Beam Deflection Calculator Formula
Understand the math behind the beam deflection calculator. Each variable explained with a worked example.
Formulas Used
Maximum Deflection
max_deflection_mm = (5 * load_per_length * pow(span, 4)) / (384 * e_pa * i_m4) * 1000Span / Deflection Ratio
span_over_deflection = span / ((5 * load_per_length * pow(span, 4)) / (384 * e_pa * i_m4))Variables
| Variable | Description | Default |
|---|---|---|
load_per_length | Distributed Load (w)(N/m) | 5000 |
span | Beam Span (L)(m) | 6 |
modulus | Elastic Modulus (E)(GPa) | 200 |
inertia | Moment of Inertia (I)(cm^4) | 8356 |
e_pa | Derived value= modulus * 1e9 | calculated |
i_m4 | Derived value= inertia * 1e-8 | calculated |
How It Works
How Beam Deflection Works
For a simply supported beam carrying a uniformly distributed load, the peak deflection occurs at midspan.
Governing Equation
delta_max = (5 w L^4) / (384 E I)
where w is the load intensity in N/m, L is the clear span, E is the material stiffness (elastic modulus), and I is the second moment of area of the cross-section. Engineers often check that the span-to-deflection ratio exceeds code limits (commonly L/360 for floor beams).
Worked Example
A W200x46 steel beam spanning 6 m carries 5 kN/m uniformly.
- 01Convert E: 200 GPa = 200 x 10^9 Pa
- 02Convert I: 8356 cm^4 = 8356 x 10^-8 m^4 = 8.356 x 10^-5 m^4
- 03Numerator: 5 x 5000 x 6^4 = 25000 x 1296 = 3.24 x 10^7
- 04Denominator: 384 x 2 x 10^11 x 8.356 x 10^-5 = 6.417 x 10^9
- 05delta = 3.24 x 10^7 / 6.417 x 10^9 = 0.00505 m = 5.05 mm
- 06Span/deflection = 6000 / 5.05 = 1188 (well above L/360 = 16.7 mm limit)
Frequently Asked Questions
What is an acceptable beam deflection?
Most building codes limit live-load deflection to L/360 for floors and L/240 for roofs, where L is the span length. More sensitive applications may require L/480 or stricter.
Does beam weight affect deflection?
Yes. The self-weight of the beam adds to the distributed load. For precise analysis, add the beam weight per unit length to the applied load w.
What if the load is a single point load at midspan?
For a central point load P, the midspan deflection is P L^3 / (48 E I), a different coefficient from the uniform-load formula used here.
Learn More
Guide
Beam Stress Calculation Guide: From Theory to Practice
Learn how to calculate beam stress step by step. Covers bending stress, shear stress, the flexure formula, stress distributions, and practical design checks for structural beams.
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Open Beam Deflection Calculator