Advances in Numerical Analysis

Volume 2016 (2016), Article ID 4815289, 12 pages

http://dx.doi.org/10.1155/2016/4815289

## Analysis of Tensegrity Structures with Redundancies, by Implementing a Comprehensive Equilibrium Equations Method with Force Densities

^{1}Department of Mathematics and Statistics, University of Cyprus, Nicosia, Cyprus^{2}Department of Civil Engineering, Frederick University, Nicosia, Cyprus

Received 26 April 2016; Revised 26 July 2016; Accepted 19 October 2016

Academic Editor: Hassan Safouhi

Copyright © 2016 Miltiades Elliotis et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A general approach is presented to analyze tensegrity structures by examining their equilibrium. It belongs to the class of equilibrium equations methods with force densities. The redundancies are treated by employing Castigliano’s second theorem, which gives the additional required equations. The partial derivatives, which appear in the additional equations, are numerically replaced by statically acceptable internal forces which are applied on the structure. For both statically determinate and indeterminate tensegrity structures, the properties of the resulting linear system of equations give an indication about structural stability. This method requires a relatively small number of computations, it is direct (there is no iteration procedure and calculation of auxiliary parameters) and is characterized by its simplicity. It is tested on both 2D and 3D tensegrity structures. Results obtained with the method compare favorably with those obtained by the Dynamic Relaxation Method or the Adaptive Force Density Method.

#### 1. Introduction

Cable networks [1–3] and tensegrity structures [4] are different from conventional structures, such as spatial steel frames or space steel trusses, in that they are lightweight structures with members which transmit only tension (cables and strings) or elements which transmit compression (bars before buckling). In this article, we are studying only the behavior of tensegrity structures. These structures are usually defined as planar or spatial trusses with a discontinuous set of members under compression, inside a continuous network of members under tension. The word tensegrity is an artificial word and it combines the words “tension” and “integrity.” This word was coined several decades ago. Professor Fuller, in the United States, was essentially involved in the invention of this technical word. In one of his last books, Fuller described the compression members as “islands of compression in a sea of tension” [4]. Using the same concept, Emmerich [5] presented, in France in 1963, his own tensegrity patent. Snelson, one of Fuller’s students, describes this type of structures as “continuous tension and discontinuous compression structures” [6].

Pretension, applied by means of tension members, plays an essential role in the structural behavior of the tensegrities. For the design of such structures, their stability is investigated under both static and dynamic loads. During the last decades, many methods have been proposed for the analysis of the tensegrities. One of the most important methods is the Dynamic Relaxation Method (DRM) which was used in this research for checking the numerical results obtained with the numerical scheme proposed in the present work. DRM is one of the classical techniques. It belongs to the family of methods under the title of three-term recursive formulae. It is an iterative procedure which is based on the fact that a system undergoing damped vibration, excited by a constant force, ultimately comes to rest in the displaced position of static equilibrium, obtained under the action of the constant force. One of the numerous first papers, written by pioneers of this method, is that of Papadrakakis [7] which proposes an automatic procedure for the evaluation of the iteration parameters and it is mentioned here (without underestimating the importance of other works in this domain) as an example of an article which presents in a strict, clear, and academic manner the way of implementing this numerical procedure. Recent research, performed in the domain of tensegrities, has given some new important techniques. However, a general review of the older and of the recent numerical schemes, developed in this area, is out of the scope of the present paper. Juan and Mirats Tur present an excellent review in their work [8] of the basic issues about the statics of tensegrity structures. Among the new methods, presented in literature during the last few years, is that of Zhang and Ohsaki [9] which is also considered in the present research for comparison purposes. Their method is an Adaptive Force Density Method (AFDM). It first finds a set of axial forces compatible with a given structure and then estimates the corresponding nodal coordinates under equilibrium conditions and constraints.

The technique implemented in the present work is a force density method which belongs to the class of the Equilibrium Equations Methods with Force Densities (EEMFD). With this method, the system of equilibrium equations is created by considering the equilibrium of forces in all the joints. The redundancies are treated by employing Castigliano’s second theorem which gives the additional equations required to have a solution [10]. The partial derivatives, which appear in the additional equations, are numerically replaced by statically acceptable internal forces acting along the members. Also, the properties of the matrix of the system of linear equations are exploited because they give a strong indication of the stability of the structure.

The assumptions adopted in the present research are the following:(1)Joints are frictionless but their mass is considered in calculations unless otherwise specified.(2)The self-weight of a member (for structures within earth’s gravity field) is not neglected unless otherwise specified. It is equally distributed at its ends.(3)Live loads and pretensions on a member are transferred to the joints. They are equally distributed at the ends of the member.(4)Displacements on the joints and deformations of the members of the structure are relatively small compared with the dimensions of the structure.(5)The axial force carried by a member is constant along its length.(6)The materials used, for all members of the structure, obey Hooke’s Law for loadings below the yield stress.The efficiency of this technique is based on its simplicity and the small number of calculations required. Thus, the objective is (i) to present the general idea and formulation and (ii) to test the method in planar and spatial tensegrity structures of any type of complexity and to compare it with the DRM or the AFDM.

The outline of the rest of this paper is as follows: in Section 2 we develop the formulation of a Comprehensive Equilibrium Equations Method with Force Densities (CEEMFD) and in Section 3 we discuss the treatment of redundancies and we investigate the stability in this type of structures. In Section 4, some applications of the method are presented on planar and on spatial tensegrity structures. The article ends with Section 5 in which conclusions and proposals for future work are given.

#### 2. A Comprehensive Equilibrium Equations Method with Force Densities

An important concept related with tensegrity structures is the “force density” [1–3]. For a member made of a material obeying Hooke’s Law, with ends at and and with a length and a longitudinal force , the force density is defined as follows:where is the strain, is the Young modulus [10], and is the effective cross-sectional area of the member. This definition will be useful in the analysis which is presented herewith. Force density has a negative sign in compression and a positive sign in tension.

Figure 1 is used to exemplify the equilibrium of a typical unconstrained node , on the* Oxy* plane, which is connected to joints and , through members which have lengths and . For the three-dimensional orthogonal Cartesian coordinate system , where joints are connected with joint , through members, the equilibrium equations, in the direction of the axes , , and are given bywhere and is the total number of nodes on the structure. In equations (2), (3), and (4) forces , , and are the external loads which include gravity loads, live loads, pretension, and even reactions in the case where the node is a support. Gravity loads are assumed to act along the negative -axis, for 2D structures or along the negative -axis for 3D structures. External loads, which appear on the right hand side of (2), (3), and (4) have the following general expressions:where is the self-weight of joint and is the weight per unit length of a member with nodes at and . Also, is the pretension per unit length of a member joining nodes and . Forces , , and are the concentrated live loads and , , and are the reaction forces (if they exist) on joint . Both live loads and reaction forces are assumed to be acting along the positive directions of axes , , and .