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 - Gere&Timoshenko

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 Mechanics of Materials
SECOND EDITION

James M. Gere
STANFORD UNIVERSITY

Stephen P. Timoshenko
LATE OF: STANFORD UNIVERSITY

CBS PUBLISHERS & DISTRIBUTORS PVT. LTD.

NEW DELHI « BANGALORE « PUNE * COCHIN « CHENNAI (INDIA)
-Brooks/Cole Engineering Division
A Division of Wadsworth, Inc.

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€ i984 by Wadsworth, Inc., Belmont, California 94002. All rights reserved.
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of this book may be reproduced, stored in a retrieval system, or transcribed
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otherwise—without the prior written permission of the publisher, Brooks/Cole
, Inc.
Engineering Division, Monterey, California 93940, a division of Wadsworth

Original English Language Edition Published by PWS Publisher, A Division of

Wadsworth Inc., 20, Park Plaza, Boston, MA-02116, USA

Copyright © 1984 by PWS Engineering, A Division of Wadsworth Inc., 20,

Park Plaza, Boston, MA-02116, USA
ISBN : 81-239-0894-6
First Indian Edition : 1986
Reprint : 2000, 2002, 2004

This edition has been published in India by arrangement with

PWS Publishers, A Division of Wadsworth Inc., USA

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 Preface

A course in mechanics of materials provides an opportunity to accom-

plish two things: first, to teach students a basic engineering subject and,
second, to develop their analytical and problem-solving abilities. While
preparing this extensive revision, I have kept both of these goals in mind.
The facts, theories, and methodologies are presented in a teachable and
éasy-to-learn manner, with ample discussions and numerous illustrative
examples, so that undergraduate students can readily master the subject
matter. At the same time, emphasis is placed on fundamental concepts
. and on how to analyze mechanical and structural systems. Many exam-
ples and problems require that students do some original thinking.
This book covers all the standard topics of mechanics of materials
and presents them at a level suitable for sophomore and junior engi-
neering students. In addition, much material of a more advanced and
specialized nature is included. Thus, this book can serve both as a text
and as a permanent reference.
A glance at the table of contents shows the topics covered and the
way..in. which they are organized. The topics include the analysis and
design of structural members subjected to axial loads, torsion, and bend-
ing, as well as such fundamental concepts as stress, strain, elastic and
inelastic behavior, and strain energy. Other topics of general interest are
the transformations of stress and strain, deflections of beams, behavior of
columns, and energy methods. More specialized topics are thermal and
prestrain effects, pressure vessels, ‘nonprismatic members, unsymmetric
bending, shear center, inelastic bending, and discontinuity functions.
Much more material than can be covered in a single course is
included in the book, hence teachers have the opportunity to select the
topics that they feel are the most fundamental and relevant. Teachers will
also appreciate the hundreds of new problems (over 1,000 probiems total)
that are available for homework assignments and classroom discussions.
y
vi Preface

Both the International System of Units (SI) and the U.S. Customary
System (USCS) are used in the numerical examples and problems. Dis-
cussions of both systems and a table of conversion factors are given in the
appendix.
References and historical notes are collected at the back of the book.
They include the original sources of the subject matter and biographical
notes about the pioneering engineers, scientists, and mathematicians who
created the subject. ;
This book is new in the sense that #t is a completely new presentation
of mechanics of materials; yet in another sense it is old because it evolved
from earlier books of Professor Stephen P. Timoshenko (1878- 1972).
Timoshenko’s first book on mechanics of materials was published in
Russia in 1908. His first American book on the subject was published in
two volumes in 1930 by D. Van Nostrand Company under the title
Strength of Materials; second éditions were published in 1940 and 1941
and third editions in 1955 and 1956. The first edition of Mechanics of
Materials, written by the present a but drawing upon the earlier
books, was published in 1972.
This second edition has been completely rewritten with expanded
and easier-to-read discussions, many more examples and problems, and
several new topics (including pressure vessels, discontinuity functions,
and inelastic buckling). Every effort has been made to eliminate errors,
but no doubt some are inevitable. If you find any, please jot them down
and mail them to the author (Department of Civil Engineering, Stanford
University, Stanford, CA 94305); then we can correct them egal
in the next printing of the book.
To acknowledge everyone who contributed to this book in some
manner is clearly impossible, but a major debt is owed my former Stan-
ford teachers (those giants of mechanics, including Timoshenko himself,
Wilhelm Fliigge, James Norman Goodier, Miklés Hetényi, Nicholas J.
Hoff, and Donovan H. Young) from whom I learned so much and my
current Stanford colleagues (especially Ed Kavazanjian, Tom Kane,
Anne Kiremidjian, Helmut Krawinkler, Jean Mayers, Cedric Richards,
Haresh Shah, and Bill Weaver) who made suggestions for the book and
provided cooperation during its writing. Several reviewers and friends
(including Jim Harp, Ian Johnston, Hugh Keedy, and Aron Zaslavsky)
provided valuable comments, and conscientious graduate students (Tha-
lia Anagnos, Joiéo Azevedo, Fouad Bendimerad, and Hassan Hadidi-
Tamjed) checked the proofs. The manuscript was carefully typed by
Susan Gere Durham, Janice Gere, Lu Ann Hall, and Laurie Yadon.
Editing and production were handled with great skill and a cooperative
spirit by Ray Kingman of Brooks/Cole and Mary Forkner of Publication
Alternatives, Palo Alto. My wife, Janice, offered encouragement and
exercised patience throughout this project. So also did other family
members—Susan and DeWitt Durham, Bill Gere, and David Gere. To all
of these wonderful peoric | am pleased to express my gratitude.

