Mechanics and Strength of Materials
Vitor Dias da Silva
Mechanics and Strength
of Materials
ABC
Vitor Dias da Silva
Department of Civil Engineering
Faculty of Science & Technology
University of Coimbra
Polo II da Universidade - Pinhal de Marrocos
3030-290 Coimbra
Portugal
E-mail:
Library of Congress Control Number: 2005932746
ISBN-10 3-540-25131-6 Springer Berlin Heidelberg New York
ISBN-13 978-3-540-25131-6 Springer Berlin Heidelberg New York
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neering who have used the first two Portuguese editions for their comments
about the text and for their help in the detection of misprints. This has greatly
contributed to improving the quality and the precision of the explanations.
The author also thanks Springer-Verlag for agreeing to publish this book
and also for their kind cooperation in the whole publishing process.
Coimbra V. Dias da Silva
March 2005
Preface to the First Portuguese Edition
The motivation for writing this book came from an awareness of the lack of
a treatise, written in European Portuguese, which contains the theoretical
material taught in the disciplines of the Mechanics of Solid Materials and
the Strength of Materials, and explained with a degree of depth appropriate
to Engineering courses in Portuguese universities, with special reference to
the University of Coimbra. In fact, this book is the result of the theoretical
texts and exercises prepared and improved on by the author between 1989-94,
for the disciplines of Applied Mechanics II (Introduction to the Mechanics of
Materials) and Strength of Materials, taught by the author in the Civil Engi-
neering course and also in the Geological Engineering, Materials Engineering
and Architecture courses at the University of Coimbra.
A physical approach has been favoured when explaining topics, sometimes
rejecting the more elaborate mathematical formulations, since the physical
understanding of the phenomena is of crucial importance for the student of
Engineering. In fact, in this way, we are able to develop in future Engineers
the intuition which will allow them, in their professional activity, to recognize
the difference between a bad and a good structural solution more readily and
rapidly.
The book is divided into two parts. In the first one the Mechanics of
Materials is introduced on the basis of Continuum Mechanics, while the second
one deals with basic concepts about the behaviour of materials and structures,
as well as the Theory of Slender Members, in the form which is usually called

finitesimal rotations when the usual methods are applied to problems where
the rotations are not small, may be mentioned. The comparison of the usual
methods for computing the deflections caused by the shear force, clarifying
some confusion in the traditional literature about the way as this deformation
should be computed, is another example. Chapter twelve contains theorems
about the energy associated with the deformation of solid bodies with appli-
cations to framed structures. This chapter includes a physical demonstration
of the theorems of virtual displacements and virtual forces, based on con-
siderations of energy conservation, instead of these theorems being presented
without demonstration, as is usual in books on the Strength of Materials and
Structural Analysis, or else with a lengthy mathematical demonstration.
Although this book is the result of the author working practically alone,
including the typesetting and the pictures (which were drawn using a self-
developed computer program), the author must nevertheless acknowledge the
important contribution of his former students of Strength of Materials for
their help in identifying parts in the texts that preceded this treatise that
were not as clear as they might be, allowing their gradual improvement. The
author must also thank Rui Cardoso for his meticulous work on the search for
misprints and for the resolution of proposed exercises, and other colleagues,
especially Rog´erio Martins of the University of Porto, for their comments
on the preceding texts and for their encouragement for the laborious task of
writing a technical book.
This book is also a belated tribute to the great Engineer and designer of
large dams, Professor Joaquim Laginha Serafim, who the Civil Engineering
Department of the University of Coimbra had the honour to have as Professor
Preface to the First Portuguese Edition IX
of Strength of Materials. It is to him that the author owes the first and most
determined encouragement for the preparation of a book on this subject.
Coimbra V. Dias da Silva
July 1995

