Displacement-Based Fragility Functions
for Low- and Mid-rise Ordinary
Concrete Buildings
Sinan Akkar,a… Haluk Sucuoğlu,a… M.EERI, and Ahmet Yakuta…
Fragility functions are determined for low- and mid-rise ordinary concrete
buildings, which constitute the most vulnerable construction type in Turkey as
well as several other countries prone to earthquakes. A hybrid approach is
employed where building capacities are obtained from field data and their
dynamic responses are calculated by response history analyses. Field data
consists of 32 sample buildings representing the general characteristics of twoto five-story substandard reinforced concrete buildings in Turkey. Lateral
stiffness, strength, and deformation capacities of the sample buildings are
determined by pushover analyses conducted in two principal directions.
Uncertainties in lateral stiffness, strength, and damage limit states are
expressed by using statistical distributions. The inelastic dynamic structural
characteristics of the buildings investigated are represented by a family of
equivalent single-degree-of-freedom systems and their seismic deformation
demands are calculated under 82 ground-motion records. Peak ground velocity
共PGV兲 is selected as the measure of seismic intensity since maximum inelastic
displacements are better correlated with PGV than peak ground acceleration
共PGA兲. Fragility functions are derived separately for different number of
stories, which is a prominent parameter influencing the vulnerability of
existing substandard concrete buildings. 关DOI: 10.1193/1.2084232兴
INTRODUCTION
Fragility functions are the essential tools for seismic loss estimation in built environments. They represent the probability of exceeding a damage limit state for a given
structure type subjected to a seismic excitation 共Shinozuka et al. 1999兲. The damage
limit states in fragilities may be defined as global drift ratio 共maximum roof drift normalized by the building height兲, interstory drift ratio 共maximum lateral displacement between two consecutive stories normalized by the story height兲, story shear force, etc. In
this study, the global drift is chosen to identify the damage limit states, as the selected
buildings do not possess soft stories that may invoke excessive local drift demands. The
ground motion intensities in the fragility functions can be spectral quantities, peak
ground motion values, modified Mercalli scale, etc. In this respect, fragility curves involve uncertainties associated with structural capacity, damage limit-state definition, and
record-to-record variability of ground motion intensity.
a兲
Earthquake Engineering Research Center, Department of Civil Engineering, Middle East Technical University,
06531 Ankara, Turkey
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Earthquake Spectra, Volume 21, No. 4, pages 901–927, November 2005; © 2005, Earthquake Engineering Research Institute
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S. AKKAR, H. SUCUOĞLU, AND A.YAKUT
Figure 1. Damage distribution in Düzce after the 12 November 1999 M7.1 Düzce earthquake,
with respect to the number of stories.
A particular structure type is considered in this study, namely, two- to five-story existing reinforced concrete buildings, which generally do not comply with modern seismic resistant design and construction practice. These buildings constitute the majority of
the vulnerable building stock in Turkey, which is revealed by the recent strong earthquakes in the last decade. Fast urban growth after the 1970s, substantiating uncontrolled
development of the physical environment, is the primary source of such existing risks.
The buildings investigated are categorized into subgroups with respect to their number
of stories. Field observations after recent damaging earthquakes have clearly indicated a
significantly increasing damage trend with the number of stories. Although this may not
be expected in buildings conforming to seismic design regulations, it is very likely otherwise. Figure 1 shows the damage distribution in 6,478 two- to five-story buildings in
Düzce after the 12 November 1999 M7.1 Düzce earthquake 共Sucuoğlu and Yilmaz
2001兲. Buildings with none/light damage were allowed to be in continuous use after the
earthquake, but retrofitting was required for moderately damaged buildings, and severely
damaged buildings had to be demolished, with no permission for further occupation. Increase in damage with the number of stories is notable.
The objective of this study is to determine the fragility functions for two- to fivestory substandard concrete buildings in Turkey. A realistic description of the fragilities
for such buildings is important because almost 75 percent of approximately one million
buildings in Istanbul are in this category, and Istanbul is under a significant seismic risk
共Parsons et al. 2000兲. The uncertainties in structural characteristics are taken into account by considering the variability in fundamental period of vibration, lateral strength,
and damage limit state that are obtained from field data. Using a set of 82 strong ground
motions spanning a broad range of intensities, on the other hand, incorporates randomness of seismic excitations. The fragility functions are determined separately for two- to
five-story concrete buildings as the probabilities of exceeding the specified
displacement-based damage limit states under ground excitations, where seismic intensities are expressed in PGV terms.
DISPLACEMENT-BASED FRAGILITY FUNCTIONS FOR LOW- AND MID-RISE CONCRETE BUILDINGS
903
Figure 2. Idealized moment-rotation relationship of a frame member-end.
FIELD DATABASE
Lateral-load deformation characteristics of typical reinforced concrete buildings in
Turkey have been determined by investigating 32 sample buildings. These buildings
were selected to represent a typical subset of a comprehensive database consisting of
nearly 500 buildings that were compiled in the city of Düzce after the 1999 earthquakes,
based on post-earthquake damage assessments 共Aydoğan 2003, Yakut et al. 2003兲. The
population of 500 buildings approximately represented the story and damage distribution in Düzce after the 1999 earthquake, shown in Figure 1. All buildings were reinforced concrete frame structures with masonry infill walls, the most common and dominant construction type in Turkey. They were further classified into four height groups
based on the number of stories. The average heights for two, three, four, and five stories
were 6.0, 8.9, 11.7, and 14.4 meters, respectively. Design drawings were available for
the selected 32 sample buildings.
SEISMIC PERFORMANCE EVALUATION BY NONLINEAR STATIC PROCEDURE
Three-dimensional models of each sample building were prepared in the SAP2000
environment 共Computers and Structures 2000兲. Nonlinear static analysis was conducted
to determine the base shear versus roof displacement relationship 共capacity curve兲. Then
modal properties were determined consistently that conform to the initial linear part of
the bilinear representation of the capacity curves. Flexural elements for beams, beamcolumn elements for columns, strut elements for infill walls, and rigid diaphragms for
floors were employed for modeling the structural components of the buildings.
Nonlinear flexural characteristics of the individual frame members were defined by
moment-rotation relationships of plastic hinges assigned at the member ends. Flexural
moment capacities were based on the section and material properties of members. Column capacities were calculated from the three-dimensional axial force-bending moment
interaction diagrams. A typical moment-rotation relationship for frame members is
shown in Figure 2. The segment AB, representing initial linear behavior, is followed by
the post-yield behavior BC. Point C corresponds to the ultimate strength, where a sudden loss of strength occurs when the associated plastic rotation level is exceeded. The
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S. AKKAR, H. SUCUOĞLU, AND A.YAKUT
Figure 3. Selected capacity curves for 2-, 3-, 4-, and 5-story buildings.
drop from C to D represents the initiation of failure in the member. It has to be noted
that certain inferior characteristics of reinforced concrete members such as bond slip or
lap splice failure cannot be incorporated directly in the model shown in Figure 2.
The infill walls were modeled by equivalent diagonal struts whose properties depend
on the infill material and wall dimensions, as given in Equation 1 共ASCE 2000兲.
k=
Tc =
Gbt
h
fm · ft
1.5共fm + ft兲
h
N y = ␶ cA v .
b
共1兲
In the above equations, ␶c denotes the shear strength of the rectangular panel corresponding to diagonal cracking, and fm and ft are the compressive and tensile strengths of
the masonry, respectively. The variable Av is the shear area, h is the height, b is the
width, and t is the thickness of the infill wall panel.
A displacement controlled nonlinear static procedure was employed with an inverted
triangular lateral-load distribution for obtaining the capacity curves of each building.
The resulting capacity curves for typical buildings, representing two-, three-, four-, and
five-story buildings, are shown in Figure 3.
All four buildings in Figure 3 have more or less similar ultimate global drift capacities, around 1.5 percent. However, their lateral strength and stiffness decrease with the
number of stories. The strength difference between the two- and three-story buildings
DISPLACEMENT-BASED FRAGILITY FUNCTIONS FOR LOW- AND MID-RISE CONCRETE BUILDINGS
905
Table 1. Statistics of the capacity curve parameters
Parameter
Story Number
Mean
Median
Standard Deviation
␩
All
2
3
4
5
All
2
3
4
5
All
2
3
4
5
0.14338
0.21990
0.16233
0.11667
0.09339
0.001150
0.001175
0.001117
0.001291
0.001084
0.01335
0.01460
0.01410
0.01390
0.01130
0.12500
0.19300
0.14100
0.10600
0.09000
0.00110
0.00100
0.00110
0.00120
0.00110
0.01400
0.01600
0.01500
0.01500
0.01200
0.06620
0.08082
0.05478
0.03438
0.02927
0.00037
0.00054
0.00035
0.00040
0.00026
0.00432
0.00334
0.00463
0.00380
0.00430
␪y
␪u
appears to be higher than the difference between three- and four-story buildings and between four- and five-story buildings. This is most likely due to the significant contribution of infill walls to the lateral load-resisting capacity of the two-story building for this
particular case. However, when the entire building inventory is considered, the mean and
median strength values for the story-based building groups do not show such abrupt
changes while passing from one group to the other. These statistics are listed in Table 1
and discussed in detail in the paper. Since higher seismic displacement demands are
placed on systems with lower stiffness and yield strength, damage vulnerability increases inevitably with the number of stories.
Figures 4 and 5 present the structural plan layout of the two- and four-story buildings
for which the capacity curves are plotted in Figure 3. Despite some variations that are
usually experienced, these layouts and plan dimensions represent the general features of
most of the low- and mid-rise reinforced concrete buildings in Turkey. The corresponding plastic hinge patterns at significant yielding and ultimate capacity states are shown
in Figures 6 and 7, respectively. The symbols used for hinges indicate whether the yielding is at an initial level 共in the vicinity of point B in Figure 2兲, major 共on portion BC in
Figure 2兲, or exceeds the failure initiation state 共on portion DE in Figure 2兲. The twostory building was pushed in the longitudinal direction, and the four-story building was
pushed in the transverse direction that is indicated in Figures 4 and 5, respectively. It is
noteworthy to state that the capacity curves for the buildings in the two principal directions may be significantly different, especially for buildings having rectangular floor
plans.
It may be observed from Figures 6 and 7 that the damage sequence in both frames
starts with the cracking of infill walls, then the yielding of beam ends at the lower sto-
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S. AKKAR, H. SUCUOĞLU, AND A.YAKUT
Figure 4. Plan layout of the 2-story building.
ries, which further propagates to upper stories, and finally with the yielding of column
bases. This is an expected sequence in achieving a ductile beam mechanism, yet the ultimate global drifts are less than 2 percent. The main reason for such low deformation
Figure 5. Plan layout of the 4-story building.
DISPLACEMENT-BASED FRAGILITY FUNCTIONS FOR LOW- AND MID-RISE CONCRETE BUILDINGS
907
Figure 6. Plastic hinge patterns for the selected 2-story building: 共a兲 status at global yield, and
共b兲 status at the global ultimate capacity.
capacity of frames is insufficient rotation capacities of unconfined yielding regions. Global failure occurs quite suddenly when no other reserve members are left that can be
activated for maintaining the inelastic capacity.
IDEALIZATION OF BUILDING RESPONSE BY A SDOF SYSTEM RESPONSE
The capacity curve of each building was approximated with a bilinear curve using
the guidelines given in FEMA-356 共ASCE, 2000兲. A typical idealization of a capacity
curve is shown in Figure 8. It is required to specify the yield and ultimate strength capacities and their associated global drift values for constructing the approximate bilinear
capacity curve. The global drifts are used here to represent the damage limit states of the
buildings because none of the buildings in the data set showed a soft-story mechanism.
The yield global drift ratio ␪y represents significant yielding of the system when the
yield base shear capacity 共Vy兲 of the building is attained, whereas the ultimate global
drift ratio ␪u corresponds to the state at which the building reaches its deformation capacity. The base shear coefficient ␩ = Vy / W in Figure 8 is the ratio of yield base shear
capacity to the building weight.
It should be noted that there is no universal consensus on how to approximate a capacity curve with a bilinear force-deformation representation. An initial stiffness targeting at the state of significant global yielding may lead to considerable variations in Vy
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S. AKKAR, H. SUCUOĞLU, AND A.YAKUT
Figure 7. Plastic hinge patterns for the selected 4-story building: 共a兲 status at global yield capacity, and 共b兲 status at global ultimate capacity.
and ␪y because there is no specific point on the capacity curve exactly describing significant yielding 共Sullivan et al. 2004兲. These variations affect the effective period 共approximate fundamental period兲 and global ductility as well. The approach employed
Figure 8. A typical bilinear capacity curve.
DISPLACEMENT-BASED FRAGILITY FUNCTIONS FOR LOW- AND MID-RISE CONCRETE BUILDINGS
909
Figure 9. Variation of fundamental period with 共a兲 the number of stories, and 共b兲 building
height.
herein is consistent with the assumptions made in the modeling and analysis phases. The
bilinear idealizations of the calculated capacity curves presented in Figure 3 are shown
in the same figure for comparison.
CAPACITY STATISTICS OF THE SAMPLE BUILDINGS
Relationships between the fundamental period, building height and story number obtained from the investigated building stock are presented in Figure 9. The fundamental
periods of the buildings in two principal directions range approximately from
0.15 s to 0.90 s. Apparent scatter for each story group reflects the natural characteristics
of the representative building stock employed. The boxes shown in Figure 9a represent
effective period intervals selected for each story group 共mean plus/minus one standard
deviation兲 where the effective period ranges for two-, three-, four-, and five-story buildings are 0.15–0.30, 0.25–0.45, 0.30–0.55, and 0.40– 0.65 seconds, respectively. Figure
9b shows the period versus height scatters for the building stock with simple fits to describe the mean and plus/minus one standard deviation trends in fundamental period as
a function of building height. If the sample buildings in this study are assumed to resemble the typical Turkish construction practice, the regressed values suggest stiffer
buildings in Turkey with respect to those in the United States that can be classified in the
same structural system category.
The variation of yield base shear coefficient with effective period for the buildings in
the database is shown in Figure 10. The spectral variations of the Turkish code yield base
shear coefficient that was promulgated in 1975 and updated in 1998 共Turkish Ministry
1975, 1998兲 are also displayed in Figure 10, and compared with the field data. There is
relevant information to support that all buildings in the database were dated from the
post-1975 era; accordingly, their seismic designs were expected to conform to the provisions of the 1975 edition of the Turkish Seismic Code 共1975兲. The inherent overstrength in the two- and three-story buildings compared to the 1975 code can be attrib-
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S. AKKAR, H. SUCUOĞLU, AND A.YAKUT
Figure 10. Comparison of the base shear capacities of sample buildings with code
requirements.
uted to different sources such as structural behavior as well as the detailing and material
properties. However, for the low-rise buildings considered in this study, the major contribution is most likely the initial lateral strength of the infill walls. Figure 10 also reveals that the minimum base shear capacity requirement of the Turkish code is not satisfied for most of the four- and five-story sample buildings. In addition to the yield base
shear coefficient ␩, two displacement parameters indicated on the capacity curve in Figure 8, namely, the yield drift ratio ␪y and the ultimate drift ratio ␪u, are of essential importance in the displacement-based performance evaluation of buildings. The statistical
properties of these three parameters are given in Table 1, both for all buildings in the
inventory, and separately for each number of stories. The approach used in these statistics is denoted as counted statistics and measure the central tendency and the dispersion
around the central tendency for each building group. There is a clear trend that ␩ and ␪u
decrease with an increase in the number of stories. The statistics are based on the model
pushover results that lack detailed modeling of bond slip and lap splice failure, which
are important in determining the ultimate drift capacity. The information provided in
Table 1 can be considered as an upper bound for the quantitative description of general
capacity trend for the building inventory presented in this study.
Representative probability density functions of ␪y and ␪u are shown in Figure 11.
When global ductility capacities 共␪u / ␪y兲 are calculated, artificially high values around
10–15 are obtained. This is misleading as the buildings investigated possess high initial
stiffness hence low ␪y due to presence of stiff infill walls and short spans 共Figures 4 and
5兲. It is more appropriate to employ ␪u in assessing the deformation capacities of such
buildings, which are fairly low as revealed in Table 1.
The peculiarities of existing Turkish buildings, which are reflected quite well in the
data set, are taken into consideration in assigning performance limit states in terms of
DISPLACEMENT-BASED FRAGILITY FUNCTIONS FOR LOW- AND MID-RISE CONCRETE BUILDINGS
911
Figure 11. Representative sketch of the probability density functions for drift ratios.
global drift ratios. Three performance limits 共immediate occupancy, life safety, and collapse prevention兲 that are specified in several other international guidelines are adopted
here. The performance limits are computed under the guidance of story-specific median
␪y and ␪u. Considering the uncertainty in modeling the structural deficiencies as described in the preceding paragraphs and the left-skewed tendency in the ultimate drift
probability density function computed from the overall building data, the collapse prevention performance limit ␪CP is taken as the 75 percent of the median ␪u computed for
each story-based building group. The proposed values are slightly higher than the corresponding mean minus one standard deviation of the ultimate drift ratios listed in Table
1. The life safety performance is assigned as the third quartile of the suggested collapse
prevention limit. The median ␪y computed for each story-based building group is accepted to be the limiting value for the immediate occupancy performance level. It is assumed that light, moderate, and severe damage states are experienced when the immediate occupancy, life safety, and collapse prevention drift limits are exceeded,
respectively. The selected performance limits that are described qualitatively in Table 2
Table 2. Assumed drift ratio limits for performance
levels
Performance Level
Collapse Prevention
Life Safety
Immediate Occupancy
Limit
␪ 艋 ␪CP
␪ 艋 43 ␪CP
␪ 艋 ␪y
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S. AKKAR, H. SUCUOĞLU, AND A.YAKUT
Figure 12. Comparisons of capacity curves for 共a兲 low-rise 共1- to 3-story兲, and 共b兲 for mid-rise
共4- to 7-story兲 buildings with other studies.
are conjectural and could be argued as subjective. However, they agree well with the
statistical comparisons made by using the actual building damage data in Turkey that is
discussed in detail in the succeeding sections of the paper.
