SIGNIFICANCE OF MODELING ERROR IN STRUCTURAL PARAMETER ESTIMATION

Transcription

1 SGNFCANCE OF MODELNG ERROR N STRUCTURAL PARAMETER ESTMATON Masoud Sanayei 1, Sara Wadia-Fascetti, Behnam Arya 3, Erin M. Santini 4 ABSTRACT Structural health monitoring systems rely on algorithms to detect potential changes in structural parameters that may be indicative of damage. Parameter estimation algorithms seek to identify changes in structural parameters by adjusting parameters of an a priori finite element model of a structure to reconcile its response with a set of measured test data. Modeling error, represented as uncertainty in the parameters of a finite element model of the structure, curtail capability of parameter estimation to capture the physical behavior of the structure. The performance of four error functions, two stiffness-based and two flexibility-based, is compared in the presence of modeling error in terms of the propagation rate of the modeling error and the quality of the final parameter estimates. Three different types of parameters are used in the parameter estimation procedure: (1) unknown parameters which are to be estimated, () known parameters assumed to be accurate, and (3) uncertain parameters that manifest the modeling error and are assumed known and not to be estimated. The significance of modeling error is investigated with respect to excitation and measurement type and locations, the type of error function, location of the uncertain parameter and the selection of unknown parameters to be estimated. t is illustrated in two examples that the stiffness-based error functions perform significantly better than the corresponding flexibility-based error functions in the presence of modeling error. Additionally, the topology of the structure, excitation and measurement type and 1 Associate Professor, Dept. of Civil & Environmental Engineering, Tufts University, Medford, MA 155 Assistant Professor, Dept. of Civil & Environmental Engineering, Northeastern University, Boston, MA Doctoral Candidate, Dept. of Civil & Environmental Engineering, Tufts University, Medford, MA Doctoral Student, Dept. of Civil & Environmental Engineering, Tufts University, Medford, MA 155 Sanayei, Wadia-Fascetti, Arya, Santini

2 locations, and location of the uncertain parameters with respect to the unknown parameters can have a significant impact on the quality of the parameter estimates. nsight into the significance of modeling error and its potential impact on the resulting parameter estimates is presented through analytical and numerical examples using static and modal data. NTRODUCTON Structural health monitoring is now a commonly used term in the structural engineering research community. Health monitoring makes it possible to monitor a structure and its systems (structural or nonstructural) to assess their condition relative to different performance characteristics. Structural identification is the process of identifying characteristic behavior of a structural system. Structural parameter estimation, a sub-field of structural identification, is a tool enabling the prediction of structural stiffness and mass parameters for finite element model (FEM) updating. The estimation is performed by adjusting the parameters of an analytical FEM in a systematic approach to reproduce measured data (static or dynamic). This adjustment makes it possible to estimate element parameters implicit in the stiffness or mass matrices describing a structural system at the component level. The updated FEM can reveal information about a structure s health. This paper investigates the challenges due to the presence of modeling error in the interpretation of the resulting parameter estimates. Liu and Yao 16 introduced the application of system identification through parameter estimation to civil engineering structures. The theory behind parameter estimation is not new but recent advances in computing capacity have made parameter estimation a realistic tool for structural condition assessment. A potential change in a structure s health or the existence of undetected damage or deterioration can be related to changes in the FEM parameters relative to a Sanayei, Wadia-Fascetti, Arya, Santini - -

3 baseline FEM. Parametric changes in a baseline model identified using a parameter estimation tool can be related to physical changes in a structure leading to an assessment of the overall health of a structure for use in condition assessment. Parameter estimation requires an a priori FEM, comprehensive experiment design, controlled testing, data acquisition, data processing, data quality assessment, minimization of an error function using nondestructive test (NDT) data, and finally, the interpretation of the results. Together the collection of data obtained through nondestructive testing and parameter estimation can give a meaningful representation of the health of a structure. Parameter estimation is a straightforward process when applied to a small-scale structure with pure simulated data. Many examples of successful identification have been documented for controlled simulated models. 5, 1 Yao et al., 31 Shinozuka and Ghanem, 8 and Hjelmstad and Shin 11 have successfully performed parameter estimation using real test data obtained from laboratory experiments on small-scale structures. However, application of parameter estimation to in-service structures introduces a number of challenges, which have been presented by Doebling et al., 8 Beck and Katafygiotis, 6 and Sanayei et al. These challenges include the possible incompleteness of the measured data, the variable quality of the test data, type and locations of available data, type of error functions and optimization algorithm used in the parameter estimation process. Sanayei et al. describe three primary challenges that are currently limiting the application of parameter estimation to full-scale structures. These challenges are summarized as: 1. Errors in NDT measurements stem from design and implementation of the experiment, which consists of: the selection of excitation sources; excitation location and magnitude; sensor types and locations; selection of data acquisition systems; data Sanayei, Wadia-Fascetti, Arya, Santini - 3 -

