Hearn E.J. Mechanics of Materials. Volume 1
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MECHANICS
OF
MATERIALS
I
An Introduction to the Mechanics of Elastic and
Plastic Deformation of Solids and Structural Materials
THIRD EDITION
E.
J.
HEARN
Ph.D., B.Sc. (Eng.) Hons., C.Eng., F.I.Mech.E., F.I.Prod.E., F.1.Diag.E.
University
of
Warwick
United Kingdom
EINEMANN
OXFORD
AUCKLAND
BOSTON JOHANNESBURG MELBOURNE
NEW
DELHI
Butterworth-Heinemann
Linacre House, Jordan Hill, Oxford OX2 8DP
225 Wildwood Avenue, Woburn, MA 01801-2041
A division
of
Reed Educational and Professional Publishing Ltd
-@A
member
of
the
Reed Elsevier
plc
group
First published 1977
Reprinted with corrections 1980, 1981, 1982
Second edition 1985
Reprinted with corrections 1988
Reprinted 1989, 1991, 1993, 1995, 1996
Third edition 1997
Reprinted 1998, 1999,2000
0
E.
J. Hearn 1977, 1985, 1997
All
rights reserved. No part of this publication
may be reproduced in any material form (including
photocopying or storing in any medium by electronic
means and whether or not transiently or incidentally to
some other use
of
this publication) without the written
permission of the copyright holder except in accordance
with the provisions of the Copyright, Designs and
Patents Act 1988 or under the terms
of
a licence issued
by the Copyright Licensing Agency Ltd, 90 Tottenham
Court Road, London, England
WIP
9HE.
Applications for the copyright holder's written
permission to reproduce any part of this publication
should be addressed
to the publishers
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN
0
7506 3265 8
Library
of
Congress Cataloguing in Publication Data
Hearn, E. J. (Edwin John)
Mechanics
of
materials
1:
an introduction to the mecahnics of elastic and plastic deformation of
solids and structural components/E. J. Hearn.
-
3rd ed.
p. cm.
Includes bibliographical references and index.
ISBN
0
7506 3265
8
1.
Strength
of
materials.
I.
Title
TA405.H3 96-49967
620.1'23-dc21 CIP
Printed and bound in Great Britain by Scotprint, Musselburgh
FOR
EVERY TITLGTHAT WE
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BU176RWORTHdlEINEMANN
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PAY
FUR
BTCV
TO
PLANT
AND
CARE
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TREE.
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INTRODUCTION
This text is the suitably revised and extended third edition of the highly successful text
initially published in
1977
and intended to cover the material normally contained in degree
and honours degree courses in mechanics of materials and in courses leading to exemption
from the academic requirements of the Engineering Council. It should also serve as a valuable
reference medium for industry and for post-graduate courses. Published in two volumes, the
text should also prove valuable for students studying mechanical science, stress analysis, solid
mechanics
or
similar modules
on
Higher Certificate and Higher Diploma courses in the
UK
or overseas and for appropriate NVQ* programmes.
The study of mechanics of materials is the study of the behaviour of solid bodies under
load. The way in which they react to applied forces, the deflections resulting and the stresses
and strains set up within the bodies, are all considered in an attempt to provide sufficient
knowledge to enable any component to be designed such that it will not fail within its service
life. Typical components considered in detail in this volume include beams, shafts, cylinders,
struts, diaphragms and springs and, in most simple loading cases, theoretical expressions are
derived to cover the mechanical behaviour of these components. Because of the reliance of
such expressions on certain basic assumptions, the text also includes a chapter devoted to the
important experimental stress and strain measurement techniques in use today with recom-
mendations for further reading.
Each chapter of the text contains a summary of essential formulae which are developed
within the chapter and a large number of worked examples. The examples have been selected
to provide progression in terms of complexity of problem and to illustrate the logical way in
which the solution to a difficult problem can be developed. Graphical solutions have been
introduced where appropriate. In order to provide clarity of working in the worked examples
there is inevitably more detailed explanation of individual steps than would be expected in the
model answer to an examination problem.
All chapters (with the exception of Chapter
16)
conclude with an extensive list of problems
for solution of students together with answers. These have been collected from various
sources and include questions from past examination papers in imperial units which have
been converted to the equivalent
SI
values. Each problem is graded according to its degree of
difficulty as follows:
A Relatively easy problem of an introductory nature.
