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.

3. The maximum shear stress theory
4. The maximum strain energy theory
5. The maximum shear strain energy theory

We will now understand here the maximum shear stress theory with the help of this article. The maximum shear stress theory is also termed as Guest and Tresca’s theory and this theory is only used for ductile materials.

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”.

Let us explain the maximum shear 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 shear stress developed in the body exceeds the value of shear stress corresponding to the point D, failure will take place.

Therefore in order to avoid the condition of failure of the component, maximum value of shear stress developed in the body must be below than the value of shear stress corresponding to the point D.

### Condition of failure

Maximum value of shear stress developed in the body > Yield strength in shear under tensile test i.e. value of shear stress corresponding to the yield point of the material

Let us consider that σ1, σ2 and σ3 are the principle stresses at a point in material and σt is the principle stress in simple tension at elastic limit.

Now as we know that maximum shear stress at a point in the material will be equal to the half of difference between maximum and minimum principle stress and therefore we will have following equation.

τMax = (1/2) x (σ1- σ3)

Let us determine the value of shear stress corresponding to the yield point of the material. In case of simple tension, Stress will be available in one direction only and therefore at elastic limit, principle stresses will be σt, 0 and 0.

Value of shear stress corresponding to the yield point of the material = (1/2) x σt

Let us write here the condition of failure
(1/2) x (σ1- σ3) > (1/2) x σt
(σ1 - σ3) > σt

### Condition for safe design

Maximum value of shear stress developed in the body  Permissible shear stress (τPer)
Permissible shear stress is basically defined as the ratio of yield strength in shear under tensile test i.e. value of shear stress corresponding to the yield point of the material to the factor of safety.

Permissible shear stress = Yield strength in shear under tensile test / F.O.S
Permissible shear stress = Value of shear stress corresponding to the yield point of the material / F.O.S

Permissible shear stress = (σt /2)/ F.O.S
Permissible shear stress = σt / (2 x F.O.S)

For tri-axial state of stress
For tri-axial state of stress, we will have following equation
Do you have suggestions? Please write in comment box.

We will now discuss the maximum strain energy theory, in the category of strength of material, in our next post.

### Reference:

Strength of material, By R. K. Bansal