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Το παρόν υλικό δημιουργήθηκε στα πλαίσια του Προγράμματος Δια Βίου Μάθησης ΑΕΙ για την Επικαιροποίηση Γνώσεων Αποφοίτων ΑΕΙ "Σύγχρονες

1 Το παρόν υλικό δημιουργήθηκε στα πλαίσια του Προγράμματος Δια Βίου Μάθησης ΑΕΙ για την Επικαιροποίηση Γνώσεων Αποφοίτων ΑΕΙ "Σύγχρονες Εξελίξεις στις Θαλάσσιες Κατασκευές" (Επιχειρησιακό Πρόγραμμα "Εκπαίδευση και Δια Βίου Μάθηση") για αποκλειστική χρήση από τους εκπαιδευόμενους του Προγράμματος. Τυχόν άλλη χρήση του υλικού ή/και αναπαραγωγή αυτού δεν επιτρέπεται, καθόσον αποτελεί πνευματικό δικαίωμα του εκάστοτε διδάσκοντος. 1 SECION 8.2 GDM Operational Programme Education and Lifelong Learning Continuing Education Programme for updating Knowledge of University Graduates: Modern Development in Offshore Structures George D. Manolis, Professor Department of Civil Engineering Aristotle University, hessaloniki GR 54124, Greece el: ( ) , Fax: ( ) E mail: gdm@civil.auth.gr

2 ΕΠΙΧΕΙΡΗΣΙΑΚΟ ΠΡΟΓΡΑΜΜΑ «ΕΚΠΑΙΔΕΥΣΗ ΚΑΙ ΔΙΑ ΒΙΟΥ ΜΑΘΗΣΗ» Πρόγραμμα Δια Βίου Μάθησης ΑΕΙ για την Επικαιροποίηση Γνώσεων Αποφοίτων ΑΕΙ Χρονική Περίοδος: ΠΕΓΑ: ΣΥΓΧΡΟΝΕΣ ΕΞΕΛΙΞΕΙΣ ΣΤΙΣ ΘΑΛΑΣΣΙΕΣ ΚΑΤΑΣΚΕΥΕΣ SECION 8: ANALYSIS AND DESIGN OF MARINE SRUCURES 8.2 ΠΛΩΤΕΣ ΚΑΤΑΣΚΕΥΕΣ 8.2.2: ΥΔΡΟΔΥΝΑΜΙΚΗ ΑΝΑΛΥΣΗ ΠΛΩΤΩΝ ΚΑΤΑΣΚΕΥΩΝ (Fluid structure interaction) (2) Γεώργιος Δ. Μανώλης, καθηγητής Α.Π.Θ. Εργαστήριο Στατικής και Δυναμικής των Κατασκευών Τηλ: , Fax: E mail: gdm@civil.auth.gr 2 SECION 8.2 GDM

3 COUPLED FIELD PROBLEMS PAR II: (SFI) Basic heory he response of geomaterials is strongly influenced by the presence of fluids in the pores, as described by the Biot theory of poroelasticity A key concept is the definition of effective stresses in a 3D porous continuum as follows: { eff } { } { I} p { I} 1,1,1, 0, 0, 0 where [σ} is the total solid fluid vector (6x1) and p is the fluid pressure. he constitutive law for the porous soil in rate form is d{ } [ D ] d{ } eff where {ε} are the solid skeleton strains and matrix [D ] contains the elastic parameters (soil shear modulus and Poison's ratio). 3 SECION 8.2 GDM

4 FEM Development he displacement field {u(x)} in the solid is interpolated by the use of shape functions [N u ] in terms of its nodal displacements {U} (see previous Figure 4 for a simple triangular solid FE with three DOF per node and three nodes): { uxt (, )} [ N]{ Ut ( )}; (3x1) (3x9)(9x1) u he solid phase equilibrium equations in discrete form are now [ B] [ ] dv [ M]{ U( t)} [ C]{ U( t)} { F( t)} {0} where [B] is the strain matrix containing derivatives of the displacement field through operator matrix [L]: { } [ L]{ u} [ L][ N ]{ U} [ B]{ U}; (6x1) (6x3)(3x9)(9x1) u By splitting the stress term into effective stresses and fluid pressure components, the equations of equilibrium become [ M]{ U( t)} [ C]{ U( t)} [ K]{ U( t)} [ Q]{ P( t)} { F( t)} {0} 4 SECION 8.2 GDM

5 FEM Development he stiffness term assumes its standard form, i.e., eff [ ]{ } [ ] { } [ ] K U B dv B [ D ][ B ] dv { U } he coupling term comes from the fluid pressure in the solid skeleton pores, interpolated as previously: pxt (, ) [ N]{ Pt ( )}; (1x1) (1x3)(3 x1) p We thus have [ Q] [ B] { I}[ N ] dv; (6x3) (6x6)(6x1)(1x3) p Finally, it is possible to use the same shape functions [N] for both displacement field and for the pressure in the simple triangular solid/fluid FE with three nodes. 5 SECION 8.2 GDM

