Beam Deflection Calculator
Reactions, diagrams and deflection for any mix of point, uniform, varying or moment loads.
Loads
Positions from the left end. Downward loads and clockwise moments are positive.Deflection limits
| Limit | Allowed | This beam | Result |
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Shear, moment and deflection diagrams
Reactions and end values
| Support | Reaction ↑ | Moment | Slope |
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Values along the beam
| x | Shear | Moment | Deflection ↓ |
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How it was calculated
Results are estimates from the formulas shown, not a professional design or certification. Have a qualified engineer verify anything safety-critical (structures, electrical installations, gas or pressure systems).
About the Beam Deflection Calculator
Analyse a single-span beam that is simply supported, a cantilever, fixed at both ends or a propped cantilever, under any combination of point loads, full or partial uniform loads, triangular or trapezoidal loads and applied moments. The calculator gives the support reactions and draws the shear-force, bending-moment and deflection diagrams with the largest values and where they occur.
Enter the section’s second moment of area I and the modulus of elasticity E (presets for steel, stainless steel, aluminium, concrete and timber, with their sources), and optionally the section modulus for the bending stress and the mass per metre for the first natural frequency and the self-weight. Deflection is compared with the usual span/180, span/240 and span/360 limits. Everything is an estimate from elastic beam theory, not a design check.
How to use it
- Choose the supports and enter the span. Pick the units (kN and m, N and mm, kip and ft, or lbf and in); changing them converts what you have typed.
- Choose the material for E, then enter I about the axis the beam bends about (I_x or I_y of a section table, or from the moment of inertia calculator). Add S for the bending stress and the mass per length for the natural frequency, if you have them.
- Add the loads. Positions are measured from the left end; downward loads and clockwise moments are positive (type a minus sign for uplift). A uniform or varying load needs where it starts and stops — 0 to L covers the whole span; a triangle has 0 at one end.
- Check the sketch above the loads, then read the maximum deflection, moment, shear and reactions, the deflection-limit table and the three diagrams.
- Download the values at 200 points along the beam as CSV, or copy a summary.
Examples
E = 200 GPa, I = 8,356 cm⁴, S = 557.1 cm³, 42.2 kg/m
Reactions 30 kN each, M = wL²/8 = 45 kN·m, Δ = 5wL⁴/384EI = 10.10 mm (L/594), σ = 80.8 MPa, f₁ = 27.5 Hz (self-weight only)
Wall moment 10 kN·m, tip deflection PL³/3EI = 0.80 mm; the limits use twice the length, 4 m
Reactions 5wL/8 = 37.5 kN (fixed end) and 3wL/8 = 22.5 kN, fixed-end moment wL²/8 = 45 kN·m, Δmax = wL⁴/185EI at 0.42L from the prop
End and mid-span moments PL/8 = 37.5 kN·m; deflection PL³/192EI — a quarter of the simply supported value
Common uses
- Sizing a floor joist, lintel or shelf bracket and checking its sag against span/360
- Drawing shear-force and bending-moment diagrams for coursework, with every maximum labelled
- Comparing simply supported, fixed and cantilever arrangements for the same load
- Estimating whether a long-span beam or footbridge is likely to feel bouncy (low natural frequency)
How the beam is solved
The bending moment is written with Macaulay’s singularity functions: every load adds a term c⟨x − a⟩ⁿ. Integrating EI·y″ = M(x) twice gives the slope and deflection in closed form, and the support conditions — zero deflection at a pin or roller, zero deflection and slope at a fixed end, zero shear and moment at a free end — give the reactions, including the extra ones of fixed and propped beams. The answer is exact for the beam model and matches the standard cases of the AISC Manual Table 3-23 and Roark Table 8.1, which this tool is tested against.
