# Static and Kinematic Formulation of Planar Reciprocal Assemblies

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## Abstract

Planar reciprocal frames are two dimensional structures formed by elements joined together according to the principle of structural reciprocity. In this paper a rigorous formulation of the static and kinematic problem is proposed and developed extending the theory of pin-jointed assemblies. This formulation is used to evaluate the static and kinematic determinacy of reciprocal assemblies from the properties of their equilibrium and kinematic matrices.

## Keywords

Reciprocal frames Static determinacy Kinematic determinacy Rigid bodies## Introduction

The use of reciprocal structures (or reciprocal frames, as they are sometimes called) as a method of construction dates back many centuries, but investigations and studies on their structural properties are from quite recent times. Historical examples, from the bridge depicted the Qingming Beijing Scroll, to the work of European architects and scientists such as Villard de Honnecourt, Leonardo da Vinci and Serlio have been described by many authors from the morphology point of view. In recent times the concept of a multi-reciprocal structure was first patented by Anthonius Bijnen in Holland (1976) and subsequently by Graham Brown in Great Britain (1989) under the title of ‘three-dimensional structure’.^{1}

While research about such structures mainly considers two themes—the study of geometrical properties of reciprocal configurations and the investigation of its structural behaviour—comprehension of the structural behaviour of reciprocal structures is still at an initial stage and mostly focused on regular configurations. Gelez, Aubry and Vaudeville (2011) analysed the behaviour of planar reciprocal configurations based on regular grid of four-element fans. Douthe and Baverel (2009) compared the structural behavior of a set of dome-like reciprocal structure (nexorades) with more conventional triangulated structures while Parigi and Kirkegaard (2013) compared the structural behavior of several two and three dimensional reciprocal structures to investigate the influence of different geometric parameters on the internal forces distribution.

In recent times, in addition to the geometry and the structural behaviour, there is an increased interest in the kinematic behaviour of reciprocal assemblies, for the analysis and discovery of novel mechanisms based on the principle of reciprocity (Parigi and Sassone 2011).

The aim of this paper is to extend the studies of Pellegrino and Calladine (1986) on pin-joint assemblies to reciprocal structures, for assessing the static and kinematic behaviour of reciprocal structures through the study of the associated equilibrium and kinematic matrix. A rigorous formulation of the static and kinematic problem in plane reciprocal grids is proposed. The influence that the number and type of internal and external constraints and the arrangement of the elements have on the static and kinematic determinacy is then assessed.

In the first part of the paper, a comparison is provided between pin-joint structures and reciprocal structures from the mechanical point of view, and an explanation is given on the reasons why the formulation used for assessing the static and kinematic determinacy of pin-jointed assemblies cannot be used for reciprocal structures.

In the second part, a more general formulation suitable for assessing the static and kinematic determinacy of both reciprocal and pin-jointed assemblies is proposed, and examples are provided for the construction of the equilibrium and kinematic matrices of pin-jointed and reciprocal assemblies with the proposed formulation.

In the last part of the paper, numerical results are shown concerning the kinematic behaviour of different planar reciprocal assemblies through the explicit formulation of their equilibrium and kinematic matrices.

## Mechanical and Kinematic Behaviour of Reciprocal Structures

### Pin-Jointed vs Reciprocal Structures

If a design solution based on the principle of structural reciprocity is analysed and compared with a similar arrangement based on a classical pin-jointed truss, it is possible to find some analogies and differences in both the static and kinematic behaviour.

When working with spatial structures, for instance, the optimal design usually leads to the use of structural elements with the same length and properties, in order to simplify the manufacturing process and the erection of the structure. This tendency is actually common to both structural systems. When the mechanical behaviour is analysed, it becomes clear that the likeness of elements and of their arrangement does not involve a corresponding analogy in the internal forces transfer mechanism. In both types of structures, forces are transferred through the internal joints, but, in reciprocal frames, joints are positioned close to element ends (but not at) as well as at intermediate points, without breaking the internal continuity of each element. As a consequence, while truss elements are only subjected to axial force, reciprocal structure elements are subjected also to shear and bending.

