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 $$\left( {{M_P},\,2{M_P}} \right)$$ as shown.

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

A propped cantilever of span $$L$$ carries a vertical concentrated load at the mid-span. If the plastic moment capacity of the section is $${M_P}$$, the magnitude of the collapse load is

For formation of collapse mechanism in the following figure, the minimum value of $${{P_u}}$$ is $$c{M_P}/L.$$ $${M_P}$$ and $$3{M_P}$$ denote the plastic moment capacities of beam sections as shown in this figure. The value of $$c$$ is ________.

A fixed end beam is subjected to a load, W at $$1/3rd$$ span from the left support as shown in the figure. The collapse load of the beam is