# One-Dimensional Consolidation Analysis of Unsaturated Soils under Cyclic Loadings

Authors: Pyol Kim; Myongchol Ri; Yonggun Kim; Gunhyang Ri; Hakbom Myong
Journal: Shock and Vibration
 (2020)

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## Abstract

This paper presents analytical solutions to Fredlund and Hasan’s one-dimensional consolidation equations for unsaturated soils subjected to various cyclic loadings. Two new variables are introduced so that the governing equations for excess pore air and water pressures can be transformed to a set of conventional diffusion equations. Based on the general solutions for two introduced variables, the analytical solutions are derived for one-dimensional consolidation of unsaturated soils under trapezoidal, rectangular, triangular, and haversine cyclic loadings. It shows through the degeneration into the existing solutions for unsaturated and saturated soils that the proposed solutions are more general ones for one-dimensional consolidation of soils from unsaturated to saturated states. A comprehensive parametric study is conducted to investigate the effects of different parameters on one-dimensional consolidation of unsaturated soils under various cyclic loadings. The proposed solutions can be effectively utilized in the analysis of consolidation of unsaturated soils subjected to various cyclic loadings.


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## Body

## 1. Introduction

In geotechnical engineering, soils beneath many structures may be subjected to complicated cyclic loadings. These cyclic loadings are induced by different vibration sources, such as filling and discharging in silos, tanks, and reservoirs; traffic loads on highways, railways, or airports; wave action for offshore structures; wind forces in high buildings; pile construction; and machine vibrations. The consolidation characteristics of soils subjected to cyclic loadings are very complex and different from those of soils under simple time-dependent loadings. Thus, consolidation behavior of subsoils under cyclic loadings has drawn considerable attention and has become the focus of academic research.Over the past few decades, numerous studies have been conducted on consolidation behavior of saturated soils under various cyclic loadings. The existing investigation of consolidation behavior of saturated soils under cyclic loadings can be divided into two groups. One focused on the consolidation based on Terzaghi’s linear consolidation theory, while the other was based on the nonlinear consolidation theory. Preceding works on the former included the efforts by Wilson and Elgohary [1], Favaretti and Soranzo [2], Conte and Troncone [3], Razouki and Schanz [4], Razouki et al. [5], and Kim et al. [6]. Wilson and Elgohary [1] presented an analytical solution for one-dimensional consolidation of saturated soil subjected to rectangular cyclic loading, whereas Favaretti and Soranzo [2] and Conte and Troncone [3] derived solutions for different types of cyclic loadings. Razouki and Schanz [4] and Razouki et al. [5] studied one-dimensional consolidation of a clay layer subjected to haversine cyclic loading with and without rest period. Kim et al. [6] investigated one-dimensional consolidation behavior of saturated clays under time-dependent loadings including haversine cyclic loading based on the lattice Boltzmann method. The latter mainly included the work by Baligh and Levadoux [7], Xie et al. [8], Geng et al. [9], Cai et al. [10], Conte and Troncone [11], Toufigh and Ouria [12], Cheng et al. [13], Hu et al. [14], and Kim et al. [15, 16]. Baligh and Levadoux [7] and Toufigh and Ouria [12] analyzed one-dimensional nonlinear consolidation of inelastic clays under rectangular cyclic loading using the superimposing rule. Based on the nonlinear consolidation theory proposed by Davis and Raymond [17], Xie et al. [8], Conte and Troncone [11], Cheng et al. [13], and Kim et al. [15] investigated one-dimensional nonlinear consolidation of saturated soil subjected to various cyclic loadings. Hu et al. [14] proposed a numerical method for modelling one-dimensional nonlinear consolidation of multilayer soil under various cyclic loadings based on the differential quadrature method. Geng et al. [9] and Cai et al. [10] developed a semianalytical method for one-dimensional nonlinear consolidation of soil with variable compressibility and permeability under cyclic loadings. More recently, Kim et al. [16] derived analytical solutions for one-dimensional nonlinear consolidation of a saturated clay layer under cyclic loadings with the consideration of variable consolidation coefficient.Most soils in nature are unsaturated and develop much more complex properties during the consolidation process than saturated soils. In the recent decades, a number of studies on the consolidation theory for unsaturated soil has been conducted, and a great progress has been achieved. Blight [18] derived a consolidation equation for the air phase of dry, rigid, unsaturated soil, whereas Barden [19] presented an analysis of the one-dimensional consolidation of compacted unsaturated clay. By assuming that the water and air phases are continuous, Fredlund and Hasan [20] presented an uncoupled one-dimensional consolidation theory for unsaturated soil, in which two partial differential equations for water and air phases were derived. Meanwhile, some scholars [21–23] proposed coupled models for consolidation in unsaturated soil. Although coupled models are theoretically more rigorous than uncoupled ones, the uncoupled theory proposed by Fredlund and Hasan [20] is now widely accepted because it is generally simple and sufficiently accurate for practical projects [24, 25]. On the basis of the consolidation theory proposed by Fredlund and Hasan [20], many researchers have investigated consolidation of unsaturated soil under different time-dependent loadings [26–43]. Among them, Feng et al. [30] presented an analytical solution for one-dimensional consolidation of unsaturated soil under sinusoidal cyclic loading. Ho and Fatahi [31] and Wang et al. [32] suggested analytical and semianalytical solutions to Fredlund and Hasan’s one-dimensional consolidation of unsaturated soils subjected to different time-dependent loadings including sinusoidal loading. On the contrary, based on the coupled consolidation model of poroelasticity, Lo et al. [44] proposed a closed-form analytical solution for one-dimensional consolidation of unsaturated soil under harmonic cyclic loading.Despite these advances, it seems that research on consolidation of unsaturated soils subjected to different cyclic loadings is very limited. Most researchers performed investigations on consolidation behavior of unsaturated soils subjected to simple time-dependent loadings, such as constant, ramp, exponential, and a few sine loadings, but little research has been done on consolidation behavior of unsaturated soils under various cyclic loadings.In this paper, based on the consolidation theory proposed by Fredlund and Hasan [20], analytical solutions are derived for one-dimensional consolidation of unsaturated soils under various cyclic loadings, such as trapezoidal, rectangular, triangular, and haversine cyclic loadings. The proposed solutions are degenerated into the existing solutions for unsaturated soils under constant and ramp loadings and for saturated soils under haversine cyclic loading. Then, a comprehensive parametric study is conducted to investigate the effects of different parameters on one-dimensional consolidation of unsaturated soils under various cyclic loadings.


