Structural analysis by kassimali

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Structural Analysis
THIRD EDITION

Aslam Kassimali
Southern Illinois University—Carbondale

Australia



Canada



Mexico



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United States

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Structural Analysis, Third Edition
by Aslam Kassimali
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IN MEMORY OF AMI

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Contents

Preface

PART ONE

1

INTRODUCTION TO STRUCTURAL ANALYSIS AND LOADS

1

Introduction to Structural Analysis 3
1.1
1.2
1.3
1.4

2

xiii

Historical Background 4

Role of Structural Analysis in Structural Engineering
Projects 6
Classification of Structures 7
Analytical Models 12
Summary 16

Loads on Structures 17
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9

Dead Loads 18
Live Loads 21
Impact 24
Wind Loads 24
Snow Loads 32
Earthquake Loads 35
Hydrostatic and Soil Pressures 37
Thermal and Other EÔects 37
Load Combinations 37
v

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vi

Contents

Summary 38
Problems 39

PART TWO

3

ANALYSIS OF STATICALLY DETERMINATE STRUCTURES

Equilibrium and Support Reactions 43
3.1
3.2
3.3
3.4
3.5
3.6
3.7

4

Equilibrium of Structures 43
External and Internal Forces 46
Types of Supports for Plane Structures 47
Static Determinacy, Indeterminacy, and Instability

Computation of Reactions 60
Principle of Superposition 78
Reactions of Simply Supported Structures Using
Proportions 78
Summary 80
Problems 82

47

Plane and Space Trusses 89
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9

5

41

Assumptions for Analysis of Trusses 91
Arrangement of Members of Plane Trusses—Internal
Stability 95
Equations of Condition for Plane Trusses 100
Static Determinacy, Indeterminacy, and Instability of Plane
Trusses 101

Analysis of Plane Trusses by the Method of Joints 106
Analysis of Plane Trusses by the Method of Sections 122
Analysis of Compound Trusses 129
Complex Trusses 134
Space Trusses 135
Summary 145
Problems 146

Beams and Frames: Shear and Bending Moment 161
5.1
5.2

Axial Force, Shear, and Bending Moment 162
Shear and Bending Moment Diagrams 168

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Contents
5.3
5.4
5.5
5.6

6

6.2
6.3
6.4

6.5
6.6

DiÔerential Equation for Beam Deection 229
Direct Integration Method 232
Superposition Method 235
Moment-Area Method 236
Bending Moment Diagrams by Parts 250
Conjugate-Beam Method 255
Summary 271
Problems 271

Deflections of Trusses, Beams, and Frames: Work–Energy Methods 277
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8

8

Qualitative Deflected Shapes 173
Relationships between Loads, Shears, and Bending
Moments 175
Static Determinacy, Indeterminacy and Instability of Plane
Frames 196
Analysis of Plane Frames 202

Summary 217
Problems 219

Deflections of Beams: Geometric Methods 228
6.1

7

vii

Work 278
Principle of Virtual Work 280
Deflections of Trusses by the Virtual Work Method 284
Deflections of Beams by the Virtual Work Method 295
Deflections of Frames by the Virtual Work Method 304
Conservation of Energy and Strain Energy 314
Castigliano’s Second Theorem 318
Betti’s Law and Maxwell’s Law of Reciprocal Deflections
Summary 328
Problems 330

Influence Lines 339
8.1
8.2

Influence Lines for Beams and Frames by Equilibrium
Method 340
Mueller-Breslau’s Principle and Qualitative Influence
Lines 355

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viii

Contents
8.3
8.4
8.5

9

9.2
9.3
9.4

Response at a Particular Location Due to a Single Moving
Concentrated Load 403
Response at a Particular Location Due to a Uniformly
Distributed Live Load 405
Response at a Particular Location Due to a Series of Moving
Concentrated Loads 410
Absolute Maximum Response 417
Summary 423
Problems 424

Analysis of Symmetric Structures 427

10.1
10.2
10.3
10.4

PART THREE

11

369

Application of Influence Lines 403
9.1

10

Influence Lines for Girders with Floor Systems
Influence Lines for Trusses 379
Influence Lines for Deflections 392
Summary 395
Problems 395

Symmetric Structures 428
Symmetric and Antisymmetric Components of Loadings
Behavior of Symmetric Structures under Symmetric and
Antisymmetric Loadings 445
Procedure for Analysis of Symmetric Structures 449
Summary 457
Problems 458

