Full text of DOI:10.1002/eqe.312

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS
Earthquake Engng Struct. Dyn. 2003; 32:2013–2025 (DOI: 10.1002/eqe.312)
An improved capacity spectrum method for ATC-40
Yu-Yuan Lin and Kuo-Chun Chang∗;†
Department of Civil Engineering; National Taiwan University; Taipei 106; Taiwan
SUMMARY
In order to account for the non-linear behavior of structures via non-linear static procedure, the capacity
spectrum method has been adopted by ATC-40 for evaluation and retroꢀt of reinforced concrete build-
ings. For elastic-perfectly-plastic SDOF systems, the accuracy of the capacity spectrum method depends
only on the acceleration response spectrum chosen to form the demand spectrum and the adopted model
for calculating the equivalent viscous damping ratios. According to this method, the pseudo-acceleration
response spectrum (PSa) is used to create the demand diagram. It is found that the ATC-40 procedure,
using its Type A hysteretic model, may be inaccurate especially for systems with damping ratios greater
than 10% and periods longer than 0:15 sec. In order to improve the accuracy of the capacity spectrum
method, this study proposes to use the real absolute acceleration response spectrum (Sa) instead of the
PSa to establish the demand diagram. The step-by-step procedure of the improved method and exam-
ples are implemented in this paper to illustrate the calculations of earthquake-induced deformations.
In addition, three selected models of equivalent viscous damping are also compared in this paper to
assess the accuracy of the model used in the ATC-40 procedure. Results show that the WJE damping
model may be used by the capacity spectrum method to reasonably predict the inelastic displacements
when the ductility demand (ꢀ) of the structures is less than 4, whereas the damping model proposed by
Kowalsky can be implemented when ꢀ¿4:0. Alternatively, the damping model proposed by Kowalsky
may be used to calculate the equivalent viscous damping for the entire range of ductility. Copyright
? 2003 John Wiley & Sons, Ltd.
KEY WORDS: capacity spectrum method; equivalent linear systems; absolute acceleration; pseudo-
acceleration
1. INTRODUCTION
The capacity spectrum method (CSM), adopted in ATC-40 [1], was ꢀrst introduced in the
1970s as a rapid evaluation procedure for assessing the seismic vulnerability of buildings
[2–4]. This procedure compares the capacity of the structure in the form of a pushover curve
∗ Correspondence to: Kuo-Chun Chang, Department of Civil Engineering, National Taiwan University, No. 1 Sec.
4, Roosevelt Road, Taipei 106, Taiwan.
† E-mail: ciekuo@ccms.ntu.edu.tw
Contract=grant sponsor: National Science Council, Taiwan; contract=grant number: NSC-90-2811-Z-002-003
Contract=grant sponsor: Sinotech Engineering Consultant of Taiwan; contract=grant number: 6120
Received 21 May 2002
Revised 30 September 2002 and 10 January 2003
Copyright
2003 John Wiley & Sons, Ltd.
Accepted 17 February 2003
?

2014
Y.-Y. LIN AND K.-C. CHANG
with demands on the structure in the form of an elastic response spectrum. The graphical
intersection of the two curves approximates the response of the structure [2; 3; 5; 6]. In order
to account for these eꢁects of non-linear behavior of structures, equivalent viscous damping has
been implemented to modify this elastic response spectrum to consider the inelastic behavior.
Implied in the capacity spectrum method is that earthquake-induced deformation of a non-
linear single-degree-of-freedom (SDOF) system can be approximately estimated by an iterative
procedure of a sequence of equivalent linear SDOF systems. Therefore, it avoids the dynamic
analysis of inelastic systems.
It should be noted that ATC-40 modiꢀed the capacity spectrum method by proposing its
own method for determining the equivalent damping ratios. The ATC-40 method was based
on an idealistic hysteretic model.
After the capacity spectrum method was proposed by ATC-40, Chopra and Goel [7; 8]
pointed out that the procedure signiꢀcantly underestimated the deformation for a wide range of
periods when using the Type A idealized hysteretic damping model. An improved method
using the inelastic response spectrum as the demand diagram was proposed by Fajfar [9] and
Chopra and Goel [7; 8]. It should be noted that in actual practice, for existing reinforced
concrete structures, the ATC-40 procedure uses the Type B and C damping values which
result in substantially smaller damping ratios.