James M. Gere
 Contents

List of Symbols xiii

CHAPTER 1

Tension, Compression, and Shear

1.1. Introduction
1.2 Normal Stress and Strain
1.3. Stress-Strain Diagrams
1.4 Elasticity and Plasticity
1.5 Linear Elasticity and Hooke’s Law
1.6 Shear Stress and Strain
1.7 Allowable Stresses and Allowable Loads
Problems
CHAPTER 2

Axially Loaded Members

2.1. ‘Introduction
2.2 Deflections of Axially Loaded Members
2.3 Displacement Diagrams .
2.4 Statically Indeterminate Structures (Flexibility Method)
2.5 Statically Indeterminate Structures (Stiffness Method)
2.6 Temperature and Prestrain Effects
2.7 Stresses on Inclined Sections
2.8 Strain Energy

vil
Vili Contents

*2.9 Dynamic Loading 93

*2.10 Nonlinear Behavior ; 10
Problems 108

CHAPTE R 3

Torsion , 131
3.1. Introduction : 131
3.2 Torsion of Circular Bars - 131
3.3 Nonuniform Torsion — 138
3.4 Pure Shear 141
3.5 Relationship Between Moduli of Elasticity E and G 146
3.6 Transmission of Power by Circular Shafts 148
3.7. Statically Indeterminate Torsional Members eddt
3.8 Strain Energy in Pure Shear and Torsion 155
3.9 Thin-Walled Tubes 160
*3.10 Nonlinear Torsion of Circular Bars 167
Problems 170
CHAPTER 4

Shear Force and Bending Moment 181

4.1 Types of Beams | 181
4.2 Shear Force and Bending Moment _ . 184
4.3 Relationships Between Load, Shear Force, and Bending Moment 188
4.4 Shear-Force and Bending-Moment Diagrams ; 192
Problems 199
CHAPTER 5

Stresses in Beams - ) 205

5.1. Introduction 205
5.2. Normal Strains in Beams 207
5.3 Normal Stresses in Beams 212
5.4 Cross-Sectional Shapes of Beams 220
5.5 Shear Stresses in Rectangular Beams 226
5.6 Shear Stresses in the Webs of Beams with Flanges 25
*5.7 — Shear Stresses in Circular Beams 236
5.8 Built-Up Beams 238

*An asterisk denotes a difficult or advanced section, example, or problem.

 Contents ix

*5.9 Stresses in Nonprismatic Beams 24,

*5.10 Composite Beams 249
5.11. Beams with Axial Loads 250
Problems 262

CHAPTER 6

Analysis of Stress and Strain 279

6.1 Introduction 279
6.2 Plane Stress 280
6.3. Principal Stresses and Maximum Shear Stresses 286
6.4 Mohr’s Circle for Plane Stress 294
6.5 Hooke’s Law for Plane Stress 303
6.6 Spherical and Cylindrical Pressure Vessels (Biaxial Stress) 306
6.7. Combined Loadings (Plane Stress) 314
6.8 Principal Stresses in Beams 316
6.9 Triaxial Stress 318
*6.10 Three-Dimensional Stress 323
6.11. Plane Strain 326
Problems 338

