CRITICAL EXCITATION METHODS FOR IMPORTANT
STRUCTURES
Izuru Takewaki 1
1

Department of Urban & Environmental Engineering, Kyoto University, Kyoto 615-8540, Japan

E-mail: takewaki@archi.kyoto-u.ac.jp

Keywords: Critical excitation, important structure, robust design, earthquake resistant design.
ABSTRACTThe role of critical excitation methods in the design of important structures is discussed
extensively. The problem of critical excitation for time-dependent performance indices istreated first. Then the problem of critical excitation for earthquake input energy is discussed.
The application of various critical excitation methods to high-rise buildings, base-isolatedbuildings, nuclear reactor facilities, etc. is presented.
1.

INTRODUCTION

There are various buildings in a city. Each building has its own natural period and its original
structural properties. When an earthquake occurs, a variety of ground motions are induced in
the city. The combination of the building natural period with the predominant period of theinduced ground motion may lead to disastrous phenomena in the city. Many past earthquake
observations demonstrated such phenomena repeatedly. Once a big earthquake occurs, somebuilding codes are upgraded. However, it is true that this repetition never resolves all theissues and new serious problems occur even recently. In order to overcome this problem, a
new paradigm has to be posed. To the author’s knowledge, the concept of "criticalexcitation" [1, 2] and the structural design based upon this concept can become one of suchnew paradigms [3, 4].
Earthquake inputs are uncertain even with the present knowledge and it does not appear
easy to predict forthcoming events precisely both in time-history and frequency contents [5,6]. For example, recent near-field ground motions (Northridge 1994, Kobe 1995, Turkey
1999 and Chi-Chi, Taiwan 1999) and the Mexico Michoacan motion 1985 have some peculiar
characteristics unpredictable before their occurrence. It is also true that the civil, mechanicaland aerospace engineering structures are often required to be designed for disturbances
including inherent uncertainties due mainly to their "low rate of occurrence". Worst-caseanalysis combined with proper information based on reliable physical data is expected to play
an important role in avoiding difficulties induced by such uncertainties. Approaches based on
the concept of "critical excitation" seem to be promising.

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The most critical issue in the seismic resistant design is the resonance. One of thepromising approaches to this phenomenon is to shift the natural period of the building throughseismic control [7] and to add damping in the building. However it is also true that theseismic control is under development and more sufficient time is necessary to respond to
uncertain ground motions.
It is believed that earthquake has a bound on its magnitude. In other words, theearthquake energy radiated from the fault has a bound [8]. The problem is to find the mostunfavourable ground motion for a building or a group of buildings (see Fig.1). The Fourier
spectrum of a ground motion acceleration has been proposed at the rock surface depending on
the seismic moment M 0 , distance R from the fault, etc. (Fig.2; for example see [9]).
A(ω ) = CM 0 S (ω , ωC ) P(ω , ωmax ) exp(−ω R /(2 β Qβ ) / R(1)

Such spectrum may have uncertainties. One possibility or approach is to specify theacceleration or velocity power and allow the variability of the spectrum (see Fig.3).
The problem of ground motion variability is very important and tough. Code-specified
design ground motions are usually constructed by taking into account the knowledge from thepast observation and the probabilistic insights. However, uncertainties in the occurrence ofearthquakes (or ground motions), the fault rupture mechanisms, the wave propagation
mechanisms, the ground properties, etc. cause much difficulty in defining reasonable design
ground motions especially for important buildings in which severe damage or collapse has tobe avoided absolutely [4, 5, 10-13].
A long-period ground motion has been observed in USA and Japan recently. This type ofground motion is told to cause a large seismic demand to such structures as high-risebuildings, base-isolated buildings, oil tanks, etc, which have long natural periods. This large
seismic demand results from the resonance between the long-period ground motion and thelong natural period of these constructed facilities.
A significance of critical excitation is supported by its broad perspective. There are twoclasses of buildings in a city (see Fig.4). One is the important buildings which play animportant role during and after disastrous earthquakes. The other one is ordinary buildings.
The former one should not have damage during an earthquake and the latter one may bedamaged partially especially for critical excitation larger than code-specified design
earthquakes. Just as the investigation on limit states of structures plays an important role inthe specification of allowable response and performance levels of structures during
disturbances, the clarification of critical excitations for a given structure or a group of
structures appears to provide structural designers with useful information in determining
excitation parameters in a reasonable way. The concept of critical excitation may enablestructural designers to make ordinary buildings more seismic-resistant.
In the case where influential active faults are known in the design stage of a structure(especially an important structure), the effects by these active faults should be taken into
account in the structural design through the concept of critical excitation. If influential active
faults are not necessarily known in advance, virtual or scenario faults with an appropriate
energy may be defined especially in the design of important and socially influential structures.
The combination of worst-case analysis [4, 14, 15] with appropriate specification of energy
levels [9] derived from the analysis of various factors, e.g. fault rupture mechanism andearthquake occurrence probability, appears to lead to the construction of a more robust andreliable seismic resistant design method. The appropriate setting of energy levels or
information used in the worst-case analysis is important and the research on this subject
should be conducted more extensively.

