２章

建築構造物のロバスト耐震設計のための極限入力解析

研究分担者
竹脇 出
Joel P. Conte

教授
Associate Professor

Enrico Spacone

Associate Professor

京都大学大学院工学研究科都市環境工学専攻
Dept. of Structural Engineering, School ofEngineering, University of California, San Diego
Dept. of Civil, Environmental and ArchitecturalEngineering, University of Colorado at Boulder

研究発表[1] I.Takewaki, Robust building stiffness design for variable critical excitations, Journal of
Structural Engineering, American Society of Civil Engineers, Vol.128, No.12, p1565-1574,
2002.
[2] I.Takewaki, Bound of Earthquake Input Energy, Journal of Structural Engineering, ASCE, inpress, 2004.
[3] I.Takewaki, Frequency Domain Modal Analysis of Earthquake Input Energy to Highly
Damped Passive Control Structures, Earthquake Engineering & Structural Dynamics, inpress, 2004.
[4] 竹脇 出，多様な減衰分布を有する構造物に入力される地震エネルギーの限界値，
日本建築学会構造系論文集, 第 572 号, pp65-72, 2003.
[5] I.Takewaki, Bound of Earthquake Input Energy to Soil-Structure Interaction Systems, Proc. of
The 11th International Conference on Soil Dynamics and Earthquake Engineering (ICSDEE)
and the 3rd International Conference on Earthquake Geotechnical Engineering (ICEGE),
Berkeley, January 7-9, 2004.
アブストラクト
都市直下の内陸型地震において顕著な通り、これまでに観測されている地震動には、
振幅・位相特性や卓越振動数特性等において大きな不確定性が存在しており、一定期間におけるその極めて小さな発生確率（長い再現期間）を考えると、近い将来にその不確定性が取り除かれ理論化されるというシナリオは描き難いと推測される。
このような状況下では、比較的不確定性のレベルが高い部分のみを未定パラメターとして含む地震動モデルを作成し、発生が予想される集合としての地震動群に対して構造物を安全に設計することが、現時点で最も期待できる信頼性の高い方法の一つといえる。
本研究では、地震動のクリティカル性を特徴付ける指標として、構造物に入力されるエ
ネルギーを新たに採用する。地震入力エネルギーの特性およびそれを用いた設計法に関しては、これまでに多くの成果が蓄積されている。本研究では、線形弾性応答に限定して、振動数領域における定式化により、あるクラスの地震動群に対してその上限値が誘導できることを示す。次に、構造物‐地盤連成系をモデル化したスウェイ・ロッキング
モデルに対して振動数領域定式化を適用することにより、基礎固定モデルと同様に地震入力エネルギーの上限値が誘導できることを示す。さらに、埋め込み基礎を有する場合の地震入力エネルギーの評価法についても論じる。

2. Worst-case Analysis for Robust Building
Seismic Resistant Design
Izuru Takewaki*, Joel P. Conte** and Enrico Spacone****Dept. of Urban & Environmental Engineering, Graduate School of Engineering, Kyoto
University, Sakyo, Kyoto 606-8501, JAPAN**Dept. of Structural Engineering, School of Engineering, University of California, San Diego,
9500 Gilman Drive, La Jolla, CA 92093-0085, USA***Dept. of Civil, Environmental and Architectural Engineering, University of Colorado,
Boulder, CO 80309-0428, USA

Abstract
A new general critical excitation method is developed for a damped linear elastic single-degreeof-freedom structure. In contrast to previous studies considering amplitude nonstationarity only,
no special constraint of input motions is needed on nonstationarity. The input energy to thestructure during an earthquake is introduced as a new measure of criticality. It is shown that theformulation of the earthquake input energy in the frequency domain is essential for solving thecritical excitation problem and deriving a bound on the earthquake input energy for a class ofground motions. It is remarkable that no mathematical programming technique is required in
the solution procedure. This enables structural engineers to use the method in their structuraldesign practice without difficulty. The constancy of earthquake input energy for various natural
periods and damping ratios is discussed based on an original sophisticated mathematicaltreatment. Through numerical examinations for four classes of recorded ground motions, thebounds under acceleration and velocity constraints (time integral of the squared base accelerationand time integral of the squared base velocity) are clarified to be meaningful in the short andintermediate/long natural period ranges, respectively. Another critical excitation method is also
developed for a building structure supported by a sway-rocking system representing a foundationground system. It is shown that the frequency-domain approach is effective in developing a
critical excitation method especially for a soil-structure interaction system.

Introduction
Earthquake ground motions involve a lot of uncertain factors in the modeling of various
aspects and it does not appear easy to predict forthcoming events precisely at a specific site bothin time and frequency contents (see, for example, Abrahamson et al. 1998, Anderson and Bertero
1987, Geller et al. 1997; PEER Center et al. 2000; Stein 2003). Some of the uncertainties may
result from the lack of information due to the low occurrence rate of large earthquakes and it does
not seem that this problem can be resolved in the near future. Especially, the modeling of nearfault ground motions involves various uncertain factors in contrast to far-fault ground motions(Singh 1984; PEER Center et al. 2000; Krawinkler et al. 2001). It is therefore strongly desirable
to develop a robust structural design method taking into account these uncertainties with limited
information and enabling the design of safer structures for a broader class of design earthquakes.

To the best of the authors' knowledge, approaches based on the concept of "critical excitation" or"worst-case input" are promising (Drenick 1970; Shinozuka 1970; Takewaki 2002b). Evidently
anticipated ground motions differ both in intensity and in character. It may be said that thecritical excitation method is aimed at identifying the critical character. Just as the investigationof response limit states of structures plays an important role in specifying allowable response andperformance limits of structures during disturbances, the clarification of critical excitations for agiven structure appears to provide structural designers with useful information in determining
excitation parameters in a reasonable and reliable way.
The critical excitation methods have a history of over 30 years. Its general review can befound in Takewaki (2002a). The previous studies have some limitations, e.g. those on treatmentof nonstationarity of ground motions, those on numerical applicability.
One of the purposes of this research is to develop a new general critical excitation method
for a damped linear elastic single-degree-of-freedom (SDOF) system. The input energy to the
SDOF system during an earthquake is introduced as a new measure of criticality. It is shownthat the formulation of the earthquake input energy in the frequency domain is essential forsolving the critical excitation problem and deriving a bound on the earthquake input energy for aclass 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 theearthquake input energy. It is remarkable that no mathematical programming technique isrequired in the solution procedure. The constancy of earthquake input energy (Housner 1956,
1959) for various natural periods and damping ratios is discussed from a new point of view basedon an original sophisticated mathematical treatment. It is shown that the constancy ofearthquake input energy is directly related to the uniformity of ‘the Fourier amplitude spectrum’of ground motion acceleration, not the uniformity of the velocity response spectrum. The
bounds under acceleration and velocity constraints (time integral of the squared base accelerationand 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. Applicability of the proposed technique to a soil-structure interaction
system is also discussed.

