Tool — Beta
Section Properties & Builder
Calculate geometric properties for common idealised section shapes, or build an arbitrary section from positioned solid and void rectangles. The tool reports area, centroid, second moments of area, product moment, elastic section moduli, radii of gyration, plastic neutral axes, plastic section moduli, geometric shape factors and principal properties. Geometric properties only — no resistance, classification or code check is carried out.
Section geometryParametric sections are generated from standard dimensions. The custom builder assembles a section from positioned solid and void rectangles. Both modes use the same geometry engine.
Properties are calculated from the idealised sharp-corner geometry shown. Rolled section catalogue properties may differ due to root radii, manufacturing geometry and published section definitions.
Section drawingScaled drawing of the geometry entered. Toggle the annotations below to show dimensions, component numbers, the centroid, centroidal axes, principal axes, plastic neutral axes and a coordinate grid.
Geometric section properties
Geometry
Centroid from origin
Elastic properties
Elastic section modulus
Plastic properties
Principal propertiesCentroidal axes for which the product moment of area is zero and the second moments of area take their principal values.
θp is measured counter-clockwise from the centroidal x-x axis to principal axis 1-1.
Extreme fibre / bounding dataBounding coordinates of the material region and the extreme fibre distances measured from the centroid.
Torsional properties
Open thin-walled idealisation: sum of rectangular plate elements (top flange, bottom flange and web), sharp corners, no root-radius contribution.
Plastic section properties describe the geometry of the full cross-section and do not establish that the section can develop its full plastic resistance. Structural resistance depends on material properties, section classification/local slenderness, stability, loading and the applicable design standard.
Show calculation
| No. | Type | A | xi | yi | A·xi | A·yi | Ix,local | Iy,local | dx | dy | A·dy² | A·dx² |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | Solid | 6,000 | 150 | 10 | 9.000 × 10⁵ | 60,000 | 2.000 × 10⁵ | 4.500 × 10⁷ | 0 | -290 | 5.046 × 10⁸ | 0 |
| 2 | Solid | 6,720 | 150 | 300 | 1.008 × 10⁶ | 2.016 × 10⁶ | 1.756 × 10⁸ | 80,640 | 0 | 0 | 0 | 0 |
| 3 | Solid | 6,000 | 150 | 590 | 9.000 × 10⁵ | 3.540 × 10⁶ | 2.000 × 10⁵ | 4.500 × 10⁷ | 0 | 290 | 5.046 × 10⁸ | 0 |
Components do not overlap, so the component summation above is equivalent to the integrated result. Voids contribute negative area.
ΣA = 18,720 mm²
x̄ = 150 mm · ȳ = 300 mm
Ix = 1.185 × 10⁹ mm⁴ · Iy = 9.008 × 10⁷ mm⁴ · Ixy = 0 mm⁴
Plastic neutral axes located for equal material area, A/2 = 9,360 mm² each side: yPNA = 300 mm, xPNA = 150 mm from the origin.
Wpl,x = ∫ |y − yPNA| dA = 4.421 × 10⁶ mm³ · Wpl,y = ∫ |x − xPNA| dA = 9.202 × 10⁵ mm³
αpl,x = Wpl,x / min(Wel,x,top, Wel,x,bottom) = 1.119 · αpl,y = 1.532
Principal check: I1 + I2 vs Ix + Iy differ by 0.0e+0 % · I1·I2 vs Ix·Iy − Ixy² differ by 0.0e+0 %
Calculation basis
- · Coordinate convention: +x to the right, +y upward. Component X and Y are the coordinates of the BOTTOM-LEFT corner; width B extends in +x and height H extends in +y.
- · Centroid: x̄ = ∫x dA / A and ȳ = ∫y dA / A, measured from the user's origin (0,0).
- · Second moments of area Ix and Iy are taken about the centroidal axes parallel to x and y, using exact integration over the resolved material region.
- · Product moment convention: Ixy = ∫ x y dA about the centroidal axes. The same convention is used in the calculation, the principal-axis equation, the drawing and every export.
- · Principal properties: I1,2 = (Ix + Iy)/2 ± √( ((Ix − Iy)/2)² + Ixy² ), with θp = ½ atan2(−2Ixy, Ix − Iy) measured counter-clockwise from the centroidal +x axis to axis 1-1.
- · Solids are combined as a geometric union — material occupying the same area is counted once. Voids are subtracted from the unioned solid material only.
- · Extreme fibre distances are taken from the bounding extents of the final material region; voids never define an external fibre.
- · Parametric sections use idealised sharp-corner geometry. Root radii, fillets and manufacturing geometry are not modelled.
- · Ip = Ix + Iy is the polar second moment of area. It is a geometric quantity and is not the Saint-Venant torsion constant J for non-circular sections.
