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Preliminary Mathematics
Published in P.K. Jayasree, K Balan, V Rani, Practical Civil Engineering, 2021
P.K. Jayasree, K Balan, V Rani
The bases of the zone are the circumference of the sections made by these two parallel planes. Since it is bounded by two parallel planes, it is called as a spherical zone with two bases. The entire 3D portion between the two parallel planes is called as a spherical segment or a spherical frustum. Figure 3.8 shows a spherical segment with altitude h, radius r, and c1 and c2 as the circumference of the two bases.
The Invisible Beauty of the Zeiss-Dywidag Domes: Topology Optimization of the Triangulated Rebar Grids
Published in International Journal of Architectural Heritage, 2023
Orsolya Gáspár, Alexandra É Kis
During the reconstruction of the optimization process, we encountered some controversies in the available resources (primary and secondary). We discuss in detail the most likely solution and any uncertainties are highlighted. We differentiate between three levels of development of the final geometry: we denote the bar dimensions the micro-scale, the arrangement of bars, i.e., the tessellation of the meso-scale and the overall geometry of the macro-scale. The key take-aways are summarized in the following points in advance to offer a short-cut to Section 4.2 and onwards. Micro-scale: The bar dimensions were chosen from a pre-determined range considering member buckling.Meso-scale: An intricate gradual reduction scheme of the number of triangles along a latitude ensured that the subdivision remained uniform enough. The location of the latitudes was determined by the vertical height of the rows. The area of a spherical segment depends linearly on its height (aka. Archimedes’ theorem, Beyer 1987; Kline 1972). By controlling the height of each row, and the number of triangles within the row, Bauersfeld was able to produce close-to-equal area triangles (see Section 5, Table 2) in the tessellation.Macro-scale: Assuming the span fixed, the opening angle of the dome was chosen to be 35º, as it resulted the best economy (lowest cost based on weight).