The buckling load $$P = {P_{cr}}$$ for the column $$AB$$ in figure, as $${K_T}$$ approaches infinity, becomes $$\alpha {{{\pi ^2}E{\rm I}} \over {{L^2}}}$$

Where $$\alpha $$ is equal to

For the section shown below, second moment of the area about an axis $$d/4$$ distance above the bottom of the area is

A thin-walled long cylindrical tank of inside radius $$'r'$$ is subjected simultaneously to internal gas pressure $$P$$ and axial compressive force $$F$$ at its ends. In order to produce 'pure shear' state of stress in the wall of the cylinder, $$F$$ should be equal to

Consider the beam $$AB$$ shown in the figure below. Part $$AC$$ of the beam is rigid while Part $$CB$$ has the flexural rigidity $$EI$$. Identify the correct combination of deflection at end $$B$$ and bending moment at end $$A,$$ respectively.