤ π σ2ρt =

ρt · g · R 2 4cos 2 ϑ − 7cosϑ + 1 6·s 1 − cos ϑ

(3.81b)

By introducing the load factors referring to hydrostatic state pˆ 1ρt (ϑ), pˆ 2ρt (ϑ) shown in Fig. 3.22, the correlations developed as above can be written in the following form: σ1ρt =

R phid,max · pˆ 1ρt (ϑ) 2s

(3.82a)

σ2ρt =

R phid,max · pˆ 2ρt (ϑ) 2s

(3.82b)

where: phid,max = ρt · g · 2R

(3.82c)

Load factors in case of 0 ≤ ϑ ≤ ϑ0

Fig. 3.22 Hydrostatic state load factors in function of angle coordinates in the meridian direction in case of support position π2 ≤ ϑ0 ≤ 2π 3

3.4 Dimensioning by Taking Hydrostatic Pressure, Mass Forces, …

71

pˆ 1ρt (ϑ) =

(1 − cos ϑ)(2 cos ϑ + 1) 6(1 + cos ϑ)

(3.82d)

pˆ 2ρt (ϑ) =

(1 − cos ϑ)(5 + 4 cos ϑ) 6(1 + cos ϑ)

(3.82e)

pˆ 1ρt (ϑ) =

5(1 − cos ϑ) + 2 cos2 ϑ 6(1 − cos ϑ)

(3.82f)

pˆ 2ρt (ϑ) =

4 cos2 ϑ − 7 cos ϑ + 1 6(1 − cos ϑ)

(3.82g)

in case of ϑ0 ≤ ϑ ≤ π

Membrane Stresses Caused by Snow Load Membrane  stresses by snow load, generated in cross-sections above the main circle ϑ = π2 can be calculated as below on the basis of Fig. 3.23. Based on the figure, distributed loads on the tank surface will be: X 1 = psnow · sin ϑ · cos ϑ

(3.83)

X 3 = − psnow · cos2 ϑ

(3.84)

where psnow = ρsnow · g · h snow is the specific snow load.  Membrane stress in the tangential direction σ1snow = N1snow resulting from the s vertical static equilibrium of the spherical calotte part pertaining to angle range 0 ≤ ϑ ∗ ≤ ϑ: σ1snow = −

R psnow · R = psnow · pˆ 1snow (ϑ) 2s 2s

(3.85a)

Membrane stress in the circumferential direction based on equilibrium Eq. (3.8c): σ2snow = −

R psnow · R cos 2ϑ = psnow · pˆ 2snow (ϑ) 2s 2s

where pˆ 1snow (ϑ) and pˆ 2snow (ϑ) are the load factors shown in Fig. 3.24.

(3.85b)

72

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

Fig. 3.23 Internal and external forces caused by snow load

Membrane Stresses Caused by Shell Mass Figure 3.25 shows the model used, together with the variables used and their signage. Based on the figure, distributed loads on the tank surface will be: X 1 = pm · sin ϑ

(3.86a)

X 3 = − pm · cos ϑ

(3.86b)

where pm = s · ρsteel · g is the mass force on a surface unit. Membrane stresses resulting from the specific mass of the spherical shell:

3.4 Dimensioning by Taking Hydrostatic Pressure, Mass Forces, …

Fig. 3.24 Snow mass load factors in function of angle coordinates in the meridian direction

Fig. 3.25 Internal and external loads caused by shell mass

73

74

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

Fig. 3.26 Shell mass load factors in function of angle coordinates in the meridian direction in case of support position π2 ≤ ϑ0 ≤ 2π 3

σ1m =

R pm · pˆ 1m (ϑ) 2s

(3.87a)

σ2m =

R pm · pˆ 2m (ϑ) 2s

(3.87b)

where the so-called load factors pˆ 1m (ϑ), pˆ 2m (ϑ) shown in Fig. 3.26 are as follows: in angle range 0 ≤ ϑ ≤ ϑ0 pˆ 1m (ϑ) = − pˆ 2m (ϑ) =

2 (1 + cos ϑ)

(3.87c)

2 − 2 cos ϑ 1 + cos ϑ

(3.87d)

2 (1 − cos ϑ)

(3.87e)

in angle range ϑ0 ≤ ϑ ≤ π pˆ 1m (ϑ) =  pˆ 2m (ϑ) = −

2 + 2 cos ϑ 1 − cos ϑ

 (3.87f)

3.4 Dimensioning by Taking Hydrostatic Pressure, Mass Forces, …

75

Membrane Stresses Caused by Wind Pressure Only membrane stresses caused by static pressure distribution will be determined in this case as well. It is expedient to examine dynamic impacts caused by wind [12] on the basis of dimensioning standards and standard-based design software programmes. As load is asymmetric, solution of the problem starts from equation system (3.88a– c) yielded on the basis of equilibrium conditions (3.8a-c) written for the shell element, by taking spherical geometry (r1 = r2 = R) into consideration [5, 14]. ∂(N1 · sin ϑ) ∂ N21 + − N2 · cos ϑ + X 1 · R · sin ϑ = 0 ∂ϑ ∂ϕ

(3.88a)

∂(N12 · sin ϑ) ∂ N2 + + N21 · cos ϑ + X 2 · R · sin ϑ = 0 ∂ϑ ∂ϕ

(3.88b)

N1 + N2 − X3 = 0 R

(3.88c)

By reason of the duality condition for shearing edge forces generated in planes perpendicular to each other, N12 = N21 = N . The following differential equations result from the equilibrium equations above for determining the edge force in the tangential direction N1 and shearing edge forces N12 = N21 = N . 1 ∂(N1 ) ∂ N + · + 2N1 · cot ϑ + R(X 1 − X 3 cot ϑ) = 0 ∂ϑ ∂ϕ sin ϑ   1 ∂ X3 1 ∂(N ) ∂ N1 − · + 2N · cot ϑ + R · + X2 = 0 ∂ϑ ∂ϕ sin ϑ ∂ϕ sin ϑ

(3.89a) (3.89b)

By setting up the spherical coordinate system according to wind direction, the direction of wind flow is characterized by the plane ϕ = 0, and the congestion point by the coordinates ϕ = 0, ϑ = π2 . In a general load case corresponding to the symmetry above (X 1 /= 0, X 2 /= 0, X 3 /= 0), distributed load on a surface unit and the edge forces generated in the structure can be stated in the form of the following function series: X1 =

n Σ

X 1i (ϑ) · cos iϕ

(3.90a)

X 2i (ϑ) · sin i ϕ

(3.90b)

i=1

X2 =

n Σ i=1

76

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

X3 =

n Σ

X 3i (ϑ) · cos iϕ

(3.90c)

N1i (ϑ) · cos i ϕ

(3.90d)

Ni (ϑ) · sin i ϕ

(3.90e)

N2i (ϑ) · cos i ϕ

(3.90f)

i=1

N1 =

n Σ i=1

N=

n Σ i=1

N2 =

n Σ i=1

By substituting member i of the Fourier series written for loads and edge forces into differential Eqs. (3.89a–b), the following equations are yielded for Fourier coefficients after executing the designated operations and reductions: i ∂ N1i + 2N1i · cot ϑ + Ni = R(X 3i · cot ϑ − X 1i ) ∂ϑ sin ϑ   i i ∂ Ni + 2Ni · cot ϑ + N1i = R X 3i − X 2i ∂ϑ sin ϑ sin ϑ

(3.91a) (3.91b)

By adding up Eqs. (3.91a–b), then subtracting them from each other, and by introducing new variables N1i + Ni = Ui , N1i − Ni = Vi , the following linear differential equations can be written for these new variables: dUi + Ui · P1i + Q 1i = 0 dϑ

(3.92a)

d Vi + Vi · P2i + Q 2i = 0 dϑ

(3.92b)

where P1i = 2 · cot ϑ +

i sin ϑ

(3.92c)

i sin ϑ   − (X 1i + X 2i ) R

P2i = 2 · cot ϑ −   Q 1i = − X 3i cot ϑ +   Q 2i = − X 3i cot ϑ −

i sin ϑ i sin ϑ



(3.92d) (3.92e)

 − (X 1i − X 2i ) R

(3.92f)

3.4 Dimensioning by Taking Hydrostatic Pressure, Mass Forces, …

77

General solutions of the differential equations above: Ui = e Vi = e

−

−





P1i dϑ

   C − Q 1i · e 

P2i dϑ

 B−

Q 2i · e



 P1i dϑ

dϑ

(3.93a) 

P2i dϑ

dϑ

(3.93b)

By assuming the simplest load function meeting the symmetry condition as per Fig. 3.27, the Fourier series X 1 = X 2 = 0, X 3 = − pst · sin ϑ · cos ϕ. Specially contain one member i = 1, X 31 = − pst · sin ϑ. In this case, P1i = P11 =

2 cos ϑ + 1 sin ϑ

(3.94a)

P2i = P21 =

2 cos ϑ − 1 sin ϑ

(3.94b)

Q 1i = Q 11 = pst · R(1 + cos ϑ)

(3.94c)

Q 2i = Q 21 = − pst · R(1 − cos ϑ)

(3.94d)

And the solutions of the differential equations will be U1 =

   cos 3 ϑ 1 + cos ϑ cos ϑ − C + R · p st sin 3 ϑ 3

Fig. 3.27 Static wind pressure used in calculations

(3.95a)

78

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

   cos3 ϑ 1 − cos ϑ B − R · pst cos ϑ − V1 = sin 3 ϑ 3

(3.95b)

out of which edge forces in the tangential direction N1 and shearing edge forces N12 = N21 = N are    C+B 1 cos4 ϑ C−B 2 cos cos ϕ ϑ − + cos ϑ + R · p st sin 3 ϑ 2 2 3 (3.96a)    C−B 1 cos3 ϑ C+B N= cos ϑ − + cos ϑ + R · p sin ϕ (3.96b) st sin 3 ϑ 2 2 3 N1 =

Edge forces in the circumferential direction can be expressed from the equilibrium equation in the normal direction. Examination of the solution functions yields C = − 23 pst · R, B = + 23 pst · R for integration constants. Finally, the edge forces sought for are described by the following correlations in function of the angle coordinates in the tangential direction (ϑ) and in the circumferential direction (ϕ) within the range above the support [4]. 0 ≤ ϑ ≤ ϑ0 R · pst cos ϑ(1 − cos ϑ)(2 + cos ϑ) cos ϕ N1 = − 3 sin ϑ(1 + cos ϑ) R · pst (1 − cos ϑ)(2 + cos ϑ) sin ϕ 3 sin ϑ(1 + cos ϑ)  R · pst (1 − cos ϑ) 2 cos2 ϑ + 4 cos ϑ + 3 cos ϕ N2 = − 3 sin ϑ(1 + cos ϑ) N =−

(3.97a) (3.97b)

(3.97c)

And in the range below the support: ϑ0 ≤ ϑ ≤ π R · pst cos ϑ(1 + cos ϑ)(2 − cos ϑ) cos ϕ N1 = 3 sin ϑ(1 − cos ϑ) R · pst (1 + cos ϑ)(2 − cos ϑ) sin ϕ 3 sin ϑ(1 − cos ϑ)  R · pst (1 + cos ϑ) 2 cos2 ϑ − 4 cos ϑ + 3 cos ϕ. N2 = − 3 sin ϑ(1 − cos ϑ) N =−

Effective membrane stresses from wind load in meridian plane ϕ = 0:

(3.97d) (3.97e)

(3.97f)

3.4 Dimensioning by Taking Hydrostatic Pressure, Mass Forces, …

79

σ1s =

R pst · pˆ 1s (ϑ) 2s

(3.98a)

σ2s =

R pst · pˆ 2s (ϑ) 2s

(3.98b)

τ12s = τ21s = 0

(3.98c)

where load factors pˆ 1s (ϑ) pˆ 2s (ϑ) as shown in Fig. 3.28 are as follows: in the range 0 ≤ ϑ ≤ ϑ0 2 cos ϑ(1 − cos ϑ)(2 + cos ϑ) 3 sin ϑ(1 + cos ϑ)  2 (1 − cos ϑ) 2 cos2 ϑ + 4 cos ϑ + 3 pˆ 2s (ϑ) = − 3 sin ϑ(1 + cos ϑ) pˆ 1s (ϑ) = −

(3.98d)

(3.98e)

in the range ϑ0 ≤ ϑ ≤ π 2 cos ϑ(1 + cos ϑ)(2 − cos ϑ) 3 sin ϑ(1 − cos ϑ)  2 (1 + cos ϑ) 2 cos2 ϑ − 4 cos ϑ + 3 pˆ 2s (ϑ) = − 3 sin ϑ(1 − cos ϑ) pˆ 1s (ϑ) =

(3.98f)

(3.98g)

Fig. 3.28 Load factors resulting from static wind effects in function of angle coordinates in the meridian direction in case of support position π2 ≤ ϑ0 ≤ 2π 3

80

3 Dimensioning of Process Equipment and Storage Tanks. Dimensioning …

Determining the Resultant Stress State; Dimensioning Spherical Tanks Exposed to Environmental Impacts In the course of dimensioning, the resultant stress state can be determined by summarizing the membrane stresses caused by the loads presented in the foregoing. It can be clearly established from the examination of the so-called load factors introduced in the solution that the wall thickness of the structure as required for strength is determined by the location of the support structure and its coordinate in the meridian direction ϑ0 . It can be further established that the smallest wall thickness results from support near the so-called main circle, to be characterized by angle coordi◦ ◦ nates 90 ≤ ϑ0 ≤ 100 By increasing the angle position of the support, in case of ◦ ϑ0 > 100 , increasing wall thickness figures are yielded. Based on Figs. 3.22, 3.26, and 3.28, it can be stated that it is expedient to place the support angle near the main . It can also be established circle, but in any case within the angle range π2 ≤ ϑ0 < 2π 3 that except for wind loads, membrane stresses do not change along the perimeter (axially symmetrical case). In case of wind loads, the effective load can be found in the meridian cross-section characterized by angle coordinate ϕ = 0, located in the direction of wind flow. Based on the above, the resultant stress state in the meridian cross-section ϕ = 0 will be as follows: in the tangential direction σ1e = σ1 p0 + σ1ρt + σ1snow + σ1m + σ1s

(3.99a)

by using load factors:   R p0 + phid,max · pˆ 1ρt (ϑ) + psnow · pˆ 1snow (ϑ) σ1e (ϑ) = + pm · pˆ 1m (ϑ) + pst · pˆ 1s (ϑ) 2s

(3.99b)

in the circumferential direction σ2e = σ2 p0 + σ2ρt + σsnow + σ2m + σ2s   R p0 + phid,max · pˆ 2ρt (ϑ) + psnow · pˆ 2snow (ϑ) σ2e (ϑ) = + pm · pˆ 2m (ϑ) + pst · pˆ 2s (ϑ) 2s

(3.99c)

(3.99d)

And the required wall thickness is yielded from the basic irregularity of dimensioning as σr ed,Mohr,max = s≥

R , · pr ed,max ≤ σallowed 2s

(3.100a)

R · pr ed,max , 2σallowed

(3.100b)

References

81

where, similarly to cylindrical columns, the following so-called reduced vapour pressure can be defined as the resultant of each load case:    pr ed,max = max pr ed p0 ; phid,max · pˆiρt (ϑ); psnow · pˆisnow (ϑ); pm · pˆim (ϑ); pst · pˆis (ϑ) i = 1.2

(3.100c)

In practice, other loads besides vapour pressure and fluid charge affect the wall thickness of the structure to a negligible degree, so reduced vapour pressure can be stated in the following simplified form:    pr ed,max = max pr ed p0 ; phid,max · pˆ iρt (ϑ) i = 1.2

(3.100d)

References 1. Fábry, Gy.: Vegyipari gépészek kézikönyve. M˝uszaki Könyvkiadó, Budapest (1987) 2. Márkus, Gy.: Körszimmetrikus szerkezetek elmélete és számítása. Budapest, M˝uszaki könyvkiadó (1964) 3. Flügge, W.: Stresses in Shells. Springer, Berlin (1960) 4. Varga, L., Szilágyi L. : Vegyipari készülékek méretezésének héjelméleti alapjai. Budapest, Tankönyvkiadó (1963) 5. Timoshenko, S., Woinowsky-Krieger, S.: Lemezek és héjak elmélete, Budapest, M˝uszaki Könyvkiadó (1966) 6. Girkmann, K.: Flächentragwerke. Springer, Wien (1959) 7. Ponomarjov, Sz.D.: Szilárdsági számítások a gépészetben. 4. kötet Budapest, M˝uszaki Könyvkiadó (1965) 8. Wlasow, W.S.: Allgemeine Schalentheorie und ihre Anwendung in der Technik. Akademie, Berlin (1958) 9. Gol’denveizer, L.: Theory of Elasic thin Shells. Pergamon Press, Oxford (1952) 10. Keresztes, J.: A vegyipari gépek és készülékek szilárdsági méretezése és szerkesztése I., II. Egyetemi jegyzet. Budapest, Tankönyvkiadó (1970) 11. Varga, L.: Nyomástartó edények tervezése. Egyetemi jegyzet Budapest, Tankönyvkiadó (1984) 12. Kollár, L.: A szél dinamikus hatása magas építményekre. Budapest, M˝uszaki Könyvkiadó (1979) 13. Brownell, L.E., Young, E.H.: Process Equipment Design. Willey, New York (1959) 14. Gill, S.S.: The stress analysis of pressure vessels and pressure vessel components. Pergamon Press, London (1970) 15. Bodor, J., Szabó, J.: Nyomástartó berendezések szilárdsági méretezése. M˝uszaki könyvkiadó, Budapest (1982)

Chapter 4

Dimensioning Equipment Loaded by External Pressure

Abstract Dimensioning cylindrical shells and conical pressure vessel ends for external pressure. Dimensioning doubly curved pressure vessel ends (spherical, torispherical, elliptical) for external pressure. Keywords External pressure · Bifurcation buckling

The behaviour of equipment operated under compression load, critical states and concomitant phenomena can be defined most easily by studying compression beams [1]. There are no detrimental consequences of disturbances in the axis shape and the central force constituting an equilibrium position in the range of small compressive forces. The load bearing capacity of a rod slightly diverted from its equilibrium position is unchanged, and it resumes its original straight shape   as the load is disconn 2 ·π 2 tinued. At Euler’s critical force F = Fcrit = E · Imin · L 2 , the situation changes because the rod assumes an increasingly curved shape, meaning that a bifurcation buckling occurs in the equilibrium to be characterized by an initial linear force versus displacement diagram, and the compression beam buckles. In case of process equipment and storage tanks, compression loads causing buckling usually arise from external overpressure. Typical cases include steam pressure in the heating chamber of equipment of dual (jacketed) pressure space or atmospheric pressure on the external surface of vacuumed equipment, and the failure of air intake valves in the course of draining the charge. Figure 4.1 shows a typical way of destruction by loss of stability, with buckling produced on the cylindrical shell of the equipment. Equilibrium state is disturbed by size and shape defects always present in thin wall thickness welded joint structures. Similarly to rods, also in the case of shells considered as structural models of equipment and storage tanks, the critical value of external pressure can be determined where the buckling position near the equilibrium state (bifurcation buckling) is produced. It is sufficient to examine the range of small displacements to determine the critical value above of external overpressure and to take first derivatives into consideration in respect of displacement changes (linear theory) [2]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Nagy, Fundamentals of Tank and Process Equipment Design, Foundations of Engineering Mechanics, https://doi.org/10.1007/978-3-031-31226-7_4

83

84

4 Dimensioning Equipment Loaded by External Pressure

Fig. 4.1 Tank buckled by vacuum

4.1 Dimensioning Cylindrical Shells and Conical Pressure Vessel Ends for External Pressure Using the assumptions mentioned, the external pressure to produce buckling to pertain to the bifurcation buckling (Fig. 4.2) in case of cylindrical shells shall be [3] pcrit

    2 λ4 1 s2 s 2 2  · λ +m  ·  =E· · 2 + R 0.5 · λ2 + m 2 12 · R 2 · 1 − ν 2 λ2 + m 2 (4.1)

where ·D • λ = n · π2·L ; • the number of half-waves in the tangential direction n = 1 (cylindrical shell with a hinged rim); • m = 2 · k; (k = 2, 3, 4, . . .) is the number of half-waves in the circumferential direction (can only be an even number).

The so-called Mises correlation (4.1) can be written in the simplified form below by assuming Ebner’s presumptions [4]:

4.1 Dimensioning Cylindrical Shells and Conical Pressure Vessel Ends …

85

Fig. 4.2 Buckled shape of cylindrical shell with a hinged rim exposed to external pressure

π D = 1; λ = ; m 2 ≫ 1; m 2 + λ2 ∼ = m2. L 2   s λ4 m2 · s2   + pcrit = E · R m6 12 · R 2 · 1 − ν 2

(4.2)

As it can be observed, the critical pressure to produce cylindrical shell buckling depends on the number of half-waves in the circumferential direction (m), in addition to geometric (s; R) and material properties (E; ν). The critical pressure stated according to Eq. (4.2) diagram 4.3 can be drawn by depicting it in the case of the geometry and material properties R = 500 mm, s = 6 mm, E = 200 GPa, ν = 0.3, in case of the number of waves in the circumferential direction (m) as a parameter. Obviously, the diagram has a physical content only in the case of even m = 2 · k; (k = 2, 3, 4, . . .) values. Out of the possible wave numbers, in case of m = 4, 6, 8 the deformed shape at the middle of the cylinder, according to the model, is illustrated by Fig. 4.4. In the case examined, the critical external pressure causing buckling pcrit,min and the number of waves in the circumferential direction produced in the course of buckling are determined by the point of the curve according to Fig. 4.3 resulting in the smallest function value. Making use of the special shape of the function to be observed in Fig. 4.3, a correlation suitable for dimensioning results from stating the boundary condition as per Eq. (4.3).

86

4 Dimensioning Equipment Loaded by External Pressure

Fig. 4.3 Critical external pressure in function of the number of waves in the circumferential direction

  −3 · λ4 ∂( pcrit ) s2 s  =0   =E· ·  + R m8 ∂ m2 12 · R 2 · 1 − ν 2

(4.3)

from which  2 m crit,min = λ ·

/ 4

  36 · 1 − ν 2 · R 2 s2

(4.4)

By substituting correlation (4.4) into (4.2), pcrit,min

π = · 9

/ 4

36 R  s  25  3 · E · · L R 1 − ν2

(4.5)

is yielded. Finally, by substituting R = D2 and after carrying out the designated operations and reductions, the so-called Ebner’s formula according to Eq. (4.6)— properly usable in design practice as well—can be obtained [4], D  s  25 pcrit,min ∼ · = 2.6 · E · L D

(4.6)

4.1 Dimensioning Cylindrical Shells and Conical Pressure Vessel Ends …

87

Fig. 4.4 Possible buckling along the circumference according to the model in the middle crosssection of the cylinder with a hinged rim

the validity range of which is L min ≤ L ≤ L max where L min = 3.46 · If L ≤ L min

(4.7)

√ (D · s) and L max = 10 · D.

pcrit,min = 0.75 · E ·

 s 2 D

(4.8a)

pcrit,min = 0.26 · E ·

 s  25 D

(4.8b)

In case of L ≥ L max

88

4 Dimensioning Equipment Loaded by External Pressure

In effect, destruction of the cylindrical part of the vessel is influenced in the negative direction by shape, material and manufacturing defects, and in the positive direction by the connection between the cylinder and the end, also making moment transfer possible compared to the presumed hinged rim. These factors resulting in uncertainties justify the fact that a relatively high safety factor z = 4 is applied in design practice for determining the allowable external pressure ( pallowed ). So the allowable external pressure is. In case of L min ≤ L ≤ L max D  s  25 · pallowed ∼ = 0.65 · E · L D

(4.9)

from which the wall thickness required for stability will be s≥

pallowed · L 0.65 · E · D

25

·D

(4.10)

In case of L ≤ L min , the allowable external pressure is  s 2 D

pallowed = 0.187 · E ·

(4.11)

the required wall thickness is / s≥ D·

pallowed 0.187 · E

(4.12)

In case of L ≥ L max , the allowable external pressure and the wall thickness to safely bear it: pallowed = 0.065 · E · s≥ D·

 s  25 D

 25  p allowed 0.065 · E

(4.13) (4.14)

The wall thickness required for the cylindrical part of the vessel can be reduced by applying reinforcing rings welded along the perimeter of the cylindrical part. In case of appropriately stiff rings, it can be achieved that buckling caused by loss of stability can occur only in the sections between the rings and between the outermost rings and the pressure vessel end. The wall thickness required for avoiding this: s ≥ max(s1 ; s2 )

(4.15a)

4.1 Dimensioning Cylindrical Shells and Conical Pressure Vessel Ends …

89

where, by assuming a fixed rim in case of the section between the reinforcing rings, the required wall thickness will be: s1 ≥

pallowed · l1 1.3 · E · D

25

·D

(4.15b)

by assuming a fixed rim on one end, and a hinged rim on the other end, the wall thickness required for the appropriate stability of the section between the outermost rings and the pressure vessel end will be: s2 ≥

pallowed · l2 0.975 · E · D

25

·D

(4.15c)

In the correlations above, l1 indicates the length of the section between the reinforcing rings, and l2 the distance between the outermost rings and the end of the cylindrical part of the pressure vessel end. In case of a conical pressure vessel end (Fig. 4.5) the correlations developed for the cylindrical part of the vessel can be applied by a substitution D = De [5], where De is a so-called equivalent cylinder diameter, which is D2 = 0 in case of a complete cone: D1 cos α

(4.16a)

0.9 · D1 + 0.1D2 cos α

(4.16b)

De = in case of a conical zone, De =

Fig. 4.5 Model applied in conical pressure vessel end stability test

90

4 Dimensioning Equipment Loaded by External Pressure

4.2 Dimensioning Doubly Curved Pressure Vessel Ends for External Pressure In contrast to cylinders, the simplest double curved form is a spherical shell, buckling as shown in Fig. 4.6a, starting from the weak point (material defect location, wall thickness reduction, local curvature radius maximum). The loss of stability of forms other than spherical—torispherical Fig. 4.6b and elliptic Fig. 4.6c used as pressure vessel ends—also occurs as local buckling to start from the weak point. In the literature, two correlations can be found for determining the critical external pressure to bring about the buckling of the forms above. The two equations, developed on the basis of principles completely different from each other (4.17) only differ in the numerical value of the coefficient: pcrit,min = c · E ·

s Re

2 (4.17)

where according to Zoelly [6] the value yielded was c = 1.21, while from the deduction of Kármán and Shen [7] c = 0.36; furthermore, Re is the so-called equivalent radius. In case of a hemisphere (Fig. 4.6a) Re = R; in case of a torisphere (Fig. 4.6b) D2 Re = R; in case of an elliptic pressure vessel end (Fig. 4.6c) Re = 4·H . Based on practical experience it can be stated that Zoelly’s constant can be used to describe buckling phenomena starting from the environment of an outlet nozzle of a jacketed pressure vessel end, while Kármán’s coefficient is suitable for describing buckling phenomena in a single pressure space case. In this case, the wall thickness required for stability and allowed external pressure can be calculated from Eq. (4.17), by taking into account a safety factor of z = 4 (4.18) and (4.19). pallowed =

a

b

c ·E· 4



s Re

2 (4.18)

c

Fig. 4.6 a, b, c Buckling of various pressure vessel ends due to loss of stability

References

91

/ s ≥ Re ·

4 · pallowed c·E

(4.19)

References 1. 2. 3. 4.

