Embed Size (px)
Citation preview
Chapter 9
Moving Force Identification based onFinite Element Formulation
9.1 Introduction
The moving load identification techniques described in previous chapters have goodaccuracy for identification but they demand extensive computation with a vehiclecrossing a multi-span bridge structure. The finite element approach is flexible whendealing with vehicle axle-loads moving on top of a bridge structure with complexboundary conditions. The method has efficient computational performance and goodidentification accuracy, especially with the orthogonal function smoothing techniqueto obtain the velocities and accelerations from the measured strains.
This chapter introduces the finite element model approach. The InterpretiveMethod I is discussed in Section 9.2.1, in which the bridge is modeled as an assembly oflumped masses interconnected by massless elastic beam elements. The Euler-Bernoullibeam model is used in the Interpretive Method II described in Section 9.2.2. The useof structural condensation technique to reduce the DOFs of the structure to have adetermined set of identification equation is revisited in Section 9.2.3. Numerical simu-lation and experimental results in Sections 9.3 and 9.4 demonstrate the efficiency andaccuracy of the method to identify the moving loads. A comparative study with theExact Solution Technique Method is also presented in Section 9.5.
9.2 Moving Force Identification
9.2.1 Interpretive Method I
The Interpretive method I (IMI) is a moving load identification approach developed byO’Connor and Chan (1988), in which a bridge is modeled as an assembly of lumpedmasses interconnected by massless elastic beam elements. It consists of two basic com-ponents in which, the predictive analysis generates the theoretical bridge response, andthe interpretive analysis identifies the original dynamic loads.
9.2.1.1 Pred ic t i ve Ana lys i s
A simply supported bridge can be modeled as a lumped mass system as shown inFigure 9.1. The nodal responses for displacements and/or bending moments at anyinstant are given by Equations (9.1) and (9.2) respectively:
{Y } = [YA]{P} − [YI ] [�m]{Y } − [YI ] [C]{Y } (9.1)
236 Mov ing Loads – Dynamic Ana lys i s and Ident i f i ca t ion Techn iques
Lumped Masses …1 2 N�1 N
Moving Loads PM PM�1 P2 P1…
Figure 9.1 Beam-element model
{M} = [MA]{P} − [MI ] [�m]{Y } − [MI ] [C]{Y } (9.2)
where {P} is the vector of wheel loads, [�m] is the diagonal matrix containing values ofthe lumped masses, [C] is the damping matrix. {M}, {Y }, {Y }, {Y } are the nodal bendingmoment, displacement, velocity and acceleration vectors respectively. [YA], [YI ] arematrices for nodal forces to obtain nodal displacements, and [MA], [MI ] are matricesfor nodal forces to obtain nodal bending moments. Here subscripts A and I representthe Acting load and Inertial force respectively.
9.2.1.2 In terpret i ve Ana lys i s
The interpretive analysis predicts the dynamic loads from the measured response data.As stated in Section 9.2.1.1, solutions can be developed using the Y (accelerations),Y (displacements) or M (bending moments). If Y is known at all times for all interiornodes, then Y and Y can be obtained using numerical differentiation. Equation (9.1)becomes an over-determined set of linear simultaneous equations in which P can besolved. Similarly, if Y is known, it can be integrated by an integration method to giveY and Y , and subsequently to get P. However, a particular difficulty arises if measuredbending moments are used as input data. Remembering that the moving load P is notalways at the nodes, and the relation between the nodal displacements and the nodalbending moments is:
{Y } = [YB]{M} + [YC]{P} (9.3)
where [YC]{P} allows for the deflections due to additional triangularly distributedbending moments that occur within elements carrying one or more point loads P.[YC] can be calculated from the known locations of the loads. Both [YB] and {M} areknown, but {Y } can not be determined without a knowledge of {P}.
9.2.2 Interpretive Method II
Chan et al. (1999) used an Euler-Bernoulli beam to model the bridge deck in theinterpretation of dynamic loads crossing the deck. If there are Np moving loads on thesingle span beam with the mode shapes:
φi(x) = sin(
iπxL
), (i = 1, 2, . . . , n)
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
Moving Force Ident i f i ca t ion based on F in i te E lement Formula t ion 237
and Equation (2.8) in Chapter 2 can be written as:
q1
q2...
qn
+
2ξ1ω1q1
2ξ2ω2q2...
2ξnωnqn
+
ω21q1
ω22q2...
ω2nqn
= 2ρL
sinπ(ct − x1)
Lsin
π(ct − x2) · · · sinπ(ct − xNp )
L
sin2π(ct − x1)
Lsin
2π(ct − x2)L
· · · sin2π(ct − xNp )
L...
......
...
sinnπ(ct − x1)
Lsin
nπ(ct − x2)L
· · · sinnπ(ct − xNp )
L
P1
P2...
