Composite Section Calculator

Purpose and model boundary

This calculator measures a composite cross-section: an outline of one material with up to five regions inside it, the inclusions, each filled with another material. You draw the outline on a fine grid, add the inclusions with the drawing tool's Add exclusion, Worked examples or Import, give each inclusion's modular ratio n (its material's elastic modulus divided by the outline material's) and a bending moment, and the spreadsheet returns the transformed 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 and its radii of gyration, the highest and lowest bending stress in each material at its extreme fibres, and each material's area.

It is a section calculator for elastic bending. It does not check a member's strength, buckling, deflection, shear or connections, it holds no table of materials or rolled shapes, and it does not compute plastic moduli, a torsion constant or a weight: a modular ratio carries only a material's stiffness, and each of those needs something else about every material (its strength, its shear stiffness or its density), which this calculator does not hold. For a section of one material, the Sectional Properties Calculator gives all three.

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 inclusions as up to 20, in the same way the Area Calculator sends its outline and exclusions: the drawing calls each inclusion an exclusion, and the form names it that way. 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 inputs beside the drawing are:

• Exclusion 1 to 5 modular ratio (n), under "Materials in the exclusions": 0 or more for each of the five slots, the inclusion material's elastic modulus divided by the outline material's. The Worked examples use 0.1243 for 4 ksi concrete in steel (1/n, n = 8.04; ACI 318-19 Ec = 57,000√f'c), 8.044 for steel bars in 4 ksi concrete, 18.125 for steel in Douglas fir No. 2 (NDS, 29,000 / 1,600 ksi) and 0.1571 for C30/37 concrete in steel (EN 1992-1-1 Ecm 33 GPa, Ea 210 GPa). A concrete slab drawn as an inclusion on a steel outline therefore takes 1/n, not the n = Es/Ec that US practice quotes: typing 8 instead of 0.1243 makes the slab about 65 times too stiff. 0, the default, leaves the inclusion a hole, and 1 makes it the outline's own material. A slot's ratio changes nothing while that inclusion is not drawn, but every ratio is required.
• Bending moment, under "Bending": in kip·ft for US or kN·m for Metric, the one for the chosen system. It is about the section's horizontal centroidal axis, and a positive moment compresses the top of the drawing. Only the stresses depend on it. The shipped 10 kip·ft and 10 kN·m are illustrative, not design loads, and not conversions of each other.

The US concrete in the Worked examples takes ACI 318-19's Ec = 57,000√f'c psi throughout, 3,605 ksi at f'c = 4,000 psi against Es = 29,000 ksi, because the reinforced concrete beam is an ACI member and a composite or filled member's designer usually quotes the same Ec. AISC 360-16 I2.1b's wc^1.5·√f'c, about 3,492 ksi for 145 pcf concrete (n ≈ 8.3), is the alternative; either is defensible, and the ratio you type decides which is used.

Draw one outline that is the union of the materials actually present, with each second material as an inclusion inside it. Anything inside the outline and outside every inclusion counts as the outline's material, so a gap between members is a notch in the outline: the air above and below a flitch plate set in from the timber's edges is notched out, not drawn as timber. A composite beam is one outline, the steel and the slab together, with the slab as inclusion 1 flush with the top, so the outline's material is the steel and the transformed figures are in steel terms, as AISC and the Eurocode report them. A reinforcing bar is drawn as a regular octagon of the bar's nominal area, which keeps its area, its centroid and its (n − 1)·As term exact.

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. It converts the moment once too: kip·ft × 12 to kip·in, and kN·m × 1,000,000 to N·mm. Every section result is in inches (in, in², in³, in⁴) for US and in millimetres (mm, mm², mm³, mm⁴) for Metric, except the bending stresses, in ksi for US and MPa 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.

Inclusions 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 inclusions, 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 inclusion 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, the outline and every drawn inclusion, 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. The transformed section is the outline's sums plus each drawn inclusion's sums weighted by n − 1: each inclusion's material is counted n times its area, less the outline material it displaces. The outline's weight is 1; an inclusion with n = 0 is a hole, subtracted as on the Sectional Properties Calculator, and one with n = 1 adds nothing. The centroid and the parallel-axis step follow, on the transformed totals:

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), all of the transformed section.

The stress comes from generalized flexure: a moment M about the horizontal centroidal axis, with no lateral restraint, so the neutral axis is inclined when the product of inertia is not 0. In the outline material's terms, tension positive and y running down,

σ(x, y) = M·[Iyy·(y − cy) − Ixy·(x − cx)] / (Ixx·Iyy − Ixy²)

and a material with modular ratio n carries n times that stress. With Ixy = 0 this is n·M·(y − cy) / Ixx, the familiar flexure formula.

