identification of moving loads on a rectangular

IDENTIFICATION OF MOVING LOADS ON A RECTANGULAR ORTHOTROPIC PLATE FROM STRAINS

X.Q.Zhu and S.S.Law Department of Civil and Structural Engineering The Hong Kong Polytechnic University Hong Kong

group of moving load: ABSTRACT A time domain method is presented to identify the moving loads on the bridge deck from measuring strains. The bridge deck is modelled as a rectangular orthotropic plate. The vehicle is modelled as two axle loads moving on the bridge deck. Like all inverse problems, the identification is an ill-posed problem, so a regularization technique is employed to stabilize the solution. This paper gives a demonstration on the robustness of the solution technique to the effects of measurement noise and in the identification of eccentric moving loads. Numerical examples show that the method is effective and practical for moving load identification.

NC NP Y,(Y)

Dirac functions: the damping coefficient of the plate; mode shape ;

4,(t)

modal coordinate;

~(X)J~Yi C

circular frequency in radians per second; the fourth and second derivatives

I 4

NOMENCLATURE The following symbols are used in this paper. D9,

flexural rigidities;

Dn a, b, h

torsional rigidity;

tiX,YJJ

displacement

EJLYJ)

strains under the orthotropic plate in x-

E>(-LY.l)

direction; strains under the orthotropic plate in y-

w,c x2 Y )

direction; vibration modes of the orthotropic plate;

P

mass density of the orthotropic plate;

4,,,(t) & &&I

length, width and thickness of the orthotropic plate;

L

y,(Y); initial moment of l-beam; distance between two l-beams;

?A A

dimensions of the l-beam;

NT N+l

the number of measuring points: the number of sampling points.

regularization

parameter;

1 INTRODUCTION

the I,th load of the /,th group of

initial location of moving load t,,,( moving

of

of the orthotropic plate:

moving load; ( t ), j,r12 ( f ) location of the moving load t,,, ( t ) ; j,?,,,,,(t)

the number of group of moving loads; the number of moving loads;

speed

of the

t);

I, th group

moving load; the number of loads for the /,th

of

Analysis of highway bridges for vehicular live loads is a major element in the design of new bridges and evaluation of existing bridges for their load-carrying capacities. O’Connor and Chan (1988) have developed a method to identify the vehicle-bridge interaction force, in which a bridge is modelled as an assembly of lumped masses interconnected by massless elastic beam elements. Law et al (1997) models the bridge deck as a simply supported Euler beam. The forces are represented as step functions in a small time interval. The moving forces are then identified using the modal superposition principle in time domain. Law et al (1999) performs Fouier transformation on the equations of motion, which are expressed in modal coordinates. The relation between the responses and the forces is constructed in frequency domain. The time histories of the forces are found by the least-square method. Chan et al (1999) uses an Euler beam to model the bridge deck in the interpretation of dynamic loads crossing the deck. The Euler beam theory associated with

158

modal analysis is used to identify moving loads from bridge responses. Law and Fang (2000) have also reported a state estimation approach in which the forces in the state-space formulation of the dynamic system is solved using dynamic programming with minimization of the errors between the measured and the reconstructed responses from the identified moving forces. A method based on the modal superposition and optimization technique is also developed by Zhu and Law (1999) to identify the moving forces in the time domain. The bridge deck is modelled as a multi-span continuous Timoshenko beam with non-uniform cross-section. The loads are moving in a group. All the above approaches are based on the beam theory, and they refer to the interaction with a simply supported beam. They all discussed the identification problem for a group moving load. A beam model cannot truly represent the three-dimensional behavior of the bridge, particularly when a moving vehicle whose path is not along the centerline of the bridge. Many types of bridge decks, including those of slab bridges, hollow-core slab bridges, and deck and girder bridges can be effectively modeled by an isotropic or orthotropic plate (Bakht and Jaege, 1985). In modeling the highway bridge as a plate, the mathematical difficulties are compounded by the complicated vehicle-bridge dynamic interaction and the difficulty in representing the behavior of the bridge to traversing multi-vehicles. Fryba (1972) have solved by analytical methods the dynamic responses of uniform flat plate under a moving load along a specified path. Gupta and Traill-Nash (1980) simplified the bridge deck as an orthotropic plate. The dynamic loading of the bridge under a moving vehicle is analyzed. A half-vehicle model is used. Wu et al (1987) analyzes the dynamic responses of a flat plate subjected to various moving loads by the finite element method. Humar and Kashif (1995) analyzed the dynamic responses of slab-type bridges under two vehicles in series moving along the centerline. The bridge deck is assumed as an orthotropic plate. The vehicle is represented by a single sprung mass. Marchesiello et al (1999) analyzes the dynamics of multi-span continuous straight bridges subject to multi-degrees of freedom moving vehicle excitation by applying the mode superposition principle. The modes are computed by means of the Rayleigh-Ritz method. Chan and Chan (1999) analyzes the dynamic behavior of slab-on-girder bridges by eccentric beam elements. Mabsout et al (1999) investigate the effect of multilanes and continuity on the wheel load distribution in steel girder bridges. Zhu and Law (2000) models the bridge deck as an orthotropic plate. The dynamic behavior of the multilane bridge deck under moving loads is analyzed basing on the modal superposition principle. In the present study, the bridge deck is modeled as a multi-lane orthotropic rectangular plate. The vehicle-bridge interaction forces are assumed as groups moving loads. Dynamic behavior of the bridge deck under multiple moving vehicles is analyzed using the orthotropic plate theory and modal superposition. A method to identify groups of moving loads based on optimal theory and modal superposition principle for multi-lane bridge deck is developed in time domain. The identification errors for the eccentric groups of moving loads and multiple groups of moving loads are discussed. The selection of the measuring locations using in the identification is also

