Bending stress is one of the first things you’ll calculate on the job. If your boss hands you a bracket, a shaft, or any beam under load, bending stress is almost always in the analysis.
The Flexure Formula
The bending stress at any point in a cross-section is:
σ = M · c / I
Where:
- σ = bending stress (psi or MPa)
- M = bending moment at the section (lb·in or N·mm)
- c = distance from the neutral axis to the outermost fiber (in or mm)
- I = second moment of area about the bending axis (in⁴ or mm⁴)
Maximum bending stress always occurs at the outermost fiber — the point farthest from the neutral axis.
Section Modulus Makes It Faster
The ratio I/c shows up so often that it has its own name: section modulus, S.
σ_max = M / S
where S = I / c (in³ or mm³)
Once you know S for a cross-section, calculating max bending stress from a moment is a one-liner. This is why McMaster-Carr lists section modulus in product specs.
Common Cross-Section Formulas
Solid Rectangle (width b, height h, bending about the horizontal axis):
- I = b·h³ / 12
- c = h / 2
- S = b·h² / 6
Solid Circle (diameter d):
- I = π·d⁴ / 64
- c = d / 2
- S = π·d³ / 32
Hollow Tube (outer diameter D, inner diameter d):
- I = π·(D⁴ − d⁴) / 64
- c = D / 2
- S = π·(D⁴ − d⁴) / (32·D)
Hollow Rectangle (outer width B, outer height H, wall thickness t):
- I = (B·H³ − b·h³) / 12 where b = B − 2t, h = H − 2t
- c = H / 2
- S = I / c
Worked Example
You have a 1-inch solid round shaft (steel, A36) spanning 12 inches, loaded at center with 200 lbs.
Step 1 — Bending moment at center: M = F·L / 4 = 200 × 12 / 4 = 600 lb·in
Step 2 — Section modulus for a 1” solid circle: S = π·(1)³ / 32 = 0.0982 in³
Step 3 — Max bending stress: σ = M / S = 600 / 0.0982 = 6,110 psi
A36 has a yield strength of 36,000 psi. Factor of safety = 36,000 / 6,110 = 5.9 — plenty of margin for a static load.
What to Watch Out For
Neutral axis location matters. For symmetric sections (circles, rectangles, hollow tubes), the neutral axis is at the geometric center. For asymmetric sections like angles or channels, it shifts — and c changes accordingly.
Moment direction matters. A rectangular bar is much stronger when loaded the tall way (large h) than the flat way (small h). The h³ term in the moment of inertia formula makes this difference dramatic.
Bending stress is not constant across the section. Stress is zero at the neutral axis and maximum at the outer fiber. Shear stress is the opposite — maximum at the neutral axis. In long beams, bending usually governs. In short, stubby beams, shear can control.
Try It With a Real Part
Paste any McMaster-Carr structural bar, tube, or round rod URL into PartCalc and it will compute I, c, S, and max bending stress for your load — with every value labeled as scraped, inferred, or computed.
Every assumption is shown. No black boxes.
Calculate it now
Paste a McMaster-Carr product URL into PartCalc to instantly get section properties, material data, and the calculations described in this article — with every value labeled as scraped, inferred, or computed.
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