Keywords: fabric, cable, tension, membrane, form finding, minimum surface area, cutting pattern.

The first tensile structures built by men were probably tents, made of animal skins and thin branches. The invention of knotting and weaving made possible fabric tents. Unfortunately, no significant progress has been made in this field since the middle age. In fact, tension loaded structures are a development of end of the last century. Large tensile structures were built all over the earth. Some important recent structure are the Sherway Gardens-Phases IV in Toronto, the Campus center of the University of La Verne and the Chene Park Amphitheater in Detroit in the USA. General development, internationally, in the field of tension-loaded surfaces structures has been noticeable since 1960. Some exceptional theoretical and practical work was done by architects and engineers, among them Otto , Hottinger and Tsuboi.

The conception of fabric structures is strongly dependent on the use of computers and many specialised softwares have been developed in the last twenty years, see Barnes (1986), Grundig (1988), Haber (1981), D'Uston and Trompette (1987), Pauli (1994) and Nouri-Baranger (1999). They enable designers to determine the shape, to perform a static analysis, and to cut and produce the fabric layers.

In this paper we deal with tensile structures made of tension-loaded fabric, cables and rigid frames such as mats and hoops. The fabric is a material which cannot support compression and bending loads effects and its constitutive law is orthotropic one. Since a membrane can only sustain tensile stresses, it must be prestressed. This can be done by tensile forces acting on membranes edges. These forces are transmitted by cables elements and rigid elements.

Tensile structures present a nonlinear mechanical behavior in response to external loading. This nonlinearity is geometric one because of the initial stress field in membrane, tension in cables and the possible large deflections. The assumption of linear constitutive law of the material and little strain are taken into account.

Because of the above particularities, the behavior of this kind of structures is complex. In addition, a great number of design parameters have to be taken into consideration. Consequently, designing a fabric structure is difficult and could require repeated analyses for successive modifications, changing material properties, cable prestresses or anchorage locations for example. All these steps involve extensive computational costs, as the numerical models are large and nonlinear. Nevertheless, three principal steps can be outlined:

• The form finding analysis: the determination of an initial equilibrium shape satisfying the architectural requirements. This phase is performed according to stress levels, cable tensions and boundary conditions,
• The structural response analysis: the response to external loads including large displacements and wrinkling effect,
• The cutting pattern: the final three dimensional shape is divided into a set of widths, which are flattened into two dimensional cutting patterns. In order to generate the ideal stress field distribution in the fabric when the structure is built, these cutting patterns have to be optimised.

Through these steps, the designer has to verify the structure stress level in its final state and under climatic loading. He has to be particularly observant of instability areas initiated by the compression strains which have a wrinkling effect, of the over-stressed area and finally of the eventual under stressed area which could cause pocket formations and therefore the collapse of the structure.

Hence, in order to reduce the computational cost, sensitivity analysis and optimisation process appear to be essential. In fact, sensitivity analysis of 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 satisfy the requirements. It also allows civil engineers to foresee the influence of inaccuracies in the building process. Also, an optimisation tool allows to determine the optimal design values of these parameters. This optimisation loop is included in a larger one that takes into account possible modifications of the cutting patterns. Wealthy work in this field has been developed by Ramm and al. and Phelan and Haber in the last decade.

After brief description of the most used methods for form finding steps and a specific and new 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 (optimal must be understood in the sense: nearest as possible) initial shape is found. A nonlinear elastic analysis is then performed using the last shape submitted to the stress field specified by the designer, in order to obtain the effective stress field distribution. Then the climatic normalised loads are applied and new nonlinear elastic analysis is done using the last initial stress distribution. The steps above have to be repeated until the results satisfy all constraints of security. The cutting pattern step is then performed.

Finally, optimisation and sensitivity analysis tools devoted to fabric structures are presented. They could be used to ease the design process in decreasing the number of back-and-forth interactions between the above steps.

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