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critical excitation ?
critical excitation Z
critical excitation A
critical excitation C
critical excitation D
critical excitation B

low-rise RC bld wood house

med-rise RC bld

base-isolated bld

high-rise bld

nuclear-powerplant

Fig.1 Structure-dependent critical excitation

ground motion
soil property
sourcecharacteristics
initiating point

rupture propagation
fault element

propagation
characteristics

wave propagation
fault
radiation energy may be bounded

Fig.2 Ground motion affected by source characteristics, propagation
characteristics and surface soil properties

∞

2

∞

2

∞

2

∞

2

∫ −∞ a(t ) dt = (1/ π )∫ 0 A(ω ) dω = C A

A(ω )

AU

∫ −∞ v(t ) dt = (1/ π ) ∫ 0 V (ω ) dω = CVAL

setting of maximum velocity

ω

A(ω )

GL
Love wave

AV (ω )

R
fault rupture

a lot of discussions in Japan for design ofsuper high-rise buildings
and base-isolated super high-rise buildings
at the location of basin (Tokyo, Osaka, …)Bounding
curve

A0

ωC

ω

Fig.3 Constraints on acceleration power and velocity power, andbounding of Fourier spectrum of the acceleration input

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Buildingresponselevel

Building
responselevel

Limit state

how to prevent
Limit state
Critical
excitation
Codespecified
groundmotionCritical
excitation
Codespecified
groundmotionIntensity of ground motion

Intensity of ground motion

Public building(base for restorationof earthquake disaster)Ordinary building

Fig.4 Relation of critical excitation with code-specified ground motion in
public and ordinary buildings
2.
CRITICAL EXCITATION METHODS FOR TIME-DEPENDENTPERFORMANCE INDICESIn this section, critical excitation methods for stationary and non-stationary random inputs are
discussed. It is natural to assume that earthquake ground motions are samples or realizationsof a non-stationary random process. Therefore critical excitation methods for non-stationary
random inputs may be desirable for constructing and developing realistic earthquake-resistantdesign methods. However it may also be relevant to develop the critical excitation methods
for stationary random inputs as the basis for further development for non-stationary random
inputs. First of all, critical excitation methods for stationary random inputs are discussed andsome fundamental and important results are derived. These results play a significant role inthe critical excitation methods for non-stationary random inputs. Secondly, critical excitation
methods for non-stationary random inputs are developed based on the theory for stationary
random inputs.
Consider a typical critical excitation problem. The response performance function may
be a storey drift [12, 16-21], a floor acceleration [22] or an earthquake energy input rate [23].
The input base acceleration model may be defined byug (t ) = c(t ) w(t )(2)where c(t ) is the prescribed deterministic envelope function and w(t ) is the stochastic
function to be obtained. The power spectral density function of w(t ) is denoted by S w (ω ) .
The critical excitation problem may be stated as:

Find S w (ω )such that max max{ f (t ; S w (ω ))} is achieved
S w (ω )t∞
S w (ω )dω ≤ S w
subject to ∫ −∞and sup S w (ω ) ≤ s( S w : given power limit)( s : given PSD amplitude limit)∞
where f = ∫ −∞H (t ; ω ) S w (ω )dω .

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s →∞
PSD function(Dirac delta
function)

F-function

F-function

power spectral
densitys
power spectral
density

F-functionΩ

circular frequency

circular frequency(a) critical excitation for infinite s(b) critical excitation for finite s

Fig.5 Power spectral density function of critical excitation for stationary inputThis problem can be solved by using the solution method for stationary problems [16]. A
critical excitation method for the corresponding stationary random inputs can be developed byintroducing a stochastic response index (mean square response) as the objective function to be
maximized. The power and the intensity of the excitations are prescribed so as to be
described in the problem just before. Fig.5 shows the schematic diagram of the solution
procedure for the corresponding stationary problem. When the restriction on the excitation
intensity does not exist, the critical PSD function is turned out to be the Dirac delta function.
In the existence of the restriction on the excitation intensity, the critical excitation is reduced
to a rectangular PSD function with the maximum intensity limit.

0.3

h=0.02

recorded
criticalSD/3.0standard deviation of drift (m)standard deviation of drift (m)

0.3

0.2

0.1

El Centro NS 1940
0
0

1

2

0.2

0.1

Taft EW 1952
0
0

3

1

natural period (s)0.3

h=0.02

recorded
criticalSD/3.0
0.2

0.1

0
0

Hyogoken-Nanbu 1995
Kobe Univ. NS
1

2

2

3

natural period (s)standard deviation of drift (m)standard deviation of drift (m)0.3

h=0.02

recorded
criticalSD/3.0

0.2

0.1

0
0

3

natural period (s)recorded
criticalSD/3.0h=0.02

Hyogoken-Nanbu 1995
JMA-Kobe. NS
123natural period (s)Fig.6 Standard deviation of interstorey drift of an SDOF model (damping ratio=0.02)
subjected to recorded earthquakes, that to the present critical excitation and
one-third of displacement response spectrum

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Interchange of order of double maximization procedures

maximization with respect to PSD function

maximization with respect to time

response f(t)to Sw(1)(ω)
overall maximum
response f(t)to Sw(2)(ω)

response f(t)to Sw(M)(ω)
t = tjt = titime

mean square of2
deformation (m )Fig.7 Schematic diagram of the procedure for finding the critical excitation for nonstationary random inputs (order interchange of double maximization procedures)
0.00350.003
0.00250.002
0.00150.001
0.000500

PSD amplitude=1
PSD amplitude=1/3
PSD amplitude=1/5
PSD amplitude=1/7

10

20

30

40

50

time (s)Fig.8 Time-dependent mean-square deformation of an SDOF model forvarious PSD amplitudes
Fig.6 shows the standard deviation of interstorey drift of a single-degree-of-freedom(SDOF) model subjected to recorded earthquakes, that to the present critical excitation and
one-third of displacement response spectrum. The level of conservativeness of the derived
critical excitation for stationary random inputs is about 2 or 3 in recorded ground motions
without conspicuous predominant frequency. On the other hand, it is close to unity in groundmotions with a conspicuous predominant frequency as shown in Fig.6. The resonant
characteristic of such ground motion can be represented suitably by the critical excitation.
The key idea for stationary random inputs can be used in SDOF models under nonstationary random inputs which can be described by a uniformly modulated excitation model.
The interchange of the order of the double maximization procedures as shown Fig.7 is taken
full advantage of with respect to time and to the PSD function. A similar algorithm can also
be devised for multi-degree-of-freedom (MDOF) proportionally damped models [17].
Fig.8 shows the time-dependent mean-square deformation of an SDOF model for various
PSD amplitudes [17].

