Constructing large-scale tensegrity structures with bar–bar connection ...

Arch Appl Mech (2015) 85:383–394 DOI 10.1007/s00419-014-0958-3

SPECIAL

Li-Yuan Zhang · Hong-Ping Zhao · Xi-Qiao Feng

Constructing large-scale tensegrity structures with bar–bar connection using prismatic elementary cells

Received: 3 January 2014 / Accepted: 26 June 2014 / Published online: 25 November 2014 © Springer-Verlag Berlin Heidelberg 2014

Abstract As a novel type of light-weight and reticulated structures, tensegrities have found many technologically important applications. In this paper, a facile method is developed to construct a class of large-scale tensegrities consisting of bar–bar connection using prismatic elementary cells. We tune the orientation of the structural axis of each cell by the affine transformation technique. Then, the cells can be assembled easily in any directions required by the structural design. The method proposed here allows us to construct various types of large-scale tensegrity structures satisfying the demands of sizes and topology. A number of representative examples are provided, including straight and curved beams, plates, shells, and three-dimensional large-scale tensegrities. Keywords Tensegrity · Structural design · Assembly method · Prismatic cell · Affine transformation 1 Introduction Tensegrity structures are composed of axial stressed elements (strings in tension and bars in compression/ tension) and appropriate to be prototypes for constructing light-weight and long-span structures with optimal material utilization [1,2]. With the increase in the scale and complexity of structures involved in practical applications, the design of large-scale tensegrities with specific shapes and functions becomes a major concern [3]. In the past decade, the design of tensegrity structures has attracted considerable attention from many researchers. Most tensegrities reported in the literature are inspired from some familiar regular geometries. Two early examples are X-shaped planar tensegrity [4] and triplex prismatic (or 3-prismatic) tensegrity [5,6]. Pugh [7] presented a number of tensegrities on the basis of truncated polyhedra in zigzag (or Z-based), diamond (or rhombic), and circuit patterns. To build more complex structures, some assembly methods using simple tensegrities as unit cells have been developed, which can be classified into three classes, namely node-onnode, node-on-cable, and cable-on-cable assembly methods [1]. These methods have been utilized to construct a variety of structures, e.g., tensegrity towers or bridges with large slenderness ratio [8,9] and quasi-twodimensional double-layer domes or grids [10,11]. More examples can be found in Motro [1], Wang [2], and Skelton and de Oliveira [12]. Recently, Feng et al. [3] and Li et al. [13] proposed a versatile method to construct tensegrity structures from one-bar elementary cells. Their scheme can design not only most conventional types of tensegrities but also some novel types, e.g., concave heart-shaped structures and carbon nanotube-like structures [13]. L.-Y. Zhang · H.-P. Zhao · X.-Q. Feng (B) AML and CNMM, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China E-mail: [email protected] L.-Y. Zhang School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China

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Prismatic elementary cells can be used to assemble many tensegrity structures. In the existing methods, the cells have the symmetric axes perpendicular to the bottom and top surfaces. Thus, the corresponding assembly process must be in a fixed direction, either along or perpendicular to the symmetric axis. Due to this limitation, these cells can produce only some simple structures, e.g., straight towers or flat plates [8,11]. For the assembly method using one-bar elementary cells, the obtained tensegrities are initially non-equilibrated [13]. Therefore, a form-finding analysis (e.g., the methods in Refs. [14,15]) is further required to determine the equilibrated configuration, which may be greatly different from the needed size and shape. In this paper, an assembly method is developed to construct a class of large-scale tensegrities consisting of bar–bar connection using prismatic elementary cells. The affine transformation technique is introduced to tune the structural axis of the cells, making it possible to assemble them in any directions. This method allows us to create tensegrities of various types, e.g., helical beams, curved plates, and three-dimensional large-scale structures. The self-equilibrium and stability of the cells ensure that the produced configurations are self-equilibrated and stable at the initial state. The paper is organized as follows. Section 2 introduces the self-equilibrated configuration and affine transformation of prismatic tensegrity. Section 3 investigates the assembly method of prismatic elementary cells. Section 4 presents a number of tensegrities to illustrate the efficiency of this assembly method. 2 Affine transformation of prismatic tensegrity 2.1 Prismatic tensegrity Prismatic tensegrities are inspired from the geometry of regular prisms. For illustration, a triplex prism and the corresponding 3-prismatic tensegrity are shown in Fig. 1. A v-prismatic tensegrity has 2v nodes, 3v strings, and v bars. Its nodes located at the bottom and top surfaces of the prism are numbered from 1 to v and from v + 1 to 2v, respectively. The strings on the bottom and top surfaces are referred to as end strings, whereas those connecting the two horizontal surfaces are named as side strings. The integer a defines the connection relation of bars. Node i on the bottom end is linked by a bar with node j on the top end, where j = i + v + a if i + a ≤ v or j = i + a if i + a > v. Invoking the concept of force density (the ratio of internal force to current length) of an element, the self-equilibrium condition of a tensegrity structure reads  qi j (pi − p j ) = 0, (1) j

where i is an arbitrary node in the structure, qi j is the force density of the element connecting nodes i and j, and pi = [xi , yi , z i ]T is the coordinate vector of node i. Assume that all elements of the same type have identical properties (i.e., force densities and lengths). In this case, the self-equilibrium condition of a prismatic tensegrity can be solved from Eq. (1) as [16] sin πa qs1 qs2 v = , = −1, qs2 qb 1 − cos 2π v (a) 6

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where qs1 , qs2 , and qb are the force densities of end strings, side strings, and bars, respectively. From Eq. (1), the relative twisting angle between the top and bottom polygons is calculated as [16]   1 a α=π − . (3) 2 v The self-equilibrated configuration of a prismatic tensegrity is fully determined by Eqs. (2) and (3). The former gives the pre-stresses in all elements and the latter describes the structural geometry. 2.2 Affine transformation Affine transformation is a linear transformation preserving the collinearity and length ratio of lines. It can transform a self-equilibrated state of tensegrity into a new self-equilibrated state without changing the force density in any element [17,18]. Thus, all affine transformations of a tensegrity are in the null-space of its stress matrix. In terms of the stress matrix, Eq. (1) can be rewritten into a matrix form as [19] S · p = 0,

(4)

where p = [xT , yT , zT ]T ∈ R3n is the nodal coordinate vector, with n being the total number of nodes. x = [x1 , x2 , . . . , xn ]T , y = [y1 , y2 , . . . , yn ]T , and z = [z 1 , z 2 , . . . , z n ]T are the coordinate components of all nodes in the x, y, and z directions, respectively. Here and in the sequel, the superscript T indicates the transpose of a vector or a matrix. S = D ⊗ I ∈ R3n×3n denotes the stress matrix, where ⊗ denotes the Kronecker product symbol, I ∈ R3×3 the identity matrix, and D ∈ Rn×n the force density matrix expressed as ⎧ if i  = j with i and j connecting, ⎪ ⎨ −qi j 0 if i  = j with i and j not connecting, (5) Di j = ⎪ Dik if i = j. ⎩− k = j

Thus, Eq. (4) has an equivalent form as [20,21] D · x = D · y = D · z = 0.

(6)

The self-equilibrium of a tensegrity requires that the nullity of its stress matrix satisfies [22] null(S) ≥ d(d + 1),

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where d denotes the dimension of the structure. Considering the self-equilibrium condition in Eq. (2), it is known that the stress matrix of a prismatic tensegrity has null(Sprism ) = 12.

(8)

This indicates that there exist twelve independent modes of nodal displacements satisfying Sprism · a I = 0 (I = 1, 2, . . . , 12),

(9)

where a I (I = 1, 2, . . . , 12) denotes the affine transformation vectors of a prismatic tensegrity. Equation (9) shows that each self-equilibrated prismatic tensegrity has twelve independent affine transformations, including three shearing, three dilation, and six rigid-body motion transformations [17,18]. The rigid-body motions are trivial for the internal properties of structures and will be ruled out in the sequel. It is known from Eq. (6) that the shearing and dilation transformations of a tensegrity can be expressed in terms of the nodal coordinate vectors x, y, and z [18]: a1 = [β1 zT , 0, β2 xT ]T , a3 = [β5 yT , β6 xT , 0]T , a5 = [0, β8 yT , 0]T ,

a2 = [0, β3 zT , β4 yT ]T , a4 = [β7 xT , 0, 0]T , a6 = [0, 0, β9 zT ]T ,

(10)

which present six independent affine transformations, namely the xz-, yz-, and xy-shearing and the x-, y-, and z-dilation transformations, respectively. The coefficients β1 , β2 , . . ., β9 can be arbitrarily chosen according to the requirements of design.