4 processing and testing economy. Errors in NDT measurements and the associated data processing are referred as measurement error.. Errors in the mathematical FEM include the stiffness of nonstructural members or members not accounted for in the a priori FEM, uncertainty in material properties, energy loss, environmental variability and nonlinear structural response. Furthermore, the FEM may not fully capture each component's behavior, due to inadequate knowledge of damping, the environment, or the boundary conditions. Errors in the mathematical model are referred as modeling error. 3. Errors in parameter estimation are affected by the type of the objective function, selection of optimization technique, possible degree of freedom mismatch due to the inability to excite and measure responses at sufficient and strategic locations, and model order which can cause the problem to be over or under defined. The first two types of error, measurement error and modeling error, influence the quality of the final parameter estimates. Measurement error typically has a zero mean and its magnitude is dependent on the experimental equipment, the test environment, and data processing. Modeling error, however, is a bias error. n other words, modeling error produces a shifted response that isn t necessarily centered about zero error. Figure 1 depicts the difference between measurement error and modeling error. The beam shown is comprised of four discretized elements with two pin supports. The mean response extracted from NDT data is denoted by the dotted line, and the deflected shape produced from a baseline FEM is denoted by the dashed-dot line. Sanayei, Wadia-Fascetti, Arya, Santini - 4 -

5 Sample status of each : Known Unknown Unknown Uncertain Modeling Error Uncertain FEM Baseline Error ={ P} adjusted by parameter estimation Measurement Error Shape Extracted from Baseline Model Shape Extracted from NDT Data Figure 1. The difference between modeling error and measurement error. Parameter estimation seeks to update the unknown parameters denoted as the moment of inertia of the two internal beam elements in Figure 1. Both measurement error and modeling error present challenges in the reconciliation between the two deflected shapes. Measurement error results in the uncertainty bounds of the deflected shape extracted from NDT data, while modeling error results in the uncertainty in the deflected shape extracted from the baseline model. n some cases, the modeling error can exceed the errors associated with measurement error. Additionally, the two different errors may be superimposed to be greater than the allowable limiting tolerances in parameter estimation. The existence of either measurement error or modeling error complicates the parameter estimation, questioning the final result quality influencing the perceived structural health and the subsequent decisions. Measurement error can be represented by a statistical dispersion about a zero mean response and can be handled in the parameter estimation solution using weight factors that are derived from the dispersion in the measured data. Significant research has been Sanayei, Wadia-Fascetti, Arya, Santini - 5 -

6 conducted to study the impact of measurement error on parameter estimates. The effect of measurement error on parameter estimation and structural identification is well understood. Sanayei et al. 4 and Sanayei and Saletnik 5 developed a heuristic method for design of NDTs to reduce the error in the geometric parameter estimates caused by noisy measurements. Beck and Katafygiotis 7 used a Bayesian probabilistic formulation to treat uncertainties, which arise from measurement error. Hjelmstad and Shin 11 proposed two damage indices and a method of establishing threshold values for these indices that would allow the detection of damage in spite of error and sparsity of measurements. Since the impact of measurement error has already been studied in depth it is not further investigated in this paper. Modeling error, a bias error, is not referenced to a zero mean. The potential bias in the modeling error presents a greater challenge to the application of parameter estimation. Without properly acknowledging modeling error the parameter estimation procedure can incorrectly estimate the unknown parameters by reconciling an incorrect or shifted baseline model to the NDT data. Modeling error, in parameter estimation, is defined by Wadia-Fascetti et al. 9 as the uncertainty in the parameters of a FEM. Major contributions of modeling error in a FEM are the existence of nonstructural members; uncertainty in material properties, geometric section properties of known parameters, incomplete information about boundary conditions; nonlinear structural response; modeling of energy losses (damping) and environmental variability for parameter estimation. Although it is possible to quantify the effect of errors in a FEM on the response of a structure, 3 these errors complicate the solution of the inverse problem and can not be avoided in the practical application of parameter estimation. At this time, limited investigations have been performed in the area of modeling error. Koh and See 14 showed that an improved version of the extended Kalman filter (EKF) in the presence of modeling error is a Sanayei, Wadia-Fascetti, Arya, Santini - 6 -

7 more reliable method compared to a recursive least-square parameter estimation method. Kim and Stubbs 13 quantified the impact of model uncertainty on the damage-prediction accuracy when applied to a plate girder model. Law and Zhang 15 proposed a method to reduce the effect of modeling error on the expansion of incomplete mode shapes to the full dimension of the FEM. The parameter estimation is then carried out using the measured and subsequently expanded mode shapes. Beck and Katafygiotis 6 presented a Bayesian statistical framework to update FEMs when uncertainties in modeling can cause discrepancies in solution uniqueness. Arya et al. 4 studied the impact of modeling error using simulated modal data for a bridge structure. Gunes et al. 9 studied the impact of modeling error on parameter estimates using simulated static data for a structural frame. Arya and Sanayei 3 investigated the existence of local minima with respect to changes in modeling error magnitude and location using modal data, by visualizing the error function surfaces. The impact of modeling error on the estimated parameters is investigated using four error functions: static stiffness-based error function, static flexibility error function, modal stiffness based error function, and modal flexibility-based error function. 4,1, While these error functions have each been evaluated separately in the literature, this is the first study that compares the performance of the stiffness-based and flexibility-based error functions in a significant presentation. Localized modeling error that can represent undetected structural damage is examined using pure simulated data (data free of measurement error) with regard to location and magnitude. The objective of this paper is to provide a thorough presentation of the impact that localized modeling error can have on the parameter estimation using the four error functions stated. A variety of test designs are investigated that mix different static load and natural modes of vibration with different measurements and error functions. nsights to the potential impact Sanayei, Wadia-Fascetti, Arya, Santini - 7 -

