Seismic Isolation of Nuclear Power Plants Using Elastomeric Bearings
Analysis of Elastomeric Bearings in...
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DEPARTMENT OF CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING UNIVERSITY AT BUFFALO Analysis of Elastomeric Bearings in Compression CIE 526: Finite Element Structural Analysis Manish Kumar 4/29/2012
Transcript of Analysis of Elastomeric Bearings in...
DEPARTMENT OF CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING UNIVERSITY AT BUFFALO
Analysis of Elastomeric Bearings in Compression
CIE 526: Finite Element Structural Analysis
Manish Kumar
4/29/2012
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Table of Contents LIST OF FIGURES ..................................................................................................................................................... 3
LIST OF TABLES ....................................................................................................................................................... 5
ABSTRACT ................................................................................................................................................................. 6
SECTION 1 INTRODUCTION ................................................................................................................................. 7
1.1 GENERAL ......................................................................................................................................................... 7 1.2 MOTIVATION AND OBJECTIVES......................................................................................................................... 7 1.3 SCOPE OF WORK .............................................................................................................................................. 8
SECTION 2 ELASTOMERIC BEARINGS IN COMPRESSION .......................................................................... 9
2.1 GENERAL ......................................................................................................................................................... 9 2.2 ANALYSIS OF SINGLE RUBBER LAYER ............................................................................................................. 9 2.3 ANALYSIS OF MULTILAYER ELASTOMERIC BEARINGS ................................................................................... 13
SECTION 3 FINITE ELEMENT FORMULATION ............................................................................................. 15
3.1 GENERAL ....................................................................................................................................................... 15 3.2 AXISYMMETRIC FORMULATION OF ELASTOMERIC BEARINGS ........................................................................ 15
3.2.1 General ................................................................................................................................................ 15 3.2.2 Mathematical Formulation .................................................................................................................. 15
3.3 THREE-DIMENSIONAL FORMULATION OF ELASTOMERIC BEARINGS ............................................................... 18 3.3.1 General ................................................................................................................................................ 18 3.3.2 Mathematical formulation ................................................................................................................... 18
SECTION 4 ANALYSIS ........................................................................................................................................... 21
4.1 GENERAL ....................................................................................................................................................... 21 4.2 GEOMETRY .................................................................................................................................................... 21
4.2.1 General ................................................................................................................................................ 21 4.2.2 Geometry of Original Bearing ............................................................................................................. 21 4.2.3 Geometry of Axisymmetric Model ........................................................................................................ 22 4.2.4 Geometry of Three-dimensional Model ............................................................................................... 23
4.3 MESH ............................................................................................................................................................. 24 4.3.1 General ................................................................................................................................................ 24 4.3.2 Meshing of Axisymmetric Model .......................................................................................................... 25 4.3.3 Meshing of Three-dimensional Model ................................................................................................. 26
4.4 MATERIAL PROPERTIES .................................................................................................................................. 27 4.4.1 Steel ..................................................................................................................................................... 27 4.4.2 Rubber ................................................................................................................................................. 27
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4.5 BOUNDARY CONDITIONS AND LOADING ........................................................................................................ 28 4.5.1 Axisymmetric Model ............................................................................................................................ 28 4.5.2 Three-dimensional Model .................................................................................................................... 28
4.6 RESULTS ........................................................................................................................................................ 30 4.6.1 General ................................................................................................................................................ 30 4.6.2 Rubber Layer ....................................................................................................................................... 30 4.6.3 Axisymmetric Model ............................................................................................................................ 30 4.6.4 Three-dimensional Model .................................................................................................................... 33
SECTION 5 DISCUSSION ....................................................................................................................................... 37
5.1 GENERAL ....................................................................................................................................................... 37 5.2 ANALYTICAL PREDICTIONS ............................................................................................................................ 37 5.3 COMPARISON OF RESULTS ............................................................................................................................. 38
5.3.1 Comparison of Vertical Stiffness ......................................................................................................... 38 5.3.2 Comparison of Normal Stress Variation .............................................................................................. 38 5.3.3 Comparison of Shear Stress Variation ................................................................................................ 39
5.4 SOURCES OF ERROR ....................................................................................................................................... 41 5.4.1 Mesh Sensitivity ................................................................................................................................... 42 5.4.2 Element Sensitivity ............................................................................................................................... 43 5.4.3 Model Sensitivity .................................................................................................................................. 43
5.5 SUMMARY ...................................................................................................................................................... 43
SECTION 6 SUMMARY AND CONCLUSIONS .................................................................................................. 45
REFERENCES .......................................................................................................................................................... 46
APPENDIX A ABAQUS INPUT FILE .................................................................................................................. 47
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List of Figures
Figure 2-1: Deformation of a constrained rubber layer under compression ................................ 10
Figure 2-2: Elastomeric bearing and sliced section of elastomeric bearing (Reproduced from
