Two ways in. Give the chord and the mid-ordinate — the two things you can measure off a drawing or off the floor — and the page finds the radius. Or give the radius and the included angle and it finds the chord. Either way you get the arc lengths you need to order material and lay out the piece, and the outer-fibre strain that tells you how hard the roller is going to have to work.
| Wanted | From chord C and rise H |
|---|---|
| Radius to the neutral axis | R = C² / (8H) + H / 2 |
| Included angle | θ = 2 · asin( C / 2R ) |
| Arc length at the neutral axis | L = R · θ, θ in radians |
| Outside arc (depth d) | (R + d/2) · θ |
| Inside arc | (R − d/2) · θ |
| Outer-fibre strain | ε = (d/2) / R, shown as a percentage |
Going the other way, from radius R and angle θ: C = 2R · sin(θ/2) and
H = R (1 − cos(θ/2)). The page checks itself by running the answer back through the
other formula — chord mode rebuilds the chord and rise, radius mode rebuilds the radius and angle —
and says so on the last line of the log. A round trip through the same formula it just used would always
agree, so it does not do that.
A rise larger than half the chord means the beam wraps past a half circle. The arcsine only ever returns the smaller angle, so in that case the page takes the reflex angle instead: θ = 2π − 2·asin(C/2R). Without that, a 20' chord with a 15' rise reads 134.8° and an arc seventeen feet short of the truth.
ε = (d/2)/R is the bending strain at the extreme fibre, the same number a roller uses as a first sanity check on whether a section can be curved the hard way. It is a geometric ratio, not a limit. Whether a given shape can actually be rolled to that radius depends on the section, the grade, the roller's equipment, and how much distortion the job will accept. Ask the roller before you promise a radius.
Chord accepts 20'-0, 20' 6 1/2, 240 (read as inches) or
20.5'. Rise and depth are read as inches unless you type a foot mark. Fractions like
6 3/16 work everywhere.
R = C² / (8H) + H / 2, where C is the chord measured end to end and H is the mid-ordinate from the chord to the arc at midspan. The included angle follows from θ = 2 · asin( C / 2R ), and the arc length from L = R · θ with θ in radians. Chord and rise are the two things you can measure off a drawing or off the floor, which is why this is the way in. Worked on this page: a 20'-0" chord with a 12" mid-ordinate gives R = 240² / (8 × 12) + 12/2 = 606" = 50'-6", included angle 22.842°, arc length 20'-1 5/8".
At the neutral axis it is L = R · θ. The outside arc is (R + d/2) · θ and the inside arc is (R − d/2) · θ, where d is the member depth. That difference is why the outside of a curved member is longer than the inside, and why ordering to the chord length comes up short.
ε = (d/2) / R is the bending strain at the extreme fibre — a geometric ratio, not a limit. Rollers use it as a first sanity check on how hard a section will be to curve. Whether a given shape can actually be rolled to a radius depends on the section, the grade, the roller's equipment and how much distortion the job will accept. Ask the roller before promising a radius.
The member wraps past a half circle. An arcsine only ever returns the smaller angle, so the reflex angle is the one wanted: θ = 2π − 2 · asin( C / 2R ). Without that correction the included angle and the arc length both come out badly short, and the error grows with the rise.
A 20'-0 chord with a 12" rise gives a 50'-6" radius, a 22.842° included angle and a 20'-1 5/8" arc. If this page says otherwise, do not use it — tell us.