- 1.The configuration obtained by the regular repetition of a four-element fan present an interesting 1 DOF (degree of freedom) folding mechanism (Fig. 1a);
- 2.
A novel 1 DOF mechanism based on the principle of reciprocity which uses internal prismatic joints (Parigi and Sassone 2011) (Fig. 1b);

- 3.
Configurations that use curved elements. A morphogenetic procedure for the design of kinetic reciprocal system (KRS) starting from the definition of the required kinematic behaviour has been developed and tested on physical models. The models’ characteristic kinematic behaviour is achieved by constraining elements to follow free-form paths, which might be coincident with the shape of elements themselves, by means of pin-slot-like internal constraints, or half-sliders. Based on this experimentation a novel mechanism that can be used to create an indefinitely large 1 DOF iris-shutter mechanism was discovered by (Parigi and Sassone 2011) (Fig. 1c).

Reciprocal structures therefore appear to offer a huge potential for the development of innovative kinetic structures. However, their behaviour is difficult to predict intuitively, and no extensive studies have been carried out on the subject. On the other hand, the behaviour of pin-jointed assemblies has been the subject of extensive researches. The assemblies’ static and kinematic behaviour can be accessed through the complete specification and evaluation of the static and kinematic matrix (Pellegrino and Calladine 1986). A similar approach would be convenient for the evaluation of the behaviour of reciprocal structures. However, the conditions for modelling a pin-jointed structure are that elements are joined at their ends, and that loads are applied at nodes only. Neither of the two conditions applies to reciprocal structures. An alternative formulation for reciprocal structures is provided in the sections of the paper that follow.

### Kinematic Determinacy of Planar Reciprocal Structures

This section will focus on the static and kinematic determinacy of planar reciprocal structures. In these structures the shape can easily be defined in plane coordinates, so that the mechanical behaviour can be studied independently from the issues that arise when the geometry of three-dimensional reciprocal configuration must be defined (Parigi and Sassone 2011). The study of plane configurations is fundamental for understanding the reciprocal mechanical behaviour of more complex structures.

If a three-element fan is considered where elements are connected with internal hinges, it is a kinematically determinate assembly in both two- and three-dimensional configurations, i.e., it exhibits only the free body motions when unconstrained to the ground. When a four-element fan where elements are connected with internal hinges is considered, it is a kinematically indeterminate assembly in both two- and three-dimensional configurations, i.e., it exhibits one additional inextensional mechanism (Fig. 2b), besides the free body motions.

The presence of infinitesimal mechanisms can be detected at an elementary level by comparing the number of DOF possessed by the assembly and the number of constrained conditions imposed by the internal and external constraints. It is well known that each rigid body in a planar configuration possesses three DOF. So the three-element assembly of Fig. 2a possesses nine DOF, and the three internal hinges each remove two DOF. Therefore, when these are subtracted from the total DOF, there remain three DOF that correspond to the three rigid body motions of the fan.

The four-element assembly of Fig. 2b possesses twelve DOF. The four internal hinges each remove two DOF. Therefore, when these are subtracted from the total DOF, there remain four DOF that correspond to the three rigid body motions of the fan plus one additional inextensional mechanism.

On the other hand, if the configuration of Fig. 1b is considered, the assembly possesses nine DOF. The three prismatic joints remove each two DOF, and so, if subtracted from the total DOF, three DOF are obtained that correspond to the three rigid body motions of the fan. However, this result contradicts the physical evidence of the additional mechanism that the assembly possesses, thanks to the translation of one element over each other.

Similarly, the assembly shown in Fig. 1c is composed of six three-armed elements and possess eighteen DOF. Each of the twelve half slider removes one DOF, and each element is constrained to an external hinge in the centre where the three ‘arms’ converge, removing two DOF. In total, there are twenty-four degrees of constraints, which is greater than the number of the DOF, and the assembly should be statically and kinematically determinate. However, the configuration possesses one DOF that corresponds to the simultaneous rotation of each element around its axis (Parigi and Sassone 2011).

