Question

An element is subjected to a tensile stress of 60 MPa and a shear stress of 40 MPa. If the material has the yield strength obtained in a simple tensile test is 320 MPa, the factor of safety based on maximum principal stress theory is

• A. 4
• B. 3
• C. (2)5
• D. 2

Hint:

Use the maximum principal stress theory for brittle materials; calculate principal stresses and compare with yield strength to find the factor of safety.

Answer 1

The Correct Option is A

Step 1: Recall the maximum principal stress theory.
The maximum principal stress theory (Rankine’s theory) states that failure occurs when the maximum principal stress in the material exceeds the yield strength in a simple tensile test. The factor of safety (FOS) is: \[ \text{FOS} = \frac{\text{Yield strength}}{\text{Maximum principal stress}}. \] Step 2: Calculate the principal stresses.
The element is subjected to a tensile stress (normal stress) and a shear stress. Assume a 2D stress state:
\( \sigma_x = 60 \, \text{MPa} \) (tensile),
\( \sigma_y = 0 \, \text{MPa} \) (since only one normal stress is given),
\( \tau_{xy} = 40 \, \text{MPa} \).
The principal stresses \( \sigma_1 \) and \( \sigma_2 \) are given by: \[ \sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}, \] \[ \frac{\sigma_x + \sigma_y}{2} = \frac{60 + 0}{2} = 30, \] \[ \frac{\sigma_x - \sigma_y}{2} = \frac{60 - 0}{2} = 30, \] \[ \left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2 = 30^2 + 40^2 = 900 + 1600 = 2500, \] \[ \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2} = \sqrt{2500} = 50, \] \[ \sigma_1 = 30 + 50 = 80 \, \text{MPa}, \] \[ \sigma_2 = 30 - 50 = -20 \, \text{MPa}. \] Step 3: Determine the maximum principal stress.
The maximum principal stress is the largest in magnitude (considering absolute values for safety):
\( \sigma_1 = 80 \, \text{MPa} \),
\( \sigma_2 = -20 \, \text{MPa} \),
Maximum principal stress = 80 MPa (since the theory typically uses the largest tensile stress for yielding in tension).
Step 4: Calculate the factor of safety.
The yield strength from the simple tensile test is 320 MPa: \[ \text{FOS} = \frac{\text{Yield strength}}{\text{Maximum principal stress}} = \frac{320}{80} = 4. \] Step 5: Select the correct answer.
The factor of safety based on maximum principal stress theory is 4, matching option (1).