Download - 2 Stability Analysis
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Push In
Basal Heave
Overall Shear Failure
Overall shear failure modes:
Stability Analysis
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Overall Shear Failure Mode: PUSH-IN
failure surface
{strut
wall
settlement
wall bottom "kick out"
It is mainly due to the unbalance between passive earth
pressure (inside) and active earth pressure (outside)
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Strut settlement
failure surface
bottom heave
Overall Shear Failure Mode: BASAL HEAVE
It is mainly due to
bearing capacity of the
soil beneath the
excavation bottom
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pp
aL
pL
sM
pP
strut
ap
aP
ANALYSIS OF PUSH-IN
The factor of safety against push-in:
aa
spp
d
rp
LP
MLP
M
MF
+==
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Distribution of earth pressures for cohesive soil:
(4.16)
(4.17)
(4.18)
(4.19)
acava cKK 2-= ss
)1(c
cKK
w
aac+=
pcpvp cKK 2+= ss
)1(c
cKK w
ppc+=
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(5.6)
Cast-in-place pile
API
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
1.2
a
vu /s s
FIGURE 4.12 Relation between adhesion and undrained shear
strength of clay
uwsc a=
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Factor of safety
(Diaphragm walls)
(Steel sheet piles)
=1.2~1.3
Cohesionless soil(sandy, gravel)
Distribution of water pressures :
Gross water pressure distribution ?
Net water pressure distribution?
uw sc 67.0=
uw sc 5.0=
pF
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(a) (b)
fu fu
Strut
Strut
id
jd
FIGURE 5.7 Distribution of water pressure due to seepage (a)
distribution of water pressure (b) net water pressure
(note: = water pressure due to seepage)f
u
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Cohesionless soil(sandy, gravel)
Distribution of earth pressures:Caquot-Kerisel's or Coulomb's active earth pressure should be adopted
for the active earth pressure.
Caquot-Kerisel's passive earth pressure should be adopted for the
passive earth pressure. When , Coulomb's passive earth
pressure coefficient is quite close.
Caquot-Kerisel's earth pressure theory's , and have
some relationship. Section 4.5.3 has summarized some findings on
values of .
Clough's research:concluded that between concrete (cast in steel
mold) and sand, is about .
2/fd <
aK pK d
d
d f0.8
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0.0
0.2
0.4
0.6
0.8
1.0
0.8 1.0 1.2 1.4 1.6 1.8
pF
eH(%
)/
eH
FIGURE 5.8 Factors of safety against push-in for excavations in
sand (all cases are safe cases; is assumed)
hm
d
hmd
hmd
fd =
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Conclusion:
Assumption that seems to be reasonable.
For conservative reason, we usually assume
=0.5
=1.2~1.3
=d
pF
d f
f
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(a)
Strut
(b)
anL
pnL
pnP
anP
sM
FIGURE 5 . 9 Analysis of- push in by the net pressure method ( a ) distribution of net earth pressure
( b ) force equilibrium of the retaining wall as a free body
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The analyses of the basal heave failure are only applicable to clayey
soils.
Like Terzaghi, Bjerrum and Eide,
Tschebotarioff, Terzaghi and Peck,
Clough and O'Rourke, etc.
But the most commonly applied of which are Terzaghi's method,
Bjerrum and Eide, and the slip circle method.
ANALYSIS OF BASAL HEAVE
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(1) Bearing capacity method (modified Terzaghi’s method)
(a) (b)
(c) (d)
d
a b c
a b c a b c
a b c
d
d d
2/1
BB =2/
1BB =
1BB 1B
1B1B
o45
eH
o45
FIGURE 5.10 Analysis of push-in by bearing capacity method
(a) a wide trial failure surface
(b) a second wide trial failure surface
(c) a third wide trial failure surface
(d) both sides of the excavation produce failure surfaces
1B
1B
1B
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FIGURE 5.11 Excavation profile of the assumed excavation case
X
2.5m
eH
62.19=satg 3
m/kN
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Bearing capacity method
Negative bearing capacity method
Slip circle method (side strength neglected)
Slip circle method (side strength considered)
0 1 2 3 40
1
2
3
eHX
Fb
)/25(2
mKNSu=
FIGURE 5.12 Relations between failure circle sizes and factors of safety against basal heave obtained by the bearing capacity method, negative bearing capacity
method, and the slip circle method
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0 1 2 3 40
1
2
3
eHX
bF
Bearing capacity method
Negative bearing capacity method
Slip circle method (side strength neglected)
Slip circle method (side strength considered)
)3.0/( =vuS s
FIGURE 5 . 13 Relations between failure circle sizes and factors of safety
against basal heave obtained by the bearing capacity method ,
negative bearing capacity method , and the slip circle method
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Terzaghi's method
Stiff soil
B
a b
c
d
1us
2us
2/B 2/B
2/B
sqsq
o45
eH
D
FIGURE 5.14 Analysis of basal heave using Terzaghi's method
2/)( BDa
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When , the formation of a failure surface is not
restrained by the stiff soil.
