Tài liệu hạn chế xem trước, để xem đầy đủ mời bạn chọn Tải xuống
1
/ 75 trang
THÔNG TIN TÀI LIỆU
Nội dung
TR I H C QU C GIA TP HCM NG I H C BÁCH KHOA L SÌU V Y FRACTURE ANALYSIS IN THIN PLATE USING KIRCHHOFF-LOVE PLATE THEORY AND AN EXTENDED MESHFREE METHOD Chuyên ngành: C k thu t Mư s : 8520101 LU N V N TH C S TP H CHÍ MINH, tháng n m 2021 CỌNG TRÌNH C HỒN THÀNH T I TR NG I H C BÁCH KHOA – HQG -HCM Cán b h ng d n khoa h c : PGS TS Tr ng Tích Thi n Cán b ch m nh n xét 1: PGS TS Nguy n Hoài S n Cán b ch m nh n xét 2: TS Nguy n Thanh Như Lu n v n th c s đ c b o v t i Tr ngày 15 tháng 07 n m 2021 ng i h c Bách Khoa, HQG Tp HCM Thành ph n H i đ ng đánh giá lu n v n th c s g m: PGS TS V Cơng Hịa PGS TS Tr ng Tích Thi n PGS TS Nguy n Hoài S n TS Nguy n Thanh Như TS Tr ng Quang Tri Xác nh n c a Ch t ch H i đ ng đánh giá LV Tr ngành sau lu n v n đư đ c s a ch a (n u có) CH T CH H I NG TR ng Khoa qu n lý chuyên NG KHOA KHOA H C NG D NG TR I H C QU C GIA TP.HCM NG I H C BÁCH KHOA C NG HÒA XÃ H I CH NGH A VI T NAM c l p - T - H nh phúc NHI M V LU N V N TH C S H tên h c viên: L Sìu V y MSHV: 1970507 Ngày, tháng, n m sinh: 16/11/1997 N i sinh: ng Nai Chuyên ngành: C k thu t Mã s : 8520101 I TÊN TÀI: Phân tích s r n n t t m m ng s d ng lý thuy t t m Kirrchhoff-Love ph ng pháp không l i m r ng Fracture analysis in thin plate using Kirchhoff-Love plate theory and an extended meshfree method NHI M V VÀ N I DUNG: Nghiên c u lý thuy t t m Kirchhoff-Love Nghiên c u ph ng pháp không l i RPIM Phát tri n công th c XRPIM dành cho t m Kirchhoff-Love b n t Phát tri n m t ch ng trình gi i tốn t m Kirchhoff-Love b ng ph ng pháp RPIM Phát tri n ch ng trình đ phân tích ng x dao đ ng t c a t m Kirchhoff-Love b n t II NGÀY GIAO NHI M V : 22/02/2021 III NGÀY HOÀN THÀNH NHI M V : 05/12/2021 IV.CÁN B H NG D N: PGS TS Tr ng Tích Thi n Tp HCM, ngày 30 tháng n m 2021 CÁN B H NG D N CH NHI M B (H tên ch ký) MÔN ÀO T O (H tên ch ký) TR NG KHOA KHOA H C (H tên ch ký) NG D NG Acknowledgement I would like to express my endless thanks and gratefulness to my supervisor Assoc Prof Dr Thien Tich Truong, Dr Nha Thanh Nguyen and Dr Minh Ngoc Nguyen for their supports and advices during the process of completion of my thesis Without their instructions, the thesis would have been impossible to be done effectively I really appreciate the lecturers of Department of Engineering Mechanics for their comments and helps while I am doing this thesis, which leads me to the right direction I also acknowledge the support of time and facilities from Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for this thesis Last but not least, a special thanks to my parents for their love, care and have most assistances and motivation me for the whole of my life i Tóm tắt luận văn Đối với tốn chịu uốn, sử dụng lý thuyết để mô tả cấu trúc mỏng tốn so với sử dụng mơ hình 3D Trong lý thuyết lý thuyết Kirchhoff-Love thích hợp để phân tích cấu trúc mỏng Nếu bỏ qua bậc tự mặt phẳng nút có bậc tự - độ võng Vì lý đó, thành phần trường chuyển vị tính theo độ võng Tuy nhiên, phương pháp phần tử hữu hạn cổ điển (FEM) cần phép biến đổi toán học phức tạp để xây dựng phần tử thỏa mãn yêu cầu lý thuyết Kirchhoff-Love Vì vậy, phương pháp nội suy điểm hướng kính (RPIM) sử dụng để mơ mỏng Kirchhoff-Love luận văn Bên cạnh đó, việc phân tích kết cấu bị nứt quan trọng liên quan đến tuổi thọ kết cấu Do đó, luận văn sử dụng phương pháp nội suy điểm hướng kính mở rộng (XRPIM) để khảo sát dao động tự Kirchhoff-Love bị nứt XRPIM phát triển dựa RPIM nên yêu cầu đạo hàm cấp hai lý thuyết Kirchhoff-Love xử lý cách dễ dàng Kết mô số từ nghiên cứu so sánh với kết tác giả khác