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Analysis of arbitrary laminated composite beams by Chebyshev series

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https://www.eduzhai.net International Journal of Composite M aterials 2012, 2(5): 72-78 DOI: 10.5923/j.cmaterials.20120205.02 Analysis of Arbitrarily Laminated Composite Beams Using Chebyshev Series A. Okasha El-Nady1,*, Hani M. Negm2 1M echatronics Dept., Faculty of Engineering, October 6 University 2Aerospace Eng. Dept., Cairo University Abstract In this work a simp le technique for the analysis of arbitrarily laminated composite beams is proposed using a higher-order shear deformation theory. The governing equations are derived by minimizing the total potential energy of arbitrarily laminated beams undergoing axial and transverse shear strains under laterally distributed load. The d isplacement and rotation of the beam center line are expanded in Chebyshev series. Using a standard procedure the governing equations are cast in matrix fo rm, which is easily handled by electronic computers. The displacements and stresses of several laminated beams are calculated and compared with published results. Keywords Higher-Order Theory, Co mposite Beams, Chebeyshev Series 1. Introduction Beam structures are among the most important structures in aerospace applications. Multilayered co mposites have gained wide applicat ion in aerospace industry due to their high strength-to-weight and stiffness-to-weight ratios. Conventional analysis of beams uses the classical beam theory based on Bernoulli-Eu ler hypotheses[1], and hence, neglects shear deformation. This theory adequately describes the behavior of slender beams, but is less adequate for thick beams in which shear deformations are important. Timoshenko[2] e xtended the classical theory to produce a first-order shear deformation theory. This is an imp rovement on the classical theory which reduces to it as the beam becomes thinner. A defect of Timoshenko theory is that the assumed displacement appro ximation violateas the "no-shear" boundary condition at the top and bottom of the beam. Levinson[3] introduced a higher-order theory to correct the drawback of Timoshenko's theory. It is based on a cubic in-pane displacement approximat ion that satisfies the no-shear condition. Bickford[4] noted that the derivation used by Levison was variationally inconsistent, and derived a corrected version fro m Hamilton's principle. In addit ion, he presented some representative solutions for simple bea ms. Hey ligher and Redd y [5] p resent ed a fin it e elemen t solut ion fo r Bickfo rd ’s th eo ry us ing po lyno mial sh ape functions. J. Petrolito[6] p resented a finite element solution * Corresponding author: rady_nady@yahoo.com (A. Okasha El-Nady) Published online at https://www.eduzhai.net Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved for isotropic beams based on a higher-order shear deformation theory. Solutions of the governing d ifferential equations are derived and used as element shape functions. For laminated beams, the classical lamination theory[7, 8, 9] is adequate to predict the global response of laminates with relatively s mall thickness. Because of the low shear to in-plane stiffness ratio, the important role of transverse shear deformation, wh ich is not contained in the classical lamination theory, cannot be neglected. S. Gopalakrishnan et al[10] derived a refined 