Free Section Builder & Custom Section Properties Calculator

Name: Optimal Beam Section Builder and Custom Section Properties Calculator
Author: Tamer Hijjawi, P.Eng.

Built-up and custom section modeling

Build the section from real components, not one flattened outline

The builder keeps every member, plate, primitive and polygon as an editable component. That makes a welded cover plate, back-to-back channel pair, perforated extrusion or irregular fabrication easier to revise than a single coordinate list. On Calculate, solid components are united, overlaps are counted once and void components are subtracted.

Choose a starting part

Search AISC, CISC, European, UK or Australian steel libraries, use an I, channel, angle, tee, plate, rectangle, circle or ring, draw a polygon, or begin from a preset.

Assemble the geometry

Drag with optional grid snapping or enter exact position, size and rotation. Duplicate or mirror selected parts about a vertical, horizontal or two-point line.

Add detail or import CAD

Edit polygon vertices and tangent corner radii, subtract openings, or import closed LINE, ARC, CIRCLE, ELLIPSE and polyline boundaries from an ASCII DXF.

Calculate and reuse

Review the calculated properties and axes, download a PDF report, open the section in the Beam Calculator, or save the editable geometry to your section library.

Optimal AI

Generate, edit or review through approved commands

Ask the assistant to build a dimensioned shape, modify selected components, check calculation readiness or explain current properties. Geometry changes are validated and previewed for your approval before they touch the canvas; the deterministic calculator remains the numerical authority.

See how Optimal AI works

Recovery & activity

Keep the working state and understand every change

Local recovery protects the current browser model. The Activity panel records manual edits, approved AI changes, calculations and saves so the section's evolution is visible instead of hidden.

Saved revisions

Create checkpoints and restore a prior version

Eligible accounts can save the full editable document to the section library. Professional history adds named checkpoints, durable revisions and restore controls for design exploration.

Need one simple shape rather than an assembly?

Use the standard section calculator for dimension-by-dimension formulas and a worked PDF calculation for a single I-section, channel, angle, tee, solid or hollow rectangle, circle or pipe.

Use the formula calculator

Worked verification · built-up section

Built-up I-beam with top and bottom cover plates

The builder combines one editable 300 × 150 mm I-section with two 220 × 12 mm cover plates. The reference values come from an independent five-rectangle hand calculation.

5 of 5 checks pass

The I-section and both cover plates remain editable components; the production analysis unions their touching faces.

Builder components · mm

I1 · d300 × bf150 × tw10 × tf15 at (0,0)
P1 · 220 × 12 at (−35,300)
P2 · 220 × 12 at (−35,−12)

Reference area

A = 2(150×15) + 10×270
+ 2(220×12)
= 12,480 mm²

Reference centroid

Symmetric about x and y
C = (75, 150) mm

Reproduce this case in the builder

Result agreement

Independent result vs. production calculator

Relative tolerance ≤ 1×10⁻⁸

Cover-plated I-section results compared with an independent five-rectangle derivation
Property	Independent	Calculator	Verdict
Area, Amm²	12,480	12,480	Pass
Centroid, C_x / C_ymm	75.000000 150.000000	75.000000 150.000000	Pass
Centroidal I_x / I_ymm⁴	2.3642244×10⁸ 2.9756000×10⁷	2.3642244×10⁸ 2.9756000×10⁷	Pass
Product of inertia, I_xymm⁴	0	0	Pass
Principal I₁ / I₂ / θ_pmm⁴ / degrees	2.3642244×10⁸ 2.9756000×10⁷ · 0°	2.3642244×10⁸ 2.9756000×10⁷ · 0°	Pass

Why this is a meaningful check: the reference treats the web, flanges and cover plates as five non-overlapping rectangles, while the calculator uses the production component-union path followed by polygon boundary integrals. Matching results exercise parametric-member geometry, plate placement, touching-face union, centroid translation and principal properties.

AI-assisted geometry · deterministic properties

Describe the section. Review every change. Then calculate.

Optimal AI turns a plain-language request into real, editable section components and builder commands, not a picture or guessed property values. Ask it to create a starting section, change selected parts, check calculation readiness or explain the results already on screen.