công bố để kiểm chứng tính xác phương pháp ii Abstract For the plate bending problems, using a plate theory for modelling thin plate structure is less computational cost than modelling it in 3D The Kirchhoff-Love plate theory is appropriate for analysing thin plate structures If the membrane deformation is ignored in the KirchhoffLove plate, each node has only one degree of freedom – the deflection For that reason, the components of the displacement field are calculated only in terms of deflection The classical finite element method (FEM), however, needs complex mathematical transformations to formulate a new element that satisfies the Kirchhoff-Love theory For this reason, the radial point interpolation method (RPIM) is used for modelling thin Kirchhoff-Love plate in this thesis Besides, the analysis of cracked structures is important because it is related to the lifetime of the structures Therefore, this thesis employs the extended radial point interpolation method (XRPIM) to investigate the free vibration of the cracked Kirchhoff-Love plate The XRPIM is based on RPIM so the requirement for calculating the second-order derivative in the Kirchhoff-Love theory is easily done The numerical results from this study are compared with other published results to verify the accuracy of the method iii Declarations I hereby declare that this master thesis represents my own work which has been done after registration for the degree of Master in Engineering Mechanics at Ho Chi Minh city University of Technology, VNU – HCM, Viet Nam and has not been previously included in a thesis or dissertation submitted to this or any other institution for a degree, diploma or other qualifications Author iv Contents List of Figures vii List of Tables ix List of Abbreviations and Nomenclatures x Introduction 1.1 State of the Art 1.2 Scope of study 1.3 Research objectives 1.4 Author’s contributions 1.5 Thesis outline Methodology 2.1 The Kirchhoff-Love plate theory 2.1.1 Equilibrium equation 2.1.2 Constitutive equation 2.1.3 Governing equation 2.1.4 Finite element approximation 10 The Radial Point Interpolation Method 14 2.2.1 Brief introduction to the RPIM 14 2.2.2 RPIM shape functions construction 14 The extended RPIM for the cracked Kirchhoff-Love plate 19 2.2 2.3 Implementation 23 3.1 Compute stiffness matrix and mass matrix 23 3.2 Compute strain computing matrix 25 v Numerical Results 27 4.1 Square plate under uniform pressure 27 4.2 Plate with a central crack 29 4.2.1 Simply supported square plate 29 4.2.2 Simply supported rectangular plate 31 4.2.3 Clamped circular plate 34 Plate containing a side crack 38 4.3.1 Simply supported square plate 38 4.3.2 Simply supported rectangular plate 38 4.3.3 Clamped annular plate 42 Square plate with an oblique crack 42 4.4.1 Central crack 44 4.4.2 Side crack 49 4.3 4.4 Conclusion and outlook 51 5.1 Conlusions 51 5.2 Future works 52 Bibliography 54 vi List of Figures 2.1 Internal and external forces on the plate 2.2 11 2.4 Plate before and after deformation ∂w ∂w , per node A rectangular element with three degrees of freedom w, ∂x1 ∂x2 Discrete nodes (gray dots) and support domains 2.5 Pascal triangle of monomials for 2D case 16 2.6 Split nodes and tip nodes around the crack curve 20 2.7 Illustration of tangential and normal direction for a crack 21 2.8 Global coordinate system and local coordinate systems 21 3.1 Algorithm for computing K and M matrices 24 3.2 Algorithm for computing B and N matrices 26 4.1 Geometry of the square plate and the coordinate system 28 4.2 Convergence of the result 29 4.3 Deflection of the simply supported square plate 30 4.4 Deflection