SLOPE AND DEFLECTION FOR A CANTILEVER BEAM SUBJECTED TO UNIFORMLY DISTRIBUTED LOAD THROUGHOUT THE ENTIRE SPAN

SLOPE AND DEFLECTION FOR A CANTILEVER BEAM SUBJECTED TO UNIFORMLY DISTRIBUTED LOAD THROUGHOUT THE ENTIRE SPAN

Fig 1

Fig 2: M/EI diagram

Consider a cantilever beam subjected PQ (shown in fig 1) of span L, subjected to uniformly distributed load of w/m throughout the entire span. Fig 2 shows bending moment diagram of the cantilever beam with uniformly distributed load throughout the span.

Slope at the free end = Area of M/EI diagram (As per 1^st moment area theorem)

Area of parabolic diagram = (1/3) (base) (height)

ϴ_Q = (1/3) (L) (-wL²/ 2EI)

Therefore,

ϴ_Q = -Wl³/ 6EI

ϴ_Q = Wl³/ 6EI rad (clockwise with tangent from P)

Consider the M/EI diagram in which O is the centroid point and X is the distance from free end to centroid (O) of the diagram.

Deflection at a point = Product of Area of M/EI diagram and its centroidal       

                                     distance from the reference point.

Here reference point is a point on which deflection has to be determined.

Therefore,

Deflection at Q = (Area of M/EI diagram)(Centroidal distance from Q to O)

Y_{Q =} (1/3) (L) (wL²/ 2EI)(X)

Y_{Q =} (1/3) (L) (wL²/ 2EI)(3/4(L))

Y_Q= -wL⁴/8EI

Y_Q= wL⁴/8EI (downward direction)