Short Answer:

The Maximum Shear Stress Theory, also known as the Tresca Theory or Guest’s Theory, states that failure of a ductile material occurs when the maximum shear stress in a complex stress system reaches the same value as the shear stress at yield in a simple tensile test.

In simple words, this theory suggests that yielding starts when the shear stress in the material becomes equal to half of the yield stress in tension. It is mainly applicable to ductile materials such as mild steel, copper, and aluminum, which fail due to shear deformation rather than direct tension.

Detailed Explanation :

Maximum Shear Stress Theory (Tresca Theory)

The Maximum Shear Stress Theory, also known as the Tresca or Guest’s Theory, was proposed by Henri Tresca and Guest in the 19th century. It is one of the most important and widely used failure theories for ductile materials. The theory is based on the assumption that yielding of ductile materials occurs when the maximum shear stress in a complex stress condition reaches the same value as the shear stress at yield in a uniaxial tension test.

This theory works well because ductile materials, such as steel or aluminum, usually fail by shear yielding rather than fracture. Therefore, it focuses on shear stress, which is responsible for plastic deformation.

Definition

According to the Maximum Shear Stress Theory (Tresca Theory):

“A material starts yielding when the maximum shear stress in a complex stress system reaches the same value as the maximum shear stress at yield in a simple tension test.”

This means that if the maximum shear stress in a loaded body equals the yield shear stress from a tensile test, the material begins to yield.

In a simple tensile test, the maximum shear stress at the yield point is given by:

where  is the yield stress in tension.

Hence, according to Tresca’s criterion, yielding occurs when the maximum shear stress in a complex stress condition equals .

Mathematical Expression

Let the three principal stresses at a point in a body be  such that .

The maximum shear stress in the system is given by:

According to Tresca’s theory, failure or yielding begins when:

Hence, the condition for failure is:

This is the mathematical form of the Maximum Shear Stress Theory.

For Two-Dimensional Stress Condition

In most engineering cases, stresses act in two directions, and the third principal stress () is zero.

Thus,

and failure occurs when:

This equation helps to find the safe stress levels for two-dimensional stress states, such as in plates, shafts, or beams.

Graphical Representation

In the principal stress diagram ( vs ), the yield surface predicted by the Tresca theory forms a regular hexagon.

• Points inside the hexagon represent safe conditions.
• Points on the boundary indicate yielding or failure.
• Points outside the hexagon represent failure conditions.

The hexagonal shape arises because the theory considers the maximum difference between any two principal stresses.

Physical Meaning

The physical meaning of this theory is that shear stress is responsible for the sliding or yielding of material layers. Ductile materials fail when the shear stress in the material reaches the same value as that required to cause yielding in a simple tension test.

In a uniaxial tension test, the material yields when the maximum shear stress is equal to half of the yield stress in tension. Hence, Tresca’s theory uses this as the failure criterion.

Applicability

• The Maximum Shear Stress Theory is best suited for ductile materials, such as:
  • Mild steel
  • Copper
  • Aluminum
  • Brass
  • Ductile iron
• It is commonly used in:
  • Design of shafts, pressure vessels, and machine components.
  • Finite Element Analysis (FEA) for yield prediction.
  • Structural design where safety against yielding is important.

Advantages

1. Accurate for ductile materials:
   It provides results close to actual experimental data for ductile materials.
2. Simple and practical:
   It uses principal stresses, making it easy to apply in engineering calculations.
3. Conservative (safe):
   The Tresca theory gives slightly lower allowable stresses than other theories, making designs safer.
4. Logical for shear-based failure:
   Since ductile materials fail by shear, this theory directly relates to their failure mechanism.

Limitations

1. Not applicable for brittle materials:
   Brittle materials fail by normal stress (fracture), not by shear.
2. Overly conservative:
   It sometimes predicts yielding earlier than what actually occurs.
3. Ignores intermediate principal stress:
   Only the maximum and minimum principal stresses are considered, neglecting the effect of the middle stress .
4. Inaccurate for multiaxial loading:
   For complex loading situations, the results may differ slightly from experimental observations.

Comparison with Other Theories

• The Rankine’s theory (maximum normal stress) is suitable for brittle materials, while Tresca’s is for ductile materials.
• The Von Mises theory (maximum shear strain energy) gives more accurate results than Tresca’s for combined stress conditions.
• However, Tresca’s theory remains popular due to its simplicity and safety.

Practical Example

In a shaft subjected to torsion, the maximum shear stress occurs at the outer surface. According to Tresca’s theory, yielding begins when this shear stress equals half of the material’s yield strength in tension.

If the material’s yield strength is , then yielding begins when:

Thus, the shaft design must ensure that the shear stress does not exceed 125 MPa to remain safe.

Conclusion

The Maximum Shear Stress Theory (Tresca Theory) states that yielding in a ductile material occurs when the maximum shear stress equals the shear stress at yield in a simple tension test. It is most suitable for ductile materials, which fail by shear yielding rather than by direct tension. Although conservative, it provides safe and reliable results for mechanical design and remains one of the most widely used theories in strength of materials and machine design.