Implementation of online model updating in hybrid simulation
M. Javad Hashemi1, Armin Masroor1 and Gilberto Mosqueda2,*,†
1Department of Civil, Structural, and Environmental Engineering, University at Buffalo, Buffalo, NY 14260, USA2Department of Structural Engineering, University of California, San Diego, CA 92093, USA
Received 12 December 2012; Revised 22 June 2013; Accepted 9 July 2013
KEY WORDS: hybrid simulation; substructuring; system identification; online model updating; unscentedKalman filter
1. INTRODUCTION
In hybrid simulation, it is often assumed that a reliable model exists for the numerical substructures,while the experimental substructures consist of components that are more difficult to model. Theapplication of hybrid simulation has, thus, been ideal for structures with complex nonlinear behaviorconcentrated in specific regions of the structure [1]. An ideal example is the seismic response of a baseisolated building where the superstructure typically remains linear and can be modeled reliably, whilethe nonlinear response is concentrated in the isolation bearings that can be physically tested [2, 3].Additionally, hybrid simulations have been shown to replicate the results of shaking table test for astructure tested to collapse that developed a mechanism in the first story [4] by testing experimentallyonly the lower story of the structure [5, 6] where the most severe damage was concentrated. In general,there are many components distributed throughout the structure with similar properties and complexseismic behavior that are not ideal for substructured hybrid simulation. In fact, an objective of seismicdesign for a conventional framed structure is to achieve distributed damage such as to avoid a weak
*Correspondence to: Gilberto Mosqueda, Department of Structural Engineering, University of California, San Diego,CA 92093, USA.†E-mail: [email protected]
story mechanism that can lead to collapse. Testing the entire structure without a substantial reduction inscale may be impractical because of the limitations of laboratory equipment and/or the expense associatedwith large-scale models.
To simulate the behavior of structures with distributed damage and improved models for numericalsubstructures, an online updating scheme is implemented in hybrid simulation. In this approach, one ormore substructures expected to be the first to experience large deformations and damage is tested in thelaboratory, while substructures with similar properties are modeled numerically with adjustablecharacteristic parameters. During the simulation, adjustable parameters corresponding to numericalsubstructures similar to the experiment are updated based on an online calibration procedure usingthe available measured data. The updated numerical model represents the behavior of the numericalsubstructure calibrated to the experimental substructure response under the specified loading as thetest progresses recognizing that they do not always experience an identical load history. In general,the restoring force depends not only on the instantaneous loading but also on the past loadinghistory. Therefore, the parameters to be updated need to be carefully selected considering thatlimited data are available early in the test.
In recent years, many techniques have been developed and evaluated for nonlinear hysteretic modelidentification, including least squares estimation [7], the extended Kalman filter [8, 9], the unscentedKalman filter (UKF) [9, 10] and Particle Filter (also known as sequential Monte Carlo methods)[11]. Also, the high gain techniques in control systems engineering have been reported to beefficient in state estimation [12]. Model updating techniques have been used in finite-elementanalysis over the last two decades to improve the predicted behavior of the actual structure byidentifying and correcting uncertain parameters. Computational methods have been developed toupdate the finite-element model to better predict measured results [13]. While modeling andidentification of hysteresis models representing structural element behavior in extreme loadingconditions is challenging for offline applications, recently developed state observer such as the UKFmakes online or even real-time model updating possible for such nonlinear behavior. The objectiveof this study is to implement this technique in online applications for hybrid simulation.
Model updating specifically in hybrid simulation has been discussed by Yang et al. [14] to improvethe consistency among substructures and has been applied to elastic systems by Wang and Wu [15].For parameter identification of nonlinear systems, Yang et al. [16] proposed to use the Nelder–MeadSimplex Method (NM) [17] to update the numerical model in their applications to hybridsimulation. Studies by Song [18] show the potential of the UKF and its performances in onlinenonlinear system parameter estimation. In this study, both the NM and the UKF method areexamined, with the UKF applied for hybrid simulations with online updating because of its fasterprocessing time and rate of convergence.
