Free vibration analysis of uniform and stepped cracked beams with circular cross sections

Transcription

1 Interntionl Journl of Engineering Science 45 (27) Free vibrtion nlysis of uniform nd stepped crcked bems with circulr cross sections Murt Kis, *, M. Arif Gurel b Deprtment of Mechnicl Engineering, Fculty of Engineering, Hrrn University, Snliurf 633, Turkey b Deprtment of Civil Engineering, Fculty of Engineering, Hrrn University, Snliurf 633, Turkey Received 3 Februry 27; ccepted 29 Mrch 27 Avilble online 22 My 27 Abstrct This pper presents novel numericl technique pplicble to nlyse the free vibrtion nlysis of uniform nd stepped crcked bems with circulr cross section. In this pproch in which the finite element nd component synthesis methods re used together, the bem is detched into prts from the crck section. These substructures re joined by using the flexibility mtrices tking into ccount the interction forces derived by virtue of frcture mechnics theory s the inverse of the complince mtrix found with the pproprite stress intensity fctors nd strin energy relese rte expressions. To revel the ccurcy nd effectiveness of the offered method, number of numericl exmples re given for free vibrtion nlysis of bems with trnsverse non-propgting open crcks. Numericl results showing good greement with the results of other vilble studies, ddress the effects of the loction nd depth of the crcks on the nturl frequencies nd shpes of the crcked bems. Modl chrcteristics of crcked bem cn be employed in the crck recognition process. The outcomes of the study verified tht presented method is pproprite for the vibrtion nlysis of uniform nd stepped crcked bems with circulr cross section. Ó 27 Elsevier Ltd. All rights reserved. Keywords: Free vibrtion; Crck; Stepped bems; Finite element nlysis. Introduction Mny engineering structures my hve structurl defects such s crcks due to long-term service, mechnicl vibrtions, pplied cyclic lods etc. Numerous techniques, such s non-destructive monitoring tests, cn be used to screen the condition of structure. Novel techniques to inspect structurl defects should be explored. A crck in structurl element modifies its stiffness nd dmping properties nd ccordingly influences its dynmicl performnce. In view of tht, the nturl frequencies nd shpes of the structure hold * Corresponding uthor. Tel.: ; fx: E-mil ddresses: mkis@hrrn.edu.tr (M. Kis), gurel@hrrn.edu.tr (M. Arif Gurel) /$ - see front mtter Ó 27 Elsevier Ltd. All rights reserved. doi:.6/j.ijengsci

2 M. Kis, M. Arif Gurel / Interntionl Journl of Engineering Science 45 (27) informtion relting to the plce nd dimension of the dmge. Vibrtion nlysis llowing online inspection is n ttrctive method to detect crcks in the structures. Investigtion of dynmic behviour of crcked structures hs ttrcted the ttention of severl reserchers in recent yers (Cwley nd Adms [], Gounris nd Dimrogons [2], Krwczuk nd Ostchowicz [3], Ruotolo et l., [4], Kis et l. [5], Shifrin nd Ruotolo [6], Kis nd Brndon [7,8], Viol et l. [9], Krwczuk [], Ptil nd Miti [], Kis [2], Kis nd Gurel [3]). Gudmundson [4] investigted the influence of smll crcks on the nturl frequencies of slender structures by perturbtion method s well s by trnsfer mtrix pproch. Yuen [5] proposed methodicl finite element procedure to estblish the reltionship between dmge loction, dmge size nd the corresponding modifiction in the eigen prmeters of cntilever bem. Rizos et l. [6] represented the crck s mssless rottionl spring, whose stiffness ws clculted by employing frcture mechnics. There re numerous studies on the vibrtion nlysis of crcked bems with circulr cross section nd shfts. Dimrogons nd Ppdopoulos [7], by using the theory of crcked shfts with dissimilr moments of inerti, investigted the vibrtion of crcked shfts in bending. Ppdopoulos nd Dimrogons [8] studied the free vibrtion of shfts nd presented the influence of the crck on the vibrtion behviour of the shfts. Kikidis nd Ppdopoulos [9] nlysed the influence of the slenderness rtio of non-rotting crcked shft on the dynmic chrcteristics of the structure. Zheng nd Fn [2] studied the vibrtion nd stbility of crcked hollow-sectionl bems. Dong et l. [2] presented continuous l for vibrtion nlysis nd prmeter identifiction of non-rotting rotor with n open crck. They ssumed tht the crcked rotor ws n Euler Bernoulli bem with circulr cross section. Studies on the vibrtion of stepped bems re crried out by number of reserchers. Jng nd Bert [22] presented the exct nd numericl solutions for fundmentl nturl frequencies of stepped bems for vrious boundry conditions. Wng [23] nlysed the vibrtion of stepped bems on elstic foundtions. Bsed on n elementl dynmic flexibility method, Lee nd Bergmn [24] studied the vibrtion of stepped bems nd rectngulr pltes. In their study, the structure with discontinues ws divided into elementl substructures nd the displcement field for ech ws obtined in terms of its dynmic Green s function. Lee nd Ng [25] computed the fundmentl frequencies nd criticl buckling lods of simply supported stepped bems by using two lgorithms bsed on the Ryleigh Ritz method. Ros et l. [26] performed the free vibrtion nlysis of stepped bems with intermedite elstic supports. Nguleswrn [27] nlysed the vibrtion nd stbility of n Euler Bernoulli stepped bem with n xil force. There re few studies on the vibrtion of stepped bems with crcks. Employing the trnsfer mtrix method, Tsi nd Wng [28] nlysed the dynmic chrcteristics of stepped shft nd multi-disc shft. Nndwn nd Miti [29] presented method bsed on mesurement of nturl frequencies for detection of the loction nd size of crck in stepped cntilever bem. Chudhri nd Miti [3] presented n experimentlly verified method for prediction of loction nd size of crck in stepped bems with crcks. Li [3,32] nlysed the vibrtory chrcteristics of multi-step bems with n rbitrry number of crcks nd concentrted msses. Abrhm nd Brndon [33] nd Brndon nd Abrhm [34] presented method utilising substructure norml s to predict the vibrtion properties of cntilever bem with closing crck. The full eigensolution of structure contining substructures ech hving lrge numbers of degrees of freedom cn be costly in computing time. A method known s component synthesis or substructure technique, proposed by Hurty [35], mde possible the problem to be broken up into seprte elements nd thus considerbly reduced its complexity. The component synthesis method ws initilly developed to ese the study of very lrge structures but in this study it is used for nother purpose. The dvntge of the method in the cse of non-liner crcked bem stems from the fct tht, when bem is split into components t the crck section, ech substructure becomes liner nd nlyticl or numericl results re vilble for their norml s. Consequently, the initil non-liner system with locl discontinuities in stiffness t the crck sections is now composed of liner segments. An importnt chrcteristic of the l developed in this study is tht it llows discontinuity in the displcement field t the crck section when the crck is open. The substructures re connected by n rtificil nd mssless spring whose stiffness coefficients re functions of the complince coefficients. To the best of the uthors knowledge, the presented method is pplied for the first time to the uniform nd stepped crcked bems of circulr cross section which commonly used in engineering structures.

