About

Incorrect arch geometry causes uneven load distribution, leading to cracking at the crown or spalling at the springers. This calculator computes critical parameters - R (radius), L (arc length), A (area), and horizontal thrust H - for seven standard arch profiles: semicircular, segmental, pointed (Gothic), flat, parabolic, elliptical, and horseshoe. Segmental radius is derived from the chord-sagitta relation R = (S² + 4h²) ÷ (8h), and arc lengths for parabolic profiles use Simpson's rule numerical integration rather than closed-form shortcuts. Results assume rigid supports and uniform self-weight. Real masonry arches require mortar joint thickness when sizing voussoirs. This tool approximates geometry assuming plane-strain conditions and does not replace finite element analysis for seismic zones or spans exceeding 12 m.

Formulas

The fundamental relationship for a segmental arch derives the radius from the clear span and rise:

R = S² + 4h²8h

where S = clear span, h = rise (sagitta), R = radius of curvature.

The arc length for a circular segment uses the central half-angle:

θ = arcsin(S2R)

L = 2Rθ

For a parabolic arch, the curve follows y = 4hS² x(S − x) and the arc length is computed by numerical integration:

L = S∫0 √1 + (dydx)² dx

Horizontal thrust under uniform distributed load w:

H = w S²8h

where w = load per unit length kN/m, H = horizontal thrust kN.

For an elliptical arch, arc length uses Ramanujan's approximation for a half-ellipse with semi-axes a = S÷2 and b = h:

L ≈ π2 (3(a + b) − √(3a + b)(a + 3b))

Area under the arch is computed per type. For a semicircular arch: A = π R²2. For parabolic: A = 23 S h. For segmental and others, numerical integration is used.

Reference Data

Arch Type	Typical Span Range	Rise-to-Span Ratio	Thrust Characteristic	Historical Use
Semicircular	1 - 25 m	0.50	Moderate, constant	Roman aqueducts, Romanesque churches
Segmental	2 - 30 m	0.10 - 0.45	Higher as rise decreases	Bridges (Pont de la Concorde), lintels
Pointed (Gothic)	3 - 20 m	0.50 - 1.50	Low, directed downward	Gothic cathedrals, Islamic architecture
Flat / Jack	0.5 - 2.5 m	≈ 0	Very high horizontal	Window heads, fireplace lintels
Parabolic	5 - 100 m	0.20 - 0.60	Uniform under UDL	Modern concrete shells, Gaudí vaults
Elliptical	3 - 30 m	0.25 - 0.50	Variable along curve	Renaissance bridges (Ponte Santa Trinita)
Horseshoe	2 - 12 m	> 0.50	Inward at base	Moorish architecture (Alhambra, mosques)
Catenary	3 - 200 m	0.30 - 0.70	Pure compression	Gateway Arch, masonry domes
Tudor	2 - 10 m	0.15 - 0.30	Moderate	English Tudor manor houses
Ogee	1 - 5 m	0.40 - 0.80	Decorative only	Venetian Gothic windows
Corbelled	0.5 - 6 m	Variable	Zero (cantilever)	Mycenaean tholos tombs, Maya arches
Three-hinged	10 - 80 m	0.20 - 0.50	Determinate	Steel bridges, exhibition halls
Lancet	1 - 8 m	> 1.0	Very low	Early Gothic (Chartres Cathedral)
Stilted	2 - 10 m	Variable + vertical legs	Similar to semicircular	Byzantine churches, Ravenna mosaics

Frequently Asked Questions

Horizontal thrust is inversely proportional to rise. The thrust formula H = wS² ÷ (8h) shows that halving the rise doubles the thrust. For segmental arches with rise-to-span ratios below 0.15, the horizontal force can exceed the vertical reaction, demanding massive abutments or tie rods. Semicircular arches (ratio 0.50) produce moderate thrust, which is why Roman engineers favored them for multi-span aqueducts where adjacent arches balance each other.

The chord-sagitta formula R = (S² + 4h²) ÷ (8h) assumes a circular arc. It produces infinite radius when rise approaches zero (flat arch) and equals S÷2 when rise equals S÷2 (semicircular). For rise values exceeding S÷2, the formula still works geometrically but the arch becomes a horseshoe form, requiring different structural analysis because the line of thrust exits the arch thickness.

Divide the arc length by the sum of voussoir width and mortar joint thickness. Traditional masonry uses voussoir widths of 200 - 400 mm with 5 - 10 mm mortar joints. An odd number of voussoirs is standard practice so that a single keystone sits at the crown rather than a joint. If the calculation yields an even number, reduce voussoir width slightly or adjust mortar thickness. For spans under 1.5 m, a minimum of 7 voussoirs is recommended per BS 5628.

A parabolic arch achieves pure compression only under uniformly distributed horizontal load (such as a bridge deck). A catenary arch achieves pure compression under its own self-weight hanging uniformly along the curve. For most masonry arches where self-weight dominates, the catenary is theoretically ideal. The difference in geometry is small for shallow arches (rise-to-span below 0.25) but becomes significant for deep arches. This calculator uses the parabolic profile y = 4hx(S − x) ÷ S², which is appropriate for most practical loading conditions.

Arch thickness must contain the thrust line under all loading conditions. The minimum thickness rule of thumb for masonry is t ≥ S ÷ 20 for semicircular arches and S ÷ 12 for segmental arches with low rise. The calculator reports the extrados arc length (outer curve), which is always longer than the intrados. The difference increases with thickness-to-radius ratio. For voussoir sizing, use the intrados length; for waterproofing or cladding, use the extrados length.

No. This tool assumes symmetric arches with equal springing points on a horizontal line. Skew arches (where the axis crosses abutments at an angle other than 90°) require helical geometry and specialized analysis. Asymmetric arches with unequal springing heights need separate calculations for each half-span. For these cases, the tool still provides useful baseline geometry if you decompose the arch into symmetric halves and compute each independently.