Wei Shi
Single-Degree-of-Freedom (SDOF) systems are widely
used to investigate the collapse resistance of building structures. A SDOF
model capable of representing the key properties associated with earthquake-induced
collapse is outlined in this study. The seismic collapse capacity of SDOF
systems is evaluated via an incremental dynamic analysis, based on which the
collapse capacity spectrum is developed. The computational procedure and engineering
significance of the collapse capacity spectrum are elaborated in some detail.
The influence of various structural
properties on the collapse resistance of SDOF structural systems is also comprehensively
discussed, which include fundamental period, ductility ratio, post-capping
stiffness, hysteretic pinching, cyclic deterioration and KEY WORDS DOI: 10.1260/1369-4332.17.9.1241 If you need the PDF version of this paper, please email to luxinzheng@sina.com |

Published records
of earthquake events indicate that the collapse of structures is the primary
cause of casualties (Zhao With rapid development of numerical simulation
techniques and computational power, the collapse fragility evaluation
approach based on incremental dynamic analysis (IDA) (Vamvatsikos
and Cornell 2002, 2004) has become increasingly popular. It provides a probabilistic-based
methodology to quantitatively evaluate the collapse capacity of structural
systems and therefore enables more accurate estimation of earthquake-induced
losses within the framework of performance-based earthquake engineering (Goulet
This study utilizes the IDA-based collapse fragility evaluation approach to
assess the seismic collapse capacity of SDOF systems and proposes the concept
of collapse capacity spectrum. The embryonic concept of collapse capacity spectrum can be dated back
20 years (Bernal 1992). Ibarra and Krawinkler (2011) discussed the variance
of collapse capacities of SDOF systems under earthquake excitations in the
context of collapse capacity spectrum. In addition, Adam and Jäger employed
the collapse capacity spectrum to study the |

An adaptive hysteretic model that can
simulate the major features of various structures is very important for collapse
assessment. The hysteretic model used to evaluate the collapse resistance
is described in this section. Noting that the cyclic strength and stiffness
deterioration are key concerns in collapse prediction (Krawinkler
and Zareian 2007), the hysteretic model proposed
by Ibarra As illustrated in Figure 1(a), the initial
backbone curve is defined by five parameters: ( i
) the elastic stiffness The pinching property is modeled in
Figure 1(c), which is described in detail by Ibarra
The cyclic deterioration
rule is based on the hysteretic energy dissipation
model proposed by Rahnama
and Krawinkler (1993), which has four cyclic deterioration
modes, including basic strength, post-capping strength, unloading stiffness
and accelerated reloading stiffness deteriorations (Ibarra In sidesway collapse, the gravity potential
energy is gradually released as the structure responds horizontally and the
gravity load is taken into account by the |

The seismic collapse capacity of SDOF systems is quantitatively evaluated by conducting IDAs. For a typical SDOF system, the IDA (Vamvatsikos and Cornell 2002, 2004) involves applying a set of earthquake ground motions to the structural system, incrementally scaling each ground motion to multiple levels of intensity and implementing nonlinear time-history analysis until dynamic instability occurs. The 22 far-field records suggested in FEMA P695 (2009), containing an aggregate of 44 horizontal ground motion components, are adopted in this study as the representative ground motion set. This ground motion set has been carefully selected, to provide a rational representation of the random nature of strong earthquakes that may cause structural collapse. Such a ground motion selection also aims to maintain its generality by avoiding potential event-based bias, specific spectrum matching and soil-structure-foundation interaction. Detailed discussions on the ground motion set is provided in FEMA P695 (2009). Figure 4(a) shows a typical IDA
result of a SDOF system, which is also referred to as the ¡®dynamic pushover
curve¡¯, on which each point is derived from a nonlinear time-history analysis
and each curve represents the structural response from elasticity to yielding
and finally to collapse when subjected to one incrementally scaled ground
motion. The horizontal axis in the ¡®dynamic pushover
curve¡¯ (Figure 4(a)) represents the engineering demand parameter (
In an IDA, the nonlinear time-history analysis of the structure subjected
to an incrementally scaled ground motion is repeated until the structure reaches
