Shock and Vibration

Volume 2017 (2017), Article ID 4071268, 12 pages

https://doi.org/10.1155/2017/4071268

## Finite Analytic Method for One-Dimensional Nonlinear Consolidation under Time-Dependent Loading

^{1}School of Environmental Science and Engineering, Chang’an University, Xi’an, China^{2}Key Laboratory of Subsurface Hydrology and Ecological Effect in Arid Region, Chang’an University, Ministry of Education, China^{3}School of Civil Engineering and Architecture, Shaanxi Sci-Tech University, Hanzhong, China

Correspondence should be addressed to Dawei Cheng

Received 23 October 2016; Accepted 2 March 2017; Published 30 March 2017

Academic Editor: Longjun Dong

Copyright © 2017 Dawei Cheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For one-dimensional (1D) nonlinear consolidation, the governing partial differential equation is nonlinear. This paper develops the finite analytic method (FAM) to simulate 1D nonlinear consolidation under different time-dependent loading and initial conditions. To achieve this, the assumption of constant initial effective stress is not considered and the governing partial differential equation is transformed into the diffusion equation. Then, the finite analytic implicit scheme is established. The convergence and stability of finite analytic numerical scheme are proven by a rigorous mathematical analysis. In addition, the paper obtains three corrected semianalytical solutions undergoing suddenly imposed constant loading, single ramp loading, and trapezoidal cyclic loading, respectively. Comparisons of the results of FAM with the three semianalytical solutions and the result of FDM, respectively, show that the FAM can obtain stable and accurate numerical solutions and ensure the convergence of spatial discretization for 1D nonlinear consolidation.

#### 1. Introduction

In geotechnical engineering, subsoil is subjected to complicated time-dependent loading paths which are induced by different vibration sources, such as wave action, groundwater level cyclic variation, filling and discharging in silos and tanks, vehicular traffic, blast, and earthquake. In these vibration sources, such as waves, changes in water table, loading and unloading in the granary and water towers, and vehicle traffic can cause regular time-dependent loading paths [1, 2]. However, time-dependent loading paths induced by blast, earthquake, and so on are irregular [1, 3]. Many discriminant methods, just like the waveform spectrum analysis, multivariate statistical methods, and soft computing techniques, can identify these vibration sources [4, 5].

Based on the cyclic loading classification discriminant criteria proposed by Zienkiewicz and Bettess [6], these relatively regular time-dependent loading paths, induced by wave action, groundwater level cyclic variation, and so forth, are considered as low frequency cyclic loadings. Three effects, including the soil movement inertia force and relative movement inertia force between pore water and soil skeleton, can be ignored in soil consolidation behavior under low frequency loadings. Many methods can be used to simulate these low frequency cyclic loadings, such as suddenly imposed constant loading, single ramp loading, rectangular cyclic loading, triangle cyclic loading, and trapezoidal cyclic loading [7–9]. A period of trapezoidal cyclic loadings includes four phases: loading phase, the maximum load phase, unloading phase, and rest phase, respectively. It is well known that when the duration time of the maximum loading phase is 0, the trapezoidal cyclic loading will be transformed into the triangle cyclic loading. However, when the duration time of loading and unloading phase is 0, the trapezoidal cyclic loading will be transformed into the rectangular cyclic loading [10]. Therefore, this paper will discuss the trapezoidal cyclic loading as a result of its flexibility. In addition, the suddenly imposed constant loading is a basic load form and the single ramp loading is the simplest time-dependent loading path. Thus, both of them are also discussed.

The problem of consolidation under time-dependent loading has received attention by various authors [11–13]. Based on the linear relationship of the soil’s constitutive relation, Terzaghi and Fröhlich [11] extended Terzaghi’s linear consolidation theory to various cases of time-dependent loading following a single ramp loading. Olson [12] presented charts for 1D consolidation for the case of simple ramp loading assuming a constant consolidation coefficient. Favaretti and Soranzo [13] derived solutions for common types of cyclic loadings. However, soil’s constitutive relationships are actually nonlinear [10, 14]. The coefficient of compressibility decreases with increasing effective stress. In addition, the permeability coefficient decreases with void ratio decrease.

