on some aspects of reliability computations in …
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ON SOME ASPECTS OF RELIABILITY
COMPUTATIONS IN BEARING
CAPACITY OF SHALLOW
FOUNDATIONS
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Some aspects of reliability computations in bearing capacity of shallow foundations
1. Bearing capacity evaluation of shallow foundation accordingly to codes.
2. Examples of reliability analysis basing on bearing capacity evaluations – some parameters studies.
3. The effect reduction of variance due to averaging.4. Averaging of soil properties along slip surfaces.5. Discussion of results. Comparison of one-dimensional
and two-dimensional cases.
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Spread foundations• The ground resistance on the sides of the
foundation does not contribute significantly to the bearing capacity resistance (pad, strip, raft foundations ).
• Drained resistance according to Eurocode EC7 and Polish Standard PN-81/B-03020. Foundation bases. Static computations and design
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Bearing capacity resistance Qf
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++= γγγγ γγ sBiNsiqNsciNLBQ qqqqcccf 2
1
( )
+=
24tantanexp 2 ϕπϕπqN
( ) ϕcot1−= qc NN
( ) ϕγ tan12 −= qNN
,
Bearing capacity factors:
EC7: PN: ( ) ϕγ tan15.1 −= qNN
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BeBB −= LeLL −=
Effective dimensions of the foundation:
Shape coefficients (rectangular shape):
EC7 PN
11
3.01
sin1
−
−=
−=
+=
q
qqc
q
NNs
s
LBs
LBs
γ
ϕ
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+=
−=
+=
LBs
LBs
LBs
c
q
3.01
25.01
5.11
γ
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Load inclination coefficients(according to DIN 4017 – Orr and Farrel: Geotechnical design to Eurocode 7):
1
tc1
m
q ocLBVHi
+
−=ϕ
H is a horizontal load, V is a vertical load
11
tc1
+
+
−=m
ocLBVHi
ϕγ
ϕtan1
c
qqc N
iii
−−=
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+
+
== 1
2
LB
LB
mm B when H acts in the direction of
+
+
== 1
2
BL
BL
mm L when H acts in the direction of
B
L
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Reliability approach - assumptions
( )nXXX ,...,, 21=X is a vector o basic random variables
gfor the safe state of the structurefor the failure state of the structure
( )x =><
00
NmQg f −= , N is a load acting on foundation
Limit state function
Probability of failure F{g( )<0}
p = f ( )dx
X x x∫
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Reliability approach - assumptions
( )nXXX ,...,, 21=X is a vector o basic random variables
gfor the safe state of the structurefor the failure state of the structure
( )x =><
00
NmQg f −= , N is a load acting on foundation
Limit state function
Probability of failure F{g( )<0}
p = f ( )dx
X x x∫
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Reliability approach - assumptions
( )nXXX ,...,, 21=X is a vector o basic random variables
gfor the safe state of the structurefor the failure state of the structure
( )x =><
00
NmQg f −= , N is a load acting on foundation
Limit state function
Probability of failure F{g( )<0}
p = f ( )dx
X x x∫
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( )Fp10−Φ−=βReliability index , provided that
21
<Fp
As a computational tool the SORM is utilised
Sensitivity parameters
**
1yyy =∂
∂=
ii y
βα
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( )Fp10−Φ−=βReliability index , provided that
21
<Fp
As a computational tool the SORM is utilised
Sensitivity parameters
**
1yyy =∂
∂=
ii y
βα
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Schematic presentation of the FORM method
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Shallow strip foundation
1=== γsss qc
Medium sand
Scheme of shallow strip foundation considered in the example
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Probabilistic characteristic of soil and loads parameters
-0.012lognormal2.25 kNm/m15.0 kNm/mMoment M
-0.055lognormal3.0 kN/m 20.0 kN/mLoad tangent to the base T
-0.205lognormal45.0 kN300 kNAxial load normal to the base N
0,021uniform0.06 m1.00 mGround water level h
nonrandom-24.0 kN/m3Unit weight of foundation material γb
0.024normal0.5889.8 kN/m3Soil Unit weight under water table γ’
0.001normal1.38 kN/m323.0 kN/m3Concrete floor unit weight γp
0.008normal1.092 kN/m318.2 kN/m3Soil Unit weight γ
0.973lognormal4.6o32°Soil friction angle ϕ
SensitivityParameters α
Probability distribution
StandardDeviation σX
Mean valueSoil property
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0.000583.254.0
0.001762.923.6
0.005232.563.2
0.014932.172.8
0.040491.752.4
0.051471.632.3
0.065141.512.2
0.157201.011.8
Probability of failure pF
Reliability index βWidth of foundation b [m]
Selected values of reliability measures obtained in the example
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It turned out that that minimal width necessary to carry the acting load is b = 2.3.
