Adjusting a lumber value: the table number is only a starting point

Hibbeler Chapter 1 sized the foundry roof beam against a bending stress number straight out of a table: 800 psi for a No. 2 Southern Pine 2×10. That number is a reference value, and the design standard for wood, the National Design Specification (NDS), says to multiply it by every adjustment factor that applies before you compare anything to it. This chapter takes the same beam through the factors that matter for it, so you can see how far a table number moves and why.

The adjusted bending stress is Fb′ = Fb × CD × CM × Cr × Ci × CL, keeping only the factors that apply. The table values are for normal load duration and dry service, so everything below is a correction away from that baseline.

The factors, one at a time

• Load duration, CD (NDS 2018 Table 2.3.2): wood carries more load for a short time than for years. Permanent load (dead) 0.9, ten years (occupancy live load) 1.0, two months (snow) 1.15, seven days (construction) 1.25, ten minutes (wind or earthquake) 1.6, impact 2.0. The rule that trips people up (section 2.3.2.2): the shortest-duration load in a combination sets the factor for that combination, and you check every combination to find the critical one.
• Wet service, CM: if the wood will stay above 19% moisture for an extended time, the table values drop. For No. 2 Southern Pine dimension lumber, bending is multiplied by 0.85 only when Fb is over 1,150 psi, so for these sizes bending is unchanged (1.0). Shear drops to 0.97 and stiffness (E) to 0.9, which matters for sag. (Southern Forest Products Association, Southern Pine Use Guide, Table A-2.)
• Repetitive member, Cr = 1.15 (section 4.3.9): joists, rafters, studs and similar 2-to-4-inch lumber that are in contact or spaced 24 inches or less apart, at least three of them, tied together by sheathing, decking or other load-distributing elements, can share load, so bending gets a 15% bump. A single roof beam is not a repetitive member and gets no bump.
• Incising, Ci = 0.80 for bending (Table 4.3.8): pressure-treated lumber is often incised, with small slits cut in so preservative soaks in, and the slits cost strength.
• Beam stability, CL (sections 3.3.3 and 4.4.1.2): a deep, narrow board loaded on its edge wants to roll sideways, so the NDS either asks you to hold it straight or takes strength away. The next section works it out.

Beam stability: is the beam held straight?

Stand a ruler on its edge and push down on the middle: it does not just bend, it flops sideways. A 2×10 on edge does the same thing, only stiffer. The NDS handles it two ways. The simple way (4.4.1.2) looks at the nominal depth-to-breadth ratio and lists what bracing earns CL = 1.0: a ratio of 2 or less needs nothing; up to 4, hold the ends; up to 5, also hold the top (compression) edge in line along its whole length, as by sheathing or subflooring; up to 6, also add bridging, blocking or cross bracing at least every 8 ft; up to 7, hold both edges. Above 7 you calculate.

• A 2×6 (ratio 3) and a 2×8 (ratio 4): hold the ends.
• A 2×10 (ratio 5): hold the ends and hold the top edge in line for its whole length.
• A 2×12 (ratio 6): also bridging or blocking at least every 8 ft, so a 10 ft span needs a row at mid-span.

The calculation way (3.3.3) is for a beam that is held only at the ends or at wide intervals. The slenderness RB = √(le × d ÷ b²), where le is an effective length from Table 3.3.3 (for a uniformly loaded single span, 2.06 × the unbraced length, or 1.63 × the length plus 3 × the depth once the length is 7 or more times the depth), d is the depth and b the breadth. The critical stress is FbE = 1.20 × Emin ÷ RB², with Emin = 510,000 psi for No. 2 Southern Pine, and CL = (1 + x)/1.9 − √(((1 + x)/1.9)² − x/0.95) where x = FbE ÷ Fb*, and Fb* is the adjusted Fb before CL. RB may not exceed 50. The longer the unbraced length, the smaller CL gets.

Worked case: the roof beam again

Same beam as Hibbeler Chapter 1: 10 ft span, 5 ft of roof depth, 7 psf dead plus 20 psf snow, so M = 1,688 lb-ft. Dead plus snow is the governing combination, and snow is the shortest-duration load in it, so CD = 1.15. Dry service, one beam (no Cr), not incised, and braced the way the NDS asks for its size (CL = 1.0), which the next list takes away.

• Single 2×10: Fb′ = 800 × 1.15 = 920 psi. Required S = (1,688 × 12) ÷ 920 ≈ 22.0 in³. A 2×10 provides 21.4 in³, so it is overstressed by about 3%. At the raw table value the overstress was about 18%, so the factor helps, but not enough.
• Single 2×12: Fb′ = 750 × 1.15 ≈ 862 psi. Required S ≈ 23.5 in³ against 31.6 in³ provided, about 74% used. Passes.
• Doubled 2×8 (side by side): Fb′ = 925 × 1.15 ≈ 1,064 psi. Required S ≈ 19.0 in³ against 26.3 in³ provided, about 72% used. Passes.
• Doubled 2×10: Fb′ = 920 psi, required 22.0 in³ against 42.8 in³, about 51% used. Passes with room to spare.
• Dead load alone (CD = 0.9) gives Fb′ = 720 psi and needs only about 7 in³, so snow, not dead load, is what governs this beam.

The same beam held only at its ends

Now take away the bracing: the beam sits on its two posts and nothing holds its top edge, so the unbraced length is the full 10 ft.

• Single 2×10: le = 1.63 × 120 + 3 × 9.25 ≈ 223 in, RB ≈ 30.3, FbE ≈ 667 psi, CL ≈ 0.66, so Fb′ ≈ 607 psi. Required S ≈ 33.3 in³ against 21.4 in³: about 156% used. Fails badly.
• Single 2×12: CL ≈ 0.58, Fb′ ≈ 499 psi, required S ≈ 40.6 in³ against 31.6 in³: about 128% used. The beam that passed at 74% when braced now fails.
• Doubled 2×8, each board on its own breadth: CL ≈ 0.73, Fb′ ≈ 772 psi, required S ≈ 26.2 in³ against 26.3 in³: right at 100%, no margin at all.
• Brace the 2×10 at mid-span only (5 ft between supports) and CL rises to about 0.90, which brings it to about 114%. That is better, but still short: the top edge held along its whole length is what the NDS rule for a 2×10 asks for, and it is what earns CL = 1.0.

Answer

The Chapter 1 verdict stands: a single 2×10 does not qualify for this roof beam, even after the snow load-duration factor, though it misses by about 3% instead of 18%. A 3% miss is not something to talk yourself past, because the things you did not count (wet wood, a heavier snow year than the 20 psf design value) push the other way. A single 2×12 or a doubled 2×8 passes with real margin, but only if the beam is braced the way the NDS asks: held at the posts, with its top edge held in line (and, for a 2×12, bridging or blocking at least every 8 ft). Held only at its ends, the single 2×10 is 156% used, the single 2×12 128%, and the doubled 2×8 exactly 100%. The bracing detail on the drawing is part of the answer.

Where to be cautious

• Do not borrow factors you do not qualify for. Cr needs at least three members at 24 inches or less on center with sheathing tying them together. Taking it on a single beam would turn a failing 2×10 into an 89% pass that the wood will not honor.
• Check every load combination. Snow governed here, but a heavy dead load alone, or a wind combination (CD = 1.6), can govern a different beam. Do the whole list and keep the worst.
• Bracing has to be real. The NDS names adequate sheathing or subflooring as the way to hold the top edge in line. Rafters or purlins fastened down to the beam's top edge may do the same job, but that is a judgment for the engineer; rafters that merely rest on the beam do not hold it. Blocking at the posts is what holds the ends.
• CD does not apply to stiffness or to compression perpendicular to the grain where a deformation limit governs, so it never makes a beam sag less.
• Pressure-treated lumber is often incised (Ci = 0.80 on bending), and load duration factors above 1.6 are not allowed for lumber treated with waterborne preservatives.
• The table values are for dry service. A roof beam that stays damp, or a beam low to the ground, needs the wet service factors, and wet wood loses stiffness and shear strength even where bending is unchanged.
• Everything here is a worked example. On anything that needs a permit, or that holds up a roof over people, a structural engineer stamps the real design.

Open items for a later pass

The real roof framing for this beam, which decides whether the top edge is actually held (the detail is not drawn yet). How two boards doubled side by side count their breadth for stability (here each board is checked alone, which is conservative; a tightly fastened pair may do better). Bearing at the posts and notches in the beam, which have their own factors. Size and flat-use factors are already built into the table for 2× lumber or do not apply to an edgewise beam. Wet service and load duration still need to be carried into the deflection and connection checks.

Sources

• American Wood Council, National Design Specification for Wood Construction 2018 with Commentary (free at awc.org): Table 2.3.2 and section 2.3.2.2 (load duration), section 4.3.9 (repetitive member), Table 4.3.8 (incising), sections 3.3.3 and 4.4.1.2, Table 3.3.3 and Appendix D (beam stability).
• Southern Forest Products Association, Southern Pine Use Guide, 2013 edition (southernpine.com): Table 1 (No. 2 reference values: Fb 1,000 / 925 / 800 / 750 psi for 2×6 / 2×8 / 2×10 / 2×12, E 1,400,000 psi, Emin 510,000 psi, Fv 175 psi) and Table A-2 (wet service).
• Written from these public documents, not from the Breyer textbook itself. If the book disagrees, the book and the current NDS win.