## 1 Introduction  
Reinforced concrete barriers on bridges play a crucial role in ensuring the safety of vehicle occupants during collisions. As essential components of bridge structures, these barriers absorb and dissipate collision energy to prevent vehicles from veering off the road [1]. However, traditional design approaches often simplify the interaction between the bridge deck and the barrier by assuming that the deck acts as a rigid body, which can lead to inaccurate assessments of barrier performance [2, 3]. The connectivity between the deck and the barrier was studied in [2, 4, 5]. Specifically, the research in [4] involved experimental investigations under static loading. In contrast, the authors of [5] compared experimental results with numerical analysis outcomes concerning stiffness, crack damage, and failure modes under static loading. In [2], the study modeled the deck and barrier to conduct a collision analysis involving truck vehicles, focusing on the failure patterns of the barrier. With increasing traffic volumes and vehicle speeds, the risk of collisions on bridges has escalated, necessitating a more precise understanding of barrier design and performance evaluation. Recognizing the flexibility of the bridge deck in analyzing barrier performance is essential for developing safer and more efficient barrier designs. Various numerical analysis methods and experimental approaches [6, 7, 8] were used in the performance evaluation of reinforced concrete barriers. In applying numerical analysis techniques, many studies frequently utilize the LS-DYNA software ([[https://lsdyna.ansys.com/manuals/](https://lsdyna.ansys.com/manuals/)]([https://lsdyna.ansys.com/manuals/](https://lsdyna.ansys.com/manuals/)), access on 1 October 2024), which is specialized for collision analysis and offers a variety of material models capable of representing concrete. In [2], the continuous surface cap model was utilized, whereas in [9], the Mat_Johnson_Holmquist model, which is primarily used for brittle materials like ceramics, was applied to concrete. In contrast, the authors of [7; 10] treated concrete barriers as rigid bodies; specifically, the authors of [7] evaluated the risk to occupants of lightweight vehicles and compared it with experimental results, while those of [10] examined the effects of friction between the vehicle and concrete. In vehicle collision analysis, the authors of [11] utilized the Mat_Brittle_Damage model to perform collision analyses for passenger cars, pickup trucks, and large trucks. Moreover, the authors of [12] conducted an analysis comparing the K&C model and the Winfrith concrete model and the authors of [13] used the K&C model to derive various parameter values that are difficult to obtain in large-scale crash tests. Additionally, the authors of [14] applied the K&C model to rubberized concrete barriers. These studies demonstrate that advanced material models in finite element analysis (FEA) for concrete barriers have significant potential in predicting barrier performance across various collision scenarios. Numerical analysis methods for reinforced concrete barriers, including material models related to concrete, fundamentally rely on finite element modeling techniques for reinforced concrete. Although numerous numerical models have been proposed for the analysis of reinforced concrete structures, none have effectively combined the desired levels of accuracy, robustness, and computational efficiency needed to predict the behavior of these structures [15]. One-dimensional beam elements based on concentrated plasticity [16] or distributed plasticity [17] struggle to predict the mechanical behavior of three-dimensional frame structures because they fail to capture local phenomena that affect shear behavior and the overall structural response.