Concrete Structures: Design and Residual Capacity Assessment [1 ed.] 0367770679, 9780367770679

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Concrete Structures

Based primarily on Eurocode 2, this book offers a comprehensive exploration of theory alongside a substantial collection of solved examples. Intended for students and professionals specializing in structural engineering, the content delves into the fundamental aspects of designing and evaluating concrete structures. Discover a unique focus on both theoretical and practical aspects, empowering engineers to unearth solutions that minimize steel usage. Step into the realm of asymmetric and environmentally appealing alternatives, redefining the way we approach modern engineering challenges. Enrique Hernández-Montes is a Professor in the School of Civil Engineering at the University of Granada, Spain. With nearly three decades of experience, he has been dedicated to research and teaching structural engineering courses. He is a member of the Spanish National group for Eurocode 2. Additionally, he has provided consultancy services for renowned engineering firms like Prointec and ERSI-Group, contributing his expertise to several globally recognized projects. Luisa María Gil-Martín is a Professor in the School of Civil Engineering at the University of Granada, Spain. With an extensive research and teaching history spanning nearly 30 years, she specializes in instructing structural engineering courses. She is a member of the Spanish National group for Eurocode 3.

Concrete Structures Design and Residual Capacity Assessment

Enrique Hernández-Montes and Luisa María Gil-Martín

Cover image: Enrique Hernández-Montes and Luisa María Gil-Martín First edition published 2024 by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN and by CRC Press 2385 NW Executive Center Drive, Suite 320, Boca Raton FL 33431 © 2024 Enrique Hernández-Montes and Luisa María Gil-Martín CRC Press is an imprint of Informa UK Limited The right of Enrique Hernández-Montes and Luisa María Gil-Martín to be identified as authors of this work has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. For permission to photocopy or use material electronically from this work, access www​.copyright​ .com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact mpkbookspermissions​@ tandf​.co​​.uk Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-367-77067-9 (hbk) ISBN: 978-0-367-77069-3 (pbk) ISBN: 978-1-003-16965-9 (ebk) DOI: 10.1201/9781003169659 Typeset in Sabon by Deanta Global Publishing Services, Chennai, India

To the faculty and practicing engineers who have contributed to enhancing our comprehension of concrete mechanics, and to our students who guide us in the teaching of this subject.

Contents

Section I  Basis 1 Principles

3

1.1 1.2 1.3 1.4

Purpose and objectives  3 Introduction 4 Reinforced and prestressed concrete as a building material  4 Unique features of the concrete–steel composite material  7 1.4.1 Durability 8 1.4.2 Anchorage and adherence  9 1.5 Normative 11 Exercises 12 Note 12

2 The limit state design method

13

2.1 2.2 2.3

Purpose and objectives  13 Deterministic and probabilistic descriptions  13 The limit state design method  14 2.3.1 Characteristic, representative, and design values  16 2.3.2 Combination of actions to verify limit states  19 2.3.3 Differences between American and European standards  25 2.4 Durability 26 2.4.1 Cover 27 Exercises 32 References 32



vii

viii Contents

Section II  Components 3 Material properties

35

3.1 3.2 3.3 3.4 3.5

Purpose and objectives  36 Concrete 36 Cement 39 Concrete strength  42 Stress–strain models for concrete  45 3.5.1 Concrete models for structural analysis  46 3.5.2 Design strength of concrete  48 3.5.3 Concrete models for strength design  49 3.6 Creep 52 3.7 Shrinkage 57 3.8 Other properties of concrete  61 3.8.1 Density, the Poisson effect, and thermal deformation  61 3.8.2 Confined concrete  62 3.9 Reinforcing steel  64 3.9.1 Anchorage of reinforcing bars  67 3.9.2 Splice 73 3.10 Prestressing steel  76 3.10.1 Anchorage 78 3.10.2 Relaxation 80 3.11 Fatigue 81 Exercises 82 Notes 82 References 83

Annex A3  Confinement and concrete softening A3.1 A3.2 A3.3 A3.4

Failure criteria in concrete  85 Strength of confined concrete: fcc 87 Stress–strain models  89 Internal arches  91 A3.4.1 Internal arches in circular sections  92 A3.4.2 Internal arches in rectangular sections  94 A3.5 Concrete softening  97 Notes 98 References 98

85

Contents 

ix

Annex B3 Tension stiffening of concrete and embedded bar models100 B3.1 Tension stiffening of concrete  100 B3.2 Embedded bar models  101 References 102

4 Prestressing force

104

4.1 4.2 4.3 4.4 4.5 4.6 4.7

Purpose and objectives  104 Introduction 105 The geometry of prestressing  105 Prestressing losses  110 Losses caused by friction (Δσp,μ) 111 Losses caused by anchorage seating (Δσp,sl) 114 Losses caused by the instantaneous deformation of concrete  119 4.8 Time-dependent prestressing losses  124 Exercises 125 Notes 125 References 125

Section III  Analysis 5 Structural analysis

129

5.1 Purpose and objectives  129 5.2 Introduction 130 5.3 Types of structural analysis  131 5.3.1 Linear elastic analysis  131 5.3.2 Linear elastic analysis with redistribution  136 5.3.3 Plastic analysis and nonlinear analysis  139 5.4 Second-order structural analysis of members and systems with axial force  143 5.4.1 Elastic buckling load of isolated members  144 5.4.2 Effective length  144 5.4.3 Approximate methods of second-order analysis  148 5.5 Compression member with biaxial bending  156 Exercises 157 Notes 157 References 158

x Contents

Annex A5  Matrix formulation of geometric nonlinearity

159

A5.1 The problem of buckling eigenvalues and eigenvectors  162 Notes 163 References 163

6 Reinforcing design of D-regions

165

6.1 6.2 6.3 6.4 6.5

Purpose and objectives​  165 Introduction 166 B- and D-regions  166 Strut-and-tie models  168 Struts, ties, and nodes  170 6.5.1 Struts 170 6.5.2 Ties 171 6.5.3 Nodes 173 6.6 Unicity of strut-and-tie models  177 6.7 Design steps  178 6.8 Partially loaded areas  185 Exercises 189 Notes 189 References 189

Section IV  Design and response 7 Bending 7.1 Purpose and objectives  193 7.2 Introduction 194 7.3 Strains and equilibrium of the cross-section  198 7.3.1 Linear strain distribution  198 7.3.2 Stress distribution  199 7.3.3 Equilibrium equations  200 7.4 Linear approximation in the pre-cracking phase  205 7.5 Fiber elements  211 7.6 Ultimate limit state of bending  212 7.6.1 Strain distribution in capacity design in accordance with ACI-318  214 7.6.2 Strain distribution in capacity design in accordance with Model Code 2010  216 7.6.3 Strain distribution in capacity design in accordance with EN 1992  218 7.6.4 Stress–strain models for strength design  220

193

Contents 

xi

7.6.5 Equilibrium equations  221 7.6.6 The N–M interaction diagram  222 7.7 Uniaxial bending  230 7.7.1 The verification problem  231 7.7.2 The design problem  232 7.7.3 Optimizing reinforcement in concrete sections: The theorem of optimal section  235 7.8 Biaxial bending  240 7.8.1 The verification problem  242 7.8.2 The sizing problem  242 7.9 Reinforcement detailing  244 7.9.1 Bar spacing  245 7.9.2 Requirements for compressed reinforcement  249 7.9.3 Member detailing  250 Exercises 253 Notes 254 References 255

8 Shear 8.1 8.2 8.3 8.4 8.5 8.6 8.7

Purpose and objectives  256 Introduction 257 Classical beam theory for the pre-cracking phase  258 Post-cracking phase  261 Inclined chords  263 Influence of prestressing force  264 Post-cracking shear behavior of members without transverse reinforcement  264 8.8 Post-cracking shear behavior of members with transverse reinforcement  267 8.8.1 Post-cracking shear behavior of members with transverse reinforcement: The truss analogy  268 8.8.2 Post-cracking shear behavior of members with transverse reinforcement: EN 1992 compression field  273 8.8.3 Other compression field theories  278 8.9 Shear–bending interaction  280 8.9.1 The “shift rule”  281 8.10 Shear between web and flanges  288 Exercises 291 Note 291 References 291

256

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9 Torsion

293

9.1 Purpose and objectives  293 9.2 Introduction 293 9.3 Thin-walled tube space truss analogy  295 9.3.1 Torsional moment in concrete members  297 9.3.2 Torsional cracking moment  298 9.4 Post-cracking torsion and torsional design  299 9.5 Combination of actions  302 Exercises 306 Notes 306 References 306

10 Membranes, shells, and slabs

307

10.1 Purpose and objectives  307 10.2 Introduction 308 10.3 Membrane design in ULS  310 Case 1: No reinforcement is required  311 Case 2: Reinforcement in both orthogonal directions is required  312 Case 3: Only y-direction reinforcement is required  314 Case 4: Only x-direction reinforcement is required  315 10.4 Compression field theories  318 10.4.1 Compatibility based on deformation  324 10.5 ULS design of shells and slabs  325 10.5.1 The sandwich method  327 10.5.2 Ultimate strain compatibility between outer layers  331 10.6 Punching 333 Exercise 338 Notes 338 References 339

11 Cracking and deformation 11.1 Purpose and objectives  341 11.2 Introduction 342 11.3 Deflection limitations  342 11.4 Simplified deflection control by span–depth ratio  343 11.5 General method for the calculation of deflections in reinforced concrete members  345 11.6 Pre-cracking deformation  347 11.6.1 Short-term pre-cracking deformation  347

341

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xiii

11.6.2 Long-term deformation in the pre-cracking phase  347 11.7 Post-cracking deformation  354 11.7.1 ACI-318 354 11.7.2 Eurocode 2  359 11.8 Crack control  374 11.9 Vibrations 378 Notes 379 References 379

12 Design of prestressed elements

381

12.1 Purpose and objectives  381 12.2 Introduction 382 12.3 Design of prestressed elements  384 12.3.1 Type of cross-section  384 12.3.2 Stress limit  385 12.3.3 Deflection limit  388 12.3.4 Strength design  388 12.4 Composite beams  389 12.4.1 Initial situation (when the prestressing force is transferred)  389 12.4.2 Casting-in-place: Pouring wet concrete  390 12.4.3 Final situation  391 12.4.4 Beam–slab shear  393 Exercise 398 References 398

13 Statically indeterminate structures

399

13.1 Purpose and objectives  399 13.2 Introduction 400 13.3 Primary and secondary moments  400 13.4 Design of the prestressing layout  410 Notes  411 References 411

Section V  Residual capacity 14 Corrosion of steel in concrete and inspection of concrete structures415 14.1 Purpose and objectives​  415 14.2 Introduction 415

xiv Contents

14.3 The reinforcement corrosion initiation phase  417 14.3.1 Carbonation 417 14.3.2 Presence of chlorides  419 14.4 The reinforcement corrosion propagation process  423 14.4.1 Reinforcement stress corrosion cracking  426 14.5 Durability of concrete structures: Inspection phase  426 References 431

15 Assessment of residual capacity in concrete structures 15.1 Purpose and objectives  433 15.2 Introduction 433 15.3 Structural modeling of corrosion effects  435 15.3.1 Loss of cross-sectional area of steel reinforcing bars  435 15.3.2 Bond strength of corroded bars  437 15.3.3 The softening effect in cracked concrete  439 15.3.4 Reduction of concrete area  441 Exercise 450 Note 450 References 450 Index 453

433

Section I

Basis

Chapter 1

Principles

Panama Canal expansion project.

1.1 PURPOSE AND OBJECTIVES This chapter presents a historical and mechanical introduction to structural concrete (reinforced and prestressed concrete). It aims to provide an understanding of why reinforced concrete is such a successful composite material, its applicability, and the range of heights and spans that it covers. The chapter considers examples of arches and domes versus frames and 3D frames. Additionally, an introduction to the main weaknesses of concrete is presented, including cracking, corrosion, and durability.

DOI: 10.1201/9781003169659-2

3

4  Concrete structures

1.2 INTRODUCTION The objective of this book is to help students understand the fundamental design principles of reinforced and prestressed concrete structures. As concrete is the most commonly used construction material today, architects and engineers need to have a solid grasp of these principles. Structural concrete, also known as reinforced and prestressed concrete, is the term used to refer to the field that examines the use of concrete in structural elements. These elements require the combined use of concrete and steel bars. Mass concrete, which resembles stone, combined with steel bars can be used to construct structural elements such as beams, columns, and slabs. These elements can withstand both compression, which concrete absorbs, and tension, which steel withstands. This technology, which is widely used in the construction industry, enables engineers and architects to construct large bridge spans and open spaces in buildings. Understanding how to work with structural concrete allows engineers and architects to create strong and sufficiently ductile structural elements. Advancements in calculation methods have led to tables and manuals being replaced with computer programs. These tools provide engineers with greater calculation capacity and freedom in structural design. However, the increased number of types of hypotheses for the analysis of structural elements – such as nonlinear analysis and dynamic analysis – have made calculations more complex, requiring a deeper understanding of the hypotheses and models used in structural concrete. Concrete is a heterogeneous material that also shrinks, creeps, and cracks. As outlined in this book, the design equations for reinforced and prestressed concrete elements are based on fundamental concepts of mechanical engineering and kinematics, supplemented by empirically derived terms. 1.3 REINFORCED AND PRESTRESSED CONCRETE AS A BUILDING MATERIAL Concrete is a material that is highly resistant to compression, with a compressive strength of around 30 N/mm 2 or MPa. However, in comparison with other materials, such as steel (which has a tensile strength of around 400 N/mm 2) and wood, relatively, its compressive strength is lower. Concrete has a very low tensile strength, which is approximately ten times lower than its compressive strength. For instance, consider a beam made entirely of concrete (without any steel), as shown in Figure 1.1. The maximum load (q) that the beam can sustain is determined by the tensile strength of the concrete. When this load is reached, the beam collapses without any warning.1 In the beam shown in Figure 1.1, failure occurs at the bottom of the beam, which experiences the maximum tensile stress. It is clear that using

P rinciples  q

Cross section

5

Normal stress diagram of the cross section at the center of the span

h b

L

Figure 1.1 Mass concrete beam.

concrete without reinforcement, also known as mass concrete, is not appropriate since the compressive strength of the concrete is not utilized effectively in this structural element. Furthermore, relying on the ability of concrete to withstand tensile stresses is unreliable as there could be cracks present in the concrete that render it incapable of resisting tension on its own. To address the aforementioned limitations, it is necessary to introduce a material that can withstand the tensile stresses that concrete alone cannot endure. Steel is a suitable material for this purpose and is typically placed, in the form of bars, into the areas where tension develops, as shown in Figure 1.2a. The combination of both materials is known as reinforced concrete, with the steel bars commonly referred to as reinforcing bars or simply “rebars”. The origins of the technique of introducing tensile steel into concrete are uncertain. Figure 1.2b shows that iron elements were already being used to withstand tension in the 15th century. However, it was the mass production of steel rods in the late 19th century, combined with the invention of Portland cement, that triggered the widespread use of reinforced concrete. As will be discussed later, with a few exceptions, the contribution that concrete makes to tension is not usually considered, as shown in the normal stress diagram in Figure 1.2a. In the case of a beam subjected to simple bending, such as the example in Figure 1.2a, the compressive and tensile stresses must balance at any given cross-section. Therefore, the resultant of the compressive stresses in concrete must be equal to the resultant of the tensile stresses in steel:



Compression zone

c dAc 



s dAs (1.1)

Tensile zone

In the equation, σ c represents the stress in the concrete (with subscript c referring to concrete), while σ s represents the stress in the steel (with subscript s referring to steel). Furthermore, dAc and dAs represent the differentials of the concrete area and steel area, respectively. The emergence of high-strength steel (with a tensile strength of approximately 2000 N/mm 2) on the market made it possible to consider its use as a

6  Concrete structures q

a) Reinforcing bar

Normal stress diagram of the cross section at the center of the span

b)

Figure 1.2 (a) Concrete beam with steel in the tension zone. (b) Dome and beam with tensile steel; The Annunciation by Fra Angelico, 1430. El Prado Museum.

reinforcing material. For both high- and low-yield strength steel, the modulus of elasticity is approximately the same, with Es = 200000 N/mm 2 . This means that large strains are required to go from an initial stress of zero to a stress equal to the maximum strength of the steel in the case of high-yield strength steel. If this type of steel is used for reinforced concrete beams, such as the one shown in Figure 1.2a, the concrete will crack excessively, and large deformations will appear before the steel reaches stresses close to its maximum capacity. Thus, working with a high elastic limit steel at its maximum strength is only possible if it is introduced into the concrete with pre-deformation (or initial stress), which is known as prestressing. This technique allows the steel to reach tensions close to its elastic limit for small deformations. Figure 1.3 shows the normal stress diagrams caused by the prestressing force. In this case, the prestressing cable (steel) is subjected to an axial load of value N. In the prestressed beam shown in Figure 1.3, the final stress state is caused by the stresses resulting from the bending moment generated by

P rinciples 

7

q

e Prestressing steel

N Normal stress diagram of the cross section at the center of the span, caused by the axial force N and the bending momento N·e

Figure 1.3 Concrete beam with prestressing steel.

Final state of stresses

Figure 1.4 State of stresses in the prestressed beam.

external load q, in addition to the initial stress state caused by the prestressing steel (axial force N and bending moment N·e). Figure 1.4 illustrates how the stresses in the final loaded state are lower than in the initial unloaded state for the beam shown in Figure 1.3, specifically in the bottom fiber. This type of structural concrete is called prestressed concrete. When the falsework is removed (the formwork and the falsework are the molds and the support required until the concrete mass hardens), the steel bars are in tension while, in the compression zone, the concrete is compressed. Thus, both in reinforced and prestressed concrete, the steel ends up in a stressed state. Sometimes, reinforced concrete reinforcement is called passive reinforcement and prestressing steel is called active reinforcement. 1.4 UNIQUE FEATURES OF THE CONCRETE– STEEL COMPOSITE MATERIAL If we examine the dome of the Pantheon of Agrippa (~123 AD), constructed using mass concrete, we can see that the concrete remains in perfect condition (see Figure 1.5). However, modern concrete structures are not expected to last for more than two centuries. The primary factor contributing to the deterioration of a well-made reinforced concrete structure is its reinforcement, which corrodes as a result of various processes, primarily carbonation and chloride attack. When the steel bars corrode, the primary

8  Concrete structures

Figure 1.5 Pantheon of Agrippa, Rome, 125 AD. (From Wikipedia, https://en​.wikipedia​ .org​/wiki​/Pantheon,​_ Rome, accessed August 12, 2023.)

mechanism of reinforced concrete, which is the bond between the steel and the concrete, is compromised.​

1.4.1 Durability As a result of its synergistic relationship, the concrete–steel composite material has some special characteristics that go beyond the mechanistic introduction we have just made. Steel, in itself, is vulnerable to fire, but when covered by a layer of concrete, it becomes well-insulated. As a result, the composite can remain exposed to high temperatures for hours without its mechanical capacity being altered. The first major advantage of concrete is that it generates free lime, Ca(OH)2 , during the setting and hardening processes, which gives it a very high pH (around 12). This alkaline environment protects the steel from corrosion. Over time, CO2 in the air penetrates the pores of the concrete and reacts with the free lime, reducing the pH. This phenomenon is the main cause of concrete degradation as it leaves the steel exposed to corrosion. The surface that separates the mass of carbonated concrete from the noncarbonated one is called the carbonation front. The contamination of concrete by chlorine salts (Cl–) creates a similar effect to that described earlier. These salts can come from water, aggregates, or appear after the manufacture of concrete (e.g., caused by the use of deicing salts on roads). Reinforcement corrosion is the main cause of deterioration of structural concrete structures. One solution to this problem is to use stainless steel reinforcement or coat the bars with epoxy resin. The service life of a structure can also be extended by increasing the cover (the thickness of concrete that separates each steel bar from the outside) and/or using less porous concrete.

P rinciples 

9

One of the great challenges of using the reinforced concrete technique is the construction of durable structures. With current techniques, no concrete structure will have as long a life as medieval cathedrals or Roman bridges. In fact, great reinforced concrete works by Eduardo Torroja, Le Corbusier, and other renowned architects and engineers are being demolished as they are impossible to maintain.

1.4.2 Anchorage and adherence The tensile forces absorbed by the steel bars are transferred to the concrete through shear stress (friction) along the perimeter of the steel bars. To ensure that these shear stresses are transferred, good adhesion between the concrete and steel is necessary. Adhesion is achieved with three mechanisms. The first is of a physical–chemical nature, originating at the concrete–steel interface on the contact surface. The second mechanism is caused by concrete shrinkage upon hardening, which provides a better reinforcement grip. The third mechanism is a forced one: passive reinforcement bars used in reinforced concrete are made with ribs to improve adhesion, as seen in Figure 1.6. Prestressed concrete is a construction material that involves steel being introduced into concrete with initial stress. There are two major technical differences between pre-tensioned and posttensioned prestressed concrete, depending on whether the steel is stretched before or after the hardening of the concrete. In pre-tensioned elements, the steel is stretched inside the element mold (in the casting bed), concreted, and then the ends of the cables are released after the concrete hardens. The cable layout in these structures must be rectilinear. In posttensioned structures, ducts or sheaths are placed inside the concrete mass, through which the prestressing cables are inserted

Figure 1.6  Rebars.

10  Concrete structures

for subsequent stressing after the concrete has hardened. The cable layout in these structures can be curved. Refer to Figure 4.1 in Chapter 4 for an illustration. The transfer of traction from steel to concrete in prestressed concrete elements is more complex than in the case of reinforced concrete. To solve this problem, different ways of carrying out this transfer have been developed, resulting in numerous prestressing systems (both posttensioning and prestressing) and multiple patents. When using reinforced concrete, there is a limit to the span that can be bridged with a horizontal beam, with the span of a straight reinforced concrete beam rarely exceeding 10 m. Large spans in reinforced concrete are achieved through structures whose geometry is such that a structure basically works in compression, with very limited bending stresses, as seen in the case of arches and vaults. Eduardo Torroja was a great master in the art of making the most of reinforced concrete. Figures 1.7 and 1.8 showcase two of his works, El Frontón de Recoletos (Figure 1.7) and El Viaducto Martín Gil (Figure 1.8). The latter is an extremely unique construction because the reinforcing cage served as support for the formwork. Unfortunately, El Frontón de Recoletos was destroyed during the Spanish Civil War. The use of prestressed concrete allows for greater spans to be bridged, and in fact, it is the preferred composite material for most large-span bridges built today. Figure 1.9 shows a bridge constructed with posttensioned prestressed concrete.

Figure 1.7 Frontón de Recoletos.

P rinciples 

11

Figure 1.8 Viduaducto Martín Gil.

Figure 1.9 Huétor-Santillán Bridge (Granada, Spain).

1.5 NORMATIVE The use of concrete is highly significant, which is why the behavioral models of concrete and steel, the calculation hypotheses, and their range of application has been standardized. In Europe, Eurocode 2 is the standard for structural concrete, while Eurocode 1 is used for the study of actions on structures. In many American countries, and parts of the world, the American Concrete Institute ACI-318 code (available in English and Spanish) is used as the standard. The CEBFIP Model Code (Euro-International Concrete Committee/International Federation of Prestressed), and the LRFD Bridge Design Specifications from

12  Concrete structures

the American Association of State Highway and Transportation Officials (AASHTO) are some other professional standards of interest. EXERCISES 1. In Figure 1.3, where would the prestressed beam fail if the prestressing force was too large and the beam was only subjected to the action of its self-weight? 2. For a continuous beam with three spans, which parts would require reinforcement and which parts would not? NOTE 1. The most common sign criterion in structural concrete is that compression is positive.

Chapter 2

The limit state design method

Partial Factors

Verification

Materials Concrete (fck, Ecm) Steel (fyk, Es)

γc γs

φ

Europe

Structural analysis

ULS SLS

America

Actions Permanent (G) Variable (Q) …

γG γQ

Combination of actions

The limit state design method.

2.1 PURPOSE AND OBJECTIVES Chapter 2 introduces both deterministic and probabilistic methods of analysis, focusing on the basics of the limit state method according to EN 1990 (EN 1990 2023) as the basis of structural and geotechnical design. The chapter includes a detailed example and highlights the main differences between European and American codes when using the limit state method. The chapter specifically analyzes the different perspectives on the safety coefficients of materials. 2.2 DETERMINISTIC AND PROBABILISTIC DESCRIPTIONS In the design of structures, data on actions, material properties, environmental conditions, and geometry are necessary for calculations. Two options can be considered for finding the values of this data: DOI: 10.1201/9781003169659-3

13

14  Concrete structures

1. Assume that the values are known and unique. 2. Suppose that each piece or set of data has a statistical distribution associated with it. The first approach is called the deterministic approach, which was used in the first concrete standards. With this approach, an entire structural calculation was carried out using fixed input values. The results obtained were then compared with the admissible values, where a safety coefficient was introduced to consider the inherent uncertainty in the process.

 admisible 

 max 

The second approach is more realistic and involves assigning a probability density function to each of the variables involved, allowing for the safety of a structure to be evaluated in terms of probability. The limit state method is a hybrid of both approaches, where a set of states is established that a structure must not exceed within a certain period so that a certain level of functionality can be guaranteed. This period is known as the design service life of a structure. Table 2.1 shows the design service life categories for buildings in EN 1990 (Table A.1.2.).​ Table 2.1  Design service life categories for buildings Category of buildings Monumental building structures Building structures not covered by another category Agricultural, industrial, and similar structures Replaceable structural parts Temporary structuresa,b a

b

Design service life,Tlife (years) 100 50 25 ≤10

For structures or parts of structures that can be dismantled in order to be reused.

For specific temporary structural members, such as anchors, Tlife ≤ 2 years can be considered.

2.3 THE LIMIT STATE DESIGN METHOD When a structure is designed and built, it must be done in such a way that the conditions of use that are required remain, without requiring operations that are more complicated than just simple maintenance, for the whole of its design service life. Different situations can occur during the design service life. The behavior of the structure has to be checked for certain situations called design situations. EN 1990 considers the following design situations (see Table 2.2).

T he limit state design method  15 Table 2.2  Design situations (Table 5.1 of EN 1990) Design situation Persistent Transient

Accidental Seismic Fatigue

Conditions Normal use and exposure Temporary use and exposure during a period much shorter than the design service life of the structure Exceptional conditions or exposure Exceptional conditions during a seismic event Conditions caused by repeated load or deformation-induced stress cycles

Examples In everyday use Execution, repair, or temporary environmental influence Flooding, fire, explosion, impact, or local failure In earthquakes Traffic loads on a bridge, wind-induced vibration of chimneys, or machineryinduced vibration

There has to be a guarantee that a limit state will not be reached in any of these design situations. Limit states are defined as those for which a structure does not satisfy the safety and/or functionality requirements of its design. Limit states are grouped into two categories: • Ultimate limit state (ULS), associated with losses of human life or injury to people or unacceptable economic or environmental losses. According to EN 1990, ultimate limit states are produced by failures of foundations or structures, or any part of these elements. In particular, EN 1992 includes ultimate limit state failures as those caused by bending, shear, torsion, punching, or failures in the discontinuity regions. • Serviceability limit state (SLS), associated with a lack of serviceability in functionality, comfort, or the aesthetic appearance of a structure. SLS can be reversible or irreversible. EN 1992 includes cracking, deformation, and vibrations as SLS limitations. The limit state design method focuses on the study of actions and structural models. Actions performed on a structure can be classified according to several criteria, including: • Manner of application: direct (loads) or indirect (imposed deformations, thermal actions, etc.). • Variation over time: – G, permanent action, such as self-weight or the weight of permanent elements. – Q, variable action, such as usage overload, wind, or snow. – A, accidental action, such as explosions or vehicle impact. – A E , seismic action.

