slope spencer formulated
DESCRIPTION
geotechTRANSCRIPT
Spencer_forSARA2003
l¢ F DumEq0.105 1.287 1.287
### ###
W H R27 5 1.73 10 15 61.518 0.2 4.91 7.95 13.40 -5.89 17.467
# x c f W u l P T E l0 17.47 3.27 ### 15 0.000
1 16.58 1.37 ### ### ### 30 ### 1.136 ### 2.10 ### ### ### 0.012 ### 12.12
2 15.70 0.00 ### ### ### 30 ### 0.997 ### 1.63 ### ### ### 0.025 ### 11.24
3 14.72 -1.18 ### ### ### 0 ### 0.879 ### 1.54 ### ### ### 0.038 ### 10.30
4 13.74 -2.14 ### ### ### 0 ### 0.770 ### 1.37 ### ### ### 0.050 ### 9.32
5 12.76 -2.92 ### ### ### 0 ### 0.673 ### 1.25 ### ### ### 0.062 ### 8.34
6 11.78 -3.56 ### ### ### 0 ### 0.582 ### 1.17 ### ### ### 0.072 ### 7.36
7 10.79 -4.10 ### ### ### 0 ### 0.496 ### 1.12 ### ### ### 0.082 ### 6.38
8 9.81 -4.53 ### ### ### 0 ### 0.415 ### 1.07 ### ### ### 0.090 ### 5.40
9 8.83 -4.87 ### ### ### 0 ### 0.336 ### 1.04 ### ### ### 0.096 ### 4.42
10 7.85 -5.13 ### ### ### 0 ### 0.259 ### 1.02 ### ### ### 0.101 ### 3.43
11 6.87 -5.31 ### ### ### 0 ### 0.184 ### 1.00 ### ### ### 0.103 ### 2.45
12 5.89 -5.42 ### ### ### 0 ### 0.110 ### 0.99 ### ### ### 0.105 ### 1.47
13 4.91 -5.46 ### ### ### 0 ### 0.037 ### 0.98 ### ### ### 0.104 ### 0.49
14 3.93 -5.42 ### ### ### 0 ### -0.037 ### 0.98 ### ### ### 0.101 ### -0.49
15 2.94 -5.31 ### ### ### 0 ### -0.110 ### 0.99 ### ### ### 0.097 ### -1.47
16 1.96 -5.13 ### ### ### 0 ### -0.184 ### 1.00 ### ### ### 0.091 ### -2.45
17 0.98 -4.87 ### ### ### 0 ### -0.259 ### 1.02 ### ### ### 0.083 ### -3.43
18 0.00 -4.53 ### ### ### 0 ### -0.336 ### 1.04 ### ### ### 0.074 ### -4.42
19 -0.98 -4.10 ### ### ### 0 ### -0.415 ### 1.07 ### ### ### 0.064 ### -5.40
20 -1.96 -3.56 ### ### ### 0 ### -0.496 ### 1.12 ### ### ### 0.053 ### -6.38
21 -2.94 -2.92 ### ### ### 0 ### -0.582 ### 1.17 ### ### ### 0.040 ### -7.36
22 -3.93 -2.14 ### ### ### 0 ### -0.673 ### 1.25 ### ### ### 0.027 ### -8.34
23 -4.91 -1.18 ### ### ### 0 ### -0.770 ### 1.37 ### ### ### 0.014 ### -9.32
24 -5.89 0.00 ### ### ### 0 ### -0.879 ### 1.54 ### ### ### 0.000 ### -10.30
Undrained shear strength profile of soft clay Embankmentdepth 0 1.5 3 5 7 10 (m)
cu 40 28 20 20 26 37 (kPa) 10 30
(kPa)
16
SM SForces Varying l
hc gw Pw Mw ru xc yc xmin xmax
ybot ytop gave arad s ' Lx
cm fm
(o)
gclay
(kN/m3)
framed cells contain equationsframed cells contain equations
-10 -5 0 5 10 15 20
-10
-5
0
5
10
Units: m, kN/m3, kN/m2, kN, or other consistent set of units.
