## 1 Introduction  
The determination of the end-bearing capacity of piles has always been a major issue in geotechnical engineering and has received a great deal of attention from researchers. Paying attention to the failure mechanism and stress field around the pile tip, several methods have been used to study the end-bearing capacity of piles, such as the limit analysis method, the characteristics method, and the cavity expansion method. In the cavity expansion method, the deformation of the soil around the pile tip is considered similar to the expansion of a spherical cavity, and the ultimate toe capacity is associated with the limited cavity expansion pressure. Through research on cavity expansion failure mechanisms and limited cavity expansion pressure, the ultimate toe capacity has been widely studied [1; 2; 3; 4; 5]. In the limit equilibrium method, the failure pattern around the pile tip and the failure surface are usually assumed directly. On the basis of the assumed failure surface, static equilibrium analysis is carried out to obtain the ultimate toe capacity and the constitutive soil model is generally not required. [PERSON] [6] established a failure pattern with the failure surface reverting to the pile shaft. [PERSON] and [PERSON] [7] established an analytical model wherein the interdependence between the shaft and tip resistance is taken into account. [PERSON] et al. [8] investigated the ultimate toe capacity of rock-socked piles based on the Hoek-Brown criterion. The limit analysis method analyzes the ultimate toe capacity based on the upper bound or the lower bound theorem and the soil is usually considered as rigid plastic or elastic perfectly plastic in the limit analysis method. [PERSON] et al. [9] studied the ultimate bearing capacity of composite pile foundations by using the finite element plastic lower bound limit method and the interior-point algorithm. [PERSON] et al. [10] applied the upper bound limit analysis method to the analysis of the ultimate toe capacity considering the splitting failure of grouted base. In the characteristics method, it is necessary to predetermine the failure pattern around the pile tip. Based on the determined failure pattern, the failure surface is defined by the stress characteristic line calculated using the equilibrium equation of the soil element around the pile tip. Considering different failure patterns and using different failure criteria for varying soil conditions, researchers studied the end-bearing capacity of piles. [PERSON] et al. [11] considered the end-bearing capacity of driven pile based on the failure pattern with the failure surface reverting to the pile shaft. [PERSON] et al. [12] focused on the ultimate bearing capacity and the load-displacement behavior of driven pile based on a non-associated flow assumption. The ultimate bearing capacity of rock-socketed piles was also studied by the characteristics method [13; 14; 15]. When it comes to the most commonly used methods, the calculated ultimate toe capacity by the cavity expansion method is influenced by the rationality of the assumption of spherical expansion and the value of the limited cavity expansion pressure. The accuracy of other methods is affected by the assumption of the failure pattern and the range of the failure surface. The form of the failure pattern and the failure surface are usually assumed directly in the limit equilibrium method. The range of the shear failure surface considered in the limit analysis method and the characteristics method usually exceeds that observed in the actual situation due to a larger assumed failure pattern. There is relatively little research on the punching shear failure in which the range of failure surface is more in line with reality. Except for the deviation in the predicted ultimate toe capacity caused by the inaccurate assumed failure pattern, the flow rule influences the predicted results by affecting the extent of the failure range.