Review ArticleSeismic Hybrid Simulation of Stiff Structures:Overview and Current AdvancesStathis N. BousiasDepartment of Civil Engineering, University of Patras, 26504 Patras, GreeceCorrespondence should be addressed to Stathis N. Bousias; sbousias@upatras.grReceived 24 September 2013; Accepted 24 February 2014; Published 27 March 2014Academic Editor: Chris G. KarayannisCopyright 2014 Stathis N. Bousias. Tis is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Advances in the area of structural testing have in recent years led to hybrid simulation, that is, the advanced structural experimentalmethod that encompasses the traditional pseudodynamic testing method and relies on substructuring to ofer the advantage ofcombining the actual experimental testing of selected parts of the structure to the numerical treatment of the rest. Te experimentalpart usually involves simplifed test setups and structural elements with few degrees of freedom. Tus, issues of cross-couplingpresent in testing MDOF structures have not been treated adequately so far. In addition, it has been realized that when it comesto testing very stif structures, in which the above phenomena are accentuated, further problems arise in relation to the quality ofactuator control (accuracy of imposed displacements and stability of the test process). Few studies have focused on these issues,thus necessitating more work in the future. Te present study provides an overview of the approaches that have been adopted sofar, reports on recent advancements, and raises the points in which more research is needed.1. IntroductionExtensiveresearchandapplicationofthepseudodynamictesting method in the 35 years that followed its inception,the computer-actuator online testing method [1, 2], havenot only established it as one of the standard methods forseismic testing, but its application has also been extendedtowards newareas. By leveragingequipment commonlyfoundat structural laboratories, themethodhas provenvery competitive against costly shaking table testing, espe-cially regarding the simulation of full-scale structures thatrespond in the nonlinear regime. With the exception, maybe,of itsapplicationtotestingrate-dependent materialsanddistributed-mass systems, manyof its initial weaknesseshave been successfully treated: implicit or explicit/implicitmethods for the integrationof the equationof motionintimearenowavailable, experimental errorshavebeenminimized/compensated thanks to the deployment of highaccuracy sensors or compensation techniques, and the stag-gered, ramp-hold-type, application of target displacementsthat introduced force-relaxation issues has been substitutedby the continuous movement of the actuators (continuouspseudodynamic testing method).Furthermore, the extension of the method based on theconcept of substructuring (i.e., the discretization of selectedpartsof thestructureundertest intoactivelyinteractingphysical and numerical models (substructures)) forming a,so-called, hybrid model has resulted in creating a valuabletool for testing structures of such large size that could notbe accommodated in present-day laboratories (e.g., bridges).It wasexactlytheconcept of substructuringthat openedup new possibilities and allowed innovative applications ofthemethod, suchas thegeographicallydistributedsub-structured tests and the real-time substructured tests. New,open-source type, sofware tools have also been developedto facilitate the coordination and execution of distributedhybrid simulations. To encompass these developments andthe fact that while in traditional pseudodynamic tests theinertia efects are treated within the analytical part,whilein real-time substructured tests part of the inertia is oferedbythephysical subsystem, thetermhybridsimulationwascoinedinrecent years, principallythroughtheNet-work for Earthquake Engineering Simulation (NEES, http://www.nees.org).Notwithstanding the progress achieved, one cannot claimthat hybrid simulation has overcome all difculties and canHindawi Publishing CorporationJournal of StructuresVolume 2014, Article ID 825692, 8 pageshttp://dx.doi.org/10.1155/2014/8256922 Journal of Structuresberoutinelyusedtoperformseismictestingoffull-scalestructures. Open issues that remain to be treated include(i) multiactuator substructured tests,(ii) hybrid real-time simulation,(iii) testing very stif structures,(iv) geographically distributed tests on MDOF structures,(v) transparency and reliability of distributed-testingemploying existing platforms.Testing stif structures is an area pinpointed as one necessi-tating further work, especially regarding the issue of force-controlled testing. Te present paper provides an overviewof the current state in the area and looks into new methodsproposed for the hybrid simulation of stif structures.2. Hybrid Simulation of Stiff Structures:Issues and ApproachesAmongst the structural elements/systems testedthroughhybrid simulation, those characterized by increased stifnessforma special class, due to a number of difculties associatedwith both their numerical and experimental treatment.One of the frst and well-known issues encountered insimulation algorithms performing the numerical integrationof theequationsof motionof stifstructuresisthat theexplicit time integration methods yield very strict (small-sized) time integration interval requirements, as determinedby the high value of the highest structural natural frequency.Tus, either unconditionallystable(implicit) schemes ormixed ones (explicit scheme for the experimental substruc-ture andimplicit scheme for the physical substructure)must be employed. Nevertheless, even if an acceptably smallintegration time step size can be defned, the