- Email: [email protected]
Road-cut slope topography and stability relationships in St Lucia, West Indies
Applied Geography (1983), 3, 105-l 14
105
Road-cut slope topography and stability relationships in St Lucia, West Indies Malcolm
G. Anderson
Department of Geography, University of Bristol, University Road, Bristol BS8 1SS, England Abstract Slope topography is shown to be a significant factor in controlling the porewater pressures in road-cut slopes in the residual volcanic soils of St Lucia, West Indies. It is shown that this control diminishes as the permeability of the material decreases. This finding is confirmed by stability envelopes in which ‘stable’ and failed slopes are examined as a function of slope topography and material properties.
Introduction
In certain undeveloped countries in the tropics there is currently a significant desire to upgrade the existing road network. The island of St Lucia in the Caribbean is no exception to this general trend. In this region of the Eastern Caribbean there is a strong economic need to establish new ‘feeder’ roads. However, empirical design criteria relating to storm precipitation, pore-water pressure, and hence stability determinations, are extremely sparse. The accurate prediction of pore-water pressures in tropical slopes is especially important given the known rapid variation in such pressures compared to temperate slopes. Thus, prior to the application of stability analysis procedures, there is a strong necessity to develop an adequate pore pressure prediction model (Fukuoka 1980). In a previous paper (Anderson 1981a) it was established that for the residual volcanic soils of the island of St Lucia a statistically significant and acceptable model could be established which was shown to be capable of predicting maximum pore-water pressure (+cm water) from total storm precipitation @-mm) and material permeability (k-cm s- ‘) alone: $ = - 324.32 + 0.85~ - 48.65 log k
(1)
Despite the success of this general formulation established for straight slopes (in both plan and profile), it was noted that evidence was available which suggested that the road-cut slope topography (slope angle, slope plan curvature and slope section form) may be a significant factor in modifying the pore-water pressures where there is a significant departure from rectilinear slope topography. It must be restated here that the overall goal is the establishment of models capable of predicting pore-water pressures in road-cut slopes incorporating as wide a range of materials, soil properties and slope topographies as possible. However, for use in the less developed world with the relative sparsity of laboratory testing facilities (Cooper 1977), a principal requirement is the ease of field calibration of 0143~228/83~020105-10$03.OOQ
1983 Butterworth
& Co (Publishers)
Ltd
106
Road-cut slope topography
and stability relationships
Plate 1. Site C (see Fig. 1).
Figure 1. Study site locations on Barre de I’Isle Road, St Lucia (see Anderson 1981b for general location map).
such models. Pursuant of this general aim, this paper seeks to establish the significance of cut-slope topographic form on pore-water pressures. Specifically, from a design standpoint, answers are required to the question: ‘given a particular permeability and storm precipitation, which slope topographies are most susceptible the principal to the worst pore-water pressure conditions. 7’ Plate 1 illustrates requirement for evaluating the cut-slope topography element in any pore-water pressure-stability analysis, for here (at site C in Fig. 1) a series of hairpin bends (convergent and divergent slope topography) is enforced by the local topography. Stability analysis procedures currently available for such cut slopes with variable corner angles reinforce the need for accurately predicting pore-water pressures if reliable slope stability determinations are to be made (Giger and Krizek 1975). Secondly, Young (1980) has stressed this need to ensure accurate determination of unsaturated soil strength, for in the tropics there is strong evidence of slope failure occurring by reductions in suction, rather than by increased positive pore pressures. Study sites
Twelve site sections were established in 1978 on the Barre de 1’Isle road in St Lucia (Fig. 1). At these sites pore-water pressure (depth 60 cm at slope base to accord with previous failure depths), Atterberg limits, permeability, storm precipitation and
Malcolm G. Anderson
107
Y= -0.51-0.206X
I
-7
-6
I
-5
-4
1
-3 Log k,
1
-2
I
-1
'0
Figure 2. (a) 2-K
(saturated) relationships based on 70 samples; (b) sample size required to estimate log k, to an accuracy of kO.25.
unconfined strength tests are available for the period October 1978-October 1981 (these data are fully reported in Anderson 198lab). The soils are all residual volcanic soils with saturated permeabilities (k,) ranging from 2 x lo- 7 to 1 x 10m3 cm s-r and unconfined strengths in the range 0540 kg cm-‘. Owing to the steepness of the cut-slope sites, permeability was best determined by taking undisturbed cores and making the necessary determinations on site with a laboratory permeameter. To test the variability of k determined in this manner 10 samples were taken from each of seven sites of differing permeability. Figure 2a illustrates the standard deviation (6) of the determined values as a function of log k.
