Beam Deflection & Bending Stress
Max deflection, moment, shear and bending stress for simply supported, cantilever and fixed beams under a point or uniform load from span, load, E and I.
Maximum deflection and where it occurs, the span-to-deflection ratio, maximum moment and shear, reactions, bending stress at the extreme fibre, and the deflected shape drawn.
Example: A 4 m simply supported steel beam (E 200 GPa, I 1 943 cm⁴) with 10 kN at mid-span deflects 3.43 mm (L/1166) with a 10 kN·m moment; at 100 mm from the neutral axis the bending stress is 51.5 MPa.
The textbook cases,
with E and I from you.
Which formulas are used, what they assume, and where the section and material numbers come from.
The seven cases
Each case is a closed-form solution of the elastic beam equation for a prismatic beam: simply supported with a centre load (δ = PL³/48EI, M = PL/4), a load anywhere (δmax = Pb(L² − b²)^1.5 ÷ 9√3·L·EI under the longer segment, M = Pab/L), or a uniform load (5wL⁴/384EI, wL²/8); a cantilever with an end load (PL³/3EI, PL at the root) or a uniform load (wL⁴/8EI, wL²/2); both ends fixed with a centre load (PL³/192EI, PL/8 at ends and centre) or a uniform load (wL⁴/384EI, wL²/12 at the ends). The deflected shape is drawn from the matching deflection function, exaggerated to be visible.
Stress and units
Bending stress at the extreme fibre is σ = M·c ÷ I, with c the distance from the neutral axis (half the depth for a symmetric section). Metric inputs are m, kN (or kN/m), GPa and cm⁴; imperial are ft, lb (lb/ft), ksi and in⁴ — converted internally to SI and back. The reported span-to-deflection ratio (L/360, L/240 …) is how serviceability limits are usually written; the page applies none, and no allowable stress: compare with your material and code.
What it assumes
Linear elastic material, small deflections (flagged when δ exceeds L/20), a constant cross-section, no shear deformation (a few percent extra on deep short beams), no self-weight unless you add it to the load, and simple supports or fully fixed ends — real connections are somewhere between. E comes from the material data sheet (steel ≈ 200 GPa, aluminium ≈ 69 GPa, timber varies widely) and I from the section-properties tool or the catalogue. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.
SOURCES
- Standard elastic beam formulas (Roark, AISC beam diagrams): δ = PL³/48EI, 5wL⁴/384EI, PL³/3EI, wL⁴/8EI, PL³/192EI, wL⁴/384EI; point load anywhere δmax = Pb(L² − b²)^1.5/(9√3·L·EI); σ = M·c/I
Last reviewed 22 September 2026. How results are checked: How we verify.