Vertical plane bc can offer shear resistance and the
factor of safety against basal heave will be:
(5.7)
(5.8)
(5.9)
2/BD
2)()1)(( 1
BqHBqHW sese
+=×+= gg
2)7.5()1(7.5 212
BsBsQ uuu
=×=
euHs
1
euse
u
eu
ub
HsBqH
Bs
HsW
QF
1
2
1 2/)(
2/7.5
-+=
-=
g
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D
D
B
D
1us
2us
sqsq
DeH
2/)( BDb <
Stiff soil
FIGURE 5.14 Analysis of basal heave using Terzaghi's method
2/)( BDb <
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When , the failure surface will be restrained by the stiff soil.
(5.10)
Clough suggested that, Terzaghi's factor of safety should be
greater than or equal to 1.5.
2/BD <
)(b
F
euse
u
eu
ub
HsDqH
Ds
HsW
QF
1
2
1 )(
7.5
-+=
-=
g
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B 1B
(a)
Bd
a b c
(b)
FIGURE 5.15 Relation between the embedded part of the retaining wall and the failure surface
(a) large penetration depth (b) small penetration depth
d
a b
90 °
90 °
eH
e
eH
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(2) Negative bearing capacity method (modified Bjerrem and Eide’s method)
1B1B
B
Assumed failure surface
(a) (b)
(c)
sqsq sqsq
eH
eH
12 B
12 B
sq
eH
sq
FIGURE 5.16 Analysis of basal heave by negative bearing capacity method (a) a wide failure
surface (b) another wide failure surface (c) Failure surface covers the whole
excavation bottom
12B
12B
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FIGURE 5.11 Excavation profile of the assumed excavation case
X
2.5m
eH
62.19=satg 3
m/kN
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Bearing capacity method
Negative bearing capacity method
Slip circle method (side strength neglected)
Slip circle method (side strength considered)
0 1 2 3 40
1
2
3
eHX
F b
FIGURE 5.12 Relations between failure circle sizes and factors of safety against basal heave obtained by the bearing capacity method, negative bearing capacity
method, and the slip circle method )/25(
2
mKNSu=
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0 1 2 3 40
1
2
3
eHX
bF
Bearing capacity method
Negative bearing capacity method
Slip circle method (side strength neglected)
Slip circle method (side strength considered)
)3.0/( =vuS s
FIGURE 5 . 13 Relations between failure circle sizes and factors of safety
against basal heave obtained by the bearing capacity method , negative bearing capacity method , and the slip circle method
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sq
eH
sq
FIGURE 5.16c
Bjerrum and Eide's method
(5.12) se
uc
bqH
sNF
+×
×=
g
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0 1 2 3 4 54
5
6
7
8
9
1=BL
23
BHe
cN
FIGURE 5.17 Skempton's bearing capacity factor
(rectangular) (square)= (5.11) cNcN )16.084.0(
L
B+
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Modified Bjerrum and Eide's method:
(5.13)
(5.14)
e
sdscu
bH
ffNsF
g
,1=
L
Bf s 2.01 +=
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D 10
9
8
7
6
5.53
2.62.42.22.01.81.61.41.2
1.00.80.60.40.25.53
5
4
3
2
1
0
0.2
0.3
0.40.5
Nc,s
12 uu ss
12 uu ss
BD /
BD /
Nc,s
112<
uu ss 112>
uu ss
undrained shear strength profile
eH
u1s
u2s
B
1
2
(a) for failure circles passing two soil layersN c,s
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If / exceeds the values in the figure, the failure circle
will be tangent to the top of the lower soil layer.