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3.

CRITICAL EXCITATION METHODS FOR EARTHQUAKE INPUT ENERGY

The purpose of this section is to explain a general critical excitation method for a damped
linear elastic SDOF system. The input energy to the SDOF system during an earthquake isintroduced as a measure of criticality. It is shown that the formulation of the earthquake input
energy in the frequency domain is essential for solving the critical excitation problem andderiving a bound on the earthquake input energy for a class of ground motions. The criticality
is expressed in terms of degree of concentration of input motion components on the maximum
portion of the characteristic function defining the earthquake input energy. It is remarkablethat no mathematical programming technique is required in the solution procedure. The
constancy of earthquake input energy [24, 25] for various natural periods and damping ratios
is discussed from a new point of view based on an original sophisticated mathematicaltreatment. It is shown that the constancy of earthquake input energy is directly related to theuniformity of ‘the Fourier amplitude spectrum’ of ground motion acceleration, not theuniformity of the velocity response spectrum as described in [24, 25]. The bounds under
acceleration and velocity constraints (time integral of the squared base acceleration and time
integral of the squared base velocity) are clarified through numerical examinations forrecorded ground motions to be meaningful in the short and intermediate/long natural period
ranges, respectively.
Consider an SDOF model of mass m as shown in Fig.9. The displacement of the mass
relative to the ground is described by x. When this model is subjected to the ground motion
acceleration ug , the earthquake input energy can be expressed by
t0

EI = ∫ m(ug + x)u g dt(3)0

The earthquake input energy can also be expressed in frequency domain by
EI / m = − ∫∞−∞ dt = − ∫xa= −(1/ 2π ) ∫
=∫
≡∫

∞
0
∞
0

∞
−∞

∞ ⎡

⎢
−∞ ⎣

(1/ 2π ) ∫∞−∞ iω t dω ⎤ adt
Xe⎦⎥A(−ω ) { HV (ω; Ω, h) A(ω )} dω

(4)A(ω ) {−Re[HV (ω ; Ω, h)]/ π } dω22

A(ω ) F (ω )dωwhere HV (ω; Ω, h) = −iω /(Ω 2 − ω 2 + 2ihΩω ) is the velocity transfer function.
The energy transfer function F (ω ) characterizing the earthquake input energy to adamped linear elastic SDOF system in the frequency domain has been proved by the Residue
Theorem to have an equi-area property, as shown in Fig.10, irrespective of natural period anddamping ratio. This property guarantees that, if the Fourier amplitude spectrum of a groundacceleration is uniform, the constancy of the input energy defined by Eq.(4) holds strictly.
Otherwise, its constancy is not guaranteed. The constancy of the earthquake input energy
defined by Eq.(4) is directly related to the constancy of ‘the Fourier amplitude spectrum’ of
the ground acceleration, not the constancy of the velocity response spectrum. Fig.11 shows
the schematic diagram of solution procedure for critical excitation problem with acceleration
constraints. The energy transfer function for velocity constraint is presented in Fig.12 and theschematic diagram of solution procedure for velocity constraints is shown in Fig.13.
The solution to this critical excitation problem provides a useful bound of the earthquake
input energy for a class of ground motions satisfying intensity constraints (see Fig.14

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including the plot for a record of Kobe Univ. NS during 1995 Hyogoken-Nanbu earthquake).
The solution with acceleration constraints can bound properly the earthquake input energy ina shorter natural period range and that with velocity constraints can limit properly the input
energy in an intermediate or longer natural period range. The applicability of this boundtheory to various actual recorded ground motions can be found in Fig.15.
4.

HIGH-RISE BUILDINGS AND BASE-ISOLATED BUILDINGSThe issue on long-period ground motions is a controversial one in Japan. This is because,
although this type of ground motion has a small intensity in the scale of conventional intensity
measure of ground motions, it could cause a large seismic demand to such structures as highrise buildings, base-isolated buildings, oil tanks, etc. In these buildings under resonant longperiod ground motions with long duration, not only large displacement demand but also large
dissipation energy demand (fatigue problem) have to be resolved [26, 27]. The occurrence
mechanisms of these long-period ground motions and its properties have been investigatedextensively. However there remain a lot of uncertainties.
Free-body diagramx(t )−m( 
x + ug ) mck−m( 
x + ug )u g (t )du g = u g dt

opposite direction

displacement increment(base motion)

Fig.9 Free-body diagram for definitionof earthquake input energy

Fig.10 Energy transfer function withequi-area property for models of
various natural periods anddamping ratios

Fig.11 Schematic diagram of solution
procedure for critical excitation
problem with acceleration
constraints

Fig.12 Energy transfer function
for velocity input

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actual input energy
credible bound (acceleration constraint)
credible bound (velocity constraint)2