Road-cut slope topography
108
and stability relationships
It is evident that a clear trend of increasing variability is associated with decreasing permeability. Field examination of samples and sites revealed that macropore (> 2 mm) occurrence most probably accounted for this trend, with the occasional macropore and fracture causing greater variability in the otherwise less permeable material. It is, of course, possible to estimate the sample size required (n) to estimate mean log k to within a specified error (d) at the 95 per cent confidence level. The appropriate relationship is: n = (S1.96/d)0’5
(2)
where c? is the sample standard deviation. Together, this relationship and the data illustrated in Fig. 2a show the significant increase in sample size required to estimate mean log k to within an accuracy of kO.25 when k,< 10m5 cm s-l (Fig. 2b). It is especially important in validating empirical conclusions that determinations of parameter variability of this sort are made (Sharma et al. 1980). In addition, slope plan curvature (c-degrees, positive concave), slope length (l-metres), and slope angle (u-degrees) were obtained. The unconfined strengths were determined by use of the Michigan penetrometer, which in the absence of more elaborate strengthtesting facilities provides an adequate method of determining unconfined strengths (Ministry of Transport, Canada 1970). Pore-water pressures were recorded on a portable transducer unit specifically designed for operation in remote locations (Anderson 1981b), whilst precipitation was recorded by an autographic raingauge located at site F (Fig. l), and by check gauges at each site. Cut-slope topographic controls on pore-water pressures
Three statistical analyses were undertaken to provide evidence of the topographic criteria giving rise to relatively high pore-water pressure conditions and to provide a model for the prediction of soil-water potentials based in part on such criteria. Such a model is a refinement of that already presented (Equation 1). The proposition here is that improved diagnostic and predictive success can be achieved by the establishment of a suite of models, one for each permeability group. Partial
correlations
Subdividing the sites into permeability groups as shown in Table 1, and undertaking a partial correlation analysis specifically designed to isolate topographic control, allows some initial general statements to be made. From the results shown in this table it is clear that for all three permeability groups, controlling for the effect of precipitation, there is a significant correlation between slope plan curvature and soil-water potentials (r+c,p). There is, therefore, firm evidence that when the effect of precipitation is controlled in the relationship, slope topographic elements are significantly associated with soilwater potentials, and that the overall dominant topographic element of the three is that of slope plan curvature. Moreover, when both p and a are controlled, the relationship between $ and c is significant at high permeability, whilst when p and c are controlled, the ~,!-a relationship is significant for less permeable material. This changing significance of topographic controls is examined below. Multiple
regression
On the basis that topographic
slope characteristics,
and slope plan curvature
in
Malcolm
G. Anderson
109
Table 1. Partial correlations Partial Permeability group (cm s-l) 10-3 (5 x 10-4-5 10-4 (5x 10-5-5x 10-6 (9x 10-7-5x
correlatior& Sample size
‘W.P
ric+
0.81***
0.81***
0.31
47
-0.22
0.66***
0.15
0.91***
36
-0.01
0.56***
0.02
r*o.P
r*l.P
0.17
0.78***
x 10-q 0.66*** 10-G) -0.64***
-0.93***
51
10-Z)
*** Significant at 99.9 per cent level. ’ For example, rj.,,, correlanon between soil-water potential ($) and slope angle (a) with precipitation controlled
(I is slope length,
c is slope plan curvature,
@)
as defined in the text).
Table 2. Significance
levels and coefficients of explanation (%) for multiple regression relationships between permeability and precipitation and site topographic indices for three permeability groups Independent
Permeability group (cm s-‘) (saturated) 1x10-3 1 x 10-4 1 x 10-6
variables
(%)
p.a
PJ
P.C
Sample size
22** 70*** 66***
83*** 70*** 43***
87*** 70*** 62***
47 36 51
** Significant at 99 per cent; *** significant at 99.9 per cent (based on F test). Nore. p = storm precipitation (mm); a = slope angle (degrees); I= slope length (m); c = slope plan curvature (degrees). Largest storm precipitation included = 203 mm, having an estimated recurrence interval of 12 years (Anderson 1981a).
particular, exert significant controls on soil-water potentials, a multiple regression analysis was undertaken on the same data to provide a predictive model. For each permeability group with I(/ as the dependent variable, multiple regressions were run for precipitation and each of the three topographic indices (c, I, a) as independent variables. Table 2 provides the explained variance and significance for each of the nine models. Whilst prediction relationships for each of the three topographic indices are significant, slope plan curvature has the highest overall explained variance in the three permeability groups. The respective prediction equations for each saturated permeability group are:
k: 1 x 10e3 cm s-l
II/= - 16578+0.75p+3.08~
(3)
k: 1 x lop4 cm SC’
i,G= -136.15+0.72~+1.69~
(4)
k:1x10-6cms-1
$= -
(5)
31.83+0.45p+O.54~
Figure 3 illustrates the three prediction equations. It is clear that the effect of slope plan curvature (c) on I,+increases in the more permeable materials. By contrast, for
Road-cut slope topography
110
k,
q
and stability relationships
Storm precipitation (mm)
Ix lO’3 cm s-l
Soil / woter potential (cm water) + 100
Storm precipitation 50 mm
Curvoture -40%
\
,
I
\
Curvature
t
50
0
-50
-----
_---
(degrees)
_---_-*
_.*_
**a-
_#xr
<-
__-= IOOmm
----i.i 0
-50
ks = 1xlO-6
cm
50
s-1
150 mm
0
0
-50 -200
50
---Soil/water
potential
(cm water)
Figure 3. Plot of equations 3-5 predicting soilwater potentials for each of three permeability
groups.
