2007 ministero dell’istruzione, dell’università e della ... oliveto.pdf · • j. a. haringx,...
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Preliminary results from 3D large displacement FE modelling for the simulation of buckling and post-buckling behaviour
of elastomeric bearings.
Nicholas D. Oliveto, Gabriela Ferraro
June 26, 2009
Università degli Studi di Catania
Ministero dell’Istruzione, dell’Università e della Ricerca
2007
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Introduction
ELASTOMERIC BEARINGS
Rubber layers
Steel shims
Vertically stiff
Horizontally flexible Reduction of Seismic Forces
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Introduction
Large horizontal displacements
Reduction of buckling load
• Reduction of horizontal stiffness due to vertical load
• Reduction of vertical stiffness due to horizontal displacement
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Stability Theory of Elastomeric Bearings
Timoshenko beam Plane sections remain plane but not normal to the deformed axis
• Linear behavior of rubber
242
ESSScr
PPPPP
++−= 2
2
hEIP S
Eπ
= SS GAP =
• Small displacementsLarge displacements
(correction factor used in design)
(Haringx)
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Kelly and Takhirov (2004)
Analytical and numerical studyon buckling of elastomeric bearings of various shape factors
• Buckling both in tension and in compression
• Influence of vertical load on horizontal stiffness
• Influence of horizontal displacement on vertical stiffness
242
ESSSC
PPPPP
++−= 2
42ESSS
TPPPP
P+−−
=
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−=
2
1cr
SH P
Ph
GAK
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Kelly and Takhirov (2004)
Buckling analysis in ABAQUS 0.1% of first buckling mode
Initial displacements equal to
Horizontal shear deformation > 3% crPand increase of vertical load
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Buckle, Nagarajaiah and Ferrell (2002)
Analytical non-linear model(Nagarajaiah et al,1999)
Non-linear behavior of rubberLarge displacements
Experimental tests on low shape factor elastomeric bearings
Numerical Finite Element analyses using ADINA
(Liu et al,1999)
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Finite Element Analysis (Liu et al, 1999)
Procedure
• Application of predetermined shear displacement by means of constant force F
• Increase of vertical load P at the top of the bearing
• Critical load is reached when a stable equilibrium configuration is no longer
Critical states
Couples of vertical loads and horizontal displacements
possible
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Buckle, Nagarajaiah and Ferrell (2002)
• Decrease of buckling load with increasing horizontal displacement
• Correction factors not conservative at small displacements andoverly conservative at larger displacements
• Decrease in horizontal stiffness with increasing shear strain
Main Results
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Objectives
• Formulation of a reliable 3D Finite Element Model for the stabilityanalysis of laminated rubber bearings under large displacements
Kelly and Takhirov • Plane strain model • Stability under moderate shear strain
Buckle et al. • Plane strain model
• Convergence problems in numerical analyses
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Modeling of Rubber
• Non-linear behavior
• Large deformations
• Almost incompressible Mixed Formulation( K>>G) (p independent variable)
Family of hyperelastic materials Strain Energy Potential U
∑∑==+
−+−−=N
i
i
i
jiN
jiij J
DIICU
1
22
11 )1(1)3()3(
Deviatoric Strain Energy Volumetric Strain Energy
POLYNOMIAL FORM
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Polynomial Forms
2
1201110 )1(1)3()3( −+−+−= J
DICICU1=N
Mooney-Rivlin form
(Liu et al.)
