The elements in conventional structures are perfectly ranked, so that load transmission is logical and follows the usual structural orders. Nevertheless, in reciprocal structures each element has to support all of the others in a less intuitive pattern of load transmission. The purpose of this paper is to understand exactly how load is transmitted between elements, quantifying this analytically by developing a new method which is applicable to a flat structure composed of a basic unit with any number of nexors. It is based on determining the increase in load to which the members in a reciprocal structure are subjected by calculating the coefficient
En las estructuras convencionales la transmisión de cargas es lógica y sigue los órdenes estructurales habituales. Sin embargo, en las estructuras recíprocas cada elemento tiene que soportar a todos los demás en un patrón de transmisión de cargas menos intuitivo. El objetivo de este trabajo es comprender exactamente cómo se transmite la carga entre los elementos, cuantificándolo analíticamente mediante el desarrollo de un nuevo método que es aplicable a estructuras planas compuestas por una unidad básica con cualquier número de nexors. Se basa en la determinación del incremento de carga al que están sometidos los miembros de la estructura recíproca mediante el cálculo del coeficiente k, o “coeficiente de transferencia”. El valor del coeficiente k, y por tanto la carga transferida entre los miembros, disminuye con el número de nexors, con la proximidad de las cargas puntuales a los apoyos exteriores y con el tamaño del espacio central.
Reciprocal structures have been used for centuries because they are an ingenious solution when the aim is to roof large spans using short elements and engagement lengths (
Wood is a perfect material for building reciprocal structures as it is very resistant against forces parallel to its fibre, very low in weight in comparison with its strength, flexible so that elements can be joined without the need to use heavy mechanical aids, and with good elastic capacity so that it can be assembled and disassembled when necessary.
As is the case in other types of ancient constructions, the basic material consisted of small pieces of wood from nearby forests, as this was the most economical solution, giving immediately available components.
According to del Rio (
Although this is not a very common structural typology, reciprocal structures fell into total disuse after the second half of the 18^{th} century because the industrial revolution brought about the development of new structural materials and the production of longer elements able to cover long spans.
However, in recent decades this type of structure has found a new market niche in what is termed ephemeral architecture. This consists of temporary works that are fully reusable, fitting very well with the current tendencies for sustainable architecture or bioconstruction. The scientific literature contains several examples of threedimensional reciprocal constructions which express this resurgence (
It is important to use the same terms when referring to the elements of these structures when studying them,
Reciprocal structures or nexorades are modular, and they are composed of basic units denominated “fans”. Each fan is composed of at least 3 nexors, which is the term used to refer to the members in a fan (
Depending on the arrangement of these basic units or fans, nexorades can be of three types: “simple” when the structure consists of a single basic unit of either 3 nexors, 4 nexors, 5 nexors, etc.; “multiple” when the structure consists of a combination of several basic units, such as a combination of several basic units of 3 nexors with several basic units of 4 nexors; and “complex” when the structure consists of the extension by repetition of the same basic unit (
A regular structure is obtained when the members of a reciprocal structure are arranged regularly around a central point of symmetry. An irregular structure results from the repetition of basic units in a disorganised way without a regular pattern (
In recent years reciprocal structures have been increasingly used by engineers and architects, due to their high degree of congruity with certain current trends. Thus some researchers undertook the laborious task of collecting the most important design aspects used to date, with the aim of encouraging their use in architecture (
Some researchers have concentrated on the study of reciprocal structures from a geometrical pointofview, establishing and optimising the parameters that determine their shape (
The elements in conventional structures are ranked perfectly, so that load transmission is logical and perfectly follows the structural orders set by the design. Nevertheless, in a reciprocal structure each element in it has to support all of the others to a greater or lesser degree, in a nonintuitive pattern of load transmissions between them. The principle of reciprocity is defined as the use of supporting elements which, resting on each other, form a spatial configuration without any clear structural hierarchy (
In a reciprocal structure, no substructure is stable until the complete structure has been assembled, and moreover if one of its elements is eliminated, then each part of the structure will be mobile respecting all of the other parts, i.e., this would create a mechanism (
Gelez analysed the behaviour of reciprocal structures using a method that is half analytical and half numerical, to obtain the maximum moment and the deformation at the centre of a basic unit (
The fact that a flat reciprocal structure is considered to be statically determined indicates that it is not necessary to undertake deformation compatibility analysis to discover the unknowns in the system. The loads which act on each connection will therefore be independent of the rigidity of the material and member crosssection.
Kohlhammer and Kotnik used a repetitive approach to discover the distribution of loads among connections, under the action of the self weight of the elements in a flat reciprocal structure (
The aim of this research work consists of understanding exactly how load transmission occurs between the members of a simple reciprocal structure and obtaining, analytically, the amount of this load in each point of the system. Once the exact amount of load on each member of a reciprocal structure and the influence of the geometry on the load transmission is known, it will be possible to design such structures more accurately and more quickly. The analytical equations obtained will be used as the basis for further research work that analyses more complex reciprocal structures.
The methodology consists of analysing how loads are transmitted between the members of a flat reciprocal structure composed of a basic unit of
The structure is considered to be subjected to perpendicular exterior loads, while ignoring the self weight of the elements. Point loads as well as loads distributed over members are analysed at any position in the structure.
The initial hypothesis considers a reciprocal structure with a point load applied on one of its members. The static balance of the member that receives the load is established, and so on successively until the final member is reached, observing that a part of the initial load returns to the first member after passing through all of the members in the structure. With this the static balance of the first member is lost, so that the extra load has to be distributed once again among all of the members. In each repetition the extra load is less each time, in a process that gradually approaches zero. The calculation methods proposed by some researchers are based on this consideration (
Based on the above argument it is possible to conclude that every time an exterior load is applied to a member in a reciprocal structure of the type studied, this will be distributed between an exterior support and an interior support, and that, in turn, the load that reaches the inner support must be supported once again by the whole structure, including the member that receives the exterior load.
The method described in this work consists of firstly establishing this increase in load on the first member in order to make the distributions to the others only once, i.e. only the final state of static equilibrium is considered. The value of the transferred load is unknown but it is clear that it will be directly proportional to the load applied,
The study takes place in 3 phases. Phase 1 considers the action of point exterior load applied to a reciprocal structure composed of a basic unit of 3 nexors. In phase 1 a generic equation is obtained to calculate the transfer coefficient,
The results obtained analytically in phases 1, 2 and 3 are compared with the results shown by calculation software to verify the validity of the method. The verification of the analytical equations obtained is carried out using the Dlubal_{©} software, version 5.21.02. For membertype elements (1D), this software uses the Matrix Stiffness Method which proposes a final equilibrium of the system to obtain the internal forces and reactions from the displacement of nodes.
In the end, an experimental demostration is carried out for a basic unit of 3 nexors. For this purpose, a 10gram weight is placed in the centre of a simple timber member of crosssection 1x10 mm and span 420 mm, and the vertical deflection in the centre of the span is measured,
This experimental verification consists of checking that the effective deflection suffered in the centre of the span by the member into reciprocal structure is the same as suffered when it is alone, but multiplied by 1 +
Firstly, study centres on how a point load
On the loaded member,
As was pointed out above, the most intuitive way of approaching the problem would firstly involve considering that the load
Applying the proposed method, the load at the centre of member 1 will be
To verify whether the approach is correct, this structure was modelled using the finite element software. In this, in a basic unit of 3 1000 mm members joined at their midpoints, a load of 1 N is placed at the centre of member 1,
The shear graph shows that the load applied by member 3, which is on member 1 at the point of application of the same, is 0.143 N, i.e., 1/7
The moments graph shows that the members are articulated at their exterior support as well as the intersection with the next member, and that in each of them the moment would correspond to an isostatic beam with two supports that supports the loads found by means of the previous procedure. We can therefore see that once the loads have been found, analysis of thestructure is simple and that there is no need to undertake deformation compatibility.
To verify the dependency of the transfer coefficient
After this point the load distributions no longer depend on
Proceeding in the same way as in the first example,
The software is used again to perform the verification, but with a load that is not centred. Using the previous model, the load
In this case, the shear graph shows that the load transferred by member 3 over member 1 is 0.214 N. If the equation
Although work took place with basic 3member units (or 3 nexors), the process would continue in the same way for a greater number of members, multiplying by
In a reciprocal structure composed of a basic unit of
is the transfer coefficient
the number of nexors
the nexor length
the distance of the load from the exterior support of the member on which it is applied
the distance from the intersection between members to the exterior support
In structures of this type it is habitual for distributed loads to appear. These usually arise due to the working load considered by regulations, the self weight of the members comprising the structure, or other elements that weigh on the same. Due to this, the way uniformly distributed load is transmitted in a reciprocal structure is analysed below.
To return to the basic 3member (or 3 nexors) unit, a uniformly distributed load
To generalise, for a reciprocal structure composed of a basic unit of
is the transfer coefficient
the number of nexors
the nexor length
the distance of the intersection between members and the exterior support
In the same way, the analytically obtained value of
Applying the analytical formula [
To verify the validity of formulas (
The distribution of loads
The transfer coefficient
The graph shown in
By applying equations [
Measurement location  Deflection 10gram weight  Deflection 20gram weight  


centre of the span  20 mm  40 mm 

end of the span  14 mm  30 mm 
centre of the span  30 mm  60 mm  
centre of the span (effective)  30  14/2 = 23 mm  60  30/2 = 45 mm  
Load increase between simple member and member into reciprocal structure  23/20 = 1.15  45/40 = 1.13  

1.15  1 = 0.15  1.13  1 = 0.13  

0.14 
Applying equation [
The analytical results obtained using equations [
Based on equations [
The exterior supports closest to the exterior load receive a higher proportion of the same. From equation [
On the other hand, it can be deduced from the method itself that the greater
This research work describes the development of a new method which makes it possible to understand and analytically quantify how loads are distributed among the members and supports of a flat reciprocal structure.
It is based on determining the increase in load to which the members of a reciprocal structure are subjected by calculating coefficient
The value of coefficient
The present research work is limited to the study of a basic unit consisting of any number of nexors of the same length and with central symmetry. This basic unit can be subjected to any system of loads, either point and/or distributed loads.
In structures with more than one basic unit (multiplex, complex, etc.) the reasoning used to determine the load transfer would be similar, but number of equations would increase considerably and will be the subject of future research.
The authors would like to express their gratitude to the Materials Laboratory of the School of Architecture of the Universidad Politécnica de Madrid for providing their facilities for this research, as well as to the company Dlubal Software GmbH for providing their software.