NoNLiNear VibratioN aNd dyNamic abstract
abstract
In the present work, the nonlinear vibrations and the corresponding dynamic fracture mechanics of cables of cablestayed bridges are studied. The cables are among the most critical components in cablestayed bridges and there are different damage sources such as corrosion, vibration, fatigue and fretting fatigue that can significantly affect them, thereby reducing the cable’s service life and even producing their failure.
CableParametric Resonance is the specific nonlinear vibration studied in this research. This type of vibration occurs due to displacements presented at the cable supports. These displacements are induced by the wind and traffic loads acting on the pylon and deck of the bridge. Under certain conditions, unstable cablevibration of significant amplitude can be registered. Therefore, numerical and experimental analyses are carried out in order to describe this phenomenon and to determine the corresponding instability conditions.
Two nonlinear models of cableparametric resonance are studied to predict the cable response.
In the simulation method, the nonlinear components are treated as external forces acting on the linear systems, which are represented by Single
Degree of Freedom systems and described by digital filters. A clear nonlinear relationship between the excitation and the cable response is observed in the simulations and the experiments.
The corresponding experimental analysis is based on a scaled model (1:200) of the Öresund bridge and a good agreement between the numerical and experimental results is found.
After obtaining the relationship between the cable response and the excitation, the cable instability conditions are determined. This is done by finding the minimum displacement required at the cable supports in order to induce nonlinear cable vibration of considerable amplitude. The instability conditions are determined within a wide range of excitation frequencies and conveniently expressed in a simplified and practical way by a curve. The determination process is rather fast and offers the possibility to evaluate all bridge cable stays in a rather short time.
Finally, the dynamic fracture mechanics of the cable is considered by studying the fracture toughness characteristics of the material under dynamic conditions. Finite Element simulations on a precracked threepoint bending specimen under impact loading are performed. The observed cable instability is equivalently considered as the associated response to impact load conditions, and a crack as a defect on the wires of a cable stay. The simulations are based on an experimental work by using the Split Hopkinson pressure bar (Jiang et al). The dynamic stress intensity factor KI(t) up to crack initiation is then obtained by different methods. The numerical estimations based on the specimen’s crack tip opening displacement
(CTOD) and midspan displacement were closest to the experimental results. It is observed that a better estimation of the dynamic stress intensity factor relies on a proper formulation of the specimen’s stiffness.
2011:02
ISSN 16502140
ISBN 9789172952010
NoNLiNear VibratioN aNd dyNamic
Fracture mechaNics oF bridge cabLes
Blekinge Institute of Technology
Licentiate Dissertation Series No. 2011:02
School of Engineering
Armando Leon
NonLinear Vibration and Dynamic
Fracture Mechanics of Bridge Cables
Armando Leon
Blekinge Institute of Technology Licentiate Dissertation Series
No 2011:02
NonLinear Vibration and Dynamic
Fracture Mechanics of Bridge Cables
Armando Leon
Department of Mechanical Engineering
School of Engineering
Blekinge Institute of Technology
SWEDEN
© 2011 Armando Leon
Department of Mechanical Engineering
School of Engineering
Publisher: Blekinge Institute of Technology
Printed by Printfabriken, Karlskrona, Sweden 2011
ISBN 9789172952010
Blekinge Institute of Technology Licentiate Dissertation Series
ISSN 16502140 urn:nbn:se:bth00488
Acknowledgements
This work was carried out at the Department of Mechanical Engineering of
Blekinge Institute of Technology, BTH in Karlskrona, Sweden, under the supervision of Professor Kjell Ahlin and Dr. Sharon KaoWalter.
I would like first to express my gratitude to my supervisors for their valuable support and guidance throughout this work, for sharing their knowledge and experiences within engineering and life.
I also want to thank Dr. Eilif Svensson for his contribution in the field of cablestayed bridges, and Dr. Fengchun Jiang for his important contribution within dynamic fracture mechanics.
Thanks to all my colleagues of the Structural Analyses group at BTH, specially, to Andreas Josefsson and Martin Magnevall for sharing their ideas, comments and advices during the development of this work.
I would also like to extend my gratitude to all my colleagues at the Department of Mechanical Engineering of BTH, for creating such a great and friendly environment here in Karlskrona.
Financial support from the Faculty Board of Blekinge Institute Technology is gratefully acknowledged.
To my family, I express all of my gratitude, for their always wonderful support and affection.
Karlskrona, February 2011
Armando Enrique León Guarena. v
vi
Abstract
In the present work, the nonlinear vibrations and the corresponding dynamic fracture mechanics of cables of cablestayed bridges are studied. The cables are among the most critical components in cablestayed bridges and there are different damage sources such as corrosion, vibration, fatigue and fretting fatigue that can significantly affect them, thereby reducing the cable’s service life and even producing their failure.
CableParametric Resonance is the specific nonlinear vibration studied in this research. This type of vibration occurs due to displacements presented at the cable supports. These displacements are induced by the wind and traffic loads acting on the pylon and deck of the bridge. Under certain conditions, unstable cablevibration of significant amplitude can be registered. Therefore, numerical and experimental analyses are carried out in order to describe this phenomenon and to determine the corresponding instability conditions.
Two nonlinear models of cableparametric resonance are studied to predict the cable response. In the simulation method, the nonlinear components are treated as external forces acting on the linear systems, which are represented by Single Degree of Freedom systems and described by digital filters. A clear nonlinear relationship between the excitation and the cable response is observed in the simulations and the experiments. The corresponding experimental analysis is based on a scaled model (1:200) of the Öresund bridge and a good agreement between the numerical and experimental results is found.
After obtaining the relationship between the cable response and the excitation, the cable instability conditions are determined. This is done by finding the minimum displacement required at the cable supports in order to induce nonlinear cable vibration of considerable amplitude. The instability conditions are determined within a wide range of excitation frequencies and conveniently expressed in a simplified and practical way by a curve. The determination process is rather fast and offers the possibility to evaluate all bridge cable stays in a rather short time.
Finally, the dynamic fracture mechanics of the cable is considered by studying the fracture toughness characteristics of the material under dynamic conditions. Finite Element simulations on a precracked threepoint bending specimen under impact loading are performed. The observed cable instability is equivalently considered as the associated response to impact load conditions, and a crack as a defect on the wires of a cable stay. The simulations are based on an experimental work by using the Split Hopkinson pressure bar (Jiang et al).
The dynamic stress intensity factor K
I
(t) up to crack initiation is then obtained vii
by different methods. The numerical estimations based on the specimen’s crack tip opening displacement (CTOD) and midspan displacement were closest to the experimental results. It is observed that a better estimation of the dynamic stress intensity factor relies on a proper formulation of the specimen’s stiffness.
Keywords
Cable Vibrations, NonLinear Vibration, Dynamic Fracture Mechanics, Cable
Stayed Bridges, Dynamic Stress Intensity Factor.
viii
Appended Papers
This thesis has three appended papers, which have been reformatted from their original publication to fit the template of this thesis. However, their contents are unchanged.
Paper A
A. León, A. Josefsson and K. Ahlin. “Simulations and identification of nonlinear models for cables of cablestayed bridges”. ICSV17,The 17
th
International
Conference on Sound & Vibration. CairoEgypt, (July 1822, 2010).
Paper B
A. León, K. Ahlin and S. KaoWalter.“On Determining Instability Conditions for
Stay Cables Subjected to Parametric Resonance”. EVACES’09 Conference on
Experimental Vibration Analysis for Civil Engineering Structures. Wroclaw,
Poland (2009).
Paper C
A. León and S. KaoWalter. “Finite Element Simulations on the determination of
the Dynamic Stress Intensity Factor”. Submitted for publication. ix
The Author’s contribution to the Appended Papers
The appended papers were planned and written with the collaboration of the coauthors. The present author’s contributions to the individual papers are as follows:
Paper A
Responsible for the planning and writing.
Responsible for the CableResponse simulations.
Paper B
Responsible for the planning and writing.
Responsible for the experimental and numerical analyses.
Paper C
Responsible for the planning and writing.
Responsible for the numerical analyses. x
Other Publications
S. KaoWalter, M. Walter, A. Dasari and A. León. “Tearing and Delaminating of a
Polymer Laminate”. Key Engineering Materials, v 465. (2011): 169174.
S. KaoWalter, Hu M., M. Walter and A. León. “A comparison of 2zone and 3zone models in tearing based on Essential Work of Fracture”. The 12
th
International Conference on Fracture. Ottawa, Canada (2009). xi
Abbreviations
CTOD Crack tip opening displacement
CMOD Crack mouth opening displacement
FEM Finite Element Method
Rms Root Mean Square
SDOF Single Degree of Freedom
3PB Threepoints bending xii
Contents
1.2. Aim and Scope .............................................................................. 4
2. Study of NonLinear Vibration in Cables .................................................. 5
2.1. Linear transverse vibration of a taut cable .................................. 5
2.2. Parametric Resonance Vibration in Cables .................................. 7
2.2.1. Uncoupled Cable Model ................................................................ 8
2.2.2. Coupled Cable Model .................................................................. 10
2.3. Numerical Simulation Results ..................................................... 11
Determining Instability Conditions for Parametric Resonance Vibration of
3.1. Identifying the Instability Conditions ......................................... 15
3.2. Numerical Estimations................................................................ 16
3.3. Experimental analysis ................................................................. 18
3.3.1. Experimental Results ................................................................... 20
4. Dynamic Fracture Mechanics ................................................................ 23
4.1. Formulations of the Dynamic Stress Intensity Factor ................ 23
(t) based on the specimen’s midspan displacement .............. 24
(t) based on CTOD ..................................................................... 25
(t) based on CMOD ................................................................... 25
4.2. FEMsimulations for determining K
(t) ....................................... 26
4.3. Numerical results ........................................................................ 27
6. Conclusions and Future Research .......................................................... 33
7. References ............................................................................................ 35 xiii
xiv
1. Introduction
Cablestayed bridges have become very popular around the world due to their construction and cost effectiveness together with their attractive aesthetics.
The significant advances and improvements on the cable material properties and construction processes bring the possibility of designing and building more cablestayed bridges with longer span distances. There are, though, still many challenges to face when designing them. In fact, the stay cables represent one of the most critical components in these structures, being in many cases vulnerable to significant vibration capable to induce, together with other factors, important damages such as their partial or total fracture.
1.1. Background
Cable bridges are mainly classified in two types: cablestayed and cablesuspension bridges. In the first ones, the cables are inclined and directly supporting the deck through the pylons; while in the second ones, there are vertical suspender cables that transmit the load from the deck to the main catenaryshaped cables, which are anchored at the two ends of the bridge.
The present work considers only the cablestayed bridges (figure 1), whose design and construction have become very popular and useful around the world due to their attractive aesthetics and their costeffectiveness.
Figure 1.1. The Öresund Bridge[1], a link that joins the cities of Malmö and
Copenhagen, in Sweden and Copenhagen, respectively
The basic concept and design of this type of bridges have existed for about 400 hundreds years. However, in the 1950’s, a new era for cablestayed bridges started with the advances in the production of highstrength cables, making it possible to design bridges with longer span distances and with significant economical benefits. Then, in 1955, the first modern cablestayed bridge is built in Sweden, the Strömsund bridge, made of steel pylons and deck, with a total
1
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges length of 332 m and with a span of 182 m. In 1962, the first major cablestayed bridge with pylons and deck made of concrete is built in Venezuela, the
Maracaibo’s bridge. It has five main spans of 235 m each, and a total length of
8.7 km.
There are different types of constructions for the stay cables (figure 1.2). They could consist of a single bar, multiple parallel bars, multiwire helical strands, a bundle of parallel wires or a bundle of parallel sevenwire strands. Only a few cablestayed bridges around the world, mainly for pedestrians, have been built with single or multiple parallel bars, whose diameter can vary from 26 mm to
36 mm. On the other hand, the parallel sevenwire strands are the most commonly used in the USA, where 75 % of the cablestayed bridges have this type of stay cable [2].
(a) (b)
(c)
Figure 1.2 Different constructions of Stay Cables [2]. (a) Helical wire strand. (b)
Bundle of parallel wires. (c) Parallel sevenwire strands.
Usually the individual strands in a stay cable are coated with wax or epoxy to avoid the interwire contact. Furthermore, to improve their protection against corrosion, the wires or arrangement of strands are located in sheathing pipes, commonly made of highdensity polyethylene (HDPE) (figure 1.3).
2
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
Figure 1.3. Protection of strands in a stay cable against corrosion [2,3]
The cables are among the most critical components in cablestayed bridges and there are many factors that can consequently cause damages on them during their operation. Corrosion, cable vibration, fatigue and fretting fatigue are the most common damage sources that can significantly affect the cables, reducing their service life and even producing their failure.
The corrosion is one of the most important damages (figure 1.4). This is, mainly, due to the surrounding environmental conditions (high humidity, salinity of the sea, etc) where bridges are usually built. Then, imperfections in the cable protection make the cables vulnerable to corrosion.
Figure 1.4 Corrosion at some strands of a stay cable [2]
In presence of any defect such as a surface deficiency due to corrosion or defects undesirably originated during the cable construction, cable vibration and fatigue also represent an important source of damages for cables in cablestayed bridges. These defects can act as stress concentrators [4] and, consequently, cable vibration and fatigue can originate the initiation and propagation of cracks possibly leading to the fracture of some or all wires of the cable.
These loading conditions can also originate what is called fretting fatigue, which refers to rubbing or interwire contact among the different wires that compound the stay cable. When the wire protection is insufficient, then wearing can be observed. This damage can be more severe when combined
3
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges with corrosion, which could lead to crack formation and the partial or total fracture of the cable [5].
Different types of cable vibrations such as raininduced and windinduced vibration, galloping, vortex shedding and parametric resonance vibration can be found in cablestayed bridges. This thesis is focused on parametric resonance as the type of vibration acting on the stay cables. Under this phenomenon, the main vibration sources are the wind and the traffic loads that excite the tower and the deck of the bridge, respectively. This causes displacements at the cable supports capable to induce, under certain conditions, cable vibration of significant amplitude. This type of vibration is characterized by a nonlinear relationship between the excitation and the cable vibration amplitude.
1.2. Aim and Scope
The present work studies the nonlinear vibration on cables of cablestayed bridges and its corresponding dynamic fracture mechanics. Parametric resonance vibration is the specific cable vibration considered here, where the cables are excited by the displacements at the cable supports due to the wind and traffic loads acting on the bridge pylon and deck. When the cable is subjected to this phenomenon, it can present an unstable vibration characterized by a sudden and significant increment in its vibration amplitude.
This could produce important damages on the cables including the fracture of some or all of their wires. Therefore, it results important to understand the corresponding dynamic fracture mechanics of the cable in order to avoid its fracture.
Numerical and experimental analyses are carried out in order to describe the cableparametric resonance vibration, as well as to determine and express in a practical way the corresponding cableinstability conditions. The dynamic fracture mechanics of the cable is considered by studying the fracture toughness characteristics of the material under dynamic conditions. This is carried out by Finite Elementsimulations and applying fundamental concepts within fracture mechanics. In this sense, the observed cable instability is equivalently considered as the associated response to impact load conditions, and a crack as a defect on the wires of a cable stay.
The aim of this work is, therefore, to develop an integrated research that considers the vibration and fracture mechanics of cables in cablestayed bridges. This type of analysis could provide, based on the observed cable behavior, important information for the monitoring and maintenance of cable stays, where many challenges are currently found since, in most cases, the visual inspections and nondestructive testing are difficult to carry out.
4
2. Study of NonLinear Vibration in Cables
There are different types of cable vibrations such as raininduced and windinduced vibrations, galloping, vortex shedding and parametric resonance vibration. This thesis is focused on the latter one, where the main vibration sources are the wind and the traffic loads that excite the tower and the deck of the bridge, respectively. This type of vibration is characterized by a nonlinear relationship between the excitation and the cable vibration amplitude.
2.1. Linear transverse vibration of a taut cable
In order to study parametric resonance vibration in cables, it is first necessary to understand the theory of lineartransverse vibration of a taut cable [6]. In this theory, it is assumed that the cable remains on a single plane with a constant tensile force during the cable motion. By applying Newton’s second law on an infinitesimal part of a cable, the following expression can be obtained:
f
(
x
,
t
)
dx
T
sin
T
sin(
d
)
m
(
x
)
dx
2
v
(
x
,
t
)
t
2
(2.1.a) where:
f(x,t) is the external transverse force per unit of length acting on the cable.
v(x,t) is the transverse displacement of the cable, depending on position and time.
T is the constant tension along the cable.
is the slope of the cable at position x.
m(x) is the cable mass per unit of length.
Since small variations in the cable slope are assumed, in Eq(2.1a), sin(
d
) can be approximated to
and
d
sin
and
, respectively. Therefore, the slope,
can be written as
v
x
and, as a consequence,
d
x
2
v
2
dx
.
Then, the corresponding governing equation that describes the transverse vibration of a taut cable is written as follows:
m
(
x
)
2
v
(
x
,
t
)
t
2
T
2
v
(
x
,
t
)
x
2
f
(
x
,
t
)
(2.1.b)
The Eq.(2.1.b), which is a partial differential equation, can be solved by using the method of separation of variables [7], where v(x,t) can be expressed as
5
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
v
(
x
,
t
)
Y
(
x
)
Q
(
t
) , the multiplication of a function depending uniquely on space, Y(x), with a function depending only on time, Q(t).
Then, by replacing this expression of v(x,t) into Eq. (2.1.b) and assuming the cable mass, m(x) constant, we obtain the following equation:
Y
(
x
)
d
2
Q
(
t
)
dt
2
c
2
d
2
Y
(
x
)
Q
(
t
)
(2.1.c)
dx
2 with
c
T
A
, where the cable mass is given by
m
A
. In this case,
represents the cable mass density and A, the cable cross section area.
By writing the timedependant functions and spacedependant functions in different members, the following expression is obtained:
Y
''
(
x
Y
(
x
)
)
1
c
2
..
Q
(
t
)
Q
(
t
)
2
(2.1.d)
The expression given in Eq.(2.1.d) is only valid if both fractions are equal to a constant, which should be negative in order to obtain an oscillatory response and avoid a nontrivial solution. From Eq(2.1.d) two equations are derived, one as a function of space and one as a function of time:
Y
" (
x
)
2
Y
(
x
)
0
(2.1.e)
(
t
)
(
c
)
2
Q
(
t
)
0
(2.1.f)
For a taut cable of length, l, the boundary conditions are given by the zero displacement condition at the cable ends, Y(0)=Y(l)=0. Then, infinite solutions are found for the spacedependant function, Y(x), which represents the different shape modes the cable describes when vibrating, as given in
Eq.(2.1.c):
Y n
(
x
)
B n
sin
n
x l
, n=1,2,3,...,
.
(2.1.g) where B
n
are constants.
Similarly, when solving Eq. (2.1.f), infinite timedependant functions are obtained:
Q n
(
t
)
C n
sin(
n t
n
)
(2.1.h)
6
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges where C n
and
n
are constants, and
n
is the natural frequency associated to each shape mode, given as:
n
n
l
T
A
, n=1,2,3,…,
.
(2.1.i)
Then, the transverse vibration of a taut cable is expressed as an infinite sum of shape modes with its corresponding generalized coordinate function as:
v
(
x
,
t
)
C n
sin
n
l x
sin(
n t
n
) (2.1.j)
If only the vibration mode with the lowest resonance frequency is considered, the time dependant equation (2.1.f) becomes as given below:
(
t
)
1
2
Q
(
t
)
0 (2.1.k)
Then, if the cable inherent relative damping,
is introduced into Eq. (2.1.k), the following equation is obtained:
(
t
)
2
1
(
t
)
1
2
Q
(
t
)
0 (2.1.l)
As will be discussed in the next section, the Eqs.(2.1.k) and (2.1.l) represent the linear equations of the cable vibration. From this linear expression and by introducing the respective nonlinear components, the corresponding governing equation for cableparametric resonance vibration is then derived.
2.2. Parametric Resonance Vibration in Cables
A general expression that describes parametric resonance vibration for a SDOF system is given in Eq. (2.2.a) [8].
p
1
(
t
)
p
2
(
t
)
x
F
(
t
)
(2.2.a)
In this equation is noticed that the coefficients, p
1
and p
2
, which for a SDOF system represent its damping and its stiffness, respectively, are in the form of functions depending on time. This functions can physically represent additional excitations to the system, besides the input force F(t). The Foquet and Mathieu
Equations are examples of parametric resonance vibration.
In cablestayed bridges, the parametric resonance vibration of the cables occurs due to the displacements at the cable supports located at the deck and the tower of the bridge. Such displacements are induced by traffic loads or wind loads acting on the bridge structure. As a result, the tension on the cable is not constant, but changing over time, which consequently represent an
7
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges additional excitation force, besides the external force that can directly act on the cable.
In this thesis two models that describe parametric resonance vibration of cables of cablestayed bridges are studied. Here, these models are called uncoupled and coupled model.
Figure 2.1 CableParametric Resonance Vibration on cablestayed bridges
2.2.1. Uncoupled Cable Model
This model, proposed by Lilien and Pinto [9], considers one cable end kept fixed and the other one free to move according to an arbitrary function, x
cs
, as seen in Fig. 2.2.
x
1 x cs
z
Figure 2.2 Cable stay model [9] where the freecable end is excited by a given displacement
The governing equation for this model can be derived from the theory of a taut cable (section 2.1). If only the first vibration mode of the cable is considered, the cable transverse displacement, x
1
can be written by considering Eq. (2.1.k) as:
8
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
1
2
T
Al
2
X
1
0
(2.2.1.a)
Under parametric resonance vibration, the tension on the cable is not constant as considered in the taut cable theory. In fact, it is considered as the sum of the initial or static tension, T
0
, plus the dynamic tension, T
d
.
In terms of the cable axial stiffness and by applying Hooke’s law, the total tension is written as [10]:
T
T
0
T d
AE
X
0
l
AE
(
X cs l
)
(2.2.1.b) where A is the cable’s cross section, E is the Young modulus of the cable’s material, and l is the cable’s length. X
0
and X
cs
correspond to the cable initial stretching and the cable support displacement, respectively; while
is the cable’s extension induced by the elastic deformation when the cable vibrates.
The cable’s extension induced by the elastic deformation,
[10,11]:
is given by
l l
/
/
2
2
dz
1
x
1
z
2
1
/ 2
dz
l
/
0
2
x
1
z
2
dz
2
X
1
2
4
l
(2.2.1.c)
Therefore, the equation for the cabletransversal displacement, X
1
(t) can be obtained by considering the total tension, according to Eqs. (2.2.1.b) and
(2.2.1.c), in Eq. (2.2.1.a) as:
1
1
2
1
X cs
X
0
X
1
2 where α is a constant, defined as follows:
X
1
0
(
2.2.1.d)
2
4lX
0
(2.2.1.e)
The firstcable natural frequency, evaluating Eq.(2.1.i)
is as in the theory of a taut cable, by with n=1. The initial stretching of the cable, x
0
, can be determined by applying Hooke’s Law as follows:
x
0
T
0
l
AE
(2.2.1.f)
9
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
According to this cable model, the governing equation for cableparametric resonance vibration is finally obtained by introducing the cable inherent relative damping,
, into Eq. (2.2.1.d) as follows:
1
2
1
X
1
1
2
1
X cs
X
0
X
1
2
X
1
0
(2.2.1.g)
2.2.2. Coupled Cable Model
The second model considers one of the cable ends fixed and the remaining one connected or coupled to the pylon or deck of the bridge. The pylon or deck is modeled as a single degree of freedom (SDOF) system, as seen in figure 2.3.
This model is based on the presented one by Sun et al.[10], with the difference that in the present research, the inherent damping of the cable and of the pylon/deck are taken into account.
Figure 2.3 Cable interacting with pylon/deck of bridge, based on model by Sun et al [10].
The interaction between cable and bridge pylon/deck leads to a model described by two degree of freedoms and consequently, two nonlinear differential equations have to be solved.
In this model, the equation for the cable is as the one for the uncoupled model,
Eq.(2.2.1.g), but, in this case, the excitation displacement at the cable support is also an unknown variable, x
2
. Therefore, x
2
must be solved through the system of equations.
On the other hand, the motion equation of the mass m can be obtained after applying Newton’s second law as follows:
T d
m
x
2
c x
2
kx
2
F
2
(2.2.2a) where is the dynamic tension acting on the cable; F
2
is the external force acting on the pylon; m the equivalent mass, c the damping and k the stiffness of the pylon/deck at the location of the respective cable support.
10
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
The dynamic tension is obtained from Eqs.(2.2.1.b) and (2.2.1.c). As a result, a coupled system of two nonlinear ordinary equations is obtained as follows:
x
1
2
1
1
x
1
1
2
1
x
2
2
2
2 2
2
2
x
2
x
2
x
0
x
1
2
x
1
AE
x
0
x
1
2
ml
0
F
2
m
(2.2.2.b) where
and
1
are the relative damping of the cable and pylon/deck,
2 respectively, and
is the constant given in Eq.(2.2.1.e).
The first natural frequency of the cable is as given by Eq. (2.1.i) with n=1 and the resulting natural frequency of the pylon is as follows:
2
k m
AE
(2.2.2.c)
ml
2.3. Numerical Simulation Results
The Eqs.(2.2.1.g) and (2.2.2.b) are solved through a method [12] carried out in
MATLAB. In this method, the nonlinear components are treated as external forces acting on the linear systems. The linear systems are represented by SDOF systems, which are described by digital filters [12]. For each time step, a nonlinear equation or equations are solved recursively.
For example, the cable expression given by Eq.(2.2.1.g) can be reorganized and written as:
x
1
2
1
1
x
1
1
2
x
1
x
2
x
0
x
1
2
x
1
(2.3.a)
In Eq.(2.3.a), the lefthand side is clearly defining a linear SDOF system. The righthand side can be seen as the external force acting on the system. Then, the response, x
1
, is calculated by using a routine where the SDOF system is described by a digital filter. In each time step, a nonlinear equation (uncoupled cable model) and a system (coupled cable model) of nonlinear equations have to be solved.
In the numerical analysis, the cable and pylon characteristics of an experimental scaled bridge model [13] are considered (Table 2.1). A description of this experimental set up is shown in section 3.3.
In the cableuncoupled model, the excitation is represented by the cablesupport displacement given by an arbitrary function, X
cs
(t) . Here, such function is estimated by the displacement of a SDOF system subjected to a given force.
11
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
The characteristics (stiffness and relative damping) of this SDOF system are given in Table 2.1.
Table 2.1 Characteristics of an experimental scaled bridge model
Cable Density, ρ, *kg/m
3
]
CableYoung Modulus, E [N/m
2
]
Cable diameter, [m]
Cable length, l [m]
CableRelative Damping,
1
[%]
Pylon equivalent stiffness, k [N/m]
Pylon relative damping,
[%]
2
7800
205x10
9
3x10
4
1.3
0.03
32848
0.22
In order to evaluate and compare the two cable models studied here (sections
2.2.1 and 2.2.2), the following common conditions were considered:
A cable to be tuned at 70 Hz.
A frequency ratio (pylon frequency/cable frequency) equal to 2.
A common excitation force used for both the coupled and uncoupled model.
The force is chosen to be random and bandpassed filtered between
110 Hz and 170 Hz, i.e. a range that includes the natural frequency of the SDOF system.
It is important to point out that for each cable model, the mass of the corresponding SDOF system is different, since the pylon frequency equations differs from each other and, for both models, it is considered a pylon frequency of 140 Hz. For the uncoupled model the pylon frequency is given by
p
k
/
m
, while for the coupled model is given by eq. (2.2.2.c).
Figure 2.4 shows the simulation’s results in terms of the root mean square
(rms) values of each input force and the corresponding cable and pylon displacements.
A clear nonlinear relationship between the cable response and the input force is observed in Figure 2.4.a. A sudden increase in the cabledisplacement amplitude is found when a particular force magnitude is reached at the pylon.
In the simulations, these forces were 0.11 N rms and 0.15 N rms for the uncoupled model and the coupled model, respectively.
12
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
The nonlinear relationship between the pylon and the input force, according to the coupled model, is not noticeable in figure 2.4.b, but it was observed when estimating the ordinary coherence [14] (see appended paper A). The force range considered in the simulations was too low for showing a clear nonlinear behavior through the pylon response.
(a) (b)
Figure 2.4 Cable and pylon response according to simulations
When comparing the two models, it is noticed that both predict instability of the cable for a similar input force. However, a lower cable amplitudedisplacement is obtained through the cablecoupled model. This amplitude difference in the cable response could lead to important results on modeling the cable parametric resonance, where a complementary experimental work might be needed in order to obtain more information about this phenomenon under the conditions considered for the simulations.
13
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
14
3. Determining Instability Conditions for
Parametric Resonance Vibration of Cables
Under certain conditions, the cableparametric resonance vibration can be characterized by instable vibrations of large amplitudes. The conditions for parametric resonance vibration in cables can be given by several factors.
Among them, the excitation amplitude and the ratio between the excitation frequency and natural frequency of the cable represent the most influential factors for this phenomenon [9].
Commonly, this phenomenon is evaluated when the excitation at the cable supports is at a frequency which is close to or twice the cable natural frequency, since under these cases minimal excitation amplitudes could induce large amplitude vibration at the cables. However, this phenomenon can also occur in many cases within a broad range of excitation frequencies, when larger excitation amplitudes are registered at the cable supports.
Since it is important to identify the cases where a stay cable can be prone to this phenomenon, the present research work has developed a method that looks for the instability conditions within a broad range of frequencies.
Usually, the instability conditions for a system subjected to parametric resonance are defined through regions on graphics. In the end, the instability conditions are found by combining the information gathered from several graphics, since each one is expressed in terms of some and not all factors that have influence on this phenomenon. This could result in a rather complicated process.
Instead, here a method has been developed for expressing in just one curve the instability conditions for a particular cable, once the characteristics of the excitation at the cable support and the characteristics of the cable (geometry, material, damping and initial tensile preload ) are known.
3.1. Identifying the Instability Conditions
The instability conditions are determined within a broad range of excitation frequencies by identifying for each of the chosen frequencies, the minimum excitation displacement at the cable support capable to induce nonlinear cable vibration of significant amplitude. The identification of the instability conditions is done through the relationship between the excitation and the cable response.
15
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
3.2. Numerical Estimations
Figure 3.1 shows as examples some numerical results of the cable response when considering the uncoupled cable model. In this case, a sinusoidal displacement swept in amplitude is exciting the cable (figure 3.1.a). The ratio
R
Freq exc
/
Freq cable
, between the excitation frequency and the cable natural frequency is 0.75, with the cable tuned at 59.5 Hz. As noticed in figures
3.1.b and 3.1.c, after reaching an excitationamplitude of 1.55 x 10
3
m the cable displacement rises significantly and suddenly from almost zero displacement. Therefore, this excitation amplitude is considered as the minimum displacement required at the cable support for inducing the instability, when the ratio R is 0.75.
(a) (b)
(c)
Figure 3.1 Numerical results when ratio R is 0.75. (a) Time history simulation of excitation amplitude at cable support. (b) Time history simulation of cable displacement at its mid point. (c) Relationship between excitation amplitude and cable displacement amplitude.
16
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
Then, by repeating this process for each of the frequency Ratios, R under consideration, the instability conditions can be determined within a frequency range and represented in just one curve.
The instability conditions are estimated here according to two different ways of varying the frequency ratio, R:
Firstly, the frequency ratio, R is varied by changing the cable frequency and keeping fixed the excitation frequency. The cable frequency is changed by only varying its initial tension, according to Eq.(2.1.i).
In the other way, the frequency ratio, R is varied by keeping the cable frequency fixed and varying the excitation frequency.
Figure 3.2 shows the corresponding instability curve for the example mentioned at the beginning of this section. In this case, the frequency ratio, R is varied according to the first way, keeping the excitation frequency fixed at 44.5
Hz. This process to vary the frequency ratio, R was also used in the experimental analysis developed here. Details on the experimental analysis are found in the following section.
Figure 3.2 Estimated instability conditions for string (L=1.33m, Ø0.3mm) when excitation frequency is kept fixed at 44.5 Hz.
On the other hand, figure 3.3 shows the instability conditions estimated according to the second way of varying the frequency ratio, R. A cable with the same geometrical and material properties as before was evaluated. Two cases were studied under this condition. In one case the cable frequency was kept fixed at 70 Hz and in the other case, it was fixed at 100 Hz.
17
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
Figure 3.3. Estimated instability conditions for string (L=1.33m, Ø0.3mm) when cable natural frequency is kept at 70 Hz and 100 Hz.
It is important to point out that from the two ways of varying the frequency ratio, R, the second one is more expected in reality than the first one. However, for practical reasons, the first way resulted convenient to carry out when developing the corresponding experiments (section 3.3).
The numerical analysis for obtaining the instability curves were done in a rather short time thanks to a computational subroutine in MATLAB. Therefore, the method developed here brings the possibility of evaluating all cables of a cablestayed bridge in a relative short time as well.
3.3. Experimental analysis
The numerical analysis described above for estimating the parametric resonance vibration of a cable and its instability conditions have been complementary evaluated by carrying out an experimental analysis.
The corresponding experimental setup used in this research is shown in figure
3.4. It is based on a simplified scaled model (1:200) of the Öresund Bridge, which joins the cities of Malmö, Sweden and Copenhagen, Denmark. The experimental setup is made of aluminum and the cables were replaced by strings made of steel. In the original bridge the largest vibration amplitudes were registered at the longest cables [15]. Then, in the experimental setup only one string, the longest one was installed.
The experimental setup is also constituted by a shaker, a signal generator, an amplifier and a force transducer, in order to introduce and measure the excitation force applied on one of the bridge towers. An accelerometer is used to estimate the cable support displacement on the bridge tower on which the
18
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges force is acting, and a laser vibrometer is used in order to estimate the displacement at the cable midpoint.
An experimental modal analysis was initially carried out on the scaled model to extract its dynamic characteristics. Then, the obtained information was used as a data for the numerical simulations (See Table 2.1).
The experiments for studying the cableparametric resonance vibration basically consisted in exciting the bridge tower where the cable is connected by introducing a sinusoidal force, swept in amplitude and with a simple frequency.
Then, the displacements of the cable midpoint and cable support at the tower were estimated by using the laser vibrometer and an accelerometer, respectively. The cable displacements are measured in plane of the cable and not out of plane as represented in figure 3.4, which corresponds to a preliminary analysis.
Figure 3.4.Experimental setup for cable parametric resonance vibration
The goal of the test is to find experimentally the displacement amplitude at the cable support capable to induce a significant and nonlinear vibration at the cable for a chosen frequency ratio, R.
For the experiments, the frequency ratio R is adjusted by tuning the cable to a chosen frequency; while the excitation frequency was kept fixed to 44.5 Hz.
This frequency represented the 1 st
bending vibration mode of the bridge tower.
With this excitation frequency, the displacement range of the pylon becomes wide enough for inducing the parametric resonance of the string under test.
19
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
3.3.1. Experimental Results
Experimental evaluations with a frequency ratio, R equal to 0.48, 0.5, 0.59,
0.64, and 0.66, were carried out to validate the corresponding numerical results. The instability conditions could be determined from the relationship between the cable response and the corresponding excitation at the cable support.
In all cases, except when the frequency ratio was 0.48 and 0.50, the cable vibration could reach the instability at certain moment. The instability was characterized by a suddenlarge amplitude displacement. However, when R was
0.48 and 0.50, the cable could also reach large amplitude vibration, but in direct proportion to the excitation amplitude.
In general, for the experiments where the instability was reached, a first stage previous to the instability was observed. In this stage, the cable used to vibrate in direct proportion to the excitation amplitude at the same frequency of the excitation and its corresponding harmonics. Then, a second stage is observed just when the instability was already reached. The vibration was characterized by multiple frequencies under this stage (see figure 3.5, with R=0.66).
In the cases where R was 0.48 and 0.50, the cable vibrates in proportion to the excitation amplitude at the same frequency of the excitation and at its 1 st harmonic. The latter frequency is coincident with the 1 st
natural frequency of the cable. Then, this could explain why the cable could easily reach large amplitude vibration with even much smaller excitation amplitudes than those required according to calculations.
(a) (b)
Figure 3.5. Experimental cable response spectrum when R=0.66. (a) Before the instability occurs. (b) When the instability already occurs.
Figure 3.6 shows the experimentalcable responses when R=0.59, R=0.64 and
R=0.66 and the excitation frequency was kept at 44.5 Hz.
20
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
(a) (b)
(c) (d)
(e) (f)
Figure 3.6. Cable response and excitation displacement for frequency ratios
R=0.59, R=0.64 and R=0.66.
As seen in figure 3.6, at certain moment and for a particular value of the excitation amplitude the instability in the cable vibration is reached.
The instability conditions can clearly be found from the relationship between the excitation and the cable response by identifying the excitation amplitude capable to induce the sudden jump in the cable vibration amplitude (see figure
3.7)
21
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
Figure 3.7. Experimental relationship between excitation amplitude and cable displacement when R=0.59, R=0.64 and R=0.66.(Cable: L=1.33m, Ø0.3mm).
According to figure 3.7, the minimum required displacements for inducing the instability of the tested string when R=0.59, R=0.64 and R=0.66 are 1.61 mm,
0.83 mm and 0.77 mm, respectively. These results show a good agreement with those obtained in the simulations (Figure 3.2)
Although, the experimental and numerical results showed that the cable is more prone to vibrate under parametric resonance for particular values of the frequency ratio R, as R=2, R=1, R=0.5, etc, it is also shown that this phenomenon can occur within a wide frequency range depending on the excitation amplitude. Therefore, the importance of obtaining the instability conditions considering a wide range of frequencies in which a cable can be subjected.
22
4. Dynamic Fracture Mechanics
In engineering, most of the failure criteria are based on the resistance properties of the component’s material, which are obtained by different characterization methods. In these criteria, the maximum expected stress on the component is compared to the resistance properties of its material.
In the case of failure due to fracture, the stress intensity factor, K
I
, a characterizing parameter around the component’s crack tip, is compared to the material’s resistance to fracture, the socalled fracture toughness. Different techniques have been developed in order to estimate the stress intensity factor, K
I
, and the fracture toughness under static and dynamic conditions. In most of these techniques, simulations and experiments are carried out on a precracked threepoint bending specimen (3PB) (see figure 4.1).
P(t)
Figure 4.1.A precracked cracked3PB specimen
In this research work, the dynamic fracture mechanics of the cable is considered by studying, under dynamic conditions, the fracture toughness characteristics of the material. This is done by studying the dynamic stress intensity factor on a highstrength steel 3PB specimen subjected to impact load.
The observed cable instability during parametric resonance is equivalently considered as the associated response to impact load conditions, and a crack as a defect on the wires of a cable stay.
4.1. Formulations of the Dynamic Stress Intensity
Factor
The dynamic stress intensity factor, K
I
(t), can be determined by different formulations based on the dynamic response of a 3PB specimen subjected to impact load. Then, K
I
(t) can be expressed in terms of the specimen’s midspan displacement, crack tip opening displacement (CTOD) and crack mouth opening displacement (CMOD).
23
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
4.1.1. K
I
(t) based on the specimen’s midspan displacement
Under the firstbending vibration mode of a precracked 3PB specimen, K
I
(t) is considered as proportional to the specimen’s midspan displacement, u(t), [16] as follows:
(4.1.1.a) where C is called the proportionality constant, which depends on the specimen’s geometry, its material and its corresponding stiffness.
In spite of the dynamic loading conditions, the formulation of the specimen’s stiffness is obtained by applying Timoshenko’s beam theory under static conditions. Then, the specimen’s inertial effects are introduced in this formulation by including the specimen’s rotary inertia and its shear deformation [17,18] (see appended paper C).
The specimen’s mid–span displacement can be described by a linear SDOF system with mass, m e
, and stiffness, K (figure 4.2).
P(t) u(t)
K
Figure 4.2.A linear massspring model describing the specimen’s midspan displacement.
Then, the corresponding governing equation is as follows:
(4.1.1.b) with the following initial conditions:
Therefore, u(t) can be expressed as:
u
(
t
)
1
1
m e t
0
P
(
) sin
1
(
t
)
d
(4.1.1.c) where
is the 1
1 st
natural frequency of the specimen given by:
24
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
1
K m e
(4.1.1.d)
4.1.2. K
I
(t) based on CTOD
The Dynamic Stress Intensity Factor can also be calculated from the crack tip opening displacement (CTOD) as [19]:
(4.1.2.a) where G is the shear modulus of the material, r is the distance between the crack tip and where the transversal displacement is measured (see figure 4.3),
v(t) is the transversal displacement and
k
3
1
, for planestress conditions
k
3
4
, for planestrain conditions
In Eq.(4.1.2.a), the displacement v(t) is half of the crack opening displacement.
CMOD
Figure 4.3. Crack Tip Opening Displacement (CTOD) and Crack Mouth Opening
Displacement (CMOD).
4.1.3. K
I
(t) based on CMOD
The K
I
(t) can also be estimated from the crack mouth opening displacement
(CMOD) [20] by the following equation:
K
Id
Ew m k
4
(
a
v
)
(4.1.3.a) where w
m
is the crack mouth opening displacement (Figure 4.3), E is the Young
Modulus, a is the crack length, α=a/W and β=S/W; while
k
(
) and
v
(
) are nondimensional functions depending on
(See paper C).
25
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
4.2. FEMsimulations for determining K
I
(t)
In order to study here the dynamic stress intensity factor, K
I
(t), simulations by
Finite Element Method (FEM) are performed on a 3PB specimen subjected to impact load. The simulations are based on the experimental work carried out on a modified Hopkinson pressure bar by Fengchun et al [18]. The specimen characteristics are as in the experiments, made of high strength steel (Fe10Cr), with Young modulus, E =210 GPa and Poisson ratio, ν=0.3. The specimen is 4 mm thick with 6.5 mm in width and 40 mm in length. The distance between the simple supports is 25 mm and the crack length is half of the specimen width
(a/W=0.5).
Figure 4.4 shows one of the measured forces in the experiments, P exp
(t). The measured force is considered as the input load in the FEMmodel.
Experimental Force
5000
4000
3000
2000
1000
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Time [s] x 10
4
Figure 4.4.Measured force, P exp
(t),[18] used for the FEMsimulations.
The simulations are carried out by using the commercial software ABAQUS. Due to symmetry, only half of the specimen is modeled (figure 4.5) and half of the input force is taken into account. A spider web mesh around the specimen’s crack tip was built. The mesh was constituted by quadrilateral linear explicit elements under plane strain conditions. The analysis is carried out by an explicit dynamic step, which is convenient [21] for short duration forces, such as impact loads. In this analysis, the crack propagation is not considered.
26
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
Figure 4.5.Modeling a 3PB specimen by FEM.
From the simulation results, K
I
(t) could be estimated in terms of the specimen’s midspan displacement, CTOD and CMOD as mentioned in section 4.1.
4.3. Numerical results
Figure 4.6 shows the estimations of the specimen’s midspan according to the
FEMsimulations and the linear massspring model described in section 4.1.1.
The predicted response according to the linear massspring model is obtained here by numerically solving the integral given by Eq.(4.1.1c) in MATLAB. In this integral, the measured force in the respective experiments is used as the input load, P(t). The respective experimental results [18] are also presented in this figure.
4 x 10
4
Midspan displacement
Massspring model
Experiment
FEM
3
2
1
0
0 1 2 3
Time [s]
4 5 x 10
5
Figure 4.6.Displacements at specimen’s midspan.
It can be seen in figure 4.6 that the simulation result by FEM and the predicted response, according to the linear massspring model, are comparable with the experimental one. However, it is still important to mention that within an initial
27
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges time period these estimations were lower than the experimental result. This may indicate the need of introducing some modifications into the described linear massspring model and into the FEMsimulations.
As described in the appended paper C of this thesis, the linear massspring model was evaluated by considering the experimental and simulation results.
This was done by finding the respective mass, m, and stiffness, K, for a linear massspring system, whose response fits the one obtained by the experiments and simulations. By this fitting process, the specimen’s stiffness calculated from the experimental and simulation results was found to be different from the theoretically estimated. This observation may indicate that the specimen’s stiffness is the variable that needs to be reformulated within the linear massspring model. Possibly, some considerations such as the contact between the specimen and the supports should be taken into account when formulating the specimen’s stiffness.
After obtaining the specimen’s dynamic response, K
I
(t) can be estimated in terms of the midspan, the crack tip opening displacement (CTOD) and the crack mouth opening displacement (CMOD), by applying the Eqs.(4.1.1.a),
(4.1.2.a) and (4.1.3.a), respectively. Figure 4.7 shows three different estimations of K
I
(t) in terms of the midspan displacement, one based on the
CTOD and one on the CMOD.
5
0
20 x 10
7 Dynamic Stress Intensity Factor, KI(t)
15
10
Midspan,Eq(4.1.1.c)
Midspan (Experiment)
Midspan (FEM)
CTOD(FEM)
CMOD(FEM)
5
0 1 2 3
Time [s]
4
Figure 4.7.Estimation of K
I
(t).
5 x 10
5
The midspan displacement was obtained as a measurement from the experiments; by numerically solving the integral in Eq.(4.1.1.c) with the measured force, P exp
(t); and by obtaining it from the FEMsimulations. The
CTOD and CMOD at the specimen are obtained from the FEMsimulations.
As seen in figure 4.7, in general, the different methods lead to a rather similar solution. It is noticed that the estimations based on the CTOD and midspan
28
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges displacement by FEM are the closest ones to the experimental results. In spite of this result, it is important to point out that in practice, measuring the CTOD is complex. In fact, measuring the CMOD is more feasible than measuring the
CTOD.
In this section it is concluded that all methods give a K
I
(t) lower than the experimental one within an initial range of time. After this period, the respective curves intersect the experimental one. This observation was also reported by Rubio et al [20] indicating that the intersection of the curves corresponded to the time of crack initiation. This difference with respect to the experimental results indicates that further work is needed in order to properly describe the dynamic stress intensity factor. Studies on the relationship between the CMOD and K
I
(t), nonlinear models for describing the specimen’s midspan displacement and crack propagation are some of the suggested topics that can contribute to obtaining a better description of the dynamic stress intensity factor.
29
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
30
5. Summary of papers
5.1. Paper A
This paper focuses on studying the cableparametric resonance vibration by considering two nonlinear models. Simulations and identification of nonlinear parameters are performed based on these two models under random excitation. In one model, the parametric excitation is treated as an arbitrary displacement introduced in one end of the cable. In the second model, such excitation is coming from an external force acting on the pylon or deck of the bridge to which the cable is coupled. The pylon or deck is modeled as a Single
Degree of Freedom System. In the simulation method, the nonlinear components are treated as external forces acting on the linear systems, which are represented by Single Degree of Freedom (SDOF) systems and described by digital filters. For each time step, a nonlinear equation or equations are solved recursively. In the identification method the nonlinearity is modeled as a feedback forcing term acting on an underlying linear system or systems and the parameter estimation is performed in the frequency domain by using conventional MI/SO techniques.
5.2. Paper B
This paper presents a method to determine, by experimental and numerical analyses, the instability conditions of cables subjected to parametric resonance vibration within a wide range of excitation frequencies. This is accomplished by finding the minimum displacement required at the cable supports in order to induce nonlinear cable vibration of considerable amplitude. Once the cable characteristics (geometry, material properties, inherent damping and initial tensile preload) are known, the instability conditions are identified and expressed in a simplified and practical way by a curve. The determination process is fast and offers the possibility to evaluate all bridge cable stays in a rather short time.
5.3. Paper C
This paper focuses on the study of the dynamic stress intensity factor, K
I
(t), as a characterizing parameter for the fracture toughness of materials under dynamic loading conditions. Finite Element simulations on a precracked threepoint bending specimen under impact loading conditions are performed. The simulations are based on two experimental results obtained by using the Split
Hopkinson pressure bar (Jiang et al). With the measured loading force as input, the midspan displacement is calculated and a good agreement with the experimental results is observed. The dynamic stress intensity factor, K
I
(t), up
31
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges to crack initiation is then obtained by different methods based on the specimen’s mid span displacement, crack tip opening displacement (CTOD) and crack mouth opening displacement (CMOD). A good agreement with the experimental results is observed by using these methods. A model for describing the specimen’s midspan displacement by a linear massspring system is also evaluated. By an unconstrained optimization procedure, it is shown that according to this model, a proper estimation of the dynamic stress intensity factor relies on the formulation of the specimen stiffness.
32
6. Conclusions and Future Research
The present thesis focuses on the nonlinear vibration and the corresponding dynamic fracture mechanics on cables of cablestayed bridges. Parametric resonance vibration is the specific nonlinear cable vibration studied here.
Under certain conditions, unstable cable vibration of significant amplitude can occur due to this phenomenon. In presence of any defect on the cable such as surface deficiency due to corrosion, this type of vibration could cause significant damages such as the partial or total fracture of the cable.
Two nonlinear models of cableparametric resonance were studied to predict the cable response. In the first model, the excitation is given by an arbitrary displacement; while in the second model, the cable is considered coupled to the bridge pylon or deck, which is excited by a force. In order to compare both models, in the first one the displacement at the cable’s free end is given by the response of a single degree of freedom (SDOF) system subjected to a given force. In the simulation method, the nonlinear components are treated as external forces acting on the linear systems, which are represented by SDOF systems and described by digital filters. A clear nonlinear relationship between the cable response and the input force was observed, characterized by a sudden increase in the cable displacement amplitude at a certain excitation force level. Even though both models predicted the instability of the cable for a similar input force, lower cable displacement amplitude was observed when the interaction with the pylon was taken into account.
From the relationship between the cable response and the excitation, the cable instability conditions were determined. This was done by finding the minimum displacement required at the cable supports in order to induce nonlinear cable vibration of considerable amplitude. The instability conditions were determined within a wide range of excitation frequencies and conveniently expressed in a simplified and practical way by a curve. The determination process was fast and offers the possibility to evaluate all bridge cable stays in a rather short time.
Although the numerical estimations and the experimental results confirmed that cables are more prone to vibrate under frequency ratios, R (excitation frequency/cable frequency) close or equal to R=2, R=1 and R=0.5 than in other cases, it is still important to consider a wide range of frequencies, since the excitation amplitudes at the cable support could be large enough for inducing the cableparametric resonance vibration under other frequency ratios.
Finally, the dynamic fracture mechanics of the cable is considered by studying the fracture behavior of the material under dynamic conditions. FiniteElement simulations were carried out to determine the dynamic stress intensity factor on a highstrength steel 3PB specimen subjected to impact load. The observed cable instability is equivalently considered as the associated response to impact
33
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges load conditions, and a crack as a defect on the wires of a cable stay. The simulations were based on an experimental work by using the Split Hopkinson pressure bar (Jiang et al) and different methods to estimate the dynamic stress intensity factor, K
I
(t), up to crack initiation were applied. The estimations based on the crack tip opening displacement (CTOD) and midspan displacement by
Finite Element Method (FEM) were closest to the experimental results. It was observed that a better estimation of the dynamic stress intensity factor relies on a proper formulation of the specimen’s stiffness.
In the future work concerning cableparametric resonance vibration, complementary experiments by using random excitation forces are needed.
The respective experimental results could bring important information to understand the difference in the predicted cablevibration amplitude according to the two models studied here. The development of a new model of cableparametric resonance, where the displacement of both cable ends is considered, could be an interesting and important contribution for finding a model closer to reality.
Regarding the dynamic fracture mechanics of the cable’s material, studies considering the relationship between the CMOD and K
I
(t), the development of a nonlinear model for describing the specimen’s midspan displacement and the consideration of crack propagation are recommended in the future work, in order to improve the estimations of the dynamic stress intensity factor.
34
7. References
[1] Öresund Bridge. www.bridgephoto.dk
[2] Tabatabai, H. “Inspection and Maintenance of Bridge Stay Cable Systems.
NCHRP synthesis 353”. Transportation Research Board of the National
Academies, USA (2005)
[3] http://www.alga.it/en/prodotto/15sistemaperstrallial200
[4] Mahmoud, K.M. ”Fracture strength for a high strength steel bridge cable wire with a surface crack.” Theoretical and Applied Fracture Mechanics
48, nr 2 (2007/10/): 15260.
[5] Perier, V., L. Dieng, L. Gaillet, C. Tessier, och S. Fouvry. ”Frettingfatigue behaviour of bridge engineering cables in a solution of sodium chloride.”
Wear 267, nr 14 (2009/06/15): 30814.
[6] de Silva, C. Vibration. Fundamentals and Practice. CRC Press, USA (2000).
[7] Ibragimov, N. A Practical Course in Differential Equations and
Mathematic Modeling. ALGA Publications, Blekinge Institute of
Technology, Sweden (2005).
[8] Genta, G. Vibration of structures and machines. Practical Aspects. Second
Edition. SpringerVerlag, Torino. (1995)
[9] Lilien, J.L., and A. Pinto Da. ”Vibration amplitudes caused by parametric excitation of cable stayed structures.” Journal of Sound and Vibration
174, nr 1 (1994): 6990.
[10] Sun, B. N., Wang, Z.G., Ko, J.M. and Ni, Y.Q. “Cable Oscillation induced by
Parametric Excitation in CableStayed Bridges”. Journal of Zhejiang
University Science. Vol.4, N.1, 1320. The Hong Kong Polytechnic
University. (2003).
[11] León A. “Study on Vibrations induced by Parametric Excitations on
Strings”. Master Thesis. Blekinge Institute of Technology, Karlskrona,
Sweden (2007)
[12] Ahlin K., Magnevall M. and Josefsson A. “Simulation of forced response in linear and nonlinear mechanical systems using digital filters”. ISMA 2006,
Leuven, Belgium (2006)
[13] León A., Ahlin K. and KaoWalter S.“On Determining Instability Conditions for Stay Cables Subjected to Parametric Resonance”. EVACES’09
Conference on Experimental Vibration Analysis for Civil Engineering
Structures. Wroclaw, Poland (2009).
35
A. Leon. NonLinear Vibration and Dynamic Fracture Mechanics of Cable Bridges
[14] León A., Josefsson A. and Ahlin K. “Simulations and identification on nonlinear models for cables of cablestayed bridges”. ICSV17,The 17
th
International Conference on Sound & Vibration. CairoEgypt (2010)
[15] Svensson, B., Emmanuelsson, L. and Svensson, E. “Øresund BridgeCable
SystemVibration IncidentsMechanism and Alleviating Measure”.
Øresundsbrokonsortiet. (2004)
[16] Nash, GE. "Analysis of the forces and bending moments generated during the notched beam impact test." International Journal of Fracture
Mechanics v5,no. 4 (1969): 269286.
[17] Kishimoto, K., M. Kuroda, S. Aoki, och M. Sakata. ”Simple formulas for dynamic fracture mechanics parameters of elastic and viscoelastic threepoint bend specimens based on Timoshenko's beam theory.”
Proceedings of the 6th International Conference on Fracture (1984)//.
317784.
[18] Jiang, Fengchun, A. Rohatgi, K.S. Vecchio, och J.L. Cheney. ”Analysis of the dynamic responses for a precracked threepoint bend specimen.”
International Journal of Fracture 127, nr 2 (2004/05/): 14765.
[19] Anderson, T. L. Fracture mechanics : fundamentals and applications. 3. ed. Taylor & Francis, 2005.
[20] Rubio, L., J. FernandezSaez, och C. Navarro. ”Determination of dynamic fractureinitiation toughness using threepoint bending tests in a modified Hopkinson pressure bar.” Experimental Mechanics 43, nr 4
(2003/12/): 37986.
[21] ABAQUS Documentation version 6.7
36
Paper A
Simulations and identification of nonlinear models for cables of cablestayed bridges
37
Paper A is published as:
A. León, A. Josefsson and K. Ahlin. “Simulations and identification of nonlinear models for cables of cablestayed bridges”. ICSV17,The 17
th
International
Conference on Sound & Vibration. CairoEgypt, (July 1822, 2010).
38
Simulations and identification of nonlinear models for cables of cablestayed bridges
Armando León, Andreas Josefsson and Kjell Ahlin
Abstract:
Simulations and identification of nonlinear parameters are applied on two models that describe the vibration due to parametric resonance in cables of cablesstayed bridges. The aim of this work is to study the dynamic response predicted by the two models under random excitation, as well as to develop a suitable strategy for system identification from random data. In one model the parametric excitation is treated as an arbitrary displacement introduced in one end of the cable. In the second model, such excitation is coming from an external force acting on the pylon or deck of the bridge to which the cable is coupled. The pylon or deck is modeled as a Single Degree of Freedom System.
In both models the cable response is obtained by a simulation method based on digital filters. The studied identification technique is based on random excitation. In this method the nonlinearity is modeled as a feedback forcing term acting on an underlying linear system or systems and the parameter estimation is performed in the frequency domain by using conventional MI/SO techniques.
Keywords:
Cable vibration, Nonlinear vibration, Parametric Resonance, Simulations,
Identification.
39
1. Introduction
Cable Parametric Resonance Vibration is a phenomenon characterized by a nonlinear relationship between the cable response and the excitation at the cable supports [1,2,3]. Under certain conditions, the displacements at the cable supports on the deck and/or the pylon of the bridge can induce tensile oscillation in the cable capable to generate large transversal vibration.
Two models are studied in the present work. In the first model, the parametric excitation is treated as an arbitrary displacement introduced at the cablefree end [1]; while in the second model, the cable end is considered to be coupled with the pylon or deck of the bridge [2], which moves as a consequence of an excitation force acting on it.
The aim of this study is to examine and compare the dynamic response predicted by the two models under random excitation, as well as to evaluate a suitable identification technique based on random data. The derivation of equation of motions for the studied system is shown in Section 2; while system identification techniques are discussed in Section 3. Simulation results are then shown in Section 4 followed by concluding remarks in Section 5.
2. Modeling cableparametric resonance vibration
Two models that describe the cable parametric resonance vibration are studied in this section: a cable uncoupled to the pylon or deck of the bridge, whose free end is moving according to an arbitrary function; and a cable coupled to the pylon or deck of the bridge, which is subjected to an excitation force.
2.1. Uncoupled Model
The first model corresponds to the one proposed by Lilien and Pinto [1], in which one cable end is kept fixed and the other one is free to move according to an arbitrary function, X
cs
, as seen in Fig. 1.
Figure 1. Cable Stay modeled by Lilien and Pinto[1] where the cablefree end is excited by a given displacement.
40
This model is described by a nonlinear ordinary differential equation, and it is written as follows:
x
1
2
1
1 1
ω
2
1
1
x cs x
0
4
3
x
1 where x
1
is the cable transversal displacement;
α
2
x
1
0
(1)
1 is the inherent relative damping of the cable; ω
1
is the first natural frequency [rad/s] of the cable; x
cs
is the given displacement at the cable end, and x
0
is the initial stretching of the cable. In Eq. (1), α is a constant and it is defined as follows:
4
where l is the originalcable length.
x
0
l
3
(2)
The firstcable natural frequency, is obtained from the theory of a taut string [4] with n=1, as follows:
1
l
T
0
A
(3) where T
0
is the initial tensile load of the cable; ρ is the cable mass density and A is the cable cross section area.
The initial stretching of the cable, x
0
can be determined by applying Hooke’s
Law using the initial tensile load, T
0 and the cableaxial stiffness expressed on terms of its original length, l, its cross section area, A and its Young Modulus, E as follows:
x
0
T
0
l
AE
(4)
2.2. Coupled Model
The second model considers the interaction between the cable and the pylon or deck of the bridge, through one of the cable ends. The pylon or deck is modeled as a Single Degree of Freedom (SDOF) system, as seen in Fig. 2. This model is based on the presented one by Sun et al.[2], with the difference that in the present work, the inherent damping of the cable and of the pylon/deck are taken into account.
41
Figure 2. Cable interacting with Pylon/Deck of bridge, based on Sun et al model
[2].
In this case, a system of two equations has to be solved. The motion equation of the mass m, can be obtained after applying Newton’s second law as follows:
T d
m
x
2
c x
2
kx
2
F
2
(5) where is the dynamic tension of the cable; F
2
is the external force acting on the pylon; m the equivalent mass, c the damping and k the stiffness of the pylon at the location of the respective cable support.
The dynamic tension of the cable is expressed as:
T d
AE l
x
2
2
x
1
2
4
l
(6)
As a result a coupled system of two nonlinear ordinary equations is obtained as follows:
x
1
2
1
1
1
1
2
1
x
2
x
0
x
2
2
2
2
x
2
2
2
x
2
x
1
2
x
1
AE ml
x
0
x
1
2
0
F
2
m
(7) where
1 and
2 are the relative damping of the cable and pylon/deck, respectively; and is a constant written as:
2
4lx
0
(8)
The cable expression in Eq. (7) is as the given one in Eq. (1), with the difference that the excitation displacement at the cable end is also an unknown variable,
x
2
which must be solved through the system of equations. The firstnatural
42
frequency of the cable is as given by Eq. (3) and the resulting natural frequency of the pylon is as follows:
2
k m
AE ml
(9)
3. System Identification Techniques
In order to obtain an accurate estimate of the parameters from input/output data from the studied system, it is necessary to take the nonlinear effects into account. A model for parameter estimation can be defined by reformulating the nonlinear equations shown in Section 2 into an equivalent MultipleInput
SingleOutput (MI/SO) as initially proposed by Bendat [5].
Starting with the cable, a linear transfer function (dynamic stiffness) can be defined as:
B
C
F
1
X
1
2
j
2
1
1
1
2
x
1
2
0
(10)
The system can then be rewritten in frequency domain as shown in Eq. (11).
Here,
F {.} denotes the Fourier transform. Hence, by creating an artificial input and output from measured signals, standard MI/SO techniques can be used to find the underlying linear system and coefficients for the nonlinear terms.
B
C
1
2
4l
F
1

F
x
2
x
1
(11)
For the pylon model, studied in Section 2.2, we can define the following linear transfer function
B
P
F
2
X
2
2
j
2
2
2
2
2
m
(12)
This transfer function can be identified with standard SI/SO (SingleInput
SingleOutput) if the excitation force can be measured or calculated, and if the interaction between the pylon and the cable can be neglected. If the latter is not true it is necessary to take the nonlinear effects into account for the pylon as well. Studying the coupled model in Section 2.2 we can formulate this as:
B
P
2
AE
x
0
F
1
F
2
l
(13)
Eq. (11) and Eq. (13) are illustrated in Fig. 3. The additional transfer functions for cable and pylon will, ideally, be realvalued and constant over the entire
43
frequency range since they represent the coefficients for the nonlinear terms.
In the proposed analysis technique, any arbitrary linear transfer function can be identified for both the cable and the pylon. However, in the present study, both cable and pylon are modeled with only one degree of freedom, respectively as shown in Eq. (10) and Eq. (12).
Cable
Pylon
X
1
F
1
B
C
X
2
B
P
2
4
l
+
F
x
2
x
1
F
AE
l
x
0
+
Figure 3. MI/SO models for identification of the linear systems and the nonlinear coefficients for the studied system. The left MI/SO model is valid for both the uncoupled system and the coupled system studied in Section 2.The right
MI/SO model can be used to identify the dynamic properties of pylon when the interaction with the cable is taken into account (Section 2.2).
F
2
Defining G
YX
as the crossspectral density matrix between output Y and inputs
{X}, and G
XX
as the autospectral density matrix between inputs (see Fig. 3.), a leastsquare solution can be determined at each frequency as shown in Eq.
(14). H
1
is the transfer matrix which in this case contains two components at each frequency; the linear system and the nonlinear coefficient.
H
1
G
YX
G
1
XX
(14)
Furthermore, a multiple coherence function [6] can be calculated as
2
m
G
YX
G
1
XX
G
YY
G
H
YX
(15)
The latter will be equal to one when the measured response can be totally explained by the measured inputs and the estimated transfer functions. Hence, the multiple coherence is useful when investigating the quality of the estimate or when searching for a possible model structure [6,7]. As an example, the ordinary coherence between the pylon force and the pylon displacement can be compared with the multiple coherence for the model in Fig. 3 to quantify what effect the interaction with the cable has on the overall behavior.
Eq. (14) is known as the H
1
estimate which implies that systematic errors due to additive noise on the output can be avoided. The physical meaning of this can
44
be difficult to interpret if the output is a synthesized signal created from quantities that are used as both inputs and outputs, as is the case with the identification of the cable properties. However, studying Eq. (11) it can be seen that any of the three terms (x
1
, x
1
3
or x
1
∙x
2
) may be used as an output in the
MI/SO model. When analyzing measurement data it may therefore be advisable to try different formulations in order to examine what effect modeling of the additive noise has on the overall result [7].
4. Simulation Results
In the numerical analysis, the cable and pylon characteristics of a bridgeexperimental scaled model used in a previous research by the authors [8] are considered. Such information is presented in Table 1.
For the uncoupled model (Section 2.1), the excitation, which is originally represented by an arbitrary function x
cs
, is given here by the displacement of a
SDOF system subjected to a force. The characteristics (stiffness and relative damping) of this SDOF system are given in Table 1.
Then, in order to establish a comparison between the results from both models, common conditions were considered: the cable was tuned at 70 Hz, the frequency ratio (pylon frequency/cable frequency) was equal to 2. The same excitation force is used for both the coupled and uncoupled model. This force was random and bandpassed filtered between 110 Hz and 170 Hz, i.e. a range that includes the natural frequency of the SDOF system.
Table 1. Cable and Pylon/deck properties
Cable Density, ρ, *kg/m
3
]
CableYoung Modulus, E [N/m
2
]
7800
205x10
9
Cable diameter, [m] 3x10
4
Cable length, l [m]
CableRelative Damping,
1
[%]
1.3
0.03
Pylon equivalent stiffness, k [N/m]
Pylon relative damping,
2
[%]
32848
0.22
45
The mass of the SDOF system is different for each model, since the pylon frequency expression differs, as seen in Eqs. (3) & (9), and it is considered a common pylon frequency (140 Hz) for the simulations.
4.1. Cable Response Simulations
The Eqs. (1) & (7) are solved through a methodology [9] carried out in MATLAB.
In this methodology the nonlinear components are treated as external forces acting on the linear systems. The linear systems are represented by SDOF systems, which are described by digital filters. For each time step a nonlinear equation or equations are solved recursively.
In Fig. 4 and Fig. 5 are shown the results from both models, given by the cable and pylon displacement and the corresponding kurtosis for several input force levels (N rms).
(a) (b)
Figure 4. Cable (a) and Pylon (b) displacement as function of pylon force rms.
(a) (b)
Figure 5. Kurtosis of Cable (a) and Pylon (b) displacement as function of pylon force rms
46
From these results, a clear nonlinear relationship between the cable response and the input force, is observed as seen in Fig. 4a. A sudden increase in the cabledisplacement amplitude is found when a particular force magnitude is reached at the pylon (0.11 N and 0.15 N; according to the uncoupled model and the coupled model, respectively). This kind of behavior was also noticed in a previous experimental work developed by the authors [8]. The high values of the kurtosis for the cable displacements in Fig. 5a could also confirm the nonlinear behavior of the cable within the force range.
On the other hand, the rms value of the pylon displacement and the corresponding kurtosis, for the coupled model, did not show a significant evidence of nonlinear behavior within the force range of the simulations, as seen in Fig. 4b and Fig. 5b.
4.2. System Identification Test
The identification techniques studied in Section 3 are here demonstrated using simulated time data. A random force signal, bandpassed filtered between 110
170 Hz and amplitude 1 N rms, is used to excite the coupled pylon model. A sampling frequency of 2000 Hz was used and enough data was simulated so that the spectral densities could be calculated with 2
15
in FFT blocksize and 72 averages with 50% overlap. The estimated linear systems are compared with the theoretical linear systems and shown in Fig. 6 together with the multiple coherence functions.
The nonlinear effects can clearly be seen when studying the raw FRF and the ordinary coherence calculated between excitation force and pylon responses.
By using the MI/SO models in Fig. 3 we can identify the true linear system for both cable and pylon (Fig. 6) and achieve a nearly perfect multiple coherence.
The small discrepancy between theory and simulation is due to spectral leakage which also explains why the multiple coherence is not exactly unity over the frequency range.
The estimated nonlinear coefficients are shown in Table 2 for some different excitation levels. These coefficients are calculated by taking the average value of the real part in the estimated transfer function (as shown in Fig. 3). As expected, a good estimate of the nonlinear coefficients can be obtained if the force level is large enough to excite the nonlinearities in the system. The result shown in Table 2 also indicates that the nonlinear coefficient for the cable can be found already at a very low excitation force.
47
10
6
Estimated Cable FRF
True Linear FRF
10
2
10
3
10
4
10
4
2
10
40 50 60 70
Frequency [Hz]
(a)
80 90
1
10
5
10
6
Estimated Pylon FRF
True Linear FRF
Raw FRF
100 120 140
Frequency [Hz]
(b)
1
0.99
0.8
0.98
Multiple Coherence
(cable)
0.6
160
0.97
0.4
0.96
0.2
Multiple Coherence, Pylon
Ordinary Coherence, Pylon
0.95
40 50 60 70
Frequency [Hz]
80 90
0
100 120 140
Frequency [Hz]
(c) (d)
160
Figure 6. Simulation results with an excitation force of 1 N rms. (a) Estimated linear system for cable compared with theory, (b) Estimated linear system for pylon compared with theory together with raw FRF, (c) Multiple coherence for
MI/SO cable model, (d) Multiple coherence for MI/SO pylon model together with ordinary coherence between excitation force and pylon response.
Table 2. Estimated coefficients for cable MI/SO model and pylon MI/SO model at different excitation levels.
Force Level
(N rms)
Nonlinear coefficient for cable, π
2
/(4l) = 1.86
Nonlinear coefficient for Pylon,
(AEβx
0
)/l = 20213
0.015
1.13
40430
0.020
1.72
22137
0.05
1.85
20302
0.25
1.86
20211
0.75
1.86
20213
48
5. Conclusions
The nonlinear interaction between a cable and a pylon/deck can lead to large cable vibrations. The pylon/deck displacements produce tensile oscillations in the cable that induce a parametric excitation of the system. This paper has studied two models of cableparametric resonance vibration. In the first model the excitation at the free cable end is given by an arbitrary displacement, while in the second model, the cable is considered coupled to the pylon/deck which is excited by a force. A comparison between the system responses from both models under random excitation was carried out and an identification technique was evaluated using numerical simulations.
A clear nonlinear relationship between the cable response and the input force is observed, characterized by a sudden increase in the cable displacement amplitude and in the corresponding kurtosis at a certain excitation force level.
Comparing the two models, the pylon response obtained is similar for a given excitation force. Even though both models predict the instability of the cable for a similar input force, a lower cable displacement amplitude was observed when the interaction with the pylon was taken into account. This amplitude difference in the cable response could lead to important results on modeling the cable parametric resonance, where a complementary experimental work might be needed in order to obtain more information about this phenomenon under the conditions considered for the simulations. The nonlinear response of the pylon in the coupled model was most clearly seen when studying the ordinary coherence between excitation force and pylon response. Furthermore, the studied parameter estimation technique proved to be capable to identify both the underlying linear system and the nonlinear coefficients when tested on numerical data. Future research and experimental testing will be carried out to validate this approach.
References
[1] J.Lilien and A.Pinto Da Costa. “Vibration Amplitudes caused by Parametric
Excitation of Cable Stayed Structures”, Journal of Sound and Vibration 174(1).
(1992) :6990.
[2] B.N.Sun, Z.G. Wang, J.M. Ko and Y.Q.Ni. “Cable Oscillation induced by
Parametric Excitation in CableStayed Bridges”
.
Journal of Zhejiang University of
Science 4(1).The Hong Kong Polytechnic University. (2003):1320.
[3] A.Dimarogonas and S.Haddad, Vibration for Engineers. Prentice Hall. New
Jersey, USA (1992).
49
[4] C.de Silva, Vibration. Fundamentals and Practice. CRC Press, USA (2000).
[5] J.S.Bendat, Nonlinear System Analysis and Identification from Random Data,
John Wiley & Sons, Canada (1990).
*6+ A.Josefsson, M.Magnevall and K.Ahlin. “Estimating the Location of Structural
Nonlinearities from Random Data”, IMACXXV Conference & Exposition on
Structural Dynamics, Orlando, USA (2008).
*7+ A. Josefsson, M.Magnevall and K.Ahlin. “On Nonlinear Parameter Estimation
With Random Noise Signals”. IMACXXV Conference & Exposition on Structural
Dynamics. Orlando, USA (2007)
[8] A.León, K.Ahlin and S.KaoWalter.“On Determining Instability Conditions for
Stay Cables Subjected to Parametric Resonance”. EVACES’09 Conference on
Experimental Vibration Analysis for Civil Engineering Structures. Wroclaw,
Poland (2009).
*9+ K.Ahlin, M.Magnevall and A.Josefsson. “Simulation of forced response in linear and nonlinear mechanical systems using digital filters”. ISMA 2006,
Leuven, Belgium.
50
Paper B
On Determining Instability Conditions for Stay Cables
Subjected to Parametric Resonance
51
Paper B is published as:
A. León, K. Ahlin and S. KaoWalter. “On Determining Instability Conditions for
Stay Cables Subjected to Parametric Resonance”. EVACES’09 Conference on
Experimental Vibration Analysis for Civil Engineering Structures. Wroclaw,
Poland (2009).
52
On Determining Instability Conditions for Stay Cables
Subjected to Parametric Resonance
Armando León, Kjell Ahlin and Sharon KaoWalter
Abstract:
Parametric Resonance Vibration in cables of cablestayed bridges is mainly studied when the excitation frequency is close to or twice the cable natural frequency. It is, however, important to consider other cases for this frequency relationship, since among other factors, cableparametric resonance vibrations are strongly depending on the displacement amplitude at the cable supports.
Consequently, the present research work is focused on determining, by experimental and numerical analysis, the instability conditions for stay cables subjected to parametric resonance within a wide range of frequency ratios. This is accomplished by finding the minimum displacement required at the cable supports in order to induce nonlinear vibration of considerable amplitude at the cable. Once the cable characteristics (geometry, material properties, inherent damping and initial tensile preload) are known, the instability conditions are identified and expressed in a simplified and practical way in a diagram. Numerical results are compared to those obtained by experimental analysis carried out on a simplified scaled model (1:200) of the Öresund Bridge.
A good agreement between numerical and experimental results is found.
Keywords:
Cable vibration, Instability Conditions, Parametric Resonance, Cablestayed bridges.
53
1. Introduction
When periodical displacements are registered at the cable supports in the tower and/or the deck of a cablestayed bridge, tensile oscillations are induced in the cables which can originate transversal vibrations characterized by large amplitudes. This phenomenon is known as CableParametric Resonance
Vibration or cable vibration due to parametric excitation and its mathematical background originally remains on the application of the Mathieu Equation [1].
Most specific numerical models describing the cable vibration under parametric excitation have been developed, showing a clear nonlinear relationship between the excitation amplitude at the cable support and the cablevibration amplitude [2, 3].
It is commonly found that CableParametric Resonance Vibration is evaluated when the excitation at the cable supports is at a frequency which is close to and/or twice the cable natural frequency; since in these cases, minimal excitation amplitudes could induce large amplitude vibration at the cables.
However, this phenomenon can also occur under other conditions. In fact, cable parametric resonance vibration can occur within a broad range of frequencies when larger excitation amplitudes are registered at the cable supports [2].
The instability conditions for a system subjected to parametric excitation are generally represented on graphics by regions as functions of nondimensional parameters [4, 5], being the frequency ratio, R (Excitation frequency normalized to the lowest natural frequency of the system) one of the most important variables to be considered. Since the studied phenomena here depends on different factors, the instability conditions can be found as a function of parameters such as the excitation amplitude at the cable supports, the cableinherent damping, ratio sag to span, etc. Therefore, to express in a general way, the instability conditions for cables is rather complicated. It implies the involvement of several parameters and as a consequence, several graphics are shown depending on those variables.
Instead, in the present work, the instability conditions are determined specifically for a cable in particular. Once the cable characteristics (geometry, material properties, inherent damping and initial tensile preload) are known, the corresponding governing equation is evaluated and, consequently, the instability conditions are identified and expressed into one curve. In such analysis, the instability conditions are given by the minimum displacement required at the cable support in order to produce nonlinear and significant amplitude vibration at the cable. This analysis is done in a rather short time thanks to a computational subroutine, offering the possibility to evaluate all cables from a cablestayed bridge in a relative short time as well.
54
The results here are obtained from numerical and experimental analyses based on a scaled model (1:200) of the Öresund Bridge, which is part of an important link that joins Sweden and Denmark through the cities of Malmö and
Copenhagen. In this bridge, large amplitude vibrations were found at the longest cable stays, and in a preliminary analysis, parametric excitation was considered as one of the vibration sources [6].
The numerical and experimental methodologies employed here are explained in section 2 and 3, respectively. The corresponding results are shown and discussed in chapter 4, while concluding remarks are presented in chapter 5.
2. Modeling cableparametric resonance vibration
2.1. Numerical Model
The numerical model employed here corresponds to the one proposed by Lilien and Pinto [2], in which, one cable end is kept fixed and the other one is free to move (See Figure 1).
Y
X cs l
Figure 1.A Cable Stay model studied by Lilien and Pinto [2].
The corresponding governing equation can be written as follows:
Y
b
ω
2
1
1
X cs
X
0
4
3
Y
K
2
Y
0
(1) where: Y cable transversal vibration; b – inherent cable damping coefficient;
ω1 first natural frequency of the cable; X cs
– displacement of cable support; X
0
– initial stretching of cable.
In equation (1), K is defined as follows:
K
4
π
X
0 l
3
(2) where: X
0
cable initial stretching; l original cable length.
55
The 1stcable natural frequency, ω1 is obtained from the theory of a taut string
[7], with n=1, as follows:
n
n
l
T
0
A
(3) where: T
0
the initial tensile load of the cable; ρ the cable density; A cable cross section area.
The initial stretching of the cable, X
0
, could be determined by Hooke’s Law by knowing the initial tensile load, T
0
, and its axial stiffness expressed on terms of the original cable length, l, its cross section area, A, and its Young Modulus, E, as follows:
X
0
T
0
L
AE
(4)
2.2. Cable characteristics, loading and excitation conditions
In order to carry out a numerical analysis based on the governing equation expressed by equation (1), it must be known first, the cable characteristics
(material, inherent damping and geometrical properties), its loading condition, which is given by the initial tension force, T
0
, and the characteristics of the excitation at the cable support, Xcs . Parameters as the 1stnatural frequency of the cable, ω
1
, and its initial stretching X
0
, are related to the cable characteristics and to the initial tension, T
0
, according to equations (3) and (4).
The geometrical, material properties and inherentrelative damping, ζ c
, of the cable are taken from an experimental model utilized here. These characteristics are shown in Table 1. Notice that, the cableinherent damping is determined as an average of the corresponding values found from experimental modal analysis.
Table 1. Cable Characteristics according to Experimental Model
Cable density,
[kg/m
3
] 7800
Cable Diameter [m] 3x10
4
Cable Elasticity Modulus, E [N/m
2
] 205x10
9
Cable Length, l [m]
Cable Relative Damping,
c
[%]
1.33
0.03
56
In the scaled model of the bridge, large displacement amplitude is expected at the cable support located near the top of the tower when the tower is vibrating under its 1stbending vibration mode (in plane), which is at 44.5 Hz, according to experimental modal analysis. Therefore, in order to conveniently have large enough displacements at the mentioned cable support, it was chosen to excite the bridge tower at its vibration mode. Such frequency represented the main excitation frequency for the parametricresonance vibration of the cable when developing the experiments.
Since the cable support motion, X cs
, is assumed to be periodical with a simple frequency, it can be written as:
X cs
X d
sin(
e t
)
(5) where: X d
 the excitation amplitude;  the excitation frequency; t the time.
The displacement amplitude, X d
, at the cable support is chosen to be swept from zero to an upper limit, which will depend on the dynamic characteristics of the bridge tower. This is in order to identify, within an amplitude range, the minimum required excitation at the cable support capable to induce a significant vibration at the cable. This is also a good choice when detecting nonlinear behavior between the excitation amplitude and the cable vibration.
The estimation of the frequency ratio, R, given by the excitation frequency and the natural frequency of the cable is also considered before carrying out the numerical analysis. Such ratio is as:
R
e
1
(6) where:  excitation frequency;  the cable1st natural frequency.
Since in the experiments the excitation frequency is chosen to be the first natural frequency (bending) of the bridge tower, then, the only way to experimentally vary the ratio R will be by changing the 1stnatural frequency of the cable, which can be reached by changing its initial tensile load, T
0
, and keeping fixed its geometrical and material properties.
However, it is important to indicate that in a practical situation, it will be expected a frequency ratio R, varying due to several excitation frequencies
(depending on the dynamic properties of the bridge and the environmental surrounding) and not because of changes at the cable natural frequency, as it was conveniently chosen for the experimental analysis carried out here.
57
2.3. Determining the instability conditions
The cable parametric vibration described by equation (1) is a NonLinear
Ordinary Differential Equation of second order. A methodology for solving this type of equations is based on recognizing the corresponding linear system and the nonlinear components, which, consequently, will be considered as external forces acting into the system [8].
Then, equation (1) can be reorganized and written as:
Y
..
b Y
.
1
2
Y
X cs
X
0
4
3
Y
K
2
Y
(7)
In equation (7), the first member is clearly defining a mechanicallinear system with a Single Degree of Freedom (SDOF), subjected to what is assumed to be an external force given by the right member of this equation.
(a) (b)
(c)
Figure 2. Numerical Results when ratio R is 0.75. (a) Time history simulation of
Excitation Amplitude at Cable Support. (b) Time history simulation of Cable displacement at its mid point. (c) Relationship between excitation amplitude and cable displacement amplitude.
58
The response, Y, to a given input, X cs
, is calculated from equation (7) using a routine where the SDOF system is described by a digital filter. In each time step, a nonlinear equation has to be solved [8].
Once the corresponding cable midpoint response is obtained, the next step is to identify the instability conditions for cable parametric resonance vibration. In this research, such conditions are represented by the minimum displacement required at the cable support in order to induce a nonlinear cable vibration of significant amplitude and the corresponding frequency ratio R.
In figure 2 is shown as example the numerical results when the ratio R is equal to 0.75, the excitation frequency is at 44.5 Hz and the geometrical and material properties of the cable are corresponding to those given in Table 1.
As seen in figure 2.a, the displacement at the cable support presents an amplitude swept from zero to an upper limit. After reaching an excitationamplitude of 1.55 x 103 m. is noticed how the cable displacement rises significantly and suddenly from almost zero displacement (see figures 2.b and
2.c). Therefore, this excitation amplitude is considered as the minimum displacement required at the cable support for inducing the instability, when the ratio R is 0.75.
Then, by repeating this process for each value of the frequency Ratio, R, the instability conditions can be determined for a frequency range in particular.
3. Experimental set up and methodology
The experimental set up used in this research is shown in figure 3. It is based on a simplified scaled model (1:200) of the Öresund Bridge, made of aluminum, whose cables were replaced by strings made of steel. Since, in the original bridge the largest vibration amplitudes were registered at the longest cables
[6], only this cable was mounted on the experimental set up.
The experimental set up is also constituted by a shaker, a signal generator, an amplifier and a force transducer, in order to introduce and measure the excitation force applied on one of the bridge towers; an accelerometer to estimate the cable support displacement on the bridge tower; and a laser vibrometer in order to estimate the displacements at the cable midpoint.
Initially, an experimental modal analysis was carried out on the scaled model.
Most of the information obtained is used as a data and reference for the numerical analysis.
The experiments for studying the cableparametric resonance vibration basically consisted in exciting the bridge tower where the cable is connected, by introducing a sinusoidal force, swept in amplitude and with a simple
59
frequency. Then, the displacements of the cable midpoint and cable support at the tower were estimated by using the laser vibrometer and an accelerometer, respectively. The cable displacements are measured in plane of the cable and not out of plane as represented in figure 3, which corresponds to a preliminary analysis [9].
Figure 3.Experimental Set up
The goal of the test is to identify the displacement amplitude at the cable support capable to induce a significant and nonlinear vibration at the cable when a particular relationship between the excitation frequency and the corresponding 1st natural frequency of the cable is obtained. This is possible by tuning the cable before the experiments to a particular frequency, while the excitation frequency is chosen to be the corresponding one to the 1stbending vibration mode of the bridge tower, for reasons that will be explained later.
4. Results & Discussion
4.1.
Numerical results
The instability conditions numerically estimated for a cable with characteristics according to table 1 are shown in figures 4 and 5. In figure 4 the frequency ratio, R, was varied through the 1st natural frequency of the cable and keeping the excitation frequency fixed at 44.5 Hz. Since in a practical situation it would be expected to have a frequency ratio, R, varying because of the excitation frequency, the instability conditions were also numerically estimated under this situation. Then, two cases were studied, when the cable natural frequency was fixed to 70 Hz and 100 Hz (Figure 5).
60
Figure 4. Estimated Instability Conditions for string (L=1.33m, Ø0.3mm) when excitation frequency is kept fixed at 44.5 Hz.
Figure 5. Estimated Instability Conditions for string (L=1.33m, Ø0.3mm) when
cable natural frequency is kept at 70 Hz and 100 Hz.
As it was expected, according to figures 4 and 5, the cable is found to be more prone to vibrate under parametric resonance when the frequency ratio, R is equal to 2 than in any other condition given by this frequency ratio. Under this case, the parametric resonance vibration is induced by the lowest excitation amplitude within the range of frequency in consideration. In figures 4 and 5 the instability conditions curve also shows dropping points around frequency ratios equals to R=2, R=1, R=0.67, R=0.5, R=0.4 and R=0.33. These values correspond
61
to the frequency ratios which define the unstable regions for parametric resonance vibration [2].
By obtaining a curve for the instability conditions as shown in figures 4 and 5, it could be possible to determine if such cable would vibrate due to parametric excitation, once the characteristics (frequencies and amplitudes) of the excitation at the cable supports are known.
Firstly, by considering the different vibration mode frequencies of the bridge, it will lead to determine the corresponding relationships (frequency ratios, R,) between the natural frequency of the cable and the different possible excitation frequencies [10, 11]. Then, by estimating a dynamic response of the bridge according to its modal properties and the expected external forces to be acting on (wind, traffic loads), the excitation displacement at the cable supports could be known. Therefore, a curve as shown in figure 5 will be a useful tool, since what remains is to check if the excitation amplitude at the cable support is large enough for inducing parametric resonance vibration of the cable, under the corresponding frequency ratio R.
One important advantage of this method, is that offers, by simply looking at one curve, the possibility to evaluate a cable under a broad range of frequency ratios and not only when R is equal to R=2, R=1 or R=0.5. The determination of the instability condition curve for each cable is rather fast, since it takes approximately 30 minutes by running a script in MATLAB on a regular PC. Then, the checking process for all cables in a bridge could also be done in a rather short time.
4.2. Experimental Results
Experimental evaluations under conditions defined by a frequency ratio, R equal to 0.48, 0.5, 0.59, 0.64, and 0.66, were carried out in order to confirm the corresponding numerical results. The instability conditions could be determined from the relationship between the cable response and the corresponding excitation at the cable support.
In all cases, except when the frequency ratio was 0.48 and 0.50, the cable vibration could reach the instability at certain moment, characterized by a large amplitude displacement. However, when R was 0.48 and 0.50, the cable could also reach large amplitude vibration, but in direct proportion to the excitation amplitude. That occurred at excitation amplitudes much smaller than those numerically estimated in order to induce the instability of the cable vibration. In the case of R=0.5, for example, the numerical calculation gives 1.65 mm as minimum excitation amplitude in order to induce such instability (See figure 4).
62
However, in this case, large amplitude vibration occurred by applying about 3 times smaller excitation amplitudes.
This could be explained by looking at the spectrum of the cable response. In general for the experiments where the instability was reached, it was observed a first stage previous to the instability, where the cable used to vibrate, in proportion to the excitation amplitude, at the same frequency of the excitation and its corresponding harmonics. Then, when the instability was already reached, the vibration was characterized by multiple frequencies (See figure 6, in which R=0.66). In the cases where R was 0.48 and 0.50, the cable vibrates, in proportion to the excitation amplitude, at the excitation frequency and at its
1st harmonic which is coincident with the 1st natural frequency of the cable.
That could explain why the cable could easily reach large amplitude vibration with much smaller excitation amplitudes than those required in order to induce the parametric resonance vibration.
(a) (b)
Figure 6. Experimental Cable Response Spectrum when R=0.66. (a) Before the instability occurs. (b) When the instability already occurs.
In figure 7 are shown the experimentalcable responses when R=0.59, R=0.64 and R=0.66 and the excitation frequency was kept at 44.5 Hz.
63
(a) (b)
(c) (d)
(e) (f)
Figure 7. Cable Response and excitation displacement for frequency ratios
R=0.59, R=0.64 and R=0.66.
As seen in figure 7, the excitation amplitude at the cable support was gradually swept. At certain moment and for a particular value of the excitation amplitude the instability in the cable vibration could be reached. Then, an experimental determination of the corresponding instability conditions could be done. These values were clearly obtained by establishing a relationship between the excitation amplitude and the cable amplitude displacement.
Then what remains is look at the excitation amplitude capable to induce the sudden jump in the cable vibration amplitude (See figure 8).
64
Figure 8. Experimental Relationship between Excitation Amplitude and Cable
Displacement when R=0.59, R=0.64 and R=0.66.(Cable: L=1.33m, Ø0.3mm).
4.3. Comparison between Numerical and Experimental
Results
A comparison between the instability conditions obtained by numerical and experimental analysis is shown in table 2.
Table 2. Experimental and Numerical Instability conditions around R=0.67,
Excitation Freq.=44.5Hz (Cable: L=1.33m, Ø0.3mm)
Freq. Ratio, R Excitation Amplitude by Experiments
1x103 [m]
0.59 1.61
0.64
0.66
0.83
0.77
Excitation Amplitude
by Numerical Analysis
1x103 [m]
1.79
1.05
0.71
Difference
[%]
10.06
20.95
8.45
In practice, it is challenging to define exactly a frequency ratio when doing the experiments, since a small variation in the measured cable resonance frequency or even in the excitation frequency can lead to evaluate a slight different value for the expected frequency ratio. From the instability curves shown in figures 4 and 5, it is noticed that the curves drop and rise sharply around the critical frequency ratios (e.g, R=2, R=1, R=0.67, R=0.5, R=0.4, etc.).
That could explain why a maximum difference of 21% between the numerical and experimental results was obtained when the ratio R was 0.64.
As a complementary test, a case when R=1 was also evaluated. In this case a cable of 1 mm in diameter was used. As in the case of R=0.48 and R=0.5, the cable vibrated reaching large amplitudes in a direct proportion to the
65
excitation. The main vibration occurred at the excitation frequency which matches the natural frequency of the cable. The excitation amplitudes applied were also much smaller than the numerically estimated in order to induce the instability of the cable vibration.
5. Conclusions
Instability conditions for a ble subjected to parametric resonance vibration could be numerically estimated for a broad range of frequency ratios and expressed into one curve. In this way, it could be known if any cable in a cablestayed bridge would vibrate due to parametric excitation, once the cable characteristics are defined, as well as the conditions of the excitation at the cable support. The process for determining the instability conditions within a broad range of frequency ratios is rather fast and offers as an advantage the possibility of evaluating, in a practical situation, all frequency ratios under which each cable is really subjected to, and not only when such ratio is R=2,
R=1 and R=0.5.
According to the experimental analysis, cables under frequency ratios equals to
0.5 and 1 could develop large vibration amplitudes without reaching any instability. In fact, a proportional relationship between the cable response and the excitation was observed where large vibration amplitudes could be obtained with a much lower excitation amplitude than that one required in order to induce the instability in the cable. This result could reinforce or justify the tendency of only evaluating cables under frequency ratios equal to 0.5 and
1, besides R=2, where is clearly known as the most critical condition for parametric resonance vibration at the cable.
Although the estimated instability conditions and the experimental results confirmed that cables are more prone to vibrate under frequency ratios close or equal to R=2, R=1 and R=0.5 than in other cases, it is still important to consider other conditions, since the excitation amplitudes at the cable support could be large enough for inducing the parametric resonance vibration of the cable for one or several frequency ratios R in which the cable could be subjected to.
6. References
[1] Dimarogonas A. and Haddad S. Vibration for Engineers. Prentice Hall.
New Jersey, USA. (1992)
66
[2] Lilien J. and Pinto Da Costa A. “Vibration Amplitudes caused by
Parametric Excitation of Cable Stayed Structures”. Journal ofSound and
Vibration, 174(1) (1994): 6990.
[3] Sun B. N., Wang Z.G., Ko J.M. and Ni Y.Q. “Cable Oscillation induced by
Parametric Excitation in CableStayed Bridges. Journal of Zhejiang
University Science. Vol.4, N.1. The Hong Kong Polytechnic University.
(2003):1320.
[4] Takahashi K. “An approach to investigate the instability of the multipledegreeoffreedom parametric dynamic systems”. Journal of Sound and
Vibration, 78(4). (1981):519529.
[5] Takahashi K. “Dynamic stability of cables subjected to an axial periodic load”. Journal of Sound and Vibration, 144(2), (1991):323330.
[6] Svensson B., Emmanuelsson L. and Svensson E. “Øresund BridgeCable
SystemVibration IncidentsMechanism and Alleviating Measure”.
Øresundsbrokonsortiet (2004).
[7] de Silva, C. Vibration. Fundamentals and Practice. CRC Press, USA.
(2000).
[8] Ahlin K., Magnevall M. and Josefsson A. “Simulation of forced response in linear and nonlinear mechanical systems using digital filters”. ISMA
2006, Leuven, Belgium. (2006)
[9] León A. Study on Vibrations induced by Parametric Excitations on
Strings. Master Thesis. Blekinge Institute of Technology, Karlskrona,
Sweden (2007).
[10] Caetano E., Cunha A., Gatulli V. and Lepidi M. “Cabledeck dynamic interactions at the International Guadiana Bridge: Onsite measurements and finite element modelling”. Structural Control and
Health Monitoring15. (2008): 237264.
[11] Wu Q., Takahashi K., Okabayashi T. and Nakamura S. “Response characteristics of local vibrations in stay cables on an existing cablestayed bridge”. Journal of Sound and Vibration, 261. (2003):403420.
67
68
Paper C
Finite Element Simulations on the determination of the Dynamic Stress Intensity Factor
69
Paper C is submitted for publication as:
A. León and S. KaoWalter. “Finite Element Simulations on the determination of the Dynamic Stress Intensity Factor”, Submitted for publication, February 2011.
70
Finite Element Simulations on the determination of the Dynamic Stress Intensity Factor
Armando León and Sharon KaoWalter
Abstract:
Finite Element simulations on a precracked threepoint bending specimen under impact loading are performed. The simulations are based on two experimental results obtained by using the Split Hopkinson pressure bar (Jiang et al). With the measured loading force as input, the mid span displacement is calculated and a good agreement with the experimental results is observed.
The dynamic stress intensity factor, K
I
(t) up to crack initiation is then obtained by different methods based on the specimen’s mid span displacement, crack tip opening displacement (CTOD) and crack mouth opening displacement (CMOD).
A good agreement with the experimental results is observed by using these methods. A model for describing the specimen’s midspan displacement by a linear massspring system is also evaluated. By an unconstrained optimization procedure, it is shown that according to this model, a proper estimation of the dynamic stress intensity factor relies on the formulation of the specimen stiffness.
Keywords:
Dynamic Stress Intensity Factor, crack initiation, threepoint bending specimen, simulations.
71
1. Introduction
In engineering, different failure criteria can be applied when designing a structural component. They are commonly based on comparing the maximum expected stress in the component to the material’s resistance properties.
Failure due to fracture is one of these engineering criteria, and in this case, the stress intensity factor, K
I
, a characterizing parameter around the component’s crack tip, is compared to the material’s resistance to fracture, the socalled fracture toughness. Different techniques have been developed in order to estimate the fracture toughness under static and dynamic conditions. In many of these, the corresponding tests and simulations are carried out on a precracked threepoints bending (3PB) specimen.
Under static conditions, there already exists an ASTM standard test (E39990), where K
I
is expressed in terms of the load and the specimen’s geometry.
However, the specimen’s response under dynamic conditions represents a much more complex case. The respective transient response is influenced by the specimen’s rotary inertia. Therefore, theories on dynamics, wave propagation and vibration should also be included in order to describe this response. As a consequence, a standard test under dynamic conditions is still not found, but different numerical and experimental methodologies are proposed by different authors.
The instrumented Charpy test has been used [1] to estimate the dynamic fracture toughness in materials when the loading rate is lower than 10
5
MPa m s
1
. For higher loading rates, most of the experimental setups have been based on the split Hopkinson pressure bar [2,3,4,5]. By this means, the dynamic load can be measured, as well as the crack initiation’s time. Then, the dynamic stress intensity factor, K
I
(t), and the material’s fracture toughness, K
Id
, can be estimated by applying different formulations that depend on the load, geometrical properties and the stiffness of the specimen.
Most of these formulations are based on static fracture mechanics and have been derived from the EulerBernoulli or Timoshenko beam theory [6].
Analyses on an alternative model, where the displacement of the specimen’s midspan is described by a linear massspring system [4], have also been done.
Other formulations which depend on the crack tip opening displacement
(CTOD) and the crack mouth opening displacement (CMOD) have been used to estimate K
I
(t). Usually, such displacements are obtained from simulations by using the Finite Element Method (FEM)[3,5]. The CMOD has also been experimentally determined with the support of highspeed cameras [3,7].
72
The aim of the present work is to describe the dynamic response of a 3PB specimen subjected to impact load and estimate its corresponding dynamic stress intensity factor. Finite Element simulations are performed based on an existing experimental work carried out on a modified Hopkinson pressure bar
[4]. Different formulations and methods are applied in order to determine K
I
(t).
Results comparable with the experimental ones are observed by using these methods. A model for describing the specimen’s midspan displacement by a linear massspring system is also evaluated. By an unconstrained optimization procedure, it is shown that according to this model, a proper estimation of the dynamic stress intensity factor relies on the formulation of the specimen stiffness.
The different formulations to estimate K
I
(t) are discussed in section 2. The characteristics of the Finite Element simulations are shown in section 3, while the results are presented and discussed in section 4. Finally, the concluding remarks are presented in section 5.
2. Dynamic Stress Intensity Factor, K
I
(t)
The Dynamic Stress Intensity Factor, K
I
(t), is estimated here according to different formulations that involve theoretical, numerical and experimental analyses .
2.1. K
I
(t) based on the specimen’s midspan displacement
Under the firstbending vibration mode of a cracked specimen, K
I
(t) is proportional to the specimen’s midspan displacement, u(t), [8]. Then, dynamic stress intensity factor can be written as:
(1) where C is the proportionality constant, based on static fracture mechanics as follows [9]:
(2)
In Eq.(2), S is the specimen’s span, a the crack length, B the specimen’s thickness, and W is the specimen’s width (figure 1); while Y(a/W) is a calibration function [4] and K(a) the stiffness of the cracked specimen.
73
Figure 1. A cracked3PB specimen.
A formulation for the stiffness K(a) of a cracked specimen under planestrain conditions, where the rotary inertia and the shear deformation of the specimen are taken into account, is as follows [4,6]:
(3) where V(a/w) is the geometrical factor of the cracked specimen [4] and
(4)
(5) where
is the material’s Poisson ratio.
2.1.1. Equivalent linear massspring model
In order to estimate the displacement, u(t), in Eq.(1), a 3PBspecimen could be modeled [4,8] as a linear massspring system, which could equivalently describe the displacement at the specimen’s mid span (Figure 2).
P(t) u(t)
K
Figure 2. Linear massspring model describing the specimen’s midspan displacement.
The dynamic response of this linear system is obtained by simply applying
Newton’s second law. The corresponding governing equation is as follows:
(6)
74
with the following initial conditions:
Then, u(t) can be expressed as:
u
(
t
)
1
1
m e t
0
P
(
) sin
1
(
t
)
d
(7) where
is the 1
1 st
natural frequency of the specimen given by:
1
K m e
(8)
In Eq.(6), the stiffness, K, can be estimated according to Eq.(3); while the equivalent mass, m
e
is given by [4]:
(9) where is the material density of the specimen.
Theoretically, an impulse force could be simplified as a half of a sine function.
Then, the applied force on the system can be written as:
, (10) where P
o
is the force amplitude and D is the corresponding duration.
Consequently, a theoretical expression of K
I
(t)[4] can be given by applying Eqs.
(1), (2) ,(7) and (10) as follows:
, when p
1
= (11)
, when p
1
≠ (12) where
p
1
and K
ISmax is the maximum staticstress intensity factor, which
D
corresponds to the force P
o
.
2.2. K
I
(t) based on CTOD
The Dynamic Stress Intensity Factor can also be calculated from the crack tip opening displacement (CTOD) as [10]:
75
(13) where G is the shear modulus of the material, r is the distance between the crack tip and where the transversal displacement is measured (see figure 3),
v(t) is the transversal displacement and
k
3
1
, for planestress conditions (14)
k
3
4
, for planestrain conditions (15)
In Eq.(13) the displacement v(t) is half of the crack opening displacement.
CMOD
Figure 3. Crack Tip Opening Displacement (CTOD) and Crack Mouth Opening
Displacement (CMOD).
2.3. K
I
(t) based on CMOD
The K
I
(t) can also be estimated from the crack mouth opening displacement
(CMOD)[3]by
K
Id
Ew m k
4
(
a
v
)
(16) where w
m
is the crack mouth opening displacement (Figure 3), E is the Young
Modulus, a is the crack length, α=a/W and β=S/W. The nondimensional functions
k
(
) and
v
(
) are written as:
k
(
)
1
1
3
1 .
9
0 .
41
0 .
51
2
0 .
17
3
(17)
v
(
)
0 .
76
2 .
28
3 .
87
2
2 .
04
3
0 .
66
1
2
(18)
It is important to mention that in practice, the CMOD is more feasible to obtain than the CTOD. The CMOD can clearly be measured on photographs taken by highspeed cameras [3,7].
76
3. Finite Element Simulations
Simulations by using the Finite Element Method (FEM) are performed to estimate the dynamic response of a 3PB specimen under impact loading conditions. The simulations are based on the experimental work done by
Fengchun et al [4].Two measured forces, P exp
(t) (figure 4) in these experiments are used as input loads in the simulations. The specimen characteristics are as in the experiments, made of a high strength steel (Fe10Cr), with Young modulus, E =210 GPa and Poisson ratio, ν=0.3. The specimen is 4 mm thick with
6.5 mm in width and 40 mm in length. The distance between the simple supports is 25 mm and the crack length is half of the specimen width
(a/W=0.5).
Experimental Force
5000
Experiment 1
Experiment 2
4000
3000
2000
1000
0
2 1 0 1 2 3 4 5 6 7 8
Figure 4. Measured force, P exp
Time [s] x 10
5
(t),[4] used for the FEMsimulations.
The simulations are carried out by using the commercial software, ABAQUS.
Due to symmetry, only half of the specimen is modeled (figure 5). A spider web mesh around the specimen’s crack tip was built. The mesh was constituted by quadrilateral linear explicit elements under plane strain conditions. The analysis is carried out by an explicit dynamic step, which is convenient [11] for short duration forces, such as impact loads. In this analysis, the crack propagation is not considered.
77
Figure 5. Modelling a 3PB specimen by FEM.
From the simulation results, K
I
(t) could be estimated in terms of the specimen’s midspan displacement, CTOD and CMOD. This is according to Eqs.(1), (13) and
(16), respectively.
4. Results and Discussion
4.1. Midspan displacement
The figures 6a and 6b show the estimated and experimental results [4] of the specimen’s midspan displacement. The estimations of the specimen’s midspan displacement are according to the linear massspring model (Section
2.1.1) and the simulations that are carried out by FEM (Section 3).
According to the linear massspring model, the specimen’s midspan displacement is given by the Eq. (7). The respective integral is here numerically solved by using MATLAB. For the specimen studied here, the stiffness and mass were K(a)=1.8873x10
7
N/m and m=0.002463 kg, calculated according to Eqs.(3) and (9). Two cases were evaluated by using the linear massspring model: by considering the force as obtained in the experiments, P exp
(t) (figure 4); and by considering these experimental forces as half of a sine function, as given by Eq.
(10).
78
2
1
4
3 x 10
4
Midspan displacement
[Force 1]
Massspring m.,Eqs(7)&(10)
Massspring m.,Eq(7)& Pexp
Experiment
FEM
2
1
4
3 x 10
4
Midspan displacement
[Force 2]
Massspring m.,Eqs(7)&(10)
Massspring m.,Eq(7)& Pexp
Experiment
FEM
0
0 1 2 3 4 5
0
0 1 2 3 4 5
Time [s] x 10
5 Time [s] x 10
5
Figure 6.Displacements at specimen’s midspan. (a) Specimen under force 1. (b)
Specimen under force 2.
It can be seen in figures 6a and 6b that the simulation results by FEM are comparable with the experimental ones. On the other hand, the predicted responses, according to the massspring model, are only comparable with the experimental results when the force is considered as the experimental one,
P exp
(t). This rather good agreement does not occur when the force is approximated as a half of sine by Eq.(10). As a consequence, this way to obtain the specimen’s midspan displacement will not be considered in further estimations of K
I
(t).
The differences or errors between the predicted and the experimental results are presented in table 1. The difference between the two results is given by the standard deviation written as:
u i est
u i
exp
2
N
(19) where N is the number of samples.
In this case, N was the number of samples within 0 to 50 x 10
6
seconds.
In spite of the comparable results, it is still important to mention that within an initial time period the predicted responses were lower than the experimental ones. This may indicate the need of introducing some modifications into the described linear massspring model and into the FEMsimulations.
79
Table 1.Differences between experimental and predicted midspan displacements.
Predicted Response
Error ,
[m]
Error ,
[m]
(With respect to Experiment 1)
Massspring m., Eqs.(7) &(10) 0.5171 x 10
4
Massspring m., Eqs.(7) & P exp
0.1619 x 10
4
FEM 0.1194 x 10
4
(With respect to Experiment 2)
0.8598 x 10
4
0.1980 x 10
4
0.1212x 10
4
4.2. Evaluation of a linear mass spring model
As mentioned in section 2.1, the specimen’s midspan displacement can also be described by a linear massspring model. The effect of the rotary inertia and shear deformation of the specimen on its own dynamic response are considered for the formulation of the specimen’s stiffness in Eq. (3).
However, it is noticed in figure 6 that the response given by this theoretical linear massspring model presents some differences with the experimental and
FEMsimulation results. Then, a reasonable question is how properly this linear model and the respective formulations of the mass and stiffness describe the dynamic response of the specimen.
A way to evaluate this linear model is by a fitting process, where the properties
m and K(a) are determined for a linear massspring system, whose response properly fits the midspan displacement estimated by FEM and by the experiments. Then, the obtained values of K(a) and m can be compared to the theoretical ones given by Eqs.(3) and (9). In this fitting process the Nelder and
Mead [12] unconstrained optimization procedure is carried out in MATLAB. The properties m and K(a) are found by iteratively solving the Eq. (6) until obtaining a minimal error between the given and the fitted response. This error is represented by the summed square of residuals.
Figure 7 shows the obtained fitting curves from the FEMsimulation and experimental results, according to this procedure. Table 2 shows the corresponding values found for the stiffness K(a) and mass m. The theoretical values are also shown in this table together with the respective fitting errors,
(
m
,
k
) estimated according to Eq.(19).
80
2
1.5
1
3.5
x 10
4
3
2.5
(Force 1)
FEM
Fitted Curve
2
1.5
1
3.5
x 10
4
3
2.5
(Force 2)
FEM
Fitted Curve
2.5
2
1.5
1
0.5
0.5
0
0
3.5
x 10
4
3
1 2 3 4
0
0 1 2
Time [s] x 10
5 Time [s]
(a) (b)
(Force 1)
3.5
x 10
4 (Force 2)
Experimental
Fitted Curve 3
2.5
2
1.5
1
3 4 x 10
5
Experimental
Fitted Curve
0.5
0.5
0
0 1 2 3 4
0
0 1 2
Time [s] x 10
5 Time [s]
(c) (d)
3 4 x 10
5
Figure 7. Fitting curves according to a linear massspring model.(a) For FEM results, with force 1. (b) For FEM results, with force 2. (c) For experimental results, with force 1. (d) For experimental results, with force 2.
As seen in figure 7, the mass values obtained from the experimental and FEMsimulation results, according to this fitting process, are in the same order as of the theoretical one. However, different stiffness values with respect to the theoretical one are found. From the simulations and experiments a higher stiffness than according to the theory given by Eq.(3) is obtained. It is also noticed that the linear massspring model better fits the FEM simulationresults than the experimental ones.
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Table 2. Mass and stiffness obtained by a fitting process.
m
[kg]
K(a)
[N/m]
Fitting Error
(
m
,
k
)
, [m]
Theory, 0.002463 1.8873x10
7

Eqs.(3)&(9)
FEM (force 1) 0.002661 2.2325 x10
7
0.1273x10
4
Experiments 0.002477 2.3943 x10
7
0.8860 x 10
4
(force 1)
FEM (force 2) 0.002691 2.1644 x10
7
0.1008 x10
4
Experiments 0.002016 2.4154 x10
7
0.6825 x10
4
(force 2)
These observations may indicate that some modifications into the linear model and FEM simulations should be introduced to get results closer to the real response. So far, it can be seen that the difference among the results relies on the estimated specimen’s stiffness. Then, this can be the variable that needs to be reformulated within the linear massspring model. It is possible that among different factors, the specimen’s stiffness can be affected by the contact conditions with the supports.
4.3. K
I
(t) estimations
After obtaining the specimen’s dynamic response, K
I
(t) can be estimated in terms of the midspan, the crack tip opening displacement (CTOD) and the crack mouth opening displacement (CMOD). These estimations are shown in figures 7a and 7b.
The estimation of K
I
(t) according to the specimen’s midspan displacement is by the Eq.(1). Three different estimations of K
I
(t) in terms of the midspan displacement, one based on the CTOD and one on the CMOD are presented in figure 7. The midspan displacement was obtained as a measurement from the experiments; by numerically solving the integral in Eq.(7) with the measured force, P exp
(t); and by obtaining it from the FEMsimulations. The corresponding proportionality constant, C, is calculated according to Eq.(2) with a stiffness,
K(a), given by Eq. (3).
82
20 x 10
7
Dynamic Stress Intensity Factor, KI(t)
[Force 1]
15
10
Midspan,Eq(7)& Pexp
Midspan (Experiment)
Midspan (FEM)
CTOD(FEM)
CMOD(FEM)
5
20 x 10
7
Dynamic Stress Intensity Factor, KI(t)
[Force 2]
15
10
Midspan,Eq(7)& Pexp
Midspan (Experiment)
Midspan (FEM)
CTOD(FEM)
CMOD(FEM)
5
0 0
5
0 1 2 3 4 5
5
0
Time [s] x 10
5 Time [s]
Figure 7. Estimation of K
I
(t).(a) Specimen subjected to force 1. (b) Specimen subjected to force 2.
1 2 3 4 5 x 10
5
The K
I
(t) estimations based on the CTOD and the CMOD are done by using the
Eqs. (13) and (16), respectively. The CTOD and CMOD at the specimen are obtained from the FEMsimulations. As mentioned in section 3, in the FEMsimulations the measured load in the experiments was used as the input force into the 3PB model.
As seen in figure 7, in general, the different methods lead to a rather similar solution. In order to compare these results, the estimation of K
I
(t) based on the measured midspan displacement is chosen as a reference. This estimation is considered here as the experimental determination of K
I
(t). Table 3 shows the error, according to Eq. (19), between the experimental results and the different estimations of K
I
(t). The error is calculated in a range from 0 till 40x10
6 s. It is noticed that the estimations based on the CTOD and midspan displacement by
FEM are the closest ones to the experimental results. In spite of this result, it is important to point out that in practice, measuring the CTOD results complex. In fact, measuring the CMOD results more feasible than measuring the CTOD.
It is observed that all methods give a K
I
(t) lower than the experimental one within an initial range of time. After that, the respective curves, except the one based on CMOD, intersect the experimental one. This observation was also reported by Rubio et al [3], also indicating that the intersection of the curves corresponded to the time of crack initiation.
83
Table 3.Differences between experimental and predicted K
I
(t).
K
I
(t) estimation method
Error ,
[m]
Error ,
[m]
(With respect to Experiment 1)
Massspring m. [Eqs.(7) & P exp
] 0.9701 x 10
7
Midspan [FEM] 0.7158 x 10
7
CMOD [FEM]
CTOD [FEM]
1.0255 x 10
7
0.7174 x 10
7
(With respect to Experiment 2)
0.7780 x 10
7
0.7594 x 10
7
1.1396 x 10
7
0.6894 x 10
7
5. Conclusions & Future work
After studying and estimating by different methods the dynamic response and the corresponding dynamic stress intensity factor of a specimen subjected to impact load, the following conclusions are obtained:
The specimen’s midspan displacements predicted by the Finite
Element simulations are comparable with the experimental results.
According to the linear massspring model, the midspan displacements results were also comparable with the experimental ones, but only when the measured force was considered in the respective estimations.
Assuming the impact load as half of sine did not lead to good results when applying this model.
The linear massspring model was evaluated by a fitting process with the experimental and simulation results. Stiffness values different to the theoretical ones were obtained from these results and by this fitting process. This difference may indicate that some modifications are needed for the formulation of the specimen’s stiffness.
From the different methods used to determine K
I
(t), the estimations based on the CTOD and midspan by FEM were the closest ones to the experimental results.
It was observed that all methods give a K
I
(t) lower than the experimental one within an initial range of time. After that, the respective curves intersect the experimental one. This observation was also reported by Rubio et al [3], indicating that the intersection of the curves corresponded to the time of crack initiation. This difference with respect to the experimental results indicates that further work is needed in order to properly describe the dynamic stress
84
intensity factor. Studies on the relationship between the CMOD and K
I
(t), nonlinear models for describing the specimen’s midspan displacement and the crack propagation are some of the suggested topics that can contribute to obtaining a better description of the dynamic stress intensity factor.
6. References
[1] Kalthoff, J.F. ”On the measurement of dynamic fracture toughnesses  a review of recent work.” International Journal of Fracture 27, nr 34
(1985): 277298.
[2] Yokoyama, T. ”Determination of dynamic fractureinitiation toughness using a novel impact bend test procedure.” Journal of Pressure Vessel
Technology, Transactions of the ASME 115, nr 4 (1993): 389397.
[3] Rubio, L., J. FernandezSaez, och C. Navarro. ”Determination of dynamic fractureinitiation toughness using threepoint bending tests in a modified Hopkinson pressure bar.” Experimental Mechanics 43, nr 4
(2003/12/): 37986.
[4] Jiang, Fengchun, A. Rohatgi, K.S. Vecchio, och J.L. Cheney. ”Analysis of the dynamic responses for a precracked threepoint bend specimen.”
International Journal of Fracture 127, nr 2 (2004/05/): 14765.
[5] RubioGonzalez, C., J.A. GallardoGonzalez, G. Mesmacque, och U.
SanchezSantana. ”Dynamic fracture toughness of prefatigued materials.” International Journal of Fatigue 30, nr 6 (2008/06/): 105664
[6] Kishimoto, K., M. Kuroda, S. Aoki, och M. Sakata. ”Simple formulas for dynamic fracture mechanics parameters of elastic and viscoelastic threepoint bend specimens based on Timoshenko's beam theory.”
Proceedings of the 6th International Conference on Fracture (1984)//.
317784.
[7] Kao, H.R."Crack growth initiation of a quarter notched 3PB ductile specimen under high velocity impact". Licentiate thesis. Lund Institute of
Technology, 1991.
[8] GE, Nash. "Analysis of the forces and bending moments generated during the notched beam impact test." International Journal of Fracture
Mechanics v5,no. 4 (1969): 269286.
[9] Bacon, C., J. Farm, and J.L. Lataillade. "Dynamic fracture toughness determined from loadpoint displacement." Experimental Mechanics 34, no. 3 (1994): 217223.
85
[10] Anderson, T. L. Fracture mechanics : fundamentals and applications. 3. ed. Taylor & Francis, 2005.
[11] ABAQUS Documentation version 6.7
[12] MATLAB Computing Software v 7.0 (R14), 2005
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abstract
In the present work, the nonlinear vibrations and the corresponding dynamic fracture mechanics of cables of cablestayed bridges are studied. The cables are among the most critical components in cablestayed bridges and there are different damage sources such as corrosion, vibration, fatigue and fretting fatigue that can significantly affect them, thereby reducing the cable’s service life and even producing their failure.
CableParametric Resonance is the specific nonlinear vibration studied in this research. This type of vibration occurs due to displacements presented at the cable supports. These displacements are induced by the wind and traffic loads acting on the pylon and deck of the bridge. Under certain conditions, unstable cablevibration of significant amplitude can be registered. Therefore, numerical and experimental analyses are carried out in order to describe this phenomenon and to determine the corresponding instability conditions.
Two nonlinear models of cableparametric resonance are studied to predict the cable response.
In the simulation method, the nonlinear components are treated as external forces acting on the linear systems, which are represented by Single
Degree of Freedom systems and described by digital filters. A clear nonlinear relationship between the excitation and the cable response is observed in the simulations and the experiments.
The corresponding experimental analysis is based on a scaled model (1:200) of the Öresund bridge and a good agreement between the numerical and experimental results is found.
After obtaining the relationship between the cable response and the excitation, the cable instability conditions are determined. This is done by finding the minimum displacement required at the cable supports in order to induce nonlinear cable vibration of considerable amplitude. The instability conditions are determined within a wide range of excitation frequencies and conveniently expressed in a simplified and practical way by a curve. The determination process is rather fast and offers the possibility to evaluate all bridge cable stays in a rather short time.
Finally, the dynamic fracture mechanics of the cable is considered by studying the fracture toughness characteristics of the material under dynamic conditions. Finite Element simulations on a precracked threepoint bending specimen under impact loading are performed. The observed cable instability is equivalently considered as the associated response to impact load conditions, and a crack as a defect on the wires of a cable stay. The simulations are based on an experimental work by using the Split Hopkinson pressure bar (Jiang et al). The dynamic stress intensity factor KI(t) up to crack initiation is then obtained by different methods. The numerical estimations based on the specimen’s crack tip opening displacement
(CTOD) and midspan displacement were closest to the experimental results. It is observed that a better estimation of the dynamic stress intensity factor relies on a proper formulation of the specimen’s stiffness.
2011:02
ISSN 16502140
ISBN 9789172952010
NoNLiNear VibratioN aNd dyNamic
Fracture mechaNics oF bridge cabLes
Blekinge Institute of Technology
Licentiate Dissertation Series No. 2011:02
School of Engineering
Armando Leon
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