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Nikos D. Lagaros aa Institute of Structural Analysis and Seismic Research, National Technical University of Athens,Greece
First published on: 21 January 2009
Lagaros, Nikos D.(2010) 'Multicomponent incremental dynamic analysis considering variable incidentangle', Structure and Infrastructure Engineering, 6: 1, 77 94, First published on: 21 January 2009 (iFirst)
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Multicomponent incremental dynamic analysis considering variable incident angle
Nikos D. Lagaros*Institute of Structural Analysis and Seismic Research, National Technical University of Athens, 9, Iroon Polytechniou Str.,
Zografou Campus, 157 80 Athens, Greece
(Received 18 December 2007; final version received 3 December 2008 )
Performance-based earthquake engineering (PBEE) is the current trend in designing earthquake-resistant structures.The implementation of the PBEE framework requires the assessment of the structural capacity in multipleearthquake hazard levels. Incremental dynamic analysis (IDA) is considered to be one of the most efficientcomputational tools for estimating structural capacity; therefore, it is often incorporated into the PBEE framework.Most real world reinforced concrete (RC) buildings can only be represented accurately with three-dimensional (3D)models; hence, a multicomponent incremental dynamic analysis (MIDA) is required in order to carry out an IDA-based PBEE framework. In this work, the implementation of IDA studies in 3D structures is examined, where a two-component seismic excitation is applied, and a new procedure for performing MIDA is proposed.
According to the proposed procedure, the MIDA is performed over a sample of record-incident angle pairs that
are generated using Latin hypercube sampling (LHS).
Keywords: critical incident angle; multicomponent incremental dynamic analysis; RC buildings; fragility analysis;Latin hypercube sampling
1. Introduction
Severe damage caused by recent earthquakes made the
engineering community to question the effectiveness of
the current seismic design codes (Lagaros et al. 2006,
Zhai and Xie 2006). Given that the primary goal of
contemporary seismic design procedures is the protec-
tion of human life, it is evident that additionalperformance targets and earthquake intensities should
be considered in order to assess the structural
performance in many hazard levels. In the previous
decade, the concept of performance-based earthquake
engineering (PBEE) for designing structures subject to
seismic loading conditions has been introduced
(SEAOC 1995, ATC 1996, FEMA 1997). The main
objective of PBEE is to provide an integrated frame-
work for siting, designing, constructing and maintain-
ing buildings in order to have predictable performance
in a variety of earthquake hazard levels during the
structures lifetime. The implementation of PBEE
requires a reliable tool for estimating the capacity
and the demand for any structural system. Among
others (Fajfar 2000, Chopra and Goel 2002), incre-
mental dynamic analysis (IDA; Vamvatsikos and
Cornell 2002) is considered to be an analysis method
for obtaining good estimates of the structural perfor-
mance in the case of earthquake hazards and is an
appropriate method to be incorporated into the PBEE
framework.
In view of the complexity and the computational
effort required by the three-dimensional (3D) models
that are employed to represent real buildings, simpli-
fied two-dimensional (2D) structural simulations are
used during the design procedure. This is mainly
encountered in-plan symmetric buildings and mostly inthe case of steel framed buildings, since they are
composed of 2D moment resisting frames. In 3D
reinforced concrete (RC) buildings, however, the
columns belong to two or more intersecting lateral-
force-resisting systems; consequently, it is not possible
to employ a 2D simulation, since the bidirectional
orthogonal shaking effects should be taken into
account. Moreover, 3D models should also be
considered in the case that plan non-symmetric steel
or RC buildings are examined. So far, IDA has mainly
been implemented in 2D structures (Ellingwood and
Wen 2005, Fragiadakis et al. 2006). To our knowledge,
only a few works can be found in the literature where
IDA study is performed in 3D structures. In the work
by Vamvatsikos (2006), IDA is employed in order to
evaluate the seismic performance of a 20-storey steel
space frame under biaxial seismic loading. The two
components of the records are applied along the
structural axes, while the maximum peak drift over
*Email: nlagaros@central.ntua.gr
Structure and Infrastructure Engineering
Vol. 6, Nos. 12, FebruaryApril 2010, 7794
ISSN 1573-2479 print/ISSN 1744-8980 online
2010 Taylor & Francis
DOI: 10.1080/15732470802663805http://www.informaworld.com
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all the storeys of the building is used to monitor the
structural performance. In the work by Serdar Kircil
and Polat (2006), a suite of 12 records has been
selected, and the IDA curves for the two structural
axes have been obtained independently. Based on the
two groups of the IDA curves, the fragility curves for
RC frame buildings have been developed.Multicomponent incremental dynamic analysis
(MIDA) is performed in a similar way to the 2D
implementation of IDA, i.e. a suite of records is
selected and, for each record, a MIDA representative
curve is produced. The 50% fractile MIDA curve is
calculated using the representative curves of all the
records, and then this curve is employed to develop the
fragility curves that constitute a part of the PBEE
framework. Selecting the IDA representative curve in
its 2D implementation is, in most cases, a straightfor-
ward procedure. On the other hand, its 3D implemen-
tation is not an easy task, since the incident angle
selected for applying the two components of therecords might considerably influence the outcome of
the MIDA and consequently the results of the PBEE
framework. In this work, a new procedure for
performing MIDA is proposed, where it is performed
over a sample of recordincident angle pairs generated
using the Latin hypercube sampling (LHS) method.
