Estimate the maximum deflection and bending moment of a simple beam. Choose a load case, then enter the load, span, material stiffness and cross-section. This is for learning and quick checks. Free, private, and runs entirely in your browser.
Euler-Bernoulli beam theory gives closed-form results for simple cases. Deflection grows with the load and the cube or fourth power of the span and shrinks with the stiffness E × I, where E is the material modulus and I the second moment of area of the cross-section.
What is a typical modulus for steel? About 200 GPa for structural steel, 69 GPa for aluminum and roughly 10 GPa for timber along the grain.
What is a typical deflection limit? Floors are often limited to span ÷ 360 and roofs to span ÷ 240, but codes vary, so check the one that applies.
A steel beam spans 3 m with a 5,000 N load at midspan, E = 200 GPa and I = 1,000 cm⁴. EI = 200 × 10⁹ × 10⁻⁵ = 2 × 10⁶ N·m². Deflection = 5,000 × 27 ÷ (48 × 2 × 10⁶) ≈ 1.41 mm, which is about L/2,133, and the maximum moment is PL/4 = 3,750 N·m.
Why does span matter so much? Deflection grows with the cube or fourth power of span, so a small increase in span has a large effect.
Does it include the beam weight? No. Add the self-weight as a uniform load if it matters.
Beam theory was developed in the 18th century by Leonhard Euler and Daniel Bernoulli, who related the curvature of a bent beam to the bending moment and the stiffness of the section. The Euler-Bernoulli formulas remain the basis of introductory structural analysis, and the simple closed-form cases are still used for quick checks.
Use this calculator to compare cases, to learn how span and stiffness affect deflection, or to check a rough estimate. For real structures, a qualified engineer must design the beam according to the applicable code. Safety is critical.
What is the second moment of area? A property of a cross-section that measures its resistance to bending. Deeper sections have much larger values.
Is a stiffer material always better? Stiffness reduces deflection, but strength, weight, cost and other factors also matter.