2

input energy per unit mass (m /s )

10

8

6

4
A

2

0

3

4

2

Hyogoken-Nanbu 1995,
JMA Kobe NS
h=0.0520
15
10

Near-fault Rock Motion

5

0

1

2

3

actual input energy
credible bound (acc)credible bound (vel)2

input energy per unit mass (m /s )25

15Cape mendocino 1992,
Petrolia NS

Near-fault Soil Motion

5

0

4

h=0.0510

0

1natural period (s)

3

4

40

0.75

2

Miyagiken-oki 1978,
Ofunato NS
h=0.05

0.5
Long-duration Rock Motion
0.25

1

2

3

actual input energy
credible bound (acc)credible bound (vel)2

input energy per unit mass (m /s )22actual input energy
credible bound (acc)credible bound (vel)00

2natural period (s)1input energy per unit mass (m /s )2

20
actual input energy
credible bound (acc)credible bound (vel)22

1

Fig.14 Credible bound of input energy
for acceleration constraint andthat for velocity constraint

30
input energy per unit mass (m /s )0natural period (s)Fig.13 Schematic diagram of solution
procedure for critical excitation
problem with velocity constraints

0

KBU NS 1995
h=0.0530

Long-duration Soil Motion
10

0

4

natural period (s)Chile 1985,
Vina del Mar NS
h=0.05

20

0

1

2

3

4

natural period (s)Fig.15 Credible bound of input energy for various types of ground motions(Near-fault rock motion, Near-fault soil motion, Long-duration rockmotion, Long-duration soil motion)

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Consider two shear building models, as shown in Fig.16, including viscous or hysteretic
dampers. It is known that both buildings have sinusoidal motions as an approximate criticalexcitation. Furtheremore, it is often the case that the long-period ground motions can bemodeled approximately by sinusoidal motions. Fig.17 shows the 3-D sweeping response
performance surfaces for the 5-storey building model with viscous dampers and that withhysteretic dampers. These building models have been reduced to two equivalent SDOF
models. The maximum interstorey drift has been plotted with respect to the maximum
acceleration amplitude and the excitation frequency. The left figure indicates the model withviscous dampers and the right figure shows the model with hysteretic dampers. The shape ofthese 3-D sweeping response performance surfaces implies the mechanical properties of thesepassive-controlled frames. Fig.18 shows the 2-D sweeping response performance curves for
5-storey models with viscous or hysteretic dampers and Fig.19 presents those for 20-storey
models. The peak frequency shift with respect to increasing input intensities can be observedin these figures.

uN

Q

mN

cN k Nframe+hysteretic damperQu(1 − β )Qu

frame

40m

2nd-branch stiffness
ratio=0.01

u18m

hysteretic damperk0δ
Story shear-deformation relation(hysteretic damper model)ugPlan view

1

β Qu

6m

36m

k f = kdψ Qumc1 k1 12mm

Shear building model

20mm

Fig.16 Passively-controlled building modelwith viscous damper （5-storey model）

with hysteretic damper (5-storey model)
0.06

circular frequency of input sine wave
（ rad/s）

0.01
04 maximum
accelerationof inputsine wave（m/s2 ）

circular frequency of input sine
wave (rad/s)

31.42

0

2

10

6.83

0

8

14.28

6

0.02

19.63

0.01

0.03

9.24

0.02

0.04

11.22

0.03

0.05

7.854

0.04

maximum interstorey drift (m)

0.05

6.283
6.83
7.48
8.267
9.24
10.47
12.08
14.28
17.45
22.44
31.42

maximum interstorey drift （m）

0.06

10
8
6
4
2 maximum
acceleration
0of inputsine wave(m/s2)

Fig.17 3-D sweeping response performance surface

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with hysteretic damper (5-storey model)

0.07
0.06

a

0.05

a

0.04

a

maximum interstorey drift (m)maximum interstorey drift (m)with viscous damper (5-storey model)2max= 1m/s2max= 5m/s2max= 10m/s

0.03
0.02
0.01
0
5
10
15
20
25
30
circular frequency of input sine wave (rad/s)

0.07
0.06

a

0.05

a

0.04

a

2

max

= 1m/s2max= 5m/s2max= 10m/s

0.03
0.02
0.01
0

5
10
15
20
25
30
circular frequency of input sine wave (rad/s)

Fig.18 2-D sweeping response performance curves for 5-storey models withviscous or hysteretic dampers
with hysteretic damper (20-storey model)

0.05

maximum interstorey drift (m)maximum interstorey drift (m)with viscous damper (20-storey model)
amax = 0.2m/s2

0.04
0.03

2

amax =

1m/s

amax =

2

2m/s

0.02
0.01
0
0
2
4
6
8
10
circular frequency of input sine wave (rad/s)

0.05
2

a

0.04

max

2

a

=

1m/s

a

=

2m/s

max

0.03

= 0.2m/s

max

2

0.02
0.01
0
0
2
4
6
8
10
circular frequency of input sine wave (rad/s)

