ABSTRACT 1

Moving load identification from the dynamic responses of bridges is a typical inverse problem that is solved to 2

estimate vehicle axle loads in motion from observed data. To reduce the ill-posedness of the problem and 3

improve solution accuracy, this paper proposes a method for reconstructing the dynamic displacement response 4

via combining the measured acceleration and strain signals for moving load identification. The identification 5

accuracy achieved by employing the reconstructed displacement under different vehicle speeds and different 6

identification algorithms was investigated via finite element (FE) analysis, and a laboratory experiment of a 7

simply supported beam model, was constructed to validate the feasibility of the proposed method. Both the 8

computation simulations and experimental results indicate that the reconstructed displacements fit the true 9

values well and that the proposed method can effectively overcome the ill-posedness of the problem in terms of 10

equation resolution and achieve a high level of accuracy. 11

12

KEYWORDS 13

Moving load identification; Acceleration; Dynamic strain; Signal reconstruction; Modal decomposition 14

INTRODUCTION 15

The amplitude of moving vehicle loads on bridges has a considerable influence on their service life. Moreover, 16

dramatically increasing traffic volumes and overweight trucks may cause fatigue and serious damage to a bridge 17

structure, even leading to bridge collapse in some extreme cases (Beizma, M. V. et al., 2007; Wang et al., 18

2016). Therefore, vehicle loads in motion must be identified by employing advanced and reliable techniques for 19

bridge design and maintenance. 20

Research on moving load identification based on measured

acceleration and strain signals

Extensive studies on moving load identification have been conducted by scholars in the last 30 years. Most of 1

these studies simplified the bridge as a uniform simply supported beam or multi-span continuous beam. 2

O’Connor and Chan (1988) proposed interpretive method I (IMI) using on-site measurements of bridge strains 3

for dynamic wheel load identification. Law et al. (1997) proposed a time domain method (TDM) based on 4

modal superposition theory to identify the dynamic interaction forces in the time domain by using bending 5

moment and acceleration measurements simultaneously. Law et al. (1999) used a new method, the 6

frequency-time domain method (FTDM), to acquire the time histories of moving forces by converting a set of 7

equations relating the Fourier transform into the time domain. Both the TDM and FTDM are based on system 8

identification theory. Interpretive method II (IMII), which differs from IMI and was introduced by Chan (1999), 9

identifies the time-varying moving loads by numerical methods based on Euler's beam theory instead of the 10

beam element model. Thereafter, Chan et al. (2000) conducted comparative studies to investigate the influence 11

of various parameters on the four methods (IMI, IMII, TDM and FTDM). The TDM was verified to be the most 12

suitable approach for incorporation into a moving force identification system (MFIS). To improve the accuracy 13

with which an over-determined set of system equations can be solved, the singular value decomposition (SVD) 14

technique was employed to obtain the identified force history (Yu and Chan, 2002, 2003, 2007). Since the 15

nature of the inverse problem is ill-conditioned (Yang Yu et al., 2016), regularization, especially Tikhonov 16

regularization, is often utilized to provide bounds to the ill-conditioned forces and effectively reduce the noise 17

corruption (Law and Zhu,2000; Zhu and Law, 2002; Law and Fang, 2001). The updated static component 18

(USC) technique has also been adopted in several studies (Asnachinda et al., 2008; Pinkaew, 2006; Pinkaew 19

and Asnachinda, 2007) to eliminate the difficulty of an optimal regularization selection and achieve a higher 20

efficiency in solving the inverse problem. Zhu (2003) and Deng and Cai (2010a, 2011) identified the dynamic 21

vehicular axle loads on a bridge deck by using measurements from a bridge-vehicle system. In terms of 22

moving loads on continuous bridges, Zhu and Law (1999) adopted a multi-span Timoshenko beam with a 1

non-uniform cross-section to illustrate a new method that carries out identification based on the modal 2

superposition and optimization technique in the time domain. In Chan’s study (2006), moving load 3

identification was conducted on a selected span of interest from a continuous bridge to achieve a higher 4

accuracy and shorter execution time. In terms of expanded applications, a case where a vehicle was travelling 5

on slab-type bridges was also studied (Zhu and Law, 2001; Law et al., 2007), in which the bridge deck was 6

modelled as an orthotropic plate and the moving loads were estimated from the dynamic responses based on 7

modal superposition. The measured time-domain responses can also be employed to identify the prestressed 8

force of a prestressed concrete bridge by a system identification approach (Chan and Yung, 2000; Law and Lu, 9

2005; Lu and Law, 2005). In addition, numerous achievements in moving load identification have been made in 10

recent years with the application of artificial neural networks (ANNs) (Lu, C. N. et al., 1993; Kowm, D. et al., 11

