Beam deflection calculator
Calculate support reactions, bending moment diagram, deflection curve, slope and bending stress of straight beams per engineering beam theory (Euler-Bernoulli). Six support conditions with 40 load cases, cross-section entered parametrically or I and W specified directly, beam self-weight included by default. The calculator performs the twin check against yield strength and deflection limit with traffic-light rating, live with every input.
Calculation
Model: straight prismatic beam per Euler-Bernoulli (first-order theory, linear-elastic, static load, bending about the major axis). No stability and no fatigue check. The beam's self-weight is superposed as a uniform load on request (on by default); the superposition is admissible under first-order theory. Sizing tool for machine and fixture design, not a structural code check (use Eurocode 3 for that). For rotating shafts use the DIN 743 shaft calculator.
Results
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Calculation in your browser, inputs go to our server only when you export or save.
Formulas and fundamentals
Euler-Bernoulli differential equation
The basis is the linearised differential equation of the deflection curve in Euler-Bernoulli theory: the curvature of the beam axis is proportional to the local bending moment. With the sign convention of this calculator - deflection w positive downwards, bending moment M positive for tension on the underside - it reads:
Integrating twice with the boundary conditions of the supports (pin: w = 0, fixed end: w = 0 and w′ = 0) yields a closed-form solution for every standard load case. For the statically indeterminate supports - fixed-fixed and fixed-pinned - the equilibrium conditions alone are not sufficient; there the compatibility condition at the redundant restraint supplies the missing equation. The calculator evaluates the resulting segment polynomials exactly - for the simply supported beam with a central point load, for example:
for the uniformly distributed load:
for the cantilever with an end load:
Section properties
The section properties are computed from the dimensions as composite areas (outer rectangle minus cut-out): for the rectangle:
for the round bar:
and analogously for tube, rectangular hollow section and doubly symmetric I-section. The parametric I-section calculation without fillet radii is about 4 to 5 percent below the catalogue values of rolled IPE sections and therefore conservative. To calculate with the catalogue value, pick the standard section instead of the parametric input: 118 rolled sections of the IPE, HEB/IPB, I/IPN, U/UPN and hollow section series with the values from the section table, fillet radii included. For IPE 240 that is 3,890 instead of 3,671 cm⁴, a difference of 5.6 percent. The calculator derives the section modulus from it - W = 2·I/h about the strong axis, W = 2·I/b about the weak one - because the printed W column occasionally carries typesetting errors; computed from I, the value matches the table to within half a percent. The role of the two properties matters: the deflection depends on the second moment of area I, the stress on the section modulus W - both are reported separately and can also be entered directly.
Stress and deflection check
The stress check compares the maximum bending stress with the allowable stress R_e/S_req:
For the beam fixed at both ends under distributed load the fixed-end moment q·L²/12 governs, not the midspan moment q·L²/24. The deflection check compares f_max with f_allow = L/k using a selectable criterion (typically L/300 for general beams, L/500 for sensitive equipment, L/150 for cantilevers). For an off-centre point load the maximum deflection is not under the load but in the longer segment at:
the calculator reports both values. Each check is rated by its utilisation, that is the actual value related to the permissible one. Above 1.0 the check is not satisfied and the indicator is red; between 0.9 and 1.0 it is satisfied but without appreciable reserve, and the indicator is amber. An actual safety factor below the required one therefore counts as not satisfied, not as borderline.
Self-weight and superposition
The self-weight of the beam is included by default. A uniformly distributed load over the full length is formed from cross-sectional area and density:
It is superimposed on the selected load case. First-order theory permits this because all quantities depend linearly on the load: shear force, bending moment and deflection curve add up point by point. The extreme values do not - for an off-centre load the sum of the two individual maxima would be too high and in the wrong place. The calculator therefore determines maximum moment, maximum deflection and moment zero crossings anew from the already superimposed curves. How much the self-weight matters does not depend on the span alone but on the ratio of beam weight to payload - for a central point load exactly G/(2·F). To calculate without it, for instance because the entered load already includes it, switch it off below the load input.
Worked example
A round bar of 50 mm diameter in S355 steel (R_e = 355 N/mm²) spans 2000 mm as a simply supported beam and carries a central point load of 5000 N. The section properties are A = 1963.5 mm², I = 306,796 mm⁴ and W = 12,272 mm³; both support reactions are 2500 N. Self-weight is left out of this calculation so the figures match the textbook formulas; in the calculator it is included by default (see final paragraph).
The maximum bending moment at midspan is M_max = F·L/4 = 2.5·10⁶ Nmm. This gives a bending stress of σ_b = M_max/W = 203.7 N/mm² and a safety factor of S = 355/203.7 = 1.74 - the stress check passes with S_req = 1.5. The maximum deflection at midspan is f_max = F·L³/(48·E·I) = 12.93 mm, the slope at the supports 19.4 mrad (1.11°).
Against the criterion L/300 = 6.67 mm, however, the deflection check fails (utilisation 194 percent). The example shows the typical situation in machine design: stiffness governs, not stress - a deeper cross-section (I grows with h³) helps far more than a stronger material.
The calculator therefore starts with the same beam but d = 65 mm: I rises to 876,240 mm⁴, the deflection drops to 4.53 mm (68 percent utilisation) and the bending stress to 92.7 N/mm² (S = 3.83) - both checks pass. With d = 50 mm instead of 65 mm the deflection check fails, because deflection decreases with the fourth power of the diameter: 30 percent more diameter cuts it to roughly a third, while the stress only falls with d³.
