Canadian Beam Design — CSA S16-19 Clause 13 Flexure and LTB

Complete reference for flexural design of steel beams to CSA S16-19. Covers moment resistance for Class 1-4 sections, lateral-torsional buckling per Clause 13.6 with the omega_2 equivalent moment factor, shear resistance for unstiffened webs, web crippling under concentrated loads, and deflection criteria per NBCC 2020.

Related pages: CSA Lateral-Torsional Buckling Guide | Canadian Compact Section Limits | Canadian Beam Sizes | Beam Capacity Calculator

CSA S16 Flexural Design Framework

CSA S16-19 Clause 13.5 provides the factored moment resistance Mr based on section class:

Class 1 and 2 (plastic design): Mr = phi x Z x Fy — full plastic moment available Class 3 (elastic design): Mr = phi x S x Fy — yield moment at extreme fibre Class 4 (slender): Mr = phi x Se x Fy — effective section accounting for local buckling

Where phi = 0.90 for all steel members in flexure, Z is the plastic section modulus, S the elastic section modulus, Se the effective elastic modulus for Class 4, and Fy is the minimum yield strength per CSA G40.21, typically thickness-adjusted.

The classification determines not only the moment resistance but also the rotation capacity. Class 1 sections can form plastic hinges with sufficient rotation for plastic analysis; Class 2 can reach the plastic moment with limited rotation; Class 3 can reach yield only; Class 4 buckles locally before yielding.

Section Classification — CSA S16 Table 1

For CSA G40.21 350W (Fy = 350 MPa): flange limit = 145/sqrt(350) = 7.75. A W610x125 has b/2tf = 229/(2x19.6) = 5.84 — well within Class 1.

For 350AT (atmospheric corrosion-resistant, Fy = 350 MPa), same limits apply. For 480W (Fy = 480 MPa), flange limit drops to 145/sqrt(480) = 6.62, which may push some lighter sections into Class 2 or 3.

Moment Resistance — Class 1 and 2 Sections

For the most common case of a Class 1 W-shape beam in strong-axis bending:

Mr = phi x Zx x Fy

This represents the full plastic moment — the entire cross-section yields in tension and compression, forming a plastic hinge. The plastic neutral axis divides the section into equal areas of tension and compression.

For a Class 3 section (e.g., deep plate girders where the slender web limits rotation): Mr = phi x Sx x Fy (yield moment only, no plastic redistribution)

Class 4 sections require computation of effective widths for all slender elements per Clause 13.5.3, reducing Se below S. For hot-rolled W-shapes in 350W, Class 4 sections are rare — they occur primarily in welded plate girders with very slender webs.

Lateral-Torsional Buckling — Clause 13.6

LTB is the governing limit state for beams without continuous lateral bracing of the compression flange. CSA S16-19 Clause 13.6 uses an equivalent moment factor omega_2 (analogous to AISC Cb) applied to the LTB moment Mu:

Mu = omega_2 x Mu_max

Where Mu_max depends on the unbraced segment length L relative to the critical lengths Lp (plastic limit) and Lr (elastic limit):

L <= Lp (plastic range): Mu = Mp (full plastic moment) Lp < L <= Lr (inelastic range):** Mu = Mp - (Mp - My) x (L - Lp) / (Lr - Lp) **L > Lr (elastic range): Mu = Mcr (elastic critical moment)

Lp = 1.76 x ry x sqrt(E/Fy) Lr depends on the section properties, torsional constant J, and warping constant Cw.

Equivalent moment factor omega_2: omega_2 accounts for the shape of the bending moment diagram between brace points. A uniform moment (omega_2 = 1.0) is the worst case. A moment gradient that reverses sign can increase omega_2 to 2.0-3.0, substantially increasing the LTB resistance.

The omega_2 method is computationally simpler than the AISC Cb formulation (which involves the quarter-point moments raised to various powers) and produces near-identical results for all practical cases.

