CSA S16 Beam Design — Flexure, Lateral-Torsional Buckling & Shear
Quick Reference: Bending resistance Mr = phi _ Zx _ Fy (Class 1-2), Mr = phi _ Sx _ Fy (Class 3). Lateral-torsional buckling per Cl. 13.6.5 with omega*2 moment gradient factor. Shear resistance Vr = phi * 0.60 _ Fy _ d _ tw. All design per CSA S16-19.
Section Classification (Cl. 11)
CSA S16-19 classifies cross-sections into four classes based on width-to-thickness ratios of the flange and web elements. Classification determines whether plastic, inelastic, or elastic resistance governs.
| Class | Description | Moment Resistance Basis | Rotation Capacity |
|---|---|---|---|
| 1 | Plastic design | Plastic moment Mp = Zx * Fy | Full plastic hinge rotation |
| 2 | Compact | Plastic moment Mp = Zx * Fy | Limited plastic rotation |
| 3 | Non-compact | Yield moment My = Sx * Fy | Elastic moment only |
| 4 | Slender (Class 4) | Reduced effective section | Elastic, local buckling limits |
Flange slenderness limits (Cl. 11.2, Table 2)
| Class | b/t limit (rolled W shapes, Fy = 350 MPa) |
|---|---|
| 1 | b/t <= 145/sqrt(Fy) = 7.8 |
| 2 | b/t <= 170/sqrt(Fy) = 9.1 |
| 3 | b/t <= 200/sqrt(Fy) = 10.7 |
| 4 | b/t > 200/sqrt(Fy) |
Web slenderness limits in flexure (Cl. 11.3, Table 2)
| Class | h/w limit (Fy = 350 MPa) |
|---|---|
| 1 | h/w <= 1100/sqrt(Fy) = 58.8 |
| 2 | h/w <= 1700/sqrt(Fy) = 90.9 |
| 3 | h/w <= 1900/sqrt(Fy) = 101.6 |
| 4 | h/w > 1900/sqrt(Fy) |
For a W310x39 (d = 310 mm, bf/2tf = 9.0, h/w = 46.0, Fy = 350 MPa): flange is Class 2, web is Class 1. Overall section class = 2.
Bending Resistance (Cl. 13.5, 13.6)
Laterally Supported (Lb <= Lu)
For sections braced against lateral-torsional buckling, the moment resistance is:
Class 1 and 2 sections: Mr = phi _ Zx _ Fy
Class 3 sections: Mr = phi _ Sx _ Fy
Class 4 sections: Mr = phi _ Se _ Fy (effective section modulus)
where:
- phi = 0.90 (Cl. 13.1)
- Zx = plastic section modulus (mm^3)
- Sx = elastic section modulus (mm^3)
- Fy = specified minimum yield strength (MPa)
For W310x39, Grade 350W: Zx = 603 x 10^3 mm^3, Fy = 350 MPa Mr = 0.90 _ 603,000 _ 350 / 10^6 = 190.0 kN.m
Unbraced Length Limit Lu
The critical unbraced length beyond which LTB must be checked:
Lu = 1.1 _ rt _ sqrt(E/Fy) (where rt is the radius of gyration of the compression flange plus one-third of the web in compression)
For W310x39: rt = 45.5 mm, Lu = 1.1 _ 45.5 _ sqrt(200,000/350) = 1.1 _ 45.5 _ 23.9 = 1,196 mm
If the unbraced length Lb > Lu, lateral-torsional buckling governs.
Lateral-Torsional Buckling (Cl. 13.6.5)
For unbraced beams, the critical elastic lateral-torsional buckling moment is:
Mu = omega2 * (pi/L) _ sqrt(E _ Iy _ G * J + (piE/L)^2 _ Iy _ Cw)
where:
- omega_2 = moment gradient factor (Table 3-2 in CISC Handbook)
- Iy = weak-axis moment of inertia (mm^4)
- J = St. Venant torsional constant (mm^4)
- Cw = warping constant (mm^6)
- G = shear modulus = 77,000 MPa
- L = unbraced length (mm)
Moment Gradient Factor omega_2
CSA S16-19 uses the omega_2 factor to account for moment gradient along the unbraced length. This is conceptually similar to AISC's C_b factor:
| Loading Case | Bending Moment Diagram | omega_2 |
|---|---|---|
| Uniform moment (worst case) | Constant M along span | 1.00 |
| Central point load | Triangular M, peak at centre | 1.35 |
| Uniformly distributed load (UDL) | Parabolic M, peak at centre | 1.14 |
| End moments M1 = -M2 (double curv) | M varies linearly to zero | 1.75 |
| End moments M1 = 0, M2 = M (cant) | M varies linearly to zero | 1.14 |
| End moments M1 = 0.5*M2 | Linear gradient | 1.30 |
LTB Moment Resistance Mr
When Mu > 0.67 * Mp: Mr = 1.15 _ phi _ Mp * (1 - 0.28*Mp/Mu) <= phi * Mp
When Mu <= 0.67 * Mp: Mr = phi * Mu
where Mp = Zx * Fy.
