## 1 Introduction  
Landslides are common geological disasters across the globe. For example, a total of 4761 geological disasters occurred in China in 2021, of which 2335 were landslides, reaching half of the total number of geological disasters [1]. Figure 1 shows the statistical chart of the number of landslide occurrences, casualties and direct economic losses from 2010 to 2021, with data from the National Bureau of Statistics (the lower limit of the scale of geological disasters calculated in this paper is three people died and disappeared or caused direct economic losses of CNY 1 million) [1]. It can be seen that the number of landslide occurrences shows a decreasing trend. However, in terms of quantity, the number of landslides is still huge. Moreover, it has become one of the most important geological disasters affecting people's lives, production and safety, contributing major threats and hazards to local human settlements and urban construction safety. The urban growth and population increase generate increasing risk as more people are exposed to such hazards. Therefore, landslide prevention and control has become a hot issue in geotechnical disaster prevention and mitigation [2, 3]. An anti-slide pile is a common landslide treatment measure, which refers to the pile that penetrates through the slide body and goes deep into the slide bed to support the sliding force of the slide body and play the role of stabilizing the slope. The thrust of the sliding body is balanced by the resistance of stable ground below the sliding surface of the anti-slide pile, and the stability of the slope is increased. Anti-slip piles have been widely used in landslide treatment because of their advantages, such as flexible pile burial, strong anti-slip ability, obvious reinforcement effect and convenient construction [4, 5]. For some accumulative landslides, when the pile body is damaged, the deformation of the pile body is \"S\"-shaped, and the deformation area is mainly concentrated near the sliding surface. The bending deformation area of a single-row pile body is about a quarter of the total length of the pile body, and the deformation area of the double-row pile group structure is about three tenths the total length of the pile body, and the deformation near the slope is relatively small [6, 7]. When the ordinary fully buried anti-slide pile is selected, the resistance is mainly borne by the pile body near the sliding surface. The middle and upper part of the load section cannot give full play to the anti-slide function, resulting in the waste of the upper part of the pile material, which makes the project investment increase. According to the survey and design specifications of various industries in China, the engineering requirements can be achieved as long as the landslide reinforcement achieves a certain stability. Therefore, the length of anti-slide piles can be reduced under the premise of meeting the design requirements so as to reduce the engineering cost and save overall costs, which requires in-depth research regarding the structural form of anti-slide piles. A buried anti-slide pile refers to an anti-slide pile with the whole pile body buried inside the slope body and a certain distance between the pile top and the slope surface [8]. The buried pile is mainly arranged near the sliding surface of the pile body. At this time, the pile body only bears part of the landslide thrust, and the remaining thrust over the top of the pile is resisted by the rock-soil body itself, so the internal force of the pile body will be much smaller than that of the full-length pile. Studies have shown that this pile structure can effectively overcome the shortcomings of full-length piles [9, 10]. Compared with full-length piles, buried piles have shorter force section length, lower maximum bending moment of pile body, and lower material usage, which can reduce costs when treating landslides with larger sliding body thickness, and they have been applied to some extent in landslide treatment in recent years [11].