2014) and genetic algorithms (GAs) (Deng and Cai, 2010b). 12

Based on previous studies, there exist two fundamental problems in moving load identification: (1) The 13

measured response may not be a part of the true response data collection induced by load excitations, which 14

may result in there being no approximate numerical solution to the motion equations; and (2) Tiny 15

measurement errors in signals may lead to critical mistakes, which are biases of the solution due to the 16

ill-conditioned matrix; therefore, the emphasis of solving the inverse problem lies in how to solve the ill-posed 17

equations, and to correctly estimate the time history of the moving loads, the dynamic algorithms must be 18

optimized and improved and the measurement errors must be eliminated. In the dynamic inspection of a bridge, 19

acceleration and dynamic strain are generally measured along the bridge, thus efficiently utilizing both of 20

these two kinds of data are desired to improve the identification accuracy. In this paper, a novel moving load 21

identification method is proposed by utilizing measured acceleration and strain signals on a bridge. The 22

dynamic displacement of a bridge subjected to moving loads consists of the vibration displacement caused by 1

inertia force as well as the static displacement caused by static force. The acceleration and strain signals were 2

measured simultaneously to improve the identification accuracy. The dynamic displacement was reconstructed 3

by a series of signal processing processes, including decomposition, optimization and conversion of the strain 4

and acceleration signals. The reconstructed vibration displacement signal was obtained by the integration of 5

measured acceleration, and the static displacement signal was deduced from the measured strain signals 6

utilizing the curvature function fitted with the least mean squares method. Finally, the reconstructed dynamic 7

displacement was used to identify the moving loads after superposition of vibration and static displacements. 8

To verify the accuracy of the proposed method, a simply supported girder bridge with a span length of 25 m 9

was set up by numerical simulation, and a set of dual-axle time-varying loads was applied to the bridge. Then, 10

the IMII and TDM algorithms were used to evaluate the accuracy and reliability of the identified loads. The 11

tested model for vehicle load identification was further constructed in the laboratory to verify the feasibility of 12

the proposed method in an experiment, and the moving loads under different working conditions were 13

successfully identified by the reconstructed displacement. 14

RECONSTRUCTION OF DYNAMIC DISPLACEMENT 15

The dynamic displacement of a beam under the moving loads is composed of not only the vibration 16

displacement caused by inertial force but also the static displacement caused by static loads. To obtain 17

acceptable results, a signal processing scheme was applied to the directly measured data, and the dynamic 18

displacement was reconstructed based on the strain and acceleration signals, as shown in Fig. 1. 19

THEORY OF MOVING LOAD IDENTIFICATION 20

Equilibrium equation of vibration 21

The bridge can be simplified as a simply supported beam model to demonstrate the method. The beam is 22

assumed to have a constant cross-section with a span length L, constant mass per unit length ρ, constant flexural 1

stiffness EI and viscous proportional damping ratio C. The beam is assumed to be a Euler-Bernoulli beam 2

without consideration of shear deformation and rotary inertia. When the time-varying point force P(t) is moving 3

on the beam from left to right at a speed of c, as shown in Fig. 2. The vibration equation of the beam can be 4

expressed as shown in Eq. (1). 5

( ) ( ) ( ) ( ) ( )2 4

2 4

, , ,

v x t v x t v x tC EI x ct P t

t t x

+ + = −

(1) 6

where ( , )v x t is the deflection at point x and time t and ( )x ct − is the Dirac delta function. 7

The nth mode shape function of the beam can be written as ( ) sin( / )n x n x L = . Hence, the solution of Eq. (1) 8

can be assumed to follow the form of Eq. (2): 9

( ) ( ) ( )1

, n nn

v x t x q t

=

=

(2) 10

where n is the mode number and ( )nq t (n=1,2,3,…∞) are the displacements of the nth mode. By substituting Eq. 11

(2) into Eq. (1), integrating the resultant equation with respect to x between 0 and l, and introducing the 12

boundary conditions of the simply supported beam, the equation of motion (EOM) can be expressed in 13

generalized modal coordinates, as shown in Eq. (3): 14

( ) ( ) ( ) ( )2 22 ( 1,2,3, )n n n n n n nq t q t q t P t n

L

+ + = =

(3) 15

where2 2

2nn EI

L

= , 2n

n

C

= and ( ) ( )sin( )n

n xP t P tL

= indicate the modal frequency, modal damping 16

and modal force of the nth mode, respectively. x is the distance from the concentrated force to the left side 17

support. If the time-varying force ( )P t is known, the modal displacements ( )nq t can be obtained from Eq. (3); 18

thus, the dynamic deflection ( , )v x t can be calculated by Eq. (2). The calculation process is a typical forward 19

problem. The unknown load ( )P t can be deduced by acquiring the displacement, velocity and acceleration 20

data at each measurement point of the structure. 21

Interpretive method II (IMII) 1

When the boundary conditions correspond to the simply supported case, the exact solution for the bridge 2

responses under dynamic forces can be deduced (Chan, 1999). If n moving loads are applied to the beam, using 3

the modal decomposition method, Eq. (3) can be written as 4

21 11 1 1 12

2 2 2 2 2 2

2

( )( ) ( )( ) ( ) ( )

22

+

2( ) ( ) ( )nn n n n n

q tt tt t t

t

q qq q q

q tq tq

+ =

5

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 2

11 2

2

1 2

ˆ ˆ ˆsin sin sin

ˆ ˆ ˆ2 2 22 sin sin sin

ˆ ˆ ˆsin sin

( )(

(sin

)

)

k

k

kk

ct x ct x ct xL L L P

ct x ct x ct x PL L LL

Pn c

tt

t x n ct x n c xL

tt

L L

− − −

− − −

− − −

(4) 6

where ˆkx is the distance from the kth load to the first load and 1̂ 0x = when k=1. 7

