Innsbruck March the 12th
An uncoupled approach for the design of rockfallprotection shelters
C. di Prisco and F. Calvetti
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Claudio di Prisco
ALERT Geomaterials Doctorate school
Mechanics of UnsaturatedGeomaterials
ALERT Olek ZienkiewiczCourse 2010
5-9 July 2010, Lausanne, Switzerland
NUMERICAL MODELLINGIN GEOMECHANICS
Madrid, 22 - 26 June 2009
R. Borja , Stanford University, L. Boutonnier , EgisR. Charlier , Université de Liège, F. Collin , Université de Liège
P. Delage , Ecole des Ponts ParisTech, A. Ferrari , EPFL
B. François , Université de Liège, A. Gens , Universitat Politecnica de
Catalunya, R. Horn , Christian�Albrechts�Universität, T. Hueckel , Duke University, C. Jommi , Politecnico di Milano, A. Koliji , STUCKY LTD
L. Laloui , EPFL, M. Nuth , EPFL, S. Salager , EPFL, P. Selvadurai , McGill University, K. Soga , University of Cambridge, A. Tarantino , Università degli
Studi di Trento
ALERT Doctoral SchoOl 20107-10 October 2009, Aussois, France"Mathematical Modelling in Geomechanics"In memory of Prof. Ioannis Vardoulakis Coordination: J. Sulem and E. Papamichos
http://alert.epfl.ch/index.htm
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Claudio di Prisco
Scheltering structures: theoretical analysis of the impact of rigidbodies on dissipative strata
Alternative applications
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Trajectory of the boulder along the slope: restitution coefficients
STONE (Crosta e Agliari (2001)
Alternative applications
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SHELTERING GALLERIES
Cantilever sheltering gallery
Standard sheltering gallery
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Impacts on shelters
1 Near field impact2 Stress propagation3 Structural response
F1(t)≠≠≠≠ F2(t)≠≠≠≠F3(t)
F1 Impact force ⇔ F2 Actions on the shelter ⇔ F3 Structural actions
Local interaction: boulder-soil surface
Stress propagation Structural response
1
2
3
http://www.prometeo.polimi.it/POSS/POSSing/POSS_CadutaMassi_eng.html
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Claudio di Prisco
1. Usually this type of structures is designed under static loads (the weight of the dissipative cushion) and the weight of the reinforced concrete structure
Introductive observations/statements
2. Sometimes, the gallery is designed according to a sort of pseudo-static approach
3. A full coupled capable numerical approach capable of simulating the structural response under dynamic induced by impact loading conditions nowadays is not yet available4. From a geotechnical/structural point of view, the correct design of the thickness of the dissipative granular cushion is a crucial item puzzling the designer, indeed this affects the structural designeven if only static/permanent loads are accounted for.
0. In principle for sheltering structures accidental loads cannot be disregarded at all, since they are designed just to reduce the risk (in particular to reduce the vulnerability and not the hazard) associated to a quite catastrophic event (rockfall )
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What could be the dream of the structural engineer?
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In the perspective of a pseudo-static approach, the maximum value of F1 should be sufficient to be known and some empirical correlations have been thus proposed in literature:
IMPACT FORCE
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E0 and n depend on both the rock mass and the relative density of the granular soil
DEM Numerical results
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PENETRATION DEPTH
These results cannot be extrapolated as these data severely depend on the boulder radius
Is the stratum thickness sufficient for preventing the direct impact of the rock on the underneath slab?
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•Experiments (Bovisa, Listolade)
•Rheological Modelling (Impact Enhanced Macroelement approach)
•Numerical simulations (Distinct Elements, Finite differences, Spectral Elements)
Research activities
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•Rheological Modelling (Impact Enhanced Macroelement approach)
What are the hypotheses of the uncoupled approach hereafter proposed?
1
2
3 3
Granular cushion
structureF1 can be calculated by assuming an infinite halfspace?
F2 can be computed by assuming the slab to be like a rigid boundary?
