This study carries out numerical simulations to identify the magnitude of prestress force in a highway bridge by making use of the dynamic responses from moving vehicular loads. The prestressed bridges are modeled using four-node isoparametric flat shell element taking into account the transverse shearing deformation in the finite element model. The vehicle is modeled as a multiple degrees-of-freedom system. An approach based on dynamic response sensitivity-based finite element model updating is proposed to identify the elemental prestress force. The identified results are obtained iteratively with the penalty function method with regularization from the measured structural dynamic responses. A single-span prestressed Tee beam and two-span prestressed box-girder bridge are studied as two numerical examples. The effects of road surface roughness, measurement noise, and speed of moving vehicle on the identification results are investigated. Studies indicate that the proposed method is efficient and robust for prestress force identification. Good identified results can be obtained from several measured acceleration responses.
1. Introduction
There are many researches in the literature which deal with the bridge vibration caused by the passing of vehicles or trains [1–11]. Akin and Mofid [1] proposed a combined analytical-numerical method to determine the dynamic behavior of beams with a moving mass. Hwang and Nowak [2] developed the models for trucks, road surface, and the bridge, and the calculated static and dynamic deflections are obtained as a function of gross vehicle weight, span, and axle distance. Yang and Lin [3] and Yang and Yau [4] studied dynamic responses of vehicle-bridge systems with the finite element method. Law and Zhu [5] investigated the bridge dynamic responses due to road surface roughness and braking of vehicle. Yau [7] investigated the interaction response of a train running over a suspension bridge. Wang et al. [9] studied the nonlinear dynamic response of a long-span suspension bridge under running train and turbulent wind. These researches can be classified into the following categories: models of the vehicle, road surface roughness, bridge-vehicle interaction, the effect of vehicle braking, and so forth. In these studies, the parameters of the bridge and the vehicles are known and the dynamic responses of the bridge and/or the vehicle-bridge system can be obtained from forward analysis.
In this study, we try to identify the prestress force from the dynamic responses induced by the passing vehicles on top of the bridges. It is known that the prestress force is one of the most important parameters in prestressed concrete bridges to control the cracks in concrete, to reduce deflection of the structure, and to add strength to the prestressed members. Therefore, a substantial difference between the desired and the in-service prestress forces can result in severe serviceability and safety problems [12, 13]. It is known that the loss of prestress force occurs due to the friction between the prestressing tendon and the surrounding concrete, creep, and shrinkage of the concrete, steel relaxation or damage of the prestress strands. Therefore, it is very important to estimate the real prestress level when the prestressed structures are in service.
Abraham et al. [13] tried to predict the loss of prestress force based on a damage index derived from the derivatives of mode shapes without success. Miyamoto et al. [14] studied the behavior of a beam with unbonded tendons, and a formula was proposed for the prediction of the modal frequency for a given prestress force with laboratory and field test verifications. Saiidi et al. [15] also reported a study on modal frequencies due to the prestress force with laboratory test results. Recently, Kim et al. [16] proposed an approach to identify the prestress loss in prestressed concrete beams using the modal information. Lu and Law [17] presented a method to identify the elemental prestress force using the structural dynamic responses.
In this paper, the dynamic behavior of the Tee beam bridge and the box-girder bridge is investigated using four-node isoparametric flat shell element; the transverse shearing deformation is taken into account in the finite element model. Then the coupling equation of motion of the bridge-vehicle system is established, and the dynamic responses of the system are obtained from numerical integration method. In the inverse analysis, we attempt to identify the elemental prestress force level making use of the dynamic responses induced by the passing vehicles. An approach based on dynamic response sensitivity-based finite element model updating is proposed to identify the prestress force. Two numerical examples are studied to illustrate the correctness of the proposed method. Some aspects such as the effects of road surface roughness, measurement noise, and speed of the moving vehicle on the identification results are investigated. Simulation studies indicate that the proposed method is efficient and robust, and good identified results can be obtained.
In this paper, the plat shell element is used to establish the finite element model of the bridge. This kind of element has five degrees-of-freedom for each node and can be obtained from superimposing the planar stress element on the plate bending element [18].
The strains of the flat shell include the inplane membrane strains εp, the curvature owing to the bending εb, and the transverse shearing strains εs.
For the inplane membrane strains
(1)εp={εxεyγxy}={∂u∂x∂v∂y∂v∂x+∂u∂y}.
For bending strains
(2)εb=-z{κxκyκxy}=-z{∂θx∂x∂θy∂y∂θx∂y+∂θy∂x}.
For transverse shearing strains
(3)εs={γxzγyz}={∂w∂x-θx∂w∂y-θy}.
