Cantilever Beams with an End Couple:

Let a cantilever beam loaded with an end couple M0 as illustrated in Figure .

Figure

Let a section X-X at a distance x from A as shown in Figure

M  =- M ₀ . x⁰

The governing equation for deflection is :

EI  (d² y  /dx² )        = M = - M ₀ x⁰

Integrating the Eq. (161), we can get

EI  (d y  /dx)        = - M ₀ x⁰  + C1

EIy =- M ₀  (x²/2)  + C₁ x + C₂

The constants C₁ and C₂ may be found from boundary conditions. The boundary conditions are :

At B, x = l, dy / dx = 0            --------- (1)

At B, x = l,      y = 0      ---------- (2)

Applying the boundary condition (1) into the Eq. (162), we may get

0 = - M ₀ l + C₁

∴          C₁ = M ₀ l

Applying the boundary condition (2) into the Eq. (163), we may get

0 =-  M ₀ l ² /2 + M 0 l . l + C₂

∴          C₂  =-  M ₀ l ² /2

The slope and deflection equations shall be :

EI dy/ dx =- M ₀ x + M ₀ l

EIy =- M ₀ x ² /2 + M₀  l x - M ₀ l²/2

Slope at A, (x = 0),

θ_A = M ₀ l / EI

Slope at C ( x = l/2) ,

EI θ_C = - M ₀ (l/2)  + M₀  l

∴          θC = + M ₀ l / 2 EI

Deflection at A, (x = 0),

y_A   =- M ₀ l²/ EI

Deflection at C, ( x = l/2)  ,

EI y _C    =- (M ₀  /2)( l /2)2 + M ₀l  . (l/2)  - M ₀ l² / 2  

y_C         =- M ₀ l²/8EI