J. Cent. South Univ. Technol. (2008) 15(s1): 351-356

DOI: 10.1007/s11771-008-379-y

Couple analysis on strength reduction theory and rheological mechanism for slope stability

LIU Zi-zhen(刘子振)¹, YAN Zhi-xin(言志信)², DUAN Jian(段 建)²

(1.School of Machine and Electronics and Architecture, Taizhou University, Taizhou 318000, China;

2. School of Civil Engineering and Machanics, Lanzhou University, Lanzhou 730000, China)

Abstract: Considering the rheological properties of rock and soil body, and exploiting the merit of strength reduction technique, a theory of couple analysis is brought forward on the basis of strength reduction theory and rheological properties. Then, the concept and the calculation procedure of the safety factor are established at different time. Making use of finite element software ANSYS, the most dangerous sliding surface of the slope can be obtained through the strength reduction technique. According to the dynamic safety factor based on rheological mechanism, a good forecasting could be presented to prevent and cure the landslide. The result shows that the couple analysis reveals the process of the slope failure with the time and the important influence on the long-term stability due to the rheological parameters.

Key words: slope stability; strength reduction; rheological properties; couple analysis; safety factor

1 Introduction

There had been many comparatively all-round analysis methods to study slope stability on theory and practice of engineering already. And, a lot of theory analyses and algorithms which are universally accepted or reliable had been obtained, i.e., traditional limit equilibrium method and finite element method^[1-6]. However, the situation of slope stability analyzed by these methods is at some time, and the safety factor is instantaneous. As we know, the stress and strain of rock and soil body in nature could change with the time, that is to say, rock and soil body had rheological properties^[7-10]. Therefore, we must consider that the strength of rock and soil body is reduced because of creep when evaluating slope stability. Furthermore, the continual and changeable real safety factor can be determined through fetching in long-term limit strength. For a long time, the slope is on moving. Its deformation is timeless to sliding. And the processing from deformation to sliding includes the sliding plastic region in slope sprouting, developing and linking up gradually, lastly, forming a sliding surface. In general, the rheological properties of slope will experience three stages including creep deformation, sliding destruction, and gradual stability. Using the strength reduction technique on finite element makes couple analysis for slope stability, which must be considered to combine the instantaneous analysis theory and rheological properties. Thus, not only the evolution of stress and strain in slope could be reflected well with the rheological properties, but also the safety factor could be calculated accurately at a certain moment. A good forecast could be presented for slope prevention and cure, which could reduce the occurrence of the landslide accident.

2 Couple theory between strength reduction and rheological properties

Considering the rheological properties of slope, the ideas of couple analysis for the stability are as follows. Firstly, on the basis of the elasto-plastic theory and the rheological mechanism of rock and soil body, the slope long-term stability is analyzed through the long-term strength behavior. Then, the indexes c(t) and φ(t) of long-term strength could be determined with the time changing. At last, the rheological formula of long-term strength indexes, strength reduction parameters and safety factor could also be reestablished, and they are unified organically, as shown in Fig.1. The calculation procedure of couple analysis is shown in Fig.2.

Fig.1 Relationship between long-term strength and safety factor

Fig.2 Calculation procedure of couple analysis

2.1 Establishment of rheological equation for slope made up of rock and soil body

The rheological properties of rock and soil body mainly include creep, relaxation, flowing, strain effect and long-term strength effect. The stress and distortion could be contacted with the time through the rheological properties. Therefore, the rheological equation must be established for rock and soil body, which shows a variety of circumstances that the rheological indexes change with the time. At the same time, the rheological properties of rock and soil material could be obtained through the experiment in laboratory. Through the long-term strength equation which is mainly under the weight of the slope itself, the equation of Mohr-Coulomb will be established to analyze the stability of rock and soil body^[7]. It is denoted as

τ(t)=c(t)+σ_n tan φ(t)                            (1)

The related references[7-8] indicated that the cohesion of rock and soil body is changeable with the time obviously, but the friction angle changes very little. So, the friction angle φ(t) is advisable to be a constant. Then Eqn.(1) becomes

τ(t)=c(t)+σ_n tan φ                              (2)

Generally, analyzing the slope stability can be done gradually from instantaneous strength to long-term limit strength of the rock and soil body. Consequently, the results below are obtained:

                          (3)

where  c₀ and c_∞ are the initial cohesion and limit time cohesion, respectively; τ₀ and τ_∞ are the initial instantaneous strength and long-term limit strength. The behavior of rock and soil body is shown in Fig.3(a).

