Morphological and Mechanical Investigation of DoubleLayer Reciprocal Structures
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Abstract
The simple connection conditions of reciprocal structures means that technological constraints become geometrical constraints and bending moments are increased. Geometrical constraints can be dealt by using formfinding methods such as a dynamic relaxation algorithm, but resisting bending moments to gain stiffness is difficult to accomplish without increasing the weight of the structure. For standard reticulated structures, common strategies consist in introducing curvature in the structure and/or modifying the structure into a doublelayer space structure. The proposed paper is thus an attempt to apply these strategies to reciprocal structures and to develop spherical domes with a structural thickness. Several configurations will be investigated and compared in term of geometrical feasibility and structural performance.
Keywords
Reciprocal structures Structural systems Design analysis Structural engineering Morphology Spatial structures Formfinding Dynamic relaxation Linear elastic behaviourIntroduction
From a technological point of view, reciprocal structures have the advantage of simplifying considerably connections in the sense that members are connected only by pairs. This constructional advantage has two main consequences: technological constrains are replaced by geometrical constrains and bending moments are increased through the non convergence of members at nodes with multiple connections. Indeed, elsewhere (Douthe and Baverel 2009) we have shown that the reciprocal structures are softer and generate higher stresses than their conventional counterparts. In order to increase the performances of reciprocal structures, here we investigate two basic strategies used for reticulated systems: the introduction of curvature and/or an increase in the structural thickness. Various configurations leading towards doublelayer systems will be presented. In “DoubleLayer Systems with Truss Elements”, simple configurations where structural thickness is obtained by the elements themselves (Vierendeel trusses or any other space trusses) will be examined and a simple construction made of ladder elements will be shown. Then, in “DoubleLayer Systems with Slender Members for Flat Configurations”, a solution for building a doublelayer reciprocal system with a flat configuration is developed based on analytical calculations (Sénéchal et al. 2011). In “DoubleLayer Systems with Slender Members for Curved Configurations”, a doublelayer dome configuration is investigated. The formfinding method using the dynamic relaxation algorithm (Douthe and Baverel 2009) is detailed and a comparison of the structural behaviour with a singlelayer configuration is made. Finally, we conclude with a discussion of the potential of doublelayer reciprocal systems.
DoubleLayer Systems with Truss Elements
Though only built for morphological purposes and not with any mechanical intention, it is worth noting that this structure has a span greater than 10 m and supports its own weight, which already proves its potential for practical applications. To go further, in (Douthe and Baverel 2009) and (Baverel 2000) it was demonstrated that, when loaded, the fans have a slight rotation due to the nonconvergence of members’ axial forces toward the fan centre. This rotation generates moments along the weak axis of the members. In the present case, moments along the weak axes might have some significant effects on the overall stiffness of the structure; thus from a structural point of view, the choice of Vierendeel trusses is not optimal. Further investigation of the mechanical behaviour should thus consider using members made of spatial trusses. However, for such members, the degree of freedom in rotation around their axis of the Vierendeel trusses will be likely lost and the connection between members will be more complicated. We can see then that there is a compromise to be found between ease of construction and efficiency.
DoubleLayer Systems with Slender Members for Flat Configurations
In “DoubleLayer Systems with Truss Elements”, the method proposed to obtain a “doublelayer” structure was to start from a singlelayer reciprocal structure and to replace slender members by space trusses. In the second method presented here, the doublelayer reciprocal system is obtained by transformation of a standard doublelayer spatial structure in a similar way to the transformations studied in (Sénéchal et al. 2011; Baverel et al. 2004; Baverel and Nooshin 2007). There are several way to compute practically the final configuration of the reciprocal system after the transformation: some analytical methods have been developed by rotation of members in (Sénéchal et al. 2011) and (Baverel and Nooshin 2007) and also some numerical methods using genetic algorithm (Baverel et al. 2004) or dynamic relaxation (Douthe and Baverel 2009). Here we will focus on the analytical method, while “DoubleLayer Systems with Slender Members for Curved Configurations” will make use of the dynamic relaxation algorithm.

Write that the direction of the eccentricity between two members is perpendicular to each member,

Write that the distance between two members is equal to the eccentricity,

Write that the distance between two contact points on a member is equal to the engagement length.