CHAPTER 7 '

Deflections of Beams 351

7.1. Introduction 35]
7.2. Differential Equations of the Deflection Curve 351
7.3 Deflections by Integration of the Bending-Moment Equation 355
7.4 Deflections by Integration of the Shear-Force and Load Equations 361
7.5 Moment-Area Method 365
7.6 Method of Superposition 377
7.7. Nonprismatic Beams 381
7.8 Strain Energy of Bending 384
*7.9 Discontinuity Functions 389
*7.10 Use of Discontinuity Functions to Obtain Beam Deflections 399
*7.11 Temperature Effects 405
*7.12 Effects of Shear Deformations\ 407
2 *7.13 Large Deflections of Beams’ | 414-
Problems 418
X Contents

CHAPTER 8

Statically Indeterminate Beams 429

8.1 Statically Indeterminate Beams 429
8.2 Analysis by the Differential Equations of the Deflection Curve 43]
8.3. Moment-Area Method ; 434
8.4 Method of Superposition (Flexibility Method) _ 439
8.5 Continuous Beams 447
*8.6 Temperature Effects 455
*8.7_ Horizontal Displacements at the Ends of a Beam 457
Problems 459

CHAPTER 9

Unsymmetric Bending 469

9.1 Introduction 469
9.2 Doubly Symmetric Beams with Skew Loads 470
9.3 Pure Bending of Unsymmetric Beams - " 474
9.4 Generalized Theory of Pure Bending 481
9.5 Bending of Beams by Lateral Loads; Shear Center 486
9.6 Shear Stresses in Beams of Thin-Walled Open Cross Sections 490
9.7 Shear Centers of Thin-Walled Open Sections 496
*9.8 General Theory for Shear Stresses . 501
Problems 507

CHAPTER 10

Inelastic Bending 515

10.1. Introduction 515
10.2 Equations of Inelastic Bending ‘D 516
10.3. Plastic Bending only
10.4 Plastic Hinges 522
10.5 Plastic Analysis of Beams 524
*10.6 Deflections 552
*10.7 Inelastic Bending 535
*10.8 Residual Stresses 541
Problems 542
 Contents xi

CHAPTER 11

Columns 551
11.1 Buckling and Stability aye!
11.2 Columns with Pinned Ends 553
11.3 - Columns with Other Support Conditions 560 ©
11.4 Columns wit’. Eccentric Axial Loads 567
. 11.5 Secant Formula 569
*11.6 Imperfections in Columns 574
11.7 Elastic and Inelastic Column Behavior ~ 576
*11.8 Inelastic Buckling 578
11.9 - Column Design Formulas 583
Problems 589

CHAPTER 12

Energy Methods 597

12.1 Introduction 597
12.2 Principle of Virtual Work 597
12.3 Unit-Load Method for Calculating Displacements 602
12.4 Reciprocal Theorems 617
12.5 Strain-Energy and Complementary Energy 623
12.6 Strain-Energy Methods 635
12.7 Complementary Energy Methods 645
12.8 Castigliano’s Second Theorem 655
*12.9 Shear Deflections of Beams 660
Problems 666

References and Histo;ical Notes 675

APPENDIX A_ Systems of Units 687

A.1 Introduction 687

A.2 SI Units 688
A.3 U.S. Customary Units 689
A.4 Conversions 693
APPENDIX B_ Significant Digits 697
B.1 Significant Digits 697
B.2 Rounding off Numbers . 699
xii Contents

APPENDIX C_ Centroids and Moments of Inertia of Plane Areas 700

C.1 Centroids of Areas 700
C.2 Centroids of Composite Areas 702
C.3 Moments of Inertia of Areas 704
C.4 Parallel-Axis Theorem for Moments of Inertia 707
C.5 Polar Moments of Inertia 710
C.6 Products of Inertia TZ
C.7 Rotation of Axes 715
C.8 Principal Axes 716
Problems 720
APPENDIX Properties of Plane Areas 724
APPENDIX Properties of Selected Structural-Stee! Shapes 729
APPENDIX Section Properties of Structural Lumber . 735
APPENDIX Defiections and Slopes of Beams 736
APPENDIX nm Mechanical
oO
zo Properties of Materials 742

