Beam Deflection Calculator

Load Cases & Deflection Formulas

Simply supported beam, point load at midspan: delta = P x L^3 / (48 x E x I).

Cantilever beam, point load at free end: delta = P x L^3 / (3 x E x I).

Simply supported beam, uniformly distributed load w (N/m): delta = 5 x w x L^4 / (384 x E x I). Use this approximation when the dominant load is distributed (floors, platforms) rather than a single point load.

Cantilever, uniformly distributed load: delta = w x L^4 / (8 x E x I).

All formulas assume elastic material behaviour, small deflections, and constant cross-section (prismatic beam).

Slope (rotation) at key points: simply supported beam with a centre point load, maximum slope at the supports theta = P x L^2 / (16 x E x I). Cantilever with an end point load, slope at the free end theta = P x L^2 / (2 x E x I). For a uniformly distributed load w on a simply supported beam, maximum slope at the supports theta = w x L^3 / (24 x E x I).

Worked Design Example: Steel Floor Beam

Check a 6 m simply supported steel beam (UB 305 x 165 x 40, I = 87,300,000 mm4) under a 12 kN centre load. E = 200,000 MPa.

delta = 12,000 x 6.0^3 / (48 x 200,000 x 87.3 x 10^-6) = 2,592,000 / 838,080,000 = 0.00309 m = 3.1 mm.

Allowable deflection L/360 = 6,000 / 360 = 16.7 mm. Actual 3.1 mm < 16.7 mm, so the beam satisfies the serviceability limit with large margin.

Additional Worked Examples: Cantilever & Distributed Load

Cantilever, point load at free end: a 2 m cantilever steel beam (E = 200,000 MPa, I = 1e7 mm4) with a 5 kN end load deflects delta = 5,000 x 2.0^3 / (3 x 200,000 x 1e-5) = 0.00667 m = 6.67 mm at the free end.

Simply supported, uniformly distributed load: a 6 m steel beam (I = 87,300,000 mm4) under w = 10 kN/m deflects delta = 5 x 10,000 x 6.0^4 / (384 x 200,000 x 8.73e-5) = 0.00967 m = 9.67 mm at midspan.

Cantilever, uniformly distributed load: a 3 m cantilever (I = 1e7 mm4) under w = 5 kN/m deflects delta = 5,000 x 3.0^4 / (8 x 200,000 x 1e-5) = 0.0253 m = 25.3 mm at the free end.

These three cases cover the four standard combinations — simply supported and cantilever, each with point or distributed load — so you can cross-check the interactive tool against hand calculation.

Deflection Limits for Structural Design

Typical serviceability limits (verify against governing code and project spec):

Floor beams, total load: L/240 to L/360. Floor beams, live load only: L/360 to L/480.

Roof purlins and girts: L/180 to L/240. Crane runway girders: L/600 to L/1000 (often governed by crane manufacturer).

Industrial platforms and mezzanines: L/360 total load is common. Precast or brittle cladding systems require tighter limits to avoid cracking.

In US practice AISC recommends L/360 for floors; Eurocode EN 1993-1-1 suggests L/300 to L/360 for floors depending on occupancy.

Unit Conversion Guide

Force: 1 N = 0.2248 lbf; 1 kN = 224.8 lbf; 10 kN ≈ 2,248 lbf.

Length: 1 mm = 0.03937 in; 1 m = 3.281 ft.

Modulus: 1 MPa = 0.145 ksi; steel E = 200 GPa ≈ 29,000 ksi; aluminum E = 69 GPa ≈ 10,000 ksi.

Moment of inertia: 1 mm4 = 2.402 x 10^-6 in4.

To use US inputs with this SI engine, convert lbf to N (x 4.4482), inches to mm (x 25.4), ksi to MPa (x 6.8948).

Worked Example: 5 m Simply Supported Steel Beam

Input: a 5 m simply supported steel beam (E = 200,000 MPa) with I = 10,000,000 mm4 (10 x 10^-6 m4), carrying a 10 kN (10,000 N) point load at midspan.

Formula: delta = P x L^3 / (48 x E x I) = 10,000 x 5.0^3 / (48 x 200,000 x 10 x 10^-6).

Result: delta = 1,250,000 / 96,000,000 = 0.01302 m = 13.02 mm deflection at midspan.

Check: allowable serviceability limit L/360 = 5,000 / 360 = 13.9 mm. Actual deflection 13.02 mm is below 13.9 mm, so this beam passes the common floor serviceability limit with a margin of about 0.9 mm. The margin is small, so for live-load-sensitive floors consider a stiffer section or a shorter span. Always confirm against AISC, ACI or Eurocode.