Find out the slope and deflection at the free end:

Discover the slope and deflection at the free end of a cantilever shown in Figure. Take EI = 200 × 10⁶ N-m².

Solution

M =- 30 x - 60 [ x - 2] - 24 ( x - 3) (( x - 3)/ 2)

=- 30 x - 60 [ x - 2] - 12 [ x - 3]²           -------- (1)

EI d 2 y/ dx2 = M

=- 30 x - 60 [ x - 2] - 12 [ x - 3]²             ----------- (2)

EI (dy/ dx) = - 15 x - 30 [ x - 2] - 4 [ x - 3] + C₁         --------- (3)

EIy =- 5 x - 10 [ x - 2] - [ x - 3] + C₁ x + C₂        ------------- (4)

 Figure

Boundary conditions are :

At B, x = 5 m, dy/ dx  = 0      ---------- (5)

At B, x = 5 m,  y = 0              --------(6)

From Eqs. (3) and (5)

0 = - 15 × 5²- 30 (5 - 2)² - 4 (5 - 3)³ + C₁

∴          C₁ = 677                               ---------- (7)

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

0 =- 5 × 5 ³- 10 (5 - 2)³ - (5 - 3) 4+ 677 × 5 + C₂

∴          C₂  =- 2474            -------- (8)

EI dy/ dx  = - 15 x²  - 30 [ x - 2]²  - 4 [ x - 3]³  + 677       --------- (9)

EIy = - 15 x³  - 10 [ x - 2]³  - [ x - 3]⁴  + 677 x - 2474 --------- (10)

At free end, i.e. at A, x = 0, From Eq. (9)

θ  = +   677 /EI =677 × 10³/200´10⁶ = 3385 × 10- 3  radians

From Eq. (10)

y_A  =- 2474/ EI =- 2474 × 10³ × 10³/200 × 10⁶ = - 12.37 mm

= - 12.37 mm ( ↓ )