The expressions for the strains-displacement matrices are deduced from (1)–(3) as
(4)εp=∑i=1N[Ni,x(m)Ni,y(m)Ni,y(m)Ni,x(m)]{uivi}=∑i=1NBimai(m),εs=∑i=1N[Ni,xw-Niθ0Ni,yw0-Niθ]{wi(θx)i(θy)i}=∑i=1NBisai(s),εb=∑i=1N[0Ni,xθ000Ni,yθ0Ni,yθNi,xθ]{wi(θx)i(θy)i}=∑i=1NBibai(b),
where N is the number of nodes of the element, Ni(m), Ni(s), and Ni(b) are the isoparametric shape functions of the plane stress, shear stress, and bending stress, respectively.
2.2. The Elemental Mass and Stiffness Matrices
The consistent elemental mass matrix can be expressed as
(5)me=∫AΘTρmΘdA,
where
(6)ρm=[ρt000000ρt000000ρt000000ρt312000000ρt3120000000],Θ=[Θ1,Θ2,…,ΘN],
in which
(7)Θi=[Nui(m)Nvi(m)Ni(w)Ni(θx)Ni(θy)].
It is known that the global stiffness is singular or ill-conditioned because of the null diagonal terms due to the drilling degrees of freedom in the transformed elemental stiffness matrix. To solve this problem, in this paper, we artificially insert a rotational stiffness coefficient as it was done by Lee and Yhim [19]; the artificial stiffness for the drilling DOF is taken as γk=107 in this study.
Elemental stiffness matrix of the concrete bridge is expressed as
(8)Ke=∫A(Bip)TDpBjpdA+∫A(Bis)TDsBjsdA+∫A(Bib)TDbBjbdA+∫AGiT[Tx0p00Ty0p]GjdA,
where Dp, Ds, and Db are the material property matrices for plane stress, shear stress, and bending stress:
(9)Gi=[Ni,xuNi,xvNi,xw000Ni,yuNi,yvNi,yw000]Tx0p and Ty0p are the two components of the prestress force in x- and y-axis, respectively.
After assembling the elemental stiffness and mass matrices into system matrices, the equation of motion of the bridge structure can be written as
(10)Mbd¨+Cbd˙+Kbd=HcPint,
where Mb and Kb are the system mass and stiffness matrices, respectively, Cb is the damping matrix; in this study, Rayleigh damping model is used; that is, C=α1Mb+α2Kb, where α1 and α2 are two constants. d¨, d˙ and d are the acceleration velocity and displacement responses of the structure, respectively. Pint is the interaction force between the bridge and the vehicle. Hc is a matrix with zero entries except at the DOFs corresponding to the nodal displacements of the shell elements on which the load is acting.
2.3. Equation of Motion for the Vehicle-5-Parameter Vehicle Model
The five-parameter vehicle model of the two degree-of-freedom system shown in Figure 1 comprises five components: an upper mass mv1 of the suspension, a lower mass mv2 of the bogie and axle connected to the suspension damper cv, and a suspension spring kv1, together with another spring kv2, which is used to represent the stiffness of the tyre. The equations of motion of the masses mv1 and mv2 are
(11)mv1z¨1+cv(z˙1-z˙2)+kv1(z1-z2)=0,(12)mv2z¨2+cv(z˙2-z˙1)+kv1(z2-z1)+kv2(z2-(w(x,y,t)+r(x)))=0,
where z¨1, z˙1, and z1 are the vertical acceleration, velocity, and displacement responses of the suspension mass of vehicle, respectively, and z¨2, z˙2, and z2 are the vertical acceleration, velocity, and displacement responses of the bogie and axle. r(x) is the road surface roughness at the location of the tyre [20], which will be given in the next subsection. It should be pointed out that, in (12), the additional velocity and acceleration due to road surface roughness may be taken into account to determine the movement of the vehicle wheel as discussed by Chang et al. [21]. The interaction force Pint between the bridge and the vehicle can be expressed as
(13)Pint=(mv1+mv2)g+kv2(z2-(w(x,y,t)+r(x)))=(mv1+mv2)g-mv1z¨1-mv2z¨2.
(a) Vehicle models and the geometry of the simply supported Tee beam (dimension in meter). (b) Element number for the flange of the Tee beam. (c) Element number for the web of the Tee beam.