ANALYSIS OF STATICALLY INDETERMINATE STRUCTURES

434

461

Introduction to Statically Indeterminate Structures 463
11.1
11.2

Advantages and Disadvantages of Indeterminate
Structures 464
Analysis of Indeterminate Structures 467
Summary 472

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Contents

12

Approximate Analysis of Rectangular Building Frames 473
12.1
12.2
12.3
12.4

13

13.2
13.3
13.4

14.2
14.3

Derivation of Three-Moment Equation 589
Application of Three-Moment Equation 594
Method of Least Work 601
Summary 608
Problems 609

Influence Lines for Statically Indeterminate Structures 611
15.1
15.2

16

Structures with Single Degree of Indeterminacy 511
Internal Forces and Moments as Redundants 533
Structures with Multiple Degrees of Indeterminacy 546
Support Settlements, Temperature Changes and Fabrication
Errors 570
Summary 579
Problems 580

Three-Moment Equation and the Method of Least Work 588
14.1

15

Assumptions for Approximate Analysis 474
Analysis for Vertical Loads 477
Analysis for Lateral Loads—Portal Method 483
Analysis for Lateral Loads—Cantilever Method 499
Summary 506
Problems 507

Method of Consistent Deformations—Force Method 510
13.1

14

ix

Influence Lines for Beams and Trusses 612
Qualitative Influence Lines by Muller-Breslau’s Principle
Summary 634
Problems 634

629

Slope-Deflection Method 637
16.1
16.2
16.3

Slope-Deflection Equations 638

Basic Concept of the Slope-Deflection Method
Analysis of Continuous Beams 653

646

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x

Contents
16.4
16.5

17

Moment-Distribution Method 709
17.1
17.2
17.3
17.4
17.5

18

Analysis of Frames without Sidesway 675
Analysis of Frames with Sidesway 683
Summary 704
Problems 704

Definitions and Terminology 710
Basic Concept of the Moment-Distribution Method
Analysis of Continuous Beams 727
Analysis of Frames without Sidesway 743
Analysis of Frames with Sidesway 746
Summary 763
Problems 764

719

Introduction to Matrix Structural Analysis 769
18.1
18.2
18.3
18.4
18.5
18.6

Analytical Model 770
Member StiÔness Relations in Local Coordinates 774
Coordinate Transformations 782
Member StiÔness Relations in Global Coordinates 788
Structure StiÔness Relations 789
Procedure for Analysis 797
Summary 815
Problems 816

Appendix A

Areas and Centroids of Geometric Shapes 818

Appendix B

Review of Matrix Algebra 821
B.1
B.2
B.3
B.4

Definition of a Matrix 821
Types of Matrices 822
Matrix Operations 824
Solution of Simultaneous Equations by the Gauss-Jordan
Method 831
Problems 835

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Contents

Appendix C

Computer Software 837
Bibliography 851
Answers to Selected Problems
Index 861

853

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xi


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Preface

The objective of this book is to develop an understanding of the basic
principles of structural analysis. Emphasizing the intuitive classical approach, Structural Analysis covers the analysis of statically determinate
and indeterminate beams, trusses, and rigid frames. It also presents an
introduction to the matrix analysis of structures.
The book is divided into three parts. Part One presents a general
introduction to the subject of structural engineering. It includes a chapter devoted entirely to the topic of loads because attention to this important topic is generally lacking in many civil engineering curricula.
Part Two, consisting of Chapters 3 through 10, covers the analysis of
statically determinate beams, trusses, and rigid frames. The chapters on
deflections (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 emphasize

topics that are consistent with the goals of the course.
Each chapter of the book begins with an introductory section defining its objective and ends with a summary section outlining its salient
features. An important general feature of the book is the inclusion of
step-by-step procedures for analysis to enable students to make an easier
transition from theory to problem solving. Numerous solved examples
are provided to illustrate the application of the fundamental concepts.
A CD-ROM containing computer software for the analysis of plane
frames, continuous beams, and trusses is attached to the back cover.
This interactive software can be used to simulate a variety of structural
and loading configurations and to determine cause versus eÔect relationships between loading and various structural parameters, thereby
enhancing the students’ understanding of the behavior of structures. The
software shows deflected shapes of structures to enhance students’ unxiii

