Analysis of Non-Rectangular Laminated Anisotropic Plates by Chebyshev Collocation Method

Transcription

1 146 Analysis of Non-Rectangular Laminated Anisotropic Plates by Chebyshev Collocation Method Chih-Hsun LIN and Ming-Hwa R. JEN The purpose of this work is to solve the governing differential equations of arbitrarily non-rectangular laminated anisotropic plates by using the Chebyshev collocation method. The four sides of the plate herein are not restricted to be straight lines and they can be curves as well. Meanwhile these four sides can be expressed in four mathematical functions. The transformation from non-rectangular boundary into rectangular type is the key point to the solution of this kind of problems. In general the research on laminated anisotropic plates is almost focused on the case of rectangular plate. It is difficult to handle the laminated anisotropic plate problems with non-rectangular borders any kind of stacking sequence and the variety of boundary conditions. However through the merits of Chebyshev collocation method such problems can be overcome as stated as follows. Two cases in EXAMPLES section are illustrated to highlight the displacements stress resultants and moment resultants of our proposed work. The preciseness is also found in comparison with the numerical results by using finite element method incorporated with the software of NASTRAN. Key Words: Anisotropic Chebyshev Collocation Method Gauss-Lobatto Laminated Composite Plate Non-Rectangular 1. Introduction Geometry material properties boundary conditions are three basic elements to construct these plate problems. Due to the complicated material properties to solve the governing differentialequationsofalaminatedanisotropic plate with the specified boundary conditions is a difficult problem. The familiar methods of Navier and Levy s solutions for boundary value problems are frequently cited in some cases of plate problems 1 3. However the semiinverse method can also be adopted. Let the solutions first be a set of Fourier series including triangular series with some unknown constants. Meanwhile they should be satisfied with the expressions of the boundary conditions. Substituting them into governing equations then the unknown constants can be determined. That is a traditional way to solve such problems. In the solving procedure two disadvantages are found as: a it is difficult to Received 4th April 003 No Dept. of Information Technology Toko University No.51 Sec. University Rd. Pu Tzu City Chai-Yi County Taiwan 603 R. O. C. ChLin@mail.toko.edu.tw Dept. of Mechanical and Electro-Mechanical Engineering National Sun Yat-Sen University Kaohsiung Taiwan 8044 R. O. C. jmhr@mail.nsysu.edu.tw achieve the theoretical solution of a plate with complicated boundary conditions and b the unknown constants can t be obtained by substituting the guess solution into governing equations with the material properties not of a simple form. It is the reason why Chebyshev collocation method is proposed herein to accomplish the composite laminated plate problem with the complexity of material properties loading and boundary conditions. Lekhnitskii 4 published early a plenty of remarkable and valuable work on the opening in composite plate subjected to bending and concentrated loads. Most