DAMAGE IDENTIFICATION OF STRUCTURES USING STOCHASTIC MODELS

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1 23 2 Vo.23 No Dec. 26 ENGINEEING MECHANIC (26) * 2 ( ) (FEM) FEM O34 A DAMAGE IDENIFICAION OF UCUE UING OCHAIC MODE * HUANG Bn, HI Wen-ha 2 (. choo of Cv Engneerng and Archtecture, Wuhan Unversty of echnoogy, Wuhan, Hube 437, Chna 2. choo of Cv Engneerng, Wenzhou Unversty, Wenzhou, Zhejang 325, Chna) Abstract: Based on recursve stochastc fnte eeent ethod a rando daage dentfcaton ethod for structures s deveoped. Contro equatons of rando daage ndex are set up after rando daage ndex s defned. tructura uncertantes and easureent noses are both consdered n the contro equatons. tatstcs of rando daage ndex are obtaned after the contro equatons are soved utzng recursve stochastc fnte eeent ethod. he resuts show that structura daages can be dentfed whe ode errors and easureent noses are both consdered. It s aso found that the ethod presented has the sae effectveness as Monte-Caro suaton ethod for dentfyng daages of rando structures n wde range of rando fuctuaton. Key words: rando structures; daage dentfcaton; stochastc ode; recursve stochastc fnte eeent ethod; nonorthogona poynoa expansons Baruch []..aw [2] W.X.en [3] Doebng [4] tubbs [5] (5286)(2532) * (968)(E-a: bwhuang@263.net) (979)(E-a: swank@sna.co.cn)

2 48 K tubbs [6] [7] Papadopouos [8] [8] Yong Xa [9] [8] [9] FEM N ( K λm) Φ () K M N N λ Φ ( K λm) Φ (2) K M N N λ Φ ()(2) 2 K λ Φ K K + K (3) M M (4) λ λ+ λ (5) Φ Φ+ Φ (6) K K α K (7) α K (3)~(7)(2)() λ M e Φ Φ α Φ K Φ (8) λ Φ α 3 ξ () K λ Φ (8) λξ M ( ) Φξ ( ) Φ α( ξ) Φξ ( ) K Φ (9) λ α ξ (8) M λ( ξη, ) Φξ ( ) Φ( η) α( ξη, ) Φξ ( ) K Φ ( η) λ 4 e α ξ η e () 4.

3 49 λ Φ λ α [] n n ξ ξξ j j j n j ξξ jξkφjk j k Φ Φ + Φ + Φ + λ λ λ + ξ λ + ξξ λ + n n j j j n j ξξ jξkλjk j k n n j j j n j ξξ jξkαjk j k ( N) () ( N) (2) α α + ξα + ξξ α + ( ) (3) ()~(3)(9) ξ α Φ Φ λ λ Φ Φ e ( ) K ( )( ) M ( N) (4) X X (5) N N e ( ) K ( λ λ )( ) Φ Φ X α Φ MΦ (6) N Moore-Penrose pseu- donverse + ξ e [ α( Φ ) + α( Φ ) ] K Φ [( λ λ )( ) λ ( ) ] M Φ Φ Φ (7) ( N,, n) X + X (8) e ( Φ ) K Φ X α λ λ λ [( )( Φ ) )( Φ ) ] MΦ (9) ξ ξ j [ α( Φ j ) + ( α ( Φ j ) + e α j ( ) )( δj /2) αj ( ) ] K [( λ λ )( Φj ) ( λ ( Φj ) + λj ( ) )( δj /2) λj ( ) ] M Φ + Φ Φ Φ Φ Φ ( N,, n, j ) X + ( X + X )( δ /2) + X 2 2 j j j j j 2 e j ( Φj ) K Φ X j αj 2 [( λ λ )( Φj ) ( λ ( Φj ) + λj δj λj ( Φ ) )( /2) ( Φ ) ] M Φ (2) (2) (22) 4.2 λ Φ η λ λ + ηλ (23) Φ Φ +ηφ (24) λ Φ λ Φ ηλ ηφ λ Φ (5)~(6) λ + ηλ λ + λ (25) Φ + ηφ Φ+ Φ (26) Φ n n ξ ξξ j j j n j ξξ jξkφjk j k Φ Φ + Φ + Φ + ( N) (27) ξ n η λ λ + ξ n λ Φ Φ +ξ n Φ λ