Fig.3 Relationship between strength and time for rock and soil body: (a) Long-term shear strength of rock and soil body;    (b) Curve of strength index and time

Taking the long-term strength and corresponding cohesion as calculation index, the relationship curve between cohesion and time can be determined through the experiment (shown in Fig.3(b)).

When the slope runs to the critical or failure state, the deformation will achieve a stable creep. It is denoted as

constant                            (4)

where  γ^c is plastic deformation, is the limit of plastic deformation.

According to the creep rule of rock and soil body, the logarithm formula of long-term strength is used for the slope stability^[7], that is

                               (5)

where  β and T are the parameters of long-term strength, t is the time, t^* is arbitrary and small time (constantly t^*=1 s).

Similarly, there is own rheological equation for cohesion c(t), and the corresponding parameters are β_c and T_c. If T_c=T, then

                              (6)

Eqn.(6) is changed into

                      (7)

According to the method of the Curve Adjustment Act, the curve is converted into a straight line by adopting a specific coordinate. Then, the rheological properties of cohesion can be got through fitting the experiment curve. And, the calculation accuracy should maintain high.

2.2 Solving rheological safety factor of slope

Using the strength reduction technique on finite element to make couple analysis for slope stability, there are criteria to reflect the destruction of failure slope. Considering the rheological properties of rock and soil body, the criteria used to evaluate the destruction of failure slope are on the basis of strength reduction technique with finite element as follows^[11-14].

The sliding destruction of slope is mainly due to the development of shear strain and displacement caused by rheological properties of rock and soil body. When the parameters, c(t) and φ(t), of the slope are continuously reduced to destruction of slope, the dangerous surface or weakest strength surface will form a sliding state in slope. Through the finite element software, the plastic region in slope will link up the slope to form a sliding band.

According to the couple theory between rheological properties of the slope and strength reduction technique, the instantaneous parameters can be got, and the instantaneous safety factor can be calculated by finite element analysis. With the time increasing, the safety factor is changeable. So,

                                   (8)

where  c is the parameter of reduction strength input to the model, and F_s(t) is the safety factor or reduction coefficient at corresponding time.

Carrying on the iteration to pilot calculation continuously, the safety factor under this time can be obtained after inputting the corresponding c and φ.  Therefore, the relational curve between the safety factor and the time on the rheological properties can be drawn, and, the secure state will be described quantitatively with the rheological properties of the slope.

3 Analysis on example

A slope is made up of rock and soil body, and the height of slope is 25 m. Fig.4 shows the slope geometrical size and boundary condition. The physical mechanical parameters are listed in Table 1 from analyzing the slope at initial time. It is supposed that the results of T=0.013 s and β_c=550 kPa could be obtained through the experiment curve fitted. The slope stability analysis is regarded as the strain question of two-dimensional solid plane.

Fig.4 Computer geometrical model of slope

Table 1 Material parameters of slope

3.1 Determining rheological model and reduced parameters of slope

Not considering the rheological influence of friction angle in the example, the rheological condition of long-term strength is mainly caused by the changeable cohesion c(t). Taking T=0.013 s and β_c=550 kPa into Eqn.(7), and simultaneously taking t^*=1 s, we can get the result as follows:

                    (9)

By using the finite element software ANSYS, the concrete model and the analysis result are as follows. By means of the front processing function of ANSYS^[15], the computation grid division is carried on for the slope, and the PLANE42 unit is selected. According to the free mapping unit of four nodes and quadrangle, the grid division results of slope model are as follows: the unit size is 2 m, the model is divided into 1 380 nodes and   1 296 units altogether (shown in Fig.5). The fixed restrain is used at the bottom surface of computer models; the horizontal direction restraint is used at the two vertical sides; and there is not any restrain at the free surface and the slope surface. The elastoplasticity and nonlinear model is used in the constitutive relationship of the rock and soil body, and the yield criterion is the D-P criterion.

