# A Multiscale Finite Element Model Validation Method of Composite Cable-Stayed Bridge Based on Structural Health Monitoring System

Authors: Rumian Zhong; Zhouhong Zong; Qiqi Liu; Haifei Zhou
Journal: Shock and Vibration
 (2015)

---

## Abstract

A two-step response surface method for multiscale finite element model (FEM) updating and validation is presented with respect to Guanhe Bridge, a composite cable-stayed bridge in the National Highway number G15, in China. Firstly, the state equations of both multiscale and single-scale FEM are established based on the basic equation in structural dynamic mechanics to update the multiscale coupling parameters and structural parameters. Secondly, based on the measured data from the structural health monitoring (SHM) system, a Monte Carlo simulation is employed to analyze the uncertainty quantification and transmission, where the uncertainties of the multiscale FEM and measured data were considered. The results indicate that the relative errors between the calculated and measured frequencies are less than 2%, and the overlap ratio indexes of each modal frequency are larger than 80% without the average absolute value of relative errors. These demonstrate that the proposed method can be applied to validate the multiscale FEM, and the validated FEM can reflect the current conditions of the real bridge; thus it can be used as the basis for bridge health monitoring, damage prognosis (DP), and safety prognosis (SP).


---

## Body

## 1. Introduction

With the development of traffic enterprise, many SHM systems have been implemented on cable-stayed bridge throughout the world; the importance of their damage detection and prognosis ability based on model has been recognized by highway administrations; thus, the efficiency and accuracy of the FEM have a significant impact on the structural safety assessment [1–6].Multiscale modeling in physics is aimed at calculating system behavior on one level using information or models from different levels, since the superiority of multiscale modeling has been the focus of many investigations and discussions in recent years. For example, Oberkampf and Roy [7] and Weng et al. [8] discussed the substructural method for multiscale modeling. Liu et al. [9] described the application of bridging scale method to the coupled atomistic/continuum simulation of dynamic fracture, where the robustness of the multiscale method was demonstrated through many benchmark problems and application examples. Takizawa and Tezduyar [10] presented a multiscale space-time technique for fluid-structure interaction (FSI) computations; a number of test computations show the performance of the method. Efendiev and Hou [11] gave a brief overview of global couplings of multiscale basis functions and performed comparisons of multiscale finite volume method, mixed multiscale finite element method, and variation multiscale methods. Ben Dhia [12, 13] offered an Arlequin method to glue two models, which allows the coexistence of incompatible models, sharing the energies of the system in the superposition regions and each part can be linked in an appropriate way in the gluing subregions. However, most of the previous studies concentrated on the laws of conservation of force, mass, and energy; it is difficult to confirm lots of coupling parameters based on these equilibrium equations in civil engineering, the efficiency and accuracy of the multiscale FEM cannot be achieved at the same time.Although several of multiscale evaluation indexes based on Runyang Bridge [14], Sutong Bridge [15], and Donghai Bridge [16] had been established, there are still a few challenges in testing accuracy of the multiscale FEM. Oberkampf and Roy [7] provided a comprehensive and systematic development of the basic concepts, principles, and procedures for verification and validation (V&V) of models and simulations. Doebling et al. [17] and George [18] discussed the response surface method; it has been applied to a wide range of applications in industry, environmental regulations and safety, product and plant safety, financial investing, and governmental regulations. Guo and Zhang [19] combined the response surface method with Monte Carlo simulation, and a thin plate experiment was used to verify the method. Zong et al. [20, 21] completed FEM updating and validation method based on the response surface for a PC continuous rigid frame bridge. However, there have been very few studies specialized in multiscale FEM updating and validation for a composite cable-stayed bridge.This paper presents a two-step response surface method for multiscale FEM updating and validation, and the organization is as follows. Section2 is devoted to the theoretical formulations of the two-step response surface method, and the framework was presented; the SHM system of a composite cable-stayed bridge is introduced in Section 3; in Section 4, the multiscale FEM coupling parameters and structural parameters are updated, respectively. Based on the measured data of SHM system and the updated multiscale FEM, the uncertainties of both the model and test are discussed, and the updated model accuracy is assessed in Section 5.


## 2. Theoretical Formulations

### 2.1. A Two-Step Multiscale Model Updating Method

The method presented here is an extension and modification of the work done by Bauman et al. [22]. The motion equation of a viscously damped system subjected to a force can be written as
(1)
M
u
¨
+
C
u
˙
+
K
u
=
F
.According to the Arlequin vision, we consider thatA is divided into two overlapping models A
1 and A
2. E denotes the gluing zone supposed to be a nonzero measured subset as E
=
A
1
∩
A
2 (see Figure 1).Figure 1
Superposed domains and the gluing zoneE.To rebuild the modelA, the weight parameter functions α and β are required to distribute energy in the gluing zone E, and they satisfy the following equalities:
(2)
α
+
β
=
1
,
In
⁡
E
,
α
=
β
=
1
,
In
⁡
A
∩
E
¯
.Stiffness matrixK
s can be used to couple the two overlapping models A
1 and A
2; the static equilibrium equations for each modal can be shown as
(3)
α
K
K
s
1
T
K
s
1
0
·
μ
1
ε
=
F
1
F
0
,
β
K
-
K
c
2
T
-
K
c
2
0
·
μ
2
ε
=
F
2
-
F
0
,
where the vectors μ
1, μ
2, F
1, and F
2 stand for node displacements and loads, respectively; vector ε is a Lagrange multiplier which is related to the stiffness matrix K
s. The discrete problems in (3) are equivalent to an overall balance equation as below which can be solved based on a mixed Arlequin formulation [22]
(4)
α
K
0
K
c
1
T
0
β
K
-
K
c
2
T
K
c
1
-
K
c
2
0
·
μ
1
μ
2
ε
=
F
1
F
2
0
.As described in the literatures [12, 13], the calculation method of the stiffness matrix K
s and the weight parameter functions α and β, which have been previously proposed, is tedious and can be affected by many factors; it determines the difficulty of applying this method to civil engineering.The traditional multiscale modeling which is based on static equilibrium equation has been the focus of many investigations and discussion in recent years. Thus, civil engineers can now contemplate simulating complex systems spanning a large range of scales based on the dynamic equilibrium equation and two-step response surface method [21].By virtue of (4), (1) is rewritten as
(5)
α
K
0
K
c
1
T
0
β
K
-
K
c
2
T
K
c
1
-
K
c
2
0
·
μ
1
μ
2
ε
+
α
C
0
0
0
β
C
0
0
0
0
·
μ
˙
1
μ
˙
2
ε
˙
+
α
M
0
0
0
β
M
0
0
0
0
·
μ
¨
1
μ
¨
2
ε
¨
=
F
1
F
2
0
,
where M, C, and K are the mass, damping, and stiffness matrices, respectively.For a single-scale model, we assume that a beam is divided into two overlapping modelsB
1 and B
2. As shown in Figure 2, the gluing zone is divided in half, and the two overlapping models B
1 and B
2 are coupled by the coupling matrix K
s
0.Figure 2
Graphical representation of a single-scale beam model.According to (5), the dynamic equilibrium equation of a single-scale model can be written as
(6)
α
0
K
0
0
K
c
0
T
0
β
0
K
0
-
K
c
0
T
K
c
0
-
K
c
0
0
·
μ
1
μ
2
ε
+
α
0
C
0
0
0
0
β
0
C
0
0
0
0
0
·
μ
˙
1
μ
˙
2
ε
˙
+
α
0
M
0
0
0
0
β
0
M
0
0
0
0
0
·
μ
¨
1
μ
¨
2
ε
¨
=
F
1
F
2
0
.From (5) and (6), the stiffness matrix K
s and the weight parameter functions α and β are approximated based on a multiscale FEM updating method. Firstly, the error between the multiscale FEM and the three-dimensional solid FEM is updated, which is so-called “first-step multiscale model updating”; and secondly combined with the visual inspection and ambient vibration testing, the error between the multiscale FE model and the real bridge is updated, which is so-called “second-step multiscale model parameters updating.”


