MODELING AND UPDATING OF CABLE-STAYED BRIDGES

by

Boris A. Zárate

Bachelor of Science Universidad del Valle, 2005

______________________________________

Submitted in Partial Fulfillment of the

Requirements for the Degree of Master of Science in the

Department of Civil and Environmental Engineering

College of Engineering and Information Technology

University of South Carolina

2007

________________________ ________________________ Department of Civil and Department of Civil and Environmental Engineering Environmental Engineering Director of Thesis 2nd Reader ________________________ ________________________ Department of Civil and Department of Civil and Environmental Engineering Environmental Engineering 3rd Reader 4th Reader

________________________ Dean of The Graduate School

ii

Acknowledgements

The author would like to thank his thesis advisor, Dr. Juan M. Caicedo, for his guidance,

encouragement, and support throughout the course of this research and graduate school.

The author also would like to thank his committee members, Dr. Ken Harrison, Dr. Paul

Ziehl, and Dr. Dimitris C. Rizos, for their valuable time and suggestions. The author

would like to acknowledge to Dr. Atanu Dutta for his important contribution to the

elaboration of the finite element model and his conversations during his visit to the

University of South Carolina.

Finally the author also would like to thank his family and girlfriend for their support

throughout his study life.

iii

Abstract

Cable-stayed bridges with longer spans and slender girder sections are constantly built

around the world, pushing their analysis and design to its limits. Therefore having a good

understanding of the structure’s behavior is of vital importance. Several methodologies

exist to model the cables of cable-stayed bridges accounting for the cable’s sag effect.

These methodologies can be classified in two distinct groups. The first is based on

polynomial interpolation of the shape and displacement field of the cable such as the

methodology of straight bar with equivalent elasticity modulus and the derivation of

Isoparametric cable formulation. The second group of methodologies uses analytical

functions that define the cable shape under certain load conditions such as the elastic

Catenary.

There are several applications that require accurate finite element models such as:

earthquake or wind simulations, health monitoring and structural control. If the results

from these numerical models are contrasted with data taken from real structures

important differences may appear. In order to reduce the difference between the

numerical results and data from the actual structure, the finite element models should be

calibrated. Such calibration is possible by identifying the dynamic characteristics of the

iv

structure and then adjusting parameters of the structure to match such dynamic

properties.

The first part of this thesis discusses the differences on the overall dynamics of a cable-

stayed bridge when modeled with three different methodologies. The study focuses on

the effect of the cable model on the dynamics of other structural members such as deck

and towers. A numerical model of the Bill Emerson Memorial Bridge over the

Mississippi river on Cape Girardeau, Missouri, which has been permanently instrumented

with a real time seismic monitoring system, is used to study the differences in each

methodology. Even though small differences are found in the natural frequencies and

mode shapes, discrepancy in the frequency response functions show a different dynamic

behavior among the methodologies, especially when the cables are subdivided.

The second part of this thesis presents a methodology to update numerical models of

complex structures such as cable-stayed bridges using Modeling to Generate Alternatives

(MGA) techniques. The goal of MGA is to use computer power to produce a few

plausible and maximally different solutions for the updating problem leaving the final

selection of the best model to a human. The whole family of solutions could also be used

for further studies depending on the use of the updating model. The methodology is

applied to the best numerical model found in the cable modeling comparison. The

Stochastic Subspace Identification (SSI) is used to calculate modal parameters of the

structure based on acceleration records of the bridge. A nonlinear variation of the Hop,

v

Skip and Jump method (HSJ) is used as the specific MGA methodology to calculate the

different solutions. The differences between the identified modal parameters and the

modal parameters of a finite element model are used as the objective function. Results

show the potential of the nonlinear HSJ method to create different solutions for the

updating of the bridge.

vi

Table of Contents

Acknowledgments………………………………………………………………………..ii

Abstract……………………………………………………………………………….....iii

List of Tables…………………………………………………………………………...viii

List of Figures …………………………………………………………………………...ix

List of Symbols…………………………………………………………………………..xi

1.  Introduction ............................................................................................................... 1 1.1.  Cable Modeling ................................................................................................... 2 

1.2.  System Identification and Updating .................................................................... 4 

1.3.  Overview ............................................................................................................. 5 

2.  Finite Element Models of Cable-Stayed Bridges .................................................... 8 2.4.  Cable Models ...................................................................................................... 8 

2.4.1.  Equivalent Elasticity Modulus .................................................................... 9 

2.4.2.  Two Node Isoparametric Lagrangian Formulation ................................... 11 

2.4.3.  Catenary Element ...................................................................................... 12 

2.4.4.  Numerical Verification (Cable Network Application) ............................. 15 

2.5.  Stability functions ............................................................................................. 17 

2.5.1.  Numerical Verification ............................................................................. 22 

2.6.  Bridge Deck Model ........................................................................................... 23 

2.7.  Comparing the Dynamic Response ................................................................... 28 

2.7.1.  State Space Representation ....................................................................... 28 

2.7.2.  Modal Assurance Criteria ......................................................................... 30 

2.7.3.  Transfer Function ...................................................................................... 32 

2.7.4.  Frequency Response Assurance Criterion ................................................ 34 

vii

3.  System Identification and Model Updating .......................................................... 35 

3.1.  System Identification ........................................................................................ 36 

3.1.1.  Stochastic Subspace Identification ........................................................... 38 

3.2.  Model Updating ................................................................................................ 45 

3.2.1.  Modeling to Generate Alternatives ........................................................... 48 

3.2.2.  Numerical Verification of the Nonlinear HSJ Method ............................. 52 

4.  Application: Cable Dynamics ................................................................................ 56 4.1.  Description of the Bill Emerson Memorial Bridge ........................................... 56 

4.1.1.  Finite Element Model ............................................................................... 58 

4.2.  Results ............................................................................................................... 59 

4.2.1.  Tension Distribution (Static Analysis) ...................................................... 59 

4.2.2.  Frequencies and Mode Shapes .................................................................. 61 

4.2.3.  Transfer Functions .................................................................................... 66 

5.  Application: Identification and Updating ............................................................. 75 5.1.  Identification Process ........................................................................................ 76 

5.1.1.  System Identification ................................................................................ 77 

5.2.  Bill Emerson’s Model Updating ....................................................................... 80 

5.2.1.  Parameters to Update ................................................................................ 81 

5.2.2.  Application of the Nonlinear HSJ Method Proposed ............................... 83 

5.2.3.  Updating Results ....................................................................................... 85 

6.  Conclusions and Future Work ............................................................................... 87 6.1.  Conclusions ....................................................................................................... 87 

6.2.  Future Work ...................................................................................................... 90 

References ........................................................................................................................ 92 

 

viii

List of Tables

Table 1. Cable net characteristics……………………..…………………………………16

Table 2. Results comparison…………………………..…………………………………17

Table 3. String characteristics ……………………………………………………………23

Table 4. First 5 natural frequencies of the string modeled as 10 and 100 beam elements………………………………………………………………………………….23

Table 5. Adjusted parameters Δk and Δm found using the nonlinear HSJ methodology proposed……………………………………………….…………………………………55

Table 6. First 20 natural frequencies (hz) of the cable-stayed bridge, using 1 element per cable……………………………………………………………………………………...61

Table 7. First 20 natural frequencies (hz) of the cable-stayed bridge, using 4 elements per cable……………………………………………………………………………………...66

Table 8. FRAC value for the transfer functions in the 0 to 1.5 hz range, among the methodologies in the same location…………………………………..………………….70

Table 9. FRAC value for the transfer functions in the 0 to 1.5 Hz range, among the methodologies in the same location, using 4 elements per cable…….………………….74

Table 10. Dynamic Characteristics Identified……………………….…………………..79

Table 11. Comparison of natural frequencies between this study and Song et al. (2006)…………………………………………………………………………………….80

Table 12. Constraints applied to the parameters to update………………………………83

Table 13. First solution found……………………………………………….…………...84

Table 14. Alternative solutions found ……………………………………………………84

Table 15. Local minima obtained ………………………………………….……………85

Table 16. Dynamic characteristics of the local minima………………………………….86

ix

List of Figures

Figure 1 . Cable structures nonlinear behavior……………………………………………9

Figure 2. Cable in its plane………………………………………………………………11

Figure 3. Cable net……………………………………………………………………….16

Figure 4. Three dimensional Euler Bernoulli elements……………………………….…17

Figure 5. String modeled…………………………………………………………………23

Figure 6. Cross section of spine model………………………………………………..…26

Figure 7. System with input, output, and no noise………………………………….……33

Figure 8. Typical bode plot………………………………………………………………33

Figure 9. System with input, disturbance and output……………………………….……36

Figure 10. Flow diagram of the primary updating process………………………………47

Figure 11. Feasible region of a two variables nonlinear programming problem…...……50

Figure 12. Flow diagram of the complete updating process……………………..………51

Figure 13 . Bill Emerson Memorial Bridge………………………………………….......52

Figure 14. Three dimensional objective function plot (Equation 90)……………………55

Figure 15. Bill Emerson Memorial Bridge………………………………………………56

Figure 16. Cross section of the deck……………………………………………………..57

Figure 17. Cross sections of the towers………………………………………………….57

Figure 18. Finite element model of the Bill Emerson Memorial Bridge………………...58

Figure 18. Cable tension distribution along the deck, using one element per cable……………...............................................................................................................60

Figure 19. MAC value for the first 20 mode shapes after nonlinear procedure among the different methodologies, using 1 element per cable……………………………….…….63

Figure 20. 13th and 14th mode shape of Catenary, Isoparametric and Equivalent formulations……………………………………………………………………………...64

Figure 21. MAC value between Isoparamteric formulation using 1 and 4 elements per

x

cables and Catenary Element using 1 and 4 elements per cable…………………………65

Figure 22. Locations at which the Transfer Functions were calculated at the Emerson Bridge…………………………………………………………………………………….67

Figure 23. Transfers Functions for all three methodologies at node 181 when ground acceleration is parallel to the deck and acceleration response is measured, using 1 element per cable………………………………………………………………………………….68

Figure 24. Transfers Functions for all three methodologies at node 93 when ground acceleration is parallel to the deck and acceleration response is measured, using 1 element per cable……………………………………………………………………….................69

Figure 25. Transfers Functions for all three methodologies at node 10 when ground acceleration is parallel to the deck and acceleration response is measured, using 1 element per cable………………………………………………………………………………….69

Figure 26. Transfers Functions for all three methodologies at node 172 when ground acceleration is parallel to the deck and acceleration response is measured, using 1 element per cable………………………………………………………………………………….70

Figure 27. Transfers Functions for all three methodologies at node 10 when ground acceleration is transverse to the deck and acceleration response is measured, using 4 cables per element………………………………………………………………………..71

Figure 28. Transfers Functions for all three methodologies at node 93 when ground acceleration is transverse to the deck and acceleration response is measured, using 4 cables per element………………………………………………………………………..72

Figure 29. Transfers Functions for all three methodologies at node 172 when ground acceleration is transverse to the deck and acceleration response is measured, using 4 cables per element………………………………………………………………………..72

Figure 30. Transfers Functions for all three methodologies at node 181 when ground acceleration is transverse to the deck and acceleration response is measured, using 4 cables per element………………………………………………………………………..73

Figure 31. Bill Emerson Memorial Bridge Instrumentation (Çelebi 2006). ……...…….75

Figure 32. Identified natural frequencies……………………………………….………..78

Figure 33. Master Masses at the deck ……………………………………………………83

xi

List of Symbols

A Cable cross section area A States matrix A Predicted states matrices B States matrix C States matrix C Predicted states matrices CD Complete damping matrix of the structure D States matrix

_E Equivalent elasticity modulus E Material’s Young modulus Es Young modulus of the steel f Natural frequency f(p) Value of the first objective function evaluated in p fi Identified natural frequencies F1, F2, F3 Cable element nodal forces G Material shear modulus Gs Shear modulus of the steel G(s) Transfer function between the input u(s) and the output y(s) g(p) Value of the second objective function evaluated in p gz Projection in the X axis of the cable weight per unit length H Resulting force at the bottom cable end hx(wl), hy(wl) Transfer Functions to correlate I Identity matrix of appropriated dimensions Imj Mass moment of inertia with respect to the centroidal j-th axis Ij Mass moment of inertia of the lumped masses about the j-th axis Imi Mass moment of inertia of the deck with respect to its centroidal

axis Iy , Iz Cross section moments of inertia about the Y and Z axis J Polar moment of inertia of the cross section k k-th step in a discrete vector K Complete stiffness matrix of the structure KT Cable tangent stiffness matrix

xii

KM Cable material stiffness matrices KM Cable geometry stiffness matrices kG Nodal cable geometry stiffness matrices kM Nodal cable material stiffness matrices kE Nodal cable stiffness matrix k1, k2, k3,k4 Stiffness of the members 1, 2, 3 and 4 J Pure torsional constant of the steel transformed cross section Jeq Equivalent pure torsional constant L Cable length Lb Main bridge span Lu Unstressed cable length lx, ly, lz Projected in the X, Y and Z axis cable length M Complete mass matrix of the structure M Number of channels used in the identification process MAC Value of the Modal Assurance Criterion Ml Lumped mass mi Mass of the i-th component m1, m2, m3 Lumped nodal masses at nodes 1, 2 and 3 M1y, M2y Element’s nodal moments 1 and 2 about the Y axis M1z, M2z Element’s nodal moments 1 and 2 about the Z axis N Number of data points used for the system identification n Torsional mode number O Bank matrix of the predicted free response p Vector of parameters to be optimized P Element’s axial load q Vector of system inputs q Step number Rc Constants for the stability functions evaluation Rcmy Constants for the stability functions evaluation Rcmz Constants for the stability functions evaluation ri Distance of the centroid of the i-th component to the shear center Rt Constants for the stability functions evaluation Rtmy Constants for the stability functions evaluation s Half of the data shift used to build the Henkel matrix S Diagonal matrix of singular values that result from the singular

value decomposition S1y to S5y Stability functions T Cable tension TT Applied torsional moment U Matrix of inputs that results from the singular value decomposition u Displacement response of the structure u1 to u12 Euler-Bernoulli beam element degrees of freedom üg Ground acceleration vector

xiii

V Matrices of outputs that result from the singular value decomposition

v(k) Measurement noise vz Constants for the stability functions evaluation

vy Constants for the stability functions evaluation w Cable weight per unit length wl Frequency range in which Transfer functions are defined w(k) Process noise x Vector of system states x Predicted vectors of state Xo Kalman state matrix

oX Predicted Kalman states matrix y Vector of system outputs y Predicted vectors of system outputs Yh Henkel matrix Γ Warping constant of the transform steel cross section Γ Predicted observability matrix Г Observability matrix Γug Matrix of degree of freedom participation in ground acceleration Γs Warping constant of the transformed steel cross section Δj Additional mass moment of inertia of the section Δk Additional stiffness of the element Δm Additional mass of the node ΔT Time between samples in the discrete data ε(k) Error in the predicted vector state ςi Identified damping ratios of the structure

