Equivalent Half Pulse (EHP) Method for Vibration Analysis under Regular Wave

Transcription

1 INERNAIONAL JOURNAL OF COASAL & OFFSHORE ENGINEERING JCOE No. / Wier 07 (-8) Equivale Half Pulse (EHP) ehod for Vibraio Aalysis uder Regular Wave ohammad Reza abeshour, Hossei Ebrahimi, ai Faemi Dowloaded from ijcoe.org a : o uesday Seember h 08 Ceer of Excellece i Hydrodyamics ad Dyamics of arie Vehicles, echaical Egieerig Dearme, Sharif Uiversiy of echology ehra, Ira; abeshour@sharif.edu Phd Cadidae, Islamic Azad Uiversiy, Arak; hossei_abrahimy@yahoo.com Dearme of arie Idusries, Sciece ad Research Brach, Islamic Azad Uiversiy ehra, Ira; m.faemi@srbiau.ac.ir ARICLE INFO Aricle Hisory: Received: Ja. 07 Acceed: ar. 07 Keywords: Resose Secrum Assessme Vibraio Aalysis ABSRAC. Iroducio Desig ad assessme of comlex srucural sysems such as fixed offshore srucures uder eviromeal loads require heavy ime cosumig comuaios. I is desirable for egieers o reduce hese cos of aalysis while maiai maximum recisio i order o obai reliable resuls. Such aroaches resul i creaio of rocedures like resose secrums for elasic ad ielasic desig of srucures ad caaciy secrum mehod for assessme of srucures subjeced o seismic loads. he mahemaical formulaio of he RS firs aeared i he docoral disseraio of.a. Bio i 9 ad i wo of his aers.his ioeerig work i he 90's o he resose of srucures o rasie disurbaces led o he key coce of resose secrum as a uiversally alied ool i earhquake roof desig, ad i may oher roblems[]. Afer e years Houser has iveed he coce of resose secrum for elasic desig of srucures i 9 []. Blume ad Newmark has develoed his coce ad creae a solid backboe o use i as a fucioal ool for rouie desig rocess of srucures []. Velesos coducs sudies o imlemeig resose secrum mehod i ielasic desig of srucures uder seismic loads []. By isiraio from Joh Blume's reserve eergy echique, Freema Fully dyamic aalysis of offshore srucures uder radom wave loads i ime domai is someimes ecessary for calculaig srucural resoses i desig issues. Such aalyses are very ime cosumig ad herefor simlified mehods for esimaio of acceable resose of hese srucures ca be very useful i iiial desig. I his aer a iovaive mehod o obai resose secrum of fixed offshore srucures caused by exreme waves is rerese based o coce of imulse resose secrum. For his urose he srucural sysem is cosidered as a simle oe degree of freedom srucure ad he differe sea saes are equalized o differe half siusoidal ulses. Resose secrum of srucure is defied as a lo of srucural resose o hese ulses for differe eriods. By usig his mehod he cos of comuaios is decreased sigificaly while he accuracy of resuls was reserved. iveed he caaciy secrum mehod [] for assessme of oliear behavior of srucures. All of aforemeioed mehods ad sudies are alicable for srucural sysems subjeced o seismic loads. A few sudies have bee coduced durig as years i order o cover hese mehods io a racical forma ha ca be used for aalysis ad desig of offshore srucures subjeced o wave loads. Velesos is ioeer i his field, i 98 he coduced feasibiliy sudy o he uilizaio of resose secrum coce for desigig offshore srucures agais radom waves by alyig he resecive simlifyig assumios [6] i oher words he ries o cover resose secrum mehod io racical ool for desig ad aalysis of fixed offshore srucure. oivaed by Velesos sudies, i 986 ug demosraed ha oe ca obai resose secrum of he offshore srucures for a equal sigle degree of freedom model, by alyig he deermiisic wave heory ad liearizaio of he drag erm [7]. I laer years oher researchers sudied he coce of resose secrum,such as Hu ad e.al [8], who has develoed a desig resose secra agais radom ocea waves.. Coce of Imulse Resose Secrum Aar from solvig differeial equaios of moio here are oher ways o fid resose of dyamic