If 1 2 kP P P、 、 are constant moving loads when ignoring the effect of damping, Eq. (1) has the following 8

closed-form solution: 9

( ) ( ) ( )( )3

2 2 21 1

ˆ ˆ1, sin sin48

ki i

i ni n

n ct x xL n xv x t P x sin tEI L L n cn n

= =

−= − −

−

(5) 10

wheren

vL

= . As long as the displacements at 2 nx x x1 、、 、 are available, the magnitude of each constant 11

moving load can be acquired from Eq. (6): 12

vPv S P= (6) 13

where

11 1 1

1

1

i k

vP m mi mk

l li lk

S S S

S S S S

S S S

=

and ( )( )3

2 2 21

ˆs

ˆin1 sin

48i i

mi nn

n ct xS sin t

L n cxL n x

EI Ln n

=

− − −

=− 14

where l and k indicate the number of displacement measurement stations and axle loads, respectively. If l k , 15

the load can be estimated by the least squares method, as shown in Eq. (7). 1

( 1)([ ] [ ]) }{[ ]T Tvp vp vpP S S S v−= (7) 2

Accordingly, this method can be applied to estimate moving time-varying axle loads from Eq. (4), in which 3

the values of the axle loads at any time can be solved using the least squares method. 4

Time domain method (TDM) 5

The TDM uses modal superposition to identify time-varying axle loads in the time domain. Based on the 6

motion equation, the modal displacement ( )nq t can be solved from Eq. (3) via a convolution integral in the 7

time domain (Law et al., 1997). 8

( ) ( ) ( )01 t

n n nn

q t h t p t dM

= − (8) 9

where ( ) ( )( )''

1 sin 0n ntn n

n

h t e t t

−= , in which ' 21 2 n n n nM

L

−= =， .

10

When substituting Eq. (8) into Eq. (2), the dynamic deflection of the beam at point x and time t can be expressed 11

as follows: 12

( ) ( )

01

1, sin sin ( )sin ( )n nt t

ni n n

n x n cv x t e t P dM L L

− −

=

= −

(9) 13

The bending moment of the beam at point x and time t can be expressed as shown in Eq. (10). 14

( ) ( ) ( )2 2 2

( )2 3 0

1

, 2, = sin sin ( )sinn nt t

nn n

v x t EI n n x n cM x t EI e t P dx L L L

− −

=

= − −

(10) 15

Assuming that the time-varying load ( )P t is a step function in a short time interval t , Eq. (10) can be 16

rewritten in discrete terms as 17

( ) ( ) ( ) ( )2 2

'3 '

1 0

2 sin sin sinn ni

t i jn

n jn

EI n n x n c tjM i e t i j P j tL L L

− −

= =

= − , 18

i =0, 1, 2, 3,…, N (11) 19

where Δt is the sampling time interval and N+1 is the number of sample points. 20

When a vehicle enters onto and leaves the bridge, assuming that (0) ( ) 0BP P N= = such that 1

( ) 0(0) 1M M= = , then Eq. (11) can be rewritten in the matrix form as shown in Eq. (12). 2

( )( )

( )

( ) ( )( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )( )

( )

11 2

121 21 2

121

1 21 2

02 0 11

11 23 0 22 1

21 1

2 1

n

nnxn

iNNn e Bn e

M PE S SE S SM PE S S

C

E S N SM N P NE S N bS

=−−

= − −−

(12) 3

Where 2 2

3 '

2 sinxnn

EI n n xC tL L

= , ( )n n t i ji j

nE e − −− = , ( ) ( )'1 sin nS i j t i j− = − , and ( )2

Δsin .n c tS j jL =

4

( )M i is the bending moment of the ith time step, ( )P i is the axle load at the ith time step, the subscript N denotes 5

the number of sampling points for the measured bending moment response, NB=L/c∆t indicates the number of 6

sampling points for the time-varying load moving through the entire bridge deck, and 7

( ) ( )11 21 1BN N

ee n B Bb E S N N S N− += − + − (13) 8

Eq. (12) can be simplified as 9

B P M= (14) 10

where [B] is a coefficients matrix relating the parameters of the vehicle-bridge system; {P} is the time-varying 11

load vector; and {M} is the vector of the bending moment responses. If N=NB and [B] is a lower triangular 12

matrix, the load vector {P} can be directly solved by Eq. (14). If N＞NB or there are Nl responses for the bending 13

moment (Nl＞1) from the measuring points, {P} can be obtained by the least squares method. In addition, Eq. 14

(14) can be utilized to perform load identification referring to a two-axle vehicle load based on the linear 15

superposition principle. 16

The above procedure is derived for single force identification. Eq. (14) can be modified for two-force 17

identification using the linear superposition principle (Law et al., 1997) 18

1

2

0a

b a

c b

BP

B B MP

B B

=

, (15) 19

where ( 1)a s BB N N − , ( 1 2 ) ( 1)b s BB N N N− − − and ( 1)c s BB N N − are submatrices of matrix B. 1

The first row of submatrices in the first matrix describes the state of having the first force on the beam after its 2

entry. The second and third rows of submatrices describe the states having two forces on the beam and one 3

force on the beam after the exit of the first force. The entire matrix has dimensions of ( )-1 ( 1)BN N − . 4

/s sN l c t= , where ls is the distance between the two forces. The two forces can be identified using more 5

than one bending moment measurement. 6

DISPLACEMENT CAUSED BY THE STATIC LOAD 7

The static displacement is obtained through the strain signal caused by the static load. After performing 8

empirical mode decomposition (EMD) on the strain signal, the high-frequency component will be filtered to 9

extract the time history curve of the static strain. Then, the curvature function of the beam can be obtained by 10

the least squares method. Finally, the vertical displacement involved with the static load is estimated by the 11

relationship between the curvature and vertical displacement. 12

Empirical mode decomposition (EMD) 13

The Hilbert-Huang Transform method (HHT) is proposed for non-stationary and non-linear signal analysis by 14

