| Plan dimensions | 30.0 m × 18.0 m | | Bay sizes | 6.0 m × 9.0 m (5 bays × 2 bays) | | Floor-to-floor height | 4.0 m (ground to level 2), 3.6 m (level 2 to roof) | | Roof | Flat roof, steel deck on OWSJ | | Floors | 75 mm concrete on 38 mm steel deck | | Lateral system | Concentrically braced frames (HSS) | | Seismic | Toronto: S_a(0.2) = 0.39, S_a(1.0) = 0.13 | | Importance category | Normal (I_E = 1.0) | | Site class | C (very dense soil, 360 ≤ v_s < 760 m/s) |

Material Specifications

Gravity Loads

Per NBCC 2020 — Ontario:

Dead Loads

Live Loads

Snow load for Toronto: S_s = 1.2 kPa (ground), C_b = 1.0, S = I_s × S_s × (C_b × C_w × C_s × C_a) = 1.0 × 1.2 × 1.0 × 1.0 × 1.0 × 1.0 × 1.2 = 1.44 kPa.

Gravity Beam Design — Typical Floor Beam

Given: Simply supported floor beam spanning 9.0 m at 3.0 m tributary width (beams at 3.0 m centres). Floor: 75 mm concrete on 38 mm steel deck. Composite action with shear studs.

Step 1 — Factored Loads:

Dead: w_D = (2.40 + 0.30 + 1.00) × 3.0 = 11.1 kN/m Live: w_L = 2.4 × 3.0 = 7.2 kN/m (office, reducible per NBCC)

Live load reduction (tributary area = 9.0 × 3.0 = 27 m² > 20 m²): Reduction = 0.3 + sqrt(9.8/27) = 0.3 + 0.603 = 0.903. Live = 0.903 × 2.4 = 2.17 kPa. w_L_red = 2.17 × 3.0 = 6.5 kN/m

Factored load (NBCC ULS 1): w_f = 1.25 × 11.1 + 1.5 × 6.5 = 13.88 + 9.75 = 23.63 kN/m

Step 2 — Bending Moment and Shear:

M_f = w_f × L² / 8 = 23.63 × 9.0² / 8 = 239.3 kN·m V_f = w_f × L / 2 = 23.63 × 9.0 / 2 = 106.3 kN

Step 3 — Section Selection (Composite Beam):

Try W360×45 (350W): A = 5,730 mm², d = 350 mm, b_f = 170 mm, t_f = 13.0 mm, t_w = 7.2 mm, I_x = 121 × 10⁶ mm⁴, S_x = 694 × 10³ mm³ (non-composite).

Composite section properties (effective slab width = min(L/4, beam spacing) = min(9,000/4, 3,000) = 2,250 mm): Concrete flange: 2,250 mm wide × 75 mm thick (above deck ribs). Assume 50% composite action with 19 mm shear studs at 300 mm centres (12 studs per half-span).

Approximate composite M_r (50% shear connection): M_r ≈ phi × (0.85 × A_s × Fy × (d/2 + h_c - a/2), where a is the depth of the concrete compression block.

For W360×45 with 50% composite: M_r,comp ≈ 390 kN·m (calculated per CSSBI design tables).

M_f = 239.3 ≤ 390 kN·m. OK. (Ratio = 0.61)

Step 4 — Deflection Check:

delta_LL = 5 × w_L_red × L⁴ / (384 × E × I_x,comp) I_x,comp (50% transformed) ≈ 240 × 10⁶ mm⁴ delta_LL = 5 × 6.5 × 9,000⁴ / (384 × 200,000 × 240 × 10⁶) = 11.6 mm

delta_allow = L/360 = 9,000/360 = 25.0 mm

delta_LL = 11.6 ≤ 25.0 mm. OK.

Result: W360×45 composite floor beam with 12-19 mm shear studs per half-span (300 mm centres). Satisfies CSA S16:24 for composite beam design.

Lateral Load Design — Braced Frame

Seismic base shear per NBCC 2020:

V = S(T_a) × M_v × I_E × W / (R_d × R_o)

T_a = 0.05 × h_n^(3/4) = 0.05 × 7.6^(3/4) = 0.05 × 4.66 = 0.233 s

S(0.233) ≈ S_a(0.2) = 0.39 (short period). M_v = 1.0.

R_d = 2.0, R_o = 1.3 (moderately ductile CBF). Total R = 2.6.

Seismic weight W: Roof = (0.30 + 0.15) × 30 × 18 = 243 kN. Floor = (2.40 + 0.30 + 1.00 + 0.25 × 2.4) × 30 × 18 = 4.30 × 540 = 2,322 kN. Cladding: 1.5 × (30 + 18) × 2 × 7.6 = 1,094 kN.

