Bonded versus unbonded strip fiber reinforced elastomeric isolators: FEA

8/9/2019 Bonded versus unbonded strip fiber reinforced elastomeric isolators: FEA

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Bonded versus unbonded strip fiber reinforced elastomeric isolators: Finite

element analysis

Hamid Toopchi-Nezhad ⇑, Michael J. Tait, Robert G. Drysdale

Department of Civil Engineering, McMaster University, 1280 Main St. W., Hamilton, ON, Canada L8S 4L7

a r t i c l e i n f o

Article history:Available online 10 August 2010

Keywords:

Strip fiber reinforced elastomeric isolators

Laminated rubber bearings

Finite element analysis

Seismic mitigation

Base isolation

a b s t r a c t

This paper presents a finite element (FE) model for the analysis of strip fiber reinforced elastomeric iso-lators (FREIs) that are subjected to any given combination of static vertical and lateral loads. The model is

able to simulate both bonded and unbonded boundary conditions at the top and bottom contact surfaces

of the isolator. Compared to bonded (B)-FREIs, the FE-analysis of stable unbonded (SU)-FREIs presents

additional analysis challenges. SU-FREI refers to unbonded FREIs that exhibit stable rollover deformation

under lateral loads. Additional analysis challenges are attributed to changes in the boundary conditions

of SU-FREI as a result of rollover type deformation. To address these challenges, the utilized FE-mesh is

updated during analysis consistent with the deformed geometry of the isolator. Using the proposed FE-

model, the lateral responses of a B-FREI and a SU-FREI were evaluated. Both isolators had the same mate-

rial and geometrical properties and were subjected to identical constant vertical loading. Comparing the

lateral responses, it was found that the SU-FREI was considerably more efficient than the B-FREI as a seis-

mic isolator. In addition, the in-service stress demands on the SU-FREI were found to be significantly

lower than the B-FREI.

2010 Elsevier Ltd. All rights reserved.

1. Introduction

Fiber reinforced elastomeric isolators (FREIs) comprise alternat-

ing bonded layers of elastomer and fiber reinforcement. The elasto-

meric layers provide lateral flexibility and the primary role of fiber

reinforcement is to constrain lateral bulging of the elastomer when

the isolator is subjected to vertical compressive loads. In terms of

connection to the isolator’s top and bottom contact supports, FREIs

can be classified as either bonded or unbonded. In a bonded (B)-

FREI, two thick steel mounting plates are bonded to the outer rub-

ber layers at the top and bottom of the isolator. During installation,

the top and bottom mounting plates are bolted to the superstruc-

ture and substructure, respectively. In an unbonded FREI, the isola-

tor is placed between the substructure and superstructure without

any bonding or fastening at its contact surfaces. During an earth-

quake, the shear loads at the contact surfaces of an unbonded

isolator are transferred through friction. In a stable unbonded

(SU)-FREI, the isolator’s geometry can be selected such that it

maintains lateral stability at extreme lateral displacements.

Over the past decade, a number of experimental investigations

on individual bonded and unbonded FREIs have been conducted

[1–10]. The common outcome of these studies is that the investi-

gated FREIs have had suitable mechanical properties to be used

as seismic isolators. Recently, a shake table study confirmed the

seismic mitigation efficiency of SU-FREI systems [11]. These stud-

ies suggest that FREIs are a viable alternative for conventional steel

reinforced elastomeric isolators (SREIs).

Current seismic codes mandate that the final mechanical design

properties of isolators should be evaluated through experiment.

This mandate applies to all isolator types. Accordingly, the main

objective of preliminary design is to provide the required informa-

tion for the fabrication of the prototype isolators. For preliminary

design, both the vertical and horizontal stiffness and damping ratio

values of an isolator should be reasonably estimated. Knowing the

vertical stiffness, one can verify that the vertical frequency of the

isolator is sufficiently greater than the target lateral base isolation

frequency to eliminate any significant contribution of rocking

modes in the total response of the base isolated structure. The esti-

mation of horizontal stiffness is critical to ensure that the designed

isolator satisfies the target base isolation frequency. Additionally,

knowing the horizontal stiffness and damping ratio, one can assess

whether the demand lateral displacements lie within the permissi-

ble displacement range of the isolator.

Since the deformation characteristic of the fiber reinforcement

is different than the steel reinforcing plates, the closed-form equa-

tions, available for the stiffness solution of conventional SREIs, are

not generally applicable to FREIs. Solutions to the vertical compres-

sion and bending stiffness of strip, rectangular, and circular FREIs

0263-8223/$ - see front matter 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.compstruct.2010.07.009

⇑ Corresponding author. Tel.: +1 905 5259140/24860; fax: +1 905 5299688.

E-mail addresses: [email protected] (H. Toopchi-Nezhad), taitm@mcmas-

ter.ca (M.J. Tait), [email protected] (R.G. Drysdale).

Composite Structures 93 (2011) 850–859

Contents lists available at ScienceDirect

Composite Structures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p s t r u c t

http://dx.doi.org/10.1016/j.compstruct.2010.07.009mailto:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.compstruct.2010.07.009http://www.sciencedirect.com/science/journal/02638223http://www.elsevier.com/locate/compstructhttp://www.elsevier.com/locate/compstructhttp://www.sciencedirect.com/science/journal/02638223http://dx.doi.org/10.1016/j.compstruct.2010.07.009mailto:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.compstruct.2010.07.009


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are available [12–15]. Also, both the horizontal stiffness and buck-

ling load of strip B-FREIs have been analytically investigated

[16,17]. Given the complex deformation characteristics of FREIs,

the analytical solutions for B-FREIs typically involve sophisticated

equations. Furthermore, the literature lacks any closed-form solu-

tion for the lateral response of SU-FREIs.

