Compressive Resistance — CSA S16 Clause 13.3

The factored compressive resistance is:

C_r = phi × A_g × F_y × (1 + lambda^(2n))^(-1/n) (CSA S16 Eq. 13.3.1)

Where lambda = sqrt(F_y / F_e) = KL/r × sqrt(F_y / (pi² × E)).

For a stub column (lambda ≤ 0.15, which corresponds to KL/r ≤ ~18 for 350W steel): C_r ≈ phi × A_g × F_y (the squash load — no buckling reduction).

Where:

• phi = 0.90 (structural steel)
• A_g = gross cross-sectional area
• F_y = minimum specified yield strength
• n = 1.34 for W-shapes (hot-rolled, CSA S16 Clause 13.3.1)

Effective Length — CSA S16 Clause 10.3

Canadian practice uses alignment charts or the direct analysis method. Common K values for braced frames:

For typical Canadian office tower with moment-connected beams providing rotational restraint: K = 0.8 is commonly used by Canadian engineers, per CISC Commentary.

Section Classification for Compression

Per CSA S16 Table 2:

For 350W steel (F_y = 350 MPa):

• Class 1 flange limit: 145/sqrt(350) = 7.75
• Class 1 web limit: 420/sqrt(350) = 22.5

Worked Example 1 — W-Shape Stub Column

Problem: W310x97 (350W) column between braced floors at 3.6 m storey height in a Toronto office tower. Factored axial load C_f = 2500 kN. K = 0.8. Check adequacy.

Section Properties (W310x97):

Step 1 — Section Classification:

Flange: b/t = (305/2)/15.4 = 9.90. For 350W: Class 2 limit = 170/sqrt(350) = 9.09. 9.90 > 9.09 → Class 3 (non-compact).

Web (uniform compression): h = d - 2t_f = 308 - 30.8 = 277.2 mm. h/w = 277.2/9.9 = 28.0. Class 3 limit = 670/sqrt(350) = 35.8. 28.0 < 35.8 → Class 3 governed by flange.

Class 3 section: use elastic section properties (S, not Z).

Step 2 — Slenderness:

KL = 0.8 × 3600 = 2880 mm. KL/r_y = 2880/77.2 = 37.3.

lambda = 37.3 × sqrt(350/(pi² × 200,000)) = 37.3 × sqrt(350/1,973,921) = 37.3 × 0.0133 = 0.496.

Step 3 — Compressive Resistance:

C_r = 0.90 × 12,300 × 350 × (1 + 0.496^(2×1.34))^(-1/1.34) / 1000 = 3,875 × (1 + 0.496^2.68)^(-0.746) = 3,875 × (1 + 0.139)^(-0.746) = 3,875 × 0.903 = 3,499 kN.

Step 4 — Check:

C_f / C_r = 2500 / 3499 = 0.714. OK (71% utilisation).

For a true stub column (KL/r < 18, lambda < 0.15), the reduction factor would be > 0.98. At KL/r = 37.3, the member is intermediate length, and the 10% reduction from squash load is consistent with the Perry-Robertson curve embedded in CSA S16 Eq. 13.3.1.

CSA S16 n-Factor and Column Curve

The CSA S16 column curve differs from the AISC 360 curve in one critical parameter: the exponent n. Clause 13.3.1 specifies n = 1.34 for W-shapes (hot-rolled), n = 1.34 for HSS (cold-formed, non-stress-relieved), and n = 2.24 for stress-relieved HSS. This exponent controls how sharply the curve transitions from the squash load plateau to the Euler buckling branch.

The lower n value (1.34 vs AISC's effective 2.0) means CSA S16 predicts slightly lower compressive resistance in the intermediate slenderness range (KL/r ≈ 40–80), which is conservative for Canadian practice. The CISC Handbook tabulates C_r for all standard sections across the full KL range — these tables are the standard Canadian design office reference and eliminate manual Eq. 13.3.1 computation.

Worked Example 2 — HSS Stub Column

Problem: An HSS 203×203×9.5 column (350W, CSA G40.21 Class C) carries a factored axial load C_f = 1600 kN between braced floors at 3.0 m storey height in a Vancouver office building. K = 0.8. Check as a stub column.

Section Properties (HSS 203×203×9.5):

Step 1 — Section Classification (CSA S16 Table 2):

Wall b/t = (203 − 3×9.5)/9.5 = 174.5/9.5 = 18.4. Class 1 limit for HSS compression = 420/sqrt(350) = 22.5. 18.4 < 22.5 → Class 1. The full cross-section is effective.

Step 2 — Slenderness:

KL = 0.8 × 3000 = 2400 mm. KL/r = 2400/78.5 = 30.6. lambda = 30.6 × sqrt(350/(pi² × 200,000)) = 30.6 × 0.0133 = 0.407.

Step 3 — Compressive Resistance (n = 1.34 for HSS):

C_r = 0.90 × 7,250 × 350 × (1 + 0.407^(2×1.34))^(-1/1.34) / 1000 = 2,284 × (1 + 0.407^2.68)^(-0.746) = 2,284 × (1 + 0.089)^(-0.746) = 2,284 × 0.937 = 2,140 kN.

