How to Analyze and Design a Two-Span Continuous Structural Glued Laminated Timber Floor Girder in ASD Using NDS 2024 (Positive and Negative Bending with Supplement Table 5A Fbx+ and Fbx−, Volume Factor Cv, and Beam Stability at an Interior Support)

This note is stand-alone for flexure, shear, and deflection design checks for a very common office and light commercial condition: a floor girder that is continuous over a column so it develops positive (sagging) moment in each span and negative (hogging) moment over the interior support. For structural glued laminated timber (glulam), Supplement Table 5A gives two reference bending stresses—F_bx⁺ for zones where the bottom of the beam is in tension (typical positive-moment regions) and F_bx⁻ where the top is in tension (typical negative-moment regions). Both must be checked against adjusted allowable stresses F′_b built from the same adjustment factors (C_D, C_M, C_t, C_L, C_V, C_fu, C_i, C_r) applied to the appropriate reference value. Internal forces, shear, and deflection used in the hand stress checks below are taken from StructSuite’s elastic beam model (Wood Beam module, load combination 2a. D + L, w = 550 plf total service, two 18 ft spans, pinned supports)—the same analysis as Section 1 figure. Classical closed-form coefficients (e.g. wL²/8 at the interior) match hogging moment and reactions; peak sagging moment is taken from the line analysis (9wL²/128 for equal spans), not the smaller wL²/16 value at midspan. This note does not replace a full package (connections, bearing, torsion, vibration, notches, fire, or load combinations beyond D + L).

Companion notes: Simple-span glulam beam ASD example. NDS Supplement Tables 1A–6B overview.

What governs in real design?

1. Problem statement and framing context

A two-bay strip of floor is framed so wood I-joists (or truss joists) span perpendicular to a built-up wood or steel column line at mid-building. A single glulam girder runs continuous over that column: two equal spans L, uniform floor dead plus live load along the full length. The girder supports a tributary width W of floor; joists bear on the girder top and continuous structural panel sheathing provides lateral support to the top edge of the glulam along most of the beam. At the interior support, the beam develops hogging moment (negative bending in the usual sagging-positive sign convention); the compression zone for that region is along the bottom of the beam at that location. Lateral bracing of that compression edge (often strapping, ceiling, knee framing, or ties to the column line) must be evaluated—this example assumes C_L = 1.0 in both zones only after the engineer confirms unbraced length and moment gradient per NDS 5.3.4 and 5.3.5; do not assume 1.0 without that check.

StructSuite — beam model, load, and diagrams (example geometry, 2a. D + L)

Uniform w on both spans; interior support is a column or wall under the beam. Sagging (M⁺) in each span uses F_bx⁺ (bottom tension); hogging (M⁻) at the interior support uses F_bx⁻ (top tension). The figure below is from StructSuite (Wood Beam module): two 18 ft spans, pinned supports at the three nodes, D = 220 plf and L = 330 plf uniform on the full length (service-level line loads from the table above), glulam 24F-1.8E, 5.125 in. × 15 in., visualization load case 2a. D + L (ASCE 7-22 ASD). It shows the beam with uniform load, then deflection, shear, and moment diagrams for that combination—the same elastic ordinates used for moments, shear, and deflection in Sections 2 and 9.

Figure. Two-span continuous glulam over three supports with uniform load on both spans; negative bending at the interior support and positive bending in the span regions—F_bx⁺ vs F_bx⁻ per Supplement Table 5A. Diagrams generated in StructSuite as described above.

Why this geometry matters in practice: Specifiers often choose continuous girders to reduce depth or meet deflection limits compared with simple spans. The penalty is a larger negative moment at the interior column than the positive moment in the span, and often a lower tabulated F_bx⁻ than F_bx⁺, so the hogging zone frequently governs bending even when compression-edge bracing is adequate.

2. Elastic internal forces (StructSuite beam solver, 2a. D + L)

For this prismatic two-span beam with equal spans L = 18 ft and uniform w = 550 lb/ft on the full length (D + L service loads combined for the strength / service diagrams), StructSuite’s elastic line analysis gives the ordinates below (visualization combination 2a. D + L). Reactions match the usual continuous-beam closed forms; peak sagging moment is the maximum M⁺ along the span (9wL²/128 for equal two-span UDL), not the wL²/16 value at midspan.

Note: Midspan sagging moment wL²/16 = 11,138 lb-ft is lower than the true peak 12,529 lb-ft; use 12,529 lb-ft for stress at the critical sagging section.

3. Section properties (gross section)

b = 5.125 in., d = 15.0 in.