at the center line y = 30 4.5 Geometries of a square plate and a rectangular plate containing a central crack 31 4.6 Mode shapes of five lowest modes of a square plate containing a central crack 33 4.7 Mode shapes of five lowest modes of a rectangular plate containing a central 2.3 15 crack 35 4.8 A circular plate containing a central crack 35 4.9 Discrete model of the circular plate 36 4.10 Mode shapes of five lowest modes of a circular plate containing a central crack 37 4.11 Geometry of square plate and rectangular plate with a side crack 38 4.12 Mode shapes of five lowest modes of a square plate with a side crack 40 4.13 Mode shapes of five lowest modes of a rectangular plate with a side crack 40 4.14 A annular plate containing two symmetric side cracks 42 vii Table 4.9: Non-dimensional frequency parameter ω ¯ of a simply supported square plate with a central oblique crack Angle c/a 0.1 0.2 0.3 30o 0.4 0.5 0.6 Ref results Mode Mode Mode Mode Mode XCS-DSG3 [60] 19.633 49.376 49.419 79.060 98.098 Ritz method [59] 19.660 49.340 49.350 78.880 97.840 This study 19.670 49.343 49.344 79.001 97.868 XCS-DSG3 [60] 19.338 49.132 49.372 78.703 95.459 Ritz method [59] 19.320 49.170 49.320 78.490 94.860 This study 19.417 49.264 49.317 78.714 95.593 XCS-DSG3 [60] 18.787 48.217 49.212 77.747 91.911 Ritz method [59] 18.830 48.440 49.200 77.630 91.710 This study 19.052 48.864 49.234 78.130 92.989 XCS-DSG3 [60] 18.257 46.307 48.947 76.359 89.866 Ritz method [59] 18.230 46.520 48.920 76.140 89.460 This study 18.508 47.685 48.994 76.983 90.393 XCS-DSG3 [60] 17.667 43.104 48.495 74.594 88.364 Ritz method [59] 17.580 42.910 48.430 74.250 88.140 This study 17.949 46.237 48.569 75.477 88.915 XCS-DSG3 [60] 17.034 38.391 47.842 72.539 85.347 Ritz method [59] 16.930 37.870 47.700 72.290 84.700 This study 17.321 44.666 47.867 73.553 87.789 46 Table 4.10: Non-dimensional frequency parameter ω ¯ of a simply supported square plate with a central oblique crack Angle c/a 0.1 0.2 0.3 45o 0.4 0.5 0.6 Ref results Mode Mode Mode Mode Mode XCS-DSG3 [60] 19.603 49.357 49.413 79.015 97.897 Ritz method [59] 19.660 49.340 49.350 78.850 97.890 This study 19.662 49.342 49.344 78.998 97.800 XCS-DSG3 [60] 19.262 49.056 49.335 78.511 95.172 Ritz method [59] 19.320 49.170 49.320 78.350 95.120 This study 19.373 49.242 49.313 78.574 95.375 XCS-DSG3 [60] 19.010 48.540 49.244 78.005 93.642 Ritz method [59] 18.820 48.410 49.190 77.270 92.310 This study 19.000 48.814 49.216 77.804 93.064 XCS-DSG3 [60] 18.396 46.626 48.949 76.364 91.287 Ritz method [59] 18.210 46.480 48.890 75.560 90.570 This study 18.404 47.348 48.943 76.233 90.956 XCS-DSG3 [60] 17.677 42.737 48.408 74.169 90.179 Ritz method [59] 17.530 42.850 48.330 73.550 89.950 This study 17.748 44.530 48.433 74.271 90.100 XCS-DSG3 [60] 16.916 37.318 47.564 71.953 83.596 Ritz method [59] 16.840 37.850 47.510 71.600 84.580 This study 17.159 41.492 47.740 72.545 89.890 47 raises, the non-dimensional frequency falls In the meantime, no observable trend is caught when the crack angle increases However, a significant deviation of non-dimensional frequency in this study is observed when the angle α increases The cause of this deviation can be explained by the node distribution The obtained results were computed in a uniform distribution of node (not aligned with the crack) Now, consider the node distribution aligned with the crack, as shown in Figure 4.18 The non-dimensional frequency of ratio c/a = 0.6 and α = 45o is shown in Table 4.11 It is seen that the result improves significantly when the node distributed aligned with the crack Figure 4.18: Node distribution Left: Not