8 that different load and measurement scenarios and various error functions can have on the quality of unknown parameter estimates are presented. PARAMETER ESTMATON FOR STRUCTURAL DENTFCATON Parameter estimation, at the element level is used to identify the implicit cross-sectional properties in the stiffness and mass matrices of a structure's FEM. The analytical response used may be static or modal measurements. 1, This procedure estimates structural stiffness parameters such as axial rigidity (EA), bending rigidity (E), and torsional rigidity (GJ), and effective mass parameters at the component level. Changes in these parameters can be indicative of deterioration or damage affecting overall structural health. A typical parameter estimation procedure considers two types of parameters: those that are known and those that are unknown. The known parameters are assumed to be accurate while the unknown parameters are not known to an acceptable level of accuracy and are to be estimated. Error function formulation must consider the inability to measure responses at all degrees of freedom. For the static case, the equilibrium equation for a series of applied sets of forces, [F], and the corresponding sets of displacements, [U], is [ ][ U] [ F] K = (1) partitioned to distinguish between displacements that are measured or not measured by Sanayei and Nelson 3 K K aa ba K K ab bb U U a b F = F a b () where the subscripts a and b denote degrees of freedom at which displacements [U a ] or [U b ], are measured or not measured, respectively, however all applied forces are assumed known and Sanayei, Wadia-Fascetti, Arya, Santini - 8 -

9 entries into [F a ] or [F b ] can be either nonzero or zero. n the "static stiffness-based error function," [E ss ], the vector of unmeasured displacements, [U b ], in () is condensed out to establish the relationship 1 1 [ E ( p) ] = ( [ K ] [ K ][ K ] [ K ] )[ U ] + [ K ][ K ] [ F ] [ F ] ss aa ab bb ba a ab bb b a (3) between the measured forces and the analytically predicted forces. t should be noted that units of [E ss ] are forces and moments depending on the DOF that is represented. Similarly, the "static flexibility-based error function," [E sf ], can be rearranged in terms of the flexibility matrix yielding the error function 1 1 [ E p) ] = ( [ K ] [ K ][ K ] [ K ] ) 1 ( [ F ] [ K ][ K ] [ F ] ) [ U ] sf ( (4) aa ab bb ba that compares measured displacements to analytically predicted displacements. Again, it should be noted that the units of [E sf ] are displacements and rotations depending on the DOF being represented. Each static error vector shown in (3) and (4) is horizontally appended for each measured load case to form static error function matrices [E ss (p)] and [E sf (p)], respectively. The characteristic equation for dynamic loading [ K]{ } j λ j [ M ]{ φ} j a ab bb φ = (5) can be partitioned in terms of mode shape coordinates at measured and unmeasured DOF for each selected natural frequency b a K K aa ba K K ab bb φ φ a b j = λ j M M aa ba M M ab bb φ φ a b j (6) n (5) and (6), [K] and [M] are the stiffness and mass matrices of the structure, {φ} j contains the j th mode shape, and λ j is the j th eigenvalue. As in the static formulation, the unmeasured mode shapes, {φ b } are condensed out of (6) yielding the modal stiffness-based error vector, {E ms Sanayei, Wadia-Fascetti, Arya, Santini - 9 -

10 (p)} j as shown in (7). Similarly, the error function can be formulated with any number of mode shapes. ms j 1 ( [ K aa ] λ j [ M aa ]) ([ K ab ] λ j [ M ab ])[ ( K bb ] λ j [ M bb ]) ([ K ba ] λ j [ M ba ])){ Φ a } j { E ( p)} = (7) The characteristic equation (5) can be rewritten in terms of the flexibility rather than the stiffness matrix 4 by using the dynamic matrix 1 [ D] [ K ] [ M ] = (8) { } j = λ j [ D]{ Φ} j Φ (9) Partitioning the equation in terms of DOF that are measured and not measured yields Φ Φ a b j = λ j D D aa ba D D ab bb Φ Φ a b j (1) The unmeasured DOF for each measured mode shape, {Φ b } is condensed out to form the modal flexibility-based error vector, {E mf (p)} j, mf j 1 (( λ j ) [ D ab ] ([ ] λ j [ D bb ]) [ D ba ] + λ j [ D aa ] [ ]){ Φ a } j { E ( p)} = (11) Each modal error vector shown in (7) and (11) is horizontally appended for measured modes to form modal error function matrices [E ms (p)] and [E mf (p)], respectively. The error functions in (3), (4), (7), and (11) can support multiple load cases or modes of vibration and are implicitly functions of all structural parameters, {p}, that describe the characteristics of the structural system at the component level. n the static error functions, each load case must produce responses that are linearly independent of other load cases. The modal error functions use mode shape values at selected DOF as the different measurements. Sanayei, Wadia-Fascetti, Arya, Santini - 1-

11 According to their orthogonality property, all mode shapes are linearly independent from each other, all mode shapes can be used in the modal error functions. The scalar objective function J in (1) is defined as the square of the Frobenius norm of the residual error matrix in (3), (4), (7), and (11). J ( p) = i j E ij (1) The square root used in the definition of the Frobenius norm is not used in (1) for computing efficiency and has no impact on the resulting parameter estimates. The respective scalar objective function (1) is minimized by updating the stiffness parameters, {p}. When there is no modeling error or measurement error, as the updated parameter estimates approach the true value of the unknown parameters J will approach zero within an acceptable tolerance limit. This implies that a global minimum has been found on the surface of the objective function, J. The program PARS 18, 1998 (PARameter dentification System), developed by Sanayei at Tufts University implements parameter estimation. The basic features of PARS include a finite element module with assumed accurate values for known parameters and initial guesses for unknown parameters, the formulation of error functions, and optimization routines to minimize the error function with respect to the unknown parameters. PARS uses the measured responses (modal and static) to obtain parameter estimates by means of minimizing error functions in (3), (4), (7), or (11). Static error functions can accommodate sparse translations, rotations, and strains. Modal error functions use measured natural frequencies and corresponding mode shapes at a subset of translational and rotational DOF. Sanayei, Wadia-Fascetti, Arya, Santini

