Renovation explicit dynamic procedures by application of Trujillo algorithm

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1 Ieraioal Joural of he Physical Scieces Vol. 6(), pp , Jauary, 0 Available olie a hp:// ISSN Academic Jourals Full egh Research Paper Reovaio eplici dyamic procedures by applicaio of Trujillo algorihm H. Hashamdar*, Z. Ibrahim,. Jameel ad H. B. ahmud Deparme of ivil Egieerig, iversiy of alaya, uala umpur, alaysia. Acceped December, 00 I his paper, a Trujillo algorihm mehod for he eac soluio of oliear eplici dyamic problems is developed. This mehod associaes umerical echiques ad i is applied o deermie he aural frequecies of srucures. The umerical predicios of he aural frequecy are compared o aalyical ad eperimeal resuls. There was o direc mehod available for solvig such a problem efficiely i he case of eplici dyamic aalysis. The Trujillo algorihm is a idirec mehod ad he approach is based upo he priciple of coservaio of eergy. I is believed ha he proposed mehod provides a powerful egieerig ool i he aalyzig of srucures. ey words: Noliear dyamic respose, opimizaio of eergy, eplici dyamic aalysis, fiie eleme mehod, vibraio, perurbaio echique, aalyzig modal, Trujillo algorihm mehod. INTRODTION The modal dyamics procedure is suiable oly for liear problems. Whe oliear dyamic respose is of ieres, he equaio of moio mus be iegraed direcly (Belyschko, 00; ksu, 009). The direciegraios of equaio of moio is performed by usig a eplici dyamic procedure. Ad his procedure ca be a effecive ool for solvig a wide variey of oliear solid ad srucural mechaism problems, however, he mehod require a small ime icreme size ha depeds solely o he highes aural frequecies of he model ad is idepede of he ype ad duraio of loadig. So ha he mehod aalyze is o well suied o lowspeed dyamic eve ad eplici mehod is compuaioally iesive ad are more epesive ha he modal mehods (Da Silva, 00). I his paper, oliear dyamic respose ca be foud by combiig a Trujillo algorihm mehod io eplici dyamic aalyzes. Whe his procedure is used, he mass, he dampig, ad siffess mari marices are assembled ad he equaio of dyamic equilibrium is solved a each poi i ime. The cable srucure is cosidered for eperimeal ad aalyical ess. *orrespodig auhor. hamidreza@siswa.um.edu.my. There are hree source of olieariy i srucural mechaics simulaio: (i) aerial olieariy (ii) Boudary olieariy (iii) Geomeric olieariy Geomeric olieariy is relaed o chages i he geomery of he model durig he aalysis (Sefaou, 009). able srucures are very ligh ad fleible ad hey udergo appreciable deflecios whe subjeced o eeral loadig. Sice all he mai load-carryig members i cable srucures are usually i esio, if he ip deflecio is small, he aalysis ca be cosidered as beig approimaely liear. However, if he ip deflecio is large, he shape of srucure ad, hece, is siffess chages, he, srucural mechaism will belog o olieariy behavior. Thus, cable srucure belogs o geomerically olieariy groups (Huu-Tai, 00). Alhough a umber of mehod have bee developed for dyamic respose aalysis of srucural sysem, bu here are oly a few mehods which ca be used for oliear dyamic respose aalysis. Tesio roof srucures are aalyzed for saic loads ad heir dyamic respose is checked o esure ha he desig provides sufficie safey ad srucural

2 56 I. J. Phys. Sci. Figure. Geeral view umber of members o cable jois. Figure. Geomeric represeaio for a fucio of wo variables i kiemaic codiio. serviceabiliy. Alhough cable srucures, uder service loads, ehibi oliear behavior, rece developmes o compuig mehods have made i possible o carry ou he aalysis wih grea accuracy (Jia, 007; aie, 00). EQATION OF OTION FOR A SYSTE The cable srucure is a muli degree sysem ad he equaio of moio for a muli degree of freedom (DOF) sysem ca be wrie as: () () P () () where ass mari () Dampig mari () Siffess mari Displaceme vecor Velociy vecor Acceleraio vecor P () oad vecor The assumpio of a cosa mass i he case of boh DOF sysems is arbirary as i could be represeed as a ime varyig quaiy. Sice m is a o-zero cosa value, boh sides of Equaio () ca be divided by m, ad for P Q F Equaio () ca be wrie as:

3 Hashamdar e al. 57 Figure. The direcio of desce vecor. P QF () The mahemaical soluio of Equaio () depeds o he values of P, Q ad F. Equaio () is a liear differeial equaio if P ad Q are idepede of ad remais so eve if P ad Q are fucios of. The soluio is ormally give i he form f () ad gives eac values of for ay. oce f () is defied ca be derived by differeiaio. Whe P ad Q are fucios of, Equaio () becomes o-liear (iqus, 00). For such equaio he soluio cao be epressed i fucioal form ad i is ecessary o plo or abulae o soluio curve poi by poi, begiig a ( 0, 0 ) ad he a seleced iervals of, usually equally spaced, uil he soluio has bee eeded o cover he required rage. Thus he soluios of o-liear equaios require a sep-by-sep approach ad are ormally based o he use of ierpolaio or fiie differece equaios. The idepede variable is divided io equal iervals, over he rage of he desired soluio. Thus he variables afer ad () iervals are give by * ad () respecively. A ime i is assumed ha he values of all he parameers are kow as well as he values for same parameers a all previous iervals (-),(- ),..,,. A ime i is assumed ha he values of he variable parameers are o kow ad ha he purpose of he aalysis is o fid he value of (i he case of DOF sysem, ) ad is derivaives which saisfy k p () A ime ad ime he codiio of dyamic equilibrium requires respecively ha: m c k ad ' p (4) c ' k p m (5) (6) Equaio (5) may be wrie as R R (7) Evaluaio of hese epressios a he ed of he ime ierval whe leads o he followig epressios for he icremeal velociy ad displaceme (Ghafari, 00): 6 6 (a)

4 5 I. J. Phys. Sci. (b) I geeral i has bee foud o be coveie o use he icremeal displaceme as he basic variable ad hece he ad i i erms of. Rearragig Equaios (a) ad (b) yields. (9a) 6 (9b) Subsiuio of Equaios (9a) ad (9b) io Equaio (7) ad assumig ad remai cosa durig he ime ierval leads o: (0) ) ( ) 6 ( ) 6 ( R m k m TRJIO'S ETHOD Trujillo preseed a eplici algorihm for he dyamic respose aalysis of srucural sysem. For liear udamped sysems he mehod was show o be ucodiioally sable. A algorihm based upo he Trujillo's mehod has bee developed for o-liear sysems by Thomas (Thomas, 9) bu does o ake io accou he effec of dampig. The Trujillo algorihm Trujillo splis he siffess ad dampig marices io upper ad lower riagular forms, as idicaed below, by he subscrips ad respecively ad preses, wihou givig he developme of he equaios, he followig algorihm which is divided io a forward ad a backward subsiuio. Forward subsiuio: ( ) [ ] P P 6 4. () 4 () Backward subsiuio: ( ) [ ] P P 6 4. () 4 (4) A advaage of his algorihm is ha, sice i is resriced o he use of diagoal mass marices oly, he coefficie marices of ad are obaied respecively i upper ad lower riagular forms. Thus he soluio of he equaios a ime is reduced o forward ad a ime o backward subsiuio oly. The Trujillo suggess wo ways of spliig he siffess ad dampig marices. The firs is a symmeric spliig which saisfies he codiios: T ; (5) ad T u l ; (6) The secod way differs from he firs oly by he maer i which he diagoal elemes are disribued. Trujillo (977), who eeded Trujillo's work o apply o oliear sysems, bu ecludes dampig ad hus reduces he equilibrium equaios a he h sep o, P R (7) where R is ieral force vecor, ad preses he followig algorihm for he middle ad he ed of sep: P R.. ()