2=N Incompressible material
ji
jiij IICU )3()3( 2
2
11 −−= ∑
=+
(Kelly and Takhirov)
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Polynomial Forms
Yeoh form
Typical S-shape of the stress-strain behavior of rubber
∑∑==
−+−=3
1
23
110 )1(1)3(
i
i
i
i
ii J
DICU
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Determination of coefficients Cij and Di
Experimental data Least Squares fitting procedure
ABAQUS can fit Poynomial forms up to order N=2
∑=
−=n
i
testi
thi TTE
1
2)/1(
User defined
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Neo-Hookean form
2nd order polynomial form ji
jiij IICU )3()3( 2
2
11 −−= ∑
=+
2
1110 )1(1)3( −+−= J
DICU
Constants from experimental test data (Treolar, 1940)
Finite Element Model
)(2 0110 CCG +=1
2D
K =
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Uniaxial stress-strain relation Neo-Hookean Material
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Finite Element Model
Steel shims thickness 2.60 mm
Rubber layer thickness 16.00 mm
Total Rubber thickness 80.00 mm
Width 160.00 mm
160 mm
160 mm
90.4 mm
Shape factor S=2.5
Strip bearing S=5
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Rubber C3D8H 8-node linear brick, hybrid with constant pressure
Steel C3D8 8-node linear brick
Finite Element Model
3 DOF/node + additional variable relating to pressure
3 DOF/node
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Finite Element Model
ABAQUS
Mesh size 8 mm
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Finite Element Model
• Top Boundary conditions
Rigid surface restrained to remain horizontal
Reference pointRigid surface
• Bottom Boundary conditions Fixed
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Eigenvalue Buckling Analysis
Mode Pcr (kN)
1
2345
-135
151-186
-194-194
• 4 Negative eigenvalues• Buckling load in tension lower than in
compression
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0.84-0.96
110-130
Square bearing
kN/mm kN
134-154
kN
Strip bearing Square bearing
151-135
kN
Haringx beam theory FE Model
Critical Loads - Haringx vs FEM
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Eigenvalue Buckling Analysis
MODE 1 (Tension)
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Eigenvalue Buckling Analysis
MODE 2 (Compression)
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Eigenvalue Buckling Analysis
MODE 3 MODE 4
MODE 5
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Modified Riks Method
Unstable postbuckling response
Loads and displacements are unknowns Arc length “l”
Controlling parameter
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Riks Analysis
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Riks Analysis
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References
• J. M. Kelly, S. M. Takhirov, Analytical and numerical study on buckling of elastomeric bearings with various shape factors, Earthquake Engineering Research Center, Report No. EERC 2004-03, December 2004.
• S. M. Takhirov, J. M. Kelly, Experimental and numerical study on vertical stiffness of elastomeric bearings with various shim thicknesses, 2004.
• H.C. Tsai, J. M. Kelly, Buckling of Short Beams with warping effects included, International Journal of Solids and Structures, 2005.
• J. M. Kelly, Tension Buckling in multilayer elastomeric bearings, Journal of Engineering Mechanics, 2003.
• K. L. Ryan, J. M. Kelly, A. K. Chopra, Nonlinear Model of Lead-Rubber Bearings,Journal of Engineering Mechanics, 2005.
• I. Buckle, S. Nagarajaiah, K. Ferrel, Stability of Elastomeric Isolation Bearings: Experimental Study, Journal of Structural Engineering, January 2002.
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References
• S. Nagarajaiah, K. Ferrel, Stability of Elastomeric Seismic Isolation Bearings: Experimental Study, Journal of Structural Engineering, September 1999.
• I. Buckle, H. Liu, Experimental Determination of Critical Loads of Elastomeric Isolators at High Shear Strain, NCEER Bull, 1994.
• A. N. Gent, Elastic Stability of Rubber Compression Springs, J. Mech. Engng. Sci., 1964.
• J. A. Haringx, On Highly Compressible Helical Springs and Rubber Rods and Their Application for Vibration-Free Mountings I, II and III, Philips Res. Rep., 1948-1949.
• G. P. Warn, A. S. Whittaker, A study of Coupled Horizontal-Vertical Behavior of Elastomeric and Rubber Seismic Isolation Bearings, MCEER-06-0011, 2006.
• M. C. Constantinou, A. S. Whittaker, Y. Kalpakidis, D. M. Fenz, G. P. Warn, Performance of Seismic Isolation Hardware under Service and Seismic Loading, MCEER-07-0012, 2007.
• ABAQUS Theory Manual, User’s Manual and Example Problems Manual.