Question

A plane frame PQR (fixed at P and free at R) is shown in the figure. Both members (PQ and QR) have length L, and flexural rigidity, EI. Neglecting the effect of axial stress and transverse shear, the horizontal deflection at free end R, is

• A. $\frac{2F{L}^3}{3EI}$
• B. $\frac{4F{L}^3}{3EI}$
• C. $\frac{F{L}^3}{3EI}$
• D. $\frac{5F{L}^3}{3EI}$

Answer 1

The correct option is B $\frac{4F{L}^3}{3EI}$

$U=U_{PQ}+U_{QP}$

$U=\frac{M^{2}L}{2EI}+{\int }_O^L\frac{\left(M_{x-x}{)}^2\right(dx)}{2EI}$

$U=\frac{(FL{)}^{2}L}{2EI}+{\int }_O^L\left(\frac{\left(Fx{)}^2\right(dx)}{2EI}\right)$

$U=\frac{F^2{L}^3}{2EI}+\frac{F^2{L}^3}{6EI}=\frac{2F^2{L}^3}{3EI}$

By Castigliano's theorem :
$({\delta }_H{)}_R=\frac{\partial U}{\partial F}=\frac{4F{L}^3}{3EI}$