Stability charts for curvilinear slopes in unsaturated soils

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Stability charts for curvilinear slopes in unsaturated soils Thanh Vo, Adrian R. Russell ⇑ Centre for Infrastructure Engineering and Safety, School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW 2052, Australia Received 20 June 2016; received in revised form 20 January 2017; accepted 23 March 2017

Abstract This paper presents charts derived from stability analyses of curvilinear slopes in non-homogeneous unsaturated soils. The charts enable quick and inexpensive stability analyses to be conducted in practice. The soils are assumed to obey the Mohr-Coulomb failure criterion. Suction effects are captured using the effective stress concept. Cohesion and the contribution of suction to the effective stress are linear functions of depth. It is shown that a stable curvilinear slope profile is uniquely governed by dimensionless parameters. The charts are presented using these dimensionless parameters and may be used in the preliminary design of stable curvilinear slopes. In general, suction has a similar influence to cohesion, and as the contribution of suction to effective stress increases, steeper curvilinear slopes become stable. An example of application is included showing the design of a curvilinear slope. Ó 2017 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society. This is an open access article under the CC BYNC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Stability analyses of slopes are important in geotechnical engineering. For newly constructed slopes they enable safe surface profiles to be determined for certain soil properties. For existing slopes they enable margins of safety to be determined so needs for remediation and stabilisation can be identified and risks assessed. Furthermore, for failed slopes, they enable a strength property of the soil at the time of failure to be back-calculated. Most detailed slope stability analyses involve computational tools, such as finite element methods (e.g. Griffiths and Lane, 1999) or software assisted limit equilibrium methods (e.g. Cheng et al., 2007). However, for simple slope geometries and soil profiles, the results of stability analyses may be presented in stability charts (for example the charts of Taylor (1948), Davis and Booker (1973), Michalowski (2002), and Li et al. (2010) among many

Peer review under responsibility of The Japanese Geotechnical Society. ⇑ Corresponding author. E-mail address: [email protected] (A.R. Russell).

others). The charts are of interest in that they enable quick and inexpensive stability analyses to be conducted in practice. They are especially useful for preliminary analyses, checking detailed design analyses and making comparisons between design alternatives. In stability charts soil and slope parameters feature inside dimensionless parameters. Most stability charts are restricted to the profiles of planar slopes. Some charts, for example those of Sokolovski (1954) and Jeldes et al. (2014b), may be applied to convex or concave profiles (collectively referred to as curvilinear in this paper), but only in special cases when the curvilinear surface has a vertical tangent at the top of the slope. Uriel Romero (1967,1969) considered one type of curvilinear slope with a non-zero vertical tangent at the top of the slope, although the results cannot be generalised to other non-vertical tangent slope profile geometries. Natural slopes subject to weathering are more likely to have a concave profile than a planar profile (Utili and Crosta, 2011). Some log-spiral concave surface has proved to be more stable than a planar profile (Utili and Nova, 2007). Concave slopes may also have superior erosion resistance compared to planar slopes (Young and Mutchler,

http://dx.doi.org/10.1016/j.sandf.2017.06.005 0038-0806/Ó 2017 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Vo, T., Russell, A.R., Stability charts for curvilinear slopes in unsaturated soils, Soils Found. (2017), http://dx.doi.org/ 10.1016/j.sandf.2017.06.005

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List of symbols c0 soil cohesion c00 c0 at the top of the slope D length scale F a dimensionless parameter H slope height kc rate of change of c0 with depth k vs rate of change of vs with depth L extent of the sliding soil mass along the horizontal surface OA q surcharge on the soil surface at the top of the slope qmin minimum surcharge on the soil surface at the top of the slope S stress scale s suction T tangent of angle of the slope profile at O ua pore air pressure

1969a, 1969b; Meyer and Kramer, 1969; Rieke-Zapp and Nearing, 2005; Schor and Gray, 2007) making them preferable within many embankments and cuttings (Jeldes et al., 2014b, 2014a). However, there is limited ability to address their stability without resorting to computational tools. Another restriction of most existing stability charts is that they require soil strength to be defined by the MohrCoulomb failure criterion, which requires the soil to have a constant cohesion and/or friction angle throughout the slope. Very few charts are relevant to cases where the soil strength varies spatially within the soil (for example Davis and Booker, 1973; Griffiths and Yu, 2015), yet this is known to be the case in many practical situations. This paper addresses these restrictions. Stability analyses of curvilinear slopes made of unsaturated and nonhomogenous soils are conducted using slip line theory, building on the work of Sokolovski (1954). The soil is assumed to be a continuous body in static equilibrium at the onset of failure. The effective stress concept for unsaturated soils and the Mohr-Coulomb failure criterion are adopted. The slip line governing equations are hyperbolic and define two families of stress characteristics (Sokolovski, 1954; Atkinson and Potts, 1975). Because the soil is assumed to have a non-zero frictional strength, the slope profiles are logarithmic spirals (Sokolovski, 1954; Davis, 1968). Cohesion and the contribution of suction to the effectives stress and strength are non-uniform and vary linearly with depth. The results of the analyses are presented in stability charts. Defining parameters appear in dimensionless form. In conducting the analyses the work of Sokolovski (1954) is extended so curvilinear slope profiles other than those with a vertical tangent at the top of the slope are considered. An example of application is given showing how a curvilinear slope can be designed.