PNp
(9.4)
in which xk is the distance between the kth load and the first load, and x1 = 0. There-fore, as mentioned above, the modal displacements at x1, x2, . . . , xl can be obtained bysolving Equation (9.4). If Equation (2.7) in Chapter 2 is expressed in a matrix form as:
w ={
sinπxL
sin2πxL
· · · sinnπxL
}{q1 q2 . . . qn}T , (9.5)
the displacements at x1, x2, . . . , xl can then be calculated from Equation (9.4) asfollows.
w1
w2...
wl
=
sinπx1
Lsin
2πx1
L· · · sin
nπx1
L
sinπx2
Lsin
2πx2
L· · · sin
nπx2
L...
.... . .
...
sinπxl
Lsin
2πxl
L· · · sin
nπxl
L
q1
q2...
qn
(9.6)
Furthermore, the accelerations on the beam at x1, x2, . . . , xl can also be obtainedfrom the second derivative of the displacements as
w1
w2...
wl
=
sinπx1
Lsin
2πx1
L· · · sin
nπx1
L
sinπx2
Lsin
2πx2
L· · · sin
nπx2
L...
.... . .
...
sinπxl
Lsin
2πxl
L· · · sin
nπxl
L
q1
q2...
qn
(9.7)
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
238 Mov ing Loads – Dynamic Ana lys i s and Ident i f i ca t ion Techn iques
Similarly, the bending moments at the corresponding locations can be obtained fromthe relationship M = −EI(∂2ν/∂x2).
M1
M2...
Ml
= −EI(π
L
)2
sinπx1
L22 sin
2πx1
L· · · n2 sin
nπx1
L
sinπx2
L22 sin
2πx2
L· · · n2 sin
nπx2
L...
.... . .
...
sinπxl
L22 sin
2πxl
L· · · n2 sin
nπxl
L
q1
q2...
qn
(9.8)
If P1, P2, . . . , PNp are known constant moving loads and the effect of dampingis ignored, the closed form solution of Equation (2.10) in Chapter 2 is given as:
w(x, t) = L3
48EI
Np∑i=1
Pi
∞∑n=1
1n2(n2 − α2)
sinnπxL
(sin
nπ(ct − xi)L
− α
nsin ωn
(t − xi
c
))
(9.9)
in which α = πc/Lωn. Therefore, if the displacements of the beam at x1, x2, . . . , xlcaused by a set of constant moving loads are known, the magnitude of each movingload can be obtained by solving the following equation:
{w} = [SνP]{P} (9.10)
or
w1...
wi...
wl
=
s11 · · · s1j · · · s1Np
.... . .
...
si1 sij siNp
.... . .
...
sl1 · · · slj · · · slNp
P1...
Pj...
PNp
(9.11)
where,
sij = L3
48EI
∞∑n=1
1n2(n2 − α2)
sinnπxi
L
(sin
nπ(ct − xj)L
− α
nsin ωn
(t − xj
c
))(9.12)
If l ≥ NP, which means that the number of displacement measuring stations is largerthan or equal to the number of axle loads, {P} can be obtained using the least-squaresmethod
{P} = ([SνP]T [SνP])−1[SνP]T{w} (9.13)
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
Moving Force Ident i f i ca t ion based on F in i te E lement Formula t ion 239
L
P2(t) v
x2(t)
x1(t)
P1(t)
Figure 9.2 A continuous beam subject to moving loads
A similar equation can be obtained using bending moments instead of displacementsas the bridge deck responses by considering the closed form solution in terms of bendingmoments.
M(x, t) = L4
Np∑i=1
Pi
∞∑n=1
8n2
π2sin
nπxL
1n2(n2 − α2)
(sin
nπ(ct − xi)L
− α
nsin ωn
(t − xi
c
))
(9.14)
It is noted that if the set of moving loads are time-varying, the method can still beapplied to determine their static equivalent values as other traditional Weigh-In-Motionmethods.
9.2.3 Regularization Method
9.2.3.1 Equat ion of Mot ion
Figure 9.2 shows two loads moving at a speed v over a bridge deck modeled with finiteelements. The elemental mass and stiffness matrices are obtained using the Hermitiancubic interpolation shape functions. The supporting beam structure is discretized intom − 1 beam element where m is the number of nodal points. The shape functions ofthe jth element in its local coordinate can be obtained as follows:
Hj ={
1 − 3(x
l
)2 + 2(x
l
)3x(x
l− 1
)23(x
l
)2 − 2(x
l
)3x(x
l
)2 − x2
l
}T
(9.15)
where l is the length of the beam element. With the assumption of Rayleigh damping,the equation of motion for the bridge can be written as:
MbR + CbR + KbR = HcPint (9.16)
where Mb, Cb, Kb are the mass, damping and stiffness matrices of the bridge struc-ture respectively. R, R, R are the nodal acceleration, velocity and displacement vectors
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
240 Mov ing Loads – Dynamic Ana lys i s and Ident i f i ca t ion Techn iques
Pi(t)
xi(t) � ( j�1)l
l
ML MR
fL fR
fLML
fRMR
� HjiPi(t)
Figure 9.3 Equivalent nodal loads for a beam element loaded by the ith bridge-vehicle interactionforce
of the bridge deck respectively, and HcPint is the equivalent nodal load vector from thebridge–vehicle interaction force with:
Hc ={
0 · · · 0 · · · H1 · · · 00 · · · H2 · · · 0 · · · 0
}T
Hc is a NN × Np matrix with zero entries except at the degrees-of-freedom correspond-ing to the nodal displacements of the beam elements on which the load is acting, andNN is the number of degrees-of-freedom of the bridge after considering the boundarycondition. The components of the vector Hi (i = 1, 2) evaluated for the ith interactiveforce on the jth finite element is given in Figure 9.3, and the shape function can bewritten in the global coordinates as:
Hj =
1 − 3(
xi(t) − (j − 1)ll
)2
+ 2(
xi(t) − (j − 1)ll
)3
(xi(t) − (j − 1)l)(
xi(t) − (j − 1)ll
− 1)2
3(
xi(t) − (j − 1)ll
)2
− 2(
xi(t) − (j − 1)ll
)3
(xi(t) − ( j − 1)l
)(( xi(t) − (j − 1)ll
)2
−(
xi(t) − (j − 1)ll
))
(9.17)
with ( j − 1)l ≤ xi(t) ≤ jl. x1(t), x2(t) are the positions of the front axle and rear axlerespectively at time t. To find the time response of the beam from Equation (9.16),
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
Moving Force Ident i f i ca t ion based on F in i te E lement Formula t ion 241
a step-by-step solution can be obtained using the Newmark direct integration method.The deflection of the bridge at position x and time t can then be expressed as:
w(x, t) = H(x)R(t) (9.18)
where H(x) = {0 · · · H(x)Tj 0 · · · 0} with ( j − 1)l ≤ x(t) ≤ jl. H(x) is a 1 × NN
vector with zero entries except at the degrees-of-freedom corresponding to the nodaldisplacements of the jth beam element on which the point x is located. The componentsof the vector H(x)j are calculated similar to Equation (9.17) with x(t) replacing xi(t).
9.2.3.2 Veh ic le Ax le Load Ident i f i cat ion f rom Stra in Measurements
According to Equation (9.18), the strain at a point x and at time t can be written asfollow:
ε(x, t) = −z∂2w(x, t)
∂x2= −z
∂2H(x) R(t)∂x2
(9.19)
where z represents the distance from the neutral axis of the beam to the location ofstrain measurements at the bottom surface. Rewriting Equation (9.19), we have:
ε(x, t) = gR (9.20)
where
g = −z{g1(x), g2(x), . . . , gNN(x)}and gi(x) is the second derivative of Hi.
The strain can be approximated by a generalized orthogonal function T(t) as:
ε(x, t) =Nf∑i=1
Ti(t)Ci(x) (9.21)
where {Ti(t), i = 1, 2, . . . , Nf } is the generalized orthogonal function (Zhu andLaw, 2001a), Nf is the number of terms in the generalized orthogonal function;{Ci(x), i = 1, 2, . . . , Nf } is the vector of coefficients in the expansion expression.The strains at the Ns measuring points can be expressed as:
ε = CT (9.22)
where
T = {T0(t), T1(t), . . . , TNf (t)}T ;
ε = {ε(x1, t), ε(x2, t), . . . , ε(xNs , t)}T ;
C =
C10(x1) C11(x1) · · · C1Nf (x1)C20(x2) C21(x2) · · · C2Nf (x2)
......
......
CNs0(xNs ) CNs1(xNs ) · · · CNsNf (xNs )
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
242 Mov ing Loads – Dynamic Ana lys i s and Ident i f i ca t ion Techn iques
and {x1, x2, . . . , xNs} is the vector of location of the strain measurements. By the least-squares method, the coefficient matrix can be obtained as:
C = ε TT (T TT )−1 (9.23)
Putting Equation (9.20) into Equation (9.22) and using the least-squares methodagain, we have:
R = (GT G)−1 GT C T (9.24)
where
G = −z
g1(x1) g2(x1) · · · gNN(x1)g1(x2) g2(x2) · · · gNN(x2)
......
......
g1(xNs ) g2(xNs ) · · · gNN(xNs )
and matrices R and R can be obtained by directly differentiating Equation (9.24) tohave:
R = (GT G)−1 GT C T,
R = (GT G)−1 GT C T.
Substituting the nodal responses R, R and R into Equation (9.16), we have:
U = HcPint (9.25)
where U = MbR + CbR + KbR.
The moving loads can then be identified from Equation (9.25) by least-squaresmethod. However, it is impractical to have measurements from all the degrees-of-freedom of the structure and Equation (9.25) would be under-determined. A structuralcondensation technique is required to reduce the unmeasured degrees-of-freedom tothe measured degrees-of-freedom of the bridge deck such that Equation (9.25) isdeterminate.
The Improved Reduced System reduction scheme O’Callahan (1989) may beadopted to condense the unmeasured degrees-of-freedom (DOFs) to the measureddegrees-of-freedom of the bridge deck. All the measured DOFs are designated asthe master DOFs and denoted by Rm(t). The remaining structural DOFs are calledthe slave DOFs, and are denoted by Rs(t). The response vector of the bridge is thenpartitioned as:
R(t) ={
Rm(t)
Rs(t)
}(9.26)
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
Moving Force Ident i f i ca t ion based on F in i te E lement Formula t ion 243
Accordingly, the bridge mass, damping, stiffness and shape function matrices arealso partitioned as:
Mb =[Mmm Mms
Msm Mss
], Cb =
[Cmm Cms
Csm Css
], Kb =
[Kmm Kms
Ksm Kss
],
H =[Hmm
Hss
]T
, Hc =[Hcmm
Hcss
] (9.27)
The total response matrix of the system R(t) can then be represented by thepartitioned master set of DOFs Rm(t) multiplied by the transformation matrix as:
R(t) = W Rm(t), (9.28)
where
W = Ws + Wi, Ws =[
I−K−1
ss Ksm
]
and
Wi =[
IK−1
ss (Msm − MssK−1ss Ksm)(WT
s MbWs)−1(WTs KbWs)
]
where I is the identity matrix. Substituting Equations (9.27) and (9.28) into Equation(9.16) and pre-multiplying WT to both sides yielding:
U = Hcr Pint (9.29)
and
U = MrRm + CrRm + KrRm
where Mr = WT ∗ Mb∗ W ,Cr = WT ∗ Cb
∗ W , Kr = WT ∗ Kb∗ W , and Hcr = WT ∗ Hc.
Substituting Equation (9.28) and H =[Hmm
Hss
]T
into Equation (9.19), matrices
Rm, Rm and Rm can be obtained by the generalized orthogonal function method.The identification with selected measuring points can then be performed using Equa-tion (9.29). Since the proposed method is based on finite element method, it could beapplied to complex structure with different boundary conditions, varying geometryand mass distribution.
9.2.3.3 Regu lar izat ion A lgor i thm
The moving forces obtained from Equation (9.25) or (9.29) using a straight forwardleast-squares solution would be unbound. A regularization technique (Law et al., 2001)is used to solve the ill-posed problem in the form of minimizing the function.
J(P, λ) = ‖B Pint − U‖2 + λ‖Pint‖2 (9.30)
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
244 Mov ing Loads – Dynamic Ana lys i s and Ident i f i ca t ion Techn iques
where B is Hc or Hcr; λ is the non-negative regularization parameter correspondingto the smallest relative percentage error calculated from Equation (9.32) or (9.33).The Generalized Cross Validation (Gorman and Heath, 1979) and L-curve method(Hansen, 1992) can be employed to determine the optimal regularization parameter.The solution of Equation (9.25) or (9.29) can be obtained by the damped least-squaresmethod as:
Pint = (HTc
∗ Hc + λI)−1HTc
∗ U (9.31)
where I is the identity matrix, and singular-value decomposition is used in the pseudo-inverse calculation.
9.3 Numerical Examples
The effects of discretization on the structure, number of measuring points, samplingfrequency, velocity of vehicle and noise level on the accuracy of the identified resultsare investigated in the following sections. The number of master DOFs is taken equalto the number of measuring points in all of the following studies.
The calculated responses are polluted with white noise to simulate the pollutedmeasurement as:
ε = εcalculated(1 + Ep∗ Noise)
where εj and εj calculated are vectors of measured and calculated responses at the jthmeasuring point; Ep is the noise level; Noise is a standard normal distribution vectorwith zero mean and unit standard deviation. The relative percentage error (RPE) in theidentified results is calculated from Equation (9.32), where ‖•‖ is the norm of matrix,Pidentified and Ptrue are the identified and the true force time histories respectively.
RPE = ‖Pidentified − Ptrue‖‖Ptrue‖ × 100% (9.32)
The bridge deck is modeled as a 30 m long simply supported beam with the physicaland material parameters same as those described in Section 8.4.1.1 with ξ = 0.02 forall modes. The two moving loads are at 4.26m spacing moving at a constant speed.They are:
P1(t) = 6268(1.0 + 0.1 sin(10πt) + 0.05 sin(40πt)) kg;
P2(t) = 12332(1.0 − 0.1 sin(10πt) + 0.05 sin(50πt)) kg.
The measuring points are evenly distributed at the bottom of the beam and theirlocations are shown in Table 9.1 for different arrangements of sensors.