From this discussion it is evident that, in general, the determination of the kinematic behaviour by merely counting of the DOF and the constrained conditions constitutes an inadequate method, because is not able to detect states of self-stress and inextensional mechanisms.

### Families of Planar Reciprocal Structures

- 1.Regular topology and joint position networks (Fig. 3a), obtained by means of the repetition of unit cells with the same shape and dimensions;
- 2.
Regular topology and irregular joint position networks (Fig. 3b), based on a regular topological scheme, but with changing in length of elements and joints position;

- 3.
Non-regular topology and joint position networks (Fig. 3c), based on non-repeating topological scheme, irregular elements length and joints positioning.

The configuration shown in Fig. 3a is obtained by the repetition of the four-element fan of Fig. 2b in a pattern of 3 rows × 3 columns. It can be described as geometrically formed by two sets of squares with the sides in a ratio of 1/2, organized in a lattice configuration.

The unit cell contains four internal constraints and four external constraints, i.e., when two cells are joined to form a more complex pattern, the external constraints become internal. The number of constraint conditions does not change during the transformation from external to internal, because the ends of elements of one cell join to intermediate points of the second cell. In fact, if the ends of first cell elements were joined to the second cell elements ends, two external constraints would be merged into one, and the total number of constraint conditions would be reduced.

The configuration shown in Fig. 3b has a similar topology of the configuration of Fig. 3a but the position of elements inside each fan is different. This configuration is therefore an example of the variety of the morphologies of reciprocal configurations that can be obtained with the variation of the intermediate support position of each element with the adjacent ones.

The configuration shown in Fig. 3c is completely non-regular, with regard to both the length of the elements and the morphology of its fans; it can only be defined through the specification of the topology of the connections.

For the determination of the static and kinematic determinacy of each of the three configurations, it is necessary to formulate the static and kinematic problem since the behaviour of the network cannot be directly determined from the properties of the unit fan.

All these configurations can be classified additionally on the basis of the use of internal constraints. As opposed to pin-jointed assemblies where the only internal constraint used is the hinge, in reciprocal systems the set of internal constraints that can be used includes sliders and prismatic joints.

Here, we will focus on a general formulation of the static and kinematic problem, and numerical examples will be provided on the three families listed that use internal hinges.

## Matrix Analysis: Pin Joint vs. Reciprocal Structures

Reciprocal frames, like truss structures, grid shells, lamella domes, cable nets and spatial frames, belong to the more general category of bar assemblies, i.e., structures formed by joining sets of thin one-dimensional elements, arranged in order to obtain stiffness and load bearing capacity. The common aspect of all these structures is that they can be geometrically described as composed by a number of elements *b*, usually straight and connected to each other by joints *J*. A further set of joints connects the structure to a fixed reference system (foundation joints). A complete analysis of the geometrical properties of such configurations was developed previously, yielding to fundamental results as the Pascal’s theorem and the corresponding dual of Brianchon, in the frame of the geometry of incidence. The general study of mechanical properties of bar assemblies, on the other hand, mainly focuses on the subcategory of pin-jointed frameworks, or truss structures, in which all the bars are connected at their ends by plane or spherical hinges. Due to their enormous diffusion as structural typologies, a number of analysis procedures, both graphical and analytical, have been developed for the study of the equilibrium condition of pin-jointed frameworks. The evaluation of static determinacy was one of the main issues, because statically determinate assemblies have great importance in applications. In fact, when assemblies are statically determinate, every structure can be solved by means of equilibrium equations without the additional compatibility equations that have to be used for statically indeterminate structures. Hence the interest has been focused on the evaluation of the static determinacy and, dually, on the kinematic determinacy of such structures.

In recent times there has been a renewed interest in the study of mechanical properties of repetitive structures, mainly as a consequence of the diffusion of micro- and nano-structured materials such as foam and micro-crystalline materials. In fact, for such regular repetitive structures extending indefinitely, the evaluation of global mechanical characteristics such as the static or kinematic determinacy, or the possible mechanisms, is not trivial. Vassart, Laporte and Motro (2000) studied the problem of the mechanism order in reticulate plane structures. They put the length variation amplitude of elements in relation with the underlying mechanism, obtaining a procedure that allows evaluation of the order of the mechanism itself. The finite mechanisms, in this approach, are then regarded as mechanisms of infinite order. Starting from two-dimensional lattices made of bars pinned to each other and following regular schemes, an interesting paradox has been underlined by Guest and Hutchinson (2003). For these systems the kinematic matrix *C* is not the inverse of the static matrix *A*, so any rigid repetitive lattice is redundant.