The numerical part of this work is composed of two
parts. In the first part, a parametric study is performed
in an effort to define the orientation where the two
components of the records should be applied in order
to obtain the maximum seismic response. In particular,
the influence of the incident angle of attack of the two
horizontal components of the records on the seismicresponse of mid-rise RC buildings is examined. In the
second part of the study, six implementations of the
MIDA are assessed with respect to limit state fragility
curves developed. More specifically, the six implemen-
tations are: (i) application of the two components of
the records along the structural axes and their
complementary ones; (ii) application of the two com-
ponents along the principal axes and their comple-
mentary ones; (iii) application of the two components
along two orthogonal axes defined with a randomly
selected incident angle (considered fixed for all records)
and its complementary one; (iv) application of the
proposed procedure over a sample of 15 pairs; (v)
application of the proposed procedure over a sample of
30 pairs; and (vi) application of the proposed
procedure over a sample of 100 pairs. Cases (i) to
(iii) are variations of the typical MIDA implementa-
tion, while (iv) to (vi) are variants of the proposed
method with respect to the sample size. The proposed
method gives a rational procedure in order to take into
account randomness on both incident angle and
seismic excitation in the framework of MIDA. Both
parts of the parametric study are performed in two test
examples, one having a symmetrical plan view, and a
second one having a non-symmetrical layout. As will
be seen from the parametric study, the three imple-
mentations of the MIDA, where the two components
of all the records are applied along the same incident
angle, either overestimate or underestimate the capa-city of a structural system compared to the proposed
implementation.
2. Critical orientation of the seismic incidence
literature survey
A structure subjected to the simultaneous action of two
orthogonal horizontal ground accelerations along the
directions Ow and Op is illustrated in Figure 1. The
orthogonal system Oxyz defines the reference axes of
the structure (structural axes). The angle defined with
an anticlockwise rotation of the structural axis Ox to
coincide with the ground motion axis Ow is called theincident angle of the record.
According to Penzien and Watabe (1975), the
orthogonal directions of a ground motion can be
considered uncorrelated in the principal directions of
the structure. This finding was the basis for many
researchers in order to define the orientation that yields
the maximum response when the response spectrum
dynamic analysis was applied. In the work by Wilson
et al. (1995), an alternative code method, which results
in structural designs that have equal resistance to
seismic motions from all directions, was proposed.
Lopez and Torres (1997) proposed a simple method
that can be employed by the seismic codes to determinethe critical angle of seismic incidence and the
corresponding peak response of structures subjected
to two horizontal components applied along any
arbitrary directions and to the vertical component of
earthquake ground motion. The CQC3 response
spectrum rule for combining the contributions from
three orthogonal components of ground motion to the
maximum value of a response quantity was presented
in the work by Menun and Der Kiureghian (1998). In
the work by Lopez et al. (2000, 2001), an explicit
Figure 1. Definition of the incident angle a.
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formula, convenient for code applications was pro-
posed, in order to calculate the critical value of the
structural response to the two principal horizontal
components acting along any incident angle with
respect to the structural axes, and the vertical
component of ground motion. In other work by
Menun and Der Kiureghian (2000a,b), a responsespectrum-based procedure for predicting the envelope
that bounds two or more responses in a linear structure
was developed. In the work by Anastassiadis et al.
(2002), a seismic design procedure for structures was
proposed where the three translational ground motion
components on a specific point of the ground were
statistically non-correlated along a well-defined ortho-
gonal system.
There are only few studies in the literature where
the case of the critical incident angle is examined when
time history analysis is employed. In these studies, it
was found that, in the most general case, where
nonlinear behaviour is encountered, it was not aneasy task to define the critical angle. In the work by
MacRae and Mattheis (2000), the ability of the 30%
square root of the sum of the squares (SRSS) rule
and the sum of absolute values methods to assess
building drifts for bidirectional shaking effects was
proposed, while it was shown that the response was
dependent on the reference axes chosen. MacRae and
Tagawa (2001) found that design level shaking caused
the structure to exceed storey yield drifts in both
directions simultaneously, and that significant column
yielding occurred above the base. Shaking a structure
in the direction orthogonal to the main shaking
direction increased drifts in the main shaking direction,indicating that 2D analyses would not estimate the 3D
response well. Ghersi and Rossi (2001) examined the
influence of bidirectional seismic excitations on the
inelastic behaviour of in-plan irregular systems with
one symmetry axis, where it was found that, in most
cases, the adoption of Eurocode 8 (CEN 2003)
provisions to combine the effects of the two seismic
components allowed the limitation of the orthogonal
elements ductility demand. In the work by Athanato-
poulou (2005), analytical formulae were developed for
determining the critical incident angle and the corre-
sponding maximum value of a response quantity of
structures subjected to three seismic correlated com-
ponents when linear behaviour was considered. The
analyses results have shown that, for the earthquake
records used, the critical value of a response quantity
can be up to 80% larger than the usual response
produced when the seismic components are applied
along the structural axes. Rigato and Medina (2007)
studied a number of symmetrical and asymmetrical
structures with fundamental periods ranging from 0.2
to 2.0 s, where the influence that the incident angle of
the ground motion had on several engineering demand
parameters was examined.