4 5 λ λ + ξ λ λ + ξ λ + n ( n ) n n j ξξ jλj ξξ jξkλjk j j k + ( N) (28) α n n j j j n j ξξξα j k jk j k α α + ξα + ξξ α + ( ) (29) (27)~(29)() ξ α Φ Φ λ λ Φ Φ e ( ) K ( )( ) M ( N) (3) N X N X (3) e ( ) K X ( )( ) λ λ Φ MΦ Φ Φ α ξ n e [ α( Φ ) + α( Φ ) ] K Φ M [( λ λ )( Φ ) λ ( Φ ) ] Φ (32) X + X (33) ( Φ ) K e Φ X α [( )( ) ( ) ] λ λ Φ λ Φ MΦ n ( Φ ) MΦ ( )( Φ ) MΦ e e α K + αn K λ + λ λ [ ( Φ ) Φ ( Φ ) Φ ] (34) * * Xn + X (35) * e ( Φ ) K Φ X n αn * λ ( Φ ) Φ + ( λ λ )( Φ ) Φ M M ξ ξ j n, j [ α( Φ j ) + ( α ( Φ j ) + α j ( Φ ) )( δj /2) + α e j ( Φ ) ] K Φ λ λ Φj λ Φj + j ( ) )( j /2) j ( ) ] M [( )( ) ( ( ) λ Φ δ λ Φ Φ (36) X + ( X + X )( δ /2) + X 2 2 j j j j j 2 ( ) e j Φj K Φ X j αj 2 [( λ λ )( Φj ) λ Φj λj Φ δj λj ( Φ ) ] M Φ ( ( ) + ( ) )( /2) n, j λ ( Φ ) M Φ + (( λ λ )( Φ ) j j λ j ( Φ ) ) M Φ e [( αn( Φj ) αnj( Φ ) ) K Φ ( α ( ) ( ) ) e j + α j K ] + + (37) Φ Φ Φ (38) * * 2* X + X + X + X (39) j n nj j j * ( ) e Φ K Φ X nj αnj 2* λ ( Φj ) MΦ (( λ λ )( Φj ) λ j ( Φ ) ) M Φ + n, j n e ( Φ ) MΦ [ αnn ( Φ ) K Φ e n( ) K ] λ + α Φ Φ (4) * 2** Xn + Xnn (4) X nn αnn 2** ( ) λ Φ MΦ α 5

5 5 [8] (PDQ-G) (PDQ-)[9] k % (PDF) k % µ µ P ( Ω x k % < ) (42) Ω k % µ [9] 95%( ) ~3 α α 2 α 3 2 Fg.2 A three DOF sprng-ass syste (a) 3 δ. FEM abe Daage cases K K 2 K 3-5% -% -5% δ. k k % P d Fg. Probabty densty functon of k and k % and probabty of daage exstence P d k (probabty of daage exstence PDE) Pd P( Ω xk < ) P( < xk Ω) (43) P d [9] 6 ( 2) N K 2 K 2 5 K 3 /% 3 K K 2 K 3 Fg.3 Mean vaues of daage ndex of K K 2 and K 3 2 α abe 2 Mean vaues of α and coeffcents of varaton of sprng stffness FEM MC α abe 3 Mean square devatons of α and coeffcents of varaton of sprng stffness FEM MC

6 52 4 α 3 abe 4 Mean vaues of α 3 and coeffcents of varaton of sprng stffness 5% FEM MC α 3 abe 5 Mean square devatons of α 3 and coeffcents of varaton of sprng stffness /% δ FEM MC (%) abe 6 Probabty of daage of sprng stffness (%) K K 2 K 3 δ δ δ δ ~ 5 MC 6 (b) Kr (+ 3 δ ) K K r2 (+ 3 δ ) K 2 Kr 3 ( 3 δ ) K 3 δ δ. 3σ Ks.7 Kr Ks 2.6 Kr 2 K s 3.5 3%4% Kr3 4 K K 2 K 3 Fg.4 Mean vaues of daage ndex of K K 2 and K 3 7 (%) abe 7 Probabty of daage of sprng stffness (%) K K 2 K δ. 7 FEM FEM FEM ( 8 )