Equations (10), (11), and (12) can be combined in a compact form and the coupled vehicle-bridge equations of motion can be written as
(14)[MbHcmv1Hcmv20mv1000mv2]{d¨z¨1z¨2}+[Cb000cv000cv]{d˙z˙1z˙2}+[Kb000kv1-kv1-HcTkv2-kv1kv1+kv2]{dz1z2}={Hc(mv1+mv2)g0kv2r(x)},
Let
(15)M-s=[MbHcmv1Hcmv20mv1000mv2],C-s=[Cb000cv000cv],K-s=[Kb000kv1-kv1-HcTkv2-kv1kv1+kv2],Rs={dz1z2},P-(t)={Hc(mv1+mv2)g0kv2r(x)}.
Equation (14) can be rewritten as
(16)M-sR¨s+C-sR˙s+K-sRs=P-(t).
The dynamic responses of the bridge and vehicle can be obtained from a step-by-step solution using the state space method [22].
2.4. Road Surface Roughness
The randomness of the road surface roughness of the bridge can be represented with a periodic modulated random process. It is specified by its power spectral density function (PSD) as [23]
(17)Sr(ωs)=Ar(ωsωs0)-2,
where Ar is the roughness coefficient in m^{2}/cycle/m, ωs is the spatial frequency in cycle/m, ωs is the discontinuity frequency equal to 1/2π (cycle/m).
The road surface roughness function r(x) can be generated from (17) using the FFT algorithm [23], which is given by
(18)r(x)=∑i=1N(4Ar(2πiLcωs0)-22πLc)1/2cos(ωsix-θi),
where ωsi=iΔωs, with Δωs=2π/Lc, in which Lc is twice the length of the bridge, θi is a random number distributed uniformly between 0 and 2π.
2.5. Dynamic Response Sensitivity with Respect to Prestress Force
Differentiating both sides of (16) with respect to the prestress force of the ith element, we have
(19)M-s∂R¨s∂Ti+C-s∂R˙s∂Ti+K-s∂Rs∂Ti+α2∂K-s∂TiRs+∂K-s∂TiRs=0.
Let D=∂Rs/∂Ti, D˙=∂R˙s/∂Ti, and D¨=∂R¨s/∂Ti; since the global matrix K-s is the function of the prestress force T, the partial derivative ∂K-s/∂Ti can be obtained directly. The fourth and fifth terms in (19) on the left-hand side can be removed to the right-hand side as the “input force.” Equation (19) can be rewritten as
(20)M-sD¨+C-sD˙+K-sD=-α2∂K-s∂TiR˙s-∂K-s∂TiRs.
Similarly, the dynamic response sensitivities D, D˙D¨ can be obtained from (20).
2.6. Identification of Prestress Force from Measured Dynamic Response
The identification problem is to find the vector of prestress force {T} such that the calculated responses Rs best match the measured responses R^; that is,
(21)[Q]{Rs}={R^},
where the selection matrix [Q] is a constant matrix with elements of zeros or ones, which maps the degrees of freedom of the system to the measured degrees of freedom. {Rs} and {R^} are the vectors of calculated and measured dynamic responses of the system, respectively. The inverse problem is to minimize the error between the calculated and measured responses as
(22){δR}={R^}-[Q]{Rs}={R^}-{Rcal}.
As we have no idea of the magnitude of the prestress force, the initial value for the elemental prestress force is set as a null vector. The corresponding response Rs0 and the response sensitivity matrix S0 are obtained from (16) and (20) with superscript “0” denoting the initial set of values.
At the jth iteration, the difference between the measured and calculated system responses can be expressed as
(23)ΔRj=R^-Rcalj,(j=0,1,2,…).
Using the penalty function method [24], the vector of the prestress force increment, ΔTj, can be obtained from the following equation:
(24)[S-j]{ΔTj}={ΔRj},
where [S-j] is a nt×N matrix selected from the sensitivity matrix [Sj]. N is the number of unknown of prestress force; nt is the number of measured data points. It is noted that nt should be greater than N to make sure that the equation is over-determined. Equation (24) can be solved by the damped least-squares method [25] with bounds to the solution
(25){ΔTj}=([S-j]T[S-j]+λI)-1[S-j]T{ΔRj},
where λ is the nonnegative damping (regularization) coefficient governing the participation of least-squares error in the solution. The solution of (24) is equivalent to minimize the function
(26)J({ΔTi},λ)=∥[S-j]{ΔRj}-{ΔTj}∥2+λ∥{ΔTj}∥2
with the second term in (26) providing bounds to the solution. When the parameter λ approaches to zero, the estimated vector {ΔTj} approaches to the solution obtained from the simple least-squares method.
The updated prestress force {Tj+1}={Tj}+{ΔTj}(j=0,1,2,…) is calculated in the next iteration, and the dynamic response Rj+1 and response sensitivity Sj+1 are also recalculated. The convergence is considered achieved when the following criterion is met:
(27)∥{Tk+1}-{Tk}∥∥{Tk}∥≤tolerance,
where k denotes the kth iteration. The tolerance is taken as 1×10-7 in this study.