12 Unknown parameters can be estimated if each structural member with unknown parameters is significantly stressed by the excitations and is observable by the sensors placed at strategic locations. At a minimum, one independent measurement for each unknown parameter is required to avoid an under defined problem. n most practical cases, the availability of excitation sources and measurement scenarios are dependent on accessibility, feasibility, and cost limitations. Without measurement error or modeling error additional excitation cases are not required for parameter estimation. However, both measurement error and modeling error can propagate throughout the estimation algorithm and potentially can have a detrimental effect on the quality of the final parameter estimates. While it is well accepted that parameter estimation is stable within prescribed limits of measurement error on each sensor, analogous conclusions related to modeling error have yet to be reported. Following is a reexamination of parameter estimation in the presence of modeling error. PARAMETER ESTMATON CONSDERNG MODELNG ERROR Typically, parameter estimation is performed with the two types of parameters presented previously (known and unknown) and modeling error is addressed by creating the best possible analytical model. Explicit consideration of modeling error in parameter estimation requires the consideration of an additional parameter. Wadia-Fascetti et al. 9 propose the use of an additional parameter referred to as an uncertain parameter that is assumed to be known and not estimated. While not explicitly considered in the parameter estimation, the uncertain parameter is known with some uncertainty. This misrepresentation must be accounted for by the unknown parameters resulting in inaccurate parameter estimates. Thus, parameter estimation implemented Sanayei, Wadia-Fascetti, Arya, Santini - 1-

13 in the presence of modeling error is performed with three types of parameters: unknown, known, and uncertain. The objective function, J, reaches a zero minimum value when no modeling error and measurement error is present and the true parameter values are used in the error functions [E ss ], [E sf ], [E ms ], or [E mf ]. The modeling error considered in parameter estimation as uncertain parameters in the stiffness matrix modifies the shape of the J surface, causing J to be nonzero when evaluated with the true values of the unknown parameters. This modification to J may move the location of the global minimum altering the quality of the final parameter estimates. 3 Thus, the challenge associated with modeling error is that the true parameter values may not be coincident with the location of the contaminated global minimum. Consider the single bay portal frame shown in Figure. E is assumed to be GPa and constant for the whole structure. Four linearly independent excitation and measurement scenarios considered in this example are summarized in Table 1. Table shows the magnitude of the applied static loads and Table 3 indicates the natural frequency of the measured modes of vibration. Sanayei, Wadia-Fascetti, Arya, Santini

14 System Stiffness Properties Member nertia (m 4 ) Area (m ) 1 & 4 7. x x 1 & x x A, A 3, System Mass Properties Node Translational Mass (kg) Rotational Mass (kg. m ) & 4, 1, 3 4, 4, 4 m 1 A 1, 1 1 A 4, Figure. Portal frame 3 m Table 1. Excitation and measurements scenarios for the portal frame in Figure. Case Node MDOF No. (a), 3, 4 1,, 3, 4, 5, 6, 7, 8, 9 (b), 3 1,, 3, 4, 5, 6 (c) 3, 4 4, 5, 6, 7, 8, 9 (d) 3 4, 5, 6 Table. Static forces applied to the portal frame in Figure. DOF FDOF magnitude 1 1, N -1, N 3 -, N.m 4, N 5-5, N 6-4, N.m 7 1, N 8-1, N 9 -, N.m Table 3. Natural Frequencies considered measured for the portal frame in Figure. Mode Frequency (Hz) Sanayei, Wadia-Fascetti, Arya, Santini - 14-

15 n the case of static testing, the measured response is obtained for the loads specified in Table at the measurement scenarios specified in Table 1. n all static cases simulated here, forces are applied at the same DOF as displacements are measured. For example, in Case (a) there are 9 linearly independent excitations and 6 sets of displacement measurements. The first excitation is a static force applied in the direction of DOF 1 (FDOF) only and response is measured at all DOF (MDOF). The second excitation is a force applied at FDOF with responses measured at all MDOF. This one-at-a-time application of the load cases ensures that the measured responses for each excitation are linearly independent. However, multiple load cases and a subset of load cases can be used if required by the test setup. n the case of modal testing, the mode shapes for the natural frequencies shown in Table 3 are used and measured at the selected DOF shown in Table 1. While practical limitations limit the use of all nine natural frequencies, the system frequencies for completeness are 1.1,.5, 6.97, 1.44, 11.16,.63, 5.49, 5.51, and 9.9 Hz. For Case (a) all DOF at three nodes are considered measured for natural frequencies 1 and. For each case, the unknown parameters are moment of inertia for members 1 and ( 1 and ) and will be estimated due to the modeling error (e) introduced to the moment of inertia of member 4, 4. n order to simplify the study of the impact of modeling error on parameter estimates and still draw useful conclusions, it was necessary to assume that for static cases FDOF is the same as MDOF in the simulations. These measurement locations have been selected to isolate the modeling error from the rest of the system. This is accomplished by measuring the three DOF at each node that contains a DOF of interest. Figures 3 and 4 show the relationship between the unknown parameters, 1 and, and e introduced by the uncertain parameter 4. Figure 3 presents the final parameter estimates using Sanayei, Wadia-Fascetti, Arya, Santini

