Copyrighted Materials 0p UyPress rm CHAPTER NINE Finite Element Analysis The finite element method offers a practical means of calculating the deformations of,and stresses and strains in,complex structures.A detailed description of the finite element method is beyond the scope of this book.Instead,we focus on those features specific to composite materials. Finite element analysis consists of the following major steps: 1.A mesh encompassing the structure is generated (Fig.9.1). 2.The stiffness matrix [k]of each element is determined. 3.The stiffness matrix [K]of the structure is determined by assembling the ele- ment stiffness matrices. 4.The loads applied to the structure are replaced by an equivalent force system such that the forces act at the nodal points. 5.The displacements of the nodal points d are calculated by k]d f. (9.1) where f is the force vector representing the equivalent applied nodal forces (Fig.9.1). 6.The vector d is subdivided into subvectors 6,each 6 representing the displace- ments of the nodal points of a particular element. 7.The displacements at a point inside the element are calculated by u=[W6, (9.2) where the vector u represents the displacements and [N]is the matrix of the shape vectors. 8.The strains at a point inside the elements are calculated by e=[B6, (9.3) where [B]is the strain-displacement matrix. 395
CHAPTER NINE Finite Element Analysis The finite element method offers a practical means of calculating the deformations of, and stresses and strains in, complex structures. A detailed description of the finite element method is beyond the scope of this book. Instead, we focus on those features specific to composite materials. Finite element analysis consists of the following major steps: 1. A mesh encompassing the structure is generated (Fig. 9.1). 2. The stiffness matrix [k] of each element is determined. 3. The stiffness matrix [K] of the structure is determined by assembling the element stiffness matrices. 4. The loads applied to the structure are replaced by an equivalent force system such that the forces act at the nodal points. 5. The displacements of the nodal points d are calculated by [K] d = f, (9.1) where f is the force vector representing the equivalent applied nodal forces (Fig. 9.1). 6. The vector d is subdivided into subvectors δ, each δ representing the displacements of the nodal points of a particular element. 7. The displacements at a point inside the element are calculated by u = [N] δ, (9.2) where the vector u represents the displacements and [N] is the matrix of the shape vectors. 8. The strains at a point inside the elements are calculated by ε = [B] δ, (9.3) where [B] is the strain–displacement matrix. 395
396 FINITE ELEMENT ANALYSIS nodal forces nodal points elements Figure 9.1:Structure and its finite element mesh. 9.The stresses at a point inside the element are calculated by =[E]E, (9.4) where [E]is the stiffness matrix characterizing the material. The element stiffness matrix,referred to in Step 2,is defined as [k6=e, (9.5) where fe represents the forces acting at the nodal points of the element.The element stiffness matrix is1 内=[A'A[Ba业 (9.6 where V is the volume of the element. The preceding steps apply to structures made of either isotropic or compos- ite materials.The only difference between isotropic and composite structures is in the material stiffness matrix [E].In the following we present expressions for [E]. 9.1 Three-Dimensional Element The stress-strain relationships for a three-dimensional element are(Eq.2.20) C11 C12 C13 C14 C15 C16 Ex 6 C21 C22 C23 C24 C25 C26 31 C32 C33 C34 C35 C36 y C C42 C43 C44 C45 C46 (9.7) C51 Cs2 C53 C54 Cs6 C61 C62 C63 C64 C65 C66 [E] where [E]is the stiffness matrix for a three-dimensional element. 1R.D.Cook,D.S.Malkus,and M.E.Plesha,Concepts and Applications of Finite Element Analysis 3rd edition.John Wiley and Sons,New York,1989.p.110
