Beam Deflection Calculator
Find simply supported beam deflection, stiffness ratio, and bending stress.
What it does
A beam deflection calculator answers a simple question: how far will this beam bend under a given load? Deflection matters because a beam can be strong enough to carry a load and still sag enough to crack ceiling finishes, feel springy underfoot, or fail a serviceability limit in a building code.
This page handles a simply supported beam. The ends can rotate but cannot move up or down. You can apply a uniform load over the full span or a point load at midspan. The tool returns the maximum deflection, the span divided by that deflection, the moment of inertia, and the maximum bending stress.
The default case is a 10 ft span with 100 lb/ft, a 1.5 x 9.25 in section, and a modulus of 1.6 million psi. It deflects 0.1421 in, or span/844, and the bending stress is 701.2 psi. Switch to metric and the same beam is 3.048 m long with a 38.1 x 234.95 mm section, 1.45939 kN/m, and 11.032 GPa. It deflects 3.61 mm, or span/844, with 4.835 MPa of bending stress.
The formulas assume linear elastic behavior, small deflections, a constant section, and a material that stays below its yield point. They do not include shear deflection, creep, composite action, lateral-torsional buckling, or the beam's own weight unless you add it to the load. Use the result as a check, then confirm the member with the governing code and a qualified designer.
How to use
- Pick imperial or metric. The visible fields change to match the system.
- Pick a uniform load over the full span or a point load at midspan.
- Enter the span, the load, the section width, the section depth, and the modulus of elasticity.
- Read the maximum deflection first. The sign is not shown because the page reports the size of the sag.
- Check the span to deflection ratio against the limit for your use, such as L/360 for many floors.
- Check the bending stress against the allowable stress for the material, then verify shear, bearing, and lateral support.
Formula
- delta
- Maximum deflection at midspan.
- w
- Uniform load per unit length.
- P
- Point load at midspan.
- L
- Clear span between supports.
- E
- Modulus of elasticity.
- I
- Moment of inertia about the bending axis.
- b
- Section width parallel to the bending axis.
- h
- Section depth perpendicular to the bending axis.
- S
- Section modulus, b h^2 / 6 for a rectangle.
- M
- Maximum bending moment.
- sigma
- Maximum bending stress.
Use consistent units. In imperial, convert the span to inches, the uniform load to lb/in, and E to psi. In metric, convert the span to meters, the load to N/m, the section to meters, and E to pascals. The span to deflection ratio is dimensionless, so it is the same in both systems. A beam loaded on its weak axis needs the other section dimension in the I formula.
Worked example
Example: 10 ft beam with 100 lb/ft uniform load
Defaults: 10 ft span, 100 lb/ft, a 1.5 x 9.25 in section, and E = 1.6 Mpsi.
- Convert the span to inches: 10 x 12 = 120 in. Convert the load to lb/in: 100 / 12 = 8.3333 lb/in.
- Moment of inertia: I = 1.5 x 9.25^3 / 12 = 98.932 in^4.
- Deflection: delta = 5 x 8.3333 x 120^4 / (384 x 1,600,000 x 98.932) = 0.1421 in.
- Span to deflection: 120 / 0.1421 = 844, shown as L / 844.
- Section modulus: S = 1.5 x 9.25^2 / 6 = 21.3906 in^3. Maximum moment: M = 8.3333 x 120^2 / 8 = 15,000 lb-in. Stress: 15,000 / 21.3906 = 701.2 psi.
- The tool shows 0.1421 in, L / 844, 98.932 in^4, and 701.2 psi. The metric defaults give 3.61 mm, L / 844, 41,178,458 mm^4, and 4.835 MPa.
Reference table
| Load case | Load | Maximum deflection | Span / deflection |
|---|---|---|---|
| Uniform | 50 lb/ft | 0.0711 in | 1,688 |
| Uniform | 100 lb/ft | 0.1421 in | 844 |
| Uniform | 150 lb/ft | 0.2132 in | 563 |
| Point at midspan | 500 lb | 0.1137 in | 1,055 |
| Point at midspan | 1,000 lb | 0.2274 in | 528 |
| Point at midspan | 2,000 lb | 0.4549 in | 264 |
Limits
- Simply supported beams only. Cantilevers, fixed ends, continuous spans, and overhangs need different formulas.
- The load is either uniform over the full span or a single point load at midspan. Partial loads and multiple point loads are not covered.
- The model is linear elastic with small deflections. It ignores shear deflection, creep, composite action, and residual stresses.
- The stress check is not a complete design check. It leaves out shear, bearing, lateral-torsional buckling, and code load combinations.
Common mistakes
- Mixing feet and inches in the same formula. Convert the span and load first.
- Using the uniform load in lb/ft without dividing by 12 when the span is in inches.
- Swapping the width and depth. The depth is cubed, so the error is large.
- Treating a span to deflection ratio as a pass or fail without checking the code and the use of the beam.
- Leaving out the beam's own weight, decking, or other permanent loads.
Try it yourself
Quick checks. Answers sit tucked under each question.
A 12 ft beam carries 80 lb/ft. The section is 1.5 x 11.25 in and E is 1.6 Mpsi. What is the deflection?
I = 177.979 in^4. Delta = 0.1311 in, or span/1,099. The bending stress is 546.1 psi.
A 3 m beam carries 2 kN/m. The section is 50 x 200 mm and E is 10 GPa. What is the deflection?
I = 33,333,333 mm^4. Delta = 6.328 mm, or span/474. The bending stress is 6.750 MPa.
What is the moment of inertia of a 1.5 x 11.25 in rectangular section?
I = 1.5 x 11.25^3 / 12 = 177.979 in^4. The section modulus is 31.641 in^3.
Questions
What does L/360 mean?
It means the maximum deflection is limited to the span divided by 360. For a 10 ft span, L/360 is 0.333 in. Many floors use L/360, while some roofs use L/240, but the governing code controls.
Can I use this for a cantilever beam?
No. A cantilever has a fixed end and a free end, so both the deflection and moment formulas differ. This page is limited to a simply supported span.
Does the calculator include the beam's own weight?
No. Add the beam weight, decking, and other permanent loads to the uniform load before calculating.
Why do I enter the span in feet but the section in inches?
Those are the usual field units for lumber. The engine converts the span and load to inches before using the formula, so the result is in inches.
Why is the metric result not exactly the same as the imperial result?
The defaults are converted and rounded. A small difference can also come from the modulus or from rounding the section dimensions. The span to deflection ratio should be close in both systems.
Does a larger moment of inertia always mean a stronger beam?
It means a stiffer beam for bending. Strength also depends on the section modulus, the material strength, the span, and the bracing. A deeper section usually helps both stiffness and strength.
Sources
- Missouri University of Science and Technology: Beam formulas
- Fairfield University: Shigley beam formulas, Table A-9
- UC Berkeley: Beam deflection notes