Beam Section Modulus

First compute the second moment of area about the centroidal axis — for a rectangle \( I = bh^3/12 \), for a solid circle \( I = \pi d^4/64 \). Then divide by the distance from the neutral axis to the outermost fibre: \( S = I/c \), where \( c = h/2 \) or \( d/2 \). For a rectangle this simplifies to \( S = bh^2/6 \). Always measure the depth in the direction of bending, and keep all dimensions in one unit system.

Work backwards from the flexure formula: the required modulus is \( S \geq M / \sigma_{allow} \), where \( M \) is the maximum bending moment and \( \sigma_{allow} \) is your material's permissible bending stress (for example, roughly 10–15 MPa for common softwood timber grades, or 165–275 MPa for structural steel, depending on the standard). If your calculated \( S_{xx} \) exceeds the required value, the section passes the bending check — but shear, deflection, and buckling still need separate verification.

Section modulus has units of length cubed, so conversions are straightforward: \( 1 \ \text{cm}^3 = 1000 \ \text{mm}^3 \) and \( 1 \ \text{in}^3 = 25.4^3 \approx 16{,}387 \ \text{mm}^3 \). For the second moment of area, the exponents are fourth powers: \( 1 \ \text{cm}^4 = 10{,}000 \ \text{mm}^4 \) and \( 1 \ \text{in}^4 \approx 416{,}231 \ \text{mm}^4 \). A frequent error is applying a linear or squared conversion factor to these values — always cube or take the fourth power of the length ratio.

The elastic modulus \( S \) assumes stress varies linearly and yielding starts at the extreme fibre, so it is used with the allowable-stress design method. The plastic modulus \( Z \) assumes the whole section reaches yield, giving a higher capacity; for a rectangle \( Z = bh^2/4 \), exactly twice the elastic value. This calculator returns the elastic value, which is appropriate for serviceability checks, brittle materials, and steel design to older allowables. Modern limit-state steel codes use \( Z \) plus shape factors and safety factors.

No — section modulus and second moment of area are purely geometric properties of the cross-section shape. A 100 × 200 mm rectangle has the same \( S_{xx} \) whether it is steel, aluminium, timber, or concrete. The material only enters when you convert to stresses: divide the applied moment by \( S_{xx} \) and compare the resulting \( \sigma \) with the material's yield or allowable stress. Stiffness-related properties like \( E I \) do involve the material's modulus of elasticity \( E \).

Not directly. Those shapes are singly symmetric, so the neutral axis does not lie at mid-height, and the extreme-fibre distances differ above and below it — you get two different section moduli. You would first locate the centroid, then compute \( I \) about it and divide by each extreme-fibre distance. As a workaround, an unsymmetric I-section can sometimes be approximated by superposing rectangles, but for real T, L, and U profiles it is safer to use tabulated catalogue values or a general-shape tool.

The classic ones are: entering the diameter where a radius is needed (or vice versa) for circular sections; using the weak-axis dimensions, which for a flat-laid rectangle cuts \( S \) by 75%; mixing units — e.g. entering millimetres with a moment in kN·m without converting; and forgetting that \( \sigma = M/S \) requires \( M \) in N·mm when \( S \) is in mm³ (1 kN·m = 10⁶ N·mm). Also remember the tube inner diameter must be strictly smaller than the outer diameter, or the formulas return nonsense.