Phân tích và chẩn đoán kết cấu dầm bằng vật liệu cơ tính biến thiên có nhiều vết nứt tt tiếng anh

MINISTRY OF EDUCATION AND TRAINING NATIONAL UNIVERSITY OF CIVIL ENGINEERING Ngo Trong Duc VIBRATION ANALYSIS AND CRACKS IDENTIFICATION ON MULTIPLE CRACKED BEAMLIKE STRUCTURES MADE OF FUNCTIONALLY GRADED MATERIALS Speciality: Engineering Mechanics Code: 9520101 SUMMARY OF DOCTORAL DISSERTATION ACADEMIC ADVISOR PROF DR TRAN VAN LIEN Ha Noi - 2019 The dissertation was completed in National University of Civil Engineering Academic advisor: Prof Dr Tran Van Lien National University of Civil Engineering Reviewer 1: Prof Sci Dr Nguyen Dong Anh Institute of Mechanics, Vietnam Academy of Science and Technology Reviewer 2: Prof Sci Hoang Xuan Luong Military Technology Academy Reviewer 3: Prof Sci Nguyen Tien Chuong ThuyLoi University The doctoral dissertation is presented at the level of the University Council of Dissertation Assessment’s meeting at the National University of Civil On 8h30,… March, 2019 Full-text of dissertation could be found out at Nation Library of Vietnam and NUCE Library 2 INTRODUCTION Reasons to choose the topic of dissertation Functionally Graded Material - FGM is one of the advanced composite materials that made from two component materials, mechanical characteristics of FGM material smoothly and continuously, so it avoids material splitting, stress concentration in contact surfaces that often happen in classic composites FGM is made for important structure components which operates in hard conditions in advanced industry such as: aerospace, manufacturing, automobile, optics, nuclear engineering,… Most of in use structures, include FGM structures, are containing errors and damages Structural damages are diversity and they occurred because of many different reasons In all structural damages, cracks are popular, its appearance decreases the local stiffness, changes dynamic characteristics and effects seriously to the performance of the structures So it is necessary and useful to monitor structural health periodically or continuously to identify damages in order to control and slow down damages improvement Then engineers can repair and maintenance structures to prevent seriously damages Recently, domestic and world researchers started to study the effects of cracks and cracks identification problem in FGM structures through NonDestructive Testing - NDT using dynamic characteristics such as: frequencies, mode shapes, vibration displacements, However, authors often study simple beam structure with limited number of cracks, the problem of complicated beam structures like multiple cracked continuous-beam hasn’t been studied yet Purpose of the research Contributing vibration modal of multiple cracked beam-like structures made of FGM using dynamic stiffness method (DSM) Therefore, offer some identification method to determine crack parameters in beam-like structures based on measured frequencies, mode shapes and vibration displacements Objects and scopes of the research Objects of the research: Beam-like structures are simple beams and continuous beams made of FGM that have one-side open cracks Scope of the research: - Beam-like structures made of FGM which mechanical characteristics varies though beam thickness (P-FGM) with predestination material, geometric parameters and boundary conditions - Cracks are one-side open cracks that perpendicular to beam axis We don’t consider the reasons and procedures of the formation and development of cracks Cracks at special positions like boundaries or joints are also out of the range of the research Methodologies of the research Theatrical methodology and numerical simulation calculation Scientific basis of the research Based on elastic theory, destructive mechanics, structural dynamic