- Email: [email protected]
Assessment of the vertical stiffness of elastomeric bearing due to displacement and rotation
Journal Pre-proof Assessment of the vertical stiffness of elastomeric bearing due to displacement and rotation Ehsan Kazeminezhad, Mohammad Taghi Kazemi, Mohammad Mirhosseini
PII: DOI: Reference:
S0020-7462(19)30515-3 https://doi.org/10.1016/j.ijnonlinmec.2019.103306 NLM 103306
To appear in:
International Journal of Non-Linear Mechanics
Received date : 31 July 2019 Revised date : 5 October 2019 Accepted date : 9 October 2019 Please cite this article as: E. Kazeminezhad, M.T. Kazemi and M. Mirhosseini, Assessment of the vertical stiffness of elastomeric bearing due to displacement and rotation, International Journal of Non-Linear Mechanics (2019), doi: https://doi.org/10.1016/j.ijnonlinmec.2019.103306. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Journal Pre-proof
Assessment of the vertical stiffness of elastomeric bearing due to displacement and rotation
13
Department of Civil Engineering, Arak Branch, Islamic Azad University, Arak, Iran.
Department of Civil Engineering, Sharif University of Technology, Tehran, Iran. Email: [email protected]; Corresponding Author
2
Email: [email protected]
3
Email: [email protected]
pro
1
re-
2
of
Ehsan Kazeminezhad1, Mohammad Taghi Kazemi2, Mohammad Mirhosseini3
Abstract
The vertical stiffness of elastomeric bearing is a dominant parameter of the base isolation
urn al P
design. Several empirical relations have been used to calculate the vertical stiffness of the elastomeric bearing systems, although, in certain conditions such as the presence of rotation, these relations are not accurate enough. In this paper, by using a nonlinear finite element program, the effect of rotation on the vertical stiffness investigated. It was observed that the vertical stiffness of the isolator could be increased or decreased depending on the amount of rotation and the value of lateral displacement limit.
Key words: bearing rotation; elastomeric bearing; finite element; lateral displacement;
Jo
vertical stiffness.
Journal Pre-proof
1. Introduction Employing elastomeric bearings, dramatically, may reduce the acceleration in the structure by increasing the natural period. In the conventional and mid-height buildings, a significant rotation will not occur at the joint between isolator and structures, but in high rise buildings and bridges
of
the rotation could affect the behavior. The laminated rubber bearing, LRB, consists of rubber and steel sheets, and has low horizontal stiffness but a high vertical stiffness. Staudacher et al. [1]
pro
reviewed the isolation history, extensively. Tarics et al. [2] reported that the first rubber with relatively high damping was manufactured in 1984. The shear stiffness of these rubbers is high in low strain; however, with the increase in strain, shear stiffness is reduced by 4 to 6 times, and
re-
when the strain reaches about 50%, stiffness is at its minimum. The building design with the base isolation was presented by Naeim and Kelly [3], revealing that the relative displacement decreases in the higher modes of vibration due to the normality of the higher modes to the first
urn al P
mode and ground motion. Ravi et al. [4], observed that regarding the combined compression and shear loading case, the numerical prediction matched well with the experimental result. The comparison was particularly good in the shear strain range of 50% to 150%, which is of a great interest in design. It means that the variation in the vertical compressive load from 0 to 200% of the design vertical load does not influence the horizontal displacement response due to the shearing loading. Kelly [5], developed a base isolation theory, providing design relationships for critical loading and created a new isolator calculation method. Warn and Whittaker [6], studied the coupled horizontal – vertical behavior of elastomeric and lead rubber seismic isolation bearing, and investigated the mechanical properties of isolators. They presented a relation to
Jo
calculate the vertical stiffness based on the springs concept. Tsai and Hsueh [7], indicated that rubber properties depend on strain amplitude and the loading frequency, hence they employed the viscoelastic treatment to indicate the effect of damping on the bearing response. Karbakhsh et al. [8], revealed that the initial geometry and slope could change the mechanical response of laminated rubber bearing. Kelly and Takhirov [9], examined the tension buckling in elastomeric
Journal Pre-proof
bearing and reported upward displacement. They also presented a theoretical relation for the tension and compression buckling loads. Chalhoub et al. [10] and Constantinou et al. [11], provided approximate expressions for the compression modulus of solid and hollow circular bonded elastomeric pads, respectively. Koh and Kelly [12], introduced the two-spring mechanical
of
model, and Naeim and Kelly [3], presented overlapping area method to calculate the vertical stiffness. Gauron et al. [13], investigated the lateral stability and shear failure limit states of
pro
bridge rubber bearings. Cancellara and Angelis [14], assessed the dynamic nonlinear behavior of different base isolation systems for a multi-storey RC building irregular in plan. Markou et al. [15], evaluated the response simulation of hybrid base isolation systems under earthquake excitation, created the new model for 1D non-linear dynamic analysis of hybrid base isolation
re-
systems, and simulated response under initial displacement for harmonic and earthquake excitations. Forcellini [16], assessed the 3D numerical simulations of elastomeric bearings for bridges and drew a comparison between the theoretical approach proposed by Forcellini and
urn al P
Kelly [17] and experimental results from Nagarajaiah and Ferrell [18] with the numerical simulations of an elastomeric bearing.
In this study, the ABAQUS [19] software has been utilized to consider the support rotation effect on the vertical stiffness, because, the existing empirical relationships are able just to calculate the vertical stiffness in non-rotational conditions.
