We were discussing Assumptions made in Euler’s theory, Derivation of beam bending equation and concept of crippling load 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.

1. The maximum principal stress theory

2. The maximum principal strain theory

3. The maximum shear stress theory

4. The maximum strain energy theory

5. The maximum shear strain energy theory

We have already discussed maximum principal stress theory, now it’s time to go ahead with the maximum principal strain theory here
with the help of this article.

According to the theory of maximum principal strain,
“The failure of a material or component will occur when the maximum value of principal strain developed in the body exceeds the limiting value of strain
i.e. value of strain corresponding to the yield point of the material”.

Maximum principal stress theory is also termed as

*. In simple we can write here the statement of maximum principal strain theory.***Saint Venant theory**
The failure of a material or component will occur
when the maximum value of principal strain developed in the body exceeds the
value of strain corresponding to the yield stress in simple tension or when the
maximum compressive strain of the material exceeds the value of strain corresponding
to the yield stress in simple compression.

Therefore in order to avoid the condition of failure
of the component, maximum value of principal strain developed in the body must
be below than the value of strain corresponding to the yield point of the
material.

###
*Condition
of failure*

*Condition of failure*

Maximum value of principal strain developed in the
body > value of strain corresponding to the yield point of the material

Ԑ

_{1}> σy/E
Ԑ

_{1}> Ԑ_{Y.P}###
*Condition
for safe design*

*Condition for safe design*

Maximum value of principal strain developed in the
body ≤ Permissible
strain

Permissible strain is basically defined as the ratio
of value of strain corresponding to the yield point of the material to the
factor of safety.

Permissible strain = Strain corresponding to the
yield point of the material / F.O.S

Permissible strain = Yielding strain /
F.O.S

Maximum value of principal strain developed in the
body ≤ Permissible
strain

Ԑ

_{1}≤ σ_{Y}/ (E x F.O.S)

*For tri-axial state of stress*
(1/E) x [σ

_{1}-μ (σ_{2}+ σ_{3})] ≤ σ_{Y}/ (E x F.O.S)
σ

_{1}-μ (σ_{2}+ σ_{3})] ≤ σ_{Y}/ F.O.S

*For bi-axial state of stress*
σ

Permissible strain = Yielding strain / F.O.S_{1}- μσ_{2}≤ σ_{Y}/ F.O.S
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We will now discuss the maximum shear stress theory, in the category of strength of material, in our next post.

### Reference:

Strength of material, By R. K. Bansal

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