2. IMPLEMENTATION OF ONLINE UPDATING IN HYBRID SIMULATION
2.1. Numerical model updating technique
The development of practical numerical model that can predict experimental nonlinear structuralbehavior is a major challenge in earthquake engineering. This is not surprising given the variety andcomplexity of structural element behavior with degradation. Good correlations between experimentaland numerical results are typically only obtained after calibration of a specific model to theexperimental data. It should be mentioned that model calibrations are often performed after theexperiment has been completed with all data available from the test. In hybrid simulation, there is anopportunity to update the numerical model parameters as the test progresses. It is more difficult toupdate the numerical model in the proposed hybrid simulation application because new data becomeavailable from the experimental substructures as the test progresses. Additionally, there may not beany initial data available to predict the structural response in the nonlinear range including simpleparameters such as yield strength.
To demonstrate the implementation of online updating for numerical models, Figure 1 compares thehysteresis of an experimental test and a simple bilinear model with initial values of 2.8 kN for the yield
strength, 0.1 for post elastic stiffness ratio and 1.015 kN/mm for the elastic stiffness. There is nostiffness or strength degradation parameter in this numerical model; thus, it has difficulty trackingthe experimentally measured response.
To demonstrate the benefits of online updating, the NM Method was used as an optimizationtechnique to update the numerical model parameters. For computation efficiency, updating is onlyapplied at specific time steps in the simulation where the difference between the numerical andcorresponding experimental force value exceeds a certain threshold. In this particular example,model updating empowers the numerical model to track the degradation behaviors while the originalnumerical model with fixed parameters is not able to do so. Figure 2 compares the updated bilinear
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Figure 2. Comparison of an updated bilinear model with experimental response.
IMPLEMENTATION OF ONLINE MODEL UPDATING IN HYBRID SIMULATION 397
numerical model with the same initial parameters to the experimental response. The evolution of theyield force, elastic and post elastic stiffness are shown in Figure 2(b). The parameters are updatedusing experimental data available only up to that point. In this example, the model parameters areupdated at nine instances through the simulation. This example shows that the numerical model isable to adapt to capture the measured behavior even though degradation is not inherent in themodel. However, it should be noted that in this particular case, it is assumed that both theexperimental and the numerical models are subjected to the same load history. Variations in loadhistory are examined later.
2.2. Model updating in hybrid simulation
Model updating can be advantageous in hybrid simulation of large structural systems with distributeddamage. To illustrate this process for the various types of substructures in hybrid simulation, anexample is presented for a bridge structure with two short piers supporting bearing Elements A andD and two long piers with bearing Elements B and C as shown in Figure 3. It is assumed that thebehavior of the isolation bearings on the short piers is more difficult to simulate numerically becausethey will experience larger deformations under the stiffer substructures. Therefore, bearing ElementD is selected for testing in the lab. The remaining bearings, two short piers, two longer piers andother components of the bridge are modeled numerically. It is assumed that the experimental bearingElement D experiences larger demands ahead or concurrent to the numerical bearing Elements A, Band C. The reason for this assumption is that the obtained information from the experimentalsubstructure is used to update the numerical substructure models. The experiment needs toexperience a particular type of behavior or damage first in order for this behavior to be calibratedinto the numerical models.
The numerical elements could be divided into three categories depending on their similarity to theexperimental elements and the relative level of demands experienced.
(1) The elements of the numerical model with properties identical to the tested substructure andexperiencing identical or very similar loading history (Bearing Element A).
(2) The elements of the numerical model with properties identical of the tested substructure but notnecessarily experiencing the same loading history. These numerical models can benefit fromupdating based on the measured experimental response assuming the experimental elementexperiences larger demands and is damaged first (Bearing Element B and C).
(3) The remaining parts of the numerical substructure, which are not updated because a similarcomponent is not tested experimentally as part of the simulation (the bridge deck and piers).
Figure 3. Illustration of the application of model updating in hybrid simulation to a complex structural model.
For the first category of numerical elements, the experimental response could be used directly toupdate the model parameters of these numerical elements because they are expected to have similarbehavior and level of damage. For instance, if Element D fails completely, the related parameterssuch as ultimate displacement is extracted from the experimental data and applied to the numericalmodel of bearing Element A, instantaneously, as it is expected that this element will failsimultaneously or immediately after bearing Element D. If the load history is completely identical,then the restoring force of Element A can be replaced with the measured force for Element D.