Array formulas
slope angle
Soft clay
Embankment
λ=λ ' sin[ (x i−x0 )( xn− x0 )
π ]
l
x0xn
Center of Rotation
xn
x0
0 25 500
2
4
6
8
10
cu (kPa)
De
pth
SPENCER method
Varying side-force angle l
Ei
liEi
li-1Ei-1
Ei-1
Ti
Pi
Wi
ai
lybot
ytop
W
A B C D E F G H I J K L M N O P Q1
23
4
567
8
9
10
11
1213
14
15
16
1718192021222324252627282930313233343536373839404142
43
44
45
464748
49
50
Spencer_forSARA2003
0 25 500
2
4
6
8
10
cu (kPa)
De
pth
A B C D E F G H I J K L M N O P Q51
Spencer_forSARA2003
1
5.63
7.26
8.54
9.61
10.47
11.19
11.78
12.26
12.64
12.95
13.17
13.31
13.39
13.39
13.31
13.17
12.95
12.64
12.26
11.78
11.19
10.47
9.61
8.54
Embankment
20
Varying l
Ly
gm
(kN/m3)
λ=λ ' sin[ (x i−x0 )( xn− x0 )
π ]x0
Varying side-force angle l
R1
23
4
567
8
9
10
11
1213
14
15
16
1718192021222324252627282930313233343536373839404142
43
44
45
464748
49
50
from Spencer_forSARA2003
l¢ F DumEq
0.103 1.253 1.253
### ### 1
W H27 5 1.73 10 15 62.321 0.2 8.00 8.00
Lever arms
# x c f W u l P T E l0 18.07 3.27 ### 15.00 0.000
1 16.88 1.68 ### ### ### 30 ### 0.931 ### 1.98 ### ### ### 0.016 ### 9.47 5.53
2 15.70 0.00 ### ### ### 30 ### 0.958 ### 2.06 ### ### ### 0.031 ### 8.29 7.16
3 14.72 -0.95 ### ### ### 0 ### 0.767 ### 1.36 ### ### ### 0.044 ### 7.21 8.47
4 13.74 -1.84 ### ### ### 0 ### 0.738 ### 1.33 ### ### ### 0.055 ### 6.23 9.39
5 12.76 -2.63 ### ### ### 0 ### 0.682 ### 1.26 ### ### ### 0.066 ### 5.25 10.24
6 11.78 -3.37 ### ### ### 0 ### 0.641 ### 1.22 ### ### ### 0.076 ### 4.27 11.00
7 10.79 -4.01 ### ### ### 0 ### 0.583 ### 1.18 ### ### ### 0.084 ### 3.29 11.69
8 9.81 -4.63 ### ### ### 0 ### 0.562 ### 1.16 ### ### ### 0.091 ### 2.30 12.32
9 8.83 -5.15 ### ### ### 0 ### 0.484 ### 1.11 ### ### ### 0.097 ### 1.32 12.89
10 7.85 -5.52 ### ### ### 0 ### 0.360 ### 1.05 ### ### ### 0.100 ### 0.34 13.33
11 6.87 -5.78 ### ### ### 0 ### 0.261 ### 1.02 ### ### ### 0.103 ### -0.64 13.65
12 5.89 -5.94 ### ### ### 0 ### 0.168 ### 1.00 ### ### ### 0.103 ### -1.62 13.86
13 4.91 -6.02 ### ### ### 0 ### 0.076 ### 0.98 ### ### ### 0.102 ### -2.60 13.98
14 3.93 -6.00 ### ### ### 0 ### -0.019 ### 0.98 ### ### ### 0.099 ### -3.58 14.01
15 2.94 -5.88 ### ### ### 0 ### -0.121 ### 0.99 ### ### ### 0.094 ### -4.57 13.94
16 1.96 -5.65 ### ### ### 0 ### -0.233 ### 1.01 ### ### ### 0.088 ### -5.55 13.76
17 0.98 -5.28 ### ### ### 0 ### -0.360 ### 1.05 ### ### ### 0.081 ### -6.53 13.46
18 0.00 -4.73 ### ### ### 0 ### -0.507 ### 1.12 ### ### ### 0.072 ### -7.51 13.00
19 -0.98 -4.08 ### ### ### 0 ### -0.587 ### 1.18 ### ### ### 0.062 ### -8.49 12.41
20 -1.96 -3.39 ### ### ### 0 ### -0.615 ### 1.20 ### ### ### 0.051 ### -9.47 11.73
21 -2.94 -2.61 ### ### ### 0 ### -0.668 ### 1.25 ### ### ### 0.039 ### -10.45 11.00
22 -3.93 -1.81 ### ### ### 0 ### -0.684 ### 1.27 ### ### ### 0.026 ### -11.43 10.21
23 -4.91 -0.93 ### ### ### 0 ### -0.734 ### 1.32 ### ### ### 0.013 ### -12.42 9.37
24 -5.89 0.00 ### ### ### 0 ### -0.757 ### 1.35 ### ### ### 0.000 ### -13.40 8.46
Undrained shear strength profile of soft clay Embankmentdepth 0 1.5 3 5 7 10 (m)
cu 40 28 20 20 26 37 (kPa) 10 30 20
(kPa)
16
SForces SM Varying l
hc gw Pw Mw ru xc yc
yb yt gave arad s ' Lx Ly
cm fm gm
(o) (kN/m3)gclay
(kN/m3)
framed cells contain equationsframed cells contain equations
Units: m, kN/m3, kN/m2, kN, or other consistent set of units.