16  Concrete structures

• Spatial variation of the action: fixed or free. • Structural response to the action: static or dynamic. The durability of a structure is a premise of each project, which means that the materials are supposed to remain in perfect condition during the design service life of a structure. To guarantee this, preventive measures are taken depending on the type of environment in which a structure is located, and the quality of the materials is ensured throughout the construction process.

2.3.1 Characteristic, representative, and design values The limit state design method relies heavily on the concept of characteristic value for both material properties and external actions. This value is closely tied to the statistical distribution of material properties, such as resistance or the modulus of elasticity, and actions. It is the most important value to consider when analyzing the statistical distribution, and it can correspond to either the mean value or a particular percentile of the distribution. 2.3.1.1 Materials When working with materials in the limit state design method, two situations need to be considered: ultimate resistance (ULS) and the limitation of deformations (SLS). For each of these options, the characteristic value of materials differs conceptually, as well as the level of safety required in calculations. For example, when studying the deformation of a concrete beam, the average value of the modulus of elasticity is crucial, as all the concrete in a beam contributes to its deformation. However, when analyzing the strength of a beam, a certain percentile must be considered, as failure occurs at the weakest point. In other words, a chain breaks at its weakest link, but its deformation is the sum of the deformations of all the links. For the ultimate strengths of materials, a normal distribution is assumed, and the characteristic value of ultimate resistance is the value that provides a 95% guarantee (i.e., only 5% of tested specimens have a resistance lower than the characteristic value). This value is determined by the mean value (fm) and the standard deviation (σ) (Hernández and Gil-Martín 2023):

fk  fm  1.65

For the stiffness of materials (the modulus of elasticity), which is necessary for calculating deformations, the characteristic value corresponds to the average value. Although the characteristic value (X k) is the most important value to consider, in the case of strength design, a safety factor must be applied to

T he limit state design method  17 Table 2.3  Partial factors for materials Design situation – Limit state Persistent and transient design situation Fatigue design situation Accidental design situation Serviceability limit state

γS for reinforcing and prestressing steel

γC and γCE for concrete

1.15

1.50

1.15 1.00 1.00

1.50 1.15 1.00

Source: From EN 1992 Table 4.3.

prevent this value from being exceeded. This is achieved through the use of a partial factor for material properties (γm), with specific values for concrete (γC), the modulus of elasticity of concrete (γCE), and reinforcing or prestressing steel (γS).

Xd  

Xk m

The design value for a material property is known as Xd and is calculated using ς, which considers size, humidity, temperature effects, and load duration (as will be seen in Chapter 3), along with the partial factors. The partial factors for material properties (γm) for concrete and steel (γC , γCE , and γ S) are presented in Table 2.3 in accordance with EN 1992. It is evident that the value of the partial factor for concrete is higher than for steel, given the greater deviation from the mean of its characteristic value. For instance, in cases where persistent situations are concerned, if the characteristic strength values are 30 MPa for concrete and 500 MPa for steel, their corresponding design values are 20 and 435 MPa, respectively. This means that, as a result of the manufacturing process, there is the same guarantee that a B-500 steel bar (characteristic strength of 500 MPa) will not exceed 435 MPa (= 500/1.15) and a C-30 concrete (characteristic strength of 30 MPa) will not exceed 20 MPa (= 30/1.5). 2.3.1.2 Actions In the case of actions, the value of the characteristic value depends on the type of action being considered, and it can be the mean value, an upper or lower value, or a nominal value. For permanent actions, a single Gk value is typically used if the variability is small. However, if the variability is large, two characteristic values, Gk,inf and Gk,sup, associated with the 5% and 95% quantile, respectively, are used. In the case of variable actions, the characteristic value corresponds to confidence levels of 2% and 98% for Qk,inf and Qk,sup, respectively, or the mean value.

18  Concrete structures

The representative value of an action is the value of the action that acts on the structure, and it is generally the characteristic value. In fact, for permanent actions, the characteristic and representative values coincide. When considering several actions, or combinations of actions, acting on a structure, the probability that they all appear simultaneously decreases. For example, several variable loads act on a bridge, such as traffic, snow, wind, and thermal loads, but the probability that all of them, their characteristic values, act at the same time is lower than the probability that only some or part of them act simultaneously. Therefore, it is logical to consider only fractions of the variable loads when they act at the same time. Considering a combination of actions means considering that the simultaneous action of all the loads is less likely, which is done by introducing the combination factors (ψ). Three representative values of variable actions appear in the standards: ψ 0Qk combination value ψ 1Qk frequent value (tied to a 5% time slice) ψ 2Qk quasi-permanent value (tied to a 50% time slice) A statistical description of the different values of a variable action is shown in Figure 2.1. The values of Ψ are determined by each standard. Table 2.4 provides a summary of the values specified in Annex A of EN 1990 for building structures. So far, this chapter has discussed two types of values for actions: the characteristic value, which is the most suitable value, and the representative value, which considers the probability of simultaneous occurrence. The next step is to determine the safety level or safety factor for these actions, which is achieved by using the partial factor for actions, γF. The design value of an action is referred to as the Fd value:

Fd   F  iFK

Value of the variable action Q

Maximum value Characteristic value Combination value Frequent value Quasi-permanent value

Time

Figure 2.1 Statistical description of a variable action.

Distribution

T he limit state design method  19 Table 2.4  Combination factors for buildings Action

Ψ0

Ψ1

Ψ2

Imposed loads in buildings: A. Domestic, residential areas B. Office areas C. Congregation areas D. Shopping areas E. Storage areas F. Traffic area vehicle weight ≤ 30 kN G. Traffic area vehicle weight from 30 to 160 kN H. Roofs Snow (H≤1000 m a.s.l.) Snow (H>1000 m a.s.l.) and Finland, Iceland, Norway, and Sweden Wind

0.7

0.5

0.3

0.7 0.7 0.7 1.0 0.7

0.5 0.7 0.7 0.9 0.7

0.3 0.6 0.6 0.8 0.6

0.7

0.5

0.3

0 0.5 0.7

0 0.2 0.5

0 0 0.2

0.6 0.6

0.2 0.5

0 0

Temperature (non-fire) Source: Adapted from EN 1990 Table A.1.7.

Table 2.5  Partial factors for actions Persistent and transient situation Type of action Permanent Prestressing Variable Accidental

Favorable effect γG = 1.00 γP = 1.00 γQ = 0.00 —

Unfavorable effect γG = 1.35 γP = 1.00 γQ = 1.50 —

Accidental situation Favorable effect γG = 1.00 γP = 1.00 γQ = 0.00 γA = 1.00

Unfavorable effect γG = 1.00 γP = 1.00 γQ = 0.00 γA = 1.00

The value of γF considers the probability of the unfavorable deviations of an action when using representative values. Factor γF can have different values depending on whether a load has favorable or unfavorable effects on a structure. For some of the γF values specified in Annex A of EN 1990, please refer to Table 2.5.

2.3.2 Combination of actions to verify limit states First, several factors must be taken into account, including all the actions to which a structure will be subjected, their design values, the design values of material properties, geometric data, and the type of structural analysis

20  Concrete structures

used. Once all of these factors have been considered, each of the limit states needs to be checked. In order to ensure that none of the limit states are exceeded during the design service life of a structure, every possible way in which actions can act on a structure (combinations) during its design life (project situations) should be considered. The limit state design method provides a series of action combinations to verify the structural reliability of each limit state and project situation. Therefore, specific combinations of actions are established for each project situation. Let Ed be the effect of actions on the structure (such as forces and moments in elements, stresses, deformations, and displacements) for a specific combination of actions, and let Rd be the resistance of the structure:

E d = E(Fd , ad , Xd ) R d = R(ad , Xd )



E(…) is the combined effect of the enclosed variables, R is resistance, Fd is the design values of the actions, ad is the design values of the geometrical properties, and Xd is the design values of the material properties. To verify the ultimate limit state (ULS), the following conditions must be satisfied: • Static equilibrium:

E d,dst ≤ E d,stb

Ed,dst is the design value of the destabilizing effects of the actions, and Ed,stb is the design value of the stabilizing effects of the actions. For example, consider the earth retaining wall in Figure 2.2. In order to ensure the stability of the wall, not only is it necessary to check that it does not break, but also that it cannot slide or overturn. • Limit state of failure (ULS) for excessive deformation:

Ed ≤ R d

Figure 2.2 Loss of stability of an earth retaining wall.

T he limit state design method  21

The limit state of failure caused by excessive deformation is considered for each ultimate limit state, except for fatigue (as defined in EN 1990). For each project situation, the value of Ed is calculated as follows: For persistent and transient design situations:



G kj   PPk   Q1Q k1 

Gj

j1



Qi

 0iQ ki (2.1)

i 1

For accidental design situations:

G

kj

 Pk  Ad  (11 or  21)Q k1 

j1

 Q 2i

ki

(2.2)

i 1

For seismic design situations:

G

kj

 Pk   I AEd 

j1



2i

Q ki (2.3)

i 1

where: + means “combines with” Σ denotes the combination of the enclosed variables Gkj characteristic value of permanent action j Pk characteristic value of a prestressing force Qk1 characteristic value of the leading variable action 1 Qki characteristic value of an accompanying variable action i Ad design value of the accidental action A Ed design value of the seismic action in an ultimate limit state γ Gj partial factor for permanent action j γ P partial factor for the prestressing forces γ Qi partial factor for variable action i γ I importance coefficient (EN 1998) • The serviceability limit state (SLS) verification can be stated as:

E d ≤ Cd

with Cd as the limit value corresponding to the serviceability limit state to be studied (e.g., Cd = 10 mm for the serviceability limit state of the deflections of a beam). For each serviceability limit state, the value of Ed is calculated for the following combinations: For the characteristic combination:

G j1

kj

 Pk  Q k1 

 Q 0i

i 1

ki

(2.4)

22  Concrete structures

For the frequent combination:

G

kj

 Q

 Pk  11Q k1 

2i

j1

ki

(2.5)

i 1

For the quasi-permanent combination:

G

kj

j1

 Pk 

 Q 2i

ki

(2.6)

i 1

For the seismic situation:

G j1

kj

 Pk  AEd,SLS 



2i

Q ki (2.7)

i 1

where A Ed,SLS is the design value of the seismic action in a serviceability limit state (EN 1998). As stated earlier, the coefficients of reduction in material resistance (γ s and γ c) for verifying the ULS are specified in Table 2.3. However, for verifying the SLS, a partial safety factor of γ m = 1 is used. Furthermore, the partial safety factors for actions are only used to verify the ultimate limit states (see Equations 2.1 to 2.7). The approaches used for ULS and SLS are fundamentally different: by using ULS we aim to stay as far away as possible from the point of failure by increasing the actions and reducing the resistance, however, by using SLS, we aim to understand how a structure will behave in reality, and therefore do not reduce or increase resistance or actions. For example, we want to know the actual deformation of a beam or the true width of a crack opening, but we want to stay as far away as possible from the point where the beam would actually collapse. The seismic situation is different from other situations as different performance objectives are established for different intensities of ground motions (Aschheim, Hernández-Montes, and Vamvatsikos 2019). EXAMPLE 2.1 LOAD COMBINATION The roof of a building consists of a series of continuous beams separated from each other by 5 m, as shown in Figure 2.3. Each beam has four pin supports that divide the beam into three spans of 6 m each. The cross-section of each beam is rectangular with dimensions of 0.5 × 0.3 m. Each beam is designed to withstand the following loads: • • • •

The weight of the roof, which is 2 kN/m2 . Its own weight. A variable snow load of 0.6 kN/m2 . A variable imposed load of 1.5 kN/m2 .

T he limit state design method  23 B

A 6m

Cross section

6m

6m 0.5 m

Side view Plan view

0.3 m

5m

Figure 2.3  Continuous beam.

The design bending moment at the inner support of an inner beam for the ultimate limit state is calculated here for both persistent and transient design situations.

SOLUTION According to Eurocode 0, the general format of the effects of actions is:

 j 1

Gkj  PPk   Q1Qk1 

Gj



Qi

 0iQki

i 1

The density of the concrete is 2500 kg/m3, so the characteristic values of the actions acting on the structure are: G = 2500 kg/m2 · 9.81 m/s2 · 0.5 m · 0.3 m + 2000 N/m2 · 5 m = = 13.7 kN/m Qsnow = 0.6 kN/m2 · 5 m = 3.0 kN/m Qimp = 1.5 kN/m2 · 5m = 7.5 kN/m ψ coefficients from Table 2.4 are used to calculate the representative values of the variable actions. The value of ψ0 is 0.5 for the snow load and 0.7 for the imposed load. In order to obtain the design value of the actions, the representative values are multiplied by the corresponding partial factor (γG or γQ):

24  Concrete structures

G Qsnow and Quse

1.35 for unfavorable effect 1.50 for unfavorable effect

1.0 for favorable effect 0 for favorable effect

The table in Figure 2.4 shows the bending moment at point B for each of the three load cases studied, where the load per unit length is represented by q, and l is the length of the beam span. The effect of each load case on the bending moment at point B can be either favorable or unfavorable, depending on which span the load acts upon. If the load is located in the first or second span, it has an unfavorable effect on the bending moment at point B. Conversely, if the load acts on the third span, it has a favorable effect on the bending moment at point B. At this stage, an engineer can either: (1) rely on intuition to estimate the most unfavorable situation, or (2) perform calculations for every possible scenario and select the worst-case scenario. For this exercise, the most unfavorable load combination is easy to identify. The imposed load is the first variable action (Q1), because it is the heaviest load. Considering the unfavorable partial factor (γQ) in the first and second spans and the favorable partial factor in the third span, the bending moment at B is: q 1 A

B Deformed shape

q 2 A

B

q 3 A

Moment in B

Figure 2.4  Load cases.

B

Case 1

Case 2

Case 3

Case 1+2+3

-0.0667 q l2

-0.0500 q l2

0.0167 q l2

-0.1000 q l2

T he limit state design method  25

MB = 1.35 · 13.7 · 62 · (–0.0667 –0.0500) + 1.00 ·13.7 · 62 · (0.0167) + 1.5 ·7.5 · 62 · (–0.0667 –0.0500) + 1.5 · 0.5 · 3.0 · 62 · ( –0.0667 –0.0500) = –126.18 kN·m However, in many situations, using intuition to identify the worst-case scenario might not be possible as there are many combinations involved. In such cases, it is necessary to calculate every possible combination to determine the worst-case scenario. The following table shows the bending moment at point B for every possible load combination in the example analyzed. Combination 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Q1

First span

Imposed load Imposed load Imposed load Imposed load Imposed load Imposed load Imposed load Imposed load Snow Snow Snow Snow Snow Snow Snow Snow

Unfavorable Unfavorable Unfavorable Unfavorable Favorable Favorable Favorable Favorable Unfavorable Unfavorable Unfavorable Unfavorable Favorable Favorable Favorable Favorable

Second span Unfavorable Unfavorable Favorable Favorable Unfavorable Unfavorable Favorable Favorable Unfavorable Unfavorable Favorable Favorable Unfavorable Unfavorable Favorable Favorable

Third span Unfavorable Favorable Unfavorable Favorable Unfavorable Favorable Unfavorable Favorable Unfavorable Favorable Unfavorable Favorable Unfavorable Favorable Unfavorable Favorable

MB (kN·m) –115.2 –126.2 –82.3 –93.2 –71.3 –82.3 –38.3 –49.3 –111.1 –121.5 –80.2 –90.5 –69.9 –80.2 –39.0 –49.3

The table shows that the initial intuitive idea was correct and the maximum bending moment (in absolute value) in the inner support is –126.18 kN·m.

2.3.3 Differences between American and European standards ACI and AASHTO are the acronyms for the American Concrete Institute and the American Association of State Highway and Transportation Officials, respectively. These organizations publish technical standards that are of great technical interest. The ACI-318 (2019) and AASHTO LRFD Bridge Design (AASHTO 2020) standards are highly influential in the field of structural concrete at an international level, so it is essential to

26  Concrete structures

understand the key differences between these American standards and the Eurocodes. Although this textbook focuses on the Eurocodes, it also considers some specific requirements of these American standards. For resistance verification (ULS), the American standards use a single resistance reduction factor (ϕ). In contrast, as we have previously seen, the Eurocodes use different resistance reduction coefficients for concrete and reinforcement, γ c and γ s, respectively (see Table 2.3). Regarding SLS, the American standards, like the Eurocodes, use ϕ = 1. Nominal strength, which refers to the strength of a structure (or element or cross-section) obtained from the strength properties of a material that are not affected by the reduction factors, is represented by a subscript n (e.g., Nn, M n, Vn). Design strength is the nominal strength reduced by ϕ. To verify each ULS, the design strength must be greater than the required strength: Design strength = ϕ (Nominal strength) ≥ Required strength Factor ϕ is a strength reduction factor, equivalent to factors γ c and γ s in Eurocode 2, which considers the contributions of steel and concrete separately. ACI-318 specifies the value of the ϕ-factor for different actions or structural elements. In the case of cross-sections under bending actions (M and/or N), if the bending capacity is more affected by concrete than by steel (i.e., compression-controlled, e.g., more concrete is involved when a crosssection is used as a column than if it is used as a beam), then the ϕ-factor is lower and the overall strength is further reduced. Eurocode 2 explicitly considers this effect dividing the strength of the concrete by γ c (usually 1.5) and the strength of the steel by γ s (typically 1.15). The value of the ϕ-factor in bending ranges from 0.65 to 0.9 (it is interesting to note that if γ c = 1.5 and γ s = 1.15, then 1/γ c = 0.67 and 1/γ s = 0.87, which are very close to the limits established for ϕ in ACI-318). The ϕ-factor for bending (N and/or M) in ACI-318 depends on the tensile strain in the extreme tension reinforcement, ε t. Note that ε t has a direct relation with the depth of the neutral fiber. For instance, if all the concrete in the cross-section is involved in the failure of a column, the value of ϕ is 0.65. In the case of a beam subjected to simple bending (N = 0), if ε t is larger than 0.005 (tension-controlled element, or, equivalently, the depth of the neutral fiber is small), the coefficient ϕ is 0.90. For structural elements subjected to bending–compression, the value of ϕ can be obtained by linear interpolation between 0.65 and 0.90. For shear and torsion, ϕ = 0.75. 2.4 DURABILITY During the design service life of a structure, no limit states should be exceeded if planned maintenance is performed. This means that it is

T he limit state design method  27

assumed that the materials will not deteriorate during the design life of a structure. To ensure the durability of a structure, preventive measures are taken. The primary cause of reinforced concrete deterioration is the corrosion of the reinforcement. In fact, unreinforced concrete structures like the Pantheon of Agrippa are still standing after almost 2000 years (see Figure 1.5). The effectiveness of corrosion protection for regular steel (carbon steel) reinforcement depends on the quality and thickness of the concrete and the extent of cracking in the cover zone. In general, higher strength concrete is more impermeable and protects the reinforcement more successfully. Current standards classify structures according to their environmental vulnerability. In EN 1992, Chapter 6 (EN 1992-1-1 2023), the environmental conditions of a structure (exposure classes) are classified according to the information in Table 2.6. According to the recommendations in EN 1992 (§6.2): “in order to achieve the required design service life of the structure, adequate measures shall be taken to protect each concrete member against the relevant environmental actions”, a series of aspects should be considered: • • • • • •

Structural conception Material selection Construction details Execution Quality management Lifetime maintenance

If necessary, special measures can be taken to prevent corrosion, such as applying coatings to concrete surfaces, using cathodic protection, employing corrosion inhibitors, or reinforcing steel with coatings such as galvanization or epoxy covers. Another option is to use stainless steel reinforcement (see additional provisions in Annex Q of EN 1992). The objective of all these requirements is to prevent or mitigate the degradation of structural elements and to guarantee the strength of both steel and concrete during the service life of a structure.

2.4.1 Cover The term “cover” refers to the distance between the reinforcement and the outer concrete surface, as indicated by c in Figure 2.5. The cover performs several important roles, including ensuring adequate stress transfer between the reinforcement and the concrete, protecting the reinforcement against corrosion, and providing fire resistance.

28  Concrete structures Table 2.6  E xposure classes related to environmental conditions (EN 1992 Table 6.1) Class

Description of the environment

Informative examples where exposure classes may occur

1.   No risk of corrosion or attack For concrete without reinforcement or embedded metal: X0 All exposure except where Pure concrete members without any there is freeze/thaw, reinforcement. abrasion or chemical attack. 2.   Corrosion of embedded metal induced by carbonation Where concrete containing steel reinforcement or other embedded metal is exposed to air and moisture, the exposure shall be classified as follows: XC1 Dry Concrete inside buildings with low air humidity, where the corrosion rate is insignificant. XC2 Wet or permanent high Concrete surfaces subject to longhumidity, rarely dry term water contact or permanently submerged in water or permanently exposed to high humidity; many foundations; water containments (not external). NOTE: Leaching could also cause corrosion (see EN 1992 §6.3(5), XA classes). XC3 Moderate humidity Concrete inside buildings with moderate humidity and not permanent high humidity. External concrete sheltered from rain. XC4 Cyclic wet and dry Concrete surfaces subject to cyclic water contact (e.g., external concrete not sheltered from rain as walls and facades). 3.   Corrosion of embedded metal induced by chlorides, excluding seawater Where concrete containing steel reinforcement or other embedded metal is subject to contact with water containing chlorides, including deicing salts, from sources other than from seawater, the exposure shall be classified as follows: XD1 Moderate humidity Concrete surfaces exposed to airborne chlorides. XD2 Wet, rarely dry Swimming pools. Concrete components exposed to industrial waters containing chlorides. NOTE: If the chloride content of the water is sufficiently low then XD1 applies. XD3 Cyclic, wet and dry Parts of bridges exposed to water containing chlorides. Concrete roads, pavements and car park slabs in areas where de-icing agents are frequently used. (Continued)

T he limit state design method  29 Table 2.6  (Continued) Class

Description of the environment

Informative examples where exposure classes may occur

4.   Corrosion of embedded metal induced by chlorides from seawater Where concrete containing steel reinforcement or other embedded metal is subject to contact with chlorides from sea water or air carrying salt originating from seawater, the exposure shall be classified as follows: XS1 Exposed to airborne salt but Structures near to or on the coast. not in direct contact with seawater XS2 Permanently submerged Parts of marine structures and structures in seawater. XS3 Tidal, splash, and spray zones Parts of marine structures and structures temporarily or permanently directly over seawater. 5.   Freeze/thaw attack Where concrete is exposed to significant attack by freeze/thaw cycles whilst wet, the exposure shall be classified as follows. An XF-classification is not necessary in cases where freeze/thaw cycles are rare. XF1 Moderate water saturation, Vertical concrete surfaces exposed to without deicing agent rain and freezing. XF2 Moderate water saturation, Vertical concrete surfaces of road with deicing agent structures exposed to freezing and airborne deicing agents. XF3 High water saturation, Horizontal concrete surfaces exposed without deicing agents to rain and freezing. XF4 High water saturation with Road and bridge decks exposed to deicing agents or seawater deicing agents; concrete surfaces exposed to direct spray containing deicing agents and freezing; splash zone of marine structures exposed to freezing. 6.   Chemical attack Where concrete is exposed to chemical attack from natural soils and ground water, the exposure shall be classified as follows: XA1 Slightly aggressive chemical Natural soils and ground water environment according to Table 6.2 of EN 1992. XA2 Moderately aggressive Natural soils and ground water chemical environment according to Table 6.2 of EN 1992. XA3 Highly aggressive chemical Natural soils and ground water environment according to Table 6.2 of EN 1992. 7. Mechanical attack of concrete by abrasion Where concrete is exposed to mechanical abrasion, the exposure shall be classified as follows: XM1 Moderate abrasion Members of industrial sites frequented by vehicles with pneumatic tires. (Continued)

30  Concrete structures Table 2.6  (Continued) Class

Description of the environment

XM2

Heavy abrasion

XM3

Extreme abrasion

Informative examples where exposure classes may occur Members of industrial sites frequented by forklifts with pneumatic or solid rubber tires. Members of industrial sites frequented by forklifts with elastomer or steel tires or track vehicles.

Ø20 Ø10 20 mm 10 mm Mechanical cover 75 mm

c =55 mm

Figure 2.5  Cover.

The nominal cover (EN 1992 §6.5) must be clearly specified in drawings, and it is calculated by adding a minimum cover value (cmin) to an allowance for deviation (Δcdev) in the design:

c nom  c min  cdev (2.8)

The allowance for deviation (Δcdev) is specified in Table 6.7 of EN 1992, with a general value of 10 mm. However, in cases where concrete is cast against unprepared ground, this value is at least 75 mm. The minimum cover (cmin) is determined by three factors: a minimum value for durability, a minimum value for bond requirements, and 10 mm. The minimum cover required for durability (cmin,dur) is dependent on the design service life, exposure class (EC), and exposure resistance class (ERC). Annex P of EN 1992 provides an alternative method for determining the minimum cover for durability without the use of exposure resistance classes.

T he limit state design method  31

EXAMPLE 2.2 DURABILITY In order to determine the required cover for a reinforced concrete beam located at the beach of La Malagueta (Málaga), the specific environmental conditions must be taken into consideration. Although the beam is not in direct contact with seawater, it is exposed to airborne salt, which can lead to corrosion over time. The exposure class for the beam, according to Table 2.6, is XS1. This information is important as the minimum cover required to ensure adequate durability and corrosion resistance of the reinforcement is given as a function of the exposure class and the design service life. The design service life for the structure is 100 years, and the execution tolerance is class 1 (EN 13670). C30 concrete with a maximum aggregate size of 25 mm is used. The bottom longitudinal reinforcement consists of 5 Ø20 bars, while the shear reinforcement is made up of Ø10@10 cm hoops. In accordance with Annex P of EN 1992, the minimum strength class required is C30/37 (Table P.4). From Table P.1, which is formulated for a S4 structural class and a design working life of 50 years, it is obtained that, for a design service life of 100 years, the structural class should be increased to class 6 (S6). So, according to Table P.2, the minimum cover required for durability (cmin,dur) is 45 mm. In this case, there is no reduction in the minimum cover because there is no additional protection (Δcdur,red = 0) and no additional cover is needed for abrasion (Δcdur,abr = 0) (§6.5.2 of EN 1992). Table 6.5 shows that the minimum cover for bond requirements is equal to the bar diameter (i.e., cmin,b = 10 mm for Ø10 and 20 mm for Ø20). The allowance for design deviation (Δcdev) for tolerance class 1 is 10 mm (see Table 6.7 of EN 1992). cnom  cmin  cdev  55 mm cmin  max  cmin,dur  cdur ,red  cdur , abr ; cmin,b ;10 mm  45 mm



cmin,b

10 mm and 20 mm

cmin,dur

 45 mm

cdur , 

0

cdur ,st  0 cdur , add  0 cdev  10 mm

32  Concrete structures

Both transverse and longitudinal reinforcements require a nominal cover of 55 mm. Therefore, the mechanical cover should be calculated by using the nominal cover of the transverse reinforcement, resulting in a required mechanical cover of 75 mm (refer to Figure 2.5)

EXERCISES

1. For Example 2.1, determine the position of a moving punctual load of 100 kN that would produce the most unfavorable effect in: a. Maximum positive (sagging) bending moment b. Maximum negative (hogging) bending moment c. Shear 2. For serviceability limit state (SLS) combinations, the load factors and the strength reduction factors are both equal to 1. Is this somehow related to the consequences of not verifying an SLS?

