## 5 Conclusions  
A study is carried out on wall pier girder bridges. Finite element models of the wall piers are established by the layered shell element and their accuracy is verified. Hysteresis analyses of the wall pier samples are carried out to obtain the drift ratio at each performance limit point. The Bayesian method is used to obtain the probabilistic in-plane performance model of the wall piers. Finite element models of the wall pier girder bridges are further established and the seismic vulnerabilities of wall pier girder bridges and their components are obtained by time-history analysis. The main conclusions are as follows:  
(1) By comparing the finite element simulation methods of the common pier columns and shear wall structures, the layered shell element is selected to carry out finite element analyses of the wall piers, and its accuracy is verified. When compared to other cases, it is found that the finite element analysis of components with small aspect ratio will cause a sharp drop in bearing capacity when the layered shell element is used, but in general the layered shell element can well simulate the different failure modes and mechanical properties of the components.  
(2) The displacement drift ratio at each performance limit point of 256 wall pier samples is obtained by numerical analysis. The Bayesian method is used to estimate and eliminate the unknown parameters in the proposed probabilistic capacity models of the wall piers. It can be seen from the probabilistic capacity models that the displacement drift ratio at each performance state limit point is mainly related to the axial compression ratio, the shear span ratio and the vertical reinforcement ratio. The displacement drift ratios of the components are positively correlated with the axial compression ratio when the components are cracked or yielded, negatively correlated with the axial compression ratio when the bearing capacities reach the peak or the limit state, and the displacement drift ratios of the components at the limit point of each damage state are all positively correlated with the shear span ratio.  
(3) Based on a reference bridge, 120 bridge samples are generated considering the uncertainty of parameters, and the seismic vulnerability model parameters of bridge components and bridge systems are obtained by time-history analysis. The seismic vulnerability analysis results of each component and bridge system show that the damage probabilities of the wall piers are the highest in the minor damage state and the damage probabilities of the bearings are the highest in other states. The wall piers are easy to enter the complete destruction state after they enter the serious damage state due to the decrease of bearing capacity. Generally, the wall pier structures perform well under the action of ground motions, and the bearings have a significant influence on the seismic vulnerability of the bridge system.  
Conceptualization, L.C.; methodology, L.C.; writing--original draft preparation, L.C. and Y.T.; writing--review and editing, L.H. All authors have read and agreed to the published version of the manuscript. This research were funded by National Natural Science Foundation of China (grant number 51308125) and China Postdoctoral Science Foundation (grant number 2014M561855). The authors want to acknowledge the comments of the reviewers, whose comments significantly improved quality and readability of the paper. The reviews from Prof. Enrico Spacone (University G. d'Anmunzio of Chieti-Pescara, Italy) are also gratefully acknowledged. The authors declare no conflict of interest.