Structural analysis by kassimali

8.3 Influence Lines for Girders with Floor Systems 3698.4 Influence Lines for Trusses 379 8.5 Influence Lines for Deflections 392Summary 395 Problems 395 9.1 Response at a Particular Locatio

Structural Analysis THIRD EDITION

Aslam Kassimali Southern Illinois University—Carbondale

Australia  Canada  Mexico  Singapore  Spain  United Kingdom  United States

Structural Analysis, Third Edition

by Aslam Kassimali Associate Vice-President and Editorial

Director:

Evelyn Veitch Publisher:

Bill Stenquist Sales and Marketing Manager:

John More Developmental Editor:

Kamilah Reid Burrell

Production Service:

RPK Editorial Services Copy Editor:

Shelly Gerger-Knechtl Indexer:

Aslam Kassimali Proofing:

Jackie Twoney Production Manager:

Renate McCloy Creative Director:

Angela Cluer

Cover Design:

Vernon Boes Compositor:

Asco Typesetters Printer:

Quebecor World

COPYRIGHT 8 2005 by Nelson, a

division of Thomson Canada Limited.

Printed in the United States

1 2 3 4 07 06 05 04 For more information contact Nelson,

1120 Birchmount Road, Toronto, Ontario, Canada, M1K 5G4 Or you can visit our internet site at http://www.nelson.com Library of Congress Control Number:

2003113087 ISBN 0-534-39168-0

ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced, transcribed, or used

in any form or by any means—graphic, electronic, or mechanical, including photocopying, recording, taping, Web distribution, or information storage and retrieval systems—without the written permission of the publisher.

For permission to use material from this text or product, submit a request online

at www.thomsonrights.com Every effort has been made to trace ownership of all copyrighted material and to secure permission from copyright holders In the event of any question arising as to the use of any material, we will be pleased to make the necessary corrections in future printings.

North America Nelson

1120 Birchmount Road Toronto, Ontario M1K 5G4 Canada

Asia Thomson Learning

5 Shenton Way #01-01 UIC Building Singapore 068808 Australia/New Zealand Thomson Learning

102 Dodds Street Southbank, Victoria Australia 3006 Europe/Middle East/Africa Thomson Learning High Holborn House 50/51 Bedford Row London WC1R 4LR United Kingdom Latin America Thomson Learning Seneca, 53 Colonia Polanco

