Nonlinear Viscous Damper Application to Arch Bridge

Transcription

1 Nonlinear Viscous Damper Application to Arc Brige lif Caga Kanemir 1, Taiji Maza Department of Civil an nvironmental ngineering, Kumamoto University, Japan 1 ecaga@yaoo.com; ma za@u ma moto-u.ac.jp Abstract- Tis paper presents nonlinear viscous amper (NVD) implementation to an existing igway arc brige structure to prevent pouning effect on abutments. Te nonlinear ynamic response analysis performe on finite element moel of brige structure ue to severe eartquae excitations sowe tat te relative isplacement response between te brige ec an te abutments excees available istance in normal conitions troug te longituinal irection. As a remey for suppressing seismic responses, nonlinear viscous ampers were approve to be installe at te en of te ec since tey are simple to implement for existing structures in construction fiel. Te transversal beavior as also been investigate to mitigate te corresponing responses by te installation of ampers in orer to control te lateral beaviour of te brige for eac irection. Te amper capacity in response to seismic responses troug transverse irection was also obtaine in te same manner as one in longituinal irection. Keywors- Higway Arc Brige; Pouning; Nonlinear Viscous Damper; nergy-uivalent Meto; Single Degree-offreeom System; inite lement Moel I. INTRODUCTION Te 1995 Hyogo-en Nanbu eartquae in Japan is te turning point wic focuses attention on seismic performance of igway briges all aroun te worl. Consierable amount of researces states te estructive effects of tis eartquae to igway briges [1]-[4]. Te innovative esign an retrofitting strategies are intensely evelope to improve seismic performance of te new an existing brige structures since ten. One of te energy issipation evices tat as wiesprea application area in structural engineering fiel is te viscous amper. Te efficiency for all moes of te structures an simple implementation in te fiel uring construction mae te viscous ampers superior an preferable among oters. In principle tey issipate energy by forcing flui troug orifices. Te force generate by flui resists te excitation force given by, ( x ) c x sign (1) in wic is amper force, c is amping coefficient, x is relative velocity between amper ens, sign is te signum function (1 for positive, -1 for negative an for zero velocity values). Te nonlinearity exponent taes 1 for linear viscous amper an < 1 for nonlinear viscous amper. or te viscous amper beaves as a friction amper [5]. Te avantage of nonlinear viscous ampers can be explaine as aving lower amper force even for excessive amounts of velocity by limiting te amper force in comparison wit linear viscous ampers. ig.1 illustrates te amper force-velocity relationsip wit regar to cange in nonlinearity exponent of. It is clear tat te nonlinear ampers yiel lower amper force even for large velocities as long as nonlinearity exponent ave smaller values wereas te issipate energy is ientical in eac sceme for a given structural system. Damper force Velocity ig. 1 Damper force-velocity relationsip Te effective usage of NVDs is abunant in literature presenting numerous metos to sow te non-linear amper efficacy an to obtain te amping coefficient e.g. energy uivalent meto [5]-[7], power consumption meto [8] etc. In [5], it is state tat proviing supplemental amping ratio by viscous ampers soul be te main purpose to reuce structural responses to esire values rater tan consiering te influence of velocity exponent. Reference [6] inclues formulas for brige structures wit linear an nonlinear viscous ampers iealize as two egree-of-freeom system base on moal caracteristics an amping ratios of structural components. In [8], te non-imensional amper capacity is referre as normalizing te amper force by te weigt of structure. Reference [1] offers formulations for friction amper by te concept of uivalent linear amper. Te retrofitting strategy of an existing igway arc brige wit nonlinear viscous ampers was te aim of tis investigation to resist strong eartquae excitations. Te escription of te normalize amping coefficient wic efines a proportion between iealize SDO system an te brige structure was propose in orer to ajust te output of SDO system to te real structure. Accoring to te time response analysis of finite element moel of te brige, te relat ive isplacement responses (target isplacement ereafter) was reuce to esire values by preventing pouning effect of te ec. JCS Volume 1, Issue 3 September 1 PP

