OpenSpecEngineering

Gives the area, moment of inertia, section modulus, radius of gyration and related properties of a cross-section, drawn to scale. Choose a rectangle (with dressed lumber sizes [3]), a circle, a tube (with steel pipe sizes [2]) or a triangle, or look up any AISC W, M, S, HP, C, MC, L, WT, HSS or pipe shape [4]. Structural and mechanical engineers use it to size beams and columns and to check stress and deflection.

Equations

For the drawn shapes, about the centroidal axis parallel to the base [1]:

Rectangle: I = bh³/12    Triangle: I = bh³/36

Circle: I = πr_o⁴/4    Tube: I = π(r_o⁴ − r_i⁴)/4

S = I/c    r = √(I/A)

b is the width, h the height, r_o and r_i the outer and inner radii, A the area and c the distance from the centroid to the outer fiber. The bending stress there is f = M/S, and column slenderness is KL/r. Solve for Required Dimension turns the I equation around to find the height, width or radius that gives a target I.

An AISC shape’s properties are the Manual’s tabulated values, including the plastic modulus Z (plastic moment M_p = F_y·Z), torsional constant J and warping constant C_w. The key properties show first, and the rest fold under “All properties”. Each value is given in US and metric units; the switch above the table chooses which.

Limits and common mistakes

• The drawn shapes are about one axis, parallel to the base. A rectangle on edge has I = hb³/12, so enter b and h as the section is loaded.
• A triangle has two section moduli. The page gives the one to the apex, which has the larger stress.
• Lumber sizes are dressed, not nominal: a 2×10 is 1.5 × 9.25 in.
• HSS properties use the design wall t_des = 0.93 t_nom. An angle’s x and y axes run along its legs; the least I is about the principal z-axis.
• Z gives the bending strength only for a compact, laterally braced section. Check the slenderness ratios and lateral-torsional buckling separately.

Worked example

A 24 ft simply supported floor beam of A992 steel (F_y = 50 ksi), braced by the deck, carries 0.4 kip/ft dead and 0.6 kip/ft live load. The factored load is 1.2 × 0.4 + 1.6 × 0.6 = 1.44 kip/ft, so M_u = 1.44 × 24²/8 = 103.7 kip·ft. The beam needs Z_x ≥ 103.7 × 12/(0.9 × 50) = 27.6 in³. Check a W12×26.

1. Choose Cross-Section Shape W-Shape.
2. Choose W-Shape (AISC Manual Table 1-1) W12×26.
3. Press Solve. The key properties include: Moment of inertia, strong axis (Ix): 204 in⁴ / 84.91 ×10⁶ mm⁴
   Plastic section modulus, strong axis (Zx): 37.2 in³ / 609.6 ×10³ mm³

Z_x = 37.2 in³ exceeds 27.6, and φM_p = 0.9 × 50 × 37.2/12 = 139.5 kip·ft. The live-load deflection is 5wL⁴/(384EI) = 0.76 in, within L/360 = 0.80 in. Check Z by hand from the flanges and web: b_ft_f(d − t_f) + t_w(d − 2t_f)²/4 = 36.7 in³. The fillets make up the rest.