Explicit geometry · centroidal horizontal axis

Centroidal x-Axis Section Properties

Calculate area A, centroidal horizontal-axis second moment of area I_X, elastic section modulus S_X and radius of gyration k_X for an idealized sharp-corner section.

area property, not mass inertiaone declared x-axisno profile catalogueno strength verdict

This calculator does not represent a real rolled, extruded, welded or machined profile unless that profile exactly matches the idealized dimensions. Fillets, corner radii, flange taper, weld metal, holes, coatings and tolerances are excluded.

Idealized geometry and record

Choose one shape and one unit. Values start blank. Changing the unit clears every dimension and result; no hidden conversion is performed.

Idealized cross-section

Shared linear unitOutputs use the corresponding unit, unit², unit³ and unit⁴.

Solid-circle dimensions

Diameter dPositive complete decimal.

Concentric-annulus dimensions

Outer diameter D

Inner diameter d_iMust be positive and smaller than D.

Rectangle dimensions

Width b, parallel to x

Height h, perpendicular to x

Double-symmetric I-section dimensions

Overall height H

Overall flange width B

Web thickness t_wMust be smaller than B.

Each flange thickness t_f2t_f must be smaller than H.

Equal-flange channel dimensions

Overall height H

Overall flange width B

Web thickness t_wMust be smaller than B.

Each flange thickness t_f2t_f must be smaller than H.

Geometry source or record IDDo not enter only a commercial profile name; this page has no catalogue lookup.

Axis and intended-use record

Required model confirmationThe entered section is intentionally idealized with sharp corners, constant listed thicknesses and the exact symmetry stated by the selected shape.I will not treat the geometric outputs as a catalogue property, bending capacity, buckling capacity, stress result or code-compliance verdict.

Geometric x-axis result

Centroidal second moment of area I_X

—

Length to the fourth power; not mass moment of inertia.

Cross-sectional area A

—

Elastic section modulus S_X=I_X/c

—

Geometric elastic section modulus about the declared x-axis.

Radius of gyration k_X=√(I_X/A)

—

Extreme-fibre distance c

—

Equal top/bottom distance for the supported symmetric-about-x shapes.

No structural verdict is produced. A stress, deflection, fatigue, yielding or buckling calculation additionally needs a verified structural model, loads and combinations, material properties, supports/effective length, imperfections, residual stress, local/global stability, connection effects and the applicable design rules.

Equations, axes and model boundary

The x-axis is horizontal and passes through the area centroid. The listed I-section is double-symmetric. The channel has equal top and bottom flanges, so its horizontal centroidal axis is at H/2; its horizontal centroid coordinate, I_y, product of inertia and principal axes are deliberately not calculated.

Supported idealizations

circle: A=πd²/4; Iₓ=πd⁴/64

annulus: A=π(D²−dᵢ²)/4; Iₓ=π(D⁴−dᵢ⁴)/64

rectangle: A=bh; Iₓ=bh³/12

sharp-corner I-section or equal-flange channel:
A=2Bt_f+t_w(H−2t_f); Iₓ=[BH³−(B−t_w)(H−2t_f)³]/12

all shapes: c=d/2, h/2 or H/2; Sₓ=Iₓ/c; kₓ=√(Iₓ/A)

For the I-section, the subtraction removes the two identical voids beside the web. For the equal-flange channel, it removes one rectangular void. Their I_X expressions are identical because the removed total width and vertical extent are identical; their y-axis properties are not identical.

Excluded geometry: root/toe radii, flange taper, unequal flanges, lips, holes, cut-outs, welds, compound or transformed sections, corrosion allowance, dimensional tolerances and non-concentric tubes. Use a verified drawing-based composite-area or CAD/section-property calculation for those cases.

Why the former profile presets were removed

The former page labelled four-number sharp-corner approximations as “IPE 200”, “IPE 300” and “Pipe DN100”. Actual catalogue/standard profiles can include radii, taper and schedule-dependent wall dimensions. The page neither cited a controlled profile table nor matched those details, so the labels could be mistaken for certified catalogue properties. Exact commercial-profile properties remain NEEDS_LICENSED_SOURCE or require a current manufacturer catalogue.

Exact numerical checks

Invoer	A	I_X	S_X	k_X	c
circle d=100 mm	7853.981633974 mm²	4908738.521234052 mm⁴	98174.770424681 mm³	25 mm	50 mm
rectangle b=50 mm; h=100 mm	5000 mm²	4166666.666666667 mm⁴	83333.333333333 mm³	28.867513459 mm	50 mm

Sources and classification

Bron	Use on this page
Penn State Mechanics Map — centroids and area moments for 2D shapes	Centroidal rectangle and circle area/second-moment formulas.
Engineering Statics — composite shapes	Addition/subtraction of simple areas and centroidal moments; basis for the idealized I/channel derivation.
Engineering Statics — radius of gyration	Definition k_X=√(I_X/A), with I and k referred to the same axis.
University of Illinois Mechanics Reference — bending	Elastic section modulus S=I/c and the conditions attached to elastic flexure; this calculator returns geometry only.

Accessed: 15 July 2026. These are general geometric/mechanics relationships, not formulas attributed to an ISO profile standard.

Independent review required. For a real product, compare against the current controlled drawing or manufacturer/standard section-property table. Do not infer compliance from numerical agreement with this idealization.