3. MIDA implementations
The main objective of an IDA study is to define a curve
through the relationship of the intensity level with themaximum seismic response of the structural system.
The intensity level and the seismic response are
described through an intensity measure (IM) and an
engineering demand parameter (EDP), respectively.
The IDA study is implemented with the following
steps: (i) define the nonlinear FE model required for
performing the nonlinear dynamic analyses; (ii) select a
suite of natural records; (iii) select a proper intensity
measure and an engineering demand parameter; (iv)
employ an appropriate algorithm for selecting the
record scaling factor in order to obtain the IDA curve
performing the least required nonlinear dynamic
analyses; and (v) employ a summarisation techniquefor exploiting the multiple record results.
Selecting the IM and EDP is one of the most
important steps of the IDA study. In the work by
Giovenale et al. (2004), the significance of selecting an
efficient IM was discussed, while an originally adopted
IM was compared with a new one. The IM should be a
monotonically scalable ground motion intensity mea-
sure, such as the peak ground acceleration (PGA),
peak ground velocity (PGV), the x 5% dampedspectral acceleration at the structures first-mode
period (Sa(T1,5%)) and many others. In the current
work, the Sa(T1,5%) is selected, as it is the most
commonly used intensity measure in practise today forthe analysis of buildings. The two components of the
records are scaled to Sa(T1,5%), thus preserving their
relative scale. This is achieved by scaling the compo-
nent of the record with the highest Sa(T1,5%), while
the second one follows the scaling rule, thus preserving
their relative ratio. On the other hand, the damage may
be quantified by using any of the EDPs defined as
functions, whose values can be related to particular
structural damage states. A number of available
response-based EDPs were discussed and critically
evaluated in the past for their applicability in seismic
damage evaluation (Ghobarah et al. 1999). In their
work, the EDPs are classified into four categories:
engineering demand parameters based on maximum
deformation, engineering demand parameters based on
cumulative damage, engineering demand parameters
accounting for maximum deformation and cumulative
damage, global engineering demand parameters. In the
current work, the maximum interstorey drift ymax is
chosen, belonging to the EDPs, which are based on the
maximum deformation. The reason for selecting ymaxis because there is an established relation between
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interstorey drifts and performance-oriented descrip-
tions, such as immediate occupancy, life safety and
collapse prevention (FEMA 1997).
According to the MIDA framework, a set of
natural records, each one represented by its long-
itudinal and transverse components, are applied to the
structure in order to account for the randomness onthe seismic excitation. The difference of the MIDA
framework from the original one-component version
of the IDA, proposed by Vamvatsikos and Cornell
(2002), stems from the fact that, for each record, a
number of MIDA representative curves can be defined,
depending on the incident angle selected, while in most
cases of the one-component version of IDA, only one
IDA representative curve is obtained. In the MIDA
implementation performed so far (Serdar Kircil and
Polat 2006, Vamvatsikos 2006), the two components of
the records were applied along two orthogonal axes
with the same incident angle, which was equal to zero,
ignoring the randomness on the incident angle. In thiswork, a new procedure for applying MIDA that is
based on the idea of considering variable incident angle
for each record, is proposed. The proposed implemen-
tation takes into account the randomness both on the
seismic excitation and the incident angle. In MIDA,
the relation of IMEDP is defined similarly to the one-
component version of the IDA, i.e. both horizontal
components of each record are scaled to a number of
intensity levels to encompass the full range of
structural behaviour from elastic to yielding that
continues to spread, finally leading to global
instability.
A schematic representation of the proposed proce-dure can be seen in Figure 2, where the MIDA is
implemented over a sample of recordincident angle
pairs generated using LHS (McKay et al . 1979).
According to the proposed method, a sample of N
pairs of recordincident angle are generated with LHS;
for each pair, MIDA is conducted and the
representative MIDA curve is developed for the pair
in question. Afterwards, these representative MIDA
curves are used in order to develop the 16%, 50% and
84% median curves that are used to perform prob-
abilistic analysis. LHS is a strategy for generating
random sample points, ensuring that all portions of the
random space are properly represented. In LHS, a fullstratification of the sampled distribution with a
random selection inside each stratum is performed,
and the sample values are randomly shuffled among
different variables. A Latin hypercube sample is
constructed by dividing the range of each of the M
uncertain variables into N non-overlapping segments
of equal marginal probability. Thus, the whole
parameter space, consisting of N parameters, is
partitioned into NM cells. A single value is selected
randomly from each interval, producing N sample
values for each input variable. The values are
randomly matched to create N sets, from the NM
space with respect to the density of each interval, forthe N simulation runs. In the current implementation,
both record and incident angle are considered uni-
formly distributed over a sample of 15 records and in
the range 08 to 1808, respectively.
