本篇論文應用RZT (Refined Zigzag Theory)理論於功能梯度三明治樑之振動與臨界屈曲之分析。在RZT的假設中，透過引入與三明治樑中各層剪力模數相關之zigzag函數來模擬三明治樑中軸向位移之zigzag效應。在三明治樑之表面FGM (Functionally Graded Material)層中，透過無限個子層來模擬FGM材料系數隨著空間的連續變化，藉此導入RZT函式來建模功能梯度三明治樑。梯度三明治樑之運動方程式、邊界條件及本構關係可藉由漢彌爾頓定理求出。自然頻率、模態形狀、正交關係、頻率響應及臨界屈曲的表示式皆以解析的方式表示。本研究考慮的邊界條件為懸臂樑、簡支樑及固定樑。各種參數如細長比(aspect ratio)、層間厚度比(layer-thickness ratio)、材料系數分布的次方與不同的三明治樑中間層材料對功能梯度三明治樑之振動與臨界屈曲的影響皆有進行討論。在振動分析中，RZT求得之自然頻率與CBT (Classical Beam Theory)、FSDT (First-order Shear Deformation Theory)、HSDT (Higher-order Shear Deformation Theory)與FEM的結果進行比較與驗證，從中發現RZT結果與FEM結果相近。在模態形狀與頻率響應的結果比較中也可以看到RZT結果與FEM結果相近。在臨界屈曲分析中，RZT求得之臨界屈曲負載與HSDT的結果進行比較。藉由上述之結果比較與討論中，可佐證本論文中RZT擴展至功能梯度三明治樑之自然頻率、模態形狀、頻率響應及臨界屈曲分析的適用性。

In this thesis, analytical solutions for vibration and critical buckling analyses in functionally graded (FG) sandwich beams based on refined zigzag theory (RZT) are presented. In RZT, the zigzag term in kinematics of axial displacements are considered by introducing a zigzag function that is related to the shear modulus of layer sequence in the sandwich beam. The face layers with FGM are modeled by infinite sub-layers and the RZT formulations can be extended to the FG sandwich beam. The equations of motion, boundary conditions and constitutive relation are derived based on the Hamilton principle. The natural frequency equations, mode shapes, orthogonal relations, frequency responses, and critical buckling analyses are presented in exact forms. Cantilevered, simply-supported, and fixed-fixed boundary conditions were considered in the case studies. Various parameters such as aspect ratios, layer-thickness ratios, powers in the material property distribution and materials for middle layer were applied to investigate their effects on the vibration and critical buckling behaviors of FG sandwich beams. In the vibration analysis, the results of natural frequencies calculated by RZT are validated by the FEM results and compared with the results calculated by CBT (classical beam theory), FSDT (first-order shear deformation theory), and HSDT (higher-order shear deformation theory). The comparisons show that the results obtained by RZT are corresponding to the FEM results. For the mode shape and frequency response comparisons, the RZT results also agree well with the FEM results. The results of critical buckling load calculations by RZT are presented and compared with those by HSDT. It is concluded that the RZT is quite applicable to calculate the natural frequencies, mode shapes, frequency responses, and critical buckling analyses of FG sandwich beams.

[1] P. Zahedinejad, C. Zhang, H. Zhang, and S. Ju, "A comprehensive review on vibration analysis of functionally graded beams," International Journal of Structural Stability and Dynamics, vol. 20, no. 04, p. 2030002, 2020.
[2] K. Swaminathan, D. Naveenkumar, A. Zenkour, and E. Carrera, "Stress, vibration and buckling analyses of FGM plates—A state-of-the-art review," Composite Structures, vol. 120, pp. 10-31, 2015.
[3] A. S. Sayyad and Y. M. Ghugal, "Modeling and analysis of functionally graded sandwich beams: A review," Mechanics of Advanced Materials and Structures, vol. 26, no. 21, pp. 1776-1795, 2019.
[4] S. Sina, H. Navazi, and H. Haddadpour, "An analytical method for free vibration analysis of functionally graded beams," Materials & Design, vol. 30, no. 3, pp. 741-747, 2009.
[5] H.-T. Thai and T. P. Vo, "A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates," Applied mathematical modelling, vol. 37, no. 5, pp. 3269-3281, 2013.
[6] T. P. Vo, H.-T. Thai, T.-K. Nguyen, A. Maheri, and J. Lee, "Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory," Engineering Structures, vol. 64, pp. 12-22, 2014.
[7] X.-F. Li, "A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams," Journal of Sound and vibration, vol. 318, no. 4-5, pp. 1210-1229, 2008.
[8] M. Mohammadi, E. Mohseni, and M. Moeinfar, "Bending, buckling and free vibration analysis of incompressible functionally graded plates using higher order shear and normal deformable plate theory," Applied Mathematical Modelling, vol. 69, pp. 47-62, 2019.