aligned with the crack, Right: Aligned with the crack Table 4.11: Non-dimensional frequency parameter ω ¯ of a simply supported square plate with a central crack The c/a ratio is equal 0.6 and α = 45o Ref results Mode Mode Mode Mode Mode XCS-DSG3 [60] 16.916 37.318 47.564 71.953 83.596 Ritz method [59] 16.840 37.850 47.510 71.600 84.580 Not aligned 17.159 41.492 47.740 72.545 89.890 Aligned 16.983 38.853 47.600 72.017 85.834 48 4.4.2 Side crack In the side oblique crack case, the relationship between a and b is b = 0.25a, a ratio of t/a = 0.001 is used In this case, the plate is discretized into a set of 50 × 50 nodes and the boundary condition is SSSS Various c/a ratios are considered in this study Tables 4.12 and 4.13 shows the first five non-dimensional frequencies obtained by different methods The results in this study show good agreement compared to other researchers The mode shapes of α = 30° are illustrated in Figure 4.19 Table 4.12: Non-dimensional frequency parameter ω ¯ of a simply supported square plate with a side oblique crack Angle c/a 0.1 0.2 0.3 15o 0.4 0.5 0.6 Ref results Mode Mode Mode Mode Mode XCS-DSG3 [60] 19.719 49.392 49.405 79.086 99.057 Ritz method [59] 19.730 49.320 49.350 78.920 98.660 This study 19.739 49.332 49.432 79.115 98.557 XCS-DSG3 [60] 19.666 49.117 49.378 78.656 98.704 Ritz method [59] 19.680 49.040 49.340 78.500 98.360 This study 19.683 49.066 49.400 78.690 98.220 XCS-DSG3 [60] 19.487 48.228 49.313 77.373 97.326 Ritz method [59] 19.510 48.240 49.290 77.230 97.170 This study 19.527 48.328 49.349 77.549 97.528 XCS-DSG3 [60] 19.122 47.016 48.845 75.058 85.930 Ritz method [59] 19.150 46.970 48.880 74.820 86.370 This study 19.201 47.175 49.070 75.728 90.262 XCS-DSG3 [60] 18.526 45.482 47.090 65.959 77.503 Ritz method [59] 18.560 45.470 47.090 65.760 77.420 This study 18.633 45.786 47.618 68.102 78.180 XCS-DSG3 [60] 17.748 40.816 45.690 59.006 76.083 Ritz method [59] 17.790 40.540 45.610 58.760 76.030 This study 17.887 41.658 45.723 59.468 76.336 It is indicated from the data that with the same crack angle, when the crack length increases, the non-dimensional frequency decreases In the meanwhile, no trend is observed when the crack angle increases 49 Table 4.13: Non-dimensional frequency parameter ω ¯ of a simply supported square plate with a side oblique crack Angle c/a 0.1 0.2 0.3 30o 0.4 0.5 0.6 Ref results Mode Mode Mode Mode Mode XCS-DSG3 [60] 19.711 49.372 49.401 79.073 98.999 Ritz method [59] 19.720 49.310 49.350 78.940 98.610 This study 19.734 49.317 49.435 79.127 98.517 XCS-DSG3 [60] 19.641 49.117 49.379 78.848 98.406 Ritz method [59] 19.640 49.030 49.350 78.690 98.030 This study 19.656 49.035 49.422 78.898 97.911 XCS-DSG3 [60] 19.413 48.318 49.301 77.847 95.991 Ritz method [59] 19.440 48.330 49.300 77.780 96.120 This study 19.466 48.347 49.381 78.107 96.465 XCS-DSG3 [60] 19.064 47.436 48.863 75.212 87.651 Ritz method [59] 19.070 47.390 48.890 75.120 87.140 This study 19.108 47.422 49.111 76.193 89.954 XCS-DSG3 [60] 18.409 45.955 46.587 65.453 79.584 Ritz method [59] 18.500 46.500 46.670 66.220 79.560 This study 18.556 46.638 47.169 67.722 80.339 XCS-DSG3 [60] 17.729 39.266 46.292 60.717 77.834 Ritz method [59] 17.780 39.990 46.380 60.770 77.660 This study 17.839 41.298 46.291 61.241 78.039 Figure 4.19: Mode shapes of five lowest mode of a square plate containing a side oblique crack 50 Chapter Conclusion and outlook 5.1 Conlusions This thesis has successfully employed the XRPIM to examine the free vibration behavior of the cracked Kirchhoff-Love plate Based on the numerical results obtained in Chapter 4, the following conclusions can