System Identification of Constructed Civil Engineering Structures and Uncertainty

Transcription

1 System Identification of Constructed Civil Engineering Structures and Uncertainty Qin Pan Ph.D. Thesis Defense Presentation Advisor: Dr. A.E. Aktan Committee Members: Drs. Aktan, Gurian, Montalto, Moon, Tan December 17, 2007

2 Key Research Objective: The impact of Epistemic Modeling Uncertainty Associated with System Identification of Constructed Systems; How to Recognize & Mitigate it

3 Outline Background & Definition Past Research: Sys-Id Applications on Civil Structures Research Motivations & Objectives Impact of Epistemic Modeling Uncertainty on the Reliability of Sys-Id Recognition & Mitigation of Epistemic Modeling Uncertainty Model Adequacy Evaluation Conclusions Recommendations & Future Work

4 Historical Outline of Characterization of Existing Systems Time Static testing Start of dynamic testing Development of FFT Modal testing ARMAV, LSCE, PTD, SSI, CMIF & etc Advanced sensing & networking Experimental Start of matrix structural analysis Development of FEM Production-level FEM program Analytical Sys-Id in control engineering Hart & Yao (1977) Liu & Yao (1978) Sys-Id in civil engineering Farrar et al (2001) Integrative

5 System Identification (Sys-Id) System identification is a general term to describe mathematical tools and algorithms that build/tune dynamical models from measured data (Wikipedia). (1) (6) Utilization of Model for Simulation, Scenario Analysis Observation and Conceptualization Evaluating Modeling Uncertainty, Model Calibration & Check Model Adequacy (5) Sys-Id (2) A-Priori Modeling Integration, Processing and Interpretation of Data (4) Measurement, Monitoring, Controlled Experimentation (3)

6 Background & Definition Outline Past Research: Sys-Id Applications on Civil Structures Research Motivations & Objectives Impact of Epistemic Modeling Uncertainty on the Reliability of Sys-Id Recognition & Mitigation of Epistemic Modeling Uncertainty Model Adequacy Evaluation Conclusions Recommendations & Future Work

7 Commodore Barry Bridge (Grimmelsman 2005) P.P. 27 P.P ft 411 ft 822 ft 411 ft 822 ft Anchor Span Cantilever Arm Suspended Span Cantilever Arm Anchor Span Before Retrofit Pin Hanger Pin Hanger 30 m Pin Cantilever Arm Suspended Span 1 3D FEM of Hanger Region

8 Commodore Barry Bridge (Grimmelsman 2005) After Retrofit Strain Response of Hanger Hanger B A Spreader LC B A Section B-B Section A-A - Vibrating Wire Strain Gage -200 Rod/Hanger Force Ratio calculated from the strain measurement data using nominal modulus of elasticity Strain Response values of Rodis Rod Spreader UC

9 Brooklyn Bridge (Grimmelsman 2006) V L V L T L T L T L L T L L L L T L L L L T L T L T Acceleration (g 2 /Hz) 4.5 x Tower Longitudinal Vibration Output Spectra B C E G H MT L L T 1.5 L L T Peaks in 0-5 Hz Band V V Frequency (Hz) the 1 st longitudinal tower mode f = Hz 50 Blong Clong Elong Glong Hlong Measures to rule out spurious results: Max MIF Shape Plausibility Shape agreement Coherence MAC values Phase factor Span modes Conceptualized Analytical model Mechanically Transparent System? Unit Normalized Modal Amplitude

10 Di3 Lab (Ciloglu 2006) Connections Reduced-Scaled Deckon-Beam Bridge Model Boundary Relationship btw Major Sources of Uncertainty & Identified Experimental Modal Parameters Structural Complexity Excitation Preprocessing Post-processing Steel Roller Steel Roller + Weight Neoprene Roller Superstructure Not Distributed Superstructure Distributed Substructure Not Distributed Substructure Distributed Random Dec. Correlation Func. Signal L-1 Signal L-2 Signal L-3 W/ Exp. Window W/o Exp. Window DFT Signal Modeling CMIF PTD SSI

11 Di3 Lab (Ciloglu 2006) Sensor in the span Deck Boundary Sensor Output: Accelerometer on the deck and support plates Output: Accelerometer under the grid Input & Output: Instrumented impact hammer, Accelerometer Support Boundary Sensor The modal properties obtained from Impact tests serve as ground truth to evaluate the effects of uncertainty Effect of Uncertainties due to Preprocessing & Post-processing Mode Number The 8th & 12th modes were only identified by CMIF- RD CMIF-RD PTD-RD SSI-RD CMIF-C PTD-C SSI-C PTD-RD PTD-RD PTD-RD PTD-RD PTD-C PTD-RD SSI RD CMIF-C CMIF-C SSI-C PTD-RD PTD-RD PTD-RD PTD-C PTD-RD SSI RD CMIF-C CMIF-C SSI-C

12 Outline Background & Definition Past Research: Sys-Id Applications on Civil Structures Research Motivations & Objectives Impact of Epistemic Modeling Uncertainty on the Reliability of Sys-Id Recognition & Mitigation of Epistemic Modeling Uncertainty Model Adequacy Evaluation Conclusions Recommendations & Future Work

13 Motivation: Lessons from Past Experience Significant obstacles for widespread implementations of Sys-Id in engineering practice primarily stem from our inability to reliably simulate, measure and interpret the actual physical behaviors of a constructed system, consequently leading to the skepticism towards the credibility of Sys-Id held by owners/stewards of constructed systems. These limitations mainly arise from various sources of uncertainty which smear into Sys-Id process through the choice of model structure, idealization of boundary and continuity conditions as well as the design, execution and interpretation of field testing and monitoring program. Current applications of Sys-Id usually lump uncertainties inherent in a constructed system as a small number of incorrect model parameters or random variables with assumed probability models. Many sources of uncertainty, as demonstrated in previous examples, stem from those unknown or less understood structural behaviors as well as their interactions with surrounding environments. They are difficult to be described with probability models and in some cases they may even not be parameterized.

14 Focus Area: Epistemic Uncertainty This type of uncertainty, as opposed to randomness-based aleatory uncertainty, arise from a lack of knowledge and is called epistemic uncertainty. Usually epistemic uncertainty is reducible when more information is available. Epistemic uncertainty has more profound impact on the reliability of system identification. However, systematic investigation on the impact of uncertainty, and epistemic uncertainty particularly, on system identification has not been made yet. Terms used in literature to describe dual meaning of uncertainty (adapted from Christian 2004) Uncertainty due to naturally variable phenomena in time or space Aleatory uncertainty Uncertainty due to lack of knowledge or understanding of nature Epistemic uncertainty Reference citation Hacking 1975; McCann 1999; Ang and De Leon 2005 Natural variability Knowledge uncertainty NRC 2000 Random or stochastic variability Functional uncertainty Stedinger et al Objective uncertainty Subjective uncertainty Chow et al External uncertainty Internal uncertainty Chow et al Statistical uncertainty Inductive probability Carnap 1936 Irreducible uncertainty Reducible uncertainty Chance Probability Poisson, Cournot (Hacking 1975)

15 DIVERGED SYS-ID RESULTS START OF SYS-ID CONVERGED SYS-ID RESULTS Additional Information CONVERGED SYS-ID RESULTS Epistemic Uncertainty Additional Information ACTUAL STATE OF A CONSTRUCTED SYSTEM: M, K, C Sys- Id Epistemic Uncertainty Ground truth Aleatory Uncertainty Aleatory Uncertainty Aleatory Uncertainty Aleatory Uncertainty

16 (3) Controlled Experimentation STRUCTURAL COMPLEXITY: Non-stationarity of boundary and continuity conditions Changes in intrinsic stresses during tests (redundancy, deterioration) Nonlinearities: Many forms of material and damping nonlinearity, friction, intermittent contact and uplift FORCE AND EXCITATION: Amplitude Spectral distribution Spatial distribution and transmissibility Directionality Dimensionality (1D, 2D or 3D) Duration and Non-stationarity DATA ACQUISITION: Spatial aliasing Time synchronization Hardware filtering options Noise & bias in signal Measurement bandwidth Cabling and installation effects (2) Preliminary Model(s) Analytical representation of physical members and connections Completeness of 3D geometry Soil-foundation, structural members, joints: stiffness and kinematics Mechanisms/Forms of Nonlinearity Sys-Id Uncertainty (4) Data Processing Real-time data quality assessment, management and warehousing Error identification/ Cleaning Different filtering, averaging and windowing options Data post-processing algorithms (1) Conceptualization Identify/Leverage Heuristics Archival of structural drawings /design calculations, inspection reports Site visits, geometry measurements, photogrammetry Material Sampling, testing, NDE Virtual Reconstruction in 3D CAD (6) Utilization Health/Performance Monitoring Damage detection, Prognosis Scenario Analysis and Vulnerability Assessment Performance-based Engineering Guidelines and Codes (5) Model Calibration Test-analysis correlation Parameter grouping Sensitivity Analysis Modality Objective function and constraints Optimization Physical interpretation of results

17 Research Objectives Influence of modeling uncertainty due to epistemic mechanisms on analytical modeling of constructed systems Feasible techniques to recognize and mitigate modeling uncertainty Adequacy of a field-calibrated model to simulate all of the critical physical mechanisms impacting a constructed system Review of model updating procedures including testanalysis correlation, error localization, sensitivity analysis, and data informativeness quantification as well as updating algorithms.

18 6 5 1 Sys-Id Laboratory Test Bed: Cantilever Beam with Two Test Configurations Research Design Epistemic Modeling Uncertainty in Sys- Id of Constructed Civil Structures Research Approach Real-Life Constructed System: the Henry Hudson Bridge Impact on Identification Results Recognizing Epistemic Uncertainty Mitigating Epistemic Uncertainty Test-Analysis Correlation Sensitivity Analysis Error Localization Index Iterative Model Updating Heuristics Recognition of Epistemic Uncertainty Recognizing Epistemic Uncertainty Mitigating Epistemic Uncertainty Model Adequacy Evaluation

19 Outline Background & Definition Past Research: Sys-Id Applications on Civil Structures Research Motivations & Objectives Impact of Epistemic Modeling Uncertainty on the Reliability of Sys-Id Recognition & Mitigation of Epistemic Modeling Uncertainty Model Adequacy Evaluation Conclusions Recommendations & Future Work Cantilever Study

20 Cantilever Beam Study Test Configuration 1 Test Configuration 2 Model A Model B

21 Analytical Simulation of MRIT Modal Contributions to Simulated Impulse Response functions Force Time (s) in Mechanical & Material Properties of the Beam Property Value Density ρ lb-f/in 3 Young s Modulus E 29x10 6 lb-f/in 2 Cross Section Area A in 2 Moment of Inertia I about weak axis in 4 Modal Parameters Estimated Based on Continuum Theory u u m( x) + ( ) = EI x 2 t x x f = Hz Simulated Frequency Response functions f = Hz f = Hz f = Hz f = Hz

22 Cantilever Beam Test Configuration Steel Pedestal Steel Tube X-section 3X1.5X0.125 in PCB Capacitive Accelerometer Model 3701G3FA3G C-shape Clamp

23 Cantilever Beam Test Configuration 2 A D C B B 24x3x3/4 in steel plate; center to center distance btw two rods connected on the same plate is 18 inch 4-inch long aluminum angle L6x6x3/8 in C High-strength steel rod with d = 3/8 inch D 16-inch long steel angle L3x3x1/2 in A 1 Steel Pedestal 2 3 Steel Tube X-section 3X1.5X0.125 in 4 PCB Capacitive Accelerometer Model 3701G3FA3G 5 6 C-shape Clamp

24 Impact Test Data Acquisition Diagram DAQ PC VXI DAC Express Software Agilent IO Lib Control DAQ PC HP VXI Impact Excitation Breakout box PCB 3701G3FA3G Capacitive Accelerometers Test Measurements HP VXI Ch1 Ch2 Ch3 Ch4 Ch5 Ch6 FRF Co-ax cable Microdot cable Ch 7 Impulse Hammer 086C02 Impact Force

25 Static Test Static Test Sensor Layout STATIC PT 5 Celesco PT101 cable-extension transducer PCB 3701G3FA3G capacitive accelerometer Time-History Displacement Response of the beam CELESCO PT101

26 Experimental Results from Configuration 1 uniform loading surface (ULS) of the cantilever beam Mode # Frequencies Initial FE Test Test Test Mean STD Diff (%) % C. I ± ± ± ± ±2.133 Mode # Damping Ratios Test Test Test Mean STD % C. I ± ± ± ± ±0.039 uniform loading surface (ULS) of the cantilever beam MODE 1 Mode Distance (in) MODE 2 Mode Distance (in) MODE 3 Mode Distance (in) MODE 4 Mode Distance (in) 1.0 MODE 5 Mode X- Modal Distance (in)

27 Experimental Results from Configuration 2 uniform loading surface (ULS) of the cantilever beam Mode # Frequencies Test Test Test Test Test Test Mean STD % C. I ±0.005 ±0.012 ±0.005 ±0.032 ±0.030 ±0.020 ±0.047 ±0.050 Mode # Damping Ratios Test Test Test Test Test Test Mean STD % C. I ±0.038 ±0.062 ±0.013 ±0.001 ±0.009 ±0.012 ±0.012 ± PTD1 PTD2 PTD3 PTD4 PTD5 PTD6 MODE 1 Mode PTD1 PTD2 PTD3 PTD4 PTD5 PTD6 DISTANCE (in) MODE 2 Mode Mode 3 DISTANCE (in) MODE 3 PTD1 PTD2 PTD3 PTD4 PTD5 PTD DISTANCE (in) MODE Mode DISTANCE (in) PTD1 PTD2 PTD3 PTD4 PTD5 PTD6 uniform loading surface (ULS) of the cantilever beam MODE 4 PTD1 MODE 5 PTD1 Mode 4 PTD2 Mode PTD2 1.2 PTD3 PTD PTD4 PTD PTD5 PTD5 0.0 PTD6 0.0 PTD DISTANCE (in) DISTANCE (in) PTD1 MODE 6 MODE 7 PTD2 Mode 6 PTD1 Mode 7 PTD3 1.2 PTD PTD4 PTD PTD5 CMIF PTD6 PTD PTD DISTANCE (in) DISTANCE (in)

28 Identification Results Using Model A Updating Parameter: E Identification Run 1: Cantilever Beam under Configuration 1 Initial value of E is 290 (x10^5) psi Identification Run 2: Cantilever Beam under Configuration 2 E (x10 5 psi) E (x10 5 psi) Change of E with iteration in Run Iterations Change of E with iteration in Run Iterations Frequency before and after calibration in Run initial updated MODE Frequency before and after calibration in Run 2 initial updated MODE

29 Observations & Discussions About Identification Results Using Initial Model A All of the five observed vibration modes from the cantilever beam under test configuration 1 could be paired up with the analytical predictions from initial model A. The test-analysis correlation were improved after calibration by adjusting the Young s modulus of steel. With a decrease of less than 10 percent of its nominal value, the difference in natural frequency was under 2 percent. The mode shapes remained because they were independent on the updating parameter. In Run 1, the information in all five modes was included in model updating, while the fourth mode was excluded in identification Run 3. However, the updated values of the model parameter as well as the natural frequencies were very similar. A total of eight modes were identified from the beam under test configuration 2. The first three and the eighth mode were paired up with the analytical first three and the fifth mode from initial model. The rest four experimental modes all demonstrate deflection shapes similar to the analytical fourth bending mode.

30 Observations & Discussions About Identification Results Using Initial Model A (Cont d) Immense uncertainty was associated with the correlation between the measured and simulated fourth bending mode. Only four pairs of modes were thus included in identification Run 2. Although the gap in natural frequency between the analytical model and experiment observation narrowed considerably after updating, relatively large difference remained, especially in the natural frequency of the first mode. The calibration process forced the pre-selected updating parameter to decrease by about 33 percent of its initial (nominal) value, which already lost its physical meaning. Significant discrepancy was observed in the estimated values for the model parameter E from the identification cases of configurations 1 and 2, which were supposed to converge to the same value because they represented the material property of the same beam.

31 Additional Test on Configuration Steel Pedestal Vertical sensor on top plate Longitudinal sensor on top plate Instrumentation Plan Vertical sensor on beam Lateral sensor on beam Frequency Response Functions (FRFs) Chan 1 Chan 2 Chan 3 Chan 4 Chan 5 Chan 6 Chan log X/F 10-4 Sensors Installed on the Boundary Assembly Frequency (Hz)

32 Additional Test on Configuration 2 Chan 1 & 2 Longitudinal sensors on top plates; Chan 3 & 4 Vertical sensors on top plates; Chan 5 & 6 Vertical sensors on cantilever; Chan 7 Transverse sensor on cantilever; Beam Mode FRFs Chan 6

33 Identification Results Using Initial Model B Updating parameters: E, K r Identification Run 4: Cantilever Beam under Configuration 1 Change of E Initial value of E is 290 (x10^5) psi; Initial value of Kr is 500 (x10^4) psi; Change of Kr Frequency before and after calibration in Run 4 initial updated E (x10 5 psi) E (x10 5 psi) Iterations Change of E Iterations Kr (x10 4 psi) Iterations Identification Run 5: Cantilever Beam under Configuration Kr (x10 4 psi) Change of Kr Iterations MODE Frequency before and after calibration in Run MODE initial updated

34 Beam with Configuration 1 (Run 4) Identification Results with Model B MODE test initial updated Distance (in) MODE test 0.8 initial 0.4 updated Distance (in) MODE test initial updated Distance (in) test initial updated MODE test initial updated Distance (in) Distance (in) MODE 5 mode # MAC Initial Updated Beam with Configuration 2 (Run 5) MODE test initial 0.2 updated DISTANCE (in) MODE test 0.8 initial 0.4 updated DISTANCE (in) MODE test initial 0.4 updated DISTANCE (in) MODE DISTANCE (in) test initial updated MODE test initial updated DISTANCE (in) Mode # MAC initial updated

35 Observations & Discussions Additional dynamic test on the beam with configuration 2 revealed the interaction between the beam and boundary assembly. This observation was conceptualized as rotational spring at the beam support in initial model B. Among the four modes which demonstrated similar deflection shapes, f = Hz was dominated by the vibrations of the beam and was only included in the calibration process. With initial model B, the beam under two test configuration 1 and 2 were calibrated in identification Run 4 and 5 respectively. In Run 4, the calibrated model converged to the similar level as that in Run 1 and 3. In Run 5 the discrepancy between the predicted natural frequencies and their experimental counterparts were significantly reduced and only ±3 percent difference remained. The values for the common model parameter in Run 4 and 5, the Young s modulus E, both decreased by about 3 percent of its nominal value, although they did not converge to the exact same value.

36 Observations & Discussions Good correlation between the values of common updated parameter as well as the modal predictions implied that the initial model B efficiently conceptualize the influences of the rotary movement of the boundary assembly on the beam without explicitly incorporating the boundary assembly in the initial model. Epistemic modeling uncertainty associated with the initial model A would seriously impair the reliability of the identification for the beam under configuration 2. The selected updating parameter tended to compensate for the influence of epistemic modeling error by distorting itself. As a result, the updated value for E lost its physical significance. The calibrated model still failed to accurately predict the experimental frequencies. Updating Run 1: Config 1 Run 2: Config 2 Run 4: Config 1 Run 5: Config 2 Nominal Parameter with model A with model A with model B with model B E ( 10 5 psi) Diff (%)

37 Outline Background & Definition Past Research: Sys-Id Applications on Civil Structures Research Motivations & Objectives Impact of Epistemic Modeling Uncertainty on the Reliability of Sys-Id Recognition & Mitigation of Epistemic Modeling Uncertainty Model Adequacy Evaluation Conclusions Recommendations & Future Work Henry Hudson Bridge

38 The Henry Hudson Bridge South Approach 409 South South Viaduct Tower Arch Span 840 North North Tower Viaduct North Approach 270 East Elevation Preliminary 3D FE Model

39 Instrumentation Plan Lower Level of Arch Span Upper Level of Arch Span Half Height of Tower DAQ Test on the north part of the bridge Upper Level L L V V V V L L T T T L L T V T T T T V T L L T T T Lower Level T T 25 T V 27 T V 30 V T V 46 T V T T South Viaduct Tower D T T T CL T T T Tower North Viaduct D T East Side Transverse Accelerometer V East Side Vertical Accelerometer L East Side Longitudinal Accelerometer T West Side Transverse Accelerometer V West Side Vertical Accelerometer L West Side Longitudinal Accelerometer V T V T Sensors Which Stay for the Tests on North & South Part of the Bridge

40 Data Processing 8 x Time-domain measurements under ambient conditions Imported in X-Modal and process with PTD Data Analysis Stationarity Check; Effect of Traffic Load Amplitude; Effect of Sampling Bandwidth; Pseudo IRF generated by spectrum estimation methods 6 x Stability plot to present consistency of poles with different model order Pseudo FRF generated by spectrum estimation methods Preprocessing Modal Parameters: ω, ζ, Φ Post-processing by PTD Imported in X-Modal and process with CMIF Post-processing by CMIF CMIF plot generated based on singular value decomposition

41 Model Calibration Model Calib bration PROCESS TOOL PURPOSE Global (Modal) Calibration: The correlation between analytical and experimental frequencies and mode shapes was achieved in modal space by adjusting the stiffness or mass or a combination of both. Local (Flexibility) Calibration: The correlation between analytical and experimental strain and/or displacement was achieved by adjusting the structural stiffness. Sensitivity Analysis Correlation: Numerical comparison of frequencies; Graphical comparison of mode shapes; Numerical comparison of mode shapes Heuristics To make sure that analytical mode order is consistent with experimental; To make sure that analytical frequency values have minimum discrepancy with those of experiment; To ensure that the analytical model is complete so that the global stiffness and force distribution mechanisms are properly simulated.

42 Test-Analysis Correlation Mode Analytical Experimental Freq (Hz) Description Freq (Hz) Description Diff (%) nd vertical bending nd vertical bending st lateral bending (arch) st lateral bending (arch) rd vertical bending rd vertical bending nd lateral bending (arch) nd lateral bending (arch) st lateral bending (global) st lateral bending (global) st vertical bending st vertical bending th vertical bending th vertical bending th vertical bending th vertical bending th vertical bending th vertical bending th vertical bending th vertical bending nd Vertical Mode f = Hz 1 st Lateral Mode f = Hz EXP MODE SHAPE 60 - SAP MODE SHAPE UNDEFORMED SHAPE EXP MODE SHAPE - SAP MODE SHAPE - - UNDEFORMED SHAPE

43 Sensitivity Analysis The model parameters/conditions selected for sensitivity analysis include: The Young s modulus of the steel; The Young s modulus of the deck concrete; Variations in boundary conditions; The continuity conditions between viaduct-deck-arch interface at both the upper and lower decks; The stiffness of the lateral translational springs located at each end of the two viaduct spans; Modal Frequency (Hz) Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Mode 8 Mode 9 Mode 10 Mode 11 Mode 12 Mode 13 Mode 14 Mode 15 Modal Frequency (Hz) Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Mode 8 Mode 9 Mode 10 Mode 11 Mode 12 Mode 13 Mode 14 Mode E 0.90 E 0.95 E 1.00 E 1.05 E 1.10 E 1.15 E Variation of E of Steel 0.4 Equal Constraint in Uy, Uz Body Constraint in Uy, Uz Body Constraint in Uy, Uz, Rx, Ry & Rz Body Constraint in all Six DOFs Variation of Continuity Condition at Deck-Tower Interface

44 Heuristics for Calibration Adjust joint rigidity factors at column-floor connections Correct joint constraints btw arch rib &lower deck; Remove frame end releases for vertical members btw arch rib & lower deck; Adjust the continuity conditions at the arch-towerviaduct interface

45 Calibration Results Mode Experimental Initial Updated Diff (%) # (Hz) (Hz) (Hz) Diff (%) Description st lateral bending (arch) nd vertical bending rd vertical bending nd lateral bending (arch) st lateral bending (global) st vertical bending th vertical bending th vertical bending th vertical bending th vertical bending 0.9 MAC Values of Vertical Modes Before After MAC Values of Lateral Modes Before After

46 Model Adequacy Check By Sensitivity Analysis of E NORM OF ERROR IS DEFINED AS: NORM = 7 i= 1 FREQFEi FREQ FREQEXPi EXPi 2 VERTICAL FREQUENCIES & NORMS OF ERROR (Hz) FROM INITIAL MODEL 0.6 MODE # 0.85 E 0.90 E 0.95 E 1.00 E 1.05 E 1.10 E 1.15E 1.20 EUPDATED 1.25 E EXP INITIAL NORM Norm of Error DIVERGENCE 0.1 DIVERGENCE 0.0 VERTICAL FREQUENCIES & NORMS OF ERROR (Hz) FROM UPDATED MODEL MODE # 0.85 E 0.90 E 0.95 E 1.00 E 1.05 E 1.10 E 1.15E EXP x E NORM DIVERGENCE DIVERGENCE

47 Observations & Discussion It is feasible to apply integrative paradigm of system identification on large-scale complex structures such as the long-span steel arch bridge. Before identifying model uncertainty, it is important to understand uncertainty due to experiment/data processing. The stationarity of recorded data, as well as the effect of the traffic load amplitude and sampling frequency bandwidth on the identified modal parameters was carefully examined to exclude possible epistemic measurement uncertainty. Global calibration in modal space was conducted based on: (1) Heuristics; (2) test-analysis correlation, and, (3) sensitivity analysis. The calibrated model was shown to accurately predict the modal properties of the bridge in the vertical direction. The test-analysis correlation in lateral direction is not as good most likely due to a lack of sufficient excitation in the lateral direction.

48 Observations & Discussion (Cont d) The sensitivity of the modal parameters with respect to the elasticity modulus of the steel was leveraged to assess the adequacy of the fieldcalibrated model. The level of discrepancy was significantly decreased after calibration: The elasticity modulus converged around its nominal value, which indicated the calibrated model was the most admissible model given the available information.

49 Conclusions: Impact of Epistemic Modeling Uncertainty Epistemic modeling uncertainty is usually closely related to the choice of model form, element type, idealization of geometry, material properties as well as boundary and continuity conditions. In real-life applications of system identification, epistemic modeling uncertainty is coupled with other sources of uncertainty. Its impacts propagate and manifest this phenomenon in various ways as identification progresses from Step 2 on. Compared with aleatory modeling uncertainty which may affect the accuracy of model parameters, the modeling uncertainty due to epistemic mechanisms could lead to divergence in identification results (large testanalysis discrepancy may remain the calibrated model, and/or the updated parameters may lose their physical significance). Epistemic modeling uncertainty may cause those updating parameters associated with little random variability (e.g Es) to appear as if they are the dominant uncertainty sources. In this study when there was epistemic modeling uncertainty, Es exhibited large variations during the updating process in order to compensate for the test-analysis discrepancy.

50 Conclusions: Recognition & Mitigation of Epistemic Modeling Uncertainty The abnormality of the updated value for the elasticity modulus of steel in the cantilever investigation served as a good indicator for the presence of unacknowledged epistemic modeling uncertainty in analytical model. Therefore a feasible and effective indicator for the presence of epistemic modeling uncertainty associated with the a priori model of a constructed system was to incorporate into updating procedure one or more of the global model parameters of little known variability. If these appear to be sensitive to the change of dynamic properties, we will use this to identify the presence of epistemic model uncertainty. Most effective solution to reduce epistemic uncertainty is to obtain additional information about the system and fully leverage heuristics. The additional dynamic test on the configuration 2 of the cantilever revealed the interactions between the beam and boundary assembly.

51 Conclusions: Evaluating Model Adequacy System identification paradigm is a powerful tool to characterize the actual behaviors of a constructed system and it is feasible to be used on large-scale complex civil engineering structures. In real-life applications, systematical utilization of engineering heuristics often played a critical role in reducing modeling uncertainty and epistemic modeling uncertain in particular embedded in the a priori model of the structure. The sensitivity analysis of the model properties predicted by analytical models before and after calibration with respect to the elasticity modulus of steel indicated that the field-calibrated model of the bridge was the most admissible one with available information. With limited information embedded in available test data, it is often extremely difficult to determine an unique and converged calibrated model. The proposed model adequacy evaluation tool was intended to assist in checking whether critical physical mechanisms of the system under study were properly incorporated in the analytical model.

52 Recommendations & Future Work A feasible technique to recognize epistemic modeling uncertainty in the analytical model are proposed in the thesis. This idea could be further investigated with designed laboratory tests and candidates for the dummy model parameters should also include global and sensitive parameters such as the mass of the system or a subsystem. Additional research is required in order to better pinpoint the sources of epistemic modeling uncertainty. This would lead to more efficient mitigation of epistemic modeling uncertainty. A stochastic framework should be incorporated into the system identification paradigm. The global calibration which mainly takes advantage of modal data from vibration tests should be utilized in conjunction with local calibration based on static data from load tests.

53 Thank You