REFERENCES AASHTO. (2020). AASHTO LRFD Bridge Design Specifications. 9th ed. Washington, DC: American Association of State Highway and Transportation Officials. ACI Committee 318. (2019). Building Code Requirements for Structural Concrete. Farmington Hills, MI: American Concrete Institute. Aschheim, M., Hernández-Montes, E., and Vamvatsikos, D. (2019). Design of Reinforced Concrete Buildings for Seismic Performance: Practical Deterministic and Probabilistic Approaches. Boca Raton: CRC Press, Taylor & Francis. EN 1990. (2023). Eurocode. Basis of Structural and Geotechnical Design. Edited by CEN. Brussels: European Committee for Standardization. EN 1998-1. (2004). Eurocode 8: Design of Structures for Earthquake resistance – Part 1: General Rules, Seismic Actions and Rules for Buildings. Brussels: European Committee for Standardization. Hernández, L., and Gil-Martín, L.M. (2023, January). “A Comparison of the Most Important Properties of Structural Concrete: European Standards versus American Standards.” European Journal of Environmental and Civil Engineering: 1–14. EN 1992-1-1. (2023). Eurocode 2: Design of Concrete Structures - Part 1–1: General Rules and Rules for Buildings, Bridges and Civil Engineering Structures EN 1992-1-1. Brussels: European Committee for Standardization.

Section II

Components

Chapter 3

Material properties

Concreting a pile. High Speed Line, Sans-La Sagrera section. Barcelona. DOI: 10.1201/9781003169659-5

35

36  Concrete structures

3.1 PURPOSE AND OBJECTIVES This chapter deals with the mechanical characteristics of the two main components of structural concrete: concrete and reinforcement. Since this book focuses on the behavior of materials, this chapter delves into the mechanical responses of these materials. Furthermore, it includes a discussion on cement, an essential component of concrete. The chapter provides an in-depth study of the mechanical models of concrete and steel, focusing on the models used in current standards such as EN 1992-1-1 (2023), also known as Eurocode 2. Additionally, it makes a clear distinction between the short-term and long-term behavior of steel and concrete and establishes the failure criteria for both of them. 3.2 CONCRETE Concrete is composed of three main components: cement, water, and aggregates. Up to 35% of the weight of cement can be replaced with additions, such as natural pozzolans, fly ash, blast furnace slag, and silica dust. Additives ( 0.5fcm).

48  Concrete structures

3.5.1.2 Nonlinear stress–strain model EN 1992 provides the short-term stress–strain model in Figure 3.13 for nonlinear calculations that can be used for the whole range of strains. The diagram in Figure 3.13, for short-term loads, corresponds to the following formulation:

c 

  2 fcm for c  cu1 1  (  2)

where

   c /  c1

(3.7)

  1.05 E cm c1 / fcm 1/ 3  2.8%o c1[%o]  0.7 fcm

c  cu1[%o]  2.8  14(1  fcm / 108)4  3.5%o with σ c Compressive stress in concrete for a given value of ε c ε c Compressive strain in concrete ε c1 Compressive strain in concrete for maximum stress ε cu1 Ultimate compressive strain in concrete4 The stress–strain model shown in Figure 3.13 is not commonly used in practice because the modulus of elasticity at one point (i.e., the slope of the curve) depends on the stress level, making it difficult to apply. The model in Equation 3.7 considers compression in only one direction, which corresponds to a test on a cylindrical specimen. When there are stresses in two or three directions, more complex models are needed to accurately represent concrete behavior. The diagram in Figure 3.12 includes two-dimensional and three-dimensional models to consider stresses in multiple directions. The effect of concrete confinement, which is discussed in detail in Annex A of this chapter, allows the three-dimensional effects to be considered.

3.5.2 Design strength of concrete When designing a concrete cross-section, a partial factor (γ c) must be applied to the characteristic strength of the concrete. Furthermore, since the characteristic strength is defined for concrete subjected to short-term loads (with a duration of one to two minutes), it is necessary to incorporate a coefficient that accounts for the effect of the duration of the load.

M aterial properties 

49

Therefore, the design value of concrete compressive strength, fcd, is defined by MC2010 (FIB 2012) as:

fcd   cc

fck (3.8) c

where: γ c concrete partial factor (see Chapter 2, Table 2.3) α cc coefficient that considers the negative effects of the duration of the load and the way it is applied; traditionally, a value of 0.85 has been adopted MC2010 recommends using α cc = 1 for new concrete structures, although values between 0.85 and 1.00 can be used. For example, the concrete of buildings in service typically experiences stress levels much lower than fcd, in this case α cc = 1, but in long-span bridges, the self-weight is the most important load, in this case α cc = 0.85 is advisable. In the latest edition of EN 1992, the definition of fcd has been revised for a more ambitious approach, as shown in Equation 3.9. Here, ktc represents the value of α cc, which was previously defined in Equation 3.8, and η cc is a coefficient that considers the size effect (EN 1992 §5.1.6). fcd  cck tc

fck c

(3.9)

13

 40  with cc     fck 

 1 .0

Regarding the design tensile strength of concrete, EN 1992 provides the value of fctd given in Equation 3.10, where ktt is a factor that considers the effect of high sustained loads and of time of loading on concrete tensile strength. The value of ktt is 0.8 for both normal and rapid strength development concretes (classes CN and CR, respectively).

fctd  k tt

fctk,0.05 (3.10) c

However, MC2010 provides fctd = α ctfctk /γ c, where α ct has a lower limit of 0.85 and a value of 1.0 for new structures.

3.5.3 Concrete models for strength design Figures 3.11 and 3.13 show that obtaining a stress–strain model for concrete is complex. Current standards only provide models for instantaneous loads

50  Concrete structures

Figure 3.14 Parabola rectangle diagram for concrete in compression (MC2010).

(Figure 3.13). However, to design structures for combinations of shortand long-duration loads, these models must be complemented. Extensive research on the subject has shown that simple stress–strain models can be used to predict the ultimate strength of concrete with an acceptable margin of error. The most widely used models are the parabola–rectangle model and the rectangular model, which are limited to the study of the ultimate limit state of bending. These models correspond to the one-dimensional “strength defined” models indicated in Figure 3.12. These models do not consider the tensile strength of concrete. The parabola–rectangle diagram (Figure 3.14) is a two-segment function. The first branch is a parabola-like function that starts at the origin and reaches strain point ε c2 , where the stress is fcd (or fck for characteristic strength), with a horizontal slope. For fck ≤ 50 MPa, this first segment is a parabola (Equations 3.11 and 3.12). ε c2 takes the constant value of 2 ‰ whenever the characteristic strength of the concrete, fck, is less than 50 MPa (Equation 3.12). The second segment of the diagram is a horizontal line connecting strain point ε c2 with the maximum strain point (ε cu). The parabola–rectangle diagram is given by the expression:



n       fcd 1   1  c   c    c 2     fcd 

para

0  c  c 2

para

c 2  c  cu

(3.11)

M aterial properties 

51

where

n

2 ì ï 4 =í æ 90 - fck ö + 1 4 23 4 . . ï ç 100 ÷ è ø î

2 ì ec 2 (‰) = í 0.53 î2.0 + 0.085(fck - 50)



3 .5 ì ï 4 ecu (‰) = í æ 90 - fck ö ï2.6 + 35 ç ÷ è 100 ø î

if

fck < 50MPa

if

fck ³ 50MPa

if if

fck < 50MPa (3.12) fck ³ 50MPa

if

fck < 50MPa

if

fck ³ 50MPa

ε c2 and ε cu are shown in Figure 3.14. EN 1992 states that n = 2 (i.e., a parabola), ε c2 = 0.002, and ε cu = 0.0035, as MC2010 does for fck of less than 50 MPa. The rectangular diagram is the most widespread diagram in ultimate bending design due to its simplicity, see Figure 3.15. The rectangular stress diagram assumes a constant stress value of ηfcd (effective resistance) over an effective compressive depth of λx, as long as λx is less than or equal to h. If λx is greater than h, the effective compressive depth is taken as h (as shown in Figure 3.15), where x denotes the distance from the most compressed fiber to the neutral fiber (depth of the neutral fiber), and h is the depth of the section. The values of η and λ that

Concrete stress diagrams Cross-section strain diagram cu

x

Compressed zone

fcd

η fcd λx

c2

h

Parable-rectangle

Rectangular

Figure 3.15 Diagrams of normal stresses in the compressed area in accordance with the parabola–rectangle and rectangular diagrams.

52  Concrete structures

Equivalent rectangular distribution True stress distribution

λ 0.80 0.70 50

90

fck (MPa)

Figure 3.16 Values of λ according to MC2010.

are needed to define the rectangular diagram provided by MC2010 (FIB 2012) are:



1 .0 ì ï h=í fck - 50 ïî1.0 - 200



0 .8 ì ï l=í fck - 50 ïî0.8 - 400

if fck £ 50 MPa if 50 < fck £ 90 MPa

(3.13)

if fck £ 50 MPa if 50 < fck £ 90 MPa

(3.14)

If the width of the cross-section decreases in the direction of the most compressed fiber (opposite to the situation in Figure 3.15), ηfcd must be decreased by 10%, according to MC2010, §7.2.3.1.5. This is the case with circular sections. Equation 3.14 shows that the effective compressive depth (λx) depends on the compressive strength of concrete (fck), as shown in Figure 3.16. There is a variation in λ because the behavior of concrete becomes more linear and less parabolic as its strength increases. Since the resultant of the compressed area in a linear stress distribution is smaller than in a parabolic distribution, the value of λ for the equivalent rectangular diagram should decrease. Figure 3.15 illustrates the stress distributions in the compressed area of the concrete cross-section obtained from the parabola–rectangle and from the rectangular diagrams. In EN 1992 (§8.1.2), the values of λ and η are set to 0.8 and 1, respectively. 3.6 CREEP In Figure 3.11, we observed how the strain obtained when applying σ1 stress varies depending on the load duration. The strain corresponding to short-term loading is ε1, but if stress σ1 is sustained for several days, the

M aterial properties 

53

Figure 3.17 Creep and relaxation.

strain increases to ε′1. This is illustrated in Figure 3.17, taken from Figure 3.11. Once the point (σ1, ε1) in Figure 3.17 has been reached, either stress σ1 or strain ε1 can be maintained. If we stress σ1 is maintained, the strain increases to ε′1 after a few days, which is known as creep. Conversely, if strain ε1 is maintained, the stress reduces to a value of σ′1 after some time, which is called relaxation. So, creep refers to the additional strain that occurs as a result of applying a load and maintaining it over time. Figure 3.18 illustrates this phenomenon in detail. When a constant load is applied to a 28-day-old concrete specimen (point A), an instantaneous AB deformation is obtained. As time passes and the load is maintained, the deformation keeps growing. This additional deformation is known as creep. At a certain time, point C (e.g., at 7 months), there are two options: to continue applying the load (solid line) or to remove it (dashed line). Choosing the second option and removing the load at point C results in an instantaneous CD recovery, followed by a deferred DE recovery over time. Figure 3.18 shows that there is a residual EF deformation that is not recovered at the end of the process. The difference between instantaneous deformation and instantaneous recovery (AB – CD) is known as residual deformation. Residual deformation occurs only during the first loading of the concrete, and its value depends on the stress applied. For subsequent loadings that are lower than the original loading, the behavior of the concrete stress–strain diagram is more linear, as depicted in Figure 3.19. The amount of creep exhibited by concrete is influenced by the age of the concrete when the load is applied (t0) and the age at which the strain is measured (t). Both t and t0 are absolute time values with the time origin set when the concrete is poured. In addition to age, other factors that can affect creep include the humidity of the surrounding environment, the dimensions of the concrete element,

54  Concrete structures Deformation for a constant loading

No load period

C

Instantaneous recovery

Loading end

D

Creep

B

E

Deferred recovery Non recovered deformation Instantaneous deformation

F

A 0

2

4

6

8

10

12

14

16

Months

Figure 3.18 Creep test and deformation types.

σc Residual strain fc

unloading

εcu

εc

Figure 3.19  Residual strain.

and the concrete composition. The creep formula adopted by EN 1992 is only suitable when the stress applied is less than 0.40fcm(t0). The stressdependent strain, εcσ , (i.e., instantaneous plus creep) can be calculated as:

c (t, t0 ) 

(t0 ) (t )    t, t0  0 (3.15) E cm (t0 ) Ec

The first term after the equal sign represents the instantaneous strain, and the second one is the creep strain. Coefficient φ is the creep coefficient. E cm(t0) and E c have been already defined in Equations 3.5 and 3.6, respectively. A more precise expression of E c is available in Annex B of EN 1992:

M aterial properties 

55

E c   cE cm

with  c 

1 f 0.8  0.2 cm,28 88

 1.0 (3.16)

fcm,28 is the mean value of concrete compressive strength at 28 days, i.e. fcm,28 = fcm if tref is 28 days. Creep coefficient φ(t, t0) can be calculated in accordance with Annex B of EN 1992 using Equation 3.17:

(t, t0 )  bc (t, t0 )  dc (t, t0 ) (3.17)

where: bc (t, t0 )  bc,fcm bc, t  t0 0.7 bc,fcm  1.8 / fcm , 28

bc, t  t0

2   30    0.035  (t  t0 )  1  ln   t0,adj    

dc (t, t0 )  dc,fcm dc,RHdc, t0 dc, t  t0 1.4 dc,fcm  412 / fcm , 28



dc,RH  dc, t0 

1  RH / 100 3 0.1hn / 100



1 0.1  t00.,20 adj

 (t  t0 )  dc, t  t0     h  (t  t0 )  (t0,adj) 

 (t0, adj )

1 2 .3 

3 .5 t0,adj

h  1.5hn  250 fcm 1500 fcm  35 

fcm     fcm,28 

0.5

56  Concrete structures

RH relative humidity (%) of the ambient environment hn = 2Ac /u, is the notional size (mm) of the member, where Ac is the cross-section area, and u is the perimeter of the cross-section area in contact with the atmosphere t0,adj adjusted age at loading in days; this adjustment is a result of the early strength development of concrete (t – t0) real (nonadjusted) duration of loading (in days) Temperature also affects creep, as high temperatures accelerate the maturity of concrete, and low temperatures delay it. The age of concrete can be adjusted to consider the effect of temperature using Equation 3.18, which is suitable for the range of 0°C to 80°C: n



tT 



13.65

e

4000 273 T(ti )

 ti (3.18)

i 1

where: tT age of temperature-adjusted concrete (in days) Δti number of days during which a temperature, T, prevails, and T(Δti) mean concrete temperature in °C during time period Δti The early strength development of concrete is another factor that affects creep, and it is considered by changing the age at loading from t0 to t0,adj using Equation 3.19:

 9  t0,adj  t0, T 1   2  t10.,2T  

SC

 0.5 (3.19)

where: t 0,T age of concrete at loading (in days) adjusted according to concrete temperatures, see Equation 3.18; for T = 20ºC: t0,T = t0 α SC coefficient that depends on the early strength development of concrete: α SC = –1 for class CS, α SC = 0 for CN, and α SC =1 for CR If the stress exceeds 0.40fcm(t0), EN 1992 provides an exponential expression to adjust the creep coefficient in the range of 0.4fcm(t0) to 0.6fcm(t0), as shown in Equation 3.20:

  c   (t, t0 )  (t, t0 )exp 1.5   0.4   (3.20)    fcm (t0 )

where φ σ (t,t0) is the nonlinear notional creep coefficient. When the stress in the concrete exhibits little variation and the load is applied after 28 days, it is possible to calculate the effective modulus of

M aterial properties 

57

elasticity of concrete, E c,eff, using Equation 3.21. This expression can be used to estimate long-term deformations in accordance with EN 1992 §9.1.

E c,eff (t, t0 ) 

1.05E cm (3.21) 1  (t, t0 )

3.7 SHRINKAGE As explained earlier in this chapter, excess water is added to concrete during mixing to make it workable. However, this water does not become part of the hardened crystalline structure of the concrete, and it will flow out unless the humidity of the atmosphere is maintained at 100%. As the concrete loses moisture and dries out, it contracts, resulting in concrete shrinkage. Conversely, when the concrete is submerged in water, it experiences swelling and increases in volume. EN 1992 (Annex B) classifies shrinkage deformation into two categories: basic shrinkage deformation (ε cbs) and drying shrinkage deformation (ε cds). Drying shrinkage develops slowly as a result of the migration of water through hardened concrete, while basic shrinkage occurs during the concrete hardening process, mostly in the days following concreting. Basic shrinkage is a linear function of concrete strength, and it is particularly important when new concrete is poured over hardened concrete. Total shrinkage (basic shrinkage plus drying shrinkage) can be expressed as: cs (t, ts )  cbs (t)  cds (t  ts ) cbs (t)  cbs,fcm bs, t



(3.22)

cds (t  ts )  cds,fcm RHds, t  ts where:



 fcm,28  cbs,fcm  bs    60  fcm,28 

2.5

106



cds,fcm   220  110ds  exp  0.012fcm,28  106 Coefficients α bs and α ds are given in Table 3.2. Table 3.2  Coefficients α bs and α ds Early strength development of concrete Low early strength (Class CS) Ordinary early strength (Class CN) High early strength (Class CR)

αbs 800 700 600

αds 3 4 6

58  Concrete structures



bs, t  1  exp 0.2 t

RH



3      1.55 1   RH     RH eq     2   RH    1.55 1       RH eq     2    RH  1.55 1     0.25   RH eq     

 35  RHeq  99    fcm,28 

for

20%  RH  RH eq

for

RH eq  RH 100%

for

RH  100%



0.1

 99

  (t  ts ) ds, t  ts    2 0 035 . ( ) h t t  n s  

0.5

ts age of the concrete at the beginning of the drying process (in days) EXAMPLE 3.1 DEFORMATION OF A CONCRETE SPECIMEN A C-35 concrete specimen with dimensions of 20 × 20 × 40 cm is subjected to a load of 500 kN after 28 days. The load is then changed to 300 kN after 100 days. The relative ambient humidity is 70%, and the specimen is steamcured for one day. The concrete strength development class is CN. Calculate the shrinkage deformation, initial strain, and total strain at 50 days, 100 days, and 2 years. The diagram of load versus time is shown in Figure 3.20. Shrinkage deformation To calculate the shrinkage deformation, Equation 3.22 (from EN 1992) is applied. In this example, ts = 1, and the notional size of the specimen is hn = 100 mm. The values of t considered here are t = 50, 100, and 730 days. The resulting shrinkage is summarized in the following table, where a distinction has been made between basic and dry shrinkage: Time t (days) εcbs (‰)

50 0.06

100 0.07

730 0.08

M aterial properties 

N

20 cm N (kN) 40 cm

500 300

days 28

50

100

730

20 cm

N

Figure 3.20 A concrete specimen under compression. εcds (‰) εcs (‰)

0.13 0.19 0.077

ΔL (mm)

0.18 0.25 0.099

0.31 0.39 0.157

Stress deformation To determine which concrete model is applicable, the maximum stress caused by the maximum load needs to be checked. In this case, the maximum stress is compared to 0.4fcm, which is used to define the modulus of elasticity (secant) of the concrete, Ecm (as shown in Figure 3.13).

c,max 

500000 N  12.5 MPa 200  200 mm2

0.4 fcm  17.2MPa As σ c,max is less than 17.2 MPa, the modulus of elasticity of the concrete can be used to calculate instantaneous deformation. Since the load is applied after 28 days, the modulus of elasticity of concrete can be calculated from Equation 3.4 as Ecm(t0) = Ecm = 33282 MPa. Therefore, instantaneous deformation and the corresponding increment in length can be calculated as follows: c 

c  0.0003756 Ecm

L 

12.5 c L 400  0.15 mm Ecm 33282

59

60  Concrete structures

So, the specimen deforms by 0.15 mm when the load is applied. If the nonlinear model (Equation 3.7) had been used, the corresponding strain for a stress of 12.5 MPa would be 0.00039. At 50 days (t = 50 days), the specimen exhibits the initial deformation that occurred at 28 days (when the load was first applied) plus the creep deformation. Since the load begins to act at 28 days, t0 = 28 days. The value of creep coefficient φ(50,28) = 0.68 can be obtained by applying the corresponding formulae (Equations 3.15 to 3.17). Therefore: c ( t, t 0 ) 

( t 0 ) ( t 0 ) 12.5    t, t 0   0.0003756  0.68  0.000606 Ecm ( t 0 )  cEcm 1.11 33282

which causes a decrease in the length of the specimen that is equal to 0.24 mm. At 100 days, two situations need to be considered: the periods before and after the change in the value of the load (see Figure 3.20). Before the load reduction, the total deformation is the sum of instantaneous deformation (at 28 days) and deformation caused by creep. The load starts acting at 28 days, so t0 = 28 days. By using the corresponding formulae (Equations 3.15 to 3.17), the creep coefficient value of φ(100,28) = 0.94 can be obtained. c ( t, t 0 ) 

( t 0 ) ( t 0 ) 12.5    t, t 0   0.0003756  0.94  0.000694 Ecm ( t 0 )  cEcm 1.11 33282

so, the specimen shortens by 0.28 mm. From this moment on, the load is reduced from 500 to 300 kN. This 5 N/ mm2 reduction in stress causes an instantaneous elongation that can be calculated using the modulus of elasticity of the concrete at 100 days, Ecm(100), which is determined from Equation 3.5 as 35999.3 MPa. Therefore, instantaneous deformation is σ c/Ecm(100) = –0.00014, resulting in a specimen elongation of 0.056 mm: L 

c 5 400  0.0555 mm L 35439 Ecm (100)

To calculate the strain after two years, the principle of superposition is applied as follows: the 500 kN creep is considered from 28 days to 2 years, and the 200 kN “tensile” creep is considered from 100 days to 2 years. The total strain is the sum of both types of creep. The corresponding creep coefficients are calculated using Equation 3.17: φ(730,28) = 1.46 for the 500 kN load and φ(730,100) = 0.99 for the second load (i.e., 200 kN). Therefore, the corresponding strains are:

M aterial properties 

61

Total deformation

Deformation (mm) 0.5 0.4 0.3 0.2 0.1

Days

0.0 0

100

200

300

400

500

600

700

Figure 3.21   Total deformation (instantaneous, shrinkage, and creep).

c (730 , 28) 

( t 0 ) ( t 0 ) 12 12.5  0.000866    t, t 0    1.46 Ecm ( t 0 )  cEcm 33282 1.11 33282

c (730 ,100) 

( t 0 ) ( t 0 ) 5 5    t, t 0    0.99  cEcm Ecm ( t 0 ) 35999.3 1.11 35999.3

 0.000263 The total strain at 730 days is the sum of the values above, which equals 0.00060. Therefore, the specimen shortens by 0.24 mm due to creep, which, when added to the shrinkage, leads to a total shortening of 0.4 mm at 730 days. Figure 3.21 shows the total deformation of the specimen as a function of time.

3.8 OTHER PROPERTIES OF CONCRETE This section covers other mechanical properties of concrete that were not addressed in the previous sections, but that need to be understood.

3.8.1 Density, the Poisson effect, and thermal deformation The approximate density of concrete is 2300 kg/m3 for mass concrete and 2500 kg/m3 for reinforced or prestressed concrete.

62  Concrete structures

There are lightweight concretes made with aggregates such as expanded blast furnace slag, which have densities ranging from 1300 to 2000 kg/m3. The characteristic strength of these concretes is extremely variable depending on the product used, with a range between 1 and 50 MPa. The modulus of elasticity is between 50% and 70% of that of normal concrete, while the values of creep and shrinkage are higher than in normal concrete. ACI-318 includes an expression that gives the modulus of elasticity of concrete as a function of its density. Poisson’s ratio is 0.2 for uncracked concrete and 0 for cracked concrete. The linear coefficient of thermal expansion, which measures the increase in volume experienced by concrete when its temperature increases, is equal to 10 –5 °C –1. Therefore, the thermal expansion strain of concrete is given by Equation 3.23, where ΔT is expressed in °C:

 T  105  T (3.23)

In order to consider the behavior of concrete at high temperatures, it can be assumed that the elastic modulus of concrete decreases by up to 400°C, but it does not lose its ultimate strength, fc. At 600°C, concrete loses onethird of its ultimate resistance, and at 800°C, only one-sixth of its initial resistance remains.

3.8.2 Confined concrete The effect of concrete confinement – a very interesting application in construction – enables the behavior of structural concrete to be better understood. Concrete can improve its compressive behavior if it is subjected to confining lateral pressure, see Figure 3.22. The ultimate compressive strength of confined concrete responds to the well-known experimental expression obtained in Illinois in the 1920s (Richart et al. 1928, 1929):

fck,conf  fck  4.1conf (3.24)

Equation 3.24 cannot be directly applied since confinement in concrete is usually achieved through reinforcement, which increases the lateral pressure as the transverse deformation of the concrete increases. Moreover, reinforcement is situated at specific positions and is noncontinuous. In concrete members, transverse reinforcement, comprising hoops or spirals, provides confinement with the aid of longitudinal reinforcement. Thus, the confining stress depends on the amount of reinforcement, its position within the cross-section, and the axial load. In studying the behavior of structures under seismic loads, suitable confined concrete models are required for the entire range of deformations.

M aterial properties 

63

Figure 3.22 Effects of confinement on the stress–strain diagram.

These models must be nonlinear, similar to that of Figure 3.11. Annex A3 of this chapter includes the model proposed by Mander (Mander, Priestley, and Park 1988), a model that has been implemented in finite element programs such as OpenSees (McKenna, Fenves, and Scott 2000). Standards such as ACI-318 and EN 1992 provide us with formulations that are only applicable in strength design. 3.8.2.1 Confined concrete in Eurocode 2 EN 1992 (§8.1.4) states that the beneficial effect of confinement increases concrete design strength. For ddg ≥32 mm and a minimum principal transverse compressive stress, σ c2d, the increase in design strength Δfcd is defined in Equation 3.25.5 If ddg is less than 32 mm, then the increase in strength Δfcd, given by Equation 3.25, should be adjusted by a factor of ddg /32 mm. fcd,c  fcd  k conf ,bk conf ,s fcd

 4c 2d fcd   3 / 4 1/ 4 3.5c 2dfcd

for c 2d  0.6fcd (3.25) or c 2d  0.6fcd

kconf,b and kconf,s are confinement reinforcement effectiveness factors. These factors are defined in Table 8.1 of EN 1992 as functions of the dimensions described in Figure 3.23. Confinement stress σ c2d can be achieved with confinement reinforcement, and EN 1992 provides different expressions to be used depending on the distribution of confinement reinforcement, as shown in Figure 3.23. Some of these expressions have been developed in Annex A3.

