Stochastic finite element analysis for free vibration of functionally graded beams

Attribution-NonCommercial-NoDerivs 2.0 KOREA You are free to :  Share — copy and redistribute the material in any medium or format Under the follwing terms : Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made You may so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use NonCommercial — You may not use the material for commercial purposes NoDerivatives — If you remix, transform, or build upon the material, you may not distribute the modified material You not have to comply with the license for elements of the material in the public domain or where your use is permitted by an applicable exception or limitation This is a human-readable summary of (and not a substitute for) the license Disclaimer Stochastic Finite Element Analysis for Free Vibration of Functionally Graded Beams Nguyen Van Thuan August 2017 Department of Civil and Environmental Engineering The Graduate School Sejong University Stochastic Finite Element Analysis for Free Vibration of Functionally Graded Beams Nguyen Van Thuan A dissertation submitted to the Faculty of the Sejong University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil and Environmental Engineering August 2017 Approved by Major Advisor Professor Hyuk Chun Noh DEDICATION I would like to dedicate this dissertation to my Parents, my Wife, and my Children who have supported me all the way ABSTRACT The objective of this research is stochastic finite element method for a free vibration characteristic of a functionally graded beam using uncertain material properties This study is divided into three parts First, we investigated the free vibration of functionally graded material (FGM) beams on an elastic foundation and spring supports Elastic modulus, the mass density and the width of the beam are assumed to vary in thickness and axial directions respectively following the exponential law The spring supports are also taken into account at both ends of the beam An analytical formulation is suggested to obtain Eigen’s solutions of the FGM beams The numerical analyses, based on finite element method by using a beam finite element developed in this study, are performed in order to show the legitimacy of the analytical solutions Some results for the natural frequencies of the FGM beams are given considering the effect of various structural parameters It is also shown that the spring supports have the most effect on the natural frequencies of FGM beams i Second, we obtained the statistical dynamic of functionally graded beam with random elastic modulus of elasticity and mass density, by using the perturbation technique in conjunction with the finite element method Governing equations of the eigenvalue and eigenvector of functionally graded beam are derived from the Hamilton's Principle The effective material properties and cross section are assumed to vary continuously through different directions according to the exponential law distribution We obtained a good agreement between the results of the first-order perturbation technique and Monte Carlo simulation on the mean values both the eigenvalues and the eigenvectors The third part deals with the stochastic analysis where both the elastic modulus and the mass density are considered