### 2.2. The Framework of Multiscale Model Validation

Firstly, the coupling matrixK
s and the weight parameters α and β were updated based on a two-step multiscale model updating method; moreover, considering the uncertainty of the model and test, such as the parameter error, shape error, discretization error and the uncertainty of the environment, and measured data, these parameters were all assumed to obey normal distribution. Then the parameters of uncertainty quantization and transmission were discussed based on Monte Carlo stochastic finite element method. Finally, the method was applied to a composite cable-stayed bridge; the procedure of multiscale FEM validation based on SHM system is shown as Figure 3.Figure 3
The multiscale FEM validation procedure based on SHM system.


## 2.1. A Two-Step Multiscale Model Updating Method

The method presented here is an extension and modification of the work done by Bauman et al. [22]. The motion equation of a viscously damped system subjected to a force can be written as
(1)
M
u
¨
+
C
u
˙
+
K
u
=
F
.According to the Arlequin vision, we consider thatA is divided into two overlapping models A
1 and A
2. E denotes the gluing zone supposed to be a nonzero measured subset as E
=
A
1
∩
A
2 (see Figure 1).Figure 1
Superposed domains and the gluing zoneE.To rebuild the modelA, the weight parameter functions α and β are required to distribute energy in the gluing zone E, and they satisfy the following equalities:
(2)
α
+
β
=
1
,
In
⁡
E
,
α
=
β
=
1
,
In
⁡
A
∩
E
¯
.Stiffness matrixK
s can be used to couple the two overlapping models A
1 and A
2; the static equilibrium equations for each modal can be shown as
(3)
α
K
K
s
1
T
K
s
1
0
·
μ
1
ε
=
F
1
F
0
,
β
K
-
K
c
2
T
-
K
c
2
0
·
μ
2
ε
=
F
2
-
F
0
,
where the vectors μ
1, μ
2, F
1, and F
2 stand for node displacements and loads, respectively; vector ε is a Lagrange multiplier which is related to the stiffness matrix K
s. The discrete problems in (3) are equivalent to an overall balance equation as below which can be solved based on a mixed Arlequin formulation [22]
(4)
α
K
0
K
c
1
T
0
β
K
-
K
c
2
T
K
c
1
-
K
c
2
0
·
μ
1
μ
2
ε
=
F
1
F
2
0
.As described in the literatures [12, 13], the calculation method of the stiffness matrix K
s and the weight parameter functions α and β, which have been previously proposed, is tedious and can be affected by many factors; it determines the difficulty of applying this method to civil engineering.The traditional multiscale modeling which is based on static equilibrium equation has been the focus of many investigations and discussion in recent years. Thus, civil engineers can now contemplate simulating complex systems spanning a large range of scales based on the dynamic equilibrium equation and two-step response surface method [21].By virtue of (4), (1) is rewritten as
(5)
α
K
0
K
c
1
T
0
β
K
-
K
c
2
T
K
c
1
-
K
c
2
0
·
μ
1
μ
2
ε
+
α
C
0
0
0
β
C
0
0
0
0
·
μ
˙
1
μ
˙
2
ε
˙
+
α
M
0
0
0
β
M
0
0
0
0
·
μ
¨
1
μ
¨
2
ε
¨
=
F
1
F
2
0
,
where M, C, and K are the mass, damping, and stiffness matrices, respectively.For a single-scale model, we assume that a beam is divided into two overlapping modelsB
1 and B
2. As shown in Figure 2, the gluing zone is divided in half, and the two overlapping models B
1 and B
2 are coupled by the coupling matrix K
s
0.Figure 2
Graphical representation of a single-scale beam model.According to (5), the dynamic equilibrium equation of a single-scale model can be written as
(6)
α
0
K
0
0
K
c
0
T
0
β
0
K
0
-
K
c
0
T
K
c
0
-
K
c
0
0
·
μ
1
μ
2
ε
+
α
0
C
0
0
0
0
β
0
C
0
0
0
0
0
·
μ
˙
1
μ
˙
2
ε
˙
+
α
0
M
0
0
0
0
β
0
M
0
0
0
0
0
·
μ
¨
1
μ
¨
2
ε
¨
=
F
1
F
2
0
.From (5) and (6), the stiffness matrix K
s and the weight parameter functions α and β are approximated based on a multiscale FEM updating method. Firstly, the error between the multiscale FEM and the three-dimensional solid FEM is updated, which is so-called “first-step multiscale model updating”; and secondly combined with the visual inspection and ambient vibration testing, the error between the multiscale FE model and the real bridge is updated, which is so-called “second-step multiscale model parameters updating.”


## 2.2. The Framework of Multiscale Model Validation

Firstly, the coupling matrixK
s and the weight parameters α and β were updated based on a two-step multiscale model updating method; moreover, considering the uncertainty of the model and test, such as the parameter error, shape error, discretization error and the uncertainty of the environment, and measured data, these parameters were all assumed to obey normal distribution. Then the parameters of uncertainty quantization and transmission were discussed based on Monte Carlo stochastic finite element method. Finally, the method was applied to a composite cable-stayed bridge; the procedure of multiscale FEM validation based on SHM system is shown as Figure 3.Figure 3
The multiscale FEM validation procedure based on SHM system.


## 3. Structural Health Monitoring System

The Guanhe Bridge is a composite cable-stayed bridge in the National Highway number G15 in China, with an overall length of 640 m and a main span of 340 m (as shown in Figure4). The concrete bridge tower, consisting of one transverse beam, is 121 m high and was constructed using sliding formwork technology. The main girder was assembled from 166 steel I-beam segments; the bridge deck is a prestressed concrete with 34 m wide and 3.08 m high. The construction of the bridge began in 2002 and was open to traffic in May, 2006.Figure 4
Guanhe Bridge panorama photo.A comprehensive structural monitoring system (Figure5) has been implemented on the Guanhe Bridge [23], and it comprises 280 sensors of different types, including temperature, anemometers, accelerometers, weigh-in-motion sensors, global positioning systems (GPS), displacement transducers, strain gauges, and CCTV (closed circuit television) video cameras. Such structural monitoring system has been continuously monitoring the environment conditions (e.g., wind, temperature, and traffic loads) and bridge response since 2013.Figure 5
Sensors setting of SHM system.