III The Strain Tensor 41
III.1 Introduction 41
III.2 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
III.3 Components of the Strain Tensor . . . . . . . . . . . . . . . . . . . . . . . . 44
III.4 Pure Deformation and RigidBodyMotion 49
III.5 Equations of Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
III.6 Deformation in an Arbitrary Direction . . . . . . . . . . . . . . . . . . . 54
III.7 VolumetricStrain 58
III.8 Two-Dimensional Analysis of the Strain Tensor . . . . . . . . . . . 59
III.8.a Introduction 59
III.8.b Components of the Strain Tensor. . . . . . . . . . . . . . . 60
III.8.c Strain in an Arbitrary Direction . . . . . . . . . . . . . . . 60
III.9 Conclusions 63
III.10 Examplesand Exercises 64
IV Constitutive Law 67
IV.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
IV.2 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
IV.3 Ideal Rheological Behaviour – Physical Models . . . . . . . . . . . . 69
IV.4 Generalized Hooke’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
IV.4.a Introduction 75
IV.4.b IsotropicMaterials 75
IV.4.c Monotropic Materials 80
IV.4.d Orthotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . 82
IV.4.e Isotropic Material with Linear Visco-Elastic
Behaviour 83
IV.5 Newtonian Liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
IV.6 DeformationEnergy 86
IV.6.a General Considerations . . . . . . . . . . . . . . . . . . . . . . . 86
IV.6.b Superposition of Deformation Energy
intheLinearElasticCase 89

V.9.c Probabilistic Approach. . . . . . . . . . . . . . . . . . . . . . . . 134
V.9.d Semi-Probabilistic Approach . . . . . . . . . . . . . . . . . . . 135
V.9.e Safety Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
V.10 Slender Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
V.10.a Introduction 137
V.10.b Definition of Slender Member . . . . . . . . . . . . . . . . . . 138
V.10.c Conservation ofPlaneSections 138
VI Axially Loaded Members 141
VI.1 Introduction 141
VI.2 Dimensioning of Members Under Axial Loading . . . . . . . . . . . 142
VI.3 Axial Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
VI.4 Statically Indeterminate Structures 143
VI.4.a Introduction 143
VI.4.b Computationof InternalForces 144
VI.4.c Elasto-PlasticAnalysis 145
VI.5 An Introduction to the Prestressing Technique . . . . . . . . . . . . 150
VI.6 CompositeMembers 153
VI.6.a Introduction 153
VI.6.b Position of the Stress Resultant . . . . . . . . . . . . . . . . 153
VI.6.c Stresses and Strains Caused by the Axial Force . . 154
VI.6.d Effects of Temperature Variations . . . . . . . . . . . . . . 155
VI.7 Non-PrismaticMembers 157
VI.7.a Introduction 157
VI.7.b Slender Members with Curved Axis . . . . . . . . . . . . 157
VI.7.c Slender Members with Variable Cross-Section . . . . 159
VI.8 Non-Constant Axial Force – Self-Weight . . . . . . . . . . . . . . . . . . 160
XIV Contents
VI.9 Stress Concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
VI.10 Examplesand Exercises 163
VII Bending Moment 189

VIII.3.e CompositeMembers 268
VIII.3.f Non-PrincipalReferenceAxes 269
VIII.4 TheShearCentre 270
VIII.5 Non-PrismaticMembers 273
VIII.5.a Introduction 273
VIII.5.b Slender Members with Curved Axis . . . . . . . . . . . . 273
VIII.5.c Slender Members with Variable Cross-Section . . . . 274
VIII.6 Influence of a Non-Constant Shear Force . . . . . . . . . . . . . . . . . 275
VIII.7 Stress State in Slender Members . . . . . . . . . . . . . . . . . . . . . . . . . 276
VIII.8 ExamplesandExercises 278
Contents XV
IX Bending Deflections 297
IX.1 Deflections Caused by the Bending Moment . . . . . . . . . . . . . . 297
IX.1.a Introduction 297
IX.1.b Method of Integration of the Curvature Equation 298
IX.1.c The ConjugateBeamMethod 302
IX.1.d Moment-AreaMethod 304
IX.2 Deflections Caused by the Shear Force . . . . . . . . . . . . . . . . . . . 308
IX.2.a Introduction 308
IX.2.b RectangularCross-Sections 311
IX.2.c Symmetrical Cross-Sections . . . . . . . . . . . . . . . . . . . 312
IX.2.d Thin-WalledCross-Sections 312
IX.3 Statically Indeterminate Frames Under Bending . . . . . . . . . . . 315
IX.3.a Introduction 315
IX.3.b Equation of Two Moments . . . . . . . . . . . . . . . . . . . . 317
IX.3.c Equation of Three Moments . . . . . . . . . . . . . . . . . . . 317
IX.4 Elasto-Plastic Analysis Under Bending . . . . . . . . . . . . . . . . . . . 320
IX.5 ExamplesandExercises 323
X Torsion 347
X.1 Introduction 347

