Arch Thrust Line 2D — Rise, Load & Middle-Third Stability

Arch geometry

Rise / span ratio 0.30

Arch thickness (× span) 0.060

Point-load position 0.50

Point-load magnitude 60 kN

Self-weight (per span length) 5.0 kN/m

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stable

Horizontal thrust H

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Max eccentricity

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Thrust-line margin (0% = edge of middle third, 100% = edge of section)

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How it works

A masonry arch cannot resist tension — every joint between voussoirs can only push, never pull. So the load path (the thrust line) must thread entirely through the stone in compression. Treating the arch as three-hinged (hinges at the two springings and the crown), the thrust-line height above the springing at any point equals the bending moment of an equivalent simply-supported beam carrying the same loads, divided by the horizontal thrust H:

M(x) = w·x·(L−x)/2 + point-load moment(x)
H = M(L/2) / h        (thrust line must pass through the crown hinge)
y_thrust(x) = M(x) / H

The eccentricity is the perpendicular distance between this thrust line and the arch's own centerline. Because stone (like any no-tension material) only stays fully in compression while the resultant stays within the middle third of the section, e ≤ t/6 means no tension anywhere. Push the thrust line past t/6 and a hairline crack opens at one face; push it all the way to t/2 and it leaves the stone entirely — a hinge forms there, and enough of those turns the arch into a collapsing mechanism (Heyman's safe theorem, the basis of modern masonry-arch assessment).

Rise/span ratio — a taller arch is closer to the funicular shape for its self-weight, so it tolerates load with less thrust-line wander.
Point-load position — a load away from the crown drags the thrust line toward one side, exactly like the eccentric flow that snapped the flat lintel in the paired 3D model.
Arch thickness — a thicker ring gives a wider middle third, so the same thrust-line wander is proportionally safer.

Thrust line in middle third Outside middle third (cracking) Outside section (hinge/collapse)

Eccentricity across the span (± half-thickness)