Section properties for composite cross-sections built from rectangles, circles, half and quarter circles and polygons, each one solid or a void. Draw the parts on the canvas, set exact sizes and positions with the Dimension tool, and read the centroid, area moments of inertia, product of inertia, polar moment and section moduli from the right panel.
| Property | Symbol | Description |
|---|---|---|
| Total area | A | Sum of all solid regions minus voids |
| Centroid | x̅, y̅ | Area-weighted average of sub-shape centroids |
| Area moment of inertia | Ix, Iy | Second moment of area about centroidal axes |
| Product of inertia | Ixy | Cross-product of area about centroidal axes |
| Polar moment of inertia | J | Ix + Iy about the centroid. Equal to the torsion constant only for circular sections |
| Section modulus | Sx, Sy | I divided by the distance to the farther extreme fiber; relates bending stress to moment |
The calculator applies the parallel axis theorem to each sub-region automatically. For a part of area A with its own centroidal moment I₀, the contribution to the composite Ix is:
I₀ + A·d², where d is the vertical distance from that sub-shape's centroid to the composite centroid.
Void regions subtract their area and their parallel axis term. This is the standard approach for I-beams, T-sections, C-channels, pipes, plates with holes, and any other built-up section you can compose from rectangles, circles, arcs and polygons.
The bending stress formula σ = M·c / I requires Ix and the distance from the centroid to the extreme fiber. The section modulus Sx = Ix / c collapses these into one value, which is why beam tables list Sx directly. The polar moment J = Ix + Iy is reported too, but τ = T·r / J only holds for circular shafts. A rectangular or built-up section warps as it twists, and its torsion constant is smaller than Ix + Iy.
Results are compared against separate calculations in Python. For rectangles, the drawn region is cut into a grid of non-overlapping cells along every rectangle edge, the solid cells are kept, and their exact properties are summed. That method does not use the parallel axis bookkeeping the tool uses, and it was itself checked against closed forms for the hollow rectangle and the I-section. The test set is 15 sections: an I-beam, T, channel, unequal angle, Z, hollow rectangle, a plate with two offset voids, and 8 random stacks. All 150 values (area, centroid, Ix, Iy, Ixy, J, Sx, Sy) agree to within 1 part in 109.
For circles, arcs and polygons, every part is integrated exactly with sympy in every orientation (each quarter, half and three quarter arc, and triangles, a trapezoid and concave polygons) and compared with the formulas the tool uses. Composite sections are then summed from exact moments about a fixed origin, again with no parallel axis step per part: Beer and Johnston's Sample Problem 5.1 (a rectangle, triangle, half circle and circular hole, x̅ = 54.8 mm, y̅ = 36.6 mm), a pipe against π(D⁴ − d⁴)/64, a slot with half circle ends, a plate with quarter circle cutouts, a T with triangular gussets and three more. That is 245 checks, and the tool's own results for the 8 composites match all 80 values to within 1 part in 109.
Limits: straight edges and circular arcs only, with arcs in whole quarter turns, so no ellipses or fillets between edges. Overlapping solids, overlapping voids, and voids that extend past the solid get a warning, because the results then no longer match the drawing.
This tool computes the area moment of inertia, the second moment of area of a cross-section about an axis, with units of length to the fourth power. It is the I in the bending stress formula and in beam deflection formulas. Mass moment of inertia is a different quantity, has units of mass times length squared, and describes resistance to angular acceleration in dynamics.
Draw your section anywhere on the canvas. The centroid is measured from the bottom left corner of the section's outline, with x to the right and y up, the same frame Hibbeler and Beer and Johnston use. Ix, Iy, Ixy, J and the section moduli are all about the centroidal axes, so where you place the section makes no difference to them. Ixy follows the y up convention, so its sign matches the textbook.
Draw the outer shape first, then draw the hole inside it. A part drawn wholly inside a solid starts as a void, and the Solid and Void buttons above a selected part switch it. The void subtracts both its area and its parallel axis term. An I-beam is three solid rectangles, a hollow box is one rectangle with one void, a pipe is a circle with a smaller circle inside, and a C-channel works either way.
Yes. The Circle tool draws a full circle, the Arc tool a quarter, half or three quarter circle in any of four orientations, and the Polygon tool any straight-edged shape, a triangle included. Each part's own centroid and centroidal moments come from the closed forms, for example a half circle's centroid at 4r/3π from its flat edge, and the Parts table shows them next to the parallel axis terms.
The farther one. Section modulus is I divided by the distance from the centroid to a fiber. For a section that is not symmetric about the bending axis, such as a T-section, the centroid sits closer to one face, so the two faces have different moduli. The tool reports the smaller, because that fiber sees the higher stress. For the other face, divide I by its own distance from the centroid.