Essential Solid Mechanics: Theory, worked examples and problems

257 Pages·1976·14.738 MB·English

by  B. W. Young (auth.)

ESSENTIAL
SOLID MECHANICS
Theory, worked examples
and problems
B.W. Young
Lecturer in Structural
and Mechanical Engineering
University of Sussex
M
©
B. W. Young 1976
All rights reserved. No part of this publication may be reproduced
or transmitted, in any form or by any means, without permission.
First published 1976 by
THE MACMILLAN PRESS LTD
London and Basingstoke
Associated companies in New York Dublin
Melbourne Johannesburg and Madras
SBN 333 16694 9
ISBN 978-1-349-02261-8 ISBN 978-1-349-02259-5 (eBook)
DOI 10.1007/978-1-349-02259-5
To my father
This book is sold subject to the standard
conditions of the Net Book Agreement.
The paperback edition of this book is sold subject to the condition
that it shall not, by way of trade or otherwise, be lent, re-sold,
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subsequent purchaser.
ii
CONTENTS
Preface vi
1. FUNDAMENTALS OF EQUILIBRIUM 1
1.1 The Equations of Statical Equilibrium 1
1.2 Statical Determinacy of Pin-jointed Frames 4
1.3 Force Analysis of Pin-jointed Plane Frames 6
1.4 Force Analysis of Pin-jointed Space Frames 11
1.5 Shear Force and Bending Moment 12
1.6 Shear Force and Bending Moment Diagrams 14
1.7 Relations between Load, Shear and Bending Moment 17
1.8 Influence Lines for Shear Force and Bending Moment 20
1.9 The Three-pinned Arch 27
1.10 Suspension Cables 30
1.11 Problems for Solution 37
42
2. THE STRESS-STRAIN RELATIONSHIP
2.1 Normal Stress and Strain 42
2.2 The Stress-Strain Relationship 43
2.3 Poisson's Ratio 48
2.4 Thin Cylinders 49
2.5 Thin Spheres 53
2.6 Bulk Modulus 55
2.7 Statically Indeterminate Systems 57
2.8 Thermal Effects 61
2.9 Problems for Solution 68
3. TORSION 72
3.1 Shear Stress 72
3.2 Complementary Shear Stress 72
3.3 Shear Strain 73
3.4 Torsion of a Solid Circular Shaft 74
3.5 Torsion of a Hollow Circular Shaft 77
3.6 Power Transmission 79
3.7 Torque and Angle-of-Twist Diagrams 79
3.8 The T/GJ Diagram 81
iii
3.9 Problems for Solution 87
4. BENDING 89
4.1 The Simple Theory of Bending 89
4.2 Composite Beams 97
4.3 Combined Bending and Direct Stress 105
4.4 Bending of Unsymmetrical Beams 111
4.5 Problems for Solution 118
5. DEFLEXION OF BEAMS 121
5.1 The Deflexion Equation 121
5.2 Superposition 125
5.3 Pure Bending 126
5.4 Bending Moments having a Discontinuous First Derivative -
the Unit Function 128
5.5 Macaulay's Method 130
5.6 Mohr's Theorems - The Moment-Area Method 133
5.7 Force Analysis of Statically Indeterminate Beams 139
5.8 Problems for Solution 144
6. STRAIN ENERGY 147
6.1 The Basic Energy Theorems 147
6.2 Expressions for Strain Energy 149
6.3 Torsion of Thin-walled Tubes 151
6.4 Direct Application of Strain Energy 157
6.5 Dynamic Loading 160
6.6 Applications of Castigliano's Second Theorem to the
Determination of Deflexions 163
6.7 Beams Curved in Plan 167
6.8 Problems for Solution 168
7. BIAXIAL STRESS AND STRAIN 172
7.1 Mohr's Circle for Stress 172
7.2 Maximum Shear Stress under Plane Stress Conditions 174
7.3 Torsion Combined with Direct Stress 177
7.4 Mohr's Circle for Strain 180
7.5 A Relationship between E, G and v 186
7.6 Strain Energy under Plane Stress Conditions 188
7.7 Theories of Elastic Failure 191
7.8 Problems for Solution 196
8. SHEAR EFFECTS IN BEAMS 200
iv
8.1 The Distribution of Shear Stress in Beams 200
8.2 Shear Flow in Thin-walled Open Sections 207
8.3 Problems for Solution 210
9. THICK CYLINDERS 212
9.1 Lame's Theory 212
9.2 Graphical Representation of Lam6's Equations - Lam6's Line 215
9.3 Strains in Thick Cylinders 216
9.4 Force Fits 219
9.5 Compound Cylinders 222
9.6 Problems for Solution 226
10. COLUMNS 228
10.1 The Euler Column 228
10.2 Real Columns 233
10.3 Beam-Columns 238
10.4 Problems for Solution 241
Appendix 244
Further Reading 246
Index 247
v
PREFACE
The analysis of force, stress and deformation in engineering com
ponents is traditionally covered in the study of strength of mater
ials and theory of structures. However, there is so much common
ground between these two disciplines, particularly in the early
stages, that it seems much more appropriate to treat them as the
single subject referred to here as Solid Mechanics.
The purpose of this book is to establish, in concise form, the
bases of solid mechanics required by mechanical, civil and struct
ural engineering undergraduates in the first half of a university
or polytechnic degree course. The format consists of the elements
of the theory for a particular topic followed by a number of worked
examples illustrating the application of the theory. Each chapter
ends with a selection of problems (with answers) which the student
can use for practice. The examples and problems are typical of
those set to first and second year undergraduates and have been
collected and adapted from a large number of sources during many
years of teaching in the field. The precise origin of the questions
is unknown but a general acknowledgement is given here. The author
alone is responsible for the solutions and answers. There are a
total of over two hundred problems in the book of which about half
have fully worked solutions.
This book presents all the necessary theory in a compact form
and for reasons of space the broader background to the subject has
obviously had to be omitted. The student is well advised to extend
his knowledge by further reading in the library and .a short list of
titles recommended for this purpose will be found at the end of the
book.
B. W. YOUNG
University of Sussex,
1976
vi
FUNDAMENTALS OF EQUILIBRIUM
Before a machine part or a structural member can be put to its