2. Materials and Methods
Base isolator bearing consists of rubber and steel sheets. These isolators have high vertical stiffness and low horizontal stiffness to resist against the vertical load and to sustain the lateral
Jo
displacement. The Steel and rubber materials, used in the isolator bearing, are connected to each other via vulcanization process. Two end plates are used on the top and the bottom of the isolator system. In the base isolator bearing, steel is linear and rubber is highly nonlinear. Full 3D models in geometry and loading conditions for each isolator are used in this study. The hybrid C3D8H element and full integration C3D8 of ABAQUS were used to model the rubber and steel shims,
Journal Pre-proof
respectively. To simulate the experimental conditions of load application, a dummy node was considered at the top of the isolator. The node is connected with all the nodes lying on the top end of the plate by multipoint constraints; thus, the end plate is constrained to translate in the horizontal direction as a whole. The vertical and horizontal forces were applied through this
of
dummy node. For modeling rubber-like material properties, there exist many options such as hyperelastic and viscoelastic materials in ABAQUS. The hyperelastic behavior is described as a
pro
function of strain energy potential defining the strain energy stored in the material per unit of reference volume as a function of strain at that point in the material. There exist several forms of stain energy potentials in ABAQUS to approximately model the elastomers: Arruda-Boyce, Marlow, Mooney-Rivilin, Neo-Hookean, Polynomial, Reduced polynomial, Yeoh and Van der
re-
Weels, described in the ABAQUS documentation and briefly explained here. Arruda-Boyce model is based on molecular chain network; it is also called Arruda-Boyce 8-chain model because it was developed based on the representative volume (hexahedron) element where 8 chains emanate from
urn al P
the center to the corners of the volume. This model is a two-parameter shear model based only on the first invariant of deformation, which works well with the limited test data. Arruda-Boyce model is shown in Eq.(1):
_ _2 _3 _4 _5 I 1 3 I 1 9 11 I 1 27 19 I 1 81 519 I 1 243 1 U μ 2 4 6 8 20λ m 1050λ m 7000λ m 673750λ m D 0.5 2
2
2
J el2 1 LnJ el 0.5
(1)
1
_ _ _ _ _ 3 99 513 42039 . In Eq.(1), I1 λ1 λ 2 λ 3 , λ i J 3 λ i and μ 0 μ 1 2 4 6 8 5λ m 177λ m 875λ m 67375λ m
Jo
Mooney-Rivilin model has two phenomenological parameters which work well for moderately large stains in uniaxial elongation and shear deformation. Mooney-Rivilin model is presented by Eq.(2):
2 _ _ 1 U C 10 I 1- 3 C 01 I 2 - 3 J el - 1 D1
(2)
Journal Pre-proof
In Eq.(2), 0 2 C 10 C 01 and K 0
2 . D1
Neo-Hookean is a special case of Mooney-Rivilin form with C01= 0 and can be used when material data is insufficient. It is simple to use and can make good approximation at relatively
2 _ 1 U C 10 I 1 3 J el 1 D1
of
small strains. Neo-Hookean model is described by Eq.(3): (3)
pro
In Eq.(3), 0 2 C 10 .
Ogden model is also a phenomenological model and is based on principal stretches instead of invariants. This model is also able to accurately capture the upturn (stiffening) of stress-strain
re-
curve and rubber models for large ranges of deformation. This model is not to be employed with limited tests such as uniaxial tension. A good agreement has been observed between Ogden model and experimental data regarding unfilled rubber for extensions of up to 700%. Ogden model is
urn al P
shown in Eq.(4):
N 1 2 i _ i _ i _ i 2i U 2 1 2 3 3 J el 1 i 1 i i 1 D i N
In Eq.(4), 0
(4)
N
i
.
i 1
Polynomial model is used for isotropic and compressible rubber. This model is shown in Eq.(5): i
j
N 1 2i _ _ U C ij I 1 3 I 2 3 J el 1 i j 1 i 1 D i N
(5)
Jo
In Eq.(5), 0 2 C 1 0 C 0 1 .
In Eq.(1) to Eq.(5), U is the strain energy per unit of initial volume, J el is elastic volume ratio, λ i is principle stretches, µ is shear modulus, ( λ m , D) is the temperature rate depending on material _
_
_
properties, I1 , I 2 , I 3 are first, second and third deviatoric strain invariants, J is the total volume
Journal Pre-proof
ratio, K 0 is the initial bulk modulus and μ 0 is the initial shear modulus. C
ij
and α i are
parameters corresponding to strain energy function to define ABAQUS material properties.
3. Model Identification
of
In this research, the finite element of the experimental model of Sanchez et al. [20] was investigated. The scaled bearing is annular with an outer bonded rubber diameter of 152 mm and
pro
an inner diameter of 30 mm. The bearing consists of 20 rubber layers, each with 3 mm thickness; 19 steel shim plates, each 3 mm thick, and two steel external end plates of 25 mm thickness. Based on Sanchez et al. [20], critical displacement is a point at the force-displacement curve where the slope of the graph reaches zero. Based on this assumption Neo-Hookean model was used for
re-
rubber. Shear modulus of rubber 0.86 MPa and Neo-Hookean parameters were C10 = 0.43 MPa
and D1 = 0.001MPa -1 . Steel elasticity modulus and poisons ratio were 200 GPa and 0.3,
urn al P
respectively. Table 1 shows the isolator dimensions and material properties. Table 1 Dimensions and material properties of isolator
Dimensions Rubber Steel shim Bonded diameter (mm) 152 152 Inner hole diameter (mm) 30 30 Number of layers 20 19 Thickness (mm) 3 3
Material properties
Rubber Shear modulus 0.86 MPa Neo-Hookean properties C10 0.43 MPa D1 0.001 MPa-1
Steel Elasticity modulus Poisons ratio
200 GPa 0.3
Jo
3.1 Mesh modeling
Since mesh size has a direct influence on the finite element responses, an appropriate mesh size was selected, according to [6]. Steel and rubber sheets were divided into one and four layers in thickness direction, respectively. The radial and circumferential direction, were divided into 10 and 36 parts, respectively. In this isolator, C3D8 and C3D8H elements were used for steel and
Journal Pre-proof
rubber, respectively. The boundary condition was considered in two states, in the first state, fixed at the base and free in vertical and lateral direction at the top of the isolator, and at the second state, fixed at the base and top is similar to first state, but with a initial rotation of 0.1 radian.
Figure 1 Rubber part
Figure 2 Isolator system
urn al P
3.2 Calibration study
re-
pro
of
Figure 1 and Figure 2 show the finite element of rubber and whole model, respectively.
Sanchez et al. [20] examined the elastomeric bearing specimen and calculated the maximum
Jo
lateral displacement corresponding to vertical load. Figure 3 shows the results.
Figure 3 Experimental and ABAQUS results of isolator showing the critical displacement corresponding to different amounts of vertical load
Journal Pre-proof
In order to ensure the accuracy of the finite element results, the analysis with ABAQUS software was repeated, and the FEM results were compared with the experimental results. Figure 4 shows the lateral force-displacement curves for the various vertical loads. The critical displacement, corresponding to the lateral load capacity for any prescribed vertical load is shown by a bold dot.
of
These points present the zero stiffness or instability condition of the isolator. The slope of the curves becomes negative after the instability limit points. Table 2 shows the summary of the
urn al P
re-
pro
results.
Figure 4 Force-displacement of the isolator based on the finite element model without consideration of the
Jo
rotation
Journal Pre-proof
Table 2 Comparison of critical displacements of experimental and FEM analysis
Buckling load (kN) 133
Critical displacement (mm) Experimental Finite element method 158 169
Relative error (Between experimental and FEM) +7%
154
149
-3.25%
178
103
138
34%
200
93
121
30%
222
92
109
267
75
101
311
65
84
353
51
67
of
153
18.5%
pro
34.6% 29% 31%
method results.
re-
Table 2, shows that a good accuracy exists between the experimental and the finite element
4. Existing models for vertical stiffness
urn al P
4.1 Vertical stiffness
Elastomeric bearing consists of multilayer rubber and steel pads. Rubber layer thickness affects the vertical stiffness. Vertical stiffness was calculated by Warn et al. [6]. This relation is shown by Eq.(6) and could be used based on the initial geometry measurements:
K vo
ECAb Tr
(6)
In Eq.(6), K vo , is vertical stiffness at zero rotation and lateral displacement, E C is compression
Jo
modulus, A b is bonded rubber area, and Tr is the total rubber thickness. Chalhoub et al. [10] and Constantinou et al. [11], provided approximate expressions for the compression modulus of solid and hollow circular bonded elastomeric pads, respectively shown by Eq.(7) and Eq.(8). The compression modulus is an important parameter for designing elastomeric bearings, and is related directly to the load carrying capacity and vertical stiffness.