This width corresponds to the value of reliability index β = 1.63, which seems to be rather small.
At same time the ISO 2394 [9] code suggests beta values equal to β = 3.1 for small, β = 3.8 for moderate and β = 4.3 for large failure consequences.
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β
B [m]
cov{ϕ} = 0,05
cov{ϕ} = 0,10
cov{ϕ} = 0,15
cov{ϕ} = 0,20
cov{ϕ} = 0,25
Reliability index β versus width of the foundation B and variation coefficient of φ
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β
B [m]
Reliability index β versus width of the foundation B for three different probability distribution of φ
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Uniform (rectangular)LognormalNormal
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β
B [m]
ρ = 0,0ρ = 0,3ρ = 0,6
Reliability index β versus width of the foundation B for three different correlation coefficients for φ and γ
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β
h [m]
An effect of water table variability h = 0.3 – 1.6 m
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Cohesive soil
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sand
clay
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cov{c} = 0,05cov{c} = 0,10cov{c} = 0,15cov{c} = 0,20
cov{ϕ} = 0,05cov{ϕ} = 0,10cov{ϕ} = 0,15cov{ϕ} = 0,20
B [m] B [m]
β β
a) b)
Reliability index β versus width of the foundation B and variation coefficients of φ (a) and c (b)
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β
B [m]
ρ(ϕ, c) = -0,8ρ(ϕ, c) = -0,6ρ(ϕ, c) = -0,4ρ(ϕ, c) = -0,2
Reliability index β versus width of the foundation B different correlation coefficients for φ and c
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Spatial averaging
Assume that a soil property X, which is consider as random, can be described by a stationary random field possessing a covariance
),,(),,( 2 zyxzyxR X ∆∆∆=∆∆∆ ρσ
and denotes its measure (volume).
The spatial averaging, introduces a new random field (moving average random field) defined by the following equation:
3R⊂VLet be a domain V
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( )∫∫∫=V
V dxdydzzyxXV
X ,,1
[ ] ( ) 22VAR XVV VX σγσ ==
( ) ( )∫∫=21
21 2222111122211121
,,),,(,,,,,1),(VV
VV zyxdVzyxdVzyxzyxRVV
XXCov
( )∫∞
∆∆=0
22 zdzR
Xσδ ( )∫ ∆∆
∆−=
L
zdzLz
LL
0
12)( ργ
One-dimensional case
( )LLL
γδ∞→
= lim L is the averaging interval
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!!!
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( )
>∆
≤∆∆
−=∆
δ
δδρ
zdla
zdlaz
z0
12
( )
>∆
≤∆=∆
20
21
1 δ
δ
ρzfor
zforz ( )
>
−
≤=
241
21
1 δδδ
δ
γLfor
LL
LforL
( )
>
−
≤−=
δδδ
δδ
γLdla
LL
LdlaL
L
31
31
2
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( )
∆−=∆
2
3 expδ
πρ zz
( ) 2
2
3
exp1erf
−+−
⋅
=
δπ
δπ
δπ
δπ
γL
LLL
L
Gaussian correlation function
( ) ( )∫ −=t
dxxt0
2exp2erfπ
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( )
∆+
∆−=∆∆
2
2
2
14 exp,
ωωρ xzzx
1 2v hδ ω π δ ω π= =
Two-dimensional Gaussian correlation function
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Example
•Cohesionless soil
•One-dimensional Gaussian correlation function
•δ = 0.8 m, the averaging area (interval) is L = 2B
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0
1
2
3
4
5
6
7
8
1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 3,0 3,2L = 2b [m]
relia
bilit
y in
dex β
z uśrednieniem
bez uśrednienia
With averaging
Without averaging
Reliability indices computed without spatial averaging and with the spatial averaging. For the mentioned width B = 2.3 m β index increases from β = 1.63 to β = 3.83.