Earthquake Input Energy to SDOF System
Much work has been accumulated on the topics of earthquake input energy (for example,
Tanabashi 1935; Housner 1956, 1959; Berg and Thomaides 1960; Goel and Berg 1968; Housner
and Jennings 1975; Kato and Akiyama 1975; Takizawa 1977; Mahin and Lin 1983; Zahrah andHall 1984; Akiyama 1985; Ohi et al. 1985; Uang and Bertero 1990; Leger and Dussault 1992;
Kuwamura et al. 1994; Fajfar and Vidic 1994; Ogawa et al. 2000; Riddell and Garcia 2001;
Ordaz et al. 2003). In contrast to most of the previous works, the earthquake input energy isformulated here in the frequency domain (Page 1952; Lyon 1975, Takizawa 1977; Ohi et al.
1985; Ordaz et al. 2003) to facilitate the introduction of critical excitation methods and thederivation of bound of earthquake input energy.
Consider a damped linear SDOF system of mass m, stiffness k and damping coefficient c.
Let Ω = k / m , h = c /(2Ωm) and x denote the undamped natural circular frequency, thedamping ratio and the displacement of the mass relative to the ground, respectively. Time
derivative is denoted by over-dot. The input energy to an SDOF system by a uni-directional
ground acceleration ug (t ) = a (t ) from t = 0 to t = t0 (end of input) can be defined by thework of the ground on the structural system and is expressed by
t0

EI = ∫ m(ug + x)u g dt
0(1)

The term − m(ug + x) indicates the inertial force and is equal to the sum of the restoring force
kx and the damping force cx in the system. Integration by parts of Eq.(1) provides
t

t

t

0
0
0
 g dt + (1/ 2)mu g 2 
EI = ∫ m( x + ug )u g dt = ∫ mxu


0
0
0

(2)

2  t0t0

  g  − ∫ mxu
g dt + (1/ 2)mu g=  mxu000
If x = 0 at t = 0 and u g = 0 at t = 0 and t = t0 , the input energy can be reduced to thefollowing form.
t0

t0

EI = − ∫ mug xdt

(3)0

It is known (Page 1952; Lyon 1975; Takizawa 1977; Ohi et al. 1985; Ordaz et al. 2003) that the
input energy per unit mass can also be expressed in the frequency domain.
∞∞∞ iω t dω  adt
 dt = − ∫ (1/ 2π ) ∫ Xe
EI / m = − ∫ xa
−∞
−∞ 
−∞


∞

= −(1/ 2π ) ∫

−∞

∞

2

=∫
≡∫

0
∞
0

A(−ω ) { HV (ω; Ω, h) A(ω )} dω

(4)A(ω ) {− Re[HV (ω ; Ω, h)]/ π } dω2A(ω ) F (ω )dωwhere HV (ω; Ω, h) is the transfer function defined by X (ω ) = HV (ω ; Ω, h) A(ω ) andF (ω ) = − Re[HV (ω ; Ω, h)]/ π .
X and A(ω ) are the Fourier transforms of x andug (t ) = a (t ) , respectively. The symbol i denotes the imaginary unit. HV (ω; Ω, h) can be
expressed by
HV (ω ; Ω, h) = −iω /(Ω 2 − ω 2 + 2ihΩω )(5)

Eq.(4) indicates that the earthquake input energy to damped linear elastic SDOF systems does notdepend on the phase of input motions and this fact is well known (Page 1952, Lyon 1975,
Takizawa 1977, Ohi et al. 1985, Kuwamura et al. 1994, Ordaz et al. 2003). It can also be
understood from Eq.(4) that the function F (ω ) plays an important role in the evaluation ofearthquake input energy and may have some influence on the investigation of constancy ofearthquake input energy for structures with various model parameters. The property of thefunction F (ω ) defined in Eq.(4) will therefore be clarified in the following section.

Property of Energy Transfer Function
Earthquake Input EnergyF(ω ) and Constancy of

The functions F (ω ) for various natural periods T=0.5, 1.0, 2.0s and damping ratios
h=0.05, 0.20 are plotted in Fig.1. It is interesting to note that the area of F (ω ) can be proved
to be constant regardless of Ω and h. This fact for any damping ratio has already been pointed
out by Ordaz et al. (2003). However, its proof has never been presented. The proof is shown
here.
The function F (ω ) , called the energy transfer function, can be expressed by

F (ω ) =2hΩω 2(6)π {(Ω2 − ω 2 ) 2 + (2hΩω ) 2}

Four singular points of the function F (ω ) in terms of complex variables can be obtained asz1 = (hi + 1 − h 2 )Ω ,z 2 = ( hi − 1 − h 2 ) Ω ,z3 = (−hi + 1 − h 2 )Ω ,z 4 = ( − hi − 1 − h 2 ) Ω .

Consider an integration path in the complex plane as shown in Fig.2. The singular points inside
the integration path are z1 and z2 . The residues for the singular points z1 and z2 can becomputed as
Res[ z = z1] =z1
2hz12Ω=π ( z1 − z2 )( z1 − z3 )( z1 − z4 ) 4π 1 − h 2 Ωi(7a)Res[ z = z2 ] =2hz2 2Ω− z2=π ( z2 − z1 )( z2 − z3 )( z2 − z4 ) 4π 1 − h2 Ωi(7b)

The integration path consists of one on the real axis and the other on the semi-circle. The
integral for the path on the semi-circle will vanish as the radius becomes infinite. On the other
hand, the integral on the real axis with infinite lower and upper limits corresponding to infinite
radius will be reduced to the residue theorem. The residue theorem provides∞∫ −∞ F (ω )dω = 2π i × ( Res[z = z1] + Res[z = z2 ])(8)

Substitution of Eqs.(7a, b) into Eq.(8) and the property of F (ω ) as an even function lead to thefollowing relation.
∞

2∫ F (ω )dω = 1
0(9)

Eq.(9) indicates that the area of F (ω ) is constant regardless of Ω and h.
It can be stated from Eqs.(4) and (9) that, if the Fourier amplitude spectrum of input groundacceleration is uniform with respect to frequency, the earthquake input energy to a damped linear
SDOF system per unit mass is exactly constant regardless of natural frequency and damping ratio.
Let SV (h = 0) denote the velocity response spectrum for null damping ratio. If A(ω ) isexactly constant with respect to frequency and an assumption A(Ω) ≅ SV (h = 0) (Hudson 1962)
holds, Eqs.(4) and (9) lead to1
EI ≅ m{SV (h = 0)}2
(10)
2
Eq.(10) is similar to the maximum total energy proposed by Housner (1956, 1959). It is notedthat Housner (1959) discussed the maximum total energy defined by EH = max t {− ∫0t mug xdt}
instead of EI defined by Eq.(3) and introduced some assumptions, e.g. slow variation of thetotal energy. SV (h ≠ 0) ≤ SV (h = 0) and a more exact relation A(Ω) ≤ SV (h = 0) can also be
shown for most cases. If SV (h ≠ 0) ≅ A(Ω) =constant holds for a specific damping ratio ha ,Eq.(10) may be replaced by EI ≅ (1/ 2)m{SV (ha )}2 in better approximation. While Housnerdiscussed the constancy of earthquake input energy (maximum total energy) only with respect tonatural period by paying special attention to the uniformity of velocity response spectrum withrespect to natural period (Housner 1956, 1959), another view point based on sophisticatedmathematical treatment has been introduced in this paper. It should be noted that the constancyof earthquake input energy defined by Eq.(3) is directly related to the uniformity of ‘the Fourier
amplitude spectrum’ of ground motion acceleration, not the uniformity of the velocity response
spectrum. This problem will be investigated numerically for some recorded ground motions
afterwards.

