Question

A solid steel cube constrained on all six faces is heated so that the temperature rises uniformly by $\Delta T$. If the thermal coefficient of the materials is $\alpha$, Young's modulus is E and the Poisson's ratio is $v$, the thermal stress developed in the cube due to heating is

• A. $-\frac{\alpha \left(\Delta T\right)E}{(1-2v)}$
  
• B. $-\frac{2\alpha \left(\Delta T\right)E}{(1-2v)}$
  
• C. $-\frac{3\alpha \left(\Delta T\right)E}{(1-2v)}$
  
• D. $-\frac{\alpha \left(\Delta T\right)E}{3(1-2v)}$
  

Answer 1

The correct option is A $-\frac{\alpha \left(\Delta T\right)E}{(1-2v)}$


Let the side of cube be a



Now since the cube is uniformly constrained to expand, the stress produced in all the three directions will be same

$\therefore$ Strain in x direction $=\alpha \left(\Delta T\right)$

$=\frac{{\sigma }_x}{E}-\frac{v{\sigma }_y}{E}-\frac{v{\sigma }_z}{E}$

${\sigma }_x={\sigma }_y={\sigma }_z=\sigma$

$\therefore \sigma =\frac{\alpha \left(\Delta T\right)E}{(1-2v)}$