If the principal stresses in a plane stress problem are $${\sigma _1}$$ $$= 100$$ MPa, $${\sigma _2}$$ $$= 40$$ MPa, the magnitude of the maximum shear stress (in MPa) wil be

A solid shaft of diameter, $$d$$ and length, $$L$$ is fixed at both the ends. A torque, $${T_0}$$ is applied at a distance, $$L/4$$ from the left end as shown in the figure below.

The maximum shear stress in the shaft is

A triangular-shaped cantilever beam of uniform- thickness is shown in the figure. The young's modulus of the material of the beam is $$E$$. $$A$$ concentrated load $$P$$ is applied at the free end of the beam.

The area moment of inertia of inertia about the neutral axis of a cross-section at a distance $$x$$ measured from the free end is

A frame of two arms of equal length $$L$$ is shown in the adjacent figure. The flexural rigidity of each arm of the frame is $$EI$$. The vertical deflection at the point of application of load $$P$$ is