Compendium of Som

Strength of Materials
Strength of materials, also called mechanics of materials, deals with the

behavior of solid objects subject to stresses and strains..
The study of strength of materials often refers to various methods of calculating the
stresses and strains in structural members, such as beams, columns, and shafts. The
methods employed to predict the response of a structure under loading and its
susceptibility to various failure modes takes into account the properties of the
materials such as its yield strength, ultimate strength, Young's modulus,
and Poisson's ratio; in addition the mechanical element's macroscopic properties
(geometric properties), such as its length, width, thickness, boundary constraints
and abrupt changes in geometry such as holes are considered.
In the mechanics of materials, the strength of a material is its ability to withstand

an applied load without failure or plastic deformation. The field of strength of
materials deals with forces and deformations that result from their acting on a
material. A load applied to a mechanical member will induce internal forces within
the member called stresses when those forces are expressed on a unit basis. The
stresses acting on the material cause deformation of the material in various
manners including breaking them completely. Deformation of the material is called
strain when those deformations too are placed on a unit basis.
The stresses and strains that develop within a mechanical member must be
calculated in order to assess the load capacity of that member. This requires a
complete description of the geometry of the member, its constraints, the loads
applied to the member and the properties of the material of which the member is
composed. The applied loads may be axial (tensile or compressive), or rotational
(strength shear). With a complete description of the loading and the geometry of
the member, the state of stress and state of strain at any point within the member
can be calculated. Once the state of stress and strain within the member is known,
the strength (load carrying capacity) of that member, its deformations (stiffness
qualities), and its stability (ability to maintain its original configuration) can be
calculated.
The calculated stresses may then be compared to some measure of the strength of
the member such as its material yield or ultimate strength. The calculated
deflection of the member may be compared to deflection criteria that are based on
the member's use. The calculated buckling load of the member may be compared
to the applied load. The calculated stiffness and mass distribution of the member
may be used to calculate the member's dynamic response and then compared to the
acoustic environment in which it will be used.
In the mechanics of the deformable bodies, the following types of loads are
commonly considered:
 Dead loads—static in nature, such as the self-weight of the roof.
 Live loads—fluctuating in nature, do not remain constant- such as a weight of a
vehicle moving on a bridge.
 Tensile loads: Tensile Load is the ability of a material to withstand a pulling
force
 Compressive loads: the capacity of a material or structure to withstand loads
tending to reduce size, as opposed to which withstands loads tending to elongate
 Shearing loads: A shear load is a force that tends to produce a sliding failure on
a material along a plane that is parallel to the direction of the force. When a paper
is cut with scissors, the paper fails in shear.
 Transverse loadings – loading perpendicular to axial loading

 Forces applied perpendicular to the longitudinal axis of a
member. Transverse loading causes the member to bend and
deflect from its original position, with internal tensile and
compressive strains accompanying the change in curvature of
the member.
Transverse loading also induces shear forces that cause shear deformation of
the material and increase the transverse deflection of the member.
 Axial loading – The applied forces are collinear with the longitudinal axis of
the member. The forces cause the member to either stretch or shorten.
 Torsional loading – Twisting action caused by a pair of externally applied
equal and oppositely directed force couples acting on parallel planes or by a
single external couple applied to a member that has one end fixed against
rotation.
 Example: Twisting a simple piece of blackboard chalk between ones fingers
until it snaps is an example of a torsional force in action. A
common example of torsion in engineering is when a transmission drive shaft
(such as in an automobile) receives a turning force from its power source (the
engine).
 Bending loading: Distortion of an object by a force (e.g., a load placed on a

beam located between two supports may cause the beam to curve).
Stress
Stress is the internal resistance offered by the body to the external load applied to it
per unit cross sectional area. Stresses are normal to the plane to which they act and
are tensile or compressive in nature.
stress is defined as the force intensity or force per unit area. Here we use a
symbol σ to represent the stress.
Units
P is expressed in Newton (N) and A, original area, in square meters (m2), the stress
σ will be expresses in N/ m2. This unit is called Pascal (Pa). As Pascal is a small
quantity, in practice, multiples of this unit is used.
1 kPa = 103 Pa = 103 N/ m2 (kPa = Kilo Pascal)
1 MPa = 106 Pa = 106N/ m2 = 1 N/mm2 (MPa = Mega Pascal)
1 GPa = 109 Pa = 109 N/ m2 (GPa = Giga Pascal)
TYPES OF STRESSES :
Only two basic stresses exists :
(1) normal stress
(2) shear stress.
Other stresses either are similar to these basic stresses or are a combination
of this e.g. bending stress is a combination tensile, compressive and shear
stresses. Torsional stress, as encountered in twisting of a shaft is a shearing
stress.
Normal stresses :
When the applied load/Force acting Normal/perpendicular to the resisting area ,it is
called normal stress
.The normal stresses are generally denoted by a Greek letter (σ)
This is also known as uniaxial state of stress, because the stresses acts only in one
direction however, such a state rarely exists, therefore we have biaxial and triaxial
state of stresses where either the two mutually perpendicular normal stresses acts
or three mutually perpendicular normal stresses acts as shown in the figures
below :
This is also known as uniaxial state of stress, because the stresses acts only in one
direction however, such a state rarely exists, therefore we have biaxial and triaxial
state of stresses where either the two mutually perpendicular normal stresses acts
or three mutually perpendicular normal stresses acts as shown in the figures
below :
Tensile or compressive Stresses:

The normal stresses can be either tensile or compressive whether the stresses acts
out of the area or into the area
Sign convections for Normal stress
Direct stresses or normal stresses
- tensile +ve
- compressive –ve
Tensile stress (σt )

If σ > 0 the stress is tensile. i.e. The fibres of the component tend to elongate due to
the external force. A member subjected to an external force tensile P and tensile
stress distribution due to the force is shown in the given figure.
Compressive stress (σc)

If σ< 0 the stress is compressive. i.e. The fibres of the component tend to shorten
due to the external force. A member subjected to an external compressive force P
and compressive stress distribution due to the force is shown in the given figure.
Shear Stresses/Tangential stress
Shear stress ( ζ )
When forces are transmitted from one part of a body to other, the stresses
developed in a plane parallel to the applied force are the shear stress. Shear stress
acts parallel to plane of interest. Forces P is applied transversely to the member AB
as shown. The corresponding internal forces act in the plane of section C and are
called shearing forces.
When the applied load/force acting parallel or tangential tio the resisting
area ,it is called shear stress.
Let us consider now the situation, where the cross – sectional area of a block of
material is subject to a distribution of forces which are parallel, rather than normal,
to the area concerned. Such forces are associated with a shearing of the material,
and are referred to as shear forces. The resulting stress is known as shear stress.
The resulting force intensities are known as shear stresses, the mean shear stress
being equal to
ζ=P/A
The Greek symbol ζ (tau, suggesting tangential) is used to denote shear stress.
Complementary shear stresses:

The existence of shear stresses on any two sides of the element induces
complementary shear stresses on the other two sides of the element to maintain
equilibrium. As shown in the figure the shear stress ζ in sides AB and CD induces
a complimentary shear stress ζ ' in sides AD and BC.
Sign convections for shear stresses:

 tending to turn the element C.W +ve.
 tending to turn the element C.C.W – ve.
Strain (ξ)
The displacement per unit length (dimensionless) is known as strain.

 Tensile strain
 Compresive strain
 Volumetric strain
 Shear strain
Tensile strain (ξ t)
The elongation per unit length as shown in the figure is known as tensile strain.
Note: It is engineering strain or conventional strain.

Here we divide the elongation to original length not actual length
Question: A rod 100 mm in original length. When we apply an axial tensile

load 10 kN the final length of the rod after application of the load is 100.1 mm.
So in this rod tensile strain is developed and is given by
Compressive strain ( ξ c)
If the applied force is compressive then the reduction of length per unit length is
known as compressive strain. It is negative. Then
Question: A rod 100 mm in original length. When we apply an axial

compressive load 10 kN the final length of the rod after application of the load
is 99 mm. So in this rod a compressive strain is developed and is given by
Longitudinal & Lateral Strain
A longitudinal strain is defined as. Change in the length to the original length of

an object. It is caused due to longitudinal stress and is denoted by the Greek letter
epsilon 𝜺l.
A lateral strain, also known as transverse strain, is defined as the ratio of the
change in diameter of a circular bar of a material to its diameter due to deformation
in the longitudinal direction. It is denoted by the Greek letter epsilon –d (𝜺d)
Shear Strain
Shear Strain ( γ ): When a force P is applied tangentially to the element shown. Its
edge displaced to dotted line. Where δ is the lateral displacement of the upper face
of the element relative to the lower face and L is the distance between these faces.
Then the shear strain is
Question: A block 100 mm × 100 mm base and 10 mm height. When we apply

a tangential force 10 kN to the upper edge it is displaced 1 mm relative to
lower face. Then the direct shear stress in the element
Volumetric strain (ξv)
The ratio of change in the volume of the body to the original volume is known as
volumetric strain.
Where V is the final volume, V0 is the original volume, and ∆V is the volume
change.
Volumetric strain is a ratio of values with the same units, so it also is a
dimensionless quantity.
Question: A square steel rod 20 mm* 20 mm in section is to carry an axial

laod (compressive) of 100 kN.Calculate the shortening in a length of 50 mm.
E= 2.14*108 kN/m2 .

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Strength of Materials

Strength of materials, also called mechanics of materials, deals with the

In the mechanics of materials, the strength of a material is its ability to withstand

 Transverse loadings – loading perpendicular to axial loading

 Bending loading: Distortion of an object by a force (e.g., a load placed on a

1 kPa = 103 Pa = 103 N/ m2 (kPa = Kilo Pascal)

1 MPa = 106 Pa = 106N/ m2 = 1 N/mm2 (MPa = Mega Pascal)

1 GPa = 109 Pa = 109 N/ m2 (GPa = Giga Pascal)

Tensile or compressive Stresses:

Tensile stress (σt )

Compressive stress (σc)

Complementary shear stresses:

Sign convections for shear stresses:

The displacement per unit length (dimensionless) is known as strain.

Note: It is engineering strain or conventional strain.

Question: A rod 100 mm in original length. When we apply an axial tensile

Question: A rod 100 mm in original length. When we apply an axial

A longitudinal strain is defined as. Change in the length to the original length of

Question: A block 100 mm × 100 mm base and 10 mm height. When we apply

Question: A square steel rod 20 mm* 20 mm in section is to carry an axial

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