Keywords: fabric, cable, tension, form finding, sensitivity analysis, optimisation.

The final design of a tensile membrane is achieved when the stress states generated by different and normalised climatic (snow, wind…) loadings have been deemed as acceptable by the official control bodies. For this reason, and since in tensile structures the shape and the stress states are strongly related, the whole design may result from a repeated iterative process between the designers, i.e. the firm of architects and the research consultants. This motivates the presentation in this paper of an optimisation tool devoted to fabric structures which could be used to ease the design process in decreasing the number of back-and-forth interactions between all the different participants. First a specific and new shape form finding procedure is proposed; taking into account the usual geometrical constraints or data requirements, but also a desired biaxial (warp and weft) non uniform stress state, an (optimal must be understood in the sense: nearest as possible) initial shape is found. Then the climatic loads are applied and the current loaded positions calculated. A sensitivity analysis of these equilibrium positions depending on different parameters such as the positions of the anchorage points and cable tensions, allows one to modify with efficiency the shape and/or the stress state in order to better satisfy the requirements. This optimisation loop is included in a larger one that takes into account possible modifications of the cutting patterns, which is an other way of optimising the stress states. The efficiency of this procedure is demonstrated on an actual example.

In the design of a tensile fabric structure the first step is called shape finding. The usual data given, by the architect are the coordinate list of several fixed points : the mast position(s) and the anchorage points. One other main data set is the tension values of all the boarder or intermediate cables and the warp and weft tensile optimal stresses in the fabric. From these data an equilibrium position is found by solving the non linear matrix equations.

is the geometric stiffness matrix, the nodal displacement vector and the optimal stress state which is defined as follows :

• in the fabric by a weft value and the ratio value between the warp and the weft tensions,
• in the cables by the desired tension.

Any initial geometry of the membrane may be chosen but a definition coupled with a finite element surface generator of the whole structure is a very convenient starting point (a simpler one is a flat mesh). The shape is easily varied in modifying the values. Constraints on maximum and minimum values of are taken into account.

As soon as the initial shape is determined a non linear finite element analysis of the structure is performed using cable and orthotropic membrane elements to evaluate the stresses and the shape of the loaded structure. From this analysis, it may be necessary to modify the initial shape if over or under stressed parts which generate wrinkles or pockets are found. In order to ease the designer work, a sensitivity analysis of this equilibrium position may be performed; the design variables are: the tensions in the cables and the positions of the anchorage points. Two objective functions have been defined: one is concerned with the nodal displacements to minimise their total norm and the second with the stresses to maximise the minimum stress areas and to minimise the maximum stress areas. The application of this optimisation procedure yields a definition of an optimal structure shape as close as possible to the architect design. It also can be used to take into account of the influence of uncertainties on the implementation in situ of the structure

When the three dimensional shape is defined there remains to define the cutting patterns. The flattening process uses a specific method. To take into account the desired stress state in the three dimensional fabric i.e. to correct the patterns obtained, an iterative stress calculation process between the and the shapes is performed.

The example presented in this paper is a swimming pool roofing. The covered surface has the dimensions: / / . The roof is made up of rows of inclined masts ( hoops) and the fabric is divided in pieces. The finite element mesh is made of triangular linear membrane elements and linear cable elements.

1

T. Nouri-Baranger, "Form-finding method of tensile fabric structures: revised geometric stiffness method", Journal of the International Association for Shell and Spatial Structures, Vol 43 No. 1, 2002.

2

T. Nouri-Baranger and T. Trompette, "Optimal cutting patterns to control Stress Field in three dimensional textile structures", The Second World Congress of Structural and Multidisciplinary Optimization. WCSMO-2, Zakopane, Poland, May 26-30 1997, pp. 953-958, Edited by W. Gutkowski and Mroz.

3

D.G. Phelan and R. B. Haber,"Sensitivity analysis of linear elastic systems using domain parametrization and mixed mutual energy principal". Computer Methods in Applied Mechanics and Engineering, Vol 77, pp. 31-59, 1989. doi:10.1016/0045-7825(89)90127-8

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