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xiv

Preface

derstanding of structural response due to various types of loadings. It
can also include the eÔects of support settlements, temperature changes,
and fabrication errors in the analysis. A solutions manual, containing
complete 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 increase the total number by about 30%. The number of problems has
also 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 the

provisions of the ASCE 7-02 Standard, and the treatment of the force
method has been expanded by including the topic of the three-moment
equation. The force method is now covered in two chapters (Chapters
13 and 14), with the new Chapter 14 containing the three-moment
equation and the method of least work. There are many other minor
revisions, including some in the computer software, which has been recompiled to make it compatible with the latest versions of Microsoft
Windows. Finally, some of the photographs have been replaced with
new ones, and some figures have been redrawn and rearranged to enhance clarity.

ACKNOWLEDGMENTS
I wish to express my thanks to Bill Stenquist of Thomson Engineering
for 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 editors
for the first and second editions, respectively, of this book. The comments and suggestions for improvement from colleagues and students
who have used previous editions are gratefully acknowledged. All of
their suggestions were carefully considered, and implemented whenever
possible. Thanks are due to the following reviewers for their careful reviews of the manuscripts of the various editions, and for their constructive suggestions:
Ayo Abatan
Virginia Polytechnic Institute and
State University
Riyad S. Aboutaha
Georgia Institute of Technology

Thomas T. Baber
University of Virginia

Osama Abudayyeh
Western Michigan University

George E. Blandford
University of Kentucky

Gordon B. Batson
Clarkson University

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Preface

Ramon F. Borges
Penn State/Altoona College

L. D. Lutes
Texas A&M University

Kenneth E. Buttry
University of Wisconsin

Eugene B. Loverich
Northern Arizona University
David Mazurek
US Coast Guard Academy

William F. Carroll
University of Central Florida
Malcolm A. Cutchins
Auburn University

Jack H. Emanuel
University of Missouri—Rolla
Fouad Fanous
Iowa State University
Leon Feign
Fairfield University
Robert Fleischman
University of Notre Dame
George Kostyrko
California State University
E. W. Larson
California State University/
Northridge

xv

Ahmad Namini
University of Miami
Arturo E. Schultz
North Carolina State University
Kassim Tarhini
Valparaiso University
Robert Taylor
Northeastern University
C. C. Tung
North Carolina State University
Nicholas Willems
University of Kansas
John Zachar
Milwaukee School of Engineering

Mannocherh Zoghi
University of Dayton

Finally, I would like to express my loving appreciation to my wife.
Maureen, for her constant encouragement and help in preparing this
manuscript, and to my sons, Jamil and Nadim, for their enormous understanding and patience.
Aslam Kassimali

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A
Areas and Centroids of
Geometric Shapes

Shape

Area

Centroid

Right-angled triangle

Aẳ

bh
2

Aẳ

bh
2

xẳ

2b
3

Triangle

xẳ

aỵb
3

818

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APPENDIX A

Areas and Centroids of Geometric Shapes

Shape

Area

Centroid

Trapezoid

A¼

bðh1 ỵ h2 ị
2

xẳ

bh1 ỵ 2h2 ị
3h1 ỵ h2 ị

Semi-parabola

Aẳ

2bh
3

xẳ

3b

8

Aẳ

bh
3

xẳ

3b
4

Aẳ

2bh
3

xẳ

b
2

Parabolic spandrel

Parabolic segment

Note: When the segment represents a
part of the bending moment diagram
of a member subjected to uniformly
distributed load w, then h ¼ wb2 =8.

Cubic

A¼

3bh
4

x¼

2b
5

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819


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820

APPENDIX A

Areas and Centroids of Geometric Shapes
Shape

Area

Centroid

Cubic spandrel

Aẳ

bh
4

xẳ

4b
5

nth-degree curve
y ẳ axn , nb1

Aẳ

bh
nỵ1

xẳ

n ỵ 1ịb
n ỵ 2ị

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B

Review of Matrix Algebra
B.1
B.2
B.3
B.4

Definition of a Matrix
Types of Matrices
Matrix Operations
Solution of Simultaneous Equations by the Gauss-Jordan Method
Problems

In this appendix, some basic concepts of matrix algebra necessary for
formulating the computerized analysis of structures are briefly reviewed.
A more comprehensive and mathematically rigorous treatment of these
concepts can be found in any textbook on matrix algebra, such as [11]
and [28].

