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Analytical Simulations of the Steel-Laminated Elastomeric Bridge Bearing
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DOI: 10.1017/jmech.2014.24
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Analytical Simulations of the Steel-Laminated Elastomeric Bridge Bearing
R.-Z. Wang, S.-K. Chen, K.-Y. Liu, C.-Y. Wang, K.-C. Chang and S.-H. Chen
Journal of Mechanics / Volume 30 / Issue 04 / August 2014, pp 373 - 382
DOI: 10.1017/jmech.2014.24, Published online: 06 June 2014
Link to this article: http://journals.cambridge.org/abstract_S1727719114000240
How to cite this article:
R.-Z. Wang, S.-K. Chen, K.-Y. Liu, C.-Y. Wang, K.-C. Chang and S.-H. Chen (2014). Analytical Simulations of the SteelLaminated Elastomeric Bridge Bearing. Journal of Mechanics, 30, pp 373-382 doi:10.1017/jmech.2014.24
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ANALYTICAL SIMULATIONS OF THE STEEL-LAMINATED
ELASTOMERIC BRIDGE BEARING
R.-Z. Wang
S.-K. Chen
National Center for Research on Earthquake Engineering
Taipei, Taiwan 10668, R.O.C.
Department of Civil Engineering
National Central University
Jhongli, Taiwan 32001, R.O.C.
K.-Y. Liu 
C.-Y. Wang
National Center for Research on Earthquake Engineering
Taipei, Taiwan 10668, R.O.C.
Department of Civil Engineering
National Central University
Jhongli, Taiwan 32001, R.O.C.
K.-C. Chang
S.-H. Chen
Department of Civil Engineering
National Taiwan University
Taipei, Taiwan 10617, R.O.C.
Department of Civil Engineering
National Central University
Jhongli, Taiwan 32001, R.O.C.
ABSTRACT
In this paper, analytical simulations of the steel-laminated elastomeric bearing (SLEB) using a threedimensional (3D) finite element model incorporating material, geometric nonlinearities, and a frictional
contact algorithm in LS-DYNA code is conducted. In order to simulate the nonlinear responses of the
elastomeric bearing under the compression and shear, a hyperviscoelastic rubber model such as The
MAT_77_H (MAT_HYPERVISCOELASTIC_RUBBER) in LS- DYNA code is adopted. Based on the
proposed material model for the SLEB, the interaction effects of the SLEB under compression, bending,
and torsion are analyzed. Analytical results are compared with the test results of the SLEBs. A set of
material parameters is proposed for 3D FEM analysis of SLEBs. The proposed material model demonstrates its accuracy.
Keywords: steel-laminated elastomeric bearing, frictional contact, LS-DYNA.
1.
INTRODUCTION
Although the bearing and joint system provides important functions to the entire bridge. However, it
costs only a small percentage of its total construction
cost (1 percent, typically). Inadequate design of these
vital elements can compromise the integrity of the entire structure. Bridge bearings have been developed to
make a positive contribution to the seismic performance
of the structure. However, observations made of the
damage modes of bridges after the 1999 Chi-Chi
earthquake in Taiwan, a special consideration regarding
the function of bearing and joint in the seismic design
of bridge is needed. Due to most of the inertial force
of superstructures to be transmitted into the bridge piers
and columns, the constraint induced movement of the
bearing system can be used to reduce the inertial forces
of the piers along the bridge transverse direction. This
load transmission mechanism may cause the bridge
destroyed such as the shear failure of bridge piers and
columns, and collapse of entire bridge. Elastomeric
*
bearings are used to support bridge girders and to accommodate the extensions, contractions, and rotation
due to changes in temperature, traffic loading, earthquakes, concrete shrinkage, creep, initial construction
tolerances, and other sources. Under service loads, the
elastomeric bearing deflects both vertically and horizontally providing a limited amount of vibration damping to the bridge superstructure. Although elastomeric
bearings can be made of plain rubber, these bearings
consist of alternating layers of elastomer and steel in
order to increase their vertical stiffness. The steel
plate constrains the lateral expansion of the rubber material and increases the vertical stiffness. This type of
design is referred to as the steel-laminated elastomeric
bearing (SLEB). The horizontal force transmitted by
the superstructure to the substructure is a function of
the elastomer’s shear modulus. In order to develop a
performance based design and to evaluate the specifications or suggestions for the seismic design and retrofit
of bridges, a research project on the seismic loading
transfer mechanism and the failure sequence of bridge
Corresponding author (kyliu@narlabs.org.tw)
Journal of Mechanics, Vol. 30, No. 4, August 2014
DOI : 10.1017/jmech.2014.24
Copyright © 2014 The Society of Theoretical and Applied Mechanics, R.O.C.
