Automated Finite Element Model Updating of the UCF …engineering.tufts.edu/cee/people/sanayei/PARIS/documents/2013... · Automated Finite Element Model Updating of the UCF Grid Benchmark

Automated Finite Element Model Updating of the UCF Grid

Benchmark Using Multiresponse Parameter Estimation

A Thesis

Submitted by

Ali Khaloo

In Partial Fulfillment of the Requirements

for the Degree of

Master of Science

in

Civil and Environmental Engineering

TUFTS UNIVERSITY

February 2014

© 2014, Ali Khaloo

ADVISOR:

Professor Masoud Sanayei

II

III

Abstract

Structural Health Monitoring (SHM) using nondestructive test (NDT) data has become very

promising for finite element (FE) model updating, model verification, structural evaluation, and

damage assessment. This research presents a multiresponse structural parameter estimation method

for the FE model updating using data obtained from a nondestructive test on a laboratory bridge

model. Having measurement and modeling errors is an inevitable part of data acquisition systems and

finite element models. The presence of these errors can affect the accuracy of the parameter

estimates. Therefore, an error sensitivity analysis using Monte Carlo simulation was used to study the

input-output error behavior of each parameter based on the load cases and measurement locations of

the nondestructive tests. Given the measured experimental responses, the goal was to select the

unknown parameters of the FE model with high observability that leads to creating a well-

conditioned system with the least sensitivity to measurement errors. A data quality study was

performed to assess the accuracy and reliability of the measured data. Based on this study, a subset of

the most reliable measured data was selected for the FE model updating. The selected subset of

higher quality measurements and the observable unknown parameters were used for FE model

updating. Three static and dynamic error functions were used for structural parameter estimation

using the selected measured static strains, displacements, and slopes as well as dynamic natural

frequencies and associated mode shapes. The measured data sets were used separately and also

together for multiresponse FE model updating to match the predicted analytical response with the

measured data. The FE model was successfully calibrated using multiresponse data. Two separate

commercially available software packages were used with real-time data communications utilizing

Application Program Interface (API) scripts. This approach was efficient in utilizing these software

packages for automated and systematic FE model updating. This method is applicable to full-scale

structures and can be used for bridge model validation and bridge management.

IV

To my parents.

V

Acknowledgements

“Far and away the best prize that life has to offer

is the chance to work hard at work worth doing.”

~ Theodore Roosevelt

This research would not have been possible without the guidance and the help of several

individuals who in one or another contributed and extended their valuable assistance in the

preparation and completion of this study.

First and foremost I would like to thank my family for giving me their unconditional love and

support through my life; I could not have done this without it. I am grateful and wish to express

my gratitude to my advisor, Professor Masoud Sanayei, for all his support and guidance

throughout this research project. Professor Sanayei has been truly selfless in dedicating his time

and effort to help me during my study. I extent sincere thanks to Professors Necati Catbas and

Mustafa Gul for their support in making the UCF gird NDT data available for my research. My

committee members, Masoud Sanayei, Necati Catbas, Babak Moaveni and Mustafa Gul provided

a wide variety of invaluable contributions to my research and I thank you for your support.

I extent deepest appreciation to my friends and officemates, Chris Paetsch, Jesse Sipple,

Peeyush Rohela, Alex Reiff, Wenjian Lin, Tianqi Qu and Mohammad Reza Saberi. Their

presence and support has made the last two years pleasantly memorable.

Finally, I would like to thank my girlfriend Bahar. Her patience and encouragement has

helped me through this process and her unending belief in me has pushed me to the best that I

can.

VI

Nomenclature

Parameter Estimation NUP Number of Unknown Parameters

NPG Number of Parameter Groups

LC Load Case

SS Static Stiffness

SF Static Flexibility

SSTR Static Strain

MF Modal Flexibility

p Vector of Unknown Parameters

e(p) Error Function Vector

NDT Nondestructive Test

FE Finite Element

f Applied Force Vector

ε Strain Vector

u Displacement Vector

M Mass Matrix

K Stiffness Matrix

D Dynamic Matrix D = K-1

M

B Mapping Matrix from Strain-Displacement Relationship ε = B q

Mode Shape

Natural Frequency

J(p) Scalar Objective Function

Σe Covariance Matrix of Error Function

NM

NOBS

MC

GM

GSD

E

DG

SG

TM

AC

IM

RMS

UCF

Total Number of Measurements

Number of the Observations

Monte Carlo Simulations

Grand Mean Percentage Error

Grand Standard Deviation Percentage Error

Percentage Error

Displacement Gage

Strain Gage

Tiltmeter

Accelerometer

Impact Hammer

Root Mean Square

University of Central Florida

VII

Table of Contents

Thesis Abstract………………………………………………………………………. III

Acknowledgements…………………………………………………………………... V

Nomenclature………………………………………………………………………… VI

Table of Contents……………………………………………………………………. VII

List of Figures………………………………………………………………………... X

List of Tables………………………………………………………………………… XIII

Chapter 1 – Introduction……………………………………………………………. 1

1.1 Overview 1

1.2 Scope of Work 2

1.3 Thesis Layout 2

Chapter 2 – Automated Finite Element Model Updating of Scale Bridge Model

Using Measured Static and Modal Test Data……………………………………… 5

2.1 Introduction 5

2.2 Error Functions 9

2.2.1 Static Flexibility (SF) 10

2.2.2 Static Strain (SSTR) 10

2.2.3 Modal Flexibility (MF) 11

2.3 Scalar Objective Function 11

2.4 Normalization 12

2.4.1 Parameter Normalization 12

2.4.2 Error Function Normalization 12

2.4.3 Objective Function Statistical Normalization 13

2.4.4 Objective Function Weighting 13

2.5 Parameter Grouping 14

2.6 Multiresponse Parameter Estimation 14

2.7 University of Central Florida Grid Benchmark 15

2.7.1 Instrumentation Plan 16

2.7.2 Finite Element Model for the UCF Grid 17

2.7.3 Boundary Stiffness Sensitivity Study 19

2.8 Nondestructive Testing of UCF Grid 20

2.8.1 Static Load Test 20

2.8.2 Dynamic Load Test 21

2.9 Error Sensitivity Analysis Using Monte Carlo Simulations 21

2.9.1 Error Sensitivity Analysis for the UCF Grid 22

2.10 Parameter Estimation for the UCF Grid Using NDT Data 24

2.10.1 NDT Data Quality Study 24

2.10.2 Multiresponse Parameter Estimation and Model Calibration of the

UCF Grid 26

VIII

2.10.3 FE Model on Pins and Rollers Used in Multiresponse Parameter

Estimation and Model Calibration 30

Chapter 3 – NDT Data Quality Analysis………………………………………….. 39

3.1 Introduction 39

3.2 Criteria Used in NDT Data Quality Analysis 40

3.2.1 Data Quality Study: Criterion 1 42



3.3 Modeling Error 43

3.4 NDT Data Used in the Parameter Estimation of the UCF Grid 44

Chapter 4 – Error Sensitivity Analysis using Monte Carlo Simulation...………. 45

4.1 Introduction 45

4.2 Monte Carlo Analysis 45

4.3 Measurements Error Types and Distributions 46

4.4 Error Sensitivity Analysis of the UCF Grid 47

4.4.1 Error Functions Used in Monte Carlo Analysis 48

4.4.2 Damage Cases and Parameter Grouping Scheme 49

4.4.2.1 Damage Case 1 50





Chapter 5 – Supplementary Parameter Estimation Cases with Analytical and

Measured Response Comparisons…………………………………….………...… 54

5.1 Introduction 54

5.2 Supplementary Parameter Estimation Cases of the UCF Grid Using NDT

Data 54

5.2.1 Additional Error Function Used in the Parameter Estimation: Static

Stiffness (SS) 55

5.2.2 FE Model on Link Elements Used in Multiresponse Parameter

Estimation 55

5.2.3 Analytical and Measured Response Comparison: Calibrated FE Model

on Link Elements 57

5.2.3.1 Strain Comparison 57

5.2.3.2 Displacements Comparison: Girder 1 59

5.2.3.3 Displacements Comparison: Girder 2 60

5.2.3.4 Slope Comparison 62

5.2.4 FE Model on Pins and Rollers Used in Multiresponse Parameter

Estimation 64

5.2.5 Analytical and Measured Response Comparison: Calibrated FE Model

on Pins and Rollers 65

5.2.5.1 Strain Comparison 65

IX

5.2.5.2 Displacement Comparison: Girder 1 67

5.2.5.3 Displacement Comparison: Girder 2 68

5.2.5.4 Slope Comparison 70

Chapter 6 – Conclusions and Recommendations for Future Works……………. 73

6.1 Conclusions 73

6.2 Recommendations for Future Works 75

Appendix A – PARIS 14.0 User Reference Manual…………………….…..……. 76

A.1 Introduction 76

A.2 SAP2000 Model 79

A.2.1 SAP2000 Model File 79

A.2.2 Unit System 79

A.2.3 Local and Global Coordinate System 81

A.2.4 Nodes and Element Labels 81

A.2.5 Frame Element Section Properties 83

A.2.5 Analysis Option: Active DOF 85

A.2.7 Load Cases 86

A.2.8 Element Grouping 87

A.3 PARIS14.0 Input File Details 87

A.3.1 SAP2000 Model Related Data 88

A.3.2 Parameter Estimation Related Data 89

A.4 Sample PARIS14.0 Input File: UCF Grid Example (FE Model on Pins and

Rollers) 100

Appendix B: Sensor Specifications of the UCF Grid…………………...………… 109

Appendix C: UCF Grid Drawings…………………………………………………. 114

X

List of Figures

Figure 2.1 UCF Grid 16

Figure 2.2 UCF Grid Member Connections (Left) and Boundary Support

(Right)

16

Figure 2.3 Instrumentation Details for Static Testing with Sensor Numbers 17

Figure 2.4 Accelerometer and Excitation Locations for Dynamics Testing 17

Figure 2.5 UCF Grid Finite Element Model 19

Figure 2.6 Parameter Grouping Scheme for UCF Grid 23

Figure 2.7 Strains, Data Load Case 1 32

Figure 2.8 Strains, Data Load Case 4 32

Figure 2.9 Displacements Along Girder 1 Load Case 5 33


Figure 2.11 Slopes, for Load Case 1 33

Figure 2.12 Slopes, for Load Case 6 33

Figure 2.13 Comparison of Natural Frequencies 35

Figure 3.1 UCF Grid Static Test L1T1 SG1 40



Figure 5.1 Strains, Load Case 1 57






















Figure 5.23 Slopes, Load Case 1 62




XI





































Figure A.1 PARIS 14.0 Flowchart 77

Figure B.1 Strain Gage Dimensions 110

Figure B.2 Tiltmeter Dimensions 111

Figure B.3 Accelerometer Dimensions 112

Figure B.4 Impact Hammer Dimensions 113

XII

List of Tables

Table 2.1 UCF Grid Initial Section Properties 18

Table 2.2 Boundary Stiffness Ranges 19

Table 2.3 UCF Grid Static Load Cases 21

Table 2.4 Sensor Numbers Used for Measured Static Responses 25

Table 2.5 MAC Values Using Initial Parameter Values 26

Table 2.6 UCF Grid Parameters Estimates 28

Table 2.7 UCF Grid Parameter Estimates Based on the Pins and Rollers Model 31

Table 2.8 MAC Values Using Parameters from Pins and Rollers Model 31

Table 2.9 Predicted vs. Experimental RMS responses 34

Table 2.10 Comparison of Analytical Mode Shapes with Experimental Mode

Shapes of the UCF Grid 36

Table 3.1 Strain Values Used in the Parameter Estimation of the UCF Grid (με) 44

Table 3.2 Displacements Values Used in the Parameter Estimation of the UCF

Grid (in) 44

Table 3.3 Slope Values Used in the Parameter Estimation of the UCF Grid

(rad×103) 44

Table 4.1 Damage Case 1 Monte Carlo Simulation Results (SF Error Function) 50



Table 4.4 Damage Case 3 Monte Carlo Simulation Results (SSTR Error

Function) 52

Table 4.5 Damage Case 4 Monte Carlo Simulation Results (MF Error Function) 52

Table 4.6 Damage Case 5 Monte Carlo Simulation Results (MF Error Function) 53

Table 5.1 UCF Grid Parameter Estimates (Including Supplementary Cases) 56

Table 5.2 MAC Values Using Parameter Values of Case No.6, SF+SSTR+MF 64

Table 5.3 UCF Grid Parameter Estimates Based on the Pins and Rollers Model

(Including Supplementary) 65

Table 5.4 MAC Values Using Parameter from Pins and Rollers Model 72

Table A.1 PARIS 14.0 Features 78

Table A.2 Load Case Nomenclature 86

Table B.1(a) Wire Strain Gage Specifications 110

Table B.1(b) Thermistor Specifications 110

Table B.2(a) Wire Tiltmeter Specifications 111

Table B.2(b) Thermistor Specifications 111

Table B.3(a) Accelerometer Specifications 112

Table B.3(b) Accelerometer Dimensions 112

Table B.4(a) Impact Hammer Specifications 113

Table B.4(b) Impact Hammer Dimensions 113

1

Chapter 1 Introduction

1.1 Overview

Finite element (FE) model updating is the process of using an experimental data obtained

from a nondestructive test (NDT) of an existing structure to update an analytical model so it can

reflects the observed behavior of the structure more accurately. Finite element model updating

using NDT data is an effective tool for FE model validation, structural evaluation, damage

assessment and bridge management.

The FE model is derived from a set of material properties and cross sectional geometry as

well as mass parameters that are used to calculate the analytical response of the structure.

Typically, a subset of uncertain structural parameters that make up the model is assumed to be

unknown and these parameters are updated in the parameter estimation process.

FE model updating is an example of an inverse problem. The typical parameter estimation

process involves using a fully defined analytical model to predict the response of a structure and

update it to match a set of response measurements.

2

1.2 Scope of Work

This research is focused on using multiresponse structural parameter estimation for the

purpose of automated FE model updating of a scale bridge model based on the NDT data. The

scale bridge model, also known as the UCF Grid benchmark, has been constructed at the

University of Central Florida (UCF). There are two longitudinal girders with two clear spans and

seven transverse beams connecting the two girders. For static and dynamic testing, 8 vibrating

wire strain gages, 4 vibrating wire tiltmeters, 4 displacement gages and 8 accelerometers were

used. Multiresponse static and modal data collected from the UCF grid was used to calibrate the

finite element model.

Based on the developments in computing systems and application programming interfaces, it

is possible to use two commercially available software packages simultaneously to solve a

complex problem. To satisfy the need of a software for automated finite element model updating

of full-scale structures using measured NDT data, the PARameter Identification System

(PARIS©

) software package has been developed at Tufts University. PARIS is a MATLAB®

-

based software which uses SAP2000® as a slave program for the finite element analysis (FEA).

MATLAB accesses SAP2000® via the API provided by SAP2000

® developers. The latest

version of this software, PARIS14.0 was developed to replace its predecessor PARIS 13.0.

Features added to PARIS were designed to fulfill the goal of this research, which was to propose

an applicable approach to full scale structures.

1.3 Thesis Layout

This thesis is divided into six main chapters with an additional three appendices. Current

chapter is providing an introduction to the materials presented in this document.

3

Chapter 2 of the thesis, which will be later submitted for peer-reviewed journal publication,

is the major contribution of this research. It presents the automated FE model updating of scale

bridge model using multiresponse parameter estimation.

Chapter 3 discusses the NDT data quality analysis which was performed using the measured

data from both static and dynamic tests. This chapter contains comprehensive information about

the criteria used in data quality analysis study prior to using the experimental data in the

parameter estimation process. The selected subsets of the measured strains, displacements and

slopes responses used for FE model updating of the UCF grid are presented at the end of this

section.

Chapter 4 presents the error sensitivity analysis using Monte Carlo simulations. The

observable parameters due to measurement locations and parameter grouping schemes were

determined based on this study.

Chapter 5 contains supplementary parameter estimation cases in addition to the typical cases

previously presented in Chapter 2. The direct comparison between the all available analytical and

measured responses was also performed to show the level of success in the FE model calibration

of the UCF grid. In this comparison calibrated FE model on link elements and FE model with

pins and rollers boundary condition were used respectively.

Appendix A is the user reference manual for PARIS14.0. It also contains a sample MATLAB

based user input file. PARIS14.0 requires to upload a previously created SAP2000 FE model and

a MATLAB based input file for FE model calibration. The detailed instructions for making a

SAP2000 model compatible with PARIS14.0 are provided in the user manual. The specific

4

format for defining model updating problem through the input file can also be found in the user

manual.

Other two appendices are included to document the specifications of the sensors used in the

non destructive static and dynamic tests and to provide supplement details of the UCF grid with

drawings.

The references of each section are provided at the end of the section. The figures and tables

are labeled and summarized in the list of figures and tables respectively. The chapter, section,

and subsection headings were kept consistent throughout this thesis to guide the reader through

the document.

5

Chapter 2 Automated Finite Element Model Updating of Scale Bridge

Model Using Measured Static and Modal Test Data

This chapter has been formatted to be suitable for peer-reviewed journal publication.

2.1 Introduction

According to the American Society of Civil Engineer’s 2013 Infrastructure Report Card,

“Over two hundred million trips are taken daily across deficient bridges in the nation’s 102

largest metropolitan regions.” One in nine, or below 11%, of the nation’s bridges are classified as

structurally deficient, while an average of 607,380 bridges are currently 42 years old. Currently,

24.9% of the nation’s bridges have been defined as functionally obsolete [1]. The Federal

Highway Administration (FHWA) estimates that the current cost to repair or replace the

deficient bridges eligible under the Federal Highway Bridge Program is almost $76 billion [2].

Damage can accumulate during the life of the structure due to normal use or overuse and

overloads. In the inspection process, many of the structure’s elements are not observable to the

inspector since they are covered by non-structural elements or are not easily accessible, which

can cause uncertainty in their final reports. To add more confidence in our current methods of

6

inspection and to improve the maintenance of infrastructures, objective and quantifiable

structural health monitoring is essential in this field.