3 366 M. Kis, M. Arif Gurel / Interntionl Journl of Engineering Science 45 (27) Theoreticl l The l chosen is stepped cntilever bem of length L nd dimeters D (D ¼ 2R Þ nd D 2 (D 2 ¼ 2R 2 Þ, hving trnsverse open edge crck of depth t vrible position L (Fig. ). Length of the first prt is L in which cn be chosen between nd. The bem is divided into two components t the crck section leding to substructure pproch. Accordingly, s mentioned before, the globl non-liner system with locl stiffness discontinuity is detched into two liner subsystems. Ech prt is lso divided into finite elements with two nodes nd three degrees of freedom t ech node s shown in Fig Locl flexibility mtrix of crcked bem with circulr cross section As result of the strin energy concentrtion t the surrounding re of the crck tip, the existence of crcks in structures is resource of locl flexibility which subsequently influences the dynmic performnce of the structures. These flexibility coefficients re expressed by stress intensity fctors derived through Cstiglino s theorem in the liner elstic rnge. The strin energy relese rte, J, represents the elstic energy in reltion to unit increse in length hed of the crck front. For plne strin, J cn be given s (Irwin [36]) J ¼ m2 E K2 I m2 þ E K2 II þ þ m E K2 III ðþ where m, E, K I, K II nd K III re Poisson s rtio, modulus of elsticity, nd the stress intensity fctors for the I, II nd III deformtion types, respectively. As torsionl effects will not be concerned bout in this study, simply I nd II types of deformtions re considered. The superposition of the stress intensity fctors gives, for the strin energy relese rte, the subsequent expression m2 J ¼ E fðk I ðp ÞþK I ðp 3 ÞÞ 2 þ K II ðp 2 Þ 2 g ð2þ where P, P 2 nd P 3 re the xil force, sher force nd bending moment, respectively, Fig. 2. E ¼ E for plne strin nd E ¼ E=ð m 2 Þ for plne stress. Bending moment P 3 nd the xil force P mke the contribution y L L L z D D 2 x Fig.. Geometry of the stepped cntilever bem with crck. X Y 2 3 m 2 n P 2 P 3 P v v 2 P 5 P 6 2 u 2 u 2 P 4 L e Fig. 2. Components of crcked bem nd their finite element l.

