Question

A propped cantilever beam of span $L$ is subjected to two equal loads of magnitude $P$ at equal distance from both ends i.e., $L/3$. The beam has plastic moment capacity $M_P$. The collapse load is:

• A. $\frac{4M_P}{L}$
  
• B. $\frac{5M_P}{L}$
  
• C. $\frac{6M_P}{L}$
  
• D. None of these

Answer 1

The correct option is A $\frac{4M_P}{L}$


Redundancy $=1$
No.of hinges to form mechanism $=(r+1)=(1+1)=2$
Mechanism -1:
$M_P\theta +M_P\left(\frac{3\theta }{2}\right)=P\cdot \left(\frac{\theta L}{3}\right)+P\cdot \left(\frac{\theta }{2}\right)\left(\frac{L}{3}\right)$
$\begin{array}{c}P=\frac{5M_P}{L}\end{array}$
Mechanism -2:
$M_P\left(\frac{\theta }{2}\right)+M_P\left(\frac{3\theta }{2}\right)=P\left(\frac{\theta }{2}\right)\left(\frac{L}{3}\right)+P\left(\frac{\theta L}{3}\right)$
$\begin{array}{c}P=\frac{4M_P}{L}\end{array}$
$\therefore$ Collapse load $=\frac{4M_P}{L}$ (minimum of above two value).