Engineering Structures 31 (2009) 2162–2170 Contents lists available at ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct An approach based on the catenary equation to deal with static analysis of three dimensional cable structures Miguel Such a,∗ , Jesus R. Jimenez-Octavio a , Alberto Carnicero b , Oscar Lopez-Garcia c a Analysis and Design Department, Institute for Research in Technology, Universidad Pontificia Comillas de Madrid, Alberto Aguilera, 23, 28015 Madrid, Spain b Escuela Superior de Ingeniería-ICAI, Universidad Pontificia Comillas de Madrid, Alberto Aguilera, 23, 28015 Madrid, Spain c Instituto Ignacio Da Riva, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros 3, 28040, Madrid, Spain article info Article history: Received 29 May 2008 Received in revised form 26 February 2009 Accepted 17 March 2009 Available online 9 April 2009 Keywords: Cable structures Catenary Non-linear analysis abstract In this paper a novel method to solve three dimensional cable structures based on the catenary equation is proposed. The method is a generalization of a previous engineering application to compute the initial equilibrium of railway overheads. The major contributions of this paper are: the extension of the previous engineering application to simulate arbitrary three dimensional cable structures; cable elasticity is incorporated into the formulation; and due to the fact that the method relies on the analytical catenary equations, high numerical efficiency is exhibited. In order to show the validity of the method, comparisons with several well reported cable structure problems are presented. The agreement between the proposed method and published results is excellent. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction Due to their efficiency and aesthetics, cable structures became quite popular from the 1950s onwards. From the mid 60s to the end of the 70s a significant number of articles dealing with cable structures were published, see for instance [1–7] among others. Nowadays, cable structures are widely used in many applications as, for example, power transmission lines, railway overheads, cable transportation systems, cable roof structures etc. Cable structures pose well known challenging problems, and the modelling of such structures has always been a subject of research and innovation. Cable members are light, very flexible and do not experience bending and compression stiffness. Therefore, cable structures exhibit a high non-linear behaviour. Another important problem of cable structures is the determination of the initial equilibrium configuration. That is, the computation of the stressed reference configuration which is an inverse structural problem. Reference [8], is one of the pioneering works dealing with the classification of the methods to solve initial equilibrium problems. Local behaviour of particular types of cables is another quite difficult problem that modelling of cable structures should ∗ Corresponding author. Tel.: +34 91 542 28 00; fax: +34 91 542 31 76. E-mail addresses: Miguel.Such@iit.upcomillas.es (M. Such), Jesus.Jimenez@iit.upcomillas.es (J.R. Jimenez-Octavio), carnicero@dim.icai.upcomillas.es (A. Carnicero), oscar.lopez.garcia@upm.es (O. Lopez-Garcia). deal with. Helically wound cables present interwire friction which influences axial stiffness [9]. Cables can show hockling or kinking phenomena as a result of torsional stability of single and double rope systems, [10]. For instance, the validity domain assessment of the mechanical behaviour of simple straight strands which are layers of helical wires wound around a central straight wire core has appeared in [11]. This paper focuses on macro- scale modelling of complex cable structures, that is, the initial equilibrium configuration computation and the cable structure response to external load equilibrium under general loading. Therefore, the modelling of the local behaviour of wire cables is beyond the scope of the paper. Broadly speaking, the methods used to model cable structures can be classified into two main groups. Following the nomencla- ture proposed in [8] these approaches are called: the non-linear displacement method and the force density method. The method of non-linear displacement is based on an iterative process that modifies step by step the geometry from one configuration to another fulfilling the equilibrium equations. Argyris’s pioneering work, [12], applies this method to the design of the cable roof of the Olympic Stadium of Munich, replacing real cables with truss elements. The dynamic relaxation method with kinetic damping is used in [13] to determine the initial equilibrium configuration and analyse prestressed nets and membranes. Based on the method of non-linear displacements, some authors, [14] and [15], modelled the cable as a series of straight linear trusses developing specific formulations to improve the method performance. Trying to improve these formulations, 0141-0296/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2009.03.018