Short Answer:

Different failure theories are used in mechanical design to predict when a material or component will fail under various types of loading. These theories help engineers choose the correct material and design dimensions to ensure safety and reliability. The most commonly used failure theories include Maximum Stress Theory, Maximum Strain Theory, Maximum Shear Stress Theory (Tresca), Distortion Energy Theory (von Mises), and Coulomb-Mohr Theory.

Each theory is based on different stress or strain conditions and is applied depending on whether the material is ductile or brittle. Choosing the right theory helps in designing components that can safely handle loads without unexpected failure.

Detailed Explanation:

Different failure theories used in design

In mechanical design, every component must be checked to ensure it can handle the loads it will face during operation. To predict whether a material will fail under stress, engineers use failure theories. These theories give mathematical methods to check if the stress in a material is safe or dangerous.

There are many types of materials and loading conditions, so different theories are used for ductile and brittle materials. These theories are also called failure criteria, and they help in determining whether a material will deform, yield, or break under applied loads.

Common failure theories

Below are the most widely used failure theories in mechanical engineering:

Maximum Normal Stress Theory (Rankine Theory)

This theory says that failure occurs when the maximum principal stress in a material reaches the material’s ultimate tensile strength.
It is mainly used for brittle materials like cast iron, ceramics, and glass.
This theory does not consider shear stress, so it may not be accurate for ductile materials.

Use: Simple and useful for brittle materials under static loads.

Maximum Shear Stress Theory (Tresca Theory)

This theory says that failure happens when the maximum shear stress in the material becomes equal to the shear stress at yielding in a simple tensile test.
Shear stress is calculated from the difference between the maximum and minimum principal stresses.
It is used mainly for ductile materials like mild steel.

Formula:

τmax=(σ1−σ3)2\tau_{\text{max}} = \frac{(\sigma_1 – \sigma_3)}{2}τmax=2(σ1−σ3)

Failure occurs when this shear stress equals half of the yield strength.

Use: Easy to apply and suitable for shafts, pressure vessels, and beams.

Distortion Energy Theory (von Mises Theory)

Also called von Mises-Hencky Theory.
It says that failure occurs when the distortion energy in a material reaches the same level as in a tensile test at yield.
It gives more accurate results than Tresca for complex stress conditions in ductile materials.

Formula:

σvon Mises=σ12+σ22+σ32−σ1σ2−σ2σ3−σ3σ1\sigma_{\text{von Mises}} = \sqrt{\sigma_1^2 + \sigma_2^2 + \sigma_3^2 – \sigma_1\sigma_2 – \sigma_2\sigma_3 – \sigma_3\sigma_1}σvon Mises=σ12+σ22+σ32−σ1σ2−σ2σ3−σ3σ1

Compare this with yield strength to check failure.

Use: Most widely used theory for ductile materials like steel under combined stresses.

Maximum Principal Strain Theory (St. Venant Theory)

This theory states that failure occurs when the maximum normal strain reaches the strain at yield point.
It is less commonly used in practice.
Not very accurate for materials with high shear sensitivity.

Use: Limited application in structural design.

Coulomb-Mohr Theory

This theory is a combination of Mohr’s Circle and stress conditions.
Used for materials with different tensile and compressive strengths.
Especially useful for brittle materials, like concrete, that behave differently under tension and compression.

Use: Common in civil and structural engineering.

How engineers choose the right theory

The correct failure theory is chosen based on:

Type of material (ductile or brittle)
Type of loading (uniaxial, biaxial, triaxial)
Direction of stress (normal, shear, combined)
Safety level required
Manufacturing and cost constraints

For example:

For ductile metals under complex loading → use von Mises Theory
For brittle materials → use Maximum Stress or Coulomb-Mohr Theory

Sometimes, more than one theory is used to cross-check safety, especially in critical parts like aircraft or turbines.

Conclusion

Failure theories are important tools in engineering design to predict when a component will fail under different types of loads. The main failure theories include Maximum Stress Theory, Tresca Theory, von Mises Theory, Principal Strain Theory, and Coulomb-Mohr Theory. Each one is suitable for different materials and loading conditions. Choosing the right failure theory ensures safety, efficiency, and durability of machines and structures. It helps engineers design components that perform well and last longer under real-life working conditions.