Question

Axial rigidity and flexural rigidity of members of frame given below is AE and EI respectively. The reaction developed at hinge would be:

• A. $\frac{7P{L}^2}{(\frac{24I}{A}+8L^2)}$
• B. $\frac{11P{L}^2}{(\frac{48I}{A}+16L^2)}$
• C. $\frac{8P{L}^2}{(\frac{32I}{A}+7L^2)}$
• D. None of these

Answer 1

The correct option is B $\frac{11P{L}^2}{(\frac{48I}{A}+16L^2)}$

From compatibility equation
Deflection at point of hinge in beam compression in vertical member
$\frac{\left(\frac{P}{L}\right)\times L^4}{8EI}+\frac{P{\left(\frac{L}{2}\right)}^3}{3EI}+\frac{P{\left(\frac{L}{2}\right)}^2}{2EI}\times \left(\frac{L}{2}\right)-\frac{R{L}^3}{3EI}=\frac{RL}{AE}$
$\frac{P{L}^3}{8EI}+\frac{P{L}^3}{24EI}+\frac{P{L}^3}{16EI}-\frac{R{L}^3}{3EI}=\frac{RL}{AE}$
$\frac{P{L}^3}{EI}\left(\frac{6+2+3}{48}\right)=R(\frac{L}{AE}+\frac{L^3}{3EI})$
$\frac{11P{L}^3}{48EI}=R(\frac{L}{AE}+\frac{L^3}{3EI})$
$R=\frac{11P{L}^3}{48EI(\frac{L}{AE}+\frac{L^3}{3EI})}$
$=\frac{11P{L}^2}{48I(\frac{I}{A}+\frac{L^2}{3l})}=\left(\frac{11P{L}^2}{\frac{48I}{A}+16L^2}\right)$