Short Answer:

Theories of failure are used in mechanical engineering to predict the failure of materials under different loading conditions. These theories help in designing safe and strong components that can withstand applied forces without breaking or deforming permanently. They provide the basis to decide whether a material will fail under given stresses.

The different theories of failure are developed based on experimental results and practical observations. The most commonly used theories include Maximum Principal Stress Theory, Maximum Principal Strain Theory, Maximum Shear Stress Theory, Total Strain Energy Theory, and Distortion Energy Theory. Each theory is suitable for different types of materials such as brittle or ductile materials.

Detailed Explanation:

Theories of Failure

When a material is subjected to various types of loads such as tension, compression, bending, or torsion, it experiences stress in different directions. To ensure safety and efficiency, it is essential to know when the material will fail. Theories of failure help engineers predict this condition. A material is said to fail when it either fractures (breaks) or yields (deforms permanently). Since it is not always possible to test every loading condition experimentally, different theories of failure have been developed to estimate failure using simple stress values.

Theories of failure are used in the design of mechanical components like shafts, beams, pressure vessels, and gears, where complex stress systems exist. These theories help engineers design structures that can withstand actual working stresses within safe limits.

1. Maximum Principal Stress Theory (Rankine’s Theory)

This theory states that failure occurs when the maximum principal stress in the material reaches the value of the ultimate stress in a simple tension test.
Mathematically, failure occurs when:

where  is the maximum principal stress and  is the tensile strength of the material.
This theory is suitable for brittle materials such as cast iron, which fail due to direct stress.

2. Maximum Principal Strain Theory (Saint-Venant’s Theory)

According to this theory, failure occurs when the maximum principal strain in the material equals the strain produced at the elastic limit in a simple tension test.
It considers both the effect of normal and Poisson’s ratio.
This theory is not much used because experimental results do not agree with it completely. It is applicable to brittle materials but less accurate compared to Rankine’s theory.

3. Maximum Shear Stress Theory (Guest’s or Tresca’s Theory)

This theory assumes that failure occurs when the maximum shear stress in the material reaches the value of the shear stress at the yield point during a simple tension test.
Mathematically,

where  is the yield stress.
This theory is most suitable for ductile materials like mild steel. It gives safe results for components that undergo yielding before failure.

4. Total Strain Energy Theory (Haigh’s Theory)

This theory states that failure occurs when the total strain energy per unit volume of the material reaches the same value as in a simple tensile test at yield point.
Although it considers the total energy due to both normal and shear stresses, it is not widely used for design because it overestimates the safe load for ductile materials.
This theory is applicable mainly to elastic materials.

5. Distortion Energy Theory (Von Mises-Hencky Theory)

This is one of the most widely used and accurate theories for ductile materials. It assumes that yielding begins when the distortion energy per unit volume in the material equals that at the yield point in a simple tension test.
Mathematically,

When , the material begins to yield.
This theory gives very close results to experimental data and is used extensively in the design of mechanical components made from ductile materials such as steel.

Comparison and Application

• Brittle materials (like cast iron, glass): Theories such as Maximum Principal Stress or Maximum Principal Strain are suitable because brittle materials fail due to normal stress without much yielding.
• Ductile materials (like steel, copper): Theories such as Maximum Shear Stress or Distortion Energy Theory are used because ductile materials yield before fracture.

Each theory provides a different safety prediction, so engineers choose one depending on the material behavior and nature of loading. For example, shafts and beams made from mild steel are usually designed using Tresca or Von Mises theory.

Conclusion

Theories of failure play a very important role in mechanical design as they help determine the safe working limit of materials under different loading conditions. Each theory provides a mathematical way to predict failure, but the selection depends on whether the material is brittle or ductile. Among all, Maximum Shear Stress Theory and Distortion Energy Theory are most reliable for ductile materials, while Maximum Principal Stress Theory works well for brittle materials. Proper use of these theories ensures safety, reliability, and efficiency in engineering design.