resulting dis-placement increments are very difcultfromthe standpointof controlto apply at the controlled DOFs both because oftheir size (being sometimes smaller than the resolution of thetransducer used for the displacement-feedback signal) and ofthe associated measurement errors.If the structure under test is a multi-degree-of-freedomone,then additionaldifculties arise from the tight crosscoupling of the DOFs. If a very stif structure is assumed witha number of displacement-controlled actuators being frmlyattached on the active structural degrees of freedom, then anyindividual actuator adjustment (due to a command signal ordue to a minimal movement around its point of equilibriumwith nil input signal) is instantly transferred through the stifstructure and sensed by the stify coupled DOFs and, hence,from the actuators. Tis minimal action is perceived by eachof the latter as an external action attempting to modify theirown state of equilibrium (note: the actuators are controlledin displacement) and, thus, they react striving to maintaintheir position. Te resulting reaction forces are of very highmagnitude due to the high structural stifness. A possibleworkaround of the actuator cross coupling problem wouldbe to operate them under force control (at least for the partof the response in which the specimen is stif), provided thatthe time-history variationof forces to be appliedat eachactiveDOF is known beforehand and operation of actuators in forceis as reliable as under displacement.An idea for facing tight stability conditions, small dis-placement increments and actuator cross coupling phenom-ena, is to provide additional fexibility in the force-transferchainfromtheactuator tothestructuretoproduceanamplifcation of the command displacement signal before it isbeing sent to the (displacement-controlled) actuator. Insteadof frmly fxing the actuator end to the structure under test, alinear elastic fexible element is provided between the actua-tor and the structure. Tis additional fexible element can berealized as in Seible et al. [3] who used elastomeric pads as thefexible link to transfer the intendedactuator force (actuator isin displacement-control mode) to a 5-story shear wall struc-ture (Figure 1). Te pads, conceived not only for displacementamplifcationbut alsoforbypassingtheproblemof tightcouplingbetweenactuatorsattachedontheactiveDOFs,were inserted between the force-transfer steel beams attachedtotheactuator andthefoor slabs of thestructureandsubjected to normal forces via prestressing bars. It was provedthat, by selecting low-stifness elastomeric pads, the accuracyin imposing the calculated target structural displacementsis indeed better than the actuator precision. Te proposedmethodology (termed sof coupling) was applied withinthe classic pseudodynamic methodanda displacement-basedintegration algorithm. However, it resulted in complicationsin the control algorithm: an iterative procedure had to beemployed seeking the appropriate displacement commandsignals to the actuators that would yield the intended (calcu-lated) structural displacements on the specimen (GeneratedSequentialDisplacementprocedureGSD).Consequently,apart fromthe questions raisedabout the linearity of responseof the elastomeric pads, their connection to the structure inthe presence of high amplitude loads and the stroke capacityof the actuators to apply the required increased displacements(especially when it comes to the sof state of response of thetested structure), the addition of an iterative procedure in thecontrol loop further complicates the solution proposed.AsimilarideawasemployedbySivaselvanetal. [4],with the diference being that in their implementation thereference quantity is the force to be applied and the actua-tor control loop is based on displacement. Tey employedthe concept of added compliance to achieve high qualitydynamic force control for the needs of real-time hybridtesting. For a single-degree-of-freedom system and with aknownforcetobeappliedbyadisplacement-controlledactuator, they employeda spring betweenthe actuator andthestructural mass (Figure 2) to play the role of the fexible link(added compliance). Te strategy, coupled with displacementfeed-forward compensation, appeared adequate for dynamicforce control of the single, lowmass (77.27 kg) structure, but itavoids sending directly forces to the actuator andhas not beenemployed for testing MDOF stif structures also presentingthe known coupling efects.3. Force-Control Hybrid Simulation ProblemsPast studies[37]haveshownthat forceinsteadof thedisplacementshould be a more appropriate commandJournal of Structures 3Reaction wallPrestressing rodsRubber padsConcrete slabActuatorFigure 1: Concept of sof coupling [3].Reference signal (force)ConverterCommand(displacement)Actuator indisplacementcontrolReaction wallFeedback(displacement)SpringelementFigure 2: Concept of added compliance by Sivaselvan et al. [4].signal for the actuator during the part of the response inwhich the specimen is stif (for the sofening branch of thespecimen response, though control has to be appropriatelyswitched to displacement to avoid instability). For employingactuators in force control the following conditions need to befulflled:(i) actuators operate in force control applying calculatedforces with minimal error;(ii) forces to be applied are known a priori, either througha force-based formulation of the hybrid simulationalgorithm(currentlynot available), or throughastifness-basedmodifcationofcalculateddisplace-ments, in case a displacement-based formulation isemployed in the fnite element analysis sofware.Regarding the frst of the conditions above, force is proved tobe difcult to use for closing