For elastic-perfectly-plastic SDOF systems, the accuracy of the capacity spectrum method
depends only on: (a) the acceleration response spectrum chosen to form the demand spectrum;
and (b) the adopted model of equivalent viscous damping. Although the errors generated
above can be eliminated by using the inelastic response spectrum [7–9], the capacity spectrum
method has the advantage of using the equivalent linear systems to predict the responses of
non-linear systems.
According to the capacity spectrum method used in ATC-40, the pseudo-acceleration (PSa),
an approximate acceleration response spectrum derived from the displacement response spec-
trum (Sd), was used to create the demand diagram based on the relationship of PSa = !2Sd,
where ! (Hz) represents the natural frequency of the systems. In this study it is found that
such a situation will obviously lose the accuracy of iteration results, especially for systems
with damping ratios greater than 10% and periods longer than 0:15 sec. To improve the ac-
curacy of the capacity spectrum method without using the inelastic response spectrum, this
paper suggests that the real acceleration response spectrum (Sa) is used instead of the pseudo-
acceleration response spectrum (PSa) to establish the demand diagram.
In addition, the equivalent viscous damping model plays another signiꢀcant role in the
capacity spectrum method. If the equivalent damping ratios of the equivalent linear systems
cannot be determined appropriately, the displacement demand of structures will not be accu-
rately obtained even when the real absolute acceleration response spectrum (Sa) is used. For
the purpose of determining the equivalent viscous damping ratios used in ATC-40 with rea-
sonable accuracy, three equivalent viscous damping models are also discussed and compared
in this paper.
2. EQUIVALENT LINEAR SYSTEMS
Figure 1 shows the force–deformation relationship of a bilinear SDOF system, where k; ꢁ; f ;
y
uy; umax and ꢀ are the elastic stiꢁness, post-yield stiꢁness ratio, yield strength, yield displace-
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AN IMPROVED CAPACITY SPECTRUM METHOD FOR ATC-40
2015
fu
ꢀk
fy
k
eq
k
uy
umax
Displacement
Figure 1. Force–displacement relationship of bilinear SDOF system.
ment, maximum displacement and ductility ratio (ꢀ = umax=uy), respectively. The maximum
force and displacement response of the bilinear system can be estimated by an equivalent
linear system [10–13] with secant stiꢁness keq, natural period Teq and equivalent viscous
damping ratio ꢂeq [8], where
ꢀ
ꢀ
Teq = Tn
(1)
(2)
1 + ꢁꢀ − ꢁ
1 ED 2 (ꢀ − 1)(1 − ꢁ)
=
ꢂeq = ꢂ0 + ꢂh; and ꢂh =
4ꢃ ES ꢃ ꢀ(1 + ꢁꢀ − ꢁ)
In Equations (1) and (2), Tn = elastic natural vibration period of the bilinear system; ꢂo =
inherent damping ratio of the bilinear system vibrating within its linearly elastic range; ꢂh =
equivalent hysteretic damping ratio; ED = energy dissipated in the bilinear system, i.e. enclosed
area of the hysteretic loops; and ES = strain energy of the system with stiꢁness keq. In ATC-40,
a modiꢀed ꢂeq is used as
ꢂeq = ꢂ0 + ꢄꢂh
(3)
where ꢂh is limited to 45%. The damping modiꢀcation factor ꢄ depends on the types of
hysteretic behavior of the system [1]. Type A denotes hysteretic behavior with stable and full
hysteresis loops. For such systems, ꢄ = 1 if ꢂh616:25%; ꢄ = 0:77 if ꢂh¿45%, and ꢄ = linear
interpolation if 16:25%6ꢂh645%. Type C represents severely pinched loops; and the hys-
teretic behavior of the Type B system is between Type A and Type C. Furthermore, ATC-40
speciꢀes three diꢁerent procedures (Procedures A, B and C) to estimate the earthquake-
induced deformation demand, all based on the same principles but diꢁerent in their methods
of implementation. Procedure A is suggested to be the best of the three procedures.