5 Hashamdar e al (9). R P 4. (0) () 4 No-lieariy is ake io accou by updaig he siffess mari a he ed ad if ecessary also a he middle of each ime sep (Trujillo, 977). Epressio oal poeial eergy The oal poeial eergy is wrie as: WV () where W he oal poeial eergy he srai eergy of he sysem, ad V he poeial eergy of he loadig. Takig he uloaded posiio of he assembly as daum, W m j j i F ji where m Toal umber of members, J Toal umber of cable jois, F ji Eeral applied load o joi j i direcio i, ad ji Displaceme of joi j i direcio i. ji () The codiio for srucural equilibrium is ha he oal poeial eergy of he sysem is a miimum, ha is, (kirsch, 006). W ji 0 ( j,,... j) & ( i,,) (4) The correc value of for which W is a miimum, ha is, g0 ca ow be foud by he ieraive process S V ji( k ) ji( k ) ( k ) ji( k ) (5) where he suffices (k) ad (k) deoe he (k)h ad (k)h ierae, respecively ad; where V ji he eleme of he direcio vecor, ad S (k ) he seplegh which defies he posiio alog V where he oal poeial eergy is a miimum. ji(k ) The epressio for V ji give by: J ji ( k ) ji ( k ) j i ji ( k ) g ji ( k ) V J ji ( k ) j i g g V (6 g ji ( k ) g ji ( k ) The saioary poi i he direcio of desce ca be foud by epressig he oal poeial eergy as a fucio of he sep legh alogv. Thus he required value of S (k ) ca be deermied by he codiio (ksu, 009). W S 0 (7) ( k ) ( k ) I he covergece eergy he ieraio ad olerace are o required ad he kiemaic codiio a oe icreme o calculae he kiemaic codiio a he e icreme. Numerical ad eperimeal esig The developme of a mahemaical corol o esure sabiliy whe usig larger ime seps is desirable. The use of larger ime seps would mea ha he sarig poi a he begiig of each ime sep would be furher removed from he posiio where he oal poeial dyamic work is a miimum ad hus probably lead o more ieraio per ime sep. The mahemaical model chose was a 7*5 fla e wih 05 degrees of freedom. The 7*5 e was also buil as a eperimeal model ad esed i order o verify he saic ad dyamic oliear heories give i his reovaio of eplici aalysis. The model, of which a diagram is give i Figure 4 ad a geeral view i Figure 5, cosised of a 7*5 cable e, wih he cables a 500 mm iervals, i which he cables were 5.4 mm diameer. A he pois of iersecio he cables were clamped ogeher wih hi wires. The cable e was coaied wihi a 4 m by m recagular seel frame made. Each seel cable was iiially esioed o aroud kn ad he lef for wo weeks o permi he idividual wires i he srads o bed i. he cables were readjused o.5 N. This esio was maiaied ad checked a iervals hroughou he es programme. The wedge ad barrel was used o hollow cylidrical seel o provide Edcaser degree of freedom o boudary codiio of ji

6 60 I. J. Phys. Sci. Figure 4. Diagram of seel frame made. Figure 5. Geeral view of seel frame mode. cables. The ereced recagular e had he specificaios show i Table. The values i Table for esile sregh ad Youg s modulus were obaied afer esig i laboraory. The mass desiy ifluece he sabiliy limi, uder some circumsace scalig he mass desiy ca poeially icrease he efficiecy of a aalysis ad he eplici dyamic uses a ceral differece rule o iegrae

