What is the deflection formula for a central point load?

For a simply supported beam of span L with a point load P at mid-span, maximum deflection at center is δ = PL³ / (48EI), where E is the modulus of elasticity and I is the second moment of area. Both support reactions equal P/2.

What formula applies for a uniformly distributed load?

For a uniform load w (force per unit length) over the full span L, maximum deflection at mid-span is δ = 5wL⁴ / (384EI). Both reactions equal wL/2. The total load carried is wL.

How is an off-center point load handled?

For load P at distance a from the left support (b = L − a from the right), the reactions are R_A = Pb/L and R_B = Pa/L. Maximum deflection occurs at x = √((L² − b²)/3) from the left support and equals δ_max = Pab(L² − a² − b²)^(3/2) / (9√3 · EIL) when a ≤ b, or its mirror image when a > b.

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, and (4) shear deformation is negligible (span-to-depth ratio above about 10). For deep beams or composite sections, use Timoshenko theory or finite element analysis.

How do I find the moment of inertia?

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

What modulus of elasticity should I use?

Common values: structural steel 200 GPa, aluminum alloy 69 GPa, timber (along grain) 11 GPa, concrete 28 GPa, titanium 110 GPa. Select a preset from the dropdown or enter a custom value.

Beam Deflection Simply Supported

Name: Beam Deflection Simply Supported
Availability: InStock
Author: Nham Vu

Enter beam length, material, moment of inertia, and load to compute maximum deflection and support reactions for a simply supported beam.

Loading Condition

Beam & Material Properties

Beam Span, 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 mid-span (L/2).

Max Deflection

—

Location: —

Reaction R₁ (left)

—

Reaction R₂ (right)

—

Fill in the inputs and click Calculate.

Active Formula

δ = PL³ / (48EI)

R₁ = R₂ = P/2

Serviceability Check

Limit: L / (common: 300–500)

Calculate first.

Summary

Enter beam length, material, moment of inertia, and load to compute maximum deflection and support reactions for a simply supported beam.

How it works

Select the loading condition: central point load, uniform distributed load, or off-center point load.
Enter beam span L in millimeters and modulus of elasticity E in GPa (use the material preset or type a value).
Enter the second moment of area I in mm⁴, or open the section helper to compute it from cross-section dimensions.
Enter the load magnitude and, for the off-center case, the distance a from the left support to the load.
Click Calculate to see maximum deflection, its location, and the reactions at both supports.
Adjust the serviceability limit (e.g. L/300) to check code compliance.

Use cases

Checking mid-span deflection of a floor beam under uniform load for serviceability.
Sizing a simply supported steel beam for an industrial platform.
Calculating reaction forces at supports for foundation design.
Verifying deflection of a bridge girder under a moving point load.
Teaching structural mechanics — comparing point vs distributed load deflection.
Preliminary beam selection before finite element or full structural analysis.
Checking deflection limits per building codes (L/300, L/360, L/500).
Estimating stiffness of test fixtures and laboratory loading frames.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu