Canadian Beam Design — Flexural Design per CSA S16-19 Clause 13

Comprehensive reference for beam flexural design per CSA S16-19 Clause 13. Covers moment resistance for Class 1-4 sections, lateral-torsional buckling (LTB) per Clause 13.6, shear resistance per Clause 13.4, deflection criteria, and full worked example for a W530x82 beam in CSA G40.21 350W steel.

Quick access: CSA S16 beam sizes → | LTB reference → | Compact section limits →

CSA S16 Flexural Design Philosophy

Per CSA S16-19 Clause 13.5, the factored moment resistance Mr for beams is:

Mr = phi × Z × Fy for Class 1 and 2 sections (full plastic moment)

Mr = phi × S × Fy for Class 3 sections (yield moment)

Mr = phi × Seff × Fy for Class 4 sections (effective section)

Where:

• phi = 0.90 (resistance factor for steel members)
• Z = plastic section modulus (mm^3)
• S = elastic section modulus (mm^3)
• Seff = effective elastic section modulus for Class 4 (mm^3)
• Fy = minimum yield strength (MPa, thickness-adjusted per CSA G40.21)

Section Classification Summary

Per CSA S16 Table 1, the width-to-thickness limits for each class depend on the element type (flange or web) and the stress distribution.

Lateral-Torsional Buckling (Clause 13.6)

For unbraced beam segments, LTB reduces the moment resistance:

Mu = omega_2 × M_u

Where omega_2 = equivalent moment factor (analogous to AISC Cb) and M_u depends on the slenderness of the unbraced segment relative to Lp and Lr.

Lp = 1.76 × ry × sqrt(E/Fy) Lr = 1.76 × ry × sqrt(E/Fy) × sqrt(1 + (Ly/Lz)^2) (simplified for doubly symmetric sections)

For a full LTB treatment with formulas, Lp/Lr, and a worked example, see the Canadian LTB reference.

Shear Resistance (Clause 13.4)

Per CSA S16-19 Clause 13.4.1, the shear resistance for unstiffened webs:

Vr = phi × Aw × Fs

Where:

• phi = 0.90
• Aw = overall depth × web thickness (d × w)
• Fs = 0.66 × Fy for h/w ≤ 1016 / sqrt(Fy) (CSA S16 Table 3 limits)
• Fs reduced for slender webs per Clause 13.4.2

Shear Resistance — Common W-Shapes

All common W-shapes have h/w ≤ 54.3 (for Fy = 350 MPa), meaning the full 0.66 × Fy shear resistance applies. Shear rarely governs for hot-rolled W-shapes except for very short spans or heavy loads near supports.

Deflection Criteria

Per NBCC 2020 and CSA S16, serviceability deflection limits:

Flange Curling and Web Buckling

For concentrated loads on beam flanges, CSA S16 Clause 14.3 requires:

1. Web crippling: Check at unframed ends: Br = phi × w × (N + 5k) × Fy
2. Web sidesway buckling: Check when compression flange is not laterally restrained at the load point
3. Flange local bending: Check for heavy concentrated loads on slender flanges

For most W-shapes with bearing stiffeners, these checks are satisfied for typical loading. Without stiffeners, web crippling may govern for heavy concentrated loads near supports.

Concentrated Load Resistance

Without bearing stiffeners, per CSA S16 Clause 14.3.2:

Br = 0.60 × phi × w^2 × Fy × (1 + 3 × N/(d_w) × (w/tf)^1.5) × sqrt(E/Fy)

For W530×82, 350W:

• w = 9.3 mm (web thickness)
• tf = 13.3 mm (flange thickness)
• N = bearing length (mm)

At an interior point with N = 150 mm: Br = 0.60 × 0.90 × 9.3^2 × 350 × (1 + 3 × 150/504 × (9.3/13.3)^1.5) × sqrt(200000/350) / 1000 = 327 kN

This exceeds typical beam reactions for this section in floor framing.

Worked Example — W530x82 Beam Design

Given: W530×82, Grade 350W steel (Fy = 350 MPa, Fu = 450 MPa). Simply supported span = 9.0 m. Uniformly distributed live load = 15.0 kN/m, dead load = 8.0 kN/m (including self-weight). Lateral bracing at supports and midspan (Lb = 4.5 m).

Step 1 — Section Properties:

• Zx = 2060 × 10^3 mm^3
• Sx = 1830 × 10^3 mm^3
• Ix = 485 × 10^6 mm^4
• ry = 43.1 mm
• h/w = 50.0

Step 2 — Factored Loads (NBCC 2020 load combination 4): wf = 1.5 × 15.0 + 1.25 × 8.0 = 22.5 + 10.0 = 32.5 kN/m

Step 3 — Maximum Moment: Mf = wf × L^2 / 8 = 32.5 × 9.0^2 / 8 = 329 kN·m

Step 4 — Section Classification (CSA S16 Table 1): Flange: b/(2tf) = 191/(2×13.3) = 7.18. Limit Class 1 = 145/sqrt(350) = 7.75. OK — Class 1. Web: h/w = 50.0. Limit Class 1 = 1100/sqrt(350) = 58.8. OK — Class 1. Section is Class 1, so full plastic moment applies.

Step 5 — Moment Resistance (Clause 13.5): Mp = Zx × Fy = 2060 × 10^3 × 350 / 10^6 = 721 kN·m Mr = phi × Mp = 0.90 × 721 = 649 kN·m

Check LTB (Clause 13.6): Lb = 4500 mm, omega_2 = 1.0 (uniform load, braced at midspan). Lp = 1.76 × 43.1 × sqrt(200000/350) = 1815 mm Lr = 1.76 × 43.1 × sqrt(200000/350) × sqrt(1 + 1^2/2) = approximately 2600 mm (depends on E/Iy/G/J) For Lb = 4500 > Lr, elastic LTB governs. Mcr = omega_2 × pi/Lb × sqrt(EIyGJ + (pi×E/Lb)^2 × Iy×Cw) Mcr ≈ 450 kN·m for this section at Lb = 4.5 m. Mu = Mcr = 450 kN·m Mr = phi × Mu = 0.90 × 450 = 405 kN·m (LTB governs over 649 kN·m plastic)

Step 6 — Check: Mf = 329 kN·m ≤ Mr = 405 kN·m. OK.

Step 7 — Shear: Vf = wf × L / 2 = 32.5 × 9.0 / 2 = 146 kN Vr = phi × Aw × 0.66 × Fy = 0.90 × (529×9.3) × 0.66 × 350 / 1000 = 787 kN Vf / Vr = 146/787 = 0.185. OK.

Step 8 — Deflection (Serviceability): Delta_live = 5 × w_L × L^4 / (384 × E × I) = 5 × 15.0 × 9000^4 / (384 × 200000 × 485×10^6) = 19.7 mm L/360 = 9000/360 = 25.0 mm. Delta_live = 19.7 < 25.0. OK.

Result: W530×82, Grade 350W is adequate for the given loading.

Frequently Asked Questions

What is the flexural resistance of a W530x82 beam in 350W steel? The plastic moment resistance per CSA S16 is Mr = phi × Zx × Fy = 0.90 × 2060×10^3 × 350 / 10^6 = 649 kN·m. However, the LTB resistance at an unbraced length of 4.5 m reduces this to approximately 405 kN·m. For a fully braced beam (Lb ≤ Lp = 1.82 m), the full 649 kN·m is available.

What is the difference between Class 1, 2, 3, and 4 sections in CSA S16? Class 1 sections can develop a plastic hinge with sufficient rotation capacity for plastic analysis. Class 2 can reach plastic moment but with limited rotation. Class 3 reaches yield moment but not plastic moment. Class 4 experiences local buckling before yielding. The classification depends on flange and web width-to-thickness ratios per CSA S16 Table 1.

When does shear govern in CSA S16 beam design? Shear rarely governs for hot-rolled W-shapes in typical construction. The h/w ratio for standard sections is well below the CSA S16 limit of 54.3 (for Fy = 350 MPa), so the full 0.66 × Fy shear resistance applies. Shear governs only for deep plate girders (h/w > 150), short spans with very heavy loads, or beams with large concentrated loads near supports.

What deflection limits apply to Canadian steel beams? Per NBCC 2020, floor beams typically require L/360 for live load deflection and L/300 for total load. For roofs with plaster ceilings, L/360 applies. For crane girders, L/600 to L/1000 depending on crane class. The structural designer selects the appropriate limit based on the supported construction type.

Related Pages

• CSA S16 Lateral-Torsional Buckling — Detailed Guide
• Canadian Compact Section Limits — Class 1-4 Table
• Canadian Beam Sizes — W-Shape Properties
• CSA S16 Column Design
• Cb Factor — Moment Modification Table
• Beam Capacity Calculator
• All Canadian References

This page is for educational reference. Beam design per CSA S16-19 Clause 13. Verify section properties against CISC Handbook and current code. Results are PRELIMINARY — NOT FOR CONSTRUCTION without independent PE/SE verification.

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