The Quantitative Woodshop

§0Notation & conventions

Unless stated otherwise: lengths and section dimensions in inches, loads in lbf, moduli in psi, angles in degrees. E is the modulus of elasticity parallel to grain; I the second moment of area (in⁴); δ deflection; w a distributed load per unit length (lbf/in) and W its total (lbf); G specific gravity on an ovendry-mass basis; MC moisture content as a percentage of ovendry mass. Wood is orthotropic — every property below is direction-dependent, and the longitudinal, radial, and tangential axes are not interchangeable.

§1Material properties

One table drives everything downstream: stiffness for the deflection cases in §2, shrinkage coefficients for the movement model in §3, specific gravity for the withdrawal equations in §5. Values are means for small clear specimens at 12% MC; real boards with knots and grain runout come in lower, which is what design safety factors are for.

Shrinkage percentages are total, green to ovendry. T = tangential (across flat-sawn width), R = radial (across quartersawn width). The T/R ratio predicts cupping and checking propensity: high-ratio species (pine, red oak) distort more per unit ΔMC. Plywood's cross-laminated plies mutually restrain movement — the reason it exists — at the cost of ~40% of solid wood's flexural stiffness.

§2Structural mechanics

Shelves, aprons, and stretchers are Euler–Bernoulli beams. For a rectangular section of width b and depth (thickness) h:

The four cases that cover nearly everything in furniture:

Serviceability limits and creep

A shelf "reads flat" below roughly 0.02 in of sag per foot of span (≈1/60° of slope, the usual Sagulator criterion); L/360 is the flooring convention. Wood is viscoelastic: under sustained load it creeps to roughly 1.5–2× the initial elastic deflection over months to years, worse at high or cycling humidity (mechano-sorptive creep) and much worse in MDF. Design against ≈2δ_elastic, not δ_elastic.

Legs as columns

Slender legs and posts fail by Euler buckling before crushing:

with effective-length factor K = 1.0 (pinned–pinned), 0.7 (fixed–pinned), 0.5 (fixed–fixed), 2.0 (fixed–free, i.e. an unbraced post). Valid only while the column stays slender and elastic; stretchers exist to cut L, which enters squared. A ¾″-square pinned oak leg 28″ tall buckles near P_cr ≈ π²(1.82×10⁶)(0.0264)/28² ≈ 600 lbf — fine for a side table, marginal for something climbed on.

§3Hygroscopic movement

Below the fiber saturation point (FSP ≈ 28% MC for most temperate species) dimensional change is approximately linear in moisture content. For a dimension D across the grain:

where S is the total (green→ovendry) tangential or radial shrinkage from §1, chosen by how the board was sawn: flat-sawn width moves tangentially, quartersawn width radially. Longitudinal movement is ~0.1–0.2% total and ignorable except in reaction wood. Equation (7) linearizes a mildly sigmoid relationship and mixes dimension bases, so treat results as ±20% — ample for sizing gaps, useless for pretending wood won't move.

Equilibrium moisture content

Wood equilibrates toward an EMC set by relative humidity and temperature. The Wood Handbook's fit is the Hailwood–Horrobin sorption model, with h = RH as a fraction and T in °F:

with temperature polynomials W = 330 + 0.452T + 0.00415T², K = 0.791 + 4.63×10⁻⁴T − 8.44×10⁻⁷T², K₁ = 6.34 + 7.75×10⁻⁴T − 9.35×10⁻⁵T², K₂ = 1.09 + 2.84×10⁻²T − 9.04×10⁻⁵T². The two bracketed terms are monolayer (Langmuir-type hydrate) and polylayer (dissolved) water — a two-population sorption model, which is why no single-exponent curve fits the isotherm.

Interiors in heated climates commonly swing from ~30% RH in winter to ~65% in summer — an EMC range of roughly 6% to 12%. That ΔMC ≈ 6% is the design condition for furniture.

§4Geometry of the cut

Plain miters

For a closed n-gon frame, each end is cut at half the exterior angle: 180°/n from square (n = 4 → 45°, 6 → 30°, 8 → 22.5°, 12 → 15°). Verify a glue-up is rectangular by comparing diagonals — d = √(L² + W²) and equality of the two diagonals is equivalent to squareness for a parallelogram.

Compound miters

For an n-sided box or frame whose sides slope at angle S from horizontal (stock flat on the saw table, S = 90° is a vertical-sided box, S = 0° a flat frame), the joint plane bisects the dihedral between adjacent faces. Working out the intersection of that bisecting plane with the board plane gives the saw settings, both measured from a square cut with a vertical blade:

Sanity checks: S = 0 recovers the flat picture frame (M = 45°, B = 0 for n = 4); S = 90° recovers the plain vertical miter (M = 0, B = 45°). Crown molding is the same problem with the spring angle as the complement of S.

