Arches & curves
Segmental arch geometry
- where
- radius of the arch (mm)
- clear span (mm)
- rise (mm)
- included angle (degrees)
- arc length (mm)
Explain this equation
This gives the radius, included angle and intrados length of a segmental arch from its clear span and rise.
S is the clear span and h is the rise from the springing line to the top of the intrados, both in mm. R is the radius of the circle the intrados is part of. θ is the angle that the arc spans, measured at the centre. a is the length along the underside of the arch.
The span is a chord of the circle and the rise is the height of the segment above it. Applying Pythagoras to the half-span gives R = S²/(8h) + h/2. Half the span over R is the sine of θ/2. The arc is R times the angle in radians, hence × π/180.
A 2000 mm span with a 500 mm rise gives a 1250 mm radius, about 106°, and a 2318 mm intrados. If the rise is half the span, the arch is a semicircle.
Watch out: this is geometry only, for rises up to half the span. It does not design the arch, its bearings or the centring.
Find the radius and intrados length of a circular segment from its span and rise.
Enter your values to see the result.
Good to know
Geometry only for rise ≤ half span. Does not size arch rings, voussoirs, bearings, abutments or centring. Obtain the arch design.
Sources
- Geometry / arithmetic
Estimates are not design approval or British Standards compliance checks. Structural arch, retaining-wall, wind, foundation, lintel and masonry-capacity design are not performed. Use the project engineer’s specification.