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

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 ______________

Considering the symmetry of a rigid frame as shown below, the magnitude of the bending moment (in $$kN$$-$$m$$) at $$P$$ (preferably using the moment distribution method) is

A prismatic beam (as shown below) has plastic moment capacity of $${{M_P}}$$, then the collapse load $$P$$ of the beam is