Multiresponse Parameter Estimation for Finite-ElementModel Updating Using Nondestructive Test Data
Erin Santini Bell1; Masoud Sanayei2; Chitra N. Javdekar3; and Eugene Slavsky4
Abstract: Structural health monitoring using field measurements has developed into a major research area, responding to an increasingdemand for evaluating the integrity of civil engineering structures. Model updating through parameter estimation is a key tool in asuccessful structural health monitoring program. A method for parameter estimation is developed for simultaneous use of static and modalnondestructive test data called the “multiresponse” parameter estimation. An error function normalization technique is also developed tofacilitate effective multiresponse parameter estimation. This normalization technique can mitigate some of the numerical issues encoun-tered during the parameter estimation procedure. However, this technique does not degrade the integrity of the parameter estimationprocedure. Multiresponse parameter estimation provides an increased level of flexibility and feasibility of model updating for structuralhealth monitoring. This paper presents full integration of static and modal nondestructive test data using both stiffness-based andmass-based error functions for structural health monitoring. A benchmark laboratory grid model of a bridge deck is utilized to illustrateapplication of both normalization and multiresponse parameter estimation for updating the stiffness and mass parameters using nonde-structive test data.
DOI: 10.1061/�ASCE�0733-9445�2007�133:8�1067�
CE Database subject headings: Parameters; Estimation; Finite element method; Nondestructive tests; Structural reliability;Monitoring; Measurement.
Introduction
It is an undisputed fact that the asset management and mainte-nance of the United States infrastructure system presents a sig-nificant set of challenges to federal, state, and local governmentagencies. Highway bridges are a key component of the transpor-tation infrastructure system. Of the approximately 590,000 high-way bridges in the United States, 27% are considered structurallydeficient or functionally obsolete �ASCE 2005�. One major chal-lenge is to find a cost effective maintenance system that providesuseful information about the infrastructure in an efficient manner�Phares et al. 2000; and Aktan et al. 2000�.
The current managerial focus for bridge systems requires theability to plan and forecast levels of structural deterioration andthe need for maintenance or rehabilitation procedures. Damagecan accumulate during the life of a structure and reach a level
1Assistant Professor, Dept. of Civil Engineering, Univ. of NewHampshire, Durham, NH 03824 �corresponding author�. E-mail:[email protected]
2Professor, Dept. of Civil and Environmental Engineering, TuftsUniv., Medford, MA 02155. E-mail: [email protected]
3Associate Professor, Dept. of Mechanical Engineering, MassBayCommunity College, Wellesley, MA 02481. E-mail: [email protected]
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such that the structure becomes deficient. Also, some forms ofdamage may remain unidentified due to the inability of visualmethods to observe the damages and can lead to component fail-ure or catastrophic failure in the absence of an effective structuralhealth-monitoring program.
Structural parameter estimation is the art of reconciling an apriori finite-element model �FEM� of the structure with nonde-structive test �NDT� data from the structure. Structural parameterestimation has a great potential for the purpose of finite-elementmodel updating for structural health monitoring of in-servicestructures, specifically as part of a current bridge managementsystem.
For finite-element based parameter estimation, the structure isfirst modeled with discrete elements assembled with known andunknown mass and stiffness properties and boundary conditions.The presented multiresponse parameter estimation procedure cansystematically adjust both the mass and stiffness of the unknownparameters using nondestructive test data. Some examples ofthe stiffness properties are axial rigidity �EA�, flexural rigidity�EI�, torsional rigidity �GJ�, support stiffness �k�, lumped mass�M�, and element distributed mass per unit length ���, whereE=modulus of elasticity; G=shear modulus; A=cross-sectionalarea; I=moment of inertia; and J=polar moment of inertia. Theestimates of mass and stiffness parameters are determined duringparameter estimation and then used to update the FEM. The dif-ference between the “design” parameters and the estimated pa-rameters reveal the condition change in the structure. Using adiscrete mathematical model, the parameter estimates reveal notonly damage location but also damage severity. Parameter estima-tion can also help determine the current load rating of an in-service bridge accounting for any loss in stiffness during the lifeof the bridge. It can also be used to predict the remaining life of
in-service structures given current loading conditions. Fig. 1
tion subject to ASCE license or copyright. Visithttp://www.ascelibrary.org
shows a comparison between direct structural analysis and struc-tural parameter estimation.
Parameter estimation is a more systematic and objective wayof assessing damage in a structure. It is achieved by determiningmass and stiffness properties of structural members in questionthat reflect the field measurements. The estimated section prop-erty values can then be compared to values from as-built struc-tural drawings and the severity of damage can be assessed.
