Keywords: inflatable structures, double-layered fabric structures, inflated beam element, plain strain element, minimization of potential energy, following forces.

The paper deals with the development of two finite elements suitable to give the solution of the response of inflatable membrane structures in two dimensions when the applied pressure is high (up to several hundred of kPa). These inflatable structures are double-layered membranes (panels). They are made of modern textile materials with important mechanical characteristics. Inflation causes tension prestressing in the walls and in the yarns of the structures. This prestressing is proportional to the pressure and ensures an important mechanical strength and makes these kinds of structures very interesting as strong building elements. Although inflatable structures are not recent, no studies have been conducted on this subject for such values of the pressure.

In the first section of the paper, we will recall the mechanics of inflatable panels [1] and construct a new inflatable beam finite element. These panels are prototypes made of two-coated fabric membranes connected by yarns, which ensure the flatness of the structure. Equilibrium equations are written in the deformed state to take into account the geometrical stiffness and the following forces. Kinematics assumptions are of Timoshenko's kind, and the constitutive law of the fabric is obtained from experimental data. Analytical study on these kind of inflatable fabric beams at high pressure in isostatic conditions show that their behaviour is a linear set of the behaviour of yarns and beams. Analytical solutions are compared with experiments and show the accuracy of the theory. The panel compliance is therefore nothing but the sum of the beam compliance and of the yarn compliance. The new finite beam element is then obtained by the equilibrium finite element method and is modified into a displacement finite element in order to be easily implemented in usual finite element software. The stiffness matrix of the free finite beam element takes into account the applied pressure. This new inflatable beam finite element is able to predict the behaviour of inflatable structures made of thin-walled beam elements.

The second section of the paper is devoted to the development of a plane strain finite element. This element takes into account finite strains, and also large deflections and following forces due to the applied pressure. Its construction provides from an energetic formulation suitable for pressurised membrane structures so the flexure effects of shell elements are avoided, and no problems of loss of stiffness can occur. The numerical computing is carried out in a different way that the classical finite element approach and is realised by means of a direct minimisation. The total potential energy of the membrane elements is calculated and then differentiated with regard to the free nodal displacements of the structure. It leads to a system of non-linear equations, which is solved by using optimisation algorithms. Theoretical comparisons of this approach with the classical finite element method show that the linear part of the previous system can be decomposed as two stiffness matrix depending respectively on the mechanical characteristics of the fabric and on the pressure. The non-linear part of the system, which represents the effect of large displacement, finite strains and following forces, involves until third degrees of displacement components. Because of the rapidly increasing of the expressions, it is not developed here. This direct energetic approach permits to model membrane structures without any artifice concerning the bending stiffness.

In the third section, these two finite element solutions are compared with experimental results obtained on a double–layered panel inflated at high pressure, submitted to bending loads and for various boundary conditions. Experiments are described and the panel is tested for three kinds of boundary conditions (isostatic and hyperstatic configurations). The beam finite element is directly used to study the behaviour of the panels. The plane strain finite element is used to discretize the membranes and the distance between the two fabrics is assumed to be constant by including rigid elements between the membranes. Comparisons are made for values of the internal pressure going from 50 kPa to 300 kPa and for external loads up to the theoretical wrinkling loads. It is shown that these two finite elements give similar numerical results, and that these numerical solutions are also close to the experiments. Errors between experimental and numerical results are within 15%. Given that the constitutive law of the fabric is obtained in the same range of error, we can assert that our finite elements are accurate to solve bending problems of inflatable double–layered structures in two dimensions.

1

C. Wielgosz, J.C. Thomas, "Deflections of inflatable fabric panels at high pressure", Thin-Walled Structures, 40, 523-536, 2002. doi:10.1016/S0263-8231(02)00010-1

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