Array formulas
slope angle
Center of Rotation
Ei
liEi
li-1Ei-1Ei-1
Ti
Pi
Wi
ai
l
-20 -15 -10 -5 0 5 10 15 20
-10
-5
0
5
10
Soft clay
Embankmentxn
x0
yb
ytyt
initial
critical
0 25 50
0
2
4
6
8
10
cu (kPa)
de
pth
SPENCER: Search for noncircular slip surface
l
x0xn
λ=λ ' sin[ (xi−x0 )( xn− x0 )
π ]
Varying side-force angle l
from Spencer_forSARA2003
0 25 50
0
2
4
6
8
10
cu (kPa)
de
pth
from Spencer_forSARA2003
Ka0.33
hc1.73205
xmin xmax-18.12 15.37 -2.944 17.028
xc yc R -15.180 -1.661
-2 8 18.00
xi ybi ytiEquilCheck 15.37 3.27 #VALUE!
### #VALUE! 14.75 1.40 #VALUE!
### #VALUE! 14.12 0.00 #VALUE!
### #VALUE! 12.66 -2.45 #VALUE!
### #VALUE! 11.19 -4.25 #VALUE!
### #VALUE! 9.73 -5.66 #VALUE!
### #VALUE! 8.26 -6.79 #VALUE!
### #VALUE! 6.80 -7.70 #VALUE!
### #VALUE! 5.33 -8.44 #VALUE!
### #VALUE! 3.86 -9.02 #VALUE!
### #VALUE! 2.40 -9.45 #VALUE!
### #VALUE! 0.93 -9.76 #VALUE!
### #VALUE! -0.53 -9.94 #VALUE!
### #VALUE! -2.00 -10.00 #VALUE!
### #VALUE! -3.47 -9.94 #VALUE!
### #VALUE! -4.93 -9.76 #VALUE!
### #VALUE! -6.40 -9.45 #VALUE!
### #VALUE! -7.86 -9.02 #VALUE!
### #VALUE! -9.33 -8.44 #VALUE!
### #VALUE! -10.80 -7.70 #VALUE!
### #VALUE! -12.26 -6.79 #VALUE!
### #VALUE! -13.73 -5.66 #VALUE!
### #VALUE! -15.19 -4.25 #VALUE!
### #VALUE! -16.66 -2.45 #VALUE!
### #VALUE! -18.12 0.00 #VALUE!
Read the Low (2003) paper "Practical Probabilistic Slope Stability Analysis", downloadable from http://alum.mit.edu/www/bklow.For hands-on:(1) Click the button "InitialCircle"(2) Invoke Solver (Menu bar: Tools/Solver); click Solve. Then click OK to accept the solution found. May run Solver one more time.
from Spencer_forSARA2003
l¢ F DumEq
0.098 1.000 1.000
### ### 1
W H27 5 1.78 10 15.869 66.455 0.2 8.00 8.00
Lever arms
# x c f W u l P T E l0 17.22 3.22 ### 15.87 0.000
1 16.19 1.66 ### ### ### 30.949 ### 0.990 ### 1.87 ### ### ### 0.015 ### 8.71 5.56
2 15.17 0.00 ### ### ### 30.949 ### 1.016 ### 1.95 ### ### ### 0.029 ### 7.68 7.17
3 14.28 -0.86 ### ### ### 0 ### 0.769 ### 1.24 ### ### ### 0.041 ### 6.72 8.43
4 13.38 -1.69 ### ### ### 0 ### 0.744 ### 1.21 ### ### ### 0.052 ### 5.83 9.27
5 12.49 -2.43 ### ### ### 0 ### 0.697 ### 1.16 ### ### ### 0.062 ### 4.94 10.06