373
structures was carried out by a task group in the National Center for Research on Earthquake Engineering
(NCREE) of Taiwan. In this project, a global-local
type analysis procedure for the simulation of nonlinear
failure behavior of a bridge system under seismic loading was developed. A SLEB including the frictional
movement and the energy dissipation functions can be
used to reduce inertial forces of the bridge piers and
columns.
Over the years, a variety of bearing designs and
analysis methods have been devised based on either
approximate or rigorous numerical procedures. Elastomeric bearing design methods are regulated by
AASHTO [1] and other specifications [2].
The
NCHRP Project 10 ~ 20, “Elastomeric Bearings Design,
Construction, and Materials,” was initiated in 1981 to
address the absence of detailed design procedures for
the use of elastomeric bearings in the AASHTO Standard Specifications for Highway Bridges. Roeder et al.
[3], Mori et al. [4], Abe et al. [5], Topkaya [6] and
Burtscher and Dorfmann [7] conducted laboratory tests
on actual bridge bearings in order to provide a correlation between bearing performance and test data, suggesting theories upon which earlier design specifications were based. Yura et al. [8] continued the
NCHRP project by further conducting a thorough study
on elastomeric bridge bearings, and developed performance-related specifications. They developed new
test methods and recommended specifications for the
testing of elastomeric bearings. A two-dimensional
finite element analysis of elastomeric bridge bearings
was conducted by Hamzeh et al. [9,10]. This investigation resulted in the following conclusions: (1) high
rubber shear stresses and strains were found to be concentrated at the layer interfaces towards the edges; (2)
shearing of the pad was found to be responsible for the
bulk of the steel stresses by bending the shims; (3)
compressive stiffness of the bearing was found to be
significantly affected by the number of laminates; shear
stiffness, on the other hand, was found to be unaffected;
(4) as long as the laminates were thick enough to remain unyielded, the thickness of the steel laminates did
not have a major effect on the pad’s behavior. Imbimbo and Luca [11] used the ABAQUS code [12] to
study the influence of shape factor on the stress distributions and stress concentrations of a laminated elastomeric bearing subjected to vertical loads. In their
analysis, the mechanical rubber behavior was modeled
by a homogeneous, isotropic and hyperelastic model; in
particular, they assumed the Mooney-Rivlin law.
Yazdani et al. [13] utilized the ANSYS code to study
the presence, extent, and effect of bearing pad restraint
on precast prestressed bridge beams. In their global
type analysis of a bridge structure, the bearing pads
were modeled with a series of spring elements,
COMBIN14, from the ANSYS code. Yoshida, Abe,
Yoshida et al. [14] conducted a very thorough study on
the laminated high damping rubber bearing. Nguyen
and Tassoulas [15] used ABAQUS to analyze the effects of shear direction on bearing behavior. They
proposed an accurate constitutive model for high
damping rubber materials and applied it into a mixed
374
three-dimensional finite element code to study complex
deformation such as torsional or rotational deformation
of the rubber bearing. So far, many analytical models
and finite element analyses of elastomeric bearing have
been presented by researchers [3-15]. However, the
performance of the rubber bearing system in the field is
complicated due to the following factors: The nonlinear
viscous character of the rubber material, the lamination
parameters of the elastomeric bearing, stress concentration at the edges, friction contact conditions between
the pad and the bridge structure, thermal effects, loading type and history. Due to these factors, researcher
used numerical method to study the local and global
behavior of steel-laminated elastomeric bearing (SLEB)
under service conditions.
In this paper, the LS-DYNA code [16,17] is used to
analyze some aspects of SLEB behavior due to its versatile capabilities on nonlinear dynamic, large deformation and contact analyses. The parameters of the
rubber material are calibrated by experimental results of
compression test, compression-shear test and compression- frictional shear test with one frictional surface.
These calibrated modeling processes under various design parameters can be further used to compute the
nonlinear dynamic responses of SLEB subjected to
compression and shear undergo complex deformations.
2.
CONSTITUTIVE MODELS
In this section, the constitutive model of the SLEB in
the LS-DYNA code is described. The SLEB consists
of the rubber and steel materials. A hyperviscoelastic
model is used to analyze deformation behavior of the
rubber. Since the strain values of the steel shims are
observed to be relatively small in all tests. An elastic
material model of steel is used in the finite element
analysis.
2.1
Constitutive Model of the Rubber Material
The rubber material used in bridge bearings requires
this hysteretic behavior to reduce the inertial force of
the bridge and absorb the energy of the earthquake.
On a macroscopic level, the behavior of the rubber exhibits certain characteristics: (1) it can undergo large
elastic deformation; (2) there is little volume change
when stress is applied. Since the rubber deformation
is related to the strengthening of the molecular chains.
Hence, elastomers are almost incompressible. Hence,
accurate modeling of the damped rubber bearings requires a joint modeling of the slight compressibility, the
kinematic nonlinearity, and the nonlinearity of the material. Many models have been proposed for the rubber material in the literatures [18-23]. Salomon et al.
[22] proposed a constitutive model of high damping
rubber, where the hysteretic behaviors of the rubber
component were modeled by viscoelastic and plastic
constitutive models. Yoshida et al. [14,21] modeled
the constitutive relations of the rubber by combining a
hyperelastic damage model with an elastoplasticity that
acts according to a strain-dependent isotropic hardening
law.
Journal of Mechanics, Vol. 30, No. 4, August 2014
A hyperviscoelastic model composed by a hyperelastic model combined in parallel with the Maxwell viscoelastic model is used for the rubber material [24]. The
mechanical illustration of this model is shown in Fig. 1.
In this model, the spring Khe represents the hyperelastic
part and the six modified Maxwell models represent the
viscous part of the rubber material. The mathematical
forms of these two major parts of the hyperviscoelastic
constitutive model are briefly explained as follows.
2.2
The Hyperelastic Part
W ( J1 , J 2 , J ) 
C
( J1  3) ( J 2  3)  WH ( J )
p
pq
q
(1)
p,q 0
with
J1  I1 J

J2  I2 J
1
3

(2)
1
3
(3)
J  1  2  3 
V
V0
(4)
where I1, I2 and I3 are the first, second and third strain
invariants, J1 and J2 are the associated deviatoric invariants, J defined by Eq. (4) is called the volume ratio, V0
and V are the initial and current volumes of a differential cell, 1, 2 and 3 are the principal stretch ratios, p
and q are the order of polynomial for J1 1 and J2 1,
respectively. In Eq. (1), WH (J) is the volumetric term
of the strain energy functional which is function of the
volume ratio J. In Eq. (1), the Cpq can be calculated
from first specifying the stress-strain relationships, then
setting the value of n (0, 1, 2 or 3) in the LS-DYNA
code for the rubber material. The polynomial form
with n  2 may be valid up to 90 ~ 100 tensile strains.
More terms may capture any inflection points in the
engineering stress-strain curve. Five or nine term
polynomials may be used up to 100 ~ 200 strains.
2.3
Viscous Part
To consider the hysteretic behavior of the rubber
component, a modified Maxwell model with frictional
damping as shown in Fig. 1 is adopted. For simple
viscoelastic behavior, the relaxation function of the
visco-elasticity theory is modeled by a 6 term Prony
series as follows:
g (t ) 
6
G e
G1
i t
i
(5)
i 1
where Gi is optional shear relaxation modulus for ith
term and i is the corresponding decaying constant with
respect to time t. In Eq. (5), twelve values of (Gi, i,
In the LS-DYNA code
i = 1 ~ 6) must be specified.