λ Elastic Catenary’s constant λi Continuous time poles of the system μi i-th eigenvalue of matrix of predicted A ρ Chord’s mass per unit length Φ Torsional mode shape of the deck Φid Matrix of identified Operating Deflection Shapes (ODS)

ife,ω i-th natural frequency of the finite element model

iid ,ω i-th identified natural frequency ω1(p), ω2(p) First and second natural frequency of the structure

1

1. Introduction

Cable-stayed bridges have increased in popularity during last decades, due to their beauty

and highly efficient use of the materials (Karoumi 1999). Cable-stayed bridges with

longer spans and slender girder sections are constantly built, pushing the analysis and

design of these structures to its limits. These structures are also exposed to dynamic event

such as hurricanes, earthquakes, and cable galloping (or excessive vibrations created by

light rain wind) the increase in cable spans and the exposure to these structures to

dynamic loads significantly increases the importance of having a good understanding of

the dynamic behavior of the structure. For instance, although cable stayed bridges have

an inherent good seismic performance, these types of structures have shown to be

damaged by seismic movements such as the case of the Gi-Lu bridge in Taiwan

(Chadwell et al. 2002, Chang et al. 2004).

One of the challenges found in modeling cable-stayed bridges is their geometrical

nonlinear behavior. This behavior mainly comes from three sources: cable sag effect, P-

Δ effect (beam-column effect) and large displacements effect. In this thesis all three

sources of nonlinearity are accounted. The sag effect is considered by the cable models,

the P-Δ effect is accounted by using stability factors that affect the stiffness of the

columns and beams as a depending upon the end moments and axial load (Shantaram G.

and Ekhande 1989), and the large displacements effect is considered by the cables

2

formulations. Although every source of geometrical nonlinearities are considered in the

numerical models, these models do not necessarily behave as the real structures. This is

due not only to the difficulty of capturing the nonlinear behavior of the structure but

because finite element models are usually developed based on idealizations of actual

structures. Therefore, modeling these structures pose some challenges to the engineer

such as the determination of the level of discretization, determination of support

conditions, section and material parameters. In order to determine a good finite element is

necessary to develop a good understanding of the behavior of the built structure and a

good understanding of the finite elements available to model the structure. The study of

the dynamic behavior of the structure can be accomplished by identifying the dynamic

characteristics of the structure (i.e. natural frequencies, mode shapes, damping rations

and transfer functions) based on data capture from the bridge. Several methods are

currently available for modal identification for instance, Ren et al. (2005) identified the

natural frequencies of the Qingzhou cable-stayed bridge in Fuzhou, China, using the Peak

Peaking method. Chang et al. (2001) identified the natural frequencies and damping

ratios, of the Kap Shui Mun cable-stayed bridge in Hong Kong, China, using a Peak

Peaking method and an ARMA model. The study of the characteristics of different finite

elements can be performed by numerical simulations and by comparing the results of

these numerical simulations to data obtained from the real structure.

1.1. Cable Modeling

The study of finite element models to model cables in cable-stayed bridges such the

dynamic behavior of the structure is correctly reproduced is one of the focuses of this

3

thesis. Many studies are available in the literature that studies the geometrical nonlinear

behavior of cables. Wang and Yang (1996) performed a parametric study of the sources

of nonlinearity, finding that for initial shape analysis the cable sag effect has the most

important contribution, but for static deflection case the cable sag becomes the least

important. In dynamic studies Abdelghaffar and Khalifa (1991) and Tuladhar (1995)

discussed the importance of cable vibration on the earthquake response of cable stayed

bridges due to the high contribution of the cable modes on the dynamic structure

response. Concerning this dynamic behavior, Abdelghaffar and Khalifa (1991) shown

that the cable motion and deck motion could be coupled due to the low modes of

vibration of the cables and the deck, even in the case of pure cable mode shape.

Nevertheless, given that the stay cable mass is light compare with mass of the deck,

vibrations induced in cables might not affect significantly the deck motion.

There are several procedures to compute the cable stiffness matrix. These methodologies

can be classified in two distinctive groups. One is based in polynomial interpolation of

the shape and displacement field of the cable such as the methodology of straight bar

with equivalent elasticity modulus, introduced by Ernst (1965) and the derivation of

isoparametric cable formulation given by Ozdemir (1979). The other type of

methodologies use analytical functions that define the cable shape under certain load

conditions. The procedure given by O’Brien and Francis (1964) Chang and Park (1992)

which use the equations that define the elastic catenary of the cable to find its stiffness

matrix are a clear example.

4

Several researchers have focused in the static cable-stayed bridge geometric nonlinearity

behavior and a better understanding of this kind of structures has been acquired trough

numerical simulations and field observations. In contrast the influence in cable dynamics

on the overall dynamics of the structure and how the different methodologies can

represent such dynamic behavior have not been widely studied.

1.2. System Identification and Updating

Accurate finite element models are needed for applications like earthquake engineering,

wind engineering, structural control and structural health monitoring. The accuracy of the

models depend not only on the type of finite element model used but also of the

properties (e.g. elasticity modulus, moment of inertia) assigned to these elements.

Differences between the dynamic behavior of a finite element model and the

corresponding real structure are common. These differences can be caused by the

discretization of the finite element model, and uncertainties in geometry, material

properties or boundary conditions. In order to reduce these discrepancies the numerical

model should be calibrated based on information from the structure.

In this thesis a new method to calibrate or update finite element models based on

Modeling to Generate Alternatives (MGA) is proposed. MGA was developed with the

goal of providing solutions to complex, incomplete problems by coupling the

computational power of computers and human intelligence (Brill et al. 1990). MGA

creates several possible good solutions from a problem by eliminating bad alternatives

using a mathematical model. These solutions are different but provide a similar outcome

5

to the problem. Here, MGA is used to provide solutions for the model updating of the

cable-stayed bridge. In particular a nonlinear variation of the Hop, Skip and Jump method

(HSJ) proposed by Brill et al. (1982) and used for a land use planning problem is used in

this paper.

Given that the number of variables used for the updating process is larger than the

number of equations available, and that there are several uncertain parameters that are not

considered in the updating process, more than one good solution may exist. The goal of

MGA here is not only to update the finite element of the structure but to create a family

of models that hold similar dynamic characteristics. Depending upon the final use, either

all the models can be used or a single model can be selected from a subsequent analysis.

For instance, updated models could potentially be used for damage detection by

comparing an updated model after an event with a dynamic model before a dynamic

event. Using MGA, some of the updated models could be discarded based on previous

experience of inspectors or based on further observations of the structure. In contrast, the

whole family of models could be considered in structural control to test the robustness of

a control strategy.

1.3. Overview

The main goal of this thesis is to develop a methodology for modeling cable stayed-

bridges under dynamic load. This thesis focuses on two of the main challenges identified

for this task: i) determine the influence of three of the most prominent cable models in

the overall dynamic behavior of the structure, and ii) develop a methodology to calibrate

6

the finite element model using data obtained from the real structure. The Bill Emerson

Bridge, spanning the Mississippi river on Cape Girardeau Missouri and permanently

instrumented is used as the model structure in this thesis.

The second chapter of this thesis presents a literature review for cable-stayed bridge

modeling, including: pylon, deck and cable modeling. The cable methodologies:

Equivalent Elasticity, Elastic Catenary and Isoparametric Formulation are presented. The

deck and tower assumptions are exposed together with the stability functions used to

represent the axial load and end moment effects at the beam elements. An example

problem is presented for the cable models and the stability functions. The criteria for the

dynamic comparisons are included at the final part of this section.

The third chapter presents the literature review for the system identification and model

updating. The SSI methodology is here exposed. MGA is presented together with a

nonlinear variation of the HSJ method. And an example problem is solved using the

updating methodology explained.

The fourth chapter discusses a comparison among different methodologies to model cable

dynamics applied to the finite element model of the Emerson Bridge and to study the

effect of the cables in the overall dynamics of the structure. The effects of the cable

methodology and cable subdivision are evaluated in terms of the dynamic response of the

structure, including cable-deck and cable-tower interaction. The theoretical background

exposed in chapter 2 is used for this task.

7

The fifth chapter presents the identification of the dynamic characteristics of the Emerson

Bridge and the posterior calibration of the finite element model of the structure. The

system identification is performed by using the Stochastic Subspace Identification (SSI)

methodology employing ambient vibration data. Once the modal parameters of the

structure are identified, the finite element model that best represent the structure obtained

from chapter fourth is updated using Model to Generate Alternatives (MGA). The

theoretical background exposed in chapter 3 is used for this task.

Finally in chapter 6 the conclusions obtained from this thesis as well as the future work to

perform are presented.

8

2. Finite Element Models of Cable-Stayed Bridges

The supper-structure of cable-stayed bridges are mainly built of three structural elements:

pylons, deck and cables. Geometrical nonlinearities are important to consider in the

modeling of cable-stayed bridges due to the structure flexibility and the use of cables.

Cable structures are characterized by a geometric hardening behavior that affects the

curvature of the force displacement curve (Figure 1). Which is produced by the increase

in cable stiffness caused by larger tension as the structure is deformed (Karoumi 1999).

This chapter provides the background information about the methodologies used to

develop the finite element models used later in this work. This chapter also describes the

methods used to compare the dynamic characteristics that result from the different

models. First, a description of the cable models compared will be presented. Then, the

stability functions will be discussed. Finally, different comparison criteria for dynamic

behavior will be introduced.

2.4. Cable Models

Modeling cables to different load conditions is not a simple task, due to the nonlinear

relationship between stress state and shape. These nonlinearities are caused because the

cable stress state is a function of the cable shape, and at the same time the cable shape is

function of the stress state. An iterative procedure based on the Newton-Raphson method

9

is implemented to address this cable nonlinear behavior. The tangent stiffness matrix of

the bridge then can be calculated at equilibrium (Buchholdt 1999). Moreover, cables are

characterized by large displacements, which come from the cable longitudinal strain and

kinematical rigid body rotation and translation (Bathe 1996). This behavior makes part of

the nonlinear relationship between stress state and shape, but it is addressed of different

ways, as part of the finite element formulation itself.

Three well established finite element methodologies to compute the cable stiffness with

three translational degrees of freedom per node are described in the following section.

Their effect on the overall behavior of cable-stayed bridge dynamics is studied and

contrasted in the following chapters.

2.4.1. Equivalent Elasticity Modulus

Modeling cables as a straight bar with equivalent elasticity modulus was first introduced

by Ernst (1965) and since then it has been adopted by several researchers for the

Displacement

Cable Structures

Non-cable Structures

Force

Figure 1 . Cable structures nonlinear behavior

10

modeling of cable structures ( Wilson and Gravelle (1991), Wang and Yang (1996), Chen

et al. (2000)). This model provides a first approximation to the geometrical non-linear

behavior of cables and due to its simplicity has been widely used. Truss elements with

equivalent elasticity modulus consider the effect given by the cable sag, but it lacks of a

way to address the large displacement effect, which gains in importance for long stay

cables. It is generally accepted for cable-stayed bridges with short spans and it is

commonly used for design (Karoumi 1999). Moreover, this approach confers rigid body

behavior to the cable and cable subdivision should be avoided.

The equivalent modulus approach is based in the use of a parabolic shape to approximate

the hanging cable catenary, as well as neglecting the cable weight component parallel to

the cable. The element is modeled as a truss element with an equivalent elasticity

modulus E expressed by the equation:

TAE

TLg

EEz

2

21211 ⎟

⎠⎞

⎜⎝⎛+

= (1)

Where E, is the material’s Young modulus; A, is the cable cross section area; T, is the

cable tension; L, is the cable length and gz is calculated as

Llwg x

z = (2)

where w, is the cable weight per unit length; and lx is the projection of the cable along the

X axis as shown in Figure 2.

11

2.4.2. Two Node Isoparametric Lagrangian Formulation

The two nodes isoparametric formulation was first introduced by Ozdemir (1979) and

used to model cable networks and cable supported roofs. Applications of this two nodes

formulation in cable-stayed bridges can be found in Wang and Yang (1996). Ali and

Abdel-Ghaffar (1991) used a four nodes formulation. In the two nodes formulation

element is assumed to be straight, so that sag effect is neglected. Some authors consider

sag effect by replacing the material modulus of elasticity for the equivalent elasticity

modulus previously described (Tibert 1999). The equations describing the body

movement in the time domain are formulated and then solved using linear function

shapes to interpolate the displacement field in the element (Bathe 1996). The large

displacement effect is modeled by using the strain Green tensor (Crisfield 1991 and

Bathe 1996).

The tangent stiffness matrix KT, is given by the sum of the material stiffness matrix KM,

Ll ly

Y

X

Figure 2. Cable in its plane.

lx

12

and the geometry stiffness matrix KG.

GMT KKK += (3)

where the material stiffness matrix is function of the cable’s material and geometry, and

the stiffness geometry matrix depends of the cable tension as well as the length. KM and

KG are described numerically by

⎥⎦

⎤⎢⎣

⎡−

−=

MM

MMM kk

kkK ; ⎥

⎦

⎤⎢⎣

⎡−

−=

GG

GGG kk

kkK (4)

where kM and kG on a three dimensional formulation are given by

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅

=000000001

LAE

Mk ; ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100010001

LT

Gk (5)

2.4.3. Catenary Element

The elastic catenary cable element is a Lagrangian formulation based on the geometric

curvature of the cable. This results in an exact treatment of the sag and the self weight as

well as the geometric nonlinear effects caused by large displacements. The two node

elastic catenary formulation was first suggested by Peyrot and Goulois (1979) based in

the expressions given by O’Brien (1967). Subsequently, Jayaraman and Knudson (1981)

13

proposed a more efficient method to compute the tangent stiffness matrix based on the

same exact analytical expressions. A more understandable and efficient presentation of

this elastic catenary cable element was given by Chang and Park (1992) which presented

the equations implemented in this study. Few applications of this catenary cable element

on cable-stayed bridges are reported in the literature, among them: Karoumi (1999) in a

plane model and Kim et al. (2004) in a three dimensional model.

Consider an elastic cable element suspended from its ends i (0, 0, 0) and j (lx, ly, lz), with

an unstressed length Lu, and weight per unit length w. The relative end node distances lx,

ly, and lz can be written in terms of the element end nodal forces F1, F2, and F3 by the

Equations (Chang and Park 1992)

( ) ( )[ ]TFHLwFwF

AELFl u

ux +−++−−= 33

11 lnln (6)

( ) ( )[ ]TFHLwFwF

AELFl u

uy +−++−−= 33

22 lnln (7)

[ ]THwAE

LwAELFl uu

z −−−−=1

2

23 (8)

where, H and T are

( )23

22

21 uLwFFFH +++= ; 2

32

22

1 FFFT ++= (9)

14

This group of expressions is a nonlinear equation system where the unknowns are the

element nodal forces. Applying the first order Taylor series, it can be shown that

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

3

2

1

321

321

321

dFdFdF

Flz

Flz

Flz

Fly

Fly

Fly

Flx

Flx

Flx

dlzdlydlx

(10)

where the Jacobian of the equation system represents the flexibility (3x3) matrix, so that

its inverse is the (3x3) stiffness matrix kE. Hence, the tangent cable stiffness (6x6) matrix

KT, is given by

⎥⎦

⎤⎢⎣

⎡−

−=

EE

EET kk

kkK (11)

A numerical solution for this set of equations can be obtained using a variety of

procedures. One of the most commonly used is the Newton Raphson method (Chapra

and Canale 1998), where initial values of the nodal forces are needed. Jayaraman and

Knudson (1981) suggested initial values for the forces based on the following equations.

λ⋅

−=21

xlwF ; λ⋅

−=22

ylwF ; ⎟

⎠⎞

⎜⎝⎛ −=

λλ

sinhcosh

21 zu lLwF (12)

15

Where λ is recommended to be 0.2 for cases when the unstressed cable length Lu is less

than the distance between the ends and 106 for vertical cables. For other cases λ is

calculated as

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−= 13 2

22

x

yu

llL

λ (13)

2.4.4. Numerical Verification (Cable Network Application)

The three cable finite element models were programmed in Matlab. In this section an

example problem is presented to verify the Elastic Catenary and Isoparametric

methodologies. The problem consists of a 300 ft x 300 ft cable net (Figure 3), originally

studied by Saafan (1970) and subsequently analyzed by Jayaraman and Knudson (1981).

16

Nodes 1, 2, 6, 7, 10, 11 and 12 are restraint in all the directions. Nodes 4, 5, 8 and 9 have

3 dof per node for a total of 12 degrees of freedom in the entire structure. The properties

of the cable net studied are presented in Table 1, and the displacements obtained at node

4 for all methodologies considered are compare in Table 2. The results show a good

agreement with differences of less than 2% among the methodologies presented in this

study and the results obtained by the others investigators. This can be attributed to

numerical errors introduced by choosing different parameters for the non-linear

procedure, such as the size of the step in the iterations or the tolerance to achieve

equilibrium in the Newton-Raphson procedure.

Characteristic Magnitude Cable cross section area 0.227 in2

Elasticity modulus 12000 kips/in2

Weight per unit length 0.0001 kip/ft Tension of cables: 1, 2, 3 and 4 5.459 kips Tension of cables: 5, 6, 7, 8, 9, 10, 11, 12 5.325 kips Vertical load at nodes: 4, 5, 8 and 9 8.0 kips

Figure 3. Cable net

17

Table 1. Cable net characteristics

Researcher Displacement of Node 4 ( ft ) X Y Z

Saafan (1970) – Isoparametric -0.1324 -0.1324 -1.4707 Jayaraman and Knudson (1981) - Elastic Catenary -0.1300 -0.1319 -1.4643 Jayaraman and Knudson (1981) - Isoparametric -0.1322 -0.1322 -1.4707 This study - Elastic Catenary -0.1339 -0.1329 -1.4695 This study – Isoparametric -0.1324 -0.1324 -1.4643

Table 2. Results comparison

2.5. Stability functions

A structural element subjected to a compression axial load and bending moments suffers

an increase in the bending moments by lateral deflection, which causes a change in the

stiffness of the member. The towers of cable stayed bridges are a typical example of

structural elements with large compression forces due to their self weight and the location

of the cables. In terms of structural analysis, the relationship between load and

deformation is no longer linear, because the stiffness matrix is affected by these bending

moments and axial loads.

Figure 4. Three dimensional Euler Bernoulli elements

18

There are two approaches to account these changes in the stiffness matrix: by using a

geometric stiffness matrix (Yang and Mcguire 1986) or the concept of stability functions

(Shantaram G. and Ekhande 1989). The stability functions are factors that multiply the

stiffness matrix in some components in order to increase or reduce the stiffness of the

elements as a consequence of the internal load and bending moment. For a three

dimensional Euler-Bernoulli beam element (dof u1 up to u12) depicted in Figure 4, the

stiffness matrix K, with such stability functions is given by

⋅

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

ΙΙ−ΙΙ

ΙΙΙΙ−

−

ΙΙΙ−

ΙΙ−Ι−

−

ΙΙ

ΙΙ−

Ι

Ι

=

llll

llll

ll

lll

lll

ll

ll

ll

l

l

l

l

Zz

Zz

Zy

Zz

Yy

Yy

Yy

Yy

Yy

Yy

Yy

Zz

Zz

Zz

Zz

Zz

Yy

Yy

Yy

Zz

ESESESES

ESESESES

JJ

ESESES

ESESES

EASEAS

ESES

ESES

J

SIMMES

ES

EAS

322422

322422

312231

312231

55

322

322

31

31

5

0006000060

060000600

6000006000

12000601200

1206000120

00000

00060

0600

6000

1200

120

K

(14)

The stability functions S1z up to S4z for a tension element are

19

t

zzz R

vvS12

sinh31 = (15)

t

zzz R

vvS

61cosh2

2−

= (16)

t

zzzzz R

vvvvS

4sinhcosh

3−

= (17)

t

zzzz R

vvvS

2sinh

4−

= (18)

where,

zzzt vvvR sinhcosh22 +−= (19)

and

z

z EIPLv = (20)

where, P, is the elements axial load. The stability functions S1y up to S4y are calculated in

a similar fashion but changing Iz for Iy. The stability function S5z for a tension member is

calculated as

( )23

5

41

1

LPRREA

Stmztmy +

−= (21)

20

where,

( )( ) ( )221

222

21 2csccoth yyyyyyyytmy MMvhvvMMvR +−++=

( )( )yyyyyy hvvvvMM csc2coth121 ++ (22)

( )( ) ( )221

222

21 2csccoth zzzzzzzztmz mMvhvvMMvR +−++=

( )( )zzzzzz hvvvvMM csc2coth121 ++ (23)

M1y and M2y are the moments at the nodes 1 and 2 about the Y axis, and M1z and M2z are

the moments at the nodes 1 and 2 about the Z axis.

For a compression member the stability functions S1z up to S4z are

c

zzz R

vvS12

sinh31 = (24)

c

zzz R

vvS6cosh12

2−

= (25)

c

zzzzz R

vvvvS4

cossin3

−= (26)

c

zzzz R

vvvS2

sin4

−= (27)

where,

21

vvvRc sincos22 −−= (28)

The stability functions S1y up to S1z for a compression member are obtained replacing Iz

for Iy. For a member in compression S5 is defined as

( )23

5

41

1

LPRREA

Scmzcmy +

−= (29)

where,

( )( ) ( )221

222

21 2csccot yyyyyyyycmy MMvvvMMvR +−++=

( )( )yyyyyy vvvvMM csc2cot121 ++ (30)

( )( ) ( )221

222

21 2csccot zzzzzzzzcmz mMvvvMMvR +−++=

( )( )zzzzzz vvvvMM csc2cot121 ++ (31)

and, G is the shear modulus, Iy and Iz are the moments of inertia of the cross section about

the Y and Z axis respectively, and J is the polar moment of inertia.

For small axial load values, S1 to S4 are numerically unstable. These values are equal to 1

when

9101 −< xEIP (32).

22

2.5.1. Numerical Verification

The stability functions were coded in Matlab. This section presents an example problem

to test the stability functions used to model the pylons and the deck. The solution is

compared with the analytical solution found in the literature. The problem consists of a

prestressed string with an axial load as is shown in the Figure 5. The natural frequencies

are computed and compare with the solution given by the string theory at equation (33)

(Kreyszing 1983).

ρT

Lnf ⋅⋅

=2

(33)

where, T, is the tension on the cable, L is the chord length, f is the n-th natural frequency

and ρ the mass per unit length of the chord. The characteristics of the string are given in

Table 3. Two numerical models of the chord where created using Euler Bernoulli beam

elements, one with 10 elements and another with 100 elements. A nonlinear static

analysis was performed by applying 0.5 kips at one side of the cord as shown in Figure 5.

The Newton-Raphson method was used in this procedure. The first 5 natural frequencies

of the string were calculated using the stiffness matrix that resulted after equilibrium was

achieved.

23

Characteristic Value Area 0.0031 in2

Inertia 7.49x10-7 in4

Mass per unit length 7.324x10-7 k/inElasticity modulus 30000 kip/in

Table 3. String characteristics

Frequency number

Analytical solution 10 Elem. 10 Elem.

Error (%) 100 Elem. 100 Elem. Error (%)

F1 74.59 Hz 73.59 Hz 1.34 73.88 Hz 0.95 F2 149.17 Hz 144.79 Hz 2.94 147.77 Hz 0.94 F3 223.76 Hz 208.45 Hz 6.84 221.67 Hz 0.93 F4 298.34 Hz 255.45 Hz 14.36 295.58 Hz 0.91 F5 372.93 Hz 265.05 Hz 28.93 369.52 Hz 0.91

Table 4. First 5 natural frequencies of the string modeled as 10 and 100 beam elements

The results shown in Table 4 show a good agreement between the finite element models

and the analytical solution for the first few natural frequencies. As expected, the model

with 10 elements performs poorly in higher frequencies while the model with 100

elements presents closer results to the analytical solution due to a higher discretization of

the structure.

2.6. Bridge Deck Model

Cable-stayed bridges present coupled translational and torsional modes due to the

100 in

0.5 kips

Figure 5. String modeled

24

presence of the stay-cables and the mass distribution on the deck. In most cases the deck

mass center does not coincide with the shear center of the section. In addition, the cables

in most bridges are attached on both sides of the deck creating rotational effects when the

deck is excited in the transverse direction of the bridge. This complex behavior can be

considered in numerical models using different approaches:

i) Idealizing the deck as a beam element or spine. The beam is massless and located

at the shear center of the section to reproduce the stiffness effects of the real deck.

Lumped masses strategically located to reproduce the distance between the mass

center and shear center are attached to the spine using rigid links. This approach

given by Willson and Gravelle (1991) has been widely adopted by several

investigators, such as Chang (1998) and Dyke et al. (2003), among others.

ii) Representing the deck as a double or triple girder model (Zhu et al. 2000) to take

into account the warping effect, has been widely accepted by different authors

especially in the case of double girder (Abdelghaffar and Nazmy 1991). In the

case of the double girder model two beam elements are used at each longitudinal

side of the bridge. The deck warping effect is accounted by using opposite vertical

bending stiffness on the girders. In the triple girder case the two side beam

elements have the same properties and the central girder has a different section

property to consider the warping stiffness. The central girder is located at the

deck’s centroid. Numerical comparisons have shown that the differences of the

25

frequencies on the first vertical mode shapes between the triple girder model and

the spine model are negligible. However, in the pure lateral and especially in the

pure torsional mode shapes the difference in the frequencies are more important

(Zhu et al. 2000).

iii) Modeling the deck as a box girder element with specific formulations of seven

degrees of freedom per node to include warping deformation. This method

drastically increases the number of degrees of freedom of the model and

intensifies the effort needed in the preprocess stage. Elements of this kind have

been used in cable-stayed bridges by Kanoknukulchai (1993), and Kim et

al.(2004)

iv) Representing the deck by shell elements with equivalent thickness. This model

considers the geometry properties of the cross section. Compared with the above

mentioned approaches this method further increases the number of degrees of

freedom of the model. Models of this nature have been developed by Brownjohn

et al.(1999) Ren and Peng (2005) among others.

It is beyond the scope of this thesis to study the effect of the different assumptions in the

deck modeling. The spine model is adopted in this study because it is the simplest

representation of the deck with smallest number of degrees of freedom and therefore

26

requires the smallest computational power.

The geometric characteristics of the main girder section were calculated to an equivalent

steel section in concordance with the bridge deck blue prints. The centroid of the main

girder was located at the shear center of the section, while the masses were placed at the

mass center, so that the eccentricity between mass and shear center could be accounted

for (Figure 6). Additional rotational masses were included at the main girder to achieve

the correct mass moment of inertia of the section. These are calculated by

jmjj II −=Δ (34)

where Imj is the mass moment of inertia with respect to the centroidal jth axis and Ij is the

mass moment of inertia of the lumped masses with respect to the jth axis and can be

calculated respectively as

( )∑=

+=n

iiimimj rmII

1

2 ; 22 rMI lj = (35)

Cable Cable

Massless Spine Beam

Figure 6. Cross section of spine model

Rigid Links

Rigid Links

27

where Imi is the mass moment of inertia of the deck with respect to its centroidal axis, mi

is the mass of this component, ri is the distance of the centroid of the component to the

shear center, Ml is the value of each lumped mass and r is the distance from the mass to

the shear center. Rigid links were used to connect the stay cables to the main girder and

the towers, so that the original lengths and angles of the cables were not affected.

In order to account for the warping stiffness Willson and Gravelle (1991), proposed an

equivalent torsional Jeq constant given by

''''' ΦΦΦ Γ−== sseqsT EJGJGT (36)

where Φ is the torsional mode shape of the deck, TT is the applied torsional moment, J is

the pure torsional constant of the steel transformed cross section, Es is the Young

modulus of the steel, Gs is the shear modulus of the steel, Γ is the warping constant of the

transformed steel cross section and the prime and triple prime denotes the first and third

order derivative with respect to the coordinate position. If the torsional mode shapes are

assumed as sine functions, the equivalent pure torsional constant can be written as

2

⎟⎟⎠

⎞⎜⎜⎝

⎛Γ+=

bs

seq L

nG

EJJ π (37)

Where Lb is the main bridge span and n represents the torsional mode numbers. Willson

and Gravelle (1991) proposed to compute this Jeq value to the three first torsional modes

28

and then to incorporate the average in the model.

2.7. Comparing the Dynamic Response

Cable structures are characterized by their low frequency vibrations because of their light

weight and high flexibility. Cable-stayed bridges are typically under low frequency load

caused by: wind, traffic or earthquakes. There are several examples of cable structures

presenting high amplitude vibrations. For instance, the Takoma Narrows Bridge, which

failed under slow wind conditions (Achkire 1998) or the Millenium Bridge, which

presented lateral vibrations for synchronization of human walking. The study of these

specific cases requires the updating of a finite element model such as the dynamics of the

structure can be represented. The differences between the real structure and the numerical

model need to be quantified to obtain this goal. This section presents different metrics

that can be used to compare with the finite element model. These metrics can also be

used to compare two numerical models.

2.7.1. State Space Representation

The dynamic behavior of a structure subjected to ground acceleration is given by the

second order differential equation (Chopra 2006):

guMuKuCuM D &&&&& −=++ (38)

29

where, u, is the displacement response of the structure, M, K, CD, are the mass, stiffness

and damping matrices respectively, gu&& , is the ground acceleration, and (.) denotes the

derivative with respect to the time.

Let, x and q be

⎥⎦

⎤⎢⎣

⎡=

uu

x& (39)

[ ]guq &&= (40).

Equation (38) can be written in terms of a first order differential equations and an

algebraic equation as follows

qDxCyqBxAx

+=+=&

(41)

where, A, B, C, and D represent the characteristics of the structure and are given by

⎥⎦

⎤⎢⎣

⎡⋅−⋅−

= −−DCMKM

I0A 11 (42)

⎥⎦

⎤⎢⎣

⎡−

=ugΓ

0B (43)

30

⎥⎦

⎤⎢⎣

⎡=

0I

C (44)

⎥⎦

⎤⎢⎣

⎡=

00

D (45)

where, Γug, is the influence matrix used to assign the ground excitation to the affected

degree of freedom. This representation of the structure is known as state space

representation, and is important for the computation of Transfer Functions and the

mathematical derivation of System Identification techniques as follows in the next

sections.

2.7.2. Modal Assurance Criteria

The Modal Assurance Criterion (MAC) was developed at University of Cincinnati in the

late 1970’s as an alternative to the orthogonality check as a quality indicator of the

assurance of an experimental found mode shape (Allemang 2003). Since then it has been

widely used for different researchers.

MAC measures the degree of consistency or linearity of the mode shapes assigning a

value of zero when the mode shapes are completely unrelated and a value of one when

they are correlated. Intermediate values represent how similar are the mode shapes. MAC

is defined as follows for the a-th and b-th mode shape

31

( ) ( )∑ ∑

∑

= =

=

⋅

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

= n

j

n

jjbja

n

jjbja

MAC

1 1

2,

2,

2

1,,

ΦΦ

ΦΦ (46)

where, n, is the number of degrees of freedom considered on the vector.

A matrix of MAC values can be calculated to explore the differences in the mode shapes

between two models. Here the MAC value is computed for all possible combinations of

calculated mode shapes, to build a matrix where the component (i,k), represents the MAC

value between the mode i-th mode of the first model and the mode k-th mode of the

second model. Discrepancies and similitude between mode shapes can be easily

identified using this method. Notice that MAC in a deeper meaning is a statistical

indicator of the coherence between two random variables. MAC is highly influenced by

the number of components of the variables or the number of degrees of freedom

instrumented in the structure, especially since MAC does not difference any sort of

distribution of the mass or stiffness along the structure.

MAC was used to compare the mode shapes of the structure among different cables

models. A minimum value of 0.95 is used in this thesis to confirm that two mode shapes

correspond to the same mode of vibration.

32

2.7.3. Transfer Function

The transfer function (TF) gives a relationship between the input and its corresponded

output in a time invariant system and it is calculated using the Laplace transform of the

differential equation that relates the input u(t) and the output y(t). The differential

equation becomes a polynomial that relates the input u(s) with the output y(s) as shown

in Figure 7 (Dorf R. C. 1998). Therefore the outputs of the system can be calculated

using:

)()()( sss uGy = (47)

where the term G(s) represents the linear transfer function between the input u(s) and the

output y(s). The transfer function is useful for many applications. In structural dynamics

the transfer function can be used to calculate the dynamic response of a linear system. In

system identification the poles (i.e. points of maximum amplitude) and zeros (i.e. points

of minimum amplitude) of the transfer function can be obtained from an experimental

frequency response function and in structural control transfer functions are commonly

used to design control strategies. In this thesis the transfer function is used to compare

the dynamic response of different structural models. The transfer function is calculated

using state space representation as

[ ][ ] )(

detdet)()( s

sssadjs u

AIDAIBAICy

−−+−

=

(48)

33

where, A, B, C and D are state matrices and I is an identity matrix of appropriated

dimensions. The TFs in this study are computed based on state space representation by

using the Matlab command bode. Bode plots are composed of two plots, a log magnitude

against log frequency plot and a phase against log frequency plot, as shown in Figure 8.

Bode plots are commonly used to visualize the transfer function of linear time-invariant

systems. Bode plots can be easily built from the transfer function shown in (48).

It is important to mention that although the cable stayed bridges studied here are highly

nonlinear when large loads are applied (i.e. dead load), it is assumed that the structure has

a linear behavior under normal traffic conditions. As shown later, this is verified with

Output: y(s) Input: u(s)

System: G(s)

Figure 7. System with input, output, and no noise

Figure 8. Typical bode plot

34

experimental data obtained from the permanent instrumentation of the Bill Emerson

Memorial Bridge.

2.7.4. Frequency Response Assurance Criterion

The Frequency Response Assurance Criterion (FRAC) is similar to MAC but it is used to

correlate two different TFs. FRAC gives value of zero when the two transfer functions

are completely different and value of one when they are the same. FRAC was proposed

by (Heylen and Lammens 1996), and it has been widely used in the literature (Zang et al

2001; Fang et al 2005). FRAC is defined as

⎟⎠

⎞⎜⎝

⎛⋅⋅⎟

⎠

⎞⎜⎝

⎛⋅

⎟⎠

⎞⎜⎝

⎛⋅

=

∑∑

∑

lyy

lxx

lyx

wlwlwlwl

wlwlFRAC

)()()()(

)()(

**

2*

hhhh

hh (49)

where hx(wl) and hy(wl) are the TFs to correlate and wl the frequency range in which they

are defined. The superscript * denotes the conjugate of the complex number.

35

3. System Identification and Model Updating

Finite element models are usually developed based on information collected from either

structural drawings or the geometry of actual structures. These models may or may not

represent the real structure because of idealizations in the behavior of structural members

and support conditions. There are several applications that require models with the

capacity to predict the dynamic behavior of structures, such as earthquake or wind

dynamic simulation, structural control and structural health monitoring. In order to obtain

such applications is imperative: i) to identify the dynamic characteristics of the real

structure (e.g. natural frequencies, mode shapes and damping ratios) and ii) to develop a

numerical model that can emulate that dynamic behavior (Zhang et al. 2000). The first

issue is part of what is known as system identification and the last section is

accomplished by model updating.

The theoretical background for the system identification and model updating procedures

used in this thesis are presented in this chapter. The system identification methodology is

the stochastic subspace identification (SSI). A methodology to generate alternative

solutions for model calibration is used for model updating.

36

3.1. System Identification

A system or structure can be represented dynamically with a black box model as shown

in Figure 9. For structures the input u corresponds to the forces applied to the structure or

ground excitation for earthquakes. The measured outputs are physical quantities such as

displacements or accelerations in specific locations of the structure. The process noise v

are other unknown or difficult to measure forces exciting the structure such as traffic or

wind and the measurement noise w is unwanted error caused by sensors or data

acquisition systems. Here the measured input u and unknown disturbances v and w are

contributing to the generation of the measured output y. It is possible to measure u or y,

but not v, nor w; likewise it is possible to manipulate u, but neither y, v, nor w.

A successful identification is achieved when a mathematical model with the same

dynamic behavior of the structure is obtained (i.e. the same relation between inputs and

outputs). This mathematical model will have the same dynamic characteristics such as

poles and zeros of the Transfer Function and can be used to determine modal parameters

of a structure.

+ ++

Known Input: u

Measured Output: y

+Output

+

Total Input

System: G

Figure 9. System with input, disturbance and output.

Process noise: v Measurement

noise: w

37

System identification can be performed based on different types of tests including: i)

forced vibration, ii) free vibration and iii) ambient vibration tests. Forced vibration test

require the measurement of both the forces acting on the structure and its response. This

is usually possible to perform in a laboratory environment. Free vibration tests do not

necessarily require the measurement of the excitation although the excitation needs to be

controlled to produce free vibration data. These two types of tests are difficult to perform

in full-scale civil structures under normal operation because they are large systems where

wind and other unknown excitation (i.e. v in Figure 9) can be of significance.

Furthermore the use of known excitation methods requires the temporary closing of the

structure while tests are performed, significantly increasing the cost related with each

test. The current state of the art in ambient vibration tests allows obtaining similar results

than those obtain with forced and free vibration tests. Evidence of this can be found in the

literature. For instance, Boroschek et al. (2005) determined the modal characteristics of a

dock at the harbor of Ventanas (Chile) by performing both free and ambient vibration

tests. A static force of 50 Tons was applied to the structure for the free vibration test

using a specially designed steel fuse. This force created an initial displacement on the

structure, and allowed free vibration after braking. The Stochastic Subspace Identification

(SSI) methodology was used for ambient vibration test in the same structure. Ambient

vibration was primarily due to the sea weaves striking the dock. Comparison of the

results shows good agreement between both tests. Basseville et al. (2001) investigated

theoretical and experimental issues in output only methodologies such as robustness

38

related to no stationary loads and handling of measured data from multiple sensor setups.

Mevel et al. (2006) found that only output identification results in reasonable approach

when compared with input-output identification for in-flight airplane structures. Ren et

al. (2005) identified some of the natural frequencies of the Qingzhou cable-stayed bridge

in Fuzhou, China, using the Peak Peaking method. Chang et al. (2001) identified the

main natural frequencies and damping ratios, of the Kap Shui Mun cable-stayed bridge in

Hong Kong, China, using a Peak Peaking method and an ARMA model. Additionally the

SHM ASCE group has performed system identification in the benchmark problem, using

a large variety of output only identification methodologies. For instance, Caicedo et al.

(2004) used the Eigensystem Realization Algorithm, Yuen et al. (2004) utilized the

MODE-ID method, Lam et al. (2004) used a Bayesian spectral-density approach, and

Yang et al. (2004) applied a technique to decompose the signal into the frequency

domain, and then calculated the natural frequencies and damping ratios.

3.1.1. Stochastic Subspace Identification

Identification algorithms such as the Eigensystem Realization Algorithm (ERA) (Juang

and Pappa 1985; Juang and Pappa 1986), and the Prediction Error Method Through Least

Squares (Andersen 1997) are based on the same assumptions: i) the structure or system

behaves linearly, ii) the structure is time invariant, and iii) the input loads are a

realization of a Gaussian white noise stochastic process uncorrelated with the system

39

response (Andersen P. 1999). Subspace identification algorithms have the same

assumption and obtain a state space representation of the system as the outcome of the

identification procedure. A common mathematical background for subspace algorithms

for linear systems is presented by Overschee and De Moor (1996). The algorithm is

called Stochastic Subspace Identification (SSI) when the input of the system is stochastic

and cannot be measured. This methodology has gained popularity among researchers in

the last few years due to its easy application. Recently, Giraldo et al. (2006) compared

the SSI with other output only identification methodologies finding that the SSI performs

as well or better than other well known algorithms and requires fewer parameters and less

experience to set up.

Consider the discrete displacement and velocity response of a system governed by

Equation (38) as

[ ]TNk )()2()1()( uuuu K= (50)

[ ]TNk )()2()1()( uuuu &K&&& = (51)

where N is the number of data points. Defining the states as

[ ]Tkkk )()()( uux &= (52)

the discrete stochastic state space system can be written as

40

)()()1( kkk wxAx +=+ (53)

)()()1( kkk vxCy +=+ (54)

where A and C are the state matrices; w(k) and v(k) are two stochastic process known as

process noise and measurement noise; and x(k) and y(k) are the state vector and the

system response at the k-th step respectively.

For a Gaussian process the optimal predictor of x(k) is given by

[ ])1:0()()(ˆ −= kkEk yxx (55)

where ( ^ ) is used to indicate prediction. Thus, the optimal predictor is the conditional

mean of x(k) given the complete measured system response y at the interval between 0

and t-1. Likewise, the error is given by

)()(ˆ)( kkk xxε −= (56)

This error is included in x(k) and cannot be predicted by )(ˆ kx .

Taking the conditional expectation on both sides of the Equation (53) and (54) we have:

[ ] [ ] [ ])1:0()()1:0()()1:0()1( −+−=−+ kkEkkEkkE ywyxAyx (57)

[ ] [ ] [ ])1:0()()1:0()()1:0()( −+−=− kkEkkEkkE yvyxCyy (58)

41

Since the process and measurement noise are assumed to be statistically independent

from the past output, their conditional expectations are zero. This yields to

)(ˆ)1(ˆ kk xAx =+ (59)

)(ˆ)(ˆ kk xCy = (60)

Now if recursively Equation (60) is evaluated in (59) from an initial step q, a total of i

times, it yields

)(ˆ)1(ˆ qq i xACy =+ (61)

Defining the matrices O, Г and Xo as

( )( )

( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−+−

++−+

=

2212ˆ

1ˆ1ˆ

iq:q

iq:qiq:q

y

yy

OM

(62)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

−1qAC

ACC

ΓM

(63)

( )( )

( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−+−

++−+

=

2212ˆ

1ˆ1ˆ

iq:q

iq:qiq:q

x

xx

XoM

(64)

where the term ( )1ˆ −+ iq:qy is the matrix of predicted outputs from the step q to the step

42

q+i-1, for each channel (i.e. ( ) ( ) ( ) ( )[ ]1ˆ...1ˆˆ1:ˆ −++=−+ iqqqiqq yyyy ). Equation (61) can

be rewritten as

oXΓO = (65)

where the matrix O represents a bank of the predicted free response ( )ky for a set of

initial unknown conditions Xo (Brincker and Andersen. 2006). The matrix Xo is known as

the Kalman states. On the other hand, the measured output is collected in a block matrix

known as the Hankel matrix (Yh):

⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡+−

−

=hf

hph

N:s

sN:sN:

YY

y

yy

Y

)2(

)122()21(

M (66)

where 2s is the shift in data among the rows. So, the Hankel matrix has a total of 2s block

rows and N-2s columns. Each one of the block rows in the Hankel matrix has a total of M

rows, which corresponds to the number of channels available for the identification. The

upper half part of the Hankel matrix Yhp is called the past and the lower part Yhf is called

the future. Now, )(ˆ ky can be predicted projecting the future in the past, so that

[ ]hphfE YYOy ==ˆ (67)

43

For a Gaussian stochastic process this can be written as (Melsa and Sague 1973)

( ) hpT

hphpT

hphf YYYYYO1−

= (68)

The singular value decomposition of the matrix O is given by

TVSUO = (69)

Where the matrices U and V are square unitary matrices that represent the “output“ and

“input” basis vector directions for O respectively, and the matrix S is diagonal positive

matrix that contains the singular values of O. So, the matrix Г and the Kalman states Xo

are defined in terms of U, S and V as

2/1ˆ SUΓ = (70)

To VSX 2/1ˆ = (71)

Removing the first and the last block from Γ :

)2:1(ˆˆ)1:2(ˆ −=− qq ΓAΓ (72)

A and C can be found as

( ) 1)2:1(ˆ)2:1(ˆ)2:1(ˆ)1:2(ˆˆ −

−−−−= TT qqqq ΓΓΓΓA (73)

44

)1:1(ˆˆ ΓC = (74)

The natural frequencies if and damping ratios iς of the structure can be calculated by

performing an eigenvalue decomposition of A , so that:

[ ] 1ˆ −= ΦΦA μ (75)

The continuous time poles λi can be found as

T

ii Δ

=)ln(μ

λ (76)

where ΔT, is the time between samples in the discrete data. Natural frequencies and the

damping ratios are found as

π

λ2

iif = (77)

i

ii λ

λς

)Re(= (78)

where Re represents the real part of the complex number.

Finally the output shapes can be calculated as

45

ΦCΦ ˆ=id (79)

where and Φid is the matrix of identified Operating Deflection Shapes (ODS) (Brincker

and Andersen, 2006). ODS corresponds to the pattern of deformation of the structure at a

specific frequency. The contents of ODSs could include the participation of several mode

shapes. Additionally ODSs could be affected by the modal damping and the input load

itself, creating small differences between ODSs and mode shapes. For structures with low

damping ratio and broad band random excitation the ODSs will be very similar to the

mode shapes of the structure. In this thesis the ODSs will be considered to be equal to

the mode shapes. The matrix Φid will have as many rows as sensors are in the structure

and as many columns as ODS are identified. The coordinate of the operating deflection

shape is identified at the location of the sensor. Notice that the matrices A and C are sM

by sM and M by sM respectively. Thus, the model order (i.e. number of poles identified)

is given by sM.

3.2. Model Updating

Considering that the assumptions made for modal identification hold and the

uncertainties (e.g. noise and signal processing errors) due to the identification process are

small, it can be said that the differences between an analytical and physical model are

caused by uncertainties in the numerical model. Adjustments in the stiffness, mass and

46

damping matrix should be performed in order to enhance the numerical response of the

finite element model and relate it to the physical structure. This process is called model

updating.

Uncertainty in finite element models can be caused by three main sources: i) not enough

discretization in the model or using elements that do not correctly describe physical

model, ii) uncertainty in model parameters (e.g. material and section properties), and iii)

indetermination of the boundary conditions (Brownjohn and Xia 2000). Model updating

can be performed by either direct changes in the mass and stiffness matrix (Caicedo et al.

2004) or by iterative parameter updating (Jaishi and Ren 2005), which with the correct

constraints result in a meaningful model (Ewins 2000). This thesis discusses only

parameters updating and boundary conditions and assumes that the mathematical model

can correctly reproduce the dynamics of the structure.

Two model updating strategies are studied in this thesis. First, a classical model updating

process is performed by adjusting the parameters of the finite element model such that the

error in the natural frequencies and mode shapes between identified and numerical

parameters is minimized. The flow diagram in Figure 10 summarizes this strategy. Either

global or local optimum that minimizes the difference between the analytical and real

model could be found as a result of this updating process. The objective function is

defined as a linear combination between the errors in the natural frequencies and the

47

mode shapes (quantified trough the MAC value), as follows

( )[ ]∑= ⎥

⎥⎦

⎤

⎢⎢⎣

⎡ −+−=

n

i iid

ifeiidifeiidf

1 ,

,,,,

)()(,MAC1)(

ωωω

φφp

pp (80)

where denotes absolute value, n is the number of identified modes, MAC( iid ,φ , )(, pifeφ )

is the modal assurance criteria between the i-th identified mode shape ( iid ,φ ) and the i-th

mode shape of the finite element model )(, pifeφ , iid ,ω is the i-th identified natural

frequency, )(, pifeω is the i-th natural frequency of the finite element model and p is the

vector of parameters to be optimized.

Constrain is applied to the procedure to assure that the variation of the parameters is not

higher than reasonable limits.

One of the challenges with this classical method is that other minima with a performance

similar to the solution found in the global optimization might exist and will be not

Figure 10. Flow diagram of the primary updating process

Real Structure Measured

Ambient Vibration

FE Model (p)

Dynamic Characteristics

[ wi øi ]

Adjust (p) J < tol

+ -

Identification: Dynamic characteristics

[ wi øi ]

YES NO

Model Updated

48

identified. One of these solutions might represent the optimal solution for the problem, if

the number of non-quantifiable variables that were not enlisted in the original objective

function, or the plain uncertainty in the model, are now considered. Depending upon the

final application of the finite element model, all the models obtained can be considered or

only one model chosen as the “best” solution. For instance, in a study of structural

control all the models can be considered to test the stability of the controller; or in

structural health monitoring, a single model can be selected based on additional

observations or the inspector’s experience. Modeling to Generate Alternatives (MGA) is

used in this thesis to determine these additional solutions and it is described in the

following section.

3.2.1. Modeling to Generate Alternatives

MGA was developed with the goal of providing solutions to complex, incomplete

problems by coupling the computational power of computers and human intelligence

(Brill et al. 1990) in a human-computer cognitive system (Baugh et al. 1997). MGA

creates several possible good solutions from a problem by eliminating alternatives with

poor performance using a mathematical model. These solutions are designed to be

maximally physically different but provide a similar outcome to the problem. MGA has

been previously applied to forest level planning (Campbell and Mendoza 1988; Sprouse

and Mendoza 1990), the seismic design and evaluation supports for pipes (Gupta et al.

49

2005), structural optimization (Baugh et al. 1997), air quality management problems

(Loughlin et al. 2001), airline route design (Zechman and Ranjithan 2004) and the design

of wastewater-treatment plants (Uber et al. 1992). In addition, a similar methodology

was used in (Rubenstein-Montano et al. 2000) where genetic algorithms were used to

create different alternatives for policy design.

A nonlinear variation of the Hop, Skip and Jump method (HSJ) proposed by (Brill et al.

1982) and used for a land use planning problem is used in this thesis and it has three main

parts: i) find a solution based on the base finite element model (i.e. traditional solution

described before), ii) Obtain alternative solutions, and iii) identify if additional local

minima are close to the solutions obtained in the previous step. For all steps a Sequential

Quadratic Programming (SQP) algorithm for non-linear programming (Shanno 1970; Gill

et al. 1981) available in the Matlab optimization toolbox was used for minimization.

These three steps are explained in the following paragraphs.

First Step

The first part of the updating process consists of finding an initial solution that will be a

start point in the search of additional solutions. The objective function is described by the

Equation (80), and repeated here for convenience

( )[ ]∑= ⎥

⎥⎦

⎤

⎢⎢⎣

⎡ −+−=

n

i iid

ifeiidifeiidf

1 ,

,,,,

)()(,MAC1)(

ωωω

φφp

pp

50

Second Step

The hypothesis in this thesis is that the results obtained from the first optimization will

provide a solution to the problem but this solution might not be unique. To find additional

solutions a search in a perpendicular direction of the original solution is performed as

shown in Figure 11. A process similar to HSJ is used where a new objective function is

minimized to obtain the l+1 solution. This function is defined as

∑= +

++

⋅=

l

j lj

ljlg

1 1

11 )(

pp

ppp (81)

where pj is the j-th solution, )(⋅ denotes dot product and denotes the norm. The value

of this function is high if the cross product between the current solution and previous

solutions is close to one and is low if the new solution is perpendicular to any other

solution found before (i.e. cross product is zero).

p1

p2

Figure 11. Feasible region of a two-variable nonlinear programming problem

First solution found

Second solution found

51

Given that any orthogonal vector does not always provide a good fit for the finite element

model, constraint are used to assure that the new solution has a similar performance to the

first solution found. The constraint used for this optimization are described by the

equation

)()( 1pp ff 1l α≤+ (82)

where ( )1pf is the value of the objective function shown in Equation (80) for the first

Figure 12 Flow diagram of the complete updating process

Dynamic Characteristics [ wi øi

]

Adjust (p) J < tol

+-

Identification: Dynamic characteristics

[ wi øi ]

YESN

J < tol

YES

N

FE Model

Angle cosine

Between pi and pi+1

pi

p i+1

Family of Solutions

52

solution found, ( )1+lf p is the value of Equation (80) for the current solution and α is 1.3.

The result from this second step provides solutions to the problem that are significantly

different but have a similar performance. A flow diagram of the procedure is shown in

Figure 12.

Third Step

This step is optional and uses the objective function in Equation (80) to refine the search

of local minima. The command fmincon in Matlab is used to minimize the objective

function using each solution found as initial conditions.

3.2.2. Numerical Verification of the Nonlinear HSJ Method

In order to demonstrate the potential of the nonlinear HSJ method, a simple test problem

is presented. The problem consists of the three degree of freedom structure shown in

Figure 13. The stiffness of the members 1, 2, 3 and 4 are given by k1, k2, k3 and k4

respectively, likewise the lumped nodal masses are m1, m2 and m3.

Let Δk and Δm be uncertain parameters in the stiffness and mass respectively. So that, the

k1 k2 k3 k4

m1 m2 m3

Figure 13 . Bill Emerson Memorial Bridge

53

member stiffness and nodal masses are given by

kk Δ+= 511 (N/m) (83)

12 =k (N/m) (84)

53 =k (N/m) (85)

kk Δ+= 14 (N/m) (86)

mm Δ+= 3101 (kg) (87)

mm Δ+= 102 (kg) (88)

mm Δ−= 3103 (kg) (89)

The complete stiffness and mass matrix of the structure are calculated as

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−−+−

−+=

433

3322

221

0

0

kkkkkkk

kkkK (90)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

3

2

1

000000

mm

mM (91)

The nonlinear HSJ is used to adjust the parameters Δk and Δm for the targeted natural

frequencies of ω1= 0.6 rad/sec and ω2 = 1 rad/sec. The first objective function to

minimize is given by the averaged error in the natural frequencies as follows

1)(1

5.06.0

)(6.05.0)( 21 pp

pωω −

+−

=f (92)

54

where ω1(p) and ω2(p) are the first and second natural frequency of the structure

calculated in terms of Δk and Δm. Figure 15 shows the plot corresponding to Equation

(92). The parameters Δk and Δm are constraint to be

60 ≤Δ≤ k (93)

3.33.3 ≤Δ≤− m (94)

The methodology found an initial solution of Δk = 6 N/m and Δm = -2.0772 kg, when the

starting point Δk = 0 N/m and Δm = 0 kg is used. Additional solutions are searched by

minimizing the objective function (step 2 of the methodology)

∑= +

++

⋅=

l

j lj

ljlg

1 1

11 )(

pp

ppp (95)

where pj is the j-th solution, )(⋅ denotes dot product and denotes the norm.

Additionally the new solutions are not allowed to increase Equation (92) more than 1.3

times the value of the first solution using the constraints:

)(3.1)( 1pp ff 1l ≤+ (96).

which yields to the new solution Δk = 3.5599 N/m and Δm = 3.3 kg, as expected

according to Figure 15.

55

Table 5 summarizes the solutions for the methodology described above, with different

initial value for the minimization. The first row in Table 5 are the solutions found when

the parameters Δk=0 N/m and Δm=0 kg were used, and the second row represents the

solutions for Δk=3 N/m and Δm=1 kg as starting point for the optimization. Notice that

the solutions are the same for both cases, although in different order and agree with the

contour plot shown in Figure 15.

Start Point First Solution Second Solution Δk (N/m) Δm (kg) f(p) (%) Δk (N/m) Δm (kg) f(p) (%) 1 6 -2.0772 8.4297 3.5599 3.3 8.4229 2 3.5599 3.3 8.4229 6 -2.0772 8.4297

Table 5. Adjusted parameters Δk and Δm found using the nonlinear HSJ methodology proposed.

Figure 15. Three dimensional objective function plot (Equation (92) Figure 15. Three dimensional objective function plot (Equation (92)

Solution

56

4. Application: Cable Dynamics

This chapter presents the modeling of the Bill Emerson Memorial Bridge using the

methodologies described in chapter two. One finite element model is created for each of

the described cable methodologies. The effect of the cable model on the decks dynamics

is explored by comparing the motion of the deck using MAC and FRAC.

4.1. Description of the Bill Emerson Memorial Bridge

The Bill Emerson Memorial Bridge opened to traffic on December of 2003. The bridge

crosses the Mississippi river on Cape Girardeau Missouri, USA and it is composed of two

towers, 128 stay cables and 4 main piers. The bridge has a main span of 350.6 m (1150

ft) and two side spans of 142.7 m (468 ft), for a total length of 1205.8 m (3956 ft), as

shown in the Figure 16. The deck has a width of 29.3 m (96 ft) and is composed by

girders of ASTM A709 grade 50W steel (fy of 50 ksi) and prestressed concrete slabs

Ben t 1

(1150’) (468’) (1870’)

Pier 2 Pier 3 Pier 4

(468’)

I llinois a pproac h

x··g

33321 64

1 Cable Number

142.7m 350.6m 142.7m 570.0m

Figure 16. Bill Emerson Memorial Bridge

57

(Figure 17) with a fc’ of 41.36 Mpa (6 ksi). Additionally the bridge deck counts with

sixteen shock transmission devices in the connection with the towers and bent. The

purpose of those devices is to restraint the longitudinal movement during seismic

excitation, behaving extremely stiff, but to permit the longitudinal expansion of the deck

caused by changes in temperature. Moreover earthquake restrainers are used in the

transverse direction at the connections between the deck and the towers, and the deck is

restraint on the vertical direction at the towers.

The cables are made of high-strength, low-relaxation steel (ASTM A882 grade 270) and

covered with a polyethylene piping to prevent corrosion. The area of the cables varies

Floor Beam

BarrierConcrete Slab

Cable(96’)

Steel girder

Rail ing Anchorage

29.3m

Figure 17. Cross section of the deck

Section ASection B

Section C

Section E

3.96 m6.71 m

4.88 m6.71 m

1.68 m2.74 m

1.68 m

Var. 2.74 m

Section A

Section B

Section C

Section D

Section E

to 2.59 m

to 3.66 m

3.96 m5.18 m

0.61 m

Var. 3.05 to 6.21 m

Section D

6.71 m

3.96 m

3.66 m 30.8 m 3.66 m 3.66 m Var. 22.0 m min 3.66 m

3.81 m

6.71 m

Dec k

Shear force response locations

Figure 18. Cross sections of the towers

58

between 28.5 cm2 (4.41 in2) and 76.3 cm2 (11.83 in2). The towers are H-shaped with a

height of 102.4 m (336 ft) at pier 2 and 108.5 m (356 ft) at pier 3. The towers are of

reinforced concrete with a resistance fc’ of 37.92 Mpa (5.5 ksi) and the cross section

varies 5 times along the towers, as shown in Figure 18.

4.1.1. Finite Element Model

The finite element model shown in Figure 19 is composed of 575 nodes, 156 beam

elements, 128 cable elements and 418 rigid links. The movements of the deck in lateral

and vertical direction as well as the rotation with respect to the X axis at bent 1 and piers

2, 3 and 4 are restricted by applying constraint equations. At pier 1 only displacement in

the X axis and rotations about the Y and Z axis are allowed. The model of the structure

does not include the Illinois approach as shown in Figure 19, because neither the

displacements nor the rotations about the X axis are restraint at pier 4 disconnecting the

dynamics of the two systems. The soil structure effects are considered negligible in this

study and the structure is assumed directly fixed to the foundation. The finite element

Figure 19. Finite element model of the Bill Emerson Memorial Bridge

59

model of the bridge was first developed in Abaqus for the benchmark problem of control

of cable-stayed bridges under earthquake excitation (Dyke et al. 2003) and later

converted to Matlab for studies in structural health monitoring (Caicedo 2003).

4.2. Results

4.2.1. Tension Distribution (Static Analysis)

The tensions of the stayed cables of the Emerson Bridge calculated using the three cable

models previously described in chapter 2. The zero displacement method given by Wang

et al. (1993) was followed to determine the final tension of the cables. The methodology

consists of the following steps:

i) Form a numerical model of the bridge using cable-tension found in the blue

prints.

ii) Perform a static nonlinear analysis, finding deck’s deformation and final cable

tensions.

iii) If the deformation of the deck is larger than a tolerance, repeat step i) using the

cable tensions found in step ii) as original tensions.

60

The distribution of the final tensions of the stayed cables along the deck is shown in

Figure 20. The methodologies of the Equivalent Elasticity, Elastic Catenary and

Isoparametric Elements required 16, 15 and 16 iterations respectively to achieve this final

set of tensions. Only half of the cable tensions appear in the Figure 20, since the

distribution of tensions is symmetrical about each side of the deck. The distribution of

cable tensions along the deck tends to be symmetric along the longitudinal axis, even

thought the structure is not entirely symmetric in this direction, mainly due to differences

in the boundary conditions at the supports (i.e. bent 1 and pier iv). Figure 20 shows a

decrement of the cable tensions related to the initial tensions assumed, indicating that

after each iteration (i.e. nonlinear procedure) cables are losing tension and therefore the

structure is gaining flexibility. The results shown in , does not show an appreciable

difference among the methodologies.

Figure 20. Cable tension distribution along the deck, using one element per cable

61

4.2.2. Frequencies and Mode Shapes

The natural frequencies and corresponding modes shapes of the Emerson Bridge are

computed using 1 and 4 elements per cable to evaluate the effect of the different

methodologies in the dynamic characteristic of the structure. Table 6 shows the first 20

natural frequencies of the Emerson Bridge with a single element per cable using the

stiffness matrix before and after the nonlinear analysis described in the previous section.

Before Nonlinear Analysis After Nonlinear Analysis

Freq. Equivalent Elas. (Hz)

Elastic Cat. (Hz)

Isopa. F. (Hz)

Equivalent Elas. (Hz)

Elastic Cat. (Hz)

Isopa. F. (Hz)

1 0.2915 0.2977 0.2999 0.2786 0.2843 0.2869 2 0.3895 0.394 0.3965 0.373 0.3766 0.3797 3 0.4897 0.5005 0.5025 0.4873 0.4971 0.4991 4 0.5433 0.5513 0.5536 0.5419 0.5487 0.5515 5 0.6106 0.6095 0.6145 0.5927 0.5896 0.5964 6 0.6762 0.6759 0.6796 0.6601 0.6579 0.6632 7 0.713 0.7159 0.7162 0.7097 0.7118 0.7127 8 0.7481 0.7476 0.7511 0.7294 0.7272 0.7322 9 0.8112 0.809 0.8177 0.8085 0.8042 0.8147 10 0.8852 0.8854 0.8926 0.8775 0.8758 0.8799 11 0.9028 0.9023 0.9053 0.8831 0.8811 0.8901 12 1.0331 1.0321 1.0368 1.0257 1.0244 1.028 13 1.0341 1.0333 1.0377 1.0301 1.0283 1.0343 14 1.05 1.0605 1.0614 1.031 1.0295 1.0351 15 1.0545 1.0612 1.0631 1.0338 1.0442 1.0435 16 1.0608 1.0656 1.0658 1.0373 1.0484 1.0477 17 1.0901 1.086 1.0973 1.0704 1.0696 1.0726 18 1.106 1.1062 1.1083 1.0898 1.0818 1.0965 19 1.1399 1.1401 1.142 1.1062 1.1055 1.1082 20 1.22 1.2303 1.2314 1.1857 1.1988 1.1966

62

Table 6. First 20 natural frequencies (hz) of the cable-stayed bridge, using 1 element per cable

The natural frequencies for the linear static analysis shown in Table 6 are similar among

the Elastic Catenary and Isoparametric Formulation for both before and after nonlinear

analysis. However, the frequencies for the Equivalent Elasticity are slightly lower than

the other methodologies, which imply a lower stiffness in the final model. These results

agree with previous analysis performed by Karoumi (1999) who found that the

Equivalent Elasticity model results in slightly lower stiffness than the Isoparametric

formulation and the Elastic Catenary for a single cable comparison. It can also be seen

that the frequencies are similar between the linear and the nonlinear analysis, with

differences no larger than 5%. This indicates that the blue drawings load state (i.e. cable

tensions before nonlinear procedure) is located in the linear section of the load-

deformation curve of the structure.

The natural frequencies of the structure are usually expected to increase after a nonlinear

analysis as a result of an increment in the stiffness of the elements due to the interaction

of the cables pretension load and the self weight of the deck. This occurs when the

original tension of the cables is less than the tension after the analysis. In this case, the

initial cables tensions of the structure are slightly higher compared with the tensions

solicited to stand the deck. Therefore, the structure looses stiffness as a result of a

reduction of the cable tensions and the natural frequencies slightly decrease.

63

The mode shapes were also compared. MACs were computed for each one of the first 20

mode shapes among the different models, generating a 20x20 matrix for each

methodology comparison, as shown in Figure 21. Here, each row represents a mode

shape of a specific methodology and each column a mode of another methodology. A

gray scale system is used in the Figure 21 to represent the values in such matrix: dark

squares signify values close to one (similar modes) and light places values close to zero

Figure 21. MAC value for the first 20 mode shapes after nonlinear procedure among the different methodologies, using 1 element per cable

64

(different modes). The first 20 mode shapes are the same and in the same order for the

Isoparametric and Catenary methodologies, but modes 13th and 14th differ from

Equivalent - Catenary and Equivalent - Isoparametric.

Figure 22 shows the 13th and 14th mode shapes of the Catenary, Isoparametric and

Equivalent Elasticity methodologies. It can be seen that both 13th and 14th modes

corresponds mainly to a torsional mode of the deck with a small movement of the towers

in all three methodologies. MAC indicates a low correlation between Equivalent

Elasticity and both Elastic Catenary and Isoparametric Formulation for both 13th and 14th

Equivalent 13 Equivalent 14

Figure 22. 13th and 14th mode shape of Catenary, Isoparametric and Equivalent formulations

Catenary 13 Catenary 14

Isoparametric 14 Isoparametric 13

65

modes. Figure 22 confirm this observation.

The same calculations were performed using 4 elements per cable. Table 7 shows the first

20 natural frequencies. A MAC matrix was also used to compare the modes of models

with four elements per cable and models with one element per cable. The first 300 modes

of models with 4 elements per cable and the first 20 modes of models with one element

per cable were used as shown in Figure 23. Since the model using 4 cables per element

have more degrees de freedom that the single element model, zeros were added to the

single cable element model in the missing degrees of freedom of the mode shape.

As expected, cable subdivision causes new frequencies to appear. Here blank spaces

represent modes that were not identified in the other models. These mode shapes

correspond to cable vibration and small deck and towers movement. Figure 23 was used

in order to identify the corresponding frequencies from the 4 cables model into the single

cable model.

Figure 23. MAC value between Isoparamteric formulation using 1 and 4 elements per cables and Catenary Element using 1 and 4 elements per cable

66

Before Nonlinear Analysis After Nonlinear Analysis

Freq.

Equivalent Elas. (Hz)

Elastic Cat. (Hz)

Isopa. F. (Hz)

Equivalent Elas. (Hz)

Elastic Cat. (Hz)

Isopa. F. (Hz)

(1 cable) (4 cables) (4 cables) (1 cable) (4 cables) (4 cables) 1 0.2915 0.2993 0.2995 0.2786 0.2869 0.285 2 0.3895 0.3959 0.396 0.373 0.3794 0.3759 3 0.4897 0.4911 0.4912 0.4873 0.4844 0.4944 4 0.5433 0.5486 0.5484 0.5419 0.5412 0.5365 5 0.6106 0.6066 0.6068 0.5927 0.5859 0.5942 6 0.6762 - - 0.6601 - 0.6617 7 0.713 - - 0.7097 - - 8 0.7481 - - 0.7294 0.373 - 9 0.8112 0.8538 - 0.8085 - 0.8213 10 0.8852 0.9386 0.9388 0.8775 0.8819 0.8797 11 0.9028 - - 0.8831 0.928 0.9024 12 1.0331 1.0579 1.0582 1.0257 - 1.0277 13 1.0341 1.0587 1.059 1.0301 - 1.0378 14 1.05 1.0683 - 1.031 - 1.0388 15 1.0545 - 1.0684 1.0338 - - 16 1.0608 - - 1.0373 - - 17 1.0901 - - 1.0704 1.0769 1.0725 18 1.106 - - 1.0898 1.1716 1.0876 19 1.1399 - - 1.1062 1.1051 1.1077 20 1.22 1.2649 - 1.1857 - -

Table 7. First 20 natural frequencies (hz) of the cable-stayed bridge, using 4 elements per cable

4.2.3. Transfer Functions

Transfer Functions were also used to compare the differences in the dynamic behavior

among the cable methodologies. In contrast to natural frequencies and mode shapes TFs

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describe the behavior of the structure in a wide range of frequencies. A state space model

with ground acceleration as inputs and displacements in the Z direction at key locations

on the deck and tower as outputs was developed for this comparison. These locations (see

Figure 24) correspond to: i) mid span between bent 1 and pier II (node 10), ii) the quarter

point of the main span (node 172), iii) the half point of the main span (node 181), and iv)

the top of the first tower (node 93). Three different angles of arrival of the earthquake

were used: i) along the X axis, ii) along the Y axis, and iii) at an angle of 45o from the X

axis. A frequency range between 0 and 50 Hz was chosen for the calculations. Figure 25

to Figure 28, show the TFs using a single element per cable when the ground acceleration

is parallel to the deck (X axis), which is the case of higher discrepancy among the

methodologies. Figure 25 to Figure 28 show no appreciable difference among the

methodologies for all the locations in the range from 0 to 50 Hz. Only a small difference

can be noticed in the range between 0 to 1.5 Hz at node 181. Here is clear that Elastic

Catenary is the methodology that produces the least displacement at 0 Hz. The FRAC

numbers were calculated for several frequency ranges. The FRACs in Table 8 are larger

than 0.96 corroborating numerically the similitude among the methodologies in the range

Figure 24. Locations at which the Transfer Functions were calculated at the Emerson Bridge

Node 10

Node 93

Node 172

Node 181

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from 0 to 1.5 Hz. Even though from Table 8 that the differences among the

methodologies are very small, the Equivalent Elasticity is slightly different than the other

methodologies (lowest FRACs).

Figure 25. Transfers Functions for all three methodologies at node 181 when ground acceleration is parallel to the deck and acceleration response is measured, using 1

element per cable

69

Figure 27. Transfers Functions for all three methodologies at node 10 when ground acceleration is parallel to the deck and acceleration response is

measured, using 1 element per cable

Figure 27. Transfers Functions for all three methodologies at node 93 when ground acceleration is parallel to the deck and acceleration response is measured,

using 1 element per cable

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Equivalent Elasticity

Catenary Element

Isoparametric Formulation

Node 10

Equivalent Elasticiy 1.0000 0.9954 0.9857 Catenary Element 0.9954 1.0000 0.9967 Isoparametric F. 0.9857 0.9967 1.0000

Node 172

Equivalent Elasticiy 1.0000 0.9956 0.9856 Catenary Element 0.9956 1.0000 0.9967 Isoparametric F. 0.9856 0.9967 1.0000

Node 181

Equivalent Elasticiy 1.0000 0.9688 0.969 Catenary Element 0.9688 1.0000 0.9875 Isoparametric F. 0.969 0.9875 1.0000

Node 93

Equivalent Elasticiy 1.0000 0.9937 0.9945 Catenary Element 0.9937 1.0000 0.9997 Isoparametric F. 0.9945 0.9997 1.0000

Table 8. FRAC value for the transfer functions in the 0 to 1.5 hz range, among the methodologies in the same location.

Figure 29 to Figure 32 show the Transfer Functions computed using the Elastic Catenary

Figure 28. Transfers Functions for all three methodologies at node 172 when ground acceleration is parallel to the deck and acceleration response is

measured, using 1 element per cable

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and the Isoparametric Formulation with 4 elements per cable and the Equivalent

Elasticity with a single element per cable. These plots show the case when the ground

excitation is transverse to the deck, which is the case of larger discrepancy among the

methodologies. The Figures show differences among the methodologies in the entire

frequency range from 0 Hz to 50 Hz. FRACs for the Transfer Functions in the 0 to 1.5 hz

range are displayed in Table 9. Here the lowest FRACs are between Equivalent elasticity

and both Elastic Catenary and Isoparametric Formulation. Hence, the deck and tower of

models using Isoparametric formulation and the Elastic Catenary, both subdivided in 4

elements have significantly different behavior from models using the Equivalent

Elasticity with a single element. Additionally in Figure 29 to Figure 32 some differences

between the Elastic Catenary and the Isoparametric Formulation can be observed.

Figure 29. Transfers Functions for all three methodologies at node 10 when ground acceleration is transverse to the deck and acceleration response is

measured, using 4 cables per element.

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Figure 31. Transfers Functions for all three methodologies at node 93 when ground acceleration is transverse to the deck and acceleration response is measured, using 4

cables per element

Figure 31. Transfers Functions for all three methodologies at node 172 when ground acceleration is transverse to the deck and acceleration response is

measured, using 4 cables per element

73

The TFs that correspond to cable-deck interaction (Figures 29, 31 and 32) showed larger

differences than those of cable-tower interaction. So that deck dynamics is more

susceptible to cable modeling tower dynamics. This can be explained due to the higher

stiffness of the towers.

The distinctive behavior of the methodologies when the cables are subdivided related to a

single element per cable in the frequency range from 0 to 50 Hz, shows that cable

subdivision not only adds new local modes of vibration to the structure but changes the

overall dynamics of the structure. Therefore, cables should be subdivided to characterize

Figure 32. Transfers Functions for all three methodologies at node 181 when ground acceleration is transverse to the deck and acceleration

response is measured, using 4 cables per element

74

the right cable-deck and cable-tower interaction. Regarding the differences between the

4 elements per cable models and Equivalent Elasticity we can say that it is caused by the

cable shear stiffness that is not taken in to account in Equivalent Elasticity, which gains

importance when the cable is subdivided. This implies that considering the cable sag is

not enough to represent the stiffness of the cable. Furthermore, some differences in the

dynamic behavior between the Elastic Catenary and the Isoparametric Formulation both

subdivided in 4 elements were found. These differences in the dynamic behavior can be

caused because Elastic Catenary considers both the sag effect and the large displacements

effect, but the Isoparametric Formulation only the large displacements effect.

Equivalent Elasticity

Catenary Element

Isoparametric Formulation

Node 10 Equivalent Elasticiy 1.0000 0.4965 0.7627 Catenary Element 0.4965 1.0000 0.7561 Isoparametric F. 0.7627 0.7561 1.0000

Node 172 Equivalent Elasticiy 1.0000 0.1703 0.2314 Catenary Element 0.1703 1.0000 0.9794 Isoparametric F. 0.2314 0.9794 1.0000

Node 181 Equivalent Elasticiy 1.0000 0.0411 0.0500 Catenary Element 0.0411 1.0000 0.9850 Isoparametric F. 0.0500 0.9850 1.0000

Node 93 Equivalent Elasticiy 1.0000 0.9008 0.9002 Catenary Element 0.9008 1.0000 0.9997 Isoparametric F. 0.9002 0.9997 1.0000

Table 9. FRAC value for the transfer functions in the 0 to 1.5 Hz range, among the methodologies

in the same location, using 4 elements per cable

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5. Application: Identification and Updating

This chapter presents an application example of the system identification and model

updating theory explained in chapter 3. Ambient vibration data obtained from the actual

Bill Emerson Memorial Bridge is used for system identification. Dynamic parameters are

obtained from this data using SSI. The finite element model of the Emerson Bridge that

includes the Elastic Catenary with 4 elements per cable described in chapter 4 is updated

to match the identified dynamic characteristics.

The Emerson Bridge is permanently instrumented with acceleration sensors distributed

along the structure and surrounding soil (Çelebi 2006). A total of 84 accelerations

channels of Kinemetrics EpiSensor have been installed in the bridge as shown in Figure

Figure 33. Bill Emerson Memorial Bridge Instrumentation (Çelebi 2006).

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33. Additionally the bridge is equipped with Q330 digitizers and data concentrator, and

mass storage devices (Balers) with wireless units. The acquisition system has the ability

to either acquire data in case of a special event as an earthquake using a trigger algorithm,

or to acquire long records of low frequency ambient vibrations in a schedule way. The

analog signal from the accelerometers is digitalized at the bridge and then sent by

wireless communication to a Central Recording System (CRS). Once at the CRS, the data

are transmitted to the Incorporated Research Institutions for Seismology (IRIS), and it is

finally broadcasted trough internet.

5.1. Identification Process

A system identification of the Emerson Bridge is performed, using the Stochastic

Subspace Identification and acceleration records obtained from the bridge

instrumentation. In this thesis a total of 25 acceleration channels were used, 17 of these

acceleration channels are located on the deck in the vertical direction and 8 acceleration

channels located on the towers oriented in the longitudinal direction. The sensor locations

were chosen to identify the dynamic characteristics of the structure corresponding to the

vertical mode shapes. Song et al. (2006) previously performed identification of the bridge

using ARMAV models. In his paper the first 5 natural frequencies and vertical mode

shapes were identified, using only 16 vertical channels of acceleration corresponding to

the sensors on the deck.

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5.1.1. System Identification

Six hours of acceleration data were used for the system identification. The first three

hours on November 2, 2005 and the last three were obtained on December 5, 2005. Data

was originally collected at 200 Hz, and resampled to 2 Hz to capture the fundamental

modes of the structure (lower than 1 Hz). This frequency was chosen based on the results

found by Song et al. (2006) and the dynamic properties of the preliminary finite element

model. Resampled was performed using the Matlab command resample which applies a

low pass filter to eliminate aliasing. The data were divided in 36 windows of 10 minutes

for a total of 1200 data points per channel per window. SSI was performed in each

window, so that a set of 36 natural frequencies, mode shapes and damping rations are

obtained for the complete record. The Hankel matrices used had a total of 750 rows and

1170 columns, which allowed determining a maximum of 375 poles.

As a result of the overestimation of the size order of the system, non-physical system

poles arise. In cable-stayed bridges is challenging to differentiate these non-physical

poles from the physical poles, because mode shapes are closely-spaced. To determine the

true poles, an additional step in the data processing is needed using stabilization diagrams

to visualize the frequencies that have been detected consistently (Brincker and Andersen.

2006). Stabilization diagrams show the natural frequency of the poles for each window

(see Figure 34). True frequencies are detected based on the consistency trough all the

windows. Previous knowledge of the structure is fundamental for the success of the

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system identification process. For instance, it is well known that damping of cable-stayed

bridges for the frequency range in consideration is under 5%. Additionally, since only the

first four modes are targeted to be identified only frequencies bellow 1 Hz are admitted.

An automated recognition system to detect the true poles from those created by noise and

numerical errors was used (Giraldo 2006; Giraldo et al. 2006). True poles are recognized

by identifying parameters within some specific characteristics. The parameters used to

identify real modes are the natural frequency and the MAC and damping values

associated with each pole as shown in chapter 3. The acceptable identified poles should

have natural frequencies within 30% of the mean of previously identified frequencies,

have damping value lower than 5%, and the MAC values of the corresponding mode

shapes should be higher than 0.98 when compared to previously accepted poles. Only

poles with more than 5 hits, or identified five times with these parameters in different

windows are assumed as real modes and the other are discarded as numerical modes. The

circles in Figure 34 show the raw identified modes and the stars indicate the acceptable

Figure 34. Identified natural frequencies

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

Frequency (Hz)

Win

dow

79

modal parameters. The standard deviation of each remaining natural frequencies (stars in

Figure 34) and the mean MAC value of the corresponding mode shapes were also

calculated.

Mode Hits Frequency Hz

Std Frequency

Hz

Damping %

Std damping % MAC Std MAC

1 31 0.324 0.0036 1.05 0.37 0.9977 0.0019 2 16 0.413 0.0026 0.64 0.09 0.9967 0.0023 3 25 0.635 0.0041 0.67 0.20 0.9983 0.0009 4 21 0.706 0.0032 0.73 0.51 0.9962 0.0046

Table 10. Dynamic Characteristics Identified

Table 10 summarizes the identified modal parameters of the Emerson Bridge. The natural

frequencies and damping ratios are calculated as the average of all the identified

parameters for each mode. The standard deviation for each parameter is also shown in the

table. The first 4 mode shapes targeted have a low covariance in terms of the natural

frequencies, damping ratios and mode shapes. Therefore, these dynamic characteristics

were consistently identified, and have a low statistical dispersion, even when using data

records from two different dates. This indicates that the system is behaving linearly for

ambient vibrations.

Table 11 shows a comparison of the natural frequencies identified by Song et al. (2006)

using an ARMAV model and the natural frequencies found in this study. Good agreement

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was found between the two methodologies.

Modal Frequency

Song et al. (2006) (Hz)

This Study (Hz)

1 0.33 0.32 2 0.42 0.41 3 0.63 0.64 4 0.71 0.71

Table 11. Comparison of natural frequencies between this study and Song et al. (2006)

5.2. Bill Emerson’s Model Updating

The finite element model of the Emerson Bridge described in chapter 4 was updated

based on the information obtained in the previous section. Cables are modeled using the

Elastic Catenary methodology subdivided in 4 elements per cable. This cable model was

chosen because it includes the lateral stiffness and therefore assumed to be more accurate

model of the bridge. As in chapter 4, the design tension obtained from construction

drawings are assigned to each cable and a non-linear static analysis of the bridge under

static load is performed to calculate cable tension after equilibrium is reached.

Translational linear springs were added to the finite element model between the deck and

the towers to represent the stiffness of the connection.

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5.2.1. Parameters to Update

Properties of the numerical model were updated such that its modal characteristics would

match as close as possible the identified natural frequencies and mode shapes. The type

of parameters to adjust were chosen because of their influence of the dynamic of the

finite element model. Three different types of parameters were considered for updating: i)

mass of the deck, ii) the rotational stiffness of the connection between the deck and the

towers, and iii) moment of inertia of the spine beam. All the parameters were lumped into

a total of six variables divided in three distinctive groups.

Mass of the Deck

A total of 66 translational lumped are used to model the deck. Assuming that the mass

distribution is symmetric due to the symmetry of the bridge this can be reduced to 33

parameters to optimize which is still a large number. Given that it is unlikely to have a

sudden change in the mass along the deck the number of parameters was reduced to 3

master masses at 3 locations along the deck. The other 30 masses were calculated using a

spline between the 3 master masses. The masses at the mid span between bent 1 and pier

2, and one quarter and one half the main span were selected as the location for the 3

master masses (Figure 35). Additional masses were constrained to be in the range of -5%

to 5% of the mass calculated from the blue prints.

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Connections between Deck and Towers

The degree of connection between the deck and the towers is highly uncertain because of

the shock absorbers and the restrainers installed at these locations. The stiffness of the 4

springs used to model these connections constitute the second set of parameters to update.

Using the symmetry of the bridge one parameter was used for bent I and Pier IV and a

second parameter was used for the connection at the towers. Thus, a total of two

parameters were used to update the connection between the deck and the supports. The

stiffness of the springs were constraint to be in the range of -100% and 100% of the

higher stiffness found in the stiffness matrix.

Moment of Inertia of the Spine Beam

In order to reduce the error contributed for this parameter, the moment of inertia about

the transverse direction (Y axis) was adjusted. The variation of the moment of inertia is

assumed to be constant along the deck, so that only one parameter was used to update the

deck’s stiffness. The moment of inertia is constrained to be in the range of -5% to 5% of

the moment of inertia calculated from the blue prints.

83

5.2.2. Application of the Nonlinear HSJ Method Proposed

First Step – First solution

The starting point for this first optimization was the original numerical model as

calculated from design drawings. Equation (80) as described in chapter 3, was used as

the objective function for the optimization. Additional weight factors of value 2000 were

applied to both natural frequency and MAC value error. These factors drastically

increased the performance of the SQP algorithm. Mathematically these new weight

factors should not skew the results toward the natural frequencies or MAC value errors

because is the same amount for both of them. Constraints were applied in this step as

summarized in Table 12.

Parameter Lower Bound 

Upper Bound 

Master Masses  ‐5%  5% Inertia of the Spine Beam  ‐5%  5% 

Stiffness of the Towers Deck Connections  ‐100%  100% 

Table 12. Constraints applied to the parameters to update

Table 13 shows the parameters of the first solution. The solution keeps the mass between

bent 1 and location 1 unchanged while decreasing the mass in the two locations of the

Figure 35. Master Masses at the deck

Master mass

84

main span. The moment of inertia is increased. The stiffness at the connection between

the deck and bent 1 is very high while the stiffness at the connection between the deck

and the main towers is low. An improvement of 28.6% in the objective function is

obtained compared to the original finite element model (f(p)= 1049.1317).

Mass (%) Inertia(%)

Stiffness (%) f(p) Eq(80) Loc1 Loc2 Mid-span Bent Tower

0.0000 -5.0000 -5.0000 5.0000 100.0000 27.3636 749.4804

Table 13. First solution found.

Second Step – Alternative Solutions

As described in chapter 3, alternative solutions are obtained using the first local minima

obtained as starting point by changing the objective function by Equation (81). Equation

(82) is placed as a new constraint with value α of 1.35 and keeping the lower and upper

bounds indicated in Table 12.

Solution Mass (%) Inertia(%)

Stiffness (%) f(p) Eq(80) Loc1 Loc2 Mid-span Bent Tower

2 ‐0.0001  0.0000  0.0000  0.0000  99.7327  0.0000  903.29893 ‐4.7430  0.0000  0.0000  0.0000  0.0000  0.0000  983.20974 0.0000  0.0000  0.0000  0.0000  0.0000  100.0000  989.8247

Table 14. Alternative solutions found

Table 14 shows the alternative solutions found. The second solution increases only the

stiffness at the connection between the Bent and the deck, while the other parameters

85

remain constant. The third solution only decreases the mass between Bent 1 and the mid

span of the first span, while the other parameters remain constant. Finally, solution

fourth increases only the stiffness of the connection between the Towers and the deck.

Third Step – Local Minima

In this final step new optimization processes are performed, using as starting points the

solutions found in the second step to detect new local minima. The objective function is

given by Equation (80) and the constraints of Table 12 are used. The procedure found a

new local minimum using the second solution as starting point for the optimization but

was not able to find a new minimum with the third solution. The second local minimum

consists on decreasing the mass of the deck between the mid pint of the first span and the

first quarter of the main span. The moment of inertia of the deck was unchanged and the

stiffness in the connections between the deck and the towers is very high.

Solution Mass (%) Inertia (%)

Stiffness (%) f(p) Eq(80) Loc1 Loc2 Mid-span Bent Tower

1 0.0000  ‐5.0000  ‐5.0000  ‐5.0000 100.0000  27.3636  749.4804 2 0.0000  ‐5.0000  0.0000  0.0000  99.9983  100.0000  807.8340 

Table 15. Local minima obtained

5.2.3. Updating Results

The final solutions are given by the results obtained in the third step (Table 15). All the

86

local minima found have an improvement larger than 20% compared to the value of f(p)

for the original model.

Solution 1ω (Hz)

2ω (Hz)

3ω (Hz)

4ω (Hz)

MAC ),( 1,1, feid φφ

(%) ),( 2,2, feid φφ

(%) ),( 3,3, feid φφ

(%) ),( 4,3, feid φφ

(%) Exp. 0.32  0.41  0.64  0.71  0  0  0  0 

Original 0.29  0.39  0.60  0.63  4.66  5.08  3.04  9.46 1 0.31  0.41  0.61  0.63  4.13  5.61  3.91  6.96 2 0.31  0.41  0.61  0.63  4.09  5.55  3.85  6.1 

Table 16. Dynamic characteristics of the local minima

Table 16 shows the first four natural frequencies identified experimentally, of the original

model and of the updated models and the percentage of difference of the MAC values

between the identified mode shapes and the numerical modes. It is clear that the updating

process greatly improves the values of the natural frequencies as compared with the

identified frequencies. The MACs were improved in all the solutions found, when

compared to the original finite element model. All three final solutions are based on

either increasing the stiffness of the structure or decreasing the mass of the structure,

because all 4 natural frequencies identified are higher than the natural frequencies of the

original finite element model.

87

6. Conclusions and Future Work

6.1. Conclusions

This thesis was divided in two main parts. The first part compares the overall dynamic

behavior of cable-stayed bridges under seismic excitation when modeled with different

cable models. The three methods used were: the Equivalent Elasticity, the Elastic

Catenary and the Isoparametric Formulation. The fundamental concepts in the

development of these techniques are discussed and finite element model of the Bill

Emerson Memorial Bridge was used for numerical evaluation. The second part of this

thesis presents the system identification and the model updating of this structure. System

identification was performed using the Stochastic Subspace Identification method using

acceleration records from ambient vibration tests on the bridge. The data was collected

using the permanent instrumentation installed on the bridge by the Missouri Department

of Transportation and the United States Geological Survey. The model that best

represents the dynamic behavior of the structure, according to the cable modeling

comparison, was updated to match the dynamic characteristics of the real structure.

Modeling to Generate Alternatives was used to find several plausible solutions for the

88

updating problem through the nonlinear HSJ method.

Statistical analyses of the Bill Emerson FE model were used to calculate the final tension

on the cables before any dynamic analysis was performed. In particular the zero

displacement method was used, including geometric nonlinearities. No significant

differences were found in the final cable tension when comparing the results of the three

cable models studied. A decrease of the cable tensions related to the assumed initial

tensions was found for all methodologies, indicating that after each iteration (i.e. static

nonlinear procedure) cables were losing tension and the structure were gaining flexibility.

In addition, numerical simulations showed no significant differences in the overall

dynamics of the FE model were presented when only one element per cable is used to

model the bridge’s cables. However, Equivalent Elasticity Modulus always produces

lower frequencies than the order methodologies, indicating a lower stiffness. When the

cables were subdivided, the differences became significant. These differences are

attributed to cable-deck interaction. Differences in the structural dynamic behavior

between the 4 elements per cable models and Equivalent Elasticity (single element) are

caused by the cable shear stiffness that is not taken in to consideration in Equivalent

Elasticity.

In terms of the Transfer Functions appreciable differences could only be found in the low

frequency range 0 – 1.5 Hz, for all the cases considered. This is due to the fact that this

frequency range is of significant cable-deck interaction. The FRACs demonstrate that

89

Equivalent Elasticity using a single element per cable is significantly different from both

the Elastic Catenary and the Isoparametric Formulation using 4 elements per cable. These

large differences can be attributed to the interaction cable-deck and cable-tower due to

new coupled modes between deck and cables. Even though there are some differences

between elastic Catenary and the Isoparametric Formulation both subdivided in 4

elements per cable, these differences are negligible. The Elastic Catenary model is

preferred than the Isoparametric Formulation because considers both the cable sag effect

and the large displacement effect.

The identification of modal parameters of the Bill Emerson Memorial Bridge was

performed using the SSI. The consistency in the identified data shows that the structure

trends to have a linear behavior under ambient vibrations. Thus a linear model can be

used to characterize the dynamics of the structure under small vibrations.

The resulting solutions from the updating process showed to improve the dynamic

performance of the numerical model, achieving both natural frequencies and mode shapes

numerically closer to these measured from the structure. Larger results were found in the

natural frequencies than in the mode shapes, even thought that both of them are equally

weighted in the objective function. Hence, the natural frequencies resulted to be more

sensitive to the adjusted parameters than the mode shapes.

A new Modeling to Generate Alternatives was proposed and named non-linear Hop Skip

90

and Jump method after the linear version proposed by Brill et al. 1990. Similar to the

linear HSJ method, the proposed technique consists of three main steps: i) Finding of a

first updated model using traditional model updating techniques, ii) Finding physically

different alternative solutions and iii) Identifying local minima of the objective function

close to the new solutions found. The proposed nonlinear HSJ method demonstrated

potential for finding alternate updated models for the Bill Emerson Bridge. A global

minimum of the objective function was obtained in the first step, greatly improving the

dynamic behavior of the FE model. The results from the second step have significantly

different physical properties with a slight decrease in the objective function when

compared with the first solution. These solutions are a good starting point for the

optimization process of the first step, because they allowed the optimization process to

identify a new local minimum. These results show the potential of the non-linear HSJ

method in the identification of a family of updated models. This new methodology

provides engineers not only with a unique solution for the updated process but with a

family of solutions. Depending on the use of the updated models engineers could pick

the most appropriated solution based on additional information known from the structure

or engineering judgment, or use the whole family of solutions for subsequent analysis.

6.2. Future Work

The work presented here shows that cable-stayed bridges tend to behave linearly for

91

small vibrations, in which the load condition tends to be constant. Future work should

focus on performing nonlinear dynamic analysis and comparing the cable-deck and

cable-towers interaction under strong earthquake or wind shake. This will lead us to a

better understanding of the cable modeling influence in the overall dynamic of the

structure.

Future work for the development of the nonlinear MGA includes:

i) More variables should be involved in future updating processes. This would

enhance the likelihood of finding better and different updated models.

i) The probabilities associated to each one of the solutions found with MGA, should

be assessed, this will provide valuable information using the updated models.

92

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