Sectional Properties Calculator

Purpose and model boundary

This calculator measures the geometric properties of a cross-section you draw: a beam, a column, a plate or any flat shape of one material, with up to five holes through it. You draw the outline on a fine grid, add holes with the drawing tool's Add exclusion, Worked examples or Import, and the spreadsheet returns the section's area, its centroid, its moments and product of inertia about the centroid, its principal moments and the angle of the major axis, its elastic section moduli to each extreme fibre, its plastic moduli, its radii of gyration, an approximate torsion constant and its weight per length in steel.

It is a geometry calculator. It does not check a member's strength, buckling, deflection or connections, it holds no table of rolled shapes, and it offers one material, steel, whose density affects the weight alone. Every figure but the torsion constant is exact for the drawn shape; the torsion constant is an approximation, estimated only where a standard approximation applies, and reads 0 with a CHECK: status where none does.

Inputs and units

The drawing is the main input. The outline, the solid, is sent as an ordered list of up to 100 corner points, and each of the five holes as up to 20, in the same way the Area Calculator sends its outline and exclusions (a hole is the drawing's exclusion). The US/Metric switch on the drawing chooses the unit the points are written in: feet for US and metres for Metric. The drawing is shown on a 1/8 in grid in inches for US, and on a 1 mm grid in centimetres for Metric. Y runs down the page, as the drawing tool draws it, so the top of the section is its smallest y.

The one input beside the drawing is:

• Material: the material the weight per length is counted at. Only steel is offered, at 490 lb/ft³; for Metric that density is converted exactly, with the exact pound (0.45359237 kg) and the exact foot (0.3048 m), to 7,849.04705 kg/m³. It changes the weight and nothing else.

The spreadsheet converts the rows once, before anything else: US rows in feet × 12 to inches, and Metric rows in metres × 1,000 to millimetres. Every section result is in inches (in, in², in³, in⁴) for US and in millimetres (mm, mm², mm³, mm⁴) for Metric, except the weight per length, in pounds per foot for US and kilograms per metre for Metric, and the angle, in degrees in both. The From the drawing table's gross, net and exclusion areas are in sq ft or m², as the next paragraph says.

Holes can be added with the drawing tool's Add exclusion as well as with Worked examples or Import: on this fine grid it places a square that fits, at most a quarter of the outline's smaller side and shrunk a grid step at a time until it lies inside the outline and clear of the other holes, and you then reshape it. The drawing's length labels follow the grid's decimals: on the 1/8 in grid a length reads to the thousandth of an inch (0.375"), and on the 1 mm grid, in centimetres, to the tenth (1.3 cm). They are measured from the drawing; the spreadsheet reads every coordinate at full precision, so a label's rounding never changes a figure. The drawing tool shows no areas of its own on this page: every area shown is one of the spreadsheet's results, in the unit that result declares and with the page's own formatting. The section's areas are in in² or mm², its section moduli in in³ or mm³ and its moments of inertia in in⁴ or mm⁴, while the From the drawing table's gross, net and exclusion areas are in sq ft or m².

Governing relationships

Before any sum, every outline and hole row is shifted by the outline's least x and least y, so the section's left edge is at x = 0 and its top edge at y = 0. A section moved across the drawing gives the same figures.

For each polygon, outline or hole, every edge from point i to the next point, the last closing to the first, contributes one cross term, and Green's theorem turns the area integrals into sums over the edges:

c = x_i·y_(i+1) − x_(i+1)·y_i

A = Σ c / 2, Qx = Σ (y_i + y_(i+1))·c / 6, Qy = Σ (x_i + x_(i+1))·c / 6

Ixx0 = Σ (y_i² + y_i·y_(i+1) + y_(i+1)²)·c / 12, Iyy0 = Σ (x_i² + x_i·x_(i+1) + x_(i+1)²)·c / 12