cases are limited to orthotropic materials and simply supported condition. Ambartsumyan 5 also solved isotropic and orthotropic rectangular plates due to various loads. Similarly most problems are bending of simply-supported two-layer and multi-layer orthotropic plates. From the survey of literature there exists little available method about solving the problem of governing differential equations of complicated material properties with the variety of boundary conditions due to general loads. That is attributed to two main reasons: a the inhomogeneity and anisotropy make the governing equations complicated and b the difficulty of finding suitable functions to satisfy simultaneously both the governing equations and boundary conditions. Most geometric shapes of related Series A Vol. 47 No. 004

2 147 problems that can be solved are limited to circular and rectangular i.e. regular shapes. Herein a methodology of Chebyshev collocation method is proposed to analyze non-rectangular anisotropic laminated plates of any stacking sequence with any type of boundary conditions such as simply supported fixed and free ends. Transforming a non-rectangular plate into a rectangular one mathematically the non-rectangular plate problems can be handled as usual. Nevertheless two drawbacks of our proposed method are: a it is suitable for any type of loadings except the concentrated load and b it is applicable only to the thin plate because the thick laminated plate is another problem. The Chebyshev polynominals formulation for laminated composite plate collocation method and the transformation of boundary conditions will be presented later. The displacements stress resultants and moment resultants will be highlighted in the example of two cases.. List of Symbols A 11 A 1 A 66 : extensional stiffness B 11 B 1 B 66 : coupling stiffness D 11 D 1 D 66 : bending stiffness E 1 E G 1 ν 1 : mechanical properties of a lamina with unidirectional fibers Ĕ F Ğ : first fundamental magnitudes of shell theory L M N : second fundamental magnitudes of shell theory N 1 N N 1 : stress resultants M 1 M M 1 : moment resultants Q 1 Q 1 : shear stress resultants in α 1 and α directions Q 11 Q 1 Q 66 : reduced stiffness r : position vector R 1 R : radii in α 1 and α directions T 1 T 1 V 1 V :Kirchoff s effective shear stress resultants T n x :thenth order Chebyshev polynominal ˆT n x : the modified nth order Chebyshev polynominal u 1 u w : displacements in α 1 α and normal to the surface directions β 1 β : rotations tangential to the reference surface ε 0 1 ε0 γ0 1 : strains at the laminate geometry mid-plane κ 1 κ κ 1 : curvatures of the laminate 3. Chebyshev Polynominals The nth order Chebyshev polynomial 6 is expressed as T n x = cosnθ x = cosθ 1 x 1 1 where n is a non-negative integer. By the trigonometric relation there exists cosn+1θ+cosn 1θ = cosnθ and the recurrence equations can be generated as T n+1 x = xt n x T n 1 x T 0 x = 1 T 1 x = x. 3 From Eq. 3 the first few Chebyshev polynomials are represented as T x = x 1 T 3 x = 4x 3 3x T 4 x = 8x 4 8x +1 T 5 x = 16x 5 0x 3 +5x 4 T n x = 1 { ] n 1 n x n x n 1 1 ] } n n 3 + x n 4 n 1 n 1 n 1! n where = 1 n 1 1!1! n! = 1 n 1!1! n n! = n!! etc. Using the weighting function 1 x 1 the orthogonality condition can be presented as 0 m n 1 T m xt n