Kollár, L., Dulácska, E.: Héjak horpadása. Akadémiai Kiadó, Budapest (1975) Wansleben, F.: Die Beulfestigkeit rechteckig begrenzter Schalen. Ing. Arch. 14, 96 (1943) Mises, R.: Der kritische Außendruck zylindrischer Rohre. Z. VDI 58, 750 (1914) Ebner, H.: Theoretische und experimentelle Untersuchung über das Einbeulen zylindrischen Tank durch Unterdruck. Stahlbau 21, 153 (1952) 5. Seide, P.: On the Buckling of Truncated Conical Shells Under Uniform Hydrostatic Pressure. North-Holland Publ., Amsterdam (1960) 6. Zoelly, R.: Über ein Knickungsproblem an der Kugelschale. Dissertation. Zürich (1915) 7. Kármán, T.H., Shen, T.-H.: The buckling of spherical shells by external pressure. J. Aeron. Sci. 7, 43 (1939)

Chapter 5

Bending Stress State of Axi-symmetrical Shells. Strength Analysis by Taking Bending Stresses into Account

Abstract Axially symmetrical bending stress state of cylindrical shells. Bending stress state of cylindrical shells loaded by shearing force at the rim. Bending stress state of cylindrical shells loaded by moment at the rim. Bending stress state of cylindrical shells loaded by shearing force at the main circle. Bending stress state of cylindrical shells loaded by moment at the main circle. Examination of shell connections. Strength tests of cylinder—cone connections. Strength tests of cylinder—hemisphere connections. Keywords Bending stress · Axially symmetrical shell · Cylindrical shell · Shell connections

The dimensioning on strength of equipment and storage tanks is carried out by taking into consideration membrane stresses primarily dangerous in respect of collapse. The pure membrane stress state serving as a basis for dimensioning is located on the cylindrical part, on the conical, hemispheric, spherical calotte, and elliptic pressure vessel ends of the structures examined, in cross-sections distant from nozzles, geometry changes, and other interfering effects. Otherwise, the stress state in the cross-sections—in the tangential and circumferential directions—of axi-symmetrical shells serving as structural models can be interpreted according to Fig. 5.1 as a result of the thin shell condition. Resulting from the basic assumption of elasticity theory (principle of superposition), the stresses along wall thickness as shown in the figure can be derived as the resultant of the membrane stress state arising from constantly changing distributed surface load and the pure bending stress state resulting from the so-called rim disturbance. Similarly to the membrane edge forces defined on the cross-section pertaining to the arc length of unit middle surface as in Chap. 3. Section 3.2.2 (N1m N2m N12m = N21m ), in case of a pure bending stress state, the following edge moments impacting the middle surface can be interpreted along the wall thickness as the moment of linearly changing stresses, to be defined by correlations (5.1a)–(5.1c):

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Nagy, Fundamentals of Tank and Process Equipment Design, Foundations of Engineering Mechanics, https://doi.org/10.1007/978-3-031-31226-7_5

93

94

5 Bending Stress State of Axi-symmetrical Shells. Strength Analysis …

Fig. 5.1 Internal shell element stresses

+ 2

5

M1h =

σ1 (x3 ) · x3 · (r2 + x3 ) ·

1 · d x3 r2

(5.1a)

σ2 (x3 ) · x3 · (r1 + x3 ) ·

1 · d x3 r1

(5.1b)

− 25

+ 2

5

M2h = − 25

+ 2

5

M12h =

τ12 (x3 ) · x3 · (r2 + x3 ) · − 25

1 · d x3 r2

+ 2

5

τ21 (x3 ) · x3 · (r1 + x3 ) ·

= M21h = − 25

1 · d x3 r1

(5.1c)

In the bending stress state mentioned in the foregoing, the shearing edge forces defined by correlations (5.2a) and (5.2b) arise, impacting the middle surface, as a resultant of shear stresses τ13 and τ23 [1–3]:

5 Bending Stress State of Axi-symmetrical Shells. Strength Analysis …

95

Fig. 5.2 Edge forces and edge moments impacting the middle surface

+ 2

5

Q 1h =

τ13 · (r2 + x3 ) ·

1 · d x3 r2

(5.2a)

τ23 · (r1 + x3 ) ·

1 · d x3 r1

(5.2b)

− 25

+ 2

5

Q 2h = − 25

When determining the resultant stress state in a general case, further edge forces N1h ; N2h ; N12h ; N21h —defined on the basis of correlations (3.5a)–(3.5c)—, generated in the tangential and circumferential directions and dying away at a short distance. By introducing the edge forces and edge moments above, the edge forces and edge moments shown in Fig. 5.2 will be associated with the resultant stress state shown in Fig. 5.1. In a general case, out of the edge forces and edge moments defined in Fig. 5.2, (X 1 /= 0; X 2 /= 0; X 3 /= 0) N1m ; N2m ; N12m = N21m pertain to the pure membrane stress state, and N1h ; N2h ; N12h ; N21h ; Q 1h ; Q 2h ; M1h ; M2h ; M12h ; M21h to the general bending stress state arising from shell rim load (X 1 = X 2 = X 3 = 0). In case of axially symmetrical loads, if X 1 = X 2 =  0 andX 3 = const = − p, p·r2 p·r2 out of the quantities above N1m = 2 ; N2m = 2 · 2 − rr21 ; N12h = N21h = 0; M12h = M21h = 0; and Q 2h = 0. And the remaining unknown quantities are N1h ; N2h ; M1h ; M2h ; and Q 1h ; in order to determine them, the equilibrium of force on the shell element in the tangential direction x1 and in the surface normal direction x3 , and the equilibrium of moment in the circumferential direction x2 can be stated. Based on

96

5 Bending Stress State of Axi-symmetrical Shells. Strength Analysis …

the number of unknown quantities and the number of equilibrium equations possible to be stated, it can be established that the problem is statically doubly indefinite [4, 5]. After determining edge forces and edge moments, the stress state of the shell can be calculated by using Eqs. (5.3a)–(5.3c). σ1 (x3 ) =

12 · M1h N1m + N1h + x3 s s3

(5.3a)

σ2 (x3 ) =

N2m + N2h 12 · M2h x3 + s s3

(5.3.b)

τ13 =

Q 1h s

(5.3.c)

The problem outlined can be solved analytically in certain special load cases of cylindrical shells and spherical shells; otherwise, numerical methods (FEM, BEM) provide solutions for shell problems with a high degree of confidence. The following part presents an analytical method for specifying the stress state of cylindrical shells under axially symmetrical loads in case of various rim loads and shell connections.

5.1 Axially Symmetrical Bending Stress State of Cylindrical Shells In case of the bending stress state of cylindrical shells, the number of both unknown quantities and the equilibrium equations to be stated decreases by one (N1h = 0), thus the problem still remains to be statically doubly indefinite. The remaining unknown quantities (N2h ; M1h ; M2h ; Q 1h ) can be defined by examining the cylindrical shell segment pertaining to a unit middle surface arc length in the circumferential direction as shown in Fig. 5.3. Out of the edge forces and edge moments indicated in the figure, Q 1h (x) and M1h (x) are generated in the crosssection in the tangential direction as a consequence of rim loads Q A and M A . Edge force N2h (x), generated in the cross-section in the circumferential direction, can be calculated on the basis of correlations (5.4a)–(5.4c) below from radial displacement w(x) of the middle surface: ε2 (x3 = 0) = ε20 = σ20 =

w(x) R

(5.4a)

N2h (x) = E · ε20 s

(5.4.b)

E ·s · w(x) R

(5.4.c)

N2h (x) =

5.1 Axially Symmetrical Bending Stress State of Cylindrical Shells

97

Fig. 5.3 Axially symmetrical bending stress state model

As a result of the axial symmetry condition, edge force M2h (x) in the circumferential direction prevents the cross-section distortion of the examined shell piece, caused by bending in the tangential direction. The equilibrium of force in the normal direction x3 and the equilibrium of moment in the circumferential direction x2 as shown in Fig. 5.3, stated for a shell piece of length d x and located at a distance x from the rim, lead to the following correlations. N2h E ·s d Q 1 (x) =− = − 2 w(x) dx R R Q 1h (x) =

d M1h (x) dx

(5.5) (5.6)

In order to solve this statically indefinite problem, the relation between the edge force and edge moments sought for and the radial displacement of the middle surface is required to be found. In the course of the solution process, first the correlation between bending stresses in the tangential and in the circumferential direction—arising from axial symmetry— is stated by applying Hooke’s law on biaxial middle surface stress state. ε2 (Mh ) =

1 (σ2 (M2h ) − ν · σ1 (M1h )) = 0 E

σ2 (M2h ) = ν · σ1 (M1h )

(5.7a) (5.7.b)

98

5 Bending Stress State of Axi-symmetrical Shells. Strength Analysis …

Using the correlations above, elongation in the tangential direction will be ε1 (Mh ) =

1 (σ1 (M1h ) − ν · σ2 (M2h )) E

(5.8a)

from which E  ε1 (Mh ) σ1 (Mh ) =  1 − ν2

(5.8.b)

is yielded. Based on the correlations above, it can be established that by introducing the E , the shell piece can also be so-called substitutive elasticity modulus E ∗ = 1−ν ( 2) considered as a beam loaded with bending moment, with free lateral surface, without constraints. In case of beams, a connection between bending moment in the tangential direction and displacement perpendicular to the longitudinal axis is made by the differential equation of elastic beam, stated in the present case as: d 2 w(x) ∼ M1h (x) = dx2 B

(5.9)

3

where B = 12 E·s is the bending stiffness of the cross-section. (1−ν 2 ) By substituting the form above of the differential equation of elastic beam into equilibrium Eqs. (5.5) and (5.6) expressing the equilibrium of the shell element, the following ordinary fourth-order defective homogeneous linear differential equation of constant coefficient is yielded for radial middle surface displacement [2, 3, 6, 7]. d 4 w(x) + 4β 4 · w(x) = 0 dx4

(5.10)

/

4 3(1−ν 2 ) where β = is a so-called shell constant. R 2 ·s 2 In the knowledge of the solution of the differential equation above, the edge forces and edge moments sought for can be calculated from correlations (5.11a)–(5.11d) below:

M1h (x) = B

d 2 w(x) dx2

M2h (x) = ν · B Q 1h (x) = B

d 2 w(x) dx2

d 3 w(x) dx3

(5.11a)

(5.11b)

(5.11c)

5.1 Axially Symmetrical Bending Stress State of Cylindrical Shells

N2h (x) =

E ·s · w(x) R

99

(5.11d)

A key to the solution is the radial displacement of the cylindrical shell, to be sought for in the form of the function w(x) = c · eλ·x , then the presumed solution function is to be substituted into the differential equation, yielding the following characteristic equation for coefficient λ. λ4 + 4 · β 4 = 0

(5.12)

Characteristic equation solutions include λ1 = β · (1 + i); λ2 = β · (−1 + i ); λ3 = β · (−1 − i); λ4 = β · (1 − i ), yielding w(x) = e−β x · (c1 · cos βx + c2 · sin βx)   ∗ + e−β x · c3 · cos β x ∗ + c4 · sin βx ∗

(5.13)

by substitution into the displacement function. Figure 5.4 can be plotted by representing the solution function yielded for radial displacement in the form of (5.13) along the cylinder’s tangential direction. On the basis of the figure it can be established that the quantities sought for are yielded in the form of trigonometric functions with exponentially decreasing amplitude. It −β (x+x p ) is characteristic of quick damping that AA12 = e e−β(x) = e12π = 1.86 × 10−3 is yielded for the quotient √ of the first two amplitudes to a so-called die-away length of x p = 2π R · s from each other. = 4.88 · β It follows from the above that rim disturbances starting from the two ends of a cylindrical shell of length l ≥ x p do not affect each other, meaning that solution function (5.13) is simplified according to (5.14). w(x) = e−βx · (c1 · cos βx + c2 · sin β x)

(5.14)

Sections 5.1.1–5.1.4 present the practical application of the solution method above for a variety of load cases.

5.1.1 Bending Stress State of Cylindrical Shells Loaded by Shearing Force at the Rim Figure 5.5 shows an outline of the cylindrical shell constituting the subject matter of this investigation, with the operating rim load, and by indicating the geometric features used and their signage.

100

5 Bending Stress State of Axi-symmetrical Shells. Strength Analysis …

Fig. 5.4 Die-away length of cylindrical shells

Fig. 5.5 Model of cylindrical shell loaded by rim shearing force

In the course of the solution process, boundary condition equations M1h (x = 0) = 0; Q 1h (x = 0) = Q A to be stated on the basis of the figure pertain to the determination of constants (c1 , c2 ) figuring in the displacement function. The equations above yield 2 c1 = 2RE·s·β Q A ; c2 = 0 for the constants sought for.

5.1 Axially Symmetrical Bending Stress State of Cylindrical Shells

101

The following is yielded for the rim displacement and angular displacement of the rim (x = 0) of the cylindrical shell: w(Q A ) = w(x = 0) = χ(Q A ) =

2R 2 · β QA E ·s

| dw(x) || 2R 2 · β 2 =− QA | d x x=0 E ·s

(5.15a)

(5.15b)

By introducing the following damping functions δi (β x), δ1 (βx) = e−βx · cos βx

(5.16a)

δ2 (βx) = e−βx · sin βx

(5.16b)

δ3 (βx) = e−βx · (cos βx + sin βx)

(5.16c)

δ4 (βx) = e−β x · (cos βx − sin βx)

(5.16d)

the correlations below are yielded for the radial displacement of the middle surface and the angular displacement of the cross-section, 2R 2 · β Q A · δ1 (β x) E ·s

(5.17a)

2R 2 · β 2 dw(x) =− Q A · δ3 (β x) dx E ·s

(5.17b)

w(x) = χ (x) =

and for edge forces and edge moments, N2h (x) = 2R · β · Q A · δ1 (βx) M1h (x) =

QA δ2 (βx) β

M2h (x) = ν · M1h (x) = ν ·

QA δ2 (β x) β

Q 1h (x) = Q A · δ4 (βx) respectively [6, 8, 9].

(5.18a) (5.18b) (5.18c) (5.18d)

102

5 Bending Stress State of Axi-symmetrical Shells. Strength Analysis …

5.1.2 Bending Stress State of Cylindrical Shells Loaded by Moment at the Rim In this case, the model used for solution is shown in Fig. 5.6. Based on equations M1h (x = 0) = M A ; Q 1h (x = 0) = 0 to be stated for the cross-section (x = 0) of 2 2 the cylindrical shell, for boundary condition constants (c1 , c2 ) c1 = 2RE·s·β · M A ;

c2 = − 2RE·s·β · M A are yielded. Displacement and angular displacement of the rim of the cylindrical shell (x = 0): 2

2

w(M A ) = w(x = 0) = χ(M A )

2R 2 · β 2 MA E ·s

| dw(x) || 4R 2 · β 3 MA = = − d x |x=0 E ·s

(5.19a)

(5.19b)

The radial displacement of the middle surface and the angular displacement of the cross-section can be calculated according to the correlations below. 2R 2 · β 2 M A · δ4 (βx) E ·s

(5.20a)

4R 2 · β 3 dw(x) =− M A · δ1 (β x) dx E ·s

(5.20b)

w(x) = χ (x) =

Fig. 5.6 Model of cylindrical shell loaded by rim moment

5.1 Axially Symmetrical Bending Stress State of Cylindrical Shells

103

And the edge forces and edge moments required for determining the stress state result from the following correlations [5, 10, 11]. N2h (x) = 2R · β 2 · M A · δ4 (β x)

(5.21a)

M1h (x) = M A · δ3 (βx)

(5.21b)

M2h (x) = ν · M1h (x) = ν · M A · δ3 (βx)

(5.21c)

Q 1h (x) = −2M A · β · δ2 (βx)

(5.21d)

5.1.3 Bending Stress State of Cylindrical Shells Loaded by Shearing Force at the Main Circle Figure 5.7 shows the model of a cylindrical shell loaded by shearing force of constant intensity along the so-called main circle far from free rims. As shown in the figure, solution of the problem leads to the examination of a| cylindrical shell loaded at its | = 0 by breaking down the rim meeting the conditions Q 1h (x = 0) = Q2A ; dw(x) d x |x=0 cylinder into three parts at the load location and using the symmetry arising from R 2 ·β R 2 ·β load input. The boundary conditions above result in c1 = 2E·s Q A ; c2 = 2E·s Q A. Radial displacement and angular displacement of the load input cross-section: w(Q A ) = w(x = 0) = χ(Q A )

R2 · β QA 2E · s

| dw(x) || =0 = d x |x=0

(5.22a) (5.22b)

In the present case, the radial displacement of the middle surface and the angular displacement of the cross-section along the tangent will change according to the correlations below. R2 · β Q A · δ3 (β x) 2E · s

(5.23a)

R2 · β 2 dw(x) =− · Q A · δ2 (βx) dx E ·s

(5.23b)

w(x) = χ (x) =

104

5 Bending Stress State of Axi-symmetrical Shells. Strength Analysis …

Fig. 5.7 Model of cylindrical shell loaded by shearing force at the main circle

And the edge forces and edge moments along the middle surface can be calculated from the following correlations. N2h (x) =

R·β · Q A · δ3 (β x) 2

M1h (x) = −

QA · δ4 (β x) 4β

M2h (x) = ν · M1h (x) = −ν · Q 1h (x) =

QA · δ4 (β x) 4β

QA · δ1 (β x) 2

(5.24a) (5.24b) (5.24c) (5.24d)

5.1.4 Bending Stress State of Cylindrical Shells Loaded by Moment at the Main Circle Similarly to the case discussed in the previous section, this problem can also be traced back to the examination of a cylindrical shell loaded at its rim by breaking down the structure into three parts at the cross-section of moment input as shown in Fig. 5.8.

5.1 Axially Symmetrical Bending Stress State of Cylindrical Shells

105

Fig. 5.8 Model of cylindrical shell loaded by moment at the main circle

In the present case, the boundary conditions are M1h (x = 0) = M2A ; w(x = 0) = 0, 2 2 ·β yielding c1 = 0; c2 = − RE·s MA. Radial displacement and angular displacement of the moment input cross-section: w(M A ) = w(x = 0) = 0 χ(M A ) =

| dw(x) || R2 · β 3 =− MA | d x x=0 E ·s

(5.25a)

(5.25b)

The radial displacement of the middle surface and the angular displacement of the cross-section along the tangent change according the correlations: w(x) = − χ (x) =

R2 · β 2 M A · δ2 (βx) E ·s

R2 · β 3 dw(x) =− M A · δ4 (βx) dx E ·s

(5.26)

(5.27)

The edge forces and edge moments sought for [5, 10, 11]: N2h (x) = −R · β 2 · M A · δ2 (βx) M1h (x) =

MA · δ1 (βx) 2

M2h (x) = ν · M1h (x) = ν · Q 1h (x) = −

MA · δ1 (βx) 2

MA · β · δ3 (βx) 2

(5.28a) (5.28b) (5.28c) (5.28d)

106

5 Bending Stress State of Axi-symmetrical Shells. Strength Analysis …

5.2 Examination of Shell Connections In the present case, a shell connection is interpreted as a welded connection between the cylindrical shell and the pressure vessel end. It is the designer’s duty to explore, in the course of the strength calculation for checking after dimensioning, the stress and strain states generated in the surroundings of the shell connection, regarded as a stress concentrating location; and in the knowledge of the stress state, to specify the local reinforcement required to avoid destruction caused by repeated loads. Obviously, various numerical (finite element and boundary element) methods are appropriate tools to be applied for the examination of shell connections. In the course of design, however, simple calculation methods for checking—included in a variety of national standards and technical specifications—are rather applied, the reliability of which has been proven in practice. In general, standards calculations are built on the basic equations of classical shell theory, the accuracy of which is also evidenced by experimental elongation measurements and finite element calculations, subject to the prevalence of certain circumstances. The shell connection problems to be presented below are also solved by analytical methods based on technical shell theory, in the course of which the internal force system, and the stress state can be determined from the fit condition of the connecting cross-section. In the course of the solution process, the cylindrical shell can be modelled according to Sect. 5.1, while the single curved (conical) or double curved (spherical or ellipsoid) pressure vessel end connected to the cylinder can be substituted by a so-called “Geckeller’s” cylinder as shown in Fig. 5.9, in a tangential position for the examination of pure rim load (Q A , M A ) in the environment of the connecting cross-section ( A). This approximation yields a result of acceptable accu. If this condition racy if the rim of the shell is within the angle range of π4 ≤ ϑ A ≤ 3π 4 is not met, the shell connection cannot be examined by the method to be presented below, therefore the numerical procedures already mentioned can be recommended for solving the problem. The practical applications of the analytical correlations stated for the shell connections studied in Sects. 5.2.1 and 5.2.2 are illustrated by numeric examples. In relation to the examples, the stress state of the connecting cross-section was determined numerically as well, using the finite element method, which provides opportunities to compare results and draw conclusions.

5.2.1 Strength Tests of Cylinder—Cone Connections In order to solve the fit problem, the structure examined was divided into two parts at the connecting cross-section “A”—(1) cylindrical shell, (2) conical shell. Figure 5.10 shows an outline of the structure split into two parts by indicating the internal force system in the connecting cross-section. In the internal force system shown in the figure, shearing edge force Q A and edge moment M A —both of them being

5.2 Examination of Shell Connections

107

Fig. 5.9 Geckeller approximation model

unknown quantities—can be determined from the following fit condition, stated for the connecting cross-section [7–9]. Δ1Ae = Δ2Ae

(5.29a)

1 2 χ Ae = χ Ae

(5.29b)

where • Δ1Ae , Δ2Ae represent the resultant displacement—perpendicular to the rotation axis—of the connecting cross-section “A” of cylindrical shell (1) and conical shell 1 2 , χ Ae represent their resultant angular displacement. (2), respectively, while χ Ae Resultant displacement and angular displacement can be broken down into components derived from internal pressure, from the shearing force perpendicular to the rotation axis, and from the bending moment to load the cross-section, resulting in the following Eqs. (5.30a) and (5.30b) in case of the directions indicated in the figure. | 1 |Δ

A( p)

| | 1 | − |Δ

A(Q A )

| | 1 | + |Δ

| 1 |χ

A(Q A )

A(M A )

| | 1 | − |χ

| | 2 | = |Δ

A(M A )

A( p)

| || 2 | + |Δ

| || 2 | = |χ

A(

A(

Q ∗A

Q ∗A

| | | | | 2 | ) | + Δ A(M A )

| | | | | 2 | ) | + χ A(M A )

(5.30a) (5.30b)

108

5 Bending Stress State of Axi-symmetrical Shells. Strength Analysis …

Fig. 5.10 Internal force system of cylinder-cone connection

In the fit equations above, the rim displacements and angular displacements of cylindrical shell (1) will be as follows based on Eqs. (3.16), (5.15a) and (5.15b), (5.19a) and (5.19b), and Fig. 5.11: | 1 |Δ

A( p)

 2 | |= p· R 1− ν E · s1 2

(5.31a)

| 2R 2 · β1 | · QA A(Q A ) = E · s1

(5.31b)

| | 1 | = |w

A( p)

| 1 |Δ

| | 1 | = |w

| 1 |Δ

| | 1 | = |w

A(Q A )

A(M A )

A(M A )

| 2R 2 · β12 |= · MA E · s1

| 1 |χ

| 2R 2 · β12 | · QA A(Q A ) = E · s1

| 1 |χ

A(M A )

| 4R 2 · β13 |= · MA E · s1

(5.31c)

(5.31d)

(5.31e)

5.2 Examination of Shell Connections

109

Fig. 5.11 Internal force system and deformation of cylindrical shell

/ where the shell constant is β1 =

4

3·(1−ν 2 ) . R 2 ·s12

The rim displacements and angular displacements of conical shell (2), using “Geckeller’s” substitute cylindrical shell as defined in Fig. 5.12: | 2 |Δ

A( p)

| | 2 | = |w

A( p)

| · cos α | =

p · R2  ν 1− E · s2 · cos α 2

| | | | 2R 2 · β   2 | 2 | | | · Q ∗A |Δ A( Q ∗ ) | = |w 2A( Q ∗ ) · cos α | = A A E · s2 | | 2 | | 2 | |Δ | | A(M A ) = w A(M A ) · cos α =

substitute cylinder.

(5.32c)

  2R 2 · β22 · Q ∗A E · s2 · cos α

(5.32d)

| 2 | |χ | A(M A ) =

4R 2 · β23 · MA E · s2 · (cos α)2

(5.32e)

/ p·R 2

(5.32b)

| | | 2 | |χ A( Q ∗ ) | = A

where Q ∗A = Q A −

2R 2 · β22 · MA E · s2 · cos α

(5.32a)

· tan α, β2 =

4

3·(1−ν 2 )(cos α)2 R 2 ·s22

is the shell constant of the

110

5 Bending Stress State of Axi-symmetrical Shells. Strength Analysis …

Fig. 5.12 Internal force system and deformation of conical shell

By substituting correlations (5.31a)–(5.31e) and (5.32a)–(5.32e) into the fit condition of connecting cross-section “A”, unknown quantities (Q A , M A ) can be specified. In the knowledge of the internal force system meeting the fit condition, the resultant stress state of the connecting cross-section as well as developments in the resultant stresses generated in the tangential and circumferential directions, respectively, along the external (K) and internal (B) surfaces of the shell can also be calculated as follows. Resultant stress state of connecting cross-section “A” of cylindrical shell (1) based on Fig. 5.11: • resultant stress in the tangential direction σ1A(K ) = (B)

N1( p) 6 · MA p·R 6 · MA ∓ = ∓ s1 2 · s1 s12 s12

• resultant stress in the circumferential direction

(5.33a)

5.2 Examination of Shell Connections

σ2 A(K ) = (B)

=

111

N2( p) N2 A(M A ) N2 A(Q A ) 6 · MA + − ∓ν· s1 s1 s1 s12 p·R 2R · β12 · M A 2R · β1 · Q A 6 · MA + − ∓ν· s1 s1 s1 s12

(5.33b)

Stress distributions along the length: • resultant stress in the tangential direction in external (K) and internal (B) surfaces 

6 · M1(Q A ) (x) − M1(M A ) (x) N1( p) σ1(K ) (x) = ± (B) s1 s12

6 · Qβ1A · δ2 (β1 x) − M A · δ3 (β1 x) p·R ± = 2 · s1 s12

(5.34a)

• resultant stress in the circumferential direction in external (K) and internal (B) surfaces 

N2( p) N2(M A ) (x) N2(Q A ) (x) 6 · M2(Q A ) (x) − M2(M A ) (x) σ2(K ) (x) = + − ± (B) s1 s1 s1 s12 p·R 2R · β12 · M A · δ4 (β1 x) 2R · β1 · Q A · δ1 (β1 x) + − s1 s1 s1

QA 6 · β1 · δ2 (β1 x) − M A · δ3 (β1 x) ±ν· s12

=

(5.34b)

Resultant stress state of connecting cross-section “A” of conical shell (2) based on Fig. 5.12: • resultant stress in the tangential direction N1A( Q ∗A ) N1A( p) 6 · MA + ∓ s2 s2 s22   p·R Q · tan α · sin α − A 2 6 · MA p·R + ∓ = 2 · s2 · cos α s2 s22

σ1A(K ) = (B)

• resultant stress in the circumferential direction N2 A( Q ∗A ) N2 A( p) N2 A(M A ) 6 · MA + + ∓ν· s2 s2 s2 s22   2R · β2 · Q A − p·R · tan α 2 p·R + = s2 · cos α s2

σ2 A(K ) = (B)

(5.35a)

112

5 Bending Stress State of Axi-symmetrical Shells. Strength Analysis …

+

6 · MA 2R · β22 · M A ∓ν· s2 · cos α s22

(5.35b)

Stress distributions along the length: • resultant stress in the tangential direction in external (K) and internal (B) surfaces N1( Q ∗A ) (x) N1( p) (x) + s2 s2

6 · M1( Q ∗A ) (x) + M1(M A ) (x) ∓ s22      p·R x · tan α · sin α · 1 − − Q A p · R · 1 − R · sin α 2 = + 2 · s2 · cos α s2    ·tan α ·cos α Q A − p·R 2 6· · δ2 (β2 x) + M A · δ3 (β2 x) β2 ∓ s22

σ1(K ) (x) = (B)

x R

· sin α



(5.36a)

• resultant stress in the circumferential direction in external (K) and internal (B) surfaces

∗ (x) + M2(M ) (x) 6 · M ∗ N (x) 2 Q A ( A) N2( p) (x) N2(M A ) (x) 2( Q A ) σ2(K ) (x) = + + ∓ (B) s2 s2 s2 s2 2    2R · β2 · Q A − p·R · tan α · δ1 (β2 x) p · R · 1 − Rx · sin α 2 + = s2 · cos α s2 2 2R · β2 · M A · δ4 (β2 x) + s2 · cos α    Q A − p·R 2 ·tan α ·cos α 6· · δ2 (β2 x) + M A · δ3 (β2 x) β2 ∓ν (5.36b) s22 The accuracy and applicability of the analytical solution based on the Geckeller approximation is demonstrated by Fig. 5.13, showing stress distribution along the length of the cylinder-cone connection examined in function of arc length. The figure shows stresses in the tangential direction σ1(K ) and in the circumferential direction (B) σ2(K ) , respectively, in case of per unit internal pressure, calculated by the finite element (B) method (dots) and by the analytical method (continuous line).

