The Section Properties tool computes the geometric and structural properties of a cross-section. These properties are fundamental inputs for calculating bending stress, shear stress, deflection, and member stability. The tool supports standard rolled steel shapes as well as user-defined custom sections.

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For any selected or defined cross-section, the tool returns:

• A — Cross-sectional area (in² or mm²). Used for axial stress calculations and for computing the self-weight of a member.
• ȳ — Distance from the reference axis to the centroid of the section. For symmetric sections this is simply half the depth; for asymmetric sections it must be computed from the weighted areas of the constituent parts.
• I_x — Moment of inertia about the horizontal centroidal axis (in⁴ or mm⁴). Governs flexural stiffness and bending stress for loads applied in the vertical plane.
• I_y — Moment of inertia about the vertical centroidal axis. Governs flexural stiffness for loads applied horizontally and lateral-torsional buckling resistance.
• S_x — Elastic section modulus about the x-axis, equal to I_x / c, where c is the distance from the centroid to the extreme fiber. Used directly to compute bending stress: f_b = M / S_x.
• S_y — Elastic section modulus about the y-axis.
• r_x — Radius of gyration about the x-axis, equal to √(I_x / A). Used to compute the slenderness ratio KL/r_x for column buckling about the x-axis.
• r_y — Radius of gyration about the y-axis. The minimum radius of gyration governs the slenderness ratio for weak-axis column buckling.

Standard rolled steel sections from the AISC Steel Construction Manual are available, including:

• W-shapes (wide flange) — the most commonly used sections for beams and columns
• S-shapes (American Standard beams)
• C-shapes (American Standard channels)
• L-shapes (angles) — equal leg and unequal leg
• HSS (hollow structural sections) — rectangular and square
• Pipe sections — standard, extra-strong, and double extra-strong

In addition to the standard library, the tool accepts user-defined rectangular, T-shaped, and built-up sections. For a custom section, enter the dimensions of each sub-element; the tool computes the combined properties using the parallel-axis theorem.

Select a W8×31 from the standard section library. The AISC tabulated properties for this section are:

• A = 9.13 in²
• d = 8.00 in (total depth)
• b_f = 7.995 in (flange width)
• I_x = 110 in⁴
• S_x = 27.5 in³
• r_x = 3.47 in
• I_y = 37.1 in⁴
• S_y = 9.27 in³
• r_y = 2.02 in

Bending stress check. Suppose this beam carries a bending moment of M = 60 kip·ft. The maximum bending stress in the extreme fiber is:

f_b = M / S_x = (60 × 12) / 27.5 = 720 / 27.5 = 26.2 ksi

For a Fy = 50 ksi section (e.g., A992 steel), the allowable bending stress under AISC ASD is 0.66 × Fy = 33 ksi. The computed stress 26.2 ksi < 33 ksi ✓.

Column slenderness check. For a column of this section with an unbraced length of KL = 10 ft = 120 in, weak-axis governs:

(KL/r)_y = 120 / 2.02 = 59.4

This slenderness ratio is used to look up the allowable compressive stress from the AISC column tables.

Flexural stiffness (EI). For deflection calculations, the flexural rigidity is:

E × I_x = 29,000 × 110 = 3,190,000 kip·in²

When a standard rolled section is not suitable — for example, a built-up plate girder or a composite concrete-steel section — the tool allows entry of custom dimensions. The parallel-axis theorem is applied to combine the inertia of each element about the combined centroid:

I_total = Σ (I_component + A_component × d²)

where d is the distance from each component's centroid to the overall centroid. This is the standard approach for computing the section modulus of plate girders, box sections, and other built-up members.

• For doubly symmetric sections (most W-shapes and HSS), ȳ = d/2 and the top and bottom section moduli are equal. For singly symmetric sections (channels, T-shapes), the tool reports separate top and bottom values of S.
• When checking lateral-torsional buckling, the radius of gyration r_ts (effective radius for LTB) differs from r_y and is calculated using the warping constant C_w. Consult the AISC Specification Chapter F for the full procedure.
• Plastic section modulus Z_x is used for LRFD plastic moment capacity (φM_n = φ × Z_x × F_y). Tabulated Z values for standard sections are available in the AISC Manual.

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