Critical Excitation Problem for Earthquake Input Energy withAcceleration Constraint
It is shown in this section that a critical excitation method for earthquake input energy canprovide upper bounds on earthquake input energy. Westermo (1985) has discussed a similar

problem for the maximum input energy to an SDOF system subjected to external forces. His
solution is restrictive because it is of the form including the velocity response quantity containing
the solution itself implicitly. A more general solution procedure will be presented here.
The capacity of ground motions is often defined in terms of the time integral of squared
ground acceleration (Arias 1970; Housner and Jennings 1975; Riddell and Garcia 2001). Thisquantity is well known as the Arias intensity measure except a difference in the coefficient. The
constraint on this quantity can be expressed by∞∞∫ −∞ a(t ) dt = (1/ π )∫ 0 A(ω ) dω = C A
2

2

(11)

where C A is the specified value of the time integral of squared ground acceleration. It is also
clear that the maximum value of the Fourier amplitude spectrum of input ground acceleration isfinite. The infinite spectrum may correspond to a perfect harmonic function or that multipliedby an exponential function (Drenick 1970) which is unrealistic as an actual ground motion. The
constraint on this property may be described byA(ω ) ≤ A ( A : specified value)
The critical excitation problem may be stated as follows:(12)

Critical excitation problem for acceleration
Find A(ω ) that maximizes the earthquake input energy per unit mass, Eq.(4), subject tothe constraints (11) and (12) on ground acceleration.
It is clear from the present author’s work (Takewaki 2001a, b, 2002b) on power spectral
2
density functions that, if A is infinite, A(ω ) turns out to be the Dirac’s delta function which
has a non-zero value at the point maximizing F (ω ) . On the other hand, if A is finite,
2
A(ω ) yields a rectangular function attaining A 2 . The band-width of the frequency can beobtained as ∆ω = π C A / A 2 . The position of the rectangular function, i.e. the lower and upper
limits, can be computed by maximizing A 2 ∫ωωU F (ω )dω . It is noted that ωU − ωL = ∆ω . ItL
can be shown that a good and simple approximation can be made by (ωU + ωL ) / 2 = Ω . The
essential feature of the solution procedure presented in this section is shown in Fig.3. It isinteresting to note that Westermo’s periodic solution (Westermo 1985) may correspond to thecase of infinite A .
The absolute maximum (absolute bound) may be computed for the infinite value A . Thisabsolute maximum can be evaluated as C A /(2hΩ) by employing a reasonable assumption that
F (ω ) attains its maximum at ω = Ω and substituting Eq.(11) into Eq.(4).

Critical Excitation Problem for Earthquake Input Energy with Velocity
Constraint
It is often argued that the maximum ground acceleration controls the behavior of structureswith short natural periods and the maximum ground velocity does the behavior of structures withintermediate or rather long natural periods (see, for example, Tanabashi 1956). On the basis of
this argument, consider the following constraint on ground motion velocity u g (t ) = v(t ) .
∞∞∫ −∞ v(t ) dt = (1/ π ) ∫ 0 V (ω ) dω = CV22

( CV : specified value)

where V (ω ) is the Fourier transform of the ground motion velocity.
A(ω ) = iωV (ω ) , Eq.(4) can be reduced to
EI / m = ∫

∞

0

2

V (ω ) ω 2 F (ω )dω(13)
From the relation(14)

It is clear that the maximum value of V (ω ) is finite in a realistic situation.
constraint on the upper limit on V (ω ) may be described by
V (ω ) ≤ V ( V :upper limit of V (ω ) )The critical excitation problem for velocity constraints may be stated as follows:

The

(15)

Critical excitation problem for velocity
Find V (ω ) that maximizes the earthquake input energy per unit mass, Eq.(14), subject tothe constraints (13) and (15) on ground velocity.
It is clear that almost the same solution procedure as for acceleration constraints can beused by replacing A(ω ) and F (ω ) by V (ω ) and ω 2 F (ω ) , respectively. The functions
ω 2 F (ω ) for various natural periods T=0.5, 1.0, 2.0s and damping ratios h=0.05, 0.20 are plottedin Fig.4. It can be observed that ω 2 F (ω ) becomes larger in peak and wider with increase innatural frequency. In case of a finite value V , the frequency band-width of the critical2
rectangular function V (ω ) can be derived from ∆ω = π CV / V 2 . The upper and lower limitsof the rectangular function can be specified by maximizing V 2 ∫ωωU ω 2 F (ω )dω whereLωU − ωL = ∆ω .
A good and simple approximation can be obtained by employing(ωU + ωL ) / 2 = Ω . The essential feature of the solution procedure presented in this section isshown in Fig.5.
The absolute maximum (absolute bound) may be computed for the infinite value V . Thisabsolute maximum can be evaluated as ΩCV /(2h) by employing an assumption that ω 2 F (ω )attains its maximum at ω = Ω and substituting Eq.(13) into Eq.(14).

Actual Earthquake Input Energy and its Bound for Recorded Ground
Motions
In order to investigate the distance from actual input energy of upper bound of earthquake
input energy presented in the foregoing sections, numerical calculation has been conducted forsome recorded ground motions. The ground motions were chosen from the PEER motions(Abrahamson et al. 1998). Four types of ground motions, i.e. (1) one at rock site in near-fault
earthquake (near-fault rock motion), (2) one at soil site in near-fault earthquake (near-fault soil
motion), (3) one of long-duration at rock site (long-duration rock motion) and (4) one of longduration at soil site (long-duration soil motion). The profile of the selected motions is shown in
Table 1. The Fourier amplitude spectra of these motions (acceleration) are plotted in Figs.6(a)(d). Amax = max A(ω ) and Vmax = max V (ω ) have been used as A and V , respectively.
Due to this treatment of A and V , the bounds, shown in the previous sections, for accelerationand velocity constraints are called ‘credible bounds’ in the following. The selection of A andV may be arguable. It is clear at least that, if A is chosen between Amax = max A(ω ) andinfinity, the corresponding bound of earthquake input energy lies between the credible bound andthe absolute maximum (absolute bound) C A /(2hΩ) . A similar fact can be stated. If V ischosen between Vmax = max V (ω ) and infinity, the corresponding bound of earthquake input
energy lies between the credible bound and the absolute maximum (absolute bound) ΩCV /(2h) .
The quantities Amax , C A , ∆ω corresponding to the critical excitation problem for acceleration
constraints are shown in Table 2 and those Vmax , CV , ∆ω corresponding to the critical excitation
problem for velocity constraints are shown in Table 3.

