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22.2 Mechanisms of Base-Isolated Systems

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Figure 22.1 shows a simplified model for a

structural system subjected to a support motion.

For this single-DoF system, the equation of

motion can be written as

mx€ þ cx_ þ kx ¼ 2mx€g ð22:1Þ

where m denotes the mass, c the damping, k the

stiffness, x the displacement of the system, and x€g

the ground acceleration. By assuming the system

to be linearly elastic, the response xðtÞ can be

obtained using Duhamel’s integral (also see

Chapter 2 and Chapter 14), as

xðtÞ ¼

ðt

x€gðtÞe2zVðt2tÞ sin Vdðt 2 tÞdt

ð22:2Þ

where the natural angular frequency, V; damped

natural frequency, Vd; and damping ratio, z; of the system are defined as follows:

V ¼

ffiffiffiffi

ð22:3aÞ

Vd ¼ V

ffiffiffiffiffiffiffiffi

1 2 z2

ð22:3bÞ

z ¼

2mV ð22:3cÞ

Correspondingly, the natural period, T; and damped period, Td; of the structure are

T ¼

V ¼ 2p

ffiffiffiffi

ð22:4aÞ

FIGURE 22.1 Model of a single-DoF system.

22-4 Vibration and Shock Handbook

Td ¼

Vd ¼

T ffiffiffiffiffiffiffiffiffi

1 2 z 2

p ð22:4bÞ

For a given support acceleration, x€g; the displacement,

x; and acceleration, x€; of the single-DoF

system can be related to the natural period, T; and

damping ratio, z; of the system. Thus, for a

specific earthquake, by first selecting a damping

ratio, z; and using Equation 22.2, one can

compute the peak displacement x; for a structure

with a period of vibration, T; with given values of

m; c; and k: Repeating the above procedure for a

wide range of periods, T; while keeping the

damping ratio, z; constant, one can

obtain response curves similar to those shown in

Figure 22.2. By varying the damping ratio, z; one

can construct the displacement response spectra

and pseudo-acceleration response spectra for all

single-DoF structures under a given earthquake,

as schematically shown in Figures 22.2 and 22.3,

respectively.

A general impression that is gained from

Figure 22.2 and Figure 22.3 is that a structure

with a shorter natural period has less displacement

response when subjected to an earthquake,

but it also has a larger acceleration response.

Specifically, let us consider a structure of a

constant damping ratio, z; with its period

increased from T1 to T2: As can be observed

from the figures, the displacement of

the structure increases from D1 to D2; while

the acceleration decreases from A1 to A2: Such a

feature is known as the period shift effect. On the other hand, by increasing damping ratio of the

structure, the displacement of the structure decreases significantly, as can be seen from Figure 22.2.

The same is also true with the acceleration response, as can be seen from Figure 22.3. Moreover, a

larger damping ratio also makes the structure less sensitive to the variation in ground vibration

characteristics, as indicated by the smoother response curves for structures having higher damping

ratios, in both figures. From the aforementioned two response spectra, one observes that the

philosophy of base isolation is to lengthen the vibration period of the structure to be protected, using

base isolators of some kind, by which the earthquake force transmitted to the structure can be greatly

reduced. In the meantime, some additional damping must be introduced on the base isolators in order

to control the relative displacements across the base isolators with tolerable limits.

To fulfill the function of lengthening the period of vibration of the structure to be protected, the base

isolators that are inserted between the structure and its foundation must be flexible in the horizontal

direction, but stiff enough in the vertical direction so as to carry the heavy loads transmitted from the

superstructure. With such devices, the natural period of vibration of the structure will be significantly

lengthened and shifted away from the dominant frequency range of the expected earthquakes.

The following is a summary of the fundamental features of four types of isolators frequently used in

engineering practice.

Period Shift

Period (s)

Acceleration (g)

T1 T2

FIGURE 22.3 Schematic of pseudo-acceleration

response spectra.

Increasing

Damping

Displacement (cm)

Period (s)

T1 T2

FIGURE 22.2 Schematic of displacement response

spectra.

Structure and Equipment Isolation 22-5

22.2.1 Elastomeric Isolation System

Elastomeric bearing is the type of base isolator most

commonly known to researchers and engineers

working on base isolation. It is usually composed

of alternating layers of steel and hard rubber and,

for this reason, it is also known as the laminated

rubber bearing. This type of bearing is stiff enough

to sustain the vertical loads, yet flexible under the

lateral forces. The ability to deform horizontally

enables the bearing to reduce significantly the

structural base shear transmitted from the ground.

