SMIP17 Seminar Proceedings UNTANGLING THE DYNAMICS
SMIP17 Seminar ProceedingsUNTANGLING THE DYNAMICS OF SOIL-STRUCTURE INTERACTION USINGNONLINEAR FINITE ELEMENT MODEL UPDATINGHamed Ebrahimian1, Domniki Asimaki2, Danilo Kusanovic3 and S. Farid Ghahari41Scientific Research Assistant, Department of Mechanical and Civil Engineering, CaltechProfessor, Department of Mechanical and Civil Engineering, Caltech3Ph.D. Candidate, Department of Mechanical and Civil Engineering, Caltech4Postdoctoral Researcher, Department of Civil and Environmental Engineering, UCLA2AbstractThe dynamic response of a building structure to an earthquake excitation is the result of acomplex interaction between the structural system and the underlying and surrounding geology.Since modeling the physics of the coupled soil-structure system is a complex undertaking, thestate-of-practice has adopted simplified modeling procedures, such as the substructure method.Nevertheless, these procedures are often empirical and/or based on idealized assumptions, suchas linear-elasticity. In this study, our objective is to develop a robust model inversion frameworkthat can be utilized to extract information from the real-world building response measurements toback-calculate the model parameters that characterize the structural response and soil-structureinteraction effects.IntroductionThe dynamic response of a building structures to an earthquake excitation is the result ofa complex interaction between the structural system and the underlying and surroundinggeology. The coupled soil-structure response is a function of seismic waves interacting with thebuilding foundation, the nonlinear structural and geologic material response, and other energydissipation mechanisms such as friction and viscous damping in the structure and soil. Therefore,the prediction accuracy of structural response quantities depends on the accuracy of theemployed numerical model in characterizing these different sources of seismic energydissipation and the dynamic soil-structure interaction.Since modeling the physics of the coupled soil-structure system in detail is a complexundertaking, especially for practical design or assessment purposes, the state-of-practice hasadopted simplified modeling procedures (e.g., [1], [2]). Soil-structure interaction effects areusually modeled using a substructure approach, where the soil flexibility and energy dissipationare modeled using distributed springs and dashpots [3]. Numerous simplified solutions exist todetermine the stiffness and damping coefficients of these elements; solutions that are nonethelessbased on idealized and restrictive assumptions. Examples of these assumptions include linearelastic soil and structural behavior, uniform soil half space (or soil profiles with stiffnessgradually varying with depth [4]), canonical foundation geometry, etc. These assumptions andthe empirical nature of mechanical analogs such as soil springs and dashpots, could potentiallylead to large error margins in predicting the seismic response of real-world building structures,even if the simplified models have demonstrated acceptable accuracy for ideal cases. The41
SMIP17 Seminar Proceedingsapplicability of these models becomes even more questionable for nonlinear response timehistory analyses. This is due to the fact that the concept of soil impedance functions and theresulting equivalent soil springs and dashpot is mainly based on the premise of linear-elasticresponse behavior. Nevertheless, the coupled soil-structure system is expected to experiencenonlinearity during strong earthquakes – both material and geometrical (e.g., foundation-soilseparation during rocking). Thus, the system may deviate substantially from the underlyingassumptions that have led to substructure modeling techniques for soil-structure interactionanalysis.On the other hand, modern seismic design and assessment codes are progressivelystirring toward nonlinear finite element (FE) modeling and response simulations for predictionsof structural and nonstructural seismic demands. In a modern seismic analysis approach, a wellcalibrated nonlinear model is required to precisely predict not only the peak values of responseparameters, but also the time histories of structural responses. Despite all the advancementsmade in the field of mechanics-based nonlinear structural modeling, the state-of-practice formodeling structural damping and soil-structure interaction is still based on empiricalassumptions. Clearly, there is an inconsistency between the mechanics-based modelingtechniques available for structural systems and the underlying assumptions guiding the structuraldamping and soil-structure interaction modeling. This could result in an "inconsistent crudeness"in state-of-the-art seismic modeling of building structures that this proposal seeks to investigate.In this study, we do not seek to develop new models of soil springs and dashpots orstructural damping. Instead, we seek developing a model inversion framework that can beutilized to extract information from the real-world building response measurements to backcalculate the model parameters that characterize the structural response and soil-structureinteraction effects. By repeating this effort for different building case studies and earthquakerecords in the long run, our objective is to compare the estimation results with the state-of-the-artrecommendations, and to provide guidelines on how to improve the state-of-practice structuralmodeling capabilities.Model Inversion through Nonlinear FE Model UpdatingSuppose that the dynamic response of a building structure is recorded during anearthquake event. To simulate the dynamic response of this building structure, a mechanicsbased (linear or nonlinear) FE model is developed. The FE model depends on a set of unknownparameters including inertia properties, damping parameters, soil spring and dashpot parameters,and parameters characterizing the nonlinear material constitutive laws used in the FE model.These parameters are referred to as the model parameters henceforth. Using the recorded inputground acceleration time history and the response of the building, the objective is to identify thebest set of unknown model parameters that minimize the discrepancy between FE predicted andmeasured structural responses. Another objective of is to utilize the measured structuralresponses to jointly estimate the model parameters and input excitation (i.e., foundation inputmotion) time history.In this study, the estimation problem is tackled by updating sequentially (i.e., for severalbatch of measurement data) the probability distribution function (PDF) of the unknown modelparameters (and input excitation) using a Bayesian inference method (e.g., [5], [6]). FE model42
SMIP17 Seminar Proceedingsupdating using the measured input excitation and output response of the structure is referred to asthe input-output model updating. Contrarily, in an output-only FE model updating, one ormultiple time histories of the dynamic input excitation are also unknown. Therefore, theobjective of the sequential Bayesian estimation is to estimate jointly the FE model parametersand the time unknown time history of the base excitation so that the discrepancies between theestimated and measured response quantities are minimized [7].BackgroundThe time-discretized equation of motion of a nonlinear FE model at time step i (i 1 k , where k denotes the total number of time steps) is expressed as i θ C θ q i θ ri q i θ , θ f i θ M θ qwhere M θ nDOF nDOF mass matrix, C θ ri q i θ , θ n DOF 1n DOF n DOF(1) damping matrix, history-dependent (or path-dependent) internal resisting force vector, i θ nDOF 1 nodal displacement, velocity, and acceleration response vectors,qi θ , q i θ , qn 1respectively, θ nθ 1 FE model parameter vector, f i θ DOF dynamic load vector, andn DOF number of degrees-of-freedom. In the case of uniform (or rigid) seismic base excitation, ig where L f i θ M θ L un DOF n g un base acceleration influence matrix, and u ig u gdenotes the seismic input ground acceleration vector. Using a recursive numerical integrationrule, such as the Newmark-beta method [8], Eq. (1) is reduced to a nonlinear vector-valuedalgebraic equation that can be solved recursively and iteratively in time to find the nodalresponse vector at each time step. In general, the response of a FE model at each time step isexpressed as a function (linear or nonlinear) of the nodal displacement, velocity, and/oracceleration response vectors at that time step. Denoting the response quantity predicted by then 1FE model at time step i by yˆ i y , it follows that 1g: i , q 0 , q 0yˆ i h i θ , u 1(2)where hi . is the nonlinear response function of the FE model at time step i. The measuredresponse vector of the structure, y i , is related to the FE predicted response, ŷ i , as 1g: iv i θ, u yin 1 1g: i yˆ i θ , u (3)in which v i y is the simulation error vector and accounts for the misfit between themeasured and FE predicted response of the structure. This misfit stems from the outputmeasurement noise, parameter uncertainty, and model uncertainties. The latter stands for themathematical idealizations and imperfections underlying the FE model, which result in aninherent misfit between the FE model prediction and the measured structural response [9]. In theabsence of model uncertainties, it is assumed here that the measurement noises are stationary,zero-mean, independent Gaussian white noise processes (i.e., statistically independent across43
