A Simple Seismic Analysis of Embankment Slopes Stabilized with a Sheet Pile

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A Simple Seismic Analysis of Embankment Slopes Stabilized with a Sheet Pile Jing-Cai Jiang*1, Tsunataka Furuya2 Department of Civil and Environmental Engineering, The University of Tokushima 2-1 Minami-josanjima-cho, Tokushima, 770-8506, Japan jiang@ce.tokushima-u.ac.jp; 2furuya@jce.co.jp

*1

Abstract A limit equilibrium approach is proposed to evaluate the seismic stability of embankment slopes stabilized with a wall such as a sheet pile wall. Different slip surface and earthquake coefficient can be assumed respectively in the upslope and downslope soil masses separated by the wall. A target value of the factor of safety is prescribed by the designer, contrary to the conventional stability analysis method where the factor of safety is usually unknown. The design for wall itself is then conducted to ensure that the target factor of safety is really reached, though the design method for wall itself is not discussed in the present paper. A practical method is developed to evaluate forces which the wall must sustain and to search for a pair of critical slip surfaces which are the most dangerous for the wall.

limit equilibrium, that interactive forces between the wall and soils are computed by giving a target value of the factors of safety for the assumed slip surfaces. It may be understood that, if the wall is designed to sustain the interactive forces between the wall and soil masses subjected an earthquake, the wall-installed slope will have, at least, the target value of the factor of safety against the same earthquake.

Method of Analysis

Embankments; Slopes; Seismic Stability; Limit Equilibrium; Critical Slip Surfaces

A cross section of an embankment slope stabilized with a sheet pile is shown in Fig. 1, where slip surface, AD and BE, is assumed respectively in the upslope and downslope soil mass of the wall. The effect of an earthquake is considered using a pseudo-static force horizontally acting at the soil masses. The proposed method consists of the following two concepts2)-3):

Introduction

1) Upslope and downslope sliding mass of the wall may have different values of the factor of safety.

Keywords

A sheet pile wall, typically installed near the toe of an embankment slope, is often used to enhance the slope stability against earthquakes. In such cases, it is rather difficult to evaluate the seismic stability of the whole slope since a single slip surface that passes through the wall cannot take place due to the high rigidity of the wall. Numerical techniques, such as FEM, are usually applied to the static stability analysis of wall-installed slopes1). However, a seismic analysis using numerical techniques is not necessarily practical/cost-effective because of the complexities of calculation procedures and complicated geological/geotechnical conditions of slopes stabilized by a sheet pile wall.

This paper proposes a simple/effective approach based on the limit equilibrium concept to evaluate the seismic stability of slopes stabilized with a sheet pile wall. A unique point of this approach is that different slip surface can be assumed separately in the upslope and downslope soil mass of the wall. The other feature of this approach is, unlike the conventional methods of

2) Interactive forces between the wall and soil masses are computed by prescribing a target value of the factor of safety.

The first concept means that the upslope and downslope sliding mass bounded by the wall are allowed to possess different factor of safety. It is quite natural to make such a consideration as different slip surface can appear respectively in the two soil masses due to the presence of the wall. In order to justify this, let's consider an example in which a sufficiently strong wall is to be installed deeply enough in an active landslide slope. The upslope sliding mass will be stopped due to sufficiently large resistance provided by the wall, and therefore become stable. However, the downslope sliding mass of the wall may still be in an unstable state, and thus it is possible that sliding movement will continue. From this it is obvious that the upslope and downslope sliding mass may have different factors of safety. The magnitude of these two factors of safety, Fu and Fd (Fig. 1), is usually unknown.

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Construction Engineering (CE) Volume 2, 2014

lower than or equal to the location of the end D of the upslope slip surface AD (Figs.1-2). Note that the thickness of the wall is not considered at this stage.

A U B

KuWu

C D

Fd

x

Ou

KdWd Pd

Embankment

Pu D E Sheet pile

Fu

ru

Soft soil layer

A

Hard soil layer lu

Ej+1

FIG. 1 EMBANKMENT SLOPE STABILIZED WITH A SHEET PILE

The second concept implies that a desired target value of the factor of safety is first specified by the designer, and then a design for the wall is made to ensure that the target value of the factor of safety are actually achieved. In this design procedure, contrary to the limit equilibrium methods where a factor of safety is calculated, Pu and Pd (see Fig. 1) acting on the upside and downside faces of the wall are calculated using a target value of the factor of safety for a given slip surface. A repeated trial procedure will be presented later to search for the critical slip surfaces in the upslope and downslope soil masses which correspond to the most dangerous situations for the wall and the associated values of Pu and Pd. Now suppose that, in Fig. 1, the interactive forces Pu and Pd for a target factor of safety Fu and Fd have been obtained. A design of the wall is performed to sustain the combined effects of Pu and Pd, so that the specified values of Fu and Fd will be ensured. In other words, in order to achieve the desired values of Fu and Fd, the wall must at least provide a resistance force which can make a balance with the combined effects of Pu and Pd. After the resistance force is obtained by the repeated trial procedure mentioned above, a detailed design for the wall itself can be conducted to determine size, strength and most suitable location of it. This paper discusses a stability analysis method for wall-installed slopes and does not deal with the design procedure for stabilizing walls themselves.