Journal Pre-proof
8GS2 E C = 6GS2 1 K
(7)
8GS2 H E C = 6GS2 F 1 K F
(8)
of
In Eq.(7) and (8), G is rubber shear modulus, S is shape factor, and F and H are defined as Eq.(11) and (12), respectively.
pro
Shape factors for solid and hollow circular bonded elastomeric pads are respectively shown by Eq.(9) and (10). In Eq.(9), D is the diameter of solid pad and tr is one rubber layer thickness.
S
D 4t r
re-
S
D o Di 4t r
2
urn al P
Do Do +1 +1 Di Di F= + 2 Do Do Do -1 1 Ln D D i Di i
D if 1 o Di
H 1 100 ; F
(9)
(10)
(11)
(12)
Do , Di are outer diameter and inner diameter, respectively.
Jo
4.2 Vertical stiffness in the presence of the lateral displacement 4.2.1 The two-spring model The two-spring mechanical model introduced by Koh and Kelly [12] was used to illustrate the derivation of an expression for the reduction in height of the system with lateral displacement, and subsequently express the vertical stiffness. Eq.(13) shows the stiffness:
Journal Pre-proof
ECAb 1 Kv = 2 2 Tr 1+ 3GA2 b Δ h π I P Tr
2 3A b Δ + 2 π I
(13)
of
In Eq.(13), Δ is lateral displacement, I is the elastomeric pad moment of inertia and P is vertical load.
pro
4.2.2 Overlapping area method
Naeim and Kelly [3] presented a widely accepted concept for the estimation of the critical buckling load of an elastomeric bearing subjected to combined compression and lateral
re-
displacement. It is adapted here for the estimation of the vertical stiffness at a given lateral displacement. This concept uses the ratio of the overlapping area between the top and bottom load plates to the bonded rubber area as a factor reducing the vertical stiffness for lateral displacements
Jo
urn al P
higher than zero, Figure 5.
Figure 5 The overlapping area concept
Eq.(14) shows the stiffness of this method.
Kv A =( r) K vo Ab
(14)
Journal Pre-proof
Where A r is the overlapping area for a solid circular bearing subjected to a lateral displacement,
Δ . A b is the bonded rubber area, and K vo is the initial vertical stiffness. The overlapping area was calculated according to Eq.(15) and Eq.(16):
(15)
Δ Φ = 2arccos D
(16)
pro
of
D 2 (Φ - sinΦ) 4
Ar =
The overlapping area method is capable to calculate the vertical stiffness at different displacements and presents Eq.(14) for computation. This relation, which is widely used in design
5. Numerical results
re-
codes, evaluated in section 5.
urn al P
5.1 The effect of lateral displacement on the vertical stiffness In this section, the vertical stiffness of the isolator is calculated. Since lateral displacement affects the vertical stiffness, first, calculated the vertical stiffness at different displacements via Eqs.(13) and (14), then, the finite-element method was used to determine the error of these relations. Using the finite-element method, determined how successful these relations are in calculating the vertical stiffness. As shown in Table 3, Eq.(14) is not capable of computing the vertical stiffness in displacements higher than the diameter of the isolator, which is a major weakness for this equation; however, Eq.(13) calculates the vertical stiffness with better accuracy. It is also observed
Jo
in other displacements the error of Eq.(14) is higher than the finite element results (The average error is more than 44%). On the other hand, Eq.(13) provides more accurate results (The average error is about 13.5%). Therefore, the overlapping area method (Eq.(14)) has to be modified and the current form does not provide exact answers.
Journal Pre-proof
Table 3 Vertical stiffness results in various conditions Finite Element Method (FEM) Vertical load (kN)
Lateral
Vertical
displacement displacement
Two spring
Overlapping area
(Eq.(13))
(Eq.(14))
Vertical
Vertical
Relative
Vertical
Relative
stiffness
stiffness
error
stiffness
error
(mm)
(kN/m)
(kN/m)
(%)
(kN/m)
(%)
133
169
11.89
11185.87
11553.87
3.29
N.A*
N.A
153
149
9.493
16117.14
18143.13
12.57
25527.21
58.39
178
138
8.801
20224.97
24091.47
19.12
34214.06
69.17
200
121
4.496
30788.18
33133.5
7.62
47396.58
53.94
222
109
5.55
33893.13
40497.84
19.49
26095.72
-23
267
101
6.63
40271.49
47081.41
16.91
24159.48
-40
311
84
5.83
53344.77
55822.41
4.64
49074.2
-8
353
67
5.11
68972.26
61850.85
-10.33
30214.81
-56.2
pro
re-
*N.A: Non Accountable
of
(mm)
5.1.1 The effect of rotation on the vertical stiffness
urn al P
In this part, which is one of the main parts of the work, the effect of rotation on the vertical stiffness is investigated. Since the equations are not applicable in the presence of rotation, the finite element method used to investigate the rotational effect of existence of the rotation of 0.1 radians. As could be observed from Table 4, in the presence of rotation, we have experienced a decrease in vertical stiffness corresponding to critical displacements of vertical load of 133, 153, 178, 200 and 222 kN and an increase in vertical stiffness corresponding to critical displacements of vertical load of 267, 311 and 353 kN. It was observed that the vertical stiffness of the isolator could be increased or decreased depending on the amount of rotation and the value of lateral
Jo
displacement limit. In other words, if the decrease in critical displacement due to rotation exceed as a certain value, the vertical stiffness increases. Fig.6 shows the variations of vertical stiffness. To determine the critical displacement in the presence of rotation, the relation of the forcedisplacement of the isolator was extracted from the numerical results. Fig.7 shows the critical lateral displacements for several values of vertical loads.