396 FINITE ELEMENT ANALYSIS nodal forces nodal points elements Figure 9.1: Structure and its finite element mesh. 9. The stresses at a point inside the element are calculated by σ = [E] ε, (9.4) where [E] is the stiffness matrix characterizing the material. The element stiffness matrix, referred to in Step 2, is defined as [k] δ = fe, (9.5) where fe represents the forces acting at the nodal points of the element. The element stiffness matrix is1 [k] = ) (V) [B] T [E] [B] dV, (9.6) where V is the volume of the element. The preceding steps apply to structures made of either isotropic or composite materials. The only difference between isotropic and composite structures is in the material stiffness matrix [E]. In the following we present expressions for [E]. 9.1 Three-Dimensional Element The stress–strain relationships for a three-dimensional element are (Eq. 2.20) σx σy σz τyz τxz τxy = C11 C12 C13 C14 C15 C16 C21 C22 C23 C24 C25 C26 C31 C32 C33 C34 C35 C36 C41 C42 C43 C44 C45 C46 C51 C52 C53 C54 C55 C56 C61 C62 C63 C64 C65 C66 % &' ( [E] x y z γyz γxz γxy , (9.7) where [E] is the stiffness matrix for a three-dimensional element. 1 R. D. Cook, D. S. Malkus, and M. E. Plesha, Concepts and Applications of Finite Element Analysis. 3rd edition. John Wiley and Sons, New York, 1989, p. 110
9.3 BEAM ELEMENT 397 9.2 Plate Element In the absence of shear deformation,the force-strain relationships for a thin-plate element are (Eq.3.21) A A1 A6 B11 B12 B16 A12 2 A26 B12 B22 B N A6 A26 A66 B16 B26 B66 Bu (9.8) B12 B16 Du D12 D16 B12 B22 B26 D12 D2 D26 Ky B16 B26 B66 D16 D26 D66 Kxy [E where [E]is the stiffness matrix for a plate element without shear deformations.In the presence of shear deformation,the force-strain relationships are (Eqs.5.13- 5.15) N A A2 A16 B11 B12 B16 0 0 A2 An 6 B12 B22 B26 0 A6 6 A66 B6 B26 B6 0 0 M Bu1 B12 B16 D11 D12 D 0 0 Kx M B12 B22 B26 D12 D22 D26 0 B16 B26 B66 D16 D26 D66 0 0 Kxy 0 0 0 0 0 0 0 0 0 0 0 0 s Yyz [目 (9.9) where [E]is the stiffness matrix for a plate element with shear deformations. Frequently,the behavior of shells can be described by replacing the curved surface with small,flat elements.The stiffness matrices above are applicable to such flat shell elements. 9.3 Beam Element For a beam element,the force-strain relationships are arbitrary layup,no shear deformation,no restrained warping (Eq.6.2) N P P2 P3 P4 P2 P2 P2s P24 Pi3 P2s P33 (9.10) 4 P2a P P 11 [可
9.3 BEAM ELEMENT 397 9.2 Plate Element In the absence of shear deformation, the force–strain relationships for a thin-plate element are (Eq. 3.21) Nx Ny Nxy Mx My Mxy = A11 A12 A16 B11 B12 B16 A12 A22 A26 B12 B22 B26 A16 A26 A66 B16 B26 B66 B11 B12 B16 D11 D12 D16 B12 B22 B26 D12 D22 D26 B16 B26 B66 D16 D26 D66 % &' ( [E] o x o y γ o xy κx κy κxy , (9.8) where [E] is the stiffness matrix for a plate element without shear deformations. In the presence of shear deformation, the force–strain relationships are (Eqs. 5.13– 5.15) Nx Ny Nxy Mx My Mxy Vx Vy = A11 A12 A16 B11 B12 B16 0 0 A12 A22 A26 B12 B22 B26 0 0 A16 A26 A66 B16 B26 B66 0 0 B11 B12 B16 D11 D12 D16 0 0 B12 B22 B26 D12 D22 D26 0 0 B16 B26 B66 D16 D26 D66 0 0 000000 S11 S12 000000 S12 S22 % &' ( [E] o x o y γ o xy κx κy κxy γxz γyz , (9.9) where [E] is the stiffness matrix for a plate element with shear deformations. Frequently, the behavior of shells can be described by replacing the curved surface with small, flat elements. The stiffness matrices above are applicable to such flat shell elements. 9.3 Beam Element For a beam element, the force–strain relationships are arbitrary layup, no shear deformation, no restrained warping (Eq. 6.2) N My Mz T = P11 P12 P13 P14 P12 P22 P23 P24 P13 P23 P33 P34 P14 P24 P34 P44 % &' ( [E] o x 1 ρy 1 ρz ϑ (9.10)