- - - _ - _ _-
k,
lx163cm
k,
lxld4 cm s-’
k,
lxlb5
cm
Figure 4. Plot of equations
se’
5.’
3-5 predicting
soil-water potentials with three storm sizes illustrated.
material with permeability of 10m6 cm s-l there are relatively smaller changes in tj for given changes in c. For a 150 mm storm in material of permeability 1 x 10d3 cm s-l a curvature (c) of +30” (strongly concave) gives tj = + 39 cm whilst c= - 30” (convex) gives $ = - 146 cm. Corresponding conditions for material of k= 1 x lO-6 cm s-l predict $ = + 52 cm and + 19 cm respectively. These established relationships can thus be used to examine the predicted soilwater potentials associated with given storm sizes for each permeability group when c is varied. Figure 4 illustrates such predictions for three storms: 50 mm, 100 mm,
Malcolm G. Anderson
111
and 150 mm. From these plots it is clear that, for a given storm size, the soil-water potentials in the more permeable (1 x lo- 3 cm s- ‘) materials approach those for the least permeable (1 x lo- 6 cm s-i) under conditions of very high slope plan curvature (+40”). For example, with a storm precipitation of 100 mm, materials with permeabilities 1 x low4 cm s-l have pore-water pressures only some 35 cm less than material of permeability 1 x lop6 cm s-i for highly concave slope plans (+30”). Whilst for a 150 mm storm the pore-water pressures these materials exhibit differ by only some 15 cm at this curvature. Beta weights
‘Beta weights’ indicate how much change in the dependent variable is produced by a standardized change in one of the independent variables when the others are controlled. In the standard notation /?ij,k,variable i is being predicted from variable j with the effect of variable k controlled. The direct utility of this procedure is that it is possible to assess the effect of changes in slope topographic elements (a and c) on soil-water potential separately from similar changes in precipitation. Moreover, this analysis can be undertaken for each of the permeability groups to ascertain whether there is any change in the relative effects of a, c and p (the independent variables) on $. Table 3 and Fig. 5 show the generalized relationships of beta weights for different controlled variables for each permeability group. The main points from this analysis are : 1. for high permeability (lo- 3 cm s- ’ ) slope plan curvature exerts as great an effect upon soil-water potential ($c.p curve at lo- 3 cm s- ’ = 0.68) as does precipitation (I(/p.c at lop3 cm s-l =0*63);
Table 3. Logarithmic regression relation-
ships of beta weights and permeability (see Fig. 5) &$ =
1.17-0.83 1nX
r= -0.60
&,,., =
1.14-0.45 1nX
r= -0.97
&,, =
0.29 +0.32 1nX
r=
0.59
&P,o = -0.03 + 0.57 1nX
r=
0.84
X = Ilog k/.
Figure 5. Beta weights showing the relative diminution of slope topography (c and a) control upon soil-water potential with decreasing permeability.
.
0
A
A
*
--e-w-
D
40
!?I
I3
0
Curvature Idegrees)
20
1
60
Concave
Figure 6. Relationship of failed and ‘stable’ cut slopes to unconfined strength, slope plan curvature and slope angle, for permeability group k=lx10-6cms-1.
Convex
I
-20
\
40
1
5 40-
$
v F 500
60-
70-
80-
go-
Convex
-20
0 Curvature (degrees)
20
Concave
60
Figure 7. Relationship of failed and ‘stable’ cut slopes to unconfined strength, slope plan curvature and slope angle, for permeabihty group k=lx10-4cms-‘.
-60
u
Convex
-40
0
‘Stable’
(degrees)
0 Curvature
20
s-r
Concove
40
.
.
60
I
Fabled
Stobilrty envelopes for each penetration test value
3- 4
2-3
k,= I x IOe3cm
-20
El
Michqan penetrameter volues
Figure 8. Relationship of failed and ‘stable’ cut slopes to unconfined strength, slope plan curvature and slope angle, for permeability group k=l~lO-~cms I.
-60
0-i
IO-
20k, = I x 10m4cm s-’
\
A
l
l
Farled
20”
-40
0 D a
’ Stohle’
Stabilrty envelopes for each penetration test value
8-11
<4 5 - 8
30-
Ei
Mlchlgon penetrameter values
30-
$ $40
0:
--
%506
60-
70-
o-4---7-60 -40
10-6cm s-1
A
n a
.
80-
go-
IO-
k,=lx
--~.-“_._
---L-_-_-o-l.
El
13
9
0 n
6 - 9
Stobrltty envelopes for each penetration test value
Mrchigan penetrameter values
.