discussed. effectiveness

simulations show Computation and the validity of the proposed method.

2 IDENTIFICATION 2.1

the

THEORY

Dynamic behavior of a multi-lane under multiple vehicles

bridge

deck

+ x

J

/

Y

I 2.

Figure 1-Orthotropic plate under a group of moving loads The bridge deck is assumed as an orthotropic plate simply supported along x=0 and x=a with the other two sides free, and the moving vehicle is modelled as a group of moving loads as shown in Figure 1. The equation of motion of the orthotropic plate under the group of moving loads can be written as follow.

D$t2D$$$+D,$ x

Dr,D, direction; D “)

where

are flexural

ngrdrtres in x-direction

is the torsional

and y-

rigidity. C is the damping

of the plate. p is the mass density of plate plate. h is thickness of the the } are the { P,,,,C f )A = 1,2,..., N,,,;Z, = 1,2;.-,N,.

coefficient material.

moving loads.

N,,,

is the number of loads for the 1, th

group of moving load. N, is the number of group of moving loads on the bridge deck. NP is the total number of moving

loads.

(2!,,,( t),$,,,,(t

moving load p[,,, ( t )

)) is the position

of the

6( x ),6( y ) are Dirac functions. If

the vehicle is moving along a line parallel to the x-direction at a constant speed, ( i,,rZ ( t ), j,,,, ( t )) can be written as follow.

%,i2 ct 1= vi12+ 4,1*0 i ?I,,,f t .,= ~i,lzO 159

where v,, is the moving speed of the 1,th group of moving load. f %,r,oj %,,,, ) Is the initial position of pl,(l ( t ) By modal superposition, the displacement of orthotropic plate can be written as follows. wfx,yJ)=

the

(3)

~wi!!fx>Y)q,“,ft)

m.n

W,, ( X, y ) = Y,, ( y )sin( y)

where

The strains under the orthotropic plate at a point (x,y) and time t is

is the

mode

shapes of the orthotropic plate (Zhu and Law, 2000). q, ( t ) is the corresponding amplitude. Substituting

Eq.(3) into Eq.(l) results to

(8) where

ii,“ft)+25,“W,,9,“ft)+W:q,“ft)=

Ex(x,y,t),&,(x,y,t)

direction and y-direction,

are

the strains

along x-

respectively.

2 At measuring

point (x,, y,),

the strain can be written in

discrete forms including the MM X NN modes.

(m,n=l,2,...) where

[,,

=

c 2phw,

(4) p~f~)‘Ym,f

a,b are the dimensions

orthotropic plate in x- and y- direction respectively.

q,,w=+-

J;H,ft-~)fmnf~w

)

mm-k if,,,,(kW

+$ml

mm, is

the natural frequency of the plate. Equation (4) can be solved in the time convolution integral, and yielding

Y, bin+

of the

t..lxl.Y,,mmi=-q~~Y~~fr(y~),i,i~i) domain

by

mm-k )f,Jk W

j+Lni mn (s = 1,2;,.,N,;mm

(5)

= 1,2;.., N.)

