Bending Stress Calculator
Check a beam, shaft or pipe for bending. Enter the bending moment, pick the cross-section and its dimensions, and choose the material — get the bending stress, section modulus (S), moment of inertia (I), the utilization against the allowable stress (Fb), and the moment the section can safely carry, plus by-moment and by-size tables and a downloadable PDF.
| Moment | Bending stress | Utilization | Status |
|---|---|---|---|
| Enter a section to compare | |||
| Size | Section modulus | Bending stress | Status |
|---|---|---|---|
| Enter a section to compare | |||
How to use the bending stress calculator
Four quick steps turn a bending moment and a cross-section into the bending stress — with the section modulus, moment of inertia, the utilization against the allowable, and the moment the section can safely carry, plus by-moment and by-size tables and a downloadable report.
Enter the moment
The maximum bending moment, in pound-feet or pound-inches.
Pick the cross-section
Rectangle, solid round or hollow round, and enter its dimensions.
Choose the material
The allowable bending stress Fb to check the stress against.
Read results & save a PDF
Get the stress, S, I, utilization and moment capacity — then download a report.
The formulas behind the stress
Bending stress is the moment divided by the section modulus; the section modulus comes from the shape. From there the utilization and the moment capacity follow.
▶ Worked example
A softwood rectangle 3.5 in wide × 9.25 in deep carries a bending moment of 1,800 lb-ft (21,600 lb-in).
S is 3.5 × 9.25² ÷ 6 ≈ 49.9 in³ (I ≈ 230.8 in⁴), so the stress is 21,600 ÷ 49.9.
σ ≈ 433 psi vs Fb 900 — 48% utilized
The moment capacity is Fb × S = 900 × 49.9 ≈ 44,900 lb-in, about 3,740 lb-ft.
Bending stress quick reference
The section modulus and moment of inertia for each shape, the allowable bending stress of common materials, and the properties of a few common sections.
| Shape | Section modulus (S) | Moment of inertia (I) |
|---|---|---|
| Rectangle | b × h² ÷ 6 | b × h³ ÷ 12 |
| Solid round | π × d³ ÷ 32 | π × d⁴ ÷ 64 |
| Hollow round | π(D⁴ − d⁴) ÷ (32 D) | π(D⁴ − d⁴) ÷ 64 |
Bending stress is the moment divided by S; a larger S means lower stress for the same moment.
| Material | Allowable Fb (psi) |
|---|---|
| Softwood framing | 900 |
| Hardwood | 1,200 |
| LVL / engineered | 2,600 |
| A36 steel | 21,600 |
Nominal guide values. Wood design uses the adjusted Fb for species, grade, size, duration and member type.
| Section | Dimensions | S (in³) | I (in⁴) |
|---|---|---|---|
| Rectangle (2×10) | 1.5 × 9.25 in | 21.39 | 98.93 |
| Rectangle (4×10) | 3.5 × 9.25 in | 49.91 | 230.84 |
| Solid round | 1 in dia | 0.098 | 0.049 |
| Solid round | 2 in dia | 0.785 | 0.785 |
| Hollow round | 2 / 1.5 in | 0.537 | 0.537 |
Enter your own section above for any rectangle, solid round or hollow round.
Everything the calculator gives you
One moment and section produce a full stress profile, not just a single number.
Bending stress reference numbers
Built for anyone checking a section
DIY builders, engineers, and students all need to turn a bending moment into a stress.
Checking whether a beam, bar or bracket is overstressed under a known moment. A quick stress check tells you if the section has margin or needs beefing up.
- Turn a bending moment into a stress in psi
- Compare the stress with the material’s allowable
- See the moment the section can safely carry
Sizing shafts, pipes and structural members, and running a fast first-pass stress check across rectangle, solid and hollow sections before the full design.
- Get S and I for three section shapes
- Compare a solid shaft with a pipe of the same size
- Hand a reviewer a clean PDF of the figures
Learning how moment, section modulus and stress relate, and why shape matters so much, with every figure worked out in one place.
- See stress, S, I and utilization together
- Watch stress fall as the section gets deeper
- Compare how each shape resists bending
Get a reliable stress check
Small habits make a bending stress estimate far more trustworthy.
Match the moment units
Pound-feet and pound-inches differ by 12. Set the unit so the stress comes out right.
Depth drives section modulus
For a rectangle, S rises with depth squared, so a deeper section cuts the stress sharply.
Use the right shape formula
Rectangle, solid round and hollow round each have their own S. Pick the section that matches.
Bending stress is one check
Staying under Fb isn’t the whole story. Shear, bearing, deflection and buckling are separate checks.
Hollow saves weight
A pipe keeps most of its strength for far less material, since the core does little work.