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Example
•Cohesive soil
•One-dimensional Gaussian correlation function
•δ = 1.0 m, the averaging area (interval) is L = 2B
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-2-1012345678
1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 3,0 3,2
B [m]
ββ4
β3
β2
β1
β1: without averaging and without correlationβ2: correlation ρ = -0.6, without averagingβ3: without correlation, with averaging δ = 1.0 mβ4: correlation ρ = -0.6, with averaging δ = 1.0 m
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Selection of the averaging area
δ [m] δ [m]
Reliability index β as a function of fluctuation scale values for three different variance functions. Fig. a shows results with spatial averaging of averaging size L = 2b. Fig. b shows results with spatial averaging of averaging size L = b.
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8.476.863.254.0
7.486.002.923.6
6.435.112.563.2
5.314.172.172.8
4.133.201.752.4
3.832.961.632.3
3.512.711.512.2
2.241.711.011.8
Reliability index βAveraging L = 2b
Reliability index βAveraging L = b
Reliability index βWithout averaging
Width of foundation b [m]
Selection of the averaging area
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Suggestions
•In order to avoid the loss of uniqueness in reliability computations the averaging area V must be carefully selectedand precisely defined among assumptions for a problem under consideration.
•To get adequate values of variance reduction in bearing capacity problems it is necessary to carry out the spatial averaging along potential slip surfaces associated with a mechanism of failure.
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Slip lines associated with the Prandtl mechanism
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•One-dimensional case (averaging in the vertical direction)
•Two-dimensional case (averaging in both vertical and horizontal dimensions)
•Separable Gaussian covariance function
( )
∆+
∆−=∆∆
2
2
2
1
2 exp,ωω
σ zxzxR X
πωδ =
anisotropic case: 21 ωω ≠
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],[)(),(
iii
ii
batfortzztxx
∈==
( ) ( ) ( ) ( ) ( )( )
jijjii
b
ajjii
b
ajiljji
ljill
dtdtdtdz
dtdx
dtdz
dtdx
tztxtztxRll
dldlzxzxRll
XXj
i
i
ii
ji
2222
2211 ,,,1,,,1),Cov(
+
+
×
×== ∫∫∫∫
Assume a parametric representation of the slip line in the form
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jijjii
b
a
jib
aji
Xll dtdt
dtdz
dtdx
dtdz
dtdxtztz
llXX
j
j
i
i
ji
22222
1
2 )()(exp),Cov(
+
+
−−= ∫∫ ω
σ
[ ]1,0;24
tan2
)(;22
)( ∈
+=−= tbttzbtbtx ϕπ
One-dimensional averaging with the Gaussian correlation function
The variance of XAB
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{ } ( ) ( )
( )∫∫
∫∫
−−=
=+
−−
+
=
1
021
2212
1
221
0
2
1
021
22
2212
1
221
02
2
2
4exp
144
exp24cos4
Var
dtdtttab
dtdtabttabb
X
X
XAB
ωσ
ω
ϕπ
σ
+=
24tan ϕπa
{ }
−
−+
=
B
B
B
B
B
XAB h
hh
hh
X21
21
221
11
2
experfVar ωω
ωω
πωσ
= exp2bhC
+=
24tan
2ϕπbhB
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Slip lines associated with the Prandtl mechanism
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= ϕπ tan
2exp
2bhC
θθθθ sin),(;2
cos),( rrzbrrx =+−=
( )
++∈=
243;
24;tanexp)( 0
ϕπϕπθϕθθ rr
+−
+
= ϕϕπϕπ
tan24
exp
24cos2
0br
The variance of XCD
The variance of XBC
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[ ] ( ) ( ) ( ) ( )
( ) ( )
3 34 2 4 2 2
22 00 1 1 2 22
4 2 4 2
1 2 1 2
Var exp sin exp tan sin exp tan
exp tan exp tan
BC XrX
d d
π ϕ π ϕ
π ϕ π ϕ
σ α θ θ ϕ θ θ ϕω
θ ϕ θ ϕ θ θ
+ +
+ +
= − − ×
×
∫ ∫
( )2
20
1tan2
exp
tan2
exptg
−
+−
=
ϕπ
ϕϕπ
ϕα
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[ ] ( )