F3 can be evaluated by disregarding the presence of the soil stratum?
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Claudio di Prisco
Experimental data (Labiouse et al. , 1994)
Parametro Valorem (kg) 100.00r (m) 0.21
V0 (m/s) 14.00Blocco roccioso
θIN (rad) 0.00K (-) 800.00n (-) 0.40
α (m-1) 0υ (-) 0.30
γ (N/m3) 18000.00Nγ (-) 40.00Dr (-) 0.90
Φ’ (rad) 0.64ω (rad) 0.00
γ0 (m/Ns) 0.00012c (m/s) 1.00
Substrato granulare
∆ (-) -0.95
Small scale experimental test results
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Claudio di Prisco
SUPPORTING STRUCTUREpillars (external) & wall (mountain
side)
Reference rockfall shelter, Listolade SS203 (BL)[length 100 m, span 10.5 m]
RoofPRC beams (H 80 cm)
+RC slab CA (H 30 cm)
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Sampling
Ultrasonic, pachometer
Details of the structure
Geometrical
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Structural monitoring
Soil - slab interface stressesload cells
Beam deflction
LVDTGefran PY2 (10 mm)Penny & Giles SLS190 (25 mm)
Accelerometer
ACTIONS
EFFECTS OF ACTIONS
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Cushioning Soil:in situ since 70s, sand/gravel/debris
Initial situation
ExcavationReady for impactsthickness 2m
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FALLING HEIGHT: 5 - 45 mIMPACT ENERGY: 40 - 400 kJ
Impact programmeImpact block
LIFTING DEVICEtelescopic crane
RC spheremass, M: 850 kg,
diameter, D: 90 cm
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Typical ImpactE = 400 kJ
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0 0.04 0.08 0.120.02 0.06 0.1
t [s]
0
200
400
600
100
300
500
∆σv [kPa]
Soil-slab stress
0 0.04 0.08 0.120.02 0.06 0.1
t [s]
4
0
-4
6
2
-2
-6
s [mm]
deflectionForza d’impatto
0 0.04 0.08 0.120.02 0.06 0.1
t [s]
0
400
800
1200
1600
200
600
1000
1400
1800
F [kN]
ResultsTypical impact (H = 36 m)
0 0.2 0.4 0.60.1 0.3 0.5
t [s]
0
400
800
1200
1600
200
600
1000
1400
1800
F [kN]
0 0.2 0.4 0.60.1 0.3 0.5
t [s]
4
0
-4
6
2
-2
-6
s [mm]
deflection
0 0.2 0.4 0.60.1 0.3 0.5
t [s]
0
200
400
600
100
300
500
∆σv [kPa]
Impact force Soil-slab stress
T1 T3 T3T2
2T2
rigorously
Impact force ⇒ Actions on the slab ⇔ Structural response
Conservative simplification
Impact force ⇒ Actions on the slab ⇒ Structural response
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Measurement point
Plan view of impacts
ResultsStructural response
0 0.04 0.08 0.12 0.160.02 0.06 0.1 0.14
t [s]
6
4
2
0
-2
5
3
1
-1
s [mm]
Impact point
0 0.04 0.08 0.12 0.160.02 0.06 0.1 0.14
t [s]
6
4
2
0
-2
5
3
1
-1
s [mm]
3.5 m from impact
0 0.04 0.08 0.12 0.160.02 0.06 0.1 0.14
t [s]
6
4
2
0
-2
5
3
1
-1
s [mm]
7 m from impact
•T2
•T2
•T2
sMAX
sMAX
sMAX
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Measurement point
Plan view of impacts
ResultsSoil structure interaction
Beam acceleration
1 g
Action on the slab
Actions ⇔
Effects of actions
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0 0.02 0.04 0.06 0.080.01 0.03 0.05 0.07
t [s]
0
100
200
300
50
150
250
a [
g]
Prova 4
Prova 15
0 0.02 0.04 0.06 0.080.01 0.03 0.05 0.07
t [s]
0
1
2
3
4
5
0.5
1.5
2.5
3.5
4.5
s [
mm
]
Prova 4
Prova 15
IMPACT FORCE STRUCTURAL RESPONSE
Structural response in dynamic conditions
First impactSecond impact
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Dynamical response
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Some remarks on the impact force evaluation
Dependence of the curves describing the evolution of the impact force with time on heterogeneity of the soil stratum: wave reflection
