## 1 Introduction  
With the development of construction technology and the increase of underground building density, the soil behind deep foundation pit retaining structures will be restricted by many other types of underground structures. For example, in an existing deep basement, underground pipe gallery, or subway station, the soil is contained within a limited width. Earthquake is also a factor that must be considered. Retaining structures in high-intensity seismic areas are constantly facing the severe test of earthquake damage. Therefore, for the safety and economy of the retaining structure, the accurate estimation of the seismic earth pressure of the limited backfill is very important. Traditional Rankine [1], Coulomb [2], and Mononobe-Okabe [3, 4] theories are based on semi-infinite space. The slip surface is assumed as a plane, and the effects of the intermediate principal stress and soil arching effects are not considered. Some scholars have also realized this problem, and many theoretical and experimental studies have been carried out. At present, the earth pressure distributions obtained by some theories commonly used in engineering are linear. However, related experimental and theoretical studies [5, 6] have shown that the distribution is actually nonlinear. For soil with limited width, the magnitude and distribution of earth pressure given by some experiments and numerical studies [7, 8, 9] are also quite different from the traditional earth pressure theory. [PERSON] and [PERSON] [10] established a general calculation model and obtained an analytical solution for limited backfill. [PERSON] et al. [11] studied the active earth pressure of the retaining wall near the rock face and proposed an analytical solution which can more accurately predict the earth pressure of the retaining wall near the rock face. Moreover, the influence of soil arching cannot be ignored. Since the earliest discovery of the arching effect by [PERSON] [12], many scholars have introduced soil arching effects into the calculation of earth pressure. [PERSON] [13] proposed the lateral earth pressure coefficient and a formula for the nonlinear active earth pressure. [PERSON] and [PERSON] [14] conducted a series of centrifugal tests on the earth pressure of a retaining wall with a limited backfill width. [PERSON] and [PERSON] [15] derived a method for the calculation of the lateral pressure coefficient based on soil arching effects. [PERSON] and [PERSON] [16] introduced the soil arching effects into limited backfill, and the results obtained were closer to the numerical analysis, which has a certain engineering reference value. As mentioned earlier, traditional earth pressure theories are based on the assumption of a planar fracture surface. However, some studies [17; 18] have shown that the slip surface of limited backfill is actually closer to a curve. [PERSON] et al. [19; 20] and [PERSON] et al. [21] conducted experimental and theoretical studies of the limited backfill slip surface. The results showed that the slip surface is a logarithmic spiral, and the theoretical results were in good agreement with the experimental results. Seismic action must be paid sufficient attention. Based on [PERSON]'s theory, [PERSON] et al. [22] analyzed the seismic active earth pressure of cohesive soil in semi-infinite space using a pseudo-static method. The rationality of the method was verified by comparison with previous studies. [PERSON] and [PERSON] [23] used the model of the combination of logarithmic spiral and tension cracks. The energy method was used to analyze the seismic active earth pressure under different crack types. In some statistics about the damage of supporting structures, the influence of earthquake action is very significant [24; 25]. However, there are few studies on the seismic earth pressure of limited backfill. Although the above works carried out studies on different aspects of earth pressure, they are all based on the traditional Mohr-Coulomb theory--that is, the effect of intermediate principal stress is ignored.