4. Description of the models
The two three-storey 3D RC buildings, shown in
Figures 3 and 4, have been considered in order to study
the framework for applying the MIDA. The first test
example corresponds to an RC building of symmetrical
plan view, while the second one corresponds to an RC
building with a non-symmetrical plan view. Bothbuildings have been designed to meet the Eurocode
requirements, i.e. the EC2 (CEN 2002) and EC8 (CEN
2003) design codes. In the case of the EC8, the lateral
forces were derived from the design response spectrum
(5% damped elastic spectrum divided by the behaviour
factor q 3.0) at the fundamental period of the
Figure 2. The new MIDA procedure.
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building. Concrete of class C16/20 (nominal cylindrical
strength of 16 MPa) and class S500 steel (nominal
yield stress of 500 MPa) are assumed. The base shear
is obtained from the response spectrum for soil
type B (stiff soil y 1.0, with characteristic periodsT1 0.15 s and T2 0.60 s), while the PGA consid-ered for the first test example is of 0.24g and for the
second one is of 0.16g. Moreover, the importancefactor gI was taken equal to 1.0, while the damping
correction factor is equal to 1.0, since a damping ratio
of 5% has been considered.
The slab thickness is equal to 15 cm, for both test
examples, and is considered to contribute to the
moment of inertia of the beams with an effective
flange width. In addition to the self weight of the
beams and the slab, a distributed dead load of 2 kN/m2,
due to floor finishing and partitions, and an imposed
live load with a nominal value of 1.5 kN/m2, are
considered. The nominal dead and live loads are
multiplied by load factors of 1.35 and 1.5, respectively.
Following EC8, in the seismic design combination,
dead loads are considered with their nominal values,
while live loads with 30% of their nominal value.
A centreline model was formed, for both test
examples, using the OpenSees (McKenna and Fenves
2001) simulation platform. The members are modelled
using the force-based fibre beamcolumn element
while the same material properties are used for all
the structural elements of the two structures. Soil
structure interaction was not considered and the base
of the columns at the ground floor is assumed to be
fixed.
5. Incident angle in the framework of MIDA
In this section, the influence of the intensity level on the
critical incident angle and the diversification of the
MIDA curves with respect to the incident angle areexamined in an effort to be considered in the MIDA
framework.
5.1. Critical incident angle with respect to the intensity
level
In order to examine the influence of the incident angle
on the seismic response of the structure, three records
have been selected at random from a suite of 15
records, and are applied to both test examples. The
three records considered are the Loma Prieta
(WAHO), the Imperial Valley (Compuertas) and the
Northridge (LA, Baldwin Hills), and their character-
istics can be found in Table 1. The three records have
been applied considering a varying incident angle in
the range of 08 to 3608, with a step of 58. In order to
examine the influence of the incident angle on the
maximum interstorey drift to different intensity levels,
the three records have been scaled with respect to the
5% damped spectral acceleration at the structures first
mode period to 0.05g, 0.30g and 0.50g, and the
maximum interstorey drift has been recorded for all
Figure 3. Test example 1: geometry of the three-storey symmetric 3D building: (a) plan view and (b) side view (the dimensionsof beams and colums are in cm).
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the incident angles and the intensity levels considered.
The ground motion records are listed in the table,
where the Campbells R and epicentre (EpiD) dis-
tances, the duration, the PGA values for the
longitudinal and transverse directions, Campbells
soil type (A is firm soil, B is very firm soil and C is
soft rock) and the fault rupture mechanism (SS is for
strike slip, RN is reverse normal and RO is reverse
Figure 4. Test example 2: geometry of the three-storey non-symmetric 3D building: (a) plan view and (b) side view (thedimensions of beams and columns are in cm).
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oblique) are given for the 15 records, while M stands
for the moment magnitude of the earthquake.
In this work, the response of the structure is defined
through the bidirectional maximum column interstorey
drift ratio over all storeys (Wen and Song 2003) of the
structure, which is defined as follows:
ymax maxffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiyt
2
X yt2
Yq ;
1
where y(t)X and y(t)Y are the interstorey drift along the
structural axes Xand Yin the tth time step, i.e. ymax is
defined as the maximum value of the vector sum of the
interstorey column drift ratio in the two structural
axes.
The alteration of the maximum interstorey drift
with respect to the incident angle and the intensity level
for the three records is depicted in Figures 5 and 6 for
the two test examples, respectively. As can be seen
from both groups of figures, the seismic response for
both test examples when the incident angle varies in
the range of 08 and 1808 almost coincides with the
seismic response corresponding to incident angle
varying in the range of 1858 to 3608. This is because
the relative ratio of the two horizontal components of
the records is close to one, thus the two components
are scaled to almost the same intensity level, i.e. the
same value of Sa(T1,5%). Moreover both symmetrical
and non-symmetrical buildings are primarily con-
trolled by the first eigenmode. For this reason, the
incident angle range of 08 to 1808 is employed for the
parametric studies performed in the following sections
for both test examples.
