Custom Cross-Section Properties

How it calculates

The Custom Cross-Section Properties calculator uses numerical mesh analysis to compute the full set of section properties for any arbitrary planar cross-section. You define the geometry using parametric shape primitives or a free-form polygon, and the calculator assembles the section, meshes it, and solves for properties that range from simple area integrals through to the torsion and warping problems that require a boundary-value solver.

Geometry input and composite sections

The calculator provides a library of built-in section types to cover the most common fabricated shapes:

• I-section and T-section - depth, flange width, flange thickness, web thickness
• Angle section - leg lengths and thickness
• Channel (C and Z sections) - height, flange width, thickness
• Rectangular and hollow rectangular - width, depth, and optional wall thickness
• Circular and hollow circular - outer diameter and optional wall thickness
• Custom polygon - vertex coordinates for fully arbitrary outlines

For composite sections (two shapes combined, such as a channel plus a plate, or back-to-back angles), you add a second shape and specify its x- and y-offset relative to the first. The calculator supports sections built from standard library members (Australian, US, or European steel database sections) combined with custom plate elements.

Area properties from direct integration

For simple geometry, first-order and second-order area properties are computed directly from the section outline by integration:

• Gross area: A = integral of dA over the section
• Centroid: x̄ = (integral of x dA) / A, ȳ = (integral of y dA) / A
• Second moments of area: Ixx and Iyy about centroidal axes
• Product moment of area: Ixy (non-zero for asymmetric sections such as angles and Z-sections)
• Principal axes: rotation angle theta_p where Ixy = 0, and principal second moments I1, I2
• Section moduli: Zx = Ixx / y_max, Zy = Iyy / x_max (elastic)
• Plastic section moduli: Sx and Sy, computed by locating the plastic neutral axis where equal areas lie above and below
• Radii of gyration: rx = sqrt(Ixx / A), ry = sqrt(Iyy / A)

Shear centre

The shear centre is the point through which a transverse shear force produces bending without twist. For doubly-symmetric sections it coincides with the centroid; for asymmetric open sections (angles, channels, Z-sections) it lies off the centroid and must be computed from the shear flow distribution.

The calculator solves the shear flow problem using the thin-wall approximation for standard section shapes and the full finite element solution for arbitrary polygons. The shear centre coordinates (x_s, y_s) are reported relative to the centroid.

Torsion constant (J)

The St Venant torsion constant J resists uniform torsion. For closed (hollow) sections it is computed from the Bredt formula:

J = 4 A_enclosed² / integral(ds / t)

where A_enclosed is the enclosed area and t is the wall thickness at each point along the perimeter.

For open sections, the thin-wall approximation gives:

J ≈ (1/3) sum(b_i × t_i³)

where each segment i has breadth b_i and thickness t_i. For irregular sections, the full Saint-Venant torsion boundary-value problem is solved on the mesh, giving the result T = G J (d theta / dz). The mesh-based result is more accurate than the thin-wall approximation for stocky flanges or re-entrant corners.

Warping constant (Cw)

The warping constant Cw (also written as Iw) governs non-uniform torsion and lateral-torsional buckling calculations. It is defined by:

Cw = integral of omega² dA

where omega is the normalised warping function at each point of the cross-section. The calculator solves the warping problem on the mesh to determine the warping function distribution, then integrates to get Cw. For doubly-symmetric I-sections the closed form is:

Cw = (Iy × h_0²) / 4

where h_0 is the distance between flange centroids. The mesh solution matches this for standard shapes and extends to arbitrary geometry where no closed-form exists.

Mesh and accuracy

The cross-section is divided into triangular elements. The calculator displays the mesh in the diagram output so you can verify that the geometry has been interpreted correctly before accepting results. More complex shapes with re-entrant corners or thin outstanding elements are handled by automatic mesh refinement where aspect ratios would otherwise degrade accuracy.

Three assumptions apply to all results:

• Entered dimensions must be physically realisable - no overlapping sub-areas within the same region
• No new enclosed areas are created by stacking shapes (each enclosed void must be explicitly modelled as a hollow)
• Properties are for the gross unreduced section - effective section properties for slender elements under compression require a separate reduction per the applicable design code