0
I
IO-
zo-
30-
a, B * 40-
a, F 50-
60-
70-
80-
go-
Furled
‘Stable’
Malcolm G. Anderson
113
2. for less permeable material (1 x 10e6 cm s- ‘) the relative effect of slope plan curvature on soil-water potential diminishes, whilst that of slope angle and precipitation increases; 3. these relationships regarding the change in significance of c on + with permeability, reinforce the relationships shown in Fig. 3, where for low permeability material c is seen to effect only comparatively small changes in \I/. The analysis of slope topographic controls on pore-water pressures has thus shown that slope plan curvature is especially important (Fig. 4) and that significant prediction equations can be established for $ based upon c and p for three material permeability groups. Thus, most importantly, it is possible to predict those topographies likely to experience the worst pore-water pressure conditions (Fig. 3, Equations 3-5), and also to show the relative diminution in the topographic control as the material permeability decreases (Fig. 5). Relationship
of failed and ‘stable’ slopes to slope topography and material strength
Throughout the duration of the project a schedule of slope failures has been kept. Since the importance of the topographic slope elements has been clearly demonstrated in the context of soil-water potential control, it is possible to plot ‘failed and ‘stable’ sites by permeability group, noting for each site the pre-failure slope angle, slope plan curvature and material strength. Permeability group, slope angle and curvature are sufficient to specify soil-water potential conditions (Fig. 3 and Equations 3-5) for given storm precipitations. Figures 6-8 illustrate the result of plotting such data by permeability group, distinguishing between slopes stable throughout the study period and those subject to failure. Envelopes are provided on these figures discriminating between stable and failed slopes for given penetrometer values at 50 or more sites. These tentative failure-topography relationships are consistent with the results of the previous discussion (Figs 3 and 5). For decreasing permeability the principal discrimination is on slope angle (Fig. 6). In other words, the envelope bounds show no significant response to slope plan curvature change. By contrast, for more permeable material, slope plan curvature provides significant discrimination (Fig. 8). In this case (k,= 1 x lo- 3 cm s- ‘), the envelope bounds are now no longer subhorizontal, implying that for k,> 1 x low5 cm s-l slope angle and slope plan curvature affect pore-water pressure, and thus stability. This finding, from plotting failed and stable slopes against morphometric slope measures, accords precisely with the conclusions on pore-water pressure controls; points 1 and 3 in the preceding section. In addition, for a given slope plan curvature, decreasing material permeability is shown to necessitate increased strength to ensure stability at a given angle. Strength values shown in Figs 6-8 are in terms of the field-determined Michigan penetrometer values. These values have a direct relationship to unconfined strength and material consistency (Ministry of Transport, Canada 1970). Michigan values in excess of 10 correspond to strengths greater than 4.0 kg cm-2 (hard consistency), those of 5-10 to strengths in the range 2-4 kg cm-* (stiff consistency). Conclusions
Prediction equations have shown those topographies likely to induce high porewater pressure conditions (Equations 3-5 and Fig. 3). Slope plan curvature is shown to exert a highly significant control on soil-water potentials when the material is of
Road-cut
114
slope topography
and stability relationships
high permeability (1 x 10-j cm SK’). This influence is shown to diminish for less permeable material (Fig. 5). The identified dominant topographic controls of slope angle and slope plan curvature on soil-water potential have been successfully used to discriminate and establish stability envelopes for slope sites during the study period in association with material strength values. Such relationships (Figs 6-8) reinforce the variable control that slope topography has on slope stability through its control of soil-water potentials. For site permeability and unconfined strength values alone it is thus possible to predict : (a) the soil-water potentials associated with a particular slope topography defined by slope angle and slope plan curvature for specified storms (Equations 3-5); (b) the likely stability condition of a slope of given plan curvature and slope angle (Figs 6-8). Since permeability and unconfined strength are both easily derived site material properties it is contended that these two relationships can provide a useful contribution to cut slope design procedures in the region. Repeated site permeability tests show that where the permeability exceeds 1 x 1O-4 cm s-l the mean log permeability can be estimated to within an accuracy of kO.25 with fewer than five samples (Fig. 2). For permeabilities as low as 1 x 10e6 although many more samples are required to achieve a similar and 1x10-‘ems-‘, accuracy in the estimation of permeability, the relatively more conservative changes in soil-water potentials in such material make such accurate estimates much less essential (Fig. 3). Acknowledgements The research project was funded by the Overseas Development Administration, London. P. Gill and P. E. Kneale assisted with the field investigation. Thanks are also due to the Government of St Lucia for permitting the work to be undertaken, and in particular to the Chief Engineer, Ministry of Communications and Works. References Anderson, M. G. (1981a) The prediction of pore water pressure conditions in road cut slopes, St Lucia, West Indies. Report 3 to Overseas Development Administration, London. Anderson, M. G. (1981b) Predicting pore-water pressures in road cut slopes in the West Indies. Applied Geography 2, 55-68. Cooper,
L. (1977) Report on road and transport planning
in tropical and subtropical
countries.
Transport and Road Research Laboratory, Supplementary Report 162. Fukuoka, M. (1980) Landslides associated with rainfall. Geotechnical Engineering
11, l-29. Giger, M. W. and Krizek, R. J. (1975) Stability analysis of vertical cut with variable corner angle. Japanese Society of Soil Mechanics and Foundation Engineering 15, 63-71. Ministry of Transport, Canada (1970) Soil survey for runways, taxiways, aprons and roads. Engineering Planning Design and Construction Manual, Section 4. Sharma, M. L., Gander, G. A. and Hunt, C. G. (1980) Spatial variability of infiltration
watershed. Journal of Hydrology 45, 101-122. Young, R. N. (1980) Some aspects of soil suction, Geotechnical
Engineering
shear
strength
11, 55-76.