be drawn: • Various numerical examples show the high accuracy of the approach The results also indicate the trend in cracked plate, it is observed that the free vibration frequency decreases when the crack length increases It means the plate structure is weaken when the crack length increases, as expected • For thin plate structure, the Kirchhoff-Love plate theory is widely used in the analysis The deflection of the plate w is the only unknown variable to find so that the computational cost and time are reduced • The main assumption of the Kirchhoff-Love theory - the transverse shear strain is neglected - leads to one major drawback of the theory that it is not appropriate for thicker plate • By using RPIM, it is not necessary to establish a new shape function (Hermite function) and deal with complicated math transformation The RPIM only employs the available shape function to satisfy the second-order derivative requirement Therefore, using the RPIM is easier and more appropriate for modelling the Kirchhoff-Love plate than using FEM • In the XRPIM, the discontinuity in the displacement field along the crack path and the singularity in the stress fields at the crack tip are described by using enrichment functions instead of defined it explicitly in the problem’s geometry 51 5.2 Future works This thesis basically investigates the free vibration of the cracked Kirchhoff-Love plate using XRPIM Based on the current work and the obtained results, the author proposes the following topics for the future works: • Evaluation of the stress intensity factors of the cracked Kirchhoff-Love plate using XRPIM • Crack propagation in the cracked Kirchhoff-Love plate using XRPIM • Fracture analysis of thin shell structure using the Kirchhoff-Love theory and XRPIM 52 List of Publications Publications are directly related to the thesis Publication A: Thien Tich Truong, Vay Siu Lo, Minh Ngoc Nguyen, Nha Thanh Nguyen, and Dinh Kien Nguyen Evaluation of fracture parameters in cracked plates using an extended meshfree method Engineering Fracture Mechanics, 247:107671, apr 2021 Publication B: Vay Siu Lo, Nha Thanh Nguyen, Minh Ngoc Nguyen, and Thien Tich Truong Free vibration analysis of cracked Kirchhoff-Love plate using the Extended Radial Point Interpolation Method Vietnam Journal of Science and Technology, 2021 Publications are not directly related to the thesis Publication C: Thien Tich Truong, Vay Siu Lo, Minh Ngoc Nguyen, Nha Thanh Nguyen, and Dinh Kien Nguyen A novel meshfree Radial Point Interpolation Method with discrete shear gap for nonlinear static analysis of functionally graded plates Engineering with Computers, Reviewing Publication D: Thien Tich Truong, Vay Siu Lo, Bang Kim Tran, Nha Thanh Nguyen and Minh Ngoc Nguyen An extended consecutive-interpolation quadrilateral element (XCQ4) applied to simulate behavior of a bimaterial interface crack Vietnam Journal of Science and Technology, 2021 Publication E: Vay Siu Lo, Nha Thanh Nguyen, Minh Ngoc Nguyen and Thien Tich Truong Impact force analysis using the B-spline material point method Science & Technology Development Journal – Engineering and Technology, 4(1):721-729, 2021 53 Bibliography [1] Michele D'Ottavio and Olivier Polit Classical, first order, and advanced theories In Stability and Vibrations of Thin Walled Composite Structures, pages 91–140 Elsevier, 2017 [2] Rudolph Szilard Theories and Applications of Plate Analysis John Wiley & Sons, Inc., jan 2004 [3] O A Bauchau and J I Craig Kirchhoff plate theory In Structural Analysis, pages 819–914 Springer Netherlands, 2009 [4] K Bhaskar and T.K Varadan Retd Classical plate theory In