The h World Conference on Earhquake Engneerng Ocober -7, 8, Beng, Chna Response-Specrum-Based Analyss for Generally Damped Lnear Srucures J. Song, Y. Chu, Z. Lang and G.C. Lee Senor Research Scens, h.d. Suden, Research Assocae rofessor, rofessor Dep. of Cvl, Srucural and Envronmenal Engneerng, Unversy a Buffalo, Buffalo, Y, USA Emal: song@buffalo.edu, ylunchu@buffalo.edu, zlang@buffalo.edu, gclee@buffalo.edu ABSTRACT : When cvl srucures are equpped wh earhquake proecve sysems such as dampng devces and base solaors, hey are lkely o be non-classcally damped. In addon, when he overall dampng s ncreased o a ceran level, some modes wll become over-damped. In such cases, he convenonal reamen by usng he classcal dampng assumpon (gnorng he non-classcal dampng effec and he over-damped modes) may resul n unaccepable desgn errors on he unsafe sde. Based on he general modal response hsory analyss formulaon wh over-damped modes developed by he auhors (Song e al. 8), hs paper furher exends o a response-specrum analyss approach and proposes a general modal combnaon rule GCQC. Examples o examne he accuracy and effecveness of hs approach are also gven. KEYWORDS: Response specrum, Generally damped lnear srucures, Sesmc analyss, Over-damped modes. ITRODUCTIO In earhquake response analyss of srucures, he response specrum mehod s commonly used as an alernave approach o he response hsory analyss for deermnng he maxmum values of he sesmc responses of classcally damped srucures. In hs mehod, he modal peak responses are obaned usng he prescrbed response specrum. These modal maxma are hen appropraely combned o esmae he peak values of he responses of neres. The convenonal response specrum mehod s deal o srucures sasfyng classcal dampng condon. For srucures ha are srongly non-classcally damped, he accuracy of he square-roo-of-sum-of-squares (SRSS) and he complee quadrac combnaon (CQC) rule becomes quesonable (Clough and Moahed 976, Warburon and Son 977 and Velesos and Venura 986). For hs reason, several modal combnaon rules accounng for he effec of he non-classcal dampng are developed (Sngh 98, Igusa e al. 98, Venura 98, Gupa and Jaw 986, Maldonado and Sngh 99 and Zhou e al. ). However, all combnaon rules developed n hese leraures dd no ncorporae he over-damped modes n he formulaon and he response quanes consdered n hese rules are lmed o deformaon-relaed response quanes. In hs paper, on he bass of he general modal response hsory analyss developed by he auhors (Song e al. 8) and he whe nose npu assumpon as well as he heory of random vbraon, a general modal combnaon rule for response specrum mehod are formulaed o deal wh he non-classcal dampng and over-damped modes. Ths general modal combnaon rule s referred o as General-Complee-Quadrac-Combnaon (GCQC) rule n hs sudy. An over-damped modal response specrum s nroduced o accoun for he correspondng peak modal responses. The accuracy of he new rule s evaluaed hrough an example by comparng o he mean response hsory resuls.. AALYTICAL FORMULATIO Accordng o he general modal response analyss mehod formulaed by Song e al. (8), he responses of a generally damped lnear srucure under sesmc excaons can be expressed as he response hsory combnaon of c complex modes and over-damped modes ( C + =, s he srucural DOFs) : C A B A = = x () = [ q () + q ()] + [ q ()] R ()
The h World Conference on Earhquake Engneerng Ocober -7, 8, Beng, Chna T where x () = [ x(), x(),..., x ()] ( R -- belongs o dmenson vecor space n real feld) represens a response vecor for mos response quany of neres for srucural sesmc evaluaon and desgn, such as relave dsplacemen and velocy, absolue acceleraon, ner-sory drf and dampng force ec. And q () R, q () R and q () R are he under-damped mode (complex mode or smply ermed as mode) dsplacemen, velocy and over-damped mode responses o sesmc acceleraon excaon x g () R, respecvely, ha s, q ( ), q ( ) and q () are he soluons of he followng dfferenal equaons, respecvely: q q q x () ( ) + ξ ( ) + ( ) = g( ) ( =,,..., C) q ( ) + q ( ) = x ( ) ( =,,..., ) () g n whch, R and ξ R are he crcular naural frequency and dampng rao of he complex mode respecvely and R s he over-damped modal crcular frequency. In Eq. (), A R, B R and A R are he coeffcen vecors assocaed wh q ( ), q ( ) and q (), respecvely. These coeffcen vecors only depend on he srucural modal parameers and are me nvaran. The expressons of hese coeffcen vecors for mos response quanes can be found n Song e al. (8).. Defnon of vecor operaon symbols For smplcy n subsequen formulaon, we defne symbol as a vecor elemen-wse operaon. For example, c= a b means ha each elemen n vecor c s he produc of he correspondng elemens n a and b, assumng ha a, b and c have he same dmenson. a means akng he square for each elemen n he vecor a.. Covarance of modal responses o saonary excaon Consder he npu ground acceleraon x g () as a wde-band saonary process. Based on he heory of random vbraon, he responses of a lnear sysem subeced o a saonary process are also saonary and he covarance or mean squares of he response x () from Eq. () s n he form of C x ( ) = E [ Aq ( ) + Bq( )] + [ Aq ( )] = = C C { AA q () q () BB q() q() AB q () q() } = + + = = C { AA q ( ) q( ) BA q( q ) ( ) } = = { A E ( ) ( ) } A q q R = = + + + () Eq. () shows ha s necessary o compue he covarance of he response produced by wo modes (e.g. q ( ) q ( ) ) n order o oban he varance of x (). Before proceedng o calculae he covarance produced by wo modes, a number of expressons are lsed as hey would be requred n he subsequen formulaons. They are:
The h World Conference on Earhquake Engneerng Ocober -7, 8, Beng, Chna q () = h ( ) x ( )d τ g τ τ R and q h xg () = ( τ) ( τ)dτ R (a,b) H ( ) = C and H v ( ) = C (6a,b) + ξ + + ξ + H ( ) = + C (7) where h( ) and h () are he un mpulse response funcon of a complex mode and an over-damped mode, respecvely; = s he magnary un; H ( ) and H ( ) v are he dsplacemen and velocy frequency response funcon of a complex mode wh respec o excaon x (), respecvely; and g H ( ) s he frequency response funcon of an over-damped mode wh respec o x () g. The dsplacemen response covarance erm q( ) q( ) n Eq. () s frs examned. Accordng o Eq. (a), hs erm may be wren as q ( ) q ( ) = h ( τ ) h ( τ ) x ( τ ) x ( τ ) dτ dτ R (8) g g Knowng ha he npu ground excaon x g () sars from zero a he me nsan = (.e. x g () = when ), s reasonable o exend he lower lm of he negraon n Eq. (8) o negave nfny as (9) q ( ) q ( ) = h ( τ ) h ( τ ) x ( τ ) x ( τ ) dτ dτ g g ow, suppose ha he ground excaon x () g s furher consdered as a whe nose process wh zero mean, descrbed by a consan power specral densy S. I follows ha he erm xg( τ) xg( τ ) n Eq. (9) becomes xg( τ) xg( τ) = π Sδ( τ τ) () where δ ( τ ) s he Drac funcon and s defned as follows. δ ( τ ) = + { τ = τ + and δ ( τ)dτ = (a,b) In lgh of he nverse of Fourer ransform, he Drac funcon also can be expressed as + τ ( τ ) e τ ( τ) δ = d π or δ( τ τ) = + e d π (a,b) Subsung Eq. (b) ogeher wh Eq. () no Eq. (9) and seng he upper negral lm o nfny o rean he seady sae response, Eq. (9) becomes
The h World Conference on Earhquake Engneerng Ocober -7, 8, Beng, Chna τ τ ( ) + + + q ( + ) q ( + ) = S h ( τ )e d τ h ( τ )e dτ d R () Denong R = q( + ) q( + ) and makng use of Eqs. () and (6) along wh conour negraon n complex plane, he dsplacemen covarance R, shown by Eq. (), may be wren as πs R = S H H = R () + ( ) ( )d ρ ξξ where ρ s he well-known dsplacemen correlaon coeffcen orgnally derved for he CQC rule (Der Kureghan 98). Furher, le = n Eq. (), can be enrely expressed n modal dsplacemen varance erms. Tha s, R = R R ρ, (, =,,..., ) () C Followng he smlar procedures for he dervaon of he modal dsplacemen response covarance R, he modal velocy response covarance