Standard results it reproduces
- Simply supported, uniform load: M = wL²/8, Δ = 5wL⁴/(384EI)
- Simply supported, central point load: M = PL/4, Δ = PL³/(48EI)
- Cantilever, end load: M = PL, Δ = PL³/(3EI); uniform load: M = wL²/2, Δ = wL⁴/(8EI)
- Fixed both ends, uniform load: end moments wL²/12, mid-span wL²/24, Δ = wL⁴/(384EI)
- Propped cantilever, uniform load: reactions 5wL/8 and 3wL/8, Δmax = wL⁴/(185EI)
- Bending stress: σ = M ÷ S
Deflection limits
IBC 2021 Table 1604.3 limits floor members to span/360 under live load and span/240 under total load, and roof members not supporting a ceiling to span/180; for cantilevers the span is taken as twice the cantilever length. Other codes and finishes (brittle partitions, glazing, machinery) can be stricter, and the deflection to check may be for a particular load case rather than all loads together.
Natural frequency
The first bending frequency of a uniform beam is f₁ = λ² ÷ (2π L²) × √(EI ÷ m), with m the mass per metre and λ = π (simply supported), 1.8751 (cantilever), 4.7300 (fixed both ends) or 3.9266 (propped), from Blevins, Formulas for Natural Frequency and Mode Shape, Table 8-1. Include the mass of anything permanently fixed to the beam in m; point masses are not modelled.
Sources
- AISC Steel Construction Manual, 16th ed., Table 3-23: shears, moments and deflections
- Young, Budynas & Sadegh, Roark’s Formulas for Stress and Strain, 9th ed. (2020), Table 8.1
- International Building Code 2021, Table 1604.3: deflection limits
- Blevins, Formulas for Natural Frequency and Mode Shape, Table 8-1
- Moduli: IS 800:2007 §2.2.4.1 (200 GPa), EN 1993-1-1 §3.2.6 (210 GPa), AISC 360 (29,000 ksi), EN 1993-1-4, EN 1999-1-1, EN 1992-1-1 Table 3.1, EN 338, EN 14080
Limitations
- Elastic Euler–Bernoulli theory with small deflections: shear deformation is ignored, which understates the deflection of short, deep beams (span less than about 10 times the depth).
- One span with a constant E and I. Multi-span continuous beams, tapered or composite sections and partial fixity are not covered.
- It does not check strength or stability: no lateral-torsional buckling, shear capacity, bearing, connections, load factors or load combinations. A qualified engineer should design anything that carries people or property.
- Concrete and timber deflect more over time (cracking, creep); the short-term elastic value shown here is a lower bound for them.
Privacy
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Frequently asked questions
How do I calculate the deflection of a beam?
For the standard cases use the textbook formulas — 5wL⁴/(384EI) for a simply supported beam with a uniform load, PL³/(48EI) with a central point load, PL³/(3EI) for a cantilever with an end load. For any other mix of loads and supports, enter them here: the calculator integrates the bending moment exactly and shows the working.
What does L/360 mean?
That the deflection must not exceed the span divided by 360 — 16.7 mm on a 6 m beam. IBC 2021 Table 1604.3 uses L/360 for floors under live load, L/240 for total load and L/180 for some roofs; for a cantilever L is twice its length.
Which I should I use?
The second moment of area about the axis the beam bends about — normally the strong axis (I_x, or I_y in Euro tables) when the load acts in the plane of the web. Steel tables list it; for other shapes use the moment of inertia calculator. Use E and I in consistent units; the calculator converts them for you.
Does this check whether my beam is strong enough?
No. It gives the elastic bending stress M ÷ S and the deflection, but a design check also needs load factors and combinations, shear, buckling, bearing and connection checks under the code that applies. Treat the results as estimates and have a qualified engineer verify anything safety-critical.
What do negative bending moments mean?
Hogging: the top of the beam is in tension. They occur over fixed supports and along cantilevers. Positive (sagging) moments put the bottom in tension, as in the middle of a simply supported span.
Why is a fixed beam so much stiffer?
Fixing the ends stops them rotating, so the beam bends in an S-shape near each support. With a uniform load the deflection falls to a fifth of the simply supported value (wL⁴/384EI against 5wL⁴/384EI), and the largest moment moves to the supports.