(9)

where At is the time step: N+i is the number of sampling points: Ns is the number of measuring points, and

Mm = phaJ,byn:nf yPy/2 H,,,,,(t)=-e

H mn( k ) = --& dmnwmnld’sin( w,, ’ kAt ) w “In

f,.(t)=

f,,,(k)= NC NP4

1 -imn~mnsin(w,“‘t),t 20 Urn”r N, NPh c E: Pi,i;ft)y~fi,~,*ft))sirlf~i,r*ft)) ,,=,I*=,

,F,,E, pl,r2 (kdt )Y,,( j,,,, (kdt ))sin(F;i,z

*=%Jl-C. wmll

m = 1,2;..,MM;n

(6)

Substituting equation (5) into equation (3) the displacement of the orthotropic plate at a point (x,y) and time tcan be found, as

(kdt ))

= 1,2;..,NN.

(10)

Equation (9) can be written in matrix form (Only the formulation on strains in x-direction is presented. The formulation on strains in y-direction is similar.)

[WP)=fExI where

[E, }

(N*N,)x(N*N,) matrix. 2.2

Load identification

from strains

160

is a

(11) (N *N, matrix;

)X

1

matrix;

[B]

(P} is a (N*N,)xl

is

a

For l-beam: I=O.i 18 m4 , J=O.O4385m* ,

m,=O.175m,n,=1.14m,~=O.3,A,

=0.299

b, = 2.25m. White noise is added to the calculated simulate the polluted measurement as

displacements

(15)

k >= I ~rnk,rlnrdIf 1 + EP * No,,, ) where

{&}

are

identification;

LD&l

[%I

the

measured

strains

E, is the noise level;

used

N,,,,

to

for the

is a standard

normal distribution vector (with zero mean value and unit standard deviation). {~,,,r,,~~,~~}are the calculated strains.

... P,,.,l]

2.125m I*

B-1 =

I

0.h

07169m

4,

=

!$

zlF,

lo

M mn

,fy/2Ymnf

y,

0.175m '

0.h

jsinf?,)

+ I f

‘LI

an

e-~mnmmn 1mm-k )*l sjn(

w,,

’ ( mm

_ k )At

)

F-VT

Y,,(jll12 f kAt))sinf7 k1,2 f kAt)I

Figure 2-A simply supported beam-deck

1= ‘5’N,

+ 1, i=l (mm=1,2,3;..,N;k=0,1,2;..,N s=1,2;..,N,;1,=1,2;..,N,,,;l,

-1;

L O.l&ll

kitil

(13)

=1,2;..,N,)

The errors in the identified following equation

J

The load identification can be formulated as a nonlineaf least-squares problem.

bridge

results are calculated

by the

f P;d,,%d 1-I P,,, 111 x loo~ llf p,,,, 111

A group of moving loads (includes two moving loads at an axle spacing of 4m) is given as follow.

minJ(IP},il)=(I&~}-tBl1P}IRl(I&~}-[BlIP})) +~(lP~,IPl) (14) where l is an optimal regularization parameter or a vector determined from the method suggested by Busby and Trujillo (1997). [PI] is a weight matrix which can be determined from the measuring information.

As an example, the group of moving load is moving along y=3/8b on the left halve of the bridge deck. The moving speed is lOm/s. The first nine vibration modes are used in the identification as suggested by Zhu and Law (199). The number of the measuring points should be determined by 3 NUMERICAL SIMULATIONS the sample theory so that the vibration modes can be reconstructed from the responses of the measuring points. 3.1 Selection of Measurement Locations The number of measuring points is determined as nine. Now the key problem is to select the measuring locations. A simply supported beam-slab bridge is shown in Figure 2. Assumed the measuring locations are even distributed on There are four 2.75 metres wide lanes on the bridge deck. the five l-beams, Table 1 shows the errors in the identified The parameters of the beam-slab bridge are listed as forces from different sets of measuring locations on the Ifollow. beams. Case 1: The measuring locations are evenly distributed on the left three l-beams; Case 2: The measuring locations are evenly distributed on the middle three l-beams; Case 3: The measuring locations are evenly distributed on the right three l-beams. From the simulations, the following results can be obtained. 161

1)

If there is no noise in the responses, the identified errors from strains are very small. It shows that the method and the algorithm proposed in the paper are correct.

2) The results show that the errors will increase as the

2)

According to the sampling theory, the number of the measuring points should not be less than the number of the vibration modes used in the identification.

3)

3)

If the number of the measuring points is not less than the number of the vibration modes, there are less identified errors when the measuring locations are near to the track of the group of moving load. The errors are sensitive to the measuring noise level. Before the responses are used to identify the group of moving load, some procedures can be used to reduce the noise level, such as filtering and smoothing of the data.