vibration, dynamic stiffness method and also recent researches about analysis and damage identification in the structures based on dynamic parameters Scientific and reality meaning of the research The research aims to solve some unsolved problems in vibration analysis and cracks identification in multiple cracked beam-like structure made of FGM The applications of the research will contribute to structural health monitoring and structural performance evaluation Therefore, we offer the suitable repairs and maintenances for the structures New achieved results of the research a) Modelling multiple cracked FGM beam as a single beam element using DSM and spring model of cracks Since then, the thesis has built dynamic stiffness matrix and nodal load vector of multiple cracked FGM beam element Timoshenko that is compressed and bended at the same time based on the spring model of crack b) Establishing the programs which has ability to analyse the changes of frequencies, mode shapes, and vibration displacement of multiple cracked FGM beam when crack parameters (number, position, depth), material parameters (Et/Eb ratio, material volume index n) or geometric parameters (L/h ratio) varies c) Appling wavelet analysis SWT and artificial neutral network ANN to propose some structural crack identification methods based on input data such as: frequencies, mode shapes or dynamic displacements Structure of the dissertation The dissertation included introduction, chapters and Conclusion The References included 169 references (16 domestic references, 153 international references) The dissertation has published 14 scientific papers included international papers (in the ISI journal list) and 10 domestic paper in domestic journals and conferences CHAPTER LITERATURE REVIEW This chapter is a literature review the most relevant technical publications related to the field of the research such as: Structural health monitoring; functionally graded material (FGM); structural damage model; crack models that are normally used in vibration analysis of frame structures; researches of the multiple cracked FGM beam-like structures; dynamic stiffness method The dissertation also review crack identification methods based on dynamic parameters; crack detection methods using wavelet analysis and artificial neutral network Besides, elastic spring model of cracks were presented to clarify multiple cracked beam element model which is used in the research Chapter conclusion evaluated the unsolved problem and set the research goals CHAPTER MULTIPLE CRACKED FGM BEAM VIBRATION MODEL 2.1 Vibration of intact Timoshenko FGM beam Considered the beam with the length L, rectangular section with dimensions A=b×h made of FGM (Fig 2.1) The FGM characteristic function is component style (P-FGM) z x u u0  w0  z w Neutral surface x h0 z Fig 2.2: Displacement of Timoshenko beam  E ( z ) G ( z )  ( z ) T  E b G b  b   E t  E b G t  G b  t   b  T Propose matrixes and vectors T n h h  z 1    ;  z 2 h 2 (2.1)   I11 0   I12  0  A11  A12   A   A12 A22  ; Π  0 A33  ; D()   2 I12 2 I 22  A33    0  A33  A33   I11  (2.23)  z  {U, ,W}T ; q  {P,0, Q}T Based on Hamilton principle, motion equation in frequency domain was Az  Πz  Dz  q (2.24) Solutions of free vibration differential equations were in the form of z0 (x,)  G(x,)C (2.32) with C was constant vector and G(x,)   G1(x,) G (x,)  1ek1x 2ek2 x 3ek3x   1ek1x 2ek2 x 3ek3x      (2.33) ek2 x ek3x  G1(x,)   ek1x ek2x ek3x  ; G2 (x,)   ek1x 1ek1x 2ek2 x 3ek3x  1ek1x 2ek2 x 3ek3x      Specific solutions of forced vibration differential equations were z q ( x ,  )  x  H ( x   ,  ) q ( ,  )d (2.34) where [H(x,)] is matrix of transfer functions that satisfied equations A H  Π H  D H  0 (2.35) With the left boundary conditions H(0)  [0] ; H (0)  A1 (2.36) Hence, general