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Y.-Y. LIN AND K.-C. CHANG
3. PSEUDO-ACCELERATION RESPONSE SPECTRUM (PSa) vs. ABSOLUTE
ACCELERATION RESPONSE SPECTRUM (Sa)
In the capacity spectrum method of ATC-40, the ordinate of the pseudo-acceleration response
spectrum (PSa) is used as the vertical axis and the ordinate of the displacement response
spectrum (Sd) is used as the horizontal axis to create the demand diagram. For a linear
SDOF system with viscous damping and subjected to ground acceleration uꢂg(t), the equation
of motion is given as
muꢂ(t) + cu˙(t) + ku(t) = −muꢂg(t)
(4)
where m; c; k; ! and ꢂ are the mass, damping coeꢃcient, stiꢁness, natural frequency and vis-
cous damping ratio of the system, respectively; u(t); u˙(t); uꢂ(t) and uꢂt(t) = uꢂ(t)+uꢂg(t) are the
relative displacement, relative velocity, relative acceleration and the absolute acceleration of
the system, respectively. The deꢀnitions of response spectra are: Sd ≡ |u(t)|max = displacement
response spectrum and Sa ≡ |uꢂt(t)|max = |uꢂ(t)+uꢂg(t)|max = (absolute) acceleration response spec-
trum. Moreover, the pseudo-acceleration response spectrum (PSa) is obtained from the dis-
placement response spectrum (Sd) based on the relationship of PSa = !2Sd.
The displacement and acceleration response spectra of Equation (4) can be obtained by
using the Duhamel’s integral as [14; 15]:
1
ꢁ
Sd ≡ |u(t)|max = −
|S(t)|max
(5)
(6)
!
1 − ꢂ2
Sa ≡ |uꢂt(t)|max = |!2(1 − 2ꢂ2)u(t) + 2ꢂ!C(t)|max
where
S(t) =
ꢂ
ꢂ
t
t
uꢂg(ꢅ) e−ꢂ!(t−ꢅ) sin !d(t − ꢅ) dꢅ; C(t) =
uꢂg(ꢅ) e−ꢂ!(t−ꢅ) cos !d(t − ꢅ) dꢅ
0
0
Comparing Sa of Equation (6) with PSa = !2Sd, it is clear that PSa is exactly equal to Sa
only for the special case of undamped systems. The diꢁerence between PSa and Sa becomes
apparent if the viscous damping ratio of the system increases.
Figure 2(a) shows the comparison of a mean acceleration response spectrum and a mean
pseudo-acceleration response spectrum for various viscous damping ratios. This ꢀgure was de-
rived from a total of 1053 earthquake acceleration time histories loaded from the strong motion
database of the Paciꢀc Earthquake Engineering Research Center (PEER; http:==peer.berkeley.
edu). All these selected ground motions have the following characteristics: (a) they are
recorded in the United States of America; (b) the range of peak ground acceleration (PGA)
is from 0.025 to 1:6 g; (c) the magnitudes of earthquakes are greater than 5.5; and (d) the
distances closest to fault rupture range from 0.1 to 180 km. Also, in order to compare the
errors between PSa and Sa in detail, the ratios of PSa=Sa are given in Figure 2(b). It is clear
from Figure 2(b) that PSa is nearly the same as Sa only for those systems where the viscous
damping ratio is less than 10% with natural periods shorter than 0:15 sec. The diꢁerences in-
crease obviously for systems with higher viscous damping ratios. For example, for the system
with a viscous damping ratio of 50% and a natural period of 4:0 sec, the ratio of PSa=Sa will
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AN IMPROVED CAPACITY SPECTRUM METHOD FOR ATC-40
2017
2.5
2
Sa
PSa
1.5
1
5% Damping
10%
20%
30%
50%
0.5
0
0
1
2
3
4
(a)
Period (sec)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
5%
10%
20%
30%
50% Damping
0
1
2
3
4
5
6
7
8
9
10
Period (sec)
(b)
Figure 2. (a) Mean elastic acceleration response spectrum (PGA = 1:0 g).
(b) PSa normalized by Sa (i.e. PSa=Sa).
decrease to 0.39, or Sa = 2:56 PSa. At this time, if PSa is used instead of Sa, the acceleration
responses of the system will be signiꢀcantly underestimated.