Fig.19 2-D sweeping response performance curves for 20-storey models
with viscous or hysteretic dampers
The resonant behaviour of base-isolated high-rise buildings as shown in Fig.20(a) isinvestigated next [26] under long-period ground motions. The long-period ground motions
are known to be induced by surface waves. While the acceleration amplitude of such longperiod ground motion is small as stated before, the velocity amplitude is fairly large. It is
expected that high-rise buildings and base-isolated buildings with long fundamental natural
periods are greatly influenced by these long-period ground motions. Especially base-isolated
high-rise buildings with friction-type bearings may have remarkable mechanicalcharacteristics unfavourable for these long-period ground motions. The restoring-force
characteristics of the isolation floor are shown in Fig.20(b). The parameter α indicates theratio of the storey shear due to the linear rubber bearing to the total one at the displacement ofinitiation of slippage of friction dampers. The purpose here is to reveal that the long-period
ground motions recorded in Japan have the intensity to make the base-isolated high-risebuildings in resonance with the long-period components and that careful treatment isinevitable in the structural design of these base-isolated high-rise buildings.
Figs.21 show the acceleration records of El Centro NS (Imperial Valley 1940), OSA NS(simulated Nankai Earthquake by Kamae et al.[28]), Tomakomai EW (Tokachi-oki 2003) anda simulated motion compatible with the response spectrum of Level 2 in the new Japanese
earthquake-resistant design code (2000). Fig.22 presents the velocity response spectra of

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these four motions considered for damping ratio 0.05. Fig.23 shows the restoring-force
characteristics in the base-isolation (BI) system under these four ground motions in the caseof α = 0.25 . It can be observed that the ground motion of Tomakomai EW with long-period
wave components has caused a dangerous result to base-isolated high-rise buildings (seeFig.24).

Q

Q

total
total
linear rubber
bearinglinear rubber
bearingfriction-type
bearingfriction-type bearing

D

D

Alpha=0.25

Alpha=0.75

(a)

(b)

OSA-NSEl Centro 1940 NS

0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4

acceleration (g)acceleration (g)Fig.20 (a) Base-isolated high-rise building, (b) Restoring-force
characteristics of isolation floor

0

100

200

300

400

0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4

0

100

0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4

Port of Tomakomai EW, Tokachi-Oki 2003.9.26

0

100

200

200

300

400

time (s)300

sim ula ted m o tio n

0 .6

acceleration (g)acceleration (g)time (sec)

0 .4
0 .2
0
-0 .2
-0 .4
-0 .6

400

0

time (s)

100

200

300

400

tim e (s )Fig.21 Acceleration records of El Centro NS (Imperial Valley 1940), OSA NS (simulated
Nankai Earthquake by Kamae, Kawabe and Irikura 2004), Tomakomai EW (Tokachioki 2003) and a simulated motion compatible with the response spectrum of Level 2 in
the new Japanese earthquake-resistant design code (2000)

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velocity response spectrum (m/s)

3Tomakomai EW
El Centro NS
JMA Kobe
OSA-NS

h=0.052.5
2？

code specified spectrum

1.5
1
0.5
0

0

2

4
6
natural period (s)810Fig.22 Velocity response spectra of four motions
considered for damping ratio 0.05

Fig.24 Limit state of rubber bearing
in the tensile region
OSA NS (alpha=0.25)

El Centro NS (alpha=0.25)

6 10 66 106

4 10 6story shear force (N)story shear force (N)

4 106
2 106

0
‑2 10 60
‑2 10 6
‑4 10 6

‑4 10 6
‑6 10 6‑0.8

2 10 6

‑0.6

‑0.4

‑0.2

0

0.2

0.4

0.6

‑6 10 6
‑0.8

0.8

‑0.6

‑0.4

6 106

4 106

4 106

story shear force (N)story shear force (N)

0

0.2

0.4

0.6

0.8

0.6

0.8

simulated motion (alpha=0.25)

Tomakomai EW (alpha=0.25)
6 106

2 106

0
‑2 10 62 106

0
-2 106-4 106‑4 10 6
‑6 10 6‑0.8

‑0.2

interstory drift (m)interstory drift (m)

‑0.6

‑0.4

‑0.2

0

0.2

0.4

0.6

-6 106-0.8

0.8

interstory drift (m)

-0.6

-0.4

-0.2

0

0.2

0.4

interstory drift (m)Fig.23 Restoring-force characteristics in the BI system under four ground motions in the caseof α = 0.25
The base-isolation systems with friction-type bearings are generally believed to beeffective in providing damping in the BI system and avoiding resonance with ground motions
by changing the equivalent natural frequency in accordance with experienced deformation.
However resonance can occur in the case where the ground motion has a large intensity ofvelocity response spectra with broad band in the long natural period range. As the drift of theBI system becomes larger, the resonance is accelerated because the modified natural period isapproaching the predominant period of the ground motion. This phenomenon is hard to occur

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in the high-rise buildings without BI system because such drastic change of the equivalentnatural period is hard to occur even in the experience of minor plastic deformation in someparts.
As the damping ratio of the additional viscous-damper system in the BI system increases,
most of the maximum interstorey drifts become smaller. However, the introduction of
sufficient amount of damping in the BI system is very hard because of the limitation on spaceand cost. Furthermore, the maximum interstorey drifts and floor accelerations in upper stories
under some ground motions can become larger in spite of the increase of the damping.
Therefore special attention should be paid in the introduction of damping in the BI system.
The empirical laws, known as the constant displacement or the constant energy, should
be discussed from the view point of the relation of the natural period of structures with thepredominant period of input ground motions. Over 40 years ago, most of the recorded groundmotions have the predominant period mostly smaller than 1-2 sec. In such case, the structureswith the fundamental natural period much larger than 1-2 sec tend to follow the constantdisplacement law. However, those structures do not necessarily follow the constantdisplacement law under the ‘long-period earthquake ground motions’. The relation of thenatural period of structures with the predominant period of input ground motions is a key tobe discussed carefully.
5.