4 M. Kis, M. Arif Gurel / Interntionl Journl of Engineering Science 45 (27) to the opening, I. The edge sliding, II, receives contribution from the sher force P 2. K I (P ), K I (P 3 ) nd K II (P 2 ) cn be written s follow (Td et l. [37]) K I ¼ P pffiffiffiffiffi p F pr 2 ð3þ where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 F ¼ p tg p :752 þ 2:2 þ :37 sin p 2 ð4þ 2 p cos 2 K I3 ¼ 4P 3 pffiffiffiffiffi p F pr 4 2 ð5þ where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 F 2 ¼ p tg p :923 þ :99 sin p 2 ð6þ 2 p cos 2 K II2 ¼ jp 2 pffiffiffiffiffi p F pr 2 3 ð7þ where 2 3 :22 :56 þ :85 þ :8 F 3 ¼ qffiffiffiffiffiffiffiffiffiffiffi ð8þ where the coefficient j is numericl fctor depending on the shpe of the cross section nd derived from Timoshenko bem theory (Cowper [38]), is the crck depth nd is the height of the strip, Fig. 3, nd written s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 R 2 x 2 ð9þ where R is the rdius of the cross section of the bem. If the stress intensity expressions re substituted into Eq. (2), the next expression is obtined 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 P 2 F 2 m2 p JðÞ ¼ E p 2 R 4 þ 6P 2 3 >< ðr 2 x 2 ÞF 2 p 2 R 8 3 þ 8P P 3 R 2 x 2 F p 2 R 6 F 3 >= þ j2 P >: 2 2 F 2 >; p 2 R 4 2 ðþ A b y x dx dy R R y x x A A - A Fig. 3. The geometry of the crcked circulr cross section.

5 368 M. Kis, M. Arif Gurel / Interntionl Journl of Engineering Science 45 (27) If U is the strin energy of crcked structure with crck re A under the lod P i, then the reltion between J nd U is J ¼ ouðp i; AÞ ðþ oa In ccordnce with the Cstiglino s theorem, the dditionl displcement cused by the crck in the direction of P i cn be given s u i ¼ ouðp i; AÞ ð2þ op i Substituting Eq. () into Eq. (2) gives the finl expression between displcement nd strin energy relese rte J s u i ¼ o Z JðP i ; AÞdA ð3þ op i A Now, the flexibility coefficients which re the functions of the crck shpe nd the stress intensity fctors cn be introduced s follows (Dimrogons nd Pipetis [39]) c ij ¼ ou Z i ¼ o2 JðP i ; AÞdA ð4þ op j op i op j A Finlly, the flexibility coefficients c, c 3, c 22 nd c 33 re obtined s c ¼ 2 Z b Z x yf 2 E pr 4 dy dx c 3 ¼ 8 E pr 6 c 22 ¼ 2j2 E pr 4 c 33 ¼ 32 E pr 8 b Z b Z x b Z b Z x yf 2 2 b Z b Z x b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y R 2 x 2 F dy dx yðr 2 x 2 ÞF 2 3 F 3 dy dx dy dx where b nd x re the boundry of the strip nd the locl crck depth, Fig. 3, respectively, nd given s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ x ¼ R 2 ðr Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 x 2 ðr Þ ð9þ Dimensionless flexibility coefficients re clculted numericlly long with the subsequent expressions nd drwn in Fig. 4. c ¼ E prc c 3 ¼ E pr 2 c 3 c 22 ¼ E prc 22 j 2 c 33 ¼ E pr 3 c 33 ð2þ Since the sher force does not contribute to the opening of the crck, the complince mtrix, in reltion to displcement vector dðu; v; hþ, cn be written s 2 3 c c C ¼ 4 c 22 5 ð2þ ð5þ ð6þ ð7þ ð8þ c 3 c 33 ð33þ The inverse of the complince mtrix C is the stiffness mtrix of the nodl point nd given s K cr ¼ C ð22þ

6 M. Kis, M. Arif Gurel / Interntionl Journl of Engineering Science 45 (27) Non-dimensionl flexibilities.e+5.e+4.e+3.e+2.e+.e+.e-.e-2.e-3.e-4.e Crck rtio (/D) Fig. 4. Non-dimensionl complince coefficients s function of the crck depth rtio /D. c c3 c22 c Coupling of the substructures by springs Consider two undmped components X nd Y joined together by mens of springs cpble of crrying xil, shering nd bending effects, Fig. 5. For this system the eqution of motion, in mtrix nottion, is given s M X qx þ K X qx ¼ f X ðtþ ð23þ M Y q Y K Y q Y f Y ðtþ where q nd f(t) re the generlised displcement nd externl force vector, respectively. M X,M Y nd K X,K Y re mss nd stiffness mtrices for the components X nd Y, respectively. Mss nd stiffness mtrices re tken from the pper of Friedmn nd Kosmtk [4]. Eq. (23) gives the eigenvlue eqution s " # K X x2 X M X! q X ¼ ð24þ K Y x 2 Y M Y q Y Eq. (24) gives eigenvlues nd modl mtrix for the components X nd Y. On prticulr spring, the exerted forces, F X nd F Y, re given by F X ¼ Kðq X q Y Þ ð25þ F Y ¼ Kðq Y q X Þ where q X nd q Y re the displcements of the connection points, Fig. 5. Then Eq. (25) cn be written s F X q X ¼ K C ð26þ F Y q Y where K C is the connection mtrix nd given s " # K C ¼ K cr K cr K cr K cr ð27þ X Y K q X q Y Fig. 5. System of the components connected by spring.

