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LIMITATIONS OF HEIGHT-TO-WIDTH RATIO FOR

BASE-ISOLATED BUILDINGS UNDER EARTHQUAKE

HONG-NAN LI1* AND XIANG-XIANG WU2

1 School of Civil and Hydraulic Engineering, Dalian University of Technology, Dalian, PR China2Department of Architectural Engineering, Tongji University, Shanghai, PR China

SUMMARY

The limitation of height-to-width ratio (HWR) for a base-isolated building with elastomeric rubber bearings is ofconsiderable concern to structural design engineers. Guidelines and codes on this type of building have to dealwith this issue. Nevertheless, until now, no systematical and quantitative studies have been done on this problemfor base-isolated buildings. For this reason, the main objective of this paper is to focus on investigations on thelimit of the HWR for the isolated building with rubber bearings under different conditions subjected to earth-

quake excitations. The simplified formulation is derived to explore the rules of seismic responses for the struc-tural system and some influential factors, such as the site soil conditions, seismic ground motion intensity, periodof the isolated system, period of the superstructure and layout of isolators, are studied and discussed. Accordingto the numerical results, it has been found that the effects of site soil conditions on the HWR limit values areimportant: the softer the site is, the smaller the HWR limit value is under different seismic intensities. The pre-dominant period of an isolated building also plays a considerable role in the HWR limit value, namely, the iso-lated building with a longer period may have a relatively large HWR value; and the stiffness of the superstructureaffects the HWR limit value little. Furthermore, an effective method to improve the HWR limit value is proposed.Copyright 2006 John Wiley & Sons, Ltd.

1. INTRODUCTION

Engineering structure has traditionally relied on its self-ability to dissipate the earthquake energy

through its design. In the last two decades, an increasing amount of attention has been given by sci-

entists and engineers all over the world to the mitigation of damage caused by strong earthquake

ground motion. A significant portion of this effort has been devoted to the studies and applications of

base-isolated systems to reduce structural damage. To date, there have been many successful imple-

mentations of this technique to practical engineering in New Zealand (Robinson, 1995), Japan (Izumi

1997), the USA (Naeim and Lew, 1995; Kelly, 1996), Italy (Martelli et al., 1996) and China (Li and

Huo, 2000). Earthquake experiences have also proven that base isolation is a successful seismic mit-

igation technique (Asher et al., 1995; Ishii et al., 1995).

With this technology, rubber isolated bearings used in anti-seismic applications offer a simple

method of isolation. Relatively easy to manufacture, isolation bearings are made by vulcanization

bonding of sheets of rubber to thin steel reinforcing plates. It is sufficient for the bearings to be stiffin the vertical direction, while quite flexible in the horizontal direction. Under seismic loading, they

act to shift the fundamental vibration period of an isolated structure from a short period range of earth-

quake horizontal excitation to a long period one, whereas the vertical component excitation is

*Correspondence to: Hong-Nan Li, School of Civil and Hydraulic Engineering, Dalian University of Technology, 2 LinggongRoad, Ganjingzi District, Dalian 116024, PR China. E-mail: [email protected]

THE STRUCTURAL DESIGN OF TALL AND SPECIAL BUILDINGSStruct. Design Tall Spec. Build. 15, 277287 (2006)Published online in Wiley Interscience (www.interscience.wiley.com). DOI: 10.1002/tal.295

Copyright 2006 John Wiley & Sons, Ltd.

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transmitted to the structure relatively unchanged. Thus, the horizontal seismic response of the isolated

structure is significantly less than the non-isolated structure. These have been extensively studied in

theory and experiment by many researchers (Kelly, 1986; Koh and Kelly, 1989; Fan and Ahmadi,

1992; Mohraz and Jian, 1994; Pan and Cui, 1994; Kikuchi and Aiken, 1997; Shin and Kim, 1997;

Malangone and Ferraioli, 1998; Chung et al., 1999; Huang and Hsu, 2000; Jangid and Kelly, 2001).

