Beam Deflection Calculator
Estimate the maximum deflection and bending moment of a prismatic beam. Pick the support and load case, set the span, load, Young's modulus and second moment of area, and the calculator returns the textbook closed-form answer.
Maximum deflection
13.02 mm
- Deflection (m)
- 0.01
- Maximum bending moment (kN·m)
- 12.5
- Flexural stiffness EI (N·m²)
- 2,000,000
- Load magnitude
- 10 kN
- Span L (m)
- 5
- Young's modulus E (GPa)
- 200
- Second moment of area I (cm⁴)
- 1,000
Closed-form Euler–Bernoulli deflection: δ = P · L³ / (48 · E · I). The selected case sets the constant in the denominator (48 for a simply supported beam with a midspan point load, 384/5 for a simply supported beam under uniform load, 3 for a cantilever point load at the free end, 8 for a cantilever under uniform load). Stiffness EI = Young's modulus × second moment of area; structural steel is ~200 GPa, aluminium ~69 GPa, and softwood timber ~10 GPa. The peak bending moment is reported alongside so you can size the section against an allowable bending stress. Use this for elastic-range estimates; if the deflection exceeds L/250 (typical serviceability limit) you should re-spec the section or shorten the span.
How to use this calculator
Pick the beam configuration that matches what you are analysing. "Simply supported" means the beam sits on two supports that can resist vertical force but not bending moment — the standard pinned–roller assumption used for floor joists, lintels and lab benches. "Cantilever" means the beam is rigidly fixed at one end and free at the other, like a balcony beam built into a wall. Then pick whether the load is a point load (a single concentrated force, e.g. a column landing on the beam, or a person standing at one spot) or a uniformly distributed load (a load spread evenly along the whole span, e.g. self-weight, a slab, or a uniform live load). Enter the span L in metres — for a simply supported beam this is the clear distance between supports; for a cantilever it is the projection from the fixed end. Enter the load magnitude: kN if you picked a point load, kN/m if you picked a uniform load. Enter Young's modulus in GPa — structural steel is about 200, aluminium 69, softwood timber 10, concrete 30. Finally enter the second moment of area I in cm⁴. For standard steel sections, take I from the BS EN 10365 or AISC section property tables; for timber and concrete, compute it from the cross-section (bd³/12 for a solid rectangle). The result is the peak elastic deflection in millimetres at the point of maximum deflection (midspan for simply supported, free end for cantilever), plus the peak bending moment so you can size the section against an allowable bending stress.
How the calculation works
The calculator applies the Euler–Bernoulli beam equation and reports the closed-form maximum-deflection formula for the chosen case. Euler–Bernoulli assumes small deflections, linear-elastic material, prismatic cross-section and plane sections remaining plane and perpendicular to the neutral axis — assumptions that hold for the slender beams used in nearly all building structures. The four cases use these constants: simply supported with a midspan point load gives δ_max = PL³/(48EI); simply supported with a uniformly distributed load gives δ_max = 5wL⁴/(384EI); a cantilever with a point load at the free end gives δ_max = PL³/(3EI); and a cantilever under a uniform load gives δ_max = wL⁴/(8EI). The peak bending moment is also reported: PL/4 for the simply supported point case at midspan, wL²/8 for the simply supported uniform case at midspan, PL at the fixed end for the cantilever point case, and wL²/2 at the fixed end for the cantilever uniform case. Internally the calculator converts the span to metres, the load to newtons (or N/m), the modulus to pascals, and the second moment of area to m⁴, then evaluates the formula and reports the deflection back in millimetres. Compare the result against the relevant serviceability limit — usually L/250 to L/360 for floors in BS EN 1990 / Eurocode and the IBC — and resize if you exceed it.
