Beam Deflection Calculator
Maximum deflection, its location, end slopes and moment for simply supported, cantilever and fixed beams, with steel and timber E values and an L/360 check.
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About this toolWhat is the Beam Deflection Calculator?
A beam can be strong enough and still be wrong. Floors that sag bounce, crack the ceiling below and pop tiles; long shelves droop; a cantilevered canopy that moves 30 mm under snow looks unsafe even when the stresses are fine. This beam deflection calculator gives the maximum deflection of a prismatic beam, where along the span it happens, the slope at each end, the reactions and the peak bending moment, and checks the answer against the L/180, L/240 and L/360 limits that building codes use.
Seven standard cases are covered: a simply supported beam with a central point load, a point load anywhere along the span, or a uniform load; a cantilever with an end load or a uniform load; and a beam fixed at both ends with a uniform load or a central point load. Enter the span, the load, the modulus of elasticity and the second moment of area, in SI or US units, and the results and the deflected shape update as you type.
The formulas are the closed-form solutions printed in the AISC Steel Construction Manual beam diagrams (Table 3-23) and in Roark's Formulas for Stress and Strain, so the numbers match a hand calculation to the last digit. Everything runs in your browser and the Share button copies a link with your inputs.
- Seven standard support and load cases, including an off-centre point load
- Maximum deflection, its location, both end slopes, reactions and peak moment
- L/180, L/240 and L/360 checks with utilisation, plus optional bending stress from S
- Typical E presets for steel, stainless, aluminium, timber grades, glulam and concrete
- SI and US units with conversion in place, deflected-shape chart, copy and share link
How to use the Beam Deflection Calculator
- Choose SI or US units. The force unit can be kN or N, or kip or lbf; switching systems converts the numbers you already typed.
- Pick the support and load case. For an off-centre point load, enter its distance a from the left support.
- Enter the span and the load. For a uniform load include the beam's self-weight and any dead load, not only the live load.
- Pick a material to fill in a typical E, or type your own, then enter the second moment of area I about the bending axis. Add the section modulus S if you also want the bending stress.
- Read the maximum deflection, its position, the end slopes and the moment, then check the L/180, L/240 and L/360 table and the deflected-shape chart.
Worked examples
Steel beam, 6 m span, 5 kN/m
Input
Simply supported UDL, L = 6 m, w = 5 kN/m, E = 200 GPa, I = 8,000 cm⁴
Output
Maximum deflection 5.27 mm at midspan (L/1,138); end slopes 0.00281 rad; M = 22.5 kN·m
This is the default example. At L/1,138 the beam comfortably passes all three limits, so a smaller section could be checked next.
Douglas fir 2x10 floor joist
Input
Simply supported UDL, L = 12 ft, w = 53.3 lb/ft (40 psf live at 16 in centres), E = 1,600 ksi, I = 98.93 in⁴
Output
0.157 in, which is L/916 against an L/360 allowance of 0.40 in
Live-load deflection is usually what sizes floor joists. The 2x10 passes with room to spare; a 2x8 (I = 47.6 in⁴) over the same span would deflect about twice as much.
Timber joist that fails the limit
Input
47 x 150 mm C24 joist, L = 3.6 m, w = 1.5 kN/m, E = 11 GPa, I = 1,322 cm⁴
Output
22.6 mm, about L/160, against L/360 = 10 mm
A 150 mm deep joist is too shallow for that span and load. Moving to 47 x 200 mm raises I by (200/150)³ = 2.37 times and brings deflection to about 9.5 mm.
Point load off centre
Input
L = 6 m, P = 20 kN at a = 2 m, E = 200 GPa, I = 8,000 cm⁴
Output
Maximum 4.84 mm at x = 2.73 m; 4.44 mm under the load; M = 26.7 kN·m
The largest deflection is not under the load but between the load and midspan, in the longer segment.
Background and insight
All of these formulas come from integrating the Euler-Bernoulli beam equation, E I y'' = M(x), twice and fixing the constants with the support conditions. That gives the familiar coefficients: 5/384 for a simply supported uniform load, 1/48 for a central point load, 1/3 and 1/8 for cantilevers, and 1/384 and 1/192 for fully fixed ends. The theory assumes a straight, prismatic beam of linear-elastic material, small deflections, and plane sections that stay plane; shear deformation is ignored, which is accurate for spans more than about ten times the depth.