Preview firstThe complete geometry proposal and change list appear before anything is applied.
You approveAccept the proposal, revise the instruction or leave the current section unchanged.
Solver-backedArea, centroid, inertia, plastic, torsion, warping and shear properties come from the section-analysis engine after you select Calculate.

Open Optimal AI in the workspace

Optimal AI is available without signing in. Public request limits protect the service from automated abuse.

Describe the geometry in plain language

“Create a 500 mm deep welded plate girder with 250 × 20 mm flanges and a 12 mm web.”

1
Model-aware proposal
AI maps the request to typed components, dimensions, positions, rotations and solid or void roles.
2
Validation and visual preview
The browser validates the entire command set and shows the proposed geometry plus exact additions, edits or removals.
3
Approve and keep editing
Only an approved proposal reaches the canvas. The applied change remains editable, undoable and visible in Activity.
4
Deterministic calculation
Calculate runs the same geometry and mesh analysis used for a manually built section.

Clear responsibility boundary

AI assists with geometry and interpretation. The analysis engine computes the properties. You remain responsible for verifying the geometry, assumptions and engineering use.

Section property definitions

Every variable in the calculation output, explained

These are geometric properties of the final material region, not a code resistance check. Length results use mm or in; area uses length²; section modulus uses length³; moments and J use length⁴; and C_w uses length⁶. Coordinate results refer to the origin and axes shown in the builder.

Geometry and centroidOverall size, net material area, centroid and distances from the centroid to the four extreme fibres.

Symbol	Output	Units	Meaning and use
W	Overall width	L	W = x_max − x_min, the horizontal extent of the analyzed material region.
D	Overall depth	L	D = y_max − y_min, the vertical extent of the analyzed material region.
A	Cross-sectional area	L²	The area of the unioned solids minus all valid openings. It is used in axial stress, mass-per-length and stiffness calculations.
C_x	Centroid x coordinate	L	The area-weighted horizontal coordinate of the geometric centroid, measured from the builder origin.
C_y	Centroid y coordinate	L	The area-weighted vertical coordinate of the geometric centroid, measured from the builder origin.
c_top	Top extreme-fibre distance	L	The vertical distance from C_y to the highest material point; used in S_x,top = I_x/c_top.
c_bottom	Bottom extreme-fibre distance	L	The vertical distance from C_y to the lowest material point; used in S_x,bottom.
c_left	Left extreme-fibre distance	L	The horizontal distance from C_x to the leftmost material point; used in S_y,left.
c_right	Right extreme-fibre distance	L	The horizontal distance from C_x to the rightmost material point; used in S_y,right.

Centroidal inertia and gyrationArea moments about the horizontal x and vertical y centroidal axes, including unsymmetric coupling.

Symbol	Output	Units	Meaning and use
I_x	Second moment of area about x	L⁴	I_x = ∫(y − C_y)²dA. With E, it forms flexural rigidity EI_x for bending and deflection about x.
I_y	Second moment of area about y	L⁴	I_y = ∫(x − C_x)²dA. It governs flexural rigidity about the vertical centroidal axis.
I_xy	Product of inertia	L⁴	I_xy = ∫(x − C_x)(y − C_y)dA. A non-zero value indicates coupling between the displayed x and y bending axes.
I_p	Polar second moment of area	L⁴	I_p = I_x + I_y about the centroid. It is an area-distribution measure and is not the torsion constant J except for circular sections.
r_x	Radius of gyration about x	L	r_x = √(I_x/A), the equivalent distance at which the area could be concentrated without changing I_x; used in L/r slenderness.
r_y	Radius of gyration about y	L	r_y = √(I_y/A), used for slenderness and buckling checks about the y-axis.

Elastic and plastic bendingElastic stress properties to each extreme fibre and full-yield geometric properties about x and y.

