Question

A square plate (a $\times$ a) rigidly held at three edges is free to move along the fourth edge. If temperature of the plate is raised by temperature 't', then the free expansion at the fourth edge will be (coefficient of thermal expansion of the material = a, modulus of elasticity of the material = E and its Poisson's ratio = μ)

• A. $a\alpha t\mu$
• B. $a\alpha t(1+\mu )$
• C. $a(\alpha t+\frac{\alpha t\mu }{E})$
• D. $a\alpha t(1-\mu )$

Answer 1

The correct option is B $a\alpha t(1+\mu )$



Free expansion in X-direction = $L\alpha t=a\alpha t$. The square plate is fixed in Y-direction. When temperature in increased, the plate will be in compression along Y-axis. Therefore lateral expansion in X-direction due to compressive stress $\sigma$ in Y-direction
$\mu \times \frac{\sigma }{E}\times a=\mu \times \frac{\alpha tE}{E}\times a$
$\therefore$ Total elongation $=a\alpha t+\mu \alpha ta=a\alpha t(1+\mu )$