With recent advancement in computational capabilities, moreadvanced methods of both parameter estimation and measurementlocation selections have been developed. Advancements in theseareas have increased the engineer’s ability to perform accuratedamage identification and model updating, creating a more effec-tive structural health monitoring program. Farrar et al. �2003� andFarrar and Jauregui �1998� summarize the current, state of the art,damage identification methods using measured modal responses.Generally, parameter estimation techniques compare the predictedanalytical response of a structure with the actual measured re-sponse. Both Aktan et al. �1997� and Jang et al. �2002� offer acomprehensive study of the integration of the analytical and theexperimental sides of parameter estimation. For this purpose, pa-rameters of the FEM are updated until the predicted response iswithin prespecified bounds of the measured response. Although
Fig. 1. Structural analys
this procedure may seem relatively simple, there are several
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sources of error that can impede the practical application of pa-rameter estimation.
Literature in the area of parameter estimation for health moni-toring of structures is extensive. Stubbs and Osegueda �1990�develop a theory using changes in modal characteristics in beams,plates, and shells to detect damage. This method was refined andapplied to offshore structures by Kim and Stubbs �2002�. Yeoet al. �2000� propose a statistical approach to static parameterestimation through hypothesis testing. Hjelmstad and Shin �1997�developed a method suited to sparsely sampled static and modalresponses used independently. Doebling et al. �1998� uses vibra-tion test data to identify a structure’s local stiffness through thedisassembly of the flexibility matrix. Frequency domain data,which are more compact than time domain data and more readilyreveal the modes of vibrations, are used for parameter estimationin Koh et al. �2000�. He and De Roeck �1997� present a method ofstructural damage detection using autoregressive modal data. Guo�2002� presents a method to improve a structural model usingvibration test data and an element-level energy error estimationmethod to identify poorly modeled regions of stiffness and massin a structure distinguishing these from the better modeledregions.
There are several optimization methods currently available for
us parameter estimation
is vers
determination of the parameter estimates �Venkataraman and
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Haftka 2004�. Arya and Sanayei �2000� integrated genetic algo-rithms �GAs� with traditional hill climbing �HC� methods toinitially locate the vicinity of global minimum using GA andswitching to HC for rapid convergence. Chou and Ghaboussi�2001� present the application of GA to structural damage detec-tion using static measurements of displacements. Lee et al. �2005�present a neural network-based damage detection method. In thismethod, mode shape ratios are used as input to the neural networkand serve to reduce the effect of the modeling error in the baselineFEM. Franco et al. �2004� present a parameter estimation tech-nique based on an evolutionary strategy. The method showspromise with simulated modal data and avoids shortcomings ofclassical optimization methods such as the need for reliable initialestimates. This paper uses the Gauss–Newton and conjugate gra-dient methods �Chong and Źak 2001� for parameter estimation.
Parameter estimation can be heavily influenced by measure-ment noise �measurement error� and uncertainties in the knownparameters �modeling error�. A significant source of error that candrastically impact the validity of the results of parameter estima-tion is modeling error or the uncertainty in the parameters of aFEM. It is caused by incorrect assumptions used in the creation ofthe initial FEM. Examples of modeling error are an overestima-tion or underestimation of the initial stiffness and mass propertiesas well as changes due to subsequent deterioration or damage.Yeun et al. �2004� use a two-step process for structural parameterestimation of Phase I benchmark studies to mitigate uncertaintyand bias error. For further information regarding impact of mod-eling errors on parameter estimation modeling error refer toSanayei et al. �2001� and Chase et al. �2005�.
The success of the parameter estimation process is also depen-dent on the quality, location, and type of NDT measurements. Thesensor location and type of sensors �input� play a major role in theaccuracy of the parameter estimates �output�. Recent advance-ment in sensor technology has improved the accuracy of the mea-surements. With proper placement and data acquisition system,the amount of noise or measurement error in the preprocessedNDT data sets can be limited to low levels. The quality of theNDT measurements is controllable in terms of measurementtechniques and measurement apparatus �Aktan et al. 1997;Sanayei et al. 1992�.
This paper focuses on combining multiple data types for mul-tiresponse parameter estimation of both stiffness and mass param-eters. Other researchers �Wang et al. 2001 and Oh and Jung 1998�have worked on combining static and modal data on a limitedscale. The proposed method uses a modified error function proto-col for statistical parameter estimation. Kiddy and Pines �1998�present a technique for simultaneous updating of mass and stiff-ness matrices using a sensitivity-based method. They propose aconstraint on the number of unknown parameters to avoid theproblem of simultaneous updating multiple parameters. Catbaset al. �2004� investigated the use of multiple inputs for modalanalysis of large-scale structures. The procedure presented in thispaper allows for elemental structural properties influencing sys-tem stiffness to be estimated independently from the propertiesthat influence system mass �Javdekar 2004�.
Parameter Estimation
A Matlab-based parameter estimation program, PARameter Iden-tification System �PARIS� �Sanayei 1997�, developed at TuftsUniversity, has been utilized to estimate the unknown stiffness
and mass parameters of the structural elements of a FEM. Typi-
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cally, structural parameters for finite-element model updating in-clude axial rigidity �EA�, flexural rigidity �EI�, and torsionalrigidity �GJ�, foundation stiffness �k�, mass per unit length �m=�AM�, and lumped mass �M�.