11560 Mexico D.F.

Mexico Spain Paraninfo Calle/Magallanes, 25

28015 Madrid, Spain

IN MEMORY OF AMI

Preface xiii

2.7 Hydrostatic and Soil Pressures 37

2.8 Thermal and Other E¤ects 37

2.9 Load Combinations 37

v

Summary 38Problems 39

3.1 Equilibrium of Structures 43

3.2 External and Internal Forces 46

3.3 Types of Supports for Plane Structures 47

3.4 Static Determinacy, Indeterminacy, and Instability 47

3.5 Computation of Reactions 60

3.6 Principle of Superposition 78

3.7 Reactions of Simply Supported Structures Using

Proportions 78Summary 80Problems 82

4.1 Assumptions for Analysis of Trusses 91

4.2 Arrangement of Members of Plane Trusses—Internal

Stability 95

4.3 Equations of Condition for Plane Trusses 100

4.4 Static Determinacy, Indeterminacy, and Instability of Plane

Trusses 101

4.5 Analysis of Plane Trusses by the Method of Joints 106

4.6 Analysis of Plane Trusses by the Method of Sections 122

4.7 Analysis of Compound Trusses 129

4.8 Complex Trusses 134

4.9 Space Trusses 135Summary 145Problems 146

5.1 Axial Force, Shear, and Bending Moment 162

5.2 Shear and Bending Moment Diagrams 168

vi Contents

5.3 Qualitative Deflected Shapes 173

5.4 Relationships between Loads, Shears, and Bending

6.1 Di¤erential Equation for Beam Deflection 229

6.2 Direct Integration Method 232

6.3 Superposition Method 235

6.4 Moment-Area Method 236

6.5 Bending Moment Diagrams by Parts 250

6.6 Conjugate-Beam Method 255Summary 271

Problems 271

7.1 Work 278

7.2 Principle of Virtual Work 280

7.3 Deflections of Trusses by the Virtual Work Method 284

7.4 Deflections of Beams by the Virtual Work Method 295

7.5 Deflections of Frames by the Virtual Work Method 304

7.6 Conservation of Energy and Strain Energy 314

7.7 Castigliano’s Second Theorem 318

7.8 Betti’s Law and Maxwell’s Law of Reciprocal Deflections 327Summary 328

8.3 Influence Lines for Girders with Floor Systems 369

8.4 Influence Lines for Trusses 379

8.5 Influence Lines for Deflections 392Summary 395

Problems 395

9.1 Response at a Particular Location Due to a Single Moving

Concentrated Load 403

9.2 Response at a Particular Location Due to a Uniformly

Distributed Live Load 405

9.3 Response at a Particular Location Due to a Series of Moving

10.2 Symmetric and Antisymmetric Components of Loadings 434

10.3 Behavior of Symmetric Structures under Symmetric and

Antisymmetric Loadings 445

10.4 Procedure for Analysis of Symmetric Structures 449Summary 457

Problems 458

11.1 Advantages and Disadvantages of Indeterminate

Structures 464

11.2 Analysis of Indeterminate Structures 467Summary 472

viii Contents

12 Approximate Analysis of Rectangular Building Frames 473

12.1 Assumptions for Approximate Analysis 474

12.2 Analysis for Vertical Loads 477

12.3 Analysis for Lateral Loads—Portal Method 483

12.4 Analysis for Lateral Loads—Cantilever Method 499Summary 506

Problems 507

13.1 Structures with Single Degree of Indeterminacy 511

13.2 Internal Forces and Moments as Redundants 533

13.3 Structures with Multiple Degrees of Indeterminacy 546

13.4 Support Settlements, Temperature Changes and Fabrication

Errors 570Summary 579Problems 580

14.1 Derivation of Three-Moment Equation 589

14.2 Application of Three-Moment Equation 594

14.3 Method of Least Work 601Summary 608

Problems 609

15.1 Influence Lines for Beams and Trusses 612

15.2 Qualitative Influence Lines by Muller-Breslau’s Principle 629Summary 634

Problems 634

16.1 Slope-Deflection Equations 638

16.2 Basic Concept of the Slope-Deflection Method 646

16.3 Analysis of Continuous Beams 653

Contents ix

16.4 Analysis of Frames without Sidesway 675

16.5 Analysis of Frames with Sidesway 683Summary 704

Problems 704

17.1 Definitions and Terminology 710

17.2 Basic Concept of the Moment-Distribution Method 719

17.3 Analysis of Continuous Beams 727

17.4 Analysis of Frames without Sidesway 743

17.5 Analysis of Frames with Sidesway 746Summary 763

18.4 Member Sti¤ness Relations in Global Coordinates 788

18.5 Structure Sti¤ness Relations 789

18.6 Procedure for Analysis 797Summary 815

Problems 816

x Contents

Appendix C Computer Software 837

Bibliography 851Answers to Selected Problems 853Index 861

Contents xi

The objective of this book is to develop an understanding of the basicprinciples of structural analysis Emphasizing the intuitive classical ap-proach, Structural Analysis covers the analysis of statically determinateand indeterminate beams, trusses, and rigid frames It also presents anintroduction to the matrix analysis of structures

The book is divided into three parts Part One presents a generalintroduction to the subject of structural engineering It includes a chap-ter devoted entirely to the topic of loads because attention to this im-portant topic is generally lacking in many civil engineering curricula.Part Two, consisting of Chapters 3 through 10, covers the analysis ofstatically determinate beams, trusses, and rigid frames The chapters ondeflections (Chapters 6 and 7) are placed before those on influence lines(Chapters 8 and 9), so that influence lines for deflections can be included

in the latter chapters This part also contains a chapter on the analysis

of symmetric structures (Chapter 10) Part Three of the book, Chapters

11 through 18, covers the analysis of statically indeterminate structures.The format of the book is flexible to enable instructors to emphasizetopics that are consistent with the goals of the course

Each chapter of the book begins with an introductory section fining its objective and ends with a summary section outlining its salientfeatures An important general feature of the book is the inclusion ofstep-by-step procedures for analysis to enable students to make an easiertransition from theory to problem solving Numerous solved examplesare provided to illustrate the application of the fundamental concepts

de-A CD-ROM containing computer software for the analysis of planeframes, continuous beams, and trusses is attached to the back cover.This interactive software can be used to simulate a variety of structuraland loading configurations and to determine cause versus e¤ect rela-tionships between loading and various structural parameters, therebyenhancing the students’ understanding of the behavior of structures Thesoftware shows deflected shapes of structures to enhance students’ un-

xiii

derstanding of structural response due to various types of loadings Itcan also include the e¤ects of support settlements, temperature changes,and fabrication errors in the analysis A solutions manual, containingcomplete solutions to text exercises, is also available for the instructor.