2 Exciaio i Base Secrum ime Hisory Exciaio i ass Secrum ime Hisory ohammad Reza abeshour e al. / Equivale Half Pulse (EHP) ehod for Vibraio Aalysis uder Regular Wave Dowloaded from ijcoe.org a : o uesday Seember h 08 sysem used by vibraio heory. Oe of hem is fidig he secral resose for exciaio or dyamic force exered i mass of he srucure. Each resose secrum rereses he aure ad characerisics of dyamic load exciaio. Resose secrum of imulse load has bee sudied ad used horoughly sice i is base for ime domai aalysis o he oher had, i he frequecy domai i ca be used o secify imac load o simulae differe scearios of loadig i order o fid he maximum resose. o illusrae his ye of aalysis i is esseial o Exciaio Force Acceleraio Geeral Exciaio ime ime SDOF SYS. Dislaceme Dislaceme Resose ime ime ackowledge subsaive differece bewee he exciaio i mass of he srucure or base of srucure. Accordig o Newo's secod law, acceleraio alied o he base of he srucure will cover io ieria force i mass of he srucure which is ime deede. Oe of he fudameal ways of sudyig ad fidig he resose o he ime deede forces is usig shor ulses of imac force. I he followig figure, differece bewee exciaio i mass ad exciaio i base of he srucure is illusraed. Exciaio Force Amliude H Velociy Amliude H Pulse Exciaio Duraio Duraio SDOF SYS. Dislaceme R d Dislaceme R d Resose ime ime Figure : Differece bewee geeral exciaio ad ulse exciaio I may loadig siuaios he exciaio is eiher harmoic or eriodic hus we mus sudy he dyamic resose of sysem which is exciaios is varied wih ime. he resose of srucure o each ulse regardless of is form is cosis of wo differe hase. Phase oe is force vibraio hase, he resose of srucure i his hase is obaied by Duhamel iegral, hase wo is free vibraio hase ad he resose of srucure i his hase is obaied by solvig he differeial equaio of moio. I he followig able he commo ye of imulse load used i dyamic aalysis is rereseed.. ehodology wo mai aroaches ca be cosidered for usig his coce i assessme of offshore srucures. I firs aroach all mechaical roeries of he srucure are cosa while he eriod of he ulse is varied. I siusoidal ulse secrum while he raio of is less ha oe, icreasig he resuls i he icreased resose. I secio where raio of is greaer ha oe, icreasig he resuls i shor decreasig resose ad afer ha resose becomes somehow cosa.

3 ohammad Reza abeshour e al. / IJCOE 07, No. ;.-8 Dowloaded from ijcoe.org a : o uesday Seember h 08 Figure : Pulse resose secrum wih variable ulse eriod ad cosa aural vibraio eriod of he srucure ye of Pulse Resose o ulse formula Pulse secrum ax codiio u () ( us ) 0 si( ) max.76 si( ) ( ) 0.8 ( )cos( ) si ( ) Siosoidal ( ) riagular Recagular u () ( us ) 0 ( si( )) 0 si ( ) si si ( ) si ( ) si SIN Pulse Secrum R d max. u () ( us ) 0 max cos 0. (si )si ( ) Figure : Differe ye of Pulse ad heir close form relaio I secod aroach he ulse eriod is cosa while he aural vibraio eriod of he srucure is variable. I his siuaio while he raio of is less ha oe, icreasig he resuls i decreased resose. I secio where raio of is greaer ha oe, icreasig he resuls i cosa resose uil he raio of becomes afer ha he resose will icrease uil he raio becomes. R d SIN Pulse Secrum Figure : Pulse resose secrum wih Cosa ulse eriod ad variable aural vibraio eriod of he srucure