( ) 21
1
0
1
0
2212
1
22
212
2
1
0
1
0
2212
1
2
2
2
2
4exp
12
1124
exp24cos4
,Cov
dtdtddtatb
dtdtaba
adbddtatbdb
XX
X
XCDAB
∫ ∫
∫ ∫
−+−=
=++
−+−
+
=
ωσ
ω
ϕπ
σ
+=
24tan ϕπa
= ϕπ tan
2expd
Covariances
[ ] ( )
( ) ( )
+
−+
−−
+
+
−+−
−−−
−=
1111
12
221
22
21
22
21
2
2
212
2erf
2erf
2erf
2erf
4exp1
4exp
4exp2,Cov
ωωωωπωσ
ωωωωσ
abbadbadadbd
ad
bd
abdabdbdab
XX
X
XCDAB
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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( )( )
( )
234 2 2
122
14 20
sinexp 1 exp tg4 4 2Cov , cos
4 2
exp tg
AB BC X
b a tX X
d dt
π ϕ
π ϕ
θ π ϕθ ϕπ ϕωσ α
θ ϕ θ
+
+
− − − − + × = + ×
∫∫
+−
+
=
ϕϕπϕϕπϕα
tan24
exptan24
3exp
tan1
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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jijjii
jib
a
jib
aji
X
ll
dtdtdtdz
dtdx
dtdz
dtdxtxtxtztz
ll
XX
j
j
i
i
ji
22222
2
2
1
2 )()(exp
)()(exp
),Cov(
+
+
−−
−−=
=
∫∫ ωωσ
Two-dimensional case with the Gaussian correlation function
{ }
−
−+
=
B
B
B
B
B
B
B
BB
B
XAB h
hh
hh
X2
02
0
220
00
2
experfVar ωω
ωω
πωσ
+=
24tan
2ϕπbhB 2
122
2
222
21
0 ωωωω
ω+
=a
aB
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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Slip lines associated with the Prandtl mechanism
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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+=
24tan
2ϕπbhB
21
222
222
21
0 ωωωω
ω+
=a
aB
{ }
−
−+
=
B
B
B
B
B
B
B
BB
B
XAB h
hh
hh
X2
02
0
220
00
2
experfVar ωω
ωω
πωσ
{ }
−
−+
=
B
B
B
B
B
XAB h
hh
hh
X21
21
221
11
2
experfVar ωω
ωω
πωσ
102
lim ωωω
=∞→ B
+=
24tan ϕπa
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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+=
24tan ϕπa
= ϕπ tan
2expd
{ }
−
−+
=
C
C
C
C
C
C
C
CC
C
XCD h
hh
hh
X20
20
220
00
2
experfVar ωω
ωω
πωσ
21
222
22
21
212
22
2
22
21
0 ωωωω
ωω
ωω
ωa
a
aC +
=+
=
102
lim ωωω
=∞→ C
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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[ ] ( )[ ( ) ( ) ( )]
( )[ ( ) ( ) ( )] ( ) ( ) 21212
221122
20
2221
243
24
121
20
243
24
02
tanexptanexptanexpcostanexpcos
tanexpsintanexpsinexpVAR
θθϕθϕθϕθθϕθθω
ϕθθϕθθω
ασ
ϕπ
ϕπ
ϕπ
ϕπ
ddr
rX XBC
−×−
−
−= ∫∫
+
+
+
+
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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−
=
24sin 1
0
ϕπbQQ
( )[ ]
( )
−
+
+
−
−+
−
=
24sin2
costanexp
tan24
sin2
11tanexp
24sin2
cos
1
323
21
221
111
ϕπϕϕπ
ϕϕπϕπ
ϕπϕ
c
ccQ
The limit state function
3210 QQQQ ++=
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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−
−+
−+=
24sin
24cos
24cos)tanexp()(
1
31
122 ϕπ
ϕϕπϕπϕπγDqQ
−−=
24cos
41 1
31ϕπγQ
The limit state function
( )2
3332313
QQQbQ ++=
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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( )
−+
−+
+−
+×
×
−+
=
24cos
24sintan3tan
23exp
24cos
24sintan3
24sin4tan912
1122
112
122
232
ϕπϕπϕϕπϕπϕπϕ
ϕπϕ
γQ
−
−+
=
24sin8
24costan
23expcos
12
31
23
33 ϕπ
ϕϕπϕπϕγQ
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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Medium sand
Example
Cohesionless soil
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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lognormal45.0 kN300 kNAxial load normal to the base N
nonrandom-24.0 kN/m3
Unit weight of foundation material
γb
nonrandom-23.0 kN/m3
Concrete floor unit weight γp
normal1.092 kN/m318.2 kN/m3
Soil Unit weight γ
lognormal4.6o32°Soil friction angle ϕ
Probability distribution
StandardDeviation σX
Mean value
Parameter
Probabilistic characteristic of parameters considered for numerical analyses – cohesionless soil