2 A. inite lement Moel II. ARCH BRIDG MODL Te arc brige investigate in tis paper is a conventional upper-ec type steel arc brige wit reinforce concrete (RC) ec slab. Te finite element moel of te brige is illustrate in ig.. Te total lengt of te ec an available wit between bearings are 9. m an 8.1 m, respectively. Te twin arc ribs tat are connecte by lateral steel bracings ave a span of 6. m an a rise at te crown of 1. m wic gives a rise-span ratio of 1:5. RC ec slab is supporte by two main longituinal girers wit transverse girers an iagonal members. Te connection between main longituinal girers an arc ribs is supporte by 11 piers at te intersection joints of te main ribs an transverse bracings wit pin connection except sie piers wic are fixe to te ec. Te steel members were all moelle as linear beam elements wereas te interaction between te ec an te abutments were assume as bilinear spring elements of wic stress-strain iagram is sown in ig. 3. K is te egrae stiffness after te member reac yiel stress. Te groun as been moelle as two noal spring elements wit 6 egree-of-freeom at eac noe of (). Table I sows bounary conitions for abutments (A), founations () an sie piers (P). (: ree, R: Restraint) can easily occur uring strong eartquae motions. Te lateral seismic movement of te ec along longituinal axis may excee available istance between te ec an te abutments uring strong eartquae motions. To protect te structure from pouning, te NVDs were installe to te eac en of ec. Terefore te seismic responses erein were presente as te relative isplacement responses of te ec an abutments subjecte to te longituinal excitations wereas te beaviour of mispan was pai attention for transversal excitations. B. Single Degree-of-reeom Iealization Te SDO iealization of large structures suc as igway briges, towers etc. is an effective approac for te sae of less computation labour. In particular regular sape structures are well-represente wit SDO system by summarizing funamental moal caracteristics of te structure into SDO system [1]. Te SDO system wit supplemental amper in ig.4 can be utilize for etermination of amper caracteristics for a given isplacement limitation as carrie out in tis paper. Te quantifie amping coefficient from te SDO system inclues te sum of all amping coefficient of ampers installe. A 9 m P A ig. 4 Iealize SDO moel Y X Z 1 m 6 m ig. Configuration of tree-imensional finit e element moel σ σ y ε y K 1 K ig. 3 Bilinear spring moel TABL I BOUNDARY CONDITIONS X Y Z x ε P y z Abutments (A) R R R R R ounation () R R R R R R Piers (P) R R R R In general te gap between ajacent structural components of te brige structures is small to provie smoot traffic flow [11]. However, at te moment pouning 8.1 m III. NONLINAR VISCOUS DAMPR C. nergy quivalent Meto for Damper Coefficient Te close-form solution for supplemental ampers can be obtaine approximately to represent aitional amping ratio in terms of structural amping ratio by uating issipate energy to inerently ampe energy over one cycle [1]. Te solution of friction amper caracteristics by energy uivalent meto is well presente in [1]. To avoi te iterative an teious calculations wic ruire computer programming to fin out te NVD amping coefficient for te structure wic as response limitations, i.e. target isplacements, energy uating is an effective solution. Te uation of motion of SDO system wit viscous amper uner armonic force is given by, ( x) P sin t m x + c x + x + c x sign ω () were P is force amplitue an ω is angular fruency of loaing. q. () can be rewritten consiering uivalent amping coefficient, ( c + c ) x + x P sinω t m x + (3) Te energy uation of te system is presente in q. 4. Since tere is no cange in inetic energy, an strain energy, s for one cycle [9], tey are neglecte. is JCS Volume 1, Issue 3 September 1 PP