A NOTE ON THE REVISED EDITION

In this third edition, 37 new solved examples have been added to crease the total number by about 30% The number of problems hasalso been increased to bring the total to over 600, of which about 40%are new problems The chapter on loads has been revised to meet theprovisions of the ASCE 7-02 Standard, and the treatment of the forcemethod has been expanded by including the topic of the three-momentequation The force method is now covered in two chapters (Chapters

in-13 and 14), with the new Chapter 14 containing the three-momentequation and the method of least work There are many other minorrevisions, including some in the computer software, which has been re-compiled to make it compatible with the latest versions of MicrosoftWindows Finally, some of the photographs have been replaced withnew ones, and some figures have been redrawn and rearranged to en-hance clarity

ACKNOWLEDGMENTS

I wish to express my thanks to Bill Stenquist of Thomson Engineeringfor his constant support and encouragement throughout this project,and to Rose Kernan for all her help during the production phase.Thanks are also due to Jonathan Plant and Suzanne Jeans, my editorsfor the first and second editions, respectively, of this book The com-ments and suggestions for improvement from colleagues and studentswho have used previous editions are gratefully acknowledged All oftheir suggestions were carefully considered, and implemented wheneverpossible Thanks are due to the following reviewers for their careful re-views of the manuscripts of the various editions, and for their construc-tive suggestions:

Ayo AbatanVirginia Polytechnic Institute andState University

Riyad S AboutahaGeorgia Institute of TechnologyOsama Abudayyeh

Western Michigan University

Thomas T BaberUniversity of VirginiaGordon B BatsonClarkson UniversityGeorge E BlandfordUniversity of Kentucky

xiv Preface

Ramon F BorgesPenn State/Altoona CollegeKenneth E Buttry

University of WisconsinWilliam F CarrollUniversity of Central FloridaMalcolm A CutchinsAuburn UniversityJack H EmanuelUniversity of Missouri—RollaFouad Fanous

Iowa State UniversityLeon Feign

Fairfield UniversityRobert FleischmanUniversity of Notre DameGeorge Kostyrko

California State University

E W LarsonCalifornia State University/

Northridge

L D LutesTexas A&M UniversityEugene B LoverichNorthern Arizona UniversityDavid Mazurek

US Coast Guard AcademyAhmad Namini

University of MiamiArturo E SchultzNorth Carolina State UniversityKassim Tarhini

Valparaiso UniversityRobert TaylorNortheastern University

C C TungNorth Carolina State UniversityNicholas Willems

University of KansasJohn ZacharMilwaukee School of EngineeringMannocherh Zoghi

University of DaytonFinally, I would like to express my loving appreciation to my wife.Maureen, for her constant encouragement and help in preparing thismanuscript, and to my sons, Jamil and Nadim, for their enormous un-derstanding and patience

Aslam Kassimali

Preface xv

Shape Area Centroid

Trapezoid

A¼bðh1þ h2Þ

2 x¼bðh1þ 2h2Þ

3ðh1þ h2ÞSemi-parabola

Shape Area Centroid

Review of Matrix Algebra

B.1 Definition of a MatrixB.2 Types of MatricesB.3 Matrix OperationsB.4 Solution of Simultaneous Equations by the Gauss-Jordan MethodProblems

In this appendix, some basic concepts of matrix algebra necessary forformulating the computerized analysis of structures are briefly reviewed

A more comprehensive and mathematically rigorous treatment of theseconcepts can be found in any textbook on matrix algebra, such as [11]and [28]

37

7ith row (B.1)

jth column m n

As Eq (B.1) indicates, matrices are usually denoted either by boldfaceletters (e.g., A) or by italic letters enclosed within brackets (e.g., [A]).The quantities that form a matrix are referred to as the elements of thematrix, and each element is represented by a double-subscripted letter,

with the first subscript identifying the row and the second subscriptidentifying the column in which the element is located Thus in Eq.(B.1), A12 represents the element located in the first row and the secondcolumn of the matrix A, and A21 represents the element in the secondrow and the first column of A In general, an element located in the ithrow and the jth column of matrix A is designated as Aij It is commonpractice to enclose the entire array of elements between brackets, asshown in Eq (B.1).

