Proposing an Equivalent Single Degree of Freedom System to Determine the Capacity Curve of Curved Bridges

Document Type : Original Research

Authors
Z. SHAMS EDDIN 1
F. Daneshjoo 2
1 Tarbiat Modares University
2 Faculty of Civil and Environmental Engineering., Tarbiat Modares Uni., Tehran, Iran
10.22034/25.4.7
Abstract
Horizontally curved bridges have been observed to suffer severe structural damage during past earthquakes so determining the seismic performance of curved bridges is crucial due to the complex dynamic behavior of these structures because of their irregular geometry and non uniform mass and stiffness distributions. Analyzing and plotting the capacity curve of these structures can be costly and time-consuming. As a result, many efforts have been made to simplify the structural models of these bridges and reduce the computational workload required for their analysis. This article presents a straightforward method to convert the multi-degree-of-freedom system of these structures into an equivalent single-degree-of-freedom system, ensuring that the capacity curve of the equivalent structure closely matches that of the original structure with minimal error. In this study, the OpenSees program was used to extract the stiffness and mass matrices of a curved bridge structure. These matrices were then condensed into one-by-one matrices for mass and stiffness using dynamic condensation equations. The characteristics of these matrices were applied to a single-degree-of-freedom stick model. In this model, the obtained mass is placed at the top of a stiff rod (stick), which is connected to the ground by a spring (zero-length element) with the equivalent stiffness obtained. A nonlinear static pushover analysis of the bridge structure was performed to obtain the capacity curve. An equivalent bilinear curve was then drawn, and the yield shear force and yield displacement were determined. The nonlinear behavior of the single-degree-of-freedom structure was modeled using the Steel02 material available in the OpenSees library by zero length element utilizing the yield shear force and yield displacement magnitudes of the curved bridge. The capacity curve of the stick model, which has a single degree of freedom, showed an error percentage of 7% compared to the bridge's capacity curve. This indicates acceptable compliance with the capacity curve of the main structure, making the stick model a viable alternative for repeated analysis of the curved bridge structure. This study also included a sensitivity analysis to investigate the effects of increasing the curvature radius and decreasing the curvature angle of the bridge on its capacity and effective mass. Due to the dynamic condensation of the curved bridge structure, the influence of all degrees of freedom was considered in the stiffness and mass matrices, unlike methods that rely solely on the first vibration mode for dynamic condensation. Comparing the capacity curve of each structure with that of an equivalent single-degree-of-freedom system revealed that the structure's capacity increases with a larger curvature radius. In contrast, the lowest capacity was observed in the straight bridge scenario. Additionally, modal analysis of the studied models showed that increasing the bridge's curvature radius leads to a longer structural period, while a decreasing curvature angle has a similar effect. However, the period of the straight bridge was longer than all the other models. Furthermore, as the curvature radius increased, the mass contribution percentage of the first mode in the translational x-direction decreased, whereas the translational mass contribution percentage in the y-direction and the rotational mass contribution around the z-axis increased.

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Modares Civil Engineering journal
Volume 25, Issue 4 - Serial Number 84
September and October 2025
Pages 71-82