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Technical Note
Bridge Influence Line Identification Based on Regularized
Least-Squares QR Decomposition Method
Downloaded from ascelibrary.org by UNIVERSITY OF NEW SOUTH WALES on 08/24/19. Copyright ASCE. For personal use only; all rights reserved.
Xu Zheng1; Dong-Hui Yang2; Ting-Hua Yi, M.ASCE3; Hong-Nan Li, F.ASCE4; and Zhi-Wei Chen5
Abstract: The bridge influence line is an important tool to study the bridge response under moving loads and contains tremendous
structural information. Structural deterioration during the bridge’s service life will induce variation in the bridge influence line, which
makes it essential to identify the influence line exactly from the field measurement data under moving vehicle loads. This article establishes the underdetermined influence line identification model for a bridge under multiaxle vehicle excitation, in which the bridge influence line is identified based on the regularized least-squares QR decomposition method. Then an example of a four-span girder bridge
is provided to prove the validity of the method, and the accuracy and computational complexity of different methods in this condition
are compared. This study thus provides an effective method for the identification of the bridge influence line with very high computational efficiency and identification accuracy. DOI: 10.1061/(ASCE)BE.1943-5592.0001458. © 2019 American Society of Civil
Engineers.
Author keywords: Influence line; Performance evaluations; Weigh-in-motion; Least-squares QR (LSQR); Tikhonov regularization.
Introduction
In the last 30 years, many large infrastructures have emerged worldwide, especially in China. However, after a long time of service, it
is vital to determine the deterioration condition of these infrastructures. Long-span bridges are the pivotal projects of many transportation systems. With increased concern about the service performance and condition of such bridges (Asadollahi and Li 2017; Yang
et al. 2017), getting bridge information in real-time becomes an important research field. The influence line is a static property of the
bridge and is directly related to bridge flexibility; it also has important applications in bridge weight-in-motion system (Moses 1979),
damage detection (Zeinali and Story 2017), model correction
(Strauss et al. 2012), and bridge evaluation (Hirachan and Chajes
2005). However, bridge influence lines will change when bridge
stiffness and constraint conditions change, and the identification of
the bridge influence line from direct measurement data becomes an
important issue in these fields.
The influence line can be extracted from structural response data
induced by moving loads in several ways. OBrien et al. (2006) first
presented a least-squares method to extract the influence line
from measurement data. Ieng (2015) presented a more direct
1
Ph.D. Candidate, School of Civil Engineering, Dalian Univ.
of Technology, Dalian 116023, China. Email: [email protected]
.edu.cn
2
Assistant Professor, School of Civil Engineering, Dalian Univ. of
Technology, Dalian 116023, China. Email: [email protected]
3
Professor, School of Civil Engineering, Dalian Univ. of Technology,
Dalian 116023, China (corresponding author). Email: [email protected]
4
Professor, School of Civil Engineering, Dalian Univ. of Technology,
Dalian 116023, China. Email: [email protected]
5
Associate Professor, Department of Civil Engineering, Xiamen Univ.,
Xiamen 361005, China. Email: [email protected]
Note. This manuscript was submitted on October 29, 2018; approved
on March 27, 2019; published online on May 20, 2019. Discussion period
open until October 20, 2019; separate discussions must be submitted for
individual papers. This technical note is part of the Journal of Bridge
Engineering, © ASCE, ISSN 1084-0702.
© ASCE
identification method using the maximum-likelihood estimator
principle, which generalizes the matrix method by simultaneously
taking into account measurements available from as many calibration trucks as needed and leads to a more robust estimation. Frøseth
et al. (2017) present a new method to extract the influence line in
the frequency domain. The method reduces the complexity of influence line extraction by several orders of magnitude compared with
the conventional matrix method.
It can be summarized that the previous methods for the identification of the influence line have already made great breakthroughs, but more development is needed. First, all of the
matrix-based methods need to take the inverse of the vehicle information matrix. That method results in calculation complexity,
and the calculation also needs a large amount of additional storage space. Second, the response equation is actually an underdetermined equation. In the previous methods, the influence lines
are extracted through an overdetermined equation. In some
cases, these methods will give rise to inaccuracy of the influence
line. To solve the problems related to the accuracy and computational efficiency of the existing methods for influence line identification, this study investigated an improved time-domain model
of the problem of bridge influence line identification and a new
algorithm focused on this model. The new mathematical model
is suitable for general test conditions, and the model can be
directly used with measured data instead of establishing a mathematical model for each test condition. The use of the regularization method makes the identified bridge influence line more
accurate, and the least-squares QR decomposition (LSQR)
method makes real-time identification of the influence line possible. The article is organized as follows: First, the complete forms
of the vehicle information matrix and response equation are
established, and the relationship of the vehicle information matrix and bridge response is illustrated. Then the normal equation
of Tikhonov regularization is presented, and the regularization
coefficient is properly chosen. After that, the iterative LSQR
method is introduced to extract the influence line. Finally, an
example of influence line identification in a four-span girder
bridge is presented to prove the efficiency of the proposed
method.
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between the first axle and the ith axle. The formula of Ci is as
follows:
Theoretical Formulations for Influence Line
Identification
Ci ¼
Complete Vehicle Information Matrix and Response
Equation
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N
X
Ai w ðkCi Þ
(2)
where f = sampling frequency of the sensor; and v = velocity of the
calibration vehicle. The sampling frequency depends on the velocity of the vehicle and the required accuracy of the influence line.
Then the response equation can be written in matrix form as
follows:
When a calibration vehicle with known axle weight Ai and known
axle distance Di passes over a bridge, it results in a series of measurements of the load effect RðkÞ. For each sampling point k, the corresponding response caused by the calibration vehicle is given by
r ðkÞ ¼
Di f
v
R ¼ LU
(1)
(3)
i¼1
where R = response vector; U = influence coefficient vector; and
L = vehicle information matrix. The following matrix shows the
detail of L:
where N = number of vehicle axles; w ðkCi Þ = influence coefficient
corresponding to the ith axle; and Ci = sampling point difference
2
6
6
6
6
6
L¼6
6
6
6
6
4
A1
0
A2
0
A3
Ak
0
0
A1
0
A2
0
A2
Ak
0
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
0
0
0
A1
0
A2
0
This vehicle information matrix has very explicit physical meaning. Each row of the matrix represents a relationship between N
influence coefficients and one bridge response. If there is a total of
m sampling points for bridge response, there will be m þ CN 1
influence coefficients involved in the response. Therefore, the complete response equation is a kind of underdetermined equation.
However, this equation can be converted to an overdetermined
equation in some test conditions. In the influence line test of shortspan bridge, the whole bridge is tested. In this condition, the influence coefficients in the starting points and ending points can be
zero. If the contribution of these coefficients is ignored, the equation
can be converted to an overdetermined one. However, the influence
coefficients in the starting points and ending points are not always
zero in field tests of bridge influence lines. In this condition, ignoring these coefficients will cause instability of the identified bridge
influence line. Therefore, to retain more influence line information,
the complete vehicle information matrix should be used.
Inverse Problems and the Regularization Method
In the real test condition, the response equation [Eq. (3)] is always
ill-posed because a small fluctuation in R may result in considerable
errors in the solution of U. Fluctuations in response are inevitable
because of the dynamic effect, rail irregularity, and other factors.
Therefore, the real measurement R can be expressed as the sum of
the following two parts:
R ¼ Rr þ e
A2
3
7
0 7
7
7
.. 7
..
. . 7
7
7
.. 7
..
. . 7
5
Ak mðmþC
(4)
N 1Þ
Commonly used regularization methods include Tikhonov regularization, sparse regularization, elastic net regularization, and truncated singular value decomposition. Among these regularization
methods, the Tikhonov regularization method can give the most acceptable results. The expression of the Tikhonov regularization
method is as follows:
h
i
(6)
min kR LUk22 þ λkTUk22
This means that to solve the ill-posed equation, a vector U that
can minimize the sum of two Euclidean norms should be chosen. In
Eq. (6), λ represents the regularization coefficient, and T represents
the regularization matrix. In this article, the regularization matrix is
chosen as follows (Reichel and Rodriguez 2013):
1
0
1 2 1
C
B
C
B
1 2 1
C
B
C
(7)
T¼B
..
.. ..
C
B
C
B
.
.
.
A
@
1 2 1 ðn2Þn
To solve this problem, the first-order derivative of the Euclidean
norms with respect to vector U is taken. After that, the Tikhonov
regularization can be expressed by
ðLT L þ λ2 TT TÞU ¼ LT R
(5)
where Rr = ideal static response of the bridge; and e = errors. To
reduce these errors, the regularization method is commonly used.
© ASCE
0
(8)
Then the regularization coefficient λ is determined. To find the
optimal regularization coefficient, the L-curve method (Morigi and
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J. Bridge Eng., 2019, 24(8): 06019004
J. Bridge Eng.
Sgallari 2001) can be used. It is shown that the plot of the curve
(logkLU Rk; logkTUk) is shaped roughly like an L. In influence line identification problems, the identified influence line is
very smooth when the regularization parameter corresponds to the
L-corner. However, some important vertex information is filtered.
To avoid this phenomenon, another regularization parameter needs
to be selected in the transition area between the horizontal and the
vertical segment.
(a)
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Influence Line Identification by LSQR Method
The LSQR method is an iterative method for solving Ax ¼ b and
mink Ax bk2 , where the matrix A is large and sparse. In this article, the LSQR method and Tikhonov regularization method are
combined, which means the method can be used to solve Eq. (8).
Compared with the least-squares method and frequency-domain
method, the LSQR method can solve underdetermined response
equation [Eq. (3)] with the complete vehicle information matrix
[Eq. (4)] because the method can return the minimal normal leastsquares solution with good numerical stability. This feature gives
the LSQR method good identification results in different test conditions. At the same time, as an iteration method, the LSQR method
saves storage space and has very fast convergence speed in the calculation (Paige and Saunders 1982).
The algorithm of LSQR is shown as follows (the iteration coefficients in the method will not be introduced in detail):
1. Initializing
(b)
1 ¼ b 1;
b 1 u1 ¼ b; a1 v1 ¼ AT u1 ; w1 ¼ v1 ; x0 ¼ 0; f
(c)
r 1 ¼ a1
2. Bidiagonalization processing
b iþ1 uiþ1 ¼ Avi ai ui ; aiþ1 viþ1 ¼ AT uiþ1 b iþ1 vi
3. Orthogonal transformation processing
1=2
r i ¼ r 2i þ b 2iþ1
; ci ¼ r i = r i ; si ¼ b iþ1 = r i ;
iþ1 ¼ si f
i; f
i
u iþ1 ¼ si aiþ1 ; r iþ1 ¼ ci aiþ1 ; f i ¼ ci f
(d)
4. Updating x, w
xi ¼ xi1 þ f i = r i wi :; wiþ1 ¼ viþ1 u iþ1 = r i wi
5. Convergence testing
Stop the iteration when the stopping criterion is satisfied.
Illustrative Example of a Four-Span Girder Bridge
Description of Bridge and Calibration Truck Condition
The four-span girder bridge is widely used in the engineering practice. A simplified model of a four-span girder bridge is presented in
Fig. 1(a). For the convenience of discussion, uniformly distributed
stiffness is assumed in this article. The calibration truck has five
axles, and the axle loads of the truck are 210, 220, 125, 125, and
115 kN, respectively. The axle distances from the first axle are 0.5,
2, 2.5, and 4 m, respectively. The calibration truck will go through
the beam with constant velocity v ¼ 1 m/s, and the bending moment
of the point is collected in the sampling rate of 10 Hz. At this
© ASCE
(e)
Fig. 1. Influence line identification for partial bridge test condition:
(a) partial bridge test condition; (b) bending-moment response of bridge;
(c) influence line identified by least-squares/RLSQR methods and baseline calculated by virtual work principle; (d) influence line identified by
FD/RLSQR methods and baseline calculated by virtual work principle;
and (e) influence line identified by filtered FD/RLSQR methods and
baseline calculated by virtual work principle.
06019004-3
J. Bridge Eng., 2019, 24(8): 06019004
J. Bridge Eng.
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velocity, the dynamic effect of the truck can be ignored. The
bending-moment responses of the point K are calculated by Eq.
(3). To simulate the responses in direct measurement, Gaussian
white-noise disturbance with a signal-to-noise ratio of 15 is added
in the bending-moment responses before calculation. The calculation accuracy will be compared among the baseline calculated
by the virtual work principle, least-squares method, frequencydomain (FD) method, and LSQR method.
Table 1. Comparison of computational complexity among three methods
Method
Complexity
Least squares
Frequency domain
LSQR
O(m2)
O(m log2m)
O(im)
Identified Influence Line Comparison among Different
Methods
In this section, the bridge influence line identified by different methods is compared in a partial bridge test condition. In this condition,
the back axles of the calibration truck are already on the bridge when
the test starts, and the front axles are also still on the bridge when the
test ends. Therefore, the bridge responses collected by the sensor are
influence coefficients. This kind of condition is used in the influence
line test of part of a long-span bridge. The arrangement of the test is
presented in Fig. 1(a), and the bending-moment responses collected
by the sensor are shown in Fig. 1(b).
In this case, the least-squares method tends to be unstable in the
two ends of the influence line. The reason for this instability is that
the contribution of the influence coefficients in the two sides of the
bridge is ignored. This simplification will cause large fluctuation in
the boundary. For the FD method, the identification result is similar
to the least-squares method because these two methods use a similar
mathematical model. After considering the regularizing filter, the
boundary and other parts of the influence line tend to be stable, but
the sharp vertex of the influence line is reduced. Conversely, the
LSQR method gives an accurate prediction of the influence line in
that condition because all of the contributions of the influence coefficients are carefully considered. A comparison of these three methods is presented in Figs. 1(c, d, and e).
Regarding the computational complexity of different algorithms
for the identification of the influence line, the FD method mainly
uses fast Fourier transformation to solve the bridge influence line,
and the computational complexity is approximately Oðm log2 mÞ.
The least-squares method can be quickly solved by a spare
matrix-vector algorithm in Oðm2 Þ. There will be 2Nm þ 4m þ
ið4Nm þ 8mÞ total multiplications in the LSQR algorithm, where
i represents the total iteration steps, and N represents the axle
number. Therefore, the computational complexity of the LSQR
method is approximately OðimÞ. When the stopping criterion has
been met, the number of iteration steps is usually less than 20 in
influence line identification problems. The computational complexity of the three methods is summarized in Table 1, and a comparison of these three methods when the signal size varies from
103 to 107 is presented in Fig. 2.
Conclusions
This study briefly introduces the development of influence line
identification technology and subsequently discusses the deficiencies of the current methods, based on which the regularized LSQR
method is proposed for influence line identification when the moving vehicle goes through the bridge. Finally, a four-span continuous
girder bridge is presented to illustrate the validity of the method.
The following conclusions can be drawn:
1. A new model for underdetermined influence line identification is
established. Compared with traditional influence line identification models, the new mathematical model can consider all of the
influence coefficients involved in the influence line test, which
© ASCE
Fig. 2. Computational complexity comparison.
can increase the accuracy of the influence line prediction. The
model is then solved by the regularized LSQR method.
2. The results of the illustrative example show that the new influence line identification model and method have especially good
accuracy in the condition when only a part of the bridge is
tested. Compared with the least-squares method, the regularized LSQR method gives more influence line information and a
more accurate influence line boundary. Compared with the FD
method, the regularized LSQR method can better predict the
peak value of the influence line. In general, the new method can
better preserve the influence line information.
3. The new method has very low computational complexity compared with other time-domain methods. The regularized LSQR
method is at least 102 times faster than the traditional matrix
method. With the improvement of sampling precision and
enlargement of signal size, the advantage will be more obvious.
Moreover, the computational complexity of the method is on
the same order as that of the FD method.
Acknowledgments
This research work was jointly supported by the National Natural
Science Foundation of China (51625802 and 51708088), the 973
Program (2015CB060000), and the Foundation for High-Level
Talent Innovation Support Program of Dalian (2017RD03).
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© ASCE
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