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 inclusion gives its cross term and its six sums; each polygon's totals are made positive by its winding, and the transformed totals weight the outline by 1 and each inclusion by n − 1.
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, the angle and the stresses.
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 stress is worked out at every vertex of every polygon. A stress that varies linearly over a flat shape is highest and lowest at its corners, so each inclusion's highest and lowest stress are the largest and smallest over its own polygon's vertices, times its n, and they are exact. The outline material's highest and lowest are read only at vertices that bound that material, by a vertex rule: a vertex counts when outline material touches that vertex. Each edge, of the outline and of every inclusion, is probed near each of its two ends: from the vertex, a point steps ε along the edge towards its other end, then δ off the edge, inward for an outline edge and outward for an inclusion's. ε is one millionth of the outline's larger dimension and δ one thousandth of ε. A probe is outline material when it lies inside the outline and in no inclusion, and a vertex counts when the probe at it on either of its two edges is outline material. Any inclusion vertex the material touches counts, wherever it lies, not only one on the outline's edge.
8. Each material's area is counted: the outline material's is the outline's area less every drawn inclusion's, and each inclusion's transformed area is n times its area. The status line is written; on a NOT VALID: status every section figure reads 0.

Outputs and interpretation

• Transformed area: the section's area in the outline material's terms, each inclusion counted n times its area.
• Centroid x, from the left edge, and centroid y, from the top edge: of the transformed section, measured from the section's own leftmost and topmost points, so the figures do not depend on where the section sits on the drawing.
• Transformed moments of inertia Ixx and Iyy, and the transformed product of inertia Ixy: about horizontal and vertical axes through the centroid. The sign of Ixy is the drawing's, with y running down.
• Transformed 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.
• Transformed elastic moduli Sx, top and bottom, and Sy, left and right: each moment divided by the distance from the centroid to that extreme fibre. M ÷ S is the outline material's stress at that fibre only when Ixy is 0 or the member is braced laterally; otherwise read the stresses, which include the inclined neutral axis. On the requirement's plate-with-an-off-centre-insert examples (protected tests), M ÷ S with the restrained formula differs from the stress this page returns by 2.4% to 6.8%.
• Radii of gyration rx and ry, of the transformed section.
• Bending stresses at the extreme fibres: the highest and lowest stress in the outline's material and in each inclusion's, tension positive. An inclusion that is not drawn reads 0. An inclusion left at n = 0 is a void: it reads 0 stress in its own material, and the outline material's stress is overstated, because the void carries nothing a real material would.
• Materials: the outline material's area, outside every inclusion, and each inclusion's area and transformed area (n times its area); 0 for a slot that is not drawn.
• 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 treat every inclusion as a hole, whatever its n, since the drawing tool measures shapes, not materials.

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.

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: an inclusion'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 inclusion lies inside another, which is also how an inclusion drawn inside another is refused
• NOT VALID: enter a modular ratio of 0 or more for exclusion 1, and the same for exclusions 2 to 5
• NOT VALID: enter a bending moment (0, or either sign)
• 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: exclusion 1 has modular ratio 1, the outline's own material, so it changes nothing, and the same for exclusions 2 to 5: the figures are those of the section without that inclusion
• 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; the drawing's own areas are still shown.

