# One-Dimensional Nonlinear Consolidation Analysis Using Hansbo’s Flow Model and Rebound-Recompression Characteristics of Soil under Cyclic Loading

Authors: Banghua Zhu; Gang Shi; Zihe Wei
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Hansbo’s flow model for one-dimensional consolidation analysis of saturated clay has been widely recognized as being the most representative for soft soils. Many studies have used the model to examine the characteristics of soil under various conditions. However, very few studies have considered soil under cyclic loading. In this study, using a Hansbo’s flow model and assuming known characteristics for soft clay deformation and rebound and recompression of soil, the one-dimensional consolidation model of soft clay under cyclic loading is established. A FlexPDE solution scheme with excess pore pressureu and void ratio e as variables is also given. The reliability of the proposed method is verified by comparing the obtained results with existing results. On this basis, the consolidation characteristics of soft clay foundations under unilateral drainage and cyclic loading are studied. The effects of soil rebound and recompression characteristics, Hansbo’s flow parameters, cyclic loading period, and cyclic loading form on the consolidation characteristics of soft clay foundation are analyzed. The results show that under cyclic loading, the effective stress, void ratio, and average consolidation degree of the foundation all present a cyclic state and gradually enter a stable cyclic state with the increase in cycles. The peak of effective stress lags behind the peak of cyclic load. The rebound and recompression characteristics of soil have little effect on the effective stress of soil but a great effect on the void ratio. In contrast to its characteristic under linear loading, the average consolidation degree of the foundation under cyclic loading finally enters a stable cyclic state. The results of the analysis can be used as a reference in the analysis of real life highways, railways, subway tunnels built on soft soil foundations subjected to periodic cyclic loading.


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## Body

## 1. Introduction

The traditional Terzaghi consolidation theory, which states that when stress is applied to a porous material, it is opposed by the fluid pressure filling the pores in the material, is based on the assumption that the soil skeleton is characterized by linear elasticity, small strain, and Darcy’s flow. The mathematical form of the theory is simple and it is easy to obtain the analytical solution. Thus, it is widely used in practical engineering. However, with the deepening of research, more and more results show that there are sometimes large errors between the predicted foundation settlement and the measured value. On the one hand, this is mainly because seepage in soft clay deviates from Darcy’s law [1–3]. On the other hand, soft clay deformation presents significant nonlinear deformation characteristics [4]. Therefore, the study of nonlinear consolidation of soft clay considering a non-Darcy’s flow has strong practical significance in engineering.Among various non-Darcy’s flow models, Hansbo’s flow model has been widely recognized by the research community [5, 6]. In order to study the influence of non-Darcy’s flow characteristics on the consolidation process of soft clay, Hansbo et al. carried out a series of model observations and theoretical studies on one-dimensional consolidation and sand well foundation consolidation problems. Hansbo’s flow model for one-dimensional consolidation analysis of saturated clay was introduced in [7, 8]. It was solved using the finite difference method. The results under Hansbo’s flow model show that the dissipation rate of pore water pressure is slower than that of Darcy’s flow. In order to consider the influence of nonlinear deformation characteristics of soft soil, one-dimensional nonlinear consolidation of soft soil based on Hansbo’s flow under variable load was studied in [9] using the finite difference method and analyzed for a constant load in [10]. A semianalytical solution was found in [11]. In addition, the model was coupled to a Merchant rheological model in [12] in order to study the one-dimensional rheological consolidation characteristics of saturated clay.None of the aforementioned studies considered cyclic loading. However, in real life, highways, railways, subway tunnels built on soft soil foundations can usually be regarded as foundations subjected to periodic cyclic loading. At this point, the consolidation characteristics of soil are obviously different from those of foundations under constant load. For the problem of cyclic loading, one-dimensional consolidation of homogeneous foundations, layered foundations, and semipermeable boundary conditions were studied in [13–15], respectively. The one-dimensional nonlinear consolidation of soft clay under cyclic loading was studied in [16–18]. An analytical solution of one-dimensional rheological consolidation of soft soil under cyclic loading was found in [19]. The Laplace transform was used to derive the solution considering the rebound-recompression characteristics of soil under cyclic loading in [20]. The one-dimensional consolidation characteristics of viscoelastic foundations were studied in [21]. The one-dimensional consolidation characteristics of saturated clay considering Hansbo’s flow under cyclic loading were determined in [22].In this article, Hansbo’s flow model and nonlinear deformation characteristics of soft clay are introduced. Considering the unloading rebound and reloading effect of soil, the nonlinear consolidation equation of soft clay under cyclic loading is established. The consolidation equation is solved by FlexPDE software (a flexible solution system for partial differential equations), and the solution scheme of the bivariate of excess pore pressure and the void ratio is given. On this basis, the one-dimensional consolidation characteristics of soil under different cyclic loads are studied.


## 2. Basic Assumptions and Model Creation

### 2.1. Mathematical Description of the Problem

As shown in Figure1, we consider a saturated soft clay layer of thickness H0. Its top is permeable and the bottom is impervious (or impermeable). We also assume that the soft clay has been consolidated under the action of soil weight and original uniform load qp. The top surface of the soft clay layer is the same height as that of the groundwater table. A uniformly distributed cyclic loading (qt) is applied on the top of the soft clay layer. The period of the load is Tc and the peak is qmax. Soil gradually begins consolidating under the uniform cyclic loading.Figure 1
One-dimensional consolidation under cyclic loading.