as random parameters The correlation between these two random parameters is considered in terms of cross-correlation function, and the magnitude of the correlation is given in terms of the correlation coefficients The spectral representation method is employed to generate numerically the random fields of the random elastic modulus and mass density The formulation for the analysis on the response variability of the eigen-values and vectors is presented, and the results are compared with those of the Monte Carlo simulation Individual two random variables are found to affect the response variability in a different way The material mass density is revealed to affect the response variability more than the randomness in the elastic modulus From the viewpoint of the correlation between these two random parameters, the negative correlation shows greater value of variability, and in particular, the perfect positive correlation is found to cause not any of response ii variability This feature can be foreseen if we take into account the equation of eigenvalue for the single degree of freedom structure In order to show the adequacy of the proposed formulation, analyses with various parameter setups are performed and the detailed discussion is given Keywords: finite element method, FGM beam, first-order perturbation technique, spectral representation, random material property iii TABLE OF CONTENTS ABSTRACT i TABLE OF CONTENTS iv LIST OF FIGURES vii LIST OF TABLES ix Chapter INTRODUCTION 1.1 Motivation 1.2 The Functionally Graded Material 1.2.1 History 1.2.2 The Areas of Application 1.3 The Objectives of This Study 1.4 The Organization of This Study Chapter EIGEN ANALYSIS OF THE FUNCTIONALLY GRADED BEAMS WITH A VARIABLE CROSS-SECTION RESTING ON ELASTIC SUPPORTS AND ELASTIC FOUNDATIONS 11 2.1 Introduction 11 2.2 The Analytical Formulation 13 2.2.1 FGM Beam Model 13 2.2.2 The Analytical Formulation 14 2.3 The Finite Element Formulation 18 2.4 The Results and the Discussions 22 2.5 Conclusions 37 iv Thus éỉ ( s ) ù (t ) ( ss ) ( tt ) ( st ) ờỗ X ,i si + X ,i ti + ( X ,ij si s j + X ,ij ti t j + X ,ij si t j ) - DX + DX ÷ ú è ø ú Var(2) ( X ) = E T ờổ ( s ) ỳ ờỗ X , j s j + X ,(jt ) t j + ( X ,(klss ) sk sl + X ,(kltt ) tk tl + X ,(klst ) sk tl ) - DX + DX ÷ ú ø ûú ëêè (5.11) Since the mean on the old number of multiplication of the random variables is zero, and DX , DX are already-established vectors, the following is derived for the second-order variance of the eigen-vector: Var(2) ( X ) = E[( X ,(is ) X ,Tj ( s ) si s j + X ,(it ) X ,Tj ( t ) ti t j + X ,(is ) X ,Tj ( s ) si t j ) - ( X ,(ijss ) si s j + X ,(ijtt ) ti t j + X ,(ijst ) si t j )(DX - DX )T æ X ,(ijss ) X ,Tkl( ss ) si s j sk sl + X ,(ijss ) X ,Tkl( tt ) si s j tk tl + X ,(ijss ) X ,Tkl( st ) si s j sk tl ữ 1ỗ + ỗ + X ,(ijtt ) X ,Tkl( ss ) si s j sk sl + X ,(ijtt ) X ,Tkl( tt ) si s j tk tl + X ,(ijtt ) X ,Tkl( st ) si s j sk tl ữ 4ỗ ữ ( st ) T ( ss ) ( st ) T ( tt ) ( st ) T ( st ) ỗ +2 X ,ij X , kl si s j sk sl + X ,ij X , kl si s j tk tl + X ,ij X , kl si s j sk tl ÷ è ø + (DX - DX )(DX - DX )T ] (5.12) If we arrange the equation for the second-order variance, Var(2) ( X ) = Var( L ) - X ,(ijss ) (DX - DX )T E[ si s j ] + X ,(ijtt ) (DX - DX )T E[ti t j ] + X ,(ijst ) (DX - DX )T E[ si t j ] æ X ,(ijss ) X ,Tkl( ss ) E[ si s j sk sl ] + X ,(ijss ) X ,Tkl( tt ) E[ si s j tk tl ] + X ,(ijss ) X ,Tkl( st ) E[ si s j sk tl ] ữ 1ỗ + ỗ + X ,(ijtt ) X ,Tkl( ss ) E[ si s j sk sl ] + X ,(ijtt ) X ,Tkl( tt ) E[ si s j tk tl ] + X ,(ijtt ) X ,Tkl( st ) E[ si s j sk tl ] ữ 4ỗ ữ ỗ +2 X ,(ijst ) X ,Tkl( ss ) E[ si s j sk sl ] + X ,(ijst ) X ,Tkl( tt ) E[ si s j tk tl ] + X ,(ijst ) X ,Tkl( st ) E[ si s j sk tl ] ÷ è ø T + (DX - DX )(DX - DX ) (5.13) 119 For k-th eigen-vector: Var(2) ( X ) = Var( L ) - X ,kij( ss ) (DX - DX )T E[ si s j ] + X ,kij( tt ) (DX - DX )T E[ti t j ] + X ,kij( st ) (DX - DX )T E[ si t j ] æ X ,kij( ss ) X ,kkl( ss )T E[ si s j sk sl ] + X ,kij( ss ) X ,kkl( tt )T E[ si s j tk tl ] + X ,(ijss ) X ,kkl( st )T E[ si s j sk tl ] ö ữ 1ỗ + ỗ + X ,kij( tt ) X ,kkl( ss )T E[ si s j sk sl ] + X ,kij( tt ) X ,kkl( tt )T E[ si s j tk tl ] + X ,kij( tt ) X ,kkl( st )T E[ si s j sk tl ] ữ 4ỗ ữ ỗ +2 X ,kij( st ) X ,kkl( ss )T E[ si s j sk sl ] + X ,kij( st ) X ,kkl( tt )T E[ si s j tk tl ] + X ,kij( st ) X ,kkl( st )T E[ si s j sk tl ] ÷ è ø k k k k T + (DX - DX )(DX - DX ) (5.14) where, E[ si s j tk tl ] = E[ si s j ]E[tk tl ] + E[ si tk ]E[ s j tl ] + E[ si tl ]E[ s j tk ] 2) Develop the stochastic finite element method for the damping of FGM beams with multiple uncertain properties, loadings, and imperfect geometry as well 3) Develop the stochastic Eigen analysis with log-normal random variable 120 REFERENCES [1] Abu-Hilal, M., Mohsen, M (2000), "Vibration of beams with general boundary conditions due to a moving harmonic load", Journal of Sound and Vibration, 232(4), 703-717 [2] Ait Atmane, H., Tounsi, A., Meftah, S.A and Belhadj, H.B (2011) “Free vibration behavior of exponential functionally graded beams with varying cross-section”, Journal of Vibration and Control, Vol 17, No 2, pp 311-318, DOI: 10.1177/1077546310370691 [3] Bekir, A and Ömer, C (2014) “Shear deformation beam models for functionally graded microbeams with new shear correction factors”, Composite Structures, Vol 112, pp 214-225, DOI:10.1016/j.compstruct.2014.02.022 [4] Alshorbagy, A.E., Eltaher , M.A., Mahmound , F.F (2011), "Free vibration characteristics of a functionally graded beam by finite element method", Applied Mathematical Modelling, 35(1), 412-425 [5] Asghar, R., Aria, A and Javad, A (2006) “Effects of Rotary Inertia and Shear Deformation on Nonlinear Free Vibration of Microbeams”, Journal of Vibration and Acoustics, Vol 125, pp 611-615, DOI: 10.1115/1.2202167 [6] Udupa, G., Rao, S.S., Gangadharan, K.V (2014), “Functionally graded Composite materials: An overvies”, Procedia Materials Science 5, pp 1291-1299 121 [7] Belabed, B., Ahmed Houari, M.S., Tounsi, A., Mahmoud, S R and Anwar, B.O (2014), "An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates", Composites: Part B, Vol 60, pp 274-283, DOI:10.1016/j.compositesb.2013.12.057 [8] Bourada, M., Kaci, A., Ahmed Houari, M., S and Tounsi, A (2015), "A new simple shear and normal deformations theory for functionally graded beams", Steel and Composite Structures, Vol 18, No 2, pp.409 – 423, DOI: http://dx.doi.org/10.12989/scs.2015.18.2.409 [9] Çalım, F.F (2009), "Free and forced vibrations of non-uniform composite beams", Composite Structures, 88(3), 413-423 [10] Caruntu, D.I (2009), "Dynamic