Journal of Mechanics, Vol. 30, No. 4, August 2014
 fric
 fric
G2
G3
K he
 fric
G6
· · ·
1
Fig. 1
In the hyperelastic part, the strain energy density
function W of a polynomial form Ogden [23] is:
n
 fric
2
3
6
Mechanical representation of the hyperviscoelastic material model
[16,17], the Gi for i 1 ~ 6 is a constant value. A limit stress (SIGF) fric for the frequency independent frictional damping is adopted. The default values of decaying constants for the five i (i  1, 3 ~ 6) can be
adopted. As described in LS-DYNA user’s guide, 1
is set to zero, 3 is 10 times 2 , 4 is 100 times greater
than 3 , and so on. In this study, the reasonable values of the limit stress fric, the shear modulus G of the
modified Maxwell’s model and decay constant 2
(BSTART) are obtained by matching the area of the
hysteretic loop in a experimental loading and unloading
response.
3.
CONSTITUTIVE PARAMETERS
The purpose of this section is to describe the constitutive parameters of the SLEB in LS-DYNA commercial
program. In the present analysis, it is assumed that the
material properties of the rubber material are the same
for all the rubber elements in the numerical model.
Testing of rubber bridge bearings, it is common practice
to take the rubber at the surface of the bearing as representative of the rubber for the whole bearing. This
practice may be misleading since the rubber properties
vary across the thickness of the laminated bearing due to
prolonged heat treatment during the molding process.
Othman [25] stated that these variations in properties
will not affect the overall performance of the bearing
provided the change of the hardness grade is small ( 5
IRHD). If the variation is large, then the compression
and the shear modulus may be affected. There are more
than 10 material models for rubber materials in the LSDYNA code. Figure 2 shows the uniaxial tension test
device and the tested elastomeric bearing by Wu [26].
In Fig. 3, the stress- strain relationship of the rubber is
required as an input for the LS-DYNA code. In this
study, setting the values of the shear modulus for frequency independent damping G  19.4635MPa and limit
stress SIGF  0.3MPa in LS-DYNA analysis, Fig. 3
shows that analytical result is very close to the result of
the tension test. In the following analysis the elastomer
is modeled as a nearly incompressible hyperviscoelastic
material
[23,27].
The
function
of
the
MAT_HYPERVISCOELASTIC_RUBBER (MAT_77_H)
in LS-DYNA code is used to analyze the deformation
The values of the
behavior of the rubber layers.
375
Fig. 2
test, compression-shear test and compression-frictional
shear test with one frictional surface. These three tests
were conducted by Wu [26]. In this study, two analytical models of the SLEB are constructed. First
model is the SLEB with two frictional surfaces subjected to compression and shear. Second model is to
consider the interaction effects of the SLEB with two
no-slip boundary conditions at the top and bottom surfaces subjected to the combined axial, shear, and bending forces. All steel laminates are assumed to be of
equal thickness; the same holds true for the rubber layers. The pads have dimensions: rubber layer thickness
 10mm; steel laminate thickness  1mm; cover thickness of rubber at lateral surface  3mm; thickness of
rubber cover at top or bottom = 2mm. Both the rubber
layers and the steel shims are modeled by fully integrated S/R solid elements (*ELEMENT_SOLOD).
For all simulation examples, the functions of the MAT_
HYPERVISCOELASTIC_RUBBER
and
MAT_ELASTIC are used to analyze the material behaviors of the hyperviscoelastic rubber and steel shims
as described section 3. The friction-contact analysis
algorithm
using
a
function
of
the
CONTCAT_AUTOMATIC in LS-DYNA is applied for
the interfaces between the bearing pad and the loading
plate.
Uniaxial tension test device and the tested
elastomeric bearing
2.5
Stress(N/m2)
2
1.5
1
Exp.
FEM
0.5
4.1
0
0
0.2
0.4
0.6
0.8
1
Strain
Fig. 3
Stress-strain curves of the rubber material [25]
material properties for MAT_ HYPERVISCOELASTIC_RUBBER used in the analysis are: Mass density
  1  103kg/m3, shear modulus for frequency independent damping G  19.4635MPa and limit stress
SIGF  0.3MPa, Poisson’s ratio v  0.4999, decay constant (BSTART)  0.0001, ramping time  0. In Eq.
(1), the Cpq values can be computed from setting the n
value to 1 in the LS-DYNA code. All the values for
these parameters are calibrated using the experimental
results from the compression test and compression-shear test to be described in section 4. The function of the MAT_ELASTIC (MAT_01) model in LSDYNA code is used to analyze the deformation behavior of the steel shims. Since the steel shim deformations in the tests are very small, the SS400 steel can
be modeled as an elastic material with the following
properties: Young’s modulus  200  103MPa, and
Poisson’s ratio v  0.29. Contact detection and displacement control functions of the LS-DYNA code are
used to analyze the frictional stick-slip phenomena of
the SLEB. The friction coefficient c between the
rubber bearing pad and a surface of different roughness
is described in section 4.
4.
NUMERICAL SIMULATIONS
A 3D FE analysis of SLEB using LS-DYNA is conducted to simulate three tests including compression
376
Compression Test
The purpose of compression test is to find the setting
of n value. The deformation characteristic of SLEB is
large vertical stiffness and large horizontal flexibility.
The evaluation of the failure conditions is an essential
step in designing the elastomeric bearing. The failure
to function of the SLEB can occur either through global
failure, buckling or roll-out of the device, because of
local rupturing due to the tensile rupture of the rubber,
or detachment of the rubber from the steel or because of
steel yielding Imbimbo and Luca [11]. In Fig. 4, a
quarter of the analytical model of the SLEB for compression test consists of a bearing pad and two loading
plates. Each node of the upper loading plate is subjected to a nodal force. Two loading plate surfaces
above and below the SLEB are slip boundary conditions. Shell elements of high rigidity are used to
model the loading plate such as upper plate and lower
plate. In Eq. (1), the Cpq values can be computed from
setting the n values to 1, 2 and 3 in the LS-DYNA code.
Figure 5 (a) shows the vertical displacements of the
SLEB versus compression force relationships. It can
find that the numerical result of SLEB computed from
setting value n  1 is close to experimental result of the
SLEB. It is seen that the hyperviscoelastic material
model can simulate the viscoelastic behavior of the
rubber bearing quite well. However, there does not
seem to be any relationship with the G value, as shown
in Fig. 5(b). In Fig. 6, due to the stress concentration
effects at the edge of the pad, it is also observed that the
deformation of the steel plate is not uniformly distributed along the vertical direction. The deformation of
the rubber material in SLEB is having the bulge deformation.
Journal of Mechanics, Vol. 30, No. 4, August 2014
P
Upper plate
Stopper
Lower plate
Fig. 7
Fig. 4
Numerical simulation model for the
compression-shear test
Finite element analysis model of the
compression test
Fig. 8
Comparisons of experimental and analytical
compression-shear test
(a) Compression forces versus vertical displacements of the SLEB
with three values (n  1, 2 and 3)
matched with the experimental results, especially in the
area of the hysteretic loop. In addition, the lateral
stiffness of SLEB is proportional to the axial load,
based on the chapter 10 in the design code [28]. This
phenomenal was confirmed in Fig. 9. One can find
that there is a threshold value for the shear force to initiate the horizontal deformation of the bearing both in
the numerical and experimental results. Though energy dissipation area is similar in each case with different axial load P, the lateral stiffness is getting higher
if SLEB was subjected to high axial load.