COMPARISON WITH OTHER STUDIES
A number of attempts have been made recently to recommend idealized capacity
curves for the common building types in Turkey. A study conducted by the Japan International Cooperation Agency 共JICA 2002兲 and the Istanbul Metropolitan Municipality
共2002兲 focused on estimating losses from future earthquakes that are likely to impact
Istanbul. The study by this joint venture idealized the capacity curves using the elastoplastic approximation that were obtained from the simplified analyses. A further endeavor by Boğaziçi University 共BU兲 dealt with the earthquake risk assessment for the
Istanbul region 共Boğaziçi 2002兲. The capacity curves were represented by elastoplastic
behavior similar to the JICA study.
In order to investigate the differences between the capacity curves of the buildings
employed herein and those proposed in other studies, the mean capacity curves obtained
for each height category are compared in Figures 12a and 12b. Included in these figures
are the capacity curves for similar buildings recommended by HAZUS 共NIBS 1997兲.
The HAZUS recommendations are valid for the U.S. buildings that are designed according to the requirements of the moderate code 共buildings designed and constructed in the
period 1940–1973兲. The significant difference between HAZUS and Turkey is quite expected due to the differences in construction practices as well as the code enforcement
and compliance efforts. Another source for the differences between this study and JICA
and BU is the simplifications implemented in modeling. However, none of these explanations justify the gross overestimation of capacities in the JICA study, which is regarded as the basic document for loss estimation in Istanbul based on an expected M7.5
earthquake in the Western Marmara Sea. The curves in this study were obtained from the
analyses that were based on 3-D modeling, whereas the JICA and BU studies are the
results of simpler analyses using certain approximations and assumptions.
DISPLACEMENT-BASED FRAGILITY FUNCTIONS FOR LOW- AND MID-RISE CONCRETE BUILDINGS
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Figure 13. Variation of PGA and PGV with respect to magnitude.
GROUND MOTION DATA
A set of 82 strong ground-motion records were used to compute the empirical building fragility curves based on the building information given in the previous sections.
Important features of the ground-motion data are listed in Table A1 in the Appendix. The
strong ground-motion data consists of dense-to-firm soil records with surface magnitudes 共Ms兲 ranging from 5.2 to 7.6. The soil profiles correspond to NEHRP C and D
sites with average shear-wave velocities of 360 m / s ⬍ vs ⬍ 750 m / s and 180 m / s ⬍ vs
⬍ 360 m / s, respectively 共ASCE 2000兲. These soil conditions are consistent with the associated local site geology of the building data set. The magnitude range represents
ground motions of moderate to large earthquakes. The site-to-source distance 共d兲 is
bounded by 20 km since the urban areas located in the near-source region of active seismic zones are more vulnerable to damage and the seismic loss estimation is a more serious concern for such locations 共Peterson et al. 2000兲. In this study, near-source records
with dominant pulse signals due to forward directivity effects were excluded from the
database as much as possible. Such records have distinct frequency and duration characteristics and dominate the structural behavior depending on the pulse duration
共Krawinkler et al. 2003, Iwan et al. 2000兲. A lack of consensus in the simplified procedures to account for such effects in estimating the seismic performance of existing
buildings led to such a compilation for the ground-motion data set. Figures 13 and 14
present the variation of PGA and PGV with respect to magnitude and distance in log
scale for the ground motions used here. Although the magnitude scatter plots in Figure
13 display a dispersive behavior, both PGA and PGV values tend to increase with increasing magnitude. The distance scatters presented in Figure 14 show an exponential
decay for PGA and PGV as the closest site-to-source distance values increase. This trend
is observed more clearly for PGV.
Among various ground motion intensity measures, PGA and PGV can be considered
as fairly robust and easy to compute. These two intensity parameters are comparatively
evaluated herein for obtaining the most consistent representation of structural damage
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S. AKKAR, H. SUCUOĞLU, AND A.YAKUT
Figure 14. Variation of PGA and PGV with respect to distance.
variation in the empirical fragility curves. The elastic 共⌬e兲 and inelastic 共⌬ie兲 spectral
displacement scatter diagrams are drawn in Figures 15 and 16, respectively, as functions
of PGA and PGV for comparison purposes. The inelastic spectral displacements in Figure 16 are presented for elastoplastic behavior with lateral capacities described by the
strength reduction factor R. This factor is defined as the elastic strength demand normalized by the yield strength of an SDOF system under a given earthquake ground motion. Equation 2 gives the definition of R
R=
mSa
Fy
共2兲
where m is the mass, Sa is the elastic pseudo-spectral acceleration, and Fy is the lateral
yield strength of the system. The product of Sa and m is the elastic strength for that particular SDOF system. Therefore, given a specific R value, the inelastic spectral displacement 共i.e., maximum absolute lateral deformation of the SDOF system兲 describes the
maximum inelastic deformation demand for the corresponding yield strength level. This
spectrum type is denoted as constant R-spectrum 共Ruiz-García and Miranda 2003兲.
The left and right columns in Figures 15 and 16 show the scatter diagrams as a function of PGA and PGV, respectively, whereas the rows correspond to the vibration periods
of 0.2 s, 0.5 s, and 1.0 s simulating relatively short, medium, and long structural periods. It can be observed from Figure 15 that the relation between the elastic 共i.e., R = 1兲
structural deformation demand and ground motion intensity is described fairly well by
PGA for short and medium period structural systems. As the structural period shifts to
longer values 共i.e., 1.0 s兲, PGV correlates well with the elastic structural deformation
demand. Inelastic deformation scatters, that are generically drawn for an elastic strength
to yield strength ratio of 4 共i.e., R = 4兲, in Figure 16 suggest that PGV is a superior indicator of deformation 共damage兲 to PGA over the period range considered. In Figure 16,
weaker correlation of PGA with the increasing deformation demand is noteworthy. Confined to the ground motions presented in this study, the above remarks are also observed
DISPLACEMENT-BASED FRAGILITY FUNCTIONS FOR LOW- AND MID-RISE CONCRETE BUILDINGS
915
Figure 15. Scatter diagrams for 5 percent damped elastic spectral displacements as functions of
PGA and PGV for all 82 earthquake records.
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S. AKKAR, H. SUCUOĞLU, AND A.YAKUT
Figure 16. Scatter diagrams for 5 percent damped elastoplastic spectral displacements as functions of PGA and PGV for all 82 earthquake records.
