scholarly journals 2D Plane Strain Consolidation Process of Unsaturated Soil with Vertical Impeded Drainage Boundaries

Processes ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 5 ◽  
Author(s):  
Minghua Huang ◽  
Dun Li
Keyword(s):  
Analytical Solution ◽  
Laplace Transform ◽  
Plane Strain ◽  
Unsaturated Soil ◽  
Analytical Solutions ◽  
Soil Layer ◽  
Permeability Ratio ◽  
Consolidation Process ◽  
The Time Domain ◽  
Excess Pore Pressures

The consolidation process of soil stratum is a common issue in geotechnical engineering. In this paper, the two-dimensional (2D) plane strain consolidation process of unsaturated soil was studied by incorporating vertical impeded drainage boundaries. The eigenfunction expansion and Laplace transform techniques were adopted to transform the partial differential equations for both the air and water phases into two ordinary equations, which can be easily solved. Then, the semi-analytical solutions for the excess pore-pressures and the soil layer settlement were derived in the Laplace domain. The final results in the time domain could be computed by performing the numerical inversion of Laplace transform. Furthermore, two comparisons were presented to verify the accuracy of the proposed semi-analytical solutions. It was found that the semi-analytical solution agreed well with the finite difference solution and the previous analytical solution from the literature. Finally, the 2D plane strain consolidation process of unsaturated soil under different drainage efficiencies of the vertical boundaries was illustrated, and the influences of the air-water permeability ratio, the anisotropic permeability ratio and the spacing-depth ratio were investigated.

The Analytical Solutions of Incompressible Saturated Poroelastic Circular Mindlin’s Plate

2012 ◽  
Vol 79 (5) ◽  
Author(s):  
P. H. Wen
Keyword(s):  
Laplace Transform ◽  
Circular Plate ◽  
Analytical Solutions ◽  
Time Domain ◽  
Numerical Algorithms ◽  
Integral Equation Method ◽  
Fundamental Solutions ◽  
Boundary Integral ◽  
Concentrated Force ◽  
The Time Domain

In this paper the fundamental solutions for an infinite poroelastic moderately thick plate and analytical solutions for a circular plate saturated by a incompressible fluid are derived in the Laplace transform domain. In order to obtain the solutions in the time domain, the Durbin’s Laplace transform inverse method has been used with high accuracy. The formulations using the boundary integral equation method can be derived directly with these fundamental solutions. In addition, the analytical solutions for a circular plate can be used to validate the accuracy of numerical algorithms such as the boundary element method and the method of fundamental solution. The deflection, moment, and equivalent moment in the time domain for a circular plate, subjected to uniform load and a concentrated force are presented, respectively. The analytical solutions demonstrate that interaction between the solid and flow is significant.

scholarly journals Analytical Solution of General Bagley-Torvik Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-4 ◽  
Author(s):  
William Labecca ◽  
Osvaldo Guimarães ◽  
José Roberto C. Piqueira
Keyword(s):  
Numerical Methods ◽  
Analytical Solution ◽  
Laplace Transform ◽  
Analytical Solutions ◽  
Empirical Data ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Polynomial Form ◽  
Inhomogeneous Term

Bagley-Torvik equation appears in viscoelasticity problems where fractional derivatives seem to play an important role concerning empirical data. There are several works treating this equation by using numerical methods and analytic formulations. However, the analytical solutions presented in the literature consider particular cases of boundary and initial conditions, with inhomogeneous term often expressed in polynomial form. Here, by using Laplace transform methodology, the general inhomogeneous case is solved without restrictions in boundary and initial conditions. The generalized Mittag-Leffler functions with three parameters are used and the solutions presented are expressed in terms of Wiman’s functions and their derivatives.

scholarly journals A General Analytical Solution for Axisymmetric Consolidation of Unsaturated Soil with Impeded Drainage Boundaries

Geofluids ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Ming-hua Huang ◽  
Chang Lv ◽  
Zheng-lin Zhou
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Laplace Transform ◽  
Unsaturated Soil ◽  
Fractured Reservoirs ◽  
Improvement Project ◽  
Soil Stratum ◽  
General Analytical Solution ◽  
Consolidation Behavior ◽  
Excess Pore