4 ohammad Reza abeshour e al. / Equivale Half Pulse (EHP) ehod for Vibraio Aalysis uder Regular Wave Dowloaded from ijcoe.org a : o uesday Seember h 08. odelig scoe I his research umerous resose secrum was obaied i order o clarify he idea. I a rouie resose secrum used i earhquake egieerig he resose of sigle degree of freedom agais he aural vibraio eriod or frequecy of srucure is loed. I formulaio of seismic load he resose of srucure is ideede of srucural geomery while i i wave load, accordig o orriso equaio he resose of srucure deeds o srucural geomery herefore resose secrums rereseed i his sudy are based o variable srucural geomery by meas of diameer. Differe geomery is defied by differe diameer of srucural member; he hickess of srucural member is defied as /0 of diameer. For he sake of simliciy diameers of 70, 80, ad 90,00,0,0 is cosidered for he models sudied. he heigh of srucural member is meer. he model is coeced o he groud by srig eleme i which heir siffess is ued so he aural vibraio eriod of.,.6,.7,.8,.9 ad secod is obaied for he srucural member. he seleced rage of aural vibraio eriod is chose based o yical aural vibraio eriod of offshore srucures locaed i Persia Gulf. Ocea Waves cosidered i his sudy are exraced from yical sea saes reored for he Persia Gulf regio wih he secific heigh ad reur eriod. Wih a variey of models meioed, a oal of 88 aalyses o simle sysems have bee erformed. abale: Sea saes iformaio for Persia gulf Sea sae secific heigh (eer) Wave Period (Secod) Reur Period (Year) Equalizaio of wave force o a ulse force I order o obai a Resose secrum of fixed offshore srucures caused by exreme waves ad based o coce of imulse resose secrum firs we mus equalize he ocea wave force o simle Siosoidal ulse wih a secific eriod ad amliude i oher word he eriod ad amliude of he ulse is comued based o hydrodyamic secificaio of waves. I order o achieve his goal firs we mus cosider a roer simlified srucural model. Assume a simle oe degree of freedom srucure wih a lumed mass subjeced o simle airy wave[9]. As show i figure, laform dislaceme i heigh z a ime is deoed by. Ocea wave is modeled by airy wave heory i he waer deh d, wave amliude a, wave umber k ad frequecy. Accordigly, he waer aricle velociy ad acceleraio are calculaed based o equaio () ad (), resecively: agk cosh kz u( z, ) cos cosh kd () cosh kz u( z, ) agk si cosh kd () he wave load is comued hrough orriso's equaio. he equaio cosiss of erms drag f ad ieria f I : f ( z, ) ( C D( u( z, ) z) u( z, ) z ) () D d D f I ( z, ) C m ( )( u( z, ) z ) () ( ) z K z f ( z, ) f ( z, ) () deck s s I D deck s D C m z kz D Cd D u ( z, ) z u ( z, ) z C m u ( z, ) D (6) I all aalyzes, added mass geeraed i he sysem due o dyamic dislaceme of he verical ie is calculaed ad cosidered is imac o he resose. I his way, he effec of he acceleraio of he aricles relaive o he ube is cosidered i he calculaio. O he oher had, he relaive velociy of he sysem wihou damig is egleced sice i has lile effec o he resose. 6. imlemeaio of mehod A o-liear differeial equaio (Equaio 8) for each of he 88 differe model develoed i alab, have bee used o lo he resuls. For his urose, accordig o orriso's equaio deedece o he heigh of he srucure he oal heigh of he ube is divided io segmes ad he oal force of he wave is obaied by umerical iegraio of orriso equaio (Force er ui legh). Figure : Wave Force is calculaed by umerical iegraio of he oriso s equaio