Plates, pages 11–32 John Wiley & Sons, Ltd, jul 2014 [5] Antonio J M Ferreira and Nicholas Fantuzzi MATLAB Codes for Finite Element Analysis Springer International Publishing, 2020 [6] J N Reddy An introduction to the finite element method McGraw-Hill Higher Education, New York, NY, 2006 [7] Eugenio O˜nate Thin plates kirchhoff theory In Structural Analysis with the Finite Element Method Linear Statics, pages 233–290 Springer Netherlands, 2013 [8] J Blaauwendraad Plates and FEM Springer Netherlands, 2010 [9] Eric Reissner On the theory of bending of elastic plates Journal of Mathematics and Physics, 23(1-4):184–191, apr 1944 [10] T J R Hughes and T E Tezduyar Finite elements based upon mindlin plate theory with particular reference to the four-node bilinear isoparametric element Journal of Applied Mechanics, 48(3):587–596, sep 1981 [11] Thomas C Assiff and David H Y Yen On the solutions of clamped reissner-mindlin plates under transverse loads Quarterly of Applied Mathematics, 45(4):679–690, dec 1987 54 [12] Douglas N Arnold and Richard S Falk A uniformly accurate finite element method for the reissner–mindlin plate SIAM Journal on Numerical Analysis, 26(6):1276–1290, dec 1989 [13] Chen Wanji and Y K Cheung Refined 9-dof triangular mindlin plate elements International Journal for Numerical Methods in Engineering, 51(11):1259–1281, aug 2001 [14] C.M Wang, G.T Lim, J.N Reddy, and K.H Lee Relationships between bending solutions of reissner and mindlin plate theories Engineering Structures, 23(7):838– 849, jul 2001 [15] Xiu Ye, Shangyou Zhang, and Zhimin Zhang A locking-free weak galerkin finite element method for reissner–mindlin plate on polygonal meshes Computers & Mathematics with Applications, 80(5):906–916, sep 2020 [16] Ibrahim Bitar and Benjamin Richard Mindlin-reissner plate formulation with enhanced kinematics: Theoretical framework and numerical applications Engineering Fracture Mechanics, 225:106839, feb 2020 [17] Noăel Challamel and Isaac Elishakoff A brief history of first-order shear-deformable beam and plate models Mechanics Research Communications, 102:103389, dec 2019 [18] J Freund Shear-corrected reissner-mindlin plate model Composite Structures, 211:144–153, mar 2019 [19] J N Reddy A simple higher-order theory for laminated composite plates Journal of Applied Mechanics, 51(4):745–752, dec 1984 [20] A Bhar, S.S Phoenix, and S.K Satsangi Finite element analysis of laminated composite stiffened plates using FSDT and HSDT: A comparative perspective Composite Structures, 92(2):312–321, jan 2010 [21] Ashok Kumar Ghosh and K.C Biswal Free-vibration analysis of stiffened laminated plates using higher-order shear deformation theory Finite Elements in Analysis and Design, 22(2):143–161, jun 1996 [22] K.C Biswal and A.K Ghosh Finite element analysis for stiffened laminated plates using higher order shear deformation theory Computers & Structures, 53(1):161–171, oct 1994 55 [23] Chien H Thai, Loc V Tran, Dung T Tran, T Nguyen-Thoi, and H Nguyen-Xuan Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method Applied Mathematical Modelling, 36(11):5657–5677, nov 2012 [24] N Moes, J Dolbow, and T Belytschko A finite element method for crack growth without remeshing International Journal for Numerical Methods in Engineering, 46:131–150, 1999 [25] Thomas-Peter Fries Extended finite element methods (XFEM) In Encyclopedia of Continuum Mechanics, pages 1–10 Springer Berlin Heidelberg, oct 2017 [26] Extended Finite Element and Meshfree Methods Elsevier, 2020 [27] Amir R Khoei Extended Finite Element Method John Wiley & Sons, 2014 [28] Amir Nasirmanesh and Soheil Mohammadi XFEM