R VV and he covarance of he h modal velocy and he h modal VD dsplacemen R can also be derved as R = R R ρ R and R VD = R R ρ VD R, (, =,,..., ) (6a,b) VV VV C VV where ρ and ρ VD are he modal velocy correlaon coeffcen and modal velocy-dsplacemen correlaon coeffcen. Ther expressons and varaons versus modal frequency rao and modal dampng rao can be found n Zhou e al. () and Song e al. (8). oed ha when =, he varance of velocy VV VD response R and he covarance of velocy and dsplacemen response R becomes R πs = = R and VV ξ R = (7a,b) VD I s clear from Eq. (7) ha he velocy varance and he dsplacemen varance of a SDOF sysem s relaed by he squares of s naural crcular frequency and he modal dsplacemen and velocy responses of a SDOF sysem are orhogonal wh each oher under he whe nose excaon assumpon. The presence of R and VD R reflecs he non-classcal dampng effec. Anoher mporan obecve of hs sudy s o consder he conrbuons of over-damped modes (f exs) o evaluae he complee srucural responses n modal response combnaon mehod. Accordng o lnear conrol sysem heory, an over-damped mode corresponds o an ndependen frs-order subsysem (negral un), whle an under-damped mode (complex mode) assocaes wh a second-order subsysem (oscllaon un). Ths s clear f we compare Eq. () wh Eq. () and Eq. (6) wh Eq. (7). The dealed naural properes regardng o he over-damped modes are descrbed n Song e al. (8). ow, consderng he over-damped modal response covarance erm q ( ) q( ) n Eq. () and followng he smlar procedures as he above, we have VV
The h World Conference on Earhquake Engneerng Ocober -7, 8, Beng, Chna + = = = R E q ( ) q ( ) S H ( ) H ( )d R, (,,,... ) (8) Subsuon of Eq. (7) no Eq. (8) and manpulaon wh conour negraon n complex plane leads o πs R = = R R ρ, (, =,,... ) + (9) n whch ρ =, (,,,..., ) R + = () s a newly derved correlaon coeffcen ha accouns for he relaonshp beween he over-damped mode responses. Smlarly, he modal dsplacemen and he over-damped modal response covarance erm q( ) q( ) and he modal velocy and he over-damped modal response covarance erm q ( ) q( ) n Eq. (6) can be derved as D D R = R Rρ and R V D = R Rρ R, ( =,,... C, =,,... ) () where ρ D ξ = R, ( =,,... C, =,,... ) () + ξ + ( ) s anoher newly developed correlaon coeffcen whch accouns for he correlaon beween he complex modal dsplacemens and he over-damped mode responses. Fg. shows he varaons of he correlaon coeffcen ρ versus, from whch s observed ha he value of ρ only depends on he over-damped modal frequences and remans o be a sgnfcan componen across he range of. Fg. D D shows he varaon of ρ wh respec o and dampng raoξ. I s seen ha he values of ρ D are sgnfcan, parcularly a large dampng level. Also, ρ grows as he rao approaches wo and decreases slowly beyond ha value. The observaons made from Fgs. and sugges ha he over-damped mode may conrbue sgnfcanly o he overall srucural response and should be consdered appropraely. Fnally, upon subsuon of he above derved covarance no Eq. (), one obans C C VV VD ρ ρ ρ R R = = x ( ) = A A + B B + A B C D ρ A A B A R R ρa A R R = = = = + + + R (). Developmen of response specrum mehod I has been shown ha he mean maxmum modal response of a lnear sysem over a specfed duraon o saonary excaons s proporonal o her respecve roo mean squares(vanmarcke 97),.e., q () = S = p R R and q () = S = p R R () max max