(b) Compression forces versus vertical displacements of the SLEB
with three shear modulus (G  200,600 and 1000)
Fig. 5
Comparisons of experimental and analytical
compression test
Fig. 6 Side view of the bulging pattern of the rubber
bearing pad under compressive loading
4.2
Compression-Shear Test
In Fig. 7, four stoppers are modeled on the top and
bottom plates. The contact analysis algorithm is activated at the contact interface between the stopper and
the bearing. The SLEB is subjected to a constant vertical compressive load simulating the dead load of the
girder during the test. In addition, a displacement
control function is applied in the horizontal direction.
In Fig. 8, it is evident that the simulation is well
Journal of Mechanics, Vol. 30, No. 4, August 2014
4.3 Compression-Frictional Shear Test with One
Frictional Surface
Figures 10 and 11 show the test setup and the assembling detail of rubber bearing in the compressionfriction shear test. One side of the SLEB was attached
with a thick steel plate, so as to bolt on the loading
frame, and sliding-friction behavior is only allowed in
the other side of the SLEB. The friction force can be
measured from the load cell in the horizontal hydraulic
actuator. Figure 12 shows the finite element modeling
of this test. The SLEB is subjected to a vertical load
of P  556kN. The contact detection function of the
CONTCAT_AUTOMATIC in LS-DYNA code to analyze the friction phenomena at the top and bottom surfaces of the SLEB is adopted. It is known that the
friction coefficient between the rubber bearing pad and
a surface of different roughness is a function of the cyclic number and the relative speed between two contact
surfaces. The value of the friction coefficient decreases with an increase in the number of loading cycles.
In LS-DYNA code, the friction coefficient c is:
377
Fig. 9
Horizontal forces versus displacements for the
compression-shear test
Fig. 10 Test setup of the rubber bearing friction test
[26]
Fig. 11
Assembling detail of the rubber bearing
specimen [26]
Fig. 12 Finite element model for the simulation of the
compression-frictional shear test
 c  FD  ( FS  FD )e
 DC Vrel
 0.17  (0.23  0.17) e
0.005 Vrel
(6a)
(6b)
where FD is the dynamic friction coefficient, FS is the
static friction coefficient, and DC is the decaying constant. Base on the compression-friction shear test Wu
378
[26] in the first cycle, we can find friction coefficients
computed from Eq. 6(b) are close to the friction coefficients of the experimental result. In Eq. 6(b), dynamic
friction coefficient, static friction coefficient and one
decaying constant are FD0.17, FS0.23 and
DC0.17, respectively. This relation will be inputted
into the code for the frictional contact analysis. In this
case, the speed and displacement amplitude during the
test are 0.05m/sec and 0.15m, respectively. In Fig. 12,
comparing the analytical and experimental results, it is
evident that the proposed analytical model using
LS-DYNA code can provide good simulation of the
nonlinear friction-sliding behavior of the SLEB including shear sliding deformation. Both initial stiffness
and restoring stiffness is determined as mentioned in
section 4.2. In Fig. 13(a), the friction force predicted
by Eq. (6) is closed to maximal force in the test. In
Figs. 13(b) and 13(c), the shear deformation of the
SLEB computed from analytical result is compared
with the SLEB deformation from the test result when
the target displacement is 0.15m. In addition, the tilt
of SLEB for analytical and test results separated from
base concrete block.
4.4 Analytical Simulations of the SLEB with Two
Frictional Surfaces Under Compression and
Shear
To investigate the performance of a rubber bearing
installed in the field, Wu [26] carried out pseudodynamic tests to understand the compression-frictional
behavior of a bearing with two frictional surfaces.
Figure 14 shows the finite element modeling of the test
setup. Both top and bottom surface of the bearing are
allowed to slide independently. In Fig. 15, a demonstration result indicated that the tilting phenomenon at
both ends of the bearing can be observed in the test.
However, for a bearing that has only one frictional surface, as shown in Fig. 13, only one edge of the rubber
bearing tilts. Both tests reveal that shearing the pad is
responsible for introducing the bulk of the steel stresses
by bending the shims. In addition, we conducted a
series of simulations with an angle  with respect to the
bridge axis as shown in Fig. 16. For the case with
angle of 0 degree denotes the case without multiple
directional loading. Adopted and modified from section 4.4, the case with loading angle  of 0, 15, 30 and
45 degree, respectively was simulated in Fig. 17. It is
evident that there is minor difference among these response curves of the bearing when the excitations come
from different directions.