9.3.1 Effect of Discretization of the Structure and Sampling Rate
The loads move on top of the beam at a constant velocity of 15 m/s. Strain at the bottomof the beam is measured with three sensors, and the sampling frequency is taken to be100, 200, 300, 400 and 500 Hz separately for the study. The simply supported beamis discretized into four, eight, 12 and 16 beam elements separately. No noise effect
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
Moving Force Ident i f i ca t ion based on F in i te E lement Formula t ion 245
Table 9.1 Sensor arrangements
Number of sensors Location
3 1/4L, 1/2L, 3/4L4 1/8L, 1/4L, 1/2L, 3/4L5 1/8L, 1/4L, 1/2L, 3/4L, 7/8L6 1/8L, 1/4L, 1/2L, 5/8L, 3/4L, 7/8L7 1/8L, 1/4L, 3/8L, 1/2L, 5/8L, 3/4L, 7/8L
Table 9.2 The percentage error of the identified forces for different discretization scheme and samplingrate
Sampling Number of elementsfrequency (Hz)
4 8 12 16
Axle-1 Axle-2 Axle-1 Axle-2 Axle-1 Axle-2 Axle-1 Axle-2
100 20.93 18.36 21.11 17.49 20.41 17.64 21.46 17.66200 15.02 13.12 11.42 9.19 11.50 9.20 11.50 9.12300 14.31 12.57 11.09 8.99 10.99 8.92 10.98 8.90400 14.64 12.92 10.92 9.03 10.94 9.04 10.95 8.94500 14.68 12.92 10.95 9.03 10.97 9.04 10.97 8.94
is included in this study. The relative percentage errors of the identified forces fordifferent number of finite elements and sampling frequency are tabulated in Table 9.2.Figure 9.4 gives the identified results from the cases of four and eight elements.
The identified force time histories from both four and eight finite elements match thetrue forces very well in the middle half of the duration. The forces from four elementshave large fluctuations after the entry of the second load and before the exit of thefirst load, while those from eight elements have slight fluctuations at these moments.Table 9.2 also shows that the relative percentage errors of the identified forces fromfour elements are much larger than those from the other discretization cases. Theseindicate discretizing the beam into eight elements would be sufficient for an accurateidentification.
The sampling frequency is shown not to have any significant effect when it is largeror equal to 200 Hz. It is noted that the first five modes are included in the measuredresponses with this sampling frequency indicating that the higher modes do not havesignificant contribution to the identification accuracy.
9.3.2 Effect of Number of Sensors and Noise Level
The loads move on top of the beam at a constant speed of 15 m/s. The bridge isdiscretized into eight finite elements, and the force identification is studied with three,five and seven sensors separately and the sampling rate is 200 Hz. The measured dataare polluted with 5 percent and 10 percent noise. The relative percentage errors of theidentified results are listed in Table 9.3. Figure 9.5 shows the identified results fromthree and five sensors with 5 percent noise level.
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
246 Mov ing Loads – Dynamic Ana lys i s and Ident i f i ca t ion Techn iques
15� 104
� 105
The first axle force
The second axle force
Time (s)
10
2
1.5
0.5
0
1
5For
ce (
N)
For
ce (
N)
00 0.5 0.5 21
0 0.5 0.5 21
Figure 9.4 Identified results from three measured points without noise (— true force, - - - 8elements, · · · · 4 elements)
Table 9.3 The percentage error of the identified forces for different sensor arrangements and noiselevel
Noise level (%) Number of measuring points
3 5 7
Axle-1 Axle-2 Axle-1 Axle-2 Axle-1 Axle-2
0 11.09 9.99 6.33 3.28 4.22 2.645 13.27 11.28 8.12 5.81 6.06 5.1110 16.59 14.81 11.27 9.85 10.34 9.97
The identified force time histories from five sensors agree with the true forces verywell with only small differences at the start and end of the time histories. Identifiedresults from three sensors are also acceptable with slight fluctuations around the trueforces in the middle half of the duration. The relative percentage errors from threesensors are significantly larger than those from five and seven sensors only becauseof the large fluctuations in the time histories at the moments close to the entry of thesecond load and exit of the first load. The relative percentage error increases with thenoise level, but it is not sensitive to the measurement noise.
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
Moving Force Ident i f i ca t ion based on F in i te E lement Formula t ion 247
15� 104
� 105
The first axle force
The second axle force
Time (s)
10
2
1.5
0.5
0
1
5For
ce (
N)
For
ce (
N)
00 0.5 0.5 21
0 0.5 0.5 21
Figure 9.5 Identified results with 5 percent noise and eight elements (— true force, - - - 5 measuredpoints, · · · · 3 measured points)
9.4 Laboratory Verification
9.4.1 Experimental Set-up
The experimental set-up of a model vehicle moving on top of a steel beam in thelaboratory described in Section 6.3.4.1 is adopted for this study.