3 inerent viscous amping energy, is uivalent issipate energy by supplemental amper, an is excitation energy (armonic force energy erein) (4) s π c ω (5) u π π c ω (6) u ( c c ) ω u π P sinφ + (7) u After substituting te amping ratios an fruency ratio ω into te q. 7, ( ω / ) r ω n ( ξ ξ ) ω u π P sinφ + (8) π r u TABL II SUPPLMNTAL DAMP ING RATIOS Target Displacement 1cm 8cm 6cm 4cm Dire cti on (un amental Moe) (4 t ) ( n ) Kobe_NS Kobe_W Taatori_NS Duzce_NS lcentro_ns ( n ) ( n ) Not e: --- means tat te brige oes not nee supplemental NVDs since te pouning effect oes not occur. ( n ) Dissipate energy by viscous amper over one cycle is obtaine by [7], π/ ω π/ ω f u f u t c u t D (9) D D πβ c ω u D (1) + 1 were te constant is + Γ ( 1 + / ) β π Γ ( + ) ro m, D πβ c ω u +1 π (11) c ω u or non-armonic excitations i.e. eartquae inuce groun motions, uivalent amping ratio an amping coefficient are calculate by, A. Moal Caracteristics β c n u ω ξ c β ω u 1 ξ (1) 1 n IV. RSULT S. (13) unamental moes troug longituinal an transversal irections are ruire to form SDO system. Since te bounary conitions are restraine along transverse irection, te stiffness between te ec an abutments were vanise in orer to be able to perform free vibration analysis of finite element moel of te brige along corresponing irection. Table II represents funamental moal caracteristics. Longituinal Axis ansverse Axis TABL III UNDAMNTAL MODAL CHARACTRISTICS Moe ruency (Hz) B. stablisment of SDO Moel Pe rio (s) ffective Mass Ratio (%) Te tree SDO moels ave been forme for eac funamental moe. ac SDO system constitute as funamental perios of te brige structure presente in Table II w ile te stiffness was calculate base on te mass an te perio ( 4π m / T ) of corresponing moe. Te weigt of te lumpe mass was assigne as 1N to provie unity. or longituinal irection te mean value of amping coefficients was utilize. C. Implementation of NVDs To perform te q. (13), ruire supplemental amping ratios to acieve target isplacements wic te SDO system soul ave were obtaine uner eac groun motion at first. Table III epicts te ruire supplemental amping ratios wile structural amping ratio is ξ.5. ( is longituinal, is transversal irection). Te target isplacements are 1 cm an lower values since te available istance between te ec an abutments is 1 cm in te investigate brige. Te amping coefficients of SDO systems were evaluate by te q. (14) so as to be ajuste for te brige structure. Te uation tat is propose erein inclues effective mass of funamental moes instea of te mass of entire brige. JCS Volume 1, Issue 3 September 1 PP

4 wereas c R M eff N (14) c is amper coefficient, R is normalize amping coefficient (amping coefficient of SDO/mass of SDO), M is effective mass of funamental moe an eff N is te amount of ampers (two ampers were implemente for eac en of te ec, i.e. four ampers in total). Te illustrations of normalize amping coefficient versus target isplacement are given in ig. 5. Bol values on te graps sow te sufficient amping coefficients wic are ajuste for te brige installation accoring to q. (14). isplacement range (1 c m). Tis arises from te istance between te location of NVDs at te en of te ec an mispan. Neverteless te reuction in te responses wic is almost alf of tose of unampe brige is satisfying from te engineering viewpoint. ig. 6 Displacement time response for te ec uner longituinal Kobe- NS eartquae ig. 7 Displacement time response for te mispan uner transversal Kobe-NS eartquae ig. 5 Normalize amping coefficient vs. target isplacement for longituinal (top) an transverse (bottom) axis Te performance inices in tis stuy were te isplacement responses of te ec an mispan of te brige structure. ig. 6 an ig. 7 illustrate te isplacement time responses of te en of te ec an mispan wit an witout NVDs. Te reuction in te isplacement responses emonstrate te efficacy of NVDs. urtermore te goo agreement between ampe systems of SDO an te brige is clear for bot longituinal an transverse irection verifying te auacy of SDO