22.6 Issues Related to Seismic Isolation Design

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22.6.1 Design Methods

Having been developing for over 30 years, the technology of seismic isolation has matured. Many

earthquake-prone countries, including the U.S., Japan, New Zealand, Taiwan, China, and European

countries, have developed their own design codes, regulations, or guidelines (Fujita, 1998; Kelly, 1998;

Martelli and Forni, 1998). Although most of the codes were developed based on the theory of structural

dynamics, the design details outlined in the codes vary from one country to another. While a

comprehensive explanation of the various design codes is not the purpose of this section, a brief overview

of the concept underlying the design codes will be given. For more details, interested readers should refer

to each code or to the books by Naeim and Kelly (1999) or Skinner et al. (1993). The design concept

introduced herein is based on the series of Uniform Building Code (UBC, 1994, 1997).

Given the fact that base isolation devices are diverse, most design codes or regulations have been

written in such a way as not to be specific with respect to the isolation systems. For instance, in the

UBC (1997), no particular isolation system is identified as being acceptable; rather, it requires that every

isolation system is stable for required displacement, has properties that do not degrade under repeated

cyclic loadings, and provides increasing resistance with increasing displacement.

The design methods for base isolation can be classified as static analysis and dynamic analysis. The

static analysis is applicable for stiff and regular buildings (in vertical and horizontal directions) that are

constructed on soil of a relatively stiff condition. On the other hand, dynamic analysis is usually required

for isolation systems with an irregular or long-period superstructure, or constructed on relatively soft

soils. For a sophisticated design case, static analysis may be used in the preliminary design phase in order

to draft or initiate the isolation design parameters, while dynamic analysis is employed in the final design

phase for tuning or finalizing the design details of the isolation system. For simple design cases, static

analysis alone is considered sufficient.

22.6.2 Static Analysis

For static analysis, a number of formulas have been specified in the design codes, so that engineers can

easily calculate the following design parameters (shown in the design sequence): maximum isolator

displacement, D; isolator total shear, Vb; total base shear, Vs; of superstructure; and seismic load, Fi;

applied on each floor. These formulas were usually derived based on a simplified isolation model,

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assuming the isolation system can be linearized (even though most isolation systems are nonlinear) and

the superstructure can be modeled as a rigid block. Such a simplified model is considered reasonable,

since the displacements of an isolated structure are concentrated at the isolation level, which implies that

the superstructure behaves as a rigid block. Based on such a model, only the first vibration mode with the

superstructure treated as a rigid body has been considered in deriving the formulas. This explains why

static analysis is suitable only for rigid and regular structures.

22.6.2.1 Computation of Maximum Isolator Displacement

An isolation design by static analysis usually starts

with the calculation of the maximum isolator

displacement, D; which depends on several factors:

D ¼ DðZ; N; S; Tef ; zefÞ ð22:68Þ

where Z denotes the earthquake zone factor, N the

near-fault factor, S the soil condition factor, Tef

the effective isolation period, and zef the effective

isolation damping. For example, in the UBC

(1994), the formula derived from the constantvelocity

spectra over the period range of 1.0 to

3.0 sec has been given in the following form:

D ¼

0:25ZNSTef

B ð22:69Þ

where B is the damping factor, given as

B ¼ Bðzef Þ < 0:25ð1 2 ln zefÞ ð22:70Þ

In the above equations, the factors Z; N; and S depend on conditions of the construction site of the isolated

structure; however, the factors Tef and zef depend solely on the properties of the chosen isolation system.