Fig. 5.13 Stress distribution of cone-cylinder connection, calculated by analytical and finite element methods

5.2 Examination of Shell Connections 113

114

5 Bending Stress State of Axi-symmetrical Shells. Strength Analysis …

5.2.2 Strength Tests of Cylinder—Hemisphere Connections In order to solve the fit problem, the structure examined was again divided into two parts at the connecting cross-section “A”—(1) cylindrical shell, (2) spherical shell. Figure 5.14 shows an outline of the structure split into two parts by indicating the internal force system in the connecting cross-section. Connecting cross-section fit condition [7–9]: Δ1Ae = Δ2Ae

(5.37a)

1 2 χ Ae = χ Ae

(5.37b)

which, broken down into components derived from internal pressure, from the shearing force perpendicular to the rotation axis, and from the bending moment to load the cross-section, and taking the directions indicated in Fig. 5.14 into account, will yield Fig. 5.14 Internal force system of cylinder-hemisphere connection

5.2 Examination of Shell Connections

| 1 |Δ

A( p)

| | 1 | − |Δ

A(Q A )

| | 1 | + |Δ

115

A(M A )

| | 2 | = |Δ

A( p)

| | 2 | + |Δ

A(Q A )

| | 2 | + |Δ

A(M A )

| 2 | 1 | | 1 | | | 2 | | |χ | | | | | | A(M A ) − χ A(Q A ) = − χ A(Q A ) − χ A(M A )

| |

(5.38a) (5.38b)

Rim displacements and angular displacements of cylinder shell (1) in the fit equations above: | 1 |Δ

A( p)

 2 | |= p· R 1− ν A( p) E · s1 2

| | 1 | = |w

(5.39a)

| 1 |Δ

| | 1 | = |w

| 2R 2 · β1 |= · QA E · s1

(5.39b)

| 1 |Δ

| | 1 | = |w

| 2R 2 · β12 |= · MA E · s1

(5.39c)

A(Q A )

A(M A )

A(Q A )

A(M A )

| 1 |χ

| 2R 2 · β12 | · QA A(Q A ) = E · s1

| 1 |χ

A(M A )

/ where the shell constant is β1 =

4

| 4R 2 · β13 |= · MA E · s1

(5.39d)

(5.39e)

3·(1−ν 2 ) . R 2 ·s12

Rim displacements and angular displacements of spherical shell (2) using “Geckeller’s” substitute cylindrical shell defined in Fig. 5.14: | 2 |Δ

A( p)

| | 2 | = |w

A( p)

| 2 |Δ

| | 2 | = |w

| 2 |Δ

| | 2 | = |w

A(Q A )

A(M A )

2 | | = p · R (1 − ν) E · s2 2

| 2R 2 · β2 | · QA A(Q A ) = E · s2

A(M A )

| 2R 2 · β22 |= · MA E · s2

| 2R 2 · β22 | · QA A(Q A ) = E · s2

| 2 |χ | 2 |χ

A(M A )

/ where β2 =

4

3·(1−ν 2 ) R 2 ·s22

| 4R 2 · β23 |= · MA E · s2

is the shell constant of the substitute cylinder.

(5.40a)

(5.40b)

(5.40c)

(5.40d)

(5.40e)

116

5 Bending Stress State of Axi-symmetrical Shells. Strength Analysis …

In the event that the wall thickness figures of the cylinder and the hemisphere are identical s1 = s2 = s, that is, β1 = β2 = β, then M A = 0, Q A = 8βp will be yielded from fit condition (5.38a) and (5.38b) (moment-free fit). Presuming identical wall thickness, the resultant stress state of connecting crosssection “A” of cylindrical shell (1) based on Fig. 5.14 will be: • resultant stress in the tangential direction σ1A(K ) = (B)

N1( p) p·R = s 2s

(5.41a)

• resultant stress in the circumferential direction σ2 A(K ) = (B)

N2( p) N2 A(Q A ) p·R 2R · β · Q A − = − s s s s

(5.41b)

Stress distributions along the length: • resultant stress in the tangential direction in external (K) and internal (B) surfaces 



σ1(K ) (x) = (B)

6 · M1(Q A ) (x) N1( p) p·R = ± ± s s2 2s

6·



QA β

· δ2 (βx) s2

(5.42a)

• resultant stress in the circumferential direction in external (K) and internal (B) surfaces

 N2( p) N2(Q A ) (x) 6 · M2(Q A ) (x) − ± σ2(K ) (x) = (B) s s s2

QA 6 · · δ (βx) 2 β 2R · β · Q A · δ1 (βx) p·R − ±ν· = (5.42b) 2 s s s Resultant stress state of connecting cross-section “A” of spherical shell (2) based on Fig. 5.14: • resultant stress in the tangential direction σ1A(K ) = (B)

N1( p) p·R = s 2s

(5.43a)

• resultant stress in the circumferential direction σ2 A(K ) = (B)

N2( p) N2 A(Q A ) p·R 2R · β · Q A + = + s s 2s s

(5.43b)

Stress distributions along the length: • resultant stress in the tangential direction in external (K) and internal (B) surfaces

References

117

Fig. 5.15 Stress distribution of hemisphere-cylinder connection, calculated by analytical and finite element methods

σ1(K ) (x) = (B)

N1( p) s



6· 6 · M1(Q A ) (x) p·R = ∓ ∓ 2 s 2s



QA β

· δ2 (βx) s2

(5.44a)

• resultant stress in the circumferential direction in external (K) and internal (B) surfaces

 N2( p) N2(Q A ) (x) 6 · M2(Q A ) (x) + ∓ σ2(K ) (x) = (B) s s s2

QA 6 · · δ (βx) 2 β 2R · β · Q A · δ1 (βx) p·R + ∓ν· = (5.44b) 2s s s2 Figure 5.15 shows the results of a numeric example associated with per unit internal pressure load. In the present case as well, stresses in the tangential and circumferential directions calculated by the finite element method are indicated by dots, and those resulting from the analytical model by continuous lines. The results yielded in two different ways are in good agreement not only in the connecting cross-section, but in the entire test range.

References 1. Girkmann, K.: Flächentragwerke. Springer, Wien (1959) 2. Ponomarjov, S.D.: Szilárdsági számítások a gépészetben. 4. kötet. M˝uszaki Könyvkiadó, Budapest (1965) 3. Timoshenko, S., Woinowsky-Krieger, S.: Lemezek és héjak elmélete. M˝uszaki Könyvkiadó, Budapest (1966) 4. Varga, L., Szilágyi, L.: Vegyipari készülékek méretezésének héjelméleti alapjai. Tankönyvkiadó, Budapest (1963) 5. Varga, L.: Nyomástartó edények tervezése. Egyetemi jegyzet. Tankönyvkiadó, Budapest (1984) 6. Fábry, G.: Vegyipari gépészek kézikönyve. M˝uszaki Könyvkiadó, Budapest (1987)

118

5 Bending Stress State of Axi-symmetrical Shells. Strength Analysis …

7. Gill, S.S.: The Stress Analysis of Pressure Vessels and Pressure Vessel Components. Pergamon Press, London (1970) 8. Márkus, G.: Körszimmetrikus szerkezetek elmélete és számítása. M˝uszaki Könyvkiadó, Budapest (1964) 9. Flügge, W.: Stresses in Shells. Springer, Berlin (1960) 10. Varga, L.: Centrifuga forgórészek méretezésének új módszere. Gép 9 (1968) 11. Bodor, J., Szabó, J.: Nyomástartó berendezések szilárdsági méretezése. M˝uszaki Könyvkiadó, Budapest (1982)

Chapter 6

Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

Abstract Pressure vessel ends. Stress states and dimensioning of elliptical pressure vessel ends. Stress states and dimensioning of torispheric pressure vessel ends. Design of openings on cylindrical shells and pressure vessel ends. Strength tests of the support environment. Strength tests of the support environment of spherical tanks. Distribution of normal force to rigid load transfer cylinders. Determining nozzle loads caused by bending moment. Distribution of tangential force to bolster plate fillet welds. Stress state caused by rigidly installed cylindrical nozzle loaded by normal force. Stress state caused by rigidly installed cylindrical nozzle loaded by shearing force. Keywords Pressure vessel ends · Nozzle environments · Support environments · Ring support · Skirt support · Bracket support · Saddle support · Shallow spherical shell

This chapter discusses pressure vessel ends, the strength tests of nozzles and their environments on cylindrical shells and pressure vessel ends, as well as various equipment support structures, and the determination of bending stresses which arise from leading through mass forces in support structure surroundings. Section 6.1 summarizes the variety of pressure vessel end types generally used in practice. Descriptions of their geometric design are followed by short presentations of their manufacturing technology and areas of application, and stress states generated in such ends due to external loads (general membrane, local bending, and general bending stresses) [1–8]. This general description is followed by special sections on the elliptical (6.1.1) and torispheric (6.1.2) pressure vessel ends applied most frequently. The internal force systems to load pressure vessel ends arising from distributed surface load (overpressure) are presented, to be followed by the numerical specification (using the finite element method) of elongation and stress states, and a detailed discussion of approximating calculation methods applied in design practice [9–15]. Section 6.2 deals with the design of nozzles welded into the equipment, and the method for determining the local reinforcement required. A presentation of the additional internal force system and the bending stress state produced as a result of such © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Nagy, Fundamentals of Tank and Process Equipment Design, Foundations of Engineering Mechanics, https://doi.org/10.1007/978-3-031-31226-7_6

119

120

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

force system in the environment of nozzles [16–23] is followed by a detailed discussion of the area method widely applied in design practice and the so-called calculation method based on weakening factor as derived from the area method [24–26]. Using the results of experimental elongation measurements, the interfering effects of welded joints at the nozzle connection are presented, which can be properly illustrated by the comparison of finite element tests and the elongations measured on the real structure. Finally, Sect. 6.3 deals with the support structures of equipment in vertical and horizontal positions, respectively. The presentation of various support constructions is followed by a detailed discussion of the additional internal force system loading the equipment shell, generated by leading through mass forces at each support. For each support construction, the force system above is discussed by assuming perfectly rigid load transfer elements. At the same time, the real elongation and stress states of the support environment can be determined by finite element analysis which takes into account the elastic interaction of the support structure and the shell, requiring the entire equipment to be modelled. The numerical analyses above, yielding accurate results, are rarely applied in the course of design by reason of their complexity. Approximating calculations based on shell theory correlations assuming rigid load input elements have spread in practice [27–32]. As a result of the existing symmetries, solution algorithms use Fourier series in case of cylindrical shells [33, 34]. Algorithms yielding complex correlations are properly programmable, and therefore calculations can be processed in the form of diagrams, thus being easy-to-use in practice. Due to limitations of scope, this section only presents the approximating procedure developed for strength calculations of the support environment of spherical tanks [35–38]. Approximating calculations of local stresses caused by supports on cylindrical shells (support brackets, feet, ring, and skirt supports) are discussed in detail in the literature, therefore they will not be treated here in detail to avoid repetitions.

6.1 Pressure Vessel Ends Various ends can be connected to the cylindrical parts of pressure vessels (see Fig. 6.1a–f). Connection to the shell can be made by a releasable flange joint or a non-releasable welded joint. The membrane displacements of shells with different curvatures (cylinder, end) differ from each other, thus additional internal forces and moments are generated in the connecting cross-section, eliminating differences in displacement and resulting in local bending stress. No welded joint is allowed to be placed in a bending stress zone, therefore pressure vessel ends terminate in cylindrical necks of at least damping length. From among the possible formations illustrated in Fig. 6.1, the spherical form would be the most advantageous one, resulting in uniform strength. Hemispheric pressure vessel ends resulting in favourable stress distribution are usually pressed from one piece; in case of large diameters, they are welded together from pre-embossed elements. In order to avoid weld contact, the top of the hemisphere

6.1 Pressure Vessel Ends

121

Fig. 6.1 a Hemispheric end, b elliptical end, c torispheric end, d dished head, e conical end, f plane end

should be a one-piece spherical calotte. Notwithstanding the above, the hemispheric pressure vessel ends shown in Fig. 6.1a are used only rarely, in case of relatively small diameters, as—due to the length of the cylindrical part—the wall thickness of the spherical part is determined by the cylinder need (see Eqs. 3.19 and 3.24). It follows from this that the spherical end is doubly overdimensioned. An elliptical pressure vessel end as shown in Fig. 6.1b results in more favourable material utilization than in the case of spherical ends. Due to the continuous curvature change of the ellipse, internal overpressure is borne in the form of pure membrane stress state. Elliptical bottom plates are pressed or rolled. Pressing tools are complicated and expensive to make; on the other hand, the engineering material can be considerably hardened by rolling. In spite of the disadvantages above, they are more and more extensively applied.

122

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

The most frequently used pressure vessel end type is the torispheric pressure vessel end shown in Fig. 6.1c, manufactured in both torispheric and semi-ellipsoidal shape. The torispheric end consists of a spherical calotte, a torus piece and the length of the cylindrical part. Torispheric pressure vessel ends are rather unfavourable in terms of stress distribution because significant levels of bending stress are generated in the connecting cross-sections of the spherical shell and the torus, and of the torus and the cylindrical part, respectively, by reason of step changes in the circle of curvature radius r1 . The bending stresses generated in the environment of connecting crosssections can be a multiple of membrane stresses, so the wall thickness required for strength considerably increases. The ensuing sections describe the determination of stresses generated in the elliptical and torispheric ends and dimensioning as discussed in the foregoing. Torispheric pressure vessel ends are generally made of one piece by cold or pre-heated pressing, depending on wall thickness and the engineering material. Over 2000 mm diameter, the starting circular disc is welded together from two or three pieces. The dished heads shown in Fig. 6.1d are intended for closing large-sized openings e.g. manholes. Advantages for their application include small structural height and stress distribution more advantageous than in case of a plane surface. A dished head consists of a spherical calotte, and a plane flange part connected by a torus transition. In case of minor loads, pressed ends are applied, and for closing higher-pressure spaces, forged ends of considerable wall thickness are applied. In case of pressed designs, the dished head is connected to the cylindrical shell by a loose flange. The conical zones shown in Fig. 6.1e are manufactured from plane plates by pressing in case of small diameters and large cone angles, and otherwise by bending into a cone and welding a plate cut according to a cutting design. The value of the cone angle is generally standardized. In respect of stresses, the plane pressure vessel ends shown in Fig. 6.1f are the most unfavourable ones. They bear internal pressure loads in the form of relatively high bending stresses distributed along the entire surface, so safe load bearing capacity can only be ensured by considerable wall thickness. Consequently, their areas of application include welded pressure vessel ends for tanks of small overpressure or atmospheric pressure, on the one hand, and for high-pressure equipment requiring large wall thickness anyway, on the other hand; they are also used as blinds to close nozzles and apertures, and as heat exchanger tube sheets. They can be manufactured from plane plates by pressing, or by forging with appropriate allowance and then turning to size.

6.1.1 Stress States and Dimensioning of Elliptical Pressure Vessel Ends The meridian curve of the pressure vessel end previously presented (Fig. 6.1b) is the ellipse with axes a and b as shown in Fig. 6.2, the curvature conditions of which are characterized by the following correlations.

6.1 Pressure Vessel Ends

123

Fig. 6.2 Structural model of elliptical pressure vessel end

a 2 · b2 r1 =   3  b2 + a 2 − b2 · sin2 ϑ 2

(6.1a)

a2 r2 =    1 b2 + a 2 − b2 · sin2 ϑ 2

(6.1b)

The membrane edge forces produced are as follows in function of angle coordinate (ϑ) in the meridian direction [2, 3]: N1 =

p · a2

   1 2 · b2 + a 2 − b2 · sin2 (ϑ) 2   b2 − a 2 − b2 · sin2 (ϑ) p · a2 N2 = b2 2 · b2 + a 2 − b2  · sin2 (ϑ) 21

(6.2a)

(6.2b)

It can be observed from Eqs. (6.2a–b) that membrane edge forces are of the highest value in the cross-section at the angle position (ϑ = 0) (at the minor axis); and that   edge forces show monotone reduction as (ϑ) is increased. Depending on the ab ratio, compression load can be generated in the circumferential direction near the major axis, as illustrated in Fig. 6.3. The edge forces mentioned are also shown in Fig. 6.2. Determining the wall thickness required on the basis of the edge forces generated in cross-section (ϑ = 0). N1max = N2max =

p · a2 2·b

(6.3)

124

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

Fig. 6.3 Membrane edge forces of an elliptical end under unit load in the case of various geometric proportions

 By applying the signage in Fig. 6.1b a ∼ = N1max = N2max

p·D · = 4



D 2

D 2· H

= R; b = H 

p·R = · 2

 

D 2· H

 (6.4)

the stresses generated and the basic irregularity of dimensioning will be   D p·R · 2·s 2· H   D p·R p·R · = · α ≤ σallowed = 2·s 2· H 2·s

σ1max = σ2max = σr ed Mohr,max

(6.5) (6.6)

out of which the wall thickness of the pressure vessel end required for strength is s, ≥

p·R ·α 2 · σallowed

(6.7)

  D is the shape factor of the elliptical end as compared to a spherical where α = 2H pressure vessel end.

6.1 Pressure Vessel Ends

125

6.1.2 Stress States and Dimensioning of Torispheric Pressure Vessel Ends Figure 6.4 shows developments in the internal forces (Q B ; Q A ) and moments (M B ; M A ) to eliminate differences between membrane displacements caused by internal overpressure [4–8] in the connecting cross-sections B (sphere-torus) and A (torus-cylinder) of the torispheric pressure vessel end, as well as the resultant displacement (w) from the interaction of structural parts, by indicating important signage. The additional internal forces and moments shown in the figure cause the considerable bending stress state—dampened within a short distance, though—near such sudden curvature transition as shown in Fig. 6.5, which needs to be taken into account for dimensioning. As a matter of fact, the results shown in Fig. 6.5 were yielded from finite element calculations performed by assuming elastic material behaviour. It needs to be mentioned that there is no analytical solution—providing results of acceptable accuracy—for determining the elastic elongation and stress states of torispheric pressure vessel ends. The approximating calculation method below—based on a so-called design factor β, compared to a hemispheric pressure vessel end of identical diameter and subject to identical load—has spread in design practice for dimensioning torispheric pressure vessel ends, to be found in national standards [9, 10]. The effective reduced stress generated in the dangerous cross-section of the pressure vessel end examined can be stated according to the correlation below, using a so-called shape factor α, and based on the stress state of a hemispheric pressure vessel end of identical diameter and subject to identical load.

w

B QB

N1B MB QB QA

N1B B

B s

p

p

r

R

A lh

p A

N1A MA

A

N1A p

QA

OD

Fig. 6.4 Structural model and internal force system of torispheric pressure vessel end

126

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

Fig. 6.5 Elastic stresses of torispheric end subject to unit load, in the tangential and circumferential directions, in function of arc length

σr ed Mohr,max =

p·D ·α 4·s

(6.8)

Besides, it is known that during the test load to follow manufacturing and stress relieving heat treatment, the stress concentrating cross-sections of the vessel— including the dangerous cross-section examined—reach an elastic–plastic state. In the course of stress relieving after test load, the elastic load carrying capacity range of material points in a plastic state will increase. Figure 6.6 shows the processes of test load and stress relieving in the critical point of the dangerous cross-section.  The figure illustrates changes in the reduced stress vs. yield stress ratio σ Redmax ReH in the point examined during  test load, in function of the load compared to elastic load carrying capacity ppE . . The model presumes that no stress larger than the yield stress can be generated in the engineering material in the course of test load (idealized material law of carbon steel), and that plastic elongation cannot exceed a required value, e.g. 0.2%, for plastic strain. After unloading, yield stress—and consequently, the elastic load carrying capacity range—increase, based on the figure, will be ∗ ReH p0,2 = ReH pE

(6.9)

By taking correlations (Eqs. 6.8–6.9) into account, the basic irregularity of dimensioning will be R∗ p·D p0,2 ReH · · α ≤ σallowed = eH = 4·s n1 pE n1

(6.10)

6.1 Pressure Vessel Ends

127

Fig. 6.6 Load-stress characteristic of critical point of torispheric end dangerous cross-section

from which p · D α · pE ReH · ≤ 4·s p0,2 n1

(6.11)

is yielded. According to Eq. (6.11), the design factor of the torispheric pressure vessel end can be stated in the form β=

α · pE p0,2

(6.12)

where α is a shape factor, p E is the elastic load carrying capacity of the pressure vessel end, p0.2 is the load in the critical point which results in 0.2% plastic strain during unloading after test load. The numerical values of the variables in Eq. (6.12) were specified on the basis of serial experiments (elongation measurements). By introducing the design factor above, the wall thickness required for strength of the torispheric end will be: s, ≥

p· D·β 4 · σallowed

(6.13)

Dimensioning requirements [9–15] include the numerical values of design factor β for pressure vessel ends of torispherical shape and semi-ellipsoidal shape in the form of diagrams (see Fig. 6.7). It can be established on the basis of the figure that the accurate value of wall thickness required for strength can only be yielded by iteration calculations performed in several steps.

128

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

Fig. 6.7 Design factors of torispherical and semi-ellipsoidal ends

6.2 Design of Openings on Cylindrical Shells and Pressure Vessel Ends In general, openings can be found on cylindrical shells (Fig. 6.8a), conical pressure vessel ends (Fig. 6.8b), and on the spherical part of torispheric pressure vessel ends (Fig. 6.8c). Designers are supposed to create a design in compliance with the applicable strength requirements, in which case the local amplitudes of stress and elongations [16–22] produced on the cylindrical shell and the pressure vessel end in the environment of the nozzle welded in the opening will remain below the allowed limit value. The amplitude of stress in the nozzle environment is caused by the shearing force and bending moment (Q A , M A ) ensuring identical displacement (we ) and angle displacement (κe ) of the shell and the nozzle. Figure 6.9 shows the internal forces of the nozzle environment [21–23, 39].   The figure shows the membrane w p and resultant (we ) displacement of the nozzle environment on the spherical part of a torispheric pressure vessel end, and the development of internal forces which eliminate differences of membrane displacements in connecting cross-section (A). The strength condition prescribed for the nozzle environment can be complied with by the appropriate strengthening of the opening, such as by local thickening of the nozzle welded in the opening (Fig. 6.10a), hanging in a thickened nozzle (Fig. 6.10b), and finally by applying a reinforcing plate of appropriate thickness (Fig. 6.10c). In case of given diameters (Db ; dk ), it is required to perform a strength analysis to determine rim thickness (sk ) and nozzle thickness (scs ), and to specify the stress and elongation states of the nozzle environment by stating the dimensions in question. The appropriacy of the wall thicknesses stated (sk ; scs ) can be decided by comparing

6.2 Design of Openings on Cylindrical Shells and Pressure Vessel Ends

129

Fig. 6.8 a Opening on cylindrical shell, b opening on conical pressure vessel end, c opening on spherical part of torispheric pressure vessel end Fig. 6.9 Internal forces of nozzle environment, indicating major geometric features

130

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

Fig. 6.10 a Opening strengthened by local thickening of nozzle welded in, b Opening strengthened by hanging in thickened nozzle, c Opening strengthened by applying reinforcing plate

6.2 Design of Openings on Cylindrical Shells and Pressure Vessel Ends

131

the calculated effective stress with the allowed stress value. It is expedient to perform stress analysis numerically, by applying the finite element method. The applicability and accuracy (inaccuracy) of the finite element method are properly demonstrated by Figs. 6.11 and 6.12. The figures show the results of the finite element calculations of a nozzle connection of a perpendicular position on the cylindrical axis of a carbon steel vessel, and of the strain gauge measurement on the internal and external surfaces, respectively (see Fig. 6.13). Figure 6.11 show the stress distributions along the length of the symmetry planes of the nozzle connection in the tangential and circumferential directions, defined from calculated and measured elastic elongation on the basis of Hooke’s law. Figure 6.12 show the distribution of calculated and measured “equivalent” elongations in the elastic–plastic load carrying capacity range. The material law used for elastic–plastic finite element tests came from experimental measurements. The example examined also properly shows that the results of the finite element calculation are in good agreement with the data from elongation measurements, except for the connecting cross-section. The difference in the connecting cross-section is caused by the welded joint as the material behaviour of the welded joint (inhomogeneity, anisotropy, etc.) may considerably differ from the properties of the engineering material of the equipment. The calculation work is significantly simplified by the introduction of the so-called weakening factor “V”, found in the applicable requirements [24–26, 40, 41], defined as the quotient of the membrane stress in the circumferential direction far from the nozzle environment (Eqs. 3.18, 3.23, and 3.27) and the local membrane stress in the circumferential direction. V =

σ2membrane σ2localmembrane

(6.14)

By introducing weakening factor V ≤ 1, the shell thickness required in the nozzle environment will be. in case of cylinder p · Db 2 · σallowed · V

(6.15a)

p · Db 2 · σallowed · cos(α) · V

(6.15b)

p · Db 4 · σallowed · V

(6.15c)

sk = in case of a cone, sk = in case of a sphere, sk =

132

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

Fig. 6.11 a Elastic stresses in the tangential and circumferential directions on the internal surface of the opening environment, yielded by measurements and FE calculations (dir. ind. = direction indicator; calc. = calculated; meas. = measured), b Elastic stresses in tangential and circumferential directions on external surface of opening, yielded by measurements and FE calculations (dir. ind. = direction indicator; calc. = calculated; meas. = measured)

In the applicable requirements [24–26, 40, 41], the numerical values of the weak sk db scs ening factor V = f Db ; Db ; sk can be drawn from diagrams (Figs. 6.14a–d and 6.15a–c) in function of the geometric conditions of the nozzle environment.

6.2 Design of Openings on Cylindrical Shells and Pressure Vessel Ends

133

Fig. 6.12 a Equivalent elongations on internal surface, resulting from elongation measurements and FE calculations in cross-section in the tangential direction, b Equivalent elongations on external surface, resulting from elongation measurements and FE calculations in cross-section in the tangential direction, c Equivalent elongations on internal surface, resulting from elongation measurements and FE calculations in cross-section in the circumferential direction, d Equivalent elongations on external surface, resulting from elongation measurements and FE calculations in cross-section in the circumferential direction

134

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

Fig. 6.12 (continued)

The values above of the weakening factor result from the simple static equilibrium in the circumferential direction relative to the opening environment, interpreted on the basis of Fig. 6.16a, b and Eq. 6.16. Based on Fig. 6.16a, b, the following static equilibrium equation in the circumferential direction can be stated in the environment of the opening delimited by the dimensions a0 and a1 :

6.2 Design of Openings on Cylindrical Shells and Pressure Vessel Ends

135

Fig. 6.13 Nozzle environment with strain gauges

p · A p = σ2localmembrane · Aσ

(6.16)

where A p indicates the projection surface loaded by pressure, and Aσ the area of the load bearing cross-section. The equivalent stress can be stated approximately in the form p σr ed ∼ = σ2localmembrane + 2

(6.17)

By substituting correlations 6.16–6.17 into the strength condition of dimensioning (σr ed ≤ σallowed ), the following Eq. 6.18 is yielded after rearrangement, which is the basic correlation of opening reinforcement calculation built on the area method.  p·

Ap 1 + Aσ 2

 ≤ σallowed

(6.18)

Therefore, it can be stated that all nozzle environments for which the irregularity above is complied with will be appropriate from the opening reinforcement aspect. The dimensions a0 and a1 , determining the size of the areas A p and Aσ , serving as a basis for calculations, was determined by experimentation, measuring elongations in the circumferential direction ε2 . By serial experimentations, it was established that the distances in question are determined by the damping length of the local bending stress (6.19a–b) caused by the force system in the connecting cross-section (see Fig. 6.9). a0 =

√

(Db + sk ) · sk

(6.19a)

136

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

Fig. 6.14 a–d Weakening factors of cylindrical shell and conical pressure vessel end in function of geometric conditions of nozzle environment

a1 = 1, 25 ·

√

(db + scs ) · scs

(6.19b)

6.2 Design of Openings on Cylindrical Shells and Pressure Vessel Ends

137

Fig. 6.15 a–c Development of spherical shell weakening factors in function of geometric conditions of nozzle environment

138

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

a

b

Fig. 6.16 a–b Structural models applied for defining weakening factor

By substituting the areas defined in Fig. 6.16a, b in Eq. (6.18), after rearrangement the correlations (6.15a–c) are yielded to define the required rim thickness. In case of a cylindrical shell (Fig. 6.16a), dependence of the weakening factor on geometric dimensions is shown by the following Eq. (6.20).