Finite element (FE) is a powerful design tool that can be em-

ployed in the preliminary design of FREIs. Analysis of elastomericcomponents requires a robust FE-model that is capable of address-

ing large deformations and accounts for the nearly incompressible

behavior of the elastomer. Although FE-analysis of B-FREIs is rela-

tively straight forward, there are challenges in the FE-analysis of

SU-FREIs due to the rollover deformation of these isolators under

lateral loads. A limited number of studies report on FE-analysis

of FREIs. These studies are either limited to the vertical compres-

sion response [18], or focus on the vertical and lateral responses

of B-FREIs only [9].

The focus of this study is on the lateral response evaluation of

both B- and SU-FREIs through FE-analysis. A FE-model, capable of

simulating both conventional bonded and unbonded boundary con-

ditions at the top and bottom contact surfaces of strip FREIs, is pre-

sented. The term strip is selected as the analysis is carried out for

the unit out-of-plane length of the isolator. The primary goal of the

presented FE-analysis is to evaluate the horizontal stiffness of the

isolators at different lateral displacements. The analysis results

are also used to assess the stress (or strain) state in the isolator’s

components. In addition to developing the FE-model, an extensive

comparative study on the lateral response of a B-FREI and a corre-

sponding SU-FREI is presented.

2. Fiber reinforced isolator

Fig. 1 shows a sketch of the cross section of the FREI investi-

gated in this study. The isolator comprises 11 layers of fiber rein-

forcement layers interleaved and bonded between 12 layers of

rubber. The physical dimensions and material properties of the iso-

lator are shown in Table 1. The isolator carries a constant vertical

compression load of P = 112 N.

3. Analytical evaluation of the horizontal stiffness for strip

FREIs

3.1. B-FREIs

The main objective of the analytical solution is to evaluate the

horizontal stiffness of a B-FREI based on the properties given in Ta-

ble 1. The analytical solutions of conventional SREIs are not directly

applicable to B-FREIs due to the different mechanical characteris-

tics of fiber reinforcement as compared to the steel reinforcing

plates. In a laterally deformed SREI, the steel plates remain planar

and are nearly rigid in both tension and flexure. On the contrary,despite their very large in-plane tensile stiffness, the fiber rein-

forcement layers show some extensional flexibility with no bend-

ing rigidity. Accordingly, in a laterally deformed B-FREI, the fiber

reinforcement layers undergo warping deformation at their ends

(see Fig. 2a) as a result of the internal moment and shear that de-

velop in the isolator.

Horizontal stiffness of a strip B-FREI can be estimated using the

closed-form elastic solution available for the horizontal stiffness

evaluation of a homogeneous short vertical beam that is subjected

to axial compression and lateral shear forces. The theory extends

the Haringx theory [19] on the stability of rubber rods by account-

ing for the shear and warping deformations of the cross section. Toreplicate the boundary conditions of a B-FREI, the lower end of the

beam is assumed to be fixed against any displacement, rotation

and warping, and the upper end is constrained against rotation

and warping but allowed to move in both lateral and axial direc-

tions. Based on these assumptions, the horizontal stiffness of the

beam (or the B-FREI given in Fig. 1) can be calculated by [16];

K H ¼ Ge A

t r

P

2P ð1þP Þþb2P ðb1 þb2 Þ

ffiffiffiffiffiffiffiffiffiffiffi2qb1

p tan

ffiffiffiffib18q

q þ 2

P bP ðb1 þb2 Þ

ffiffiffiffiffiffiffiffiffiffiffi2qb2

p tan

ffiffiffiffib28q

q 1

when ð1 þ P Þkc kb P 0 ð1Þ

K H ¼ Ge A

t r P

2P ð1þP Þþb2

P ðb1 þb2 Þ ffiffiffiffiffiffiffiffiffiffiffi2

qb1p tan ffiffiffiffib1

8qq

2P b

P ðb1 þb2 Þ ffiffiffiffiffiffiffiffiffiffiffi2

qb2p tan ffiffiffiffib2

8qq

1

when ð1 þ P Þkc kb < 0 ð2Þ

where A is the plan cross section area of the isolator, P ¼ P =Ge A rep-

resents a dimensionless compression force, and coefficients b1 and

b2 are defined as follows;

b1 ¼ ½P ð1 þ P Þ þ kb kc

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffi½P ð1 þ P Þ þ kb kc

2þ 4P ½ð1 þ P Þkc kb

q ð3Þ

b2 ¼ ½P ð1 þ P Þ þ kb kc

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffi½P ð1 þ P Þ þ kb kc

2 þ 4P ½ð1 þ P Þkc kb

q ð4Þ

Parameters kb and kc , which are attributed to the cross sectionwarping of a laterally deformed B-FREI, can be estimated using

[17];

K b 20 1 2

21 ðaaÞ2

h i 1 1

210 ðaaÞ2 þ P 3

10 þ 8

525 ðaaÞ2

h in o21 2

77 ðaaÞ2 þ 3675

64kS 3 ða=t f Þ3

ð5Þ

kc

3 1 221

ðaaÞ2h i

23 463

ðaaÞ2 þ P 9 þ 26105

ðaaÞ2 þ 245S 12kða=t f Þ

3

1 277

ðaaÞ2 þ 367564kS 3 ða=t f Þ

3

ð6Þ

In Eqs. (5) and (6), aa, the extensional rigidity of the fiber reinforce-ment is given by

t e

t f

2a

R1

R12

R6

Fig. 1. Cross section of the strip FREI used in the finite element analysis.

