# Seismic Response Analysis and Evaluation of Laminated Rubber Bearing Supported Bridge Based on the Artificial Neural Network

Authors: Bingzhe Zhang; Kehai Wang; Guanya Lu; Weizuo Guo
Journal: Shock and Vibration
 (2021)

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## Abstract

Laminated rubber bearings are commonly adopted in small-to-medium span highway bridges in earthquake-prone areas. The accurate establishment of the mechanical model of laminated rubber bearings is one of most critical steps for the bridge seismic response analysis. A new constitutive model of bearing based on the artificial neural network (ANN) technique is established through the static cyclic test of laminated rubber bearings, considering the bearing initial stiffness, friction coefficient, and other parameters such as the bearing sectional area, height, loading velocity, vertical load, and aging time. Combined with the ANN method, the ANN-based bridge seismic demand model is built and applied to the rapid evaluation of the bridge seismic damage. The importance of the bearing affecting design factors in the bridge seismic demands are ranked. The results demonstrated that the dimensions of the bearing and vertical load are the main factors affecting the bearings constitutive model. Based on the partial dependency analysis with the ANN-based bridge seismic demand model, it is concluded that the height of bearing is the key design parameter which affects the bridge seismic response the most. The ANN seismic demands model can fit the complex function relationship between various factors and bridge seismic response with high precision, so as to achieve the rapid evaluation of bridge seismic damage.


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## Body

## 1. Introduction

LRBs (laminated rubber bearings) technique, as a prevailing and efficient seismic mitigation technique, is commonly employed in small-to-medium span bridges constructed in countries having the active seismic zones such as USA, Mexico, Japan, and China. As an important part of the seismic isolation system, LRB can effectively support loads and accommodate motions in different directions including vertical and shear displacements to protect the structure and surrounding infrastructure [1, 2]. Through the engineering investigation after the Wenchuan earthquake which happened in 2008 in Sichuan, China, the occurrence of plastic hinge and plastic damage in simply supported beam bridges with LRBs is much lower than in bridges with fixed bearings [3–5], and similar circumstances have been observed in the Northridge earthquake and the Chi-Chi earthquake as well [6]. An exemplar comparison of earthquake damage on bridges with steel bearing and those with LRB is shown in Figure 1; the piers of Hanwang bridge, which was using steel bearing, experienced serious plastic hinge damages during the Wenchuan earthquake, to reinforce and reuse the bridge which was not economical. Meanwhile, the seismic damage of Huilan bridge included bearings sliding away and shifting, with negligible damage in piers. After the minor repairs and reinforcements, traffic was quickly resumed. It was found that the LRBs play a critical role on the bridge seismic response and damage evaluation. Establishing a reasonable constitutive model of bearing was the prerequisite for analyzing the bridge seismic response accurately. At present, there is no multi-affection LRBs analysis model based on machine learning [7–10], so it is necessary to establish an accurate constitutive model based on the LRBs test datasets, further, to simulate and predict earthquake response of LRBs supported bridges.Figure 1
Typical earthquake damages on bridges supported by different types of bearings. (a) Mianzhu Huilan bridge (LRB). (b) Mianzhu Hanwang bridge (steel bearing).
(a)(b)In recent years, many studies based on the seismic response of bridges with LRBs have been conducted all over the world. The California DOT and Illinois DOT analyzed the key design factors which affect the seismic performance of LRBs supported bridge, and the conclusion revealed that seismic motion intensity, pier height, and bearing type are the keys which affect the isolation system performance [11, 12]. Deng et al. and Billah et al. compared the impact of geometric parameters of the bearing and vertical load on the bridge seismic response by using FEM (finite element method) to model the rubber bearing supported bridge [13, 14]. Mitoulis et al. studied the influence on the characteristics of steel plate on the performance of rubber bearing under the combined effects of shear, rotation, and axial load [15, 16]. Xiang et al. investigated the shear and sliding behaviors of rubber bearing with different vertical loads and concluded that the sliding friction coefficient of the bearing was inversely proportional to the vertical load and was ranged between 0.11 and 0.57 [17]. Wu et al. proposed a trilinear mechanical model based on a series of quasistatic tests of LRBs [18]. Bhuiyan et al. established a speed-dependent mechanical model based on the test data [19]. Compared with the abovementioned studies, there are bearing constitutive models of mechanical experimental and finite element method, but the samples of bearings in other research studies are less in quantity, and the distribution has limitations. And several regression methods have been used in fitting the mechanical behavior of the test of rubber bearing, such as linear regression and exponential regression, but the ANN technique which is the nonparametric model has the ability to simulate the complex constitutive model of the bearing with high accuracy. Although these studies took into account one or several factors on the performance of LRBs, such as vertical load, loading speed, and geometry, the LRBs mechanical model integrating multiple factors simultaneously is not yet established, and the uncertainty analysis of the parameters of bearing has not been evaluated previously.The booming machine learning (ML) technologies have been proven to be a successful classification and prediction tool in some traditional engineering applications. Some successful adoptions of ML in multifactor seismic response analysis of bridges have been published recently. Ghosh et al. and Guo et al. used linear models and ANN models to establish seismic response models for cable-stayed bridges and small-to-medium span bridges [20, 21]. Jia et al. solved a ranking problem of the importance of structure type, bearing type, and other 9 features based on the random forest model using the datasets collected during Wenchuan earthquake damage investigations, and the artificial neural network model is adopted to predict the earthquake damage of bridges [22]. Hwang et al. and Kiani et al. implemented certain algorithms such as the k-nearest neighbor, simple Bayes, random forest, and decision tree to analyze and predict the seismic response, and it was concluded that the dataset is the key to the learning performance, and accuracy of the classification models was better than those of the regression models in seismic response prediction [23, 24]. Figueiredo et al. explored a hybrid approach for damage detection with machine learning techniques [25]. Wang et al. and Alwanas et al. proposed a methodology which employs the machine learning methods to establish the constitutive models of structural member by using the test data [26–29]. The above research studies’ progress shows that the ML can effectively fit the complex functioning relationship in seismic response analysis, but yet, it is not seen to discuss and order the effect of the bearing geometric parameters and nongeometric parameters on the bridge seismic demands by using ANN technique.It is necessary to establish a reasonable constitutive model of the bearing which is the foundation for analysis of the bridge seismic response and evaluating the seismic damage. The constitutive model of LRBs in the Chinese specification only considers the influence of geometry on the initial stiffness of bearing [9, 10] but does not considers other factors such as vertical load, aging time, and loading speed. The ANN method can effectively simulate the mechanical constitutive model of rubber bearing integrating multiple factors with high accuracy and fit the complex functioning relationship in seismic response analysis. A laboratory shear testing program of LRBs is proposed in this study. Based on the experimental results and the ANN method, a constitutive model is developed to accurately fit the bearing mechanical constitution. A large-scale nonlinear seismic analysis is carried out on a typical three-span continuous bridge, and a seismic demands model is established through the ANN to evaluate the seismic damage. In addition, by analyzing the partial dependence relationship of the bearing design parameters and seismic demands model, the order of affecting the bearing design parameters on the bridge seismic response is determined.


## 2. Test Procedures

### 2.1. Specimen Preparation

In this study, the behavior of eight circular laminated rubber bearings with 370 mm diameters, with full-scale bearings commonly used at T-girder small and medium bridges, is investigated. The laminated rubber bearings without thick end plates are covered by rubber material. The structure diagram of bearing is shown in Figure2, and the bearing conformation and material mechanics parameters satisfied the design specifications of highway bridge elastic bearing of China (MOTC) [10].Figure 2
Configuration details of the bearing specimens.The test specimens consisted of two batch of LRBs; the first batch specimens are the new LRBs, and the relevant research conclusions can be found in [18, 30]; the second batch specimens are the batch one specimens after four years of natural atmospheric aging. Load the weight on the bearing during the aging time to ensure that it conforms to the normal load state. Then, the actuator exerted cyclic horizontal displacement to the bottom steel plate of the bearing. The parameters for vertical load, loading velocity, rubber material, and aging time of the testing program form are given in Table 1.Table 1
Testing plan.
Test specimenVertical load (MPa)Loading velocity (mm/s)Rubber materialResearch parametersAging time (years)Y1-8-30-N830Neoprene materialLoading velocity0Y1-8-3-N30Y1-8-45-N450Y1-4-30-N430Vertical load0Y1-6-30-N6300Y2-8-30-N830Natural rubberRubber material and loading velocity0Y2-8-3-N30Y2-8-45-N450Y1-8-30-A830Neoprene materialLoading velocity3.5Y1-8-3-A33.5Y1-8-45-A453.5Y1-4-30-A430Vertical load3.5Y1-6-30-A6303.5Y2-8-30-A830Natural rubberRubber material and loading velocity3.5Y2-8-3-A33.5Y2-8-45-A453.5


### 2.2. Test Procedure

The LRBs quasistatic shear testing was carried out in the Experimental Lab of Hengshui China Railway Construction Engineering Rubber Co., Ltd. The setting up of the experimental device shown in Figure3 included 30,000 kN capacity of the coordinated loading system. The example specimen of bearing that was placed between the top and the bottom steel plates is shown in Figure 3. The steel plate was kept smooth to ensure that sliding response will take place between the bearing and steel pad. There were no constraint components to set in the pad surface for simulating the true state of the bridge bearing placement.Figure 3
Testing setup.First, the dead load was implemented on the bearing specimens through a vertical servo actuator. Second, the dead load is maintained at the given value by the vertical servo actuator automatically adjusted, ignoring the bearing horizontal deformation. Finally, the target shear circular horizontal displacement is applied to the bottom supporting plate by the horizontal executor. The equivalent shear strain (ESS) is adopted as the loading displacement measurement, in which the test program has set a target of 400% ESS as shown in Figure4 [17]. The horizontal and vertical executors operate independently as they are controlled by two separate hydraulic systems. Record the bearing’s horizontal force-replacement curve during the test. The loaded displacements included the bearing shear deformation and the sliding displacement. Each steel plate is marked with a 5 cm × 5 cm white gird and specimen profile on the supporting surface for more accurate measurements of the sliding displacement. The vertical white lines are carried out on the side of the bearing to measure the extent of the warped during the testing.Figure 4
Cycle loading testing protocol.


## 2.1. Specimen Preparation

In this study, the behavior of eight circular laminated rubber bearings with 370 mm diameters, with full-scale bearings commonly used at T-girder small and medium bridges, is investigated. The laminated rubber bearings without thick end plates are covered by rubber material. The structure diagram of bearing is shown in Figure2, and the bearing conformation and material mechanics parameters satisfied the design specifications of highway bridge elastic bearing of China (MOTC) [10].Figure 2
Configuration details of the bearing specimens.The test specimens consisted of two batch of LRBs; the first batch specimens are the new LRBs, and the relevant research conclusions can be found in [18, 30]; the second batch specimens are the batch one specimens after four years of natural atmospheric aging. Load the weight on the bearing during the aging time to ensure that it conforms to the normal load state. Then, the actuator exerted cyclic horizontal displacement to the bottom steel plate of the bearing. The parameters for vertical load, loading velocity, rubber material, and aging time of the testing program form are given in Table 1.Table 1
Testing plan.
Test specimenVertical load (MPa)Loading velocity (mm/s)Rubber materialResearch parametersAging time (years)Y1-8-30-N830Neoprene materialLoading velocity0Y1-8-3-N30Y1-8-45-N450Y1-4-30-N430Vertical load0Y1-6-30-N6300Y2-8-30-N830Natural rubberRubber material and loading velocity0Y2-8-3-N30Y2-8-45-N450Y1-8-30-A830Neoprene materialLoading velocity3.5Y1-8-3-A33.5Y1-8-45-A453.5Y1-4-30-A430Vertical load3.5Y1-6-30-A6303.5Y2-8-30-A830Natural rubberRubber material and loading velocity3.5Y2-8-3-A33.5Y2-8-45-A453.5


## 2.2. Test Procedure

The LRBs quasistatic shear testing was carried out in the Experimental Lab of Hengshui China Railway Construction Engineering Rubber Co., Ltd. The setting up of the experimental device shown in Figure3 included 30,000 kN capacity of the coordinated loading system. The example specimen of bearing that was placed between the top and the bottom steel plates is shown in Figure 3. The steel plate was kept smooth to ensure that sliding response will take place between the bearing and steel pad. There were no constraint components to set in the pad surface for simulating the true state of the bridge bearing placement.Figure 3
Testing setup.First, the dead load was implemented on the bearing specimens through a vertical servo actuator. Second, the dead load is maintained at the given value by the vertical servo actuator automatically adjusted, ignoring the bearing horizontal deformation. Finally, the target shear circular horizontal displacement is applied to the bottom supporting plate by the horizontal executor. The equivalent shear strain (ESS) is adopted as the loading displacement measurement, in which the test program has set a target of 400% ESS as shown in Figure4 [17]. The horizontal and vertical executors operate independently as they are controlled by two separate hydraulic systems. Record the bearing’s horizontal force-replacement curve during the test. The loaded displacements included the bearing shear deformation and the sliding displacement. Each steel plate is marked with a 5 cm × 5 cm white gird and specimen profile on the supporting surface for more accurate measurements of the sliding displacement. The vertical white lines are carried out on the side of the bearing to measure the extent of the warped during the testing.Figure 4
Cycle loading testing protocol.