Ixy0 = Σ (x_i·y_(i+1) + 2·x_i·y_i + 2·x_(i+1)·y_(i+1) + x_(i+1)·y_i)·c / 24

Each polygon's sums are multiplied by the sign of its own area, which normalises the winding, so either direction round gives the same figures, and the holes' sums are subtracted from the outline's. The centroid and the parallel-axis step follow:

cx = Qy / A, cy = Qx / A

Ixx = Ixx0 − A·cy², Iyy = Iyy0 − A·cx², Ixy = Ixy0 − A·cx·cy

The principal moments are in closed form, and the angle of the major axis follows the product of inertia:

I1, I2 = (Ixx + Iyy)/2 ± √(((Ixx − Iyy)/2)² + Ixy²)

θ = 0 when Ixy = 0, otherwise θ = ½·atan2((Ixx − Iyy)/2, −Ixy), in degrees

The elastic moduli divide each moment by the distance from the centroid to the outline's extreme fibre on that side, and the radii of gyration are rx = √(Ixx / A) and ry = √(Iyy / A). The plastic modulus about each axis is the first moment of the whole area about the axis that halves the area, Zx = ∫ |y − y_p| dA, with Zy likewise in x. The weight per length is the area times the density: A ÷ 144 × 490 lb/ft for US, with A in in², and A ÷ 1,000,000 × 7,849.04705 kg/m for Metric, with A in mm².

Calculation sequence

1. The rows are read, each blank row as empty, never as zero, and each point converted once, × 12 or × 1,000, after the shift that puts the outline's left edge and top edge at zero.
2. Each edge of the outline and of every drawn hole gives its cross term and its six sums; each polygon's totals are made positive by its winding, and the holes' are subtracted from the outline's.
3. The centroid is measured from the left edge and from the top edge; the moments are moved to it by the parallel-axis step.
4. The product of inertia reads exactly 0 when negligible: when its size is no more than one billionth of Ixx + Iyy, which is rounding left by the sums, not a property of the section. The snapped value feeds the principal moments and the angle.
5. The principal moments and the angle follow in closed form. The angle is 0 whenever the product of inertia is 0, and I1 may be about either axis: it is about whichever of x and y has the larger moment. Otherwise it runs from −90° to 90°, positive toward +y, which is clockwise on the page because y runs down.
6. The extreme fibres are the outline's own bounds: the left and right edges and the top and bottom. The elastic moduli are the moments divided by the centroid's distance to each.
7. The axis that halves the area is found by bisection over the outline's height (and, for the other axis, its width), repeated for 50 steps, each step halving the bracket; the plastic modulus is then the sum over the edges of the area's first moment about that axis.
8. The torsion constant is estimated by the rule below, the weight is counted from the area and the density, and the status line is written. On a NOT VALID: status every section figure reads 0.

The torsion constant is approximate, and one estimate is used for each kind of drawing:

• No hole and a convex outline: Saint-Venant's approximation, J = A⁴ / (4π²·(Ixx + Iyy)), exact for a circle or an ellipse and within about ±10% for other compact convex solids (an equilateral triangle reads about 14% high).
• No hole and an outline that is not convex (an I, a channel, a tee, an angle): the section is treated as one strip of uniform wall with the section's own area and perimeter, J = L·t³ / 3, with t = P/4 − √(P²/16 − A) and L = P/2 − t. It reads low, because it averages walls of different thicknesses into one, and by an amount that varies with the member's flange and web thicknesses. On the Worked examples it reads 3% low for the L4×4×1/2, 11% to 19% for the W12×26, W14×90, IPE 300 and HEB 200, 31% for the C10×15.3, and 44% below the plates' sum of b·t³ / 3 for the 48 in welded girder with 1 in flanges on a 3/8 in web. When no strip has that area and perimeter (a compact, dented shape), J is not estimated.
• One hole, with a thin wall (no thicker than a tenth of the mean perimeter) of uniform wall thickness (the four gaps between the outline's and the hole's bounding boxes within 10% of the largest): Bredt's single-cell formula, J = 4·Am²·t / s, where Am is the mean of the outline's and the hole's areas, s the mean of their perimeters and t = A / s.
• A thick or uneven single cell, and two or more holes: J is not estimated; it reads 0 and the status is the CHECK: line.