*s*is the number of independent states of self-stress and

*m*the number of independent inextensional mechanisms. The simultaneous presence of self-stress states, as an effect of static indeterminacy and of inextensional mechanisms due to kinematic indeterminacy, is no longer regarded as an anomaly but rather as a specific aspect of the structural behaviour of bar assemblies.

The presence of finite or infinitesimal inextensional mechanisms *m* (kinematic indeterminacy) or the possibility of self-stress states *s* (static indeterminacy) is related to the spatial disposition of elements and the topology of the connections, while the only constraint present is the internal hinge. Although for given assemblies it might be possible to detect the number of *s* and *m* by physical intuition, it is clearly desirable to have a general algorithm for the direct computation of *s* and *m* so as to form the geometrical data of any given assembly. The values of *m* and *s* may be determined by Linear Algebra techniques from the study of the equilibrium matrix or the kinematic matrix set for the initial configuration, and in particular starting from the determination of the rank *r* of such matrices (Pellegrino and Calladine 1986).

Compared with the large number of studies about pin-joined assemblies, so far almost no research has been undertaken regarding reciprocal assemblies, in particular plane reciprocal assemblies. The specificity of being joined near the ends as well as at intermediate points makes these structures behave in a different way from pin-joined assemblies. Furthermore, their plane geometrical configurations must satisfy topological rules that provide these structures with a recognizable rotational symmetry. In the next part of the paper the problem of static and kinematic determinacy of reciprocal plane unbounded grids is introduced, and some considerations made about the role played by internal constraints.

In order to extend the theory of pin-joint assemblies to planar reciprocal frames, it is necessary to remove two assumptions that are at the basis of such a theory: the use of pins as the unique kind of constraints, and the position of joints only at the ends of the bars. In fact planar reciprocal frames can be described exactly in such a way. The general concepts of the theory, however, can be maintained, i.e., the static determination as the possibility to find the internal state of stress only with equilibrium equations, the kinematic determinacy as the possibility to define the shape of the structure from the length (and other geometrical properties) of bars, the self-stress states as self-equilibrated stress states that can be present in the structure and the inextensional mechanism as the free movement that can occur to the structure. According to the formulation proposed by (Pellegrino and Calladine 1986), it is possible to distinguish between rigid-body motions and internal mechanisms, and between finite and infinitesimal mechanisms.

### Static and Kinematic Determinacy of Reciprocal Assemblies

When dealing with an assembly of bars, the concept of kinematic equations refers to the description of the mutual constraints between the assembly elements. On the other hand the static equations express the fact that forces present inside the assembly elements must be in equilibrium. These two sets of equations involve four groups of variables, according to (Pellegrino and Calladine 1986).

When setting up the problem for pin-jointed assemblies, the kinematic variables are the displacements of nodes and the elongations of bars. If bars are rigid, elongations are null, and so bars can be regarded as ‘constraints’ linking the nodes.

The presence of an infinitesimal mechanism corresponds to the presence of non-trivial solutions of the homogeneous kinematic system, and the presence of a self-stress state corresponds to a non-trivial solution of the homogeneous static system.

According to the theory of pin-jointed assemblies, the equilibrium can be assessed writing *3J* (*2J* for planar structures) equations, stating the equilibrium of each joint under the external load and the internal forces (axial) in the bars. In such equations, the *b* axial forces {*N*} and the *2J* external loads {*F*} are put in relation through a \( 2J \times b \) matrix of coefficients, called the ‘equilibrium matrix’. The properties of such a matrix will govern the static determinacy of the structure.