DISPLACEMENT-BASED FRAGILITY FUNCTIONS FOR LOW- AND MID-RISE CONCRETE BUILDINGS
917
Table 3. Assumed thresholds for performance levels and the associated damage limit states 共in terms
of global drift兲
Story
Number
Immediate Occupancy, ␪IC
共Light Damage兲
Life Safety, ␪LS
共Moderate Damage兲
Collapse Prevention, ␪CP
共Severe Damage兲
2
3
4
5
0.0011
0.0011
0.0012
0.0011
0.0090
0.0080
0.0080
0.0068
0.012
0.011
0.011
0.009
for other R and period values. Thus PGV appears to be a more suitable ground motion
intensity parameter for describing deformation demands in structures that deform beyond the elastic range.
The observations highlighted in the preceding paragraph confirm the conclusions derived by Wald et al. 共1999兲; in their study, Wald et al. indicated that low levels of structural damage identified by the modified Mercalli scale less than VII correlate well with
PGA. As structural damage increases 共i.e., modified Mercalli scale greater than VII兲,
PGA values level off and PGV is more indicative for defining the correlation between
structural damage and ground motion intensity. Under the guidance of these observations, PGV is selected as the ground motion intensity measure in the derivation of empirical fragility curves. A detailed description of the analytical method for obtaining the
fragility functions is presented in the next section.
ANALYTICAL METHOD
Generically, fragility curves are conditional cumulative distribution functions that
define the exceedance probability of a damage state for a given ground motion intensity
level. The probability distribution function is the standard lognormal distribution in most
cases and the curves represent median fragility values. The lognormal distribution fit is
assured by certain optimization algorithms and goodness-of-fit tests 共Shinozuka et al.
2000, Kircher et al. 1997兲. The following sections summarize the methodology used in
deriving the fragility curves.
DISPLACEMENT-BASED DAMAGE LIMIT STATES
Table 3 lists the damage threshold levels used for defining the performance levels of
building groups with different number of stories. The details of the pertinent statistics
and the computation of these thresholds were given in the previous sections.
The values of the median life safety and collapse prevention drift thresholds have a
decreasing trend with increase in the number of stories. On the other hand, the threshold
for immediate occupancy that can be considered as a transition boundary between global
elastic to inelastic behavior practically attains the same level regardless of the number of
stories. It is noteworthy that all of the above drift-based performance levels are smaller
than the ones proposed in the building rehabilitation standards of the United States or
Japan 共ASCE 2000, JICA 2002, NIBS 1997兲. This has to be recognized as a serious con-
918
S. AKKAR, H. SUCUOĞLU, AND A.YAKUT
Figure 17. Effective period and base shear capacity ranges for different building groups.
cern for addressing the country specific vulnerabilities and, accordingly, loss estimation
policies. The damage levels presented in Table 3 are consistent with other individual
studies that investigate the drift capacity of reinforced concrete frame buildings in Turkey 共Sucuoğlu et al. 2004兲.
STORY-BASED PERIOD AND STRENGTH RANGES
The relationship between the fundamental period of vibration, base shear capacity,
and the number of stories was established by using the field data presented previously.
For each building group identified with different number of stories, the relevant mean
and standard deviation statistics were computed for the associated base shear capacity
and fundamental period of vibration distribution. These statistics were then employed to
find the effective base shear and period ranges that represent the group of buildings investigated. The corresponding ranges cover the intervals of mean plus/minus one standard deviation for the parameters of interest to account for their central dispersion. Figure 17 illustrates these intervals with respect to the number of stories with the
superimposed actual trend of the building data. Each block in Figure 17 represents the
overall distribution of period and base shear for a building group with a particular story
number. It can be observed that the variation in base shear capacity decreases whereas
the dispersion in period increases with the increasing number of stories.
NONLINEAR DYNAMIC RESPONSE HISTORY ANALYSES
The set of 82 records comprising the ground motion data was used to compute the
inelastic displacement response histories of equivalent SDOF systems representing the
base shear capacities and fundamental vibration periods of building groups discussed
above. The variations in building periods and base-shear coefficients were described
more accurately by dividing the rectangular blocks in Figure 17 into finer meshes. For
DISPLACEMENT-BASED FRAGILITY FUNCTIONS FOR LOW- AND MID-RISE CONCRETE BUILDINGS
919
the sake of uniformity, the period range of each rectangular block was divided into
equally spaced intervals of 0.05 seconds. Similarly, the corresponding base shear coefficient ranges were divided into four equal intervals. The global drift values for each
yield base shear coefficient 共␩兲 versus period 共T兲 pair was computed by using the approximate procedure described in Equations 3a and 3b.
R=
␪=
␦top
H
=
Sa
␩
Cm
1
关C ⌬ 共T,R兲兴.
H 0 ie
共3a兲
共3b兲
This approximate procedure is similar to the displacement coefficient method described in the FEMA-356 共ASCE 2000兲 document except for the C1 coefficient that relates the elastic spectral displacement to inelastic spectral displacement for an idealized
bilinear hysteretic model. Instead, the inelastic spectral displacements were computed directly from response history analyses. The base shear coefficient 共␩兲 was converted to
strength reduction factor R using Equation 3a, where Sa designates the elastic pseudospectral acceleration at period T for a specific ground motion. The coefficient Cm is the
effective mass modification term to account for the effective modal mass computed for
the fundamental mode. This parameter was taken as 1.0 and 0.9 for the two- and three-,
four-, and five-story building groups, respectively. These values are consistent with the
FEMA-356 document. The two modification coefficients C2 and C3 in FEMA-356,
which account for the hysteretic model and dynamic P-delta effects, respectively, are not
included in Equation 3b since they were taken as 1.0. The modification factor C0 relates
the spectral displacement of the equivalent SDOF system to the approximate roof displacement ␦top and values between 1.0 and 1.1 were assigned depending on the story
number that approximately represents the first-mode participation factors of the building
stock described. The global drift values were calculated by normalizing the approximate
top-story displacements with the average building heights H defined for the building inventory.