(Revised
manuscript
received
26 May
1982)
and
soil
in a
stability.
105
Road-cut slope topography and stability relationships in St Lucia, West Indies Malcolm
G. Anderson
Department of Geography, University of Bristol, University Road, Bristol BS8 1SS, England Abstract Slope topography is shown to be a significant factor in controlling the porewater pressures in road-cut slopes in the residual volcanic soils of St Lucia, West Indies. It is shown that this control diminishes as the permeability of the material decreases. This finding is confirmed by stability envelopes in which ‘stable’ and failed slopes are examined as a function of slope topography and material properties.
Introduction
In certain undeveloped countries in the tropics there is currently a significant desire to upgrade the existing road network. The island of St Lucia in the Caribbean is no exception to this general trend. In this region of the Eastern Caribbean there is a strong economic need to establish new ‘feeder’ roads. However, empirical design criteria relating to storm precipitation, pore-water pressure, and hence stability determinations, are extremely sparse. The accurate prediction of pore-water pressures in tropical slopes is especially important given the known rapid variation in such pressures compared to temperate slopes. Thus, prior to the application of stability analysis procedures, there is a strong necessity to develop an adequate pore pressure prediction model (Fukuoka 1980). In a previous paper (Anderson 1981a) it was established that for the residual volcanic soils of the island of St Lucia a statistically significant and acceptable model could be established which was shown to be capable of predicting maximum pore-water pressure (+cm water) from total storm precipitation @-mm) and material permeability (k-cm s- ‘) alone: $ = - 324.32 + 0.85~ - 48.65 log k
(1)
Despite the success of this general formulation established for straight slopes (in both plan and profile), it was noted that evidence was available which suggested that the road-cut slope topography (slope angle, slope plan curvature and slope section form) may be a significant factor in modifying the pore-water pressures where there is a significant departure from rectilinear slope topography. It must be restated here that the overall goal is the establishment of models capable of predicting pore-water pressures in road-cut slopes incorporating as wide a range of materials, soil properties and slope topographies as possible. However, for use in the less developed world with the relative sparsity of laboratory testing facilities (Cooper 1977), a principal requirement is the ease of field calibration of 0143~228/83~020105-10$03.OOQ
1983 Butterworth
& Co (Publishers)
Ltd
106
Road-cut slope topography
and stability relationships
Plate 1. Site C (see Fig. 1).
Figure 1. Study site locations on Barre de I’Isle Road, St Lucia (see Anderson 1981b for general location map).
such models. Pursuant of this general aim, this paper seeks to establish the significance of cut-slope topographic form on pore-water pressures. Specifically, from a design standpoint, answers are required to the question: ‘given a particular permeability and storm precipitation, which slope topographies are most susceptible the principal to the worst pore-water pressure conditions. 7’ Plate 1 illustrates requirement for evaluating the cut-slope topography element in any pore-water pressure-stability analysis, for here (at site C in Fig. 1) a series of hairpin bends (convergent and divergent slope topography) is enforced by the local topography. Stability analysis procedures currently available for such cut slopes with variable corner angles reinforce the need for accurately predicting pore-water pressures if reliable slope stability determinations are to be made (Giger and Krizek 1975). Secondly, Young (1980) has stressed this need to ensure accurate determination of unsaturated soil strength, for in the tropics there is strong evidence of slope failure occurring by reductions in suction, rather than by increased positive pore pressures. Study sites
Twelve site sections were established in 1978 on the Barre de 1’Isle road in St Lucia (Fig. 1). At these sites pore-water pressure (depth 60 cm at slope base to accord with previous failure depths), Atterberg limits, permeability, storm precipitation and
Malcolm G. Anderson
107
Y= -0.51-0.206X
I
-7
-6
I
-5
-4
1
-3 Log k,
1
-2
I
-1
'0
Figure 2. (a) 2-K
(saturated) relationships based on 70 samples; (b) sample size required to estimate log k, to an accuracy of kO.25.
unconfined strength tests are available for the period October 1978-October 1981 (these data are fully reported in Anderson 198lab). The soils are all residual volcanic soils with saturated permeabilities (k,) ranging from 2 x lo- 7 to 1 x 10m3 cm s-r and unconfined strengths in the range 0540 kg cm-‘. Owing to the steepness of the cut-slope sites, permeability was best determined by taking undisturbed cores and making the necessary determinations on site with a laboratory permeameter. To test the variability of k determined in this manner 10 samples were taken from each of seven sites of differing permeability. Figure 2a illustrates the standard deviation (6) of the determined values as a function of log k.
