Beam Deflection Calculator

Reviewed by Hamza Gujjar, Algorithm Architect & Software Developer Last updated: August 19, 2026

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Where the Load Sits Changes Everything About How Much a Beam Bends

The same total load produces wildly different deflection depending on how it's applied and how the beam is supported. A cantilever carrying a point load at its free end deflects roughly 16 times more than a simply supported beam of the same span carrying the same point load at midspan — not because the material is different, but because the boundary conditions change the governing equation entirely. This calculator applies the correct linear-elastic formula for four common beam and loading configurations.

The Formulas

Simply supported, point load at center: δ = (P × L³) / (48 × E × I)
Simply supported, uniform load: δ = (5 × w × L⁴) / (384 × E × I)
Cantilever, point load at free end: δ = (P × L³) / (3 × E × I)
Cantilever, uniform load: δ = (w × L⁴) / (8 × E × I)

P is a point load, w is a distributed load per unit length, L is the span, E is the modulus of elasticity, and I is the moment of inertia of the cross-section.

Same Load, Four Very Different Outcomes

Deflection for a 120 in (10 ft) steel span, E = 29,000,000 psi, I = 100 in⁴, with a total load of 1,000 lb applied as either a point load or an equivalent total uniform load
Configuration	Deflection
Simply supported, point load at center	0.0124 in
Simply supported, uniform load	0.0078 in
Cantilever, point load at free end	0.1986 in
Cantilever, uniform load	0.0745 in

The cantilever point-load case deflects roughly 16 times more than the simply supported point-load case for identical span, material, section, and total load — a direct result of the cantilever having only one support instead of two.

Where This Gets Used

Checking serviceability limits — building codes typically cap allowable deflection as a fraction of span (e.g. L/360 for floors under live load), and this gives the raw deflection figure to compare against that limit.
Comparing section options — since deflection is inversely proportional to I, doubling the moment of inertia halves the deflection, which helps evaluate whether a deeper or wider section solves a deflection problem.
Early-stage sizing — a quick check before running a full structural analysis, useful for cantilevered balconies, shelving brackets, or simply supported joists.

How to Use This Calculator

Choose Unit system — Imperial (in, psi, lb) or Metric (m, Pa, N).
Choose Load Type — Simply Supported with a point load or uniform load, or Cantilever with a point load or uniform load.
Enter Load, Span/Length, Modulus of Elasticity E, and Moment of Inertia I, using one consistent unit system throughout.
Select Calculate to see the deflection.

Note: this is a linear-elastic estimate and does not account for shear deformation or material yielding.

Related Calculations

Check the load feeding into this beam with the Structural Load Calculator, or size a concrete pour with the Concrete Mix Calculator.

Principles of Euler-Bernoulli Beam Deflection Theory

In civil, structural, and mechanical engineering, beam deflection calculations determine the elastic vertical displacement (δ) of a structural beam subjected to transverse bending loads. Governed by the classical Euler-Bernoulli Beam Equation, deflection modeling ensures that building floors, bridge girders, and machinery shafts satisfy structural serviceability deflection limits without excessive sagging, vibration, or drywall cracking.

The Governing Fourth-Order Differential Equation

E × I × (d⁴w / dx⁴) = q(x) | M(x) = -E × I × (d²w / dx²)

Where E is the material's Modulus of Elasticity (Young's Modulus) (e.g., 200 GPa for structural steel, 11 GPa for Douglas Fir timber), and I is the cross-sectional Area Moment of Inertia about the bending axis. The product E × I represents the overall flexural rigidity of the beam.

Standard Moment of Inertia (I) Formulas

Solid Rectangular Beam (Width b, Height h): I = (b × h³) / 12 (vertical height h dominates stiffness by power of 3).
Solid Circular Shaft (Diameter d): I = (π × d⁴) / 64.
Hollow Structural Tube (Outer D, Inner d): I = π × (D⁴ - d⁴) / 64.

Classical Beam Deflection Standard Load Formulas

Beam Support & Loading Condition	Maximum Deflection (δ_max) Formula	Location of Max Deflection
Simply Supported — Center Point Load P	δ_max = (P × L³) / (48 × E × I)	Center Span (x = L/2)
Simply Supported — Uniform Distributed Load w	δ_max = (5 × w × L⁴) / (384 × E × I)	Center Span (x = L/2)
Cantilever Beam — End Point Load P	δ_max = (P × L³) / (3 × E × I)	Free End Tip (x = L)
Cantilever Beam — Uniform Distributed Load w	δ_max = (w × L⁴) / (8 × E × I)	Free End Tip (x = L)
Fixed-Fixed Beam — Center Point Load P	δ_max = (P × L³) / (192 × E × I)	Center Span (x = L/2)

Step-by-Step Worked Calculation Example

Example: Calculating Center-Span Deflection of a Steel I-Beam Girder

Problem: A W8×31 structural steel floor girder spans L = 6.0 meters (6,000 mm) simply supported at both ends. It supports a central point load P = 24.0 kN (24,000 N). For steel, Young's Modulus E = 200 GPa (200,000 N/mm²). The beam has an Area Moment of Inertia I = 45.8 × 10⁶ mm⁴. Calculate: (1) The maximum central deflection δ_max in millimeters; and (2) Verify if it meets the IBC building code floor live-load serviceability limit of L/360.

Step 1: Calculate central deflection using standard point load formula:

δ_max = ( P × L³ ) / ( 48 × E × I )

Numerator = 24,000 N × (6,000 mm)³ = 24,000 × 2.16 × 10¹¹ = 5.184 × 10¹⁵ N·mm³

Denominator = 48 × 200,000 N/mm² × (45.8 × 10⁶ mm⁴) = 4.3968 × 10¹⁴ N·mm²

δ_max = (5.184 × 10¹⁵) / (4.3968 × 10¹⁴) = 11.79 mm (approx. 0.464 inches)

Step 2: Check IBC L/360 deflection limit:

δ_allowable = Span / 360 = 6,000 mm / 360 = 16.67 mm

Conclusion: Because actual deflection δ_max (11.79 mm) is strictly less than allowable limit (16.67 mm), the beam satisfies IBC structural serviceability standards.

Building Code Deflection Serviceability Limits (IBC)

L/360 Limit: Floor live loads supporting plaster/drywall ceilings (prevents brittle plaster cracking).
L/240 Limit: Total floor load (dead + live load) or roof members supporting non-plaster ceilings.
L/180 Limit: Secondary roof rafters and purlins supporting non-plaster ceilings.

The Principle of Superposition for Combined Loadings

In linear elastic structural analysis, when a beam is subjected to multiple simultaneous loads (e.g., a uniform dead load w combined with two concentrated live point loads P₁ and P₂), total elastic deflection is calculated using the Principle of Superposition:

δ_total(x) = δ_uniform(x) + δ_point1(x) + δ_point2(x)

Dynamic Natural Resonance Frequencies

In structural dynamics, static deflection predicts the fundamental natural vibration frequency (f_n, Hz) of a floor girder: f_n ≈ 0.18 × √(g / δ_static), preventing footfall-induced floor bounce resonance.

Thermal Deflection from Linear Temperature Gradients

When a structural bridge girder experiences solar radiant heating on its top flange while the bottom flange remains in shadow (ΔT gradient), differential thermal expansion induces vertical thermal bowing deflection: δ_thermal = α × ΔT × L² / (8 × h).