numerical and physical modeling of tensegrity structures

NUMERICAL AND PHYSICAL MODELING OF TENSEGRITY STRUCTURES T.E.C. Deifeld and R.M.O. Pauletti Escola Politécnica da Universidade de São Paulo, Av. Professor Almeida Prado – Travessa 2, no. 83 Cidade Universitária – CEP: 05508-900, São Paulo, Brasil. [email protected], [email protected]

Abstract This work outlines a research on tensegrity structures, currently under development at the Department of Structural Engineering of the Polytechnics School of the University of São Paulo. Initially, the paper briefly discuss some proper definitions to tensegrity, since there still is some controversy about aspects as disconnection between compressed bars, kinematic deficiency, presence of infinitesimal mechanisms and of self-equilibrated internal stress fields, adding geometrical stiffness to the structure. Nevertheless, tensegrity structures fit well into the broader classification of tension structures, and also to them the design process generally includes a form finding process. This work presents alterative procedures to form finding, that were implemented in the PEFSYS finite element program [PIM98], including a variable-initial length cable element, capable of sustaining a constant prescribe force, previously described in [DEI02]. The program PEFSYS was then employed to find equilibrium shapes and to simulate the erection process of tensegrity models, and to develop a preliminary design of a large cable dome.

1. Introduction Richard Buckminster Füller defined ‘tensegrity’ as “a structural-relationship principle in which structural shape is guaranteed by the finitely closed, comprehensively continuous, tensional behaviors of the system and not by the discontinuous and exclusively local compressional member behaviors” [FUL75]. Structures which adhere to this principle are called tensegrity structures and belong to the larger group of tension structures, i.e., those structures that are capable of withstanding external loads only in presence of a self-equilibrated state of initial stresses. Controversy arise, both in academic and professional fields, when some structures are included in the definition of tensegrity structures. One point of disagreement relates to the way compressive loads are transmitted from one member to the others. Some researcher admit only structures where this is done exclusively by cables in traction, while others include structures where compression members connected to each other, at some points. Another point of divergence among some researcher relates to the necessity of a self-equilibrium state. In this sense, definitions found in literature can be grouped in two broad categories: those that adhere to a restrict definition, that contemplates only self-equilibrated cable networks, with a number of isolated struts working in compression, and a broad definition, that admits also those structures that transfer the prestressing loads to supports (not self-equilibrated, therefore). Fall in this definition one of the most successful subclass of tensegrity structures –the cable domes– that are usually anchored to external elements, normally a single steel or concrete ring. However, this discussion is relatively innocuous, since one could also propose a compression ring built up from the repetition of tensegrity modules [DEI04]. It is also known that tensegrity structures are statically and kinematically indeterminate. Therefore, their design requires consideration of the infinitesimal mechanisms associated to the system, and the corresponding states of self-stress capable to stabilize them [PEL86], [VAS00]. Also, like in many other types of tension structures, the initial equilibrium configuration of a tensegrity structure will probably be an unknown, rather than a given design data. In these cases, the process of finding an initial equilibrium form (closely related to the infinitesimal mechanisms intrinsic to the system) becomes an important part of the design methodology, and influences also the building processes, since adequate measures are required to impose the self-stress state associated with the desired form.

2. The form finding process A very important phase in the design of a tensegrity structure is the determination of its geometric configuration under self-stress, a process known as form finding. Pugh [PUG76] grouped the previous works of Füller, Snelson and Emmerich about the form of simplex tensegrity structures, based mainly in regular convex polyhedra, as a starting point to find feasible equilibrium configurations, and identified three patterns of form: diamond, circuit and zigzag. However, a simple verification of nodal equilibrium allows understanding that regular polyhedra are not equilibrium configurations, and thus adequate methods, whether physical or numerical, are generally required to find some feasible, equilibrated configuration, even in the case of simple modules. Several form finding methods have been studied by different authors. In 1994, Motro has classified these methods in geometric, analytical and numerical [MOT94]. More recently, Tibert & Pellegrino [TIB01], in a state-of-the-art review, have proposed an alternative classification, among kinematic and static methods, identifying advantages and limitations of each class, and establishing the essential relationships between them. Broadly speaking, kinematic methods are characterized by determination of the tensegrity geometry by means of the maximization (or minimization) of the length of some members, while keeping some other members with constant length. On the other hand, static methods search the geometric variations of a given topology, which are able to equilibrate imposed forces in some of its members. The present work proposes a methodology to form finding based in the finite element method, in particular the use a special truss element capable of sustaining a constant normal load (in tension or compression). A similar finite element was previously proposed by Meek [MEE71], and was used by the author to adjust geometry and loads of a classical example of cable-truss (the Poskit truss). The element was called ‘variable initial length element’ (VIL), nomenclature that is kept in the present work.

