Improving Seismic Design Through Equivalent Linearization for Self-Centering Structures


As seismic events continue to pose a significant threat to the built environment, the pursuit of innovative engineering solutions to mitigate their impact remains paramount. Among these solutions, self-centering (SC) structures have emerged as a promising approach. SC structures are designed to withstand seismic forces while maintaining their original position after the earthquake subsides. This property not only enhances the structural resilience but also reduces the downtime and repair costs associated with seismic events. However, to effectively design and analyze SC structures, engineers must navigate the complex realm of nonlinear behavior, introducing challenges that necessitate innovative solutions.

In a new study published in the peer-reviewed Journal Engineering Structures led by Dr. Gang Xu, Professor Tong Guo, and Professor Aiqun Li from the Key Laboratory of Concrete and Prestressed Concrete Structures of the Ministry of Education at Southeast University, they investigated and discussed the cutting-edge field of seismic design for self-centering structures, focusing on the development and application of an empirical method known as equivalent linearization. This method, as proposed by a recent study, aims to address the intricate interplay between nonlinear behavior and seismic performance in SC structures. By employing dynamic nonlinear analysis and regression techniques, this approach seeks to bridge the gap between accurate modeling and computational feasibility. Their exploration encompasses various aspects of this method, its validation, and its implications for the seismic design of self-centering structures.

To appreciate the significance of equivalent linearization, it is imperative to grasp the unique characteristics of self-centering structures and the nonlinear behavior they exhibit during seismic events. Unlike conventional structures, SC structures are designed to dissipate energy through controlled yielding elements, such as friction dissipators and prestressed steel strands. This intentional nonlinear behavior allows them to absorb seismic energy while returning to their original position post-earthquake. However, this nonlinear behavior poses a formidable challenge for engineers. Accurately simulating the response of SC structures under varying seismic conditions is a computationally intensive task, often necessitating the use of dynamic nonlinear analysis. Such analyses consider a multitude of parameters, including the ductility of materials (μ), post-yield stiffness (α), energy dissipation coefficient (β), and period (T0). The sheer volume of combinations makes it practically impossible to conduct comprehensive dynamic nonlinear analyses for each design scenario, calling for innovative approaches like equivalent linearization.

In view of these challenges, the proposed equivalent linearization method by the authors offers a promising avenue for addressing the complexities of SC structures’ nonlinear behavior. This method leverages dynamic nonlinear analysis to identify optimal equivalent linear models that closely replicate the response of inelastic SC structures. The key steps in their approach can be summarized as follows:

  1. Identifying Optimal Equivalent Linear Models

Through dynamic nonlinear analysis, the maximum inelastic displacement (Die) for a specific SC model is determined. This process involves varying parameters such as α, β, and μ, as well as considering different ground motion records. The goal is to create an accurate representation of the inelastic behavior. Simultaneously, linear elastic models with varying parameters (Fps, ξeq) are analyzed to determine their maximum elastic displacement (De). The linear elastic model that best approximates the inelastic behavior is considered the optimal equivalent linear model.

  1. Statistical Analysis and Regression

The parameters of these optimal equivalent linear models are subjected to statistical analysis and regression to derive empirical equations. This step ensures that the equivalent linear models capture the essence of the inelastic behavior and can be efficiently incorporated into design procedures.

  1. Validation and Application

The proposed method is validated using a diverse set of earthquake ground motions, and its accuracy is assessed by comparing the predicted responses with those obtained through dynamic nonlinear analysis. This validation process is essential to demonstrate the method’s efficacy and reliability. Once validated, the method can be applied to streamline the seismic design of self-centering structures.

Validation: Ensuring Accuracy and Reliability

To assess the accuracy of the proposed equivalent linearization method, a comprehensive validation process is essential. This involves subjecting SC models to a range of earthquake ground motions and comparing the results obtained using the equivalent linear models with those derived from dynamic nonlinear analysis.

The validation results presented by the researchers reveals several key findings. Firstly, the proposed method demonstrates superior accuracy compared to classic methods. The absolute mean displacement errors for SC models analyzed using classic methods typically fall within the 20% to 30% range, with a standard deviation ranging from 10% to 20%. In contrast, the proposed method yields mean displacement errors that are generally lower and standard deviations comparable to classic methods. This improvement in accuracy is crucial for reliable seismic design and performance assessment. Additionally, the validation process highlights the influence of hysteretic parameters on inelastic displacement. Parameters such as μ, α, and β significantly impact the response of SC models. The proposed method accounts for these parameters, resulting in better estimation accuracy. Furthermore, the standard deviation of the proposed method is within acceptable engineering limits, underscoring its reliability for practical applications.