7 37 M. Kis, M. Arif Gurel / Interntionl Journl of Engineering Science 45 (27) where K cr is the stiffness mtrix of the nodl point nd given by Eq. (22). The force vector f(t) comprises pplied forces g(t) nd forces resulting from the springs d(t), such tht f ðtþ ¼gðtÞþdðtÞ ð28þ From the equilibrium dðtþ ¼ F X q X ¼ K C ð29þ F Y q Y Substituting Eqs. (28) nd (29) into Eq. (23) gives M X qx þ K X qx q X ¼ ½K C Š þ g X ðtþ M Y q Y K Y q Y q Y g Y ðtþ or M X qx K X qx þ þ½k C Š ¼ g X ðtþ M Y q Y K Y q Y g Y ðtþ If modl vector / ij is normlised by the mss, the following expression is given ð3þ ð3þ w ij ¼ / ij pffiffiffiffiffiffi m jj ð32þ where w ij is mss normlised vector. By using the trnsformtion q X ¼ w T X sx ð33þ q Y w Y s Y where s X nd s Y re the principl coordinte vectors. By premultiplying w T nd substituting Eqs. (32) nd (33) into Eq. (3), gives s " X I þ x2 X # sx þ w T X w X sx ½K s Y x 2 C Š ¼ w T X g X ðtþ ð34þ Y s Y w Y w Y s Y w Y g Y ðtþ where I ¼ w T X M X w X ¼ w T Y M Y w Y x 2 X ¼ wt X K X w X ð35þ x 2 Y ¼ wt Y K Y w Y Eq. (34), for free vibrtion, gives the eigenvlue eqution s " # " # x 2 X þ wt X w X I! s ½K x 2 Y w T C Š x 2 X ¼fg ð36þ w Y Y I s Y From Eq. (36) the eigenvlues nd eigenvectors of the system cn be determined. After solving this eqution, the displcements for ech component re clculted by using Eq. (33). 3. Numericl exmples nd discussion 3.. Uniform cntilever bem with crck Presented method, initilly, hs been pplied to uniform crcked cntilever bem with circulr cross section, Fig. 6. The geometricl properties of the bem re length L = 2 m, slenderness rtios R/L =.,.6 nd.4. Clcultion hs been performed with the numericl vlues, Young s modulus E = 26 9 Nm 2, the Poisson s rtio m =.33 nd mss density q = kg m 3.

8 M. Kis, M. Arif Gurel / Interntionl Journl of Engineering Science 45 (27) Fig. 7 illustrtes the first non-dimensionl frequencies of the crcked bem s function of the crck depth rtio (/D) for severl slenderness rtios R/L (.,.6 nd.4). In the nlysis, the crck loction is chosen s L /L =.2. In this study, non-dimensionl nturl frequencies re normlised ccording to next eqution x ¼ x cr x nc ð37þ y L L z D x Fig. 6. Geometry of the uniform cntilever bem with crck. st non-dimensionl nturl L /L =.2 Present study R/L=. Present study R/L=.6 Present study R/L=.4 Ppdopoulos R/L=. Ppdopoulos R/L=.6 Ppdopoulos R/L= Crck rtio ( /D) Fig. 7. First non-dimensionl nturl frequencies s function of crck depth rtio, for severl slenderness rtios R/L =.,.6,.4 nd crck position L /L =.2. st non-dimensionl R/L =. L/L=.2 L/L=.4 L/L=.6 L/L= Crck rtio (/D) st non-dimensionl R/L =.6 L/L=.2 L/L=.4 L/L=.6 L/L= Crck rtio (/D) st non-dimensionl R/L =.4 L/L=.2 L/L=.4 L/L=.6 L/L= Crck rtio (/D) Fig. 8. First non-dimensionl nturl s function of crck depth rtio, for severl slenderness rtios R/L =.,.6,.4, nd crck loctions L /L =.2,.4,.6, nd.8.

9 372 M. Kis, M. Arif Gurel / Interntionl Journl of Engineering Science 45 (27) where x cr nd x nc refer to the nturl of the crcked nd non-crcked cntilever bems, respectively. The nturl frequencies of the crcked bem re lower thn those of corresponding intct bem, s expected. Differences in the frequencies get higher s the depth of the crck increses. Becuse of the bending moment long the bem, which is concentrted t the fixed end, crck closer to the free end will hve smller effect on the fundmentl thn crck closer to the fixed end. It cn be obviously seen from the Fig. 7 tht when the slenderness rtio (R/L) increses, the reduction gets higher, too. The results obtined by the current pproch re compred with those of Ppdopoulos nd Dimrogons [4] nd s 2nd non-dimensionl R/L =. L/L=.2 L/L=.4 L/L=.6 L/L= Crck rtio (/D) 2nd non-dimensionl R/L =.6 L/L=.2 L/L=.4 L/L=.6 L/L= Crck rtio (/D) 2nd non-dimensionl R/L =.4 L/L=.2 L/L=.4 L/L=.6 L/L= Crck rtio (/D) Fig. 9. Second non-dimensionl nturl s function of crck depth rtio, for severl slenderness rtios R/L =.,.6,.4, nd crck loctions L /L =.2,.4,.6, nd.8. R/L =., L /L =.2 R/L =., L /L =.4 st nturl bending Intct /D=.2 /D=.4 /D= st nturl bending Intct /D=.2 /D=.4 /D= st nturl bending R/L =., L /L =.6 Intct /D=.2 /D=.4 /D= Fig.. First nturl bending s function of crck depth rtio, for slenderness rtio R/L =., nd crck loctions L / L =.2,.4, nd.6.