Symbol	Output	Units	Meaning and use
S_x,top	Elastic section modulus (top)	L³	I_x/c_top. In elastic bending, the top extreme-fibre stress magnitude is M_x/S_x,top.
S_x,bottom	Elastic section modulus (bottom)	L³	I_x/c_bottom. It can differ from the top value when the section is not symmetric about x.
S_y,left	Elastic section modulus (left)	L³	I_y/c_left, for elastic extreme-fibre stress on the left side under bending about y.
S_y,right	Elastic section modulus (right)	L³	I_y/c_right, for elastic extreme-fibre stress on the right side under bending about y.
Z_x	Plastic section modulus about x	L³	The first moment of area about the horizontal plastic neutral axis: ∫\|y − y_PNA\|dA. For one uniform yield stress, M_p,x = F_yZ_x.
Z_y	Plastic section modulus about y	L³	The corresponding first moment about the vertical plastic neutral axis, used for full-plastic bending about y.

Shear and torsionFinite-element warping results for a connected section. These are not obtained from the polygon area integrals alone.

Symbol	Output	Units	Meaning and use
A_s,x	Shear area for x-direction loading	L²	The effective shear area returned by the warping analysis for transverse shear in the x direction; it is a stiffness/stress parameter, not a literal cut area.
A_s,y	Shear area for y-direction loading	L²	The effective shear area for transverse shear in the y direction. It may be substantially smaller than gross area A.
S_c,x	Shear-centre x coordinate	L	The horizontal coordinate of the point through which transverse load can act without inducing twist in the connected section.
S_c,y	Shear-centre y coordinate	L	The vertical coordinate of that no-twist loading point, in the same coordinate system as C_x and C_y.
J	Saint-Venant torsion constant	L⁴	The geometric constant in uniform torsional rigidity GJ. J is generally not equal to I_p. For disconnected regions, the reported J is a component sum assuming equal twist; connection effects are excluded.
C_w	Warping constant	L⁶	A measure of resistance to non-uniform or restrained warping, used in torsion and lateral-torsional buckling formulations. It is reported only for a connected analyzable region.

Principal and plastic axesThe rotated axes where product of inertia is zero, plus the equal-area axes used for plastic bending.

Symbol	Output	Units	Meaning and use
I₁	Major principal second moment	L⁴	The larger centroidal second moment after rotating to the principal axes, where I₁₂ = 0.
I₂	Minor principal second moment	L⁴	The smaller principal second moment. For any rotation, I₁ + I₂ = I_x + I_y.
θ_p	Principal-axis angle	degrees	The displayed rotation from the global x-axis to the major principal 1-axis; its sign follows the builder coordinate convention.
PNA_x	Horizontal plastic neutral axis	L	The distance from the section bottom to the horizontal line that divides the net area into equal areas above and below.
PNA_y	Vertical plastic neutral axis	L	The distance from the section left edge to the vertical line that divides the net area into equal areas on each side.

Disconnected built-up regions: area, centroid, inertia, elastic/plastic and principal properties include all regions. J is the sum of the disconnected component torsion constants under an equal-twist assumption. C_w, A_s,x, A_s,y and the shear centre are not reported because the physical connection and load transfer between regions are undefined.

Calculation method

Boolean geometry first, then geometric and mesh analysis

1. Compose the material region. Parametric and catalogue components are converted to closed boundaries. Solid shapes are geometrically united so overlaps are counted once; valid voids are subtracted. This final region, rather than the visible sum of the input parts, is what the calculator analyzes.

2. Solve geometric and plastic properties. Area integration provides A, centroid, I_x, I_y, I_xy, elastic moduli, radii and principal properties. The plastic solver locates equal-area axes and obtains Z_x and Z_y.

3. Mesh the connected section. A triangular finite-element mesh is used to solve the Saint-Venant torsion and warping problem for J, C_w, shear areas and shear centre. If the hosted analysis service is unavailable, the browser can fall back to polygon boundary integrals for non-touching geometric regions; advanced warping results are then left blank.

C_x = Σ(A_ix_i) / ΣA_i

r_x = √(I_x / A)

S_x = I_x / c

I_p = I_x + I_y ≠ J

DXF section properties

Prepare a CAD cross-section for reliable import

1
Create closed 2D outlines
Join the outside perimeter and every opening into separate closed boundaries on a flat XY plane.
2
Remove non-geometry entities
Delete dimensions, annotations, hatches, centre lines, title blocks, duplicate edges, and construction geometry.
3
Match model-space and DXF units
Set the CAD model-space unit and DXF $INSUNITS declaration to the same physical unit. After import, compare the reported width × depth with one known model-space dimension before calculating.

