Influence of spatial variations in soil properties on

International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada

Influence Influence ofof spatial variations spatial variations in in soilsoil propertiesproperties on structural on structural reliabilityreliability

Denys BREYSSE, Sidi Mohamed ELACHACHI, Denys BREYSSE, Sidi Mohamed ELACHACHI, HalidouHalidou NIANDOU, NIANDOU, Christian LA BORDERIEChristian LA BORDERIE

CDGA, UniversitCDGA, Universitéé Bordeaux I, FranceBordeaux I, Franceandand LasagecLasagec, UPPA, France, UPPA, France


Variations in the longitudinal profile of a pavement: IRI

Buried pipe collapseLongitudinal control of railway track shape

The longitudinal variation of soil properties has a major influence for many types of

structures

(pavements, buried pipes, foundations, railways...)

since it induces stresses and/or displacements that cannot be predicted

when assuming homogeneity.


pressure on ground [q] soil’s stiffness

stiffness – geometry

of the interface

settlements [s]

Geotechnical study

4

Stresses in soil

1

Stresses in the structure

2

3

External loads [p]

boundary conditions

Stiffness - geometry

pressure on ground [q]

Structural study


non linear behaviour of soil and building materials

non linear behavior of interface

effect of boundary conditions on load redistributions

3

4

1-2

Ground spatial variability Soil and structuralstiffnesses


The simplest case :when variability reduces to uncertainty

p

E(x)

E(x) ?

Estimate of a design (or « risky ») value of EEstimate of a design (or « risky ») value of settlement

It is the role of the engineer, who must define design values (« cautious estimate... » - Eurocodes)

Duncan (2000) : 3-sigma rule

“a conscious effort should be make the range between higher conceivable value and lower conceivable value as wide as seemingly possible, or even wider, to overcome the natural tendancy to make the range too small”


λB

D

Influence of correlation length:(a) when it reduces to purely geometrical causes

Variance reduction belowshallow foundation Γ(B, λ)

Distance between supportsD

Correlation coefficient between the two settlements s and s’ (Tang, 1984)ρ(s, s’) = [Σk (-1)k Lk² Γs²(Lk)] / [2 B B’ Γs(B) Γs(B’)]

k = 0 - 3 L0 = D – B/2 – B’/2, L1 = D + B, L2 = D + B + B’, L3 = D + B’Γ²(x) variance reduction function

(depends on the shape of the autocorrelation fct)


Variance reductionbelow the foundation

Γ(B, λ)

Correlationbetween supports

fct (D, λ)

0

0,4

0,8

1,2

1,6

2

0,01 0,1 1 10 100 1000 10000

lc/B

L/B = 2L/B = 5L/B = 10L/B = 20L/B = 50

Var(Δ) = 4 (λ/B)² (B/λ – 1 + e-B/λ) . (1- ρ(s, s’)) . Var (so)

When it reduces to purely geometrical causes: differential settlement

Var (so) comes fromthe magnitude of local

scatteringof soil properties

3 dimensions (B, D and λ) govern the problem

Var (so) = 1

Mag

nitu

de o

fdiff

. set

tlem

ent

λ/B

increa

sing D/B

D/B


When the influence of correlation length reduces to purely geometrical causes

Monte-Carlo simulations for 32 values of λ and 4 values of D (250 simulations, q = 300 kPa, E = 10MPa)

tass.diff. caractéristique (m)

0,000

0,002

0,004

0,006

0,008

0,010

0,012

0,014

0,016

0,018

0,020

0,10 1,00 10,00 100,00 1000,00

D/B = 2

D/B = 5

D/B =10

D/B = 20


Γ(B, λ )


fct (D, λ)

λ/B

95% fractile of differential settlement


When it reduces to purely geometrical causes: about reliability


Γ(B, λ ) for bothindiv. and diff. settl.


fct (D, λ)

λ/B

moy. tass. diff. / e-t (so)

0,00

0,20

0,40

0,60

0,80

1,00

1,20

1,40

0,10 1,00 10,00 100,00 1000,00

D/B = 2

D/B = 5

D/B =10

D/B = 20

mean differentialsettlement / s.d. individual settlement

dmean ≤ (2/√π) s.d. ≈ 1.128 s.d.

tass. diff. caract. / e-t (so)

0,00

0,50

1,00

1,50

2,00

2,50

3,00

3,50

0,10 1,00 10,00 100,00 1000,00

D/B = 2

D/B = 5

D/B =10

D/B = 20

95% fractile of differential settlement/ s.d. individual settlement

d95 ≤ (1.96√2) s.d. ≈ 2.77 s.d.