uw y; x c g; n h hs r0 r0m r ryy ; rxx ryx u0 v vs ðvsÞ0

pore water pressure vertical and horizontal coordinates soil total unit weight families of characteristic curves angle between the vertical axis and the major principal stress direction angle h along the slope profile effective stress effective mean stress total stress normal stresses in the y and x directions respectively shear stress soil friction angle effective stress parameter contribution of suction to the effective stress vs at the top of the slope

2. Problem set-up A non-homogeneous unsaturated soil slope is shown in Fig. 1 along with the coordinates system adopted, with y and x representing vertical and horizontal directions, respectively. The failed part of the slope is denoted as the ‘‘sliding soil mass” in Fig. 1 which intercepts the curvilinear surface at y ¼ H and the horizontal surface OA at x ¼ L. A stress q is applied at y ¼ 0 over the distance L extending from x ¼ 0 to L. It is assumed that soil shear strength (s) is governed by the Mohr-Coulomb failure criterion: s ¼ c0 þ r0 tan u0

ð2:1Þ

in which c0 is the soil cohesion and u0 is the soil friction angle. r0 is the effective stress and for an unsaturated soil is defined as (Bishop, 1959): r0 ¼ r  ua þ vðua  uw Þ

ð2:2Þ

where r  total stress, ua  pore air pressure, uw  pore water pressure and v  effective stress parameter. Bishop (1960) suggested that v and thus r0 depend on many factors including degree of saturation, soil structure, the dryingwetting history and the stress history. The dependencies L q O

A

sliding soil mass

x

H soil B

y

Fig. 1. Geometry and parameters of the curvilinear slope problem.

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of v and r0 on these factors have been studied by many authors (Bolzon et al., 1996; Khalili and Khabbaz, 1998; Gallipoli et al., 2003; Lu and Likos, 2006; Khalili et al., 2008; D’Onza et al., 2011; Lu et al., 2014). In these studies slightly different terminologies have been adopted for v and r0 . When ua is equal to atmospheric pressure and is taken as the pressure datum, Eq. (2.2) can be rewritten as: r0 ¼ r þ vs

ð2:3Þ

where s ¼ ua  uw  soil suction. vs is the contribution of suction to the effective stress. The analyses and results presented here apply to when the changes to ua are negligible. There may be examples when this is not the case and fully coupled numerical analyses may be warranted, e.g. Oka et al. (2010) and Xiong et al. (2014). For simplicity it is assumed that u0 is constant and that 0 c and vs vary linearly with depth according to the functions: c0 ¼ c00 þ k c y

ð2:4Þ

vs ¼ ðvsÞ0 þ k vs y

ð2:5Þ

where c00  cohesion at y ¼ 0 (that is at the top of the slope), ðvsÞ0  contribution of suction to the effective stress at y ¼ 0 and k c ; k vs are constants. Vo and Russell (2016) showed that c0 and vs cot u0 have similar but independent effects on an unsaturated soil’s strength. Different combinations of c0 and u0 may define peak strength, critical state strength and residual strength. There is experimental evidence that c0 and u0 are independent of s at the critical state, meaning the same values apply to saturated, dry and unsaturated conditions (Escario, 1980; Gan et al., 1988; Likos, 2010). When soils exhibit significant suction hardening the values of c0 and u0 which define the peak strength may have a secondary dependence on s, and may be slightly larger for unsaturated conditions than saturated or dry conditions. For an effective stress analysis in a saturated soil v ¼ 1 and uw appears in Eqs. (2.3) and (2.5) in place of vs. For a total stress analysis in a saturated soil the contribution of uw is ignored and Eq. (2.4) can be used to express a linear variation of undrained shear strength with depth. A linear variation of undrained shear strength with depth is often assumed for normally consolidated clays and layered soils (Gibson and Morgenstern, 1962; Bishop, 1966; Lump and Holt, 1968; Hunter and Schuster, 1968). 3. Governing equations and boundary conditions

3

where ryy ; rxx  normal stresses in the y and x directions, respectively, r0m  effective mean stress, ryx  shear stress and h  angle between the vertical axis and the major principal stress direction. The stress components in Eqs. (3.1)–(3.3) are scaled by a constant stress quantity S and differentiated with respect to lengths which are scaled by D. The resulting expressions are substituted into the static equilibrium equation to obtain:
dx ¼ tanðh þ lÞdy hgi  ð3:4Þ dr0m þ 2 tan u0 r0m dh ¼ F ðdy  tan u0 dxÞ
dx ¼ tanðh  lÞdy ð3:5Þ hni  dr0m  2 tan u0 r0m dh ¼ F ðdy þ tan u0 dxÞ in which
  D @ðc0 cot u0 þ vsÞ cþ F ¼ S @y

ð3:6Þ 0

and c  total unit weight of soil, l ¼ p4  u2 and an overbar 0

indicates a scaled quantity (r0m ¼ rSm , y ¼ Dy and x ¼ Dx ). Eqs. (3.4) and (3.5) express two families of characteristic curves (g; n) which are inclined at angles l to the major principal stress direction. When the stress scale is set to S ¼ q þ ðvsÞ0 þ c00 cot u0 and the length scale is set to D ¼ L it follows that Lðcþk cot u0 þk Þ F ¼ qþðvsÞc þc0 cot vsu0 . For these choices of S and D the numer0

0

ator of F captures the combined influence of geometry (through D ¼ L) and an ‘‘equivalent total unit weight” c þ k c cot u0 þ k vs . The denominator of F captures the influence of an ‘‘equivalent applied pressure” q þ ðvsÞ0 þ c00 cot u0 . For a dry soil, F reduces to the parameter defined by Martin (2004) used when applying this theory to the analysis of a different boundary value þc tan u0 Þ problem, i.e. F ¼ Lðkc0 cþq . The problem addressed by tan u0 0

Martin (2004) was the bearing capacity of shallow footings, where L is the footing width and acts as the length scale. Whatever the choice of S and D a solution ðy ; x; r0m ; hÞ of Eqs. (3.4) and (3.5) depends on u0 and the non-dimensional quantity F . A solution also depends on the type of boundary value problem and the boundary conditions. A standard finite difference method and an iterative procedure are used to solve Eqs. (3.4) and (3.5). The method is adapted from Sokolovski (1954). The solution procedures used here were validated by setting vs ¼ 0 and c0 ¼ constant and reproducing one set of Sokolovski’s (1954) results.

3.1. Governing equations 3.2. Boundary conditions The Mohr-Coulomb failure criterion can be written as: ryy ¼ ½ð1 þ sin u0 cos 2hÞr0m  c0 cot u0   vs

ð3:1Þ

rxx ¼ ½ð1  sin u0 cos 2hÞr0m  c0 cot u0   vs

ð3:2Þ

With reference to the coordinates system shown in Fig. 1, two boundary conditions can be derived at the soil surface OA:

ð3:3Þ

h¼0

0

ryx ¼ sin u

sin 2hr0m

ð3:7Þ

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r0m ¼

T. Vo, A.R. Russell / Soils and Foundations xxx (2017) xxx–xxx

1 1 þ sin u0

ð3:8Þ

unsaturated soil, i.e. when u0 ; c0 and vs are constant with depth (k vs ¼ 0 and k c ¼ 0). The lower bound of y 0 is: 2½ðvsÞ0 þ c00 cot u0  sin u0 cð1  sin u0 Þ

Two boundary conditions can be derived at the slope profile OB:

y0 ¼

dx ¼ tan hs dy

A work rate dissipation function for unsaturated soils (Vo et al., 2016) was adopted to derive the upper bound of y 0 and it is:

r0m ¼

ð1 

ð3:9Þ vs þ c0 cot u0 þ ðvsÞ0 þ c00 cot u0 Þ

sin u0 Þðq

ð3:10Þ

where hs  angle between the major principal stress direction and the vertical axis at a point on the slope profile OB. When c þ k c cot u0 þ k vs – 0, hs varies along the slope profile resulting in a curvilinear shape (Eq. (3.9)). In particular, when hs reduces with depth along the slope profile, it has a concave shape and when hs increases with depth along the slope profile, it has a convex shape. In this paper the shape of curvilinear slopes are obtained for when c þ k c cot u0 þ k vs > 0, as c þ k c cot u0 þ k vs can be thought of as an equivalent total unit weight. The governing equations and boundary conditions in Sections 3.2 and 3.3 are valid for any constant stress scale and length scale. 3.3. A particular case for a curvilinear slope

