Stability coefficients for highway cutting slope design
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Stability coefficients for highway cutting slope design by GE Barnes'bstract A new method is presented which gives the minimum factor of safety for long-term stability of a homogeneous slope of height H using effective stress parameters in the form F = a + b tang'. The stability coefficients a and b are presented in tabular and graphical form for slope inclinations of 1: 1, 2: 1, 3:1 and 4:1and have been found to be related to the cohesion soil parameter, c'/yH, and the water table level parameter, h JH. Pore pressures are represented as a quasi steady state condition by a water table at toe level beyond the toe and then inclined at various angles within the slope given by the depth below crest level, h„. This is considered to be of more practical use than the arbitrary pore pressure coefficient, r„, which has been used previously in slope stability charts. When the critical circle lies below the water table the slope is considered 'wet'. The situation when the critical circle lies above the water table and the slope can be considered 'dry'as been determined. This enables the assessment of whether the presence of a water table is important or not. The coefficient a was found to depend on slope inclination, c'/yH, and whether the slope is 'wet'r 'dry'. The coefficient b was found to depend on slope inclination, c'/yH, and water table level, h„. The location of the critical slip circle in terms of depth factor, D, is also presented for the case when P' 30'. An example calculation is given on the use of the coefficients. H D.H Water table DH H Introduction Bishop and Morgenstern (1960) published the coefficients, m and n, to provide the factor of safety for a circular slip surface in a 'Senior lecturer, Geotechnical Engineering, School of Civil Engineering A Building, Bolton Institute. 1.4 Figure 1:Notation. homogeneous slope assuming the linear relationship, F = m — n. r„, and this method has been used extensively in practice ever since. However, despite its simplicity this method has two major disadvantages. First, it was mainly intended for earth dam design and relies on the pore pressure coefficient, r„, to represent the pore Figure 2: Coefficient a. 1.2 0.8 0.6 0.4— 0.6 — 0.4 0 0.05 0. 10 0 0.15 2 6 GROUND ENGINEERING MAY 1992 L c'/yH
Transcript of Stability coefficients for highway cutting slope design
Stability coefficients for highway cutting slope design
by GEBarnes'bstract
A new method is presented which gives the minimum factor of safetyfor long-term stability of a homogeneous slope of height H using
effective stress parameters in the form F = a + b tang'. The stability
coefficients a and b are presented in tabular and graphical form for
slope inclinations of 1:1, 2:1, 3:1 and 4:1and have been found to berelated to the cohesion soil parameter, c'/yH, and the water tablelevel parameter, hJH. Pore pressures are represented as a quasi
steady state condition by a water table at toe level beyond the toe and
then inclined at various angles within the slope given by the depthbelow crest level, h„. This is considered to be of more practical usethan the arbitrary pore pressure coefficient, r„, which has been usedpreviously in slope stability charts. When the critical circle lies below
the water table the slope is considered 'wet'. The situation when thecritical circle lies above the water table and the slope can beconsidered 'dry'as been determined. This enables the assessmentof whether the presence of a water table is important or not. Thecoefficient a was found to depend on slope inclination, c'/yH, and
whether the slope is 'wet'r 'dry'. The coefficient b was found todepend on slope inclination, c'/yH, and water table level, h„. Thelocation of the critical slip circle in terms of depth factor, D, is also
presented for the case when P' 30'. An example calculation is given
on the use of the coefficients.
H D.H
Water table
DH H
IntroductionBishop and Morgenstern (1960)published the coefficients, m and n,
to provide the factor of safety for a circular slip surface in a
'Senior lecturer, Geotechnical Engineering, School of Civil
Engineering A Building, Bolton Institute.
1.4
Figure 1:Notation.homogeneous slope assuming the linear relationship, F = m —n. r„,and this method has been used extensively in practice ever since.However, despite its simplicity this method has two majordisadvantages. First, it was mainly intended for earth dam design and
relies on the pore pressure coefficient, r„, to represent the pore
Figure 2: Coefficient a.