XI.4.b Safety Evaluation 414
XI.4.c Composed Bending with a Tensile Axial Force . . . 416
XI.5 ExamplesandExercises 416
XI.6 Stability Analysis by the Displacement Method . . . . . . . . . . . 439
XI.6.a Introduction 439
XI.6.b SimpleExamples 440
XI.6.c Framed Structures Under Bending . . . . . . . . . . . . . 445
XI.6.c.i Stiffness Matrix of a Compressed Bar . . . 445
XI.6.c.ii Stiffness Matrix of a Tensioned Bar . . . . . 451
XI.6.c.iiiLinearization of the Stiffness Coefficients 452
XI.6.c.ivExamples of Application . . . . . . . . . . . . . . . 455
XII Energy Theorems 465
XII.1 GeneralConsiderations 465
XII.2 Elastic Potential Energy in Slender Members . . . . . . . . . . . . . . 466
XII.3 Theorems for Structures with Linear Elastic Behaviour . . . . . 468
XII.3.a Clapeyron’s Theorem 468
XII.3.b Castigliano’sTheorem 469
XII.3.c Menabrea’s Theorem or Minimum Energy
Theorem 473
XII.3.d Betti’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
XII.3.e Maxwell’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
XII.4 Theorems of Virtual Displacements and Virtual Forces . . . . . 479
XII.4.a Theorem of Virtual Displacements. . . . . . . . . . . . . . 479
XII.4.b Theorem of Virtual Forces. . . . . . . . . . . . . . . . . . . . . 482
XII.5 Considerations About the Total Potential Energy . . . . . . . . . . 485
XII.5.a Definition of Total Potential Energy . . . . . . . . . . . . 485
XII.5.b Principle of Stationarity of the Potential Energy . 486
XII.5.c Stability of the Equilibrium . . . . . . . . . . . . . . . . . . . 486
XII.6 ElementaryAnalysisofImpact Loads 489
XII.7 ExamplesandExercises 491

proach.
From these considerations we conclude that, in Mechanics of Materials,
a phenomenological approach must almost always be used to quantify the
rheological behaviour of a solid, a liquid or a gas. Furthermore, as the consid-
eration of the discontinuities that are always present in the internal structure
of materials (for example the interface between two crystals or two granules,
micro-cracks, etc.), substantially increases the degree of complexity of the
problem, we assume, whenever possible, that the material is continuous.
4 I Introduction
From a mathematical point of view, the hypothesis of continuity may be
expressed by stating that the functions which describe the forces inside the
material, the displacements, the deformations, etc., are continuous functions
of space and time.
From a physical point of view, this hypothesis corresponds to assuming
that the macroscopically observed material behaviour does not change with
the dimensions of the piece of material considered, especially when they tend
to zero. This is equivalent to accepting that the material is a mass of points
with zero dimensions and all with the same properties.
The validity of this hypothesis is fundamentally related to the size of the
smallest geometrical dimension that must be analysed, as compared with the
maximum dimension of the discontinuities actually present in the material.
Thus, in a liquid, the maximum dimension of the discontinuities is the
size of a molecule, which is almost always much smaller than the smallest
geometrical dimension that must be analysed. This is why, in liquids, the
hypothesis of continuity may almost always be used without restrictions.
On the other side, in solid materials, the validity of this hypothesis must
be analysed more carefully. In fact, although in a metal the size of the crystals
is usually much smaller than the smallest geometrical dimension that must
be analysed, in other materials like concrete, for example, the minimum di-
mension that must be analysed is often of the same order of magnitude as