The second category of numerical elements is different from the first in that the loading history canbe different from the experiment and there may be slight physical difference. In general, numericalmodel parameters can be updated with some conversion or modification of parameters necessary tomake the experimental substructure data applicable in the case that the numerical element is slightlydifferent or scaled. In addition, time of updating should occur at a proper time. For instance, if theshort pier isolator (Bearing Element D) fails, this phenomena can be considered instantaneously forthe bearing Element A. However, for bearing Elements B and C, the ultimate displacement shouldbe updated in the model such that failure can be experienced at a similar deformation.
An important consideration in the implementation of online updating, particularly for elements witha different loading history, is that the updating parameters should be independent of the instantaneousdeformation of the elements. For example, the tangent stiffness of the numerical element cannot beinstantaneously updated based on the current tangent stiffness of the experimental element. Instead,the hysteretic model parameters for stiffness degradation can be updated so that the numerical modelexperiences similar behavior at similar demand levels. Likewise, the backbone parameters can beupdated such that the numerical model displays similar strength degradation.
2.3. Experimental challenges
In post-experiment parameter identification, there are usually no constraints on computational and dataprocessing time. However, the online identification process as applied here should instantaneously andautomatically track the critical and desirable characteristics of the components. Note that the hybridsimulation would need to be paused if the computations time is slow or any manual input isrequired. Also, there are concerns related to experimental measurements and noise contaminationthat can substantially influence the accuracy of the identification result. In addition, contrary tooffline approaches, it is difficult to manipulate the input–output data in online schemes.
2.4. Numerical challenges
For effective online identification schemes, it is necessary to apply appropriate nonlinear models thatare able to represent the measured system behavior. In addition, problems related to under- andover-parameterization exist as some parameters may not be reliably calibrated based on availabledata. Also, independent of the system to be identified, the online identification algorithm must becapable of converging smoothly and rapidly to the proper parameter values in order to captureparameter changes as time progresses. Therefore, it will be necessary to use novel techniques toconstruct algorithms and identify parameters that are important to capture structural behavior.
3. ALGORITHM FOR UPDATING IN HYBRID SIMULATION
The implementation of online updating in hybrid simulation requires only software modifications to aconventional setup in the laboratory. Figure 4 shows a hybrid test framework and identifies newsoftware modules to accommodate updating. Specifically, four modules are added as discussed next.Note that there could be multiple substructures of each type in a hybrid simulation.
3.1. Auxiliary numerical model
Auxiliary numerical models can be used for different purposes in model updating during a hybridsimulation. They experience a displacement history identical to the corresponding experimental
IMPLEMENTATION OF ONLINE MODEL UPDATING IN HYBRID SIMULATION 399
substructure, and the model parameters can be directly calibrated to capture the experimental behavior.The resisting forces and hysteretic behavior of the auxiliary model is passed on to the Inspector toevaluate this model relative to the measured response. Required updates are then applied on theauxiliary numerical model, and selected parameter values can be passed on to numerical models ofsimilar components in other parts of structure.
3.2. Inspector
The Inspector determines if updating of parameters is necessary based on error monitoring between theauxiliary model and experimental measurements. Its main role is to identify when the modelparameters need to be updated. For example, suppose an unexpected change in the response of anexperimental element is recognized during the test, such as a fracture that leads to a significant lossof strength. This will trigger a large error between both models, indicating the need for updating,and the analysis can be continued until sufficient information is obtained for proper updating.
3.3. Parameter identification
Parameter identification is a critical part of the updating process. The parameter identification toolboxfinds updated parameters based on comparing the auxiliary numerical model and experimental elementresponse data. The objective of parameter identification is to extract the best-fit parameters from theexperimental response history. An optimization process, which typically involved iteration, is usedto find the best parameters.
3.4. Reporter
The Reporter identifies the time of updating and modifies the parameters if needed. The time ofupdating is the instant that the adjusted parameters are applied to the numerical model, which is notnecessarily identical to the time that inspector triggers the updating process or current step of theanalysis. For instance, the numerical model may be updated only at the zero-force crossing to avoidnon-smooth hysteresis.
Figure 4. Framework for model updating in hybrid simulation.