solution forced vibration differential equations were ~z c ( x ,  )  z c ( x ,  )  z q ( x ,  ) (2.39) 2.2 Continuous condition at crack positions Two equivalent springs model keY a h a) keX b) Fig 2.3: FGM beam with open crack and two equivalent springs model The beam has crack at position e The crack is modelled as equivalent springs: axial spring with the stiffness of keX and rotational spring with the stiffness of keY (Fig 2.3) Continuous conditions at crack positions are [16] U(e  0) U(e  0)  N(e) / keX ;(e  0)  (e  0)  M(e) / keY ; W(e  0) W(e  0) (2.40) N(e)  N(e  0)  N(e  0) ;Q(e  0)  Q(e  0) ; M(e  0)  M(e  0)  M(e) In vibration analysis of multiple cracked FGM beam, we use formula [7, 55] 1  F1( z)  2 (1 )h1 f1(s)   F2 ( z)  6 (1 )h f2 (s) (2.48) (2.49) These functions will be used in spring stiffness calculation based on the depth of cracks  RE  n  24  3RE  n RE  n R n 2  ( RE , n)  ;  ( RE , n)    E    RE   3(3  n) 2n 1 n  RE  11  n   f1 ( z )  s (0.6272  0.17248s  5.92134 s  10.7054 s  31.5685s  67.47 s   139.123s  146.682 s  92.3552 s ) f ( z )  s (0.6272  1.04533s  4.5948s  9.9736 s  20.2948s  33.0351s   47.1063s  40.7556 s  19.6 s ) 2.3 Vibrations of multiple cracked FGM Timoshenko beam   2.3.1 Crack function matrix G(x) and displacement expression z c ( x, ) Propose 3×3 matrix [Gc(x,)] as bellowed [G c ( x,  )]  [L( x,  )][ Σ] (2.58) where  0 1 cosh k1x 2 cosh k21x 3 cosh k3x 11 12 13  [L(x,)]   cosh k1x cosh k21x cosh k3x   21 22 23  , Σ    0   0  1 sinh k1x 2 sinh k2 x 3 sinh k3x  31 32 33  We defined the crack function matrix as G ( x, )  : x  G ( x, ) : x  G(x,)   c ; G(x,)   c 0 : x  0 : x    (2.60) Solutions of free vibration equations of the beam with n cracks were n zc ( x, )  z0 ( x, )   G( x  e j , ).μ j (2.62) j 1 where z ( x) is determined from (2.32) and  j  is 3×1 retriveal vector j 1 μ   z (e ,)   G(e  e ,) μ  ; j  1,2,3, , n j j j k k (2.63) k 1 2.3.2 Frequencies and mode shapes of multiple cracked Timoshenko beam For single span beam, boundary conditions on beam ends can be written B0 z c  x0   0 ; B L z c  xL   0 (2.64) Apply boundary conditions (2.64), general solutions of beam with n cracks were n   z ( x ,  )  G ( x ,  )  c     G( x  e j , ) χ j  CL   G L  x,   CL  j 1   (2.74) Frequency equations of multiple cracked FGM Timoshenko beam were ()  det[BLL ()]  0,[BLL ()]  BL  GL ( x, ) xL  (2.79) Each natural frequency j related to a mode shape  (x)  c G  ( x,  j ) C j  2.3.3 Forced vibration of multiple cracked Timoshenko beam j j L (2.80) Solutions of equations (2.24) can be written in the form of n   ɶ z ( x ,  ) G ( x ,  )   c     G( x  e j , ) χ j  CL   zq ( x, ) j 1   (2.83) Apply boudary conditions to equations (2.83), the full solutions of forced vibration equations were n  ɶ z ( x ,  )  G ( x ,  ) C  c     L  G(x  ej ,) χ j  .CL zq (x,) (2.87)  j1  2.4 Dynamic stiffness matrix and nodal load vector of multiple cracked FGM Timoshenko beam element : 2.4.1 Dynamic stiffness matrix and nodal load vector Considered beam element made of FGM which is bended and compressed at the same time Defined node coordinates and nodal forces as Fig 2.4 We obtained  Kˆ e  and Fˆ e  respectively are dynamic stiffness matrix and nodal load vector of multiple cracked FGM beam element z Q2 Q1 L N1 x N2 j i M M1 W W2 U1 U2 1 2 Fig 2.4 Node coordinates, nodal load of beam element     (2.97) ˆ ˆ K  ˆ  e ( )  U e  Pe ( )  Fe where Uˆ   {U ,  ,W ,U ,  ,W } ˆ  ; Pe   {N1 , M , Q1 , N , M , Q2 }T ;  K e 1  B Ψ ɶ    Ψ ɶ  F x0     (0,  )    ˆ  [K e ]    ɶ ɶ B F Ψ     Ψ    ( L,  )  xL     T e 1 2 Fˆ  e         B  Ψɶ      B  Ψɶ   B  z  F q x 0  ˆ {Fe }    BF  z q xL  F F 1    Ψ ɶ (0,  )    0  x0          ɶ ( L,  )   z q ( L)     Ψ      xL   (2.98) (2.99) with BF is free boundary condition matrix operator ej is position of