Keywords: coated woven fabrics, lightweight structures, NURBS, response surface, numerical models, biaxial stress, data fitting, non-linear behaviour.

This paper investigates the use of Bézier functions, B-splines and NURBS (Non-Uniform Rational B-Splines) for the representation of the stress-strain response of coated woven fabrics. These fabrics are used in a wide range of structural applications to provide lightweight, architecturally striking solutions. The design of fabric structures is complicated by the complex response of coated woven fabrics to biaxial loads in the plane of the fabric. A better understanding of the behaviour of architectural fabrics may significantly reduce levels of uncertainty in the design process and enable more ambitious architectural forms to be generated.

Fabric structures resist environmental loads as tensile stresses in the plane of the fabric. Under biaxial tensile loading the response of coated woven fabrics is highly non-linear. Geometric non-linearity occurs in the yarns (due to the complex twisted fibre structure) and in the finished fabric (interaction of orthogonal warp and weft yarns under biaxial in-plane stress leads to fundamental non-linearities, compounded by the effect of the coating). Material non-linearity is evident in the load-extension characteristics of both the yarn fibres and the fabric coating. The material response is also time-dependent and hysteretic due to the material properties and frictional effects (inter-fibre and inter-yarn friction).

NURBS (Non-Uniform Rational B-Splines) are mathematical functions used for curve and surface definition [1]. They are a development of piece-wise Bézier curves and B-splines. NURBS are used as the basis for surface definition in a range of three dimensional computer graphics applications (CAD and animation) and computer controlled machining, and for data representation in signal processing applications. The strength of these parametric functions is an ability to describe any continuous curve or surface with a smooth function whilst allowing local modification of the surface.

Minami et al [2] used response surfaces to represent biaxial fabric behaviour. Orthogonal stresses (, ) and strains ( , ) from biaxial fabric tests form surfaces in the , , and , , and coordinate systems. Elastic constants are established using a multi-step linear approximation. The surface is divided into smaller quadrilaterals and for each quadrilateral the elastic constants are determined. The size of the small quadrilaterals is critical in ensuring discontinuities in the fabric behaviour are accurately captured.

In this paper the utility of Bézier functions, B-splines and NURBS (Non-Uniform Rational B-Splines) is investigated for the represention of fabric biaxial test data. Bézier surfaces interpolating all data points with continuous curvature are developed, using data from the work of Day [3]. The Bézier surface is manipulated by changing the positions and weights of a grid of control points (the control net). The fit is optimised by minimising the mean squared error of the surface from the data points. The Bézier surface is not a unique solution to the data fitting problem; a large number of Bézier surfaces could be found to interpolate any data set. Further optimisation criteria which are required to give a unique solution are discussed.

NURBS have similar characteristics to Bézier functions whilst allowing greater local control of the surface shape including the representation of discontinuities. Further testing of fabrics is proposed to establish whether the response is discontinuous. Future work aims to extend these Bézier, B-spline and NURBS representations to include shear, hysteresis and time dependent behaviour.

The lack of a unique solution without complex optimisation suggests that Bézier curves, B-splines and NURBS might not be the best method for the representation of biaxial tensile fabric behaviour. It is therefore the authors' intention to investigate other surface fitting techniques: function derivation using genetic programming, function parameter optimisation using neural networks, and Response Surface Methodology.

Acknowledgements

This research is jointly funded by the EPSRC (Engineering and Physical Sciences Research Council), Arup and Architen-Landrell.

1

Dierckx, P. "Curve and surface fitting with splines", Oxford Science Publications, Clarendon Press, Oxford, 1993.

2

Minami, H., Yamamoto, C., Segawa, S., Kono, Y. "A method for membrane material nonlinear stress analysis using a multi-step linear approximation", in "IASS International Symposium on Shell and Spatial Structures", Singapore, 595-602, 1997.

3

Day, A.S. "Stress strain equations for non-linear behaviour of coated woven fabrics", in "IASS symposium proceedings: shells, membranes and space frames", Osaka, Elsevier, Amsterdam, 2, 17-24, 1986.

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