10 M. Kis, M. Arif Gurel / Interntionl Journl of Engineering Science 45 (27) is noticed from the Fig. 7, n excellent concurrence hs been found between the results. Figs. 8 nd 9 demonstrte the first nd second non-dimensionl nturl frequencies s function of crck depth rtio for severl slenderness rtios R/L =.,.6, nd.4, nd crck loctions L /L =.2,.4,.6, nd.8. As perceptible from the figures, the first reduction is higher when the crck loction L /L is equl to.2, while the difference is higher in the second when the crck loction L /L is between.4 nd.6. Figs. 2 illustrte the first, second nd third nturl bending shpes s function of crck depth rtio for different slenderness rtios nd crck loctions L /L =.2,.4, nd.6. From the shpes the position of the crck cn be clerly seen. 2nd nturl bending R/L =., L /L =.2 Intct /D=.2 /D=.4 /D= nd nturl bending R/L =., L /L =.4 Intct /D=.2 /D=.4 /D= nd nturl bending R/L =., L /L =.6 Intct /D=.2 /D=.4 /D= Fig.. Second nturl bending s function of crck depth rtio, for slenderness rtio R/L =., nd crck loctions L / L =.2,.4, nd.6. 3rd nturl bending R/L =.4, L /L =.2 Intct /D=.2 /D=.4 /D= rd nturl bending R/L =.4, L /L =.4 Intct /D=.2 /D=.4 /D= rd nturl bending R/L =.4, L /L =.6 Intct /D=.2 /D=.4 /D= Fig. 2. Third nturl bending s function of crck depth rtio, for slenderness rtio R/L =.4, nd crck loctions L / L =.2,.4, nd.6.

11 374 M. Kis, M. Arif Gurel / Interntionl Journl of Engineering Science 45 (27) st non-dimensionl nturl L/L=. L/L=.3 L/L=.5 L/L=.7 L/L= Crck rtio (/D) 2nd non-dimensionl nturl L/L=. L/L=.3 L/L=.5 L/L=.7 L/L= Crck rtio (/D) 3rd non-dimensionl nturl L/L=. L/L=.3 L/L=.5 L/L=.7 L/L= Crck rtio (/D) Fig. 3. First, second nd third non-dimensionl nturl frequencies s function of crck depth rtio for severl crck loctions L / L =.,.3,.5,.7, nd.9. st nturl bending 2nd nturl bending L /L =.2 Intct /D=.2 /D=.4 /D= L /L =.2 Intct /D=.2 /D=.4 /D= st nturl bending 2nd nturl bending L /L =.5 Intct /D=.2 /D=.4 /D= L /L =.5 Intct /D=.2 /D=.4 /D= rd nturl bending L /L =.2 Intct /D=.2 /D=.4 /D= rd nturl bending L /L =.5 Intct /D=.2 /D=.4 /D= Fig. 4. First, second nd third nturl bending shpes s function of crck depth rtio for crck loctions L /L =.2 nd.5.

12 M. Kis, M. Arif Gurel / Interntionl Journl of Engineering Science 45 (27) Stepped cntilever bem with crck Second exmple is selected s stepped cntilever bem with crck, Fig.. The mteril properties of the bem for the present nd subsequent instnces re identicl to the bem in the previous exmple. The length of the bem is L = 4 m nd the step prt is t the middle of the bem ( =.5). In the first nd second prts of the bem the slenderness rtios re chosen s R /L =. nd R 2 /L =.4, respectively. Fig. 3 displys the first, second nd third non-dimensionl nturl frequencies s function of crck depth rtio for severl crck loctions L /L =.,.3,.5,.7, nd.9. From Fig. 3, one cn discern tht the greter drops in the first nd third nturl frequencies re occurred when the crck is locted just t the step prt, L /L =.5, s the stiffness of the bem decreses due to the stepped vrition in dimeter nd the presence of crck. If the crck is locted ner the fixed end the reduction in the second nturl frequencies is the highest. When the first non-dimensionl nturl is considered, the crck locted t the step prt of the bem cuses more reductions in the nturl frequencies compred to crck situted ner to the fixed end. Fig. 4 illustrtes the first, second nd third nturl bending shpes s function of crck depth rtio for crck loctions L /L =.2,.5. From the Fig. 4, it cn be seen tht when the crck is closer to step prt of the bem, L /L =.5, the differences in the shpes of crcked nd intct bems re getting higher. Exploring the shpes, it cn be observed tht t the crck section the devitions of shpes of crcked nd intct bems re greter. Accordingly, utilising the shpes the position of the crck cn be detected esily. y L D D2 D x z L/3 L/3 L/3 L Fig. 5. Geometry of two-step simply supported bem with crck. st non-dimensionl nturl L/L=.8 L/L=.6 L/L=.25 L/L=.33 L/L=.4 L/L= Crck rtio (/D) 3rd non-dimensionl nturl L/L=.8 L/L=.6 L/L=.25 L/L=.33 L/L=.4 L/L=.5 2nd non-dimensionl nturl Crck rtio (/D) L/L=.8 L/L=.6 L/L=.25 L/L=.33 L/L=.4 L/L= Crck rtio (/D) Fig. 6. First, second nd third non-dimensionl nturl frequencies s function of crck depth rtio for severl crck loctions L / L =.8,.6,.25,.33,.4, nd.5.