## 4. Multiscale FEM Updating

### 4.1. FEM of the Guanhe Bridge

The single-scale FEM and multiscale FEM were developed on the basis of the engineering drawings and implemented using the ANSYS software package. As shown in Figure6, the cables were modeled using linear elastic link elements (LINK 8), secondary dead load was approximated as a mass element (MASS21), and the restriction effect of the rubber supports was simulated using linear elastic spring elements (COMBIN 14). For the single-scale FEM, both the bridge towers and the bridge deck were modeled using a 3D solid element (SOLID 45), and the main girder and small girder were modeled using 3D shell elements (SHELL 43). However, for the multiscale FEM, both the bridge towers and the main girder of the large-scale element were simulated using three-dimensional beam elements (BEAM188), and the coupling elements of the two submodels in gluing zone E were modeled using linear elastic spring elements (COMBIN 14). The single-scale FEM and multiscale FEM, respectively, consists of 46228 and 2253 elements, respectively.The multiscale FE model of Guanhe Bridge.
(a)
The single-scale FEM
(b)
The multiscale FEM


### 4.2. Multiscale Model Coupling Parameters Updating

#### 4.2.1. Parameters Selection

As shown in Table1, the weight parameters α (β
=
1
-
α) and the coupling parameters k
v, k
t, and k
l which are related to coupling matrix K
s were selected as the independent variables; the trial results of the multiscale model show that the mode shapes and these independent variables have a close relationship, especially in the gluing zone. Therefore, the model assurance criterion (MAC) values of the 1st vertical, transverse, and longitudinal mode shapes were selected as the outputs in the multiscale model updating.Table 1
Parameters selection.
Parameters
k
v
k
t
k
l
α
The vertical stiffness of spring elements in the gluing zone
The transverse stiffness of spring elements in the gluing zone
The longitudinal stiffness of spring elements in the gluing zone
The weight parameter functions in the gluing zone


#### 4.2.2. Model Updating Based on Response Surface Method

The 3rd order response surface method is suited for the multiscale model updating, and the procedure is organized as follows: (1) the MAC values of each mode shape was calculated through the single-scale FEM; (2) the experimental samples were selected by D-optimal design method, whereas the MAC values of each mode shape under different experimental sample were calculated through the multiscale FEM, as shown in Table2; (3) the least square regression analysis and 3rd order response surface method were used to fit these experimental samples. Then, the mathematical relationship between input and output variables can be obtained, and the response surface model of the first vertical mode shape is shown in Figure 7.Table 2
Experimental samples.
N
Input variables
MAC
k
v(107)
k
t(107)
k
l(107)
α
V
1
T
1
L
1
1
1.00
1.00
100.00
0.57
0.72
0.98
0.84
2
79.71
43.57
22.86
0.57
0.66
0.78
0.76
3
100.00
68.25
100.00
0.69
0.93
0.32
0.89
⋮
⋮
⋮
⋮
⋮
⋮
⋮
28
76.74
1.00
1.00
0.81
0.63
0.97
0.97
29
75.25
32.19
62.88
0.81
0.64
0.63
0.75
30
1.00
26.74
73.77
0.64
0.84
0.66
0.85
Note: “N” is the number of the samples.Figure 7
The response surface model of first vertical mode shape.The value ofR
2 and RMSE can be used to verify accuracy of the response surface model; they are calculated based on (7a) and (7b), and the results are shown in Table 3
(7a)
R
2
=
1
-
∑
j
=
1
N
y
RS
j
-
y
j
2
∑
j
=
1
N
y
j
-
y
-
2
,
(7b)
RMSE
=
1
N
·
y
-
·
∑
y
j
-
y
RS
j
2
,where y
RS
(
j
) is the calculated value by response surface model; y
(
j
) is the relative calculated result by finite element analysis of the multiscale model; y
- is the mean value by finite element analysis of the multiscale model. Table 3 shows that the values of R
2 are close to 1, and the values of RMSE are near zero. Therefore, the response surface model can show the mathematical relationship between input and output parameters. In other words, it can be used for multiscale model updating.Table 3
R
2 and RMSE values.
Mode
1st vertical model shape
1st transverse model shape
1st longitudinal model shape
R
2
0.9996
0.9998
0.999997
RMSE
8.78 × 10−6
3.11 × 10−6
5.25 × 10−7The parameters can be updated based on the response surface model which has been built and the updated MAC values can also be calculated. In general, it is generally acceptable when the MAC is greater than 80% [21]. The MAC values in this study are listed in Table 4: the lowest MAC is 93.51% which implies an excellent match between the mode shapes from the two methods. Therefore, it can be concluded that the mode shapes of the multiscale FEM are consistent with the single-scale FEM, and the weight parameter functions α (β
=
1
-
α) and the coupling parameters k
v, k
t, and k
l are well suited to the multiscale model.Table 4
The updated parameters and MAC value.
Parameters
Initial values
Updated values
MAC (%)
V
1
T
1
L
1
k
v(107 N/m)
1
91.32
93.51
97.11
95.43
k
t(107 N/m)
1
2.13
k
l(107 N/m)
1
1.59
α
0.5
0.71
β
0.5
0.29


### 4.3. Structural Parameter Updating

#### 4.3.1. Ambient Vibration Testing

Excitation by ambient vibration sources (i.e., passages of vehicles and trains) is conveniently utilized as input to the structure. To obtain the natural frequencies and mode shapes, 56 measure points and a reference point were arranged as shown in Figure8, and the reference sensor was placed on the middle of the bridge. Data was sampled at 200 Hz, and each setup for all tests was recorded for duration of 15 minutes; the frequencies and mode shapes were obtained based on peak picking (PP) and stochastic subspace identification (SSI) method.Ambient vibration testing of Guanhe Bridge.
(a)
(b)


#### 4.3.2. Parameters Selection

There are no visual cracks by means of the visual inspection of Guanhe Bridge; therefore the elasticity modulus of the concrete bridge deck and spring stiffness of the rubber supports were selected as the input parameters, as shown in Table5.Table 5
Parameters selection.
Parameters
K
1
K
2
K
3
E
1
The longitudinalstiffness of spring elements
The transverse stiffness of spring elements on the bridge tower
The transverse  stiffness of spring elements on the bridge pier
Elasticity modulusof the concrete bridge deck


