SLOPE AND DEFLECTION FOR A CANTILEVER BEAM SUBJECTED TO POINT LOAD AT FREE END

SLOPE AND DEFLECTION FOR A CANTILEVER BEAM SUBJECTED TO POINT LOAD AT FREE END

Fig 1

Fig 2: Deflected shape of beam

Fig 3: M/EI diagram of a beam

Consider a cantilever beam PQ (fig 1) of span L subjected to point load of magnitude W KN at free end. Fig 2 shows the deflected shape of the beam.Fig 3 shows bending moment diagram of the cantilever beam with concentrated load. Let ϴ be the slope and y is the deflection for the deflected beam.

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

ϴq= ½ (L) (-WL/EI)

Therefore,

ϴ_q = (-WL²/2EI)

ϴ_q = (WL²/2EI)(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 =} ½ (L) (-WL/EI)(X)

Y_Q= ½ (L) (-WL/EI)( 2/3(L))

Y_{Q =  -} WL³/3EI

Y_{Q =}  WL³/3EI(downward direction)

Note: Always for a cantilever beam slope and deflection is maximum in free end.