## 1 Introduction  
Reinforced concrete (RC) buildings typically exhibit highly non-linear hysteresis behavior with stiffness and strength degradation during damaging earthquakes, as well as pinching effects, which all adversely affect seismic performance [1, 2, 3]. Pinching plays a main role in defining RC column failure modes [3, 4, 5, 6]. According to the pinching severity and spread observed in force-deformation hysteresis loops, the failure modes are classified as flexural failure (FF) [7, 8, 9] with negligible pinching region, flexural-shear failure (FSF) [10, 11, 12] with slight pinching region, and shear failure (SF) [13, 14] with significant pinching effect [4, 8, 15, 16]. To better evaluate and optimize design for seismic performance, all typical damage mechanisms in RC buildings should be considered. This requirement highlights the importance of a robust and efficient analytical model accurately simulating strength and stiffness degradation and pinching effects with accuracy. Numerous hysteresis models have been developed to simulate the non-linear behavior and dynamic response of RC-frame buildings. These models can be broadly categorized [17; 18] into piecewise linear polygonal models (PMs) [19; 20; 21; 22; 23; 24] or smooth models (SMs) [17; 25; 26; 27; 28; 29; 30; 31]. Piecewise linear functions/models describing different behavioral stages of structures, such as initial elastic, post-yielding, cracking and gap closure regimes, are generally used to define PMs. In contrast, non-linear differential equations for SMs are used to define continuous stiffness changes in structures as a function of the load in SMs [15; 32; 33]. Many simplifying assumptions made in PM approaches generally result in an unreliable and inaccurate assessment of structural response to describe degradation and damage mechanisms [15]. Moreover, the lack of mathematical versatility in PM formulations makes them inappropriate for simulating complex hysteretic behaviors, like the commonly observed pinching phenomenon in RC structures. This issue is more apparent in dynamic simulation, which is increasingly used in design and design optimization [3]. In contrast, SMs can more realistically simulate the non-linear responses of RC buildings and their mechanical characteristics. A wide range of damage mechanisms in RC buildings can be efficiently simulated just by defining more differential equation parameters in SMs, making them versatile, accurate, and computationally intensive approaches [15; 34]. However, as model complexity increases, so does the need for accurate model parameters, as well as the difficulty in identifying these nonlinear models from relatively limited data compared to the number of dynamics modelled [35; 36]. The Bouc-Wen-Baber-Noori (BWBN) model is one of the most widely used and comprehensive smooth hysteresis models, and includes all major structural damage mechanisms of strength, stiffness degradation, and shear pinching deteriorating effects [4; 17; 37; 38; 39; 40; 41]. However, the post-yield and pinching region stiffness have constant values even when structural damage occurs [42], which contradicts experimental results for RC columns [42] and buildings [17]. In particular, a six-story RC building study using measured hysteresis loops demonstrates how the pinching region length increases as the pinching stiffness decreases during a damaging earthquake [43]. Thus, the model. while widely accepted, is not necessarily accurate. Generally, [PERSON]'s original model and its extensions, including the BWBN model, have been developed to simulate the hysteretic behavior of experimental or real systems by tuning parameters to minimize differences between measured and simulated responses [44]. However, these hysteresis models are typically non-phenomenological, so their parameters often have no explicit physical meaning, and some parameters are mathematically redundant and unidentifiable [45; 46].