Description of the research
The buckling of gridshells is of practical interest for engineers who want to assess the safety and reliability of structures. Some post-buckling analysis have been applied to the gridshell of the Solidays festival. They demonstrate that these structures have a ductile behaviour, even with the assumption of accidental yield of some members (Tayeb 2013). These calculations are however time-consuming. Simpler indicators can be used to assess the stability of gridshells during early stages of design. The current literature provided the designers with little insight on the actual influence of the forces induced by the beams deformation on the buckling capacity of gridshells.
Studies by Lefevre et al. and Mesnil et al. have shown that the pre-stress due to the shaping process of elastic gridshells has little impact on their stability. In pseudo-funicular grid-shell, the pre-stress consists of compression forces, and the bearing capacity decreases (Mesnil et al. 2013&2015). In gridshells mapped on arbitrary surfaces, both tension and compression can result from the form-finding process. The pre-stressed structures can thus in some cases be more stable than unstressed ones (Lefevre et al. 2015).
The methods developed can be extended to other kind of residual stress fields. As an example, residual stresses are present in some steel structures because of the fabrication process (think of cold bending techniques). The stability of steel structures is currently under investigation for non-conventional patterns. The first results concern the kagome grid pattern, which is shown to have promising properties, as it outperforms quadrangular gridshells without cable bracing (Mesnil et al. 2017).
- Architectural Geometry
- Steel Structures
- Active Bending
Non-‐standard patterns for gridshell structures: fabrication and structural assessment – R. Mesnil et al. (2017)
Mesnil R., Douthe C., Baverel O.
This paper investigates the potential offered by non-regular patterns for the design of gridshell structures focusing on two aspects: fabrication and structural performance. Studies on non-regular patterns are scarce, with a few notable exceptions . We propose a framework for shape generation in real-time of gridshells with non-standard patterns with planar facets and the assessment of the proposed grid patterns. The possibilities of the framework are demonstrated through a detailed study on the kagome grid pattern, which is made of hexagons and triangles.
We propose two intuitive methods to generate kagome gridshells with planar facets. The first method is based upon the marionette technique, a method for shape generation inspired by descriptive geometry and proposed by the authors in . The marionette technique constitutes the most general framework for the shape generation of gridshells with planar facets. We re-interpret this method with an algorithm of conversion from Planar-Quadrilateral (PQ) mesh to a Planar-Kagome (PK) mesh.
We study then the structural performance of the kagome grid patterns by comparing the linear buckling load of gridshells covered with quadrangular or kagome grid pattern. We summarized the main results of a parametric study presented in  for various geometries and grid densities showing that the linear buckling load of kagome gridshells can be up to twice as high as the one of quadrangular gridshells. Finally, we propose a method of homogenization inspired by Winslow  for the evaluation of the stiffness of non-standard periodic patterns.
Linear buckling of quadrangular and kagome gridshells: a comparative assessment – R. Mesnil et al. (2017)
Mesnil R., Douthe C., Baverel O. and Léger B.
The design of gridshells is subject to strong mechanical and fabrication constraints, which remain largely unexplored for non-regular patterns. The aim of this article is to compare the structural performance of two kind of gridshells. The first one is the kagome gridshell and it is derived from a non-regular pattern constituted of triangles and hexagons. The second one results from a regular pattern of quadrangles unbraced by diagonal elements. A method is proposed to cover kagome gridshells with planar facets, which reduces considerably the cost of fabrication of the cladding.
The sensitivity of kagome gridshells to geometrical imperfections is discussed. The linearised buckling load of kagome gridshells is then compared to the one of quadrilateral gridshells. The most relevant design variables are considered in the parametric study. Two building typologies are studied for symmetrical and non-symmetrical load cases: dome and barrel vault. It reveals that the kagome gridshell outperforms quadrilateral gridshell for a very similar construction cost.
Buckling of Elastic gridshells – B. Lefevre et al. (2015)
Lefevre B., Douthe C. and Baverel O.