W_total ≈ 243 + 2,322 + 1,094 = 3,659 kN (total seismic weight)

V = 0.39 × 1.0 × 1.0 × 3,659 / 2.6 = 549 kN

Distribute to roof and floor per NBCC: F_roof = 549 × (243 × 7.6) / (243 × 7.6 + 2,322 × 4.0) = 549 × 1,847 / 11,135 = 91 kN. F_floor = 458 kN.

Two braced frames in the short direction (18 m): V_per_frame = 549/2 = 274.5 kN.

Brace design:

Brace force at ground level (diagonal brace in 6.0 m bay × 4.0 m storey height): Brace length = sqrt(6.0² + 4.0²) = 7.21 m. Angle = tan^(-1)(4.0/6.0) = 33.7°.

Horizontal force = 274.5 kN. Brace axial = 274.5 / cos(33.7°) = 274.5 / 0.832 = 330 kN (tension or compression).

Try HSS 127×127×8 (350W): A = 3,710 mm², r = 48.2 mm. KL/r = 1.0 × 7,210 / 48.2 = 149.6.

Fe = pi² × 200,000 / 149.6² = 88.2 MPa. lambda = sqrt(350/88.2) = 1.992.

C_r = 0.9 × 3,710 × 350 × (1.0 + 1.992^2.68)^(-0.746) / 1,000 = 0.9 × 3,710 × 350 × 0.314 / 1,000 = 367 kN.

Brace compression = 330 ≤ 367 kN. OK. (Ratio = 0.90)

Note: CSA S16 seismic requirements (Clause 27) apply. For MD CBF, KL/r ≤ 100 for HSS — HSS 127×127×8 with KL/r = 149.6 does NOT meet MD slenderness limit. Increase to HSS 152×152×10: A = 5,640 mm², r = 57.4 mm. KL/r = 7,210/57.4 = 125.6 — still exceeds 100 for MD. Increase to HSS 178×178×10: A = 6,540 mm², r = 67.8 mm. KL/r = 7,210/67.8 = 106.3 — close to limit. Use HSS 178×178×13: A = 8,310 mm², r = 66.7 mm. KL/r = 7,210/66.7 = 108.1. For LD CBF (R_d = 1.5, R_o = 1.3, KL/r ≤ 120): KL/r = 108.1 ≤ 120. OK for LD.

Revised seismic — LD CBF:

R_d = 1.5, R_o = 1.3. V = 0.39 × 1.0 × 3,659 / (1.5 × 1.3) = 0.39 × 3,659 / 1.95 = 732 kN. V_per_frame = 366 kN. Brace force = 366 / 0.832 = 440 kN.

C_r = 0.9 × 8,310 × 350 × (1.0 + 1.273^2.68)^(-0.746) / 1,000 = 0.9 × 8,310 × 350 × 0.487 / 1,000 = 1,274 kN.

440 ≤ 1,274 kN. OK. (Ratio = 0.35 — capacity governed by analysis, not brace member.)

Column Design — Ground Floor Interior Column

Tributary area: 6.0 m × 9.0 m = 54 m² per floor. Two floors + roof.

Axial loads:

Factored: C_f = 1.25 × 224.1 + 1.5 × 195.0 = 280.1 + 292.5 = 572.6 kN (snow as companion — check NBCC load combinations. For dead + live dominant: C_f = 1.25 × 224.1 + 1.5 × 117.2 + 0.5 × 77.8 = 280.1 + 175.8 + 38.9 = 494.8 kN.)

Use C_f = 573 kN (snow dominant combination for roof + live for floor — conservative).

Column selection: Ground floor interior, unbraced length = 4.0 m. K = 1.0 (braced frame).

Try W250×58 (350W): A = 7,420 mm², r_y = 46.5 mm. KL/r = 4,000/46.5 = 86.0. Fe = pi² × 200,000 / 86.0² = 266.8 MPa. lambda = sqrt(350/266.8) = 1.145. C_r = 0.9 × 7,420 × 350 × (1.0 + 1.145^2.68)^(-0.746) / 1,000 = 0.9 × 7,420 × 350 × 0.545 / 1,000 = 1,274 kN.

C_f = 573 ≤ 1,274 kN. OK. (Ratio = 0.45 — significant reserve for lateral load effects.)

Combined axial + bending from lateral:

Column moment from braced frame action (estimated from analysis): M_f ≈ 45 kN·m at ground level from frame action.

W250×58: M_rx = 235 kN·m (Class 1). Interaction per Clause 13.8:

C_f/C_r + 0.85 × M_f/M_r = 573/1,274 + 0.85 × 45/235 = 0.450 + 0.163 = 0.613 ≤ 1.0. OK.

