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How to use
A Blender-built nave with four bays, aisles, clerestory windows, flying buttresses, piers and pinnacles, driven by the same statics as the 2D page. Drag to orbit, wheel to zoom, use the camera presets or Guided tour. The glowing line is the line of thrust; the pier colour shows the safety factor (green safe, amber cracked, red failing) and a failing pier tips about its toe. Storm test ramps the wind until a pier fails; Auto-size pinnacle sets the ballast that keeps the line in the middle third. Wind streaks, stained-glass light shafts and the dust motes are visual only. Things to try: remove the flyers; use Storm 40 m/s; use Light pinnacles; compare Wide shallow vault.
The science
A Gothic nave is a stone vault carried on thin walls. A stone arch or vault does not only push down: because the stones can only push (they carry almost no tension), the vault also pushes outwards on its supports. The structure stands if a line of thrust, the path of the resultant force, can run through the stones without leaving them. For a uniformly loaded vault of span L, rise f and load w per metre, the line of thrust is a parabola and the horizontal thrust at the springing is
Hv = w L2 / (8 f), w = qv B (kN/m), per bay of width B
so a flatter vault (smaller f) pushes harder: the thrust varies as 1/f. With the reference bay (L = 10 m, f = 4 m, qv = 6 kN/m2, B = 6 m) Hv = 112.5 kN per bay on each side.
The wall alone. A wall of thickness t, height Hw and weight Ww = t Hw B γ (with γ = ρ g) tips about its outer toe when the overturning moment exceeds the restoring moment:
SFwall = (Ww + Vv) (t/2) / (Hv hs ± (W/2) yw), Vv = qv (L/2) B
For the reference numbers this is about 0.93: the wall by itself is just beyond its limit. The builders of the 12th century solved this by moving the support outside, as a flying buttress: an inclined arch that carries the thrust over the aisle roof to a massive pier.
Wind. The dynamic pressure is q = ½ ρair v2 (ρair = 1.25 kg/m3); the force on the exposed roof and clerestory is W = q Cp Hexp B. In this model the nave acts as a rigid box that shares the wind shear equally between its two flyers, so the windward flyer carries Hv − W/2 (the wind pushes the wall onto its support, relieving it) and the leeward flyer Hv + W/2.
The pier and the middle third. The flyer arrives at the pier at height hc = ha − sf tanβ with a horizontal force Hf and a vertical component Hf tanβ + Wflyer/2. Together with the weight of the pier (b d Hp γ) and the pinnacle (m g), the reaction at the base lies at a distance from the inner face
xR = ( Σ Vi xi + Hf hc ) / Σ Vi, SF = Σ Vi (b − xi) / (Hf hc)
The pier is safe against overturning if SF > 1 (xR < b). The masonry rule is stricter: if xR exceeds 2b/3 the reaction leaves the middle third, the inner face goes into tension, and mortar joints open (cracks). A heavy pinnacle adds vertical load and pulls the line back: the pinnacle mass that puts the reaction exactly at a chosen position X is P = (X ΣVo − ΣVoxo − Hf hc) / (b/2 − X). That is why Gothic piers carry pinnacles: they are not decoration, they are ballast.
Numerical checks run in the page (independent re-implementations). (1) The vault thrust from a numerical integration of the shear of a simply supported beam divided by the rise against wL2/(8f); (2) the pier reaction from moments about the inner face against moments about the toe; (3) the pinnacle mass for SF = 1 from the closed form against bisection on the full solver; (4) the pinnacle mass for the middle third, same way; (5) the dynamic pressure at 20 m/s (250 Pa); (6) the vertical flyer load growth with tanβ.
Simplifications, stated honestly. Rigid-body statics per bay: no masonry deformation, no sliding, no vault cracking, no ribs; the nave wall is treated as rigid between the vault level and the flyer contact (no bending of the wall); the wind coefficient Cp = 1.2 is a typical bluff-body value, not a code value; the vault load of 6 kN/m2, the stone density of 2300 kg/m3 and all member sizes are typical illustrative numbers, not measurements of a real cathedral; the load is shared equally between the two sides. Real cathedrals (Chartres, Amiens, Reims) were stabilised by many additional effects, including the mortar tensile strength and the stiffness of the tracery. This lab teaches why the structure is shaped as it is; it is not a safety assessment of any building.