jth crack and n ɶ  x,     G( x, )  G( x  e ) χɶ  Ψ  j   j  j 1 j 1 χɶ j   G(e j )   G(e j  ek )  χɶ k  ; j  1, 2,3, , n k 1 2.4.2 Matrix Assembling and boundary conditions Dynamic stiffness matrix, nodal load vector in general coordinate system ˆ ˆ ( )  ; {Fˆ }   {Fˆ } K  K e e  ( )      e e (2.102) Dynamix matrix and nodal load vector assembling with be operated through direct stiffness method [115] After applying boundary conditions, we obtained equations for structural analysis 2.4.3 Structural analysis using dynamic stiffness method a) Static analysis problem is in the form of ˆ (0) U ˆ  Fˆ (0) K (2.104) b) Free vibration problem is in the form of Kˆ (  ) Φ   (2.105) Where natural frequencies j are obtained from equations det Kˆ ( )  (2.106) Mode shape  j  corresponded to natural frequency j           (x)  C Ψˆ  x, Uˆ  j j j (2.107) j c) Forced vibration problem with harmonic load Then dynamic displacement of e element is  0 ˆ ˆ ˆ     zˆ e (x,)  Ψ(x,)  Ue  Ψ(x,)    zq (L)       zq (x,)     (2.110) 2.5 Block diagram of algorithms and programs 2.5.1 Block diagram of structural analysis using DSM (Diagram 2.1) 2.5.2 Block diagram of obtained programs (Diagram 2.2) 2.6 Conclusions of Chapter Establishing vibration differential equations of Timoshenko FGM beam in frequency domain taking into account real position of neutral Using spring model of cracks, the dissertation obtained frequency equations, mode shape expressions and dynamic displacement of multiple cracked Timoshenko FGM with different boundary conditions through DSM Establishing dynamic stiffness matrix and nodal load vector of multiple cracked Timoshenko FGM beam element which is bended and compressed at the same time through dynamic stiffness method Therefore, the dissertation obtained frequency equations, mode shape expressions and dynamic displacement of multiple cracked beam-like structure for structural analysis using dynamic stiffness method Establishing block diagram and algorithms of the programs that can determine frequencies, mode shapes and dynamic displacements of multiple cracked FGM structures The obtained results allowed us to study the effects of crack parameters (number, position, depth), geometric parameter, FGM material parameters and boundary conditions to dynamic characteristics of FGM beam-like structures (direct problem of damage structure analysis) These results also are basic data to solve inverse problem to identify crack parameters on FGM beam-like structures using dynamic characteristic (inverse problem of cracks identification on beam) CHAPTER VIBRATION ANALYSIS OF MULTIPLE CRACKED FGM BEAM-LIKE STRUCTURES 3.1 Result verification of proposed programs 3.1.1 Verifying natural frequency results Comparing the obtained natural frequencies from programs incase of homogeneous beam (Et=Eb=E, volume ratio index n=0) without or with cracks, and cracked FGM beam to results of previous studies, the obtained results are closed to the research results of Khiem & Lien [53], Aydin [24], Yu & Chu [104] and Su & Banerjee, they proved that the programs were reliability 3.1.2 Verifying mode shapes results Comparing the obtained mode shapes form the programs to the result of homogeneous beam in Lien and Hao research [162] and FGM beam in Su & Banerjee [91] research, the matched data proved the reliability of programs 3.2 Vibration analysis of intact FGM Timoshenko beam 3.2.1 The effects of neutral axis position to natural frequencies Considered simple supported FGM Timoshenko beam with material parameter as [7] We surveyed the effects of volume index n and ratio Et/Eb to the difference between neutral axis and middle axis (Fig 3.5), and the difference between natural frequencies calculated with neutral axis (NA) and with middle axis (MA) (Fig 3.6) 3.2.2 The effects