7 Hashamdar e al. 6 Table. The ereced recagular e specificaios. Specificaio Values Overall dimesios 000*4000 mm Spacig of he cables 500 mm Number of free jois 5 Number of fied boudary jois 4 Number of liks ables: ross-secioal area 4.9 mm Youg s modulus 9.60 N/ mm Breakig load 7.9 N Y/Sregh N Preesio 500 N/lik Diameer 5.4 mm Table. Degree of symmery abou he major, mior aes, deflecios due o coceraed load o Node. oad(n) 400 Theoreical (T) Eperimeal (E) ( T E ) / T*00 Z Ais deflecios(m), Node (VDT) 7.6E E Z Ais deflecios(m), Node (VDT) 9.E-0 7.9E-0.0 Srai gage(µe),eleme 6 (verical) o cable 6.46E E-06.5 Srai gage(µe),eleme 44 (verical) o cable.e E-06. srai gage(µe),eleme 5 (verical) o cable 6.4E-06 6.E srai gage(µe), Eleme 45 (verical) o cable.5e E-06.6 Z ais deflecios(m),node 4 (VDT ) 50.75E-0 50.E-0.6 Z Ais deflecios(m),node 5 (VDT) 7.9E-0 7.5E Z Ais deflecios(m),node (VDT) 50.75E E-0. Z Ais deflecios(m),node 5 (VDT) 5.E-0 4.E-0 5. Z Ais deflecios(m),node 6 (VDT) 74.46E E-0.55 Z Ais deflecios(m),node 7 (VDT) 5.7E-0.5E-0. Z Ais deflecios(m),node 9 (VDT) 5.7E E Z Ais deflecios(m),node 0 (VDT) 74.46E-0 7.5E-0.6 Z Ais deflecios(m),node (VDT) 5.E E-0.47 Z Ais deflecios(m),node (VDT).9E-0.E-0.4 Z Ais deflecios(m),node 7 (VDT).9E-0.E-0.05 Z Ais deflecios(m),node 9 (VDT).9E-0.56E Z Ais deflecios(m),node 5 (VDT).9E-0.05E-0.9 he equaio of moio eplicily hrough ime. The deflecios due o coceraed load o Joi are show i Figure 7. Degrees of symmery abou he major, mior aes are ivesigaed by deflecios due o coceraed load o Node is show i Table. The ivesigaio cosised of checkig he degree of symmeric behavior abou he major ad mior aes. The degree of symmeric behavior abou he mior ais was ivesigaed by firs placig a icreasig load o Joi ad he comparig he resula displaceme wih hose obaied by placig similar loads o Joi 5. The degree of symmeric behavior abou he mior ais was similarly sudied by loadig firs Joi 6 ad he Joi 0. The resuls idicae ha he perceage differece bewee he heoreical ad eperimeal displacemes decreases wih icreasig loadig. For he maimum loadig a joi he perceage differeces bewee he heoreical ad eperimeal displacemes a he jois measured raged bewee. ad 5.%. Figure shows he relaioship bewee loads ad deflecio abou he major ais. Whe he coceraed load is placed o Node 0, he deflecio gradually icreased from 5.5 E- uis of deflecio o Node 5 uil i

8 6 I. J. Phys. Sci. Figure 7. Deflecios due o coceraed load o Joi. Figure. Degree of symmeric abou major ais whe he load is placed o Nodes 6 ad 0.

9 Hashamdar e al. 6 Figure 9. Degree of symmeric abou mior ais whe he load is placed o Nodes 7 ad 9. reached a peak of.7e- o Node 0. From his poi owards, i is projeced o drop sharply uil i reached 6.07 E- o Node 5. Whe coceraed load is placed o Node 6, he deflecio from abou 6.07 E- uis o Node 5 rapidly rose o reach a peak of.7 E-0 o Node 6. From his poi owards, i is projec o fell slighly uil i reached 5.5 E- uis o Node. The dampig mari for he proposed heory is calculaed separaely ad is calculaio does o affec he formulaio of he heory i prese form. The eergy los i vibraio is due o work doe by forces resisig he moio. A geeral orhogoal dampig mari ca be show o be of he from ε b ω a b b [ k ] b a b () ω b where ε he dampig raio for mode, ad ω he frequecy of mode. Rayleigh dampig which is give by 0 b b (9) a a (0) ad i which a ad 0 a are arbirary proporioaliy facors, as coaied i Equaio (0) ad saisfies he orhogoaliy codiios wih regard o boh he mass ad siffess. The use of value Rayleigh dampig assumes ha he dampig raios i all modes ca be epressed by he relaioship ( a ω a ω ) ε 0 / () Figure 9 shows ha whe he coceraed load was placed o Node, he deflecio soared bewee Nodes 4 ad from 5.0E- uil.79e-. From his poi owards, i is projeced o declie slighly o reach 5.0E- o Node. Differece of deflecio i mior ad major aes is sychroizaio. A mode shape describes he epeced curvaure of a surface vibraig a a paricular mode. odes shape ad are show i Figures0 ad. Figure shows ha icreases i degree of freedom i Trujillo algorihm are slowly ad chages i degree of freedom are reasoable.. Newo-Raphso mehod (liear). Newo-Raphso mehod (oliear). Trujillo algorihm mehod (liear) 4. Trujillo algorihm mehod (oliear) The perceage differeces bewee he heoreical ad umerical esig resuls did o i ay case eceed 0% ad his is accepable. The ime icreme used i his reovaio was beig smaller ha he sabiliy limi of he ceral-differece operaor. The use a small eough ime icreme will resul i a usable soluio. Whe he soluio becomes usable, he ime hisory respose of soluio variable such as displacemes will usually