Elastic gridshells are doubly curved grids with a shell-like behaviour, obtained by elastic deformation and bracing of a flat grid with no shear rigidity. After a brief review of their design process, the buckling of elastic gridshells is numerically investigated through a geometrically nonlinear analysis; then a formula for predicting the local buckling load is derived from the results of a parametric study. The numerical study required the development of an accurate form-finding method named " the advanced compass method " and of a specific algorithm ensuring the quick convergence of the non-linear finite element simulations, both presented in this paper. The influence of the prestress, the eccentricity and the anisotropy of the shell are discussed, giving hence indications for correct modelling and efficient design of elastic gridshells.
Stability of pseudo-funicular elastic grid shells – R. Mesnil et al. (2015)
R.Mesnil, J. Ochsendorf, C. Douthe
The paper presents some results on the influence of the pre-stress induced by the erection method of elastic grid shells on their buckling capacity. It starts with the numerical methods and their validation with the study of a pre-buckled arch. Then, a form-finding scheme using low-speed dynamics is used to generate automatically a family of elastic grid shells, and their buckling capacity is compared to the one of grid shells with the exact same geometry, but without any pre-stress. The paper demonstrates finally that the pre-stress decreases by a few percent the buckling capacity of elastic grid shells.
Stability of elastic grid shells – R. Mesnil (2013)
The elastic grid shell is a solution that combines double curvature and ease of mounting. This structural system, based on the deformation of an initially at grid without shear stiffness was invented more than fifty years ago. The apparition of new materials such as GFRP increased the potential of such structures whose properties depend on the deformation, or equivalently pre-stress of an initial structure. Elastic grid shells seem particularly promising as shelters, lightweight roofs, or kinetic structures. Although fundamental to the behavior of the strucure, the influence of the pre-stress on the stability of elastic grid shells has yet to be studied. Understanding this phenomenon could allow engineers to design more efficiently elastic grid shells. This thesis studies the influence of pre-stress on the stability of elastic grid shells. The research conducts a parametric study that focuses both a pre-buckled arch and initially at circular elastic grid shells with dierent grid spacing and levels of pre-stress. Realistic values of the parameters are determined from existing projects. The buckling analysis as well as the form-finding of the different structures are performed using finite element analysis. The tools are validated with comparison of the shape and buckling capacity of a pre-buckled arch with existing experiments. The parametric studies lead to recommendations aiming to facilitate the design of elastic grid shells. Keywords Elastic grid shell, Low-Speed Dynamics, form-finding, linear buckling analysis
Stability and robustness of a 300 m2 composite gridshell structure – F.Tayeb et al. (2013)
Tayeb F., Caron J.-F., Baverel O., Léger B., Du Peloux L.
In this paper, ductility aspects of a light-weight composite gridshell are developed. A gridshell is a very light structure that can support relative high loads. For many reasons, the materials used by the Navier laboratory are glass fibre reinforced polymers (GFRPs) that have elastic brittle behaviour. To ensure the safety of people these structures have to behave in a ductile way, that is to say they must not collapse without showing signs of weakness. This paper deals with the pseudo-ductile behaviour of the GFRP gridshells designed by the Architected Structures and Materials research unit of Navier laboratory.
After a reminding context about gridshells, the buckling of the Solidays’ festival gridshell prototype (June 2011) is considered. It is demonstrated that buckling has to be avoided carefully because it fosters high stresses in the beams and may lead to a brutal collapse of the structure. Then it is shown that, under Ultimate Limit State (ULS), the gridshell prototype is far from buckling. Finally, a simulation of accident is performed: from the ULS, several elements are broken in order to understand the behaviour of the structure in case of accident. The pseudo-ductility of the structure is demonstrated thanks to the redundancy of the structural concept of the gridshell.
The articles [Mesnil 2013 &2015] are the result of a research pursued at MIT within John Ochsendorf’s Structural Design Lab.