buckling analysis of cracked composite plates Composite Structures, 131:333–343, nov 2015 [29] Soheil Mohammadi XFEM Fracture Analysis of Composites John Wiley & Sons, Ltd, sep 2012 [30] John Dolbow, Nicolas Moăes, and Ted Belytschko Modeling fracture in mind- lin–reissner plates with the extended finite element method International Journal of Solids and Structures, 37(48-50):7161–7183, nov 2000 [31] Jun Li, Zahra Sharif Khodaei, and M.H Aliabadi Dynamic dual boundary element analyses for cracked mindlin plates International Journal of Solids and Structures, 152-153:248–260, nov 2018 [32] J Useche Fracture dynamic analysis of cracked reissner plates using the boundary element method International Journal of Solids and Structures, 191-192:315–332, may 2020 [33] J´er´emie Lasry, Yves Renard, and Michel Salaăun Stress intensity factors computation for bending plates with extended finite element method International Journal for Numerical Methods in Engineering, 91(9):909–928, jun 2012 [34] S.J Rouzegar and M Mirzaei Modeling dynamic fracture in kirchhoff plates and shells using the extended finite element method Scientia Iranica, jan 2013 56 [35] J´er´emie Lasry, Julien Pommier, Yves Renard, and Michel Salaăun eXtended finite element methods for thin cracked plates with kirchhoff-love theory International Journal for Numerical Methods in Engineering, 84(9):1115–1138, oct 2010 [36] Maurice Petyt Introduction to Finite Element Vibration Analysis Cambridge University Press, 2010 [37] N Nguyen-Thanh, N Valizadeh, M.N Nguyen, H Nguyen-Xuan, X Zhuang, P Areias, G Zi, Y Bazilevs, L De Lorenzis, and T Rabczuk An extended isogeometric thin shell analysis based on kirchhoff–love theory Computer Methods in Applied Mechanics and Engineering, 284:265–291, feb 2015 [38] Nhon Nguyen-Thanh, Weidong Li, and Kun Zhou Static and free-vibration analyses of cracks in thin-shell structures based on an isogeometric-meshfree coupling approach Computational Mechanics, 62(6):1287–1309, mar 2018 [39] Vitor M A Leit˜ao A meshless method for kirchhoff plate bending problems International Journal for Numerical Methods in Engineering, 52(10):1107–1130, dec 2001 [40] Y Liu, Y C Hon, and K M Liew A meshfree hermite-type radial point interpolation method for kirchhoff plate problems International Journal for Numerical Methods in Engineering, 66(7):1153–1178, 2006 [41] G R Liu and Y T Gu A point interpolation method for two-dimensional solids International Journal for Numerical Methods in Engineering, 50(4):937–951, 2001 [42] Nha Thanh Nguyen, Tinh Quoc Bui, Chuanzeng Zhang, and Thien Tich Truong Crack growth modeling in elastic solids by the extended meshfree galerkin radial point interpolation method Engineering Analysis with Boundary Elements, 44:87–97, jul 2014 [43] Nha Thanh Nguyen, Tinh Quoc Bui, and Thien Tich Truong Transient dynamic fracture analysis by an extended meshfree method with different crack-tip enrichments Meccanica, 52(10):2363–2390, nov 2016 [44] Nha Thanh Nguyen, Tinh Quoc Bui, Minh Ngoc Nguyen, and Thien Tich Truong Meshfree thermomechanical crack growth simulations with new numerical integration scheme Engineering Fracture Mechanics, 235:107121, aug 2020 [45] Thien Tich Truong, Vay Siu Lo, Minh Ngoc Nguyen, Nha Thanh Nguyen, and Dinh Kien Nguyen Evaluation of fracture parameters in cracked plates using an extended meshfree method Engineering Fracture Mechanics, 247:107671, apr 2021 57 [46] G.R LIU and X.L CHEN A MESH-FREE METHOD FOR STATIC AND FREE VIBRATION ANALYSES OF THIN PLATES OF COMPLICATED SHAPE Journal of Sound and Vibration, 241(5):839–855, apr 2001 [47] S Shojaee, E Izadpanah, N