Foundry roof post — buckling & compression

Continues Chapter 1's open item — does the treated wood post holding up the foundry roof beam actually hold, or does it buckle or crush first?

Concepts pulled in

Euler buckling (Pᴄᵣ = π²EI/Lₑ²)
Axial compressive stress (σ = P/A)

The problem: continues Chapter 1's open item — does the treated wood post holding up the foundry roof beam actually hold, or does it buckle or crush first? Load: 675 lb per post (half the beam's total reaction — each beam end sits on one post).

Assumed: nominal 4×4 treated post (actual 3.5" × 3.5"), 8 ft (96 in) tall, pinned-pinned end conditions (conservative first pass). Concepts pulled in: Euler buckling (Pᴄᵣ = π²EI/Lₑ²) and simple axial compressive stress (σ = P/A) — the two failure modes that govern a slender compression member.

Buckling check

I = bd³/12 = 3.5 × 3.5³ ÷ 12 ≈ 12.5 in⁴
E ≈ 1,400,000 psi (typical treated Southern Yellow Pine)
Pᴄᵣ = π² × 1,400,000 × 12.5 ÷ 96² ≈ 18,760 lb

Compression (crushing) check

σ = P/A = 675 ÷ (3.5 × 3.5) ≈ 55 psi, vs. an allowable of roughly 1,000+ psi for treated SYP.

Answer

Both pass with wide margin — buckling capacity is ~28× the actual load, compressive stress is under 6% of allowable. At this load, the 4×4 post isn't sized by strength at all; it's sized by practical minimums (standard stock, impact resistance, code minimum sizes). Not every member in a structure is governed by the same failure mode — sometimes "already way oversized" is the honest answer.

Open items for a later pass

Wind uplift/lateral load on the post (closed in Chapter 3), and a real NDS column-stability-factor check. This chapter uses the simpler Euler form Hibbeler teaches; Breyer Chapter 4 now does the NDS check, and the allowed load comes out about a third of the Euler number (6,270 lb against 18,750 lb for this post).

Try your own numbers

Starts on this chapter's foundry roof post. Change anything to test your own post. Straight downward load only: wind on the post (Chapter 13) is separate. This is the plain Euler form; Breyer Chapter 4 redoes this post with the NDS column stability factor, which allows far less load.

Load on the post (lb)e.g. half the beam's total reactionPost height (ft)unbraced lengthModulus of elasticity, E (psi)default is typical treated Southern Yellow Pine, as in the chapterAllowable compression (psi)chapter says roughly 1,000+ for treated SYP. The No. 2 Southern Pine table value for a 4×4 is 1,450 psi (SFPA Southern Pine Use Guide, Table 1), and incised treated lumber takes ×0.8 of that, so 1,000 is conservative; use your lumber's valuePost sizeEnd conditions

Holds against bucklingHolds against crushing27.8× buckling margin

Weak-axis I = 12.5 in⁴, area A = 12.3 in²
Pcr = π²EI/(KL)² = π² × 1,400,000 × 12.5 ÷ (1 × 96)² = 18,749 lb
Buckling margin: 18,749 ÷ 675 = 27.8×
σ = P/A = 675 ÷ 12.3 = 55.1 psi, which is 5.5% of the 1,000 psi allowable
Slenderness KL/r = 95

Pcr is the load where the post fails, not a safe load, so you want a wide margin. Euler buckling assumes a long slender post; on a short stocky one, the crushing check is the one that matters.

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