Possibility to adoptconservative rules, without knowing λ


Requires a detailed modelling of both:- soil heterogeneity

- materials response and soil/structure interaction

Ex. 1: buried pipes Ex. 2: piled-raft foundation

(b) Explaining the influence of correlation length:when it does not reduce to purely geometrical causes

Several dimensions (defining the structure geometry and the soil

correlation scale) have an influence

Several stiffnesses(soil Es and

structure [Ec, J, L]) have an influence

Statistics andgeostatistics

Mechanics andmaterial modelling


Example 1 : buried pipes (sewer)

0

2

4

68

0

20

40

6080

0

100

200

300

400

500

0

100

200

300

400

500M95 (kN.m)

λ/Dk (MN/m3)

Statistical analysisof

bending moments

M95

The problem is driven by two ratios :

- length ratio : λ / D

- stiffness ratio k (soil) / EI (pipe)

connection soil (continuous Winkler springs)

uniform loads

pipe



connection soil (continuous elastic springs)

uniform loads

pipe

statistical analysisof

bending moments (ULS) andof

misalignements (SLS)

focus on the connection stiffness influence (stiffness fictitious length)

PfULS (Mmax < Mlim)

PfSLS (CSmax < CSlim)

random variables : k (c.v., λ), Mlim, CSlim

βULS

βSLS


0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

1.00E-10 1.00E-05 1.00E+00 1.00E+05 1.00E+10

0.10.20.40.8Eurocode


with λ = 3 m = D, c.v. (Mlim) = 0.15, c.v. (Slim) = 0.2

0.0

1.0

2.0

3.0

4.0

5.0

1.00E-10 1.00E-05 1.00E+00 1.00E+05 1.00E+10

0.10.20.40.8Eurocode

βSLS

soft connection stiff connection

βULS

soft connection stiff connection

It is difficult to satisfy target reliability levels both at SLS and ULS.

It demands a not too heterogeneous soil, and a stiff enough connection


Example 2 : piled-raft foundation

L=4mL=4mL=4mL=4mL=4m

k6k5k4k3

EI

L=4m

k2k1k0

q

2 reference solutions (q = 400 kN/m) :- k 0 (stiff raft)

uniform load in all piles = 1.38 MN- k infinity (stiff piles)

load R3 = 1.63 MN

λ = 10 m1

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,8

1,9

1E-06 0,00001 0,0001 0,001 0,01 0,1 1

h3/k (m 4/MN)

Load

on

cent

ral p

ile (R

3) M

N

h=0.30m 0.60m 0.90m 1.20m 1.50m


Example 2 : raft foundation on piles

The response is statistically driven by:- a length ratio λ/D

- a stiffness ratio EI h3/k

Results from engineer’s estimates highly underestimate stresses

Worst value of the ratio

1,21,3

1,41,51,61,7

1,81,9

2

2,12,2

0,1 1 10 100

correlation length Lc (m)

Load on central pile (MN)

5% 95% fractile

Ref. stiff raft

+ 52 %

λ (m)


Example 2 : raft foundation on piles

reliability index (Cornell)

μ is mean, σ is c.o.v.E regards load effect and R regards pile strength

β depends on:- soil variability (c.v. and λ),- pile variability (c.v.),- structural model (interaction)

deterministic F.S. = 2.7

2E

2R

ER

σ+σ

μ−μ=β

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

0 1 2 3 4 5 6

dd

Load effect

Resistance

Prob

abili

ty D

ensi

ty F

unct

ion

( )b

1,75

2

2,25

2,5

2,75

3

3,25

3,5

3,75

4

0,1 1 10 100

correlation length

relia

bilit

y in

dex

cv=0,10cv=0,15cv=0,20cv=0,25cv=0,30cv=0,35cv=0,40

effect of soil variability

effect of corr. length and interaction (λ = 4m)


Trying to synthetizeinteraction ?

no

Influence ofstatistics

cv(E), pdf(E)

yes

full interaction ?

Influence of variability reduces to uncertaintychoice of design values

(the challenge is to link design values and saftey level)


According theproblem complexity

Trying to synthetizeFull interaction ?

ratio λ/D’

ratio λ/D

no

Influence ofgeometry/scales

yes

Influence of geometryAND stiffness

ratio « EcI’/EsJ »

ratio « EcI/EsJ »

According theproblem complexity

« Worst » value of λ/D = fct (stiffnesses, type of output)enables to reduce the costs of geotechnical investigations

and of Monte-Carlo simulations (λ is no longer a variable or an unknown)


(1) Disorders/damage depend on- soil properties (scatter, correlation length λ)

- structural geometry (D, B…)- mechanical behaviour of the system

(2) Accounting for spatial variability of soil enables the description of main phenomena: differential settlements, load redistributions

(3) Predicting safety/reliability levels requires to reproduce these effects andto input data representative of soil randomness

(4) Identifying « worst cases» (worst ratios) can enablethe use of deterministic models

while including the statistical effects of randomness

Accounting for damage sensitivitymust be the following step

Conclusions

Documents

Influence of spatial variations in soil properties on