Reinforced Concrete Beams, Columns and Frames E-Book

Jostein Hellesland

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 352

Veröffentlichungsjahr: 2013

Beschreibung

This book is focused on the theoretical and practical design of reinforced concrete beams, columns and frame structures. It is based on an analytical approach of designing normal reinforced concrete structural elements that are compatible with most international design rules, including for instance the European design rules – Eurocode 2 – for reinforced concrete structures. The book tries to distinguish between what belongs to the structural design philosophy of such structural elements (related to strength of materials arguments) and what belongs to the design rule aspects associated with specific characteristic data (for the material or loading parameters). A previous book, entitled Reinforced Concrete Beams, Columns and Frames – Mechanics and Design, deals with the fundamental aspects of the mechanics and design of reinforced concrete in general, both related to the Serviceability Limit State (SLS) and the Ultimate Limit State (ULS), whereas the current book deals with more advanced ULS aspects, along with instability and second-order analysis aspects. Some recent research results including the use of non-local mechanics are also presented. This book is aimed at Masters-level students, engineers, researchers and teachers in the field of reinforced concrete design. Most of the books in this area are very practical or code-oriented, whereas this book is more theoretically based, using rigorous mathematics and mechanics tools.

Leseprobe

Contents

Preface

Chapter 1. Advanced Design at Ultimate Limit State (ULS)

1.1. Design at ULS – simplified analysis

1.2. ULS – extended analysis

1.3. ULS – interaction diagram

Chapter 2. Slender Compression Members – Mechanics and Design

2.1. Introduction

2.2. Analysis methods

2.3. Member and system instability

2.4. First- and second-order load effects

2.5. Maximum moment formation

2.6. Local and global slenderness limits

2.7. Effect of creep deformations

2.8. Geometric imperfections

2.9. Elastic analysis methods

2.10. Practical linear elastic analysis

2.11. Simplified analysis and design methods

2.12. ULS design

Chapter 3. Approximate Analysis Methods

3.1. Effective lengths

3.2. Method of means

3.3. Global buckling of unbraced or partially braced systems

3.4. Story sway and moment magnification

Appendix 1: Cardano’s Method

A1.1. Introduction

A1.2. Roots of a cubic function - method of resolution

A1.3. Roots of a cubic function - synthesis

A1.4. Roots of a quartic function - principle of resolution

Appendix 2: Steel Reinforcement Table

Bibliography

Index

First published 2013 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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© ISTE Ltd 2013

The rights of Jostein Hellesland, Noël Challamel, Charles Casandjian and Christophe Lanos to be identified as the author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2012954708

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-569-6

Preface

The authors have written two books on the theoretical and practical design of reinforced concrete beams, columns and frame structures. The book entitled Reinforced Concrete Beams, Columns and Frames – Mechanics and Design deals with the fundamental aspects of the mechanics and design of reinforced concrete in general, related to both the Serviceability Limit State (SLS) and the Ultimate Limit State (ULS). This book, entitled Reinforced Concrete Beams, Columns and Frames – Section and Slender Member Analysis, deals with more advanced ULS aspects, along with instability and second-order analysis aspects. The two books are complementary, and, indeed, could have been presented together in one book. However, for practical reasons, it has proved more convenient to present the material in two separate books with the same preface in both books.

The books are based on an analytical approach for designing usual reinforced concrete structural elements, compatible with most international design rules, including for instance the European design rules Eurocode 2 for reinforced concrete structures. The presentations have tried to distinguish between what belongs to the philosophy of structural design of such structural elements (related to strength of materials arguments) and the design rules aspects associated with specific characteristic data (for the material or the loading parameters). The Eurocode 2 design rules are used in most of the examples of applications in the books. Even so, older international rules, as well as national rules such as the old French rules BAEL (“Béton Armé aux Etats Limites”, or Reinforced Concrete Limit State in English) will sometimes be mentioned, at least for historical reasons. Whatever the design rules considered, the fundamental concept of Limit State will be detailed, and more specifically, the Serviceability Limit State (SLS) and Ultimate Limit State (ULS) both in bending and in compression will be investigated.

