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