Beam Deflection Calculator
What is Beam Deflection Calculator?
Beam deflection is the vertical displacement of a beam under load, governed by the Euler-Bernoulli (or Timoshenko for short/deep beams) differential equation. Controlling deflection is a key serviceability limit state (SLS) requirement in structural design.
The Beam Deflection Calculator for Structural/Civil Engineers is a fast and accurate online tool that instantly computes deflection δ(x), slope θ(x), maximum deflection δ_max, bending moment M(x), shear V(x), and deflection ratio L/δ for simply supported, cantilever, fixed-fixed, continuous, propped, and overhanging beams under any combination of point loads, UDL, triangular loads, moments, self-weight, and support settlements. It supports both Euler-Bernoulli and Timoshenko theory, effective stiffness for cracked concrete, creep adjustment, and code-compliant serviceability checks (ACI, Eurocode, IS, BS, AS/NZS). Perfect for beam deflection calculator online, deflection formula calculator, SLS check, cantilever deflection, simply supported beam deflection, and quick structural serviceability verification.
This beam deflection calculator provides relevant visualizations, a dedicated section for comments, analysis and recommendations, full step-by-step calculation with every integration constant shown, CSV export/download of results (δ, θ, M, V at any interval), and a Colorblind view mode to improve accessibility.
How to use Beam Deflection Calculator
Purpose: Calculate actual deflection and slope at any point x, maximum values, and check against code limits so you can verify serviceability before final design.
Inputs you will enter:
- Beam type / support conditions (simply supported, cantilever, fixed-fixed, propped, continuous, overhanging)
- Span length L (m)
- Section properties: Ix (m⁴), A (m² for Timoshenko), E (GPa), G (GPa), shear correction k
- Loads: point load P at a, UDL w (full/partial), triangular load, moment M, self-weight
- Optional: support settlement, creep coefficient φ, cracked-section effective EI, evaluation point x
Beam Deflection Formula
Simply Supported – Central Point Load \(\delta_{max} = \frac{P L^3}{48 E I}\)
Simply Supported – Full UDL \(\delta_{max} = \frac{5 w L^4}{384 E I}\)
Cantilever – End Point Load \(\delta_{max} = \frac{P L^3}{3 E I}\)
Cantilever – Full UDL \(\delta_{max} = \frac{w L^4}{8 E I}\)
Fixed-Fixed – Full UDL \(\delta_{max} = \frac{w L^4}{384 E I}\)
Where:
- P = point load (kN)
- w = UDL intensity (kN/m)
- L = span (m)
- E = modulus of elasticity (GPa)
- I = second moment of area (m⁴)
- δ = deflection (mm)
How to Calculate Beam Deflection (Step-by-Step)
- Select support conditions and enter geometry/material properties.
- Add all loads with their positions.
- Choose theory (Euler-Bernoulli or Timoshenko).
- Calculator integrates the load → shear → moment → slope → deflection (or uses closed-form formulas).
- Applies superposition for multiple loads.
- Applies creep/effective stiffness if selected.
- Compares δ_max and L/δ against code limits (ACI L/360, Eurocode L/250, etc.).
- Shows deflected shape, warnings, and recommendations.
Examples
Example 1 – Simply Supported Beam Span L = 6 m, UDL w = 25 kN/m, E = 200 GPa, I = 250×10⁻⁶ m⁴ \(\delta_{max} = \frac{5 \times 25 \times 6^4}{384 \times 200 \times 10^9 \times 250 \times 10^{-6}} = 0.0169\ \text{m} = 16.9\ \text{mm}\) L/δ = 355 → OK for ACI L/360 floor beam.
Example 2 – Cantilever with End Point Load L = 4 m, P = 50 kN at free end, E = 25 GPa (concrete), I = 120×10⁻⁶ m⁴ \(\delta_{max} = \frac{50 \times 4^3}{3 \times 25 \times 10^9 \times 120 \times 10^{-6}} = 0.0356\ \text{m} = 35.6\ \text{mm}\) L/δ = 112 → exceeds ACI L/180 → increase section.
Beam Deflection Categories / Normal Range (Common Code Limits)
| Support Condition | Load Type | Typical δ_max Formula | ACI Limit (Floor) | Eurocode Limit |
|---|---|---|---|---|
| Simply Supported | Central point | PL³/48EI | L/360 | L/250 |
| Simply Supported | Full UDL | 5wL⁴/384EI | L/360 | L/250 |
| Cantilever | End point | PL³/3EI | L/180 | L/200 |
| Cantilever | Full UDL | wL⁴/8EI | L/180 | L/200 |
| Fixed–Fixed | Full UDL | wL⁴/384EI | L/360 | L/250 |
| Propped Cantilever | Full UDL | wL⁴/185EI | L/360 | L/250 |
Limitations
- Small-deflection theory only (δ ≪ L).
- Prismatic sections; variable EI requires segmentation.
- No dynamic, thermal, or shrinkage effects unless manually added.
- Timoshenko option only for short/deep beams (L/h < 10).
- Creep is approximate (effective E); long-term camber not included.
Disclaimer
This calculator is provided for educational purposes, learning, and preliminary serviceability checks only. All final structural designs must be reviewed and certified by a qualified professional structural engineer. The developer and platform are not liable for any errors, misinterpretations, or consequences arising from the use of these results in actual construction projects.