The h World Conference on Earhquake Engneerng Ocober -7, 8, Beng, Chna.8.6.8.6 ξ =.8 ξ =. ξ =. ξ =..... 6 8 6 8 Fgure Varaon of correlaon coeffcen ρ Fgure Varaon of correlaon coeffcen D ρ where he S s he ordnae of he mean dsplacemen response specrum and S s he ordnae of he mean over-damped mode response specrum. The dea of he over-damped mode response specrum wll be nroduced n a laer secon. The numercal value of p, n general, does no dffer grealy n magnude from mode o mode. Thus, for pracce applcaons, s reasonable o assgn he same value o p for each mode as well as for he combned responses. As a resul, he followng General-Complee-Quadrac-Combnaon rule (GCQC rule) applcable o srucural sysems wh non-classcal dampng and over-damped modes s derved. x () max / C C ρ μ A A + B B + υa B SS = = = C R () D + ρ + SS + ρ S S A A B A A A = = = = where ρ ξ + ξ γ μ = = ρ ξ ξγ VV + and ρ γ υ = =, (, =,,..., C ). ρ γ ( ξ ξ γ ) VD + As a specal case of he GCQC, f he correlaons beween each mode are gnored; ha s, when ρ =, ρ = as well as υ = and ρ D = for all and, Eq. () s reduced o C x () = max ( A + B ) S + ( A ) ( S ) R (6) = = Eq. (6) s ermed as General-Square-Roo-of-Sum-of-Square combnaon rule (GSRSS rule). If he dampng marx of a srucure sasfes Caughey Creron and all over-damped modes are gnored, Eqs. () and (6) can be reduced o he convenonal CQC and SRSS rules for classcally-damped srucures. However, he formulaon resulng from hs sudy can be used o evaluae mos peak response quanes of neres, such as relave dsplacemen and velocy, absolue acceleraon, ner-sory drf, sory shear and dampng force ec. The mos reduced form of Eq. (6), for example, can be used o evaluae he peak absolue acceleraon of a lnear SDOF sysem, ha s xa() = + ξ max ns( Tn, ξ) = + ξ SA( Tn, ξ) R (7) where n, T n and ξ are he SDOF sysem s naural frequency, perod and dampng rao; and SA( Tn, ξ ) s he ordnae of pseudo-acceleraon specrum. If ξ value s small, say, less han %, xa() SA( Tn, ξ ). max
The h World Conference on Earhquake Engneerng Ocober -7, 8, Beng, Chna. OVER-DAMED MODE RESOSE SECTRUM The over-damped mode response specrum follows a smlar defnon as he convenonal response specrum used n earhquake engneerng. The obecve of he over-damped mode response specrum s o accoun for he peak over-damped mode response conrbuons. Rewrng Eq. () as a general form of a frs-order dfferenal equaon wh excaon npu x () g, we have () + () = () (8) q q xg Smlar o he concep of convenonal response specrum, he over-damped mode response specrum s defned as a plo of peak over-damped mode responses q (), as a funcon of over-damped modal frequency or over-damped modal perod T = π (acually, T s ermed as me consan of a frs-order sysem n conrol heory) under a gven ground acceleraon va Eq. (8). Unlke he convenonal response specrum, he over-damped mode response specrum has only one parameer,, nfluencng he response and he over-damped response, q (), whch has velocy dmenson. The procedure o drecly consruc he over-damped mode response specrum consss of he followng hree seps: () Selec he ground moon o be consdered; () Deermne he peak over-damped mode responses represened by Eq. (8) usng he seleced ground moon for dfferen over-damped modal frequences; and () The peak over-damped modal response obaned offers a pon on he over-damped mode response. As a resul, s found ha he consrucon of over-damped mode response specrum reles on he avalably of he ground acceleraon hsores. However, when usng he response specrum approach, he se response specrum specfed n desgn codes s used, whch may vary from se o se, raher han ground acceleraon hsores. Therefore, he over-damped mode response specrum canno be drecly generaed due o he unavalably of ground acceleraon records. A converson approach o consruc an over-damped mode response specrum based on he % dampng dsplacemen specrum (or pseudo-acceleraon specrum) s also esablshed o address hs ssue. The cenral dea s derved from he fac ha he ground moon power specral densy (SD) ha serves for npu o eher second-order subsysem or frs-order subsysem s he same. Thus, afer esablshng he relaonshp beween SD and he peak response for boh subsysems, we can furher consruc he connecon beween wo peak responses and hen use % dampng dsplacemen specrum o predc he compable over-damped mode response specrum. The dealed procedure can be found n Song e al. (8).. EVALUATIO OF THE GCQC RULE The accuracy and applcably of