Beam Deflection Calculator

For a simply supported beam carrying a single point load P at its midspan, the maximum deflection occurs at the center and is given by δ = PL³ / (48EI), where P is the load in newtons, L is the span length in meters, E is the modulus of elasticity in pascals, and I is the moment of inertia of the beam's cross-section in meters to the fourth power. For example, a 1000 N load on a 2 m steel beam (E = 200 GPa) with I = 1×10⁻⁵ m⁴ produces a deflection of about 0.0833 mm — small because steel is stiff and the moment of inertia is large relative to the load.

The maximum bending moment for a simply supported beam with a single point load P at midspan is M = PL/4, and it occurs directly under the load at the center of the span. This is the highest internal bending stress point along the beam, so it is the location engineers check first when verifying a beam won't yield or fail under the applied load.

Modulus of elasticity (also called Young's modulus) measures a material's stiffness — how much it resists elastic (non-permanent) deformation under load. It appears in the denominator of the deflection formula, so a stiffer material (higher E) produces less deflection for the same load and geometry. Typical values are about 200 GPa for structural steel, 69 GPa for aluminum, and roughly 11 GPa for wood, which is why a wood beam deflects far more than a steel beam of identical dimensions under the same load.

Moment of inertia (technically the second moment of area) describes how a beam's cross-sectional area is distributed relative to its neutral bending axis — the further material is placed from that axis, the higher I becomes and the stiffer the beam is against bending. This is why I-beams and hollow tubes resist bending so efficiently: they concentrate material away from the center. I is usually expressed in m⁴ or mm⁴, and because it is raised to the first power in the denominator of δ = PL³/(48EI), doubling I halves the deflection.

A simply supported beam rests on two supports — typically a pin support at one end (resists vertical and horizontal movement) and a roller support at the other (resists only vertical movement) — allowing the beam ends to rotate freely under load. This is the most common textbook and real-world case for bridges, floor joists, and shelving. It differs from a cantilever beam (fixed at only one end, free at the other) or a fixed-fixed beam (both ends rigidly clamped), both of which have different deflection and moment formulas because rotation at the supports is restrained.

Deflection is proportional to the cube of the span length (L³) in the formula δ = PL³/(48EI), so even a modest increase in span causes a dramatic rise in deflection. Doubling the length of a beam while keeping the load, material, and cross-section the same increases the deflection eightfold (2³ = 8). This is why longer spans require disproportionately deeper or stiffer beams to keep deflection within safe, serviceable limits.