CRACK DETECTION IN EULER-BERNOULLI BEAMS ON ELASTIC FOUNDATION USING GENETIC ALGORITHM BASED ON DISCRETE ELEMENT TECHNIQUE

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1 Idia J.Sci.Res.() : 48-5, 04 ISSN:50-08(Olie) ISSN : (Pit) CRACK DEECION IN EULER-BERNOULLI BEAMS ON ELASIC FOUNDAION USING GENEIC ALGORIHM BASED ON DISCREE ELEMEN ECHNIQUE MOJABA GHASEMI a, ALIREZA ARIAEI b ab Depatmet of Mechaics, College of Egieeig, Uivesity of Isfaha ABSRAC Diffeet models of elastic foudatio have bee studied by eseaches. Oe of the most commo is Wikle foudatio i which foce is diectly popotioal to the beam tasvese displacemet. Cack is the most commo defects i stuctues. Geeally it educes the atual fequecies ad stuctual stiffess. I this pape, Eule-Beoulli beam model is cosideed. Fist, Eule-Beoulli fomulatio is calculated usig Discete Elemet echique (DE) ad the the fomulatios is applied to Wikle elastic foudatio. I he discete elemet method, flexible cotiuous beam ca be split up to a system of igid bas ad flexible oits which esist elative otatio. hus, the paametes of cack icludig the size ad locatio ae set to detemie. o esue this, cack detectio issue is modeled to a oliea optimizatio poblem usig a Geetic Algoithm to solve i Matlab (vesio 0b, he Mathwoks, Ic.). Fially, the effect of vaious paametes such as depth ad cack legth fo diffeet bouday coditios o the atual fequecies, ae ivestigated. he accuacy of this method is sigificat due to its speed. KEYWORDS: Eule-Beoulli Beam; Wikle Elastic Foudatio; Cack; Discete Elemet; Geetic Algoithm ypically, cacks iduce chages i the stuctue s stiffess, also educig its atual fequecy. Beams o elastic foudatio ae widely used i may egieeig applicatios such as ailway applicatios, cocete stuctues o soil i civil egieeig applicatios, isolate diffeet costitute pats of machiey, etc. Due to such applicatios, vibatios chaacteistics of beams o elastic foudatios ad cack detectio have bee subect of eseach i may studies. Also effect of elastic foudatio o atual fequecies of cacked ad ucacked beams has bee ivestigated i may eseaches. I geeal, elastic foudatios cotiuously suppot the beams alog thei spa ad they povide eactio foces o momets which ae popotioal to the displacemets o otatios. Foudatios model ae mostly oe paamete. Oe paamete foudatio model is also efeed to as Wikle elastic foudatio i which foce is diectly popotioal with flexual displacemet of the beam. I two paamete foudatio models aothe paamete is take ito cosideatio i additio to Wikle paamete (Ma et al., 009; De Rosa, 995; Wag & Stephes, 997; Razaqpou & Shah, 99; Kagaovi & Youesia, 004). he Eule-Beoulli beam with liea foudatio model was cosideed by Hsu (005). I this, spig foced is diectly popotioal to the tasvese displacemet of the beam. his is a wikle type foudatio model. Fo the cack model a compliace value was obtaied by factue mechaics methods. Hsu (005) used diffeetial quadatue method fo modelig the cacked Eule-Beoulli beam o Wikle elastic foudatio. By doig so, the equatio of motio fo a beam is tasfomed to a discete fom. Hsu (005) cocluded that the fist atual fequecy iceased sigificatly as the Coespodig autho foudatio stiffess iceased. Also the depth ad locatio of cack affected the atual fequecies. I aothe of cacked beams o elastic foudatios, compaiso of diffeet foudatio models was caied out by Shi et al. (006). I this eseach cacks wee assumed to be ope ad the Eule-Beoulli beam theoy was used. Both Wikle foudatio model ad two-paamete foudatio model wee used. I the, the cacks wee eplaced with massless spigs. Spig costat was calculated by usig cack compliace value which depeds o cack chaacteistics such as cack legth. Effect of foudatio spig