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Bending of a Cantilever

When one end of a horizontal beam is rigidly clamped and the other end is free, it is called a cantilever. Suppose a mass M is loaded at the free end of a beam of negligible mass and length l. The beam is consequently bent, with curvature changing along the length of the beam.

Let us consider the equilibrium of a transverse section of the beam at a point P whose coordinates are (x, y). Since weight of the beam is neglected, at equilibrium, the external torque by load Mg at P is balanced by the bending moment there. That is, we have

where the radius of curvature ρ at P is given by

Hence, we have

Integrating, we get

Constants of integrating are fixed from boundary condition : at x = 0, y = 0, dy/dx = 0.

The depression at the free end (x = l) of the beam is given as

For a rectangular beam of breadth b and depth d, the geometrical moment of inertia I_g = bd³/12 about the axis passing through the C.M. and perpendicular to plane of bending. For a beam of circular cross-section of radius r, I_g = π r⁴/4.