the servo loop, due to a numberof reasons. At frst, assume that it is possible for the fniteelement analysis sofware to provide the forces to be appliedat thecontrolledDOFs or that calculateddisplacementsare(somehow)translatedintoreferenceforces. Teforcesignal measured by the actuator load cell and subsequentlyusedas the feedbacksignal inthe control algorithmisnot noise-free. A number of sources contribute to the lowqualityoftheforcesignal (friction,break-awayforcesonpiston seals, and stick slip) leading to a degraded actuatorperformance. It is also obvious that tuning of a controllerwhich commands an actuator in force is stifness-dependentand requires the availability of a very stif elastic test piece(other than that used for the purposes of the research). Evenif this condition is satisfed, the results are valid only for thatstifness the controller was tuned for and will sufer fromlossof smoothness if the specimen under study is characterizedby changing stifness.Another difculty for performing force-controlled hybridtests stems fromthe fact that evenif forces could bedetermined via a force-based integration scheme, they wouldbe then expressed in the same coordinate system in whichequations have been formulated, that is, the global coordinatesystem. However, actuators can realize command forces (anddisplacements) only along their longitudinal axis, with thelatternotnecessarilybeingalignedtoanyof theaxesoftheglobal coordinatesystemduringatest. It isobviousthat a transformation is needed in this case, similar to thatemployed for obtaining the actuator displacement commandfrom a displacement-based algorithm. While displacementtransformation is straightforward using kinematic compati-bility at nodes, when it comes to the case of forces, however,such transformation does not provide a unique solution forthe forces to be applied to the physical (sub)structure.Anevenharderproblemtofaceisthecontrol oftheactuators themselves in force mode.With the mechanicalimpedanceof asystembeingdefnedastheratioof thepotential (e.g., force) at a point to the resulting velocity atthat point, actuators(asvelocitysources)are, bydesign,high impedance systems (mechanically stif) ofering goodposition control [4].Te high structural stifness leads totheselectionofalowproportionalgaininthecontrollerand with an increasing structure-to-actuator stifness ratiothe response is characterized by increasing risk of oscillatorynature, that is, instability (the poles of the systemmove closerand upwards to the imaginary axis).4 Journal of StructuresTe second condition for performing hybrid simulationvia force-driven actuators, that is, that of the a priori avail-ability of the time history of forces to be applied at eachcontrolled DOF,requires the formulation of aforce-basedtimeintegrationalgorithmforthehybridsimulationandthe transformation of the forces determined in the globalcoordinate system to forces in the actuator reference system.Suchformulationof general applicationis completely lacking,possibly because the development of schemes directly pro-viding target forces is compromised by the difculties arisingin running the actuators under force control. Te only eforttowards a force-based method is that developed by Kim andreported by Whyte and Stojadinovic [8]; detailed informationon the formulation is lacking and no data regarding actualapplication of this approach are available. With the exceptionof this study, the preferred testing methodology remains thatof employing existing displacement-based formulations todetermine target forces to be applied via displacement- orforce-driven actuators. As fnite element analysis codes andall currently available hybrid simulation platforms (Simcor,OpenFresco, Netslab, ISEE, etc.) aredisplacement-based,focus is directed on methods of obtaining target forces fromcalculateddisplacements. Teavailableapproachesfallinginthiscategoryareknownascompatibilitymethodsforobtaining target forces and are summarized in Figure 3, witha more detailed description given in the following.4. Target Force Determination4.1. Stifness-Based Approaches. Tese approaches comprisethose by Pan et al.[9],Nakata et al.[10],Elkhoraibi andMosalam[6], Igarashi et al. [5], Ahmadizadeh and Mosqueda[11], and Hung and El-Tawil [12]all are based on the deter-minationof the stifness matrix to be used withdisplacementsobtained from the time integration process to yield targetforces.Pan et al. [9] performed hybrid simulation on a base-isolated building by substructuring the isolators as physicalspecimens. Loading along the vertical direction was realizedon the isolator by a force-control loop only for the stif stateof the isolator, that is, the compression, while for tension (sofstate) the control switched to displacement. Vertical forcesto be applied were determined from displacements via theelastic stifness of the bearing.Tis defnes also the maindrawback of the method; that is, it is not valid for the caseof nonlinear (and unknown) variation of stifness.Nakata et al. [10] studied the problem of concurrentlycontrolling a number of actuators used to load specimenswith prescribed target displacements and forces along dif-ferent axes. As the servo loops of all actuators were closedwith displacement feedback, a procedure was developed toderive displacement commands for the actuators in load-controlled axes. Specifcally, the study examined the responseof reinforced concrete columns under imposed cyclic lateraldeformations at the presence of constant axial compression.In the load-controlled axis (direction in which a prescribedforce should be achieved), the displacement command signalto the actuator during the th iteration of step is obtainedby