V =

+

scs sk

 · 1+

+ 0.5 ·

db Db

·

a0 sk a0 sk

+

scs sk

Db sk

a1 sk



 + 1+

a1 sk



·

db Db

(6.20a)

where a0 = sk / a1 = 1.25 · sk

/ 1+ scs sk



Db sk

 (6.20b)

scs db Db · + D b sk sk

 (6.20c)

It can be established on the basis of correlations (6.20a–c) that the numerical values of the weakening factor really depend on the geometric ratios below of the nozzle environment already presented.  V = f

sk scs db ; ; D b sk D b

 (6.21)

Correlations (6.20–6.21) actually result in the curves shown in Fig. 6.14a–d.

6.2 Design of Openings on Cylindrical Shells and Pressure Vessel Ends

139

By repeated application of the calculation method presented (iteration solution), the wall thickness values (sk ; scs ) pertaining to the given diameters are yielded as follows. 1.

Determining the required rim thickness (sk ) in case of fixed nozzle thickness (scs ). Initial data: constants : Db ; dk ; db ; scs ; s starting rim thickness value : sk (0) = s

1.1 Based on the skD(0) quotient, selecting the curve set to be used in the calculation b (see Figs. 6.14a–d and 6.15a–c). cs 1.2 After calculating factors Ddbb and sks(0) , the first approximating value V (1) of the weakening factor is yielded (intersection of parameter lines e1 and e2 on Figs. 6.14a and 6.15a). 1.3 In accordance with the geometry of the cylindrical shell and the pressure vessel end, the first approximation of rim thickness sk (1) results from Eqs. (6.15a–c). By using the calculated rim thickness, the iteration can be continued up to an arbitrary number of steps by repeating points 1.1, 1.2, and 1.3. It is proven that after 4 to 5 steps, the calculation method described converges to the rim thickness sk required for strength. If the calculated rim thickness is smaller than the wall thickness of the cylindrical shell or of the pressure vessel end sk < s, then no reinforcing plate is required to be welded in the nozzle environment (Fig. 6.10c). In this case, either the cylindrical shell or the pressure vessel end is of overstrength, meaning that the wall thickness (s) is much larger than required for strength (Eqs. 3.19, 3.24, and 3.28), or the wall thickness (scs ) of the nozzle welded in the opening properly reinforces it. If sk > √ s then the thickness of

h = sk − s, a reinforcing plate with a width of bt = max (Db + sk ) · sk ; 3 · sk is required to be applied (see Fig. 6.10c). In the reinforcing plate design it should be noted that its thickness should be h ≤ s. The dimensions of the reinforcing plate required for strength (h;bt ) can be changed by taking the following condition into consideration. h · bt ≤ h 1 · bt1 2.

(6.22)

Determining the required nozzle thickness (scs ) in case of fixed rim thickness (sk ). Initial data: constants : Db ; dk ; sk starting nozzle thickness value : scs (0).

140

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

2.1 Based on the Dskb quotient, selecting the curve set to be used in the calculation (Figs. 6.14a–d and 6.15a–c), and determining the weakening factor V = const. pertaining to rim thickness sk = const. from Eqs. (6.15a–c). 2.2 The intersection of the line V = const. and the curve of scss(0) parameter (see k Figs. 6.14b and 6.15b) provides a first approximation of nozzle wall thickness. cs (1) , from which scs (1) = 0, 5 · (dk − e1 · Db ). since e1 = Ddbb = dk −2·s Db By using the calculated nozzle thickness, the iteration can be continued up to an arbitrary number of steps by repeating point 2.2. Calculation convergence applies in this case as well, so the wall thickness (scs ) required for strength is yielded after 4 to 5 steps. Required length of the calculated nozzle thickness: lcs = 1.25 ·

√

(db + scs ) · scs

(6.23)

The dimensions required for strength (lcs ;scs ) of the nozzle welded in the opening can be changed by taking the following condition into consideration. lcs · scs ≤ lcs1 · scs1

(6.24)

In case of pressure vessel ends, it occurs frequently that nozzles are located close to each other. The calculation method presented can only be applied if the impact of adjacent nozzles to each other is negligible. This is satisfied if the shortest distance between nozzles (the ligament between two nozzles) (l), l≥

√ (Db + sk ) · sk

(6.25)

The strength control of the environment of nozzles closer than the distance determined by Eq. (6.25) is feasible by using the area method based on correlation (6.18), where the interpretation of the projection areas Aσ and A p is shown in Fig. 6.17.

6.3 Strength Tests of the Support Environment In support environments, mass forces from equipment charge and body can cause considerable additional load besides internal overpressure as primary load. The nature, distribution, and extent of such additional load primarily depends on support design, which may even affect the required wall thickness of the structure. Consequently, selection of the support design requires great circumspection, in the course of which the following should be taken into account: equipment load parameters (extent of overpressure, equipment mass), equipment shape (cylinder, sphere, etc.), position (vertical, horizontal), and support location (cylindrical part, pressure vessel end). It can be stated as a general rule that in case of a support on the cylindrical shell, it should be endeavoured to use a support that causes increased stress in the

6.3 Strength Tests of the Support Environment

141

Fig. 6.17 Interpretation of projection areas in case of nozzles closer than damping length

tangential direction if possible because in this direction there is nearly 50% stress reserve compared to the circumferential direction as a result of the overpressure of the internal medium. In this case, the wall thickness of the structure does not need to be increased by reason of mass forces as long as stress in the tangential direction (resultant of general membrane and local membrane stress) does not exceed the resultant membrane stress in the circumferential direction. In case of supports on pressure vessel ends and spherical tanks, designers cannot take the phenomenon above into account. Solution of the problem is helped by the fact that in determining the required wall thickness of the structure, the allowable stress was taken into consideration with a n = 1.5 safety factor compared to the yield stress of the engineering material. In the support environment, local stresses generated from the transfer of mass forces are damped within a relatively short distance, therefore a safety factor smaller than the yield stress can be stated for the resultant membrane stress in the course of dimensioning, so the allowable stress can be higher. Based on equipment position (vertical, horizontal) and shape, and location of the support, the following support designs have spread in practice: In case of vertical cylindrical equipment where the support is located on the cylindrical part of the equipment, • ring; • skirt; and • feet. supports can be applied.

142

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

If the support is on the pressure vessel end of vertical cylindrical equipment and in case of spherical tanks, feet can be applied along the equipment perimeter. For horizontal equipment, saddle support is to be used. Out of the supports listed, Fig. 6.18 shows the load transfer process in case of ring support, indicating the main geometric features and their signage. The support outlined on the figure is made of hollow sections in case of major loads, and of open segments in case of minor loads. The segments are pre-heated and bent to fit the cylindrical mantle radius, and then welded to each other and to the equipment along the perimeter. Therefore the support, as shown in the figure, forms a closed ring in itself, encircling the entire perimeter. In case of ring support, circular seating of the equipment can be ensured. It follows from the ring-type closed formation encircling the equipment that the support shares a significant part in bearing the distributed reaction force along the perimeter Ft of the circle with diameter dt . The shell is only loaded by edge forces and edge moments (Q A , M A , Q B , M B ) generated in the connecting cross-sections “A” and “B” of the ring, and below the support by edge force N1Mg in the tangential direction, resulting from mass force. Figure 6.19 shows the interpretation of the force and moment Q d , M ds distributed along the perimeter loading the ring, resulting from mass force and internal pressure, as defined by correlations (6.26a–b). The distributed force system interpreted according to the figure results in normal force N T and bending moment MT , causing normal stress in the tangential direction in the cross-sections of the ring support.

M ds =

Q d = p · h + (Q B − Q A )

(6.26a)

  d QA + QB M ·g·e d − (M A + M B ) · ·h· + ds · π ds 2 ds

(6.26b)

The shearing forces and bending moments generated in the connecting crosssections are determined by the interaction of the ring and the cylindrical part. Such interaction comes from the inhibited expansion of the shell and its angular displacement caused by the distributed moment loading the ring. In case of a ring crosssection of adequately high angular stiffness and small internal overpressure, the above shearing forces and bending moments (Q A , M A , Q B , M B ) are negligible. In this case, the cylindrical shell is only loaded in the cross-section below the support by local edge force N1Mg in the tangential direction, resulting from mass force transfer, as the rest of the loads are borne by the ring. For the above reasons, ring support is reasonable to apply mainly for atmospheric fluid tanks resulting in large mass force. The skirt support shown in Fig. 6.20 also results in favourable stress distribution in the cylindrical shell; it is generally located near the bottom pressure vessel end of vertical cylindrical equipment. The support is a reinforced circular cylinder with a base ring which is tightly fitted to the external equipment surface, connected to it by a circular weld. The support design results in favourable mass force transfer along the perimeter of the circular weld, generating stress increase mainly in the tangential

6.3 Strength Tests of the Support Environment

143

Fig. 6.18 Internal force system of ring support environment

direction in the connection environment. The shearing force and bending moment (Q A , M A ,) shown in the figure result from the inhibited expansion of the mantle caused by the skirt, while bending moment M Mg from the reduction of edge force N1Mg to the middle surface. In general, the edge forces and edge moments above do not cause considerable additional load in the equipment wall. The local axially symmetrical bending stresses caused by the supports shown in Figs. 6.18 and 6.20 and loading the equipment shell can be determined numerically on the one hand using the finite element method, and on the other hand by applying analytical correlations as discussed in Chap. 5.

144

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

Fig. 6.19 Load on closed ring encircling cylindrical shell

The equipment brackets shown in Figs. 6.21 and 6.22 are frequent support structures of vertical cylindrical equipment. Contrary to the support designs presented in the foregoing, support brackets are connected to the cylindrical part locally along the perimeter. It comes from the local connection that the reaction force Ft per bracket, resulting from the entire equipment mass, is transferred to the equipment mantle accompanied by a moment Mt = Ft · e. In general, z = 3 support brackets are placed to an angle of ϕ = 120◦ from each other along the perimeter, and more rarely z = 4 brackets in a ϕ = 90◦ angle position. It is not expedient to apply more support brackets because equal load bearing cannot be ensured among brackets due to static uncertainty. Basically the two types of bracket design widely used in practice include the one with vertical load input ribbing as shown in Fig. 6.21, and the one with horizontal load input ribbing as per Fig. 6.22, but it is not infrequent to apply so-called closed frame brackets made of hollow sections. Bracket load carrying capacity can be considerably increased by welding a bolster plate between the bracket structures shown in the figures above and the cylindrical shell, of the same material and thickness as those of the equipment shell. It can be stated irrespective of the bracket design that the impact of the bending moment Mt resulting from reaction force Ft arises on the shell in the form of a distributed force system along the line perpendicular to the surface operating in the median line of

6.3 Strength Tests of the Support Environment

145

Fig. 6.20 Internal force system of skirt support environment

load transfer ribbing (see Q(x1 ) in Fig. 6.21, and ±Q max in Fig. 6.22). Reaction force Ft is transferred in the form of distributed load impacting along the line of welded joints between the load transfer ribbing and the shell (see V (x1 ) in Fig. 6.21, and V a , V f in Fig. 6.22). Determination of the local stresses generated in the shell as a result of the load input above is discussed in detail in the literature [27–30, 33, 34], so it will not be presented here. A frequent method for horizontal cylindrical equipment support is shown in Fig. 6.23. The figure shows the structural outline of a saddle support along the seating surface of the saddle, by indicating the distributed surface load q N resulting from load transfer. In general, two saddles positioned symmetrically along the longitudinal axis serve for equipment support. Basically, the position of supports determines the extent of additional loads resulting from mass forces. For determining support locations, horizontal equipment is simply considered as a two support beam subject to distributed load. The optimal support location yielded by the support model is at a distance of L ∗ = 0.207L from the end cross-section. In practice, supports are placed

146

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

Fig. 6.21 Structural design and internal force system of bracket with vertical load input ribbing

Fig. 6.22 Structural design and internal force system of bracket with horizontal load input ribbing

6.3 Strength Tests of the Support Environment

147

Fig. 6.23 Saddle support of horizontal cylindrical equipment

a bit more outwards due to the cross-section factor reduction arising from oval deformation indicated by a broken line in Fig. 6.23. In order to accurately determine the stress state of the support environment, the finite element modelling of the entire structure is required in this case. An approximating solution for the problem can be yielded on the basis of analytical calculations in certain standards [42, 43].

6.3.1 Strength Tests of the Support Environment of Spherical Tanks In the course of the strength tests of the support environment of spherical tanks, the finite element calculation of the stress state generated in the feet environment is a difficult task. In addition to modelling the entire structure, it also requires to solve the contact problem between the spherical shell and the connecting bolster plate. In the course of engineering calculations, approximating solutions resulting from the assumption of rigid load transfer feet yield an approximation of the real situation with proper accuracy, so their current application is also justified and accepted. Development of the approximating calculation method to be presented was preceded by theoretical and experimental research work at the Mechanical Engineering Department of Manchester Institute of Science and Technology in the 1950s, giving rise to a number of publications in the subject [31, 32, 35–37]. The analytical calculation method disclosed in the publications examined the stress states generated in the support location environment of spherical tanks for the case of thin rib-like support structures. The calculation method mentioned above

148

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

can also be extended to the examination of load transfer caused by feet considered to be rigid, and connected by bolster plate insertion [38]. In this case, load transfer between the spherical shell and the feet can be modelled by rigidly installed cylindrical nozzles. In case of a structural element of arbitrary shape connected to the spherical shell, the impact of any arbitrary force transfer surface can be calculated by covering the contact area with elemental cylindrical nozzles. The distribution of reaction components (normal and shearing force, bending moment) to cylinders can be solved by using displacement conditions due to the rigidity of the connecting elemental cylinders. The resultant stress state is yielded by the aggregation of stresses caused by the loads impacting each cylinder. The determination of stresses caused by different nozzle loads is based on the theory of shallow spherical shells [31, 32]. Figure 6.24 shows an outline of the feet environment of the spherical tank examined, indicating loads resulting from the reaction force. As shown in the figure, the feet are connected to the spherical shell by the interposition of a bolster plate delimited by an ABCD square. Such connection is ensured by welded joints along the lines AB, BC, C D, and D A. By reducing the per feet load indicated by Ft , impacting along diameter Dt shown in the figure, to median line E E , pertaining to basic circle diameter Ds , normal force FN , shearing force FV , and bending moment Mt are yielded, which can be calculated by correlations 6.27a–c.  π FN = Ft · sin ϑ0 − 2  π FV = Ft · cos ϑ0 − 2 Mt = Ft ·

(Ds − Dt ) 2

Fig. 6.24 Force system loading the shell in the feet environment

(6.27a) (6.27b) (6.27c)

6.3 Strength Tests of the Support Environment

149

Fig. 6.25 Model of nozzle loads caused by reaction force

According to the model assumption, the shell is loaded by normal force FN and bending moment Mt along the entire bolster plate surface, and by tangential shearing force FV along lines AB, BC, C D, and D A, through the welded joints. The model according to Fig. 6.25 is yielded by approximating the entire surface of the bolster plate of dimensions (2l1 , 2l2 ) by rigid load transfer cylinders, then by substituting normal force FN and bending moment Mt with radial nozzle loads FN k and FMk , impacting in the median line of elemental cylinders. The figure shows a case of approximation by 4 load transfer cylinders.

Distribution of Normal Force FN to Rigid Load Transfer Cylinders In a general case, by covering the entire surface (2l1 , 2l2 ) of the bolster plate by an i number of nozzles of radius r0 and by approximating nozzle load distribution with Fourier series, and furthermore by assuming that middle surface displacement in the normal direction is constant (w(x1 ) = const) under the bolster plate, the following are yielded: In case of an even number of nozzles (i = 2m), the load function in the normal direction will be:

150

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments i−2

2 

FN (x1 ) =

F N j cos

j=0

j · π · x1 2l1

(6.28a)

in case of an odd number of nozzles (i = 2m + 1): i−1

2 

FN (x1 ) =

F N j cos

j=0

j · π · x1 2l1

(6.28b)

Loads on each nozzle:   FN k = FN x1,k

(6.29)

The numerical values of Fourier coefficients F N j are provided by the following boundary conditions:       w x1,1 = w x1,2 = · · ·· = w x1,i i 

  FN x1,k = FN

(6.30a)

(6.30b)

k=1

Determining Nozzle Loads Caused by Bending Moment Mt Assuming that middle surface displacement in the normal direction changes according to the function w(x1 ) = k1 · x1 under the bolster plate, in case of an even number of nozzles (i = 2m), the load function in the normal direction will be: i−2

FM (x1 ) =

2 

F M( j+1) sin

j=0

( j + 1) · π · x1 2l1

(6.31a)

in case of an odd number of nozzles (i = 2m + 1): i−3

FM (x1 ) =

2 

j=0

F M( j+1) sin

( j + 1) · π · x1 2l1

(6.31b)

Loads on each nozzle:   FMk = FM x1,k

(6.32)

6.3 Strength Tests of the Support Environment

151

The numerical values of Fourier coefficients F M( j+1) are provided by the following boundary conditions:

i 

  w x1,k x1,k  = x1,1 w x1,1

(6.33a)

  FM x1,k · x1,k = Mt

(6.33b)

k=1

Distribution of Tangential Force FV to Bolster Plate Fillet Welds Tangential shearing force FV loads the shell along welded joints AB, BC, C D, and D A. Based on the foregoing, by substituting such welded joints with a series of cylinders located close to each other, and by approximating shearing force along longitudinal welds AB, C D with the following function (6.34), the forces shown in Fig. 6.26 are yielded. FV (x1 ) = F V 0 −

2  j=1

F V j sin

j · π · x1 2l1

(6.34)

Assuming that along bolster plate side welds AB and C D, middle surface displacement in the normal direction changes according to the function w(x1 ) = k2 · x1 + c, the numerical values of coefficients F V 0 and F V j are provided by the following boundary conditions:       l1 l1 − w(x1 = 0) = w x1 = − w(x1 = 0) 2 w x1 = 4 2     l1 − w(x1 = 0) = w(x1 = l1 ) − w(x1 = 0) 2 w x1 = 2 ⎤ ⎡ +l1 2  j · π · x 1 ⎦d x1 + 4l2 · F V 0 F V j sin FV = 2 ⎣ F V 0 − 2l 1 j=1 −l1

(6.35a) (6.35b)

(6.35c)

152

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

Fig. 6.26 Distribution of shearing force to bolster plate fillet welds

Stress State Caused by Rigidly Installed Cylindrical Nozzle Loaded by Normal Force Analytical solution of the problem shown in Fig. 6.27 is provided by applying the theory of axially symmetrical bending of axially symmetrical shells [32] to shallow spherical shells. Edge forces and edge moments generated on the spherical shell as a result of load N1 (r ) =

1 dF r dr

(6.36a)

N2 (r ) =

d2 F dr 2

(6.36b)

6.3 Strength Tests of the Support Environment

153

Fig. 6.27 Model of rigidly installed nozzle loaded by normal force

 M1 (r ) = B  M2 (r ) = B

d 2 w ν dw + dr 2 r dr



d 2w 1 dw +ν 2 r dr dr

Q 1 (r ) = B ·

d (Δw) dr

(6.36c)  (6.36d) (6.36e)

where: is the so-called stress function, w is middle surface displacement in the normal direction x3 ; is shell bending stiffness;

F E·s 3 12(1−ν 2 ) 2 Δ(· · · ) = d dr(···2 )

B=

+

1 d(··· ) r dr

is a differential operator.

Equations to define the stress function and middle surface displacement in the normal direction: ΔΔF −

E ·s Δw = 0 R

(6.37a)

ΔΔw +

1 ΔF = 0 B·R

(6.37b)

/ √ 2 By introducing the dimensionless coordinate r˜ = 2 · β · r , β = 4 3(1−v) R 2 ·s 2 the problem requires the solution of a Bessel-type differential equation of complex variables, resulting in the following correlations for the quantities sought for: π π · ψ3 − c4 · · ψ4 + c5 + c7 · ln(˜r ) 2 2  E · s2 π F(˜r ) = /   −c2 · ψ1 − c1 · ψ2 + c3 · 2 · ψ4 12 1 − ν 2

w(˜r ) = c1 · ψ1 − c2 · ψ2 − c3 ·

154

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

−c4 ·

 π · ψ3 + c6 + c8 · ln(˜r ) 2

(6.38a–b)

where ψi = ψi (˜r ) (i = 1, 2, 3, 4) indicates the Schleicher functions defined by Eqs. 6.41–6.42. Boundary conditions to determine integration constants (c1 − c8 ): • for large values of r˜ , w(˜r ) = 0 from which c1 = c2 = c5 = c7 = 0; • In stress function F(˜r ), c6 = 0; • for large values of r˜ , M1 = M2 = Q 1 = 0, therefore edge force in the tangential direction N1 maintains equilibrium with loading force FN , from which

c8 = − • in case of r = r0 ,

|

dw(r ) | dr |r =r

/   FN · R 12 1 − ν 2

(6.39a)

2π · E · S 2

= 0; ε2 (r = r0 ) = 0, from which 0

, ψ40 (6.39b) , ψ30 /   , 2FN · R · ψ30 (1 + ν) · 3 1 − ν 2    (6.39c)  c4 =  , , ,2 ,2 E · S 2 · π 2 r˜0 r˜0 ψ30 · ψ40 − ψ40 · ψ30 − (1 + ν) ψ30 + ψ40

c3 = −c4

where , ψi0 =

|

dψi (˜r ) | d r˜ |r˜ =˜r

, ψi0 = ψi (˜r0 ), r˜0 =

√

2 · β · r0 (see Eqs. 6.41–6.42).

0

In the knowledge of the boundary condition constants above, the quantities sought for will be: π w(˜r ) = − [c3 · ψ3 + c4 · ψ4 ] (6.40a) 2   2 · c8 E ·s·π , , c3 · ψ4 − c4 · ψ3 + (6.40b) N1 (˜r ) = 2R · r˜ π · r˜   2 · c8 ψ3 · r˜ + ψ4, ψ , − ψ4 · r˜ E ·s·π (6.40c) −c3 + c4 3 − N2 (˜r ) = 2R r˜ r˜ π · r˜ 2   ψ3 · r˜ + (1 − ν) · ψ4, E · s2 · π (1 − ν) · ψ3, − ψ4 · r˜ /  + c4 M1 (˜r ) =  c3 r˜ r˜ 2R 12 1 − ν 2 

ν · r˜ · ψ4 + (1 − ν) · ψ3, ν · r˜ · ψ3 − (1 − ν) · ψ4, /  + c −c M2 (˜r ) = 4 3  r˜ r˜ 2R 12 1 − ν 2 E · s2 · π



(6.40d) (6.40e)

6.3 Strength Tests of the Support Environment

155

The Schleicher functions and their derivatives included in these correlations above: r˜ 8 r˜ 12 r˜ 4 + − + ... 2 2 (2 · 4) (2 · 4 · 6 · 8) (2 · 4 · 6 · 8 · 10 · 12)2 4 · r˜ 3 8 · r˜ 7 12 · r˜ 11 dψ1 (˜r ) =− ψ1, (˜r ) = + − + ... 2 2 d r˜ (2 · 4) (2 · 4 · 6 · 8) (2 · 4 · 6 · 8 · 10 · 12)2 r˜ 6 r˜ 10 r˜ 2 ψ2 (˜r ) = − 2 + − + ... (2) (2 · 4 · 6)2 (2 · 4 · 6 · 8 · 10)2 2 · r˜ 6 · r˜ 5 10 · r˜ 9 dψ2 (˜r ) =− 2 + − + ... ψ2, (˜r ) = d r˜ (2) (2 · 4 · 6)2 (2 · 4 · 6 · 8 · 10)2     r˜ 2 1 R1 + C + ln ψ2 (˜r ) ψ3 (˜r ) = ψ1 (˜r ) − 2 π 2     r˜ r 2 1 dψ 1 (˜ ) 3 , , , , R + ψ2 (˜r ) C + ln + ψ2 (˜r ) · ψ3 (˜r ) = = ψ1 (˜r ) − d r˜ 2 π 1 2 r˜     r˜ 2 1 R2 + C + ln ψ1 (˜r ) ψ4 (˜r ) = ψ2 (˜r ) + 2 π 2     r˜ 1 r 2 1 dψ (˜ ) 4 , , , , ψ4 (˜r ) = = ψ2 (˜r ) + R + ψ1 (˜r ) C + ln + ψ1 (˜r ) · d r˜ 2 π 2 2 r˜ (6.41a–h) ψ1 (˜r ) = 1 −

where  2 1 + 1 + 1 + 1 + 1  10 1 + 1 + 1  6 r˜ r˜ r˜ − 1 2 23 + 1 2 3 42 5 − ···· 2 2 2 (2 · 3) (2 · 3 · 4 · 5)     5 1 + 1 + 1 1 d R1 (˜r ) r˜ r˜ = − 1 2 23 ·6 = 2 d r˜ 2 2 2 (2 · 3)   9 1 + 1 + 1 + 1 + 1 r˜ + 1 2 3 4 2 5 · 10 − ···· 2 (2 · 3 · 4 · 5) 1 + 1  4 1 + 1 + 1 + 1  8 r˜ r˜ = 1 22 − 1 2 3 2 4 2 2 (2) (2 · 3 · 4) 1 + 1 + 1 + 1 + 1 + 1  12 r˜ − ···· + 1 2 3 4 52 6 2 · 3 · 4 · 5 · 6) (2   3  7 1 + 1 + 1 + 1 1 11 + 21 r˜ d R2 (˜r ) 1 2 3 4 · 8 r˜ = · 4 − = d r˜ 2 (2)2 2 2 (2 · 3 · 4)2    1 + 1 + 1 + 1 + 1 + 1 r˜ 11 − ···· (6.42a–d) + 1 2 3 4 52 6 · 12 2 (2 · 3 · 4 · 5 · 6)

R1 = R1,

R2

R2,

C = 0.57722 is the so-called Euler’s number.

156

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

Stress State Caused by Rigidly Installed Cylindrical Nozzle Loaded by Shearing Force The basic correlations of general bending theory [31, 32, 35–37] serve as a starting point for determining the stress state caused by the nozzle loaded by shearing force as shown in Fig. 6.28. In this case as well, similarly to the previous section, determination of the edge forces and edge moments generated in the shallow spherical shell can be traced back to specifying the so-called stress function F and the displacement in the normal direction x3 of middle surface w as follows. Edge forces and edge moments on the spherical shell resulting from load: N1 (r, ϕ) =

1 ∂2 F 1 ∂F + 2 r ∂r r ∂ϕ 2

(6.43a)

∂2 F ∂r 2

(6.43b)

N2 (r, ϕ) =

  ∂ 1 ∂F N12 (r, ϕ) = N21 (r, ϕ) = − ∂r r ∂ϕ    2 1 ∂ 2w 1 ∂w ∂ w + M1 (r, ϕ) = B + ν ∂r 2 r ∂r r 2 ∂ϕ 2   1 ∂ 2w ∂ 2w 1 ∂w + 2 +ν 2 M2 (r, ϕ) = B r ∂r r ∂ϕ 2 ∂r

(6.43c)

(6.43d)

(6.43e)

Equations independent of each other for determining stress function (F) and the displacement in the normal direction x3 of middle surface (w): Fig. 6.28 Model of rigidly installed nozzle loaded by shearing force

6.3 Strength Tests of the Support Environment

157

ΔΔΔw +

E ·s Δw = 0 R2 · B

(6.44a)

ΔΔΔF +

E ·s ΔF = 0 R2 · B

(6.44b)

where Δ(· · · ) =

∂ 2 (··· ) ∂r 2

1 ∂ 2 (··· ) r 2 ∂ϕ 2 E·s 3 B = 12 1−ν 2 ( )

+

1 ∂(··· ) r ∂r

+

is a differential operator; is shell bending stiffness.