Huang, N. E. (2000). The key step of HHT is to use the EMD method that decomposes the complex signal into 15

a collection of intrinsic mode functions (IMFs). Since the decomposition is based on the local characteristic 16

time scale of the data, mean or zero reference is not required in the EMD method, which is adaptive and 17

efficient for analysing non-stationary and non-linear time series. 18

The EMD method is based on the following assumptions (Huang, N. E. et al., 1998): 19

(1) The signal has at least two extremes, one maximum and one minimum; 20

(2) The local characteristic time scale is defined by the time lapse between the extrema; 21

(3) If the data were totally devoid of extrema but contained only inflection points, then they can be 22

differentiated once or more times to reveal the extrema. The final results can be obtained by integrating the 1

components. 2

Each IMF extracted by the sifting process satisfies two conditions: (1) the number of extrema and the number 3

of zero crossings must either equal or differ at most by one; and (2) the envelope is defined by the local 4

minima. An IMF involves only one mode of oscillation embedded in the vibration signal and gives rise to a 5

well-defined instantaneous frequency. By using the EMD method, the original multicomponent signal is 6

decomposed into a series of mono-components, making it accessible for further analysis, while some 7

problems remain in the decomposed result, such as end effects and the mode mixing problem. 8

Displacement by the curvature function 9

For the beam, the displacement v is defined as the deflection of the cross-section in the direction perpendicular 10

to the neutral axis during bending deformation. The neutral axis forms a plane curve in the central principle 11

plane of inertia, which is called the deflection curve. The strain ε represents the deformation of the infinitesimal 12

element per unit length subjected to stress, and the curvature k indicates the bending extent of the beam. An 13

element with length dx is randomly selected from the bending beam, as shown in Fig. 3, where O1O2 is the 14

neutral axis of the beam element; y is the distance from the edge to the neutral axis; ρ is the radius of curvature; 15

and the length of the neutral axis remains constant before and after the deformation. 16

Thus, the length change s of the beam edge can be expressed as 17

1 2= =( )ab o os l l y d d yd − + − = (16) 18

Accordingly, the magnitude of strain can be calculated as 19

= s yd y

dx d

= =

(17) 20

In Eq. (17), the strain and curvature at the same location are linearly correlated. According to the differential 21

theorem and ignoring the effect of second-order terms, the deflection-curvature relationship can be expressed as 22

2

2

1( )

d vx d x=

(18) 1

The deflection curve can be obtained by integrating the beam curvature function twice, and the strain and 2

curvature at the same cross-section are linearly correlated; thus, the strain-deflection relationship can be formed 3

through the curvature. 4

DISPLACEMENT CAUSED BY THE INERTIA FORCE 5

Vibration displacement can be obtained by integrating the measured acceleration signal twice in the frequency 6

domain. As shown in Fig. 1, in the signal processing procedure, the signal should be filtered and the trend term 7

should be eliminated before each step of integration, so the adaptive least-mean-square (LMS) algorithm for 8

noise reduction is utilized. 9

LMS adaptive noise reduction 10

On the basis of the Wiener filter, the LMS algorithm developed with the minimum mean square error (MMSE) 11

and the steepest-descent algorithm, which simulates a desired signal by finding the filter weights, is related to 12

producing the least mean square of the error signal. Based on the gradient decent algorithm, the adaptive LMS 13

filter updates the weights to approach the optimum values for minimizing the error criterion. 14

The main procedure of the adaptive LMS algorithm (Zhen Zhu et al.,2016) is as follows: 15

(1) Calculate the output signal ( )y n from the adaptive filter: 16

( ) ( ) ( )Ty n X n W n= (19) 17

where n indicates the number of iterations of the algorithm, ( )X n is the input signal in vector form and ( )W n18

is the filter weights in vector form; 19

(2) By calculating the difference between the reference signal ( )d n and the filter’s output ( )y n , the error 20

signal ( )e n is acquired as follows: 21

( ) ( ) ( )e n d n y n= − (20) 22

(3) Update the filter weights by the steepest-descent algorithm: 23

( 1) ( ) 2 ( ) ( ) W n W n e n X n+ = + (21) 1

where ( 1)W n+ indicates the filter weights for the next iteration and is the convergence factor, which can 2

be used to control the filtering rate. 3

(4) Repeat steps (1) to (3) until the algorithm converges, continuous iteration and updating the filter weights 4

will contribute to achieving the minimum mean square error 2[ ( )]E e n . Meanwhile, the output signal ( )y n5

approaches the reference signal ( )d n . 6

The LMS filter carries out filtering using only the observable signals x(n), d(n) and e(n) without involving any 7

matrix operations, so it has low computational complexity for adaptive noise reduction. The flowchart of 8

adaptive LMS algorithm for noise reduction is shown in Fig. 4. 9

Integration in the frequency domain 10

Fourier transform can be performed to convert the signals from the time domain to the frequency domain. 11

Meanwhile, the signal can be restored to the time domain by inverse Fourier transformation. Then, the 12

displacement value in the time domain can be obtained from the signal by reserving its real part. The integration 13

method is shown in the flowchart in Fig. 5. 14

Assuming a discrete acceleration sequence ( )a n in the time domain with a data length N, the discrete Fourier 15

transform (DFT) is performed according to Eq. (22). 16

( ) ( )

( )( )

( ) ( ) ( )

