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.

The maximum principal stress theory
The maximum principal strain theory
The maximum shear stress theory
The maximum strain energy theory
The maximum shear strain energy theory

### We will first understand here the maximum principal stress theory

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

Let us explain the maximum principal stress theory by considering here one component which is subjected with an external load and we have drawn here the stress-strain curve as displayed in following figure.
Point A – It is proportionality limit; up to this point hooks law will be followed.
Point B – Elastic limit, up to this point the deformation will be elastic.
Point C – Lower yield stress.
Point D – Ultimate stress, it is the maximum value of stress in stress – strain diagram.
Point E-  It is the fracture point, up to this point the material will have only elastic & plastic deformation ,but at this point fracture or rupture take place.

If maximum value of principal stress developed in the body exceeds the point D, failure will take place.

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.

### Condition of failure

Maximum value of principal stress developed in the body > Failure stress
σ1 > σy or σul

### Condition for safe design

Maximum value of principal stress developed in the body Permissible stress or allowable stress

Permissible stress is basically defined as the ratio of failure stress i.e. ultimate stress or yield stress to the factor of safety.

Permissible stress = Ultimate stress or yield stress / F.O.S

Maximum principal stress theory is also termed as Rankine’s theory

Maximum principal stress theory is quite suitable for securing the safe design of machine component made of brittle material as brittle materials are weak with respect to tension.

Maximum principal stress theory is not suitable for securing the safe design of machine component made of ductile material as shear failure may take place.

Maximum principal stress theory may be suitable for securing the safe design of machine component made of ductile material under following three situations.

1. Uniaxial state of stress
2. Biaxial state of stress when principal stresses are like in nature
3. Under hydrostatic stress

Do you have suggestions? Please write in comment box.

We will now discuss t, in the category of strength of material, in our next post.

### Reference:

Strength of material, By R. K. Bansal