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 Saint Venant theory. In simple we can write here the statement of maximum principal strain 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

Maximum value of principal strain developed in the body > value of strain corresponding to the yield point of the material
Ԑ1 > σy/E
Ԑ1 > ԐY.P

### 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
Ԑ1  σY/ (E x F.O.S)

For tri-axial state of stress
(1/E) x [σ1-μ (σ2 + σ3)]  σY/ (E x F.O.S)
σ1-μ (σ2 + σ3)]  σY/ F.O.S

For bi-axial state of stress

σ1- μσ2  σY/ F.O.S
Permissible strain = Yielding strain / 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