- · Plastic neutral axes are located on the final Boolean material region so that equal material area lies each side: area above yPNA = area below yPNA = A/2, and likewise either side of xPNA. Voids contribute no material to the balance.
- · Plastic section moduli are integrated exactly as Wpl,x = ∫ |y − yPNA| dA and Wpl,y = ∫ |x − xPNA| dA over the resolved material region. No shape factor is assumed and Wel is never scaled to obtain Wpl.
- · Geometric shape factors are αpl,x = Wpl,x / min(Wel,x,top, Wel,x,bottom) and αpl,y = Wpl,y / min(Wel,y,left, Wel,y,right). They are geometric ratios only and are unrelated to Eurocode partial factors or section classification.
- · Principal elastic moduli are taken from the true extreme material fibres measured perpendicular to each principal axis after rotating the resolved geometry, not from the global bounding box.
- · Plastic section moduli about the principal axes (Wpl,1 and Wpl,2) are not calculated in this release.
- · Perimeter is the total boundary length of the resolved material region, including the boundaries of internal voids.
- · Mass per metre = A × density, with density entered by the user (default 7850 kg/m³ for steel). It is a material quantity only.
- · The Saint-Venant torsion constant J is reported only for parametric shapes where a recognised closed-form idealisation applies. It is not the polar second moment of area Ip, and it is not calculated for arbitrary custom geometry.
Calculation record
Produce a formatted Bebbington Engineering Studio calculation sheet showing the inputs, working, results and engineering basis. These fields are optional and are printed on the document only.
BES branded PDF
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The full calculation sheet under the Bebbington Engineering Studio logo. No account and no payment required.
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PDF is the principal formal output; Word is an editable convenience copy. Outputs are marked as preliminary / informational and are not a BES project-specific structural design.
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A section properties calculator for real fabricated geometry
Most steel section properties calculators only handle catalogue profiles. This one covers both: pick a parametric I beam, channel, angle, tee, RHS, SHS or plate and enter the dimensions, or build the section from positioned rectangles when the shape is fabricated, notched, plated or partly removed.
The I section input accepts a symmetric rolled profile or an unequal-flange I section, with the top and bottom flange widths and thicknesses entered independently. That covers fabricated I section properties and asymmetric I section properties, where the centroid sits away from mid-depth and the elastic moduli at the top and bottom fibres differ. Anything more irregular, including eccentric flanges, is handled by the custom section builder.
It works as a second moment of area calculator, a moment of inertia calculator, an elastic section modulus calculator and a plastic section modulus calculator in one place. Area, centroid, Ix, Iy, Ixy, elastic moduli at all four extreme fibres, radii of gyration, plastic neutral axes, Wpl, geometric shape factors and principal properties are recalculated as you type, with a scaled drawing showing the origin, centroid, centroidal axes, plastic neutral axes and principal axes.
Typical uses are checking I beam section properties for an altered floor, comparing RHS and SHS section properties for a goalpost frame, working out angle section properties where the principal axes matter, or establishing the properties of a plated beam before a capacity check. Every result can be issued as a BES calculation sheet in PDF, print or Word format.
Section properties explained
+What are section properties?
Section properties are the geometric quantities of a cross-section that govern how a member responds to bending, axial load and deflection. The common set is area, centroid position, second moments of area, section moduli and radii of gyration. They depend only on the shape of the section, not on the material grade or the loading.
In practice they sit between the geometry you draw and the member check you carry out: you establish the properties first, then combine them with material strength and a design code to assess resistance. Our independent design checks cover that second step.
+Understanding Ix and Iy
Ix is the second moment of area about the horizontal centroidal axis and governs bending in the vertical plane, which for most beams is the strong axis. Iy is the equivalent value about the vertical centroidal axis. A deep, narrow section has a large Ix and a small Iy, which is why slender beams need restraint against lateral movement.
The product moment Ixy is zero for any section with an axis of symmetry. When it is not zero the section bends about axes inclined to those you drew, and the principal values become the meaningful ones.
+What is elastic section modulus Wel?
Wel is the second moment of area divided by the distance to an extreme fibre, so it converts a bending moment into a peak elastic stress. An asymmetric section has a different value at the top and bottom fibre, and the smaller of the two normally governs. This calculator reports all four values — top, bottom, left and right — rather than a single figure.
+What is plastic section modulus Wpl?
Wpl assumes the whole cross-section has yielded, with equal material area either side of the plastic neutral axis. It is calculated here as the absolute first moment of area about that axis. Wpl is a geometric quantity only: whether a section can actually develop it depends on cross-section classification, restraint and the design code, none of which this tool assesses.
