S3-1 1 PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation SECTION 3 ELASTIC BEHAVIOR OF MATERIALS AND CLASSICAL LAMINATION THEORY

S3-2 2 PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation

S3-3 3 PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation ELASTIC BEHAVIOR OF MATERIALS

S3-4 4 PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation

S3-5 5 PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation n The most general form of the linear elastic stress- strain relationship is u The stress and strain tensors both have 9 components, while the stiffness 4 th order tensor has 81 terms. n For elastic materials the stiffness tensor is symmetric. So, using only 6 components of stress and strain the above equation can be simplified to the following expression. GENERALIZED MATERIAL i,j=1,2,3; k,l=1,2,3

S3-6 6 PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation ANISOTROPIC MATERIAL n No planes of material property symmetry n 21 independent elastic constants n The S in the lower left corner denotes that the matrix is symmetric about its main diagonal.

S3-7 7 PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation ORTHOTROPIC MATERIAL n Three orthogonal planes of material property symmetry n Nine independent elastic constants (E 1, E 2, E 3, 12, 23, 13, G 12, G 23, G 13 ) n No interaction between normal and shear stresses

S3-8 8 PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation ORTHOTROPIC MATERIAL (Cont.) n Solved for the strains. Used engineering constants.

S3-9 9 PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation ISOTROPIC MATERIAL n Infinite number of planes of material property symmetry n Two independent elastic constraints (E and )

S PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation n Solved for the strains. Used engineering constants. ISOTROPIC MATERIAL (Cont.)

S PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation CLASSICAL LAMINATION THEORY

S PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation

S PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation LAMINATE SHELL ELEMENT DEFINITION n To specify the stiffness of a laminated shell element it is necessary to use the relationship between force and displacement response for the plate n Assumptions u Lamina are in a state of plane stress (σ z = 0) u Lamina are perfectly bonded u Linear variation of strain through the thickness

S PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation FORCES AND MOMENTS DEFINITION x z y MyMy F xy VyVy MyMy FyFy M xy F xy M xy FxFx MxMx VxVx F xy M xy FyFy VyVy FxFx MxMx VxVx F xy Warning: some composites texts use a different system, per R.M. Jones (1999)

S PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation APPROACH TO SOLVING FOR STRESS n Calculate the stiffness of each lamina in their coordinate systems n Transform the lamina stiffnesses into the laminate coordinate system n Add the stiffness for each lamina to obtain the laminate stiffness n Perform a stress analysis, e.g. MSC.Nastran n Calculate stress in lamina as a function of reference strain and curvature

S PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation STIFFNESS RELATION FOR ANISOTROPIC MATERIAL n The starting point is the stiffness matrix, [C], of an anisotropic material

S PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation INTRODUCE PLANE STRESS n Using the assumption of plane stress yields the following matrix equation for a lamina u This is valid for one lamina in its local lamina coordinate system. n Q ij are the equivalent stiffnesses of the lamina for plane stress

S PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation TRANSFORMING EQUIVALENT STIFFNESS n The lamina stiffness, [Q], must be transformed so it is expressed in the coordinate system of the laminate. This is done using an ordinary coordinate system transformation. X (lamina) X (laminate) Y Y Z, Z - -

S PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation LAMINATE STIFFNESS n The contributions from the different laminas to the total laminate stiffness is classified using three groups A ij – Extensional Laminate Stiffness B ij – Coupling Stiffness D ij – Bending Stiffness n The stiffness relationship is

S PAT325, Section 3, February 2004 Copyright 2004 MSC.Software Corporation CALCULATING STRESSES IN LAMINA n The local in-plane stresses anywhere in the plate element can now be found as a function of the global strains Where Z is the distance from the neutral plane to the relevant point in the ply, and is the curvature.