Formulations Based on the Bishop Method Based on the Bishop method4), this section derives the equations required to calculate values of Pu and Pd. We assume that, as shown in Fig. 2, both Pu and Pd act in a horizontal direction by neglecting the friction between the wall faces and the upslope/downslope soil masses.

It should be noted that such a simplified assumption is not always effective and needed to be improved in future research. In addition, the location of the end E of the downslope slip surface BE is constrained to be 22

KuW Xj+1

Ei

Pu

W

Xi

hu D

α

T N=N′+ul

(a) Forces acting on a slice in Upslope sliding mass x

Od

rd

ld B

KdW

Xj

Xj+1

Ej

Pd Ej+1

W T

hd E

N=N′+ul

(b) Forces acting on a slice in downslope sliding mass FIG.2 SLIP CIRCLES IN UPSLOPE & DOWNSLOPE SOIL MASSES

Equations for Computing Interactive Forces Since the slip surfaces AD and BE are assumed to be circles, a moment equilibrium for the sliding mass on the upslope slip surface is described by summing moments about the center Ou of rotation:

ru ∑ T + lu Pu ∑ Wx + ∑ K uW yu = u

u

[1]

u

where W is the total weight of a slice, T is the shear force mobilized on the base of slice, ru is the radius of the upslope slip circle, lu is the moment arm associated with Pu, and Ku is a horizontal seismic coefficient. See Fig. 2 for other symbols.

The shear force, T, can be expressed as

= T

R f c′l + N ′ tan ϕ ′ = Fu Fu

[2]

where N′ is the normal effective force at the base of

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slice, l is the length of the base of slice, and c′, φ′ are the strength parameters of soil.

Considering the force equilibrium of a slice in vertical direction yields

N′ =

W + X i +1 − X i − ul cos α − cl sin α / Fu cos α + tan ϕ ′ sin α / Fu

[3]

in which α is the inclination angle of the base of slice.

By submitting Eq. [3] into [2] and then into [1] and assuming Xi+1-Xi =0, we obtain the following.

c′l cos α + (W − ul cos α ) tan ϕ ′ cos α (1 + tan α tan ϕ ′ / Fu ) u Fu = y l ∑ W sin α + K u ∑ W u − u Pu ru ru u u ∑

[4]

Rearranging Eq. [4], the equation of Pu can be obtained.

ru  yu −  ∑ W sin α + K u ∑ W lu  u ru u

Pu=

1 c′l cos α + (W − ul cos α ) tan ϕ ′  − ∑  Fu u cos α (1 + tan α tan ϕ ′ / Fu ) 

surfaces are defined as a pair of slip circles which give the maximum value ∆Pmax of ∆P=Pu –Pd (see Figs. 2-3), subject to Pu >0 and Pd >0. Note that the wall should not be installed at positions in which Pd<0.

2) ∆Mmax approach: This approach defines the critical slip surfaces to be a pair of slip circles which provide the maximum value ∆Mmax of ∆M=Pu hu-Pd hd (see Figs.

2-3), subject to Pu >0 and Pd >0.

In a theoretical manner, we cannot conclude which approach is more adequate and is better to use. It is thus suggested that both approaches are applied and results are chosen in which a stronger wall is required for the same target values of the factor of safety. A repeated trial procedure is used to find the most dangerous situation for the wall and the associated critical slip surfaces in the upside and downside soil masses, as shown in Fig. 3.

Od Ou

[5]

A

Similarly, the factor of safety equation and the force Pd for the downslope sliding mass can also be derived.

c′l cos α + (W − ul cos α ) tan ϕ ′ cos α (1 + tan α tan ϕ ′ / Fu ) d Fd = y l ∑ W sin α + K d ∑ W d − d Pd rd rd d d ∑

Pd

rd ld

d

B

[6]

d

yd   rd 

D

KdWd Pd

Embankment

KuWu

C

Fd

 1 c′l cos α + (W − ul cos α ) tan ϕ ′ −  ∑  Fd d cos α (1 + tan α tan ϕ ′ / Fd )

− ∑ W sin α − K d ∑ W

U

Pu D E Sheet pile

Fu

Soft soil layer

Hard soil layer

FIG.3 METHOD FOR DEFINITION OF CRITICAL SLIP SURFACES

[7]

When the factor of safety is prescribed for the upslope and downslope sliding mass respectively, Pu and Pd can easily be computed from Eqs. [5] and [7].