The consolidation of soil is one of the most common phenomena in geotechnical engineering. Previous studies for the axisymmetric consolidation of unsaturated soil have usually idealized the boundary conditions as fully drained and absolutely undrained, but the boundaries of unsaturated soil are actually impeded drainage in most practical situations. In this study, we present a general analytical solution for predicting the axisymmetric consolidation behavior of unsaturated soil that incorporates impeded drainage boundary conditions in both the radial and vertical directions simultaneously. The impeded drainage boundary is modeled using the third kind boundary, and it can also realize fully drained and absolutely undrained ones by changing the drainage parameter. A general analytical solution is developed to predict the excess pore-air and pore-water pressures as well as the average degree of consolidation in an unsaturated soil stratum using the common methods of eigenfunction expansion and Laplace transform. The newly developed solution is expressed in the product of the terms of time, depth, and radius, which are derived using Laplace transform, usual Fourier, and Fourier-Bessel series, respectively. The eigenfunctions and eigenvalues are evaluated from the impeded drainage boundaries in both radial and depth dimensions. Then, the correctness of the proposed analytical solution is verified against the existing analytical solution for the case of traditional boundaries and against the finite difference solution for the case of general impeded drainage boundaries, and excellent agreements are obtained. Finally, the axisymmetric consolidation behavior of unsaturated soil involving impeded drainage boundaries is demonstrated and analyzed, and the effects of the drainage parameters are discussed. The results indicate that the larger drainage parameter generally expedites the dissipations of the excess pore pressures and further promotes the soil settling process. As the drainage parameter increases, its influence gradually diminishes and even can be neglected when it is larger than 100. The general analytical solution and findings of this study can help for better understanding the axisymmetric consolidation behavior of the unsaturated soil stratum in the ground improvement project with vertical drains as well as the gas-oil gravity drainage mechanism in the naturally fractured reservoirs.

scholarly journals Semi-analytical solution for consolidation of vertical drain to unsaturated soils considering well resistance and smear effect under time-varying loadings

2020 ◽  
Vol 198 ◽  
pp. 02033
Author(s):  
Aifang Qin ◽  
Lianghua Jiang ◽  
Linzhong Li ◽  
Xinhao Li
Keyword(s):  
Analytical Solution ◽  
Laplace Transform ◽  
Unsaturated Soils ◽  
Inverse Laplace Transform ◽  
Loading Frequency ◽  
Time Varying ◽  
Numerical Inverse Laplace Transform ◽  
Vertical Drain ◽  
Special Cases ◽  
The Time Domain

In this paper, based on equal-strain assumption a semi-analytical solution, considering well re-sistance, smear effect and time-varying loading, is deduced for radial consolidation aided by vertical drain (VD) to unsaturated soils. Firstly, by employing the general integration, Laplace transform, decoupling methods and numerical inverse Laplace transform, the semi-analytical solution in the time domain is ob-tained. Then, its validity is verified by the special cases of the proposed solution under instantaneous loading. Finally, the case analysis show that the dissipation of excess pore pressures is accelerated with the decrease of smear coefficients (αa or αw) or well resistance factors (Ga or Gw). In addition, when the well resistance factor is less than 1, the barrier of VD material to flow can be ignored. Furthermore, a smaller value of the loading frequency of cyclic loading, the bigger the amplitude, and the less fluctuation period in the dissipa-tion rates. Moreover, the current solution can analyse the consolidation characteristics of unsaturated soils with VDs under arbitrary time-varying loadings, including cyclic loadings.

scholarly journals New Analytical Solutions for Longitudinal Vibration of a Floating Pile in Layered Soils with Radial Heterogeneity

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1294 ◽  
Author(s):  
Zhimeng Liang ◽  
Chunyi Cui ◽  
Kun Meng ◽  
Yu Xin ◽  
Huafu Pei ◽  
...  
Keyword(s):  
Analytical Solution ◽  
Analytical Solutions ◽  
Longitudinal Vibration ◽  
Three Dimensional ◽  
Complex Impedance ◽  
Vibration Characteristics ◽  
Layered Soils ◽  
Transform Method ◽  
Floating Pile ◽  
The Time Domain

Based on the theory of wave propagation in three-dimensional (3D) continuum, a new analytical approach for the longitudinal vibration characteristics of a floating pile in layered soils with radial heterogeneity is developed by employing a viscous-type damping model. Firstly, an analytical solution for the longitudinal complex impedance at the pile head is deduced by employing the Laplace transform and complex stiffness technique with the compatibility conditions of the pile and radially inhomogeneous surrounding soil. Secondly, a semi-analytical solution in the time domain is further acquired by using the inverse Fourier transform method. Furthermore, the corresponding analytical solutions are validated through contrasts with previous solutions. Finally, parametric analyses are underway to investigate the effect of radial heterogeneity of surrounding soils on longitudinal vibration characteristics of floating piles. It is indicated that the proposed approach and corresponding solutions can provide a more wide-ranging application than the simple harmonic vibration for longitudinal vibration analysis of a floating pile in soils.

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