A total of 9,020 equivalent SDOF inelastic response history analyses were carried
out for the associated period and base shear combinations. The capacity curves computed from the building data were represented by a bilinear hysteretic model with 3
percent strain hardening, which corresponds to the median post yielding stiffness ratio
of the building data set. The global drift ratios were computed via Equations 3a and 3b
as described in detail in the above paragraph. This process assumes a fundamental mode
dominant structural behavior, which can be considered as a reasonable assumption for
the existing low- to mid-rise buildings in the data set.
The proposed procedure only offers an approximation to the global drift values and
clearly has limitations confined to the certain simplifications described in the previous
paragraphs. These limitations might be surmounted by using exact nonlinear response
history analyses of the building data at the expense of significant computational time
that requires a considerable effort, which is not efficient for the loss estimation of large
building stocks in big metropolitan areas like Istanbul in Turkey.
920
S. AKKAR, H. SUCUOĞLU, AND A.YAKUT
Figure 18. The scatter diagram and the corresponding fragility curves for the 3-story buildings.
COMPUTATION OF FRAGILITY CURVES
The maximum global drift values computed by the approximate procedure were assumed to represent the seismic performance of the investigated concrete frames. Using
the damage threshold levels defined in Table 3, the exceedance probabilities of that particular fragility curve were computed from the PGV versus maximum global drift scatters specific to each building group 共i.e., two-, three-, four-, and five-story buildings兲.
The scatter diagrams were clustered for different PGV intervals and the global drift percentiles greater than a given damage threshold level were computed by using the lognormal distribution to estimate the exceedance probabilities of the fragility curves. Exponential functions were fit over the jaggedly varying exceedance probability points to
achieve smooth fragility curves for that specific damage state and building group. A representative sketch for the above procedure is shown in Figure 18.
RESULTS AND DISCUSSIONS
The fragility curves produced by the presented methodology are shown in Figures
19a–d for two-, three-, four-, and five-story concrete buildings, respectively. The three
curves in each figure represent the probability of exceeding the immediate occupancy
共light damage兲, life safety 共moderate damage兲, and collapse prevention 共severe damage兲
limit states, respectively, from left to right. They are also grouped separately in Figures
20a–c for the three limit states, respectively, to compare the effect of the number of stories on fragility. It can be observed from Figure 20 that the number of stories has a significant effect on the probability of exceeding the moderate and the severe damage limit
states. Furthermore, the moderate and severe damage fragility patterns of the two- and
three-story building groups and the four- and five-story building groups follow a closer
trend to each other. These two distinct groups can be designated as low- and mid-rise
concrete frame buildings for the construction practice in Turkey.
The fragilities presented in Figure 19 can be compared and verified with the damage
distribution observed in Düzce, which is shown in Figure 1. Before this comparison, the
DISPLACEMENT-BASED FRAGILITY FUNCTIONS FOR LOW- AND MID-RISE CONCRETE BUILDINGS
921
Figure 19. Fragility curves for 共a兲 2-, 共b兲 3-, 共c兲 4-, and 共d兲 5-story buildings.
Figure 20. Fragility curves for 共a兲 light, 共b兲 moderate, and 共c兲 severe damage limit states.
922
S. AKKAR, H. SUCUOĞLU, AND A.YAKUT
Table 4. Comparison of calculated and observed damages in 32 sample
buildings
Story
Number
␪LS
␪CP
␪DD
Damage
Decision
Observed
Damage
2
3
0.0090
0.0084
0.0120
0.0113
0.0061
0.0075
Light
Light
4
0.0084
0.0113
0.0081
Light
5
0.0068
0.0090
0.0080
Moderate
5 Light
5 Light
4 Moderate
3 Light
4 Moderate
4 Light
3 Moderate
3 Severe
difference in damage definitions in Figures 1 and 19 is assessed. The global drift demands were calculated for the two- to five-story buildings by assuming bilinear hysteretic model with 3 percent post-yielding stiffness and with strength and period ranges defined in Figure 17 under the horizontal components of the Düzce record. The results are
given in Table 4, where the calculated damages based on average global drift demands
␪DD are in good agreement with the observed damages in the 32 sample buildings.
A comparison of the calculated probabilities of exceeding moderate and severe damage limit states, and the corresponding observed damage ratios for the Düzce case given
in Figure 1 are listed in Table 5 for two- to five-story buildings. It is appropriate to exclude undamaged and lightly damaged buildings since they are not separated in Figure 1.
The PGV values recorded along the two horizontal components in Düzce are listed in
Table A1 in the Appendix, and their geometric mean is 70.8 cm/ s. Accordingly, the fragilities associated with this PGV value can be read from the related curves in Figure 19.
Fragilities exceed the observed damages by 10 and 23 percent in the two-story buildings
Table 5. Comparison of predicted and observed damage
distributions
Story
Damage Limit Prediction
Observed
Number
Statea
共Median兲 共Damage Ratio兲
2
3
4
5
a
⬎M
⬎S
⬎M
⬎S
⬎M
⬎S
⬎M
⬎S
37.7
15.1
47.8
20.5
81.4
43.1
93.6
69.8
34.1
12.3
51.9
19
86.4
39.4
97.4
67.1
M and S denote moderate and severe damage, respectively
DISPLACEMENT-BASED FRAGILITY FUNCTIONS FOR LOW- AND MID-RISE CONCRETE BUILDINGS
923
at moderate and severe damage limit states, respectively. The differences between predicted and observed fragilities are less than 10 percent at both damage limit states in the
three-, four-, and five-story buildings. Considering the uncertainties inherent in the fragilities and randomness of ground intensities, these results can be accepted as satisfactory, suggesting that the proposed procedure can be implemented to large building
stocks in Turkey for loss estimation.
CONCLUSIONS
Fragility functions are derived for groups of existing concrete buildings in Turkey, as
a function of number of stories. These curves can be used for regional loss estimation
studies in different seismic prone zones of Turkey. The parameters that were considered
as uncertain in the analysis are the ground motion intensity, fundamental period of vibration, yield strength, and global drift that is used to identify the damage limit states.