The model vehicle has two axles at a spacing 0.557 m and it runs on four steel wheelswrapped with rubber band on the outside. The mass of the model vehicle is 12.55 kgwith the front axle load and rear axle load weigh 8.725 and 3.825 kg respectively. Thefirst three measured natural frequencies of the beam are: 3.67, 16.83 and 37.83 Hz.
9.4.2 Identif ication from Measured Strains
The average velocity of the model vehicle at 0.787 m/s is used for the identification.The beam is discretized into eight Euler-Bernoulli beam elements. The measured strainsare re-sampled at 100 Hz. The sensor arrangements are shown in Table 9.1.
Since the true force Ptrue is not known, both the norm of residuals E of the forcesand the semi-norm E1 of the estimated forces are calculated as:
E = ‖B Pidentify − U‖
E1 = ‖Pidentifyj+1 − Pidentify
j ‖ (9.33)
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
248 Mov ing Loads – Dynamic Ana lys i s and Ident i f i ca t ion Techn iques
Table 9.4 Correlation coefficients of measured and reconstructed strains at 5/8L from different numberof measured points
Number of measured points Number of master degree-of-freedom
3 4 5 6 7
3 0.935 0.142 0.105 0.101 0.0844 0.936 0.943 0.107 0.234 0.0925 0.939 0.946 0.947 0.329 0.1156 0.940 0.946 0.954 0.960 0.1347 0.940 0.947 0.956 0.962 0.963
150
100
50
80
200
150
100
50
0
60
40
20
0
00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
The first axle force
NN
N
The second axle force
The resultant force
Time (s)
Figure 9.6 Experimental identified axle loads (— static loads; - - - from 3 measured strains;· · · · from 7 measured strains)
where Pidentifyj , Pidentify
j+1 are the identified forces with λj and λj + �λ, and L-curve method(Hansen, 1992) is used to find the optimal regularization parameter λopt.
Since it is practically not possible to measure large number of responses, the numberof master DOFs is varied to study its effect on the experimental identification. Thedynamic strain at 5/8L is reconstructed by inputting the identified forces into thestructure. The correlation coefficients of the measured strain and the reconstructedstrain at 5L/8 for different number of master DOFs and measured points are tabulatedin Table 9.4. Figure 9.6 shows the identified axle loads and the resultant loads from
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
Moving Force Ident i f i ca t ion based on F in i te E lement Formula t ion 249
using three, five and seven master DOFs. The number of sensors is taken equal to thenumber of master DOFs in this study.
Table 9.4 shows that when the number of sensors is larger than or equal to thenumber of master DOFs, the correlation coefficients between the measured andreconstructed strains at 5L/8 are all larger than 0.935, even with as few as threemaster DOFs. The total load identified from three, five and seven master DOFs is rel-atively stable in the middle part of the time history, but those from three master DOFsonly have much smaller value than those from five and seven master DOFs. This con-firms the observations from the simulations that accepted results can be obtained fromonly a few measuring points by this method. The accuracy increases with the num-ber of master DOFs particularly in the region towards the start and end of the timehistories.
9.5 Comparative Studies
Methods have been developed in Chapters 5 to 8 to identify moving loads on top ofa continuous bridge using measured vibration responses. A comparison is presentedas follows to illustrate the robustness and accuracy of two time domain method, i.e.the Exact Solution Technique (EST) Method in Section 6.3 and the Finite ElementMethod based on regularization in Chapter 9. Numerical studies with a single-spanbridge deck are presented in this section. Parameters that may influence the accuracyof moving load identification, such as sampling frequency, number of vibration modesand measuring points in the identification are discussed.
A single span simply supported beam with two forces p1(t) and p2(t) moving on topis studied.
{p1(t) = 20000[1 + 0.1 sin(10πt) + 0.05 sin(40πt)] Np2(t) = 20000[1 − 0.01 sin(10πt) + 0.05 sin(50πt)] N
(9.34)
The parameters of the beam are: EI = 2.5 × 1010 Nm2, ρA = 5000 kg/m, L = 30 m.The distance between the two moving forces is 4.27 m. The first six natural frequenciesof the beam are 3.90 Hz, 15.61 Hz, 35.13 Hz, 62.45 Hz, 97.58 Hz and 140.51 Hz.White noise is added to the calculated responses of the beam to simulate the pollutedmeasurements.
9.5.1 Effect of Noise Level
The first six modes are used in the simulation. The time interval between adjacentdata points is 0.002 s. Six measuring points are evenly distributed on the beam at1/7L spacing. The moving velocity of forces is 30 m/s, and 20 terms are used in theorthogonal function to approximate the measured responses. Monte Carlo methodis used to simulate the noise in 20 sets of responses, and the noise level varies from1 percent to 10 percent. Figure 9.7 shows the mean and standard deviation of the errorsin the identified moving loads using the method based on the EST in Section 6.3, andFigure 9.8 shows those from using the method based on finite element formulation inSection 9.2.3.
