Elastic section modulus S = I/c, plastic modulus Z with shape factor, and a two-way bending stress solver — for six common cross-sections.
S = I / c · Bending stress: σ = M / SBending about the horizontal centroidal (strong) axis. For a T-section the neutral axis is offset, so both Stop and Sbottom are reported; the smaller governs. Getting the bending moment from your loads? Use the beam load calculator.
σ_allow should be the allowable stress for your material and code (e.g. a fraction of yield σy), not the yield itself. All three directions are exact round-trips: σ → M → σ returns the original value.
Z is the first moment of area above plus below the equal-area axis. For symmetric sections this is the horizontal centroidal axis; for a T-section the plastic neutral axis shifts so the areas balance. Shape factor f = Z/S.
| Section | Moment of inertia I | Elastic modulus S | Plastic modulus Z | Shape factor f |
|---|---|---|---|---|
| Rectangle b×h | bh³/12 | bh²/6 | bh²/4 | 1.50 |
| Solid circle D | πD⁴/64 | πD³/32 | D³/6 | 1.70 |
| Hollow circle D/d | π(D⁴−d⁴)/64 | π(D⁴−d⁴)/(32D) | (D³−d³)/6 | ~1.2–1.7 |
| I-beam / channel, strong axis (B, H, tf, tw; hw=H−2tf) | [BH³−(B−tw)hw³]/12 | I/(H/2) | B·tf(H−tf)+twhw²/4 | ~1.10–1.20 |
| T-section (asymmetric) | parallel-axis sum about ȳ | I/ctop, I/cbot (min governs) | about equal-area axis | ~1.7–1.9 |
Unit consistency: with dimensions in mm, I is in mm⁴ and S, Z in mm³; 1 cm³ = 1,000 mm³; 1 in³ = 16,387.064 mm³; 1 in⁴ = 416,231.43 mm⁴. Bending stress σ = M/S needs M in N·mm and S in mm³ to give N/mm² = MPa directly. Rolled steel sections vary by manufacturer — for certified W/UB/IPE properties use the mill table; the formulas here are for idealized geometry.
Companions: torsion of shafts — angle-of-twist calculator · loads, shear and deflection of the beam itself — beam load calculator · material elastic constants E/G/K/ν — elastic constants calculator.
| Case | Inputs | Result |
|---|---|---|
| Rectangle section | b=50 mm, h=100 mm | S = 83,333.33 mm³, Z = 125,000.00 mm³, f = 1.50 |
| Solid circle | D=40 mm | S = 6,283.19 mm³, f = 1.6977 |
| T-section (asymmetric) | flange 200×20, stem 12×280 mm | S_top = 804,965.47 mm³, S_bot = 285,174.22 mm³ (governs), Z = 509,888.00 mm³, f = 1.7880 |
| Bending stress | M=10 kN·m, S=200 cm³ | σ = 50.0 MPa |
| Required section | M=50 kN·m, σ_allow=250 MPa | S_req = 200.00 cm³ |
Methodology: 100% client-side, no server, no external requests. Formulas are the standard mechanics-of-materials results (S=I/c about the horizontal centroidal axis); worked numbers are recomputed from the frozen high-precision oracle at build time.
Section modulus S measures a beam cross-section's resistance to bending. It is defined as S = I/c, where I is the second moment of area about the neutral axis and c is the distance from that axis to the outermost fiber. Bending stress follows directly: σ = M/S. Example: an 80 mm × 120 mm rectangle has I = 11,520,000.00 mm⁴ and c = 60 mm, giving S = 192,000.00 mm³ (about 11.7166 in³).
For a rectangle, S = b·h²/6. A 50 mm × 100 mm rectangle gives S = 50 × 100² / 6 = 83,333.33 mm³ = 83.3333 cm³ (about 5.0853 in³). Note the height is squared: doubling the beam depth quadruples S. Enter your dimensions in Mode 1 for any of the six profiles.
The elastic section modulus S = I/c governs first yield: the extreme fiber reaches yield when M = σ_y·S. The plastic modulus Z is the first moment of the fully yielded section about the equal-area axis, and the full plastic moment is M_p = σ_y·Z. Their ratio f = Z/S, the shape factor, measures reserve strength beyond first yield: 1.50 for a rectangle, 1.6977 for a solid circle. Mode 3 computes Z and f for all six profiles.
An asymmetric section has two elastic moduli, one per side: S_top = I/c_top and S_bot = I/c_bot. The smaller (governing) value controls the maximum elastic stress. For a T with a 200×20 mm flange and a 12×280 mm stem: S_top = 804,965.47 mm³, S_bottom = 285,174.22 mm³, so the bottom fiber governs at 285,174.22 mm³. The plastic modulus is 509,888.00 mm³, giving a shape factor of 1.7880.
σ = M/S with consistent units. Example: M = 10 kN·m = 10,000,000 N·mm and S = 200 cm³ = 200,000 mm³ gives σ = 50.0 MPa. The Mode 2 solver works all three ways: stress from moment and section, moment capacity M = σ·S, or the required section S_req = M/σ_allow — e.g. a 50 kN·m moment at 250 MPa allowable needs S_req = 200.00 cm³.
Length cubed: mm³, cm³, m³, or in³. The key rule is consistency: with M in N·mm and S in mm³, stress comes out in N/mm² = MPa. Conversions: 1 cm³ = 1,000 mm³ and 1 in³ = 16,387.064 mm³ (so 83,333.33 mm³ is about 5.0853 in³). This calculator converts every input automatically.
No. The second moment of area I (mm⁴) measures bending stiffness — deflection is proportional to 1/I. Section modulus S = I/c (mm³) measures bending strength — stress is M/S. Two sections can share the same I but have very different S: a deep, narrow beam has a much larger S than a shallow, wide one with identical I. For the deflection side of the problem, see the beam load calculator.
No — they are completely different quantities that happen to share the word modulus. Section modulus S = I/c is a pure GEOMETRIC property of a beam cross-section, measured in mm³; it says nothing about the material. Bulk modulus K is a MATERIAL property, measured in GPa: a solid's resistance to uniform volumetric compression (K = −V₀·ΔP/ΔV). This tool needs no material data at all. For volumetric compression, sound speed, and K/E/G/ν relations, use the bulk modulus calculator.
No. S, Z, and I are computed from dimensions alone — the same cross-section has the same S whether it is steel or timber. Material properties enter only afterward: moment capacity M = σ_y·S needs the yield stress, and deflection δ ∝ 1/(E·I) needs Young's modulus. For the relations between E, G, K, and Poisson's ratio ν, use the elastic constants calculator. Limits of this tool: linear-elastic bending about the tabulated axis only — it does not cover shear stress, torsion, or lateral-torsional buckling.
Free tool by NRTechStudio
Explore similar science tools and calculators. 100% free, accurate, and runs completely in your browser without sign-up.
2 taps, no typing needed — diagnostics are sent automatically.