Fig.7(a) presents the actual earthquake input energy for various natural periods and its
corresponding credible bounds for near-fault rock motions. The damping ratio is fixed to 0.05.
It can be observed that, since the Fourier amplitude spectrum of ground acceleration is not
uniform even in the frequency range of interest in almost all the ground motions, the constancy ofearthquake input energy is not seen in the present case. As for the bound of input energy, it isinteresting to note that the monotonic increase of credible bound for acceleration constraints inthe shorter natural period range results mainly from the characteristic of the function F (ω ) as amonotonically increasing function with respect to natural period (Fig.1 is arranged with respect tonatural frequency). This fact explains mathematically actual phenomena for most groundmotions. It can also be observed that the actual input energy in the shorter natural period range
is bounded properly by the bound for acceleration constraints and that in the intermediate andlonger natural period range is bounded properly by the bound for velocity constraints. These
properties correspond well to the well-known fact (Tanabashi 1956) that the maximum groundacceleration influences the behavior of structures with shorter natural periods and the maximum
ground velocity controls the behavior of structures with intermediate or longer natural periods.
From another point of view, it may be said from Fig.7(a) that, while the behavior of structureswith shorter natural periods is governed by an hypothesis of ‘constant energy’, that of structureswith intermediate or longer natural periods is governed by a hypothesis of ‘constant maximum
displacement’. In the previous studies on earthquake input energy, this property of ‘constantmaximum displacement’ in the longer natural period range has never been considered explicitly.
Fig.7(b) shows the actual earthquake input energy for various natural periods and its
corresponding credible bounds for near-fault soil motions. As in near-fault rock motions, theactual input energy is bounded properly by the two kinds of bound. It is also clear that most of
the bounds in the intermediate natural period range are nearly constant. This fact can be2
explained by Eqs.(4), (9), (11) and the critical shape of A(ω ) as rectangular one. It is notedthat the frequency limits ωL and ωU are varied so as to coincide with the peak of F (ω ) forvaried natural period.
Fig.7(c) shows those for long-duration rock motions and Fig.7(d) illustrates those for longduration soil motions. It can also be observed that the actual input energy is bounded properly
by the two kinds of bound pointed out earlier. It may be concluded that two kinds of boundproposed in this paper provide a physically meaningful unified limit on earthquake input energy
for various types of recorded ground motions.

Robust Design Problem
Consider an n-story shear building model supported by swaying and rocking springs anddashpots. The set of story stiffnesses is denoted by k = {ki } . The design problem treated here
may be stated as:
Design problem for critical state
Find k = {ki } and A(ω )(such that min max EIS
k

A(ω )

)

(16)

subject to∞∞2∫−∞ a(t ) dt = (1/ π ) ∫0 A(ω ) dω = C(17)A(ω ) ≤ A

(18)

∑ in=1 ki = K
ki > 0 (i = 1," , n)

(19)
(20)

2

Solution Procedure for Robust Design Problem = ∆ω denote the frequency bandwidth in the positive frequency range of the criticalLet Ω
rectangular function A(ω ) . It is assumed here that the upper and lower frequency limits of therectangular Fourier amplitude spectrum of the input acceleration can be expressed by , ω = ω − (1/ 2)Ω

ωU = ω1 + (1/ 2)Ω(21)
L1 = π C . The input energy to the structure corresponding to the critical input may be
where A 2Ω
described by
2
Eˆ S = ∫ ∞ F (ω ; k ) A(ω ) d ω ≅ A 2 ∫ωU F (ω; k )d ω = A2 {Φ ( ω ; k ) − Φ ( ω ; k )}
(22)
I

0

S

ωL

S

U

L

where Φ(ω ; k ) = ∫0ω FS (ω ; k )dω .
This insightful approximate manipulation enables ananalytical treatment of the present complicated strongly nonlinear problem.
Let us define the following Lagrange function in terms of a Lagrange multiplier λ .
n

n

(23)
L = Eˆ IS + λ  ∑ ki − K  = A 2 {Φ (ωU ; k ) − Φ ( ωL ; k )} + λ  ∑ ki − K  i =1 i =1The stationarity condition of the Lagrange function with respect to story stiffnesses may bedescribed by∂ L / ∂ ki = Eˆ IS ,i + λ= A2 (∂ω1 / ∂ ki ) {FS (ωU ; k ) − FS ( ωL ; k )}(24)

+ ∫ωωU (∂ FS (ω; k ) / ∂ ki )d ω  + λ = 0LSS
where Eˆ I ,i ≡ ∂ Eˆ I / ∂ ki and Eq.(24) represents the optimality condition.

This robust design problem can be solved by almost the same procedure developed in thereference (Takewaki 2002b) which is based on the incremental inverse problem due to the present
author.

Earthquake Input Energy to SDOF System with Embedded Foundation
Consider a linear elastic SDOF super-structure of story stiffness k and story damping
coefficient c , as shown in Fig.8, with a cylindrical rigid foundation embedded in the uniform
half-space ground. Let r0 and e denote the radius and the depth of the foundation,
respectively. Let m and I R denote the mass of the super-structure and the mass moment ofinertia of the super-structure and let m0 and I R 0 denote the mass of the embedded foundationand the mass moment of inertia of the embedded foundation “around its top center node”. The
height of the super-structure mass from the ground surface is denoted by H .
U 0* and Θ*0 are the effective input motions in the frequency domain for horizontal androtational components, respectively, at the top center of the foundation. The corresponding
effective input motions in the time domain may be expressed by
1 ∞ *
1 ∞
iω t
iω t
u0* (t ) =(25a)∫−∞ U 0 (ω )e dω =∫−∞ S HT (ω )Ug (ω )e dω
2π
2π
1 ∞  *
1 ∞
iω t
iω t
θ0* (t ) =(25b)∫−∞ Θ0 (ω )e dω =∫−∞ S RT (ω )Ug (ω )e dω2π
2π
S HT (ω ) and S RT (ω ) are the ratios of the effective input motions, U 0* and Θ*0 , in thefrequency domain for horizontal and rotational components, respectively, at the top center of thefoundation to the Fourier transform U g (ω ) of the free-field horizontal ground-surface
displacement. Let us assume that only a vertically incident shear wave (SH wave) is considered.