In addition, according to Equation (2), a relationship between ductility (ꢀ) and equivalent
viscous damping ratio is plotted in Figure 3. From this ꢀgure, it is found that a small ductility
ratio always corresponds to a large equivalent viscous damping ratio. For the case of the
elastic-perfectly-plastic systems, the viscous damping ratio will be as high as 37% for the
system with a ductility ratio of 2. Even for the case of a ductility ratio of 1.5, the viscous
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Y.-Y. LIN AND K.-C. CHANG
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
α
=0
ξ
α=5%
2 (µ −1)(1− α)
µ(1+
ξeq = 0.05+
π
)
αµ − α
1
2
3
4
5
6
Ductility,µ
Figure 3. Ductility vs. equivalent viscous damping.
damping ratio also will be 26.2%. These values are obviously too large to aꢁect the iterative
results of the capacity spectrum method if the pseudo-acceleration response spectrum (PSa)
was used to construct the demand diagram.
4. IMPROVEMENT OF THE CAPACITY SPECTRUM METHOD
Shown in Figure 4 are the demand diagrams plotted by Sa and PSa with damping ratios
of 5% and 30%, respectively, for the north–south component of the 1940 El Centro ground
motion. Included in the ꢀgure are the capacity diagrams of two elastic-perfectly-plastic sys-
tems. System I has an elastic natural vibration period Tn = 1:0sec, a normalized yield strength
f =w = 0:1032, and a yield displacement uy = 2:562 cm. System II has Tn = 0:4 sec; f =w =
y
y
0:22, and uy = 0:9 cm. It can be seen from the ꢀgure that the displacement demands (D) of
System I are 4:43 cm and 11:0 cm corresponding to PSa and Sa, respectively. Similarly, the
displacement demands for System II are 3:17cm (PSa) and 3:74cm (Sa). The two systems get
signiꢀcant errors if PSa is used to form the demand diagram. Therefore, in order to improve
the accuracy of the capacity spectrum method, it is suggested that the real absolute acceler-
ation response spectrum (Sa) be used instead of the pseudo-acceleration response spectrum
(PSa) to generate the demand diagram of the capacity spectrum method.
The improved capacity spectrum method for ATC-40 Procedure A is described herein as a
sequence of steps:
1. Plot the force–deformation diagram and the 5%-damped elastic response (or design)
spectrum, both in the A–D format (Acceleration–Displacement) to obtain the capac-
ity diagram and 5%-damped elastic demand diagram, respectively. Notice that for the
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Earthquake Engng Struct. Dyn. 2003; 32:2013–2025

AN IMPROVED CAPACITY SPECTRUM METHOD FOR ATC-40
2019
1
Sa vs. Sd
5% Damping
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
PSa vs. Sd
30% Damping
System II
System I
3.17 3.74 4.43
11.0
0
5
10
15
Sd (cm)
Figure 4. Demand diagram for 1940 El Centro earthquake.
5%-damped systems, the pseudo-acceleration response spectrum (PSa) is almost equal
to the acceleration response spectrum (Sa). Therefore, in this step, both the ordinate of
the pseudo-acceleration response spectrum and that of the acceleration response spectrum
can be used as the vertical axis of the 5%-damped elastic demand diagram.
2. Estimate the peak deformation (displacement) demand Di and determine the correspond-
ing acceleration Ai for the capacity diagram. Initially, assume Di = Sd (Tn; ꢂ = 5%),
determined for period Tn from the elastic demand diagram.
3. The ductility ratio (ꢀ) can be computed from ꢀ = Di=uy.
4. By Equation (3), the equivalent damping ratio (ꢂeq) can be obtained.
5. Plot the elastic demand diagram for ꢂeq determined in Step 4. Notice that in this step,
the ordinate of acceleration response spectrum (Sa) instead of PSa should be used as the
vertical axis to form the elastic demand diagram with high damping ratio.
6. Read-oꢁ the displacement Dj where the demand diagram obtained from Step 5 intersects
the capacity diagram.