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(a)

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We take a 3-prismatic tensegrity as an example to illustrate the shearing and dilation affine transformations. Refer to the Cartesian coordinate system (O–xyz) established at the original configuration, as shown in Fig. 1, where the xy-plane and z-axis are parallel to and perpendicular to the bottom surface, respectively. The structural shapes after the xz-, yz-, xy-shearing and x-, y-, z-dilation transformations are shown in Fig. 2, in which we specify the coefficients β1 = β3 = 0.6, β2 = β4 = β5 = 0, and β6 = β7 = β8 = β9 = 0.5. It is emphasized that because of the property of affine transformations, all these new configurations with the force densities of elements in Eq. (2) remain self-equilibrated. For conciseness, the shearing transformations in the bottom and top surfaces (e.g., xy-shearing in Fig. 2c and its linearly dependent transformations) will be referred to as in-plane shearing, and the other shearing transformations (e.g., xz-, yz-shearing in Fig. 2a, b and their linearly combined transformations) will be referred to as out-of-plane shearing.

2.3 Sheared prismatic tensegrity In this paper, the structural axis of a prismatic tensegrity refers to the line connecting the centroids of the top and bottom polygons. In a regular configuration (see, e.g., Fig. 1) and those after dilation and in-plane shearing transformations (Fig. 2c–f), the structural axis is always perpendicular to the bottom surface. Therefore, the assembly of these cells can only be made along a fixed direction, either along or perpendicular to the structural axis. In a configuration induced by out-of-plane shearing transformation, in contrary, the structural axis can be adjusted from its original vertical orientation to an arbitrary direction, as shown in Fig. 2a, b. The corresponding cells can be further assembled in the transformed directions, yielding new tensegrity structures. The prismatic tensegrities obtained by out-of-plane shearing transformations will be referred to as sheared prismatic tensegrities in the sequel. Four parameters are required to describe the sizes and shape of a sheared prismatic tensegrity. Here, they are chosen as the radius of the bottom circumcircle (r ), the distance between the bottom and top surfaces (h), the angle measured from the bottom surface to the structural axis (ϕ), and the angle made by the projection of structural axis on the bottom surface and the radius passing a specified node (θ ), as shown in Fig. 3a. Among them, r and h describe the structural sizes, and θ and ϕ define the orientation of the structural axis. We set 0 ≤ θ < 2π/v due to the v-fold symmetry of a v-prismatic tensegrity, and 0 < ϕ ≤ π/2 to preserve the relative rotation between the bottom and top surfaces. Refer to a local Cartesian coordinate system (O  − x  y  z  ), with the origin O  located at the bottom center,  x -axis passing through the specified node (say, node 1), and z  -axis perpendicular to the bottom surface. The coordinate vector of the top center is expressed as

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Fig. 3 a Definitions of the parameters r , h, θ and ϕ describing the sizes and structural axis orientation of a sheared prismatic tensegrity. b A 3-prismatic structure with r = 1.0, h = 3.0, θ = π/6, and ϕ = π/4

⎧  ⎫ ⎧ ⎫ x ⎪ ⎨ tc ⎪ ⎬ ⎨ h cot ϕ cos θ ⎬ = h cot ϕ sin θ . ptc = ytc ⎭ ⎪ ⎩  ⎪ ⎭ ⎩ h z tc

(11)

Then, the structural configuration can be described in terms of nodal coordinates. The nodes on the bottom and top surfaces have ⎧ ⎫ ⎧ ⎫ x ⎪ ⎬ ⎨ r cos 2π(i−1) ⎨ i⎪ ⎬ v pi = yi = r sin 2π(i−1) (i = 1, 2, . . . , v), (12) v ⎪ ⎭ ⎭ ⎩ ⎩ ⎪ 0 zi ⎫

 ⎧ ⎫ ⎧ 2π(i−1)  ⎪ xi ⎪ ⎪ r cos + α + x ⎪ ⎪ tc ⎪ ⎨ ⎬ ⎨ ⎬

v   2π(i−1)  (i = v + 1, v + 2, . . . , 2v), (13) pi = yi = r sin + α + y tc ⎪ ⎪ v ⎪ ⎩ ⎪ ⎪ ⎭ ⎪ ⎩ ⎭  zi z tc  ) are defined in Eq. (11). For illustration, respectively, where the value of α is given in Eq. (3) and (xtc , ytc , z tc Fig. 3b shows the shape of a sheared 3-prismatic tensegrity with r = 1.0, h = 3.0, θ = π/6, and ϕ = π/4, in which the nodal coordinates are determined according to Eqs. (11)–(13). Finally, we emphasize that by varying the values of r , h, θ , and ϕ, different sheared prismatic tensegrities can be obtained, which will be used as elementary cells in structural assembly.