There are two basic conclusions that can be derived from this study. First, fragility
functions taking into consideration the basic regional characteristics of an investigated
building stock, based on field data, serve for reliable estimates of expected loses in similar buildings from strong ground shaking. Here, the field data is collected from a group
of representative existing concrete buildings in Turkey that sustained various degrees of
damage during the 1999 Düzce earthquake. The number of stories in buildings was employed as a dominant structural parameter influencing their seismic performance. The
results have revealed that the fragility curves for different number of stories are well
separated, hence they are informative on the distribution of expected damages. Furthermore, they were successful in reproducing the damage distribution in 6,478 two- to fivestory buildings in Düzce observed after the 1999 Düzce earthquake.
These results are readily applicable to Istanbul where an earthquake with a M7.5 is
considered as the scenario event 共Note: a smaller magnitude event is more probable than
the M7.5 event兲. There are approximately 120,000 concrete buildings housing 2 million
residents in nine subprovinces of Istanbul that are located along the western Marmara
coast, at a distance of 10 to 15 km from the major Marmara Sea segment of the North
Anatolian Fault. This fault was last broken with a M7.6 earthquake in 1766. Microzonation studies conducted recently 共JICA 2000兲 indicated that the expected PGV values
in this region during the scenario earthquake varies between 50 and 60 cm/ s, excluding
isolated pockets of high site amplification. When the results of this study are applied
conservatively, for a PGV of 50 cm/ s, the probabilities of exceeding severe damage in
two-, three-, four-, and five-story buildings are obtained as 3, 5, 7, and 20 percent, respectively. It has to be noted that almost 25 percent of all buildings in this region are
five-story, and 60 percent range in height between two- and five-story concrete construction according to the latest building census data. There are no calibrated tools other than
the one proposed herein for a reliable damage estimation of large urban building stocks
in Turkey.
The second major conclusion relates to the use of PGV as a measure of strong motion intensity in fragility functions developed for large building stocks. It is commonly
accepted that PGV has a good correlation with MMI for large magnitude earthquakes.
924
S. AKKAR, H. SUCUOĞLU, AND A.YAKUT
The results of this study have confirmed this observation and revealed that the inelastic
dynamic response displacements are significantly better correlated with PGV than PGA
across the structural period range from 0.2 s to 1.0 s.
ACKNOWLEDGMENTS
The research work presented in this study is supported in part by the Scientific and
Research Council of Turkey 共TUBITAK兲 under Grant YMAU-ICTAG-1574, and by
NATO Scientific Affairs Division under Grant NATO SfP977231. The authors would like
to express their gratitude to Dr. Altug Erberik for discussing some specific issues in the
computation of fragility curves presented in this study. The careful and constructive
comments of two anonymous reviewers are acknowledged that led to significant improvements in the text.
APPENDIX
Table A1. List of ground motions used in the study
Earthquake
Record and Component
Cape Mendocino 04/25/92
Cape Mendocino 04/25/92
Chi-Chi 09/20/99
Chi-Chi 09/20/99
Chi-Chi 09/20/99
Chi-Chi 09/20/99
Chi-Chi 09/20/99
Chi-Chi 09/20/99
Chi-Chi 09/20/99
Chi-Chi 09/20/99
Chi-Chi 09/20/99
Chi-Chi 09/20/99
Coalinga 05/02/83
Coyote Lake 08/06/79
Coyote Lake 08/06/79
Coyote Lake 08/06/79
Coyote Lake 08/06/79
Coyote Lake 08/06/79
Duzce 11/12/99
Duzce 11/12/99
Gazli 05/17/76
Gazli 05/17/76
Imperial Valley 10/15/79
Imperial Valley 10/15/79
Imperial Valley 10/15/79
Imperial Valley 10/15/79
Petrolia, 000
Rio Dell Overpass, 360
WNT, E
TCU076, N
TCU049, W
TCU049, N
TCU082, W
CHY028, W
TCU051, W
TCU074, W
CHY006, E
TCU070, N
Pleasant Valley P.P.—Yard, 045
Gilroy Array #3, 050
Gilroy Array #2, 140
Gilroy Array #3, 140
San Juan Bautista, 213
SJB Overpass Bent 3 G.L., 067
Duzce Meteorology Sta, 180
Duzce Meteorology Sta, 270
Karakyr, 000
Karakyr, 090
El Centro Array 6, 230
Calexico Fire Sta, 315
Parachute Test Site, 315
El Centro Array #1, 230
1
M d 共km兲 Site
7.1
7.1
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
6.5
5.7
5.7
5.7
5.7
5.7
7.3
7.3
7.3
7.3
5.2
6.5
6.5
6.5
9.5
18.5
1.2
2.0
4.5
4.5
5.7
7.3
8.3
13.7
14.9
19.1
8.5
6.0
6.0
6.0
15.6
17.2
8.2
8.2
3.0
3.0
13.1
10.6
14.2
15.5
D
C
D
D
D
D
D
D
D
D
D
C
D
D
D
D
C
C
D
D
D
D
D
D
C
D
2
Fault
RN
RN
RN
