Section Modulus Calculator

The section modulus (S) is a geometric property of a cross-section used to calculate the maximum bending stress a beam will experience under a given bending moment. It is defined as S = I / c, where I is the moment of inertia and c is the distance from the neutral axis to the extreme fiber. A higher section modulus means the beam can resist more bending stress with less material.

For a solid rectangle with width b and height h, the elastic section modulus is S = (b × h²) / 6. The moment of inertia about the centroidal axis is I = (b × h³) / 12, and the neutral axis is located at c = h / 2 from the centroid.

For a square section with side length a, the elastic section modulus is S = a³ / 6. Since both width and height equal a, this is a simplified form of the rectangular section formula. The neutral axis sits at the center of the square.

The elastic section modulus has units of length cubed. In SI units this is typically mm³ or m³, and in imperial units it is in³. These units arise from dividing the moment of inertia (length⁴) by the distance to the extreme fiber (length).

The second moment of area (moment of inertia) has units of length to the fourth power — mm⁴ in SI units or in⁴ in imperial units. It quantifies how a cross-section's area is distributed relative to a reference axis, and directly influences the bending stiffness of a beam.

For a hollow circular pipe with outer diameter D and inner diameter d, the moment of inertia is I = π(D⁴ − d⁴) / 64 and the elastic section modulus is S = π(D⁴ − d⁴) / (32D). This calculator handles hollow rectangular sections; for circular pipes, use those specific formulas directly.

The elastic section modulus (S or Z_e) applies when the material behaves elastically — stress is proportional to strain. The plastic section modulus (Z or Z_p) assumes the entire cross-section has yielded and is used in plastic analysis of steel structures. The plastic modulus is always larger than the elastic modulus, and their ratio is called the shape factor.

For an I-section, the plastic section modulus Z = A_flange × ȳ_flange + A_web_half × ȳ_web_half, where each term represents the first moment of area of half the section above and below the plastic neutral axis. For symmetric I-sections, the plastic neutral axis coincides with the centroidal axis, simplifying the calculation.