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Typical bending members are straight and long with a pure bending couple, M . The figure

below shows various cross-sectional possibilities all of which are symmetrical about the y-axis.

In a Nut Shell: Deformations (Deflections) of Beams, y

Under the assumption that plane sections before bending remain plane afterwards it can be

shown that the curvature of the beam (1/ρ) is directly proportional to the couple (bending

moment) and inversely proportional to the flexural rigidity (EI) of the beam where E is the

modulus of elasticity and I_zz is the moment of inertia of the entire cross-sectional area about

the centroidal (neutral) axis.

Curvature = 1/ρ = M(x) / EI_zz

From calculus the curvature of the neutral axis of the beam is

1/ρ = d²y/dx² / [ 1 + (dy/dx)² ]^3/2

Typically the slopes of the beam are very small so that 1/ρ = d²y/dx²

which gives the differential equation describing the beam deflection y = y(x) as

follows:

d²y/dx² = M(x) / EI

Double integration results in an expression for y(x), the equation of the elastic curve of the

beam’s neutral axis. So you need two boundary conditions in order to determine the

constants of integration. Typical boundary conditions include:

y(0) = 0 or specified, y(L) = 0 or specified, dy/dx = slope specified at given point

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