Section properties

Geometric properties of a cross-section: area, centroid, moments and product of inertia, principal axes, elastic and plastic section moduli, radii of gyration and the torsion constant, for common shapes and for sections built up from rectangles. The tool has no design code and no LRFD / ASD choice; the values feed any member calculation.

Inputs of the section properties #

• Shape: rectangle, circle, hollow rectangle, hollow circle (tube or pipe), I-shape from plates, T-shape, channel, angle, or built up from rectangles.
• Dimensions (only those of the chosen shape are shown):
  • rectangle: width b and height h; circle: outside diameter D;
  • hollow rectangle: width, height, wall thickness t and the outside corner radius (0 = sharp corners; the AISC Manual HSS tables use twice the design thickness, and the design thickness is 0.93 times the nominal for ERW HSS, AISC 360-22 B4.2);
  • tube: D and t;
  • I-shape: depth d, flange width bf and thickness tf, web thickness tw, the web-to-flange fillet radius, and with Equal flanges off a different bottom flange (width and thickness);
  • T-shape and channel: d, bf, tf, tw and the fillet radius (a channel's flanges are taken with a constant thickness, so for a rolled channel use the average flange thickness);
  • angle: the vertical leg h, the horizontal leg b, the thickness t and the inner-corner fillet radius;
  • built up: a table of rectangles, each solid or a hole, by width, height and centre (x, y). Solid rectangles must not overlap and each hole must lie inside one solid rectangle.
• Unit weight: only for the weight per length (steel 490 pcf, concrete 150 pcf).

For a rolled W-shape the fillet radius is kdes − tf from the AISC Shapes Database.

Results and checks of the section properties #

The x axis is horizontal and the y axis vertical, both through the centroid; the centroid is measured from the left and bottom edges of the bounding box.

The drawing (on screen, in the report and on the PDF / DXF drawing sheet) shows the section to scale with the solid parts and holes as integrated, the centroid, the centroidal x and y axes and, when Ixy ≠ 0, the principal axes 1 and 2, the overall dimensions with the centroid located from the left and bottom edges, and the thickness callouts of the shape. The sheet's design summary lists A, Ix, Iy, Sx, Zx, Zy, rx, ry and J (and I1, I2, θ when Ixy ≠ 0).

Limitations #

• Homogeneous sections only (no transformed sections of two materials; enter a transformed width yourself).
• J of open sections ignores the fillets, so it is a little below the AISC tabulated value of a rolled shape; J of a built-up section is the sum of its rectangles and does not apply to closed cells or to plates with holes.
• No warping constant Cw, shear centre, or effective (reduced) section for slender elements.
• Plastic moduli assume the whole section yields at one stress (Mp = Fy Z).
• Channels and the rolled flange-tip rounding are modelled with flat plates.

Verification examples #

Closed-form hand calculations and the AISC Shapes Database v16.0 (tests/test_verify_section_properties.py):

The W14X90 values are within 0.7 % of the database, which also includes the rolled flange-tip rounding.

Published section properties of the AISC Shapes Database v16.0, each shape built from its tabulated dimensions (W and WT: fillet radius kdes − tf; HSS: tdes with outside corner radius 2 tdes; L: heel fillet kdes − t):

\* Documented deviations: AISC-J-FILLETS (J of open shapes by Σ b t³/3 without the fillet term, 6.5 % low), AISC-C-SLOPE (channel flanges modelled parallel at the average thickness: weak-axis values differ), AISC-L-TOES (square angle toes: Iz 3 % high). Use the database values for those properties of rolled shapes.