Soil-Structure Interaction - University of Colorado Boulder

Soil-Structure Interaction James Lewis FSI-ASEN 5519 University of Colorado April 27, 2004

Overview z

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 z

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: z z z

Field Elimination Simultaneous Solution Partitioned Solution Procedure

Part I: Problematic Solution Techniques z

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 z

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 z

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 – – –

Time Domain Frequency Domain Cyclic Dynamic/Seismic Excitation

Part II: Typical Soil-Structure Problem Formulations z

Element Discretization –

Structure z

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Finite Elements (FEM)

Soil z

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 – – – –

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 z

Applications – – – – – – –

Foundations Bridge Abutments Driven Piles Dams Retaining Walls Offshore Structure Consolidation

Part III: Simple Soil-Structure Problem Setup z

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 z

Motion Equations of Any Structure – – – – –

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 z

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 z

z

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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 z

Partition the nodal displacement vector and EOM of the structure in the frequency domain –

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 z

Equations of Motion of the Soil in the frequency domain – b b (θ ) – d b (θ ) –

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 z z

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 z z

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 z

Problems with this solution Procedure –

Non-linear properties z z z

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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 z

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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.