# One-Dimensional Consolidation Properties of Soft Clay under Multi-Stage Loading
## 1 Introduction  
Soft clay typically has a high water content, poor permeability, high compressibility, and low strength [1, 2, 3, 4, 5, 6]. When subjected to an external load, the excess pore water pressure gradually dissipates as the free water in the soil is discharged. This leads to an increase in effective stress and the development of settlement deformation until the soil reaches a stable state. While consolidation theory provides an idealized concept of soil settling, it is important to note that the actual process is highly complex in practical engineering. This complexity is a major factor contributing to errors in [PERSON]'s classical consolidation theory [7]. For instance, in highway construction, preloading is often employed to mitigate post-construction settlement and reduce the construction period [8, 9, 10]. Accurately predicting embankment settlement is crucial for ensuring the stability of the foundation [11, 12]. The traditional Terzaghi theory assumes a constant consolidation coefficient and a uniformly distributed and continuously applied external load. However, this is rarely the case in practical engineering conditions, leading to a significant discrepancy between the settlement estimated using the Terzaghi theory and the measured value. Therefore, it has become crucial in soil mechanics research to enhance the consolidation theory of soft soil to account for the complexities encountered in practical situations. [PERSON] and [PERSON] [13] improved the linear relationship between deformation and stress by incorporating the nonlinear stress-deformation relationship of \(e\)-lg\(P\). However, they still made the assumption that the coefficient of consolidation remains constant andthat the permeability is directly proportional to the compressibility. They successfully derived the analytical solution for one-dimensional non-linear consolidation. In a study by [PERSON] et al. [14], indoor and field tests were conducted to investigate the thermal conductivity of Bangkok saturated soft clay. The findings indicated that the thermal conductivity of clay increases with soil density. The study also discussed the reliability of the results obtained through different test methods and provided insights for enhancing the consolidation process of soft soil using heat treatment technology. [PERSON] et al. proposed that the permeability coefficient, k, and the volume compressibility coefficient, mv, are polynomial and exponential functions of depth, respectively. They utilized the finite difference method to solve the one-dimensional problem of soft soil consolidation under instantaneous loading [15]. [PERSON] and [PERSON] [16] propose that the amount of soil deformation is equivalent to the volume of water discharged from the soil pores. They theoretically derive a calculation method for the soil consolidation degree based on water content. [PERSON] et al. [17] demonstrate that the consolidation coefficient of soft soil is influenced by the pre-consolidation coefficient and varies with the consolidation stress. [PERSON] et al. [18] investigate the compression and consolidation behavior of structured natural clay and propose formulas for settlement and consolidation with greater sensitivity. [PERSON] et al. [19] propose a three-dimensional thawing consolidation theory that combines [PERSON]'s consolidation theory with conduction equations, taking into account the effects of the ice-water phase change through specific heat. [PERSON] et al. [20] develop an axisymmetric large strain consolidation model with the void ratio as the variable under equal strain conditions. They provide difference schemes for the model equation, initial condition, and boundary condition, and evaluate the influence of different consolidation theories and self-weight stress on average consolidation. [PERSON] et al. investigated the impact of pumping velocity, loading pressure, pumping and loading type on consolidation behavior.