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## Q. 1.22

A thin rectangular steel plate 100 mm by 50 mm undergoes deformations 0.05 mm and 0.01 mm in x and y directions due to stresses a,$\sigma _x$ and $\sigma _y$ as shown in Fig. 1.32. Determine the magnitude of $\sigma _x$ and $\sigma _y$ if modulus of elasticity is 200 GPa and Poisson’s ratio is 0.3.

## Verified Solution

It is plane stress problem where,

$\varepsilon_x=\frac{0.05}{100}=0.005 mm / mm$ , and

$\varepsilon_y=\frac{0.01}{50}=0.0002 mm / mm$

Substituting in Eq. (1.231),

$\sigma_x =\frac{E}{1-\mu^2}\left(\varepsilon_x+\mu \varepsilon_y\right)$

$=\frac{200000}{1-0.3^2}(0.0005+0.3 \times 0.0002)$ =123.07  MPa
$\sigma_y =\frac{E}{1-\mu^2}\left(\varepsilon_y+\mu \varepsilon_x\right)$

$=\frac{200000}{1-0.3^2}(0.0002+03 \times 0.0005)$ =76.92  MPa