the designation internal force usually corresponds to the first kind (internal
surface forces).
– External forces – Forces exerted by external entities on a solid body or
liquid mass. The forces may also be sub-divided into surface external forces
and mass external forces. The corresponding definitions are:
– External surface forces – External forces acting on the boundary surface
of a body. Examples of these include the weight of non-structural parts
of a building, equipment, etc., acting on its structure, wind loads on
a building, a bridge, or other Civil Engineering structure, aerodynamic
pressures in the fuselage and wings of a plane, hydrostatic pressure on
the upstream face of a dam or on a ship hull, the reaction forces on the
supports of a structure, etc.
– External mass forces – External forces acting on the mass of a solid body
or liquid. Examples of external mass forces are: the weight of the material
a structure is made of (earth gravity force), the inertial forces caused
by an earthquake or by other kinds of accelerations, such as impact,
vibrations, traction, braking and curve acceleration in vehicles and planes,
and external electromagnetic forces.
– Rigid body motion – displacement of the points of a body which do not
change the distances between the points inside the body.
– Deformation – Variation of the distance between any two points inside the
solid body or the liquid mass.
These definitions are general and valid independently of assuming that the
material is continuous or not. In the case of continuous materials two other
very useful concepts may be defined:
– Stress – Physical entity which allows the definition of internal forces in a
way that is independent of the dimensions and geometry of a solid body or
a liquid mass. There are several definitions for stress. The simplest one is
used in this book, which states that stress is the internal force per surface
unit.


3
←→ strain

2
←→ displacement
1–Force-stress relations – Group of relations based on force equilibrium
conditions. Defines the mathematical entity which describes the stress –
the stress tensor – and relates its components with the external forces. This
set of relations defines the theory of stresses. This theory is completely
independent of the properties of the material the body is made of, except
that the continuity hypothesis must be acceptable (otherwise stress could
not be defined).
2–Displacement-strain relations – Group of relations based on kinematic
compatibility conditions. Defines the strain tensor and relates its compo-
nents to the functions describing the displacement of the points of the body.
This set of relations defines the theory of strain. It is also independent of
the rheological behaviour of material. In the form explained in more detail
in Chap. III, the theory of strain is only valid if the deformations and the
rotations are small enough to be treated as infinitesimal quantities.
I.3 Subdivisions of the Mechanics of Materials 7
3–Constitutive law – Defines the rheological behaviour of the material, that
is, it establishes the relations between the stress and strain tensors. As men-
tioned above, the material rheology is determined by the complex physical
phenomena that occur in the internal structure of the material, at the level
of atom, molecule, crystal, etc. Since, as a consequence of this complexity,
the material behaviour still cannot be quantified by deductive means, a
phenomenological approach, based on experimental observation, must used
in the definition of the constitutive law. To this end, given forces are ap-
plied to a specimen of the material and the corresponding deformations are

components. Higher order tensors may also be defined. An n
th
order tensor will
have 3
n
components in a three-dimensional space (or 2
n
in a two-dimensional
space). As will be seen later, the tensor components are not necessarily all
independent.
Below, the stress tensor is defined and some of its properties are analysed.
II.2 General Considerations
Consider a solid body under a system of self-equilibrating forces, as shown
in Fig. 1-a. Imagine that the body is divided in two parts by the section
represented in the same Figure. Internal forces act in the left surface of the
section, representing the action of the right part of the body on the left part.
Similarly, as a consequence of the equilibrium condition, in the right surface
forces act with the same magnitude and in opposite directions, as shown
in Fig. 1-b. The force F and the moment M represent the resultant of the
internal forces distributed in the section, which generally vary from point to
10 II The Stress Tensor
point. However, by considering an infinitesimal area, dΩ, in the surface (Fig.
2-a), we may consider a homogeneous distribution of the internal force in this
area. Dividing the infinitesimal force dF , which acts in the infinitesimal area
dΩ, we get the internal force per unit of area or stress.
T =
dF
dΩ
. (1)
F