Recently, some design guidelines and codes including the regulations of base isolated buildings, such

as the Uniform Building Code (UBC) (ICBO, 1997), International Building Code (IBC) (ICC, 1998)

and China Design Code for Aseismic Buildings (China Standard and Code Committee, 2000), have

been applied in engineering practice.

In the case of a seismic isolated building with a large height-to-width ratio (HWR), the overturn-

ing moment at the level of the seismic isolated layer could be sufficient to exceed the overturning

resistance supplied by gravity. As a result, the edge of the building base lifts up. Although such

an uplift lasts only a very short while, it could result in disconnection of bearings from the super-

structure so as to produce internal damage of the rubber layers, even leading to destruction of the

entire building. Accordingly, special attention should be paid to the overturning of large HWR iso-

lated buildings owing to the low tensile strength of elastomeric isolators. Nevertheless, to date, no

systematical and quantitative investigation has been made concerning this problem for base-isolatedbuildings with laminated rubber bearings. Li et al. (1997) had given quantitative results of the limits

of HWR for sliding base-isolated buildings, rather than dealing with buildings with laminated rubber

bearings. Takayama and Tada (1995) provided a relationship of base shear coefficient with HWR, but

they did not systematically study the limit values of HWR with change in conditions, such as the

ground motion intensity, site condition and periods of the isolated building. Therefore, the limit of the

HWR for a base-isolated building with rubber bearings is of considerable concern to structural design

engineers.

The main objective of this paper is to focus on investigations on limits of the HWR for an isolated

building with laminated rubber bearings under different conditions subjected to earthquake excita-

tions. Some influential factors on HWR are given graphically, providing visual references to structural

design engineers.

2. DYNAMIC EQUILIBRIUM EQUATION OF BASE-ISOLATED STRUCTURE

Figure 1 shows the simplified diagram of a base-isolated building with laminated rubber bearings. Its

dynamic equilibrium equation of motion is derived as follows:

(1)

where [MS], [KS] and [CS] are the mass, stiffness and damper matrices of the superstructure,respectively; mb, cb and Rb represent the mass, damper and restoring force of the isolated bearings;

ui means the relative displacement of the ith floor with respect to the ground; xb implies the relative

displacement of the base slab to the ground; g is the ground acceleration; and {I} denotes the unit

vector.

Generally, the isolated layer consists of combinations of laminated rubber pads and absorbed-energy

dampers. At the design-permitted range, the relationship between the hysteretic loops at the various

strain levels indicates it is possible to use a linear model to analyse its response for laminated rubber

x

m

M

x

u

c I C I I C

C I C

x

u

I K I I K

K I

b

S

b b

T

S

T

S

S S

b

T

S

T

S

S

0

0

[ ]

{ } [ ]

{ }

+ + { } [ ]{ } - { } [ ]-[ ]{ } [ ]

{ }

+ { } [ ]{ } -{ } [ ]-[ ]{ }

KK

x

u

R m

MI x

S

b b b

S

g

[ ]

{ }

+{ }

= -[ ]

{ } [ ]

{ }0

0

0

278 H.-N. LI AND X.-X. WU

Copyright 2006 John Wiley & Sons, Ltd. Struct. Design Tall Spec. Build. 15, 277287 (2006)

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bearings, while dampers yield quickly after some deformation and then enter a plastic stage. Thus, the

restoring force model of an isolator can be shown in Figure 2 when the two parts work together. Mean-

while, the seismic force of the superstructure is so small that its deformation can be considered to be

within the elastic range in computation, according to previous studies (Kelly, 1993).

3. DERIVATIONS OF A SIMPLIFIED FORMULATION

The two key conditions, which determine the HWR limits for an isolated structure, are:

(1) the outermost rubber pads of the isolated layer cannot bear a tensile force;

(2) the compressive force that the outermost rubber pads bear cannot exceed their ultimate anti-pressure strength.

