## 1 Introduction  
Post-earthquake investigations following the 2008 Wenchuan earthquake revealed that a significant proportion of gravity retaining walls and cantilever retaining walls exhibited RB (rotation about the base) or RB+T (combined rotation and translation about the base) failure modes during the seismic event [1]. The RB displacement pattern represents a common deformation mode of retaining walls during operational service. Under seismic loading, the backfill soil behind the wall does not fully reach the limit state, and the seismic earth pressure is influenced by wall displacement. The calculation of dynamic earth pressure induced by ground motion acting on retaining walls constitutes a fundamental aspect of seismic design for retaining structures. Although extensive research has been conducted on this subject, satisfactory results have yet to be achieved. Current methods for calculating seismic earth pressure can be classified into two categories: (1) methods considering soil-wall interaction and actual stress-strain behavior of soil [2; 3]; (2) methods assuming sufficient relative movement between the wall and backfill to reach limit or failure states, with the Mononobe-Okabe (M-O) theory being the most representative achievement [4; 5; 6]. This approach, based on [PERSON]'s earth pressure theory, applies seismic inertial forces to the sliding soil wedge behind the wall and derives earth pressure formulas through equilibrium conditions for stability analysis. Also known as the pseudo-static method, it remains widely adopted in engineering practice despite the actual stress-strain characteristics of soil-wall systems during earthquakes being far more complex than the M-O assumptions. Developed from [PERSON]'s theory, the M-O theory assumes that seismic earth pressure is generated by a triangular soil wedge bounded by the wall back and a failure surface passing through the wall heel. During earthquakes, the entire wedge moves as a rigid body with uniform acceleration. While the M-O theory reveals the resultant seismic earth pressure, the pressure distribution and point of application currently lack theoretical solutions, being supported only by experimental studies [7; 8; 9]. Extensive model tests indicate that the resultant force application point varies between 0.45 \(H\) and 0.60 \(H\) (where \(H\) is the backfill height), closely related to wall stiffness and movement patterns [10], highlighting the need for further research on seismic earth pressure distribution. Traditional earth pressure theories ([PERSON] and [PERSON]) assume limit-state conditions. However, in practical engineering, backfill soils behind walls typically exist in non-limit states. Non-limit earth pressures lie between at-rest and limit earth pressures, depending on wall displacement and movement patterns. Through model tests, [PERSON] [11] proposed that soil transitions gradually from at-rest to active limit states, introducing the concept of \"intermediate active earth pressure\" between these states. This work established the theoretical basis for displacement-dependent earth pressure calculations by deriving expressions for active earth pressure under RB displacement patterns. While active earth pressure represents the minimum resultant force, practical constraints often prevent backfill soils from reaching limit equilibrium states due to restricted wall displacements. Consequently, the soil stress state remains between at-rest and active limit conditions--referred to as the non-limit active state--with corresponding pressures defined as non-limit active earth pressures [12; 13; 14; 15; 16]. Current research on non-limit earth pressure primarily focuses on theoretical approaches: (1) Models incorporating variations in the soil internal friction angle and wall-soil friction angle with wall displacement, integrated into limit-state calculation methods [17; 18; 19; 20; 21; 22]; (2) formulations establishing non-limit earth pressure calculations based on stress-strain displacement relationships [23; 24; 25; 26].