Modal Analysis of Generally Damped Linear Structures Subjected to Seismic Excitations

Transcription

1 ISS 5-95X Modal Analyss of Generally Damped Lnear Structures Subjected to Sesmc Exctatons by Janwe Song, Y-Lun Chu, Zach Lang and George C. Lee echncal Report MCEER-8-5 March 4, 8 hs research was conducted at the Unversty at Buffalo, State Unversty of ew York and was supported by the Federal Hghway Admnstraton under contract number DFH6-98-C-94.

2 OICE hs report was prepared by the Unversty at Buffalo, State Unversty of ew York as a result of research sponsored by MCEER through a contract from the Federal Hghway Admnstraton. ether MCEER, assocates of MCEER, ts sponsors, the Unversty at Buffalo, State Unversty of ew York, nor any person actng on ther behalf: a. makes any warranty, express or mpled, wth respect to the use of any nformaton, apparatus, method, or process dsclosed n ths report or that such use may not nfrnge upon prvately owned rghts; or b. assumes any labltes of whatsoever knd wth respect to the use of, or the damage resultng from the use of, any nformaton, apparatus, method, or process dsclosed n ths report. Any opnons, fndngs, and conclusons or recommendatons expressed n ths publcaton are those of the author(s) and do not necessarly reflect the vews of MCEER or the Federal Hghway Admnstraton.

3 Modal Analyss of Generally Damped Lnear Structures Subjected to Sesmc Exctatons by Janwe Song, Y-Lun Chu, Zach Lang 3 and George C. Lee 4 ublcaton Date: March 4, 8 Submttal Date: February, 8 echncal Report MCEER-8-5 ask umber 94-D-. FHWA Contract umber DFH6-98-C-94 Senor Research Scentst, Department of Cvl, Structural and Envronmental Engneerng, Unversty at Buffalo, State Unversty of ew York h.d. Student, Department of Cvl, Structural and Envronmental Engneerng, Unversty at Buffalo, State Unversty of ew York 3 Research Assocate rofessor, Department of Cvl, Structural and Envronmental Engneerng, Unversty at Buffalo, State Unversty of ew York 4 Specal asks Drector, MCEER; Samuel. Capen rofessor of Engneerng, Department of Cvl, Structural and Envronmental Engneerng, Unversty at Buffalo, State Unversty of ew York MCEER Unversty at Buffalo, he State Unversty of ew York Red Jacket Quadrangle, Buffalo, Y 46 hone: (76) ; Fax (76) E-mal: mceer@buffalo.edu; WWW Ste:

4

5 reface he Multdscplnary Center for Earthquake Engneerng Research (MCEER) s a natonal center of excellence n advanced technology applcatons that s dedcated to the reducton of earthquake losses natonwde. Headquartered at the Unversty at Buffalo, State Unversty of ew York, the Center was orgnally establshed by the atonal Scence Foundaton n 986, as the atonal Center for Earthquake Engneerng Research (CEER). Comprsng a consortum of researchers from numerous dscplnes and nsttutons throughout the Unted States, the Center s msson s to reduce earthquake losses through research and the applcaton of advanced technologes that mprove engneerng, pre-earthquake plannng and post-earthquake recovery strateges. oward ths end, the Center coordnates a natonwde program of multdscplnary team research, educaton and outreach actvtes. MCEER s research s conducted under the sponsorshp of two major federal agences, the atonal Scence Foundaton (SF) and the Federal Hghway Admnstraton (FHWA), and the State of ew York. Sgnfcant support s also derved from the Federal Emergency Management Agency (FEMA), other state governments, academc nsttutons, foregn governments and prvate ndustry. he Center s Hghway roject develops mproved sesmc desgn, evaluaton, and retroft methodologes and strateges for new and exstng brdges and other hghway structures, and for assessng the sesmc performance of hghway systems. he FHWA has sponsored three major contracts wth MCEER under the Hghway roject, two of whch were ntated n 99 and the thrd n 998. Of the two 99 studes, one performed a seres of tasks ntended to mprove sesmc desgn practces for new hghway brdges, tunnels, and retanng structures (MCEER roject ). he other study focused on methodologes and approaches for assessng and mprovng the sesmc performance of exstng typcal hghway brdges and other hghway system components ncludng tunnels, retanng structures, slopes, culverts, and pavements (MCEER roject 6). hese studes were conducted to: assess the sesmc vulnerablty of hghway systems, structures, and components; develop concepts for retrofttng vulnerable hghway structures and components; develop mproved desgn and analyss methodologes for brdges, tunnels, and retanng structures, whch nclude consderaton of sol-structure nteracton mechansms and ther nfluence on structural response; and develop, update, and recommend mproved sesmc desgn and performance crtera for new hghway systems and structures.

6 he 998 study, Sesmc Vulnerablty of the Hghway System (FHWA Contract DFH6-98-C-94; known as MCEER roject 94), was ntated wth the objectve of performng studes to mprove the sesmc performance of brdge types not covered under rojects 6 or, and to provde extensons to system performance assessments for hghway systems. Specfc subjects covered under roject 94 nclude: development of formal loss estmaton technologes and methodologes for hghway systems; analyss, desgn, detalng, and retrofttng technologes for specal brdges, ncludng those wth flexble superstructures (e.g., trusses), those supported by steel tower substructures, and cable-supported brdges (e.g., suspenson and cable-stayed brdges); sesmc response modfcaton devce technologes (e.g., hysteretc dampers, solaton bearngs); and sol behavor, foundaton behavor, and ground moton studes for large brdges. In addton, roject 94 ncludes a seres of specal studes, addressng topcs that range from non-destructve assessment of retroftted brdge components to supportng studes ntended to assst n educatng the brdge engneerng professon on the mplementaton of new sesmc desgn and retrofttng strateges. Motvated by the need for a systematc approach for sesmc evaluaton and desgn of structures wth supplemental dampng, a general modal analyss method consderng over-damped modes s developed and descrbed n ths report. he method deals wth a unfed formulaton used to evaluate most structural response quanttes of nterest, such as dsplacements, velocty, nterstory drfts, story shear, dampng forces and absolute acceleratons, etc. In addton, a novel general real-valued transformaton matrx s establshed, whch can be used to decouple the equatons of moton of a generally damped structure n terms of real-valued modal coordnates. he propertes related to ths transformaton are dscussed n detal to explan the dynamc nature of the generally damped structural system. Also, on the bass of the general modal response hstory analyss, two general modal combnaton rules for the response spectrum analyss, GCQC and GSRSS, are formulated. o enable the new rules to be applcable to the practcng earthquake engneerng communty, a converson procedure to construct an over-damped mode response spectrum compatble wth the gven 5% standard desgn response spectrum s establshed. he adequacy of ths converson procedure s also valdated. Examples are gven to demonstrate the applcaton of the modal analyss method, assess the accuracy of the new modal combnaton rules, and show that over-damped modes may develop n structures wth supplemental dampng, whch can provde sgnfcant response contrbutons to certan response parameters. v

7 ABSRAC Motvated by the need for a systematc approach for sesmc evaluaton and desgn of cvl engneerng structures wth supplemental dampng, a general modal analyss method, n whch over-damped modes are taken nto account, s developed and descrbed n ths report. hs general modal analyss method deals wth a unfed formulaton used to evaluate most structural response quanttes of nterest, such as dsplacements, velocty, nter-story drfts, story shear, dampng forces and absolute acceleratons etc. In addton, a novel general real-valued transformaton matrx s establshed, whch can be utlzed to decouple the equatons of moton of a generally damped structure n terms of real-valued modal coordnates. on-sngularty of ths matrx and other propertes related to ths transformaton, such as modal responses to ntal condtons, modal energy dstrbuton, modal effectve masses and modal truncaton etc., are dscussed n detals to explan the dynamc nature of the generally damped structural system. Furthermore, on the bass of the general modal response hstory analyss and the whte nose nput assumpton as well as the theory of random vbraton, two general modal combnaton rules for the response spectrum analyss, GCQC and GSRSS are formulated to handle non-classcal dampng and over-damped modes. o enable the new rules applcable to the practcal earthquake engneerng, a converson procedure to construct an over-damped mode response spectrum compatble wth the gven 5% standard desgn response spectrum s establshed. he adequacy of ths converson procedure s also valdated. Examples are gven to demonstrate the applcaton of the modal analyss method, to assess the accuracy of the new modal combnaton rules, and to show that over-damped modes may develop n structures wth supplemental dampng whch can provde sgnfcant response contrbutons to certan response parameters. v

8

9 ACKOWLEDGEME hs study s orgnally motvated by the need for a more comprehensve and systematc approach for the desgn of structure wth added earthquake response control devces, especally for large, unusual and complex brdges and buldngs. he authors greatly acknowledge the support of the Federal Hghway Admnstraton (Contract umber: DFH6-98-C-94) and the atonal Scence Foundaton through MCEER (CMS 97-47) for the development of a modal analyss approach for generally damped lnear MDOF system reported heren. hs fundamental study wll be further developed for the desgn of sesmc response modfcaton devces and systems for hghway brdges and other structures under the sponsorshp of FHWA. v

10

11 ABLE OF COES Chapter tle age Introducton.... Overvew.... Research Objectves Scope of Work Organzaton of the Report... 7 Equaton of Moton and Egen Analyss Introducton Equaton of Moton Egen Analyss....4 Orthogonalty Modal Decomposton and Superposton of Modal Responses Expanson of System Matrces n terms of Modal arameters A Specal roperty of System Modal Shapes Expanson of the Mass Matrx M and Its Inverse Expanson of the Dampng Matrx C Expanson of the Stffness Matrx K and Flexblty Matrx K Expanson of the otal Mass of the System M Σ....7 Reducton to Classcally-Damped System... 3 General Modal Response Hstory Analyss Introducton Analytcal Formulaton Laplace ransform Operaton Frequency Response Functons Response Solutons to Dsplacement, Velocty and Absolute Acceleraton Dsplacement Response Vector Velocty Response Vector Absolute Response Vector A Unfed Form for Structural Responses Inter-Story Drft x

12 ABLE OF COES (CO D) Chapter tle age 3.3. Inter-Story Shear General Inter-Story Shear Inter-Story Moment General Inter-Story Moment Damper Forces Generalzaton Reducton to Classcally Damped Systems General Modal Coordnate ransformaton and Modal Energy Introducton General Modal ransformaton Matrx roof of Modal Decouplng on-sngularty Analyss for General ransformaton Matrx umercal Example for General Modal Responses General Modal Responses to Intal Condtons General Modal Energy Energy Integral for Arbtrary Ground Moton Exctaton Energy Integral for Snusodal Ground Moton Exctaton Reducton to Classcally Damped System Reducton of Modal ransformaton Matrx Reducton of Modal Responses to Intal Condtons Reducton of Energy Integral Dual Modal Space Approach and Structural DOFs Reducton Formulas Development umercal Example runcaton of Modes Introducton Effectve Modal Mass for Classcally Damped Systems w/o Over-Damped Modes Effectve Modal Mass for Generally Damped Systems Example... 7 x

13 ABLE OF COES (CO D) Chapter tle age 6 Response Spectrum Method Introducton Analytcal Formulaton Defnton of Vector Operaton Symbols Covarance of Responses to Statonary Exctaton Development of Response Spectrum Method Investgaton of the Correlaton Factors Reducton to Classcally under-damped Structures Over-Damped Mode Response Spectrum he Concept Constructon of Over-Damped Mode Response Spectrum Consstent wth 5% Dsplacement Response Spectrum Response Spectrum Consstent SD G ( ) x g ω rocedures η Factor Determnaton Valdaton of the Over-Damped Mode Response Spectrum Analyss Applcaton Examples Introducton Example Buldng Frames Response Hstory Analyss usng Modal Superposton Method Ground Motons Comparson of the Analyss Results Response Spectrum Analyss Ground Motons Comparson of the Analyss Results Summary, Conclusons and Future Research Summary Conclusons Future Research x

14 ABLE OF COES (CO D) Chapter tle age 9 REFERECES... 7 AEDIX A on-sngularty of Matrces A, B, â and ˆb A. on-sngularty of A and B A. on-sngularty of â and ˆb AEDIX B roof of Caughey Crteron x

