Question

$U_1$ and $U_2$ are the strain energies stored in a prismatic bar due to axial tensile forces $P_1$ and $P_2$, respectively. The strain energy U stored in the same bar due to combined action of $P_1$ and $P_2$ will be

• A. $U=U_1+U_2$
• B. $U=U_1{U}_2$
• C. $U<U_1+U_2$
• D. $U>U_1+U_2$

Answer 1

The correct option is D $U>U_1+U_2$

Strain energy stored in a prismatic bar due to axial load $P_1$,
$U_1=\frac{P_1^{2}L}{2AE}$
Due to load, $P_2,U_1=\frac{P_2^{2}L}{2AE}$
Due to combined action of $P_1$ and $P_2$
$U=\frac{{(P_1+P_2)}^{2}L}{2AE}$
$=\frac{P_1^{2}L}{2AE}+\frac{P_2^{2}L}{2AE}+\frac{2P_1{P}_{2}L}{2AE}$
$U=U_1+U_2+\frac{2P_1{P}_{2}L}{2AE}$
$\Rightarrow U>U_1+U_2$