7. Check for convergence. If (Dj −Di) ÷ Dj6tolerance, the earthquake-induced deformation
demand D = Dj. Otherwise, set Di = Dj (or another estimated value) and repeat Steps
3–7 until convergence occurs.
5. EXAMPLES
The improved procedure is applied to compute the earthquake-induced deformation of six
elastic-perfectly-plastic SDOF example systems listed in Table I. The six systems were origi-
nally used by Chopra and Goel [7] to discuss the capacity spectrum method of ATC-40. Two
values of natural period Tn are considered. One is 0:5sec in the acceleration-sensitive spectral
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Y.-Y. LIN AND K.-C. CHANG
Table I. Properties and response of example systems to the 1940 El Centro earthquake.
System properties
System response (cm)
Tn (s)
Fy=w
uy (cm)
ꢀ
DChopra;1999
Dthis paper
Dexact
System 1
System 2
System 3
System 4
System 5
System 6
0.5
0.5
0.5
1.0
1.0
1.0
0.1257
0.1783
0.3411
0.0714
0.1032
0.1733
0.781
1.106
2.117
1.773
2.562
4.302
6
4
2
6
4
2
3.534
3.072
Unconverged
7.192
4.458
Unconverged
4.88
3.65
4.65
4.40
3.31
4.21
11.71
8.31
5.367
10.55
10.16
8.53
Table II. The WJE damping model.
ꢀ
1.0
1.25
1.5
2.0
3.0
4.0
ꢂeq (%) (median + 1ꢆ)
ꢂeq (%) (median)
5.0
5.0
7.5
8.5
10
12
14
16
21
26
26
35
region, and the other is 1:0 sec in the velocity-sensitive region. Furthermore, three levels of
yield strength and an inherent damping ratio ꢂ0 of 5% for all systems are used in this study.
The excitation chosen is the north–south component of the 1940 El Centro ground motion
[15]. Detailed steps are presented next for System 1, whereas Table I tabulates the ꢀnal results
of displacement demand (i.e., the earthquake-induced deformation) for all systems.
The improved procedure is illustrated for System 1 as follows:
1. Figure 5 shows the capacity diagram and the 5%-damped elastic demand diagram.
2. Assume an initial displacement demand Di = Sd (Tn = 0:5 sec; ꢂ = 5%) = 5:69 cm.
3. Ductility ꢀ = Di=uy = 5:69=0:781 = 7:30.
4. For Type A systems, according to Equation (3), ꢂeq = 0:05 + 0:77 × 0:45 = 0:40.
5. The 40%-damped elastic demand diagram intersects the capacity diagram at Dj = 4:88cm
(Figure 5). As mentioned previously, at this step, the vertical axis of the 40%-damped
elastic demand diagram should be the ordinate of the absolute acceleration response
spectrum (Sa).
6. Diꢁerence = 100 × (Dj − Di)=Dj = 100 × (4:88 − 5:69)=5:69 = −14:2%¿5% tolerance.
Therefore, set Di = 4:88 and repeat Steps 3 to 6.
For the second iteration, Di = 4:88cm, ꢀ = 4:88=0:781 = 6:26; ꢂeq = 0:05+0:77 × 0:45 = 0:40.
The intersection point is Dj = 4:88 cm and the diꢁerence between Di and Dj is 0% which is
smaller than the 5% tolerance. Therefore, the iteration process converged at the second cycles.
If the pseudo-acceleration response spectrum (PSa) was used to create the 40%-damped elastic
demand diagram, the result computed by Chopra and Goel [7] was that the displacement
demand of System 1 equals 3:534cm (Table I). Furthermore, the inelastic time history analysis
results give Dexact = 4:65cm and ꢀ = 6 for System 1 (Table I). It can be seen from this example
Copyright ? 2003 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2003; 32:2013–2025

AN IMPROVED CAPACITY SPECTRUM METHOD FOR ATC-40
2021
1
0.9
5%-Damped Demand Diagram
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
40%-Damped Demand Diagram
Capacity Diagram of System 1
4.88
0
2
4
6
8
10
12
14
D or Sd (cm)
Figure 5. Demand diagram of system 1 for El Centro earthquake.
that the iterative results derived from the acceleration response spectrum (Sa) have a much
better accuracy than those derived from the pseudo-acceleration response spectrum (PSa).