Nc,s
5
10
20
30
40
50
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
3.02.5
2.01.45
1.251.2
BD
Nc,s
12 uu ss
(b) for failure circles tangent to the top of the lower soil layer
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DM7.2 suggested that Bjerrum and Eide's factor of safety
should be greater than or equal to 1.5.
FIGURE 5.18 Extended Bjerrum and Eide's method
(a) for failure circles passing two soil layers
(b) for failure circles tangent to the top of the lower soil layer
(c) width modification factorNc,s
df
df
(c)
0 1 2 3 4 5 61.0
1.1
1.2
1.3
1.4
1.5
BH e /
Nc,s
bF
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(3) Slip circle method
O
W
r
us
Lowest level of struts
A
B
FIGURE 5.19 Location of the center of a failure circle for the slip circle method
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(5.15)
X
O
ab
d f
e c
(a)
W
OsM
(b)
Lowest level of struts
N
N
N N
N
N
N
us
us
X
s
g
s
d
a
FIGURE 5.20 Analysis of basal heave by the slip circle method
(a) the failure surface (b) balance of the a free body
2
)(2
0
XW
MdXsX
M
MF
su
d
r
b
×
+
==
+ap
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Bearing capacity method
Negative bearing capacity method
Slip circle method (side strength neglected)
Slip circle method (side strength considered)
0 1 2 3 40
1
2
3
eHX
bF
FIGURE 5.12 Relations between failure circle sizes and factors of safety against basal heave obtained by the bearing capacity method, negative bearing capacity
method, and the slip circle method )/25(
2
mKNSu=
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FIGURE 5.13 Relations between failure circle sizes and factors of safety against basal heave obtained by the bearing capacity method,
negative bearing capacity method, and the slip circle method
)3.0/( =vuS s
0 1 2 3 40
1
2
3
eHX
bF
Bearing capacity method
Negative bearing capacity method
Slip circle method (side strength neglected)
Slip circle method (side strength considered)
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Lowest level
of struts
FIGURE 5.21 Factor of safety increasing due to the failure circle exceeding
the excavation width
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Lowest level of struts
u1s
u2s
u2u1 SS >
FIGURE 5.22 Analysis of basal heave in layered soft soils
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Bearing capacity method
Negative bearing capacity method
Slip circle method (side strength neglected)
Slip circle method (side strength considered)
0 1 2 3 40
1
2
3
eHX
bF
FIGURE 5.12 Relations between failure circle sizes and factors of safety against basal heave obtained by the bearing capacity method, negative bearing capacity
method, and the slip circle method )/25(
2
mKNSu=
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0 1 2 3 40
1
2
3
eHX
bF
Bearing capacity method
Negative bearing capacity method
Slip circle method (side strength neglected)
Slip circle method (side strength considered)
)3.0/( =vuS s
FIGURE 5 . 13 Relations between failure circle sizes and factors of safety
against basal heave obtained by the bearing capacity method , negative bearing capacity method , and the slip circle method
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(5) Applicability to sandy soils
B
a b
c
d
1us
2us
2/B2/B
2/B
sqsq
o45
eH
D
Stiff soil
2/)( BDa
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D
D
B
DStiff soil
1us
2us
sqsq
DeH
FIGURE 5.14 Analysis of basal heave using Terzaghi's method
2/)( BDb <
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X
O
ab
d f
e c
(a)
W
OsM
(b)
Lowest level of struts
N
N
N N
N
N
N
us
us
X
s
g
s
d
a
FIGURE 5.20 Analysis of basal heave by the slip circle method
(a) the failure surface (b) balance of the a free body
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北投自強路 excavation failure case
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5.5.3 Case Study of Overall Shear Failure
The excavation case was located in Taipei. The width of the
excavation was 17.6m; the length was 100.1m ;the depth was
13.45m. The excavation adopted a 70cm thick , 34, deep
diaphragm wall as the retaining wall. There four levels of struts
and the excavation was carried out in 5 stages.