## 3. Test Results and the LRB Mechanical Model

### 3.1. Bearing Response

As the vertical pressure of laminated rubber bearings is preserved on the target value, the horizontal displacement of the bearings increased gradually. The deformation of the bearing is observed (Figure5). There are three stages during the loading process, which represent different characteristics of the displacement response, and each stage is as follows:Stage I (ESS <75%). The increasing displacement demands of the test are from the pure shear deformation in the elastomer layer rather a sliding response. The force-displacement relationship is basically linear.Stage II (75%< ESS <100%). With the increasing of horizontal actuator displacement loading, the shear deformation of the rubber layer of specimen continued to increase as well. At the same time, the differences from Stage I that slight sliding displacement and warping are observed at the bearing interface. The initial stiffness begins to decrease, and an inflection point of the line is observed from the force-displacement curve. At this stage, the bearing sliding response and elastic shear deformation constitute the bearing recorded displacement by the horizontal actuator.Stage III (ESS >100%). The bearing performed the sliding response almost, and the ratio of shear deformation and warping are no longer rising in this stage. The horizontal force is equal to the friction force during the sliding response approximately. As the number of loading circle increases, a large amount of rubber clastic appears between the steel plate surfaces and bearings. Sliding displacement increases at a high rate, resulting in an increase in the amount of displacement imposed. Satisfactorily, the LRBs did not occur as delamination and instability between the steel plate and the rubber layer during the test, and the mechanical properties are stable even under the seismic demands of long distance.Figure 5
Bearing response in different stages.In this study, the sliding LRB constitutive model is developed with the bilinear elastic-plastic model, which is defined by two parameters: sliding friction coefficient (μ) and initial shear stiffness (Ke). It is supposed that the displacement response behavior of LRB is divided into elastic response stage and sliding response stage; the first stage is the elastic behavior before sliding (Stage I), corresponding to the first red line of Figure 6, which is characterized by initial shear stiffness (Ke) mainly. When the horizontal loading is larger than the maximum friction force (Fmax) of the bearing (Stage III), the plastic deformation will be developed, corresponding to the horizontal red line in Figure 6. According to Coulomb’s friction criteria, after the sliding resistance is reached, the maximum friction force (Fmax) is characterized by the sliding friction coefficient (μ). The sliding LRB model is computed by the following formula.Figure 6
Specimen Y1-8-45 force-displacement curve.Whenx≤x1,(1)F=Kex.Whenx>x1,(2)F=Fmax=μN,where x1=Fmax/Ke.Based on the experimental results, the sliding friction coefficient (μ) is developed. The sliding friction under different vertical loads is analyzed (Figure 7). When the vertical loads are 4 MPa, 6 MPa, and 8 MPa, the surface friction coefficients of LRB are 0.42, 0.26, and 0.2, respectively. It can be concluded that the vertical load negatively correlates to the friction coefficient. The reason for this change in friction coefficient is with the sliding of bearing generates heat, and the rubber of the contact surface melts and adheres to the steel plate, which changes the friction coefficient of the steel plate surface. Therefore, investigating the friction sliding characteristics and laws of the LRB under vertical earthquake and the influence on the structural dynamic response is necessary.Figure 7
Observed friction (μ) versus ESS.


### 3.2. Bearing Analytical Model Based on the ANN

In this study, 140 sets of LRB force-displacement curves data of foreign scholars were collected [31–34]. The parameters of bearing such as the sectional area (A), height (H), loading velocity (V), vertical load (L), and aging time (T) were counted, which maintain the range of the 30 m span highway bridge bearings in Table 2. The peak point of displacement (blue dot in Figure 6) in each cyclic curve is found by Matlab software; the initial stiffness (Ke) and coefficient of friction (μ) of the bilinear model are determined. The data preprocessing of the above parameters is normalized to reduce the influence of data units.Table 2
The distribution of bearing parameters.
FactorRange maximumRange minimumA (m)0.360.07065H (m)0.2040.048V (m/s)0.8350.003L (MN/m)101.5T (year)200Controlling the number of input variables optimizes the structure of the neural network, and more input variables may be redundant. So, it is necessary to analyze the correlation of the research data and determine the key design parameters that affect the bearing mechanics performance.The parameters ofA, H, V, L, and T are used as the input variables, and the Ke and μ are used as the output variables. First, the correlation analysis is performed on the input variables, and the correlation matrix is shown in Figure 8. The results show that the sectional area and height have a certain correlation with loading velocity, and the correlation coefficients are 0.85 and 0.47, respectively, so the loading velocity cannot be used as an independent variable. In addition, the correlation analysis is performed of input and output variables too. The result shows that the initial stiffness is obviously related to the sectional area and height, and the correlation coefficients are 0.45 and −0.56, respectively. The friction coefficient is obviously related to the vertical load, the correlation coefficient is −0.83, but the correlation with other parameters is not strong. The conclusion conforms to Huang’s research results [35, 36]. Therefore, sectional area (A), height (H), and vertical load (L) are determined as independent input variables.Figure 8
Correlation matrix for bearing parameters with (a)Ke and (b) μ.
(a)(b)There are numerous factors affecting the seismic response of bearing and containing a large number of nonlinear processes. It is difficult to simulate the relationship accurately by the linear regression model, but it can be achieved by the nonparameter model such as the ANN with high precision. The ANN calculates the gradient termgj of the output layer neurons and the gradient term eh of the hidden layer neurons, with the average variance Ek minimum of the network at (xk, yk) as the target.(3)Ek=12∑j=1ly^jk−yjk2,gj=−∂Ek∂y^jk⋅∂y^jk∂βj,eh=−∂Ek∂bh⋅∂bh∂αh.In this study, the fitting of the LRB model is carried out by the ANN, and the control group is multivariable linear regression (MLR). Based on the Python computer programming language, the LRB mechanical model was developed by the multilayer perceptron regressor model and the linear regressor model, respectively. The ANN model consisted of several layers: an input layer and two hidden layers with 50 neurons in one hidden layer to ensure smooth decision boundary and an output layer. The inlayer and interlayer connections are implemented through activation functions. The fitted goodnessR2 is used as the evaluation index for the performance of the model. As the ANN is a nonparameter model, the LRB common types are extracted from the ANN model presented in Table 3.Table 3
LRB model by the ANN.
A (m2)TypeH (mm)Ke (MN/m)μP1P2P3P4P50.07065d300520.9900.2320.2010.1710.1410.108630.8940.2270.1950.1650.1350.102740.7970.2210.1900.1590.1300.096850.7000.2150.1840.1540.1240.0900.09300 × 300521.1210.2340.2030.1730.1420.110631.0240.2290.1970.1680.1360.105740.9270.2230.1920.1620.1310.099850.8300.2180.1860.1560.1250.0940.0962d350631.0660.2350.2040.1740.1420.110740.9690.2290.1980.1680.1370.105850.8720.2240.1920.1630.1310.100960.7750.2190.1870.1580.1250.0940.1225350 × 350631.2420.2370.2050.1740.1430.111741.1460.2310.1990.1680.1370.105851.0490.2250.1930.1620.1310.099960.9520.2190.1870.1560.1250.0930.1256d400541.3420.2370.2060.1740.1430.112691.2100.2310.2000.1680.1370.106841.0790.2250.1940.1620.1310.100990.9470.2190.1880.1560.1250.0940.16400 × 400541.5740.2450.2130.1820.1510.119691.4420.2370.2050.1740.1420.111841.3100.2280.1970.1660.1340.103991.1780.2200.1890.1570.1260.0950.1589d450691.4350.2360.2050.1740.1420.111841.3030.2280.1970.1660.1340.103991.1710.2200.1890.1570.1260.0951141.0390.2120.1800.1490.1180.0860.2025450 × 450691.7280.2400.2090.1770.1460.114841.5960.2320.2000.1690.1380.106991.4650.2230.1920.1610.1290.0981141.3330.2150.1840.1530.1210.090Note:Pi, vertical load. P1 = 5 MPa; P2 = 6 MPa; P3 = 7 MPa; P4 = 8 MPa; P5 = 9 MPa.The control group LRB constitutive model by MLR is expressed as(4)Ke=0.297A−0.623H−0.031L+0.344,μ=−0.056A−0.106H−0.602L+0.707.The results demonstrate thatR2 of friction coefficient of the LRB model by the ANN method is 0.902, and R2 of initial stiffness is 0.852. The R2 of friction coefficient of the LRB model by the MLR method is 0.86, and R2 of initial stiffness is 0.82. It can be concluded that the ANN method is better than the MLR method in fitting the LRB model.By comparing the initial stiffness of the bearing, theKe value of the ANN model is lower than the recommended value of the MOTC [9], such as the GYZ d400 × 54 model, and the value of the ANN model is 1342 kN m, which is much smaller than the recommended value of the MOTC; it is 3770 kN m. The maximum friction coefficient of the ANN model is 0.245, which decreases with the increase of vertical load, and the minimum is 0.09, which is different from the MOTC value of 0.2.


