soil mechan - dot.ny.gov...in the field knowing the angle of internal friction of the soil, the...
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STATE OF NEW YORK DEPARTMENT OF TRANSPORTATION
SM 238b (1/73)
SOIL MECHAN
INFINITE SLOPE ANALYSIS
February, 1972
7 .41-6, SEM 3/72
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INFINITE SLOPE ANALYSIS
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TABLE OF CONTENTS
Introduction 2
Derivation of Equation 3
Typical Section (Fig . 1) 5
Force Diagram (Fig . 2) 5
Infinite Slope Curves 6
Variance of SF with Total Unit Weight 20
Appendix A The Computer Program 21
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IN'FRQDU(--:['I ON
The infinite slope analysis for cut slopes is designed to allow the engineer to approximate the factor of safety existing in the field knowing the angle of internal friction of the soil, the existing slope angle, the depth to the actual or potential failure plane and the depth to the water table .
The Hl/Z ratio is the ratio of the depth to water divided by the depth to the actual or potential failure plane (Fig . 1) and ra.nges from 0 for the water table at the original ground surface to 1 .0 for the water table existing at the failure plane .
This analysis assumes that the original ground surface and the water table are parallel and that the ratio of depth to length of the failure is very small, therefore eliminating the active and passive wedges .
The analysis and resulting stability curves was based on a total unit weight of 125 pcf . Stability curves are also presented in Fig. 3 showing the variance of safety factor with changes in total unit weight .
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From Fig . 2
Overturning
1)
OT -_ Weight and seepage
2)
F -_ W sin(a) (bbl CT + bh2Gb ) sin (a) = b sin (a) (hIGT + h2GT
3)
Seepage = i Gwbh2 = sin(a) bh2Gw
OT = bsin(a)h2Gw +- bsin(a)(hiGT + h2GT - h2Gw)
= bsin(a) (h2Gw + h1GT - ;- h2GT - h2Gw)
= bsin(a) (h1GT + h2GT)
4)
Resisting Forces = N tan
N - W cos (a) = b cos (a) (hlGT + h2GT - h2Gw)
R = tan 0 b cos (a) (hlGT + h2GT - h2Gw)
5)
SF = tan 0 b cos (a) (hlGT + h2GT - h2Gw)
DERIVATION OF THE EQUATION
bsin (a) (hlGT + h2GT)
= tan 0 IhlGT + h2GT - h2Gw
tan(a) ~ hIGT + h2GT
- tan 0
hlGT + h2GT - h2Gw
tan(a) ( hlGT + h2GT h 1GT + h2GT
tan 0
h2Gw 1 -
tan(a) GTZ v
- tan 0
l
(Z - hl) Gw
tan(a)
GT Z
FS = Resisting
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Definition of Terms
GT = total unit wt .
Gw - unit wt . of water
Gb - bouyant unit weight
Z
hl }- h2
a = slope angle
For this analysis GT -= 125 was used .
tan 0 1ihlGw Gw
tan a)
= tan
CT Z
(
1+hlGw
GT
Gw
tan(a) Z GT GT
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CROSS - SECTION FIG . I
FORCE DIAGRAM FIG, 2
~T = 125 PCF
= 62.4 PC F
h, t h2 = Z
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u
W
d
2.0
1 :1 .5
1 :2
1 :2.5
1 :3
1 :3.5 1 :4 1 :4 .5 1 :5
0
6 z2 I
14
SLOPE ANGLE AND HYDRAULIC GRADIENT
.
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a 0 V 4
H W
4 N
2 .0
1 :2.5 1'.3 1 :3.5 1 :4 1 :4.51 :5
0
6 2'2 113 14
sl nPF ANGLE AND HYDRAULIC GRADIENT
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0 U d
f-W a
1 :1 .5 2 .3f
I : -2 1'.2 .5 1'.3 1 :3.5 1 :4 1 :4.51:5
-- j ~ p 6 22
14
ENT ANGLE AND.-HYDRAULIC GRADINT
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0
2.4
1 :2 .5
1 :3 1 :3.5 1 :4 1 :4.51 :5
O b e_2 I SLOPE ANGLE AND HYDRAULIC GRADIENT
4
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a
1 :1 .5
1 :2
1 :2.5
1 :3
1 :3.5 1 :4 1 :4.5 1 :5 3.0
2.7
2.4
0
6 22 I
14
SLOPE ANGLE AND HYDRAULIC GRADIENT
0. k .300 .