δv = 0.8 mApplications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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2.9172.7312.5762.4442.3302.2302.1422.0641.9941.9301.8721.8191.7701.725
3.9593.8123.6783.5563.4463.3453.2533.1683.0903.0182.9512.8892.8312.777
3.5873.3913.2203.0712.9402.8242.7202.6262.5412.4642.3932.3282.2682.213
4.84.84.84.84.84.84.84.84.84.84.84.84.84.8
1.21.41.61.82.02.22.42.62.83.03.23.43.63.8
3.1112.9162.7562.6182.4982.3952.3032.2172.1432.0742.0111.9601.9081.856
4.0143.8693.7373.6173.5073.4073.3143.2293.1513.0783.0112.9482.8892.835
3.6103.4153.2433.0942.9622.8422.7392.6472.5612.4812.4122.3492.2922.229
4.84.84.84.84.84.84.84.84.84.84.84.84.84.8
1.21.41.61.82.02.22.42.62.83.03.23.43.63.8
CDBCABCDBCAB
SegmentSegment
Reduced standard deviation ϕ [°]
Standard deviationof ϕ [°]
Width of the foundationb [m]
Reduced standard deviation of ϕ [°]
Standard deviationof ϕ [°]
Width of the foundation
b [m]
ω2 = 3ω1One-dimensional
πωδ 1=v πωδ 2=h
Comparison of standard deviations
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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3.1072.9152.7532.6152.4962.3912.2982.2152.1412.0732.0121.9561.9031.855
4.0073.8623.7303.6103.5013.4003.3083.2233.1453.0723.0052.9422.8842.829
3.6073.4123.2413.0922.9602.8442.7392.6452.5602.4822.4112.3462.2862.230
4.84.84.84.84.84.84.84.84.84.84.84.84.84.8
1.21.41.61.82.02.22.42.62.83.03.23.43.63.8
3,0892,8982,7362,5992,4802,3762,2832,2012,1272,0591,9981,9421,8901,843
4.0033.8583.7263.6063.4963.3963.3033.2183.1403.0673.0002.9382.8792.824
3.6053.4103.2403.0902.9592.8422.7382.6442.5582.4812.4102.3452.2842.228
4.84.84.84.84.84.84.84.84.84.84.84.84.84.8
1.21.41.61.82.02.22.42.62.83.03.23.43.63.8
CDBCABCDBCAB
SegmentSegment
Reduced standard deviation of ϕ [°]
Standard deviationof ϕ [°]
Width of the foundationb [m]
Reduced standard deviation of ϕ [°]
Standard deviationof ϕ [°]
Width of the foundation
b [m]
ω2 = 30ω1ω2 = 10ω1
Comparison of standard deviations
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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3.1072.9152.7532.6152.4962.3912.2982.2152.1412.0732.0121.9561.9031.855
4.0073.8623.7303.6103.5013.4003.3083.2233.1453.0723.0052.9422.8842.829
3.6073.4123.2413.0922.9602.8442.7392.6452.5602.4822.4112.3462.2862.230
4.84.84.84.84.84.84.84.84.84.84.84.84.84.8
1.21.41.61.82.02.22.42.62.83.03.23.43.63.8
3.1112.9162.7562.6182.4982.3952.3032.2172.1432.0742.0111.9601.9081.856
4.0143.8693.7373.6173.5073.4073.3143.2293.1513.0783.0112.9482.8892.835
3.6103.4153.2433.0942.9622.8422.7392.6472.5612.4812.4122.3492.2922.229
4.84.84.84.84.84.84.84.84.84.84.84.84.84.8
1.21.41.61.82.02.22.42.62.83.03.23.43.63.8
CDBCABCDBCAB
SegmentSegment
Reduced standard deviation of ϕ [°]
Standard deviationof ϕ [°]
Width of the foundationb [m]
Reduced standard deviation of ϕ [°]
Standard deviationof ϕ [°]
Width of the foundation
b [m]
ω2 = 30ω1One-dimensional
Comparison of standard deviations
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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Comparison of correlation coefficients
3.74701E-051.43088E-063.66429E-086.24374E-107.02954E-125.20151E-142.51961E-167.96581E-191.63978E-212.19363E-241.90399E-271.07077E-303.89732E-349.17226E-38
4.30754E-052.83633E-061.29973E-074.11650E-098.97937E-111.34662E-121.38730E-149.81440E-174.76713E-191.58971E-213.63950E-245.72077E-276.17475E-304.57769E-33
4.77876E-023.58498E-022.78413E-022.20502E-021.76622E-021.42342E-021.15013E-029.29340E-037.49600E-036.02630E-034.82360E-033.84080E-033.04100E-032.39130E-03
0.896470.886630.878170.871060.866010.861790.857250.854540.852120.851240.849110,844480.841620.84442
