## 1 Introduction  
Analyzing 3D masonry structures under dynamic loads is highly demanding in terms of time and computational resources, mainly due to the complex dissipative behaviors of masonry walls. This complexity necessitates 3D nonlinear transient dynamic analyses, which are both sophisticated and resource-intensive. For instance, [PERSON] conducted, in 2022, 3D nonlinear dynamic analyses on two historical fortified masonry places using 455,000 and 345,000 four-node tetrahedral finite elements (FE) [1]. Similarly, in 2023, [PERSON] performed analyses on two masonry churches with 266,000 and 145,000 FE [2]. These recent studies are notable as they enabled nonlinear dynamic analyses of complex historical structures; however, the computational demands constrained the refinement of discretization, and this limitation resulted in a FE size ranging between 20 cm and 40 cm, which is comparable to the characteristic dimensions of the masonry elements. In addition, these FE analyses were carried out using the software code Abaqus [3], which requires complex developments to implement constitutive laws specific to masonry in user subroutines. Consequently, employing a simplified model based on a macro-element with nonlinear constitutive behavior appears particularly relevant, especially for the seismic analysis of large complex masonry constructions. To address these challenges, researchers have developed various modeling techniques specifically designed for masonry structures, especially infill walls made of unreinforced masonry (URM). These innovative methods aim to optimize the analytical process by preserving critical insights into structural behavior while significantly reducing computational time. Among the different proposed strategies, the pier/spandrel partition is the most common, also known as the equivalent frame model (EFM) [4; 5; 6; 7; 8; 9], which is recommended by international standards such as Eurocode 8 [10] or FEMA 356 [11] (Federal Emergency Management Agency). Although these models have been used to simulate complex structures, they mainly focus on determining the in-plane (IP) behavior of the structure. Only a few studies have proposed the determination of out-of-plane (OOP) behavior [12]; these studies model the OOP behavior using a strategy similar to that of the IP direction, with adjusted values for the parameters. Another significant advancement in 3D URM structural modeling involves the integration of the rigid body spring model (RBSM) and the rigid element model (REM) [13; 14; 15; 16]. These models consist of rigid elements interconnected through a combination of springs to simulate IP behavior or hinges for OOP behavior. These springs and hinges are characterized by their inelastic behavior, and they are defined by studying a representative volume. Despite demonstrating impressive results, these models face a limitation; they cannot comprehensively address both IP and OOP behaviors simultaneously. Consequently, inertial forces due to perpendicular walls are incorrectly taken into account under dynamic loads with these models. To streamline the examination of the 3D URM structures under dynamic loads, without sacrificing crucial insights into their structural behavior, the rigid macro-element model (RMEM) emerges as a significant advancement [17; 18; 19]. This model is adept at simultaneously analyzing IP and OOP behaviors, employing a system of rigid frames linked by both normal and tangential springs. These springs facilitate the simulation of tension/compression and sliding movements within the masonry, while diagonal springs within the frames mimic shear behavior. To support modal and seismic analyses, the RMEM incorporates a mass matrix, enhancing its applicability in dynamic scenarios [20]. However, RMEM encounters challenges in correctly capturing the OOP bending of the wall, which necessitates a fine mesh and, consequently, increases computation time significantly. Moreover, the inclusion of additional structural elements such as frames in the openings can be cumbersome [21].