7 8 : [] Achenbach J D, Z. Pane stress crack ne feds for crack growth n an eastc-perfecty pastc atera [J]. Engneerng Fracture Mechancs, 98, 2(3): 534~544. [2] Achenbach J D, Dunayevsky V. Crack growth under pane stress condton n an eastc-perfecty pastc atera [J]. Journa of the Mechancs and Physcs of ods, 984, 32(2): 89~. [3] Guo Q, K. he denson of pastc zone ahead of a ode pane stress crack [J]. Engneerng Fracture Mechancs, 988, 29(): 7~2. [4] Y Z. he new and anaytca soutons for ode cracks n an eastc-perfecty atera [J]. Engneerng Fracture Mechancs, 992, 42(5): 833~84. [5] Y Z. Precse soutons of eastc-pastc crack ne feds for crack oaded by antpane pont forces [J]. Engneerng Fracture Mechancs, 997, 57(): 75~83. [6],. [J]., 996, 7(2): 59~64. Wu Chengpng, Wang Cheng. Near crack ne eastc-pastc anayss for a crack oaded by antpane pont forces [J]. Apped Matheatc and Mechancs, 996, 7(2): 59~64. (n Chnese) [7] Zhou Xaopng, Wang janhua. Near crack ne eastc-pastc anayss for an nfnte pate oaded by a par of pont shear forces [J]. Journa of hangha Jaotong Unversty, 23, E-8 (2): 5~7. [8] Wang Janhua, Zhou Xaopng. Near crack ne eastc-pastc anayss for an nfnte pate oaded by two pars of pont tense forces [J]. Mechancs esearch Councatons, 24, 3(4): 45~42. [9] Zhou Xaopng, Wang Janhua. tudy on the coaescence echans of spttng faure of crack-weakened rock subjected to copressve oads [J]. Mechancs esearch Councatons, 25, 32(2): 6~7. [],,. [J]., 25, 26(3): 267~272. Zhou Xaopng, Zhang Yongxng, Zhu Keshan. Eastc-pastc anaytca souton for an eccentrc crack oaded by a par of tense pont forces [J]. Acta Mechanca oda nca, 25, 26(3): 267~272. (n Chnese) ( 52 ) [] Baruch M. Optzaton procedure to correct stffness and fexbty atrces usng vbraton test [J]. AIAA Journa, 978, 6(): 28~2. [2] aw, h Z Y, Zhang M. tructura daage detecton fro ncopete and nosy oda test data [J]. J. of Engr. Mech, ACE, 998, 24(): 28~288. [3] en W X, oeck G D. tructura daage dentfcaton usng oda data I: uaton verfcaton [J]. J. of truc. Engr, ACE, 22, 28(): 87~95. [4] Doebng W, Farrar C, Pre M B, hevtz D W. Daage dentfcaton and heath ontorng of structura and echanca systes fro changes n ther vbraton characterstcs: A terature evew []. esearch ep. os Aaos Natona ab., A-37-M, UC-9, os Aaos, NM, May, 996. [5] tubbs N, kkorsky C, Park, Cho, Boton. A ethodoogy to nondestructvey evauate the structura propertes of brdges [A]. Proc. IMAC7, the 7 th Int. Moda Anayss Conf, Kss-ee, Fa, 999, 26~ 268. [6] K J, tubbs N. Mode-uncertanty pact and daage detecton accuracy n pate grder [J]. J. tructura Engneerng, ACE, 995, 2: 49~47. [7],. [J]. (), 2, 23(2): 6~63. Zhang Be, Yn Xuegang. Probabty dagnoss of structura faure [J]. Journa of Chongqng Unversty, 2, 23(2): 6~63. (n Chnese) [8] Papadopouos, Garca E. tructura daage dentfcaton: A probabstc approach [J]. AIAA J., ~245. [9] Yong Xa, Hong Hao, Jaes M W Brownjohn. Daage dentfcaton of structures wth uncertan frequency and ode shape data [J]. Earthquake Engneerng and tructura Dynacs, 22, 3: 53~ 66. []. [J]., 24, 26(5): 42~45. Huang Bn. A new spectra rando fnte eeent ethod [J]. Journa of Wuhan Unv. of ech., 24, 26(5): 42~45. (n Chnese)

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