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 and ICR 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 (in^3 per inch of throat).
• 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).

Weld group property formulas

Rectangular weld group (width b, height d)

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

Two parallel vertical welds (height d, separation b)

Single horizontal line weld (length L)

L-shape weld (vertical d, horizontal b)

Ip values for common weld group sizes

Rectangular weld groups (Ip in in^3)

Ip = d^2(3b + d)/6 + b^2(3d + b)/6. Higher Ip = lower peak stress for a given torque.

Sw values for out-of-plane bending (in^2)

Sw = d(3b + d)/3.

Fillet weld capacity per unit length (E70XX, FEXX = 70 ksi)

phi*Rn = 0.75 * 0.60 _ 70 _ (weld_size * 0.707) per inch of weld.

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 10 in tall + one horizontal weld 6 in at bottom). Fillet weld size = 5/16 in (throat = 0.221 in). E70 electrode. Applied load P = 20 kips at eccentricity e = 8 in from the weld group centroid. Load acts vertically.

Step 1: Weld group properties. Aw = 2(10) + 6 = 26 in. y_bar = 10^2/(2*10 + 6) = 3.85 in from bottom.

Step 2: Ip (approximate for C-shape): Ix = 10^2(410 + 36)/12 - 3.85^2 * 26 = 483.3 - 385.1 = 98.2 in^3. Iy = 2*10*(6/2)^2 + 6^3/12 = 180 + 18 = 198 in^3. Ip = 98.2 + 198 = 296.2 in^3.

Step 3: Direct shear: f_v = 20/26 = 0.77 kip/in (uniform). Torque T = 20 x 8 = 160 kip-in.

Step 4: Maximum torsional shear at the farthest point (top of left vertical): r_max = sqrt((6)^2 + (10 - 3.85)^2) = sqrt(36 + 37.8) = sqrt(73.8) = 8.59 in. f_torsion = 160 x 8.59 / 296.2 = 4.64 kip/in.

Step 5: Vector sum at critical point. f_r = sqrt((0.77 + 4.64cos(40))^2 + (4.64sin(40))^2) = sqrt((0.77 + 3.56)^2 + 2.98^2) = sqrt(18.8 + 8.9) = sqrt(27.7) = 5.26 kip/in.

Step 6: Capacity check. phi*Rn = 6.95 kip/in. 5.26 < 6.95. Utilization = 5.26/6.95 = 0.76 OK.

Elastic method vs ICR method

The elastic method assumes the weld is rigid and fully elastic. The ICR method accounts for ductile load redistribution among weld segments. AISC Manual Tables 8-4 through 8-11 provide C-coefficients for standard geometries.

ICR C-coefficients for rectangular weld groups

P*capacity = C * phi _ 0.60 _ FEXX _ throat * d (for eccentricity e). Higher C = higher capacity.

Code comparison

Cross-code capacity comparison: 1/4" fillet weld, E70/Fuw=490 MPa

CSA uses phi = 0.67 vs AISC's 0.60 * phi = 0.75, giving CSA approximately 6% higher capacity for the same weld.

AISC Weld Group Tables 8-4 to 8-11: C-Coefficients

AISC Steel Construction Manual Tables 8-4 through 8-11 provide C-coefficients for common weld group configurations using the Instantaneous Center of Rotation (ICR) method. These tables eliminate the need for iterative ICR calculations and give the coefficient C directly, from which the total weld group capacity is:

phi * Rn = phi * C * C1 * D * l     (kips)

Where:
  C  = Coefficient from AISC Tables 8-4 through 8-11
  C1 = Electrode strength coefficient (1.0 for E70XX)
  D  = Weld size in sixteenths of an inch (e.g., 5 for 5/16 in)
  l  = Weld length (inches)
  phi = 0.75 (LRFD)

Available Weld Group Configurations

The parameter a = eccentricity (ex) / weld length (l). The parameter kl = distance between weld lines (k) / weld length (l). Both are dimensionless ratios used to look up C in the appropriate table.

C-Coefficient Reference Table (Selected Values)

The following table shows representative C values from AISC Table 8-4 (double angle, in-plane eccentric shear, E70 electrode). Values of C vary with the ratio a/l (eccentricity normalized to weld length).

Higher kl means welds are farther apart, which increases Ip and improves capacity. Lower a/l means less eccentricity, which also increases capacity.