With its accuracy validated, the proposed equivalent linearization method holds substantial promise for the seismic design of self-centering structures. By providing a computationally efficient means to capture the nonlinear behavior of SC structures, it enables engineers to make informed design decisions with confidence. This advancement is particularly significant in regions prone to seismic activity, where the safety and resilience of structures are of paramount importance. The implications of this method extend beyond mere accuracy and efficiency. It empowers engineers to explore design scenarios with greater depth and precision, enabling the optimization of SC structures for specific performance objectives. Whether the goal is to minimize structural damage, reduce repair costs, or enhance post-earthquake functionality, the proposed method equips engineers with the tools to tailor SC structures to meet these objectives.

To illustrate the practical application of the proposed equivalent linearization method, a case study involving the seismic design of a self-centering structure the authors presented in their study. The designed structure, a 6-story concrete frame-shear wall structure, incorporates key elements of SC technology, including friction dissipators and prestressed steel strands. The case study underscores the effectiveness of the updated displacement-based design approach for SC structures. The designed structure successfully meets stringent design targets, including elastic behavior and limited story drift ratios under both frequently occurred earthquake and rarely occurred earthquake levels. The structure’s self-centering and energy dissipation capabilities are clearly demonstrated through comprehensive dynamic analysis. Furthermore, the case study emphasizes the role of prestressed steel strands in maintaining the structure’s elastic behavior under large earthquakes. The utilization of steel strands at 50% of their ultimate strength ensures that the structure retains its self-centering capacity.

In conclusion, the development and application of the proposed equivalent linearization method represent a significant stride in the field of seismic design for self-centering structures. By addressing the complexities of nonlinear behavior through empirical modeling, this method offers engineers a valuable tool to streamline the design process while maintaining accuracy and reliability. The method’s validation results demonstrate its superiority over classic methods, highlighting its potential to revolutionize seismic design practices. Its ability to account for hysteretic parameters and efficiently capture the nonlinear behavior of SC structures positions it as a valuable asset for engineers tasked with designing structures resilient to seismic events.

Improving Seismic Design Through Equivalent Linearization for Self-Centering Structures - Advances in Engineering

About the author

Dr. Gang Xu received his PhD in 2020 and later carried out research work as a Faculty Postdoc and Assistant Research Fellow of School of Civil Engineering at Southeast University, China. He won the U35 Exploration Award of Jiangsu Youth Scientific and Technological Innovation in 2021 and the Gold Award at the 122nd Paris International Invention Exhibition. He has secured more than 10 scientific research projects, including the National Natural Science Foundation of China, Natural Science Foundation of Jiangsu Province, Postdoctoral Science Foundation of China, and Postdoctoral Research Foundation of Jiangsu Province.

He is mainly engaged in the research and development of new prefabricated structural systems, seismic resilience enhancement of building structures, and structural vibration control. He has published more than 30 scientific papers as the first author in high-level journals. His nine China invention patents has been authorized. He also served as Chief guest editor of internationally renowned journals ‘Structures’ and ‘Sustainability’, editorial board member of ‘Advances in Civil Engineering’, and reviewers for several well-known international journals in the field of civil engineering.

About the author

Professor Tong Guo is the dean of School of Civil Engineering at Southeast University, China. He works in multi-disciplinary research areas with the specific research interests in the development and application of structural condition assessment and upgrade methods to civil structures. Since 2007, he has secured thirty-five funded research projects, including one National Key Research and Development Project and six projects funded by the National Natural Science Foundation of China. He is the winner of the National Science Fund for Distinguished Young Scholars of China.

So far, He has published over 160 research articles, with them generating an h-index of 28. In addition to serving as Associate Editor of two ASCE journals, ‘Journal of Bridge Engineering’ and ‘Journal of Performance of Constructed Facilities’. He was honored as ASEC fellow in 2020 and currently is Education Committee Member of ASCE Forensic Engineering Division. He is also in the editorial board of various international journals. He also authored three books in engineering technologies, as well as led or participated into the regulations of fifteen national or provincial standards. His innovation and creativity have led to forty-two invention patents. His research achievements have been applied in more than forty different engineering projects.

About the author

Professor Aiqun Li is a National Famous Teacher and Distinguished Professor of Southeast University, China. He has been Vice President of Beijing University of Civil Engineering and Architecture since April 23, 2015. He has successively served as the director of School of Civil Engineering at Southeast University, the dean of School of Civil Engineering, the executive vice president and secretary of Chien-Shiung Wu College, and the director of Key Laboratory of Concrete and Prestressed Concrete of Ministry of Education.

He is mainly engaged in the research of disaster prevention and mitigation, structural vibration control, structural health monitoring, performance assessment and promotion of building structures. He has completed and undertaken more than 10 national projects such as National Science Fund for Distinguished Young Scholars of China, National Science and Technology Support Plan, etc. He has trained or supervised more than 30 doctoral students and more than 50 master students. The projects presided over or participated in have won two National Science and Technology Progress Awards, ten Provincial Science and Technology Progress Awards. He has obtained more than 20 China invention patents, edited five books and published more than 150 research articles as the first or corresponding author.


Gang Xu, Tong Guo, Aiqun Li, Equivalent linearization method for seismic analysis and design of self-centering structures, Engineering Structures, Volume 271, 2022, 114900.

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