Determination of Critical Slip Surfaces In the conventional slope stability analysis, the critical slip surface is defined as a surface on which the factor of safety is smallest. However, this definition is not valid for the present study as the factor of safety is assigned in advance. In this paper, the critical slip surfaces are defined as a pair of slip circles which are most dangerous (or critical) for the wall. Two possible approaches can be considered in order to search for the critical slip surfaces: 1) ∆Pmax approach: In this approach, the critical slip

Example ad Results The proposed method is used to analyze the stability of a homogeneous slope reinforced by a wall installed at a location of DIS=0.0m, where DIS represents the horizontal distance from the toe of the slope, as shown in Fig. 4. The slope in Fig. 4 has a height of 4.0m, an inclination of 1:2 with the soil parameters of c′=4.9kPa, φ′= 10.0° and γ=15.68kN/m3. The minimum factor of safety of the slope without a wall is calculated to be 1.10 by the Bishop method on the critical slip surface shown in Fig. 4. By specifying a target value of Fu=Fd =1.2 for both upside and downside sliding masses with Ku =Kd = 0.0 (no earthquakes), calculation results for the wall-installed slope in Fig. 4 are shown in the same figure. It is seen that for achieving the target values of the factor of safety, the ∆Pmax, needed to be provided by the wall, is found to be 9.0kN/m from the

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Construction Engineering (CE) Volume 2, 2014

∆Pmax, approach and to be 6.8kN/m using the ∆Mmax approach. The associated critical slip surfaces in the upslope and downslope soil masses are also illustrated in Fig. 4.

By assuming Ku =Kd = 0.1 for both upside and downside sliding masses and specifying a target value of Fu=Fd =1.2, the wall-installed slope in Fig. 4 is reanalysed using the proposed method, and results obtained are illustrated in the same figure. In this case, the values of ∆Pmax are computed to be 74.4kN/m from the ∆Pmax, approach and 71.3kN/m from the ∆Mmax approach. The associated critical slip surfaces in the upslope and downslope soil masses are also shown in Fig. 5.

y-axis (m)

From the results shown in Fig. 4 and Fig. 5 it is seen that 1) in order to ensure a same target value of Fu and Fd, a much more stronger wall is needed to sustain the effect of ∆Pmax due to earthquakes; 2) The critical slip surfaces for the case of Ku=Kd=0.1 are deeper than those for Ku =Kd =0.0 (no earthquakes).

c′=4.9kPa, φ′=10° γ=15.68kN/m 3

It is of interest to note that the point of intersection of the downside critical slip surface with the wall almost coincides with that of the upslope critical slip surface (see Figs. 4-5), regardless of the analysis in terms of the ∆Pmax and ∆Mmax approaches. The same results were also found for other cases when the wall is inserted at other positions (DIS ≠0.0m).

Conclusions A limit equilibrium-based method is proposed to evaluate the seismic stability of embankment slopes stabilized with a sheet pile wall. Different slip surface and earthquake coefficient can be assumed in upslope and downslope soil masses separated by the wall. A unique point of the method is that a target factor of safety is first specified by the designer, and the design for wall itself is then conducted to ensure the target value of the factors of safety, though the details of design method for walls is not discussed in the present paper. A practical method is developed to evaluate forces which the wall should sustain and to search for critical slip surfaces which are the most dangerous for the wall. The development of this study makes it possible to perform a satisfactory design for slopes stabilized by a wall which not only meets a specified target factor of safety of the slope but also ensures the structural integrity of the wall itsself. In future research, it is necessary to involve a structural analysis of the wall in the procedure of stability calculations.

REFERENCES

x-axis (m) FIG.4 RESULTS OF ANALYSIS FOR Ku =Kd = 0.0 CASE

Bishop, A. W. The use of the slip circle in the stability analysis of slopes, Geotechnique, Vol. 5, No. 1, pp. 7-17, 1955.

Kanda, Yamagami, and Jiang. A limit equilibrium stability analysis of slopes with a stabilizing wall, Journal of the

y-axis (m)

Japan Landslide Society, Vol. 40, No. 3, pp. 1-7, 2003. (in Japanese) Kanda, Yamagami, Jiang and Nguyen. A limit equilibrium stability analysis of slopes reinforced with two sheet pile c′=4.9kPa, φ′=10° γ=15.68kN/m 3

walls, Journal of the Japan Landslide Society, Vol. 41, No. 5, pp. 37-45, 2005. (in Japanese)

x-axis (m)

Ugai, K. and Cai, F. Evaluation of global safety factor of stabilized slopes based on elasto-plastic FEM, Tsuchi to

FIG. 5 RESULTS OF ANALYSIS FOR Ku =Kd = 0.1 CASE

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Kiso, Vol. 49, No. 4, pp. 16-18, 2001. (in Japanese)