2-node, 4-DOF co mposite beam element based on a higher-order shear deformation theory in asymmetrically stacked laminates. V. G. Mokos and E. J. Saountzakis[11] developed a boundary element method for the solution of the general transverse shear loading of composite beams o f arbitrary constant cross section. Exact solution for the bending of thin and thick cross-ply laminated beams was presented by Khedir and Reddy[12 and 13] using the state space concept. Exact solution for arbirarily-laminated beams based on a higher-order shear deformation theory was presented by A. Okasha[14]. The exact analytical solution is restricted to simple geometry and loading. For general analysis, it is preferable to use a numerica l approach. In practice, some care needs to be taken with numerical solutions to avoid difficult ies, such as locking with increasing beam aspect ratio. This is of major concern when using higher order theories for beams and plates[6, 10]. To avoid these difficult ies of exact and numerical solutions of differential equations based on the higher-order theory, approximate analytical solution in the form of Chebyshev series is proposed. In the present work the analysis of arbitrarily laminated composite beams is presented based on a higher-order shear deformation theory using Chebyshev series. The governing equations are derived by min imizing the total potential 73 International Journal of Composite M aterials 2012, 2(5): 72-78 energy of arbitrarily laminated beams undergoing axial and beam, b is the width and L is the length of the beam. Taking transverse shear strains under laterally distributed load. The into consideration that variation in the potential energy is due displacement of the beam w and rotation θ are expanded in to variation in the displacements and strains, then the first Chebyshev series. Using a standard procedure the governing variation of the potential energy, δ Π , can be written as: equations are cast in matrix form, wh ich is easily handled by h/2 b L L ∫ ∫ ∫ ∫ electronic co mputers. Using this method it is possible = to δ Π (σ x δε x +τ xz δγ xz ) dx dy dz − qδ w dx (5) analyze beam structures with s mall aspect ratios. The −h/2 0 0 0 displacements and stresses of several laminated beams are Substituting equations (1)-(3) into equation (5) and calculated and compared with published results. The effect integrating over the width and depth of the beam, equation (5) of ply stacking in symmetric and asymmetric laminated takes the form beams with d ifferent boundary conditions and aspect ratio= s is δ Π in v es tig ated . ∫L [EA ∂uo 0 ∂x ∂δ uo ∂x + (B1 + B2 ) ( ∂uo ∂x ∂δθ ∂x + ∂θ ∂x ∂δ uo ) ∂x 2. Mathematical Formulation 2.1. Kinematic Relati ons Assuming that the beam is subjected to lateral load only as shown in Fig. (1); the deformation of the beam is described + B2 ( ∂uo ∂x ∂2δ w + ∂2w ∂x2 ∂x2 ∂δ uo ∂x ) + EIθ ∂θ ∂x ∂δθ ∂x + EIθ w ( ∂θ ∂x ∂2δ w ∂x2 + ∂δθ ∂x ∂2w ∂x2 ) + EIw ∂2w ∂x2 ∂2δ w ∂x2 (6) ∫ + GA∗(θδθ +θ ∂δ w + δθ ∂w + ∂w ∂δ w)]dx − L qδ w dx ∂x ∂x ∂x ∂x 0 by two displacements, U and W, and a rotation, θ. These where EA is the axial stiffness, B1 and B2 are the displacements are assumed to be of the form[6, 10]: cross-coupling terms due to axial-flexu ral deformation of the U (x, y, z) =U (x, z) = u + zθ − 4 z3 3 h2 (θ + ∂w) ∂x composite laminated beam. EIθ , EIθw , EI w are the bending stiffnesses and GA∗ is the shear stiffness of the θ = θ (x) (1) laminated composite beam, and all are defined in Appendix W (x, y, z) = w(x) A. Integration by parts and equating to zero g ives the where h is the depth of the beam. 2.2. Strain-Displ acement Rel ati ons equilibriu m equations of arbitrarily laminated beam [EIθ wθ ' + EIw w"]" −[GA∗(θ + w' )]' + [B2 u' ]'' − q = 0 The beam is considered as a wide beam. So, the only non-zero strains are[6] ε x = ∂U ∂x = εxo + ∂θ z ∂x −4 3 z3 h2 ∂θ ( ∂x + ∂2w ∂x 2 ) γ xz = ∂U ∂z + ∂W ∂x = (1 − 4 z h 2 2 ) (θ + ∂w ) ∂x (2) 2.3. Stress-Strain Relations The laminate stresses are [EIθθ ' + EIθw w"]' − GA∗(θ + w' ) + [(B1 + B2 ) u' ]' (7) = 0 [B2 w'' + (B1 + B2 ) θ ' + EA u' ]' = 0 d where a prime denotes . The procedure also leads to the dx definit ion of the generalized forces used in expressing the beam boundary conditions F = − [EIθ wθ ' + EIw w"]' + GA∗(θ + w' ) − [B2 u' ]' σ x = Q11 ε x τ xz = Q 55 γ xz (3) M=1 EIθθ ' + EIθw w" + (B1 + B2 ) u' (8) where Q11 and Q55 are given in Appendix A. M 2 = EIθ wθ ' + EIw w" + B2 u' 3. Governing Equations 3.1. Differential Equati ons Minimizing the total potential energy of the beam can lead to the governing equations of static analysis of the beam. In the present case, the total potential energy, Π is ∫ ∫ ∫ ∫ Π = 1 h/ 2 b 2 −h / 2 0 L (σ x 0 εx +τ xz γ xz ) dx dy dz − L q w dx 0 (4) where q is the applied transverse load per unit length of the N= B2 w'' + (B1 + B2 ) θ ' + EA u' The force F can be interpreted as a generalized shear force, while M 1 and M 2 are generalized mo ments. Also, the force N can be interpreted as a generalized axial force. With these definitions, the appropriate boundary conditions for the beam are as fo llo ws: 1) either w or F is specified; 2) either θ or M 1 is specified; 3) either w' or M 2 is specified; A. Okasha El-Nady et al.: Analysis of Arbitrarily Laminated Composite Beams Using Chebyshev Series 74 4) either u or N is specified; expanding w(ξ), θ(ξ) and u(ξ) in (N+1)-term Chebyshev For most practical problems the properties of the beam are series we have a total of 3N+3 unknown coefficients. Using constant along the length of the beam. In this case, equations matrix formu lation for the functions and function derivatives (7) and (8) reduce to and applying the ru le of mat rix mu ltiplication as explained in EIw w''" − GA∗w'' + EIθwθ ''' − GA∗θ ' + B2 u''' = q reference[15], equations (9) can be written as a system of algebraic equations in the following matrix form: EIθw w'" − GA∗w' + EIθθ '' − GA∗θ + (B1 + B2 ) u'' =0(9) B2 w'" + (B1 + B2 ) θ '' + EA u'' = 0 and F = − EIw w"' + GA∗w' − EIθ wθ '' + GA∗ θ − B2 u'' = M1 EIθ w w" + EIθθ ' + (B1 + B2 ) u' (10) M 2 = EIw w" + EIθ wθ ' + B2 u' N = B2 w'' + (B1 + B2 ) θ ' + EA u' Therefore, the higher-o rder beam theory is represented by a system of ord inary differential equations of order six. 3.2. Boundary Conditions [ 256 EIw L4 [ A04] − 16GA∗ L2 [ A02] ] {wi} +[ 64 EIθ w L3 [ A03] − 4 GA∗ L [ A01] ] {θi } (15a) + [ 64 B2 L3 [ A03]] {ui} = {q j} [ 64 EIθ L3 w [ A03]− 4 GA∗ L [ A01] ] {wi} +[ 16 EIθ L2 [ A02] − GA∗[I ]] {θi } (15b ) + 16 [ (B1 + L2 B2 ) [ A02] ]{ui } = {0} [ 64 B2 L3 [ A03] ]{wi } + [ 16 (B1 + L2 B2 ) [ A02]] (15c) {θi } + [ 16 EA L2 [ A02] ]{ui } = {0} Fixed end w = 0; θ = 0; w' = 0; u = 0 (11) Hinged end at ξ = 0 w = 0; M 1 = 0; M 2 = 0; u = 0 (12) Roller end at ξ = 1 w = 0; M 1 = 0; M 2 = 0; N = 0 (13) Free end F = 0; M 1 = 0; M 2 = 0; N = 0 (14) 4. Solution of the