Problem Statement

A floor beam spans 8.0 m simply supported between columns. The beam supports a concrete slab on steel deck providing continuous lateral restraint to the top flange. Spacing between beams is 3.0 m. The slab provides full lateral restraint to the compression flange for positive moment.

Loads (NBCC 2020):

• Dead load: 3.5 kN/m^2 (slab, deck, ceiling, services)
• Live load: 2.4 kN/m^2 (office occupancy, NBCC Table 4.1.5.3)
• Live load reduction: not applicable for tributary area < 20 m^2

Tributary width: 3.0 m per beam

Service loads: w_D = 3.5 x 3.0 = 10.5 kN/m, w_L = 2.4 x 3.0 = 7.2 kN/m

Material: CSA G40.21 350W (Fy = 350 MPa, Fu = 450 MPa). E = 200,000 MPa, G = 77,000 MPa.

Step 1 — Factored Load Combinations (NBCC 2020)

Per NBCC 2020 Table 4.1.3.2-A, the governing load combinations for this beam:

• Case 1: 1.4D = 1.4 x 10.5 = 14.7 kN/m
• Case 2: 1.25D + 1.5L = 1.25 x 10.5 + 1.5 x 7.2 = 13.13 + 10.80 = 23.93 kN/m (governs)

Factored design load: w_f = 23.93 kN/m

Step 2 — Design Forces

Maximum factored moment at midspan:

• M_f = w_f x L^2 / 8 = 23.93 x 8.0^2 / 8 = 191.4 kNm

Maximum factored shear at support:

• V_f = w_f x L / 2 = 23.93 x 8.0 / 2 = 95.7 kN

Step 3 — Section Properties (W310x39, 350W)

From CISC Handbook of Steel Construction, 12th Edition:

Step 4 — Section Classification (CSA S16:24 Table 1)

Flange check (W-shape, Flange, Class 1 limit): b/2t = 165 / (2 x 9.7) = 8.51 Limit: 145 / sqrt(Fy) = 145 / sqrt(350) = 7.75

8.51 > 7.75 — flange exceeds Class 1 limit. Check Class 2: Class 2 flange limit: 170 / sqrt(350) = 9.09. 8.51 < 9.09 — Class 2 flange.

Web check (W-shape, Web in bending, Class 1 limit): h/w (approximate: (d - 2t) / w) = (310 - 2 x 9.7) / 5.8 = 50.1 Class 1 web limit: 1100 / sqrt(350) = 58.8. 50.1 < 58.8 — Class 1 web.

Section classification: Class 2 (governed by flange). Full plastic moment applies but with limited rotation capacity.

Step 5 — Moment Resistance

Plastic moment: M_p = Zx x Fy = 592 x 10^3 x 350 / 10^6 = 207.2 kNm

Factored moment resistance (CSA S16:24 Clause 13.5): phi = 0.90 for steel in flexure M_r = phi x M_p = 0.90 x 207.2 = 186.5 kNm

Utilization: M_f / M_r = 191.4 / 186.5 = 1.026 > 1.0 — exceeds capacity!

The W310x39 is slightly inadequate in pure flexure. Two options:

1. Upgrade to W310x44.5 (Zx = 697 x 10^3 mm^3, M_r = 219.6 kNm)
2. Recheck with composite action if slab is positively connected

Composite option (W310x39 + 75 mm slab on 38 mm deck): With 19 mm diameter shear studs at 300 mm centres and effective slab width b_eff = min(L/4, beam spacing) = min(2000, 3000) = 2000 mm, the composite plastic neutral axis lies in the slab and the full composite moment resistance typically reaches approximately 320-350 kNm.

Proceeding with the W310x39 for illustration, using improved bracing (lower floor, beam spacing adjacent provides restraint at 2.67 m):

Step 6 — Lateral-Torsional Buckling Check

For buildings with concrete slab providing continuous restraint to the top flange: LTB of the positive moment region is eliminated (CSA S16:24 Clause 13.6.1). For the bare steel before composite action, check the construction stage:

Construction stage L_b = 8.0 m (unbraced during deck placement).

Plastic length L_p: L_p = 1.76 x ry x sqrt(E / Fy) = 1.76 x 38.7 x sqrt(200000 / 350) = 1,630 mm

Elastic length L_r (approximate for W310x39): L_r ~ 4,800 mm for this section.

L_b = 8,000 mm > L_r — elastic LTB governs.

