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GATE 2022 ME (ME2) – Question 18

Machine Design · Design for static and dynamic loading, failure theories · 1 mark · Multiple choice

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 $\tau_{xy}=\sqrt2P$, 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 is

  1. 70 MPa
  2. 90 MPa
  3. 120 MPa
  4. 75 MPa

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Correct answer: (A) 70 MPa

Explanation

**Step 1: the criterion.** The maximum distortion energy (von Mises) theory says yielding starts when the equivalent stress equals the yield strength. For plane stress:
$$\sigma_e=\sqrt{\sigma_x^2+\sigma_y^2-\sigma_x\sigma_y+3\tau_{xy}^2}.$$

**Step 2: substitute** $\sigma_x=3P$, $\sigma_y=-2P$, $\tau_{xy}=\sqrt2P$:
$$\sigma_e^2=9P^2+4P^2-(3P)(-2P)+3(2P^2)=9P^2+4P^2+6P^2+6P^2=25P^2,$$
so $\sigma_e=5P$.

**Step 3: set $\sigma_e=S_y$.**
$$5P=350\;\Rightarrow\;P=\mathbf{70\ MPa}.$$