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
02468
10121416
1 2 3 4 5 6 7 8 9 10Noise level (%)
1 2 3 4 5 6 7 8 9 10Noise level (%)
Err
or (
%)
(a) Errors in the identified first moving load
0
2
4
6
8
10
12
14
16
18
Err
or (
%)
(b) Errors in the identified second moving load
Figure 9.7 The mean (circles) and standard deviation (error bars) of the errors in the identifiedmoving loads using EST Method
0
1
2
3
4
5
6
1 2 3 4 5 6 7 8 9 10Noise level (%)
1 2 3 4 5 6 7 8 9 10Noise level (%)
Err
or (
%)
Err
or (
%)
(a) Errors in the identified first moving load
0
1
2
3
4
5
6
(b) Errors in the identified second moving load
Figure 9.8 The mean (circles) and standard deviation (error bars) of the errors in the identifiedmoving loads using FEM-based Method
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
Moving Force Ident i f i ca t ion based on F in i te E lement Formula t ion 251
0
5
10
15
20
25
30
35
0 1 2 3 4 5 6 7 8 9 10Noise Level (%)
0 1 2 3 4 5 6 7 8 9 10Noise Level (%)
Err
or (
%)
Err
or (
%)
6 modes
5 modes
4 modes
3 modes
2 modes
(a) Error in the identified first moving load
0
5
10
15
20
25
30
35
6 modes
5 modes
4 modes
3 modes
2 modes
(b) Errors in the identified second moving load
Figure 9.9 Errors in the identified moving loads using EST Method
The errors from using EST method vary approximately linearly with the noise levelsin the responses. The standard deviation in the errors is largest with 6 percent noisein the responses. The errors from using FEM-based Method exhibit little change withthe noise level in the responses. This is because the orthogonal function approachin the identification reduces the effect of noise by its own filtering effect. When thenoise level in the responses increases the standard deviation in the errors also increases.This indicates the EST method could give very accurate results at low noise level, but itmay be badly influenced by the noise effect. On the other hand, the orthogonal functionapproximation in the FEM-based Method reduces consistently the noise effect to giveaccuracy results in all cases studied.
9.5.2 Effect of Modal Truncation
The first six modes are included in the responses, and six measuring points are evenlydistributed on the beam at 1/7L spacing. The first two, three, four, five and six vibra-tion modes are used in the identification in turn. Other parameters are the same asdescribed in last section. Figures 9.9 and 9.10 show the errors in the identified results
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
252 Mov ing Loads – Dynamic Ana lys i s and Ident i f i ca t ion Techn iques
02468
101214161820
0 1 2 3 4 5 6 7 8 9 10Noise Level (%)
0 1 2 3 4 5 6 7 8 9 10Noise Level (%)
Err
or (
%)
Err
or (
%)
6 modes
5 modes
4 modes
3 modes
2 modes
6 modes
5 modes
4 modes
3 modes
2 modes
(a) Error in the identified first moving load
02468
101214161820
(b) Error in the identified second moving load
Figure 9.10 Errors in the identified moving loads using FEM-based Method
with different number of modes using EST Method and FEM formulation, respectively.Figure 9.11 shows the errors in the identified results with different number of termsin the orthogonal function in FEM-based Method when six modes are included in theresponses.
The errors derived from EST Method increase roughly proportional to the noiselevel in the responses and with similar rate of change for different number of modes.The errors from using FEM-based Method exhibit little change with noise. This showsthat the errors in the identified results using FEM-based Method are mainly governedby the efficiency of the filtering effect in the orthogonal function approach. FEM-basedMethod is, in general, much better than the EST Method in the identification.
The error shown in Figures 9.9 and 9.10 increases by a large extent when the numberof the modes in the identification is less than three. This is because the first three naturalfrequencies of the beam cover the frequency range of the moving loads, and there isa loss of measured information in the identification when only two vibration modesare used. The errors in the identified forces in Figure 9.11 remain relatively constantfor different noise levels when the number of terms in the orthogonal function in theFEM-based Method is less than 20. And the noise level would have a negative effect onthe errors when there are more terms in the orthogonal function. This is because thefrequency range in the orthogonal function increases with increasing number of terms,
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
Moving Force Ident i f i ca t ion based on F in i te E lement Formula t ion 253
0
1
2
3
4
5
6
7
5 10 15 20 25 30 35 40 45 50Number of terms
5 10 15 20 25 30 35 40 45 50Number of terms
Err
or (
%)
Err
or (
%)
1% noise
5% noise10% noise
1% noise
5% noise10% noise
(a) Error in the identified first moving load
0
1
2
3
4
5
6
7
(b) Error in the identified second moving load
Figure 9.11 Error in the identified moving loads from FEM-based formulation with differentnumber of terms in the orthogonal function
and the high frequency components in the noise would be retained in the calculationand thus affecting the final results.