x π dx= m = n x 1 π m = n = 0 4. Formulation for Laminated Anisotropic Plate The strain-displacement relations and equilibrium equations to shell problems can be obtained by the method cited in Ref. 7. Even though the formulas are derived for thin shells they are also applicable to thin plates. Many formulas in Ref. 7 are quoted in this section and only the final results are listed for simplicity. First of all the position vector equation of the parametric curves of surface can be represented as rα 1 α = r 1 α 1 α i+r α 1 α j+r 3 α 1 α k 6 where α 1 and α are the curvilinear coordinates of the surface. The first fundamental magnitudes are Ĕ = r 1 r 1 F = r 1 r 7 Ğ = r r where r i = r i = 1. Take the square roots of the first α i and the last equations in Eq. 7 to obtain A 1 = Ĕ 8 A = Ğ. The unit normal vector is defined as nα 1 α = r 1 r r 1 r. 9 The second fundamental magnitudes are L = r 1 n 1 Series A Vol. 47 No. 004

3 148 M = r 1 n + r n 1 10 N = r n where n i = n i = 1. The curvatures of the surface are α i K 1 = 1 = L R 1 Ĕ K = 1 = N 11 R Ğ. where R 1 and R are the radii in the directions of α 1 and α respectively. According to Love s first approximation to the theory of thin elastic shells 8 there are four postulates as follows: a the shell is thin b the deflections of the shell are small c the transverse normal stress is negligible and d normals to the reference surface of the shell remain normal to it and undergo no change in length during deformation. Therefore the strain-displacement relations in a thin elastic shell 7 are given by ε 1 = ε 0 1 +zκ 1 ε = ε 0 +zκ 1 γ 1 = γ 0 1 +zκ 1 where ε 0 1 = 1 u 1 + u A 1 + w A 1 A 1 A R 1 ε 0 = 1 u + u 1 A A A 1 A γ 1 0 = A A 1 u A + w R + A 1 A κ 1 = 1 β 1 + β A 1 A 1 A 1 A κ = 1 β + β 1 A A A 1 A κ 1 = A A 1 β + A 1 A u1 β1 A A A 1 The quantities of β 1 and β in Eq. 13 are the rotations tangential to the reference surface oriented along the parametric lines α 1 and α theyare β 1 = u 1 1 w R 1 A 1 β = u 1 14 w. R A The following equilibrium equations are derived by Hamiltons s principle 7 some terms in the equations are omitted in the consideration of a static case. N 1 A + N 1A 1 A 1 A Q 1 + N 1 N + A 1 A = 0 R 1 N 1 A + N A 1 A A 1 Q + N 1 N 1 + A 1 A = 0 R Q 1 A + Q A 1 x N1 + N A 1 A q n A 1 A = 0 R 1 R Series A Vol. 47 No. 004 M 1 A + M 1A 1 A 1 A + M 1 M Q 1 A 1 A = 0 M 1 A + M A 1 A A 1 + M 1 M 1 Q A 1 A = Because of the symmetry of stress tensor i.e. τ 1 = τ 1 and the characteristics of thin shell N 1 = N 1 and M 1 = M 1 in Eq. 15 can be obtained. The natural boundary conditions are: Along the edge of constant α 1 : N 1 = N 1 or u 1 = u 1 T 1 = T 1 or u = u 16.a V 1 = V 1 or w = w M 1 = M 1 or β 1 = β 1. Along the edge of constant α : N = N or u = u T 1 = T 1 or u 1 = u 1 16.b V = V or w = w M = M or β = β where T nt = N nt + M nt R t V n = Q n + 1 nt = M nt A t α t The symbols of n and t denote normal and tangential directions on a designated boundary edge. Now let position vector of a plate problem be rα 1 α = α 1 i+α j+0 k. 18 Substituting Eq. 18 into Eqs and manipulating those results provide A 1 = 1 A = 1 R 1 = R =. 19 Substituting Eq. 19 into Eqs. 14 and 13 it yields β 1 = w 1 β = w 0 and ε 0 1 = u 11 ε 0 = u γ 0 1 = u 1 +u 1 1 κ 1 = w 11 κ = w κ 1 = w 1. Again substituting Eq. 19 into the last two equations of Eq. 15 and after rearrangement it becomes Q 1 = M 11 + M 1 Q = M 11 + M. The substitution of Eqs. 19 and into the first three equations of Eq. 15 yields the simplified equilibrium equations as N 11 + N 1 = 0 N 11 + N = 0 3 M 111 +M 11 + M = q n. Finally substituting Eqs. 19 and into Eq. 17 pro-

4 149 vides T 1 = N 1 T 1 = N 1 4 V 1 = M 11 +M 1 V = M +M 11. Next the relations of stress resultants moment resultants and displacements for a laminated anisotropic plate 9 are 0 N 1 A 11 A 1 A 16 B 11 B 1 B 16 ε 1 0 N A 1 A A 6 B 1 B B 6 ε 0 N 1 A = 16 A 6 A 66 B 16 B 6 B 66 γ 1 M 1 B 11 B 1 B 16 D 11 D 1 D 16 κ 1 M B M 1 1 B B 6 D 1 D D 6 κ B 16 B 6 B 66 D 16 D 6 D 66 κ 1 5 where A ij B ij D ij = h/ h/ Q ij 1zz dz i j = Now substitute Eq. 1 into Eq. 5 to generate six equations. For space saving the abbreviation of those results can be expressed as N 1 = A 11 u 11 + A 1 u + B 16 w 1 N = A 1 u 11 + A u + B 6 w 1 N 1 = A 16 u 11 + A 6 u + B 66 w 1 7 M 1 = B 11 u 11 + B 1 u + D 16 w 1 M = B 1 u 11 + B u + D 6 w 1 M 1 = B 16 u 11 + B 6 u + D 66 w 1. Moreover the substitution of Eq. 7 into Eq. 3 yields A 11 u A 1 u 1 + A 16 u 11 + A 6 u + B 16 w 11 B 66 w 1 = 0 8.a A 16 u A 6 u 1 + A 1 u 11 + A u + B 66 w 11 B 6 w 1 = 0 8.b B 11 u B 1 u 11 +B 16 u 111 +B 6 u 1 + B 1 u 11 + B u + D 16 w 111 4D 66 w 11 D 6 w 1 = q n. 8.c To solve the systemof partial differential equations of Eq. 8.a b c Chebyshev collocation method is considered to be one of the best choices. 5. Boundary Conditions The edges of a non-rectangular plate are not guaranteed to be rectangular to the directions of α 1 or α. Hence the boundary conditions must be reconsidered further. By referring to Fig. 1 the rotation angles θ 1 θ ϑ 1 and ϑ are illustrated as df1 α df α θ 1 = arctan θ = arctan dα dα dg1 α 1 dg α 1 ϑ 1 = arctan ϑ = arctan. dα 1 dα 1 9 By the transformation of rotation axes the relations can α 1 Fig. 1 Contour of a non-rectangular plate be expressed as α 1 = α 1 cosθ α sinθ α = α 1 sinθ+α cosθ. 30 By the chain rule for differentiation it provides = cosθ +sinθ α = sinθ +cosθ. 31 Referring to Ref. 5 the displacements stress and moment resultants with respective to the boundary edges α 1 = f 1 α andα 1 = f α can be expressed as follows: u 1a = u 1 cosθ i u 1 sinθ i u a = u 1 sinθ i +u 1 cosθ i N 1a = N 1 cos θ i + N sin θ i +N 1 sinθ i cosθ i T 1a = N 1a = N N 1 sinθ i cosθ i + N 1 cos θ i sin θ i 3 M 1a = M 1 cos θ i + M sin θ i +M 1 sinθ i cosθ i M 1a = M M 1 sinθ i cosθ i + M 1 cos θ i sin θ i where i = 1 and the sub-index a denotes α 1 referring to functions f i. Referring to Eqs. 0 4 and 31 it yields β 1a = cosθ i β 1a sinθ i β 1a Q 1a = cosθ i M 1a sinθ i M 1a V 1a = cosθ i M 1a +sinθ i M 1a +cosθ i M 1a 33 +sinθ i M 1a sinθ i M 1a +cosθ i M 1a where i = 1. Similarly the displacements stress and moment resultants with respective to the boundary edges α = g 1 α 1 Series A Vol. 47 No. 004