2 1 1

1

0

2

1 1

2

k n iNN

k ka k a n e a b iN

n ff k k N

N

− − −

= = +

−=

(22) 17

where ( )a k is the Fourier transform of the acceleration sequence ( )a n , ( )f k is the corresponding frequency 18

and 0f is the sampling frequency. The amplitude kA of the simple harmonics, the modal frequency k and 19

the initial phase k , corresponding to ( )a k , can be obtained by Eq. (23). 20

2 2

arctan( )

2

k k

kk

k

k

k

k

A a bba

f

= + = =

(23) 1

The simple harmonic wave is expressed as 2

cos( ) ( )k k k ka t A = + (24) 3

The displacement harmonic wave is available after double integration for ( )ka t , as shown in Eq. (25). 4

( ) ( )cossk sk skkS t A t = + (25) 5

Where 2/sk k k sk k sk kA A = = − =，， . 6

Based on the signal superposition principle, the displacement curve can be obtained as shown in Eq. (26). 7

( ) ( ) ( )1 1

cosN N

sk sk skkk k

S t S t A t = =

= = + (26) 8

By performing inverse Fourier transformation, the equation above can be rewritten as 9

( )( ) ( ) ( )( ) ( )( ) ( ) ( )( )12 2

1 1F s t H F a t F a t s t F F a t

− = = − = −

(27) 10

where ( )F x indicates performing Fourier transform on the function, ( )H is the transfer function, ( )a t is 11

the acceleration signal in the time domain and ( )s t is the displacement signal obtained by integration. 12

Eq. (27) shows that low-frequency noise contained in the acceleration signal will be amplified during the 13

integration and cause a large error due to the amplitude-frequency characteristic 21/ . Therefore, the 14

integration error in calculating the displacement from the acceleration response can be constrained by filtering 15

the low-frequency noise from the observed signals. 16

NUMERICAL SIMULATION 17

To validate the feasibility of the proposed method, a simply supported beam model was built in ANSYS for FE 18

model analysis. The beam element selected was BEAM3, where each element node had three degrees of 19

freedom (DOFs). The following parameters were chosen for the modelled bridge: EI=2.87×109 N·m2，ρ=2,303 20

kg/m and L=25 m. The beam was divided into 100 elements, and the first three natural frequencies of the 21

bridge were determined to be f1=4.79 Hz, f2=19.18 Hz and f3=43.15 Hz by calculation. Assuming that dual-axle 1

time-varying loads move on the bridge deck, the expressions of the load are as follows: 2

1 28.75(1 0.25s( in(2 )))f t t− += (28) 3

2 57.5(1 0.25s( in(2 )))f t t− += (29) 4

The spring stiffness was chosen as k=1,595×103 N/m, and the damping was 334,200 kg/s. 5

Three measuring points were assumed to be located at 1/4, 1/2, and 3/4 of the span along the beam, as shown in 6

Fig. 6. The acceleration and strain responses acquired from the measuring points were used for moving load 7

identification, and the first three modes of the beam were of interest in the computation. The sampling frequency 8

was set as 200 Hz to cover the first three modes as well as meet the requirement of the Nyquist–Shannon 9

sampling theorem. Three load cases were designed, while the moving load speeds were set as 10, 20, and 30 10

m/s. 11

In practical application and bridge tests, the magnitude of moving loads cannot be obtained by direct 12

measurement. To assess the identified results, FE software is used to generate the true output signal from the 13

measuring points, and then, the loads obtained from the true and reconstructed signals are compared for error 14

evaluation using Eq. (30). 15

100%ture reconerror

ture

f fP

f −

=

(30) 16

where errorP represents the relative percentage error and turef and reconf represent the loads identified from the 17

true signal and reconstructed signal, respectively. 18

Static displacement identification 19

When the loads are moving across the bridge at a speed of 20 m/s, the dynamic responses of the three measuring 20

points can be calculated directly from ANSYS, as shown in Fig. 7. 21

When a car is travelling on the beam at a low speed, noises caused by the environment and instrument during 22

data acquisition may corrupt the signals. To simulate the influence of environment noise, 10 dB of white noise 1

was added to the strain generated from the FE software, as shown in Fig. 8. 2

The dynamic strain signal was decomposed by the EMD method to acquire the corresponding IMFs at three 3

points, one of which is illustrated in Fig. 9, and the calculated static strain response caused by the static load is 4

shown in Fig. 10. As shown by the decomposed results, the EMD method decomposed the complex signal into 5

multiple IMFs, where the low-frequency component IMF11 reflects the strain caused by static vehicle loads. 6

7

The neutral axis is at the middle height of each beam section, the curvature of multiple points can be derived 8

from the nodal strain responses according to Eq. (17), and the curvature function of the entire beam is obtained 9

by the least-squares method. Then, the deflection curve can be obtained by integrating the curvature function 10

twice, as shown in Eq. (18). The static displacements at points 1, 2, and 3 due to the static forces are shown in 11

Fig. 11. 12

Vibration displacement identification 13

The true value and the value after adding 5 dB of white noise to the acceleration signals are shown in Fig. 12. 14

Since the signal after adding white noise contained many interference frequencies, the adaptive LMS algorithm 15

was utilized to reduce the noise. The filter order was set as 100, and the step factor was set as 0.005. The 16

frequency domain characteristics of the filtered signal are shown in Fig. 13, illustrating that the filtered signal is 17

largely consistent with the true signal except for a small amount of noise mixed in the high-frequency 18

components. 19

The vibration displacement was calculated by integrating the acceleration twice in the frequency domain, which 20

can ensure the reliability of the signal and achieve a high precision. There are two equilibrium conditions to 21

meet the application: the velocity and displacement at the measuring points should be idealized as zero before 22

the moving loads enter onto the beam, and the beam vibration will be decay to the initial equilibrium state after 1

the moving loads leave the beam. The two conditions can be used to eliminate the trend term in the integration 2

process; thus, the vibration displacement is calculated as shown in Fig. 14 (a). 3