The reasons why the rubber pads cannot bear tensile force have two aspects: firstly, the tensile

strength of the rubber pads is relatively small; secondly, the study on such tensile strength is insuffi-

cient at present. Hence, it can be taken for certain that the compression of rubber pads must be posi-

tive; and once it becomes negative the isolators will no longer work.

Thus, if these two conditions are satisfied, the isolated building will not overturn under seismic

excitation.

LIMITATIONS OF HEIGHT-TO-WIDTH RATIO 279


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Figure 1. Simplified diagram of base-isolated building

Y Y Y

k2

k1 k2 kd

= +

X X X

(a) Rubber Isolator (b) Rubber pads (c) Damper

Figure 2. Restoring model of rubber isolator

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3.1 Forces which rubber pads bear

The forces that the rubber pads bear are comprised of two parts: one is caused by vertical load; the

other is produced by horizontal seismic load, in which the first part is uniform, while the second part

is uneven, as shown in Figure 3.

The above two conditions can be expressed by the following equations:

(2)

where [f] is the design compressive strength of the rubber bearing, usually 1520 Mpa;fg andfm are

depicted in Figure 3.

3.2 Force on isolator originating from horizontal seismic load

The overturning moment of an isolated structure due to horizontal seismic load is given by

(3)

where mi is the mass of the ith storey; i is the relative acceleration of the ith floor with respect to

ground; and hi denotes distance from the ith floor to the pad bottom. Let Kvl represent the vertical stiff-

ness of the lth rubber pad, then

(4)

whereEl,Al and hl are Youngs modulus, cross-sectional area and height of the lth rubber pad, respec-tively. Thus, the rotational stiffness of an isolated layer is expressed by

(5)

where ris the total number of pads within half the width of the structural plan; bl implies the distance

from the lth pad to centre of the isolated layer. Inserting Equation (4) into Equation (5), one can obtain

K K bvll

r

lq ==2

1

2*

KE A

hvl

l l

l

=

Mov m u x hi

n

i g i= +( )1

f f f g m+ [ ]

f fg m- 0



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Figure 3. External forces of isolation system

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(6)

In practical engineering, the dimensions of most of rubber pads are normally the same. Accord-

ingly, their vertical stiffness may be assumed to be the same in the following derivations of formula-tions for the sake of simplification. Accordingly, Equation (5) can be rewritten as

(7)

With the known overturning moment and rotational stiffness of isolators, the rotational angle of the

isolated layer can be expressed by

(8)

In general, rubber pads are uniformly distributed in the structural plan of a practical project, and

can then be further simplified as (if the rubber pads are not uniformly distributed, a equivalent

formula can also be obtained easily)

(9)

in which aa is obtained from the following equation:

(10)

Now, Equation (8) is rewritten as

(11)

Using D andfm to represent the vertical deformation and force of the outermost rubber pads, onecan then obtain

(12)

and

(13)f KMov

aa bm vl= =*

* *D

2

D = = =qbvl l vl

bMov

K aa bb

Mov

K aa b*

* * **

* * *2 22

qq

b

vl l

Mov

K

Mov

K aa b= =

2 2* * *

aai

ri

r

=

=

2

1

b aa bll

r

2

1

2

= = *

bll

r

2

1=

qq

b

vl l

l

r

Mov

K

Mov

K b

= =

=2 2

1* *

K K bvl ll

r

q ==2 2

1

* *

Kb E A

h

l l l

ll

r

q ==2

2

1

*



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3.3 Forces on isolators originating from gravity and vertical seismic load

If there is vertical seismic ground excitation, the total vertical load that the structure has to bear

includes two parts: the gravity of the structure itself and the vertical inertial force. Thus the total ver-

tical force exerted on the structure equals

(14)

Thus, the internal force,fg, of each rubber isolator caused by vertical external loads is given by

(15)

3.4 Computational formulation for HWR limit

Substitution of Equations (13) and (15) in the first condition of Equation (2) leads to

(16)

It is then easy to obtain from Equation (16)

(17)

On the other hand, substituting Equations (13) and (15) into the second condition of Equation (2),one can obtain

(18)

in which

Choosing the smaller of Equations (17) and (18), one may obtain the limitation value of half the

width for the base-isolated building plan expressed by

(19)

Finally, the computational formulation of the HWR limit for the base-isolated building is given by

(20)

whereHandB are the height and width of the base-isolated building, respectively.