Implementing a damage identification strategy for aerospace, civil, and mechanical

engineering infrastructures is referred to as Structural Health Monitoring (SHM). There are two

main approaches to SHM: (a) non-model-based approach and (b) model-based approach [3].

Both approaches have been successfully used for damage detection in structural applications.

The non-model-based approach relies on the signal processing of experimental data, while the

model-based approach relies on mathematical descriptions of structural systems [4]. The

alternatives to the non-model-based method include: modal analysis, dynamic flexibility

measurements, matrix update methods and wavelet transform technique, which are used to

determine changes in structural vibration to identify damage [5].

The model-based approach is usually implemented by using a computer model of the

structure of interest, such as a Finite-Element Method (FEM), to identify structural parameters

based on the measured test data. In the model-based method, the first step is to create the FE

model based on the design calculations. In most cases the initial model cannot accurately predict

the actual parameters and responses of the structure; this may arise from the simplification in the

modeling process. The model-based methods are often used for structural evaluations at the

element level. The recent advancements in data acquisition systems (DAQ) and sensor

technology have significantly improved our confidence in nondestructive test (NDT) data.

Measured responses of the structure based on the NDT data can be used to validate the initial

structural model and to update the parameters of the assumed model. The model calibration

process involves selecting a small number of model parameters that have uncertainty so that their

values cannot be known a priori and using various procedures to find the values for which their

7

measurements best match the model predictions [6]. A promising method is to minimize the

residual between the predicted response from the initial model and the measured response of the

actual structure according to the observed data sets. Approaches, methods, and technologies for

effective practice of structural health monitoring were surveyed in [7]. In this book, model

calibration was classified based on the selection of parameters to estimate, available measured

data, formulation of the objective function, an optimization approaches, and the level of

uncertainties. Using a model-based approach allows modeling and estimating the physical

properties using commercially available software to create and maintain the structural model.

Robert-Nicoud et al. [8] performed the model calibration of the Lutrive Highway Bridge in

Switzerland, a 395 m three span bridge with a maximum span of approximately 130 m, based on

static measurements such as deflections, rotations, and strain. Bodeux and Golinval [9]

successfully applied the Auto-Regressive Moving Average Vector (ARMAV) method to identify

frequencies and mode shapes of the Steel-Quake benchmark, a two story steel frame structure.

Koh et al. [10] presented a combination of genetic algorithms (GA) and local search techniques

to identify structural parameters using vibration measurements. Hybrid structural identification

methods were performed to estimate the structural parameters of a structure with 52 unknown

parameters. Huang et al. [11] performed the system identification of the dynamic properties of a

three-span box-girder concrete bridge using the Ibrahim Time-Domain (ITD) technique based on

the free vibration test results of the bridge.

Along with the analytical studies, a feasible way to verify a new methodology for structural

health monitoring and system identification is to apply it first to a scale model structure. As

stated by Catbas et al. [12], “the main purpose of constructing this laboratory model was to close

the gap between very simple laboratory tests and the field tests.” To provide an abstract

8

representation of a short to medium span highway bridges, the University of Central Florida

(UCF) grid benchmark was designed to have the lower natural frequencies in the range of 1 to 50

Hz, similar to full scale bridges.

In the present work, NDT data quality criteria was established and implemented to reduce the

measurement errors in the model calibration of the UCF grid. To lessen modeling error, the most

accurate FE model was created with all the details including geometry, section properties, and

boundary conditions. Error sensitivity analysis was performed using simulated damage cases to

find the parameters with least sensitivity to measurement errors and also to determine the most

observable parameters in presence of measurement error. In these simulations, various grouping

of elements with the same properties were used to reduce the number of unknowns, and then

ungrouping was utilized to reduce the modeling errors. In this process a set of observable and

error tolerant unknown parameter groups was selected that were for the purpose of FE model

calibration using measured NDT data. Multiresponse static and modal data collected from the

UCF grid was used to calibrate the finite element model so that it can more closely reflect the

actual behavior of the structure. The modal data includes mode shapes and corresponding natural

frequencies as well as static data including displacement, rotation and strain, which have been

used in this study. Selecting an observable and error tolerant set of unknown structural

parameters and using the best subset of measured data led to successful simultaneous estimation

of stiffness and mass parameters.

The present study is based on the parameter estimation formulations developed by

researchers at Tufts University used in PARIS (PARameter Identification System) for automated

FE model calibration of full scale structures. PARIS is a MATLAB [13] based program which

uses SAP2000 [14] as a slave FE analysis engine using Application Programming Interface

9

(API) for finite element analysis (FEA). APIs allow real-time exchange of information between

MATLAB and SAP2000. PARIS is designed for calibrating FE models comprised of frame,

quadrilateral planar shell, and cuboid solid elements using static and/or modal data. FE model

updating in PARIS is based on minimizing the residual between the predicted response of the FE

model and the measured data from the NDT. In the parameter estimation process of the UCF grid

that was based on measured NDT data, both mass-based and flexibility-based error functions

have been used. PARIS is an open source research software and has been posted at the SHM

research website of the Civil and Environmental Engineering Department at Tufts University at

http://engineering.tufts.edu/cee/shm/software.asp.

2.2 Error Functions

In the parameter estimation process, error functions play a pivotal role. In general, error

functions define the difference between analytical and experimental measurements. The

estimation of an unknown parameter is the result of finding the global minimum of the error

function. Error functions represent a residual between the predicted responses from the computer

model, such as finite element model (FEM) and the NDT measured data. Error functions are

generally defined as the residuals between predicted and measured system responses, in which

‘q’ represents the response quantity, e(p) is the error function, and p represent the unknown

structural parameters.

measuredpredicted qqpe )( (2.1)

The response of the structure can be quantified under a set of known static loads or modal

excitations that categorize error functions as Static-based and Modal-based. To find the

http://engineering.tufts.edu/cee/shm/software.asp

10

discrepancy between the analytical and experimental response, the main categories of error

functions have been subdivided into stiffness and flexibility based error functions. The stiffness-

based error function measures the residual between applied forces, and the flexibility-based error

function determines the difference between experimental and analytical displacements estimation

of the unknown structural parameters. These error functions are implicitly a function of the

structural parameters {p}. Examples of the unknown parameters include axial rigidity (EA),

bending rigidity (EI) and torsional rigidity (GJ) for frame elements. For plates and shells,

modulus of elasticity and plate thickness are used, and for isoperimetric solids, the modulus of

elasticity is used as unknown parameters. For dynamic systems, lumped mass and/or distributed

mass can be used as modeled in the FE model. Link elements are also used to represent axial,

bending and torsional springs in 2D and 3D spaces.

2.2.1 Static Flexibility (SF)

The static flexibility-based error function was developed by Sanayei et al. [15] and it is based

on the residual between measured and analytical displacements only for measured responses.

1[ ( )] [ ( )] [ ] [ ]SFe p K p F U

(2.2)

Where [F] and [U] represent the applied live loads cases and the corresponding measured

displacements per load case. The error function eSF(p) is calculated based on the analytical

stiffness matrix [K(p)] and known applied live load cases.

2.2.2 Static Strain (SSTR)

Sanayei and Saletnik [16] developed a strain-based error function that is the residual of

predicted and measured strains at selected observation points. The B matrix was used to map the

nodal displacements to strains along the length of each member.

11

][][)](][[)]([ 1 FpKBpeSSTR (2.3)

In Eq (2.3), is the measured strain matrix for load cases.

2.2.3 Modal Flexibility (MF)

Sanayei et al. [17] developed the modal flexibility-based error function that is the residual

between analytical and measured modal displacements. The inverse form of the analytical

stiffness matrix [K(p)] was used in the formulation of the modal-flexibility error function and

analytical mass matrix [M(p)].

jjjMF pMpKpe }{})]{([)]([)]([ 12 (2.4)

Mode shapes }{ and natural frequencies }{ can be used as the input data based on the

identified modal parameters from the NDT data.

2.3 Scalar Objective Function

The scalar objective function J(p) is created in Eq (2.5). The value of J(p) is minimized

during the parameter estimation process using numerical sensitivities. In this work, the

constrained nonlinear multivariable function (fmincon), which is available in MATLAB

Optimization Toolbox, was used as the minimization technique. When neither the modeling error

nor the measurement error is a part of the parameter estimation, the scalar objective function will

approach zero.

2

( ) ( )ij

i j

J p e p (2.5)

12

where i represents a measurement at a selected DOF and j represents the number of sets of

forces.

To verify the final result of the J(p) minimization, which should have resulted in the global

minimum, multiple initial starting points were selected. Since the final results from multiple

initial points were converged on the same answer, it was concluded with high possibility that the

global minimum had been reached in the minimization procedure.

2.4 Normalization

In solving any system of equations, there is the possibility of facing ill-conditioning resulting

in poor estimation of the unknown parameters; therefore four types of normalizations were

proposed to prevent such numerical challenges while using a combination of error functions with

different types of measurements.

2.4.1 Parameter Normalization

In normalization at the parameter level, the estimated value was divided by the initial value

of the parameter. Ratios less than one indicate degradation in the value of the parameter in

comparison with the initial guess, and ratios larger than one represent that the actual value of the

parameter is larger than the initial estimate.

2.4.2 Error Function Normalization

To ensure that all the measurements have a reasonable influence on the estimation process, a

normalization technique was needed to apply to the error functions, while multiple sets of

measurements, each in different units and scales, were used in the parameter estimation

procedure. In the case of combining multiple measurements without error function

normalization, the final result could not be robust since measurements with different magnitudes

13

were used in the scalar objective function. The larger measurements could potentially

overshadow the smaller measurements during the parameter estimation process. For instance,

displacements were measured on the order of millimeters (10-1

), while strains were measured on

the order of micro strains (10-6

). In order to overcome this issue, each error function was divided

by its initial value as the optimization iterations progress. This method was used to make the

error functions normalized and unitless to reduce ill-conditioning in the system of equations.

2.4.3 Objective Function Statistical Normalization

In the normalization method presented by DiCarlo [18], the static-based error functions were

normalized with respect to their standard deviations, which make the objective function

weighted, normalized and unitless. This method also resolved the different order of magnitudes

issue for multiple measurement types. When objective function normalization was applied, each

measurement was weighted proportional to the inverse of its uncertainty in the estimation

procedure.

)()()(1

pepepJ e

T (2.6)

In Eq (2.6), e is the covariance matrix of the measured response data. To find the parameter

that makes the measured data the “most probable,” the maximum likelihood estimator was used.

Using this normalization technique was also a very efficient approach to smooth the objective

function surface.

2.4.4 Objective Function Weighting

In using multiresponse parameter estimation, to prevent the domination of a scalar objective

function by another one, there is a need to normalize the objective functions. Each objective

function was normalized with respect to their initial value, rendering a weight of 1.0 to each

14

objective functions. If needed, each scalar objective function can be multiplied by a suitable

weight factor based on the measurement types used and the number of measurements used

2.5 Parameter Grouping

Parameter grouping is a technique that allows assigning parameters with the same structural

parameters (e.g., mass, stiffness and area) into one group. Grouping reduces the total number of

unknown parameters, which results in fewer of the required measurements for the parameter

estimation process.

2.6 Multiresponse Parameter Estimation

Multiresponse parameter estimation is an effective method by which to simultaneously use

different error functions that are based on different types of measurements, which allows the

combination of static and modal data in the model updating procedure (Bell et al. [19]).

Usingerror function stacking would increase the number of measurements in the parameter

estimation process that can result in estimating unknown parameters. To have the information of

each error function in this technique, the vector of eStack(p) has been created as follows:

1

2

( )

( )( )

( )

Stack

n

e p

e pe p

e p

(2.7)

where n = number of error functions.

15

2.7 University of Central Florida Grid Benchmark

The University of Central Florida (UCF) Grid benchmark is a scale bridge model which was

designed, constructed and used for experimental studies at UCF [20], see Figure (2.1). As stated

in Catbas et al. [12], this model is a multi-purpose specimen, enabling researchers to use it as a

test bed for different techniques, algorithms and methods of the model updating procedure. The

UCF grid has two longitudinal girders which are run continuously for the whole length of the

structure, each with a length of 5.48 m (18 ft), with two clear spans and seven transverse beams

connecting the two girders. As stated in Burkett [22], the transverse beams are connected into

girders by two angle clips and two cover plates to provide shear and moment transfer, which

makes it a partially restrained connection, see Figure (2.2). Both girder and beams are A36 steel

S3x5.7 standard sections. These sections were found to be the best fit to make the modal

frequencies, deflections and slopes represent the short to medium span bridges. To reduce the

effect of support vertical movements, W12x26 sections were used as piers, each with a length of

1.07 m (42 in) and fixed at the base.

The UCF grid has complex boundary conditions as shown in Figure (2.2). The pier to girder

connection was designed to behave as an ideal roller support. It consists of a cylinder

sandwiched between two curved bearing plates which allow the translation and rotation in the

plane of bending for the girders.

16

Figure 2.1 UCF Grid

Figure 2.2 UCF Grid Member Connections (Left) & Boundary Support (Right)

2.7.1 Instrumentation plan

For static testing, the structure was instrumented with 8 vibrating wire strain gages (Geokon

model 4000), 4 vibrating wire tiltmeters (Geokon model 6350) and 4 displacement gages (Space

Age Control) [21]. Figure (2.3) shows the instrumentation details and sensor numbering for

static testing. Displacement gages (DG) were connected in between the floor and underneath four

of the connections. Tiltmeters (TM) were mounted on the longitudinal members with the typical

distance between each end of the girders and the closest tiltmeter (TM) was 178 mm (7 in).

Strain gages (SG) were welded to the top and/or bottom flanges of the longitudinal members and

placed in the 266.7 mm (10.5 in) from the closest transverse beam.

17

Figure 2.3 Instrumentation Details for Static Testing with Sensor Numbers

Eight accelerometers (PCB 393C) were used for dynamic testing on the UCF grid. Each

accelerometer was placed vertically and the grid was excited at four different locations. Figure

(2.4) shows the accelerometer and excitation locations for dynamic testing using an impact

hammer (IPC 086D20). Based on the instrumentation of the structure, a multi-input multi-output

data set with 4 sets of impact forces and 8 sets of acceleration time histories was created.

Figure 2.4 Accelerometer and Impact Locations for Dynamic Testing

2.7.2 Finite Element Model for the UCF-Grid

A finite element model was created in SAP2000©

based on the design calculations and CAD

drawings in Burkett [22]. Table 1 shows the initial section properties for the finite element

model. All of the structure’s elements were modeled as A36 steel with a Young’s modulus of

18

200 GPa (29,000 ksi). For the girder and beams, S3×5.7 standard sections have been used, and

for supports, W 12×26 sections have been used as piers, each with a length of 1.07 m (42 in) and

fixed at the base. Section properties for these standard sections can be found easily in the AISC

manual [23]. The longitudinal, transverse, and support members are modeled as frame elements

with their respective structural properties based on structural shape dimensions. The support

columns are assumed to have fixed boundary conditions at the base. To have the best a priori

model for the purpose of finite element model updating, the authors tried to create the geometry

of the UCF grid FE model as accurately as possible. To achieve this goal, the support columns

have been extended to the center of the rotation of roller supports, and the connections between

the grid and the support members were modeled using vertical link elements; the six stiffness

values of the boundary links can be adjusted to accurately represent the boundary conditions.

Figure (2.5) shows the finite element model of the UCF grid.

Table 2.1 UCF Grid Initial Section Properties

Model

Element

AISC

Classification

Ak

[m2 (ft

2)]

Am

[m2 (ft

2)]

Iz

[m4 (ft

4)]

Iy

[m4 (ft

4)]

Elz

[N.m2 (lbf.ft

2)]

Girder S3x5.7 1077×10

-6

(115.92×10-4

)

1077×10-6

(115.92×10-4

)

1.05×10-6

(1.22×10-4

)

0.19×10-6

(0.22×10-4

)

20.98×104

(5075×106)

Beam S3x5.7 1077×10

-6

(115.92×10-4

)

1077×10-6

(115.92×10-4

)

1.05×10-6

(1.22×10-4

)

0.19×10-6

(0.22×10-4

)

20.98×104

(5075×106)

Girder

Connection built-up

1077×10-6

(115.92×10-4

)

1077×10-6

(115.92×10-4

)

1.05×10-6

(1.22×10-4

)

0.19×10-6

(0.22×10-4

)

20.98×104

(5075×106)

Beam

Connection built-up

1077×10-6

(115.92×10-4

)

1077×10-6

(115.92×10-4

)

1.05×10-6

(1.22×10-4

)

0.19×10-6

(0.22×10-4

)

20.98×104

(5075×106)

Pier W12x26 4935×10

-6

(531.19×10-4

)

4935×10-6

(531.19×10-4

)

84.91×10-6

(98.38×10-4

)

7.2×10-6

(8.34×10-4

)

1698.22×104

(410833×10

6)

19

Figure 2.5 UCF Grid Finite Element Model

2.7.3 Boundary Stiffness Sensitivity Study

To determine the range for the boundary link elements in the UCF grid, Sipple [24]

performed a sensitivity study. Boundary condition stiffness has been divided into free, partially

restrained and fixed boundary conditions. This study was only performed for the active boundary

stiffness directions, which are Kx, Kz and Kry. Table 2.2 shows the boundary stiffness ranges.

Table 2.2 Boundary Stiffness Ranges

Stiffness Direction Free Partially Restrained Fixed

Kx [N/m (Kip/in)] 0 to 2×107(114) 2×10

7(114) to 2×10

9(11420) 2×10

9(11420)

+

Kz [N/m (Kip/in)] 0 to 5×104(0.28) 5×10

4(0.28) to 2×10

9(11420) 2×10

9(11420)

+

Kry [rad/m (rad/in)] 0 to 4×105(10) 4×10

5(10) to 4×10

10(10

6) 4×10

10(10

6)

+

In this work, only Kz stiffness direction was used since gravity loading and data acquisition

was performed only in the vertical direction.

Boundary

Link (Kz)

Fixed Boundary

Condition

Connection Elements W12x26

S3x5.7

Y X

Z

20

2.8 Nondestructive Testing of UCF Grid

Nondestructive tests were designed and performed on the UCF grid by Gul and Catbas [21]

by applying static gravity loadings and dynamic loadings using an impact hammer. For static

testing, vertical displacements, rotations about y-axis, and longitudinal strains on beams were

measured. Three identical static NDTs were performed on the UCF grid using 8 different load

cases, and static responses were measured.

Dynamic tests were used to determine the modal parameters of the UCF grid, i.e.,

frequencies and mode shapes. An impact hammer was used to excite 4 nodes, and time history

accelerations were measured with all 8 accelerometers. The modal measured data were processed

at UCF by Catbas et al. [12] using the Complex Mode Indicator Function (CMIF) algorithm to

identify modal parameters, such as natural frequencies and mode shapes of the structure.

2.8.1 Static Load Test

The main rationale for the measurement and loading locations was to stress all the main

structural elements that affect the characteristic deflections of the system. To mitigate uplift at

the supports due to individual live loads, an additional dead weight of 178 N (40 lbf) was

permanently placed on the grid at all support points during the static tests. There were a total of 8

load cases for static testing; each of them includes 2 or 4 point loads, which were about 670 N

(150 lbf). Table 2.3 shows the location of each load case. The structure was tested three times

independently under each load case which involved unloading and reloading the structure for

each measurement. Ideally, to have the true independency of measurements, it would have

required to re-instrumenting the structure for each trial; since this would have taken a

prohibitively long time, it was not requested.

21

Table 2.3 UCF Grid Static Load Cases

Load Magnitude and Location

LC 1 672.8 N @ B1 671.9 N @ E1 671.9 N @ C3 671.1 N @ F3

LC 2 672.8 N @ B1 671.9 N @ E1

LC 3 671.9 N @ C3 671.1 N @ F3

LC 4 671.9 N @ E1 671.1 N @ F3

LC 5 672.8 N @ B1 671.9 N @ C3

LC 6 672.8 N @ B2 671.9 N @ C2 671.9 N @ E2 671.1 N @ F2

LC 7 672.8 N @ B2 671.9 N @ C2

LC 8 671.9 N @ E2 671.1 N @ F2

2.8.2 Dynamic Impact Test

The dynamic test plan for the UCF grid consisted of four vertical impact locations with 8

vertical acceleration response measurement locations. The test consisted of hitting the structure

with an impact hammer at a specific point of the structure. The test was designed only to capture

the vertical modes of the structure because these could have given more information about the

strong axis stiffness of each member, similar to the static tests under gravity loadings. During the

instrumentation process of the UCF grid, it was believed that measuring the vibrations of the

middle connections in comparison with the connections above the piers would have resulted in

the better characteristic mode shape of the scale bridge model; this was because there was not

enough vibration to measure with accelerometers since the vertical direction was restrained by

the model supports.

2.9 Error Sensitivity Analysis Using Monte Carlo Simulations

Monte Carlo (MC) analysis is a powerful tool for performing statistical analysis of a complex

system in which numerous mathematical operations create a nonlinear system of equations.

Using Monte Carlo simulations can determine how error in measurements can affect the

parameter estimation process, since accuracy of parameter estimates is one of the essential

components for a successful finite element model calibration. This method was used by selecting

22

a large number of observations (NOBS) and performing an independent parameter estimation for

each of them in the presence of simulated measurement error. When parameter estimation was

completed for each observation, the final contaminated parameter estimates were saved. Using

all MC simulations, statistical properties of parameter estimates, such as mean and standard

deviation, were calculated. Sanayei and Saletnik [25] used Monte Carlo simulations to study the

input-output error relationship of a 2D truss and a 2D frame based on the noise in static strain

measurements. The mean and standard deviation of the final estimations became more stable and

accurate as more observations were performed.

2.9.1 Error Sensitivity Analysis for the UCF Grid

The purpose of the error sensitivity analysis was to determine the effect of measurement error

in the parameter estimation process of the UCF grid. In each case of Monte Carlo simulation, 50

observations (NOBS) were chosen for different simulated parameter estimation scenarios.

Various numbers of MC experiments were performed and it was determined the 50 observations

were sufficient to produce reasonable input-output error estimates. In this process, unknown

mass and stiffness parameters were identified in different simulated scenarios. In each simulated

scenario, all of the structural parameters except the damage ones were assumed to be known with

a high level of confidence. The number and location of the measured responses were selected as

it was in the nondestructive test of the UCF grid. The response data were simulated to have a

uniform-proportional (Sanayei et al. [25]) random error in the range of 5%. This range was

believed to provide a reasonable and practical level of measurement error. In order to find the

influence of the measurement error based on different sensors and the respective error function,

stacking was not used in Monte Carlo simulations.

23

To find the strong axis bending rigidity (EIz) and mass of the main structural members and

connection members, three member groups were defined: (G1) L shape corner connections, (G2)

T shape middle connections and (G3) transverse beams and longitudinal girders. Figure (2.6)

shows the parameter grouping scheme which was used in all parameter estimations. Both the

stiffness and mass parameter values for all the boundary links in the vertical direction (z-

direction) were grouped, creating six parameter groups for the UCF grid error sensitivity

analysis.

Figure 2.6 Parameter Grouping Scheme for UCF Grid

The parameter grouping scheme was designed based on the results of the Monte Carlo

simulations for multiresponse parameter estimation using NDT data. All excitations and sensors

were applied to longitudinal members 1 and 3 and none on the transverse members A through G.

According to the error sensitivity analysis for the given loadings and sensor locations, both T and

L shape connections should be grouped as one for better observability and estimation of

connections stiffness and mass parameters. It was also necessary for the longitudinal and

transverse beams to be grouped together for stiffness and mass parameter estimations. Given the

current set of measured excitation and response locations, this study resulted in forming five

highly observable and error tolerant parameter groups consisting of stiffness and mass of all

24

beam members and stiffness and mass of all connections along with the boundary link stiffness.

Rolled beam section properties are known with a high confidence. Their stiffness and mass

properties are kept as unknown for quality control of the unknown connection stiffness and mass

parameters. Based on MC simulation results, feasible parameter estimation cases were

determined to be used with the experimental NDT data from the UCF grid.

2.10 Parameter Estimation of the UCF-Grid Using NDT Data

Since the precision of the measurements has a major impact on the final results of the model

updating procedure, data quality analysis was performed on the measured data from

nondestructive tests. The goal of this analysis was to minimize the influence of the measurement

noise by finding the best subset of the measured data and dropping the noisy measurements

before using them in the parameter estimation process. In general, measurement error is the

difference between actual and measured values in measurements. The source of the noise in

measurements can be from electronic devices (e.g., sensor types, signal conditioning, electronic

random noise picked up by unshielded wires), installation faults and/or ambient noise. In

addition, ambient vibrations are always present, adding to the signal noise.

2.10.1 NDT Data Quality Study

Based on the level of noise to signal ratio and confidence in the measured values, some

measurements were eliminated from the measured test dataset. To select the noisiest

measurements, standard deviation of measured data in the unloaded and loaded phases were

calculated. The larger values of the standard deviation of the measured data were used for

selecting the nosiest sensors. For each sensor, values smaller than ten times of standard deviation

of the measurements were eliminated from the dataset. For example, the static strain

measurements were all in the order of 0 to 60 με, and since the standard deviation of data in most

25

strain gages were around 0.5 με, it was decided to eliminate the measurements that were smaller

than 5 με. Eliminating the measured values based on the noise floor was also applied to

measured displacements and slopes which ensured that smaller measured values were eliminated

since they were mainly delivering more noise to the objective function. For example, slope

values less than 0.01 degrees were eliminated since it has been assumed to be contaminated by a

high level of error; slopes were in the order of 0 to 0.06 degrees. Displacements were in the order

of 0 to 0.762 mm (0.03 in), where measurements less than 0.254 mm (0.01 in) were eliminated

based on the noise floor.

Since three trials for each measurement set were conducted, one other criterion was to use

only the values that are based on exactly three independent tests and no less. The reason for this

was to have consistency in the level of confidence in the measured data values. The average

value of three sets of tests was used in the parameter estimation process.

Table 2.4 presents the final outcome of the data quality analysis of the measured responses

from the static test indicated by the sensor numbers. These data were used in the final parameter

estimation process of the UCF grid.

Table 2.4 Sensor Numbers Used for Measured Static Responses

Sensors Disp. Gage Tiltmeter Strain Gage

LC 1 2, 4 1, 2, 3, 4 1, 3, 4, 5, 6, 7

LC 2 1, 2 1, 2 1, 3, 4

LC 3 4 3, 4 5, 6, 7

LC 4 2, 4 1, 2, 3, 4 1, 2, 3, 4, 5, 7

LC 5 1, 2, 3 1, 2, 3, 4 1, 3, 4, 5, 7

LC 6 1, 2, 3, 4 1, 2, 3, 4 1, 3, 4, 5, 7, 8

LC 7 1, 2, 3 1, 2, 3, 4 1, 2, 3, 4, 5, 7, 8

LC 8 2, 4 1, 2, 3, 4 1, 3, 4, 5, 7, 8

Total Data Points 19 of 32 28 of 32 42 of 64

26

All the modal data delivered from the UCF grid was also examined for reliability.

Experimental mode shapes were compared to analytical mode shapes using the Modal Assurance

Criterion (MAC). Allemang [26], presented the following formulation for MAC:

2

MAC

T

ei aj

ij T T

ei ei aj aj

(2.8)

In the Eq (2.8), ei is the ith

experimental mode shape and aj is the j

th analytical mode shape.

Table 2.5 MAC Values Using Initial Parameter Values

Analytical Mode

Modes A1 A2 A3 A4 A5 A6 A7 A8 A9 A10

Ex

per

imen

tal

Mo

de

E1 0.998 0.002 0.001 0.000 0.065 0.003 0.001 0.986 0.011 0.000

E2 0.000 0.995 0.000 0.004 0.000 0.000 0.000 0.000 0.000 0.000

E3 0.000 0.000 0.999 0.000 0.151 0.004 0.001 0.013 0.988 0.000

E4 0.000 0.005 0.000 0.995 0.000 0.000 0.000 0.000 0.000 0.000

E5 0.000 0.000 0.000 0.000 0.714 0.927 0.031 0.000 0.001 0.000

E6 0.000 0.000 0.001 0.005 0.102 0.043 0.969 0.001 0.000 0.000

E7 0.000 0.000 0.004 0.150 0.009 0.000 0.033 0.000 0.004 0.002

E8 0.997 0.002 0.000 0.000 0.070 0.002 0.001 0.983 0.014 0.001

E9 0.001 0.000 0.997 0.001 0.144 0.004 0.001 0.020 0.979 0.000

E10 0.000 0.001 0.000 0.000 0.001 0.000 0.000 0.000 0.000 0.996

Since the number of measurements used to determine the mode shapes of the structure was a

small subset of the available degrees of freedom, MAC values cannot perfectly show the

correlation between the analytical and experimental mode shapes. In the model calibration of the

UCF grid, only the first 5 mode shapes were used. The reason for this was the inaccuracy in the

measured modal responses in higher frequencies and the calculated MAC values in Table 2.5.

2.10.2 Multiresponse Parameter Estimation and Model Calibration of the UCF Grid

The scope of the parameter estimation of the UCF grid was to update both mass and stiffness

properties. Parameters were selected for the estimation process due to their observability. These

parameters were: (a) Strong axis bending rigidity (EIz), (b) Area mass (Am) and (c) Boundary link

stiffness Kz. EIz and Am were estimated for both main structural members and connection

27

members. Model updating was not performed on parameters such as weak axis bending rigidity

and shear rigidity because of their low or no observability. Boundary link elements that were not

observable, such as rotation in all three directions and translation in x and y direction, were also

not included as unknown parameters.

The results of the model calibration are shown in Table 6. In this table, estimated bending

rigidity (EIz) of elements about their strong axis, and the cross-sectional area of elements (Am)

used for mass calculation in dynamic analysis, were presented as the normalized values with

respect to their initial guesses. Blank cells denote “not applicable” because the static error

function is not for estimation of mass parameters. All initial conditions for longitudinal,

transverse, and connection groups are assigned as the properties of S3x5.7 presented in Table

2.1. The initial Kz value for the vertical supports’ spring stiffnesses in all cases was 2.1 ×109

N/m

(12,000 Kip/in). In the experimental grid nondestructive tests, the static loads were applied in the

vertical direction (gravity loadings) and all dynamic impacts were applied in the vertical

direction. In the process of parameter estimation using the static and modal data, seven

combinations of error functions were used to examine their capabilities individually and together

for multiresponse parameter estimation in four different parameter estimation cases. Column 1 of

Table 2.6 shows the error function combinations used. In each case the parameter estimation was

performed simultaneously for all the unknown parameters shown in column 2. Other parameters

not included in column 2 were assumed to be known in each case using the initial parameter

values. Parameter grouping was also applied to all of the structural parameters using the same

modifier. Estimated values in column 3, 4 and 6, demonstrate the usage of error functions SF,

SSTR and MF individually. The rest present use of multiresponse parameter estimation using

more than one set of measured data with the associated error functions.

28

Table 2.6 UCF Grid Parameter Estimates

Case Estimated

Parameters SF SSTR

SSTR

+SF MF

MF

+SF

MF

+SSTR

MF+SF

+SSTR

1 EIz Long & Trans Beams 1.01 0.98 0.99 1.00 1.00 1.00 1.00

EIz Connection 2.09 1.75 1.84 1.88 1.88 1.88 1.88

2

EIz Long & Trans Beams - - - 1.00 1.00 1.00 1.00

EIz Connection - - - 1.78 1.79 1.80 1.78

Am Long & Trans - - - 1.00 1.00 1.00 1.00

Am Connection - - - 1.50 1.50 1.51 1.50

3


EIz Connection - - - 1.80 1.78 1.78 1.78

Am Long & Trans - - - 1.01 1.00 1.00 1.00

Am Connection - - - 1.51 1.50 1.50 1.50

Kz (Grouped) - - - 1.45 1.49 1.40 1.40

4


EIz Connection - - - 1.78 1.80 1.78 1.79

Am Long & Trans - - - 1.00 1.00 1.00 1.00

Am Connection - - - 1.51 1.51 1.50 1.50

Kz 1 - - - 1.49 1.46 1.46 1.48

Kz 2 - - - 1.49 1.50 1.46 1.50

Kz 3 - - - 1.50 1.59 1.47 1.64

Kz 4 - - - 1.50 1.65 1.47 1.69

Kz 5 - - - 1.49 1.15 1.45 1.08

Kz 6 - - - 1.49 1.28 1.45 1.20

In Case 1, the EIz parameter of the main structural members and connection members were

estimated. Due to complex shape of connection stiffener plates and angle clips with bolt holes,

their effective bending rigidities were unknown. As a result, EIz of the standard beam sections

were used as the initial value for both the beams and connections. All cases using individual

error functions or multiresponse error functions converged. Overall, the estimated bending

rigidities for beam sections were close to the values of the standard section. The values for

connection bending rigidities were much larger than the beam values. These larger values are

reasonable since the connections were built up with plates connecting the flanges and webs of

the longitudinal and transverse members shown in Figure (2.2). The parameter estimates

29

improved as more data was added, especially modal data, using multiresponse parameter

estimation.

In Case 2, rigidities and mass parameters of beams and connections of the UCF grid were

estimated. It was observed that by adding measured modal data to the estimation process, there

was more consistency in the final parameter estimates. Since mass parameters were involved, the

static error functions were not applicable.

In Case 3, there was interest to determine if there was any vertical flexibility due to floor,

column, roller, etc. For this purpose, the Kz stiffness values for all boundary links were

considered as one unknown group. Four groups of parameters from members and connections

stiffness and mass properties, plus one vertical boundary stiffness group, resulted in five

unknowns for this case.

In Case 4, the total of ten unknown parameters, including four groups of member stiffness

and mass, added to six individual Kz values for estimation. To reduce the effect of modeling on

parameter estimates, all significant structural parameters engaged in resisting vertical

deformations were assumed to be unknown in cases 3 and 4. Based on the estimated parameters

in the last two cases, there was no real change in stiffness and mass parameters of the

longitudinal girders and transverse beams.

In summary, there was an increase of about 80% in bending rigidity and 50% in mass of the

connections. This was expected since cover plates, angle clips, bolt holes, and bolts were used in

these connections. Boundary link stiffness Kz, remained unchanged in the range of fully fixed,

based on Table 2.2. Estimated bending rigidity of the connection members in Case 1 had larger

values in comparison with other cases. Based on the error sensitivity analysis performed for SF

30

and SSTR error functions individually, bending rigidity parameter of longitudinal and transverse

beams was more error tolerant than connection members. MC simulations were showed larger

estimated values for EIz of connection members than the true values. Since mass parameters were

assumed to be known in Case 1, due to a modeling error caused by this assumption, it was

predictable to have errors in finding the stiffness parameters alone.

2.10.3 FE Model on Pins and Rollers Used in Multiresponse Parameter Estimation and Model

Calibration

According to the estimated values of Kz stiffness presented in Table 2.6, and in comparison

with the boundary stiffness ranges in Table 2.2, it was realized that the Kz values were all in the

“fixed” range. Based on this, it has been decided to create a new FE model with “Pin-Roller-

Roller” boundary conditions. The boundaries at A1 and A3 were assumed to be pin supports and

the rest were modeled as roller supports. The parameter estimation was also performed using the

pins and rollers FE model and the same experimental static and modal data from the UCF grid.

The values that were presented in Table 2.1 were used to define the EIz and Am parameter’s

initial values. The parameter estimates are summarized in Table 2.7. The estimates for the pins

and rollers boundary conditions presented in Table 2.7 closely matched with the estimates in

Table 2.6 that were based on the previous larger model. The small differences in parameter

estimates are due to using a different size FE model whose sensitivities to measurement and

modeling errors can be different.

31

Table 2.7 UCF Grid Parameter Estimates Based on the Pins and Rollers Model

Case Estimated

Parameters SF SSTR

SSTR

+SF MF

MF+

SF

MF+

SSTR

MF+SF

+SSTR

1 EIz Long & Trans Beams 1.02 1.00 0.99 1.00 1.00 1.00 1.00

EIz Connection 1.82 1.79 1.85 1.88 1.88 1.88 1.88

2


EIz Connection - - - 1.78 1.81 1.80 1.80

Am Long & Trans - - - 1.00 1.00 1.00 1.00

Am Connection - - - 1.50 1.50 1.50 1.50

MAC matrix presented in Table 2.8 shows an excellent match between the first 6 mode

shapes of the calibrated model and measured data based on the Case 2 of Table 2.7 using MF, SF

and SSTR error functions simultaneously. From these results it is inferred that the pins and

rollers model is a better representation of the measured data.

Table 2.8 MAC Values Using Parameters From Pins and Rollers Model

Analytical Mode


Ex

per

imen

tal

Mo

de

E1 0.998 0.001 0.000 0.000 0.000 0.000 0.498 0.000 0.000 0.002

E2 0.000 0.999 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.007

E3 0.000 0.000 0.999 0.000 0.000 0.000 0.000 0.693 0.307 0.000

E4 0.000 0.000 0.000 0.999 0.000 0.000 0.000 0.000 0.000 0.005

E5 0.000 0.000 0.000 0.000 0.998 0.002 0.493 0.000 0.001 0.000

E6 0.000 0.000 0.001 0.005 0.000 0.988 0.000 0.319 0.669 0.288

E7 0.000 0.000 0.003 0.159 0.000 0.035 0.000 0.003 0.036 0.394

E8 0.997 0.001 0.000 0.000 0.000 0.000 0.487 0.000 0.000 0.003

E9 0.002 0.000 0.996 0.001 0.000 0.000 0.001 0.687 0.310 0.000

E10 0.000 0.001 0.000 0.000 0.001 0.000 0.000 0.000 0.000 0.086

To verify the estimated values and assess the physical meaning of the parameters, a direct

comparison with the NDT data provides an objective and quantifiable basis for comparison. The

predicted responses based on the calibrated model in Case 2 (with 4 unknown parameters) were

compared with experimental responses. Both the static and modal responses for the updated

models were compared with the NDT data. Displacements, slopes, strains, frequencies and mode

shapes were graphed to compare the predicted response from the calibrated model with the

measured response from the nondestructive tests. It is important to remember that static

displacements, slopes, and strains are impacted only by the stiffness parameters and not by the

32

mass parameters. Both stiffness and mass parameters affect modal frequencies and mode shapes

of a structure that present the global model calibration.

Figures (2.7) and (2.8) show typical strains from LC1 and LC4 for 6 gages on the grid, which

were used in the parameter estimation of the UCF Grid and show the analytical prediction of the

uncalibrated and calibrated models for comparison with the measured strains. These plots show

an excellent match between the calibrated model response and NDT measurements.

Figure 2.7 Strains, Load Case 1


Figures (2.9) and (2.10) show typical displacements from 2 gages along longitudinal girders

(one per girder) for LC5, showing the analytical prediction of the uncalibrated and calibrated

models for comparison with the measured displacements. These plots show an improvement to

the match between the calibrated model response and NDT measurements. Although the

predicted responses moved toward the measurements, some of them did not match with the

measured displacements as well as other displacements did. This might be due to errors in

displacement measurements.

33

Figure 2.9 Displacements Along Girder 1,

Load Case 5


Load Case 5

Figures (2.11) and (2.12) show typical slopes from LC1 and LC6 for 4 tiltmeters on the grid

which were used in the parameter estimation of the UCF Grid and show the analytical prediction

of the uncalibrated and calibrated models for comparison with the measured strains. These plots

show an excellent match between the calibrated model response and NDT measurements.

Figure 2.11 Slopes, Load Case 1


Along with the direct comparison between the predicted and experimental responses of the

UCF grid, Root Mean Square (RMS) values for the predicted and experimental measurements

were also calculated in all load cases for each type of measurements. Table 2.9 presents the RMS

values.

34

Table 2.9 Predicted vs. Experimental RMS Responses

Measurement

Type

Uncalibrated FE

Model on Link

Support RMS

Calibrated FE

Model on Link

Supports RMS

Calibrated FE

model on Pins &

Rollers RMS

Measured

RMS

Strain (με) 37.89 32.65 31.29 31.72

(+19.42%) ( +2.92% ) (-1.36%)

Disp. (mm) 0.52299 0.47625 0.47498 0.44907

(+16.46%) ( +6.05%) (+5.77%)

Slope (Degree) 0.03829 0.03281 0.03282 0.03322

(+15.24%) (-1.98%) (-1.22%)

Avg. RMS Error 17.04% 3.65% 2.78%

RMS values of the predicted responses were calculated for both calibrated and uncalibrated

FE models. Percentage error of the predicted RMS responses with respect to the experimental is

data also presented in Table 2.9. The RMS percentage errors show great improvements for all

three types of predicted responses, more so for strains and slopes. This might have to do with

lower noise to signal ratios in measured strains and slopes. The UCF grid model supported on

pins and roller has a slightly lower average percentage error compared to the model supported on

link elements (springs). Overall, both of the calibrated models with measured NDT data showed

great improvements in predicting responses compared to the uncalibrated model.

Figure (2.13) shows the five natural frequencies used in parameter estimation. Generally all

natural frequencies moved towards the measured data. Frequencies of the calibrated model show

a great match with the measured frequencies in modes 1, 2, 3 and 5.

35

Figure 2.13 Comparison of Natural Frequencies

Table 2.10 presents the comparison of the mode shapes and frequencies of the UCF Grid. It

compares the uncalibrated and calibrated FE model modal data with the measured data for the

first 5 mode shapes of the UCF grid. Since each mode shape used only 8 measured degrees of

freedom, it is believed that the most efficient observability occurred only in the first few modes.

Percentage errors and the average errors of the predicted natural frequencies of the

uncalibrated and calibrated FE models were calculated with respect to the measured frequencies.

The average percentage errors show great improvements for both calibrated models. The UCF

grid model supported on pins and roller has a slightly lower average percentage error compared

to the model supported on link elements (springs). Overall both of the calibrated models showed

great improvements in predicting responses compared to the uncalibrated model, demonstrating

the robustness of the FE model updating method used.

36

Table 2.10 Comparison of Analytical Mode Shapes with Experimental Mode Shapes

Mode Uncalibrated FE Model

on Link Elements

Calibrated FE Model on

Link Elements

Calibrated FE Model on

Pins and Rollers Measured

Mo

de

1

f1 = 20.41 Hz f1 = 22.23 Hz f1 = 21.12 Hz f1 = 22.22 Hz

(-8.11%) (+0.06%) (-4.96%)

Mo

de

2

f2 = 24.44 Hz f2 = 27.84 Hz f2 = 26.76 Hz f2 = 26.83 Hz

(-8.91%) (+3.76%) (-0.28%)

Mod

e 3

f3 = 31.32 Hz f3 = 33.60 Hz f3 = 33.15 Hz f3 = 33.30 Hz

(-5.96%) (+0.90%) (-0.47%)

Mod

e 4

f4 = 38.07 Hz f4 = 43.00 Hz f4 = 42.57 Hz f4 = 40.63 Hz

(-6.30%) (+5.83%) (+4.76%)

Mo

de

5

f5 = 63.20 Hz f5 = 64.25 Hz f5 = 64.61 Hz f5 = 64.57 Hz

(Error = -2.12%) (Error = -0.50%) (Error = +0.06%)

Avg.

Error 6.28% 2.21% 2.11%

37

Although, by using the pins and rollers as the boundary conditions the percentage error of the

frequencies improved for f2, f3, f4 and f5, the FE model supported on the link elements had closer

predicted f1 with respect to the experimental frequency. Overall, the mode shapes and

frequencies using the calibrated UCF grid on pins and rollers supports demonstrate a better

match with the measured modal responses.

References

[1] American Society of Civil Engineer’s (ASCE), (2013) “Infrastructure Report Card”.

http://www.infrastructurereportcard.org/

[2] Federal Highway Administration (FHWA), (2012) “Deficient Bridges by State and

Highway Systems”

[3] Farrar, C. R. and Worden, K. (2006) “Introduction to Structural Health Monitoring,” Phil.

Trans. R. Soc. A 15 February 2007 vol. 365 no. 1851 303–315

[4] Barthorpe, R. (2011) “On Model- and Data-based approaches to Structural Health

Monitoring,” PhD Thesis – The University of Sheffield

[5] Saadat, S., Noori, M. N., Buckner. G. D., Furukawa, T., Suzuki, Y. (2004) “Structural

health monitoring and damage detection using an intelligent parameter varying (IPV)

technique,” International Journal of Non-Linear Mechanics 39(10), 1678–1697

[6] Aktan, A. E., Farhey, D. N., Helmicki, A. J., Brown, D. L., Hunt, V. J., Lee, K. L., and

Levi, A. (1997) "Structural identification for condition assessment: Experimental arts,"

Journal of Structural Engineering, 123(12), 1674–1684

[7] Catbas, N., Kijewski-Correa, T., Aktan, A. E., (2013) “Structural Identification (St-Id) of

Constructed Systems: Approaches, Methods, and Technologies for Effective Practice of St-

Id,” ASCE Books, ISBN: 9780784411971

[8] Robert-Nicoud, Y., Raphael, B., Burdet, O. and Smith, I. (2005) ‘‘Model identification of

bridges using measurement data,’’ Computer-Aided Civil and Infrastructure Engineering,

20(2), pp. 118–131

[9] Bodeux, J. and Golinval, J. (2001) ‘‘Application of ARMAV models to the identification

and damage detection of mechanical and civil engineering structures,’’ Smart Materials and

Structures, 10(3), pp. 479–489

[10] Koh, C. G., Chen, Y. F. & Liaw, C.-Y. (2003), “A hybrid computational strategy for

identification of structural parameters,” Computers and Structures, 81 107–17

[11] Huang, C.S., Yang, Y., Lu, L. and Chen, C. (1999) ‘‘Dynamic testing and system

identification of a multi-span highway bridge,’’ Earthquake Engineering and Structural

Dynamics, 28(8), pp. 857–878

http://www.infrastructurereportcard.org/

38

[12] Catbas, N., Gul, M., Burkett, J. (2008) “Damage assessment using flexibility and

flexibility-based curvature for Structural Health Monitoring,” Smart Materials and

Structures 17(1) 015024

[13] MATLAB R2012b (2012), MathWorks, Inc.

[14] SAP2000v15 (2012), Computers and Structures, Inc.

[15] Sanayei, M., Imbaro, G. R., McClain, J. A. S., and Brown, L. C. (1997), "Structural Model

Updating Using Experimental Static Measurements," ASCE, Journal of Structural

Engineering, 1997, 123(6), pp. 792–798

[16] Sanayei, M. and Saletnik, M.J. (1996) “Parameter Estimation of Structures from Static

Strain Measurements I: Formulation,” ASCE, Journal of Structural Engineering, Vol. 122,

No. 5

[17] Sanayei, M., Wadia-Fascetti, S., Arya, B., and Santini, E. M. (2001) "Significance of

Modeling Error in Structural Parameter Estimation," Computer-Aided Civil and

Infrastructure Engineering, 16(1), 12–26

[18] DiCarlo, C. (2008) “Applications of Statistics to Minimize and Quantify Measurement

Error in Finite Element Model Updating,” Masters Thesis – Tufts University

[19] Bell, E. S., Sanayei, M., Javdekar, C. N., Slavsky, E. (2007). "Multiresponse Parameter

Estimation for Finite-Element Model Updating using Nondestructive Test Data," J. Struct.

Eng., 133(8), 1067–1079

[20] Catbas, N. (2013) Benchmark Problem on Health Monitoring of Highway Bridges.

[Online]. http://www.cece.ucf.edu/people/catbas/benchmark.htm

[21] Gul, M., and Catbas, N. (2008). Test Details for the Preliminary Data for International

Bridge Health Monitoring Benchmark Study, 2nd Ed., Structures and Systems Research

Group, University of Central Florida, Orlando, FL

[22] Burkett, J. (2003) “Benchmark Studies for Structural Health Monitoring Using Analytical

and Experimental Models,” Masters Thesis – University of Central Florida

[23] Steel Construction Manual (2011), 14th Ed., Third Printing

[24] Sipple, J. D., “Frequency Response Function Based Finite Element Model Updating,” PhD

Thesis – Tufts University

[25] Sanayei, M. and Saletnik, M.J., (1996) “Parameter Estimation of Structures from Static

Strain Measurements. II: Error Sensitivity Analysis,” Journal of Structural Engineering,

Vol. 122, No.5

[26] Allemang, R. J. (2003). "The Modal Assurance Criterion - Twenty Years of use and

Abuse," Sound and Vibration, 37(8), 14–23

http://www.cece.ucf.edu/people/catbas/benchmark.htm

39

Chapter 3 NDT Data Quality Analysis

3.1 Introduction

Since the precision of the measurements has a major impact on the final results of the model

updating procedure, data quality analysis was performed on the measured data from

nondestructive tests. The goal of this analysis was to minimize the influence of the measurement

noise by finding the best subset of the measured data and dropping the noisy measurements

before using them in the parameter estimation process. Since error functions, as the heart of the

parameter estimation, are sensitive to errors, it is important to minimize the number of noisy

measurements. In general, measurement error is the difference between actual and measured

values in measurements. Noise in measurements can be from electronic devices (e.g., sensor

types, signal conditioning, electronic random noise picked up by unshielded wires), installation

faults and/or ambient noise. In addition, ambient vibrations are always present, adding to the

signal noise. Generally it is impossible to have a set of perfect measurements in any set of

experiments.

40

In the presence of measurement errors, the parameter estimation errors are normally

magnified, especially in ill-conditioned systems of equations used in optimization. By removing

the highly error contaminated measurements, the errors in parameter estimates were reduced.

This chapter presents the NDT quality analysis of the data from the University of Central

Florida based on the nondestructive test of the benchmark grid structure.

3.2 Criteria Used in NDT Data Quality Analysis

Pre-processing of the recorded data had to be performed in order to make successful

parameter estimation more likely. The data delivered from the static test consisted of time

histories of the sensors during the test. In each time history there was a recorded section prior to

applying load, the “unloaded” measurements, and there was a section that recorded with the

loads in place, “loaded” measurements. Figure (3.1) shows these two phases of measurements

clearly. For each section the averaged values were calculated. Subtracting the unloaded value

from the loaded value gives the measurement for that test from the specific load case. Since three

trials for each measurement set were conducted, the average value of the three independent tests

was used for the parameter estimation process.

Figure 3.1 UCF Grid Static Test L1T1 SG1

Unloaded

Loaded

41

In Figure (3.1), L1T1 SG1 stands for “Load Case 1-Test 1-Strain Gage 1”. Some of the time

histories were better than others and this was applicable for all types of measurements. Figures

(3.1), (3.2) and (3.3) are examples of good, acceptable, and unacceptable time histories,

respectively. The best time histories had clear unloaded and loaded sections separated by a steep

section representing the time it took to physically load the model.


The acceptable time histories had the unloaded, loaded sections contaminated by some noisy

data points. Care had been taken not to average the corrupt data points.


Figure (3.3) is an example of a time history that was unusable because the sensor did not

record enough acceptable data to clearly pick out unloaded and loaded data points.

Unloaded

Loaded

42

3.2.1 Data Quality Study: Criterion 1

Based on the level of noise to signal ratio and confidence in the measured values, some

measurements were eliminated from the measured test dataset. To select the noisiest

measurements, standard deviation of measured data in the unloaded and loaded phases were

calculated. The larger values of the standard deviation of the measured data were used for

selecting the nosiest sensors. For each sensor, values smaller than ten times of standard deviation

of the measurements were eliminated from the dataset. For example, the static strain

measurement based on the SG2 from L1T2 was 35.4 με; since the standard deviation of the data

in loaded phase was 32.18 με, ten times it was 321.8, this record was eliminate form the set of

measurements.

The static strain measurements were all in the order of 0 to 60 με, and since the standard

deviation of data in most strain gages was around 0.5 με, it was decided to eliminate the

measurements which were smaller than 5 με. Similarly eliminating the measured values based on

the noise floor was also applied to measured displacements and slopes. This Criterion also

ensured that smaller measured values were eliminated since they were mainly delivering more

noise to the objective function. For example, slopes were in the order of 0 to 0.06 degrees. Slope

values less than 0.01 degrees were eliminated since they have been assumed to be contaminated

by a high level of error. Displacements were in the order of 0 to 0.03 in (0.762 mm), where

measurements less than 0.01 in (0.254 mm) were eliminated based on the noise floor. It was

observed that in one or two cases, the measured responses were not eliminated based on the

Criterion 1; however, they were removed from the dataset due to their small values.

43


To have high confidence in the measured data, one other criterion was established to only

use the values that are based on exactly three independent tests and no less. This created

consistency in the level of confidence in the measured data values. The average value of three

sets of tests was used in the parameter estimation process.


Segments of data points from unacceptable time histories were eliminated from the measured

values in each record for use in the parameter estimation process. Figure (3.3) is an example of

the unacceptable time histories.

3.3 Modeling Error

In FE model updating, usually an a priori model is updated. In this process, the geometry and

connectivity of the structure is assumed to be known and only the parameters (e.g., bending

rigidity, mass, and damping) that define the model are updated. Although modeling error is an

unavoidable part of the FE model calibration, it was attempted to be reduced it by carful

verification of the FE model using different mesh sizes, modeling the boundary condition as

accurately as possible, comparing the mode shapes and frequencies with the measured modal

data and using engineering judgment. If an incorrect value of a structural parameter is use as

known, then it remains in the system as a modeling error. Also if all the key stiffness and mass

parameters are considered as unknowns, then there is a smaller possibility of having

measurement errors.

44

3.4 NDT Data Used in the Parameter Estimation of the UCF Grid

Table 3.1 presents the final outcome of the data quality analysis of the measured strain

responses from the static test indicated by the sensor numbers. These data were used in the final

parameter estimation process of the UCF grid.

Table 3.1 Strain Values Used in the Parameter Estimation of the UCF Grid (με)

Load Case LC1 LC2 LC3 LC4 LC5 LC6 LC7 LC8

SG1 -38.66 -40.14 -- 13.94 -54.43 -31.65 -44.50 12.56

SG2 -- -- -- 23.99 -- -- 6.14 --

SG3 -40.59 -40.47 -- -29.12 -12.64 -36.21 -9.95 -25.99

SG4 -25.04 -24.23 -- -35.60 10.80 -32.39 12.51 -44.32

SG5 -24.99 -- -24.30 11.17 -36.76 -32.97 -45.55 12.25

SG6 39.11 -- 37.69 -- -- -- -- --

SG7 -40.45 -- -39.64 -11.81 -28.49 -34.64 -25.41 -9.51

SG8 -- -- -- -- -- -32.46 14.28 -44.95

The measured displacements data used in the FE model updating of the UCF grid are

summarized in Table 3.2.

Table 3.2 Displacement Values Used in the Parameter Estimation of the UCF Grid (in)


DG1 -- -0.0172 -- -- -0.0223 -0.0122 -0.0203 --

DG2 -0.0159 -0.0139 -- -0.0211 0.0091 -0.0124 0.0108 0.0208

DG3 -- -- -- -- -0.0196 -0.0991 -0.0203 --

DG4 -0.0203 -- -0.0143 -0.0234 -- -0.0155 -- -0.0203

Table 3.3 presents the measured slope data from the static test of the UCF grid. The total of

four slopes’ data point was eliminated from LC2 and LC3.

Table 3.3 Slope Values Used in the Parameter Estimation of the UCF Grid (rad×103)


TM1 0.6547 0.6519 -- -0.2923 0.9833 0.5664 0.8574 -0.2696

TM2 -0.4337 -0.4346 -- -0.6562 0.2361 -0.5289 0.2720 -0.7857

TM3 0.4603 -- 0.4446 -0.2589 0.6940 0.5424 0.7985 -0.2826

TM4 -0.6424 -- -0.6211 -0.9402 0.2922 -0.5533 0.2687 -0.8179

All the modal data delivered from the UCF grid was also examined for reliability. This study

was presented in Chapter 2.

45

Chapter 4 Error Sensitivity Analysis Using Monte Carlo Simulations

4.1 Introduction

Prior to the parameter estimation process of the UCF grid using NDT data, an error

sensitivity analysis was performed to determine the effect of the measurement errors in the

parameter estimation procedure of the UCF grid. Simulated measurements were used at the

selected DOFs, at the exact location of the sensors used in the NDT of the UCF grid. This study

was helpful in finding the observability of each unknown parameter. Along with the accuracy of

the measurements, the number of unknown parameters, type of parameters, and parameter

grouping scheme used in the estimation process has a major impact on the estimated values. The

final unknown parameters and grouping scheme used in the FE model updating of the UCF grid

was determined based on the error sensitivity analysis.

4.2 Monte Carlo Analysis

The Monte Carlo analysis is a powerful statistical tool used for examining the complex

problems where numerous variables are related through an algebraically nonlinear system of

46

equations. In this study, Monte Carlo analysis was used to simulate data as a representation of

actual NDT data. This method was used by selecting a large number of observations (NOBS) and

performing independent parameter estimation for each of them in the presence of simulated

measurement error. When completed, the final results are summarized and statistical properties

such as mean and standard deviations of the estimated parameters can be calculated. The

aggregate results, such as mean and standard deviation of final estimates, can be accurate if a

large number of MC observations are used. This data gives the user information about the

reliability of the estimation procedure. Monte Carlo analysis was also used to compare how

effectively the data from different nondestructive test strategies can be used to estimate a set of

unknown parameters

4.3 Measurement Error Types and Distributions

This study also used the two common error distributions, uniform and normal, to simulate

error in static and modal response data. Uniform distribution represents error with equal

probability of occurrence through the predefined range. In the normal distribution, error level has

a higher probability of occurrence closer to the mean (Sanayei et al. [1]).

Measurement errors were simulated and added to response values in two types, proportional

and absolute. Absolute errors were added to the measurements regardless of their magnitudes.

They represent a baseline bias in the measurements. Absolute errors were added to the simulated

data based on Eq (4.1) in which u is the measured response data; eu represents the percentage

error which was predefined to apply to the response data; Ru is a fully populated matrix of

uniformly or normally distributed random numbers.

47

uu

m Reuu 2

1

(4.1)

In this equation, represents the element by element matrix multiplication (scalar). The

proportional error represents a calibration error and on average they get larger with larger

response magnitudes. It was added to the simulated response data as:

uu

m Reuu2

11

(4.2)

4.4 Error Sensitivity Analysis of the UCF Grid

Monte Carlo simulations (MC) were used to study the algebraically nonlinear input-output

system error analysis for different damage cases and parameter estimation procedures. In this

process, unknown mass and stiffness parameters were identified in various simulated damage

cases. In each simulated case, all of the structural parameters, except the damage ones, were

assumed to be known with high level of confidence. The number and location of the measured

responses were simulated as it was in the nondestructive test of the UCF grid. Since it was

possible to have measurement errors in either of the excitation and response data, it was

simulated to have a uniform-proportional random error in the range of 5% in both of the

excitation and the response simultaneously. This range was believed to provide a reasonable and

practical level of errors.

When the MC simulation was completed, each unknown parameter had a total of 50 values,

equal to NOBS, which produced the total of NUP x NOBS estimated parameters. Eq (4.3)

48

defines the percentage error of the kth

observation of the jth

parameter based w.r.t. the true value

of parameter j.

tj

tjkj

kjP

PPE

,

,,

, 100

(4.3)

To compare different damage cases and to better understand the input-output error behavior

of various sets of unknown parameters and parameter grouping schemes, grand mean percentage

error (GM) and grand standard deviation percentage error (GSD) were defined as in Eq (4.4) and

Eq (4.5), respectively [1]. To eliminate the possibility of one parameter overshadowing the

statistics of the others, percentage error was used as in Eq (4.3).

NUP

j

NOBS

k

kjENUPNOBS

GM1 1

,

1

(4.4)

NUP

j

NOBS

k

kj GMENOBSNUP

GSD1 1

2

,1

1

(4.5)

4.4.1 Error Functions used in the Monte Carlo Analysis

In order to find the influence of the measurement error based on different sensors and its

respective error function, stacking was not used in Monte Carlo simulations. SF, SSTR and MF

error functions were used individually in different damage cases.

49

4.4.2 Damage Cases and Parameter Grouping Scheme

Five damage cases were used in this study based on the type of the unknown parameters,

measurement type and error functions. The stiffness parameters for all structural members were

estimated with both SF and SSTR error functions. For estimating the mass parameters, MF error

function was used in the Monte Carlo simulations.

As it was presented in Figure (2.6), the strong axis bending rigidity (EIz) was categorized in

three member groups: (a) L shape corner connections, (b) T shape middle connections and (c)

Transverse beams and longitudinal girders. The same parameter grouping scheme was also

applied to the mass parameter. The stiffness values for all the boundary links in the vertical

direction (z-direction) were also grouped together for the UCF grid error sensitivity analysis. In

the case of using static-based error functions, three groups from member stiffness plus one group

of boundary link stiffness result in 4 NUP. In the case of using modal-based error function, three

groups of mass parameters were added to the total 4 unknown groups of stiffness parameters,

which resulted in 7 unknown parameters.

In each case of the Monte Carlo simulations (MC), 50 observations (NOBS) were chosen for

different damage cases and parameter estimation procedures. Higher and lower number of

observations was used to assure that 50 produce accurate enough error levels in the parameter

estimation. Since estimating the parameters for 50 NOBS individually was a time consuming

process, prior to using Monte Carlo simulations, each simulated damage case was estimated in

the absence of the measurement errors. This method helped in eliminating the low or

unobservable parameters in different damage cases which could also add ill conditioning into the

system of equations and result in non reliable estimated values. Up to 10% error in parameter

estimation was considered to be acceptable in simulated cases.

50

4.4.2.1 Damage case 1

In the first damage case, loss of 15% in the strong axis rigidity (EIz) in all steel beams and

connection of the grid and 30% in the boundary link element stiffness Kz were estimated using

static-based error functions. The totals of 4 unknown grouped parameters were estimated in Case

1. Table 4.1 shows the result of the MC simulations with 50 observations for a 5% uniform-

proportional random error in both applied static loads and measured responses. Since SSTR error

function was defined as the difference between predicted and measured strains at selected

observation points, it was incapable of finding the Kz parameter for the vertical link support. The

static flexibility error function was able to converge for all 50 observations. The mean of the

normalized estimated values are shown below with the percentage error w.r.t the True values.

The longitudinal and transverse beams group EIz was estimated with very low errors, where

the rest were error sensitive. Since there are no strain or displacement measurements, it is

inferred that the L shape and T shape connection stiffnesses cannot be estimated simultaneously

and accurately.

Table 4.1 Damage Case 1 Monte Carlo Simulation Results (SF Error Function)

Unknown Parameters

Groups

Normalized

True Value

Mean of the

Normalized

Estimated Value

Error NOBS

Converged

EIz Long & Trans Beams 0.85 0.84 -1.18%

47 EIz L Shape Connections 0.85 0.49 -42.35%

EIz T Shape Connections 0.85 1.09 +28.23%

Kz 0.70 1.00 +42.85%

4.4.2.2 Damage Case 2

Based on the Monte Carlo analysis in the Damage case 1, the L shape corner connections

have low observability due to the location of sensors in the static test. To overcome this issue, all

connection members, both L shape and T shape, were grouped as one unknown parameter. In

51

this case, the total number of estimated parameters was reduced into 3 groups. Strong bending

rigidity (EIz) for connection members, EIz for transverse and longitudinal beams, and Kz were

estimated in this damage case. Damage percentage was assumed to be 15%, the same as it was in

Case 1. Table 4.2 summarized the MC results based on the SF error function. Now the EIz of the

longitudinal and transverse members convereged with a low error. However Kz of the supports

shows very high error sensitivity.


Unknown Parameters

Groups

Normalized

True Value

Mean of the

Normalized

Estimated Value

Error NOBS

Converged

EIz Long & Trans Beams 0.85 0.84 -1.18%

50 EIz Connections 0.85 1.03 +21.17%

Kz 0.70 1.19 +70.00%


A loss of 15% of the strong bending rigidity was assumed to occur at the connection

members and main members of the UCF grid. According to the error sensitivity analysis in the

Damage Case 2, simultaneous estimation of member’s stiffness and link stiffness at the supports

is highly sensitive to the measurements error. In Case 3, only the EIz parameter was estimated

using SF and SSTR error functions individually. Both SF and SSTR error functions estimated the

EIz of beams and connections with low errors.


Unknown Parameters

Groups

Normalized

True Value

Mean of the

Normalized

Estimated Value

Error NOBS

Converged

EIz Long & Trans Beams 0.85 0.85 0.00% 50

EIz Connections 0.85 0.86 +1.17%

52

Table 4.4 Damage Case 3 Monte Carlo Simulation Results (SSTR Error Function)

Unknown Parameters

Groups

Normalized

True Value

Mean of the

Normalized

Estimated Value

Error NOBS

Converged

EIz Long & Trans Beams 0.85 0.85 0.00% 50

EIz Connections 0.85 0.90 +5.88%


A damage case that considers a theoretical 15% reduction in the mass and bending capacity

of the main members and connections, and 30% loss in the boundary link stiffness was

considered first. In this parameter estimation problem, all of the observable parameters,

including both the stiffness and mass, were estimated using MF error function. L shape corner

connections and T shape middle connections were separated in this damage case and grouped as

two unknown parameters. Seven unknown parameters were estimated in Damage Case 4.

Table 4.5 Damage Case 4 Monte Carlo Simulation Results (MF Error Function)

Unknown Parameters

Groups

Normalized

True Value

Mean of the

Normalized

Estimated Value

Error NOBS

Converged

EIz Long & Trans Beams 0.85 0.85 +0.61%

50

EIz L Shape Connections 0.85 0.77 -9.53%

EIz T Shape Connections 0.85 0.80 -5.48%

Am Long & Trans Beams 0.85 0.85 +0.42%

Am L Shape Connections 0.85 0.76 -10.09%

Am T Shape Connections 0.85 0.85 +0.39%

Kz 0.70 1.52 +116.43%


In the Damage Case 5, the number of unknown parameters was reduced to 5 by assuming

that the L shape and T shape connections were one group of members. Table 4.5 shows the final

results of the MC simulations using 50 NOBS. Simulated damages were assumed to be 15%, the

same as it was in Damage Case 4.

53

Table 4.6 Damage Case 5 Monte Carlo Simulation Results (MF Error Function)

Unknown Parameters

Groups

Normalized

True Value

Mean of the

Normalized

Estimated Value

Error NOBS

Converged

EIz Long & Trans Beams 0.85 0.86 +1.18%

49

EIz Connections 0.85 0.81 -4.71%

Am Long & Trans Beams 0.85 0.85 0.00%

Am Connections 0.85 0.85 0.00%

Kz 0.70 0.74 +5.71%

Based on the error sensitivity analysis presented in this section, it was decided to estimate

parameters using the NDT data only in the cases with low sensitivity to measurement errors.

Also it was derived that both T and L shape connections should be grouped as one for better

estimation of element mass and stiffness parameters, given the current set of measurements.

Static-based error functions were used in the same case as proposed in Damage Case 3 and

modal-based error function was used to estimate the parameters in similar method as Damage

Case 5.

References

[1] Sanayei, M. and Saletnik, M.J., (1996) “Parameter Estimation of Structures from Static

Strain Measurements. II: Error Sensitivity Analysis,” Journal of Structural Engineering, Vol.

122, No.5

54

Chapter 5 Supplementary Parameter Estimation Cases with Analytical and

Measured Response Comparisons

5.1 Introduction

This chapter provides supplementary parameter estimation cases with analytical and

measured response comparisons in all load cases for each type of measurements based on the

two FE models described in Chapter 2.

5.2 Supplementary Parameter Estimation Cases of the UCF Grid Using NDT Data

Along with the parameter estimation cases presented in Table 2.6 and 2.7 of Chapter 2, two

more cases were used for the FE model updating of the UCF grid based on the NDT data. In each

of these cases, both the stiffness and mass properties of the main structure members and

connection members were estimated independently.

55

5.2.1 Additional Error Function Used in the Parameter Estimation: Static Stiffness (SS)

The static stiffness-based error function was developed by Sanayei et al. [1]. This error

function is based on the residual between measured and analytical forces at a subset of DOF.

FUpKpeSS )()( (5.1)

Where [F] and [U] represent the applied live load cases and the corresponding measured

displacements per load case. The error function eSS(p) is calculated based on the analytical

stiffness matrix [K(p)] and known applied live load cases. The complete formulation using a

subset of measurements that can be different from applied load is presented by Sanayei et al. [1].

5.2.2 FE Model on Link Elements used in Multiresponse Parameter Estimation

In total, six parameter estimation cases were performed based on the experimental data of the

UCF grid. Table 5.1 summarized the estimated values.

In Case 1, the strong bending rigidity (EIz) of the main structural members and connection

members were also estimated by using the static stiffness (SS) error function. The final results

were close to the previous values estimated by SF error function. The results of the SS error

function were not included in Chapter 2 because it was mainly focused on flexibility-based error

functions that seem to be more error tolerant.

In Case 2, rigidity and mass parameters of the transverse and longitudinal beams were

estimated. Since standard sections were used in the main members of the UCF grid, the

estimated values show an excellent match with the AISC Manual.

56

Table 5.1 UCF Grid Parameter Estimates (Including Supplementary Cases)

Case Estimated

Parameters SF SS SSTR

SSTR

+SF MF

MF

+SF

MF

+SSTR

MF+SF

+SSTR

Iterations for

Convergence

1 EIz Long & Trans Beams 1.01 1.00 0.98 0.99 1.00 1.00 1.00 1.00 47,57,19, 24,

71, 55,67,61 EIz Connection 2.09 2.05 1.75 1.84 1.88 1.88 1.88 1.88

2 EIz Long & Trans Beams - - - - 1.00 1.00 1.00 1.00

24, 7, 25, 6 Am Long & Trans - - - - 1.00 1.00 1.00 1.00

3 EIz Connection - - - - 1.88 1.80 1.80 1.80

19, 18, 29, 22 Am Connection - - - - 1.44 1.50 1.50 1.50

4

EIz Long & Trans Beams - - - - 1.00 1.00 1.00 1.00

101,133,103,

72

EIz Connection - - - - 1.78 1.79 1.80 1.78

Am Long & Trans Beams - - - - 1.00 1.00 1.00 1.00

Am Connection - - - - 1.50 1.50 1.51 1.50

5


94, 66,142,

106

EIz Connection - - - - 1.80 1.78 1.78 1.78


Am Connection - - - - 1.51 1.50 1.50 1.50

Kz (Grouped) - - - - 1.45 1.49 1.40 1.40

6


100,117,

139,112

EIz Connection - - - - 1.78 1.80 1.78 1.79


Am Connection - - - - 1.51 1.51 1.50 1.50

Kz 1 - - - - 1.49 1.46 1.46 1.48

Kz 2 - - - - 1.49 1.50 1.46 1.50

Kz 3 - - - - 1.50 1.59 1.47 1.64

Kz 4 - - - - 1.50 1.65 1.47 1.69

Kz 5 - - - - 1.49 1.15 1.45 1.08

Kz 6 - - - - 1.49 1.28 1.45 1.20

In Case 3, the EIz and Am parameters of the connection members were estimated. It was

expected to have increase in both stiffness and mass parameters. The estimated values were close

to the estimated values in Case 6 where the totals of 10 unknown parameters were used in the

estimation process based on the simultaneous use of three different error functions. Cases 1, 4, 5

and 6 were previously presented and discussed in Chapter 2.

57

5.2.3 Analytical and Measured Response Comparison: Calibrated FE Model on Link Elements

The predicted responses based on the calibrated model in Case 6 of Table 5.1 using

MF+SF+SSTR error functions were compared with the experimental response data. This

comparison was used to verify the estimated values. In Chapter 2, typical plots were presented.

The current chapter provides comprehensive comparisons of all load cases for each type of

measurements

5.2.3.1 Strain Comparison

Figures (5.1) to (5.8) show strain data from LC1to LC8 for 8 strain gages on the grid, which

were used in the parameter estimation of the UCF Grid that shows the analytical prediction of the

uncalibrated and calibrated models for comparison with the measured strains. These plots show

an excellent match between the calibrated model response and NDT measurements.



58




Figure 5.6 Strain, Load Case 6



59

5.2.3.2 Displacement Comparison: Girder 1

In Figures (5.9) to (5.14), displacements from 2 gages along longitudinal girder 1 were

compared with the analytical prediction of the uncalibrated and calibrated FE models. Since data

from LC3 was never used in the parameter estimation process based on the NDT data quality

analysis presented in Chapter 3, the following figures did not show the data from this load case.


Load Case 1


Load Case 2


Load Case 4


Load Case 5

60

Figure 5.13 Displacements Along Girder 1.

Load Case 6


Load Case 7


Load Case 8


Figures (5.16) to (5.22) show comparison between the analytical and experimental

displacements along longitudinal girder 2 for seven load cases. Data from the LC2 test was

eliminated based on the data quality study. In general, the predicted responses moved toward the

measurements. Some of these plots did not match with the measured displacements as well as

other displacements did. This might be due to errors in displacement measurements.

61


Load Case 1


Load Case 3

Figure 5.18 Displacements Along Girder 2

Load Case 4


Load Case 5


Load Case 6


Load Case 7

62


Load Case 8

5.2.3.4 Slope Comparison

Figures (5.23) to (5.30) show slopes from LC1 to LC8 based on the 4 tiltmeterson the grid,

which were used in the parameter estimation of the UCF grid and show the analytical prediction





63







Figure (5.31) shows the five natural frequencies used in parameter estimation based on the

FE model on the link elements.

64


Table 5.2 presented the MAC matrix, which shows an excellent match between the first 4

mode shapes of the calibrated model and measured data based on the Case 6 of Table 5.1 using

MF, SF and SSTR error functions simultaneously.

Table 5.2 MAC Values Using Parameter Values of Case No. 6, SF+SSTR+MF

Analytical Mode


Ex

per

imen

tal

Mo

de

E1 0.997 0.003 0.001 0.000 0.135 0.001 0.000 0.988 0.010 0.000

E2 0.001 0.995 0.000 0.004 0.000 0.000 0.000 0.000 0.000 0.000

E3 0.000 0.000 0.998 0.000 0.346 0.001 0.000 0.011 0.989 0.000

E4 0.000 0.004 0.000 0.995 0.000 0.000 0.000 0.000 0.000 0.000

E5 0.000 0.000 0.000 0.000 0.464 0.977 0.013 0.000 0.001 0.000

E6 0.000 0.000 0.001 0.005 0.084 0.009 0.984 0.001 0.001 0.000

E7 0.000 0.000 0.004 0.153 0.009 0.000 0.032 0.000 0.003 0.002

E8 0.996 0.003 0.000 0.000 0.143 0.000 0.000 0.985 0.012 0.001

E9 0.001 0.000 0.996 0.002 0.332 0.001 0.000 0.018 0.981 0.000

E10 0.000 0.001 0.000 0.000 0.001 0.000 0.000 0.000 0.000 0.996

5.2.4 FE Model on Pins and Rollers used in Multiresponse Parameter Estimation

As presented in Chapter 2, since the estimated values of Kz stiffness were all in the “fixed”

range, it was decided to create a new FE model with the pins and rollers boundary conditions. As

in the first FE model, two supplementary cases were added to the parameter estimation of the

UCF grid based on the pins and rollers FE model. In summary, 4 parameter estimation cases

were performed based on the NDT data and pins and rollers FE model. Normalized estimated

parameters w.r.t their initial values are summarized in Table 5.3.

65

Table 5.3 UCF Grid Parameter Estimates Based on the Pins and Rollers Model (Including Supplementary Cases)

Case Estimated

Parameters SF SS

SST

R

SSTR

+SF MF

MF

+SF

MF+

SSTR

MF+

SF+

SSTR

Iterations for

Convergence

1 EIz Long & Trans Beams 1.02 1.02 1.00 0.99 1.00 1.00 1.00 1.00 39,22,27,20,29

, 34,49,47 EIz Connection 1.82 1.82 1.79 1.85 1.88 1.88 1.88 1.88

2 EIz Long & Trans Beams - - - - 1.00 1.00 1.00 1.00

9, 14,8, 7 Am Long & Trans Beams - - - - 1.00 1.00 1.00 1.00

3 EIz Connection - - - - 1.81 1.81 1.81 1.81

24, 29, 29,28 Am Connection - - - - 1.50 1.50 1.50 1.50

4


88,87, 77, 58 EIz Connection - - - - 1.78 1.81 1.80 1.80


Am Connection - - - - 1.50 1.50 1.50 1.50

5.2.5 Analytical and Measured Response Comparison: Calibrated FE Model on Pins and Rollers

In Chapter 2, typical plots were presented. The current section provides comprehensive

comparisons of all load cases for each type of measurements.

5.2.5.1 Strain Comparison

Figures (5.32) to (5.39) present comparison of the predicted strain data from the uncalibrated

and calibrated FE model with the measured data in all load cases for 8 strain gages. These plots

show a better match between the calibrated pins and rollers model response and NDT

measurements in comparison with the previous FE model.



66







67


In Figures (5.40) to (5.46), displacements from 2 gages along longitudinal girder 1 were

compared with the analytical prediction of the uncalibrated and calibrated FE models. This

comparison was performed for all load cases except the LC3, which was eliminated from the

data set due to poor quality.


Load Case 1


Load Case 2


Load Case 4


Load Case 5

68


Load Case 6


Load Case 7


Load Case 8


Figures (5.47) to (5.53) show a comparison between the analytical and experimental

displacements along longitudinal girder 2 for seven load cases. Data from LC2 test was

eliminated based on the data quality study. Although, the strain data matches better with the

experimental data based on the pins and rollers FE model, the analytical displacements did not

show much improvement in terms of moving towards the measured data.

69


Load Case 1


Load Case 3


Load Case 4


Load Case 5


Load, Case 6


Load Case 7

70


Load Case 8

5.2.5.4 Slope Comparison

Figures (5.54) to (5.61) show slopes from LC1 to LC8 based on the 4 tiltmeters on the grid

which were used in the parameter estimation of the UCF grid and show the analytical prediction





71







Figure (5.62) shows the comparison between the five natural frequencies used in parameter

estimation based on the pins and rollers FE model.

72


The MAC matrix presented in Table 5.4 shows an excellent match between the first 6 mode

shapes of the calibrated model and measured data based on Case 4 of Table 5.3 using MF, SF

and SSTR error functions simultaneously.

Table 5.4 MAC Values Using Parameters From Pins and Rollers Model

Analytical Mode


Ex

per

imen

tal

Mo

de

E1 0.998 0.001 0.000 0.000 0.000 0.000 0.498 0.000 0.000 0.002

E2 0.000 0.999 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.007

E3 0.000 0.000 0.999 0.000 0.000 0.000 0.000 0.693 0.307 0.000

E4 0.000 0.000 0.000 0.999 0.000 0.000 0.000 0.000 0.000 0.005

E5 0.000 0.000 0.000 0.000 0.998 0.002 0.493 0.000 0.001 0.000

E6 0.000 0.000 0.001 0.005 0.000 0.988 0.000 0.319 0.669 0.288

E7 0.000 0.000 0.003 0.159 0.000 0.035 0.000 0.003 0.036 0.394

E8 0.997 0.001 0.000 0.000 0.000 0.000 0.487 0.000 0.000 0.003

E9 0.002 0.000 0.996 0.001 0.000 0.000 0.001 0.687 0.310 0.000

E10 0.000 0.001 0.000 0.000 0.001 0.000 0.000 0.000 0.000 0.086

From these results it is inferred that the pins and rollers model is a better representation of the

measured data.

References

[1] Sanayei, M. and Onipede, O., (1991) “Damage Assessment of Structures Using Static Test

Data,” AIAA Journal, Vol. 29, No.7, pp.1174-1179

73

Chapter 6 Conclusions and Recommendations for Future Work

6.1 Conclusions

FE model updating using NDT data is an effective tool for FE model validation, structural

evaluation, damage assessment and bridge management. In this research, the FE model of a scale

model bridge deck was successfully calibrated using the multiresponse structural parameter

estimation. Flexibility-based error functions were utilized for parameter estimation of the UCF

grid. Measured data, including strains, displacements, slopes, mode shapes and frequencies, were

used in the FE model updating.

Having measurement errors in experimental data is unavoidable, and the presence of these

errors can affect the accuracy of the parameter estimates. An error sensitivity analysis was

performed using Monte Carlo simulations to find the parameters with the least sensitivity to

measurement errors and to remove them from the parameter estimation process. In these

simulations, various parameter groupings of elements with the same properties were used to

reduce the number of unknowns and then ungrouping was utilized to reduce the modeling errors.

74

In any event, changes in such parameters have minimal effects on the analytical responses for the

stated problem.

A data quality study was conducted to assess the accuracy and reliability of the measured

data. Based on this study, a subset of most reliable measured data was used for the parameter

estimation and FE model updating of the UCF grid. Using the subset of data improved the

convergence and the quality of estimated structural parameters.

Element rigidities and mass properties of the finite element model were successfully updated

using the measured multiresponse data, which demonstrates the feasibility of simultaneously

using static and modal NDT data by employing various error functions. In this process using a

set of the most observable and error tolerant unknown parameter groups, had a substantial role in

this success. The initial grid model used vertical springs to model the supports. Since these

springs were identified to be in the fixed stiffness range, the model was modified to use pins and

rollers supports between the columns and the UCF grid. Both calibrated models showed great

improvements in predicting responses compared to the uncalibrated model. A great match was

observed between the analytical and measured strain, slopes, mode shapes and frequency

responses. Some cases of displacement predictions showed a better match with the measured

displacements compared to other cases. This can be due to measurement and/or modeling errors

present in the system.

In this process two commercial software packages were used simultaneously in real time for

automated and systematic optimization and FE analysis.

75

6.2 Recommendations for Future Work

This approach of FE model calibration is applicable to full scale structures and can be used

for bridge model validation, and bridge management. Calibrated FE models can be used for

realistic modeling of deteriorated bridges to assess the damage in these bridges and to assess

retrofitting options for effective rehabilitation.

The Powder Mill Bridge (PMB) is a three continuous steel girder bridge with a reinforced

concrete deck in composite action, which has been instrumented by researchers from Tufts

University and the University of New Hampshire. Further research would be to use the real NDT

data from the PMB for FE model updating of the full scale bridge using multiresponse parameter

estimation.

Since development of PARIS14.0 was a part of this research, the following can be possible

paths that future researchers may take in order to improve PARIS software and eventually make

parameter estimation an accessible tool for engineers in the industry.

i. Implementation of Genetic Algorithms for parameter estimation

ii. Developing the statistical error function normalization scheme for modal error functions

iii. Environmental effects such as ambient temperature change should be considered to

improve the reliability of the estimated parameters in long-term continuous SHM

iv. Applying weight factor to each error function in multiresponse parameter estimation

v. Enabling different normalization techniques in static-based and model-based error

functions simultaneously

76

Appendix A PARIS14.0 User Reference Manual

A.1 Introduction

PARIS (PARameter Identification System) is the acronym for a custom MATLAB based

computer program for parameter estimation and FE model updating. The initial software was

developed by Sanayei (1997) with his graduate students at Tufts University. Using static and

modal measurements as input data, it could be used for stiffness and mass parameter estimation

at the element level. However, its use was limited to FE model updating for only planar and

spatial truss and frame structures. The in-house FEA solver for PARIS was also written in

MATLAB. Text based finite element modeling and restricted FEA capabilities rendered PARIS

unsuitable for full-scale model updating. PARIS14.0 is the new program that succeeds PARIS.

Figure (6.1) illustrates the FE model updating process in PARIS14.0. The flowchart connectors

labeled as API describe the two-way automatic data exchange between MATLAB and SAP2000.

This section contains description of the parameter estimation process using PARIS14.0.

77

Figure A.1 PARIS14.0 Flowchart

Table 6.1 summarizes the capabilities of PARIS14.0 in comparison to the original PARIS.

PARIS14.0 has the most advantage in the broader range of finite elements available through

SAP2000. In PARIS14.0, six out of seven error functions have been kept. Its emphasis is on

using MATLAB Optimization Toolbox because of its better performance for full-scale FE model

updating. Analytical sensitivity based optimization option has also been retained in PARIS14.0

but only with Gauss Newton and Steepest Descent methods for solving inverse problems.

78

Table A.1 PARIS14.0 Features

Feature Description PARIS11.0 PARIS14.0

Finite Elements

Truss

Frame

Shell (4 Node – Quadrilateral – Planar)

Solid (8 Node – Cuboid)

Link Element (2 Node)

Joint Spring

Error Functions

Static Stiffness (SS)

Static Flexibility I (SF1)

Static Flexibility II (SF2)

Static Strain (SSTR)

(frames, shells)

Modal Stiffness (MS)

Modal Flexibility I (MF1)

Modal Flexibility II (MF2)

Normalization

Parameter

Measured Data

Initial Value Error Function

Previous Step Error Function

Standard Deviation Error Function

Maximum Likelihood Estimator

Multi-response

Parameter Estimation Error Function Stacking

Optimization Techniques

Hill Climbing Methods

i) Generalized Inverse

ii) Least Squares/ Gauss Newton

iii) Steepest Descent

iv) Standard Conjugate Gradient

v) BFGS Quasi Newton

vi) Gauss Newton w/ Line Search

MATLAB Optimization Toolbox

Genetic Algorithms

Modeling Error

Measurement Error

Absolute and Proportional Errors

Uniform and Normal Errors

Monte Carlo Simulation

Reading in NDT data

Parameter grouping

79

PARIS14.0 program reads the SAP2000 model data into MATLAB using SAP2000’s API

and then employs SAP2000 program again during the FEA stage of model updating process. It is

required that the SAP2000 model is created under specific guidelines to be used with PARIS14.0

program. PARIS14.0 also requires a MATLAB based user input file (PARIS14InputFile.m),

which essentially supplies information regarding unknown parameters and parameter estimation

options. This chapter is also the reference manual for PARIS14.0 program.

A.2 SAP2000 Model

In this section the input file for making and using SAP2000 model is shown.

A.2.1 SAP2000 Model File

The SAP2000 model file (.SDB) should be stored at this location on the computer’s hard disk

at: C:\API\

Before running PARIS14.0, it is required that the user creates the API folder in the C:\ drive

and save the SAP2000 model at the location mentioned above.

A.2.2 Unit System

The units in SAP2000 can be classified as ‘Present Units’ and ‘Database Units’. While the

former can be changed at any time by the user, the latter can only be set before the user starts

creating a FE model in SAP2000. The unit system chosen at the beginning cannot be altered by

the user and the entire analysis in SAP2000 is carried out in ‘Database Units’. However, for

result output, any system of units can be chosen as ‘Present Units’. It is recommended (not

required) to select the Database Units as desired in your parameter estimations.

SAP2000 Procedure (Database Units):

i. File -> New Model

80

ii. Select the system of units from the drop down menu in the dialogue box as shown below.

SAP2000 Procedure (Present Units):

i. In the right bottom corner of main window in SAP2000, select any system of units from the

drop down list to set as present units.

81

A.2.3 Local and Global Coordinate System

PARIS14.0 performs all internal calculations in the global coordinate system. Therefore, it is

important that the SAP2000 model should comply with the following rules:

1. Local axes of all nodes in the model should be oriented similar to global axes.

2. Care should be taken that stiffness values are entered for link elements in local direction that

correspond to desired stiffness values in global axes.

A.2.4 Nodes and Element Labels

1. Nodes / Joints: All nodes in the model should be numbered from 1 through n where n is

the total number of nodes/joints elements in the FE model.

2. Frame Elements: All frame elements in the model should be numbered from 1 through n

where n is the total number of frame elements in the FE model.

3. Shell Elements: All shell elements in the model should be numbered from 1 through n

where n is the total number of shell elements in the FE model.

4. Solid Elements: All solid elements in the model should be numbered from 1 through n

where n is the total number of solid elements in the FE model.

5. Link Elements: All link elements in the model should be numbered from 1 through n

where n is the total number of link elements in the FE model.

SAP2000 Procedure:

i. Select all objects/elements of a particular type

ii. Edit -> Change Labels

82

iii. In the ‘Interactive Name Change’ form: Edit -> Copy All Data in Grid

iv. Paste the data on clipboard on to MS Excel and reorder the second columns in ascending

order without any gaps. The numbering should be 1 through n for a particular element/object

type.

v. Copy the data in the second column.

vi. In the ‘Interactive Name Change’ form: Single click on the first cell of the second column ‘New

Name’.

Edit -> Paste

vii. Repeat this procedure for each type of object/element in the FE model.

Step (vi)

83

A.2.5 Frame Element Section Properties

All frame section properties must be defined under ‘General’ category. If a section from

SAP2000 library is to be used, the user should first define a general section in SAP2000 with the

same section properties as of the desired predefined section.

SAP2000 Procedure:

i. Define -> Section Properties -> Frame Sections

ii. Add New Property

iii. In the Add Frame Section Property dialogue box, select ‘Other’ from the drop down menu.

iv. Select ‘General’ and enter section properties manually.

84

v. Click ‘Ok’

vi. Enter the ‘Section Name’ in the dialogue box that appears next.

85

A.2.6 Analysis option: Active DOF

2D FE model can be created in any of the three planes i.e. XY, YZ, or XZ. Gravity loads in

SAP2000 can be applied along any axes, thus not restricting the model to be in XZ or YZ plane

should gravity loads be considered.

Any combination of DOFs can be chosen for analysis. The inactive DOFs are not considered

in PARIS14.0. This option proves to be valuable in use with dynamic error functions where all

mode shapes are not always useful like out of plane modes for 2D FE model.

SAP2000 Procedure:

i. Analysis -> Set Analysis Option

ii. Select the DOFs to be made active

86

A.2.7 Load Cases

PARIS14.0 provides the user with an option of defining and applying 999 load cases for each

of the five available error functions. However, the load cases are required to be defined in a

particular format as described in the table below.

1. At least one load case for Static Stiffness (SS) error function: LCsst001 must be defined

before running PARIS14.0.

2. Load Pattern and Load Cases should have the same label for corresponding load cases i.e.

if a load case LCsst001 is to be defined, the corresponding load pattern should be

LCsst001 as well. Nomenclature scheme for Load Cases is shown in Table 6.2.

Table A.2 Load Cases Nomenclature

Error Function Load Case (first) Load Case (last)

Static Stiffness (SS) LCsst001 LCsst999

Static Flexibility (SF) LCsf1001 LCsf1999

Static Strain (SSTR) LCstr001 LCstr999

Modal Stiffness (MS) MODAL

Modal Flexibility (MF)

SAP2000 Procedure:

i. Define -> Load Patterns

87

ii. Define -> Load Cases

A.2.8 Element Grouping

PARIS14.0 is not designed to update models with elements grouped in SAP2000. The user is

required to ascertain that element grouping is not done in the SAP2000 model. Instead,

parameter grouping in PARIS14.0 is achieved by defining the following in the input file, which

is explained in detail under ‘Standardized Input File’:

1. pgflag: Flag for existence of parameter grouping

2. pg: Parameter grouping matrix

A.3 PARIS14.0 Input File Details

Different components of the MATLAB based the input file are described in this section in

the order in which they appear in the input data file (PARIS13InputFile.m file). The input file

itself contains detailed instructions for entering data.

88

A.3.1 SAP2000 Model related data

1. SAP2000 File Name

This is the name of the SAP2000 finite element model file located in the C:\ of computer as

described in section 5.2.1

filename = ‘C:\API\InputFile.SDB’;

2. Finite Elements in SAP2000 Model

framemodel

Defines whether SAP2000 model consists of frame elements or not.

0: Model consists of frame elements

1: Model does not consist of frame elements

shellmodel

Defines whether SAP2000 model consists of shell elements or not.

0: Model consists of shell elements

1: Model does not consist of shell elements

solidmodel

Defines whether SAP2000 model consists of solid elements or not.

0: Model consists of solid elements

1: Model does not consist of solid elements

linkmodel

Defines whether SAP2000 model consists of link elements or not.

0: Model has link elements

1: Model does not have link elements

89

Stype

Defines whether SAP2000 model consists of standard sections or general sections

0: Model consists of general sections

1: Model consists of standard sections

A.3.2 Parameter Estimation Related Data

1. Parameter Grouping

Supplying parameter grouping data to PARIS14.0 requires the input for two variables:

pgflag

Defines whether parameter grouping exists or not.

0: No grouping

1: Parameter grouping exists

pg

This is the grouping matrix. This matrix should be a rectangular matrix. If the number of

elements grouped is different in each row, pad the rows with lesser number of elements with

zeros. The values are entered in the format described here.

Column 1: Group number

Column 2: Parameter type

There are a total of 20 parameter types that can be updated using PARIS14.0.

Column 3 onwards: Grouped element numbers

2. Error Function(s)

errorfunction

Error function(s) selected for a particular run of PARIS14.0 is (are) chosen using this row vector.

0: Do not use corresponding error function

90

1: Use corresponding error function

3. Unknown Parameters

PARIS14.0 has the capability to update twenty types of parameters in a finite element model.

Their description is provided in the Input File. These parameters are further classified under

different categories based on to which type of finite elements they belong.

0: Do not consider unknown parameter

1: Consider unknown parameter

upframe

This is the matrix for unknown parameters for frame elements in the SAP2000 model.

Column 1: Frame element number (Entries must be in ascending order in column 1)

Column 2: Area (Area) parameter of frame element

Column 3: Moment of Inertia (Iz) parameter of frame element

Column 4: Moment of Inertia (Iy) parameter of frame element

Column 5: Torsional Constant (J) parameter of frame element

Column 6: Area Mass (AreaMass) mass parameter of frame element

upshell

This is the matrix for unknown parameters for shell elements in the SAP2000 model.

Column 1: Shell element number (Entries must be in ascending order in column 1)

Column 2: Modulus of Elasticity (E) parameter of frame element

Column 3: Shell Mass (ShellMass) parameter of frame element

upsolid

This is the row vector containing solid element numbers for which Modulus of Elasticity (E) is

an unknown parameter in the SAP2000model.

91

uplink

This is the matrix for unknown parameters for 2 node link elements in the SAP2000 model.

Column 1: Link element number (Entries must be in ascending order in column 1)

Column 2: Translational Stiffness in local ‘x’ direction (kxx)

Column 3: Translational Stiffness in local ‘y’ direction (kyy)

Column 4: Translational Stiffness in local ‘z’ direction (kzz)

Column 5: Rotational Stiffness about local ‘x’ direction (krx)

Column 6: Rotational Stiffness about local ‘y’ direction (kry)

Column 7: Rotational Stiffness about local ‘z’ direction (krz)

upspring

This is the matrix for unknown parameters for nodal spring assignments in the SAP2000 model.

Column 1: Node number (Entries must be in ascending order in column 1)

Column 2: Translational Stiffness in local ‘x’ direction (ksxx)

Column 3: Translational Stiffness in local ‘y’ direction (ksyy)

Column 4: Translational Stiffness in local ‘z’ direction (kszz)

Column 5: Rotational Stiffness about local ‘x’ direction (ksrx)

Column 6: Rotational Stiffness about local ‘y’ direction (ksry)

Column 7: Rotational Stiffness about local ‘z’ direction (ksrz)

4. Nondestructive Test Data

PARIS14.0 is able to use measured test data for parameter estimation process of full scale

structures. The user can create an input file with “.txt” format in the program directory. For each

error function user should create a separate input file in these formats:

92

SS error function: “utdofss.txt”

SF error function: “utdofsf1.txt”

SSTR error function: “utdofsstr1.txt”

MS error function: “utdofms.txt”

MF1 error function: “utdofmf1.txt”

MF2 error function:”utdofmf2.txt”

inflag

0: Do not use test data, use simulated data from the ‘true’ section properties

1: Do use test data. Read in .txt file

5. Data to Simulate Measurements

PARIS14.0 uses this data to simulate a model to represent the true parameters. The user can

input the factor for each parameter type with respect to its initial guess which is representative of

the true value of the parameter. The initial guess of parameter value is always 1.

factorforframe

This is a row vector for five parameter types available for frame element.

Column 1: Factor for frame cross-sectional area representing true value

Column 2: Factor for frame Iz representing true value

Column 3: Factor for frame Iy representing true value

Column 4: Factor for frame J representing true value

Column 5: Factor for frame mass representing true value

factorforshell

This is a row vector for two parameter types available for shell element.

Column 1: Factor for material modulus of elasticity (E) area representing true value

93

Column 2: Factor for shell mass representing true value

factorforshell

This is a single value for factor for material modulus of elasticity (E) for solid element.

factorforlink

This is a row vector for six parameter types available for link element.

Column 1: Factor for link translational stiffness along global X (kxx)

Column 2: Factor for link translational stiffness along global Y (kyy)

Column 3: Factor for link translational stiffness along global Z (kzz)

Column 4: Factor for link rotational stiffness about global X (krx)

Column 5: Factor for link rotational stiffness about global Y (kry)

Column 6: Factor for link rotational stiffness about global Z (krz)

outflag: Save the simulated data to an external “.txt” file

0: Do not create an output file

1: Do write an output file

unitflag: The processed NDT data should be in either of these units:

1: lb_in_F

2: lb_ft_F

3: kip_in_F

4: kip_ft_F

5: kN_m_C

10: N_m_C

94

6. Observation Points for Measurements

The observations or measurements can be classified into two categories, displacement and

strains. The stiffness and flexibility based error functions requires input of measured nodes

location as a matrix of nodes where displacement is measured either under the associated load

case or for modal analysis.

mnodesss

This is the matrix of measured nodes for SS error function. In columns 3-8, enter:

0: Do not measure translation/rotation along/about this DOF

1: Measure translation/rotation along/about this DOF

Column 1: Load Case corresponding to LCsst001 to LCsst999 load case. (Entries should be

in ascending order)

Column 2: Node Number (Entries should be in ascending order for each load case defined in

column 1)

Column 3: Measure translation along local ‘x’ direction

Column 4: Measure translation along local ‘y’ direction

Column 5: Measure translation along local ‘z’ direction

Column 6: Measure translation about local ‘x’ direction

Column 7: Measure translation about local ‘y’ direction

Column 8: Measure translation about local ‘z’ direction

mnodessf

This is the matrix of measured nodes for SF error function. Data entry is similar to [mnodesss]

described above except that the load cases in column 1 correspond to load case LCsf1001

through LCsf1999 in the SAP2000 model.

95

Modal error functions, in addition to measurement locations for each mode, also require another

input matrix for providing data as to which modes are measured.

melsstr

This is the matrix of measured nodes for SSTR error function. In columns 1-5, enter:

Column 1: Load Case corresponding to LCstr001 to LCstr999 load case. (Entries should be

in ascending order)

Column 2: Measured Element Number (Entries should be in ascending order for each load

case defined in column 1)

Column 3: The distance form node i to the gage in local ‘x’ direction

Column 4: The distance form node i to the gage in local ‘y’ direction (required for 2D and

3D frames)

Column 5: The distance form node i to the gage in local ‘z’ direction (required for 3D

frames)

Modal error functions, in addition to measurement locations for each mode, also require

another input matrix for providing data as to which modes are measured.

fms

This is the matrix of set of modes measured for MS error function

Column 1: Set number / Serial number

Column 2: Measured mode number

mnodesms

This is the matrix of measured nodes for MS error function. Data entry is similar to [mnodesss]

described above except that the load cases in column 1 correspond to MODAL case in the

SAP2000 model described in column 1 of [fms] matrix.

96

fmf

This is the matrix of set of modes measured for MF error function

Column 1: Set number / Serial number

Column 2: Measured mode number

mnodesmf

This is the matrix of measured nodes for MS error function. Data entry is similar to [mnodesss]

described above except that the load cases in column 1 correspond to MODAL case in the

SAP2000 model described in column 1 of [fmf] matrix

7. Normalization Methods

scaledataflag

‘0’: Do not scale data

‘1’: Scale data with respect to maximum static displacement

‘2’: Scale data with respect to maximum static strain

nepflag

‘0’: Do not normalize

‘1’: Normalize error function with respect to its covariance matrix

‘2’: Normalize error function with respect to its initial value

8. Measurement Errors

erdflag

‘0’: Uniform error distribution

‘1’: Normal error distribution

ertflag

‘0’: Proportional error type

97

‘1’: Absolute error type

fmeflag

‘0’: No measurement error in force measurement

‘1’: Error in force measurement

Absolute error in force measurement

faetss: Force/excitation Measurement error level for stiffness and flexibility error

functions

faetsstr: Force/excitation Measurement error level for static strain error functions

faetms: Force/excitation Measurement error level for modal error functions

Proportional error in force measurement

fpetss: Force/excitation Measurement error level for stiffness and flexibility error

functions

fpetsstr: Force/excitation Measurement error level for static strain error functions

fpetms: Force/excitation Measurement error level for modal error functions

umeflag

‘0’: No measurement error in displacement or strain measurement

‘1’: Error in displacement/strain measurement

Absolute error in displacement/strain measurement

uaetss: Measurement error level for stiffness and flexibility error functions

98

uaetsstr: Measurement error level for static strain error functions

uaetms: Measurement error level for modal error functions

Proportional error in displacement/strain measurement

upetss: Measurement error level for stiffness and flexibility error functions

upetsstr: Measurement error level for static strain error functions

upetms: Measurement error level for modal error functions

9. Monte Carlo Simulation

montecarlo

‘0’: Do not use Monte Carlo Analysis

‘1’: Do use Monte Carlo Analysis

nobs: Number of total Monte Carlo experiments

The final results of the parameter estimation stores in “parahistory.mat” file. The number of

convergence iterations for each observation also stores in “iterations.mat” file. These “.mat” can

find in the PARIS program directory.

10. Solution Techniques

SolveMethod

The variable requires a string to choose between two optimization options available in

PARIS14.0. One is based on analytical sensitivities while the other uses constrained function

minimization technique available in MATLAB optimization toolbox.

‘SensitivitySol’: Choose analytical sensitivity solution

99

‘MatlabSol’: Constrained function minimization in MATLAB toolbox

11. Under-relaxation

This is a flag to activate under relaxation.

UnderRelaxFlag

‘0’: Do not under relax change in parameter

‘1’: Under relax change in parameters

12. Convergence Limits

rdjlim

Convergence limit relative change in the value of quadratic objective function J(p)

rdpnlim

Convergence limit relative change in the normalized value of parameter(s)

13. Output Parameter Estimation Results

OutResults

Write the results of parameter estimation to a text file with the name same as the SAP2000

model filename (.txt file)

‘1’: Create the Output File

‘0’: Do not write results to the Output File

100

A.4 Sample PARIS14.0 Input File – UCF Grid Example (FE Model on Pins and Rollers)

% ************************************************************************* % PARIS14.0 % ************************************************************************* tic; clc clear all close all

% -------------------Global Variable Declaration--------------------------- % SAP2000 Model file path global filename unitflag % SAP2000 Model related variables global framemodel shellmodel solidmodel linkmodel % Parameter estimation data global pgflag pg errorfunction lsepflag upframe upshell upsolid uplink

factorforframe ... factorforshell factorforsolid factorforlink global SolveMethod % Matrices of measured nodes for static error function global mnodesss mnodessf1 mnodessf2 melsstr melsstr1 melsstr2 % Matrices of standard deviations in displacement measurements for static % error functions global mnodesssds mnodessf1ds mnodessf2ds mnodessstrds % Matrices of measured frequencies for dynamic error function global fms fmf1 fmf2 % Matrices of measured nodes for dynamic error function global mnodesms mnodesmf1 mnodesmf2 global mni inflag outflag scaledataflag nepflag global rdplim rdpnlim rdjlim global fmeflag umeflag ertflag erdflag global fpetss fpetsstr fpetms faetss faetsstr faetms global upetss upetsstr upetms upetmstr uaetss uaetsstr uaetms global montecarlo nobs

% ************************************************************************* % INPUT DATA FILE % *************************************************************************

% ---------------------SAP2000 MODEL RELATED DATA--------------------------

% Enter the SAP2000 file name. Complete file path should be specified: % filename = C:\API\3DFrameProblem.SDB' % NOTE: The file location should be: C:\API\ % Please create a folder in C:\ on your computer and rename it to API % if it does not already exist

filename='C:\API\UCF_GRID.SDB';

% Type of Finite Elements Used framemodel = 1; % 0 - Not include frame element

101

% 1 - Include frame element shellmodel = 0; % 0 - Not include shell element % 1 - Include shell element solidmodel = 0; % 0 - Not include solid element % 1 - Include solid element linkmodel = 0; % 0 - Not include link element % 1 - Include link element Stype = 1; % 0 - Not include standard sections % 1 - Include standard sections

% --------------ENTER DATA FOR PARAMETER ESTIMATION------------------------

% -----------------------Parameter Grouping data--------------------------- pgflag = 1; % 1 - Group parameters % 0 - Do not group parameters % Enter grouping matrix % pg : Grouping Matrix % NOTE : A parameter group should contain more than one parameter % - [pg] should be a rectangular matrix, empty spaces should be filled % with zeros % - Column 1 contains the group number % - Column 2 represents the parameter type as per the following % classification % Type Parameter % 1 ---> Frame Area % 2 ---> Frame Iz % 3 ---> Frame Iy % 4 ---> Frame J % 5 ---> Frame AreaMass % 6 ---> Shell 'E' % 7 ---> Solid 'E' % 8 ---> Link 'kxx' % 9 ---> Link 'kyy' % 10 ---> Link 'kzz' % 11 ---> Link 'krx' % 12 ---> Link 'kry' % 13 ---> Link 'krz' % - The elements of each row from column 3 onwards represent contain % parameters in that group % - Only unknown parameters may be grouped % - An unknown parameter can belong to only one group

% Group # Parameters Type Grouped Element Numbers pg = [ 1 2 7 8 9 10 11 12 ]; % ----------------------------Error Function------------------------------- % Abbreviations for the error functions that are currently available are: % SS : Static Displacement Data - Stiffness Error Function % SF1 : Static Displacement Data - Flexibility Error Function (Sanayei) % SF2 : Static Displacement Data - Flexibility Error Function (Hjemstad) % SSTR : Static Strain Data and Error Function % MS : Modal Displacement Data - Stiffness Error Function % MF1 : Modal Displacement Data - Flexibility Error Function (Sanayei) % MF2 : Modal Displacement Data - Flexibility Error Function (Hjemstad)

% Enter matrix for the type of parameter estimation to be performed.

102

% Several error functions can be used simultaneously. If the column

corresponding % to an individual error is '1' then that error function will be used in the % parameter estimation process, if it is '0', that error function will not be

used. % 1 2 3 4 5 6 7 8 % SS SF1 SF2 SSTR MS MF1 MF2 MSTR errorfunction = [ 0 1 0 1 0 1 0 0 ];

% -----------------------Error Function Stacking--------------------------- % LSEPFLAG is used to turn stacking on. Similar to ERRORFUNCTION, each

column % represents a particular load case. These load cases can be used

simultaneously % in the formulation of the [E] and [S] matrices or they can be used % individually to calculate the [E] and [S] and then stacked prior % to solving for the change in P. The individual calculation method is not % appropriate for orthogonality, strain, modal frequency or pseudo-linear % error functions. % 1 2 3 4 5 6 7 8 % SS SF1 SF2 SSTR MS MF1 MF2 MSTR lsepflag = errorfunction; % 0 = use the load cases simultaneously, no stacking for each error function % 1 = use the load cases individually and stack for each error function used

% ---------------------------Unknown Parameters---------------------------- % NOTE: Mass parameters must be considered unknown if the modal error % functions is not activated

% Measured unknown parameters for FRAME ELEMENTS % NOTE: Enter the unknown parameters in the matrix form % Column 1 : Frame element number % Column 2 : 1 if 'Area' is unknown parameter, else 0 % Column 3 : 1 if 'Iz ' is unknown parameter, else 0 % Column 4 : 1 if 'Iy ' is unknown parameter, else 0 % Column 5 : 1 if 'J ' is unknown parameter, else 0 % Column 6 : 1 if 'AreaMass' is unknown parameter, else 0

% Frame Element # Area Iz Iy J AreaMass upframe = [ 7 0 1 0 0 0 8 0 1 0 0 0 9 0 1 0 0 0 10 0 1 0 0 0 11 0 1 0 0 0 12 0 1 0 0 0]; % Measured unknown parameters for SHELL ELEMENTS % NOTE: Enter the unknown shell parameters in the matrix form % Column 1 : Shell element number % Column 2 : 1 if 'E' is unknown parameter, else 0 % Column 3 : 1 if 'mass' is unknown parameter, else 0

% Shell Element # E Mass upshell = [];

% Measured unknown parameters for SOLID ELEMENTS % NOTE: Enter the solid element numbers for which 'E' is unknown in a row

103

% vector form

upsolid = [];

% Measured unknown parameters for LINK ELEMENTS % NOTE: Enter the link element numbers for which stiffness is unknown in % along/about different degrees of freedom. % NOTE: Enter the link parameters in the matrix form % Column 1 : Frame element number % Column 2 : 1 if 'k' in 'X' direction is unknown parameter,else 0 % Column 3 : 1 if 'k' in 'Y' direction is unknown parameter,else 0 % Column 4 : 1 if 'k' in 'Z' direction is unknown parameter,else 0 % Column 5 : 1 if 'k' about 'X' direction is unknown parameter,else 0 % Column 6 : 1 if 'k' about 'Y' direction is unknown parameter,else 0 % Column 7 : 1 if 'k' about 'Z' direction is unknown parameter,else 0

% Link Element # kXX kYY kZZ krX krY krZ uplink = []; % ------------Data related to Measured Static Displacements----------------

inflag = 1; % Measured static displacement data % 0 - Do not use test data, use simulated data from the % 'true' section properties % 1 - Do use test data. Read in .txt file outflag = 0; % Save Simulated Static Displacement Data to an external file % 0 - Do not create an output file % 1 - Do write an output file unitflag = 3; % The processed NDT data should be in either of the two % of units: % lb_in_F = 1 % lb_ft_F = 2 % kip_in_F = 3 % kip_ft_F = 4 % kN_m_C = 5 % N_m_C = 10

% If inflag == 0 i.e. simulated data from 'true' section properties are used % NOTE: This factor is only multiplied with unknown paramters to simulate % their section properties.

% Enter a factor for each of the 5 properties of frame elements to % simulate true properties for parameter estimation. % Frame Elements % Area Iz Iy J AreaMass factorforframe = [1.00 1.00 1.00 1.00 1.00];

% Shell Elements factorforshell = [1.00];

% Solid Elements factorforsolid = [1.00];

% Link (spring) Elements % kX kY kZ kXX kYY kZZ factorforlink = [1.00 0.70 0.70 0.70 0.70 0.70];

104

% Matrices of measured nodes (mnodesss,mnodessf1,mnodessf2) % Column # 1 load case number. if lsepflag=0 enter 0 for load

case # % Column # 2 measured node number. % Columns # 3-8 measured dof code % 0 -not measured (or nonexisting in 2d

cases) % 1 -measured % NOTE : If in SAP2000 Finite Element Model, a node is associated % exclusively with a solid element (i.e. no frame or shell element is % connected to it) then for that node 'rx','ry','rz' should be equal to 0

% Matrix of nodes at which displacements are measured for SS errorfunction % Load Case node # ux uy uz rx ry rz mnodesss= [1 16 0 0 0 0 1 0 % 1 20 0 0 1 0 0 0 1 40 0 0 1 0 0 0 1 51 0 0 0 0 1 0 1 93 0 0 0 0 1 0];

% Matrix of nodes at which displacements are measured for SF1 errorfunction mnodessf1 = mnodesss;

% Matrix of nodes at which displacements are measured for SF1 errorfunction % Load Case node # ux uy uz rx ry rz mnodessf2 = mnodesss;

% Matrices of measured nodes (mnodesss,mnodessf1,mnodessf2) % Column # 1 load case number. if lsepflag=0 enter 0 for load case # % Column # 2 measured node number. % Columns # 3-8 the standard deviation of measurement for ux, uy, uz, uxx,

uyy, uzz. % Standard deviations are always positive. All measured displacements must % have a corresponding standard deviation if this option is used.

% NOTE : If in SAP2000 Finite Element Model, a node is associated % exclusively with a solid element (i.e. no frame or shell element is % connected to it) then for that node standard deviations related with % 'rx','ry','rz' should be equal to 0

% Matrix of standard deviations at nodes at which displacements are % measured for SS errorfunction % Load Case node # ux uy uz rx ry rz mnodesssds = [];

% Matrix of standard deviations at nodes at which displacements are % measured for SF1 errorfunction mnodessf1ds = mnodesssds;

% Matrix of standard deviations at nodes at which displacements are % measured for SF2 errorfunction mnodessf2ds = mnodesssds;

%-------------- Static Strain Measurement Related Data --------------------

105

% Matrix of strain measurements % Column 1 Load case number: If lsepflag=0 enter 0 for all % If lsepflag = 1 enter load case number % Column 2 Measured element number % Columns 3-5 Define location of strain gage on element as follows: % --------------------------------------------------------------- % Column 3 the distance from node i to the gage in local x % direction % Column 4 the distance from the NA to the gage in the % local y direction (required for 2D or 3D frames) % Column 5 the distance from the NA to the gage in the % local z direction (required for 3D frames only) % --------------------------------------------------------------- % NOTE: Enter element numbers in order % Enter load cases in order when using stacking

% Load Case Element # xbar (ybar) (zbar) melsstr = [ 1 16 0 1.50 0 1 24 6 1.50 0 1 24 6 -1.50 0 1 37 0 1.50 0 1 99 6 1.50 0 1 112 6 1.50 0 1 112 6 -1.50 0];

% Matrix of standard deviations for strain measurements % only used for nepflag = 5 with manual sd input % Column 1 load case number. if lsepflag=0 enter 0 for load

case # % Column 2 measured element number. % Column 3 standard deviation of strain gage mesasurement % % Load Case Element # st.dev sgsd = [];

%------- Selected Modes of Vibration for Modal Error Functions ------------ % Column #1 - set number (similiar to load case number in static cases) % Column #2 - mode of vibration number

% Matrix of set of mode shapes for MS errorfunction % Set# Mode# % fms = [ 1 1 % 2 2 % 3 3 % 4 4 % 5 5]; Matrix of set of mode shapes for MF1 errorfunction % Set# Mode# fmf1 = fms; % Matrix of set of mode shapes for MF2 errorfunction % Set# Mode# fmf2 = fms; % Number of selected measured frequencies for modal error functions nsfms = size(fms,1); % number of selected measured frequencies for MS nsfmf1 = size(fmf1,1); % number of selected measured frequencies for MF1

106

nsfmf2 = size(fmf2,1); % number of selected measured frequencies for MF2

% Matrix of measured nodes for MS errorfunction % Load Case node # ux uy uz rx ry rz mnodesms =[ 1 20 0 0 1 0 0 0 1 27 0 0 1 0 0 0 1 40 0 0 1 0 0 0 1 47 0 0 1 0 0 0 1 97 0 0 1 0 0 0 1 104 0 0 1 0 0 0 1 117 0 0 1 0 0 0 1 124 0 0 1 0 0 0 2 20 0 0 1 0 0 0 2 27 0 0 1 0 0 0 2 40 0 0 1 0 0 0 2 47 0 0 1 0 0 0 2 97 0 0 1 0 0 0 2 104 0 0 1 0 0 0 2 117 0 0 1 0 0 0 2 124 0 0 1 0 0 0 3 20 0 0 1 0 0 0 3 27 0 0 1 0 0 0 3 40 0 0 1 0 0 0 3 47 0 0 1 0 0 0 3 97 0 0 1 0 0 0 3 104 0 0 1 0 0 0 3 117 0 0 1 0 0 0 3 124 0 0 1 0 0 0 4 20 0 0 1 0 0 0 4 27 0 0 1 0 0 0 4 40 0 0 1 0 0 0 4 47 0 0 1 0 0 0 4 97 0 0 1 0 0 0 4 104 0 0 1 0 0 0 4 117 0 0 1 0 0 0 4 124 0 0 1 0 0 0 5 20 0 0 1 0 0 0 5 27 0 0 1 0 0 0 5 40 0 0 1 0 0 0 5 47 0 0 1 0 0 0 5 97 0 0 1 0 0 0 5 104 0 0 1 0 0 0 5 117 0 0 1 0 0 0 5 124 0 0 1 0 0 0]; % Matrix of measured nodes for MF1 errorfunction % Load Case node # ux uy uz rx ry rz mnodesmf1 = mnodesms;

% Matrix of measured nodes for MF2 errorfunction % Load Case node # ux uy uz rx ry rz mnodesmf2 = mnodesms;

% --------------------NORMALIZATION METHODS-------------------------------- scaledataflag = 0; % 0: do not scale data % 1: scale data with respect to maximum static

displacement

107

% 2: scale data with respect to maximum static strain.

nepflag = 0; % 0: do not normalize ep, % 1: normalize ep w.r.t the standard deviation of ep

(Recommended) % 2: normalize ep w.r.t the covariance matrix of ep % for nepflag = 1 or 2: % -Requires fmeflag, umeflag =1 or manual % input of standard deviations (see above) % -Only static error functions supported

% ----------------------MEASUREMENT ERRORS--------------------------------- erdflag = 1 ; % error distribution % 1: uniform % 2: normal ertflag = 1 ; % error type % 1: proportional % 2: absolute seednum = 234286 ; % seed for random number generator

% --------------------Measurement Error in Forces-------------------------- fmeflag = 0; % force measurement error flag

% Proportional Error - Enter % force error at all dof fpetss = 0.000; % For static displacement data fpetsstr = 0.000; % For static strain data fpetms = 0.000; % For dynamic displacment data

% Absolute Error - Enter absolute force error at all dof faetss = 0.000; % For static displacement data faetsstr = 0.000; % For static strain data faetms = 0.000; % For dynamic displacment data

% -----------------Measurement Error in Displacement----------------------- umeflag = 0; % displacement error measurement flag

% Proportional Error - Enter % Disp error at all lvdt,strain gage,tilts,

etc. upetss = 0.00000000001; % For static displacement data upetsstr = 0.2; % For static strain data upetms = 0.00000000001; % For dynamic displacment data upetmstr = 0.00000000001; % For dynamic strain data

% Absolute Error - Enter absolute force error at all dof uaetss = 0.00000000001; % For static displacement data uaetsstr = 0.00000000001; % For static strain data uaetms = 0.00000000001; % For dynamic displacment data % -------------------- Monte Carlo Activation flag ------------------------ montecarlo = 0 ; % 0 - Do not use Monte Carlo Analysis % 1 - Do use Monte Carlo Analysis nobs = 10 ; % no. of total monte carlo experiments. %************************************************************************** % ----------------------SOLUTION TECHNIQUES-------------------------------- % Choose between MATLAB Optimization Method and Sensitivity Analsyses SolveMethod = 'MatlabSol'; % Enter 'MatlabSol' for MATLAB Optimization

108

% Enter 'SensitivitySol' for Sensitivity % Analysis

% If 'SensitivitySol' is chosen above, choose solution method solflag = 1; % Various hill climbing solutions available are: % 1 = use direct inv. / least squares - Gauss-Newton % 2 = use generalized inverse option % 3 = use gradient method - Steepest Descent % 4 = use gradient method with switch to Gauss-Newton % 5 = use Standard Conjugate Gradient Algorithm % 6 = use BFGS Quasi-Newton Method % 7 = use Gauss-Newton with line search algorithim

swlim = 0.1; % limit for switching from gradient method(4) or % under-relaxation option to Gauss-Newton's method (1) % compared to relative change in j (rdj) and % relative change in parameters (rdp)

mni = 300; % max. no. of iterations (used with Hill Climbing methods) % ---------------------UNDER-RELAXATION FACTORS---------------------------- % CHECK WHY UNDER-RELAXATION FACTOR IS NOT USED FOR SOLID ELEMENTS?

UnderRelaxFlagframe = 1; % 0 - Not using UnderRelaxation % 1 - Using UnderRelaxation UnderRelaxFlagshell = 0; % 0 - Not using UnderRelaxation % 1 - Using UnderRelaxation

UnderRelaxframe = 0.25; % Under-relaxation factor for frame elements UnderRelaxshell = 0.20; % Under-relaxation factor for shell elements

% ---------------------------CONVERGENCE LIMITS---------------------------- % The following limits all deal with the relative change in parameters % associated with the parameter estimation procedure. Therefore at % convergence their value should be 1 or close to 1. The limit entered here % is the range above and below 1 (epsilon + or - 1 )that is acceptable for % convergence. rdplim = 1e-06; % convergence limit for deltap/p(initial) rdjlim = 1e-10; % convergence limit for relative change in j rdpnlim = 1e-10; % convergence limit for relative change in normalized % value of deltap/p(initial) % ************************************************************************* % END OF INPUT FILE % ************************************************************************* paris132; disp('--------------------------------END----------------------------------

'); toc;

______________________________________________________________________________

109

Appendix B Sensor Specifications of the UCF Grid

110

B.1 Geokon Vibrating Wire Strain Gage

Model 4000

Table B.1 (a) Wire Strain Gage Specifications

Strain Gage English SI

Range 3000

Resolution 0.1

Accuracy 0.1 % FS

Operating Temp. -5 to 175ºF -20 to 80ºC

Length 5.875 in 150 mm

Table B.1 (b) Thermistor Specifications

Thermistor English SI

Range -112 to 302ºF -80 to 150ºC

Accuracy 0.9ºF +/- 0.5ºC

Figure B.1 Strain Gage Dimension (http://www.geokon.com/arc-weldable/)

http://www.geokon.com/arc-weldable/

111

B.2 Geokon Vibrating Wire Tiltmeter

Model 6350

Table B.2 (a) Wire Tiltmeter Specifications

Tiltmeter English SI

Range +/-15º

Resolution 10 arc seconds

Accuracy 0.1 % FSR

Operating Temp. -40 to 90ºF -40 to 90ºC

Length 7.375 in 187 mm

Diameter 1.250 in 32 mm

Weight 1.5 lb 0.7 kg

Table B.2 (b) Thermistor Specifications

Thermistor English SI

Range -112 to 302ºF -80 to 150ºC

Accuracy 0.9ºF +/- 0.5ºC

Figure B.2 Tiltmeter Dimension (http://www.geokon.com/vibrating-wire-tiltmeter/)

http://www.geokon.com/vibrating-wire-tiltmeter/

112

B.3 ICP® Accelerometer

Model 393C

Table B.3 (a) Accelerometer Specifications

Accelerometer Performance English SI

Sensitivity 1000 mV/g 101.9 mV/(m/s2)

Measurement Range 2.5 g pk 24.5 m/s2 pk

Frequency Range (± 5%) 0.025 to 800 Hz 0.025 to 800 Hz

(± 10%) 0.01 to 1200 Hz 0.01 to 1200 Hz

Table B.3(b) Accelerometer Dimensions

Accelerometer English SI

Size (Diameter x Height) 2.25 x 2.16 in 57.2 x 54.9 mm


Figure B.3 Accelerometer Dimensions (http://www.pcb.com/)

http://www.pcb.com/

113

B.4 086 Series ICP® Hammer

Model 086D20

Table B.4(a) Impact Hammer Specifications

Hammer Performance English SI

Sensitivity (± 15%) 1 mV/lbf 0.23 mV/N

Measurement Range ± 5000 lbf pk ± 22000 N pk

Frequency Range (-10 dB) (Hard Tip) 1 kHz

Frequency Range (-10 dB) (Medium Tip) 700 Hz

Frequency Range (-10 dB) (Soft Tip) 450 Hz

Frequency Range (-10 dB) (Super Soft Tip) 400 Hz

Table B.4(b) Impact Hammer Dimensions

Hammer English SI

Head Diameter 2.0 in 5.1 cm

Tip Diameter 2.0 in 5.1 cm

Hammer Length 14.5 in 37 cm


Electrical Connector BNC Jack

Figure B.4 Impact Hammer Dimensions (http://www.pcb.com/)

http://www.pcb.com/

114

Appendix C UCF Grid Drawings

Drawings By Nimisha Rao

115

116

117

118

119

120

121

122

123

124

125

126

Documents

Automated Finite Element Model Updating of the UCF …engineering.tufts.edu/cee/people/sanayei/PARIS/documents/2013... · Automated Finite Element Model Updating of the UCF Grid Benchmark