5 150 and α = g α 1 are u 1b = u 1 cosϑ i u 1 sinϑ i u b = u 1 sinϑ i +u 1 cosϑ i N b = N 1 sin ϑ i + N cos ϑ i N 1 sinϑ i cosϑ i T 1b = N 1b = N N 1 sinϑ i cosϑ i + N 1 cos ϑ i sin ϑ i M b =M 1 sin ϑ i + M cos ϑ i M 1 sinϑ i cosϑ i M 1b = M M 1 sinϑ i cosϑ i + M 1 cos ϑ i sin ϑ i β b = cosϑ i w sinϑ i w Q b = cosϑ i M 1b sinϑ i M b V b = cosϑ i M 1b +sinϑ i M 1b +cosϑ i M b +sinϑ i M 1b sinϑ i M 1a +cosϑ i M b. 34 where i = 1 and the sub-index b denotes α referring to functions g i. 6. Chebyshev Collocation Method Consider a non-rectangular plate with four boundaries combined as f α f α ] g 1 α 1 g α 1 ] and it implies that f 1 α α 1 f α 35 g 1 α 1 α g α 1. In order to match the characteristics of Chebyshev polynomials in Eq. 1 let 1 η η 1 where 1 { η 1 = α1 f α + f 1 α 1 ]} f α f 1 α 1 1 { η = α g α +g 1 α 1 ]} 37. g α g 1 α 1 Rearrange Eq. 37 to have α 1 = f α f 1 α 1 α = g α g 1 α 1 η 1 + f α + f 1 α 1 η + g 38 α +g 1 α 1. Now separate the equilibrium Eq. 8.a b c with the boundary conditions of Eqs into three groups of the problem as follows: Group 1: Eq. 8.a and N 1a = N 1a or u 1a = u 1a 39.a on the edges of α 1 = f 1 α andα 1 = f α T 1b = T 1b or u 1b = u 1b. 39.b on the edge of α = g 1 α 1 andα = g α 1 Series A Vol. 47 No. 004 Group : Eq. 8.b and T 1a = T 1a or u a = u a 40.a on the edges of α 1 = f 1 α andα 1 = f α N b = N b or u b = u b. 40.b on the edge of α = g 1 α 1 andα = g α 1 Group 3: Eq. 8.c and V 1a = V 1a or w = w M 1a = M 1a or β 1a = β 1a 41.a on the edges of α 1 = f 1 α andα 1 = f α V b = V b or w = w M b = M b or β b = β b. 41.b on the edge of α = g 1 α 1 andα = g α 1 Note: In Group 1 and 3 means the quantity of on the relative boundary edge. Let the solutions to the system of partial differential equations be u 1 η 1 η = M N a mn T m η 1 T n η u η 1 η = M wη 1 η = M N N b mn T m η 1 T n η 4 c mn T m η 1 T n η where a mn b mn and c mn are unknown constants. Substituting Eq. 37 into Eq. 4 to replace the independent variables η 1 and η by α 1 and α wehave u 1 α 1 α = M N a mn ˆT m α 1 ˆT n α u α 1 α = M wα 1 α = M N N b mn ˆT m α 1 ˆT n α 43 c mn ˆT m α 1 ˆT n α where ˆT m α 1 = T m η 1 and ˆT n α = T n η. Let the Chebyshev-extrema points 10 i.e. also named Gauss-Lobatto points be πi ˆη i = cos i = 01 M M π j 44 η j = cos j = 01 N. N In order to transform the coordinates ˆη i η j back to the original coordinates ˆα i α j substitute Eq. 44 into Eq. 38 to yield ˆα i = f ˆα j f 1 ˆα i ˆη i + f ˆα j + f 1 ˆα i α j = g ˆα j g 1 ˆα i η j + g 45 ˆα j +g 1 ˆα i. Eq. 45 is a system of equations with two variables ˆα i and α j that the solutions of ˆα i α j can be obtained easily. Although there are many sets of solutions of Eq. 45 just

6 151 only one set is reasonable i.e. ˆα i α j is located within the region of the border of the plate. By substituting ˆα i α j into Eq. 35 it is easy to distinguish which one of the solutions is correct. For example if a plate is subjected to the transverse load q n with four edges simply supported the boundary conditions are { u1a = 0 α Group 1 : 1 = f 1 α α 1 = f α u 1b = 0. α = g 1 α 1 α = g α 1 46.a { u = 0 α Group : 1 = f 1 α α 1 = f α 46.b u = 0. α = g 1 α 1 α = g α 1 { w = 0 M1 = 0 α Group 3 : 1 = f 1 α α 1 = f α w = 0 M = 0. α = g 1 α 1 α = g α 1 46.c By collocation method 11 substituting Eq. 43 into Eq. 8.a and Eqs. 46.a b c and then substituting ˆα i α j i = 01...M j = 01...N into the results it yields Group 1: M N { amn A11 ˆT m ˆα i ˆT n α j + ] +b mn A1 ˆT m 1 ˆα i ˆT n 1 α j + ]+ c mn B16 ˆT m ˆα i ˆT n 1 α + ]} j = 0 i = 1 M 1 j = 01 N a u 1a ˆα 0 α j = M N a mn ˆT m ˆα 0 ˆT n α j = 0 j = 01 N. u 1a ˆα M α j = M N a mn ˆT m ˆα M ˆT n α j = 0 j = 01 N. u 