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.