398 FINITE ELEMENT ANALYSIS orthotropic,no shear deformation,no restrained warping (Eq.6.8) N EA 0 0 0 就介 0 Elyy 0 0 (9.11) 0 0 0 0 [间 orthotropic,no shear deformation,restrained warping(Eqs.6.8 and 6.233) N EA 0 0 0 0 0 Elyy Ely: 0 0 0 Elyz El. 0 0 1101 (9.12) 0 0 0 0 熙 0 0 0 0 29 [可 orthotropic,shear deformation,restrained warping (Eqs.7.30,7.32,7.34,7.36) N EA 0 0 0 0 0 0 0 e 0 Elyy Elyz 0 0 0 0 0 dx: d 0 Elye El 0 0 0 0 0 dxy d 0 0 EL 0 0 0 0 doB dx 0 0 GI 0 0 0 29 0 0 0 0 Snt Yy 0 0 0 0 0 0 0 0 0 0 5 (E] (9.13) where [E]is the stiffness matrix for a beam element. 9.4 Sublaminate A laminate consisting of several plies may be analyzed by either plate(flat shell) or three-dimensional elements (Fig.9.2).For thick laminates neither of these Figure 9.2:Thick laminate (top),analysis with plate elements (left).analysis with three-dimensional elements(right)
398 FINITE ELEMENT ANALYSIS orthotropic, no shear deformation, no restrained warping (Eq. 6.8) N My Mz T = EA 0 00 0 EI yy EI yz 0 0 EI yz EI zz 0 00 0 GI t % &' ( [E] o x 1 ρy 1 ρz ϑ (9.11) orthotropic, no shear deformation, restrained warping (Eqs. 6.8 and 6.233) N My Mz Mω T sv = EA 0 0 00 0 EI yy EI yz 0 0 0 EI yz EI zz 0 0 00 0 EI ω 0 00 00 GI t % &' ( [E] o x 1 ρy 1 ρz −dϑ dx ϑ (9.12) orthotropic, shear deformation, restrained warping (Eqs. 7.30, 7.32, 7.34, 7.36) N My Mz Mω T sv V y V z T ω = EA 0 0 0 0000 0 EI yy EI yz 0 0000 0 EI yz EI zz 0 0000 00 0 EI ω 0000 00 00 GI t 000 00 0 00 Syy Syz Syω 00 0 00 Syz Szz Szω 00 0 00 Syω Szω Sωω % &' ( [E] o x −dχz dx −dχy dx −dϑB dx ϑ γy γz ϑS , (9.13) where [E] is the stiffness matrix for a beam element. 9.4 Sublaminate A laminate consisting of several plies may be analyzed by either plate (flat shell) or three-dimensional elements (Fig. 9.2). For thick laminates neither of these Figure 9.2: Thick laminate (top), analysis with plate elements (left), analysis with three-dimensional elements (right).
9.4 SUBLAMINATE 399 Laminate Sublaminates FE Mesh plies sublaminates Figure 9.3:Thick laminate(left),sublaminates(middle),and the finite element mesh(right). is practical.Plate elements give inaccurate results.Three-dimensional elements require that the material be uniform throughout the element,and,hence,an ele- ment must contain a single layer or adjacent identical layers.This may result in a very large number of elements,making the numerical computation difficult and often infeasible. We can overcome these difficulties by dividing the laminate into sublaminates (Fig.9.3).Each layer in the sublaminate may be monoclinic,orthotropic,trans- versely isotropic,or isotropic.The thickness of each element is the same as the thickness of the corresponding sublaminate.The stiffness matrix [E]of such a sublaminate is defined by the relationship Ex =[ (9.14) 7y2 气xy The bar denotes average stresses and strains.It is convenient to represent this expression in terms of the compliance matrix [ J11 J12 J13 J14 J15 J16] x Jy J24 J25 J26 J31 J32 /33 J54 J35 J36 J41 J J 5 J4 (9.15) J51 J52 J53 J54 J55 56 J62 163 J64 165 J66 元y where [=]-1 (9.16)
9.4 SUBLAMINATE 399 Laminate Sublaminates FE Mesh plies sublaminates Figure 9.3: Thick laminate (left), sublaminates (middle), and the finite element mesh (right). is practical. Plate elements give inaccurate results. Three-dimensional elements require that the material be uniform throughout the element, and, hence, an element must contain a single layer or adjacent identical layers. This may result in a very large number of elements, making the numerical computation difficult and often infeasible. We can overcome these difficulties by dividing the laminate into sublaminates (Fig. 9.3). Each layer in the sublaminate may be monoclinic, orthotropic, transversely isotropic, or isotropic. The thickness of each element is the same as the thickness of the corresponding sublaminate. The stiffness matrix [E] of such a sublaminate is defined by the relationship σ x σ y σ z τ yz τ xz τ xy = [E] x y z γ yz γ xz γ xy . (9.14) The bar denotes average stresses and strains. It is convenient to represent this expression in terms of the compliance matrix [J ] x y z γ yz γ xz γ xy = J11 J12 J13 J14 J15 J16 J21 J22 J23 J24 J25 J26 J31 J32 J33 J34 J35 J36 J41 J42 J43 J44 J45 J46 J51 J52 J53 J54 J55 J56 J61 J62 J63 J64 J65 J66 % &' ( [J ] σ x σ y σ z τ yz τ xz τ xy , (9.15) where [E] = [J ] −1 . (9.16)