S HT (ω ) and S RT (ω ) are expressed in terms of the ratios S HB (ω ) and S RB (ω ) , given in
Meek and Wolf (1994) and Wolf (1994), of the effective input motions in the frequency domain
for horizontal and rotational ( × r0 ) components at the bottom center of the foundation to U g (ω ) .
S HT (ω ) = S HB (ω ) + (e / r0 ) S RB (ω )(26a)
S RT (ω ) = (1/ r0 ) S RB (ω )(26b)
The frequency-domain formulation of the earthquake input energy to the soil-structureinteraction system can be found in the reference (Takewaki and Fujimoto 2003).
Examples of time histories of earthquake input energies (no embedment of foundation) andtheir final values E IA , E IS for El Centro NS (Imperial Valley 1940) and Kobe University NS(Hyogoken-Nanbu 1995) are shown in Fig.9 for natural period of the structure T=0.5(s) andground shear wave velocity Vs=50, 100(m/s). A similar concept has been proposed by Yang andAkiyama (2000).
Fig.10 shows the earthquake input energies to the structure-foundation-soil system E IA
and the structure only EIS with various degrees of foundation embedmente / r0 = 0.0, 0.5,1.0, 2.0 for the ground equivalent shear wave velocity=100(m/s) to El Centro NS
of Imperial Valley 1940. It can be observed from Fig.10 that, while the input energy to thestructure alone is smaller than that to the structure-foundation-soil system in all the natural period
range up to 2.0(s) for e / r0 = 0.0, 2.0 , that relation does not exist for e / r0 = 0.5,1.0 . Moredetailed examination for a broader range of parameters will be necessary to clarify the effect ofdegree of foundation embedment on the input energies to a structure and to the corresponding
structure-foundation-soil system.
It should be remarked that only once computation of the Fourier amplitude spectra ofground motion accelerations is necessary and the input energy can be evaluated by combining
those, through numerical integration in the frequency domain, with the energy transfer function.
The structural designers can understand easily approximate input energies from the relation of theFourier amplitude spectra of ground motion accelerations with the energy transfer functions bothof which are expressed in the frequency domain.
The solid lines in Fig.11 show the plots of earthquake input energies by the ground motionof El Centro NS to the overall system (structure plus surrounding soil) without embedment offoundation and the structure alone for Vs=50, 100, 200(m/s) with respect to natural period of the
fixed-base structure. The damping ratio of the super-structure is 0.05. It can be observed from
Fig.11 that the input energy to stiff structures with short natural periods is governed primarily by
the energy dissipated by the ground (surrounding soil) and the input energy to flexible structureswith intermediate natural periods (around 1(s)) is governed mainly by the energy dissipated by thedamping of super-structures. This phenomenon corresponds well to the well-known fact that thesoil-structure interaction effect is notable in the stiff structures on flexible ground. The dottedlines in Fig.11 show the credible and absolute bounds of earthquake input energies by the groundmotion of El Centro NS to the overall system and the structure alone. As the shear wave
velocity of the ground becomes larger, the input energy is governed mainly by the energy
dissipated by the damping of super-structures. It should also be pointed out that the groundmotion of El Centro NS does not have a notable predominant period and the distance between theactual input energy and the credible bound is almost constant with respect to natural priod of thesuper-structure.

Conclusions
The conclusions may be stated as follows.

(1) The function F (ω ) characterizing the earthquake input energy in the frequency domain to adamped linear elastic SDOF system has been proved to have an equi-area property regardlessof natural period and damping ratio. This property guarantees that, if the Fourier amplitude
spectrum of ground motion acceleration is uniform with respect to frequency, the constancy ofearthquake input energy defined by Eq.(3) holds strictly. Otherwise, its constancy is notguaranteed. It should be remarked that the constancy of earthquake input energy defined by
Eq.(3) is directly related to the uniformity of ‘the Fourier amplitude spectrum’ of groundmotion acceleration, not the uniformity of the velocity response spectrum.
(2) A new critical excitation method has been formulated which has the earthquake input energy
as a new measure of criticality and has acceleration and/or velocity constraints (time integralof squared base acceleration and time integral of squared base velocity). No mathematicalprogramming technique is needed in this method and structural engineers can find the solution
without difficulty.
(3) The solution to the aforementioned 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 ashorter natural period range and that with velocity constraints can bound properly theearthquake input energy in an intermediate or longer natural period range.
(4) A new critical excitation method has been developed for soil-structure interaction systems.
Definition of two input energies, one to the overall system (structure plus surrounding soil)
and the other to the structure alone is very useful in understanding the mechanism of energy
input and the effect of soil-structure interaction under various conditions of soil properties andnatural period of structures.
(5) Even soil-structure interaction systems including embedded foundations can be treated in a
simple way and effects of the foundation embedment on the earthquake input energy to thesuper-structure can be clarified systematically by the proposed frequency domain formulation.
It can be stated from a limited analysis that the input energy to the sway-rocking model
without embedment is almost the same as that to the fixed-base model. As the degree of
embedment becomes larger, the input energy is decreased regardless of the natural period
range. The ratio of the input energy to the structure alone to that to the structure-foundationsoil system is affected in a complicated manner by the degree of embedment.
The evaluation of the earthquake input energy in the time domain is suitable for theevaluation of the time history of the input energy, especially for non-linear systems. Dual use of
the frequency-domain and time-domain techniques may be preferable in the advanced seismic
analysis for robuster design.
Appendix I. References
Abrahamson, N., Ashford, S., Elgamal, A., Kramer, S., Seible, F., and Somerville, P. (1998).
Proceedings of 1st PEER Workshop on Characterization of Special Source Effects.
Akiyama, H. (1985). Earthquake Resistant Limit-State Design for Buildings. University of Tokyo
Press, Tokyo, Japan.
Anderson, J.C., and Bertero, V.V. (1987). "Uncertainties in establishing design earthquakes."
Journal of Structural Engineering, ASCE, 113(8), 1709-1724.
Arias, A. (1970). "A measure of earthquake intensity." In Seismic Design for Nuclear Power
Plants (Ed. R.J.Hansen), The MIT Press, Cambridge, MA, 438-469.
Berg, G.V., and Thomaides T.T. (1960). "Energy consumption by structures in strong-motion
earthquakes." Proceedings of 2nd World Conf. on Earthquake Engineering, Tokyo and Kyoto,
681-696.