#### 4.3.3. Model Updating Based on Response Surface Method

In the previous section, the response surface method has been introduced. Table6 and Figure 9 have shown experimental samples and response surface model for the model parameter updating, respectively. The value of R
2 and RMSE can also be calculated based on (8), as shown in Table 7.Table 6
Experimental samples.
N
Input parameters
Frequencies
A
B
C
D
R
1
R
2
R
3
R
4
R
5
R
6
R
7
1
3.45
20.71
15.00
12.08
0.347
0.505
0.797
0.377
0.531
0.878
0.585
2
3.57
25.00
5.00
18.42
0.355
0.508
0.797
0.411
0.615
0.886
0.585
3
3.80
19.37
5.00
5.00
0.362
0.512
0.799
0.311
0.423
0.875
0.585
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
43
3.80
17.22
15.00
8.98
0.361
0.512
0.799
0.360
0.492
0.869
0.585
44
3.80
15.00
5.00
20.00
0.362
0.512
0.799
0.423
0.636
0.861
0.585
45
3.80
25.00
5.00
11.34
0.362
0.512
0.799
0.380
0.526
0.886
0.585
Note:A
,
B
,
C
,
and
D are the values of  k
v
,
k
t
,
k
l, and α; R
1
–
R
3
,
R
4
-
R
5
,
R
6
,
and
R
7 are the frequencies of 1st–3rd vertical mode, 1st-2nd transverse mode, 1st longitudinal mode, and 1st torsion mode.Table 7
R
2 and RMSE values.
Mode
1st vertical model shape
1st transverse model shape
1st longitudinal model shape
R
2
0.99999926
0.9999989
0.99999998
RMSE
1.55 ×  10−6
1.96 × 10−6
2.44 × 10−7The response surface model of mode shape.
(a)
The response surface model of first vertical mode shape
(b)
The response surface model of first transverse mode shape
(c)
The response surface model of first longitudinal mode shapeTable7 shows that the R
2 value is near to 1, and the RMSE value is near zero. Therefore, the response surface model can show the mathematical relationship between input and output parameters, so that it can be used to the multiscale model parameters updating.Based on the response surface model, the input parameters can be updated, and the updated frequencies and MAC values of the main bridge can be calculated, as shown in Tables8 and 9. It can be observed that the calculated frequencies from the updated multiscale FEM based on the two-step response surface method are in good agreement with the measurements, as the maximum error is not larger than 8% and the values of MAC are over 90%.Table 8
Parameters change before and after updating based on the response surface method.
Parameters
Initial values
Updated values
Error/%
E
1(×1010 Pa)
3.45
3.6
4.2
K
1(×108 N/m)
15
17.54
14.5
K
2(×109 N/m)
5
6.47
22.7
K
3(×106 N/m)
10
10.77
7.1Table 9
Comparison of updated frequencies and measured frequencies, MAC values (unit: Hz).
Direction
N
Frequencies (Hz)
MAC values (%)
Measured frequencies ①
Calculated frequencies of single-scale FEM
Updated  frequencies of multiscale FEM ②
Relative error(② − ①)/① (%)
Vertical
1
0.38
0.36
0.35
7.80
91.01
2
0.50
0.50
0.51
2.00
91.21
3
0.77
0.76
0.80
3.90
93.15
Transverse
1
0.39
0.28
0.37
5.13
91.11
2
0.50
0.39
0.52
4.00
92.25
Torsion
1
0.63
0.59
0.59
6.35
92.11
Longitudinal
1
0.87
0.15
0.87
0.00
95.23


## 4.1. FEM of the Guanhe Bridge

The single-scale FEM and multiscale FEM were developed on the basis of the engineering drawings and implemented using the ANSYS software package. As shown in Figure6, the cables were modeled using linear elastic link elements (LINK 8), secondary dead load was approximated as a mass element (MASS21), and the restriction effect of the rubber supports was simulated using linear elastic spring elements (COMBIN 14). For the single-scale FEM, both the bridge towers and the bridge deck were modeled using a 3D solid element (SOLID 45), and the main girder and small girder were modeled using 3D shell elements (SHELL 43). However, for the multiscale FEM, both the bridge towers and the main girder of the large-scale element were simulated using three-dimensional beam elements (BEAM188), and the coupling elements of the two submodels in gluing zone E were modeled using linear elastic spring elements (COMBIN 14). The single-scale FEM and multiscale FEM, respectively, consists of 46228 and 2253 elements, respectively.The multiscale FE model of Guanhe Bridge.
(a)
The single-scale FEM
(b)
The multiscale FEM


## 4.2. Multiscale Model Coupling Parameters Updating

### 4.2.1. Parameters Selection

As shown in Table1, the weight parameters α (β
=
1
-
α) and the coupling parameters k
v, k
t, and k
l which are related to coupling matrix K
s were selected as the independent variables; the trial results of the multiscale model show that the mode shapes and these independent variables have a close relationship, especially in the gluing zone. Therefore, the model assurance criterion (MAC) values of the 1st vertical, transverse, and longitudinal mode shapes were selected as the outputs in the multiscale model updating.Table 1
Parameters selection.
Parameters
k
v
k
t
k
l
α
The vertical stiffness of spring elements in the gluing zone
The transverse stiffness of spring elements in the gluing zone
The longitudinal stiffness of spring elements in the gluing zone
The weight parameter functions in the gluing zone


### 4.2.2. Model Updating Based on Response Surface Method

The 3rd order response surface method is suited for the multiscale model updating, and the procedure is organized as follows: (1) the MAC values of each mode shape was calculated through the single-scale FEM; (2) the experimental samples were selected by D-optimal design method, whereas the MAC values of each mode shape under different experimental sample were calculated through the multiscale FEM, as shown in Table2; (3) the least square regression analysis and 3rd order response surface method were used to fit these experimental samples. Then, the mathematical relationship between input and output variables can be obtained, and the response surface model of the first vertical mode shape is shown in Figure 7.Table 2
Experimental samples.
N
Input variables
MAC
k
v(107)
k
t(107)
k
l(107)
α
V
1
T
1
L
1
1
1.00
1.00
100.00
0.57
0.72
0.98
0.84
2
79.71
43.57
22.86
0.57
0.66
0.78
0.76
3
100.00
68.25
100.00
0.69
0.93
0.32
0.89
⋮
⋮
⋮
⋮
⋮
⋮
⋮
28
76.74
1.00
1.00
0.81
0.63
0.97
0.97
29
75.25
32.19
62.88
0.81
0.64
0.63
0.75
30
1.00
26.74
73.77
0.64
0.84
0.66
0.85
Note: “N” is the number of the samples.Figure 7
The response surface model of first vertical mode shape.The value ofR
2 and RMSE can be used to verify accuracy of the response surface model; they are calculated based on (7a) and (7b), and the results are shown in Table 3
(7a)
R
2
=
1
-
∑
j
=
1
N
y
RS
j
-
y
j
2
∑
j
=
1
N
y
j
-
y
-
2
,
(7b)
RMSE
=
1
N
·
y
-
·
∑
y
j
-
y
RS
j
2
,where y
RS
(
j
) is the calculated value by response surface model; y
(
j
) is the relative calculated result by finite element analysis of the multiscale model; y
- is the mean value by finite element analysis of the multiscale model. Table 3 shows that the values of R
2 are close to 1, and the values of RMSE are near zero. Therefore, the response surface model can show the mathematical relationship between input and output parameters. In other words, it can be used for multiscale model updating.Table 3
R
2 and RMSE values.
Mode
1st vertical model shape
1st transverse model shape
1st longitudinal model shape
R
2
0.9996
0.9998
0.999997
RMSE
8.78 × 10−6
3.11 × 10−6
5.25 × 10−7The parameters can be updated based on the response surface model which has been built and the updated MAC values can also be calculated. In general, it is generally acceptable when the MAC is greater than 80% [21]. The MAC values in this study are listed in Table 4: the lowest MAC is 93.51% which implies an excellent match between the mode shapes from the two methods. Therefore, it can be concluded that the mode shapes of the multiscale FEM are consistent with the single-scale FEM, and the weight parameter functions α (β
=
1
-
α) and the coupling parameters k
v, k
t, and k
l are well suited to the multiscale model.Table 4
The updated parameters and MAC value.
Parameters
Initial values
Updated values
MAC (%)
V
1
T
1
L
1
k
v(107 N/m)
1
91.32
93.51
97.11
95.43
k
t(107 N/m)
1
2.13
k
l(107 N/m)
1
1.59
α
0.5
0.71
β
0.5
0.29