This two-region equation accounts for inelastic buckling. The 0.67*Mp threshold roughly corresponds to Lb = Lu, the boundary between inelastic and elastic LTB.
Worked Example: LTB Check
Given: W310x39 Grade 350W, 6 m simple span, UDL, unbraced length Lb = 3,000 mm (supports at ends, one intermediate brace at mid-span).
Step 1 — Section and material properties:
- Zx = 603 x 10^3 mm^3, Sx = 549 x 10^3 mm^3
- Iy = 8.41 x 10^6 mm^4
- J = 125 x 10^3 mm^4
- Cw = 165 x 10^9 mm^6
- E = 200,000 MPa, G = 77,000 MPa
- Fy = 350 MPa
Step 2 — Mp and My:
- Mp = Zx _ Fy = 603,000 _ 350 / 10^6 = 211.1 kN.m
- My = Sx _ Fy = 549,000 _ 350 / 10^6 = 192.2 kN.m
- phi _ Mp = 0.90 _ 211.1 = 190.0 kN.m
Step 3 — Elastic LTB moment Mu (Cl. 13.6.5): For UDL, omega_2 = 1.14.
Mu = omega2 * (pi/L) _ sqrt(EIyGJ + (pi*E/L)^2 _ Iy _ Cw)
First term: EIyG*J = 200,000 * 8.41e6 _ 77,000 _ 125e3 = 1.619e25
Second term: (pi _ E / L)^2 _ Iy _ Cw = (pi _ 200,000 / 3,000)^2 _ 8.41e6 _ 165e9
= (209.44)^2 _ 8.41e6 _ 165e9 = 43,870 * 1.388e18 = 6.089e22
Combined: sqrt(1.619e25 + 6.089e22) = sqrt(1.625e25) = 4.031e12
Mu = 1.14 _ (pi/3,000) _ 4.031e12 = 1.14 _ 0.001047 _ 4.031e12 = 4,809 kN.m
Step 4 — LTB resistance: Mu = 4,809 kN.m >> 0.67 _ Mp = 0.67 _ 211.1 = 141.4 kN.m
Since Mu > 0.67 * Mp:
Mr = 1.15 _ phi _ Mp _ (1 - 0.28 _ Mp/Mu)
= 1.15 _ 190.0 _ (1 - 0.28 * 211.1/4,809)
= 218.5 _ (1 - 0.0123) = 218.5 _ 0.9877
= 215.9 kN.m
But Mr cannot exceed phi * Mp = 190.0 kN.m, so:
Mr = 190.0 kN.m — the section is fully braced and plastic moment governs.
Step 5 — Check against applied moment: Mf = w _ L^2 / 8 = 20 kN/m _ (6.0)^2 / 8 = 90.0 kN.m (factored)
Utilisation = Mf / Mr = 90.0 / 190.0 = 0.47 — OK.
In this case, with a 3 m unbraced length, the W310x39 is fully braced and Mr is governed by the plastic moment capacity. The LTB check is not critical because the unbraced length is short relative to Lu.
Shear Resistance (Cl. 13.4)
Vr = phi _ 0.60 _ Fy _ d _ tw (for h/w <= 439 * sqrt(kv/Fy))
where kv = 5.34 for unstiffened webs.
For W310x39: d = 310 mm, tw = 6.1 mm, h/w = (310-2*9.7)/6.1 = 47.6
Limit: 439 _ sqrt(5.34/350) = 439 _ 0.1235 = 54.2
Since 47.6 < 54.2, shear buckling does not govern.
Vr = 0.90 _ 0.60 _ 350 _ 310 _ 6.1 / 1,000 = 0.90 _ 0.60 _ 350 * 1.891 = 357 kN.
Applied shear Vf = 20 * 6.0 / 2 = 60 kN (simple beam, UDL)
Utilisation = 60 / 357 = 0.17 — OK. Shear rarely governs in W shapes.
Combined Bending and Shear (Cl. 14.6)
When Vf > 0.60 * Vr, interaction must be checked. For W shapes, this threshold is typically not reached — shear utilisation is usually below 0.60 so no interaction check is required.
Combined Bending and Axial Compression (Cl. 13.8)
For beam-columns (members subject to both bending and axial compression), the interaction check is:
Cf/Cr + 0.85 _ U1x _ Mfx/Mrx <= 1.0 (Cl. 13.8.2 — cross-section strength)
Cf/Cr + U1x * Mfx/Mrx <= 1.0 (Cl. 13.8.3 — member stability)
where:
- Cf = factored axial compression
- Cr = compressive resistance (Cl. 13.3)
- U1x = omega_1 / (1 - Cf/Ce) for strong-axis bending
- omega_1 = 0.60 for Class 1 and 2 sections
- Ce = elastic buckling load = pi^2 _ E _ I / (K*L)^2
Beam-column Worked Example
W310x39 Grade 350W, 4 m column height, K = 1.0, Cf = 300 kN, Mfx = 60 kN.m.