The relative error in identified results in each element is defined as
(28)RE=(Ti)identified-(Ti)true(Ti)true×100%,(i=1,2,…,N).
3. Numerical Simulations3.1. A Prestressed Tee Beam
The prestressed Tee beam studied by Figueiras and Póvoas [26] was used as the first numerical example in this paper. The beam is simply supported and prestressed with a parabolic tendon. The geometry of the beam is shown in Figure 1(a). The mechanical properties of the concrete are Young’s modulus Ec=3.4×1010Pa, mass density ρc=2800kg/m3, and Poisson ratio υ=0.18. The mechanical properties of the prestressing tendon are Young’s modulus Ep=210GPa, mass density ρp=7800kg/m3. The magnitude of the prestress force is 3 MN, and the prestressing tendon is assumed to be perfectly bonded, and no prestress loss is taken into account.
In the finite element of the beam, the flange is discretized into two 4-node isoparametric flat shell elements in the transverse direction and 20 elements in the longitudinal direction. The web is discretized into 20 elements in the longitudinal direction. Figures 1(b) and 1(c) show the finite element number sequence of Tee beam. From the free vibration analysis, the first nine natural frequencies of the prestressed beam are 6.30, 7.441, 8.268, 21.545, 21.923, 24.243, 35.075, 36.847, and 47.075 Hz, respectively. The modal damping ratio is taken as 0.01 for the first two modes to obtain the two coefficients in Rayleigh damping.
The parameters of the five-parameter vehicle model are mv1=3.6×103kg, mv2=0.25×103kg, cv=1.0×103Ns/m, kv1=6.0×105N/m, and kv2=8.5×105N/m. The traveling speed of the vehicle is assumed to be 20 m/s. Classes A to D road surface roughness is included. Figure 2 gives a comparison on the displacement responses of the vehicle and that at the middle point of the flange for different road roughness coefficients. From this figure one can find that the road roughness coefficient has significant effect on the responses for both the bridge and the vehicle.
Effect of road roughness on the displacement responses of the vehicle and the beam ((a) the vehicle; (b) middle span point of the beam).
To simulate the effect of measurement noise, a normally distributed random error with zero mean and a unit standard deviation is added to the calculated acceleration as
(29)R¨^=R¨cal+Ep×Noise×var(R¨cal),
where R¨^ is the vectors of measured structural acceleration response; Ep is the noise level; Noise is a standard normal distribution vector with zero mean and unit standard deviation; var(R¨cal) is the variance of the time history.
Study Case 1: Effect of Different Road Surface Roughness. In this case, we make use of the noise-free acceleration response of the bridge induced by the vehicle passing on top of the bridge at a speed of 10m/s to identify the prestress force with different road surface roughness. The sampling rate is 100 Hz which is high enough to include the first nine frequencies of the bridge, and the measurement time duration is 2 seconds in the identification. Three acceleration measurements located at L/4, L/2, and 3L/4 of the bridge at the bottom of the web are used for the prestress force identification. The maximum errors in identified result for each road surface roughness are listed in Table 1. One can find the prestress force in each element has been identified with very high accuracy for each road surface roughness. Figure 3 shows the identified results for road roughness Class B.
Identified results for different road surface roughness.
Road surface roughness
Class A
Class B
Class C
Class D
Max error (%)
0.01
0.02
0.02
0.05
Iteration number
12
12
13
18
λoptimal
1.21×10-9
1.23×10-9
1.30×10-9
1.49×10-9
Identified elemental prestress force (without noise).
Study Case 2: Effect of Different Vehicle Speed. Three different vehicle speeds are studied, that is, 10 m/s, 20 m/s, and 40 m/s, and the corresponding measurement time duration is 2 seconds. The sampling rate is 100 Hz. The same vehicle and measurement locations as Study Case 1 are used. Class B road surface roughness is assumed. The maximum errors in identified results for each speed are summarized in Table 2. This table indicates that different vehicle speed has little effect on the identified results.
Identified results for different vehicle speed.
Vehicle speed (m/s)
10
20
40
Max error (%)
0.02
0.02
0.02
Iteration number
12
13
13
λoptimal
1.23×10-9
1.24×10-9
1.24×10-9
Study Case 3: Effect of Different Measurement Noise. The same three acceleration measurements as Study Case 1 contaminated with different noise levels, that is, 1%, 5%, and 10%, are studied. The maximum errors in identified result for each noise level are listed in Table 3. From this table, one can find that the effect of measurement noise on the identified results is not significant. Even for 10% noise level, the maximum identification error is 4.11%. This indicates the identified results are insensitive to the measurement noise. Figure 4 shows the identified results in which the noise level is 10%.