4)

distance between the measuring locations and the moving track of the group of moving load becomes larger.

The method proposed in the paper is effective to identify the eccentric group of moving load, and acceptable results can be obtained from strains.

3.3 Multiple

Groups

of Moving

Loads

According to Nowak et al (1993) in his study of the dynamic response of bridge deck, three cases of multiple vehicles are considered. In-lane: two vehicles following headway distance less than 15 m.

each

other

with

Side by side in tandem: two vehicles in adjacent lanes travelling with front axles on the same line. Table 1 Error of identification from with different sets of measurement locations Noise Errors (%) Locations Level Set First Force Second Force 0% Case I

Case II

Case III

Identification

0.49

1%

16.58

18.55

5%

28.79

30.92

10%

36.53

38.73

0%

0.00

0.49

I%

16.60

18.40

Table 2 Errors in the identified forces moving at for different eccentricity Noise Errors (%) Eccentricity Level First Force Second Force 0% 0

0.49

1%

16.46

18.25

5%

28.22

30.13

5%

28.51

30.48

35.97

38.13

10%

35.55

37.73

0%

0.00

0.49

0%

0.00

0.49

1%

16.09

17.56

1%

16.60

18.48

5%

27.59

29.54

34.80

of Eccentric

118b

37.13

Group of Moving

Load

3/8b

The group of moving loads in Eq. (26) is moving at an eccentricity of 0, 118b and 318b separately. The first nine vibration modes are used in the identification. The nine measuring points are even distributed on the three middle l-beams. The identified results for different noise levels are shown in Table 2. Figure 3 shows the time histories of the identified forces from measured strains with 1% noise level. From the simulations, the following results can be obtained. 1I)

0.00

10%

10%

3.2

0.00

Side by side and behind: two vehicles in adjacent lanes but one behind the other with the distance between front axles less than 15m.

5%

28.51

30.48

10%

35.97

38.13

0%

0.00

0.49

1%

16.90

18.96

5%

29.15

31.22

10%

36.92

39.19

A group of moving loads is assumed as follows (at an axle spacing of 4m). p,(t)=75000*(1+0.1sin10m+0.05sin4O7rf)N 1 pl(r)=75000*(1-0.1sin10m+0.05sin50m)N

There are large errors in the beginning and the end of the identified result. This is because of the optimization on the total time duration. If only the beginning and the end time segments are considered in the optimization, the optimal parameter for these time segments should be less than the optimal parameter for the total time duration.

Now two groups of these moving loads are moving along y=l/Bb and 318b. Three configurations of the loads are discussed. Case 1: Two groups of loads get on the bridge at the same time with the same speed of IOm/s; Case 2: Two groups of loads get on the bridge at the same time with different moving speeds of lOm/s and 12rn/s; Case 3: Two groups of loads get on the bridge with a longitudinal spacing of lm and with different moving speeds of 1Omi.s 162

and 12mis. Note that the in-lane problem of Nowak et al (1993) is a special case of Case 3. Table 3 shows the errors in the identified forces obtained from signal with different noise level. The following observations are made.

1)

2)

3)

The errors are very sensitive to the noise level of the responses. So some procedures should be used to improve the ratio of the signal to noise when the noise level is large. In order to improve the identified accuracy, some procedures can be used, such as optimization for each load in the identification (multi-objective optimization). The total time length can be divided into several time intervals, and optimization for each time interval can be performed separately in the identification (Multi-variable optimization). The method proposed in the paper is effective to identify multiple groups of moving loads, and acceptable results can be obtained from strains when the noise level in the responses is not large.

length. New objective function in the optimization be used to reduce the errors. 4)

In order to improve the accuracy, the multi-objective optimization and the multi-variable optimization can be adopted in the identification.

5)

The method proposed in the paper is effective to identify multiple groups of moving loads from the responses of the multi-lane bridge deck, and acceptable results can be obtained from strains.

ACKNOWLEDGEMENTS The work described in this paper was supported by a grant from the Hong Kong Polytechnic University Research Funding Project No. V653.

REFERENCES

111 Bakht, B. and Jaeger, Table 3 Errors in the identified forces from multiple groups of moving loads Errors (%) Case

Noise level

0% 1

2

3

First Force Group First Second Force Force 0.995 0.991

Second Force Group First Second Force Force 0.995 0.991

1%

19.0

19.3

18.2

1a.3

5%

30.8

32.2

34.0

35.0

0%

0.995

0.991

0.597

0.001

1%

19.3

20.1

18.5

18.8

5%

30.6

33.4

33.7

34.0

0%

0.995

0.991

0.540

0.001

1%

19.4

19.9

18.6

18.8

5%

30.9

33.3

33.4

34.1

fied, The MCGRAW-HILL,

Busby, H.R. and Trujillo, D.M., Optimal regularization of an inverse dynamic problem. Computers and Structures Vol.63, No.2, pp.243-248, 1997.