y0 ¼

4½ðvsÞ0 þ c00 cot u0  sin u0 cð1  sin u0 Þ

ð3:12Þ

ð3:13Þ

The bounds on y 0 can be rewritten as: qmin 2q < y 0 < min c c

ð3:14Þ

Eqs. (3.11) to (3.13) show that qmin is related to y 0 . qmin may be thought of as the load applied to the top of a slope by a layer of a soil with a vertical face. This soil layer does not feature in the stability analysis other than through qmin . The Chen and Scawthorn (1970) approach was also used to compute the lower bound of y 0 for a nonhomogeneous unsaturated soil slope with u0 ; c0 defined by Eqs. (2.4) and (2.5). The lower bound of y 0 is: pffiffiffiffiffiffi ðvsÞ0 ð1  K a Þ þ 2c00 K a pffiffiffiffiffiffi y0 ¼ ð3:15Þ cK a þ k vs ðK a  1Þ  2k c K a 0

In the boundary value problem shown in Fig. 1, point O is a singular point because of the jump in h between OA (Eq. (3.7)) and OB (Eq. (3.9)). Point O is treated as a n characteristic curve in the limit by the slip line theory. At O, h is allowed to numerically transit  from 0 at the soil sur ððvsÞ0 þc00 cot u0 Þð1þsin u0 Þ  cot u0 face to 2 ln ðqþðvsÞ þc0 cot u0 Þð1sin u0 Þ at the slope profile. The 0 0

tangent of angle of the slope profile hs at O is denoted T i.e. 0

ððvsÞ þc0 cot u0 Þð1þsin u0 Þ

T  cot2u ln j ðqþðvsÞ0 þc0 0 cot u0 Þð1sin u0 Þ j. 0

u is the Rankine active earth pressure where K a ¼ 1sin 1þsin u0 coefficient. qmin and y 0 can also be related to the maximum depth of a vertical tension crack because crack depth is governed by the stability of the vertical crack boundaries (Michalowski, 2013; Utili, 2013). When designing a curvilinear slope it is recommended that the lower bound of y 0 be applied since it results in the smallest overall slope height and thus a safer slope.

0

Because the condition of a non-overlapping stress field at O requires T 6 0, the surcharge q must be larger than or equal to a minimum value qmin : 2½ðvsÞ0 þ c00 cot u0  sin u0 qP  qmin 1  sin u0

ð3:11Þ

Curvilinear slope profiles were obtained by Sokolovski (1954) for the case q ¼ qmin . When q ¼ qmin , the curvilinear slope is tangential to the vertical axis at O, i.e. T ¼ 0 at O. There are reasons to consider the case q ¼ qmin separately. Using limit analysis, Chen and Scawthorn (1970) developed a method for computing the upper and lower bounds on the maximum height of a vertical slope in a homogeneous soil. Here the maximum height is denoted as y 0 . A vertical slope need not be at limiting equilibrium everywhere for heights inside these bounds as stress and displacement increment discontinuities are permitted in limit analysis. The Chen and Scawthorn (1970) approach was used here to compute the bounds of y 0 in a homogenous

4. Results 4.1. The case when q = qmin Sokolovski (1954) presented analysis results in a chart for the case when q ¼ qmin , c0 is constant with depth, vs ¼ 0 and u0 ¼ 20 ; 30 ; 40 . Sokolovski (1954) used 0 S ¼ c0 and D ¼ cc as the stress and length scales, respectively, so that F ¼ 1 in Eqs. (3.4) and (3.5) (from Eq. (3.6)). The chart produced by Sokolovski (1954) is reproduced here in Fig. 2a. Sokolovski’s (1954) data is plotted using dash lines. New data generated by the authors is plotted using continuous lines. A different presentation of Sokolovski’s (1954) results is shown in Fig. 2b, but with vs now existing in the soil and being constant with depth. The stress scale 0 0 S ¼ q þ vs þ c0 cot u0 and length scale D ¼ qþvsþcc cot u have been used to form dimensionless quantities y ¼ ðqþvsþcy0 cot u0 Þ=c and x ¼ ðqþvsþcx0 cot u0 Þ=c. It is demonstrated