### 2.2. Fundamental Assumption

Soft soil deformation usually presents obvious nonlinear characteristics in saturated soft clay foundations. Therefore, the following assumptions are made to establish an analytical model for the flow and aforementioned consolidation of the soft clay foundation .(1)
The flow of water in the soil satisfies Hansbo’s flow model shown in Figure2. Moreover, Hansbo’s flow model parameters (m and i1) remain constant in the consolidation process. Hansbo’s flow model can be expressed as follows:(1)v=K˜im,i≤i1,Ki−i0,i>i1,wherev is the seepage rate of pore water in soil; m is a constant determined by experiments; i is hydraulic gradient; i1 is the initial hydraulic gradient in the linear seepage stage; i0 is the calculated initial hydraulic gradient of linear seepage; K˜ is the permeability coefficient at the beginning of the exponential seepage stage.(2)i0=i1m−1mK˜=Kmi1m−1.(2)
Soil particles and pore water in the soil layer are incompressible and the soil layer is in the normal consolidation state.(3)
Nonlinear deformation and flow law of soft soil can be expressed in the classical form ase−lgσ′ and e−lgK [4]. The format is as follows:(3)e−e1=cclgσ1′σ′,(4)e−e1=cklgKK1,wheree is the pore ratio of soil; σ1′ is the reference effective stress; K1 and e1 are the permeability coefficient and void ratio corresponding to the reference effective stress (σ1′); cc is the compression index of soil; ck is the permeability index of soil; σ′ is effective stress; K is the permeability coefficient corresponding to the void ratio (e).(4)
Considering the influence of unloading rebound and recompression of soil, the unloading curve and recompression curve are a straight line in thee−lgσ′ coordinate system and the two straight lines overlap. As shown in Figure 3, the rebound index ce (or recompression index) remains unchanged during consolidation.Figure 2
Hansbo’s flow model.Figure 3
Compression and springback recompression characteristics of soil.


## 2.1. Mathematical Description of the Problem

As shown in Figure1, we consider a saturated soft clay layer of thickness H0. Its top is permeable and the bottom is impervious (or impermeable). We also assume that the soft clay has been consolidated under the action of soil weight and original uniform load qp. The top surface of the soft clay layer is the same height as that of the groundwater table. A uniformly distributed cyclic loading (qt) is applied on the top of the soft clay layer. The period of the load is Tc and the peak is qmax. Soil gradually begins consolidating under the uniform cyclic loading.Figure 1
One-dimensional consolidation under cyclic loading.


## 2.2. Fundamental Assumption

Soft soil deformation usually presents obvious nonlinear characteristics in saturated soft clay foundations. Therefore, the following assumptions are made to establish an analytical model for the flow and aforementioned consolidation of the soft clay foundation .(1)
The flow of water in the soil satisfies Hansbo’s flow model shown in Figure2. Moreover, Hansbo’s flow model parameters (m and i1) remain constant in the consolidation process. Hansbo’s flow model can be expressed as follows:(1)v=K˜im,i≤i1,Ki−i0,i>i1,wherev is the seepage rate of pore water in soil; m is a constant determined by experiments; i is hydraulic gradient; i1 is the initial hydraulic gradient in the linear seepage stage; i0 is the calculated initial hydraulic gradient of linear seepage; K˜ is the permeability coefficient at the beginning of the exponential seepage stage.(2)i0=i1m−1mK˜=Kmi1m−1.(2)
Soil particles and pore water in the soil layer are incompressible and the soil layer is in the normal consolidation state.(3)
Nonlinear deformation and flow law of soft soil can be expressed in the classical form ase−lgσ′ and e−lgK [4]. The format is as follows:(3)e−e1=cclgσ1′σ′,(4)e−e1=cklgKK1,wheree is the pore ratio of soil; σ1′ is the reference effective stress; K1 and e1 are the permeability coefficient and void ratio corresponding to the reference effective stress (σ1′); cc is the compression index of soil; ck is the permeability index of soil; σ′ is effective stress; K is the permeability coefficient corresponding to the void ratio (e).(4)
Considering the influence of unloading rebound and recompression of soil, the unloading curve and recompression curve are a straight line in thee−lgσ′ coordinate system and the two straight lines overlap. As shown in Figure 3, the rebound index ce (or recompression index) remains unchanged during consolidation.Figure 2
Hansbo’s flow model.Figure 3
Compression and springback recompression characteristics of soil.


## 3. Mathematical Model Formulation

### 3.1. Initial Effective Stress and Initial Void Ratio

According to [23], effective stress at any time can be expressed as follows:(5)σ′=qt−u+σ0′,where u is excess pore water pressure; qt is the cyclic uniform load acting on the soil surface; σ0′ is initial amplitude effective stress.(6)σ0′=σ0′z,0=∫0zγwGs−11+e0dz′+qp,where Gs is the relative density of particle; γw is the gravity of water; e0 is the pore ratio corresponding to initial effective stress (σ0′); qp is original uniform load.According to the aforementioned assumptions, the initial effective stress (σ0′) and initial void ratio (e0) meet the requirements of (3). Bringing (3) into (6) and differentiating results in the following equation:(7)dσ0′dz=γwGs−11+e1+cclgσ1′−lgσ0′.Equation (7) is a complex nonlinear ordinary differential equation that cannot be solved by conventional methods. In engineering practice, a numerical method is often used to find an approximate solution; boundary conditions σ0′t=0,z=0=qp are added for solution analysis.


### 3.2. Consolidation Process of Soil under Cyclic Loading

Taking the triangular cyclic load given in Figure4 as an example, the soil layer is fully consolidated under the action of self-weight stress and original uniform load (qp), that is, point a on the compression curve shown in Figure 3. Cyclic uniform load (qt) began to act on the soil surface, and under this load, the effective stress (σ′) gradually increased, whereas the void ratio (e) gradually decreased. As the soil is normally consolidated clay, σ′ and e follow the normal compression curves shown in Figure 3. When the time t exceeds T1 but is less than Tc, the external load decreases gradually from peak qmax and the excess pore pressure u decreases. However, due to the drainage consolidation effect, the effective stress σ′ does not decrease immediately; it begins to decrease after a period of time exceeding T1. It decreases from point b in Figure 3 to point c along the rebound recompression curve. At this time, σ′ and e fall on the rebound recompression curve. When the cyclic load enters the second cycle, the external load gradually increases. At this point, with the drainage consolidation of soil, the effective stress σ′ changes from point c along the rebound-recompression curve to point b. When the effective stress σ′ is greater than σb′, σ′ and e fall along the normal compression curve.Figure 4
Triangle cycle loading.Therefore, when considering the rebound-recompression characteristics of soil, it is necessary to distinguish the normal compression section from the rebound-recompression section and establish the corresponding consolidation differential equations.


