J. Cent. South Univ. (2015) 22: 2322−2327 DOI: 10.1007/s11771-015-2757-6

Three-dimensional analysis of slopes reinforced with piles

GAO Yu-feng(高玉峰), YE Mao(叶茂), ZHANG Fei(张飞)

Key Laboratory for Geomechanics and Embankment Engineering of Ministry of Education (Hohai University), Nanjing 210098, China

© Central South University Press and Springer-Verlag Berlin Heidelberg 2015

Abstract: Based on the upper bound of limit analysis, the plane-strain analysis of the slopes reinforced with a row of piles to the 3D case was extended. A 3D rotational failure mechanism was adopted to yield the upper bound of the factor of safety. Parametric studies were carried out to explore the end effects of the slope failures and the effects of the pile location and diameter on the safety of the reinforced slopes. The results demonstrate that the end effects nearly have no effects on the most suitable location of the installed piles but have significant influence on the safety of the slopes. For a slope constrained to a narrow width, the slope becomes more stable owing to the contribution of the end effects. When the slope is reinforced with a row of piles in small space between piles, the effects of group piles are significant for evaluating the safety of slopes. The presented method is more appropriate for assessing the stability of slopes reinforced with piles and can be also utilized in the design of plies stabilizing the unstable slopes. Key words: three-dimensional rotational failure mechanism; stability of slopes; limit analysis; landslides; piles

1 Introduction

Installing retaining structures such as piles has been proved to be an effective way to remedy unstable slopes in many engineering practices. The piles have been used to stabilize slopes or to improve slope stability since the 1950’s. Numerous analyses of the stability of the slopes reinforced with piles were conducted [1], including limit equilibrium (LE) methods [2−4], limit analysis (LA) methods [1, 5], and finite element (FE) methods [6−8]. However, these methods are limited to the two- dimensional (2D) plane-strain analysis. Some attempts have thus been made to extrapolate the application of numerical methods for slope stability analysis from the 2D case to 3D case [6−8]. Although the effects of the group piles are incorporated into the 3D analysis, the end effects of 3D slopes are still neglected, which has been proved to be significant for the assessment of slope stability. As a result, the 3D effects should be considered in the stability analysis of the slope reinforced with a row of piles.

Based on the upper bound of limit analysis, LI et al [1] adopted the method of prediction of the lateral force acting on a row of piles by ITO and MATSUI [9] and presented a 2D approach to estimate the safety of slopes reinforced with piles.

The aim of this work is to extend such 2D analysis to 3D case. A 3D rotational failure mechanism presented by MICHALOWSKI and DRESCHER [10] is adopted here to obtain the upper-bound solution. Parametric studies are carried out to investigate the end effects and the effects of group piles on the safety of the reinforced slopes. 2 Limit analysis of 3D slopes reinforced with

piles 2.1 3D rotational failure mechanism

In the framework of limit analysis, a kinematically admissible velocity field (failure mechanism) needs to be established to obtain an upper bound. MICHALOWSKI and DRESCHER [10] proposed a class of 3D admissible rotational failure mechanisms for slopes in both frictional/cohesive and purely cohesive soils, as shown in Fig. 1(a). The postulated mechanism is adopted here to analyze the stability of 3D slope reinforced with a row of piles. As proposed, the 3D mechanism for the slope limited in a certain width of B is modified with a cylinder with width of b to allow the transition to the 2D failure of CHEN [11] as b approaches infinity, as shown in Fig. 1(b). The details of the 3D admissible rotational failure mechanism are presented in Refs. [10, 12].

Foundation item: Projects(51278382, 51479050) supported by the National Natural Science Foundation of China; Project(2015CB057901) supported by the

National Key Basic Research Program of China; Project(201501035-03) supported by the Public Service Sector R&D Project of Ministry of Water Resource of China; Project(2014B06814) supported by the Fundamental Research Funds for the Central Universities, China; Project(B13024) supported by the “111” Project; Project (YK913004) supported by the Open Foundation of Key Laboratory of Failure Mechanism and Safety Control Techniques of Earthrock Dam of the Ministry of Water Resources, China

Received date: 2014−04−30; Accepted date: 2014−10−15 Corresponding author: GAO Yu-feng, Professor, PhD; Tel: +86−25−83787287; E-mail: yfgao66@163.com

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Fig. 1 Schematic diagrams of 3D rotational failure mechanism

(a) and 3D failure mechanism with plane insert (b)

