Beam Bending Calculator – Stress & Deflection for Structural Beams
Analyze beam performance under load with our beam bending calculator. Calculate maximum bending stress, deflection, and moment for simply supported and cantilever beams.
Select support and load type
Choose simply supported or cantilever beam. Select point load (single force) or distributed load (force spread across the beam).
Enter beam dimensions and load
Input beam length in meters and total load in newtons. For point loads, specify the position from the left support.
Enter material properties and calculate
Input Young's modulus (E) and moment of inertia (I). Click Calculate to see maximum bending moment and deflection.
| Material | Young's Modulus (GPa) | Young's Modulus (Pa) |
|---|---|---|
| Structural Steel (A36) | 200 GPa | 200 × 10⁹ Pa |
| Stainless Steel (304) | 193 GPa | 193 × 10⁹ Pa |
| Aluminum (6061-T6) | 69 GPa | 69 × 10⁹ Pa |
| Copper | 117 GPa | 117 × 10⁹ Pa |
| Titanium (Grade 5) | 114 GPa | 114 × 10⁹ Pa |
| Concrete | 25-30 GPa | 25-30 × 10⁹ Pa |
| Wood (Douglas Fir) | 11-13 GPa | 11-13 × 10⁹ Pa |
Note: Values are approximate and vary with alloy, grade, and manufacturing process. Always use manufacturer specifications for critical applications.
What Is Beam Bending?
When a beam carries a load, it bends. The top fibers compress while the bottom fibers stretch (for a simply supported beam). This creates internal stresses. Engineers calculate bending moment and deflection to ensure beams can safely support their loads without excessive deformation.
Bending Moment Explained
Bending moment measures the internal force that causes bending. It is calculated in newton-meters (N·m). The maximum moment occurs where the beam experiences the greatest bending stress. For a simply supported beam with a center point load, maximum moment equals (P × L) / 4.
Beam Deflection
Deflection is how much the beam bends under load, measured in meters or millimeters. Excessive deflection causes cracks in ceilings, sagging floors, and structural concerns. Building codes typically limit deflection to L/360 for floors and L/240 for roofs, where L is the span length.
Moment of Inertia (I)
The moment of inertia describes a beam's resistance to bending based on its cross-sectional shape. A tall, narrow beam resists bending better than a short, wide beam of the same area. For a rectangle, I = (b × h³) / 12, where b is width and h is height.
Simply Supported Beam, Center Point Load
Max Moment: M = (P × L) / 4
Max Deflection: δ = (P × L³) / (48 × E × I)
Simply Supported Beam, Distributed Load
Max Moment: M = (w × L²) / 8
Max Deflection: δ = (5 × w × L⁴) / (384 × E × I)
Cantilever Beam, End Point Load
Max Moment: M = P × L
Max Deflection: δ = (P × L³) / (3 × E × I)
Cantilever Beam, Distributed Load
Max Moment: M = (w × L²) / 2
Max Deflection: δ = (w × L⁴) / (8 × E × I)
Where: P = point load, w = distributed load per unit length, L = beam length, E = Young's modulus, I = moment of inertia
Check both stress and deflection
A beam might be strong enough but still deflect too much. Always verify both bending stress is within limits and deflection meets code requirements.
Apply safety factors
Real-world loads have uncertainties. Building codes specify load combinations and safety factors. Never design to the theoretical limit — always include a margin of safety.
Consider beam orientation
A 2×10 joist is much stronger on edge than flat. The moment of inertia changes dramatically with orientation. Always install beams with the strong axis resisting the load.
What is Young's modulus?
Young's modulus (E) measures material stiffness. It describes how much a material stretches under tension. Steel has E ≈ 200 GPa, aluminum ≈ 70 GPa. Higher E means less deflection for the same load.
How do I calculate moment of inertia for a rectangular beam?
For a rectangle: I = (b × h³) / 12, where b is the width (base) and h is the height. The height is cubed, so increasing beam depth has a much bigger effect than increasing width.
What is an acceptable deflection limit?
Building codes specify limits based on application. Floors: L/360 (span divided by 360). Roofs: L/240. For a 10-foot (120 inch) floor span, maximum deflection is 120/360 = 0.33 inches.
What's the difference between simply supported and cantilever beams?
Simply supported beams rest on supports at both ends (like a bridge). Cantilever beams are fixed at one end and free at the other (like a diving board). Cantilevers experience higher moments and deflections.
Can I use this calculator for wood beams?
Yes, but wood is anisotropic — properties vary with grain direction. Use appropriate E values for the wood species and grade. Also check shear stress, which can govern for short, deep wood beams.
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