I
r
n
I
..
1 .
-
16 ,
/
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1 :1 .5
1 :2
1 :2.5
1 :3
1 :3.5 1 :4 1 :4.5 1 :5 3 .0
2.7
2.4
o
s 22 1
14
SLOPE ANGLE AND HYDRAULIC GRADIENT
_ !
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0 u Q
F-W Q
1 :1 .5 3.3
3.0
2.7
2.4
1 :2 .5
1 : 3
1 :3.5
1 :4
1 :4.5 1 :5
6 2'2
1 14 0
SLOPE ANGLE AND HYDRAULIC GRADIENT
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a 0 U d
H W Q
3.6
3.2
2 .8
2.4
2.0
1 :2 .5
1 .3 1 :3.5 1 :4 1 :4.51 :5
0 6
2'2 113 14
SLOPE ANGLE AND HYDRAULIC GRADIENT
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O
a
3.6
3 .2
2 .8
24
2 .
1 : 1 .5
1 : 2
1 :2.5
1'. 3
1 :3.5
1 :4
1 :4.5 1 :5
is
i'2
ig
i a SLOPE ANGLE AND HYDRAULIC GRADIENT
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0 V
1 :1 .5 4.0
3.6
3.2
2 .8
ir 2 .4
2 .
15 1 :2 .5
1 :3
1 :3.5 1 :4 1 :4 .5 1 :5
0 6 22 114 c I nor
AKJr1_F
AKIrt WY_nRAIII IC- _naeni_FN_T -
6 ' . .__ . . 1 _ . _~ _. _ __..._ _ . . . . . .
. . . . : . . ._ . . . .fi :_ . . . . . . _ .~- . . . . . .} . . ~ . . .
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H W Q N
1 :1 .5 4.2
3.8
3.4
3.0
2.2
1 :2.5 1 :3
1 :3.5 1 :4 1 :4.5 1 :5
0
6 22 I
14
SLOPE ANGLE AND HYDRAULIC GRADIENT
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0 U
4.5
4.0
3.5
3.0
2.5
2.0
1 :2 .5
I:3
1 :3.5 1 :4 1 :4.5 1 :5
0
6 22 I
14 ci rnpr &mr.i r earn NYnpem ir rannIFNT
y~ " 142°~,~
___4___-__
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O
Q
1 :1 5 5 .
4.5
4.
3.
3 .
2 .
1 :2 .5 1 --3
1 :3.5 1 :4 1 :4 .5 1 :5
0
6 22 1
14
s, npF a.Nrl F ANn HYDRAULIC GRADIENT
. . _ . . . _ _ . _ . ._
. _~- - 41 V ,
. . 6
5 . . . . . . . . . . . ;
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Ix 0
u
W
Q
1 :1 .5
1 :2
1 :2.5
1 :3
1 :3.5 1 :4 1 :4 .5 1 :5 5 .5
5.0
4.5
4.0
3.
3 .
2.
2,
14 0 6 22 I
SLOPE ANGLE AND HYDRAULIC GRADIENT
OF-
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W W Q
TOTA
UNIT WEIGHT VS SAFETY FAC
R FOR 1 :3 SLOPE
LEG'E ND
HI / Z = RATIO DEPTH OF WATER TABLE TO DEPTH OF OVERBURDEN .
-H,/Z= .5
H, /Z = 1 .0
440
o
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PRINT TAB (25) "INFINITE SLOPE ANALYSIS"
FOR A-20 TO 4 6 STEP 2
PRINT A
111 :311,
111 : 3 .-5 11 .0 111 :4") 11 1 :4 .5 11 1 11 1 :5 11
FOR B=0 TO 1 STEP .2
FOR C=3 TO 5 STEP .5
G1=62 .4
G2-125
F=TAN(A*3 .141/180)*(1-Gl/G2+Gl*B/G2)*C
PRINT F,
NEXT C
NEXT B
NEXT A
PRINT TAB (25) "DEFINITION OF TERMS"
PRINT "A=PHI" , 11 B=H1/Z" I " C zl/SLOPE"," GlnWATER", 11G2t.TOTAL WT ."
END
THE COMPUTER PROGRAM