15 LIS OF FIGURES Fgure tle age 3. FRF of modal dsplacement response (perods equal to.s,.4s,.6s,.8s,.s,.s, 3.s and 5.s; dampng rato=5%) FRF of modal dsplacement response (dampng ratos equal to %, 5%, %, %, 5% and 8%; perod= sec) FRF of over-damped modal (perods equal to.s,.4s,.6s,.8s,.s,.s, 3.s and 5.s) DOFs Symmetrc Structural Model Modal Shapes EL Centro Earthquake Acceleraton me Hstory General Modal Responses Modal Structural Dsplacement Responses of the st DOF Modal Structural Velocty Responses of the st DOF Modal Structural Absolute Acceleraton Responses of the st DOF Structural Dsplacement and Velocty Responses of the st DOF Four ypcal Examples of Frst Order Subsystem Modal Responses to Modal Intal Condtons Structural Dsplacement and Velocty Responses of the st and nd DOF Structural Dsplacement and Velocty Responses of the st and nd DOF hyscal Interpretaton of Effectve Modal Mass Story Shear Frame Structure Model ( DOFs) A planar -DOFs multstory frame Illustraton of the statc structural response subjected to s Conceptual explanaton of the expanson of s= MJ and the resultng base shear forces Cumulatve effectve modal mass Correlaton coeffcent 6. Correlaton coeffcent 6.3 Correlaton coeffcent 6.4 Correlaton coeffcent ρ for responses to whte nose exctatons... 4 ρ for responses to whte nose exctatons... 4 ρ for responses to whte nose exctatons ρ for responses to whte nose exctatons DD j VV j VD j D j x

16 LIS OF FIGURES (CO D) Fgure tle age 6.5 Correlaton coeffcent ρ j for responses to whte nose exctatons Generaton of over-damped modal response spectrum Varaton of η factor for over-damped modal response Over-damped modal response spectrum constructon procedures Mean 5% dampng dsplacement response spectrum (a) ensemble A (b) ensemble B Comparsons of exact and estmated over-damped modal response spectrum (a) ensemble A (b) ensemble B Confguratons of example buldng frames A, B and C Estmated errors of forced classcal dampng assumpton and excluson of over-damped mode n example A Estmated errors of forced classcal dampng assumpton and excluson of over-damped modes n example B Estmated errors of forced classcal dampng assumpton and excluson of over-damped modes n example C Estmated errors due to GCQC,CDA and EM n example A Estmated errors due to GCQC,CDA and EM n example B Estmated errors due to GCQC,CDA and EM n example C xv

17 LIS OF ABLES able tle age 4. Modal Frequency, erod and Dampng Rato Orgnal Structural Modal arameters Modal erods Estmaton Results and Error Comparson Dampng Rato Estmaton Results and Error Comparson Summary of the expressons of the effectve modal mass Whte-nose-determned η factor for over-damped modal response Far-feld ground motons used n AC Far-feld ground motons used by Vamvatskos and Cornell Modal perods of example buldngs A, B and C Modal dampng ratos of example buldngs A, B and C Results of mean peak responses of example A va modal superposton methods Results of mean peak responses of example B va modal superposton methods Results of response hstory analyss of example C va modal superposton methods Comparsons of results by GCQC, CDA and EOM to mean results of response hstory analyss for examples A, B and C xv

18

19 CHAER IRODUCIO. Overvew Modal analyss to a lnear structural system may be explaned as a method for decouplng the equatons of moton by means of modal coordnate transformaton matrx as well as evaluatng modal responses and further combned modal responses to estmate the structural responses. By the ad of modal analyss, the structural solutons, especally to a complcated structure wth large degrees of freedom (DOFs), can be sgnfcantly smplfed and the computaton efforts can be largely reduced. Meanwhle, the resultng analyss accuracy can stll be ensured wthn the range that s reasonable or acceptable n engneerng applcatons. In addton, usng ths method, certan of the structural nherent propertes may be much easer to be exposed. he decouplng coordnate transformaton can be determned by the soluton of an algebrac egenvalue problem of the system. In earthquake engneerng, the classcal modal analyss method s consdered as a powerful approach to analyze the sesmc responses of classcal damped lnear structures. wo approaches of ths method are: the modal response hstory analyss, whch gves the complete response hstory of the structures, and the response spectrum analyss. When the structures satsfy the crteron specfed by Caughey and O Kelly (965), the modes of the structure are real-valued and are dentcal to those of the assocated undamped systems. hs lnear vbratng structure s sad to be classcally damped and possesses normal modes as well as can be decoupled by the same modal transformaton that decouples the assocated undamped structures. hose structures that do not satsfy the Caughey and O Kelly crteron are sad to be nonclasscally damped; consequently, ther equatons of moton cannot be decoupled by the classcal modal transformaton. In prncple, the couplng arses from the dampng term. ypcal examples nclude the structures wth added dampng devces and base-solated structures as well as prmary-secondary systems.

20 Bascally, responses of the non-classcally damped systems may be evaluated by usng the decoupled method suggested by Foss (958). However, t s generally beleved that, concurrent wth the classcal dampng assumpton, the structural responses calculated by the classcal modal superposton method are acceptable. For example, current methods for sesmc desgn of structures enhanced wth dampng devces are developed based on the classcal dampng assumpton (BSSC 3). hs may not always be true due to the uncertanty of the nature and magntude of the dampng n structures. hs phenomenon can be further magnfed when the structure s rregularly shaped. here are nstances that the structures can be hghly non-classcally damped (Warburton and Son 977) and, n some occasons, develop over-damped modes (Inman and Andry 98), whch n turn result n a chance of the naccuracy of the response estmatons. For example, akewak (4) has demonstrated that the structural energy transfer functon and dsplacement transfer functon wll be underestmated f the over-damped modes are neglected. o advance the applcatons of the classcal modal analyss to the non-classcally damped systems, a number of researchers have conducted extensve studes on developng complex modal superposton methods for systems not satsfyng the classcal dampng condton. Igusa et al. (984) studed the statonary response of mult-degreesof-freedom (MDOF) non-classcally damped lnear systems subjected to statonary nput exctatons. Veletsos and Ventura (986) presented a crtcal revew of the modal superposton method of evaluatng the dynamc response of non-classcally damped structures. Sngh and Ghafory-Ashtany (986) studed the modal tme-hstory analyss approach for non-classcally damped structures subjected to sesmc forces. Yang et al. (987 and 988) used real-valued canoncal transformaton approach to decouple nonclasscally damped system from a set of second order dfferental equatons to a set of frst order ones, and then performed the tme hstory analyss as well as response spectrum analyss. Zhou et al. (4) provded a refned complex mode superposton algorthm to evaluate the sesmc responses of non-classcally damped systems. All the above are mportant contrbutons but none addressed the over-damped modes.