Arcs and ellipses

Given a chord c and rise h (the usual situation: you know the opening and how much curve you want), the radius follows from the intersecting-chords theorem:

An ellipse with semi-axes a > b has foci at ±c on the major axis with c = √(a² − b²) — set two pins and a string of length 2a for layout, or use a trammel (a stick with pins at distances a and b from the pencil, riding in two perpendicular slots) for a cleaner curve.

Dovetail slopes

The mechanical difference across this range is small; short-grain fragility at the pin tips grows with angle, which is the real argument for 1:8 in brittle hardwoods.

Proportioning

For a graduated drawer stack of total opening H, n drawers in geometric progression with ratio r (1.15–1.3 reads well), the smallest drawer height is:

E.g. H = 24″, n = 4, r = 1.2 → heights 4.47″, 5.37″, 6.44″, 7.73″. The golden ratio φ = (1+√5)/2 ≈ 1.618 and the √2 rectangle are conventional starting proportions for casework elevations; the empirical evidence that φ is perceptually privileged is weak, so treat it as a defensible default rather than a law.

§5Joinery & fasteners

Withdrawal resistance

Wood Handbook empirical fits, per inch of thread (or shank) penetration into side grain, with G ovendry-basis specific gravity and D the diameter in inches:

These are ultimate short-term values on clean specimens: divide by ~6 for design. Note the G² and G^(5/2) dependence — density dominates. A #8 screw (D = 0.164″) in hard maple (G ≈ 0.56 ovendry) holds ≈ 800 lbf per inch of engagement ultimate; the same screw in white pine (G ≈ 0.34) about 300. End-grain withdrawal is roughly 75% of side-grain for screws and unreliable for nails — hence pocket screws angle into side grain.

Pilot holes

Principle: pilot ≈ root (minor) diameter in hardwood so threads cut rather than split; one size under root in softwood so fibers compress into the threads. Clearance hole in the top piece ≥ major diameter, or the screw can't clamp the joint.

Mortise & tenon proportions

• Tenon thickness ≈ ⅓ of the mortised member, rounded to the nearest chisel/bit — it balances tenon shear against mortise-cheek splitting, the two competing failure modes.
• Tenon width ≤ ~5–6× its thickness; beyond that, cross-grain movement inside the mortise builds enough stress to split — use twin tenons with a gap.
• Depth ≥ 5× thickness where racking loads matter; haunch the tenon at frame corners to keep glue area without weakening the stile end.
• Modern PVA shear strength (3,000–4,500 psi) exceeds wood's own shear parallel to grain (≈1,000–2,000 psi at 12% MC), so a properly fitted long-grain glue joint fails in the wood. Joint design is therefore geometry: its job is converting loads into long-grain shear on adequate area, because end-grain bonding and cross-grain tension are the modes glue cannot save.

§6Stock dimensions

The pattern: subtract ¼″ below nominal 2″, ½″ from 2–6″, ¾″ above 6″ (widths ≥ 8″ lose ¾″).

Imported hardwood ply and Baltic birch are milled to metric thicknesses that never line up with these nominal inch sizes — a nominal ½″ Baltic sheet measures about 0.47″, ¾″ about 0.71″. The operational rule: never cut a dado to the nominal size — measure the actual sheet with calipers, or use an undersized-plywood router bit.

§7Cutting & yield

Board feet

Hardwood is billed on rough thickness (4/4 counts as 1″ even after surfacing to 13/16″) and never below 1″. Budget a waste factor of 1.15–1.4× over the cut list: defect boxing, grain matching, and kerf all eat stock, and the factor climbs with lower lumber grades and pickier grain matching.

Pieces per stick, with kerf

Cutting pieces of length ℓ from stock of length L with kerf k, the last piece needs no trailing kerf:

Full-kerf blades cut ⅛″, thin-kerf 3/32″. The general cut-list problem is 1-D bin packing (NP-hard); first-fit-decreasing — allocate longest parts first — is the sensible heuristic by hand and is what optimizer software runs anyway. Leave 1″+ of trim on rough stock ends: end checks travel farther than they show.

Blade kinematics

with feed F in in/min, spindle speed N in rpm, tooth count T, diameter D in inches. A 10″ blade at 4,000 rpm runs ≈ 10,500 sfpm (~120 mph) at the rim. Hand-feeding 20 ft/min through a 24-tooth rip blade gives c ≈ 0.0025″ per tooth — healthy. The same feed through an 80-tooth blade gives 0.00075″: teeth rub instead of cutting, and rubbing is burning. That single equation explains most burn marks: more teeth demand faster feed, not slower.