Bending Stress Calculator
Calculate the peak bending stress in a beam from its bending moment and rectangular section.
Reviewed by the ToolNestr Editorial Team — July 2026
How the bending stress calculator works
When a beam bends, one face is squeezed in compression and the opposite face is stretched in tension, with a neutral axis of zero stress in between. The flexure formula gives the peak stress at the outer fiber: σ = M·c ÷ I, where M is the bending moment, c is the distance from the neutral axis to the extreme fiber, and I is the moment of inertia. Because the section modulus S equals I ÷ c, the formula reduces to the simpler σ = M ÷ S.
For a solid rectangle, the section modulus is width × depth² ÷ 6. Depth is squared here, so a deep, narrow beam carries a given moment at much lower stress than a shallow, wide one of the same area — the reason floor joists and rafters stand tall on edge. The calculator computes S and I from the width and depth you enter, then divides the moment by S to return the stress.
Moment is entered in pound-feet and converted to pound-inches (×12) so the units match a section measured in inches, giving stress in pounds per square inch. If you enter an allowable bending stress (Fb) for your material, the tool flags whether the section is within it. This is a single-equation estimate — it does not check shear, deflection, buckling, or load combinations, and allowable stresses must come from the governing code and a qualified engineer.
Allowable stress design methodology follows the AISC Steel Construction Manual and the NDS (National Design Specification for Wood Construction).
Enter the moment
The maximum bending moment on the beam.
Give the section
Rectangular width and depth in inches.
Read the stress
Compare it to your allowable Fb if entered.
The formula explained
Section modulus
S = b·d² ÷ 6. A 1.5 × 7.25 in section: 1.5 × 7.25² ÷ 6 = 13.14 in³.
Bending stress
σ = M ÷ S with M in lb·in (lb·ft × 12).
Moment of inertia
I = b·d³ ÷ 12, and c = d ÷ 2, so S = I ÷ c as expected.
Worked example
A 2×8 (1.5 × 7.25 in) carrying a 1,800 lb·ft moment, Fb = 900 psi.
Real-world context
Floor joists are the everyday case. A 2×10 joist (actual 1.5 × 9.25 in) carrying a 3,000 lb·ft moment has a section modulus S = 1.5 × 9.25² ÷ 6 = 21.39 in³. Converting the moment to 36,000 lb·in and dividing gives a peak stress of about 1,683 psi. Against a typical Douglas fir Fb around 900-1,000 psi, this joist would be flagged as overstressed — a reminder that span tables already account for the actual expected moments, and increasing depth or reducing spacing is the usual fix.
Steel beams tolerate far higher stress before failing. A steel section approximated as a 6 × 12 in rectangle carrying a 15,000 lb·ft moment has S = 6 × 12² ÷ 6 = 144 in³, giving a stress of 180,000 ÷ 144 = 1,250 psi. Structural steel (A36) has an allowable bending stress well above 20,000 psi, so this beam runs at only a small fraction of its capacity — steel's headroom is why long-span commercial framing favors it over wood.
Engineered lumber often lands in between. An LVL beam 3.5 × 11.875 in carrying a 6,000 lb·ft moment has S = 3.5 × 11.875² ÷ 6 = 82.26 in³, giving a stress of 72,000 ÷ 82.26 = 875 psi. LVL is typically rated for Fb around 2,600 psi, so this beam has a healthy margin — which is exactly why LVL headers can span openings that would overstress an equivalent sawn-lumber beam.
Common misconceptions
"If the bending stress is under the allowable Fb, the beam is fully safe to build." Bending stress is only one of several checks. Deflection, shear at the supports, bearing at the ends, lateral-torsional buckling, and connection design all have to pass independently. This calculator only computes peak flexural stress for one section and moment — a licensed engineer must verify the complete design.
"A wider beam resists bending about as well as a deeper one of the same cross-sectional area." Depth wins by a wide margin. The section modulus for a rectangle is width × depth² ÷ 6 — depth is squared, width is linear. Reorienting a given piece of lumber from flat to on-edge can cut its bending stress dramatically without adding any material.
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Frequently asked questions
What is the bending stress formula?
The flexure formula is σ = M·c ÷ I, where M is the bending moment, c is the distance from the neutral axis to the outermost fiber, and I is the moment of inertia. Using the section modulus S = I ÷ c, it simplifies to σ = M ÷ S.
What is the section modulus?
The section modulus S combines shape and size into one number that resists bending. For a solid rectangle it is width × depth² ÷ 6. A larger S means lower stress for the same moment, which is why beams are made deep rather than wide.
How do I find the bending moment?
For a simply supported beam with a uniform load, the maximum moment is w·L² ÷ 8; for a center point load it is P·L ÷ 4. The moment must be in consistent units — pound-inches when the section is in inches — before dividing by the section modulus.
How do I know if the stress is safe?
Compare the calculated stress to the material’s allowable bending stress (Fb) with its safety factor applied. If σ exceeds the allowable value the section is overstressed. This tool computes the stress only; a licensed engineer must confirm allowable values and the full design.
Sources & references
This tool uses standard formulas and reference values from:
- • American Concrete Institute — ACI 318, Building Code Requirements for Structural Concrete. concrete.org
- • ICC — International Residential Code (IRC), span, footing and framing tables. codes.iccsafe.org
- • APA – The Engineered Wood Association, allowable span and load guidance.
Estimates for planning only. Span, load and code values vary by jurisdiction — verify against your local adopted code and a licensed engineer before building.