The structure can be excited either statically with appliedloads, F, measuring displacements and rotations, U, or dynami-cally measuring frequency response functions of lightly dampedsystems and extracting natural resonance frequencies, �, andassociated mode shapes, �, for linear parameter estimation. Aselected number of measurements gathered sparsely at certainstrategically selected degrees of freedom �DOF� can be used forparameter estimation. PARIS can utilize both static and modaldata such as displacements and rotations under stationary loads,or extracted natural frequencies and associated mode shapes forparameter estimation. PARIS can simulate and also receive bothcomplete and sparse static and modal data sets.
Four distinct error functions are used for multiresponse param-eter estimation in this paper. However, the combining techniquepresented is applicable to several other error functions withinPARIS such as those based on strain measurements �Sanayei et al.1997� among others. A brief formulation of each error functionused in this paper is presented here for clarity in the example.
PARIS also allows for grouping the several parameters thathave same stiffness and mass properties and also are expected tohave similar final estimated values together. Grouping allows fora smaller total number of unknown parameters and consequentlya smaller number of required measurements for estimating thoseparameters, both of which increase the computational speed. Theunknown parameter grouping schemes can be refined based onthe change in the grouped parameter estimates. A significantchange in a grouped parameter estimate indicates that one ormore of the parameters in that group have experienced damage. Itis feasible to methodically subdivide the unknown parametergroup to locate the damaged member.
Static Stiffness-Based Error Function
The static stiffness-based error function, �Ess�p��, was developedby Sanayei and Nelson �1986�
This error function is based on the residual forces at a subset ofDOF. It is essentially the difference between the predicted andmeasured forces. �Fpredicted� is calculated using the analytical stiff-ness matrix and the set of measured displacement, �U�, from anNDT at a subset of DOF. �Fmeasured�=applied set of live loadcases. The complete formulation using a subset of measurementsthat can be different from the applied load is presented by Sanayeiand Onipede �1991�.
Static Flexibility-Based Error Function
The static flexibility-based error function, �Esf�p��, was developedby Sanayei et al. �1997�
This error function is based on the residual displacements at asubset of DOF. Similar to the formulation of �Ess�p��, �Upredicted� iscalculated using the analytical stiffness matrix and the set of ap-
plied live load cases. The measured displacement set, �Umeasured�
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=response to the set of applied live load cases from an NDT at asubset of DOF.
Modal Stiffness- and Mass-Based Error Function
Using the basic modal analysis theory, the modal stiffness-basederror function, �Ems�p��, was developed by Gornshteyn �1992�and enhanced by Sanayei et al. �1999�
�Ems�p�� j = �K�p����� j − � j2�M�p����� j �3�
This error function is based on the residual modal elastic andinertia forces predicted at a subset of DOF. It uses the modeshapes, ���, and natural frequencies, ���, extracted from an NDTdata set. The natural mode of vibration used is labeled withsubscript j. �K�p�� and �M�p��=analytical stiffness and massmatrices.
Modal Flexibility- and Mass-Based Error Function
The modal flexibility-based error function, �Emf�p�� was devel-oped by Arya �2000� and enhanced by Sanayei et al. �2001�
�Emf�p�� j = � j2�K�p��−1�M�p����� j − ��� j �4�
Similar to the formulation of �Esf�p��, this error function is basedon residual modal displacements predicted at a subset of DOF.
Minimization of the above error functions leads the search forparameter estimation of the objective function, J�p�, that is theFrobenius norm of any error function
J�p� = �i
�j
E�p�ij2 �5�
where i=measured DOF; and j=measured load case or mode ofvibration.
Error Function Normalization in MultiresponseParameter Estimation
When working with multiple error functions, some numerical dif-ficulties are encountered. A wide range of numerical values isprocessed when using multiple sets of measurement types due tousage of different units and scale. For example, displacements aremeasured in “inches,” rotations are measured in “radians,” andstrains are measured in “�inch/ inch.” In multiresponse parameterestimation, it is essential that one measurement type does notovershadow another, therefore error function normalization�EFN� is required.
When normalizing the error function, the parameter estimationprocess is manipulated at its core. The error function �residual� isthe basis for the parameter estimation procedure. The disparitybetween cell values in the error matrix can magnify the peaks andvalleys that are inherent to the objective function surface �costfunction�, complicating the parameter estimation. The role ofEFN is to smooth the objective function surface without mutingthe actual global minimum. Normalization is applied both at theparameter level �Sanayei et al. 1999� and at the error functionlevel �Santini-Bell and Sanayei 2005�.
In addition to smoothing the variation in magnitudes withinthe �E�p��, EFN also prepares the different error functions to be
used simultaneously. For example, the units associated with the
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�Ess�p�� are forces and moments while �Esf�p�� has units of dis-placement and rotations. Using EFN would make both �Ess�p��and �Esf�p�� unitless and decrease the chance that one error func-tion would overshadow the other �Santini 2003�. This paper usesthe initial value of the error functions �E0�P�� calculated usinginitial estimates of parameters for normalizing the error functions.Typically, information from existing as-built structural drawingsand field measurements is the basis for the calculation of theinitial parameter values.
