Question

Find the ratio of strain energies stored in bars A and B of the same axial tensile loads. The bar A is of 50mm diameter through out its length, while bar B though of same length as of A but has diameter of 25 mm for the middle one - third of its length and the remainder is of 50 mm . Assume gradual load

Answer 1

The strain energy formula

$U=F\delta/2$,

where $\delta=$ compression, $F=$ force applied. So,

$U_2/U_1=(F\delta_2/2)/(F\delta_1/2)=\delta_2/\delta_1.$

$\delta_1=\frac{FL}{A_1E}$ and $\delta_2=\frac{F(1/3)L}{A_2E}+\frac{F(1/3)L}{A_1E}+\frac{F(1/3)L}{A_1E}=$

$=\frac{F(1/3)L}{A_2E}+\frac{F(2/3)L}{A_1E}.$

$U_2/U_1=\delta_2/\delta_1=\frac{\frac{F(1/3)L}{A_2E}+\frac{F(2/3)L}{A_1E}}{\frac{FL}{A_1E}}=\frac{A_1+2A_2}{3A_2}=$

$=\frac{\pi d_1^2/4+2\pi d_2^2/4}{3\pi d_2^2/4}=\frac{d_1^2+2d_2^2}{3d_2^2}=\frac{0.05^2+2\cdot0.025^2}{3\cdot0.025^2}=2(J)$