alt="figure_6.jpg"/>
Fig. 6 Resulting structure
An alternative method has been proposed by the Navier laboratory to achieve this form-finding stage by computer. Based on a dynamic relaxation algorithm which considers the elements bending stiffness, it leads to new shapes where free outlines express the grid natural stiffness (Douth, Baverel and Caron 2006 & 2007).
Grid-Finding
The compass method is used to develop the initial shape in a quadrangular mesh. Rebuilt on a plane, the mesh leads to a regular grid suitable to materialise the studied shape by a gridshell. An alternative method, taking into account the grid’s mechanical properties, was also proposed by the Navier Laboratory (Bouhay, Baverel and Caron 2009). This method uses explicit dynamic algorithms to pin an initially flat grid on a given shape, with a system of fictive forces.
2 Case Study
2.1 The Project
In June 2011, six students from École des Ponts ParisTech (French engineering school) supported by the Navier Laboratory gave birth to a structure for the association Solidarité Sida: a tent unlike any other, reminding blob architecture with its curved and rounded shape (Fig. 7).
Fig. 7 Photo of the final structure (http://vimeo.com/31341461)
Description
This structure of 300m² was the first gridshell in composite materials (GFRP) to receive an audience. It had to get a certificate of approval that involved administrative requirements and EUROCODE justifications, a first for such a structure. Beyond the technical performance, this large scale project designed to house up to 500 people at a time has shown the economic relevance of this concept. It became reality thanks to key partnerships including T/E/S/S and Viry.
Solidays Festival
Each year in June, Solidarité Sida organizes the Solidays Festival. It is a music festival that attracted around 160,000 people during four days and raised about 1.7 million Euros in 2011. It is also a forum. Its purpose is to raise awareness about AIDS and raise funds for medical research and outreach initiatives. The gridshell structure was designed to house the forum during the festival.
2.2 Design Process
The opposite diagram summarizes the design process from the initial architectural intent (a 2D sketch) to the final gridshell. Each step is then detailed.
2.3 3D Modelling
From the initial architectural intent we have modelled the shape as a wireframe (Fig. 9) including the outline and some sections. With those curves we can control the shape in-plane and drive its volumetry. The surface is then derived from the wireframe using a NURBS interpolation (Fig. 10). All along the design process adjustments were carried out on this surface until it reached the architectural and structural requirements. In this process, the initial surface was mostly deformed or sculpted by handling its control points.
Fig. 8 Step by step design process including different levels of structural control, based on curvatures and stresses checks (1,2,3)
Fig. 9 Wireframe geometry
Fig. 10 NURBS patch
2.4 Shape Optimisation
For now, we built a space sketch of the project. However, this NURBS surface has no reason to lead directly to a structural shape. Thus, the structural elements have to be checked to make sure they will support the stress field induced by grid shaping.
Stress & Curvature
Stresses in elements are mainly due to grid bending, that is to say geometric curvature imposes the grid stress state (Eq. 1). Thus, principal curvatures of the surface describing the shape are good indicators to evaluate if the structural members have the required mechanical properties. This preliminary control can be completed by an analysis of the curvature of the mesh elements. Finally, nothing but a true structural analysis considering members mechanical properties will allow us to find the exact relaxed shape and the stress field in the structure.
(1)
Where σ represents the total stress (compressive plus bending) induced by the shaping. E, v and I are respectively the longitudinal young modulus, the radius and the bending inertia of the profile. R is the radius of curvature of the profile.
Surface Optimisation
Before any attempt to mesh the shape, it is recommended to optimize the sketch shape regarding its minimal principal curvatures (Eq. 2). Using the curvature-analysis built-in function in Rhino it is easy to identify and smooth areas that are initially too curved (Fig. 11).
(2)
Different sketches are compared according to this criterion to smooth areas where curvature is excessive.
Sketch n°1
Sketch n°2
Sketch n°3
Fig. 11 Geometric curvature minimisation in three steps. Rmin ϵ [blue = 3,00m; red = 10,00m]
2.5 Shape Meshing
Following a decade of research on this topic at the Navier Laboratory, a specific tool has been developed on Rhino & Grasshopper for the design of such shape-driven gridshells (Fig. 12). This tool gathers several components that process basic operations (meshing with the compass method, grid-processing, structural analysis) required for the generation of a suitable grid for the materialization of a 3D shape by a gridshell structure.
Fig. 12 Grasshopper canvas (compass method, grid processing, structural model generation)
Compass Method
This process propagates a two-way mesh of constant pitch on any NURBS surface.
Fig. 13 Compass method principle
Two crossing guide-curves are drawn on the surface to mesh. These curves mark the boundary of four quarters. Each half guide-curve is then subdivided with a compass