modal characteristics of transverse vibrations of cantilevers of parabolic thickness", Mechanics Research Communications, 36(3), 391-404 [11] Chakraborty, A., Gopalakrishnan, S., Reddy, J.N (2003), "A new beam finite element for the analysis of functionally graded materials", International Journal of Mechanical Sciences, 45(3), 519-539 [12] Deodatis, G (1991) "Weighted integral method I: stochastic stiffness matrix." J Engng Mech, 117, pp 185-1864 [13] Ece, M.C., Aydogdu, M., Taskin, V (2007), "Vibration of a variable cross-section beam", Mechanics Research Communications, 34, 78-84 122 [14] Ferrante, F J and Graham, L (2001) "Randomness effects on the properties of a functionally graded plate using probabilistic methods", Structural Safety and Reliability: ICOSSAR '01, Newport Beach, USA [15] Ghanem, R (1991) "Ingredients for a general purpose stochastic finite elements implementation." Comput methods Appl Mech Engrg, 168, 19-34 [16] Ghanem, R G and Spanos, P D (1991) "Spectral stochastic finite element formulation for reliability analysis." J Engng Mech, ASCE, 10, pp 2351-2372, http://dx.doi.org/10.1061/(ASCE)0733-9399(1991)117:10(2351) [17] Gupta, A (1985), "Vibration of Tapered Beams", Journal of Structural Engineering, 111(1), 19-36 [18] Gutierrez, R.H., Laura, P.A.A., Rossi, R.E (1991), "Vibrations of a timoshenko beam of non-uniform cross-section elastically restrained at one end and carrying a finite mass at the other", Ocean Engineering, 18(1–2), 129-145 [19] Graham-Brady, L.L (2005) "Stochastic simulation of non-Gaussian/nonstationary properties in a functionally graded plate", Computer methods in applied mechanics and engineering, Vol 194, Nos 12-16, pp 1675–1692, DOI: 10.1016/j.cma.2004.03.020 [20] Gururaja, U., Rao, S S., and Gangadharan, K V (2014) "Functionally graded composite materials: An overview." Procedia Materials Science, Vol 5, pp 12911299, doi:10.1016/j.mspro.2014.07.442 123 [21] Haasen, A.A., Abdelouahed, T., Sid, A.M., Hichem, A.B (2011), "Free Vibration Behavior of Exponential Functionally Graded Beams with Varying Crosssection", Journal of Vibration and Control, 17(2), 311-318 [22] Hien, T D., Thuan, N V., and Hyuk, C N (2014) "Eigen analysis of functionally graded beams with variable cross-section resting on elastic supports and elastic foundation." Structural Engineering and Mechanics, Vol 52, No 5, DOI: http://dx.doi.org/10.12989/sem.2014.52.5.000 [23] Jagtap, K.R., Lal, A and Singh, B.N (2011) "Stochastic nonlinear free vibration analysis of elastically supported functionally graded materials plate with system randomness in thermal environment", Composite Structures, Vol 93, No 12, pp 3185–3199, DOI: 10.1016/j.compstruct.2011.06.010 [24] Kar, K K (2017) "Composite materials: Processing, Applications, Characterizations" Springer-Verlag Berlin Heidelberg 2017, ISBN 978-3-66249514-8 (eBook), pp 1–694, DOI 10.1007/978-3-662-49514-8 [25] Ke, L.