2.1. Formulation of the geometrically exact truss element The tangent stiffness matrix of a truss (or cable) element is defined as

kt =

∂p , ∂u

(1)

where p is the vector of internal loads and u is the vector of nodal displacement for the element. Nodes i and j in the global structural system corresponds to nodes 1 and 2, respectively, in the element system (Figure 1). Consider the derivation of the tangent stiffness matrix

k t of this element.

Figure 1. A truss finite element, with local and global identification for the nodes.

The vector of internal loads is given by

⎡ − vN ⎤ p=⎢ (2) ⎥ = CN , ⎣ vN ⎦ Where the scalar N is the internal normal load and C is a geometric vector operator, given by ⎡− v ⎤ (3) C = ⎢ ⎥. ⎣ v⎦ and v is a unit vector directed from node 1 to node 2.

Substituting (2) in (1), the elastic (disregarding material nonlinearities) and geometric parts of the tangent stiffness matrix are obtained:

∂C ⎛ ∂N ⎞ kt = ke + k g = C⎜ . ⎟ +N ∂u ⎝ ∂u ⎠ T

(4)

Consider initially the derivation of the elastic part of (4):

N=

EA (A − Ar ) . Ar

(5)

Where A is the undeformed length (or reference length) of the element, that in an initial configuration, 0 where the element is already under a normal force N , is given by r

Ar =

EA A0 . 0 EA + N

(6)

After some algebraic operation, it results that

∂N EA = r C. A ∂u

( 7)

The elastic part of the tangent stiffness is, therefore,

EA T ⎛ ∂N ⎞ ke = C⎜ ⎟ =C r C A ⎝ ∂u ⎠ T

(8)

where the symmetry of k e becomes obvious. Taking into account the definition of the geometric operator C , it results that

ke =

EA ⎡ vvT ⎢ A r ⎣ − vvT

− vvT ⎤ ⎥ vvT ⎦

(9)

Consider now the derivation of the geometric part of k t :

⎡ ∂v ⎤ ⎢ − ∂u ⎥ ∂C =N⎢ kg = N ⎥. ∂u ⎢ ∂v ⎥ ⎣⎢ ∂u ⎦⎥

( 10)

After some algebraic operations, it results that

T T ∂C N ⎡⎢ ( I 3 − vv ) − ( I 3 − vv ) ⎤⎥ = kg = N ∂u A ⎢ − ( I 3 − vvT ) ( I3 − vvT ) ⎥⎦ ⎣

(11)

Where I 3 is the unit matrix of order 3. Finally, combining (9) and (11) in (4), the tangent stiffness matrix for the geometrically exact truss element is written:

EA ⎡ vvT kt = ke + kg = r ⎢ A ⎣ − vvT

T T − vvT ⎤ N ⎡ ( I 3 − vv ) − ( I3 − vv ) ⎤ ⎥ ⎥+ ⎢ T ⎥ vvT ⎦ A ⎢ − ( I3 − vvT ) − I vv ( ) 3 ⎣ ⎦

(12)

In the above equation A is the length of the element in the current configuration, A is the length in é the reference configuration, N is the normal force acting on the element, E is the elastic modulus, A is the cross section and v is the unit vector direct along the element axis. r

Now, the VIL element is a specialization of the truss element defined above, characterized by a constant normal load, physically correspondent, for instance, to the action on an ideal hydraulic actuator. In such a case, since N does not vary, one gets k e = 0 , and the tangent matrix is given solely by k g . The initial length of the VIL element is calculated, for every equilibrium configuration, recasting equation (5), so that

Ar =

E A. E + N0

( 13)

2.2. Application of VIL elements to the form finding of tensegrity structures In order to find equilibrium configuration of tensegrity modulus, the above described truss and VIL elements were uses to model cables and struts in tensegrity topologies. Numerical computations were undertaken with the aid of the PEFSYS code [PIM98] –a finite element program for non linear, static and dynamical analyses of structures. The program uses exact Newton’s method to solve the equilibrium equations, thus requiring, for every iteration, the assembling of the elements’ load vector, equation (2), and stiffness matrices, equation (12), into the global load vector and tangent stiffness matrix of the structure. The procedure devised to form finding consists now in the specification of normal loads (tension or compression) in some of the members of the structure, in an arbitrary initial geometry, but fixed topology. In this paper, only polyhedral initial configurations were considered, but the method has also been successfully applied to other kinds of tensegrity structures [DEI04]. However, for more complex geometries, it may be not easy to find an adequate set of members to define as VIL elements, nor the ration between the normal loads acting on them [DEI02]. By the classification of form finding methods proposed by Tibert & Pellegrino [TIB01], the present methodology could be classified either as kinematic, since the length of some elements are kept practically constant, while the VIL elements vary in length, or as static, since the normal force on the VIL elements are imposed, and an equilibrium configuration is sought, resulting in the internal loads for every element, capable of sustaining equilibrium at every node of the structure.