Fig.5 Computer model of ANSYS

3.2 Determining F_s(t) on strength reduction technique

According to the basic theory of strength reduction technique, the strength indexes c(t) and φ will be reduced unceasingly. When the slope achieves the critical destruction state, the surface will have the remarkable distortion and the displacement. Moreover, the displacement of horizontal direction at the toe of slope and the vertical direction at the top of slope will increase suddenly. So, the plastic region in slope will link up the slope to form an obvious sliding band that is the most weak or dangerous surface. In the computation analysis process, firstly, the parameters c=30 kPa and φ=20? are carried on to reduce at initial time, then, the reduction coefficient F_s(t) is increased unceasingly by taking 0.1 as the increasing length of stride. With F_s(t) increasing, the convergence rate of the finite element gets much slow. When F_s=1.25, the plastic region in slope does not link up the slope (shown in Fig.6(a)). When F_s=1.26, the plastic region in slope already links up the slope to form an obvious sliding band (shown in Fig.6(b)). If the increased length of stride is larger, the critical state of failure slope may jump over the critical point. So, in order to get the quite precise reduction coefficient, the increased length of stride needs reducing. Before the slope borders on the failure state, the length of stride is reduced as 0.01, and the number of computer iteration is 600 times. Simultaneously, the characteristic points (for example, at the top and the toe of slope) may also be chosen to analyze through the displacement changing suddenly. According to the above analyses, the reduction coefficient (or safety factor) can be obtained at initial time, and the result is F_s=1.257. The sliding surface is shown in Fig.6(c).

As a result of the rheological properties of rock and soil body, the strength index c(t) is changed along with the time. After a half year, the index c(t) is smaller than before. According to the analysis method above, the corresponding safety factor can be obtained which is F_s(1/2)=1.102 (shown in Fig.7(a)). Until the 4th year, the

Fig.6 Failure sliding surface and plasticity region of slope:   (a) Plasticity region at F_s=1.250; (b) Plasticity region at F_s= 1.260; (c) Failure sliding surface at F_s=1.257

corresponding safety factor can also be obtained: F_s(4)=0.999 (shown in Fig.7(b)). When the time is at the fortieth year, the corresponding safety factor can also be obtained: F_s(40)=0.922. With the time prolonging, the solution will be done unceasingly to show that the cohesion c(t) and safety factor of the slope become small. The calculation results of couple theory between rheological properties and strength reduction technique are listed in Table 2 for the slope. From the results of Table 2, the rheological curve of strength indexes cohesion c(t), friction angle φ(t) and safety factor F_s(t) can be drawn well (the relatively instantaneous results are shown in Fig.8).

From Table 2 and Fig.8, we can discover the safety factor F_s(4) =0.999 that is below 1 after the 4th year, which shows that the slope will be on failure with the time increasing. Therefore, the long-term stability tendency will be forecasted through considering the couple theory between the rheological properties and strength reduction technique of rock and soil body.

Fig.7 Failure sliding surface under reheological properties of slope: (a) Sliding surface at F_s(1/2)=1.102; (b) Sliding surface at F_s(4)=0.999

Table 2 Results of couple analysis for slope

Fig.8 Indexes c(t), φ(t) and F_s(t) of soil falling with time

3.3 Influence analysis of safety factor

As we know, the traditional limit equilibrium method and the finite element method of strength reduction can be used to evaluate the instantaneous slope stability quantitatively by instantaneous safety factor. However, the couple analysis considered the rheological properties. It evaluated the dynamic stability through the long-term safety factor of the slope. The results of all kinds of analyses are listed in Table 3, and the corresponding safety factors based on rheological properties are shown in Fig.9.

Table 3 Results of stability analysis for slope

Fig.9 Relationship between safety factor and time for slope analysis: 1—Sweden method; 2—Bishop method; 3—Janbu method; 4—Strength reduction technique; 5—Couple analysis

Therefore, considering the rheological properties and the strength reduction technique, the couple analysis can manifest the actual dynamic process of the slope.

For the quantitative analysis of the slope stability, its influence analysis of the result precisions is mainly decided by the strength indexes c(t) and φ(t) of the slope. On the basis of rheological properties, the cohesion influence is discussed mainly. Because the cohesion c(t) is an important parameter at instantaneous or dynamic stability analysis, there is tremendous influence on the result of evaluating stability at different c(t). For this example, the cohesion at the initial time is taken as 10, 20, 30, 40 and 50 kPa, respectively. If the parameters and the conditions except for cohesion are invariable, the calculation results are shown in Fig.10.

When the cohesion increases, the sliding force of the slope is not changeable by means of the limit equilibrium method or the strength reduction technique. However, the anti-sliding force increases. So, the safety factor obtained increases correspondingly. Because the couple analysis has considered the rheological process, the influence of safety factor includes the different cohesion at initial time as well as the rheological cohesion. Therefore, the changing rate of safety factor, which is calculated by the limit equilibrium method or the strength reduction technique, is smaller with the changeable cohesion c(t). But the changing rate calculated by couple analysis is dynamic because of its rheological properties.

