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Maximum shear stress theory provides failure criteria of mechanical components made of a ductile material. This failure criterion is developed by the French mechanical engineer, Henri Tresca and based on his name maximum shear stress theory is also known as the Tresca theory of failure. Due to the enormous contribution in the field of plasticity, Henry Tresca is popular as the father of the field of plasticity.

Maximum shear stress theory is one of the two main failure criteria that are widely used in recent times for predicting the failure of ductile materials. To establish failure criteria of material, all failure theories compare a specific parameter with the same parameter for the uniaxial tension test. The maximum shear stress theory is no exception and the parameter for comparison in Tresca theory is maximum shear stress.

“The maximum shear stress theory states that the failure or yielding of a ductile material will occur when the maximum shear stress of the material equals or exceeds the shear stress value at yield point in the uniaxial tensile test.”

## Maximum shear stress theory formula

Let’s deduce the mathematical form of the above-mentioned Tresca theory statement.

Considering principal stresses, at the yield point, the principal stresses in a uni-axial test, σ_{1} =σ_{y}; σ_{2} = 0 and σ_{3} = 0.

So the maximum shear stress at yielding: σ_{sy} =σ_{1}/2. Therefore σ_{sy} = σ_{y}/2

Now assuming σ_{1} >σ_{2} >σ_{3}; the maximum shear stress for the material is given by:

**τ**_{max}=(σ_{1} -σ_{3} )/2

_{max}=(σ

_{1}-σ

_{3})/2

Now comparing these to maximum shear stresses following Tresca theory, failure will happen when **τ _{max}>=σ_{sy} **

## Safe Design Condition as per Tresca theory of failure

So the design of a mechanical component should be based on the following maximum shear stress theory equation

### τ_{max}<=σ_{sy} or **(σ**_{1} -σ_{3})<=σ_{y}

_{1}-σ

_{3}

The factor of safety (N) can also be calculated based on maximum shear stress theory and given by **N=σ _{sy} /τ_{max}**

Hence, maximum permissible shear stress for designing a component as per maximum shear stress theory is given by **τ _{max} =σ_{sy} /N**

The failure envelope for Tresca theory of failure is provided in Fig. 1 below:

## Steps for using the Maximum Shear Stress Theory

To use the maximum shear stress theory in problem solving the following steps are necessary to be followed:

- Step 1: Determine the three principal stresses (σ
_{1},σ2, and σ3) from the tri-axial stress system using principal stress equations or Mohr’s circle method. - Step 2: Find out the maximum (σ1) and the minimum (σ3) principal stresses.
- Step 3: Determine the value of the maximum shear stress τmax=(σ1 -σ3 )/2.
- Step 4: Find out the allowable stress value of the material; allowable stress= σsy /N or σy /2N as mentioned above (N=Factor of safety)
- Step 5: Compare the value calculated at step 3 with the allowable value found at step 4. If the Value at step 3 is less than the allowable value at step 4, then the design is safe as per the maximum shear stress theory.

## Maximum Shear Stress Theory vs Von Mises Stress Theory

Von Mises and Tresca failure criteria are normally presented jointly with little discrimination between them. However, there are a few differences between them which are tabulated below:

Maximum Shear Stress Theory | Von Mises Stress Theory |

Maximum Shear Stress theory or Tresca theory of failure relates to the maximum shear stress of ductile materials. | Von Mises stress theory represents the maximum distortion energy of a ductile material. |

This theory is considered to be more conservative. | Considered less conservative when compared with Tresca theory. |

Component cost increases. | Optimized Component Cost. |

Required only two principal stress equations (σ_{max} and σ_{min}) to calculate the maximum shear stress. | Use all the three principal stresses (σ_{1} ,σ_{2} , and σ_{3}) in its equation for calculating Von Mises Stress. |

**Table: Tresca theory vs Von Mises theory**

The failure envelope for Tresca theory and Von Mises theory is given in Fig. 2:

## Limitations of Maximum Shear Stress Theory

- The maximum shear stress theory does not give accurate results for the state of pure shear stresses developed by the Torsion test.
- The Tresca theory provides conservative results leading to an increase in component cost.
- This theory is not applicable to brittle materials.

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