Drenick, R.F. (1970). "Model-free design of aseismic structures." Journal of Engineering
Mechanics Division, ASCE, 96(EM4), 483-493.
Fajfar, P., and Vidic, T. (1994). "Consistent inelastic design spectra: hysteretic and input energy."
Earthquake Engineering and Structural Dynamics, 23(5), 523-537.
Geller, R.J., Jackson, D.D., Kagan, Y.Y., and Mulargia, F. (1997). "Earthquakes cannot be
predicted." Science, 275, 1616.
Goel, S.C., and Berg, G.V. (1968). "Inelastic earthquake response of tall steel frames." Journal of
Structural Division, ASCE, 94, 1907-1934.
Housner, G.W. (1956). "Limit design of structures to resist earthquakes." Proceedings of the FirstWorld Conference on Earthquake Engineering, Berkeley, CA, 5:1-11.
Housner, G.W. (1959). "Behavior of structures during earthquakes." Journal of the Engineering
Mechanics Division, ASCE, 85(4), 109-129.
Housner, G.W., and Jennings, P.C. (1975) "The capacity of extreme earthquake motions todamage structures." Structural and geotechnical mechanics: A volume honoring
N.M.Newmark edited by W.J. Hall, 102-116, Prentice-Hall Englewood Cliff, NJ.
Hudson, D.E. (1962). "Some problems in the application of spectrum techniques to strong-motion
earthquake analysis." Bulletin of Seismological Society of America, 52, 417-430.
Kato, B., and Akiyama, H. (1975). "Energy input and damages in structures subjected to severe
earthquakes." Journal of Structural and Construction Eng., Archi. Inst. of Japan, 235, 9-18
(in Japanese).
Krawinkler, H., Medina, R., and Alavi, B. (2001). “Seismic drift and ductility demands and theirdependence on ground motions.” Proc. of US-Japan seminar on advanced stability andseismicity concepts for performance-based design of steel and composite structures, Kyoto,
Japan, July 23-26.
Kuwamura, H., Kirino, Y., and Akiyama, H. (1994). "Prediction of earthquake energy input fromsmoothed Fourier amplitude spectrum." Earthquake Engineering and Structural Dynamics,
23, 1125-1137.
Leger, P., and Dussault, S. (1992). "Seismic-energy dissipation in MDOF structures." Journal of
Structural Engineering, ASCE, 118(5) 1251-1269.
Lyon, R.H. (1975). Statistical energy analysis of dynamical systems, The MIT Press, Cambridge,
MA.
Mahin, S.A., and Lin, J. (1983). "Construction of inelastic response spectrum for single-degreeof-freedom system." Report No. UCB/EERC-83/17, Earthquake Engineering Research Center,
Univ. of California, Berkeley, CA.
Meek, J.W., Wolf, J.P. (1994) "Cone models for embedded foundation." J Geotechnical Engng,
ASCE, 120(1), 60-80.
Ohi, K., Takanashi, K., and Tanaka, H. (1985). "A simple method to estimate the statisticalparameters of energy input to structures during earthquakes." Journal of Structural andConstruction Eng., Archi. Inst. of Japan, 347, 47-55 (in Japanese).
Ogawa, K., Inoue, K., and Nakashima, M. (2000). "A study on earthquake input energy causing
damages in structures." Journal of Structural and Construction Eng., Archi. Inst. of Japan,
530, 177-184 (in Japanese).
Ordaz, M., Huerta, B., and Reinoso, E. (2003). "Exact computation of input-energy spectra from
Fourier amplitude spectra" Earthquake Engineering and Structural Dynamics, 32, 597-605.
Page, C.H. (1952). "Instantaneous power spectra." Journal of Applied Physics, 23(1), 103-106.
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.
Riddell, R., and Garcia, J.E. (2001). "Hysteretic energy spectrum and damage control."Earthquake Engineering and Structural Dynamics, 30, 1791-1816.

Shinozuka, M. (1970). "Maximum structural response to seismic excitations." Journal ofEngineering Mechanics Division, ASCE, 96(EM5), 729-738.
Singh, J.P. (1984). “Characteristics of near-field ground motion and their importance in building
design.” ATC-10-1 Critical aspects of earthquake ground motion and building damage
potential, ATC, 23-42.
Stein, R.S. (2003). "Earthquake conversations." Scientific American, 288(1), 72-79.
Takewaki, I. (2001a). "A new method for nonstationary random critical excitation." Earthquake
Engineering and Structural Dynamics, 30(4), 519-535.
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Takewaki, I. (2002a). "Critical excitation method for robust design: A review." Journal of
Structural Engineering, ASCE, 128(5), 665-672.
Takewaki, I. (2002b). "Robust building stiffness design for variable critical excitations." Journalof Structural Engineering, ASCE, 128(12) 1565-1574.
Takewaki, I. and Fujimoto, H. (2003). "Earthquake input energy to soil-structure interaction
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Tanabashi, R. (1956). "Studies on the nonlinear vibrations of structures subjected to destructive
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Wolf, J.P. (1994) Foundation vibration analysis using simple physical models. Prentice-Hall,Englewood Cliffs, NJ.
Yang, Z and Akiyama, H. (2000). "Evaluation of soil-structure interaction in terms of energy
input." Journal of Structural and Construction Eng., Archi. Inst. of Japan, 536, 39-45 (inJapanese).
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Fig.1 Energy transfer function F (ω ) for natural periods T=0.5, 1.0, 2.0s and damping ratios
h=0.05, 0.20

Fig.2 Integration path in complex plane and singular points z1, z2 of function F (ω ) inside
the path and those z3 , z4 outside the pathFig.3 Schematic diagram of solution procedure for critical excitation problem with acceleration
constraints

Fig.4 Function ω 2 F (ω ) for natural periods T=0.5, 1.0, 2.0s and damping ratios h=0.05, 0.20

10

10

Loma Prieta 1989,
Los Gatos NS
8

6

4

2

0

0

20

40
60
frequency (rad/s)

80

100

Fourier amplitude of acceleration (m/s)

Fourier amplitude of acceleration (m/s)

Fig.5 Schematic diagram of solution procedure for critical excitation problem with velocity
constraints

Hyogoken-Nanbu 1995,
JMA Kobe NS

8

6

4

2

0

0

20

Fig.6(a) Near-fault rock motion
Fig.6 Fourier amplitude spectrum of ground motion acceleration:(a) Near-fault rock motion,(b) Near-fault soil motion,(c) Long-duration rock motion,(d) Long-duration soil motion

40
60
frequency (rad/s)

80

100

8

6

4

2

20

40
60
frequency (rad/s)

80

100

10

Northridge 1994,
Rinaldi NS
8

6

4

2

0

0

20

40
60
frequency (rad/s)

80

100

Fourier amplitude of acceleration (m/s)

Fourier amplitude of acceleration (m/s)

10Northridge 1994,
Sylmar NS

8

6

4

2

0

0

20

40
60
frequency (rad/s)

80

100

5

Imperial Valley 1979,
Meloland NS
4

3

2

1

0

0

20

40
60
frequency (rad/s)

80

Fourier amplitude of acceleration (m/s)

0Fourier amplitude of acceleration (m/s)