Due to load symmetry, by seeking for the displacement in the normal direction x3 of middle surface (w) and stress function (F) in the form of the functions below: w(r, ϕ) =

∞ 

wi (r ) · cos(i ϕ)

(6.45a)

Fi (r ) · cos(iϕ)

(6.45b)

i=1

F(r, ϕ) =

∞  i=1

then by taking into consideration specifically only one member from the Fourier series and substituting it into the equations above, the following solution is yielded.  π π c6  cos ϕ w(r, ϕ) = c1 · ψ1, − c2 · ψ2, − c3 · · ψ3, − c4 · · ψ4, + c5 · r˜ + 2 2 r˜   π c12 π cos ϕ F(r, ϕ) = c7 · ψ1, − c8 · ψ2, − c9 · · ψ3, − c10 · · ψ4, + c11 · r˜ + 2 2 r˜ (6.46a–b) where ψi, = dψdir˜(˜r ) (i = 1, 2, 3, 4) denotes the derivatives of Schleicher functions as defined by Eqs. 6.41–6.42. In this case, out of boundary condition constants (c1 − c12 ): • c1 = c2 = c5 = c6 = c7 = c8 = c11 = 0; • c9 = c4 ; c10 = −c3 ;

• c12 = •

/   R · β 6 1 − ν2 E · s2 · π

, c4 r˜0 · ψ40 − ψ30 = , ; c3 r˜0 · ψ30 − ψ40

FV √

r2 R2 − r 2

;

(6.47a) (6.47b)

158

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

/   R · β · 4 · (1 + ν) 6 1 − ν 2 r2 • c3 =   FV √   ; R2 − r 2 E · s 2 · π 2 · r˜03 cc43 f˜4 + g˜ 3 + f˜3 − g˜ 4

(6.47c)

where

, ψi0 =

ψ , · r˜0 − (1 + ν) · ψ30 ; f˜3 = 30 r˜0

(6.48a)

ψ , · r˜0 − (1 + ν) · ψ40 ; f˜4 = 40 r˜0

(6.48b)

g˜ 3 =

, 2(1 + ν)ψ30 ; r˜02

(6.48c)

g˜ 4 =

, 2(1 + ν)ψ40 ; r˜02

(6.48d)

| √ dψi (˜r ) || , ψi0 = ψi (˜r0 ), r˜0 = 2 · β · r0 , (i = 3, 4). | d r˜ r˜ =˜r0

In case of small nozzle size (r0 ) (welded joint model), boundary condition constants result from the following correlations: • c4 = 0;

• c3 =

(6.49a)

R·β ·

/   6 1 − ν2

E·

s2

• c12 = c3 .

·π

FV √

r2 R2 − r 2

;

(6.49b) (6.49c)

In the knowledge of the boundary condition constants above, the quantities sought for will be:  π c3 · ψ3, + c4 · ψ4, cos ϕ 2       ψ3 ψ4 4 c12 2ψ , 2ψ , E ·s·π cos ϕ c3 + 24 + c4 − 23 + N1 (˜r , ϕ) = − 2R r˜ r˜ r˜ r˜ π r˜ 3    ψ3 2ψ , E ·s·π −c3 − ψ3, + 24 N2 (˜r , ϕ) = − 2R r˜ r˜   , ψ4 4 c 2ψ 12 cos ϕ + ψ4, + 23 − +c4 − r˜ r˜ π r˜ 3 N12 (˜r , ϕ) = N21 (˜r , ϕ) = N1 (˜r , ϕ) tan ϕ w(˜r , ϕ) = −

6.3 Strength Tests of the Support Environment

159

   2(1 − ν)ψ3, E · s2 · π (1 − ν)ψ4 , / + ψ4 + M1 (˜r , ϕ) =  c3 −  r˜ r˜ 2 2R 12 1 − ν 2   2(1 − ν)ψ4, (1 − ν)ψ3 +c4 cos ϕ − ψ3, + + r˜ r˜ 2    2(1 − ν)ψ3, E · s2 · π (1 − ν)ψ4 , /  + ν · ψ4 − M2 (˜r , ϕ) =  c3 r˜ r˜ 2 2R 12 1 − ν 2   2(1 − ν)ψ4, (1 − ν)ψ3 , − νψ3 − − cos ϕ (6.50a–b) +c4 − r˜ r˜ 2 ψi = ψi (˜r ), ψi, =

dψi (˜r ) , d r˜

(i = 3.4) (see Eqs. 6.41–6.42).

Numerical Tests and Results Using the analytical model disclosed in the previous sections, numerical tests were performed on the following geometric and load data [38]. External diameter of the spherical tank examined: Dk = 12,450 mm; Wall thickness of the spherical tank: s = 16 mm; Number of feet: z = 9; Angle coordinate in the meridian direction of central load input: ϑ0 = 97◦ ; Feet circle diameter: DT = Ds = 12,357 mm; Load input size in the meridian direction (bolster plate): 2l1 = 1900 mm; Load input size in the circumferential direction (bolster plate): 2l2 = 500 mm; N Internal pressure: p = 0.94 mm 2;

Total equipment mass: Mo¨ = 1,092,492 kg; Load per feet from equipment mass FT = 1,190,816 N. Normal force from load per feet:

160

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

 π = 1,190,816 · sin 7◦ = 145,124 N; FN = FT · sin ϑ0 − 2 Tangential force from load per feet:  π = 1,190,816 · cos 7◦ = 1,181,940 N; FV = FT · cos ϑ0 − 2 Using the data above, first the stress state caused by normal load FN = 145,124 N was determined. Out of the calculation results,Fig. 6.29  shows the development of resultant stresses in the tangential direction σ1K ,B and in the circumferential   direction σ2K ,B in the line of welded joint BC on the external (K) and internal (B) surfaces of the spherical shell.   τ12K  Figure  6.30 shows normal and shear stresses   ,B in the tangential direction σ1K ,B and in the circumferential direction σ2K ,B , also in the line of welded joint BC on the external (K) and internal (B) surfaces of the spherical shell, as a result of tangential load FV = 1,181,940 N. Finally, Figs. 6.31 and 6.32 show the final results of the calculation that takes into account the joint effect of internal pressure andfeet load.  The figures show the developments of Tresca Mohr’s equivalent stresses σeK ,B on the external (K) and internal (B) surfaces of the spherical shell. Based on the results, it can be established that effective stresses are generated (dangerous cross-section) at the bottom corner points “B” and “C” of the bolster plate, which coincides with the statements in the relevant literature [31, 32, 35–37].

Fig. 6.29 Stresses in the tangential and circumferential directions caused by normal force along welded joint BC

6.3 Strength Tests of the Support Environment

161

Fig. 6.30 Stresses in the tangential and circumferential directions and shear stress along welded joint BC, caused by shearing force

Fig. 6.31 Equivalent stresses on external and internal surfaces along welded joint BC in support environment

162

6 Strength Tests of Pressure Vessel Ends, Nozzle and Support Environments

Fig. 6.32 Equivalent stresses on external and internal surfaces in support environment

References 1. Watts, G.W., Burrows, W.R.: The basic elastic theory of vessel heads under internal pressure . Trans. ASME, J. Appl. Mech. 55 (1949) 2. Galletly, G.D.: Bending of 2:1 and 3:1 open-crown ellipsoidal shells. Weld. Res. Counc. Bull. 54 (1959) 3. Kraus, H., Bilodeau, G.G., Langer, B. F.: Stresses in thin-walled pressure vessels with ellipsoidal heads. Trans. ASME 83(2)(1), 29–42 (1961) 4. Kraus, H.: Elastic stresses in pressure vessel heads. Weld. Res. Counc. Bull. 129 (1968) 5. Shield, R.J., Drucker, D.C.: Limit strength of thin–walled pressure vessels with an ASME Standard torispherical head. In: Proceedings of 3rd U.S. National Congress for Applied Mechanics, ASME, pp. 665–672 (1958) 6. Shield, R.J., Drucker, D.C.: Design of thin-walled torispherical and toriconical pressure vessel heads. J. Appl. Mech. 28, 292–297 (1961) 7. Adachi, J., Benicek, M.: Buckling of torispherical shells under internal pressure. Exp. Mech. 217–222 (1964) 8. Galletly, G.D.: A simple design equation for preventing buckling in fabricated torispherical shells under internal pressure. J. Press. Vessel Technol. 108, 521–525 (1986) 9. AD- Merkblätter B3 1995 július 10. MSZ 13822/4-80 11. Siebel, E., Schwaigerer, S.: Die Berechnung von Kesselböden. BWK 2, 37 (1950) 12. Esslinger, M.: Statische Berechnung von Kesselböden, Berlin, Göttingen. Springer, Heidelberg (1952) 13. Wellinger, K., Krägeloh, E.: Versuche an dünnwandigen Klöpperböden, Techn. Mitt. Deutscher Behälter-Verb. (1965) 14. Siebel, E., Schwaigerer, S.: Beanspruchungsverhältnisse in Mannloch-Böden, Techn. Mitt. GWK-Verb. (1952) 15. Siebel, E.: Über die Festigkeitsverhältnisse von Böden mit Aushalsungen, Mitt. GWK-Verb. (1955) 16. PennY, R.K., Leckie, F. A.: Solutions for the stresses at nozzles in pressure vessels. Weld. Res. Counc. Bull. 90 (1963) 17. Rose, R.T.: New design method for pressure vessel nozzles. The Engineer 124, 90–93 (1962)

References

163

18. Taylor, C.E., Lind, N.C., Schweiker, J.W.: A three dimensional photoelastic study of stresses around reinforced openings in pressure vessels. Weld. Res. Counc. Bull. 51 (1959) 19. Leckie, F.A., Penny, R.K.: Stress concentration factors for the stresses at nozzle intersections in pressure vessels. Weld. Res. Counc. Bull. 90 (1963) 20. van Dyke, P.: Stresses about a circular hole in a cylindrical shell. Technical Report No. 21, Division Engineering Harvard University (1964) 21. Cloud, R.L.: The limit pressure of radial nozzles in spherical shells. Nucl. Struct. Eng. 1, 403–413 (1965) 22. Dinno, K.S., Gill, S.S.: Limit pressure for a protruding cylindrical nozzle in a spherical pressure vessel. J. Mech. Eng. Sci. 7(3), 259–270 (1965) 23. Calladine, C.R.: On the design of reinforcement for openings and nozzles in spherical pressure vessels. J. Mech. Eng. Sci. 8(1), 1–14 (1966) 24. Siebel, E., Hauser, H.: Versuche über die Beanspruchung von Zylindern mit eingeschweißten Stutzen. Techn. Mitt. GWK-Verb. (1955) 25. Siebel, E., Schwaigerer, S.: Untersuchungen über das Festigkeitsverhalten ausgehalster Abzweigstücke. VRB-Mitt. WKV (1960) 26. Wellinger, K., Krägeloh, E., Beckmann, G.: Innendruckversuche an dickwandigen Stutzenrohren. Techn. Mitt. WKV (1960) 27. Bijlaard, P.P.: Stress from local loadings in cylindrical pressure vessels. J. Appl. Mech. 77(6), 805 (1955) 28. Bijlaard, P.P.: Stresses from radial loads in cylindrical pressure vessels. Weld. J. Res. Supp. 33, 615 (1954) 29. Bijlaard, P.P.: Stresses from radial loads and external moments in cylindrical pressure vessels. Weld. J. Res. Supp. 34 (1955) 30. Bijlaard, P.P.: Additional data on stresses in cylindrical shells under local loading. Weld. Res. Counc. Bull. 50 (1959) 31. Bijlaard, P.P.: Stresses in spherical vessels from radial loads and external moments acting on a pipe. Weld. Res. Counc. Bull. 49 (1959) 32. Bijlaard, P.P.: Stresses in spherical vessels from local loads transferred by a pipe. Weld. Res. Counc. Bullet. 50 (1959) 33. Nabil Ibri, G.: Készülékpata okozta feszültségek meghatározása. Kandidátusi értekezés, MTA, Budapest (1977) 34. Józsa, I.: Hengeres héjak er˝obevezetési problémái. M˝uszaki doktori értekezés. Budapest (1986) 35. Kitching, R., Olsen, B.E.: Pressure vessel support brackets: stresses due to dead loads. J. Strain Anal. 2(1), 1 (1967) 36. Kitching, R., Olsen, B.E.: Discrete tubular supports on pressure vessels. J. Strain Anal. 2(1), 17 (1967) 37. Kitching, R., Olsen, B.E.: Pressure stresses at discrete supports on spherical shells. J. Strain. Anal 2(4), 298 (1967) 38. Nagy, A.: Gömbtartályok támaszhelyeinek szilárdsági és konstrukciós értékelése, Diplomaterv. BME (1986) 39. Flügge, W.: Statik und Dynamik der Schalen. Springer, Berlin, Göttingen, Heidelberg (1957) 40. AD-Merblätter B9 1995 július 41. MSZ 13822/8-80 42. AD-Merblätter S 3/2 (1995) 43. MI 13822/16-84

Chapter 7

Strength Test and Dimensioning on Leak Tightness of Flange Joints

Abstract Engineering design of pipeline and equipment flange joints. Internal loads of flange structure. Flange joint operation under simultaneous heat and mechanical load. Stress state of the shell connected to the flange ring. Determining gasket load drop in case of ideally elastic gasket model. Determining gasket load drop in case of non-linear gasket model. Viscoelastic model for determining gasket load reduction. Tests and calculations to verify the model. Tests to verify the accuracy of the viscoelastic gasket model. Numerical examination of the loosening process caused by internal pressure. Dimensioning on leak tightness of flange joints exposed to simultaneous heat and mechanical load. Keywords Flange joints · Gaskets · Internal loads · Viscoelastic gasket model · Gasket load reduction · Loosening factor

7.1 Engineering Design of Pipeline and Equipment Flange Joints The basic materials, semi-finished and finished products of numerous chemical and food industry technologies are corrosive mediums of fluid or gaseous state, of high pressure and high temperature. Technological mediums are transported and stored in pipelines and pressure vessels. It is required for the ease of transport, installation, and cleaning that the structural parts of the equipment (pipe sections, fittings, and pressure vessel ends) should be connected to each other by releasable joints. Flange joints are the most widespread releasable joints of pipelines and tanks. In line with requirements, a number of flange designs have been developed in practice, the common components of which include gaskets made of soft metal or non-metallic engineering material, bolt fittings made of high-strength steel, and flange sections (see Fig. 7.1a–c). The main function of the flange structure consisting of bolt fitting, gaskets and flange sections is to ensure hermetic closure at the given pressure and temperature, and consequently to safely bear the forces and moments required for closure. The bolt and gasket load required for hermetic closure give rise to considerable additional loads in the flange section (flange ring, conical neck, co-working part of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Nagy, Fundamentals of Tank and Process Equipment Design, Foundations of Engineering Mechanics, https://doi.org/10.1007/978-3-031-31226-7_7

165

166

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

connecting cylindrical part of damping length) compared to other equipment parts. Such additional load is the reason for the mandatory strength control of flange joints, requirements for which are included in national standards and technical directives [1–3]. In terms of structural design, the joint may consist of weld-on flanges (Fig. 7.1a), welding neck flanges (Fig. 7.1b), and loose flanges (Fig. 7.1c), while in terms of the implementation method of leak tightness, conditionally releasable and releasable joints are involved. In case of releasable joints, the gasket load required is generally provided by bolts or other tensioners. Also, there are solutions where some part of the gasket load required for hermetic closure comes from the pressure of the medium to be sealed (self-sealing joint). Figure 7.2 shows an example of a conditionally releasable joint where leak tightness is ensured by welded membrane gasket rings. a

b

Fig. 7.1 a Weld-on flange joint. b Welding neck flange joint. c Loose flange joint

7.1 Engineering Design of Pipeline and Equipment Flange Joints

167

c

Fig. 7.1 (continued)

Fig. 7.2 Flange joint with welded membrane gasket ring

The advantage of the solution shown in Fig. 7.2 is the relatively small bolt force requirement, and a disadvantage is that the joint can be released by cutting the welded joints of metal rings (destruction). Releasable joints without gaskets are applied in practice only infrequently, in case of extreme loads (high temperature and pressure; flange joints of high-pressure hot steam pipelines). Leak tightness requires good

168

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

Fig. 7.3 a Leak tightness by line contact. b Leak tightness by double surface contact

surface quality (mirroring) and high gasket load, which can be produced by line contact (Fig. 7.3a) and surface contact (Fig. 7.3b). In both cases, gasket pressure along the contact zone is provided by the so-called Hertz stress (σh ) of metal contact. In case of releasable joints with gaskets, the main role of gaskets between gasket surfaces is to equalize the dislevelments of gasket surfaces by elastic or elastic–plastic elongation (see Fig. 7.4), hermetic closing with possibly small restraint load. Requirements for gasket materials: • • • •

good elongation ability (in respect of compression and spring back); good strength properties (pressure resistance); heat resistance; resistance to chemicals.

Most frequently applied materials which comply with the requirements above: • for soft gaskets: cellulose, paper, cartoon soaked with oil, (asbestos∗ ), rubber, artificial rubber, Neoprene, Thioplast, silicone; Fig. 7.4 Optimal gasket operation (filling gasket surface dislevelments)

7.1 Engineering Design of Pipeline and Equipment Flange Joints

169

Fig. 7.5 Spiral—“asbestos” gasket

• for hard gaskets: lead, aluminium, soft copper, steel (in case of very high pressure and temperature). Gaskets applied in joints are previously selected in the knowledge of the highest temperature (Tmax ). and the highest pressure ( pmax ), based on a so-called ( p − T ). diagram and a so-called “corrosion table” indicating resistance to charge. Such gasket selection does not substitute for the sealing engineering dimensioning of joints. The expectations of good elongation ability and proper strength can be simultaneously provided by applying so-called multi-material gaskets. Multi-material gaskets are made by associating non-metallic soft materials with good elongation ability, thus aduately fling the dislevelments of the gasket surface, and metal inlets or metal covers of appropriate strength properties. Among multimaterial gaskets, the so-called “spiral asbestos” gasket (today already asbestos-free, of course) shown in Fig. 7.5 should be mentioned first, which is mainly applied for sealing gaseous mediums. * Not applied any longer due to its health impairment effects. As shown in the figure as well, a gasket consists of a profiled metal band and a soft space filler material in between, which is closely wound up on a drum of the required diameter in the course of manufacturing and fixed by spot welding both on the inside and the outside. In the present case, the otherwise radial leakage route will be a helix in the tangential direction as defined by the metal band pressed into the gasket surface. This is why it is applied mainly for sealing gaseous mediums. The gasket wound up in a spiral form is protected from destruction by being overstressed by external and internal support rings. Multi-material gaskets also include the corrugated gasket and the gasket with metal cover shown in Figs. 7.6 and 7.7a, b. The external metal cover also ensures heat and chemical resistance to the medium to be sealed. The shape of the gasket applied (flat, profiled) also determines the front surface of the flange ring.

170

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

Fig. 7.6 Gasket reinforced by metal band

a

b

Fig. 7.7 a Open metal cladding. b Closed metal cladding

As regards flat gaskets, in case of low pressures, flat raised face gasket surfaces are applied as outlined in Fig. 7.1a, c, while in case of higher pressures groove-tongue gasket surfaces as per Fig. 7.1b are usually applied. As regards profiled gaskets, the socalled joint ring gasket of large load carrying capacity should be highlighted together with the self-sealing convex gasket. The shapes of the gasket surfaces pertaining to the gaskets mentioned—together with the gaskets installed—are shown in Figs. 7.8 and 7.9. Fig. 7.8 Joint ring gasket structure

7.2 Internal Loads of Flange Structure

171

Fig. 7.9 Convex gasket design

The high load carrying capacity of the joint ring gasket construction can be explained by the nearly hydrostatic stress state in the gasket enclosed by the gasket surface of special shape. In the course of the sealing engineering and strength examination of the joint, the following loads and characteristic load states to impact the flange structure are required to be taken into consideration. As regards load: • the bolt force (Fc ). and gasket load (FG ) required for hermetic closure, • the internal pressure to be sealed ( p), • the temperature that causes internal forces and moments due to the difference in heat expansion of the bolt fitting and the flange section, on the one hand, and may cause a deterioration in mechanical features (yield stress, tensile strength) on the other hand, including laxation, creeping and relaxation of non-metallic gaskets, • any chemical impact, which may result in a reduction of the load-bearing crosssection in case of inappropriate material section. In respect of load state: • installation state ( p = 0; FC0 = FG0 ; T = 20 ◦ C),  • trial and test state p = p pr ; FC pr /= FGpr ; T = 20 ◦ C , • operational state ( p = pU ; FCU /= FGU ; T = TU /= 20 ◦ C). For further examination, it is required to know internal forces, which can be most simply determined by the equilib.ium of the flange section connected to both the bolt fitting and gaskets. Operational and test states can be collectively treated as pressurized states.

7.2 Internal Loads of Flange Structure The research of flange structures dates back to more than 100 years and goes on in our days as well, except for minor interruptions, revealing more and more details [4–7]. The first publication on flange joints is linked with the name of Bach [8]; and afterwards, in the period between 1930 and 1940, the research team led by Waters,

172

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

Rossheim, Westrom and Williams developed the calculation process [9–11] which forms the basis of numerous national standards even today [1, 2]. In the course of their deductions, they only assumed elastic elongations and stresses in the flange section. They did not take into consideration internal pressure as surface distributed load, they only took account of the moment caused by the bolt and the gasket load. Quantities used in the course of calculations were referred to the internal flange surface, and radial displacement was neglected at the connection of the flange ring and the cylindrical shell. For the welding neck flange, displacement of the conical shell was sought for in the form of a polynomial that approximately satisfied boundary conditions, instead of solving the relevant differential equation. The unknown coefficients of the polynomial were provided by the minimization of conical neck strain energy. Solution of the problem was specified in a dimensionless form for the ease of treatment, in curve sets and tables. Although the loosening effect of internal pressure was recognized [12], its determination was disregarded. The dimensioning method developed by Schwaigerer (1950–1955) represented a paradigm shift and a leap forward, revealing considerable load carrying reserves by allowing for plastic elongations in the flange [13]. For force calculations, the elastic interaction of flanges, bolts, and gaskets are taken into consideration, though approximately [14]. The limit of flange load carrying capacity is interpreted as a product of the yield stress of the engineering material of the flange and a so-called plastic cross-section coefficient. The cross-section coefficient thus defined refers to a possible plastic limit state of the flange. When the joint is loaded by pressure, the moment allowed for the flange is stated by a safety factor referring to the limit state above. In the late 1950s and early 1960s, the analysis of shell connections came to the fore again on the basis of the results of the American school. Thus, by taking the elastic limit state as a starting point, the plasticity condition of Lake and Boyd [15] was developed, together with the elastic stress analysis method developed by Murray and Stuart, also taking into account the effect of the conical neck [16]. Finally, since 1970s to the present day, the finite element and the boundary element methods have become increasingly widespread besides analytical methods in the research of flange structures; their accuracy has been demonstrated by a number of experimental tests [17–19]. Figure 7.10 shows the internal forces of the joint. In a state loaded by pressure, the force equilibrium FC0 = FG0 produced at installation (see Fig. 7.10a) is modified according to Fig. 7.10b as a result of simultaneous heat and mechanical loads derived from the medium to sealed. The loosening process in the joint needs to be known for determining the gasket load reduction ΔFG also outlined in the figure; modelling thereof is discussed in detail in the literature. Apart from few exceptions [4–6, 15, 20– 23], the publications to be found [14, 17, 24–30] only examine the effect of internal pressure as mechanical load, tracing back the gasket load reduction ΔFG P to the elastic deformation of the joint. In the course of developing the calculation model, the concept of the loosening factor—defined by correlation (7.1)—was introduced [17, 28], which—as an engineering feature—can be considered to be of constant value in case of a joint of given geometry and material.

7.2 Internal Loads of Flange Structure

  ∗ ∗ kC + z G K 1 · z 1P + K 2 · z 2P ΔFG P D= = FP kC + k G + z 2G (K 1 + K 2 )

173

(7.1)

  K 1 , K 2 Nrad , appearing in correlation (7.1), denote the torsion spring constants mm   the spring constants of the gasket of flanges for couples of forces, k G and kC mm N ring and the bolt fitting, respectively, and z i∗P (i = 1, 2) substitutive arms of load. By introducing substitutive arms of load, the distributed force system arising from internal pressure can be substituted by the couple of forces consisting of loosening forces FP impacting on lever z i∗P , in accordance with Fig. 7.11. In the correlation mentioned, calculation requirements apply the following approximations for a closed shape solution. A gasket ring of non-linear characteristics [22, 30–32] is treated as an ideally elastic element (k G = const); as regards flange spring constant (K 1 , K 2 ), the effect of the shell connected to the flange ring is either neglected or taken into consideration in a simplified manner in most cases [24, 25, 27]; and substitutive arms of load z i∗P (i = 1, 2) are approximatively substituted by the arm of force F1 : z i∗P ∼ = z 1 (see Fig. 7.10) [14, 24]. The latter approximations cause inaccuracies particularly in the case of neck flange profiles applied at higher operating pressures. Accurate calculations taking account of gasket non-linearity and the effect of the shell connected to the flange ring can be performed by complicated numerical procedures and by the finite element method [17, 32]. In case of flange joints operated at high temperatures TU > 380 ◦ C, gasket load is further reduced in operational state, in addition to the loosening effect caused by internal pressure. Reasons include the reduction of the elasticity modulus and the Fig. 7.10 Internal force system of flange in the installation (a) and operational (b) states of the joint

174

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

Fig. 7.11 Substitution of distributed flange force system by couple of forces

yield stress of engineering materials due to temperature rise, creeping and relaxation in flanges, bolts and gaskets, as well as heat expansion differences between structural parts. Out of the effects mentioned, internal pressure results in gasket load reduction immediately, without any time delay. Other phenomena, arising from high temperatures, exert their impact with delay, in proportion to the heating up of the joint; creeping and relaxation exert their impact continuously during operation. The aggregate impact of the phenomena mentioned complicates the sealing engineering dimensioning of joints operating at high temperatures, as well as the determination of the bolt pre-load to be adjusted at the time of installation. In the absence of a model based on analytical correlations, aggregately taking into consideration the impacts mentioned, joints operating at high temperatures are dimensioned for lifetime by applying semi-empirical correlations based on experience from experiments [5, 7, 20]. At lower temperatures 50 ◦ C ≤ TU ≤ 300 ◦ C, the process in flange joints is much simpler and more transparent, as the creeping and relaxation of stainless or carbon steel bolts and flanges, and the plastic elongation resulting from yield stress and elasticity modulus reduction are of negligible proportions [15, 20, 21]. In this case, gasket load reduction is caused by the loosening effect due to internal pressure as already mentioned, and by the creeping and relaxation of the non-metallic gasket material. Considerable efforts were made as early as the 1950s to explore the time-dependent properties of flat flange gaskets and determine them experimentally [33–36]. It was then that a test method [37, 38] was developed and standardized, which is suitable for stating the long-time loadability of flat gaskets and is currently in use as well. However, a strength factor determined by this method cannot be considered as a material property of the gasket entirely, since its value is also affected by the interaction

7.2 Internal Loads of Flange Structure

175

of the bolted joint and the gasket ring. Therefore, the test method mentioned is only suitable for comparing different gasket materials, although it properly approximates the phenomenon occurring in the real flange joint. The model presented in the following sections, describing the internal loosening process in flange joints of medium high temperatures [22, 23, 39] takes into consideration, in addition to the stiffening effect of the shell connected to the flange ring, the non-linearity, creeping and relaxation of the non-metallic gasket. The model based on analytical correlations was established in several steps. First, by assuming an ideally elastic (σG − εG ) gasket model, an iteration process was developed which provided accurate  ∗ values  for the torsion spring constants (K 1 , K 2 ) and substitutive ∗ , z 2P included in correlation (7.1). In connection therewith, the arms of load z 1P flange model developed by Murray and Stuart [16] was supplemented and specified. In case of the welding neck flange as per Fig. 7.1b, the so-called reduction middle surface—bisecting the wall thickness AB , shown in Fig. 7.12—was introduced for the conical neck connected to the flange ring, together with the calculation surface AB as a continuation of the middle surface of the cylindrical shell of constant wall thickness. In the course of solution—similarly to [16]—, the conical neck can be approximately modelled as a cylindrical shell of linearly varying wall thickness with middle surface AB; then the edge forces and edge moments pertaining to the middle surface AB thus calculated can be reduced to the middle surface bisecting the real wall thickness AB , . The quantities occurring in the course of calculating the stiffness of the shell connected to the flange ring can be stated in function of the variables shown in Fig. 7.12 and of the dimensionless coordinate interpreted by Eq. (7.2). 0.25 −0.5 −0.5 0.5  x˜ = 5.264 1 − ν 2 · d1 · α ·x

(7.2)

in which α = tan γ and ν = 0.3 Poisson ratio are indicated. The next step is to take gasket non-linearity into account. The basis for the solution is provided by the gasket versus compression diagram (σG − εG ) stated by measurement, a converted form of correlation (7.1), and an iteration process. Finally, the model suitable for describing the loosening process resulting from gasket laxation takes into consideration, in addition to the viscoelastic behaviour of non-metallic

Fig. 7.12 Conical neck model

176

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

gasket, the interaction of the flange and the gasket ring. Thus, the model developed makes the gasket test method mentioned [37, 38] suitable for determining gasket material properties, meaning that the time-dependent features (creeping, relaxation) of flat gaskets to be installed can be estimated in advance in case of arbitrary joints. The following sections present the equations that describe the analytical joint model, convergence of the iterations applied in the course of solution, as well as the calculations and experimental tests that demonstrate the accuracy of the model.