The perimeters are the spreadsheet's own, measured from the coordinates.

Outputs and interpretation

• Area: the solid's area, the outline's area less every hole's.
• Centroid x, from the left edge, and centroid y, from the top edge: measured from the section's own leftmost and topmost points, so the figures do not depend on where the section sits on the drawing.
• Moments of inertia Ixx and Iyy, and the product of inertia Ixy: about horizontal and vertical axes through the centroid. The sign of Ixy is the drawing's, with y running down.
• Major and minor principal moments I1 and I2, and the principal axis angle θ: the largest and smallest moments about any axis through the centroid, and the angle of the I1 axis.
• Elastic moduli Sx, top and bottom, and Sy, left and right: each moment divided by the distance from the centroid to that extreme fibre. The smaller of each pair governs the extreme-fibre stress.
• Plastic moduli Zx and Zy: about the axes that halve the area.
• Radii of gyration rx and ry.
• Torsion constant J (approximate): the estimate above, or 0 with the CHECK: status.
• Weight per length: in steel, at 490 lb/ft³.
• Outline area and hole areas 1 to 5: the outline's gross area and each hole's, in the section's unit.
• From the drawing: the drawing's net and gross areas and each exclusion's area, in square feet or square metres, the figures the drawing tool's own area panel reads. They repeat the area, the outline area and the hole areas in the drawing's own unit.

A curve is represented by the straight edges drawn for it, and the calculator works from the drawn polygon, not the curve. A circle drawn as a regular 32-gon with its corners on the circle has moments of inertia about 1.3% from the circle's, and a 64-gon about 0.3%. On a section with more than one curve the differences can offset, so drawing more points on one curve alone can move a figure further from the true section.

Worked examples

The Worked examples menu holds the default, an 8 by 12 in I-section, and nine members an engineer designs with:

• a W12×26 beam, a W14×90 column, an HSS 8×8×1/2 square tube column, a C10×15.3 channel and an L4×4×1/2 angle, drawn from the AISC Shapes Database v16.0;
• a welded plate girder 48 in deep, 16 by 1 in flanges on a 46 by 3/8 in web, whose area, moments of inertia and moduli match a hand calculation exactly; its torsion constant is the strip estimate, 44% below Σ b·t³ / 3 for this girder;
• the W14×90 boxed with a 1/2 in plate welded across each side's flange tips, whose two closed cells give the CHECK: status: its torsion constant needs a numerical solution this calculator does not make;
• an IPE 300 beam and an HEB 200 column in Metric, from ArcelorMittal's tables to EN 10365.

How they are drawn. Each rolled I-shape has its table's depth, flange width and thicknesses, with its four root fillets drawn as quarter circles of 4 chords each whose ends lie on the arc; the fillet radius is AISC's kdes less the flange thickness, or ArcelorMittal's root radius. The HSS is drawn at its design wall of 0.465 in, 0.93 of the nominal 1/2 in, as AISC tabulates it, with 0.93 in outside and 0.465 in inside corners of 4 chords each. The channel's inner flange faces slope 1 in 6, with the table's flange thickness at mid-flange and no fillet, since AISC gives no fillet radius for channels. The angle is drawn sharp: its toe radii take away about what its root fillet adds. The welded plates are sharp rectangles, as welded plates have no fillets.

The tolerance. Drawn this way, the rolled I-shapes and the HSS land within 1.0% of their tables in area, moments of inertia, elastic and plastic moduli and radii of gyration, and the channel and the angle within 2.5%. Part of each difference is the table's own rounding to three figures. The HSS weight per length is the drawn steel's, while AISC's table weighs the nominal wall, so the table's 48.85 lb/ft is higher. The approximate torsion constant is not compared: it reads 11% to 19% below the table for the W12×26, W14×90, IPE 300 and HEB 200, 31% for the C10×15.3, and 44% below Σ b·t³ / 3 for the plate girder, whose flanges and web differ in thickness (1 in against 3/8 in). The HSS's Bredt estimate is within 0.4% of its table.