Before implementing online updating in hybrid simulation, the user should carefully study thesensitivity of the numerical models, the parameters that are to be updated, the time of updating andthe criteria to be used to trigger the updating among others, and should be customized for eachspecific test based on the test objectives. In this study, a smooth hysteretic model with degradationis updated online using the data measured from the test specimen.
4. HYSTERESIS MODEL
A smooth hysteretic material model developed by Bouc [19] and Wen [20] with degrading behavior asimplemented by Baber and Noori [21] is considered in this study. This model is highly nonlinear andhas nine control parameters including stiffness and strength degradation.
The stress is defined as the sum of a linear part and a hysteretic part:
σ ¼ αk0 εþ 1� αð Þk0 z (1)
In the previous equation, ε is the strain, z represents the hysteretic deformation, k0 is the elasticstiffness and α is the ratio of the post-yielding to elastic stiffness. To accommodate degradation,Baber and Noori formulated the rate of hysteretic deformation in the form:
z ¼A ε� β εj j z zj jn�1 þ γ ε zj jn
n oν
η(2)
where, β, γ and n are parameters that control the shape of the hysteretic loop, while A, ϑ and η arevariables that control the material degradation. The model may be rewritten as
z ¼ A� zj jn βsgn ε zð Þ þ γf g νη
ε ¼ ∂z∂ε
∂ε∂t
(3)
This leads to the following expression for stiffness:
k ¼ ∂σ∂ε
¼ αk0 þ 1� αð Þk0 A� zj jn βsgn ε zð Þ þ γf g νη
(4)
It is seen that the stiffness is composed of a linear term and a hysteretic contribution. The evolutionof material degradation is governed by the following choice of equations:
A ¼ Ao � δAe
ν ¼ 1þ δνe
η ¼ 1þ δηe
(5)
where e is defined by the rate equation
e ¼ 1� αð Þk0 ε z (6)
In the above equations Ao, δA, δν and δη are user-defined parameters. This hysteresis model possessesimportant features for this study, which include inherent strength and stiffness degradation parametersthat can be calibrated for models experiencing different load histories or changes in behavior atdifferent times. The role of each parameter is summarized in Table I.
IMPLEMENTATION OF ONLINE MODEL UPDATING IN HYBRID SIMULATION 401
The objective of online model updating is to better predict structural behavior by reducing thedifferences between the numerical model and the experimental substructure response of similarelements. The measured response of the experimental elements in each step is a reliable data sourcethat can be utilized to update numerical models and improve their accuracy. System identificationtechniques implemented in this study are parametric identification procedure for which themathematical model of the system is assumed to be known from theoretical considerations, and onlythe values of model parameters are to be determined from experimental data.
Using a parametric approach, several researchers have used a smooth hysteretic model forsimulating and identifying hysteretic systems [22]. Yang et al. [16] used NM to update thenumerical model in their applications to hybrid simulation. However, in evaluations implemented aspart of this study, it was found that NM was not ideal for online model updating in hybridsimulation because of the required computational time. For every updating step, the actuators in thehybrid tests had to pause or hold until the computational tasks were completed. For this reason, onlya limited number of steps were updated while using the NM, which could lead to a non-uniformhysteresis. It was also observed that the computation time increased significantly as the number ofupdating parameters increased. Therefore, this method was found to be appropriate only for basicparameters in very simple numerical models with a limited number of updating steps and whenthere is no computational time constrains.
Previous studies [18] show the potential of the UKF and its performances in nonlinear systemestimation for updating parameters in a modified Bouc–Wen model with degrading stiffness andstrength. UKF is a recursive algorithm for estimating the optimal state of a nonlinear system fromnoise-corrupted data. This method averages a prediction of a system state with a new measurementusing a weighted average. The weights are calculated from the covariance that shows the estimateduncertainty of the system state prediction. As a result, the weighted average is a new state estimatethat lies between predicted and measured states, having a better estimated uncertainty. Therefore,this method requires only the last estimate of the system parameters rather than the entire history tocalculate the new parameters. To accommodate this need, the UKF is applied in every time step,although the numerical model parameters are not necessarily updated in each step.
