Gothic Cathedral Flying Buttress: Vault Thrust Line, Wind Load, Pinnacle Ballast and Pier Stability

Presets

Live readouts

Graphs

Safety factor against wind speed (pier + flyer and the wall alone)

Thrust line inside the pier: height against position (middle third lines)

Vault thrust against rise f

Reaction position at the leeward pier base against pinnacle mass

What is happening now

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Numerical checks

running...

How to use

Pick a preset or drag any slider. The yellow dashed line is the line of thrust; the coloured bar under the leeward pier is the base split into thirds (amber, green = middle third, red) and the white dot is where the reaction lands. Storm test raises the wind by 4 m/s every second until something fails. Auto-size pinnacle finds the pinnacle mass that keeps the leeward thrust line in the middle third. Things to try: (1) switch the flyers off and see the wall fail; (2) shrink the vault rise f to 3 m and watch the thrust grow; (3) remove the pinnacles and watch the dot move into the red third; (4) increase the flyer inclination and see the vertical load on the pier grow; (5) compare Wide shallow vault with Tall slender 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

H_v = w L² / (8 f), w = q_v 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, q_v = 6 kN/m², B = 6 m) H_v = 112.5 kN per bay on each side.

The wall alone. A wall of thickness t, height H_w and weight W_w = t H_w B γ (with γ = ρ g) tips about its outer toe when the overturning moment exceeds the restoring moment:

SF_wall = (W_w + V_v) (t/2) / (H_v h_s ± (W/2) y_w), V_v = q_v (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 ρ_air v² (ρ_air = 1.25 kg/m³); the force on the exposed roof and clerestory is W = q C_p H_exp 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 H_v − W/2 (the wind pushes the wall onto its support, relieving it) and the leeward flyer H_v + W/2.

The pier and the middle third. The flyer arrives at the pier at height h_c = h_a − s_f tanβ with a horizontal force H_f and a vertical component H_f tanβ + W_flyer/2. Together with the weight of the pier (b d H_p γ) and the pinnacle (m g), the reaction at the base lies at a distance from the inner face

x_R = ( Σ V_i x_i + H_f h_c ) / Σ V_i, SF = Σ V_i (b − x_i) / (H_f h_c)

The pier is safe against overturning if SF > 1 (x_R < b). The masonry rule is stricter: if x_R 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 ΣV_o − ΣV_ox_o − H_f h_c) / (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 C_p = 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.