Finite Element Analysis of Consolidation of Homogeneous Ground under Interval Drainage Boundary

FAN He-bin1, MEI Guo-xiong1,2

1. College of Transportation Science & Engineering, Nanjing Tech University, China 2. College of Civil Engineering and Architecture, GuangXi University, China

E-mail:meiguox@163.com

Abstract: Based on the characteristics of the great displacement of foundation consolidation in the early state and small displacement during the late stage, this paper presents an interval drainage boundary condition. On the basis of Terzaghi’s one-dimensional consolidation theory, the simulation calculation of soil consolidation under the interval drainage boundary is realized by using ABAQUS finite element software. The effect of drainage spacing and the size of drainage outlet on the ground consolidation is analyzed, and then the influence of soil permeability coefficient and thickness of the soil on the consolidation of homogeneous foundation under interval drainage boundary is discussed. The research result shows that: compared to complete drainage boundary, the interval drainage boundary cannot meet the requirements of the water drainage in the early stage, but it can completely meet the requirements of the water drainage in the middle and later period of consolidation. At the same time, interval drainage boundaries about the overall of soil consolidation have little effect. The research results will provide a basis for reducing the laying of sand cushion on saturated clay drainage consolidation. Keywords: foundation treatment; interval drainage boundary; finite element analysis; consolidation efficiency; drainage outlet 1. Introduction

In engineering construction, the method of laying a large area of sand cushion on the surface of saturated cohesive foundation is often used to accelerate the soil consolidation[1]. It makes full use of the sand cushion as the foundation of the complete drainage boundary. However, during the middle and later stage of the drainage consolidation, the drainage capacity of sand cushion is far greater than the required drainage requirements due to the reduction of water displacement and drainage rate, which causes the waste of resources. Thus, it is necessary to optimize the drainage boundary conditions.

Not only in theoretical aspect but also in practical aspect, the soil consolidation has always been an important issue in geotechnical engineering field, and it has been widely and thoroughly studied. Terzaghi[2] in 1925 was the first person to establish the one-dimensional consolidation theory of saturated soils which is composed of consolidation equation, boundary conditions, and initial conditions; then MC Namee[3] et al. solved the consolidation problems under some special conditions. However, the application of Terzaghi one-dimensional consolidation theory has been the most widely used, and many other scholars[4] have modified and promoted the theory which is combined with the characteristics of soil and load. At present, researches on the soil consolidation theory are mainly in the constitutive relation and the initial conditions, while researches on the boundary drainage conditions have little to be studied. For the boundary drainage conditions, Terzaghi has divided them into two extreme types: completely drainage and completely undrained; Gray[5] was the earliest one who considered foundation consolidation under the semi-permeable boundary; Mei Guoxiong[6] proposed a continuous drainage boundary, which takes into account the continuity of the boundary conditions between completely drainage and completely undrained. These are the optimization of the drainage boundary, and these different drainage boundaries have different drainage capacities.

In this paper, based on the one-dimensional consolidation of Terzaghi’s theory[7][8], the complete drainage boundary is optimized for interval drainage boundary. Using the ABAQUS finite element numerical calculation, the consolidation process of soil mass in different drainage spacing and size of drainage outlet are calculated, and the calculation results are compared with the consolidation process under the complete drainage boundary. Then, the influence of soil permeability coefficient and elastic modulus on the compression and consolidation of soil mass is analyzed by the calculated the optimal interval drainage boundary. 2. Reasons of proposing interval drainage boundary

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Without violating the Terzaghi consolidation theory and having little effect on the consolidation time as much as possible, the interval drainage boundary condition presented in this paper is to reduce the laying sand cushion on the foundation surface. The idea is based on the size of water displacement changes over time of the foundation consolidation. According to the elastic analysis, with the lateral deformation of consolidation not considered in this paper, the amount of the discharged water should be equal to the volume deformation of the foundation (settlement). As shown in Figure 1, during the early process of consolidation, the amount of the discharged water is very large and the speed is fast. Laying large area of sand cushion as complete drainage boundary can be very effective, but along with the conduct of consolidation, amount of discharged water and rate of drainage decrease, and complete drainage capacity of the drainage boundary will be far beyond the need of soil foundation drainage consolidation. Especially at the later stage of the consolidation, the amount of discharged water is very small, and there is not much difference in effects between complete drainage boundaries and complete undrained boundaries in the foundation consolidation. Therefore, in order to take into account the characteristics of both early stage and later stage of the consolidation, the interval drainage boundary is put forward.