The reconstructed displacement signal shown in Fig. 14(b) generally matched the true signal, and deviations 4

occurred mainly in the higher frequency range and at both ends of the beam. The deviations were caused by 5

error accumulation in the integration process as well as the end effects, which refers to the divergence 6

phenomenon in the signal decomposition, occurring during the EMD decomposition. When the deviation or 7

uncertainty occurred in the calculation of maximum or minimum values of the signal, errors may arise in the 8

calculated envelope curve and further lead to a divergence of the EMD results. 9

Moving load identification 10

The modal displacement can be obtained using the reconstructed displacement signal through Eq. (2), and the 11

modal velocity and acceleration can be obtained via numerical differentiation. By substituting the modal 12

displacement, velocity and acceleration into Eq. (3), the value of the moving load can be identified by the IMII 13

method as well as the TDM. Three cases, in which the load moved on the beam at speeds of 10, 20, and 30 m/s, 14

were considered for load identification of the front and rear axles, and the identified results are shown in Fig. 15. 15

In the figure, ‘True’ is the load identified by the TDM from the true displacement signal generated by the FE 16

simulation, and ‘Recon1’ and ‘Recon2’ are the results identified by the IMII method and TDM. 17

The time-varying loads can be identified by the IMII method and TDM when moving across the bridge and the 18

results certify a high reliability for load identification from the reconstructed displacement, by which the noise 19

corruption problem is overcome and the identification error is constrained within a reasonable range of 6%. 20

LABORATORY EXPERIMENT VALIDATION 21

Experiment introduction 22

A simply supported RC slab was constructed for the experiment to validate the proposed theory in application. 1

The length of the slab was 3.6 m, and the width was 0.4 m with a thickness 0.055 m. The cross-section and 2

reinforcement layout of the test specimen are shown in Fig. 16(a). Two leading beams with a length of 0.5 m at 3

both ends, which were utilized as the speed guiding way for the model car getting on and off the slab, are 4

shown in Fig. 16(b). 5

Structural dynamic test 6

Before the moving load identification, a multireference impact test (MRIT) was conducted to obtain the modal 7

parameters. Due to the limitations of equipment channels, the measurement points were divided into four testing 8

cases by arranging four reference points, as shown in Fig. 17. LMS Cadax-8 data acquisition was used to 9

acquire the hammer impact force and acceleration signals. A sampling frequency of 2,048 Hz was selected to 10

capture the impact force at high resolution in the time domain. Each point was hammered three times, and the 11

complex mode indicator function (CMIF) was utilized in the modal analysis. The singular value curves of the 12

frequency response functions (FRFs) produced by the CMIF method are shown in Fig. 17, including four curves 13

for each of the reference points. 14

The FE model was established through Midas software to obtain analytical modal information, which was used 15

to compare with the modal analysis results obtained by the CMIF method in Table 1. The measured and 16

calculated dynamic characteristics are similar; thus, the FE simulating structure is capable of reflecting the 17

performance of the actual structure. 18

Moving load experiment 19

The top of the leading beam was connected to the slab top, and the edge of the beam was very close to the edge 20

of the slab without direct contact. The acceleration and dynamic strain were measured in the experiment, and 8 21

measurement points were set up on the structure along the longitudinal direction, as shown in Fig. 19(a). 22

According to the vehicle speed and loaded mass, 12 cases were scheduled in the experiment, as shown in Table 1

2. The cases included three sets of vehicle speeds, i.e., v1=0.12 m/s, v2=0.24 m/s, and v3=0.32 m/s, and four sets 2

of vehicle loaded masses, i.e., m1=5 kg, m2=10 kg, m3=15 kg, and m4=20 kg. To verify the monitored dynamic 3

displacement at point 4, a high-speed camera (SONY FDR-AX700) was used to accurately measure the 4

dynamic displacement. The flexural stiffness of the bridge model is estimated to be 188.28 kN/m2 by loading 5

weights and monitoring the deflection. 6

Experiment results 7

According to the dynamic strain signals measured when the vehicle is travelling on the bridge, the raw signal 8

contained a large amount of noise interference, and the high-frequency part was mainly composed of the 9

vibration strains induced by the noise and inertial force. The static strain in the low-frequency domain and the 10

dynamic strain in the high-frequency domain can be separated through EMD. In Case 4, for example, the 11

decomposed results at instrumentation point #4 in Case 4 is shown in Fig. 20, in which the decomposed signal 12

becomes smooth and the strain reaches the peak when the load passes the instrumentation point. The static strain 13

obtained by the EMD method was used to derive the displacement responses caused by the static loads. After 14

using the data acquired from eight strain sensors and calculating the curvature function by the iterative 15

algorithm, the static displacement under the moving loads could be identified as shown in Fig. 21. 16

The identified vibration displacement was calculated from the acceleration signal according to the integration 17

flowchart in Fig. 5. Noise reduction was performed on the acquired signal using the LMS algorithm, in which 18

the filter order was set as 100 and the step factor was set as 0.005. Then, the velocity and displacement were 19

obtained by integrating the signal in the frequency domain via MATLAB. The calculated vibration 20

displacement in Case 12 is shown in Fig. 22. By comparing the results in various cases, the model car moving 21

at a low speed resulted in more obvious high-frequency vibrations of the displacement due to a longer 22

duration on the bridge model than the high-speed model car. The reconstructed displacements obtained by 1

combining the static displacement and vibration displacement are shown in Fig. 22 (b). 2

