Module 2: SIMPLE STRAIN

Simple Strain

Whenever a force is applied to a body, it will tend to change the body’s shape and size. These changes
are referred to as deformation, and they may be either highly visible or practically unnoticeable. For example,
a rubber band will undergo a very large deformation when stretched, whereas only slight deformations of
structural members occur when a building is occupied by people walking about. Deformation of a body can
also occur when the temperature of the body is changed. A typical example is the thermal expansion or
contraction of a roof caused by the weather.

Strain is a measure of the deformation of the body, also known as a unit deformation. It is the ratio of the
change in length caused by an applied force to the original length.

𝛿
𝜀=
𝐿

Where: δ = elongation, mm

L = original length, mm

ε= strain, dimensionless (unit less)

A. STRESS – STRAIN DIAGRAM

Suppose that a metal specimen be placed in a tension – compression testing machine. As the axial
load is gradually increased in increments, the total elongation over the gage length is measured at each
increment of the load and this is continued until failure of the specimen takes place. Knowing the cross
sectional area and length of the specimen, the normal stress σ and the strain ε can be obtained. The graph of
these quantities with the stress σ along the y – axis and the strain ε along the x – axis is called the stress –
strain diagram. The stress – strain diagram differs in form for various materials. The diagram shown is that
for a medium – carbon structural steel. Metallic engineering materials are classified as either ductile or brittle
materials. A ductile material is one having relatively large tensile strains up to the point of rupture like
structural steel and aluminum, whereas brittle materials has a relatively small strain up to the point of
rupture like cast iron and concrete. An arbitrary strain of 0.05mm/mm is frequently taken as the dividing
line between these two classes.

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Module 2: SIMPLE STRAIN

PROPORTIONAL LIMIT (HOOKE’S LAW)

From the origin O to a point called proportional limit, the stress strain curve is a straight line. This
linear relation between elongation and the axial force causing it was first noticed by Sir Robert Hooke in
1678 and is called Hooke’s Law, that within the proportional limit, the stress is directly proportional to strain
or

𝝈𝜶𝜺 or 𝝈 = 𝑲𝜺

The constant of proportionality K is called the modulus of elasticity E or Young’s modulus and is
equal to the slope of the stress strain diagram from O to P. then,

𝝈 = 𝑬𝜺

ELASTIC LIMIT

The elastic limit is the limit beyond which the material will no longer go back to its original shape
when the load is removed, or it is the maximum stress that maybe developed such that there is no permanent
or residual deformation when the load is entirely removed.

ELASTIC AND PLASTIC RANGES

The region in the stress – strain diagram from O to P is called the elastic range. The region from P to
R is the plastic range.

YIELD POINT

The point at which the material will have an appreciable elongation or yielding without any increase
of load.

ULTIMATE STRENGTH

The maximum ordinate in the stress – strain diagram is the ultimate strength or the tensile strength.

RAPTURE STRENGTH

The strength of material at rapture. This is also known as the breaking strength.

MODULUS OF RESILIENCE

Modulus of resilience is the work done on a unit volume of material as the force is gradually increased
from O to P, in N-m/m3. This may be calculated as the area under stress – strain curve forms the origin O up
to the elastic limit E. The resilience of a material is its ability to absorb energy without creating a permanent
distortion.

MODULUS OF TOUGHNESS

Modulus of toughness is the work done on a unit volume of material as the force is gradually
increased from O to R in N-m/m3. This may be calculated as the area under the entire stress –strain curved
(from O to R). The toughness of a material is the ability to absorb energy without causing it to break.

WORKING STRESS, ALLOWABLE STRESS AND FACTOR OF SAFETY

Working stress is defined as the actual stress of a material under a given loading. The maximum safe
stress that a material can carry is termed as the allowable stress. The allowable stress should be limited to
the values not exceeding the proportional limit. However, since the proportional limit is difficult to
determine accurately, the allowable stress is taken as either the yield point or ultimate strength divided by
the factor of safety. The ratio of this strength (ultimate strength or yield strength) to the allowable strength
is called the factor of safety.

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