Write geometric compatibility between upper and lower fan (which is equivalent to adjusting the diagonal length so that upper and lower fans can effectively be connected).
DoubleLayer Systems with Slender Members for Curved Configurations
FormFinding of DoubleLayer Grids
In “DoubleLayer Systems with Slender Members for Flat Configurations”, the system was extremely symmetric and the irregularities of the system at boundaries were neglected, so that an analytic solution for the transformation could be found. This time, symmetries are more complex and practical solutions for the interruption of the grid at the boundary are desired. The transformation into a reciprocal system will thus be done numerically with a formfinding method based on the dynamic relaxation algorithm (Douthe and Baverel 2009). The principle of this method relies on the introduction of a fictitious mechanical problem where members initially converging at nodes are bent to form a reciprocal system. The solution of the problem is a structure in equilibrium where the bending energy of all members has been dissipated and where members are straight. The fictitious mechanical properties are defined according to the designer’s needs as will be illustrated in “FormFinding of a SingleLayer Equivalent Grid”. Then, as in the final configuration members are straight, there is no need to finely model the curvature. Convergence speed is thus increased by using the smallest number of nodes required to describe the members’ geometry, which is only four nodes per member (one node at each connection). Convergence is also improved by the use of a routine which has been added to the method presented in (Douthe and Baverel 2009) and which consists in gradually increasing the bending stiffness of members and updating their reference length as they straighten.
The geometrical parameters of the formfinding procedure consist in a set of two configurations: a reference configuration which is stress free and used to evaluate at each time step the stresses in the members, and an initial configuration from which calculations are started. Since by definition the members are straight, the reference configuration reduces to a set of reference lengths of the members. The reference length for the members of the hexagonal grid is thus set to 100 cm with an engagement length of 10 cm, that of the members of the larger triangular grid is consequently 173.2 cm, while their engagement lengths are the same, 10 cm. Then, since the intermediate members connecting the upper and lower grid will be used as spacers between the two layers, their length is set as 122.1 cm (100 cm in projection on the grids planes and 70 cm vertically); thus, after formfinding, the structural height should be close to 70 cm.
Finally the fictitious mechanical properties of the members are set so that their bending stiffness is much larger than their axial stiffness, so that the lengths of members are susceptible of finite variations. They can hence adapt to the various geometrical constraints and still stay straight (the outof straightness after formfinding should be <10^{−3}).
Considering the number of parameters that enter into the formfinding procedure (namely, the geometry of members, the disposition and style of the fans, the initial configuration and the boundary conditions), the reciprocal doublelayer structure shown in Fig. 10 is only one among an infinity of possible structures. At this stage, where only geometrical issues have been discussed, the criterion for choosing a suitable structure is constructability. Practical experience from previous experiments with regular polyhedra (Sénéchal et al. 2011; Baverel and Nooshin 2007) have proved that engagement lengths larger than 1.5 times the eccentricity between members axes are sufficient to insure feasibility and placement of connectors. As this is the case here, the reciprocal structure seems suitable. Further optimisation should thus be made relying on the mechanical behaviour, which will be investigated in section “Mechanical Behaviour of the Two Configurations”.
FormFinding of a SingleLayer Equivalent Grid
In the final configuration of Fig. 11, the reciprocal structure has a height of 1.94 m for a 7 m span which corresponds to a spherical dome with a radius of 4.1 m. As for the doublelayer structure, during the formfinding, the lengths of the members have changed and vary between 143 cm at the crown to 116.5 cm close to the supports. The engagement lengths vary between 34.5 cm at crown to 21.5 cm close to the supports. This dome structure is indubitably constructible, and due to its doublecurved shape, should prove to have good mechanical behaviour despite a relatively large engagement length.
Mechanical Behaviour of the Two Configurations
During the formfinding step, fictitious mechanical properties had been used to find the geometry of the previous reciprocal systems. In the geometries shown in Fig. 10 or 11, the members are straight but stretched (elongated) because initially the lengths of the members are unknown. Before conducting any mechanical analyses under external load, it is thus necessary to relieve the systems from inner axial stresses induced by these elongations. This is done by setting each member’s length at rest (the reference length) equal to its length in the final configuration obtained by formfinding. The removing of these stresses in either the singlelayer or doublelayer structure has no effect on the geometry of the structure (or, more precisely, the changes in the form are smaller than the outofstraightness tolerated in the formfinding).
Then, for the study of the mechanical behaviour, realistic mechanical and geometrical properties are introduced. Members are thus taken to be steel circular hollow sections with an external diameter of 40 mm and a thickness of 3 mm. Boundary conditions consist in blocking the positions of nodes located on the periphery of both structures. For the singlelayer dome, the boundary conditions are thus identical to those used for the formfinding. For the doublelayer structure, it is not easy to precisely define external nodes considering the number of members touching the periphery, so that it was chosen to support the structure on the additional members shown in Fig. 8, whose six degrees of freedom have been fixed.
Concerning external loads, two cases have been investigated: one symmetric, uniformlydistributed load and one asymmetrical distributed load on half of the structure. To ease comparison between systems, these distributed loads have been introduced via vertical point loads applied at each end of every member of the smaller grid (i.e., every member of the singlelayer system and only the shorter members of the doublelayer system). The intensity of each point load is 0.5 kN, which corresponds approximately to 3.5 kN/m^{2} (an acceptable density for a roof structure).
From a numerical point of view, nonlinear analyses are conducted using the same algorithm as for the formfinding procedure (reproducing the methodology used in [Douthe and Baverel 2009)]. Convergence is reached when the initial kinetic energy of the structure has been dissipated (practically when the current kinetic energy peak is below 1/100th of the initial/maximum peak). It must be noted here that the poor meshing of the formfinding has been slightly improved for the accuracy of results and that the density of the mesh has been refined to reach six nodes per member: one for each end connection, one for each engagement connection and then two intermediate nodes. Mechanical behaviour is assessed through: the overall stiffness of the structure (measured through the average and maximal displacement of the loaded points and the rotation of the fans), the intensity of reactions and inner stresses in the members (essentially axial forces and bending moments).
Comparison of displacements of single and doublelayer reciprocal systems
Symmetric loading  Asymmetric loading  

