The given propped cantilever beam can be assumed to be consisting of two types of loads. First part is simple cantilever with UDL and second part is cantilever beam with point load of R₂ at end.

Balancing forces we get

R₁ + R₂ = qL            ----------(1) 

Balancing deflection at end point as net deflection at end is zero.

Deflection at B due to UDL alone

${\delta }_B=\frac{q{L}^4}{8EI}$

Deflection at B due to R₂ alone, 

${\delta }_B^{{}^{\prime }}=\frac{R_2{L}^3}{3EI}$

$\therefore {\delta }_B={\delta }_B^{{}^{\prime }}$

$\Rightarrow \frac{q{L}^4}{8EI}=\frac{R_2{L}^3}{3EI}\Rightarrow R_2=\frac{3qL}{8}$

From 1) 

$R_1=qL-R_2$

$=qL-\frac{3qL}{8}=\frac{5qL}{8}$

$Moment,M=R_{2}L-qL\times \frac{L}{2}$

$=\frac{3q{L}^2}{8}-\frac{q{L}^2}{2}=-\frac{q{L}^2}{8}$

At end supports, moment (couple) developed is zero, because re is no distance to take perpendicular acting load. As distance is zero, moment is obviously zero.

The shear force at cross section of a beam may also be defined as unbalanced vertical force to left or right of section. It is also algebraic sum of all forces I get to left to right of section.

The relationship between slope and maximum bending moment is inversely proportional because, For example in simply supported beams slope is maximum at supports and zero at midspan of a symmetrically loaded beam where as bending moment is zero at supports and maximum at mid span. Hence we conclude that slope and maximum bending moment are inversely proportional to each or in a case of simply supported beam.

As train?s wheels (rolling stock) move in rolling way. The upcoming load will be considered as rolling load.

Point of contraflexure in a beam is a point at which bending moment changes its sign from positive to negative and vice versa. In case of overhanging beam, re will be two points of contraflexure.

SF @ AB is 10 kN
FA = 10 kN
FB = 10 kN.

At support of a member, re is no distance prevailing to take upcoming load. As we know moment is a product of force and perpendicular distance, but at hinge (end support) distance is zero. Hence moment developed is zero.

In above figure we can observe that beam is supported at both ends so beam is fixed at both ends. Hence support is a fixed support.

A beam which is fixed at both supports is called fixed beam or encastered beam. All framed structures are examples of fixed beams.