-L., Yang, J., Kitipornchai, S (2010), "An analytical study on the nonlinear vibration of functionally graded beams", Meccanica, 45(6), 743-752 [26] Klein, L (1974), "Transverse vibrations of non-uniform beams", Journal of Sound and Vibration, 37(4), 491-505 [27] Kaminski, M (2006) "On generalized stochastic perturbation-based finite element method." Commun Nimer Meth Engng 22, 23-31 [28] Kleiber, M and Hien, T D (1992) "The stochastic finite element method." John Wiley & Sons, New York 124 [29] Koizumi, M (1997) "FGM activities in Japan." Composites Part B: Engineering, Vol 25, Nos 1-2, pp 1-4, DOI: 10.1016/S1359-8368(96) 00016-9 [30] Koizumi, M (1997) "FGM activities in Japan", Composites Part B: Engineering, Vol 25, Nos 1-2, pp 1-4, DOI: 10.1016/S1359-8368(96) 00016-9 [31] Kitipornchai, S., Yang, J and Liew, K.M (2006) "Random vibration of the functionally graded laminates in thermal environments", Computer methods in applied mechanics and engineering, Vol 195, Nos 9-12, pp 1075–1095, DOI: 10.1016/j.cma.2005.01.016 [32] Kumar, A U., Boopathi, S, Boobalan, R and Kavin, R (2016) "An overview of functuonally graded materials", ICSSCCET 2016, , Vol 02, ISBN 978-81- 929866-6-1, icssccet.org [33] Lardner, T.J (1968), "Vibration of beams with exponentially varying properties", Acta Mechanical, 6(2-3), 197-202 [34] Lee, S.Y., Lin, S.M (1995), "Vibrations of elastically restrained non-uniform timoshenko beams", Journal of Sound and Vibration, 184(3), 403-415 [35] Lal, A., Jagtap, K.R and Singh, B.N (2013) "Post buckling response of functionally graded materials plate subjected to mechanical and thermal loadings with random material properties", Applied Mathematical Modelling, Vol 37, No 5, pp 2900–2920, DOI: 10.1016/j.apm.2012.06.013 [36] Mohanty, S.C., Dash, R.R., Rout, T (2011), "Parametric instability of a functionally graded Timoshenko beam on Winkler's elastic foundation", Nuclear Engineering and Design, 241(8), 2698-2715 125 [37] Noh, H.C (2006) "Effect of multiple uncertain material properties on the response variability of in-plane and plate structures", Computer methods in applied mechanics and engineering, Vol 195, Nos 19-22, pp 2697-2718, DOI: 10.1016/j.cma.2005.06.026 [38] Noh, H.C and Park, T (2011) “Response variability of laminate composite plates due to spatially random material parameter”, Computer Methods in Applied Mechanics and Engineering, Vol 200, Nos 29-32, pp.2397-2406, DOI: 10.1016/j.cma.2011.03.020 [39] Noh, H C (2011) “Plate response variability due to triple random parameters", KSCE Journal of Civil Engineering, Vol 15, No 3, pp 517-526, DOI: 10.1007/s12205-011-1081-4 [40] Noh, H C and Choi, C K (1999) "Stochastic finite element analysis of plate structures by weighted integral method." Structural Engineering and Mechanics, Vol 4, No 6, pp 703-715 [41] Pakar, M B (2012), "Accurate analytical solution for nonlinear free vibration of beams", Structural Engineering and Mechanics, 43(No ), 337-347 [42] Papadopoulos, V and Papadrakakis, M (1997) "Stochastic finite element- based reliability analysis of space frames." Prob Eng Mech 13, pp 53-65 [43] Ramalingerswara Rao, S., Ganesan, N (1995), "Dynamic response of tapered composite beams using higher order shear deformation theory", Journal of Sound and Vibration, 187(5), 737-756 126 [44] Shaker, A., Abdelrahman, W., Tawfik, M.,and Sadek, E (2008) "Stochastic Finite element analysis of the free vibration of functionally graded material plates." Comput Mech, Vol 41, No 5, pp 707–714 [45] Shahistha, A C., Binol V and Anjali, B (2014) "A review on functionally graded materials", The International Journal of Engineering and Science, Kannur, India, Vol 