Fifth International Congress on Advances in Civil Engineering, 25-27 September 2002
Istanbul Technical University, Istanbul, Turkey

Buckling of Rubber Bearings for Seismic Isolated Structures
Tomoki Furuta
Bando Chemical Industries LTD., Civil Engineering and Construction Products
Division, Japan

Yutaka Furukawa
Bando Chemical Industries LTD., Civil Engineering and Construction Products
Division, Japan

Toshiyuki Kanakubo
University of Tsukuba, Institute of Engineering Mechanics and Systems, JapanAbstract
The characteristics of rubber bearings used for seismic isolated structures involve themany unknown factors such as a mechanism at an ultimate stage. In case of design ofthemselves, experimental results are commonly used. For example, the results ofregression analysis of test data are utilized for the design of buckling point, surfacepressure, temperature and strain dependence for shear stiffness. Especially for buckling
behavior, it is very important to estimate the precise point for more safety.
In this paper, a prediction method of buckling point is proposed by considering therelationship between the stress of interlayer steel plate and shear strain of interlayer
rubber. In this prediction method, it is considered that the buckling of rubber bearings iscaused by the yield of steel plate around the center hole. To verify the method, finite
element analysis (FEA) is carried out for whole the rubber bearing for several cases.
From the results of FEA, it is also recognized that the yield of steel plate around center
hole causes the buckling of rubber bearing. It is also confirmed that the buckling point
predicted by proposed method show a good agreement with the load decreasing point ofprevious experimental results.

Introduction
Rubber bearings are set between the upper structures and the base to not transmit theearthquake vibration for seismic isolated structures. In these structures, not only
damages of themselves but also secondary damages such as stumbles and destructionsof furniture are restrained. The rubber bearing is made from the laminates of 4-5mm

thick rubbers and 3-6mm thick steel plates. The stiffness of rubber bearing for verticaldirection is very high, and one for horizontal direction is small. It can be possible tosupport large axial forces of columns and to minimize shear force acting to the structure.
For example, in 7 story buildings subjected to 500gal acceleration, the maximum
response acceleration on first story and 7th story decreases 150gal and 200gal,respectively. The response of maximum drift angle of first story becomes less than
1/1000 radians. The value of base shear coefficient is 0.06, which is less than 1/5 ofordinary buildings. In Japan, seismic isolated structures have been popular especiallyafter Hyougo-ken Nanbu Earthquake (Kobe Earthquake) in 1995. And the performanceof rubber bearings has been improved dramatically.
The characteristics of rubber bearings, however, involve the many unknown factors
such as a mechanism at an ultimate stage. In case of design of themselves, experimental
results are commonly used. For example, the results of regression analysis of test data
are utilized for the design of buckling point, surface pressure, temperature and strain
dependence for shear stiffness. Especially for buckling behavior, it is very important toestimate the precise point for more safety.
In this paper, a prediction method of buckling point is proposed by considering therelationship between the stress of interlayer steel plate and shear strain of rubber. In thisprediction method, it is considered that the buckling of rubber bearings is caused by theyield of steel plate around the center hole. To verify the method, finite element analysis(FEA) is carried out for whole the rubber bearing for several cases.

Predicting Method for Buckling of Rubber Bearings
Factors for Load Decreasing of Rubber Bearings
The factors for load decreasing of rubber bearings are assumed that the impossibility ofequilibrium of forces or yielding (buckling) of internal steel plates (Fig. 1).
Considering the mechanism for the former case, rubber bearings are subjected to shear
force and vertical compressive force at the same time. As shown in Fig. 2, compression
strut is formed in the bearings. If the horizontal deformation (δ) exceeds outside
diameter of bearings (D), carrying part for vertical force disappears and then loaddecreasing takes place. In the experimental results, however, rupture of rubbers often
recognized after deformation of δ > D. The reason for this phenomenon is assumed thatedge of outer rubber deforms toward expansion by local compressive stress andsectional area for vertical force increases outwardly. After that, compression strut can
not be formed and load decreasing is observed. In the most experiments, rupture ofrubbers takes place before load decreasing.
Considering the mechanism for the later case, the compression strut makes high stressin steel plates around the center hole. It is considered that the load decrease is caused byyielding (buckling) of interlayer plates. Some experimental results indicate thisphenomenon as shown in Photo. 1.