Based on the upper bound of limit analysis, an

energy balance equation of work rate is written for the kinematically admissible failure mechanism and an upper-bound solution can be derived from the balance equation. To account for the presence of the piles, an additional rate of energy dissipation done by the resistance of the piles Dp must be counted in the balance equation of work rate as

ends cylinder ends cylinder ends cylinderp pW W D D D D

(1) where superscript “ends” denotes the work rates for a section of the curvilinear cone at the two ends of the failure mechanism and “cylinder” relates to the plane insert in the center of the mechanism. The expressions of

endsW and Dends for the two end parts, and cylinderW and Dcylinder for the cylindrical parts can be found in Refs. [11−12]. 2.2 Dissipation due to resistance of piles

Based on the theory of plastic deformation, ITO and MATSUI [9] presented theoretical equations for estimating the lateral forces acting on a row of piles as follows.

For frictional/cohesive soils (φ>0, c>0), 1/ 2tan 1

1 1 2(

2 2

)

11

( ) exptan

N ND D D

p z cDD N D

1/ 2πtan tan 2 tan 1

8 4N N

1/ 2 1/ 2

1/ 2

2tan 2

tan 1

N N

N N

1/ 2 1/ 21/ 2

1 21/ 2

2tan 22

tan 1

N Nc D D N

N N

1/ 2tan 11

12

N NDz

DN D

1 22

2

πexp tan tan

8 4

D DN D

D

(2)

For purely cohesive soils (undrained shear strength

φ>0, c>0),

1 1 21 1 2

2 2

π( ) 3lg tan 2( )

8

D D Dp z c D D D

D D

1 2( )z D D (3) where c is the cohesion of soil; D1 is the center-to-center spacing between piles; D2 is the opening between piles; (D1−D2) is the pile diameter; γ is the unit weight of soil; z is the depth of soil layer from the ground surface; Nφ=tan(π/4+φ/2). For the details of the theory and equations, see Ref. [9]. The rate of energy dissipation due to the resistance of the piles, Dp, equals the pile lateral force p(z) in the active area (Fig. 1(b)) multiplied by the instantaneous velocity at the point of p(z). And the instantaneous velocity at the point of p(z) equals the product of the angular velocity ω and the arm of lateral force p(z) to point O. The rate of dissipation for the two end parts ends

pD and the central cylindrical part cylinderpD can be obtained from:

0ends

p 1

( )2 ( ) ( )d

h

p zD x z l z z

D (4)

0cylinder

p 1

( )( )d

h

p zD bl z z

D (5)

where x(z) is the width of failing soil at depth z in the position of piles, as shown in Fig. 1(a); l(z) is the vertical distance between the position of lateral force p(z) and the rotational center point O. Functions x(z) and l(z) can be expressed as

22 2 2e p mp e( ) ( cos )x z R l r r (6)

h h F( ) sin tanl z r X z (7) And the integration limit along z is

p h Fp hsin sin tanH r r X (8)

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where XF is the horizontal distance between the piles position and slope toe, as shown in Fig. 2; Re=(re−re)/2 and rme=(re+re)/2; angles e and p are derived from the trigonometric relation, as

p pe

2 2p p

cosarccos

c( )os

r

r l

(9)

F p p h hcos cos (cot cot )X r r H (10)

Fig. 2 Maximum cross section of 3D failure surface for slopes

reinforced

2.3 Factor of safety

The factor of safety (FS, Fs) for slopes is defined as the ratio of the soil strength parameters to those necessary to maintain limit equilibrium as `

sm m

tan

tan

cF

c

(11)

where cm and φm are the soil strength parameters necessary only to maintain the structure in limit equilibrium. It should be emphasized that, the strength parameters c and φ in the equation of the lateral force of piles p(z) also need to be reduced by FS.

To obtain the minimum FS, an iteration procedure is carried out with respect to the unknown parameters (angles 0, h and ', and ratios r0/r0 and b/B) describing the slip surface as

s s 0 0 0min ( , , , ', '/ , / , , , , , )hF f F r r b B c H B (12) where H is slope height and is slope angle. The maximum of the length of slope failures should be limited to the feasible length of collapse B, as shown in Fig. 1(b). 3 Results and discussion 3.1 Most suitable location of piles in 3D slope