the collapse state when the ground motion is scaled up to the intensity level
of
A tentative study has led to the recognition of a strong correlation between the collapse capacity of SDOF systems and their fundamental period of vibration. A flexible structural system (i.e., one with a longer period of vibration) tends to have a higher capacity to resist earthquake-induced collapse. Figures 5 (a) to (d) display the dynamic pushover
curves for four SDOF systems with fundamental periods of 0.2 s, 2.0 s, 4.0
s and 6.0 s, respectively. All of the four SDOF systems have identical modeling
parameters except for the elastic
stiffness. It is observed from Figure 5 that the
Because of the fact that the collapse capacity of SDOF systems is strongly correlated with its fundamental period of vibration, the median value and dispersion of the collapse capacity corresponding to the fundamental period of interest are quantified by carrying out IDAs on SDOF systems (Figure 6), which is defined as the collapse capacity spectrum. This spectrum is a rational extension of the classical inelastic response spectrum (Riddell 2008; Chopra 2001), which is used to study the strength demand of SDOF systems with given ductility capacity for various periods of vibration (Chakraborti and Gupta 2005), or the ductility demand of SDOF systems with given strength capacity (Miranda and Ruiz-Garcia 2002). Under both conditions, SDOF systems are limited in the nonlinear range without excessive strength and stiffness deterioration (Krawinkler and Zareian 2007), and consequently the collapse state is not considered in the classical inelastic spectrum. In contrast, the collapse capacity spectrum focuses more on the ultimate state of collapse, which advances the investigation technique for collapse resistance of building structures. As will be discussed in the following
two sections, (i) a comprehensive parametric analysis of collapse capacity
spectrum will offer insight into the collapse mechanism of SDOF systems and
( ii ) the collapse capacity spectrum can be used to study the
engineering characteristics of earthquake ground motions, similar to the elastic
response spectrum. Additionally, the collapse capacity spectrum has the potential
to facilitate the static pushover-based approximate procedure to quantify
the collapse resistance of MDOF systems, as stated in the opening section,
and also to provide a reference for seismic design purposes. The
collapse capacity spectrum uses the elastic strength demand normalized
by the yield strength of the structure to measure the collapse capacity, which
has a similar significance to the strength reduction factor from the perspective
of seismic designs. According to the
2003 NEHRP provisions (FEMA 2004), a reduction
factor 2/3 is multiplied to the maximum considered earthquake (MCE) intensity
to derive the design earthquake intensity. This reduction
factor of 2/3 is the reciprocal of a lower bound estimate of the inherent
safety margin against collapse of the structures that are designed following
the NEHRP provisions (Luco |

This section discusses the influences
of several major structural
properties on the collapse capacity spectrum through a series of parametric
analysis, including
the ductility ratio, post-capping stiffness, hysteretic pinching,
cyclic deterioration and the
In this study, the ductility ratio is defined as the ratio of the capping deformation to the yield
deformation (Figure 1(a)). The collapse
capacity spectra of the SDOF systems with various ductility ratios (Models
1 and 2-5) are shown in Figure 7. It is observed from Figure 7(a) that
( i ) structural systems with larger ductility ratios have much higher
collapse resistance; ( ii ) the ductility also affects the pattern of the collapse capacity spectrum,
which may be less obvious. For the SDOF systems with smaller ductility ratios,
(e.g., It is interesting to note that when analyzing the dispersion of collapse capacity spectrum (Figure 7(b)), SDOF systems with larger ductility ratios will systematically have larger dispersion, indicating that the SDOF systems with larger ductility ratios are more sensitive to the RTR uncertainty.
The post-capping stiffness is governed
by the softening
ratio
The hysteretic
pinching behavior is controlled by two parameters Traditionally, structures with less pinching behaviors are considered
to have better energy dissipation capacities, so they should be better able
to resist collapse. However, a comparison of the collapse capacity spectra
for the SDOF systems with various pinching properties reveals that
the hysteretic pinching behavior generally has little effect on the collapse
resistance of the SDOF system, neither on the median value nor on the dispersion.
