S9-1 PAT325, Section 9, February 2004 Copyright 2004 MSC.Software Corporation SECTION 9 DOUBLY CURVED SURFACES

S9-2 PAT325, Section 9, February 2004 Copyright 2004 MSC.Software Corporation

S9-3 PAT325, Section 9, February 2004 Copyright 2004 MSC.Software Corporation COMPOSITES ON CURVED SURFACES n Composites achieve maximum efficiency when used to manufacture curved surfaces n For curved surfaces the fiber orientations and flat pattern profiles of a ply is not obvious n MSC.Laminate Modeler has extensive tools to simulate the application of plies to flat, single curved, and double curved surfaces

S9-4 PAT325, Section 9, February 2004 Copyright 2004 MSC.Software Corporation n Flat surfaces u Orientation is specified by a constant normal vector, N, in space u Specifying a fiber orientation at one point of the surface gives a unique solution for the fiber orientation at every point on the surface, and a unique flat pattern u No shearing of material during application of plies, and the flat pattern has the same shape as the surface DEFINITION OF SURFACE TYPES N

S9-5 PAT325, Section 9, February 2004 Copyright 2004 MSC.Software Corporation n Singly curved surfaces u Orientation is specified by a variable normal vector, N, in space. It is constant, when going from a point, in axial direction. It varies when going in circumferential direction. u Specifying a fiber orientation at one point of the surface gives a unique solution for the fiber orientation at every point on the surface, and a unique flat pattern u No shearing of material during application of plies, and the flat pattern has the same shape as the map of the surface to 2D space N DEFINITION OF SURFACE TYPES (Cont.)

S9-6 PAT325, Section 9, February 2004 Copyright 2004 MSC.Software Corporation n Doubly curved surfaces u Orientation is specified by a variable normal vector, N, in space. It varies, when going from a point, in either of two orthogonal directions. u Surface can be covered by a fabric in an infinite number of ways u Shearing happens when ply is drawn over the surface. The amount of shearing is greatly dependent on the application starting point and fiber direction, and of course the amount of curvature of the surface. DEFINITION OF SURFACE TYPES (Cont.) N

S9-7 PAT325, Section 9, February 2004 Copyright 2004 MSC.Software Corporation n A factor called the Gaussian Curvature is used to describe the degree and type of curvature of any surface n The Gaussian Curvature(GC) is the product of the principal curvatures at a point. So, GC = K 1 K 2 = 1/(R 1 R 2 ) u Where, the curvature, K, is equal to 1/R, where R is the corresponding radius of curvature of the surface GAUSSIAN CURVATURE K1K1 K2K2 Z 2 1

S9-8 PAT325, Section 9, February 2004 Copyright 2004 MSC.Software Corporation GAUSSIAN CURVATURE (Cont.) n How is the radius of curvature defined ? u Surface must be smooth at the point where the radius is to be determined u Create a curve at intersection of the surface and a plane perpendicular to tangent plane at point of interest u The radius of curvature of the surface is equal to the radius of curvature of the curve at the point of interest, and is in the perpendicular plane Tangent plane Perpendicular plane Surface Intersection curve

S9-9 PAT325, Section 9, February 2004 Copyright 2004 MSC.Software Corporation n Dome-like surfaces have positive Gaussian Curvature because the principal curvatures correspond to the same direction (have the same sign). n Geodesic lines of the surface, parallel at some point, will tend to converge together POSITIVE GAUSSIAN CURVATURE

S9-10 PAT325, Section 9, February 2004 Copyright 2004 MSC.Software Corporation ZERO GAUSSIAN CURVATURE n Cylindrical surfaces have zero Gaussian Curvature because there is zero curvature in one direction n Geodesic lines of the surface, parallel at some point, will remain parallel

S9-11 PAT325, Section 9, February 2004 Copyright 2004 MSC.Software Corporation n Saddle like surfaces have negative Gaussian Curvature because the principal curvatures correspond to opposite directions (have different signs). n Geodesic lines of the surface, parallel at some point, will tend to diverge NEGATIVE GAUSSIAN CURVATURE

S9-12 PAT325, Section 9, February 2004 Copyright 2004 MSC.Software Corporation VISUALIZATION OF GAUSSIAN CURVATURE Positive Gaussian Curvature: insufficient material to form the disc Zero Gaussian Curvature: exactly enough material to form the disc Negative Gaussian Curvature: excess material for forming the disc Drape several disc surfaces with ply......and then flatten the ply Re-flatened ply Dome discFlat discWarped disc