Conversely, the internal compatibility of the structure is assessed by writing *b* equations in which the vector of bar elongations {*e*} is put in relation with the vector of joints displacements {*d*}. The size of the coefficients matrix, in this case, is \( b \times 2J \), and it takes the name ‘kinematic matrix’. Due to the static-kinematic duality, the kinematic matrix is the transpose of the equilibrium matrix. The properties of this matrix rule the kinematic determinacy of the structure.

In order to write the equilibrium and compatibility equations for reciprocal assemblies that differ from pin-jointed assemblies in the kind and position of joints, an alternative formulation has been developed. The static and kinematic equations are written for the bars rather than for the joints. In order to clarify the approach, the two alternatives are shown here in the case of a pin-jointed structure.

*a*(linking bars

*i, j, k*) and one bar

*j*(linking points

*a, b*).

The equilibrium equations can be written in two different ways: considering all the forces acting on the joint or, alternatively, on the bar.

*F*

_{ xa }and

*F*

_{ ya }, and the axial forces inside the bars,

*N*

_{ i },

*N*

_{ j },

*N*

_{ k }. The equilibrium equation is then, in vector form:

*H*

_{ ia },

*V*

_{ ia },

*H*

_{ ib },

*V*

_{ ib }, exchanged by the bars through the joints, and the external forces

*F*

_{ xi },

*F*

_{ yi },

*C*

_{ i }, considered to be acting on the bar itself. The internal reactions of joints then play the same role as the axial forces in the elements, and they represent the unknown in the equations. For the sake of simplicity, the external forces are considered to be applied on the first end of the bar. The equations are then the following:

*e*

_{ i }of the bars and the components

*u*

_{ ia }, via,

*u*

_{ ib },

*v*

_{ ib }of the displacements of two bar ends are taken into account.

*e*

_{ i }, and nodal displacements

*u*

_{ ia },

*v*

_{ ia },

*u*

_{ ib },

*v*

_{ ib }.

In case of perfectly rigid bars, elongations are null.

*g*

_{ a′ },

*g*

_{ a″ }of the joint are taken into account, i.e., the separation of the points where bar ends are joined, and the equations assess the dependency of dislocations on nodal displacements {

*u*}

_{ ai }, {

*u*}

_{ aj }, {

*u*}

_{ ak }.

In case of perfectly rigid joints, dislocations are null.

When setting up the kinematic problem, it is convenient to express the nodal displacements in terms of the three generalized coordinates. In the plane, the DOF of a rigid body are three, and its position can be determined with the values of the three generalized coordinates; two of translation and one of rotation, each one corresponding to one DOF with respect to a reference point on the rigid body itself. Any motion of the rigid body can therefore be interpreted as a combination of two translations u and v, and one rotation φ.

The kinematic problem can be expressed with a system of *c* equations of the constrained conditions in *3b* generalized coordinates, i.e., the three generalized coordinates *u, v, φ* for each bar.

*R*is therefore given by the following relations:

*B*is the Kinematic matrix of size [

*J*,

*3b*],

*d*is the vector of bar displacements (size

*3b*), and

*e*is the vector of joint dislocation (size

*J*).

*A*is the Equilibrium matrix of size [

*3b,J*],

*d*is the vector of the forces carried in the joints (size

*J*), and

*e*is the vector of the external forces applied to the bars (size

*3b*).

### Static and Kinematic Determinacy of Pin-Jointed Assemblies

The kinematic matrix, in the context of infinitely rigid bars and small deflection theory, is the transpose of the equilibrium matrix. This may be deduced by the application of the principle of virtual work.

### Static and Kinematic Determinacy of a Three-Element Reciprocal Fan

### Discussion on the Rank of the Kinematic Matrix B

*c*is the number of constrained conditions,

*r*is the rank of the kinematic matrix,

*s*is the number of states of self-stress and

*m*is the number of inextensional mechanisms.

*c*= 3

*b*and rank is maximum

*r*=

*c*=3

*b*, all constraints conditions are effective, and therefore:

*c*> 3

*b*and rank is maximum

*r*=

*c*= 3

*b*, the structure is statically indeterminate:

*< 3*

**c***and rank is maximum*

**b***=*

**r***<3*

**c***, the structure is kinematically indeterminate with:*

**b**The discussion of the rank allows detection of whether or not there are in extensional mechanisms. However, it does not detect whether they are of first, second or higher order.

