We were discussing the concept of stress
and strain and also we have discussed
the different
types of stress, different
types of strain 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.

####
*Let us see here various theories of failure one by one……*

*Let us see here various theories of failure one by one……*

###
**Maximum principal stress theory (Rankine’s theory)**

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

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.

For detailed information about maximum principal
stress theory, please find the respective post “The
maximum principal stress theory ”.

###
**Maximum
principal strain theory (Saint Venant theory) **

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”.

For detailed information about maximum
principle strain theory, please find the respective post “The
maximum principal strain theory ”.

###
**Maximum shear stress theory (Guest and Tresca’s theory)**

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

For detailed information about maximum shear
stress theory, please find the respective post “The
maximum shear stress theory ”.

###
**Maximum strain energy theory (Haigh’s theory)**

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

For detailed information about maximum strain
energy theory, please find the respective post “The
maximum strain energy theory ”.

###
**The maximum shear strain energy theory**
(Maximum distortion energy theory)

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.

For detailed information about maximum strain
energy theory, please find the respective post “The maximum shear strain energy theory ”.

Do you have suggestions? Please write in
comment box.

We will now discuss the Principal
planes and principal stresses, 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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