21 In addton, n earthquake engneerng, the response spectrum method s commonly used as an alternatve approach to response hstory analyss for determnng the maxmum values of the sesmc responses of classcally damped structures. In ths method, the modal peak responses are obtaned usng the prescrbed response spectrum. hese modal maxma are then approprately combned to estmate the peak values of the responses of nterest. here are several combnaton rules proposed by varous researchers. Among whch, the smplest s the square-root-of-sum-of-squares (SRSS) modal combnaton rule (Rosenblueth 95). hs rule gnores the correlatons between vbraton modes and can provde excellent estmates for structures wth well-separated modal frequences. o further consder the correlatons between each vbraton mode, Der Kureghan (98 and 98) proposed a ratonal rule, known as complete quadratc combnaton (CQC) rule, n whch the correlatons among modes are connected by correlaton coeffcents. Both rules deal wth classcally damped structures. he conventonal response spectrum method s deal to structures satsfyng classcal dampng condton. For structures that are strongly non-classcally damped, the accuracy of SRSS or CQC rule becomes questonable (Clough and Mojtahed 976, Warburton and Son 977 and Veletsos and Ventura 986). For ths reason, several modal combnaton rules accountng for the effect of the non-classcal dampng are developed. Sngh (98) formed a modfed conventonal SRSS approach where nonproportonal dampng effects can be ncluded properly. Igusa et al. (984) descrbed the responses n terms of spectral moments and provded the formulatons of correlaton coeffcents among modes usng fltered whte nose process as nputs. Ventura (985) stated that the peak modal responses can be obtaned by takng square roots of the sum of squares of the ndvdual modal maxma contrbutng from dsplacement and velocty responses, assumng harmonc exctatons. Gupta and Jaw (986) developed the response spectrum combnaton rules for non-classcally damped systems by usng the dsplacement and velocty response spectrum. Vllaverde (988) mproved Rosenblueth s rule (95) by ncludng the effect of modal velocty responses. Maldonado and Sngh (99) proposed an mproved response spectrum method for non-classcally damped systems. It reduces the error assocated wth the truncaton of the hgh frequency modes wthout explctly usng them n the analyss. Zhou et al. (4) generalzed the CQC rule for ts applcaton 3

22 to non-classcally damped systems. However, all above-mentoned combnaton rules dd not ncorporate the over-damped modes n the formulaton and the response quanttes consdered n these rules are lmted to deformaton-related response quanttes. In general, when usng modal superposton method, the response contrbutons of all modes should be ncluded to obtan the exact results. At the same tme, t s observed that lmted amount of modes can usually gve suffcently accurate results. he number of modes requred s well-defned n the classcal dampng cases through the use of the cumulatve effectve modal mass. he correspondng crtera, however, for the nonclasscally damped structures wth or wthout over-damped modes are not well addressed. o address ths ssue, an approprate expresson of the effectve modal mass for nonclasscal dampng structures s formulated. Intally motvated by the need for a systematc approach for the desgn of structures wth added dampng devces, a general modal response hstory analyss method s developed and presented n ths report. he method advances the complex modal analyss to be applcable to structure wth over-damped modes. In addton, a unfed form that s able to express any response quanttes of the systems, ncludng the veloctes and absolute acceleratons, s establshed. hs unfed form s made possble by several novel modal propertes found n ths study. Also, on the bass of the general modal response hstory analyss and the whte nose nput assumpton as well as the theory of random vbraton, a general modal combnaton rule for response spectrum method s formulated to deal wth the non-classcal dampng and over-damped modes. hs general modal combnaton rule s referred to as General-Complete-Quadratc-Combnaton (GCQC) rule n ths report. o enable the new rules applcable to the practcal earthquake engneerng, an over-damped modal response spectrum, followng a smlar defnton as the conventonal response spectrum, s ntroduced to account for the peak modal responses of the over-damped modes. A converson procedure to construct an overdamped mode response spectrum compatble wth the gven 5% standard desgn response spectrum s establshed. he adequacy of ths converson procedure s also valdated. In addton to the dsplacement correlaton coeffcent gven n the CQC rule, new correlaton coeffcents to account for the cross correlatons between modal dsplacement, 4

23 modal velocty and over-damped modal responses are also provded. It s shown that ths rule s also sutable to estmate the velocty-related and absolute acceleraton-related response quanttes. For example, the absolute acceleraton of a sngle-degree-of-freedom system can be approxmated more accurately by ths rule nstead of usng the correspondng pseudo-acceleraton values. he applcablty of the general modal response hstory analyss method s demonstrated by two numercal examples. Also, the errors n structural response estmatons arsng from the classcal dampng assumpton are dentfed, and the effect of the over-damped modes on certan response quanttes s observed. he accuracy of the GCQC rule s also evaluated through the two examples by comparng t to the mean response hstory results. For engneerng applcatons, a procedure to convert the gven 5% desgn spectrum to the over-damped mode response spectrum s gven. Its accuracy s also verfed. In addton, a general real-valued modal coordnate transformaton matrx whch can decouple the equatons of moton of generally damped structures s derved n the process of theoretcal formulaton. A rgorous proof of the modal decouplng by usng ths general modal coordnate transformaton s gven. Further, based on ths transformaton matrx, the formulaton of the general modal responses subjected to structural ntal condtons and the formulaton of the general modal energy for arbtrary ground moton are establshed. Fnally, a dual modal space approach to reduce the scale of the modelng and computaton burden s proposed.. Research Objectves he prmary objectves of ths research are: () o explore the modal propertes of the non-classcally damped systems wth over-damped modes and to further expand/complement current lnear structural modal analyss theory. () o mprove the present modal analyss procedure to accurately evaluate the peak sesmc responses of damped lnear structures wth over-damped modes. 5

24 (3) o extend the present response spectrum method to be applcable to the nonclasscally damped systems wth over-damped modes and to establsh a sold foundaton for structural dampng desgn. (4) o propose an easy and reasonable crteron to determne the number of modes requred to be ncluded n the modal analyss of non-classcally damped systems wth over-damped modes to acheve an acceptable level of accuracy..3 Scope of Work he work has proceeded as follows: () Examne the theory presently beng used for analyzng the non-classcally damped lnear systems. () Formulate the equaton of moton of a MDOF system by means of the state space method, as well as perform an egen analyss and explore the modal propertes. All the formulatons are presented n the form of matrx. (3) Clarfy the modal energy dstrbutons of non-classcally damped lnear MDOF structures. (4) Formulate the response hstory analyss procedure n the manner of modal superposton and offer nterpretaton of the physcal meanng n the formulaton. Man effort focuses on the analytcal formulaton about the overdamped modes. (5) Formulaton for the response hstory analyss procedure s extended for the use of response spectrum method. Much of ths effort focuses on the treatment of the over-damped modes. (6) Revew the crteron that s used to determne the number of modes requred for the classcally damped structures and develop a correspondng crteron for the generally damped lnear MDOF structures. 6