B.1 DEFINITION OF A MATRIX
A matrix is a rectangular array of quantities arranged in rows and columns. A matrix containing m rows and n columns can be expressed as:
2
3
A11 A12 Á Á Á
Á Á Á A1n
6A
Á Á Á A2 n 7
6 21 A22
7
A ẳ ẵA ẳ 6
(B.1)

7ith row
Á Aij Á Á Á
4 ÁÁÁ
5

Á Á Á Am n
Am1 Am2 Á Á Á 
jth column
mÂn
As Eq. (B.1) indicates, matrices are usually denoted either by boldface
letters (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 the
matrix, and each element is represented by a double-subscripted letter,
821

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822

APPENDIX B

Review of Matrix Algebra

with the first subscript identifying the row and the second subscript
identifying the column in which the element is located. Thus in Eq.
(B.1), A12 represents the element located in the first row and the second
column of the matrix A, and A21 represents the element in the second
row and the first column of A. In general, an element located in the ith

row and the jth column of matrix A is designated as Aij . It is common
practice to enclose the entire array of elements between brackets, as
shown in Eq. (B.1).
The size of a matrix is measured by its order, which refers to the
number 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 of
order m  n (m by n). As an example, consider a matrix B given by
2
3
5 21
3 À7
6
7
B ¼ 4 40 À6 19 23 5
À8 12 50 22
The order of this matrix is 3 Â 4, and its elements can be symbolically
represented by Bij , with i ¼ 1 to 3 and j ¼ 1 to 4; for example, B23 ¼ 19,
B31 ¼ À8, B34 ¼ 22, etc.

B.2 TYPES OF MATRICES
Row Matrix
If all the elements of a matrix are arranged in a single row (i.e., m ¼ 1),
then the matrix is called a row matrix. An example of a row matrix is
C ẳ ẵ50

3

27

35

Column Matrix
A matrix with only one column of elements (i.e., n ¼ 1) is called a
column matrix. For example,
2
3
À10
6 33 7
6
7
D ¼ fDg ¼ 6
7
4 À6 5
15
Column matrices are also referred to as vectors and are sometimes
denoted by italic letters enclosed within braces (e.g., fDg).

Square Matrix
A matrix with the same number of rows and columns m ẳ nị is called a
square matrix. An example of a 3 Â 3 square matrix is

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B.2

2

6 5

6
A¼6
6 40
4
À8

21
À6

Types of Matrices

823

3

3 7
7
19 7
7
5
50

(B.2)

12
ƒƒ!
ƒ
ƒ
ƒ
ƒƒƒMain diagonal

The elements with the same subscripts—that is, A11 ; A22 ; . . . ; Ann —form
the main diagonal of the square matrix A. These elements are referred to
as the diagonal elements. As shown in Eq. (B.2), the main diagonal extends from the upper left corner to the lower right corner of the square
matrix. The remaining elements of the matrix (i.e., Aij with i 0 j) that
are not along the main diagonal are termed the oÔ-diagonal elements.

Symmetric Matrix
If the elements of a square matrix are symmetric about its main diagonal (i.e., Aij ¼ Aji ), the matrix is called a symmetric matrix. An example
of a 4 Â 4 symmetric matrix is
2

À12
6 À6
6
A ¼6
4 13
5

À6
13
7 À28
À28
10
31 À9

3
5
31 7
7

7
À9 5
À2

Diagonal Matrix
If all the oÔ-diagonal elements of a square matrix are zero (i.e., Aij ¼ 0
for i 0 j), the matrix is referred to as a diagonal matrix. For example,
2
3
3
0
0
6
7
A ¼ 4 0 À8
05
0
0 14

Unit or Identity Matrix
A diagonal matrix with all its diagonal elements equal to 1 (i.e., Iii ¼ 1
and Iij ¼ 0 for i 0 j) is called a unit, or identity, matrix. Unit matrices
usually are denoted by I or [I ]. An example of a 4 Â 4 unit matrix is
2

1
60
6
I ¼6
40

0

0
1
0
0

0
0
1
0

3
0
07
7
7
05
1

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824

APPENDIX B

Review of Matrix Algebra

Null Matrix
When all the elements of a matrix are zero (i.e., Oij ¼ 0), the matrix is
called a null matrix. Null matrices are commonly denoted by O or [O].
For example,
2
3
0 0 0 0
6
7
O ¼4 0 0 0 0 5
0 0 0 0

B.3 MATRIX OPERATIONS
Equality
Two matrices A and B are equal if they are of the same order and if
their corresponding elements are identical (i.e., Aij ¼ Bij ). Consider, for
example, the matrices
2
3
2
3
À3 5 6
À3 5 6
6
7
6
7
A ¼4 4 7 9 5
and
B ¼4 4 7 9 5

12 0 1
12 0 1
Since both A and B are of order 3 Â 3 and since each element of A is
equal to the corresponding element of B, the matrices are considered to
be equal to each other; that is, A ¼ B.

