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Amongst the most fabrication-friendly structures we then identified the roof versions with the best structural capacities by rating the maximal deflections of single bars or regions of the roof structure.

The most promising configurations were then used for further structural optimisation. A custom-made VB Script in MS Excel links the geometry in Rhinoceros with RSTAB, a structural analysis software from Dlubal. Every lamella, spanning between two points on the roof perimeter, is subdivided at every intersection point. The algorithm, starting from a homogeneous configuration, differentiates height and thickness of every piece and tapers the flat steel bars.

The goal of the discrete optimisation problem was a minimum weight design respecting local stress and global deflection constraints using a set of available cross sections. For solving the optimisation problem, we choose an algorithm based on the CAO algorithm proposed by Matthek and Burkhardt (Matthek, et al. 1990). The CAO seeks an optimal design by simulating the growing pattern of biological load bearing structures. So areas with high stress concentrations are strengthened and those with light stresses are degenerating.

The algorithm reduces the lamella cross sections by iteratively analysing their maximum von Mises Stresses. Tapering the pieces lead to an undulating bottom edge of every lamella. To guarantee this rolling effect every intersection point needed profiles with identical height. Hence, we constantly analysed the von Mises Stresses of all lamellas intersecting in one point. The lamella with the maximum grade of utilisation at each intersection point defined the cross section. To rationalise the structure the cross section dimensions were limited to ten different types in between 20 to 40mm thickness and 150 to 600mm height.

Fig. 9 Structural optimisation process from a homogenous structure to an undulating, material efficient, differentiated system.

Each lamella cross section change results in a redistribution of stiffness’s, which induces in a hyperstatic structure a shifting of load paths. Therefore, it was necessary to recalculate at each iteration step the von Mises Stresses for the whole structure to consider this load path shifting. The stopping criterion for the iteration was chosen as a determined percentage of lamella cross sections changing between iteration step n and n+1. So shifting of load paths due to the redistribution of stiffness’s could be considered as negligible. In a subsequent step, the lamella cross section thicknesses between each intersection point were controlled. In case of lamellas with different thicknesses at starting and ending point the higher thickness was chosen for the full length.

The global deflection constraint was implemented in the algorithm by reducing the maximum stress limit for the lamellas. It was decided to limit the deflection of the outer edge of the canopy at the side with the longest cantilevering to 50mm due to variable loads. This corresponds to l/200 for the cantilevering length of 10m. This deflection limit was achieved by limiting the von Mises Stresses to 85% of the admissible stresses.

This intuitive approach was chosen as the analysed structures showed a direct and almost linear correlation between the maximum deflections and the stresses of the heavy-duty girders located nearby the supporting columns. That way the optimum girder configuration with the appropriate section size was found.