If $$E$$ is the modulus of elasticity in $$GPa$$, $$G$$ is the shear modulus in $$GPa$$ and $$v$$ is the Poisson's ratio of a linear elastic and isotropic material, the three terms are related as

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

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).

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