Table 1

Material and geometrical properties of the strip FREI.

Material properties

Ge = Shear modulus of the elastomer (rubber) = 0.4 MPa

t f = Poisson’s ratio of the fiber reinforcement = 0.2E f = Young’s modulus of the fiber reinforcement = 137 GPa

Geometrical properties

2a = Width of the isolator = 70 mm

t e = Thickness of a single elastomer layer = 1.587 mmt r = Total thickness of the rubber layers = 19.044 mm

t f = Thickness of the fiber reinforcement = 0.55 mm

h = Height of the isolator = 25.094 mm

S = Shape factor (=a/t e) = 22

H. Toopchi-Nezhad et al. / Composite Structures 93 (2011) 850–859 851


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aa ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12kS

a

t f

s ð7Þ

where

k ¼ð1 t2 f ÞGe

E f ð8Þ

In Eqs. (1) and (2), q ¼ ðEI Þeff =Ge At 2r indicates the ratio of the effec-

tive bending to shear rigidity of the isolator. For a homogeneous

beam (EI )eff = EI , however, for a laminated strip rubber isolator, itis defined as follows [17];

ðEI Þeff ¼ a3GeS

2 24

ðaaÞ4 1 þ

1

3ðaaÞ2

aatanhðaaÞ

ð9Þ

3.2. SU-FREIs

As shown in Fig. 2b, when a SU-FREI is laterally displaced, its

upper and lower faces roll off the contact supports and the isolator

exhibits rollover deformation. The rollover deformation results in a

non-linear load–displacement relationship. Therefore, the horizon-

tal stiffness varies with isolator lateral displacement. As such, the

analytical stiffness solution for this type of isolators is more com-

plex and Eqs. (1) and (2) are no longer applicable. Currently, noclosed-form solution has been reported in the literature to esti-

mate the horizontal stiffness of SU-FREIs.

3.3. Limitations of closed-form solutions

Any closed-form stiffness solution only serves for the prelimin-

ary design of FREIs. There are a number of features which are not

addressed by the simplified equations used in the closed-form

solutions. Among these features, one can refer to the dependency

of the rubber material properties on the amplitude of the shear

strain, which is ignored in closed-form linear solutions. Addition-

ally, due to the static nature of the solution, no information can

be obtained on the influence of rate and history of the lateral dis-

placement on the lateral response of the isolator. Analytical evalu-

ation of the effective damping of the isolator, which is an important

design parameter, is not possible. Therefore, the final design prop-

erties of any isolation device should be evaluated through experi-

mental testing on prototype samples of the isolator [20].

4. Finite element modeling

4.1. Objectives

In the FE-analysis presented in this paper, the strip isolator, un-

der a constant axial compression load, is subjected to a static lat-

eral load acting at its top support. Due to the non-linear nature

of the analysis, the lateral load is applied incrementally in multiple

steps from zero to its target value. The target lateral load is selectedsuch that a 200% t r lateral deformation in the isolator is achieved.

The primary objective of the FE-analysis is to evaluate the lateral

load–displacement relationship of a B-FREI or a SU-FREI with given

geometric and material properties as listed in Table 1. From the lat-

eral load–displacement curve, the effective (secant) horizontal

stiffness corresponding to each level of lateral displacement can

be assessed to verify whether the isolator meets the seismic isola-

tion design requirements. The second objective of the FE-analysis

is to evaluate the stress state within the isolator. Estimating the

maximum stress demand in the rubber and fiber reinforcement

layers and evaluating the required bond strength between the

alternating layers are needed for manufacturing requirements.

4.2. General description of the model

Modeling and analysis of the isolators were carried out using

MSC. Marc [21], a commercially available general purpose FE-soft-

ware package. In the model presented, the rubber layers were dis-

cretized using quadrilateral plane-strain solid elements, and 2D-

truss elements were selected to represent the fiber reinforcement.

The truss elements in the model were characterized with zero

physical thickness. A previous experimental study [5] revealed that

the ratio of width to the total thickness of isolator (i.e., the aspect

ratio), plays a crucial role in the lateral response of SU-FREIs. To ac-

count for the real physical thickness of the fiber reinforcement in

the model, and to accurately simulate the isolator aspect ratio, ri-gid gaps were defined between the top and bottom faces of the fi-

ber reinforcement and the adjacent rubber layers.

Two horizontal rigid wires were defined at the top and bottom

of the isolator to represent the contact supports. All degrees of

freedom of the bottom support and the rotational degree of free-

dom of the top support were constrained. Since both the vertical

load and lateral load were applied to the top support, it was al-

lowed to move in both the vertical and horizontal directions.

Given the available contact models in the utilized software, the

‘‘touching” contact model was selected for the SU-FREI model. In

this contact model, the nodal points of the exterior rubber ele-

ments at the contact surface were constrained in the direction nor-

mal to the contact support. No tension stress was resisted at the

contact surfaces meaning that when the compression contactstress approached zero, the nodal points were allowed to detach

from the contact support. Shear force between the contact sup-

ports and the isolator was transferred through a Coulomb friction

mechanism. To prevent slip at the contact support, the coefficient

of friction at the contact surfaces of the isolator and the rigid sup-

ports was selected to be one in the FE-model. Based on a previous

experimental study [6,7] that was conducted on square SU-FREIs

having the same characteristics as listed in Table 1, slip at the con-

tact surfaces of the isolators was found to be negligible.

