本研究考慮側向剪切變形之拉伸彎矩偶合的複材疊層厚樑結構分析。根據Timoshenko樑理論我們建立了任意疊層樑之數學模型，藉由狀態空間法(state-space approach)推得以矩陣指數形式呈現的通解，可應用於各種負載和邊界條件，如常見樑問題之明示解析解抑或是無窮域複材疊層厚樑的格林函數(Green’s function)。為解決更複雜的樑結構問題，我們以所推得之格林函數求取基本解，並應用至邊界有限元素法(Boundary-based finite element method)。透過互易定理(Reciprocal theorem of Betti and Rayleigh)可推導出邊界積分方程式(Boundary integral equations)。而所有涉及的積分項皆以解析積分的方式解出，不涉及任何奇異積分或數值積分。我們提出的邊界有限元素法只需兩端節點的單一元素即可為均勻厚度的直樑提供明示解。對於非均勻厚度的曲線梁，即使邊界有限元素法只能提供近似解，其收斂速度也遠比傳統有限元素法快。此研究也進一步延伸至自由振動分析，建構出以矩陣形式呈現之統御方程式，利用狀態空間法以及分離變數法求得自然頻率之通解，再透過施加邊界條件來建立特徵方程式，並利用二分法(Bisection method)和最小平方法(Least-squares technique)求解自然頻率和模態形狀。在數值範例中我們將其數值結果與有限元素商用軟體ANSYS進行比對，並驗證了明示解析解、邊界有限元素法以及分析自由振動方法之適用性。

This research focuses on the structural analysis of laminated composite thick beams with coupled stretching-bending and transverse shear deformation. A mathematical model based on Timoshenko beam theory is established for general laminated beams. With the state-space approach, a general solution in terms of matrix exponential is obtained, which is applicable to various loading and boundary conditions. With the obtained general solutions, several analytical solutions are derived explicitly such as the Green’s functions for an infinite laminated composite thick beam. To cover more complicated beam structures, we develop the boundary-based finite element method (BFEM) using the obtained Green's functions. Since the boundary integral equations (BIEs) derived through the reciprocal theorem of Betti and Rayleigh do not involve singular integrals or numerical integration, our proposed BFEM provides an exact solution for a straight beam with uniform thickness using only one element with two end nodes. Even only an approximate solution is obtained for a curvilinear beam with non-uniform thickness, BFEM converges faster than the conventional finite element method. The research further extends to free vibration analysis, constructing displacement-based governing equations in matrix form. By enforcing boundary conditions, the eigen-relation is established to solve for natural frequencies and mode shapes using the bisection method and least-squares technique. Numerical examples validate the applicability of newly derived explicit solutions, the BFEM and the proposed free vibration method by comparing the results with commercial finite element software ANSYS.

Abramovich, H. and Livshits, A., 1994, “Free vibrations of non-symmetric cross-ply laminated composite beams,” Journal of Sound and Vibration, Vol. 176(5), pp. 597-612.
Antes, H., 2003, “Fundamental solution and integral equations for Timoshenko beams,” Composite Structures, Vol: 81, pp. 383-396.
Antes, H., Schanz, M. and Alvermann, S., 2004, “Dynamic analyses of plane frames by integral equations for bars and Timoshenko beams,” Journal of Sound and Vibration, Vol. 276, pp. 807-836.
Andrilli, S. and Hecker, D., 2010, Elementary Linear Algebra (4th edition), Academic Press, Massachusetts.
Ahmed, A.M. and Rifai, A.M., 2021, “Euler-Bernoulli and Timoshenko beam theories analytical and numerical comprehensive revision,” European Journal of Engineering and Technology Research, Vol. 6(7), pp. 20-32.
Barbero, E.J., Lopezanido, R. and Davalos, L.F., 1993, “On the mechanics of thin-walled laminated composite beams,” Journal of Composite materials, Vol. 27(8), pp. 806-829.
Boay, C.G. and Wee, Y.C., 2008, “Coupling effects in bending, buckling and free vibration of generally laminated composite beams,” Composites Science and Technology, Vol. 68, pp. 1664-1670.
Brebbia, C.A., Telles, J.C.F. and Wrobel, L.C., 1984, Boundary Element Techniques: Theory and Applications in Engineering, Springer-Verlag, Berlin.
Cowper, G.R., 1966, “The Shear Coefficient in Timoshenko’s Beam Theory,” Journal of Applied Mechanics, Vol. 3, No. 2 pp. 335-340.
Chandrashekhara, K., Krishnamurthy, K. and Roy, S., 1990, “Free vibration of composite beams including rotary inertia and shear deformation,” Composite Structures, Vol. 14, 269-279.