A comparison between the reconstructed displacements and the displacements measured via a high-speed 3

camera at point #4 is illustrated in Fig. 23. The measured displacement and reconstructed displacement at point 4

#4 fit well, and the errors at the peak values are less than 6% when the loads are moving across the measuring 5

point. The comparison results show that the vibration displacement was accurately restored by the integration 6

of the acceleration signal in the frequency domain. The deflection matched the actual deformation, which 7

indicated that it is capable of reaching the requirement of the high accuracy measurement by using the strain 8

values for structural curvature reconstruction. 9

Load identification was carried out with the reconstructed displacement under various cases, and a small portion 10

of the identified results is shown in Fig. 24 and Fig. 25, where ‘IMII’ denotes the results using the IMII method 11

after noise reduction, ‘TDM(1)’ presents the results using SVD after noise reduction and TDM(2) presents the 12

results for the raw signal with the pseudo inverse method. Considering the mass of the entire model car and the 13

loaded weights, the actual loads under Cases 1, 4, 8 and 11 are identified as 72.8 N, 122.8 N, 172.8 N and 222.8 14

N, respectively, which are consistent with the true values. The reconstructed displacement illustrates that the 15

information regarding the single-axle load when moving across the measuring point cannot be adequately 16

identified from the trend of the entire curve. A direct comparison of the identified results illustrates that both 17

the IMII method and TDM are capable of estimating the moving axle loads accurately under different vehicle 18

speed and mass cases after signal processing and optimization. The IMII computation required less time than 19

the TDM and was more sensitive to interference noise. 20

CONCLUSIONS 21

Moving load identification aims to resolve the inverse problems that occur when estimating wheel loads in 22

motion from bridge responses. In this paper, a methodology was proposed to reduce the ill-conditioned 1

problem and improve the accuracy of the identification by reconstructing the dynamic displacement of the 2

bridge. The reconstructed displacement is a superposition of the static displacement acquired from the 3

decomposed strain signal and the vibration displacement calculated by denoising the acceleration signal. The 4

following conclusions can be drawn: 5

(1) By employing the measurements from the acceleration and strain sensors, the static and vibration 6

displacements can be calculated to reconstruct the dynamic displacements for further moving load 7

identification of the bridge. As indicated from the load identification results of the FE analysis and laboratory 8

experiment, the reconstructed displacement signal can accurately identify the moving loads under different 9

testing cases, which is effective for overcoming the ill-conditioned problem and achieving a high level of 10

accuracy. 11

(2) The adaptive LMS algorithm for noise reduction can directly filter the noise through signal processing and 12

improve the signal-to-noise ratio. However, the efficiency of noise reduction and the computation rate may be 13

influenced by the convergence ratio. 14

(3) Since the low-frequency noise causes greater corruption on the observed acceleration signal during 15

integration, integration of the denoised signal in the frequency domain can achieve better restoration of the 16

vibration displacement than that in the time domain. Moreover, the trend generated in the integration can be 17

eliminated by simulating it with the least squares method. 18

19

20

21

22

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5

List of Tables 1

Table 1 Dynamic test results obtained by the MRIT 2

Table 2 Load cases of the moving load experiment 3

4

Table 1 Dynamic test results obtained by the MRIT 1

Mode FEM simulation CMIF measurement

Frequency (Hz) Mode shape Frequency

(Hz) Mode shape

1 7.49

7.50

2 29.98

30.25

3 67.33

67.75

4 81.53

82.01

5 119.62

118.75

2

Table 2 Load cases of the moving load experiment 1

Case Speed/(m/s) Mass/(kg) Case Speed/(m/s) Mass/(kg)

1 0.12 5 7 0.12 15

2 0.24 5 8 0.24 15

3 0.32 5 9 0.32 15

4 0.12 10 10 0.12 20

5 0.24 10 11 0.24 20

6 0.32 10 12 0.32 20

2

3

List of Figures 1

Fig.1 Simply support beam for moving load identification 2

Fig.2 Reconstruction of dynamic displacement signal 3

Fig.3 Curvature deformation of beam element 4

Fig.4 Flowchart of LMS adaptive noise reduction 5

Fig.5 Flowchart of integration method 6

Fig.6 Instrumentation layout of bridge model 7

Fig.7 Moving load response at instrument at v=20m/s (a) Acceleration signals; (b) Dynamic displacement 8

signals; (c) Dynamic moment signals 9

Fig.8 Calculated strain when adding noise 10

Fig.9 EMD decomposition of strain signals at point 2 11

Fig.10 Strain signal decomposition at point 2 12

Fig.11 Identification results of static load caused displacement 13

Fig.12 (a) Acceleration signals at point 1with noise; (b) Fourier spectrum of acceleration at point 1 14

Fig.13 (a) Acceleration denoising signals at point 1; (b) Fourier spectrum of denoising acceleration at point 1 15

Fig.14 (a) Calculated vibration displacement signal at point 2; (b) Reconstructed dynamic displacement signal 16

at point 2 17

Fig.15 Identified results of time-variable moving load (a) Front-axle at v1=10m/s; (b) Rear-axle at v1=10m/s; (c) 18