NUCLEAR REACTOR FACILITIESThe ground surface motion during an earthquake consists of multiple components. It is often
assumed that there exists a set of principal axes in the ground motions [29]. This set ofprincipal axes varies in time. In the current seismic structural design practice, the effect ofthe multi-component inputs is often taken into account by use of the SRSS (square root of thesum of the squares) method in which the maximum responses under respective input motions
are combined by the rule of SRSS. The SRSS method assumes the statistical independence
among the maximum responses under respective input motions. However, the multicomponent inputs have some statistical dependence (e.g. [30]). It seems that the discussion
from the viewpoint of worst-case analysis is appropriate in such a case.
A problem of critical excitation is considered here for a rigid block subjected tohorizontal and vertical simultaneous base inputs [31]. The rigid block is supported by a set ofa horizontal spring and a dashpot and rests on another set of two vertical springs and dashpots
as shown in Fig.25. These springs and dashpots are the representatives of the stiffness anddamping of ground. The motion of the block consists of a horizontal-rotational (swayingrocking) motion under a horizontal base input and a vertical motion under a vertical baseinput. Both the horizontal-rotational motion and the vertical motion clearly contribute to the
uplift of the edge of the block. In the seismic structural design practice, the design spectra forthe horizontal and vertical motions are often provided independently and it does not appear
that their relationship is discussed in detail except a few (for example see [32]). This
relationship will be discussed here from the viewpoint of critical excitation.
It seems natural and appropriate to assume that the horizontal and vertical base inputs
are to be described by a non-stationary random process whose power spectra are prescribed.
A critical excitation problem is considered such that the worst cross spectrum (allowableregion is shown in Fig.25(b)) and the corresponding cross-correlation function of thehorizontal and vertical inputs are searched for the maximum mean-squares response of the
uplift. It is found that the real part (co-spectrum, e.g. see [33]) and the imaginary part (quadspectrum) of the worst cross spectrum can be obtained by a devised algorithm, as shown in
Fig.26, including the interchange of the double maximization procedure with respect to thetime and cross-spectrum (functional: function of frequency) domains.
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2L

Qwv wu (ω )S wv wv (ω ) S wu wu (ω )Centroid C

cH

vg

2H

u

m, Ig

θ

v

Cwv wu (ω )kH
(1/2)cV(1/2)kV(1/2)cV(1/2)kV− S wv wv (ω ) S wu wu (ω )

ug

(a)

(b)

Fig.25 (a)Elastically supported rigid block subjected to horizontal-verticalsimultaneous inputs and (b) allowable domain defined from theproperty of the cross spectrum and the power spectra
The mean-squares uplift of the foundation edge may be described as
E[ D(t ) 2 ] = ∫∞−∞Fuu (t ; ω ) S wu wu (ω )dω + ∫∞−∞Fvv (t ; ω ) S wv wv (ω )dω + ∫∞F (t ; ω ) S wu wv (ω )dω
−∞ uv
(5)

The cross spectrum is defined by
S wu wv (ω ) = S wv wu (ω )* = Cwv wu (ω ) + iQwv wu (ω )(6)maximization with respect to cross spectrum

maximization with respect to time

Uplift to crossspectrum 1

Uplift to crossspectrum 2

Uplift to crossspectrum Mt = tit = tjtime

Cwv wu (ω )

ω1 ω2

Qwv wu (ω )

ω1

ω2

ω

ω

Sub-problem optimization
for a given frequency

Fig.26 Schematic diagram of solution procedure for critical excitation problem with unknown
cross-spectrum

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Fig.27 3-D view of the function, part of integrand for computation of mean-squareuplift under vertical input, with respect to time and frequency(a) co-spectrum(b) quad-spectrum

Fig.28 3-D view of temporal characteristics of critical co-spectrum and quad-spectrum

30
2 4

10
0
-10
-20
hu=0.01, hv=0.014
hu=0.05, hv=0.025
hu=0.05, hv=0.071
hu=0.10, hv=0.071

-30
-40

cross-correlation function (m /s )2 4cross-correlation function (m /s )20

-2 -1.5 -1 -0.5 0 0.5
time-lag (s)

1

1.5

20

hu=0.01, hv=0.014
hu=0.05, hv=0.025
hu=0.05, hv=0.071
hu=0.10, hv=0.071

10
0
-10
-20
-30
-2 -1.5 -1 -0.5 0 0.5
time-lag(s)

2

1

1.5

2

(b)

(a)

Fig.29 Critical cross-correlation function for TH = 0.25(s) and TH = 0.50(s)