Elastic critical moment (omega_2 = 1.0 for uniform moment, conservative): M_y = Sx x Fy = 546 x 10^3 x 350 / 10^6 = 191.1 kNm

M_cr for doubly symmetric section: M_cr = (pi / L) x sqrt(E x Iy x G x J + (pi x E / L)^2 x Iy x Cw) = (pi / 8000) x sqrt(200000 x 6.54 x 10^6 x 77000 x 142 x 10^3 + ...) = approximately 105 kNm (computed via CISC Beam Tables)

omega_2 factor (CISC Table 2-4, UDL on simply supported beam): omega_2 = 1.13 for simply supported beam with UDL (moment gradient factor)

M_u = omega_2 x max(M_cr, ...) = 1.13 x 105 = 118.7 kNm

M_r_LTB = 0.90 x 118.7 = 106.8 kNm << 191.4 kNm — NOT adequate during construction.

Construction requirement: Either shore the beam during concrete placement or provide temporary bracing at midspan. With midspan brace: L_b = 4.0 m, L_p < 4.0 m < L_r, inelastic LTB applies, M_r ~ 155 kNm. Still marginal — consider shoring or a heavier beam.

Final design for bare steel condition: W310x44.5, or shore at third points during construction.

Step 7 — Shear Verification (CSA S16:24 Clause 13.4)

Web shear area: A_w = d x w = 310 x 5.8 = 1,798 mm^2

Shear buckling coefficient: k_v = 5.34 (unstiffened web) Web slenderness: h/w = 50.1

Shear resistance — stocky web (h/w <= 439 x sqrt(k_v / Fy) = 439 x sqrt(5.34/350) = 54.2): V_r = phi x A_w x 0.66 x Fy = 0.90 x 1798 x 0.66 x 350 / 1000 = 373.9 kN

373.9 kN >> 95.7 kN. Shear OK with substantial reserve.

Step 8 — Deflection (NBCC 2020 Serviceability)

Live load deflection (office floor, L/360 limit): delta_L = 5 x w_L x L^4 / (384 x E x Ix) = 5 x 7.2 x 8000^4 / (384 x 200000 x 84.8 x 10^6) = 22.6 mm

L/360 = 8000 / 360 = 22.2 mm. 22.6 mm > 22.2 mm — marginally exceeds limit.

Total load deflection (L/240 limit for floors with brittle finishes): delta_total = 5 x (10.5 + 7.2) x 8000^4 / (384 x 200000 x 84.8 x 10^6) = 55.6 mm

L/240 = 33.3 mm. Total load deflection exceeds limit!

Recommendations:

• Pre-camber beam 25 mm upward (approximately 75% of dead load deflection)
• Dead load deflection = 5 x 10.5 x 8000^4 / (384 x 200000 x 84.8 x 10^6) = 33.0 mm
• With 25 mm camber, net total deflection ~ 30.6 mm < L/240. Acceptable.
• For open-plan office, consider L/480 vibration limit — check natural frequency.

Step 9 — Summary

Final specification: W310x44.5, CSA G40.21 350W, camber 25 mm upward. Provide 2 rows of 19 mm dia. shear studs at 300 mm centres for composite action (reduces LTB concern entirely once slab is cured).

CISC Handbook References

• Table 5-4: Beam selection tables, W-shapes, Fy = 350 MPa
• Table 2-4: omega_2 moment gradient factors for beams
• Part 5, Table 5.2: Section classification limits
• Part 3, Table 3.2: W-shape section properties

For production design, use the CISC Beam Selection Tables directly rather than hand calculation — the tables pre-compute M_r for all standard sections at common unbraced lengths.

Frequently Asked Questions

When can I assume continuous lateral restraint from a concrete slab? Continuous restraint can be assumed when: (a) shear studs are provided at spacing <= 600 mm, (b) the deck is positively connected to the beam (puddle welds or shot-fired pins at <= 450 mm), and (c) the slab is at least 65 mm thick over the deck. The restraint is valid for the positive moment region only; negative moment regions over supports have the bottom flange in compression and must be checked separately.

What is the difference between omega_2 (CSA) and Cb (AISC)? Both are moment gradient factors. omega_2 applies directly to the factored moment resistance M_r; Cb applies to the elastic buckling moment M_cr. For the same loading, the numerical values differ by typically 5-10%. The CISC Handbook provides omega_2 values for common restraint and loading conditions in Table 2-4.

How do I select a trial beam section for a given span? For floor beams: start with depth = span/24 to span/20 (8 m span = 333 to 400 mm depth). For roof beams: span/30 to span/24. The CISC Handbook provides preliminary beam selection tables by span and loading. A W310 (nominally 310 mm deep) at 8 m span is span/25.8 — appropriate for a moderately loaded floor beam.

Is composite action always beneficial for beam design? Composite action increases moment resistance by 50-100% compared to bare steel for typical floor beams. However, it adds fabrication cost (shear studs, deck attachment) and increases floor depth. For spans under 6 m, bare steel is often more economical. For spans over 8 m, composite action is standard practice in Canadian building construction.

This page is for educational reference. Beam design per CSA S16:24 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.