Two plates of $$8$$ $$mm$$ thickness each are connected by a fillet weld of $$6$$ $$mm$$ thickness as shown in the figure.

The permissible stresses in the plate and the weld are $$150$$ $$MPa$$ and $$110$$ $$MP,$$ respectively. Assuming the length of the weld shown in the figure to be the effective length, the permissible load $$P$$ (in $$kN$$) is _____________.

Two prismatic beams having the same flexural rigidity of $$1000\,kN$$-$${m^2}$$ are shown in the figures.

If the mid-span deflections of these beams are denoted by $${\delta _1}$$ and $${\delta _2}$$ (as indicated in the figures), the correct option is

A hollow circular shaft has an outer diameter of $$100$$ $$mm$$ and inner diameter of $$50mm.$$ If the allowable shear stress is $$125$$ $$MPa$$, the maximum torque (in $$kN$$-$$m$$) that the shaft can resist is ___________

Consider the three prismatic beams with the clamped supports $$P,Q$$ and $$R$$ as shown in the figures.

Given that the modulus of elasticity, $$E$$ is $$2.5 \times {10^4}\,\,MPa;$$ and the moment of inertia, $${\rm I}$$ is $$8 \times {10^3}\,\,m{m^4},$$ the correct comparison of the magnitudes of the shear force $$S$$ and the bending moment $$M$$ developed at the supports is