13 376 M. Kis, M. Arif Gurel / Interntionl Journl of Engineering Science 45 (27) Two-step simply supported bem with crck Third exmple is selected s two-step simply supported crcked bem with circulr cross section, Fig. 5. The length of the bem is L = 6 m nd the step loctions re t the /3 nd 2/3 of the bem length. In the first, st nturl bending L /L =.6 Intct /D=.2 /D=.4 /D= st nturl bending L /L =.33 Intct /D=.2 /D=.4 /D= st nturl bending L /L =.5 Intct /D=.2 /D=.4 /D= nd nturl bending L /L =.6 Intct /D=.2 /D=.4 /D=.6 2nd nturl bending L /L =.33 Intct /D=.2 /D=.4 /D= nd nturl bending L /L =.5 Intct /D=.2 /D=.4 /D=.6 3rd nturl bending L /L =.6.4 Intct /D=.2 /D=.4 /D= rd nturl bending rd nturl bending L /L = Intct /D=.2 /D=.4 /D= L /L =.33 Intct /D=.2 /D=.4 /D= Fig. 7. First, second nd third nturl bending shpes s function of crck depth rtio for crck loctions L /L =.6,.33, nd.5.

14 M. Kis, M. Arif Gurel / Interntionl Journl of Engineering Science 45 (27) second nd third prts of the bem, the slenderness rtios re chosen s R /L =.4, R 2 /L =. nd R 3 / L =.4, respectively. Fig. 6 demonstrtes the first, second nd third non-dimensionl nturl frequencies s function of crck depth rtio for severl crck loctions L /L =.8,.6,.25,.33,.4 nd.5. Due to the symmetry, only the results belong to the one hlf of the bem is shown in this figure. It is cler from the Fig. 6 tht the lrger flls in the first, second nd third nturl frequencies re observed when the crck is locted t the step, L / L =.33. Fig. 7 illustrtes the first, second nd third nturl bending shpes s function of crck depth rtio for crck loctions L /L =.6,.33, nd.5. Similr to the nturl frequencies, the lrgest chnges in the shpes of crcked nd intct bems occur when crck is locted t the step, Fig Two-step simply supported bem with two crcks The lst exmple is two-step simply supported bem with two crcks, Fig. 8. The geometricl properties of the bem re chosen identicl to the one given in the former exmple. In the first, second nd third prts of the bem the slenderness rtios re chosen s R /L =.4, R 2 /L =. nd R 3 /L =.4, respectively. y L L 2 z D 2 2 D D x L/3 L/3 L/3 L Fig. 8. Geometry of two-step simply supported bem with two crcks. st non-dimensionl nturl L/L=.8, L2/L=.6 L/L=.25, L2/L=.33 L/L=.33, L2/L=.4 L/L=.4, L2/L=.5 L/L=.33, L2/L=.66 L/L=.6, L2/L= Crck rtio (/D) 3rd non-dimensionl nturl nd non-dimensionl nturl L/L=.8, L2/L=.6 L/L=.25, L2/L=.33 L/L=.33, L2/L=.4 L/L=.4, L2/L=.5 L/L=.33, L2/L=.66 L/L=.6, L2/L= Crck rtio (/D) L/L=.8, L2/L=.6 L/L=.25, L2/L=.33 L/L=.33, L2/L=.4 L/L=.4, L2/L=.5 L/L=.33, L2/L=.66 L/L=.6, L2/L= Crck rtio (/D) Fig. 9. First, second nd third non-dimensionl nturl frequencies s function of crck depth rtio for severl crck loctions L / L =.8 L 2 /L =.6, L /L =.25 L 2 /L =.33, L /L =.33 L 2 /L =.4, L /L =.4 L 2 /L =.5, L /L =.33 L 2 /L =.66, L /L =.6 L 2 /L =.83.

