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
Introduction
ELASTOMERIC BEARINGS
Rubber layers
Steel shims
Vertically stiff
Horizontally flexible Reduction of Seismic Forces
Introduction
Large horizontal displacements
Reduction of buckling load
• Reduction of horizontal stiffness due to vertical load
• Reduction of vertical stiffness due to horizontal displacement
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)
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
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
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)
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
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
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
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
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)
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
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
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 =
Uniaxial stress-strain relation Neo-Hookean Material
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
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
Finite Element Model
ABAQUS
Mesh size 8 mm
Finite Element Model
• Top Boundary conditions
Rigid surface restrained to remain horizontal
Reference pointRigid surface
• Bottom Boundary conditions Fixed
Eigenvalue Buckling Analysis
Mode Pcr (kN)
1
2345
-135
151-186
-194-194
• 4 Negative eigenvalues• Buckling load in tension lower than in
compression
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
Eigenvalue Buckling Analysis
MODE 1 (Tension)
Eigenvalue Buckling Analysis
MODE 2 (Compression)
Eigenvalue Buckling Analysis
MODE 3 MODE 4
MODE 5
Modified Riks Method
Unstable postbuckling response
Loads and displacements are unknowns Arc length “l”
Controlling parameter
Riks Analysis
Riks Analysis
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.
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.