16 the static stiffness-based and flexibility-based error functions of equations (3) and (4). Parameter estimates obtained using (7) and (11), the modal error functions are presented in Figure 4. n each case, 'o' denotes estimates obtained using the stiffness-based error function and 'x' denotes results from the flexibility-based error functions. These graphs clearly demonstrate that the stiffness-based error functions are superior to the flexibility-based error functions in the presence of modeling error. The error in the estimated 1 is typically less than the error in estimated. This is especially apparent when comparing the results from the flexibility-based error functions. However, even within the same error function formulation the different test scenarios can have a significant impact on the quality of the resulting parameter estimates. n some cases, such as (a) and (c) the results appear to be immune from the error, while in others (b) and (d) the final parameter estimates are contaminated. While the flexibility error function produces results for 1 that may be considered acceptable within ± % error, the errors in have magnified to errors on the order of % and 4% rendering the results useless. t is worth noting that the large vertical scale in Fig. 3(b) and 4(b) suggests that the stiffness error function produced near perfect results, however, these errors are masked by the scale and are typically within ± 1%. Thus, this initial reaction to the results shown in Figures 3 and 4 would lead one to conclude that the effect of modeling error can be extreme. Following is an analytical discussion supporting the observations related to Figures 3 and 4. nitially, explanation is provided for Fig. 3(a), however the conclusions will be generalized to the other cases. n this example, J is symbolically computed using the [E ss ] (3) for Case (a), when all DOF are measured and written in terms of 1,, and e J = e (13) Sanayei, Wadia-Fascetti, Arya, Santini - 16-

17 % error in estimated % modeling error in 4 % error in estimated % modeling error in 4 Case (a) FDOF & MDOF = All (1,, 3, 4, 5, 6, 7, 8, 9) % error in estimated % modeling error in 4 % error in estimated 3 1 Case (b) FDOF & MDOF = 1,, 3, 4, 5, %modeling error in 4 % error in estimated % modeling error in 4 % error in estimated 6 4 Case (c) FDOF & MDOF = 4, 5, 6, 7, 8, %modeling error in 4 % error in estimated % modeling error in 4 % error in estimated Case (d) FDOF & MDOF = 4, 5, %modeling error in 4 Figure 3. Error sensitivity for estimated parameters 1 and using the static data. o = stiffness-based error function; x = flexibility-based error function. Sanayei, Wadia-Fascetti, Arya, Santini

18 % error in estimated % modeling error in 4 % error in estimated % modeling error in 4 Case (a) Measured Modes = 1 & ; MDOF = All (1,, 3, 4, 5, 6, 7, 8, 9) % error in estimated % modeling error in 4 % error in estimated % modeling error in 4 Case (b) Measured Modes = 1 & ; MDOF = 1,, 3, 4, 5, 6 % error in estimated % modeling error in 4 % modeling error in 4 Case (c) Measured Modes = 1 & ; MDOF = 4, 5, 6, 7, 8, 9 % error in estimated % error in estimated % modeling error in 4 % error in estimated % modeling error in 4 Case (d) Measured Modes = 1 & ; MDOF = 4, 5, 6 Figure 4. Error sensitivity for estimated parameters 1 and using the modal data. o = stiffness-based error function; x = flexibility-based error function. Sanayei, Wadia-Fascetti, Arya, Santini - 18-

19 The modeling error, denoted by e, represents the percentage deviation of the uncertain parameters from its true value. Since 4 is an uncertain parameter that will not be estimated, its influence on (13) is only evidenced by e. The nonlinear relationship between J and the governing parameters is clear by inspection of (13). For a given e a pair of 1 and that describe the location of the global minimum exists. The modeling error alters the shape of the J surface and its potential influence on the resulting parameter estimates can be assessed by the sensitivity matrix of the residual error matrix [ E( p)] [ S( p) ] = T { p} (14) where [S] is the sensitivity matrix, [E] is the residual error matrix, such as (3), (4), (7), and (11), and {p}is the vector of unknown parameters. n this example, {p} is the vector { 1 } T. n parameter estimation, the search for the global minimum is complete when the relative changes in { p}, [E], and J are within prescribed tolerances. The symbolic objective function shown in (13) is a scalarized version of (3), however it is still possible to correlate the relevant relationship between J and e in the sensitivity matrix. Equation (1) is the sum of the squares implemented for each term in the residual error matrix and doesn t mix the unknown parameters ( 1 and ) with the modeling error, e. The second partial derivative of (14) with respect to e yields S( p) [ E( p)] = e { p} T e (15) and is equal to zero if the convexity or shape of the J surface and consequently the location of the global minimum is independent of e. nspection of (13) confirms that e doesn t move the global minimum. This is due to the fact that e is an additive constant resulting in only a vertical Sanayei, Wadia-Fascetti, Arya, Santini