6 11.60 -3.11 ### ### ### 0 ### 0.648 ### 1.12 ### ### ### 0.071 ### 4.04 10.77
7 10.71 -3.72 ### ### ### 0 ### 0.600 ### 1.08 ### ### ### 0.079 ### 3.15 11.41
8 9.81 -4.30 ### ### ### 0 ### 0.578 ### 1.07 ### ### ### 0.086 ### 2.26 12.01
9 8.92 -4.79 ### ### ### 0 ### 0.505 ### 1.02 ### ### ### 0.091 ### 1.37 12.55
10 8.03 -5.14 ### ### ### 0 ### 0.370 ### 0.96 ### ### ### 0.095 ### 0.47 12.97
11 7.14 -5.37 ### ### ### 0 ### 0.256 ### 0.92 ### ### ### 0.097 ### -0.42 13.26
12 6.24 -5.53 ### ### ### 0 ### 0.174 ### 0.91 ### ### ### 0.098 ### -1.31 13.45
13 5.35 -5.62 ### ### ### 0 ### 0.097 ### 0.90 ### ### ### 0.097 ### -2.20 13.57
14 4.46 -5.63 ### ### ### 0 ### 0.016 ### 0.89 ### ### ### 0.094 ### -3.10 13.63
15 3.57 -5.57 ### ### ### 0 ### -0.069 ### 0.89 ### ### ### 0.090 ### -3.99 13.60
16 2.67 -5.43 ### ### ### 0 ### -0.162 ### 0.90 ### ### ### 0.084 ### -4.88 13.50
17 1.78 -5.18 ### ### ### 0 ### -0.268 ### 0.93 ### ### ### 0.077 ### -5.77 13.30
18 0.89 -4.78 ### ### ### 0 ### -0.424 ### 0.98 ### ### ### 0.069 ### -6.66 12.98
19 0.00 -4.17 ### ### ### 0 ### -0.600 ### 1.08 ### ### ### 0.059 ### -7.56 12.47
20 -0.90 -3.41 ### ### ### 0 ### -0.703 ### 1.17 ### ### ### 0.048 ### -8.45 11.79
21 -1.79 -2.59 ### ### ### 0 ### -0.744 ### 1.21 ### ### ### 0.037 ### -9.34 11.00
22 -2.68 -1.79 ### ### ### 0 ### -0.732 ### 1.20 ### ### ### 0.025 ### -10.23 10.19
23 -3.57 -0.91 ### ### ### 0 ### -0.777 ### 1.25 ### ### ### 0.013 ### -11.13 9.35
24 -4.47 0.00 ### ### ### 0 ### -0.796 ### 1.28 ### ### ### 0.000 ### -12.02 8.46
Undrained shear strength profile of soft clay Embankmentdepth 0 1.5 3 5 7 10 (m)
cu 34.07 22.74 15.91 15.85 22.46 34.82 (kPa) 10.47 30.95 20.76
(kPa)
16
SForces SM Varying l
hc gw Pw Mw ru xc yc
yb yt gave arad s ' Lx Ly
cm fm gm
(o) (kN/m3)gclay
(kN/m3)
framed cells contain equationsframed cells contain equations
Units: m, kN/m3, kN/m2, kN, or other consistent set of units.
Array formulas
slope angle
Center of Rotation
Ei
liEi
li-1Ei-1Ei-1
Ti
Pi
Wi
ai
l
-10 -5 0 5 10 15 20 25
-10
-5
0
5
10
Soft clay
Embankmentxn
x0
yb
yt
yt
initial
critical
0 25 50
0
2
4
6
8
10
cu (kPa)
de
pth
SPENCER: Search for noncircular slip surface
l
x0xn
λ=λ ' sin[ (xi−x0 )( xn− x0 )
π ]