http://www.columbia.edu) ............................................................................................................ 13
Figure 2-3: Internal construction of an elastomeric bearing ........................................................ 14
Figure 3-1: Internal construction and face considered for axisymmetric formulation ................ 16
Figure 3-2: Co-ordinate system used for axisymmetric formulation ........................................... 16
Figure 3-3: 20 node brick element ............................................................................................... 19
Figure 4-1: Low damping rubber bearing used for finite element analysis(dimensions are in mm)
....................................................................................................................................................... 22
Figure 4-2: Axisymmetric model of elastomeric bearing ............................................................ 22
Figure 4-3: Three-dimensional model of elastomeric bearing ..................................................... 23
Figure 4-4: Mesh used for axisymmetric model .......................................................................... 25
Figure 4-5: Mesh used for three-dimensional model ................................................................... 26
Figure 4-6: Hyperelastic model of rubber .................................................................................... 28
Figure 4-7: Loads and boundary condition for axisymmetric model ........................................... 29
Figure 4-8: Loads and boundary condition for three-dimensional model ................................... 29
Figure 4-9: State of stress(Mises) in the rubber layer .................................................................. 30
Figure 4-10: State of logarithmic shear strain in the rubber layer ............................................... 30
Figure 4-11: First 3 modes of axisymmetric model ..................................................................... 31
Figure 4-12: State of stress(Mises) in the axisymmetric model ................................................... 32
Figure 4-13: State of logarithmic shear strain in the axisymmetric model .................................. 32
Figure 4-14: Eigenproperties of three-dimensional model .......................................................... 33
Figure 4-15: Modes of vibration of three-dimensional model ..................................................... 34
Figure 4-16: State of stress(Mises) in the three-dimensional model ........................................... 35
Figure 4-17: State of logarithmic shear strain in the three-dimensional model ........................... 35
Figure 4-18: Resultant force and moment at a section mid-height of bearing ............................. 36
Figure 5-1: Normal stress variation along the radius of three dimensional model ...................... 39
Figure 5-2: Shear stress variation along the radius of three dimensional model ......................... 40
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Figure 5-3: Shear stress variation along the radius of axisymmetric model ................................ 40
Figure 5-4: Shear stress variation along height of axisymmetric bearing model ......................... 41
Figure 5-5: Mesh sensitivity for shear stress variation of 3D model ........................................... 42
Figure 5-6: Mesh sensitivity for shear stress variation of 3D model ........................................... 42
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List of Tables
Table 4-1: Shape metrics of meshes used in axisymmetric model .............................................. 25
Table 4-2: Shape metrics of meshes used in three-dimensional model ....................................... 26
Table 4-3: Eigenproperties of axisymmetric model ..................................................................... 31
Table 5-1: Vertical stiffness values (all values in N/m) ............................................................... 38
Table 5-2: Vertical stiffness values (all values in N/m) ............................................................... 43
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ABSTRACT
Elastomeric bearings are composite elements made up of natural or synthetic rubber
layers bonded to reinforcing steel shims in alternate layers. Rubber, owing to its low shear
modulus, accommodates large horizontal displacements, and steel shims combined with almost
incompressible rubber provides high vertical stiffness. This behavior of elastomeric bearings is
desired in isolation of civil engineering structures. Rubber bearings are used in varieties of
applications including seismic isolation, bridge expansion bearings, vibration isolation etc.
In analysis and design of elastomeric bearings, it is crucial to predict the compressive and
shear stress and strains. Existing analytical solutions predict the desired quantities using
“pressure solution” approach, which simplifies the complex problem to a relatively simple one
using many assumptions. Although simplified, these solutions are in terms of infinite series and
are not practical for use in design calculations. Recent work has been done to obtain simple
analytical solution that can be used for practical calculations.
This work is proposed to model the composite elastomeric element using finite element
method and assess the accuracy of simplified closed form solutions by comparing it with the
results obtained from finite element analyses. A review of limitations and advantages of both
solutions methods is to be presented.
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SECTION 1
INTRODUCTION
1.1 General
Finite element analysis provides an effective method to assess the behavior of a physical system
for which the analytical closed form solution does not exist or deemed to be very complicated in
practical use. Accuracy of analysis depends on analyst’s ability to model the structure and their
understanding of the problem. If behavior of the structure is well understood at material level,
and created model reasonably represents the actual structure, finite element analysis provides
good results.
Elastomeric bearings are analyzed in different software programs using springs in vertical and
horizontal direction. Stiffness values of these springs are most important parameter required for
analysis of base isolated structures. Elastomeric bearings have a complex construction made up
of alternate layers of rubber and steel. Simplified expressions suggested by researchers are used
to estimate the vertical stiffness of elastomeric bearings. However, accuracy of these expressions
is contested. Attempts have been made to analyze the elastomeric bearings using finite element
analysis using different software programs and tools (Simo and Kelly, 1984). Development of
new finite element analysis programs has enabled researchers to model the structure more
accurately using graphical user interface and increase the accuracy of analysis using smaller
elements.
1.2 Motivation and objectives
Elastomers are conventionally modeled as hyperplastic material. Hyperelastic material models
are characterized by highly nonlinear behavior and usually difficult to implement in software
programs. Almost all proposed hyperelastic constitutive laws are energy based formulation.
Accuracy of these constitutive laws in capturing the actual behavior of elastomer depends on
loading type and range of strain.
Steel is modeled as an isotropic linear elastic material. Sudden change of properties at layer
interface of rubber and steel make it difficult to capture the behavior, and errors resulting from
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this transition are not known. Also, effect of internal and external bearing plate on the vertical
stiffness of bearing is not known.
Simplified analytical expressions that have been suggested for analysis of elastomeric bearings
make many assumptions and it is not known how these assumptions affect the accuracy of
analytical solution.
The objectives of this study are: (1) to investigate the validity of analytical solutions suggested
for analysis of elastomeric bearings (2) asses the error resulting from inappropriate material
modeling, and (3) to check the ability of finite element methods in modeling and analysis of
elastomeric bearing.
1.3 Scope of Work
The scope of work of this study is as follows:
1. Build a finite element model of a circular elastomeric bearing in Abaqus.
2. Use a static analysis to find the compressive stiffness.
3. Investigate the validity of analytical formulations and finite element analysis.
4. Check the sensitivity of finite element results using different models, element types, and
mesh sizes.
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SECTION 2
ELASTOMERIC BEARINGS IN COMPRESSION
2.1 General
Rubber is considered as an almost incompressible material. Elastomeric bearings are constructed
using rubber layers. Vertical capacity of a rubber layer is inversely proportional to its thickness,
and horizontal flexibility is directly proportional to its thickness. Reinforcing steel shims in
alternating layers with rubber provide constraints which help in reducing shear, and hence
minimize the bulging or rubber layer. Alternate layers of rubber and steel in elastomeric bearing
provide a very high vertical stiffness, while still maintaining high flexibility in horizontal
direction required for lengthening the time period of structure in seismic isolation applications.
