Canadian Combined Loading — Axial + Bending Interaction per CSA S16 Clause 13.8

Complete reference for beam-column design per CSA S16-19 Clause 13.8. Covers interaction equations, moment magnification (U1/U2 factors), biaxial bending, interaction diagrams, and a step-by-step worked example for a W310x107 column-beam.

Quick access: Column design → | Beam design → | Column capacity calculator →

CSA S16 Interaction Equations

Per CSA S16-19 Clause 13.8.3, members subject to combined axial compression and bending must satisfy:

For Cf/Cr ≥ 0.2: Cf/Cr + (U_x × Mfx)/Mrx + (U_y × Mfy)/Mry ≤ 1.0

For Cf/Cr < 0.2: Cf/(2 × Cr) + (U_x × Mfx)/Mrx + (U_y × Mfy)/Mry ≤ 1.0

Where:

• Cf, Cr = factored axial load and axial resistance (Cr from Clause 13.3)
• Mfx, Mfy = factored moments about x and y axes
• Mrx, Mry = moment resistances about x and y axes (Mr from Clause 13.5-13.6)
• U_x, U_y = moment magnification factors

Moment Magnification

Per CSA S16 Clause 13.8.4, the moment magnification factor U accounts for P-delta effects:

U1 (Non-Sway Magnification — Braced Frame)

U1 = omega_1 / (1 - Cf/Ce) ≥ 1.0

Where:

• omega_1 = equivalent moment factor (= 1.0 for members without transverse loads; = 0.6-1.0 for other cases)
• Ce = pi^2 × E × I / (KL)^2 (Euler buckling load for the axis of bending)

U2 (Sway Magnification — Unbraced Frame)

U2 = 1 / (1 - sum(Cf) × Delta_s / (sum(H) × h_s))

Where:

• sum(Cf) = total factored load for the storey
• Delta_s = first-order lateral deflection
• sum(H) = total storey shear
• h_s = storey height

Alternatively, the simplified method: U2 = 1/(1 - sum(Cf)/sum(Ce_sway))

Omega_1 Factor

For M1/M2 = -1.0 (double curvature): omega_1 = 0.6 - 0.4(-1.0) = 1.0 For M1/M2 = 0 (one end zero): omega_1 = 0.6 For M1/M2 = +1.0 (single curvature): omega_1 = 0.6 - 0.4(1.0) = 0.2, but minimum is 0.4

Biaxial Bending

Per CSA S16 Clause 13.8.5, for biaxial bending (both Mx and My present):

Cf/Cr + (U_x × Mfx)/Mrx + (U_y × Mfy)/Mry ≤ 1.0

Additionally, the interaction must satisfy:

Mfx/Mrx' + Mfy/Mry' ≤ 1.0 (section capacity check, where Mr' = phi × Mp without LTB reduction)

Both equations must be satisfied for biaxial bending.

Interaction Diagram for W310x107 (350W)

The interaction is linear for Cf/Cr ≥ 0.2 and bilinear below 0.2. This is simpler than the AISC H1-1a parabolic interaction.

Section Classification Under Combined Loading

The interaction classification rules per CSA S16 Table 1 for combined axial and bending:

When Cf/Cr > 0.15, use axial compression limits even for the web. This can cause the classification to shift from Class 1 (flexure) to Class 3 (axial), requiring effective section properties.

Worked Example — Combined Loading Check

Given: W310×107 Grade 350W, KLx = 5000 mm (x-x, braced), KLy = 5000 mm (y-y). Factored loads: Cf = 1800 kN, Mfx = 180 kN·m (from eccentric gravity load, with transverse load on member), Mfy = 0 (no weak-axis moment).