### 3.3. Consolidation Equation of the Normal Compression Section

In order to simplify the analysis, the following equation can be obtained based on the continuity condition of seepage with reference to the derivation process of the traditional Terzaghi one-dimensional consolidation equation.(8)∂vz∂z=−11+e0∂e∂t,where vz is vertical seepage velocity of pore water.The following equation can be obtained by combining (1) and (2) and referring to literature [7]:(9)∂vz∂z=∂∂zsgnKMγwMi1M−1∂u∂zM,where M is the design conditions; when i is greater than i1, M=1; when i is less than or equal to i1, M=m. sgn is the symbolic function, when i is less than 0, sgn is negative; conversely, sgn is positive when i is greater than 0. Owing to the negative hydraulic gradient, the Hansbo seepage equation becomes ambiguous; hence, this paper extends the equation to include the case of the negative hydraulic gradient by introducing the symbolic function.Substituting (5) into (3) and differentiating time, we get the following equation:(10)∂e∂t=ccσ′ln10∂u∂t−dqdt.According to (3) and (4), the relationship between the permeability coefficient K and the effective stress σ′ is obtained.(11)K=K1σ1′σ′cc/ck.By substituting (9)–(11) into (8), the consolidation differential equation of the normal compression curve segment can be obtained.(12)11+e0ccσ′ln10∂u∂t−dqdt+∂∂zsgnK1MγwMi1M−1∂u∂zMσ1′σ′cc/ck=0.


### 3.4. One-Dimensional Consolidation Equation of the Rebound-Recompression Section

The relationship between effective stressσ′ and void ratio e in the rebound-recompression section can be expressed using (3) as follows:(13)e−ex=celgσx′σ′,where σx′ and ex is a point on the springback recompression curve.The effective stress levels are different when the soil at different positions starts unloading at different times. Therefore,σx′ and ex in (13) cannot be expressed explicitly.Similar to the normal compression section, the following equation can be obtained by taking (5) into (13) and differentiating time on both sides of the equation:(14)∂e∂t=ceσ′ln10∂u∂t−dqdt.The one-dimensional consolidation equation of the rebound-recompression section can be obtained by introducing (14) and (11) into (8). Its form is similar to that of (12) if cc is replaced with ce.


### 3.5. Initial and Boundary Conditions

To solve the aforementioned consolidation equation, the corresponding initial and boundary conditions are given as follows:Initial condition:(15)u=0,t=0,0≤z≤H0.Boundary condition:(16)u=0,t>0,z=0,∂u∂z=0,t>0,z=H0,.


### 3.6. Simulation of Consolidation Process under Cyclic Loading

Because the consolidation equations of the normal compression section and rebound-recompression section are different, corresponding judgment criteria should be set in the calculation process; hence, the values of the soil parameters at different positions can be calculated by different consolidation equations at different time steps.The judgment criteria of soil unloading is simple: according todσ′/dt<0, the consolidation equation of the rebound-recompression section is used to calculate unloading.However, the judgment criteria of soil when the load is applied is relatively complex. It is necessary to distinguish the rebound-recompression section and normal compression section, which can be judged according to the following method:Normal compression stage:(17)dσ′dt≥0,σ′<σmax′.Recompression stage:(18)dσ′dt≥0,σ′=σmax′,whereσmax′ is the maximum effective stress of the soil element before the current moment.


## 3.1. Initial Effective Stress and Initial Void Ratio

According to [23], effective stress at any time can be expressed as follows:(5)σ′=qt−u+σ0′,where u is excess pore water pressure; qt is the cyclic uniform load acting on the soil surface; σ0′ is initial amplitude effective stress.(6)σ0′=σ0′z,0=∫0zγwGs−11+e0dz′+qp,where Gs is the relative density of particle; γw is the gravity of water; e0 is the pore ratio corresponding to initial effective stress (σ0′); qp is original uniform load.According to the aforementioned assumptions, the initial effective stress (σ0′) and initial void ratio (e0) meet the requirements of (3). Bringing (3) into (6) and differentiating results in the following equation:(7)dσ0′dz=γwGs−11+e1+cclgσ1′−lgσ0′.Equation (7) is a complex nonlinear ordinary differential equation that cannot be solved by conventional methods. In engineering practice, a numerical method is often used to find an approximate solution; boundary conditions σ0′t=0,z=0=qp are added for solution analysis.


## 3.2. Consolidation Process of Soil under Cyclic Loading

Taking the triangular cyclic load given in Figure4 as an example, the soil layer is fully consolidated under the action of self-weight stress and original uniform load (qp), that is, point a on the compression curve shown in Figure 3. Cyclic uniform load (qt) began to act on the soil surface, and under this load, the effective stress (σ′) gradually increased, whereas the void ratio (e) gradually decreased. As the soil is normally consolidated clay, σ′ and e follow the normal compression curves shown in Figure 3. When the time t exceeds T1 but is less than Tc, the external load decreases gradually from peak qmax and the excess pore pressure u decreases. However, due to the drainage consolidation effect, the effective stress σ′ does not decrease immediately; it begins to decrease after a period of time exceeding T1. It decreases from point b in Figure 3 to point c along the rebound recompression curve. At this time, σ′ and e fall on the rebound recompression curve. When the cyclic load enters the second cycle, the external load gradually increases. At this point, with the drainage consolidation of soil, the effective stress σ′ changes from point c along the rebound-recompression curve to point b. When the effective stress σ′ is greater than σb′, σ′ and e fall along the normal compression curve.Figure 4
Triangle cycle loading.Therefore, when considering the rebound-recompression characteristics of soil, it is necessary to distinguish the normal compression section from the rebound-recompression section and establish the corresponding consolidation differential equations.