Shear force
Q

Buckling

γ=300%

線形領域
Lineararea
Shear strain

γ=400%γ

非線形領域
Hardeningarea（ハードニング）

Fig. 1 Buckling pointQExpansionpart
膨張部
Centralhole
中心孔

Compression
圧縮ストラット
strut

QFig. 2 Compression strut
Loading direction

Perpendicular direction

Photo. 1 Deformation of interlayer plates around center hole
Predicting Method for Stress of Interlayer Plates
Wrapping area of each interlayer steel plate after deformation is defined as effective
acting area, A0 as shown in Fig. 3. Von-Mises stress at the effective acting area can becalculated by Eq. (1). In this equation, the ratio of rubber thickness (tr) to steel plate
thickness (ts) is multiplied by the assumption that the effective acting area transmutesvertical force for whole bearing and shear force for only this area. Where, shear force(Q) is given by horizontal stiffness (Kh) (AIJ 2001).

σ A0 =

σ 1 2 + σ 2 2 + (σ 1 − σ 2 ) 2 t r
2

⋅

ts

1
σ 1 =  σ N + σ N 2 + 4τ 2 2



(1)

(2)

1
σ 2 =  σ N − σ N 2 + 4τ 2 2(3)

where,σ A0σNPA0τ

QA: Von-Mises stress of effective acting area
: compressive stress of effective acting area = P / A0
: vertical force: sectional area for effective acting area
: shear stress of effective acting area = Q / A: shear force: effective area

Fig. 3 Effective acting area
Relationship between Interlayer Plate Stress and Shear Strain

From the previous experimental test results, the relationships between interlayer platestress for effective acting area and shear strain are shown in Fig. 4 for the specimens
failed by buckling (No.2, 3, 4, 5 × 2, 6 and 7), failed by rubber rupture (No.8) andwithout fail (No.1). The filled circles show buckling points and the triangle shows themaximum point. It is recognized that if the stress of effective acting area exceeds
270N/mm2, buckling takes place for the specimens with second shape factor (S2) of lessthan 4.0 (No.1, 2, 5, 6 and 7), and specimens subjected to high vertical stress with valueof S2 of 4.5 (No.3 and 4). For the specimen No.8 having value of S2 of 5.0, theinterlayer plate stress is very small comparing with other specimens at same shear strain.
Predicting of Buckling Point

Based on the assumption that the buckling of rubber bearings caused by yielding ofinterlayer plates, buckling takes place in the region surrounded by dotted lines shown inFig. 4. In the region of γ >320%, it becomes difficult to cause buckling because stress ofinterlayer plate decreases by the hardening of rubber. Therefore, the prediction ofbuckling of rubber bearing is can be calculated by Eq. (4).

σ A0 ≥ σ sa = 270 N / mm 2(γ < 320%)

where,

σ sa: tensile strength of interlayer plate (SPHC)

(4)

500
①②③④
⑤⑥⑦⑧

450
400

2σ×(tr/ts)[N/mm ]350

G0.39- φ600, S1=33.1, S2=3.0：σ= 7.0 N/mm2G0.37-φ 700, S1=29.5, S2=3.5：σ= 9.8 N/mm2G0.36-φ 900, S1=30.0, S2=4.4：σ=21.2N/mm2G0.38-φ 900, S1=30.4, S2=4.4：σ=24.5N/mm2
G0.35-φ1000, S1=25.0, S2=4.0：σ=12.7N/mm2G0.39-φ 800, S1=40.0, S2=4.0：σ=18.0N/mm2
G0.39-φ 600, S1=37.5, S2=3.0：σ=11.0N/mm2
G0.39-φ 700, S1=35.4, S2=5.0：σ=20.0N/mm2300

Buckling area

250

No damage⑤④⑦⑧②③⑥①200
150
100: Buckling outbreak: Load keeping

5000.00.51.0

1.5

2.0

2.5
γ[×100%]

3.0

3.5

4.0

4.5

5.0

Fig. 4 Relation between interlayer plate stress for effective acting area and shear strain

Finite Element Analysis for Rubber Bearing
Finite Element Model
Rubber : Rubber is modeled by using strain energy density function (W). In thisanalysis, the cubic polynomial expression considering compressive characteristics ofrubber materials is used as shown in Eq. (5).
3W = ∑ C ij ( I 1 − 3) i ( I 2 − 3) j +i + j =1

1( J el − 1) 2 j
D1(5)where,222I1: primary unbiased of deviation strain = λ1 + λ 2 + λ3

I2: secondary unbiased of deviation strain = λ1λi

: deviation elongation ratio = J −1 / 3 λi
: volume ratio: mechanical elastic volume ratio

JJ el( −2 )+ λ2( −2 )+ λ3( −2 )Table 1 shows the values of constants, Cij and D1. These values are intended for natural
rubber with shear modulus (G) of 0.4N/mm2. These are decided by the regression
analysis using the least square method from the stress – strain curves obtained by tensile
test (JIS K6251, #3 test coupones) and shear test for 220 × 15 × 1mm rectangle testcoupones.

