Design of cable-membrane structures
  
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Introduction
The design of a tension structure is a complex procedure consisting of many parts which results in an iterative process. This process may include: conceptual design; physical modelling; computer modelling; formfinding; analysis; and cutting pattern generation.  
  
The design usually commences with a sketch drawn by the architect. This sketch provides the basis for further conceptual design, which at the beginning reduces to a ``simple'' task to identify and fix the structural form. Because in tensile structures the form is mainly governed by the stresses either physical modelling or computer simulation of the structure is required to help the designer to find realistic shapes.  
  
From the historical point of view, the tension cable and membrane structures were analysed with great difficulties and great effort and even then only by approximate methods (Frei Otto). For this reason the structures have been developed in model form. One method was to create a design model (usually small scale), a measuring model (large scale, made of strings, paper, fabric or chains for measuring the contour, the size and the shape of the structure) and a structural testing model (for loading tests.)  
  
In recent years computer simulation is available as a replacement of physical models. Several methods have been developed for computational simulation of shape (form) finding, like constrained force density method, Newton-Raphson iteration, optimization methods, direct shape fitting and dynamic relaxation. Here the dynamic relaxation method will be discussed. But even when computers are used for the design of membrane structures the three basic steps remain the same:  
  
form-finding  
cutting-pattern generation  
analysis  
  
The first two steps give the detailed geometric description of the structure. The form-finding provides the exact shape. At this stage fictitious values can be used to accelerate the convergence of the dynamic relaxation algorithm. The term "cutting pattern generation" contains the determination of the distortion- and stress-free lengths (slack length) of all cable and membrane pieces as a prerequisite for the cutting of cable or membrane materials and their knotting and sewing together to a structure. In the cutting-pattern generation real material values are used. Moreover this step becomes important because in the analysis step where the structure is checked under different loading conditions the slack lengths determine the prestress in the structure.  
  
Dynamic Relaxation
It has been shown by several people (ref. not included now) that the dynamic relaxation method is suitable for computer simulation of tensile structures. The method can easily take account for non-linear behaviour resulting from large deformations or material non-linearities. Moreover, in the refinement or adjustment of the form the method's iterative nature can be utilised because the change in the topology or boundary conditions does not imply that form-finding must necessarily commence from the beginning, but the computation may usually continue with the changed parameters producing a new equilibrium form.  
  
The basis of the method is a step-by-step algorithm tracing the motion of a structure until the structure reaches equilibrium due to damping. Some of the most important features of this method:  
  
The method is basically for static problems using fictitious dynamic analysis.  
No assembled structural stiffness matrix is required due to the use of natural stiffness. Hence it is suitable for highly nonlinear problems.  
The results always expressed in terms of the current coordinates of the structure, hence analysis of large displacements is possible.  
Instabilities during the procedure show problems with the structure to design and they are not numerical instabilities !  
Element library
Line elements with two connecting node  
Truss elements allowing tension and compression stresses in the element  
Cable elements allowing only for the building of tension stresses  
Constant force element, where the tension stress is pre-specified  
Constant force density element  
Membrane elements  
"Solid" triangular element allowing tension and compression stresses  
"Membrane" triangular element allowing only tension stresses  
"Formfinding" (constant stress) triangular element, which can be isotrop or anisotrop with different specifications for the weft and warp directions  
  
Modell generation
The procedure of the generation of a three dimensional model of a tension membrane contains the following steps  
  
Generating a flat triangular finite element mesh. The finite element mesh is structured and uniform.  
Attaching the the edge and ridge cables. Edge and ridge cables are modelled as a sequence of line elements between each succesive pair of nodes.  
Adding the geodesic strings.  
Applying the boundary conditions.  
Specifying stress levels in all the elements.  
   
Modell generation of a concert hall  
  
Geodesic strings
In the case of a membrane structure idealised as a finite element mesh containing triangular membrane elements and edge cables the elements will spread themselves over the surface during the form-finding without any respect for the distribution of the cloth seams. This results, during the cutting pattern generation, in irregular pieces of cloth. Irregular means, that the edges of the strips are not straight lines. These irregular strips are not suitable for fabrication since the cloth wastage will be great. To avoid these badly shaped strips geodesic strings must be included into the form-finding model. These geodesic strings describe lines that follow a minimum distance over the surface. Hence if they are in equilibrium they must stay in a straight line.  
  
Form-finding
In the form-finding analysis an equilibrium form is obtained from a model. The basic control at this stage are the forces or stress levels of the structure. Although elastic properties can be assigned to cable or other line elements, properties should be chosen on the basis of convergence rate than physical values. At the initial stage the data may well be crude and inaccurate and several analyses with adjustments of geometry and stress will be necessary before the desired equilibrium form is established.  
  
Once the equilibrium form has been obtained, the same model can be used for the analysis under loadings. During the form-finding, there will be a mixture of elastically controlled and stress controlled elements. Moreover, the values in the form-finding might not be real values. Hence some ``conversion'' of the properties of the elements is required.  
   
Cutting pattern generation
At this stage the spatial geometry is transformed into planar pattern geometry by folding. By the use of geodesic strings it can be ensured that the edge of the strips of clothes will be straight. (This produces less material wastage.) In the case, when the strip is only one element wide the folding procedure is obvious, but when the strip is several element wide dynamic relaxation must be used for flattening. This step also provides the stress free lengths of the geometry which will be used in the analysis step as pretensioning.