Road-cut slope topography
108
and stability relationships
It is evident that a clear trend of increasing variability is associated with decreasing permeability. Field examination of samples and sites revealed that macropore (> 2 mm) occurrence most probably accounted for this trend, with the occasional macropore and fracture causing greater variability in the otherwise less permeable material. It is, of course, possible to estimate the sample size required (n) to estimate mean log k to within a specified error (d) at the 95 per cent confidence level. The appropriate relationship is: n = (S1.96/d)0’5
(2)
where c? is the sample standard deviation. Together, this relationship and the data illustrated in Fig. 2a show the significant increase in sample size required to estimate mean log k to within an accuracy of kO.25 when k,< 10m5 cm s-l (Fig. 2b). It is especially important in validating empirical conclusions that determinations of parameter variability of this sort are made (Sharma et al. 1980). In addition, slope plan curvature (c-degrees, positive concave), slope length (l-metres), and slope angle (u-degrees) were obtained. The unconfined strengths were determined by use of the Michigan penetrometer, which in the absence of more elaborate strengthtesting facilities provides an adequate method of determining unconfined strengths (Ministry of Transport, Canada 1970). Pore-water pressures were recorded on a portable transducer unit specifically designed for operation in remote locations (Anderson 1981b), whilst precipitation was recorded by an autographic raingauge located at site F (Fig. l), and by check gauges at each site. Cut-slope topographic controls on pore-water pressures
Three statistical analyses were undertaken to provide evidence of the topographic criteria giving rise to relatively high pore-water pressure conditions and to provide a model for the prediction of soil-water potentials based in part on such criteria. Such a model is a refinement of that already presented (Equation 1). The proposition here is that improved diagnostic and predictive success can be achieved by the establishment of a suite of models, one for each permeability group. Partial
correlations
Subdividing the sites into permeability groups as shown in Table 1, and undertaking a partial correlation analysis specifically designed to isolate topographic control, allows some initial general statements to be made. From the results shown in this table it is clear that for all three permeability groups, controlling for the effect of precipitation, there is a significant correlation between slope plan curvature and soil-water potentials (r+c,p). There is, therefore, firm evidence that when the effect of precipitation is controlled in the relationship, slope topographic elements are significantly associated with soilwater potentials, and that the overall dominant topographic element of the three is that of slope plan curvature. Moreover, when both p and a are controlled, the relationship between $ and c is significant at high permeability, whilst when p and c are controlled, the ~,!-a relationship is significant for less permeable material. This changing significance of topographic controls is examined below. Multiple
regression
On the basis that topographic
slope characteristics,
and slope plan curvature
in
Malcolm
G. Anderson
109
Table 1. Partial correlations Partial Permeability group (cm s-l) 10-3 (5 x 10-4-5 10-4 (5x 10-5-5x 10-6 (9x 10-7-5x
correlatior& Sample size
‘W.P
ric+
0.81***
0.81***
0.31
47
-0.22
0.66***
0.15
0.91***
36
-0.01
0.56***
0.02
r*o.P
r*l.P
0.17
0.78***
x 10-q 0.66*** 10-G) -0.64***
-0.93***
51
10-Z)
*** Significant at 99.9 per cent level. ’ For example, rj.,,, correlanon between soil-water potential ($) and slope angle (a) with precipitation controlled
(I is slope length,
c is slope plan curvature,
@)
as defined in the text).
Table 2. Significance
levels and coefficients of explanation (%) for multiple regression relationships between permeability and precipitation and site topographic indices for three permeability groups Independent
Permeability group (cm s-‘) (saturated) 1x10-3 1 x 10-4 1 x 10-6
variables
(%)
p.a
PJ
P.C
Sample size
22** 70*** 66***
83*** 70*** 43***
87*** 70*** 62***
47 36 51
** Significant at 99 per cent; *** significant at 99.9 per cent (based on F test). Nore. p = storm precipitation (mm); a = slope angle (degrees); I= slope length (m); c = slope plan curvature (degrees). Largest storm precipitation included = 203 mm, having an estimated recurrence interval of 12 years (Anderson 1981a).
particular, exert significant controls on soil-water potentials, a multiple regression analysis was undertaken on the same data to provide a predictive model. For each permeability group with I(/ as the dependent variable, multiple regressions were run for precipitation and each of the three topographic indices (c, I, a) as independent variables. Table 2 provides the explained variance and significance for each of the nine models. Whilst prediction relationships for each of the three topographic indices are significant, slope plan curvature has the highest overall explained variance in the three permeability groups. The respective prediction equations for each saturated permeability group are:
k: 1 x 10e3 cm s-l
II/= - 16578+0.75p+3.08~
(3)
k: 1 x lop4 cm SC’
i,G= -136.15+0.72~+1.69~
(4)
k:1x10-6cms-1
$= -
(5)
31.83+0.45p+O.54~
Figure 3 illustrates the three prediction equations. It is clear that the effect of slope plan curvature (c) on I,+increases in the more permeable materials. By contrast, for
Road-cut slope topography
110
k,
q
and stability relationships
Storm precipitation (mm)
Ix lO’3 cm s-l
Soil / woter potential (cm water) + 100
Storm precipitation 50 mm
Curvoture -40%
\
,
I
\
Curvature
t
50
0
-50
-----
_---
(degrees)
_---_-*
_.*_
**a-
_#xr
<-
__-= IOOmm
----i.i 0
-50
ks = 1xlO-6
cm
50
s-1
150 mm
0
0
-50 -200
50
---Soil/water
potential
(cm water)
Figure 3. Plot of equations 3-5 predicting soilwater potentials for each of three permeability
groups.
- - - _ - _ _-
k,
lx163cm
k,
lxld4 cm s-’
k,
lxlb5
cm
Figure 4. Plot of equations
se’
5.’