While the major function of elastomeric bearings

is to reduce the transmission of shear forces to the

superstructure by lengthening the vibration period

of the entire system, they must also provide sufficient rigidity under vertical loads. Let us consider a

structure installed with elastomeric bearings, which is subjected to a support acceleration, x€g; as in

Figure 22.4. By representing the isolated structure as a single-DoF system, based on the assumption that

the superstructure is rigid in comparison with the stiffness of the elastic bearings, the equation of motion

for the entire system can be written as

m 0

0 mb

" #

x€

x€b

( )

c 2c

2c c þ cb

" #

x_b

( )

k 2k

2k k þ kb

" #

( )

¼ 2

mx€g

mbx€g

( )

ð22:5Þ

where m; c; and k denote the mass, damping, and stiffness of the superstructure, respectively, mb; cb; and

kb denote the mass, damping, and stiffness of the base raft, respectively, and x and xb denote the

displacements of the superstructure and the base, respectively.

In reality, the reduction in the seismic forces transmitted to a superstructure through the installation of

laminated rubber bearings is achieved at the expense of large relative displacements across the bearings. If

substantial damping can be introduced into the bearings or the isolation system, then the problem of

large displacements can be alleviated. It is for this reason that the laminated rubber bearing with a central

lead plug inserted has been devised (Yang et al., 2002). To simulate the dynamic properties of the lead –

rubber bearing (LRB) system, an equivalent linearized system has been proposed, for which the equation

of motion is

m 0

0 mb

" #

x€

x€b

( )

c 2c

2c c þ ceq

" #

x_b

( )

k 2k

2k k þ keq

" #

( )

¼ 2

mx€g

mbx€g

( )

ð22:6Þ

where ceq and keq respectively represent the equivalent linearized damping and stiffness coefficients of the

LRB system. The dynamic behavior of a structure – equipment system isolated by elastomeric bearings

with linearized damping and stiffness coefficients, when subjected to harmonic and earthquake

excitations, will be investigated analytically and numerically, respectively, in Section 22.3.

22.2.2 Sliding Isolation System

Another means for increasing the horizontal flexibility of a base-isolated structure is to insert a sliding or

friction surface between the foundation and the base of the structure. The shear force transmitted to the

superstructure through the sliding interface is limited by the static frictional force, which equals the

product of the coefficient of friction and the weight of the superstructure. The coefficient of friction is

usually kept as low as is practical. However, it must be high enough to provide a frictional force that can

sustain strong winds and minor earthquakes without sliding. Since the sliding system has no dominant

c c b

FIGURE 22.4 Model for base-isolation systems with

elastic bearing.

22-6 Vibration and Shock Handbook

natural period, it is generally frequency-independent

when the structure is subjected to earthquakes

with a wideband frequency content. As

mentioned previously, when a sliding structure is

subjected to a ground motion, transitions may

occur repeatedly between the sliding and nonsliding

phases. To take into account such a phase

transition, Yang et al. (1990) proposed the use of a

fictitious spring between the structural base raft

and the underlying ground to simulate the static –

dynamic frictional force of the sliding device. With

reference to Figure 22.5, the equation of motion

for the structure with sliding base can be written as

follows:

m 0

0 mb

" #

x€

x€b

( )

c 2c

2c c

" #

x_b

( )

k 2k

2k k

" #

( )

¼ 2

mx€g

mbx€g

( )

ð22:7Þ

where kf is the stiffness of the fictitious spring and the frictional force, fr; can be represented as

fr ¼

kf ðxb 2 xb0Þ for non-sliding phase;

^mðm þ mbÞg for sliding phase

(

ð22:8Þ

with xb0 indicating the initial elongation of the fictitious spring in the current nonsliding phase, m the

coefficient of friction, and g the acceleration of gravity. The fictitious spring concept will be incorporated

in the analysis of sliding structures in Section 22.4 of this chapter, when considering both harmonic and

seismic excitations.

22.2.3 Sliding Isolation System with Resilient Mechanism

One particular problem with a sliding structure is

the occurrence of residual displacements after

earthquakes. To remedy such a drawback, the

sliding surface is often made concave, so as to

provide a recentering mechanism for the isolated

structures. This is the idea behind the friction

pendulum system (FPS), shown in Figure 22.6,

which utilizes a spherical concave surface to

produce a recentering force for the superstructure

under excitations. To guarantee that a sliding

structure can return to its original position, other

mechanisms, such as high-tension springs and

elastomeric bearings, can be used as an auxiliary

system for providing the restoring forces. Previously,

the sliding isolation systems have been

successfully applied in the protection of important

structures, such as nuclear power plants, emergency

fire water tanks, large chemical storage tanks, and

so on, from the damaging actions of severe

earthquakes.

To improve the performance of sliding isolators under strong earthquakes, Mostaghel (1984)

and Mostaghel and Khodaverdian (1987) proposed the resilient-friction base isolator (RFBI) for

FIGURE 22.5 Model for base-isolation systems with

sliding support.