## 4.2.1. Parameters Selection

As shown in Table1, the weight parameters α (β
=
1
-
α) and the coupling parameters k
v, k
t, and k
l which are related to coupling matrix K
s were selected as the independent variables; the trial results of the multiscale model show that the mode shapes and these independent variables have a close relationship, especially in the gluing zone. Therefore, the model assurance criterion (MAC) values of the 1st vertical, transverse, and longitudinal mode shapes were selected as the outputs in the multiscale model updating.Table 1
Parameters selection.
Parameters
k
v
k
t
k
l
α
The vertical stiffness of spring elements in the gluing zone
The transverse stiffness of spring elements in the gluing zone
The longitudinal stiffness of spring elements in the gluing zone
The weight parameter functions in the gluing zone


## 4.2.2. Model Updating Based on Response Surface Method

The 3rd order response surface method is suited for the multiscale model updating, and the procedure is organized as follows: (1) the MAC values of each mode shape was calculated through the single-scale FEM; (2) the experimental samples were selected by D-optimal design method, whereas the MAC values of each mode shape under different experimental sample were calculated through the multiscale FEM, as shown in Table2; (3) the least square regression analysis and 3rd order response surface method were used to fit these experimental samples. Then, the mathematical relationship between input and output variables can be obtained, and the response surface model of the first vertical mode shape is shown in Figure 7.Table 2
Experimental samples.
N
Input variables
MAC
k
v(107)
k
t(107)
k
l(107)
α
V
1
T
1
L
1
1
1.00
1.00
100.00
0.57
0.72
0.98
0.84
2
79.71
43.57
22.86
0.57
0.66
0.78
0.76
3
100.00
68.25
100.00
0.69
0.93
0.32
0.89
⋮
⋮
⋮
⋮
⋮
⋮
⋮
28
76.74
1.00
1.00
0.81
0.63
0.97
0.97
29
75.25
32.19
62.88
0.81
0.64
0.63
0.75
30
1.00
26.74
73.77
0.64
0.84
0.66
0.85
Note: “N” is the number of the samples.Figure 7
The response surface model of first vertical mode shape.The value ofR
2 and RMSE can be used to verify accuracy of the response surface model; they are calculated based on (7a) and (7b), and the results are shown in Table 3
(7a)
R
2
=
1
-
∑
j
=
1
N
y
RS
j
-
y
j
2
∑
j
=
1
N
y
j
-
y
-
2
,
(7b)
RMSE
=
1
N
·
y
-
·
∑
y
j
-
y
RS
j
2
,where y
RS
(
j
) is the calculated value by response surface model; y
(
j
) is the relative calculated result by finite element analysis of the multiscale model; y
- is the mean value by finite element analysis of the multiscale model. Table 3 shows that the values of R
2 are close to 1, and the values of RMSE are near zero. Therefore, the response surface model can show the mathematical relationship between input and output parameters. In other words, it can be used for multiscale model updating.Table 3
R
2 and RMSE values.
Mode
1st vertical model shape
1st transverse model shape
1st longitudinal model shape
R
2
0.9996
0.9998
0.999997
RMSE
8.78 × 10−6
3.11 × 10−6
5.25 × 10−7The parameters can be updated based on the response surface model which has been built and the updated MAC values can also be calculated. In general, it is generally acceptable when the MAC is greater than 80% [21]. The MAC values in this study are listed in Table 4: the lowest MAC is 93.51% which implies an excellent match between the mode shapes from the two methods. Therefore, it can be concluded that the mode shapes of the multiscale FEM are consistent with the single-scale FEM, and the weight parameter functions α (β
=
1
-
α) and the coupling parameters k
v, k
t, and k
l are well suited to the multiscale model.Table 4
The updated parameters and MAC value.
Parameters
Initial values
Updated values
MAC (%)
V
1
T
1
L
1
k
v(107 N/m)
1
91.32
93.51
97.11
95.43
k
t(107 N/m)
1
2.13
k
l(107 N/m)
1
1.59
α
0.5
0.71
β
0.5
0.29


## 4.3. Structural Parameter Updating

### 4.3.1. Ambient Vibration Testing

Excitation by ambient vibration sources (i.e., passages of vehicles and trains) is conveniently utilized as input to the structure. To obtain the natural frequencies and mode shapes, 56 measure points and a reference point were arranged as shown in Figure8, and the reference sensor was placed on the middle of the bridge. Data was sampled at 200 Hz, and each setup for all tests was recorded for duration of 15 minutes; the frequencies and mode shapes were obtained based on peak picking (PP) and stochastic subspace identification (SSI) method.Ambient vibration testing of Guanhe Bridge.
(a)
(b)


### 4.3.2. Parameters Selection

There are no visual cracks by means of the visual inspection of Guanhe Bridge; therefore the elasticity modulus of the concrete bridge deck and spring stiffness of the rubber supports were selected as the input parameters, as shown in Table5.Table 5
Parameters selection.
Parameters
K
1
K
2
K
3
E
1
The longitudinalstiffness of spring elements
The transverse stiffness of spring elements on the bridge tower
The transverse  stiffness of spring elements on the bridge pier
Elasticity modulusof the concrete bridge deck


### 4.3.3. Model Updating Based on Response Surface Method

In the previous section, the response surface method has been introduced. Table6 and Figure 9 have shown experimental samples and response surface model for the model parameter updating, respectively. The value of R
2 and RMSE can also be calculated based on (8), as shown in Table 7.Table 6
Experimental samples.
N
Input parameters
Frequencies
A
B
C
D
R
1
R
2
R
3
R
4
R
5
R
6
R
7
1
3.45
20.71
15.00
12.08
0.347
0.505
0.797
0.377
0.531
0.878
0.585
2
3.57
25.00
5.00
18.42
0.355
0.508
0.797
0.411
0.615
0.886
0.585
3
3.80
19.37
5.00
5.00
0.362
0.512
0.799
0.311
0.423
0.875
0.585
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
43
3.80
17.22
15.00
8.98
0.361
0.512
0.799
0.360
0.492
0.869
0.585
44
3.80
15.00
5.00
20.00
0.362
0.512
0.799
0.423
0.636
0.861
0.585
45
3.80
25.00
5.00
11.34
0.362
0.512
0.799
0.380
0.526
0.886
0.585
Note:A
,
B
,
C
,
and
D are the values of  k
v
,
k
t
,
k
l, and α; R
1
–
R
3
,
R
4
-
R
5
,
R
6
,
and
R
7 are the frequencies of 1st–3rd vertical mode, 1st-2nd transverse mode, 1st longitudinal mode, and 1st torsion mode.Table 7
R
2 and RMSE values.
Mode
1st vertical model shape
1st transverse model shape
1st longitudinal model shape
R
2
0.99999926
0.9999989
0.99999998
RMSE
1.55 ×  10−6
1.96 × 10−6
2.44 × 10−7The response surface model of mode shape.
(a)
The response surface model of first vertical mode shape
(b)
The response surface model of first transverse mode shape
(c)
The response surface model of first longitudinal mode shapeTable7 shows that the R
2 value is near to 1, and the RMSE value is near zero. Therefore, the response surface model can show the mathematical relationship between input and output parameters, so that it can be used to the multiscale model parameters updating.Based on the response surface model, the input parameters can be updated, and the updated frequencies and MAC values of the main bridge can be calculated, as shown in Tables8 and 9. It can be observed that the calculated frequencies from the updated multiscale FEM based on the two-step response surface method are in good agreement with the measurements, as the maximum error is not larger than 8% and the values of MAC are over 90%.Table 8
Parameters change before and after updating based on the response surface method.
Parameters
Initial values
Updated values
Error/%
E
1(×1010 Pa)
3.45
3.6
4.2
K
1(×108 N/m)
15
17.54
14.5
K
2(×109 N/m)
5
6.47
22.7
K
3(×106 N/m)
10
10.77
7.1Table 9
Comparison of updated frequencies and measured frequencies, MAC values (unit: Hz).
Direction
N
Frequencies (Hz)
MAC values (%)
Measured frequencies ①
Calculated frequencies of single-scale FEM
Updated  frequencies of multiscale FEM ②
Relative error(② − ①)/① (%)
Vertical
1
0.38
0.36
0.35
7.80
91.01
2
0.50
0.50
0.51
2.00
91.21
3
0.77
0.76
0.80
3.90
93.15
Transverse
1
0.39
0.28
0.37
5.13
91.11
2
0.50
0.39
0.52
4.00
92.25
Torsion
1
0.63
0.59
0.59
6.35
92.11
Longitudinal
1
0.87
0.15
0.87
0.00
95.23