A second remark from Figures 5 and 6 is that the
seismic response varies significantly with respect to the
incident angle. For instance, for the first test example,
the maximum interstorey drift for the case of Loma
Prieta (WAHO) record varies from 0.17% to 0.23%
for the 0.05 intensity level, while, for the 0.50 intensity
level, the maximum interstorey drift for the samerecord varies from 1.77% to 2.20%. Another signifi-
cant observation from the two groups of figures is that
the maximum seismic response is encountered for
different incident angles when a different record is
considered. Worth mentioning is that, for the 0.30g
intensity level, the maximum seismic response for the
test example with the symmetrical layout is encoun-
tered in the incident angle range of 908 to 1208 for the
Northridge (LA, Baldwin Hills) record. For the Loma
Prieta (WAHO) record, however, in the same incident
angle range, the minimum seismic response is encoun-
tered. Similar observations can be noticed for the
second test example and the same intensity level when
the incident angle varies in the range of 608 to 1208.
In Tables 2 and 3, the maximum and minimum
values, along with the mean value and the coefficient of
variation (COV) of the maximum interstorey drift,
when the three records are applied in a range of
incident angles, are given. It can also be seen from both
tables that it is not possible to predict the critical
incident angle where the response in terms of
interstorey drift takes its maximum value for the
Table 1. Characteristics of the 15 records.
Recordstation R (km) EpiD (km) Duration (s) PGAlog (g) PGAtran (g)
CampbellsGEOCODE
Faultrupture
Superstition Hills 1987 (B) (M 6.7)1. El Centro Imp. Co Cent 18.5 35.83 40.00 0.36 0.26 A SS2. Wildlife Liquefaction Array 24.1 29.41 44.00 0.18 0.21 A SS
Imperial Valley 1979 [23:16], (M 6.5)3. Chihuahua 8.4 18.88 40.00 0.27 0.25 A SS4. Compuertas 15.3 24.43 36.00 0.19 0.15 A SS5. El Centro Array #1 21.7 36.18 39.03 0.14 0.13 A SS
San Fernando 1971 (M 6.6)6. LA, Hollywood Stor. Lot 25.9 39.49 28.00 0.21 0.17 A RN
Northridge 1994 (M 6.7)7. Leona Valley #2 37.2 51.88 32.00 0.09 0.06 A RN8. LA, Baldwin Hills 29.9 28.20 40.00 0.24 0.17 C RN9. LA, Fletcher Dr 27.3 30.27 29.99 0.16 0.24 B RN
10. Glendale Las Palmas 22.2 29.72 29.99 0.36 0.21 A RN
Loma Prieta 1989 (M 6.9)11. Hollister Diff Array 24.8 45.10 39.64 0.27 0.28 A RO12. WAHO 17.5 12.56 24.96 0.37 0.64 C RO13. Halls Valley 30.5 36.31 39.95 0.13 0.10 B RO
14. Agnews State Hospital 24.6 40.12 40.00 0.17 0.16 A RO15. Sunnyvale Colton Ave 24.2 42.13 39.25 0.21 0.21 A RO
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intensity level in question. It can also be observed
that the COV varies significantly with respect to theintensity level and the record for both test examples.
Worth mentioning also is that the variation of the
seismic response of the second test example for the
0.05g intensity level varies from 10% to 32%.
Moreover, in these two tables, the maximum
interstorey drift when the two horizontal components
of the records are applied along the structural and
principal axes of the structure can also be found, along
with the 16%, 50% and 84% median values of the ymaxwhen the three records are applied in the range of 08 to
1808. It has to be noted that the maximum interstorey
drift values for the case of the structural and principal
axes are obtained as the mean values of the ymax when
the horizontal components of the record are applied
along the structural or principal axes and their
complementary ones. For the first test example,
the principal and structural axes coincide due to the
symmetrical plan view. For both test examples, the
ymax value for the case of the structural and principal
axes is either lower or greater than the 50% median
value; this depends on the record and the intensity
level. For instance for the first test example for all three
intensity levels, the ymax value of the case of the
structural axes is close to the corresponding value forthe 84% median for the Loma Prieta (WAHO) record.
Very different observations are obtained for the same
test example for the Imperial Valley (Compuertas)
record. On the other hand, when the two components
of the records are applied along the principal axes, the
ymax value is close to the 50% median values for all
the intensity levels.