1.2
0.8
0.6
0.4— 0.6—0.4
0 0.05 0.100
0.15
26GROUND ENGINEERING MAY 1992
L
c'/yH
hv/H C /yH = 0.005 Ci/yH = 0.025 C'/yH = 0.050 C'yH = 0.100 Ci/yH = 0.150 c'/yH = 0.005 c'/yH = 0.025 c'/yH = 0.050 c'yH = 0.100 c lyH = 0.150
a b aI
b a b a b a b a b a b a i b a b
o
0.10
O.20
0.16
0.51
0.71
0.27
0.54
0.73
0.60
0.78
, 0.45
0.67
0.85
0.52
0.74
0.91
0.10
0.20
1.24
1.46
1.31
1.51
i1.38
1.57 1.69 1.78
0 0.88 1.01 1.11 1.27 1.37
025 0.06 0.79 0.22 0.82 0.38 0.87 0.68 0.93 0.97 1.00
0.30 0.87
0.40 1.01
0.90 0.95 1.01 1.08
1.06 1.11 1.17 1.24
0.60 1.29(35) 1.36 1.42
0.50 1.15(35) 1.21 1.27 1.33 1.40
025 1.56 1.60 1.66 1.78 1.87
0.30 1.64 1.69 1.75 1.86 1.95
0.40 '1.81 1.85 1.91 2.02 2.11
0.50 0.06 1.96 0.24 2.00 0.42 2.06 0.75 2.17 1.07 2.26
0.60 2.07 2.13 2.22 2.33 2.41
0.70 1.41(40) 0.70 2.17(45) 2.22 2.31 2.52
0.75
0.80
0.90
1.43(35) 1.54(45) 1.63(45)
1.56(40) 1.65(40)
1.70(30) 0.90 2.38(30) 2.44 2.60 2.70
0.75 2.20(30) 2.27 2.35 2.48 2.57
0.80 2.22(25) 2.30(45) 2.38 2.52 2.62
2.39(25) 2.50(40) 2.67 2.79
ORY 0.08 1.12 0.27 1.21 0.44 1.32 0.75 1.46 1.04 1.56
Where tr is greater than the value shown in brackets treat the slope as 'dry'.
Table 1:Stability coefficients a and b for slope 1:1.
ORY 0.10 2.08 0.30 2.24 0.50 2.37 0.86 2.56 1.20 2.69
Where 9 is greater than the value shown in brackets treat the slope as 'dry'.
Table 2:Stability coefficients a and b for slope 2:1.
hwrH c'/yH = 0.005 c /yH = 0.025~
ci/yH = 0.050
a b a b
c'yH = 0100
a I
c'lyH = 0.150
a b a b a b a b a b a b
c'/yH = 0.005 c /yH = 0.025 c /yH = 0.050 c'yH = 0.100 c'/yH = 0.150
0
0.10 '.870.20 2.11
0.25 241
1.65
1.95
j2.16 2.24
1.95
2.18
2.37
0.10
0.20
0.25
os
2.46 2.54
2.72 2.84
2.94
2.78
2.96
3.07
2.67
3.18
0.30 i 2.31 2.42 2.65
0.40 2.49 2.59 2.71 , 2.81
0.60
0.70
0.75
0.07I
2.66
2.79
,'2.91
2.97
0.25 2.69
, 2.65
3.01
0.441 2.75 0.79 2.87 1.12 2.97
2.92 3.02 3.13
3.14 3.25
3.31
0.80 3.02 3.14
0.90i 3.11
i3.16
1.00 ' 3.25
OR 011 310 0331327 056 341
3.23
3.32
3.25
/
3.36
! 3.45
o.g5 I 3.63
3.36
'.47
3.57
1.31i 3.80
Table 3:Stability coefficients a and b for slope 3:1.
0.70 3.62 3.66 3.70 3.81 3.90
0.75 3.69 3.72 3.76 3.87 3.97
0.80 3.74 3.78 3.82 3.93 4.03
0.90 3.86 3.89 3.94 4.05 4.15
1.00 3.97 4.00 4.05 4.16 4.26
ORY 0.12 4.11 0.37 4.29 0.60 4AS 1.01 4.70 1.38 4.90
Table 4: Stability coefficients a and b for slope 4:1.
0.30 2.93 2.98 3.04 3.16 3.27
0.40 3.13 3.17 3.22 3.34 3.44
0.50 0.07 3.32 0.26 3.35 0.47 3.40 0.84 3.51 1.18 3.61
0.60 3.49 3.52 3.57 3.67 3.77
pressure conditions. This can only be arbitrarily determined since itsrelationship with likely water table levels will be uncertain (Bromhead1986).Secondly, the coefficients are restricted to circles passingthrough the slope with fixed depth factors which does not alwaysproduce the lowest factor of safety. It has been shown (O'onnor andMitchell 1977, Barnes 1991)that the lowest factor of safety is mostoften obtained for toe circles which Bishop and Morgenstern did notconsider as special cases.