[9] E. F. Erdurcan and Y. Cunedioğlu, "Free vibration analysis of a functionally graded material coated aluminum beam," AIAA Journal, vol. 58, no. 2, pp. 949-954, 2020.
[10] J.-S. KAO and R. J. ROSS, "Bending of multilayer sandwich beams," AIAA Journal, vol. 6, no. 8, pp. 1583-1585, 1968.
[11] A. K. Murty, "Analysis of short beams," AIAA Journal, vol. 8, no. 11, pp. 2098-2100, 1970.
[12] M. Şimşek and T. Kocatürk, "Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load," Composite Structures, vol. 90, no. 4, pp. 465-473, 2009.
[13] S. Timoshenko, "On the correction for shear of the differential equation for transverse vibration of prismatic bars," Philosophical Magazine, voL41, pp744, vol. 746, 1921.
[14] R. A. Shanab, S. A. Mohamed, N. A. Mohamed, and M. A. Attia, "Comprehensive investigation of vibration of sigmoid and power law FG nanobeams based on surface elasticity and modified couple stress theories," Acta Mechanica, vol. 231, no. 5, 2020.
[15] I. Katili, T. Syahril, and A. M. Katili, "Static and free vibration analysis of FGM beam based on unified and integrated of Timoshenko’s theory," Composite Structures, vol. 242, p. 112130, 2020.
[16] M. Levinson, "An accurate, simple theory of the statics and dynamics of elastic plates," Mechanics Research Communications, vol. 7, no. 6, pp. 343-350, 1980.
[17] Q. Jin, X. Hu, Y. Ren, and H. Jiang, "On static and dynamic snap-throughs of the imperfect post-buckled FG-GRC sandwich beams," Journal of Sound and Vibration, vol. 489, p. 115684, 2020.
[18] H.-T. Thai and T. P. Vo, "Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories," International Journal of Mechanical Sciences, vol. 62, no. 1, pp. 57-66, 2012.
[19] M. Di Sciuva, "Bending, vibration and buckling of simply supported thick multilayered orthotropic plates: an evaluation of a new displacement model," Journal of Sound and Vibration, vol. 105, no. 3, pp. 425-442, 1986.
[20] H. Murakami, "Laminated composite plate theory with improved in-plane responses," 1986.
[21] A. Tessler, M. Di Sciuva, and M. Gherlone, "A refined zigzag beam theory for composite and sandwich beams," Journal of Composite Materials, vol. 43, no. 9, pp. 1051-1081, 2009.
[22] L. Iurlaro, A. Ascione, M. Gherlone, M. Mattone, and M. Di Sciuva, "Free vibration analysis of sandwich beams using the Refined Zigzag Theory: an experimental assessment," Meccanica, vol. 50, no. 10, pp. 2525-2535, 2015.
[23] A. Treviso, D. Mundo, and M. Tournour, "Dynamic response of laminated structures using a Refined Zigzag Theory shell element," Composite Structures, vol. 159, pp. 197-205, 2017.
[24] C.-D. Chen, "A distributed parameter electromechanical model for bimorph piezoelectric energy harvesters based on the refined zigzag theory," Smart Materials and Structures, vol. 27, no. 4, p. 045009, 2018.
[25] C.-D. Chen and W.-L. Dai, "The analysis of mode II strain energy release rate in a cracked sandwich beam based on the refined zigzag theory," Theoretical and Applied Fracture Mechanics, vol. 107, p. 102504, 2020.
[26] F. Farhatnia and M. Sarami, "Finite element approach of bending and buckling analysis of FG beams based on refined zigzag theory," Uni. J. Mech. Eng, vol. 7, pp. 147-158, 2019.
[27] I. Kreja and A. Sabik, "Equivalent single-layer models in deformation analysis of laminated multilayered plates," Acta Mechanica, vol. 230, no. 8, pp. 2827-2851, 2019.
[28] L. Iurlaro, M. Gherlone, and M. Di Sciuva, "Bending and free vibration analysis of functionally graded sandwich plates using the refined zigzag theory," Journal of Sandwich Structures & Materials, vol. 16, no. 6, pp. 669-699, 2014.
[29] M. S. Marco Di Sciuva, "Bending and free vibration analysis of functionally graded sandwich plates: An assessment of the Refined Zigzag Theory," Journal of Sandwich Structures & Materials, vol. 0, pp. 1-43, 2019.
[30] M. Di Sciuva and M. Sorrenti, "Bending, free vibration and buckling of functionally graded carbon nanotube-reinforced sandwich plates, using the extended Refined Zigzag Theory," Composite Structures, vol. 227, p. 111324, 2019.