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Fig 1 Require discharge the volume of water due to consolidation

Although the interval drainage boundary cannot meet the need of drainage on early period of consolidation,

it can satisfy meet the requirements of the drainage on the middle and later period of consolidation, even its capacity of drainage is surplus in the late of soil consolidation. This not only saves the cost, but also has little influence on the consolidation time of the foundation soil without affecting the progress of the project.

3. Verifying the ABAQUS model

In this paper, a numerical calculation method is used to establish the model of soil consolidation under interval drainage boundary by ABAQUS finite element software[9], as shown in Figure 2.

complete drainage boundary

interval drainage boundary L D

Homogeneous foundation

Fig. 2 Calculation model

The soil model is one-dimensional homogeneous foundation, the thickness is 8.0 m, the initial void ratio is

1.5, soil surface moment applied constant uniform load is P=100kPa, the elastic modulus of soil is E=10MPa, the Poisson's ratio is v=0.3, and the permeability coefficient is k =1×10-9 m/s.

3.1 Fundamental assumption

The following basic properties of the soil are assumed: (1) The soil is isotropy clays of saturate with water and constitute a porous material with elastic properties, the water contained in the pores is incompressible; (2) The discharge of the porous water in the soil occurs only in the direction of the exterior normal direction of the drainage boundary; (3) No consideration on the weight of the soil, soil surface receives a constant load; (4) Void

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ratio and permeability coefficient of the soil are not changed under the process of the soil consolidation, and the water flows through the porous skeleton according to Darcy's law.

3.2 Processing of model boundary

Boundary conditions of the soil model are: The left and right side are undrained boundary, and they only have vertical displacement and no horizontal displacement; the bottom surface of the model is undrained and no displacement boundary, in this way the foundation has only conducted vertical consolidation, so the foundation is one-dimensional. Top surface of the soil is set to the interval drainage boundary, which makes the permeability of the pore water in the soil is not only the vertical direction, so the drainage boundary is two-dimensional. Uniform load is applied to the entire surface of soil in a moment, which is one-dimensional. Therefore, it is a two-dimensional problem for the consolidation model under the discontinuous drainage boundary.

3.3 Model Validation

The interval drainage boundary in the soil model degenerates to the complete drainage boundary, and the consolidation simulation is carried out. Such foundation, load, and drainage boundary are one-dimensional, and the calculation results can be compared with the one-dimensional consolidation analytical solution of Terzaghi, and the rationality of the model is verified. The results are shown in Figure 3.

As can be seen from the chart 3, the ABAQUS finite element calculation results and Terzaghi’s consolidation analytical curve are all exponential decreasing along with time, and the results are quite consistent. Therefore, the finite element simulation of this paper has a high accuracy, and can be further studied with foundation consolidation under the interval drainage boundary.

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Fig. 3 Result comparison of Terzaghi and ABAQUS

4. Design and analysis of interval drainage boundary parameters

By the uniform layout of different drainage mouth distance and the size of drainage outlet on the surface of

the soil, the optimal interval drainage boundary was computed that the results has little effect on the whole consolidation of soil.

4.1 Effect of drainage spacing L on consolidation of soil

The size of the drainage outlet is D, and L is the spacing between the drainage outlets. About drainage spacing L for the following instructions: (1) When n=0, the drain spacing is L=0D=0; the drainage boundary becomes a completely drained boundary. Therefore, the complete drainage boundary is an ideal condition of interval drainage boundary. (2) In this paper, L=1D, L=2D, L=3D, and L=4D are compared with the complete drainage boundary (L=0D) to study the effect of the changes of n value on soil consolidation.

Under a fixed the size of drainage outlet D. By layout different drainage spacing L on the surface of soil, the simulation of the process of consolidation is calculated, results as shown in Figure 4. As can be seen from the chart 4: With the increase of the n value, the consolidation curve of soil is gradually outward, which indicates that the consolidation efficiency decreases gradually. When the average degree of soil consolidation is in 5%~60%, the difference of consolidation curve is larger, and the average degree of soil consolidation is in 60%~100%, the consolidation curve is gradually close, and the difference is small.

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L=0D L=1D L=2D L=3D L=4D

Fig. 4 U-t curves along different Permeable port spacing

In order to observe the changes of soil consolidation more conveniently, the extension rate of consolidation

time St (relative to the ideal completely drained state, interval drainage boundary consolidation time extension ratio) is adopted in this paper, and its definition is as follows:

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Where tn is the interval drainage consolidation time under a certain degree of consolidation, and t0 is the

complete drainage consolidation time under a certain degree of consolidation. Combining formula (1), under different consolidation degrees, the interval drainage consolidation time and

the extension rate of consolidation time are arranged as shown in Table 1 and Table 2.