required use, the designer has to satisfy himself that it is strong
and stiff enough to withstand the loads it is likely to meet during
its lifetime.
The purpose of solid mechanics is to describe the way in which
applied forces are distributed in a component or structure and to
determine the resulting deformations. With this information the
designer is able to decide the correct geometry for cross-sections
and to select suitable materials.
Forces may be statically or dynamically applied. We shall be con
cerned almost exclusively with static forces in this book although
mention will be made of the effects of suddenly applied loads. The
study of dynamic loading in so far as it is associated with time
dependent phenomena requires a separate text of its own.
The techniques and principles we shall be exam1n1ng are applicable
to all materials provided their load-deformation characteristics are
known. However, the subject matter of this chapter is quite in
dependent of material behaviour. We shall be concerned here with the
force analysis of statically determinate systems. In later chapters
the relationships between forces and deformations will be investi
gated so that the force analysis of statically indeterminate systems
and the deformation analysis of both statically determinate and in
determinate systems will be possible. The concept of statical de
terminacy, which may be unfamiliar, is explained in the following
sections.
1.1 THE EQUATIONS OF STATICAL EQUILIBRIUM
We shall make a start by looking at the requirements for equilibrium
of a body subjected to a general load system consisting of forces and
moments.
The concept of force is readily understood. We recognise that a
force has magnitude, direction and a definite line of action. The
concept of moment is less obvious. The moment of a force about a
point is defined simply as the product of the force and the perpen
dicular distance of its line of action from that point.
An unrestrained body that is acted upon by a system of forces and
moments will move. In the three spatial dimensions this movement
will consist in general of three components of translation along
three axes mutually at right angles and three components of rotation
about these axes. The body therefore has six degrees of freedom of
movement. If we confine our attention to two dimensions, the situa
tion is simpler since the body now has three degrees of freedom: two
1
components of translation along two axes at right angles and one
rotation in the plane.
An obvious requirement for a satisfactory structure is that it
should be supported in such a way that movement as a whole under the
action of applied forces is prevented.
The three basic types of structural support are shown in figure
1.1. The roller support in figure l.la allows the body to rotate
about a pin or hinge (shown as a small circle) and to move horizon
tally. Vertical movement is prevented by the reaction Rv· The
pinned support in figure l.lb allows the body to rotate but the
horizontal and vertical components of translation are prevented by
the two independent reactions Rv and Rh. The three independent re
actions (two forces, Rh and Rv and the moment M) provided by the
built-in support shown in figure l.lc, prevent rotation of the body
and translation in the vertical and horizontal directions. To res
train a body completely in two dimensions we therefore require a
minimum of three independent reactions which may consist of two
forces at right angles and a moment, or three forces that combine to
give the same effect as two forces and a moment.
Figure 1.1
The body is said to be in statical equilibrium if the applied
forces and the reactions balance each other. This situation is
satisfied for a two-dimensional body if
(i) the sum of the forces in the x-direction is zero
(ii) the sum of the forces in the y-direction (at right angles to
the x-direction) is zero, and
(iii) the sum of all the moments about any point in the xy-plane is
zero.
These three conditions give rise to three equations of statical equi
librium.
If a body is supported in such a way that three independent react
ions are required, it is said to be statically determinate with res
pect to the supports since the three equations of statical equilib
rium are ·sufficient to determine the magnitudes of the reactions. If
more than three independent reactions are present their values cannot
be found solely from the three equations and the body is then said to
be statically indeterminate with respect to the supports.
The following example illustrates the application of the equilib-
2
riurn equations to the determination of reactive forces in two dimen
sions for a statically determinate body.
Example 1.1
Figure 1.2 shows a box-girder bridge section being gradually raised
into position using two lifting-cables attached at corners A and B
and a tethering cable attached at corner C. The purpose of the
tethering cable is to maintain the box in a horizontal position.
~e box is 10 m long, 2 m deep and the centre of mass is at 7 m from
the left-hand end. The weight of the box is 200 kN. Determine the
cable tensions when the cables attached at A, B and C make angles
with the horizontal of 60°, 30° and 45° respectively.
Wm
7m
B
2m
D
200kN
Figure 1.2
(i) Summing horizontal forces
pl cos 60° + p3 cos 45° - p2 cos 30° 0
therefore
(1)
(ii) Summing vertical forces
P1 sin 60° + P2 sin 30° - P3 sin 45° - 200 0
therefore
13Pl + P2 - I2P3 - 400 = 0 (2)
(iii) Summing moments about A
(200) (7) - lOP2 sin 30° + 2P3 cos 45° 0
therefore
1400 - 5P2 + I2P3 = 0 (3)
Eliminating P1 from equations 1 and 2 we have
400 - 4P2 + 12(1 + 13)P3 0 (4)
Solving equations 3 and 4 simultaneously gives
3