7.2.1 Flange Joint Operation Under Simultaneous Heat and Mechanical Load The model to be presented—expressing connections between forces and displacements—is based on the following assumptions. • the joint is only loaded by a medium to be sealed, of internal pressure p and temperature TU ; • in a state loaded by pressure, the impact of yield stress and elasticity modulus reduction as a result of temperature rise is negligible in the flange and in the bolt fitting; • in the state examined, the creeping and relaxation of the flange and the bolt fitting are also negligible. Within the meaning thereof, in the joint shown in Fig. 7.13, the spring constants K 1 , K 2 of the flanges and kC of the bolts, respectively, do not depend on time and joint elongation. The model assumes that the internal forces and stresses that eliminate differences of heat expansions between structural parts in the course of the heat equalization process do not affect gasket laxation, meaning that the two phenomena can be examined independently of each other. The model disregards the effective gasket width change resulting from flange ring rotation, thus the width according to the original geometry is taken into consideration for the spring constant of the gasket. This approximation is acceptable for thin gaskets of large diameter. Figure 7.14 illustrates displacement and force changes by simultaneous heat and mechanical load in the flange joint shown in Fig. 7.13. The figure shows displacements ΔW and angular displacements χ causing gasket load changes. As can be observed, flanges are substituted by rigid corner skeletons of torsion spring constants K 1 and K 2 , marked by thick lines. As shown in the figure, resultant displacement and angular displacement can be divided into parts and can be examined approxid 2 ·π matively independently of each other. Loosening force FP = G4 · p in the axial direction causes a spring back ΔWGP of the gasket and elongation ΔWCP of the bolts. As a result, the load equilibrium FC0 = FG0 , characteristic of the installation state, is modified as the gasket load changes by ΔFG P and the bolt load by a value of ΔFC P = FP − ΔFG P . Function relations between displacements and load changes

7.2 Internal Loads of Flange Structure

177

Fig. 7.13 Outline of the flange joint examined

are as follows (7.3) and (7.4). P P ΔWCP = ΔW1C + ΔW2C = kC0 (FP − ΔFG P )

    P P ΔWGP = ΔW1G + ΔW2G = ΔWCP + χ1P − χ10 z G + χ2P − χ20 z G

(7.3) (7.4)

where the angular displacement of flanges comes from the formula 

   χiP − χi0 = K i0 FP · z i∗P − ΔFG P · z G

(7.5)

(i = 1, 2). As force and displacement changes already occur at the beginning of the heat equalization process in the state of the joint loaded by pressure, it is expedient to determine the spring constants kC0 , K 10 , K 20 appearing in the correlations above by taking into consideration the elasticity moduli pertaining to ambient temperature

178

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

T = T0 . As a consequence of temperature rise, creeping and relaxation in the gasket material result in further loosening of the joint and a gasket load and bolt load reduction in the value of ΔFGV . The following correlations apply to the loosening process caused by gasket laxation. V V ΔWCV = ΔW1C + ΔW2C = kCU · ΔFGV

    V V ΔWGV = ΔW1G + ΔW2G = ΔWCV + χ1P − χ1V z G + χ2P − χ2V z G

(7.6) (7.7)

where the angular displacement of flanges comes from the formula  P  χi − χiV = K iU · ΔFGV · z G

(7.8)

(i = 1, 2). By taking Eqs. (7.6), (7.7) and (7.8) into account, gasket creeping can be stated in the simple form below.     ΔWGV = ΔFGV kCU + K 1U + K 2U z 2G

(7.9)

In the event that temperature TF of the flange and temperature TC of the bolt fitting are unknown, spring constants kCU of the bolt fitting and K 1U , K 2U of the flanges, respectively, can be calculated with good approximation by taking into account the stationary temperature (T = TU ) of the medium to be sealed. However, heat distribution within the joint needs to be known for  precisely  determining the internal T T ΔW = ΔF displacements , ΔW forces (ΔF ), CT iG iC and angular displacements  T GT χi − χiV that eliminate heat expansion differences. Determining Gasket Load Drop FG P in Case of Ideally Elastic Gasket Model The forces between the flange ring and the cylindrical shell connected to it are shown in Fig. 7.15 in the installation state (Fig. 7.15a) and in the state loaded by pressure (Fig. 7.15b) of the joint. Further tests and statements refer to one of the flanges arbitrarily selected from the joint, so the distinctive marking of the flanges (i = 1, 2) will be eliminated for the sake of being more concise. According to correlation (7.1), in this case the numerical value of load reduction ΔFG P can be obtained through the loosening coefficient, so the task is modified to the determination of the so-called loosening coefficient D. Out of the forces the flange ring interpreted in the figure is subject to, FC0 and FCU represent bolt loads in the installation state and in the state loaded by pressure, while FG0 and FGU represent gasket loads. By stating the gasket load required in a state loaded by pressure FGU ≥ FGmin the bolt load to be produced in the course of prestressing the joint can be calculated on the basis of the expression.

7.2 Internal Loads of Flange Structure

179

Fig. 7.14 Displacement and internal force changes under simultaneous heat and mechanical load

FC0 = FG0 = D · FP + FGU

(7.10)

Out of the forces the flange ring is subject to, the effect of the shell is expressed by shearing forces Q B0 , Q BU and moments M B0 , M BU , M F1 , indicated in the connecting cross-section. Out of these, Q B0 , Q BU , M B0 , M BU are internal forces and moments ensuring the continuity of displacements, while M F1 is the moment resulting from the reduction of force F1 in the axial direction to the surface AB. Based on Fig. 7.11, the angular displacement of the flange ring caused by internal pressure can be calculated on the basis of the expression.   Δχ P = K FP · z ∗P − ΔFG P · z G

(7.11)

In case of flanges of large diameter, the flange ring can be modelled as an elastic ring, so the angular displacement as per Eq. (7.11) can also be specified by the

180

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

Fig. 7.15 Internal force system of flange ring and connecting shell in the installation state (a) and state loaded by pressure (b) of the joint

equation below on the basis of Fig. 7.15: Δχ P = χ P − χ 0 = [F1 · z 1 + F2 · z 2 − ΔFG P · z G − d1 · π (ΔM B + h S · ΔQ B )]

dS 4π · I · E (7.12)

in which ΔM B = M BU + M F1 − M B0

(7.13a)

ΔQ B = Q BU − Q B0

(7.13b)

Furthermore, I is the secondary moment of inertia of the flange ring cross-section, calculated for the bending axis, and E is its elasticity modulus. Based on Eqs. (7.11) and (7.12), the following correlations are yielded for the torsion spring constant of the flange and for the substitutive arm of load. K =

dS 4π · I · E

(7.14)

7.2 Internal Loads of Flange Structure

z ∗P =

181

F1 · z 1 + F2 · z 2 (ΔM B + h S · ΔQ B ) − d1 · π FP FP

(7.15)

On the basis of Eqs. (7.14) and (7.15), it can be established that in respect of the loosening coefficient defined by Eq. (7.1), the impact of the connecting shell is expressed by substitutive arm of load z ∗P in this case. Within the meaning of Eq. (7.15), the internal forces and moments at the flange ring connection need to be known to be able to determine the former. Such internal forces and moments result from the fit conditions shown in Fig. 7.15. w B0 = M B0 · w M B + Q B0 · w Q B

(7.16a)

χ B0 = M B0 · χ M B + Q B0 · χ Q B

(7.16b)

w BU = M BU · w M B + Q BU · w Q B + p · w P

(7.17a)

χ BU = M BU · χ M B + Q BU · χ Q B + p · χ P

(7.17b)

In Eqs. (7.16a), (7.16b), (7.17a) and (7.17b), w B0 and w BU represent the radial displacement of point B of the flange ring, while χ B0 and χ BU indicate the angular displacement of the fit cross-section. These quantities can be calculated from the equations below, in function of the forces the flange ring as an elastic ring is subject to. w B0 =

M S0 · d S2 Q S0 · d S · d1 hS + 4· I ·E 4· A·E

(7.18a)

M S0 · d S2 4· I ·E

(7.18b)

M SU · d S2 Q SU · d S · d1 hS + 4· I ·E 4· A·E

(7.19a)

M SU · d S2 4· I · E

(7.19b)

χ B0 = w BU =

χ BU =

in which Q S0 , Q SU , M S0 , M SU are the resultant forces and moments on the joint in its installation state (Fig. 7.15a) and state loaded by pressure (Fig. 7.15b), calculated from the forces on the flange ring for its cross-section centre of gravity using Eqs. (7.20a), (7.20b), (7.21a) and (7.21b),  Q S0 = −Q B0

d1 dS

 (7.20a)

182

M SU

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

    d1 d1 FC0 · z G − Q B0 · h S M S0 = − M B0 (7.20b) dS · π dS dS     d1 dB − Q BU (7.21a) Q SU = p · h dS dS     d1 d1 F1 · z 1 + F2 · z 2 + FGU · z G − (M BU + M F1 ) − Q BU · h S = dS · π dS dS (7.21b)

and A represents the area of the flange ring cross-section. In the case shown in Fig. 7.1a, w β and χ β , β = (M B, Q B, P), figuring in Eqs. (7.16a), (7.16b), (7.17a) and (7.17b), can be calculated on the basis of the correlations in Chap. 5 (5.15a), (5.15b), (5.19a) and (5.19b) and in Chap. 3 (3.16), by stating the unit rim load and the internal pressure. In case of the cylindrical shell with a neck of varying thickness as shown in Fig. 7.16, the following equations can be used for calculating the radial displacement and angular displacement of the connecting cross-section caused by unit bending moment (Fig. 7.16a), shearing force (Fig. 7.16b), and internal pressure (Fig. 7.16c): ,

w β = e1 · c jβ · ψ j (x˜ B ) + e2

(7.22a)

  χ β = e3 c(2i+1)β · ψ(2i+2) (x˜ B ) − c(2i+2)β · ψ(2i+1) (x˜ B ) ,

− e4 · c jβ · ψ j (x˜ B ) − e5

(7.22b)

in which i = (0.1); j = (1, . . . 4); β = (M B, Q B, P); furthermore, e1 = x B−0.5

(7.23a)

in the cases of β ≡ M B and β ≡ Q B, respectively, e2 = 0; and in the case of β ≡ P, e2 = 0.125 · E −1 (2 − ν)α −1 [d1 + α(x B − x A )]2 x B−1

(7.23b)

 0.25 −0.5 −0.5 −1 e3 = 2.632 1 − ν 2 ·α · d1 · x B

(7.23c)

e4 = x B−1.5

(7.23d)

in the cases of β ≡ M B and β ≡ Q B, respectively, e5 = 0; and in the case of β ≡ P, e5 = 0.125 · E −1 (2 − ν)α −1 d12 · x B−2

(7.23e)

7.2 Internal Loads of Flange Structure

183

Fig. 7.16 Stiffness of cylindrical shell neck loaded by bending moment, shearing force and internal pressure ,

dψ

in addition, ψ j and ψ j = d x˜ j denote the Schleicher functions and their derivatives, to be calculated by correlations (6.41a)–(6.41h) and (6.42a)–(6.42d). Boundary condition constants c jβ figuring in Eqs. (7.22a) and (7.22b) are obtained by solving the following linear equation system: bβi = ai j · c jβ

(7.24)

in which (i, j ) = (1, . . . , 4). The values of bβi appearing in equation system (7.24) can be taken from Table 7.1, while coefficients ai j can be calculated as a linear combination of Schleicher functions interpreted by Eqs. (7.25a) and (7.25b). in case of i = (1, 2); ai j = ωi jk · φk (x˜ B );

j = (1, . . . , 4), k = (1, . . . , 8),

(7.25a)

j = (1, . . . , 4), k = (1, . . . , 8),

(7.25b)

in case of i = (3, 4); ai j = ωi jk · φk (x˜ A );

where the non-zero elements of ωi jk  are included in Table 7.2; furthermore,  [φ1 , . . . , φ8 ] = ψ1 , . . . , ψ4, ψ1, , . . . , ψ4, . Within the meaning of Eq. (7.10), shearing force Q B0 and moment M B0 , produced in the connecting cross-section of the flange ring in the installation state, can only be calculated in the knowledge of the loosening coefficient of the joint. Consequently, the precise value of the loosening coefficient according to Eq. (7.1) can only be obtained through several steps in the course of solution. First in the operational state, by stating the gasket load required for

184

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

Table 7.1 Values of bβi figuring in equation system (7.24)

β

i

bβi

MB

1

1

2

0

3

0

4

0

1

0

2

1

3

0

4

0

1

M F1 +

2

0

QB

P

3 4

0.02(2−ν)α 2 ·d12 1−ν 2

0.036(2−ν)d13

E (1−ν 2 )

0.5

(2−ν)d12 E·α 0.5 ·x 2A



·x 2A 0.134d1 (1−ν 2 )0.25 ·x 0.5 A

+

0.125 α 0.5

hermetic closure FGU ≥ FGmin , Eqs. (7.17a) and (7.17b) should be used for determining the forces of the connecting cross-section of the flange ring as outlined in Fig. 7.15b. Then, by stating the initial values of M B0 and Q B0 arbitrarily in Eqs. (7.13a), (7.13b) and (7.15), the first approximation of the loosening coefficient defined by correlation (7.1) will be yielded. M B0 and Q B0 can be calculated from Eqs. (7.16a) and (7.16b) using the approximating value of the loosening coefficient, then—through Eqs. (7.13a), (7.13b) and (7.15) already mentioned—the subsequent approximating value of the loosening coefficient will be yielded. By repeating the calculation above in several steps, the final values of the loosening coefficient and of the gasket load reduction ΔFG P sought for can be obtained. Process convergence will be demonstrated in connection with a numerical example.

Stress State of the Shell Connected to the Flange Ring The stress state of the connecting shell can be determined in the knowledge of the internal forces loading the flange ring. In the case according to Fig. 7.1a, the calculation can be performed as set out in Chap. 5. In respect of the flange with a neck of varying thickness as per Fig. 7.1b, first the internal force system under rim load—outlined in Fig. 7.15—should be determined, consisting of edge forces and edge moments on calculation surface AB. Then, in order to determine the stresses generated on the external and internal surfaces of the neck of varying thickness, the force system above should be reduced to the middle surface AB , , bisecting the wall thickness shown in Fig. 7.12. Using this method, errors arising from the asymmetry of the conical neck can be reduced and the correlations in the literature, deduced

7.2 Internal Loads of Flange Structure Table 7.2 Non-zero elements of ωi jk figuring in Eqs. (7.25a) and (7.25b)

185

ω112 = −ω121 = ω134 = −ω143 = −0.75 2.5 −0.5  α · d1 · xB 0.438E 1 − ν 2 −ω115 = −ω126 = −ω137 = −ω148 = −1  0.166E 1 − ν 2 α 3 · x B0.5 −ω116 = ω125 = −ω138 = ω147 = −0.5 2 −1 1.5  · α · d1 · x B 0.577E 1 − ν 2 ω211 = ω222 = ω233 = ω244 = −0.25 1.5 −1.5  · α · d1 · xB 1.52E 1 − ν 2 ω216 = −ω225 = ω238 = −ω247 = −0.5 2 −1 0.5  · α · d1 · x B −0.577E 1 − ν 2 −ω311 = −ω322 = −ω333 = −ω344 = 1.414x −0.5 A ω312 = −ω321 = ω334 = −ω343 =  −0.25 0.5 0.5 −1 0.76 1 − ν 2 · α · d1 · x A ω315 = ω326 = ω337 = ω348 = −0.5  · α · d1 · x −1.5 + x −0.5 −0.288 1 − ν 2 A A −ω316 = ω325 = −ω338 = ω347 =  −0.25 0.5 0.5 −1 · α · d1 · x A + x −0.5 0.537 1 − ν 2 A ω411 = ω422 = ω433 = ω444 = 0.25 −0.5 −0.5 −1  ·α · d1 · xA −2.632 1 − ν 2 ω412 = −ω421 = ω434 = ω443 =  0.25 −0.5 −0.5 −1 ·α · d1 · x A + 2.828x −1.5 2.632 1 − ν 2 A ω415 = ω426 = ω437 = ω448 = −0.25 0.5 0.5 −2  · α · d1 · x A − x −1.5 −1.074 1 − ν 2 A ω416 = ω425 = ω438 = −ω447 =  0.25 −0.5 −0.5 −1 ·α · d1 x A − x −1.5 −3.722 1 − ν 2 A

for the stress state of the neck flange [9–11, 16] can be specified. Following the line of thought above, in the installation state (7.26a)–(7.26c) and in a state loaded by pressure (7.27a)–(7.27c) of a welding neck flange joint, the following correlations are yielded for the stress state of a shell of varying thickness connected to the flange ring: in the installation state, σ1R K 0 =

6M1R0 √ 1 + α2 α2 · x 2

(7.26a)

186

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

σ1R B0 = − σ2R KB 0 =

6M1R0 α2 · x 2

6M1R0 N2R0 ±ν 2 2 α·x α ·x

(7.26b) (7.26c)

in a state loaded by pressure, 

σ1R K U

√ 6M1RU (d1 − α · x A )2 = +p 1 + α2 α2 · x 2 4[d1 + α(x − x A )]α · x   6M1RU (d1 − α · x A )2 σ1R BU = − 2 2 + p α ·x 4[d1 + α(x − x A )]α · x σ2R KB U =

6M1RU N2RU ±ν 2 2 α·x α ·x

(7.27a)

(7.27b) (7.27c)

where, in accordance with the signage of Fig. 7.15, σ1R K 0 , σ1R B0 , σ2R KB 0 , σ1R K U , σ1R BU , σ2R KB U represent stresses in the tangential direction (1) and in the circumferential direction (2), generated in the outside surfaces (K = exter nal, B = internal) of the neck of varying thickness in the installation state (0) and in a state loaded by pressure (U ), respectively. Edge moments in the tangential direction M1R0 , M1RU and edge forces in the circumferential direction N2R0 ,N2RU , interpreted on middle surface AB , of the neck of varying thickness and appearing in correlations (7.26a)–(7.26c) and (7.27a)–(7.27c), are defined by the following correlations: M B0 · ξ M B + Q B0 · ξ Q B d1 + α(x − x A )

(7.28)

M BU · ξ M B + Q BU · ξ Q B + p(ξ P − g5 (x) − g4 ) d1 + α(x − x A )

(7.29)

M1R0 = M1RU = where

  , , ξβ = g1 c(2i+2)β · ψ(2i+1) ˜ − c(2i+1)β · ψ(2i+2) ˜ x 1.5 (x) (x)     + g2 c(2i+1)β · ψ(2i+2) (x) ˜ − c(2i+2)β · ψ(2i+1) (x) ˜ x − g3 c jβ · ψ ,j (x) ˜ x 0.5 (7.30) in which −0.5 2  g1 = 0.577E 1 − ν 2 α

(7.31a)

−0.75 2.5 0.5  g2 = 0.438E 1 − ν 2 α · d1

(7.31b)

−1  g3 = 0.166E 1 − ν 2 α 3 · d1

(7.31c)

7.2 Internal Loads of Flange Structure

187

−1  g4 = 0.02(2 − ν) 1 − ν 2 α 2 · d13

(7.31d)

g5 = 0.125α(d1 − α · x A )2 (x − x A )

(7.31e)

i = (0, 1), j = (1, . . . , 4), β ≡ (M B, Q B, P) furthermore,

N2RU

  2E · α · x 0.5 M B0 · δ M B + Q B0 · δ Q B (7.32) N2R0 = d1 + α(x − x A )   2E · α · x 0.5 M BU · δ M B + Q BU · δ Q B + p · δ P + 0.5 p(d1 + α(x − x A )) = d1 + α(x − x A ) (7.33)

where δβ = c jβ · ψ ,j (x) ˜

(7.34)

j = (1, . . . , 4), β ≡ (M B, Q B, P). Determining Gasket Load Drop FG P in Case of Non-linear Gasket Model By further developing the model presented in the preceding section, gasket nonlinearity can also be taken into consideration in determining the internal force system of the flange. The model uses the compressive diagram (σG − εG ) outlined in Fig. 7.17—possible to be determined by experiments—for characterizing nonmetallic gaskets, which diagram shows the compressive stress produced in the gasket in the unloading phase in function of specific compression. In the course of solution, the compressive diagram of the gasket can be substituted with lines by segment as shown in Fig. 7.17, which will follow the real curve with sufficient accuracy in case of divisions of appropriate density. Pursuant to the approximation above, the gasket load of FG0 value, created in the installation state, will be reduced as a result of internal pressure along the characteristic substituted with lines in accordance with Fig. 7.17. In the course thereof, gasket j load reduction ΔFG P , pertaining to the straight segment characterized by elasticity j modulus j, E G , will be produced as a result of pressure rise Δp j . Figure 7.18 shows changes in the internal force system of the flange as a result of pressure rise Δp j . As opposed to the force system shown in Fig. 7.15, in this case the unknown quantity j is pressure rise Δp j , generating the given gasket load reduction ΔFG P . Δp j can be determined by the iteration calculation to be presented below, in the course of j j which, in the knowledge of the approximating value of pressure rise i, Δpi , Δpi+1 will result from the following form of the transformed correlation (7.1):

188

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

Fig. 7.17 Characteristic compression diagram σG − εG of non-metallic flange gaskets

j

Δpi+1

 j j ΔFG P kC + k G + z 2G (K 1 + K 2 ) 4

 · 2 = ∗j ∗j d G ·π kC + z G K 1 · z 1Pi + K 2 · z 2Pi j

(7.35)

j

where, taking account of elasticity modulus E G pertaining to segment j, k G is the gasket spring constant interpreted by the following equation: j

kG = ∗j

hG j

dG · π · bG · E G

,

(7.36)

∗j

furthermore, z 1Pi , z 2Pi is the substitutive arm of load pertaining to the approximating j value of pressure rise i, Δpi . By eliminating the distinctive marking of the flanges, the approximating value i of the substitutive arm of load can be calculated using the following correlation on the basis of Eq. (7.15) and Fig. 7.18: ∗j

z Pi =

d B2

  · z 1 + dG2 − d B2 z 2 dG2 j

j

j

−



j j j d1 · π ΔM Bi + ΔM(F1)i + h S · ΔQ Bi j

ΔFPi

(7.37)

where ΔM Bi , ΔM(F1)i , ΔQ Bi are the moment and shearing force increments j j produced at the connecting cross-section of the flange ring. ΔM Bi , ΔQ Bi , appearing in correlation (7.38a) and (7.38b), is yielded as a solution of the equation system that can be stated using Eqs. (7.17a) and (7.17b):

7.2 Internal Loads of Flange Structure

189

j

j

j

j

(7.38a)

j

j

j

j

(7.38b)

Δw Bi = ΔM Bi · w M B + ΔQ Bi · w Q B + Δpi · w P Δχ Bi = ΔM Bi · χ M B + ΔQ Bi · χ Q B + Δpi · χ P

in which w β , χ β ; β ≡ (M B, Q B, P) are the displacement and angular displacement j j interpreted on the basis of Fig. 7.16; furthermore, Δw Bi and Δχ Bi are the radial displacement and angular displacement of the connecting cross-section of the flange ring as determined by Eqs. (7.39a) and (7.39b). j

j

j

Δw Bi =

ΔQ Si · d S · d1 ΔM Si · d S2 hS + 4· I ·E 4· A·E

(7.39a)

j

j

Δχ Bi = j

j

ΔM Si · d S2 4· I · E

(7.39b)

Finally, ΔM Si and ΔQ Si , figuring in Eqs. (7.39a) and (7.39b), can be calculated by correlations (7.40a) and (7.40b), yielded by using Fig. 7.18 and Eqs. (7.21a) and (7.21b): Fig. 7.18 Internal force system change in flange ring and connecting shell as a result of pressure increment Δp j

190

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints j ΔM Si

  j j j ΔF1i · z 1 + ΔF2i · z 2 + ΔFG P · z G d1 j = − ΔQ Bi · h S dS · π dS  d 

1 j j − ΔM Bi + ΔM(F1)i (7.40a) dS     dB d1 j j j ΔQ Si = Δpi · h − ΔQ Bi (7.40b) dS dS

By stating the initial value of Δp j arbitrarily, the iteration process above can be used for determining the value of pressure increment Δp j producing a given gasket j load reduction ΔFG P in respect of each segment of the compression diagram shown in Fig. 7.17. And in the knowledge thereof, the total gasket load reduction pertaining to the final state can be calculated by taking Fig. 7.17 into consideration. Convergence of the iteration applied in the process above will also be demonstrated in connection with a numerical example. Viscoelastic Model for Determining Gasket Load Reduction FGV Analysis of the internal loosening process due to gasket material laxation requires gasket ring modelling and examination of the interaction of the flange, the bolt fitting, and the gasket. When creating a viscoelastic model, the characteristic nonlinear compression diagram of non-metallic gasket materials should be taken into consideration together with the fact that remaining stresses are produced in the course of the relaxation testing of the most frequently applied gasket materials [37, 38]. The rheological analogue model shown in Fig. 7.19 satisfies both criteria. Noni built in parallelly connected linear gasket characteristics are provided by springs E G1 elementary Maxwell models. Connections between the gasket and metallic parts (flange sections, bolt fitting) are symbolized by spring connected to the gasket model, the rigidity K F of which can be calculated from the following correlation: KF =

  εGV AG  U  U = kC + K 1 + K 2U z 2G σGV hG

(7.41)

in which A G denotes the front surface, and h G the thickness of the gasket ring. Otherwise, Fig. 7.19 illustrates the state pertaining to initial compressive stress σGN . in f (N ) · A G resulting from gasket laxation Total gasket load reduction ΔFGV = ΔσGV can only be determined in several steps, by using the following correlations valid in cases N ≥ 1 pertaining to the linearized segments of the gasket compression diagram. In range j of the laxation process (Fig. 7.20) the time dependence of compressive j stress σGV (t) generated on the gasket is yielded by the following differential equation:

7.2 Internal Loads of Flange Structure

191

Fig. 7.19 Rheological analogue model of flange joint

 1  j j j 2 + E G0 1 + K F E G0 1 + K F · E G d 2 σGV dσGV =0 · + · Σ Σj N i i dt 2 dt i= j ηG i=1 E G1

(7.42)

the solution of which will be, by taking boundary conditions into account: 

 1 2 E Gi − E G0 · ΔεGi − E G0  1  − = 2 1 + K F E G0 + E G0 Σj j 

 i j i=1 E G1 · εG  1  1 − exp λ − · t 2 2 1 + K F E G0 + E G0 ΣN

j σGV (t)

j

where coefficient εG =

i=1+ j

σGN

Σj

j

i=1

ΔεGi , λ2 can be calculated from the following equation:

Σj j λ2

(7.43)

=−

  1  i 2 1 + K F E G0 E G1 + E G0  ΣN i j i= j ηG 1 + K F · E G

i=1

(7.44)

The real validity range of solution function (7.43) is the period 0 ≤ t ≤ t j according to Fig. 7.20, where

 j i j−1 1 + K F · EG ηG E tj = −Σj · ln Gj    i 1 2 EG i=1 E G1 1 + K F E G0 + E G0 ΣN

i= j

(7.45)

192

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

Fig. 7.20 Range j of laxation process j j j   The gasket relaxation ΔσGV and creeping ΔεGV = εGV t j produced during the period 0 ≤ t ≤ t j can be calculated from Eqs. (7.46) and (7.47).

