Bending Stress Calculator

Maximum bending stress σ = M·c/I = M/S in a beam for a rectangle, round bar, tube or any section, with section modulus and utilization.

Maximum moment in the beam, e.g. wL²/8 for a uniformly loaded simple span.
Use actual (dressed) lumber sizes, e.g. a 2×10 is 1.5 × 9.25 in.
Shows utilization. Structural steel yield ≈ 250 MPa (36 ksi); use your code’s allowable or design stress. Enter 0 to skip.
Maximum bending stress σ
60.00
Tension on one face, compression on the other
Section modulus S = I/c
83.333
Moment of inertia I
416.667 cm⁴
10.010 in⁴
Utilization σ / allowable
24.0 %
Within the entered stress limit

Estimate generated with SortedSums: https://sortedsums.com/bending-stress-calculator/. For informational purposes only; verify before relying on it.

Estimate only. This tool is for informational and educational purposes. Results depend on your inputs and simplifying assumptions, and are not a substitute for professional engineering, design or financial advice. Always verify with a qualified professional and applicable codes before purchasing, building or making decisions.

How it’s calculated

In a beam bending elastically, normal stress varies linearly from zero at the neutral axis to a maximum at the fiber farthest from it. The section modulus S = I/c packages the shape’s resistance to bending into one number.

σ = M × y / I σ_max = M × c / I = M / S Rectangle: I = b h³ / 12, S = b h² / 6 Solid round: I = π d⁴ / 64, S = π d³ / 32 Tube: I = π (D⁴ − d⁴) / 64, c = D / 2

Example: a 50 × 100 mm rectangular section with M = 5 kN·m. S = 50 × 100² ÷ 6 = 83,333 mm³ = 83.3 cm³, so σ = 5 × 10⁶ N·mm ÷ 83,333 mm³ = 60 MPa. A 2×10 joist (1.5 × 9.25 in) has S = 21.39 in³; at M = 1,000 lbf·ft = 12,000 lbf·in the stress is 12,000 ÷ 21.39 = 561 psi.

Assumes linear-elastic, homogeneous material, a straight prismatic beam and bending about a principal axis (no twisting or lateral buckling). Check shear, deflection and lateral stability separately.

Frequently asked questions

What is the difference between I and S?

I (second moment of area, length⁴) measures stiffness and governs deflection. S = I/c (length³) governs bending stress.

Which way should a rectangular beam be oriented?

On edge. S grows with the square of depth, so a 50 × 100 mm section on edge has twice the strength of the same piece laid flat.

How do I get the bending moment?

From statics or standard beam formulas: wL²/8 for a uniform load on a simple span, PL/4 for a central point load.

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Sources

Formulas are taken from the free public references above. Results are provided “as is” for informational and educational purposes only. See our disclaimer.

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