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.

### 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”.

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.

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