In order to understand the mechanics of elastomeric bearing, it is necessary to first understand
the behavior of a constrained rubber layer under compression. Once the stiffness of a single
constrained rubber layer is obtained, vertical stiffness of elastomeric bearing can be predicted. It
should be noted that steel shims are considered as rigid layers in elastomeric bearing, having
negligible contribution to stiffness values in vertical and horizontal direction.
2.2 Analysis of Single Rubber Layer
Linear elastic theory is used to predict the compression stiffness of a rubber layer and is based on
the work of Chalhoub and Kelly (1990) and Constantinou et al. (1992). Mathematical
formulation presented here has been adopted from Constantinou et al. (2007) and Kelly and
Konstantinidis (2011).
Assumptions made for the analysis have been outline below:
1. Any element inside the constrained layer is in hydrostatic state of stress, i.e., normal
stresses are equal at all faces of a differential square element ( )xx yy zz pσ σ σ= = = − .
2. No shear stresses are experienced in the horizontal plane ( 0xyτ = ).
3. Points lying on a vertical line in undeformed state lie on a parabola after loading.
A schematic of a rubber layer deforming under compression has been shown in Figure 2-1.
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Figure 2-1: Deformation of a constrained rubber layer under compression
If u, v, and w are deformations along x, y, and z direction respectively; deformation fields in
three directions are given by
2
0 2
4( , , ) ( , )(1 )zu x y z u x yt
= −
2
0 2
4( , , ) ( , )(1 )zv x y z v x yt
= −
( , , ) ( )w x y z w z=
These deformation fields are based on the parabolic state of deformation assumed for the
analysis. Since rubber was assumed to be incompressible, it implies the following condition:
0xx yy zzε ε ε+ + =
Substituting the values of normal strains in terms of u, v, and w we obtain the governing
differential equation:
2
0 02
4( )(1 ) 0u v z dwx x t dz
∂ ∂+ − + =
∂ ∂
Rearranging the terms above, equation is rewritten as:
P
x
z y
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0 02
2
14(1 )
u v dwzx x dz
t
∂ ∂+ = −
∂ ∂ −
We see that left hand side is a function of x and y, and right hand side is function of z only. As
this relationship must hold everywhere in the domain of rubber layer, both sides of equation must
be identically equal to a constant k. To determine constant k, we solve the differential equation
using the boundary conditions:
2
2
4(1 ), ( / 2) / 2, ( / 2) / 2dw zk w t w tdz t
= − − = −∆ − = ∆
k is obtained as 3 / 2c
k ε= , where c
ε is the compressive strain and is given by the expression
( / 2) ( / 2)c
w t w tt
ε − −= −
Also we have
0 0 32
cu vx x
ε∂ ∂+ =
∂ ∂
Key parameter for analysis of elastomeric bearing is its stiffness value in vertical direction which
depends on the compression modulus of rubber. Compression modulus cE , is calculated as
cc
PEAε
=
Equations of equilibrium for the stresses are:
0
0
0
xyxx xz
xy yy yz
yzxz zz
x y z
x y z
x y z
τσ τ
τ σ τ
ττ σ
∂∂ ∂+ + =
∂ ∂ ∂∂ ∂ ∂
+ + =∂ ∂ ∂
∂∂ ∂+ + =
∂ ∂ ∂
Since any point inside the rubber layer is assumed to be in state of hydrostatic stress, we have
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xx yy zz pσ σ σ= = = −
This simplifies the equations of equilibrium as:
xz
yz
pz x
pz y
τ
τ
∂ ∂=
∂ ∂∂ ∂
=∂ ∂
Complete set of differential equations involving shear stresses and normal stresses are solved,
which has not been shown here but user is referred to Constantinou et al. (1992) and Kelly and
Konstantinidis (2011) for detailed derivation. Solution of differential equations produces the
expression for pressure inside the rubber layer as
2 22
3 ( )cGp R rtε
= −
Where 2 2 2r x y= +
Total force P is calculated as
4
20
32 ( )2
R cG RP p r rdrt
ε ππ= =∫
Compression modulus is thus obtained as 26cE GS=
It should be noted that rubber was assumed as incompressible; however, natural rubber possesses
some degree of compressibility. In that case, expression for compression modulus is modified
accordingly. Final expression for compression modulus cE of a circular compressible rubber
layer, with shear modulus G and bulk modulus K, is given as:
1
2
1 46 3cEGS K
− = +
This is the expression that would be used to calculate the vertical stiffness of elastomeric
bearing.
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Shear stresses are calculated as
2 2
6 6,c cxz yz
G Gxz yzt tε ετ τ= − = −
2.3 Analysis of Multilayer Elastomeric Bearings
Figure 2-2 shows an elastomeric bearing and its internal construction.
Figure 2-2: Elastomeric bearing and sliced section of elastomeric bearing (Reproduced from
http://www.columbia.edu)
A schematic of internal construction of an elastomeric bearing has been shown in Figure 2-3.
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Figure 2-3: Internal construction of an elastomeric bearing
Vertical stiffness of a single rubber layer is given by expression:
ci
r
AEKt
=
Under the application of vertical load all these layers behave as springs of stiffness iK in series.
Stiffness of elastomeric bearing can then be calculated as:
1 1 r
i c
tK K AE
∑= ∑ =
Hence,
c
r
AEKT
=
Where, rT is the total thickness of rubber layers in elastomeric bearing.
This is the most important equation for our analysis and would be used as a benchmark to
calculate the accuracy of finite element analysis.