To identify the unknown parameters of a system, these parameters should be added to the states ofthe system to be estimated using measurement data. In what follows, the UKF formulation isintroduced to explain the procedure used in online parameter updating. Consider a discrete timenonlinear stochastic system:
xkþ1 ¼ f xk; uk;ωk; kð Þyk ¼ h xk; kð Þ þ νk
(7)
where the vectors xk, yk are states and measurements, and ωk, νk are process noise and measurementnoise. Unlike the extended Kalman filter, which linearizes a process function around the estimatedstate, UKF approximates mean and covariance of a random vector obtained from a nonlineartransformation. For this purpose, the state vector is converted into 2L+ 1 vectors (L is dimension ofthe state vector) and processed into the function. Then the mean and covariance are calculated using
Table I. Numerical model parameters.
Parameter Parameter description
k0 Elastic stiffnessα The ratio of the post-yielding to elastic stiffnessA, δA, β, γ Basic hysteresis shapen Sharpness of the yieldδv, δη Strength and stiffness degradation
weighted transformed points. The UKF algorithm used to identify parameters in this study is presentedin what follows. A state vector is defined as
x ¼ z e k0 α β γ n δν δη� �
(8)
The first two arrays of this vector are states of the Bouc–Wen constitutive material model, and therest are parameters defined before. The initial state covariance matrix P0 is computed consideringinitial variation in each state and parameter. The UKF uses a deterministic sampling techniqueknown as the unscented transform to pick a minimal set of sample points, called sigma points,around the mean. At time step k, the sigma points are computed by
X k ¼ xk xk þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLþ λð ÞPk
in which a and κ are constant parameters that determine the spread of the sigma points. Parameter awas taken as 1e� 4, and κ is set to 3� L. The time update step is performed using calculated sigmapoints
X kþ1jk ¼ f X k; uk;ωk; kð Þ (11)
The function f is the discrete form of the Bouc–Wen constitutive material model presented inEquation 1. The predicted state vector x�kþ1 and its predicted covariance P�
kþ1 are calculated usingcorresponding weight factors
x�kþ1 ¼ ∑2L
0Wm
i X i;kþ1jk
P�kþ1 ¼ ∑
2L
0Wc
i X i;kþ1jk � x�kþ1
� � X i;kþ1jk � x�kþ1
� �T þ Q
(12)
in which Wmi and Wc
i are weight factors for mean and covariance and Q is process noise covariancematrix (ωkωk
T). The weight factors are given by
Wm0 ¼ λ
Lþ λ; i ¼ 0ð Þ
Wc0 ¼
λLþ λ
þ 1� a2 þ b� �
; i ¼ 0ð Þ
Wmi ¼ Wc
i ¼1
2 Lþ λð Þ; i > 0ð Þ
(13)
where b is another scale constant used to incorporate prior knowledge of the distribution of thepredicted state vector. For the Guassian distribution, b = 2 is optimal.
IMPLEMENTATION OF ONLINE MODEL UPDATING IN HYBRID SIMULATION 403
The predicted measurement vector and its predicted Pyykþ1 are calculated as
Ykþ1jk ¼ h X i;kþ1jk; k� �þ νk
y�kþ1 ¼ ∑2L
0Wm
i Ykþ1jk
Pyykþ1 ¼ ∑
2L
0Wc
i Yi;kþ1jk � y�kþ1
� � Yi;kþ1jk � y�kþ1
� �T þ R
(14)
in which R is measurement noise covariance matrix (νkνkT). The measurement update step is
Pxykþ1 ¼ ∑
2L
0Wc
i X i;kþ1jk � x�kþ1
� � Yi;kþ1jk � y�kþ1
� �T
Kkþ1 ¼ Pxykþ1 Pyy
kþ1
� ��1
xkþ1 ¼ y�kþ1 þKkþ1 ykþ1 � y�kþ1
� �
Pkþ1 ¼ P�kþ1 �Kkþ1P
yykþ1KT
kþ1
(15)
In this study, force feedback from actuators is used as measurement; therefore, yk+ 1 is the forceobtained from the experimental substructure.
6. TEST SETUP DESCRIPTION
A hybrid simulation with physical and updated numerical substructures was conducted. A simplestructural model is selected to demonstrate the features of the online model updating algorithms andtheir effects on the simulation results. The hybrid model is solved using the explicit form ofNewmark’s integration method to avoid iterations. For the experimental substructure, it is assumedthat actuator control errors are small, and reliable force measurements are obtained in each step.Experimental errors and their potential effect on the simulation response are considered negligible.
