If the following equation establishes equilibrium in slightly bent position, the mid-span deflection of a member shown in the figure is
$${{{d^2}y} \over {d{x^2}}} + {P \over {EI}}y = 0$$

If $$'a'$$ is amplitude constant for $$y,$$ then

For the cantilever beam of span $$3$$ $$m$$ (shown below), a concentrated load of $$20$$ $$kN$$ applied at the free end causes a vertical displacement of $$2$$ $$mm$$ at a section located at a distance of $$1$$ $$m$$ from the fixed end. If a concentrated vertically downward load of $$10$$ $$kN$$ is applied at the section located at a distance of $$1$$ $$m$$ from the fixed end (with no other load on the beam), the maximum vertical displacement in the same beam (in $$mm$$) is ________

The possible location of shear center of the channel section, shown below, is

A box of weight 100 kN shown in the figure is to be lifted without swinging. If all forces are coplanar, the magnitude and direction ($$\theta$$) of the force (F) with respect to x-axis should be