0.365300.336980.314400.296240.281510.269080.258770.250600.243430.237500.232470.227470.223710.22071
5.53350E-024.13260E-023.28861E-022.73803E-022.35114E-022.07034E-021.84270E-021.65867E-021.50881E-021.38416E-021.27558E-021.18205E-021.10017E-021.03310E-02
1.21.41.61.82.02.22.42.62.83.03.23.43.63.8
ρ(φAB, φCD)ρ(φBC, φCD)ρ(φAB, φBC)ρ(φAB, φCD)ρ(φBC, φCD)ρ(φAB, φBC)
ω2 = 3ω1One-dimensional
Correlation coefficientsWidth of the
foundationb [m]
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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Slip lines associated with the Prandtl mechanism
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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0.7846800.7373650.6889600.6398380.5904430.5412620.4928110.4455870.4000590.3566430.3156820.2774410.2421080.209779
0.316720.278790.247070.219880.196210.175400.156960.140540.125830.112610.100690.089920.080180.07137
5.53572E-024.13126E-023.29574E-022.74770E-022.35903E-022.06753E-021.83940E-021.65543E-021.50365E-021.37618E-021.26758E-021.17397E-021.09243E-021.02078E-02
0.896470.886630.878170.871060.866010.861790.857250.854540.852120.851240.849110,844480.841620.84442
0.365300.336980.314400.296240.281510.269080.258770.250600.243430.237500.232470.227470.223710.22071
5.53350E-024.13260E-023.28861E-022.73803E-022.35114E-022.07034E-021.84270E-021.65867E-021.50881E-021.38416E-021.27558E-021.18205E-021.10017E-021.03310E-02
1.21.41.61.82.02.22.42.62.83.03.23.43.63.8
ρ(φAB, φCD)ρ(φBC, φCD)ρ(φAB, φBC)ρ(φAB, φCD)ρ(φBC, φCD)ρ(φAB, φBC)
ω2 = 30ω1One-dimensional
Correlation coefficientsWidth of the foundation
b [m]
Comparison of correlation coefficients
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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1 1.5 2 2.5 3 3.5 4
2
4
6
8
10
β5
β6
β7
β8
B1
Reliability indices versus width of the foundation. One-dimensional case.
Particular curves are addressed to the following cases: β5: the friction angle of the subsoil is modelled by single random variable without spatial averaging; β6 : the friction angle of the subsoil is modelled by three independent random variables ϕ1, ϕ2, ϕ3 but spatial averaging is not incorporated; β7 : modelling by three independent random variables ϕ1, ϕ2, ϕ3 with incorporating spatial averaging; β8: three correlated random variables ϕ1, ϕ2, ϕ3 with incorporating spatial averaging.
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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1
2
3
4
5
6
7
8
9
10
1 1,5 2 2,5 3 3,5 4
b [m]
β9
β10
β11
β12
1
2
3
4
5
6
7
8
9
10
1 1,5 2 2,5 3 3,5 4
b [m]
β13
β14
β15
β16
Reliability indices versus width of the foundation. Cohesionless soil. Two-dimensional case. Particular curves are addressed to the following cases: β9: three independent random variables ϕ1, ϕ2, ϕ3 employed with two-dimensional spatial averaging, where ω2 = 3ω1; β10: three independent random variables ϕ1, ϕ2, ϕ3 involved, with two-dimensional spatial averaging, where ω2 = 10ω1; β11: three independent random variables ϕ1, ϕ2, ϕ3 with two-dimensional spatial averaging, where ω2 = 30ω1; β12 = β7: three independent random variables ϕ1, ϕ2, ϕ3 with one-dimensional spatial averaging;β13: three correlated random variables ϕ1, ϕ2, ϕ3 with two-dimensional spatial averaging, where ω2 = 3ω1; β14: three correlated random variables ϕ1, ϕ2, ϕ3 with two-dimensional spatial averaging, where ω2 = 10ω1; β15: three correlated random variables ϕ1, ϕ2, ϕ3 with two-dimensional spatial averaging, where ω2 = 30ω1; β16 = β8: three correlated random variables ϕ1, ϕ2, ϕ3 incorporated with one-dimensional spatial averaging.