Davis and Raymond [14] developed a nonlinear theory of consolidation assuming the relationship between void ratio and the logarithm of effective stress to conform to linear law, and the decrease in permeability coefficient is proportional to the compressibility. Xie et al. [10] derived a semianalytical solution for trapezoidal cyclic loading by assuming the validity of Davis’s nonlinear theory of consolidation. Razouki et al. [15] derived an analytical solution for consolidation under haversine cyclic loading based on the same assumption. Geng et al. [7] developed a semianalytical method to solve 1D consolidation behavior taking into account the relationship between void ratio and the logarithm of linear effective stress responses. Research has revealed that a stress-strain relation curve is more aligned with the hyperbolic model for certain types of soil, such as soft clay [16, 17]. Shi et al. [18] derived a semianalytical solution for consolidation under suddenly imposed constant loading, taking into account the stress-strain relation curve with a hyperbolic model and the decrease in permeability coefficient being proportional to the decrease in compressibility. Zhang et al. [19] adopted the same assumption, deriving a semianalytical solution for consolidation under trapezoidal cyclic loading. But, for simplifying nonlinear consolidation model, the assumptions of constant initial effective stress are often used, such as Davis’s model and Zhang’s model, which does not conform to practical condition and limits the application of those models in various initial conditions.

It is worth noting that the governing partial differential equation is nonlinear partial differential equation. It is difficult to obtain analytical solution, except under specific conditions. So developing numerical solutions for solving complex realistic problems is necessary. As a methodology, the finite analytic method (FAM) was first introduced to mainly solve heat conduction and Navier–Strokes equations [20, 21]. The combination, under FAM of the numerical method and analytical method, gives higher precision, good numerical stability, and fast convergence and is widely employed in fluid mechanics and groundwater dynamics [22–25].

In this paper, FAM is first developed to solve the governing partial differential equation of 1D nonlinear consolidation, taking into account the stress-strain behavior expressed by a hyperbolic model and where the permeability coefficient is proportional to compressibility. Then three corrected semianalytical solutions undergoing suddenly imposed constant loading, single ramp loading, and trapezoidal cyclic loading are, respectively, obtained, without the assumption of constant initial effective stress. Finally, the numerical solution of FAM is compared with these three semianalytical solutions and the numerical solution of finite difference method (FDM), respectively.

#### 2. The Governing Partial Differential Equation of 1D Nonlinear Consolidation

Modify Terzaghi’s hypothesis as follows [19]:

(a) According to results of oedometer tests, the stress-strain relation curve is fitted with a hyperbolic model [16, 17]:where is the effective stress; is the vertical strain; is the initial elastic modulus; / is the reciprocal of the final vertical strain. When equals zero, the stress-strain relationship is linear.

(b) According to results of oedometer tests, the coefficient of consolidation varies much less than the compressibility coefficient and may be taken as constant; that is, the decrease in permeability coefficient is proportional to the decrease in compressibility coefficient [14]. The coefficient of consolidation is constant; that is,where is the initial void ratio.

(c) The initial effective stress is constant; that is, where is the soil effective gravity and is the depth of the calculation.

(d) Imposed loadings change with time.

Other assumptions are the same as those in Terzaghi’s theory.

Based on the abovementioned assumptions, except hypothesis (c), the governing partial differential equation of 1D consolidation for time-dependent loading [19] can be established as follows:where is excess pore-water pressure and is the height beneath the upper boundary.

Note that hypothesis (c) means that the initial effective stress is the same for every point in depth, which does not conform to practical condition and limits the application of (4) in various initial conditions. Thus, hypothesis (c) will not be considered in the paper.

By the principle of effective stress, this becomeswhere* q*(*t*) is cyclic loading.

By defining a new parameter ,

(4) can be simplified to the following form:where .

#### 3. Finite Analytic Scheme

In the FAM, the modeling domain of (7) can be discretized into small elements by using the spatial discretization (Figure 1). A typical FAM element with the space step in a given time interval is shown in Figure 1. The origin of local coordinates is symbolized by . Therefore, the local element is