Σj

j

j

ΔεGV

j

E i · ΔεG  G1  1 2 1 + K F E G0 + E G0 j   j = εGV t j = K F · ΔσGV

ΔσGV =

i=1

(7.46) (7.47)

In the course of the entire laxation process, the resultant relaxation and creeping of the gasket can be calculated from the following expressions, taking account of correlations (7.46) and (7.47). in f (N ) ΔσGV

=

N Σ

ΣN Σ j j ΔσGV

=

j=1 in f (N )

ΔεGV

=

N Σ

j

i i=1 E G1 1 K F E G0 +

j=1

1+



in f (N )

ΔεGV = K F · ΔσGV

j

· ΔεG  2 E G0

(7.48)

(7.49)

j=1 i 1 2 The features E G0 , E G0 , E G1 of the springs shown in Figs. 7.19 and 7.20, and the i features ηG of the damping elements representing viscosity can be determined on the i basis of the following considerations. The features ηG are provided by the conditions below, ensuring the continuity of the resultant relaxation curve:

7.2 Internal Loads of Flange Structure



j

dσGV dt

193



 = t=t j

j−1

dσGV dt

 (7.50) t=0

By substituting the relaxation curve segments defined by correlation (7.43) into Eq. (7.50) and performing the operations specified, the following recursion formula will be provided: N Σ

j

i ηG

=

i= j−1

1 + K F · EG j−1

1 + K F · EG

 Σ j−1 i=1 Σj i=1

i E G1

2

i E G1

N Σ

i ηG

(7.51)

i= j

i From the correlation above, in the knowledge of ηGN the values of ηG can be deterN i 1 2 , serving mined. The initial damping factor ηG and spring constants E G0 , E G0 , E G1 as a starting point, can be determined experimentally by setting gasket compression diagram (σG − εG ) and by jointly applying the long-time sealing test already mentioned [37, 38]. For this purpose, first the static compression diagram (σG − εG ) of the gasket material (see Fig. 7.17) should be determined for the unloading phase. By the segmental linearization of the diagram, the numerical values of ΔσGi , ΔεGi , E Gi can also be determined. The long-time test mentioned can also be used for deterin f (1) in f (2) mining relaxations Δσˆ GV , Δσˆ GV , pertaining to initial prestresses σG1 = ΔσG1 and σG2 = ΔσG1 + ΔσG2 , respectively. Within the meaning of the analogy between the test equipment and the flange joint, connections between the values measured and the elasticity features can be written in the following form according to Fig. 7.19 and correlation (7.48): 1 · Δε1 E G1  1 G 2  1 + Kˆ F E G0 + E G0  2  1 1 2 ΔεG · ΔεG1 + E G1 + E G1 E G1 =  1  2 1 + Kˆ F E G0 + E G0

in f (1)

Δσˆ GV in f (2)

Δσˆ GV

=

(7.52)

(7.53)

1 1 E G1 = E G0 + E G1

(7.54)

1 2 1 2 E G2 = E G0 + E G0 + E G1 + E G1

(7.55)

where Kˆ F denotes test equipment rigidity [37, 38] referred to the gasket ring. In possession of the data measured, the elasticity characteristics sought for can be calculated from the correlations below, to be deduced on the basis of Eqs. (7.52)– (7.55). In case of a solution fitting to two measurement points:

194

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

1 E G0



in f (2) in f (1) E G2 · ΔεG2 − Δσˆ GV − Δσˆ GV 1 2

 E G0 + E G0 = in f (2) in f (1) ΔεG2 + Kˆ F Δσˆ GV − Δσˆ GV

 ⎡ ⎤ in f (1) ΔεG2 1 + Kˆ F · E G2 Δ σ ˆ GV ⎣

⎦ = E G1 − in f (2) in f (1) 2 ΔεG1 ˆ ΔεG + K F Δσˆ GV − Δσˆ GV

(7.56)

(7.57)

In case of a solution fitting to one measurement point: in f (1)

E G2 · ΔεG1 − Δσˆ GV in f (1) ΔεG1 + Kˆ F · Δσˆ GV

 in f (1) Δσˆ GV 1 + Kˆ F ΔE G2

1 2 E G0 + E G0 =

1 E G0 = E G1 −

(7.58)

(7.59)

in f (1) ΔεG1 + Kˆ F · Δσˆ GV

As a matter of fact, the solution fitting to one measurement point resulted from 2 1 2 = 0. In the knowledge of elasticity factors E G0 , E G0 , the the assumption that E G1 i features E G1 of the springs connected to the damping elements can be calculated on the basis of Fig. 7.19. Finally, in the knowledge of elasticity features, the numerical values of initial damping factors ηGN result from correlation (7.60).

Σ N ηGN = −

i=1

i E G1

2

· εGN  d σˆ

1 + Kˆ F · E GN

 GV

(7.60)

dt

t=0

Interpretation and determination of the derivative formula is included in the ensuing sections.

−1

d σˆ GV dt

 t=0

featuring in the

7.2.2 Tests and Calculations to Verify the Model Tests to Verify the Accuracy of the Viscoelastic Gasket Model A number of experimental measurements have been performed to demonstrate the reliability and applicability of the viscoelastic model. In the framework thereof, the material characteristics and time-dependent properties of flat gaskets of a variety of manufacture and thickness were first examined. The numerical values of the elasticity and damping factors defined by Eqs. (7.56)–(7.60) were also determined on the basis of the measurement data. Calculations and experiments were performed in the knowledge of material characteristics, by stating various initial prestresses σGN . In the first step, calculations referred to a joint of rigidity K F = Kˆ F , identical with

7.2 Internal Loads of Flange Structure

195

that of the test equipment used for specifying gasket material characteristics, thus the accuracy of the model can also be checked by such standard gasket test equipment. The major experimental results presented below are associated with flat gaskets of measured thickness h G = 2.0 mm, of type REINZ AFM34, KLINGER C4430, and KLINGER C4300, all asbestos-free, and asbestos-based IT300 flat gaskets of thickness h G = 1.5 mm, h G = 2.0 mm, and h G = 3.0 mm, respectively. Compression diagrams (σG − εG ) were recorded on the ZWICK 1464 compressive and tensile testing machine with programme controlled heating shown in Photo 7.1. Table 7.3 contains the numerical data of compression diagrams, linearized per segment measured in the course of unloading. The data included in the table resulted from tests on specimens of 20 × 20 mm2 , placed between steel plates with free lateral surfaces, at a temperature of TU = 25,090 ◦ C and an unloading velocity of . 0.0166 mm s Long-time tests to specify material characteristics were performed on the BOSSERT type standard [37, 38] gasket test equipment shown in Photo 7.2.

Photo 7.1 ZWICK 1464 compressive and tensile testing machine with programme controlled heating

Table 7.3 Numerical data of gasket compression diagrams linearized per segment i (MPa) ΔσGi (MPa) E G σG (MPa) i (−) Reinz KLINGER AFM C4430 34 hG = 2.0 mm hG = 2.0 mm

0–5

1

5

5–10

2

5

66.03 136.9

62.3 133.4

KLINGER IT 300 IT 300 IT 300 C4300 hG = hG = hG = 1.5 mm 2.0 mm 3.0 mm hG = 2.0 mm 53.92 134.1

65.67 124.0

80.6

98.95

166.6

221.92

10–20

3

10

177.05

175.4

175.4

156.7

205.5

290.7

20–30

4

10

209.4

216.07

221.04

182.7

246.0

353.8

30–40

5

10

245.3

253.16

253.3

214.5

280.3

405.5

40–50

6

10

287.7

317.45

305.25

258.6

327.4

477.8

196

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

Bolted joint rigidity Kˆ F of the equipment changes in function of gasket thickness according to correlation (7.41). Its value at gasket thickness h G = 1.5 mm 2 2 is Kˆ F = 4.8553 × 10−3 mm ; in case of h G = 2.0 m, Kˆ F = 3.6415 × 10−3 mm , and N N 2 mm −3 at gasket thickness h G = 3.0 mm, Kˆ F = 2.4276 × 10 . These values resulted N from the finite element analysis of the bolted joints, by taking account of material N characteristics E = 190,000 mm 2 and ν = 0.3 related to test temperature. In the in f (1) in f (2) knowledge of relaxations Δσˆ GV and Δσˆ GV , pertaining to initial compressive N N 2 stresses σG1 = 5 mm 2 and σG = 10 mm2 shown in Table 7.3, the gasket material characteristics sought for can be calculated by correlations (7.56)–(7.59). Pursuant to correlations (7.56)–(7.57) resulting from the solution fit to two measurement points, 1 2 , E G0 pertaining to temperature the numerical values of elasticity characteristics E G0 ◦ TU = 250 C were as shown in Table 7.4. Figures 7.21 and 7.22 show the stress reductions measured and calculated. Out of these, Fig. 7.21 shows stress reductions measured and calculated on gaskets of a variety of manufacture and identical thickness h G = 2.0 mm in function of prestress; and Fig. 7.22 shows stress reduction changes measured and calculated on IT300 gaskets in case of various thickness and prestress values. In the figures,

Photo 7.2 BOSSERT type gasket test equipment

1 , E 2 , determined by correlations (7.56) and (7.57) Table 7.4 Elasticity characteristics E G0 G0

Reinz AFM KLINGER 34 C4430 h G = 2.0 mm h G = 2.0 mm

KLINGER C4300 hG = 2.0 mm

IT 300 hG = 1.5 mm

IT 300 hG = 2.0 mm

1 (MPa) E G0

1.96

41.85

2.12

34.85

20.8

2 (MPa) E G0

67.66

52.79

40.31

44.29

74.79

IT 300 hG = 3.0 mm 1.207 96.63

7.2 Internal Loads of Flange Structure

197

measured results are indicated by continuous lines, and calculated values by broken lines. Figure 7.23 shows the calculated and measured stress reduction in function of prestress in REINZ AFM34 gasket material of h G = 2.0 mm thickness, pertaining to temperatures TU = 250 ◦ C and TU = 150 ◦ C. The calculated results pertaining to the temperature TU = 150 ◦ C and indicated by a broken line were yielded by taking into consideration the elasticity characteristics determined by correlations (7.58)–(7.59). In the course of the experiments, the time dependence of the relaxation process in the test equipment of rigidity Kˆ F was also recorded. Figures 7.24 and 7.25 show the measured and calculated relaxation curves of REINZ AFM34 and IT300 gaskets of h G = 2.0 mm thickness. The tests revealed that regardless of gasket type, the exponential reduction characteristic of ideal relaxation processes only occurs in the N constant temperature range. Upon increasing initial prestress σGN > 20 mm 2 , larger and larger compressive stress reduction—different from the exponential type—can be seen in the heat-up phase. The difference between calculated and measured results in the initial test phase can probably be explained by this phenomenon. For that matter, the relaxation curves indicated by broken lines in the figures and departing from the starting point of the temperature hold phase were yielded by taking account of the initial damping factors ηGN determined from correlation (7.60). The numerical value of the derivative in correlation (7.60) is provided by the slope of the tangent to the relaxation curve defined by measurement at the starting point of the temperature hold phase. Finally, Fig. 7.26 illustrates the impact of heat-up speed on relaxation processes in the temperature hold phase. The figure shows relaxation curves measured at three different heat-up speeds. In this case as well, measurements were performed on REINZ AFM34 gaskets of h G = 2.0 mm thickness, subject to the parameters N TU = 250 ◦ C and σGN = 30 mm 2 . It can be established on the basis of the figure that heat-up speed does not have a considerable effect on total stress reduction since there is hardly any change in the compressive stress reduction value produced during the test period upon a more than threefold increase of the heat-up speed. Figure 7.27 shows the test measurement station developed for size impact testing [39]. In the course of the experiments, mechanical load on the insulated atmospheric flange joint with programme controlled heating and closed with flat cover was only caused by bolt and gasket load. Gasket load reduction due to gasket laxation was indicated by 4 displacement meters of 1 µm accuracy, fixed to bolt shanks, and arranged symmetrically along the perimeter. The test temperature (TU = 250 ◦ C) and the gaskets placed in the joint (R E I N Z AF M34, K L I N G E RC4430, andK L I N G E RC4300) were identical with those of the small sample test. Figure 7.28 shows flange and bolt temperatures during the test in the heat-up range and the ensuing stationary range. The broken line shows the temperature of the insulated flange body, and the continuous line shows the temperature of the bolt shank measured by a contact thermometer. Symmetrical bolt loads are illustrated in Fig. 7.29, showing measuring bolt elongations (thin line) and calculated average values (thick line) in the function of time in the course of testing R E I N Z AF M34 gaskets.

198

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

Fig. 7.21 Measured and calculated compressive stress reduction of gaskets of a variety of manufacture and h G = 2.0 mm thickness at TU = 250 ◦ C, in function of prestress

Fig. 7.22 Measured and calculated compressive stress reduction of IT-300 gaskets of various thickness at TU = 250 ◦ C, in function of prestress

 N Upon performing the tests

by setting  various initial prestress values σG and in f (N ) with model calculation results, Figs. 7.30, comparing the test results ΔσGV 7.31 and 7.32 were produced, demonstrating the applicability of the viscoelastic joint model presented.

7.2 Internal Loads of Flange Structure

199

Fig. 7.23 Measured and calculated compressive stress reduction of REINZ AFM34 gaskets of h G = 2.0 mm thickness at temperatures of TU = 250 and 150 ◦ C

Fig. 7.24 Relaxation of REINZ AFM34 gaskets of h G = 2.0 mm thickness, prestressed to various degrees, at TU = 250 ◦ C

Numerical Examination of the Loosening Process Caused by Internal Pressure The purpose of the numerical example below is to show the convergence of the iterations applied in the calculation process published in the previous sections and

200

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

Fig. 7.25 Relaxation of IT300 gaskets of h G = 2.0 mm thickness, prestressed to various degrees, at TU = 250 ◦ C

Fig. 7.26 Effect of heat-up speed on relaxation process

the accuracy of the procedure. The calculation results disclosed refer to the flange joint shown in Fig. 7.13, with the following dimensions: d K 1 = d K 2 = 1057 mm; dC = 1009 mm; dG = 953 mm; d B1 = d B2 = 895 mm; s B1 = s B2 = 24.6 mm; s A1 = s A2 = 10.3 mm; h 1 = h 2 = 51 mm; L 1 = L 2 = 43.5 mm; γ1 = γ2 = 18.2◦ . In the flange joint examined, the prestress force is produced by n = 44 bolts of d0 = 18.9 mm diameter, the resultant spring constant of which is kC0 =

7.2 Internal Loads of Flange Structure

201

Fig. 7.27 Outline and image of test equipment

Fig. 7.28 Flange body and bolt fitting temperature in heat-up phase

at T0 = 20 ◦ C ambient temperature. Table 7.5 contains the data 4.05 × 10−8 mm N of the compression diagram, linearized per segment, of the asbestos-free gasket of bG = 15.8 mm width, h G = 3.0 mm thickness, and dG = 953 mm mean diameter, ensuring hermetic closure. The calculation results to be presented below, referring N to T0 = 20 ◦ C ambient temperature, were yielded by setting p = 1.6 mm 2 operating

202

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

Fig. 7.29 Measuring bolt elongation in function of time

Fig. 7.30 Measured and calculated gasket load reduction in function of prestress for KLINGER C4300 gaskets

pressure and the associated FGU = 189,930 N gasket load required in a state loaded by pressure. For the calculation performed by assuming an ideally elastic gasket, the spring constant—considered to be of constant value—of the gasket resulted in the N by using elasticity modulus E G11 = 700 mm value of k G0 = 9.06×10−8 mm 2 pertaining N to linearized segment 11 of the compression diagram. The loosening coefficient of

7.2 Internal Loads of Flange Structure

203

Fig. 7.31 Measured and calculated gasket load reduction in function of prestress for REINZ AFM 34 gaskets

Fig. 7.32 Measured and calculated gasket load reduction in function of prestress for KLINGER C4430 gaskets

204

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

Table 7.5 Non-linear gasket characteristics linearized per segment

j

j

σG (MPa)

j (−)

ΔσG (MPa)

E G (MPa)

36.11

11

2.78

700

10

5.56

570

9

2.77

530

8

2.78

460

7

5.56

445

6

2.78

370

5

5.55

300

4

2.78

265

3

2.78

130

2

1.39

66

1

1.38

57

33.33 33.33 27.77 27.77 25 25 22.22 22.22 16.66 16.66 13.88 13.88 8.33 8.33 5.55 5.55 2.77 2.77 1.38 1.38 0

the joint was determined by the iteration process presented in Section “Determining Gasket Load Drop ΔFG P in Case of Ideally Elastic Gasket Model”. Convergence of the calculation process is demonstrated by Fig. 7.33, showing the development of loosening coefficient pertaining to various initial values ΔM B0 and ΔQ 0B at each iteration step. It can be established on the basis of the figure that the calculation according to Section “Determining Gasket Load Drop ΔFG P in Case of Ideally Elastic Gasket Model”—regardless of the initial values—converges relatively rapidly in the case examined to the value D = 1.216, pertaining to the given joint parameters. This was followed by the determination of pressure increments Δp j pertaining to the segments of the measured gasket compression diagram substituted by lines (see Table 7.5), by using the correlations disclosed in Section “Determining Gasket Load Drop ΔFG P in Case of Non-linear Gasket Model”. Convergence of the iterations applied in the calculation is demonstrated by Fig. 7.34, showing the pressure increments pertaining to various initial values Δp011 at each iteration step on linearized segment 11 of the gasket characteristic.

7.2 Internal Loads of Flange Structure

205

Fig. 7.33 Loosening coefficient at each iteration step

In the knowledge of the pressure increments Δp j pertaining to the segments of the gasket characteristic substituted by lines, the gasket load reduction caused by internal pressure can be determined, the course of which is shown in Fig. 7.35 in the present case. The curves in the figure depict the gasket load values pertaining to bolt pre-loads of various degrees in function of loosening force FP in the axial direction, calculated from internal pressure. By displaying the minimum gasket load FGmin pertaining to the leakage state in the figure (broken line), the maximum overpressure value pertaining to the given bolt preload can be determined. By displaying the broken straight line yielded by assuming an ideally elastic gasket pertaining to bolt pre-load FG0 = 17.1 × 105 N, it can be stated that gasket non-linearity has a considerable impact on the value of the gasket load generated in the operational state. The accuracy of the correlations specifying the rigidity of the welding neck flange and the stress state of the shell connected to the flange ring—presented in Sections “Determining Gasket Load Drop ΔFG P in Case of Ideally Elastic Gasket Model” and “Stress State of the Shell Connected to the Flange Ring”, respectively—is verified by Figs. 7.36, 7.37 and 7.38, illustrating the results of a finite element comparison test [40, 41]. These figures show the angular displacement of the flange ring and the maximum Mises equivalent stress of the

206

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

Fig. 7.34 Numerical pressure increment values pertaining to segment 11 of the compression diagram at each iteration step

Fig. 7.35 Gasked load in function of loosening force in the axial direction in case of non-linear gasket model

7.2 Internal Loads of Flange Structure

207

Fig. 7.36 Angular displacement of flange ring and effective Mises equivalent stress of connecting shell in function of cone angle

shell connected to the flange ring in a state of the flange joint loaded by pressure, in function of the cone angle (Fig. 7.36) and flange ring thickness (Figs. 7.37 and 7.38). In the figures, continuous lines show analytical calculation results, while dots represent the results of finite element calculations.

7.2.3 Dimensioning on Leak Tightness of Flange Joints Exposed to Simultaneous Heat and Mechanical Load Dimensioning on leak tightness is based on the assumptions presented in Sect. 7.2.1 and the explanatory notes to Fig. 7.14. As a consequence thereof, the internal loosening process results in the stress changes shown in Fig. 7.39 on the gasket surface. The curve indicated by a bold dash-dot line shows changes in the remaining compressive stresses in a state loaded by pressure. Leakage occurs when it intersects the limit curve of leakage σG L = m U · p, pertaining to temperature TU . Internal pressure

208

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

Fig. 7.37 Effective Mises equivalent stress of shell connected to flange ring in function of flange ring thickness

Fig. 7.38 Angular displacement of flange ring in function of flange ring thickness

7.2 Internal Loads of Flange Structure

209

Fig. 7.39 Compressive stress change on gasket caused by simultaneous heat and mechanical load

pmax giving rise to leakage designates one point (σGmin ) of the new modified limit curve required for dimensioning on leak tightness from curve σG ( p) pertaining to ambient temperature T0 . By changing initial prestress σG0 , an arbitrary number of further points of such modified limit curve can be drawn on the basis of Fig. 7.39. And linking such points results in a new limit curve. The task is considerably simplified by introducing the modified limit curve, since a joint of medium high temperature can be dimensioned as a joint of ambient temperature loaded by pure internal pressure according to Section “Numerical Examination of the Loosening Process Caused by Internal Pressure”. Practical application of the dimensioning on leak tightness presented is illustrated by the following numerical example. The calculations, the results of which are shown in Fig. 7.40, refer to a joint consisting of the welding neck flanges and bolts of the geometries already mentioned, and a REINZ AFM34 flat gasket ring of bG = 15.8 mm width, h G = 2.0 mm thickness, and dG = 953 mm mean diameter. The compression diagrams of the gasket ensuring hermetic closure, referring to ambient temperature T0 = 20 ◦ C and operating temperature TU = 250 ◦ C, resulted from experimental measurements. The characteristics of compression diagram (σG − εG ) referring to operating temperature are included in Table 7.3, and ambient temperature data—in case of identical N N N 2 3 divisions—are as follows: E G1 = 72.75 mm 2 ; E G = 131.5 mm2 ; E G = 169.5 mm2 ; N N N 5 6 4 E G = 217.4 mm2 ; E G = 253 mm2 ; E G = 403.9 mm2 . From correlation (7.41), the 2 . Figure 7.40 rigidity of the flange and the bolt fitting is K F = 4.3834 × 10−3 mm N shows the end result drawn by using the data above, where the modified limit curve was obtained by assuming ΔσGT = 0 and m U = 2.5. The stress versus pressure curve σG ( p), indicated by a continuous line, was specified according to Section “Numerical Examination of the Loosening Process Caused by Internal Pressure”, and stress in f (N ) due to gasket laxation was provided by the viscoelastic model reduction ΔσGV

210

7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

Fig. 7.40 Dimensioning on leak tightness of flange joint exposed to simultaneous heat and mechanical load

presented in Section “Viscoelastic Model for Determining Gasket Load Reduction ΔFGV ”. In the knowledge of gasket load reduction ΔFG , the joint can already be dimensioned on leak tightness, meaning that the bolt pre-load value FC0 ensuring the required leak-tightness—to be set at the time of installation—can be determined. In addition to the above, when determining bolt pre-load, gasket load carrying capacity (FC0max ) should be taken into account together with the fact that the gasket should fall into the correct operating range (FC0min ) in accordance with Fig. 7.4. So, the bolt pre-load to be set at installation should satisfy the following condition: FC0max > FC0 ≥ FGmin + ΔFG > FC0min

(7.61)

FGmin = σGmin · dG · π · bG

(7.62)

where

In the relation above, gasket load carrying capacity (FC0max ) and the minimum gasket load to be set (FC0min ) are determined by gasket type, gasket surface design, and the features of the medium to be sealed, so they can be calculated independently of the joint [31, 33–35]. Their method of calculation based on sealing experiments can be found in certain national standards and technical specifications [1–3]. Finally, Fig. 7.41 illustrates the correct operating range of the joint in terms of strength and sealing engineering, showing bolt loads within the joint in function of

References

211

Fig. 7.41 Correct operating range of flange joint in terms of strength and sealing engineering

loosening force FP in the axial direction, arising from the internal pressure to be sealed. The figure shows bolt load changes by assuming that gaskets are considered as ideally elastic (D = constans). Obviously, by taking account of gasket nonlinearity, bolt load changes in function of the loosening force will be modified in accordance with Fig. 7.35, and load values belonging together are not arranged linearly. In the figure, broken lines indicate the limits of the correct operating range, which is affected by the load carrying capacity of the gasket, the bolt fitting and the flange section, together with the gasket type/sealing surface and the effect of joint laxation on leakage.

References 1. ASME BPV Code: Section VIII, Division 2. The American Society of Mechanical Engineers, New York (1989) 2. BS 5500: Pressure Vessels Design, 1 Jan 1994 3. DIN 2505-1: 1990-04. Berechnung von Flanschverbindungen 4. Bouzid, A.-H., Nechache, A.: Thermally induced deflection in bolted flanged connections. J. Press. Ves. Tech. 127(4), 394–401 (2005) 5. Nassar, S.A., Alkelani, A.A.: Clamp load loss due to elastic interaction and gasket creep relaxation in bolted joints. Trans. ASME 128, 394–401 (2006) 6. Bouzid, A.-H., Nechache, A.: On the use of plate theory to evaluate the load relaxation in bolted flanged joints subjected to creep. Int. J. Press. Ves. Pip. 85, 486–497 (2008)

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7 Strength Test and Dimensioning on Leak Tightness of Flange Joints

7. Mourya, R.K., Banerjee, A., Sreedhar, B.K.: Effect of creep on the failure probability of bolted flange joints. Eng. Fail. Anal. 50, 71–87 (2015) 8. Bach, C.: Die Maschinen elemente. 10. Auflage, Leipzig (1908) 9. Modern Flange Design. Taylor Forge and Pipe Works, Chicago 10. Waters, E.O., Westrom, D.B., Rossheim, D.B., Williams, F.S.G.: Formulas for stresses in bolted flanged connections. Trans. ASME 59, 161–169 (1937) 11. Waters, E.O., Rossheim, D.B., Westrom, D.B., Williams, F.S.G.: Development of General Formulas for Bolted Flanges. Taylor Forge and Pipe Works, Chicago 12. Westrom, D.B., Bergh, S.E.: Effects of internal pressure on stresses and strains in bolted-flanged connections. Trans. ASME 73, 121–136 (1951) 13. Schwaigerer, S.: Die Berechnung der Flanschverbindunge im Behälter-und Rohrleitungsbau. Z. VDI. 96, 1 (1954) 14. Schwaigerer, S.: Festigkeitsberechnung von Bauelemeneten des Dampfkessel, Behälter und Rohrleitungsbaues. Springer, Berlin (1961) 15. Lake, G.F., Boyd, G.: Design of bolted flanged joints of pressure vessels. Proc. Int. Mech. E 171, 843–856 (1957) 16. Murray, N.W., Stuart, D.G.: Behaviour of large taper hub flanges. In: Proceedings of the Symposium on Pressure Vessel Research Towards Better Design. J. Mech. Eng. 133 (1961) 17. Varga, L.: Untersuchung von Flanschkonstruktionen. Konstruktion 33(H.9), 361–365 (1981) 18. Reuss, P., Szász, A.: Karimás kötések teherbírása. GÉP, 167–173 (1980) 19. Sima és toldatos készülékkarimák vizsgálata és takarékos kialakítása. Kutatási Jelentés BME. GSZI (1988) 20. Bernhard, H.J.: Flange theory and revised standard B.S. 10: 1962. Flanges and bolting for pipes valves and fittings. Proc. Inst. Mech. E 178, 107–130 (1963–1964) 21. Downey, S.C., Draper, J.H.M.: Creep test data in relation to high-temperature bolt design. In: Conference on Thermal Loading and Creep in Structures and Components. Int. Mech. Eng. 32 (1964) 22. Nagy, A.: Determination of the gasket load drop at large size welding neck flange joints in the case of nonlinear gasket model. Int. J. Pres. Ves. Pip. 67, 243–248 (1996) 23. Nagy, A.: Time depending characteristics of gasket at flange joints. Int. J. Pres. Ves. Pip. 12 219–229 (1997) 24. Haenle, S.: Beiträge zum Festigkeitsverhalten von Vorschweissflanschen and zur Ermittlung der Dichtkräfte für einige Flachdichtungen auf Asbestbasis. Forsch. Geb. Ingenieurwes. 23(H.4), 113–134 (1957) 25. Wölfer, J., Rabisch, W.: Berechnung und Stadardisierung von Flanschverbindungen. Chem. Techn. 27(H.8), 470–478 (1975) 26. Donald, M.B., Salomon, J.M.: The behaviour of narrow-faced bolted flanged joints under the influence of internal pressure. Proc. Int. Mech. E. 173, 459–466 (1959) 27. Singh, K.P., Soler, A.I.: Mechanical Design of Heat Exchangers and Pressure Vessel Components, pp. 81–159. Arcturus Publishers Inc., Cherry Hill, New Jersey (1984) 28. Varga, L., Barátossy, J.: Optimal prestressing of bolted flanges. Int. J. Pres. Ves. Piping. 63, 25–34 (1995) 29. Nagy, A.: Toldatos karimaszerkezetek szilárdsági vizsgálata. M˝uszaki Doktori Értekezés. Budapest (1991) 30. Nagy, A.: Determination of the loosening coefficient at large size welding neck flange joints. Period. Polytech. Ser. Mech. Eng. 38, 179–199 (1994) 31. Irving, R., Jeanette, PA.: Gasket and bolted joints. J. Appl. Mech., 169–179 (1950) 32. Soler, A.I.: Analysis of bolted joints with nonlinear gasket behaviour. ASME J. Press. Vessel Technol. 102, 249–256 (1980) 33. Siebel, E., Krägeloh, E.: Untersuchungen an Dichtungen für Rohrleitungen. Konstruktion 7(H.9), 123–137 (1955) 34. Krägeloh, E.: Die Wesentlichen Prüfmethoden für IT Dichtungen. Gumi Asbest. 8, 628–636 (1955) 35. Schwaigerer, S., Krägeloh, E.: Prüfung von Weichdichtungen. BWK 4, 404–407 (1952)