Average  Maximum  Average  Maximum  
Vertical displ. (mm)  
Doublelayer  7.5  11.5  3.0  6.0 
Singlelayer  12  30  5.5  29 
Fan rotation (deg)  
Doublelayer  ≈0.0  ≈0.1  ≈0.0  0.1 
Singlelayer  0.2  0.5  ≈ 0.0  0.5 
Comparative of forces in single and doublelayer reciprocal systems
Max R_{H} (kN)  Max R_{V} (kN)  N_{min} (kN)  N_{max} (kN)  M_{max} (kN.m)  σ_{min} (MPa)  σ_{max} (MPa)  

Symmetric loading  
Doublelayer  9.8  13.1  −6.0  4.3  0.57  −188  189 
Singlelayer  6.0  8.0  −7.2  2.2  0.71  −245  228 
Asymmetric loading  
Doublelayer  9.4  14.4  −9.5  9.0  0.59  −194  199 
Singlelayer  6.0  8.0  −8.3  2.6  0.82  −277  269 
Conclusions
Different configurations to improve the efficiency of reciprocal systems have been investigated in this paper. The first configuration showed the potential of using spatial members to create reciprocal configurations with structural thickness. The second configuration then illustrated how planar doublelayer reciprocal systems can be obtained through the transformation of standard spatial grids. The results of analytical developments on a twoway square grid have been presented. However the toorestrictive hypotheses made for the analytical calculations of the geometry led to engagement lengths that are quite small compared to the diameter of the members, thus making construction difficult. Further development with more design parameters are thus currently under progress.
In the final section, two spatial configurations of reciprocal systems were investigated. The first one is a doublelayer dome based on the superposition of two triangular grids, the second one a singlelayer dome based on a triangular grid. A formfinding method for the definition of geometries compatible with reciprocal arrangement of members was illustrated. Then a mechanical study of the two structures was conducted. On the one hand, the doublelayer reciprocal structure works effectively like a spatial structure, like a thick plate in bending with high bending stiffness and low sensitivity to asymmetric loading; on the other hand, the singlelayer dome works like a thin shell dominated by membrane forces and is therefore more sensitive to asymmetric loading. However, we have seen that both structures are efficient and capable of satisfying standard design criteria for roofing structures.
The structural typologies which can be achieved with reciprocal systems seem thus to be as varied as those obtained with standard typologies. This means that, for designers, there is thus a very wide universe of morphologies to explore, and ample room for optimisation and creativity.
Footnotes
 1.
An image of one such structure is shown on the webpage http://designmuseum.org/design/rbuckminsterfuller.
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