S = bd²/6 = 5.125 × 225 / 6 = **192.2 in.³**

I = bd³/12; **d**³ = 3,375 in.³; **I** = 5.125 × 3,375 / 12 = **1,441.4 in.⁴**

Rounding in this note: S and I use one decimal in.³ / in.⁴; bending stresses to whole psi unless a tighter value is needed for the comparison; deflection to two decimal places of an inch in summaries. Use full precision on contract calculations.

4. Reference design values (Supplement Table 5A)

Table 5A (bending members) for 24F-1.8E (values consistent with the NDS Supplement):

Key point: F_bx⁻ is not a “reduced F_b for the same fiber”; it is the reference stress for the stress state associated with negative bending about the strong axis. Apply the same C factors (with C_L and C_V rules) to each reference value to obtain F′_b⁺ and F′_b⁻.

5. Adjustment factors (ASD)

C_D: D + L with floor live controlling duration → C_D = 1.0 (NDS load duration table; shortest-duration load in the combination).

C_M, C_t: Dry service, T ≤ 100 °F → 1.0 for strength and E (unless project-specific otherwise).

C_fu, C_i, C_r: Strong-axis bending, single girder → C_fu = 1.0, C_i = 1.0, C_r = 1.0 per NDS applicability.

C_L: Required for both positive and negative moment regions. Positive zones: compression on top—often fully braced by floor sheathing. Negative zone at interior support: compression on bottom—often not braced the same way; unbraced length l_u between points of lateral support to the compression edge must be used in NDS 5.3.4. This example assumes C_L = 1.0 in both zones for illustration after the engineer verifies bracing; if C_L < 1.0 in the hogging region, recalculate F′_b⁻—that often controls continuous glulam design.

C_V (NDS 5.3.6, Equation 5.3-1): Use span length L = 18 ft (each span), d = 15 in., b = 5.125 in., x = 10 (Douglas Fir–Larch, not Southern Pine):

C_V = (21/18)^0.1 × (12/15)^0.1 × 1 ≈ **0.993**

C_V vs C_L for glulam F_bx: C_V is computed per NDS 5.3.6 (Equation 5.3-1). C_L is computed per NDS 5.3.4 / 5.3.5 (beam stability). These are different phenomena; do not assume one formula applies to the other. For structural glued laminated timber reference bending design values from Supplement Table 5A, the Supplement states that C_V shall not be applied simultaneously with C_L—use the lesser of C_V and C_L in the F_bx adjustment path (verify the exact sentence in your NDS/Supplement edition). Sawn dimension lumber uses size factor C_F, not C_V; the Table 5A C_V/C_L rule is glulam-specific.

6. Flexure — positive moment region (sagging, F_bx⁺)

Bending stress magnitude (M⁺ = 150,348 lb-in from Section 2):

f_b⁺ = M⁺ / S = 150,348 / 192.2 ≈ **782 psi**

Adjusted allowable (same factor product applied to F_bx⁺; min(C_L, C_V) = min(1.0, 0.993) = 0.993):

F′_b⁺ ≈ 2,400 × 1.0 × 1.0 × 1.0 × 0.993 × 1.0 × 1.0 × 1.0 × 1.0 ≈ **2,380 psi**

Check: 782 psi ≤ 2,380 psi → OK (positive zone is not governing here).

7. Flexure — negative moment region (hogging, F_bx⁻) — usually governing

Bending stress magnitude at interior support (|M⁻| = 267,288 lb-in from Section 2):

f_b⁻ = \| M⁻ \| / S = 267,288 / 192.2 ≈ **1,391 psi**

Adjusted allowable (apply adjustments to F_bx⁻; min(C_L, C_V) = 0.993):

F′_b⁻ ≈ 1,450 × 1.0 × 1.0 × 1.0 × 0.993 × 1.0 × 1.0 × 1.0 × 1.0 ≈ **1,440 psi**

Check: 1,391 psi ≤ 1,440 psi → OK for this trial section if C_L = 1.0 in the hogging region.

7.1 Beam stability at interior support — sensitivity (why C_L = 1.0 is often too optimistic)

At the interior support, hogging puts compression in the bottom of the beam. Floor sheathing on the top does not brace that compression edge. Unless strapping, ties, ceiling, blocking, or column details provide lateral support to the bottom flange near the support, C_L can be well below 1.0 and often governs continuous glulam design. The baseline calculation above passes only because f_b⁻ ≈ 1,391 psi is barely below F′_b⁻ ≈ 1,440 psi (demand is ~97% of capacity).

