Phân tích đánh giá ổn định động của hệ thống điện khi có tác động của nguồn năng lượng điện gió

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 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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