Valizadeh, and J Kiendl Free vibration analysis of thin plates by using a NURBS-based isogeometric approach Finite Elements in Analysis and Design, 61:23–34, nov 2012 [48] J.G Wang and G.R Liu Radial point interpolation method for elastoplastic problems pages 703–708, 01 2000 [49] G R Liu Meshfree methods : moving beyond the finite element method CRC Press, Boca Raton, 2010 [50] Tan-Van Vu, Ngoc-Hung Nguyen, Amir Khosravifard, M.R Hematiyan, Satoyuki Tanaka, and Tinh Quoc Bui A simple FSDT-based meshfree method for analysis of functionally graded plates Engineering Analysis with Boundary Elements, 79:1–12, jun 2017 [51] H.S Yang and C.Y Dong Adaptive extended isogeometric analysis based on PHTsplines for thin cracked plates and shells with kirchhoff–love theory Applied Mathematical Modelling, 76:759–799, dec 2019 [52] Stephen Timoshenko Theory of plates and shells McGraw-Hill, New York, 1959 [53] Robert L Taylor and Ferdinando Auricchio Linked interpolation for reissner-mindlin plate elements: Part II—a simple triangle International Journal for Numerical Methods in Engineering, 36(18):3057–3066, sep 1993 [54] Abderahim Belounar, Sadok Benmebarek, Mohamed Nabil Houhou, and Lamine Belounar Static, free vibration, and buckling analysis of plates using strain-based reissner–mindlin elements International Journal of Advanced Structural Engineering, 11(2):211–230, may 2019 [55] T Nguyen-Thoi, P Phung-Van, H Nguyen-Xuan, and C Thai-Hoang A cell-based smoothed discrete shear gap method using triangular elements for static and free vibration analyses of Reissner-Mindlin plates International Journal for Numerical Methods in Engineering, 91(7):705–741, may 2012 [56] B Stahl and L.M Keer Vibration and stability of cracked rectangular plates International Journal of Solids and Structures, 8(1):69–91, jan 1972 58 [57] K.M Liew, K.C Hung, and M.K Lim A solution method for analysis of cracked plates under vibration Engineering Fracture Mechanics, 48(3):393–404, jun 1994 [58] M Bachene, R Tiberkak, and S Rechak Vibration analysis of cracked plates using the extended finite element method Archive of Applied Mechanics, 79(3):249–262, apr 2008 [59] C.S Huang and A.W Leissa Vibration analysis of rectangular plates with side cracks via the ritz method Journal of Sound and Vibration, 323(3-5):974–988, jun 2009 [60] T Nguyen-Thoi, T Rabczuk, T Lam-Phat, V Ho-Huu, and P Phung-Van Free vibration analysis of cracked mindlin plate using an extended cell-based smoothed discrete shear gap method (XCS-DSG3) Theoretical and Applied Fracture Mechanics, 72:150– 163, aug 2014 59 PH N LÝ L CH TRÍCH NGANG H tên: L Sìu V y Ngày, tháng, n m sinh: 16/11/1997 N i sinh: ng Nai a ch liên l c: 106B4, 268 Lý Th ng Ki t, ph ng 14, qu n 10, Tp.HCM QUÁ TRÌNH ÀO T O T 2015 đ n 2019: H c i h c t i: Tr ng k thu t T 2019 đ n 2021: H c Th c s t i: Tr ng k thu t i h c Bách Khoa Tp.HCM, chuyên ngành: C i h c Bách Khoa Tp.HCM, chun ngành: C Q TRÌNH CƠNG TÁC T 2020 đ n nay: Làm vi c t i B môn C k thu t, khoa Khoa h c ng d ng, tr h c Bách Khoa Tp.HCM ng i ... n H i đ ng đánh giá lu n v n th c s g m: PGS TS V Cơng Hịa PGS TS Tr ng Tích Thi n PGS TS Nguy n Hoài S n TS Nguy n Thanh Như TS Tr ng Quang Tri Xác nh n c a Ch t ch H i đ ng đánh giá LV Tr ngành... ch ng trình đ phân tích ng x dao đ ng t c a t m Kirchhoff-Love b n t II NGÀY GIAO NHI M V : 22/02/2021 III NGÀY HOÀN THÀNH NHI M V : 05/12/2021 IV.CÁN B H NG D N: PGS TS Tr ng Tích Thi n Tp HCM,... dụng mơ hình 3D Trong lý thuyết lý thuyết Kirchhoff-Love thích hợp để phân tích cấu trúc mỏng Nếu bỏ qua bậc tự mặt phẳng nút có bậc tự - độ võng Vì lý đó, thành phần trường chuyển vị tính theo