ON SOME ASPECTS OF RELIABILITY COMPUTATIONS IN …

ON SOME ASPECTS OF RELIABILITY

COMPUTATIONS IN BEARING

CAPACITY OF SHALLOW

FOUNDATIONS

Applications of Computation Mechanics in Geotechnical Engineering – 5th International Workshop

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.


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


Bearing capacity resistance Qf


++= γγγγ γγ sBiNsiqNsciNLBQ qqqqcccf 2

1

( )

+=

24tantanexp 2 ϕπϕπqN

( ) ϕcot1−= qc NN

( ) ϕγ tan12 −= qNN

,

Bearing capacity factors:

EC7: PN: ( ) ϕγ tan15.1 −= qNN

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


+=

−=

+=

LBs

LBs

LBs

c

q

3.01

25.01

5.11

γ

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

−−=


+

+

== 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


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∫





( )x =><

00




p = f ( )dx

X x x∫





( )x =><

00




p = f ( )dx

X x x∫


( )Fp10−Φ−=βReliability index , provided that

21

<Fp

As a computational tool the SORM is utilised

Sensitivity parameters

**

1yyy =∂

∂=

ii y

βα


( )Fp10−Φ−=βReliability index , provided that

21

<Fp

As a computational tool the SORM is utilised

Sensitivity parameters

**

1yyy =∂

∂=

ii y

βα


Schematic presentation of the FORM method


Shallow strip foundation

1=== γsss qc

Medium sand

Scheme of shallow strip foundation considered in the example



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


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

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.


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


β

B [m]

Reliability index β versus width of the foundation B for three different probability distribution of φ


Uniform (rectangular)LognormalNormal

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


β

h [m]

An effect of water table variability h = 0.3 – 1.6 m


Cohesive soil


sand

clay

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)


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


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


( )∫∫∫=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


!!!

( )

>∆

≤∆∆

−=∆

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


( )

∆−=∆

2

3 expδ

πρ zz

( ) 2

2

3

exp1erf

−+−

⋅

=

δπ

δπ

δπ

δπ

γL

LLL

L

Gaussian correlation function

( ) ( )∫ −=t

dxxt0

2exp2erfπ


( )

∆+

∆−=∆∆

2

2

2

14 exp,

ωωρ xzzx

1 2v hδ ω π δ ω π= =

Two-dimensional Gaussian correlation function


Example

•Cohesionless soil

•One-dimensional Gaussian correlation function

•δ = 0.8 m, the averaging area (interval) is L = 2B


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.


Example

•Cohesive soil

•One-dimensional Gaussian correlation function

•δ = 1.0 m, the averaging area (interval) is L = 2B


-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


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.


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


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.


Slip lines associated with the Prandtl mechanism


•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 ≠


],[)(),(

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


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


{ } ( ) ( )

( )∫∫

∫∫

−−=

=+

−−

+

=

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




= 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


[ ] ( ) ( ) ( ) ( )

( ) ( )

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


[ ] ( )

( ) 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


( )( )

( )

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


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




+=

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


+=

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


[ ] ( )[ ( ) ( ) ( )]

( )[ ( ) ( ) ( )] ( ) ( ) 21212

221122

20

2221

243

24

121

20

243

24

02

tanexptanexptanexpcostanexpcos

tanexpsintanexpsinexpVAR

θθϕθϕθϕθθϕθθω

ϕθθϕθθω

ασ

ϕπ

ϕπ

ϕπ

ϕπ

ddr

rX XBC

−×−

−

−= ∫∫

+

+

+

+


−

=

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 ++=


−

−+

−+=

24sin

24cos

24cos)tanexp()(

1

31

122 ϕπ

ϕϕπϕπϕπγDqQ

−−=

24cos

41 1

31ϕπγQ

The limit state function

( )2

3332313

QQQbQ ++=


( )

−+

−+

+−

+×

×

−+

=

24cos

24sintan3tan

23exp

24cos

24sintan3

24sin4tan912

1122

112

122

232

ϕπϕπϕϕπϕπϕπϕ

ϕπϕ

γQ

−

−+

=

24sin8

24costan

23expcos

12

31

23

33 ϕπ

ϕϕπϕπϕγQ


Medium sand

Example

Cohesionless soil


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



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

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 [°]


Width of the foundation

b [m]

ω2 = 3ω1One-dimensional

πωδ 1=v πωδ 2=h

Comparison of standard deviations


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







b [m]

ω2 = 30ω1ω2 = 10ω1



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







b [m]




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)


Correlation coefficientsWidth of the

foundationb [m]




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)


Correlation coefficientsWidth of the foundation

b [m]

Comparison of correlation coefficients


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.


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.

Example

Cohesive soil

Sandy clay

sand


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



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

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







b [m]


Comparison of standard deviations of φ


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







b [m]


Comparison of standard deviations of φ


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]


Reduced standard deviation of c [kPa]

Standard deviation

of c[kPa]


b [m]


Comparison of standard deviations of c


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.


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.

• 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).


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