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5

(a) -100

-80

-60

-40

-20

0 0

φ'=5° 10

10° 15° 20° 25° 30° 35°

20

30

40° 45° 50° 55° 40

(b) -40

-30

-20

-10

0

φ'=5°

0

5

10° 10 15° 15 20° 25°

35°

30°

40°

45° 50° 55° 20

Fig. 2. Sokolovski’s (1954) and additional results. (a) F ¼ 1, T ¼ 0 at O, q ¼ qmin , c0 ¼ constant with y ¼ c0y=c and x ¼ c0x=c (stress scale is c0 , length scale is c0 =c); (b) F ¼ 1, T ¼ 0 at O, q ¼ qmin , c0 ¼ constant and vs ¼ constant with y ¼ ðqþvsþc0ycot u0 Þ=c and x ¼ ðqþvsþc0xcot u0 Þ=c (stress scale is q þ vs þ c0 cot u0 , length scale is ðq þ vs þ c0 cot u0 Þ=c).

-6

-4.5

-3

-1.5

0

-6

-4.5

-3

-1.5

0

-6

-4.5

-3

-1.5

0 0

0.5

1

10° 20° 30° φ'=40°

1.5

10° 20° 30°

10° 20°

1.5 3

30° 40°

4.5

40°

6 -24

-18

2

-12

-6

0

-24

-18

-12

10° 20° 30° 40°

-6

0

-24

-18

10° 20° 30°

-12

10° 20° 30°

40° 40°

-6

0 0 6 12 18 24

Fig. 3. Results for when F ¼ 0:5; 1; 1:5; 2; 3; 4; T ¼ 0 at O, q ¼ qmin , c0 ¼ constant and vs ¼ constant with y ¼ Ly and x ¼ Lx (stress scale is q þ vs þ c0 cot u0 , length scale is L). Please cite this article in press as: Vo, T., Russell, A.R., Stability charts for curvilinear slopes in unsaturated soils, Soils Found. (2017), http://dx.doi.org/ 10.1016/j.sandf.2017.06.005

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later that when dimensionalised, Fig. 2a and b give the same curvilinear slope profile for a given set of material parameters. Fig. 2a and b both correspond to F ¼ 1 as S=D ¼ c and c0 and vs are constant with depth. It is only possible to define S and D in this way because q ¼ qmin , T ¼ 0 and the solutions for a given slope profile are independent of the slope height and size of the sliding mass. For a given slope profile, there are an infinite number of sliding soil masses that may be in a state of limiting equilibrium, each having a different size defined by L and H . This is not true of the more general case when q > qmin , T < 0 and/or c0 and vs vary with depth, and different choices of S and D are required, as will be demonstrated in Section 4.2. -4

-3

The general case, or more specifically the case when q > qmin , has not been considered by Sokolovski (1954) or Jeldes et al. (2014b). It is considered here for when c0 and vs vary linearly with depth and T < 0. qmin may be thought of as the load applied to the top of a slope by a layer of a soil with a vertical face of height y 0 ¼ qmin =c. The quantity q  qmin represents a uniform surcharge on top of that layer. The curvilinear slope profile commences at O located at the base of the vertical face. For q > qmin it is necessary to represent the analysis results in several charts. An example of what a series of charts will look like for the more general case is shown in Fig. 3, albeit restricted to when q ¼ qmin , T ¼ 0 and c0 -2

-1

0

T=

(a)

4.2. The general case

0

0.5

1

1.5

2 -6

-4

-2

0

0

1

T=

-8

2

3

4 - 16

-12

-8

-4

0

0

T=

2

4

6

8

10

Fig. 4. The general case for unsaturated soil where T < 0 at O (stress scale is q þ ðvsÞ0 þ c00 cot u0 , length scale is L). (a) F ¼ 0:01; (b) F ¼ 0:05; (c) F ¼ 0:1; (d) F ¼ 0:5; (e) F ¼ 1; (f) F ¼ 5. Please cite this article in press as: Vo, T., Russell, A.R., Stability charts for curvilinear slopes in unsaturated soils, Soils Found. (2017), http://dx.doi.org/ 10.1016/j.sandf.2017.06.005

T. Vo, A.R. Russell / Soils and Foundations xxx (2017) xxx–xxx

(b)