The FE-model of the B-FREI was the same as the SU-FREI model,

however, the rigid supports were connected to the isolator using

the ‘‘glue” contact model in MSC. Marc [21]. This contact model

prevents any detachment or slip between the isolator and the rigid

supports by constraining the nodal points in the directions normaland tangential to the contact support.

(a) B-FREI (b) SU-FREI

Warping at

the ends of fiber

reinforcement layers

Fig. 2. Sketch of laterally deformed FREIs with different boundary conditions.

852 H. Toopchi-Nezhad et al. / Composite Structures 93 (2011) 850–859


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4.3. Material models used for the rubber and the fiber reinforcement

Since the FE-analysis developed in this study serves as a tool for

the preliminary design of FREIs, it relies on a minimum set of re-

quired material data properties to minimize the cost of the preli-

minary design. As such, the presented FE-analysis employs a

Neo-Hookean model as the simplest available hyperelastic material

model for elastomeric materials. In this model, the rubber is trea-ted as an elastic isotropic material with a strain energy function

that is characterized by shear (Ge) and bulk (ke) moduli of the rub-

ber. The nominal shear modulus at 100% elongation can be pro-

vided, as routine standard product information, by many rubber

suppliers. If the rubber in an isolator performs as a nearly incom-

pressible material with a Poisson’s ratio of approximately 0.5, an

appropriate value of the bulk modulus can be estimated using

the following well-known elasticity relationship.

keGe

¼ 2ð1 þ teÞ3ð1 2teÞ

ð10Þ

The bulk modulus of rubber in the presented FE-analysis is ta-

ken as ke = 1900 MPa. With this value and the shear modulus of

Ge = 0.4 MPa (see Table 1), the use of Eq. (10) results in a Poisson’s

ratio of 0.4999 which is sufficiently close to 0.5 to simulate the

incompressible behavior of the rubber in the FE-analysis.

The fiber reinforcement in the presented FE-model is treated as

a linear elastic isotropic material with material properties given in

Table 1.

4.4. Rubber incompressibility

The incompressibility of rubber can result in serious numerical

errors if standard isoparametric elements, in which the stress state

is determined from the strain state, are used in the FE-analysis.

Since the volumetric strain in the finite elements is nearly zero

in a nearly incompressible material, determining the mean stress

or pressure based on the volumetric part of the strain is challeng-ing [22]. Additionally, standard isoparametric elements in an

incompressible material show a pathological behavior entitled vol-

umetric mesh-locking that is attributed to the inaccurate perfor-

mance of an element due to an over-constrained condition and

insufficient active degrees of freedom [23]. To avoid these prob-

lems, modern FE-techniques utilize so called mixed-methods for

incompressible materials [22]. In mixed-methods both the strains

and stresses are assumed as the unknowns. In the FE-model pro-

posed in this paper, a commonly used mixed-method developed

by Hermann [24] has been used for the rubber elements.

4.5. Large deformation and element distortion

Elastomeric isolators may undergo very large deformations un-der extreme lateral loads. Therefore, in the FE-analysis of these iso-

lators, the use of conventional total Lagrangian formulation is not

recommended. As such, in the FE-analysis presented in this study,

an updated Lagrangian approach has been used. In this approach,

the orientation of the local coordinate system is updated during

the analysis based on the deformed configuration of the element.

In a laterally deformed SU-FREI, the rollover deformation, in

addition to the large deformation of the isolator, adds to the com-

plexity of the FE-analysis. When rollover deformation occurs, one

end of the outer rubber-layers rolls off the contact support while

the opposite end is pressed against the contact support (see

Fig. 2b). Accordingly, in the FE-analysis, the rubber elements at

the pressed ends may experience severe distortion such that they

no longer accurately discretize the problem. As the lateral load isincreased, the intermediate rubber layers of the isolator may also

experience local excessive distortion. When the FE-software de-

tects any excessive distortion, it automatically terminates the anal-

ysis to preserve the accuracy of the results. To resolve this

termination, prior to excessive element distortion in the original

mesh, the FE-solution can be mapped onto a new mesh that is

adapted with the current deformed geometry of the model. The

new mesh arrangement results in a new FE-problem. The analysis

is then continued by treating the solution from the previous mesh,at the point of mapping, as the initial conditions of the new FE-

problem.

A global remeshing procedure available in MSC. Marc [21] was

used in the presented FE-analysis. The remeshing procedure uti-

lized the updated Lagrangian formulation. The remeshing criterion

was based on monitoring the deviations of the inner angles of the

elements from their undeformed configuration. The default limit of

the software was used as the threshold angle-change. For the top

and bottom rubber layers, the edge length of the quadrilateral ele-

ments was selected to be approximately 0.3 mm in the original

mesh. The element edge length for the intermediate rubber layers

was set to be approximately 0.4 mm due to the reduced distorted

pattern of deformation in these layers as compared to the outer

rubber layers. If required, the updated mesh could be finer in the

regions of high-strain gradients in the rubber layers.

