A rod of length $$L$$ having uniform cross-sectional area $$A$$ is subjected to a tensile force $$P$$ as shown in the figure below. If the Young’s modulus of the material varies linearly from $${E_1}$$ to $${E_2}$$ along the length of the rod, the normal stress developed at the section $$-$$ $$SS$$ is

A simply supported beam of length $$L$$ is subjected to a varying distributed load $$\sin \left( {3\pi x/L} \right)N{m^{ - 1}},$$ where the distance $$x$$ is measured from the left support. The magnitude of the vertical reaction force in $$N$$ at the left support is

A long thin walled cylindrical shell, closed at both the ends, is subjected to an internal pressure. The ratio of the hoop stress (circumferential stress) to longitudinal stress developed in the shell is

For a ductile material, toughness is a measure of