## 2. Governing Equations

An unsaturated soil layer of thicknessH subjected to time-dependent loading qt is considered as shown in Figure 1. In order to obtain the analytical solution for one-dimensional consolidation of the unsaturated soil layer subjected to time-dependent loading, the main assumptions are summarized as follows:Figure 1
An unsaturated soil layer subjected to time-dependent loading.(1)
The flows of air and water phases are continuous and independent
(2)
The soil particles and water phase are incompressible
(3)
The effects of temperature change, air diffusing through water and air dissolving in the water, and the movement of vapor are ignored
(4)
The coefficients of permeability and volume change for both air and water phases remain constant during the consolidation process
(5)
The deformation takes place only along the vertical directionThe governing equations for one-dimensional consolidation of unsaturated soils subjected to time-dependent loading are as follows [19, 23]:(1)∂ua∂t=−Ca∂uw∂t+Cva∂2ua∂z2+Cσa∂q∂t,(2)∂uw∂t=−Cw∂ua∂t+Cvw∂2uw∂z2+Cσw∂q∂t,where ua and uw are the excess pore air and pore water pressures, respectively. The consolidation parameters can be expressed as follows:(3)Ca=m2am1a−m2a−n01−Sr0/u^a,Cva=kaRTabsgu^aMam1a−m2a−n01−Sr0/u^a,Cσa=m1am1a−m2a−n01−Sr0/u^a,Cw=1−m2w/m1wm2w/m1w,Cvw=kwγwm2w,Cσw=m1wm2w,where m1a and m1w are the coefficients of air and water volume change with respect to a change in the net normal stress σ−ua, respectively; m2a and m2w are the coefficients of air and water volume change with respect to a change in the matric suction ua−uw, respectively; ka and kw are the coefficients of permeability for air and water phases, respectively; g is the gravitational acceleration; u^a is the absolute pore air pressure, u^a=ua+u^atm, in which u^atm is the atmospheric pressure, 101 kPa; when the pore air pressure ua is small or rapidly dissipates during consolidation, u^a can be considered constant, and therefore, in this study, u^a=u^atm; Ma is the molecular mass of air, 0.029 kg/mol; R is the universal gas constant, 8.314 J/mol/K; Tabs is the absolute temperature, (20 + 273.16)K; n0 is the initial porosity; Sr0 is the initial degree of saturation; and γw is the unit weight of water.The initial excess pore air and pore water pressures can be expressed as follows:(4)uaz,0=ua0,uwz,0=uw0,where ua0 and uw0 are the initial excess pore air and pore water pressures, respectively.The boundary conditions for the excess pore air and pore water pressures can be given by the following equations:(5)ua0,t=0,uw0,t=0,∂uaH,t∂z=0,∂uwH,t∂z=0for single drainage condition,uaH,t=0,uwH,t=0for double drainage condition.Equations (1) and (2) can be presented in nondimensional form by using nondimensional parameters:(6)∂u¯a∂T=Aa∂2u¯a∂z¯2+Aw∂2u¯w∂z¯2+Aσ∂q¯∂T,(7)∂u¯w∂T=Wa∂2u¯a∂z¯2+Ww∂2u¯w∂z¯2+Wσ∂q¯∂T,where u¯a=ua/qu, u¯w=uw/qu, q¯=q/qu, z¯=z/H, qu = ultimate value of loading q, T=kw/γwm1st/H2, m1s = coefficient of volume change with respect to a change in net normal stress σ−ua, m1s=m1a+m1w,(8)Aa=Cva1−CwCa⋅γwm1skw,Aw=−CaCvw1−CwCa⋅γwm1skw,Aσ=Cσa−CaCσw1−CwCa,Wa=−CwCva1−CwCa⋅γwm1skw,Ww=Cvw1−CwCa⋅γwm1skw,Wσ=Cσw−CwCσa1−CwCa.The dimensionless initial and boundary conditions can be expressed as follows:(9)u¯az¯,0=u¯a0,u¯wz¯,0=u¯w0,u¯a0,T=0,u¯w0,T=0,∂u¯a1,T∂z¯=0,∂u¯w1,T∂z¯=0for single drainage condition,u¯a1,T=0,u¯w1,T=0for double drainage condition,where u¯a0=ua0/quandu¯w0=uw0/qu.


