A cantilever beam of span L, flexural rigidity El is subjected to a hogging bending moment M at the free end. Calculate the maximum deflection by direct integration method.

Answer 1

The deflection of a cantilever beam under a load can be calculated using the equation:

δ(x) = (Mx^2)/(2EI)

where:
δ(x) = deflection at distance x from the fixed end
M = bending moment at the free end
E = Young's modulus of the beam material
I = moment of inertia of the beam cross-section

To calculate the maximum deflection, we need to differentiate the deflection equation with respect to x and set it to zero to find the point of maximum deflection. The maximum deflection occurs at the free end of the beam (x = L).

δ'(x) = Mx/(EI)
δ'(L) = ML/(EI) = 0

Since ML ≠ 0, the deflection at the free end is maximized when x = L.

Substitute x = L into the deflection equation:

δ(L) = (ML^2)/(2EI) = (ML^2)/(2EI)

Therefore, the maximum deflection of the cantilever beam is:

δ_max = (ML^2)/(2EI)