The reciprocal frame roof results in a very strong self-supporting structure with very unique features.


This drawing illustrates a reciprocal frame designed with 3 beams. The following parameters define the structure:

  • the number of beams (n)
  • the radius through the outer supports (ro)
  • the radius through the beam intersection points (ri)
  • the vertical rise from the outer supports to the beam intersection points (H)
  • the vertical spacing of the centrelines of the beams at their intersection points (h2)
  • the length of the beams on the slope (L)

It can be seen that two polygons are formed. The inner polygon is formed by the intersection of the beams and the outer by the beam ends.  The number of sides to the polygon is equal to the number of beams used.

The parameters ro, ri and H are specified as design requirements.  These are dependent upon both structural and architectural factors. The remaining defining parameters can be calculated as shown.

  Sector angle between the beams.
θ = 360/n

Overall plan length of beam.
x = x1 + x2

Plan length to first intersection.
x2 = 2risin(θ/2)

Plan length between intersections.
x1 = {ro2 -[ricos(θ/2)]2}½ - x2/2

Rise to first intersection.
h1 = H( x1/x)

Rise between intersections.
h2 = H - h1

Slope length of beam.
L = (x2 + H2)½
L = (ro2 - ri2 +H2)½

Some interesting observations can be made that must be considered during the design process:
  • If h2 is less than the depth of the beam the upper beam will require notching on the bottom to maintain the beam spacing. This will weaken the upper beam.
  • If h2 is large there may be a space between the beams and require a beam larger than necessary to meet loading requirements just to allow the two beams to meet.
  • To avoid excessively deep beams where a small number of beams are used the central opening should be small.
  • By increasing the number of beams while keeping H constant a larger central opening is possible with beams of given depth. This will maintain beam contact at the intersection point.
  • If the central opening is not increased as the number of beams are increased deeper notches will be required. This is particularly the case when H is small.


Additional information for Reciprocal roof (self supporting)


A small wood hut

A small wood hut

Something I'd been planning to do for a while. Not a big project, just needed to sit down and work out the basic dimensions and materials then throw it together. The main features are a reciprocal roof consisting of 6 beams supported on 6 poles for the walls. Some metal sheeting on the rainy, windward side and chip limestone on the floor. Rubber roof membrane with ferocement to top it all off.
May 08, 2019