CSA S16:24 Column Design Framework — Clause 13.3

The factored compressive resistance Cr is given by:

Cr = phi x A x Fy x (1 + lambda^(2 x n))^(-1/n)

Where:

• phi = 0.90 for steel in compression
• A = gross cross-sectional area (Class 1-3) or effective area Ae (Class 4)
• Fy = minimum specified yield strength (MPa)
• lambda = KL/r x sqrt(Fy / (pi^2 x E)) — non-dimensional slenderness parameter
• n = 1.34 for W-shapes (hot-rolled, buckling about strong or weak axis)
• n = 1.34 for HSS (hot-formed or cold-formed per CSA G40.20 Class C)
• n = 1.34 for welded HSS per CSA G40.20 Class H
• n = 1.34 for angles

This is the SSRC column curve adopted by CSA S16, identical in form to AISC 360 Chapter E. The parameter n = 1.34 corresponds to an initial out-of-straightness of L/1000 — appropriate for hot-rolled sections. For fabricated members, CSA S16 allows n = 1.00 (higher imperfection, corresponding to L/500).

Key Parameter: lambda — Non-Dimensional Slenderness

lambda = (KL/r) x sqrt(Fy / (pi^2 x E))

For Fy = 350 MPa, E = 200,000 MPa: sqrt(Fy / (pi^2 x E)) = sqrt(350 / (pi^2 x 200000)) = 0.01332

So for 350W steel: lambda = 0.01332 x (KL/r)

Worked Example 1: W250x73 Interior Column

Problem: An interior column in a 4-storey office building, height 4.0 m floor-to-floor, pinned-pinned (K = 1.0). Design axial load Cf = 1,200 kN. Steel grade: CSA G40.21 350W. Check W250x73.

Section Properties (W250x73, 350W, CISC Handbook):

• A = 9,280 mm^2
• d = 253 mm, b = 254 mm, t = 14.2 mm, w = 8.6 mm
• rx = 111 mm, ry = 64.8 mm
• Section class: Class 2 (flange), Class 1 (web) — Class 2 overall

Step 1 — Effective length: Kx = 1.0, Ky = 1.0. L = 4,000 mm. KLx/rx = 1.0 x 4000 / 111 = 36.0 (strong axis) KLy/ry = 1.0 x 4000 / 64.8 = 61.7 (weak axis — governs)

Step 2 — Non-dimensional slenderness: lambda = 61.7 x 0.01332 = 0.822

Step 3 — Compressive resistance: Cr = 0.90 x 9280 x 350 x (1 + 0.822^(2 x 1.34))^(-1/1.34) / 1000

lambda^(2n) = 0.822^2.68 = 0.822^2.68. Let's compute: ln 0.822 = -0.196, x 2.68 = -0.525, exp = 0.592

(1 + 0.592)^(-1/1.34) = 1.592^(-0.7463) = exp(-0.7463 x ln 1.592) = exp(-0.7463 x 0.465) = exp(-0.347) = 0.707

Cr = 0.90 x 9280 x 350 x 0.707 / 1000 = 2,067 kN

Utilization: Cf / Cr = 1200 / 2067 = 0.58. OK with substantial reserve.

The W250x73 is more than adequate for this load. For 1200 kN, a W250x58 (A = 7,400 mm^2, ry = 63.4 mm, Cr ~ 1,640 kN) or W200x59 (A = 7,560 mm^2, Cr ~ 1,680 kN) would also work. The lighter section saves approximately 15 kg/m.

Worked Example 2: HSS 127x127x6.4 Brace

Problem: Diagonal brace in a concentrically braced frame. Length = 5,200 mm, K = 1.0 (pinned ends). Compressive load Cf = 185 kN. Try HSS 127x127x6.4, CSA G40.21 350W Class C (cold-formed).

Section Properties (HSS 127x127x6.4):

• A = 2,930 mm^2
• r = 49.2 mm (both axes, square)
• t = 6.35 mm, b/t = (127 - 3 x 6.35) / 6.35 = 16.9

Step 1 — Section classification (CSA S16:24 Table 1, HSS): Flange b/t limit (Class 1): 335 / sqrt(Fy) = 335 / sqrt(350) = 17.9 16.9 < 17.9 — Class 1 section. Full area effective.

Step 2 — Slenderness: KL/r = 1.0 x 5200 / 49.2 = 105.7 lambda = 105.7 x 0.01332 = 1.408

Step 3 — Compressive resistance: lambda^(2n) = 1.408^2.68. ln 1.408 = 0.342, x 2.68 = 0.917, exp = 2.502

(1 + 2.502)^(-1/1.34) = 3.502^(-0.7463) = exp(-0.7463 x ln 3.502) = exp(-0.7463 x 1.253) = exp(-0.935) = 0.393

Cr = 0.90 x 2930 x 350 x 0.393 / 1000 = 362.7 kN

Utilization: 185 / 362.7 = 0.51. OK with substantial reserve.

