Form finding for large-span pneumatic roof cushions

With particular emphasis on cable net form finding using Oasys GSA, Eindhoven University masters students Arjen Deetman and Thijs de Geode determined the optimum design. Their ultimate goal was to complement architectural design with stable structures creating minimal visual obstruction.

Pneumatic cushions are generally accepted as the best solution for covering courtyards. Structurally, however, the most effective shape for a pneumatic cushion is circular and courtyards are, of course, predominantly rectangular. Deetman and de Goede wanted to explore other options, and to optimise design for rectangular buildings.

Their final design proposal is a pneumatic cushion combined with a steel frame to avoid large horizontal prestress forces on the existing building. The flexible skin is an ETFE Ethylene tetrafluoroethylene layer strengthened with an orthogonal cable net. A diagonal cable net spreads the horizontal forces more uniformly over the edge of the steel frame.

Combining the four circular shaped arches results in a compression force in the arch with edge beams that are loaded in tension. By applying four corner bars a tension belt is designed that closes the system (this belt is holding the four arches together). This shape led to a minimal visual coverage of the courtyard.

 

 

 At the end of this phase of the project, a final GSA model combined the two separate models for the flexible skin and the compression elements. With this combined model the behavior of the structure is more realistic and can be further improved.  

 

Determining the span of  ETFE

The tensile strength of ETFE is relatively low, so adding a cable net is necessary for large span structures. 

During the preliminary design of the cable net, four different configurations were considered. Two orthogonal cable net grids with a grid size of 2500mm and 5000mm, and two diagonal grids with a grid size of 1750mm and 3550mm. The minimum curvatures are shown in the figure. The grid size of 5000mm was rejected since the curvature is to large.

The preliminary design of the square cushion required a determinination of the stresses in the ETFE, which are related to internal pressure [p], radius [R] and ETFE thickness. However, known factors for circular (0.5) and cylindrical (1.0) domes not applicable for square ground shapes. So, with the GSA model a factor of 0.63 was approximated for a square cushion. This was based on geometrical linear theory, and only used for preliminary design calculations. 

Form finding of the cable net for the flexible skin

Three different form finding methods were then explored  for defining the geometry of the cable net.

  • Force density (force divided by length)
  • Soap film property (constant pre-stress)
  • Manipulation of a deformed shape

 

The soap film model resulted in the most constant cable forces and spread the horizontal reaction forces more uniformly. The figures show the cable forces of the three different methods that were used. Uniform loading  (air pressure) was assumed for these calculations.

Cable nets with a grid size of 1750mm, 2500mm and 3550mm were modeled in GSA with a soap film property (the 5000mm option having already bene eliminated in the preliminary design phase of the proejct) . The models were loaded with three different load cases as shown in the figure.

The analyses showed that the diagonal cable net with a grid size of 3550 mm was the best. This cable net loads the compression element inside the cushion with a more constant load.   With an orthogonal cable net the compression arch would be loaded asymmetrically.

 

Diagonal

  Orthogonal

 

Compression element

The proposed compression element consists hollow steel rectangular hollow elements that load the edge beams in tension. By applying four bars in the corners a tension belt is designed that closes the system. This shape led to a minimal visual coverage of the courtyard.

 

Arches shaped with graphical form finding were compared with circular shaped arches, and different methods of connecting them at intersections were considered. Important criteria here were the stresses in the arch beam (largest profile), buckling loads, horizontal displacements and construction.

The researchers concluded that a circular shaped arch gives the lowest stresses in the arches beam. Fixed (welded) connections give by far the largest buckling load of the total structure.

 

What’s next?

For this project the assumption was that the internal air pressure of pneumatic cushions was constant. In reality the air pressures varies. For further research, it will now be interesting to study the dynamic behavior of this large span pneumatic cushion, including a parameter study.