## 3.3. Consolidation Equation of the Normal Compression Section

In order to simplify the analysis, the following equation can be obtained based on the continuity condition of seepage with reference to the derivation process of the traditional Terzaghi one-dimensional consolidation equation.(8)∂vz∂z=−11+e0∂e∂t,where vz is vertical seepage velocity of pore water.The following equation can be obtained by combining (1) and (2) and referring to literature [7]:(9)∂vz∂z=∂∂zsgnKMγwMi1M−1∂u∂zM,where M is the design conditions; when i is greater than i1, M=1; when i is less than or equal to i1, M=m. sgn is the symbolic function, when i is less than 0, sgn is negative; conversely, sgn is positive when i is greater than 0. Owing to the negative hydraulic gradient, the Hansbo seepage equation becomes ambiguous; hence, this paper extends the equation to include the case of the negative hydraulic gradient by introducing the symbolic function.Substituting (5) into (3) and differentiating time, we get the following equation:(10)∂e∂t=ccσ′ln10∂u∂t−dqdt.According to (3) and (4), the relationship between the permeability coefficient K and the effective stress σ′ is obtained.(11)K=K1σ1′σ′cc/ck.By substituting (9)–(11) into (8), the consolidation differential equation of the normal compression curve segment can be obtained.(12)11+e0ccσ′ln10∂u∂t−dqdt+∂∂zsgnK1MγwMi1M−1∂u∂zMσ1′σ′cc/ck=0.


## 3.4. One-Dimensional Consolidation Equation of the Rebound-Recompression Section

The relationship between effective stressσ′ and void ratio e in the rebound-recompression section can be expressed using (3) as follows:(13)e−ex=celgσx′σ′,where σx′ and ex is a point on the springback recompression curve.The effective stress levels are different when the soil at different positions starts unloading at different times. Therefore,σx′ and ex in (13) cannot be expressed explicitly.Similar to the normal compression section, the following equation can be obtained by taking (5) into (13) and differentiating time on both sides of the equation:(14)∂e∂t=ceσ′ln10∂u∂t−dqdt.The one-dimensional consolidation equation of the rebound-recompression section can be obtained by introducing (14) and (11) into (8). Its form is similar to that of (12) if cc is replaced with ce.


## 3.5. Initial and Boundary Conditions

To solve the aforementioned consolidation equation, the corresponding initial and boundary conditions are given as follows:Initial condition:(15)u=0,t=0,0≤z≤H0.Boundary condition:(16)u=0,t>0,z=0,∂u∂z=0,t>0,z=H0,.


## 3.6. Simulation of Consolidation Process under Cyclic Loading

Because the consolidation equations of the normal compression section and rebound-recompression section are different, corresponding judgment criteria should be set in the calculation process; hence, the values of the soil parameters at different positions can be calculated by different consolidation equations at different time steps.The judgment criteria of soil unloading is simple: according todσ′/dt<0, the consolidation equation of the rebound-recompression section is used to calculate unloading.However, the judgment criteria of soil when the load is applied is relatively complex. It is necessary to distinguish the rebound-recompression section and normal compression section, which can be judged according to the following method:Normal compression stage:(17)dσ′dt≥0,σ′<σmax′.Recompression stage:(18)dσ′dt≥0,σ′=σmax′,whereσmax′ is the maximum effective stress of the soil element before the current moment.


## 4. Solving Consolidation Equation Based on FlexPDE

In contrast to the methods employed by other studies for solving consolidation equations; in this paper, FlexPDE software is used to solve the consolidation equation. FlexPDE software uses the finite element method to solve partial differential equations. It is widely used to solve one-dimensional to three-dimensional development problems, steady-state problems, and eigenvalue problems; high-precision numerical solutions of partial differential equations can be obtained. In addition, FlexPDE software usually uses a very simple programming language to define the partial differential control equation, solution domain, and boundary conditions of the analyzed problem, which greatly reduces the difficulty of obtaining mathematical solutions to complex problems.However, FlexPDE software cannot record the maximum effective stressσmax′ of each finite element mesh node before the current calculation step. To address this issue, the following approaches are available:(1)
Define two variables of excess pore pressureu and void ratio e.(2)
Add the following initial and boundary conditions.The initial condition ise=e0,t=0,0≤z≤H0 and the boundary condition is ∂e/∂z=0,t>0,z=H0.(3)
The unloading criteria remain unchanged; the loading criteria are defined by the following equations:Normal compression stage:(19)dσ′dt≥0,et−ee<D.Recompression stage:(20)dσ′dt≥0,et−ee≥D,whereD is the threshold value. It can be selected according to the calculation accuracy; et is the pore ratio calculated by (3).