Supported geometry

Closed LWPOLYLINE and POLYLINE entities, connected LINE and ARC paths, CIRCLE entities, and closed ELLIPSE entities. Curves are converted into short polygon segments for calculation.

Remove unsupported entities

Blocks, non-planar geometry, hatches, splines, text and dimensions are not section boundaries. Explode or convert required profile geometry to supported planar entities, and remove duplicates before export.

Local parsing, analyzed coordinates

JavaScript reads the DXF locally; the original drawing file is not uploaded or stored. When you click Calculate, the derived boundary coordinates are sent to Optimal Beam's analysis endpoint to obtain the section properties.

Engineering applications

Use one builder for fabricated, irregular and asymmetric sections

Use the section builder when a handbook row or one closed-form shape no longer represents the member. The result describes the geometry you assembled and can supply properties to later beam, frame, buckling, stress or code checks.

Built-up steel

Welded plate girders, back-to-back channels, cover-plated beams, boxed members and other fabrications assembled from catalogue shapes and plates.

Extrusions and profiles

Aluminum extrusions, machined shapes, cold-formed outlines, and proprietary product profiles.

Openings and cut-outs

Sections with service penetrations, internal voids, notches, copes, or material removed during retrofit.

Unsymmetric geometry

Angles, offset webs, rotated shapes, and profiles where I_xy and the principal axes matter.

Choose the fastest way to define the section
Starting point	Best for	Builder workflow	Use
Steel library	A built-up section based on a regional catalogue member	Search a designation, add it as a component, then attach plates or other members	Open Library
Parametric parts	Plate girders, boxes, rings and assemblies of known dimensions	Add shapes, edit dimensions and use exact transforms, copy and mirror	Open Add
Drawn polygon	Tapered, notched or irregular geometry	Place ordered points visually or by x-y coordinate; add radii and voids	Draw polygon
DXF import	Detailed CAD profiles with many edges, arcs or internal openings	Import clean closed 2D boundaries from an ASCII DXF, then inspect and edit	Open DXF

Continue the structural model

Send the section to a beam or frame analysis

Use area and I_x for axial and bending stiffness, or send the calculated values directly to the Optimal Beam beam calculator. For connected beams, columns, trusses, and frames, continue in the 2D structural analysis workspace.

Open beam calculator Open 2D structural analysis

Scope and accuracy

Know what the section result represents

These are gross geometric and mesh-derived cross-section properties for the boundary you define. Numerical precision cannot correct an inaccurate outline, wrong units, omitted holes, unrealistic connection assumptions or coarse curve segmentation. Verify critical values independently and apply the governing design standard.

Included

Area, centroid, centroidal and principal inertia, elastic and plastic moduli, J, plus C_w, shear areas and shear centre for connected sections when mesh analysis succeeds.

Outside this tool

Material transformations, local buckling, code resistance checks, and full stress-field contour plots.

Engineering calculators

Take the section into your next calculation

Use the area and inertia in a beam or frame model, switch to a standard shape calculator, or look up catalogue steel properties without leaving the Optimal Beam toolkit.

Beam Calculator Calculate reactions, shear, bending moment, slope, deflection and stress for a straight beam. Open Beam Calculator 2D Frame & Truss Analysis Model and analyze frames and trusses with member releases, load cases and combinations. Open Frame Calculator Section Properties Calculate area, centroid, moments of inertia, section modulus and radius of gyration for standard shapes. Open standard section calculator Steel Beam Sizes Find dimensions and properties for AISC, CISC, UK, European and Australian steel shapes. Browse Steel Beam Sizes

FAQ

Cross-section builder and properties calculator questions

What is a section builder?+

A section builder assembles a structural or mechanical cross-section from catalogue members, parametric parts, plates, polygons, openings or imported CAD geometry. It combines those components into the material region used to calculate the section properties.

How do I calculate the moment of inertia of an arbitrary cross-section?+

Assemble the cross-section from components, define each custom boundary with ordered x-y vertices, or import a clean 2D DXF. Calculate composes the net material region and obtains centroidal I_x, I_y, I_xy and the principal properties.