ICR Method: How It Works

The Instantaneous Center of Rotation method accounts for the nonlinear, ductile behavior of weld groups under eccentric loading:

1. Find the ICR location: The instantaneous center is the point about which the weld group rotates at the instant of failure. It is not at the centroid of the weld group.
2. Compute deformation at each weld element: Each weld element has a deformation that depends on its distance from the ICR and its angle to the line connecting it to the ICR.
3. Apply the load-deformation relationship: The strength of each weld element follows the formula R = Rult * [1 - e^(-0.417 * delta/delta_u)]^0.382 (refined from AISC research).
4. Iterate until equilibrium: The ICR location is adjusted iteratively until the sum of all weld element forces equals the applied load and all moments balance.
5. Read C directly: The AISC tables provide pre-computed C values, so iteration is not needed in practice.

The elastic method (simpler) assumes all weld elements have equal stress. It is conservative by 30-50% compared to the ICR method. Use elastic for quick checks and ICR for final design.

Worked Example: Eccentrically Loaded Weld Group

Problem: A 5/16 in fillet weld (E70XX) connects a gusset plate to a beam web using a double-line weld pattern. Each weld line is 10 in long, spaced 5 in apart (kl = 5/10 = 0.5). The applied shear is P = 40 kips with an eccentricity ex = 4 in (a/l = 4/10 = 0.4).

Step 1: Determine C coefficient

From AISC Table 8-4 with kl = 0.5 and a/l = 0.4 (interpolating between 0.3 and 0.5):

• C(0.3, 0.5) = 1.33
• C(0.5, 0.5) = 0.96
• C(0.4, 0.5) ≈ (1.33 + 0.96) / 2 = 1.15 (approximate)

Step 2: Check weld capacity

phi * Rn = phi * C * C1 * D * l
         = 0.75 * 1.15 * 1.0 * 5 * 10
         = 0.75 * 57.5 = 43.1 kips

Step 3: Compare to demand

phi * Rn = 43.1 kips > Pu = 40 kips  --> OK

The 5/16 in fillet weld, 10 in long on each side, is adequate for the 40 kip load at 4 in eccentricity. The utilization ratio is 40/43.1 = 0.93 (93%). If the eccentricity were 6 in (a/l = 0.6), C would drop to approximately 0.85, and capacity would be 0.75 _ 0.85 _ 5 * 10 = 31.9 kips -- inadequate, requiring a larger weld or longer weld length.

Common mistakes

1. Using the elastic method when the ICR method is available. The elastic method is conservative by 30-50%. For final design, use AISC Tables 8-4 through 8-11 C-coefficients.

2. Forgetting to include the weld return at corners. A C-shape weld that wraps around the top corners by 2x weld size prevents stress concentration at the corner.

3. Mixing up Ix and Ip. Ix is for out-of-plane bending. Ip is for in-plane eccentric shear. Using Ix for torsion ignores the Iy component and under-predicts stress.

4. Not checking base metal capacity. The weld may be adequate, but base metal shear rupture along the weld line may govern (AISC J4.2: 0.60 _ Fu _ Agv).

5. Ignoring directional strength increase. AISC Eq. J2-5 gives a strength increase of 1.0 + 0.50*sin^1.5(theta) for transversely loaded fillet welds. This can add up to 50% capacity for welds loaded at 90 degrees.

Frequently asked questions

What is the difference between the elastic and ICR methods? The elastic method assumes rigid, elastic behavior. The Instantaneous Center of Rotation (ICR) method accounts for ductile redistribution. ICR gives 30-50% higher capacity. Use elastic for quick checks, ICR for final design.

What weld group shape has the highest Ip? A full rectangular box (welded on all four sides). For a given weld length, spreading the weld farther from the centroid increases Ip and reduces peak stress.

Do I need to check weld group properties for simple shear connections? No. Simple shear connections (single plate, double angle, shear tab) have minimal eccentricity. The weld is designed for direct shear only, using phi _ 0.60 _ FEXX _ throat _ length.

What is the minimum weld size for a given connection? Per AISC Table J2.4, the minimum fillet weld size depends on the thinner part: 1/8" for material up to 3/16" thick, 3/16" for 3/16" to 1/4", 1/4" for 1/4" to 1/2", and so on.

Can I use the ICR method for C-shape weld groups? Yes, but AISC only tabulates C-coefficients for specific geometries. For other configurations, use the elastic method or derive the ICR solution iteratively.

How do I account for weld throat size? Calculate forces per unit length of weld (kip/in), then divide by the effective throat to get stress. For fillet welds, throat = weld size _ cos(45 deg) = weld size _ 0.707.

Run this calculation

• Welded Connections Calculator
• Section Properties Database

Related references

• Minimum Weld Size
• Weld Electrodes
• Weld Joint Types
• Connection Checks
• Eccentric Connection
• Steel Fasteners
• Bolt Bearing & Tearout
• 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.

Design Resources

Calculator tools

• Weld Capacity Calculator
• Weld Symbol Generator

Design guides

• Weld Design Checklist