## 3. Derivation of Analytical Solutions under Cyclic Loadings

According to the simple analytical solutions presented by Zhou et al. [29], (8) and (9) can be transformed to a set of conventional diffusion equations by introducing two new variables ϕ1 and ϕ2:(10)∂ϕi∂T=ξi∂2ϕi∂z¯2+ηi∂q¯∂T,i=1,2,where(11)ϕ1=u¯a+α21u¯w,ϕ2=α12u¯a+u¯w,ξ1,2=12Aa+Ww±Aa−Ww2+4AwWa,η1=Aσ+α21Wσ,η2=α12Aσ+Wσ,α12=Waξ2−Aa,α21=Awξ1−Ww.The corresponding initial and boundary conditions for excess pore air and pore water pressures can be transformed as follows:(12)ϕiz¯,0=ϕi0,i=1,2,(13)ϕi0,T=0,i=1,2,(14)∂ϕi1,T∂z¯=0,i=1,2for single drainage condition,(15)ϕi1,T=0,i=1,2for double drainage condition,where ϕ10=u¯a0+α21u¯w0andϕ20=α12u¯a0+u¯w0.For the single drainage condition, the general solutions of (10) subjected to the solution conditions of (12)–(14) can be obtained as follows:(16)ϕi=∑m=1∞2MsinMz¯e−βmTTmT,where M=2m−1π/2,m=1,2,3,…, βm=M2ξi, TmT=∫0TeβmτRτdτ, and RT=ηi∂q¯/∂T, in which i=1,2.The solutions for the double drainage condition can be easily obtained by replacingz¯ in (16) with 2z¯ considering the drainage height (i.e., H/2).The complete solutions ofϕi for various cyclic loadings can be calculated by substituting the different loading functions into (16).On the basis of the definition of the average degree of consolidation, the degree of consolidation,Ui, with respect to ϕi for different cyclic loadings can be expressed as follows:(17)Ui=ηiq¯−∫01ϕidz¯,i=1,2.The detailed mathematical process of obtaining the degree of consolidation,Ui, can be found in Appendix.


### 3.1. Trapezoidal Cyclic Loading

Trapezoidal cyclic loading shown in Figure2(a) can be expressed as follows:(18)qt=quαt0t−N−1βt0,N−1βt0≤t≤N−1β+αt0,qu,N−1β+αt0≤t≤N−1β+1−αt0,−quαt0t−N−1βt0−t0,N−1β+1−αt0≤t≤N−1β+1t0,0,N−1β+1t0≤t≤Nβt0,where t0 is the period of one loading cycle; α and β are the loading parameters corresponding to the rate of loading increment or decrement and the rest period of loading, respectively; and N is the cycle number.Figure 2
Various cyclic loadings: (a) trapezoidal cyclic loading; (b) rectangular cyclic loading; (c) triangular cyclic loading; (d) haversine cyclic loading.
(a)
(b)
(c)
(d)Transforming (18) into nondimensional form and substituting into (16), the complete solutions of (10) can be obtained as follows:(19)ϕi=∑m=1∞2C1iMsinMz¯e−M2ξiT,N−1βt0≤t≤N−1β+αt0,∑m=1∞2C2iMsinMz¯e−M2ξiT,N−1β+αt0≤t≤N−1β+1−αt0,∑m=1∞2C3iMsinMz¯e−M2ξiTN−1β+1−αt0≤t≤N−1β+1t0,∑m=1∞2C4iMsinMz¯e−M2ξiT,N−1β+1t0≤t≤Nβt0,where(20)C1i=DiN−1+D1i,C2i=DiN−1+D2i,C3i=DiN−1+D2i+D3i,C4i=DiN,Din=∑N=1nD2i+D4i,D1i=ηiM2ξiTceM2ξiT−eM2ξiTb,D2i=ηiM2ξiTceM2ξiTbeM2ξiTc−1,D3i=−ηiM2ξiTceM2ξiT−eM2ξiTf−Tc,D4i=−ηiM2ξiTceM2ξiTf1−e−M2ξiTc,Tc=kwγwm1s⋅αt0H2,Tb=kwγwm1s⋅N−1βt0H2,Tf=kwγwm1s⋅N−1β+1t0H2,i=1,2.Substituting (19) into (17), the solutions of Ui for trapezoidal cyclic loading can be given as follows:(21)Ui=ηiTcT−Tb−∑m=1∞2C1iM2e−M2ξiT,N−1βt0≤t≤N−1β+αt0,ηi−∑m=1∞2C2iM2e−M2ξiT,N−1β+αt0≤t≤N−1β+1−αt0,ηiTcTf−T−∑m=1∞2C3iM2e−M2ξiT,N−1β+1−αt0≤t≤N−1β+1t0,−∑m=1∞2C4iM2e−M2ξiT,N−1β+1t0≤t≤Nβt0,where i=1,2.


### 3.2. Rectangular and Triangular Cyclic Loadings

Rectangular and triangular cyclic loadings can be degenerated from trapezoidal cyclic loading. Whenα⟶0, trapezoidal cyclic loading reduces into rectangular cyclic loading (Figure 2(b)). In addition, trapezoidal cyclic loading reduces to triangular cyclic loading (Figure 2(c)) when α=0.5. Therefore, the solutions of ϕi and Ui for rectangular and triangular cyclic loadings can be given by substituting α=0 and α=0.5 into (19) and (21), respectively.