The books are devoted mainly to the bending (flexural) behavior of reinforced concrete elements, including geometrical nonlinear effects (in this book). However, two major aspects of reinforced concrete design are not treated. These are shear force effects and the calculation of crack width as dealt with in the Crack Opening Limit State in Eurocode 2. The latter represents a major new contribution as compared to some older European rules such as BAEL. The readers are referred to the very good monographs devoted to the general presentation of Eurocode 2 for these additional parts (see for instance [CAL 05]; [DES 05]; [MOS 07]; [EUR 08]; [PAI 09]; [PER 09]; [ROU 09a]; [ROU 09b]; [THO 09]; [PER 10]; [SIE 10]; [PAU 11]).

We would also like to point out that the calculation of crack widths, even under a simple loading configuration, such as uniform tension loading, still remains a difficult topic. Besides, the authors are even convinced that meaningful efforts should be addressed in the future, for facilitating the transfer of knowledge from theoretical research in fracture or damage mechanics, to applied, practical design rules. In connection with this, cohesive crack models were introduced in the 1970s to investigate the crack opening in mode I of failure [HIL 76], whereas non-local damage mechanics models were developed in the 1980s for efficient computations of damage softening materials [PIJ 87]. Both appear to belong to the families of nonlocal models which contain an internal length, for the control of the postfailure process [PLA 93]. Non-local damage mechanics is now widely used in the research community for the study of reinforced concrete structures (see for instance [BAZ 03]; [MAZ 09]). The authors of these books have also conducted some research in this field to better understand the failure of some simple reinforced concrete structural elements (research at INSA, Rennes, University of Rennes I, University of South Brittany or University of Oslo – see for instance [CHA 05]; [CHA 06]; [CHA 07]; [CHA 08]; [CHA 09]; [CHA 10]; [CHA 11]; [CHA 12]). However, the engineering community has not yet necessarily integrated these results into the design process or even into the rules. The gap between the research activity and the engineering methodology is probably too large at present, and researchers will probably have some responsibility in the future to make their results more tractable to the engineering community. With respect to these books, some very simple concepts of non-local mechanics will be presented when necessary. However, the books are mainly devoted to the design of a reinforced concrete structure at a given limit state, the cracking evolution problem often being considered as a secondary problem. We have chosen to concentrate our efforts on the bending design based on the pivot concept, at both the Serviceability Limit State (SLS) and the Ultimate Limit State (ULS). The last part of this book deals with the design of columns against buckling, and how to take into account second-order effects will be presented for stability design. In particular, some engineering approaches practiced by engineers will be detailed, to replace efficiently, when possible, the nonlinear evolution problem associated with micro-cracking and failure.

The books are aimed at both undergraduate and graduate (Licence and master) students in civil engineering, engineers and teachers in the field of reinforced concrete design. In addition, researchers and PhD students can find something of interest in the books, including the presentation on elementary applications of non-local damage or plasticity mechanics applied to the ultimate bending of reinforced concrete beams (and columns). We hope that the basic ideas presented in the books can contribute to stimulating the links between the research community in this field (computational modeling and structural analysis) and the design community with practical structural cases. The principles of limit state design will be introduced and developed first, both at both the Serviceability limit state (SLS) and the Ultimate Limit State (ULS), illustrated by some detailed examples to illustrate the introduced methodology.

Older books (see for instance [HOG 51]; [BAK 56]; [SAR 68]; [ROB 74]; [PAR 75]; [FUE 78]; [LEO 78]; [ALB 81]; [LEN 81]; [BAI 83]; [GYO 88]; [WAL 90]; [PAU 92]; [MAC 97]) have been used in some portions of the books (for establishing familiar and well known equations on section design (in particular equations based on the simplified rectangular stress-strain diagram for concrete in compression). In particular, the authors want to acknowledge the very exhaustive work of Professor Robinson, at Ecole Nationale des Ponts et Chaussées, whose reinforced concrete teaching book published in 1974 can still be considered as a main reference with modern insights into reinforced concrete design [ROB 74]. We have also been inspired by the more recent and very exhaustive works of Professor Thonier (see for instance [THO 09]), also at Ecole Nationale des Ponts et Chaussées.

