How to use the beam deflection calculator
- Choose US units (feet, pounds, psi, in^4) or metric units (meters, newtons, GPa, cm^4).
- Choose how the beam is supported and loaded: resting on two supports, fixed at both ends, or held at one end as a cantilever, with a load spread along it or concentrated at a point.
- Enter the span length, the load and the stiffness of the material (modulus of elasticity). The 1,400,000 psi shown is an example for a softwood, so use the value for your material.
- Describe the section: enter its width and depth, or enter the moment of inertia if you already know it. Read the largest sag, the span divided by sag, the bending moment and the reactions.
Formula
simple span, even load w: deflection = 5 w L^4 / (384 E I), moment = w L^2 / 8
simple span, center load P: deflection = P L^3 / (48 E I), moment = P L / 4
cantilever, end load P: deflection = P L^3 / (3 E I), moment = P L
cantilever, even load w: deflection = w L^4 / (8 E I), moment = w L^2 / 2
fixed ends, even load: deflection = w L^4 / (384 E I); fixed ends, center load: deflection = P L^3 / (192 E I)
rectangle: I = width x depth^3 / 12
E is the modulus of elasticity of the material and I is the moment of inertia of the cross section about the bending axis. The results are for a uniform beam with small deflection.
A 2x10 over 12 feet
A nominal 2x10 measures 1.5 by 9.25 inches, so I = 1.5 x 9.25^3 / 12 = 98.93 in^4. Take E = 1,400,000 psi and an even load of 40 pounds per foot (3.333 pounds per inch) on a 12 foot (144 inch) span resting on two supports.
The sag is 5 x 3.333 x 144^4 / (384 x 1,400,000 x 98.93) = 0.135 inch, which is the span divided by 1,069. The largest bending moment is 40 x 144 / 8 = 720 pound-feet per beam, split evenly between the supports as 240 pounds each. With a 500 pound load at the center the sag is 0.225 inch and the moment is 1,500 pound-feet.
In metric units, a 45 by 195 mm member over 3 meters with 2,000 newtons per meter and E = 12 GPa sags 6.32 mm.
What the sag tells you
Sag is the vertical drop at the worst point. Engineers often compare it with the span, written as L divided by a number. A bigger number means a stiffer beam. Which ratio is acceptable depends on what the beam carries and on the rules that apply where you build, so this page reports the ratio and leaves the limit to you and your local code.
Doubling the depth of a beam makes it eight times stiffer, because I grows with depth cubed, while doubling the span makes the sag 16 times larger for an even load. Compare joist sizes with the Joist Count Calculator and price lumber with the Board Foot Calculator.
What these numbers do not cover
The weight of the beam itself, shear deflection, creep over time, notches, holes, connections, buckling and the strength of the material are not included. Bending strength and sag are separate checks, and a beam can sag acceptably and still fail in bending or shear. Anything that carries people or part of a building should be sized by a qualified engineer.
More layout tools
For other building and cutting questions, try the Arch Template Calculator, Cut List Optimizer and Plank Stagger Planner.
Assumptions and limits
- The beam has one material and one cross section along its whole length.
- The load is either spread evenly over the whole span or concentrated at the center or end.
- Deflection is small compared with the span, and the beam stays elastic.
- The weight of the beam, shear deflection and long term creep are not included.
Frequently asked questions
How do I calculate the deflection of a simply supported beam?
For an even load w, deflection is 5 w L^4 divided by 384 E I. For a single load P at the center it is P L^3 divided by 48 E I. This page applies those formulas and shows the steps.
What is the moment of inertia of a rectangular beam?
It is width times depth cubed divided by 12, with the depth measured in the direction of bending. A 1.5 by 9.25 inch board has 98.93 in^4 standing on its edge.
What modulus of elasticity should I use?
Use the value published for your exact material and grade. The 1,400,000 psi shown is only an example, and values for wood, steel and aluminum are very different.
Is this enough to size a floor beam or a deck beam?
No. It estimates sag only. A real design also checks bending strength, shear, bearing, connections and the loads and rules that apply to your project, which is work for a qualified engineer.