Mechanical response of shallow foundations - Some experimental/theoretical and numerical issues: monotonic loading

IntroductionProf. ing. Claudio di Prisco

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Prof. ing. Claudio di Prisco

Experimental device

10 20 30–800

–400

0

400

800

Time (sec)

Acc

.　(g

al) Max= 712 gal

20 40 60 80 100 120–600–400–200

0200400600

Time (sec)

Acc

.　(g

al) Max= 608 gal

Shear stacksShear stacks

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Prof. ing. Claudio di Prisco

Experimental evidence

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Prof. ing. Claudio di Prisco

The elasto-perfectly plastic approach

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Prof. ing. Claudio di Prisco

Coupled elastoplastic CONSTITUTIVE RELATIONSHIP

qqQQDQ pttt ))(,,(

dq C dQ

Defined　in　quasi　static　conditions:　　

THE MACRO-ELEMENT THE MACRO-ELEMENT CONCEPTCONCEPT

333231

232221

131211

CCC

CCC

CCC

C

Incremental constitutive relationship

FOUNDATION

NEAR FIELD

FAR FIELD

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Prof. ing. Claudio di Prisco

The interaction domain for rigid footings

Interaction domain for rigid strip Interaction domain for rigid strip footings and elasto-perfectly plastic footings and elasto-perfectly plastic approachapproach

m = M/ψBVMAX,

h = H/μVMAX

ξ = V/VMAX

1. To each point belonging to the failure locus a distinct failure mechanism corresponds

2. Difficulty in defining the failure locus 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

MONOTONOUSLY INCREASING LOADING

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Prof. ing. Claudio di Prisco

Settlements

-0.002 -0.001 0 0.001 0.002horizontal displacement (m )

-120

-80

-40

0

40

80

120

ho

rizo

nta

l lo

ad

(k

N/m

)

Cyclic Load

Centered vertical load Centered inclined load

Cyclic inclined loading

anisotropic hardening

Cremer et al, 2001

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Prof. ing. Claudio di Prisco

Hardening

22 2

2 1 0c

M H Vf V

B V

Isotropic hardening (Nova and Montrasio, 1991)

Swipe testsButterfield and Gottardi, 2003

H

V

V

M/B

c=Vc / VMAX

Hardening　parameter

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Prof. ing. Claudio di Prisco

Inclined-eccentric loads

Nova and Montrasio, 1991

Inclined and eccentric loadingInclined and eccentric loading

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Prof. ing. Claudio di Prisco

How can R0 be determined?

Some hints about settlement evaluation

Homogeneous stratigraphic profiles

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Prof. ing. Claudio di Prisco

Settlement evaluation

• Irreversibility of the material mechanical response

• Non-linearity

• Dependency of stiffness on confining pressure

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Prof. ing. Claudio di Prisco

The elastic solution

Poulos e Davis (1974)

Gibson (1973)

Winkler (1867)

Dependency of stiffness on confining pressure:

0( )E z E z

Eo/λD

1

0

0.5

r/D0 1

= 0.5

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Prof. ing. Claudio di Prisco

Settlement evaluation

The elastic solutionThe elastic solution

21q BI

E

Linear loading distribution Strip loading distribution

Rectangular loading distribution

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Prof. ing. Claudio di Prisco

Generalised plasticity

dq I)pl

( ( , ) ciclig

Q

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Prof. ing. Claudio di Prisco

Numerical validation

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Prof. ing. Claudio di Prisco

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Prof. ing. Claudio di Prisco

Cyclic loads

c)

b)

a)

Loose sandLoose sand Dense sandDense sand

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Prof. ing. Claudio di Prisco

Numerical validation

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Prof. ing. Claudio di Prisco

Numerical results

1E-005 0.0001 0.001 0.01 0.1

rocking angle (rad)

0

0.2

0.4

0.6

0.8

1

K

/ K 0

(-)

dense sandVM A X /V = 2VM A X /V = 4VM A X /V = 6VM A X /V = 8VM A X /V = 10

1E-005 0.0001 0.001 0.01 0.1

rocking angle (rad)

0

0.2

0.4

0.6

0.8

1

K

/ K 0

(-)

medium density sandVM A X /V = 2VM A X /V = 4VM A X /V = 6VM A X /V = 8VM A X /V = 10

1E-005 0.0001 0.001 0.01

rocking angle (rad)

0

0.1

0.2

0.3

0.4

dam

pin

g (

-)

dense sandV M A X /V = 2V M A X /V = 4V M A X /V = 6V M A X /V = 8V M A X /V = 10

1E-005 0.0001 0.001 0.01

rocking angle (rad)

0

0.1

0.2

0.3

0.4

dam

pin

g (

-)

m edium density sandV M A X /V = 2V M A X /V = 4V M A X /V = 6V M A X /V = 8V M A X /V = 10

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Prof. ing. Claudio di Prisco

Cyclic consequences of the mechanical coupling

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Prof. ing. Claudio di Prisco

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Prof. ing. Claudio di Prisco

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Prof. ing. Claudio di Prisco

A dynamic/non seismic problem

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Prof. ing. Claudio di Prisco