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SECTION 3
FINITE ELEMENT FORMULATION
3.1 General
Symmetrical geometry of circular elastomeric bearing allows to use different types of finite
element formulations. In Abaqus, elastomeric bearings can be analyzed using either
axisymmetric elements or 3-dimensional elements. In order to make a relative comparison of
accuracy of solutions, ease of modeling, cost of computation, and results visualization, both type
of elements have been considered here for the analysis.
Abaqus library provides option of using first(linear) or second-order(quadratic) elements. First
order elements include 4 node quadrilateral element for plane and axisymmetric elements and
second order elements include 8 node brick element for three-dimensional analysis. Generally,
better results are obtained with second order elements but at higher cost of computations.
3.2 Axisymmetric Formulation of Elastomeric Bearings
3.2.1 General First formulation presented here makes use of fact that circular elastomeric bearing is symmetric
about its central axis and loading is applied symmetrically in only radial and/or axial direction.
Axisymmetric formulation considers only half of the face about the axis of symmetry.
Deformation state at any point in rz plane completely defines the state of strain and stress in the
body. Internal construction of an elastomeric bearing and face that would be considered for
axisymmetric model has been shown in Figure 3-1.
3.2.2 Mathematical Formulation
Kinematic and static variables used in axisymmetric formulation have been shown below.θ
1 01 0
, , , 1 0(1 )(1 2 )1 20 0 0
2
rr rr
zz zz
r r
u EU Cw
θθ θθ
θ θ
υ υ υε τυ υ υε τ
ε τ υ υ υε τ υ υυγ τ
− − = = = = −+ − −
Where, symbols have their usual meaning. Please note that here 0z zrθτ τ= =
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Figure 3-1: Internal construction and face considered for axisymmetric formulation
Figure 3-2: Co-ordinate system used for axisymmetric formulation
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For the problem considered in this report, CAX8R-An 8-node biquadratic axisymmetric
quadrilateral, reduced integration, was used.
Interpolation functions in terms of generalized co-ordinates have been given below.
1
2
3
4
25
26
27
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1 (1 )(1 )( 1)41 (1 )(1 )( 1)41 (1 )(1 )( 1)41 (1 )(1 )( 1)41 (1 )(1 )21 (1 )(1 )21 (1 )(1 )21 (1 )(1 )2
N r s r s
N r s r s
N r s r s
N r s r s
N r s
N s r
N r s
N s r
= + + + −
= − + − + −
= − − − − −
= + − − −
= − +
= − −
= − −
= − +
Distinction between generalized co-ordinate axis r and radius r should be kept in mind. Strain
matrix [B] is obtained by differentiating the deformation vector and element stiffness matrix;
nodal forces are found out using the formulae below.
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1 1
1 11 1
1 11 1
1 1
det
det
T
T BB
T SS
K B CB JdrdsR
R N f JdrdsR
R N f ds
− −
− −
− −
=
=
=
∫ ∫
∫ ∫
∫ ∫
An analytical solution for element stiffness matrix has not been presented here because of
complexity. Finite element software programs calculate element stiffness matrix using numerical
integration techniques like Gauss integration.
3.3 Three-dimensional Formulation of Elastomeric Bearings
3.3.1 General Three-dimensional formulation is the most generalized formulation in finite element analysis. It
can be seen as an extension of familiar 4-node quad element into three-dimensional 8-node or
20-node brick element. Brick element is referred to as “Hex” element in Abaqus. Eight-node first
order brick element makes use of linear interpolation functions; whereas, 20 node second order
brick elements makes uses quadratic interpolation functions.
3.3.2 Mathematical formulation For the three-dimensional model considered in this report, C3D20R-20-node, biquadratic,
reduced integration, brick element was used. Schematic of a 20 node brick element have been
shown in Figure 3-3.
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Figure 3-3: 20 node brick element
Interpolation functions used for 20-noded brick element have been shown below.
11 (1 )(1 )(1 )(2 )8
N r s t r s t= − − − − + + + , 21 (1 )(1 )(1 )(2 )8
N r s t r s t= − + − − − + +
31 (1 )(1 )(1 )(2 )8
N r s t r s t= − + + − − − + , 41 (1 )(1 )(1 )(2 )8
N r s t r s t= − − + − + − +
51 (1 )(1 )(1 )(2 )8
N r s t r s t= − − − + + + − , 61 (1 )(1 )(1 )(2 )8
N r s t r s t= − + − + − + −
71 (1 )(1 )(1 )(2 )8
N r s t r s t= − + + + − − − , 81 (1 )(1 )(1 )(2 )8
N r s t r s t= − − + + + − −
91 (1 )(1 )(1 )(1 )4
N r r s t= − + − − , 101 (1 )(1 )(1 )(1 )4
N r s s t= + − + −
111 (1 )(1 )(1 )(1 )4
N r r s t= − + + − , 121 (1 )(1 )(1 )(1 )4
N r s s t= − − + −
131 (1 )(1 )(1 )(1 )4
N r r s t= − + − + , 141 (1 )(1 )(1 )(1 )4
N r s s t= + − + +
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151 (1 )(1 )(1 )(1 )4
N r r s t= − + + + , 161 (1 )(1 )(1 )(1 )4
N r s s t= − − + +
171 (1 )(1 )(1 )(1 )4
N r s t t= − − − + , 181 (1 )(1 )(1 )(1 )4
N r s t t= + − − +
191 (1 )(1 )(1 )(1 )4
N r s t t= + + − + , 201 (1 )(1 )(1 )(1 )4
N r s t t= − + − +
Strain matrix [B] is obtained by differentiating the deformation vector. Element stiffness matrix
and nodal forces are found out using the formulae below:
1 1
1 11 1
1 11 1
1 1
det
det
T
T BB
T SS
K B CB JdrdsR
R N f JdrdsR
R N f ds
− −
− −
− −
=
=
=
∫ ∫
∫ ∫
∫ ∫
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SECTION 4
ANALYSIS
4.1 General
Axisymmetric and three dimensional formulations were considered for elastomeric bearing as
discussed in previous section. Use of different formulations was to gain the understanding the
finite element methods and find out if results differ by using different formulations. In order to
understand behavior of constrained rubber layer under compressive loading, analysis of single
rubber layer was done as well. Finite element models were built to imitate all features of actual
bearing; however, some features, which were understood not to affect the behavior of bearing
and were difficult to model, were omitted. Material properties and element formulations were
chosen in order to capture high nonlinear stress- strain behavior of rubber. Eigen value analysis
was done for each model and each model was checked for negative frequencies or inconsistent
behavior arising out of insufficient constraints. Finally, a static analysis was done to find out the
behavior of bearing under compressive loading.
