Keywords: medial-axis, mesh generation, cable-membrane structures, conceptual design tool.

Cable-membrane structures are generally very attractive architectural landmarks. All aspects of their design have been investigated throughout the years [1,2,3,4,5]

The design of cable-membrane structures is different from "standard" steel and concrete structures as the shape of the structure is dictated by the internal forces and external loading. This behaviour is due to the fact, that cables and membranes have no out-of-line and out-of-surface stiffnesses therefore they undergo large displacements under loading which is perpendicular to the line of the cable or to the surface of the membrane. This behaviour results in geometric non-linear behaviour. Furthermore this structural behaviour has a great influence on the shape of the structure which should be considered during the design.

The main aim of this paper is to present a new idea for the conceptual design of cable-membrane structures. During the conceptual design the client specifies all requirements for the structure. The designer, an architect or a civil-engineer, tries to satisfy these requirements and prepares some drawings of the shape of the structure. It is possible to use the shapes of structures that have been already built, or the designer may try to find new shapes. In this step an automatic conceptual design tool can be very useful. This paper shows how the medial-axis construction method with some modifications can be used as a conceptual design tool for cable-membrane structures with antyclastic shapes.

The medial axis can be defined as follows: if a two dimensional domain is given, then the medial axis of this domain is the locus of the centers of the maximal, empty circles inside the domain [6]. In this paper the medial-axis of the two dimensional domain is contructed from the Delaunay triangulation of the domain. The method follows the approach of Damrong and Erickson [6] but with a modified set of rules. For the Delaunay triangulation only the boundary is supplied and no internal points are generated. If the triangulation is fine enough (not coarse), then the dual of the Delaunay triangulation (the Voronoi diagram) can approximate the medial axis. The algorithm to determine the medial-axis of a two dimensional domain contains the following steps:

• Classification of the triangles according to the number of sides that lie on the boundary: internal, corner, boundary triangles.
• Tracing medial-axis lines: Starting from a triangle with three internal sides the algorithm basically walks through adjacent elements and connects the centre points of the circumscribed circles of the triangles into a line.
• Using the new set of rules in the medial-axis contrsution may result in unconnected boundaries. unconnected means, that there is no edge between the medial-axis lines and the boundary lines. At this stage these missing connections are created.
• Cleaning up the medial-axis: removing duplicate nodes and lines with two nodes between two internal triangles.

The above described medial-axis contruction methodology can be used as a conceptual design tool for cable membrane structures. The medial-axis lines determined by the algorithm provide some suggestions as to the shape of the structure covering the given domain. The paper includes some examples to demonstrate the concepts discussed.

1

University of Stuttgart. Proceedings of the International Symposium Conceptual design of structures, volume Volume I. Institut für Konstruktion und Entwurf II, October 1996.

2

M. R. Barnes. Form-finding and analysis of prestressed nets and membranes.
Computers and Structures, 30(3):685-695, 1988. doi:10.1016/0045-7949(88)90304-5

3

E. Moncrieff and B. H. V. Topping. Computer methods for the generation of membrane cutting patterns. Computers and Structures, 37(4):441-450, 1990. doi:10.1016/0045-7949(90)90034-Y

4

B. Tabarrok and Z. Qin. Nonlinear analysis of tension structures. Computers & Structures, 45(5/6):973-984, 1992. doi:10.1016/0045-7949(92)90056-6

5

B. H. V. Topping and P. Iványi. Computer Aided Design of Cable-Membrane Structures. Saxe-Coburg Publications, Stirling, 2006.

6

Guoy Damrong and Jeff Erickson. Automatic blocking scheme for structured meshing in 2D multiphase flow simulation. In Proceedings, 13th Internation Meshing Roundtable, pages 121-132. Sandia National Laboratories, 2004.

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