LTB Resistance — W530x82 Example (350W)

Note that at Lb = 6.0 m, the beam is in the elastic LTB range (Lb > Lr ~ 4.5 m for this section), and the capacity drops sharply. Adding a single midspan brace (reducing Lb to 3.0 m) more than doubles the capacity from 173 to 551 kNm — far more cost-effective than upsizing the section.

Shear Resistance — Clause 13.4

For unstiffened webs, the shear resistance:

Vr = phi x Aw x Fs

Where Aw = d x w (total depth x web thickness) and Fs depends on the web slenderness h/w.

For h/w <= 1016/sqrt(Fy) = 54.3 (for 350W): Fs = 0.66 x Fy — shear yielding governs. For h/w > 54.3: Fs = 180,000 / (h/w)^2 — shear buckling governs (per Clause 13.4.2).

Most hot-rolled W-shapes comfortably satisfy the 54.3 limit. For a W610x125 in 350W: h/w = (610-2x19.6)/11.9 = 48.0 < 54.3. Fs = 0.66 x 350 = 231 MPa.

Vr = 0.90 x (610 x 11.9) x 231 / 1000 = 1,511 kN.

Shear rarely governs for hot-rolled beams except in very short spans with heavy concentrated loads (e.g., transfer girders, crane runway beams).

Web Crippling and Web Yielding — Clause 14.3.2

For concentrated loads or reactions applied to beam flanges without bearing stiffeners:

Web yielding: Br = phi x w x (N + 5k) x Fy Where N = bearing length, k = distance from flange outer face to web toe of fillet (from CISC Handbook).

Web crippling (local buckling): Br = 0.60 x phi x w^2 x sqrt(E x Fy x tf/w) x (1 + 3 x (N/d) x (w/tf)^1.5)

Web crippling is the more critical check for slender webs (h/w > 80) or for short bearing lengths. For typical beam bearing seats (N >= 75 mm), web crippling rarely governs provided the web is not slender.

When web crippling governs, the remedy is either: (a) increase the bearing length N (wider bearing plate at the support), or (b) provide bearing stiffeners (pair of plates welded to the web and flanges).

Deflection Criteria — NBCC 2020

The National Building Code of Canada 2020 specifies serviceability deflection limits in Commentary C, Table C-2:

Canadian deflection limits are marginally more generous than AISC recommendations (L/500 vs L/600 for floors with partitions) due to the expectation of pre-cambering of beams in Canadian practice.

Pre-camber: For beams spanning more than 8 m, pre-camber of 50-75% of the calculated dead load deflection is standard specification on Canadian fabrication drawings. This eliminates the visual sag of the beam under dead load without affecting the live load behaviour.

Worked Example — W610x125 Roof Beam

Problem: A W610x125 beam (CSA G40.21 350W) spans 15.0 m simply supported, supporting roof purlins at 1.5 m centres. Dead load = 4.5 kN/m, snow load (S = 2.0 kPa x 6 m tributary) = 12.0 kN/m. Beam is braced at supports and at 5.0 m centres by fly braces from purlins. Steel roof deck — no continuous lateral restraint.

Section properties — W610x125: Zx = 3670 x 10^3 mm^3 | Sx = 3220 x 10^3 mm^3 | Ix = 1030 x 10^6 mm^4 | ry = 48.9 mm | h/w = 48.0

Step 1 — Factored loads (NBCC 2020): Case 1: 1.4D = 1.4 x 4.5 = 6.3 kN/m Case 3: 1.25D + 1.5S = 1.25 x 4.5 + 1.5 x 12.0 = 5.63 + 18.0 = 23.63 kN/m (governs)

Step 2 — Design moment: Mf = 23.63 x 15.0^2 / 8 = 664.6 kNm

Step 3 — Section classification (Class 1): Flange: b/2tf = 7.18 <= 7.75. OK. Web: h/w = 48.0 <= 58.8. OK. Class 1 — full plastic moment applies.