Drift Check

Lateral drift under factored seismic: estimate from braced frame stiffness.

Frame stiffness (approximate, brace axial deformation dominant): K_frame ≈ (A_brace × E × cos²theta) / L_brace for tension brace. K_frame ≈ (8,310 × 200,000 × 0.692) / 7,210 = 159,000 N/mm = 159 kN/mm.

Drift at roof (from 91 kN): delta_roof = 91/159 = 0.6 mm. Negligible — okay.

Inter-storey drift (from 458 kN at floor): delta_floor = 458/159 = 2.9 mm.

Drift ratio = 2.9/4,000 = 1/1,379 ≤ 1/500 per NBCC. OK.

Foundation Interface

Interior column C_f = 573 kN. Use base plate 350 × 350 mm on spread footing.

Concrete bearing: B_r = 0.85 × 0.65 × 25 × 350 × 350 × sqrt(4.0) / 1,000 (limit 2.0) = 0.85 × 0.65 × 25 × 122,500 × 2.0 / 1,000 = 3,384 kN > 573 kN. OK.

Base plate thickness: t_p = 350 × 0.272 (from previous building design logic) → Use 350 × 350 × 20 mm base plate.

4-M20 anchor rods, Grade 55, embedded 300 mm into footing. OK for minimum constructability.

Frequently Asked Questions

What is the typical design sequence for a steel building per CSA S16? The design sequence is: (1) establish column grid and framing layout based on architectural requirements; (2) determine dead, live, snow, wind, and seismic loads per NBCC 2020; (3) select the gravity framing system (composite or non-composite beams); (4) select and design the lateral force resisting system (braced frame, moment frame, or shear wall); (5) run structural analysis to determine member forces for all load combinations; (6) size all members for strength (flexure, shear, axial, combined) and serviceability (deflection, vibration, drift); (7) design connections (shear connections, moment connections, brace connections, splices); (8) design foundation interface (base plates, anchor rods). Each step must be verified before proceeding.

How do I choose between a braced frame and a moment frame for the lateral system? Braced frames are more efficient for low-to-mid-rise buildings (up to about 12 storeys) because braces carry axial loads, which is stiffer and more economical than bending in moment frames. Braced frames also have lower drift — typically 40-60% less than moment frames for the same member weight. However, braces obstruct architectural openings (windows, doors, corridors). Moment frames provide clear open bays but require heavier columns and beams, and connection detailing is more expensive. For the example building, a braced frame at the building perimeter (stairwells or exterior walls) is the economical choice.

What live load reduction can I use for Canadian office buildings? Per NBCC 2020 Table 4.1.5.3, live load reduction factor = 0.3 + sqrt(9.8/A) where A is the tributary area in m² (limited to reduction ≥ 0.5 for most occupancies). For office floors: the reduction applies to beams, girders, and columns. Example: interior column with 54 m² tributary area per floor, two floors = 108 m²: reduction = 0.3 + sqrt(9.8/108) = 0.3 + 0.301 = 0.601. Live load on column = 0.601 × 2.4 = 1.44 kPa (per floor). Snow load is not reduced.

Does the building example include fire protection requirements? The member sizing presented above addresses structural strength and serviceability only — fire protection is a separate design requirement per the National Building Code and CSA S16 Annex K. For a 2-storey office building: (a) columns typically require 1-hour fire resistance rating (spray-applied fireproofing or intumescent coating); (b) floor beams supporting a rated floor assembly require 1-hour protection if the floor is a fire separation; (c) roof members supporting only the roof typically do not require fire protection unless the roof is an occupancy or contains combustible construction. Fire protection thickness is determined by the UL directory based on the W/D ratio (heated perimeter / cross-sectional area) of each section.

Related Pages

• CSA S16 Beam Design
• CSA S16 Column Design
• Canadian Braced Frame Design
• CSA S16 Load Combinations
• Canadian Base Plate Design
• Canadian Framing Systems
• Canadian Seismic Design Guide
• All Canadian References

This page is for educational reference. Complete building design per CSA S16:24 and NBCC 2020. All member sizes, connection details, and foundation designs must be verified by a licensed Professional Engineer for the specific building configuration, site conditions, and local building code requirements. Seismic design parameters must be confirmed from the actual site geotechnical report. Results are PRELIMINARY — NOT FOR CONSTRUCTION without independent PE/SE verification.

Design Resources

Reference pages

• CA Steel Building Example

Disclaimer: This content is for educational purposes only. Results must be verified by a licensed professional engineer. Steel Calculator provides preliminary design tools — NOT a substitute for professional engineering judgment.