Governing Equations The exact analytical solution is restricted to simple geometry and loading. The exact solution of equation (9) is limited as it contains sinh and cosh terms, which tend to infinity as the length to the depth ratio of the beam increases . For general analysis, it is preferable to use a nu merical approach. In practice, some care needs to be taken with numerical solutions to avoid difficulties, such as locking with increasing beam aspect ratio. This is of major concern when using higher order theories, not only for beams but also for p lates[6, 10]. To avoid these difficult ies of exact and numerical solutions of differential equations based on the higher-order theory, an appro ximate analytical solution in the form of Chebys hev series is propos ed. Using the nondimensional coordinate ξ=x/ L and where[A01],[A 02],[A 03] and[A04] are matrices of o rder N x N+1, N-1 x N+1, N-2 x N+1 and N-3 x N+1 respectively. They are derived in reference[15] and are given in Appendix B. The highest derivative expressed by equation (15a) is of order 4, so the number of algebraic equations in it is N-3. On the other hand, the highest derivative expressed by equations (15b) and (15c) are o f order 3, so the number of algebraic equations is N-2 in each. Hence, the total number of algebraic equations is 3N-7 in 3N+3 unknown wi, θi and ui coefficients. Hence, the total number of algebraic equations is 3N-7 along with 8 boundary conditions at ξ=0 and ξ=1. This leads to 3N+1 equations in 3N+3 unknowns, which have an infinite number of solutions. To overcome this difficulty θ(ξ) and u(ξ) are expanded in N-term Chebyshev series, while w(ξ) is expanded in an N+1-term Chebyshev series. This way, the number of unknown Chebeyshev coefficients is reduced to 3N+1, and the system of equations (15) along with the boundary conditions can be easily solved It is important to note that all matrices in each of equations (15) take the order of the matrix corresponding to the highest derivative. That is, in (15a) all matrices are of order (N-3 x N+1), wh ile all matrices in (15b) and (15c) are of order (N-2 x N). 5. Results and Discussions To study convergence of Chebyshev solution, the nondimensional deflection of symmetric and asymmetric 75 International Journal of Composite M aterials 2012, 2(5): 72-78 cross-ply laminated beams with different boundary conditions are calculated and compared in Table (1) and Table (2) with the published results[10, 13]. The beams have the follo wing d imensionless orthotropic material properties E11/E22 = 25; G12 = 0.5E22; G23 = 0.2E22; ν12 = 0.25, and the deflections are non-dimensionalized as w*= w A E22 h 2 102 . q L4 Table 1. Non-dimensional central displacement, w∗ of symmetric[0/90] laminates exh ibit lo wer transverse deflections than asymmetric ones. The axial-flexu ral coupling doesn't exist in the symmetric laminates, and its effect increases as the degree of asymmetry increases to reach the highest value in cross-ply laminates. Also, the axial-flexu ral coupling, which may cause delamination, varies along the beam span according to the boundary conditions. Delamination starts at the end which is free to move in the axial direction irrespective the boundary conditions at the other end. In case of fixed-fixed beam, delamination starts nearly at the quarter-span from the two ends. cross-ply laminat ed beams with different boundary condit ions L/h H-H C-H C-C C-F Ref.