Theoretical Study and Experimental Verification of the ... · u 3）（ In the formula, σ' and u...

Date of reception: 2013-1 Funded projects: NSFC(Natural Science Foundation of China) (No. 51178290), Tianjin natural science fund (No. 11 jcybjc02900) Brief introduction to the author: Li Shunqun, male, born in 1971, Ph.D., professor, research field including basic theories of soil mechanics, geotechnical microstructure and unsaturated soil. E-mail: [email protected]

Theoretical Study and Experimental Verification of the Consolidation Stage of Triaxial Test

1LI Shunqun, 2CHENG Xuelei, 3Yang Sen

1, First Author School of Civil Engineering, Tianjin Chengjian University, Tianjin Key Laboratory of Soft Soil Characteristics and Engineering Environment,

Tianjin 300384, China, Email:[email protected] *2, Corresponcing Author School of Civil Engineering, Tianjin Chengjian University, Tianjin 300384,

China. Email:[email protected]

3, School of Foreign language, Tianjin Chengjian University, Tianjin 300384, China. Email:[email protected]

Abstract The consolidation stage in conventional triaxial test is of a three-dimensional compression under

spherical stress but seepage occurring only in the vertical direction. Therefore, Terzaghi’s one-dimensional consolidation, in which strain only induced in vertical direction, is very different from the consolidation in triaxial test and can’t be directly utilized to solve the problem. In this paper, equations to describe the water pressure dissipation in consolidation phase of the conventional triaxial test are constructed based on the effective stress principle, Darcy's law and linear materials assumption. Furthermore, the power series solution based on the method of separation of variables, the numerical result from the method of the finite element, is provided to solve the constructed equations. Analysis of numerical results shows that the convergences of the power series solution at the initial stage of the consolidation is relatively poor and will become more stable with the consolidation progress continue. Comparing with the triaxial test of WF effective stress path completed in double pressure chamber shows that the constructed equation can reveal the main features of the consolidation stage in the triaxial test.

Keywords: Triaxial test consolidation, Spherical stress, Elastic assumption, The finite element method

CLC：TU441+.8，TU411.5， Document code：A

1 Introduction

The consolidation progress is an essential topic in soil mechanics. The consolidation of natural soil layer with no lateral deformation is called K0 consolidation, which is one of the most common forms of consolidation. This solution to the topic was given by Terzaghi in 1925. Along with the effective stress principle proposed by Terzaghi, it constructed the foundation of the modern soil mechanics [1].

After study the relationship between three-dimensional deformation and pore water pressure, Biot derived a more reasonable consolidation theory of fluid-solid coupling [2]. Biot consolidation theory satisfies equilibrium condition, material constitutive relation, deformation compatibility condition and flow continuity conditions, and realizes the direct coupling of the seepage field and displacement field. It can demonstrate the change of the average total stress in the soil with time in the consolidation process, and reflect the Mandel-Cryer effect.

Biot consolidation theory is widely utilized, which reveals the essence of soil consolidation more reasonably than Terzaghi consolidation theory does. It involves more parameters and determining these parameters accurately is very difficult. But for the commonly used calculation methods such as finite element method, the finite layer method and other numerical methods [3-6], whether the result from calculation is reasonable or not depends mainly on the value of the parameters. Usually, the analytical solution of Biot consolidation theory can be achieved only in some special cases [7, 8]. The complexity and uncertainty of the numerical methods, along with the limitation of analytical methods, lead to serious application limitation of Biot consolidation theory in the engineering.

As Terzaghi consolidation theory basically reflects the physical process of consolidation, the parameters can be provided by conventional indoor test, and some needed data can be achieved by

Journal of Convergence Information Technology(JCIT) Volume8, Number13, August 2013

Theoretical Study and Experimental Verification of the Consolidation Stage of Triaxial Test LI Shunqun, CHENG Xuelei, Yang Sen

113

hand computation. Presently, Terzaghi one-dimensional consolidation theory is used most in the engineering practice. Some consolidation problems in some special cases can be solved by reformed Terzaghi consolidation theory [9-11].

