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Soil-Structure Interaction
James Lewis
FSI-ASEN 5519
University of Colorado
April 27, 2004
Overview
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Part I: Introduction
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Part II: Typical Soil-Structure Problem Formulations
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Simple Soil-Structure Problem Setup
Structure Equations
Foundation Equations
Soil Structure System
Part IV:
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Assumptions
Fields
Time Integration
Element Discretization
Applications
Part III:
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Coupled Problems
Solution Techniques
Conclusion
Further Difficulties
Direction of the Future
References
Part I: Introduction
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Coupled Field Problems
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In the age of modern engineering computers have
expanded the range and complexity of the problems that
can be practically handled. Of particular interest is in the
area of multi-physics dynamic interaction problems, using
coupled fields.
Numerical solutions of coupled field equations was
traditionally achieved with three different approaches:
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Field Elimination
Simultaneous Solution
Partitioned Solution Procedure
Part I: Problematic Solution Techniques
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Field Elimination
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Simulation Solution
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By eliminating one of the coupled fields a time integration
solution of a single system of increased order is applicable,
but maybe more complicated due to the raised order of the
resulting single system.
Setting up the system of coupled equations into a system of
simulation equations to be solved, leads to the loss of the
sparseness and solution difficulties.
These solution approaches result in systems that are
difficult to implement and solve, and are rendered
almost useless in practical applications.
The use of preexisting single field software is not
possible with these solution procedures.
Part I: Partitioned Solution
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Partitioned Solution Procedure
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In this procedure the solution of each field is done
separately, making use of preexisting single field
software, in a staggered (alternating) procedure
where the interaction of fields is accomplished with
predicting external forcing quantities, extrapolated
from the solution of the previous step.
In this solution procedure the modularity of the
separate fields is utilized, as well as preserving the
sparseness of the originals systems, simplifying
solution procedures for efficient computation, with
the possible use of parallel processors.
Part II: Typical Soil-Structure Problem
Formulations
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Assumptions
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Fields
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Linear elastic material behavior of structure and foundation
Constant contact/No uplift condition
Near Field (nf) Structure
Far Field (ff) Soil
Time Integration
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Time Domain
Frequency Domain
Cyclic Dynamic/Seismic Excitation
Part II: Typical Soil-Structure Problem
Formulations
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Element Discretization
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Structure
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Finite Elements (FEM)
Soil
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Infinite Elements
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Elements extend to infinity
Acts as a continuous medium for the propagation of waves
Will not allow a false boundary reflection
Boundary Elements (BEM) for the Soil
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For linear problems of homogeneous media
Requires only boundary discretization of considered domain
Acts as a continuous medium for the propagation of waves
Will not allow a false boundary reflection
Part II: Typical Soil-Structure Problem
Formulations
Part II: Typical Soil-Structure Problem
Formulations
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Applications
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Foundations
Bridge Abutments
Driven Piles
Dams
Retaining Walls
Offshore Structure
Consolidation
Part III: Simple Soil-Structure Problem
Setup
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Seismic excitation of structure
embedded in foundation
ground. Incident seismic wave
is refracted and reflected as it
encounters discontinuities in
the soil strata. This reflected
portion of the wave will be used
as the base rock motion.
Part III: Simple Soil-Structure Problem
Setup (FEM-BEM formulation)
Part III: Simple Soil-Structure
Problem Setup
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Motion Equations of Any Structure
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M mass matrix
C damping matrix
K stiffness matrix
d nodal displacement vector
a(t) seismic acceleration
&& + Cd& + Kd = −M1a(t )
Md
1T = [1 1 L 1 1]
Part III: Simple Soil-Structure Problem
Setup
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Equations of Motion of the Structure in the time domain
– bt soil-structure interaction forces
– t
subscript refers to all the DOF of the structure
&& + C d& + K d = −M 1a (t ) + b
Mtd
t
t t
t t
t
t
Part III: Simple Soil-Structure
Problem Setup
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Equations of Motion of the Structure in the frequency
domain
θ
frequency of the excitation
d t (θ )
the Fourier Transform of
d t (t )
b t (θ )
the Fourier Transform of
b t (t )
at (θ )
the Fourier Transform of
at (t )
(− θ
2
)
M t + iθ C t + K t d t (θ ) = −M t 1a (θ ) + b t (θ )
Part III: Simple Soil-Structure
Problem Setup
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Partition the nodal displacement vector and EOM of the
structure in the frequency domain
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db
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d
the displacement vector corresponding to the soil-structure
interface
the displacements of the remaining nodes of the structure

M 0 
C 0   K
 − θ 2 
θ
i
+
 0 Cb  +  '
b

0
M



 K

K '    d (θ )  M 0 
 0 

(
)
θ
1
a
=
+
 
 
 b b (θ )
b
K    d b (θ )  0 M 


 d (θ ) 
d
d t =  b  ⇒ dt (θ ) =  b 
d 
 d (θ )
b=0
Part III: Simple Soil-Structure
Problem Setup
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Equations of Motion of the Soil in the frequency domain
– b b (θ )
– d b (θ )
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interaction forces which act on the soil, at the soil-structure interface
nodal displacements, at the soil- structure interface, due to interaction
forces ,
Yb (θ ) square matrix whose elements are the dynamic stiffnesses
corresponding to the DOF of the soil-structure interface. This complex
coefficient matrix must be established by analyzing the second
substructure, i.e. the foundation ground (soil).
b b (θ ) = Yb (θ )d b (θ )
Part III: Simple Soil-Structure
Problem Setup
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Equations of Motion of the Soil-structure system in
the frequency domain
Conditions for the nodes of the interface of the two
substructures
db = db
the compatibility condition
bb + bb = 0
the equilibrium condition
b b = − Yb × d b
the two above conditions provide the following relationship
Part III: Simple Soil-Structure Problem
Setup
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Now the EOM for the structure
can be expressed as follows
Finally, the complex linear
system is solved for all the
values of the excitation
frequency q for which the
matrix has previously been
calculated. Thus, the vectors of
the displacement are obtained
in the frequency domain and a
discrete inverse Fourier
transform provides the
response in the real time
domain of the structure in
interaction with the foundation
ground.
G = −θ 2 M + iθ C + K
G' = K'
G b = −θ 2 M b + iθ C b + K b
G
 '
G
  d (θ ) 
M 0 
=
−



 0 M b 1a (θ )
G b + Yb (θ )  d b (θ )


G'
Part III: Simple Soil-Structure Problem
Setup
Conclusions
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Problems with this solution Procedure
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Non-linear properties
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Cracking in the Concrete is non-linear
Plastic deformation of the Soil is non-linear
Rock and soil layers produce discontinuities
Uplift
Future work
References
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Beskos, D. E. & S. A. Anagnostopoulos, Computer Analysis and
Design of Earthquake Resistant Structure: A Handbook.
Computational Mechanics Publications; Southhampton, UK:
1997.
Cakmak, A.S. Soil-Structure Interaction. Elsevier; Amsterdam:
1987.
Chopra, Anil K. Dynamics of Structures: Theory and Applications to
Earthquake Engineering. 2nd ed. Prentice Hall; Upper Saddle
River, New Jersey: 2001.
Hinton, E, P. Bettess & R.W. Lewis, Numerical Methods for Coupled
Problems. Pineridge Press Limited; Swansea, UK: 1981.
Hinton, E, P. Bettess & R.W. Lewis, Numerical Methods in Coupled
Systems. Wiley-Interscience Publication; Chichester, UK: 1984.