### 3.3. Haversine Cyclic Loading

Haversine cyclic loading shown in Figure2(d) can be expressed as follows:(22)qt=qusin2πtt0.Transforming (22) into nondimensional form and substituting into (16), the complete solutions of (10) can be obtained as follows:(23)ϕi=∑m=1∞2CiMsinMz¯e−M2ξiT,i=1,2,where(24)Ci=ηiπM4ξi2T02+4π2M2ξiT0sin2πTT0eM2ξiT−2πcos2πTT0eM2ξiT−1,i=1,2,T0=kwγwm1st0H2.Substituting (23) into (17), the solutions of Ui for haversine cyclic loading can be given as follows:(25)Ui=ηisin2πTT0−∑m=1∞2CiM2e−M2ξiT,i=1,2.


### 3.4. Excess Pore Pressures and Average Degree of Consolidation

Based on the solutions ofϕi, the normalized excess pore air and pore water pressures can be given by following relations:(26)u¯a=α21ϕ2−ϕ1α12α21−1,(27)u¯w=α12ϕ1−ϕ2α12α21−1.Based on the solutions ofUi for different cyclic loadings, the average degrees of consolidation for both air and water phases can be expressed as follows:(28)Ua=α21U2−U1α12α21−1Aσ⋅100%,Uw=α12U1−U2α12α21−1Wσ⋅100%,where Ua and Uw are the average degrees of consolidation for air phase and water phase, respectively.


## 3.1. Trapezoidal Cyclic Loading

Trapezoidal cyclic loading shown in Figure2(a) can be expressed as follows:(18)qt=quαt0t−N−1βt0,N−1βt0≤t≤N−1β+αt0,qu,N−1β+αt0≤t≤N−1β+1−αt0,−quαt0t−N−1βt0−t0,N−1β+1−αt0≤t≤N−1β+1t0,0,N−1β+1t0≤t≤Nβt0,where t0 is the period of one loading cycle; α and β are the loading parameters corresponding to the rate of loading increment or decrement and the rest period of loading, respectively; and N is the cycle number.Figure 2
Various cyclic loadings: (a) trapezoidal cyclic loading; (b) rectangular cyclic loading; (c) triangular cyclic loading; (d) haversine cyclic loading.
(a)
(b)
(c)
(d)Transforming (18) into nondimensional form and substituting into (16), the complete solutions of (10) can be obtained as follows:(19)ϕi=∑m=1∞2C1iMsinMz¯e−M2ξiT,N−1βt0≤t≤N−1β+αt0,∑m=1∞2C2iMsinMz¯e−M2ξiT,N−1β+αt0≤t≤N−1β+1−αt0,∑m=1∞2C3iMsinMz¯e−M2ξiTN−1β+1−αt0≤t≤N−1β+1t0,∑m=1∞2C4iMsinMz¯e−M2ξiT,N−1β+1t0≤t≤Nβt0,where(20)C1i=DiN−1+D1i,C2i=DiN−1+D2i,C3i=DiN−1+D2i+D3i,C4i=DiN,Din=∑N=1nD2i+D4i,D1i=ηiM2ξiTceM2ξiT−eM2ξiTb,D2i=ηiM2ξiTceM2ξiTbeM2ξiTc−1,D3i=−ηiM2ξiTceM2ξiT−eM2ξiTf−Tc,D4i=−ηiM2ξiTceM2ξiTf1−e−M2ξiTc,Tc=kwγwm1s⋅αt0H2,Tb=kwγwm1s⋅N−1βt0H2,Tf=kwγwm1s⋅N−1β+1t0H2,i=1,2.Substituting (19) into (17), the solutions of Ui for trapezoidal cyclic loading can be given as follows:(21)Ui=ηiTcT−Tb−∑m=1∞2C1iM2e−M2ξiT,N−1βt0≤t≤N−1β+αt0,ηi−∑m=1∞2C2iM2e−M2ξiT,N−1β+αt0≤t≤N−1β+1−αt0,ηiTcTf−T−∑m=1∞2C3iM2e−M2ξiT,N−1β+1−αt0≤t≤N−1β+1t0,−∑m=1∞2C4iM2e−M2ξiT,N−1β+1t0≤t≤Nβt0,where i=1,2.


## 3.2. Rectangular and Triangular Cyclic Loadings

Rectangular and triangular cyclic loadings can be degenerated from trapezoidal cyclic loading. Whenα⟶0, trapezoidal cyclic loading reduces into rectangular cyclic loading (Figure 2(b)). In addition, trapezoidal cyclic loading reduces to triangular cyclic loading (Figure 2(c)) when α=0.5. Therefore, the solutions of ϕi and Ui for rectangular and triangular cyclic loadings can be given by substituting α=0 and α=0.5 into (19) and (21), respectively.


## 3.3. Haversine Cyclic Loading

Haversine cyclic loading shown in Figure2(d) can be expressed as follows:(22)qt=qusin2πtt0.Transforming (22) into nondimensional form and substituting into (16), the complete solutions of (10) can be obtained as follows:(23)ϕi=∑m=1∞2CiMsinMz¯e−M2ξiT,i=1,2,where(24)Ci=ηiπM4ξi2T02+4π2M2ξiT0sin2πTT0eM2ξiT−2πcos2πTT0eM2ξiT−1,i=1,2,T0=kwγwm1st0H2.Substituting (23) into (17), the solutions of Ui for haversine cyclic loading can be given as follows:(25)Ui=ηisin2πTT0−∑m=1∞2CiM2e−M2ξiT,i=1,2.


