  # Maximum Shear Stress Theory

By BYJU'S Exam Prep

Updated on: September 25th, 2023 When a material is subjected to a variety of complex stresses, it becomes challenging to determine the point of yielding or fracture for the material. Various theories of failure have been developed to define the criteria for failure of the material. One such theory of failure is the maximum shear stress theory of failure.

All failure theories compare a certain parameter with the same parameter for the uniaxial tension test in order to establish material failure criteria. As the name suggests, the maximum shear stress theory of failure compares the maximum shear stress parameter. In this article, we will study the maximum shear stress theory in detail.

## What is Maximum Shear Stress Theory?

According to the maximum shear stress theory, a material will fail or yield when its maximum shear stress equals or exceeds the shear stress value at the yield point in the uniaxial tensile test. This theory is suitable for ductile materials. The maximum shear stress theory was given by Henry Tresca. Hence Maximum shear stress theory is also known as Tresca’s theory of failure.

## Maximum Shear Stress Theory Formula

In a uniaxial tensile test, the principal stress at the yield point will be σ1y, σ20, σ30

where σy= yield stress

So, maximum shear stress at yielding for the uniaxial test, τuniaxial1/2=σy/2

Then according to maximum shear stress theory, for no failure:

τmax(biaxial or triaxial) ≤ τuniaxial

τmax≤ σy/2

For design, the maximum shear stress formula is given as

τmax≤ (σy/F.O.S)/2

where F.O.S = factor of safety

In this section, we will understand the failure criteria for biaxial and triaxial loading conditions, according to the maximum shear stress theory.

If σ1, σand σare the principal stresses, then the maximum shear stress under triaxial stress condition will be,

τmax = maximum of [(σ12)/2, (σ23)/2 and (σ31)/2]

Then according to maximum shear stress theory, for no failure:

|(σ12)/2| ≤ σy/2⇒|(σ12)|≤ σy

|(σ23)/2| ≤ σy/2⇒|(σ23)|≤ σy

|(σ31)/2| ≤ σy/2⇒|(σ31)|≤ σy

In the biaxial loading condition, one of the principal stresses is zero. Here, let’s assume that 3=0. The maximum shear stress under biaxial stress conditions will be-

τmax=maximum of [(σ12)/2, σ2/2 and σ1/2]

Then according to maximum shear stress theory, for no failure:

|(σ12)/2| ≤ σy/2⇒|(σ12)|≤ σy

2/2| ≤ σy/2⇒|σ2|≤ σy

1/2| ≤ σy/2⇒|σ1|≤ σy

## Failure Envelope for Maximum Shear Stress Theory

According to the maximum shear stress formula for the biaxial loading condition given in the previous section, a failure envelope can be drawn. The failure envelope for Tresca’s theory of failure is hexagonal in shape as shown below. ## Limitation of Maximum Shear Stress Theory

For forecasting the failure of ductile materials, the maximum shear stress theory has become one of the popular failure theories in recent years. However, the maximum shear stress theory has certain limitations which are listed below –

• It is not applicable to brittle materials.
• Maximum shear stress theory provides overly safe results which may make the component expensive.
• It cannot be applied to components subjected to hydrostatic loading. Under hydrostatic loading, all the principal stresses are equal so maximum shear stress will be zero and the component will never fail which is impossible.

## Maximum Shear Stress Problems

Maximum shear stress MCQ is provided in this section. These Maximum Shear Stress problems can give a brief overview of the types of questions expected from the maximum shear stress theory in various competitive exams.

1. The maximum shear stress theory is used for which of the following

1. Ductile materials
2. Brittle materials
3. Plastic materials
4. Non-ferrous materials

Answer: (a) The maximum shear stress theory is used for ductile materials.

2. The maximum shear stress theory is also known as

1. Rankine’s theory
2. St. Venant’s theory
3. Tresca’s theory
4. Von Mises theory

Answer: (c) The maximum shear stress theory is also known as Tresca’s theory.

3. According to maximum stress criteria, at what ratio of maximum shear stress to yield stress of the material, does yielding of material take place?

1. 2
2. 2/3
3. 1/3
4. 1/2

Answer: (d) According to maximum stress criteria, the ratio of maximum shear stress to yield stress of material at which yielding of material takes place is 1/2.

4. The shape of the failure envelope for maximum shear stress theory is

1. Rhombic
2. Hexagonal
3. Elliptical
4. Rectangular

Answer: (b) The shape of the failure envelope for maximum shear stress theory is hexagonal.

5. According to the maximum shear stress theory of failure, yielding occurs in the material when

1. Maximum shear stress = yield stress
2. Maximum shear stress = 2 x yield stress
3. Maximum shear stress = ½ x yield stress
4. Maximum shear stress = 2 x yield stress

Answer: (c) According to the maximum shear stress theory of failure, yielding occurs in the material when maximum shear stress = ½ x yield stress

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