25 .4 Organzaton of the Report Chapter detals the egen-analyss of a generally damped lnear system, stressng on the treatment of the over-damped modes. Several fundamental modal propertes are explored and presented. Chapter 3 presents the formulaton of the modal analyss procedures for the generally damped lnear MDOF systems wth hghlght on the treatment of the overdamped modes. hs chapter also presents a unfed form sutable for any response quanttes, whch s obtaned based on the modal propertes found n Chapter. It should be noted that Laplace transform approach s utlzed n ths Chapter, whch smplfes the process of the formulaton wth certan level and may make t possble that new attractve formulas can be developed. Chapter 4 descrbes a general modal coordnate transformaton resulted from the unfed formulaton obtaned n Chapter 3. Based on ths transformaton matrx, the structural modal responses subjected to structural ntal condtons are gven. A smplfed numercal example model wth 4-DOFs s provded to further clarfy some propertes related to modal transformaton matrx, over-damped modes and modal responses. Also, the general modal energy for structural ntal condtons, snusodal exctatons and sesmc exctatons s derved, leadng to a deep physcal nsght of the formulaton work. A dual modal space approach to reduce the scale of the modelng and computaton effort s also gven. Chapter 5 proposes a crteron on determnng the number of modes that should be ncluded n the modal analyss for the generally damped lnear MDOF system. Chapter 6 shows the rgorous formulaton of the response spectrum method for the analyss of the generally damped lnear MDOF system wth over-damped modes. hs chapter focuses on the development of a manner to handle the over-damped when usng the prescrbed ste response spectra. Chapter 7 demonstrates the use of the proposed modal analyss method and response spectrum method through three example buldngs. he results obtaned by usng the 7

26 classcal dampng assumpton and the exact solutons are compared. he effect of the over-damped modes on the peak response estmaton s examned and dscussed. Fnally, Chapter 8 presents the summary as well as conclusons, and provdes some suggestons for future research needs. 8

27 CHAER EQUAIO OF MOIO AD EIGE AALYSIS. Introducton hs chapter presents the mathematcal modelng of a generally damped planar lnear MDOF structure subjected to a dynamc loadng. Correspondng egen analyss of the system s performed and ts modal propertes are explored, n whch the case wth real-valued egenvalues s also addressed. he real-valued egenvalues correspond to the presence of over-damped modes are usually assumed to be unlkely to occur n the engneerng practce. However, ths may not be always true for systems wth added earthquake protectve systems. Also, the orthogonalty of the modal vectors s examned and the system mass, dampng and stffness matrces are expanded n terms of modal parameters. he results shown n ths chapter are shown to be useful for the analytcal formulatons n the subsequent chapters.. Equaton of Moton For a lnear, dscrete, generally damped planar structure wth degree-of-freedom (DOF) subjected to a dynamc loadng f () t, whch has dmensonal real feld vector,.e., f( t) R, the equaton of moton can be descrbed as Mx () t + Cx () t + Kx() t = f() t (.) n whch x( t) R, x ( t) R and x ( t) R are the relatve structural nodal dsplacement, velocty and acceleraton vectors, respectvely. M R, C R and K R are real and symmetrc mass, vscous dampng and stffness matrx, respectvely. M and K are postve-defnte matrces when the structure s completely constraned, whle C s a sem-postve defnte matrx. It s noted that no further restrcton s mposed on the form of the dampng matrx. 9

28 .3 Egen Analyss In general, when the structure s non-classcally damped, the formulaton n the dmensonal state space s essental for solvng the equaton of moton va the modal analyss approach. In turn, Equaton (.) can be reduced to a set of frst-order dmensonal equatons as Ay () t + By() t = f () t (.) S where M ( ) R, B M M C ( K) t = x { ()} R fs t = t { () t } R x f A= = R y( t), ( ) (.3) It can be proved that A and B are non-sngular, mplyng that both A and B exst. he proof s gven n Appendx A. he assocated egen-equaton wth Equaton (.) s gven by ( λ A+ B) ψ = (.4) he soluton to the above egenvalue problem leads to a set of total egenvalues (also known as egen-roots or characterstc roots) λ C (belong to complex feld) and complex egenvectors ψ C. For a conventonal structure or a structure enhanced wth passve dampng devces, a stable system s expected. In other words, the egenvalues are ether complex-valued wth negatve real parts or negatve real-valued. When the egenvalues are complex-valued, the correspondng modes are underdamped and the egenvalues and egenvectors appear n complex-conjugated pars, whch can be easly proved as follows. Supposng λ and ψ are the th complex egenvalue and correspondng egenvector, respectvely, λ and ψ therefore satsfy Equaton (.4). hat s,

29 ( ) λ A+ B ψ = (.5) akng conjugate operaton to both sdes of Equaton (.5) and notng that A and B are both real-valued matrces, operaton), we have A = * A and B = * B (the superscrpt * denotes a conjugate ( ) λ A+ B ψ = (.6) * * * * whch means that λ and ψ also satsfy Equaton (.4) and they are one par of egensoluton of Equaton (.4). ow assumng that there are egenvalues and egenvectors can be expressed as C pars, the correspondng * λ, λ = ξω ± j ωd ( =,,3 C) (.7) λ ϕ { ϕ } * * * λϕ ψ =, ψ = * ϕ (.8) where ϕ C or ϕ C s the th complex modal shape; ω R and ξ R are called * the th modal crcular frequency and th dampng rato, respectvely, and ω d ξ ω = R s called the th damped modal crcular frequency. When the egenvalues are real-valued, the correspondng modes are the propertes of the over-damped frst order subsystems whch are no longer second-order oscllatory subsystems. For the sake of smplcty, all related varables, such as perods, modal shape and modal responses, to the over-damped subsystems are denoted by superscrpt or subscrpt ( O may be a better choce, but t s easly confused wth ) s to dstngush them from the varables assocated wth complex modes. Mathematcally speakng, over-damped modes appear n pars. However, based on the control theory, each over-damped mode must be consdered as an ndependent basc unt. here are no functonal relatonshps among all over-damped modes, mathematcally or physcally. hus t would not be necessary to group them n pars n