## 5. Algorithm Verification

The saturated clay samples were prepared in advance; each group was subjected to three repeated tests, and the weighted average was taken. As shown in Figure5, the saturated clay consolidation test was carried out by a WG single lever consolidation instrument (main technical: 1. Loading pressure is 4000 kPa/30 cm2 or 2000 kPa/50 cm2; 2. Load arm is 20 : 1 or 24 : 1; 3. Specimen area is 30 cm2 or 50 cm2; especially, the sample area of this paper is 50 cm2). The WG-type series consolidation instrument is used for soil compression test to detect the relationship between soil pressure and deformation, calculate the unit deposition of soil, compression index, rebound index, and consolidation coefficient; the cyclic loading and unloading process was simulated by a Kamoer peristaltic pump (Its flow rate is 2.6–41.5 ml/min). The loading period can be modified by changing the pumping speed of the peristaltic pump. The relevant test parameters are as follows: the linear load loading period is 10,800 s, which remains unchanged after reaching the peak; the loading and unloading cycle of the triangle cyclic load is 2700 s, and the test is conducted for six cycles; the loading time of the trapezoidal cyclic load is 1440 s, the peak load is 480 s, the unloading time is 1440 s, and the no-load time is 480 s. The test is carried out for 5 cycles. The peak value of the cyclic load is 24,000 s, and the sedimentation is assumed to be the final sedimentation.Figure 5
WG single lever consolidation instrument (a), the type of F01A-STP Kamoer peristaltic pump (b), and manufacturing details (c).
(a)(b)(c)As shown in Figure6, the test results are in good agreement with the simulation results; the average consolidation degree of the test results is higher than that of the simulation. The reason is that the test instrument is limited, and the quality of the permeable stone and the metal cover above the soil sample is large, resulting in the error of the test results. Under linear load, the mass effect of the metal cover will be more obvious, resulting in the settlement in the early stage of the test being significantly higher than that of the simulation analysis. During triangular and trapezoidal cyclic loading, the experimental results are basically consistent with the simulation results, indicating that the algorithm has high reliability.Figure 6
Comparison of the results by the FlexPDE software to those of experimental results.We can define the time factorTv as follows:(21)Tv=K11+e1σ1′ln10γwccH2t.If there is no special description below, the dimensionless time factorTv is used as the measurement of time. Reference [10] used the finite difference method to calculate and analyze the one-dimensional consolidation problem considering the nonlinear deformation characteristics of soil under Hansbo’s flow condition. This study does not consider the deposition effect of the soil layer. The initial void ratio e0 and initial effective stress σ0′ of the whole soil layer are considered equal, which is suitable for the case of small soil thickness. The relevant calculation parameters are as follows: m=2.0, I1=2.0 or 0.5; an external load q=4σ1′ is applied instantaneously and remains unchanged; the ratio of the compression index to permeability index is 1.2. The calculation results are shown in Figure 7. It can be seen from the figure that the calculation results in this paper have a high coincidence degree with the calculation results in [10]. This shows that the calculation results in this paper have high reliability.Figure 7
Comparison of the results by FlexPDE software to those in [10].However, our study considers the soil layer deposition effect and nonlinear deformation characteristics, as well as Hansbo’s flow and variable load conditions. In order to verify the reliability of the algorithm under the aforementioned conditions, we consider a verification example that involves large consolidation deformation of soft soil considering the influence of Hansbo’s flow and variable load found in [24]. In order to obtain results that correspond with the calculation of large consolidation deformation in [24], the program in the current study is modified simply to consider the case of large consolidation deformations. In fact, if 1+e replaces 1+e0 and ∂/∂ξ replaces ∂/∂z in equation (12), the expression for nonlinear consolidation equation in the flow coordinate system can be obtained. By transforming this equation into the Lagrangian coordinate system, the nonlinear consolidation equation under large deformation can be obtained. When verifying the calculation, the external load is loaded in single-stage loading mode. The time factor of loading duration is Tvc=0.1 (the actual loading time is about 1000 d), the time factor Tv=0.4. The other indexes of soil are the same as those in [24], and the calculation results are shown in Figure 8.Figure 8
Comparison of the results by FlexPDE to those of the literature [24].It can be seen from Figure8 that the numerical solution in this paper is consistent with the calculation results in [24]. This shows that the one-dimensional consolidation algorithm considering Hansbo’s flow condition, soft soil deposition effect, and nonlinear deformation characteristics of soil in this research is reliable.


## 6. Analysis of Consolidation Behavior

Based on the aforementioned one-dimensional consolidation analysis program under cyclic loading, the consolidation characteristics of soft soil foundation under cyclic loading and under different conditions are studied by taking unilateral drainage as an example. The physical and mechanical parameters of the soil are shown in Table1. Some typical forms of cyclic loading are shown in Figures 9(a)–9(c); among them, Tc is a dimensionless period. At the same time, for the comparative analysis, the linear load is given, as shown in Figure 9(d). The relevant parameter of the cyclic load is qmax=100kPa and the Hansbo flow parameters are m=1.5, I1=2.0. The soil rebound index is ce=0.52 and its thickness is 20 m.Table 1
Physical and mechanical properties of soils.
σ1′ (kPa)e1ccckH0 (m)GsK1 (m/s)qp (kPa)501.570.640.75202.751 × 10−810Figure 9
Typical forms of loading.
(a)(b)(c)(d)There are two ways to define the average degree of consolidation, that is, average consolidation degreeUp defined by excess pore pressure u and the average consolidation degree Us defined by surface settlement st, as follows:(22)Upt=∫0H0qt−uqmaxdz,Ust=∫0H0e0−ee0−efdz,where e is the void ratio at t time and ef is the final thin void ratio. Other parameters are the same as before.