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力霸百老匯excavation failure
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力霸百老匯 excavation failure case
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SM
OL-ML
SP-SM
CL
¡E
GL-4.5 m
GL-8.7 m
GL-10.7 m
GL-13.45 m
GL-2.8 m
GL-10.15 m
GL-24.0 m
14.7 2m/kN
FIGURE 5.23 Stability analysis of an excavation case history
(a) excavation and geological profiles
o18=uf
2/7.16 mkNcu
=
3/8.18 mkNt
=g
0=c
3/7.19 mkNt
=g
o10=uf
2/98.0 mkNcu
=
3/6.15 mkNt
=g
o33=f0=c
3/3.20 mkNt
=g
o32=f
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and
will be 1.5
will be 2.3
were the total stress strength parameters of the clay soils,
obtained from the triaxial CU test
adopt by the original designer
We assume the soil below the lowest level of struts (GL-10.15 m)
to be a clayey layer, the adhesion between the retraining wall and
the soil
and the normalized undrained shear strength
(4.16)
(4.18)
pF
bF
3/2 uw sc =
22.0/ =vus s
acava cKK 2-= ss
pcpvp cKK 2+= ss
ucuf
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At the depth of GL-10.15 m
2/5.18545.17.192.46.155.43.20 mkNv
=×+×+×=s
2/1.7281.9)8.215.10( mkNu =×-=
2/3.113 mkNuvv
=-= ss
2/9.243.11322.022.0 mkNs vu
=×== s
2
, /2.1213
21)9.24(25.185)1(2 mkN
s
cKsK
u
wauavah
=+-=+-= ss
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At the depth of GL-13.45 m
Before excavation─
after excavation was started, on the passive side, but
value stayed unchanged.
Thus,
2/0.24875.28.1855.07.195.185 mkNv
=×+×+=s
2/5.10481.9)8.245.13( mkNu =×-=
2/5.143 mkNuvv
=-= ss
2/6.315.14322.022.0 mkNs vu
=×== s
0=vs us
2
, /5.813
21)6.31(20)1(2 mkN
s
cKsK
u
wpupvph
=++=++= ss
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At the depth of GL-24.0 m
The active side─2
/3.44655.108.180.248 mkNv=×+=s
2/0.20881.9)8.224( mkNu =×-=
2/3.238 mkNuvv
=-= ss
2/4.523.23822.022.0 mkNs vu
=×== s
2
, /0.3113
21)4.52(23.446)1(2 mkN
s
cKsK
u
wauavah
=+-=+-=ss
![Page 53: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/53.jpg)
At the depth of GL-24.0 m
The passive side─
After excavation was start stayed constant,us2
/3.198)45.130.24(8.18 mkNv=-×=s
2
, /6.3333
21)4.52(23.198)1(2 mkN
s
cKsK
u
wpupvph
=++=++= ss
![Page 54: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/54.jpg)
The factor of safety against push-in as
(b)
81.5
121.2
311.0333.62
mkN
2mkN
2mkN
2mkN
GL-10.15 m
GL-13.45 m
GL-24.0 m
FIGURE 5.23 Stability analysis of an excavation case history
(b) distribution of earth pressure for the push-in analysis
89.03/285.135.085.13)2.1210.311(2/85.1385.132.121
)3.33/255.10(5.055.10)5.816.333()3.32/55.10(55.105.81=
××××-+××
+××××-++××=pF
![Page 55: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/55.jpg)
Compute the factor of safety against basal heave
according to Slip circle method :Similarly, assuming the soil below the lowest level of
struts is clay, the average value of the undrained shear
strengths (the active side) of the soil between GL-10.15 m
and GL-24.0 m would be
The average value of the undrained shear strengths of
the soil between GL-13.45 m and GL-24.0 m would be
2
, /7.382
4.529.24mkNs au
=+
=
2
, /0.422
4.526.31mkNs pu
=+
=
![Page 56: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/56.jpg)
The radius of the failure circular arc would be
The central angle of the failure circular arc on the active side
would be
The central angle of the failure circular arc on the passive
side would be
The factor of safety against circular arc failure would be
m85.1315.1024 =-
57.12==
p
33.1)85.13
3.3(cos
1 == -
94.023786
22370
2/85.1385.13
85.137.3857.185.1385.130.4233.185.13
)45.13(
==+×
=-GLv
bFs
× × × × ×
××
![Page 57: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/57.jpg)
Computing the factor of safety against basal heave
following Terzaghi's method:
The width of the excavation , was larger than the
penetration depth (10.55 m). Assumed failure surface will pass below the
bottom of the retaining wall.