30 the analytcal formulaton process. In ths report all over-damped modes are handled ndvdually. Assumng that there are [ ( )] = real negatve-valued egenvalues: C λ = ω = R (,, 3 ) (.9) where ω >, whch has dmenson rad/sec, s defned as th over-damped modal crcular natural frequency. Each real egenvalue λ, the correspondng egenvector must be a real-valued vector, that s, magnary part of ψ, easly derved after substtutng λ and Im( ψ ) = ψ, whch can be ψ nto Equaton (.4) and takng magnary parts operaton for both sdes of the consequent equaton. hus, and ψ R ( =, ) ψ λ ϕ = R ϕ (.) where ϕ R s the th over-damped modal shape. he egenvalue matrx, whch s the assembly of all egenvalues, s a dagonal matrx and s denoted as ( λ λ λ λ λ λ λ λ λ ) * * * Λ = dag,,,,, C (.) C C he egenvector matrx, whch s the assembly of all egenvectors, s denoted as Ψ = * * * ( ψ, ψ ψ, ψ, ψ ψ, ψ, ψ ψ ) Φ Λ ( ) = Φ C C n whch (,,,,, ) C (.) * * * Φ= ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ C s the egenvector C C matrx assocated wth the dsplacement vector (modal shape matrx) and ΦΛ s the egenvector matrx assocated wth the velocty vector.

31 .4 Orthogonalty he egenvectors correspondng to dfferent egenvalues can be shown to satsfy the followng orthogonalty condtons, n whch no repeated egenvalues condton s assumed. For smplcty of the proof of the orthogonalty, denote ( λ, ψ ) ( r =, ) or ( λ, ψ ) ( s =, ) to express all general egen-pars. hus, ( λ, ψ ) and s s ( λ, ψ ) satsfy the followng two equatons, respectvely: s s r r r r re-multplyng gves ψ s and ( ) λ ra+ B ψr = C (.3) ( ) λ sa+ B ψs = C (.4) ψ r to both sdes of Equatons (.3) and (.4), respectvely, ( ) ψ λ A+ B ψ = C (.5) s r r ( ) ψ λ A+ B ψ = C (.6) r s s Snce A and B are symmetrc matrces, ψ Aψ Further, subtractng (.5) from (.6) gves = ψ Aψ and s r r s ψ Bψ = ψ Bψ. s r r s ( λ λ ) ψ Aψ = (.7) s r r s Assume that there are no repeated egenvalues n the structural system, that s, f r = s, λ r λ s. hus When r ψraψs ψrbψs, f r s = s, the followng relatonshps exst = = (.8) ψ Aψ = a C (.9) r r r ψ Bψ = b = λ a C (.) r r r r r 3

32 Equatons (.8) to (.) show that all egenvectors are orthogonal wth respect to the matrces A and B, respectvely, under the condtons that no repeated egenvalues exst. ote that a complex mode as denoted wth subscrpt as prevously defned s composed of two egen-pars whch are conjugated wth each other. However, they actually belong to dfferent egen-pars because ther egenvalues are unequal and ther egenvectors are orthogonal each other. Combnng Equatons (.8) to (.) and expandng the consequent equatons wth the help of Equaton (.3) as well as rewrtng Equaton (.8) for r and s, respectvely, as ψ r λrϕ r { } C ϕ = r (.) s s and { } ψ s λ ϕ = C ϕ s (.) we can have ar f r = s ψraψs = ( λ r+ λ s) ϕrmϕs+ ϕrcϕ s= (.3) f r s = f br λ rar r = s ψ rbψ s = λλ r sϕrmϕs+ ϕrkϕ s= (.4) f r s ( rs, =,,3 ) Supposng that r and s express the same complex mode (the th order), that * s, λ = λ, r s * * ϕ r= ϕ s, and notng that ( ) λ + λ = ξ ω and λλ ( =,,3 ), * = ω C from Equatons (.3) and (.4), we can obtan two useful formulas for calculatng modal natural frequency ω and modal dampng rato ξ of the system, respectvely: ω = ϕ ϕ H H K ϕ Mϕ R (.5) and 4

33 H H ϕ Cϕ ϕ Cϕ ξ = = R (.6) H ( ω ) H H ϕ Mϕ ( ϕ Mϕ)( ϕ Kϕ) ( =,,3 ) C where superscrpt H denotes Hermtan transpose, whch s equvalent to conjugate and transpose operaton. ote that snce M and K are assumed postve-defnte matrces and H H the structure s completely constraned, ( ϕ Mϕ ), ( K ) ϕ ϕ and ω. Consderng the orthogonalty shown n Equatons (.3) and (.4), the general orthogonal property can be re-expressed as where ( a a a a a a a a a ) aˆ = A = dag,,,,, * * * Ψ Ψ C (.7) C C ( b b b b b b b b b ) bˆ = B = dag,,,,, * * * Ψ Ψ C (.8) C C b a ( λ ) ψ Aψ M C = = ϕ + ϕ C (.9) ( λ ) = ψ Bψ = ϕ M+ K ϕ = aλ C (.3) ( =,,3 for complex modes) C and a ( λ ) = ( ψ ) Aψ = ( ϕ ) M+ C ϕ R (.3) b = ( ψ ) Bψ = ( ϕ ) ( λ ) M + K ϕ = a λ R (.3) ( =,, 3 for pseudo modes) In Appendx A, the non-sngular propertes of â and ˆb have been proven, whch means that each element ( a, non-zero parameter. * a, b, * b a and b ) n dagonal matrces â and ˆb s 5

34 .5 Modal Decomposton and Superposton of Modal Responses Equaton (.) can be decoupled nto ndependent modal equatons, after takng congruent transformaton to coeffcent matrces n Equaton (.) based on the orthogonalty property shown n the prevous secton. Let n whch x t { ()} z t R x t y() t = = Ψ () (.33) * * * C C z () t = z (), t z (), t z (), t z (), t z () t z (), t z (), t z () t z () t C (.34) s the complex-valued modal coordnate vector n tme doman. Substtutng Equaton (.33) nto Equaton (.) and pre-multplyng Ψ to both sdes of the resultng equaton as well as makng use of Equatons (.9) to (.3), the followng equatons can be obtaned az t bz t ϕ t C (.35) ( ) + ( ) = f( ) ( =,... C ) az ( t) + bz( t) = ϕ f( t) C ( =,... ) (.36) * * * * H C and az () t+ bz() t = ( ϕ ) f() t R ( =, ) (.37) Equatons (.35) to (.37) are all frst-order complex numbered dfferental equatons that can be readly solved usng the standard dgtal algorthms. he solutons of (.35) to (.37) can be expressed as (Hart and Wong 999) = ϕ f τ C (.37a) t λ ( t τ) () e ()d t a z t z () t = e ϕ f ()d t τ C (.37b) t * λ * ( t τ) H * a z () t = e ( ϕ ) ()d t (.37c) a t λ ( ) t τ f τ R 6