Using the same F_bx⁻ adjustment path as Section 7 with min(C_L, C_V) and C_V = 0.993 fixed, F′_b⁻ ≈ 1,450 × min(C_L, 0.993) psi (other factors 1.0):

To just satisfy f_b⁻ ≤ F′_b⁻ with f_b⁻ = 1,391 psi and F_bx⁻ = 1,450 psi, you need min(C_L, C_V) ≥ 1,391 / 1,450 ≈ 0.96 (so C_L ≥ ~0.96 when C_V controls at 0.993). Calculate C_L from NDS 5.3.4 using the actual l_u, l_e, and C_b for the hogging segment—do not copy C_L from the positive-moment zones.

Engineering takeaway: The ratio of demand to capacity is much higher here than in the positive zone (~0.97 vs ~0.33 on stress basis for this set of numbers). In practice, continuous glulam often sizes from F_bx⁻ or from deflection or from C_L at the interior support.

8. Shear (maximum considered here — interior support)

For a rectangular section, NDS 3.4.2 is used in many wood design workflows in the form V_n = (2/3) F′_v A with A = bd, or equivalently compare 1.5V/A to F′_v. With V = 6,142 lb from Section 2, b = 5.125 in., d = 15 in.:

f_v = 3V / (2bd) = 3 × 6,142 / (2 × 5.125 × 15) ≈ **120 psi**

f_v = 1.5V / A with A = bd = 76.875 in.² → 1.5 × 6,142 / 76.875 ≈ **120 psi**

F′_v = F_vx × C_D × C_M × C_t × C_i × C_vr = 265 psi (dry, C_vr = 1.0 without notches).

Check: 120 psi ≤ 265 psi → OK.

8.1 Bearing and tension perpendicular to grain (qualitative)

Strong-axis bending puts extreme-fiber tension and compression parallel to grain along the beam length; the F_bx⁺ / F_bx⁻ checks above address that. Separately, supports and connections can introduce compression perpendicular to grain (F_c⊥, bearing per NDS bearing provisions) and, for some details, tension or splitting perpendicular to grain (e.g. concentrated reactions, hangers, notches, or fastener groups). Wood is relatively weak in tension perpendicular to grain; check tension perpendicular to grain at supports and connections where applicable to the load path and detailing, per NDS and manufacturer data. This hand example does not size bearing length, hardware, or connection capacities.

9. Deflection (serviceability — elastic δ from StructSuite)

For serviceability, compare immediate elastic deflection from the same E′I model as the analysis. StructSuite reports peak magnitude δ for the selected visualization case; for 2a. D + L (total w = 550 plf on both spans), δ_max = 0.2083 in. (peak magnitude along the span; E′ = E × C_M × C_t × C_i for stiffness, E′ ≈ E here).

δ_D+L = **0.2083 in.**

Live-load-only deflection for the same beam and supports scales linearly with load intensity in elastic range: δ_L ≈ δ_D+L × (w_L / w) = 0.2083 × (330 / 550) ≈ 0.125 in.

δ_L ≈ **0.125 in.**

Compare to IBC Table 1604.3 limits (e.g. L/360 for live on floors with flexible finishes): L/360 = 216/360 = 0.60 in. → OK for δ_L vs L/360.

Long-term deflection (creep): The elastic δ above is immediate. Sustained loads cause creep; total deflection over time can exceed these values. Confirm NDS, IBC, and project rules for creep factors or stricter serviceability criteria where applicable.

10. Governing summary (this example)

11. Relation to analysis software and StructSuite

Any tool you use should: (1) produce M⁺ and M⁻ for continuous framing; (2) apply F_bx⁺ and F_bx⁻ in the correct regions; (3) apply C_V (5.3.6) and C_L (5.3.4/5.3.5) per NDS, and for glulam Table 5A follow the Supplement rule that C_V and C_L are not used together—use the lesser where that provision applies. StructSuite’s framing verification path for glulam uses Supplement Table 5A for F_bx⁺/F_bx⁻ and computes C_V from NDS Equation 5.3-1; continuous beam moments and shears come from the beam solver for the loads you enter.

12. References (NDS 2024 and Supplement)

• NDS 2024 Chapter 4 — Load duration (C_D).
• NDS 2024 Chapter 5 — ASD adjustments; Equation 5.3-1 (C_V); beam stability (C_L).
• NDS Supplement Table 5A — F_bx⁺, F_bx⁻, F_vx, E, E_min.
• IBC — Deflection limits; live load.
• Elastic continuous-beam theory — line analysis for moments/shear/deflection; peak M⁺ for equal two-span UDL is 9wL²/128 (not wL²/16 at midspan).

This technical note is educational. Confirm bracing, C_L, reference stresses from the Supplement edition in use, reactions, connections, and deflection criteria with the governing NDS, IBC, and the authority having jurisdiction.