# Advancing Bridge Time-dependent Reliability Analysis Using Probability Density Evolution Theory

### Significance

The dynamic reliability of existing bridges has become a significant concern in the field of bridge engineering and structural health monitoring. Understanding the dynamic reliability of bridges is essential for making informed decisions about maintenance and preventive measures. Traditionally, several methods have been used to assess structural reliability, including the Poisson process method, first-order reliability method (FORM), second-order reliability method (SORM), and importance sampling Monte Carlo method (ISMCM). However, these methods have limitations in dealing with time-varying systems, complex structures, and correlated random variables. A recent study published in the Journal Structures by Mr. Heng Zhou, Associate Professor Xueping Fan, and Associate Professor Yuefei Liu from Lanzhou University developed the Probability Density Evolution Method (PDEM) for analyzing the time-varying reliability of existing bridges. PDEM is based on the concept of probability density evolution theory and offers a promising approach to address the challenges associated with dynamic reliability analysis.

The foundation of the PDEM lies in probability density evolution theory. This theory provides a new perspective on structural reliability analysis by considering the propagation of randomness and uncertainty in physical systems. It is based on the principle of probability conservation, which asserts that in conservative stochastic systems, probability is conserved during the system’s evolution. The probability conservation principle can be described in terms of random events or state space. In the context of dynamic systems, it means that the probability measure of the same random event does not change over time. This principle is expressed as an equation involving the derivative of an event and the probability density function of the system state variable.

The generalized probability density evolution equation (GDEE) is a fundamental component of this theory. It describes the evolution of the probability density function of the system state variable over time. The equation takes into account the fundamental parameter affecting the system state, the dimension of the system state variable, and time. In practice, when analyzing specific physical quantities, a one-dimensional partial differential equation can be derived from the GDEE.

Solving the GDEE analytically is often challenging, especially for complex systems. Therefore, numerical methods are employed to obtain solutions. The authors outlined a Dirac sequence solution method, which utilizes Dirac delta functions to approximate probability density functions. This method involves selecting representative points, dissection of probability space, and the assignment of probabilities to each point. By approximating the Dirac delta function with a smoothed version, numerical integration can yield the probability density solution of the system response.

Associate Professor Xueping Fan and colleagues applied the PDEM to analyze the time-varying reliability of a reinforced concrete bridge. It considers both the serviceability limit state and load-carrying capacity limit state of the bridge. The primary variables of interest are deflection and section bending moment. The analysis involves multiple fundamental random variables, including the effects of vehicle loads, shrinkage, creep, and self-weight. They derived the global functional function for each time period, and the probability density evolution equation is solved numerically. The results demonstrated the evolution of the probability density of the functional function over time. The method was compared to Monte Carlo simulation (MCS), and the accuracy and efficiency of PDEM are highlighted. The results show that PDEM offered a high level of accuracy with a significantly reduced computational burden.

In conclusion, the authors’ PDEM and probability density evolution theory present a promising approach to address the challenges of time-varying reliability analysis in existing bridges. This method offers enhanced accuracy and computational efficiency compared to traditional MCS. It is applicable to complex loading conditions and correlated random variables, making it a valuable tool for bridge engineers and researchers. However, there are limitations to the method, particularly in choosing the smoothing parameter σ in the Dirac sequence solution. Further research and refinement of the method are needed to address these limitations and improve its applicability in practical engineering scenarios. Nevertheless, PDEM represents a significant step forward in the field of structural reliability analysis, providing a new avenue for advancing the safety and maintenance of existing bridges.

### About the author

Mr. Heng Zhou is a research assistant and a postgraduate pursing the master’s degree from School of Civil Engineering and Mechanics at Lanzhou University, China. He held a B.S. in Civil Engineering from Southwest Jiaotong University in China in 2019. His research is focused on bridge reliability. He has published 6 journal papers, 3 of which were in SCI-index journals.

### About the author

Dr. Xueping Fan is an Associate Professor of Disaster Prevention and Mitigation Engineering from School of Civil Engineering and Mechanics at Lanzhou University, China. He serves as the members of ASCE IALCCE and IABMAS. Xueping Fan held a B.S. in Civil Engineering from Shijiazhuang Railway Institute (Now renamed Shijiazhuang Tiedao University) in China in 2008; a M.S. in Structural Engineering from Harbin Institute of Technology in China in 2010 and Ph.D. in Engineering Mechanics from Harbin Institute of Technology in 2014. His research interests include Bridge monitoring data processing, Information fusion of bridge reliability prediction and assessment, Dynamic prediction and abnormal monitoring of bridge dynamic responses, Information fusion of building (group) resistance assessment under earthquake and Bridge optimization design theory. He has published 70+ journal/conference papers, more than 20 of which were in SCI-index journals. He is an Academic Editor of an international journal (Advances in Civil Engineering), a Youth editorial board member of Journal of Jilin University (Engineering and Technology Edition) and Reviewers of more than 50 peer-reviewed journals in the areas of civil engineering, engineering mechanics and artificial intelligence.

### About the author

Dr. Yuefei Liu is an Associate Professor of Disaster Prevention and Mitigation Engineering from School of Civil Engineering and Mechanics at Lanzhou University, China. Yuefei Liu hold a B.S. in Applied Mathematics from Central South University in China in 2006; a M.S. in Fundamental Mathematics from Changsha University of Science & Technology in China in 2009 and Ph.D. in Engineering Mechanics from Harbin Institute of Technology in China in 2015. Her research interests include Numerical algorithm for structural dynamic differential equations, Bridge reliability, Bridge monitoring data processing, Bridge safety assessment and prognosis and Structural Seismic vulnerability and safety assessment. She is Reviewers of more than 30 peer-reviewed journals in the areas of Civil Engineering, Engineering Mechanics and Mathematics.

### Reference

Heng Zhou, Xueping Fan, Yuefei Liu, Dynamic reliability analysis of the existing bridges based on probability density evolution method, Structures, Volume 57, 2023, 105245,