1b ˆα i α 0 = M N a mn ˆT m ˆα i ˆT n α 0 = 0 i = 1 M 1. u 1b ˆα i α N = M N a mn ˆT m ˆα i ˆT n α N = 0 i = 1 M 1. Group : M N { amn A16 ˆT m ˆα i ˆT n α j + ] 47.b 47.c 47.d 47.e +b mn A6 ˆT m 1 ˆα i ˆT n 1 α j + ]+ c mn B66 ˆT m ˆα i ˆT n 1 α + ]} j = 0 i = 1 M 1 j = 1 N a u a ˆα 0 α j = M N b mn ˆT m ˆα 0 ˆT n α j = 0 j = 01 N. u a ˆα M α j = M N j = 01 N. b mn ˆT m ˆα M ˆT n α j = 0 48.b 48.c u b ˆα i α 0 = M N b mn ˆT m ˆα i ˆT n α 0 = 0 i = 1 M 1. u b ˆα i α N = M N b mn ˆT m ˆα i ˆT n α N = 0 i = 1 M 1. Group 3: M N { amn B11 ˆT m 3 ˆα i ˆT n α j + ] 48.d 48.e +b mn B1 ˆT m ˆα i ˆT n 1 α j + ]+ c mn D16 ˆT m 3 ˆα i ˆT n 1 α + ]} j = q n i = 3 M j = 3 N. 49.a wˆα 0 α j = M N c mn ˆT m ˆα 0 ˆT n α j = 0 j = 01 N. wˆα M α j = M N c mn ˆT m ˆα M ˆT n α j = 0 j = 01 N. wˆα i α 0 = M N c mn ˆT m ˆα i ˆT n α 0 = 0 i = 1 M 1. wˆα i α N = M N c mn ˆT m ˆα i ˆT n α N = 0 49.b 49.c 49.d i = 1 M e M 1a ˆα 0 α j = M N { amn B11 ˆT m 1 ˆα 0 ˆT n α j + ] +b mn B1 ˆT m ˆα 0 ˆT n 1 α j + ]+ c mn D16 ˆT m 1 ˆα 0 ˆT n 1 α + ]} j = 0 j = 1 N f M 1a ˆα M α j = M N { amn B11 ˆT m 1 ˆα M ˆT n α j + ] +b mn B1 ˆT m ˆα M ˆT n 1 α j + ]+ c mn D16 ˆT m 1 ˆα M ˆT n 1 α + ]} j = 0 j = 1 N g M ˆα i α 0 = M N { amn B1 ˆT m 1 ˆα i ˆT n α 0 + ] +b mn B ˆT m ˆα i ˆT n 1 α 0 + ]+ c mn D6 ˆT m 1 ˆα i ˆT n 1 α + ]} 0 = 0 i = 3 N. 49.h M ˆα i α N = M N { amn B1 ˆT m 1 ˆα i ˆT n α N + ] +b mn B ˆT m ˆα i ˆT n 1 α N + ]+ c mn D6 ˆT m 1 ˆα i ˆT n 1 α + ]} N = 0 i = 3 N. 49.i Series A Vol. 47 No. 004

7 15 In Eqs and 49 ˆT m i α 1 and ˆT n i α imply d i ˆT m α 1 and di ˆT n α respectively. dα i 1 dα i Both of the numbers of the total equations in three groups and all the unknown constants a mn b mn c mn are 3M + 1N + 1. Through this the unique solution of the unknown constants a mn b mn c mn can be received. Other problems with any different boundary conditions can be solved in the same way. 7. Examples Case 1: Consider a four-layered cross-ply laminated plate which is subjected to a uniformly distributed load q n = 0.1Pa and the four edges of the plate are clamped. The contour of the plate is shown in Fig.. Material: Graphite/Epoxy T300/508. Mechanical properties of a lamina with unidirectional fibers: E 1 = 181Gpa E = 10.3Gpa 50 G 1 = 7.17Gpaν 1 = 0.8. Thickness of each layer: 0.15 mm. Stacking sequence: 0/90/90/0]. Dimension: 1 α +] 11]m. i.e. 1 α 1 α + and 1 α 1 For clamped edges the boundary conditions are a For α 1 = 1andα 1 = α +: u 1a = 0 u a = 0 w= 0 β 1a = For α = 1andα = 1: u b = 0 u 1b = 0 w= 0 β b = 0. 5 Solution procedure: 1. By using the method of Ref. 9 calculate A ij B ij and D ij i j = 1.6 in Eq. 6 and then input the results into Eq. 8.a.. Substitute Eq. 43 into Eq. 8.a and boundary conditions of Eqs. 51 and Separate these equations into three groups as previously described in Eqs and After the manipulation by Chebyshev collocation method as above-mentioned the unknown constants a mn b mn c mn in Eq. 43 will be obtained. 