Beam bending and deflection

beam formulasbending stressMc/Isection modulusbeam deflection formulassimply supported beam

Maximum moment, bending stress by Mc/I or M/S, and the deflection of simply supported and cantilever beams under point and uniform loads.

Max Bending Moment — Centre Point Load

M = \frac{P L}{4}

Maximum bending moment in a simply supported beam carrying one point load at midspan, M = PL/4, occurring under the load.

Max Bending Moment — Uniform Load

M = \frac{w L^{2}}{8}

Maximum bending moment at midspan of a simply supported beam under a uniformly distributed load, the classic M = wL²/8.

Area Moment of Inertia — Rectangle

I = \frac{b h^{3}}{12}

Area moment of inertia of a rectangle about its centroidal axis, I = bh³/12, returned in m⁴ and entered as a plain number.

Elastic Section Modulus (S = I/c)

S = \frac{I}{c}

Elastic section modulus S = I/c, in m³, the single number that turns a bending moment straight into a bending stress.

Bending Stress (σ = Mc/I)

\sigma = \frac{M c}{I}

Bending stress at a distance c from the neutral axis of a beam, with the area moment of inertia I entered in m⁴.

Bending Stress from Section Modulus (σ = M/S)

\sigma = \frac{M}{S}

Bending stress straight from the moment and a tabulated section modulus S in m³, the everyday form used with steel tables.

Beam Deflection — Simply Supported, Centre Load

\delta = \frac{P L^{3}}{48 E I}

Maximum midspan deflection of a simply supported beam with a central point load, δ = PL³/48EI, with I entered in m⁴.

Beam Deflection — Simply Supported, Uniform Load

\delta = \frac{5 w L^{4}}{384 E I}

Maximum midspan deflection of a simply supported beam under a uniform load, δ = 5wL⁴/384EI, with I entered in m⁴.

Cantilever Deflection — End Load

\delta = \frac{P L^{3}}{3 E I}

Tip deflection of a cantilever carrying a point load at its free end, δ = PL³/3EI, with I entered in m⁴ as a plain number.

How they fit together

Beam design runs in a fixed order: find the maximum bending moment from the load and the supports, find the section's second moment of area from its shape, then divide. σ = Mc/I and σ = M/S are the same equation — S is simply I/c precomputed, which is why steel tables list section modulus and you rarely need I explicitly for a stress check. Deflection is the separate question, and it depends on the same I but on the fourth power of span rather than the first.

Choose the moment formula by load and support: PL/4 for a point load at midspan, wL²/8 for a uniform load, and note the different deflection coefficients that follow — 1/48 for the point load, 5/384 for the uniform, 1/3 for a cantilever with the load at its tip. That cantilever number is the warning: the same load at the end of a cantilever deflects sixteen times as far as it would at the centre of a simply supported beam of the same span. The deep mistake is checking stress and stopping. Long spans are almost always governed by deflection, not strength, because δ grows with L⁴ while σ grows with L — a floor joist that passes its stress check by a wide margin can still bounce badly enough to fail its serviceability limit. Depth is the lever worth pulling: I goes as the cube of depth, so a joist turned on edge is dramatically stiffer than the same joist laid flat.