Question

An infinitesimal square element \(PQRS\) is shown in the figure. The x and y axes are also marked in the figure. The strains on the element are given by: \[ \varepsilon_{xx} = 500 \times 10^{-6}, \varepsilon_{yy} = 100 \times 10^{-6}, \varepsilon_{xy} = 0 \]

Which of the following statements is/are CORRECT?

• A. Percentage change in length of the diagonal PR is 0.03
• B. Change in angle between PR and QS is \(4 \times 10^{-4}\) rad
• C. Change in angle between PR and QS is \(2 \times 10^{-4}\) rad
• D. Percentage change in length of the diagonal QS is 0.03

Hint:

Diagonal strain is the average of normal strains. Angular distortion is the difference of normal strains, halved for diagonals.

Answer 1

The Correct Option is A, B, D

Step 1: Direct strain along PR.
Diagonal PR makes \(\theta = 45^\circ\). \[ \varepsilon_{PR} = \varepsilon_{xx}\cos^2\theta + \varepsilon_{yy}\sin^2\theta = \tfrac{1}{2}(500 \times 10^{-6} + 100 \times 10^{-6}) = 300 \times 10^{-6} \] Percentage change in length = \(300 \times 10^{-6} \times 100 = 0.03%\). Hence, (A) is correct.

Step 2: Direct strain along QS.
Diagonal QS makes \(\theta = 135^\circ\). Using same relation: \[ \varepsilon_{QS} = 300 \times 10^{-6} \Rightarrow 0.03% \, \text{change} \] So (D) seems possible, but the real check is angular distortion.

Step 3: Change in angle between diagonals.
Angular change: \[ \Delta \phi = \varepsilon_{xx} - \varepsilon_{yy} = (500 - 100) \times 10^{-6} = 400 \times 10^{-6} \] Half of this is the change in angle between diagonals: \[ \Delta \phi = 2 \times 10^{-4} \, rad \] So (C) is correct. Final Answer:
\[ \boxed{(A) \; \text{and} \; (C)} \]