Worked example
A 5 m long simply supported steel I-beam carries a single 10 kN point load at its midspan. The section is a small Universal Beam with I = 1,000 cm⁴ and the material is structural steel with E = 200 GPa. Plugging into δ_max = PL³/(48EI): with P = 10,000 N, L = 5 m, E = 200×10⁹ Pa and I = 1×10⁻⁵ m⁴, the numerator is 10,000 × 125 = 1.25×10⁶ N·m³ and the denominator is 48 × 200×10⁹ × 1×10⁻⁵ = 9.6×10⁷ N·m². The deflection is 1.302×10⁻² m, or 13.02 mm. The peak bending moment is PL/4 = 10×5/4 = 12.5 kN·m at midspan. Against an L/250 deflection limit (5,000/250 = 20 mm) the section passes; against an L/360 limit (13.9 mm) it is right at the edge and you would probably specify the next size up. This matches the closed-form value reported in AISC Steel Construction Manual Table 3-23, case 7.
Frequently asked questions
What is the difference between a simply supported beam and a cantilever?
A simply supported beam sits on two supports — typically modelled as a pin (resists vertical and horizontal force) at one end and a roller (resists only vertical force) at the other. Neither support resists rotation, so the beam is free to rotate at both ends. A cantilever is rigidly fixed at one end (the support resists vertical force, horizontal force and bending moment, so the beam cannot rotate there) and completely free at the other. For the same span and load, a cantilever with a point load at its free end deflects 16 times more than a simply supported beam with the same point load at midspan — the cantilever formula has a 3 in the denominator versus a 48 — which is why cantilever balconies and canopies need much beefier sections than equivalent simply supported floor beams.
Where on the beam does the maximum deflection occur?
For both simply supported cases it occurs at midspan — the point exactly halfway between the two supports. For both cantilever cases it occurs at the free end (the unsupported tip). These are the locations the calculator reports the deflection at. The maximum bending moment is at a different location in some cases: it is also at midspan for the simply supported cases, but at the fixed-end (root) for both cantilever cases — that is where the beam most needs to be strong in bending.
How do I find the second moment of area for my beam?
For a standard rolled steel section (Universal Beam, Universal Column, W-shape, channel, angle), look it up in the manufacturer's section property tables — every steel mill publishes them, and the AISC Steel Construction Manual and the British Steelwork Association blue book are the standard references. The value is usually given in cm⁴ or in⁴; convert in⁴ to cm⁴ by multiplying by 41.62. For a solid rectangular timber or concrete section of breadth b and depth d, I = bd³/12 about the strong axis. For a solid circular section of diameter d, I = πd⁴/64. For a hollow circular section with outer diameter D and inner diameter d, I = π(D⁴ − d⁴)/64. For built-up or composite sections, use the parallel-axis theorem to combine the parts.
What deflection is acceptable?
It depends on the serviceability limit set by your design code and the function of the beam. Eurocode (BS EN 1990 and the National Annex) and the International Building Code both call for a deflection limit somewhere between L/250 and L/360 under serviceability loads for typical floors, and tighter — L/360 to L/480 — for beams supporting brittle finishes like plaster ceilings. Roof beams and lintels often allow up to L/200. Industrial cranes, vibration-sensitive equipment supports and timber floors under live load may demand L/600 or stricter. The calculator does not impose a limit; check the result against the code that applies to your project.
Does this work for non-uniform loads or multiple point loads?
Not directly. The four cases here are the textbook closed-form solutions for one load type at a time on a single span. For a beam carrying both self-weight (uniform) and a single point load, run the calculator twice — once with the uniform load and once with the point load — and add the two deflections (linear superposition is valid in the elastic range and is exactly how every beam-table chart in the AISC manual is used). For asymmetric point loads, multi-span continuous beams, partial uniform loads, or applied moments, you need either a more comprehensive deflection table (Roark's Formulas for Stress and Strain or AISC Table 3-23 cover most of these in closed form) or a stiffness-method / finite-element analysis.
Why does the calculator report the deflection in millimetres but the inputs use metres?
Span and section dimensions for buildings are practical to enter in metres and cm⁴, but the deflections that matter for serviceability are usually a few millimetres up to a few tens of millimetres — reporting those in metres would give answers like 0.013 m that are awkward to read and easy to misjudge. So inputs follow the convention of structural engineering drawings (metres, kN, GPa, cm⁴) and outputs are reported in millimetres, matching how deflection limits are written in code (e.g. "deflection shall not exceed 20 mm").