0100Fourier amplitude of acceleration (m/s)

Fourier amplitude of acceleration (m/s)

Fourier amplitude of acceleration (m/s)

Cape Mendocino 1992,
Petrolia NS

Fourier amplitude of acceleration (m/s)

1010Cape Mendocino 1992,
Petrolia EW
8

6

4

2

0

0

20

40
60
frequency (rad/s)

80

100

10

Northridge 1994,
Rinaldi EW
8

6

4

2

0

0

20

40
60
frequency (rad/s)

80

100

10

Northridge 1994,
Sylmar EW
8

6

4

2

0

0

20

40
60
frequency (rad/s)

80

100

5

Imperial Valley 1979,
Meloland EW
4

3

2

1

0

0

20

Fig.6(b) Near-fault soil motion

40
60
frequency (rad/s)

80

100

Michoacan 1985,
Caleta de Campos NS
2

1.5

1

0.5

0

0

20

40
60
frequency (rad/s)

80

Fourier amplitude of acceleration (m/s)

Fourier amplitude of acceleration (m/s)

2.5

2.5

Miyagiken-oki,
Ofunato NS
2

1.5

1

0.5

100

0

0

20

40
60
frequency (rad/s)

80

100

Fourier amplitude of acceleration (m/s)

10Chile 1985,
Vina del Mar NS
8

6

4

2

0

0

20

40
60
frequency (rad/s)

80

100

Fourier amplitude of acceleration (m/s)

2.5

Olympia 1949,
Seattle Army Base NS
2

1.5

1

0.5

0

0

20

40
60
frequency (rad/s)

80

100

Fourier amplitude of acceleration (m/s)

Fourier amplitude of acceleration (m/s)

Fig.6(c) Long-duration rock motion

10

Chile 1985,
Vina del Mar EW
8

6

4

2

0

0

20

40
60
frequency (rad/s)

80

100

2.5

Olympia 1949,
Seattle Army Base EW
2

1.5

1

0.5

0

0

20

40
60
frequency (rad/s)

Fig.6(d) Long-duration soil motion

80

100

30

30

Loma Prieta 1989,
Los Gatos NS

25
20

∞

15
∞

∫ −∞ a (t ) dt = C A

102

h=0.05∫ −∞ v (t ) dt = CV
2

5
0

0

1

2

3

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

2
2

35

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

input energy per unit mass (m /s )

40

4

25

Hyogoken-Nanbu 1995,
JMA Kobe NS
h=0.05

20
15
10
5
0

0

natural period (s)

1

2

3

4

natural period (s)Fig.7(a) Near-fault rock motion
Fig.7 Actual earthquake input energy (damping ratio 0.05), credible bound for acceleration
constraints and credible bound for velocity constraints:(a) Near-fault rock motion,(b) Near-fault soil motion,(c) Long-duration rock motion,(d) Long-duration soil motion

20
2

15
Cape mendocino 1992,
Petrolia NS
h=0.0510∞∫ −∞ a (t ) dt = C A
5

0

2

∞

∫ −∞
0

1

2

v (t ) dt = CV2

3

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

input energy per unit mass (m /s )actual input energy
credible bound (acc)credible bound (vel)2

input energy per unit mass (m /s )

20

15

10

5

0

4

Cape mendocino 1992,
Petrolia EW
h=0.050

1

2

Northridge 1994,
Rinaldi NS
h=0.05
10

5

0

1

2

3

15

10

5

0

4

Northridge 1994,
Rinaldi EW
h=0.050

1natural period (s)

4

2

5

0

1

2

3

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

input energy per unit mass (m /s )22

input energy per unit mass (m /s )Northridge 1994,
Sylmar NS
h=0.05

10

Northridge 1994,
Sylmar EW
h=0.05

10

5

0

4

0

1

2

3

4

natural period (s)

10

10

8

2

Imperial Valley 1979,
Meloland NS
h=0.05

6

4

2

0

1

2

3

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

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

2
2

3

15
actual input energy
credible bound (acc)credible bound (vel)natural period (s)input energy per unit mass (m /s )2natural period (s)

15

0

4

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

input energy per unit mass (m /s )

2

15

0

3

20
actual input energy
credible bound (acc)credible bound (vel)2

input energy per unit mass (m /s )

20

0

2

natural period (s)natural period (s)48Imperial Valley 1979,
Meloland EW

h=0.05

6

4

2

0

0

natural period (s)Fig.7(b) Near-fault soil motion

1

2

natural period (s)

3

4

1
2

Michoacan 1985,
Caleta de Campos NS
h=0.051∞∞∫ −∞ a (t ) dt = C A

0.5

0

0

∫ −∞ v(t ) dt = CV
2

2

1

2

3

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

2

2

1.5

input energy per unit mass (m /s )actual input energy
credible bound (acc)credible bound (vel)2

input energy per unit mass (m /s )

2

0.75

Miyagiken-oki 1978,
Ofunato NS
h=0.05

0.5

0.25

0

0

4

1

2

3

4

natural period (s)natural period (s)Fig.7(c) Long-duration rock motion

10
2
2

2

input energy per unit mass (m /s )actual input energy
credible bound (acc)credible bound (vel)2

input energy per unit mass (m /s )

40

30
∞

∫ −∞

Chile 1985,
Vina del Mar NS
h=0.0520∞∫ −∞ v(t ) dt = CV

10

0

2

a ( t ) dt = C A

0

1

2

2

3

7.5
actual input energy
credible bound (acc)credible bound (vel)

5

2.5

0

4

Chile 1985,
Vina del Mar EW
h=0.05

0

1

3

4

1
2

Olympia 1949,
Seattle Army Base NS1

h=0.05

0.5

0

1

2

3

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

2

2

1.5

input energy per unit mass (m /s )actual input energy
credible bound (acc)credible bound (vel)2

input energy per unit mass (m /s )

2

0

2

natural period (s)natural period (s)4Olympia 1949,
Seattle Army Base EW
0.5
h=0.050

01natural period (s)Fig.7(d) Long-duration soil motion

2

natural period (s)34m, I RH

ug (t ) = a (t )

*
*
k HH
, kHR

c*HH , c*HRek,c

c*RH , c*RR

H
m0 , I R 0

*
*
k RH
, kRR

r0
Fig.8 Linear elastic SDOF super-structure of story stiffness k and story damping coefficient
c with a cylindrical rigid foundation embedded in the uniform half-space ground

4

6 10

4

T=0.5(s), h=0.05,
Vs=50(m/s)

4

input energy (J)input energy (J)5 10overallstructure

4 10

EIA

4

3 10

4

2 10

EIS

4

4

2.5 1042 10

4

1.5 10

4

1 10

4

Kobe University NS 1995
T=0.5(s), h=0.05,
Vs=100(m/s)
overallstructure

EIA

5000

1 10

00

3 10
El Centro NS 1940

10

20
time (s)