Pick the right Fb
Use the adjusted allowable for your actual material, grade and conditions, not just a round number.
Confirm structural members
For anything affecting safety, have a licensed engineer and the building code confirm the design.
Frequently asked questions
Common questions about bending stress, section modulus, and moment capacity.
Bending stress is the moment divided by the section modulus: stress = M divided by S. The bending moment comes from the load and span; the section modulus comes from the shape and size of the cross-section.
Work in consistent units. This calculator uses pound-inches for the moment and cubic inches for the section modulus, so the stress comes out in pounds per square inch. If your moment is in pound-feet, multiply by 12 first.
For example, a moment of 21,600 pound-inches on a section with a section modulus of 49.9 cubic inches gives a bending stress of about 433 psi. Enter the moment, pick the cross-section and material, and this calculator returns the stress and checks it against the allowable.
The section modulus (S) measures how well a cross-section resists bending, and it is the link between the moment and the stress: bending stress equals the moment divided by S. A larger section modulus means lower stress for the same moment.
For a rectangle it is the width times the depth squared, divided by 6. Because the depth is squared, a deeper section has a much larger modulus and so much lower stress — doubling the depth cuts the stress to a quarter.
This calculator works out S from the cross-section you choose — rectangle, solid round or hollow round — and uses it to find the bending stress and the moment the section can carry.
For a solid round bar, the section modulus is pi times the diameter cubed, divided by 32. So a 2-inch solid shaft has a section modulus of pi times 8 over 32, about 0.785 cubic inches.
For a hollow round — a pipe or tube — it is pi times the outer diameter to the fourth power minus the inner diameter to the fourth power, all divided by 32 times the outer diameter. The hollow centre removes material that was doing little work, so a pipe is efficient for its weight.
This calculator handles solid and hollow rounds as well as rectangles, and reports the section modulus and moment of inertia for each.
The bending moment is the external bending the load applies to the beam, in pound-feet or pound-inches; the bending stress is the internal force per unit area that moment creates in the material, in pounds per square inch. One causes the other.
The link between them is the section modulus: stress equals moment divided by section modulus. The same moment produces a high stress in a small section and a low stress in a large one.
This calculator takes the bending moment you enter and the section you choose, and works out the resulting bending stress — then compares it with the material’s allowable.
Compare the bending stress with the material’s allowable bending stress, Fb. The stress is the moment divided by the section modulus, and the section is adequate in bending when that stays at or below Fb.
Utilization shows how close you are — the stress divided by Fb, as a percentage. At 100 percent you are right at the allowable; below that you have margin, and above it the section is overstressed.
For example, a stress of 433 psi against a softwood Fb of 900 is about 48 percent utilized. This calculator shows the stress, the allowable and the utilization together, and the moment the section can safely carry.
The allowable bending stress (Fb) is the stress a material can safely carry in bending, in pounds per square inch. Typical values are about 900 psi for softwood framing lumber, 1,200 for hardwoods, 2,600 for engineered lumber like LVL, and about 21,600 for A36 structural steel.
These are nominal figures. Real wood design values also depend on the species, the grade, the member size, the load duration and whether the member is repetitive, and the published Fb is adjusted for those.
Use the design value for your actual material from a span table or supplier. This calculator offers softwood, hardwood, LVL and steel so you can compare, but a real design uses the adjusted Fb for the exact product and conditions.
In bending, the material farthest from the center of the section does the most work, and the material near the center does very little. A hollow round removes that lightly loaded core, so it keeps most of its bending resistance while shedding weight.
That is why pipes and tubes are so common for shafts, handrails, frames and masts — they give a high section modulus for the material used. A solid bar of the same outer diameter is heavier but only a little stronger in bending.
This calculator computes the section modulus of a hollow round from the outer and inner diameters, so you can compare it with a solid section of the same size.
The moment capacity is the bending moment that brings the stress up to the allowable Fb. It is the allowable stress times the section modulus: moment capacity equals Fb times S.
A larger section modulus or a stronger material raises the capacity. For example, a softwood section with a section modulus of 49.9 cubic inches has a capacity of 900 times 49.9, about 44,900 pound-inches, or roughly 3,740 pound-feet.
This calculator reports the moment capacity for your section and material, so you can see how much margin a given cross-section has against the moment you entered.
Disclaimer: US units (inches, pounds). Bending stress = M ÷ S, where S is the section modulus (rectangle b·h²/6; solid round π·d³/32; hollow round π(D⁴−d⁴)/32D). Allowable Fb values are nominal and do not include span, load duration, size or repetitive-member adjustments, shear, bearing, deflection, lateral or local buckling, load factors, or connection design. A planning aid only — for any structural member affecting safety, confirm with a licensed professional engineer and the governing building code.