Question

A frame is subjected to a load P as shown in the figure. The frame has a constant flexural rigidity EI. The effect of axial load is neglected. The deflection at point A due to the applied load P is

• A. $\frac{1}{3}\frac{P{L}^3}{EI}$
• B. $\frac{2}{3}\frac{P{L}^3}{EI}$
• C. $\frac{P{L}^4}{EI}$
• D. $\frac{4}{3}\frac{P{L}^3}{EI}$

Answer 1

The correct option is D $\frac{4}{3}\frac{P{L}^3}{EI}$


$U_1={\int }_o^L\frac{M^{2}dx}{2EI}=\frac{M^{2}L}{2EI}$
$=\frac{(PL{)}^2.L}{2EI}=\frac{P^2{L}^3}{2EI}$
$U_2={\int }_o^L\frac{(Px{)}^{2}dx}{2EI}=\frac{P^2{L}^3}{6EI}$
$U=U_1+U_2=\frac{4P^2{L}^3}{6EI}$
By Castigliano's theorem,
${\delta }_A=\frac{\partial U}{\partial P}=\frac{\partial }{\partial P}\left(\frac{4P^2{L}^3}{6EI}\right)=\frac{8P{L}^3}{6EI}$
${\delta }_A=\frac{4P{L}^3}{3EI}$