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Bending Moment

Let us consider a horizontal beam of uniform cross-section clamped rigidity at one end and a load W placed at the other end. Under the load, the beam bends. The filaments in the upper portion of the beam are stretched while those in the lower portion are compressed. In between, there is a neutral surface NN’ whose length does not change.

Consider a longitudinal section ABCD of the beam as shown in fig. 1. Take a transverse sector EF and consider the equilibrium of part EBCF of the beam. Load W acts vertically downwards at end BC. Hence there acts a reaction force W upwards at end EF. (We are neglecting the weight of beam.) These two forces constitute a clockwise torque on the part EBCF of the beam. This is balanced by a counter-clockwise torque produced by forces of elastic stresses T in the beam. As shown in the fig. 1, elastic tension in upper part and compression in lower part produces a counter-clockwise torque. This is called bending moment at the face EF due to section EBCF.

In order to determine bending moment, let us first find out longitudinal strain in the beam. Consider a small portion of the beam bounded by two transverse section PQ and P’ Q’.

Line NN’ represents neutral surface.

Now consider a filament GH at distance y from NN’. The length of filament GH is equal to length NN’ before bending. Due to bending, change in length of filament GH is given by

GH – NN’ = (ρ + y) Ø – ρ Ø = y Ø

where ρ is the radius of curvature of neutral surface NN’. Hence, longitudinal strain us

If Y is Young’s modulus, then longitudinal stress is

If a denotes the area of cross-section of the given filament, tensile force in it is (Y ya/ρ). The moment of this force about the tangential line passing through center of gravity of the section is (this line lies on neutral surface NN’):

Hence the sum of the moments of all the forces of tension and compression acting over the entire area of cross-section of the beam is

The above is the required value of bending moment. The expression Σ ay² is called geometrical moment of inertia; it is analogous to moment of inertia about the line y = 0 except that instead of mass of the element, its area has been taken. Denoting geometrical M.I. by I_g, we get

The quantity YI_g is known as flexural rigidity.