Table 1 Time consuming under a certain degree of consolidation

Table 2 Extension rate of consolidation time under a certain degree of consolidation

It can be obtained from Table 2 that under the same degree of consolidation, the extension rate of

consolidation time increases along with the increase of n value, it shows that the effect of interval drainage consolidation becomes worse and worse with layout spacing increasing. Under the same n value, the extension rate of consolidation time decreases with the rising of average degree of consolidation, it shows that the interval drainage boundary has great influence on the soil pre-consolidation, while it has little effect on the whole process of the soil consolidation. This is due to the large displacement during the early state of consolidation, so interval drainage boundary drainage capacity cannot meet the drainage requirements; while during the later period of consolidation, with the decrease of displacement, the drainage capacity can be better adapted.

When U=60% and L=1D, extension rate of consolidation time is St=3.95%<5%. Meanwhile, the consolidation time is only more than 581003s (about 6.7 days). Therefor, the interval drainage spacing is more reasonable being as big as the size of drainage outlet.

Degree of consolidation U

Time consuming (s) L=0D L=1D L=2D L=3D L=4D

20% 1551244 1741442 2107681 2707899 3310271 40% 6402935 6785603 7534883 8805641 10105999 60% 14695573 15276576 16427036 18427207 20537442 80% 29336774 30241761 32050350 35254427 38705741

Degree of consolidation U

Extension rate of consolidation time (%) L=1D L=2D L=3D L=4D

20% 12.26 35.87 74.56 113.39 40% 5.98 17.68 37.53 57.83 60% 3.95 11.78 25.39 39.75 80% 3.08 9.25 20.17 31.94

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4.2 Influence of size of drainage outlet on soil consolidation

According to the calculation of drainage spacing L=1D, the effect of interval drainage boundary on soil consolidation is achieved by controlling the size of different drainage outlets. In order to facilitate research and engineering application, the size of the drainage outlet is selected by 0.5m, 1.0m and 2.0m. Their calculation results are compared with the consolidation process under complete drainage boundary, as shown in Figure 5. As can be seen from the chart 5: In the case of determining the drainage spacing, consolidation curves with different size of drainage outlet are similar to each other.

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Fig. 5 U-t curves along different opening proportion

Combining formula (1), under different consolidation degrees, the extension rate of consolidation time is

calculated under the different drainage outlet, results as shown in figure 6. As can be seen from the chart 6: (1) At the same consolidation degree, the extension rate of consolidation time increases along with the size of drainage outlet, they are a linear relationship. (2) When the average degree of soil consolidation is in 20%, the size of drainage outlet has a great influence on the consolidation time, and the average degree of soil consolidation is in 40%~80%, the size of drainage outlet has little effect on the consolidation time. It is proved again that the interval drainage boundary has a great influence on the early consolidation of soil, and has little effect on middle and later stage of the drainage consolidation. (3) When the average degree of soil consolidation is in 60% and the extension rate of consolidation time is less than 5%, the size of the drainage outlet is less than 0.5m.

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Fig. 6 St-D curves along different soil consolidation contrast

4.3 Design parameters of the best interval drainage boundary

According to the above research the influence law of the drainage spacing and the size of drainage outlet on the soil consolidation process, the following conclusions can be drawn: Under this parameter of foundation, the

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interval drainage consolidation with drainage spacing L=1D and the size of drainage outlet D=0.5m is appropriate, which has little effect on the overall consolidation of soil.

5. Analysis of the influence of soil parameters change on the interval drainage boundary

5.1 Sensitivity analysis on permeability coefficient

Permeability coefficient is an important index that influence the effectiveness in soil consolidation. In this paper, three kinds of different permeability coefficients are selected, the calculation and the analysis are carried out under the interval drainage boundary with L=1D and D=0.5m. The permeability coefficients are 1×10-7, 1×10-8, and 1×10-9 m/s, and the calculation results are shown in Figure 7.

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k=110-7m/s complete drainage

k=110-7m/s interval drainage

k=110-8m/s complete drainage

k=110-8m/s interval drainage

k=10-9m/s complete drainage

k=110-9m/s interval drainage

Fig. 7 U-t curves along different coefficient of permeability

According to the data in figure 7, the extension rate of consolidation time under different permeability

coefficients is shown in Table 3. As can be seen from the Table 3: (1) The extension rate of consolidation time decreases with the rising of the average degree of consolidation under a certain permeability coefficient. This accords with the conclusion drawn above. (2) Under the certain average degree of soil consolidation, permeability coefficient has little effect on extension rate of consolidation time with interval drainage boundary. The reason may be that the extension rate of consolidation time is only about the relative error quantity between the interval drainage consolidation and the complete drainage consolidation.