costat, cack legth, locatio of the cack ad umbe of cacks wee ivestigated. Also compaiso betwee two foudatio models; Wikle ad two-paamete foudatio models wee doe. he cocluded that fo the case of fixed-fixed bouday coditios, the atual fequecies of a beam which ests o two-paamete foudatio came out to be highe tha those of a beam o Wikle foudatio. his pape aims to develop ad optimize the existig methods based o locatio ad depth of cacks i beam with Wikle foudatio that pefom by usig DE. Fist, the DE will be modified ad the exteded to coside effects of cack o the dyamic espose of a beam. Fially, the Wikle foudatio is peseted ad the effects ae ivestigated i fomulatio. DE FORMULAION I the aalytical method due to diffeetial legth elemet, by igoig the poduct of ρi which ρ is the desity of the beam ad I is the momet of ietia of coss sectio, the otatioal ietia tem ca be eglected. Howeve, ulike the aalytical

2 appoach, the otatioal ietia tem of the DE ca ot be totally igoed because the legth of elemets ca ot be abadoed compaed with the thickess of the beam. Fo this easo, i the followig sectios oly the otatioal ietia of the beam cotaiig the tem ρi ca be igoed ad the othe pat should be cosideed. he DE divides a cotiuous flexible beam ito a system of igid bas ad oits, which esist elative otatio of attached bas (Fig. ). heefoe, the kietic eegy is calculated fo igid bas ad potetial eegy is stoed i flexible oits. he potetial eegy of the beam due to bedig is give by whee U θ Ι M θ K θ (5) h [( y y ) + ( y y )] + EI L M Kθ, K, m mh, h h (6) Figue. Appoximatio of DE fo beam ude bedig Fo the bedig deflectio, the total kietic eegy of the beam ca be expessed as the sum of the taslatioal ad otatioal compoets (Khiem et al., 004; Yavai et al., 00): dy Ι m ( ) + dt I ω () whee is the umbe of elemets, m the mass of the elemet, I the mass momet of ietia at the cete of the elemet ad y the deflectio at the cete of each elemet. he deflectio, the momet of ietia, ad the otatioal velocity ca be appoximated as ( y y ) y + () I above equatio, L is the legth of the beam, m the mass pe uit legth, EI the bedig stifess, M ad K ae the bedig momet ad the otatioal stifess of oit ad the elative otatio of the attached bas. he tem fo otatioal stiffess of oit is coect oly if the elemet - does ot have a cack. Othewise, the otatioal stiffess of oit should be calculated sepaately. I that case, the otatioal stiffess of oit ca be witte as (Mahmoud & Zaid, 00) θ 0 + D M θ (7) whee is the elative otatio of oit i the pesece of a cack, 0 the elative otatio of oit if the cack was abset, D the cack compliace ad M the bedig momet at oit. he cack compliace D may be detemied by modellig the cacked sectio as a otatioal spig coectig two udamaged beam segmets. he stiffess of the otatioal spig is detemied usig factue mechaics which fo a ectagula sectio of height H with a cack of depth a, it ca be show to be a H ( ) H D / EI a/ H 4 [ ( a/ H) + 7.4( a/ H) 5.84( a/ H) +.( a/ H) ] Also the bedig momet ca be expessed as h I ρ Ih+ ρa () h ( y& y& ) ω (4) whee h is the legth of the elemet, ρ the desity of the beam, A the coss sectioal aea ad I the costat coss sectio momet of ietia. Idia J.Sci.Res.() : 48-5, M M k θ (9) 0 0 k θ (0) whee K is the otatioal stifess of oit i the pesece of a cack ad K 0 is the otatioal stifess of oit if the cack was abset. By substitutig Eq. (9) ad Eq. (0) ito Eq. (7), the otatioal stiffess at oit ca be calculated as