The study on the engineering properties of the rock and soil under various confining pressures is of great significance, the results from which are often used to determine the soil strength and deformation characteristics [12, 13]. The consolidation stage of conventional triaxial tests is a special process in which under isobaric pressure or spherical stress, seepage occurs only vertically. What it has in common with Terzaghi one-dimensional consolidation process is that seepage occurs vertically [14.] The difference lies in that the consolidation in triaxial tests is condensation under spherical stress. Therefore, Terzaghi one-dimensional consolidation theory cannot be directly used to deal with the consolidation of triaxial samples. Biot consolidation theory can be applied in the study of consolidation of soil under loads of rotational symmetry. Thus, it is certainly applicable to the study of consolidation process in triaxial tests. But the application of Biot consolidation theory is too complex to be grasped by many workers. Therefore, seeking a simple and practical method to reasonably describe the consolidation process in triaxial tests is an important issue [15-16].

In the paper, based on the effective stress principle, water-spring model and Darcy's law, an equation is established to deal with the consolidation stage in triaxial tests, from which the analytical solution and the numerical solution can be obtained. And a comparison drew between these solutions and the results from tests shows that the equation is correct and reliable.

2 Theoretical Calculation Model

Soil consists of particles and pores, which is often presented as Water-Spring Model. When the triaxial sample is under confining pressure, with water seepage through the porous stone, the stress will shift from pores to particles. Namely, the pore water stress fades and the effective stress increase gradually. When there is no the pore water stress, consolidation process is finished.

Conventional triaxial test is the test of soil mechanics in rotational symmetry in which water seeps along the axis in a one-dimension way. Similar to Terzaghi one-dimensional consolidation theory, assumptions for the triaxial test consolidation stage are as follows. ① The soil is homogeneous, isotropic and fully saturated. ② The soil compression is completely caused by pore shrinkage while soil particle and pore water are incompressible. ③ Drainage occurs only vertically. ④ The water seepage follows Darcy’s law. ⑤ In the osmotic consolidation process, soil permeability coefficient, the elastic modulus and Poisson’s ratio are constant. ⑥external load is exerted once and suddenly and it remains unchanged in the consolidation process. ⑦The soil deformation is entirely caused by the dissipation of extra pore water pressure.

The assumptions are different from those for Terzaghi one-dimensional consolidation theory in two aspects. Firstly, in Terzaghi’s theory, consolidation is assumed to occur only vertically while in the triaxial test consolidation of the triaxial sample is three-dimensional isobaric under spherical stress. Secondly, in Terzaghi’s assumptions, compression coefficient is constant as the elastic modulus assumed to be constant.

In the Terzaghi one-dimensional consolidation model as well as in triaxial test, the total stress at a given point in the soil is identical. In the Terzaghi one-dimensional consolidation model, there must be

22 33 0 11 0 0k k （1）

In the formula, σ11, σ22 and σ33 represent respectively the first principal stress, the second principal stress and the third principal stress. k0 is the static soil pressure coefficient and σ0 is the total pressure. Whereas, in conventional triaxial test, the confining pressure is a spherical stress namely identical in all direction.

11 22 33 （2）

σ is the confining pressure exerted. In the one dimensional consolidation theory and the triaxial test, the effective stress principle applies has same form, namely


114

u （3）

In the formula, σ' and u represent respectively the effective stress and the water pressure. In the triaxial test, after exerting confining pressure on the sample, valves are opened for drainage to the benefit of consolidation. Water is discharge gradually through the porous stone at the bottom of the sample. The water pressure, equivalent to confining pressure at the beginning, decreases gradually while the effective stress increases gradually from 0. When consolidation completed, the water pressure fades away and the effective stress increases to confining pressure. The change process is shown in Fig.1.

In Fig.1, H, r represents respectively the length and the radius of the cylinder sample. h is height of water pressure for the super hydrostatic pressure and t is the duration for the drain valve keeping open. A unit is taken from the sample which is as much as z from the boundary of drainage. And its length, width and height at certain time are assumed to be dx, dy, dz respectively. In the triaxial test, because of restriction of the rubber mold, water seepage will only occur vertically.

Fig. 1 Stress change and flow continuity condition in the consolidation stage of Triaxial test Therefore, the amount of water that flows into the unit is represented as q'

'q kiA （4）

In the formula, k, i, A are permeability coefficient, hydraulic gradient and horizontal seepage area of the unit dxdy respectively, and

d

d

ui

z （5）

wu h （6）

h and γw are height of water and unit weight of water respectively. Therefore,

w

' d dk u

q x yz

（7）

Similarly, the outflow of water from the unit q'' is

2

2w

'' d d dk u u

q z x yz z

（8）

Therefore, the loss of water in the unit will be

2

2w

'' ' d d dk u

q q z x yz

（9）

dx

dz

dy

q'

q''

t

u

σ'

h

H

z

r

u

σ'

透水石

橡皮膜Rubber membrane

Permeable stone


115

In addition, the change rate of the pore volume in the unit is

d d dV e

x y zt t n

（10）

In the formula, V, e, n and t respectively represent the volume of the pore in the unit, soil porosity, pore ratio and time elapsed.