35 Obvously, only one of Equatons (.37a) and (.37b) s ndependent and needs to be solved. After the modal responses are solved, the total responses are back-calculated by the superposton of the modal responses. Consderng Equaton (.) n Equaton (.33) gves the followng two expressons: C * * = = x() t = ϕ z () t + ϕ z () t + ϕ z () t R (.38) C * * * λ λ λ = = x () t = ϕ z () t + ϕ z () t + ϕ z () t R (.39).6 Expanson of System Matrces n terms of Modal arameters.6. A Specal roperty of System Modal Shapes akng the tme dervatve of Equaton (.38) gves C * * = = x () t = ϕ z () t + ϕ z () t + ϕ z () t (.4) Substtutng Equatons (.35) to (.37) nto Equaton (.4) accordngly leads to C * H ϕϕ f() t ϕϕ f() t * * * ϕ( ϕ) f() t x () t = + ϕλ z() t + + ϕ () () * λ z t + + λ z t = a a ϕ = a (.4) After comparng Equatons (.39) and (.4), t can be observed that ϕϕ ϕϕ ϕ ( ϕ ) + f() + f() t = R C * H t * = a a = a (.4) As f () t s an arbtrary exctaton force vector, t mples that ϕϕ ϕϕ ϕ ( ϕ ) + + = C * H * = a a = a R (.43) 7

36 whch can be assembled n a smple matrx form as ˆ Φ a Φ = (.44) Actually, Equatons (.4) and (.44) are the result of the summaton of the system resdual matrx, whch wll be defned n detals n the next chapter. hs property s a key element n the dervaton of other modal propertes..6. Expanson of the Mass Matrx M and Its Inverse Substtutng Equaton (.) nto Equaton (.8) yelds M ΦΛ M ΦΛ Ψ BΨ = = + = ˆ ΛΦ MΦΛ Φ KΦ b = aλ ˆ Φ K Φ (.45) that s, ΛΦ MΦΛ Φ KΦ = aλ (.46) ˆ leads to re-multplyng Φ aˆ and post-multplyng ˆ Τ a Φ to both sdes of Equaton (.46) ( ˆ ) ( ˆ ) ( ˆ Τ ) ( ˆ ) ( ˆ a Λ M a Λ a K a = Λa ) Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ (.47) Substtutng Equaton (.44) nto Equaton (.47), t can be smplfed as ( ˆ ) ( ˆ Τ ) ( ˆ a Λ M Λa = Λa ) Φ Φ Φ Φ Φ Φ (.48) hus, M = ( aˆ Λ ) Φ Φ (.49) and M = Φaˆ ΛΦ = λ ϕϕ λ ϕϕ + + λ ϕ ( ϕ ) C * * H * = a a = a = λ ϕϕ Re + λ ϕ ( ϕ ) C = a = a (.5) 8

37 .6.3 Expanson of the Dampng Matrx C Substtutng Equaton (.) nto Equaton (.7) yelds ΦΛ M ΦΛ Ψ AΨ = Φ M C Φ = Φ MΦΛ+ ΛΦ MΦ+Φ CΦ = a ˆ (.5) that s, Φ MΦΛ+ ΛΦ MΦ+Φ CΦ = a (.5) ˆ ost-multplyng ˆ Λa Φ to both sdes of Equaton (.5) leads to ˆ ˆ ˆ Φ MΦΛ a Φ + ΛΦ MΦΛa Φ +Φ CΦΛa Φ = ΛΦ (.53) Usng Equaton (.5), Equaton (.53) becomes ˆ Φ MΦΛ a Φ + ΛΦ +Φ CM = ΛΦ (.54) hat s, ( ˆ ) Φ MΦΛ a Φ + CM = (.55) Snce Φ C and Τ det[ ΦΦ ], we have ˆ MΦΛ a Φ + CM = (.56) he dampng matrx C therefore can be expressed as ˆ C= MΦΛ a Φ M (.57) Equaton (.57) can be further expanded as C * * H λ Mϕϕ M ( λ ) Mϕϕ M ( λ ) Mϕ (ϕ) M C = + * = a a = a (.58) 9

38 .6.4 Expanson of the Stffness Matrx K and Flexblty Matrx o obtan the expanson of K, we assemble the egen-equaton shown n Equaton (.4) as K gves M ΦΛ + CΦΛ + KΦ = (.59) Substtutng the expanson of C shown n Equaton (.57) nto Equaton (.59) M ΦΛ MΦΛ aˆ Φ MΦΛ + KΦ = M ΦΛ I a Φ MΦΛ + KΦ (.6) = ( ˆ ) where I s a dentty matrx. ost-multplyng (.6) and usng Equaton (.49) gve ( ) ( ˆ ) M ΦΛ I aˆ Φ MΦΛ Λaˆ Φ + KΦΛaˆ Φ Τ Τ = M Λ I a M Λ Λaˆ + KM = ˆ Λa Φ to both sdes of Equaton Τ Φ Φ Φ Φ (.6) As a result, the stffness matrx K can be expanded as ( ˆ ) Φ Φ Φ ˆ Φ (.6) K = M Λ I a M Λ Λa Τ M o obtan the expanson of the structural K, pre-multplyng ˆ Φ a Λ to both sdes of Equaton (.46) leads to ( ) ( ) ˆ ˆ ˆ ˆ Φa Λ ΛΦ M ΦΛ Φa Λ Φ KΦ = Φ Λ a aλ (.63) Usng the resdual matrx ˆ Φa Φ = shown n Equaton (.44), Equaton (.63) becomes ˆ Φa Λ Φ KΦ = Φ (.64)