Front-axle at v1=20m/s; (d) Rear-axle at v1=20m/s; (e) Front-axle at v1=30m/s; (f) Rear-axle at 19

v1=30m/s. 20

Fig.16 (a) Tested plate cross-sectional view; b) Structural dimension and instrumentation layout 21

Fig.17 Instrumentation layout of modal test 22

Fig.18 Mode extraction by CMIF method 23

Fig.19 (a) Instrumentation layout for data acquisition; (b) Picture of instrumentation; (c) Moving load vehicle 1

and target displacement 2

Fig.20 EMD decomposition of strain signal at point 4 in case 12 (a) IMF signals; (b) Static strain 3

Fig.21 Identified static displacement results in case 12 4

Fig.22 (a) Calculated vibration displacement identification at point 4 in Case 12; (b) Different reconstructed 5

displacements at different points in Case 12 6

Fig.23 Comparison of displacement at point 4 in Case 12 7

Fig.24 Moving load identification results (a) Case 10; (b) Case 12; (c) Case 24; (d) Case 31 8

Fig.25 Histogram of moving load identification (a) IMII method in Case 10; (b)TDM method in Case 10; (c) 9

IMII method in Case 12; (d) TDM method in Case 12; (e) IMII method in Case 24; (f) TDM method in 10

Case 24; (g) IMII method in Case 31; (h) TDM method in Case 31 11

12

1

Fig. 1 Reconstruction of a dynamic displacement signal 2

3

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

-75-60-45-30-15

015

Calculate acceleration signal

ckck

Weight: Static load Inertia force：Dynamic load

Displacement response:

Vehicle load

Calculate static load caused strain

Original strain

0.0 0.5 1.0 1.5 2.0 2.5

-60

-50

-40

-30

-20

-10

0

10

20

30

0 100 200 300 400 500-5

0

5

0 100 200 300 400 500-2

0

2

0 100 200 300 400 500-2

0

2

0 100 200 300 400 500-5

0

5

0 100 200 300 400 500-5

0

5

0 100 200 300 400 500

-40

-20

0

0 50 100 150 200 250 300 350 400 450 500-2.5

-2

-1.5

-1

-0.5

0x 10

-3

0 50 100 150 200 250 300 350 400 450-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5x 10

-4

0 50 100 150 200 250 300 350 400 450 500-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

-3

0 50 100 150 200 250 300 350 400 450 500-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 50 100 150 200 250 300 350 400 450 500

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Signal processing

Integration Eliminate trend

0 50 100 150 200 250 300 350 400 450 500-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-3

Acquire static strain Curvature function Double integration

EDM decompose

Static displacement

Denoising signal

Velocity signal

Vibration displacement

Acceleration with noise

Integration Eliminate trend

1

Fig. 2 Simply supported beam for moving load identification 2

3

P(t)x

y c

ctL

�̅� = 𝑐𝑡

1

Fig. 3 Curvature deformation of the beam element 2

3

dx

d

a1o1 o2ab1

b

O

No

Yes

Initialize ( )iw n

Input ( )x n

Calculate1

( ) ( ) ( 1)M

ii

y n w n x n i=

= − +

Calculate error ( ) ( ) ( )e n d n y n=

Update weight coefficient ( 1) ( ) ( ) ( )i iw n w n e n x n+ = −

2[ ( )]E e n reaches minimum?

Output

1

Fig. 4 Flowchart of LMS adaptive noise reduction 2

3

1

Fig. 5 Flowchart of the integration method 2

3

Acceleration signal a(t)

Determine target frequency

Amplitude-frequency characteristic

displacement signal s(t)

FFT

IFFT

Amplitude spectrum

Phasespectrum

Displacement spectrum

1

Fig. 6 Instrumentation layout of the bridge model 2

3

(a) (b) (c) 1

Fig. 7 Moving load response at the instrument for v=20 m/s: (a) acceleration signals; (b) dynamic 2

displacement signals; (c) dynamic moment signals 3

4 5

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-0.4

-0.2

0.0

0.2

0.4

0.6

Acc

eler

atio

n(m

/s2 )

Time(s)

Point 1 Point 2 Point 3

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-2.4

-1.8

-1.2

-0.6

0.0

Dis

plac

emet

(mm

)

Time(s)

Point1 Point2 Point3

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-400

-300

-200

-100

0

Mom

ent(k

N*m

)

Time(s)

Point 1 Point 2 Point 3

1

Fig. 8 Strain calculated after adding white noise 2

3

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6-75-60-45-30-15

015

Stra

in(μ

ε)Time(s)

Point 1 Point 2 Point 3

1

Fig. 9 EMD decomposition of the strain signals at point 2 2

3

1

Fig. 10 Strain signal decomposition at point 2 2

3

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-60

-40

-20

0

20

40

60

Stra

in(μ

ε)

Time(s)

Total strain+noise Static strain Dynamic strain+noise

1

Fig. 11 Identification results for the displacement caused by the static load 2

3

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

Dis

plac

emen

t(mm

)Time(s)

Point 1 Point 2 Point 3

(a) (b) 1

Fig. 12 (a) Acceleration signals at point 1 with and without noise; (b) Fourier spectrum of 2

acceleration at point 1 3

4

0.0 0.2 0.4 0.6-2

-1

0

1

2

3

Acc

eler

atio

n(m

/s2 )

Time(s)

Signal adding noise True signal

0 20 40 60 80 100 120 1400.00

0.05

0.10

0.15

0.20

0.25

0.30 FFT-Adding noise FFT -Ture

Am

plitu

de

Frequency(Hz)