Figures 3 and 4 show the effects of the pile location on FS for 3D slopes in purely cohesive soils (=0) and frictional/cohesive soils (0), respectively, where D1 is the center-to-center distance between the piles, and dp is the diameter of piles. As expected, the 2D solutions are

conservative to analyze the slope stability when being compared with 3D solutions. The difference of the factors of safety between 2D and 3D solutions can even exceed 20% when a slope is constrained to a narrow width of B/H=2.0. The most effective location of the piles (i.e., the maximum factor of safety) can be observed between the middle and top of the slope. It should be noted that, the most effective location of the piles nearly keeps in the same location as the ratio of B/H decreases. This implies that the 3D analysis has insignificant effects on the most effective location of the piles but appropriately assesses the safety of 3D slopes. Furthermore, Fig. 5 illustrates the critical slip surfaces for 3D slopes reinforced with a row of piles in different locations and the 3D slope without piles. It can be seen that the pile location has significant effect on the geometry and location of the critical slip surface. According to the width (y-axis in Fig. 5) of the curved surface at the end of the 3D failure surface, the 3D end effects in the slope without piles are the most significant. However, for the slope reinforced with a row of piles, the installed piles can provide more resistance than the 3D end effects. Therefore, the width of the curved surface for the reinforced slope is smaller than that for the slope without piles.

Fig. 3 Effects of pile location on factor of safety for 3D slopes

when =0° (D1/dp=2.5), c=23.94 kPa and γ=19.63 kN/m3:

(a) β=2° and H=13.7 m; (b) β=45° and H=10.0 m

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Fig. 4 Effects of pile location on factor of safety for 3D slopes when =10°, c=23.94 kPa and γ=19.63 kN/m3 (D1/dp=2.5): (a) β=30°

and H=13.7 m; (b) β=45° and H=10.0 m

3.2 Effect of D1/dp on 3D slope reinforced by piles Figure 6 shows the factors of safety for 3D slopes

reinforced with a row of piles with different ratios of D1/dp when β=30°, H=13.7 m, c=23.94 kPa and γ=19.63kN/m3. As expected, the factor of safety increases with the ratios of B/H and D1/dp decrease. It can be also found that the effects of the ratio D1/dp on the factor of safety are most pronounced for D1/dp2.5. As D1/dp increases, FS gradually approaches to the FS for

slopes without piles. It is noticed that D1/dp should not be very large, because the assumption of a plastic state around the piles, as used in the derivation of Eq. (11), is not fulfilled for excessive spacing between the piles. Figure 7 demonstrates the effects of the ratio of D1/dp on the critical slip surfaces. The critical slip surface becomes shallower as D1/dp decreases. It should be noted that, the width of the central cylinder decreases as D1/dp increases. The 3D end effect becomes more significant,

Fig. 5 Effects of pile location on critical

slip surfaces for 3D slopes: (a) Without

piles; (b) XF/Lx=0.3; (c) XF/Lx=0.7;

(d) XF/Lx=0.9; (e) Maximum cross

section of 3D failure surface

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Fig. 6 Effects of ratio of D1/dp on factor of safety for 3D slopes: (a) φ=0°, XF=8.0 m; (a) φ=0°, XF=8.0 m; (b) φ=0°, XF=13.7 m; (c)

φ=10°, XF=8.0 m; (d) φ=10°, XF=13.7 m

Fig. 7 Effects of ratio of D1/dp on

critical slip surfaces for 3D slopes:

(a) D1/dp=2; (b) D1/dp=2.5; (c) D1/dp=

3; (d) D1/dp=4; (e) Maximum cross

section of 3D failure surface

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but it does not make the slopes more stable. This implies that when D1/dp is small, the effects of group piles provide more contributions to the stability of slopes. Whereas, for large value of D1/dp, the 3D end effects predominate due to the insignificant effects of the group piles on the stability of slopes. Therefore, the significant end effects make the 3D critical slip surface deeper and the end part of the curved surface larger in width. 4 Conclusions

Based on the upper bound of limit analysis, a

method is presented to assess the stability of 3D slopes reinforced with a row of piles. The factor of safety and its corresponding critical slip surface are obtained from the optimization procedure. Some specific slope examples are presented to investigate the 3D effects on the safety of the reinforced slopes. The ratio of B/H nearly has no effect on the most effective location of the installed piles but has significant effect on the safety of the slopes, especially for slopes constrained to a narrow width. Furthermore, the factor of safety increases as the ratio of D1/dp decreases owing to the significant effects of the group piles. The critical slip surface becomes shallower as the ratio of D1/dp decreases. The presented 3D analysis method can be more appropriately used in the design of slopes reinforced by a row of piles compared with the traditional 2D analysis.

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(Edited by FANG Jing-hua)