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

Prestress-Force Estimation in PSC Girder Using Modal
Parameters and System Identification
Duc-Duy Ho1, Jeong-Tae Kim2,*, Norris Stubbs3 and Woo-Sun Park4
1Faculty

of Civil Engineering, Ho Chi Minh City University of Technology, Vietnam
of Ocean Engineering, Pukyong National University, Korea
3Department of Civil Engineering, Texas A&M University, College Station, USA
4Coastal Engineering & Ocean Energy Research Department, Korea Ocean Research & Development Institute, Korea
2Department

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 labscaled 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.
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 prestressforce 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).

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: ; Fax: +82-51-629-6590; Tel: +82-51-629-6585.

Advances in Structural Engineering Vol. 15 No. 6 2012

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Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification

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 sensitivitybased 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 vibrationbased 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 prestressforce 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 prestressforces 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

998

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 zeroprestress 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 Lr) 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 − Lr), and the
expansion in the cross-section by Poisson effect.

Concrete beam

Steel tendon

x

Lr = L(1 − δL /L)

y

(a) Prestressed beam with a parabolic tendon

T

Anchor
force T

f(x)

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

e(0)
ε

T
y

x

e(x) e(Lr /2)
f(x)

Ls

Tendon
force T

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

Figure 1. Effective flexural rigidity model of PSC beam with an
eccentric tendon

Advances in Structural Engineering Vol. 15 No. 6 2012


Duc-Duy Ho, Jeong-Tae Kim, Norris Stubbs and Woo-Sun Park

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:
2

2

2

∂ y
∂ 
∂ y
+ mr 2 = 0

EI
2 r r
2
∂t
∂x 
∂x 

(1)

where Er Ir is the effective flexural rigidity of PSC beam
section which is assumed constant along the entire
length of the beam and mr is 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 Er Ir and the
effective mass mr of the PSC beam can be estimated,
respectively, as follows:
Er I r = Ec I c + E p I p

(2)

mr = ρc Ac + ρ p Ap

(3)

where Ec is the elastic modulus of concrete, Ic is the
second moment of concrete beam’s cross-section area,
Ep is the elastic modulus of steel tendon, and Ip is the

second moment of tendon’s cross-section area. Also,
ρc Ac is the concrete mass per unit length and ρp A p is the
tendon mass per unit length.
The equivalent flexural rigidity of tendon is derived
from analyzing flexural vibration of tendon of arc-length
Ls, as shown in Figure 1(c). The arc-length Ls is
calculated as Ls = βLr, in which the geometric constant β
is computed approximately as β ≈ (Lr /4ε) sin−1(4ε/Lr )
and ε = e(Lr /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 Ls and the mass
property ρp A p as the tendon, as shown in Figure 2(a), the
cable subjected to tension force T leads the nth natural
frequency ω nc . By setting a corresponding beam with a
span length Lr which produces the same nth natural
frequency ω nc , as shown in Figure 2(b), the equivalent
flexural rigidity EpIp to the tension force T is obtained as:
2

ω nc 2

4

 nπ  E p I p
 nπ 
T
=
=
 β Lr  ρ p Ap  Lr  ρ p Ap

T L 
E pI p = 2  r 
β  nπ 

(4a)

2

Advances in Structural Engineering Vol. 15 No. 6 2012

(4b)

ρp Ap

f (x)

ωcn

T

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

Ep Ip

f (x)

ρp

Ap

ωnc

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

Figure 2. Flexural rigidity model of tendon subjected to
tension force 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 nth natural frequency of the effective flexural
rigidity model of the PSC beam can be obtained as:
4

ω n2

 nπ  1 
T  Lr  2 
E
I
+
= 
  
 c c
 Lr  mr 
β 2  nπ  

(5)

Once the nth natural frequency ω n of the PSC beam is
known, the prestress force can be identified from an
inverse solution of Eqn 5, as follows:
2
2

 nπ  
 Lr 
2
Tn = β  ω n mr   − Ec I c   
 nπ 

 Lr  
2

(6)

where Tn is the identified prestress force by using the nth
natural frequency and structural properties. By assuming
the mass property (mr) and the span length (Lr) remain
unchanged due to the change in prestress force, the first
variation of the prestress force can be derived as:
2
2

 nπ  
 Lr 
2
δ Tn = β  δ ω n mr   − δ Ec I c   

 nπ 

 Lr  
2

(7)

where δTn is the change in the prestress force that is
identified from the nth mode and δω 2n is the change in ω 2n
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 nth mode is obtained as:

δ Tn
=
Tn

 nπ 
L 2
δ ω n2 mr  r  − δ Ec I c  
 nπ 
 Lr 
 nπ 
L 2
ω n2 mr  r  − Ec I c  
 nπ 
 Lr 

2

2

(8)

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Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification

On dividing both numerator and denominator by
mrL2r /(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., δEc Ic ≈ 0), Eqn 8
is simply rearranged as:

δ Tn
δ ω2
= 2 n 2
Tn ω n − ϖ n

(9)

where ϖn is the nth natural frequency of the beam with
zero prestress force and is given by:

where Zi is the fractional change in the ith eigenvalues
between two different structural systems (e.g., an
analytical model and a real structure); M is the number
of known eigenvalues. Also, Sij is the dimensionless
sensitivity of the ith eigenvalue ωi2 with respect to the jth

structural parameter pj (Stubbs and Osegueda 1990;
Zhang et al. 2000).
Sij =

δ ω i2 p j
δ p j ω i2

(13a)

δ ω i2
ω i2

(13b)

4

ϖ n2

 nπ  E I
=  c c
 Lr  mr

(10)

From Eqn 9, the relative change (estimated by the nth
mode) in prestress force between a reference prestress
state (Tn, ref ) and a prestress-loss state (Tn, los) can be
estimated as δ T n T n = (Tn ,ref − Tn ,los ) T n ,ref by
measuring the corresponding nth natural frequencies ωn,ref
and ωn,los, from which the reference eigenvalue is defined

as ω n2 = ω n2,ref and the eigenvalue changes is computed as
δω n2 = ω n2,ref − ω n2,los . Unless measured at as-built state,
the zero-prestress ϖn should 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 jth member of
a structure. Also, suppose pj is a known parameter of the
jth member of a FE model. Then, relative to the FE
model, the fractional structural parameter change of the
jth member, αj ≥ –1, and the structural parameters are
related according to the following equation:

(

p*j = p j 1 + α j

)

(11)

The fractional structural parameter change α j can be

estimated from the following equation (Stubbs and
Osegueda 1990):
M

Zi = ∑ Sijα j
i =1

1000

(12)

Zi =

The term δ pj is the first order perturbation of pj which
produces the variation in eigenvalue δωi2 .
The fractional structural parameter change of NE
members may be obtained using the following equation:

{α } = [ S ]−1 { Z }

(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 jth member;
(2) Determine the eigenvalues of the initial FE model
(ωi2o , i = 1, M); (3) Determine the eigenvalues of the
damaged structure (ωi2 , i =1, M); (4) Calculate the
2
2
fractional changes in eigenvalues by Zi = ω i / ω io − 1 ;
(5) Calculate the individual sensitivity components from
Sij = Zi / αj ; 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 substructural members are combined together with complex
response motions in the vibration modes. In order to

(

)

Advances in Structural Engineering Vol. 15 No. 6 2012

Duc-Duy Ho, Jeong-Tae Kim, Norris Stubbs and Woo-Sun Park

Perform model update phase-by-phase (K = 1, NP )
Select target structure:
Select a model update phase (K )

Measure experimental modes: φ i,m, ω 2i,m (i = 1, M )

Compute numerical modal parameters of FE model
φ ∗i,a, ω ∗2i,a (i = 1, M )

Establish initial FE model:
Analyze numerical modes: φi,a, ω 2i,a (i = 1, M )

Compute sensitivity and fractional eigenvalue change

Sij =
Select NE model-updating parameters:

δω 2i,a p ∗j
δ pj

ω∗2i,a

& Zi =

ω 2i,m

=1

ω∗2i,a

Group FE model into NE sub-structures
Fine-tune structural parameters

Analyze modal sensitivities of NE parameters up
to M modes

{α} = [S ]−1{Z }

Check
{α } ≅ 0

Determine multi-phase for model update:
Decide number of phases: NP = NE/M
Arrange model-updating parameters (pj, j = 1, M )
for each phase

No

Yes
No

Update parameters
p ∗ = pj (1 + αj )
j

Check
K = NP

( j = 1, M )

Yes
Identify the baseline model

Figure 3. Multi-phase system identification (SID) scheme

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
parameters
, a multi-phase SID is
designed as schematized in Figure 3. First, an initial FE
model is established to numerically analyze modal
parameters
. Second, NE structural
parameters ( pj, 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 ( pj, 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;

Advances in Structural Engineering Vol. 15 No. 6 2012

(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 posttension 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

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Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification

Stressing jack
Load cell

Accelerometer

1m

Anchor plate
Sensor 1

Sensor 2

Sensor 3

Sensor 4

0.2 m

0.95 m

Sensor 5

RC beam

Tendon

Wedge

Impact

Sensor 6

Sensor 7

Rubber support

6m

0.2 m

(a) Experimental setup for PSC girder

(b) Test girder

(c) Stressing jack and load cell

Figure 4. Vibration test on the lab-scaled PSC girder

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 24hour 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

1002

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 PXI4472 DAQ, a PXI-8186 controller with LabVIEW
(2009) and MATLAB (2004).

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Duc-Duy Ho, Jeong-Tae Kim, Norris Stubbs and Woo-Sun Park

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
Prestress force
Prestress case

Natural frequency (Hz)

(kN)

T1
T2
T3
T4
T5

Mode 1

Mode 2

Mode 1

Mode 2

23.72
23.60
23.39
23.23
23.08

102.54
101.70
101.65
101.39
98.73

–
0.51
1.39
2.07
2.70

–
0.82
0.87
1.12
3.72

117.7
98.1
78.5
58.9
39.2

10−4

0.1

Power spectrum

Acceleration (g)

0

−0.05

−0.1
0

T5

4

6

8

10−10

10−12

10

200

300

400

(a) Acceleration signal

(b) Frequency responses

T4

T3

T2

0.7

T1

0.5

0.3

0.4
0.3
0.2
0.1
3

4
5
Sensor location

T5

T4

6

7

T3

T2

T1

4
5
Sensor location

6

500

0.1
−0.1
−0.3
−0.5

Mode 1
2

100

Frequency (Hz)

0.5

1

0

Time (s)

Mode value

Mode value

2

10−8

0.6

0

T5
T4
T3
T2
T1

10−6

0.05

0.7

Variation of frequency (%)

−0.7

Mode 2
1

2

3

7

(c) Bending mode shapes

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

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Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification

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 PRESTRESSFORCES
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 threedimensional 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 Ec = 2 × 1010 N/m2, the second
moment of area Ic = 4.9 × 10−3 m4, mass density ρc =
2500 kg/m3, and Poisson’s ratio vc = 0.2; (2) for the steel
tendon, elastic modulus Ep = 3 × 1011 N/m2, the second

moment of area Ip = 1.9 × 10–5 m4, mass density ρp =
7850 kg/m3, and Poisson’s ratio vp = 0.3 ; and (3) the
stiffness of vertical and horizontal springs kv = kh = 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 modelupdating 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 (EcIc) in the simple-span domain, (2) flexural
rigidity of steel tendon (EpIp) in the overall structure,
(3) flexural rigidity of the left overhang zone (EloIlo),
(4) flexural rigidity of the right overhang zone
(EroIro), (5) vertical spring stiffness (kv) at the left and
right supports, and (6) horizontal spring stiffness (kh)

71 cm
8 cm

2 cm
4 cm
27 cm

4 cm 9 cm 4 cm

27 cm

Springs
32 cm
Concrete

Tendon
14 cm
7 cm

Springs
18 cm

Figure 6. Initial FE model of the PSC girder

1004

Advances in Structural Engineering Vol. 15 No. 6 2012


Duc-Duy Ho, Jeong-Tae Kim, Norris Stubbs and Woo-Sun Park

(a) Mode 1

(b) Mode 2

EcIc

EloIlo
kh

EroIro
kv

EpIp

kv

0.2 m

0.07 m
0.6 m

Figure 7. Numerical mode shapes of initial FE model

6m

0.2 m

Figure 8. Six model update parameters for the PSC girder
Table 2. Eigenvalue sensitivities of six model update
parameters
Mode
No.

1
2

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: EcIc = 9.81 × 107 Nm2, EpIp = 5.73 × 106 Nm2,
EloIlo = EroIro = 9.81 × 107 Nm2, and kv = kh = 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
(EcIc) and steel tendon (EpIp);
(2) Phase II: flexural rigidities of left overhang

(EloIlo) and right overhang (EroIro); and
(3) Phase III: vertical spring stiffness (kv) and
horizontal spring stiffness (kh).

Eigenvalue sensitivities
Eclc

Eplp

Elollo

Erolro

kv

kh

0.8855
0.8817

0.1039
0.0537

0.0064
0.0223

0.0034
0.0105

0.0029

0.0150

0.0007
0.0268

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

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-afterphase 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.

Table 3. Natural frequencies (Hz) during model update iterations for prestress case T1 (117.7 kN)
Updated frequencies (Hz) at each iteration
_______________________________________________________________________________
Initial
Phase I

Phase II
Phase III
Target
Mode Freqs. _______________________________________________________________________________
(Girder & Tendon)
(Overhang zones)
(Spring supports)
Freqs.
No.
(Hz)
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
10th
11th
12th 13th
14th
15th
(Hz)
1
2

23.65
97.77

24.70
104.08

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
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

Advances in Structural Engineering Vol. 15 No. 6 2012

23.91
101.80

23.72
102.54

1005


Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification

5.0
Phase I
(Girder & Tendon)

Error (%)

4.0

Phase III
(Support springs)

Phase II
(Overhang zones)

3.0
Mode 1

Mode 2

2.0
1.0

0.0
Initial

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Iteration

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

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 EcIc = 1.12 × 108 Nm2 and
EpIp = 5.68 × 105 Nm2. The flexural rigidities of
overhang zones were identified as EloIlo = 4.15 × 108 Nm2

and EroIro = 1.38 × 106 Nm2, respectively. Also, the
stiffness parameters of support springs were identified
as kv = 3.38 × 108 N/m and kh = 2.12 × 1012 N/m. System
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 modelupdating 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

Table 4. Natural frequencies (Hz) of updated FE models and target structures for five prestress cases
Prestress
case
T1
T2
T3
T4
T5

1st Frequency (Hz)

2nd Frequency (Hz)

Prestress
force
(kN)

Experiment

FEM

Error (%)

Experiment

FEM

Error (%)

117.7
98.1
78.5
58.9
39.2

23.72
23.60
23.39
23.23
23.08

23.91
23.74
23.62
23.50
23.09

0.80
0.59
0.98
1.15
0.06

102.54
101.70
101.65
101.39
98.73

101.80
101.11
100.64
100.13
98.54

0.72
0.58
0.99
1.24
0.19

Table 5. Identified values of model update parameters for five prestress cases
Prestress
case
T1
T2
T3
T4
T5

1006

Prestress
force
(kN)

Updated model parameter
EcIc (Nm2)

EpIp (Nm2)

117.7
98.1
78.5
58.9
39.2

1.12E+8
1.11E+8
1.10E+8

1.09E+8
1.06E+8

5.68E+5
4.78E+5
3.79E+5
2.88E+5
1.92E+5

EloIlo (Nm2)
4.15E+8
4.14E+8
4.14E+8
4.14E+8
4.13E+8

EroIro (Nm2)

kv (N/m)

kh (N/m)

1.38E+6
1.37E+6
1.37E+6
1.36E+6
1.35E+6

3.38E+8
3.32E+8

3.28E+8
3.20E+8
3.17E+8

2.12E+12
2.12E+12
2.12E+12
2.12E+12
2.12E+12

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Duc-Duy Ho, Jeong-Tae Kim, Norris Stubbs and Woo-Sun Park

1.2

δ EcIc = 7.0 ×

10−4

T + 0.921

Relative change of EpIp

Relative change of EcIc

1.2
1.1

1.0
0.9

0.8
39.2

58.9
78.5
98.1
Prestress force (kN)

δ EpIp = 8.45 × 10−3 T + 0.007

1.1

1.0
0.9

0.8
39.2

117.7

(a) Concrete girder’s EcIc
1.2
δ EloIlo = 5.73 × 10−5 T + 0.993

Relative change of EroIro

Relative change of EloIlo

117.7

(b) Steel tendon’s EpIp

1.2
1.1

1.0
0.9

0.8
39.2

58.9
78.5
98.1
Prestress force (kN)

δ EroIro = 2.88 × 10−4 T + 0.967

1.1

1.0
0.9

0.8
39.2

117.7

(c) Left overhang’s EloIlo

117.7

1.2
Relative change of kh

δ kv = 8.07 × 10−4 T + 0.904

1.1

1.0
0.9

0.8
39.2

58.9
78.5
98.1
Prestress force (kN)
(d) Right overhang’s EroIro

1.2
Relative change of kv

58.9
78.5
98.1

Prestress force (kN)

58.9
78.5
98.1
Prestress force (kN)

117.7

(e) Vertical spring’s kv

δ kh = 1.63 × 10−5 T + 0.998

1.1

1.0
0.9

0.8
39.2

58.9
78.5
98.1
Prestress force (kN)

117.7

(f) Horizontal spring’s kh

Figure 10. Relative changes in updated model parameters due to changes in prestress forces

reference) due to the changes in prestress forces. For
both concrete girder and steel tendon, their flexural
rigidities (δEcIc and δEpIp) changes almost linearly as
the prestress force changes. The steel tendon’s stiffness
was greatly influenced by the prestress forces, but the
change in concrete girder’s stiffness was relatively
small. The change in the right overhang’s stiffness
(δEloIlo and δEroIro) was relatively larger than the left
overhang which remained almost unchanged. The
change in the vertical spring’s stiffness (δkv) was
relatively small and the horizontal spring’s stiffness
(δkh) was remained nearly unchanged as the prestress

Advances in Structural Engineering Vol. 15 No. 6 2012

force changes. To estimate the relationships between the
prestress-forces and the relative changes in the six
structural parameters (i.e., δEcIc, δEpIp, δEloIlo, δEroIro,
δkv, and δkh), six empirical equations were established
as follows:
Ec I c = 1.12 × 108 × δ Ec I c = 78.62T + 1.03 × 108 Nm 2

(15a)
E p I p = 5.68 × 105 × δ E p I p = 4.8T + 3.99 × 103 Nm2
(15b)

1007

Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification

Elo I lo = 4.15 × 108 × δ Elo I lo = 23.79T + 4.12 × 108 Nm2
(15c)
Ero I ro = 1.38 × 106 × δ Ero I ro = 0.4T + 1.33 × 106 Nm2
(15d)
kv = 3.38 × 108 × δ kv = 272.61T + 3.05 × 108 N/m (15e)
kh = 2.12 × 1012 × δ kh = 34, 594T + 2.12 × 1012 N/m
(15f)
From the empirical equations, values of the six model
parameters can be identified with respect to the required
amount of prestress force. That is, the updated FE model
represents the six model parameters corresponding to
dynamic responses for a certain prestress-force state of
the PSC girder. The identified baseline model can be the
role of the reference structure to make diagnosis and
prognosis on the structure of interest.

5. PRESTRESS-FORCE MONITORING OF
PSC GIRDER
5.1. Estimation of Natural Frequencies for
Various Prestress Forces
Natural frequencies of the PSC girder are predicted by
an updated FE model, from which a zero-prestress state
model was also identified for prestress-force estimation.
By noticing that the updated FE model is represented by
the six model parameters (i.e., EcIc, EpIp, EloIlo, EroIro,
kv, and kh), natural frequencies for various prestress
forces can be estimated by using the six empirical

equations. So the FE model parameters corresponding to
certain prestress-forces can be estimated from Eqns
15(a) to 15(f), in which values of the six model
parameters are linearly related to required amounts of
prestress forces.
As plotted in Figure 11 and also listed in Table 6, the
FE analysis produced natural frequencies of the PSC
girder for the five prestress-force cases and a zeroprestress (i.e., T = 0) state. Compared to the
experimental results, the natural frequencies of the FE
model, fn,f, show very small estimation errors: 0.5−0.9%
104

25

Mode 2
Natural frequency (Hz)

Natural frequency (Hz)

Mode 1

24

23

22

102

100

98

96
0

20

40
60
80
Prestress force (kN)

100

120

Experiment

0

20

40
60
80
Prestress force (kN)

100

120

FE model by empirical equations

Figure 11. Prediction of natural frequencies of the PSC girder by two zero-prestress models

Table 6. Estimation of natural frequencies for five prestress cases

Prestress
case
T1
T2
T3
T4
T5
Zero-Prestress

1008

Prestress
force
(kN)

Experiment
fn,e (Hz)
Mode 1

Mode 2

117.7

98.1
78.5
58.9
39.2
0.0

23.72
23.60
23.39
23.23
23.08
N/A

102.54
101.70
101.65
101.39
98.73
N/A

FE model by
empirical equations
fn,f (Hz)
Mode 1
Mode 2
23.94
23.76
23.57
23.39
23.20

22.81

101.94
101.20
100.46
99.71
98.94
97.38

Advances in Structural Engineering Vol. 15 No. 6 2012


Duc-Duy Ho, Jeong-Tae Kim, Norris Stubbs and Woo-Sun Park

5.2. Evaluation of Prestress-Force Monitoring
The relative change in prestress force with reference to
the full prestress force is identified by measuring the
relative change in the nth natural frequency with
reference to the frequency margin between the full
prestress state and the zero prestress state. In order to
predict the prestress-loss for the PSC girder, Eqn 9 is
rewritten in a convenient form as follows:
 δT 
fn2,ref − fn2,los
δ fn2
=
=


2

2
fn2,ref − ζ n2
 Tref  n fn ,ref − ζ n

(16)

From Eqn 16, the relative change in prestress force,
δT / Tref = (Tref – Tlos) / Tref, between a reference
prestress state (Tref) and a prestress-loss state (Tlos) can
be estimated by measuring the corresponding nth natural
frequencies of the reference state fn,ref and the prestressloss state fn,los.
In this study, we selected Tref = 117.7 kN as the
reference state, and f1,ref = 23.72 Hz for mode 1 and
f2,ref = 102.54 Hz for mode 2, accordingly, as listed in
Table 1 or Table 7. Next, we selected two zero-prestress
models as follows: the initial FE model and the updated
FE model by the six empirical equations. Then, the nth
natural frequency for the zero-prestress state, ζ n, were
estimated from the two zero-prestress models. The first
two natural frequencies ζ1 and ζ2 were estimated as
follows (also listed in Table 7): (1) ζ1 = 23.65 Hz and
ζ2 = 97.77 Hz for the initial FE model and (2) ζ1 = 22.81
Hz and ζ2 = 97.38 Hz for the updated FE model for T = 0
state. Here, all five prestress cases in Table 1 were
examined to detect the prestress-loss. Table 7 shows

prestress-loss prediction results for the PSC girder using
Tref = 117.7 kN and the two zero-prestress models.
The predicted prestress-loss results from the updated
FE model were compared with the measured

experimental prestress-losses. The predicted prestresslosses versus the inflicted experimental prestress-losses
were plotted in Figure 12. Depending on the accuracy of
the experimental natural frequencies, the prediction by
mode 1 was more accurate than by mode 2. In mode 1,
the correlation between those sets was good; however,
in mode 2, the correlation was relatively low due to
measurement errors which might affect the
experimental natural frequencies.
By substituting the reference state Tref = 117.7 kN
into the prestress-loss prediction results (i.e., Table 7),
prestress forces were predicted by the updated FE
model. As shown in Figure 13, compared to the
experimental results, the accuracy of prestress-force
prediction was relatively high in the updated FE model

1.0

Predicted prestress-loss

in mode 1 and 0.2−1.7% in mode 2. It is observed that
the updated FE model shows very accurate estimation
results of natural frequencies.

0.8

0.6

0.4
Mode 1

0.2

Mode 2
0.0
0.0

0.2

0.4
0.6
0.8
Experimental prestress-loss

1.0

Figure 12. Predicted prestress-losses versus inflicted prestresslosses for updated FE model

Table 7. Prestress-loss prediction using Tref = 117.7 kN and two zero-prestress models
Initial

Experiment
Prestress
case
T1
T2
T3
T4
T5

Updated

FE model

FE model

ζ1 = 23.65 Hz
ζ 2 = 97.77 Hz

ζ1 = 22.81 Hz
ζ2 = 97.38 Hz

T (kN)

δT
Tref

f1 (Hz)

f2 (Hz)

 δT 
 T 
1

 δT 
 T 
2

 δT 
 T 

1

 δT 
 T 
2

117.7
98.1
78.5
58.9
39.2

0.0
0.17
0.33
0.50
0.67

23.72
23.60
23.39
23.23
23.08

102.54
101.70
101.65
101.39
98.73

0.0
N/A
N/A
N/A
N/A

0.0
0.18
0.19
0.25
0.80

0.0
0.13
0.37
0.54
0.71

0.0
0.17
0.18
0.23
0.74

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1009


Prestress force (kN)

120
Experiment
Mode 1
Mode 2

100
80
60
40
20
0

T2 (98.1 kN) T3 (78.5 kN) T4 (58.9 kN) T5 (39.2 kN)
Prestress case

Figure 13. Predicted prestress forces by updated FE model and
Tref = 117.7 kN

(i.e., 0.1−54.6% error). It is also observed that the
accuracy of the prestress-loss monitoring depends on
the accuracy of measured experimental frequencies
and the accuracy of the baseline modeling of the zero
prestress state as well. For the vibration test,
acceleration signals were measured in t = 10 seconds
with the sampling frequency of fs = 1 kHz, as shown in
Figure 5(a). The number of data points used for the fast
Fourier transform (FFT) was N = 213 = 8192. As a
result, the frequency resolution was ∆f = fs / N =
1000/8192 = 0.122 Hz. The accuracy of measured

experimental frequencies depends on frequency
resolution. The more accurate natural frequencies were
obtained as the smaller frequency resolution was used
for the data process. Furthermore, as described in
section 2.1, the natural frequencies with zero prestress
force are required in order to predict prestress-loss for
the PSC girders. However, in reality, the field
measurement of most existing structures is almost
impossible; therefore, they should be estimated from a
baseline model updated from well-established SID
process. Due to the uncertainties in structural and
environmental parameters, the initial FE model may
not truly represent all the physical aspects of an actual
structure. For this reason, it must be updated by using
experimental results. As listed in Table 8, the errors in
Table 8. Effect of error in zero-prestress state’s
natural frequencies (ζ) on prestress-loss prediction
accuracy for prestress case T5
Error in ζ

Mode 1

Mode 2

(%)

ζ1

 δT 
 T 

1

ζ2

 δT 
 T 
2

0
1.0
5.0
10.0
25.0

22.81
22.58
21.67
20.53
17.11

0.71
0.57
0.32
0.21
0.11

97.38
96.40
92.51
87.64

73.03

0.74
0.63
0.39
0.27
0.15

1010

Predicted prestress-loss

Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification

0.8

Inflicted prestress-loss (δ T/ Tref) = 0.67

0.6

Mode 1
Mode 2

0.4
0.2
0

0

1

5

10

25

Error inζ (%)

Figure 14. Effect of error in zero-prestress state’s natural
frequencies (ζ) on prestress-loss prediction accuracy for
prestress case T5

zero-prestress state’s natural frequencies were
simulated for five error levels (e.g., 0, 1, 5, 10 and
25%). Figure 14 shows the effect of error in zeroprestress state’s natural frequencies (i.e., ζ ) on the
accuracy of prestress-loss prediction for prestress case
T5. From the results of mode 1 and mode 2, it is
observed that the more accurate prediction was
obtained as the smaller error was inflicted in ζ. Note
that the zero-prestress state’s natural frequencies of the
updated FE model (i.e., ζ1 = 22.81 Hz and ζ2 = 97.38 Hz)
were set as the error-free baseline state. Note also that
the errors in natural frequencies were simulated by
percentage reduction.
In reality, the mechanical properties of concrete vary
with time and also by temperature, which would also
cause the change in natural frequencies. Besides,
typical damage types of PSC girder bridges are not only
tendon damage but also stiffness-loss in concrete girder

and failure of support or connection. Hence, the
proposed method using the change in vibration
characteristics alone may not be able to distinguish or
isolate the change in prestress-loss from other damage
types in the PSC girder. To detect multiple damage
types such as tendon damage and girder damage, Kim
et al. (2010) proposed the combined global vibrationbased and local impedance-based methods. In their
approach, the multiple damage types were classified
into either tendon or girder damage by recognizing
patterns of impedance features. To deal with the real
damage situations, therefore, global and local damage
detection methods should be applied in conjunction
with the presented method to identify the type, damage,
and severity of damage in the PSC girder structures.
6. SUMMARY AND CONCLUSIONS
In this study, a vibration-based method to estimate
prestress-forces in a PSC girder by using vibration
characteristics and SID approaches was presented. The
following approaches were implemented to achieve the

Advances in Structural Engineering Vol. 15 No. 6 2012


Duc-Duy Ho, Jeong-Tae Kim, Norris Stubbs and Woo-Sun Park

objective. Firstly, a prestress-force monitoring method
was formulated to estimate the change in prestress-force
by measuring the change in modal parameters of a PSC
beam. Secondly, a multi-phase SID scheme was
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 model update
scheme were evaluated from controlled experiments on
a lab-scaled PSC girder.
On the PSC girder, a few natural frequencies and
mode shapes were experimentally measured for various
prestress forces, 117.7−39.2 kN. The corresponding
modal parameters were analyzed from a baseline FE
model, from which structural parameters were identified
with respect to prestress forces from the proposed SID
method. From multi-phase SID, good correlations of
natural frequencies between updated FE models and the
target PSC girder were obtained for the various prestress
forces. From linear regression analysis of the results, the
linear relationships between updated model parameters
and prestress forces were established to estimate the
influence of prestress forces on the performance of
structural subsystems (and also to identify values of the
six model parameters with respect to the required
amount of prestress forces). Natural frequencies of the
PSC girder under the various prestress forces were
estimated by FE models, from which zero-prestress state
models of the PSC girder were identified. As a result,
prestress-losses were accurately predicted by using the
measured natural frequencies and the identified zeroprestress state models.
ACKNOWLEDGEMENT
The authors would like to acknowledge the financial
support of the project “Development of inspection
equipment technology for harbor facilities” funded by

Korea Ministry of Land, Transportation, and Maritime
Affairs.
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