RN
RN
RN
RN
RN
RN
RN
RN
RN
RO
SS
SS
SS
SS
SS
SS
SS
RN
RN
SS
SS
SS
SS
PGV
PGA
共cm/ s2兲 共cm/s兲
578
539
940
408
287
246
219
641
183
586
357
166
581
267
333
224
106
95
341
525
596
704
359
198
200
132
48.4
42.1
68.8
64.2
47.9
61.2
58.4
72.8
49.3
73.3
55.4
62.3
60.2
18.7
24.9
28.8
7.6
5.9
60.0
83.5
65.4
71.6
20.8
16.0
16.1
10.7
DISPLACEMENT-BASED FRAGILITY FUNCTIONS FOR LOW- AND MID-RISE CONCRETE BUILDINGS
925
Table A1. „cont.…
Earthquake
Imperial Valley 10/15/79
Imperial Valley 10/15/79
Imperial Valley 10/15/79
Imperial Valley 10/15/79
Imperial Valley 10/15/79
Imperial Valley 10/15/79
Imperial Valley 10/15/79
Imperial Valley 10/15/79
Imperial Valley 10/15/79
Imperial Valley 10/15/79
Imperial Valley 10/15/79
Imperial Valley 10/15/79
Imperial Valley 10/15/79
Imperial Valley 10/15/79
Imperial Valley 10/15/79
Kocaeli 08/17/99
Landers 06/28/92
Landers 06/28/92
Livermore 01/24/80
Livermore 01/27/80
Record and Component
PGA
PGV
M d 共km兲 Site1 Fault2 共cm/ s2兲 共cm/s兲
EC Meloland Overp FF, 000 6.9
El Centro Array #7, 140
6.9
El Centro Array #5, 140
6.9
Bonds Corner, 140
6.9
Bonds Corner, 230
6.9
El Centro Array #8, 140
6.9
El Centro Array #4, 140
6.9
El Centro Diff Array, 230
6.9
EC Co Center FF, 002
6.9
Aeropuerto Mexicali, 315
6.9
Aeropuerto Mexicali, 045
6.9
El Centro Array #2, 140
6.9
Sahop Casa Flores, 270
6.9
El Centro Array #11, 230
6.9
El Centro Array #12, 230
6.9
Arcelik, 000
7.4
Morango Valley, 000
7.3
22170 Joshua Tree
7.4
San Ramon Kodak Bldg, 270 5.8
Livermore Morgan Terr Park, 5.4
265
Loma Prieta 10/18/89
Gilroy Array #6, 090
6.9
Loma Prieta 10/18/89
Corralitos, 000
7.1
Loma Prieta 10/18/89
LGPC, 000
7.1
Loma Prieta 10/18/89
Gilroy Array #2, 000
7.1
Loma Prieta 10/18/89
Gilroy Array #2, 090
7.1
Loma Prieta 10/18/89
Saratoga W Valley Coll, 270 7.1
Loma Prieta 10/18/89
Gilroy Array #3, 000
7.1
Loma Prieta 1989/10/18 47006 Gilroy—Gavilan Coll., 7.1
067
Loma Prieta 1989/10/18 47006 Gilroy—Gavilan Coll., 7.1
337
Morgan Hill 04/24/84
Halls Valley, 240
6.1
Morgan Hill 04/24/84
Gilroy Array #6, 000
6.2
Morgan Hill 04/24/84
Gilroy Array #7, 090
6.2
Morgan Hill 04/24/84
Gilroy Array #3, 090
6.2
Morgan Hill 04/24/84
Gilroy Array #2, 000
6.2
Morgan Hill 04/24/84
Gilroy Gavilan Coll, 067
6.2
N. Palm Springs 07/08/86 Fun Valley, 045
6.0
N. Palm Springs 07/08/86 Palm Springs Airport, 000
6.0
Northridge 01/17/94
Sylmar-Converter Sta-East, 2886.7
Northridge 01/17/94
Sylmar—Hospital, 090
6.7
0.5
0.6
1.0
2.5
2.5
3.8
4.2
5.3
7.6
8.5
8.5
10.4
11.1
12.6
18.2
17.0
19.3
11.6
17.6
8.0
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
C
C
C
D
C
SS
SS
SS
SS
SS
SS
SS
SS
SS
SS
SS
SS
SS
SS
SS
SS
SS
SS
SS
SS
308
331
509
577
760
590
476
471
209
255
321
309
496
372
114
215
184
279
75
194
71.8
47.6
46.9
45.2
45.9
54.3
37.4
40.8
37.5
24.9
42.8
31.5
31.0
42.1
21.8
17.7
16.6
43.2
6.1
11.7
19.9
5.1
6.1
12.7
12.7
13.7
14.4
11.6
C
C
D
D
D
C
D
C
RO
RO
RO
RO
RO
RO
RO
RO
167
632
553
360
316
326
544
350
14.2
55.2
94.8
32.9
39.1
61.5
35.7
28.6
11.6
C
RO
319
22.3
3.4
11.8
14.0
14.6
15.1
16.2
15.8
16.6
6.1
6.4
D
C
D
D
D
C
C
D
D
D
SS
SS
SS
SS
SS
SS
RO
RO
RN
RN
306
218
111
197
159
112
126
155
484
593
39.4
11.4
6.0
12.7
5.1
3.6
6.4
12.4
74.6
78.2
926
S. AKKAR, H. SUCUOĞLU, AND A.YAKUT
Table A1. 共cont.兲
Earthquake
Northridge 01/17/94
Northridge 01/17/94
Northridge 01/17/94
Northridge 01/17/94
Northridge 01/17/94
Northridge 01/17/94
Northridge 01/17/94
Northridge 01/17/94
Northridge 01/17/94
Northridge 01/17/94
Northridge 01/17/94
Parkfield 06/28/66
Superstition Hills
11/24/87
Superstition Hills
11/24/87
Westmoreland 04/26/81
Whittier 10/01/87
Whittier 10/01/87
1
2
Record and Component
Newhall, 090
Newhall, 360
Pacoima Kagel Canyon, 360
Sepulveda VA, 360
Northridge—Saticoy, 180
90009 N. Hollywood—CWC,
180
24688 LA—UCLA Grounds,
090
24688 LA—UCLA Grounds,
360
Canoga Park-Topanga Canyon,
196
Tarzana—Cedar Hill Nursery
A, 360
638 Brentwood V.A. Hospital,
195
Cholame #5, 085
El Centro Imp Co Center, 090
PGA
PGV
M d 共km兲 Site1 Fault2 共cm/ s2兲 共cm/s兲
6.7
6.7
6.7
6.7
6.7
6.7
7.1
7.1
8.2
8.9
13.3
14.6
D
D
C
D
D
C
RN
RN
RN
RN
RN
RO
572
579
424
921
468
292
75.5
97.3
51.5
76.6
61.5
25.0
6.7
14.9
C
RN
273
22.0
6.7
14.9
C
RN
465
22.2
6.7
15.8
D
RN
412
60.8
6.7
17.5
B
RN
971
77.6
6.9
16.3
C
RO
183
23.7
6.1
6.6
5.3
13.9
D
D
SS
SS
433
253
24.7
40.9
El Centro Imp Co Center, 000 6.6
13.9
D
SS
351
46.4
Fire Station, 090
Bell Gardens-Jaboneria, 207
Brea-S. Flower Av, 020
13.3
9.8
17.9
D
D
D
SS
RN
RN
361
214
113
48.7
18.9
7.1
5.8
6.0
6.0
NEHRP site classification 共ASCE 2000兲
SS, RN, and RO designate strike normal, reverse normal, and reverse oblique, respectively.
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共Received 28 February 2004; accepted 25 January 2005兲