Journal Pre-proof
Table 4 Effect of rotation on the vertical stiffness results
Rotation at the top = zero Vertical load (kN)
Lateral
Vertical
displacement displacement
Rotation at the top = 0.1 radian Lateral
Vertical
Vertical
stiffness
displacement displacement
Vertical stiffness
(mm)
(kN/m)
(mm)(Fig.6)
(mm)
(kN/m)
133
169
11.89
11185.87
157
14.074
9420.78
153
149
9.493
16117.14
139
11.351
13446.3
178
138
8.801
20224.97
125.5
9.805
18117.1
200
121
4.496
30788.18
107
7.750
25768
222
109
5.55
33893.13
97
7.029
31548.1
267
101
6.63
40271.49
72
5.284
50495.8
311
84
5.83
53344.77
58
4.831
64340.4
353
67
5.11
68972.26
0
3.191
110624
Jo
urn al P
re-
pro
of
(mm)
Fig.6 Change in vertical stiffness due to rotation and decrease in critical lateral displacement
pro
of
Journal Pre-proof
re-
Figure 7 Lateral force-displacement curves bases on finite element model ,with the initial rotation of 0.1 radian
Fig.8, shows the comparison between results of critical lateral displacements in presence of displacement.
urn al P
rotation with one without initial rotation and indicated that the rotation reduces the critical
Jo
Figure 8 Effect of the rotation on the critical displacement
6. Conclusion
This study indicated that a good accuracy existed between experimental and finite element results. The two-spring model has a better accuracy than the overlapping area method and it could be
Journal Pre-proof
considered to estimate the vertical stiffness in the presence of the lateral displacement, but not the rotation. It was observed that the vertical stiffness of the isolator could be increased or decreased depending on the amount of rotation and the value of lateral displacement limit. In the other words, rotation can increase the vertical stiffness of the isolator when the critical lateral
of
displacement is reduced to a certain value.
pro
7. References
[1] Staudacher, E., Habacher, C., Siegenthaler, R., 1970. Erdbebensicherung in Baum. Zürich, Neue Zürcher Zeitung. Tech.
[2] Tarics, A G., Way, D., Kelly, J M., 1984. The implementation of base isolation for the Foothill Communities Law and Justice Center. Rep. to Natl. Sci. Found. Cty. San Bernardino.
re-
[3] Naeim, F., Kelly, J M., 1999. Design of seismic isolated structures: From theory to practice. John Wiley & Sons, DOI:10.1002/9780470172742.
[4] Ravi, R., Selvaraj, T., Chellapandi, P. Chetal, S C., Bhoje, S B. 1998. Finite element analysis of laminated rubber bearings-verification with KAERI HDRB, ALMR HDRB and
urn al P
CRIEPI LRB data. International Atomic Energy Agency, International Working Group on Fast Reactors, Vienna.
[5] Kelly, J M. 1997. Earthquake-Resistant Design with Rubber . Switzerland, Springer. [6] Warn, G. P., Whittaker, A. S., 2006. A study of the coupled horizontal-vertical behavior of elastomeric and lead-rubber seismic isolation bearings. Tech. Rep. MCEER-06-001, University at Buffalo, State University of NewYork.
[7] Tsai, H C., Hsueh, S J., 2001. Mechanical properties of isolation bearings identified by a viscoelastic
model.
International
Journal
of
Solids
and
Structures,
Elsevier,
https://doi.org/10.1016/S0020-7683(00)00010-X.
[8] Karbakhsh Ravari, A. and Bin Othman, I. and Binti Ibrahim, Z., Ab-Malek, K., 2011. P-Δ
Jo
and end rotation effects on the influence of mechanical properties of elastomeric isolation bearings. Journal of Structural Engineering, American Society of Civil Engineering, DOI: 10.1061/(ASCE)ST.1943-541X.0000503. [9] Kelly, J M., Takhirov S., 2007. Tension buckling in multilayer elastomeric isolation bearings. Journal of Mechanics of Materials and Structures, Mathematical Sciences Publishers, DOI: 10.2140/jomms.2007.2.1591.
Journal Pre-proof
[10] Chalhoub, M. S., Kelly, J. M., 1990. Effect of Bulk Compressibility on the Stiffness of Cylindrical Base Isolation Bearings. International Journal of Solids Structures, 26(7): 743-760. [11] Constantinou, M. C., Kartoum, A., Kelly, J. M., 1992. Analysis of compression of hollow circular elastomeric bearings. Engineering Structures, 14(2): 103-111. [12] Koh, C. G., Kelly, J. M., 1987. Effects of axial load on elastomeric isolation bearings. UBC/EERC-86/12, Earthquake Engineering Research Center, College of Engineering,
of
University of California, Berkeley, California.
[13] Gauron, O., Saidou. A., Busson,A., Siqueira, G. H., 2018. Experimental determination of
pro
the lateral stability and shear failure limit states of bridge rubber bearings. Journal of Engineering Structures 174. P.39-48, Elsevier, https://doi.org/10.1016/j.engstruct.2018.07.039. [14] Cancellara, D., Angelis, F. D., 2017. Assessment and dynamic nonlinear analysis of different base isolation systems for a multi-storey RC building irregular in plan, Journal of Computer and Structures. P.74-88, Elsevier. https://doi.org/10.1016/j.compstruc.2016.02.012.
re-
[15] Markou, A. A., and Oliveto, G., and Athanasiou, S., 2016. Response simulation of hybrid base isolation systems under earthquake excitation. Journal of Soil Dynamics and Earthquake Engineering. 84. P.120-133, Elsevier. https://doi.org/10.1016/j.soildyn.2016.02.003.
urn al P
[16] Forcellini, D., 2016. 3D Numerical simulations of elastomeric bearings for bridges. Journal of Innovative Infrastructure Solutions.1:45, Springer. DOI:10.1007/s41062-016-0045-4. [17] Forcellini, D., Kelly, J. M., (2014). Analysis of the large deformation stability of elastomeric bearings. Journal of Engineering Mechanics. DOI: 10.1061/(ASCE)EM.19437889.0000729.
[18] Nagarajaiah, S., Ferrell, K., 1999. Stability of elastomeric seismic isolation bearings. Journal
of
Structural
Engineering,
American
Society
of
Civil
Engineering.
DOI:10.1061/(ASCE)07339445(1999)125:9(946). [19] Hibbett, Karlsson, Sorensen, 1988. ABAQUS/standard: User’s Manual, vol. 1. Hibbitt, Karlsson & Sorensen.
Jo
[20] Sanchez, J., Masroor A., Mosqueda G., Ryan K., 2013. Static and dynamic stability of elastomeric bearings for seismic protection of structures. Journal of Structural Engineering, American Society of Civil Engineers, DOI: 10.1061/(ASCE)ST.1943-541X.0000660. Conflict of Interest and Authorship Conformation Form
Journal Pre-proof
Please check the following as appropriate: o
All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.
o
This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue.
o
The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript
o
The following authors have affiliations with organizations with direct or indirect financial interest in the subject matter discussed in the manuscript:
of
Affiliation
re-
Author’s name
pro
urn al P
Ehsan Kazeminezhad Department of Civil Engineering, Arak Branch, Islamic Azad University, Arak, Iran
Mohammad Taghi Kazemi Department of Civil Engineering, Sharif University of Technology, Tehran, Iran
Jo
Mohammad Mirhosseini Department of Civil Engineering, Arak Branch, Islamic Azad University, Arak, Iran
Journal Pre-proof
Highlights:
urn al P
re-
pro
of
Good accuracy exists between the experimental and the finite element method. The two-spring model has a better accuracy than the overlapping area method. Vertical stiffness of the isolator could be increased or decreased by rotation.