64  Concrete structures

Figure 3.23 Dimensions used to describe the different confinement reinforcement configurations. (From EN 1992.)

3.8.2.2 Confined concrete in ACI-318 The American standard allows improvement in the behavior of elements with circular cross-sections to be considered if these elements are properly confined. To achieve this, the standard considers an increase in the value of the strength reduction factor (ϕ). The transverse reinforcement of such elements should be spirals that satisfy specific minimum diameter, volumetric quantity, and separation requirements, as set out in ACI-318-19 §25.7.3 (ACI Committee 318 2019). 3.9 REINFORCING STEEL In most structural elements, such as beams and slabs, a part of the crosssection is in tension. Due to the low tensile strength of concrete, steel bars are used as reinforcement to hold the tension that concrete cannot withstand. Reinforcing steel placed in the formwork without any tension is called passive reinforcement. It can also be used in compressed areas, working together with concrete under compression. Another way of placing reinforcement into concrete is with pretensioning. In this case, the reinforcement passes its tension to the concrete, and it is called active reinforcement or prestressing steel. Active reinforcement alters the tension or compression zones, as shown in Figures 1.3 and 1.4 (see Chapter 1). Although not widely used, passive reinforcement composed of fiberglass or carbon bars also exists. Additionally, other materials such as glass, steel, carbon, or basalt fibers can be mixed into the concrete mass to control cracking. The primary function of reinforcing steel is to absorb the tractions that concrete cannot withstand. To guarantee the functionality of reinforced

M aterial properties 

65

Figure 3.24  Ribbed bar.

concrete, the tensile bars must be anchored properly, preferably in compressed areas. To improve the adhesion between the steel and the concrete, the reinforcing steel bars are generally ribbed, as shown in Figure 3.24. Reinforcing steel can be presented as straight ribbed bars, coil ribbed bars, electrowelded mesh, and electrowelded lattice reinforcement. In Europe, ribbed bars are categorized according to their nominal diameter, which can be 6, 8, 10, 12, 14, 16, 20, 25, 32, or 40 mm. Thus, an Ø20 bar is a bar whose nominal diameter is 20 mm. EN-10080 is the European standard that specifies the characteristics of steel for reinforced concrete. Table 6 of this standard includes the usual nominal diameters, nominal areas, and weight per linear meter for different types of passive reinforcement. However, there may be other steel products that meet the requirements for use but do not appear in this table. For example, although Table 6 specifies that the maximum diameter of steel in coils is Ø16, Ø20 coils are currently available on the market. In America, bars are named according to the number of eighths of an inch that make up the diameter of a bar. Therefore, a #8 bar has a diameter of 8 eighths of an inch, which is one inch (1 inch = 25.4 mm). To apply EN 1992, the steel must have a characteristic elastic limit ranging from 400 to 700 MPa (f yk or f y0.2k, corresponding to a residual deformation of 0.2%). The most common types of steel are 400 and 500 MPa. Annex C of EN 1992 presents three classes of steel for bars and coils: A, B, and C, depending on their ductility. Table 3.3 shows the characteristic strain at maximum force Agt[%], ε uk (%), for the 10% quantile of each class. Reinforcing and prestressed steel only work axially, so a one-dimensional stress state is suitable for describing their behavior. For the design of concrete elements, the stress–strain diagram of reinforcing steel obtained from experimental data can be used (provided that Table 3.3  Types of steel for bars and coils Ductility Class Characteristic strain at maximum force Agt[%], εuk (%) (10 % quantile)

A

B

C

≥2.5

≥5.0

≥7.5

66  Concrete structures

Figure 3.25 Stress–strain diagrams for reinforcing steel (for tension and compression).

the tests comply with current regulations) or, alternatively, the diagram represented in Figure 3.25 (taken from EN 1992). For ULS bending design, the diagram represented in Figure 3.25 can be used for bending in both compression and tension. This is because concrete and steel deform together, and steel cannot buckle freely (Gil-Martín et al. 2006). In the diagram in Figure 3.25, f yk is the characteristic value of yield strength of reinforcement, k is a coefficient that considers the strain hardening of reinforcement (its value must be between 1.05 and 1.35, according to EN 1992), and ε uk is the characteristic strain. Regarding ε ud, EN 1990 recommends that ε ud = 0.9 ε uk. Figure 3.25 also shows the design yield strength of the reinforcement, denoted by f yd, which is calculated as f yk /γ s, where f yk is the characteristic yield strength of the steel and γ s is the partial safety factor for the material. The design value of the modulus of elasticity, Es, is assumed to be 200000 MPa. If the horizontal line of the design diagram in Figure 3.25 (i.e., no strain hardening so k = 0) is adopted, EN 1992, like ACI-318, does not limit the strain. This hypothesis, which could seem excessive at first, is quite logical when the difference between the ultimate strain values of steel (>0.025) and concrete (≈0.003) is considered. The mean density of the steel is 7850 kg/m3, and its linear coefficient of thermal expansion is 10 –5 °C –1. According to current regulations, such as EN 1992 §11, groups of contacting parallel bars (or bundles of bars) can be used. These groups consist of bars that are joined together and treated as a single unit. Up to three bars can be put together, or four if they are vertical compression bars.

M aterial properties 

67

3.9.1 Anchorage of reinforcing bars As mentioned in the previous section, rebars (short for reinforcing bars) must be anchored in a way that allows stresses in the bars to be transmitted back to the concrete through friction, a tangential stress called bond stress (τ). As seen in Chapter 1, rebar–concrete adhesion is achieved using three mechanisms: a chemical reaction at the concrete–steel interface, steel–concrete friction aided by concrete shrinkage, and the reaction force of concrete against the ribs of the rebars. In the past, using the first two processes was enough to ensure adherence, which is why smooth bars were used. However, due to the improvement in steel quality, which has meant that it can resist higher tension stresses, adhesion improved by introducing ribs to the rebars. If a steel bar is inserted into a concrete block and an attempt is made to extract it once the concrete has hardened (pull-out test, Figure 3.26), the further the bar has been inserted into the concrete (Figure 3.26a), the greater the extraction force (P). This force P has a limit value that corresponds to the tensile strength of the bar (Pmax = Asf y). In Figure 3.26a, L0 is the minimum length needed for tensile force P to be equal to Pmax. If the embedded length (lb) is less than L0, then failure is caused by the sliding of the bar with respect to the concrete (sliding failure) or to the splitting of the concrete block (splitting failure), see Figure 3.27. On the contrary, if lb is greater than L0, failure occurs because the tensile capacity of the rebar is exceeded (bar break failure), see Figure 3.27. When a sliding failure occurs, a maximum bond stress (τ max) develops between the bar and the concrete. Splitting failure may happen before sliding failure, in this case the bond stress is called ultimate bond stress (τ u).

P P

lb ≤ L0

Embedment length

P=Asσs

Pmax τmax

b) a)

L0

Embedment length, lb

Figure 3.26  Pull-out test.

68  Concrete structures

a) Bar break

rebar

concrete

Failure surface

b) Splitting c) Sliding

Figure 3.27 Pull-out failures: (a) bar break failure, (b) splitting failure, (c) sliding failure.

Mechanically, a simple equilibrium equation connects the anchorage length (lb), the axial stress in the bar (σ sd), and the maximum bond stress (τ max):

P  sd As  sd 

2  sd    lb max  lb  (3.26) 4 4 max

The value of the maximum and ultimate bond stresses is influenced by several factors, including the quality of the concrete (the higher the fck, the greater the ultimate and maximum bond stresses), rib geometry, cover thickness (a thin cover will provoke concrete splitting, i.e., delamination cracking parallel to the anchored reinforcement, see Figure 3.27b), confinement (if stirrups enclose the longitudinal bars) or nonconfinement, and the position of the bar during casting (i.e., depending on the potential segregation of the aggregate during concreting, good bond conditions or poor bond conditions are generated). As mentioned before, the pull-out test of a deformed bar can result in three modes of failure: bar breakage (Figure 3.27a), splitting of concrete (Figure 3.27b), and the sliding of the bar caused by the concrete shearing off between ribs (Figure 3.27c). Bond action tends to cause the concrete cover around the rebar to split (Figure 3.27b). When enough concrete cover, or stirrups, are provided, bond failure is usually characterized by crushing or the concrete shearing off between ribs. The transfer of force between the reinforcing bar and the concrete is mainly produced by the bearing of the concrete against the ribs of the reinforcing bar, and so the bond strength is mainly governed by the shear strength of the portions of concrete between ribs. Once the concrete has sheared off, the transmission of force occurs only through friction along the failure surface (Figure 3.27c). Figure 3.28 (adapted from in MC2010) illustrates the relationship between bond stress (τ) and the relative slip between the concrete and

M aterial properties 

τ

Shear off of concrete by ribs

τmax

Pull-Out

τu,split2 τu,split1 τf

69

stirrups

Splitting

unconfined

s1

s2

s3

Slip s

Figure 3.28 Bond stress–slip relationship, for good conditions. (Adapted from MC2010.)

the rebar. The figure shows that concrete only shears off between the ribs (which corresponds to the plateau for τ max) if splitting does not occur earlier. For ribbed reinforcing steel that complies with international regulations and good bond conditions, MC2010 recommends s1 = 1.0 mm, s2 = 2.0 mm, and τ max = 2.5fck0.5. In the case of poor bond conditions, the recommended values are s1 = 2.0 mm, s2 = 3.6 mm, and τ max = 1.25fck0.5. In both cases, s3 represents the distance between ribs. MC2010 provides a formulation of τ u,split for unconfined concrete and for passive confinement from transverse reinforcement, as shown in Figure 3.28. EN 1992 (§11.4.2) gives a formula for determining the design anchorage length in tension (lbd). The value of lbd in Equation 3.27 considers the three possible failure modes represented in Figure 3.37. This equation is applicable to ribbed bars with a diameter of ϕ ≤ 32 mm and indented bars with a diameter of ϕ ≤ 14 mm. n



1/ 2

    25  lbd  k lbk cp  sd     435   fck 

1/ 3

    20   

1/ 2

 1.5   c   d 

 10 (3.27)

Where: cd is the nominal cover (see Figure 3.29) nσ = 1.0 for σ sd ≤ 435 MPa nσ = 1.5 for σ sd > 435 MPa kcp is a coefficient that considers the casting effects on bond conditions: = 1.0 for bars with good bond conditions in accordance with Figure 3.30 = 1.2 for poor bond conditions and for all bars used in slipform construction unless it is shown that the vertical bars cannot move during casting = 1.4 for all bars executed under bentonite or similar slurries unless data is available for the specific slurry to be used

70  Concrete structures

cs

cx

cs cd=min{0.5cs;cx; cy}

cy Exterior bar Interior bar

Figure 3.29  Nominal cover.

A 300 mm

B

300 mm

C

A Top surface during concreting B Zone with poor bond conditions for bars with an inclination less than 45° to the horizontal C Zone with good bond conditions

Figure 3.30 Description of bond conditions as a function of member depth. (Adapted from EN 1992.)

Rebars are considered to have good bond conditions if they incline 45° to 90° to the horizontal during concreting or if they have an inclination of under 45° to the horizontal and are located at a distance of at least 300 mm from the free surface or 300 mm from the bottom of the formwork, as shown in Figure 3.30. Rebars that do not meet these conditions are considered to have poor bond conditions. klb = 50 for persistent and transient design situations klb = 39 for accidental design situations Ratios involving bar diameters in Equation 3.27 shall be limited as follows: (ϕ/20mm) ≥ 0.6 and (1.5ϕ/cd) ≥ 0.4. The latter condition marks the boundary between the two failure modes: concrete shearing off between ribs and concrete splitting. Therefore, if cd > 3.75ϕ, slip failure occurs, while if cd < 3.75ϕ, split failure happens. The design anchorage length (lbd) can be reduced by the presence of confinement pressure or transverse reinforcement. This is achieved by replacing the value of cd with a new value: cd,conf (refer to §11.4.2 of EN1992). Bundles of bars can be treated as one single bar whose equivalent diameter is

b 

4 As (3.28) 

M aterial properties 

71

where A s is the total area of all bars contained in the bundle (see also Hernández-Montes and Gil-Martín 2020).

EXAMPLE 3.2 DESIGN ANCHORAGE LENGTH OF A STRAIGHT REBAR The design anchorage length (lbd) for Ø20 B500 steel bars (σ sd = f yd = 500/1.15 = 435 MPa → nσ = 1.0) in C-30 concrete is calculated by assuming a persistent situation (klb = 50) and good bond conditions (kcp = 1.0) (refer to Figure 3.31). According to Figures 3.29 and 3.31, we get cd = Min{32/2;65;65} = 16 mm. As cd < 3.75ϕ = 75 mm, splitting failure is expected to occur. Figure 3.32 shows the relationship between the design anchorage length (lbd) and the stress of a bar (σ sd). The thinnest line in the graph corresponds to this specific example. Note that the nominal cover has not taken into account transverse reinforcement. In this case, the calculated value of lbd is 1250 mm. The thick line in Figure 3.32 corresponds to the case where cd = 3.75ϕ = 75 mm, which is the cover limit between the slip and split failure modes. From Equation 3.27, the calculated value of lbd for this case is 577 mm. The presence of transverse reinforcement can reduce the design anchorage length. In accordance with EN 1992, the current case falls into the category of case a of Figure 3.33. Figure 3.31 shows that the transverse reinforcement in this case consists of Ø10 stirrups spaced at 200 mm (i.e., ϕ t = 10 and s = 200). According to EN 1992 §11.4.2(5):

32 mm

Ø20 Ø10@200 mm 20 mm

Mechanical cover 75 mm

Figure 3.31 Anchorage length example.

10 mm c =55 mm

Design anchorage length mm

72  Concrete structures

1400 1200 1000

cd=16 mm

800 cd=21.9 mm

600 400

cd=75 mm

200 0 0

100 200 300 Design stress in the bar MPa

400

Figure 3.32 Design anchorage length versus design stress in bar.

Potential splitting surface φt cs ≤ 5φ

cs ≥ 8cy

φ

φ cy

φ φt

σccd a) nc=1, nb=2. Confinement reinforcement at spacing sc

b) Transverse reinforcement at spacing st

c) Extermal design confinement pressure σccd

Figure 3.33 Cases where the design anchorage length (lbd) may be reduced by confinement or transverse reinforcement. (Adapted from Figure 11.5 of EN 1992.)

 8  cd,conf  cd   30k conf conf  ctd    21.89  75 fck   conf 

nt 2t  0.00982 4nbs

k conf  1.0 ctd  0

As shown in Figure 3.32, the presence of confinement reinforcement changes the nominal cover to cd,conf = 21.89 mm, resulting in a reduced design anchorage length (lbd) of 1069 mm for σ sd = f yd = 435 MPa.

M aterial properties 

≥ 5φ α bend ≥ 135º

α bend = 90º

φmand,min πφ 2 4

lbd a) hook

c yb

φ

φ

≥ 5φ

73

cy

φmand,min πφ 2

σ sd

4

lbd b) 90º bend

σ sd cd = min {cs / 2; cx ; c y ; c yb } c) cd

Figure 3.34 Anchorage with (a) standard hook and (b) standard 90º bend anchorage in tension. (c) Nominal cover cd is defined.

3.9.1.1 Bend bars, hooks, and head bars The design anchorage length in tension (given by Equation 3.27) can be reduced by 15ϕ, as long as the design anchorage length (lbd) is greater than or equal to 10ϕ, when standard hook or bend anchorages are used and comply with Figure 3.34. To prevent damage to both the reinforcement and the concrete, bars must be bent over a minimum diameter known as the mandrel diameter (ϕ mand,min). This diameter is equal to 4ϕ if ϕ ≤ 16 mm, and 7ϕ if ϕ > 16 mm, as specified in Section 11.3 of EN1992. If the bend or hook anchorage in tension does not comply with Figure 3.34, the total design anchorage length (lb,tot) measured along the center line of the bar can be calculated as if it were a straight bar. For bend or hook anchorage in compression, only the first straight segment contributes to anchorage, except when the cover is sufficient (≥3.5ϕ) (see §11.4.4 of EN 1992). In this case, the same procedure as that used in tension is applicable. The anchorage length can be reduced using welded transverse reinforcement, U-bar loops, or headed bars. ​ If properly designed (EN 1992 §11.4.7), the head could be big enough to anchor the rebar without any additional anchorage length, see Figure 3.35. The anchorage of headed bars may present four different failure modes (Gil-Martín and Hernández-Montes 2019). The anchorage of shear reinforcement is considered to be correctly designed if it complies with the corresponding standards (Figure 3.36) and no further calculations are needed. A longitudinal bar with a minimum diameter equal to and no less than the diameter of the stirrup or link should be provided at each corner of the stirrup and inside the end hooks.

3.9.2 Splice The passive reinforcement bars are usually supplied in maximum lengths of 12–14 m, so often it is necessary to splice the bars (EN 1992 §11.5).

74  Concrete structures

Figure 3.35  Headed bars.

Figure 3.36 (a–e) Anchorage of links and (f–h). (Adapted from EN 1992.)

There are different types of splices: lap splices, weld splices, and mechanical splices. 3.9.2.1 Lap splices The design lap length (lsd) is the length of overlap needed to maintain the continuity of bars in lap splice. It ensures that forces are transferred from one bar to the next while preventing concrete spalling and minimizing the

M aterial properties 

75

development of large cracks parallel to the lapped bars that may affect the performance of a structure. In the overlapping splice bars are placed next to each other, either touching or at a distance that is preferably shorter than the smaller of 4ϕ and 50 mm to ensure force transfer and avoid cover spalling. Lap length (lsd) is directly linked to design anchorage length (lbd) and they are usually the same (see §11.5 of EN 1992 for further details). ACI-318 does not allow for lap splicing for bars with diameters exceeding 35 mm because there is a lack of experimental data. 3.9.2.2 Mechanical splices Mechanical splices are generally more expensive than other types of splices, but they are also more reliable as they do not require forces to be transferred through the concrete in the splice zone. There are various types of mechanical splices on the market. One of the most commonly used is the half-thread full-thread (HT-FT) splice, as shown in Figure 3.37. This splice uses a sleeve with dimensions of D × L to thread together the ends of two bars (that need to have been previously threaded). The advantage of this system is that only the sleeve is rotated, not the bars themselves.

Figure 3.37  Mechanical coupler.

76  Concrete structures

3.10 PRESTRESSING STEEL As seen in previous sections, concrete has very little resistance to traction and, for this reason, steel bars have to be placed into the tensioned zones of the element (passive reinforcement). Another way to overcome the inconvenience of the low tensile strength of concrete is to force all the concrete to work under compression by introducing prestressing forces (Figure 1.3). These forces are introduced by active reinforcement. There are three types of prestressing reinforcing steel: bars, strands, and wires.6 A tendon is a set of several parallel active reinforcements housed within the same duct. Specific properties of prestressing steel include strength class, product type (wire, strand, or bar), diameter, and execution class. Table 3.4, from EN 1992, shows the strength classes of prestressing steel and their corresponding properties: The modulus of elasticity of prestressing steel can be taken as Ep = 205000 MPa for wires and bars, and Ep = 195000 MPa for strands. ​ Prestressing wires are available in diameters ranging from 3 to 10 mm and are made of high-carbon steel (0.7% to 0.85% carbon). The wire rod used in the production of wires is first heated to approximately 900°C and then slowly cooled to enhance the homogeneity of the steel. Subsequently, the wire rod undergoes a cold drawing process, which is carried out in several phases to gradually reduce the diameter of the wire and enhance its strength.

Table 3.4  Strength classes of prestressing steel Properties in stress–strain diagram (Figure 3.38) (characteristic values) 0.1% proof test (MPa) Tensile strength fpk (MPa)

0.1% proof test (MPa) Tensile strength fpk (MPa)

0.1% proof test (MPa) Tensile strength fpk (MPa)

Wiresa Y1570 1380 1570 Strandsa Y1770 1550 1770 Barsa Y1030 830 1030

Y1670

Y1770

Y1860

1470 1670

1550 1770

1650 1850

Y1860 1650 1860

Y1960 1740 1960

Y2060 1810 2060

Y1050 950 1050

Y1100 890 1100

Y1230 1080 1230

In all strength classes, ductility value of k = (fpk/fp0,1k) = 1.1 and characteristic strain at maximum force εuk = 3.5%. a

M aterial properties 

77

Figure 3.38 Stress–strain diagrams for prestressing steel. (Adapted from EN 1992.)

Wires can be either smooth or indented. Both types are produced using the same drawing process, but indents can be made in the surface of the wires to improve adhesion, as shown in Figure 3.39. An example of how prestressing wire is categorized is “EN 10138-2 Y 1770 C 5.0 I”, which means that this wire conforms to the EN 10138-2 standard, Y denotes prestressing steel, 1770 is its nominal tensile strength in MPa, C indicates that it is a cold-drawn wire, 5.0 is its nominal diameter, and I indicates that it is indented. For all types of prestressing steel, the nominal resistance can be considered equal to the characteristic resistance. The most commonly used wires are summarized in Table 3.5, which shows the types of wires available, their nominal diameters in mm, and their characteristic strengths (fpk) in MPa. Strands are made up of several wires that are helically wound together. The most widely used type is the 7-wire strand (S7), which can be seen in

Figure 3.39 Detail of an indented wire. (Adapted from EN 10138-2.)

78  Concrete structures Table 3.5  Types of wires Designation

Nominal diameters (in mm)

fpk (in MPa)

Y 1570 C

9.4 – 9.5 – 10.0

1570

Y 1670 C

6.9 – 7.0 – 7.5 – 8.0

1670

Y 1770 C

3.2 – 5.0 – 6.0

1770

Y 1860 C

3.0 – 4.0 – 5.0

1860

Figure 3.40  Strand S7.

Figure 3.40. After being wound, the strand is subjected to a thermomechanical process that involves heating it up to 350°C while simultaneously applying a tensile stress to obtain low-relaxation strands. The advantage of using low-relaxation steel is that it reduces relaxation losses, which in turn results in a smaller required tendon area (the relaxation phenomenon is discussed in Section 3.10.2). EN 1992 only considers low-relaxation wires and strands. Table 3.6 lists the most commonly used strands. Prestressing bars are manufactured by cold drawing, and their diameters typically range from 20 to 40 mm. According to EN 1992, these bars are classified as class 3 relaxation bars, and their maximum characteristic strength cannot be less than 980 MPa. The bars can be either threaded or smooth, but even smooth bars must have threaded ends as their prestressing system uses thread: the bar is tensioned, and the nut is tightened. Figure 3.41 illustrates a prestressing bar.

3.10.1 Anchorage In elements with adherent prestressed steel, the wires or strands transfer the prestressing force to the concrete through the contact surface. Unlike passive reinforcement, the wires and strands are smooth and do not have ribs.

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79

Table 3.6  Types of strands Designation Strands of 2, 3, and 7 wires Y 1770 S2 Y 1860 S3 Y 1960 S3 Y 2060 S3 Y 1770 S7 Y 1860 S7

Nominal diameters mm (area in mm2) 5.6 – 6.0 6.5 – 6.8 – 7.5 5.2 5.2 16.0 9.3 – 13.0(99) – 15.2(140) –16.0(150)

Characteristic strength fpk (in MPa) 1770 1860 1960 2060 1770 1860

Figure 3.41  Prestressing bar.

To improve adherence, the wires or strands may have notches. In addition, the strands are curly. Thus, notches and curling, along with the Poisson effect (i.e., the increase in cross-sectional area at lower stress, as shown in Figure 3.42, where the stress of a strand is zero at the end of a beam, resulting in a wider cross-section) and the other mechanisms described above (concrete shrinkage and chemical reaction at the steel–concrete interface) ensure that the wires and strands adhere to the concrete in prestressed elements. In Section 3.9.1, the concept of anchorage length was defined for passive reinforcement. In the case of active reinforcement, the anchorage can be concentrated in a very short length (by means of wedges or nuts) or extended along a greater length, as in the case of passive reinforcement. Localized anchoring is typically used in active posttensioned reinforcement, while anchoring in length is common in the case of adherent pretensioned active reinforcement.

80  Concrete structures

Transmission length

Figure 3.42 The Poisson effect and transmission length.

EN 1992 (§13.5) defines three types of lengths to describe the process of anchorage of prestressing steel: • The transmission length (lpt), over which the prestressing force is fully transmitted to the concrete (see Figure 3.42) • The dispersion length (ldisp), over which the concrete stresses gradually disperse to a linear distribution across the concrete section • The anchorage length (lbpd), over which the tendon force, at the ultimate limit state, is fully anchored in the concrete

3.10.2 Relaxation Relaxation in prestressing steel (Δ σ pr/σ pi) refers to stress loss (Δ σ pr) under constant strain (as shown by the vertical line in Figure 3.43). This phenomenon is particularly significant in prestressing steel since it is assumed that the deformation of the prestressing steel remains constant in the concrete element. Relaxation is affected by time, temperature, and stress levels. When stress values are below 0.5fpk (where fpk is the characteristic tensile strength of the prestressing steel), relaxation can be considered negligible. The stress losses caused by relaxation (in %) after 1000 hours of tensioning with an initial stress (σ pi) of 0.7 or 0.8 of its tensile strength at a temperature of 20°C are represented by ρ 1000. The values of ρ 1000 can be obtained from Table 3.7 and/or from test certificates provided by the supplier. The evolution of relaxation loss over time may be calculated as (Annex B EN 1992):

Figure 3.43 Relaxation of the prestressing steel.

M aterial properties 

81

Table 3.7  Relaxation of prestressing steel ρ1000

Maximum relaxation at 1000 hoursa

Type of prestressing steel

Wires and strands

Bars ϕ ≤ 15 mm

Bars ϕ > 15 mm

0%

0%

0%

2.5%

6.0%

4.0%

4.5%

—

—

For initial stress σpi of 50% of actual tensile strength fp For initial stress σpi of 70% of actual tensile strength fp For initial stress σpi of 80% of actual tensile strength fp Notes: - Values apply for all prestressing steel strength classes.

- In absence of more detailed information, the relaxation loss can be interpolated linearly between the initial stress values. - fp = fpk can be assumed if the actual strength of prestressing steel is known. - Relaxation losses are very sensitive to the temperature of the prestressing steel. a



Relaxation losses apply at a mean temperature of 20°C.

pr  24 t   1000   pi  1000 

0.16

(3.29)

where t is the time in days, and Δ σ pr is the absolute value of loss of stress in the prestressing steel. Due to the relaxation effect, the design value of the modulus of elasticity of prestressing steel varies from Ep to Ep,eff (Figure 3.43). EXAMPLE 3.3 RELAXATION LOSS By applying Equation 3.29, the loss of prestressing caused by relaxation of a cable at an age of 10 years and tensioned at 0.7fpk is 5.1%. If the cable had been tensioned at 0.8fp,k the loss caused by relaxation would be 9.2%.