## 3.4. Excess Pore Pressures and Average Degree of Consolidation

Based on the solutions ofϕi, the normalized excess pore air and pore water pressures can be given by following relations:(26)u¯a=α21ϕ2−ϕ1α12α21−1,(27)u¯w=α12ϕ1−ϕ2α12α21−1.Based on the solutions ofUi for different cyclic loadings, the average degrees of consolidation for both air and water phases can be expressed as follows:(28)Ua=α21U2−U1α12α21−1Aσ⋅100%,Uw=α12U1−U2α12α21−1Wσ⋅100%,where Ua and Uw are the average degrees of consolidation for air phase and water phase, respectively.


## 4. Degeneration of the Proposed Solution to Special Cases

The proposed solutions can be degenerated into the existing solutions for one-dimensional consolidation of both unsaturated and saturated soils subjected to different time-dependent loadings.


### 4.1. Constant Loading

Whenα=0,β=1, and N=1, trapezoidal cyclic loading reduces to constant loading (Figure 3(a)) expressed by qt=qu. Then, the solutions of ϕi and Ui can be given by(29)ϕi=∑m=1∞2ϕi0MsinMz¯e−M2ξiT,i=1,2,(30)Ui=ϕi0−∑m=1∞2ϕi0M2e−M2ξiT,i=1,2.Figure 3
Constant loading (a) and ramp loading (b).
(a)
(b)Equations (29) and (30) are the solutions of ϕi and Ui for one-dimensional consolidation of unsaturated soil under constant loading and constant initial condition proposed by Zhou et al. [29].It can be seen from Figure4 that the degenerate solutions from this paper are consistent with the existing solutions and the proposed solutions are accurate.Figure 4
Comparison between the degenerate solution and the existing solution for constant loading.
(a)
(b)


### 4.2. Ramp Loading

WhenN=1 and t0 tends to infinity, trapezoidal cyclic loading reduces to ramp loading (Figure 3(b)) given by(31)qt=ttcqu,t≤tc,qu,t≥tc,where tc=αt0.The solutions ofϕi and Ui can be obtained by(32)ϕi=∑m=1∞2ηiM3ξiTcsinMz¯1−e−M2ξiT,t≤tc,∑m=1∞2ηiM3ξiTcsinMz¯e−M2ξiTeM2ξiTc−1,t≥tc,(33)Ui=ηiTTc−∑m=1∞2ηiM4ξiTc1−e−M2ξiT,t≤tc,ηi−∑m=1∞2ηiM4ξiTce−M2ξiTeM2ξiTc−1,t≥tc,where i=1,2 and Tc=kw/γwm1s⋅tc/H2.Equations (32) and (33) are the solutions of ϕi and Ui for one-dimensional consolidation of unsaturated soil subjected to ramp loading proposed by Zhou et al. [29].Figure5 shows the comparison between the degenerate solutions and the existing solutions by Zhou et al. [29] for ramp loading. It can be found that the degenerate solutions from this paper show good agreement with the existing results, which proves that the proposed solutions are more general ones for one-dimensional consolidation of unsaturated soil under time-dependent loading.Figure 5
Comparison between the degenerate solution and the existing solution for ramp loading.
(a)
(b)


### 4.3. Degeneration to Saturated Condition for Haversine Cyclic Loading

The consolidation coefficients for unsaturated soil can be reduced to those for saturated soil: the coefficients of water volume change (m1w and m2w) are equal to the coefficient of volume change (mv); the coefficients of air volume change (m1a and m2a) are zero; air pressure may be assumed to be equal to water pressure, that is, ua−uw=0; Cw=0, Cvw=Cv=kw/γwmv, Cσw=1, Ca=0, Cva=0, and Cσa=0. Then, the corresponding parameters in (8) and (9) become Aa=0, Aw=0, Aσ=0, Wa=0, Ww=1, and Wσ=1. In addition, ξ1=0, ξ2=1, α12=0, α21=0, η1=0, and η2=1. It follows from (26) and (27) that u¯w=ϕ2. The solution for the excess pore water pressure of saturated soil with single drainage condition under haversine cyclic loading can be obtained as follows:(34)u¯w=ϕ2=∑m=1∞2C2MsinMz¯e−M2T,where(35)C2=πM4T02+4π2M2T0sin2πTT0eM2T−2πcos2πTT0eM2T−1,T0=kwγwm1st0H2.Equation (34) is the solution for one-dimensional consolidation of saturated soil subjected to haversine cyclic loading proposed by Razouki et al. [5].As observed in Figure6, identical results are obtained from both analytical solutions, suggesting that the proposed solutions are reliable as well as more general for one-dimensional consolidation of a soil from unsaturated to saturated states.Figure 6
Comparison between the degenerate solution and the existing solution for haversine cyclic loading.