In addition, as indicated by Chopra and Goel [7], System 3 and System 6 failed to converge
in their study (Table I). However, using the improved method, the two systems can obtain
converged values (Table I) and all the results of the six systems calculated by the improved
procedure are more accurate when compared to the exact results obtained from non-linear
time history analysis.
6. ACCURACY OF THE CAPACITY SPECTRUM METHOD
For a SDOF elastic-perfectly-plastic system, the accuracy of the iterative results of the capacity
spectrum method will depend only on the chosen acceleration response spectrum to form the
demand spectrum and the model of the equivalent viscous damping. If Sa is used to form
the demand diagram, then the only factor to aꢁect the accuracy of iterative outcomes is the
model of equivalent viscous damping. Although the factor can be thoroughly eliminated by
using the inelastic response spectrum [8,9], the equivalent linear systems can still be used
to approximately predict the responses of non-linear systems. To respond to the linearization
methods, many equivalent viscous damping models have been proposed based on harmonic
or earthquake responses [10; 13; 16–18].
In 1979, Iwan and Gates [13] compared the accuracy of nine damping models for deꢀning
equivalent linear systems by considering a bilinear hysteretic oscillator. He found that all the
existing approximate methods overestimate the equivalent viscous damping for most of the
range of ductility considered. However, the average stiꢁness and energy method (ASE) [13]
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Y.-Y. LIN AND K.-C. CHANG
gives a relatively uniform low error for the entire range of ductility investigated. Moreover,
in 2001, Xue [19] assessed the accuracy of the damping model used in the displacement-
based seismic demand evaluation and design of inelastic structures. Five damping models
were considered, including the WJE damping model [4; 20] where the error was within 10%.
Among others, for structures with a ductility ratio less than 4, the damping model based on
the average stiꢁness and energy method seems to give the most accurate results.
Based on the above descriptions, the ATC-40 damping model for the Type A system, as
shown in Equation (2), and the following three damping models will be used to assess the
accuracy of six example systems listed in Table I.
1. Average stiꢁness and energy (ASE) damping model [13]. Based on the earthquake re-
sponse, the equivalent linear system of this method is deꢀned in terms of average values
of stiꢁness and energy dissipated.
1
2
2(1 − ꢁ)(ꢀ − 1)2 + ꢃꢂ0[(1 − ꢁ)(ꢀ2 − ) + ꢁꢀ3]
3
2ꢃꢀ2
3
3
ꢂeq =
×
(7)
(1 − ꢁ)(1 + ln ꢀ) + ꢁꢀ
where, ꢁ = strain hardening ratio; ꢀ = ductility ratio; ꢂ0 = inherent damping.
2. The WJE damping model [4; 19; 20], as listed in Table II, is based on the maximum
displacement determined from the elastic response spectrum being equal to that obtained
from the inelastic response spectrum.
3. The Kowalsky hysteretic damping model [21] was derived based on the Takeda hysteretic
model [22], which considers stiꢁness degradation and energy dissipation in a vibration
cycle of the inelastic system and of the equivalent linear system.
ꢃ
ꢄ
ꢅꢆ
1
ꢃ
1 − ꢁ
ꢂeq = ꢂ0 +
1 − ꢀn
+ ꢁ
(8)
ꢀ
where n = stiꢁness degradation factor, with a suggested value of 0 for steel structures and
0.5 for RC structures.
A comparison of ꢂeq corresponding to the various damping model is plotted in Figure 6
for systems with ꢁ = n = 0 and ꢂ0 = 5%. It is shown that the equivalent viscous damping
ratios derived from the ATC-40 damping model are obviously larger than those obtained
from the ASE and Kowalsky models. For the six systems (Table I) subjected to the north–
south component of the 1940 El Centro earthquake, Table III shows the displacement demand
calculated from the ATC-40 Procedure A by using four diꢁerent damping models. Table IV
lists a summary of errors with respect to the solutions of non-linear time-history analyses.