Jo
PII: DOI: Reference:
S0020-7462(19)30515-3 https://doi.org/10.1016/j.ijnonlinmec.2019.103306 NLM 103306
To appear in:
International Journal of Non-Linear Mechanics
Received date : 31 July 2019 Revised date : 5 October 2019 Accepted date : 9 October 2019 Please cite this article as: E. Kazeminezhad, M.T. Kazemi and M. Mirhosseini, Assessment of the vertical stiffness of elastomeric bearing due to displacement and rotation, International Journal of Non-Linear Mechanics (2019), doi: https://doi.org/10.1016/j.ijnonlinmec.2019.103306. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Journal Pre-proof
Assessment of the vertical stiffness of elastomeric bearing due to displacement and rotation
13
Department of Civil Engineering, Arak Branch, Islamic Azad University, Arak, Iran.
Department of Civil Engineering, Sharif University of Technology, Tehran, Iran. Email: [email protected]; Corresponding Author
2
Email: [email protected]
3
Email: [email protected]
pro
1
re-
2
of
Ehsan Kazeminezhad1, Mohammad Taghi Kazemi2, Mohammad Mirhosseini3
Abstract
The vertical stiffness of elastomeric bearing is a dominant parameter of the base isolation
urn al P
design. Several empirical relations have been used to calculate the vertical stiffness of the elastomeric bearing systems, although, in certain conditions such as the presence of rotation, these relations are not accurate enough. In this paper, by using a nonlinear finite element program, the effect of rotation on the vertical stiffness investigated. It was observed that the vertical stiffness of the isolator could be increased or decreased depending on the amount of rotation and the value of lateral displacement limit.
Key words: bearing rotation; elastomeric bearing; finite element; lateral displacement;
Jo
vertical stiffness.
Journal Pre-proof
1. Introduction Employing elastomeric bearings, dramatically, may reduce the acceleration in the structure by increasing the natural period. In the conventional and mid-height buildings, a significant rotation will not occur at the joint between isolator and structures, but in high rise buildings and bridges
of
the rotation could affect the behavior. The laminated rubber bearing, LRB, consists of rubber and steel sheets, and has low horizontal stiffness but a high vertical stiffness. Staudacher et al. [1]
pro
reviewed the isolation history, extensively. Tarics et al. [2] reported that the first rubber with relatively high damping was manufactured in 1984. The shear stiffness of these rubbers is high in low strain; however, with the increase in strain, shear stiffness is reduced by 4 to 6 times, and
re-
when the strain reaches about 50%, stiffness is at its minimum. The building design with the base isolation was presented by Naeim and Kelly [3], revealing that the relative displacement decreases in the higher modes of vibration due to the normality of the higher modes to the first
urn al P
mode and ground motion. Ravi et al. [4], observed that regarding the combined compression and shear loading case, the numerical prediction matched well with the experimental result. The comparison was particularly good in the shear strain range of 50% to 150%, which is of a great interest in design. It means that the variation in the vertical compressive load from 0 to 200% of the design vertical load does not influence the horizontal displacement response due to the shearing loading. Kelly [5], developed a base isolation theory, providing design relationships for critical loading and created a new isolator calculation method. Warn and Whittaker [6], studied the coupled horizontal – vertical behavior of elastomeric and lead rubber seismic isolation bearing, and investigated the mechanical properties of isolators. They presented a relation to
Jo
calculate the vertical stiffness based on the springs concept. Tsai and Hsueh [7], indicated that rubber properties depend on strain amplitude and the loading frequency, hence they employed the viscoelastic treatment to indicate the effect of damping on the bearing response. Karbakhsh et al. [8], revealed that the initial geometry and slope could change the mechanical response of laminated rubber bearing. Kelly and Takhirov [9], examined the tension buckling in elastomeric
Journal Pre-proof
bearing and reported upward displacement. They also presented a theoretical relation for the tension and compression buckling loads. Chalhoub et al. [10] and Constantinou et al. [11], provided approximate expressions for the compression modulus of solid and hollow circular bonded elastomeric pads, respectively. Koh and Kelly [12], introduced the two-spring mechanical
of
model, and Naeim and Kelly [3], presented overlapping area method to calculate the vertical stiffness. Gauron et al. [13], investigated the lateral stability and shear failure limit states of
pro
bridge rubber bearings. Cancellara and Angelis [14], assessed the dynamic nonlinear behavior of different base isolation systems for a multi-storey RC building irregular in plan. Markou et al. [15], evaluated the response simulation of hybrid base isolation systems under earthquake excitation, created the new model for 1D non-linear dynamic analysis of hybrid base isolation
re-
systems, and simulated response under initial displacement for harmonic and earthquake excitations. Forcellini [16], assessed the 3D numerical simulations of elastomeric bearings for bridges and drew a comparison between the theoretical approach proposed by Forcellini and
urn al P
Kelly [17] and experimental results from Nagarajaiah and Ferrell [18] with the numerical simulations of an elastomeric bearing.
In this study, the ABAQUS [19] software has been utilized to consider the support rotation effect on the vertical stiffness, because, the existing empirical relationships are able just to calculate the vertical stiffness in non-rotational conditions.
2. Materials and Methods
Base isolator bearing consists of rubber and steel sheets. These isolators have high vertical stiffness and low horizontal stiffness to resist against the vertical load and to sustain the lateral
Jo
displacement. The Steel and rubber materials, used in the isolator bearing, are connected to each other via vulcanization process. Two end plates are used on the top and the bottom of the isolator system. In the base isolator bearing, steel is linear and rubber is highly nonlinear. Full 3D models in geometry and loading conditions for each isolator are used in this study. The hybrid C3D8H element and full integration C3D8 of ABAQUS were used to model the rubber and steel shims,
Journal Pre-proof
respectively. To simulate the experimental conditions of load application, a dummy node was considered at the top of the isolator. The node is connected with all the nodes lying on the top end of the plate by multipoint constraints; thus, the end plate is constrained to translate in the horizontal direction as a whole. The vertical and horizontal forces were applied through this
of
dummy node. For modeling rubber-like material properties, there exist many options such as hyperelastic and viscoelastic materials in ABAQUS. The hyperelastic behavior is described as a
pro
function of strain energy potential defining the strain energy stored in the material per unit of reference volume as a function of strain at that point in the material. There exist several forms of stain energy potentials in ABAQUS to approximately model the elastomers: Arruda-Boyce, Marlow, Mooney-Rivilin, Neo-Hookean, Polynomial, Reduced polynomial, Yeoh and Van der
re-
Weels, described in the ABAQUS documentation and briefly explained here. Arruda-Boyce model is based on molecular chain network; it is also called Arruda-Boyce 8-chain model because it was developed based on the representative volume (hexahedron) element where 8 chains emanate from
urn al P
the center to the corners of the volume. This model is a two-parameter shear model based only on the first invariant of deformation, which works well with the limited test data. Arruda-Boyce model is shown in Eq.(1):
_ _2 _3 _4 _5 I 1 3 I 1 9 11 I 1 27 19 I 1 81 519 I 1 243 1 U μ 2 4 6 8 20λ m 1050λ m 7000λ m 673750λ m D 0.5 2
2
2
J el2 1 LnJ el 0.5
(1)
1
_ _ _ _ _ 3 99 513 42039 . In Eq.(1), I1 λ1 λ 2 λ 3 , λ i J 3 λ i and μ 0 μ 1 2 4 6 8 5λ m 177λ m 875λ m 67375λ m
Jo
Mooney-Rivilin model has two phenomenological parameters which work well for moderately large stains in uniaxial elongation and shear deformation. Mooney-Rivilin model is presented by Eq.(2):
2 _ _ 1 U C 10 I 1- 3 C 01 I 2 - 3 J el - 1 D1
(2)
Journal Pre-proof
In Eq.(2), 0 2 C 10 C 01 and K 0
2 . D1
Neo-Hookean is a special case of Mooney-Rivilin form with C01= 0 and can be used when material data is insufficient. It is simple to use and can make good approximation at relatively
2 _ 1 U C 10 I 1 3 J el 1 D1
of
small strains. Neo-Hookean model is described by Eq.(3): (3)
pro
In Eq.(3), 0 2 C 10 .