The EFN based on initial value, �E0�P��, uses the a prioriparameter values, �pI� for creating an error function matrix�E0�pI��. Each entry of this matrix can have contributions fromseveral unknown parameters. The error function �E�p�� and thesensitivity matrix �S�p��, which is the partial derivative of theerror function with respect to each unknown parameter, in eachiteration, including the first iteration, is divided element by ele-ment by �E0�p�� for normalization at the matrix cell level. TheEFN must be applied to both the �E�p�� and �S�p��, since �S�p�� iswhat directs the search for the global minimum using both theGauss–Newton and conjugate gradient optimization methods pre-sented in this paper
�EN�p�� =�E�p��
�Eo�pI��
�SN�p�� =�S�p��
�Eo�pI���6�
This type of normalization creates both a unitless error functionand a unitless sensitivity matrix. Matrices �E�p��, �S�p��, and�E0�pI�� are of the same sizes NMDOF�NSF �number of mea-sured DOF�number of sets of forces or natural modes�. �p�=vector of unknown parameters and �pI� represents its initial val-ues of size NUP�1 �number of unknown parameters�. By usingEFN, each component that contributes to the error function matrixis affected. In order to account for different sensors, PARIS hasthe ability to apply weight factors. Because this was a laboratoryexperiment under controlled circumstances, the weighting optionwas not used for this scenario.
Multiresponse Parameter Estimation
When different error functions that are based on different mea-surement types are used it is necessary to combine such data in asystemic and appropriate fashion to ensure that the integrity of thedata is not compromised. A stacking procedure can be applied tothe combination of load cases or modes of vibration using differ-ent error functions. The stacking allows the use of several differ-ent measurement types simultaneously, increasing the number ofunknown parameters that can be estimated and allowing the com-bination of the static and modal data.
Within each active normalized error function �E�p��, a vector�E�p�� of size NM�1 is calculated where NM=NMDOF�NSF.Then each �E�p�� is stacked to create a �EStack�p�� which containsthe information from each active error function for the entire
multiresponse parameter estimation scenario
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�EStack�p�� =��E�p��1
]
�E�p��n−1
]
�E�p��n
�7�
where n=number of error functions.The stacked error function Eq. �7� is used to form the objective
function in Eq. �5� and then optimized using any standard opti-mization techniques such as the Gauss–Newton or conjugate gra-dient method.
This type of data stacking to one error function vector willallow experimentalists to use all sensor types at their disposal forthe NDT with the knowledge that these data can all be used to-gether for parameter estimation. Using multiresponse parameterestimation, the user has more control over the information inputinto the parameter estimation procedure and can make the mostefficient use of information collected from a NDT.
Laboratory Grid Model of Bridge Deckwith Measured Responses
The UCII grid �University of Cincinnati Infrastructure Institute�was designed as part of a collaborative research effort involving,UCII, Northeastern University, and Tufts University. The UCIIgrid was intended to capture the behavior of a highway steel-stringer bridge deck in order to assist in the validation of themethod �see Fig. 2�. The connections were designed to supportthe predetermined static load, 666 N �150 lb� without failure. TheUCII grid was also designed so that the significant modal infor-mation for parameter estimation could be measured within the0–100 Hz range �Ciloglu et al. 2001�. This is a practical fre-quency range because most typical steel-stringer bridge measur-able modes are within this range. The testing procedure for theUCII grid is also designed to excite the structure within the linear
Fig. 2. UCII grid �Ciloglu et al. 2001, with permission�
elastic range only.
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The UCII grid was a 3.65 m �12 ft��1.83 m �6 ft� grid pro-viding enough laboratory space for the required monitoring anddata collection equipment. The UCII grid members were7.62 cm �3 in.��5.08 cm �2 in.��0.476 cm �3/16 in.� structuralsteel tubing, in both the transverse and longitudinal directions.Fig. 3 represents the drawing of the UCII grid’s final design. Theconnection zones between the tubing were constructed using0.476 cm �3/16 in.� steel plates with A307 0.635 cm �1/4 in.�diameter bolts. Detailed drawings for each type of connection areshown in Fig. 4. To simulate the performance of a bridge deck,four neoprene pads supported the UCII grid on the sawhorses atthe four corners as shown in Fig. 5.
Type and Method of Nondestructive Testingon UCII Grid
Both static and modal NDT were performed on the UCII gridusing static loads at predetermined locations on the UCII grid andusing an impact hammer. Three identical static NDT were per-formed on the UCII grid using 222 N �50 lb�, 444 N �100 lb�, and666 N �150 lb� loadings. For the static tests, vertical displace-ments, rotations, and strains were measured. The measurementlocations for each date type were the same for all three tests �seeFig. 6�. Based on the data quality analysis performed by North-eastern University �Wadia-Fascetti et al. 1999� only data fromTest 3 were used for parameter estimation.