Evaluation of the function F1 versus time
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Claudio di PriscoClaudio di Prisco
29Modelling the near-field impact and IMPACT FORCE
BIMPAM MODEL (boulder kinematics & impact force)derived from MACROELEMENT for shallow foundations
boulder - soil interaction
deformability of the layer and 3D damping
visco-plastic slider
ξ=V/VMAX
V
B
qqQQDQ ptttɺɺɺ ))(,,( ψ=
dq C dQ= ⋅
=
333231
232221
131211
CCC
CCC
CCC
C
Defined in quasi static conditions: INTERACTION CONSTITUTIVE RELATIONSHIP
THE MACRO-ELEMENT CONCEPT as an homogeneisation theory
Incremental constitutive relationship
1
FOUNDATION
NEAR FIELD
FAR FIELD
Interaction domain for rigid strip footings
and elasto-perfectly plastic approach
m = M/ψBVMAX,
h = H/µVMAX
ξ = V/VMAX
1. To each point belonging to the failure locus a distinct failuremechanism corresponds
2. Difficulty in defining the failurelocus when loose sand strata are concerned
3. Non-associated flow rule
4. Within the failure locus the mechanical behaviour is assumed to be elastic and uncoupled
5. Extension to rectangular footings
2
MONOTONOUSLY INCREASING LOADING
H/M
Penetration mechanismSliding / toppling mechanism
coupling
V
The coupling is essentially due to the deformability/limited strenghtof the soil stratum
31 2 2 22 2
1 2 1 2
( ) ( ) (1 ) 0m m
m m m m
V VF Q n h m
V V V Vδδ δ γ βξ ξ= + + − − − + =
− −
Soil-pipe interaction
Role of the footing embedment
3
Pipelines passing across active landslides
Failure mechanisms
22 2
2 1 0c
M H Vf V
B V
β
ψ µ = + − − ≤
Isotropic hardening (Nova and Montrasio, 1991)
Swipe testsButterfield and Gottardi, 2003
H
V
V
M/B
ρc=Vc / VMAX
Hardening parameter
Plastic potential
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( )2
2 2 2 2 2, 1 0gg
g Q h m
βξρ λ χ ξρ
= + − ⋅ − =
0, , , ,Rλ χ α γ Constitutive parameters
Innsbruck March the 12th
Nova and Montrasio, 1991
Inclined and eccentric loading
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Dynamic model equations:
Mass
Elastic Spring
(K)Viscous Damper
(C)
Visco - Plastic
Slider
( ) ( )⋅ + ⋅ − + ⋅ − =ɺɺ ɺ ɺvp vp
ij j ij j j ij j j iM u C u u K u u b
y y
x x
u u
u u
Bk 0
G
B0 k
G
ηρ
ηρ
⋅ ⋅ = ⋅ ⋅
Cm 0
0 m
=
M N
T
k 0K
0 k
=
Uncoupled matrices
mg=b t
nt
v
v
ωωωω
In this case fast loading are considered and neither the i mpact force nor the genralised strain variables (displacements) are a priori known:
they derive from the solution of this differential equati ons system
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Viscous Nucleus ΦΦΦΦ
d d( , f )= l
( )vp
vpN
T
guR d,
u2
lξ
∂ = = Φ ⋅ ∂
ɺ
ɺ
ɺvpu
FOR FAST LOADING:
1 2 3F F F F= + + Jaeger and Nagel (1992
dissipative, the dynamic and the static componentsBoguslavskiiet al., 1996
( )22F c v t Aρ= ⋅ ⋅ ⋅ Sedov (1959)
Distance d
Φ(d
) / ξ
cV
∆
γV / (∆1)0.5
slidingDistance d
P
Image point
Radial mapping
γ
d≈
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Pseudo-elastic response
Pseudo-elastoplastic response
Elasto-viscoplasticresponse
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Claudio di PriscoFrancesco Calvetti, Claudio di Prisco
38Modelling the near-field impact and IMPACT FORCE
block deceleration, velocity and penetration into the soil layer as a function of time
model vs. experiments
model:dense and loose sand
layers
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Claudio di PriscoClaudio di Prisco
39Modelling the near-field impact and IMPACT FORCE
impact force as a function of timevarious falling height and block mass
