What is maximum shear stress theory?

Short Answer:

The Maximum Shear Stress Theory is also known as the Guest or Tresca’s Theory of Failure. It states that failure of a material begins when the maximum shear stress in a complex stress system reaches the shear stress at yield point in a simple tensile test. In simple words, a material fails when the shear stress produced in it equals the maximum shear stress observed during the test of that same material.

This theory is mainly used for ductile materials, such as mild steel and copper, where failure occurs due to yielding rather than fracture. It is one of the most widely accepted theories for the design of ductile components under combined loading.

Detailed Explanation:

Maximum Shear Stress Theory

The Maximum Shear Stress Theory (Tresca’s Theory) is one of the most important theories used to predict the failure of materials under complex stress conditions. It was proposed independently by Guest (1890) and Tresca (1864). The theory is based on the concept that yielding of a material starts when the maximum shear stress in a material reaches the shear stress at yield obtained from a simple tension test.

In a simple tension test, the yield point of a material is defined by a certain stress value called the yield strength (σy). At this point, the material begins to deform permanently. The corresponding maximum shear stress in the material is given by:

This relationship forms the basis of the Maximum Shear Stress Theory.

Mathematical Expression

Let the principal stresses acting on an element be:

where  is the maximum principal stress and  is the minimum principal stress.

The maximum shear stress is given by:

According to this theory, yielding or failure occurs when:

or,

Simplifying this,

This is the failure condition according to the Maximum Shear Stress Theory.

It means that failure will begin when the difference between the maximum and minimum principal stresses in the body equals the yield strength of the material.

Concept Explanation

When a body is subjected to combined stresses, such as tensile, compressive, or shear stresses acting in different directions, the total stress condition can be represented by three principal stresses. The material at a particular point experiences a tendency to deform due to shear.

According to this theory, failure does not occur due to normal stresses alone, but because of the maximum shear stress developed inside the material. The maximum shear stress causes one layer of the material to slide over another, leading to yielding.

In a simple tensile test, when a ductile material reaches its yield point, the ratio of maximum shear stress to applied tensile stress is 0.5. Thus, Tresca assumed that any combination of stresses that produces the same maximum shear stress will cause the material to yield.

This theory assumes that materials yield when the maximum shear stress reaches a critical value, regardless of the type of loading (uniaxial, biaxial, or triaxial). Hence, it provides a simple but effective way to design mechanical parts that experience multiple stress conditions.

Applications

  1. Used for ductile materials: It accurately predicts yielding in metals like mild steel, aluminum, and copper.
  2. Design of pressure vessels: Helps in analyzing stresses caused by internal pressure.
  3. Shaft design: Used in calculating torque and bending stresses in rotating shafts.
  4. Machine components: Useful for determining safe dimensions of beams, connecting rods, and crankshafts.
  5. Structural design: Applied in mechanical and civil design for ductile materials.

Assumptions

  1. The material is homogeneous and isotropic, meaning it has the same properties in all directions.
  2. The material obeys Hooke’s Law up to the yield point.
  3. The material fails by yielding due to shear, not by cracking or fracture.
  4. The effect of hydrostatic stress (equal stress in all directions) does not cause yielding.

Advantages

  1. It gives accurate results for ductile materials, which fail by yielding.
  2. It is easy to apply for combined loading conditions.
  3. It closely matches the experimental results of ductile materials under multi-axial stresses.
  4. Provides a simple and safe design criterion for most engineering materials.

Limitations

  1. It is not suitable for brittle materials such as cast iron or glass because they fail due to normal stress and not shear stress.
  2. It ignores the influence of intermediate principal stress (σ2), which may be significant in some cases.
  3. For complex loading conditions, it may not be as accurate as more advanced theories like the Distortion Energy Theory.
  4. It does not account for strain energy or energy absorption during yielding.

Example

Let a material be subjected to two principal stresses:

and the yield stress in a simple tension test is:

Then, according to the theory:

The yield shear stress is:

Since , the material is safe according to the Maximum Shear Stress Theory.

Conclusion

The Maximum Shear Stress Theory (Tresca’s Theory) states that yielding of a ductile material starts when the maximum shear stress in the system equals the maximum shear stress at yield in a simple tension test. It is best suited for ductile materials and is simple to apply in real-world engineering problems. Although it neglects the intermediate principal stress and strain energy, it remains one of the most reliable theories for safe design of components subjected to combined stresses.