C10
3.67×10-2
C30-2.08×10-7

Table 1 Constants for rubber modelC01
C20C11-2-4-1.92×10
6.55×10-1.09×10-3
C21C12C0300
1.07×10-6C02
7.96×10-4
D1
5.70×10-3

Steel Plate : Elastic-plastic model are used for interlayer steel plate as shown in Fig. 5.
Elastic modulus (E) and Poisson’s ratio (ν) is 210000N/mm2 and 0.3, respectively.
Perfect plastic occurs at 2% strain.
3002σ（N/mm ）250
20015010050000.01

0.02

0.03

0.04

0.05

εFig. 5 Steel plate model
Boundary Conditions : Freedom of nodes at upper and lower flange unified into two
representative nodes. Compressive and shear force are applied between the two nodes.
Elements and Mesh: Analyzed specimens are listed in Table 2. Specimen No.1
corresponds to No.2 in Fig. 4, which failed by buckling. Specimen No.2 corresponds toNo.6, which failed by rubber rupture. Fig. 6 shows the mesh model for specimen No.1.
Elements are divided in nine for radius direction, six for circumference direction andrubber and steel plate are divided two elements for one laminate. Utilized model is solid

Table 2 Shape of specimens
Total Thickness
Outer
Inside Thickness
First Second
Number of thickness of a steel
Specimen diameter diameter of rubber
shape shapelaminating of rubberplate
(mm)
(mm)
(mm)
factor factor
(mm)
(mm)
No.1
700
110
5.0
40
200.0
2.5
29.5 3.50
No.2
700
354.7
30
141.0
3.1
35.4 4.96

Fig. 6 Mesh for specimen No.1

model with eight nodes. Compressive stress of 10N/mm2 was applied, and then shear
displacement was applied monotonously.
Results of Finite Element Analysis

Fig. 7 shows effective strain contours of interlayer steel plates of the upper and center
part of specimens No.1 and No.2 at shear strain (γ) of 300%. It is recognized that strainsof two specimens are different even if each specimen is applied same vertical stress(10N/mm2) and same shear strain (300%). In specimen No.1, which is inferior tobackling, strains at the top edge and the center around the hole are bigger than No.2.
These agree with the stresses shown in Fig. 4.

UpperCenter
No.1

Upper

CenterNo.2
Fig. 7 Effective strain contour of interlayer plate
Deformations of specimens No.1 and No.2 at the central sections are shown in Fig. 8.
Deformations of specimen No.1 at the top and bottom edges and center around the hole
are big comparing with those of specimen No.2. Fig. 9 indicates the compression strut
overdrawn on FEA results of specimen No.1.

No.1

No.2
Fig. 8 Deformation at the section

Fig. 9 Compression strut from FEA
Shear force versus shear strain curves are compared between analytical results andexperimental results for each specimen as shown in Fig. 10. In specimen No.1, bucklingpoint obtained from FEA almost agrees with experimental result.

Shear force

No.2 analysis
No.2 experimentNo.1 experimentNo.1 analysis

Shear strain（γ：×100 %）Fig. 10 Comparison between experimental results and FEA results

Conclusions
A prediction method of buckling point of rubber bearings is proposed by considering
the compression strut, of which sectional area is defined by wrapping area of eachinterlayer steel plate as effective acting area. From the relationship between the VonMises stress of effective acting area and shear strain of rubber, the stress exceeds tensile
strength of steel plate when rubber bearings fail by buckling. FEA results also show theformation of compression strut and concentration of stress around the center hole. Thesephenomena are observed in case of inferior shape of rubber bearing. It is considered thatthe buckling of rubber bearings is caused by the yield of steel plate around the centerhole.

References
Architectural Institute of Japan (AIJ) (2001). Design Guidelines for Isolated Structures,Tokyo, Japan.