# Rankine Active Earth Pressure of Unsaturated Filling under Earthquake
## 1 Introduction  
The earth pressure of retaining structures has remained an important issue in geotechnical engineering [1, 2]. Meanwhile, the evaluation of earth pressure under seismic or unsaturated conditions is an important part of the design for retaining structures. As a classical theory widely used in earth pressure calculation, the Rankine theory has been extended to the field of unsaturated soil in recent years. [PERSON] analyzed the lateral earth pressures of a retaining wall produced by saturated clays with negative pore-water pressures and unsaturated expansive clays with positive matric sections [3]. [PERSON] et al. [4] took the vertical rigid retaining wall with a narrow unsaturated backfill as the research object. By assuming that the backfill behind the wall forms a circular soil arch, the analytical solution of the active earth pressure of narrow unsaturated soil was derived based on the thin layer element method. Based on [PERSON]'s shear strength criterion for unsaturated soils and the Rankine earth pressure theory, [PERSON] and [PERSON] proposed the calculation methods for unsaturated earth pressure [5, 6]. [PERSON] was concerned with the nonlinearity of unsaturated soils strength and its influence on passive earth pressure, and proposed the parameter determination approach [7]. By combining the Rankine earth pressure formula of unsaturated soil and [PERSON]'s analytic solution of rainfall infiltration, the formula for the earth pressure of unsaturated soils under rainfall was concluded [8]. Furthermore, in order to consider the effect of intermediate principal stress, the Rankine earth pressure calculation method was modified according to the twin shear unified strength theory [9]. The unifiedsolution of unsaturated Rankine earth pressure under rainfall conditions was put forward with the consideration of intermediate principal stress and the tension-compression ratio of the soil [10]. Additionally, based on the unified strength theory and the shear strength criterion of unsaturated soils with two stress state variables, the unified solutions of shear strength and earth pressure of unsaturated soil were obtained [11]. In recent years, significant progress has been achieved in the research of seismic earth pressure. For instance, Rankine earth pressure was extended to seismic circumstance by the use of the quasi-static method, though it was only applicable for the case of non-cohesive backfill and vertical border with the retaining wall [12]. [PERSON] and [PERSON] adopted the horizontal layer analysis method to acquire the stress distribution and the location of resultant earth pressure under seismic conditions, limited to the horizontal backfill surface and saturated soils [13; 14]. Based on the classic [PERSON] earth pressure theory, the seismic active earth pressure in the pseudo-static state was converted to non-seismic active earth pressure by using the transformation approach of rotating the retaining wall model [15]. In order to investigate the response spectrum characteristics of the dynamic soil pressure along the elevation and sliding surface of loess slopes under the action of earthquake, large-scale shaking table experiments were performed by progressively loading X-direction and Z-direction seismic waves until failure of the slope. The dynamic soil pressure curves of five sensors along the elevation and near the sliding surface were obtained, and the time-domain curve of the dynamic soil pressure was decomposed by using wavelet transform [16]. Based on the plane strain unified strength theory formula, while also considering the soil arching effects and tension cracks, the analytical solutions of the lateral earth pressure coefficient and the active earth pressure under the earthquake action were deduced by [PERSON] et al. [17], and the mechanism and distribution of seismic active earth pressure with a limited width were discussed in terms of some relevant parameters.