Ogden model is also a phenomenological model and is based on principal stretches instead of invariants. This model is also able to accurately capture the upturn (stiffening) of stress-strain
re-
curve and rubber models for large ranges of deformation. This model is not to be employed with limited tests such as uniaxial tension. A good agreement has been observed between Ogden model and experimental data regarding unfilled rubber for extensions of up to 700%. Ogden model is
urn al P
shown in Eq.(4):
N 1 2 i _ i _ i _ i 2i U 2 1 2 3 3 J el 1 i 1 i i 1 D i N
In Eq.(4), 0
(4)
N
i
.
i 1
Polynomial model is used for isotropic and compressible rubber. This model is shown in Eq.(5): i
j
N 1 2i _ _ U C ij I 1 3 I 2 3 J el 1 i j 1 i 1 D i N
(5)
Jo
In Eq.(5), 0 2 C 1 0 C 0 1 .
In Eq.(1) to Eq.(5), U is the strain energy per unit of initial volume, J el is elastic volume ratio, λ i is principle stretches, µ is shear modulus, ( λ m , D) is the temperature rate depending on material _
_
_
properties, I1 , I 2 , I 3 are first, second and third deviatoric strain invariants, J is the total volume
Journal Pre-proof
ratio, K 0 is the initial bulk modulus and μ 0 is the initial shear modulus. C
ij
and α i are
parameters corresponding to strain energy function to define ABAQUS material properties.
3. Model Identification
of
In this research, the finite element of the experimental model of Sanchez et al. [20] was investigated. The scaled bearing is annular with an outer bonded rubber diameter of 152 mm and
pro
an inner diameter of 30 mm. The bearing consists of 20 rubber layers, each with 3 mm thickness; 19 steel shim plates, each 3 mm thick, and two steel external end plates of 25 mm thickness. Based on Sanchez et al. [20], critical displacement is a point at the force-displacement curve where the slope of the graph reaches zero. Based on this assumption Neo-Hookean model was used for
re-
rubber. Shear modulus of rubber 0.86 MPa and Neo-Hookean parameters were C10 = 0.43 MPa
and D1 = 0.001MPa -1 . Steel elasticity modulus and poisons ratio were 200 GPa and 0.3,
urn al P
respectively. Table 1 shows the isolator dimensions and material properties. Table 1 Dimensions and material properties of isolator
Dimensions Rubber Steel shim Bonded diameter (mm) 152 152 Inner hole diameter (mm) 30 30 Number of layers 20 19 Thickness (mm) 3 3
Material properties
Rubber Shear modulus 0.86 MPa Neo-Hookean properties C10 0.43 MPa D1 0.001 MPa-1
Steel Elasticity modulus Poisons ratio
200 GPa 0.3
Jo
3.1 Mesh modeling
Since mesh size has a direct influence on the finite element responses, an appropriate mesh size was selected, according to [6]. Steel and rubber sheets were divided into one and four layers in thickness direction, respectively. The radial and circumferential direction, were divided into 10 and 36 parts, respectively. In this isolator, C3D8 and C3D8H elements were used for steel and
Journal Pre-proof
rubber, respectively. The boundary condition was considered in two states, in the first state, fixed at the base and free in vertical and lateral direction at the top of the isolator, and at the second state, fixed at the base and top is similar to first state, but with a initial rotation of 0.1 radian.
Figure 1 Rubber part
Figure 2 Isolator system
urn al P
3.2 Calibration study
re-
pro
of
Figure 1 and Figure 2 show the finite element of rubber and whole model, respectively.
Sanchez et al. [20] examined the elastomeric bearing specimen and calculated the maximum
Jo
lateral displacement corresponding to vertical load. Figure 3 shows the results.
Figure 3 Experimental and ABAQUS results of isolator showing the critical displacement corresponding to different amounts of vertical load
Journal Pre-proof
In order to ensure the accuracy of the finite element results, the analysis with ABAQUS software was repeated, and the FEM results were compared with the experimental results. Figure 4 shows the lateral force-displacement curves for the various vertical loads. The critical displacement, corresponding to the lateral load capacity for any prescribed vertical load is shown by a bold dot.
of
These points present the zero stiffness or instability condition of the isolator. The slope of the curves becomes negative after the instability limit points. Table 2 shows the summary of the
urn al P
re-
pro
results.
Figure 4 Force-displacement of the isolator based on the finite element model without consideration of the
Jo
rotation
Journal Pre-proof
Table 2 Comparison of critical displacements of experimental and FEM analysis
Buckling load (kN) 133
Critical displacement (mm) Experimental Finite element method 158 169
Relative error (Between experimental and FEM) +7%
154
149
-3.25%
178
103
138
34%
200
93
121
30%
222
92
109
267
75
101
311
65
84
353
51
67
of
153
18.5%
pro
34.6% 29% 31%
method results.
re-
Table 2, shows that a good accuracy exists between the experimental and the finite element
4. Existing models for vertical stiffness
urn al P
4.1 Vertical stiffness
Elastomeric bearing consists of multilayer rubber and steel pads. Rubber layer thickness affects the vertical stiffness. Vertical stiffness was calculated by Warn et al. [6]. This relation is shown by Eq.(6) and could be used based on the initial geometry measurements:
K vo
ECAb Tr
(6)
In Eq.(6), K vo , is vertical stiffness at zero rotation and lateral displacement, E C is compression
Jo
modulus, A b is bonded rubber area, and Tr is the total rubber thickness. Chalhoub et al. [10] and Constantinou et al. [11], provided approximate expressions for the compression modulus of solid and hollow circular bonded elastomeric pads, respectively shown by Eq.(7) and Eq.(8). The compression modulus is an important parameter for designing elastomeric bearings, and is related directly to the load carrying capacity and vertical stiffness.