Carrer, J.A.M., Fleischfresser, S.A., Garcia, L.F.T. and Mansur, W.J., 2013, “Dynamic analysis of Timoshenko beams by the boundary element method,” Engineer Analysis with Boundary Elements, Vol. 37, pp. 1602-1616.
Carrer, J.A.M., Mansur, W.J., Scuciato, R.F. and Fliischfresser S.A., 2014, “Analysis of Euler-Bernoulli and Timoshenko beams by the boundary element method,” The 10th World Congress on Computational Mechanics, São Paulo.
Doeva, O., Masjedi, P.K. and Weaver, P.M., 2020a, “Exact solution for the deflection of composite beams under non-uniformly distributed loads,” Proceedings of the AIAA Scitech Forum, Florida.
Doeva, O., Masjedi, P.K. and Weaver, P.M., 2020b, “Static deflection of fully coupled composite Timoshenko beams: An exact analytical solution,” European Journal of Mechanics, Vol. 81, 103975.
Franklin, J.N., 1968, Matrix Theory, Prentice-Hall, New Jersey.
Gere, J.M. and Goodno, B.J., 2012, Mechanics of materials, Cengage learning, Australia.
Hwu, C. and Hu, J.S., 1992, “Buckling and Postbuckling of Delaminated Composite Sandwich Beams,” AIAA Journal, Vol. 30, pp. 1901-1909.
Hutchinson, J.R., 2001, “Shear coefficients for Timoshenko beam theory,” Journal of Applied Mechanics, Vol. 68(1), pp. 87-92.
Hwu, C., 2010, Anisotropic Elastic Plates, Springer, New York.
Hwu, C., Huang, S.T. and Li, C.C., 2017, “Boundary-based Finite Element Method for Two-Dimensional Anisotropic Elastic Solids with Multiple Holes and Cracks,” Engineering Analysis with Boundary Elements, Vol.79, pp. 13-22.
Huang, W.Y., Hwu, C. and Hsu, C.W., 2023a, “Boundary-based finite element method for curvilinear non-uniform thickness laminated composite beams,” submitted for publication.
Huang, W.Y., Hwu, C. and Hsu, C.W., 2023b, “Explicit analytical solutions for arbitrarily laminated composite beams with coupled stretching-bending and transverse shear deformation,” submitted for publication.
Huang, W.Y., Hwu, C. and Hsu, C.W., 2023c, “Free vibration analysis of arbitrarily laminated composite thick beams,” submitted for publication.
Kapania R.K. and Raciti, S., 1989, “Recent advances in analysis of laminated beams and plates, part I: shear effects and buckling, AIAA Journal, Vol. 27(7), pp. 923-935.
Krishnaswamy, S., Chandrashekhara, K. and Wu, W.Z.B., 1992, “Analytical solutions to vibration of generally layered composite beams,” Journal of Sound and Vibration, Vol. 159(1), pp. 85-99.
Khdeir, A.A. and Reddy, J.N., 1994, “Free vibration of cross-ply laminated beams with arbitrary boundary conditions,” International Journal of Engineering Science, Vol. 32(12), pp. 1970-1980.
Khdeir, A.A. and Reddy, J.N., 1997, “An exact solution for the bending of thin and thick cross-ply laminated beams,” Composite Structures, Vol 37, pp. 195-203.
Kant, T. and Swaminathan, K., 2001, “Analytical solutions for free vibration of laminated composite and sandwich plates based on a higher-order refined theory,” Composite Structures, Vol. 53(1), pp.73-85.
Kim, W. and Reddy, J.N., 2011, “Nonconventional finite element models for nonlinear analysis of beams,” International Journal of Computational Methods, Vol. 8(3), pp. 349-368.
Kahya, V., 2016, “Buckling analysis of laminated composite and sandwich beams by the finite element method,” Composites Part B-Engineering, Vol. 91, pp. 126-134.
Khodabakhshpour-Bariki, S., Jafari-Talookolaei, R.A., Attar, M. and Eyvazian, A., 2021, “Free vibration analysis of composite curved beams with stepped cross-section,” Structures, Vol. 33, pp. 4828-4842.
Lou, P., Dai, G.L. and Zeng, Q.Y., 2007, “Dynamic analysis of a Timoshenko beam subjected to moving concentrated forces using the finite element method,” Shock and Vibration, Vol. 14(6), pp. 459-468.
Li, J., Hua, H.X. and Shen, R.Y., 2008, “Dynamic finite element method for generally laminated composite beams,” International Journal of Mechanical Sciences, Vol. 50(3), pp. 466-480.