## Two Equivalent Kinematic Matrices for a Four-Element Reciprocal Fan

In the following example the two equivalent systems are explicitly written for a four-element fan where elements are connected through hinges.

*b*with 3

*b*kinematic unknowns. The constrained conditions are described by eight equations that describe the compatibility of displacements at the hinges in twelve kinematic unknowns (three for each bar):

The four inextensional mechanisms include the three rigid body motions given by the absence of external constraints, and one additional inextensional mechanism.

*e*with 3

*e*kinematic unknowns, where each bar

*b*is composed with two elements

*e*. In this system, in addition to the eight equations that describe the compatibility of displacements at the hinges, it is necessary to add three equations that describe the compatibility of

*u*,

*v*, and each couple of elements that compose the bars. The kinematic problem is therefore described by twenty equations (eight equations for the displacements at the hinges and twelve equations for the fixed joint in the bars) in twenty-four kinematic unknowns (three for each element):

The four inextensional mechanisms comprise three rigid body motions given by the absence of external constraints, and one additional inextensional mechanism. From the study of the kinematic matrix, it is possible to confirm that the two systems are equivalent. However, the first system might be preferable due to the smaller number of unknowns.

## Numerical Results and Discussion

From the study of the rank *r* of the kinematic matrix the kinematic behaviour of single fans and different reciprocal networks belonging to the families listed in Sect. 2 can be assessed.

*m*always comprises the three rigid body motions, so there is an additional inextensional mechanism for all the configurations shown except the three-elements fan (Fig. 12a), and the value of

*s*is always 0.

*m*(three rigid body motion + one inextensional mechanism). This behaviour is explained by the simultaneous presence of sixteen states of self-stress

*s*which correspond to an equal number of ill-disposed constraints.

In the second case (Fig. 13b), the regular topology and irregular joint position networks although the number of constrained condition and the topology of the assembly is the same the value of *m* is 3, and the value of *s* decreases to 15. This indicates that the irregular network loses one inextensional mechanism, because one conditions of ill-disposition is lost, and therefore it only possesses the three rigid-body motions. From the analysis of the third case (Fig. 13c), it is possible to deduce that the network loses the inextensional mechanism, even if only a single fan has a non-regular joint disposition.

The non-regular topology and joint position networks (Fig. 13d), based on non-repeating topological scheme, irregular bar length and joint positioning, also possess only the rigid body motions. In this case the kinematic behaviour highly depends on the topology of the assembly, and no considerations of a general character can be made.

## Further Work

The paper presents a formulation in the context of small-deflection theory of the static and kinematic problem of reciprocal assemblies. The aim of this work is to propose a formulation that extends the studies of Pellegrino and Calladine on pin-joint assemblies to reciprocal assemblies, for assessing the static and kinematic behaviour of reciprocal structures through the study of the associated equilibrium and kinematic matrix.

More generally, the present formulation can be adopted for any kind of geometric assembly, and it can be interpreted as the starting point for a complete extension of matrix analysis to assemblies that do not fit within the conditions imposed by pin-jointed assemblies. A rigorous formulation of the static and kinematic problem in plane reciprocal grids is proposed. The influence that the number of internal and external hinges and the arrangement of the elements have on the static and kinematic determinacy of planar assemblies has been assessed.

Further work is required for the complete specifications of all the possible constrained conditions, i.e., with the inclusion of configurations with slider and sliding hinges. Additionally, the analysis of the four fundamental vector subspaces associated with the kinematic matrix or the equilibrium matrix should be carried out in order to distinguish between first-order infinitesimal mechanisms associated with second-order changes of bar lengths from higher-order infinitesimal or finite mechanisms associated with no change in the length of bars. In order to understand the effect of more complex geometries and topologies, a necessary step is the extension of the formulation to three-dimensional configurations.

## Footnotes

- 1.
For a complete list of references see [Pugnale and Sassone 2014].

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