A/B Generally suitable for first-year studies.
B
Generally suitable for second or third-year studies.
C More difficult problems generally suitable for third year studies.
*National Vocational Qualifications
xv
xvi
Introduction
Gratitude is expressed to the following examination boards, universities and colleges who
have kindly given permission for questions to be reproduced:
City University
East Midland Educational Union
Engineering Institutions Examination
Institution of Mechanical Engineers
Institution
of
Structural Engineers
Union of Educational Institutions
Union of Lancashire and Cheshire Institues
University of Birmingham
University of London
C.U.
E.M.E.U.
E.I.E. and C.E.I.
1.Mech.E.
1.Struct.E.
U.E.I.
U.L.C.I.
U.Birm.
U.L.
Both volumes
of
the text together contain 150 worked examples and more than
500
problems for solution, and whilst it is hoped that no errors are present it is perhaps inevitable
that some errors will be detected. In this event any comment, criticism
or
correction will be
gratefully acknowledged.
The symbols and abbreviations throughout the text are in accordance with the latest
recommendations
of
BS 1991 and PD
5686t.
As mentioned above, graphical methods
of
solution have been introduced where appro-
priate since it is the author’s experience that these are more readily accepted and understood
by students than some
of
the more involved analytical procedures; substantial time saving
can also result. Extensive use has also been made
of
diagrams throughout the text since in the
words of the old adage “a single diagram is worth 1000 words”.
Finally, the author is indebted to all those who have assisted in the production of this
volume; to Professor H.
G.
Hopkins, Mr R. Brettell, Mr R.
J.
Phelps for their work asso-
ciated with the first edition and to Dr
A.
S.
Tooth’, Dr N. Walke?, Mr R. Winters2 for their
contributions to the second edition and to Dr M. Daniels for the extended treatment
of
the
Finite Element Method which is the major change in this third edition. Thanks also go to the
publishers for their advice and assistance, especially in the preparation
of
the diagrams and
editing, to Dr.
C.
C. Perry (USA) for his most valuable critique of the first edition, and to Mrs
J.
Beard and Miss
S.
Benzing for typing the manuscript.
E.
J.
HEARN
t
Relevant Standards for
use
in Great Britain: BS 1991; PD 5686 Other useful
SI
Guides:
The Infernational
System
of
Units,
N.P.L. Ministry
of
Technology,
H.M.S.O.
(Britain). Mechty,
The International System
of
Units
(Physical Constants and Conversion Factors),
NASA, No SP-7012, 3rd edn. 1973 (U.S.A.)
Metric Practice
Guide,
A.S.T.M. Standard E380-72
(U.S.A.).
1.
$23.27.
2.
$26.
3.
$24.4
Dr. A.
S.
Tooth, University
of
Strathclyde, Glasgow.
D.
N.
Walker and Mr.
R.
Winters, City
of
Birmingham Polytechnic.
Dr M.
M.
Daniels, University of Central England.