3 β is oly elated to oit, is equal to k β k () 0 + D k β () 0 he total mechaical eegy of the beam is the sum of kietic ad potetial eegies: Φ+V+U () We ca defie the momet of ietia of the beam aea as the poduct of the aea ad the squae of gyatio adius of the beam s sectio about z-axis: EI V y + y y + y y y + y y y { δ ( ) ( ) ( ) ( ) β Ι + + L ( y y y ) ( y y δ y ) δ ( y δ y ) + } K h K h w + δ y + 4 w y he geeal dyamic equatio ca ow be abtaied fom Lagage s equatios. d V + 0 dt y& y (9) (0) Afte some eaagemet, a matix equatio of motio may be obtaied at each istat: I Az (4) Q Y, tt A Y 0 + () By substitutig Eq. (4) ito Eq. () ad eglectig the tem ρi due to eglectig otatioal ietia, we have mh I (5) Now the kietic eegy of the beam ca be witte as whee Q ad A ae the costat symmetic matices. he deivatio of matices ae give i Appedix. Eq. () ca be witte i momet i time, each cosistig of equatios. I this equatio we have Y { y y y }... () ml I y& + y& y& + + δ y& + δ y& y& 6 (6) Y,tt { y y y }, tt, tt..., tt () Let us defie the Wikle foudatio equatio as Now we defie the followig equatio (, ) K y( x, t) P x t (7) w A - Qw 0 (4) whee P( x,t ) is the foce pe uit legth exeted by the foudatio ad K w is the Wikle foudatio costat which has the dimesio of foce pe uit legth pe uit displacemet. hus, the equatio of Eule-Beoulli beam with Wikle foudatio becomes ( ) ( ) 4 y x, t y x, t K y( x t) ρ A 4 w EI +, + 0 x t (8) Expadig the detemiat leads us to a equatio ivolvig atual fequecy, size ad locatio of the cack. he equatio ca be witte as whee D ( ω, l, γ ) 0 (5) his will ot chage the kietic eegy of the beam but the potetial eegy will be tasfomed as followig l γ { l,l,...,l } { γ, γ,..., γ } (6a) (6b) Eq. (6a) ad Eq. (6b) epeset positio vecto of cacks which divided ito the legth of the beam ad depth vecto of cacks which divided ito the thickess of the beam, espectively. I fowad poblem with the values of size ad Idia J.Sci.Res.() : 48-5,

4 locatio of cacks, omal values of atual fequecies should be foud due to establish Eq. (5). I ivese poblem, this ted exactly happes i the opposite way. Now, the measued ad kow values of the atual fequecies ca be defied by followig vecto f ( y 0 y y,y,...,y, ) mi,,,...,. () ω { ω, ω,..., ω } m (7) he poblem should ow be solved iclude fidig () he positio of cacks, Eq. (6a), that fits i l l l + () he depth of cacks, Eq. (6b). he ukow paametes of cack cotaied i the ivese fom of poblem ae as follows y { y,...,y } { l,...,l, γ γ },..., he iput values ae the atual fequecies as Eq. (7). he obective fuctio defied as (Khiem et al., 004) m [ ] ω ( l, γ ω (8) f ( y ) ) (9) Hee, he obective fuctio is defied i the umeical fom obtaied each time by solvig Eq. (5) egadig ω. he Geetic algoithm is used to solve. his algoithm has a oticeable advatage ove othe methods because it does ot eed to seach the etie solutio space. NUMERICAL EXAMPLE he esult ae obtaied by usig simple bouday coditios. he effects of cack ae ivestigated fo diffeet values. he popeties of the beam ae as follows L 0 m, E.06 0 N m -, I m 4, m kg m -, A m, height 0.5 m, width 0.5 m, K w N m - But the impact of lowe fequecies decease due to the isigificat values, while we kow these fequecies ae ofte a detemiat paamete i vibatio poblems. heefoe, the obective fuctio is defied as m ω ( ( ) l, γ ) f y (0) ω I this equatio, by dividig to the size of fequecy, lowfequecy effects ae simila to highe fequecies. I view of the pevailig costait, the optimizatio poblem will ed up to able. Fist atual fequecies (Hz) fo fixed-fixed beam o Wikle foudatio whee K w is the Wikle foudatio costat. ables ad espectively show the fist fequecies of fixed-fixed beam ad suppoted-suppoted beam fo the locatio ad the depth of a cack. he cause of the obseved diffeece betwee the fequecies epoted by Shi et al. (006) ad esults of the cuet is due to the diffeece i the cack compliace defied by Eq. (8) as well as the solutio method. It is