The HSS 127x127x6.4 has significant reserve — a HSS 127x127x4.8 (A = 2,260 mm^2, r = 49.7 mm, Cr ~ 280 kN) would also work and saves 30% weight.

Worked Example 3: W310x158 Heavy Column (6-storey)

Problem: Ground-floor column in 6-storey moment frame. Height 4,500 mm. Base fixed, top rotationally restrained (K = 0.80 braced frame). Cf = 5,800 kN. Fy = 350 MPa (50 mm flange, thickness-reduced to 345 MPa per CSA G40.21).

Section Properties (W310x158, CISC Handbook):

• A = 20,100 mm^2
• rx = 139 mm, ry = 79.0 mm
• Flange: b/2t = 310/(2 x 25.0) = 6.20
• Web: h/w = 38.0
• Class 1 in both flange and web — full plastic design permitted.

Step 1 — Fy thickness reduction: Flange 25 mm < 40 mm: no reduction for 350W. Use Fy = 350 MPa.

Step 2 — Effective length (braced frame, K per CSA S16 Annex E): Weak axis governs: KLy/ry = 0.80 x 4500 / 79.0 = 45.6

Step 3 — Non-dimensional slenderness: lambda = 45.6 x 0.01332 = 0.607

Step 4 — Compressive resistance: lambda^(2n) = 0.607^2.68. ln 0.607 = -0.499, x 2.68 = -1.338, exp = 0.262 (1 + 0.262)^(-0.7463) = 1.262^(-0.7463) = 0.839

Cr = 0.90 x 20100 x 350 x 0.839 / 1000 = 5,314 kN

Utilization: 5800 / 5314 = 1.09 — EXCEEDS capacity!

Options: (a) W310x179 (A = 22,800 mm^2, Cr ~ 6,020 kN), (b) W360x162 (A = 20,600 mm^2, ry = 80.2 mm, Cr ~ 5,460 kN — still marginal), or (c) W310x158 with K = 0.65 (fixed base + stiff beam connections, Cr ~ 5,770 kN). Option (c) requires confirming that the beam-to-column connections provide adequate rotational restraint per Annex E.

Effective Length Factors — CSA S16:24 Annex E

CSA S16 provides two methods for determining K:

Alignment chart (Annex E, Figure E.1): For braced frames, K ranges from 0.5 (fully fixed ends) to 1.0 (pinned). The G-factor ratio at each end determines K.

Simplified K values (Clause 13.3.1): For preliminary design of typical braced frames:

• Interior columns, beams connected to both flanges: K = 0.90
• Exterior columns, beams connected to one flange: K = 1.0
• Corner columns, beams one way: K = 1.0
• Braced frame diagonals: K = 1.0

For unbraced (sway) frames, K > 1.0 — governed by the sidesway buckling mode.

Interaction with Flexure (Clause 13.8)

Columns in moment frames must be checked for combined axial compression and bending. CSA S16:24 Clause 13.8.2 provides the interaction equation:

**Cf/Cr + 0.85 x U1x x Mfx / Mrx + 0.60 x U1y x Mfy / Mry <= 1.0** (for Cf/Cr >= 0.20)

Where U1x = omega_1 / (1 - Cf/Cex) is the moment amplification factor accounting for P-delta effects. Cex = pi^2 x E x Ix / (Kx x Lx)^2 is the Euler buckling load.

For Cf/Cr < 0.20, a simpler equation applies: Cf/(2 x Cr) + Mfx/Mrx + Mfy/Mry <= 1.0.

Frequently Asked Questions

How does the CSA column curve compare to AISC 360? CSA S16:24 uses the identical SSRC column curve (n = 1.34) as AISC 360-22 Chapter E for hot-rolled W-shapes. The only difference is phi: CSA uses 0.90, AISC uses 0.90 for compression — identical. A column designed to CSA S16 will have the same nominal resistance as one designed to AISC 360, but CSA specifies different load factors (NBCC vs ASCE 7).

When does the column buckling curve actually govern over yielding? The column curve (Cr < A x Fy) governs when lambda exceeds approximately 0.15, corresponding to KL/r > 11 for 350W steel. In practice, all columns longer than 300 mm will be governed by the column curve rather than pure crushing. The transition from inelastic to elastic buckling occurs around lambda = 1.5 (KL/r ~ 113 for 350W).

What reduction applies for Class 4 (slender) columns? For Class 4 sections, use Ae (effective area) instead of A in the Cr formula. Ae accounts for local buckling of slender elements (flanges, webs). Additionally, the column curve uses the gross area for slenderness calculation but the effective area for capacity. CISC Handbook Part 4 provides effective width formulas per CSA S16 Annex H.

This page is for educational reference. Column design per CSA S16:24 Clause 13.3. Verify column capacities against the current CISC Handbook Column Tables. All structural designs must be independently verified by a licensed Professional Engineer. Results are PRELIMINARY — NOT FOR CONSTRUCTION.