Experimental modal analysis was used to determine the modalparameters �frequencies and mode shapes� of the assembled UCIIgrid. An impact hammer was used to excite al 21 nodes andaccelerations were measured with all 21 accelerometers. Next, themeasured data were processed at UCII using frequency and timedomain techniques to obtain the modal shapes and natural fre-quencies of the structure.
Primarily, the stiffness and mass of connections, structural tub-ing elements, and the stiffness of the pads against the verticaldeflection of the UCII grid were the target of parameter estima-tion for model updating. Thus, vertical input and measuringdevices �accelerometers� were placed at the intersection of longi-tudinal and transverse members �at connection points�. Measuringresponse at all connection points provided ample information forparameter estimation.
Static NDT Displacement Data Selection
The analysis of the data revealed that many of the load cases,
Fig. 3. UCII grid, plan view �Ciloglu et al. 2001, with permission�
where the load was located toward the edges of the UCII grid,
tion subject to ASCE license or copyright. Visithttp://www.ascelibrary.org
inted w
created uplift at some locations along the opposite edges. Sincethe FEM of the UCII grid could not accurately account for theuplift, these load cases were discarded. Each static loading andmeasurements was repeated ten times. Basic statistical analysis onthe ten sets of measurements was performed and those measure-ments with high standard deviations were discarded. For consis-tency, when a measurement was discarded, the entire load casewas also discarded to maintain a constant 11 static displacementmeasurements for each load case used �Fig. 6�. As a result ofthese elimination techniques, three 666 N �150 lb� load cases re-mained to be used as data for parameter estimation. These threeindependent load cases were located at E3, G3, and I3; the middleloading points along Line 3. The three load cases were applied atthe UCII grid connections individually and yielded a total of 33vertical displacement measurements.
Fig. 4. UCII grid, connection types �Repr
Fig. 5. UCII grid, support pad positioned between grid and support�Reprinted with permission from Ciloglu et al. 2001�
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Dynamic NDT Measurement Location Selection
Modal identification was performed using data from time domain�TD� and frequency domain �FD� tests, which was in the form offrequency response function �FRF�, was postprocessed, and natu-ral frequencies and mode shapes were extracted. The modal pa-rameters acquired from the TD and FD tests were then comparedto confirm their consistency by Ciloglu et al. �2001� and Slavsky�2005�. The FD tests generated a repeating frequency for Modes 4and 5. In contrast, TD tests generated distinct frequencies �how-ever, with some degree of modal coupling�. The resulting modeshapes from TD Test 3 are shown in Fig. 7. The TD modal datawere used for the parameter estimation presented.
Parameter Estimation for UCII Grid Experiment
The first step in parameter estimation is to create a FEM that cancapture the general behavior of the structure. The FEM of the
ith permission from Ciloglu et al. 2001�
Fig. 6. Static displacement measurement locations
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UCII grid uses separate elements for the connection zones andstructural tubes. Parameter groupings are used to reduce the num-ber of unknown parameters for similar structural elements. Bychanging the grouping configuration of the unknown parameters,rather than changing the elements in the FEM itself, differentparameter estimation scenarios were created for the UCII grid.
Given the purely vertical nature of both the static loadings andmodal excitations, the measured NDT data indicate that there isno horizontal motion �in the X and Y direction� or twisting motion�about the Z axis�. In order to both simplify the FEM, in terms ofthe number of DOF, and create a FEM that would best reflect theNDT data using a smaller number of DOF; UX, UY, and �Z DOFof the FEM were restrained. Based on the nature of the load casesand excitations, only vertical displacements were expected for theUCII grid. Static displacements and modes of vibration from boththe unrestrained and restrained FEM were compared to ensurethat there were no adverse affects caused by the restrained DOF.
Finite-Element Model for UCII Grid
The FEM for the UCII grid was created with 85 nodes, 96 frameelements, and four linear spring elements. This FEM was createdin both PARIS and SAP2000. Three-dimensional beam elementswere used to represent both the beams and the connections zones.For the four support pads, linear springs with stiffness only in thetranslation directions, UX, UY, and UZ, �KUX, KUY, and KUZ� wereused.
In order to account for the additional weight of the bolts, theclip angles, gusset plates, and sensors or the loss of weight due todrilled holes, the cross-sectional area “Am” represents the area-
Table 1. Initial Section Properties of the Structural Members
Structural member typeA
�in.2 �cm2��Am
�in.2 �cm2
Tubes �TS 3�2�3/16� 1.64 �10.6� 1.64 �10.
Connection elements 1.64 �10.6� 3.70 �23.
