What is maximum stress theory?

Short Answer:

The Maximum Stress Theory is also known as Rankine’s Theory of Failure. It states that a material fails when the maximum principal stress in a complex stress condition reaches the value of the ultimate stress obtained from a simple tensile test. This means that if the maximum stress at any point in a material becomes equal to the material’s ultimate tensile strength, the material will fail.

This theory is mainly used for brittle materials such as cast iron and concrete, which generally fail suddenly without much deformation. The Maximum Stress Theory is simple and easy to apply but not suitable for ductile materials like steel because they fail due to yielding, not direct breaking.

Detailed Explanation:

Maximum Stress Theory

The Maximum Stress Theory, also called Rankine’s Theory, is one of the oldest and simplest theories of material failure. It is used to predict when a material will fail under different combinations of stresses such as tension, compression, or shear. The theory is based on the assumption that failure of a material occurs when the maximum principal stress in a complex state of stress equals the maximum stress obtained in a simple uniaxial tension test.

In simple terms, the material is considered to fail when the largest normal stress acting on it reaches the same value as its ultimate strength. This theory assumes that material failure is caused only by the maximum normal stress and ignores the influence of shear or intermediate stresses.

Mathematical Expression

Let:

be the three principal stresses in a body (in tension or compression). According to the Maximum Stress Theory, failure occurs when:

where:

  •  = maximum principal stress
  •  = ultimate tensile stress of the material

If the material is under compression, then the failure occurs when:

where  is the ultimate compressive stress.

Hence, the condition for safety can be written as:

This means that for safe design, the maximum stress developed in the component must always be less than the material’s ultimate tensile strength.

Graphical Representation

If the stresses are represented on coordinate axes, the failure envelope forms a rectangle bounded by the tensile and compressive strengths of the material. This is because failure is assumed to occur when the stress reaches these limiting values. The graphical representation is simple and helps engineers visualize the limit of stress for safe operation.

Assumptions of the Theory

  1. Material fails when the maximum principal stress equals the ultimate stress in simple tension.
  2. It ignores the effect of shear stress and intermediate stresses.
  3. The material is assumed to be homogeneous and isotropic (having same properties in all directions).
  4. It is applicable mainly for brittle materials which fail due to direct fracture.

These assumptions make the theory simple, but they also limit its accuracy for materials that yield before breaking.

Applications

The Maximum Stress Theory is generally used for:

  • Brittle materials such as cast iron, glass, and concrete.
  • Components subjected to static or low variable loads.
  • Design of structural members where failure is sudden and brittle.

It is not used for ductile materials like mild steel because such materials fail due to shear stress and plastic deformation before reaching ultimate tensile stress.

Limitations

  1. The theory does not consider shear stress, which is significant in ductile materials.
  2. It provides unsafe results for ductile materials like steel and aluminum.
  3. It overestimates the strength of materials that yield before breaking.
  4. It does not match well with experimental results for ductile materials.

Due to these reasons, this theory is mainly used as an approximation for brittle materials, while other theories like Maximum Shear Stress Theory (Tresca) or Distortion Energy Theory (Von Mises) are preferred for ductile materials.

Example

Suppose a cast iron bar is subjected to two-dimensional stresses of

and the tensile strength of the material is 150 MPa.

The maximum principal stress is approximately 100 MPa.
Since , the material is safe according to the Maximum Stress Theory.

Conclusion

The Maximum Stress Theory (Rankine’s Theory) is one of the simplest methods to predict material failure. It assumes that failure occurs when the maximum normal stress reaches the ultimate strength of the material. This theory works well for brittle materials, which fail suddenly without yielding. However, it is not suitable for ductile materials where shear stress plays a major role in failure. Despite its limitations, it provides a quick and conservative estimate for the design of brittle components and is often used for preliminary analysis.