3-5 predicting
soil-water potentials with three storm sizes illustrated.
material with permeability of 10m6 cm s-l there are relatively smaller changes in tj for given changes in c. For a 150 mm storm in material of permeability 1 x 10d3 cm s-l a curvature (c) of +30” (strongly concave) gives tj = + 39 cm whilst c= - 30” (convex) gives $ = - 146 cm. Corresponding conditions for material of k= 1 x lO-6 cm s-l predict $ = + 52 cm and + 19 cm respectively. These established relationships can thus be used to examine the predicted soilwater potentials associated with given storm sizes for each permeability group when c is varied. Figure 4 illustrates such predictions for three storms: 50 mm, 100 mm,
Malcolm G. Anderson
111
and 150 mm. From these plots it is clear that, for a given storm size, the soil-water potentials in the more permeable (1 x lo- 3 cm s- ‘) materials approach those for the least permeable (1 x lo- 6 cm s-i) under conditions of very high slope plan curvature (+40”). For example, with a storm precipitation of 100 mm, materials with permeabilities 1 x low4 cm s-l have pore-water pressures only some 35 cm less than material of permeability 1 x lop6 cm s-i for highly concave slope plans (+30”). Whilst for a 150 mm storm the pore-water pressures these materials exhibit differ by only some 15 cm at this curvature. Beta weights
‘Beta weights’ indicate how much change in the dependent variable is produced by a standardized change in one of the independent variables when the others are controlled. In the standard notation /?ij,k,variable i is being predicted from variable j with the effect of variable k controlled. The direct utility of this procedure is that it is possible to assess the effect of changes in slope topographic elements (a and c) on soil-water potential separately from similar changes in precipitation. Moreover, this analysis can be undertaken for each of the permeability groups to ascertain whether there is any change in the relative effects of a, c and p (the independent variables) on $. Table 3 and Fig. 5 show the generalized relationships of beta weights for different controlled variables for each permeability group. The main points from this analysis are : 1. for high permeability (lo- 3 cm s- ’ ) slope plan curvature exerts as great an effect upon soil-water potential ($c.p curve at lo- 3 cm s- ’ = 0.68) as does precipitation (I(/p.c at lop3 cm s-l =0*63);
Table 3. Logarithmic regression relation-
ships of beta weights and permeability (see Fig. 5) &$ =
1.17-0.83 1nX
r= -0.60
&,,., =
1.14-0.45 1nX
r= -0.97
&,, =
0.29 +0.32 1nX
r=
0.59
&P,o = -0.03 + 0.57 1nX
r=
0.84
X = Ilog k/.
Figure 5. Beta weights showing the relative diminution of slope topography (c and a) control upon soil-water potential with decreasing permeability.
.
0
A
A
*
--e-w-
D
40
!?I
I3
0
Curvature Idegrees)
20
1
60
Concave
Figure 6. Relationship of failed and ‘stable’ cut slopes to unconfined strength, slope plan curvature and slope angle, for permeability group k=lx10-6cms-1.
Convex
I
-20
\
40
1
5 40-
$
v F 500
60-
70-
80-
go-
Convex
-20
0 Curvature (degrees)
20
Concave
60
Figure 7. Relationship of failed and ‘stable’ cut slopes to unconfined strength, slope plan curvature and slope angle, for permeabihty group k=lx10-4cms-‘.
-60
u
Convex
-40
0
‘Stable’
(degrees)
0 Curvature
20
s-r
Concove
40
.
.
60
I
Fabled
Stobilrty envelopes for each penetration test value
3- 4
2-3
k,= I x IOe3cm
-20
El
Michqan penetrameter volues
Figure 8. Relationship of failed and ‘stable’ cut slopes to unconfined strength, slope plan curvature and slope angle, for permeability group k=l~lO-~cms I.
-60
0-i
IO-
20k, = I x 10m4cm s-’
\
A
l
l
Farled
20”
-40
0 D a
’ Stohle’
Stabilrty envelopes for each penetration test value
8-11
<4 5 - 8
30-
Ei
Mlchlgon penetrameter values
30-
$ $40
0:
--
%506
60-
70-
o-4---7-60 -40
10-6cm s-1
A
n a
.
80-
go-
IO-
k,=lx
--~.-“_._
---L-_-_-o-l.
El
13
9
0 n
6 - 9
Stobrltty envelopes for each penetration test value
Mrchigan penetrameter values
.