Table 3 Extension rate of consolidation time under different permeability coefficients

Degree of consolidation U Extension rate of consolidation time (%)

k=1×10-7 m/s k=1×10-8 m/s k=1×10-9 m/s 20% 12.33 12.29 12.26 40% 5.90 5.98 5.98 60% 3.96 3.91 3.95 80% 3.11 3.08 3.08

5.2 Sensitivity analysis on the thickness of soil layer

In this paper, five kinds of different soil thicknesses are selected, the calculation and the analysis are carried out under the interval drainage boundary with L=1D and D=0.5m. The soil thicknesses are 4, 6, 8,10 and 12m, and the calculation results are shown in Figure 8. As can be seen from chart 8: (1) With the increase of soil thickness, the consolidation curve under the interval drainage boundary is getting closer to the consolidation curve under the complete drainage boundary. This indicates that the effect of the interval drainage boundary on the soil consolidation is decreasing. (2) With the increase of the soil thickness, regardless of the soil surface in complete drainage boundary or interval drainage boundary, the interval between the consolidation curve is constantly decreasing. This indicates that the consolidation rate of soil decreases.

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H=4m complete drainage boundaryH=4m interval drainage boundaryH=6m complete drainage boundaryH=6m interval drainage boundaryH=8m complete drainage boundaryH=8m interval drainage boundaryH=10m complete drainage boundaryH=10m interval drainage boundaryH=12m complete drainage boundaryH=12m interval drainage boundary

Fig 8 U-t curves along different soil thickness

According to the data in figure 8, the extension rate of consolidation time under different soil thicknesses are

shown in chart 9. By chart 9, it can be seen that extension rate of consolidation time decreases with the increase of the average degree of consolidation under certain soil thickness. Under a certain degree of consolidation, extension rate of consolidation time decreases with the increase of the soil thickness in interval drainage consolidation. Therefore, for the whole consolidation, the analysis show that that within certain thickness range, the thicker the clay layer, the smaller the consolidation effect under the interval drainage boundary condition. So when the foundation reaches a certain degree, its influence can even be neglected.

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Fig 9 Extension rate of consolidation time in different soil thickness

6. Conclusions

In this paper, the consolidation of saturated cohesive soil under interval drainage boundary is studied by ABAQUS finite element software. The preliminary conclusions are as follows:

(1) The computational model of ABAQUS has a direct influence on the numerical simulation results, and it is necessary to compare with the classical analytical solution before the simulation analysis. The rationality of the model is verified.

(2) For the homogeneous soil, when the size of drainage outlet reaches certain value, the consolidation rate decreases with the increase of soil surface drainage spacing; when the drainage spacing of the soil surface reaches a certain value, the consolidation rate decreases with increase of the size of drainage outlet.

(3) For homogeneous soil, under the condition of a certain size of drainage outlet and drainage spacing, permeability coefficient has little effect on soil consolidation under interval drainage boundary. The extension rate of consolidation time decreases with the increase of soil thickness.

(4) Interval drainage consolidation has little effect on the overall consolidation of soil, and it can be used to replace of soil consolidation under the complete drainage boundary. In the construction, can be appropriate to reduce the laying of sand cushion, so as to reduce the cost of the project.

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7. References [1] WANG Tian-hong, ZHANG Hong-ling. Sand cushion Application in Soft Foundation Treatment. Science and Technology Information, 2007: 301-302. [2] Terzaghi K. Theoretical Soil Mechanics. John Wiley and Sons Inc, 1943. [3] MC NAMEE, GIBSON R E. Plan strain and axially symmetric problems of the consolidation of a semi-infinite clay stratum. Journal of Mechanical Applied Mathematics, 1960: 124-135. [4] Xie Kang-he. Theory of one dimensional consolidation of double-layered ground and its applications. Chinese Journal of Geotechnical Engineering, 1994: 25-35. [5] Gray H. Simultaneous consolidation of contiguous layers of unlike compressible. JSMFD ASCE, 1970: 1499-1504 [6] MEI Guo-xiong, XIA Jun, MEI Ling. Terzaghi’s one-dimensional Consolidation Equation and its Solution based on Asymmetric Continuous Drainage Boundaries. Chinese Journal of Geotechnical Engineering, 2011, 28-31. [7] Znidarcic D, Schiffman R L, et al. Theory of one-dimensional consolidation of saturated clays. Geotechnique, 1986: 227-237. [8] Tekinsoy M Arslan. One-dimensional consolidation of unsaturated fine-grained soils. Journal of geotechnical engineering, 1990:838-850. [9] Fei kang, Zhang jian-wei. ABAQUS is applicated in geotechical engineering. China water power press. 2010.

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