Weld Group Properties — Section Modulus, Polar Moment & Elastic Method

Weld group geometric properties: Ix, Iy, Ip (polar moment), Sw (section modulus per unit length), for common weld configurations. Elastic vector method for eccentric shear on weld groups.

Why weld group properties matter

When a weld group resists moment or eccentric shear, the force is not uniformly distributed along the weld length. Instead, the force per unit length varies with position — welds farther from the group centroid carry more force. Calculating the peak weld force requires the geometric properties of the weld group, analogous to how Ix and Sx are used for beam bending.

Weld group properties are calculated treating the weld as a line (zero width) with unit throat thickness. The actual weld stress is obtained by dividing the force per unit length by the effective throat dimension.

Key properties (per unit throat)

• Aw = total weld length (sum of all weld segments).
• x_bar, y_bar = centroid of the weld group.
• Ix = moment of inertia about horizontal centroidal axis (mm^3 per mm of throat, i.e., mm^3).
• Iy = moment of inertia about vertical centroidal axis.
• Ip = polar moment of inertia = Ix + Iy (used for in-plane eccentric shear).
• Sw = section modulus = Ix / c_max (used for out-of-plane bending).

Common weld group property formulas

Two parallel vertical welds (height d, separation b)

Aw = 2d. Ix = d^3 / 6 (per line, total = 2 x d^3/12 = d^3/6). Iy = b^2 x d / 2. Ip = d^3/6 + b^2 x d/2. Sw_x = d^2/3.

C-shape weld (two verticals d, one horizontal b at bottom)

Aw = 2d + b. Centroid from bottom: y_bar = d^2 / (2d + b). Ix = 2 x d^3/12 + b x y_bar^2 + correction terms. These are tabulated in AISC Manual Table 8-8.

Rectangular weld (two verticals d, two horizontals b)

Aw = 2d + 2b. Ix = d^2(3b + d) / 6. Iy = b^2(3d + b) / 6. Ip = Ix + Iy. Sw_x = d(3b + d) / 3.

Single line weld (length L)

Aw = L. Ix = L^3 / 12 (about centroid). Sw = L^2 / 6.

Worked example — eccentric shear on a C-shape weld group

A bracket welded to a column flange with a C-shape weld (two vertical welds 250 mm tall + one horizontal weld 150 mm at bottom). Fillet weld size = 8 mm (throat = 8 x 0.707 = 5.66 mm). E70 electrode (FEXX = 482 MPa). Applied load P = 90 kN acting at eccentricity e = 200 mm from the weld group centroid. Load acts vertically.

Step 1: Find weld group centroid. x_bar from the left vertical = b^2 / (2 x 2d + 2b) — but for a C-shape (open on top), x_bar = b^2 / (2(2d + b)) = 150^2 / (2(500 + 150)) = 22,500 / 1,300 = 17.3 mm from the vertical welds.

Step 2: Weld group properties. Aw = 2(250) + 150 = 650 mm.

Ip (about centroid): For the C-shape, AISC Table 8-8 gives: Ix = d^2(2d + b)/6 - ... Alternatively compute component by component. Two vertical welds: each contributes Ix_v = 250^3/12 = 1,302,083 mm^3 per line about its own centroid, Iy_v about group centroid = 250 x (150 - 17.3)^2/2 and 250 x 17.3^2/2 (left and right verticals are both at x = 0 from the web, but one at x = (150 - 17.3)... Let me use the published formula.

Using AISC Manual Table 8-8 for C-shape: Ip = (8d^3 + 6bd^2 + b^3) / 12 - b^4/(4(2d+b)) for the standard orientation. With d = 250 mm, b = 150 mm:

Ip = (8 x 250^3 + 6 x 150 x 250^2 + 150^3) / 12 - 150^4 / (4 x 650) = (125,000,000 + 56,250,000 + 3,375,000) / 12 - 506,250,000 / 2,600 = 15,385,417 - 194,712 = 15,190,705 mm^3.

Step 3: Force components on critical weld. The direct shear is uniform: f_v = P / Aw = 90,000 / 650 = 138 N/mm. The torsional shear from eccentricity: torque T = P x e = 90,000 x 200 = 18,000,000 N-mm. The maximum torsional shear occurs at the point farthest from the centroid. The farthest corner is at the top of the vertical weld opposite the horizontal: r_max = sqrt((150 - 17.3)^2 + 125^2) = sqrt(17,617 + 15,625) = sqrt(33,242) = 182 mm.

f_torsion = T x r_max / Ip = 18,000,000 x 182 / 15,190,705 = 216 N/mm.

Step 4: Vector sum. The direct shear acts vertically downward. The torsional shear acts perpendicular to r_max. The resultant at the critical point: f_r = sqrt((f_v + f_t_y)^2 + f_t_x^2). Breaking the torsional component: angle of r from horizontal = arctan(125/(150-17.3)) = arctan(0.942) = 43.3 degrees. f_t_x = 216 x sin(43.3) = 148 N/mm. f_t_y = 216 x cos(43.3) = 157 N/mm.

f_r = sqrt((138 + 157)^2 + 148^2) = sqrt(295^2 + 148^2) = sqrt(87,025 + 21,904) = sqrt(108,929) = 330 N/mm.

Step 5: Weld capacity. phi*Rn per mm = 0.75 x 0.60 x FEXX x throat = 0.75 x 0.60 x 482 x 5.66 = 1,229 N/mm. Since 330 < 1,229, the weld group is adequate with utilization ratio = 330/1,229 = 0.27.

Code comparison

AISC and CSA both offer the instantaneous center of rotation (ICR) method as an alternative to the elastic method. ICR gives 15-40 percent higher capacity because it accounts for load redistribution among weld segments. AISC Manual Tables 8-4 through 8-11 provide C-coefficients for standard geometries.

Common pitfalls

• Using the elastic method when the ICR method is available. The elastic method is conservative — it assumes the weld is rigid and fully elastic. For eccentric shear, the ICR method (AISC Tables 8-4 to 8-11) gives 15-40 percent higher capacity. Use elastic method for quick checks, ICR for final design.
• Forgetting to include the weld return at corners. A C-shape weld that wraps around the top corners by 2 x weld size adds negligible length but prevents stress concentration at the corner. More importantly, the return prevents the weld from terminating at the corner — a location of high stress concentration.
• Mixing up Ix and Ip. Ix is used for out-of-plane bending (moment about the weld group axis). Ip is used for in-plane eccentric shear (torque about the centroid). Using Ix for an in-plane torsion problem under-predicts the weld stress by ignoring the Iy component.
• Not checking base metal capacity. The weld may be adequate, but the base metal at the fusion face (plate shear rupture along the weld line) may govern. AISC J4.2 requires checking shear rupture of the base metal = 0.60 x Fu x Agv.

Run this calculation

• Welded Connections Calculator
• Section Properties Database

Related references

• Minimum Weld Size
• Weld Electrodes
• Weld Joint Types
• Connection Checks
• Eccentric Connection
• How to Verify Calculations

Disclaimer

This page is for educational and reference use only. It does not constitute professional engineering advice. All design values must be verified against the applicable standard and project specification before use. The site operator disclaims liability for any loss arising from the use of this information.