D[ ] =[ ]

=H

b

H

B2

b b b[ ] = { }min ,1 2

tt fG

r= [ ] -

2 *.

bMov

aa tt 2

2

* *

br Mov

G aa1

*

*

G

r

Mov

aa b2 2* * *

fG

rg =

2

G m g y yii

n

g= + +( )=

1



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4. ANALYSES OF INFLUENCED FACTORS ON HWR LIMIT VALUES

A time-history calculational method, i.e., the Newmarkbmethod (with coefficients a= 1/2; b= 1/4),is applied to quantitatively analyse the factors influencing the HWR limit of the base-isolated build-

ing with rubber bearings. Theses factors include the site soil conditions, seismic ground motion inten-

sity, period of the isolated system, period of the superstructure and layout of isolators. In computations,

different structures have been chosen within the range of periods of the practical mediumlow-storey

buildings.

In order to explore the effects of different site soil conditions on the HWR limits of the isolated

building, nine seismic acceleration records have been employed, listed in detail in Table 1, which

belong to three different types of soil site conditions: hard site, medium site and soft site. Each case

of HWR limit at the different sites is calculated with three seismic acceleration inputs. The mean value

and variation coefficient (standard variance/mean value) of the results will be given for each site.

4.1 Effect of site condition and seismic intensity

In this section, the analysis is based on a base-isolated building, whose predominant period is 14s,

while 065 s before the corresponding building was isolated. Here, the effect of peak accelerations on



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Figure 4. Effect of site condition and seismic peak acceleration

Table 1. Earthquake acceleration records

Site type Earthquake name Magnitude Time Record place Acceleration peak (gal)

Solid site Tangshan aftershock 63 1976.11.16 Qian An 11891Landers 75 1992.6.28 Baker Fire 10558

Landers 75 1992.6.28 Fort Irwin 11985Medium site Imperial Valley 67 1940.5.18 El Centro 3417

Kern County 77 1952.7 Taft Lincoln school 1527ImperialValley 56 1951.1.23 El Centro 3035

Soft site Tangshan aftershock 71 1976.11.16 Tianjin Hospital 10418San Fernando 66 1971.2.9 Navy Laboratory 2591San Fernando 66 1971.2.9 University Avenue 5636

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the HWR limit values is studied at the three different sites. The mean values of the results are shown

in Figure 4, in which the variation coefficients are 022 for the hard site, 0136 for the medium site,

and 025 for the soft site. It is noted from Figure 4 that:

(1) The limit values of the HWR for a base-isolated building decrease with the increase of seismic

ground accelerations in all cases. Yet, the limit values on the hard site are much larger than thosefor the building on the other sites, and the softer the site, the smaller is the limit value.

(2) It is worth noting that the limit value has gone down to 10 on the soft site when the intensity of

ground motion is over 07 g.

(3) According to the calculated results, relatively tall buildings may be built on the hard site.

4.2 Effect of natural period of isolated system

The natural period of an isolated system mainly reflects its stiffness. In computations, the stiffness of

an isolated system was changed by adjusting the number of the rubber bearings within the permitted

range of compressive stress from 11Mpa through 70Mpa. At the same time, the maximum peaks of

the nine seismic records are all tuned to be 04 g, similar to midstrong motion earthquake. The data-

processing method here is the same as above. The curves of the mean values of calculated results varyas shown in Figure 5, in which the variation coefficients are 022 for the hard site, 026 for the medium

site and 046 for the soft site.