Can I import a DXF to calculate section properties?+

Yes. Import a 2D ASCII DXF containing closed polylines or supported connected line, arc, circle and ellipse boundaries. Set the CAD model-space unit and DXF $INSUNITS declaration to match, then compare the imported size with one known dimension. The original file is parsed locally in your browser. When you click Calculate, the derived boundary coordinates, not the DXF file, are sent to the analysis endpoint.

How do I represent a hole or opening?+

Add a void/opening component (parametric rectangle or polygon). Voids are subtracted from the unioned solids on Calculate. For local results, every void must remain completely inside material; the editor reports an invalid opening instead of silently subtracting one outside the section.

What custom section properties are calculated?+

Results include overall size, area, centroid and extreme-fibre distances; I_x, I_y, I_xy and I_p; radii of gyration; elastic and plastic moduli; principal and plastic axes; J; and, for connected sections, C_w, shear areas and shear centre. Every output is defined in the property guide above.

Does this calculate torsion constant J or warping constant C_w?+

Yes. A connected material region uses a mesh-based sectionproperties analysis for J, C_w, shear areas and shear centre. For disconnected regions, J is a component sum under an equal-twist assumption; C_w and shear results are left unavailable because the connection is not defined.

Can section components touch or overlap?+

Yes. Touching and overlapping solid components are united before analysis, so shared material is counted once. That geometric union assumes full connection; it does not model weld flexibility, bolt slip or partial composite action.

Can different materials be assigned to different components?+

Not in this calculation. All components are treated as one homogeneous geometric section. A material and modulus can be attached when saving for downstream beam or frame analysis, but the builder does not calculate transformed multi-material section properties.

Can I use the calculated section in the beam calculator?+

Yes. After calculating, send the area, depth, centroid, and I_x directly to the Optimal Beam beam calculator without re-entering those properties.

Can I save a custom section and reuse it?+

Yes. Registered users with section-saving access can save the complete editable workspace before or after calculating. Drafts reopen in Section Builder; sections with current calculated properties also appear in the Beam Calculator and 2D Structural Analysis libraries.

Can Optimal AI build or edit a section?+

Yes, without signing in. Ask it to propose a dimensioned section, change selected components, review calculation readiness or explain the deterministic results. Public request limits protect the service from automated abuse. AI geometry arrives as typed builder commands: the browser validates the complete proposal and shows a preview that you must approve before it is applied. AI does not invent the calculated property values.

Does the section builder keep recovery and version history?+

The current model has local browser recovery and an Activity panel for edits, AI-applied changes, calculations and saves. Eligible cloud-saved sections can also create named checkpoints, retain durable revisions and restore an earlier version; history features depend on the account plan.

In what order should I enter polygon coordinates?+

Move around each boundary in sequence without jumping across the section. Clockwise or counter-clockwise order is accepted; the last point connects back to the first automatically. Edges may not cross.

Can I add rounded corners to coordinate geometry?+

Yes. Select a point and enter its tangent corner radius in the point inspector. The preview shows the curved boundary and the calculator prevents adjacent radii from overlapping along one edge.

Engineering authorship

Prepared and reviewed by Tamer Hijjawi, P.Eng.

Component composition, polygon geometry, centroidal and principal properties, plastic-modulus logic, section-analysis result handling, DXF import scope and the published verification cases below were technically reviewed for this builder. Last technical review: August 8, 2026.

Read engineering biography

Sources and verification

Verification you can audit, not just matching numbers

Each published check identifies the independent reference, runs the production calculation path, measures the error and applies a declared tolerance. Exact polygon properties, discretized DXF curves and mesh-dependent warping results are kept separate because they do not have the same error behavior.

01
Establish the oracle
Derive the expected value independently from closed-form geometry or the governing file-format definition.
02
Run the production path
Use the same polygon, Boolean, DXF and result-normalization code exercised by the section builder.
03
Apply the release gate
Compare relative error with the case-specific threshold; invalid geometry must fail explicitly rather than return a number.