5 ohammad Reza abeshour e al. / IJCOE 07, No. ;.-8 Dowloaded from ijcoe.org a : o uesday Seember h 08 Oce agai, he same sysem has bee aalyzed agais he half sie ulse wih secific amliude ad eriod. I all cases, half si ulse wih eriod equal o half of he airy wave eriod is cosidered. he amliude of he half si ulse is obaied so he maximum resose of he srucure agais he airy wave load becomes equal o maximum resose of he srucure agais he half si ulse. Wih his aroach we have relaced a wave force wih a shor ulse from he ulse secrum wih kow aswer. Real Sea Sae Simle Wave Secrum Regular Wave EHP (Equivale Half ulse) H Comlex measureme rasformed io mahemaical odel Hours wave ime hisory rasforms io Simlified Regular wave Simle Regular wave rasforms io simle half si ulse 7. resuls ad discussio Due o he regular wave heory, he resose ime hisory is cosis of wo domia eriods, oe cocerig he aure of he srucure ad he oher eriod due o he variey of sea wave. Dis. (cm) Sea Sae: D=90cm s=.7sec (sec) Figure 7: Resose ime hisory of he aural eriod of.7 secods, he ie diameer of 70 cm ad a wave of he sea sae I rocess of srucural assessme, he maximum resose is always i favor[0]. I secral oi of view he followig diagrams ca be exraced from maximum resose obaied from dyamic aalysis. For examle, hree-dimesioal diagram of dislaceme resose secrum versus ie diameer ad wave eriod, for srucures wih aural vibraio eriod of, 9 is show i figure 8. We ca coclude ha he maximum resose i iese sea saes is more sesiive o srucural member s geomery. his subjec is Jusifiable cosiderig he imorace of ieria erm i deermiaio of force exered o srucural member wih large diameer. Regardig he equaio he ieria load is roorioal o square of diameer ad comared o drag force ha has roorioaliy o diameer, i has more deedecy o geomery. Figure 6: Coversio of Comlex eviromeal loadig io simle half si ulse s=.9 sec 7 6 Dela(cm) D(cm) w(sec) Figure 8: Rage hree-dimesioal srucures of he ormal eriod of 9. secods.

6 ohammad Reza abeshour e al. / Equivale Half Pulse (EHP) ehod for Vibraio Aalysis uder Regular Wave Dowloaded from ijcoe.org a : o uesday Seember h 08 By more comrehesive aroach, he resose ca be deiced as raio of aural vibraio eriod o eriod of wave versus raio of diameer of srucural member o wave heigh. I figure 9 his aroaches has bee show for sea sae. ore resose flucuaios ca be see i his diagram for srucures wih smaller diameers while he chagig rae of resose for srucures wih larger diameers is somehow liear. Variey of secrum diagram ca be defied for each grou of daa caegorized based o diameer, aural vibraio eriod, wave eriod, wave heigh or ay Dela(cm) s/w D Secrum-Sea Sae: 0.6 Figure 9: hree-dimesioal secrum of he waves geeraed by he Sea x 0 Fo & Srucure eriod Sea Sae:.6. meaigful raio of hem serve as iu agais resose like dislaceme, srig force, eergy ad RAO ec. I his secio a secral aroached has bee used i order o exlai he amliude of half si ulse. A firs diagram of F0 versus aural vibraio eriod of srucure i differe sea saes has bee showed. F0 is half si ulse amliude ha will equalize he maximum resose obaied from wave aalysis o maximum resose obaied from half si ulse D/Hw D=70cm D=80cm D=90cm D=00cm D=0cm D=0cm 0.. Fo(N) s(sec) Figure 0: Half he ulse amliude of he waves geeraed by he Sea sae 6

