What is the deflection formula for a cantilever with a point load?

For a point load P at the free end of a cantilever of length L, maximum deflection is δ = PL³ / (3EI) and slope is θ = PL² / (2EI), where E is the modulus of elasticity and I is the second moment of area.

What formula is used for a uniformly distributed load?

For a uniform load w (force per unit length) over the full span L, maximum deflection at the free end is δ = wL⁴ / (8EI) and slope is θ = wL³ / (6EI).

What formula applies when a bending moment is applied at the free end?

For an applied moment M₀ at the free end, deflection is δ = M₀L² / (2EI) and slope is θ = M₀L / (EI). This case produces a constant bending moment along the entire beam.

What are the assumptions of Euler-Bernoulli beam theory?

The formulas assume: (1) linear elastic homogeneous material, (2) small deflections relative to span, (3) plane sections remain plane after bending, and (4) shear deformation is negligible (span-to-depth ratio above about 10). For short deep beams, use Timoshenko beam theory.

How do I find the moment of inertia of my cross-section?

Rectangle: I = bh³/12. Solid circle: I = πd⁴/64. Hollow circle (pipe): I = π(D⁴ − d⁴)/64. Use the section helper built into this tool, or enter the value directly in mm⁴.

What modulus of elasticity value should I use?

Common values: structural steel 200 GPa, aluminum alloy 69 GPa, timber (along grain) 10–12 GPa, concrete 25–30 GPa. Use the material preset dropdown or enter a custom value.

Beam Deflection Cantilever Calculator

Enter beam length, modulus of elasticity, moment of inertia, and load to instantly compute maximum deflection and slope at the free end of a cantilever beam.

Loading Condition

Beam & Material Properties

Beam Length, L (mm)

Modulus of Elasticity, E (GPa)

Moment of Inertia, I (mm⁴)

Enter directly or use the section helper below.

Applied Load

Point Load, P (N)

Concentrated force at the free end.

Max Deflection at Free End

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δ = PL³ / (3EI)

Free End Slope

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θ = PL² / (2EI) (rad)

Fill in the inputs and click Calculate.

Active Formulas

δ = PL³ / (3EI)

θ = PL² / (2EI)

Serviceability Check

Limit: L / (common: 300–500)

Calculate first.

Summary

This cantilever beam deflection calculator applies classical Euler-Bernoulli beam theory to find maximum deflection and slope at the free end. Three standard loading cases are supported: a point load at the free end (δ = PL³/3EI), a uniformly distributed load over the full span (δ = wL⁴/8EI), and an applied bending moment at the free end (δ = ML²/2EI). Moment of inertia can be entered directly or computed from rectangular, circular, or hollow circular cross-sections. All calculations run entirely in your browser.

How it works

Select the loading type: point load at free end, uniform distributed load, or applied end moment.
Enter beam length L in millimeters and modulus of elasticity E in GPa.
Enter the second moment of area I in mm⁴, or use the section helper to compute it from cross-section dimensions.
Enter the load value (force in N, load intensity in N/mm, or moment in N·mm).
Click Calculate to see maximum deflection and free-end slope.
Adjust the L/n serviceability limit to check code compliance.

Use cases

Checking free-end deflection of a steel cantilever bracket under a point load.
Sizing a cantilevered balcony beam for serviceability deflection limits.
Verifying tip deflection of a diving board under design load.
Calculating deflection of a machine arm or robotic linkage under tip forces.
Teaching Euler-Bernoulli beam theory in structural mechanics courses.
Preliminary beam selection before detailed finite element analysis.
Checking deflection in overhanging crane girders and jib arms.
Estimating springback and stiffness of cantilever-type sensors.

Frequently Asked Questions

Last updated: 2026-05-23 · Reviewed by Nham Vu