Rigorous Axially Symmetric Consolidation Solution of Vacuum Combined Linear Loading Surcharge Preloading

PENG Jie( )， JIANG Rui( )， TIAN Yanmei()

Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering， Hohai University， Nanjing 210098， China

Abstract： Vacuum combined surcharge preloading is an effective method which has been used in ground treatment. This study presents the analytical solution of vertical drains with vacuum combined surcharge preloading. The surcharge was considered to be time-dependent preloading due to its reflection of the real situation. Moreover， the free strain， instead of equal strain hypothesis of the surface， was adopted for the mathematical rigor. Both vertical and horizontal drainages， along with the well resistance were considered. Besides， the time-dependent surcharge preloading was resolved with the impulse theorem which has been presented in the study， and the solutions for pore water pressure and degree of consolidation were derived by applying the approach of separation of variables. Furthermore， the solution is compared with previous solutions and numerical solutions， and the results verify the correctness of the proposed model.

Key words： vacuum combined surcharge preloading; time-dependent surcharge; consolidation; analytical solution

Introduction

Prefabricated vertical drains (PVDs) with vacuum combined surcharge preloading have been widely used to accelerate the consolidation rate and eliminate the instability problem.More and more studies have been done on vacuum preloading.The effective stress generated by vacuum preloading increases equiaxially， while the corresponding lateral movement is compressive and converse to the lateral movement generated by surcharge preloading. As a result， the risk of shear failure can be reduced even at a high rate of embankment construction[1-6].

The theory of radial drainage and consolidation was initially presented to analyze the surcharge preloading with PVDs[7-17]. Subsequently， several analytical solutions based on radial consolidation theory have been proposed， in which the equal strain hypothesis is adopted to analyze the behavior of soil improved by PVDs with vacuum preloading[18-22]. Moreover， Mohamedelhassan and Shang[23]initially developed a combined one-dimensional vacuum and surcharge consolidation model based on Terzaghi’s consolidation theory， followed by analytical and numerical models of vertical drains incorporating vacuum preloading in axisymmetric and plane strain conditions， which was proposed by Indraratnaetal[18， 24-25].

The solutions mentioned above assume that the surcharge loading is applied instantaneously and kept constant during the consolidation process; meanwhile， equal strain hypothesis is adopted to reduce the difficulty of solving. In the practical engineering， with the gradual application of the surcharge loads over time， the dissipation rate of excess pore water pressure and settlement may be affected by the load variation over time apparently. Furthermore， the solutions were not mathematically rigorous due to the adoption of equal strain hypothesis. There are some solutions considering the ramp loading for surcharge preloading[26-32]; Gengetal.[20-21]presented a solution of vacuum combined time-dependent surcharge preloading， but the solution was complicated and the equal strain hypothesis was still kept.

This study presents a rigorous solution for vacuum combined with time-dependent surcharge preloading with PVDs. Since the free strain of surface is considered， the equal strain hypothesis is abandoned. Besides， the well resistance of the drain is also considered; the accuracy of the proposed solutions is verified through the use of previous and numerical solutions.

1 Governing Equations

To obtain the governing equation for the consolidation of soil with drains， the following assumptions are made: (1) the soil is fully saturated and Darcy’s law is valid; (2) water and soil are incompressible; (3) strains are small and the coefficient of compressibility and permeability of soil is constant.

The basic partial differential equation for excess pore water pressure by vertical and radial drainage is shown as

(1)

whereu(r，z，t) represents the excess pore pressure，ris the radial distance from the center of the drain well，zrefers to the vertical distance from the soil surface，tis the time，quis the surcharge load，t0is the time to maximum surcharge load， andchandcvare the radial consolidation coefficient and vertical consolidation coefficient， respectively.

The boundary and initial conditions for Eq. (1)are expressed as

(2)

(3)

(4)

u(r，z， 0)=uw(r，z， 0)=0,

(5)

and the continuity at the interface between the drain and soil can be expressed by

uw(rw，z，t)=u(rw，z，t)，

(6)

,

(7)

wherep0is the vacuum pressure at the top soil surface. Along the drain，reis the radius of the soil cylinder，t1is the total time of vacuum preloading,His the soil thickness equaling the length of vertical drains,rwis the equivalent radius of the drain，uwis the pore pressure in the drain，khis the horizontal permeability of soil， andkwis the permeability of drain.

2 Proposed Analytical Solution

Firstly, the solutions of pore water pressure and consolidation degree are solved considering instantaneous preloading in this study. Then the pulse function is used to simulate the time-dependent surcharge preloading， and the amendment is conducted to get the solutions considering time-dependent surcharge preloading. Finally， the linear superposition principle is used to combine the solutions of vacuum preloading with time-dependent surcharge preloading.

2.1 Solution about instantaneous surcharge preloading

The boundary and initial conditions about instantaneous surcharge preloading include Eqs.(3)-(7)， and then the governing equations of instantaneous surcharge preloading are shown as

(8)

u(r， 0，t)=0.

(9)

The separation variables method is used to solve the equations above， with the solution shown as

(10)

where，nis the ratio of the radius of the model to the drain；αmiis the dimensionless eigenvalues of Bessel equation, which is solved by transcendental equation.