20 shift in the J value. Thus, the parameter estimates obtained with the stiffness-based error function and the experiment scenario described in Case (a) will not be contaminated by the modeling error on 4. n other words, the modeling error will not contaminate the final parameter estimates. Two different perspectives of the function J ( 1,, e) for the stiffness-based formulation in Case (a) are presented in Figure 5. J is plotted in Figure 5(a) with the true values of the unknown parameters ( 1 and ) and with respect to e. Fig. 5(a) shows the value of the objective function at the true values of the unknown parameters. The contour plots in Fig. 5(b), adapted from Arya and Sanayei, 3 describe the surface of J at two different levels of modeling error. J surface with -6% error in 4 x E+9 J values E+9 J 8.E+8 4.E+8.E % Modeling error in x 1-4 J surface with +% error in 4 x x 1-4 (a) (b) Figure 5. Relationship between the objective function (J), parameter estimates ( 1 and ), and modeling error (e). (a) J with respect to modeling error and true values for 1 and. (b) Contour plots describing relationship between the surface and unknown parameters. Sanayei, Wadia-Fascetti, Arya, Santini - -

21 The surface shown in Figure 5(b) is described by J(e, 1,, ) where e is a prescribed level of modeling error on 4. Since (15) is equal to zero, the location of the global minimum is not dependent on e, and J at the completion of the parameter estimation will be nonzero and equal to the J obtained with the true values of the unknown parameters shown in Figure 5(a). n this case, the error only alters the magnitude of J at the location of the global minimum. The only difference in the two contour plots in Figure 5(b) is the magnitude of the minimum value of J. The discussion will now be expanded to consider the Case (a) using [E sf ], the flexibilitybased formulation, that involves an unavoidable inversion independent of the number of measurements. This inversion forces the multiplication between the unknown parameters and e creating a situation where (15) will never equal zero and the true parameter values will not be coincident with the global minimum of the surface in (16). The objective function for [E sf ] and Case (a) is of the form J 4 3 e f1( 1, ) + e f ( 1, ) + e f 3( 1, ) + e f 4 ( 1, ) + f5( 1, ) = (16) g ( e,, ) 1 where f i ( 1, ) is a polynomial function of the two unknown parameters. The denominator is a polynomial g (e, 1, ) that is a function of the modeling error in addition to the two unknown parameters. t should be noted that the power of e in (16) is related to the size of the matrix and that larger practical problems will lead to higher powers of e. nspection verifies that the second partial derivative of (16) with respect to e is nonzero demonstrating that the final parameter estimates will be contaminated by the modeling error. The flexibility error function formulation guarantees that any error regardless of where it is in the system will be spread throughout the Sanayei, Wadia-Fascetti, Arya, Santini - 1 -

22 error matrix. Thus, the flexibility-based error functions (4) and (11) will never produce parameter estimates that are immune from error. Given that measuring all DOF is an impractical expectation, scenarios such as those presented in Cases (b), (c), and (d) should be assessed considering the presence of modeling error. More practical examples of static measurement will be using only forces (no moments) and rotation measurements that can be easily be measured using reference independent tiltmeters. For dynamic testing, the mode shapes at translational DOF can be identified using reference independent accelerometers.,1 The static stiffness-based error function, [E ss ], in (3), consists of sub-matrices of the stiffness matrix, applied loads, and measured displacements at a subset of DOF. The submatrices presented in (17), (18), (19), and () are for Case (c) of Figure 3. The sensitivity matrix [S(p)] shown in (14) is a function of both the unknown parameters and the modeling error. The following discussion will use the sub-matrices to illustrate how modeling error can propagate. [ ] K aa 6.67 E = E E 1 sym 1.33 E.67 E E E E 6.63 E e E E 1.73 E 5.1 E E E E e E E e E (17) Sanayei, Wadia-Fascetti, Arya, Santini - -

23 [ ] K ab 3.33 E = E 1.33 E E 1.33 E (18) [ ] K ba 3.33E = E 1.33E E 1.33E (19) [ ] K bb E E = 1 7.5E E + 5.E E 8.E 7.5E 1.33E E 11 () n Case (c), DOF 4, 5, 6, 7, and 8 are measured. Since the unmeasured DOF 1,, and 3 are condensed out of [E ss ], [K aa ] contains the unknown parameter and the modeling error e introduced by the uncertain parameter 4 in an additive form. [K ab ], [K bb ], and [K ba ] are not a function of e. Therefore, the partial derivative of [K bb ] -1 is not a function of e. [K ab ] and [K ba ] are only a function of and their derivatives are not a function of e either. As a result, [E ss ] and the sensitivity matrix associated with the [E ss ] are linear functions of e, where e is never a coefficient to 1 or. Since the search direction determined by the sensitivity matrix is not contaminated by e, the optimization routine locates the actual value of the unknown parameters in spite of existence of modeling error. Also, J defined in (13) is a quadratic function of e where 1 or is never multiplied by e. The only impact of e is that J values are shifted by e such that 1 or are never affected. This condition is shown clearly in Case (c) of Figure 3 where 1 and are identified accurately in presence of high percentages of modeling error that does not alter the shape of the J surface, rather it shifts the surface up increasing the magnitude of the minimum Sanayei, Wadia-Fascetti, Arya, Santini - 3 -

