Question

A fixed-end beam is subjected to a concentrated load (P) as shown in the figure. The beam has two different segments having different plastic moment capacities Mp and 2Mp as shown.


The minimum value of load (P) at which the beam would collapse (ultimate load) is _______.

Answer 1

First mechanism :

$2M_p\theta +2M_p\theta +2M_p\varphi +M_p\varphi =P\times \frac{2L}{3}\theta$

$4M_p\theta +3M_p\varphi =\frac{2PL}{3}\theta (\because \frac{2L}{3}\theta =\frac{4L}{3}\varphi \Rightarrow \theta =2\varphi \text{)}$

$P=\frac{8.25M_p}{L}$

Second mechanism:


$5M_p\theta =P\times \frac{2L}{3}\theta$ $\Rightarrow P=\frac{7.5M_p}{L}$

$P=min\{\frac{8.25M_p}{L},\frac{7.5M_p}{L}\}$

$P=\frac{7.5M_p}{L}$