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Distance d
reboundimpact Penetration failure
mechanism
Interface
mechanism
lN/VMAX
lT/µµµµVMA
X ωlN
lT
Inclined trajectories and inclined planes
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Impact on Impact on inclinedinclined granular granular stratastrata
VIN
VOUT
θIN
θOUT
Restitution coefficient
Impact forces
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DESIGN APPROACH: impact force (BIMPAM MODEL)
Time
Imp
ac
t L
oa
d
T1 T2 T4T3
FMAX
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1 2
22 3
2 4
3 4
se 0 t<T
se T T
se T T
MAX
MAX
MAX
MAX
FF t
T
F F t
TF F t
t
T T tF F
T T T
α
α
= ≤
= ≤ <
= ⋅ ≤ <
−= ⋅ ⋅ −
3 43
se T Tt
≤ <
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DESIGN APPROACH: impact force (BIMPAM MODEL)
0 20 40 60 80 100Kinetic energy E/E0 (-)
0
5
10
15
20
25
Imp
ac
t L
oa
d L
/L0 (
-)
L/L0 = a (E/E0)n
E0 = 100kJ
L0 = 527.1 kN
a = 1.00
n = 2/3
FMAX as a function of impact energy
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DESIGN APPROACH: impact force (BIMPAM MODEL)
T1,2,3&4 as a function of falling height and block mass
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DESIGN APPROACH: impact force (BIMPAM MODEL)
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Claudio di PriscoFrancesco Calvetti, Claudio di Prisco
46Modelling stress propagation
a = B(T1)/2h = thickness of soil stratum
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Claudio di Prisco
r
Francesco Calvetti, Claudio di Prisco
47Modelling stress propagation
NUMERICL ANALYSIS (FLAC, GEOELSE)finite differences - spectral element codes
wave propagation
stress increment on structure
∆σ(t,r)
F(t) σ(t) = F(t)/A(t)
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Claudio di PriscoClaudio di Prisco
48Modelling stress propagation
stress increment at r = 0, as a function of timeexperimental and numerical results
elastic soil layer elasto-plastic soil layer elasto-viscoplastic soil layer
Only the elastoviscoplastic solution seems to be capable satisfactorily simulating the propagation of the dynamic wave within the soil stratum by keeping unaltered the frequency content of the wave
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Claudio di PriscoClaudio di Prisco
49Modelling stress propagation, geometric effect
maximum stress increment, as a function of rexperimental and numerical results
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Claudio di PriscoClaudio di Prisco
50Modelling stress propagation, dynamic effect
dynamic amplification factor, fa
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Claudio di PriscoFrancesco Calvetti, Claudio di Prisco
51SYNTHETIC STRESS INCREMENT
tarr is a function of distance d and wave velocity
σMAX(r)
If an elastic numerical analysis is performed, the frequency content of the perturbation must be evaluated empirically from the trend of F1 versus time already defined
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Claudio di PriscoFrancesco Calvetti, Claudio di Prisco
52CONCLUSIONS
Uncoupled, one-way, design approachDesign charts and simple formulae are provided in order to obtain impact force, and stress increments on the structureDesign steps implemented in a spreadsheetNumerical dynamic analysis of the structure under the stress increments previously determined has to be performed (simplifieddynamic analysis is possible, by evaluating 1D equivalent stiffness and mass)
Design charts for structural response are under development