Section Properties:

• A = 13,600 mm^2, Zx = 1,720 × 10^3 mm^3
• Ix = 230 × 10^6 mm^4, rx = 136 mm
• Sx = 1,510 × 10^3 mm^3, ry = 77.5 mm

Step 1 — Axial Resistance Cr: KL/ry = 5000/77.5 = 64.5 (weak axis governs — check both) KL/rx = 5000/136 = 36.8 lambda_y = 64.5 × sqrt(350/(pi^2 × 200000)) = 64.5 × 0.01332 = 0.859 Cr = 0.90 × 13,600 × 350 × (1 + 0.859^2.68)^(-0.746) = 2866 kN (from standard column design)

Step 2 — Moment Resistance Mrx: Flange b/2tf = 6.28 → Class 1 Web h/w = 24.3 → Class 1 (axial classification; but Cf/Cr = 1800/2866 = 0.63 > 0.15, so use axial limits) h/w = 24.3 ≤ 35.8 (Class 1 axial limit) → Class 1 overall Mrx = phi × Zx × Fy = 0.90 × 1,720 × 10^3 × 350 / 10^6 = 542 kN·m Check LTB: Lb = 5000 mm, ry = 77.5 mm Lp = 42.1 × 77.5 = 3263 mm Lb = 5000 > Lp → LTB may reduce Mr (assume braced at 3000 mm centres for full capacity)

Step 3 — Moment Magnification (U1, braced frame): Ce_x = pi^2 × E × Ix / (KLx)^2 = pi^2 × 200000 × 230e6 / 5000^2 = 18,160 kN Cf/Ce_x = 1800/18160 = 0.099 omega_1 = 1.0 (transverse load on member with unrestrained ends) U1 = 1.0/(1 - 0.099) = 1.11

Step 4 — Interaction Check: Cf/Cr = 1800/2866 = 0.63 ≥ 0.2, so: Cf/Cr + (U1 × Mfx)/Mrx = 0.63 + (1.11 × 180)/542 = 0.63 + 0.37 = 1.00

Result: W310×107 is adequate. Ratio = 1.00. The combined loading check is satisfied.

P-Delta Effects

Second-order (P-delta) effects are critical in combined loading:

Per CSA S16 Clause 8.4, second-order effects must be considered when:

• Delta_s/h_s > 0.0015 (sway frames)
• Cf/Cr > 0.15 (significant axial load)
• Any case where P-delta increases moments by more than 5%

Frequently Asked Questions

When does Cf/(2Cr) apply instead of Cf/Cr in the interaction equation? The equation Cf/(2Cr) + U×Mf/Mr ≤ 1.0 applies when Cf/Cr < 0.2 (low axial load relative to capacity). This bilinear transition provides a more accurate interaction curve — at very low axial loads, the axial term only contributes half, recognising that the member behaves more like a beam than a column.

How does moment magnification (U factor) work in CSA S16 beam-column design? U1 accounts for P-delta within the member: U1 = omega_1/(1 - Cf/Ce). When the axial load approaches the Euler buckling load Ce, the magnification becomes very large. U2 accounts for storey P-delta in sway frames. The magnified moment U×Mf is used directly in the interaction equation.

Do I need to check biaxial bending for columns? Yes, all columns with moments about both axes must satisfy the full interaction equation: Cf/Cr + Ux×Mfx/Mrx + Uy×Mfy/Mry ≤ 1.0. In many practical cases, the weak-axis moment Mfy is small (from eccentric beam reactions or wind), but biaxial effects can reduce capacity significantly when both moments are substantial.

What is the Cf/Cr threshold for using axial vs flexure classification limits? Per CSA S16 Table 1 and Clause 13.8.3, when Cf/Cr > 0.15, use axial compression limits for web classification. When Cf/Cr ≤ 0.15, use flexure limits. This is because a low axial load does not significantly affect the web buckling behaviour, allowing the more generous flexure limits.

Related Pages

• CSA S16 Column Design
• CSA S16 Beam Design
• CSA S16 Effective Length Factor K
• Canadian Compact Section Limits
• CSA S16 Cb (Omega_2) Factor
• Column Capacity Calculator
• Beam Capacity Calculator
• All Canadian References

This page is for educational reference. Combined loading per CSA S16-19 Clause 13.8. Verify interaction equations and moment magnification against CISC Handbook. Results are PRELIMINARY — NOT FOR CONSTRUCTION without independent PE/SE verification.