6.1. Hybrid simulation architecture
The hybrid simulation control systems at the University at Buffalo Structural Engineering andEarthquake Simulation Laboratory using xPCTarget comprised a three-loop architecture as shown inFigure 5. The innermost Servo-Control Loop contains the MTS STS controller that sends commanddisplacements to the actuator-specimen and reads back measured displacements and forces. Themiddle loop runs the Predictor–Corrector actuator command generator on the xPC-Target real-timedigital signal processor and delivers the command displacements to the STS controller in real-timethrough the shared memory SCRAMNet [23]. Finally, the outer Integrator Loop runs on the xPC-Host PC and includes OpenSees [24], MATLAB [25], and OpenFresco [26] that can communicatewith the xPC-Target through TCP/IP. The xPC-Host numerical model converges to the nextcommand displacement at large and somewhat variable time intervals. On the other hand, the STScontroller is required to send commands to the specimen at a fast rate of 1024Hz. In order to bridgethese two time scales, the xPC-Target’s Predictor–Corrector model predicts target displacements forthe STS controller until the next displacement arrives from the computational driver. Then thePredictor–Corrector model corrects the command to achieve the new target displacement. Also, ifthe parameter identification needs more time than the prediction phase limit of the simulation timestep, the actuator is automatically slowed down to zero velocity (zero displacement increment) untilthe next target is achieved [27].
The structural model used to conduct hybrid simulations with online updating is illustrated in Figure 6.The one-bay frame model consists of two elastic columns with nonlinear rotational springs (Elements 1and 2) at the base and an elastic truss element connecting the top of both columns (Element 3). Alumped mass is placed at the top of each column. The column on the left including the basenonlinear spring is the experimental substructure, and the remaining elements of the frame, inertiaforces and damping are modeled numerically. This simple model is selected in order to evaluate theidentification techniques and to simulate general conditions in earthquake loading tests.
The experimental specimen pictured in Figure 5 consists of a steel column bolted to a clevis at thebase with sacrificial steel coupons, providing for low-cost nonlinear hybrid simulation. The steelcolumn remains elastic, while the steel coupons in the clevis were designed to provide repeatableductile behavior in the specimen.
The nonlinear rotational spring on the right is considered as the adjustable part of the numericalsubstructure that can be updated. The objective of test is to update the numerical hysteretic model
parameters for the rotational spring based on the measured response of the rotational spring in theexperimental element. To simulate the behavior of the experimental rotational spring, the modifiedBouc–Wen model with stiffness and strength degradation is used.
By assigning different mass values to the dynamic degrees of freedom and elastic stiffness toElement 3, the relative deformation demands between both nonlinear rotational springs can bemodified. The different test conditions that can be realized are listed in Table II. This allows for anexamination of the updating techniques when considering a numerical element with demands thesame or different from the experiment.
In order to evaluate the performance of the parameter identification techniques, the first testcondition was conducted in the laboratory. In this test, Mass 1 and Mass 2 are equivalent, andElement 3 is considered rigid to produce identical deformation demands. Here, the numericalelement response could simply be substituted with the measured force. However, the main objectiveis to evaluate the identification methods and determine accurate parameter values of the numericalmodel to represent the experimental response. The differences between the experimental andnumerical hysteresis is mainly due to the performance of the identification technique.
In addition, the third test condition in Table II was conducted in the lab to simulate a more generalcondition where updating can be beneficial in hybrid simulation. In this case, the numerical elementsmay have similar properties to the experiment but may experience different deformation demandsand damage at different times. An example of this could be beam-to-column connections within thesame floor of the building.
MATLAB is used as the computational driver for the hybrid simulation with all numerical modelsand parameter identification codes programmed within. OpenSees also can be used as the computationprogram for updating the parameters at element and material level in addition to having real-timeaccess to nodal displacements and element forces during the test. Previous to running the hybridsimulation, the experimental substructure behavior is simulated using OpenSees as the finite-elementprogram and OpenFresco as the middleware between MATLAB and OpenSees to validate theonline updating hybrid simulation framework. Afterwards, the finite-element model in OpenSees isreplaced with the physical model in the lab connected through OpenFresco and xPC target.