[10] 4.750 2.855 1.924 15.334 5 Ref.[13] 4.777 2.863 1.922 15.279 Present N=20 4.7768 2.8627 1.927 15.2788 Ref.[10] 3.668 1.736 1.007 12.398 10 Ref.[13] 3.688 1.740 1.005 12.343 Present N=20 3.6883 1.7401 1.0054 12.3417 Ref.[10] 3.318 1.343 0.681 11.392 50 Ref.[13] 3.336 1.346 0.679 11.337 Present N=20 3.3362 1.3493 0.6827 11.1213 Table 2. Non-dimensional central displacement, w∗ of symmetric[0/90/0] cross-ply laminat ed beams with different boundary condit ions L/h H-H C-H C-C C-F z,w h 21 α L x,u θ Figure 1. Beam Geometry w*E22*b*h3*100/(q*L4) 6 0.2 u*E22*b*h3*100/(q*L4) 0 4 -0.2 2 -0.4 0 0/90 0/90/0 -0.6 -0.8 0/90 0/90/0 -2 -1 0 0.5 1 0 0.5 1 ξ ξ Fi gure 2. Axial Variat ion of non-dimensional displacement s of symmet ric and asymmetric cross ply laminat ed H-H beams (L/h=5) Ref.[10] 2.398 1.946 1.538 6.836 Ref.[13] 2.412 1.952 1.537 6.824 5 Present N=14 2.4124 1.9517 1.5369 6.8236 N=12 2.4124 1.9521 1.5374 6.8215 N=10 2.4116 1.9480 1.5307 6.7885 Ref.[10] 1.090 0.738 0.532 3.466 Ref.[13] 1.096 0.740 0.532 3.455 10 Present N=14 1.0963 0.7395 0.5307 3.4492 N=12 1.0966 0.7434 0.5366 3.4352 N=10 1.0966 0.7216 0.5022 3.3047 Ref.[10] 0.661 0.279 0.147 2.262 Ref.[13] 0.665 0.280 0.147 2.251 50 Present N=20 0.6645 0.2804 0.1480 2.2293 N=14 0.6643 0.2745 0.1365 2.0484 N=12 0.6644 0.3082 0.1787 1.8090 It is clear fro m the results that Chebyshev solution converges to the exact solution given in reference[13], and that 14:20 terms are sufficient to get good results. The proposed procedure is then used to study the effect of ply stacking of sy mmetric and asymmetric laminated beams with different boundary conditions and aspect ratio, L/h=5, on the central displace ment response. The results are shown in Figs. 2-9. It is clear fro m the figures that symmetric u*E22*b*h3*100/(q*L4) w*E22*b*h3*100/(q*L4) 5 0/θ 4 0/θ/0 3 0.5 0/θ 0/θ/0 0 2 1 0 20 40 60 80 θ -0.5 0 20 40 60 80 θ Figure 3. Variation of non-dimensional mid-span displacements with the orientation of symmetric and asymmetric laminated H-H beams (L/h=5) w*E22*b*h3*100/(q*L4) 3 0.2 u*E22*b*h3*100/(q*L4) 2 0.1 0 1 -0.1 0 0/90 0/90/0 -0.2 0/90 0/90/0 -1 -0.3 0 0.5 1 0 0.5 1 ξ ξ Fi gure 4. Axial Variat ion of non-dimensional displacement s of symmet ric and asymmetric cross ply laminat ed C-H beams (L/h=5) A. Okasha El-Nady et al.: Analysis of Arbitrarily Laminated Composite Beams Using Chebyshev Series 76 w*E22*b*h3*100/(q*L4) 3 0/θ 2.5 0/θ/0 2 1.5 1 0 20 40 60 80 θ u*E22*b*h3*100/(q*L4) 0.1 0 -0.1 -0.2 -0.3 0/θ 0/θ/0 -0.4 0 20 40 60 80 θ Figure 5. Variations of non-dimensional mid-span transverse displacement and full-span axial displacement with the orientation of symmetric and asymmetric laminated C-H beams (L/h=5) 2 0.1 0/90 1.5 0.05 0/90/0 1 0 0.5 6. Conclusions An approximate analytical method using Chebyshev series is proposed for the analysis of arbitrarily laminated composite beams based on the higher order shear deformation theory. The method is powerful in the analysis of short beams and quickly converges to the exact solution. The method can be easily applied to different loading and boundary conditions. APPENDIX A: Bending and Torsion Stiffnesses of Laminated Beam According to the Higher-order theory The stress-strain constants appearing in equation (3) are u*E22*b*h3*100/(q*L4) w*E22*b*h3*100/(q*L4) 0 0/90 0/90/0 -0.05 Q11 = C 4 Q11 + S 4 Q22 + 2S 2C 2 (Q12 + 2Q33 ) ; -0.5 -0.1 0 0.5 1 0 0.5 1 ξ ξ Q 55 = C 2 Q55 + S 2 Q44 Fi gure 6. Axial