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2.2 Analytical form finding of membrane structuresUntil the 70ties of the last century physical form finding was used for engineering tensile structures. Theimplementation of the model has to be done as precise as possible. It is a time consuming process with therisk of many inaccuracies. After the knowledge gained during the construction of the Olympic Stadiumin Munich (1972) computer programs were introduced for the engineering and form finding of membranestructures.In 1974, H.J. Schek published the paper “The force density method for form-finding and computations ofgeneral networks”, the fundamental theory for the analytical approach of form finding, the force densitymethod. The computer program Easy and, more recently, other programs use this method for form findingmembrane structures. In the analytical form findings process, according to Klaus Linkwitz, two phases canbe identified.2.2.1 Phase1A number of design studies, non-materialized equilibrium models, have to be done. The process is almostanalogous to the soap film method. Within the boundaries a minimal surface will be generated. Themembrane can be seen as the discretization of a cable net. There are a number of parameters that influencethe geometry of the equilibrium surface. By varying these parameters the geometry (curvature) of thesurface can be influenced. The parameters are:a) the position of boundaries, the high points and low points;b) the curvature of the surface;c) proportional force density in the different parts of the surface;d) orientation of the network; ande) external forces/load cases.In the first phase the equilibrium geometry of the surface is found. In this phase it is possible to dopreliminary studies to load cases, deformation, and stress contribution.2.2.2 Phase 2The second phase consists of the materialization of the equilibrium surface. By introduction of the materialproperties it is possible to have a complete analysis of the load cases, deformation and stress contribution.The geometry of the surface will be influenced by the material properties.Phase 1 Phase 2conditions closed boundaries and/or points equilibrium (Phase 1)shape equilibrium materialised equilibriumshape influence a) boundaries, high points, low material propertiespointsb) the curvature of the surfacec) proportional force densityd) orientation of the networke) external forces/load casesTable 4: Process for the analytical form finding of membrane structures2.2.3 Dynamic relaxationIn the force density method there is a ratio between the force and the length of a line. This is specified bythe designer. Beside the force density method it is possible to calculate the deformation of membranes withthe dynamic relaxation method. The dynamic relaxation method is based on discretizing the continuumunder consideration by lumping the mass at nodes and defining the relationship between nodes in terms32