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 2 nd Reader ________________________ ________________________ Department of Civil and Department of Civil and Environmental Engineering Environmental Engineering 3 rd Reader 4 th Reader ________________________ Dean of The Graduate School
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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
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
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
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 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
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
67
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
70
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
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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).
76
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.
77
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
78
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
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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