4.2 Geometry
4.2.1 General In this report, a low damping natural rubber bearing, tested at University at Buffalo
(Constantinou et al., 2007), was used. Two finite element models were considered in Abaqus-1)
Axisymmetric model, and 2) three-dimensional model. Each model was a simplification of actual
rubber bearing and has been discussed below separately.
4.2.2 Geometry of Original Bearing Geometrical features of the original bearing have been outlined below and schematic has been
shown in Figure 4-1.
Diameter of bearing D = 250 mm
Bonded area of rubber = 49,087 mm2
Total Rubber Thickness Tr = 82.5 mm
Shape factor S = 9.8
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Figure 4-1: Low damping rubber bearing used for finite element analysis(dimensions are in mm)
4.2.3 Geometry of Axisymmetric Model Axisymmetric model was created by drawing the half cross-section and then partitioning in
different regions to facilitate material assignment of rubber and steel. Schematic of the cross-
section and co-ordinate system used have been shown in Figure 4-3.
Figure 4-2: Axisymmetric model of elastomeric bearing
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4.2.4 Geometry of Three-dimensional Model Three dimensional model was created by first sketching a half cross-section and then revolving it
to 360 degrees. Three-dimensional region obtained after 360 degree revolution was partitioned in
multiple layers in order to assign different material properties for rubber and steel in different
layers. Rubber cover around the cylindrical cross-section was not considered in the model, as it
is known not to affect the overall behavior. Schematic of the cross-section and co-ordinate
system used have been shown in Figure 4-3.
Figure 4-3: Three-dimensional model of elastomeric bearing
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4.3 Mesh
4.3.1 General Meshing capability of Abaqus is highly dependent on geometry. Element shapes available for
two dimensional meshing are: 1)Quadrilateral, 2) Quadrilateral-dominated, and 3)Triangular.
Element shapes available for three-dimensional meshing are: 1)Hex, 2)Hex-dominated, and
3)Tet.
Triangular and Tet element provides more flexibility in meshing the complicated geometry
compared to Quad and Tet elements; however, Triangular and Tet elements have less number of
degrees of freedom and hence more stiff which leads to loss of information while capturing the
behavior of structure. A careful consideration should be given when choosing Triangular and Tet
elements which are mainly popular in modeling crack problems.
Only Quad and Hex elements have been used in this report. Complex geometrical features were
divided in simple regions which could be meshed using simple techniques. Abaqus provides
primarily three meshing techniques:
1. Free meshing
2. Swept meshing
3. Structured meshing
In the mesh module of Abaqus, different regions of model are color coded according to the
default method it will use to generate the mesh:
• Green color indicates that a region can be meshed using structured methods
• Yellow color indicates that a region can be meshed using sweep methods
• Pink color indicates that a region can be meshed using the free method
• Orange color indicates that a region cannot be meshed using default element options and
must be divided further into simpler regions to allow meshing
Mesh shapes and meshing techniques were chosen for axisymmetric and three-dimensional
model and have been discussed in following sections
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4.3.2 Meshing of Axisymmetric Model Axisymmetric model considers a plane cross-section of model to analyze the whole geometry.
Owing to its simple geometry, it was meshed using structured meshing technique. Quadratic
quadrilateral, type CAX8R, element was used. Mesh bias was used to provide denser meshes on
the edges to capture sudden drop of shear stress.
Figure 4-4: Mesh used for axisymmetric model
Meshed model was checked for mesh quality using “verify mesh” tool in Abaqus. Mesh quality
parameters have been summarized in the table
Minimum angle Maximum angle Aspect ratio
Average 86.73 93.27 1.98
Worst 48.96 131.04 5.86
Table 4-1: Shape metrics of meshes used in axisymmetric model
Ideally, angle between included edges should be 90 degrees and aspect ratio should be less than
10. As can be seen here, aspect ratio confirms the guidelines but included angle deviates a little
bit due to non-uniform seeding of edge. It can be seen that although mesh bias provides the
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flexibility obtain accurate results with less cost of computation; it leads to large variation in mesh
aspect ratios.
4.3.3 Meshing of Three-dimensional Model Three-dimensional model had a complicated geometry that was not possible to mesh using
default meshing techniques. Model was partitioned in four quadrants, which enabled use of Hex
elements using sweep meshing technique. An advancing front algorithm using mapped meshing
was used to generate the meshes. Quadratic hexahedral element of type C3D20R was used.
Three-dimensional meshed model has been shown in Figure 4-5.
Figure 4-5: Mesh used for three-dimensional model
Mesh quality parameters for three-dimensional model have been summarized in Table 4-2.
Minimum angle Maximum angle Aspect ratio
Average 79.52 103.2 5.73
Worst 58.29 129 11.86
Table 4-2: Shape metrics of meshes used in three-dimensional model
Included angle for the meshes used was satisfactory; however, aspect ratios could be improved.
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4.4 Material Properties
4.4.1 Steel Steel was assigned the property of a linear elastic material, with Young’s modulus of 210 GPa
and Poisson’s ratio of 0.3. Assumption of linear elasticity is valid for range of loading considered
in this problem and hence biaxial model was not considered. It was assigned a density of 7850
kg/m3.
