# Bearing Capacity of Karst Cave Roof under File Foundation Load Using Limit Analysis
## 1 Introduction  
The stability of karst foundations is a crucial area of study within the field of geotechnical engineering. Research has revealed that the roof's thickness, cave size, cave span, and location of the karst cave are the primary factors that influence the ultimate bearing capacity of the karst foundation [1, 2]. Therefore, conducting an analysis of the correlation between these various factors and the bearing capacity of the karst cave roof is of utmost importance in guaranteeing the safety of karst foundation engineering projects [3]. Currently, scholars within this field are primarily concerned with addressing the issue of determining the ultimate bearing capacity of karst foundations. Drawing upon the fundamental principles of mechanical analysis, [PERSON], [PERSON] has produced an analytical solution to ascertain the ultimate bearing capacity of the karst cave roof, of which the calculated outcomes closely correspond with those produced by numerical simulation [4, 5]. However, more complex strata distribution will usually be encountered in practical projects. This causes situations where karst foundations cannot be simplified as axisymmetric models, which inevitably brings great difficulties in deriving analytical solutions for the bearing capacity of karst foundations. The establishment of a geotechnical structure's constitutive model through the finite element method (FEM) allows for predicting the karst foundation'sdeformation and stability, rendering it highly applicable in resolving intricate load and boundary issues [6]. However, the accuracy of the FEM calculation's outcomes can be influenced by multiple factors, such as meshing, type of element, and convergence criteria. Furthermore, the imprecise approximation of the displacement function poses a challenge in accurately assessing whether the FEM results are predisposed towards safety or hazard; precisely, it is difficult to determine whether it is an upper-bound solution or a lower-bound solution [7; 8]. According to the upper-bound method (UBM), the real solution of the ultimate load is located below the upper-bound solution. Therefore, the UBM can provide an upper-bound prediction for the bearing capacity of karst foundations. At present, certain investigations employing UBM have been conducted on the ultimate bearing capacity of karst foundations. However, these inquiries have omitted the impacts of three-dimensional spatial factors and have streamlined karst foundations to a two-dimensional plane strain problem [9; 10; 11; 12; 13; 14]. Although this strategy has successfully curbed computational expenses, it unavoidably leads to computational inaccuracies in complex three-dimensional circumstances [14]. The key to extending the UBM of finite element limit analysis from two-dimensional problems to three-dimensional problems lies in treating the yield function and selecting calculation methods. In two-dimensional cases, the Mohr-Coulomb yield function can be linearized through inscribed polygons, thereby, reducing the difficulty of solving and improving convergence speed [15]. However, the calculation workload will significantly increase if this method is applied to three-dimensional cases. Thus, the nonlinear programming solution of three-dimensional problems is one of the difficulties to be overcome by the UBM of finite element limit analysis. Many scholars have done much work in this regard. For example, the two-stage quasi-Newton algorithm and alternating direction method of multipliers are effective methods for solving high degree of freedom nonlinear programming problems [16; 17]. Thus far, research on the ultimate bearing capacity of a three-dimensional karst foundation has predominantly relied on experimental observations [18; 19; 20]. Moreover, earlier investigation indicates that the primary contributor to cave failure is tensile stress [21]. Despite these efforts, there remains a scarcity of studies investigating the limit analysis of three-dimensional karst foundations and the stress experienced by karst cave roofs.