Question

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 shown in the figure below. If these two sets of loading are applied simultaneously, then the net normal component of stress $\sigma_{xx$ is ________________.}

• A. $3\sigma/2$
• B. $\sigma(1 + 1/\sqrt{2})$
• C. $\sigma/2$
• D. $\sigma(1 - 1/\sqrt{2})$

Hint:

For rotated stress states, always use the plane stress transformation equations. Superposition applies directly for linear elastic systems.

Answer 1

The Correct Option is A

Under Loading–I, the stress state is a uniaxial normal stress along the $x$ direction. Thus, \[ \sigma_{xx}^{(I)} = \sigma. \] Step 1: Stress transformation for Loading–II.
The second loading applies the same magnitude $\sigma$, but the element is rotated by $45^\circ$. For a uniaxial stress $\sigma$ rotated by angle $\theta$, the transformed normal stress is: \[ \sigma_{xx}^{(II)} = \sigma \cos^2\theta. \] For $\theta = 45^\circ$, \[ \cos^2 45^\circ = \frac{1}{2}, \] so \[ \sigma_{xx}^{(II)} = \sigma \cdot \frac{1}{2}. \] Step 2: Superposition of the two loadings.
Since the structure is linear elastic, stresses add: \[ \sigma_{xx} = \sigma_{xx}^{(I)} + \sigma_{xx}^{(II)} = \sigma + \frac{\sigma}{2} = \frac{3\sigma}{2}. \] Step 3: Final conclusion.
The combined normal stress is: \[ \sigma_{xx} = \frac{3\sigma}{2}. \] Final Answer: $3\sigma/2$