Fig. 3 The grid, prominently exposed when approaching the gate, is both an undulating ornament and an expression of the relationship between matter and force.
Fig. 4 Existing infrastructure and street layout defines the location of the new columns and the area that needs shelter.
The design team selected upstanding flat steel lamellas to form the structural grid of the roof. Being strong in y-axis but very weak in z-axis the elements needed to be positioned in a system where they mutually brace each other: A network of intricate interrelations replaces hierarches of primary and secondary structure. The roof is currently under construction and will be erected in August 2013.
It was prefabricated in seven parts in the workshop (Fig. 5) and assembled on the ground next to its final position on the bridge. After applying the final coatings of the anticorrosion paint the wooden covering plate and the waterproofing were mounted, so that the number of outstanding work steps after lifting the whole structure on the top of the bridge is minimised.
Fig. 5 Prefabrication of the canopy in the workshop
Fig. 6 The completed project in August 2013 © Messe Frankfurt/ Bach
4 Geometry Generation
We automated the generation process of the lamella structure through a Rhino script to create a large variety of design solutions with different lamella configurations. The four triangular, chamfered columns define the position of the first twelve lamellas which sweep along the intersection of column face and roof surface. Every lamella extends to the roof perimeter. Intersections between these lamellas form a first fragile system of interconnected bars that needs additional elements to support the cantilevering roof. A generative algorithm placed the second set of lamellas. Two random points on the roof perimeter define the end points of a beam. The lamellas interweave into a coherent structure which is capable to cantilever, span and transfer the horizontal forces.
Fig. 7 Lamella configuration based on column position and perimenter subdivision.
After generating a multitude of different configurations the results are analysed and evaluated. First, we looked for lamella intersections with angles smaller than 30 degrees. In a previous project these small angles created problems during assembly and welding as working space becomes too small for the tool at work. We rated therefore all generated solutions by the number of those problematic intersections.
Fig. 8 Bundles of lamellas with spatial proximity create complex nodes. Here: the Sphere project, Deutsche Bank Frankfurt, Mario Bellini Architects, 2011 (left). Small angle make welding difficult (right).
5 Structural Optimisation
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.
6 Conclusion
The developed structure answers to different and irregular column positions, geometries, and its high level of recognition in an optimal way to the design task given by the Messe Frankfurt. The roof structure sought to be an alternative approach to a hierarchical organization of structure or conventional proportional systems. Structural purity would have been the wrong answer to the design challenge since the particular requirements of a complex site needed to be taken into account. The developed algorithms for the girder configurations with subsequent structural optimisation is intuitive but cope with the given optimisation problem, as the behaviour of the given structure - a cantilevering canopy - is quite predictable. The design process blurred the boundaries between design development and design-tool development. The lacking automated feedback loop between geometry generation and its structural and geometrical analysis prevented exploring a larger solution space and showed the need for further development of design tools. A series of comparable projects lead to a collection of tools, all of them highly project specific and therefore of limited use for following endeavours. The experience of linking parametric and structural design for this and several other similar projects designed at this time showed Bollinger + Grohmann Ingenieure the need to consolidate and formalise the acquired knowledge