DSpace at VNU: Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification

DSpace at VNU: Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification tài liệu, giáo...

1 INTRODUCTION

The interest on the safety assessment of existing

prestressed concrete (PSC) girders has been increasing

For a PSC girder, typical damage types include loss of

prestress-force in steel tendon, loss of flexural rigidity

in concrete girder, failure of support, and severe

ambient conditions Among them, the loss of

prestress-force is an important monitoring target to secure the

serviceability and safety of PSC girders against external

loads and environmental conditions (Miyamoto et al.

2000; Kim et al 2004) The loss of prestress-force

occurs along the entire girder due to elastic shortening

and bending of concrete, creep and shrinkage in

concrete, relaxation of steel stress, friction loss and

anchorage seating (Collins and Mitchell 1991; Nawy

1996)

Parameters and System Identification

Duc-Duy Ho1, Jeong-Tae Kim2,*, Norris Stubbs3and Woo-Sun Park4

1 Faculty of Civil Engineering, Ho Chi Minh City University of Technology, Vietnam

2 Department of Ocean Engineering, Pukyong National University, Korea

3 Department of Civil Engineering, Texas A&M University, College Station, USA

4 Coastal Engineering & Ocean Energy Research Department, Korea Ocean Research & Development Institute, Korea

Abstract: In this paper, a vibration-based method to estimate prestress-forces in a

prestressed concrete (PSC) girder by using vibration characteristics and system identification (SID) approaches is presented Firstly, a prestress-force monitoring method is formulated to estimate the change in prestress forces by measuring the change in modal parameters of a PSC beam Secondly, a multi-phase SID scheme is designed on the basis of eigenvalue sensitivity concept to identify a baseline model that represents the target structure Thirdly, the proposed prestress-force monitoring method and the multi-phase SID scheme are evaluated from controlled experiments on a lab-scaled PSC girder On the PSC girder, a few natural frequencies and mode shapes are experimentally measured for various prestress forces System parameters of a baseline finite element (FE) model are identified by the proposed multi-phase SID scheme for various prestress forces The corresponding modal parameters are estimated for the model-update procedure As a result, prestress-losses are predicted by using the measured natural frequencies and the identified zero-prestress state model.

Key words: prestress concrete girder, presstress-loss, modal parameters, system identification, structural health

monitoring

Unless the PSC girder bridges are instrumented at the time of construction, the occurrence of damage can not be directly monitored and other alternative methods should

be sought Since as early as 1970s, many researchers have focused on the possibility of using vibration characteristics of a structure as an indication of its

structural damage (Adams et al 1978; Stubbs and Osegueda 1990; Doebling et al 1998; Kim et al 2003).

Recently, research efforts have been made to investigate the dynamic behaviors of prestressed composite girder

bridges (Miyamoto et al 2000), and to identify the

change in prestress forces by measuring dynamic

responses of prestressed beams (Kim et al 2004).

However, to date, no successful attempts have been made

to estimate the relationship between the loss in prestress forces and the change in geometries, material properties,

*Corresponding author Email address: idis@pknu.ac.kr; Fax: +82-51-629-6590; Tel: +82-51-629-6585.

and boundary conditions of the PSC girder bridges.

Hence, it is necessary to develop a system identification

(SID) method that can identify the change in structural

parameters due to the change in prestress forces

An accurate finite element (FE) model is prerequisite

for civil engineering applications such as damage

detection, health monitoring and structural control For

complex structures, however, it is not easy to generate

accurate baseline FE models for the use of structural

health monitoring, because material properties,

geometries, boundary conditions, and ambient

temperature conditions of those structures are not

completely known (Kim and Stubbs 1995; Kim et al.

2007) Due to those uncertainties, an initial FE model

based on as-built design may not truly represent all the

physical aspects of an actual structure Consequently,

there exists an important issue that how to update the FE

model using experimental results so that the numerically

analyzed structural parameters match to the real

experimental ones

Many researchers have proposed model update

methods for SID by using vibration characteristics

(Friswell and Mottershead 1995; Kim and Stubbs 1995;

Zhang et al 2000; Jaishi and Ren 2005; Yang and Chen

2009) Among those methods, the eigenvalue

sensitivity-based algorithm has become one of the most popular and

effective methods to provide baseline models for

structural health assessment (Brownjohn et al 2001; Wu

and Li 2004) The FE model update is a process of

making sure that FE analysis results better reflect the

measured data than the initial model For the

vibration-based SID, this process is conducted in the following

steps: (1) measuring vibration data to be utilized;

(2) determining structural parameters to be updated;

(3) formulating a function to represent the difference

between the measured vibration data and the analyzed

data from FE model; and (4) identifying parameters to

minimize the function (Friswell and Mottershead 1995;

Kwon and Lin 2004)

The objective of this paper is to present a

prestress-force estimation method for PSC girders by using

changes in vibration characteristics and SID approaches

The following approaches are implemented to achieve

the objective Firstly, a prestress-force monitoring

method is formulated to estimate changes in

prestress-forces in a PSC girder by measuring changes in modal

parameters Secondly, a multi-phase SID scheme is

designed on the basis of eigenvalue sensitivity concept

to estimate a baseline model which represents the target

structure Thirdly, the proposed prestress-force

monitoring method is evaluated from controlled

experiments on a lab-scaled PSC girder On the PSC

girder, a few natural frequencies and mode shapes are

experimentally measured for various prestress forces System parameters of a baseline FE model are identified

by the proposed multi-phase SID scheme for various prestress forces The corresponding modal parameters are estimated for the model-update procedure As a result, prestress-losses are predicted by using the measured natural frequencies and the identified zero-prestress state model

2 THEORY OF APPROACH

2.1 Vibration-Based Prestress-Force Monitoring Method

Based on the previous study by Kim et al (2004), an

effective flexural rigidity model of a simply supported PSC beam with an eccentric tendon is schematized as shown in Figure 1 The curved tendon is initially stretched and anchored to introduce prestressing effect Then, as shown in Figure 1(b), the structure is in axial compression due to the prestress loads applied at the anchorage edges The beam is also subjected to the

upward distributed load, f(x), which is induced by the

prestressed tendon That is, the structure is initially deformed in compression (e.g., up to the deformed span

length L r) and the tendon is still in tension due to the constraint after elastic stretching as shown in Figure 1(c) The tendon is also subjected to the downward distributed

force, f(x) The initial deformation of the beam results in the reduction of span length, δL(= L − L r), and the expansion in the cross-section by Poisson effect

(a) Prestressed beam with a parabolic tendon

(b) Upward distributed force and anchor force T on beam

(c) Tension force T on pin-pin ended tendon of arc-length Ls

Tendon force T

f(x)

Ls T

e(Lr /2) e(0)

ε

Lr T

Steel tendon Concrete beam

y

x

y

Anchor force T

Lr= L(1 − δ L /L)

f(x)

Figure 1 Effective flexural rigidity model of PSC beam with an

eccentric tendon

The governing differential equation of the effective

flexural rigidity model of the PSC beam with the

curved tendon [as shown in Figure 1(a)] is expressed

by:

(1)

where E rIris the effective flexural rigidity of PSC beam

section which is assumed constant along the entire

length of the beam and m ris the effective mass per unit

length of the beam The effective flexural rigidity of

PSC beam can be evaluated as the combination of the

flexural rigidity of concrete beam section and the

equivalent flexural rigidity of tendon As shown in

Figure 1, the effective flexural rigidity E r I r and the

effective mass m r of the PSC beam can be estimated,

respectively, as follows:

(2) (3)

where E c is the elastic modulus of concrete, I c is the

second moment of concrete beam’s cross-section area,

E p is the elastic modulus of steel tendon, and I p is the

second moment of tendon’s cross-section area Also,

ρc A cis the concrete mass per unit length and ρp A pis the

tendon mass per unit length

The equivalent flexural rigidity of tendon is derived

from analyzing flexural vibration of tendon of arc-length

L s , as shown in Figure 1(c) The arc-length L s is

calculated as L s= βLr, in which the geometric constant β

is computed approximately as β ≈ (L r/4ε) sin−1(4ε/L r)

and ε = e(L r /2) –e(0) with e(x) is the eccentric distance

between the neutral axis of beam and the center of

tendon section at x location By analyzing a pin-pin

ended cable with the same span length L s and the mass

property ρp A pas the tendon, as shown in Figure 2(a), the

cable subjected to tension force T leads the n th natural

frequency ωn c By setting a corresponding beam with a

span length L r which produces the same n th natural

frequency ωn c, as shown in Figure 2(b), the equivalent

flexural rigidity E p I p to the tension force T is obtained as:

(4a)

(4b)

E I T L

n







2

π ρ

n

c

p p

p p

n L

T A

n L

E I A

2

=













m r =ρc c A +ρp A p

E I r r =E I c c+E I p p

∂

∂

∂

∂









 +

∂

2

2

2 2

2

2 0

x E I

y

y t

where n is mode number and T is tension force of cable.

On substituting Eqn 4(b) into Eqn 2 and furthermore applying Eqn 2 with appropriate boundary conditions to

Eqn 1, the n thnatural frequency of the effective flexural rigidity model of the PSC beam can be obtained as:

(5)

Once the n thnatural frequency ωnof the PSC beam is known, the prestress force can be identified from an inverse solution of Eqn 5, as follows:

(6)

where T n is the identified prestress force by using the n th

natural frequency and structural properties By assuming

the mass property (m r ) and the span length (L r) remain unchanged due to the change in prestress force, the first variation of the prestress force can be derived as:

(7)

where δT n is the change in the prestress force that is

identified from the n thmode and δω2nis the change in ω2n due to the change in prestress-force From Eqns 6 and 7, the relative change in the prestress force that can be

identified from the n thmode is obtained as:

(8)

π

ω

T T

m L

n L m

n n

r

n r

=















2

n E I

n L

r

c c r

π

π















π

n L

r





 −















2









n E I

n L

r





 −





















π

π

2





n

n

L m E I

T L n

2

4

2

2

1

=

























(a) Pin-pin ended cable of span-length Ls subjected to tension force T

(b) Equivalent beam of span-length Lr with flexural rigidity EpIp

T

f (x)

f (x)

Ap

ρp

ρ p

ω c

Ip

Lr

Ap

Ls= L βr

Figure 2 Flexural rigidity model of tendon subjected to

tension force T

On dividing both numerator and denominator by

m r L2r /(nπ)2 and by further assuming that the change in

concrete beam’s flexural rigidity due to changes in the

prestress force is negligibly small (i.e., δE c I c ≈ 0), Eqn 8

is simply rearranged as:

(9)

where ϖn is the n thnatural frequency of the beam with

zero prestress force and is given by:

(10)

From Eqn 9, the relative change (estimated by the n th

mode) in prestress force between a reference prestress

state (T n, ref ) and a prestress-loss state (T n, los) can be

measuring the corresponding n thnatural frequencies ωn,ref

and ωn,los, from which the reference eigenvalue is defined

as and the eigenvalue changes is computed as

Unless measured at as-built state, the zero-prestress ϖnshould be estimated from numerical

modal analysis In most existing structures, its field

measurement is almost impossible and we should rely on

a baseline model updated from well-established SID

process

2.2 Multi-Phase System Identification (SID)

Scheme

To identify a realistic theoretical model of a structure,

Kim and Stubbs (1995) proposed a model update method

based on eigenvalue sensitivity concept that relates

experimental and theoretical responses of the structure

(Adams et al 1978; Stubbs and Osegueda 1990).

Suppose p j*is an unknown parameter of the j th member of

a structure Also, suppose p jis a known parameter of the

j th member of a FE model Then, relative to the FE

model, the fractional structural parameter change of the

j th member, αj ≥ –1, and the structural parameters are

related according to the following equation:

(11)

The fractional structural parameter change α jcan be

estimated from the following equation (Stubbs and

Osegueda 1990):

(12)

Z i S ij j

i

M

=

=

1

p*j =p j(1 α+ j)

δωn2 =ωn ref2, −ωn los2,

ωn2=ωn ref2,

δT n T n=(T n ref, −T n los, ) T n ref,

r

c c r

n L

E I m

2

4

=







T T

n n

n

=

−

2

where Z i is the fractional change in the i th eigenvalues between two different structural systems (e.g., an

analytical model and a real structure); M is the number

of known eigenvalues Also, S ij is the dimensionless

sensitivity of the i theigenvalue ωi2 with respect to the j th structural parameter p j (Stubbs and Osegueda 1990;

Zhang et al 2000).

(13a)

(13b)

The term δp j is the first order perturbation of p jwhich produces the variation in eigenvalue δωi2

The fractional structural parameter change of NE

members may be obtained using the following equation:

(14)

where {α} is a NE × 1 matrix, which is defined by Eqn

11, containing the fractional changes in structural parameters between the FE model and the target

structure; {Z} is defined as Eqn 13(b) and it is a M × 1

matrix containing the fractional changes in eigenvalues

between two systems; and [S] is a M × NE sensitivity

matrix, which is defined by Eqn 13(a), relating the fractional changes in structural parameters to the fractional changes in eigenvalues The sensitivity

matrix, [S], is determined numerically in the following

procedure (Stubbs and Osegueda 1990): (1) Introduce a known severity of damage (αj , j = 1, NE) at j thmember; (2) Determine the eigenvalues of the initial FE model (ωi2

o , i = 1, M); (3) Determine the eigenvalues of the

damaged structure (ωi2, i =1, M); (4) Calculate the

(5) Calculate the individual sensitivity components from

S ij = Z i / α j ; and (6) Repeat steps (2)−(5) to generate the

M × NE sensitivity matrix.

If the number of structural parameters is much larger

than the number of modes, i.e., NE >> M, the system is ill-conditioned and Eqn 14 will not work properly,

which is a typical situation for civil engineering structures To produce stable solution, therefore, the number of structural parameters should be equal to or

less than the number of modes, NE ≤ Μ In addition, for

most complex structures, only a few vibration modes can be measured with good confidence and many sub-structural members are combined together with complex response motions in the vibration modes In order to

Z i =(ωi2/ωio2 −1)

α

Z i i

i

=δ ω ω

2 2

S p

p

j

j i

=δ ω

2 2

overcome these problems, a multi-phase model update

approach is needed to be implemented for updating the

FE models of the complex structures

For a target structure which has experimental modal

designed as schematized in Figure 3 First, an initial FE

model is established to numerically analyze modal

parameters ( p j , j = 1,NE ) are selected by grouping the

FE model into NE sub-structures and analyzing modal

sensitivities of the NE parameters up to M modes Third,

the number of phases NP is determined by computing

NP = NE/ M and arrange the M number of structural

parameters ( p j , j = 1, M) for each phase Finally,

the following five sub-steps are performed for phase

K (i.e., K =1, NP):

(1) Compute numerical modal parameters of a

selected FE model;

(2) Compute sensitivities of structural parameters

and the fractional change in eigenvalue between

the target structure and the updated FE model

(i.e., M × 1 {Z} matrix);

(3) Fine-tune the FE model by first solving Eqn 14

to estimate fractional changes in structural

parameters (i.e., NE × 1 {α} matrix) and then

solving Eqn 11 to update the structural

parameters of the FE model;

(4) Repeat the whole procedure until {Z} or {α}

approach zero when the parameters of the FE model are identified; and

(5) Estimate the baseline model after the parameters

are identified from phase K.

In each phase, the selection of structural parameters is based on the eigenvalue sensitivity analysis and the number of available modes Primary structural parameters which are more sensitive to structural responses will be updated in the prior phases It is also expected that the error will be reduced phase after phase, and, as a result, the accuracy of the baseline model will be improved consequently Note that numerical modal analysis is performed by using commercial FE analysis software such as SAP2000 (2005)

3 VIBRATION TEST ON LAB-SCALED PSC GIRDER

Dynamic tests were performed on a lab-scaled post-tension PSC girder to determine the experimental modal parameters for a set of prestress cases The schematic of the test structure is shown in Figure 4 The PSC girder was simply supported with the span length of 6 m and installed on a rigid testing frame Two simple supports

of the girder were simulated by using thin rubber pads

as interfaces between the girder and the rigid frame The

Select target structure:

Measure experimental modes: i,m, 2

i ,m (i = 1, M )

Establish initial FE model:

Analyze numerical modes: i,a, 2

i ,a (i = 1, M )

Select NE model-updating parameters:

Group FE model into NE sub-structures

Analyze modal sensitivities of NE parameters up

to M modes

Determine multi-phase for model update:

Decide number of phases: NP = NE/M

Arrange model-updating parameters (pj, j = 1, M )

for each phase

Check { } ≅ 0

Fine-tune structural parameters { } = [S ] − 1 { Z }

Compute sensitivity and fractional eigenvalue change

&

No

Yes

Select a model update phase (K )

Compute numerical modal parameters of FE model

Sij = δω

Update parameters p

j ∗ = pj (1 + j ) ( j = 1, M ) Check

K = NP Yes

No Perform model update phase-by-phase (K = 1, NP )

Identify the baseline model

φ ∗ ω

i,a , ∗ 2 i,a (i = 1, M )

2 i,a

p ∗

j

i,a

Zi = ω 2 = 1

i,m

i,a

α

α

α

ω φ

ω φ

Figure 3 Multi-phase system identification (SID) scheme

T-section was reinforced in both longitudinal and

transverse direction with 10 mm diameter reinforcing

bars (equivalent to Grade 60) The stirrups were used to

facilitate the position of the top bars A seven-wire

straight concentric mono-strand with 15.2 mm diameter

(equivalent to Grade 250) was used as the prestressing

tendon The tendon was placed in a 25 mm diameter

duct that remained ungrouted The structure was tested

in Smart Structure engineering Lab located at Pukyong

National University, Busan, Korea

During the test, temperature and humidity in the

laboratory were kept close to constant as 18−19oC and

40−45% by air conditioners, respectively, in order to

minimize the effect of those ambient conditions that, if

not controlled, might lead to significant changes in

dynamic characteristics Recently, the interest on

variability of dynamic properties of bridges (i.e., natural

frequency, mode shape, damping ratio) caused by

environmental effects (i.e., temperature, humidity, wind)

has been increasing Cornwell et al (1999) reported that

the natural frequencies of the Alamosa Canyon Bridge in

southern New Mexico were varied by up to 6% over a

24-hour period The results of almost one year monitoring of

the Z24-Bridge located in Switzerland were presented by

Peeters and De Roeck (2001) During the monitoring

period, the frequency differences ranged from 14−18%

due to normal environmental changes To study the

environmental effects on modal parameters, a long term

monitoring test was carried out during 8 months on the Romeo Bridge which is a prestressed concrete box girder

bridge located in Switzerland (Huth et al 2005) Due to

the temperature change of 40oC, the variations of natural frequencies of the first three bending modes were 0.3 Hz,

0.35 Hz, and 0.5 Hz, respectively In addition, Kim et al.

(2007) proposed a vibration-based damage monitoring scheme to give warning of the occurrence, the location, and the severity of damage to a model plate-girder bridge under temperature-induced uncertainty conditions For the test bridge, natural frequencies went down as the temperature went up and bending modes were more sensitive than torsional modes

As shown in Figure 4(a), seven accelerometers (Sensors 1–7) were placed on top of the girder with a constant 1 m interval The impact excitation was applied

in vertical direction by an electromagnetic shaker VTS100 at a location 0.95 m distanced from the right edge Seven ICP-type PCB 393B04 accelerometers with the nominal sensitivity of 1 V/g and the specified frequency range (± 5%) of 0.06–450 Hz were used to measure dynamic responses with the sampling frequency of 1 kHz The accelerometers were mounted

on magnetic blocks which were attached to steel washers bonded on the top surface of the girder The data acquisition system consists of a 16-channel

PXI-4472 DAQ, a PXI-8186 controller with LabVIEW (2009) and MATLAB (2004)

Load cell

Stressing jack

Accelerometer

1 m

6 m

Impact 0.95 m

Wedge Anchor plate

(a) Experimental setup for PSC girder

Figure 4 Vibration test on the lab-scaled PSC girder

Axial prestress forces were introduced into the

tendon by a stressing jack as the tendon was anchored at

one end and pulled out at the other A load cell was

installed at the left end to measure the applied prestress

force Each test was conducted after the desired

prestress force has been applied and the cable has been

anchored During the measurement, the stressing jack

was removed from the girder to avoid the influence of

the jack weight on dynamic characteristics of the test

structure The prestress force was applied to the test

structure up to five different prestress cases (i.e., T1−T5

as indicated in Table 1) The maximum and minimum

prestress forces were set to 117.7 kN and 39.2 kN, respectively The force was uniformly decreased by 19.6 kN for each prestress-loss case Figure 5(a) shows acceleration response signals measured from Sensor 5 when the prestress force was 117.7 kN Figure 5(b) shows frequency response curves measured from Sensor

5 for the five prestress cases, T1–T5 Frequency domain

decomposition (FDD) technique (Brincker et al 2001;

Yi and Yun 2004) was implemented to extract natural frequencies and mode shapes from the acceleration signals For the five prestress cases, natural frequencies

of the first two modes were extracted as summarized in

Table 1 Experimental natural frequencies of test structure for five prestress cases

− 0.1

− 0.05

0

0.05

0.1

Time (s)

10 − 12

10 − 10

10 − 8

10 − 6

10 − 4

Frequency (Hz)

T5 T4 T3 T2 T1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Mode 1

− 0.7

− 0.5

− 0.3

− 0.1 0.1 0.3 0.5 0.7

Mode 2

(c) Bending mode shapes

Figure 5 Acceleration signal, frequency responses and mode shapes from experimental measurement

Table 1 Also, the variation of natural frequencies with

respect to the maximum prestress force T1 as the

reference are given in Table 1 The corresponding

mode shapes of the first two bending modes were

extracted as shown in Figure 5(c) Note that mode

shapes were not changed significantly due to the change

in prestress forces From the frequency response plots,

Figure 5(b), there are several peaks between first and

second bending modes These modes are torsional

modes, axial modes and horizontal bending modes

However, only vertical bending modes were considered

in this study As shown in Figure 4(a), seven

accelerometers were placed on top of the girder with a

constant 1 m interval Also, the impact excitation was

applied in vertical direction by an electromagnetic

shaker VTS100 For this reason, only vertical bending

modes were extracted exactly from the experimental

setup

4 SYSTEM IDENTIFICATION OF PSC

GIRDER WITH VARIOUS

PRESTRESS-FORCES

4.1 Initial FE Model and Model-Updating

Parameters

A structural analysis and design software, SAP2000

(2005), was used to model the PSC girder As shown in

Figure 6, the girder was constructed by a

three-dimensional FE model using solid elements For analysis

purpose, we divided the girder into 11,264 block

elements The dimensions of the FE model were

described in Figure 6 For the boundary conditions,

spring restraints were assigned at supports: horizontal

and vertical springs for the left support and vertical

spring for the right support Initial values of material,

geometric properties and boundary conditions of the FE model were assigned as follows: (1) for the concrete

girder, elastic modulus E c = 2 × 1010 N/m2, the second

moment of area I c = 4.9 × 10−3 m4, mass density ρc =

2500 kg/m3, and Poisson’s ratio v c = 0.2; (2) for the steel tendon, elastic modulus E p = 3 × 1011 N/m2, the second

moment of area I p = 1.9 × 10–5 m4, mass density ρp =

7850 kg/m3, and Poisson’s ratio v p = 0.3 ; and (3) the stiffness of vertical and horizontal springs k v = k h = 109N/m Numerical modal analysis was performed on the initial

FE model and initial natural frequencies of the first two bending modes were computed as 23.65 Hz and 97.77

Hz, respectively Figure 7 shows mode shapes of the two modes analyzed from the FE model

Choosing appropriate structural parameters is an important step in the FE model-updating procedure All parameters related to structural geometries, material properties, and boundary conditions can be potential choices for adjustment in the model-updating procedure For the PSC girder, therefore, structural parameters which were relatively uncertain

in the FE model due to the lack of knowledge on their properties were selected as model update parameters Also, structural parameters which are relatively sensitive to vibration responses were considered as prior choices As shown in Figure 8, for the present PSC girder, six model update parameters were selected as follows: (1) flexural rigidity of concrete

girder (E c I c) in the simple-span domain, (2) flexural

rigidity of steel tendon (E p I p) in the overall structure,

(3) flexural rigidity of the left overhang zone (E lo I lo), (4) flexural rigidity of the right overhang zone

(E ro I ro ), (5) vertical spring stiffness (k v) at the left and

right supports, and (6) horizontal spring stiffness (k h)

Springs

Springs

Concrete

Tendon

18 cm

8 cm

2 cm

4 cm

32 cm

14 cm

71 cm

7 cm

27 cm 4 cm 9 cm 4 cm 27 cm

Figure 6 Initial FE model of the PSC girder

at the left support Note that the left overhang zone

includes stressing-jack, load-cell, tendon anchor, and

0.2 m girder section at the left edge, as shown in

Figure 4(a) Also, the right overhang zone includes

tendon anchor and 0.2 m girder section at the right

edge Both overhang zones were selected due to the

uncertainty in the stiffness due to the effect of tendon

anchors and concrete sections on dynamic responses under varying prestress forces

On estimating the initial FE model, the initial values

of the six model update parameters were assumed as

follows: E c I c = 9.81 × 107 Nm2, E p I p = 5.73 × 106 Nm2,

E lo I lo = E ro I ro = 9.81 × 107 Nm2, and k v = k h = 109 N/m Then, the eigenvalue sensitivity analysis for the six model update parameters was carried out, as summarized

in Table 2 From the results, the flexural rigidity of concrete girder was the most sensitive parameter for both mode 1 and mode 2 The flexural rigidity of steel tendon was the second sensitive parameter Those high sensitive parameters were expected to contribute more intensively

on the model update The stiffness of overhang zones and the stiffness of support springs were relatively less sensitive parameters That is, those less sensitive parameters were expected to contribute less intensively

on the model update

Due to the availability of the two modes, three model update phases were chosen to treat the six model update parameters In each phase, two structural parameters were chosen for adjustment Based on their sensitivities

as listed in Table 2, the order of model update was arranged as follows:

(1) Phase I: flexural rigidities of concrete girder

(E c I c ) and steel tendon (E p I p);

(2) Phase II: flexural rigidities of left overhang

(E lo I lo ) and right overhang (E ro I ro); and

(3) Phase III: vertical spring stiffness (k v) and

horizontal spring stiffness (k h)

4.2 System Identification Results for Various Prestress-Forces

After selection of vibration modes and model-updating parameters, an iterative procedure schematized in Figure 3 was carried out for model update It should be noted that three phases were performed phase-after-phase and two model-updating parameters were updated iteratively at each phase Consequently, the analytical natural frequencies determined at the end of iterations gradually approached those experimental values For prestress case T1 (117.7 kN), SID results are summarized in Table 3 and also shown in Figure 9

(a) Mode 1

(b) Mode 2

Figure 7 Numerical mode shapes of initial FE model

EpIp

EcIc

kv

kh

kv

0.2 m

Figure 8 Six model update parameters for the PSC girder

Table 2 Eigenvalue sensitivities of six model update

parameters

No. E c l c E p l p E lo l lo E ro l ro k v k h

1 0.8855 0.1039 0.0064 0.0034 0.0029 0.0007

2 0.8817 0.0537 0.0223 0.0105 0.0150 0.0268

Table 3 Natural frequencies (Hz) during model update iterations for prestress case T1 (117.7 kN)

Updated frequencies (Hz) at each iteration _

Mode Freqs _(Girder & Tendon) (Overhang zones) (Spring supports) Freqs.

No (Hz) 1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 10 th 11 th 12 th 13 th 14 th 15 th (Hz)

1 23.65 24.70 23.22 23.80 23.95 24.00 24.01 24.01 24.04 24.01 23.99 23.97 23.97 23.94 23.92 23.91 23.72

2 97.77 104.08 98.16 100.47 101.05 101.22 101.26 101.28 102.29 102.02 101.79 101.52 101.47 102.15 102.01 101.80 102.54

Table 4 Natural frequencies (Hz) of updated FE models and target structures for five prestress cases

Table 5 Identified values of model update parameters for five prestress cases

case (kN) E c I c (Nm 2 ) E p I p (Nm 2 ) E lo I lo (Nm 2 ) E ro I ro (Nm 2 ) k v (N/m) k h (N/m)

Table 3 shows natural frequencies during 15 iterations

of multi-phase model update Figure 9 shows

convergence errors of updated natural frequencies with

compared to target natural frequencies which were

experimentally measured at the prestress force of 117.7 kN

Natural frequencies were converged with 1.2% error at

Phase 1 (when concrete girder and steel tendon

members were updated), 1.0% error at Phase 2 (when

overhang members were updated), and less than 0.8 %

error at the end of Phase 3 (when support spring

members were updated) Meantime, the flexural

rigidities of concrete girder and steel tendon were

identified, respectively, as E c I c = 1.12 × 108 Nm2 and

EpIp = 5.68 × 105 Nm2 The flexural rigidities of

overhang zones were identified as E lo I lo = 4.15 × 108 Nm2

and E roIro = 1.38 × 106 Nm2, respectively Also, the stiffness parameters of support springs were identified

identification results for all five cases (i.e., T1–T5) are summarized in Table 4 and Table 5 Table 4 shows natural frequencies of updated FE models with compared to those of the target structure For all five prestress cases, natural frequencies were converged with 0.1−1.2% error range Meanwhile, the six model-updating parameters were identified as listed in Table 5

As listed in Table 5, the updated model parameters were changed as the prestress forces were changed from T1 (117.7 kN) to T5 (39.2 kN) Figure 10 shows the relative changes in updated model parameters (with respect to the maximum prestress force T1 as the

0.0 1.0 2.0 3.0 4.0 5.0

Iteration

Phase I (Girder & Tendon)

Phase III (Support springs)

Phase II (Overhang zones)

Figure 9 Convergence errors of natural frequencies for prestress case T1 (117.7 kN)