Variat ion of non-dimensional displacement s of symmet ric and asymmetric cross ply laminat ed C-C beams (L/h=5) Q11 = 1 E11 −ν 12ν 21 ; Q12 = ν 21E11 1 −ν 12ν 21 ; w*E22*b*h3*100/(q*L4) 2 1.8 1.6 1.4 1.2 1 0 0/θ 0/θ/0 20 40 60 80 θ u*E22*b*h3*100/(q*L4) 0.08 0/θ 0.06 0/θ/0 0.04 0.02 0 -0.02 0 20 40 60 80 θ Q 22 = 1 E22 −ν 12ν 21 Q33 = G12 ; Q44 = G23 ; Q55 = G13 C = cosα ; S = sin α α is the angle between the fiber axis and the global laminate axis. The bending stiffnesses appearing in equation (6) are Figure 7. Variation of non-dimensional mid-span transverse displacement and quarter-span axial displacement with the orientation of symmetric and EIθ = EI e + 2EI s1 + EI s2 asymmetric laminat ed C-C beams (L/h=5) EIθw = EI s1 + EI s2 (A 1) u*E22*b*h3*100/(q*L4) w*E22*b*h3*100/(q*L4) 40 0/90 0/90/0 2 1.5 EI w = EI s2 1 0/90 0/90/0 where 20 0.5 0 h/2 ∫ (EI e , EI s1, EI s2 ) = b Q11 (z 2 , z 4 , z 6 )dz (A2) −h / 2 ∑ 0 -0.5 0 0.5 1 0 0.5 1 ξ ξ = EIe Fi gure 8. Axial Variat ion of non-dimensional displacement s of C-F beams N k b Q11 [(Z k k =1 )2 tk + (tk )3 12 ] (L/h=5) w*E22*b*h3*100/(q*L4) 16 14 0/θ 0/θ/0 12 10 8 6 4 0 20 40 60 80 θ u*E22*b*h3*100/(q*L4) 2 0/θ 1.5 0/θ/0 1 0.5 0 -0.5 0 20 40 60 80 θ ∑ EIs1 = − 4b 15h2 N k Q11 [5(Z k )4 tk k =1 +2.5(Z k )2 (tk )3 + (tk )5 16 ] (A 3) ∑ EIs2 = 16b 63h4 N k Q11 [7(Z k )6 tk + 35 4 (Z k )4 (tk k =1 + 21 16 (Z k )2 (tk )5 + (tk )7 64 ] )3 The shear stiffness GA∗ appearing in equation (6) is Figure 9. Variation of non-dimensional mid-span displacements with the orientation of symmetric and asymmetric cross ply laminated C-F beams (L/h=5) where GA∗ = GA1 + GA2 + GA3 (A 4) 77 International Journal of Composite M aterials 2012, 2(5): 72-78 h/2 ∫ (GA1, GA2 , GA3 ) = b Q55 (1, z 2 , z 4 )dz (A5) −h / 2 ∑ GA1 = b N k Q55 tk k =1 ∑ ∑ GA2 = = − 8hb2 kN1 Q= 55k kN1 [(Z k )2 tk + (tk )3 ] 12 (A6) ∑ ∑ GA3 = 16b N = 5h4 k 1 kN Q55 =k 1 [5(Z k )4 tk +2.5(Z k )2 (tk )3 + (tk )5 16 ] The a xia l stiffness EA appearing in equation (6) a re h/2 ∫ EA = b Q11 dz (A7) −h / 2 ∑ EA =b N k Q11 tk (A8) k =1 The cross coupling terms B1 and B2 appearing in equation (6) is h/2 ∫ (B1, B2 ) = b Q11 ( z, z 3 )dz (A9) −h / 2 ∑ B1 = b N k Q55 Zk tk k =1 (A10) ∑ ∑ B2 = −= 3bh2 kN1 Q= 11k [kN1 (Z k )3tk + Z k (tk )3 ] APPENDIX B: Matrix Formulation for Functions and Function Derivatives Any continuous function f(ξ) in the interval 0 ≤ ξ ≤ 1 and its derivatives can be written in Chebyshev series as follows: ∞ N −1 ∑ ∑ f(ξ) = + arTr (ξ ) f \ (ξ) = + ar(1)Tr (ξ ) r =0 r =0 (B1) N −2 N −n ∑ ∑ f \\ (ξ) = + ar(2)Tr (ξ ) f n (ξ) = + ar(n)Tr (ξ ) r=0 r=0 The matrices[A01] to[A04] relate the 1st, 2nd, 3rd and 4th derivative coefficients of a function to the original function coefficients. The first-order-derivative coefficients { ar(1) } as defined in Ref.[15] can be written in terms of the original function coefficients {ai} using matrix notation as follo ws: { a (1) r } = 4[A01] {a i} ; r = 0,1, 2, …..........., N -1 i = 0,1, 2, 3……............, N (B2) where[A01] is of order N x N+1. It is composed of an N x N matrix designated as[A] matrix and an N x 1 colu mn with zero entries at the left of matrix[A ] Matrix[A] is an upper triangular matrix. Its elements aij are defined as: 0 i > j Or i + j odd aij =   j i≤ j and i+ j even The form of[A] and[A01] for N=5 for examp le is: 1 0 [A]= 0 0 0 2 0 0 3 0 3 0 0 4 0 4 5 0 5 0 [A01]= 0 0 0 0 1 0 0 0 0 2 0 0 3 0 3 0 0 4 0 4 5 0 5 0 0 0 0 0 5 0 