24 value of J. t should also be noted that Equation (15) for Case (c) has been evaluated symbolically and verified as zero. Case (b) and Case (d) each represent a situation that [K bb ] is a function of e. Case (b) represents a measurement scenario that only modeling error contributes to [K bb ] and spreads through out [E ss ] in a nonlinear form. [K ab ] and [K ba ] are functions of. This causes [S(p)] to be a nonlinear function of e due to the triple product [K ab ][K bb ] -1 [K ba ]. As a result, the parameter estimates are contaminated by e nonlinearly. Case (d) is a more complex scenario that is due to poor selection of the measured DOF, the unknown parameters are combined with e in [K bb ]. The modeling error in Case (d) spreads nonlinearly through out [E ss ] generating products of the unknown and uncertain parameters both in the nominator and denominator of [E ss ]. This relationship between the error and the unknown parameters in the different submatrices produces a highly nonlinear error function and sensitivity matrix with respect to e. The modal stiffness-based error function [E ms ], (7), consists of sub-matrices of the stiffness matrix, mass matrix, measured frequencies, and measured modes at a subset of DOF. For the portal frame example, the mass matrix is assumed known. n [E ms ], the mass matrix is multiplied by the measured λ j and does not play any role in modeling error propagation. For the known mass matrix case, [E ms ] simplifies to a formulation similar to the first term of [E ss ]. This clearly explains that all of the cases of the modal measurements shown in Figure 4 behaved similarly to the corresponding cases for the static measurements shown in Figure 3. Using [E ms ], the unknown parameters 1 and in Case (a) and Case (c) were not contaminated by the modeling error and identified both parameters exactly. Case (b) resulted in some degree of Sanayei, Wadia-Fascetti, Arya, Santini - 4-

25 contamination by modeling error and Case (d) resulted in a higher degree of sensitivity to modeling error using [E ms ]. Strategic selection of measurement points can be used to avoid the contribution of uncertain parameters to [K bb ] resulting in a modeling error-tolerant stiffness-based error function. Additionally, the stiffness-based error functions, [E ss ] and [E ms ], performance was consistently superior to the flexibility-based error functions, [E sf ] and [E mf ], in presence of modeling error. LLUSTRATVE BRDGE BENT EXAMPLE Local modeling error that can be caused by undetected damage is illustrated on a bridge bent shown in Figure 6. The structure shown is the pier for a 5.5-meter span concrete bridge deck supported by concrete cap beams and columns for Trinity Bridge in Liberty County, Texas.,6,7 The marked nodes shown in Figure 6 indicate the location of the bi-directional accelerometers or static tiltmeters on the bridge bent. Parameter estimation in the presence of simulated modeling error is performed and the error in the estimated parameters is quantified to demonstrate the influence of undetected local damage (.95 m) 76 (1.93 m) 76 (1.93 m) 76 (1.93 m) 37.5 (.95 m) K s 84 (.13 m) b c Ground Level b b Figure 6. Bridge bent model. b b1 K s Damaged Foundation Sanayei, Wadia-Fascetti, Arya, Santini - 5 -

26 A 3D FEM of the bridge was created in the finite element analysis software, ANSYS 1 to calibrate the behavior of the simplified D FEM. n the D FEM, moments of inertia of the structural components are grouped and labeled as b1, b and c for parameter estimation. The mass density of the beams was adjusted to reflect the tributary mass of the slab. Horizontal springs were added to the D bent to account for the in-plane stiffness of the slab. This pair of springs ensures that the mode shapes of the D model are equivalent to the 3D FEM by preventing the sidesway mode of the bent from becoming the first mode of vibration. The stiffness of the spring (K s ) was determined to be 1.7x1 9 N/m. The modulus of elasticity (E) of the structure was calibrated through ultra-sonic pulse velocity tests, which reflects the tangent modulus at low stress levels. Thus, E is greater than the values typically used in design. Using the blueprints and field data provided by Olson Engineering, section and material properties for the cap beam and piers of the bridge bent are calculated and shown in Table 4. Table 4. Material and section properties of the D bridge bent model. Parameter A (m ) (m 4 ) E (GPa) ρ (kg/m 3 ) b x1 4 b x1 4 c x1 3 The foundation system consists of single concrete piles.41 m in diameter and 5.49 m long. The soil shear modulus was estimated at MPa from in-situ shear wave velocity tests. Each pile was modeled with lumped springs to reflect the soil-structure interaction. A Poisson's ratio of.4 was used to calculate the horizontal stiffness (K hh ) vertical stiffness (K vv ) and the rotational stiffness (K θθ ) using Poulos and Davis. 19 To reduce the loss in accuracy of the parameter estimates due to boundary condition modeling error, a soil-substructure super-element Sanayei, Wadia-Fascetti, Arya, Santini - 6-

27 has been used to capture the lumped stiffness and mass properties of the soil-structure interface (Sanayei et al. 1999, McClain 1996). The numerical values are shown in 6E7N / m [ K = E N m sss ] 11 7 / (1) 7.E7N * m / Rad The same section and material properties will be used for both the dynamic and static parameter estimations. Both types of error functions (Stiffness-based and flexibility-based) are used for parameter estimation. Element cross sectional area (A), length (L), modulus of elasticity (E), and density (ρ) of the cap beam and columns are assumed to be known with a high degree of certainty for the entire bridge bent. Parameter estimation is performed using PARS to update the structural element section properties: moments of inertia for the cap beam and columns ( b1, b and c ) and the lumped spring constants used for the foundation elements (K hh, K vv and K θθ ). The impact of the location and magnitude of modeling error is quantified to determine the level of influence on the estimated parameters. Local undetected damage is simulated on the bridge bent by adding modeling error to K hh, K vv and K θθ of the far right side foundation, indicated in Figure 6 by gray shading. The right foundation stiffnesses K hh, K vv and K θθ are assigned the same level of modeling error from -1% to -1% where -1% indicates complete failure of the right foundation. Two "Estimation Scenarios," shown in Table 5, are investigated with modal and static data in the following subsections. Parameter estimation is used to find the unknown parameters and the modeling error is located at the right foundation for both scenarios. n the first scenario, moment of inertia Sanayei, Wadia-Fascetti, Arya, Santini - 7 -