3.11 FATIGUE Fatigue is the phenomenon of material failure due to cyclic loading, typically characterized by the relationship between the difference in maximum and minimum stress (σ s,max – σ s,min) and the number of cycles endured before failure. In reinforced and prestressed concrete, fatigue typically occurs in steel earlier than in concrete.

82  Concrete structures

Figure 3.44  Fatigue curve.

Figure 3.44 illustrates the relationship between the range of stresses and the number of load cycles required until failure occurs. EN 1992 addresses fatigue in Section 10. EXERCISES





1. To estimate deflections in concrete, a precise value of the secant modulus of elasticity of concrete (E cm) is required. If the range of strains to be considered is from 0 to 0.001, can the slope of the curve in the parabola–rectangle model shown in Figure 3.14 be used as an approximation of E cm? 2. Compare the definition of the modulus of elasticity of concrete (E c) in ACI-318 with the definitions of E cm and E c in EN 1992. Additionally, examine how both standards address the consideration of aggregates like limestone or basalt. 3. In Example 3.1, calculate the deformation by considering the effect of temperature when it changes from 20°C to 40°C after 50 days. Redraw Figure 3.21. 4. Briefly explain the beneficial effect of confining concrete. Why do some regulations not consider this improvement in sizing concrete columns? How can confinement affect the deformation of columns? 5. Based on current regulations, assess whether passive reinforcement, which has been exposed to the open environment and shows considerable rusting, can be utilized as is, or if further evaluation is necessary after cleaning with steel bristles.

NOTES 1. In other countries, other types of specimens are used, such as prismatic ones. 2. For example, evaluation of old concrete only using the sclerometer test can lead to incorrect conclusions being drawn. Sclerometer tests are based on a

M aterial properties 

83

rebound rate of a mass against a concrete surface to measure its modulus of elasticity and, using this, estimate its resistance. Since the carbonation process (Chapter 14) is associated with a hardening phenomenon, old concretes will be harder on the outside than on the noncarbonated inside. 3. σ c in EC2 is the same concept as fc in ACI-318, and fck in EC2 is equivalent to f′ c in ACI-318, (Hernández and Gil-Martín 2023). 4. The subscript “cu1” refers to the model in Figure 3.13. This subscript is “cu” for the parabola–rectangle model and for the rectangular model. 5. ddg is a size parameter describing the failure zone roughness, which depends on the concrete type and its aggregate size. The value of ddg may be taken (in mm) as: 16 mm + Dlower ≤ 40 mm for concrete with fck ≤ 60MPa 16 mm + Dlower(60/fck)4 ≤ 40 mm for concrete with fck > 60 MPa Dlower is the smallest value of upper sieve size D in an aggregate for the coarsest fraction of aggregates in concrete permitted by the specification of concrete (EN 206). 6. The European standard that regulates prestressing steel is EN 10138.

REFERENCES ACI 318. (2019). Requisitos de Reglamento Para Concreto Estructural (ACI 318S-19). FIB. (2012). Model Code 2010 – Final Draft, Vol. 1. Fib Bulletin No. 65. Lausanne. EN 1992-1-1. (2023). Eurocode 2: Design of Concrete Structures - Part 1–1: General Rules and Rules for Buildings, Bridges and Civil Engineering Structures EN 1992-1-1. Brussels: European Committee for Standardization. Gil-Martín, L.M., and Hernández-Montes, E. (2019). “Reinforcement Anchored in Tension by Heads: Review of Capacity Formulation and Applicability Limits.” Engineering Structures, 184, 186–193. Gil-Martín, L.M., Hernández-Montes, E., Aschheim, M.A., and Pantazopoulou, S.J. (2006). “Slenderness Effects on the Simulated Response of Longitudinal Reinforcement in Monotonic Compression.” Structural Engineering and Mechanics 23 (4), 369–386. Hernández, L., and Gil-Martín, L.M. (2023). “A Comparison of the Most Important Properties of Structural Concrete: European Standards versus American Standards.” European Journal of Environmental and Civil Engineering 1–14. Hernández-Montes, E., and Gil-Martín, L.M. (2020). “Tolerancia En El Empalme Por Solape (Traslapo) de Grupos (Paquetes) de Barras Según El EC2 y La ACI-318.” Hormigón y Acero 71 (292): 59–62. Mander, J.B., Priestley, M.J.N., and Park, R. (1988). “Theoretical Stress-Strain Model for Confined Concrete.” Journal of Structural Engineering 114 (8): 1804–26. McKenna, F., Fenves, G.L., and Scott, M.H. (2000). Open System for Earthquake Engineering Simulation. Berkeley, CA: University of California.

84  Concrete structures Richart, F.E., Brandtzæg, A., and Brown, R.L. (1928). A Study of the Failure of Concrete under Combined Compressive Stresses. Urbana, IL. http://hdl​.handle​.net​/2142​/4277. Richart, F.E., Brandtzæg, A., and Brown, R.L. (1929). The Failure of Plain and Spirally Reinforced Concrete in Compression. Urbana, IL. http://hdl​.handle​ .net​/2142​/4073.

Annex A3

Confinement and concrete softening

A3.1 FAILURE CRITERIA IN CONCRETE The stress state of a point inside a deformed solid is defined by the Cauchy stress tensor, T. The expression for T, referring to the axes defined by the principal stresses (σ 1, σ 2 , σ 3), is given by Equation A3.1: 0 0  1   T  0 2 0  (A3.1)  0 3  0 There are many failure models in the literature, each of which is best suited for a particular type of material. The simplest failure model for brittle materials such as rock and concrete is the maximum stress or Rankine model. This model assumes that if the maximum stress (σ 1) reaches the capacity level, either in tension (σ 1 = fct) or in compression (σ 1 = fc), the material will be considered to have failed, regardless of the values of the stresses in the other axes (σ 2 and σ 3). Although it is not the only failure model used for concrete, it is the most commonly used one. Either compression or tension can cause concrete failure. There are various failure modes in reinforced concrete (RC) members. For instance, common types of concrete failure modes in anchor zones are splitting, lateral blowout, breakout, spalling, and bursting. To gain a better understanding of these types of failure, see current regulations such as ACI-318 or EN 1992. Since the beginning of the 20th century (Considère 1902), the improvement in the compression capacity of concrete when it is laterally confined has been studied. The existence of a confining transverse stress σ conf means that σ 2 ≠ 0 and/or σ 3 ≠ 0, as shown in Figure A3.1.1 The simplest failure model for the effect of confining stresses is the Mohr–Coulomb (M-C) model. On the σ-τ axes, this model establishes the relationship where a material fails, following the equation:



  c   Tan() (A3.2)

DOI: 10.1201/9781003169659-6

85

86  Concrete structures

Figure A3.1 Effects of confinement on the stress–strain diagram.

where τ is the tangential stress, σ is the normal stress, c is the cohesion, and ϕ is the internal friction angle. Equation A3.2 is graphically represented by bold lines in Figure A3.2 for specific values of c and ϕ. σ 1, σ 2 , and σ 3 are the principal stresses. The compression failure of unconfined concrete occurs when σ 1 = fc, with σ 2 = σ 3 = 0. In this case, the stress state is graphically represented by the gray Mohr’s circle in Figure A3.2. The M-C model predicts that failure will occur at point A, which

Figure A3.2 Mohr–Coulomb failure criteria and stress states (Mohr’s circle) of confined and unconfined concrete.

C onfinement and concrete softening  87

connects angle ϕ with angle θ of the struts (or the inclination of cracks) of the concrete. In the presence of confining stress, and assuming equal stresses in the other two principal directions (i.e., σ 1 > σ 2 = σ 3), the stress state at failure predicted by the M-C criterion is indicated by a Mohr’s circle with a larger diameter, as shown in Figure A3.2. In this case, the main compressive stress is fcc, and failure occurs at point B. The M-C criterion can be graphically represented by a prism with a hexagonal base if the three principal stresses are different. Figure A.3.3 shows an example of this representation for the case of c = 2 MPa and ϕ = 20°. For a more detailed explanation of this model, please refer to Wikipedia. The Willam–Warnke model (Willam and Warnke 1975) is a failure surface model that applies to concrete with smoother edges than those of the M-C criterion (refer to Figure A3.3). This model is used in the majority of finite element software packages. A3.2 STRENGTH OF CONFINED CONCRETE: fcc As noted earlier, the compressive strength of concrete can be improved by providing lateral confinement. This effect can be analyzed using models such as M-C or Willam–Warnke. The most common approach is to use Rankine’s theory, which assumes that confined concrete behaves similarly to unconfined concrete but with a higher compressive capacity (fcc). Note that the M-C criterion is equivalent to the Rankine criterion when ϕ = 90°. Lateral confinement can be achieved with the direct application of pressure or by using transverse reinforcement such as hoops, stirrups, spirals,

Figure A3.3 3D failure surface of the M-C criterion for c = 2 MPa and ϕ = 20°.

88  Concrete structures

or rings to prevent the transverse expansion of concrete. The interior concrete in thick structures is typically more confined than the exterior, resulting in greater compressive strength caused by the size effect. Figure A3.4 displays one of the columns tested by Frank Richart in the 1920s. To provide confinement to the concrete, Richart used two different systems, as documented in his work (Richart et al. 1928, 1929): lateral pressure applied with a triaxial test and wire spirals, as shown in Figure A3.4. The following expressions were derived from Richart’s work:

fcc  fc  k1conf (A3.3)



   cc  c0  1  k 2 conf  (A3.4) f c  

where the confined concrete compressive strength (fcc) and its corresponding strain (ε cc) are expressed as functions of lateral confinement stress (σ conf), unconfined concrete strength (fc), and its corresponding strain (ε c0), as shown in Figure A3.5. 2 The coefficients k1 and k 2 in Equations A3.3 and A3.4 depend on both the lateral pressure and the type of concrete. In the work of Richart et al., experimental values of k1 = 4.1 and k 2 = 5k1 were obtained. Fifteen years earlier, Considère (1902) came up with a similar expression for fcc.

Figure A3.4 Confinement with helicoidal wires.

C onfinement and concrete softening  89

Figure A3.5 Stress–strain model of confined and unconfined concrete for monotonic loading (Popovics 1973).

A3.3 STRESS –STRAIN MODELS In the structural analysis of concrete structures, the maximum stress criterion is the most commonly used failure criterion. This means that an element fails when stress σ c reaches strength value fck. However, more precise failure criteria such as M-C or Willam–Warnke are used in detailed studies. For the strength design of structural elements like beams, columns, and beam–column elements, one-dimensional σ c–ε c models are commonly used. The rectangular diagram, in both EN 1992 (Figure 3.15) and ACI318, is the most widely used model, even though the European Standard includes other diagrams like the parabola–rectangle. The increased strength that accounts for the confinement effect in EN 1992 is given in §3.8.2.1, and §3.8.2.2 presents the ACI-318 procedure to consider the confinement effect in circular sections confined with spirals. As previously mentioned, for instantaneous loads, lasting between 1 and 2 minutes and which compressive stress that does not exceed 0.4 fcm, the modulus of elasticity (secant) of concrete (E cm) can be used to study strains, in other words, σ c = E cm · ε c. However, for greater strains (i.e., σ c > 0.4 fcm), the general nonlinear diagram given by Equation 3.7 (for unconfined concrete; see Chapter 3) needs to be used, which is suitable for the entire range of strains. Similarly, the most commonly used one-dimensional model for studying the behavior of confined concrete across the whole range of strains is that created by Popovics (1973) (see Figure A3.5).

90  Concrete structures

The stress–strain model (σ c–ε c) in Figure A3.5 for both unconfined and confined concrete under monotonic loading corresponds to the following formulation, proposed by Popovics (1973): c 

fcc xr r  1  xr

x

c cc

 f  cc  c0 1  5  cc  1   (A3.5)  fc   r

Ec E c  E sec

E sec 

fcc cc

where E c is the tangent modulus (the authors of this model used E c = 5000 fc MPa). To define the behavior of unconfined concrete, Popovic’s model assumes that there is a straight extension of the curve beyond the strain of 2ε c0 (i.e., ε c > 2ε c0) up to the intersection with the abscissa axis at ε c = ε sp, as shown in Figure A3.5. The first studies on confined concrete were conducted on specimens with circular sections. With these specimens, straightforward calculations of the maximum confining stress (σ conf) provided by circular ties or spirals can be made. Assuming that the circular ties (or spirals) are separated by a distance of s, have a diameter of ds, a yield strength of f y,90, and a cross-sectional area of As,90, the maximum lateral confining pressure can be obtained from the equilibrium of half a circular section, as illustrated in Figure A3.6:

2As,90fy,90  conf sds  conf 

2As,90fy,90 (A3.6) sds

Figure A3.6 Confinement pressure in a circular section with circular ties (or spirals).

C onfinement and concrete softening  91

A3.4 INTERNAL ARCHES In RC structural elements, the confinement of the core of a concrete crosssection is provided by the transverse and longitudinal reinforcements. Due to the Poisson effect, when a concrete element is compressed longitudinally, it tends to expand transversely. However, transverse reinforcement prevents this expansion and creates internal arches supported at the most immobilized points of the reinforcement. Therefore, Figure A3.7 shows that panel a is less effectively confined than panels b and c, despite having the same reinforcement diameter and transverse reinforcement spacing. Mander et al. (1988) proposed a method to explain the confinement effectiveness of circular and rectangular sections in structural analysis. The gross area of the core (Ac), as shown in Figure A3.8, is defined as the area enclosed within the midline of the transverse reinforcement (i.e., circular ties or spirals). This definition is needed so that the confined concrete model (σ c–ε c) can be simplified, given the difficulty of considering the effectively confined area, which varies longitudinally and transversely.

Figure A3.7 Internal arches, unconfined zone, and effective and ineffective confined zones.

Figure A3.8 Gross area of the core, Ac.

92  Concrete structures

Figure A3.9 Concrete core confined by circular transverse reinforcement.

A3.4.1 Internal arches in circular sections Figure A3.9 shows a confined circular section where unconfined concrete (cover) and both ineffectively and effectively confined concrete can be observed. The gross area of the core of the section is Ac = πds²/4, where ds is the diameter of the midline of the circular ties or spirals. The circular ties (or spirals) are separated by a distance of s (clear distance s′) and have a cross-sectional area of As,90. If it is assumed that the internal arch is longitudinally supported by the circular ties forming a parabola with an inclination of 45 degrees at its ends, then the height of the vertex of the parabola is s′/4. The minimum effectively confined area (Ae) is located midway between two consecutive circular ties and has a value given by: 2



Ae 

2

   s’  s’  ds    ds2  1  (A3.7) 4  2 4  2ds 

C onfinement and concrete softening  93

The core area (Acc) is obtained by subtracting the area of the longitudinal reinforcement (As) from the gross area of the core (Ac):

Acc  Ac  As  Ac 1  cc  

 2 ds 1  cc  (A3.8) 4

where ρ cc is the geometric ratio of the longitudinal reinforcement with respect to the gross section enclosed by the midline of the circular ties or spirals (i.e., ρ cc = As/Ac). The term σ conf in expression A3.6 does not account for the effect of ineffectively confined concrete. Therefore, it is assumed that the confined area (Ac = πds2 /4) remains constant along the length of the element. If the circular ties or spirals are so far apart that the arches between parabolas come together (i.e., ds – s′/2 = 0 in Figure A3.9) then the value of Ae is zero, and the confinement effect disappears. To address this, Mander et al. (1988) defined the effective lateral confining pressure as the product of the lateral pressure from the transverse reinforcement and the confinement effectiveness coefficient ke (see Equation A3.9). They proposed a simple linear interpolation between two extreme cases: Ae = 0 (i.e., ke = 0) and Ae = Acc (i.e., ke = 1). When Ae = 0 the effect of confinement expressed as internal arches no longer makes sense (ke = 0). conf ,ef  conf k e

ke 

Ae (A3.9) Acc

If ρ s is defined as the volumetric ratio of the transverse reinforcement with the core of the gross section: As,90 ds 4As,90  (A3.10)  2 dss ds s 4 then the values of both the confining lateral pressure and the effective confining lateral pressure can be obtained from expressions A3.6, A3.9, and A3.10 as:

s 

conf 

1 sfy,90 2

(A3.11) 1 conf ,ef  k esfy,90 2 Once the effective confinement stress is known, the strength values of the confined concrete (fcc) and the corresponding strain (ε cc) can be calculated from expression A3.3. These values of fcc and ε cc can be introduced into the expression A3.5 to obtain the stress–strain model for confined concrete.

94  Concrete structures

Expressions A3.10 and A3.11 are equivalent to the ones proposed by EN 1992 (§8.1.4).

A3.4.2 Internal arches in rectangular sections In rectangular cross-sections, as in circular cross-sections, it is assumed that the parabolas (internal arches) form an initial tangent of 45° and are supported at the immobilized points of the cross-section. In rectangular crosssections, the parabolas are both vertical and horizontal. If the parabola has a w′ chord (as shown in Figure A3.10) and an initial tangent of 45°, the height of the vertex of the parabola is w′/4 and the area inside the parabola is equal to (w′)²/6. Therefore, the inefficiently confined area at the level of the transverse reinforcement for n parabolas formed between n immobilized longitudinal bars can be calculated using the following expression: n



Ai 

 i 1

 w i ’ 6

2

(A3.12)

where w′ i is the chord of each of the discharge parabolas (i.e., the clear distance between adjacent longitudinal bars). If the existence of vertical parabolas between the different levels of transversal reinforcement is also considered, the minimum effectively confined area can be calculated (Ae) as:

Figure A3.10 Confined concrete core for rectangular transverse reinforcement.

C onfinement and concrete softening  95 2 n   w i ’   1  s ’   1  s ’  (A3.13) A e   b cd c   6   2bc   2d c  i 1   where bc and dc are the dimensions of the core of the section, that is, the area enclosed by the median line of the outer stirrups, where bc ≥ dc. Note that Ae is located midway between the transversal reinforcement. Accounting for the definition given in expression A3.9, the confinement effectiveness coefficient ke can be formulated as:







 1   ke  

n

 i 1

 w i ’

 s’   s’   1  1   6bcd c   2bc   2d c   (A3.14) 1  cc 2

Balancing of forces in the x and y directions, respectively, as was done in Figure A3.7 for circular sections, the following is obtained:

Asx,90 fy,90  conf , x sd c Asy,90 fy,90  conf , y sbc

(A3.15)

where Asx,90 and Asy,90 are the areas of transverse reinforcement in the x and y directions, and σ conf,x and σ conf,y are the confining lateral pressures in the x and y directions, respectively. From the previous expressions, both the confining lateral pressure and the effective confining lateral pressure in the x and y directions can be calculated as: conf , x 



Asx,90fy,90  xfy,90 sd c

conf , x,ef  k exfy,90 conf , y

A f  sy,90 y,90  y fy,90 sbc

(A3.16)

conf , y,ef  k ey fy,90 where ke is given by = Equation A3.14. When the effective confining stresses in both directions are known, a three-dimensional failure model can be used to obtain an expression for fcc. The graph presented in Figure A3.11 was made using the Willam–Warnke model (Willam and Warnke 1975; Mander et al. 1988). So, once the value of fcc is obtained from Figure A3.11, the stress–strain model for confined concrete described in expression A.3.5 can be applied.

96  Concrete structures

Figure A3.11 Increase in compressive strength of rectangular sections confined by two lateral confining stresses, (Mander et al. 1988).

In the case of a confined core in triaxial compression with equal effective lateral confining stresses from spirals or circular hoops, the following expression can be used (Mander et al. 1988):

 7.94e,conf  fcc  fc  1.254  2.254 1   2 e,conf f fc c 

  (A3.17) 

The ultimate strain of concrete (ε cu, confinement failure), see Figure A3.5, can be approximated by (Paulay and Priestley 1992):

cu  0.004  1.4s

fy,90 sm (A3.18) fcc

where ε sm is the steel strain corresponding to the maximum tensile stress and ρ s is the volumetric ratio of the transverse reinforcement. For rectangular sections, ρ s = ρ x + ρ y, where ρ x and ρ y are defined in A3.16. The values of ε cu range between 0.012 and 0.05. EXAMPLE A3.1 MOMENT–CURVATURE GRAPH USING CONFINED CONCRETE3 The square section with sides of 0.5 m depicted in Figure A3.12a is considered here. The concrete is C-25 and the steel is B-400. The longitudinal reinforcement consists of 5 Ø20 bars on each side (a total of 16 Ø20 bars). Closed stirrups and Ø12 ties have been placed at 8 cm intervals, as recommended for areas with high seismic risk by ACI-318. Figure A3.12b shows the

C onfinement and concrete softening  97

Figure A3.12 Cross-section moment–curvature analysis when confined concrete is considered.

moment–curvature diagram for this section under a constant axial compression force of 1500 kN. The confined concrete model has been applied to the core of the section, while an unconfined concrete model has been used for the outer part of the section. Point 1 on the diagram corresponds to the failure of the outer zone as a result of cover spalling, while point 2 represents the failure in tension of the confining transverse reinforcement.

A3.5 CONCRETE SOFTENING The inverse effect of concrete confinement is softening, which refers to the decrease in the compression capacity of concrete caused by tensile lateral strain (ε 1). One of the models for concrete softening found in the literature is presented in Equation A3.19 (Collins and Mitchell 1991) and has been illustrated in Figure A3.13 for C=30 concrete (fck = 30 MPa).4

98  Concrete structures

Figure A3.13 Concrete softening model of concrete, for C-30 concrete.



      2  c  fc max 2  c    c     c1   c1  

(A3.19)

f 1 where c max   1 .0 fck 0.8  170 1 NOTES 1. Failure criteria such as Tresca and Von Mises are based on shear failure and do not consider the effect of confinement. 2. Values of ε c0 can be obtained from current regulations (e.g., EN 1992). fc and ε c0 used in Figure A3.5 are equivalent to fcm and ε c1 in §3.4.1. 3. To understand this exercise, it is necessary to have studied Chapter 7. 4. Equation A3.19 has been adapted to the European notation, assuming that f′ c is equivalent to fck.

REFERENCES Collins, M.P., and Mitchell, D. (1991). Prestressed Concrete Structures. Hoboken, NJ: Prentice Hall. Considère, A. (1902). “Résistance à La Compression Du Béton Armé et Du Béton Fretté.” Génie Civil, 82–86. EN 1992-1-1. (2023). Eurocode 2: Design of Concrete Structures - Part 1–1: General Rules and Rules for Buildings, Bridges and Civil Engineering Structures EN 1992-1-1. Brussels: European Committee for Standardization. Mander, J.B., Priestley, M.J.N., and Park, R. (1988). “Theoretical Stress-Strain Model for Confined Concrete.” Journal of Structural Engineering 114 (8): 1804–26. Paulay, T., and Priestley, M.J.N. (1992). Seismic Design of Reinforced Concrete and Masonry Buildings. New York: John Wiley & Sons.

C onfinement and concrete softening  99 Popovics, S. (1973). “A Numerical Approach to the Complete Stress-Strain Curves for Concrete.” Cement and Concrete Research 3 (5): 583–99. Richart, F.E., Brandtzæg, A., and Brown, R.L. (1928). A Study of the Failure of Concrete under Combined Compressive Stresses. Urbana, IL. Engineering Experiment Station, http://hdl​.handle​.net​/2142​/4277. Richart, F.E., Brandtzæg, A., and Brown, R.L. (1929). The Failure of Plain and Spirally Reinforced Concrete in Compression. Urbana, IL. http://hdl​.handle​ .net​/2142​/4073. Willam, K.J., and Warnke, E.P. (1975). “Constitutive Model for the Triaxial Behavior of Concrete.” International Association for Bridge and Structural Engineering 19: 1–30.

Annex B3

Tension stiffening of concrete and embedded bar models

B3.1 TENSION STIFFENING OF CONCRETE The phenomenon of concrete contributing to the tensile behavior of reinforced concrete elements is known as tension stiffening. Despite concrete’s low tensile strength, its contribution, though small, should be considered in the study of deformations of structural concrete elements when the concrete strain in tension exceeds the strain corresponding to the tensile strength of the concrete (fct). Figure B3.1a shows the graph of axial force N versus longitudinal strain for a concrete specimen with a square cross-section and a steel bar located in the center. The figure represents both the response of the bare bar and the response of the reinforced concrete specimen (i.e., concrete–steel set). After the formation of the first crack, the concrete stops contributing at the crack, but it continues to contribute between cracks. This helps to reduce the deformation of the element. Even with numerous cracks, the total deformation experienced by the embedded bar is lower than that experienced by the bare steel bar. In Figure B3.1a, the maximum axial load is Nmax (= f yAs). Some authors consider that the tensile stiffness effect disappears when ε = ε y (Wu and Gilbert 2008), while others suggest that there is a small amount of residual tensile stiffness (Gil-Martín et al. 2009; Lee et al. 2011). Figure B3.1b represents the distribution of the external action N along the element for the values of the axial tensile force N1 and N2 indicated in Figure B3.1a. For N1 (before cracking), the stress distributions in the steel and concrete (σ s and σ c, respectively) are constant along the bar. However, for N2 (after cracking), both stress distributions vary. Just before the first crack appears, the concrete stress and strain are fct and ε ct, respectively. After cracking, the mean tensile stress of the concrete (σ ct,mean) decreases as the mean strain value of the concrete increases. Equation B3.1, proposed by Bentz (2005), provides the mean value of the tensile concrete stress when the mean concrete strain is greater than ε ct:

100

DOI: 10.1201/9781003169659-7

Tension stiffening of concrete and embedded bar models  101

Figure B3.1 Contribution of concrete in tension.



ct,mean 

fctm 1  3.6Mc

where

M

Ac,eff

d 

(in mm) (B3.1)

b

where db is the diameter of the bar, and Ac,eff is the area of concrete effectively contributing to tension stiffening. Tension stiffening does not affect the entire cross-sectional area of the concrete but only to a part in the vicinity of the steel bar, called the effective area. Traditionally, the effective area (Ac,eff) has been assumed to be the rectangular area (perpendicular to the bar) extending over a distance no greater than 7.5 times the bar diameter. However, EN 1992 has changed 7.5 to 5.0. Figure B3.2 shows Equation B3.1 represented against experimental data taken from Wu and Gilbert (2008). The EN 1992 model (Equation 3.7) can be complemented with Equation B3.1 for short-term loads. However, since tension stiffening has a relatively small effect, it is often approximated by a line that becomes zero at ε y. This linear approximation of concrete tension stiffening is illustrated in Figure B3.2 (Hdz-Gil and Hernández-Montes 2023). B3.2 EMBEDDED BAR MODELS Figure B3.1b shows that when using the tension stiffening model for concrete in tension after cracking (σ ct,mean), a behavioral model for the average stress of the steel along the bar can be used (σ s,mean). This steel bar model

102  Concrete structures

Figure B3.2 Tension stiffening and steel bar models. Experimental data and formulation for fctm = 2.04 MPa,Ecm = 22400 MPa,db = 12 mm,and concrete cross-section of 100 × 100 mm.

is called the embedded bar model, as opposed to the bare bar model shown in Figure B3.2. The embedded bar model and the tensile concrete model must be mathematically related by equilibrium, as noted by Gil-Martín et al. (2009). For small steel strains, the behavioral model is linear elastic: σ s = Es ε s. The linear relationship remains valid until yielding of the steel occurs in a crack, known as the apparent yield, as seen in Figure B3.2. Beyond the apparent yield strain, Equation B3.1 is no longer applicable and there is a sudden drop in the concrete stress. Different assumptions have been made about this post-apparent yield behavior, such as those proposed by Collins and Mitchell (1991) and Hsu (1988). REFERENCES Bentz, E.C. (2005). “Explaining the Riddle of Tension Stiffening Models for Shear Panel Experiments.” Journal of Structural Engineering 131 (9): 1422–25. https://doi​.org​/10​.1061/(asce)0733-9445(2005)131:9(1422).