## 3.1. Bearing Response

As the vertical pressure of laminated rubber bearings is preserved on the target value, the horizontal displacement of the bearings increased gradually. The deformation of the bearing is observed (Figure5). There are three stages during the loading process, which represent different characteristics of the displacement response, and each stage is as follows:Stage I (ESS <75%). The increasing displacement demands of the test are from the pure shear deformation in the elastomer layer rather a sliding response. The force-displacement relationship is basically linear.Stage II (75%< ESS <100%). With the increasing of horizontal actuator displacement loading, the shear deformation of the rubber layer of specimen continued to increase as well. At the same time, the differences from Stage I that slight sliding displacement and warping are observed at the bearing interface. The initial stiffness begins to decrease, and an inflection point of the line is observed from the force-displacement curve. At this stage, the bearing sliding response and elastic shear deformation constitute the bearing recorded displacement by the horizontal actuator.Stage III (ESS >100%). The bearing performed the sliding response almost, and the ratio of shear deformation and warping are no longer rising in this stage. The horizontal force is equal to the friction force during the sliding response approximately. As the number of loading circle increases, a large amount of rubber clastic appears between the steel plate surfaces and bearings. Sliding displacement increases at a high rate, resulting in an increase in the amount of displacement imposed. Satisfactorily, the LRBs did not occur as delamination and instability between the steel plate and the rubber layer during the test, and the mechanical properties are stable even under the seismic demands of long distance.Figure 5
Bearing response in different stages.In this study, the sliding LRB constitutive model is developed with the bilinear elastic-plastic model, which is defined by two parameters: sliding friction coefficient (μ) and initial shear stiffness (Ke). It is supposed that the displacement response behavior of LRB is divided into elastic response stage and sliding response stage; the first stage is the elastic behavior before sliding (Stage I), corresponding to the first red line of Figure 6, which is characterized by initial shear stiffness (Ke) mainly. When the horizontal loading is larger than the maximum friction force (Fmax) of the bearing (Stage III), the plastic deformation will be developed, corresponding to the horizontal red line in Figure 6. According to Coulomb’s friction criteria, after the sliding resistance is reached, the maximum friction force (Fmax) is characterized by the sliding friction coefficient (μ). The sliding LRB model is computed by the following formula.Figure 6
Specimen Y1-8-45 force-displacement curve.Whenx≤x1,(1)F=Kex.Whenx>x1,(2)F=Fmax=μN,where x1=Fmax/Ke.Based on the experimental results, the sliding friction coefficient (μ) is developed. The sliding friction under different vertical loads is analyzed (Figure 7). When the vertical loads are 4 MPa, 6 MPa, and 8 MPa, the surface friction coefficients of LRB are 0.42, 0.26, and 0.2, respectively. It can be concluded that the vertical load negatively correlates to the friction coefficient. The reason for this change in friction coefficient is with the sliding of bearing generates heat, and the rubber of the contact surface melts and adheres to the steel plate, which changes the friction coefficient of the steel plate surface. Therefore, investigating the friction sliding characteristics and laws of the LRB under vertical earthquake and the influence on the structural dynamic response is necessary.Figure 7
Observed friction (μ) versus ESS.


## 3.2. Bearing Analytical Model Based on the ANN

In this study, 140 sets of LRB force-displacement curves data of foreign scholars were collected [31–34]. The parameters of bearing such as the sectional area (A), height (H), loading velocity (V), vertical load (L), and aging time (T) were counted, which maintain the range of the 30 m span highway bridge bearings in Table 2. The peak point of displacement (blue dot in Figure 6) in each cyclic curve is found by Matlab software; the initial stiffness (Ke) and coefficient of friction (μ) of the bilinear model are determined. The data preprocessing of the above parameters is normalized to reduce the influence of data units.Table 2
The distribution of bearing parameters.
FactorRange maximumRange minimumA (m)0.360.07065H (m)0.2040.048V (m/s)0.8350.003L (MN/m)101.5T (year)200Controlling the number of input variables optimizes the structure of the neural network, and more input variables may be redundant. So, it is necessary to analyze the correlation of the research data and determine the key design parameters that affect the bearing mechanics performance.The parameters ofA, H, V, L, and T are used as the input variables, and the Ke and μ are used as the output variables. First, the correlation analysis is performed on the input variables, and the correlation matrix is shown in Figure 8. The results show that the sectional area and height have a certain correlation with loading velocity, and the correlation coefficients are 0.85 and 0.47, respectively, so the loading velocity cannot be used as an independent variable. In addition, the correlation analysis is performed of input and output variables too. The result shows that the initial stiffness is obviously related to the sectional area and height, and the correlation coefficients are 0.45 and −0.56, respectively. The friction coefficient is obviously related to the vertical load, the correlation coefficient is −0.83, but the correlation with other parameters is not strong. The conclusion conforms to Huang’s research results [35, 36]. Therefore, sectional area (A), height (H), and vertical load (L) are determined as independent input variables.Figure 8
Correlation matrix for bearing parameters with (a)Ke and (b) μ.
(a)(b)There are numerous factors affecting the seismic response of bearing and containing a large number of nonlinear processes. It is difficult to simulate the relationship accurately by the linear regression model, but it can be achieved by the nonparameter model such as the ANN with high precision. The ANN calculates the gradient termgj of the output layer neurons and the gradient term eh of the hidden layer neurons, with the average variance Ek minimum of the network at (xk, yk) as the target.(3)Ek=12∑j=1ly^jk−yjk2,gj=−∂Ek∂y^jk⋅∂y^jk∂βj,eh=−∂Ek∂bh⋅∂bh∂αh.In this study, the fitting of the LRB model is carried out by the ANN, and the control group is multivariable linear regression (MLR). Based on the Python computer programming language, the LRB mechanical model was developed by the multilayer perceptron regressor model and the linear regressor model, respectively. The ANN model consisted of several layers: an input layer and two hidden layers with 50 neurons in one hidden layer to ensure smooth decision boundary and an output layer. The inlayer and interlayer connections are implemented through activation functions. The fitted goodnessR2 is used as the evaluation index for the performance of the model. As the ANN is a nonparameter model, the LRB common types are extracted from the ANN model presented in Table 3.Table 3
LRB model by the ANN.
A (m2)TypeH (mm)Ke (MN/m)μP1P2P3P4P50.07065d300520.9900.2320.2010.1710.1410.108630.8940.2270.1950.1650.1350.102740.7970.2210.1900.1590.1300.096850.7000.2150.1840.1540.1240.0900.09300 × 300521.1210.2340.2030.1730.1420.110631.0240.2290.1970.1680.1360.105740.9270.2230.1920.1620.1310.099850.8300.2180.1860.1560.1250.0940.0962d350631.0660.2350.2040.1740.1420.110740.9690.2290.1980.1680.1370.105850.8720.2240.1920.1630.1310.100960.7750.2190.1870.1580.1250.0940.1225350 × 350631.2420.2370.2050.1740.1430.111741.1460.2310.1990.1680.1370.105851.0490.2250.1930.1620.1310.099960.9520.2190.1870.1560.1250.0930.1256d400541.3420.2370.2060.1740.1430.112691.2100.2310.2000.1680.1370.106841.0790.2250.1940.1620.1310.100990.9470.2190.1880.1560.1250.0940.16400 × 400541.5740.2450.2130.1820.1510.119691.4420.2370.2050.1740.1420.111841.3100.2280.1970.1660.1340.103991.1780.2200.1890.1570.1260.0950.1589d450691.4350.2360.2050.1740.1420.111841.3030.2280.1970.1660.1340.103991.1710.2200.1890.1570.1260.0951141.0390.2120.1800.1490.1180.0860.2025450 × 450691.7280.2400.2090.1770.1460.114841.5960.2320.2000.1690.1380.106991.4650.2230.1920.1610.1290.0981141.3330.2150.1840.1530.1210.090Note:Pi, vertical load. P1 = 5 MPa; P2 = 6 MPa; P3 = 7 MPa; P4 = 8 MPa; P5 = 9 MPa.The control group LRB constitutive model by MLR is expressed as(4)Ke=0.297A−0.623H−0.031L+0.344,μ=−0.056A−0.106H−0.602L+0.707.The results demonstrate thatR2 of friction coefficient of the LRB model by the ANN method is 0.902, and R2 of initial stiffness is 0.852. The R2 of friction coefficient of the LRB model by the MLR method is 0.86, and R2 of initial stiffness is 0.82. It can be concluded that the ANN method is better than the MLR method in fitting the LRB model.By comparing the initial stiffness of the bearing, theKe value of the ANN model is lower than the recommended value of the MOTC [9], such as the GYZ d400 × 54 model, and the value of the ANN model is 1342 kN m, which is much smaller than the recommended value of the MOTC; it is 3770 kN m. The maximum friction coefficient of the ANN model is 0.245, which decreases with the increase of vertical load, and the minimum is 0.09, which is different from the MOTC value of 0.2.