1n e （11）

As the sample is saturated, the change rate of pore water volume of the unit must be equal to that volume, namely,

2

2w

d d d d d dk u e

z x y x y zz t n

（12）

As the volume of soil particles remains constant, that is, d d dx y z

nis constant, therefore,

2

2w

1k u e

z n t

（13）

d de V （14）

The strain of the unit is equal to the sum of the three principal strains, namely

d x y zV （15）

In the formula, εx, εy and εz are respectively the three principal strains of the unit. For homogeneous and isotropic materials under spherical stress, the three principal strains are identical. According to the elastic assumption as well as formula (14) and formula (15), we can get

1 2d 3 de

E

（16）

In the formula, E and μ are respectively elastic modulus and Poisson’s ratio. From formula (16) and formula (13), formula (17) can be obtained.

2

2w3 1 2

knE u

z t

（17）

In addition, from the formula (3), we can get the formula (18).

d du （18）

From the combination of formula (18) and formula (17) 2

a 2

u uc

z t

（19）

In which,

aw3 1 2

knEc

（20）


116

In the formula, ca is the consolidation coefficient in the consolidation test under spherical stress. It is easy to find that this equation is a second order parabolic partial differential equation and it is very similar to Terzaghi one-dimensional consolidation equation in terms of the form. It can be solved by analytical method and numerical method with initial conditions and boundary conditions given.

3. Solution to the Equation

The subsequent experimental verification will be conducted on a saturated cylindrical sample with the size r×H=3.5cm×14cm. Thus, calculation here is done in accordance with the test. And the power series solution in the method of separation of variables and finite element methods will be used respectively in the equation solving.

3.1 Power series solution

With Fig.1, boundary conditions and initial conditions can be obtained, namely ①t=0, 0≤z≤H, u=σ. Initially, pore water pressure is equal to the total confining pressure. ②0<t<∞, z=0, u=0. The bottom of the sample is boundary for drainage, pore water pressure is

0 all the time from the beginning of consolidation. ③0<t<∞, z=H, ∂h/∂z=0. At the top of the sample drainage is not allowed, hydraulic gradient is

0. ④t=∞, 0≤z≤H, h=0. When consolidation has been completed, pore water pressure at any part

of the sample will be 0. Thus, triaxial sample consolidation equation and Terzaghi one-dimensional consolidation

equation have similar boundary conditions and initial conditions. The power series solution obtained by the method of separation of variables is

2 2

, a1

4 1 π m πsin exp -

π 2 4z tm

m zu p T

m H

（21）

In the formula, uz, t is the pore water pressure at the point of z from the boundary at t after the consolidation begins. m and π is narrower and PI. z is the distance between the given point and the boundary of drainage. Ta represents coefficient factor of time, and

aa 2

c tT

H （22）

The subsequent test will be conducted on some remolded samples of salty clay with the plastic limit of 18 and liquid limit of 29. Undisturbed soil should be dried for 12 hours and then sieved with the aperture at 2mm and then to make wet soil with water content of 22.5%. The wet soil should be sealed and kept tranquilly for more than 2 days until the moisture is completely balanced. And then the samples will be made by the pressure method with the size as stated earlier.

The dry density of the sample is 1.72 g/cm3, pore ratio 0.7 and saturation 100%. The permeability coefficient k, the elastic modulus E and Poisson’s ratio ν is measured by the parallel sample and the value are 10-8m/s, 8 MPa and 0.35. It is easy to obtained that ca=1.5x10-6 m2/s, Ta=0.000078t. First, the power series solution will be adopted to calculate the process of consolidation in the triaxial test when confining pressure is 100 kPa.

To obtain a higher precision，the first three items are taken as the power series solution for calculation the dissipation of the water pressure and the results is shown in Fig.2. In which c1, c2, c3, c4, c5, c6 and c7 represent respectively the curves of pore water pressure change at the points from which it is 2 cm, 4 cm, 6 cm, 8 cm, 10 cm, 12 cm, 14 cm to the drainage boundary.