10 64 I. J. Phys. Sci. Figure 0. 7*5 fla e, ode. oscillae wih icreasig ampliude ad also he mass scalig was corolled by ime icrimiaio. ONSIONS The objec of his work was pricipally o develop a eplici dyamic aalysis heory ad verify he heory by umerical ad ecremeal esig. The proposed mehod was foud o be sable for ime seps equal o less or less ha half he smalles periodic ime of he sysem. The umerical eperimeaio ad eperimeal esig c by saic ad dyamic esig of fla e showed a good agreeme bewee he umerical eperimeaio resuls ad he heoreically prediced values. The Turjillo mehod fouds ha mehod coverged rapidly ear he soluio. Fially he compariso of eperimeal ad heoreically prediced values showed ha he deflecio calculaed by he proposed oliear mehod gives

11 Figure. 7*5 fla e, ode. Hashamdar e al. 65

12 66 I. J. Phys. Sci. Ampliude (mm) Time (s) Figure. Ampliude diagram i compariso wih resuls from dyamic esig. reasoably accurae resuls. REFERENES Belyschko T, Egelma BE (00). Eplici ime iegraio wih ehaced sabiliy for srucural dyamics. ompu. Sruc., 9: Da Silva GS, Vellasco SA (00). Noliear dyamic aalysis of seel poral frames wih semi-rigid coecios. Eg. Sruc., 0: Ghafari OG (00). Dyamic Aalysis of able Roofs der rasie Wid: A ompariso bewee Time Domai ad Frequecy Domai Approaches. Tsighua Sci. Techol., : Huu-Tai TSE (00). Noliear saic ad dyamic aalysis of cable srucures. Fiie Elem. Aal. Des., 47: Jia-Big JO (007). The ereme value disribuio ad dyamic reliabiliy aalysis of oliear srucures wih ucerai parameers. Sruc. Safey, 9: irsch B (006). Noliear dyamic reaalysis of srucures by combied approimaios. omp. ehods Appl. ech. Eg., 95: aier JE (0). Specral aalysis of a high-order Hermiia algorihm for srucural dyamics. Appl. ah. odel., 5: iqus J (00). Noliear elasoplasic dyamic aalysis of siglelayer reiculaed shells subjeced o earhquake eciaio. omp. Sruc., : 77-. Sefaou GF (009). Noliear dyamic aalysis of frames wih sochasic o-gaussia maerial properies. Eg. Sruc., : Thomas JR, Hughes RAS (9). overgece of implici-eplici algorihms i oliear rasie aalysis. I. J. Eg. Sci., 9: Trujillo D (977). A ucodiioally sable eplici algorihm for fiieeleme hea coducio aalysis. Nuclear Eg. Des., 4: ksu, Theodore RT, Galambos V (009). -D oliear dyamic behavior of seel jois girder srucures. Eg. Sruc., : 6-74.