The Metric axis names. The Eurocode calls the strong axis y–y. This calculator's Ixx, Sx, Zx and rx are ArcelorMittal's Iy, Wel,y, Wpl,y and iy, and its Iyy is the weak axis, ArcelorMittal's Iz.

A column read on both axes needs one drawing: Ixx, Sx, Zx and rx are about the strong axis and Iyy, Sy, Zy and ry about the weak one. The same W14×90 drawn on its side, turned 90°, swaps them, and its area, weight and torsion constant do not change; a protected test holds that pair. The L4×4×1/2's principal axes lie at 45°, and θ reads −45° because y runs down the drawing.

Not offered: a round HSS or a pipe, because a hole is drawn with at most 20 points, and a 20-point hole has 1.6% less area than its circle, which changes a thin pipe's figures by about 9% on Pipe 6 Std; and double angles, because two separate parts cannot form one outline.

Validation and status logic

The status line is the first of these that applies:

• NOT VALID: choose US or Metric
• NOT VALID: draw at least three outline points
• NOT VALID: every outline point needs both coordinates
• NOT VALID: every point of exclusion 1 needs both coordinates, and the same sentence for exclusions 2, 3, 4 and 5 in that order
• NOT VALID: each exclusion needs zero or at least three complete points
• NOT VALID: outline and exclusion points must fill consecutive rows, with no blank row between them
• NOT VALID: the outline encloses no area
• NOT VALID: exclusion 1 encloses no area, and the same for exclusions 2 to 5
• NOT VALID: exclusion 1 is not inside the outline, and the same for exclusions 2 to 5: a hole's bounds must lie within the outline's, and a point just inside its first edge must lie inside the outline
• NOT VALID: exclusions 1 and 2 overlap, and the same for every pair: that point of one hole lies inside another
• NOT VALID: choose a material from the list
• NOT VALID: the exclusions leave no section area
• NOT VALID: the drawing gives impossible figures (a negative area, a principal moment of inertia that is not positive, or a centroid outside the outline's bounds); check that the outline and the exclusions do not cross themselves, leave the outline or overlap
• CHECK: the torsion constant J is not estimated for this section, so it reads 0; every other figure is exact for the drawn shape
• OK

If the drawing's numbers are too large for the spreadsheet to compute with, the status reads NOT VALID: the drawing's numbers are out of range. On any NOT VALID: status every section figure reads 0.

Assumptions and limitations

• The section is one material, flat and of constant shape along its length; the outline and each hole are simple polygons whose edges are straight, so a curve is represented by the straight edges drawn for it.
• Only steel is offered, and only its density is used; no elastic modulus or yield strength is held or applied.
• The torsion constant is an approximation, estimated only for a convex solid, one open strip of wall, or one thin cell of uniform wall, and it is labelled approximate everywhere it appears.
• The drawing's edges are straight, so a rolled shape's curved fillets and corners are drawn as short chords, and a shape drawn with sharp corners does not carry them. The Worked examples draw root fillets and HSS corners as short chords; an I-shape drawn with sharp corners reads 1% to 6% low against its table, and a tube drawn with sharp corners reads 4% to 7% high.
• Holes can come from the drawing tool's Add exclusion, Worked examples or Import, and the drawing's length labels are rounded to the grid's decimals (0.001 in on the 1/8 in grid, 0.1 cm on the 1 mm grid); the spreadsheet reads every coordinate at full precision.

Trust boundary. On this page the drawing tool keeps the outline and every hole (the drawing's exclusions) simple, keeps each hole inside the outline and keeps the holes apart, and the spreadsheet trusts it. A direct API or MCP caller must meet the same guarantees. The spreadsheet adds its own cheap check: each hole's bounds must lie within the outline's, a point just inside its first edge must lie inside the outline and inside no other hole, and the figures must be possible for a real section (no negative area, a positive principal moment of inertia, the centroid within the outline's bounds). A drawing that fails is NOT VALID and every section figure reads 0. The containment check never refuses a drawing this page accepts. It does not catch every partial overlap, every hole that crosses the outline's edge, or every outline or hole that crosses itself, and those can give wrong figures with an OK status. Send holes that lie wholly inside the outline and apart, in feet (US) or metres (Metric), not inches, with y running down.