On start-up the calculator shows slightly higher figures than just calculated, because it includes the bar's self-weight: the round bar d = 65 mm weighs about 52 kg over 2000 mm, which corresponds to a distributed load of 0.256 N/mm. Support reactions therefore rise from 2500 N to 2755 N, the deflection from 4.53 mm to 4.82 mm (72 instead of 68 percent utilisation) and the bending stress from 92.7 to 97.5 N/mm². For a simply supported beam with a central load the self-weight share follows the ratio G/(2·F) and grows linearly with the span - at a 4 m span with the same load it already accounts for a third of the bending moment. To calculate without it, for instance because the load entered already includes the self-weight, clear the checkbox below the load input.
Frequently asked questions
What is the difference between second moment of area I and section modulus W?
I describes the stiffness of the cross-section and governs deflection (f ~ 1/I); W describes the utilisation of the outer fibre and governs bending stress (σ = M/W). A common mistake is σ = M/I. For a doubly symmetric section, W = 2·I/h with the section depth h.
Which rolled sections are included, and when should I use the parametric input instead?
118 sections are included: IPE (DIN 1025-5), HEB/IPB (DIN 1025-2), I/IPN (DIN 1025-1), U/UPN (DIN 1026-1) plus hot-finished and cold-formed hollow sections - each with area and second moment of area from the section table. Use the standard section whenever you are installing a catalogue profile: the parametric input assembles the section from four dimensions and omits the fillet radii at the web-to-flange transition, which costs a good 5 percent of the second moment of area for IPE 240. The parametric input remains for welded or otherwise non-standard beams, the direct input for values from CAD or a manufacturer's data sheet. The HEA series is not included: the available source skips HEA 220, 260 and 300, and a series with holes in the most common sizes would be worse than none - please use the direct input there.
Why is the U section available for the strong axis only?
Because a pure bending check would be on the unsafe side there. The U section is not symmetric about the weak axis, the centroid does not sit at mid-width - so the section modulus cannot be obtained from the second moment of area and half the width as usual; for U 100 that would give 11.7 instead of 8.49 cm³, 38 percent too much. More importantly, the shear centre of a U section lies outside the section: a load through the centroid also twists the beam, and engineering beam theory does not capture that torsion. For bending about the weak axis a U section is the wrong component; if it has to be one, a check for bending with warping torsion belongs with it.
Why does deflection often govern instead of stress?
Because deflection grows with L³ or L⁴ while stress grows only with L. For longer beams the stiffness check is therefore almost always more critical. A stronger material does not help then, since the elastic modulus of all steels is practically identical - only a larger second moment of area (more depth) effectively reduces deflection.
Where is the maximum deflection for an off-centre point load?
Not under the load but in the longer span segment at x = √((L²−b²)/3) from the support remote from the load. The maximum, however, never lies farther than about 0.077·L from midspan, so the midspan deflection deviates by at most roughly 3 percent. The deflection directly under the load can be considerably smaller; the calculator reports both values.
Which beams does the calculation apply to?
Straight prismatic beams with constant cross-section under static load, linear-elastic with small deformations (first-order theory). Euler-Bernoulli theory neglects shear deformation; as a rule of thumb L/h should be at least 10, below that the calculator warns. If f_max exceeds L/50 the small-angle assumption is violated and a warning appears as well.
How realistic is the ideal fixed end?
Ideal fixed ends are rare: bolted end plates or short clamping lengths behave somewhere between fixed and pinned. The real deflection therefore lies between the two limiting cases - up to a factor of 5 for distributed load (q·L⁴/384 versus 5·q·L⁴/384). When in doubt, calculate both cases and use the less favourable one.
Does the calculator include the beam's self-weight?
Yes, by default. A uniformly distributed load q_eg = ρ·g·A is formed from cross-sectional area and density and superimposed on the selected load case. What matters is not the span alone but the ratio of beam weight to payload: for a central point load the additional share of the bending moment is exactly G/(2·F). A 60×50 tube therefore contributes 6 per cent, a solid rectangle of the same length 20 per cent - there is no rule of thumb “from X metres onwards”. It can be switched off below the load input, for instance when the entered load already includes it. With direct input of I and W without an area it cannot be formed; the calculator says so.
What do green, amber and red mean?
Each check is rated by its utilisation, that is the actual value related to the permissible one. Red means the check is not satisfied - the bending stress exceeds R_e/S_req or the deflection exceeds the criterion L/k. Amber means satisfied, but with less than 10 per cent reserve. An actual safety factor below the required one is therefore red, not amber: a check that is not satisfied is not borderline, it has failed. For more reserve, raise S_req or tighten the deflection criterion instead of interpreting the indicator.
Does the fixed-end caveat also apply to the cantilever?
It applies, but it acts differently. The fixed-fixed and fixed-pinned beams are statically indeterminate: a flexible connection shifts moments into the span and increases the deflection, and the real value lies between the two limiting cases. The cantilever is statically determinate - there a flexible connection changes neither shear force nor bending moment, it rotates the beam as a whole. This rigid-body rotation adds to the calculated bending and grows linearly with the cantilever length: 0.1 degrees of rotation amount to about 1.7 mm per metre of cantilever, so for a 2 m cantilever a good third of a typical deflection.
Does the calculator replace a structural code check?
No. It is a sizing tool for machine and fixture design with an elastic stress check and a global safety factor against yield. Civil structures must be verified to Eurocode 3 with partial safety factors, stability checks (lateral-torsional buckling, plate buckling) and regulated load assumptions.
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