of boundary conditions to natural frequencies So sánh tần số khơng thứ ngun i tính tốn theo lý thuyết với kết Su, Banerjee (S&B) [91] với dầm nguyên vẹn FGM ứng với L/h , số n điều kiện biên khác nhau: Dầm đơn giản (SS), hai đầu ngàm (CC) công xôn (CF) Ta thấy kết tính tốn gần với nghiên cứu S&B Error(%) 10 Fig 3.5 Effects of ratio Et/Eb and index n to neutral axis position Fig 3.6 Variation of 1 calculated with NA and MA 3.2.3 The effects of FGM material parameters to natural frequencies Analyze the variation of the first non-dimensional frequencies i of simple supported FGM Timoshenko beam with different ratio L/h , index n We saw that all frequencies decreased when n increased from with boundary conditions, when n1 (hoặc Eb/Et=75dB The dissertation also survey the effects of material parameters (index n and ratio Et/Eb), crack parameters (position, depth), and noise level SNR to detail coefficients of SWT Using ANN with input datas of the mode shapes or dynamic displacements to determine the number, position and depth of cracks on beam The detection results are highly accurate To decrease variable numbers of inverse problem, the dissertation combines ANN and SWT of the mode shapes and dynamic displacements to determine the number, position and depth of cracks on beam The results proved that this method has higher accuration with lower operation time GENERAL CONCLUSIONS The main new achieved results of the dissertation are: Modeling multiple cracked FGM beam as a single beam element by DSM with spring model of cracks Therefore, the dissertation established dynamic stiffness matrix and nodal load vector of multiple cracked Timoshenko FGM beam element that is compressed and bended at the same time while previous researches only established the dynamic stiffness model of intact FGM beam or finite element model of multiple cracked FGM beam Then we obtained frequency equations, mode shape expresstions and dynamic displacements to analyse vibration of multiple cracked beam-like structures by dynamic stiffness method Contributing programs to determine the natural frequencies, mode shapes and dynamic displacements of uncracked and cracked FGM beam using dynamic stiffness method taking into account of neutral axis position Then the dissertation verify the results from programs to other author’s results to 21 prove the reliable of programs Hence, the dissertation analyzed the variation of natural frequencies, mode shapes and dynamic displacements of the FGM beam-like structures (simply supported and continuous beam) with different crack parameters (number, postion, depth), geometric parameters, FGM parameters (volume ratio index n, ratio Et/Eb) and boundary conditions These results are new and reliable Applying wavelet analysis SWT and artificial neural network ANN to contribute some crack identification methods on beam based on measured input datas:  Using artificial neural network (ANN) of measured frequencies  Stationary wavelet transformation (SWT) of the mode shapes and dynamic displacements to determine the number and position of cracks on the structures  Using ANN with the input datas of the mode shapes or dynamic displacements to determine the number, position and depth of cracks on the structures  Combining ANN and SWT of the mode shapes or dynamic displacements to determine the number, position and the depth of cracks on the structures These are new results of crack identification on cracked FGM beam-like structures when we have mesured datas of natural frequencies, mode shapes or dynamic displacements These results offer a crack identification method on FGM beam-like structures SCIENCE PAPERS RELATED TO DISSERTATION HAVE BEEN PUBLISHED Trần Văn Liên, Ngô Trọng Đức, Nguyễn Tiến Khiêm (2016) Phân tích dao động tự dầm Timoshenko làm