The ultimate collapse load $$(P)$$ in terms of plastic moment $${M_P}$$ by kinematic approach for a propped cantilever of length $$L$$ with $$P$$ acting at its mid-span as shown in the figure, would be

For the truss shown below, the member $$PQ$$ is short by $$3$$ $$mm.$$ The magnitude of vertical displacement of joint $$R$$ (in $$mm$$) is ______________________

In a beam of length $$L,$$ four possible influence line diagrams for shear force at a section located at a distance of $${L \over 4}$$ from the left end support (marked as $$P,Q,R$$ and $$S$$ ) are shown below. The correct influence line diagram is

The degree of static indeterminacy of a rigid jointed frame PQR supported as shown in the figure is