Bd
a b c
(b)
90°
eH
FIGURE 5.15 Relation between the embedded part of the retaining wall and the failure surface
(b) small penetration depth
mB 6.17= 2/B
![Page 58: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/58.jpg)
The average undrained shear strength of soil within the range of
the failure circle can be calculated as follows:
of soil deep below the ground surface--mB 9.252/45.13 =+
2/1.482)45.1390.25(8.180.248 mkNv
=-+=s ×
2/5.25581.9)8.290.25(1.482 mkNuvv
=--=-= ss ×
2/2.565.25522.022.0 mkNs vu
=== s ×
![Page 59: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/59.jpg)
The average undrained shear strength within the range of the
failure circle would be
As computed earlier, the total stress outside the excavation zone
at the depth equaling the excavation surface would be
2/9.432/)2.566.31( mkNsu
=+=
2/0.248 mkNv
=s
![Page 60: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/60.jpg)
To simplify the analysis and be conservative, we assume the
soil above the excavation surface is clay and has soil shear
strength expressed as . The average undrained shear
strength of the soil outside the excavation zone and the excavation
surface would be
The factor of safety according to Terzaghi's method would be
22.0/ =vus s
2)45.13(/8.15
2
6.31
2
22.0mkNs
GLv
u==
=
-s
08.12874
3118
45.138.152/6.170.248
2/6.179.437.5
1
==-
=-
=eu
u
bHsW
QF
× ×
× ×
![Page 61: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/61.jpg)
The factor of safety following Bjerrum and Eide's method would be
10.10.248
2.69.43==
+=
se
cub
qH
NsF
g
×
![Page 62: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/62.jpg)
Undrained shear strength and the depth:
)(2mkN
Active side ),( uuc f
0.22/ =vus s
Passive side ),(uuc f
(c)
FIGURE 5.23 Stability analysis of an excavation case history
(c) the undrained shear strength used in the analysis
25
2
0
15
10
5
00 50 100 150 200
dept
h(m
)
us
![Page 63: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/63.jpg)
【Example 5.1】Assume a 9.0 m deep excavation in a sandy
ground and the lowest level of struts is 2.5m above the
excavation surface. The level of groundwater outside the
excavation zone is ground surface high while that within the
excavation zone is as high as the excavation surface. The
unit weight of saturated sandy soils , the
effective cohesion and the effective angle of friction
. Because of the difference between the levels of
groundwater , seepage will occur. Assume that the friction
angles ( ) between the retaining wall and soil on both the
active and passive sides are and the factor of safety
against push-in, . Compute the required penetration
depth( ).
20=satg 3
m/kN
o30=f
0=c
d
f5.05.1=
pF
pH
![Page 64: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/64.jpg)
jd
id
x
x
a
b
ccu
wjep dHH g)( -+
(a)
bu
(b)
FIGURE 4.22 Simplified analysis method for seepage
(a) distribution of water pressure (b) net water pressure
z
z
wip dH g)( -
eH
pH
![Page 65: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/65.jpg)
1.determine the coefficient of the earth presure
Compute both the active and passive earth pressures following
Caquot-Kerisel's earth pressure theory. When , the
coefficients of active and passive earth pressure can be found from
Figure 4.9 and Figure 4.10 to be 0.3 and 4.6 separately. Thus, the
coefficients of the horizontal active and passive earth pressure
would be
【Solution】
f=d 5.0
29.05.0cos3.0cos3.0,=== fd
haK
4.45.0cos6.4cos6.4,=== fd
hpK