Answers to Selected Problems 748

Name Index 763:
Subject Index 764
 List of Symbols

area, action (force or couple), constant

= = dimensions, distances, constants
centroid, constant of integration, compressive force
distance from neutral axis to outer surface of a beam
displacement (translation or rotation)
diameter, .dimension, distance
modulus of elasticity, elliptic integral.of the second kind
reduced modulus of elasticity
tangent modulus of elasticity
>a eccentricity, dimension, distance, unit volume change (dilatation,
Ua
®m@M@ma
Qa
volumetric strain)
force, discontinuity function, elliptic integral of the first kind, flexibility
shear flow, shape factor for plastic bending, flexibility, frequency (Hz)
form factor for shear
modulus of elasticity in shear
acceleration of gravity
distance, force, reaction, horsepower
height, dimension
moment of inertia (or second moment) of a plane area
moments of inertia with respect to x, y, and z axes
product of inertia with respect to the x and y axes
polar moment of inertia
principal moments of inertia
torsion constant
bulk modulus of elasticity, effective length factor for a column
spring constant, stiffness, symbol for V P/EI
length, distance, span length
effective length of a column
bending moment, couple, mass
plastic moment for a. beam
yield moment for a beam
nwe
ne
SEO
sS moment per unit length, mass per unit length

xiii
XIV List of Symbols

axial force
factor of safety, number, ratio, integer, revolutions per minute (rpm)
origin of coordinates
center of curvature
force, concentrated load, axial force, power
allowable load (or working load)
critical load for a column
reduced-modulus load for a column
tangent-modulus load for a column
ultimate load r
yield load
pressure 3
force, concentrated. load, first moment (or static moment) of a plane area
intensitv of distributed load (load per unit distance),
intensity of distributed torque (torque per unit distance)
ultimate load intensity
yield load intensity
reaction, radius; force
radius, distance, radius.of gyration (r = VI/A)
= section
Amie modulus of the cross section of a beam, shear center, stiffness,
force
distance, length along a curved line
twisting couple or torque. temperature, tensile force
ultimate torque
yield torque
thickness, time
strain energy
strain energy density (strain energy per unit volume)
modulus of resilience
modulus of toughness
ee
eecomplementary energy
complementary energy density (complementary energy per unit volume)
shear force, volume
deflection of a beam, velocity
dv/dx, d*v/dx?, etc.
weight, work
complementary work
statical redundant
rectangular coordinates, distances
coordinates of centroid
plastic modulus of the cross section of a beam
 List ot Symbols XV

angle, coefficient of thermal expansion, nondimensional

ratio, spring constant, stiffness
as shear coefficient
angle, nondimensional ratio, spring constant, stiffness
y shear strain, specific weight (weight per unit volume)
Yxys Yyzs Vex shear strains in the xy, yz, and zx planes
Ye shear strain for inclined axes
Yxays shear strain in the x, y, plane
6,A deflection, displacement, elongation
normal strain
normal strains in the x, y, and 7 directions
normal strain for inclined axes
normal strains in the x, and y, directions
principal normal strains
yield strain
angle, angle of twist per unit length angle of rotation of beam axis
angle to a principal plane or to a principal axis
angle to a plane of maximum shear stress
curvature (« = 1/p)
yield curvature
distance
radius, radius of curvature, radial distance in polar coordinates, mass
density (mass per unit volume, specific mass)
Poisson’s ratio
normal stress
normal stresses on planes perpendicular to the x, y, and z axes
normal stress on inclined plane
normal stresses on planes perpendicular to the rotated x, y, axes
principal stresses
allowable stress (or working stress)
critical stress for a column (¢,, = P,,/A)
proportional limit stress
residual stress
ultimate stress
yield stress
shear stress
Uxys Ty29 Tzx shear stresses on planes perpendicular to the x, y, and z axes and parallel
to the y, z, and x axes
T9 shear stress on inclined plane
Tx
shear stress on piane perpendicular to the rotated x, axis and parallel to
the y,; axis
XVi List of Symbols

Tallow allowable stress (or working stress) in shear

a ultimate stress in shear
t, yield stress in shear”
g angle, angle of twist
“yw nondimensional ratio
@ angular velocity, angular frequency (w = 2zf)

e *An asterisk denotes a difficult or advanced section, example, or

problem.