## 4. Seismic Response of LRB Bridge

### 4.1. Bridge Finite Element Modeling

A typical example was proposed to study and compare the effects of different LRB models on the seismic response demands of highway bridges. The typical prestressed concrete continuous bridge was located in the near-fault seismic area of Wenchuan province of China. The span was 90 m in total; one single span was 30 m long, consisted of 1.7 m high five T-girders with a total width of 10 m. The substructure was composed of a 10 m high double column constructed of C30 concrete with a circular cross-section of 1.5 m in diameter, as shown in Figure9. The piers were reinforced longitudinal with HRB 400 rebar with a diameter of 28 mm, and the reinforcement rate was 1.5%. Circle steel bars with a diameter of 10 mm were used as hooping, each steel bar was spaced 100 mm apart, and the thickness of the covered layer of concrete was 50 mm. The girder was supported by one bearing at the bent top, where the type was GJZ400 × 400 × 69. The bearing in this model was subjected to the superstructure gravity loading, maintaining an average pressure of 5 MPa. For the regular service demands, five LRBs were placed in each span. The concrete shear keys were positioned to limit the horizontal displacement of the girder, in which the gap between the beam and shear keys was 0.05 m.Figure 9
Finite element model of prototype bridge.An integrated nonlinear finite element model was established in OpenSees software, which was considered as the diverse material performance. In this model, seismic masses of the structure are assumed to be lumped among the components. The beam, pier cap, and diaphragms are simulated by linear elastic beam elements for they will not be damaged in the earthquake. However, for the small and medium span bridges, the top and bottom of the piers often appeared as plastic hinge deformation, and nonlinear fiber elements were used for simulation. The circular pier section was divided into three parts, including reinforcement steel fibers, confined concrete fibers, and nonconstrained concrete fibers. The concrete area was simulated by the Concrete01 material, which was based on the Kent–Scott–Park model [37], and the reinforced fibers were simulated by the Steel02 material accounting for the Bauschinger effect [38]. The sliding behavior of LRB was similar to the mechanical characteristic of Steel02 in OpenSees. The constitutive model of the LRB component was defined by the initial shear stiffness and the friction coefficient, which specifies the behavior of sliding on the steel plate.The bridge seismic response may be influenced by the pounding effects of girders and shear keys, especially the high global motion intensity. In this study, a shear keys model consisting of two hysteretic elements was developed in OpenSees, which was taken into the contribution of concrete and steel. The pounding effect was matched by the connection of the gap element and shear keys, and the parameters of the model were determined by the material and geometric dimensions [39]. The interaction between soil and foundation was ignored in the whole bridge model, and fixed constraints on the column foundation were imposed.


### 4.2. Ground Motion Selection and Bearing Parameters Distribution

Vertical earthquake motion will increase the vertical load of the bearing and thus affect the coefficient friction. In this study, vertical earthquake intensity is selected as the index of the vertical load on the bearing, and the vertical seismic load of the bearing is assumed to be the maximum force response value under vertical earthquake movement. The 1994 Northridge earthquake is obtained from the PEER Ground Motion Database because it is the characteristic of near-fault ground motions. Table4 provides the details of these two direction ground motions. The peak ground acceleration (PGA) is used as an indicator of seismic intensity measurement (IM) [9]. The PGA of the horizontal ground motions is scaled from 0.2 g to 0.8 g in increments of 0.1 g and input in the transverse direction. The ratio of the horizontal component to the vertical component is 157%, adjusted vertical ground motions at the same scale to ensure the integrity of the near-fault ground motion characteristics [40]. In addition, the setting of PGA-vertical to 0 adds to simulate none of the vertical components.Table 4
Ground motion.
EarthquakeYearStationMagnitudePGA-horizontal (g)PGA-vertical (g)Rjb (km)Rrup (km)“Northridge-01”1994“Sylmar—OVM FF”6.690.8430.5361.745.3According to the normal used loading requirements and the design demand of the highway bridge, the basic design parameter is generated by possible values of the Latin hypercube sampling (LHS) simulation probability distribution function. The LHS is a hierarchical random process that provides an effective method for sampling from variable distribution [41]. In the independent variables X1, X2,…, Xk, the LHS established a dataset of 2916 bearings design parameters with a 95% guarantee rate, by dividing each variable into N equal probability intervals and obtaining each variable paired with other variables randomly. The bearing parameters are divided into two categories to clearly express the effect of the bearing design parameters on the bridge seismic response. The sectional area (A) and height (H) are divided into the bearing geometry parameters; the earthquake vertical component (PGA-V) and the bearing number (B), which is bearing number of one span, are divided into the bearing nongeometric parameters. The ANN bearing models are introduced in Section 3.2 of getting the initial stiffness and coefficient of friction. Table 5 provides the distribution of 2916 bearings design parameters.Table 5
Bearing parameters distribution.
FactorRange maximumRange minimumA0.36 m20.1225 m2H0.15 m0.054 mB52PGA-V00.508 gPGA-horizontal (PGA-H)0.2 g0.8 g