4.4.2 Rubber Rubber is a hyperelastic material with low shear modulus and very high bulk modulus. Poison’s
ratio of rubber is close to 0.5 and it is considered almost incompressible. High nonlinearity of
rubber behavior cannot be captured with linear elastic models.
An almost incompressible Neo-Hookean model(Rivlin, 1948) was used for rubber. Constitutive
law for this model is based on energy potential formulation.
210 1
1
1( 3) ( 1)elU C I JD
= − + −
Where U is the strain energy per unit reference volume; 10C and 1D are temperature-dependent
material parameters; 1I is the first deviatoric strain invariant; elJ is the elastic volume ratio that
accounts for finite compressibility.
The initial shear modulus and bulk modulus are related to material parameters as:
10 12;
2GC D
K= =
G and K are the shear modulus and bulk modulus of the rubber. Shear modulus of 0.65 MPa and
bulk modulus of 2000 MPa was used for the natural rubber used in this report.
28
Figure 4-6: Hyperelastic model of rubber
4.5 Boundary Conditions and Loading
4.5.1 Axisymmetric Model Two boundary conditions were applied to the axisymmetric model: 1) Encastre (U1= U2= U3=
UR1= UR2= UR3 =0) at the support to fix all degrees of freedom, and 2) XSYMM (U1= UR2=
UR3= 0) at the axis of symmetry to assign symmetry about the center.
A load of 1000 kN was applied at the top surface as pressure of 14154 kN/m2 distributed over the
area. Boundary conditions for axisymmetric model have been shown in Figure 4-7.
4.5.2 Three-dimensional Model Two boundary conditions were applied-1)Fixed boundary condition was applied at the base of
the model (U1=U2=U3=UR1=UR2=UR3=0) and 2)Uniform downward displacement of 1 mm at
the top surface while keeping all other degrees of freedom fixed(U1=U3=UR1=UR2=UR3=0,
U2=-0.001).
Boundary conditions for axisymmetric model have been shown in Figure 4-8.
Stress
Strain
29
Figure 4-7: Loads and boundary condition for axisymmetric model
Figure 4-8: Loads and boundary condition for three-dimensional model
30
4.6 Results
4.6.1 General Results have been presented in separate sections for rubber layer, axisymmetric model, and
three-dimensional model. Although, formulation for single rubber layer has not been shown,
results have been presented here to understand the behavior of constrained rubber layer. For
eigenvalue analysis, only relevant first few modes have been shown.
4.6.2 Rubber Layer An axisymmetric model of rubber layer was imposed a downward displacement of 1 mm at the
top surface. State of stress (Mises) and logarithmic shear strain has been shown in figures below.
Figure 4-9: State of stress(Mises) in the rubber layer
Figure 4-10: State of logarithmic shear strain in the rubber layer
4.6.3 Axisymmetric Model 4.6.3.1 Eigen-value Analysis Analysis was checked for any negative eigenvalues and only 10 modes were extracted from the
analysis, summary of which has been presented in Table 4-3. As horizontal degree of freedom
was constrained at the axis of symmetry (U1=0), axisymmetric model doesn’t have 1st mode as
shear of bearing. Shear mode can be extracted at the expense of losing the symmetry which, off
course, is not the aim of axisymmetric modeling.
31
Mode Number Eigenvalues(×106) Eigenfrequencies(×103)
1 8.32 0.460
2 20.20 0.715
3 47.14 1.093
4 58.01 1.212
5 75.68 1.385
6 96.51 1.563
7 101.56 1.60
8 102.18 1.61
9 122.17 1.76
10 129.67 1.81
Table 4-3: Eigenproperties of axisymmetric model
First 3 modes of model, superimposed on undeformed state, have been shown in Figure 4-11.
Figure 4-11: First 3 modes of axisymmetric model
4.6.3.2 Static Analysis State of stress (Mises) and logarithmic shear strain has been shown in figures below
Mode 1 Mode 2 Mode 3
32
Figure 4-12: State of stress(Mises) in the axisymmetric model
Figure 4-13: State of logarithmic shear strain in the axisymmetric model
33
4.6.3.3 Calculation of vertical stiffness Our main focus of the analysis was to calculate the vertical stiffness of circular elastomeric
bearing. For axisymmetric model, total force of 1000 kN was applied and deformation was
measured at the top surface. Top surface was restrained against any deformation other than
vertical deformation. After analysis deformation was measured at two corner points and
averaged. Vertical displacement was calculated as U2=5.164×10-6 m. Hence, stiffness was
calculated as:
3
66
1000 10 193.6 10 /2 5.164 10
FK N mU −
×= = = ×
×
4.6.4 Three-dimensional Model 4.6.4.1 Eigen-value Analysis Eigenproperties for three-dimensional model has been presented in Table 4-3.
Mode Number Eigenvalues(×103) Eigenfrequencies
1 16.0 20.13
2 18.8 21.83
3 18.8 21.84
4 434 104.87
5 438 105.37
6 438 105.38
7 1610 201.71
8 1610 201.77
9 1610 201.79
10 2946 273.18
Figure 4-14: Eigenproperties of three-dimensional model
As can be seen from Table 4-3, few modes are identical. Modes 1, 2, 4, and 7 is shown in Figure
4-15.
34
Figure 4-15: Modes of vibration of three-dimensional model
Mode 1 Mode 2, 3
Mode 4, 5, 6 Mode 7, 8, 9
35
4.6.4.2 Static Analysis State of stress (Mises) and logarithmic shear strain has been shown in Figure 4-16, and Figure
4-17.