The previous book, Reinforced Concrete Beams, Columns and Frames – Mechanics and Design, is organized as follows. Chapters 1 and 2 deal with the Serviceability Limit State, for both the design and the cross-section verification. The French school of reinforced concrete design have commonly used the concept of “Pivot”, which is related to the limit behavior of the cross-section with respect to the steel and concrete material characteristics. The Pivot A (where the steel material characteristics control the behavior of the cross-section at the Limit State), and Pivot B (where the concrete material characteristics control the behavior of the cross-section at the Limit State) concepts are introduced with the Serviceability Limit State in Chapter 1. Chapter 1 is mainly focused on the design aspects, whereas Chapter 2 deals with the verification of the reinforced concrete section with both the bending and the normal forces effects. The general theory presented in these first two chapters is valid for arbitrary shapes of the reinforced concrete cross-section including for instance rectangular, triangular, trapezoidal or T-cross-sections. Chapter 2 ends with the presentation of a cubic equation for the determination of the neutral axis in the general loading configuration, including the normal force effects. This elegant equation is also known as the cubic equation of the French reinforced concrete design rules dating from 1906 (Circulaire du 20 Octobre 1906) (and reported in the book by Magny, [MAG 14]) or those dating from 1934 (Règlements des marchés de l’état de 1934 – also in French), also recently reported by Professor Thonier for T-cross-sections [THO 09]. Finally, the tension stiffening phenomenon is introduced in terms of a nonlinear bending moment-curvature constitutive law and some verification examples are given to illustrate the theoretical results obtained in the fundamental parts.

Chapters 3 and 4 focus on the fundamental aspects of the Ultimate Limit State. Chapter 3 starts with a brief introduction to the concept of the Ultimate Limit State for the bending of a reinforced concrete beam. The need to use some non-local theory to correctly model the post-failure behavior of reinforced concrete structural elements is shown in the presence of global curvature softening. The material characteristics of the steel and concrete allowed by Eurocode 2 are listed, and compared with each other. It is possible to derive analytically the normal forces and the resultant bending moment in the compression block for each considered concrete law, including the parabolic-rectangle constitutive law, the simplified rectangular constitutive law, the bilinear constitutive law or Sargin’s nonlinear constitutive law. These preliminaries will be used later for the design of reinforced concrete sections at Ultimate Limit State. Chapter 4 discussed some possible bending moment – curvature law of typical reinforced concrete sections. These cross-sectional behaviors can be deduced from the local characteristics of the steel and concrete constituents. The relevancy of a bilinear approximation for the moment-curvature constitutive law is discussed, with possible tractable analytical results for engineering purposes. Chapter 4 concludes with some buckling and post-buckling results obtained for a reinforced concrete column modeled with a simplified nonlinear bending-curvature constitutive law. It is shown that reinforced concrete columns typically behave like imperfection-sensitive structural systems.

The current book, Reinforced Concrete Beams, Columns and Frames – Section and Slender Member Analysis, is organized as follows. The advanced design of general reinforced concrete sections is treated in Chapter 1. The reinforced concrete section can be optimized for a given loading (in term of minimization of the steel quantity for instance), with some constrained equations. Also discussed is how the Serviceability and the Ultimate Limit States can be compared, depending on the material and loading features of the problem. A design of the cross-section in biaxial bending is also proposed. More generally in this chapter, the reinforced concrete section is designed for various constitutive laws for concrete and the steel behavior, including possible steel hardening, with possible analytical solutions for the optimized design. Some design examples are included for the various solicitations including simple bending, bending combined with normal forces or bi-axial bending. The last part of Chapter 1 discusses the possible use of moment-normal forces interaction diagrams available in international codes, and some new possible improvements of these simplified diagrams.

