The inside and outside radii of a thick- walled cylindrical pressure vessel are denoted by $$a$$ and $$b,$$ respectively. If the vessel is subjected to an internal pressure $$P,$$ then the magnitude of the radial stress $${\sigma _{rr}}$$ is

The simply-supported beam shown in the Figure is loaded symmetrically using two equal point loads $$P.$$ The radius of curvature of the deflection-curve is $$15$$ m for the portion of the beam that is subjected to pure bending. The vertical deflection (in mm) at point $$M,$$ equidistant from both the supports is _______ (up to two decimal places).

For a two dimensional state-of-stress defined as $${\sigma _{xx}} = {\sigma _{yy}} = {\tau _{xy}} = S,$$ the Mohr's circle of stress has

A reversible heat engine $$(E)$$ operating in a cycle interacts with three reservoirs $$1,$$ $$2$$ and $$3$$ maintained at temperature $${T_1} = 500K,$$ $${T_2} = 400K,$$ and $${T_3} = 300K,$$ respectively. The engine receives $$10$$ $$kJ$$ of heat from reservoir $$1$$ and rejects $$3$$ $$kJ$$ to reservoir $$3.$$ The net work output, $${W_{net}}$$ (in $$kJ$$) from the engine is