Question

A rigid beam is hinged at one end and supported on linear elasitc spring (both having a stiffness of 'k') at points '1' and'2', and an inclined load acts at '2', as shown in figure

which of the following option represents the deflection ${\delta }_{1}and{\delta }_2$ at points 1 and 2?

• A. ${\delta }_1=\frac{2}{5}\left(\frac{2P}{k}\right)and{\delta }_2\frac{4}{5}\left(\frac{2P}{k}\right)$
• B. ${\delta }_1=\frac{2}{5}\left(\frac{P}{k}\right)and{\delta }_2\frac{4}{5}\left(\frac{P}{k}\right)$
• C. ${\delta }_1=\frac{2}{5}\left(\frac{P}{\sqrt{2k}}\right)and{\delta }_2\frac{4}{5}\left(\frac{2P}{\sqrt{2k}}\right)$
• D. ${\delta }_1=\frac{2}{5}\left(\frac{\sqrt{2}P}{k}\right)and{\delta }_2\frac{4}{5}\left(\frac{\sqrt{2}P}{k}\right)$

Answer 1

The correct option is B ${\delta }_1=\frac{2}{5}\left(\frac{P}{k}\right)and{\delta }_2\frac{4}{5}\left(\frac{P}{k}\right)$


Deflected shape of rigid beam,
$R_1=k\delta ;R_2=2k\delta$
Taking moment about O,
$\sum M=0$
$R_{1}l+R_2.2l-P.2l=0$
$R_1+2R_2=2P$
$\Rightarrow k\delta +4k\delta =2P$
$\Rightarrow \delta =\frac{2P}{5k}$
Deflection at point 1,
${\delta }_1=\delta =\frac{2P}{5k}$
Deflection at point 2,
${\delta }_1=2\delta =\frac{4P}{5k}$