Nguyen, T.K., Nguyen, N.D., Vo, T.P. and Thai, H.T., 2017, “Trigonometric-series solution for analysis of laminated composite beams,” Composite Structures, Vol. 160(15), pp. 142-151.
Nguyen, N.D., Nguyen, T.K., Vo, T.P. and Thai, H.T., 2018, “Ritz-Based Analytical Solutions for Bending, Buckling and Vibration Behavior of Laminated Composite Beams,” International Journal of Structural Stability and Dynamics, Vol. 18(11), 1850130.
Nascimento Júnior, P.C. and Mendonca, A.V., 2021, “Fundamental solutions and integral equations of first order laminated composite beams,” Journal of the Brazilian Society of Mechanical Sciences and Engineering, Vol. 43, 13.
Ozturk, H., Yesilyurt, I. and Sabuncu, M., 2006, “In-plane stability analysis of non-uniform cross-sectioned curved beams,” Journal of Sound and Vibration, Vol. 296(1-2), pp. 277-291.
Ozutok, A. and Madenci, E., 2013, “Free vibration analysis of cross-ply laminated composite beams by mixed finite element formulation,” International Journal of Structural Stability and Dynamics, Vol. 13(2), 1250056
Providakis, C.P. and Beskos, D.E., 1986, “Dynamic analysis of beams by the boundary element method,” Composite Structure, Vol. 22, pp. 957-964.
Qatu, M.S., 1992, “In-plane vibration of slightly curved laminated composite beams,” Journal of Sound and Vibration, Vol. 159(2), pp. 327-338.
Rand, O., 1994, “Nonlinear analysis of orthotropic beams of solid cross-sections,” Composite Structures, Vol. 29(1), pp. 27-45.
Reddy, J.N., 1993, An Introduction to the Finite Element Method (2nd ed.), McGraw-Hill, New York.
Reddy, J.N., 2003, Mechanics of Laminated Composite Plates and Shells (2nd ed.), CRC Press, Boca Raton.
Reddy, J.N., 2017, Energy Principles and Variational Methods in Applied Mechanics (3rd ed.), John Wiley & Sons, New York.
Sokolnikoff, I.S., 1956, Mathematical Theory of Elasticity (2nd Edition), McGraw Hill, New York.
Sankar, B.V., 1993, “A Beam Theory for Laminated Composites and Application to Torsion Problems,” ASME Journal Applied Mechanics, Vol. 60(1), pp. 246-249.
Swanson, S.R., 1997, Introduction to design and analysis with advanced composite materials, Prentice-Hall, Tokyo.
Salim, H. A. and Devalos, J.F., 2005, “Torsion of open and closed thin-walled laminated composite sections,” J Compos Mater, Vol. 39(6), pp.497-524.
Sheikh, A.H. and Thomsen O.T., 2008, “An efficient beam element for the analysis of laminated composite beams of thin-walled open and closed cross sections,” Composites Science and Technology, Vol. 68(10-11), pp. 2273-2281.
Staab, G.H., 2015, Laminar Composites (2nd Edition), Butterworth-Heinemann, New York.
Timoshenko S.P., 1921, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, Vol 41 (245), pp. 744-746.
Timoshenko, S.P., 1922, “On the transverse vibrations of bars of uniform cross section,”
Philosophical Magazine, Vol 43, pp. 125-131.
Timoshenko, S.P., 1953, History of strength of materials., McGraw-Hill., New York.
Tong, X., Mrad, N. and Tabarrok B., 1998, “In-plane vibration of circular arches with variable cross-sections,” Journal of Sound and Vibration, Vol. 212(1), pp. 121-140.
Tufekci, E. and Ozdemirci, O., 2006, “Exact solution of free in-plane vibration of a stepped circular arch,” Journal of Sound and Vibration, Vol. 295(3-5), pp. 725-738.
Tufekci, E. and Yigit, O.O., 2013, “In-plane vibration of circular arches with varying cross-sections,” International Journal of Structural Stability and Dynamics, Vol. 13(1), 1350003.
Vinson, J.R, Sierakowski, R.L., 1987, The behavior of structures composed of composite materials, Martinus Nijhoff, The Netherlands.
Wieckowski, Z. and Golubiewski, M., 2007, “Improvement in accuracy of the finite element method in analysis of stability of Euler–Bernoulli and Timoshenko beams,” Thin-walled Structures, Vol. 45 (10-11), pp. 950-954.
Xie, X., Zheng, H. and Yang, H.S., 2016, “Vibration Analysis of Laminated and Stepped Circular Arches by a Strong Formulation Spectral Collocation Approach,” International Journal of Applied Mechanics, Vol. 8(3), 1650036.