What to do with the computed stress
The flexure formula gives a stress, not a verdict. Adequacy is a code-based comparison against the material's allowable or design stress, alongside checks the formula does not cover.
| Check | Governing question | Where it comes from |
|---|---|---|
| Bending stress vs. allowable | Is σ = Mc/I below the permitted value? | Material standard and design code (e.g., AISC for steel, NDS for timber) |
| Lateral-torsional buckling | Can the compression flange buckle sideways before yielding? | Design code member checks; not captured by Mc/I |
| Shear and deflection | Do shear stress and serviceability limits also pass? | Separate formulas and code limits |
- The flexure formula assumes linear-elastic material, a slender prismatic beam, bending about a principal axis, and stresses below the proportional limit; it does not account for buckling, shear, stress concentrations, holes or combined loading.
- This calculator is educational. Sizing or verifying any load-bearing member is structural engineering work governed by the applicable building code and must be confirmed by a qualified engineer.
What is bending stress?
When a beam bends, its material stretches on one face and compresses on the other, with stress varying linearly through the depth and passing through zero at the neutral axis. The flexure formula of Euler–Bernoulli beam theory, σ = M·c / I, gives the maximum stress at the extreme fiber: M is the internal bending moment at the section, c the distance from the neutral axis to the outermost fiber, and I the section's second moment of area about the bending axis. The formula is a cornerstone of mechanics of materials and applies to slender beams behaving linearly elastically.
The section modulus S = I ÷ c packages the geometry into a single number, so the formula collapses to σ = M ÷ S — which is why steel and timber section tables list S directly and designers select sections by comparing the required S against the table. A computed bending stress means nothing in isolation: design verdicts come from comparing it against the allowable or design stress for the material and design method (allowable-stress or load-and-resistance-factor design), which accounts for safety factors, lateral-torsional buckling, shear and deflection — checks that belong to a qualified engineer.
How to use this bending stress calculator
- Enter the bending moment M at the section of interest in kN·m (from a beam analysis — e.g., wL²/8 for a uniformly loaded simple span).
- Enter c, the distance from the neutral axis to the extreme fiber, in millimeters — half the depth for symmetric sections.
- Enter the section's moment of inertia I in cm⁴, from section tables or the shape formulas (bh³/12 for a rectangle).
- Read the maximum bending stress in MPa and the section modulus, then compare the stress against the material's allowable or design value — a check a qualified engineer must confirm for any real structure.
The formula behind bending stress
The flexure formula is σ = M·c / I. With M converted to N·m, c to meters and I to m⁴, the result is in pascals; the calculator reports MPa (N/mm²). The section modulus S = I ÷ c is reported in cm³, matching how steel and timber tables present it, so σ = M ÷ S gives the same answer.
Worked example: a section carrying M = 12.223 kN·m with c = 92.5 mm and I = 10,406 cm⁴ has σ = (12,223 N·m × 0.0925 m) ÷ (10,406 × 10⁻⁸ m⁴) ≈ 10.87 MPa, and a section modulus of 10,406 ÷ 9.25 = 1,124.97 cm³. Whether 10.87 MPa is acceptable depends entirely on the material's allowable stress and the governing design code.
Common mistakes
- Mixing unit systems — M in kN·m, c in mm and I in cm⁴ must all be converted to a consistent base (this calculator does it internally, but hand checks often slip a factor of 10).
- Using the moment of inertia about the wrong axis — bending about the weak axis of an I-beam uses a far smaller I, and the stress is correspondingly higher.
- Treating a low computed stress as proof the beam is adequate — lateral-torsional buckling, shear and deflection frequently govern before bending strength does.
- Using half the depth for c on non-symmetric sections — for T-shapes and other asymmetric sections the neutral axis is not at mid-depth, and the two faces see different stresses.
자주 묻는 질문
What is the flexure formula?
σ = M·c / I: the maximum bending stress in a beam equals the bending moment times the distance from the neutral axis to the extreme fiber, divided by the section's second moment of area. It comes from Euler–Bernoulli beam theory for linear-elastic, slender beams.
What is section modulus and why does it matter?
Section modulus S = I ÷ c condenses a section's bending geometry into one number, so maximum stress is simply M ÷ S. Steel and timber section tables list S directly, and designers select members by comparing the required S = M ÷ σ_allowable against the tables.
Is 10.87 MPa a safe bending stress?
That cannot be answered from the stress alone. Safety is a comparison against the allowable or design stress for the specific material and design method — very different for timber, steel or aluminum — plus separate checks for buckling, shear and deflection. A qualified engineer makes that determination under the governing code.
Where does the bending moment M come from?
From a structural analysis of the beam's loads and supports. Standard cases have closed-form results — a uniformly loaded simple span has a midspan maximum of wL²/8, a central point load PL/4 — and general cases come from beam analysis or software.
What are the limits of the flexure formula?
It assumes linear-elastic behavior, a slender prismatic beam, bending about a principal axis and plane sections remaining plane. It does not cover lateral-torsional buckling, shear stress, local buckling, stress concentrations, holes or combined axial-plus-bending loading — all of which the design codes check separately.
참고 자료
- Hibbeler, R.C. — Mechanics of Materials: derivation and assumptions of the flexure formula σ = Mc/I and section modulus.
- Beer, Johnston, DeWolf & Mazurek — Mechanics of Materials: pure bending, the elastic flexure formula and neutral-axis location for asymmetric sections.
- American Institute of Steel Construction (AISC) — Steel Construction Manual: section property tables (I, S) and flexural member design checks including lateral-torsional buckling.
- American Wood Council — National Design Specification (NDS) for Wood Construction: allowable bending design values for timber members.