5. Substitute a mn b mn and c mn into Eq. 43 to receive u 1 u and w. 6. Substitute u 1 u and w into Eq. 7 to obtain the stress and moment resultants. In the case the numerical results of u 1 u N 1 N and N 1 are zeros. The numerical results of w M 1 M and M 1 are listed in tables for reference. At the same time the numerical results obtained by finite element method incorporated with NASTRAN software are demonstrated to validate the correctness of the proposed method of Chebyshev polynomials. The meshes as shown in Fig. 3 are adopted for the finite element method in this case. For M = N = 1 in Eq. 43 the numerical results of w M 1 M and M 1 are listed in Tables 1 4 and depicted in Figs. 4 7 respectively. Case : Consider a four-layered cross-ply laminated plate which is subjected to a uniformly distributed load q n = 0.1 Pa and the four edges of the plate are clamped. The contour of the plate is shown in Fig. 8. Material: Graphite/Epoxy T300/508. Dimension: 1 α +] 11]m. i.e. 1 α 1 α + and 1 α 1 The mechanical properties thickness of each layer and the stacking sequence are the same as those in Case 1. b Fig. a Contour of the plate in Case 1 b Loading q n = 0.1PainCase1 Fig. 3 Meshes of FEM NASTRAN of the plate in Case 1 Series A Vol. 47 No. 004

8 153 Table 1 The displacements of w at some selected position points with u 1 = 0andu = 0inCase1 Table 3 The moment of resultant M at some selected position points in Case 1 Table 4 The moment of resultant M 1 at some selected position points in Case 1 Table The moment of resultant M 1 at some selected position points in Case 1 For the clamped edge the boundary conditions are For α 1 = 1andα 1 = α +: u 1a = 0 u a = 0 w= 0 β 1a = For α = 1andα = 1: u 1b = 0 u b = 0 w= 0 β 1b = The solution procedure is also similar to that in Case 1. The meshes of this case are shown in Fig. 9. For M = N = 1 in Eq. 43 the numerical results of w M 1 M and M 1 are depicted in Figs respectively. Series A Vol. 47 No. 004

9 154 Fig. 4 Displacement of w in Case 1 a Fig. 5 Moment resultant M 1 in Case 1 b Fig. 8 a Contour of the plate in Case b Loading q n = 0.1PainCase Fig. 6 Moment resultant M in Case 1 Fig. 9 Meshes of FEM NASTRAN of the plate in Case Fig. 7 Moment resultant M 1 in Case 1 Comparing the results M = N = 1 in Eq. 43 with those obtained by NASTRAN all the errors of the results are very small. For space saving the numerical result lists are no listed. 8. Discussion Through the features of formulation and the whole procedure as stated Chebyshev collocation method is found to be capable of solving the problems of nonrectangular and any stacking sequence laminated composite plates with complicated boundary conditions. The solutions to most problems in this field of anisotropic laminated plates can be accomplished by our proposed method however there are two groups of problems which still left unsolved. They are a the problem of anisotropic laminated plate subjected to concentrated loading and b the problem of thick laminated plate. In group a it is hard to handle the problem if the position of a concentrated load is not directly located at one of the collocation points. From our point of view that may be overcome by selecting larger values of M and N e.g.m and N are 1 in the two cases as illustrated. The larger values of M and N will result in Series A Vol. 47 No. 004