Figure 4-16: State of stress(Mises) in the three-dimensional model
Figure 4-17: State of logarithmic shear strain in the three-dimensional model
36
4.6.4.3 Calculation of Vertical Stiffness A uniform displacement of 1 mm was applied to top surface of three-dimensional model. Total
result force generated at any surface perpendicular to vertical axis was measured using “Free
Body Cut” tool in Abaqus as shown in Figure 4-18.
Figure 4-18: Resultant force and moment at a section mid-height of bearing
5
63
1.806 10 180.6 10 /2 1 10
FK N mU −
×= = = ×
×
37
SECTION 5
DISCUSSION
5.1 General
Finite element method was used to gain a better understanding of behavior of circular
elastomeric bearings under compression, and to investigate the validity of analytical solutions.
Different finite element formulations might converge to different results. Two finite element
models of bearing, axisymmetric and three-dimensional model, was created and analyzed to
understand the effectiveness of different modeling techniques. A model of single constrained
rubber layer was created as well to see how global behavior differs from local behavior. Models
were also checked for mesh sensitivity and element sensitivity. Summary of comparisons and
discussions on results have been presented in following sections.
5.2 Analytical Predictions
Analytical solutions were obtained for vertical stiffness, normal stress, and shear stress using the
theory presented in section 2.2 and section 2.3.
Vertical stiffness 6162.5 10c
r
AEKT
= = ×
Variation of normal stress is given by 2 22
3 ( )cxx yy zz
Gp R rtεσ σ σ= = = − = − −
Total force at a horizontal plane was obtained as4
2
32
cG RPt
ε π= . We can normalize the stress
quantities as:
2
2
, ,2(1 )
/xx yy zz r
P A Rσ σ σ
= − −
Variation in shear stress is given by:
2 2
6 6,c cxz yz
G Gxz yzt tε ετ τ= − = −
38
If the variation is considered at rubber-steel interface(z=-t/2), and shear stress is normalized,
expression is given as
/S r
P A Rτ
=
Which is a linear variation with respect to radius(x,y direction).
5.3 Comparison of Results
Analytical predictions were compared with results from different finite element models. Values
were normalized for better representation.
5.3.1 Comparison of Vertical Stiffness Values of vertical stiffness calculated from analytical solutions, axisymmetric model and three-
dimensional model has been shown in Table 5-1.
Analytical Axisymmetric Three-dimensional
162.5×106 193.6×106 167.5×106
Table 5-1: Vertical stiffness values (all values in N/m)
As can be seen from Table 5-1, values obtained from finite element are close to the analytical
predictions with results from three-dimensional model being closer. Difference in values may be
attributed to many factors like analytical predictions neglect the contribution to overall stiffness
of bearing by steel shims. Difference in results between finite element models is attributed to
use of different meshes, elements, and constraints.
Using analytical solution as a reference value, three-dimensional model produced an error of 3%
and axisymmetric model produced an error of 19%. Hence based on this study it can be
concluded that three-dimensional model produces better result than axisymmetric model.
Generalization of this conclusion needs to be verified.
5.3.2 Comparison of Normal Stress Variation Graphical comparison of normal stress distribution obtained from analytical solution and three-
dimensional finite element model has been presented in Figure 4-17.
39
Figure 5-1: Normal stress variation along the radius of three dimensional model
Normal stress distribution obtained from finite element analysis shows a very good match with
the analytical solutions.
5.3.3 Comparison of Shear Stress Variation Graphical comparison of shear stress distribution along radial direction obtained from analytical
solution with three-dimensional and axisymmetric finite element model have been presented in
Figure 5-2 and Figure 5-3.
Figures suggest that finite element analysis matches with analytical predictions quite accurately
in the inner regions of the bearing; but at the circumference where there is a sudden drop in shear
stress from maximum value to zero; neither analytical, nor finite element model could capture
that. This was because very coarse mesh was used for the analysis. Sudden drop in shear stress
can be captured upto an extent by providing very fine mesh closed to the boundary, however;
that would increase the computational cost significantly. Efficiency of axisymmetric model is
evident from graphs which produced better results using less number of elements compared to
0
0.5
1
1.5
2
2.5
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Stre
ss/(
P/A)
r/R
Compressive stress distribution: 3D Model
Analytical
S11/(P/A)
40
three-dimensional model. A convergence could be obtained for three-dimensional model as well
at the boundary but further analysis could not be done because of computing limitations.
Figure 5-2: Shear stress variation along the radius of three dimensional model
Figure 5-3: Shear stress variation along the radius of axisymmetric model
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.2 0.4 0.6 0.8 1 1.2
Stre
ss*S
/(P/
A)
r/R
Shear Stress Variation: 3D Model
Analytical
s12*S/(P/A)
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Stre
ss*S
/(P/
A)
r/R
Shear Stress Variation: Axisymmetric
Analytical
s12*S/(P/A)
41
Distribution of shear stress along height of bearing has been shown in Figure 5-4.
Figure 5-4: Shear stress variation along height of axisymmetric bearing model
Figure validates the analytical solution which dictates that shear variation along height(z) of
bearing should be linear 2 2
6 6( , )c cxz yz
G Gxz yzt tε ετ τ= − = − . A saw-tooth pattern is obtained due to
alternate layers of steel and rubber. Value of shear stress at mid height of each layer is zero and
magnitudes of shear stress in layers vary around zero due to very low shear modulus of rubber.
However, external and internal steel bearing plate provides sufficient shear resistance and
experience significant shear stress. Variation in the bearing plates at the top is parabolic due to
beading and linear in the bearings plates at the bottom as it’s restrained and experiences no
bending.
5.4 Sources of Error
Finite element analysis results presented here contained errors that were produced by many
factors. Contribution to error by these factors is problem specific. Some of these factors, which
played an important role in analyses presented in this report, have been discussed in following
sections.