The plane structure is subjected to the north–south component of the ground motion recorded atEl Centro, California, during the Imperial Valley earthquake of 18 May 1940. The acceleration dataare sampled at 0.02 s with a PGA of 0.32 g.
7. TEST RESULTS
The first and third model configurations listed in Table II were tested experimentally and presentedhere. For each configuration, an experimental hybrid simulation was conducted as the reference test,which serves as the reference ‘exact’ response of the model. Then, an adjustable numerical model isassigned to Element 2, and the test is repeated to reproduce the reference test response using modelupdating.
7.1. Online calibration test
The first test is the online calibration test. In this test series, the objective is to evaluate the performanceof the identification techniques in the real experiment. In addition, the numerical element is calibrated
Table II. Test configurations of experimental and numerical rotational spring models.
Test condition Assigned mass Assigned elastic stiffness
1 Identical deformation demands Mass 1 =Mass 2 K(Element 3) = inf2 Different deformation demands, similar amplitude Mass 1 =Mass 2 K(Element 3)< inf3 Different deformation demands, larger amplitude in Element 1 Mass 1>Mass 2 K(Element 3)< inf
to the experimental element response online, and the smooth hysteretic model is evaluated in terms ofrepresenting the measured experimental behavior.
7.1.1. Reference test. To provide the ‘exact’ response of the one-bay frame model as the reference,Element 3 is considered rigid and equal masses are assigned to each degree of freedom to make thestructure symmetrical. Hence, identical displacement histories for each degree of freedom andconsequently identical hysteresis for each rotational spring are expected. In this configuration, thereis no need to use two experimental setups. To perform the test, the command displacement is sent tothe Experimental Element 1, and the measured resisting force value is considered as the response ofboth Elements 1 and 2. Figure 7 illustrates the hysteresis of the reference test, which is identical forboth elements.
7.1.2. Updating test. In the updating test, all four steel coupons were replaced in the experimentalsubstructure (Element 1) to start with an undamaged specimen. Also, an adjustable numerical modelis assigned to Element 2 to be updated while the test is repeated. For the numerical model, initialvalues should be provided. Also, the parameters should be restricted to regions imposed by physicalconstraints to ensure that the overall continuity and differentiability of the model are not affected.The numerical model was calibrated offline to the experimental hysteresis obtained from thereference test, using NM method to provide meaningful interval constraint for updating parameters.Figure 8 shows both the hysteresis from the offline calibrated numerical model and experimentalresponse. Accordingly, an interval constraint is specified for each parameter as listed in Table III.
In this test, all the numerical model parameters except A and δA are updated by online calibration ofthe auxiliary numerical model using the UKF method. All the calibrated parameters are extracted fromthe auxiliary model and applied to Element 2 instantaneously. For computation efficiency, the updatingalgorithm does not need to be applied at all integration time steps, rather only at specific steps when ithas been detected that the errors between the numerical model with the latest parameters and theexperimental response exceed a specific threshold value. However, the UKF algorithm is fast andefficient, and therefore, updating is performed within each integration time step.
Figure 9 illustrates the comparison of the experimental response and the updated numerical model.The performance of the parameter identification method is verified as the numerical model wassuccessfully calibrated online without any major issue in performing the test.
The history of updated parameter values during the test are shown in Figure 10. Generally,numerical models are able to capture some or all of the critical behavior of the experimental elementresponse. If the parameter corresponding to a specific behavior has already been defined in thenumerical model, it starts to converge once the experimental element experiences that specificbehavior. For example, the yield strength could only be estimated once the experimental specimenhas yielded. As illustrated in Figure 10, all the parameters of the smooth hysteretic model convergeto a constant value, which can be used to tune the numerical model parameters representing theexperimental element behavior. The large initial fluctuations in parameter estimation indicate thatthere may not have been sufficient data to accurately calibrate a parameter by that time.
7.2. Implementation in general condition
To simulate a more general case, the third model configuration in Table II is examined. The massassigned to DOF 2 is divided by 2 to reduce the deformation amplitude experienced by Element 2.Also, an elastic truss element is used as Element 3 to produce different displacement demands onboth rotational springs. The reference test, without updating, and the updating test are conductedexperimentally for this configuration. The objective of these test series is to use the informationobserved from the experimental measurement for another element in the structure, which is modelednumerically and has similar properties compared with the experimental element but does notexperience the same load history. Note that the experimental element is assumed to experiencelarger deformation and associated damage prior to the numerical element.