It can be seen from Figure 5 that with the increase of the period of the isolated system, i.e., decrease

in stiffness of the isolated system, the limit values of the HWR gradually increase, but the curve rises

quite slowly, inclining almost to a horizontal line on the soft site. Concerning engineering design,

when the periods of isolated systems vary from 11 to 23, the HWR limit values increase from 8 to

170 on hard site, from 20 to 50 on medium site and from 10 to 20 on the soft site.

As a consequence, the limit values can be further raised according to the above results if the higher

compressive ability of the rubber bearings is made full use of. On the other hand, the stiffness of the

isolated layer cannot be designed to too small a degree in practical engineering in order to maintain

the stability of the system.



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Figure 5. Effect of natural period of isolation system

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4.3 Effect of natural periods of superstructure

Also, the natural period of the superstructure mainly reflects its stiffness. In this section, the effect of

stiffness of the superstructure on the limit value of the HWR is studied based on the preceding method

and earthquake records. The curves of the HWR limit values vary with the period of the superstruc-

ture as depicted in Figure 6, in which the variation coefficients are 021 for the hard site, 005 for the

medium site and 026 for the soft site.

As shown in Figure 6, the stiffness of the superstructure affects the HWR limit values little forbuildings of common height below around 1012 storeys almost to the period of 08 s in the figure.

The explanation is that although the periods of the superstructures are within the range of

mediumlow-storey structures (0408s), they are still much more rigid than those after isolation,

namely the predominant periods of the superstructures are far from those of the isolated systems. That

is to say, what plays a significant role is the predominant period determined by the stiffness of the iso-

lated system, not by stiffness of the superstructure. Therefore, the variation in stiffness of the super-

structure cannot produce a large effect on the HWR limit value of an isolated building.

4.4 Effect of layout of rubber bearings

If the number of rubber bearings along the direction of width of a structural plan is decreased, typi-

cally for instance from 4 to 2, with the total number fixed, the HWR limit values increase significantly.Figure 7 shows the varied relations of the HWR limit values with two and four pads against the periods

of the isolated system on the medium site, in which the variation coefficients are 022 for the case

with four bearings and 020 for the case with two bearings. Also, there are the same rules for isolated

buildings on other sites. It is obvious that the reduction in number of bearings along with the build-

ing width direction can effectively improve the HWR limit value; that is to say, to place the rubber

pads as near an edge of the structural plan as possible will lead to a considerable increase in the HWR

limit value.



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Figure 6. Effect of natural periods of superstructure

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5. CONCLUSIONS

A simplified formulation has been derived to compute the HWR limit values of an isolated building

with rubber bearings subjected to earthquake ground motions. Some influential factors, such as the

site soil conditions, seismic ground motion intensity, period of the isolated system, period of the super-

structure and layout of isolators, have been studied and discussed. According to the numerical results,

some conclusions with practical significance may be drawn as follows:

(1) The effects of site soil conditions on the HWR limit values are important. The softer the site, the

smaller is the HWR limit value under different seismic intensities. For instance, if the earthquake

peak acceleration is 063g, the HWR value reaches 10 for the isolated building on the hard site,

while the value is 25 on a mid-hard site and only 10 on soft site. Consequently, it is dangerous

to build an isolated structure on the soft site subjected to a strong motion earthquake if it has a

large ratio of height to width.

(2) The predominant period of an isolated building plays a considerable role in the HWR limit value.

The isolated building with a longer period may have a relatively large HWR value, namely, the

height of an isolated system with a smaller stiffness could be relatively taller within the design-

permitted range of its isolated layers stiffness.

(3) The stiffness of the superstructure affects the HWR limit value little.

(4) From the computational and analytical results, it has been found that a reasonable layout of rubberbearings will improve the HWR limit value; that is, the HWR value when the rubber pads are laid

along the structural plan sides is larger than when they are not.

ACKNOWLEDGEMENT

This research is supported by the Department of Construction of Liaoning Province. This support is

greatly appreciated.



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Figure 7. Effect of layout of rubber bearings

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