he proposed GCQC rule s evaluaed by conducng response specrum analyses of a seel frame example buldng shown n Fg.. The dealed nformaon of hs buldng and he ground moon acceleraon ensemble used can be referred o Song e al. (8). Ths example buldng s amed o represen a hghly non-classcally damped srucure wh over-damped modes. I s noed ha n order o evaluae he errors arsng from he combnaon rule self, he acual mean peak values of modal dsplacemen responses o he acceleraon ensemble (consdered as he dsplacemen response specra) are used n he modal response combnaons. The example buldng frame s analyzed by usng lnear response hsory analyss o each ground moon record lsed n he ensemble. The mean response analyss resuls (consdered as he exac values) are hen used o examne he accuracy of he GCQC rule, ncludng a comparson of he effec of: () usng he forced classcal dampng assumpon, and () gnorng he over-damped modes when hey are presen. Three ses of resuls are obaned and compared wh he exac values. These hree ses are obaned under he followng condons: (a) resuls of he frs se are obaned based on he proposed GCQC rule. The sae space approach s used o derve he mode shapes, modal frequences and modal dampng raos. These modal parameers are hen used o generae he correlaon coeffcens and peak modal responses requred n he GCQC rule. The conrbuons from he over-damped modes are consdered; (b) resuls of he second se are based on he modal parameers obaned under he forced classcal dampng assumpon. Smlar o he GCQC rule, hese properes are used o generae he daa requred n he modal combnaon rule. The over-damped modes are gnored when hey are presen. Ths process s ofen used for he desgn and analyss of srucures
The h World Conference on Earhquake Engneerng Ocober -7, 8, Beng, Chna wh added dampng devces. Ths rule s referred o as he CDA (forced classcal dampng assumpon); and (c) resuls of he hrd se are dencal o he GCQC rule excep ha he over-damped modes are no aken no accoun n he modal combnaon process. Ths consderaon s amed o examne he effecs of he over-damped modes n erms of response quanes. Ths rule s referred o as he EOM (exclude over-damped modes). Fgure Confguraon of he example buldng Sory - - Esmaon error (%) a. Iner-Sory drf Sory - - Esmaon error (%) b. Iner-Sory velocy Sory - - Esmaon error (%) c. Floor acceleraon Sory - - Esmaon error (%) d. Sory shear a max. drf Sory - - Esmaon error (%) e. Max. sory shear GCQC CDA EOM GCQC : ew combnaon rule CDA : Classcal dampng assumpon EOM : ew combnaon rule exludng over-damped modes Fgure Esmaed errors due o GCQC,CDA and EOM Fg. shows he esmaon errors of each combnaon rule. I s shown ha he GCQC provded excellen resuls excep overesmaed he frs floor acceleraon by abou % where he damper s added. The CDA, however, consderably underesmaed he peak responses wh one excepon (overesmaed he peak nersory velocy of he frs floor by %). The error ncreases as he level of sory ncreases. Ths overesmaon s more profound for ner-sory velocy and floor acceleraon. On he oher hand, EOM overesmaed he ner-sory velocy whle underesmaed he floor acceleraon a he frs floor. For he res of he response quanes, he EOM provded conservave esmaes. In general, he resuls show ha usng GCQC, n whch he over-damped modes, f exs, are consdered, can esmae he peak responses more accuraely. I s found ha he ner-sory velocy and floor acceleraon are sgnfcanly nfluenced by he over-damped modes. Ths s parcularly rue for he floors a whch dampers are nsalled. The responses esmaed by usng he forced classcal dampng assumpon devae subsanally from he exac values. Mos of he responses are underesmaed, whch s undersandable, because he complex modal effecs and over-damped modal conrbuons are gnored by usng hs mehod. Ths mples ha he uly of he forced classcal dampng assumpon should be furher examned n he desgn and analyss of srucures supplemened wh dampers.