Step 4 — Check:

C_f / C_r = 1600 / 2140 = 0.748. OK (75% utilisation).

The HSS section is more efficient than an equivalent W-shape for pure compression due to the uniform radius of gyration about both axes — there is no weak axis to govern. For the same mass per metre, an HSS column typically provides 10–15% higher compressive resistance than a W-shape in the stub and intermediate slenderness range.

Stub Column vs Slender Column — When the Distinction Matters

The distinction between stub and slender columns is a continuous spectrum governed by the non-dimensional slenderness lambda. For practical Canadian design, three regimes are recognised:

Most building columns in Canadian practice fall in the intermediate range. True stub columns are relatively rare outside low-rise construction where brace points are closely spaced. The term "stub column" is most useful as a teaching concept — it establishes the upper-bound capacity that no column can exceed regardless of how closely it is braced.

Practical Design Considerations

Base plate interaction: A stub column delivering its full squash load demands a base plate sized for concrete bearing at phi_c × 0.85 × f'_c × sqrt(A2/A1). For a 350W W310×97 delivering C_r = 3,499 kN, a 450×450 mm base plate on C30 concrete is required. Increasing the stub column capacity without checking the base plate is a common error in Canadian design offices.

Splice design: Stub columns between floors typically require a bearing splice (machined ends) or a bolted web splice. CSA S16 Clause 22 requires splices to develop 50% of the column capacity in shear for braced frames. For a 3,499 kN stub column, the splice must resist 1,750 kN in bearing — a substantial connection that often governs the splice plate thickness and bolt count.

Brace force accumulation: In Canadian braced frames, the bracing system accumulates axial load as it descends the building. A stub column at ground level may carry compression from 10+ storeys above. The "stub" assumption must remain valid at all levels — if brace forces increase the effective length by attracting load through the bracing diagonals, the column may transition from stub to intermediate behaviour. The CISC Commentary recommends verifying KL/r at each floor level, not just at the column mid-height.

Fire resistance: Canadian building codes often require 1–2 hour fire resistance ratings for columns. Stub columns with higher utilisation (0.90+) may require additional fire protection thickness compared to slender columns at lower utilisation. The CISC fire design guide provides section-specific fire resistance tables for intumescent coatings and board systems.

FAQ

What is the difference between a stub column and a short column?

In Canadian practice, the terms are synonymous — both refer to a compression member where buckling does not reduce the cross-section capacity. CSA S16 does not formally define "stub column" — the distinction is pedagogical. The code uses the continuous column curve (Eq. 13.3.1) which naturally captures stub behaviour when KL/r is small.

Does CSA S16 permit using the simple squash load phi × A_g × F_y for stub columns?

The code requires Eq. 13.3.1 for all compression members regardless of slenderness. However, when lambda ≤ 0.15, the equation reduces to within 2% of the direct squash load. Canadian engineers often use phi × A_g × F_y as a quick upper-bound check, then refine with Eq. 13.3.1 if utilisation exceeds 0.90.

What happens if the column section is Class 4 (slender elements)?

Class 4 sections require the use of effective area A_eff per CSA S16 Clause 13.3.5, replacing A_g in Eq. 13.3.1. The effective area accounts for local buckling of individual plate elements. For Canadian W-shapes, few rolled sections are Class 4 in compression, but slender HSS walls and welded built-up sections commonly are.

How does the CISC Handbook differ from raw code equations?

The CISC Handbook tabulates C_r values for all standard Canadian sections at KL values from 0 to 12,000 mm. These tables incorporate the full CSA S16 Eq. 13.3.1 with n = 1.34 for W-shapes. Using the handbook tables is faster and less error-prone than manual computation — they are the standard Canadian design office reference.

When should I use 350WT steel instead of 350W for stub columns?

350WT (notch-tough) steel is required when the column is subject to dynamic loading, seismic ductility demands (CSA S16 Clause 27), or when the minimum service temperature drops below -20°C per NBCC requirements. For typical enclosed building columns in Toronto or Vancouver, 350W is acceptable. For exposed columns in Edmonton or Winnipeg, 350WT is typically specified.

Related Pages

• CSA S16 Column Design — Full Worked Example
• CSA S16 Column Buckling — Euler & CSA Curves
• CSA S16 Beam Design — Flexure & LTB
• CSA S16 Base Plate Design — Canadian Guide
• Canadian Steel Grades — G40.21 350W & 350WT
• CSA S16 Load Combinations — NBCC ULS & SLS
• Canadian Compact Section Limits — CSA S16 Table 2
• CSA HSS Column Design — Hollow Structural Sections
• CSA Column Splice Design — Clause 22
• CSA Anchor Bolt Design — Base Plate Connections
• Column Capacity Calculator — Free Tool

Educational reference only. Verify compressive resistance against CSA S16:24 and the current CISC Handbook of Steel Construction. Stub column capacity assumes adequate bracing at both ends. For seismic applications, additional ductility requirements per CSA S16 Clause 27 may apply. Results are PRELIMINARY — NOT FOR CONSTRUCTION without independent P.Eng. verification.