Step 4 — Moment resistance (full bracing assumed — check LTB next): Mp = Zx x Fy = 3670 x 10^3 x 350 / 10^6 = 1,285 kNm Mr = 0.90 x 1285 = 1,156 kNm > 664.6. OK without LTB check.

Step 5 — LTB at Lb = 5.0 m: Lp = 1.76 x 48.9 x sqrt(200000/350) = 1,993 mm (plastic range) Lr ~ 5,800 mm for this section. Lb = 5,000 mm is in the inelastic range (Lp < Lb < Lr).

My = Sx x Fy = 3220 x 10^3 x 350 / 10^6 = 1,127 kNm Mu = Mp - (Mp - My) x (5000 - 1993) / (5800 - 1993) = 1285 - (1285 - 1127) x 3007/3807 Mu = 1285 - 158 x 0.79 = 1285 - 125 = 1,160 kNm

omega_2 for a UDL segment with end moments = 0 (simply supported): omega_2 ~ 1.13. Mr = 0.90 x 1.13 x 1160 = 1,180 kNm > 664.6. LTB OK with substantial reserve.

Step 6 — Shear: Vf = 23.63 x 15.0 / 2 = 177.2 kN Vr = 0.90 x (611 x 11.9) x 0.66 x 350 / 1000 = 1,511 kN >> 177.2. OK.

Step 7 — Deflection (serviceability, snow = live load): Delta_snow = 5 x 12.0 x 15000^4 / (384 x 200000 x 1030 x 10^6) = 38.7 mm L/180 (roof, no ceiling) = 83.3 mm. 38.7 < 83.3. OK.

Consider pre-camber of 20 mm upward for dead load deflection (approximately 9.5 mm).

Result: W610x125, 350W is adequate. Substantial reserve in bending (1,180 vs 665 kNm) suggests W610x101 or W530x92 could also work, but the W610x125 provides stiffness for serviceability and accommodates future roof equipment.

Frequently Asked Questions

When does LTB actually govern over section capacity? LTB governs when Lb exceeds approximately Lp (typically 1.5-2.0 m for W-shapes with Fy = 350 MPa). For Lb = 3 m (floor beam with intermediate restraint at third points): ratio Mr_LTB / Mr_section ~ 0.75-0.90. For Lb = 6 m (roof beam with end restraints only): ratio drops to 0.30-0.50. For composite floor beams with continuous slab restraint to the top flange, LTB of the positive moment region is eliminated.

What is the difference between CSA S16 omega_2 and AISC Cb? Both are equivalent moment factors that increase LTB resistance for non-uniform moment. omega_2 is applied directly as a multiplier on Mu, while Cb is applied to the elastic buckling moment Mcr. For the same moment diagram, the numerical values differ slightly but produce results within 5% of each other. The CISC Handbook tabulates omega_2 values for common loading and restraint conditions.

Should I specify camber for Canadian steel beams? Yes, for beams spanning more than 8 m where dead load deflection exceeds L/500. Specify 50-75% of the calculated dead load deflection as upward camber on the fabrication drawings. The beam is fabricated with the specified camber (cold-cambered by the fabricator or ordered as mill-cambered). Caution: camber is less effective for composite beams where the deck and concrete mass may be supported by the bare steel; the camber should account for the pre-composite and post-composite load paths separately.

What governs beam design — strength or deflection? For floor beams with L/360 deflection limit and typical office live load (2.4 kPa), deflection governs for spans exceeding approximately 8-9 m for 350W steel. For roof beams with L/180 limit and snow load, strength (LTB) typically governs. For heavy industrial floors with L/600 or L/800 limits (crane rails, vibratory equipment), deflection is always the controlling design criterion.

This page is for educational reference. Beam design per CSA S16-19 Clause 13. Verify section properties against the current CISC Handbook of Steel Construction. All structural designs must be independently verified by a licensed Professional Engineer. Results are PRELIMINARY — NOT FOR CONSTRUCTION.