gular Cartesian reference frame xyz may be defined by a unit vector n
→
,which
is perpendicular to the facet and points to the outside direction in relation to
the part of the body considered (Fig. 2-b). This vector n
→
, is the semi-normal
of the facet and, as a unit vector, its components are the cosines of the angles
between the vector and the coordinate axes – the direction cosines of the facet
⎧
⎨
⎩
n
x
=cos(n, x)=l
n
y
=cos(n, y)=m
n
z
=cos(n, z)=n.
As the vector has a unit length, we have
l
2
+ m
2
+ n
2
=1. (2)
The stress acting on the facet may be decomposed into two components: a

and the second one the direction of the shearing stress vector. For example τ
yz
denotes the shearing stress component which is parallel to the z coordinate
axis and acts in a facet whose semi-normal is parallel to the y axis.
External force components are positive if they have the same direction as
the coordinate axes to which they are parallel.
Figure 3 shows the stresses acting in a rectangular parallelepiped defined
by three pairs of facets, which are perpendicular to the three coordinate axis
and are located in an infinitesimal neighborhood of point P .
σ
x
τ
xy
τ
xz
σ
x
τ
xy
τ
xz
τ
yx
σ
y
τ
yz
τ
yx
σ

The static equilibrium of a body, or a part of it, under the action of a system of
forces demands that both its resulting force and its resulting moment vanish.
If the resulting moment is zero, we have rotation equilibrium; if the resulting
force is zero, equilibrium of translation is attained.
The forces acting in the rectangular parallelepiped defined by the three
pairs of facets in Fig. 3 are in equilibrium of translation, since the stress
vectors in each pair of facets are equal (more precisely, the difference between
them is infinitesimal) and have opposite directions. The external body forces
are therefore equilibrated by the infinitesimal difference between the stresses
in the negative and positive facets of the pair. The corresponding expressions
are presented later. We will first analyse the rotation equilibrium conditions.
Equilibrium of Rotation
Assuming that the translation equilibrium is guaranteed, the resulting mo-
ment will be zero or a couple. The latter will vanish if the moments of the
forces in relation to three axes, which have a common point, are non-parallel
and do not lie along to the same plane, are zero. For simplicity, we consider
axes, which are parallel to the reference system and contain the geometrical
center of the infinitesimal parallelepiped (Fig. 3). Considering, for example,
the axis x

parallel to x, the only forces which have a non-zero moment in
relation to this axis are the resultants of τ
yz
and τ
zy
, as it can be confirmed
by looking at Fig. 3 and as represented in Fig. 4.
The condition of zero moment of the forces which result from the stresses
represented in Fig. 4, around the axis x


yx
and τ
xz
= τ
zx
. These expressions, together with expression 3,
represent the so-called reciprocity of shearing stresses in perpendicular facets.
Since the reference axes may have any spatial orientation, the reciprocity may
be expressed in the following way, which is independent of reference axes:
considering two perpendicular facets, the components of the shearing stresses
II.3 Equilibrium Conditions 13
τ
zy
τ
zy
τ
yz
τ
yz
dx
dy
dz
x
x
+
Fig. 4. Equilibrium of rotation around axis x

which are perpendicular to the common edge of the two facets have the same
magnitude and either both point to that edge or both diverge from it.
1

zx
dx dy + X dx dy dz =0. (4)
1
If the external loading were to include moments M
X
, M
Y
, M
Z
, distributed in
the volume of the body, instead of equation (3) we would obtain the expression
τ
yz
− τ
zy
+ M
X
= 0 and there would be no reciprocity of the shearing stresses.
However, this kind of loading does not usually have physical significance, except in
problems which are beyond the scope of this text, such as the case of the influence
of a strong magnetic field on the stress distribution in a magnetized body. For this
reason, in the discussion below, the reciprocity of the shearing stresses will will
always be considered valid.