4.5 Interaction Effects of the SLEB Under Axial
Force, Shear Force, and Bending
It is reported that torsion deformation of the bearing
appears especially in base-isolated bridges with Lshaped piers Yoshida et al. [14]. The torsion test is to
discuss the interaction effects of the SLEB under Compression, Bending, and Torsion. The purpose of torsion test is to discuss what causes would influence the
performance of the rubber bearing’s torsion under various loading actions. Figure 18 shows the numerical
Journal of Mechanics, Vol. 30, No. 4, August 2014
Horizontal Force(kN)
200
Direction 0o
Direction 15o
Direction 30o
Direction 45o
100
0
-100
-200
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
Displacement(m)
(a) Horizontal forces versus displacements for compression-frictional shear test
Fig. 17 Hysteretic loops of the bearing under horizontal actions with different angles
(b) Deformation of bearing in FEM model
(c) Deformation of bearing in experiment
(a) One axial load and one torsion force (Case A)
Fig. 13 Deformation of the elastomeric bearing in
compression-frictional shear sliding
Fig. 14 Finite element model for the simulation of the
double-sided frictional shear test
(b) One bending moment, two axial forces and one torsion
force (Case B)
(a) Deformation of bearing in FEM model
(c) One eccentric axial loading and one torsion force
(Case C)
(b) Deformation of bearing in experiment
Fig. 15 Deformation of the elastomeric bearing in
compression-frictional shear sliding
Fig. 16 Top view of the finite element model for the
simulation of the double-sided frictional shear
test with loading direction of  angle
Journal of Mechanics, Vol. 30, No. 4, August 2014
Fig. 18
Loading conditions of torsion test
rubber bearing model. In order to exclude the frictional effects in torsion test of the SLEB, all the nodes
at the top rigid body and the rubber bearing interface
are constrained. Similarly, all the nodes at the bottom
surface of the rubber bearing are assumed fixed. The
bearing is subjected axial forces, shear forces, and
bending moments. Three cases of the loading conditions of torsion test are cases A, B and C, respectively.
These three cases are used to test the influence of the
rubber bearing’s torsion performance. The rubber
bearing’s torsion behavior follows the upper rigid
plate’s torque angle control curve with a loadingunloading period T  16 sec. In case A, the bearing is
subjected to an axial loading P at the centroid of the
379
30000
P0=0.5P
P=179kN
P2=2P
P3=3P
Torsion (N-m)
20000
10000
0
-10000
-20000
-30000
-0.6 -0.4
In this study, three-dimensional (3D) finite element
model incorporating material, geometric nonlinearities,
and a frictional contact algorithm in LS-DYNA code is
conducted to simulate the responses of the SLEB under
the compression and shear. The proposed material
model for simulating the nonlinear responses of the
SLEB is MAT_77_H (MAT_HYPERVISCOELASTIC
_RUBBER) in LS-DYNA code. In addition, a set of
380
0.2
0.4
0.6
0.2
0.4
0.6
0.2
0.4
0.6
(a) Case A
M=10919N-m
M1=2M
M2=3M
Torsion (N-m)
20000
10000
0
-10000
-20000
-30000
-0.6
-0.4
-0.2
0
Angle(rad)
(b) Case B
Torsion (N-m)
30000
L=0.0315m
L1=2L
L2=3L
L3=4L
20000
10000
0
-10000
-20000
-30000
-0.6 -0.4 -0.2
0
Angle (rad)
(c) Case C
Fig. 19 Simulation results of the hysteretic loops of
the bearing under three loading cases
0.6
T=16sec
T1=0.5T
T2=0.25T
T3=0.125T
0.4
0.2
0
-0.2
-0.4
-0.6
0
4
8
12
16
20
Time (sec)
(a) Loading periods versus time
20000
Torsion (N-m)
CONCLUSIONS
0
30000
T=16sec
T1=0.5T
T2=0.25T
T3=0.125T
10000
5.