![Page 66: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/66.jpg)
2. Compute the effective active active earth pressure on the wall
According to Eq. 4.51the porewater pressure at x away from upstream
water level would be
At the lowest level of strut (z=6.5m, x=6.5m) --
havha K ,,ss = 2
kN/m1305.620 ==vs ×
5.4
77.63
92
81.95.62
2
)(2
+=
+=
--+
-=
p
p
p
p
jiep
wip
H
H
H
H
ddHH
dHxu
g × × ×
5.4
49.187.3729.0)
5.4
77.63130()( ,,
+-=
+-=-=
p
p
P
p
hahaH
H
H
HKuss ×
![Page 67: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/67.jpg)
At the bottom of the retaining wall( , )─pHz +=9 pHx +=9
)5.4
60.2584.28.52.52(
29.0)5.4
29.8881.920180(
2
2
,
+
+-+=
×+
+-+=
p
pp
p
p
pp
pha
H
HHH
H
HHHs
ppv HH 20180)9(20 +=+=s ×
5.4
29.8881.9
92
81.9)9(2
2
)(22
+
+=
+
+=
--+
-=
p
pp
p
pp
jiep
wip
H
HH
H
HH
ddHH
dHxu
g × × ×
![Page 68: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/68.jpg)
3. Compute the lateral effective passive earth pressure on the wall
At the bottom of the retaining wall( )─pHz =
ppv HH 2020 =×=s
5.4
29.8881.92
+
+=
p
pp
H
HHu
5.4
48.38816.4388
4.4)5.4
29.8881.920(
2
2
,
+
+-=
×+
+-=
p
pp
p
p
pp
php
H
HHH
H
HHHs
![Page 69: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/69.jpg)
4. Compute the maximum net water pressure (at the
excavation surface)
According to Eq. 4.53, the maximum net water pressure
would be
5.4
29.88
92
81.992
2
))((2
+=
+
×××
=--+
--+=
p
p
p
p
jiep
wipjie
bH
H
H
H
ddHH
dHddHu
g
![Page 70: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/70.jpg)
5. The effective earth pressure on both sides of the wall and the
distribution of the net water pressure are as shown in Figure
5.24
Lowest level
of struts
FIGURE 5.24 Distribution of lateral earth pressure
o
b
2.5
9
pH
bu9
5.6 bu
netu
hp ,s
ha,s
5.4
48.38816.4388
2
+
+-
p
pp
pH
HHH
5.4
60.2584.28.52.52
2
+
+-+
p
pp
pH
HHH
5.4
49.187.37
+-
p
p
H
H
![Page 71: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/71.jpg)
6. Compute the driving moment ( ) and the resistant moment
( ) for the free body below the lowest level of strutsrM
dM
bppp
p
pp
p
bbppb
p
p
pp
p
p
p
p
ahad
uHHHH
HHH
uuHHu
H
H
HHH
H
H
H
LPM
]84.225.117.0[)5.2)(5.4
62.1195.093.168.23(
32
5.22
9
5.2
2
5.2
9
5.6)
35.2(
2
32
)5.2(2)
5.4
11.784.28.55.14(
2
)5.2()
5.4
49.187.37(
22
2
22
222
,
+++++
+-+=
×
××+×++×+
×
+×
+
+-++
+×
+-=
=
bppp
p
pp
p
bbppb
p
p
pp
p
p
p
p
ahad
uHHHH
HHH
uuHHu
H
H
HHH
H
H
H
LPM
]84.225.117.0[)5.2)(5.4
62.1195.093.168.23(
32
5.22
9
5.2
2
5.2
9
5.6)
35.2(
2
32
)5.2(2)
5.4
11.784.28.55.14(
2
)5.2()
5.4
49.187.37(
22
2
22
222
,
+++++
+-+=
×
××+×++×+
×
+×
+
+-++
+×
+-=
=
![Page 72: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/72.jpg)
)3
25.2)(
5.4
24.19458.2144(
)3
25.2(
2)
5.4
48.38816.4388(
23
2
2
,
p
p
pp
p
pp
p
pp
p
phpr
H
H
HHH
HH
H
HHH
LPM
++
+-=
+××
+
+-=
=
![Page 73: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/73.jpg)
7. determine the penetration depth
Then we have m
pH
5.1==
d
rp
M
MF
25.7=pH
![Page 74: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/74.jpg)
【Example 5.2】An excavation in clay goes 9.0m in to the
ground . The groundwater outside the
excavation zone is at the ground surface level while
that within the excavation zone is at the level of the
excavation surface. . The
undrained shear strength .
Suppose the excavation width B =10m and the
excavation length L =30m. Compute the factor of
safety against basal heave according to Terzaghi's
method and Bjerrum and Eide's method,
respectively.
