Gothic Rib Vault Thrust Line - Limit Analysis, Hinges, Flying Buttress and Pier (2D)

Presets

Live readouts

Graphs

Thrust H min and H max vs rise / span (kN/m)

Geometric safety factor vs rise / span

Pier overturning factor vs buttress angle

Overturning factor vs wind speed

Numerical checks

running...

How to use

Presets jump between Romanesque, Gothic and over-ambitious designs. Drag the crown up or down to change the rise, drag the outer edge of the pier to change its thickness (2D). Space pauses, R resets.

Start with Romanesque barrel, then press Early Gothic: the pointed arch needs less depth and the thrust becomes a point load.
Pick Too bold (Beauvais-like) and raise the ring depth until the vault stands: the graph shows how much is missing.
Untick Flying buttresses present and see the wall overturn; tick it again and raise the storm wind.
Set the pinnacle weight to 0 and watch the resultant leave the middle third.

The science

A Gothic vault is built of stones that carry compression but almost no tension. Jacques Heyman's safe theorem says: if any line of thrust (the path of the resultant force through the joints) can be drawn that stays inside the masonry, the structure will stand. This lab finds the whole family of such lines for one transverse rib, per metre run, with the vault cut into 36 voussoirs per half.

M_P = H (y_c - y_P) - Σ_i≤k W_i (x_P - x_i) ≥ 0 at every intrados point, ≤ 0 at every extrados point

These inequalities are linear in the crown thrust H and its height y_c. The solver maximises the margin over H (a concave function), then bisects for the smallest and largest admissible thrust, H_min and H_max. The real thrust is somewhere in between and depends on tiny movements of the supports, which is why the buttress is designed for the maximum.

Geometric safety factor. Shrink the ring depth about its centre line until no line of thrust fits any more: that is the critical depth t_min. Then GSF = t / t_min. At t_min the line touches the ring at four hinge joints (crown, haunches, springings), the stonework becomes a four-bar mechanism and the vault would crack open. A semicircular arch of uniform depth needs t_min/R = 0.1075; the page re-computes this as a check.

pointed arch: R = (S²/4 + f²) / S, centres at x = ±(S/2 - R) on the springing line

A pointed arch is steeper at the crown, so it carries more of its load through the haunches as vertical force: its horizontal thrust H falls as the rise f grows (see the first graph), while a taller wall makes the overturning arm longer. That trade-off is why Gothic builders went up and thin at the same time.

Buttress and pier. The thrust of a bay is F_h = H · ell (plus the wind on the roof, F_w = ½ ρ_air C v² A). The flying buttress strut, sloped at angle α, passes F_h to the outer pier and adds a vertical part F_h tan α. The pier is a rigid block that tips about its outer toe:

FOS = (W_p t/2 + P_pin t/2 + F_v t) / (F_h h_a), x_R = (M_rest - M_over) / N

The resultant should stay in the middle third of the base (t/3 ≤ x_R ≤ 2t/3) so that no joint opens. A heavy pinnacle adds weight at no cost in width, which is exactly why medieval piers are crowned with pinnacles.

Honest simplifications. The rib is treated as a plane strip with a uniform ring depth (rib, web and haunch fill lumped); real vaults are three-dimensional shells with diagonal ribs, and the web carries load by shell action. The stones are rigid with infinite compressive strength and no sliding (Heyman's assumptions); the strut stress shown is only compared with a typical limestone crushing strength of 20+ MPa. Wind is a static, conservative gust on the roof area (rise + 4 m) per bay, passed entirely to the leeward buttress. The preset names are inspired by the proportions of well-known churches and are illustrative, not surveys of those buildings.