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Rigorous solution for 1D consolidation of a clay layer under haversine cyclic loading with rest period
 Nina Müthing^{1}Email authorView ORCID ID profile,
 Sabah S. Razouki^{2},
 Maria Datcheva^{3} and
 Tom Schanz^{1}
 Received: 21 September 2016
 Accepted: 7 November 2016
 Published: 17 November 2016
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 steadystate at a time factor T _{ v } 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 wellknown 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 timedependent 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 quasistatic 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 1D 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 1D heat diffusion problems, Carslaw and Jaeger (1959) considered, among others, periodic boundary conditions in a rectangular wave form or a sinewave form only. This means that they focused their attention only on the homogeneous 1D heat equation with periodic boundary conditions.
The problem of onedimensional 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 onedimensional 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 onedimensional 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.
Governing differential equation and clay deposit boundary conditions
The initial condition is given as u(z, 0) = 0.
Fourier representation of rate of imposition of the haversine load
Solution of governing differential equation
Excess pore water pressure
For the special case of zero rest period (i.e. α = 0), all the terms in the first series \(\sum \limits _{m=1}^{\infty }\dfrac{\sin \dfrac{\alpha m \pi }{1+\alpha }}{(1+\alpha )^2m^2}\) in Eq. (20) vanish except the first term for m = 1. The limit of this term for \(\alpha \rightarrow 0\) is \(\lim \limits _{\alpha \rightarrow 0}\dfrac{\sin \dfrac{\alpha \pi }{1+\alpha }}{(1+\alpha )^21} = \dfrac{\pi }{2}\)
Effective stress
Verification and parametric study
For the purpose of calculating the time variation of excess pore water pressure at any depth in the clay deposit based on the analytical solution, a computer program using the software Matlab2010 was written. To achieve high accuracy of results, one hundred terms of each series was covered through the analysis to ensure convergence of each infinite series. Moreover, the analytical solution was compared to the approximate numerical solution obtained via finite element method (FEM) employing the FE code PLAXIS.
Effect of rest period
To study the effect of rest period on the consolidation process due to repeated haversine loading, the solution was obtained for the PTIB case and the following αvalues of α = 1, 2, 3 and 4.
The average of the applied normalized cyclic loading is 0.25, 0.167, 0.125 and 0.1 for α = 1, 2, 3 and 4 respectively.
Accordingly, Fig. 5 shows the decrease in effective stress due to increase in αvalues. This means that an increase in the rest period for the applied haversine cyclic loading causes a decrease in the settlement of the deposit.
Effect of frequency of cyclic loading
To study the effect of frequency of cyclic loading on the consolidation process, the solution was obtained for the PTIB case of α = 1 for the following T _{0} values namely T _{0} = 0.5, 1, 2, 5 and 10.
Comparison with the results of analysis via finite element method (FEM)
The analytical solution was compared with the numerical solution of the considered consolidation problem obtained using the finite element (FE) software PLAXIS (Brinkgreve et al. 2010). However, the problem to be solved with help of the FEM is posed slightly differently. To avoid a singular system of equations, a finite stiffness is assigned to the water. By combining the linear elastic behaviour of the soil skeleton and the flow of the water through the pore system a system of equations for displacements and pore pressure as unknowns in the FE nodes is defined. Details of the numerical model can be found in Razouki et al. (2013).
Summarizing the results presented in this section it can be concluded that the analytical and numerical solutions coincide perfectly at least for pore water pressure and effective stress evolution at the considered three locations and the fit remains good for different rest periods and load frequencies.
Conclusions

Although the loading function for imposed excess porewater is always positive, the excess pore water pressure at any depth in the clay deposit (PTPB or PTIB case) changes sign during the consolidation process causing both positive and negative “excess” porewater pressure to develop.

Due to the haversine cyclic loading, the transient state of the consolidation process is almost completed at a dimensionless time factor T _{ v } of about 2.0. Therefore, the steady state of undamped oscillation takes place and continues with continuous cyclic loading.

For a given dimensionless time factor T _{0}, an increase in the rest period causes the final average effective stress to decrease and to converge to the average of the loading function. For T _{0} = 0.15 and α = 1, 2, 3 and 4, the average effective stress converges to \(\frac{q}{4}\), \(\frac{q}{6}\), \(\frac{q}{8}\), and \(\frac{q}{10}\) respectively. The decrease in effective stress causes the settlement of the deposit to decrease.

A decrease in the frequency of the loading function (T _{0} increases) causes the maximum effective stress to increase converging to a steady state after a few number of cycles.
Declarations
Authors' contributions
The analytical solution presented in this study was derived by SSR. The evaluation and comparison of the analytical results were performed by all authors in equal shares. All authors read and approved the final manuscript.
Acknowledgements
Visiting Prof. Razouki wishes to express his thanks and gratitude to Prof. Schanz, Chair of Foundation Engineering, Soil and Rock Mechanics, for the invitation to the Ruhr University Bochum. Thanks are due to Mr. Kavan Khaledi and Mr. Usama AlAnbaki for assisting in computer programming.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis 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.
Authors’ Affiliations
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