9.5.3 Effect of Number of Measuring Points
Again the first six modes are used in the simulation. The number of measuring pointsis selected as six, seven, eight, nine, ten in turn. The measuring points are evenly dis-tributed on the beam. Other parameters are the same as in the last section. Figures 9.12and 9.13 show the errors in the identified results with different number of measuringpoints as the noise level in the responses is increased. The number of measuring pointsis shown insignificant to the identified results. It should be noted that the number ofmeasuring points used are all larger than the number of the modes in the identification.
9.5.4 Effect of Sampling Frequency
The first six modes are used in the simulation. The responses are calculated with0.001 s time interval between data points, and they are re-sampled with a time interval
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
0
2
4
6
8
10
12
14
16
1 2 3 4 5 6 7 8 9 10Noise level (%)
Err
or (
%)
Err
or (
%)
6 sensors
7 sensors
8 sensors
9 sensors
10 sensors
6 sensors
7 sensors
8 sensors
9 sensors
10 sensors
(a) Error in the identified first moving load
1 2 3 4 5 6 7 8 9 10Noise level (%)
(b) Error in the identified second moving load
02468
1012141618
Figure 9.12 Errors in the identified moving loads using EST Method with different number ofmeasuring points
3.83.823.843.863.883.9
3.923.943.963.98
4
1 2 3 4 5 6 7 8 9 10Noise level (%)
Err
or (
%)
Err
or (
%)
(a) Error in the identified moving load
1 2 3 4 5 6 7 8 9 10Noise level (%)
(b) Error in the identified second moving load
3
3.05
3.1
3.15
3.2
3.25
3.3
3.35
3.4
6 sensors7 sensors8 sensors9 sensors10 sensors
6 sensors7 sensors8 sensors9 sensors10 sensors
Figure 9.13 Errors in the identified moving loads using FEM-based formulation with differentnumber of measuring points
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
Moving Force Ident i f i ca t ion based on F in i te E lement Formula t ion 255
0
2
4
6
8
10
12
14
16
18
1 2 3 4 5 6 7 8 9 10Noise level (%)
Err
or (
%)
Err
or (
%)
0.001s
0.002s
0.003s
0.001s
0.002s
0.003s
(a) Errors in the identified first moving load
1 2 3 4 5 6 7 8 9 10Noise level (%)
(b) Errors in the identified second moving load
0
2
4
6
8
10
12
14
16
18
20
Figure 9.14 Errors in the identified moving loads with different sampling frequencies using ESTMethod
of 0.002 s and 0.003 s in turn. The moving velocity of vehicle is 30 m/s. Figures 9.14and 9.15 show the errors in the identified results with different sampling frequenciesand noise levels using these two methods.
The errors in the identified results from EST Method are largest when the noise levelis above 2 percent and the sampling time interval is 0.001 s. This is again due to theinclusion of the high frequency components of noise in the calculation with a highersampling frequency. The errors from using FEM-based Method are smaller than thosefrom using EST Method and with smaller variations.
9.6 Summary
A general moving load identification method basing on finite element formulationand the Improved Reduced System condensation technique has been presented in this
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
256 Mov ing Loads – Dynamic Ana lys i s and Ident i f i ca t ion Techn iques
3.4
3.6
3.8
4
4.2
4.4
4.6
1 2 3 4 5 6 7 8 9 10Noise level (%)
1 2 3 4 5 6 7 8 9 10Noise level (%)
Err
or (
%)
Err
or (
%)
0.001s
0.002s
0.003s
0.001s
0.002s
0.003s
(a) Errors in the first identified moving load
2.5
2.7
2.9
3.1
3.3
3.5
3.7
3.9
(b) Errors in the second identified moving load
Figure 9.15 Errors in the identified results with different sampling frequencies using FEM-basedMethod
chapter. The approximated finite element formulations, developed earlier, are alsodescribed. The measured displacement is expressed as the shape functions of the finiteelements of the structure without the modal coordinate transformation as required inmethods described in earlier chapters. Numerical simulation and experimental resultsdemonstrate the efficiency and accuracy of the method to identify the moving loads orinteraction forces between the vehicle and the bridge deck.
The number of master degrees-of-freedom of the system should be selected smallerthan or equal to the number of measured points, and the identified results are relativelynot sensitive to the sampling frequency, velocity of vehicle, measurement noise level androad surface roughness when a minimum of eight finite elements are used to model thesimply supported bridge deck with measured information from at least three measuringpoints.
Dow
nloa
ded
by [
IST
AN
BU
L T
EK
NIK
UN
IVE
RSI
TE
SI]
at 0
6:52
12
Janu
ary
2014
