We were discussing moment of resistance of beam section, Differential equation of elastic curve of beam and Rankine's formula for columns in our previous posts.

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.

5. The maximum shear strain energy theory

We will now understand here the maximum
shear strain energy theory of failure with the help of this article.

The maximum shear strain energy theory
of failure is also termed as maximum distortion energy theory or von mises
criterion of failure. This theory is the best theory for failure of ductile
material.

According to the maximum
shear strain energy theory of failure, “The failure of a material or
component will occur when the total shear strain energy per unit volume exceeds
the limiting value of shear strain energy per unit volume i.e. value of shear strain
energy per unit volume corresponding to the yield point of the material under
tension test”.

Therefore in order to avoid the condition of failure
of the component, total shear strain energy per unit volume must be below than
the value of shear strain energy per unit volume corresponding to the yield
point of the material under tension test.

Let us consider the condition of three dimensional
stress systems. Where σ

_{1}, σ_{2}and σ_{3}are the principal stresses developed in the material under tension test.
Total shear strain energy per unit volume will be
given as mentioned here

**U = (1/12E) x [(**

**σ**

_{1 }– σ_{2})^{ 2 }+ (σ_{2 }– σ_{3})^{ 2}+ (σ_{3 }– σ_{1})^{ 2}]
At the elastic limit under tension test,
σ

_{t}, 0 and 0 will be the principal stresses developed in the material.
Let us determine the value of
shear strain energy per unit volume corresponding to the yield point of the
material under tension test will be given by following equation as mentioned
here

**U**

_{t}= (1/2E) x [2 x**σ**

_{t}^{2}]
Where σ

_{t}is the principle stress at elastic limit under tension test###
*Condition
of failure*

*Condition of failure*

Total shear strain energy per unit volume > Value
of shear strain energy per unit volume corresponding to the yield point of the
material under tension test

(1/12E) x [(σ

_{1 }– σ_{2})^{ 2 }+ (σ_{2 }– σ_{3})^{ 2}+ (σ_{3 }– σ_{1})^{ 2}] > (1/2E) x [2 x σ_{t}^{2}]
[(σ

_{1 }– σ_{2})^{ 2 }+ (σ_{2 }– σ_{3})^{ 2}+ (σ_{3 }– σ_{1})^{ 2}] > [2 x σ_{t}^{2}]###
*Condition
for safe design*

*Condition for safe design*

Total shear strain energy per unit volume ≤ Permissible value of shear strain
energy per unit volume

Permissible principle stress = Principle stress at
elastic limit / F.O.S

Permissible principle stress = σ

_{t}/ F.O.S
σ

_{P}= σ_{t}/ F.O.S

*For three dimensional stress systems*
For tri-axial state of stress, we will
have following equation

**[(**

**σ**≤

_{1 }– σ_{2})^{ 2 }+ (σ_{2 }– σ_{3})^{ 2}+ (σ_{3 }– σ_{1})^{ 2}]**[2 x**

**σ**

_{t}^{2}]

*For two dimensional stress systems*
For two dimensional state of stress, σ

_{3}will be zero and hence we will have following equation
[(σ

_{1 }– σ_{2})^{ 2 }+ (σ_{2})^{ 2}+ (– σ_{1})^{ 2}] ≤ [2 x σ_{t}^{2}]
[2σ

_{1}^{ 2}+ 2 σ_{2}^{ 2}– 2 σ_{2}σ_{1}] ≤ [2 x σ_{t}^{2}]**σ**

_{1}

^{ 2 }**+**

**σ**

_{2}^{ 2}– σ_{2}σ_{1}≤ σ_{t}2
Do you have suggestions? Please write in comment
box.

We will now discuss the assumptions made in the theory of simple bending, 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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