Strength of Materials | Introduction to Strength of Materials
Strength of Materials
INTRODUCTION
* Basic assumptions:
1. Materials are continuous
2. Material is homogeneous
3. Material is isotropic
4. Material is free from internal forces prior to the loading considered (residual stresses are zero).
* Fundamental laws
1. Elastic behaviour
2. Law of superposition holds good.
3. St. Venant principle holds good. In other words stress concentrations at the points of load and
geometric discontinuities are not considered in the analysis.
5.2 SIMPLE STRESSES AND STRAINS
* Stress is resistance per unit area.
* Unit of stress is N/m2 or N/mm2
.
* 1 MPa = 1 N/m2
\ 1 MPa = 1 N/mm2
.
* In case of direct forces,
Stress =
Strain: Strain is change in dimension for unit of original dimension.
Linear strain =
Lateral strain =
* Behaviour of mild steel in tension:
1. Limit of proportionality: It is the limiting value of the stress up to which stress is proportional
to strain.
2. Elastic limit: It is the limiting value of the stress upto which if the material is stressed and then
released, the strain disappears completely and the original shape and size is regained.
3. Upper yield point: At this stress the load starts reducing and extension increases.
4. Lower yield point: This is the stress at which the load starts reducing and the extension
increases.
5. Ultimate stress: This is the maximum nominal stress the material can resist. At this stage
formation of neck starts.
6. Breaking point: The nominal stress at which the specimen finally breaks into two parts is called
breaking point.
* If unloading is made within elastic limit, the stress-strain curve follows the original straight portion.
If unloading is made after elastic limit, the stress-strain curve traced is parallel to original curve
with a certain amount of permanent set.
* In case of aluminium and high strength steel, there is no yield point. The stress at which if unloading
is made, there going to be 0.2 % permanent set, is treated as yield point.
* In case of brittle materials, there is no appreciable strain. There is no yield point and no necking.
The ultimate and breaking points are one and the same.
* Percentage elongation: It is the ratio of the final extension at rupture to the original length, expressed
as percentage.
% elongation =
In case of steel it is 20 to 25%
* Percentage reduction in area: It is the ratio of maximum changes in the cross-sectional area to the
original cross-sectional area, expressed as percentage.
% reduction in area =
* Nominal stress =
True stress =
* The maximum stress at which even a billion reversal of stress cannot cause failure of the material is
called endurance limit of the material.
Factor of safety =
In case of elastic materials, instead of ultimate stress, yield stress or 0.2% proof stress is considered
in defining factor of safety.
* Hooke’s law states, stress is proportional to strain within elastic limit.
p µ e, within elastic limit.
* Elongation D =
where P = axial load, L = length
A = CS area and E = modulus of elasticity
* The extension of the bar of uniform thickness tapering from width b1
ro b2
in a length L, and subject
to axial load
=
* The extension of a bar tapering uniformly from diameter d1
to d2
in length L and subject to axial pull
P
=
* Extension of a bar of uniform cross section A and length L, due to self weight
=
where g = unit weight E = Young’s modulus.
* The extension of a conical bar of diameter D at one end to zero at the other end in length L due to
self-weight only
=
where g = unit weight and E = Young’s modulus.
* If a compound bar of two materials is subjected to axial force, the conditions to be satisfied are
P1 + P2 = P and =
* If a is the coefficient of thermal expansion, t is change in temperature and L is the length, free
expansion of the box is
D = a t L.
* Due to change in temperature stresses induced in a bar are so as to cause change in length equal to
free expansion prevented.
Simple Shear
* A material is said to be in a state of simple shear, if it is subjected to only shearing stresses.
* Simple shear gives rise to tensile and compressive stresses across planes inclined at 45° to the
shearing planes, the intensity of direct stresses being same as the shearing stresses.
* Poisson’s ratio: Poisson’s ratio is the ratio of lateral strain to linear strain within elastic limit.
m =
* For most of metals Poisson’s ratio is between 0.25 to 0.33. For steel its value is 0.3. For concrete
its value is 0.15.
* Volumetric strain: The ratio of change in volume to original volume is known as volumetric strain.
ev =
* Volumetric strain is equal to the sum of strains in three mutually perpendicular directions ev = ex +
ey + ez
.
Elastic Constants
* Modulus of elasticity, modulus of rigidity and bulk modulus are the elastic constants.
* Modulus of rigidity is defined as the ratio of shearing stress to the shearing strain within elastic
limit.
G =
where q = shearing stress and f = shearing strain
* Bulk modulus: Bulk modulus is the ratio of identical pressure p acting in three mutually
perpendicular directions (hydrostatic pressure) to the corresponding volumetric strain.
K =
* E, G, K and m have the following relations:
E = 2G(1 + m) = 3K(1 – 2m)
=
E =
Strain Energy
* The energy which is stored in a body duty to straining, is called strain energy.
* Strain energy =
=
* Strain energy per unit volume is known as resilience. Thus,
Resilience =
* The maximum strain energy which can be stored by a body without undergoing permanent
deformation is called proof résistance. In other words, proof resilience is the strain energy in the
body when it is strained to elastic limit.
Proof resilience =
* Stress developed due to application of a load p suddenly is twice that due to same load applied
gradually.
p =
* The stress developed in the material of a bar due to free falling load W from a height h is
p =
where A = Cross-section of the bar
and E = Young’s modulus.
Since the term is very large compared to unity, the above expression for instantaneous stress may
be approximated as,
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