Question

Three principal stresses at a point in a material are 300 MPa, 250 MPa, and 100 MPa. If the yielding just starts at that point, the yield strength (in MPa) of the material as per Tresca criterion is ............
Answer: The yield strength lies between 199 to 201 MPa.

Hint:

In the Tresca criterion, the yield strength is determined by twice the maximum shear stress, which is the difference between the maximum and minimum principal stresses divided by 2.

Answer 1

Correct Answer: 199

To determine the yield strength of the material using the Tresca criterion, we identify the maximum shear stress based on the principal stresses. The Tresca criterion states that yielding begins when the maximum shear stress in the material reaches a critical value, equal to half the yield strength in pure shear. The principal stresses given are σ₁ = 300 MPa, σ₂ = 250 MPa, and σ₃ = 100 MPa.

First, calculate the absolute differences between the principal stresses:

σ₁ - σ₂ = 300 - 250 = 50 MPa

σ₂ - σ₃ = 250 - 100 = 150 MPa

σ₁ - σ₃ = 300 - 100 = 200 MPa

The maximum shear stress is defined as half the maximum difference among these:

τ_max = 1/2 * max(50,150,200) = 1/2 * 200 = 100 MPa

According to the Tresca criterion, τ_max = Syield/2, where Syield is the yield strength. Solving for Syield gives us:

Syield = 2 * τ_max = 2 * 100 = 200 MPa

Thus, the yield strength of the material per the Tresca criterion is 200 MPa.

This value of 200 MPa lies within the provided answer range of 199 MPa to 201 MPa, confirming it to be correct.