SMIP17 Seminar Proceedingstime and measurement channels) [10]. Therefore, the probability distribution function (PDF) ofthe simulation error in Eq. (3) is expressed asp v i 1 2π ny / 2R1/ 21 1 vTi R vi2ein which R denotes the determinant of the diagonal matrix R (4)ny ny , which is the (time- Tinvariant) covariance matrix of the simulation error vector (i.e., R E v i v i , i ).In an output-only FE model updating problem, the FE model parameter vector ( θ ) and thetime history of the seismic input ground acceleration at each time step ( u 1g:k ) are unknown and g ,modeled as random variables (the corresponding random variables are denoted by Θ and U1: krespectively). Using Bayes’ rule, the posterior probability distribution of the unknowns can beexpressed as 1g: kp θ, u where p y1: k θ,u 1g: k p v1:ky1: k 1g: k p θ, u 1g: kp y1: k θ, u p y1: k likelihood function, 1g: kp θ,u (5) joint prior distribution of the g , and p y normalizing constant independent of Θ and U g .random variables Θ and U1:k1: k1: kThe objective of the output-only FE model updating problem is to estimate jointly the unknownmodel parameters and the ground acceleration time history such that their joint posterior PDFgiven the measured response of the structure is maximized, i.e., θˆ , u ˆ g1:k MAP in which y1: k y1 , y 2 ,., y kTT T T 1g:k arg max p θ, u g θ, u 1:k y1:k (6) time history of the measured response of the structure, andMAP stands for the maximum posterior estimate. The estimation uncertainty is quantified byevaluating the parameter estimation covariance matrix at the MAP estimate.Since the model inversion problem is highly nonlinear, a sequential estimation approach,referred to as the sequential Bayesian estimation method, is used in this study to improve thecomputational efficiency and convergence rate. In this approach, the estimation time interval isdivided into successive overlapping time windows, referred to as the estimation windows. Theestimation problem is solved at each estimation window to estimate the posterior PDF ofunknown parameters. The first two moment of distribution (i.e., mean vector and covariancematrix) of the parameters are then transferred to the next estimation window and used as priorinformation. The sequential Bayesian estimation method approach is schematically shown inFigure 1. Two approaches are developed to find the posterior mean vector and covariance matrixof the unknown parameters and from the prior estimates: (i) a FE model linearization approach,44
SMIP17 Seminar Proceedingsand (ii) an unscented transformation approach. These two methods are described in the next twosections, respectively.Figure 1: Schematic presentation of the sequential Bayesian FE model updating method.Sequential Bayesian Estimation using FE Model LinearizationFollowing Eq. (5) and the sequential estimation logic described above, the naturallogarithm of the posterior joint PDF of the FE model parameters and base acceleration timehistory at the mth estimation window, spanning from time step t1m to time step t 2m , can be derivedas gm,m m y m m c log p y m m θ, u gm,m m log p θ, u gm,m m log p θ, ut1 : t 2t1 : t 2t1 : t 2t1 : t 2t1 : t 2 (7) in which c log p y t m: t m is a constant. In this equation, the time history of the base 1 2 acceleration from time step 1 to t1m 1 , (i.e., u 1g: t m 1 ), is assumed to be deterministic and equal to1the mean estimates obtained from previous estimation sequences. For notational convenience, anTTextended parameter vector at the mth estimation window is defined as ψ m θ T ,u tgm,m: t m , where1 2 n t n 1θ lg uψm . Since the simulation error is modeled as an independent Gaussian white noiseprocess, the likelihood function is given by k 1g: k p v i p y 1: k θ, ui 145(8)
SMIP17 Seminar Proceedings p y 1: k θ, u 1kg1: ki 1 2π ny / 2R1/ 2eT1 g ,q 0 ,q 0 R 1 y i h i θ ,u g ,q 0 ,q 0 y i h i θ ,u 1: i 1: i 2 By substitution of Eq. (8) into Eq. (7) and assuming a Gaussian distribution for the prior jointPDF, it follows thatT1 ˆ g m R 1 y m m h m m ψ m , u ˆ g m log p ψ m y t m : t m k 0 y t m : t m h t m :t m ψ m , utttt1:1::1: t1 1 t1 2 1 2 11 2 2 1 2 1 2 1 T1ˆ m Pˆ ψ ψ m ψˆ m ψm ψ2 (9) where k 0 is a constant, and ψ̂ m and P̂ψ are the prior mean vector and covariance matrix of the t n t n extended parameter vector at the mth estimation window. R l y l y is a block diagonalmatrix, in which the diagonals are the simulation error covariance matrix R . To find the MAPestimate of ψ m , the posterior PDF in Eq. (9) is maximized, i.e., log p ψ m y m mt :t1 2 ψm 0 T ˆ g m h t m : t m ψ m , u1: t1 1 1 1 2 R ˆ g m Pˆ ψ y t m:t m h t m:t m ψ m , u1: t1 11 2 ψm 1 2 ψ 1mˆ m 0 ψ(10)Eq. (10), which is a nonlinear algebraic equation in ψ m can be solved using an iterative first g m at ψ̂ m asorder approximation of the FE response function ht m :t m ψ m ,u1:t 112 1 ˆ g m h t m :t m ψ m , u1: t1 11 2 ˆ g m h m m ψˆ m , u ˆ g m h t m : t m ψ m , utttt 1:1:1:11 2 11 2 1 ψm ψm ψˆ m H .O.T .(11)ˆ ψ m ψm g 1:t m 1 h t m :t m ψ m , u1 1 2 The matrix ψ mrepresents the FE response sensitivities with respect to theˆ ψ m ψmextended parameter vector, evaluated at the prior mean values of the extended parameter vector,ψ̂ m . This matrix is denoted by C hereafter for notational convenience. Substituting Eq. (11) intoEq. (10) and neglecting the higher order terms results in the following (first order approximate)equation for the MAP estimate of ψm : 1 1 T 1 ˆ g m ψˆ m ψˆ m C T R 1C Pˆ ψ C R y m m h m m ψˆ m , ut1 : t 2 t1 : t 21: t1 1 46(12)
