Beam Deflection & Bending Stress Calculator
Analyze maximum displacement, internal bending moments, and outer fiber stress under standard static load cases.
Analysis Summary
Max Deflection (δ)
0.000 in
Max Bending Stress (σ)
0 PSI
Max Internal Moment
0 ft-lbs
Span-to-Deflection Ratio
L / --
Engineering Formulas & Governing Equations
Beam deflection is governed by classical Euler-Bernoulli beam theory, which assumes small deflections and linear elastic material behavior.
1. Bending Stress Equation (Flexure Formula)
The maximum flexural stress occurring at the extreme outer fiber of the beam is calculated as:
σ = (M_max * c) / I
M_max= Maximum internal bending moment (in-lbs)c= Distance from neutral axis to extreme outer fiber (in)I= Second moment of area (Moment of Inertia) (in⁴)
2. Deflection Equations
| Support Condition | Load Configuration | Max Moment (M_max) | Max Deflection (δ_max) |
|---|---|---|---|
| Simply Supported | Center Point Load | (P * L) / 4 |
(P * L³) / (48 * E * I) |
| Simply Supported | Uniform Distributed Load (UDL) | (P * L) / 8 |
(5 * P * L³) / (384 * E * I) |
| Cantilever | End Point Load | P * L |
(P * L³) / (3 * E * I) |
| Cantilever | Uniform Distributed Load (UDL) | (P * L) / 2 |
(P * L³) / (8 * E * I) |
3. Common Deflection Limits for Construction & Machine Design
Standard architectural and structural codes specify allowable limits based on the span length (L):
- L / 360: Standard floor joists under live load conditions.
- L / 240: General roof framing with no plaster or brittle ceiling attached.
- L / 500: Precision machine bases and overhead crane runways.