Table III. Updating parameters boundaries.
Par. ko α n β γ A δA δν δη
Value 0.7–1.4 0.001–0.05 1.0–3.0 2.5–5.0 0–2.5 1.0 0.0 0.001–0.1 0.001–0.1
7.2.1. The reference test. In order to simulate the ‘exact’ response of the one-bay frame system withthe aforementioned configuration, two experimental setups are needed to simulate the behavior ofElements 1 and 2. However, as an alternative, experimental substructure (Element 2) was modelednumerically using the calibrated parameters obtained from the previous test. Figure 11 illustratesboth hysteresis of the reference model for this test condition.
7.2.2. Test without and with updating. Two experimental tests are conducted without and withupdating. In the experimental test without utilizing the updating technique, Element 1 isexperimentally tested in the lab, while Element 2 is numerically modeled using predefined initialparameters that remain constant during the test. The parameters are intentionally selected to bedifferent from the calibrated model of Element 2 in Figure 11 in order to challenge the performance
of the parameter identification technique. Afterwards, an updating test is conducted similar to the initialtest with UKF updating the parameter values of Element 2 based on the instantaneously-measuredresponse of the experimental substructure. The objective of this test is to reproduce the Element 2hysteresis obtained from the reference test illustrated in Figure 11. The comparison of the hysteresisof Element 2 in three different experimental tests are presented in Figure 12. UKF gives relativelyaccurate estimates of model parameters to capture the behavior of the experimental response duringthe hybrid simulation. Figure 13 shows the comparison of Element 2 resisting moment history forall three experimental tests.
In addition to the force values, the global response of the structure should also be evaluated.Figure 14 shows the global displacement history at DOF 2. It is noteworthy that the global response
-0.06 -0.04 -0.02 0 0.02 0.04-12
-8
-4
0
4
8
12
Mom
ent (
kN-m
)
Rotation (rad)
ReferenceInitialUpdated
Figure 12. Comparison of rotational springs hysteresis in different experimental tests.
0 5 10 15 20 25 30-12
-8
-4
0
4
8
12
Mom
ent (
kN-m
)
Time (s)
ReferenceInitialUpdated
Figure 13. Comparison of rotational springs moment history in different experimental tests.
0 5 10 15 20 25 30-50
-25
0
25
Dis
plac
emen
t (m
m)
Time (s)
ReferenceInitialUpdated
Figure 14. Comparison of the displacement history of DOF 2 in different experimental tests.
of the structure can also be used to evaluate the identification process as the calibrated force serves asfeedback for the structural system model. It can be seen that the smooth hysteretic model parametersare successively calibrated as both local and global responses are accurately reproduced.
8. CONCLUSION
A basic objective of this research is to extend the capabilities of hybrid simulation with substructuringby developing online updating techniques for numerical models to better predict the response ofstructures with distributed damage. Model updating can be used as an effective alternative allowingfor tests to be performed using a simpler numerical model with adjustable parameters. In this study,the UKF was implemented successfully to calibrate a smooth hysteretic model parameters byminimizing an objective function defined as the error between numerical and experimentalsubstructure resisting force values. Updating was applied to numerical substructures with similarproperties as experimental substructure, and experienced similar or different loading histories. In thisapplication, it is important that the experimental substructure corresponds to elements experiencingthe largest demands first such that a particular behavior can be observed and sufficient data obtainedto calibrate related parameters before the numerical elements experience similar demands.Algorithms for model updating, time to implement the updated parameters in numerical model andothers can be coded into existing software for hybrid simulation.
ACKNOWLEDGEMENTS
This research was primarily supported by the National Science Foundation under grants CMMI-0936633and CMMI-0748111. Additional support was provided from CMS-0402490 for shared-use of NEESequipment at the University at Buffalo. This support is gratefully acknowledged. Any opinions, findings,and conclusion or recommendation expressed in this paper are those of the authors and do not necessarilyreflect the views of the National Science Foundation.
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