### 4.3. Comparison between Different Bearing Models

Three LRB analysis models were proposed to compare the effects of different bearing models on the seismic response of bridges, so as to determine the bearing key parameters that affect the seismic response of the small-to-medium spanning highway bridges. The models were as follows:Model 1. An LRB analysis model based on the ANN was established, considering the influence of vertical earthquake component on the coefficient of frictionModel 2. An LRB analysis model based on MLR was established, considering the influence of vertical earthquake component on the coefficient of frictionModel 3. An LRB analysis model based on MOTC was established, regardless of the influence of vertical seismic motion on the friction coefficient, and the friction coefficient was a fixed value of 0.2.Considered that the seismic damages such as excessive displacement of bearing and plastic hinges in the pier were of the higher level of harmfulness, the displacement of the pier top in first span was selected as the pier seismic response indicator, and the bearing ESS was selected as the bearing response indicator, as shown in Figure10.Figure 10
Comparison of (a) bearing equivalent shear strain and (b) pier displacement response using various bearing models.
(a)(b)Figure10 describes the peak displacements of the bearing and pier under ground motion of three models. For the different model, considering that vertical seismic motion will greatly affect the seismic response of bridges. As the PGA increases, the displacement response of bearing and pier gradually increases. The displacement responses of bearing of the ANN model and MLR model are larger than the MOTC model; since the characteristics of the coefficient of friction with vertical load changes are accounted, the bearings are more prone to sliding which increases the bridge seismic response; thus, the pounding effect will greatly increase the seismic response of the bridge. For displacement response on the pier top, the result of the MOTC model is smaller than the ANN model and MMLR model when PGA ≤0.6 g. While at PGA >0.6 g, the pier displacement response of the MOTC model is much greater than the ANN model and MLR model, due to the friction of the MOTC model after the bearing sliding is higher than other two models.


### 4.4. Seismic Response Demands Analysis

In the traditional seismic demand analysis, only one single variable is considered each time, and the relative importance of the different bearing properties in the seismic demands model could not be determined. To amend such a deficiency, a seismic demand model using the ANN technique was proposed as depicted in the following steps:Step 1: The input variables were sampled by the Latin hypercube sampling technique across the range of parameters. Then, 2916 statistically significant 3D bridge finite element models with different bearing parameters were built for the bridge nonlinear dynamic response analysis. After the nonlinear dynamic response analysis, the initial seismic demands samples were generated.Step 2: The initial seismic demand samples were divided into a training dataset (75 percent) and a test dataset (25 percent). The training dataset was entered into the ANN regressor model for training.Step 3: The connection rights and thresholds were randomly initialized in the network, calculating the outputy^k of the current sampleStep 4: By comparing the expectedEk and network output, the gradient gj of the output layer and gradient eh of the hidden layer were calculated (forward process)Step 5: The ANN model within the expected error range was generated by continuously adjusting connection rightωhj, υih, and threshold θj, γh in calculation of the multilayer hidden layerStep 6: The test dataset was entered into the ANN bridge seismic demands model to evaluate its performance and accuracyThrough theR2 of the initial sample set and the output of the ANN demands model, the validity of the ANN seismic demand model was verified. The seismic damage criteria of the small-to-medium span highway bridge was introduced [30, 42], which were three levels of noninjury, minor injury, and serious injury, and three tags are defined as shown in Figure 11. The displacement of bearing and pier top are selected as the measurements of the bridge seismic demands. The ratio of accurate tag allocation number to total quantity is defined as the accuracy of the ANN model, indicating that the model has the capability to determine the bridge damage level accurately.Figure 11
The seismic damage criteria of the bridge.The results show that theR2 of the bearing parameter and pier seismic demands in the ANN model was 0.985, and the R2 of the bearing seismic demands is 0.978, which shows that the ANN method can predict the seismic demands with high precision. Figure 12 shows the comparison of damage distribution obtained by the ANN model results and those obtained from nonlinear dynamic analysis (NDA), with 98% accuracy of tag allocation (98/100), 2 errors in 46 green labels, and 2 errors in 34 red labels.Figure 12
Damage distribution of the test dataset. (a) NDA results. (b) ANN model results.
(a)(b)The partial dependence plots show the relationship between the ANN seismic demand model and bearing features of attention, downsizing other features to express the relative importance of each feature. It is appropriate to select a small size of feature datasets, generally choosed from the most important features. Thus, they are generally chosen among the most important features. Figures13 and 14 show the dependences between the ANN seismic demand model and bearing geometry parameters, bearing nongeometry parameters.Figure 13
The partial dependence of sectional area (A) and bearing height (H) with (a) pier’s top displacement and (b) bearing ESS.
(a)(b)Figure 14
The partial dependence of vertical PGA (PGA-V) and bearing number (B) with (a) pier top displacement and (b) bearing ESS.
(a)(b)From the Figure13, it can be concluded that the original values of partial dependency function with bearing height to the seismic demands of pier and bearing are 22.59–28.32 and 1.42–2.89, which are greater than the values of the bearing area for 22.35–25.47 and 1.81–2.17. With the bearing’s cross-sectional area (A) increased, the pier’s seismic demands increased correspondingly, as shown in Figure 13(a). This observation is reasonable because the increase of the cross-sectional area leads to the rise of the bearing stiffness, leading to the enhancement of the overall seismic response of the bridge. The bearing height (H) shows a significant negative correlation with the pier’s seismic demands and bearing ESS. One explanation is that, on the one hand, because the increase of bearing height results in the decrease of the bearing stiffness, and it leads to the loss of the overall seismic response of the bridge, consequently reducing the pier’ seismic demands. On the other hand, bearing ESS is inversely proportional to the bearing height, and the downward trend in Figure 13(b) confirms this correlation. Therefore, raising the height of bearing within an appropriate range can not only reduce the pier’ seismic demands but also decrease the bearing ESS. The discovery of this phenomenon can provide a new thought for bridge seismic design.Figure14 shows the partial dependence relationship of seismic demands with the bearing’s nongeometry parameters. The vertical seismic motion (PGA-V) is positively related to the pier’ seismic demands and negatively associated with the bearing seismic demands. As the vertical seismic motion (PGA-V) increases, the bearing’s friction coefficient reduces, and the bearing is more prone to sliding, so increasing indirectly the risk of hit between the beam and shear key, which increases the pier seismic demands in turn, the difference is about 11%. However, the vertical seismic motion (PGA-V) changing had less effect on the bearing seismic demands. Another interesting observation is that increasing the bearing number (B) will increase the pier seismic demands but decrease the bearing ESS. The reason is that the increasing the number of bearing (B) may change the stiffness distribution of the whole bridge, higher constraints in the transverse direction in a way, resulting in the higher seismic demands of the bridge. However, the seismic force shared by each bearing is reduced, leading to the decrease of bearing ESS. It is different from the conclusion of the traditional design method for bearing bridges.By comparing the effects of vertical seismic strength (PGA-V), bearing number (B), sectional area (A), and bearing height (H) on bridge seismic demands, the factors sorted based on the value of the partial dependency function of pier seismic demands are as follows: H > A > PGA-V > B, and the order of impact on the bearing ESS is H > B > A > PGA-V. It can be concluded that the bearing height (H) is the most significant factor on the seismic demands of this type of bridge.