obvious fom the esults, whateve the depth of the cack iceases, the diffeece becomes moe. I additio, by iceasig the depth of the cack, the eductio ate of the atual fequecies chages faste. As expected, the fixed-fixed beam shows the highe values of atual fequecy owig to highe esistat agaist a otatig ad theeby, the eductio of atual fequecies is gate. I fowad fom of poblem, the fist thee atual fequecies of fixed-fixed beam fo the specific depth ad positio of cack (depth 00 mm, positio 50 mm) ae peseted (able ). Dimesioless legth Dimesioless depth Peset Shi et al (006) Peset Shi et al (006) Peset Shi et al (006) Idia J.Sci.Res.() : 48-5,

5 able. able. Fist atual fequecies (Hz) fo suppoted beam o Wikle foudatio Dimesioless legth Dimesioless depth Peset Shi et al (006) Peset Shi et al (006) Peset Shi et al (006) able. able. Fist thee atual fequecies (Hz) Fequecy Peset Shi et al (006) f (Hz) f (Hz) f (Hz) I the ivese poblem, with the atual fequecy values, we fid out the paametes of cack (depth ad positio) by takig a cack ad thee atual fequecies. Numeical esults obtaied fom solvig the optimizatio ae pezeted (able 4). able 4. able 4. he cack paametes Cack Peset Shi et al (006) Positio (mm) Depth (mm) CONCLUSION I this pape, ivese ted of poblem fo Eule-Beoulli beam with a cack has bee examied by DE appoach. A optimizatio poblem has defied i ode to calculate the positio ad depth of cack usig atual fequecies. Cosideig the esults, we ealized that the umbe of measued fequecies at least shall be two times geate tha umbe of cacks because havig N cacks, thee should be N δ EI β 4 4 β β KL w A + L β 4+ 8 β 4 4 β K β 4 4 β 0+ β 8... δ 8... δ δ+ 0 4 δ( + δ) δ 4 δ( + δ) δ( + δ) ukow paametes which iceasig umbe of fequecies we could get moe pecise esults. It should be metioed that this matte is ust coect i ideal situatios. Howeve, i eal wold i pesece of oise, thee is cosideable eo i the esult, as it is ot possible to achieve to exact aswe ust by peseted method. I additio, ivese ted does t have a uique esult always which this coditio shall be cosideed especially i beams with symmetic bouday coditio. Compaig the esults, it ca be udestood that the positio ad depth of cack ae aoud the efeece values. Appedix I this appedix, matices A ad Q ae peseted. Idia J.Sci.Res.() : 48-5,

6 REFERENCES é4 ù ml Q d êë d d úû Ma X., Buttewoth J., Clifto G.; 009. Static Aalysis of a Ifiite Beam Restig o a esioless Pasteak Foudatio, Euopea Joual Of Mechaics A/Solids 8: De Rosa M.; 995. Fee Vibatios of imosheko Beams o wo Paamete Elastic Foudatio, Computes ad Stuctues 57(): Wag., Stephes, J.; 997. Natual Fequecies of imosheko Beams o Pasteak Foudatios, Joual of Soud ad Vibatios 5(): Razaqpou A., Shah, K.; 99. Exact Aalysis of Beams o wo Paamete Elastic Foudatios, Iteatioal Joual of Solids Stuctues 7(4): Kagaovi M., Youesia, D.; 004. Dyamics of imosheko Beams o Pasteak Foudatio Ude Movig Load, Mechaics Reseach Commuicatios : 7-7. Hsu M.; 005. Vibatio Aalysis of Edge Cacked Beam o Elastic Foudatio with Axial Loadig Usig Diffeetial Quadatue Method, Compute Methods iapplied Mechaics ad Egieeig 94: -7. Shi Y., Yu J., Seog K., Kim J., Kag, S.; 006): Natual Fequecies of Eule-Beoulli Beam with Ope Cacks o Elastic Foudatios, Joual of Mechaical Sciece ad echology (KSME It. J.) 0(4): Khiem N., Lie., Leo D.; 004): Multi-Cack Detectio fo Beam by the Natual Fequecies, Joual of Soud ad Vibatio 7: Yavai A., Noui M., Mofid M.; 00.Discete Elemet Aalysis of Dyamic Respose of imosheko Beams Ude Movig Mass, Advaces i Egieeig Softwae : 4-5. Mahmoud ad, M., Zaid M.; 00.Dyamic Respose of a Beam with a Cack Subect to a Movig Mass, Joual of Soud ad Vibatio 56: Idia J.Sci.Res.() : 48-5,