For Model 22, whose hysteretic pinching behavior is the most severe among
the investigated cases, its collapse capacity spectrum is slightly lower during
the short period range and almost the same during the median and long period
ranges, relative to the collapse capacity spectrum of Model 1, which has no
pinching effect. This outcome differs from the results based on traditional
inelastic response spectrum available in existing literature, in which the
hysteretic pinching behavior will amplify the ductility demand under given
strength capacity (Lee
To reduce the number of parameters,
the following four cyclic deterioration modes, including the basic strength, post-capping strength, unloading stiffness
and accelerated reloading stiffness deteriorations,
are considered simultaneously and controlled
by the common parameter The cyclic deterioration effect
on the collapse capacity spectrum obviously interacts with the ductility capacity
(Figure 10). For non-ductile systems (e.g.,
The |

To investigate the potential influence of various ground motions, two additional ground motion record sets suggested by FEMA P695 (2009) are used as input for Model 1, and the corresponding collapse capacity spectra are calculated via IDA. The two sets of ground motions are both recorded on near-field sites with site-to-rupture distance being less than 10km, of which one set contains rupture directivity pulses (Malhotra 1999; Chopra and Chintanapakdee 2001) while the other set does not. The collapse capacity spectra of Model 1 derived from various sets of ground motion records are shown in Figure 12. It is observed that structures with fundamental periods ranging from 0.4 s to 1.8 s are more vulnerable to the near-field pulse-like ground motions, which coincides with previous earthquake damage investigations and theoretical analysis by other researchers (Malhotra 1999; Champion and Liel 2012). Previous work had indicated that near-field ground motions containing directivity pulse generally impose higher ductility demand on structures with short-to-median fundamental periods (Malhotra 1999; Chopra and Chintanapakdee 2001). The pattern of collapse capacity spectrum derived from the near-field pulse-like ground motions is still similar to the previous discussion. The collapse resistance increases abruptly within the acceleration-sensitive range and reaches a plateau within the velocity-sensitive range. The fact that near-field pulse-like ground motions have a wider acceleration-sensitive range and narrower velocity-sensitive range explains the relatively low collapse capacity over the period range from 0.4 s to 1.8 s (Chopra and Chintanapakdee 2001). Additionally, the structures with fundamental periods ranging from 2.6 s to 5.0 s are more vulnerable to the far-field set relative to both near-field sets because the far-field ground motions generally contain more components with long period. |

This paper utilizes the IDA-based collapse fragility evaluation approach to quantitatively assess the seismic capacity of SDOF systems to resist sidesway collapse and proposes the concept of collapse capacity spectrum, which is a rational extension of the classical inelastic response spectrum. The calculation procedure and engineering significance of the collapse capacity spectrum are illustrated. The collapse capacity spectrum provides a quantitative estimation for the collapse capacity of SDOF systems. This spectrum will be helpful to provide a direct analytical basis for collapse safety margin and also to facilitate the static pushover-based approximate methods of collapse evaluation. Through a systematic parametric discussion on the collapse capacity spectrum, the major findings of this study are summarized below: (i) Fundamental periods are important
factors for the collapse capacity, and ductile structures generally have high
collapse capacity but are more sensitive to RTR uncertainty.
The influence of the post-capping stiffness on the collapse capacity spectrum
is generally insignificant compared to that of the ductility ratio. When the softening ratio (ii) The ductility capacity and
(iii) The influence of earthquake ground motions on the collapse resistance capacity of SDOF systems depends on the fundamental period. Structures with fundamental periods ranging from 0.4 s to 1.8 s are more vulnerable to the near-field pulse-like ground motions and structures with fundamental periods ranging from 2.6 s to 5 s are more vulnerable to the far-field ones.
The authors are grateful for the financial support received from the National Key Technology R&D Program (No. 2013BAJ08B02), and the National Nature Science Foundation of China (No. 51222804, 51178249, 51378299).
