Beam Deflection Calculator Formula
Understand the math behind the beam deflection calculator. Each variable explained with a worked example.
Formulas Used
Max Deflection
max_deflection = elastic_modulus > 0 && moment_of_inertia > 0 ? 5 * load_pli * pow(span_in, 4) / (384 * elastic_modulus * moment_of_inertia) : 0Span/Deflection Ratio
span_ratio = elastic_modulus > 0 && moment_of_inertia > 0 && (5 * load_pli * pow(span_in, 4) / (384 * elastic_modulus * moment_of_inertia)) > 0 ? span_in / (5 * load_pli * pow(span_in, 4) / (384 * elastic_modulus * moment_of_inertia)) : 0L/360 Limit
limit_360 = span_in / 360Variables
| Variable | Description | Default |
|---|---|---|
span | Beam Span(ft) | 12 |
load_plf | Uniform Load(lb/ft) | 200 |
elastic_modulus | Modulus of Elasticity (E)(psi) | 1700000 |
moment_of_inertia | Moment of Inertia (I)(in^4) | 98 |
span_in | Derived value= span * 12 | calculated |
load_pli | Derived value= load_plf / 12 | calculated |
How It Works
Simply Supported Beam Deflection
Delta_max = 5wL^4 / (384EI)
Where w is the load per inch, L is the span in inches, E is the modulus of elasticity, and I is the moment of inertia. Common deflection limits are L/360 for floors and L/240 for roofs.
Worked Example
12 ft beam, 200 lb/ft load, E = 1,700,000 psi, I = 98 in^4.
- 01w = 200/12 = 16.67 lb/in
- 02L = 144 in
- 03Delta = 5 x 16.67 x 144^4 / (384 x 1,700,000 x 98)
- 04Delta = 0.283 in
- 05L/360 = 0.40 in; deflection is within limits.
When to Use This Formula
- Checking whether a floor joist or roof rafter meets the building code deflection limit (typically L/360 for live load or L/240 for total load) before construction.
- Selecting a steel beam size for a structural span by verifying that deflection under the expected load stays within acceptable limits.
- Designing shelving, workbenches, or machine frames where excessive sag would cause functional problems or look unprofessional.
- Comparing the stiffness of different beam materials (steel, aluminum, wood, composite) for the same span and loading conditions.
- Verifying that a bridge or walkway design meets serviceability requirements, where the deflection limit is driven by user comfort rather than strength.
Common Mistakes to Avoid
- Using the wrong formula for the support and loading conditions — a simply supported beam with a uniform load uses 5wL⁴/384EI, but a cantilever with a point load uses PL³/3EI. Using the wrong case produces wildly incorrect results.
- Mixing unit systems within the formula — E (modulus of elasticity) in psi, I (moment of inertia) in in⁴, w (load) in lb/ft, and L (span) in feet will not work unless everything is converted to consistent units (typically all inches and pounds, or all SI).
- Using the moment of inertia (I) for the wrong axis — beams resist bending about their strong axis (Ix), not their weak axis (Iy). A beam oriented flat instead of on edge has a much smaller I and will deflect far more.
- Ignoring the difference between total load and live load deflection limits — building codes specify separate limits for each. A beam that passes the total-load check at L/240 may still fail the live-load check at L/360.
- Assuming the formula accounts for long-term creep — wood beams sag more over time under sustained load due to creep. Design codes require a creep factor (typically 1.5-2.0x) for permanent loads on wood members.
Frequently Asked Questions
What is L/360?
L/360 means the allowable deflection is the span divided by 360. For a 12 ft (144 in) span, the limit is 144/360 = 0.40 inches. This is the typical limit for floor beams under live load.
Where do I find E and I values?
E depends on the material (wood: 1.2-1.8M psi, steel: 29M psi). I depends on the cross-section; look it up in beam tables or calculate from the section dimensions.
Ready to run the numbers?
Open Beam Deflection Calculator