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0 0.05 0.1 0.15 0.2Shea
r Str
ess(
MPa
)
Height(m)
Shear Stress: Axisymmetric Model
42
5.4.1 Mesh Sensitivity Results obtained were highly sensitive to the mesh size. In finite element it’s always a
compromise between size of mesh and cost of analysis. Shear stress variation near the
circumference, in particular, was highly sensitive to mesh size. Shear stress value goes from
maximum to zero value at the circumference and it requires very fine meshes to capture this
behavior.
Figure 5-5: Mesh sensitivity for shear stress variation of 3D model
Figure 5-6: Mesh sensitivity for shear stress variation of 3D model
0
0.25
0.5
0.75
1
1.25
1.5
1.75
0 0.2 0.4 0.6 0.8 1 1.2
Stre
ss*S
/(P/
A)
r/R
Mesh Sensitivity: 3D Model Analytical
5158 elements
9390 elements
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.2 0.4 0.6 0.8 1 1.2
Stre
ss*S
/(P/
A)
r/R
Mesh Sensitivity: Axisymmetric
Analytical4424 elements5954 elements20569 elements46043 elements
43
As shown in Figure 5-5, as number of meshes is increased, shear stress variation obtained from
finite element analysis starts to capture the sudden drop at the circular edge of the bearing. It can
be seen that axisymmetric model captured the behavior more accurately with less cost of
computation comparing to three-dimensional model.
5.4.2 Element Sensitivity Element sensitivity is not independent of mesh sensitivity. A large number of first order elements
can produce better result than a low number of second order elements; however, computation
costs would be higher. In this report, quadratic elements produced much better result than linear
elements. In order to assess the relative performance of first-order(linear) and second-
order(quadratic) elements, three-dimensional model with same mesh sizes were analyzed with
two type of elements. Model with first order element, with distortion control, second-order
accuracy, and enhanced hourglass control, produced constant warnings of excessive distortion
and converged after few attempts. Model with quadratic elements had no convergence issues.
Values of stiffness obtained from two models with different elements have been shown in Table
5-2.
Analytical Linear Quadratic
162.5×106 810.0×106 180.5×106
Table 5-2: Vertical stiffness values (all values in N/m)
As values reported in table suggests, linear elements produced absurd results. Hence, it can be
claimed that for large deformation analysis, like rubber, quadratic elements produce better
results.
5.4.3 Model Sensitivity Three-dimensional model produced the results that were closer to analytical predictions. If
analytical solution is assumed to be correct, it can be claimed that three-dimensional model
produces better results than the axisymmetric model.
5.5 Summary
Comparison of results obtained from finite element analyses showed good agreements with
analytical predictions. Sudden drop in shear stress at edges could not be captured with models of
coarse meshes; however, as the mesh size was decreased, models begin to capture the drop to a
44
reasonable extent. Sources of error were identified, and biggest factor was mesh size. Rubber
shows highly nonlinear behavior and generally it’s difficult to capture high non-linearity, but
finite element model presented here captured the actual behavior quite accurately.
45
SECTION 6
SUMMARY AND CONCLUSIONS
Finite element analysis(FEA) of elastomeric bearing was done under compressive loading. Two
finite element models of elastomeric bearing were created-1) Axisymmetric model, and 2)Three-
dimensional model. Complex internal geometry and highly nonlinear material behavior of rubber
were the main issues in modeling. A Neo-Hookean model of rubber was used and partitioning
was used to divide the cylindrical bearing in small regions which could be easily meshed for
analysis. Vertical stiffness, radial variation of compressive stress, and radial variation of shear
stress were used as benchmarks in assessing the result of a particular analysis. In order to
understand the local behavior, a separate model was created for constrained rubber layer.
A modal analysis followed by the static analysis was done. Finite element analyses produced
very good results and captured the behavior predicted by the analytical solutions. Three-
dimensional model produced results that were closer to analytical solutions compared to
axisymmetric model. However axisymmetric model captured the sudden drop in shear stress at
the boundaries more accurately at less cost of computation. Accuracy of finite element results
increased with increasing number of elements. A very fine meshing was able to capture sudden
drop of shear stress at the boundary. Quadratic interpolation functions were more effective
compared to linear interpolation functions. Analysis with limited number of linear elements
produced very erroneous results. Neo-Hookean hyperelastic model was found to be very
effective in capturing the nonlinear behavior of rubber.
Finite element analysis is an effective tool to analyze complex structures, which otherwise
cannot be done using simplified methods of analysis. However, accuracy of FEA results depends
on ability to model the actual structure and loading conditions. In this study, FEA produced good
results except at singularity point where shear stress drops from maximum to zero. Such
problems in FEA is attempted using asymptotic analysis; however it was out of scope for this
project.
46
REFERENCES Chalhoub, M. S., and Kelly, J. M. (1990). "Effect of bulk compressibility on the stiffness of cylindrical base isolation bearings." International Journal of Solids and Structures, 26(7), 743-760.
Constantinou, M. C., Kartoum, A., and Kelly, J. M. (1992). "Analysis of compression of hollow circular elastomeric bearings." Engineering Structures, 14(2), 103-111.
Constantinou, M. C., Whittaker, A. S., Kalpakidis, Y., Fenz, D. M., and Warn, G. P. (2007). "Performance of Seismic Isolation Hardware Under Service and Seismic Loading."United States, 472p.
Kelly, J. M., and Konstantinidis, D. (2011). "Mechanics of Rubber Bearings for Seismic and Vibration Isolation." John Wiley & Sons, Hoboken, 1 online resource (240 p.).
Rivlin, R. S. (1948). "Large elastic deformations of isotropic materials." Royal Society of London -- Philosophical Transactions Series A, 240(823), 509-525.
Simo, J. C., and Kelly, J. M. (1984). "Finite element analysis of the stability of multilayer elastomeric bearings." Engineering Structures, 6(3), 162-174.
47
APPENDIX A
Abaqus Input File Submitted separately due to excessive size of the file