Published benchmark snapshot

All 6 checks are within their declared tolerance

Reviewed August 8, 2026 · metric reference geometries · production calculation routines

Passed: 6 / 6
Polygon gate: 1×10⁻⁸
DXF area gate: 0.05%

Swipe to compare reference, result, error and verdict →

Published section calculator verification results and acceptance thresholds
Case / behavior	Property	Independent reference	Production result	Measured error	Acceptance	Verdict
GEO-01Solid rectangle200 × 300 mm · base polygon	Area, A mm²	200 × 300 60,000	60,000	0	≤ 1×10⁻⁸ rel.	✓ Pass
GEO-01Solid rectanglecentroidal distribution	I_x mm⁴	bh³ / 12 4.500000×10⁸	4.500000×10⁸	0	≤ 1×10⁻⁸ rel.	✓ Pass
BOOL-01Hollow box200×300 outer − 160×260 void	I_x mm⁴	(BH³−bh³) / 12 2.156533×10⁸	2.156533×10⁸	0 to shown precision	≤ 1×10⁻⁸ rel.	✓ Pass
ASYM-01Unsymmetric L-sectioncomposite centroid	C_x = C_y mm	ΣAᵢxᵢ / ΣAᵢ 32.222222	32.222222	0	≤ 1×10⁻⁸ rel.	✓ Pass
XFORM-01Rotated rectangle200 × 100 mm · invariant principals	I₁ / I₂ mm⁴	6.666667×10⁷ 1.666667×10⁷	6.666667×10⁷ 1.666667×10⁷	0 to shown precision	≤ 1×10⁻⁸ rel.	✓ Pass
DXF-01Imported CIRCLEr = 50 mm · adaptive 0.01 mm chord error · 158 segments	Area, A mm²	πr² 7,853.982	7,851.912	0.0264%	≤ 0.05% rel.	✓ Pass

How error is measured: |calculated − reference| / max(|reference|, 1).

Values shown in mm-based units.

Coverage beyond the snapshot

The suite checks behavior, not only happy-path values

The six rows are compact, reproducible examples. Automated release tests also exercise the transformations and failure modes that commonly expose geometry bugs.

Composition & voids

Solid unions, overlaps, internal voids, nested DXF rings and disconnected result contracts.

Invariance

Translation, rotation, mirroring, scaling laws and metric-imperial unit conversion.

Invalid geometry

Self-intersections, open boundaries, non-planar DXF entities and invalid hole relationships must be rejected.

DXF interpretation

LINE, ARC, CIRCLE, ELLIPSE, legacy and lightweight polylines, bulges and drawing units.

Read the green checks correctly

A pass means the result met the declared test threshold. It does not imply every analysis mode is exact.

Exact polygon geometry

Floating-point agreement

Straight-edged area, centroid and inertia use boundary integrals. Closed-form benchmarks should agree near machine precision.

Curved boundaries

Discretization tolerance

DXF curves use adaptive chord-error sampling before becoming straight segments. Larger radii receive more segments to hold the maximum deviation to 0.01 mm (0.000394 in).

Warping analysis

Mesh sensitivity applies

J, C_w, shear areas and shear centre come from a finite-element mesh. The exact-value rows above do not establish convergence for an arbitrary section.

Closed-form and format references

Engineering Statics, Parallel Axis Theorem Centroidal translation and the I = Ī + Ad² relationship used after origin-based polygon integrals.
Engineering Statics, Products of Inertia I_xy, centroidal axes and principal-direction transformation for unsymmetric sections.
Autodesk, DXF entity reference Entity and group-code definitions used by the browser-side DXF importer.
sectionproperties analysis documentation Documented basis for geometric, plastic, Saint-Venant torsion, warping, shear-area and shear-centre results returned by the hosted mesh solver.

How to interpret this evidence

The benchmark oracle does not reuse the polygon implementation it is testing. Rectangle formulas, composite-area equations and principal-property invariants provide the independent values; DXF behavior is checked against both the format definition and analytic geometry.

Review the calculation method

Scope boundary: these checks verify declared idealized geometry cases, transformations, import behavior and software result contracts; they are not design certification. Independently review critical geometry, mesh sensitivity, material assumptions, load transfer, connections, stability and governing design requirements.