It can be observed from Fig.2 that at the initial stage of consolidation, the power series solution


117

shows very poor convergence. It can be seen from the fact that even we take the first three items of Formula (21), the process of dissipation of pore water pressure derived from calculation is in disorder at the initial stage. Firstly, it appears abnormally that the pore water pressure exceeds the confining pressure. Secondly, the pressure at the point which is far from the boundary of drainage is less than that at the point which is closer. When t>500s, the power series solution will show better convergence. Thus, only when the consolidation develops to a certain degree will it be reasonable. Since Formula (19) and Terzaghi one-dimensional consolidation equation have the same form, the conclusion above also is the same with the power series solution of Terzaghi one-dimensional consolidation equation.

Fig.2 The process of dissipation of the pore water pressure based on the power series solution to the

consolidation equation

It is important to stress that the abnormal increase of pore water pressure is not Mandel-Cryer effect. The reason is that the most striking features of Mandel-Cryer effect is that the pore water pressure at the center of the sample increasing gradually from equivalent of consolidation stress to the peak value before it declines and fades away. Yet in Fig 2, it can be found that the initial pore water pressure calculated from power series solution is greater than the confining pressure. 3.2 Finite element solution

Formula (19) is a constant coefficient parabolic partial differential equation, which can be solved by finite element method. In Fig.3 (a), it is the grid division for the triaxial sample with boundary condition and initial condition as follows.

(1)t=0, 0≤z≤H, u=σ. (2)0<t<∞, z=0, u=0, Z =H and the boundaries on the left and right are non-drainage boundaries

with hydraulic gradient at 0.

（a）（b）（c）（d）（e）（f）（g）

Fig.3. Meshing for the finite element method of Partial differential equation and the relative flow distribution in the sample by calculation

c1

c2

c3

c4

c5

c6 c7

u /×

104 P

a

Drainage boundary


118

t /s

u /×

104 P

a

l2

l1

l3

l4l5

l6

l7

In Fig.3, (b), (c), (d), (e), (f) and (g) are respectively the hydraulic gradient distribution when it is 10s, 100s, 1000s, 5000s, 10000s and 30000s from the beginning of consolidation respectively. Here, the length of the arrows only implies the relation between values within the same figure but not different figures. It can be found that with its distance from the drainage boundary increases, the hydraulic gradient decreases gradually. In addition, the drainage of pore water is from the proximity to the distance. That is to say the redundant water closed to the drainage boundary will flow out earlier than that far from the boundary.

Fig.4 Dissipation of pore water pressure based on the finite element method of consolidation equation

In Fig4, it is the dissipation of pore water pressure at various positions derived from the finite

element method, in which l1, l2, l3, l4, l5, l6 and l7 represent respectively the curves for pore water pressure change at the points from which it is 2 cm, 4 cm, 6 cm, 8 cm, 10 cm, 12 cm, 14 cm to the drainage boundary respectively.

Fig.5 Comparison between the two algorithms for consolidation partial differential equation

for triaxial test

Compared with the fig.2, the method has overcome the problem of poor convergence at the initial stage of consolidation. Therefore, it is theoretically reasonable.

In order to further compare the two solutions to Formula (17), the curves for pore water pressure change at various points are given in Fig.5 when t is valued at 50 s, 100 s, 500 s, 1000 s, 3000 s, 6000 s, 10000 s, 30000 s. There, the solid line is the result from power series solution while the dotted line is that from the finite element solution.

From Fig.5, it can be seen that at the intial stage of consolidation (when t<1000s), results from the two methods there have some difference. Especially at the beginning of the consolidation, because of the poor convergence of power series solution, there are apparent differences between the two solutions.

As consolidation develops viz. when t>3000s, results from the two methods gradually tend to converge. Thus, in the paper it is proposed that at the initial stage of consolidation, the finite

t=30000

t=10000

t=6000

t=3000 t=1000 t=500

t=100 t=50

z /cm

u /×

104 P

a

0.02 0.04 0.06 0.08 0.1 0.12 0.140

2

4

6

8

10

12

10

8

6

4

2

0 2 4 6 8 10 12 14


119

Vw

/cm

3

100kPa

200kPa

20

10

0 0 20000 40000 60000 80000 100000

t /s

element solutions to the consolidation equation should be adopted, while power series solutions should be selected when it is needed to calculate the pore water pressure or degree of consolidation after the consolidation has developed for a while.

