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Two and Three-Dimensional Contact Element Implementation for Geotechnical Applications in OpenSees
Kathryn Petek
Pedro Arduino
Peter Mackenzie-HelnweinUniversity of Washington
August 24, 2005OpenSees Developer Symposium
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Presentation Overview
� Background
� Contact Element & Interface Material
Formulations
� OpenSees Implementation
� Element Features
� Examples
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Objectives
1. Realistic soil-pile interaction
2. Consideration of complex
soil models
3. Alternative pile modeling
approaches
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Background: Interface Behavior
Pile-soil interaction: stick, slip, debonding, and rebonding behavior
(Desai et al., 1988)
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Background: Interface Behavior
Pile-soil interaction: stick, slip, debonding, and rebonding behavior
Finite Element approaches:
� Zero-length elements
� Joint and thin-layer elements
� Gap elements Body
A Body
B
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Contact Element Model Development
Node-to-Surface ElementNode-to-Segment Element
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Contact Element Model Development
Node-to-Segment Element
t
n
α tn
(1−α) tn
g
Master
body Slave
body
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Contact Element Formulation
Using the method of Lagrange Multipliers, the element utilizes the Hertz-Signori-Moreau conditions for contact: tn
g
0≥g 0≥nt 0=⋅ gtn
Contact element applies a geometric constraint to the system that relates a slave node to a master contact line segment or surface.
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Contact Material Formulation
� The geometric constraints are related with an interface constitutive law:
– Mohr-Coulomb Friction Law
– Can also use non-linear and history dependent material models, including specific models for concrete structures on soil
0 ≤−⋅−= cttf ns µtn
ts
µ
c
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Contact Material Formulation
0 ≤−⋅−= cttf ns µ
s
ts
G
µ tn
f = 0
sticking sliding
f < 0
Elastic slip
Plastic slip
s
ts
tn
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Variational Contact Formulation
� Expression for Virtual Work:
� Linearization:
s )( dtsdgtdtgWd nn δδδδ −+=
stgtgtW nn δδδδ s−+=
)( qgg =
)( qss =
nsnssn
n
sss dtCdsCdt
t
tds
s
tdt +=
∂
∂+
∂
∂= :
Note: Css & Csn depend on the state: sticking, sliding
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2D Contact Formulation
Linearization and 2D Tangent Stiffness Matrix:
nT
g Bqδδ :=
[ ]
⋅
−−⋅=
nTn
snTs
Tn
Tssss
nT
td
dtWd
0 )(
q
B
CBBBCBq δδδ
sT
s Bqδδ :=
−
−=
n
n
n
B )1(
α
α
n
−
−=
t
t
t
B )1(
α
α
s
TK
s )( dtsdgtdtgWd nn δδδδ −+=
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Implementation in OpenSees
New element and material classes
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Implementation in OpenSees
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Implementation in OpenSees
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3D Contact Element
s
ts
tng1 g2
s
ts
tn
n
xξ_n xξ_n+1
2D Node-to-Line Element 3D Node-to-Surface Element
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3D Geometric Pseudo-Nonlinearity
� Project xs_n on to master surface patch & determine tangent plane
� Slip, snn+1,
moves along tangent plane of step n
� Converges to nonlinear solution
x3
g1 g2
snn+1
n
xξ_n xξ_n+1
xs_n
ηξ
x2
x4
x1
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Solution Strategy w/ Lag Step
� No contact search algorithm
� Contact Conditions:
Determines: - in contact
- not in contact
- should be released
� Added lag step for stability near boundary of in and out of contact
( )
⋅=≤
>
0
0 ? nxx ξ -g s
<
≥
0
0 ?
nt
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Solution Strategy w/ Lag Step
tn < 0
g ≤ 0
should_release = true
was_in_Contact = true
to_be_released = false
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GiD Development
Developed pre- and post-processing tools using commercial software GiD
- Model creation
- Mesh generation
- Results visualization
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GiD Contact Element Generation
� No native support for this type of element
� GiD creates contact pairs for all nodes within
range that can go in and out of contact.
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Example 1: Simple 3D Blocks
Block moving across surface:
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Example 2: 3D Friction Pile
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Example 2: 3D Friction Pile
Parameter Testing and Calibration: Evaluate frictional forces developed in contact element
zzQ
D
hs d tan)(B
0
δσπ ∫=
And compare with conventional β-method used for pile analysis :
∑=
=N
i
scontact ifQ
1
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Example 2: 3D Friction Pile
Frictional contact element gives good approximation:
Typical Values of µ = tan δ :
δ / φ = 1.0 – 0.5
Concrete-sand: µ = 0.35 – 0.6
4%88091810002
5%44046210001
14%11012810000.5
13%556310000.25
Gµµµµ zzQ
D
hs d tan)(B
0
δσπ ∫=∑=
=N
i
scontact ifQ
1
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Example 3: 3D Pushover Analysis
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Example 3: 3D Pushover Analysis
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Example 3: 3D Pushover Analysis with Cohesive Contact Material
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Example 3: 3D Pushover Analysis with Plastic Soil
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Summary
� Contact elements are implemented in OpenSees using a stable, pseudo-nonlinear approach
� Examples demonstrate element capability to describe interface behavior for pile analysis
� Further validation and testing is underway prior to submission to OpenSees repository.