We were discussing Slope and deflection of beam,
Rankine’s formula for columns, bending stress in beam and different types of load acting on beam in our previous post.

Today we will understand here the theories of
failure, in strength of material, with the help of this post.

As we know very well that when a body or component
or material will be subjected with an external load, there will be developed
stresses and strains in the body or component.

As per hook’s law, stress will be directionally
proportional to the strain within the elastic limit or we can say in simple
words that if an external force is applied over the
object, there will be some deformation or changes in the shape and size of the
object. Body will secure its original shape and size after removal of external
force.

Within the elastic limit, there will be no permanent deformation in the body i.e. deformation will be disappeared after removal of load.

If external load is applied beyond the
elastic limit, there will be a permanent deformation in the body i.e. deformation
will not be disappeared after removal of load. Component or material or body
will be said to be failed, if there will be developed permanent deformation in
the body due to external applied load.

Theories of failure help us in order to calculate
the safe size and dimensions of a machine component when it will be subjected
with combined stresses developed due to various loads acting on it during its
functionality.

There are following theories as listed
here for explaining the causes of failure of a component or body subjected with
external loads.

The maximum principal stress theory

The maximum principal strain theory

The maximum shear stress theory

The maximum strain energy theory

The maximum shear strain energy theory

###
**We will first understand here the maximum principal stress theory**

According to the theory of maximum
principal stress, “The failure of a material or component will occur when the
maximum value of principle stress developed in the body exceeds the limiting
value of stress”.

Let us explain the maximum principal stress theory by considering here one component which is subjected with an external load and we have drawn here the stress-strain curve as displayed in following figure.

Point
A – It is proportionality limit; up to this point hooks law will be followed.

Point
B – Elastic limit, up to this point the deformation will be elastic.

Point C – Lower yield stress.

Point D – Ultimate stress, it is the maximum value of stress in stress – strain
diagram.

Point E- It is the fracture point, up to this point the material will have only
elastic & plastic deformation ,but at this point fracture or rupture take
place.

If
maximum value of principal stress developed in the body exceeds the point D, failure
will take place.

Therefore
in order to avoid the condition of failure of the component, maximum value of
principal stress developed in the body must be below than the failure stress
i.e. ultimate stress or yield stress.

###
*Condition of failure *

*Condition of failure*

Maximum
value of principal stress developed in the body > Failure stress

σ

_{1}> σy or σ_{ul}###
*Condition for safe
design *

*Condition for safe design*

Maximum
value of principal stress developed in the body ≤ Permissible stress or allowable stress

Permissible
stress is basically defined as the ratio of failure stress i.e. ultimate stress
or yield stress to the factor of safety.

Permissible
stress = Ultimate stress or yield stress / F.O.S

###
*
Important points in maximum
principal stress theory*

*Important points in maximum principal stress theory*

Maximum
principal stress theory is also termed as Rankine’s theory

Maximum principal stress theory is quite suitable for securing the safe design of
machine component made of brittle material as brittle materials are weak with
respect to tension.

Maximum principal stress theory is not suitable for securing the safe design of machine
component made of ductile material as shear failure may take place.

Maximum principal stress theory may be suitable for securing the safe design of machine
component made of ductile material under following three situations.

1. Uniaxial
state of stress

2. Biaxial state of stress when principal stresses are
like in nature

3. Under hydrostatic stress

Do you have suggestions? Please write in
comment box.

We
will now discuss the maximum principal strain theory, in the category of
strength of material, in our next post.

###
**Reference:**

Strength
of material, By R. K. Bansal

Image
Courtesy: Google

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