Fig. 7. Experimental mode shapes and frequencies from time domainNDT data, Test 3 �Ciloglu et al. 2001, with permission�
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mass for each element. Therefore, mass per unit length of eachconnection member is “�Am.” Also to account for bending stiff-ness changes due to cuts in the transverse direction, clip angles,gusset plates, bolt holes, and bolt tightness, Ixx represents theeffective moment of inertia of the connections. The initial valuesfor the connection element section properties are shown in Table1. A refers to effective cross-sectional area used for axial stiffnesscalculation. Am refers to effective cross-sectional area of mass,which is used for mass based parameter estimation. This arrange-ment allows for separate estimation of mass and stiffness param-eters for the connection elements �Javdekar 2004�. Ixx�momentof inertia about the primary bending axis. Iyy�moment of inertiaabout the secondary bending axis. J�polar moment of inertia.The values for the tubular steel �TS� sections were obtained fromthe AISC Manual.
The bearing pads are modeled as linear springs. The verticaldisplacement tests on the bearing pads conducted at UCII yieldeda range of stiffness values from 4.37 to 17.5 kN/cm�2.5–10 kips/ in.� for the four stiffness pads �Ciloglu et al. 2001�.Axial stiffness of 4.37 kN/cm �2.5 kips/ in.� is used as the initialparameter of all four neoprene bearing pads. The translationalstiffness of the springs in the x and y directions were set at highmagnitudes to effectively make the springs rigid in the plane ofthe UCII grid. The unknown parameter for the springs is thevertical stiffness, Kz, only.
Parameter Estimation for Model Updating UsingNDT Data
For the UCII grid model updating, two model cases were created.Model Case 1 estimated individual bearing pad stiffnesses andtwo grouped connection stiffness and mass parameters. ModelCase 2 used the pad stiffness estimates and refined the connec-tions to four groups of stiffness and mass parameters �Slavsky2005�. In both model cases, the TS section properties were con-sidered known. In Model Case 1, the two categories of unknownparameters were: �1� the bearing pads and �2� the connectionzones. The four bearing pads have a major impact on the overallresponse of the UCII grid, and as a result the vertical springstiffness parameters for the bearing pads were not grouped andestimated individually. Since there are a large number of connec-tion elements, those that were expected to behave similarly weregrouped together to a single unknown parameter which reducesthe total number of unknown stiffness and mass parameters. ForModel Case 1, the exterior connections were grouped togetherand interior connections to a second group.
Eight parameters were used in Model Case 1: K1, K2, K3, K4,I1, I2, Am1, and Am2. Connections were grouped into “external”�Parameters I1 and Am1� and “internal” �Parameters I2 and Am2�.Parameter estimation was performed using only static test data togain a better understanding of the differences in the FEM pre-dicted responses versus and the measured responses. It revealedthat the majority of the static stiffness changes occur in the fourcorner pad stiffness values. The estimates of the vertical stiffness
tion subject to ASCE license or copyright. Visithttp://www.ascelibrary.org
of the bearing pads are: K1=12.66 kN/cm �7.23 kip/ in.�,K2=14.8 kN/cm �8.43 kip/ in.�, K3=13.1 kN/cm �7.50 kip/ in.�,and K4=13.4 kN/cm �7.65 kip/ in.� �Slavsky 2005�. The FEMwas updated using the vertical stiffness values of the springs.
In Model Case 1, the connection elements were subdividedinto two groups �interior and exterior�. From a comparison ofdisplacements at bearing pads with NDT data, it was clear that adisplacement match with NDT data was achieved. Since the lon-gitudinal elements of the UCII grid was observed to play a majorrole in the load transfer and controlling the displacements, threeconnection groups were created for the longitudinal elements ofthe UCII grid and the transverse connection elements weregrouped into a single group for Modal Case 2.
Fig. 8 illustrates how the 64 connection elements logically andeffectively are separated into four meaningful groups for param-eter estimation of Model Case 2. Group 1 represents the 28 cross�transverse� connection elements grouped as one. Groups 2, 3,and 4 each represent 12 longitudinal connection elements of Lines1, 3, and 5, respectively. Each group represents one unknownstiffness and one unknown mass parameter each creating eightunknown parameters to be estimated using multiresponse param-eter estimation.
Multiresponse parameter estimation requires EFN. A compari-son of the performance of the normalization methods presented inthis paper was completed using both the Gauss–Newton and con-jugate gradient optimization techniques. This comparison showsthe very effective performance of EFN. All multiresponse param-eter estimation cases presented will use normalization with re-spect to the initial parameter values, �Eo�p��.
Multiresponse Parameter Estimation Results
The scope of this parameter estimation is to update both mass andstiffness properties of the connection elements, 4I+4Am. As dis-cussed above, the four pad stiffness values are considered knowndue to their successful estimation using only the static data lead-ing to a close match of displacements at the four corner pads withNDT data �Model Case 1�. Therefore, the unknown parametergroups are the moment of inertia and mass parameter of the fourconnection groups leading to a total of eight unknown parameters�Model Case 2�. The surface plots, Fig. 9, show the complexity of
Fig. 8. Connection grouping scheme for UCII Grid Model Case 2
the flexibility-based error surfaces. The right graphs in Fig. 9
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show the path of the parameter estimation from the initial value tothe final parameter estimates. The parameter estimates from theflexibility-based cases are more reflective of the laboratory NDTdata, as shown in Figs. 10 and 11, therefore they are the onlysurfaces that are presented.
