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+  + C₁          ------------------ (3)

 EI y     =- (w/6)[x- (l/4)]³+  + C₁  + C₂                ------- (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           + C₁

C₁  =+ 9 wl ² / 32                          ------------. (7)

From Eqs. (4), (6) and (7)

0 =- (w/6)[l-(l/4 ) ³+ 9 wl ³/32+C₂

C₂  = (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 wl³/128           ---------- (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 wl³/ 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