0 0 0 0 5 The matrices[A02],[A 03],[A04] and[A0n] which relate the 2nd, 3rd, 4th and nth derivative coefficients to the original function coefficients are obtained as follo ws: [A02]=[A]-1,-1 –1[A01] [A03]=[A]-2,-2 –1[A02] [A04]=[A]-3,-3 –1[A03] [A0n ]=[A]1-n, 1-n –1[A0(n-1)] where, n … the order of derivative [A]1-n, 1-n … mat rix[A] after deleting the last (n-1) rows and (n-1) colu mns. -1[ ] … matrix[ ] after deleting the first row. Nomenclature E11, E22 Young's moduli in 1 and 2 directions respectively. G12, G13, G23 Shear moduli in 1-2, 1-3 and 2-3 planes res p ectiv ely . L, b, h Beam length, width and height respectively. U, W A xial and transverse global-beam displacements res p ectiv ely . u, w A xial and transverse displacements along the beam reference plane. q Lateral d istributed load per unit length α Angle between the fiber axis and the laminate axis. θ Beam rotation about y-axis ν12 Poisson ratio for transverse strain in the 2-direction when stressed in the 1-d irection. σ1, σ2, τ12 In-plane stresses in local lamina coordinates. REFERENCES [1] Z. M . Elias, Theory and M ethods of Structural Analysis. Wiley-Interscience, New York (1986). [2] S. P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Phil. M ag. Vol. 41, pp. 744-746, (1951). [3] M . Levinson, A new rectangular beam theory, J. Sound Vibr. Vol. 74, pp. 81-87, (1981). [4] W. B. Bickford, A consistent higher order beam theory, Devl. Theor. Appl. M ech., vol. 11. 137-150, (1982). [5] P. R. Heyliger and J. N. Reddy, A higher order beam finite element for bending and vibration problems, J. Sound Vib. A. Okasha El-Nady et al.: Analysis of Arbitrarily Laminated Composite Beams Using Chebyshev Series 78 Vol. 126, pp. 309-326, (1988). [6] J. Petrolito, Stiffness analysis of beams using a higher-order theory, Computers & Structures vol. 55, pp. 33-39, (1995). [7] Ever J. Barbero, Introduction to Composite M aterials Design, Taylor Frances, Philadelphia, PA19106, USA, (1999). [8] Lee R. Calcote, The Analysis of Laminated Composite Structures, VAN Reinhold Company, (1969). [9] Stephen R. Swanson, Introduction to Design and Analysis with Advanced Composite M aterials, Prentice Hall, Inc., (1997). [10] M . V. V. S. M urthy, D. Roy M ahapatra, K. Badarinarayana and S. Gopalakrishnan, A refined higher order finite element for asymmetric composite beams, Composite Structures, vol. 67, issue 1, pp. 27-35, January (2005). [11] V.G. M okos and E.J. Sapountzakis, A BEM solution to transverse shear loading of composite beams, International Journal of Solids and Structures, vol. 42, issue 11-12, pp. 3261-3287, January (2005). [12] A. A. Khedir and J. N. Reddy, Free vibration of cross-ply laminated beams with arbitrary boundary conditions. Int. J. Eng. Sci. vol. 32, 12, pp. 1971-1980, (1994). [13] A. A. Khedir and J. N. Reddy, An exact solution for the bending of thin and thick cross-ply laminated beams. Composite Structures, vol. 37, pp. 195-203, (1997). [14] A. Okasha, Exact solution of arbitrarily laminated composite beams using a higher-order theory. Eleventh International Conference on Aerospace Science and Aviation Technology, M ilitary Technical College, Cairo, 2005. [15] A. Okasha and H. M . Negm, M atrix formulation of Chebyshev solution to shell problems. Tenth International Conference on Aerospace Science and Aviation Technology, M ilitary Technical College, Cairo, 2003.

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