10 155 Fig. 10 Displacement of w in Case much more collocation points that make every collocation point close to the adjacent collocations points. Hence the collocation point will match the position of a concentrated load asymptotically. Next in group b the stresses σ z γ xz and γ yz z-axis is in the direction normal to the middle surface are neglected owing to the thin plate theory i.e. generalized plane stress problem. Nevertheless the terms of σ z γ xz and γ yz interlaminar stresses should be considered in thick laminated plate for the further research. From the detailed demonstration as shown in Example we find that the numerical results by Chebyshev collocation method are mostly close to those by NASTRAN in cases 1 and. So far as we know NASTRAN is a precise and reliable finite element analysis software that has been widely used in many fields such as science physics and engineering. It is found that the errors expressed in parentheses in Tables are satisfactorily acceptable that implies our proposed Chebyshev collocation method is a correct and acceptable one even if some large errors occur at the most irregular region. Thus the transformation of boundary conditions at that region is thought of as the origin of large errors. The situation may be improved by enlarging M and N values just like fine meshes in finite element analysis. 9. Conclusion Fig. 11 Moment resultant M 1 in Case Fig. 1 Moment resultant M in Case The methodology of Chebyshev collocation method is proposed herein to efficiently solve the problems of non-rectangular laminated anisotropic thin plates with any boundary conditions due to any type of loadings except the concentrated load. The method possesses two merits: a it is more efficient and applicable than Fourier series method to handle the problems of complicated material properties and boundary conditions and b the results of the method are expressed in a group of functions that is more useful than those of finite difference and finite element methods. Most part of the analytical solution by Chebyshev collocation method is very close to the data obtained by NASTRAN except the irregular region which still within acceptable errors. Thus our proposed method is a reliable and precise one that is capable of solving complicated problems powerfully. References Fig. 13 Moment resultant M 1 in Case 1 Reissner E. and Stavsky Y. Bending and Stretching of Certain Type of Heterogeneous Aeolotropic Elastic Plates J. Appl. Mech. Vol pp Whitney J.M. and Leissa A.W. Analysis of a Simply Supported Laminated Anisotropic Rectangular Plate AIAA J. Vol.8 No pp Whitney J.M. Structural Analysis of Laminated Anisotropic Plates 1987 Technomic Publishing Company Inc. 4 Lekhnitskii S.G. Anisotropic Plates 1986 Gordon and Breach Science Publishers N.Y. 5 Ambartsumyan S.A. Theory of Anisotropic Plates 1970 Technomic Publishing Company Inc. 6 Fox L. and Parker I.B. Chebyshev Polynomials in Numerical Analysis 1968 Oxford University Press. 7 Kraus H. Thin Elastic Shells: An Introduction to the Theoretical Foundations and the Analysis of Their Static and Dynamic Behavior 197 John Wiley & Sons Inc. Series A Vol. 47 No. 004

11 156 8 Love A.E.H. A Treatise on the Mathematical Theory of Elasticity 1944 Dover Publications. 9 Jones R.M. Mechanics of Composite Materials 1975 Scripta Book Company. 10 Boyd J.P. Chebyshev & Fourier Spectral Methods 1989 Springer-Verlag. 11 Karageorghis A. A Note on the Satisfaction of the Boundary Conditions for Chebyshev Collocation Methods in Rectangular Domains J. Sci. Comput. Vol.6 No pp.1 6. Series A Vol. 47 No. 004