Post-Formfinding Determination of Geodesic Lines in Cutting Pattern Design for Membrane Structures

Abstract

This paper describes a numerical procedure for determining geodesies on a surface. The surface is defined by a mesh of planar triangular finite elements. Examples are presented to illustrate the relevance of this procedure in the development of cutting patterns for membrane structures.

Computer Cutting Pattern Generation of Membrane Structures

“…From differential geometry, it is known that a surface can only be developed onto a plane if it is generated by a family of straight lines such that along each of these generators it has the same tangent plane. Hence it is unavoidable that the attempt to fabricate the doubly curved surface from flat strips only gives an approximation to the true shape [1].…”

“…Ishii [7] obtained the differential equation of geodesic lines for a surface given in the form z = f (x, y) , as (1) Equation 1is a non-linear differential equation and requires numerical solution by techniques such as the finite difference method. A geodesic line on the surface is uniquely defined either by specifying its direction through a point on the surface, or by giving two points on the surface through which it passes.…”

“…A geodesic line on the surface is uniquely defined either by specifying its direction through a point on the surface, or by giving two points on the surface through which it passes. However, Equation (1) is only applicable to an analytically defined surface [1] [8].…”

“…Based on the Struik's theory [6], Meek and Tan [1] proposed a method for generating geodesics on predefined surface and proved that, when cable elements of constant uniform pre-stress are stretched between two points, the equilibrium position of the assembly also corresponds to the shortest path between these points. For a single such element with a pre-stress force F, length L an axial rigidity EA, the strain energy can be expressed as (2) If F and EA are constant along the member, Equation (2) becomes 3For n such elements stretched between two points, the total strain energy is given by 4If all the members have the same pre-stress and sectional properties, and the internal member forces are also stretch independent, then (5) The strain energy is equal to the total potential energy since no external forces are applied.…”

Shape Design of Curved Surface of Membrane Structure Using Developable Surface

“…Since the membrane structure can be realized by connecting several planar sheets, it is desirable to design the cutting pattern so that uniform in-plane tension stress is generated in the membrane. The most conventional approach utilizes the geodesic line [5] to obtain approximate cutting patterns from the target curved surface [6,7]. Punurai et al [8] proposed an optimization approach using genetic algorithm.…”