Question

A structural member under loading has a uniform state of plane stress which in usual notations is given by $\sigma_x = 3P$, $\sigma_y = -2P$ and $\tau_{xy} = \sqrt{2}\,P$, where $P>0$. The yield strength of the material is 350 MPa. If the member is designed using the maximum distortion energy theory, then the value of $P$ at which yielding starts (according to the maximum distortion energy theory) is

• A. 70 MPa
• B. 90 MPa
• C. 120 MPa
• D. 75 MPa

Hint:

For plane stress problems, the von Mises stress formula simplifies calculations by combining normal and shear stresses into a single equivalent stress.

Answer 1

The Correct Option is A

According to the maximum distortion energy (von Mises) theory, yielding starts when the equivalent stress satisfies \[ \sigma_{vm} = \sqrt{\sigma_x^2 + \sigma_y^2 - \sigma_x \sigma_y + 3\tau_{xy}^2} = \sigma_y^{(yield)}. \] Here, $\sigma_x = 3P$, $\sigma_y = -2P$, and $\tau_{xy} = \sqrt{2}P$.
Step 1: Substitute into the von Mises equation.
\[ \sigma_{vm} = \sqrt{(3P)^2 + (-2P)^2 - (3P)(-2P) + 3(\sqrt{2}P)^2}. \] Step 2: Simplify the expression.
\[ = \sqrt{9P^2 + 4P^2 + 6P^2 + 3(2P^2)} \] \[ = \sqrt{9P^2 + 4P^2 + 6P^2 + 6P^2} \] \[ = \sqrt{25P^2} = 5P. \] Step 3: Apply yielding condition.
\[ 5P = 350 \quad \Rightarrow \quad P = 70\ \text{MPa}. \] Final Answer: 70 MPa