The roof post as an NDS column: stability factor and combined loading

Hibbeler Chapter 2 checked the foundry roof post two separate ways: Euler buckling with the average stiffness of the wood, and crushing against a flat allowable of about 1,000 psi. Chapter 13 then added wind by summing the two stresses. The NDS does all of this in one method, and it is more cautious. This chapter redoes the same post, an 8 ft treated 4×4 carrying the 675 lb roof load, the NDS way.

How the NDS treats a column

• One formula for every length. A short post fails by crushing, a long thin one by buckling, and in between by a mix of the two. The column stability factor CP (NDS 3.7.1.5) blends them: Fc′ = Fc* × CP, where Fc* is the compression value with every adjustment except CP.
• A slenderness cap. The effective length divided by the smaller face dimension, le/d, may not exceed 50 (75 during construction). Beyond that the NDS does not allow a solid sawn column at all.
• Effective length. le = Ke × the real length, where Ke depends on how the ends are held (NDS Appendix G): 1.0 pinned at both ends, 0.8 fixed at the base and pinned at the top, 0.65 fixed at both, and 2.1 for a post held only at its base.
• A lower stiffness. The buckling stress is FcE = 0.822 × Emin′ ÷ (le/d)². The 0.822 is π² ÷ 12, the Euler constant for a rectangle, but the stiffness used is Emin, about a third of the average E, because it carries a lower-bound allowance and a safety factor of 1.66. For No. 2 Southern Pine Emin is 510,000 psi against 1,400,000 psi for E.

The post, step by step

• Pinned at both ends, Ke = 1.0: le/d = 96 in ÷ 3.5 in = 27.4, well under 50.
• FcE = 0.822 × 510,000 ÷ 27.4² ≈ 557 psi. Compare Euler with the full E: about 1,530 psi.
• Fc* = 1,450 psi (No. 2 Southern Pine, 4×4) × 1.15 (snow load duration) ≈ 1,668 psi. CP ≈ 0.31, so Fc′ ≈ 512 psi.
• The post can carry about 512 × 12.25 in² ≈ 6,270 lb straight down. The roof puts 675 lb on it, about 11% of that. Plain Euler gave 18,750 lb; the NDS number is about a third of that, because of the lower stiffness and its built-in safety factor.

Height matters fast. Pinned at both ends, the same 4×4 can carry about 4,160 lb at 10 ft, 2,935 lb at 12 ft and 2,180 lb at 14 ft, and 14.6 ft is the tallest the slenderness cap allows. It holds a 675 lb roof at every one of those heights, so the post is not close to its limit, but the capacity is falling much faster than the height rises.

End fixity: the free-topped post

Chapter 13 put the wind moment at the base, which is the moment of a post held only at its base. Under the NDS that post is not allowed: with Ke = 2.1 the 8 ft 4×4 has le/d = 57.6, over the cap of 50, and the tallest a free-topped 4×4 can be is 6.9 ft. At 6 ft it passes (about 2,670 lb). The practical reading: the post has to be held at the top, which in the foundry the roof beam does, or the post has to be bigger. The pinned model (the beam holds the top, the base is held) is the one that matches the build, and it is also the less severe one for wind.

Weight and wind together

For bending plus compression the NDS (3.9-3) uses (fc ÷ Fc′)² + fb ÷ (Fb′ × (1 − fc ÷ FcE)) ≤ 1.0. The first term is the axial load against the stability-reduced capacity, squared. The second is the bending stress against the bending capacity, magnified by 1 ÷ (1 − fc ÷ FcE) because a post that is bending and pushed down bends further. A second check, (3.9-4), must stay under 1.0 as well.

• The roof post with 14.5 psf of sideways wind on its face (Chapter 13 with Cf = 2.0), pinned both ends: the wind makes 4.2 lb/ft, a bending moment of about 406 lb-in and a bending stress of about 57 psi against Fb′ ≈ 1,265 psi. The combined value is about 0.06, far under 1.0.
• Chapter 13's simple sum of the two stresses came to about 11% of a 1,000 psi limit for the same post. The two methods agree that this post is nowhere near its limit; the NDS one just shows it as 6% of its own, stricter, equation.
• Taller posts squeeze the margin: the same combined value is about 0.11 at 10 ft, 0.18 at 12 ft and 0.29 at 14 ft.

Answer

Under the NDS the roof post is fine with a wide margin: it uses about 11% of its column capacity for weight alone and about 6% of the combined limit with a stiff wind on its face. The Euler number in Chapter 2 was about three times too generous for design, and the free-topped model in Chapter 13 was not an allowed configuration. Neither changes the verdict for this post, but they matter for a taller or slimmer one, where capacity falls quickly and the slenderness cap ends the discussion at 14.6 ft for a 4×4.

Where to be cautious

• Ke values assume the ends are really held. A post whose top is not tied to the beam is not pinned; treat it as free and the slenderness cap bites at 6.9 ft for a 4×4. A post set in the ground is not reliably fixed at its base, so do not count on fixity for strength.
• The load duration factor 1.6 is for a combination that includes wind. The snow factor 1.15 is for the roof weight with snow. Pick the combination that matches the loads you are checking.
• The slenderness limit is a hard stop, not a warning: over 50 the NDS gives no capacity.
• These formulas assume the load is centered on the post. A beam resting off-center adds bending (NDS 15.4).
• Treated lumber is often incised (×0.8 on strength), and wet service lowers the compression value (×0.8 here) and the stiffness (×0.9).
• Everything here is a worked example. A post that holds up a roof over people is for a structural engineer to design and stamp.

Open items for a later pass

Posts and timbers 5 in and thicker (6×6 and 8×8) use a different table of reference values that was not read here, so only 4×4 and 4×6 are included; the Ogopogo fire tower posts (Hibbeler Chapters 7 and 12) will need it. The buckling length coefficients come from the NDS recommended values as listed in a university course table, since Table G1 itself is a figure that could not be read. Eccentric loads, tapered or round columns, and the post's own base and top connections.

Sources

• American Wood Council, NDS 2018 with Commentary (free at awc.org): 3.6 and 3.7 (column stability factor, Equation 3.7-1, slenderness limit), 3.9.2 (Equations 3.9-3 and 3.9-4), Appendix G (effective length) and Appendix H (background).
• Southern Forest Products Association, Southern Pine Use Guide, 2013: Table 1 (No. 2 Fc 1,450 and 1,400 psi, Fb 1,100 and 1,000 psi, Emin 510,000 psi), Table A-2 (wet service) and Table A-4 (flat use).
• Purdue University CE 479 wood design notes: the worked NDS example used to check this chapter's calculation (CP 0.792, combined value 0.884). Michigan Tech's course page for the recommended Ke values.
• Written from these public documents, not from the Breyer textbook itself. If the book disagrees, the book and the current NDS win.