SMIP17 Seminar Proceedingsin whichψ̂ mis the updated (or the posterior) mean estimate of ψ m . It can be shown that the 1 term K CT R 1C Pˆ ψ 1 CT R 1 is similar to the Kalman gain matrix, as used for Kalmanfiltering [11].The updated ψ̂ m from Eq. (12) is iteratively used as the new point for the linearization ofthe nonlinear FE model in Eq. (11) to find an improved estimation. This iterative predictioncorrection procedure at each estimation window is equivalent to an iterative EKF method forparameter-only estimation [11]. Following the EKF procedure, the prior covariance matrix of theextended parameter vector Pˆ ψ ,m is updated to the posterior covariance matrix Pˆ ψ ,m after eachprediction-correction iteration. Moreover, it is assumed that both the prior and posterior jointPDF of the extended parameter vector are Gaussian. The updated estimation covariance matrix,can be derived as T TPˆ ψ , m E Ψ m ψˆ m Ψ m ψˆ m I KC Pˆ ψ , m I KC K R K T(13)Furthermore, to improve the convergence of the iterative prediction-correction procedure, aconstant disturbance matrix is added to the posterior covariance matrix at each iteration to providethe prior covariance matrix for the next iteration, i.e.,Pˆ ψ , i 1 Pˆ ψ , i Q(14)where Q is a constant diagonal matrix with small positive diagonal entries (relative to the diagonalentries of matrix Pˆ ψ , i ). The matrix Q is referred to as process noise covariance matrix in theKalman filtering world. The subscript i in Eq. (14) denotes the iteration number.Sequential Bayesian Estimation using the Unscented TransformationWhile the terms P̂ψy cross-covariance matrix of Ψ and Y , and P̂yy covariancematrix of Y in are derived by linearizing the nonlinear FE model in the previous section, anunscented transformation (UT) method (e.g., [12], [13]) can also be used to derive the P̂ψy andP̂yy . UT is a deterministic sampling approach to propagate the uncertainty in Ψ through thenonlinear FE model; thus, circumventing the linearization of the FE model. Therefore, it resultsin a more accurate estimation of the P̂ψy and P̂yy , especially for highly nonlinear models.Indeed, using the UT method is at the cost of evaluating the FE model at multiple samples of thevector Ψ ; nevertheless, the additional FE computations can be performed in parallel ( [14], [15]).In this approach, the nonlinear FE model is evaluated separately at a set ofdeterministically selected realizations of the extended parameter vector Ψ , referred to as thesigma points (SPs) denoted by j , which are selected around the prior mean estimate Ψ̂ . Inthis study, a scaled UT based on 2 nΨ 1 sigma points (i.e., j 1, 2 , , 2 nΨ 1 ) is used,where n Ψ denotes the size of the extended parameter vector. The mean and covariance matrix of47
SMIP17 Seminar Proceedingsthe FE predicted structural response Y , and the cross-covariance matrix of Ψ and Y arerespectively computed using a weighted sampling method asy Pˆ yy j 1Pˆ Ψy j 1W ej yˆ j y yˆ j y2 n Ψ 1 W mj yˆ jj 1(15) 2 n Ψ 1 2 n Ψ 1 T R ˆ yˆ j y TW ej j Ψ(16)(17)where Wmj and Wej denote the mean and covariance weighting coefficients, respectively [13].With this approach, the Kalman gain matrix can be computed and the sequential parameterestimation can be pursued following Eqs. (12) and (13).FE Model Updating of the Millikan Library BuildingThe Millikan LibraryThe Millikan Library is a reinforced concrete shear wall building with a basement leveland nine stories above the ground. It is located on the California Institute of Technology(Caltech) campus in Pasadena, and was constructed from 1966 to 1967. Millikan library has beenthe subject of several studies, especially in the fields of system identification and structuralhealth monitoring. The building is a unique case for soil-structure interaction studies, due to itsunique structural and soil properties (Figure 2).The Millikan Library structure is 43.9 m tall above ground including the roof level. It hasa 4.3 m deep basement below the ground level. Except for the first and the roof levels, which are4.9 m high, all floors are 4.3 m high. The basement is encased by surround
SMIP17 Seminar Proceedings 41 UNTANGLING THE DYNAMICS OF SOIL-STRUCTURE INTERACTION USING NONLINEAR FINITE ELEMENT MODEL UPDATING Hamed Ebrahimian1, Domniki Asimaki2, Danilo Kusanovic3 and S. Farid Ghahari4 1 Scientific Research Assistant, Department of Mechanical and Civil Engineering,
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