Identified results for different noise level.
Measurement noise
1%
5%
10%
Max error (%)
1.65
2.24
4.11
Iteration number
14
15
17
λoptimal
5.63×10-9
9.12×10-9
1.04×10-8
Identified elemental prestress force (with 10% noise).
3.2. A Prestressed Box-Girder Bridge
A two-span prestressed box-girder bridge is studied in this numerical example as shown in Figure 5(a). The dimension of the bridge is shown in Figure 5(a). Physical parameters of the bridge are Young’s modulus E=3.4×1010Pa, mass density ρ=2.5×103kg/m3. The bridge is assumed to be jacked at both ends and the magnitude of the prestress force at the jack is assumed to be T=4MN in the left rib and T=6MN in the right rib. The layout of prestressing tendon is parabolic. The prestress loss is taken into account. Table 4 shows the prestress force in each of the 20 elements in the rib.
Magnitude of prestress force in each element.
Element number
Prestress (MN)
81
4.0
82
3.88
83
3.76
84
3.64
85
3.52
86
3.40
87
3.28
88
3.12
89
2.96
90
2.8
91
2.8
92
2.96
93
3.12
94
3.28
95
3.40
96
3.52
97
3.64
98
3.76
99
3.88
100
4.0
101
6.0
102
5.82
103
5.64
104
5.46
105
5.28
106
5.10
107
4.92
108
4.68
109
4.44
110
4.20
111
4.20
112
4.44
113
4.68
114
4.92
115
5.10
116
5.28
117
5.46
118
5.64
119
5.82
120
6.0
(a) Two-span prestressed box-girder bridge (dimension in meter). (b) Element number for the upper plate of the box-girder bridge. (c) Element number for the lower plate of the box-girder bridge. (d) Element number for the left rib of the box-girder bridge. (e) Element number for the right rib of the box-girder bridge.
In the finite element of the bridge, the upper plate of bridge is discretized into three 4-node isoparametric flat shell elements in the transverse direction and 20 elements in the longitudinal direction. The bottom plate of the bridge is discretized into 1 element in the transverse direction and 20 elements in the longitudinal direction. The rib is discretized into 20 elements in the longitudinal direction. Figures 5(b)–5(e) show the finite element number sequence of the bridge. The first ten natural frequencies of the prestressed beam are 7.942, 11.510, 13.369, 17.086, 18.773, 23.529, 23.604, 29.427, 33.345, and 34.134 Hz, respectively. The modal damping ratio is taken as 0.01 for the first two modes to calculate the two coefficients in Rayleigh damping.
Study Case 4: Identification of Different Distributions of Prestress Force. The traveling speed of the vehicle is 20 m/s, which moves in the global y direction on the top of the left rib. The sampling rate is 100 Hz. Class B road surface is adopted. Six acceleration measurements M1–M6 as shown in Figure 5(a) are used for prestress force identification. The identified results converge to the true values after 15 iterations with a maximum relative identified error of 0.008% at element 91. The optimal regularization parameter λopt is found to be 2.24×10-12. Figure 6 shows the identified results.
Identification of distributed prestress force in box-girder bridge (noise free).
Study Case 5: Effect of Measurement Noise. The effect of measurement noise on the identified results is studied. Case 5 is restudied, but it is assumed the “measured” acceleration responses are contaminated with noise. Ten percent noise level is studied. The identified results converged after 27 iterations as shown in Figure 7 with a maximum identified error 3.58% at element 92. The optimal regularization parameter λopt is found to be 3.28×10-12.
Identification of distributed prestress force in box-girder bridge (10% noise).
4. Conclusions
An approach making use of the dynamic responses of the bridge under moving vehicular load is proposed to identify the prestress force in the bridge. Four-node isoparametric flat shell element with the transverse shearing deformation is used to model the prestressed bridges. And the equation of motion of the bridge-vehicle system is established. The prestress forces are identified iteratively from a dynamic response sensitivity analysis based model updating using the measured dynamic responses of the bridge. Two numerical simulations indicate that the proposed method is correct and efficient for prestress force identification. Study shows that the speed of the moving vehicle has little effect on the identified results. Studies also show that artificial measurement noise does not have significant effect on the identified results, but larger identified error is observed under a weak road condition.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (11172333, 11272361), the Fundamental Research Funds for the Central Universities (13lgzd06), and the General Financial Grant from the China Postdoctoral Science Foundation (2013M531893). Such financial aids are gratefully acknowledged.
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