## 4.1. Constant Loading

Whenα=0,β=1, and N=1, trapezoidal cyclic loading reduces to constant loading (Figure 3(a)) expressed by qt=qu. Then, the solutions of ϕi and Ui can be given by(29)ϕi=∑m=1∞2ϕi0MsinMz¯e−M2ξiT,i=1,2,(30)Ui=ϕi0−∑m=1∞2ϕi0M2e−M2ξiT,i=1,2.Figure 3
Constant loading (a) and ramp loading (b).
(a)
(b)Equations (29) and (30) are the solutions of ϕi and Ui for one-dimensional consolidation of unsaturated soil under constant loading and constant initial condition proposed by Zhou et al. [29].It can be seen from Figure4 that the degenerate solutions from this paper are consistent with the existing solutions and the proposed solutions are accurate.Figure 4
Comparison between the degenerate solution and the existing solution for constant loading.
(a)
(b)


## 4.2. Ramp Loading

WhenN=1 and t0 tends to infinity, trapezoidal cyclic loading reduces to ramp loading (Figure 3(b)) given by(31)qt=ttcqu,t≤tc,qu,t≥tc,where tc=αt0.The solutions ofϕi and Ui can be obtained by(32)ϕi=∑m=1∞2ηiM3ξiTcsinMz¯1−e−M2ξiT,t≤tc,∑m=1∞2ηiM3ξiTcsinMz¯e−M2ξiTeM2ξiTc−1,t≥tc,(33)Ui=ηiTTc−∑m=1∞2ηiM4ξiTc1−e−M2ξiT,t≤tc,ηi−∑m=1∞2ηiM4ξiTce−M2ξiTeM2ξiTc−1,t≥tc,where i=1,2 and Tc=kw/γwm1s⋅tc/H2.Equations (32) and (33) are the solutions of ϕi and Ui for one-dimensional consolidation of unsaturated soil subjected to ramp loading proposed by Zhou et al. [29].Figure5 shows the comparison between the degenerate solutions and the existing solutions by Zhou et al. [29] for ramp loading. It can be found that the degenerate solutions from this paper show good agreement with the existing results, which proves that the proposed solutions are more general ones for one-dimensional consolidation of unsaturated soil under time-dependent loading.Figure 5
Comparison between the degenerate solution and the existing solution for ramp loading.
(a)
(b)


## 4.3. Degeneration to Saturated Condition for Haversine Cyclic Loading

The consolidation coefficients for unsaturated soil can be reduced to those for saturated soil: the coefficients of water volume change (m1w and m2w) are equal to the coefficient of volume change (mv); the coefficients of air volume change (m1a and m2a) are zero; air pressure may be assumed to be equal to water pressure, that is, ua−uw=0; Cw=0, Cvw=Cv=kw/γwmv, Cσw=1, Ca=0, Cva=0, and Cσa=0. Then, the corresponding parameters in (8) and (9) become Aa=0, Aw=0, Aσ=0, Wa=0, Ww=1, and Wσ=1. In addition, ξ1=0, ξ2=1, α12=0, α21=0, η1=0, and η2=1. It follows from (26) and (27) that u¯w=ϕ2. The solution for the excess pore water pressure of saturated soil with single drainage condition under haversine cyclic loading can be obtained as follows:(34)u¯w=ϕ2=∑m=1∞2C2MsinMz¯e−M2T,where(35)C2=πM4T02+4π2M2T0sin2πTT0eM2T−2πcos2πTT0eM2T−1,T0=kwγwm1st0H2.Equation (34) is the solution for one-dimensional consolidation of saturated soil subjected to haversine cyclic loading proposed by Razouki et al. [5].As observed in Figure6, identical results are obtained from both analytical solutions, suggesting that the proposed solutions are reliable as well as more general for one-dimensional consolidation of a soil from unsaturated to saturated states.Figure 6
Comparison between the degenerate solution and the existing solution for haversine cyclic loading.


## 5. Parametric Study

In this section, a comprehensive parametric study is conducted to investigate the effects of the coefficients ratio of permeability for air and water phases and the other loading parameters on consolidation behavior of unsaturated soils with single drainage condition under various cyclic loadings. The material properties adopted in this study are as follows:(36)n0=0.50,Sr0=0.80,kw=10−10m/s,H=10m,qu=100kPa,m1s=−2.5×10−4kPa−1,m2s=0.4m1s,m1w=0.2m1s,m2w=4m1w.The initial excess pore air and pore water pressures are assumed to be zero (i.e.,ua0=0 and uw0=0) because of q0=0 for cyclic loadings except for rectangular cyclic loading, while ua0=20kPa and uw0=40kPa (constant with depth) for rectangular cyclic loading and constant loading.


### 5.1. Effects ofka/kw

Figures7 and 8 represent the variation in the average degrees of consolidation for both water and air phases with time at different values of ka/kw under various cyclic loadings. It can be found that the average degrees of consolidation Uw and Ua increase but fluctuate with the increase in ka/kw, indicating that the dissipation rate of excess pore pressures increase with the fluctuation when ka/kw increases. Moreover, it can be noticed that the higher values of ka/kw lead to the bigger amplitudes of fluctuation in the average degrees of consolidation Uw and Ua.Figure 7
The effects ofka/kw on consolidation of unsaturated soils under trapezoidal cyclic loading (a) and rectangular cyclic loading (b).
(a)
(b)Figure 8
The effects ofka/kw on consolidation of unsaturated soils under triangular cyclic loading (a) and haversine cyclic loading (b).
(a)
(b)It can be observed through the locally enlarged diagrams with different cyclic loading forms that there is similar characteristics for the dissipation process of excess pore pressures and the change in applied loading. The variation in the average degrees of consolidation with time shows the oscillation with the same frequency and the similar forms in response to the applied cyclic loadings. It can also be seen that the dissipation process of excess pore pressures does not complete after a long time but rather continues oscillating with a certain amplitude. It is due to repetitive loading and unloading stages of cyclic loadings, which results in squeezing out of pore water and air during the loading stage and then absorbing during the unloading stage.