+Difference between centroidal and principal axes
Centroidal axes pass through the centroid parallel to the axes you drew. Principal axes also pass through the centroid but are rotated to the orientation at which the product moment Ixy is zero, giving the maximum and minimum second moments I1 and I2. For a symmetric section the two coincide. For angles and other asymmetric shapes they do not, and using only Ix and Iy understates the true behaviour.
+How custom section properties are calculated
A custom section is defined as positioned rectangular solid and void components, each located by the coordinates of its bottom-left corner. Overlapping solids are resolved as a union so material is never double counted, and voids are subtracted from the result. Properties are then integrated exactly over the resolved material region rather than summed component by component.
That approach suits fabricated and reinforced sections — the kind that come up regularly in structural steelwork packages and alteration work.
+Limitations of geometric section property calculations
Sections are idealised with square corners: root radii, fillet welds and rolling tolerances are not modelled, so values differ slightly from published section tables. No resistance, cross-section classification, buckling, shear area, shear centre or warping constant is calculated, and the Saint-Venant torsion constant is reported only where a recognised closed-form idealisation applies.
The tool is in beta and has not yet been independently reviewed. Results should be checked by a competent engineer before use in design. If you need calculations issued for construction, get in touch.
Tool version 0.9 (beta) · Last reviewed Beta — not yet independently reviewed
Engineering basis & limitations — Section Properties & Builder (beta)v0.9 (beta) · reviewed Beta — not yet independently reviewed
Design standard
- · Geometric section properties calculated using classical engineering mechanics and exact integration of the resolved section geometry.
- · Structural notation generally follows BS EN 1993-1-1 conventions where applicable (A, Ix, Iy, Ixy, Wel, i / r, I1, I2). This calculation is not a BS EN 1993-1-1 structural design.
Analysis and design method
- · Geometry is defined as axis-aligned rectangular components: solids are combined as a geometric union and voids are subtracted from that union, so overlapping material is never double-counted.
- · The resolved material region is integrated exactly, region by region: A = ΣA_i, first moments ΣA_i·x_i and ΣA_i·y_i, and second moments by the parallel-axis theorem about the centroid.
- · Ixy = ∫ x y dA about the centroidal axes; principal values I1,2 = (Ix + Iy)/2 ± √(((Ix − Iy)/2)² + Ixy²) with θp = ½ atan2(−2Ixy, Ix − Iy), counter-clockwise from the centroidal x-x axis to axis 1-1.
- · Elastic moduli use the extreme fibre distances of the final material region; voids never define an external fibre.
- · Plastic neutral axes are located on the resolved material region so equal material area lies each side (A/2), and plastic section moduli are integrated exactly as Wpl = ∫ |s − sPNA| dA. Geometric shape factors use the smaller extreme-fibre elastic modulus for the axis considered.
- · Principal elastic moduli use the true extreme material fibres after rotating the resolved geometry into the principal coordinate system.
Material assumptions
- · No material is assigned. All outputs are geometric quantities and are independent of grade, strength and modulus.
Not checked by this tool
- · Cross-section classification, and all resistance checks (bending, shear, axial, buckling, lateral-torsional buckling and combined effects).
- · Plastic section moduli about the principal axes (Wpl,1 and Wpl,2), shear area, shear centre, torsion constant J and warping constant Iw.
- · Demonstration that a section can develop its plastic moduli: plastic properties are geometric only and do not establish plastic resistance, which depends on material, classification, stability and the design standard.
- · Root radii, fillets, welds, tapers and any non-rectangular, curved or arbitrary polygonal geometry.
- · Composite action, transformed sections and any material-dependent property.
- · Connections, bearings, fixings and their detailing.
- · Fire resistance, corrosion protection and durability requirements.
- · Overall structural stability, robustness and disproportionate collapse.
- · Vibration, dynamic response and fatigue.
- · Temporary works, construction sequence and buildability.
Where this tool does not apply
- · Sections that cannot be represented by axis-aligned rectangles, including circular, tapered, curved and cold-formed profiles with significant corner radii.
- · Comparison against published catalogue properties without allowing for root radii and manufacturing geometry.
- · Any situation where a checked, code-compliant section property is required — this tool is in beta and has not been independently reviewed.
Preliminary / informational calculation — not a BES project-specific structural design. Within the checks and assumptions listed above, only the stated utilisations have been assessed; a satisfactory utilisation does not confirm overall structural adequacy.
Disclaimer
Indicative result only. The calculation depends on the assumptions and information entered and does not constitute structural design or professional advice. Do not use the result for construction, procurement, Building Regulations submissions or safety-critical decisions. All results must be independently verified by a suitably qualified structural engineer before being used for design, construction or regulatory purposes. Bebbington Engineering Studio Ltd accepts no responsibility for reliance placed on unverified results.