Deflection depends on the product E I, the flexural stiffness. E is a material constant (about 200 GPa or 29,000 ksi for any structural steel, whatever its strength grade) and I depends only on the shape of the cross-section. That is why a higher-strength steel does nothing for deflection and why doubling a beam's depth cuts deflection by a factor of eight. Timber E values in the presets are mean values: EN 338 lists 11 GPa for C24 and 8 GPa for C16, and the NDS Supplement gives 1,600,000 psi for Douglas fir-larch No.2 and 1,400,000 psi for spruce-pine-fir No.2.
The limits come from the International Building Code, Table 1604.3: L/360 for floor live load, L/240 for floors under total load and for roofs with a plaster ceiling, and L/180 for roofs without a ceiling. Eurocode practice is similar, with L/250 for quasi-permanent load and L/300 or stricter where finishes can crack.
Common use cases
Sizing floor joists and beams
Check that a timber joist, steel beam or glulam meets L/360 under live load before ordering material.
Shelves, mezzanines and racks
Estimate how far a shelf or platform beam will sag with a known load so it looks and feels solid.
Cantilevered brackets and canopies
Get the tip deflection and slope of a bracket, sign arm or balcony beam under a point or uniform load.
Machine frames and test rigs
Keep aluminium or steel frame members stiff enough for alignment-sensitive equipment.
Checking hand calculations
Students and engineers can confirm a textbook answer and see the full deflected shape.
Tips and limitations
- Use the I about the axis the beam actually bends about. A plank laid flat has a far smaller I than the same plank on edge.
- Include self-weight. A long steel beam or a glulam can carry a large share of its own deflection.
- Timber creeps. For long-term loads, Eurocode 5 multiplies deflection by (1 + kdef), about 1.6 in a heated building, and the NDS applies a factor of 1.5 for seasoned lumber.
- Fixed-end results assume perfectly rigid supports. Real bolted or welded connections rotate a little, so the true deflection lies between the fixed and simply supported values.
- For loads that combine several cases, deflections add (superposition), as long as the material stays elastic. Use the shear and moment calculator for arbitrary load patterns.
Frequently asked questions
What is the formula for deflection of a simply supported beam with a uniform load?
The maximum is 5wL⁴ / (384EI) at midspan, where w is load per unit length, L the span, E the modulus of elasticity and I the second moment of area. Keep the units consistent: in N and mm, use w in N/mm, E in N/mm² and I in mm⁴.
What does L/360 deflection mean?
It is a serviceability limit: the deflection must not exceed the span divided by 360. A 12 ft (144 in) floor joist may deflect up to 0.40 in under live load. IBC Table 1604.3 uses L/360 for floor live load, L/240 for total load and L/180 for roofs without a ceiling.
How much does a cantilever deflect compared with a simply supported beam?
A lot more. For the same span, load and section, a cantilever with an end point load deflects PL³/3EI, which is 16 times the PL³/48EI of a simply supported beam with a central load. Cantilevers need deeper sections or shorter reach.
Does a stronger steel grade reduce deflection?
No. Deflection depends on stiffness, E times I, and all structural steels have E close to 200 GPa (29,000 ksi). A higher yield strength lets the beam carry more load before yielding, but it bends the same amount under the same load. To reduce deflection, increase I with a deeper section.
Where is the maximum deflection for an off-centre point load?
In the longer segment, at x = √((L² − b²)/3) from the support farther from the load, where b is the shorter distance. It is never more than about 0.077L from midspan, which is why midspan deflection is often a close estimate.
What E value should I use for timber beams?
Use the mean modulus for serviceability checks: 11 GPa for C24 and 8 GPa for C16 under EN 338, or 1,600 ksi for Douglas fir-larch No.2 and 1,400 ksi for SPF No.2 in the NDS Supplement. Grade and species matter, so check your stamp.
Why does doubling the beam depth reduce deflection by eight times?
For a rectangle, I = bh³/12, so I grows with the cube of depth. Doubling h multiplies I by 8 and divides deflection by 8. Doubling the width only doubles I, which is why joists are deep and narrow.
Is there an in-depth guide to the Beam Deflection Calculator?
Yes. “Beam Deflection Formulas: Where 5wL⁴/384EI Comes From and When L/360 Applies” covers the background, how it works, common pitfalls and worked examples. Read the full guide.
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