Chapter 2 is devoted to general aspects of instability of and second-order effects in slender compression members, and in frames that include such members. For such cases, it is necessary to consider second-order load effects in the analysis and design. The concepts of braced, unbraced and partially braced systems as well as associated moment formulations are presented, and the useful distinction between local and global second-order effects discussed. The general principles of analysis and design of individual reinforced concrete columns and frame systems are reviewed in order to provide a general understanding of the problem area. This includes a presentation and discussion of fundamental concepts and theory behind approximate analysis and design methods to provide a reasonable complete basis for relevant analysis and design requirements as given in existing design rules, such as in Eurocode 2. This also includes a discussion of the applicability of equivalent elastic analysis as an approximation to nonlinear analyses (accounting for both material and geometric nonlinear effects). Local and global slenderness limits, allowing second-order effects to be neglected, are presented and discussed. Chapter 3 deals with approximate analysis methods used for efficient and practical elastic stability calculations, and second-order elastic sway and moment calculations. Included in this chapter are different methods for computing effective lengths, and methods employing the widely used effective length concept in frame analysis. Basic concepts are explained and simple and more complex engineering examples are included to provide a better understanding of the methods.

Reinforced Concrete Beams, Columns and Frames – Mechanics and Design along with the first chapter of the current book were mainly written by Charles Casandjian, Noël Challamel and Christophe Lanos, whereas Jostein Hellesland mostly contributed to the final two chapters of this book.

Finally, an appendix is provided that gives further developments on the theoretical background of Cardano’s method, useful for the resolution of a cubic equation, often encountered in the designing of a reinforced concrete section at both Serviceability and Ultimate Limit States. An appendix giving a table of steel diameters is also provided for the quick and efficient selection of reinforcement sizes in design calculations.

Charles CASANDJIAN, Noël CHALLAMEL, Christophe LANOS and Jostein HELLESLAND December 2012

Chapter 1

Advanced Design at Ultimate Limit State (ULS)

1.1. Design at ULS – simplified analysis

1.1.1. Simplified rectangular behavior–rectangular cross-section

1.1.1.1. Simplified rectangular behavior– rectangular cross-section with only tensile steel reinforcement

In this section, the design of a reinforced concrete section at the ultimate limit state (ULS) is considered by using a rectangular simplified law for the compression concrete block, and a bilinear law for the steel that accounts for the hardening behavior. This design is compatible with Eurocode 2 material parameters. The rectangular cross-section is shown in Figure 1.1. The steel reinforcement area has to be designed for this given concrete section.

Figure 1.1. Rectangular cross-section at ultimate limit state

Pivot AB is characterized for the rectangular cross-section by the neutral axis position:

[1.1]

For a general reinforced concrete section, the bending moment and normal force equilibrium equations are written with respect to the center of gravity of the tensile steel reinforcement as:

[1.2]

where Nc and Mc are the normal force and moment in the compression concrete block calculated from the simplified rectangular constitutive law.

[1.3]

For a reinforced concrete section with only steel reinforcement, the bending moment equilibrium equation is written in a dimensionless format:

[1.4]

We recognize a second-order equation with respect to the position of the neutral axis α:

[1.5]

whose solution of interest is given by:

[1.6]

Note that this equation is independent of the pivot considered (pivot A or pivot B). Once the position of the neutral axis is calculated, the tensile steel area is obtained from the normal force equilibrium equation:

[1.7]

In the case of pivot A, for α ≤ α AB, the strain capacity of the tensile steel reinforcement εs1 is equal to εud, and the steel stress σs1 is equal to (see [equation 3.91] of [CAS 12]), leading to:

[1.8]

In the case of pivot B, for α ≤ α AB the strain of the tensile steel reinforcement ε depends on the position of the neutral axis. If the tensile steel reinforcement behaves in elasticity, the tensile steel area is calculated from: