Structures 33 (2021) 2061–2065
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Structures
journal homepage: www.elsevier.com/locate/structures
An investigation of bridge influence line identification using time-domain
and frequency-domain methods
Samim Mustafa a, *, Ikumasa Yoshida b, Hidehiko Sekiya b
a
b
Advanced Research Laboratories, Tokyo City University, 1-28-1 Tamazutsumi, Setagaya-ku, Tokyo 158-8557, Japan
Department of Urban and Civil Engineering, Tokyo City University, 1-28-1 Tamazutsumi, Setagaya-ku, Tokyo 158-8557, Japan
A R T I C L E I N F O
A B S T R A C T
Keywords:
Bridge influence line
Time-domain method
Frequency-domain method
Displacement response
Steel girder bridge
Calibration trucks
The method to obtain an accurate influence line (IL) from the direct measurement is an important research topic
for structural condition assessment, model correction and bridge weigh-in-motion (BWIM) system. The two most
common approaches used for the identification of IL are the time-domain (TD) method and the frequencydomain (FD) method. Despite having a similar mathematical framework, the TD and the FD methods are
treated as two different methods by the researchers working on this field. This paper presents a detailed theo­
retical demonstration to show that the two methods discussed above are nothing but the same. The two methods
were compared experimentally by using field measurement data on an existing steel girder bridge which were
obtained by using three calibration trucks (CTs) with different axle weights and axle configurations. Although the
ILs identified by the two methods were apparently different, but a theoretical insight into the frameworks
revealed that the TD and FD methods are basically the same and a seeming difference between the two methods
is due to the inherent assumptions involved in the discrete Fourier transform (DFT) such as the assumption of
cyclic nature of analysis interval. Finally, a method to obtain an accurate influence line has been outlined.
1. Introduction
Due to the ever increasing nature of traffic loads, many existing
bridges which are in-operation for two decades or more are under
enormous strain already. Therefore, it is indispensable to know the
actual in-service condition of bridges for the evaluation of their per­
formances, safety and the remain life. To achieve this goal, the bridge
influence line (IL) can play a key role as it represents a unique charac­
teristic of a bridge. The bridge IL describes an important static property
of the bridge that shows the variation of reaction or any internal forces
at a certain location when a bridge is subjected to a moving unit load. As
an IL gives a direct relationship between the load and the response, it is
widely used in design of bridges [1], structural condition assessment [2],
model correction [3] and bridge weigh-in-motion system (BWIM) sys­
tem [4,5].
There are several ways to extract the bridge IL from measured re­
sponses such as from measured strains, bending moments or displace­
ments. One of the very first methods was proposed by the McNulty and
O’Brien [6] which is a point-by-point graphical method. Later, O’Brien
et al. [7] proposed an improved method using least-squares solutions to
extract the IL from direct measurements. This method is often referred as
the matrix method. Ieng [8] proposed a more robust method for esti­
mating an IL that uses the maximum likelihood estimation and takes into
account multiple measurements from as many CTs as needed. Froseth
et al. [9] proposed a frequency-domain method for IL identification by
representing the bridge response as a convolution of the load function
and the IL. However, the proposed method is ill-posed for certain types
of vehicle configurations and a regularization technique based on a
stabilization filter was applied to stable the solution. The main reason of
this instability is due to the inherent assumptions involved in DFT which
the authors failed to recognize and address in their paper. Zheng et al.
[10] presented a comprehensive review and comparison between
different IL identification methods. They have also considered the
identification of IL using TD and FD formulations as two separate
methods and presented a criterion for selecting suitable IL identification
method under different conditions.
Based on the above discussion, it is evident that the TD and the FD
methods for the IL identification are treated as two different methods by
the researchers working on this field, despite having a similar mathe­
matical model. This paper presents a detailed theoretical demonstration
to show that the two methods discussed above are nothing but the same.
The two methods were compared experimentally by using field
* Corresponding author.
E-mail address: samim@tcu.ac.jp (S. Mustafa).
https://doi.org/10.1016/j.istruc.2021.05.082
Received 18 May 2021; Accepted 29 May 2021
Available online 7 June 2021
2352-0124/© 2021 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
S. Mustafa et al.
Structures 33 (2021) 2061–2065
Fig. 1. Test bridge with the location of a sensor: (a) Plan view; (b) Sectional view.
measurement data on an existing steel girder bridge. The bridge ILs were
estimated by the two methods from the measured displacement re­
sponses corresponding to three CTs with different axle weights and axle
configurations.
of VIM H is given below
⎡
w1 0 ⋯ ⋯
⎢ 0 w1 0 ⋯
⎢
⎢ ⋮
⋮ ⋱ ⋯
⎢
⎢ w2 0 ⋯ w1
⎢
⎢ 0 w2 0 ⋯
⎢
⋮ ⋱ ⋯
H=⎢
⎢ ⋮
⎢ wN 0 ⋯ ⋱
⎢
⎢ 0 wN 0 ⋯
⎢
⎢ ⋮
0 ⋱ ⋯
⎢
⎣ ⋮
⋮ ⋯ ⋱
0
0 ⋯ ⋯
2. Theoretical formulations for bridge IL identification
2.1. Time-domain (TD) method
By definition, an IL represents the response of a bridge at a specific
location due to a unit load placed at any location along its length.
Assuming a linear response and independent actions of vehicle axles on
the bridge, the response due to a travelling vehicle can be represented as
a summation of the contributions from individual axles
N
∑
zt (t) =
(3)
Then, the IL vector Is of the test bridge can be determined by the
least-square solution of Eq. (2) which is given below as
(
)− 1
Is = HT H Hz
(4)
(1)
wi Is (si (t)); si (t) = tv − di
⎤
0
⋮ ⎥
⎥
⋮ ⎥
⎥
⋮ ⎥
⎥
w1 ⎥
⎥
⋮ ⎥
⎥
⋮ ⎥
⎥
w2 ⎥
⎥
⋮ ⎥
⎥
0 ⎦
wN
i=1
2.2. Frequency-domain (FD) method
where zt (t) represents the vehicle-induced response at sensor location, N
is the number of axles, Is (s) represents the bridge IL corresponding to the
ith axle, wi represents the weight of ith axle, di represents the distance
between the first and the ith axles and v represents the vehicle speed.
The response equation due to a calibration vehicle, shown in Eq. (1), can
be rewritten in the following matrix form
By applying the Fourier transform to the measured bridge response
due a calibration vehicle as given in Eq. (1), the following response
equation in FD can be obtained
∫
(2)
z = HIs
∫
∞
zt (t)e−
zf (f ) =
i2π ft
∞
N
∑
dt =
− ∞
wi × Is (tv − di )e−
i2π ft
dt
− ∞ i=1
(5)
By simplifying Eq. (5), the expression for If (f/v) can be obtained as
where z ∈ RNr ×1 represents the measured response vector with Nr sam­
pling points, Is ∈ RNl ×1 represents the IL vector with Nl influence co­
efficients, and H ∈ RNr ×Nl represents the vehicle information matrix
(VIM) or loading matrix which is constructed based on the information
of axle weights and axle spacings of a calibration vehicle. It is worthy to
mention here that the matrix H is a non-square matrix. This is because of
the fact that only the first axle is present on the bridge at the time of
entry whereas only the last axle is present on the bridge at the time of
exit of a vehicle. Therefore, the length of IL vector Nl will be shorter than
the length of response vector Nr by a number pN , where pN represents the
sampling point difference between the first and the last axles. The detail
If (f /v) = ∑N
vzf (f )
i=1 wi e
− i2π fdi
(6)
Therefore, the IL in FD can be determined by solving Eq. (6) where
zf (f) represents the Fourier transform of the measured bridge response
zt (t) at sensor location. By applying the inverse Fourier transform to Eq.
(6), the IL Is (s) ∈ RNr ×1 can be obtained. It should be noted that the
length of the IL obtained from the FD method is equal to the length of the
response vector Nr in contrast to the TD method where Nl < Nr .
3. IL identification from measured data
3.1. Description of test bridge and calibration trucks (CTs)
Table 1
Specifications of contact displacement gauge.
Model
Sampling
frequency
(Hz)
Capacity
(mm)
Nonlinearity
(mm)
Sensitivity
(×10–6 strain/
mm)
CDP-25
(Tokyo Sokki
Kenkyujo Co.,
Ltd.)
100
0–25
0.1% of rated
output
500
The test bridge is a single-span slab-on-girders steel plate girder
bridge system with three main girders each of having a height of 1.6 m.
Fig. 1 shows the sectional and plan views of the test bridge along with
the location of a sensor used in this study [11]. A displacement gauge
was instrumented at the longitudinal centre of G2 to measure the bridge
displacement response caused by a traversing vehicle. The specification
of the contact displacement gauge is listed in Table 1. A total of three CTs
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Structures 33 (2021) 2061–2065
Table 2
Specifications of three CTs used in this study.
Run no.
Driving lane
Calibration truck (CT)
1
2
3
Traveling
Passing
Passing
4-axle
3-axle
2-axle
Axle weights and GVW (kN)
Axle spacing (m)
Speed (m/s)
1st axle
2nd axle
3rd axle
4th axle
GVW
1st-2nd
2nd-3rd
3rd-4th
44.7
53.6
22.8
44.2
34.9
24.1
53.8
35.0
–
52.5
–
–
195.2
123.5
46.9
1.85
3.23
2.87
4.18
1.31
–
1.2
–
–
10.87
14.29
13.88
Fig. 2. Measured displacement responses for each of the CTs used in this study.
Fig. 3. Comparison of displacement ILs identified by the TD and FD methods using measured data for each CT.
with different axle weights and axle configurations, namely those with
two, three, and four axles, were used in this study as listed in Table 2.
Although a large number of test runs was conducted considering
different test conditions in our previous research works [12], only three
test runs, one for each CT as listed in Table 2, are considered in this
study. The test bridge was having a smooth road profile at the time of
measurements.
3.2. IL identification results
Fig. 2 shows the measured displacement responses for three test runs
using three types of CTs. As can be observed, the dynamic effect in
measured displacement response of the 4-axle CT is much lower than
that of the 3-axle and 2-axle CTs. Fig. 3 shows the comparison of
displacement-based ILs identified by both the methods using data for
each CT. The two sets of curves agreed well for the case of 4-axle CT
whereas a disparity was observed between the two for the case of 3-axle
and 2-axle CTs. Therefore, despite having a similar mathematical model,
the ILs identified by the TD and the FD methods showed some agreement
and disagreement depending on the use of different CTs. The possible
sources of this disparity between the two ILs are believed to be due to the
inherent assumptions involved in DFT such as the cyclic nature of
analysis interval and the use of the window length of power of two (e.g.,
2n ). By incorporating such assumptions in the TD method, the two
methods should produce identical results for the IL identifications which
have been investigated in the following section.
Fig. 4. Illustration of a measured bridge response and a representation of
vehicle positions on the bridge according to the shapes of vehicle informa­
tion matrix.A
3.3. Time-domain method with cyclic assumption (TD-WCA)
Previously, we have observed that the VIM given in Eq. (3) is a nonsquare or rectangular matrix. However, in order to incorporate the cyclic
assumption and to make the TD formulation consistent with the FD
formulation, it is necessary to consider a square matrix for the VIM.
Fig. 4 illustrates a measured bridge response and a representation of
vehicle positions on the bridge according to the shapes of VIM H. The
matrix H is a square matrix when it is considered that only the front axle
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Structures 33 (2021) 2061–2065
Fig. 5. Comparison of ILs identified by the three methods using measurement data for each CT.
of the CT is present on the bridge at the start of the measurement, and
the front axle is about to exit the bridge when the last scan of the
measurement is done. Based on the above consideration, the matrix Hsq
can be expressed in the following form
⎡
⎤
w1 0 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ 0
⎢ 0 w1 0 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋮ ⎥
⎥
⎢
⎢ ⋮
0 ⋱ 0 ⋯ ⋯ ⋯ ⋯ ⋯ ⋮ ⎥
⎢
⎥
⎢ w2 ⋮ ⋯ w1 0 ⋯ ⋯ ⋯ ⋯ ⋮ ⎥
⎥
⎢
⎢ 0 w2 0 ⋯ ⋱ ⋯ ⋯ ⋯ ⋯ ⋮ ⎥
⎥
Hsq = ⎢
(7)
⎢ ⋮
0 ⋱ 0 ⋯ ⋱ 0 ⋯ ⋯ ⋮ ⎥
⎢
⎥
⎢ wN ⋮ ⋯ w2 0 ⋯ w1 0 ⋯ ⋮ ⎥
⎢
⎥
⎢ 0 wN 0 ⋯ ⋱ ⋯ ⋯ ⋱ ⋱ ⋮ ⎥
⎢
⎥
⎣ ⋮ ⋯ ⋱ ⋯ ⋯ ⋱ ⋯ ⋯ ⋱ 0 ⎦
0
0 ⋯ wN 0 ⋯ w2 0 ⋯ w1
Therefore, matrix Hsq becomes a lower triangular matrix when it is
expressed in a square form. Now, incorporating the cyclic assumption,
the VIM Hc can be expressed in the following mathematical form
⎤
⎡
w1 0 ⋯ wN 0 ⋯ w2 0 ⋯ 0
⎢ 0 w1 0 ⋯ wN 0 ⋯ w2 0
⋮ ⎥
⎥
⎢
⎢ ⋮
0 ⋱ 0 ⋯ ⋱ ⋱ ⋯ ⋱ 0 ⎥
⎥
⎢
⎢ w2 ⋮ ⋯ w1 0 ⋯ ⋱ ⋱ ⋯ w2 ⎥
⎥
⎢
⎢ 0 w2 0 ⋯ ⋱ ⋯ ⋯ ⋱ ⋱ 0 ⎥
⎥
(8)
Hc = ⎢
⎢ ⋮
0 ⋱ 0 ⋯ ⋱ 0 ⋯ ⋱ ⋮ ⎥
⎥
⎢
⎢ wN ⋮ ⋯ w2 0 ⋯ w1 0 ⋯ wN ⎥
⎥
⎢
⎢ 0 wN 0 ⋯ ⋱ ⋯ ⋯ ⋱ ⋱ 0 ⎥
⎥
⎢
⎣ ⋮ ⋯ ⋱ ⋯ ⋯ ⋱ ⋯ ⋯ ⋱ ⋮ ⎦
0
0 ⋯ wN 0 ⋯ w2 0 ⋯ w1
Fig. 7. Static ILs for three calibration runs obtained after low-pass filtering
with a passband frequency of 0.9 Hz.
3.4. Results and discussion
The ILs for the three test runs using different CTs were estimated
again by TD-WCA method. Fig. 5 shows a comparison among the ILs
identified by the TD, TD-WCA and the FD methods using measured data
for each CT. As can be seen, the results of TD-WCA, which is a TD
method with cyclic assumption, are identical to the results of FD method
for all the cases. This means, the TD method and the FD method are
basically the same and the difference in IL identification results between
the two methods is due to the inherent assumptions involved in the DFT.
Therefore, while identifying an IL by FD method, one should take into
account appropriately the effects of inherent theoretical assumptions
involved in DFT. One of the ways to reduce the effect of cyclic
assumption in the FD method is by considering a longer time-series data
for the measured bridge response. Fig. 6 shows the calculated influence
lines by the TD and FD methods using a longer displacement response for
each test run. The two sets of curves agreed well thereby reducing the
effect of cyclic assumption in the FD formulation. Ideally, an IL of a
bridge should be unique and independent of the loading conditions but
the identified ILs using the three CTs differed greatly because of the
varying level of dynamic effects in their measured responses. This is why
The matrix Hc ∈ RNr ×Nr is a square Toeplitz matrix. In this case, the
solution for bridge IL can be obtained just by doing matrix inversion of
Eq. (2), as given below
Ics = H−c 1 z
(9)
where Ics ∈ RNr ×1 represents the IL identified by the TD method with
cyclic assumption (TD-WCA). Therefore, the identified IL using TD-WCA
has same number of coefficients as that of the measured response vector
z.
Fig. 6. Comparison of identified influence lines by the TD and the FD methods using a longer measurement data for each CT.
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Structures 33 (2021) 2061–2065
interests or personal relationships that could have appeared to influence
the work reported in this paper.
many researchers working on BWIM use a heavier and slow running
truck during calibration runs so as to minimize the effect of dynamics in
the measured responses. Another way of removing the dynamic response
from the total response is by using a low-pass filter. Fig. 7 shows the
static ILs for three calibration runs which were obtained after low-pass
filtering at 0.9 Hz. Unlike the ILs with dynamic components, the static
ILs obtained using different CTs agree well with each other’s.
References
[1] Fiorillo G, Ghosn M. Application of ILs for the ultimate capacity of beams under
moving loads. Eng Struct 2015;103:125–33.
[2] Chen ZW, Zhu S, Xu YL, Li Q, Cai QL. Damage detection in long suspension bridges
using stress ILs: a case study. J Bridge Eng 2015;20(3):05014013.
[3] Xiao X, Xu YL, Zhu Q. Multiscale modeling and model updating of a cable-stayed
bridge. II: model updating using modal frequencies and ILs. J Bridge Eng 2015;20
(10):04014113.
[4] Moses F. Weight-in-motion system with instrumented bridge. Transportation
Engineering Journal 1979;105(3):233–48.
[5] Mustafa S, Sekiya H, Hirano S, Miki C. Iterative linear optimization method for
bridge weigh-in-motion systems using accelerometers. Struct Infrastruct Eng 2020;
1–12. https://doi.org/10.1080/15732479.2020.1802490.
[6] McNulty P, O’Brien EJ. Testing of bridge weigh-in-motion system in a sub-Arctic
climate. J Test Eval 2003;31(6):1–10.
[7] O’Brien EJ, Quilligan MJ, Karoumi R. Calculating an IL from direct measurements.
Proc Inst Civ Eng 2006;159(1):31–4.
[8] Ieng S. Bridge IL estimation for bridge weigh-in-motion system. J Comput Civil Eng
2015;29(1):06014006.
[9] Frøseth GT, Rønnquist A, Cantero D, Øiseth O. IL extraction by deconvolution in
the frequency domain. Comput Struct 2017;189:21–30.
[10] Zheng X, Yang DH, Yi TH, Li HN. Development of bridge IL identification methods
based on direct measurements data: A comprehensive review and comparison. Eng
Struct 2019;198:109539.
[11] Sekiya H, Kubota K, Miki C. Simplified portable bridge weigh-in-motion system
using accelerometers. J Bridge Eng 2018;23(1):04017124.
[12] Yoshida I, Sekiya H, Mustafa S. Bayesian bridge-weigh-in-motion and uncertainty
estimation. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems,
Part A: Civil Engineering 2021;7(1):04021001.
4. Conclusions
The two most common approaches used for the identification of IL
are the TD and the FD methods. Despite having a similar mathematical
framework, the TD and the FD methods are treated as two different
methods by the researchers working on this field. This paper presents a
detailed theoretical demonstration to show that the two methods dis­
cussed above are nothing but the same. For the validation, the influence
lines were extracted by the two methods from measured displacement
responses on an existing steel girder bridge by using three CTs with
different axle weights and axle configurations. It was observed that the
results of TD-WCA, which is a TD method with cyclic assumption, are
identical to the results of FD method for all the cases which means that
the TD method and the FD method are basically the same and the dif­
ference in IL identification results between the two methods is due to the
inherent assumptions involved in the DFT such as the cyclic nature of
analysis intervals.
Declaration of Competing Interest
The authors declare that they have no known competing financial
2065