NOTATION
Quantity
Angle
Length
Area
Volume
Time
Angular velocity
Velocity
Weight
Mass
Density
Force
Moment
Pressure
Stress
Strain
Shear stress
Shear strain
Young's modulus
Shear modulus
Bulk modulus
Poisson's ratio
Modular ratio
Power
Coefficient
of
linear expansion
Coefficient
of
friction
Second moment
of
area
Polar moment
of
area
Product moment
of
area
Temperature
Direction cosines
Principal stresses
Principal strains
Maximum shear stress
Octahedral stress
A
V
t
0
V
W
m
P
F
or
P
or
W
M
P
0
E
z
Y
E
G
K
m
V
Si
Unit
rad (radian)
m (metre)
mm (millimetre)
mz
m3
s
(second)
rad/s
m/s
N (newton)
kg (kilogram)
kg/m3
N
Nm
Pa
(Pascal)
N/m2
bar
(
=
lo5
N/m2)
N/m2
N/m2
N/m2
N/m2
N/m2
-
-
-
-
W (watt)
m/m
"C
m4
m4
m4
"C
N/m2
N/mz
N/mZ
-
-
-
xvii
xviii
Notation
Quantity Symbol
Deviatoric stress
Deviatoric strain
Hydrostatic or mean stress
Volumetric strain
Stress concentration factor
Strain energy
Displacement
Deflection
Radius of curvature
Photoelastic material fringe value
Number of fringes
Body
force stress
Radius of gyration
Slenderness ratio
Gravitational acceleration
Cartesian coordinates
Cylindrical coordinates
Eccentricity
Number of coils or leaves of spring
Equivalent
J
or effective polar moment
of area
Autofrettage pressure
PA
Radius
of
elastic-plastic interface
RP
Thick cylinder radius ratio
R2/R1
K
m
Ratio elastic-plastic interface radius to
internal radius of thick cylinder
R,/R1
Resultant stress on oblique plane
Normal stress on oblique plane
Shear stress on oblique plane
Direction cosines of plane
Direction cosines of line of action of
resultant stress
Direction cosines of line of action of
shear stress
Components of resultant stress on
oblique plane
Shear stress in any direction
4
on
oblique plane
Invariants of stress
Invariants of reduced stresses
Airy stress function
SI
Unit
N/m2
N/mz
-
-
-
J
m
m
m
N/m2/fringe/m
N/m3
-
m4
N/m2 or bar
m
-
N/m2
N/m2
N/m2
-
N/m2
Notation
xix
Quantity
‘Operator’ for Airy stress function
biharmonic equation
Strain rate
Coefficient of viscosity
Retardation time (creep strain recovery)
Relaxation time (creep stress relaxation)
Creep contraction
or
lateral strain ratio
Maximum contact pressure (Hertz)
Contact formulae constant
Contact area semi-axes
Maximum contact stress
Spur gear contact formula constant
Helical gear profile contact ratio
Elastic stress concentration factor
Fatigue stress concentration factor
Plastic flow stress concentration factor
Shear stress concentration factor
Endurance limit for
n
cycles of load
Notch sensitivity factor
Fatigue notch factor
Strain concentration factor
Griffith‘s critical strain energy release
Surface energy of crack face
Plate thickness
Strain energy
Compliance
Fracture stress
Stress Intensity Factor
Compliance function
Plastic zone dimension
Critical stress intensity factor
“J”
Integral
Fatigue crack dimension
Coefficients of Paris Erdogan law
Fatigue stress range
Fatigue mean stress
Fatigue stress amplitude
Fatigue stress ratio
Cycles to failure
Fatigue strength for
N
cycles
Tensile strength
Factor of safety
SI
Unit
S
S
-
N/mz
(N/m2)-
m
N/mZ
N/mZ
Nm
m
Nm
mN-’
N/m2
N/m3I2
m
N/m3I2
m
N/m2
N/m2
N/m2
-
-
N/m2
N/m2
-
xx
No
tu
t
ion
Quantity
Elastic strain range
Plastic strain range
Total strain range
Ductility
Secondary creep rate
Activation energy
Universal Gas Constant
Absolute temperature
Arrhenius equation constant
Larson-Miller creep parameter
Sherby-Dorn creep parameter
Manson-Haford creep parameter
Initial stress
Time to rupture
Constants
of
power law equation
Symbol
SI
Unit
-
N/m2
S
-
CONTENTS
Introduction
Notation
xv
XVii
1
Simple
Stress
and Strain
1.1
Load
1.2
Direct or normal stress
(a)
1.3
Direct strain
(E
)
1.4
Sign convention for direct stress and strain
1.5
Elastic materials
-
Hooke’s law
1.6
Modulus of elasticity
-
Young’s modulus
1.7
Tensile test
1.8
Ductile materials
1.9
Brittle materials
1.10
Poisson’s
ratio
1.1
1
Application of
Poisson’s
ratio to a two-dimensional stress system
1.12
Shear stress
1.1 3
Shear strain
1.14
Modulus of rigidity
1,15
Double shear
1.16
Allowable working stress -factor
of
safety
1.17
Load
factor
1.18
Temperature stresses
1.19
Stress concentrations -stress concentration factor
1.20
Toughness
1.21
Creep and fatigue
Examples
Problems
Bibliography
2
Compound Bars
Summary
2.1
Compound bars subjected to external load
V
1
1
2
2
2
3
3
4
8
8
9
10
11
11
12
12
12
13
13
14
14
15
17
25
26
27
27
28