6 ransient Seepage he equation describing transient seepage (υπόγεια ροή) in a porous solid (πορώδες υλικό) is p ( kp) ii 0 q We now define the following variables: Soil permeability (Διαπερατότητα): k ( cm / sec) Fluid compressibility rate parameter (Συμπιεστότητα υγρού): q ( MPa / sec) Volumetric strain (Παραμόρφωση όγκου): ii xx yy zz he volumetric strain is interpolated in terms of nodal values of the displacement field as ii { I} [ B]{ U}; { I} 1,1,1 ; (1x1) (1x3)(3 x9)(9x1) 6 SECION 8.2 GDM

7 ransient Seepage he discrete FEM form of the above seepage equation comes from use of the Galerkin method, which is essentially an error minimization between the true BVP solution and its FEM approximation: [ Q] { U( t)} [ S]{ P( t)} [ H]{ P( t)} { F ( t)} {0} Definitions for the 'damping' and 'stiffness' matrices: 1 [ S] [ Np] ( )[ Np] dv; (3x3) q [ H] [ N ] ( k) [ N ] ds; (3x3) P p he forcing function and boundary terms are contained in the load term { FFL}; (3x 1) FL 7 SECION 8.2 GDM

8 he Coupled System he coupled equations of motion for flow in a porous solid are now collected as: [ M] 0 { Ut ( )} [ C] 0 { Ut ( )} [ K] [ Q] { Ut ( )} { Ft ( )} 0 0 [ Q] [ S] 0 [ H] { P( t)} { FFL ( t)} { Pt ( )} { Pt ( )} Comments: he above system of equations is non symmetric he free vibration problem for flow in porous media is of minor interest he transient problem is solved by using time stepping algorithms, as in FSI problems 8 SECION 8.2 GDM

9 ime Stepping Finite difference method expansions (aylor series) at time nδτ, n=1,2,3,...n: 2 t { U} n1 { U} n { U} nt{ U}(1 2)... 2 { U} n 1 { U} n { U} n(1 1) t... { P} { P} { P} t{ P} t...; { P} { P} { P} n1 n n n1 n he choice of weight functions in the above expansion is dictated from numerical stability analysis considerations: ; 0.5; SECION 8.2 GDM

10 ime Stepping he final form of the time stepping algorithm for SFI is condensed as 2 [ M] [ C] { } { } 1t[ K] 2t /2 [ Q] t U n F n 2 [ Q] { } 1t [ S] 1t[ H] F 1 t FL n { P)} n Special Cases: (1) If the fluid phase compresibility (συμπιεστότητα) is negligible: [S] 0 (2) If the soil permeability (διαπερατότητα) (as in clays) is negligible: [H] 0 (3) Soil sonsolidation (στερεοποίηση) implies slow (quasi static) development of the phenomenon with no inertia effects [ M]{ U} 0 10 SECION 8.2 GDM

11 Soil FIuid Interaction Example Figure 9: SFI example involving a dyke (ανάχωμα) foundation subjected to a simulated earthquake: (a) Centrifuge test model; (b) FEM mesh; (c) comparison between experimental and numerical results for excess pore pressure at the right edge 11 SECION 8.2 GDM

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15 COUPLED FIELD PROBLEMS REFERENCES ANSYS (2008), Structural Mechanics Finite Element Software, Version 10.0, Canonsburg, Pennsylvania. Bardet J.P. (1992) A viscoelastic model for the dynamic behaviour of saturated poroelastic soils. ransactions of the ASME, Vol. 59, pp Biot, M. (1956) heory of propagation of elastic waves in a fluid saturated porous solid. Journal of the Acoustical Society of America, Vol. 28(4), pp Cook, R.D., Malkus, D.S., Plesha, M. and Witt, R. (2002) Concepts and Applications of Finite Element Analysis, 4 th Edition, John Wiley, New York. Hardjants, F.A. and assoulas, J.L. (2002) Numerical Simulation of Dynamic Response of Floating Structures, Computational Mechanics, Vol. 32, pp Johnson J.J., editor (1981) Soil Structure Interaction: he Status of Current Analysis Methods and Research, Research Report NUREG CR 1780, Nuclear Regulatory Commission, Washington, DC. Ohayon, R. and Felippa, C., editors (2001) Advances in Computational Methods for Fluid Structure Interaction, Dedicated Issue in Computer Methods in Applied Mechanics and Engineering, Vol. 190, Issues 24 25, pp Sarpkaya,. and Isaacson, M. (1981) Mechanics of Wave Forces on Offshore Structures, Cambridge University Press, Cambridge. Zienkiewicz, O.C. and aylor, R.L. (1991) he Finite Element Method, 4 th Edition, Vol. 2, McGraw Hill, London. 15 SECION 8.2 GDM

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Operational Programme Education and Lifelong Learning. Continuing Education Programme for updating Knowledge of University Graduates:

Operational Programme Education and Lifelong Learning Continuing Education Programme for updating Knowledge of University Graduates: Modern Development in Offshore Structures 1 - SECTION 8.2 - GDM George