An uncoupled approach for the design of rockfall ... · An uncoupled approach for the design of rockfall protection shelters C. di Prisco and F. Calvetti. 2 Claudio di Prisco

Innsbruck March the 12th

An uncoupled approach for the design of rockfallprotection shelters

C. di Prisco and F. Calvetti

2

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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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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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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Claudio di Prisco

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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Claudio di Prisco

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


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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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

11

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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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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T1,2,3&4 as a function of falling height and block mass

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46


46Modelling stress propagation

a = B(T1)/2h = thickness of soil stratum

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r

Francesco Calvetti, Claudio di Prisco


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)

48



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

49


49Modelling stress propagation, geometric effect

maximum stress increment, as a function of rexperimental and numerical results

50


50Modelling stress propagation, dynamic effect

dynamic amplification factor, fa

51


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

52


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

Documents

An uncoupled approach for the design of rockfall ... · An uncoupled approach for the design of rockfall protection shelters C. di Prisco and F. Calvetti. 2 Claudio di Prisco