Assumptions and limitations

• Each material is linear elastic and the section uncracked, with plane sections remaining plane: concrete in tension is counted as if uncracked, so a reinforced or composite concrete section that has cracked is stiffer here than it is. A cracked section cannot be drawn, because it needs the neutral axis before the drawing exists. The reinforced concrete beam example is loaded at 40 kip·ft, below its 45.5 kip·ft cracking moment (fr·Ig/yt with fr = 7.5√f'c, ACI 318-19), so the uncracked section is the right one for it.
• Composite deck with its ribs perpendicular to the beam cannot be drawn, because the slab above the deck would be separated from the steel, so the composite-beam examples use a solid slab.
• For deflection, the AISC 360-16 Commentary uses a lower-bound moment of inertia, I_LB, not the full transformed moment of inertia this page returns.
• A concrete-filled column is designed by AISC 360-16 I2 with an effective stiffness EIeff and its axial load, not by this calculator. For the filled HSS 10×10×1/2 example, I2.2b's EIeff/Es = Is + C3·(Ec/Es)·Ic with C3 = 0.9 gives 317.4 in⁴ on the drawn figures, against its transformed Ixx of 324.4 in⁴.
• The moment is about the section's horizontal centroidal axis, with no lateral restraint and no axial load, shear or torsion.
• The materials are bonded, so they bend together, and each is flat and of constant shape along the member; the outline and each inclusion are simple polygons whose edges are straight.
• Inclusions cannot nest: an inclusion lies in the outline's material, never inside another inclusion. To draw a region inside an inclusion, such as a bar in a concrete-filled tube, split the surrounding region into inclusions that do not contain one another.
• The outline material's highest and lowest stresses are read only at vertices that bound that material, by the vertex rule of step 7 of the calculation sequence. A corner of the outline that a flush inclusion covers along both its edges, such as a composite beam's top corners, which lie in its slab where there is no steel, is not a fibre of the outline's material and is not counted, and neither is the covered end of an edge that an inclusion covers only in part. The rule can misread only a feature within ε (one millionth of the outline's larger dimension) of a vertex, or a corner sharper than about 0.057°.
• Plastic moduli, a torsion constant and a weight are not computed, for the reason given at the top of this page.
• Inclusions 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 inclusion (the drawing's exclusions) simple, keeps each inclusion inside the outline and keeps the inclusions 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 inclusion's bounds must lie within the outline's, a point just inside its first edge must lie inside the outline and inside no other inclusion, 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 inclusion that crosses the outline's edge, or every outline or inclusion that crosses itself, and those can give wrong figures with an OK status. Send inclusions 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 inclusions of none or at least three complete points each, in consecutive rows, with a modular ratio of 0 or more for every slot and a bending moment of either sign, 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 unchanged inputs 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, such as a negative modular ratio, 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 of the moment from kip·ft to kip·in.
• "Shoelace formula," Wikipedia. en.wikipedia.org/wiki/Shoelace_formula. The area of a simple polygon from the coordinates of its corners, negative when the corners run the other way round, which is why each polygon's sums are normalised by the sign of its own area.
• "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 parallel axis theorem, by which the moments are moved to the centroid.
• "Flitch beam," Wikipedia. en.wikipedia.org/wiki/Flitch_beam. A timber beam with a steel plate let into it, calculated by treating the plate as an equally stiff piece of wood, its width changed by the ratio of the two materials' moduli of elasticity: the transformed section this calculator forms with each inclusion's modular ratio.
• "Bending," Wikipedia, section "Complex or asymmetrical bending". en.wikipedia.org/wiki/Bending. The normal stress in a beam bent about an axis that is not a principal axis, from the moments, both second moments and the product of inertia, under the Euler-Bernoulli assumptions (plane sections remain plane; linear elastic material): with a moment about one axis only, the stress formula this calculator uses.

The Worked examples take their section dimensions and their concrete and timber moduli from these sources:

• American Institute of Steel Construction, AISC Shapes Database v16.0, its spreadsheet of section dimensions and properties. aisc.org. The W16×26's depth, flange and web, and its kdes, from which the root fillet is drawn; the HSS 10×10×1/2's design wall of 0.465 in and its 0.93 in corner radius.
• ArcelorMittal, Orange Book, "Parallel flange I sections - IPE - 1. Section properties". orangebook.arcelormittal.com/node/6. The IPE 330's depth, flange, web and 18 mm root radius, to EN 10365.
• American Concrete Institute, ACI CODE-318-19, Building Code Requirements for Structural Concrete, section 19.2.2.1(b). concrete.org. Ec = 57,000√f'c psi for normalweight concrete, which gives 3,605 ksi at f'c = 4,000 psi and the ratios 0.1243 (concrete in steel) and 8.044 (steel bars in concrete) against Es = 29,000 ksi.
• EN 1992-1-1:2004, Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings, Table 3.1, as listed by the European Commission's Joint Research Centre. eurocodes.jrc.ec.europa.eu. Ecm = 33 GPa for C30/37 concrete, which gives the ratio 0.1571 against steel at 210 GPa.
• American Wood Council, National Design Specification (NDS) Supplement: Design Values for Wood Construction, 2024 Edition, Tables 4A and 1B. awc.org. E = 1,600,000 psi for Douglas Fir-Larch No. 2 (Table 4A), which gives the flitch plate's ratio 29,000 / 1,600 = 18.125, and a 2×12 dressed to 1.5 by 11.25 in (Table 1B).

These sources describe the units, the geometry, the method and the Worked examples' inputs; they do not certify a member, a section or a result. Take each material's elastic modulus from its producer or the governing standard, and check the figures before relying on them.