Journal Pre-proof
8GS2 E C = 6GS2 1 K
(7)
8GS2 H E C = 6GS2 F 1 K F
(8)
of
In Eq.(7) and (8), G is rubber shear modulus, S is shape factor, and F and H are defined as Eq.(11) and (12), respectively.
pro
Shape factors for solid and hollow circular bonded elastomeric pads are respectively shown by Eq.(9) and (10). In Eq.(9), D is the diameter of solid pad and tr is one rubber layer thickness.
S
D 4t r
re-
S
D o Di 4t r
2
urn al P
Do Do +1 +1 Di Di F= + 2 Do Do Do -1 1 Ln D D i Di i
D if 1 o Di
H 1 100 ; F
(9)
(10)
(11)
(12)
Do , Di are outer diameter and inner diameter, respectively.
Jo
4.2 Vertical stiffness in the presence of the lateral displacement 4.2.1 The two-spring model The two-spring mechanical model introduced by Koh and Kelly [12] was used to illustrate the derivation of an expression for the reduction in height of the system with lateral displacement, and subsequently express the vertical stiffness. Eq.(13) shows the stiffness:
Journal Pre-proof
ECAb 1 Kv = 2 2 Tr 1+ 3GA2 b Δ h π I P Tr
2 3A b Δ + 2 π I
(13)
of
In Eq.(13), Δ is lateral displacement, I is the elastomeric pad moment of inertia and P is vertical load.
pro
4.2.2 Overlapping area method
Naeim and Kelly [3] presented a widely accepted concept for the estimation of the critical buckling load of an elastomeric bearing subjected to combined compression and lateral
re-
displacement. It is adapted here for the estimation of the vertical stiffness at a given lateral displacement. This concept uses the ratio of the overlapping area between the top and bottom load plates to the bonded rubber area as a factor reducing the vertical stiffness for lateral displacements
Jo
urn al P
higher than zero, Figure 5.
Figure 5 The overlapping area concept
Eq.(14) shows the stiffness of this method.
Kv A =( r) K vo Ab
(14)
Journal Pre-proof
Where A r is the overlapping area for a solid circular bearing subjected to a lateral displacement,
Δ . A b is the bonded rubber area, and K vo is the initial vertical stiffness. The overlapping area was calculated according to Eq.(15) and Eq.(16):
(15)
Δ Φ = 2arccos D
(16)
pro
of
D 2 (Φ - sinΦ) 4
Ar =
The overlapping area method is capable to calculate the vertical stiffness at different displacements and presents Eq.(14) for computation. This relation, which is widely used in design
5. Numerical results
re-
codes, evaluated in section 5.
urn al P
5.1 The effect of lateral displacement on the vertical stiffness In this section, the vertical stiffness of the isolator is calculated. Since lateral displacement affects the vertical stiffness, first, calculated the vertical stiffness at different displacements via Eqs.(13) and (14), then, the finite-element method was used to determine the error of these relations. Using the finite-element method, determined how successful these relations are in calculating the vertical stiffness. As shown in Table 3, Eq.(14) is not capable of computing the vertical stiffness in displacements higher than the diameter of the isolator, which is a major weakness for this equation; however, Eq.(13) calculates the vertical stiffness with better accuracy. It is also observed
Jo
in other displacements the error of Eq.(14) is higher than the finite element results (The average error is more than 44%). On the other hand, Eq.(13) provides more accurate results (The average error is about 13.5%). Therefore, the overlapping area method (Eq.(14)) has to be modified and the current form does not provide exact answers.
Journal Pre-proof
Table 3 Vertical stiffness results in various conditions Finite Element Method (FEM) Vertical load (kN)
Lateral
Vertical
displacement displacement
Two spring
Overlapping area
(Eq.(13))
(Eq.(14))
Vertical
Vertical
Relative
Vertical
Relative
stiffness
stiffness
error
stiffness
error
(mm)
(kN/m)
(kN/m)
(%)
(kN/m)
(%)
133
169
11.89
11185.87
11553.87
3.29
N.A*
N.A
153
149
9.493
16117.14
18143.13
12.57
25527.21
58.39
178
138
8.801
20224.97
24091.47
19.12
34214.06
69.17
200
121
4.496
30788.18
33133.5
7.62
47396.58
53.94
222
109
5.55
33893.13
40497.84
19.49
26095.72
-23
267
101
6.63
40271.49
47081.41
16.91
24159.48
-40
311
84
5.83
53344.77
55822.41
4.64
49074.2
-8
353
67
5.11
68972.26
61850.85
-10.33
30214.81
-56.2
pro
re-
*N.A: Non Accountable
of
(mm)
5.1.1 The effect of rotation on the vertical stiffness
urn al P
In this part, which is one of the main parts of the work, the effect of rotation on the vertical stiffness is investigated. Since the equations are not applicable in the presence of rotation, the finite element method used to investigate the rotational effect of existence of the rotation of 0.1 radians. As could be observed from Table 4, in the presence of rotation, we have experienced a decrease in vertical stiffness corresponding to critical displacements of vertical load of 133, 153, 178, 200 and 222 kN and an increase in vertical stiffness corresponding to critical displacements of vertical load of 267, 311 and 353 kN. It was observed that the vertical stiffness of the isolator could be increased or decreased depending on the amount of rotation and the value of lateral
Jo
displacement limit. In other words, if the decrease in critical displacement due to rotation exceed as a certain value, the vertical stiffness increases. Fig.6 shows the variations of vertical stiffness. To determine the critical displacement in the presence of rotation, the relation of the forcedisplacement of the isolator was extracted from the numerical results. Fig.7 shows the critical lateral displacements for several values of vertical loads.