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Example
Cohesive soil
Sandy clay
sand
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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lognormal50.0 kN500 kNAxial load normal to the base N
nonrandom-24.0 kN/m3Unit weight of foundation material γb
nonrandom-23.0 kN/m3Concrete floor unit weight γp
normal1.092 kN/m318.2 kN/m3Unit weight of sand in the vicinity of foundation γz
normal1.9 kN/m319.0 kN/m3Soil Unit weight γlognormal4.65 kPa 31 kPaCohesion clognormal2.7o18°Soil friction angle ϕ
Probability distribution
StandardDeviation σX
Mean valueParameter
Probabilistic characteristic of parameters considered for numerical analyses – cohesive soil
δv = 1.0 mApplications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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2.1712.0671.9731.8891.8131.7451.6841.6281.5781.5311.4891.4501.4131.379
2.4702.4062.3422.2772.2142.1542.0952.0401.9861.9351.8871.8411.7971.755
2.3272.2372.1522.0721.9981.9301.8671.8101.7571.7081.6631.6211.5821.546
2.72.72.72.72.72.72.72.72.72.72.72.72.72.7
1.21.41.61.82.02.22.42.62.83.03.23.43.63.8
2.2352.1312.0401.9601.8791.8111.7531.6961.6441.5991.5531.5461.4731.438
2.5732.5342.4932.4502.4072.3642.3212.2792.2382.1992.1612.1252.0902.056
2.3432.2572.1722.0912.0171.9481.8911.8281.7761.7301.6841.6391.6101.564
2.72.72.72.72.72.72.72.72.72.72.72.72.72.7
1.21.41.61.82.02.22.42.62.83.03.23.43.63.8
CDBCABCDBCAB
SegmentSegment
Reduced standard deviation ϕ [°]
Standard deviationof ϕ [°]
Width of the foundationb [m]
Reduced standard deviation of ϕ [°]
Standard deviationof ϕ [°]
Width of the foundation
b [m]
ω2 = 3ω1One-dimensional
Comparison of standard deviations of φ
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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2.2322.1322.0401.9561.8811.8121.7501.6941.6421.5951.5511.5111.4741.439
2.5732.5332.4922.4492.4052.3612.3182.2762.2352.1952.1572.1212.0852.052
2.3422.2542.1702.0912.0181.9501.8871.8301.7771.7231.6831.6401.6011.564
2.72.72.72.72.72.72.72.72.72.72.72.72.72.7
1.21.41.61.82.02.22.42.62.83.03.23.43.63.8
2.2352.1312.0401.9601.8791.8111.7531.6961.6441.5991.5531.5461.4731.438
2.5732.5342.4932.4502.4072.3642.3212.2792.2382.1992.1612.1252.0902.056
2.3432.2572.1722.0912.0171.9481.8911.8281.7761.7301.6841.6391.6101.564
2.72.72.72.72.72.72.72.72.72.72.72.72.72.7
1.21.41.61.82.02.22.42.62.83.03.23.43.63.8
CDBCABCDBCAB
SegmentSegment
Reduced standard deviation ϕ [°]
Standard deviationof ϕ [°]
Width of the foundationb [m]
Reduced standard deviation of ϕ [°]
Standard deviationof ϕ [°]
Width of the foundation
b [m]
ω2 = 30ω1One-dimensional
Comparison of standard deviations of φ
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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3.8433.6723.5133.3693.2393.1213.0152.9172.8282.7462.6722.6022.5382.478
4.4314.3634.2924.2174.1424.0673.9933.9203.8493.7813.7153.6523.5913.533
4.0333.8823.7383.6013.4753.3583.2503.1513.0602.9762.8982.8252.7582.694
4.654.654.654.654.654.654.654.654.654.654.654.654.654.65
1.21.41.61.82.02.22.42.62.83.03.23.43.63.8
3.8443.6733.5153.3713.2413.1233.0162.9192.8302.7482.6732.6622.5392.479
4.4314.3644.2934.2204.1454.0713.9973.9253.8553.7873.7223.6593.5993.541
4.0333.8833.7383.6023.4763.3593.2513.1523.0612.9772.8982.8262.7732.695
4.654.654.654.654.654.654.654.654.654.654.654.654.654.65
1.21.41.61.82.02.22.42.62.83.03.23.43.63.8
CDBCABCDBCAB
SegmentSegment
Reduced standard deviation of c [kPa
Standard deviation
of c[kPa]
Width of the foundationb [m]
Reduced standard deviation of c [kPa]
Standard deviation
of c[kPa]
Width of the foundation
b [m]
ω2 = 30ω1One-dimensional
Comparison of standard deviations of c
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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β
1 1.5 2 2.5 3 3.5 4
2
4
6
8
10
12
β1
β2
β3
β4
B
Reliability indices versus width of the foundation. Cohesive soil. One-dimensional case.
