# Phyllotactic domes:design & optimization

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- Technische Universiteit Eindhoven
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## Abstract

The head of a sunflower has a very interesting arrangement. When looked closely, many spirals in both directions can be distinguished. These spirals show up in many other flowers and plants as well. The structural properties of these so called Phyllotactic spirals are studied in this research, by using them as a basis for the creation of domes. These so called Phyllotactic domes are calculated and optimized. By quantifying their structural properties, it is possible to compare them with Geodesic domes. The creation of Phyllotactic domes originates from the research conducted by Dr. ir. Frank M.J. van der Linden. He studied the formation of phyllotactic patterns, and created a computer program; named Apex. This program simulates the stacking and growing of units: without making use of any mathematical rules, the program creates the same phyllotactic patterns as are found in nature. By adjusting the parameters, like the sensitivity and canalization, various patterns can be made. Four of these patterns are used for the creation of Phyllotactic domes. A design procedure is set up to convert each stacking of units in Apex to a calculable structure. Furthermore, the procedure ensures that all boundary conditions are equal for the different domes. Therefore, all calculated differences are completely due to pattern differences, which are unique for every dome. A challenging dome shape with a span of 100 meter and a height of only 13.5 meters is applied to each domes. The created domes are calculated non-linear static in GSA. The final domes that are calculated and compared, are completely hinged, except for four nodes in the middle. Moment fixed beam connections are not desired as they are harder to build in practise and, in addition, more expensive. Furthermore, the Geodesic domes perform very well when completely hinged as well. The few fixed nodes in the middle are are caused by the possibility of snap through buckling. The Phyllotactic domes have a number of short beams in the middle, caused by some not fully grown units in the stacking. Short beams are more sensitive to snap through; making a few connections fixed, solves this problem. The domes are calculated in two steps. First, the calculations are done with oversized beams. The resulting force and moment distribution is used for the optimization process. The optimization is performed using formulas based on the Unity Checks. Each beam is checked on various Cross Section Optimization Checks, resulting in one overall minimum cross section for each individual beam. The resulting cross section redistribution is calculated again. The main problem of Phyllotactic domes that became clear during these calculations, is snap through buckling. To deal with this problem, it is made possible to change the Optimization Safety Factors from 1.0 to a lower, stricter, number. Various Factors provided different cross section distributions. Using trial and error, the optimal one is determined. The instability problem of snap through buckling is prevented by using larger cross sections. When more beams are applied in a structure, the overall beam length decreases. Shorter beams are more likely to snap through. Therefore, a dome with more beams needs larger cross sections, resulting in a higher total weight of the structure. This relation is exponential. The results are based on four total different phyllotactic patterns made in Apex. The patterns are based on two main variables, canalization and sensibility. Each pattern is calculated with various number of beams as well. It turned out that the patterns that are created with more regularity perform better than the freely formed patterns, although in different areas. Patterns with a high canalization factor result in smaller deformations, while patterns with a high sensitivity factor have a lower total weight. Comparing the results of both types of domes showed that Phyllotactic domes do not perform as well as Geodesic domes. More steel is required (Geodesic domes use 25% less steel) and the deformations are larger (2-3 times as large). However, a deformation of 200 mm on a span of 100 meters is still acceptable. Furthermore, the Phyllotactic domes are more intriguing due to the spirals that can be followed throughout the structure. This study has been carried out in a complete new research field. Much has been explored, though there are still questions unanswered. The first results are promising, however additional research needs to be done. Further research can continue from the created basis; the easy to use design procedure and the useful Cross Section Optimization method.

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