Visualization of the objective function surfaces plots �J plots�is a powerful method to gain insight to the nature of the indi-vidual parameter estimation scenario and optimization process.For plotting the objective function J�p� for “n” unknown param-eters, the surface will be in n+1 dimensional space. Since it is notpossible to plot in more than the three dimensions �3D�, the sur-face plots in 3D are created using only two unknown parameters.The rest of the unknown parameters are maintained at their finalestimated parameter values. This is analogous to slicing a multi-dimensional space to observe the objective function surface ver-sus only two unknown parameters.
The flat surface of the error surface shown in Fig. 9 indicatesthat this parameter estimation scenario is suited for the conjugategradient optimization technique. Due to the relatively large stepsize of the Gauss–Newton technique, surfaces with many localminima and a flat surface in the vicinity of the global minimumare better served by the conjugate gradient due to the small stepnature of the algorithm �Slavsky 2005�.
For Model Case 2, the moment of inertia and area mass pa-rameter estimate values of both the stiffness-based and flexibility-based case are summarized in Table 2. The flexibility-based caseresults indicate the highest parameter estimates occur in the UCIIgrid Line 3 group �I3 and Am3�. The second highest area massesoccur in Lines 1 and 5, and the lowest area mass values occur inthe area mass of the cross/transverse connection elements �Am1�.These results make sense physically, since the largest gussetplates with the most bolts are used along the middle line �Line 3�.
The parameter estimates from the stiffness-based case do notreflect visual observations of the UCII grid and engineering judg-ment based on the known construction techniques of the UCIIgrid. This is an early indication that these parameter estimates donot reflect the condition of the UCII grid. This observation isproved accurate when the response from the updated model usingthe parameter estimates from both the stiffness and flexibilitybased error functions are compared with the NDT data in Figs. 10and 11.
It is important to assess the physical meaning of parameterestimates derived from the stiffness and flexibility-based cases.Even though inspection of the numerical results reveal significantinformation regarding the fitness of the parameter estimation, di-rect comparison with the NDT data provides an objective andquantifiable basis for comparison. Each set of parameter estimatesfor the benchmark UCII grid were used to update the FEM. Boththe static and modal responses for the updated models were thencompared with the NDT data. The tools of simulated displace-ment plots, frequency matching, and modal assurance criterion�MAC� matrices are used to compare the predicted response tothe NDT data. Based on the comparisons detailed below, the flex-ibility based 4I+4Am case is considered the best multiresponsecase.
Displacement plots for each of the three longitudinal UCII gridlines were created using the estimated parameter values, shown inTable 2, and the measured NDT data. All three plots show excel-lent match between the NDT measurements and updated modelresponse for the flexibility-based case and not such a good matchfor its stiffness-based counterpart �Fig. 10�. It is important to re-member that static displacements are impacted by the stiffness
parameters alone and not by the mass parameters.
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tion subject to ASCE license or copyright. Visithttp://www.ascelibrary.org
Fig. 9. Objective function surface plots and convergence paths for 4I+4Am, SF+MF case, conjugate gradient
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Both stiffness and mass parameters affect modal frequenciesand mode shapes of a structure. A visual chart of NDT versuspredicated frequencies for both stiffness based and flexibilitybased cases is shown in Fig. 11. It can be seen in the chart that theflexibility based case produces a closer match to the NDT fre-quencies. Table 3 shows the MAC values measuring the degree offitness between the mode shapes extracted during NDT and thecorresponding mode shapes obtained from �a� the initial FEM and�b� the updated FEM. Note that the NDT modes are shown in thevertical column and the FEM modes are shown in the horizontalrow.
Fig. 10. Static displacements usi
Using results of Model Case 2, Table 3�a� indicates a strong
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coupling of Modes 3 and 4 and also problems with Modes 7 and8 being out of order. This was discovered by comparing the modeshapes from the analytical model using the initial parameter val-ues with the experimentally identified mode shapes. As a resultonly Modes 1, 2, 3, 6, and 9 were used for parameter estimationmeasured by accelerometers at 21 vertical DOF �all 21 connec-tions�. Comparison of the MAC values indicate improvement inMAC values �closer to 1.0� for measured Modes 1, 2, 3 6, and 9.Based on a comparison with both static and modal NDT data, theflexibility-based 4I+4Am case is considered the best multire-sponse parameter estimation case. Fig. 11 shows that the correla-
ameter estimates, Load Case G3
ng par
tion between the modal frequencies has significantly improved.
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tion subject to ASCE license or copyright. Visithttp://www.ascelibrary.org
JOURNA
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Fig. 10 illustrates that the static vertical displacements are signifi-cantly closer to the NDT data as compared with that of thestiffness-based case. The superiority of the flexibility-based errorfunction parameter estimates as compared to the results from thestiffness-based error function is unique to this structure and theassociated parameter estimation scenario.
Conclusions
This research demonstrates that it is feasible to simultaneouslyestimate stiffness and mass parameters using multiresponse pa-rameter estimation with static and modal NDT data. The necessityand effectiveness of error function normalization with multire-sponse parameter estimation due to varying scale were presentedin this paper and illustrated in the benchmark UCII grid labora-tory model. The importance of mode shape behavior matchingwas demonstrated with respect to the results from mass-based
4I+4M, SS+MS 4I+4M, SF+MF
72.8 �1.75� 129.9 �3.12�
65.8 �1.58� 106.6 �2.56�
62.9 �1.51� 124.9 �3.00�
64.9 �1.56� 74.5 �1.79�
8.9 �1.38� 15.5 �2.38�
26.2 �4.06� 24.9 �3.87�
28.5 �4.41� 30.9 �4.80�
24.8 �3.84� 22.9 �3.56�
7 102
117.8 35.29
F+MF Case
5 6 7 8 9
al parameter values
0.000 0.000 0.000 0.062 0.000
0.002 0.006 0.000 0.000 0.001
0.017 0.007 0.004 0.000 0.032
0.371 0.024 0.020 0.000 0.000
0.544 0.001 0.009 0.033 0.003
0.000 0.001 0.840 0.000 0.108
0.000 0.000 0.000 0.953 0.000
0.000 0.904 0.000 0.009 0.023
0.000 0.005 0.048 0.000 0.909
es of case 4I+4Am, SF+MF
0.000 0.000 0.001 0.083 0.000
0.002 0.000 0.002 0.000 0.001
0.016 0.001 0.006 0.000 0.075
0.495 0.004 0.019 0.000 0.000
0.454 0.015 0.001 0.023 0.003
0.003 0.954 0.000 0.000 0.042
0.000 0.000 0.012 0.965 0.000
0.000 0.007 0.919 0.048 0.029
Table 2. Parameter Estimate Values from Multiresponse, 4I+4Am Scenario
Initial parameter values
I1-Cross �cm4�in.4�� 77.4 �1.86�
I3-Line 3�cm4�in.4�� 77.4 �1.86�
I3-Line 3 �cm4�in.4�� 77.4 �1.86�
I4-Line 5 �cm4�in.4�� 77.4 �1.86�
Am1-Cross �cm2�in.2�� 23.9 �3.70�
Am2-Line 1 �cm2�in.2�� 23.9 �3.70�
Am3-Line 3 �cm2�in.2�� 23.9 �3.70�
Am4-Line 5 �cm2�in.2�� 23.9 �3.70�
No. of iterations for convergence
J�p�
Table 3. MAC Values �a� Using Initial Parameter Value; �b� Using 4I+4Am, S
tion subject to ASCE license or copyright. Visithttp://www.ascelibrary.org
parameter estimation. The objective function surface plots andconvergence path plots were both used as a means to assess thebehavior of the optimization algorithms used in parameter estima-tion. These plots provide a visual insight and serve to unlocksome of the black-box feeling associated with parameter estima-tion in a multidimensional parameter space.
Since static displacements and modal data were used simulta-neously there is more NDT data available for parameter estima-tion. As a result, more structural parameters can be estimated perset of field measurements. This paper illustrated that multiple datatypes can be successfully combined to perform parameter estima-tion. A larger and more assorted data set can provide flexibility tothe experimenters when performing data quality analysis and se-lection of the subset of measurements. Two model cases wereused for parameter estimation and model updating. Since theUCII grid bearing pads were much more flexible than the grid,only static data were used to identify the four bearing pad stiff-nesses in Model Case 1. Using this updated model a match at thebearing pad locations was achieved between the predicted andmeasured responses. Model Case 2 used a combination of staticand model NDT data for estimation of stiffness and mass param-eters of the connections. Although only eight parameters wereestimated in Model Case 2, those eight parameters represented 49structural parameters, condensed through grouping. The param-eter estimates using the flexibility-based error functions resultedin a better correlation between the updated model responses andlaboratory NDT data for the UCII grid.
There are several examples of bridges that are heavily instru-mented for structural health monitoring �Cuelho et al. 2006�. Inmost cases this instrumentation is a combination of strain gaugesand tiltmeters �Riad et al. 2006�. However, the common problemthat face bridge owners and decision makers is how interpret thisdata into a decision making, assessment management tool. Mul-tiresponse parameter estimation, as presented in this paper, is apostprocessing tool that can use field measurements to obtainmeaningful locations of stiffness and mass changes of the testedstructures. This information can then be used to allocate fundsand manpower for further visual inspection and rehabilitation ifrequired. In closing, regardless of the mathematical tools avail-able for model updating, the engineering judgment of the struc-tural engineer is paramount to successful parameter estimation formeaningful model updating and structural condition assessment.
Acknowledgments
The writers are grateful for partial funding of this research byNSF Grant No. CMS-9622067. They would like to thank Profes-sors Aktan �Drexel University� and Helmicki �The University ofCincinnati� and their students for performing the experiments onthe physical model and for access to this experimental data.
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