## 4.3.1. Ambient Vibration Testing

Excitation by ambient vibration sources (i.e., passages of vehicles and trains) is conveniently utilized as input to the structure. To obtain the natural frequencies and mode shapes, 56 measure points and a reference point were arranged as shown in Figure8, and the reference sensor was placed on the middle of the bridge. Data was sampled at 200 Hz, and each setup for all tests was recorded for duration of 15 minutes; the frequencies and mode shapes were obtained based on peak picking (PP) and stochastic subspace identification (SSI) method.Ambient vibration testing of Guanhe Bridge.
(a)
(b)


## 4.3.2. Parameters Selection

There are no visual cracks by means of the visual inspection of Guanhe Bridge; therefore the elasticity modulus of the concrete bridge deck and spring stiffness of the rubber supports were selected as the input parameters, as shown in Table5.Table 5
Parameters selection.
Parameters
K
1
K
2
K
3
E
1
The longitudinalstiffness of spring elements
The transverse stiffness of spring elements on the bridge tower
The transverse  stiffness of spring elements on the bridge pier
Elasticity modulusof the concrete bridge deck


## 4.3.3. Model Updating Based on Response Surface Method

In the previous section, the response surface method has been introduced. Table6 and Figure 9 have shown experimental samples and response surface model for the model parameter updating, respectively. The value of R
2 and RMSE can also be calculated based on (8), as shown in Table 7.Table 6
Experimental samples.
N
Input parameters
Frequencies
A
B
C
D
R
1
R
2
R
3
R
4
R
5
R
6
R
7
1
3.45
20.71
15.00
12.08
0.347
0.505
0.797
0.377
0.531
0.878
0.585
2
3.57
25.00
5.00
18.42
0.355
0.508
0.797
0.411
0.615
0.886
0.585
3
3.80
19.37
5.00
5.00
0.362
0.512
0.799
0.311
0.423
0.875
0.585
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
43
3.80
17.22
15.00
8.98
0.361
0.512
0.799
0.360
0.492
0.869
0.585
44
3.80
15.00
5.00
20.00
0.362
0.512
0.799
0.423
0.636
0.861
0.585
45
3.80
25.00
5.00
11.34
0.362
0.512
0.799
0.380
0.526
0.886
0.585
Note:A
,
B
,
C
,
and
D are the values of  k
v
,
k
t
,
k
l, and α; R
1
–
R
3
,
R
4
-
R
5
,
R
6
,
and
R
7 are the frequencies of 1st–3rd vertical mode, 1st-2nd transverse mode, 1st longitudinal mode, and 1st torsion mode.Table 7
R
2 and RMSE values.
Mode
1st vertical model shape
1st transverse model shape
1st longitudinal model shape
R
2
0.99999926
0.9999989
0.99999998
RMSE
1.55 ×  10−6
1.96 × 10−6
2.44 × 10−7The response surface model of mode shape.
(a)
The response surface model of first vertical mode shape
(b)
The response surface model of first transverse mode shape
(c)
The response surface model of first longitudinal mode shapeTable7 shows that the R
2 value is near to 1, and the RMSE value is near zero. Therefore, the response surface model can show the mathematical relationship between input and output parameters, so that it can be used to the multiscale model parameters updating.Based on the response surface model, the input parameters can be updated, and the updated frequencies and MAC values of the main bridge can be calculated, as shown in Tables8 and 9. It can be observed that the calculated frequencies from the updated multiscale FEM based on the two-step response surface method are in good agreement with the measurements, as the maximum error is not larger than 8% and the values of MAC are over 90%.Table 8
Parameters change before and after updating based on the response surface method.
Parameters
Initial values
Updated values
Error/%
E
1(×1010 Pa)
3.45
3.6
4.2
K
1(×108 N/m)
15
17.54
14.5
K
2(×109 N/m)
5
6.47
22.7
K
3(×106 N/m)
10
10.77
7.1Table 9
Comparison of updated frequencies and measured frequencies, MAC values (unit: Hz).
Direction
N
Frequencies (Hz)
MAC values (%)
Measured frequencies ①
Calculated frequencies of single-scale FEM
Updated  frequencies of multiscale FEM ②
Relative error(② − ①)/① (%)
Vertical
1
0.38
0.36
0.35
7.80
91.01
2
0.50
0.50
0.51
2.00
91.21
3
0.77
0.76
0.80
3.90
93.15
Transverse
1
0.39
0.28
0.37
5.13
91.11
2
0.50
0.39
0.52
4.00
92.25
Torsion
1
0.63
0.59
0.59
6.35
92.11
Longitudinal
1
0.87
0.15
0.87
0.00
95.23


## 5. Multiscale FEM Validation of Guanhe Bridge

### 5.1. Sample Data

A total of 720 sample data were measured under the temperature of 30~35°C for the purpose of keeping ambient temperatures constant, data were sampled at 50 Hz, and all tests were recorded for duration of 30 minutes. The frequencies can be obtained based on SSI method; and the measured frequencies are assumed to obey the normal distributions based on a descriptive statistical analysis, as shown in Tables10 and 11.Table 10
Sample data of measured frequencies.
N
Verticalfrequencies
Transversefrequencies
Longitudinal frequencies
1
2
3
1
2
1
1
0.378
0.503
0.757
0.378
0.481
0.874
2
0.376
0.503
0.759
0.376
0.481
0.874
3
0.376
0.503
0.757
0.376
0.481
0.857
⋮
⋮
⋮
⋮
⋮
⋮
⋮
717
0.365
0.482
0.746
0.376
0.489
0.845
718
0.376
0.503
0.767
0.376
0.503
0.815
719
0.376
0.503
0.762
0.376
0.503
0.84
720
0.376
0.503
0.767
0.376
0.503
0.835Table 11
The statistics of measured frequencies.
Model
Mean value
Standard deviation
Variable coefficient (%)
Vertical
1st
0.379
0.0085
2.243
2nd
0.501
0.0165
3.293
3rd
0.778
0.0089
1.144
Transverse
1st
0.381
0.0068
1.785
2nd
0.511
0.0095
1.859
Longitudinal
1st
0.861
0.0087
1.010