15 378 M. Kis, M. Arif Gurel / Interntionl Journl of Engineering Science 45 (27) Fig. 9 illustrtes the first, second nd third non-dimensionl nturl frequencies s function of crck depth rtio for severl crck loctions L /L =.8 L 2 /L =.6, L /L =.25 L 2 /L =.33, L / L =.33 L 2 /L =.4, L /L =.4 L 2 /L =.5, L /L =.33 L 2 /L =.66, L /L =.6 L 2 /L =.83. 2nd nturl st nturl bending st nturl bending bending 3rd nturl bending 3rd nturl bending 2nd nturl bending L /L =.8, L 2 /L =.6 Intct /D=.2 /D=.4 /D= L /L =.33, L 2 /L =.4 Intct /D=.2 /D=.4 /D= L /L =.8, L 2 /L =.6 Intct /D=.2 /D=.4 /D= L /L =.33, L 2 /L =.4 Intct /D=.2 /D=.4 /D= L /L =.8, L 2 /L =.6 Intct /D=.2 /D=.4 /D= L /L =.33, L 2 /L =.4 Intct /D=.2 /D=.4 /D= st nturl bending st nturl bending 2nd nturl bending 2nd nturl bending 3rd nturl bending 3rd nturl bending L /L =.6, L 2 /L =.83 Intct /D=.2 /D=.4 /D= L /L =.33, L 2 /L =.66 Intct /D=.2 /D=.4 /D= L /L =.6, L 2 /L =.83 Intct /D=.2 /D=.4 /D= L /L =.33, L 2 /L =.66 Intct /D=.2 /D=.4 /D= L /L =.6, L 2 /L =.83 Intct /D=.2 /D=.4 /D= L /L =.33, L 2 /L =.66 Intct /D=.2 /D=.4 /D= Fig. 2. First, second nd third nturl bending shpes s function of crck depth rtio for crck loctions L /L =.8 L 2 / L =.6, L /L =.6 L 2 /L =.83, L /L =.33 L 2 /L =.4, L /L =.33 L 2 /L =.66.

16 It is cler from the Fig. 9 tht the lrger decreses in the first, second nd third nturl frequencies re seen when the crcks re locted t the step prts, L /L =.33 L 2 /L =.66. Fig. 2 shows the first, second nd third nturl bending shpes s function of crck depth rtio for crck loctions L /L =.8 L 2 /L =.6, L /L =.6 L 2 /L =.83, L /L =.33 L 2 /L =.4, L / L =.33 L 2 /L =.66. From the Fig. 2, the effects of the step prts of the bem where the crcks re locted on the shpes cn be noticed. 4. Conclusions In this pper new pproch for the vibrtion nlysis of uniform nd stepped crcked bems with circulr cross sections is presented. In the method, the component synthesis technique ccompnied by the finite element method is used nd non-liner problem seprted into liner subsystems. As the whole structure is detched from the crck section, mking use of the present pproch is believed to offer n efficient method cpble of investigting the non-liner interfce effects such s contct nd impct tht occur when crck closes. It is reveled tht the knowledge of modl dt of crcked bems forms n importnt spect in ssessing the structurl filure. Presented four numericl exmples verified tht the proposed method is effective end verstile. Besides, it is shown tht the crck loctions nd sizes cn notbly influence the modl fetures, i.e. nturl frequencies nd shpes, of the crcked bems especilly when the crcks re locted t the step prts of the bems. It is evident tht, the ppliction of the present study is restricted to the vibrtion nlysis of bems with non-propgting open crcks. Some potentil extensions, which re left for future works, of the present study re the investigtion of the bems with other cross sections nd inclusion of contct nd impct effects when the crck brethes. References M. Kis, M. Arif Gurel / Interntionl Journl of Engineering Science 45 (27) [] P. Cwley, R.D. Adms, The loction of defects in structures from mesurements of nturl frequencies, J. Strin Anl. 4 (979) [2] G. Gounris, A.D. Dimrogons, A finite element of crcked prismtic bem for structurl nlysis, Comput. Struct. 28 (988) [3] M. Krwczuk, W.M. Ostchowicz, Trnsverse nturl vibrtions of crcked bem loded with constnt xil force, J. Vib. Acoust. 5 (993) [4] R. Ruotolo, C. Surce, P. Crespo, D. Storer, Hrmonic nlysis of the vibrtions of cntilevered bem with closing crck, Comput. Struct. 6 (996) [5] M. Kis, J.A. Brndon, M. Topcu, Free vibrtion nlysis of crcked bems by combintion of finite elements nd component synthesis methods, Comput. Struct. 67 (998) [6] E.I. Shifrin, R. Ruotolo, Nturl frequencies of bem with n rbitrry number of crcks, J. Sound Vib. 223 (999) [7] M. Kis, J.A. Brndon, Free vibrtion nlysis of multiple open edge crcked bems by component synthesis, Struct. Eng. Mech. (2) [8] M. Kis, J.A. Brndon, The effects of closure of crcks on the dynmics of crcked cntilever bem, J. Sound Vib. 238 (2) 8. [9] E. Viol, L. Federici, L. Nobie, Detection of crck loction using crcked bem element method for structurl nlysis, Theor. Appl. Frct. Mech. 36 (2) [] M. Krwczuk, Appliction of spectrl bem finite element with crck nd itertive serch technique for dmge detection, Finite Elem. Anl. Des. 38 (22) [] D.P. Ptil, S.K. Miti, Detection of multiple crcks using mesurements, Eng. Frct. Mech. 7 (23) [2] M. Kis, Free vibrtion nlysis of cntilever composite bem with multiple crcks, Compos. Sci. Technol. 64 (24) [3] M. Kis, M.A. Gurel, Modl nlysis of multi-crcked bems with circulr cross section, Eng. Frct. Mech. 73 (26) [4] P. Gudmundson, The dynmic behviour of slender structures with cross-sectionl crcks, J. Mech. Phys. Solids 3 (983) [5] M.M.F. Yuen, A numericl study of the eigenprmeters of dmged cntilever, J. Sound Vib. 3 (985) 3 3. [6] P.F. Rizos, N. Asprgthos, A.D. Dimrogons, Identifiction of crck loction nd mgnitude in cntilever bem from the vibrtion s, J. Sound Vib. 38 (99) [7] A.D. Dimrogons, C.A. Ppdopoulos, Vibrtions of crcked shfts in bending, J. Sound Vib. 9 (983) [8] C.A. Ppdopoulos, A.D. Dimrogons, Coupled longitudinl nd bending vibrtions of rotting shft with n open crck, J. Sound Vib. 7 (987) [9] M.L. Kikidis, C.A. Ppdopoulos, Slenderness rtio effect on crcked bem, J. Sound Vib. 55 (992). [2] D.Y. Zheng, S.C. Fn, Vibrtion nd stbility of crcked hollow-sectionl bems, J. Sound Vib. 267 (23)

