## 1 Introduction  
Seismic reconnaissance surveys [1, 2, 3, 4] consistently reveal that conventional structures lacking self-centering mechanisms often sustain excessive residual displacements following strong earthquake shaking, even when collapse is prevented. The large permanent deformations frequently necessitate structural demolition due to two critical factors: (1) pronounced visual deformations that jeopardize structural functionality and (2) technical challenges and economical impracticality in rehabilitating significantly tilted structures. The 1995 Kobe earthquake [2] exemplifies this phenomenon, where 88 reinforced concrete piers with tilt angles exceeding 1\({}^{\circ}\) were irreparable and required replacement despite minimal structural damage. Such observations have elevated residual displacement as a pivotal performance metric in modern performance-based earthquake engineering frameworks. Extensive research has been conducted on post-earthquake residual displacements of structures that can be simplified to single-degree-of-freedom (SDOF) oscillators. Some studies [5; 6; 7; 8; 9; 10; 11; 12; 13] have focused on analyzing the factors that influence residual displacement response, while others [14; 15; 16; 17; 18; 19; 20; 21; 22; 23] have aimed to develop computational models for estimating residual displacements. For instance, [PERSON] and [PERSON] [5] and [PERSON] et al. [6] identified the post-yield stiffness ratio as the dominant factor governing residual displacements and proposed a constant-ductility residual displacement spectrum for seismic design. [PERSON] et al. [7] observed that the residual displacement of oscillators with softening behavior (negative post-yield stiffness) is more significant than in the case of elastic-perfectly plastic and hardening SDOF systems. [PERSON] and [PERSON] [14; 15] derived simplified equations to estimate residuals considering site conditions, natural periods, and lateral strength, extending their framework to soft-soil site motions. [PERSON] and [PERSON] [9; 10] linked hysteresis loop shapes to residual displacement magnitudes. Their analyses revealed that the ratio of residual to maximum inelastic displacement remains approximately constant, even for ground motions containing velocity pulses. [PERSON] and [PERSON] [18], [PERSON] and [PERSON] [19], and [PERSON] and [PERSON] [20] generated several ensembles of ground motions based on the control variate technique to isolate the effect of different seismological parameters on structural response. A predictive model was developed for structures with bilinear and pinching hysteretic characteristics. [PERSON] et al. [21] investigated the constant-damage residual displacement ratios of SDOF systems with different damage levels. While these prior works advanced the understanding of residual displacement mechanics, a fundamental limitation persists: existing models are predominantly calibrated to single-earthquake scenarios. In reality, mainshock-damaged structures often endure aftershocks that exacerbate residual displacements through cumulative damage. Studies in [14; 15; 16; 17; 18; 19; 20; 21; 22; 23], though pioneering in methodology, neither explicitly account for ground motion sequences nor evaluate interactions between structural damage progression (e.g., ductility) and aftershock properties (e.g., duration, intensity ratios, spectral compatibility). This critical gap between conventional analytical assumptions and real seismic scenarios highlights the necessity to account for sequential ground motions in evaluating post-earthquake residual displacements. To the knowledge of the authors, the present research on seismic sequences remains focused on sequence characteristics [24; 25; 26], displacement ductility [27; 28; 29; 30; 31], seismic vulnerability [25; 32; 33; 34; 35; 36; 37], cumulative damage [38; 39; 40], and structural resilience [41; 42; 43], with residual displacement evaluation receiving scant attention. Notably, [PERSON] et al.