Question

For a state of plane strain, the normal strains are given by \(\epsilon_{xx} = 1000 \times 10^{-6}, \epsilon_{yy} = 200 \times 10^{-6}, \) and the maximum shear strain is \( \gamma_{\text{max}} = 1000 \times 10^{-6}. \)The value of shear strain\( \, \gamma_{xy} \, \)for this strain state is

• A. \( 600 \times 10^{-6} \)
• B. \( 183 \times 10^{-6} \)
• C. \( 1000 \times 10^{-6} \)
• D. \( 800 \times 10^{-6} \)

Hint:

To find shear strain \( \gamma_{xy} \), use the relationship between the maximum shear strain and normal strains in the plane strain state.

Answer 1

The Correct Option is A

We are given the normal strains and the maximum shear strain. The relationship between the maximum shear strain and the shear strain \( \gamma_{xy} \) is given by: \[ \gamma_{\text{max}} = \frac{1}{2} \sqrt{(\epsilon_{xx} - \epsilon_{yy})^2 + 4\gamma_{xy}^2}. \] Substitute the given values into the equation: \[ 1000 \times 10^{-6} = \frac{1}{2} \sqrt{(1000 \times 10^{-6} - 200 \times 10^{-6})^2 + 4\gamma_{xy}^2}. \] Solving for \( \gamma_{xy} \), we find: \[ \gamma_{xy} = 600 \times 10^{-6}. \] Final Answer: \text{(A) \( 600 \times 10^{-6} \)}