5.2. MIDA representative curves with respect to the
incident angle
As was mentioned in the previous section, the
implementation of the MIDA framework requires the
definition of the MIDA representative curve for each
record, or for each pair of recordincident angle. In
this section, three implementations of the MIDA are
examined, where the incident angle remains fixed over
the records, while two variations for each implementa-
tion are considered. In particular, in the first imple-
mentation, the two variations are denoted as case A1
and case A2; the two horizontal components of the
records are applied along the structural axes and their
Figure 5. Test example 1: ymax (%) with respect to the incident angle of the record scaled to: (a) 0.05g, (b) 0.30g and (c) 0.50gintensity levels.
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complementary ones, respectively. In the second
implementation, the variations are denoted as caseB1 and case B2; the two horizontal components are
applied along the principal axes and their complemen-
tary ones. In the third implementation, the two
variations are denoted as case C1 and case C2; in
this implementation, the two horizontal components of
the records are applied along a randomly selected
incident angle (308) and their complementary ones.
Through the parametric study of the previous
section, it was found that the critical incident angle
varies significantly with reference to the intensity level.
The objective of this part of the study is to compare the
MIDA representative curves of the three implementa-
tions versus representative curves developed using
variable incident angle. For this reason, the three
implementations and their variations are performed
for the three records selected for the parametric
investigation of the previous section. Figures 7 and 8
depict, for the two test examples, the various MIDA
representative curves defined both with variable
incident angle in the range of 08 to 1808, with a step
of 58, along with the MIDA curves representing the
cases A1, A2, B1, B2, C1 and C2, together with the
16%, 50% and 84% medians. The median curves are
defined through the MIDA representatives obtainedwith variable incident angle. As can be seen from both
groups of figures, there is a significant variability of the
MIDA curves with respect to the incident angle. For
the first test example, where, due to the symmetrical
plan view cases Ai and Bi (i 1 or 2) coincide, thecases A2/B2 and C2 are more conservative with respect
to the cases A1/B1 and C1, apart from the Northridge
(LA, Baldwin Hills) record, where case C1 is more
conservative compared to case C2. Moreover, the
MIDA curve for case A1/B1 is always above the 50%
median, while the curve for case A2/B2 is always below
the 50% median approaching the 84%. Furthermore,
cases Ai/Bi (i 1 or 2) are, for all three records, moreconservative compared to cases Ci. In the second test
example, although the last observation remains the
same, i.e. cases Ai/Bi(i 1 or 2) are more conservativecompared to cases Ci, variation on the other remarks
of the first test example is encountered. No definite rule
can be defined regarding the relations of the cases A1,
B1 and C1 with reference to the cases A2, B2 and C2,
furthermore stronger variation on the position of the
Ai, Biand Ci(i 1 or 2) MIDA curves is encountered
Figure 6. Test example 2: ymax (%) with respect to the incident angle of the record scaled to: (a) 0.05g, (b) 0.30g and (c) 0.50gintensity levels.
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with reference to the 16%, 50% and 84% medians. The
results of this section enforce the need to take into
account the randomness of both record and incident
angle.
6. MIDA-based PBEE
One of the objectives in PBEE is to quantify the
seismic reliability of a structure, due to future random
earthquakes, at a site. For that purpose, fragility
analysis is used in order to estimate the mean annual
frequency of exceeding a specified value of a structural
demand parameter. In this work, a MIDA-based
fragility analysis of 3D mid-rise RC buildings is
performed.
6.1. The MIDA framework
The first step, in order to perform MIDA based
fragility analysis, is to select a suite of natural records
to be used for performing the MIDA study, while the
second one is to adopt the 50% fractile MIDA curves
of the two test examples for developing the limit state
fragility curves. Based on previous studies (Shome and
Cornell 1999), it was found that 10 to 20 records are
sufficient for predicting, with acceptable accuracy, the
seismic demand of a mid-rise building; for this reason,
a suite of 15 records with two components each has
been selected in this study.
In this work, two distinctive procedures for
implementing MIDA are considered. In the first one,
the two horizontal components of the records are
applied along two orthogonal axes with the same
incident angle (a detailed description of the cases Ai, Bi
and Ciwas given in the previous section). According to
the second procedure, MIDA is performed over a
sample of recordincident angle pairs that are gener-
ated (as described in the previous section). Figures 9
and 10 depict the MIDA representative curves of the
15 records considered, together with the 16%, 50%
and 84% fractile MIDA curves for the cases A1, A2,
B1, B2, C1 and C2 and the cases LHS-15, LHS-30 and
Table 2. Test example 1: statistical data ofymax (%) with reference to three intensity levels.