(a)

(b)

(c)

Figure 2. (a) Initial, non-equilibrated configuration of a three-bar simplex tensegrity model, (b) equilibrated configuration and (c) ‘Monumento à Forma Fútil II’ (sculpture by Pauletti, Titotto and Deifeld [TIT04]).

(a)

(b)

Figure 3. (a) ‘Monument à la Forme Futile’ [EMM66], a six-bar simplex tensegrity model and (b) the same structure, modeled with PEFSYS.

Some examples of the form finding for some simplex tensegrity modules are presented in Figures 1 and 2. Figure 1 shows, in (a), a regular prism with triangular base, in (b) the equilibrium configuration found by PEFSYS and in (c) a sculpture (‘Monumento à forma Fútil II’) composed of a lycra membrane stretched in-between the PVC bars of the tensegrity module, without touching them. Figure 2, by its turn, depicts in (a) the ‘Monument à la Forme Futile’ and in (b) the corresponding structure, whose geometry was again determinate by PEFSYS.

3. Assembling of tensegrity structures One of the key phases in the project of a tensegrity structure is the definition of its assembling process. Each step in this process has to be rigorously planned, in order to achieve the desired form. One way to determine the step sequence is to start with the structure fully assembled and imagine its disassembling, inverting the process during construction. This method was suggested by David Geiger to determine the assembling sequence of his first cable domes [TUC86].

3.1. Use of VIL elements to simulate the assembling of tensegrity structures The VIL element is found to be efficient also in the design of assembling procedures. Starting from and equilibrium configuration, a gradual reduction of the normal forces on a set of VIL elements is imposed, and the new equilibrium configurations are sought. After determination of the non-stressed configuration, its assembling process can be determined, inverting the sequence. As an example, Figure 4 shows the assembling sequence for the ‘Monument à la Forme Futile’. The cable used to stress the module is depicted in read. Starting from the unstressed configuration, the whole structure is fitted in place, simply shortening the length of this cable.

Figure 4. Assembling process for the ‘Monument to the Futile Form’, simulated with PEFSYS. The red cable is used to stretch the rest of the structure to the final form.

Figure 5. Assembling process for a truncated tetrahedral tensegrity module, fully assembled in a horizontal plane, according to the geometry determined by PEFSYS, and stretched by a single cable.

4. Numerical simulation of a Tensegrity dome PEFSYS allows analysis of tensegrity structures under a variety of loads. During the prestressing phase, VIL elements are employed to stretch the cables. Then, in the other phases of the analysis, the VIL elements are automatically replaced by normal truss elements. As an example, the analysis of a 100m radius, circular tensegrity dome under static wind loads is presented. The dome, anchored to a trussed steel compression ring, is composed by a central mast and two tension rings, each one with 24 flying posts and a triangulated cable net, as shown in Figure 5. The height of the cable dome, measured from the center of the compression ring is 22.5m (or 28.5m, measured from the bottom of compression ring).

Z Y X

Z Y X

(b) Z Y

X

(a) (c) Figure 5. Tensegrity cable dome modeled with PEFSYS: (a) elements (thicknesses increased 6 times); (b) inclined and (c) lateral views of the model. Cables connecting the central post to the first intermediate flying posts, and these to the second ones, have a cross-section area of 50.10-4m2; cables connecting the second intermediate flying posts to the external compression ring have a cross-section area of 113.10-4m2. All cables have an elastic modulus of 195GPa and an ultimate stress of 1900MPa. Determination of cross sections considered a minimum safety factor of 1.15 for all elements, taking into account loads produced for several load cases, as discussed ahead. Table 1 shows geometric properties of the flying posts. Elastic modulus and ultimate stress for the assumed steel are, respectively, 210GPa and 500MPa. Flying Post

External Thickness Length diameter (mm) (mm) (m) Central 610 40 15 first intermediate posts 305 13 15 Second intermediate posts 394 13 15 Table 1. Geometric properties of the flying posts

Mass (kg) 4271 705 945

The compression ring with a cross section of 12m both in width and height, is composed by 96 threedimensional modules composed by steel pipes, with 0.61m of diameter and 60mm of thickness. Table 2 shows the total mass of the different set of dome components.

Mass (tons) Compression ring 2942 Flying posts 44 Cables 662 Total 3648 Table 2. Total mass of dome components Component

Four loading cases have been considered. The first case considers the prestressing loads action on the dome, whist the second load case corresponds to a uniform 1kN/m2 uplift wind pressure, acting on the

whole surface of the dome. The third load case considers the same uplift pressure acting on half the dome surface, and the fourth case combines the uplift of half the dome with a downward pressure with the same magnitude acting on the other half. Self-weight is added to all four load cases. Figure 6 shows resulting displacements, largely amplified. Maximum displacements amounts to 1.20m downward, due to the first load case; 0.45m upward, due to the second load case; 1.01m upward, due to the third load case; and 1.62m upward, due to the fourth one.