### 6.1. Consolidation Characteristics of Soil under Cyclic Loading

Taking the triangular cyclic load shown in Figure9(a) as an example with Tc=0.2, we analyze the consolidation behavior at different positions of the soft soil foundation and the variation of average consolidation degree U with time under cyclic loading. The calculation results are shown in Figures 10–13.Figure 10
σ′−Tv curves for triangle cyclic loading of different z.Figure 11
ccorce−Tv curves for z=H0/4.Figure 12
ccorce−Tv curves for z=3H0/4.Figure 13
U−Tv curves for triangle cyclic loading.Figure10 shows the variation curves of effective stress σ′ with time factor Tv at different positions of the foundation under cyclic loading. It can be seen from the figure that the effective stress in the foundation also has a cyclic effect under cyclic loading. Moreover, due to the drainage consolidation on the surface of the foundation, the cyclic variation range of effective stress at different depths of the foundation is different under the condition of unilateral drainage. The closer to the surface, the greater is the impact of surface drainage and the greater the amplitude of σ′ cycle change. The peak of cyclic load does not appear simultaneously with the peak of effective stress, and the peak of effective stress has a certain degree of hysteresis. In addition, with the increase of cycle times, the effective stress σ′ gradually enters a stable cycle state. At this point, the peak effective stress almost does not increase with the increase of cycle times.Figures10–12 show the variation curves of effective stress σ′ and cc or ce with time factor Tv at z=H0/4 and z=3H0/4, respectively. It can be seen from the figure that under cyclic loading, the effective stress σ′ of the shallow foundation reaches its maximum σmax′ quickly after the previous several cycles. Subsequently, the soil σ′,e falls on the rebound-recompression curve in Figure 3. The deep part of the foundation is less affected by surface drainage; hence, at the early stage of the cycle, with the increase of the cycle number, the peak value of the effective stress σ′ of the soil increases gradually. σ′,e of the soil constantly changes from the compression curve to the rebound compression curve and then to the compression curve.Figure13 is the curve of average consolidation degree U versus time factor Tv under cyclic loading. It can be seen from the figure that there are some similarities and differences between the average consolidation degree Up defined by excess pore pressure u and the average consolidation degree Us defined by surface settlement st. Under cyclic loading, Up and Us also present cyclic characteristics. At the early stage of the cycle, the wave peaks of Up and Us increase with the increase of cycle times; wave peaks of Up and Us tend to be stable after more than a certain number of cycles. In addition, Us is greater than Up from the beginning of cyclic loading. This shows that the consolidation degree defined by the surface settlement is higher than that defined by excess pore pressure, which should be considered in practical engineering.


### 6.2. Influence of Cyclic Load Parameters

This section mainly analyzes the influence of the cyclic loading cycle and loading form on the consolidation characteristics of soft soil foundations. When analyzing the influence of the cyclic loading period, cyclic loading is the same as presented in the previous section. The cycle periodsTc are 0.4, 0.6, 0.8, and 1.0, respectively. When the influence of the cyclic loading form is analyzed, the cyclic loading period is 0.6. The form of cyclic load is shown in Figures 9(a)–9(c); the linear load is also considered for comparative analysis.Figures14 and 15 show the variation curves of average consolidation degree with time factor Tv at different cyclic periods of triangular cyclic loading. The figure shows that the cyclic loading period Tc has a great influence on the average consolidation degree of foundation. With the increase of cyclic loading period, the variation range of average consolidation degrees Up and Us (the difference between peak and trough) of the foundation within the same cyclic cycle increases. The greater the cycle period of cyclic loading is, the fewer cycle times the average consolidation degree Up and Us need to enter the stable cycle state. In addition, the average consolidation degree Us is more affected by the cyclic loading period than Up. Moreover, for the average consolidation degree Us, the wave peak and wave trough of Us in the long cycle of the same cycle are higher than those in the short cycle.Figure 14
Up−Tv curves for different Tc under triangular loading.Figure 15
Us−Tv curves for different Tc under triangular loading.Figures16 and 17 show the variation curves of average consolidation degree Up and Us with time factor Tv under different cyclic loading forms. It can be seen from the figure that the form of cyclic load also has a great influence on the average consolidation degree of foundation. The variation range of the average consolidation degree of the foundation under trapezoidal cyclic loading is the largest in the same cycle. The average consolidation degree under cyclic loading of the triangle is similar to that under semisinusoidal cyclic loading. However, the peak and trough values of the average consolidation degree under semisinusoidal cyclic loading are greater than those under triangular cyclic loading. That is, the consolidation degree of the foundation under semisinusoidal cyclic loading is higher than that under triangular cyclic loading. Compared with cyclic loading, the average consolidation degree of foundation under linear loading increases gradually and will reach 100% at some time. Under cyclic loading, the average consolidation degree of the foundation gradually tends to be stable after several cycles, but it can never reach 100%.Figure 16
Up−Tv curves for different cycle loadings.Figure 17
Us−Tv curves for different cycle loadings.


### 6.3. Effect of the Rebound Recompression

The cyclic load adopts the triangular cyclic load in Figure9(a), and Tc=0.6, T1=0.3. When the effect of rebound compression and recompression is not considered and cc=ce, the calculation results are shown in Figures 16–18.Figure 18
σ′−Tv curves when taking into account the swelling characteristic of soil vs. disregarding it.Figure18 shows the variation curve of effective stress σ′ with time factor Tv when considering and not considering the influence of rebound and recompression. The figure shows that the effective stress difference between the two cases is very small and can be ignored.Figure19 shows the variation curves of void ratio e with time factor Tv when considering and not considering the influence of rebound and recompression. It can be seen from the figure that compared with the effective stress, there is a certain difference in the void ratio in the two cases. In particular, the wave peaks are quite different. The void ratio is relatively small when rebound and recompression are considered. This is mainly because when considering rebound and recompression, the slope ce of the rebound-recompression section is less than cc, and the increment of void ratio generated during unloading is less than that without considering rebound and recompression.Figure 19
e−Tv curves when taking into account the swelling characteristic of soil vs. disregarding it.Figure20 shows the variation curves of average consolidation degree with time factor Tv when considering and not considering the influence of rebound and recompression. From the aforementioned analysis, it can be seen that the effect of rebound and recompression on effective stress σ′ is much smaller than that on the void ratio e. Therefore, on the average degree of consolidation Up, the difference between the two cases is very small. The difference of Us is relatively large when considering the effect of springback and recompression.Figure 20
U−Tv curves when taking into account the swelling characteristic of soil vs. disregarding it.


