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 ________

A simply supported beam is subjected to a uniformly distributed load of intensity $$w$$ per unit length. On half of the span from one end. The length of the span and the flexural stiffness are denoted as/and $$EI$$, respectively. The deflection at mid-span of the beam is

In the cantilever beam $$PQR$$ shown in figure below, the segment $$PQ$$ has flexural $$EI$$ and the segment $$QR$$ has infinite flexural rigidity

The deflection of the beam at $$' R '$$ is

In the cantilever beam $$PQR$$ shown in figure below, the segment $$PQ$$ has flexural $$EI$$ and the segment $$QR$$ has infinite flexural rigidity

The deflection and slope of the beam at $$'Q'$$ are respectively