Restrictions and non-computing states

This calculator accepts an outline of at least three complete points and holes of none or at least three complete points each, in consecutive rows, and refuses a drawing its status line names. The drawing tool refuses any edit that would leave a shape outside its limits. Nothing is sent until you press Calculate, and pressing it again on an unchanged drawing with an unchanged material sends nothing.

Errors and warnings

A blank required input keeps Calculate from sending, and pressing it names the input and moves to it. If the calculation service refuses an input, the status line names it and the input is marked until you change it. If the service does not answer, the page says so, keeps your drawing, your inputs and the last result, and lets you press Calculate again. A CHECK: status means the figures are returned with a reason to look again; a NOT VALID: status means they cannot be computed and none are shown.

Sources and further reading

• National Institute of Standards and Technology, Guide for the Use of the International System of Units (SI), NIST Special Publication 811, Appendix B.8. nist.gov. The exact foot (0.3048 m) and inch (0.0254 m), behind the conversion of the drawing's rows from feet to inches (× 12) and the Metric figures, and the pound and the cubic foot, behind the weight per length.
• "Pound (mass)," Wikipedia. en.wikipedia.org/wiki/Pound_(mass). The international avoirdupois pound, defined in 1959 as exactly 0.45359237 kg, with which the steel density of 490 lb/ft³ is converted exactly to kg/m³.
• Bureau International des Poids et Mesures, "SI base units." bipm.org. The metre, the SI base unit of length, from which the Metric section results in millimetres (mm, mm², mm³, mm⁴) are formed, and the kilogram, the base unit of mass, in which the Metric weight per length is given.
• "Second moment of area," Wikipedia, section "Any polygon". en.wikipedia.org/wiki/Second_moment_of_area. The second moments and the product of inertia of a polygon from its corner points, one term per edge with the last point closing to the first, as the spreadsheet sums them, and the note that a clockwise order gives the same figures with the opposite sign, which is why each polygon is normalised by its winding; and the parallel axis theorem, by which the moments are moved to the centroid.
• "Torsion constant," Wikipedia, section "Thin walled open tube of uniform thickness". en.wikipedia.org/wiki/Torsion_constant. J = U·t³/3 for a thin open wall of uniform thickness t and median length U, the strip estimate the spreadsheet uses for an outline that is not convex.

The Worked examples take their section dimensions and tabulated properties from these tables:

• American Institute of Steel Construction, AISC Shapes Database v16.0, its spreadsheet of section dimensions and properties. aisc.org. The W12×26's, W14×90's and C10×15.3's depth, flange width and thicknesses, the kdes from which each W-shape's root fillet is drawn, the HSS 8×8×1/2's design wall of 0.465 in and its 0.93 in corner radius, the L4×4×1/2's legs and thickness, and each shape's tabulated properties.
• ArcelorMittal, Orange Book, "Parallel flange I sections - IPE - 1. Section properties". orangebook.arcelormittal.com/node/6. The IPE 300's depth, flange, web and 15 mm root radius, to EN 10365, and its properties, including its torsion constant It of 19.9 cm⁴.
• ArcelorMittal, Orange Book, "Wide flange beams - HE - 1. Section properties". orangebook.arcelormittal.com/node/16. The HEB 200's depth, flange, web and 18 mm root radius, to EN 10365, and its properties.

Saint-Venant's approximation for a convex solid, Bredt's single-cell formula for a thin tube and the limits the spreadsheet sets on each (a wall no thicker than a tenth of the mean perimeter, four wall gaps within 10% of each other) are stated here as the spreadsheet uses them; no textbook or design guide is cited for them, because none was read for this page. The worked examples this calculator was built against reproduce with the formulas above. These sources describe the units, the geometry and the Worked examples' shapes; they do not certify a member, a section or a result. Check the figures, and take a rolled shape's properties from its producer's tables, before relying on them.