-0.2
Angle (rad)
Angle (rad)
upper rigid plate as shown in Fig. 18(a). In case B, the
bearing is subjected to two axial loadings P  179kN at
the distance L on both sides of the line ab as shown in
Fig. 18(b).
The bending moment (M  P
2L10919N-m ) can be computed. In case C, the
bearing is subjected to an axial loading P at a distance L
= 0.0305m on the centroid of the upper rigid plate with
the four distances are L = 0.0305m, 0.061m, 0.0916m
and 0.122m as shown in Fig. 18(c), respectively. Figure 19(a) shows that slopes of the torsions versus angles
curves, and shows that the effect of the torsion force for
the deformation of the bearing subjected to four axial
forces (P0, P, P2, P3) is small. Each slope of the torsion-angle curve is increases with the increase of the
axial load in Case A. Figures 19(b) and 19(c) show
that slop amplitudes of the torsion-angle curve for the
bearing are similar under three bending moments (M,
M1, M2) and eccentric axial loading P with four distances (L, L1, L2, L3). In Fig. 19(b), each slope of the
torsion-angle curve is decreases with the increase of the
bending moment in Case B. In Fig. 19(c), each slope
of the torsion-angle curve is the same with the increase
of the eccentric axial loading in Case C. Figure 20(a)
shows the torsion angle versus time curves. In this
case, the bearing is subjected to an amplitudes of the
torsion angle 0.5 (rad) under four loading periods (T, T1
~ T3). Figure 20(b) shows that the performance of the
rubber bearing model is unstable under the period of the
displacement loading cycles of the torsion decreases.
It is seen that the torsion rigidity increased with the
strain rate effect and rubber bearing’s warping deformation occurs at the outside of the vertical plane. In
addition, the distribution of the shear stress XY on the
inside steel laminates are concentrated on the four corners. In Fig. 21, the bearing is subjected to four axial
loading (P1, P2, P3, P4) and an amplitudes of the torsion
angle 0.5 (rad) under loading period T=16sec. Figure
21 plots that the maximum shear stress XY is at the
corner of the four steel laminates and that the value of
the third steel layer is larger than the other steel layers.
In addition, the maximum shear stress XY of the second,
third and fourth steel layers increases with the increase
in axial loading. In order to show the shear stress XY
at each steel layer, Fig. 22 plots that the shear stress XY
changes with the height of the rubber at the corner.
The maximum shear stress XY occurs in the third steel
laminate. Therefore, we can predict that the damage
to the rubber would occur first in the third steel laminate.
0
-10000
-20000
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Angle (rad)
(b) Torsions versus angles
Fig. 20 Simulation results of hysteretic loops of the
bearing under torque actions with four loading
periods
Journal of Mechanics, Vol. 30, No. 4, August 2014
P=179kN
P1 =2P
P2 =3P
P3 =4P
Height (m)
0.03
0.02
0.01
0
0
10
20
30
40
Shear StressXY(MN/m 2)
Fig. 21 The distribution of shear stress XY at the
corner of elastomeric bearing under compressive-torsion loadings
Shear Stress XY (MN/m2 )
40
30
20
4st Steel Layer(Botton)
3st Steel Layer
2st Steel Layer
1st Steel Layer(Top)
10
0
0
1
2
3
4
5
nP (n=1~4) (N)
Fig. 22 The maximum shear stress XY of the steel
plate at the corner of elastomeric bearing under compressive-torsion loadings
material parameters is proposed. The hyperviscoelastic material model calibrated by experimental results
can describe the nonlinear viscous, hysteretic behavior
of the rubber material well under various loading conditions. The proposed material model demonstrates its
accuracy.
ACKNOWLEDGEMENTS
This material is based upon work supported by National Science Council, Taiwan, R.O.C. under Grant No.
NSC93-2625-Z002-025 and NSC93-2625-Z008-019.
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(Manuscript received May 12, 2013,
accepted for publication December 19, 2013.)
Journal of Mechanics, Vol. 30, No. 4, August 2014