Fig.10　Curves of safety factor and cohesion at different states: 1—Bishop method; 2—Strength reduction technique; 3—Couple analysis at t=0 a; 4—Couple analysis at t=1 a; 5—Couple analysis at t=4 a; 6—Couple analysis at t=10 a

4 Conclusions

1) The distortion destruction of the slope is mainly due to the rheological properties and the strength reduction of the rock and soil body simultaneously.

2) The method of couple analysis evaluates the slope stability by means of combining the instantaneous and dynamic characteristic of the slope, which manifests the elastoplasticity behavior and the rheological properties of the rock and soil on the strength reduction technique simultaneously.

3) The destruction standard of the couple analysis is judged by the rheological properties of the slope and the changed tendency of the distortion plastic region. According to the theory of strength reduction technique, the rheological relation of the safety factor can be obtained. And the concept of this method is clear to judge the slope failure.

4) The example indicates that the rheological index c(t) of shearing strength is very important to the safety factor of the slope, and the influencing process is dynamic. So, a better effect could be obtained to forecast the possibility of landslide and reinforce the slope.

References

[1] SWAN C C, SEO Y K. Limit state analysis of earthen slopes using dual continuum/FEM approaches [J]. Int J Numer Anal Meth Geomech, 1999, 23: 1359-1371.

[2] CHEN Zu-yu. Soil slope stability analysis—Theory?methods? program [M]. Beijing: China Water Press, 2003. (in Chinese)

[3] DUNCAN J M. State of the art: Limit equilibrium and finite element analysis of slopes [J]. Journal of Geotechnical Engineering, 1996, 22(7): 577-596.

[4] HUANG Run-qiu, XIAO Hua-bo, JU Neng-pan, ZHAO Jian-jun. Deformation mechanism and stability of a rocky slope [J]. Journal of China University of Geosciences, 2007, 18(1): 77-84.

[5] GRIFFITHS D V, LANE P A. Slope stability analysis by finite elements [J]. Geotechnique, 1999, 49(3): 387-403.

[6] JIANG Jing-cai, YAMAGAMI T. Charts for estimating strength parameters from slips in homogeneous slopes [J]. Computers and Geotechnics, 2006, 33(6/7): 294-304.

[7] BЯЛОВ C C. Rheological theory of soil mechanics [M]. Beijing: Science Press, 1987. (in Chinese)

[8] SUN Jun. Rheological behavior of geomaterials and its engineering applications [M]. Beijing: China Architecture and Building Press, 1999. (in Chinese)

[9] YE Zhou-yuan, HONG Liang, LIU Xi-ling, YIN Tu-bing. Constitutive model of rock based on microstructures simulation [J]. J Cent South Univ Technol, 2008, 15(2): 230-236.

[10] SHIOTANI T. Evaluation of long-term stability for rock slope by means of acoustic emission technique [J]. NDT&E International, 2006, 39(8): 217-228.

[11] CHEN Wei-bing, ZHENG Ying-ren, HONG Xia-ting, ZHAO Shang-yi. Study on strength reduction technique considering rheological property of rock and soil medium [J]. Rock and Soil Mechanics, 2008, 29(1): 101-105. (in Chinese)

[12] DAWSON E M, ROTH W H, DRESCHER A. Slope stability analysis by strength reduction [J]. Geotechnique, 1999, 49(6): 835-840.

[13] LUAN Mao-tian, WU Ya-jun, NIAN Yan-kai. A criterion for evaluating slope stability based on development of plastic zone by shear strength reduction FEM [J]. Journal of Seismology, 2003, 23(3): 1-8. (in Chinese)

[14] LIU Jin-long, LUAN Mao-tian, ZHAO Shao-fei, YUAN Fan-fan, WANG Ji-li. Discussion on criteria for evaluating stability of slope in elastoplastic FEM based on shear strength reduction technique [J]. Rock and Soil Mechanics, 2005, 26(8): 1345-1348. (in Chinese)

[15] LIU Tao, YANG Feng-peng, LI Gui-ming, ZHANG Jian, WANG Jian-guo. Make yourself master of ANSYS [M]. Beijing: Tsinghua University Press, 2001. (in Chinese)

(Edited by YANG Bing)

Received date: 2008-06-25; Accepted date: 2008-08-05

Corresponding author: YAN Zhi-xin, Professor, PhD; Tel: +86-931-8632418; E-mail: yzx10@163.com