Adam, C. and Jäger, C. (2012). ¡°Seismic collapse capacity
of basic inelastic structures vulnerable to the P-delta effect¡±, Araki, Y. and Hjelmstad, K.D. (2000). ¡°Criteria for assessing
dynamic collapse of elastoplastic structural systems¡±, Asimakopoulos, A.V., Karabalis, D.L. and Beskos, D.E. (2007).
¡°Inclusion of P-¦¤ effect in displacement-based seismic design of steel moment
resisting frames¡±, Bernal, D. (1987). ¡°Amplification factors for inelastic
dynamic p-¦¤ effects in earthquake analysis¡±, Bernal, D. (1992). ¡°Instability
of Buildings Subjected to Earthquakes¡±, Brookshire, D.S., Chang, S.E., Cochrane, H., Olson, R.A.,
Rose, A. and Steenson, J. (1997). ¡°Direct and indirect economic losses from
earthquake damage¡±, Chakraborti, A. and Gupta, V.K. (2005). ¡°Scaling of strength
reduction factors for degrading elasto-plastic oscillators¡±, Champion, C. and Liel, A. (2012). ¡°The effect of near-fault
directivity on building seismic collapse risk¡±, Chopra, A.K. (2001). Chopra, A.K. and Chintanapakdee, C. (2001). ¡°Comparing response
of SDF systems to near-fault and far-fault earthquake motions in the context
of spectral regions¡±, Clough, R.W. and Johnston, S.B. (1966). ¡° DesRoches, R., Comerio, M., Eberhard, M., Mooney, W. and
Rix, G.J. (2011). ¡°Overview of the 2010 Haiti Earthquake¡±, Fajfar, P. and Dolšek, M. (2011). ¡°A practice-oriented
estimation of the failure probability of building structures¡±, FEMA (2004). FEMA (2009). Goulet, C.A., Haselton, C.B., Mitrani-Reiser, J., Beck,
J.L., Deierlein, G.G., Porter, K.A. and Stewart, J.P. (2007). ¡°Evaluation
of the seismic performance of a code-conforming reinforced-concrete frame
building-from seismic hazard to collapse safety and economic losses¡±, Han, S.W., Moon, K.H. and Chopra, A.K. (2010). ¡°Application
of MPA to estimate probability of collapse of structures¡±, Haselton, C.B., Liel, A.B., Deierlein, G.G., Dean, B.S.
and Chou, J.H. (2010). ¡°Seismic collapse safety of reinforced concrete buildings.
I: Assessment of ductile moment frames¡±, Ibarra, L.F., Medina, R.A. and Krawinkler, H. (2005). ¡°Hysteretic
models that incorporate strength and stiffness deterioration¡±, Ibarra, L.F. and Krawinkler, H. (2011). ¡°Variance of collapse capacity of SDOF systems under earthquake excitations¡±, Earthquake engineering & structural dynamics, Vol. 40, No. 12, pp. 1299-1314. Jäger, C. and Adam, C. (2013). ¡°Influence of Collapse Definition
and Near-Field Effects on Collapse Capacity Spectra¡±, Kato, B.A. (Ye, L.P. and Pei, X.Z., Trans.) (2010). Kircher, C.A., Reitherman, R.K., Whitman, R.V. and Arnold,
C. (1997). ¡°Estimation of earthquake losses to buildings¡±, Kiureghian, A.D. and Ditlevsen, O. (2009). ¡°Aleatory or
epistemic? Does it matter? ¡±, Krawinkler, H. and Zareian, F. (2007). ¡°Prediction of collapse-How
realistic and practical is it, and what can we learn from it? ¡±, Lee, L.H., Han, S.W. and Oh, Y.H. (1999). ¡°Determination
of ductility factor considering different hysteretic models¡±, Liel, A.B., Haselton, C.B. and Deierlein, G.G. (2010). ¡°Seismic
collapse safety of reinforced concrete buildings. II: Comparative assessment
of nonductile and ductile moment frames¡±, Lu, X.Z., Ye, L.P., Ma, Y.H. and Tang, D.Y. (2012). ¡°Lessons
from the collapse of typical RC frames in Xuankou School during the great
Wenchuan Earthquake¡±, Luco, N., Ellingwood, B.R., Hamburger, R.O., Hooper, J.D.,