### 6.4. Influence of Hansbo Flow Parameters

This section mainly studies the influence of Hansbo seepage parameters on the consolidation characteristics of soft clay foundations under cyclic loading. The cyclic load is the same as specified in the section “Influence of cyclic load parameters”:m is 1.0, 1.2, 1.5, and 1.8, respectively; I1 is 0.5, 1.0, 2.0, and 5.0, respectively. The calculation results are shown in Figures 17 and 18.Figures21 and 22 show the curves of average consolidation degree versus time factor Tv when m or I1 is different. It can be seen from the figures that when the seepage in soil satisfies Darcy flow (m=0), the drainage consolidation rate of the foundation is the fastest. The wave peak value is the largest in the same cycle, whereas the wave trough is the smallest. That is, the variation range is the largest in the same cycle. With the increase of m, the variation range of the average consolidation degree gradually decreases in the same cycle. It can be seen from Figures 23 and 24 that the influence of I1 on the consolidation characteristics of the foundation is similar to that of m. With the increase of I1, the variation range of the average consolidation degree decreases gradually in the same cycle. In addition, the influence of Hansbo flow parameters on Up and Us is basically the same.Figure 21
Up−Tv curves for different m.Figure 22
Us−Tv curves for different m.Figure 23
Up−Tv curves for different I1.Figure 24
Us−Tv curves for different I1.


## 6.1. Consolidation Characteristics of Soil under Cyclic Loading

Taking the triangular cyclic load shown in Figure9(a) as an example with Tc=0.2, we analyze the consolidation behavior at different positions of the soft soil foundation and the variation of average consolidation degree U with time under cyclic loading. The calculation results are shown in Figures 10–13.Figure 10
σ′−Tv curves for triangle cyclic loading of different z.Figure 11
ccorce−Tv curves for z=H0/4.Figure 12
ccorce−Tv curves for z=3H0/4.Figure 13
U−Tv curves for triangle cyclic loading.Figure10 shows the variation curves of effective stress σ′ with time factor Tv at different positions of the foundation under cyclic loading. It can be seen from the figure that the effective stress in the foundation also has a cyclic effect under cyclic loading. Moreover, due to the drainage consolidation on the surface of the foundation, the cyclic variation range of effective stress at different depths of the foundation is different under the condition of unilateral drainage. The closer to the surface, the greater is the impact of surface drainage and the greater the amplitude of σ′ cycle change. The peak of cyclic load does not appear simultaneously with the peak of effective stress, and the peak of effective stress has a certain degree of hysteresis. In addition, with the increase of cycle times, the effective stress σ′ gradually enters a stable cycle state. At this point, the peak effective stress almost does not increase with the increase of cycle times.Figures10–12 show the variation curves of effective stress σ′ and cc or ce with time factor Tv at z=H0/4 and z=3H0/4, respectively. It can be seen from the figure that under cyclic loading, the effective stress σ′ of the shallow foundation reaches its maximum σmax′ quickly after the previous several cycles. Subsequently, the soil σ′,e falls on the rebound-recompression curve in Figure 3. The deep part of the foundation is less affected by surface drainage; hence, at the early stage of the cycle, with the increase of the cycle number, the peak value of the effective stress σ′ of the soil increases gradually. σ′,e of the soil constantly changes from the compression curve to the rebound compression curve and then to the compression curve.Figure13 is the curve of average consolidation degree U versus time factor Tv under cyclic loading. It can be seen from the figure that there are some similarities and differences between the average consolidation degree Up defined by excess pore pressure u and the average consolidation degree Us defined by surface settlement st. Under cyclic loading, Up and Us also present cyclic characteristics. At the early stage of the cycle, the wave peaks of Up and Us increase with the increase of cycle times; wave peaks of Up and Us tend to be stable after more than a certain number of cycles. In addition, Us is greater than Up from the beginning of cyclic loading. This shows that the consolidation degree defined by the surface settlement is higher than that defined by excess pore pressure, which should be considered in practical engineering.


## 6.2. Influence of Cyclic Load Parameters

This section mainly analyzes the influence of the cyclic loading cycle and loading form on the consolidation characteristics of soft soil foundations. When analyzing the influence of the cyclic loading period, cyclic loading is the same as presented in the previous section. The cycle periodsTc are 0.4, 0.6, 0.8, and 1.0, respectively. When the influence of the cyclic loading form is analyzed, the cyclic loading period is 0.6. The form of cyclic load is shown in Figures 9(a)–9(c); the linear load is also considered for comparative analysis.Figures14 and 15 show the variation curves of average consolidation degree with time factor Tv at different cyclic periods of triangular cyclic loading. The figure shows that the cyclic loading period Tc has a great influence on the average consolidation degree of foundation. With the increase of cyclic loading period, the variation range of average consolidation degrees Up and Us (the difference between peak and trough) of the foundation within the same cyclic cycle increases. The greater the cycle period of cyclic loading is, the fewer cycle times the average consolidation degree Up and Us need to enter the stable cycle state. In addition, the average consolidation degree Us is more affected by the cyclic loading period than Up. Moreover, for the average consolidation degree Us, the wave peak and wave trough of Us in the long cycle of the same cycle are higher than those in the short cycle.Figure 14
Up−Tv curves for different Tc under triangular loading.Figure 15
Us−Tv curves for different Tc under triangular loading.Figures16 and 17 show the variation curves of average consolidation degree Up and Us with time factor Tv under different cyclic loading forms. It can be seen from the figure that the form of cyclic load also has a great influence on the average consolidation degree of foundation. The variation range of the average consolidation degree of the foundation under trapezoidal cyclic loading is the largest in the same cycle. The average consolidation degree under cyclic loading of the triangle is similar to that under semisinusoidal cyclic loading. However, the peak and trough values of the average consolidation degree under semisinusoidal cyclic loading are greater than those under triangular cyclic loading. That is, the consolidation degree of the foundation under semisinusoidal cyclic loading is higher than that under triangular cyclic loading. Compared with cyclic loading, the average consolidation degree of foundation under linear loading increases gradually and will reach 100% at some time. Under cyclic loading, the average consolidation degree of the foundation gradually tends to be stable after several cycles, but it can never reach 100%.Figure 16
Up−Tv curves for different cycle loadings.Figure 17
Us−Tv curves for different cycle loadings.


