If the following equation establishes equilibrium in slightly bent position, the mid-span deflection of a member shown in the figure is
$${{{d^2}y} \over {d{x^2}}} + {P \over {EI}}y = 0$$

If $$'a'$$ is amplitude constant for $$y,$$ then

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

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