The B-FREI was modeled using the same original FE-mesh that

was employed for the SU-FREI. Since the boundary conditions of

the isolator remained unchanged throughout the analysis, no

remeshing was required. The lower-order finite elements were

used in modeling of both isolators as the performance of these ele-

ments is superior to higher-order elements in simulating large dis-

tortions [25]. For both isolators, the length of the 2D-truss

elements, representing the fiber reinforcement, was selected to

be 0.28 mm.

5. Finite element model validation

5.1. B-FREI

Table 2 contains the horizontal stiffness values of the B-FREI,

which are calculated using closed-formed equations and the FE-

analysis. For conventional SREIs, when the vertical load carried

by the isolator is significantly lower than the isolator’s buckling

load, the horizontal stiffness can be calculated using the simple

formula of K H = Ge A/t r [26]. This simple formula is expected to over-

estimate the horizontal stiffness of a B-FREI due to the flexibility of

the fiber reinforcement layers that are employed in the isolator.

Given the properties cited in Table 1, under a vertical load of

P = 112 N, the control parameter ð1 þ P Þkc kb acquires a positive

value. Therefore, Eq. (1) is used to calculate the horizontal stiffness

of the B-FREI. As can be seen in Table 2, for the B-FREI considered in

this study, the stiffness value that is estimated by the simple for-

mula of Ge A/t r is in close agreement with the value calculated byEq. (1). This close agreement suggests that one can use the simple

formula, with less calculation effort, for the preliminary design of

B-FREIs. However, caution should be used as the error of employ-

ing the simple formula rises with increased flexibility of the fiber

reinforcement in the isolator.

From the FE-analysis, a linear lateral load–displacement rela-

tionship was evaluated for the B-FREI. The slope of this line was

Table 2

Horizontal stiffness (N/mm) of the B-FREI.

Analytical Finite element

Ge A/t r Eq. (1)

1.470 1.455 1.453



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calculated as the horizontal stiffness of the isolator. According to

Table 2, excellent agreement is observed between both the finite

element and analytical stiffness solutions. This confirms the accu-

racy of the presented FE-model and its suitability for the prelimin-

ary design of the B-FREIs.

5.2. SU-FREI

As described earlier, both B- and SU-FREI models employed the

same FE-assemblage. However, in the SU-FREI model, no tension is

transmitted between the isolator and its top and bottom contact

supports. Due to the model similarity, validation of the FE-model

for B-FREI provided sufficient confidence that the SU-FREI model

would give satisfactory results. However, for further validation,

the lateral load–displacement curve of the SU-FREI that was ob-

tained from the FE-analysis was compared with the results from

a previous experimental research study on similar SU-FREIs [6,7].

This experimental study was conducted on square SU-FREIs having

the same characteristics as listed in Table 1.

Fig. 3 shows the lateral load–displacement relationships from

both the experimental study and the FE-analysis. It should be

noted that the FE-results in Fig. 3 are obtained by multiplying

the lateral load of the strip isolator by 70, to account for the

70 mm out-of-plane length of the square isolator. In Fig. 3, the lat-

eral deformations of the isolator are described in terms of % t r lat-

eral displacement not percent of shear strain, which is commonly

used to express the level of lateral deformation in conventional

SREIs. The normalized lateral displacement is employed as shear

deformation is not the dominant deformation mode (see Fig. 2b).

As seen in Fig. 3, reasonable agreement was found between the

experimental and FE-results at displacements ranging from 100%

to 200% t r . However, for lower displacements, the accuracy of the

FE-analysis was reduced as the presented FE-model assumed a

constant value for Ge (corresponding to 100% elongation in the rub-

ber) throughout the analysis, and neglected any strain dependency

of the rubber material.

6. Finite element analysis results and discussion

6.1. Lateral load–displacement relationship

Fig. 4 contains the lateral load–displacement relationships for

both the B- and SU-FREIs up to 200% t r lateral displacement. As

shown in this figure, while the lateral response of the B-FREI re-

mains nearly linear, the SU-FREI behaves nonlinearly due to the

rollover deformation. As a result of the rollover deformation, the

horizontal effective secant stiffness decreases with increased lat-

eral displacement. This decrease results in an increase in the base

isolated period of the isolator, increasing its seismic mitigation effi-

ciency. Accordingly, for a given FREI, unbonded application leads to

superior seismic isolation provided that lateral stability of the iso-

lator is maintained.

It has been shown that unbonded FREIs with inappropriate

geometry may exhibit lateral instability [5]. Inthedesign ofan unb-

onded FREI, the isolator must maintain positive incremental load-

resisting capacity throughout the entire range of lateral displace-

ments that are imposed on the isolator. The presented FE-model

can be employed to verify the achievement of stable rollover (SR)

deformation in the preliminary design of an unbonded FREI. As

can beseen in Fig. 4, the unbonded isolator maintained positive tan-

gent stiffness over the range of lateral displacement investigated.

In a base isolated system, the lateral shear load resisted by the

isolators is transmitted to the superstructure. According to Fig. 4,

for any given lateral displacement above 50% t r , the SU-FREI is sub-

jected to a significantly lower shear compared to the B-FREI. As the

lateral displacement is increased, the difference between the lat-

eral (shear) loads, resisted by the B- and SU-FREIs is found to in-

crease (see Fig. 4). At 200% t r displacement, the shear load that is

transmitted to the superstructure by the SU-FRE is approximately40% less than that of the B-FREI. Therefore, with the same physical

dimensions and material properties, the response attenuation of

the SU-FREI is superior to that of the B-FREI.