() =

(1) + (1)(

(1)) , (1)where

represents the command displacement signal, is again matrix, (1) is the updated stifness afer the ( 1)thiteration,

is the target force, and

is the measured forceat ( 1)th iteration. Afer the th iterative ramp is realized,displacements and forces are measured and their respectiveincrements are determined:

() =

()

(1),

() =

()

(1). (2)Afer the th iterative ramp, the secant stifness is estimatedvia the Broyden update of the Jacobian [13]:

() = (1)+ (

(1) (1)

()) (

())

()

2,(3)where

represents the measured displacement afer the thiterative ramp. Convergence to the prescribed force targ

isachieved when

targ

meas(1)

< tolerance. (4)Temethodhas beensuccessfullyincorporatedintothecontrol sofware of the NEES MUST-SIM facility at UIUC[10] to cope with issues related to their mixed displacement-load-controlled specialized equipment.Another secant-stifness-based methodology yieldingtarget forces is that by Elkhoraibi and Mosalam [6]. For asystem with uncoupled DOFs, secant stifness at each DOFwasdetermineddirectlyusingtheincremental measuredforce and displacement during the iterations within a timestep ( + 1) and the respective converged values at the endof previoustimestep (). Toovercometheproblemsofthe resolution of measuring sensors and random noise inmeasurements, they used the average of a number (100) ofconsecutiveforce/displacementmeasurementstocalculatethe average stifness. Tey reported satisfactory results, butthe method has not been further verifed for stif MDOFstructures presenting coupling between the degrees of free-dom.Te approaches by Pan et al. [9], Nakata et al. [10], andElkhoraibi andMosalam[6]aresecant-stifness-basedasopposed to those by Igarashi et al. [5], Ahmadizadeh andMosqueda [11], and Hung and El-Tawil [12] which target todetermine the tangent stifness. To perform hybrid simula-tion on a stif5-story full-scale building Igarashi et al. [5] useddisplacement feedback for the servo loop of the actuatorsand the Broyden-Fletcher-Goldfarb-Shanno (BFGS) methodfor tangent stifness updating during the test. At step , theincremental displacement and force measured at each DOFare related to the tangent stifness matrix act

