Question

Explain the theory of bending of the beam.

Answer 1

When a beam is bent, the strain produced is longitudinal and so elastic modulus involved is Young's modulus.

The bending moment is the algebraic sum of moments of all restoring forces developed in the filaments of the bent beam about a neutral axis. If Y is Young's modulus, R radius of curvature of neutral filament and I, the geometrical moment of inertia, then longitudinal strain at a distance Z from neutral filament = Z/R

Bending moment = YI/R

For a beam of circular cross-section of radius r,

I = πr⁴/4

For a beam of rectangular cross-section

I = bd³/12

where b is the breadth and d its depth.

For a beam supported at ends loaded in the middle by a load W = Mg, the depression at the centre is given by

δ = Wl³/48YI

For a beam of rectangular cross-section

I = bd³/12 and δ = Wl³/4Ybd³