Z Y

Z Y

X

X

(c) (a) Y

Z X

Z Y

X

(d)

(b)

Figure 6. Cable dome displacements due to (a) first, (b) second, (c) third and (d) fourth load caser.

Figure 7 shows the normal loads acting on the different elements, for each load case. A symmetric distribution of normal loads is observed in the two first load cases, with larger loads in the second case. In the third case, a release in the normal loads is observed in the loaded half of the dome, and a yet larger release is obtained in the fourth case. -.156E+08 -.116E+08 -.750E+07 -.344E+07 Y Z

623802 X .469E+07

-.175E+08

-.117E+08

-.126E+08

-.847E+07

-.792E+07

-.553E+07

-.409E+07

Z

X

-.779E+07 -.295E+07 Y

-271056

-.258E+07

Z

X

.189E+07 X

360886

.355E+07

.673E+07

.331E+07

.738E+07

.116E+08

.128E+08

.625E+07

.112E+08

.164E+08

.169E+08

.920E+07

.150E+08

.212E+08

.121E+08

.188E+08

.261E+08

.875E+07

.209E+08

1st load case

-.156E+08

-.114E+08

Y

Y Z

-.144E+08

2nd load case

3rd load case

4th load case

Figure 7. Distribution of normal loads along the tensegrity dome and the compression ring, for the considered load cases.

Also stability of each linear structural element was considered for all the four load cases. The minimum multiplier to local bucking was 6.4. Global stability of the compression ring was determined with the aid of the Ansys program. The four lower buckling modes are depicted Figure 8. The reference load was given by the loads transferred from the dome to the ring, taking an average uniform load for each load case. Lower bucking multiplier (λ) obtained was equal to 2.2, for the first buckling mode.

Y

Y

Y

Z

X

1st mode λ=2.22

Z

X

2nd / 3rd mode λ=4.10

Z

Y X

4th mode λ=5.90

Figure 8. First buckling modes of the compression ring

Z

X

5th mode λ=6.90

6. References [FUL75] R.B. Füller (1975), “Synergetics, Explorations in the Geometry of Thinking”. Macmillan Publishing Co. Inc., V1, 372. [PEL86] S. Pellegrino (1986), “Analysis of prestressed mechanisms”. International Journal of Solids and Structures, V. 22, 409-428. [VAS00] N. Vassart and R. Motro (2000), “Determination of mechanism’s order for kinematically and statically indetermined systems”. International Journal of Space Structures, V. 37, 3807-3839. [DEI04] T.E.C. Deifeld and R.M.O. Pauletti (2004), “Sobre o projeto e a construção de estruturas tensegrity”. XXXI Jornadas Sud-Americanas de Ingeniería Estructural, Mendoza, Argentina. In CDROM. [PUG76] A. Pugh, "An Introduction To Tensegrity", ed University of California Press Berkeley.U.S.A. [MOT94] R. Motro, S. Belkacem and N. Vassart (1994), “Form finding numerical methods for tensegrity systems”. Spatial, Lattice and tension structures – proceeding of the IASS-ASCE International Symposium 1994, New York, USA, 704-713. [DEI02] T.E.C. Deifeld and R.M.O. Pauletti (2002), “Um breve estudo sobre as estruturas tensegrity”. I Simpósio Nacional sobre Tensoestruturas, São Paulo, Brasil, May, 5-6, Proceedings of the Symposium, In CD-ROM. [PAU03] R.M.O. Pauletti, (2003) “História, Análise e Projeto de Estruturas Retesadas”. Tese de LivreDocência. Escola Politécnica da Universidade de São Paulo, São Paulo, Brasil. [MEE71] J.L. Meek (1971), “Matrix Structural Analysis”. McGraw-Hill Kogakusha, LTD. Tokyo, Japan. [PIM98] P.M. Pimenta, C.E.M. Maffei, H.H.S. Gonçalves and R.M.O. Pauletti (1998), “A programming system for nonlinear dynamic and static analysis of tall buildings” Computational Mechanics New Trends and Applications. Barcelona, Spain, 1-17. [TIT04] S.L.M.C. Titotto and R.M.O. Pauletti (2004). “Spaciality in Membrane Structures”. To be presented at this Conference. [EMM66] D.G. Emmerich (1966). “Réseaux”. International Conference on Space Structures. University of Surrey, 1059-1072. [TUC86] J. Tuchman and Shin Ho-Chul, "Olympic Domes First of Their Kind", Engineering News Record, March 6, 1986, pp. 24-27. Disponível em: . Acesso em 15.11.2003.