Varying side-force angle l
from Spencer_forSARA2003
Ka hc
0.32 1.78151
mean StDev
DistName s
lognormal 10.472107 10 1.50 ### ###
lognormal 30.948505 30 3.00 ### ###
lognormal 20.757609 20 1.00 ### ###
lognormal 34.072981 40 6.00 ### ###
lognormal 22.735506 28 4.20 ### ###
lognormal 15.906807 20 3.00 ### ###
lognormal 15.852003 20 3.00 ### ###
lognormal 22.463107 26 3.90 ### ###
lognormal 34.81741 37 5.55 ### ###
EquilCheck Array formula: Ctrl+Shift, then Enter
### #VALUE!
### #VALUE!
### #VALUE!
### #VALUE!
### #VALUE! DeterministicCritical Normals, reliability-based critical### #VALUE! x yb yt x yb yt
### #VALUE! 18.07 3.27 5.00 17.2543 3.2144 5.0000
### #VALUE! 16.88 1.68 5.00 16.2395 1.6480 5.0000
### #VALUE! 15.70 0.00 5.00 15.2246 0.0000 5.0000
### #VALUE! 14.72 -0.95 5.00 14.3227 -0.8758 5.0000
### #VALUE! 13.74 -1.84 5.00 13.4208 -1.7092 5.0000
### #VALUE! 12.76 -2.63 5.00 12.5188 -2.4603 5.0000
### #VALUE! 11.78 -3.37 5.00 11.6169 -3.1408 5.0000
### #VALUE! 10.79 -4.01 5.00 10.7150 -3.7526 5.0000
### #VALUE! 9.81 -4.63 5.00 9.8131 -4.3359 5.0000
### #VALUE! 8.83 -5.15 4.50 8.9111 -4.8270 4.5404
### #VALUE! 7.85 -5.52 4.00 8.0092 -5.1699 4.0809
### #VALUE! 6.87 -5.78 3.50 7.1073 -5.3990 3.6213
### #VALUE! 5.89 -5.94 3.00 6.2054 -5.5524 3.1618
### #VALUE! 4.91 -6.02 2.50 5.3034 -5.6343 2.7022
### #VALUE! 3.93 -6.00 2.00 4.4015 -5.6452 2.2427
### #VALUE! 2.94 -5.88 1.50 3.4996 -5.5803 1.7831
### #VALUE! 1.96 -5.65 1.00 2.5976 -5.4312 1.3236
### #VALUE! 0.98 -5.28 0.50 1.6957 -5.1821 0.86400.00 -4.73 0.00 0.7938 -4.7721 0.4045-0.98 -4.08 0.00 -0.1081 -4.1579 0.0000
-1.96 -3.39 0.00 -1.0101 -3.4079 0.0000
-2.94 -2.61 0.00 -1.9120 -2.5921 0.0000
-3.93 -1.81 0.00 -2.8139 -1.7930 0.0000
-4.91 -0.93 0.00 -3.7158 -0.9136 0.0000
-5.89 0.00 0.00 -4.6178 0.0000 0.0000
vi mi mN sN
cm
fm
gm
cu1
cu2
cu3
cu4
cu5
cu6
=SQRT(MMULT(TRANSPOSE(vn),MMULT(MINVERSE(crmat),vn)))
EqvN(…, 1) EqvN(…, 2)
Read the Low (2003) paper "Practical Probabilistic Slope Stability Analysis", downloadable from http://alum.mit.edu/www/bklow.
For hands-on: Read the section in paper related to Figure 6.
from Spencer_forSARA2003
vn
#VALUE! 1 -0.3 0.5 0 0 0 0 0 0
#VALUE! -0.3 1 0.5 0 0 0 0 0 0
#VALUE! 0.5 0.5 1 0 0 0 0 0 0
#VALUE! 0 0 0 1 0.6065 0.3679 0.1889 0.097 0.0357 0
#VALUE! 0 0 0 0.6065 1 0.6065 0.3114 0.1599 0.0588 1.5
#VALUE! 0 0 0 0.3679 0.6065 1 0.5134 0.2636 0.097 3
#VALUE! 0 0 0 0.1889 0.3114 0.5134 1 0.5134 0.1889 5
#VALUE! 0 0 0 0.097 0.1599 0.2636 0.5134 1 0.3679 7
#VALUE! 0 0 0 0.0357 0.0588 0.097 0.1889 0.3679 1 10
0 1.5 3 5 7 10 dv
b PrFail
Array formula: Ctrl+Shift, then Enter ### ###
LogNormals, reliability-based critical
17.22 3.218 5
16.19 1.655 5
15.17 0 5
14.28 -0.86 5
13.38 -1.69 5
12.49 -2.43 5
11.6 -3.11 5
10.71 -3.72 5
9.813 -4.3 5
8.921 -4.79 4.545
8.028 -5.14 4.091
7.136 -5.37 3.636
6.243 -5.53 3.181
5.351 -5.62 2.726
4.458 -5.63 2.272
3.566 -5.57 1.817
2.674 -5.43 1.3621.781 -5.18 0.908
0.889 -4.78 0.453
0 -4.17 0
-0.9 -3.41 0
-1.79 -2.59 0
-2.68 -1.79 0
-3.57 -0.91 0
-4.47 ### 0
Correlation matrix, "crmat"
(v-mN)/sN cm fm gm cu1 cu2 cu3 cu4 cu5 cu6
-10 -5 0 5 10 15 20
-8
-6
-4
-2
0
2
4
6
=SQRT(MMULT(TRANSPOSE(vn),MMULT(MINVERSE(crmat),vn)))
Equation 9 autofilled
Probability of failure
EqvN(…, 2)
deterministic critical
=normsdist(-b)
Case1
Page 10
2 12
Case1
Page 11
6 21 24
Case1
Page 12
1.940582171
Case2
Page 13
2 14
Case2
Page 14
9 19 24
Case2
Page 15
1.230582584