Journal Pre-proof
Table 4 Effect of rotation on the vertical stiffness results
Rotation at the top = zero Vertical load (kN)
Lateral
Vertical
displacement displacement
Rotation at the top = 0.1 radian Lateral
Vertical
Vertical
stiffness
displacement displacement
Vertical stiffness
(mm)
(kN/m)
(mm)(Fig.6)
(mm)
(kN/m)
133
169
11.89
11185.87
157
14.074
9420.78
153
149
9.493
16117.14
139
11.351
13446.3
178
138
8.801
20224.97
125.5
9.805
18117.1
200
121
4.496
30788.18
107
7.750
25768
222
109
5.55
33893.13
97
7.029
31548.1
267
101
6.63
40271.49
72
5.284
50495.8
311
84
5.83
53344.77
58
4.831
64340.4
353
67
5.11
68972.26
0
3.191
110624
Jo
urn al P
re-
pro
of
(mm)
Fig.6 Change in vertical stiffness due to rotation and decrease in critical lateral displacement
pro
of
Journal Pre-proof
re-
Figure 7 Lateral force-displacement curves bases on finite element model ,with the initial rotation of 0.1 radian
Fig.8, shows the comparison between results of critical lateral displacements in presence of displacement.
urn al P
rotation with one without initial rotation and indicated that the rotation reduces the critical
Jo
Figure 8 Effect of the rotation on the critical displacement
6. Conclusion
This study indicated that a good accuracy existed between experimental and finite element results. The two-spring model has a better accuracy than the overlapping area method and it could be
Journal Pre-proof
considered to estimate the vertical stiffness in the presence of the lateral displacement, but not the rotation. It was observed that the vertical stiffness of the isolator could be increased or decreased depending on the amount of rotation and the value of lateral displacement limit. In the other words, rotation can increase the vertical stiffness of the isolator when the critical lateral
of
displacement is reduced to a certain value.
pro
7. References
[1] Staudacher, E., Habacher, C., Siegenthaler, R., 1970. Erdbebensicherung in Baum. Zürich, Neue Zürcher Zeitung. Tech.
[2] Tarics, A G., Way, D., Kelly, J M., 1984. The implementation of base isolation for the Foothill Communities Law and Justice Center. Rep. to Natl. Sci. Found. Cty. San Bernardino.
re-
[3] Naeim, F., Kelly, J M., 1999. Design of seismic isolated structures: From theory to practice. John Wiley & Sons, DOI:10.1002/9780470172742.
[4] Ravi, R., Selvaraj, T., Chellapandi, P. Chetal, S C., Bhoje, S B. 1998. Finite element analysis of laminated rubber bearings-verification with KAERI HDRB, ALMR HDRB and
urn al P
CRIEPI LRB data. International Atomic Energy Agency, International Working Group on Fast Reactors, Vienna.
[5] Kelly, J M. 1997. Earthquake-Resistant Design with Rubber . Switzerland, Springer. [6] Warn, G. P., Whittaker, A. S., 2006. A study of the coupled horizontal-vertical behavior of elastomeric and lead-rubber seismic isolation bearings. Tech. Rep. MCEER-06-001, University at Buffalo, State University of NewYork.
[7] Tsai, H C., Hsueh, S J., 2001. Mechanical properties of isolation bearings identified by a viscoelastic
model.
International
Journal
of
Solids
and
Structures,
Elsevier,
https://doi.org/10.1016/S0020-7683(00)00010-X.
[8] Karbakhsh Ravari, A. and Bin Othman, I. and Binti Ibrahim, Z., Ab-Malek, K., 2011. P-Δ
Jo
and end rotation effects on the influence of mechanical properties of elastomeric isolation bearings. Journal of Structural Engineering, American Society of Civil Engineering, DOI: 10.1061/(ASCE)ST.1943-541X.0000503. [9] Kelly, J M., Takhirov S., 2007. Tension buckling in multilayer elastomeric isolation bearings. Journal of Mechanics of Materials and Structures, Mathematical Sciences Publishers, DOI: 10.2140/jomms.2007.2.1591.
Journal Pre-proof
[10] Chalhoub, M. S., Kelly, J. M., 1990. Effect of Bulk Compressibility on the Stiffness of Cylindrical Base Isolation Bearings. International Journal of Solids Structures, 26(7): 743-760. [11] Constantinou, M. C., Kartoum, A., Kelly, J. M., 1992. Analysis of compression of hollow circular elastomeric bearings. Engineering Structures, 14(2): 103-111. [12] Koh, C. G., Kelly, J. M., 1987. Effects of axial load on elastomeric isolation bearings. UBC/EERC-86/12, Earthquake Engineering Research Center, College of Engineering,
of
University of California, Berkeley, California.
[13] Gauron, O., Saidou. A., Busson,A., Siqueira, G. H., 2018. Experimental determination of
pro
the lateral stability and shear failure limit states of bridge rubber bearings. Journal of Engineering Structures 174. P.39-48, Elsevier, https://doi.org/10.1016/j.engstruct.2018.07.039. [14] Cancellara, D., Angelis, F. D., 2017. Assessment and dynamic nonlinear analysis of different base isolation systems for a multi-storey RC building irregular in plan, Journal of Computer and Structures. P.74-88, Elsevier. https://doi.org/10.1016/j.compstruc.2016.02.012.
re-
[15] Markou, A. A., and Oliveto, G., and Athanasiou, S., 2016. Response simulation of hybrid base isolation systems under earthquake excitation. Journal of Soil Dynamics and Earthquake Engineering. 84. P.120-133, Elsevier. https://doi.org/10.1016/j.soildyn.2016.02.003.
urn al P
[16] Forcellini, D., 2016. 3D Numerical simulations of elastomeric bearings for bridges. Journal of Innovative Infrastructure Solutions.1:45, Springer. DOI:10.1007/s41062-016-0045-4. [17] Forcellini, D., Kelly, J. M., (2014). Analysis of the large deformation stability of elastomeric bearings. Journal of Engineering Mechanics. DOI: 10.1061/(ASCE)EM.19437889.0000729.
[18] Nagarajaiah, S., Ferrell, K., 1999. Stability of elastomeric seismic isolation bearings. Journal
of
Structural
Engineering,
American
Society
of
Civil
Engineering.
DOI:10.1061/(ASCE)07339445(1999)125:9(946). [19] Hibbett, Karlsson, Sorensen, 1988. ABAQUS/standard: User’s Manual, vol. 1. Hibbitt, Karlsson & Sorensen.
Jo
[20] Sanchez, J., Masroor A., Mosqueda G., Ryan K., 2013. Static and dynamic stability of elastomeric bearings for seismic protection of structures. Journal of Structural Engineering, American Society of Civil Engineers, DOI: 10.1061/(ASCE)ST.1943-541X.0000660. Conflict of Interest and Authorship Conformation Form
Journal Pre-proof
Please check the following as appropriate: o
All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.
o
This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue.
o
The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript
o
The following authors have affiliations with organizations with direct or indirect financial interest in the subject matter discussed in the manuscript:
of
Affiliation
re-
Author’s name
pro
urn al P
Ehsan Kazeminezhad Department of Civil Engineering, Arak Branch, Islamic Azad University, Arak, Iran
Mohammad Taghi Kazemi Department of Civil Engineering, Sharif University of Technology, Tehran, Iran
Jo
Mohammad Mirhosseini Department of Civil Engineering, Arak Branch, Islamic Azad University, Arak, Iran
Journal Pre-proof
Highlights:
urn al P
re-
pro
of
Good accuracy exists between the experimental and the finite element method. The two-spring model has a better accuracy than the overlapping area method. Vertical stiffness of the isolator could be increased or decreased by rotation.
Jo