3 Assembly of two sheared prismatic tensegrities In the node-on-node assembly method, elementary cells are connected by simply jointing at least three pairs of non-collinear nodes in two neighboring cells together [1]. The combined cells will have no possible relative rigid-body motion, and the resulted structure does not have any finite mechanism. In this paper, we mainly consider the assembly method in which sheared prismatic cells connect with each other via their end surfaces. The surface circumcircle radii of the cells to be jointed are identical. On the connecting surfaces, each pair of nodes will be joined together and the corresponding strings will overlap each other. To clearly reveal the above assembly method, we here discuss the assembly of two sheared prismatic cells in detail. For illustration, the assembly process of two sheared 3-prismatic tensegrities is shown in Fig. 4. According to the assembling direction, the assembly modes can be divided into two classes, parallel and series. In the former, the two cells will be on the same side of the connecting surface after assembly, as shown in Fig. 4a, while in the latter, they will be on the different sides, as shown in Fig. 4b. As mentioned in Sect. 2.3, varying the values of parameters r , h, θ , and ϕ will lead to different prismatic cells, which can be assembled using either the parallel or the series method, indicating a large number of possible combinations of cells. Even for two identical v-prismatic cells (having the same values of r , h, θ ,

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Combine paired nodes

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Fig. 4 Node-on-node assembly of two sheared 3-prismatic tensegrities: a the parallel scheme and b the series scheme

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Fig. 5 a Ten possible combinations of two identical 3-prismatic cells with r = 1.0, h = 3.0, θ = π/6, and ϕ = π/4. (1)–(4) are the four configurations of parallel assembly, and (5)–(10) are the six configurations of series assembly, where R and R denote two nodes specified to indicate the structural axis orientations of the two cells, respectively. b Top views of the structural axis orientations in the ten assembled tensegrities

and ϕ), there exist 4v-2 different connecting modes, including 2v-2 parallel modes and 2v series modes. For example, Fig. 5 shows ten configurations assembled from two 3-prismatic cells given in Fig. 3b. It is worth mentioning that in all structures constructed using the node-on-node assembly method, each originally self-equilibrated and stable elementary cell [23] maintains its structural integrity. This indicates some significant advantages in engineering applications. Firstly, these structures can keep their self-equilibrated and stable configurations without any external constraints, regardless of their pre-stress levels and constitutive relations of element materials. This benefits the modification of the structural initial stiffness. Secondly, both the geometric sizes and shape of these tensegrities can be easily modulated by changing the number of cells or replacing some elementary cells in structural engineering, without affecting their overall equilibrium. This advantage facilitates the design of tensegrities satisfying various functional demands. In addition, the assembled structures have multiple independent pre-stress states. When one or more cells have been damaged or broken, the structure can still be self-equilibrated, stable and bear external loads; in other words, it will not collapse

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Fig. 6 Straight tensegrity beams assembled from 3-prismatic cells using a the parallel scheme and b the series scheme

Fig. 7 Curved tensegrity beams obtained from the straight beam in Fig. 6a via affine transformation, in which the structural axes of cells are linked as a a curve in the xy plane, b a curve in the yz plane, and c a spring-like helical curve, respectively

provided that no possible finite mechanism has appeared due to local flaws or damage. The damage-tolerance property helps improve the safety of engineering structures such as domes, bridges, and antenna.

4 Examples In this section, a number of examples, including straight and helical beams, flat plates, curved shells, and threedimensional tensegrities of large scale, will be provided to illustrate the efficiency of the assembly method we

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Fig. 8 Curved tensegrity beams obtained from the straight beam in Fig. 6b via affine transformation: a a planar-curved beam and b a helical structure

Fig. 9 Square flat plates assembled from the straight beam in Fig. 6a using a the parallel scheme and b the series scheme

have described above. For illustration, only 3-prismatic cells will be utilized in the following examples. The same procedure can also be used to assemble tensegrities from other prismatic cells. 4.1 Tensegrity beams In this subsection, we assemble sheared prismatic cells into some tensegrity beams. Both straight and curved beams can be constructed using either the parallel or series assembly scheme. A few examples are given as follows. First, a straight beam is built from 3-prismatic cells using the parallel assembly scheme, as shown in Fig. 6a, where the top/bottom surface of a cell connects only the top/bottom surface of its neighboring cell. The length direction of the beam is parallel to the end surfaces of the cells, while their structural axes form a zigzag curve.