### 5.2. Effects oft0

Figure9 demonstrates the variation in average degrees of consolidation for both water and air phases with time under trapezoidal cyclic loading (α=0.2 and β=1.5) and haversine cyclic loading with different loading parameters t0 for ka/kw=1. It can be seen that a smaller value of the loading parameter t0 induces a smaller amplitude and more cycles of fluctuation in the dissipation rate of excess pore water and air pressures. In addition, the loading parameter t0 has a significant influence on the amplitude and the cycles of fluctuation in the dissipation rate of excess pore pressures but does not affect the mean values of the dissipation rates. The loading parameter t0 is the period of loading, and thus, it can be concluded that the longer the period of loading, the bigger the amplitude, and the less the cycles of fluctuation in the dissipation rates.Figure 9
The effects oft0 on consolidation of unsaturated soils under trapezoidal cyclic loading (a) and haversine cyclic loading (b).
(a)
(b)


### 5.3. Effects ofα

Figure10 presents the variation in average degrees of consolidation for both water and air phases with time under different types of cyclic loadings with β=1.5 and different loading parameters α for ka/kw=1. It can be seen that the smaller the loading parameter α is, the more quickly the dissipation of excess pore water and air pressures develops. Since the loading parameter α reflects the rate of loading increment or decrement, it can be found that the higher the rate of loading increment or decrement is, the faster the dissipation rates of excess pore pressures are. Moreover, the dissipation rates of excess pore pressures under rectangular cyclic loading are the biggest and the dissipation rates under triangular cyclic loading are the smallest. For the comparison with the results for static loading, the variation in average degrees of consolidation under constant loading is also shown in Figure 10. It can be observed that the average degrees of consolidation under cyclic loadings increase but fluctuate with a certain amplitude, whereas the average degrees of consolidation under constant loading increase monotonically with time. This phenomenon implies that consolidation of unsaturated soil under cyclic loadings does not complete after a long time, but rather continues oscillating with a certain amplitude, generating continuous volume change of the soil. Thus, neglecting the effect of cyclic loadings may result in overestimated settlement and cannot consider the continuous volume change.Figure 10
The effects ofα on consolidation of unsaturated soils under various cyclic loadings.
(a)
(b)


### 5.4. Effects ofβ

Figure11 depicts the variation in average degrees of consolidation for both water and air phases with time under different cyclic loadings with different loading parameters β for ka/kw=1. It can be found that the dissipation rates of excess pore water and air pressures increase with the decrease in the value of loading parameter β. Furthermore, a smaller value of the loading parameter β results in more cycles and a smaller amplitude of fluctuation in the dissipation rates. The loading parameter β is related to the rest period of loading. Thus, it is quite obvious that a longer rest period of loading induces the decrease in the dissipation rates, less cycles, and a bigger amplitude of fluctuation in the dissipation rates of excess pore pressures. As a result of that the dissipation rates of excess pore water and air pressures under the cyclic loading without rest period (β=1) are the biggest for each case. Specifically, rectangular cyclic loading without rest period reduces into constant loading, and the dissipation rates of excess pore water and air pressures are the highest and increase monotonically with time in that case, as shown in Figure 11(b).Figure 11
The effects ofβ on consolidation of unsaturated soils under various cyclic loadings: (a) trapezoidal cyclic loading; (b) rectangular cyclic loading; (c) triangular cyclic loading.
(a)
(b)
(c)It should be noted that the solutions presented in this paper do not consider nonlinear characteristics of soil properties such as the coefficients of permeability and volume change for both air and water phases under time-dependent loading. Even though they are limited for a linear elastic stress-strain model of unsaturated soil based on Fredlund and Hasan’s one-dimensional consolidation theory, the proposed solutions are more general ones for one-dimensional consolidation of soils from unsaturated to saturated state under time-dependent loading.


## 5.1. Effects ofka/kw

Figures7 and 8 represent the variation in the average degrees of consolidation for both water and air phases with time at different values of ka/kw under various cyclic loadings. It can be found that the average degrees of consolidation Uw and Ua increase but fluctuate with the increase in ka/kw, indicating that the dissipation rate of excess pore pressures increase with the fluctuation when ka/kw increases. Moreover, it can be noticed that the higher values of ka/kw lead to the bigger amplitudes of fluctuation in the average degrees of consolidation Uw and Ua.Figure 7
The effects ofka/kw on consolidation of unsaturated soils under trapezoidal cyclic loading (a) and rectangular cyclic loading (b).
(a)
(b)Figure 8
The effects ofka/kw on consolidation of unsaturated soils under triangular cyclic loading (a) and haversine cyclic loading (b).
(a)
(b)It can be observed through the locally enlarged diagrams with different cyclic loading forms that there is similar characteristics for the dissipation process of excess pore pressures and the change in applied loading. The variation in the average degrees of consolidation with time shows the oscillation with the same frequency and the similar forms in response to the applied cyclic loadings. It can also be seen that the dissipation process of excess pore pressures does not complete after a long time but rather continues oscillating with a certain amplitude. It is due to repetitive loading and unloading stages of cyclic loadings, which results in squeezing out of pore water and air during the loading stage and then absorbing during the unloading stage.