(11)

where

V0(αmin)=N1(αmin)J0(αmin)-J1(αmin)N0(αmin)，

V0(αmi)=N1(αmin)J0(αmi)-J1(αmin)N0(αmi)，

whereJ0(x)andJ1(x)are 0- and 1-order Bessel functions.N0(x)andN1(x)are 0- and 1-order Neumann functions.

The average pore water pressure is

(12)

The consolidation degree is defined as

and then

(13)

Since

V1(αmi)=N1(αmin)J1(αmi)-J1(αmin)N1(αmi),

then

The consolidation degree can also be calculated by formula(14).

(14)

2.2 Solution about time-dependent surcharge preloading

Considering the time-dependent surcharge， the governing equation is shown as

(15)

The boundary conditions include Eqs.(3)-(7) andu(r， 0，t)=0.

(16)

(17)

And

(18)

Equation (17) is only applicable to the condition of 0≤τ≤t0. The response functionw(r，z，t；τ) equals 0 whenτ>t0， so the segmented results ofu(r，z，t) are shown as

(1) When 0≤t≤t0，

(19)

(2) Whent>t0，

(20)

The average pore water pressure and consolidation degree are shown as follows.

(1) When 0≤t≤t0，

(21)

It should be noted that Carrillo theorem can’t be applied to variable loads in the period of 0≤t≤t0， so only the consolidation degree in the period oft>t0is presented.

(2) Whent>t0，

(22)

(23)

2.3 Solution about vacuum preloading

The loading time of vacuum in practical engineering is very short， so the vacuum preloading can be considered as instantaneous load. The governing equation is shown in Eq.(8).

The boundary conditions include Eqs.(3)-(7) and

u(r，0，t)=-p0.

(24)

The solving process of vacuum preloading is similar to the instantaneous surcharge preloading and the results are shown after variable separation as

(25)

The average pore water pressure is presented as

(26)

The average consolidation degree is presented as

(27)

Whent>t1， the pore water equals 0 (u(r，z，t)=0) because the vacuum is removed. Therefore， the consolidation degree is shown as

(28)

2.4 Solution about vacuum combined time-dependent surcharge preloading

The average pore water pressure of vacuum combined time-dependent surcharge preloading can be solved according to the linear superposition principle.

(1) When0

(29)

It should be noted that Carrillo theorem can’t be applied to variable loads in the period of 0≤t≤t0， so only the consolidation

degree in the period oft>t0is presented.

(2) Whent0≤t≤t1， the average pore water pressure is

(30)

The consolidation degree is

(31)

(3) Whent>t1，

(32)

3 Verification of the Proposed Model

The proposed model was validated by comparing the predictions for settlement and excess pore pressure with numerical solution and analytical solution proposed by Rujikiatkamjornetal.[19]and Zhu and Yin[33].

3.1 Analysis of vacuum preloading using presented solution

Considering only the vacuum preloading， the solution by Rujikiatkamjornetal.[19]and a numerical solution were presented to verify the solution of the study. Some parameters used in the analysis were provided in Table 1， andp=-80 kPa.

Table 1 Parameters used in the analytical solution and the numerical model

In the solution proposed by Rujikiatkamjornetal.[19]， resistance was not considered. A numerical model based on Eq. (1) was developed to analyze the consolidation of soil treated by vacuum preloading with a central PVD. The equal strain hypothesis is not necessary in the numerical model， while the parameters adopted in the numerical model are the same with what have been displayed in Table 1.

The comparison is shown in Fig. 1. The results from the analytical solution derive in this study correlate well with those from the numerical simulation， and there are apparent differences between the solution of Rujikiatkamjorn and Indraratna[34]and numerical solution.

Fig.1 Comparison of analytical solutions

3.2 Analysis of time-dependent surcharge preloading using presented solution

With vacuum combined time-dependent surcharge preloading considered， the numerical solution is presented to verify the solution of this study. The loading curve is shown in Fig. 2. The vacuum loadp=-80 kPa， was applied instantaneously， while the surcharge loadqu=20 kPa， was applied linearly with timet0=15 d andt1=200 d.

Fig.2 Loading curve

Figure 3 presents the calculated consolidation degree over time， in which the consolidation degree curve of 0

The calculated result is shown in Fig. 3. The results from the analytical solution derived in this study correlate well with those from the numerical simulation， and there are apparent differences between solution of Zhu and Yin[33]and numerical solution. Based on the results， the presented solution agrees well with numerical solution， indicating the correctness and the reasonableness of the presented solution.

Fig.3 Comparison of analytical solutions

4 Conclusions

The rigorous analytical solutions of vertical drains incorporating the vacuum preloading under time-dependent surcharge loading are presented in this paper. The free strain of surface is adopted instead of equal strain hypothesis， and the resistance is also considered. The general solution of consolidation degree of pore water under pressure is derived by separating variables. With the use of the impulse theorem， the time-dependent surcharge preloading can be considered instead of using instantaneously applied constant loading， which cannot simulate the construction time of the surcharge. The solution is compared with previous solutions and numerical solution， and the results verify the correctness of the proposed model.