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The mean-squares response of uplift has proven to be the sum of the term due to thehorizontal input, that due to the vertical input and that due to their correlation as shown inEq.(5). Since the power spectra of the horizontal and vertical ground motions are given andprescribed, the maximization in the critical excitation problem means the maximization of thecorrelation term of the horizontal and vertical ground motions.
The root-mean-squares value of uplift to the critical combination of the horizontal andvertical motions becomes about several through fifteen percent larger than that by the SRSS
estimate of uplift due to the horizontal input and that due to the vertical input. The present
method including the critical correlation of the horizontal and vertical inputs should be usedin the design of important structures, e.g. nuclear reactor facilities, super high-rise buildings
etc., in place of the well-known SRSS method.
In order to demonstrate the validity of the proposed critical excitation method, several
numerical examples have been presented. Four damping cases have been treated which
include proportional and non-proportional damping cases. The numerical examples indicatethat the proposed algorithm can work very well and the critical cross spectrum and criticalcross-correlation function can be understood from the view point of physical meaning. Fig.27
shows the 3-D view of the function, i.e. part of integrand for computation of mean-squareuplift under vertical input, with respect to time and frequency and Fig.28 presents the 3-Dview of temporal characteristics of critical co-spectrum and quad-spectrum. When thedamping ratio in the horizontal and vertical vibration modes is small, the critical crosscorrelation function has its maximum at the zero time-lag as shown in Fig.29(a). On the other
hand, when such a damping ratio is relatively large, the critical cross-correlation function has
its maximum at a non-zero time-lag (see Fig.29(b)).
The present method has been applied to a problem of foundation uplift. However, themethod can be applied to a broader class of critical excitation problems. For example, thepresent method has been applied to a moment-resisting frame of large span as shown in
Fig.30(a) [34]. The horizontal and vertical stochastic ground accelerations are input to theframe and the mean-squares of the sum of the beam-end bending moments due to both inputs
is adopted as the objective function for a critical excitation problem. The frame has beenmodelled into an SDOF system as shown in Fig.30(b).
Fig.31 shows the comparison of the maximum root mean squares of the sum of the beamend bending moments among the SRSS estimate, the proportional input case (quad spectrum
Q=0) and the critical input due to the present author [34]. It can be observed that, while thecritical input is close to the proportional input case in the model (span length =17m) with analmost equal horizontal and vertical natural frequency, the critical input is far from the
proportional input case in the other models.
v
L

B

C

u

Ib
Ic
A

Ic

H

beam

k u , cuk v , cv

column(a)(b)
Fig.30 Moment-resisting frame of large span subjected to horizontal and vertical stochastic
ground accelerations

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1.0

800

Niigataken-Chuetsu-Oki Earthquake July 16, 2007
Kashiwazaki NS (K-net, NIG018)

600

acceleration (gal)of the bending moment (*108 Nm)

max value of root mean square of sum

1.2

0.8
0.6

including criticalcorrelation
Q=0
without correlation(SRSS)

0.4
0.2
10

15

20

25
30
span (m)

35

400
200
0
-200
-400
-600

40

-800
10

20

30

40

time (s)Fig.31 Comparison of the maximum RMS
values of sum of beam-end bending
moments among SRSS estimate,proportional input case (Q=0) andthe critical input

Fig.32 Resonant critical input in recent
earthquake (near nuclear reactor
facilities)

It may be said that the present theory opens the door for critical excitation problems for
multi-component inputs. Although it is a rare case to include the effect of multi-component
inputs in the current structural design practice, simultaneous consideration of bi-directional
ground motion components is inevitable in the reliable design of structures. It has been
demonstrated that the present model allowing arbitrary specification of the cross spectrum
between respective-direction ground motion components includes the conventional PenzienWatabe model assuming the existence of the principal axes without correlation and is suitable
for seismic-resistant design of building structures. This is because the usual structural design
practice requires the specification of design response spectra or power spectra in the building
structural axes (usually one axis). The present theory satisfies this requirement and facilitates
the introduction of the concept of critical excitation.
Recently (July 16, 2007), a severe ground motion attacked the city of Kashiwazaki,
Niigata Prefecture in Japan and many old wood houses were destroyed. It has been reported
that a peculiar ground motion as shown in Fig.32 has been observed and the ground motion
had a predominant period of 2.5 (s). This period is thought to be resonant with the natural
period of old wood houses with heavy roofs. This ground motion is very similar to one
predicted in [35]. It should be noted that a large nuclear reactor facility is located in the cityof Kashiwazaki and that facility had relatively minor damage. Further damage investigation
is being conducted even now. While the IAEA reported that the damage is relatively minor,
further extensive damage has been found at some areas. It is hoped that a critical excitation
method is used in the design and upgrading of these nuclear reactor facilities.
6.

CONCLUSIONS(1) In critical excitation problems with time-dependent performance indices as objectivefunctions, a frequency-domain approach is appropriate and a solution procedure
consisting of the double maximization procedure can be developed. The double
maximization procedure is with respect to the power spectrum density function and the
time.

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(2) In critical excitation problems for earthquake input energy, a frequency-domain approachis effective. The solution to this critical excitation problem provides a useful bound of theearthquake input energy for a class of ground motions satisfying intensity constraints.
The solution with acceleration constraints can bound properly the earthquake input energy
in a shorter natural period range and that with velocity constraints can limit properly the
input energy in an intermediate or longer natural period range.
(3) Critical correlation among multi-component ground motions can be defined. This
criticality reveals the amplification of the response over the conventional square root of
the sum of the squares (SRSS) response.
ACKNOWLEDGEMENTS

