Investigation of the elastic behavior of reciprocal systems using homogenization techniques – IASS2013

This is the paper I presented at the International Association for Shell and spatial Structures 2013. Also you can find the full paper here.


“Reciprocal system” is an expression that designates a specific family of structures which has its roots in very old traditional construction systems in various regions of the world. Some examples can be found in traditional 12th century roofing structures in Japan as well as in manuscripts from the French architect Villard de Honnecourt or the Italian scholar Leonardo da Vinci. This particular structural system can be seen as a construction technique to cover large spaces with short members assembled together with very simple connections: traditionally, for stone architecture, connections are considered as compressed and unidirectional joints but, in many recent applications, connections are made of spherical joints.

The reciprocal character of the system comes from the fact that locally every member is supported by and supports its neighbours in a cyclic way as illustrated in the figure. To describe this kind of assembly, Baverel suggested that a reciprocal system, also called a ”nexorade”, can be seen as a traditional structure in which members converging at a node would have been engaged within this node and thus proposed to call ”engagement length” the length of the member which has entered the connection and ”engagement window” the opening formed by the engagement of the members within the node. They then defined some other geometrical properties to characterise end dispositions but it is not necessary to recall them here as the only configuration that will be investigated in this paper is that of the figure.

Considering structural behaviour, research has been mainly dedicated to the investigation of some particular configurations or some particular structures. In these studies, the authors observed many interesting phenomena and relationships which are quite characteristic of the behaviour of reciprocal systems. For instance, about the way the transverse load is ”circulating” from the interior of the surface to the supports, Gelez pointed out that global bending of the “grid” generates additional local bending and large shear forces and that, more surprisingly for a simply supported structure, the shear forces in the members were maximum at mid-span whereas it is classically acknowledged that shear forces are maximum close to the supports. Likewise Douthe and Baverel or Sanchez and Escrig underlined a strong link between the engagement length, inner bending moments and bending stiffness of the system.

However, from a design point of view, there are actually no practical recommendations or formulas giving the influence of the various parameters on, for example, the global deflection of a reciprocal system or inner forces in the members. So, as the generation of complete reciprocal system (especially its geometry) is time consuming, the possibilities for optimisation and the variety of construction systems are practically reduced.

It is possible to give partial answers to these questions by means of homogenization techniques. The basic idea of homogenization consists in “seeing from afar” the beam assembly which constitutes the N. Indeed, even if the assembly is made of a well-defined periodic pattern at local scale, when zooming out, the pattern blurs until only a surface is seen. In mathematical terms, this procedure consists in separating scales: macroscopically the engineer wants to see an equivalent shell or plate model and microscopically he does not want to miss the detail of the stress state. Scale separation was rigorously established mathematically. Based on this assumption in the linear elasticity framework, it is possible to derive a macroscopic model with an homogenized constitutive equation and also localization fields which enable the reconstruction of fields inside the unit cell which generates the whole structure.

Here, the N is made of few beams arranged in order to create a planar periodic pattern. The unit-cell which generates it, is constituted of only two beams (see red members in the above figure). It will thus be possible to derive closed-form solutions of the localized resultants and moments which will be induced by the macroscopic plate fields. The corresponding plate model, that of the macroscopic surface, will be a Kirchhoff-Love plate model whose equivalent bending stiffness will be derived mathematically and investigated.

It must be made clear that, if generally homogenization techniques are used to reduce the computational burden by separating scales, in the present case, it is obviously straightforward and accurate to compute directly the full structure, so that a gain is only expected in terms of parametric study: homogenization techniques will avoid the necessity for generating numerous models with different geometries. And, apart from this aspect, our intention is to reveal some interesting features about the intrinsic behaviour of these periodic structures and to infer their consequences on preliminary design.