Beam Deflection Calculator
A 1,000 lb load at the center of a 10 ft beam with a modulus of elasticity of 1,500,000 psi and a moment of inertia of 40 in⁴ deflects 0.6 inches. Use this beam deflection calculator to estimate the maximum deflection of a simply supported beam under a single center point load. Enter the load, span length, modulus of elasticity, and moment of inertia to get the deflection in inches along with a span-to-deflection ratio you can compare against common design limits.
Quick answer
This calculator covers one load case only, a simply supported beam with a single point load at midspan.
What this tells you
- •This calculator covers one load case only, a simply supported beam with a single point load at midspan.
- •Deflection grows with the cube of the span length, so a longer span deflects far more than a heavier load.
- •The span-to-deflection ratio shows how the result compares to common limits like L/360.
- •Higher modulus of elasticity or a larger moment of inertia both reduce deflection.
How to Use
- 1Enter the point load in pounds-force (lbf), applied at the center of the span.
- 2Enter the span length in feet, measured between the two supports.
- 3Enter the modulus of elasticity in psi for the beam material. Steel is roughly 29,000,000 psi and typical dimensional lumber is roughly 1,200,000 to 1,900,000 psi.
- 4Enter the moment of inertia in in⁴ for the beam cross section. This value comes from a span or beam table for the actual size and species or grade you are using.
- 5Click Calculate to see the maximum deflection, the span in inches, and the span-to-deflection ratio.
How It Works
Formula
Deflection = (P x L^3) / (48 x E x I)
P = center point load (lbf)
L = span length (in)
E = modulus of elasticity (psi)
I = moment of inertia (in^4)This is the standard closed-form deflection formula for a simply supported beam carrying one point load at the exact center of the span. The calculator converts the span length from feet to inches, then applies the formula directly. Deflection scales with the cube of the span, so doubling the span multiplies deflection by roughly eight if nothing else changes.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
1,000 lb load on a 10 ft span
The 10 ft span converts to 120 inches. Plugging in P = 1,000, L = 120, E = 1,500,000, and I = 40 gives (1,000 x 120^3) / (48 x 1,500,000 x 40) = 0.6 inches. The ratio of span to deflection is 120 / 0.6 = 200, or L/200. That is a larger deflection than the common L/360 live load limit, so a stiffer beam or shorter span would be needed to meet that target.
1,000 lb load on a 10 ft steel beam
Steel has a much higher modulus of elasticity than wood, so the same load and span produce a far smaller deflection. Here the ratio of 120 / 0.0124 works out to about L/9667, well inside typical deflection limits.
Example Deflection Results
Sample results for a center point load on a simply supported beam. These are formula outputs for the listed inputs, not a substitute for checking your actual beam and load.
| Load | Span | Modulus (E) | Moment of Inertia (I) | Deflection | Ratio |
|---|---|---|---|---|---|
| 500 lbf | 8 ft | 1,600,000 psi | 30 in⁴ | 0.1920 in | L/500 |
| 1,000 lbf | 10 ft | 1,500,000 psi | 40 in⁴ | 0.6000 in | L/200 |
| 2,000 lbf | 12 ft | 1,600,000 psi | 60 in⁴ | 1.2960 in | L/111 |
| 1,500 lbf | 16 ft | 1,900,000 psi | 100 in⁴ | 1.1641 in | L/165 |
Deflection limits are usually written as a fraction of the span, such as L/360 for live load or L/240 for total load. Check the applicable building code and your engineer's specification before treating any ratio as a pass or fail.
Why span length matters more than load
Deflection is proportional to the cube of the span length, but only proportional to the load itself. That means a 20% longer span increases deflection by roughly 73%, while a 20% heavier load only increases deflection by 20%. Small changes in span length have an outsized effect on how much a beam bends.
The moment of inertia depends on the shape and depth of the beam cross section, not just its overall size. A deeper beam resists bending far more effectively than a wider one with the same cross-sectional area, which is why beams are almost always installed with their long dimension oriented vertically.
Common mistakes
- Entering the span length in inches when the field expects feet
- Using a load case that includes distributed load or multiple point loads with a formula built for one center point load
- Pulling the moment of inertia from the wrong axis of a non-symmetric cross section
- Comparing the deflection result against a deflection limit without checking which limit (L/360, L/240, or another) applies to the load type and building code