4 Comparison with the data from the test

Fig. 6 WF stress path triaxial test system The corresponding test will be done in WF stress path triaxial test system. As mentioned earlier,

the sample is a cylinder with specification of r×H=3.5cm×14cm. WF stress path triaxial test system consists mainly of double pressure chamber, load sensor, real-time control system, data acquisition system, servo hydraulic pressure controller, automatic volume change tester and triaxial loading frame, etc, as shown in figure 6.

The equipment, with double pressure chamber and catamaran variable instrument, can measure change in the amount of water in internal and external chambers to avoid the effect of deformation of internal chamber under pressure on the precision of water displacement measurement. For better illustration, consolidation tests are conducted under the confining pressure at 100 kPa and 200 kPa respectively. The drainage process in the test is shown in Fig.7, in which Vw is for displacement.

According to the concept of average degree of consolidation, the actual and theoretical degree of consolidation at different time can be obtained, the result of which is shown in Fig.8.

The curve for the theory of degree of consolidation can at large reflect the main features of the consolidation stage in triaxial test. The inconspicuous difference between the theoretical curve and the result from the test is caused mainly by the assumption that the soil is homogeneous, isotropic and fully saturated as well as by the assumptions about soil permeability coefficient, the elastic modulus and Poisson’s ratio.

Fig.7 Drainage under confining pressure at 100kPa and 200kPa in triaxial test

PC数采仪

液压加压装置

三轴加载架

气压转换装置

体变仪Double

Triaxial loading frame

Double pressure chamber

Num

ber by instrument

Pressure conversion device

Hydraulic pressure device

Body becom

ing nstrument


120

Fig.8 Comparison between theoretical algorithm and the average degree of consolidation derived from

WF stress path triaxial test

From Fig.8, it is found that at the initial stage (when t<10000s), when the confining pressure is 200 kPa , the actual degree of consolidation is quite close to its theoretical value. When t>20000s and the confining pressure is 100 kPa, the actual degree of consolidation is quite close to its theoretical value. Besides, when the confining pressure is low, the sample consolidates at a higher rate than when it is under greater confining pressure. The reason mainly lies in the fact that when it is under less pressure, the amount of pore water discharged is less.

5. Conclusion

Based on the Effective Stress Principle, Darcy's Law and Linear Elastic Assumption, equation for the consolidation stage of triaxial test is established according to conditions for deformation and water flow continuity. Power series solution and finite element solution are employed to solve the pore pressure change at different time. Finally the deformation and distribution of effective stress are obtained. It is found that even to take the first four items for calculation, finite element solution still shows very poor convergence at the initial stage of consolidation with the abnormal phenomenon that pore pressure is greater than confining pressure, or the value of pore pressure at different points are reversed. With the consolidation develops, the power series solution shows good stability and precision. By comparison between the result from the two theoretical algorithms and that from WF stress path triaxial test, it is found that they are very close and the theoretical curve can reflect the main features of the consolidation stage in triaxial test. The difference may mainly result from the assumption that the soil is homogeneous, isotropic and fully saturated as well as by the linear elastic assumptions and Darcy's law, etc.

6. Acknowledgments

The authors gratefully acknowledge the support of Chinese Nature Science Foundation under the Grant No. 51178290 and Tianjin Nature Science Foundation under Grant No. 11JCYBJC02900. Our thanks also go to School of Civil Engineering, Tianjin Chengjian University, to my friends and colleagues, without whose help the paper never come into being.

Reference

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100

80

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0

/%

200kPa

100kPa理论值

0 20000 40000 60000 80000 100000

t /s

Theoretical value


121

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Theoretical Study and Experimental Verification of the ... · u 3）（ In the formula, σ' and u...

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Transcript of Theoretical Study and Experimental Verification of the ... · u 3）（ In the formula, σ' and u...

Theoretical Study and Experimental Verification of the ... · u 3） （ In the formula, σ' and u...

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Transcript of Theoretical Study and Experimental Verification of the ... · u 3） （ In the formula, σ' and u...

Theoretical Study and Experimental Verification of the ... · u 3）（ In the formula, σ' and u...

Transcript of Theoretical Study and Experimental Verification of the ... · u 3）（ In the formula, σ' and u...