7 ohammad Reza abeshour e al. / IJCOE 07, No. ;.-8.6 x 0 Fo & Srucure eriod Sea Sae:8. D=70cm D=80cm D=90cm D=00cm D=0cm D=0cm. Fo(N) Dowloaded from ijcoe.org a : o uesday Seember h s(sec) Figure : Half he ulse amliude of he waves geeraed by he Sea sae 8 By lookig a above diagram, he daa ca be assessed from wo ois of view. he firs oe is wha is he effec of variable diameer o he EHP for he srucures wih same aural vibraio eriod i he same sea saes? From his diagram i is clear ha i he fixed aural vibraio eriod (a lie arallel o Y- axis) icreasig he diameer resuls i larger EHP. Sice he icreasig of diameer meas higher wave force (icreasig he drag ad ieria load), his red is romed. he secod oe is wha he effec of variable aural vibraio eriod o he srucures wih same diameer i he same sea sae is? Aswerig his quesio requires more accurae reviewig. I geeral he icrease of aural vibraio eriod resuls i larger EHP bu here are some ois ha are disruig he red of chage. ha s because, wih icrease of eriod ad decrease of siffess, he wave force is icreased wih oliear aer ad his heomeo resuls i larger dislaceme i he model ad for his reaso larger EHP is eeded for equalizaio. Ceralizaio of exreme values relaed o all srucures wih differe diameers a cerai eriods could idicae he eed for sesiiviy i he equalizaio of a o-liear aalysis ad a liear aalysis i secific ois. 8. Summary ad coclusios Aalyses offshore srucures uder eviromeal loads are very ime cosumig ad herefor simlified mehods for esimaio of acceable resose of hese srucures ca be very useful i iiial desig. Coce of resose secrum is oe of hose simlified mehods ha is used for seismic aalysis of srucures secially fixed offshore srucures. his mehod is fully alicable for srucures sice i is srucural ideede ad oly deeds o dyamic secificaio of srucure bu his coce cao be used for wave loads sice he amliude of he load is deed o srucural geomery ad herefor secrum obaied for offshore srucures mus be adjused for variey of geomery ye, loadig level ad ec. I his aer a ew mehod for reducig he cos of aalysis for srucural desig ad assessme of offshore srucures uder wave loads is roosed ad i was show ha by equalizig he wave force o half si ulse ad usig ulse resose secrum we ca obai scoe of resose secrums adjused for srucure ha we wa o desig. I order o obai hese resose secrums firs we simlified he srucural sysem he we equalize he regular wave force o half si ulse by fidig amliude ha equalizes he maximum resose of half si ulse o maximum resose obaied from regular wave aalysis. Refereces ] [ rifuac D, odorovska. Origi of he resose secrum mehod. Proceedigs of he h world coferece o earhquake egieerig ] [ Houser GW. Calculaig he resose of a oscillaor o arbirary groud moio. Bullei of he Seismological Sociey of America. 9;:-9. ] [ Blume JA, Newmark N, Corig LH. Desig of mulisory reiforced cocree buildigs for earhquake moios. Cocree ieraioal. 96;6. ] [ Velesos A, Newmark N. Effec of ielasic behavior o he resose of simle sysems o earhquake moios. Proceedigs of he d world coferece o earhquake egieerig ] [ Freema S, Nicolei J, yrell J. Evaluaios of exisig buildigs for seismic risk A case sudy of Puge Soud Naval Shiyard, Bremero, Washigo. Proceedigs of he s US Naioal Coferece o Earhquake Egieerig: Earhquake Egieerig Research Isiue Oaklad, CA;

8 ohammad Reza abeshour e al. / Equivale Half Pulse (EHP) ehod for Vibraio Aalysis uder Regular Wave Dowloaded from ijcoe.org a : o uesday Seember h 08 ] 6[ Velesos A, Hah G, Bordihao R. Resose secrum aroach o desig for waves. Rece Advaces i Egieerig echaics ad heir Imac o Civil Egieerig Pracice: ASCE; ] 7[ ug C. A sudy of resose ad resose secra of offshore srucures o regular waves. Alied ocea research. 986;8:8. - ] 8[ Hu S-LJ, adao J. Desig resose secra for offshore srucures. he Firs Ieraioal Offshore ad Polar Egieerig Coferece: Ieraioal Sociey of Offshore ad Polar Egieers; 99. ] 9[ Borgma LE. Secral Aalysis of Ocea Wave Forces o Pilig (Coasal Egieerig Coferece i Saa Barbara, Califoria, Ocober 96). Joural of he waerways ad harbors divisio. 967;9:9-6. ] [ Bruce R. Quasi-saic resose of jacke laforms subjec o o-liear wave loadig. Proceedigs of he h Coferece o he Behavior of Offshore Srucures