Tension stiffening of concrete and embedded bar models  103 Collins, M.P., and Mitchell, D. (1991). Prestressed Concrete Structures. Hoboken, NJ: Prentice Hall. EN 1992-1-1. (2023). Eurocode 2: Design of Concrete Structures - Part 1–1: General Rules and Rules for Buildings, Bridges and Civil Engineering Structures EN 1992-1-1. Brussels: European Committee for Standardization. Gil-Martín, L.M., Hernández-Montes, E., Aschheim, M.A., and Pantazopoulou, S.J. (2009). “Refinements to Compression Field Theory, with Application to Wall-Type Structures.” In American Concrete Institute, ACI Special Publication. Hdz-Gil, L., and Hernández-Montes, E. (2023). “Linear Concrete Tension Stiffening Model for Reinforced Concrete Elements.” Hormigón y Acero, published online. https://doi​.org ​/10​. 33586​/ hya​. 2023​. 3097. Hsu, T.T.C. (1988). “Softened Truss Model Theory for Shear and Torsion.” ACI Structural Journal 85 (6): 624–35. Lee, S.C., Cho, J.Y., and Vecchio, F.J. (2011). “Model for Post-Yield Tension Stiffening and Rebar Rupture in Concrete Members.” Engineering Structures 33 (5): 1723–33. https://doi​.org ​/10​.1016​/j​.engstruct​. 2011​.02​.009. Wu, H.Q., and Gilbert, R.I. (2008). An Experimental Study of Tension Stiffening in Reinforced Concrete Members under Short-Term and Long-Term Loads. Sydney: University of New South Wales.

Chapter 4

Prestressing force

Inside Kantutani Bridge, La Paz, Bolivia.

4.1 PURPOSE AND OBJECTIVES This chapter explains how prestressing force is transferred to concrete. It presents the various prestressing systems and their corresponding losses, as well as the layout of prestressing force.

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P restressing force 

105

Figure 4.1 (a) Pretensioned prestressed beam. (b) Posttensioned prestressed beam.

4.2 INTRODUCTION Prestressing force is so important in the field of structural concrete that it requires its own chapter. This chapter shows how prestressing force is introduced and how its value varies along the length of the prestressing tendon. The introduction and value of prestressing force along a structural element depend on whether the element is posttensioned or pretensioned. Figure 4.1a illustrates the manufacturing process of two prestressed beams on a casting bed using pretensioned prestressing. In this process, the steel is stretched before the concrete is poured, as shown in Figure 4.2. Figure 4.1b shows a simply supported beam manufactured using posttensioned prestressing. In this case, the concrete is cast with the sheath, and the steel is tensioned after the concrete has hardened. Initially, the jack introduces a tensioning force (Pmax) into the active reinforcement (both in pretensioning and posttensioning procedures). The maximum value of the tensioning force must agree with Equation 4.1, where σ p,max is given by Table 4.1 (Table 7.1 EN1992-1-1 2023).

Pmax  A pp,max (4.1)

with Ap as the cross-sectional area of the prestressed reinforcement and σ p,max as the maximum prestressing stress imposed at the active end of the prestressed reinforcement by the jack. Compressive stress in concrete also needs to be limited. A recommended limit is 0.6fck. 4.3 THE GEOMETRY OF PRESTRESSING In posttensioned structures, the geometry of the cable profile is usually parabolic. In simply supported beams, the cable profile takes the form of a

106  Concrete structures

Figure 4.2 Prestressed pretensioned beam with the prestressed reinforcement stressed before concreting. Table 4.1  Limits to short-term prestressing stresses Limited stress

Stress limit

σp,max

≤0.8 fpk ≤0.9 fp0.1k 0.95 fp0.1k ≤0.75 fpk ≤0.85 fp0.1k

Overstressinga (e.g., in the case of unexpectedly high friction) Maximum stress after prestress transfer/anchoringb σp,m0 (x)

 verstressing is permitted only if the force in the jack can be measured to an accuracy O of ±5% of the final value of the prestressing force. b The stress after transfer/anchoring is determined by subtracting the immediate losses (see 7.6.3 of EN 1992) from the stress on the active end. a

simple parabola, as shown in Figure 4.1b. In the case of continuous beams, the cable profile consists of a series of connected parabolas that are convex at the supports and concave at the center of the spans, as illustrated in Figure 4.3. The parabola is formulated as shown in Figure 4.4. In Figure 4.4., the parabola passes through the origin and has a horizontal slope at the origin. The angle α(x) represents the inclination of the

P restressing force  Maximum eccentricity in the span

107

Eccentricity at the support

Figure 4.3 Continuous beam with three spans.

Figure 4.4  Parabola formulation.

tangent to the parabola with respect to the horizontal, and for x = L0, the value of y is e0. The radius of curvature R and the variation of angle α, both as functions of x, are:



1 y¢¢ = R 1 + y¢2

(

)

3

2

2e dy = y¢ = tan ( a(x)) = 20 x dx L0

L20 2 e0

Þ

R(x = 0) =

Þ

2e a ( x = L0 )  0 L0

(4.2)

In the case of connected parabolas, such as those in Figure 4.5, the approach is similar. Let the horizontal projections of the parabolas within the first span be called L1, L 2, and L3. Parabolas 1 and 2 are connected at their lowest point, where both have horizontal slopes. If L1 and e1 are known, it is possible to determine the radius of curvature at the lowest point and the variation of the angle using Equation 4.2. Using the notations in Equation 4.2 and Figures 4.4 and 4.5, e0 = h – h 2 and L0 = L 2 for parabola 2. With these values, the radius of curvature at the lowest point and the variation of the angle can be calculated using Equation 4.2. A similar approach can be taken for parabola 3, where L0 = L3 and e0 = h 2 .

108  Concrete structures

Figure 4.5  Connected parabolas.

If two consecutive parabolas have the same slope (that is, α 2 = α 3) at their common point, then the initial and final points of these parabolas (parabolas 2 and 3 in this case) are aligned:

2 

2  h  h2  2 h2  3  (4.3) L2 L3

Solving for h 2:

h2  h

L3 L 2  L3

or  

h2 h  L3 L 2  L3

In the last equality of the previous equation, the term on the left corresponds to the slope of the line that joins the inflection point with the upper point of parabola 3, while the term on the right is the slope of the line that joins the upper point of parabola 3 with the lower point of parabola 2. As a result, the lower point of parabola 2, the inflection point, and the upper point of parabola 3 form a straight alignment. This alignment property is utilized to streamline the design process. In Figure 4.5, the upper point of parabola 3 has a horizontal slope and coincides with the upper point of parabola 4. The angle variation along the entire profile of the tendon is determined using the same procedure as the one employed for parabolas 2 and 3. Convex parabolas (e.g., parabolas 3 and 4 in Figure 4.5), whose main mission is to give continuity to the profile, are usually adjusted with minimum radii. Table 4.2 shows the minimum recommended radii based on the inner diameter of the sheath.

Table 4.2  Minimum values of the radii of curvature Diameter of the sheath (mm) Minimum radius of curvature (m)

45–55 3.5

65–80 4.5

85–95 5.0

100–110 7.0

P restressing force 

109

Figure 4.6 Eccentricities at end points.

Another factor to consider in the design of the prestressing profile is the eccentricity between the centers of gravity (cg) of the tendon and the sheath (Figure 4.6). Table 4.3 indicates the approximate values of this eccentricity, which have been taken from technical catalogs. Figures 4.7 and 4.8 show details of strands, sheaths, trumpets, wedges, and wedge plates. Figure 4.9 shows a jack for prestressing a single strand.

Table 4.3  Eccentricity values Number of S7 strands of 13.0 mm 3 4 7 12 19 22 31 55 Number of S7 strands of 15.2 mm 3 4 7 12 19 31 55

Diameter of the sheath (mm)

Eccentricity (mm)

32 41 51 64 80 86 102 140

7 7 8 11 13 12 14 23

38 51 57 76 100 125 165

5 5 10 13 19 22 30

110  Concrete structures

Figure 4.7 Anchor detail of posttensioned prestressing.

Figure 4.8 Trumpet and sheath, before and after concreting.

4.4 PRESTRESSING LOSSES The value of the prestressing force transferred by the jack to the tendons (= σ p,mt(x)Ap, with σ p,mt(x) as the mean value of the prestressing stress at

P restressing force  111

Figure 4.9 Jack used to tighten a single strand.

time t > t0) decreases as a result of different types of losses. Losses can be immediate (Δ σ p,i(x)) or time-dependent (Δ σ p,c+s+r(x)) and they might or might not appear, depending on the prestressing system.

p,m t (x)  p,m0 (x)  p,c  s  r (x) (4.4)

with σ p,m0(x) as the mean value of the prestressing stress after accounting for immediate losses. Immediate losses occur: • During the stressing process: • As a result of friction between the prestressing steel and the duct or the deviation devices Δ σ p,μ (x) • As a result of anchorage seating (e.g., wedge draw-in) Δ σ p,sl • At the transfer of prestress to concrete, caused by the instantaneous deformation of concrete Δ σ p,el • Because of the short-term relaxation of the steel during the period that elapses between the tensioning of the tendons and the prestressing of the concrete (e.g., in the fabrication of railway sleepers1). 4.5 LOSSES CAUSED BY FRICTION (Δσ p,μ) Frictional losses along the tendon only affect posttensioned reinforcements. In Figure 4.1a, it is clear that the force (Pmax) introduced before concreting

112  Concrete structures

Figure 4.10 Angle friction losses.

has the same value along the entire tendon. In Figure 4.1b, the beam is tensioned from the left end and, as a consequence of friction along the sheath, the prestressing force at the right end is lower. Figure 4.10 shows a tendon segment with a length of dx and an angle variation of dα. The tendon is subjected to prestressing forces P and P-dP at its ends, which produce a vertical force with a value of 2Psin(dα/2). In turn, the sheath exerts an equal and opposite vertical force on the tendon, N. If the coefficient of friction between the tendon and the sheath that houses it is μ, the friction loss is μN. Because α is a small angle, the following approximation can be made: sin(dα/2) ≈ dα/2, and therefore, the friction loss caused by the angle variation is μPdα:

mN = m(2P Sin[da / 2])  mPda (4.5)

In addition to the friction caused by angle variation, there is another type of friction along the tendon that is independent of the angle variation. This friction depends on the diameter of the sheath that houses the tendon, the type of tendon, and the way the tendon is housed inside the sheath. The loss associated with this type of friction in a length of dx is µκ µPdx, where κ µ is the coefficient of friction in a straight line or parasitic friction, and P is the prestressing force. The origin of this parasitic friction can be found in the relative undulation that exists between the sheath and the tendon (Figure 4.11). Adding the two types of friction losses:

Figure 4.11 Parasitic effect on straight segments.

P restressing force 

113

dP   P d  P dx (4.6)



Integrating: Pmax



Pm0 ( x )



x

dP   d   dx P

 0





 Ln[Pmax ]  Ln[Pm0 ( x)]     x 

0

Pm0 ( x)  exp[(   x)]  Pmax  Pm0 ( x)  Pmax  Pmax exp Pmax [(   x)]

 P ( x,  )  Pmax (1  exp[(   x)]) or in terms of stresses:

p,  p,max 1  exp[(   x)]

(4.7)

where (as stated in EN 1992, §7.6.3.2): α μ sum of the absolute values of angular deviations over distance x µ coefficient of friction between the tendon and its sheath or deviation device (see Table 4.4) κ µ unintentional angular deviation for internal posttensioning tendons per unit length (curvature), (0.005 < κ μ < 0.01) x distance along the tendon from the point where the prestressing stress is equal to σ p,max (the force at the active end during tensioning) Table 4.4  Friction coefficients, μ, of internal posttensioning tendons and external tendons to be used in the absence of more precise data Internal tendons

Greased and sheathed strands

External tendons

Type of prestressing steel

Metal duct

Polymer duct

Polyethylene duct

Polyethylene duct

Cold drawn wire Strand Deformed bar

0.17 0.19 0.65 0.33

0.12 0.14 NA NA

NA 0.05 NA NA

0.10 0.12 NA NA

Smooth round bar

114  Concrete structures

4.6 LOSSES CAUSED BY ANCHORAGE SEATING (Δσ P,SL) Anchoring with wedges is normally used in posttensioned prestressing (Figure 4.7). A wedge is inserted together with a strand, and the wedge immobilizes the strand in the wedge plate. When the wedges penetrate into the wedge plate, pliering the strand, the strand shortens by approximately 5 mm, producing the anchorage seating loss of Δσp,sl. In this type of prestressing, once the jack has applied the tensioning force to the tendon, the tendon is anchored and the jack is removed. The anchorage seating loss (Δσp,sl) affects only the length of the tendon that is called the influence length (X), see Figure 4.12. The value of X depends on the friction (p) between the tendon and the sheath, which is assumed to be constant per unit length. The value of p does not depend on how the cable moves in the sheath (inward or outward), so the solid line (AB) and the dashed line (CB) have equal and opposite slopes (Figure 4.12a). Figure 4.12b shows that the initial value of the force in the anchor is Pmax, passing to Pmax–ΔPsl when wedge penetration occurs (ΔPsl = Ap Δσp,sl). At a distance of X, the prestressing force has a value P′ which, by definition of X, is the same before and after the wedge penetration loss, ΔPsl. By examining the equilibrium in the two situations shown in Figure 4.12b (before and after wedge penetration) the value of the prestressing force and the ΔPsl can be obtained as:



Pmax  P  p  X   Pmax  Psl  p  X  P



p,sl,meam 



P ’  Pmax 

Psl 2

(4.8) Psl  pX 2 Note that the friction developed in influence length (X) changes sign (i.e., orientation) when passing from the prestressing phase (without wedge penetration) to the anchorage seating phase (with wedge penetration), as shown in Figure 4.12b. As a result of anchorage seating, the greatest variation of the prestressing force is where the wedge is (=ΔPsl; see Figure 4.12b), and it is zero at any distance greater than X from the anchorage. Thus, the average variation of the prestressing force in distance X is ΔPsl /2. The average loss of stress can be obtained by dividing the loss of prestressing force by the area of the tendon (Ap) and, therefore: Psl (4.9) 2A p

Since the variation of the stress in the tendon along X goes from a maximum value to zero (with a variation that is assumed to be linear), the

Figure 4.12 (a) Prestressing force. (b) Equilibrium in the influence length.

P restressing force  115

116  Concrete structures

deformation corresponding to the penetration of the wedge (a) is the same as if the entire length of X is subjected to an average loss of stress of:

a

p,sl,mean X (4.10) Ep

with Ep as the modulus of elasticity of prestressing steel. By substituting Δ σ p,sl,mean for ΔPsl /2Ap (Equation 4.9) and ΔPsl /2 for pX (second Equation 4.8) in Equation 4.10, the value of the influence length (X) is obtained:

X=

a E pAp (4.11) p EXAMPLE 4.1 FRICTION AND SEATING LOSSES CALCULATION

Figure 4.13 shows a posttensioned beam with four spans of 25 m each. The eccentricities of the tendon profile are indicated in the figure (0.6 m in the middle of the span and 0.5 m at the supports). The posttensioned reinforcement is composed of a tendon of 19 Y-1860-S7 strands with a nominal diameter of 15.2 mm. All the strands in the tendon are stressed simultaneously. The tendon is anchored at the axis of symmetry by using a joint, and the tensioning of the tendon is carried out from both ends. The wedge penetration is 5 mm. The values of µ and κ are given by the supplier (µ = 0.21 and κ μ = 0.006). A 100 mm diameter sheath is needed to house the 19 strands (according to technical catalogs, Table 4.4). Ap = 140 · 19 = 2660 mm2 Ep = 190000 N/mm2 As there is wedge penetration, we have chosen to tighten up to 85% of fpk, which means Pmax = 0.85 · 1860 · 2660/1000 = 4205 kN.

Figure 4.13 Beam with four spans, 25 m each.

P restressing force 

117

Friction losses are calculated by applying Equations 4.2 and 4.7. In section 2 of Figure 4.13: 2 

2e 2  0.6   0.107 rad L 11.25

(    x )  0.21( 0.107  0.006 11.25)  0.03658 P  Pmax 1  e ( (   x ))   151.02 kN To locate the following sections, the alignment of the inflection points (sections 3, 5, and 7) with the points of the adjacent sections (Equation 4.7) is considered. Table 4.5 has been created for the eight sections indicated in Figure 4.13. The value of Pmax–ΔPµ is represented in Figure 4.14. To determine the influence length of wedge penetration loss (from Equation 4.11) it is necessary to know the value of p (friction per unit length), which is obtained between sections 1 and 3 as: p

340000  15.11 N / mm 2 11250

and therefore the value of X is: X

aEp A p  p

5 190000  2660  12930 mm  12.93 m 15.11

The value of the anchorage seating loss is determined from the second part of Equation 4.8: Psl  2  p  X  2 15.12 12930  391003 N  391.00 kN Calculating the elongation that the tendon experiences during the stressing phase is a good way to check if the stressing process has proceeded as planned. Assuming that the prestressing force varies linearly between the ends of each parabola, and assuming that it is measured after anchorage seating, the value of the average prestressing force is: Pmean 

1  4005  3809 4005  3865 12.93    2 11.25 12.93  50  2 2

3725  3579 3579  3406 3406  3241 3865  3725  2.5   2.5  10  10  2 2 2 2 3241  3114   2.5  3657.8 kN 2 

ΔPsl (kN)

e (m) L (m) θ (rad) x (m) ΔPµ (kN) Pmax – ΔPµ (kN) Anchorage seating losses a (mm) X (m)

Section

Friction losses

5 12.93 390.91

0 0 0 4205

1

Table 4.5  Friction losses, accumulated values

0.6 11.25 0.107 11.25 151 4054

2 0.9 11.25 0.267 22.5 340 3865

3

p=

0.2 2.5 0.427 25 480 3725

4

15.12

0.22 2.5 0.603 27.5 626 3579

5

kN/m

0.88 10 0.779 37.5 799 3406

6

0.88 10 0.955 47.5 964 3241

7

0.22 2.5 1.131 50 1091 3114

8

118  Concrete structures

P restressing force 

119

Figure 4.14 Variation of prestressing force.

and, therefore, the elongation experienced by the tendon during tensioning must be: L 

50000  3657800  361.9 mm 190000  2660

4.7 LOSSES CAUSED BY THE INSTANTANEOUS DEFORMATION OF CONCRETE In pretensioned elements, before the loosening phase, the cable is stressed and the concrete is hardened but not stressed. In the loosening phase, stresses are transferred to the concrete. These stresses cause a shortening of the structural element as a result of the compressive stress introduced into the concrete by the cable (Figures 4.15 and 4.16). In turn, the shortening of the concrete beam affects the tendon since this tendon also shortens, which produces a new loss in the tensioning force: ΔPel (ΔPel = Δ σ el Ap). According to the plane section hypothesis, strain at a certain height is the same in both concrete and steel:

120  Concrete structures

Figure 4.15 Losses caused by instantaneous deformation of concrete.

Figure 4.16 Loosening process of prestressed steel.





Ep c (t) p   p  c (t) (4.12) E cm (t) Ep E cm (t)

In the case of posttensioning, if the tension is transferred at the same time to all the tendons, elastic shortening does not cause any losses. This is because the jack leans on the concrete. In this type of prestressing, the loss caused by elastic shortening appears when the tendons are successively stressed.

P restressing force 

121

Figure 4.17 Losses in a wedge plate of six strands.

The 6-strand wedge plate shown in Figure 4.17, in which strands 1, 2, and 3 have already been tensioned, needs to be looked at in detail. When tensioning strand 4, there is no loss of elastic shortening in that strand, but there are losses in strands 1, 2, and 3 (see the table in Figure 4.17). If, when tensioning strand 4, there is a variation in the stress in concrete Δ σ c (t), the loss in strands 1, 2, and 3 is:

A pi

Ep c (t) (4.13) E cm (t)

where Api is the area of strand i. The loss corresponding to the elastic deformation of the concrete can be calculated by considering the order in which the strands and tendons are tensioned, and the number of strands tensioned at the same time. EXAMPLE 4.2 CALCULATION OF THE ELASTIC SHORTENING IN A SIMPLY SUPPORTED PRESTRESSED BEAM The prestressed beam shown in Figure 4.18 is 20 m long. It is made of C-45 concrete and Y-1860-S7 steel. On the casting bed, prestressing steel was subjected to an initial tension of 75% of fpk. Inputs: Ac = 4.53 × 105 mm2; Ap = 2850 mm2; Ic = 9.43 × 1010 mm4, and the self-weight is 10 kN/m. To calculate the elastic shortening, the following aspects must be considered: • Longitudinally, steel and concrete located at the same height shorten to the same extent. As a result of this shortening, the prestressing force, which was initially Pmax, becomes Pmax – ΔPel. • The beam is subjected to the prestressing force when the cables or strands are released, and the beam tends to rise holding its own weight.

Figure 4.18 20 m span pretensioned beam.

122  Concrete structures

P restressing force 

123

It is assumed that all the strands are located at e = 560 mm (this hypothesis would not be valid if strands were not located close enough to each other). The value of the maximum bending moment caused by self-weight is M 0: M0 

10  202  500 kN  m  5 108 Nmm 8

and the concrete stress at the height of the center of gravity of the prestressing steel in the section located at the center of the span is:2 c 

Pmax  Pel (Pmax  Pel ) e  e M0  e   Ic Ac Ic

By introducing Equation 4.12 into the previous equation (note that initial stress in concrete is zero, so Δ σ c = σ c) and considering that Pmax = 0.75fpk Ap, the following system of two equations with two unknowns is obtained: Pmax  Pel (Pmax  Pel )e  e M0  e  Ecm ( t )    E p  Ic Ac Ic  p  Pmax  Pel  A p  ( po  p ) When solving the previous system of equations (Ecm = 36283 MPa, Ep = 190000 MPa), we obtain Δ σ p = 92.05 N/mm2 and ΔPel = 262328 N. At the ends of the beam, the bending moment caused by self-weight is zero, and it is obtained that Δ σ p = 106.41 N/mm2 and ΔPel = 303262 N. Therefore, the average loss in the active reinforcement is 283 kN, which represents a loss caused by an elastic shortening of 7.1%.

EXAMPLE 4.3 ELASTIC SHORTENING IN A POSTTENSIONED SIMPLY SUPPORTED BEAM The beam from the previous example is considered here but with a parabolic prestressing profile (see Figure 4.19). The posttensioned technique has been used and the beam has been stressed in three tensioning sessions. The expression of the concrete stress at the height of the center of gravity of the prestressing steel together with Equation 4.13 form the following system of equations: Pmax  Pel (Pmax  Pel )  e  e M0  e    c  3c  Ic Ac Ic   A p Ep  Pel  ( 0  1  2) 3 Ecm c 

124  Concrete structures

Figure 4.19  Posttensioned beam.

(0 + 1 + 2) is the losses in each tensioning session (see table in Figure 4.17). σ c = 18.52 N/mm2 and ΔPel = 92128 N are obtained from the previous system of equations. At the ends of the beam, the moment caused by self-weight is zero, and the eccentricity is also zero, obtaining σ c = 6.68 N/mm2 and ΔPel = 43187 N. The average loss caused by elastic shortening in prestressing reinforcement can be taken as the average between both values, ΔPel = 68 kN, which represents a loss of 1.7%.

4.8 TIME-DEPENDENT PRESTRESSING LOSSES Once the prestressing steel has been anchored, and after the instantaneous losses have occurred, new losses appear over time. These losses are caused by the shortening of the concrete (shrinkage and creep) and by the relaxation of the prestressed steel.​ The creep and relaxation phenomena, in turn, depend on the value of the deferred losses, so an iterative process must be carried out. The timedependent losses can be approximately calculated by using the expression:



Pc  s  r  A p p,c  s  r  A p

Ep (t, t0 )cp,QP E cm (4.14) 2  Ap  Ac z cp 1   1  0.8(t, t0 ) Ac  Ic 

csE p  0.80pr  1

Ep E cm

where: Δ σ pr absolute value of the variation of stress in the tendons at location x, at time t, caused by the relaxation of the prestressing steel; it should be determined for the initial stress in the tendons caused by initial prestressing and the quasi-permanent combination of actions σ pr = σ pr(G+Pm0 +ψ 2Q) σ cp,QP stress in the concrete adjacent to the tendons, caused by selfweight and initial prestressing and other quasi-permanent actions.

P restressing force 

125

The value of σ cp,QP can be the effect of part of the self-weight and the initial prestressing or the effect of a full quasi-permanent combination of actions [σ cp(G + Pm0 + ψ 2Q)], depending on the stage of construction being considered (see B.8 of EN 1992) zp distance between the centroid of the concrete section and the tendons A conservative of 1.0 can be adopted for the denominator of Equation 4.14. EXERCISES 1. Calculate the time-dependent losses in Example 4.1. Make a graph similar to the one in Figure 4.14.

NOTES 1. In cases like this, where heat curing is used, losses caused by shrinkage and relaxation should be considered, as well as thermal effects. 2. In the stress expression, σ c, the bending moment caused by the self-weight of the gross section instead of the real section has been considered. Although this is not entirely accurate, the difference is small.

REFERENCES EN1992-1-1. (2023). Eurocode 2: Design of Concrete Structures - Part 1–1: General Rules and Rules for Buildings, Bridges and Civil Engineering Structures EN 1992-1-1. Brussels: European Committee for Standardization.

Section III

Analysis

Chapter 5

Structural analysis

Arch evolution and historical precedence: Robert Hooke’s notion and Farghan’s catenary concept in Taq-i Kisra palace design.

5.1 PURPOSE AND OBJECTIVES The analysis of structures involves determining the distribution of internal forces and moments (diagrams of axial forces, shearing forces, bending moments, and torsional moments), and the stresses, deformations, and displacements of the elements of a structure when it is subjected to external loads or actions. This chapter focuses on the different types of structural analysis commonly used for reinforced concrete structures. A comprehensive study of second-order analysis using the stiffness geometric matrix and buckling analysis will be presented.

