Analytical Solution For One-dimensional Consolidation Research Articles

Analytical solution for one-dimensional consolidation of structured soft soils with continuous drainage boundary

A closed form analytical solution is proposed to analyze the one-dimensional consolidation behavior of structured saturated clay soils under continuous drainage boundary. Firstly, the soil structural properties and arbitrary loading conditions are taken into account in the established mathematical model. Using the finite sine Fourier transform and characteristic function methods, analytical calculation is conducted, and the solution effectiveness is evaluated against the degradation of solutions and the results of finite difference analysis. Finally, the influences of interface parameter, properties of soil structure and structural yield stress on consolidation behaviors are discussed. Results show that the larger the interface parameter is, the easier the soil structure enters the failure stage, and the smaller the consolidation degree is. Furthermore, the time for initiating the failure stage does not change, when the interface parameter and structural yield stress keep unchanged; in the meanwhile, the greater the consolidation coefficient is, the greater the consolidation degree of foundation becomes when the soil structure begins to fail, and the earlier the soil enters the complete failure stage. When the interface parameter is determined, the ratio of structural yield stress to load can accelerate the soil consolidation rate, but all consolidation curves coincide for the same ratio.

Analytical solution for one-dimensional consolidation of unsaturated soils under dynamic load

Analytical solution for one-dimensional consolidation of unsaturated soils under dynamic load

One-dimensional consolidation of multilayered aquifer systems with viscoelastic properties induced by time-dependent groundwater drawdown

Most analytical and semi-analytical models for pumping-induced land subsidence invoke the simplifying assumptions regarding characteristics of geomaterials, as well as the pattern of drawdown response to pumping. This paper presents an analytical solution for one-dimensional consolidation of the multilayered soil due to groundwater drawdown, in which viscoelastic property and time-dependent drawdown are taken into account. The presented solution is developed by using the boundary transformation techniques. The validity of the proposed solution is verified by comparing with a degenerated case for a single layer, as well as with the numerical solutions and experimental results for a two-layer system. The difference between the average consolidation degree U p , defined by hydraulic head, and U s , defined by total settlement, is discussed. Detailed parametric studies are conducted to reveal the effects of viscoelastic properties and drawdown patterns on the consolidation process. It is revealed that while the effect of different drawdown response patterns is significant during the early–intermediate stages of consolidation, the viscoelastic properties may have a more dominant influence on long-term consolidation behaviour, depending on the values of the material parameters, which are reflected in both the deformation process of soil layers and the dissipation of excess pore-water pressure.

Analytical Solution for One-Dimensional Consolidation of Soil with Exponentially Time-Growing Drainage Boundary under a Ramp Load

By introducing the exponentially time-growing drainage boundary, this paper investigated the one-dimensional consolidation problem of soil under a ramp load. Firstly, the one-dimensional consolidation equations of soil are established when there is a ramp load acting on the soil surface. Then, the analytical solution of excess pore water pressure and consolidation degree is derived by means of the method of separation of variables and the integral transform technique. The rationality of this solution is also verified by comparing it with other existing analytical solutions. Finally, the consolidation behavior of soil is studied in detail for different interface parameters or loading scheme. The results show that the exponentially time-growing drainage boundary can reflect the phenomenon that the excess pore water pressure at the drainage boundaries dissipates smoothly rather than abruptly from its initial value to the value of zero. By adjusting the values of interface parametersbandc, the presented solution can be degraded to Schiffman’s solution, which can compensate for the shortcoming that Terzaghi’s drainage boundary can only consider the two extreme cases of fully pervious and impervious boundaries. The significant advantage of the exponentially time-growing drainage boundary is that it can be applied to describe the asymmetric drainage characteristics of the top and bottom drainage surfaces of the actual soil layer by choosing the appropriate interface parametersbandc.