Kimball, J.K. and Kircher, C.A. (2007). ¡° Mahin, S.A. and Bertero, V.V. (1976). ¡°Nonlinear seismic
response of a coupled wall system¡±, Malhotra, P.K. (1999). ¡°Response of buildings to near-field
pulse-like ground motions¡±, Mehanny, S.S. (1999). Miranda, E. and Akkar, S.D. (2003). ¡°Dynamic instability
of simple structural systems¡±, Miranda, E. and Ruiz-Garc¨ªa, J. (2002). ¡°Evaluation of approximate
methods to estimate maximum inelastic displacement demands¡±, Rahnama, M. and Krawinkler, H. (1993). Riddell, R. (2008). ¡°Inelastic response spectrum: early
history¡±, Shafei, B., Zareian, F. and Lignos, D.G. (2011). ¡°A simplified
method for collapse capacity assessment of moment-resisting frame and shear
wall structural systems¡±, Takizawa, H. and Jennings, P.C. (1980). ¡°Collapse of a model
for ductile reinforced concrete frames under extreme earthquake motions¡±,
Tang, B., Lu, X., Ye, L.P. and Shi, W. (2011). ¡°Evaluation
of collapse resistance of RC frame structures for Chinese schools in seismic
design categories B and C¡±, Tong, G. and Zhao, Y. (2007). ¡°Seismic force modification
factors for modified-Clough hysteretic model¡±, Vamvatsikos, D. and Cornell, C.A. (2002). ¡°Incremental dynamic
analysis¡±, Vamvatsikos, D. and Cornell, C.A. (2004). ¡°Applied incremental
dynamic analysis¡±, Vamvatsikos, D. and Cornell, C.A. (2005). ¡°Direct Estimation
of seismic demand and capacity of multidegree-of-freedom systems through incremental
dynamic analysis of single degree of freedom approximation¡±, Vian, D. and Bruneau, M. (2003). ¡°Tests to structural collapse
of single degree of freedom frames subjected to earthquake excitations¡±, Williamson, E.B. (2003). ¡°Evaluation of damage and P-¦¤ effects
for systems under earthquake excitation¡±, Ye, L.P., Lu, X.Z. and Li, Y. (2010). ¡°Design objectives
and collapse prevention for building structures in mega-earthquake¡±, Ye, L.P., Ma, Q.L., Miao, Z.W., Guan, H. and Zhuge, Y. (2013).
¡°Numerical and comparative study of earthquake intensity indices in seismic
analysis¡±, Zareian, F. and Krawinkler, H. (2007). ¡°Assessment of probability
of collapse and design for collapse safety¡±, Zareian, F. and Krawinkler, H. (2010). ¡°Structural system
parameter selection based on collapse potential of buildings in earthquakes¡±,
Zareian, F., Krawinkler, H., Ibarra, L. and Lignos, D. (2010).
¡°Basic concepts and performance measures in prediction of collapse of buildings
under earthquake ground motions¡±, Zhao, B., Taucer, F. and Rossetto, T. (2009). ¡°Field investigation
on the performance of building structures during the 12 May 2008 Wenchuan
earthquake in China¡±,
Table 1. Model Identifier and Modeling Parameters
Figure 1. Description of the hysteretic model Figure 2.
Hysteretic curves for various Figure 3.
Simulation of the Figure 4. Collapse capacity assessment of SDOF systems Figure 5. Dynamic pushover curves for SDOF systems with various fundamental periods Figure 6. Collapse Capacity Spectrum Figure 7.
Influence of the ductility ratio Figure 8.
Influence of the softening ratio Figure 9.
Influence of the pinching property on collapse capacity spectrum
( Figure 10.
Effect of the cyclic deterioration property on collapse capacity
spectrum ( Figure 11.
Effect of the Figure 12. Collapse capacity spectrum with different ground motion input Table 1. Model Identifier and Modeling Parameters
[*] Corresponding author. Tel: +86-10-62795364; fax: +86-10-62795364 E-mail address: luxz@tsinghua.edu.cn |