39 we have ost-multplyng ˆ Λa Φ to both sdes of Equaton (.64) and usng Equaton (.5), ˆ Φa Λ Φ KM = M (.65) hus, K = Φaˆ Λ Φ (.66) Equaton (.66) can be further expanded as K C * H ϕϕ ϕϕ ϕ( ϕ) = Φaˆ Λ Φ = + * * (.67) = λ a λ a = λ a.6.5 Expanson of the otal Mass of the System M Σ For a planar lumped-mass buldng system whch may or may not have rotaton moment of nerta, the total translatonal mass (or the total mass n the drecton of sesmc exctaton) of the system s denoted as M Σ, whch can be represented by the followng form. M Σ = JMJ R (.68) n whch, J s the ground moton nfluence vector. ote that when the system s a shear frame structure, J s a -dmensonal ones vector (all elements are untary) whle for a general planar structure, the elements correspondng to the rotaton DOFs n J are zero because n most cases rotaton exctatons from ground moton are gnored. he expanson of the total mass M Σ can be shown n the followng manner. M Σ = J MJ = J MM = MJ ( MJ) M ( MJ) (.69) Substtutng the nverse of the mass matrx (.69) gves the expanson as M shown n Equaton (.5) nto Equaton

40 M Σ ( ) λ ( ) C MJ ϕϕ MJ MJ λ ϕ( ϕ) MJ = Re + = a = a ( ϕ MJ) λ (( ϕ ) MJ ) C λ = Re + a = = a (.7).7 Reducton to Classcally-Damped System When a structure satsfes the Caughey crteron CM K = KM C (Caughey and O Kelly 965), all system mode shapes are real-valued and are consstent wth those of an undamped system. In Appendx B, Caughey crteron has been extended to be used as the crteron for a system wth over-damped subsystems. hus, the modal shape proportonal factors a for real modes be smplfed as where ( =, ) expressed by Equaton (.9) can ( M C) ( ) d C a = ϕ λ + ϕ = m λ + ξ ω = jmω (.7a) m = ϕ Mϕ s the th real modal mass whle a ( =, ) remans the same format. Substtutng Equaton (.7a) nto Equatons (.5), (.58), (.6), (.67) and (.7) we have M ϕϕ ω ϕ ( ϕ ) = C = m = a (.7b) ( ξω ) C Mϕϕ M ( ω ) Mϕ (ϕ) M C = = m = a (.7c) ω Mϕϕ M ω Mϕ ( ϕ ) M K = + C = m = a (.7d) K ϕϕ ϕ ( ϕ ) = + C = ωm = ωa (.7e)

41 M Σ C ( ϕ MJ) ω ( ( ϕ) MJ) = m = = a (.7f) If there are no over-damped modes exstng n the system ( = and C Equatons (.7a) to (.7f) can be further smplfed as = ), the M C ϕϕ = m = (.7g) C C = = ( ξω ) Mϕϕ M m (.7h) C ω K = M ϕϕ M = m (.7) K ϕϕ = C = ω m (.7j) M Σ ( ϕ ) C = MJ = m (.7k) In Equaton (.7), ( MJ) ϕ / m s the th modal partcpaton mass (Wlson 4). 3

42

43 CHAER 3 GEERAL MODAL RESOSE HISORY AALYSIS 3. Introducton In Chapter, the dynamc response of a generally damped lnear MDOF structure has been expressed by means of the superposton of ts modal responses. However, t s expressed n complex-valued form wthout physcal meanng. An mproved general soluton, completely expressed n real-valued form, for calculatng sesmc response hstory of the MDOF structure, s deduced n ths chapter and the physcal explanaton of each resultng terms are gven. In ths formulaton, the Laplace transformaton approach s employed, by whch the system orgnal dfferental equatons n tme doman can be converted to algebrac equatons n Laplacan doman, to show the ntrnsc relatonshp among the system s parameters. 3. Analytcal Formulaton A soluton to the problem of obtanng the dynamc response of generally damped lnear MDOF systems va the modal analyss approach s presented n ths secton. hs formulaton takes nto consderaton the presence of the over-damped modes (.e., the egenvalues assocated wth these modes are real-valued rather than complex-valued). 3.. Laplace ransform Operaton o smplfy further development, the Laplace transformaton (Greenberg 998) s employed frst to transform the dfferental equatons to the lnear algebrac equatons n Laplacan doman, by whch the system response n Laplace doman can be easly expressed as the combnaton of the complete orders of the modal subsystems (composed by the correspondng modal parameters). he responses n the tme doman for the complete system and all sub-systems are easly retreved through the nverse-laplace transform. Applyng Laplace transform to Equatons (.33) and (.35) to (.37) under zero ntal condtons, respectvely, yelds 5

44 X () s sx() s { X() s } { X() s } Y() s = = = () Ψ Z s C (3.) ( as + b) Z( s) = F( s) C ( =, C) ϕ (3.) * * * H ( as + b ) Z () s = F() s ( =, C) ϕ C (3.3) and ( as + b ) Z () s = ( ) F() s ( =, ) ϕ C (3.4) where s s the Laplace parameter and * * * = C C Z() s Z (), s Z () s Z (), s Z (), s Z() s Z (), s Z (), s Z() s Z () s (3.5) s the modal coordnates vector expressed n Laplace doman and Z() s C. () s X s the Laplace transformaton of the dsplacement vector x () t, X ( s) s the Laplace transformaton of the velocty vector x () t and F () s s the Laplace transformaton of the force vector f () t. Solvng Z ( s ), Z * () s and Z from Equatons (3.) to (3.4), respectvely, and substtutng the resultng solutons nto Equaton (3.) as well as usng Equatons (.3) and (.3) result n C * H ϕϕ ϕϕ ϕ( ϕ ) X() s = + () s * * = a( s λ ) a( s λ ) + F = a ( s λ ) R R R = C * + * + () s = s λ s λ F C = s λ (3.6) and C * * H λ λ λ ( ) X ϕϕ ϕϕ ϕ ϕ () s = sx() s = + + () s * * = a( s λ ) a ( s λ ) F = a ( s λ ) C * * λ R λ R λ R = + * + F() s C = s λ s λ = s λ (3.7) 6

45 where R I R = R + j R = ϕϕ C ( =, C ) (3.8) a R = R j R = ϕϕ C ( =, ) (3.9) * R I * H * a C R ϕ( ϕ ) = R ( =, ) (3.) a R, * R and * R are the system resdual matrces correspondng to the egenvalues λ, λ and λ, respectvely. ote that all resdual matrces are ntrnsc and nvarant to a lnear structural system, although modal shapes may be vared along wth the correspondng changes of modal normalzed factors (proportonal multplers) a, a or a. In addton, referrng to Equatons (.43) or (.44), t has been proved that * the summaton for all resdual matrxes s a zero matrx, that s, C * R + R + R = R (3.a) = = or C R R + R = R (3.b) = = ow suppose that the structure s excted by the earthquake ground moton acceleraton x () t g, the force vector F () s can be descrbed as F() s = MJX () s C (3.) g 7