Prefabricated vertical drains (PVDs) are highly effective in hastening the consolidation process of soil and enhancing the strength of the foundation. Enhanced computational precision is achieved by utilizing a twodimensional (2D) plane strain model throughout the analytical procedure. The pronounced layering characteristic of saturated soils, coupled with the obstruction of pore water drainage across interfaces, results in a pronounced flow contact resistance effect. A comprehensive investigation into the 2D plane strain consolidation behavior of layered saturated soils under continuous drainage boundary conditions is facilitated by the presentation of the interfacial flow contact model. Subsequently, semi-analytical solutions for pore water pressure and the degree of consolidation are derived using the Laplace transform and the Crump inverse method. The proposed solution is analyzed for its degradation and compared against the experimental results and numerical solutions, to ascertain the accuracy and reliability of the presented solution. The research delves into the effects of flow contact resistance on parameters, including the permeability coefficient ratio (kv / kh) and boundary coefficients (rt and rb) throughout the consolidation process. Additionally, the impact of the flow contact resistance on the degree of consolidation is discussed. The results indicate that both the permeability coefficient ratio and boundary parameters have a close association with the flow contact resistance effect. Ignoring this effect may lead to inaccurate predictions of pore water pressure distribution and an overestimation of the soil consolidation.

Introduction Many theories of consolidation for soils have been proposed in the past, but most of them have ignored the structural characteristics of clay, yet the natural layered soils are widely distributed around the world.Methods A theoretical model is established to analyze the one-dimensional consolidation behavior of layered soils, in which a time-dependent drainage boundary and the structural characteristics of the soil are taken into account. Using the integral transform and characteristic function methods, the analytical solution is derived, the effectiveness of which is evaluated against the degradation of solutions and the numerical results calculated using the finite element method.Results and discussion Finally, the influences of interface parameter, soil permeability coefficient and soil compressibility on consolidation behaviors are discussed. Results show that in structured soils, early dissipation of excess pore water pressure and consolidation rates are predominantly influenced by interface parameters, permeability, and volume compression coefficients. Higher values of these parameters accelerate early stages of consolidation, which is especially evident in the upper soil layers. Over time, the distinct effects of interface and permeability coefficients on consolidation diminish. Higher volume compression coefficients, while initially beneficial, eventually slow down the consolidation process, indicating an interaction with the ongoing soil structural changes.

Permeable pipe piles accelerate the bearing capacity of the pile foundation by releasing the excess pore water pressure (EPWP) of the soil around the pile through appropriate openings in the pile body. This study couples the Material Point Method (MPM) and the Finite Element Method (FEM) to establish a full-process model of pile driving and consolidation of permeable piles, and proposes a continuous drainage boundary condition that can reflect the plugging effect of permeable holes. The correctness of the model and boundary conditions are verified by comparison with experiments, and then the effects of soil properties, opening characteristics, and boundary permeability on the accelerated consolidation effect of permeable piles are analyzed. The results show that: the permeable pile with a permeable area ratio greater than 50% and a local opening ratio greater than 5% can save more than 60% of the consolidation time compared to conventional piles; the proposed boundary conditions can accurately describe the permeability of the permeable hole under the influence of plugging; in addition, the calculation formulae for the accelerated consolidation effect of permeable piles and the variation of continuous drainage boundary interface parameters with permeable area ratio are given, which can provide references for engineering design.

Laboratory one-dimensional consolidation tests were conducted to measure the variation trend of the soil pore pressure at the drainage boundary with time under different magnitudes of loads. Based on the test data, continuous drainage boundary interface parameters under arbitrary loads were inversely derived, the reasonableness of which was verified by comparing the theoretical values of the boundary pore pressure with the experimental results. Moreover, the one-dimensional consolidation model of the layered foundation was established with a continuous drainage boundary. The semianalytical solution of the corresponding model under an arbitrary load was given by using the boundary transformation method. A comparison with degraded results and the finite-element calculation results verified the correctness of the present solutions. Finally, the influences of the interface parameters and loading rate on the soil consolidation behavior were studied, where three different types of loads (i.e., linear, exponential, and simple harmonic) were considered. The results revealed that the consolidation rate reaches the peak value for the linear loading pattern when the loading is completed. Moreover, the exponential load used to describe the surcharge preloading method also positively influenced the theoretical analysis due to its concise expression form. When the simple harmonic load was applied, the excess pore-water pressure in the soil element presented stable periodic vibration after the first cyclic load. In addition, the loading rate and interface parameters exhibited different influences on the consolidation behaviors. The research results of this paper can provide a theoretical reference for the settlement calculation of subgrades during the construction and operation phases.

To comprehensively consider the impacts of stratification, residual pore water pressure, soil nonlinearity, and boundary permeability on consolidation settlement of soft soil foundations for accurate prediction, a continuous drainage boundary condition is proposed in this study that reflects the residual pore pressure under multistage loading, and a nonlinear elastic constitutive model based on the double logarithmic model is adopted to account for the nonlinear consolidation behaviour of soils. A UMAT subroutine is developed based on the proposed boundary condition and nonlinear elastic constitutive model. Subsequently, the developed subroutine is compared with the built-in linear elastic soil constitutive model in ABAQUS and engineering examples. The application of continuous drainage boundaries in stratified foundations is analysed, as well as the influence of factors such as the loading rate and soil nonlinearity on consolidation settlement. The results indicate that, compared to the built-in model, the subroutine developed in this study can be employed to more accurately calculate the nonlinear consolidation of multilayered foundations under multistage loading. By adjusting the loading rate parameter alpha k, consolidation under different loading conditions can be predicted. Additionally, the proposed boundary condition simplifies the calculations for soft soil foundations with sand layers, providing a novel computational approach for the design of construction loading schemes and long-term settlement predictions in soft soil foundations.

Conventionally, drainage boundaries are often assumed to be either perfectly permeable or completely impermeable. However, a more realistic approach considers continuous drainage boundaries. In this context, an analytical solution for double drainage consolidation in vertical drains is derived. The proposed method is evaluated against existing solutions and finite element simulations. The study investigates the impact of drainage capacity, soil nonlinearity, smear effect, and well resistance. The results show that the continuous drainage boundary parameters (i.e., b and c) significantly affect the distribution of excess pore water pressure and the consolidation rate. Increasing b and c allows realistic modeling of drainage capacity variations from impermeable to permeable boundaries. Notably, when b not equal c, the maximum excess pore water pressure plane shifts from the mid-height of the foundation soil, diverging from conventional consolidation theory. Soil nonlinearity (Cc/Ck) and boundary permeability (b and c) jointly affect consolidation. Higher Cc/Ck values correlate with more detrimental consolidation effects. Minimizing disturbance around vertical drains during construction is crucial due to well resistance and smear zone effects, which can significantly slow down consolidation. This study provides an analytical solution considering soil nonlinearity for predicting consolidation in actual engineering scenarios involving vertical drainage trenches.