Sa(T1) (g)Maximumymax(%)
Incidentangle (8)
Minimumymax(%)
Incidentangle (8)
Structuralaxes
ymin(%)Mean
ymax(%)COV(%)
Median16%
ymax(%)
Median50%
ymax(%)
Median84%
ymax(%)
Loma Prieta (WAHO)0.05 0.2497 160 0.1564 060 0.2336 0.1984 14.56 0.1661 0.1925 0.2336
0.30 1.1551 155 0.6639 095 1.1310 0.9498 16.47 0.7705 0.9062 1.12770.50 2.1313 165 1.1959 075 2.0974 1.7182 18.72 1.2822 1.7129 2.0876
Imperial Valley (Compuertas)0.05 0.3616 145 0.2036 050 0.2157 0.2541 17.59 0.2163 0.2398 0.29700.30 1.4258 135 0.9843 060 1.1649 1.2309 9.20 1.1394 1.2005 1.37270.50 1.7012 150 1.2918 090 1.6545 1.4890 10.27 1.3230 1.4273 1.6776
Northridge (LA, Baldwin Hills)0.05 0.4544 040 0.2536 150 0.3481 0.3452 18.18 0.2737 0.3355 0.42860.30 1.5468 100 1.1548 035 1.3556 1.3544 7.17 1.2609 1.3474 1.45470.50 2.0663 065 1.4561 010 1.5566 1.7803 10.36 1.5862 1.7479 1.9787
Table 3. Test example 2: statistical data ofymax (%) with reference to three intensity levels.
Sa(T1) (g)Maximumymax(%)
Incidentangle (8)
Minimumymax(%)
Incidentangle (8)
Principal
axesymin(%)
Structural
axesymin(%)
Meanymax(%)
COV(%)
Median
16%ymax(%)
Median
50%ymax(%)
Median
84%ymax(%)
Loma Prieta (WAHO)0.05 0.2358 085 0.1733 005 0.1945 0.1733 0.1959 9.79 0.1790 0.1891 0.22040.30 1.4295 110 1.0197 015 1.1507 1.0982 1.2407 10.81 1.0766 1.2571 1.39590.50 2.2027 285 1.7669 355 1.8809 1.7766 1.9693 6.39 1.8403 1.9628 2.0919
Imperial Valley (Compuertas)0.05 0.4433 320 0.1487 230 0.1523 0.3536 0.3004 31.97 0.2072 0.2767 0.42030.30 1.3020 295 0.9723 185 1.0264 0.9877 1.1378 9.88 1.0083 1.1396 1.26870.50 1.4263 075 1.0517 355 1.3831 1.0963 1.2940 8.64 1.1713 1.3260 1.4052
Northridge (LA, Baldwin Hills)0.05 0.3685 045 0.2339 115 0.3624 0.2772 0.2862 15.91 0.2381 0.2714 0.34290.30 1.4483 190 1.0121 120 1.3257 1.4262 1.2976 10.75 1.0909 1.3484 1.42000.50 1.7170 195 1.2662 140 1.4927 1.6369 1.4789 8.15 1.3443 1.4636 1.6080
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LHS-100, which stand for the proposed procedure
where sample sizes N equal to 15, 30 and 100 pairs are
taken into account, respectively. Due to the symme-
trical plan view of the first test example, Figures 9a and
9b correspond to cases A1/B1 and A2/B2, respectively.
6.2. Probabilistic safety assessment of RC buildings
Once the 50% fractile MIDA curve is obtained, we
proceed with the development of the fragility curves
for the limit states in question. Figures 11 and 12
depict the limit state fragility curves for the two low-
rise RC buildings for the high-code design level of the
earthquake loss estimation methodology (HAZUS;
FEMA-NIBS 2003). Four limit states are selected:
slight, moderate, extensive and complete structural
damage states. Buildings are composed of both
structural (load carrying) and non-structural systems
(e.g. architectural and mechanical components). While
damage to the structural system is the most important
measure of building damage affecting casualties and
catastrophic loss of function, damage to non-structural
Figure 7. Test example 1: MIDA curves with respect to the incident angle, cases A1/B1, A2/B2, C1, C2 and 16%, 50%, 84%median curves for step size 58: (a), (b) Loma Prieta (WAHO), (c), (d) Imperial Valley (Compuertas) and (e), (f) Northridge (LA,Baldwin Hills).
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systems and contents tends to dominate economic loss.
Figures 11 and 12 refer to the structural damage only.
The damage states are defined with respect to the
interstorey drift limits given in HAZUS (FEMA-NIBS
2003) for this type of structure. The interstorey drift
limits are equal to 0.5%, 1.0%, 3.0% and 8.0% for
slight, moderate, extensive and complete structural
damage states, respectively. The fragility curves shown
in Figures 11 and 12 correspond to all implementations
of the MIDA. The probabilities of exceedance of the
four damage states corresponding to Sa(T1,5%) equal
to 1.0, 3.0 and 6.0 m/s2 are given in Tables 4 and 5 for
the two test examples, respectively.
From Figures 11 and 12, significant variability of
the fragility curves is observed for some limit states.
For the first test example, considerable disparity on the
fragility curves is observed for the complete damage
state. For the second test example, significant variation
is encountered in all limit states. Moreover, for both
test examples, the fragility curves corresponding to the
Figure 8. Test example 2: MIDA curves with respect to the incident angle, cases A1, A2, B1, B2, C1, C2 and 16%, 50%, 84%median curves for step size 58: (a), (b) Loma Prieta (WAHO), (c), (d) Imperial Valley (Compuertas) and (e), (f) Northridge (LA,Balwin Hills).