Bearing

Superstructure

Concave Sliding

Surface

FIGURE 22.6 Friction pendulum system.

Structure and Equipment Isolation 22-7

controlling the transmission of shear force to the

superstructures, while keeping the residual displacements

within an allowable level. The RFBI

device is basically made of a central rubber core

and Teflon-coated steel plates, and offers a friction

resistance for keeping the system in the nonsliding

mode under wind excitations and small earthquakes,

and a restoring force by the rubber

ingredient for limiting the maximum sliding

displacements. The equation of motion for a

structure installed with RFBI, as shown in

Figure 22.7, can be written as

m 0

0 mb

" #

x€

x€b

( )

c 2c

2c c þ cb

" #

x_b

( )

k 2k

2k k þ kb

" #

( )

¼ 2

mx€g

mbx€g

( )

ð22:9Þ

The interfacial frictional force, fr; existing in the RFBI and appearing in Equation 22.9 serves as the

outlet for energy dissipation. The behavior of a structure – equipment system supported by sliding

isolators with resilient mechanism subjected to both harmonic and earthquake excitations will be

investigated in Section 22.5.

22.2.4 Electricite de France System

To limit effectively the acceleration of base-isolated

structures and internal secondary systems, such as

those of nuclear power plants, when subjected to

strong earthquakes, the Electricite de France (EDF)

system was proposed by Gueraud et al. (1985). The

design concept of an EDF system is to arrange

the elastomeric bearing and sliding device at the

base of a structure in series. For low-level ground

motions, the EDF system will behave as an elastomeric

bearing and return to the original position

after support motions, while for strong earthquakes,

the EDF system will behave as a sliding

device. The EDF system may have a residual

displacement after some major earthquakes. Because of the sliding mechanism of the EDF system, the

maximum horizontal acceleration of the superstructure is kept within a certain range (Gueraud et al.,

1985; Park et al., 2002), while the shear force transmitted to the superstructure through the frictional

interface is smaller than the static frictional force. For the mathematical model shown for the EDF system

in Figure 22.8, the equations of motion for the nonsliding and sliding phases can be written as

(a) nonsliding phase:

mx€

mbx€b

8>><

>>:

9>>=

>>;

c 2c 0

2c c 0

0 0 cEDF

664 3 775

x_b

x_EDF

8>><

>>:

9>>=

>>;

k 2k 0

2k k þ kf 2kf

0 2kf kf þ kEDF

664

775

xEDF

8>><

>>:

9>>=

>>;

2mx€g

2mbx€g

8>><

>>:

9>>=

>>;

ð22:10aÞ

FIGURE 22.7 Model for base-isolation systems with

RFBI device.

kEDF

cEDF

xg x

xEDF

FIGURE 22.8 Model for base-isolation systems with

EDF device.

22-8 Vibration and Shock Handbook

(b) sliding phase:

mx€

mbx€b

8>><

>>:

9>>=

>>;

c 2c 0

2c c 0

0 0 cEDF

664

775

x_b

x_EDF

8>><

>>:

9>>=

>>;

k 2k 0

2k k 0

0 0 kEDF

664

775

xEDF

8>><

>>:

9>>=

>>;

2mx€g

2mbx€g 7mðm þ mbÞg

^mðm þ mbÞg

8>><

>>:

9>>=

>>;

ð22:10bÞ

where cEDF and kEDF; respectively, denote the damping and stiffness of the EDF system, and xEDF denotes

the displacement of the system.

22.2.5 Concluding Remarks

To mitigate the transmission of earthquake forces to a structure, and the potentially earthquakeinduced

damage to the equipment attached to the structure, base isolation is an effective structural

design philosophy. With the installation of base isolators, the natural period of vibration of the

structure will be significantly lengthened and shifted away from the dominant frequency range of the

expected earthquakes. In accordance, the earthquake force transmitted to the structure can be

significantly reduced. In this section, the mechanisms of four types of base isolator frequently used in

engineering practice are introduced. Since the base isolators, such as the elastomeric bearings or

sliding isolations, have relatively flexible stiffness in the horizontal direction, the occurrence of

residual displacements after earthquakes may cause certain problems on the structure to be protected.

To remedy such a drawback and to further guarantee that a base-isolated structure can return to its

original position, the RFBI is implemented for controlling the transmission of shear force to the

superstructure, while keeping the residual displacement within an allowable level. On the other hand,

to limit the acceleration level of internal secondary systems housed in a base-isolated structure under

strong earthquakes, such as those of the nuclear power plants, the EDF system can be used as an

alternative device for base isolation, even though some residual displacements may be induced after

the earthquakes.

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