The secant horizontal stiffness of the SU-FREI at 200% t r dis-

placement (38 mm) is calculated to be 0.85 (N/mm)/mm. To

achieve this horizontal stiffness in the B-FREI, using the simple for-

mula of K H = Ge A/t r , the minimum required total thickness of rub-

ber layers in the isolator would be t r = 33 mm. To assemble such

an isolator with materials given in Table 1, 21 rubber layers and

20 interleaved bonded fiber reinforcement layers are required. This

implies a 75% increase in the volume of utilized materials. Addi-

tionally, the significant increase in the isolator height may force

the designer to increase the isolator’s width to prevent Euler buck-

ling [26], which may occur in the B-FREI under large lateraldisplacements.

An examination of Fig. 4 indicates that for any given level of lat-

eral load, the SU-FREI undergoes larger lateral displacement than

the B-FREI. A well recognized way of dealing with excessive lateral

displacements in elastomeric isolators is to provide supplementary

damping in the isolation system. Traditionally, a lead core or a

high-damped rubber compound is employed in order to dissipate

a larger portion of the input excitation and limit isolator deforma-

tion. Results of previous studies [2,5–7] suggest that in FREIs that

utilize sufficient relative volume of fiber to elastomer, the interac-

tion between fiber reinforcement and the rubber layers provides a

new source of energy dissipation in addition to the intrinsic damp-

ing of the elastomer. This phenomenon is more pronounced for SU-

FREIs due to the increased distorted pattern of lateral displacementas compared to B-FREIs (compare Fig. 2a and b). Even though the

Fig. 3. Lateral load–displacement relationship of the square SU-FREI; comparisonbetween finite element and experimental results.

Fig. 4. Lateral load–displacement of the bonded and stable unbonded FREIs.



6/10

SU-FREIs are more flexible than their corresponding bonded isola-

tors, the presence of increased damping may aid in restricting ex-

treme lateral displacement under a strong earthquake event.

6.2. Stress/strain state in the isolators

For each rubber element, the local axes are denoted by Axis 1

and Axis 2, which are parallel and perpendicular to the orientation

of the fiber reinforcement layers, respectively. Fig. 5 contains the

stress contour for normal stress S 22 corresponding to 200% t r dis-

placement. The normal stress S 22 acts perpendicular to the orienta-

tion of fiber reinforcement layers. The stress contours shown in

Fig. 5 reflects the average analysis output at all integration points

of the elements.

For the B-FREI, as can be seen in Fig. 5a, the axial load is carried

through a compression core within the isolator, which acts as an

equivalent column. The cross section area of the equivalent column

is limited to the overlap region between the top and bottom faces

of the laterally deformed B-FREI. In a B-FREI, the boundary condi-tions and the point of application of the vertical load resultant at

the top and bottom of the isolator, remain constant regardless of

the level of lateral deformation. Therefore, to establish equilibrium

in the isolator, the balancing moments are generated at the top and

bottom surfaces of a laterally deformed B-FREI (see Fig. 6a). As a re-

sult, the regions outside the central compression core undergo sig-

nificant tensile stresses which are normal to the bonding interface

between the outer rubber layers and the steel mounting plates and

between the inner rubber and the fiber reinforcement layers. The

tensile regions (yellow1 triangles) and the compression core (equiv-

alent column) can be clearly seen in Fig. 5a.

Due to unbonded application, no tensile stress is transferred to

a laterally deformed SU-FREI at the isolator’s contact supports. As a

result, no balancing moment develops at the top and bottom sur-faces of a SU-FREI. As the SU-FREI is deformed laterally, one end

of the contact surface rolls off the support, and the opposite end

is pressed against the support. Therefore, as shown in Fig. 5b, the

stress S 22 at the contact surfaces ranges from its maximum value,

at the pressed end, to zero, at contact regions that are at the onset

of rolling off the support. Due to the non-uniform stress distribu-

tion, the point of application of the vertical load resultant, at each

contact surface, shifts toward the pressed corner of the isolator. As

can be seen in Fig. 6b, the offset of vertical resultant loads at the

top and bottom of the isolator produces a couple that balances

the overturning moment caused by the shear loads.

A close examination of Fig. 5a and b indicates that tension stres-

ses S 22 in the SU-FREI are negligible compared to the B-FREI. Ten-

sion regions in the SU-FREI include the outer rubber layers at

regions where the bearing has rolled off the supports, and the rub-

ber layers next to these layers where tension stresses are localizedonly near the maximum curvature in the layer. Other than these

regions, the remainder of the SU-FREI carries compression along

local Axis 2 of the rubber elements. From Fig. 5a and b, one can

conclude that the peeling stress demand on the bond between rub-

ber and fiber reinforcement layers in a B-FREI is an order of mag-

nitude higher than that of the SU-FREI.

According to Fig. 5a and b the magnitudes of peak compression

S 22 that develop in both isolators are comparable. In the B-FREI the

maximum pressure occurs at the centre of the compression core

within the isolator and remains nearly unchanged along the isola-

tor’s height. As the lateral deformation increases, the cross section

area of the compression core decreases and the magnitude of peak

pressure increases. Accordingly, similar to conventional SREIs, in

the design of a B-FREI, one should ensure that the isolator willnot encounter Euler buckling [26] at extreme lateral deformations.