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Fig. 10 Square curved shells obtained from the flat plate in Fig. 9a via affine transformation, which are curved about a the x axis and b two axes, respectively

Fig. 11 Square curved shells obtained from the flat plate in Fig. 9b via affine transformation, which are curved about a the z axis, b the y axis, and c the two axes, respectively

Fig. 12 a A circular flat tensegrity plate and b a circular tensegrity shell

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Fig. 13 a A three-dimensional tensegrity assembled from sheared prismatic cells, b its top view, and c side view

Second, we use the series assembly scheme to build a straight beam from 3-prismatic cells, as shown in Fig. 6b, where the top surface of a cell is connected with the bottom surface of its neighboring cell and vice versa. All structural axes of the assembled cells are approximately along the length direction of the beam, which is perpendicular to the end surfaces of the cells. In addition, curved beams are created by assembling 3-prismatic elementary cells. The structural axis orientations of cells are tuned through affine transformation. Based on the straight beam in Fig. 6a, we construct three curved beams, as shown in Fig. 7a–c, where the structural axes of cells are linked as a curve in the xy plane, a curve in the yz plane, and a spring-like helical curve, respectively. Similarly, on the basis of the straight beam in Fig. 6b, a planar-curved beam and a helical beam are built (Fig. 8a, b).

4.2 Tensegrity plates Then, we address the construction of tensegrity plates or shells whose sizes in two directions are much larger than that in the third direction. Such tensegrity plates and shells can be used as load-bearing systems like platforms and walls. A few flat and curved plates assembled from sheared prismatic cells are as follows. By assembling some straight beams obtained in Fig. 6a, two flat square plates are designed, as shown in Fig. 9, in which the prismatic cells are linked along the width direction via the parallel assembly scheme and along the height direction via the series assembly scheme, respectively. To create curved shells, we adjust the structural axes of cells in the above flat plates via affine transformation. Transformed from the flat structure in Fig. 9a, a bridge-like tensegrity curved about the x axis and a dome-like tensegrity are derived, as shown in Fig. 10a, b, respectively. Similarly, based on the flat structure in Fig. 9b, three shells are given in Fig. 11, which are curved about the z axis, the y axis, and the two axes, respectively. Besides the above flat and curved structures of square shape, other quasi-two-dimensional plates or shells can also be constructed from sheared prismatic cells. For illustration, two flat and curved plates of circular shape are given in Fig. 12.

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4.3 Tensegrity bulks Finally, we construct three-dimensional large-scale tensegrities with sheared prismatic cells. The parallel scheme is first used to assemble the cells into a series of flat or even curved plates, and then, the series scheme is used to stack up the plates layer by layer along their thickness direction. A cubic tensegrity bulk consisting of six layers is taken as an example, as shown in Fig. 13a. Its top view in Fig. 13b is similar as the plate in Fig. 9a, and its multilayered structure can be seen in Fig. 13c. Such reticulated three-dimensional tensegrities hold promise for potential applications as ultra-light materials and structures. These structures have some unique mechanical properties, which will be further investigated in another paper.

5 Conclusions We have developed a facile and versatile assembly method to construct large-scale tensegrity structures with bar–bar connection using sheared prismatic elementary cells. The cells with well-designed affine transformations can be connected in any directions via the parallel and/or series assembly scheme. Our analysis shows that the sheared prismatic cells allow one to design various types of tensegrities that hold potential applications in a diversity of fields, e.g., civil engineering, smart structures and devices, and biomechanics. A number of beams, plates, shells, and three-dimensional tensegrities have been made to illustrate the efficiency of this method. Finally, it is mentioned that in the adopted assembly process, the node-on-node scheme is used to connect the adjacent cells, which renders bar–bar connection. This class of structures fulfill the generalized definition of tensegrity of Pugh [7] and Motro [1], but not the strict definition of Snelson [4] and Fuller [5]. The tensegrities with bar–bar connections have good performance to bear bending loads and are preferred in some engineering applications, e.g., beams, bridges, and platforms [11,12,24–26]. The basic idea of the presented assembly method can also be used to construct structures without bar–bar connection by invoking the node-on-string assembly scheme. Acknowledgments Supports from the National Natural Science Foundation of China (Grant Nos. 31270989 and 11372162), Tsinghua University (20121087991), and the 973 Program of MOST (2012CB934101) are acknowledged.

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