It has been found by the author that a linear relationship betweenthe minimum factor of safety, F, and tant))'xists for effective stressanalysis and this has been applied to produce a simple, rapid method ofdetermining F for various slope angles and soil parameters. Thisenables the groundwater conditions to be represented by a morerelevant and understandable water table level and the effect onstability of fluctuations in water table can be readily determined.
Stability coefficientsAdopting the linear relationshipF = a+ btant/(r'1)the stability coefficients, a and b, have been determined as describedbelow. The coefficients a and b are presented in Tables 1-4and
plotted on Figures 2-6. Values of F for c'/yH values and slope anglesintermediate between those given can be obtained with a sufficientdegree of accuracy by linear interpolation of the factor of safetybetween the slope or cot/(I values: see the example page??.
Where the critical circle existed below the water table('wet'ondition)the values of a were found to be very similar for all levels of
h„and have, for simplicity, been combined into one value withminimum loss of accuracy, typically less than 1%.Different values of awere obtained for the 'dry'ondition, when the critical circle liesabove the water table.
For the steeper slopes it has been found that the 'dry'onditionoften applies even when the water table lies above the toe level. For aparticular value ofh /H a lower factor of safety may be obtained forthe 'dry'ondition with higher values of t/(r'o that equation 1 then onlyapplies up to a certain (I(r'alue. These values of ())', above which theslope should be considered as 'dry're given in brackets in Tables 1and 2.
Water tableThe water table surface is represented as a quasi steady statecondition (Mitchell 1983)by an inclined straight line between the toe 27
GROUND ENGINEERING MAY 1992
1.8
1.6
1.4——
2.6
2.6—
1.2~—1.0—--
b0.8
0.6——0.4 ———
0.2
0 0.05c'/yH
Figure3: Coefficient bf 1ors ope1:1.
0.15
1.8
1.6
1.4
1.2-
1.O—0.8/'.6
0 0.05g/ H
0 10
Figure4: Coefficientbf 1ors ope2:1.
0.16
3.8
3.6
3.4—
5.0
4.8
4.6 -—4.4
4.2
2.6
2.4
2 ~ 2
2.0
1.8hJH
3.8
3.6
3.4b
.2
3.0
1.20 0.05 0.10 0.15c'/yH
Figure 5:Coefficient bfor slope 3:1.
andcrestoftheslope. T thhorizontal and beyo d th
e. o e rear of then e toe the water
e crest the water tabl 'is
d th th t bl'easuredat the crest. F
e is representede. e
entirn the water table may lieres . or the situation whe
I W I
urf d thh b l
The assum tie ow crest le
p 'onofnosee a eievel, see Figure l.
p ge is made in the method of analysis,e as the height between th
e ase of each circlM d ease pore water res
thentheFvaluesobt'
s o ained using e uati
8 dth f h
'p s ace
mew atconservativ .ere stimated
GROUND ENGINEERING .
m ive.
MAY 1992
2.8
..8
1.80 0.05
c'/yH0.10 0.15
Figure 6:Coefficient b for slope 4:1.
Method
The slope and water table condi'n
Figure l.U'h l 'icon 'tions used in the an
pgas en determin
e minimumo gl
e coe cients a and b. The 'ee analyses were carried
0.5Note: critical circles
I
pass through toe
0 025
0.05
Figure 9:Location ofcriticalslip circle —slope 1:1.
0.9
1.0
0 10
0 150
1.10 0.4 0.6 0.8h+H
1.0 1.2
Figure 10:Location ofcritical slip circle —slope2:1.
0.6
0.7
0.8
(0.9
1.1
CIr'VH
gtL i
1.2
1.3 0.4 0.6 0.8 1.0 1.2hJH
Figure 11:Location ofcritical slip circle —slope3:1.
0.8
O.S
1.0
C
1.5
Figure 12:Location ofcri ti cal slip circle —slope4:1.
' 0.2 0.4 0.6 0.8 1.0 1.2 1.4
h+H
O.S
1.0
1.6
1.7—
30GROUND ENGINEERING MAY 1992
0.2 0.4 0.6 0.8 1.0h+H
1.2 1.4 1.6
presence of a particular water table level needs to be considered ornot. It was found in most cases that a critical circle passing throughthe toe was obtained for the dry condition.
