2.8 Analysis methods
In this section, some techniques used for analysis of cable structures will be men-tioned. None of the numerical techniques described here are applicable to only cable structures. Therefore, only a short historical review is considered necessary.
Early analyses were done by applying membrane shell theory to cable nets. Appli-cation of the membrane shell theory results in a set of differential equations. Except for special cases, these equations are difficult to solve in closed form. In most cases, the equations are solved by numerical techniques, such as the finite difference method [112]. Shore and Bathish [107] used double Fourier series to transform the differential equations to a system of algebraic equations. One flat and one hyperbolic paraboloid prestressed cable net, both square in plan, were analysed numerically and experimentally. The agreement between the results was acceptable. Recently, T¨arno performed a parametric study of saddle-shaped networks with elliptic plane layouts and stiff contours [116]. Such a study serves as an aid in choosing the dimensions of the roof and the structural elements. In general, membrane shell theory is less accurate if the cable mesh in a net is coarse; it is inadequate for complicated roof shapes.
Since the introduction of computers in the 1960’s, several numerical methods have been developed for the general analysis of structures. Among these methods, the stiffness technique (finite element method) have been widely adopted. Originally, the method was developed to analyse structures with small displacements. Under the action of external loads cable nets undergo large displacements and it became evident that the stiffness technique was not applicable to such structures in its original form. Therefore, the original method was modified and applicable structures with geometrically non-linear characteristics. Several iterative methods have been applied to the non-linear stiffness method. The most popular is the Newton-Raphson technique, which has proven to be accurate, efficient and applicable to the majority of cable structures [1]. A comprehensive description of the finite element method and the Newton-Raphson technique can be found in, for example, [28]. Other authors, e.g. [16, 111] have used a method based on the minimisation of the total potential energy of the structure. The minimisation was done using the the conjugate gradient method. The dynamic relaxation technique has been used by several authors for both form-finding [8, 66] and load analysis [68, 69].
Approximate methods for the preliminary design of cable trusses and simple cable nets, can be found in references 16, 57 and 76. For elliptical cable nets the method described by T¨arno [116] is recommended.
The following chapters will investigate some earlier reported analysis methods and some new variants of them, aiming at accurate analyses of the form-finding and normal usage stages of some cable roof structures. Failure stage analysis will be identified as a topic for further research.
Chapter 3
The initial equilibrium problem
3.1 Introduction
For structural analysis, the equilibrium configuration of a structure is generally known in advance. This is not the case for tension structures, i.e. cable and mem-brane structures. Due to the low flexural stiffness of the cables and the fabric these structures have to be constructed so that they will experience a significant prestress at all times. Thus, there is no compatible unstressed configuration for a tension structure, even if no external loads are applied and its self-weight is neglected.
Therefore, the designer must specify a reference configuration for the structure that is stressed. The shape of the reference configuration depends upon the internal stresses and forces. Hence, the load bearing behaviour and the shape of the struc-ture cannot be separated and cannot be described by simple geometric models. In addition to satisfying the equilibrium conditions, the initial configuration must ac-commodate both architectural, structural and constructional requirements [44, 114].
Finding the stressed initial configuration is an inverse structural problem, in which the specified force distribution is the driving parameter in the process. This is inverse to standard problems where the forces are the structural response to the deformations of the structure [13].
The problem of finding a configuration that satisfies the laws of equilibrium is usu-ally called form-finding or shape-finding. Haber and Abel [44] thought that this nomenclature was inappropriate to use when describing methods in which variables besides the shape were adjusted to satisfy equilibrium. Therefore, they used the term initial equilibrium problem instead. Throughout the present chapter and the rest of the thesis this term will mainly be used.
The objectives of this chapter are to describe the initial equilibrium problem and review the existing computer methods for solving it. All the methods that are to be described are applicable to mainly cable structures, membrane structures and bar frameworks. Among the methods, one is especially interesting, namely the force density method. This method will further on be described in detail and applied to a number of different problems. First, a brief description of the methods that were
CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM
used before computer methods were available will be given.
3.1.1 Physical modelling
Due to the mathematical difficulty and spatial complexity of the structural forms of tension structures, physical modelling was the primary method to solve the initial equilibrium problem of tensile structures until 1969 [44], when the cable roofs for the 1972 Munich Olympic Games were about to be built [62]. A pioneer in the physical modelling field was Frei Otto, who performed extensive experiments using a variety of different media, including soap films, fabric, and wire models.
The most simple modelling tools are soap film models, which are obtained by dipping wire frames into soap water. These models are used as a first check if the curvature of the roof surface is appropriate. As is well known, a soap film always contracts to the minimal surface. The minimal surface may be the most aesthetic shape, but it is not always the best structural form, as the minimal surface approach tends to produce very flat areas, which may induce flutter (see also section 3.2.1). After the soap film models, working models of larger scale and of other materials were built for further processing within the design process [87]. The Institute for Lightweight Structures in Stuttgart has worked with physical modelling techniques for several decades and a large number of structures with complex structural shapes of different scales have been realised during the years, e.g. [31] or [87]. The Institute was involved in the construction of the some of the largest and most complicated cable nets built in the world: the German pavilion at the 1967 World’s fair in Montreal and the Olympic cable roofs in Munich.
Models give many useful insights into the the behaviour of tension structures. A large number of configurations can in a short time be studied if for example soap film models are used. For practical design work, however, they do not provide sufficient accuracy and have to be replaced by computational methods [31, 44]. Nevertheless, physical models of tension structures will always be made, because of the excellent structural visualisation they provide [77].