Determine the slopes at free end - cantilever beam:
A cantilever beam of length l carries a point load W at a distance l /4 from free end.
Determine the slopes at free end and under load and the maximum deflection and deflection under the load.
Solution
M =- w [x - l/4 ]------------- (1)
EI (d 2 y /dx2) =- w [x - l /4] ------------ (2)
EI dy/ dx =- (w/2)[x- (l/4)]^{2}+ + C_{1 }------------------ (3)
EI y =- (w/6)[x- (l/4)]^{3}+ + C_{1 } + C_{2} ------- (4)
Figure
Boundary conditions are :
At B, x = l, dy/ dx = 0 ---------- (5)
At B, x = l, y = 0 --------- (6)
From Eqs. (3) and (5),
0 =- w/2 [l -l/4]^{2} + C_{1}
C_{1} =+ 9 wl ^{2} / 32 ------------. (7)
From Eqs. (4), (6) and (7)
0 =- (w/6)[l-(l/4 ) ^{3}+ 9 wl ^{3}/32+C_{2}
C_{2} = (wl^{ 3} / (6 × 64)) [- 108 + 27]
=-81wl ^{3}/ (6 × 64) = - 27 w l ^{3}/128 ---------- (8)
The Eqs. (3) and (4) will be :
∴ EI (dy/ dx) =- (w/2) [x-(l/4)]^{2} + 9 wl ^{2}/32 ------------ (9)
EIy = (w/6)[l-(l/4 ) ^{3}+ 9 wl ^{3}/32-27 wl^{3}/128 ---------- (10)
Slope at A, (x = 0),
θ_{A }= + 9 wl ^{2} / 32EI ---------------(11)
Slope at C, (x = l/4) ?,
Θ_{c }= + 27 wl ^{2} / 128EL
Deflection at A, (x = 0),
y A =-27 wl^{3}/ 128 EI -------------- (12)
Deflection under load at C, (x = l/2) ,
=(9 wl ^{2} /32 ).(l/4) -27 wl ^{3} /128 = - 18 wl ^{3}/128------------ (13)
∴ y_{C} =- 18 wl ^{3} / 128 EI