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36. Farnam, R.G.: Studies of relaxation characteristics of non-metalic gasket materials. J. Rubber World, 679–685 (1951) 37. DIN 52913: Druchstandversuch an IT Dichtungsplatten (1990) 38. BS 7531: 1992 Specification for Compressed Non-Asbestos Fibre Joining. Appendix B 39. Nagy, A., Dudinszky, B.: Determination of gasket load drop at large size welding neck flange joints operating at medium high temperature. In: Proceedings of the EXPRES 2016, pp. 60–65, 31 Mar–02 Apr 2016 40. Nagy, A.: Hegeszt˝otoldatos készülékkarimák rugalmas alakváltozási és feszültségi állapotának meghatározása analitikus módszerrel. GÉP XL 12, 463–473 (1988) 41. Varga, L., Nagy, A.: Optimale Form und neue Analyse von Flanschkonstruktionen. Konstruktion 49(H.9), 25–30 (1997)

Chapter 8

Investigation of Stress Concentrating Cross-Sections Using the Finite Element Method

Abstract Defining material law to serve as a basis for finite element calculations. Specimen-level tests. Structural tests. Evaluation method for elastic–plastic finite element calculations. Numerical tests and results. Keywords Finite element method · Stress concentrating cross-sections · Elastic–plastic finite element calculations

In case of normal loads, dimensions and requirements, the basic level design of storage tanks and process equipment can be carried out with great safety on the basis of applicable dimensioning standards. As regards equipment so designed, it can be stated that they resist effective static loads constant in time, meaning that neither internal, nor external overpressure cause catastrophic destruction (no burst, no buckling). In case of loads other than usual, changing in time, or even extreme, it is no longer sufficient to apply standard calculation correlations, so expensive preliminary experiments are required to be performed or the finite element method—already referred to several times—must be applied in the course of the design process. The same applies to the examination of various equipment stress concentrating locations (cross-sections close to leaping curvature changes of nozzles, flanges, supports, and pressure vessel ends). The problem is aggravated by the fact that in the course of the pressure test following equipment manufacture and stress relieving heat treatment, local plastic elongation is produced in such stress-concentrating cross-sections, releasing elastic stress reserve—as already mentioned in Sect. 6.1.2—in the material points that underwent plastic elongation in the course of unloading subsequent to the test pressure [1–8]. At the same time, the finite element algorithms mentioned only yield true and accurate results if initial calculation data (material behaviour, load, boundary condition) are true and correct. In the course of strength analyses based on the numerical method mentioned, the loadability limit is mostly defined in static cases as pertaining to 0.2% plastic strain or in case of repeated loads, assuming the invariance and uniformity of tensile and compressive yield stresses as loads pertaining to double the yield stress (2ReH ), representing the limit of shakedown [9]. In the course of earlier research [3, 6, 10] (elongation measurement of structures, see Photo 8.1), elongation © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Nagy, Fundamentals of Tank and Process Equipment Design, Foundations of Engineering Mechanics, https://doi.org/10.1007/978-3-031-31226-7_8

215

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8 Investigation of Stress Concentrating Cross-Sections Using the Finite …

Photo 8.1 Test equipment

measurements of stress concentrating cross-sections showed, [10, 11], that in static cases they can bear loads causing much larger elongation than the required 0.2% plastic strain without getting damaged. Cyclic tests (loading, unloading) carried out at increasing loads have demonstrated that the compressive yield stress of materials built into the structure and the utilizable elastic stress range pertaining to shakedown are not constant: their value changes considerably in function of elongation under load [10–13]. The conclusions above, drawn from experiments, coincide with the findings of other researchers who clearly trace this phenomenon back to the socalled “Bauschinger effect”, apparent in engineering materials. Having studied the relevant special literature, it can be established that a number of researchers investigate the Bauschinger effect as a phenomenon even today [14–23], but in practice its impact is taken into consideration only rarely in structural strength tests [2, 3, 5, 7, 12, 24–27]. The procedure to be presented in the following sections is based on the elastic and elastic–plastic finite element analysis of stress concentrating cross-sections. In order to describe processes affecting loadability in stress concentrating cross-sections during test and operating loads, values of operating and test pressure—and obviously, accurate geometric dimensions—are required in addition to the material law determined by measurements. In the course of structural design, the procedure facilitates the interpretation and evaluation of finite element calculation results, thereby assisting structure loadability assessment.

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217

Section 8.1 discusses the definition and interpretation of the material law forming a basis for the evaluation of calculation results; Sect. 8.2 deals with the processing and evaluation of finite element results. Finally, Sect. 8.3 presents the practical application of the procedure in relation to the examination of a pressure vessel end connected to the equipment cylindrical shell by a flange joint.

8.1 Defining Material Law to Serve as a Basis for Finite Element Calculations 8.1.1 Specimen-Level Tests The measurement results to be presented are derived from the tests carried out at ambient temperature on the MTS 810 programme controlled compressive and tensile testing machine of TUB Department of Material Science and Technology, as shown in Photo 8.2 [13]. The geometry of the carbon steel and stainless steel specimens used in material tests are shown in Fig. 8.1. Specimens were machined parallelly with the rolling direction of the steel plate, then—after stress relieving heat treatment—their surface was made suitable for fine elongation measurements by way of polishing. In determining the material law, specimens were loaded up to a variety of tensile side elongations, and after unloading they were subjected to compressive loads in the opposite direction. Stress versus elongation relations were recorded by a fine elongation meter of 8 mm measurement basis. Fine elongation meter type: MTS 632.26 C-20.

Photo 8.2 MTS 810 tensile testing machine and the fine elongation meter

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8 Investigation of Stress Concentrating Cross-Sections Using the Finite …

Fig. 8.1 Tensile specimen geometry

Stress versus elongation relations σe (εe ) of the specimens—loaded up to a variety of tensile side elongations ε, unloaded and subjected to compressive loads—were recorded via a data logger. The course of measurement evaluation is shown in Fig. 8.2. In order to eliminate measurement uncertainties, compressive side yield stresses σ P and utilizable elastic stresses σ E were determined as shown in the figure, Fig. 8.2 Measurement evaluation

8.1 Defining Material Law to Serve as a Basis for Finite Element Calculations

219

Fig. 8.3 Curves σe (εe ) of specimens made of engineering material 1.0345, loaded up to various tensile side elongations

that is, in function of deviation from the unloading straight line, pertaining to differences Δε = 50 × 10−6 , Δε = 100 × 10−6 , Δε = 200 × 10−6 , in the E E E values σ P50 (ε), σ50 (ε), σ P100 (ε), σ100 (ε), σ P200 (ε), and σ200 (ε). Figure 8.3 shows the curves (σe − εe ) measured in the case of specimens machined from carbon steel plates of engineering material 1.0345. By fitting approximating functions to the linked measurement data points σe (εe ) of specimens loaded up to various elongations εe according to Figs. 8.3, 8.4 and 8.5 can be drawn where the horizontal straight lines indicated by thin broken lines represent compressive side yield stress independent of elongation, resulting from the assumption of idealized material behaviour. The following statements can be made based on test results. In case of carbon steels, compressive side yield stress values change (decrease) considerably in function of tensile side elongation: the approximating function fitted to the measurement points is exponentially decreasing. As can be observed, the approximating polynomial fitted to tensile side measurement points is properly aligned with the linearly elastic, perfectly plastic material model resulting from the assumption of ideal material law, except for the close environment of the proportionality limit. By supplementing the stresses outlined in Fig. 8.5 by the utilizable elastic stress interpreted according to Figs. 8.2, 8.6 can be drawn by dividing the numerical values of the stresses above by tensile yield stress (+ReH ),and dividing  the elongation by the elongation pertaining to tensile side yield stress εeE = +REeH . The figure shows     the dimensionless stress versus elongation function relationships σˆ e εˆ e , σˆ P εˆ e ,      σˆ eE εˆ e = σˆ e εˆ e + σˆ P εˆ e resulting from the tensile-compressive testing of the engineering material, which form a material test basis for the procedure to be hereinafter presented.

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8 Investigation of Stress Concentrating Cross-Sections Using the Finite …

Fig. 8.4 Measurement points and approximating functions fitted to such points in case of engineering material 1.0345

Fig. 8.5 Tensile curve σe (εe ) and compressive side yield stress σ p (εe ) in function of tensile side elongation in case of engineering material 1.0345

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221

Fig. 8.6 Material law determined by measurements in dimensionless coordinate system (1.0345)

Out of the test results of stainless steel specimens, Figs. 8.7, 8.8, 8.9 and 8.10 show the results yielded in case of engineering material of (X5 CrNi 18 10) composition. By analyzing the figures, it can be established again that compressive yield stress values differ from (are lower than) the values of ideal material law. Measurement points are fitted to a straight line (thick broken line), which is parallel with the straight line of the theoretical kinematic material law of hardening, marking compressive yield stress (thin broken line). It can further be established that the approximating

Fig. 8.7 Curves σe (εe ) of specimens made of engineering material X5 CrNi 18 10, loaded up to various tensile side elongations

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8 Investigation of Stress Concentrating Cross-Sections Using the Finite …

Fig. 8.8 Measurement points and approximating functions fitted to such points in case of engineering material (X5 CrNi 18 10)

Fig. 8.9 Tensile curve σe (εe ) and compressive side yield stress σ p (εe ) in function of tensile side elongation in case of engineering material (X5CrNi1810)

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223

Fig. 8.10 Material law determined by measurements in dimensionless coordinate system (X5 CrNi 18 10)

polynomial fitted to tensile side measurement points is properly aligned in this case as well with the bilinear material model resulting from the assumption of ideal material law, except for the close environment of the proportionality limit, as well as that the utilizable elastic stress is not constant: its numerical values change in function of tensile side elongation.

8.1.2 Structural Tests Elongation measurements carried out on structures [11] served to demonstrate the material law derived from specimen-level experiments, presented in the previous section. The purpose of the measurements is to decide to what extent the behaviour of material points in structures differs from specimen-related experiences as a consequence of the multiaxial stress state and the inhomogeneous stress field (interaction of purely elastic and elastic–plastic parts). Photo 8.3 shows the test equipment enclosed by circular plates, developed for these tests. Internal pressure induces a primarily dangerous bending stress state along the entire surface of the circular plate, therefore elongation and stress states change considerably along the radius, starting from the centre of the circular plate. This, and the fact that strain gauges are relatively easy to place, make it suitable for performing the experiments above. The dimensions of the manufactured and gauged equipment (diameter and thickness of circular plate, wall thickness of connected cylinder) were

224

8 Investigation of Stress Concentrating Cross-Sections Using the Finite …

Photo 8.3 Test equipment developed for structural testing

specified so as to enable yield in the opposite direction—after reaching the compressive yield stress—on the largest surface part possible along the circular plate radius within the unloading range in the course of the experiment. Elongation measurement gauges were stuck on the external surface of the circular plate in the radial and tangential directions, starting from the centre. The engineering material of the equipment thus set up was stainless steel as specified in Sect. 8.1.1 [11, 27]. In the course of the finite element calculations preceding measurements, the elastic–plastic elongation and stress states of the entire circular plate were determined at various load levels ( p). Calculations were performed by taking into consideration the material law as per the preceding section. Figure 8.11 shows equivalent elongations calculated at various pressures in function of the radial coordinate. It can be established on the N basis of the figure that at p = 2.2 mm 2 overpressure, the external surface of the plate gets into an elastic–plastic state within a radius of r = 100 mm. In the course of unloading, as also shown in Fig. 8.12, material points within r = 70 mm radius will most probably yield in the opposite direction as well, meaning that compressive side yield stress can be measured. As a matter of fact, Fig. 8.12 shows the values of utilizable elastic stress determined according to the previous section (broken line) and N Mises equivalent stress at p = 2.2 mm 2 load, by assuming elastic material behaviour (continuous line). In the course of unloading, plastic elongation (yield) is expected where the value of utilizable elastic stress is lower than that of elastic stress reduction (r ≤ 70 mm). In the test equipment shown in Photo 8.3, pressure drop Δp pertaining to yield in the opposite direction can be determined by increasing internal pressure up to N the calculated value of p = 2.2 mm 2 and followed by unloading at the measurement location, using the strain gauges fastened. Knowing the value of the pressure drop

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225

Fig. 8.11 Equivalent elongations pertaining to various loads, resulting from finite element calculations

Fig. 8.12 Utilizable elastic stress and elastic stress reduction under load p = 2.2

N mm2

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8 Investigation of Stress Concentrating Cross-Sections Using the Finite …

above, compressive side yield stress can be determined at each measurement location using the correlation below. σ P (εe ) = σe (εe ) − Δp · ΔσeE

(8.1)

In correlation (8.1), σe (εe ) is the tensile stress resulting from specimen-level testing, pertaining to equivalent elongation εe produced at the measurement location (material law), and ΔσeE is the Mises equivalent stress calculated for load per unit in the point examined. By comparing the compressive side yield stress values calculated on the basis of correlation (8.1) with specimen test results, Fig. 8.13 can be drawn. In the figure, compressive yield stresses resulting from the elongation measurement of the stainless steel test equipment examined are marked by black squares and points, while empty circles indicate the values yielded by specimen measurements as described in the preceding Sect. 8.1.1. By comparing the yield stresses determined in two ways, it can be stated that structural and specimen tests are in good agreement [11, 26, 27], meaning that the data obtained from specimen tests can be used for the elastic–plastic examination of structures.

Fig. 8.13 Compressive yield stresses resulting from structural tests and specimen measurements

8.2 Evaluation Method for Elastic–Plastic Finite Element Calculations

227

8.2 Evaluation Method for Elastic–Plastic Finite Element Calculations Figure 8.14 illustrates the essence of the method, showing the elongation character  point of the stress concentrating istic curve pˆ εˆ e of the dangerous    location examined law ε ˆ εˆ e , and utilizable elastic σ ˆ , compressive side yield stress σ ˆ [1–8], material e e P   stress σˆ eE εˆ e in function of tensile side elongation εˆ e . The dangerous point of the stress concentrating location is the point that first reaches the tensile side yield stress (+ReH ) of the engineering material subject to test pressure. The pressure at which the dangerous point of the stress concentrating location reaches yield stress +ReH , is termed as the elastic load carrying capacity of such location, indicated by p E . Interpretation of variables in Fig. 8.14: p pE

(8.2a)

σe +ReH

(8.2b)

pˆ = σˆ e =

σP +ReH       σˆ eE εˆ e = σˆ e εˆ e + σˆ P εˆ e σˆ P =

Fig. 8.14 Elastic–plastic finite element calculation evaluation method

(8.2c) (8.2d)

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8 Investigation of Stress Concentrating Cross-Sections Using the Finite …

εˆ e =

εe ( p) εeE

(8.2e)

εeE =

+ReH E

(8.2f)

Out of  curves shown in the figure, the so-called elongation characteristic  the curve pˆ εˆ e can be obtained from the elastic–plastic finite element analysis of the stress concentrating location examined, using the tensile side material law σe (εe ) of the engineering material. The so-called equivalent elongation εe ( p) (see Figs. 6.12 and 8.11) can be determined by the following correlations, using the calculated elongation components εi j ( p), and based on Fig. 8.15, the generalized Hooke’s law, and Prandtl-Reuss equations:   p εiEj ( p) = εi j p E · E p

(8.3a)

εiPj ( p) = εi j ( p) − εiEj ( p) i, j = 1, 2, 3.

(8.3b)

/         

   1  E 1 E 2 + εE − εE 2 + εE − εE 2 + 3 εE 2 + εE 2 + εE 2 εeE ( p) = ε11 − ε22 22 33 11 33 12 13 23 1+ν 2 4

/ εeP ( p) =

(8.3c) /  P 2  P 2  P 2 1  P 2  P 2  P 2  2 ε11 + ε22 + ε33 + · ε12 + ε13 + ε23 3 2 (8.3d) εe ( p) = εeE ( p) + εeP ( p)

(8.3e)

    As shown in Fig. 8.14, the horizontal distances between curves pˆ εˆ e and σˆ e εˆ e (the densely hatched area) illustrate the remaining stress to follow unloading after  test pressure, the yield stress of which is shown by curve σˆ P εˆ e . In the optimal case indicated [12, 24–27], intersection “S”—representing the shakedown limit—pertains to the dangerous point of the stress concentrating location examined. If the extent of load does not change in a construction so established and pressure tested, the stress concentrating location will always be in an elastic stress state, and no accumulating plastic elongation can occur in the course of a repeated load. In case of test pressure exceeding the intersection “S”, the possibility of so-called low-cycle fatigue can exist at the stress concentrating location examined, caused by accumulating plastic elongations due to repeated test loads.  Incase  of loads not reaching the intersection “S”, unutilized stress reserve σˆ eE εˆ e − pˆ εˆ e is produced in the engineering material (the sparsely hatched area). Obviously, the above involves the assumption that the largest elongation under test pressure is produced in the dangerous point (8.3e), meaning that the elongation characteristic curves are not intersected.

8.3 Numerical Tests and Results

229

Fig. 8.15 Division of elongations into purely elastic and plastic components

The method presented can be used for the strength analysis of the construction designed: processes affecting loadability in stress concentrating cross-sections during test and operating loads can be examined.

8.3 Numerical Tests and Results Figure 8.16 shows an outline of the pressure vessel end subject to examination, by indicating the main dimensions. According to the figure, a Klöpper-type pressure vessel end of torispherical shape is connected to the cylindrical shell of the equipment by a plain flange joint. On the pressure vessel end, there is a DN 200 cylindrical nozzle of 5 mm wall thickness, in an axial position coinciding with the rotation axis thereof. Load was exerted by internal pressure p operating on the internal surface; by reduced gasket load, with a circular impact on the mean diameter dG = 1022.5 mm of the gasket ring; and by membrane stresses in the tangential direction generated at the nozzle, in at least damping distance from the connecting cross-section. Static equilibrium was provided by roller support applied at the bolt circle. The calculation results presented in the following refer to the carbon steel and stainless steel materials discussed in Sect. 8.1.1.

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8 Investigation of Stress Concentrating Cross-Sections Using the Finite …

Fig. 8.16 Outline of pressure vessel end examined

In the calculation, first the elastic load carrying capacity p E of the nozzle environment, of the torus transition, and of the flange connection were determined. This was followed by an elastic–plastic test, in the course of which finite element calculations were performed on the basis of the material laws determined from measurements. As equipment loads are proportionate to internal pressure p, the elastic–plastic state of the vessel end can be tracked within the range p ≥ p E by gradually increasing the pressure. Figures 8.17, 8.18, 8.19, 8.20, 8.21, 8.22, 8.23 and 8.24 [25–27], modelled on Fig. 8.14, show the elongation characteristic curves referring to the external and internal surfaces at stress concentrating locations. Figures 8.17, 8.18, 8.19 and 8.20 show carbon steel, while Figs. 8.21, 8.22, 8.23 and 8.24 show stainless steel results. Parameter (s) of the characteristic curves marks the distance of the point examined from the point of the highest reduced stress, measured in arc length, in the elastic state of the stress concentrating location. The figures indicate the measured compressive side yield stress (thick broken lines), as well as utilizable elastic stresses resulting from ideal material law (thin broken lines) and determined from measurements (thick continuous lines). The intersections of calculated elongation characteristic curves     pˆ εˆ e and the thick continuous lines denoting the utilizable elastic stress σˆ eE εˆ e determined from measurements define the load pertaining to shakedown in respect

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231

Fig. 8.17 Elongation characteristic curves calculated on nozzle environment external surface in case of engineering material 1.0345

of the points examined. By analyzing the figures, it can be stated that the elongation characteristic curves were not intersected in the cases examined. This means that the material point producing the highest reduced stress in the elastic state went through the largest elongation in the elastic–plastic load carrying range p ≥ p E .

Fig. 8.18 Elongation characteristic curves calculated on nozzle environment internal surface in case of engineering material 1.0345

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Fig. 8.19 Elongation characteristic curves calculated on torus transition internal surface in case of engineering material 1.0345

Fig. 8.20 Elongation characteristic curves calculated on flange connection external surface in case of engineering material 1.0345

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233

Fig. 8.21 Elongation characteristic curves calculated on nozzle environment external surface in case of engineering material (X5 CrNi 18 10)

Fig. 8.22 Elongation characteristic curves calculated on nozzle environment internal surface in case of engineering material (X5 CrNi 18 10)

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Fig. 8.23 Elongation characteristic curves calculated on torus transition internal surface in case of engineering material (X5 CrNi 18 10)

Fig. 8.24 Elongation characteristic curves calculated on flange connection external surface in case of engineering material (X5 CrNi 18 10)

References 1. Varga, L.: Szerkezetek rugalmas-képlékeny teherbírása. GÉP, 11 (1985) 2. Varga, L.: Untersuchung der Belastbarkeit und elastisch-plastischen Zustands von Korbogenböden. Konstruktion, vol. 38, p. 12. Springer (1986) 3. Varga, L., Nagy, A.: Szerkezettervezés a rugalmas-képlékeny teherbírás kihasználásával. 5-108 OTKA zárójelentés (1986–1991) 4. Varga, L.: Optimale Konstruktion von Druckbehältern unter Berücksichtigung der nach dem Probedruck zurückbleibenden Spannungen. Konstruktion, vol. 42, p. 6. Springer (1990)

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5. Varga, L.: Design of optimum high-pressure monobloc vessels. Int. J. Press. Ves. Pip. 48, 93–110 (1991) 6. Varga, L., Nagy, A., Gara, P.: Szerkezettervezés a rugalmas-képlékeny teherbírás kihasználásával. OTKA Zárójelentés, T 20980 (1994) 7. Varga, L., Nagy, A.: Optimale form und neue Analyse von Flanschkonstruktionen. Konstruktion 49(9), 25–30 8. Varga, L.: Design of pressure vessels taking plastic reserve into account. Int. J. Press. Ves. Pip. 75, 331–341 (1998) 9. Kaliszky, S.: Képlékenységtan. Akadémiai Kiadó Budapest (1975) 10. Nagy, A.E. Székelyhidi, E. Balogh, Á.: Nyomástartó edények csonk környezetének szilárdsági vizsgálata. In: IX. MAMEK, pp 105–108. Miskolc, Hungary (2003) 11. Nagy, A., Székelyhidi, E., Tasnádi, P.: An experimental study of the Bauschinger effect on a pressure vessel with flat head. In: Gépészet 2006. Proceedings of the Fifth Conference on Mechanical Engineering. Budapest (2006) 12. Nagy, A., Székelyhidi, E.: Dimensioning on repeated load stress concentrating places by taking into consideration the Bauschinger effect. In: Gépészet 2004. Proceedings of the Fourth Conference on Mechanical Engineering, pp. 588–592. Budapest (2004) 13. Székelyhidi, E., Nagy, A.: Experimental investigation of the Bauschinger effect using fatigue specimens made of different engineering materials. In: Gépészet 2004. Proceedings of the Fourth Conference on Mechanical Engineering, pp. 186–190. Budapest (2004) 14. Yoshida, F., Uemori, T.: A model of large-strain cyclic plasticity and its application to springback simulation. Int. J. Mech. Sci. (2003) 15. Chun, B.K., Jinn, J.T., Lee, J.K.: Modeling the Bauschinger effect for sheet metals, part I: theory. Int. J. Plast. 18, 571–595 (2002) 16. Chun, B.K., Jinn, J.T., Lee, J.K.: Modeling the Bauschinger effect for sheet metals, part II: applications. Int. J. Plast. 18, 597–616 (2002) 17. Yoshida, F., Uemori, T., Fujiwara, K.: Elastic-plastic behaviour of steel sheets under in-plane cyclic tension-compresson at large strain. Int. J. Plast. 18, 633–659 (2002) 18. Yoshida, F., Uemori, T.: A model of large-starin cyclic plasticity describing the Bauschinger effect and workhardening stagnation. Int. J. Plast. 18, 661–686 (2002) 19. Thumser, J., Bergmann, J.W., Vormwald, M.: Residual stress fields and fatigue analysis of autofrettaged parts. Int. J. Press. Ves. Pip. 79, 113–117 (2002) 20. Mollica, F., Rajagopal, K.R., Srinivasa, A.R.: The enelastic behaviour of metals subject to loading reversal. Int. J. Plast. 17, 1119–1146 (2001) 21. Gau, J.T., Kinzel, G.L.: An experimental investigation of the influence of the Bauschinger effect on springback predictions. J. Mater. Process. Technol. 108, 369–375 (2001) 22. Jiang, N., Zhen, L., Xu, B.P.: Study on control limits of secondary stress strength in pressure vessels. Int J. Press. Ves. Pip. 76, 711–714 (1999) 23. Yong, L., Naije, S.: Residual stresses analysis for actual material model of autofrettaged tube by non-linear boundary element method. Int. J. Press. Ves. Pip. (1991) 24. Nagy, A., Molnár, K.: Dimensioning pressure vessels considering Bauschinger effect. In: Gépészet 2002 Proceedings of the Third Conference on Mechanical Engineering, pp. 664–668. Budapest (2002) 25. Nagy, A., Dudinszky, B.: Dimensioning pressure vessels considering Bauschinger effect. In: Gépészet 2008. Proceedings of the Sixth Conference on Mechanical Engineering. Budapest G2008-A-26 26. Nagy, A., Dudinszky, B.: Próbanyomás és az anyagtörvény hatása nyomástartó edények terhelhet˝oségére OGÉT 2015. In: 23rd International Conference on Mechanical Engineering Csíksomlyó, pp. 237–240 27. Nagy, A., Dudinszky, B.: Experimental and investigating of different engineering materials. In: Proceedings of 7th International Symposium on Exploitation of Renewable Energy Sources, pp.116–120. Subotica, Serbia (2015). ISBN 978-86-82621-15-7

Annex

Strategic Storage Tank Production Technical Specifications Nominal volume:

20,000 m3

Tank internal diameter:

36,000 mm

Tank shell height:

22,000 mm

Protective ring internal diameter:

41,000 mm

Protective ring shell height:

17,600 mm

Bottom designs:

Double bottom with vacuum leakage monitoring

Empty weight:

670,000 kg

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Nagy, Fundamentals of Tank and Process Equipment Design, Foundations of Engineering Mechanics, https://doi.org/10.1007/978-3-031-31226-7

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Reinforced concrete ring foundation construction, ring foundation reinforcement installation

Reinforced concrete ring foundation completed

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In the foreground: layout of laid bottom plate, adjusted lifting devices, with steel structure to support crown ring in the centre

In the foreground: installation of tank shell plates onto lifting devices, with crown ring lifted to position in the centre

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In the foreground: installation of steel support girders to support dish plate, with completed dish plate in the background

In the foreground: installation of tank shell zone rows, with lifted tank in the background

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Tanks completed, with steel walkway structures and fire protection piping installed on them

Commencement of protective ring shell construction

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Protective ring shell construction, with steel walkway structures and fire protection piping installed

In the foreground: completed unpainted tank; in the background: preparation for painting of tank plate surfaces by sand spraying

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In the foreground: preparation for painting of tank plate surfaces by sand spraying, with complete painted tank in the background

Finished tanks with fire protection piping (red piping: extinguishing by foam; green piping: shell cooling) and technology (white) piping

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Spherical Tank Production

Tank plate pressing

Excavation works of spherical tank ring foundation body

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Ring foundation reinforcement

Completed tank foundation body

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Erection of support columns

Pairing of tank segment plates in assembly workshop

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Tank hemisphere assembly on site

Production of welded joints on site

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Joining hemispheres

Heat insulated spherical tank of butadiene completed