The factors Tef and zef are called the “effective” period and damping of the isolation system, because they

are frequently obtained by linearizing a nonlinear isolation system. The way to linearize an isolation system

will be explained below, along with the formulas for computing Tef and zef : Suppose that for a nonlinear

isolation system, the force-displacement relation (hysteresis loop) obtained from a component test is

shown in Figure 22.57. The effective stiffness of this isolation system can be computed by

Kef ¼

Fþ 2 F2

2D ð22:71Þ

where Fþ and F2; respectively, denote the largest positive and negative forces in the test. After the linearized

stiffness is obtained from Equation 22.71, the corresponding effective quantities Tef and zef can be

computed from the dynamic theory for a single DoF oscillation system; that is

Tef ¼ 2p

ffiffiffiffiffiffiffi

W

Kef g

s

ð22:72Þ

zef ¼

1

2p

A

Kef D2

􀀏 􀀐

ð22:73Þ

where W is the structural weight, g the gravitational acceleration, and A the total area enclosed by the

hysteresis loop in Figure 22.57.

Displacement

Force

F

−

−D

D

F +

Kef

FIGURE 22.57 Typical force – displacement diagram

for an isolation system.

Structure and Equipment Isolation 22-51

© 2005 by Taylor & Francis Group, LLC

22.6.2.2 Computation of Maximum Isolator Shear

After the maximum isolator displacement, D; is obtained, the maximum isolator shear, Vb; can be

estimated by the following formula:

Vb ¼ Kef D ð22:74Þ

Obviously, the above equation represents an equivalent static force exerted on the isolation system, when

the system is displaced by an amount, D: In some design codes, Vb has also been referred to as the design

force beneath the isolation system.

22.6.2.3 Computation of Total Base Shear

The total base shear, Vs; of the superstructure can be given as

Vs ¼

Kef D

RI ð22:75Þ

where RI is a reduction factor (ductility factor) to account for structural ductility, which will be

developed when the structure is subjected to an earthquake with intensity above the design level. In some

codes, Vs has also been referred to as the design force above the isolation system.

22.6.2.4 Computation of Shear Force for Each Floor

Having computed the above total base shear, Vs; a formula is employed to distribute this total shear to

each floor of the isolated structure. For instance, in the, UBC (1997), the shear force, Fi; exerted on each

floor is computed by

Fi ¼ Vs

hiwi

Xn

j¼1

hjwj

ð22:76Þ

where n denotes the number of floors, wi the weight of the ith floor, and hi the height of the ith floor

above the isolation level. Note that the sum of Fi ði ¼ 1 to nÞ must be equal to Vs:

The general procedure for static analysis was illustrated in Figure 22.58. Once the design parameters,

D; Vb; Vs; and Fi; are all determined according to the code, they can be used in the detailed design of

structural elements as well as of isolator elements. Nevertheless, in most applications, because the test

data of the isolation system may not be available in the beginning of design, the values of Kef ; Tef; and zef;

which are required in computing D; are not known to the designer. If this is the case, the design can begin

with assumed values of Kef ; Tef ; and zef ; which may be obtained from experience or previous test data on

similar isolators. After the preliminary design is completed, prototype isolators will be fabricated and

tested. The actual values of Kef ; Tef ; and zef ; obtained from the tests will be used in the aforementioned

code formulas to update the design parameters D; Vb; Vs: Moreover, one observes from Equation 22.71

that the linearized isolator stiffness, Kef ; is a function of the design parameter, D; itself, and so are Tef ; and

zef ; obtained from Equation 22.72 and 22.73. In order to obtain Kef ; as well as Tef ; and zef ; an initial guess

of D is required at the beginning of design. As a result, the design procedure may have to be repeated

iteratively until the difference between the final value of D and the value D0 computed in the last iteration

is less than a preset tolerance. Such an iterative process is illustrated in Figure 22.58.

22.6.3 Dynamic Analysis

The dynamic analysis may be carried out in one of the two forms: response spectrum analysis and timehistory

analysis. Response spectrum analysis usually involves application of the concepts of response

spectrum and modal superposition, and so on. Since these concepts primarily come from the dynamics

of linear systems, the response spectrum analysis is only suitable for isolation systems with linear

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properties. For the case when the isolation system or the superstructure appears to be highly nonlinear, a

time-history analysis is generally required.