It is found that: (a) the ATC-40 damping model underestimates the displacement demand
for ductility ꢀ¡5:0 since it apparently overestimates the equivalent viscous damping caused
by non-linear behavior of structures; (b) the WJE damping model gives the most accurate
results because it is derived based on the mapping relationship between the elastic and the
inelastic response spectrum. Although the WJE damping model has the smallest errors, it is
not available for ductility ꢀ¿4:0 (Table II). Thus, it may be suggested that the WJE damping
model be used to calculate the equivalent viscous damping when structures have a ductility
ratio ꢀ64:0. On the other hand, the Kowalsky damping model may be implemented when
ꢀ¿4:0. Alternatively, the Kowalsky damping model may be used to calculate the equivalent
viscous damping ratios for the entire range of ductility due to its relatively smaller errors.
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AN IMPROVED CAPACITY SPECTRUM METHOD FOR ATC-40
2023
0.45
ATC-40, Type A System
0.4
0.35
WJE
Takeda
0.3
0.25
0.2
ASE
ξ
0.15
0.1
α
=0, ξ 0 =5%,
0.05
1
2
3
4
5
6
Ductility,µ
Figure 6. Equivalent viscous damping for various damping models.
Table III. Displacement demands from ATC-40 Procedure A analysis by
various damping models (cm).
ATC-40
ASE
WJE
Kowalsky
Exact
System 1
System 2
System 3
System 4
System 5
System 6
4.88
3.65
5.27
4.53
5.07
4.32
4.65
4.40
4.10
4.22
3.31
4.12
4.00
4.21
11.71
8.31
5.367
14.32
13.06
6.62
12.92
11.17
6.01
10.55
10.16
8.53
9.81
6.56
Table IV. Errors for ATC-40 Procedure A analysis by various damping models.
ATC-40
ASE
WJE
Kowalsky
System 1
System 2
System 3
System 4
System 5
System 6
5%
−17%
−21%
11%
13%
3%
9%
−2%
−5%
22%
−7%
−2%
0%
36%
29%
−22%
−18%
−3%
10%
−37%
−23%
−30%
Copyright ? 2003 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2003; 32:2013–2025

2024
Y.-Y. LIN AND K.-C. CHANG
7. CONCLUSIONS
To trace the iterative procedure of the capacity spectrum method, it is found that the accu-
racy of its iterative results depends on the chosen acceleration response spectrum to form
the demand spectrum and the model of equivalent viscous damping for the SDOF elastic-
perfectly-plastic systems. Although these errors can be eliminated by using the inelastic re-
sponse spectrum, it is still acceptable to use the equivalent linear systems to estimate the
responses of non-linear systems. From the results of this study, it has been shown that the
iterative results are more accurate when compared to the exact values if Sa is adopted instead
of PSa to create the demand diagram. In fact, for a linear system, Sa is the exact value of
the acceleration response but PSa is not. Therefore, in order to improve the accuracy of the
capacity spectrum method, it is suggested that the real absolute acceleration response spectrum
(Sa) be used instead of the pseudo-acceleration response spectrum (PSa) to create the demand
diagram, especially for systems with ꢂeq¿10% and Tn¿0:15 sec.
The equivalent viscous damping model plays another signiꢀcant role in the capacity spec-
trum method. If the equivalent damping ratios of the equivalent linear systems are not used
appropriately, the displacement demand of structures also will not be accurately estimated
even if the real absolute acceleration response spectrum (Sa) is used. Since the damping
model used by ATC-40 apparently overestimates the equivalent viscous damping ratio caused
by the non-linear behavior of structures, this paper recommends that the WJE damping model
be used to calculate the equivalent viscous damping ratio when the ductility demand of the
structures is less than 4 (ꢀ64:0), whereas the Kowalsky damping model can be implemented
when ꢀ¿4:0. Alternatively, the Kowalsky damping model may be used to calculate the equiv-
alent viscous damping for the entire range of ductility due to its relatively smaller overall
errors.
ACKNOWLEDGEMENTS
This study was supported by the National Science Council (NSC-90-2811-Z-002-003) and the Sinotech
Engineering Consultant (Grant No. 6120) of Taiwan, R.O.C. The ꢀnancial support is greatly acknowl-
edged. Moreover, the careful review and comments of two anonymous reviewers is also appreciated.
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