Cantilever along a Gradually Varying Load:

Let a cantilever beam AB of length l, subjected to a slowly varying load from zero at the free end to w per unit length at the fixed end, as shown in Figure.

Reaction at the support A,

R_A  = Total downward load

= + (1 /2 )× l × w (area of loading diagram)

=+ wl / 2  (plus sign denote right downward)

Figure

Let a section XX at a distance 'x' from the free end.

Shear force at section XX

F _x  = + 1/2   × (w/l) × x × x   (area of loading between B and xx)

=+ wx²/2l

SF at B, F_B = 0, Whereas x = 0,

 SF at A, F_A   =+ wl/ 2  , whereas x = l.

The shear force diagram is in form of parabolic curve like the shear force equation is parabolic equation.

Bending moment at section XX

M_x  =- (½) × (w/l) x (1/3) x (Moment of the area loading between B and XX)

=- wx³/6l     (that means area × CG of loading diagram around XX)

BM at B, M_B = 0 (Whereas x = 0), and

BM at A,

M_x =- wl²/6 (when x = l).

The bending moment diagram is in form of parabolic curve.