Beam Deflection Calculator

How to use this beam deflection calculator

What this calculator does

This calculator estimates the small, elastic vertical deflection of a prismatic beam under one of four idealized loading cases: a simply supported beam with a centered point load, a simply supported beam with a uniform load, a cantilever with an end point load, or a cantilever with a uniform load. It combines the beam span, load, Young's modulus, and second moment of area into the classical Euler – Bernoulli beam equations. It is useful for preliminary sizing, classroom checks, comparing materials or sections, and checking how a load change affects serviceability. It does not verify strength, shear deformation, local buckling, connection behavior, vibration, creep, cracking, or code compliance.

When to use it

Use it to compare two candidate beam sections, estimate bench or shelf sag, check a simple machine member, or create a transparent hand-calculation checkpoint before using a full structural model. The underlying assumptions and common beam formulas are summarized in the Air Force Stress Manual beam-bending reference.

How to calculate

1. The calculator opens with a complete demonstration: a 1.5 m simply supported timber beam, a 400 N midspan point load, E = 6.8 GPa, and I = 160 cm⁴. The result and a validated XLSX workbook are immediately available.
2. Choose the Beam and load case, then replace the sample values with your span, load, material stiffness, and section inertia. The load unit changes between N for point loads and N/m for uniform loads.
3. Read Maximum deflection, Flexural rigidity, Span / deflection, Maximum slope, and the selected formula. Review the curve and checkpoint table for the distribution along the span.
4. Select Download Excel to export the current canonical inputs and results. Reset clears the demonstration values, results, chart, table, and workbook state; Excel remains disabled until a complete valid state is entered again.

Input guide

Beam and load case is required and selects both support condition and load pattern. Choose the idealization that best matches the real member; a common mistake is treating a fixed support as a simple pin or representing a distributed load as a point load. Span length (L) is required in metres, accepts a plain decimal such as 1.5, and must be greater than zero. Because deflection varies with L³ for point loads and L⁴ for uniform loads, modest span increases can cause large deflection increases. Load magnitude is required as newtons for point loads or newtons per metre for uniform loads; use 400 N for a person-sized point force or 500 N/m for a light distributed load. Do not enter kilonewtons without converting to newtons. Modulus of elasticity (E) is required in gigapascals; 6.8 GPa is a representative softwood value, while structural steel is roughly 200 GPa. Higher E reduces deflection in direct proportion. Area moment of inertia (I) is required in cm⁴ and describes cross-section geometry about the bending axis. For a rectangle, I = bh³/12, so rotating a section changes stiffness dramatically. Higher I reduces deflection directly; do not confuse second moment of area with mass moment of inertia.

Output guide

Maximum deflection (δmax) is the largest downward displacement in millimetres under the selected idealization. It is an elastic estimate, not an allowable limit. Flexural rigidity (EI) is the product of material and geometric stiffness in kN·m². Span / deflection reports an intuitive L/n ratio; a larger denominator means less relative sag. Maximum slope is the largest tangent rotation in milliradians. Deflection direction states the assumed direction of the entered positive load. The Deflected shape along the beam plot and Deflection checkpoints table use the same model values, with x in metres, deflection in millimetres, and relative deflection as a percentage of the maximum.

Worked example

For the startup example, P = 400 N, L = 1.5 m, E = 6.8 GPa = 6.8×10⁹ Pa, and I = 160 cm⁴ = 1.6×10⁻⁶ m⁴. For a centered point load on a simply supported beam, δmax = PL³/(48EI). Substitution gives 400×1.5³ ÷ (48×6.8×10⁹×1.6×10⁻⁶) = 0.002585 m, or 2.585 mm. The same values produce EI = 10.880 kN·m² and a span-to-deflection ratio of approximately L/580.