Analysis of Arbitrarily Laminated Composite Beams Using Chebyshev Series

Abstract In this work a simple 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.


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] extended 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 beams.
Hey ligher and Redd y [5] p resented a fin ite 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 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 numerical 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 energy of arbitrarily laminated beams undergoing axial and transverse shear strains under laterally distributed load. The displacement of the beam w and rotation θ are expanded in Chebyshev series. Using a standard procedure the governing equations are cast in matrix form, wh ich is easily handled by electronic co mputers. Using this method it is possible to analyze beam structures with s mall aspect ratios. The displacements and stresses of several laminated beams are calculated and compared with published results. The effect of ply stacking in symmetric and asymmetric laminated beams with d ifferent boundary conditions and aspect ratios is investigated.

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 by two displacements, U and W, and a rotation, θ. These displacements are assumed to be of the form [6,10]: where h is the depth of the beam.

Strain-Displ acement Rel ati ons
The beam is considered as a wide beam. So, the only non-zero strains are [6] ) ( 3

Stress-Strain Relations
The laminate stresses are where 11 Q and 55 Q are given in Appendix A.

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 where q is the applied transverse load per unit length of the beam, b is the width and L is the length of the beam. Taking into consideration that variation in the potential energy is due to variation in the displacements and strains, then the first variation of the potential energy, Π δ , can be written as: Substituting equations (1)-(3) into equation (5) and integrating over the width and depth of the beam, equation (5) takes the form where EA is the axial stiffness, B 1 and B 2 are the cross-coupling terms due to axial-flexu ral deformation of the composite laminated beam. M 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: For most practical problems the properties of the beam are constant along the length of the beam. In this case, equations (7) and (8) reduce to (9) and (10) Therefore, the higher-o rder beam theory is represented by a system of ord inary differential equations of order six.

Boundary Conditions
Fixed end Roller end at ξ = 1 Free end

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 Chebyshev series is proposed.
Using the nondimensional coordinate ξ=x/ L and expanding w(ξ), θ(ξ) and u(ξ) in (N+1)-term Chebyshev series we have a total of 3N+3 unknown coefficients. Using matrix formu lation for the functions and function derivatives and applying the ru le of mat rix mu ltiplication as explained in reference [15], equations (9) can be written as a system of algebraic equations in the following matrix form:  [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 w i , θ i and u i 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).

Results and Discussions
To study convergence of Chebyshev solution, the nondimensional deflection of symmetric and asymmetric 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 displacement response. The results are shown in Figs. 2-9. It is clear fro m the figures that symmetric 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.

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)  The bending stiffnesses appearing in equation (6)  The shear stiffness GA * appearing in equation (6) The axial stiffness EA appearing in equation (6) are The cross coupling terms B 1 and B 2 appearing in equation (6) is

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:   θ Beam rotation about y-axis ν 12 Poisson ratio for transverse strain in the 2-direction when stressed in the 1-d irection.