Joint geometry, proportion and deflection equations.
δ = (5 × w × L⁴) / (384 × E × I)
δ = (P × L³) / (48 × E × I)
δ = (P × L³) / (3 × E × I)
I = (b × h³) / 12
Miter = arctan( sin(S) x tan(C/2) ) [S = spring, C = corner angle]
Bevel = arcsin( cos(S) x sin(C/2) )
Dihedral = arccos( cos2(S) x cos(C) + sin2(S) )
Polygon corner = 180 - (360 / N_sides)
φ = (1 + √5) / 2 = 1.6180339887…
Given width a → height b = a × φ
Given height b → width a = b / φ
Diagonal d = a × √(1 + φ²) = a × 1.9021…
theta = 90 - (180 / N) where N = number of segments
L = 2 x R_outer x sin(PI / N)
dL = kerf / sin(theta) [material lost per cut end]
L_cut = outer_chord - kerf_correction
V = PI x (Rout2 - Rin2) x H / N
FH = (OH - bot_reveal - reveal x (N+1)) / N
FW = OW - 2 x reveal
BW = OW - 3mm
BW = OW - 2 x slide_clearance
Yn = bot_reveal + (n-1) x (FH + reveal) + reveal
N (odd) = nearest odd to (W / finger_width)
Actual fw = W / N
Glue area = 2 × N × fw × thickness
Strength ratio = glue_area / (W × thickness) [typically 3–6×]
Tenon thickness = Rail thickness / 3
Tenon width = Rail width − 2 × shoulder_allowance
Tenon length = 4.5 × tenon_thickness (max = 80% of stile thickness)
Mortise depth = Tenon length + 2mm clearance
Cheek = (Rail thickness − Tenon thickness) / 2
theta = arctan(1 / ratio)
TW = T_narrow + 2 × thickness × tan(theta)
round((W − min_pin) / (TW + min_pin))
S = (W − pin_width) / N_tails
Movement = W × S × ΔMC (W=width, S=coeff/MC%, ΔMC=change %)
Radial coeff ≈ Tangential × 0.55
Gap per board = max(shrink, swell) × 1.15 (15% safety factor)
2016 — W = k × d × L (k = 12/10/7/5 by density group)
NDS 2018 — W = 124.9 × G^1.5 × D^0.75 (lbs/in penetration)
Eurocode 5 — f_ax,k = 0.52 × d^-0.5 × L^-0.1 × ρ^0.8
Group factor — Cg = max(0.4, 1 − 0.05×(n−1))