To determine the equation for the elastic curve of a beam th

To determine the equation for the elastic curve of a beam that is subjected to an applied moment and distributed load and to use this equation to determine the deflection of the beam at given locations along the beam’s length. The cantilever beam shown below has an applied moment of M = 11.0 kip?in at point B and a distributed load of w0 = 5.0 kip/in centered at point C. The beam is fixed to a wall at point A, has a modulus of elasticity of E = 28000 ksi , and a moment of inertia of I = 57.0 in4 . Assume EI is constant. The measurements corresponding to the figure are a = 10.5 in , b = 11.0 in , c = 10.0 in , and d = 6.0 in .(Figure 1)

Determine the deflection at point D.

Determine the slope at point D.

Solution

solution:

1)here elastic curve means plot of deflected point with respected to longitidinal axis when load is applied over beam.

2)elastic curve equation is given by

M/EI=1/R=d^2y/dx^2

on arranging we get equation for moment as

EId2y/dx2=M=-[11+5(10)(10.5+11+(10/2))]=-1336 kip in

here on integrating we get equation for slope and on gain integrating we get equation for deflection

dy/dx=(1/EI)[-1336x+c1]

y=(1/EI)[-1336x^2/2+c1x+c2]

here on apply boundary condition for cantilever beam,x=0 as free end and x=L as fixed support

x=L,y=dy/dx=0 as at support deflection and slope is zero.

on putting we get constant c1 and c2 as

c1=1336L

c2=-1336L^2/2

on putting we get equation of slope and deflection as follows

dy/dx=[1336/EI][L-x]

y=-[1336/EI][Lx-x^2/2-L^2/2]

here slope and deflection at point D where x=0 is given as follows from above formulas

dy/dx=.03139 rad

y=-.5885 in