[31

Chan, T.H.T. and Chan, J.H.F., The use of eccentric beam elements in the analysis of slab-on-girder bridges. Structural Engineering and Mechanics, Vol.8, No.1, pp.85-102, 1999.

t41 Chan, T. H. T., Law, S. S., Yung, T.H. and Yuan, X. R.. An interpretive method for moving force identification. Journal of Sound and Vibration Vol.219, No.3, pp.503-524, 1999. Fryba, L. Vibration of Solids and Structures Moving Loads. Groningen: Noordoff, 1972.

The number of the measuring points should not less than the number of the number of the vibration modes in the identification. It can be determined by the sampling theory.

2)

The errors in the identified forces are more sensitive to the measuring noise level as the number of the moving loads increases. Therefore some procedures can be used to reduce the noise level of the responses before they are used to identify the groups of moving load.

3)

under

El

Gupta, R. K. and Traill-Nash, R.W., Bridge dynamic loading due to road surface irregularities and braking of vehicle, Earthquake Engineering and Structural Dynamics, Vol.& No.1, pp.83-96, 1980.

j71

Humar, J. L. and Kashif, A. H., Dynamic response analysis of slab-type bridges, Journal of Structural Engineering ASCE, Vol.1 21, No.1, pp.48-62, 1995.

PI

Law, S. S., Chan, T. H. T. and Zeng, Q.H., Moving force identification: a time domain method. Journal of Sound and Vibration, Vol.201, No.1, pp.l-22, 1997.

PI

Law, S. S., Chan, T. H. T. and Zeng, 0. H., Moving force identification: A frequency and time domains analysis, Journal of Dynamic Systems, Measurement and Control ASME, Vol.12, No.3, pp.394-401, 1999.

the following

1)

L.G., Bridge Analysis Simpli1985.

PI

4 CONCLUSIONS From the above simulations and discussions, conclusions can be obtained.

can

[I 01 Law, S.S. and Fang, Y.L., Moving force identification: Optimal state estimation approach, Sound and Vibration, 2000. (In press)

There are large errors at the beginning and the end of the identified force time histories. This is due to the optimization in the solution is over the total time

Dll

163

Journal of

Mabsout, M. E., Tarhini, K.M., Frederick, G.R. and Kesserwan, A. Effect of multilanes on wheel load

distribution in steel girder bridges, Journal of Bridge Engineering ASCE, Vol.4, No.1, pp.99-106, 1999. [12] Marchesiello, S., Fasana, A., Garibaldi, L. and Piombo, B. A. D., Dynamics of multi-span continuous straight bridges subject to multi-degrees of freedom moving vehicle excitation. Journal of Sound and Vibration, Vol.224, No.3, pp.541-561, 1999. [13]

[14]

[15]

Wu, J. -S., Lee, M.-L. and Lai, T. -S., The dynamic analysis of a flat under a moving load by the finite element method. International Journal for Numerical Methods in Engineering, Vol.24, No.4, pp.743-762, 1987. [16] Zhu, X. Q. and Law, S. S. Moving forces identification on a multi-span continuous bridge. Journal of Sound and Vibration, Vol.228, No.2, pp.377-396, 1999.

Nowak, A. S., Nassif, H. and Defrain, L., Effect of truck loads on bridges. Journal of Transportation Engineering ASCE, Vol.1 19, No-l, pp.853-867, 1993.

[17]

Zhu, X. Q. and Law, S. S., Dynamic behavior of orthotropic plates under moving loads. Journal of Engineering Mechanics ASCE, 2000(Under review).

O’Connor, C. and Chan, T. H. T., Dynamic wheel loads from bridge strains. Journal of Structural Engineering ASCE, Vol.1 14, No.& pp.l703-1723. 1988.

x IO5

The first axle force

31

0' 0

0.5

X105 21

1

1.5

The second

axle force

I 2

I 2.5 I

1 .5 z

I0.5 0

0

0.5

1

1 .5

2

2.5

Figure 3-Time histories of identified axle forces moving at different eccentricity (_

True Loads; ---- e=1/8b;

164

e=3/8b.)