DOI: 10.1201/9781003169659-10

129

130  Concrete structures

5.2 INTRODUCTION When selecting a type of structural analysis, it is useful to consider both traditional and current trends. The traditional trend that is most commonly mentioned in standards is the use of first-order analysis, which does not consider possible instabilities. Therefore, it is necessary to supplement this approach with additional studies of local and/or global instabilities (buckling). When using nonlinear and plastic analysis, instability analysis does not need to be carried out if the equilibrium of the deformed structure is considered.1 The steps involved in designing a concrete structure include making a model of the structure using nodes and elements, assigning properties to each member, choosing a type of structural analysis, calculating the structure, and sizing the structure (i.e., provide reinforcement) so that it complies with the corresponding limit states. Making a model of a structure with nodes and elements is a simple process in which experience plays a fundamental role. A useful recommendation for beginners is to obtain drawings of a similar structure and try to understand every detail before designing a new one. Additionally, imperfections caused by deviations in the geometry of the structure and in the position of loads must be considered (§7.2.1 of EN 1992-1-1 2023). The next step is to define the properties of the elements of the structure, which involves deciding on the area, the inertia, and the material properties. This step is traditionally referred to as predimensioning. There are numerous procedures for predimensioning a concrete structure, with the simplest being to take the sections of a similar structure and modify them accordingly. Once the model of the structure and the sections of the elements have been determined, a type of structural analysis needs to be chosen. Each type of structural analysis has unique features, which are examined in the following section. After selecting the type of analysis to be used, the internal forces and moments of the elements of the structure, as well as their deformations, are determined. The final step involves sizing the concrete elements (providing reinforcement) so that they can withstand the loads required (forces and moments) and comply with the deformation limitations stated by the standards (deflections and crack widths). In concrete structures, rectangular or circular cross-sectional shapes are commonly used for columns, while rectangular, T, or double-T shapes are commonly used for beams2 (see Figure 5.1). Other cross-sections are also used, and the geometry of the cross-section sometimes varies along the axis of the element. In concrete structures, nodes transmit forces and moments, causing columns to typically function as beam-columns that experience a diverse range of loads, including axial, bending, shearing, and torsional loads.

S tructural analysis 

131

Figure 5.1 Square and circular column sections and T-beam section.

Soil–structure interaction is often an important factor that must be considered. One way to address this is by considering an elastic foundation using the Winkler coefficient. This method has proven to be effective in the case of common structures. 5.3 TYPES OF STRUCTURAL ANALYSIS Chapter 7 of EN 1992-1-1 considers the following types of structural analysis: • • • •

Linear elastic analysis Linear elastic analysis with redistribution (or elasto-plastic analysis) Plastic analysis, including strut-and-tie models, and Nonlinear analysis

The need for local analysis to complement global analysis in D-regions will be discussed in Chapter 6.

5.3.1 Linear elastic analysis Linear elastic analysis, also known as first-order analysis, is based on the theory of elasticity and considers the equilibrium conditions of the undeformed structure. Additionally, this method uses the plane section hypothesis (PSH). This type of analysis can be used to calculate internal forces and moments in both serviceability and ultimate limit states if P-delta effects are not significant. Moreover, this type of analysis can be used to calculate deformations if materials are properly characterized (e.g., creep).3 The main advantage of this type of analysis is that the governing equations can be written in a linear matrix formulation. Structural analysis courses have shown that a two-dimensional bar has six degrees of freedom (DOFs). Displacements in the nodes (Figure 5.2 left), collected in vector δ , are related to the forces acting on the nodes (Figure 5.2 right), grouped in vector f, throughout the stiffness matrix of bar k. The components of k are kij, where kij represents the value of the ith component of force vector

132  Concrete structures

Figure 5.2 Displacements and forces in an element with six DOFs.

f when a unitary value is imposed on the jth DOF in vector δ , while the remaining displacements are held equal to zero. The stiffness matrix (k) for the beam-column element shown in Figure 5.2 (which has six degrees of freedom per node) is provided next. (f = kδ ): 0 0 A 0 0  A  12I 6I 12I 6I   0 0   ui   Hi  L  L    L2 L2 V   vi    i 6I 6I 4I 0  2I     Mi  E  0 L L  i    (5.1) H 0 0 A 0 0   uj   j  L  A     V  12I 6I   v j   0  12I  6I 0  j   Mj  L L   j  L2 L2       6I 6I  0 2I 0  4I    L L 14444444 4244444444 3 k with A, I, and L as the cross-sectional area, moment of inertia, and length of the element, respectively. The standard matrix expression of linear elastic analysis is achieved by assembling all the elements of a structure and making the changes required by the reference system, resulting in the following equation:

F = KD (5.2)

In Equation 5.2, F represents the vector of external forces acting on the degrees of freedom considered, D is the displacement vector of all the DOFs, and K is the stiffness matrix of the structure. This analysis is linear because if an F vector leads to D displacements, then an α F vector produces α D displacements for any real number α. This hypothesis, however, is no longer suitable when plasticizations occur in the structure, as some components of the K matrix are no longer constant. According to EN 1992-1-1 (§7.3.1(2)), a linear elastic calculation can be carried out by using the characteristics of uncracked cross-sections, linear stress–strain relationships, and mean values of the modulus of elasticity. The area and moment of inertia of the uncracked and gross sections are

S tructural analysis 

133

very similar. Using the gross-section properties has an added benefit, as knowing what reinforcement has been used for the elements is not necessary for the structural analysis. While considering gross-section characteristics provides good approximations for internal forces and moments, it might result in inaccurate values for deformations. The equilibrium of the undeformed structure is considered in a linear elastic analysis. However, when a structure undergoes significant deformation, the bending moments change as a result of variations in the geometry of the structure. Therefore, the hypothesis of establishing the equilibrium of an undeformed structure, also known as the hypothesis of geometric linearity, can no longer be accurate. EXAMPLE 5.1 LINEAR ELASTIC ANALYSIS OF A FRAME The frame in Figure 5.3 is based on the dimensions and loads of the Femern tunnel, between Germany and Denmark, which is a submerged box that withstands 40 m of water above it. Linear elastic analysis is used to determine the internal actions and displacements at points 3, 4, and 5, using the mechanical characteristics of the gross sections and a modulus of elasticity of 32837 MPa. The results presented in Table 5.1 were obtained using OpenSees software (McKenna et al., 2000). Table 5.1 presents the results obtained with and without considering the effect of creep Ecm and Eceff. The comparison of the results in Table 5.1 indicates that although creep affects deformation, it hardly affects the values of axial forces, shearing forces, and bending moments. In this example, a linear elastic analysis was performed. As already seen, it must be verified that there are no significant displacements that, in conjunction with the axial force on the side walls of the tunnel (elements 1–3 and 2–5), could generate major additional moments.

Figure 5.3  Frame example.

ux (m)

uy (m)

θ (rad)

0.00508 –0.000478 –0.001803 0.00489 –0.008883 0.000245 0.00470 –0.000566 0.000777 Using the effective modulus of elasticity, Eceff=10946 MPa 3 0.01525 –0.001434 –0.005408 4 0.01467 –0.026649 0.000736 5 0.01410 –0.001698 0.002332 0 0 0 1

3 4 5

Node

Table 5.1  Results of linear elastic analysis

1.374×106 –1.374×106 –1.374×106 1.374×106 –1.374×106 –1.374×106 2.747×106

3–4 3–4 4–5 1–3

N (N)

3–4 3–4 4–5

Ecm = 32837 MPa

Element

2.747×106 0.253×106 3.253×106 –0.374×106

2.747×106 0.253×106 3.253×106

V (N)

–3.189×106 4.293×106 –6.226×106 0.195×106

–3.189×106 4.293×106 –6.226×106

M (N·m)

134  Concrete structures

S tructural analysis 

135

Figure 5.4  P-delta effect.

EXAMPLE 5.2 EQUILIBRIUM ON A DEFORMED STRUCTURE Figure 5.4 shows a column with height h subjected to axial force P and horizontal force V. In linear elastic analysis, the moment at point A (MA) is given by MA = Vh (since load P does not produce any bending moments in the undeformed structure). However, if the column deforms as a result of the forces applied, V and P, as shown in the figure, displacement Δ occurs and the additional PΔ moment must be taken into account. If the equilibrium in the deformed configuration is considered, the moment at point A becomes MA = Vh + PΔ. In a linear elastic analysis, the hypothesis of small displacements is made, which means that the equilibrium of the undeformed structure is considered and so Δ displacement and the additional stresses associated with it are not taken into account. This is equivalent to not considering the PΔ effects, or in other words, geometric nonlinearity is not considered. In Annex A5, the matrix formulation of geometric nonlinear analysis is presented. In concrete structures, P-delta (PΔ) effects are typically negligible as the displacements are usually small.

136  Concrete structures

EXAMPLE 5.3 ELASTIC ANALYSIS CONSIDERING GEOMETRIC NONLINEARITY Example 5.1 is considered again, but geometric nonlinearity (or P-delta effects) is considered this time. Table 5.2 shows the results obtained with OpenSees (McKenna et. al, 2000) for the frame of Example 5.1 when P-delta effects and creep (Ec,eff ) are taken into account. Comparing Tables 5.1 and 5.2 reveals that the P-delta effect is insignificant in the beam (elements 3–4 and 4–5). However, the bending moment at node 1 is 195 kN·m without P-delta effects (as shown in Table 5.1) and 219 kN·m with P-delta effects (as shown in Table 5.2). So, due to the displacement of the structure, the columns experience a 12% increase in the value of the bending moment. Since this example involves a submerged structure (40 m under the sea), the lateral walls bear huge axial forces, making the P-delta effects especially important. In accordance with EN1992 (§7.4.1(3)), second-order effects do not need to be considered if they make up no more than 10% of the corresponding first-order effects. Thus, in this example, second-order effects cannot be ignored.

5.3.2 Linear elastic analysis with redistribution Eurocode 2 allows for the redistribution of moments in beams (EN 1992, §7.3.2). This redistribution is possible because of the plasticization of certain zones. EN 1992 presents two types of linear analysis with redistribution: with and without an explicit check of rotation capacity. This analysis is typically used in continuous beams (as shown in Figure 5.5), where the negative bending moment at the inner supports is greater (in absolute value) than the bending moment at the center of the spans. In these cases, plasticization enables the moment at the supports to be reduced, resulting in an increase in the bending moment at the center of the spans. EXAMPLE 5.4 MOMENT REDISTRIBUTION Figure 5.6 shows a built-in beam with a constant cross-section that is subjected to a uniform load of q. A linear elastic analysis provides the moment at the ends as qL 2/12 and at the center of the span as qL2/24 (dashed line in Figure 5.6a). However, if the rotation capacity of the section is sufficient and Mpl is less than qL 2/12 (Figure 5.6b), then plasticization can occur at the beam

0.01542 0.01484 0.01426 0

3 4 5

1

ux (m)

Node

–0.001433 –0.026704 –0.001700 0

uy (m) –0.005432 0.000744 0.002322 0

θ (rad) 3–4 3–4 4–5 1–3

Element

Table 5.2  Elastic analysis results, considering material linearity and P-delta effects 2.748×10 0.252×106 3.260×106 –0.370×106

1.376×10 –1.376×106 –1.376×106 2.744×106

6

V (N) 6

N (N)

–3.180×106 4.306×106 –6.252×106 0.219×106

M (N·m)

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138  Concrete structures

Figure 5.5 Redistribution of bending moments in a three-span beam.

Figure 5.6  Built-in beam.

ends and cause a redistribution of the bending moment diagram (continuous line in Figure 5.6a). But, if the plastic moment (Mpl in Figure 5.6b) of the cross-section of the beam is greater than qL2/12, then the bending moment diagram will be the same as the one obtained by linear elastic analysis without redistribution (dashed line in Figure 5.6a).

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Figure 5.7 Plastic models of beam-column elements. (Adapted from NEHRP Seismic Design Technical Brief No 4, Deierlein, Reinhorn, and Willford 2010.)

5.3.3 Plastic analysis and nonlinear analysis The term plastic analysis refers to the process of putting a material through plastic deformation, as introduced in the previous section and Example 5.4. Plastic analysis is based on the successive formation of hinges (in the case of concentrated plasticity) or plastic zones (in the case of distributed plasticity), in which large rotations can occur until a structure collapses when it becomes a mechanism. Figure 5.7 shows the different structural approaches used in plastic analysis. In the case of concentrated plasticity (Figures 5.7a and 5.7b), bilinear moment–rotation relationships or other plastic models, such as empirical and polylinear relationships, are used (FIB 2012). The yield-line theory of concrete slabs is an example of concentrated plasticity (Johansen 1962). In reinforced concrete beam-column elements, plasticity is usually assumed to be distributed over a certain length. The three most widely used models for this type of plasticity are plastic zones, fiber elements, and finite elements (Figures 5.7c, 5.7d, and 5.7e, respectively). Another type of plastic analysis used in concrete structures is the strutand-tie model (or STM), which will be studied in Chapter 6. The term nonlinear analysis, when referring to a material, indicates that the stress–strain relationship of a material is nonlinear, whether it is plasticized or not. This type of analysis can be used in serviceability limit state (SLS) and ultimate limit state (ULS). EXAMPLE 5.5 GEOMETRIC AND MATERIAL NONLINEAR ANALYSIS The structure in Example 5.1 is being studied again, this time considering geometric nonlinearity and material nonlinearity. Two types of concrete models

140  Concrete structures

Figure 5.8 Frame in Figure 5.3. Column section at support and beam section mid-span.

are used: one for the cover and the other for the confined core. The effect of creep is not taken into account on this occasion. Fiber-section-type elements are used, and four different reinforcement sections are defined: two for the lateral walls and two for the beams. Figure 5.8 displays the section of the columns at the supports and the section of the beam at mid-span. The parameters given by EN 1992 (see Chapter 3, Figure 3.13) for C-30 are considered for the concrete model (nonlinear σ-ε diagram for instantaneous load). The results of considering tension stiffening of concrete are summarized in the lower part of Table 5.3 (in this case, a linear model is used for tension stiffening, see Annex B3). Table 5.3 indicates that the effect of tension stiffening is significant in both the distribution of bending moments and deformations.

Several hypotheses and simplifications can be used in structural analysis:

i. Shear deformation: The coefficients kij of matrix k in Equation 5.1 usually consider axial and bending deformations but not shear deformations. However, shear deformations could also be considered, as will be explored in Chapters 8 and 11. ii. Small deformations: Considering small deformations is equivalent to assuming that the bending curvature is d 2y/dx 2 .

5

Considering tension stiffening of concrete 0.00919 0.00968 0.01023

0.04932 0.05045 0.05655

3 4 5

3 4

ux (m)

Node

–0.000322 –0.014585 0.000393

0.003646 –0.027115 0.013792

uy (m)

Table 5.3  Results of the nonlinear analysis obtained with Opensees

–0.000304 0.000579 0.001149

–0.011369 0.002517 0.006317

θ (rad)

3–4 3–4 4–5

3–4 3–4 4–5

Element

1.245×106 –1.245×106 –1.245×106

2.757×106 0.247×106 3.251×106

2.760×10 0.252×106 3.261×106

1.058×10 –1.058×106 –1.058×106

6

V (N) 6

N (N)

–3.275×106 4.257×106 –6.248×106

–2.349×106 5.170×106 –5.387×106

M (N·m)

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142  Concrete structures

iii. Large displacements or second-order analysis (geometric nonlinearity): This hypothesis involves formulating the equilibrium on the deformed structure. In Example 5.2, the moment at point A is M A = Vh + PΔ (instead of Vh, which is obtained from a linear or first-order elastic analysis). Because Δ is not known a priori, an iterative procedure is required to calculate M A. Refer to Annex A5. iv. Creep: In Equation 5.1, the value of E corresponds to the modulus of elasticity of the concrete, E cm. If the effects of creep are considered, the value of E cm will vary as a function of time and loading time. In this case, the effective modulus of elasticity of the concrete is given by Equation 5.3:

E c,eff (t, t0 ) 

1.05E cm (5.3) 1  (t, t0 )

where φ(t,t0) is the creep coefficient studied in Section 3.5 (see Chapter 3). The effect of creep in the frame of Example 5.1 is shown in Table 5.1. To account for the effect of the duration of the load, it is necessary to know which part of the load causes creep and which part of the load, because of its short duration, does not. The effect of load duration can be taken into account in a simplified way by introducing an effective creep coefficient (EN 1992 §7.4.2(2)): φ eff,s, for global second-order effects, and φ eff,b for isolated members and local second-order effects.

v. Cross-section variation: The cross-section and its mechanical properties can change along the element. In reinforced concrete elements, this variation is mainly caused by cracking. EN 1992 (§7.3.1(2)) only allows the characteristics of the uncracked sections to be used when the internal actions are obtained from a linear elastic analysis. Additionally, EN 1992 (§O8.1(5)) and ACI-318-19 (§6.6.3.1.1) allow for a reduction in the inertia of the cross-section as a way to consider cracking in global analysis. This method provides an approximate value of internal actions and lateral displacements. EN 1992 proposes:

0.4E cd I c for walls and columns  EI  0.3E cdI c for reinforced concrete beams and slabs  E I for uncracked concrete beams and slabs caused by to prestressing  cd c

(5.4) E cd is the design value of the modulus of elasticity of concrete (= E cm /γ cE , being γ cE = 1.3), and Ic = Ig.

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143

ACI-318 modifies the second moment of area, considering 0.7Ig for columns and uncracked walls, 0.35Ig for cracked walls and beams, and 0.25Ig for flat plates and flat slabs. When using this approximation and to prevent false force redistributions, the stiffness of restraining elements should be reduced by the same proportion. vi. Tension stiffness of concrete: The influence of concrete in tension in the post-cracking phase is considered by using concrete tension stiffening models, as described in Annex B3.

5.4 SECOND-ORDER STRUCTURAL ANALYSIS OF MEMBERS AND SYSTEMS WITH AXIAL FORCE In this section, the additional second-order moments are considered by using the data obtained from a first-order linear analysis. This widely used procedure is also included in current standards. Compression causes instability in structural elements. However, firstorder analysis (Equation 5.2) does not detect the instability inherent to compression. In other words, Equation 5.2 can produce displacement values regardless of the magnitude of axial force, even if the force exceeds the buckling load. Therefore, when performing a first-order analysis, it is necessary to consider second-order effects,4 either directly or indirectly. To achieve this, concepts such as elastic buckling load (N B) and effective length (10) must be used. Figure 5.4 illustrates the importance of second-order analysis when axial compressive forces are high, as in the case of columns or bridge piers. These structural elements differ from others because axial loading can significantly affect bending moments. Determining the bending moment diagram is usually a simple process for elements such as beams, which are not subjected to axial loads. However, for elements subjected to axial loads, such as slender piles, as shown in Figure 5.4, the determination of the maximum moment requires knowledge of the horizontal displacement of a pile, which can make the process more complex. A structure is defined as translational when its horizontal displacement must be taken into account, and therefore the additional effects, known as second-order effects or PΔ effects, must be considered. As mentioned earlier, EN1992 (§7.4.1(3)) states that second-order effects do not need to be considered when they constitute less than 10% of the corresponding first-order effects.

144  Concrete structures

Figure 5.9 Biarticulated nontranslational column.

5.4.1 Elastic buckling load of isolated members The maximum load that a simply supported (biarticulated) bar can withstand when subjected to a centered compression load applied at its ends, as shown in Figure 5.9, is the critical elastic buckling load (N B):

NB 

2EI (5.5) l2

where EI is the representative effective stiffness in the plane of bending.

5.4.2 Effective length The effective length or buckling length, denoted by l0, is defined as “the distance between two consecutive inflection points, real or fictitious, of the deformed shape of an element, caused by elastic instability”. The term “buckling length” is used when applying Equation 5.5 to bars with end conditions other than simply supported. In such cases, the critical elastic buckling load is expressed as:

NB 

2 E I (5.6) l02

where α is the buckling length coefficient and l0 = αl. For instance, in the case of a simply supported (biarticulated) bar like the one shown in Figure 5.9, α = 1. The value of α for the most common cases of isolated bars is provided in Figure 5.10. The cases shown in Figure 5.10 are very simple and can be considered extreme cases. In practice, while it is easy to construct a pin-jointed (or

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145

Figure 5.10 Simple cases of buckling length coefficients.

Figure 5.11  Non-sway structure.

articulated) node, it is highly unlikely that the rotation of a node will be exactly zero, even if the node is built-in (or encastred). A structure that does not undergo horizontal translations (this is called a nontranslational structure, a structure with braced members, or a nonsway structure) (Figure 5.11). An intermediate column connects to beams and other columns at its upper and lower ends. The ends of the column are neither pin-joints (case 1 of Figure 5.10, α = 1) nor built-in connections (case 4, α = 0.5). Therefore, the buckling length coefficient of the column must be an intermediate value between these two extreme cases (i.e., 0.5 < α < 1). In practice, for buildings where horizontal translation is not considered (i.e., nontranslational structures), it is common to assume that α = 1. This is a safe choice and avoids the calculation of α, which can be somewhat tedious and imprecise. Section 5.4.2.1 will explain this. The expression for N B in Equation 5.6 can be formulated in terms of stress as:

B 

2 E (5.7) 2

146  Concrete structures

Figure 5.12  Buckling curve.

where λ is the slenderness ratio (λ = l0/i), with i as the radius of gyration of the uncracked concrete cross-section:

i=

Ic (5.8) Ac

Figure 5.12 shows the concrete stresses from Equation 5.7 as a function of the slenderness ratio. The intersection point of the horizontal line corresponding to the concrete compressive strength (σ c = fcm) with Equation 5.7 is represented by point A in Figure 5.12. Point A sets a value of λ called λ A , so that if λ < λ A , column failure will be caused by concrete strength, whereas if λ > λ A , failure will be caused by instability before concrete strength is reached. EN 1992 (§O.6) defines a threshold value for the slenderness ratio, known as λ lim, so that the second-order effects do not have to be considered for values lower than this limit. 5.4.2.1 Buckling length of a member belonging to a structure To determine the buckling length of a member of a structure, it is necessary to isolate the member and establish the equilibrium equations. Once this has been carried out, the critical buckling load value can be determined. Then, the buckling length value can be calculated from Equation 5.6. A member with generic support conditions as shown in Figure 5.13, where k1 is the stiffness5 to rotation at the lower support, k 2 is the stiffness to rotation at the upper support, and k3 is the stiffness to the relative transverse displacement between both ends. Traditionally, the calculation of the buckling length of a bar in a structure (Hernández-Montes and Gil-Martín 1996) has been carried out for the following two limit cases: • Considering that k3 = 0 • Considering that k3 = ∞

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Figure 5.13 Study of a column of a structure.

When k3 = ∞, a structure is nontranslational (also known as a braced or non-sway structure), as shown in Figure 5.13. The nontranslationality of the structure is ensured by additional elements, such as centered stiffeners (Saint Andrew’s crosses), eccentric stiffeners, shear walls, and contiguous structures. When k3 = 0, a structure is classified as a translational structure (or unbraced or sway structure), as shown in Figure 5.13. In this case, the structure itself (with no additional elements) must ensure that horizontal displacements are small enough. For translational structures, the buckling length coefficient is always greater than 1 (α ≥ 1), with α = 1 being the fifth case of Figure 5.10. Thus, the value of α for an element of a structure depends on whether the structure is considered braced or unbraced (i.e., k3 = 0 or k3 = ∞). The value of the buckling length coefficient of a column is a function of the stiffness of each element that converges at the ends of this column. To simplify the calculation, EN1992 (§O.5) provides the following approximate expressions of l0 for columns in regular frames. For braced members:

fr1 fr 2    l0  0.5l  1  1 (5.9)   0.45  fr1   0.45  fr 2  

For unbraced members:

l0  l

1  2.4fr1  2.4fr2  1  2.4fr1  1  2.4fr2  1  1.2fr1  1.2fr 2

(5.10)

In the preceding equations, the relative flexibilities of rotational restraints at ends 1 and 2 are denoted by fr1 and fr2 , respectively. Since fully rigid

148  Concrete structures

restraints are uncommon in practice, a minimum value of 0.1 should be taken for fr1 and fr2 . The relative flexibility is defined as:

fr 

 EI M l

where θ is the rotation of restraining members for bending moment M, EI is the bending stiffness of the compression member, and l is the clear height of the compression member between end restraints. If an adjacent compression member (column) in a node is likely to contribute to rotation at buckling, the definition of fr should be modified to replace (EI/l) with [(EI/l)a + (EI/l)b], where a and b represent the compression members above and below the node (EN 1992 §O.5(4)). The theoretical limit for rigid rotation restraint is fr = 0, while the limit for free rotational support is fr = ∞. The traditional classification of structures as translational or nontranslational was introduced to simplify the buckling problem of columns belonging to structures (see k3 in Figure 5.13). However, nowadays it might not be necessary to resort to this classification to solve either the problem of global instability or to calculate the buckling length (Gil-Martín and HernándezMontes 2012). In addition to column instability, the phenomenon of the lateral buckling of slender members can occur in concrete structures, although this phenomenon is more common in steel beams (EN 1992 §7.5). EXAMPLE 5.6 BUCKLING LENGTH CALCULATION In the column (wall) of the frame in Figure 5.3, the relative flexibilities fr1 and fr2 are: fr 

 EI l    EI l f

fr1  0.1

columns

r2

beams



1.43 / 8  1.87 1.33 / 12

In the built-in support, a value of 0.1 is advisable. From Equation 5.10: l0 = 1.86 · l = 14.85 m.

5.4.3 Approximate methods of second-order analysis EN 1992 provides two simplified methods for determining internal forces that account for second-order effects (i.e., N Ed, V Ed, and M Ed), based on the internal forces obtained from first-order analysis (i.e., N Ed, V Ed, and M0Ed).

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Figure 5.14 Approximate second-order analysis.

The first method is the moment magnification method (EN 1992 §O.8), and the second method is based on the nominal curvature (EN 1992 §O.7). Figure 5.14 provides a summary of both methods. To account for geometric nonlinearity, the moment magnification method amplifies the first-order moments, while the nominal curvature method introduces an additional eccentricity that, when combined with the axial force, yields the second-order moments. Both methods are discussed in detail next.

150  Concrete structures

5.4.3.1 Moment magnification method (MMM) In this method, the design moments (M Ed) are obtained by magnifying the first-order moments6 (M0Ed):

   MEd  M0Ed 1   (5.11)  (N B / N Ed )  1 

where M0Ed first-order moment β coefficient that depends on the distribution of first- and second-order moments, see EN 1992 §O.8.2(3 and 4); if these sections are not applicable, a reasonable approximation is β = 1 N Ed design value of the axial load N B buckling load (Equation 5.6) based on the effective stiffness Alternatively, for a global analysis, second-order effects can be obtained by the fictitious magnification of the horizontal forces:

  1 FEd,2  FH,0Ed   (5.12) 1  ( N / N ) Ed B  

Both the moment magnification method (MMM) and the nominal curvature method require the use of effective stiffness. This stiffness can be conservatively determined by considering the situation where yielding occurs. When calculating global effects in situations where yielding takes place successively at different locations, the stiffness corresponding to the situation where the last plastic hinge has developed (before mechanism) can be conservatively used, as shown in Figure 5.15. Alternatively, the simplification given in Equation 5.4 can be used. 5.4.3.2 Simplified analysis of isolated members based on nominal curvature For isolated members subjected to a constant normal force and whose effective (or buckling) length is known, total eccentricity etot is defined as:

e tot  e0  e2

By multiplying etot by axial force N Ed, a M Ed design moment is obtained that includes the first-order moment (with the effects of imperfections) and the second-order effect, as shown in Figure 5.16:

MEd  N Ed e tot  N Ed e0  N Ed e2  MEd  M0Ed  N Ed e2 (5.13)

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Figure 5.15 Possible effective stiffness. (Adapted from Figure 7.4 of EN 1992.)