via

= act

. (5)An incremental stifness is determined by the algorithm toupdate the stifness matrix and a parameter representing theJournal of Structures 5Hybrid simulationDisplacement-based formulationExperimentalsubstructureNumericalsubstructureForces determined viaForce-based formulationmodifed dynamic equilibrium(Whyte and Stojadinovic)Actuator in forcefeedbackActuator indisplacement feedbackActuator in forcefeedbackStifness-free approaches- Krylov subspace (Scott and Fenves)- Palios et al.Determination of forcesStifness-freeapproachesStifness-basedapproaches- Krylov subspace (Scott and Fenves)- Palios et al.Secant stifness methodsBroyden update (Nakata et al.)Tangent stifness methods- BFGS method (Igarashi et al.)- Transpose method (Hung et al.)- Intrinsic coordinate system method (Ahmadizadeh and Mosqueda)Figure 3: Force determination methods in displacement-based hybrid simulation.fdelity of the testing system is employed, so that spuriousupdating of stifness is avoided. Te incremental stifness isobtained as1act

= [1 +

act

]

act

act

,2act

= [1 +

act

]

act

act

,

act+1 = act

+ 1act

, if

1act

<

2act

,

act+1 = act

+ 2act

, if

1act

>

2act

.(6)As has been shown by Ahmadizadeh and Mosqueda [11], theBroyden update approach may result in a nonsymmetric ornonpositive defnite stifness matrix and with a slow stifnessupdate rate, while the BFGS method sufers from consider-able distortions being introduced by noise. In general, resultsfrom simulations indicated that both the Broyden and theBFGS methods do not exhibit sufcient accuracy over thatresulting from the use of the initial stifness matrix. To thatend, AhmadizadehandMosqueda[11]oferedadiferentapproachinestimatingthetangentstifnessmatrix. Teyuse the notion of intrinsic coordinate system (determinedfor each experimental setup and specimen) to express themeasured displacements/forces (in actuator coordinate sys-tem) and ultimately obtain a (diagonal) stifness matrix in theintrinsic coordinate system,

:

=

act

,

=

+

(act1)1

,

act

=

.(7)Te latter is then expressed in the global coordinate systemthroughanappropriate transformation. Althoughthe authorsprovidedanexampleof applicationof themethodona6 Journal of Structuressimple, two-DOFspecimen, the procedure may become moreintricate for more complex setups.Based on small-scale tests on a simplifed setup (NEES),it was shown that the method ofers a far better estimationof the tangent stifness from the point of view of accuracycompared to the Broyden or the BFGS approach. To present,themethodhasnotbeenverifedfortestingstifMDOFstructures.Althoughnotdevelopedforhybridsimulationofstifstructures but rather to improve the integration schemes (aswith [11]),the methodology by Hung and El-Tawil [12] isa nonsecant approach for determining the tangent stifnessmatrix of MDOF structures. It employs past -steps duringthetest toformulatearelationshipbetweenincrementalforces and displacements, based on the assumption that thestifness does not change substantially within a number of previous steps ( is defned by the user, but in general DOF). A unique solution is obtained if is equal to thenumber of DOFs, while least-squares approximation must beemployed when >nDOFs. Te method has been examinedonly numerically (no physical testing was performed) and forstructures with hardening response.4.2. Stifness-Free Methods. As stifness-based methods suferfrom noise in the measuring sensors causing contaminationin force/displacement measurements and spurious updatesof stifness,alternative approaches have been investigated.Scott and Fenves [14] used a Krylov subspace as the basis fordetermining target forces fromdisplacements available by theintegration process. Nonetheless, the method uses the initialstifness at some point of the process (or can employ tangentstifness obtainedbyoneof theprevious methods). Temethod has been implemented in the OpenFresco platform[15] but no actual application has been reported.Te approach by Scott and Fenves [14] along with someof the tangent-stifness-based approaches reported above hasbeenimplementedinOpenFrescobyKimetal. [15]andthe NEES experimental setup was used for verifcation andcomparison. Indicative results seem to sustain that the BFGSmethod yields better results than those by the rest of themethods, aslongasthecontrollingvariableisnotforce.Nonetheless, the results of Ahmadizadeh and Mosqueda [11]obtained from the same test setup indicated that both BFGSand Broyden methods do not ofer clear advantage over usingthe initial stifness matrix, as far as accuracy is concerned.A more recent, stifness-free method, is that by Palioset al.[16]the method strives to tackle concurrently twoproblems, that of determining the appropriate target forceand that of actuator coupling efects. Te method functionsunder the usual displacement-based formulationand is basedon the use of two controllers per actuator, in an inner-outerloop confguration: the frst controller is assigned with theconversionoftargetdisplacementintotargetforce, whilethe second one is responsible for driving the actuator underforce control. At each sampling time interval of the controlsystem, new target displacement increments are calculatedby the integration scheme and sent to the frst controller.Te displacement-feedback signal is compared to the targetdisplacement and the error is used within the frst controllerto produce a signal that is sent as command to the secondcontroller. As the latter is the one actually driving the actuatorin force, the signal received from the frst controller is scaledso that its maximum value corresponds to the full capacityof the actuator used. Te resulting signal is the servovalvecommand to drive the actuator in force control. Althoughthe method is the only one to have been applied for testingaverystifreinforcedconcretewall andshowntoyieldaccurate results, further research regarding its stability limitsis required.4.3. Applications inActual Testing. Most of themethodspresented above have been applied to rather idealized spec-imens of few degrees of freedom, as focus was mainly on thedevelopment and analysis of the relevant methods. Te fewcases of actual application of seismic hybrid simulation totesting real-scale stif structures are those by Seible et al. [3],Igarashi et al. [5], Whyte and Stojadinovic [8], and Palios et al.[16]. In the frst two studies (the only ones on actual MDOFstifstructures) the tangent stifness approachwas usedandwhiletheexperimental error (thediferencebetweencommanded and measured displacements) was shown to bewithinacceptablelimitsforthelowexcitationlevel tests,increasing seismic input level induced higher displacementerror, owing to the nonlinearity of the sof coupling system.Several of the difculties encountered in these tests due tothe analog controlsystemand the poor resolution of thedisplacement-feedback transducers are deemed to have beensolvedbythetechnological evolutionsintheperiodthatfollowed these tests.Te physical components in the hybrid simulation testsperformed by Whyte and Stojadinovic [8] were representedby squat shear walls subjected to seismic input of increasinglevel. Te tests were actually performed under displacement-feedback control employing high-resolution digital displace-ment transducer. As the specimencomprisedonly onedegree-of-freedom, nostifcouplingwaspresent andtheuseof forcecontrol wasnot necessary. Accuratecontrolwas partially compromised by the equipment employed (slipinactuator clevis) andwas solvedintroducingaspecialprocedure (constant velocity control).Te approach by Nakata et al. [10] was employed by Hartet al. [17] in hybrid simulation testing of a series of three-storyshear walls in 