Bending Stress Calculator

Bending stress is the internal stress that resists the bending moment applied to a beam. It varies linearly across the cross-section: maximum compressive stress on the top fiber (for a simply supported beam), maximum tensile stress on the bottom fiber, and zero stress at the neutral axis (centroid). The stress distribution follows σ = My/I, where M is the moment, y is the distance from neutral axis, and I is the moment of inertia. The maximum bending stress occurs at the outermost fibers (top and bottom) where y is largest.

Section modulus (S = I/c) is a geometric property that directly relates to a beam's bending strength, where I is moment of inertia and c is the distance from neutral axis to extreme fiber. For a rectangular section: S = bd²/6. A larger section modulus means the beam can resist more bending moment with less stress. Doubling beam depth increases section modulus by 4× (d² effect). This is why deeper beams are dramatically stronger in bending than wider ones of the same cross-sectional area.

Compare the actual bending stress (fb) to the allowable bending stress (Fb). For design to be adequate: fb ≤ Fb × adjustment factors. Actual stress: fb = M / S, where M is the maximum moment and S is the section modulus. Allowable stress depends on material: Douglas Fir #2: ~900 psi, Southern Pine #2: ~1000 psi, LVL: ~2600 psi, Steel: ~24,000 psi. The ratio Fb/fb gives the safety factor. A safety factor of 1.0+ is required, with 1.5-2.0 recommended for most applications.

For bending strength, I-beams and wide-flange shapes are most efficient because they concentrate material at the extreme fibers (top and bottom flanges) where bending stress is highest, while keeping the web thin to reduce weight. The order of efficiency (strength-to-weight ratio for same area): I-beam > rectangular > square > circle. However, rectangular beams are most common in wood construction due to manufacturing simplicity. For steel, W-shapes (wide flange) are standard. The section modulus per unit weight is the key efficiency metric.