Analytical solution for one-dimensional consolidation of double-layered soil with exponentially time-growing drainage boundary

In this article, the exponentially time-growing drainage boundary is introduced to study the one-dimensional consolidation problem of double-layered soil. First, the one-dimensional consolidation equations of soil underlying a time-dependent loading are established. Then, the analytical solution of excess pore water pressure and average consolidation degree is obtained by utilizing the method of separation of variables when the soil layer is separately undergone instantaneous load and single-stage load. The validity of the present solution is proven by the comparison with other existing analytical solution. Finally, the influence of soil properties and loading scheme on the consolidation behavior of soil is investigated in detail. The results indicate that, the present solution can be degraded to Xie’s solution utilizing Terzaghi’s drainage boundary by adjusting the interface parameter, that is to say, Xie’s solution can be regarded as a special case of the present solution. The interface parameter has a significant influence on the excess pore water pressure of soil, and the larger interface parameter means the better drainage capacity of the soil layer.

A practical analytical solution for one-dimensional consolidation

In this work a four-step strategy to derive an analytical solution to the one-dimensional consolidation equation under a general time-dependent loading is presented. The strategy is based on the eigenfunction expansion method. Most existing solutions in the specialised literature are developed for a particular type of loading, whereas the proposed strategy can be easily applied to obtain an accurate response for a general loading. In order to assess the validity of the proposed strategy, it is applied to constant, single ramp, cyclic square and cyclic haversine loading profiles. The results are in agreement with the analytical solution previously obtained by other researchers.

Analytical Solution for One-Dimensional Consolidation of Soil Layer Induced by Time-Dependent Groundwater Drawdown

The governing equation was formulated for one-dimensional consolidation of the soil layer induced by time-dependent groundwater drawdown. Using Duhamel's theorem and method of separation of variables, analytical solutions were developed for the problem. Based on the solutions, the consolidation behaviors of the soil layer were studied and the factors influencing the consolidation were investigated. The results show that it is necessary to consider both the changes of total stress and pore water pressure on the boundary. The greater initial time factor Tvc is, the slower the rate of the consolidation is.

One dimensional consolidation under cyclic loading

When cyclic loads are applied to the soil, the deformation may increase significantly depending upon the magnitude of load and nature of soil. Cyclic loads such as those induced by traffic on motorways and railways, ocean waves are long-term, sometimes including drainage during or after cyclic loading, resulting consolidation. Analytical solutions for one-dimensional consolidation of saturated cohesive soil are available under slow cyclic loading where pore water pressure builds up during loading and dissipates during unloading. In the proposed paper, an analytical solution for one-dimensional consolidation under rapid long term cyclic loading has been developed. It has been found that rate of consolidation is faster and stable condition is reached earlier in comparison with the monotonic loading.

Analytical solution for one-dimensional consolidation of clayey soils with a threshold gradient

General approximate analytical solutions are developed for one-dimensional consolidation with consideration of the threshold gradient under a time-dependent loading. A comparison is made between the present solution and some available numerical solutions for a particular case, and the results show that the approach employed in this article is reasonable. The influence of the threshold gradient and the loading period on consolidation behaviour is investigated, and the results show that the moving boundary of seepage moves downward gradually. The greater the threshold gradient is, the slower the boundary moves. The excess pore pressure will not be completely dissipated at the end of consolidation, and the larger the threshold gradient is, the greater the residual excess pore pressure is. The average degree of consolidation considering the threshold gradient defined by settlement is different from that defined by pore pressure. Moreover, the greater the threshold gradient is, the larger the average degree of consolidation in terms of strain, whereas the smaller the average degree of consolidation in terms of stress. It is also shown that the longer the loading period is, the longer the time moving boundary takes to reach the bottom of the layer, and the greater the average degree of consolidation is.

Nonlinear analytical solution for one-dimensional consolidation of soft soil under cyclic loading

This paper presents an analytical solution for one-dimensional consolidation of soft soil under some common types of cyclic loading such as trapezoidal cyclic loading, based on the assumptions proposed by Davis and Raymond (1965) that the decrease in permeability is proportional to the decrease in compressibility during the consolidation process of the soil and that the distribution of initial effective stress is constant with depth. It is verified by the existing analytical solutions in special cases. Using the solution obtained, some diagrams are prepared and the relevant consolidation behavior is investigated.