## 5.2. Effects oft0

Figure9 demonstrates the variation in average degrees of consolidation for both water and air phases with time under trapezoidal cyclic loading (α=0.2 and β=1.5) and haversine cyclic loading with different loading parameters t0 for ka/kw=1. It can be seen that a smaller value of the loading parameter t0 induces a smaller amplitude and more cycles of fluctuation in the dissipation rate of excess pore water and air pressures. In addition, the loading parameter t0 has a significant influence on the amplitude and the cycles of fluctuation in the dissipation rate of excess pore pressures but does not affect the mean values of the dissipation rates. The loading parameter t0 is the period of loading, and thus, it can be concluded that the longer the period of loading, the bigger the amplitude, and the less the cycles of fluctuation in the dissipation rates.Figure 9
The effects oft0 on consolidation of unsaturated soils under trapezoidal cyclic loading (a) and haversine cyclic loading (b).
(a)
(b)


## 5.3. Effects ofα

Figure10 presents the variation in average degrees of consolidation for both water and air phases with time under different types of cyclic loadings with β=1.5 and different loading parameters α for ka/kw=1. It can be seen that the smaller the loading parameter α is, the more quickly the dissipation of excess pore water and air pressures develops. Since the loading parameter α reflects the rate of loading increment or decrement, it can be found that the higher the rate of loading increment or decrement is, the faster the dissipation rates of excess pore pressures are. Moreover, the dissipation rates of excess pore pressures under rectangular cyclic loading are the biggest and the dissipation rates under triangular cyclic loading are the smallest. For the comparison with the results for static loading, the variation in average degrees of consolidation under constant loading is also shown in Figure 10. It can be observed that the average degrees of consolidation under cyclic loadings increase but fluctuate with a certain amplitude, whereas the average degrees of consolidation under constant loading increase monotonically with time. This phenomenon implies that consolidation of unsaturated soil under cyclic loadings does not complete after a long time, but rather continues oscillating with a certain amplitude, generating continuous volume change of the soil. Thus, neglecting the effect of cyclic loadings may result in overestimated settlement and cannot consider the continuous volume change.Figure 10
The effects ofα on consolidation of unsaturated soils under various cyclic loadings.
(a)
(b)


## 5.4. Effects ofβ

Figure11 depicts the variation in average degrees of consolidation for both water and air phases with time under different cyclic loadings with different loading parameters β for ka/kw=1. It can be found that the dissipation rates of excess pore water and air pressures increase with the decrease in the value of loading parameter β. Furthermore, a smaller value of the loading parameter β results in more cycles and a smaller amplitude of fluctuation in the dissipation rates. The loading parameter β is related to the rest period of loading. Thus, it is quite obvious that a longer rest period of loading induces the decrease in the dissipation rates, less cycles, and a bigger amplitude of fluctuation in the dissipation rates of excess pore pressures. As a result of that the dissipation rates of excess pore water and air pressures under the cyclic loading without rest period (β=1) are the biggest for each case. Specifically, rectangular cyclic loading without rest period reduces into constant loading, and the dissipation rates of excess pore water and air pressures are the highest and increase monotonically with time in that case, as shown in Figure 11(b).Figure 11
The effects ofβ on consolidation of unsaturated soils under various cyclic loadings: (a) trapezoidal cyclic loading; (b) rectangular cyclic loading; (c) triangular cyclic loading.
(a)
(b)
(c)It should be noted that the solutions presented in this paper do not consider nonlinear characteristics of soil properties such as the coefficients of permeability and volume change for both air and water phases under time-dependent loading. Even though they are limited for a linear elastic stress-strain model of unsaturated soil based on Fredlund and Hasan’s one-dimensional consolidation theory, the proposed solutions are more general ones for one-dimensional consolidation of soils from unsaturated to saturated state under time-dependent loading.


## 6. Conclusion

In this paper, analytical solutions are derived for one-dimensional consolidation of unsaturated soils subjected to various cyclic loadings. It shows through the degeneration into the existing solutions for both unsaturated and saturated soils that the proposed solutions are reliable and more general for one-dimensional consolidation of soils from unsaturated to saturated states under time-dependent loading. The effects of different parameters on one-dimensional consolidation of unsaturated soils under various cyclic loadings are investigated. The main conclusions can be drawn as follows:(1)
The coefficients ratio of permeability for air phase and water phase,ka/kw has sufficient effects on consolidation of the unsaturated soil layer under cyclic loadings. The dissipation rates of excess pore water and air pressures increase with the fluctuation when ka/kw increases. Moreover, the higher values of ka/kw lead to the bigger amplitudes of fluctuation in the dissipation rates.(2)
Under cyclic loadings, the dissipation process of excess pore pressures does not complete after a long time, but rather continues oscillating with a certain amplitude, generating continuous volume change in the soil. Thus, neglecting the effect of cyclic loadings may result in overestimated settlement and cannot consider the continuous volume change.(3)
The cyclic loading parameters such as the period of loading (t0), the rate of loading increment or decrement (α), and the rest period of loading (β) have considerable impacts on consolidation of unsaturated soil. The longer the period of loading, the bigger the amplitude, and the less the cycles of fluctuation in the dissipation rates of excess pore water and air pressures. The higher the rate of loading increment or decrement is, the faster the dissipation rates of excess pore pressures are. A longer rest period of loading induces the decrease in the dissipation rates, less cycles, and a bigger amplitude of fluctuation in the dissipation rates of excess pore pressures.



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