28 of all the beams and columns are unknown while the parameters of the left three foundations are known. The unknown and known parameters are switched in the second scenario. Table 5. Unknown, uncertain and known parameters. Estimation Scenarios Unknown Parameters Uncertain Parameters (assumed known) Known Parameters 1 b1, b, c Right K hh, K vv, K θθ Left Three K hh, K vv, K θθ Left Three K hh, K vv, K θθ Right K hh, K vv, K θθ b1, b, c Parameter Estimation Using Modal Data n this example, parameter estimation is performed using natural frequencies and mode shapes at a subset of DOF. Two different measurement cases are considered for each uncertainty scenario presented in Table 5. Case (a) uses modes 1 and and Case (b) uses modes 3 and 4. n both cases, the translational DOF at the marked nodes in Figure 6 are measured. The four mode shapes representing (a) axial, (b) rocking, (c) bending, and (d) sway modes are shown in Figure 7 and are simulated using ANSYS. Table 6 summarizes the two simulated measurement cases used in this example. (a) Mode 1, f = Hz (b) Mode, f = Hz (c) Mode 3, f = Hz (d) Mode 4, f = Hz Figure 7. Mode shapes for the D FEM of the bridge bent. Sanayei, Wadia-Fascetti, Arya, Santini - 8-

29 Table 6. Simulated measurement cases. Measurement Cases Modes Measured DOF at Selected Nodes Case (a): Modes & 6 DOF 1, All Translations of Nodes 1,7,13,19,5,9,3,37,4,45,48,53,56 Case (b): Modes & 6 DOF 3,4 All Translations of Nodes 1,7,13,19,5,9,3,37,4,45,48,53,56 The resulting parameter estimates are reported in Figures 8 and 9 for "Estimation Scenarios" 1 and, respectively. There are three types of parameter estimate behavior: convergence, partial convergence, and divergence. As shown in Figures 8 and 9, several cases using the modal flexibility-based error function were not reported due to divergence. There are some cases of partial convergence where one or more of the unknown parameters converged while the rest diverged. Even though some parameters were stable in each case, they were not reported because other parameters in the same case had diverged and therefore the partial convergence was not trusted with the high degree of certainty. Only cases with full convergence are reported. Sanayei, Wadia-Fascetti, Arya, Santini - 9 -

30 % error in estimated b % error in estimated b Case (a) Modes 1 & All translational DOF at bold nodes % error in estimated c % error in estimated b % error in estimated b Case (b) Modes 3 & 4 All translational DOF at bold nodes Figure 8. "Estimation Scenario 1:" Error sensitivity for estimated parameters ( b1, b and c ) using modal data. Modeling error in the right foundation (K hh, K vv and K θθ ). o = stiffness-based error function; x = flexibility-based error function. % error in estimated c 1-1 % error in estimated Khh % error in estimated Kvv % error in estimated K θ θ Case (a) Modes 1 & All translational DOF at bold nodes % error in estimated Khh % error in estimated Kvv % error in estimated K θ θ Case (b) Modes 3 & 4 All translational DOF at bold nodes Figure 9. "Estimation Scenario :" Error sensitivity for estimated parameters (K hh, K vv and K θθ ) of the three left foundations using modal data. Modeling error in the right foundation. o = stiffness-based error function; x = flexibility-based error function. Sanayei, Wadia-Fascetti, Arya, Santini - 3-

31 n Figure 8 the moments of inertia of the cap beam and columns ( b1, b and c ) were unknowns and the right foundation parameters (K hh, K vv and K θθ ) are uncertain. The modal stiffness-based error function, [E ms ], more accurately estimated the unknown parameters when compared with the modal flexibility-based error function, [E mf ]. For measurement Case (a), [E mf ] did not converge when modeling error was introduced to the system. However [E ms ] did converge with relatively low error in the parameter estimates even for modeling error level as high as -1%. For measurement Case (b), [E mf ] converged only with low modeling error levels and even then produced a higher level of error than the [E ms ]. [E ms ] converged for all error levels with low error in the parameter estimates. A comparison of Cases (a) and (b) of Figure 8 indicates that modes 1 and provided more useful information for the estimation of b1, b and c. n Figure 9 the foundation stiffness parameters (K hh, K vv and K θθ ) of the three left foundations are unknown and the right foundation parameters (K hh, K vv and K θθ ) are uncertain. Again, [E ms ] performs consistently better than [E mf ]. As in Figure 8(a), [E mf ] does not converge successfully, with any level of modeling error. Case (b), [E mf ] exhibited convergence for cases with up to 6% modeling error even though the resulting error in the parameter estimates was high and unacceptable for K θθ. The convergence for [E ms ] is excellent. Even though the estimated error in K hh and K θθ, using [E ms ] appear to be zero in the graphs they are not, due to the high error in the [E mf ] estimates For both error functions when modes 3 and 4 are measured, foundation stiffness parameters K hh, K vv and K θθ are estimated more accurately. Modes 3 and 4 excited the foundations in this example far more than modes 1 and. Sanayei, Wadia-Fascetti, Arya, Santini