Research Article

Innovative Slope Stability and Displacement Analyses

Department of Civil Engineering, National Cheng Kung University, Tainan City, Taiwan

***Corresponding author: Ching-Chuan Huang**,
Department of Civil Engineering,
National Cheng Kung University,
Address: No. 1, University Rd., Tainan City,
Taiwan,
Tel: 886-6-2757575 ext. 63160,
Fax: 886-6-2383042,
E-mail: samhcc@mail.ncku.edu.tw

**Received:** November 9, 2018
**Accepted:** December 21, 2018
**Published:** January 2, 2019

**Citation:** Huang CC. Innovative Slope Stability
and Displacement Analyses. *Madridge J Agric Environ Sci*. 2019; 1(1): 7-13.
doi: 10.18689/mjaes-1000102

**Copyright:** © 2019 The Author(s). This work
is licensed under a Creative Commons
Attribution 4.0 International License, which
permits unrestricted use, distribution, and
reproduction in any medium, provided the
original work is properly cited.

Abstract

Conventional methods of slope stability provides a constant value of safety factor for the slope, providing no information of slope displacements and possible variations of safety margins along the potential failure surface. To overcome this drawback, an innovative approach is proposed here, which takes into account all limit equilibrium requirements originally adopted in the conventional slope stability analyses, with a displacement compatibility function and a hyperbolic shear stress-displacement soil model. The new method provides incremental slope displacements induced by internal or external stress (or safety status) variations. A case study on a well-monitored slope during a rainstorm showed that the measured slope displacement caused by an elevated groundwater table can be simulated using the proposed method along with hyperbolic soil parameters obtained in large-scale direct shear tests. The proposed method substantially strengthened the weakness associated with conventional slice methods, providing useful information of slope displacement induced by the elevated groundwater table.

**Keywords:** Slope failure; Slope displacement; Stability analysis; Disaster mitigation; Force
equilibrium.

Introduction

The slice method of slope stability was pioneered by Fellenius in 1920’s [1]. The
original Fellenius’ method and the following updates constitute a major contribution to
the practice and development of geotechnical engineering [2-6]. It is a well-known fact
that the sliced potential failure mass is a statically indeterminate system [7,8]. Table 1
summarizes unknowns and equations for a potential failure surface with a total of ns
slices. Figure 1 schematically shows the force acting on a potential failure mass with ns
vertical slices, in which, *>W _{i}, N_{i}* and

Where,

*i*: slice number (*i*=1, 2, …, *ns*)

*W _{i}*: self-weight of slice

α* _{i}*: inclination angle of slice base

*c*: cohsion intercept of soil

φ: internal friction angle of soil

*u _{i}*: porewater pressure acting at slice base

*ℓ _{i}*,

The static determinate conditions for the Fellenius′ slice method is summarized in table 2. Note that the force equilibrium in the direction normal to the slice base (Σ *F _{N}*=0) does not take into account the influence of inter-slice forces. This method is based on an implicit assumption that the resultant inter-slice force acts parallel to the slice base, as pointed out by Whitman and Bailey [7].

Derivation of Displacement-Based Fellenius′ Method

In the following, local force-based safety factors (*FS _{i}*) and a hyperbolic stress-displacement model will be incorporated in the Fellenius′ method. According to Σ FN=0, the effective normal force

According to Mohr-Coulombs′ failure criterion and the definition of local stress-based safety factor *FS _{i}*:

where,

τ*f _{i}* , τ

*S _{fi}, S_{i}* : ultimate shear resistance and shear force, respectively, for slice

*FS _{i}*: local force-based safety factor As shown in figure 2, where the shear stress (τ

where,

*k _{initial}*: initial shear stiffness

*K, n*: material constants

*R _{f}*: failure ratio

σ* _{ni}*′ : normal stress acting at the base of slice

*P _{a}*: atmospheric pressure

Normalizing Eq. (8) using τ* _{fi}*:

Based on the definitions of local safety factors in Eqs. (5), Eq. (13) can be re-written as:

Introducing a displacement diagram [9] that satisfy displacement compatibility as schematically shown in figures 3a and 3b:

where,

ψ: agle of dilatancy

The displacement of slice _{i} can be related to the vertical displacement at the top of slice No. 1 (Δ_{o}) using the following equation:

Equation (18) can be expressed as:

where,

Substitute Eq. (20) into Eq. (16),

Based on the principle of moment equilibrium at the center of circle, i.e.,ΣMo_{o}=0:

Rewriting Eq. (22):

Substitute Eqs. (4) and (21) into Eq. (23), and re-arrange to solve for Δ_{o}:

It can be seen that Eq. (24) is basically an inverted expression of Eq. (1), with additional displacement-related known parameters *‘a’,‘b’,‘ƒ(α _{i})’*, and an unknown ‘Δ

Local Displacement-Based Safety Factors

The displacement at failure (Δ* _{f}*) can be obtained by using

Re-arrange the above equation to obtain Δ* _{f}*:

A displacement-based safety factor, *FD _{i}* can be defined as:

Substituting Eqs. (19), (27) into Eq. (28):

Analytical Procedure in Computer Program

A computer program in Visual Basic 2010 (Microsoft, 2010) was coded based on the following algorithm:

1. Input analytical parameters, including slope profile, circular arc failure surface (rotation center, coordinates, and radius), and displacement-related parameters, *K, n*, and *R _{f}*.

2. Perform a conventional slope stability analysis using Eq. (1) to derive a constant value of *F _{s}*.

3. Calculate preliminary values of *N _{i}′* using Eqs. (4), or calculate σ′

4. Calculate preliminary values of *‘a’, ‘b’*, and *f(α _{i}*) using Eqs. (14), (15) and (20), respectively.

5. Calculate preliminary value of ‘Δ_{o}’ using Eq. (24).

6. Calculate preliminary values of *FS _{i} (i=1,---, ns*) using
Eq. (21).

7. Calculate improved values of N′_{i} and ‘*a*’ using Eqs. (4), and (14), respectively.

8. Calculate improved value of ‘Δ_{o}’ using Eq. (24).

9. Check the convergence of ‘Δ_{o}’ using Eq. (25). If not satisfied, repeated from step (6).

10. Calculate final values of *FS _{i}* and

11. Calculate final values of *∆ _{i}* using Eq. (19).

12. Calculate final values of internal stresses σ′_{ni} and τ_{i}, using Eqs. (12) and (8), respectively.

Increments of Slope Displacement

In calculating slope displacements induced by external and internal condition changes (e.g., loading, water table, and porewater pressure variations), two values of *∆ _{i}* (or Δ

Case Study and Discussions

The studied slope locates in south-west foothill area of Taiwan. The slope is a part of highway No.18 which winds through a chronic landslide area. The landslide caused property losses and traffic problems in rainfall seasons, and therefore, was well monitored and studied [10]. Figure 5 shows the studied slope with points of inclinometer measurements and possible locations of slip surface [11]. The observed slip surface is simulated using seven segments of straight lines which are also shown in the figure. Underground water table observations were conducted in a borehole adjacent to the slope during the rain strom, and the recorded water table height is also shown in figure 5. The slope displacement during typhoon Herb in 1995 using the data of inclinometer measurements is 30 mm in the downward direction. Site exploration has shown that the slope mass consisted of colluviums, and the fragments of rock frequently showed high blow counts (*N*-value) of standard penetration tests. The *N*-value for the matrix material on-site was about 10. Probable values of φ in the range of 25°-30° are estimated for the focused slope. The cohesion intercept (*c*) for the slope mass has been back-calculated, and a probable range of *c*=30-40 kPa has been reported for the studied slope by ERRL [11]. Figure 6 shows the changes of conventional safety factors (*F _{s}*) for the studied slope due to the elevated groundwater table. For a range of internal friction angle (ϕ) ranging between 25° and 30°, and a range of cohesion intercept (

Figure 8a shows analytical values of horizontal displacements for at *x*=800 m where the inclinometer was installed. Three groups of curves are characterized by their *R _{f}* values, i.e., curves with higher

Figure 12a shows typical examples of mobilized shear and effective normal forces along the sliding surface based on identical soil parameters for figures 9a and 10a. Distributions of *σ′ _{ni}* and

Conclusions

A novel improvement of a conventional slice method of slope stability is proposed here, providing significant information regarding the displacement of the slope subjected to internal and/or external environmental changes. The proposed method satisfies force and moment equilibrium criteria adopted in the original slice method, with additional displacement compatibility requirement and a hyperbolic shear stress-displacement soil model. A new static determinate system was attained by introducing displacement compatibility functions and a hyperbolic shear stress-displacement model for the Fellenius’ method. As a result, local displacement-based and stress-based safety factors along the potential failure surface are parts of the analytical solution. Based on the case study of a well-monitored slope during a rainstorm, the effect of groundwater table rise during the rainstorm, expressed as an incremental slope displacement, was computed using the proposed method. It was shown that the slope displacement measured during the focused rainstorm can be closely simulated using stress vs. displacement relationships obtained from a large-scale direct shear test, revealing the potential of the present method for further applications.

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