17 38 M. Kis, M. Arif Gurel / Interntionl Journl of Engineering Science 45 (27) [2] G.M. Dong, J. Chen, J. Zou, Prmeter identifiction of rotor with n open crck, Eur. J. Mech. A: Solids 23 (24) [22] S.K. Jng, C.W. Bert, Free vibrtion of stepped bems: higher frequencies nd effects of steps on, J. Sound Vib. 32 (989) [23] J.I. Wng, Vibrtion of stepped bems on elstic foundtions, J. Sound Vib. 49 (99) [24] J. Lee, L.A. Bergmn, The vibrtion of stepped bems nd rectngulr pltes by n elementl dynmic flexibility method, J. Sound Vib. 7 (994) [25] H.P. Lee, T.Y. Ng, Vibrtion nd buckling of stepped bem, Appl. Acoust. 42 (994) [26] M.A. De Ros, P.M. Belles, M.J. Murizi, Free vibrtions of stepped bems with intermedite elstic supports, J. Sound Vib. 8 (995) [27] S. Nguleswrn, Vibrtion nd stbility of n Euler Bernoulli bem with up to three-step chnges in cross-section nd in xil force, Int. J. Mech. Sci. 45 (23) [28] T.C. Tsi, Y.Z. Wng, Vibrtion nlysis nd dignosis of crcked shft, J. Sound Vib. 92 (996) [29] B.P. Nndwn, S.K. Miti, Detection of the loction nd size of crck in stepped cntilever bems bsed on mesurements of nturl frequencies, J. Sound Vib. 23 (997) [3] T.D. Chudhri, S.K. Miti, Experimentl verifiction of method of detecting of crcks in tper nd segmented bems bsed on lling of trnsverse vibrtion, Int. J. Frct. 2 (2) [3] Q.S. Li, Vibrtory chrcteristics of multi-step bems with n rbitrry number of crcks nd concentrted msses, Appl. Acoust. 62 (2) [32] Q.S. Li, Free vibrtion nlysis of non-uniform bems with n rbitrry number of crcks nd concentrted msses, J. Sound Vib. 252 (2) [33] O.N.L. Abrhm, J.A. Brndon, The lling of the opening nd closure of crck, J. Vib. Acoust. 7 (995) [34] J.A. Brndon, O.N.L. Abrhm, Counter-intuitive qusi-periodic motion in the utonomous vibrtion of crcked Timoshenko bems, J. Sound Vib. 85 (995) [35] W.C. Hurty, Dynmic nlysis of structures using substructure s, AIAA J. 3 (965) [36] G. Irwin, Frcture Mechnics, Structurl Mechnics, Pergmon Press, 96. [37] H. Td, P.C. Pris, G.R. Irwin, The Stress Anlysis of Crcks Hndbook, Pris Production Incorported nd Del Reserch Corportion, 985. [38] G.R. Cowper, The sher coefficient in Timoshenko s bem theory, J. Appl. Mech. (966) [39] A.D. Dimrogons, S.A. Pipetis, Anlyticl Methods in Rotor Dynmics, Applied Science Publishers., 983. [4] Z. Friedmn, J.B. Kosmtk, An improved two-node Timoshenko bem finite element, Comput. Struct. 47 (993) [4] C.A. Ppdopoulos, A.D. Dimrogons, Coupled longitudinl nd bending vibrtions of crcked shft, J. Vibr. Acous. Stress Rel. Des. (988) 8.