Rigorous solution for 1-D consolidation of a clay layer under haversine cyclic loading with rest period

Müthing et al. SpringerPlus (2016) 5:1987 DOI 10.1186/s40064-016-3660-9 Open Access RESEARCH Rigorous solution for 1‑D consolidation of a clay layer under haversine cyclic loading with rest period Nina Müthing1* , Sabah S. Razouki2, Maria Datcheva3 and Tom Schanz1 *Correspondence: nina. 1 Chair of Foundation Engineering, Soil and Rock Mechanics, Ruhr-Universität Bochum, Bochum, Germany Full list of author information is available at the end of the article Abstract Presented in this paper is a rigorous solution of the conventional Terzaghi onedimensional consolidation under haversine cyclic loading with any rest period. The clay deposit is either permeable at both top and bottom or permeable at the top and impermeable at the bottom. This exact analytical solution was achieved using Fourier harmonic analysis for the periodic function representing the rate of imposition of excess pore water pressure. The double Fourier series in the rigorous solution was found to be rapidly convergent. The analysis of excess pore water pressure and effective stress is done in the Matlab 2010 environment. Both the effects of rest period and frequency of cyclic loading are investigated. The analysis reveals that the excess pore water pressure arrives the steady-state at a time factor Tv of about 2. Furthermore, finite element method (FEM) is applied to solve numerically the corresponding consolidation problem and the FEM solution is compared to the analytical solution showing a good match. Keywords: Analytical solution, Consolidation, Cyclic loading, Haversine repeated loading, Rigorous solution Background It is well-known that cyclic loading of soils may result from natural phenomena or human activities such as wind and water waves, vehicular traffic, reciprocating machinery and others (Mitchell 1993; Zhang et al. 2009). Special structures such as silos and fluid tanks that undergo filling and discharging subject their foundation soils to loading unloading stages that repeat themselves more or less periodically over time (Conte and Troncone 2006). Many forms of time-dependent behaviour of repeated loading such as sinusoidal, rectangular, triangular, trapezoidal and haversine waves were suggested by various authors as the type and duration of loading to be used in any repeated load test or analysis should simulate that actually occurring in the field (Zimmerer 2009; Huang 1993; Barksdale 1971; Razouki and Schanz 2011). Zienkiewicz et al. (1980) studied a soil layer subject to a periodic surface force represented by a function in complex form containing both real and imaginary parts (Kreyszig 2006) to find out under what conditions such extremes as undrained or quasi-static © The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Müthing et al. SpringerPlus (2016) 5:1987 Page 2 of 13 assumptions can be safely used. They used their solution for earthquake analysis of an earth dam and they carried out a parametric analysis of pore pressure distribution in a seabed due to the passage of a surface wave. Due to the fact that many problems of 1-D consolidation of cohesive soils have an equivalent problem in the heat condition in solids, it is necessary to review the wave forms considered by Carslaw and Jaeger (1959) in the field of heat diffusion. Using either the Fourier series approach or the Laplace transforms approach for solving 1-D heat diffusion problems, Carslaw and Jaeger (1959) considered, among others, periodic boundary conditions in a rectangular wave form or a sine-wave form only. This means that they focused their attention only on the homogeneous 1-D heat equation with periodic boundary conditions. The problem of one-dimensional consolidation under cyclic loading (rectangular, triangular, sinusoidal and trapezoidal waves) has received attention by various authors, Baligh and Levadoux (1987), Favaretti and Soranzo (1995), Guan et al. (2003), Geng et al. (2006) and Hsu and Lu (2006). However, the problem of haversine repeated loading in one-dimensional consolidation analysis has received attention for the first time by Razouki and Schanz (2011). They applied a numerical implicit finite difference method to obtain the solution of the Terzaghi conventional consolidation theory under haversine cyclic loading and investigated the effect of rest period on the time variation of excess pore water pressure and effective stress. They concluded that an increase in rest period reduces the final average effective stress and hence the settlement. Razouki et al. (2013) derived an analytical solution of the Terzaghi one-dimensional consolidation under haversine cyclic loading without rest period and analysed the main features of the process based on that solution. The comparison of the analytical solution with a corresponding finite element solution shows excellent agreement. The goal of this paper is to obtain the analytical solution in the general case of haversine cyclic loading with any rest period which is of greater importance and benefit in practice of geotechnical engineering. Haversine loading waveform with rest period to be considered here is the same as that reported by Razouki and Schanz (2011) which is a periodic function L(t) whose definition in the fundamental period is given by    0    � �  R 2 π t− L(t) = q sin  d 2     0 R 0≤t≤ 2 R R ≤t≤d+ 2 2 R d+ ≤t ≤d+R 2 (1) where q is the amplitude of loading, d is the duration of loading within a loading cycle and R is the duration of rest period. Figure 1 illustrates the form of the haversine loading function for a chosen rest period of R/d = 1. Due to the fact that only one-dimensional consolidation is of interest in this paper, the haversine loading is considered uniformly distributed and applied to the top surface of the clay deposit and as a result the loading rate is given as Normalized Loading L(t)/q Müthing et al. SpringerPlus (2016) 5:1987 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0 2 Page 3 of 13 4 6 8 Dimensionless Time t/d 10 12 14 Fig. 1 Haversine cyclic loading with rest period for the case of dR = 1    0    � �  2π R dL qπ = sin t−  dt d d 2     0 R 0≤t≤ 2 R R ≤t≤d+ 2 2 R d+ ≤t ≤d+R 2 (2) Governing differential equation and clay deposit boundary conditions For the case of time-dependent loading, the governing differential equation for onedimensional consolidation analysis becomes (Verruijt 1995; Coussy 2004) ∂ 2u dL ∂u (3) = Cz 2 + η ∂t ∂z dt γ , γ and β are the compressibility of the solid and the water and φ is where η = γ +φβ the porosity; Cz is the coefficient of consolidation in vertical direction z (Fig. 2) and u(z, (...truncated)