Figure 5.16 Nominal curvature model.

where e0 first-order eccentricity including imperfections e2 eccentricity caused by second-order effects (see EN 1992 §O.7) N Ed value of the design axial force M Ed design moment considering second-order effects M0Ed first-order design moment including imperfections (M0Ed = N Ede0) Additionally, the second-order effects of slender columns have to be considered (see EN 1992 Figure O.1). This method assumes that a deformed element has a sinusoidal shape. When placing the x and y axes as indicated in Figure 5.16, the deformed shape can be expressed as:

152  Concrete structures



 x  y(x)  y maxsin    l0 

with l0 as the buckling length (l0 = 2L in Figure 5.16). The preceding expression satisfies the boundary conditions at x = 0, where y = 0, and at x = l0/2 = L, where y = ymax = e2 , as shown in Figure 5.16. If the preceding equation is derived twice:

y(x)  (x) 

 x  1 2 2  y max 2 sin    2 y(x)  l0 r(x) l0  l0 

At x = L, the maximum moment and curvature occur. If π 2 ≈ 10, the following value of e2 (i.e., ymax) is obtained from the earlier equation:

e2  

l02 max (5.14) 10

EN 1992 provides the following value for e2:

e2 =

1 l02 (5.15) r c1/ r

where: l0 effective length (or buckling length) 1/r curvature c1/r factor that depends on the curvature distribution along the element; for single unbraced members with constant cross-sections, c1/r = 10 (≈ π 2) can be used; for braced members, c1/r = 8 can be adopted EXAMPLE 5.7 BRIDGE COLUMN Figure 5.17 shows the pier of a bridge with an annular cross-section, constructed using C35 concrete, and reinforced with type B 400 steel. The design loads acting on the pier are a centered vertical axial load of 4500 kN, a horizontal head load of 275 kN, and a shaft load of 9.0 kN/m. It is assumed that the loads are acting 28 days after construction, that the relative humidity is 70%, and that the quasi-permanent moment represents 20% of the maximum moment. In this example, the maximum second-order moment is going to be calculated by applying the two methods proposed by EN 1992: the moment magnification method (MMM) and the nominal curvature method.

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Figure 5.17 Circular bridge column.

In this case, the buckling length factor is 2, resulting in a buckling length of l0 = α · l = 60 m. The mechanical characteristics of the gross section and the slenderness ratio are:





A c    rext 2  rint 2  2.54 m2 Ic  

  rext 4  rint 4  2.347 m4 4



l0  ic



l0  62.42 Ic Ac

The geometric imperfection proposed by EN 1992 (§7.2.1.2(5)) can be considered as an initial eccentricity: = ei

l0 = 150 mm 400

Using this initial eccentricity, and for the loads specified in the statement, the first-order moment distribution can be calculated (see Figure 5.17). The maximum first-order moment at embedment is M 0Ed = 12975 kN·m.

Magnified moment method (MMM) The design moment can be calculated using Equation 5.11:    MEd  M0Ed 1   ( N / N )  1 B Ed  

154  Concrete structures

The coefficient β can be approximated as 1.0, but for this example, the value provided by EN 1992 has been adopted, with β = π 2/c1/r = 1.03, where c1/r depends on the distribution of first-order moments along the member, which is 9.6 for parabolic distribution. NB is the buckling load based on the effective stiffness according to Equation 5.4: NB 

2EI 2EI  2  65893 kN 60 l20 EI  0.4EcdIc  2.4035 107 kN  m2 Ecd 

1 1 1/ 3 Ecm  9500  fcm   25602 MPa  25.6 106 kN / m2 1.3 1.3

Once the values of the β and NB are known, the total design moment at embedment can be calculated as:      1.03 MEd  M0Ed 1    13955kN  m   12975 1  ( N / N )  1 65893 4500  1 B Ed    

Nominal curvature method As previously mentioned, the nominal curvature method introduces secondorder effects through the addition of an extra eccentricity. According to EN 1992 §O.7.3: MEd  M0Ed  NEde2  12975  4500 e2  15334 kN  m e2 

1 l20 602  0.001456  0.524 r c1/r 10

1 1  k rk   1.0 1.04  0.0014  0.001456 r r0

k   1  fck eff ,b  1  0.11 0.36  1.04 k r  1.0  fck  0.35 

 35 62.42 fck   0.11   0.35  200 150 200 150

2 yd 1   0.0014 r0 (d  d ’) c1/r  10

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The second-order moment calculation in ACI-318 is simpler than the methods provided by EN 1992. The ACI-318 code provides the following equation for the design moment including the second-order effects: M2 order  M1order ,non sway 

1

N 1

Ed



M1order , sway

Vh

where V is the shear in the column or the total shear per story, Δ is the interstory drift, and h is the height of the column. For small values of NΔ/Vh:7

1 N  1 N Vh 1 Vh So, the expression of the second-order moment can be rewritten as:8

 N  M2 order  M1order , non sway  1  Ed  M1order , sway Vh   NEd/NB can be approximated by NEdΔ/Vh as long as the factor that increases the translational moments is less than about 1.5 (Lai and MacGregor 1983), then:

NEd NEd 1   Nb Vh 2 When applying the preceding expressions to the previous example:  N   4500  0.2  M2 order  1   M0Ed  1  12975  13689 kN  m V h 545  30    



275 h3 9.0 h4   0.20 m 3EI 8EI

V  9.0  30  275  545 kN

The advantage of the ACI-318 method is that it calculates the second-order moment by using only the data obtained from the first-order analysis, and the buckling lengths of the elements do not need to be calculated.

156  Concrete structures

5.5 COMPRESSION MEMBER WITH BIAXIAL BENDING A cross-section is said to be subjected to biaxial bending when the loading causes bending in both principal directions simultaneously. As far as the second-order analysis is concerned, if a fully nonlinear analysis of the structure (i.e., the general method) is carried out, then no additional verification is necessary. However, when simplified methods are used, additional considerations must be taken into account. EN 1992 (§7.4.4) allows, as a first step, the second-order effects in each principal direction to be checked. Imperfections should only be considered in the direction with the most unfavorable effects. Moreover, further checks can be omitted if the slenderness ratio satisfies the following two conditions:

i) 0.5 

ii)

y 2 z

ey  0 .2 ez

or

ey 5 ez

with (see Figure 5.18): λ y, λ z slenderness ratios l0/i with respect to y- and z-axis, respectively iy, iz radii of gyration with respect to y- and z-axis, respectively e′ z = M Edy/(N Ed·b) is the dimensionless eccentricity along the z-axis e′ y = M Edz /(N Ed·h) is the dimensionless eccentricity along the y-axis For nonrectangular sections, b and h can be replaced by beq = iy 12 and heq = iz 12 .

Figure 5.18 Definition of eccentricities ey and ez.

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Figure 5.19  Concrete frame.

If the two aforementioned conditions are not fulfilled, biaxial bending should be considered, including second-order effects in each direction. EXERCISES 1. The frame shown in Figure 5.19 can withstand a distributed load of 45 kN/m and a wind load of 35 kN acting at node f. Using OpenSees, determine the axial and bending moment for which column a–f needs to be reinforced. Consider the reduced inertia values provided by ACI-318 that were mentioned in this chapter Section 5.3.3(v). 2. Repeat exercise 1 by applying the MMMs given by EN 1992 and ACI-318. NOTES 1. However, even though the use of nonlinear analysis meaning that the calculation of buckling can be skipped, engineers have to understand this concept (elastic instability) as a fundamental part of their training. 2. If the width of the flanges of the T-section is greater than a certain value, a reduced (or effective) width should be used (beff), see §7.2.3 of EN 1992. 3. See Example 5.1. 4. Geometric nonlinearity and elastic instability are related concepts, see Annex A5. 5. Stiffness is the relationship moment/rotation or force/displacement. 6. The MMM provided by EN 1992 differs from that provided by ACI-318. Whereas EN 1992 magnifies the first-order moments, ACI-318 divides a

158  Concrete structures structure into the sway part and the non-sway part. Sway moments (Ms) and non-sway moments (M ns) are amplified by different magnification coefficients (see §6.6.4.5 and §6.6.4.6 of ACI318-19).

7. If a is smaller than1.0, then 1/(1 – a) = (1 + a)/((1 + a)(1 – a)) = (1 + a)/(1 – a2) ≈ 1 + a. 8. Note that the second term at the right of the equality coincides with the final expression of Equation A5.1 (see Annex A5).

REFERENCES Deierlein, G., Reinhorn, A., and Willford, M. (2010). “NEHRP Seismic Design Technical Brief No. 4 - Nonlinear Structural Analysis for Seismic Design: A Guide for Practicing Engineers.” https://www​.nist​.gov​/publications​/nehrp​ -seismic​- design​-technical​-brief​-no​- 4 ​-nonlinear​- structural​- analysis​- seismic​ -design. EN 1992-1-1. (2023). Eurocode 2: Design of Concrete Structures - Part 1–1: General Rules and Rules for Buildings, Bridges and Civil Engineering Structures EN 1992-1-1. Brussels: European Committee for Standardization. FIB. (2012). Model Code 2010. Model Code 2010 – Final Draft, Vol. 1. Fib Bulletin No. 65. Internatio. Lausanne. Gil-Martín, L.M., and Hernández-Montes, E. (2012). “Unified Buckling Length Coefficient for Sway and Non-Sway Structures.” Engineering Structures 40: 436–44. https://doi​.org ​/10​.1016​/j​.engstruct​. 2012​.03​.008. Hernández-Montes, E., and Gil-Martín, L.M. (1996). “Análisis Comparativo Del Cálculo de Las Longitudes de Pandeo Según El Eurocódigo 3 y La NBE-EA-95.” Hormigón y Acero 201: 13–26. Johansen, K.W. (1962). Yield Line Theory. London: Cement and Concrete Association. Lai, S.M., and MacGregor, J.G. (1983). “Geometric Nonlinearities in Unbraced Multistory Frames.” ASCE Structural Engineering 109 (11): 2528–45. McKenna, F., Fenves, G.L., and Scott, M.H. (2000). Open System for Earthquake Engineering Simulation. Berkeley, CA: University of California. Vamvatsikos, D. (2008). “Opensees Fiber Section Plotting.” http://users​.ntua​.gr​/ divamva​/software​/plotSection​.zip.

Annex A5

Matrix formulation of geometric nonlinearity

In order to incorporate geometric nonlinearity into matrix analysis, a new set of external forces, referred to as second-order forces,1 have been introduced. Figure A5.1 illustrates this concept, where a new force F2order is introduced on the right, acting in conjunction with forces N and V. By including the second-order forces in the external actions, the first-order analysis in the undeformed configuration can be used to obtain secondorder internal forces and moments. In Figure A5.1, the value of the second-order force can be obtained by setting the first-order moment in A (considering F2order) equal to the secondorder moment. 2

M2,order  (V  F2,order )h  V h  N   F2,order 

N h

and so:

 N  M2,order  V h  1   (A5.1) Vh  

 N  The coefficient  1   is known as the moment amplification factor Vh   since multiplying the first-order moment by this coefficient results in the second-order moment. In traditional strength of materials books (such as Ortiz-Berrocal 1996), this coefficient is denoted as η and its theoretical expression is: 

1 N 1 NB

1    M2,order  V h    1  N / NB  (A5.2) where N B 

2

 EI l02

with l0 as the buckling length and N B as the critical elastic buckling load. DOI: 10.1201/9781003169659-11

159

160  Concrete structures

Figure A5.1 Second-order calculation using second-order forces.

Equation A5.1 includes the second-order transverse displacement, Δ, which can only be calculated iteratively. However, Equation A5.2 is independent of any displacement, although it involves the buckling length, l0, which could be challenging to evaluate. As the value of Δ is small, Equation A5.1 can be written as:

 N  1 M2,orden  V h  1  (A5.3)   Vh N Vh   1 Vh

Equations A5.2 and A5.3 can be compared for the simple case illustrated in Figure A5.1, which represents an axially loaded cantilever. If Δ is the horizontal first-order displacement induced by force V (= h3V/(3EI)) then: Vh

1 Vh Vh Vh    3 N N N N h V 1 1 1 2 1  EI Vh N V h 3EI B 2 l0

Vh Vh   2 Nh Nh2 1 1 3EI 2.47 EI



Vh Vh   N h2 1  N 1 2  EI 3EI 4h2



M atrix formulation of geometric nonlinearity  161

Figure A5.2  Second-order forces.

The expression on the left of the symbol ≈ in the preceding equation corresponds to the approximate expression given by Equation A5.3, while the expression on the right is the exact expression given by Equation A5.2. The results from both expressions are very similar, but the expression of the amplification coefficient given by Equation A5.3 is easier to use because the data needed to perform the second-order analysis can be obtained from the results of a first-order analysis. The axial forces hold the greatest responsibility for the second-order effects. Although the displacement of the structure causes shearing forces and bending moments that also influence the second-order effects, their contribution is relatively low. The effect of axial forces on a deformed structure can be examined. Figure A5.2 illustrates an element that is part of a structure. The displacements of the ends (i and j), measured perpendicular to the axial force direction, are vi and vj, respectively, with respect to the initial position. The effects of the axial force on the deformed shape of the element (i.e., ΔM = N(vj – vi)) can be expressed by fictitious second-order forces (i.e., ΔM = fGjL) applied to the undeformed structure. Since, by definition, both express the same phenomenon, the value of the fictitious second-order forces can be calculated as: vi  vj N 1  v i  fGi  N  1 L matrix formulation        (A5.4)  fGj  L  1 vj  vi 1   v j   fGj  N L The matrix expression of the second-order forces needs to be expanded to include all the degrees of freedom of the beam-column element as follows (see Chapter 5, Equation 5.1): fGi 

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0 0 0 0 0   ui  0 0   f  0 1 0 0 1 0  v i   Gi    0  N 0 0 0 0 0 0   i         (A5.5) 0 0 0 0 0  uj   0  L 0  fGj  0 1 0 0 1 0  v j        0 0 0 0 0  j   0  0 14444442444444 3 kG

Symbolically, the matrix expression for the second-order forces can be written as fG = kG δ . So, the general matrix formulation (i.e., including second-order analysis) of the isolated element shown in Figure A5.2 is:

f   k  kG  (A5.6)

where k is the stiffness matrix (f = k δ ) and kG is the geometric stiffness matrix of the element. As in the first-order analysis, the stiffness matrices and geometric stiffness matrices of all the elements in a structure must be assembled, with corresponding changes in reference systems, to obtain the structural matrix formulation for second-order analysis:

F   K  KG   D (A5.7)

where K is the stiffness matrix of the structure, KG is the geometric stiffness matrix of the structure, F is the vector of external forces on the degrees of freedom (DOFs) considered, and D is the displacement vector of these DOFs. The system in Equation A5.7 is solved iteratively, as the KG matrix contains internal axial force values. The resolution of the problem begins with a first-order calculation (Equation 5.2) to obtain the initial approximation of the internal axial forces. A5.1 THE PROBLEM OF BUCKLING EIGENVALUES AND EIGENVECTORS The problem of geometric nonlinearity, also known as second-order analysis, can be approached from two perspectives. The first approach involves calculating the displacement vector D using the matrix formulation provided by Equation A5.7. The second approach involves determining the load (the critical elastic or buckling load) for which there are infinite solutions of the displacement vector D, i.e., infinite displacements. This second approach is known as global instability.

M atrix formulation of geometric nonlinearity  163

Equation A5.5 indicates that kG (and thus KG) varies linearly with the force vector. Therefore, if a F0 vector corresponds to a KG0 geometric stiffness matrix, then a κ F0 vector (κ is a scalar) will correspond to κ KG0, so that:

F0   K  K G0   D F0   K   K G0   D ’



where D' is a displacement vector that is different from vector D. The situation in which there are multiple solutions for vector D' that satisfy the preceding matrix expression (i.e., the mathematical problem is indeterminate) corresponds to the global instability problem. Therefore, it is necessary to determine the value of the scalar κ that satisfies:

det  K   K G0   0 (A5.8)

The mathematical expression given in Equation A5.8 is commonly known as the buckling eigenvalue problem. Loading vector κ F0, which causes the global instability of a structure, can be obtained from this equation. By definition, κ F0 represents the loading limit that cannot be exceeded. However, the practical significance of determining the eigenvalues (and their associated eigenvectors) is limited since they are obtained under the assumption that the material is linear and elastic. And, in practice, yielding often occurs before the global buckling of a structure. Nonetheless, global buckling has been extensively studied, and numerous research works on structural optimization use the buckling eigenproblem (e.g., Hjelmstad and Pezeshk 1991; Hernández-Montes et al. 2004; Gil-Martín et al. 2006). NOTES 1. There are other more exact approaches that can be used to address nonlinear geometrical analysis (e.g. Clough and Penzien 2003; Cook et al. 2002), but the authors have chosen this one for its conceptual clarity. 2. Traditionally, axial force has been defined with the letter P, and so the second-order effects of a geometric nature have been called PΔ effects. European standards use the letter N for axial load, while American standards keep the letter P. In this volume, the authors will use both the letters N and P interchangeably for axial load.

REFERENCES Clough, R.W., and Penzien, J. (2003). Dynamics of Structures. 3rd ed. Berkeley, CA: Computers & Structures.

164  Concrete structures Cook, R.D., Malkus, D.S., Plesha, M.E., and Witt, R.J. (2002). Concepts and Applications of Finite Element Analysis. 4th ed. New York: John Wiley & Sons. Gil-Martín, L.M., Hernández-Montes, E., and Aschheim, M. (2006). “Optimal Design of Planar Frames Based on Stability Criterion Using First-Order Analysis.” Engineering Structures 28 (13). https://doi​.org​/10​.1016​/j​.engstruct​ .2006​.03​.007. Hernández-Montes, E., Gil-Martín, L.M., and Aschheim, M. (2004). “Optimal Design of Planar Frames Based on Approximate Second-Order Analysis.” Engineering Optimization 36 (3). https://doi​.org​/10​.1080​/030​5215​0 410​ 001657578. Hjelmstad, K.D., and Pezeshk, S. (1991). “Optimal Design of Frames to Resist Buckling Under Multiple Load Cases.” Journal of Structural Engineering 117 (3): 914–935. Ortiz-Berrocal, L. (1996). Resistencia de Materiales. Madrid: McGraw-Hill.

Chapter 6

Reinforcing design of D-regions

Strut-and-tie model for a pile cap.

6.1 PURPOSE AND OBJECTIVES​ This chapter introduces the strut-and-tie method, a structural analysis method used for designing the reinforcement of parts of structures where the plane section hypothesis is not applicable: the discontinuous regions (D-regions). The chapter also covers the design of partially loaded areas. DOI: 10.1201/9781003169659-12

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166  Concrete structures

6.2 INTRODUCTION Designing a structure involves more than just making calculations. Engineers must have complete confidence in the structural design and the calculations made as these factors will ensure the safety and functionality of the structure being designed. Understanding the function of a structure is critical to its design. The strut-and-tie method (STM) is an intuitive modeling technique that can assist designers in quickly and easily determining the suitability of a structural solution. Engineers can use the STM to complement other analytical methods. This method enables designers to draw imaginary struts and ties to ensure that reinforcement is provided where it is needed. This chapter focuses on situations where the STM is the most reliable and effective method for analysis. 6.3 B- AND D-REGIONS The plane section hypothesis (PSH), also known as the Bernoulli hypothesis, states that a plane section before bending remains plane after bending. Therefore, if the strains at two points of a cross-section are known (e.g., the maximum and minimum strains, ε top and ε bottom in Figure 6.1), the strains at any point of the cross-section can be calculated.1 In practice, the strain distribution of the cross-section is usually expressed as a function of the strain of the center of gravity of the section and the curvature (ε cg and ϕ), see Figure 6.1. This hypothesis (i.e., beam theory) is a fundamental assumption for the study of structural elements. However, the PSH is not always suitable as plane sections do not always remain plane after bending or, as in the case of solid prisms, a clear predominant cross-section might not exist. The PSH is not suitable for

Figure 6.1 Strain distribution in the cross-section according to the PSH.

Reinforcing design of D-regions  167

beam-column elements in the vicinity of sudden changes in geometry or the proximity of concentrated loads. To design concrete structures, it is necessary to differentiate between B-regions (i.e., Bernoulli regions) and D-regions (i.e., discontinuous regions). B-regions are the zones of a structure where the PSH can be applied. D-regions are regions where this hypothesis is not suitable, as nonlinear strain distribution exists (see Figure 6.2). The Saint-Venant principle states that at a sufficient distance from the point of the application of loads on a structural element, the effects primarily depend on the resultant (net effect) of the loads rather than the specific distribution of those loads. In other words, stresses and strains are practically the same for statically equivalent systems beyond a certain distance from the loads. Figure 6.2 shows how a concentrated load produces a stress distribution that is different from that of a distributed load, and how the stresses become constant after a certain distance. By using this principle, the D-region can be estimated to extend up to one height (h) from the discontinuity. Figures 6.3 and 6.4 depict examples of zonification in B- and D-regions for two structures, where D-regions are shown in gray and B-regions in white. The heights of the structural elements (h1, h 2 , h3, h4, and h5) and the breadths (or widths) of the flanges (b1 and b2) are also specified. The design of B-regions is based on the beam theory for calculating longitudinal reinforcement and the truss analogy (or compression field theory) for shear design. In contrast, D-regions, commonly found in elements like deep beams, supports, beam-column connections, pile caps, and footings, are designed using the STM.

h Cross section

h

B-region

D-region

h D-region

N

b

N

Stress distributions

σ =N/hb

Figure 6.2 Saint-Venant principle. Boundaries of the D-region (b is the breadth).

168  Concrete structures

h1

B

D h2

h2 h2

B h1

D B h2

h3 h4

h2

B

D

h1

D

B

D

B

D

h3 h3

h1

D

B

B

D

B

h4 h4 h4

D D h5

h3

h5

B h3

h1

h5

D

D

D

Figure 6.3 Division between B- and D-regions in a building structure. (Drawing provided by Professor Daniel Kuchma, Tufts University.) b1

b1>h1

b2

h1 b1

D D

b1

Sección -1-1

1

B

D D

h1 h2

B

D

B

D

h2

Sección - 2-2

B

D D

b1

1

b2>h4 h4

b2

B

D b2

b2

2

D b2

D

h3

h3 h2

h4

2

B

D

h3

D

Figure 6.4 Division between B- and D-regions in a bridge structure. (Drawing provided by Professor Daniel Kuchma, Tufts University.)

6.4 STRUT-AND-TIE MODELS The STM models the D-region using statically determinate truss structures2 consisting of articulated elements that work only in tension or compression. These structures are made up of nodes, compression elements (or struts), and tension elements (or ties), and they cannot withstand bending moments. The concrete in the D-region forms the struts, while the reinforcement forms the ties. The angle between the axes of the elements should be greater than 25° (ACI-318 2019).

Reinforcing design of D-regions  169

The application of a strut-and-tie model involves establishing the geometry of a truss structure inserted into the D-region. Once the truss structure is inserted, it has to be calculated, and the suitability of the dimensions for the struts, nodes, and ties must be verified. If they are not suitable, the geometry of the truss structure has to be changed. This is why the application of the strut-and-tie model requires a trial-and-error process. D-regions can be studied using the finite element method (FEM), considering the D-region as a continuous medium made up of concrete. Once the state of stresses is obtained with FEM, reinforcement can be added to absorb the tensile stresses that cannot be withstood by the concrete. In other words, the stress distribution marks the principal directions, and these can be used to define the geometry of a strut-and-tie model. Nevertheless, several different strut-and-tie models can be used for one specific D-region, and the model chosen might not coincide with the model that follows the stress distribution marked by the FEM. The strut-and-tie method is particularly reliable in the post-cracking and yielding phases. When cracking is significant, the principal directions of compression tend to become straight lines, making the strut-and-tie model ideal for such situations (Muttoni et al. 1997). Figure 6.5 shows

Figure 6.5 Flat strut-and-tie flat models.

170  Concrete structures

some examples of two-dimensional D-regions modeled using strut-and-tie models, and Figure 6.6 shows a spatial model. The dashed lines represent the compression elements (struts), while the solid lines represent the tensile elements (ties). 6.5 STRUTS, TIES, AND NODES Traditionally, there have been two approaches to modeling the strut-andtie method within a D-region: starting with the maximum compressive stress allowed in the struts or starting with detailed nodal configurations from which the node dimensions are known. Traditionally, EN 1992 (EN 1992-1-1 2023) has been applied in the first approach, while the AASHTO Bridge Design (American Association of State Highway and Transportation 2017) uses the second approach. Example 6.1 compares both perspectives.

6.5.1 Struts The struts in D-regions are compression members of internal trusses, representing compression fields within the concrete. The principal compressive stress aligns with the struts in the model. In two-dimensional models, the shape of the compression field in the strut can be prismatic-, bottle-, or fan-shaped, as depicted in Figures 6.7a, 6.7b, and 6.7c, respectively. Struts can also be reinforced with reinforcing bars. Figure 6.8 shows the typical two-dimensional strut shapes. EN 1992 states that the stresses in the concrete struts can be assumed to be uniformly distributed across their cross-sections. The stress value in concrete is given by Equation 6.1:

cd 

Fcd   fcd (6.1) bc t

where Fcd represents the compressive force acting on the strut, t is the thickness of the strut limited by the thickness of the member, bc is the width of the strut, and ν is the strength reduction factor. EN 1992 §8.5.2 presents various methods for calculating the strength reduction factor (ν). Table 6.1 provides a possible value for the strength reduction factor (ν) based on the presence of a tie and the minimum angle between the strut and any tie (θ cs). The compressive force (Fcd) is obtained by solving the truss structure inserted into the D-region, and for a given value of ν, the minimum thickness of the strut can be calculated using Equation 6.1. In contrast, AASHTO assumes that strut failure occurs at the interface with the nodal zone, so verifying the nodes indirectly verifies the struts. This is discussed in Example 6.1.

Reinforcing design of D-regions  171

(a)

(b)

(c) Figure 6.6 Example of a spatial strut-and-tie model: (a) perspective of the beam, (b) model of struts and ties in the web of the beam, (c) model of struts and ties in the flange of the beam.

6.5.2 Ties Ties are the elements of the strut-and-tie model that can withstand tension. Generally, they are made of passive reinforcement bars, but occasionally they can also be made of active reinforcement or concrete compression

172  Concrete structures

σc1< σc

σc

σc

σc

σc (a)

(b)

(c)

Figure 6.7 Basic strut shapes in two dimensions: (a) prismatic, (b) bottle, (c) fan.

(f) (d)

(e)

(c)

(a)

(b)

Figure 6.8 Types of struts in a D-region. (a) Prismatic strut in a non-cracked area. (b) Prismatic strut in a cracked area that is parallel to the cracks. (c) Prismatic strut in a cracked area that is not parallel to the cracks. (d) Bottle-shaped strut with reinforcement for cracking control. (e) Bottle-shaped strut without cracking reinforcement. (f) Confined concrete strut. Table 6.1  Strength reduction factor For compression fields and struts crossed or deviated by a tie at an angle of θcs 20°≤θcs