3, No 6, pp 2319-1805 [46] Shaker, A., Abdelrahman, W., Tawfik, M and Sadek, E (2008) "Stochastic Finite element analysis of the free vibration of functionally graded material plates", Comput Mech, Vol 41, No 5, pp 707–714, DOI: 10.1007/s00466-007-0226-2 [47] Shegokar, N.L and Lal, A (2014) "Stochastic finite element nonlinear free vibration analysis of piezoelectric functionally graded materials beam subjected to thermo-piezoelectric loadings withmaterial uncertainties", Meccanica, Vol 49, No 5, pp 1039–1068, DOI: 10.1007/s11012-013-9852-2 [48] Shinozuka, M and Deidatis, G (1991) "Simulation of stochastic processes by spectral representation", AppliedMechanics Reviewers, Vol 44, No 1, pp 191204, DOI: 10.1115/1.3119501 [49] Sen Yung, L., Huel Yaw, K (1990), "Free vibrations of non-uniform beams resting on non-uniform elastic foundation with general elastic end restraints", Computers & Structures, 34(3), 421-429 [50] Şimşek, M., Cansız, S (2012), "Dynamics of elastically connected doublefunctionally graded beam systems with different boundary conditions under action of a moving harmonic load", Composite Structures, 94(9), 2861-2878 127 [51] Suppiger, E., Taleb, N (1956), "Free lateral vibration of beams of variable cross section", Journal of Applied Mathematics and Physics (ZAMP), 7(6), 501-520 [52] Takada, T (1990) "Weighted integral method in stochastic finite element analysis." Probabilistic Engineering Mechanics, Vol 5, No 3, pp 146-156, doi: 10.1016/0266-8920(90)90006-6 [53] Tong, X., Tabarrok, B., Yeh, K.Y (1995), "Vibration analysis of timoshenko beams with non-homogeneity and varying cross-section", Journal of Sound and Vibration, 186(5), 821-835 [54] Talha, M and Singh, B N (2014) "Stochastic perturbation-based finite element for buckling statistics of FGM plates with uncertain material properties in thermal environments." Composite structures, 108, pp 823-833 [55] Wang, R.T (1997), "Vibration of multi-span timoshenko beams to a moving force", Journal of Sound and Vibration, 207(5), 731-742 [56] Yamazaki, F., Shinozuka, M and Dasgupta, G (1988) "Neumann expansion for stochastic finite element analysis." J Engng Mech, 114, pp 1335-1354, http://dx.doi.org/10.1061/(ASCE)0733-9399(1988)114:8(1335) [57] Yang, J., Liew, K M., and Kitipornchai, S (2005) "Stochastic analysis of compositionally graded plates with system randomness under static loading." International Journal of Mechanical Sciences, Vol 47, No 10, pp 1519–1541, DOI: 10.1016/j.ijmecsci.2005.06.006 128 [58] Yang, J., Liew, K.M and Kitipornchai, S (2005) "Stochastic analysis of compositionally graded plates with system randomness under static loading", International Journal of Mechanical Sciences,Vol 47, No 10, pp 1519–1541, DOI: 10.1016/j.ijmecsci.2005.06.006 [59] Yang, J., Liew, K.M and Kitipornchai, S (2005) "Second-order statistics of the elastic buckling of functionally graded rectangular plates", Composites Science and Technology, Vol 65, Nos 7-8, pp 1165–1175, DOI: 10.1016/j.compscitech.2004.11.012 [60] Ying, J., Lu, C.F., Chen, W.Q., (2008), "Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations", Composite Structures, 84, 209-219 [61] Zhou, D (1993), "A general solution to vibrations of beams on variable winkler elastic foundation", Computers & Structures, 47(1), 