In the SU-FREI, the peak pressure S 22 is locally concentrated at the

corners of the isolator where the outer rubber layers are pressed

against the contact supports. Similar to the B-FREI, the vertical load

is carried through a compression zone within the SU-FREI. How-

ever, excluding the local regions at top and bottom corners of the

SU-FREI, the peak pressure within the body of the compression

zone in the SU-FREI is significantly lower than that of the B-FREI.

Excluding the outer rubber layers, the compression zone in the

SU-FREI is subjected to a relatively more uniform stress as opposed

to the B-FREI.

Fig. 7a–d shows the distribution of the normalized stress S 22/ pnalong the length of the 6th rubber layer (R6) located at the mid-

height of the isolator (see Fig. 1), at lateral displacements of 50%,


Fig. 5. Contour of normal stress S 22 (MPa) in the rubber layers of the isolators at lateral displacement of 200% t r (positive values indicate tension).


P M

V a

V a

P

M V b

P

P

V b

Fig. 6. Free body diagram in laterally deformed FREIs with different boundaryconditions.

1 For interpretation of color in Fig. 5, the reader is referred to the web version of this article.



7/10

100%, 150%, and 200% t r , respectively. Under a vertical load of

112 N, and given isolator width of 70 mm, the nominal vertical

pressure per unit length of both isolators is pn = 1.6 MPa. According

to Fig. 7, as the lateral displacement in the B-FREI increases the

peak compression stress increases and the length of the compres-sion region decreases. However, the peak compression stress in the

SU-FREI remains approximately constant regardless the level of

lateral displacement. Additionally, no tension develops in the rub-

ber material. Since the peak pressure at the middle of the SU-FREI

is approximately 38% lower than in the B-FREI at extreme lateral

displacement of 200% t r (Fig. 7d), the likelihood of Euler buckling

in the SU-FREI is significantly lower than in the B-FREI.

Fig. 8a and b contain the contours of normal stress S 11 in the iso-

lators corresponding to 200% t r lateral deformation. The normal

stress S 11 acts parallel to the orientation of fiber reinforcement lay-

ers. An examination of Figs. 5a and 8a indicate that the overlap re-

gion between the top and bottom supports of the B-FREI is

subjected to biaxial compression. But, the regions outside the over-

lap (yellow triangles) are under biaxial tension. In the tension re-

gion, the bond between rubber and fiber reinforcement layers

must have sufficient strength to resist large biaxial tensile stressesat extreme lateral deformations.

Similar to the B-FREI, the central region of the SU-FREI resists

biaxial compression (see Figs. 5b and 8a). According to Fig. 8a

and b, the peak values of compression stress S 11 for both isolators

are similar. However, unlike the B-FREI, the peak compression

stresses in the SU-FREI are only localized at the compressed cor-

ners of the isolator. Beyond these corners, the peak compressive

stress S 11 within the central compression region of the SU-FREI is

approximately 50% lower than that of the B-FREI. The peak tensile

stresses S 11 in the SU-FREI are significantly (40%) lower than for

the B-FREI. The maximum tensile stress is localized in a few

(a) 50% t r (b) 100% t r

(c) 150% t r (d) 200% t r

Fig. 7. Distribution of normalized stress S 22/ pn along the length of the 6th bottom rubber layer of the isolators at different lateral displacement amplitudes; (Notes: nominal

vertical pressure pn = 1.6 MPa; negative stress values indicate compression).


2.55

1.92

1.30

0.67

0.05

- 0.58

- 1.20

- 1.83

- 2.45

- 3.08

- 3.70

1.50

0.98

0.46

- 0.06

- 0.58

- 1.10

- 1.62

- 2.14

- 2.66

- 3.18

- 3.70

Fig. 8. Contour of normal stress S 11 (MPa) in the rubber layers of the isolators (positive values indicate tension).



8/10

interior rubber layers near the maximum curvature of the layer be-

yond the central compression zone (see Fig. 8b).

In a laterally deformed SU-FREI, the compressive stresses are

concentrated at the compressed corners of isolator (see Figs. 5b

and 8b). In these regions, the rubber material and bond between

rubber and the fiber reinforcement layers are subjected to large

biaxial compressive stresses. Nonetheless, damage in the corner

areas is not a common failure mode in SU-FREIs as the materialsustains a confined state of stress. In previous experimental studies

that were conducted on SU-FREIs [5–7], no damage or delamina-

tion was observed between the rubber and fiber reinforcement lay-

ers at the corners of the tested isolators. In these tests, the isolators

were repeatedly subjected to large lateral displacements.

The distribution of the normalized stresses S 11/ pn along the

length of the rubber layer R6 (see Fig. 1) is shown in Fig. 9a–d

for lateral displacement magnitudes of 50%, 100%, 150%, and

200% t r , respectively. As seen in these figures, overall, peak stress

values (compression or tension) in the SU-FREI are significantly

lower than those in the B-FREI. The peak compression in the SU-

FREI remains nearly constant at different lateral displacements.

As the lateral displacement increases the length of compression re-

gion within the rubber material decreases in both isolators.

For efficient operation of an elastomeric isolator, the shear

strain in the rubber material must be extremely large. Fig. 10aand b illustrates the contour of shear strain within the isolators

and Fig. 11a–d contains the distribution of shear strain along the

6th rubber layer (R6 in Fig. 1), at different lateral displacements.

