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Comparison with hand calculations
1
Introduction
In the textbook, Soil Mechanics by Lambe and Whitman (1969), the authors present a hand-calculated
factor of safety for a simple slope with an underdrain. The purpose of this example is to verify SLOPE/W
by comparing its solution with the hand calculations. Features of this simulation include:
2
•
Analysis method: Bishop
•
Use of a piezometric line
•
Use of a single point Grid and Radius slip surface
•
Use of points on regions to control slice discretization
Configuration and set-up
Figure 1 illustrates a simple slope with an underdrain used by Lambe and Whitman (1969). The slope is
20 feet high, with a slope of 1 vertical to 1.5 horizontal. The material of the slope is homogenous with c =
90 psf, φ = 32 and γ = 125 pcf. The slip surface is assumed to be circular, with a radius of 30 feet from the
center, and the pore-water pressure conditions for the slope are characterized by a flow net.
Center of failure circle
Radius of circle = 30 ft
7
γ = 125 lb/ft
c = 90 lb/ft
6A
6
φ = 32
5
1.5
1
20 ft
4
3
2A
2
1
Drain
Surface of firm stratum
Figure 1 Stability of slope with an underdrain (after Lambe and Whitman)
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Lambe and Whitman’s Hand Calculations
Lambe and Whitman divide the entire sliding mass into nine slices with each slice width, average height
and weight calculated as shown in Table 1. The total weight of the sliding mass is about 26,500 lbs.
Table 1 Lambe and Whitman weight computations
Slice
Width (ft)
Average Height (ft)
Weight (kips)
1
4.5
1.6
0.9
2
3.2
4.2
1.7
2A
1.8
5.8
1.3
3
5.0
7.4
4.6
4
5.0
9.0
5.6
5
5.0
9.3
5.8
6
4.4
8.4
4.6
6A
0.6
6.7
0.5
7
3.2
3.8
1.5
W=2.65
Table 2 presents Lambe and Whitman’s calculation for determining the Ordinary factor of safety. The
hand-calculated factor of safety is 1.19.
Table 2 Lambe and Whitman calculation of the Ordinary factor of safety
cosθi
Wi cosθi
(kips)
ui
(kips/ft)
Δl
(ft)
ui
(kips)
Ni
(kips)
0
1.00
0.9
0
4.4
0
0.9
0.1
1.00
1.7
0
3.2
0
0.1
0.2
1.99
1.3
0.03
1.9
0.05
0.25
0.25
1.2
1.97
4.5
0.21
5.3
1.1
3.4
0.42
2.3
1.91
5.1
0.29
5.6
1.6
3.5
5.8
0.58
3.4
1.81
4.7
0.25
6.2
1.55
3.15
Slice
Wi
(kips)
sinθi
1
0.9
-0.03
2
1.7
0.05
2A
1.3
0.14
3
4.6
4
5.6
5
Wi sinθi
(kips)
6
4.6
0.74
3.4
1.67
3.1
0.11
6.7
0.7
2.4
6A
0.5
0.82
0.4
1.57
0.3
0
1.2
0
0.3
7
1.5
0.87
1.3
1.49
0.7
0
1.3
0
12.3
F=
0.09 ( 41.8 ) + 17.3 tan32o
12.3
=
0.7
41.8
17.3
3.76 + 10.82 14.58
=
= 1.19
12.3
12.3
Lambe and Whitman also compute the Bishop’s Simplified factor of safety using a trial and error
approach. The computations and results are presented in Table 3.
Table 3 Lambe and Whitman calculation of the Bishop Simplified factor of safety
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
Slice
Δx
cΔxi
uiΔxi
(5)tan φ
(3)+(6)
Mi
(7) + (8)
(ft)
(kips)
(kips)
Wi uiΔxi
(kips)
(kips)
(kips)
F=
1.25
F=
1.35
1
4.5
0.40
0
0.9
055
0.95
0.97
0.97
1.0
1.0
2
3.2
0.29
0
1.7
1.05
1.35
1.02
1.02
1.3
1.3
SLOPE/W Example File: Comparison with hand calculations.doc (pdf) (gsz)
F=
1.25
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F=
1.35
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2A
1.8
0.16
0.05
1.25
1.80
1.95
1.06
1.05
0.9
0.9
3
5.0
0.45
1.05
3.55
2.25
2.70
1.09
1.08
2.5
2.5
2.75
4
5.0
0.45
1.45
4.15
2.55
3.00
1.12
1.10
2.7
5
5.0
0.45
1.25
4.55
2.7
3.15
1.10
1.08
3.85
2.9
6
4.4
0.40
0.50
4.1
2.63
3.05
1.05
1.02
2.9
2.95
6a
0.6
0.05
0
0.5
2.30
0.35
0.98
0.95
0.35
0.4
7
3.2
0.29
0
1.5
2.95
1.25
0.93
0.92
F = 1.25
F=
15.8
= 1.29
12.3
F = 1.35
F=
16.05
= 1.31
12.3
1.3
1.35
15.8
16.05
As shown in the above calculations, a trial factor of safety of 1.25 results in a computed factor of safety of
1.29, and a trial factor of safety of 1.35 results in a computed value of 1.31. Since the trial value of 1.25 is
too low and the trial value of 1.35 is too high, the correct value using the Bishop Simplified method is
between 1.25 and 1.35.
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SLOPE/W Solutions
The same problem is analyzed using SLOPE/W. Figure 2 shows the same slope as modeled by
SLOPE/W. The single circular slip surface is modeled with a single grid point and a single radius point.
Some extra points along the lower region boundary are added to control the slice discretization.
SLOPE/W computes the factor of safety to be 1.208 using the Ordinary method and 1.344 using the
Bishop method.
Nine slices are also used in the analysis. The computed slice width, average height and weight of the
sliding mass are tabulated in Error! Reference source not found.. The total weight of the sliding mass is
26,040 lbs. The slices modeled by SLOPE/W are very similar, but not exactly the same as those used by
Lambe and Whitman.
In SLOPE/W, a straight line is assumed at the base of a slice and the vertical height of a slice is computed
from the ground surface to the base center on the straight line. As a result, the total sliding mass is
slightly less than the total sliding mass used by Lambe and Whitman.
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Figure 2 Factor of safety for the Ordinary method using SLOPE/W
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Figure 3 Factor of safety for the Simplified Bishop method using SLOPE/W
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Conclusion
SLOPE/W gives essentially the same factor of safety as the hand calculated solution by Lambe and
Whitman. The small difference is due to the difference in the way the slice weight is computed. This
simple example confirms that SLOPE/W is formulated correctly.
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