-4

-3

-2

-1

7

0

0

T=

0.5

T=

1

1.5

2 -8

-6

-4

-2

0

0

2

T=

T=

1

3

4 -16

- 12

-8

-4

0

0

T=

2

4

T=

6

8

10

Fig 4. (continued)

and vs are constant with depth so comparisons with Fig. 2a and b can be made. Each chart corresponds to a certain F value. Here the charts are for F ¼ 0:5; 1; 1:5; 2; 3; 4 and u0 ¼ 10 ; 20 ; 30 ; 40 . The analysis was conducted using S ¼ q þ vs þ c0 cot u0 and D ¼ L as the stress and length scales, respectively, so y ¼ Ly and x ¼ Lx in the charts. As described above, because of the restriction q ¼ qmin , T ¼ 0 and c0 and vs being constant with depth, a slope profile corresponds to an infinite number of sliding soil masses that may be in a state of limiting equilibrium, whatever their size. In other words, although not obvious, the slope profiles in Fig. 3 are insensitive to the value of L used, and depend only on the material parameters u0 , c0 , vs

and c. Different L values can be chosen to obtain different F values although Fig. 3 will give the same curvilinear slope profiles as Fig. 2a and b, for a given set of material parameters, when dimensionalised. The most general results are now presented for when q > qmin , T < 0 and/or c0 and vs vary with depth. A slope profile is unique for any combination of ðF ; u0 ; T Þ. Fig. 4a–f plot the slope geometries in the y  x plane for F ¼ 0:01; 0:05; 0:1; 0:5; 1; 5, for u0 ¼ 20 ; 30 ; 40 and for T ¼ 0; 0:1 p2 ; 0:2 p2 ;    ; 0:9 p2 ;  p2. These charts apply to when the stress and length scales used were S ¼ q þ ðvsÞ0 þ c00 cot u0 and D ¼ L, respectively, so again y ¼ Ly and x ¼ Lx in the charts.

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(c)

-4

-3

-2

-1

0

0

0.5

T=

1

T=

1.5

2 -8

-6

-4

-2

0

0

1

T=

2

T=

3

4 -16

-12

-8

-4

0

0

T=

2

4

6

T=

8

10

Fig 4. (continued)

In applying the charts in Fig. 4 it is important to note that L (which is used to define D) now defines a specific sliding soil mass of a certain size. The uniform surcharge q  qmin applies over this L (Fig. 1). It can be observed that as T transitions from zero to a negative value, the slope profile transitions from a concave shape to a convex shape. When T ¼ 0, k c ¼ 0 and k vs ¼ 0 it follows that q ¼ qmin and the slope geometries are given in Fig. 3. It can also be observed that as F ! 1, for a given u0 , the results become independent of T and the surface profiles become unique and planar. These results are valid for any value of c00 ; k c ; ðvsÞ0 ; k vs ; c; L provided that q P qmin and c þ k c cot u0 þ k vs > 0.

5. Example of application A hillside slope, 53 m in height, is to be landscaped to have a curvilinear profile to limit erosion and sediment runoff. The following three cases are considered in this example. Case 1: u0 ¼ 30 , c ¼ 15 kN/m3 and c0 ¼ 30 kPa at all depths. The slope is dry. No surcharge acts on the ground surface at the top of the hill meaning q ¼ qmin applies. This is an extended version of the example considered by Jeldes et al. (2014b). Case 2: u0 ; c; c0 of case 1. The slope is dry. A uniform surcharge of 30 kPa is applied across the ground surface at the top of the hill extending a distance of 6.2 m from the slope face.

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(d)

-4

-3

-2

9

-1

0

0

T=

1

T=

2

3

4 -6

-4

-2

0

0

3

T=

-8

T=

6

9

12 -16

-12

-8

-4

0

T=

-20

0

6

12

T=

18

24

30

Fig 4. (continued)

Case 3: u0 ; c; c0 of case 1. The slope is unsaturated. The contribution of suction to the effective stress can be expressed as vs ¼ 20  0:1y (kPa). A uniform surcharge is applied as in case 2. For simplicity, the curvilinear slopes at limiting condition will be obtained for cases 1–3. Design for nonlimiting conditions will be then discussed. Case 1 0 cos u0 The soil properties correspond to q ¼ qmin ¼ 2c ¼ 1sin u0 104 kPa and y 0 ¼ qmin ¼ 6:93 m. c When a pressure scale S ¼ c0 ¼ 30 kPa and a length 0 scale D ¼ cc ¼ 2 m are chosen, the curvilinear profile for u0 ¼ 30 in Fig. 2a can be adopted. This profile is

dimensionalised and shown in Fig. 5a. As expected, this profile is identical to the one obtained by Jeldes et al. (2014b). This slope profile corresponds to many possible sliding soil masses which may be a state of the limiting equilibrium. The one which passes through the bottom of the hillside extends along the horizontal surface OA by a distance L ¼ 15:2 m and is shown in Fig. 5b. When a pressure scale S ¼ qmin þ c0 cot u0 ¼ 156 kPa 0 0 and a length scale D ¼ qmin þcc cot u ¼ 10:4 m are chosen, the curvilinear profile for u0 ¼ 30 in Fig. 2b can be adopted. When dimensionalised, this profile is identical to the one shown in Fig. 5a.

