1. STRAIN AND STRESS

1.1 Introduction. When a load is applied to a member of a machine or
structure, the material distorts. The stress intensity (usually abbreviated to
stress) is the load transmitted per unit area of cross-section and the strain
is a measure of the resulting distortion.
Assuming that the load is insufficient to cause rupture, it is resisted by
the force of attraction between the molecules of the material and the
deformation is the result of the slight re-orientation of the molecules.
If the material returns to its former shape when the load is removed, it
is said to be elastic; if the strain is permanent, it is said to be plastic. Most
engineering materials are elastic up to a certain stress (referred to as the
elastic limit), after which they are partly elastic and partly plastic. The
transition is not always abrupt, but for the purposes of calculation it is
usually assumed to be so, an assumption which is reasonably justified for
common mild steel.
In the simple theory of Strength of Materials, it is assumed that the
material is isotropic (i.e. displays the same properties in all directions) and
that it is equally rigid in tension and compression. It is further assumed
that the stress is uniformly distributed over the area resisting the load;
this is approximately true, except in the near vicinity of the point of
application of the load or a sudden change of section (St Venant's Prin
ciple).
1.2 Tensile and compressive stress and strain. If a piece of material
of cross-sectional area a is subjected to equal and opposite forces P, either
tensile, as in Fig. 1.1(a) or compressive, as in Fig. 1.1(6), then
force
stress = cross-sectional area
i.e. σ= - (1.1)
a
If the original length of the bar is I and under the effect of the force Ρ it
extends or compresses a distance x, then
change in length
strain = original length
i.e. e= | (1.2)
1 2
STRENGTH OF MATERIALS
The deformed shapes of the bars are as shown dotted in Fig. 1.2; the
strain in directions perpendicular to that of the load is proportional to that
in the direction of the load and is of the opposite sign.
The ratio ^
e r
^
s
^
r a m
j
s ca
j j
e (
j p
0
isson's Ratio and is denoted by v.
axial stram
Thus if the axial strain is ε, the lateral strain is —νε.
(a)
FIG. 1.1
(b)
(a)
Τ
(b)
FIG. 1.2
1.3 Shear stress and strain. If a piece of material of cross-sectional
area a is subjected to equal and opposite forces Ρ which produce a state of
shear, as shown in Fig. 1.3, then
force
shear stress :
i.e.
=
cross-sectional area
Ρ
(1.3)
If the deformation in the direction of Ρ is χ and the perpendicular dis
tance between the applied forces is Z, then
deformation
shear strain :
couple arm
i.e.
(1.4)
is the angular displacement in radians, since - is very small.
V
i
/
/
/
1/ p
FIG. 1.3
τ
FIG. 1.4 SIMPLE STRESS AND STRAIN
3
When a shear stress r is applied to the faces AB and CD of an element of
the material, Fig. 1.4, a clockwise couple ( X AB χ t) χ BC is applied to
the element, t being the thickness of the material. Since it does not rotate,
however, an equal anticlockwise couple must be applied by means of shear
stresses induced on faces AD and BC.
If the magnitude of these stresses is τ', then for equilibrium,
(r χ AB χ t) X BC = (τ' X BC χ t) X AB
τ
' = τ
Thus a shear stress in one plane is always accompanied by an equal shear
stress (called the complementary shear stress) in the perpendicular plane.
1.4 Hooke's Law. Hooke's Law states that when a load is applied to
an elastic material, the deformation is directly proportional to the load
producing it. Since the stress is proportional to the load and the strain is
proportional to the deformation, it follows that the stress is proportional to
the strain, i.e. the ratio stress/strain is a constant for any given material.
For tensile or compressive stresses, this constant is known as the Modulus
of Elasticity (or Young's Modulus) and is denoted by E.
Thus Ε = - = -L- = — . . . . (1.5)
x/l αχ
For shear stress, this constant is known as the Modulus of Rigidity and
is denoted by G.
Thus 0 * ...
. (1.6)
x/l ax
Instead of basing this factor on the stress at failure, it is sometimes based
on the stress at the yield point (where the material suddenly becomes
plastic) or, for materials which have no well-defined yield point, on the
stress at which the extension is a certain percentage (e.g. 0-1 per cent) of
the original length.
1.5 Factor of safety. The maximum stress used in the design of a
machine or structure is considerably less than the ultimate stress (i.e. the
stress at failure), to allow for .possible overloading, non-uniformity of stress
distribution, shock loading, faults in material and workmanship, corrosion,