1 : 3 scale, via the secant-stifness approach. Tetest results have shown that accurate response is obtainedalong the load-controlled axes of the specimen.In the seismic hybrid simulation of four-story shear walltests performed by Palios et al. [16] at a 3 : 4 scale (Figure 4),the dual controller approach was used with a target displace-ment resulting from a displacement-based formulation andthe actuator control loop closing in force. Testing of this stifspecimen was performed with an accurate force control asevidenced by the negligible diference between target andmeasured forces (error) observed.5. ConclusionsTe objective of this study is to address the issue of testingverystifstructureswithinthecontext of seismichybridJournal of Structures 7(a)0 2 4 6 8 10 120.0250.020.0150.010.00500.0050.010.0150.020.025Time (s)DoF2 force error (kN)(b)Figure 4: (a) Test setup and (b) force error in force-controlled actuator.simulation. A number of currently employed methods werereviewedtoidentify their pros andcons. Te commongroundis that, while force should be the controlling variable of theactuator servo loop, it has not been so far possible to performaccurately controlled testing with actuators in force control. Itis obvious that more efort is needed along this direction andinvestigation of alternatives within this particular applicationarea, for both low and high force amplitude/frequency.Regardless of the current difculties in testing very stifstructures in control of force, the issue of how to obtain theforces to be applied from present-day integration schemes(which are all displacement-based) remains. Te approachesin this area can be broadly categorized as stifness-based orstifness-free, with the majority falling into the frst category.Te fact that these methods have been primarily developedfor improving the time integration schemes and not so muchfor being used for force-controlled testing has not allowedthe thorough investigation of their potential on actual stifstructures. Another point requiring further work regards thedevelopment of morestifness-freemethods, asawaytoovercome problems associated with measurement errors thatinfuence negatively all stifness-based approaches.Conflict of InterestsTeauthordeclaresthat thereisnoconfict of interestsregarding the publication of this paper.References[1]M. Hakuno, M. Shidawara, and T. Hara, Dynamic destructivetest of a cantilever beam controlled by an analog computer,Proceedings of the Japan Society of Civil Engineers, vol. 1969, no.171, pp. 19, 1969.[2]K. Takanashi, K. Udagawa, and H. Tanaka, Earthquakeresponse analysis of steel frames by computer-actuator on-linesystem, in Proceedings of the 5th Japan Earthquake EngineeringSymposium, pp. 13211328, Architecture Institute of Japan (AIJ),Tokyo, Japan, 1978.[3]F. Seible, G. Hegemier, andA. Igarashi, Simulatedseismiclaboratory loadtestingof full-scale buildings, EarthquakeSpectra, vol. 12, no. 1, pp. 5786, 1996.[4]M. V. Sivaselvan, A. M. Reinhorn, X. Shao, and S. Weinreber,Dynamic force control with hydraulic actuators using addedcompliance and displacement compensation, Earthquake Engi-neering and Structural Dynamics, vol. 37, no. 15, pp. 17851800,2008.[5]A. Igarashi, F. Seible, and G. A. Hegemeier, Development ofthe pseudodynamic technique for testing a full-scale 5-storyshear wall structure, in Proceedings of the U.S. Japan Seminaron the Development and Future Directions of Structural TestingTechniques, Honolulu, Hawaii, USA, 1990.[6]T. Elkhoraibi and K. M. Mosalam, Towards error-free hybridsimulation using mixed variables, Earthquake Engineering andStructural Dynamics, vol. 36, no. 11, pp. 14971522, 2007.[7]H. Kim, Extending hybrid simulation methods in OpenFrescosofware framework, Tech. Rep. CE-299, University of Califor-nia, Berkeley, Calif, USA, 2009.[8]C. A. Whyte and B. Stojadinovic, Hybrid simulation of theseismicresponseof squat reinforcedconcreteshear walls,PEER Report 2013/02, 2013.[9]P.Pan,M.Nakashima,and H.Tomofuji,Online test usingdisplacement-force mixedcontrol, Earthquake Engineering andStructural Dynamics, vol. 34, no. 8, pp. 869888, 2005.[10]N. Nakata, B. F. Spencer, and A. S. Elnashai, Mixedload/displacement control strategy for hybrid simulation, inProceedings of the 4th International Conference on EarthquakeEngineering, Taipei, Taiwan, 2006.[11]M. Ahmadizadeh and G. Mosqueda, Hybrid simulation withimprovedoperator-splitting integrationusing experimentaltangent stifness matrix estimation, Journal of Structural Engi-neering, vol. 134, no. 12, pp. 18291838, 2008.[12]C.-C. Hung and S. El-Tawil, Amethod for estimating specimentangent stifness for hybrid simulation, Earthquake Engineeringand Structural Dynamics, vol. 38, no. 1, pp. 115134, 2009.[13]C. G. Broyden, Aclass of methods for solvingnonlinearsimultaneous equations, Mathematics of Computation, vol. 19,pp. 577593, 1965.[14]M. H. Scott and G. L. Fenves, A krylov subspace acceleratornewtonalgorithm, inProceedings of the 4thInternationalConference on Earthquake Engineering, Taipei, Taiwan, 2006.[15]H. Kim, B. Stojadinovic, T. Y. Yang, and A. Schellenberg, Alter-native control strategies in hybrid simulation, in Proceedingsof the 2nd EFAST Workshop & 4th International Conference onAdvances in Experimental Structural Engineering, JRC, ELSA,Ispra, Italy, 2011.8 Journal of Structures[16]X. Palios, E. Strepelias, and S. N. Bousias, A novel strategyfor the hybrid simulation of stif structures, in Proceedings ofthe 5th International Conference on Advances in ExperimentalStructural Engineering, Taipei, Taiwan, 2013.[17]C. R. Hart, D. A. Kuchma, L. N. Lowes, D. E. Lehman, K. P.Marley, and A. C. 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