## 6.3. Effect of the Rebound Recompression

The cyclic load adopts the triangular cyclic load in Figure9(a), and Tc=0.6, T1=0.3. When the effect of rebound compression and recompression is not considered and cc=ce, the calculation results are shown in Figures 16–18.Figure 18
σ′−Tv curves when taking into account the swelling characteristic of soil vs. disregarding it.Figure18 shows the variation curve of effective stress σ′ with time factor Tv when considering and not considering the influence of rebound and recompression. The figure shows that the effective stress difference between the two cases is very small and can be ignored.Figure19 shows the variation curves of void ratio e with time factor Tv when considering and not considering the influence of rebound and recompression. It can be seen from the figure that compared with the effective stress, there is a certain difference in the void ratio in the two cases. In particular, the wave peaks are quite different. The void ratio is relatively small when rebound and recompression are considered. This is mainly because when considering rebound and recompression, the slope ce of the rebound-recompression section is less than cc, and the increment of void ratio generated during unloading is less than that without considering rebound and recompression.Figure 19
e−Tv curves when taking into account the swelling characteristic of soil vs. disregarding it.Figure20 shows the variation curves of average consolidation degree with time factor Tv when considering and not considering the influence of rebound and recompression. From the aforementioned analysis, it can be seen that the effect of rebound and recompression on effective stress σ′ is much smaller than that on the void ratio e. Therefore, on the average degree of consolidation Up, the difference between the two cases is very small. The difference of Us is relatively large when considering the effect of springback and recompression.Figure 20
U−Tv curves when taking into account the swelling characteristic of soil vs. disregarding it.


## 6.4. Influence of Hansbo Flow Parameters

This section mainly studies the influence of Hansbo seepage parameters on the consolidation characteristics of soft clay foundations under cyclic loading. The cyclic load is the same as specified in the section “Influence of cyclic load parameters”:m is 1.0, 1.2, 1.5, and 1.8, respectively; I1 is 0.5, 1.0, 2.0, and 5.0, respectively. The calculation results are shown in Figures 17 and 18.Figures21 and 22 show the curves of average consolidation degree versus time factor Tv when m or I1 is different. It can be seen from the figures that when the seepage in soil satisfies Darcy flow (m=0), the drainage consolidation rate of the foundation is the fastest. The wave peak value is the largest in the same cycle, whereas the wave trough is the smallest. That is, the variation range is the largest in the same cycle. With the increase of m, the variation range of the average consolidation degree gradually decreases in the same cycle. It can be seen from Figures 23 and 24 that the influence of I1 on the consolidation characteristics of the foundation is similar to that of m. With the increase of I1, the variation range of the average consolidation degree decreases gradually in the same cycle. In addition, the influence of Hansbo flow parameters on Up and Us is basically the same.Figure 21
Up−Tv curves for different m.Figure 22
Us−Tv curves for different m.Figure 23
Up−Tv curves for different I1.Figure 24
Us−Tv curves for different I1.


## 7. Conclusion

In this paper, the nonlinear consolidation equation of soft clay under cyclic loading is established by considering Hansbo’s flow model and the nonlinear deformation characteristics of soft clay, as well as the deposition effect of the soil layer and the unloading-rebound-reloading effect of soil. The numerical solution of the consolidation equation was solved by FlexPDE software, and the one-dimensional nonlinear consolidation characteristics of soil under cyclic loading were studied. The main results are as follows:(1) Under cyclic loading, the effective stressσ′, void ratio e, and average consolidation degree U of the foundation show a cyclic state. They gradually enter a stable cycle state as the cycle number increases. The wave peak of effective stress σ′ does not occur simultaneously with the wave peak of cyclic load but shows a corresponding hysteresis phenomenon.(2) Under the action of periodic cyclic loading, considering or not considering the rebound-recompression characteristics of soil has little effect on the effective stressσ′ and the average consolidation degree Up of soil. However, it has a great influence on the soil void ratio e and the average consolidation degree Us defined by surface settlement. Without considering the rebound-recompression characteristics of soil, the average consolidation degree of the foundation will be underestimated.(3) When the flow law conforms to Hansbo’s flow, the variation amplitude of average consolidation of the foundation in the same cycle is smaller than that of Darcy’s flow. With the increase of seepage parametersm and I1, the variation range of the average consolidation degree of the foundation decreases gradually.(4) With the increase of load cycle period, the average consolidation degree of the foundation requires fewer cycle times to enter the stable cycle state. Moreover, the variation range of the average consolidation degree of the foundation also gradually increases within the same cycle. In addition, within the same cycle, the variation range of average consolidation degree of foundation under trapezoidal cyclic load is the largest; the variation range of average consolidation degree under semisinusoidal load and triangular load is basically the same. However, the wave peak and wave trough of the average consolidation degree under semisinusoidal load are greater than those under triangular cyclic load. It will eventually enter a stable cyclic state.(5) The results of the analysis can be used as a reference in the analysis of real life highways, railways, subway tunnels built on soft soil foundations subjected to periodic cyclic loading. Specifically, the analysis shows that the consolidation degree defined by the surface settlement is higher than that defined by excess pore pressure, which should be considered in practical engineering.



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*Source: 9919203-2021-09-08.xml*