As seen in Fig. 11 the peak values of shear strain in the rubber

material of both isolators are comparable. For a displacement of

200% t r (38 mm) the shear strain in the isolator can be approxi-

mated as (38 mm/25 mm) 1.5, which is in reasonable agreement

with the values given in Figs. 10 and 11d. According to Fig. 10a and

(a) 50% t r

(c) 150% t r

(b) 100% t r

(d) 200% t r

Fig. 9. Distribution of normalized stress S 11/ pn along the length of the 6th bottom rubber layer of the isolators at different lateral displacement amplitudes; (Notes: nominal

vertical pressure pn = 1.6 MPa; negative stress values indicate compression).


1.65

1.49

1.32

1.26

0.99

0.03

0.66

0.50

0.33

0.17

0.00

1.65

1.39

1.12

0.85

0.59

0.33

0.06

- 0.20

- 0.47

- 0.73

- 1.00

Fig. 10. Contour of shear strain in the rubber layers of the isolators at 200% t r lateral displacement.



9/10

dashed curves in Fig. 11a–d, the profile of shear strain is approxi-

mately uniform within the B-FREI. However, in the SU-FREI, uni-

form shear strains are found to occur only in the central region

of the isolator, which remains in contact with the top and bottom

supports. In the regions of the SU-FREI that are not in contact withthe supports, the shear strain is found to decrease from its peak po-

sitive value (see Fig. 10b and solid curves in Fig. 11a–d). At extreme

displacements the shear strain acquires negative values in the SU-

FREI at regions close to the ends of the rubber layer (see Fig. 11d).

The main purpose of fiber reinforcement in a FREI is to restrain

the lateral bulging of the rubber layers when the isolator is sub-

jected to vertical loads. When a FREI is loaded vertically, the rubber

layers, which are confined by the reinforcement layers, undergo

compression and the fiber reinforcement layers in turn experience

tension. At zero lateral displacement, the profile of fiber tensile

stress along the length of the reinforcement layer can be described

with a parabola with the peak value of tensile stress occurring at

the mid-length of the reinforcement layer. Lateral displacement

of an FREI can affect the fiber stress profile and alter the magnitudeand location of the peak stress developed in the fiber reinforce-

ment layer. Fig. 12 shows the variation of peak fiber tensile stress

as a function of lateral displacement. The fiber reinforcement layer

in this figure is located at the mid-height of the isolator. As can be

seen in Fig. 12, the peak fiber tensile stress in the B-FREI increases

significantly with increased lateral displacement. However, for the

SU-FREI, the influence of increased lateral displacements on the

peak fiber tensile stress is negligible.

From the FE-analysis, it is found that the fiber reinforcement

layers in the SU-FREI are generally subjected to significantly lower

tensile stresses as compared to the B-FREI. According to Fig. 12, at

200% t r displacement, the maximum fiber tensile stress at the mid-

height of the SU-FREI is nearly 40% lower than that of the B-FREI.

Also, the peak shear stress demand on the bond between fiber rein-forcement and rubber layers in the SU-FREI is lower. At 200% t r

displacement, the FE-analysis indicates that in the B-FREI the mag-

nitude of peak tensile stress for different reinforcement layers is

approximately equal. However, in comparison with the inner

layers, the peak tensile stress in the extreme reinforcement layers

of the SU-FREI is largely reduced as a result of rollover response

behavior.

7. Conclusions

Finite element analyses were conducted on two identical FREIs,

one assuming conventional bonded application (the bonded (B)-

FREI) and the other one assuming unbonded contact surfaces that

remained laterally stable (the stable unbonded (SU)-FREI). TheFE-model was found to be sufficiently accurate for the preliminary

(a) 50% t r (b) 100% t r

(c) 150% t r (d) 200% t r

Fig. 11. Distribution of shear strain along the length of the 6th bottom rubber layer of the isolators at different lateral displacement amplitudes.

Fig. 12. Influence of isolator displacement amplitude on the peak tensile stress in

the middle fiber reinforcement layer.



10/10

design of the isolators. In addition, it could be used for effective

determination of the peak stress and strain demands on isolator’s

components. Both B- and SU-FREIs were deformed up to 200% t r (t r = total thickness of rubber layers in the isolator) lateral displace-

ment under identical constant vertical loads. Results of the FE-

analyses documented the following advantages in the lateral re-

sponse of the SU-FREI compared to that of the B-FREI:

i. Significantly lower horizontal stiffness, and hence signifi-

cantly higher seismic isolation efficiency.

ii. Considerably lower stress demand on rubber material.

iii. Negligible peeling stress demand on the bond between rub-

ber and the fiber reinforcement layers.

iv. Reduced tensile stress demand on the fiber reinforcement,

and subsequently a significantly lower shear stress demand

on bond between rubber and the fiber reinforcement layers.

There is significant potential for SU-FREIs to be cost effective

elastomeric isolators for the seismic protection of ordinary build-

ings and structures. To achieve similar seismic mitigation, a SU-

FREI requires a significantly shorter operational height than a B-

FREI. This indicates considerable material saving in the isolator.

Due to the lower in-service demands on SU-FREIs, they can be fab-

ricated using a simple manufacturing process. Additional cost sav-

ing exists due to the elimination of thick steel mounting plates,

which are typically bonded to the top and bottom faces of isolators

employed in bonded applications.

Acknowledgements

The authors would like to gratefully acknowledge the support

provided by the Ontario Ministry of Research and Innovation

(MRI), and the Natural Sciences and Engineering Research Council

of Canada (NSERC). Additionally, the authors are grateful for the

support provided by MSC Incorporation.

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