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T. Vo, A.R. Russell / Soils and Foundations xxx (2017) xxx–xxx

(e)

-6

-4.5

-3

-1.5

0

0

T=

1.5

T=

3

4.5

6 -9

-6

-3

0

0

3

T=

-12

6

T=

9

12 -24

-18

-12

-6

0

T=

-30

0

6

12

18

T=

24

30

Fig 4. (continued)

The analyses used to present the family of charts in Fig. 3 can also be used to obtain the same slope profile. The sliding mass which passes through the bottom of the hillside has an L ¼ 15:2 m. Using L ¼ 15:2 m as the length scale and S ¼ qmin þ c0 cot u0 ¼ 156 kPa as the pressure scale corresponds to F ¼ q þcLc0 cot u0 ¼ 1:47. A slope profile min identical to the one shown in Fig. 5a is recovered. Case 2 The soil properties and surcharge condition correspond 0 cos u0 to q ¼ qmin þ 30 ¼ 2c þ 30 ¼ 134 kPa and y 0 ¼ qmin ¼ 1sin u0 c 6:93 m. The pressure scale S ¼ q þ c0 cot u0 ¼ 186 kPa and length scale D ¼ 6:2 m correspond to F ¼ qþc0Dccot u0 ¼ 0:5.

 0  0  ðc cot u0 Þð1þsin u0 Þ  p Also, T ¼ cot2u ln ðqþc 0 cot u0 Þð1sin u0 Þ ¼ 0:153 0:1 2 . The slope profile corresponding to F ¼ 0:5; u0 ¼ 30 ; T 0:1 p2 can be obtained from Fig. 4. It is dimensionalised and shown in Fig. 5c. When the 30 kPa surcharge is applied on top of the hill between the edge and a point at a distance L ¼ 6:2 m from the edge, the failure plane will intersect the face of the slope rather than extend all the way to the toe. The extent of soil mass at limiting equilibrium for case 2 is shown in Fig. 5d. For a failure plane to intersect the toe the surcharge of 30 kPa must extend a distance L ¼ 14:6 m from the edge. This is also indicated in Fig. 5c and d. The extent of the surcharge and L control the size of the sliding mass and the depth of the failure plane.

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T. Vo, A.R. Russell / Soils and Foundations xxx (2017) xxx–xxx

(f)

-12

-9

-6

-3

11

0

0

T=

3

T=

6

9

12 - 18

-12

-6

0

0

6

T=

- 24

T=

12

18

24 -60

-48

-36

-24

-12

0

0

T=

12

24

T=

36

48

60

Fig 4. (continued)

Case 3 The soil properties and surcharge condition correspond 2½ðvsÞ0 þc0 cot u0  sin u0 þ 30 ¼ 174 kPa. Eq. to q ¼ qmin þ 30 ¼ 1sin u0 (3.15) is applied for nonhomogeneous unsaturated soil to pffiffiffiffi ðvsÞ ð1K Þþ2c0 K obtain y 0 ¼ cK0 a þkvsaðK a 1Þ a ¼ 9:47 m. The pressure scale S ¼ q þ ðvsÞ0 þ c0 cot u0 ¼ 246 kPa and length scale D ¼ 6:2 m correspond to F ¼ 0 Dðcþk vs Þ ððvsÞ þc0 cotu0 Þð1þsinu0 Þ ¼ 0:376. Also, T ¼ cotu lnj ðqþðvsÞ0 þc0 cotu0 Þð1sinu0 Þ j ¼ qþðvsÞ0 þc0 cot u0 2 0 0:113 0:0717 p2. The slope profile corresponding to F ¼ 0:376; u0 ¼ 30 ; T 0:0717 p2 can be obtained from Fig. 4. It is dimensionalised and shown in Fig. 5e. The influence of vs is evident through the formation a steeper curvilinear slope for case

3 than case 2 (Fig. 5e). If the surcharge is applied on top of the hill between the edge and a distance L ¼ 6:2 m from the edge, the failure plane intersects the face of the slope. The extent of soil mass at limiting equilibrium for case 3 is shown in Fig. 5f. Also, it has been determined that the failure plane would intersect the toe of the slope if the surcharge extended across the top of the slope by a distance L ¼ 15:9 m. The extent of the sliding mass for this condition is also indicated in Fig. 5e and f. 6. Non-limiting conditions In practice, it is often necessary that preliminary designs be carried out for non-limiting conditions. This may be