=17.0
=45
9.0m)( =e
H
satg 3
m/kN
2m/kNus
![Page 75: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/75.jpg)
【Solution】
In this example, the surcharge
According to Terzahi's method,
0=sq
7.2
107.0
4517
457.5
9
1
7.0
7.51=
×-
×=
-
××=
B
s
s
HF
u
u
e
b
g
![Page 76: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/76.jpg)
According to Bjerrum and Eide's method,
According to Figure 5.17, we have
0.310
30==
B
L
9.010
9==
B
He
1.7=cN
09.2917
1.745=
×
×==
et
cub
H
NsF
g
![Page 77: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/77.jpg)
Upheaval
Impermeable layer
Permeable layerw
HWater pressure
1h
2h
1tg
2tg
The factor of safety against upheaval should be larger than or equal to 1.2
FIGURE 5.31 Analysis of upheaval
(5.17)
ww
iiti
upH
h
Fg
g
×
×
=
upF
![Page 78: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/78.jpg)
(CL)
(ML)
(SM)
GL-0.0m
GL-8.0m
GL-25.0m
GL-32.5m
GL-36.0m
GL-40.5m
Grouting
Gravel
2.9m -Stage 1
5.9m -Stage 2
8.8m -Stage 311.9m -Stage 4
14.4m -Stage 5
17.5m -Stage 6
20.0m -Stage 722.5m -Stage 825.1m -Stage 9
27.55m -Stage 10
30.6m -Stage 11
(SF)
GL-58.5m GL-58.0m
GL-63.0m
33.7m -Stage 12
36.75m -Stage 13
40.5m -Stage 14
GL-10.0m(CL)
(SP-SM)GL-15.0m
(CL)
(SM)GL-43.0m
Case Study
ww
iiti
upH
h
Fg
g
×
×
=
ww
i
iti
upH
h
Fg
g
=
Upheaval failure:
![Page 79: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/79.jpg)
1H
2H
z
Upward water flow
u
0
Saturated soil
A
C
B
2H
hzh
FIGURE 5.32 Total stresses, effective stresses, and change of porewater pressure in
sandy soils acted on by an upward water flow
wH g1 wH g
1
satw HH gg21
+whHH g)( 21
++whH gg -
2
s s
Sand Boiling --- Factor of safety and failure mechanism
![Page 80: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/80.jpg)
The critical hydraulic gradient is then
Besides, according to the phase relationship of soil, the
submerged unit weight is
(5.24)
(5.25)
-w
s
e
Ggg |
+
=1
1
e
Gi
s
cr+
-=
1
1
)max(exit
crs
i
iF = (5.26)
![Page 81: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/81.jpg)
(5.26)
A
wHD
pH
FIGURE 5.33 Seepage in soil below sheetpiles
Impermeable layer
Wall
Sand boiling zone
)max(exit
cr
si
iF =
![Page 82: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/82.jpg)
Terzaghi's method:
=(the volume of the soil column)
The factor of safety is
(5.27)
(5.28)
(5.29)
wavgpwaviHi gg 2
2
1)( =×
ggg =-= 22
2
1)(
2
1pwsatp HHW
wavg
siU
WF
g
g =
=
U
![Page 83: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/83.jpg)
Provided the computed factor of safety is too small, we can
consider placing filters at the exits of seepage. Assuming the
weight of filters is Q, the factor of safety will be
(5.30)
In general, the required for the above equation should
be greater than of equal to 1.5sF
U
QWFs
+=
![Page 84: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/84.jpg)
Marsland's method:
DM7.1 suggested that the reasonable factor of safety against piping
in an excavation be around 1.5 to 2.0.
Retaining wall
2.0
1.5
1.0
0.5
00.5 1.0 1.5 2.0
2.0
1.5
2.0
1.5
1.0
1.0sF
Loose sandDense sand
wH
PH
B
(a)
)2/(w
HB
wp
HH
/
FIGURE 5.34 Relations between wall penetration depths and factors of safety against sand boiling
(a) dense and loose sands with the impermeable layer located at the infinite depth
![Page 85: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/85.jpg)
FIGURE 5.34 Relations between wall penetration depths and factors of safety against sand boiling
(b) dense sand with the impermeable layer located at a finite depth
2.0
1.5
1.0
0.5
00.5 1.0 1.5 2.0
2.0
1.5
1.01.0
2.0
1.5
wH
B
D
Impermeable layer
(b)
sF
PH
)2/( wHB
2=wHD
1=wHD
wp
HH
/
![Page 86: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/86.jpg)
One dimension seepage method:
id
FIGURE 5.35 Analysis of sand boiling
wH
Sandy soil
a
bc
d
pH
eH
jd
D
![Page 87: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/87.jpg)
The total head at the upstream elevation (point a) will be
The difference of the total heads between upstream and
downstream levels will be
If we assume the datum is at the downstream level, the total
head at the elevation of downstream (point d) will be
(5.31)
(5.32)
(5.33)
000,=+=+=
pedt hhh
jiejiepeat ddHddHhhh -+=+-+=+= 0,
jiedtatw ddHhhH -+=-=,,
D
![Page 88: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/88.jpg)
Suppose the seepage is one dimensional and the hydraulic
gradients for each depth along the flow path abcd are equal. the
hydraulic gradient will be
The factor of safety against boiling will be
The required for the above equation should be greater than
or equal to 1.5.