The present work is partially supported by the Grant-in-Aid for Scientific Research of Japan
Society for the Promotion of Science (No.18360264), the SECOM Science and Technology
Foundation and the Japan Steel Federation Research Grant. This support is greatly
appreciated. The author is also grateful to NISEE (USA), JMA (Japan) and NIED (Japan) for
providing recorded ground motions
REFERENCES[1] Drenick, R. F. (1970). "Model-free design of aseismic structures." J. Engrg. Mech. Div.,ASCE, 96(EM4), 483-493.
[2] Shinozuka, M. (1970). "Maximum structural response to seismic excitations." J. Engrg.
Mech. Div., ASCE, 96(EM5), 729-738.
[3] Takewaki, I. and Ben-Haim, Y. (2005). "Info-gap robust design with load and model
uncertainties." J. Sound and Vibration, 288(3), 551-570.
[4] Takewaki, I. (2006). Critical Excitation Methods in Earthquake Engineering, Elsevier,268pages.
[5] Anderson, J. C., and Bertero, V. V. (1987). "Uncertainties in establishing design
earthquakes." J. Struct. Engrg., ASCE, 113(8), 1709-1724.
[6] PEER center, ATC, Japan Ministry of education, science, sports, and culture, US-NSF(2000). Effects of near-field earthquake shaking, Proc. of US-Japan workshop, San
Francisco, March 20-21, 2000.
[7] Johnson E, Smyth A (eds.). Proc. of Fourth World Conference on Structural Control andMonitoring (4WCSCM), San Diego, 11-13, July, 2006.
[8] Trifunac, M.D. (2008). "Energy of strong motion at earthquake source." Soil Dyn. Earthq.
Engrg., 28, 1-6.
[9] Boore, D.M. (1983). "Stochastic simulation of high-frequency ground motions based onseismological models of the radiated spectra", Bull. Seismological Society of America,
73(6), 1865-1894.
[10] Singh, J.P. (1984). "Characteristics of near-field ground motion and their importance inbuilding design." ATC-10-1 Critical aspects of earthquake ground motion and building
damage potential, ATC, 23-42.
[11] Geller, R.J., Jackson, D.D., Kagan, Y.Y., and Mulargia, F. (1997). "Earthquakes cannot
be predicted." Science, 275, 1616.
[12] Takewaki, I. (2002). "Seismic critical excitation method for robust design: a review." J.
Struct. Engrg, ASCE, 128(5), 665-672.
[13] Stein, R.S. (2003). "Earthquake conversations." Scientific American, 288(1), 72-79.

19

ESP5

[14] Takewaki, I. (2004). "Bound of earthquake input energy." J. Struct. Engrg., ASCE,
130(9), 1289-1297.
[15] Takewaki, I. (2005). "Bound of earthquake input energy to soil-structure interaction
systems." Soil Dyn. Earthq. Engrg., 25(7-10), 741-752.
[16] Takewaki, I. (2000). Optimal damper placement for critical excitation, Probabilistic
Engrg. Mech., 15(4), 317-325.
[17] Takewaki, I. (2001). "A new method for nonstationary random critical excitation."
Earthq. Engrg. Struct. Dyn., 30(4), 519-535.
[18] Takewaki, I. (2001). "Nonstationary random critical excitation for nonproportionallydamped structural systems." Comput. Meth. Appl. Mech. Engrg., 190(31), 3927-3943.
[19] Takewaki, I. (2001). "Critical excitation for elastic-plastic structures via statisticalequivalent linearization." Probabilistic Engrg. Mech., 17(1), 73-84.
[20] Takewaki, I. (2001). "Probabilistic critical excitation for MDOF elastic-plastic structureson compliant ground." Earthq. Engrg. Struct. Dyn., 30(9), 1345-1360.
[21] Takewaki, I. (2002). "Robust building stiffness design for variable critical excitations." J.
Struct. Engrg., ASCE, 128(12), 1565-1574.
[22] Takewaki, I. (2001). "Nonstationary random critical excitation for acceleration
response." J. Engrg. Mech., ASCE, 127(6), 544-556.
[23] Takewaki, I., (2006). "Probabilistic critical excitation method for earthquake energy
input rate." J. Engrg. Mech., ASCE, 132(9), 990-1000.
[24] Housner, G.W. (1959). "Behavior of structures during earthquakes." J. Engrg Mech. Div.,ASCE, 85(4), 109-129.
[25] Akiyama, H. (1985). Earthquake Resistant Limit-State Design for Buildings. Universityof Tokyo Press, Tokyo, Japan.
[26] Ariga, T., Kanno, Y. and Takewaki, I. (2006). "Resonant behavior of base-isolated highrise buildings under long-period ground motions." The Structural Design of Tall andSpecial Buildings, 15(3), 325-338.
[27] Heaton, T., Hall, J., Wald, D. and Halling, M. (1995). Response of high-rise and baseisolated buildings to a hypothetical M 7.0 blind thrust earthquake, Science, 267, 206-211.
[28] Kamae, K, Kawabe, H and Irikura, K. (2004). Strong ground motion prediction for huge
subduction earthquakes using a characterized source model and several simulation
techniques, Proc. of the13th WCEE, Vancouver.
[29] Clough, R.W., and Penzien, J. (1993), Dynamics of Structures, second edition, PrenticeHall.
[30] Smeby, W., and Der Kiureghian, A. (1985), "Modal combination rules for multicomponent earthquake excitation." Earthq. Engrg. Struct. Dyn., 13, 1-12.
[31] Fujita, K., Yoshitomi, S., Tsuji, M. and Takewaki, I. (2007). "Critical cross-correlation
function of horizontal and vertical ground motions for uplift of rigid block." Engrg.
Struct. (Online, July 2007).
[32] Menun, C., and Der Kiureghian, A. (1998). "A replacement for the 30%, 40%, and SRSS
rules for multicomponent seismic analysis." Earthq. Spectra, 14(1), 153-163.
[33] Nigam, N.C. (1981), Introduction to Random Vibrations, MIT Press.
[34] Fujita, K., Takewaki, I. and Nakamura, N. (2008), Critical disturbance for stress resultant
in long-span moment-resisting frames subjected to horizontal and vertical simultaneous
ground inputs, J. Struct. Construction Eng., No.626, AIJ (in Japanese).
[35] Takewaki, I. (2004). "Critical envelope functions for non-stationary random earthquake
input." Computers & Structures, 82(20-21), 1671-1683.

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