Because dynamic analysis depends generally on the usage of computer programs, relatively few

formulas have been given in the dynamic analysis sections of design codes. Nevertheless, for a successful

time-history analysis, the designer must prepare the following three basic elements: (1) a set of

representative input ground motions, (2) accurate mathematic models for isolators and superstructures,

and (3) a computer program that is capable of performing the nonlinear time-history analysis. These

three elements are explained below.

22.6.3.1 Input Ground Motions

The response of an isolated system depends greatly on the chosen input ground motions, which are

usually expressed in the form of ground accelerations. Each ground motion is called one record event.

(D0 = D computed in

previous design

iteration)

Given structure properties and site condition: Z, N, S

Assume isolator parameters: Kef, Tef, zef

Choose smaller Tef Compute D by design code

(stiffer system)

D too large

Compute Vb and Vs by design code

Order or fabricate and test prototype isolators

Vb or Vs too large

Choose larger Tef

(softer system)

Compute actual Kef, Tef, zef

Computer D by design code using actual

Kef, Tef, zef

|D−D0| < tolerance

Finalize Kef, Tef, zef and D, Vb, Vs

End

Design iteration

yes

no

yes

no

yes

Start

no

FIGURE 22.58 Flow chart of static analysis.

Structure and Equipment Isolation 22-53

© 2005 by Taylor & Francis Group, LLC

The chosen events must be representative of the site conditions and soil characteristics. Design codes

usually specify the minimum number of events required for analysis. Each ground motion event must

be scaled so that all events are compatible with each other and also with the code specified target

spectrum. In the UBC (1997), the scaling factor for each event is obtained in response spectra, and then

applied to the time domain of the record data. In particular, site specific ground motions are required

in the UBC for the following cases: (1) an isolated structure located on a soft soil, (2) an isolated

structure located within certain distance (e.g., 10 km) of an active fault, (3) an isolated structure with

very long period of vibration (e.g., greater than 3 sec).

22.6.3.2 Mathematic Models

Before any time-history analysis can be carried out, a mathematic model that can accurately reflect

the mechanical behavior of the isolation system and the superstructure must be constructed. If the

isolation system is nonlinear, the nonlinear parameters must be identified so that the constructed

mathematic model can correctly describe the hysteretic behavior of the isolation system. In

many cases, the isolation system is assumed to be nonlinear, but the superstructure linear.

Establishing an accurate mathematic model is curial for obtaining reliable results in a time-history

analysis.

22.6.3.3 Computer Programs

In practice, the task of time-history analysis is executed through the use of a computer program. The

mathematic model properties mentioned above will be input to the program for analysis. The computer

program selected should be capable of simulating the three-dimensional behavior of structures with

selected nonlinear elements. To serve the purpose of isolation design and analysis, several structural

analysis programs running on the platform of personal computers have been developed for easy access.

Some of the widely used programs include (but are not limited to): ETABS (ETABS, 2004), SAP-2000

Nonlinear (SAP, 2000), and 3D-BASIS (Nagarajaiah et al., 1993). Most of these programs provide a set of

imbedded mathematic models for the widely used isolator elements with linear or nonlinear parameters.

The designers using these programs can easily build up the mathematic model for the isolated structure

considered, specify the parameters of the isolator elements selected, and execute a nonlinear time-history

analysis on a personal computer.

22.6.4 Concluding Remarks

In this section, the design concept of seismic isolation for structures was briefly reviewed. The design

methods can be based either on static or dynamic analysis. The fundamental issues that should be

considered in each design method were highlighted, along with some relevant formulas for computing

the relevant parameters. It is believed that, with the concepts and procedures presented in this section, the

readers should have a general knowledge of the procedure for base isolation design of structures and

equipment.

Acknowledgments

The authors are indebted to the graduate student, Cheng-Yan Wu, at the Department of Construction

Engineering, National Kaohsiung First University of Science and Technology, for preparing some of the

graphs presented in this chapter.

22-54 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

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