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
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
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.
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
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
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
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
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
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”
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
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)
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
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
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
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
Variance reductionbelow the foundation
Γ(B, λ )
Correlationbetween supports
fct (D, λ)
λ/B
95% fractile of differential settlement
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
When it reduces to purely geometrical causes: about reliability
Variance reductionbelow the foundation
Γ(B, λ ) for bothindiv. and diff. settl.
Correlationbetween supports
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
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
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
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
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
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
Example 1 : buried pipes (sewer)
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
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
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
Example 1 : buried pipes (sewer)
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
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
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
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
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)
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
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)
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
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)
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
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)
International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada
(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