approac. However one can observe tat te transversal response excees te target V. CONCLUSIONS Te SDO approac to fin out supplemental amper caracteristics is very effective an te irect approac for preliminary analysis of seismic performance improvement of large structures. In aition to tis, fast an efficient retrofitting strategy for large structures is very limite in literature wit lac of etaile explanation. Terefore te nonlinear viscous amper implementation to an existing arc brige structure as been investigate in tis paper. Te amping coefficient of nonlinear viscous ampers was propose to be compute from te effective mass of funamental moe instea of wole mass of te brige. Te reuce isplacement responses for given target isplacement confirme te auacy of usage of effective mass. Te amping coefficient of eac NVD installe between te en of te ec an abutment is calculate by te square root of te sum of te squares (RSS). Te conclusions can be summarize as follows: a) Te superiority of nonlinear viscous ampers on constraining te amper force on structure even uring large velocities provies better protection tan linear amper wile tey bot yiel same issipate energy. Base on tis property, energy uivalent meto wic as wiesprea application area offers an accurate solution to figure out nonlinear viscous amper caracteristics. b) Single egree-of-freeom moels establise for eac funamental moe was utilize to figure out te JCS Volume 1, Issue 3 September 1 PP

5 amping coefficients wic are essential to perform te uation of motion of finite element moel. c) Te ajustment of amping coefficients obtaine from te SDO systems were carrie out by te proportion of mass of SDO system an funamental effective masses of te brige. ) Te SDO iealization an aapting way of amping coefficient succeee in terms of reuce isplacement responses ue to severe groun motions. ACKNOWLDGMNT Te autors gratefully acnowlege Mr. Nurui from Ku ma moto Pre fecture, Mr. Miyamoto an Mr. Cou from igt-japan ngineering Consultants Incorporation for teir contributions to provie information about te brige. RRNCS [1] K. Kawasima, Seismic Isolation of Higway Brige, Journal of Japan Association for artquae ngineering, vol. 4, pp.83-97, 4. [] T. Usami, Z. Lu, H. Ge an T. Kono, Sesimic performance of steel arc briges against major eartquaes. Part I: Dynamic analysis approac, artquae ngineering an Structural Dynamics, vol.33, pp , 4. [3] O. T. Cetinaya, S. Naamura an K. Taaasi, A static analysis-base meto for estimating te maximum out-ofplane inelastic seismic response of steel arc briges, ngineering Structures, vol. 8, pp , 6. [4] W.. Cen an L. Duan, Brige ngineering Seismic Design, loria, U.S.: CRC Press LLC, 3. [5] W. H. Lin an A. K. Copra, artquae response of elastic SD systems wit non-linear flui viscous ampers, artquae ngineering an Structural Dynamics, vol. 31, pp ,. [6] J. S. Hwang an Y. S. Tseng, Design formulations for supplemental viscous ampers to igway briges, artquae ngineering an Structural Dynamics, vol. 34, pp , 5. [7] W. H. Lin an A. K. Copra, Asymmetric one-story elastic systems wit nonlinear viscous an viscoelastic ampers: Simplifie analysis an supplemental amping system esign, artquae ngineering an Structural Dynamics, vol. 3, pp , 3. [8] G. Pecan, J.B. Maner an S.S. Cen, unamental consierations for te esign of non-linear viscous ampers, artquae an ngineering Dynamics, vol. 8, pp , [9] J. M. Kim, M. Q. eng an M. Sinozua, nergy issipating restrainers for igway briges, Soil Dynamics an artquae ngineering, vol. 19, pp ,. [1] K. W. Min, J. Y. Seong an J. Kim, Simple esign proceure of a friction amper for reucing seismic responses of a single-story structure, ngineering Structures, vol. 3, pp , 1. [11] K. Bi, H. Hao an N. Couw, Ruire separation istance between ecs an at abutments of a brige crossing a canyon site to avoi seismic pouning, artquae ngineering an Structural Dynamics, vol. 39, pp , 9. [1] H. Kuramoto, M. Tesigawara, T. Ouzono, N. Kosia, M. Taayama an T. Hori, Preicting te eartquae response of builings using uivalent single egree of freeom system, 1 t WC,. JCS Volume 1, Issue 3 September 1 PP