distance w (the pitch). Finally, from two consecutive half guide-curves, quadrants are meshed with the same compass distance (Fig. 13).
Meshed Surface
The compass method does not allow meshing the entire meshing domain. Only a smaller part could be meshed and its area varies according the chosen set of guide-curves (Figs. 14-15). Thus, it is not possible to rely exclusively on the shape to be realized with the lattice. An extended surface - chosen carefully - has to be considered as the meshing domain.
Fig. 14 Two different meshes are obtained from two distinct sets of guide-curves. The meshed area never takes on the whole surface. Convergence phenomena could be observed (right picture).
Overall Process
To overcome this difficulty we propose a methodology, which relies both on the creation of a meshing domain (domainSrf) from the targeted surface to materialise (gsSrf) and on the identification of a suitable set of guide-curves.
Step One
We consider the gridshell surface (gsSrf) a part of a larger surface (domainSrf). Trimmed by a clipping plane or surface (cuttingSrf), this domain surface should give back the intended shape to build (Figs. 15-16).
Fig. 15 gsSrf
Fig. 16 domainSrf and cuttingSrf
Step Two
A set of guide-curves is chosen (Fig. 17) and the mesh is propagated on the domain surface according to the compass method (Fig. 18). The guide-curves have to be chosen so that the whole gridshell surface (gsSrf) is meshed. Several trials can be necessary to get a suitable mesh.
Fig. 17 Guide-curves set
Fig. 18 Resulting mesh on domainSrf
Step Three
The mesh is trimmed by the clipping surface (Fig. 19). The resulting mesh lays on the whole initial intended surface to mesh. The gridshell support-outline is given by the intersection of the clipping plane and the domain surface (Fig. 20).
Fig. 19 Trimmed mesh
Fig. 20 Final mesh and support outline
Mesh Optimisation
Since the form has been known and the procedure to mesh the surface is known now, an optimisation of the mesh can be performed (Fig. 21). The aim is to find a mesh that can mesh the entire surface and that creates acceptable stresses in the beams. To this end, the curvature of the elements is checked using the following equation (Eq. 3):
(3)
Different sets of guide-curves are chosen and the resulting meshes are compared according to this criterion:
Mesh n°1
Mesh n°2
Mesh n°3
Fig. 21 Mesh testing
Grid Processing
The generated mesh can be shaped in a matrix. This allows both its transformation in a planar grid of regular pitch (Fig. 22) and automatic definition of triangulation elements for bracing.
Fig. 22 Developed surface and derived grid.
Generation of Analysis Model
A procedure gathers and processes all the geometric information. It creates an import file for the automatic generation of an analysis model in GSA, a third-party software dedicated to structural analysis. Additional components assist the designer in the definition of complex load cases, such as non-uniform wind and snow loads, directly in Rhino & Grasshopper.
2.6 Structural Analysis
Once the structural model is built by our tools, it can be loaded in the structural analysis software to perform:
Computation of the permanent flexural stress and the relaxed shape using a dynamic relaxation algorithm (Fig. 23)
Loading analysis according to the Eurocode (self-weight, snow, wind ...)
Fig. 23 Final compass mesh and corresponding relaxed mesh (stress diagram).
3 Conclusion
This paper has presented the different steps for the design of a gridshell in composites materials built for the Solidays Festival in 2011 in Paris. The first step was the optimisation of the shape in order to avoid concentrations of curvature locally. The second step showed a tool to automatically mesh a surface using the compass method. With this tool the optimum orientation of the mesh is studied. The last step showed the details of the structural analysis of the gridshell. This construction demonstrated the technical feasibility and also the economical feasibility of the gridshell in composites materials.
Acknowledgements
The authors would like to thank the students E. Roux, E. Blache, J.-R. Nguyen, A. Grandi, G. Frambourt, T. Perarnaun, their university supervisor R. Mège and the association Solidarité Sida for their initiative and confidence. Special thanks also to T/E/S/S and Viry for their technical and financial support, which permitted this project to become real. Thanks also to all our partners who provided significant material assistance: Serge Ferrari, Top Glass & Solutions Composites, Owens Corning Reinforcement, DSM Resins, ENSG, Esmery Caron, Axmann, Chastagner and ¨Paris Voile.
References
Douthe, C; Baverel, O.; Caron, J.-F., 2007: Gridshell in Composite Materials: Towards Wide-Span Shelters. Journal of the I.A.S.S, Volume 48 Issue 155, pp. 175-180.
Douthe,