Greek Alphabet

A «a Alpha Ney Ng
B Bp Beta Si Go Xi
IT y Gamma O o Omicron
A 6 Delta Tis, - uel
E e€ Epsilon P p_ Rho
aC -Leta x oa Sigma
He’ 7" Eta Tt “lau
© @ Theta Y v_ Upsilon
I 1: Iota ® @¢ Phi
K « Kappa X x Chi
A A Lambda | a
~M yp Mu Q mw Omega
 CHAPTER 1

Tension, Compression,
and Shear

1.1 INTRODUCTION
Mechanics of materials is a branch of applied mechanics that deals
with the behavior of solid bodies subjected to various types of load-
ing. This field of study is known by several names, including “strength
of materials” and “mechanics of deformable bodies.” The solid bodies
considered in this book include axially loaded members, shafts in tor-
sion, thin shells, beams, and columns, as well as structures that are as-
semblies of these components. Usually the objectives of our analysis
will be the determination of the stresses, strains, and deflections pro-
duced by the loads. If these quantities can be found for all values of
load up to the failure load, then we will have a complete picture of the
mechanical behavior of the body,
A thorough understanding of mechanical behavior is essential for
the safe design of all structures, whether buildings and bridges, machines
and motors, submarines and ships, or airplanes and antennas. Hence,
mechanics of materials is a basic subject in many engineering fields. Of
course, statics and dynamics are also essential, but they deal primarily
with the forces and motions associated with particles and rigid bodies.
In mechanics of materials, we go one step further by examining the
stresses and strains that occur inside real bodies that deform under
loads. We use the physical properties of the materials (obtained from
experiments) as well as numerous theoretical laws and Conorpts, which
are explained.in succeeding sections of this book.
Theoretical analyses and experimental results have equally impor-
tant roles in the study of mechanics of materiafs. On many occasions, we
will make logical derivations to obtain formulas and equations for pre-
dicting mechanical behavior, but_we must recognize that these formulas
cannot be used in a realistic way unless certain properties of the mate-
rials are known. These properties are available to us only after suitable
2 Chapter1 Tension, Compression, and Shear

experiments have been carried out in the laboratory. Also, because many
practical problems of great importance in engineering cannot be hanuled
efficiently by theoretical means, experimental measurements become a
necessity. :
The historical development of mechanics of materials is a fasci-
nating blend of both theory and experiment; experiments have pointed
the way to useful results in some instances, and theory has done so in
others. Such famous men as Leonardo da Vinci (1452-1519) and Galileo
Galilei (1564-1642) performed experiments to determine the strength
of wires, bars, and beams, although they did not develop any adequate
theories (by today’s standards) to explain their test results. Such theories
came much later. By contrast, the famous mathematician Leonhard
Euler (1707-1783) developed the mathematical theory of columns and
calculated the theoretical critical load of a column in 1744, long before
any experimental evidence existed to show the significance of his results.
Thus, for want of appropriate tests, Euler’s results remained unused for
many years, although today they form the basis of column theory.*
When studying mechanics of materials from this book,. you will
find that your efforts are divided naturally into two parts: first, under-
standing the logical development of the concepts, and second, applying
those concepts to practical situations. The former is accomplished by
studying the derivations, discussions, and examples, and the latter by
solving problems. Some of the examples and rvc ‘lems are numerical
in character, and others are algebraic (or symbolic). An advantage of
numerical problems is that the magnitudes of all quantities are evident
at every stage of the calculations. Sometimes these values are needed to
ensure that practical limits (such as allowable stresses) are not exceeded.
Algebraic solutions have certain advantages, too. Because they lead to
formulas, algebraic solutions make clear the variables that affect-he
final result. For instance, a certain quantity may actually cancel out of
the solution, a fact that would not be evident from a numerical problem
Also apparent in algebraic solutions is the manner in which: variahies-
affect the results; such as the appearance of one variable in the numera-
tor and another in the denominator. Furthermore, a symbolicsolution
provides the opportunity to check the dimensions at any stage of. the
work.’ Finally, the most important reason for obtaining an algebraic
solution is to obtain a general formula that can be programmed on a
‘computer and used for many different problems. In contrast, a numeri-
cal solution applies to only one set of circumstances. Of course, you
must be adept at both kinds of solutions, hence you will find a mixture
of numerical and algebraic problems throvciivut the book.
Numerical problems require that you work with specific units of
‘measurements. This book utilizes both the International System of Units
(SI) and the U.S. Customary System (USCS). A discussion of both of
* The history of mechanics of materials, beginning with Leonardo and Galileo, is
given in Refs. 1-1, 1-2, and 1-3.
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