### 5.2. Uncertainty Analysis

A probability statistics method was used for the uncertainty quantification and transmission of the multiscale model parameters. To study the influence of structural parameters on the frequencies, we considered these three cases.Case 1.
Study on the influence of parameterN (N
=
E
1
/
3.45
×
10
10) to the vertical frequencies, while keeping parameters K
1, K
2, and K
3 constant.Case 2.
Study on the influence of parametersK
2 and K
3 to the transverse frequencies, while keeping the others constant.Case 3.
Study on the influence of parameterK
1 to the longitudinal frequencies, while keeping the others constant.When doing probability/statistics-based uncertainty quantification and transmission, probability distributions of the parameters should be known. However, in this study we know only the measured frequencies but not the probability distributions of the parameters. For this reason, the authors used the combination of response surface model and optimization theory. The statistical characteristics of the parameters were obtained by inverse transformation of the model and assumed to obey normal distributions. In the following the procedure of uncertainty quantification and transmission is outlined: (1) the updated parameters can be obtained based on two-step multiscale model updating method and used as the mean values of the structural parameters; (2) based on the sample data of the measured frequencies and optimization theory, the parameters can also be calculated, and the standard deviation of these calculated values were used as the standard deviation of the structural parameters; (3) assuming that the design parameters obey normal distributions, the random numbers which conform to the probability distributions of structural parameters can be obtained based on the Monte Carlo simulation method; then the frequencies can be calculated based on the response surface model, and the probability distributions of calculated values can be compared with the measured values, as listed in Tables12 and 13.Table 12
Distribution characteristics of uncertainty parameters for Guanhe Bridge.
Parameters
Mean value
Standard deviation
Variable coefficient (%)
Distribution pattern
N
1.040
0.171
16.44
Normal distribution
K
1(×108 N/m)
17.540
0.004
0.03
Normal distribution
K
2(×109 N/m)
6.470
3.520
54.41
Normal distribution
K
3(×106 N/m)
10.770
0.484
4.49
Normal distributionTable 13
The statistics of the frequencies in different conditions.
Case
Mode
Vertical frequencies
Transverse frequencies
Longitudinal frequencies
1
2
3
1
2
1
1
Mean value (Hz)
0.37036
0.50987
0.78325
0.38117
0.52315
0.87115
Standard deviation (Hz)
0.00825
0.01464
0.00742
0.00153
0.00452
0.00156
Variable coefficient (%)
2.22702
2.87191
0.94695
0.40140
0.86400
0.17907
2
Mean value (Hz)
0.37012
0.50526
0.78112
0.38225
0.52465
0.87119
Standard deviation (Hz)
0.00356
0.00421
0.00125
0.00435
0.00854
0.00985
Variable coefficient (%)
0.96185
0.83323
0.16003
1.13748
1.62832
1.13064
3
Mean value (Hz)
0.37052
0.50891
0.78411
0.38115
0.52122
0.87335
Standard deviation (Hz)
0.00289
0.00394
0.00156
0.00098
0.00115
0.02740
Variable coefficient (%)
0.77998
0.77420
0.19895
0.25712
0.22064
3.13689It can be concluded from Table13 that (1) the transverse and longitudinal restriction effect of the rubber supports have a great influence on the transverse and longitudinal frequencies, respectively, and the variable coefficient is more than 2%, (2) the elasticity modulus of the concrete bridge deck has a larger influence on the vertical frequencies, for the variable coefficient is between 0.9% and 2.9%, and (3) the mean value of each frequency has been less affected by the uncertainty of each parameter.


### 5.3. Model/Test Correlation Analysis

To analyze the correlation between model and test results, overlap ratio criterion (ORC) can be used. As shown in (8), J
p is an overlap ratio index of PDF (probability density of frequency), and it can be calculated by comparing calculated frequencies of multiscale FEM with measured frequencies of ambient vibration testing. If J
p
=
1, it shows that multiscale FEM is completely consistent with the bridge
(8)
J
p
=
PDF
test
∩
PDF
.The overlap ratio index (Figure10) has been calculated, as shown in Table 14, and Table 15 shows the overlap ratio index of each modal frequency without the average absolute value of relative errors between calculated and measured values. It can be concluded that the average absolute value of relative errors between calculated and measured values have an influence on the overlap ratio index, without the average absolute value of relative errors, the overlap ratio index of each modal frequencies are over 80%. Thus, the updated multiscale FEM can well represent the dynamic characteristics of Guanhe Bridge.Table 14
Overlap ratio index of each modal frequency with the average absolute value of relative errors.
Mode
Verticalmodel shape
Transversemodel shape
Longitudinal model shape
1
2
3
1
2
1
J
(
p
)
0.736
0.782
0.821
0.778
0.451
0.573Table 15
Overlap ratio index of each modal frequency without the average absolute value of relative errors.
Mode
Verticalmodel shape
Transversemodel shape
Longitudinal model shape
1
2
3
1
2
1
J
(
p
)
0.896
0.857
0.816
0.805
0.810
0.821The probability distribution of Guanhe Bridge frequency.
(a)
The probability distribution of 1st vertical frequency
(b)
The probability distribution of 2nd vertical frequency
(c)
The probability distribution of 3rd vertical frequency
(d)
The probability distribution of 1st transverse frequency
(e)
The probability distribution of 2nd transverse frequency
(f)
The probability distribution of 1st longitudinal frequency


### 5.4. Model Accuracy Evaluation

Considering the uncertainty of multiscale FEM, the mean values of the calculated frequencies were obtained based on the Monte Carlo simulation and response surface method. It can be seen from Table16 that the relative errors between the calculated and measured frequencies are less than 2% with the uncertainty factors fully considered, which indicates that the multiscale FEM can reflect current conditions of the real bridge; thus it can be used as the basis for bridge health monitoring, damage prognosis (DP), and safety prognosis (SP).Table 16
Comparison of average values of each modal frequency.
Mode
Verticalfrequencies
Transversefrequencies
Longitudinal frequencies
1
2
3
1
2
1
Calculated frequency (Hz)
0.379
0.501
0.778
0.381
0.511
0.861
Measured frequency (Hz)
0.378
0.503
0.765
0.378
0.512
0.86
Relative error (%)
0.26
0.40
1.67
0.79
0.20
0.12


## 5.1. Sample Data

A total of 720 sample data were measured under the temperature of 30~35°C for the purpose of keeping ambient temperatures constant, data were sampled at 50 Hz, and all tests were recorded for duration of 30 minutes. The frequencies can be obtained based on SSI method; and the measured frequencies are assumed to obey the normal distributions based on a descriptive statistical analysis, as shown in Tables10 and 11.Table 10
Sample data of measured frequencies.
N
Verticalfrequencies
Transversefrequencies
Longitudinal frequencies
1
2
3
1
2
1
1
0.378
0.503
0.757
0.378
0.481
0.874
2
0.376
0.503
0.759
0.376
0.481
0.874
3
0.376
0.503
0.757
0.376
0.481
0.857
⋮
⋮
⋮
⋮
⋮
⋮
⋮
717
0.365
0.482
0.746
0.376
0.489
0.845
718
0.376
0.503
0.767
0.376
0.503
0.815
719
0.376
0.503
0.762
0.376
0.503
0.84
720
0.376
0.503
0.767
0.376
0.503
0.835Table 11
The statistics of measured frequencies.
Model
Mean value
Standard deviation
Variable coefficient (%)
Vertical
1st
0.379
0.0085
2.243
2nd
0.501
0.0165
3.293
3rd
0.778
0.0089
1.144
Transverse
1st
0.381
0.0068
1.785
2nd
0.511
0.0095
1.859
Longitudinal
1st
0.861
0.0087
1.010