83-90 [62] Jha, D.K., Tarun, K., Singh, R.K., (2013) "A critical review of recent research on functionally graded plates." Composite Structures, 96, 833–849 129 국문초록 기능경사재료 보의 추계론적 자유진동 유한요소해석 세종대학교 대학원 건설환경공학과 Nguyen Van Thuan 본 연구의 목적은 기능경사재료보의 섭동법을 이용하여 고유치 및 고유 벡터에 대한 추계론적 유한요소해석법을 제시하는 것이다 이를 위하여 재 료상수인 탄성계수와 재료의 질량밀도를 두 개의 불확실인수로 다룬다 본 연구는 세 부분으로 나눌 수 있다 먼저, 확정론적 해석으로서 기능경사재료보에 대한 자유진동을 다른다 해당 보는 탄성지반 상에 있으며 양단은 스프링으로 지지되어 있다 탄성계 수, 질량밀도, 그리고 보의 폭 등이 위치에 따라 변화되는데, 탄성계수와 질 량 밀도의 경우 두께 방향을 따라 지수적으로 변화되며, 보의 폭은 기하학 적 인수에 따라 보의 축을 따라 지수함수로 변화한다 해당 기능경사보의 고유치 해석을 위하여 해석적 정식화가 제안되었다 유한요소법에 근간한 130 수치적 해석은 본 연구에서 개발된 유한요소를 사용하여 수행되었다 수치 해석은 제안된 해석적 정식화를 확인하기 위하여 수행하여으며, 상호 잘 일 치하는 결과를 나타내었다 기능경사보를 표현하는 다양한 구조인수를 고려 한 고유치 해석을 수행하여 그 결과를 제시하였다 특히 양단의 스프링 지 지점의 영향에 대한 해석을 통하여 이들 지지점 또는 스프링 강성이 보의 고유치에 미치는 영향에 대하여 고찰 하였다 두번째 부분은 추계론적 해석에 대한 것으로서, 본 연구에서는 기능경 사 보의 고유치에 대한 변화도를 섭동법을 이용하여 제시하였다 확률변수 로는 탄성계수를 택하였다 해석에는 유한요소법이 적용되었고, 유한요소법 의 범주 내에서 기능경사재료 보의 고유치 및 고유벡터에 대한 응답변화도 를 제시하였다 기능경사보의 지배방정식은 Hamilton 원리로부터 유도하였 다 제시한 정식화에 의한 해석결과의 확인을 위하여 수치적으로 생성한 추 계장 표본을 적용하여 몬테카를로 해석을 수행하였다 제안한 정식화에 의 한 결과와 몬테카를로 해석에 의한 결과는 상호 매우 잘 일치하는 결과를 보여주어 본 연구에서 제시한 정식화의 타당성을 알 수 있었다 세번째 부분은 구조의 동적 거동 및 고유치 해석에서 주된 인수인 재료 탄성계수와 재료밀도를 동시에 확률변수로 한 추계론적 해석부분이다 두 재료상수 사이의 상관관계는 상호상관함수를 가정하여 적용하였고, 이들 사 이의 상관관계의 크기는 상관관계 계수로 적용하였다 몬테카를로 해석에서 131 는 스펙트럼모사법으로 수치생성된 표본을 사용하였다 두 확률변수에 의한 기능경사보의 고유치응답변화도 산정을 위한 정식화를 제시하였고, 이를 유 한요소법을 이용하여 해석하고 그 결과를 몬테카를로 해석과 비교하였다 개별적인 두 변수는 고유치 및 고유벡터의 응답변화도에 미치는 영향의 특 성이 다르게 나타났는데, 재료밀도의 불확실성이 재료탄성계수의 그것보다 크게 나타났으며, 이들 사이 상관관계의 경우 음의 상관관계에서 최대의 응 답변화를 나타냈으며, 양의 상관관계 특히 완전양의 상관관계에서는 응답변 화도가 나타나지 않았다 이러한 특성은 1자유도에 대한 고유치 식에서 그 특성을 예견할 수 있다 제안된 정식화의 타당성을 제시하기 위하여 기능경 사재료 보의 다양한 구조인수에 대한 해석을 수행하고 이들에 대한 해석결 과를 제시하였다 핵심 용어: 유한요소법, 기능경사재료 보, 1차 섭동법, 스펙트럼모사법, 임 의재료상수 132 ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my advisor, Professor Hyuk Chun Noh for his tremendous support throughout my Ph.D research, for his motivation, patience and his vast knowledge His guidance helped me have more confidence to solve any difficult problems during my time of studying engineering In addition to my advisor, I would like to thank my entire dissertation committee members that include: Professor Jong Min Kim, Professor Jong Jae Lee, Professor Jung Joong Kim and Professor Dong Kyu Lee for devoting time to my dissertation I would like to thank Dr Ta Duy Hien, Ph.D student Lieu Xuan Qui for the close discussion I had with them and my Vietnamese friends at Sejong University Last but not the least; I would like to express my thanks to my family, my dear wife Pham Thi Lien, my lovely children Nguyen Pham Thanh Thanh, and Nguyen Viet Tung for their continual Seoul, August 2017 Nguyen Van Thuan 133 ... University Stochastic Finite Element Analysis for Free Vibration of Functionally Graded Beams Nguyen Van Thuan A dissertation submitted to the Faculty of the Sejong University in partial fulfillment of. .. Stochastic Finite Element Analysis for Free Vibration of Functionally Graded Beams Nguyen Van Thuan August 2017 Department of Civil and Environmental Engineering... vibration characteristics of functionally graded beams Chapter 1: Introduction Deterministic analysis Probabilistic analysis Chapter 2: Eigen analysis of the functionally graded beams with variable