The beam of an overall depth $$250$$ $$mm$$ (shown below) is used in a building subjected to two different thermal environments. The temperatures at the top and bottom surfaces of the beam are $${36^ \circ }C$$ and $${72^ \circ }C$$ respectively. Considering coefficient of thermal expansion $$\left( \alpha \right)$$ as $$1.50 \times {10^{ - 5}}\,\,$$ per$${}^ \circ C,$$ the vertical deflection of the beam (in $$mm$$) at its mid-span due to temperature gradient is ______________

The first moment of area about the axis of bending for a beam cross-section is

The values of axial stress $$\left( \sigma \right)$$ in $$kN/{m^2},$$ bending moment $$(M)$$ in $$kNm,$$ and shear force $$(V)$$ in $$kN$$ acting at point $$P$$ for the arrangement shown in the figure are respectively.

For the state of plane stress (in $$MPa$$) shown in the figure below, the maximum shear stress (in $$MPa$$) is ___________