jipe
w
ipije
wavg
ddHH
H
dHddH
Hi
--+=
-++-=
2)(2
DD
ww
jipe
avg
cs
H
ddHH
i
iF
Dg
g )2( --+==
sF
![Page 89: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/89.jpg)
Case Study
FIGURE 5.36 Excavation of Siemen Station of Taipei Rapid Transit System
(a) excavation and geological profiles
SM
CL
SM
CL
SM
EL.+72.5m
EL.+69.4m
EL.+63.4m
EL.+59.6m
EL.+56.4m
Sungshan III formation
Sungshan II formation
. . .. . . .. . . .. . . . . ... .. . .. .... .. .... .
EL.+81.8m
EL.+87.2m
EL.+95.0m
EL.+104.5m
Sungshan IV formation
Sungshan VI formation
Groundwater level EL.+98.0mEL.+98.0m
CL
SM
CL
Chingmei formation
EL.+88.0m
EL.+80.0m
Sand boiling
23.25m
(a)
Sungshan V formation
Sungshan I formation
![Page 90: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/90.jpg)
FIGURE 5.36 Excavation of Siemen Station of Taipei Rapid Transit System
(b) plan view
6.2m
5.0m
23.25m
sump
bore
Diaphragm wall
N
(b)
![Page 91: 2 Stability Analysis](https://reader033.vdocuments.mx/reader033/viewer/2022051415/55cf9aad550346d033a2dc31/html5/thumbnails/91.jpg)
SM
CL
SM
CL
SM
EL.+80.0m
EL.+72.5m
EL.+69.4m
EL.+63.4m
EL.+59.6m
EL.+56.4m
Casing
5m 6m
4m
Sump
8m9m
. . . . . .
. .
. . . . . . . .. .. .. . . . .. . . . .
. .
. . . . .. . . .
(c)
. . . .
. .
FIGURE 5.36 Excavation of Siemen Station of Taipei Rapid Transit System
(c) process of sand boiling
Sungshan III
Sungshan I
Sungshan II
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Marsland’s method {Fig. 5.34,1.5}(1, 5)
Terzaghi’s method {Eq. 5.30
assuming Ms=0}(1)
Overall shear failure
Push-in Basal heave
Gross pressure method
{Eq. 5.5, }(1)
Short term behaviors can
be ignored while long
term behaviors may need
consideration. The
analysis methods are the
same as those for sand
and gravel.
Sand boiling Upheaval
San
d o
r g
rav
e A
lter
nat
ed l
ayer
s of
sand
(or
gra
vel
) an
d c
lay
Cla
y ˍˍ
ˍˍ ˍˍ
ˍˍ
TABLE 5.2 Stability analysis methods for strutted walls and the required minimum factors of safety
Gross pressure method
{Eq.5.5, 1.2,
Harza’s method {Eq. 5.26,
2.0}(4)
1.5~2.0}(4)
}(1, 5)
Simplified 1-D seepage
method {Eq. 5.35,
Terzaghi’s method
Bjerrum and Eide’s
Slip circle method {Eq. 5.15,
}(1)
Gross pressure method
{Eq. 5.5, }(1)
assuming Ms=0}(1)
assuming Ms=0}(1)
Terzaghi’s method {Eq. 5.9 or
5.10, }(1,2,3)
Bjerrum and Eide’s method{Eq.
5.12 or 5.13, }(4,3)
Slip circle method {Eq. 5.15,
}(1,3)
{Eq. 5.17, Fup≧1.2}(1)
pF
sF
sF
sF
0.2sF
2.1bF
2.1pF
2.1pF
5.1bF
2.1bF
2.1bF
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NOTE:
(1) The methods and factors of safety are suggested by TGS (2001)
and JSA (1988)
(2) The factor of safety is suggested by Mana and Clough (1981)
(3) It is only when clay is the dominant soil layer that the analysis of
basal heave is required
(4) The factor of safety is suggested by NAVFAC DM 7.1 (1982)
(5) TGS (2001) and JSA (1988) suggest the conservative value
obtained by Terzaghi's method or the simplified 1-D seepage be
adopted for design.