###### Abstract  
To reasonably analyze the bearing characteristics of curved beam bridge pile foundation under the combined action of vertical force (\(V\)) and torque (\(T\)) in expansive soil, a method of determining \(T_{\text{u}}\) (limit torque) under the action of \(V\) is proposed. Considering the effects of expansive force and ground heave after immersion, the load transfer function at the pile-soil interface with positive and negative friction resistance is established. The nonlinear solution of a single pile under vertical load is achieved by the finite difference method. Subsequently, the circumferential limit friction resistance is modified, and a loop iteration program is compiled through MATLAB. Thus, the bearing capacity of single pile under the loading path of \(V\)\(\rightarrow\)\(T\) is obtained. The corresponding failure envelope curve is drawn and verified by laboratory model tests. Based on the verified solution, the bearing capacity envelope curve and deformation characteristics of single pile are analyzed. The influence of the expansion rate on the bearing capacity envelope curve is discussed, and reasonable engineering suggestions are put forward. single pile; expansive soil; bearing characteristics; \(V\)\(\rightarrow\)\(T\) loading path; model test +
Footnote †: journal: Applied sciences  
immersion in expansive soil would raise the single pile and generate an axial internal force in the pile. In theory, [PERSON] and [PERSON] [9] considered the influence of the expansion force, while the internal force of the single pile was solved by the elastic theory method after immersion. However, there is a significant error between the calculated results and the experimental data. [PERSON] and [PERSON] [10] proposed a hyperbolic model to predict the uplift force of single pile in expansive soil based on the model pile uplift test, pile-soil interface shear test, and consolidation test. Nonetheless, test process is cumbersome. [PERSON] et al. [11] proposed the analytical solution of internal force and displacement for the single pile after immersion, considering the variety of soil expansion rates and stiffnesses with depth. The drawback is that calculation parameters are difficult to determine. For the vertically loaded piles in expansive soil, [PERSON] et al. [12] used the shear displacement method and superposition principle to solve the analytical solution. There were errors in the calculation results still. To solve the displacement response of a single pile in expansive unsaturated soil under vertical load, [PERSON] and [PERSON] [13] considered the influence of matric suction on the ultimate frictional resistance of the pile-side. It should be known that obtaining the water-soil characteristic curve takes much time. In conclusion, the current theories for determining the axial force and the displacement response under vertical load in expansive soil after immersion are not perfect. For _V-T_ combined loaded piles, [PERSON] et al. [14; 15] used independent nonlinear axial and torsional springs to simulate the interaction of the pile-soil interface, and the numerical solution of pile top load-displacement was obtained in a clay foundation. [PERSON] and [PERSON] [16] analyzed the difference in the bearing capacity envelopes of the _V-T_ combined loaded piles under different loading paths using three-dimensional finite element software in clay foundation. [PERSON] et al. [17] obtained the nonlinear solution of _V-T_ combined loaded piles in clay by finite difference. [PERSON] et al. [5] obtained the elastoplastic analytical solution of _V-T_ combined loaded piles in the Gibson foundation. However, the above studies were carried out in non-expansive soil. Due to axial force and displacement of the pile after immersion in expansive soil, the mechanical mechanism of _V-T_ combined loaded piles in expansive soil foundations is significantly different from the above studies. Therefore, the above theories are not fully applicable to expansive soil foundations.