Step 1 — Compression resistance:
lambda = (KL/r) * sqrt(Fy/(pi^2E)) = (4,000/135) * sqrt(350/(pi^2*200,000))
lambda = 29.63 _ sqrt(350/1,973,900) = 29.63 _ sqrt(1.773e-4) = 29.63 * 0.0133 = 0.394
For CSA S16, the column curve uses n = 1.34:
Cr = phi _ A _ Fy * (1 + lambda^(2n))^(-1/n) = 0.90 _ 4,960 _ 350 * (1 + 0.394^(21.34))^(-1/1.34) / 1,000 = 1,562.4 _ (1 + 0.394^2.68)^(-0.746) / 1,000 = 1,562.4 _ (1.292)^(-0.746) / 1,000 = 1,562.4 * 0.826 / 1,000 = 1,291 kN
Step 2 — Beam-column interaction: Cf/Cr = 300/1,291 = 0.232
Ce = pi^2 _ 200,000 _ 84.0e6 / (4,000)^2 / 1,000 = 10,370 kN
U1x = 0.60 / (1 - 300/10,370) = 0.60 / 0.971 = 0.618
Mfx/Mrx = 60 / 190.0 = 0.316
Cl. 13.8.2: 0.232 + 0.85 _ 0.618 _ 0.316 = 0.232 + 0.166 = 0.398 — OK.
Cl. 13.8.3: 0.232 + 0.618 * 0.316 = 0.232 + 0.195 = 0.427 — OK.
The beam-column passes both cross-section strength and member stability checks.
Deflection Limits
CSA S16-19 does not prescribe deflection limits directly — these are in NBCC 2020 and project specifications. Common limits per NBCC and industry practice:
| Member Type | Live Load Deflection Limit | Total Load Deflection Limit |
|---|---|---|
| Floor beams (typical) | L/360 | L/240 |
| Roof beams (no ceiling) | L/240 | L/180 |
| Roof beams (with plaster ceiling) | L/360 | L/240 |
| Floor beams (vibration-sensitive) | L/480 | L/360 |
| Crane runway girders | L/600 (vertical) | L/400 (lateral) |
For the W310x39, 6 m simple span, unfactored live load = 12 kN/m:
Deflection = 5 _ w _ L^4 / (384 _ E _ I) = 5 _ 12 _ (6,000)^4 / (384 _ 200,000 _ 84.0e6)
= 5 _ 12 _ 1.296e15 / (384 _ 200,000 _ 84.0e6) = 77.76e15 / 6.451e15 = 12.1 mm
Limit = L/360 = 6,000/360 = 16.7 mm. 12.1 < 16.7 — OK.
CSA S16 vs AISC 360 — Beam Design Comparison
| Feature | CSA S16-19 | AISC 360-22 |
|---|---|---|
| Design philosophy | Limit states | LRFD |
| phi for flexure | 0.90 | 0.90 |
| Section classification | Class 1, 2, 3, 4 | Compact, Noncompact, Slender |
| Plastic moment | Mr = phi _ Zx _ Fy (Class 1,2) | Mn = Mp = Fy * Zx (compact) |
| LTB moment | Mr per Cl. 13.6.5 | Mn per Chapter F (F2-F7) |
| Moment gradient | omega_2 factor | C_b factor |
| Column-buckling curve | Single (n = 1.34) | Two curves per E3 |
| Shear | Vr = phi _ 0.60 _ Fy * Aw | Vn = 0.6 _ Fy _ Aw * Cv1 |
| Combined bending+axial | Cl. 13.8 | Chapter H |
| omega_1 (beam-column) | 0.60 (Class 1,2) | Equivalent to C_m (typically 0.85) |
The resistance factor for flexure is identical (phi = 0.90), and both use Zx * Fy for compact/Class 1-2 sections. The key differences are:
- LTB: CSA S16 uses omega_2 for moment gradient with a single formula, while AISC 360 uses C_b with separate equations for each section type (F2, F3, F4, etc.)
- Column curve: CSA S16's single curve (n = 1.34) is slightly more conservative than AISC's two-curve system for intermediate slenderness
- Beam-column interaction: CSA S16's omega_1 = 0.60 is analogous to AISC's C_m but without the "no transverse loading" restriction. AISC uses a two-equation format (H1-1a and H1-1b)
Related Pages
- Canadian Steel Beam Sizes — W Shapes, HSS, G40.21 — Complete section tables
- CSA S16 Code Overview — Full CSA S16-19 design code reference
- Load Combinations — CSA S16 & NBCC — Canadian load combinations
- EN 1993-1-1 Beam Design Guide — Eurocode beam design with worked example
- AISC 360 Beam Design — The Complete Guide — US beam design reference
- Beam Capacity Calculator — Free multi-code beam calculator
This page is for educational reference. All formulae per CSA S16-19. For section properties, refer to the current CISC Handbook of Steel Construction. Results are PRELIMINARY — NOT FOR CONSTRUCTION without independent PE/SE verification.