## 4.1. Bridge Finite Element Modeling

A typical example was proposed to study and compare the effects of different LRB models on the seismic response demands of highway bridges. The typical prestressed concrete continuous bridge was located in the near-fault seismic area of Wenchuan province of China. The span was 90 m in total; one single span was 30 m long, consisted of 1.7 m high five T-girders with a total width of 10 m. The substructure was composed of a 10 m high double column constructed of C30 concrete with a circular cross-section of 1.5 m in diameter, as shown in Figure9. The piers were reinforced longitudinal with HRB 400 rebar with a diameter of 28 mm, and the reinforcement rate was 1.5%. Circle steel bars with a diameter of 10 mm were used as hooping, each steel bar was spaced 100 mm apart, and the thickness of the covered layer of concrete was 50 mm. The girder was supported by one bearing at the bent top, where the type was GJZ400 × 400 × 69. The bearing in this model was subjected to the superstructure gravity loading, maintaining an average pressure of 5 MPa. For the regular service demands, five LRBs were placed in each span. The concrete shear keys were positioned to limit the horizontal displacement of the girder, in which the gap between the beam and shear keys was 0.05 m.Figure 9
Finite element model of prototype bridge.An integrated nonlinear finite element model was established in OpenSees software, which was considered as the diverse material performance. In this model, seismic masses of the structure are assumed to be lumped among the components. The beam, pier cap, and diaphragms are simulated by linear elastic beam elements for they will not be damaged in the earthquake. However, for the small and medium span bridges, the top and bottom of the piers often appeared as plastic hinge deformation, and nonlinear fiber elements were used for simulation. The circular pier section was divided into three parts, including reinforcement steel fibers, confined concrete fibers, and nonconstrained concrete fibers. The concrete area was simulated by the Concrete01 material, which was based on the Kent–Scott–Park model [37], and the reinforced fibers were simulated by the Steel02 material accounting for the Bauschinger effect [38]. The sliding behavior of LRB was similar to the mechanical characteristic of Steel02 in OpenSees. The constitutive model of the LRB component was defined by the initial shear stiffness and the friction coefficient, which specifies the behavior of sliding on the steel plate.The bridge seismic response may be influenced by the pounding effects of girders and shear keys, especially the high global motion intensity. In this study, a shear keys model consisting of two hysteretic elements was developed in OpenSees, which was taken into the contribution of concrete and steel. The pounding effect was matched by the connection of the gap element and shear keys, and the parameters of the model were determined by the material and geometric dimensions [39]. The interaction between soil and foundation was ignored in the whole bridge model, and fixed constraints on the column foundation were imposed.


## 4.2. Ground Motion Selection and Bearing Parameters Distribution

Vertical earthquake motion will increase the vertical load of the bearing and thus affect the coefficient friction. In this study, vertical earthquake intensity is selected as the index of the vertical load on the bearing, and the vertical seismic load of the bearing is assumed to be the maximum force response value under vertical earthquake movement. The 1994 Northridge earthquake is obtained from the PEER Ground Motion Database because it is the characteristic of near-fault ground motions. Table4 provides the details of these two direction ground motions. The peak ground acceleration (PGA) is used as an indicator of seismic intensity measurement (IM) [9]. The PGA of the horizontal ground motions is scaled from 0.2 g to 0.8 g in increments of 0.1 g and input in the transverse direction. The ratio of the horizontal component to the vertical component is 157%, adjusted vertical ground motions at the same scale to ensure the integrity of the near-fault ground motion characteristics [40]. In addition, the setting of PGA-vertical to 0 adds to simulate none of the vertical components.Table 4
Ground motion.
EarthquakeYearStationMagnitudePGA-horizontal (g)PGA-vertical (g)Rjb (km)Rrup (km)“Northridge-01”1994“Sylmar—OVM FF”6.690.8430.5361.745.3According to the normal used loading requirements and the design demand of the highway bridge, the basic design parameter is generated by possible values of the Latin hypercube sampling (LHS) simulation probability distribution function. The LHS is a hierarchical random process that provides an effective method for sampling from variable distribution [41]. In the independent variables X1, X2,…, Xk, the LHS established a dataset of 2916 bearings design parameters with a 95% guarantee rate, by dividing each variable into N equal probability intervals and obtaining each variable paired with other variables randomly. The bearing parameters are divided into two categories to clearly express the effect of the bearing design parameters on the bridge seismic response. The sectional area (A) and height (H) are divided into the bearing geometry parameters; the earthquake vertical component (PGA-V) and the bearing number (B), which is bearing number of one span, are divided into the bearing nongeometric parameters. The ANN bearing models are introduced in Section 3.2 of getting the initial stiffness and coefficient of friction. Table 5 provides the distribution of 2916 bearings design parameters.Table 5
Bearing parameters distribution.
FactorRange maximumRange minimumA0.36 m20.1225 m2H0.15 m0.054 mB52PGA-V00.508 gPGA-horizontal (PGA-H)0.2 g0.8 g


## 4.3. Comparison between Different Bearing Models

Three LRB analysis models were proposed to compare the effects of different bearing models on the seismic response of bridges, so as to determine the bearing key parameters that affect the seismic response of the small-to-medium spanning highway bridges. The models were as follows:Model 1. An LRB analysis model based on the ANN was established, considering the influence of vertical earthquake component on the coefficient of frictionModel 2. An LRB analysis model based on MLR was established, considering the influence of vertical earthquake component on the coefficient of frictionModel 3. An LRB analysis model based on MOTC was established, regardless of the influence of vertical seismic motion on the friction coefficient, and the friction coefficient was a fixed value of 0.2.Considered that the seismic damages such as excessive displacement of bearing and plastic hinges in the pier were of the higher level of harmfulness, the displacement of the pier top in first span was selected as the pier seismic response indicator, and the bearing ESS was selected as the bearing response indicator, as shown in Figure10.Figure 10
Comparison of (a) bearing equivalent shear strain and (b) pier displacement response using various bearing models.
(a)(b)Figure10 describes the peak displacements of the bearing and pier under ground motion of three models. For the different model, considering that vertical seismic motion will greatly affect the seismic response of bridges. As the PGA increases, the displacement response of bearing and pier gradually increases. The displacement responses of bearing of the ANN model and MLR model are larger than the MOTC model; since the characteristics of the coefficient of friction with vertical load changes are accounted, the bearings are more prone to sliding which increases the bridge seismic response; thus, the pounding effect will greatly increase the seismic response of the bridge. For displacement response on the pier top, the result of the MOTC model is smaller than the ANN model and MMLR model when PGA ≤0.6 g. While at PGA >0.6 g, the pier displacement response of the MOTC model is much greater than the ANN model and MLR model, due to the friction of the MOTC model after the bearing sliding is higher than other two models.