The size of a matrix is measured by its order, which refers to thenumber of rows and columns of the matrix Thus the matrix A in Eq.(B.1), which consists of m rows and n columns, is considered to be oforder m n (m by n) As an example, consider a matrix B given by

37

The order of this matrix is 3 4, and its elements can be symbolicallyrepresented by Bij, with i¼ 1 to 3 and j ¼ 1 to 4; for example, B23¼ 19,

615

266

377

Column matrices are also referred to as vectors and are sometimesdenoted by italic letters enclosed within braces (e.g.,fDg)

2664

5 21 3

40 6 19

8 12 50

377

377

37

Unit or Identity Matrix

A diagonal matrix with all its diagonal elements equal to 1 (i.e., Iii¼ 1and Iij¼ 0 for i 0 j) is called a unit, or identity, matrix Unit matricesusually are denoted by I or [I ] An example of a 4 4 unit matrix is

377

B.2 Types of Matrices 823

Null Matrix

When all the elements of a matrix are zero (i.e., Oij¼ 0), the matrix iscalled a null matrix Null matrices are commonly denoted by O or [O].For example,

37

B.3 MATRIX OPERATIONS

Equality

Two matrices A and B are equal if they are of the same order and iftheir corresponding elements are identical (i.e., Aij¼ Bij) Consider, forexample, the matrices

3

7 and B¼ 34 57 69

12 0 1

26

37

Since both A and B are of order 3 3 and since each element of A isequal to the corresponding element of B, the matrices are considered to

be equal to each other; that is, A¼ B

Addition and Subtraction

The addition (or subtraction) of two matrices A and B, which must be ofthe same order, is carried out by adding (or subtracting) the corre-sponding elements of the two matrices Thus if Aþ B ¼ C, then

Cij¼ Aijþ Bij; and if A B ¼ D, then Dij¼ Aij Bij For example, if

3

7 and B¼ 106 47

9 2

26

37then

37and

37Note that matrices C and D have the same order as matrices A and B

824 APPENDIX B Review of Matrix Algebra

cB¼ 21 9

3 12

Multiplication of Matrices

The multiplication of two matrices can be carried out only if the number

of columns of the first matrix equals the number of rows of the secondmatrix Such matrices are referred to as being conformable for multipli-cation Consider, for example, the matrices

BA does not exist, because now the first matrix, B, has three columnsand the second matrix, A, has two rows The product AB is usually re-ferred to either as A postmultiplied by B or as B premultiplied by A.Conversely, the product BA is referred to either as B postmultiplied by

A or as A premultiplied by B

When two conformable matrices are multiplied, the product matrixthus obtained will have the number of rows of the first matrix and thenumber of columns of the second matrix Thus, if a matrix A of order

m n is postmultiplied by a matrix B of order n  s, then the productmatrix C will be of order m s; that is,

37

7 ¼



Cij

ith row

jth column

B.3 Matrix Operations 825

As illustrated in Eq (B.4), any element Cij of the product matrix C can

be evaluated by multiplying each element of the ith row of A by thecorresponding element of the jth column of B and by algebraicallysumming the resulting products; that is,

Cij¼ Ai1B1jþ Ai2B2jþ    þ AinBnj (B.5)Equation (B.5) can be conveniently expressed as

To illustrate the procedure of matrix multiplication, we compute theproduct C¼ AB of the matrices A and B given in Eq (B.3) as

of B and adding the resulting products; that is,

C21¼ 7ð2Þ  3ð4Þ ¼ 2The remaining elements of C are determined in a similar manner:

C12¼ 1ð3Þ þ 5ð8Þ ¼ 43

C22¼ 7ð3Þ  3ð8Þ ¼ 45

C13¼ 1ð6Þ þ 5ð9Þ ¼ 51

C23¼ 7ð6Þ  3ð9Þ ¼ 69Note that the order of the product matrix C is 2 3, which equals thenumber of rows of A and the number of columns of B

A common application of matrix multiplication is to express taneous equations in compact matrix form Consider the system of si-multaneous linear equations:

simul-826 APPENDIX B Review of Matrix Algebra

A11 A12 A13

A21 A22 A23

A31 A32 A33

26

375¼

P1

P2

P3

26

It is, therefore, necessary to maintain the proper sequential order ofmatrices when computing matrix products Although matrix multiplica-tion is generally not commutative, as indicated by Eq (B.10), it is asso-ciative and distributive, provided that the sequential order in which thematrices are to be multiplied is maintained Thus

ABC¼ ðABÞC ¼ AðBCÞ (B.11)and

AðB þ CÞ ¼ AB þ AC (B.12)Multiplication of any matrix A by a conformable null matrix Oyields a null matrix; that is,

Inverse of a Square Matrix

The inverse of a square matrix A is defined as a matrix A1 with ments of such magnitudes that the multiplication of the original matrix

ele-A by its inverse ele-A1 yields a unit matrix I; that is,