0
I
IO-
zo-
30-
a, B * 40-
a, F 50-
60-
70-
80-
go-
Furled
‘Stable’
Malcolm G. Anderson
113
2. for less permeable material (1 x 10e6 cm s- ‘) the relative effect of slope plan curvature on soil-water potential diminishes, whilst that of slope angle and precipitation increases; 3. these relationships regarding the change in significance of c on + with permeability, reinforce the relationships shown in Fig. 3, where for low permeability material c is seen to effect only comparatively small changes in \I/. The analysis of slope topographic controls on pore-water pressures has thus shown that slope plan curvature is especially important (Fig. 4) and that significant prediction equations can be established for $ based upon c and p for three material permeability groups. Thus, most importantly, it is possible to predict those topographies likely to experience the worst pore-water pressure conditions (Fig. 3, Equations 3-5), and also to show the relative diminution in the topographic control as the material permeability decreases (Fig. 5). Relationship
of failed and ‘stable’ slopes to slope topography and material strength
Throughout the duration of the project a schedule of slope failures has been kept. Since the importance of the topographic slope elements has been clearly demonstrated in the context of soil-water potential control, it is possible to plot ‘failed and ‘stable’ sites by permeability group, noting for each site the pre-failure slope angle, slope plan curvature and material strength. Permeability group, slope angle and curvature are sufficient to specify soil-water potential conditions (Fig. 3 and Equations 3-5) for given storm precipitations. Figures 6-8 illustrate the result of plotting such data by permeability group, distinguishing between slopes stable throughout the study period and those subject to failure. Envelopes are provided on these figures discriminating between stable and failed slopes for given penetrometer values at 50 or more sites. These tentative failure-topography relationships are consistent with the results of the previous discussion (Figs 3 and 5). For decreasing permeability the principal discrimination is on slope angle (Fig. 6). In other words, the envelope bounds show no significant response to slope plan curvature change. By contrast, for more permeable material, slope plan curvature provides significant discrimination (Fig. 8). In this case (k,= 1 x lo- 3 cm s- ‘), the envelope bounds are now no longer subhorizontal, implying that for k,> 1 x low5 cm s-l slope angle and slope plan curvature affect pore-water pressure, and thus stability. This finding, from plotting failed and stable slopes against morphometric slope measures, accords precisely with the conclusions on pore-water pressure controls; points 1 and 3 in the preceding section. In addition, for a given slope plan curvature, decreasing material permeability is shown to necessitate increased strength to ensure stability at a given angle. Strength values shown in Figs 6-8 are in terms of the field-determined Michigan penetrometer values. These values have a direct relationship to unconfined strength and material consistency (Ministry of Transport, Canada 1970). Michigan values in excess of 10 correspond to strengths greater than 4.0 kg cm-2 (hard consistency), those of 5-10 to strengths in the range 2-4 kg cm-* (stiff consistency). Conclusions
Prediction equations have shown those topographies likely to induce high porewater pressure conditions (Equations 3-5 and Fig. 3). Slope plan curvature is shown to exert a highly significant control on soil-water potentials when the material is of
Road-cut
114
slope topography
and stability relationships
high permeability (1 x 10-j cm SK’). This influence is shown to diminish for less permeable material (Fig. 5). The identified dominant topographic controls of slope angle and slope plan curvature on soil-water potential have been successfully used to discriminate and establish stability envelopes for slope sites during the study period in association with material strength values. Such relationships (Figs 6-8) reinforce the variable control that slope topography has on slope stability through its control of soil-water potentials. For site permeability and unconfined strength values alone it is thus possible to predict : (a) the soil-water potentials associated with a particular slope topography defined by slope angle and slope plan curvature for specified storms (Equations 3-5); (b) the likely stability condition of a slope of given plan curvature and slope angle (Figs 6-8). Since permeability and unconfined strength are both easily derived site material properties it is contended that these two relationships can provide a useful contribution to cut slope design procedures in the region. Repeated site permeability tests show that where the permeability exceeds 1 x 1O-4 cm s-l the mean log permeability can be estimated to within an accuracy of kO.25 with fewer than five samples (Fig. 2). For permeabilities as low as 1 x 10e6 although many more samples are required to achieve a similar and 1x10-‘ems-‘, accuracy in the estimation of permeability, the relatively more conservative changes in soil-water potentials in such material make such accurate estimates much less essential (Fig. 3). Acknowledgements The research project was funded by the Overseas Development Administration, London. P. Gill and P. E. Kneale assisted with the field investigation. Thanks are also due to the Government of St Lucia for permitting the work to be undertaken, and in particular to the Chief Engineer, Ministry of Communications and Works. References Anderson, M. G. (1981a) The prediction of pore water pressure conditions in road cut slopes, St Lucia, West Indies. Report 3 to Overseas Development Administration, London. Anderson, M. G. (1981b) Predicting pore-water pressures in road cut slopes in the West Indies. Applied Geography 2, 55-68. Cooper,
L. (1977) Report on road and transport planning
in tropical and subtropical
countries.
Transport and Road Research Laboratory, Supplementary Report 162. Fukuoka, M. (1980) Landslides associated with rainfall. Geotechnical Engineering
11, l-29. Giger, M. W. and Krizek, R. J. (1975) Stability analysis of vertical cut with variable corner angle. Japanese Society of Soil Mechanics and Foundation Engineering 15, 63-71. Ministry of Transport, Canada (1970) Soil survey for runways, taxiways, aprons and roads. Engineering Planning Design and Construction Manual, Section 4. Sharma, M. L., Gander, G. A. and Hunt, C. G. (1980) Spatial variability of infiltration
watershed. Journal of Hydrology 45, 101-122. Young, R. N. (1980) Some aspects of soil suction, Geotechnical
Engineering
shear
strength
11, 55-76.
(Revised
manuscript
received
26 May
1982)
and
soil
in a
stability.