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

A prismatic beam (as shown below) has plastic moment capacity of $${{M_P}}$$, then the collapse load $$P$$ of the beam is

A propped cantilever made of a prismatic steel beam is subjected to a concentrated load $$P$$ at mid span as shown.

If load $$P=80$$ $$kN,$$ find the reaction ܴ$$R$$ (in $$kN$$) (correct to $$1$$-decimal place)using elastic analysis ___________