Addition and Subtraction
The addition (or subtraction) of two matrices A and B, which must be of
the same order, is carried out by adding (or subtracting) the corresponding elements of the two matrices. Thus if A ỵ B ẳ C, then
Cij ẳ Aij ỵ Bij ; and if A À B ¼ D, then Dij ¼ Aij À Bij . For example, if
2
3
2
3
2 5
10 4
6
7
6
7
A ¼4 3 0 5
and
B ẳ4 6 7 5
8 1
9 2
then

2

12

6
A ỵ B ẳ C ¼4 9
17
and

3
9
7
75
3

2

3
À8
1
6
7
A À B ¼ D ¼ 4 À3 À7 5
À1 À1

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

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B.3

Matrix Operations

825

Multiplication by a Scalar
To obtain the product of a scalar and a matrix, each element of the
matrix must be multiplied by the scalar. Thus, if
!
7 3
B¼
and
c ¼ À3
À1 4
then
cB ¼

À21
3

À9
À12

!

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 second
matrix. Such matrices are referred to as being conformable for multiplication. Consider, for example, the matrices
!
!
À1

5
2
3 À6
A¼
and
B¼
(B.3)
7 À3
4 À8
9
in which A is of order 2 Â 2 and B is of order 2 Â 3. Note that the
product AB of these matrices is defined, because the first matrix, A, of
the sequence AB has two columns and the second matrix, B, has two
rows. However, if the sequence of the matrices is reversed, the product
BA does not exist, because now the first matrix, B, has three columns
and the second matrix, A, has two rows. The product AB is usually referred 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 matrix
thus obtained will have the number of rows of the first matrix and the
number 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 product
matrix C will be of order m  s; that is,
A
mÂn

ith row

B
À equal À! n  s

Ai1 ! Ain

2
!6 B1j
6
6 #
4
Bnj

¼

C

#
mÂs
"

3
7
7
7
5

!
¼

Cij

ith row

jth column
jth column

(B.4)

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826

APPENDIX B

Review of Matrix Algebra

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 the
corresponding element of the jth column of B and by algebraically
summing the resulting products; that is,
Cij ¼ Ai1 B1j ỵ Ai2 B2j ỵ ỵ Ain Bnj

(B.5)

Equation (B.5) can be conveniently expressed as

Cij ¼

n
X

Aik Bkj

k ¼1

(B.6)

in which n represents the number of columns of the matrix A and the
number of rows of the matrix B. Note that Eq. (B.6) can be used to determine any element of the product matrix C ¼ AB.
To illustrate the procedure of matrix multiplication, we compute the
product C ¼ AB of the matrices A and B given in Eq. (B.3) as
!
!
!
À1
5 2
3 À6
18 À43
51
C ¼ AB ¼
¼
7 À3 4 À8
9
2
45 À69
2Â2
2Â3
2Â3
in which the element C11 of the product matrix C is obtained by multiplying each element of the first row of A by the corresponding element
of the first column of B and summing the resulting products; that is,

C11 ẳ 12ị ỵ 54ị ẳ 18
Similarly, the element C21 is determined by multiplying the elements of
the second row of A by the corresponding elements of the first column
of B and adding the resulting products; that is,
C21 ẳ 72ị 34ị ẳ 2
The remaining elements of C are determined in a similar manner:
C12 ẳ 13ị ỵ 58ị ẳ 43
C22 ẳ 73ị 38ị ẳ 45
C13 ẳ 16ị ỵ 59ị ẳ 51
C23 ẳ 76ị 39ị ẳ 69
Note that the order of the product matrix C is 2 Â 3, which equals the
number of rows of A and the number of columns of B.
A common application of matrix multiplication is to express simultaneous equations in compact matrix form. Consider the system of simultaneous linear equations:

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