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GATE 2022 ME (ME2) – Question 38

Mechanics of Materials · Transformation of stress and strain, Mohr circle · 2 marks · Multiple choice

A linear elastic structure under plane stress condition is subjected to two sets of loading, I and II. The resulting states of stress at a point corresponding to these two loadings are as shown in the figure below. If these two sets of loading are applied simultaneously, then the net normal component of stress $\sigma_{xx}$ is ________.

loading I is uniaxial tensile stress σ along x; loading II is uniaxial tensile stress σ along an axis rotated 45°.
  1. $3\sigma/2$
  2. $\sigma(1+1/\sqrt2)$
  3. $\sigma/2$
  4. $\sigma(1-1/\sqrt2)$

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Correct answer: (A) $3\sigma/2$

Explanation

The structure is linear elastic, so the stresses from the two loadings **add** (superposition). We want the net normal stress in the $x$ direction.

- **Loading I:** a uniaxial tension $\sigma$ along $x$ contributes $\sigma_{xx}=\sigma$.
- **Loading II:** a uniaxial tension $\sigma$ acting along an axis at $45^\circ$ to $x$. Resolve it onto the $x$ face with the stress-transformation formula
$$\sigma_{xx}=\sigma\cos^2\theta=\sigma\cos^2 45^\circ=\frac\sigma2 .$$

**Net:**
$$\sigma_{xx}=\sigma+\frac\sigma2=\frac{3\sigma}2 .$$

Answer **A**.