The h World Conference on Earhquake Engneerng Ocober -7, 8, Beng, Chna. SUMMARY AD COCLUSIO There are many desgn and analyss approaches for dampng devce applcaons n new consrucon and rehablaon of cvl engneerng srucures. The response specrum mehod s one of he mos common approaches. When dampng devces are added o complex and rregular srucures, he srucures are, n general, heavly non-classcally damped and some over-damped modes may develop. Under such crcumsances, he convenonal CQC or SRSS rules for he response specrum analyss mehod, assumng he srucures are classcally-damped, may no provde accurae resuls. A general modal combnaon rule for he response specrum mehod, denoed as GCQC, s developed o accommodae he presence of non-classcal dampng and over-damped modes. Ths GCQC rule reans he concepual smplcy of he convenonal CQC rule and offers an effcen and accurae esmaon of he peak responses of srucures wh added dampng devces. In addon, a ransformaon prncple o consruc he over-damped mode response specrum from he gven desgn specrum s also nroduced brefly. Ths ensures he applcably of he GCQC rule n engneerng pracce. Example sudy shows ha srucures wh added dampers should be modeled as non-classcally damped and he over-damped modes should be ncluded n he analyss n order o acheve more relable esmaes. In hs paper, he formulaon s focused on a planar srucure subeced o sngle dreconal excaon. Ths formulaon has been exended o he D generally damped srucures under mul-componen excaon by he auhors (Chu e al. 8). 6. ACKOWLEDGEMET The auhors express her sncere apprecaon for he suppor of he aonal Scence Foundaon hrough MCEER (CMS 97-7) and he Federal Hghway Admnsraon (Conrac umber: DTFH6-98-C -9). REFERECES Clough, R.W. and Moahed, S. (976). Earhquake response analyss consderng non-proporonal dampng. Earhquake Engneerng and Srucural Dynamcs, 89-96. Chu, Y.-L., Song, J., Lang, Z. and Lee, G.C. (8). A Unfed Form for Response of D Generally Damped Lnear Sysems under Mulple Sesmc Loads hrough Modal Analyss. roceedng of h World Conference on Earhquake Engneerng, Ocober -7, 8, Beng, Chna. Der Kureghan, A. (98). A response specrum mehod for random vbraon analyss of MDF sysems. Earhquake Engneerng and Srucural Dynamcs 9, 9-. Gupa, A.K. and Jaw, J-W. (986). Response specrum mehod for nonclasscally damped sysems. uclear Engneerng and Desgn. 9, 6-69. Igusa, T, Der Kurghan, A and Sackman, J.L. (98). Modal decomposon mehod for saonary response of non-classcally damped sysems. Earhquake Engneerng and Srucural Dynamcs., -6. Maldonado, G.O. and Sngh, M.. (99). An mproved response specrum mehod for calculang sesmc desgn response. ar : on-classcally damped srucures. Earhquake Engneerng and Srucural Dynamcs, :7, 67-69. Sngh, M.. (98). Sesmc response by SRSS for nonproporonal dampng. Journal of he Engneerng Mechancs Dvson. (ASCE) 6:6, -9. Song, J., Chu, Y.-L., Lang, Z. and Lee, G.C. (8). Modal analyss of generally damped lnear srucures subeced o sesmc excaons. Repor o. MCEER-8-, February, 8, MCEER, Buffalo, Y. Vanmarcke, E.H. (97). roperes of specral momens wh applcaons o random vbraon. Journal of he Engneerng Mechancs Dvson, 98, -6. Velesos, A.S. and Venura, C.E. (986). Modal analyss of non-classcally damped lnear sysems. Earhquake Engneerng and Srucural Dynamcs., 7-. Venura, C.E. (98). Dynamc analyss of nonclasscally damped sysems. h.d. Thess. Rce Unversy, Houson, Texas. Warburon, G.B. and Son, A.R. (977). Errors n response calculaons for non-classcally damped srucures. Earhquake Engneerng and Srucural Dynamcs, 6-76. Zhou, X., Yu, R. and Dong, D. (). Complex mode superposon algorhm for sesmc responses of non-classcally damped lnear MDOF sysem. Journal of Earhquake Engneerng, 8:, 97-6.