## 5.2. Uncertainty Analysis

A probability statistics method was used for the uncertainty quantification and transmission of the multiscale model parameters. To study the influence of structural parameters on the frequencies, we considered these three cases.Case 1.
Study on the influence of parameterN (N
=
E
1
/
3.45
×
10
10) to the vertical frequencies, while keeping parameters K
1, K
2, and K
3 constant.Case 2.
Study on the influence of parametersK
2 and K
3 to the transverse frequencies, while keeping the others constant.Case 3.
Study on the influence of parameterK
1 to the longitudinal frequencies, while keeping the others constant.When doing probability/statistics-based uncertainty quantification and transmission, probability distributions of the parameters should be known. However, in this study we know only the measured frequencies but not the probability distributions of the parameters. For this reason, the authors used the combination of response surface model and optimization theory. The statistical characteristics of the parameters were obtained by inverse transformation of the model and assumed to obey normal distributions. In the following the procedure of uncertainty quantification and transmission is outlined: (1) the updated parameters can be obtained based on two-step multiscale model updating method and used as the mean values of the structural parameters; (2) based on the sample data of the measured frequencies and optimization theory, the parameters can also be calculated, and the standard deviation of these calculated values were used as the standard deviation of the structural parameters; (3) assuming that the design parameters obey normal distributions, the random numbers which conform to the probability distributions of structural parameters can be obtained based on the Monte Carlo simulation method; then the frequencies can be calculated based on the response surface model, and the probability distributions of calculated values can be compared with the measured values, as listed in Tables12 and 13.Table 12
Distribution characteristics of uncertainty parameters for Guanhe Bridge.
Parameters
Mean value
Standard deviation
Variable coefficient (%)
Distribution pattern
N
1.040
0.171
16.44
Normal distribution
K
1(×108 N/m)
17.540
0.004
0.03
Normal distribution
K
2(×109 N/m)
6.470
3.520
54.41
Normal distribution
K
3(×106 N/m)
10.770
0.484
4.49
Normal distributionTable 13
The statistics of the frequencies in different conditions.
Case
Mode
Vertical frequencies
Transverse frequencies
Longitudinal frequencies
1
2
3
1
2
1
1
Mean value (Hz)
0.37036
0.50987
0.78325
0.38117
0.52315
0.87115
Standard deviation (Hz)
0.00825
0.01464
0.00742
0.00153
0.00452
0.00156
Variable coefficient (%)
2.22702
2.87191
0.94695
0.40140
0.86400
0.17907
2
Mean value (Hz)
0.37012
0.50526
0.78112
0.38225
0.52465
0.87119
Standard deviation (Hz)
0.00356
0.00421
0.00125
0.00435
0.00854
0.00985
Variable coefficient (%)
0.96185
0.83323
0.16003
1.13748
1.62832
1.13064
3
Mean value (Hz)
0.37052
0.50891
0.78411
0.38115
0.52122
0.87335
Standard deviation (Hz)
0.00289
0.00394
0.00156
0.00098
0.00115
0.02740
Variable coefficient (%)
0.77998
0.77420
0.19895
0.25712
0.22064
3.13689It can be concluded from Table13 that (1) the transverse and longitudinal restriction effect of the rubber supports have a great influence on the transverse and longitudinal frequencies, respectively, and the variable coefficient is more than 2%, (2) the elasticity modulus of the concrete bridge deck has a larger influence on the vertical frequencies, for the variable coefficient is between 0.9% and 2.9%, and (3) the mean value of each frequency has been less affected by the uncertainty of each parameter.


## 5.3. Model/Test Correlation Analysis

To analyze the correlation between model and test results, overlap ratio criterion (ORC) can be used. As shown in (8), J
p is an overlap ratio index of PDF (probability density of frequency), and it can be calculated by comparing calculated frequencies of multiscale FEM with measured frequencies of ambient vibration testing. If J
p
=
1, it shows that multiscale FEM is completely consistent with the bridge
(8)
J
p
=
PDF
test
∩
PDF
.The overlap ratio index (Figure10) has been calculated, as shown in Table 14, and Table 15 shows the overlap ratio index of each modal frequency without the average absolute value of relative errors between calculated and measured values. It can be concluded that the average absolute value of relative errors between calculated and measured values have an influence on the overlap ratio index, without the average absolute value of relative errors, the overlap ratio index of each modal frequencies are over 80%. Thus, the updated multiscale FEM can well represent the dynamic characteristics of Guanhe Bridge.Table 14
Overlap ratio index of each modal frequency with the average absolute value of relative errors.
Mode
Verticalmodel shape
Transversemodel shape
Longitudinal model shape
1
2
3
1
2
1
J
(
p
)
0.736
0.782
0.821
0.778
0.451
0.573Table 15
Overlap ratio index of each modal frequency without the average absolute value of relative errors.
Mode
Verticalmodel shape
Transversemodel shape
Longitudinal model shape
1
2
3
1
2
1
J
(
p
)
0.896
0.857
0.816
0.805
0.810
0.821The probability distribution of Guanhe Bridge frequency.
(a)
The probability distribution of 1st vertical frequency
(b)
The probability distribution of 2nd vertical frequency
(c)
The probability distribution of 3rd vertical frequency
(d)
The probability distribution of 1st transverse frequency
(e)
The probability distribution of 2nd transverse frequency
(f)
The probability distribution of 1st longitudinal frequency


## 5.4. Model Accuracy Evaluation

Considering the uncertainty of multiscale FEM, the mean values of the calculated frequencies were obtained based on the Monte Carlo simulation and response surface method. It can be seen from Table16 that the relative errors between the calculated and measured frequencies are less than 2% with the uncertainty factors fully considered, which indicates that the multiscale FEM can reflect current conditions of the real bridge; thus it can be used as the basis for bridge health monitoring, damage prognosis (DP), and safety prognosis (SP).Table 16
Comparison of average values of each modal frequency.
Mode
Verticalfrequencies
Transversefrequencies
Longitudinal frequencies
1
2
3
1
2
1
Calculated frequency (Hz)
0.379
0.501
0.778
0.381
0.511
0.861
Measured frequency (Hz)
0.378
0.503
0.765
0.378
0.512
0.86
Relative error (%)
0.26
0.40
1.67
0.79
0.20
0.12


## 6. Conclusions

A novel method termed two-step response surface method for multiscale FEM updating and validation has been developed based on the structural health monitoring system of Guanhe Bridge in this paper. The following conclusions can be drawn from the analysis and the measurements.(1)
The calculated frequencies of the updated FEM based on the two-step response surface method are in good agreement with the measured frequencies, with the maximum error being not larger than 8% and the values of MAC being over 90%. Moreover, the updated parameters of the FEM model still keep their physical signification.
(2)
The results of the multiscale FEM/test correlation analysis indicate that the overlap ratio indexJ
(
p
) of each modal frequency is over 80% without the average absolute value of relative errors, which means that the updated multiscale FEM can reflect well on the dynamic characteristics of Guanhe Bridge.
(3)
Considering the influence of structural parameters uncertainties, the relative errors between the calculated and measured frequencies are less than 2%. It can be concluded that the multiscale FEM after validation can reflect well on the current conditions of the real bridge; thus it can be used as the basis for bridge health monitoring, damage prognosis (DP), and safety prognosis (SP).



---

*Source: 817281-2015-03-02.xml*