Particular curves are addressed to the following cases: β1: the friction angle φ and the cohesion c of the subsoil are modelled by two independent random variables without spatial averaging; β2 : soil strength parameters are modelled by six independent random variables ϕ1, c1,ϕ2, c2, ϕ3, c3 that corresponds to segments AB, BC i CD of the slip line (ϕ1, c1 corresponds to AB, etc.), but spatial averaging is not incorporated; β3 : soil strength parameters are modelled by six independent random variables ϕ1, c1,ϕ2, c2,ϕ3, c3 that corresponds to segments AB, BC i CD of the slip line (ϕ1, c1 corresponds to AB, etc.) with incorporating spatial averaging; β4: soil strength parameters are modelled by six correlated random variables ϕ1, c1,ϕ2, c2, ϕ3, c3 that corresponds to segments AB, BC i CD of the slip line (ϕ1, c1 corresponds to AB, etc.) with incorporating spatial averaging.
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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1 1.5 2 2.5 3 3.5 4
2
4
6
8
10
12
β1
β2
β3
β4
B
1 1.5 2 2.5 3 3.5 4
2
4
6
8
10
12
β5
β6
β7
β8
B1
Reliability indices versus width of the foundation. Cohesive soil. Comparison of cases. Fig. a: Independent random variables: β1: ϕ1, c1,ϕ2, c2, ϕ3, c3 with two-dimensional spatial averaging, where ω2 = 3ω1 ; β2: ϕ1, c1,ϕ2, c2, ϕ3, c3 with two-dimensional spatial averaging, where ω2 = 10ω1; β3: ϕ1, c1,ϕ2, c2, ϕ3, c3 with two-dimensional spatial averaging, where ω2 = 30ω1; β4 : ϕ1, c1,ϕ2, c2, ϕ3, c3 with incorporating one-dimensional spatial averaging.Fig. b: Correlated random variables: β5: ϕ1, c1,ϕ2, c2, ϕ3, c3 with two-dimensional spatial averaging, where ω2 = 3ω1; β6: ϕ1, c1,ϕ2, c2, ϕ3, c3 with two-dimensional spatial averaging, where ω2 = 10ω1; β7: ϕ1, c1,ϕ2, c2, ϕ3, c3 with two-dimensional spatial averaging, where ω2 = 30ω1 ; β8 : ϕ1, c1,ϕ2, c2, ϕ3, c3 with incorporating one-dimensional spatial averaging.
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• Remarks• The numerical studies have shown that by incorporating spatial
averaging one can significantly reduce standard deviations of soil strength parameters, which leads to a significant increase in reliability indices (decrease in failure probabilities). This is a step forward in making reliability measures more realistic in the context of well-designed (according to standards) foundations.
• Another important aspect is the correlation of the area of averaging with the potential failure mechanism.
• The results of numerical computations have also demonstrated that for reasonable, from practical point of view, values of horizontal scale of fluctuation (about 10 to 20 times greater than values of vertical fluctuation scale), the reliability measures obtained from two-dimensional averaging are almost the same as those corresponding to one-dimensional averaging. This means that in the case of shallow strip foundations one-dimensional (along the depth) averaging can be sufficient, which simplifies computations and requires a smalleramount of statistical data (vertical fluctuation scale instead of both vertical and horizontal scales).
Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop
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»hank
you
Thank you