When the water table lies close to this critical level the location ofthe slip surface can be rather unstable, rising rapidly as the watertable drops from just above its critical location (wet condition) to justbelow (dry condition). This is illustrated on Figures 9-12.
Location ofcritical slip circle
The depth factor, D, of the critical slip surface was determined in eachcase in relation to slope height, H. Where the critical circle lies belowthe toe of the slope this can be simply represented by the lowesthorizontal tangent level of the circle. Where the critical circle existedabove toe level its location was denoted by the dimension DH,measured at the crest, representing the lowest inclined tangent levelof the critical circle, as shown on Figure 1.Values of D for il)'
30're
plotted on Figures 9-12.For P' 45'he depth factor issomewhat smaller and for (fi' 15't is larger.
The depth factor should be used to check whether a rigid stratumexists within this depth. A rigid stratum would then produce ashallower slip circle in which case a higher factor of safety than givenby this method could be obtained.
Effect ofdensity
A single value of unit weight y = 20kN/m was used throughout withno difference in values above and below the water table. A checkcarried out on the effect of unit weight showed that the factor of safetyvaried by less than + 5% within the range of values between 18kN/msand 22kN/m . Therefore, for most soil types the present methodgives sufficient accuracy, within this range ofunit weights. Evenoutside this range the effect was small, typically less than + 5% forthe shallower slopes (4:1)but the effect of unit weight was larger forthe steeper slopes. An exception was found to occur with a low
c'alue(1kN/m ) and a very high water table (h„/H = 0) when the effectwas more significant.
Example
For 3:1 slope —use Table 3
for c'/yH = 0.025 a = 0.25 b = 2.69 F = 0.25 + 2.69tan30' 1.80
fore'/yH = 0.050a = 0.44b =2.75F= 0.44+ 2.75tan30' 2.03
Therefore, for c'/yH = 0.03
F = 1.80+ 0.23(0.03—0.025) = 1.850.025
For the slope of 2.5:1
F = 1.44+ 1.85= 1.652
From charts
Fore'/yH = 0.03
slope 2:1
a = 0.28 (Fig 2) b = 2.02 (Fig 4) F = 0.28+ 2.02tan30 = 1.45
slope 3:1
a = 0.29 (Fig 2) b = 2.70 (Fig 5) F = 0.29 + 2.70tan30' 1.85
Therefore, for slope 2.5:1
F= 1.45+1.85= 1.652
To determine the depth factor, interpolate between Figures 10and11,D is approximately 1.2.This is above the bedrock surface so thecritical slip circle lies wholly within the soil. A computer analysis of thesame data gave F = 1.65and D = 1.19.
An example calculation is given for a slope of cot(tJ = 2.5 (slope 2.5:1),height 15m, with a water table 7.5m below crest level and bedrock at5m below toe level. The soil is a clay with properties of c' 9kN/m,9)' 30"and y = 20kN/ms.
c'/yH = 9 = 0.0320x15
h„/H = 0.5 ..slope 'wet'(from Figures 7and 8)
From tables
For2:1slope-use Table2
fore'/yH = 0.025a = 0.24b = 2.00F = 0.24+ 2.00tan30' 1.40
for c'/yH = 0.050a = 0.42 b = 2.06 F = 0.42 + 2.06tan30' 1.61
Therefore, for c'/yH = 0.03F = 1.40+0.21(0.03—0.025) = 1.44
0.025
ReferencesBarnes GE (1991).'A simplified version of the Bishop and Morgenstern slope stabilitycharts.'anadian Geotechnical journal, 28, 630-637.Bishop AW and Morgenstern NR (1960). 'Stability coefficients for earth slopes'.Geotechnirfuevol. 10, 129-150.Bromhead EN (1986). 'The stability of slopes'. Surrey University Press. Blackie andSons, UK.Chandler RJ and Peiris TA (1989). 'Further extensions to the Bishop and Morgensternslope stability charts'. Ground Engineering Vol. 22, No.4, 33-38.Mitchell RJ (1983).'Earth structures engineering'. Allen and Unwin, USA.O'onnor MJ and Mitchell RJ (1977). 'An extension of the Bishop and Morgenstern slopestabiTity charts.'anadian Geotechnicaf Journal, 14, 144-151.
GROUND ENGINEERING MAY 1992