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T. Vo, A.R. Russell / Soils and Foundations xxx (2017) xxx–xxx

-70

-60

-50

-40

-30

-20

-10

0

10 -10

6.93 m

(a)

(b)

0

H = 53 m

10 20 30 40 50 60

(c)

-70

-60

-50

-40

-30

-20

-10

0

10

-20

L*

-10

H = 53 m

6.93 m

L

(d)

20

0 10

face failure point

20 30 40

L = 6.2 m for face failure L* = 14.6 m for toe failure

(e)

-70

-60

-50

-40

-30

-20

-10

0

50 60

10

-20

L*

-10

H = 53 m

9.47 m

L

0 10

face failure point design of case 2

(f)

20

20 30

L = 6.2 m for face failure L* = 15.9 m for toe failure

40 50 60

Fig. 5. Example of a hillslope design for limiting condition. (a) Case 1 at limiting condition; (b) Extent of soil mass at limiting equilibrium in case 1; (c) Case 2 at limiting condition; (d) Extent of soil mass at limiting equilibrium in case 2; (e) Case 3 at limiting condition; (f) Extent of soil mass at limiting equilibrium in case 3.

done by reducing soil strength parameters by a factor of safety (FS). The reduced soil strength parameters are denoted (c0m ; u0m ) and defined as: FS ¼

c0 tan u0 ¼ 0 cm tan u0m

ð6:1Þ

where FS is a constant greater than 1. Eq. (6.1) is the definition of Bishop (1955) and is widely used (Matsui and San, 1992; Dawson et al., 1999; Griffiths and Lane, 1999). Adopting the reduced strength parameters, a curvi-

linear slope design may be obtained which is not at a limiting condition of failure and may have superior resistance compared to planar slopes. It is also possible to design for non-limiting conditions by increasing body forces and external forces by a factor in a way that drives failure in a particular problem. This approach has been employed in many limit equilibrium types of analyses although it is dependent on the boundary conditions of the problem (Sloan, 2013; Tschuchnigg et al., 2015).

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T. Vo, A.R. Russell / Soils and Foundations xxx (2017) xxx–xxx

It is important to note that the above two approaches are distinct, and it needs to be unambiguously stated in design which of these is adopted. 7. Conclusion Stability analyses were completed for slopes comprising unsaturated soils with convex or concave surface profiles, with soil having a strength defined by the MohrCoulomb failure criterion, a constant friction angle (u0 ), and profiles of cohesion ðc0 Þ and the contribution of suction to the effective stress (vsÞ that vary linearly with depth. Once the constants defining the linearly varying c0 and vs profiles are included in the stress characteristic curves, the solution procedure required for the analysis of the problem is well established. The analysis results have been presented in a series of stability charts, which show surface profiles at the onset of instability depend only on u0 and the dimensionless parameters F;T. Of these parameters Lðcþk c cot u0 þk vs Þ F ¼ ðqþðvsÞ þc 0 cot u0 Þ tan u0 contains a characteristic length for 0

0

the slope (L), an equivalent total unit weight of the soil (contained in the parenthesis on the numerator, being the total unit weight adjusted to account of the linear variations of c0 and vs profiles) and an equivalent cohesion (the denominator, being the actual cohesion at the top of the slope plus the effect of the vs profile and a surcharge q at the top of the slope). T captures the inclination of the slope profile at the top of the slope. vs tan u0 has a very similar influence to c0 on slope stability, and as vs increases, steeper curvilinear slopes become stable. As F ! 1 the results become independent of T and the surface profiles become unique and planar. The inclusion of T in the governing dimensionless parameters has not been previously considered. Its inclusion enables charts to be produced of wide practical applicability to assist with preliminary design. A design example has been given on how to use the stability charts. Acknowledgements Thanks also goes to the Australian Research Council for funding through grant DP140103142. References Atkinson, J.H., Potts, D.M., 1975. A note on associated field solutions for boundary value problems in a variable phi–variable v soil. Ge´otechnique 25, 379–384. Bishop, A.W., 1955. The use of the slip circle in the stability analysis of slopes. Ge´otechnique 5, 7–17. Bishop, A.W., 1959. The principle of effective stress. Teknisk Ukeblad 106, 859–863. Bishop, A.W., 1960. The measurement of pore pressure in the triaxial test. In: Proceedings of Conference on Pore Pressure and Suction in Soils, 1960 London, UK. Butterworth, pp. 38–46. Bishop, A.W., 1966. The strength of soils as engineering materials. Ge´otechnique 16, 89–130.

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