Question

At a point in a steel member, a major principal stress is 200 MPa (tensile) and minor principal stress is compressive. If uniaxial tensile yield stress is 250 MPa, then according to maximum shear stress theory, the magnitude of the minor principal stress (compressive) at which yielding will commence is ...........

• A. 200 MPa
• B. 100 MPa
• C. 50 MPa
• D. 25 MPa

Hint:

According to the maximum shear stress theory (Tresca's criterion), yielding begins when the maximum shear stress reaches half the yield stress in simple tension.

Answer 1

The Correct Option is C

According to the maximum shear stress theory (also known as Tresca's criterion), yielding occurs when the maximum shear stress reaches half of the yield stress in simple tension. The relationship is given by: \[ \tau_{max} = \frac{\sigma_{y}}{2} \] Where \( \sigma_{y} \) is the yield stress in simple tension. In this case, the uniaxial tensile yield stress \( \sigma_{y} \) is given as 250 MPa. Therefore, the maximum shear stress is: \[ \tau_{max} = \frac{250}{2} = 125 \text{ MPa} \] The maximum shear stress in a biaxial stress system is given by: \[ \tau_{max} = \frac{\sigma_1 - \sigma_2}{2} \] Where \( \sigma_1 \) is the major principal stress (200 MPa) and \( \sigma_2 \) is the minor principal stress (which is compressive). Now, substituting the values: \[ 125 = \frac{200 - \sigma_2}{2} \] Solving for \( \sigma_2 \): \[ 250 = 200 - \sigma_2 \quad ⇒ \quad \sigma_2 = 200 - 250 = -50 \text{ MPa} \] Thus, the magnitude of the minor principal stress is 50 MPa (compressive).