Safety and soil-structure interaction: Safety and soil-structure interaction: a question of scalesa question of scales

Denys BREYSSEDenys BREYSSECDGA, Université Bordeaux ICDGA, Université Bordeaux I

Probamat – JHU Baltimore – 5-7/01/2005

Variations in the longitudinal profile of a pavement: IRI

Buried pipe collapseLongitudinal control of railway track (Mauzin)

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

types of structures (pavements, buried pipes, raft 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 structure

settlements [s]

Geotechnical study

Stresses in soilStresses in the structure

External loads [p]

boundary conditions

Stiffness - geometry

pressure on ground [q]

Structural study

pressure on ground [q] soil’s stiffness

stiffness – geometry

of the structure

settlements [s]

Geotechnical study

3

Stresses in soil

2

Stresses in the structure

4

1

External loads [p]

boundary conditions

Stiffness - geometry

pressure on ground [q]

Structural study

non linear behavior of interface non linear behaviour of soil

effect of boundary conditions on load redistributions

effect of loading on structural stiffness

1

4

3

2

Ground spatial variability Structural stiffness

First case :when variability reduces to uncertainty

p

E(x)

E(x) ?

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

It is the role of the engineer, who must choose design values

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”

3,2

3,4

3,6

3,8

4

4,2

4,4

4,6

0 5 10 15 20

Xi (m)

lc=2m

15m

200m

Second case : when spatial correlation has an influence

p

E(x)

E(x)

Role of lc

which explains and influences differential settlements:

existence of a « critical range » of lc values

20 m

0

0,00005

0,0001

0,00015

0,0002

0,00025

0,0003

0,00035

0,0004

0,00045

0,0005

0 50 100 150 200

ABS(Rota tion) 95%

ABS(Rota tion) 99%

Ecart-type(Rotation)

Lc (m)

rotation

50 %

95 %

99 %

0

0,00005

0,0001

0,00015

0,0002

0,00025

0,0003

0,00035

0,0004

0,00045

0,0005

0 50 100 150 200

ABS(Rota tion) 95%

ABS(Rota tion) 99%

Ecart-type(Rotation)

Lc (m)

rotation

50 %

95 %

99 %

Various correlation lengths lc

lc

B

L

Explaining the influence of correlation length: when it reduces to purely geometrical causes

Variance reduction below shallow foundation (B, lc)

Distance between supports L

Correlation coefficient between the two settlements s and s’ (Tang, 1984)

(s, s’) = [k (-1)k Lk² s²(Lk)] / [2 B B’ s(B) s(B’)]

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

(depends on the shape of the autocorrelation fct)

lcB

L

Variance reduction below the foundation

(B, Lc)

Correlation between supports

fct (L, Lc)

0,01 0,1 1 10 100 1000 10000

lc/B

L/B = 2

L/B = 5

L/B = 10

L/B = 20

L/B = 50

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

Three dimensions (B, L et Lc) drive the problem

Explaining the influence of correlation length: when it reduces to purely geometrical causes

Var (so) comes from the magnitude of local scatter of soil properties

Var (so) = 1

Third (general) case: stiffness(es) play a role

p

E(x)

p

E(x)

KR = (EcJ)/EsL3 relative structure/soil stiffness ratio

KR > 0.5 “stiff” structure f.i., with Es = 20 MPa and Ec = 30 000 MPa

- cylindric pipe L = 3 m, width R/10, stiff if R > 0,5 m,- slab L = 8 m stiff if depth > 1m30, - multi-storey building (wall thickness 20 cm), stiff if H>L/3

EN 1992 (Annex G - informative): must account for interactions and its consequences on loads (on soil and structure) and displacements

structure very soft loads on soil easy to computestructure very stiff displacements easy to compute

Reality intermediate

Needs a detailed modelling of:- soil heterogeneity

- materials response and soil/structure interaction

Ex. 1: beam on three supports (coll. G. Frantziskonis)Ex. 2: buried pipe (coll. SM. Elachachi)Ex. 3: poutre sur 3 appuis (coll. H. Niandou)

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

Two stiffnesses (soil Es and

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

Statistics and geostatistics

Mechanics and material modelling

Example 1: beam on three supportsE(soil) = 10 MPa (coll. G. Frantziskonis)

Moments on

supports and at

midspan(kN.m)

fixed supports

elastic supports

mean st.-dev. fractile 95 %

fractile 99 %

centralsupport

midspan

- 250 - 163 25.2 - 196 - 205

+ 125 + 169 12.4 +191 (+53%)

+208 (+66%)

0

50

100

150

200

250

300

-300 -200 -100 0 100M at central support (kN.m)

count

mea

n

0

50

100

150

200

250

300

350

400

100 150 200 250

M at halfspan (kN.m)

me

an

Example 2 : buried pipe (sewer)

connection soil (continuous elastic springs)

uniform loads

pipe

0

2

4

6

8

0

20

40

60

80

0

100

200

300

400

500

0

100

200

300

400

500

M95 (kN.m)

Lc/Lk (MN/m3)

Statistical analysis of bending moments

The problem is driven by ratios :- length ratio : L / lc - stiffness ratio k (soil) / EI (pipe)

Example 3 : raft foundation on piles

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

Lc = 10 m

1

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)

Lo

ad

on

ce

ntr

al

pil

e (

R3

) M

Nh=0.30m 0.60m 0.90m 1.20m 1.50m

Example 3 : raft foundation on piles

The response is statiscally driven by:- a length ratio L/Lc

- a stiffness ratio h3/k (= cte x KR)

1,2

1,3

1,4

1,5

1,6

1,7

1,8

1,9

2

2,1

2,2

0,1 1 10 100

correlation length Lc (m)

Load on central pile (MN)

5% 95% fractile

Ref. stiff raft

Consequences: different loads in piles and in raft safety estimate ?

+ 52 %

Critical value of the ratio

Example 3 : raft foundation on piles

2R

2E

2R

2E

E

R

V1V1ln

V1

V1ln

safety index

is mean, V is c.o.v.E regards load effect and R regards strength

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

11

Load effect

Pro

bab

ilit

y D

en

sity

Fu

ncti

on

Resistance

a

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

0 1 2 3 4 5 6d

d

Load effect

Resistance

Pro

bab

ilit

y D

ensi

ty F

un

ctio

n

b

Vr = 0,2, deterministic F.S. = 2,7

Deterministic load effect = 4,97

Pf = 0,34 10-6

Interaction and probabilistic load effect = 3,20

Pf = 0,69 10-3

Critical value of L/Lc = fct (stiffnesses and behavior, studied output) enables to reduce the cost of Monte-Carlo simulations

(Lc is not a variable or an unknown)

Trying to synthetizeFull interaction ?

yesno

Influence of geometry/scales

Influence of geometry AND stiffness

ratio L’/Lc ratio « EcI’/EsJ »

ratio « EcI/EsJ »ratio L/LcAccording the problemcomplexity

According the problemcomplexity

• Disorders/damage will depend on- soil properties (scatter, correlation length lc)

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

• Accounting for spatial variability of soilsand settlements of supports makes possible the description of main phenomena:

differential settlements, load redistribution

• Predicting safety levels requiresinputs representative of soil randomness

• Identifying « worst cases» (worst ratios)can enable •the use of deterministic models

while including the statistical effects of randomness

• Accounting for damage sensitivitymust be the following step

Conclusions