## 4.4. Seismic Response Demands Analysis

In the traditional seismic demand analysis, only one single variable is considered each time, and the relative importance of the different bearing properties in the seismic demands model could not be determined. To amend such a deficiency, a seismic demand model using the ANN technique was proposed as depicted in the following steps:Step 1: The input variables were sampled by the Latin hypercube sampling technique across the range of parameters. Then, 2916 statistically significant 3D bridge finite element models with different bearing parameters were built for the bridge nonlinear dynamic response analysis. After the nonlinear dynamic response analysis, the initial seismic demands samples were generated.Step 2: The initial seismic demand samples were divided into a training dataset (75 percent) and a test dataset (25 percent). The training dataset was entered into the ANN regressor model for training.Step 3: The connection rights and thresholds were randomly initialized in the network, calculating the outputy^k of the current sampleStep 4: By comparing the expectedEk and network output, the gradient gj of the output layer and gradient eh of the hidden layer were calculated (forward process)Step 5: The ANN model within the expected error range was generated by continuously adjusting connection rightωhj, υih, and threshold θj, γh in calculation of the multilayer hidden layerStep 6: The test dataset was entered into the ANN bridge seismic demands model to evaluate its performance and accuracyThrough theR2 of the initial sample set and the output of the ANN demands model, the validity of the ANN seismic demand model was verified. The seismic damage criteria of the small-to-medium span highway bridge was introduced [30, 42], which were three levels of noninjury, minor injury, and serious injury, and three tags are defined as shown in Figure 11. The displacement of bearing and pier top are selected as the measurements of the bridge seismic demands. The ratio of accurate tag allocation number to total quantity is defined as the accuracy of the ANN model, indicating that the model has the capability to determine the bridge damage level accurately.Figure 11
The seismic damage criteria of the bridge.The results show that theR2 of the bearing parameter and pier seismic demands in the ANN model was 0.985, and the R2 of the bearing seismic demands is 0.978, which shows that the ANN method can predict the seismic demands with high precision. Figure 12 shows the comparison of damage distribution obtained by the ANN model results and those obtained from nonlinear dynamic analysis (NDA), with 98% accuracy of tag allocation (98/100), 2 errors in 46 green labels, and 2 errors in 34 red labels.Figure 12
Damage distribution of the test dataset. (a) NDA results. (b) ANN model results.
(a)(b)The partial dependence plots show the relationship between the ANN seismic demand model and bearing features of attention, downsizing other features to express the relative importance of each feature. It is appropriate to select a small size of feature datasets, generally choosed from the most important features. Thus, they are generally chosen among the most important features. Figures13 and 14 show the dependences between the ANN seismic demand model and bearing geometry parameters, bearing nongeometry parameters.Figure 13
The partial dependence of sectional area (A) and bearing height (H) with (a) pier’s top displacement and (b) bearing ESS.
(a)(b)Figure 14
The partial dependence of vertical PGA (PGA-V) and bearing number (B) with (a) pier top displacement and (b) bearing ESS.
(a)(b)From the Figure13, it can be concluded that the original values of partial dependency function with bearing height to the seismic demands of pier and bearing are 22.59–28.32 and 1.42–2.89, which are greater than the values of the bearing area for 22.35–25.47 and 1.81–2.17. With the bearing’s cross-sectional area (A) increased, the pier’s seismic demands increased correspondingly, as shown in Figure 13(a). This observation is reasonable because the increase of the cross-sectional area leads to the rise of the bearing stiffness, leading to the enhancement of the overall seismic response of the bridge. The bearing height (H) shows a significant negative correlation with the pier’s seismic demands and bearing ESS. One explanation is that, on the one hand, because the increase of bearing height results in the decrease of the bearing stiffness, and it leads to the loss of the overall seismic response of the bridge, consequently reducing the pier’ seismic demands. On the other hand, bearing ESS is inversely proportional to the bearing height, and the downward trend in Figure 13(b) confirms this correlation. Therefore, raising the height of bearing within an appropriate range can not only reduce the pier’ seismic demands but also decrease the bearing ESS. The discovery of this phenomenon can provide a new thought for bridge seismic design.Figure14 shows the partial dependence relationship of seismic demands with the bearing’s nongeometry parameters. The vertical seismic motion (PGA-V) is positively related to the pier’ seismic demands and negatively associated with the bearing seismic demands. As the vertical seismic motion (PGA-V) increases, the bearing’s friction coefficient reduces, and the bearing is more prone to sliding, so increasing indirectly the risk of hit between the beam and shear key, which increases the pier seismic demands in turn, the difference is about 11%. However, the vertical seismic motion (PGA-V) changing had less effect on the bearing seismic demands. Another interesting observation is that increasing the bearing number (B) will increase the pier seismic demands but decrease the bearing ESS. The reason is that the increasing the number of bearing (B) may change the stiffness distribution of the whole bridge, higher constraints in the transverse direction in a way, resulting in the higher seismic demands of the bridge. However, the seismic force shared by each bearing is reduced, leading to the decrease of bearing ESS. It is different from the conclusion of the traditional design method for bearing bridges.By comparing the effects of vertical seismic strength (PGA-V), bearing number (B), sectional area (A), and bearing height (H) on bridge seismic demands, the factors sorted based on the value of the partial dependency function of pier seismic demands are as follows: H > A > PGA-V > B, and the order of impact on the bearing ESS is H > B > A > PGA-V. It can be concluded that the bearing height (H) is the most significant factor on the seismic demands of this type of bridge.


## 5. Conclusions

A series of tests confirmed the rules of laminated rubber bearings shear and sliding response on steel shims. Then, analytical models for the bearing and seismic demands model using the ANN were proposed. During the bridge seismic demands model, the influence of the various bearings design parameters was validated by partial dependency analysis. Moreover, the ANN bridge seismic demand model was applied to the bridge earthquake damage rapid assessment. The conclusions are as follows:(1)
The surface friction coefficients were found to increase with the increases in the vertical load by the performance test of LRB. The test data’s correlation analysis determines that the bearing’s critical characteristics are initial stiffness and friction coefficient. The geometric parameters were a vital factor to the initial stiffness of the bearing, and the friction coefficient is negatively related to the vertical load, which is different from the specified value of 0.2 of the coefficients of friction in Chinese specifications.(2)
By comparing the impact bearing design parameters to the seismic demands model, the order is sorted based on the value of the partial dependency function of pier seismic demands:H > A > PGA-V > B, and the order of bearing factors to the bearing seismic demands is H > B > A > PGA-V. The height of bearing is the most significant factor on the seismic demand model, which provides a new focus for the bridge seismic design.(3)
The seismic response of bridges varies greatly under different bearing models. The bearing response in the ANN model is larger than the MOTC model since the characteristics of the coefficient of friction with vertical load changes are accounted for. For the pier response, the result of the MOTC model is smaller than the ANN model when PGA ≤0.6 g, while PGA >0.6 g, the result of the MOTC model is far greater than the ANN model.(4)
The ANN method could be applied in the estimate of the seismic demand model with great performance after training. The results show that theR2 of the bearing parameter and pier seismic demand in the ANN is 0.985, and the R2 of the bearing seismic demand is 0.978 and with 98% accuracy of tag allocation, which means that the ANN method can complete the damage evaluation of seismic demands with high precision.



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*Source: 5566874-2021-04-26.xml*