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

Consider a propped cantilever $$ABC$$ under two loads of magnitude $$P$$ each as shown in figure below. Flexural rigidity of the beam is $$EI$$.

The reaction at $$C$$ is

Consider a propped cantilever $$ABC$$ under two loads of magnitude $$P$$ each as shown in figure below. Flexural rigidity of the beam is $$EI$$.

The rotation at $$B$$ 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