Some of this information is based on articles received from Dr. Olga Popovic of the School of Architecture at the University of Nottingham, Nottingham, UK. In particular the following paper is referenced:

Title: Reciprocal Frame 3-Dimensional Grillage Structures
Authors: J.C. Chilton, B.S. Choo, O. Popovic

WHAT IS A RECIPROCAL FRAME?

The reciprocal frame is a roof structure where each beam both supports and is supported by other beams in the roof structure. A minimum of 3 beams is required to create a reciprocal frame roof. As each beam supports the next in a reciprocal manner no internal support structure is required.  Only the outer end of each beam requires support which will normally be a post used for the wall.  The roof loads are transferred to these posts and in turn to the supporting foundation. The beams can be fabricated from timbers, laminated wood, steel or reinforced concrete. A very inexpensive roof structure can be made from logs.

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

RECIPROCAL FRAME DESIGN

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 (r_o)

• the radius through the beam intersection points (r_i)

• 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 r_o, r_i 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.

Overall plan length of beam.
x = x₁ + x₂

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

Plan length between intersections.
x₁ = {r_o² -[r_icos(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
or
L = (r_o² - r_i² +H²)^½

• 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.

These are but a few of the factors that affect the design of a reciprocal frame. You can easily set up a spreadsheet that would perform the above calculations and see the results of varying the design parameters.

In many cases it will be necessary to notch the bottom of the beams so that they will fit properly. This notch is complex both in its design and execution. The following sketch shows a typical notch. The notch shown at the right does not actually exist. This shows how the next beam would rest upon the beam shown.

The following drawing shows the design of the notch for the Pavilion. It illustrates that considerable effort is required to do this once all the parameters discussed in the previous section have been established.

Typically the reciprocal frame roof is covered with flat triangular panels. These panels are attached to the top of each beam and to the side of the adjacent beam upon which the first beam rests. As a result the panels are inclined and a step is introduced from one panel to the next. The resulting effect is quite striking as can be seen from the photographs in the construction section.

The Pavilion is an octagonal structure and the reciprocal frame roof is constructed using 8 solid wood beams 3 inches wide and 11 inches thick. The diagonal distance between the supporting posts is 16 feet. The posts are 6 inch square hollow steel and are bolted to a concrete foundation. Hollow steel is welded between the posts near the to to help resist any twisting forces that the posts may experience. This was required as part of this particular design. The following drawing illustrates the layout.

The roof panels were framed with lumber, sheeted with plywood and finished with a red metal roof. A cupola was built over the inner polygon and covered with Plexiglas. Two of the walls were left open and are used to enter and exit the Pavilion. The back three walls that face the edge of the rock ridge are covered for safety reasons. A display case is mounted on each of these walls and hold brass plaques with the names requested to be displayed by contributors to the "Buy-a-Board" fundraising campaign. The other three walls are relatively short with benches mounted on the inside and out of two and on the outside only on the third. Cedar was used for all this construction.