Section Modulus Calculator

Elastic section modulus is a geometric property that connects bending moment to the maximum elastic bending stress in a cross-section. For bending about the horizontal centroidal axis, the relationship is σmax = M/Z, where M is bending moment and Z is section modulus. A larger section modulus means lower extreme-fiber stress for the same applied moment and material, making Z a convenient measure when comparing beam shapes and orientations. In general, elastic section modulus is Z = I/c. I is the second moment of area about the relevant neutral axis, and c is the distance from that axis to the most distant fiber. This calculator assumes bending about the centroidal horizontal axis and symmetric dimensions. With millimeter inputs, I has units of mm⁴ and division by distance produces Z in mm³. Unit consistency is essential when combining this result with moments and stresses. For a solid rectangle, Z = bh²/6, where b is width and h is the bending depth. A solid circle uses Z = πd³/32. A hollow rectangle with uniform wall thickness uses Z = [bh³ − (b − 2t)(h − 2t)³]/(6h). The hollow-section formula subtracts the centered interior void from the outer second moment of area, then divides by h/2. Thickness must be less than half both outside dimensions. For a symmetric I-beam, the calculator builds inertia from two flanges and the web. Each flange contributes its own centroidal inertia plus an area-times-distance-squared parallel-axis term. The centered web contributes tw(h − 2tf)³/12. Their sum is divided by h/2. The input assumes equal top and bottom flanges, constant flange width, centered web, and no fillets or tapers. Published rolled-section properties may differ slightly because real profiles include root radii and manufacturing details. Section modulus depends strongly on depth because dimensions are squared or cubed in the underlying equations. Rotating a rectangular section therefore changes its capacity dramatically. Z describes elastic first yield at the extreme fiber; it is not plastic section modulus and does not by itself determine deflection, buckling, shear behavior, fatigue, or ultimate capacity. Engineering design must also consider material strength, load combinations, stability, connections, code safety factors, and the correct bending axis. Use tabulated manufacturer values for standard structural shapes when those data are available.