30

40

00

EIS
5

10
time (s)15

20

Fig.9 Examples of time histories of earthquake input energies (no embedment of foundation)and their final values E IA , E IS for El Centro NS (Imperial Valley 1940) and Kobe
University NS (Hyogoken-Nanbu 1995)

5

5

1 10

1 10

structure
structure + soil

structure
structure + soil
4

actual input energy (J)actual input energy (J)

4

8 10

Vs=100(m/s)
e/r=0.0

4

6 10

4

4 10

4

8 10

Vs=100(m/s)
e/r=0.5

4

6 10

4

4 10

4

2 10

2 10

El Centro NS 1940

El Centro NS 1940
0

0

0.5

1

1.5

0

2

0

0.5

1

1 10

structure
structure + soil

structure
structure + soil
4

8 10

actual input energy (J)4actual input energy (J)

2

5

5

1 10

Vs=100(m/s)
e/r=1.0

4

6 10

4

4 10

4

8 10

Vs=100(m/s)
e/r=2.0

4

6 10

4

4 10

4

2 10

2 10

El Centro NS 1940

El Centro NS 1940
0

1.5

natural period (s)natural period (s)

0

0.5

1

1.5

0

2

0

0.5

1

1.5

2

natural period (s)natural period (s)Fig.10 Earthquake input energies to the structure-soil system and the structure only with various
degrees of foundation embedment for ground equivalent shear velocity 100 (m/s) to El
Centro NS of Imperial Valley 1940

actual input (overall)credible bound (overall)absolute bound (overall)
actual input (structure)credible bound (structure)absolute bound (structure)

actual input (overall)credible bound (overall)absolute bound (overall)
actual input (structure)credible bound (structure)absolute bound (structure)
2 105

1.5 105

1 10

5

Vs=100(m/s)

El Centro NS 1940

El Centro NS 1940

h=0.05

5 10

4

0

0

0.2

0.4

5

Vs=50(m/s)

input energy (J)input energy (J)

2 10

0.6

0.8

natural period (fixed base) (s)1

1.5 10

5

1 10

5

5 10

4

Fig.11: Earthquake input
energies by theground motion of
El Centro NS
1940 to overallsystem andstructure alone;(a) Vs=50(m/s),(b) Vs=100(m/s)

h=0.05

0

0

0.2

0.4

0.6

0.8

natural period (fixed base) (s)1Table 1 Ground motions selected from PEER motions (Abrahamson et al. 1998)
magnitude
earthquakeug max (g) u g max (m/s) u g max (m)site and component

Mw

Los Gatos NS
JMA Kobe NS

6.9
6.9

0.570
0.833

0.988
0.920

0.379
0.206

Petrolia NS
Petrolia EW
Rinaldi NS
Rinaldi EW
Sylmar NS
Sylmar EW
Meloland NS
Meloland EW

7.0

0.589
0.662
0.480
0.841
0.842
0.604
0.317
0.297

0.461
0.909
0.795
1.726
1.288
0.778
0.711
0.943

0.265
0.268
0.505
0.487
0.306
0.203
1.242
3.124

8.1
7.4

0.141
0.211

0.255
0.131

1.464
0.163

8.0

0.362
0.214
0.06780.06730.337
0.267
0.07850.0777

2.400
1.212
0.192
0.0278(Near fault motion/rock records)

Loma Prieta 1989
Hyogoken-Nanbu 1995
(Near fault motion/soil records)
Cape Mendocino 1992
Northridge 1994

Imperial Valley 1979

(Long duration motion /rock records)
Michoacan 1985
Caleta de Campos NS
Miyagiken-oki 1978
Ofunato NS(Long duration motion /soil records)
Chile 1985
Vina del Mar NS
Vina del Mar EW
Olympia 1949
Seattle Army Base NS
Seattle Army Base EW

6.7

6.5

6.5

Table 2 Maximum Fourier amplitude spectrum of ground motion acceleration, time integralof squared ground motion acceleration and frequency band-width of critical
rectangular Fourier amplitude spectrum of ground motion acceleration
earthquakesite and component

Amax (m/s) C A (m 2 /s3 )∆ω (rad/s)(Near fault motion/rock records)

Loma Prieta 1989
Hyogoken-Nanbu 1995
(Near fault motion/soil records)
Cape Mendocino 1992
Northridge 1994

Imperial Valley 1979

Los Gatos NS
JMA Kobe NS

6.80
5.81

49.5
52.3

3.36
4.87

Petrolia NS
Petrolia EW
Rinaldi NS
Rinaldi EW
Sylmar NS
Sylmar EW
Meloland NS
Meloland EW

4.49
3.85
2.98
4.70
3.92
2.95
2.01
3.09

21.5
23.9
25.0
46.3
31.3
16.3
5.43
6.93

3.35
5.07
8.84
6.58
6.40
5.88
4.22
2.28

1.33
1.03

3.97
2.35

7.05
6.96

7.87
4.14
1.57
1.12

34.3
18.7
1.28
0.877

1.74
3.43
1.63
2.20

(Long duration motion /rock records)
Michoacan 1985
Caleta de Campos NS
Miyagiken-oki 1978
Ofunato NS(Long duration motion /soil records)
Chile 1985
Vina del Mar NS
Vina del Mar EW
Olympia 1949
Seattle Army Base NS
Seattle Army Base EW

Table 3 Maximum Fourier amplitude spectrum of ground motion velocity, time integral ofsquared ground motion velocity and frequency band-width of critical rectangularFourier amplitude spectrum of ground motion velocity
earthquakesite and component

Vmax (m)CV (m 2 /s) ∆ω (rad/s)(Near fault motion/rock records)

Loma Prieta 1989
Hyogoken-Nanbu 1995
(Near fault motion/soil records)
Cape Mendocino 1992
Northridge 1994

Imperial Valley 1979

Los Gatos NS
JMA Kobe NS

1.81
0.746

1.49
0.854

1.43
4.82

Petrolia NS
Petrolia EW
Rinaldi NS
Rinaldi EW
Sylmar NS
Sylmar EW
Meloland NS
Meloland EW

0.531
0.697
1.01
1.02
1.22
0.968
0.738
1.44

0.253
0.509
0.62
1.13
0.858
0.45
0.356
1.06

2.82
3.29
1.90
3.42
1.81
1.51
2.05
1.61

0.408
0.087

0.07590.0119

1.44
4.89

0.865
0.563
0.224
0.189

0.455
0.199
0.02320.0154

1.91
1.97
1.45
1.35

(Long duration motion /rock records)
Michoacan 1985
Caleta de Campos NS
Miyagiken-oki 1978
Ofunato NS(Long duration motion /soil records)
Chile 1985
Vina del Mar NS
Vina del Mar EW
Olympia 1949
Seattle Army Base NS
Seattle Army Base EW