vật liệu chức có nhiều vết nứt Tuyển tập Hội nghị khoa học toàn quốc Vật liệu Kết cấu Composite – Cơ học, công nghệ ứng dụng, Trường Đại học Nha Trang, 391-399, 28-29/7/2016 Tran Van Lien, Nguyen Tien Khiem, Ngo Trong Duc (2016) Free vibration analysis of functionally graded Timoshenko beam using dynamic stiffness method Journal of Science and Technology in Civil Engineering, National University of Civil Engineering, 31 (10/2016), 19-28 Tran Van Lien, Ngo Trong Duc, Nguyen Tien Khiem (2016) Mode shape analysis of multiple cracked functionally graded Timoshenko beam Proceeding of the International Conference on Sustainable Developement in Civil Engineering, Hanoi, 15-16/11/2016, 213-223 Tran Van Lien, Ngo Trong Duc, Nguyen Tien Khiem (2017) Mode shape analysis of functionally graded Timoshenko beams Latin American Journal of 22 Solids and Structures, 14 (7), 1327-1344 (ISI Journal list) Trần Văn Liên, Nguyễn Tiến Khiêm, Ngô Trọng Đức (2017) Phân tích dao động cưỡng dầm Timoshenko vật liệu FGM có nhiều vết nứt Tạp chí khoa học công nghệ xây dựng, Trường Đại học Xây dựng, 11(3), 10-19 Trần Văn Liên, Ngô Trọng Đức, Nguyễn Tiến Khiêm (2017) Xây dựng ma trận độ cứng động lực véc tơ tải trọng quy nút phần tử dầm FGM Timoshenko có nhiều vết nứt ứng dụng vào phân tích dao động tự dầm liên tục nhiều nhịp Tạp chí khoa học công nghệ xây dựng, Trường Đại học Xây dựng, 11(3), 20-29 Tran Van Lien, Ngo Trong Duc, Nguyen Tien Khiem (2017) Free vibration analysis of functionally graded Timoshenko beams Latin American Journal of Solids and Structures, 14 (9), 1752-1766 (ISI Journal list) Tran Van Lien, Ngo Trong Duc, Nguyen Tien Khiem (2017) Mode shape analysis of multiple cracked functionally graded beam-like structures by using dynamic stiffness method Vietnam Journal of Mechanics, 39(3), 215-228 Trần Văn Liên, Ngô Trọng Đức, Dương Thế Hùng (2017) Xác định vết nứt dầm FGM phân tích wavelet với dạng dao động riêng Tuyển tập Hội nghị Cơ học toàn quốc lần thứ X Tập 3: Cơ học vật rắn Quyển 1, 702-709 10.Trần Văn Liên, Nguyễn Tiến Khiêm, Ngô Trọng Đức (2017) Phân tích dao động cưỡng dầm FGM liên tục nhiều nhịp phương pháp độ cứng động lực ứng dụng Tuyển tập Hội nghị Cơ học toàn quốc lần thứ X Tập 3: Cơ học vật rắn Quyển 1, 710-717 11.Ngô Trọng Đức, Trần Văn Liên, Nguyễn Thị Hường (2018) Xác định vết nứt dầm FGM mạng trí tuệ nhân tạo Tuyển tập Hội nghị Cơ học vật rắn biến dạng lần thứ XIV Thành phố Hồ Chí Minh 19-20/7/2018 12.Tran Van Lien, Ngo Trong Duc, Nguyen Tien Khiem (2018) A new form of frequency equation of functionally graded Timoshenko beams with arbitrary number of open transverse crarks Iranian Journal of Science and Technology: Transactions of Mechanical Engineering, 42(1), 1-18 (ISI Journal list) 13.Tran Van Lien, Ngo Trong Duc (2018) Crack identification in multiple cracked beams made of functionally graded material by using stationary wavelet transform of mode shapes Vietnam Journal of Mechanics (accepted) 14.Tran Van Lien, Ngo Trong Duc, Nguyen Tien Khiem (2019) Free and forced vibration analysis of multiple cracked FGM multi span continuous beams using dynamic stiffness method Latin American Journal of Solids and Structures, 16(2), e157 (ISI Journal list)  ... Đức, Nguyễn Tiến Khiêm (2016) Phân tích dao động tự dầm Timoshenko làm vật liệu chức có nhiều vết nứt Tuyển tập Hội nghị khoa học toàn quốc Vật liệu Kết cấu Composite – Cơ học, công nghệ ứng dụng,... Trần Văn Liên, Nguyễn Tiến Khiêm, Ngô Trọng Đức (2017) Phân tích dao động cưỡng dầm Timoshenko vật liệu FGM có nhiều vết nứt Tạp chí khoa học cơng nghệ xây dựng, Trường Đại học Xây dựng, 11(3),... trận độ cứng động lực véc tơ tải trọng quy nút phần tử dầm FGM Timoshenko có nhiều vết nứt ứng dụng vào phân tích dao động tự dầm liên tục nhiều nhịp Tạp chí khoa học công nghệ xây dựng, Trường