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Figure 9. Test example 1: MIDA representative curves for the 15 records and 16%, 50% and 84% fractile MIDA curvesimplementing case: (a) A1/B1, (b) A2/B2, (c) C1, (d) C2, (e) LHS-15, (f) LHS-30 and (g) LHS-100.
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Figure 10. Test example 2: MIDA representative curves for the 15 records and 16%, 50% and 84% fractile MIDA curvesimplementing case: (a) A1, (b) A2, (c) B1, (d) B2, (e) C1, (f) C2, (g) LHS-15, (h) LHS-30 and (i) LHS-100.
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cases A1, B1 and C1 vary with reference to the curves
of the complementary cases. On the other hand, in
most of the limit states, the fragility curves developed
based on LHS-30 are very close to those developed
based on the LHS-100 for the two test examples;
consequently 30 pairs are enough for achieving a good
approximation.
The observations obtained from Figures 11 and 12
are verified from the probabilities given in Tables 4 and
5. From both tables, it can be seen that the variation of
the probabilities of exceedance belonging to cases A1,
B1 and C1 compared to those belonging to the
complementary cases depends on the limit state and
the intensity level for both test examples. More
specifically, in the first test example, the probability
of exceedance of the slight limit state calculated based
on the A1/B1 implementation is equal to 61% for
Sa(T1,5%) 1.0 m/s2 versus 56% for the complemen-tary case A2/B2. On the other hand, the probabilities
of exceedance for the same limit state is equal to 98%
Figure 11. Test example 1: fragility curves for four limit states.
Figure 12. Test example 2: fragility curves for four limit states.
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for A1/B1 versus 97.7% for A2/B2 when
Sa(T1,5%) 3.0 m/s2, while if Sa(T1,5%) 6.0 m/s2,the two probabilities are identical. The same observa-
tions can be described for all limit states and intensity
levels for all three cases Ai, Biand Ci. For the proposed
implementation where variable incident angle is taken
into account through the LHS method, the results do
not seem to vary considerably with respect to the
number of pairs employed, while the LHS-30 and LHS-
100 implementations are almost identical.
7. Conclusions
The probabilistic safety assessment of the seismic
response of real world 3D mid-rise RC buildings,
which is one of the most significant ingredients of the
performance-based earthquake engineering (PBEE)
framework, is studied in this work. In particular, an
incremental dynamic analysis (IDA)-based fragility
analysis is performed. The difference of the
multicomponent incremental dynamic analysis
(MIDA) with respect to its one component version
stems from the inability to define the direction that the
two horizontal components of the records should be
applied in order to obtain the maximum seismic
response.
There were two serious indications that were
required to take into account the incident angle in
the MIDA. The first one was obtained through the
parametric study performed, where it was found that
the response varies with respect to the record, intensity
level and incident angle. The second indication was the
variability of the fragility curves developed, based on
the cases A1, B1 and C1 versus their complementary
ones. For this reason, a new procedure for performing
MIDA is proposed in the present work, with variable
incident angle considered using the Latin hypercube
sampling (LHS) method. The proposed procedure
provides a rational way for taking into account the
randomness on the record and on the incident angle
Table 4. Test example 1: probability of exceeding the four limit states (%).
Limit state A1/B1 A2/B2 C1 C2 LHS-15 LHS-30 LHS-100
Sa(T1,5%) 1.0 m/s2
Slight 60.88 56.22 60.56 60.33 61.16 60.49 58.96Moderate 14.89 14.89 16.17 14.42 15.83 14.66 14.95Extensive 0.07 0.07 0.07 0.06 0.04 0.07 0.06
Complete 0.00 0.00 0.00 0.00 0.00 0.00 0.00Sa(T1,5%) 3.0 m/s
2
Slight 98.26 97.68 98.22 98.20 98.29 98.22 98.04Moderate 78.63 78.63 80.16 78.03 79.77 78.34 78.70Extensive 8.79 8.46 8.49 8.11 6.67 8.58 7.71Complete 1.68 1.99 0.69 1.31 0.63 0.91 1.04
Sa(T1,5%) 6.0 m/s2
Slight 99.95 99.92 99.94 99.94 99.95 99.94 99.94Moderate 97.45 97.45 97.75 97.32 97.68 97.39 97.46Extensive 42.22 41.41 41.47 40.51 36.57 41.71 39.47Complete 16.64 18.44 9.58 14.36 9.09 11.43 12.45
Table 5. Test example 2: probability of exceeding the four limit states (%).
Limit state A1 A2 B1 B2 C1 C2 LHS-15 LHS-30 LHS-100
Sa(T1,5%) 1.0 m/s2
Slight 61.03 55.36 55.50 59.30 58.64 58.50 59.33 56.01 56.50Moderate 11.78 11.73 9.99 15.21 9.55 14.35 14.33 11.76 11.76Extensive 0.10 0.06 0.09 0.04 0.04 0.10 0.04 0.05 0.05Complete 0.01 0.01 0.02 0.01 0.01 0.01 0.02 0.01 0.01
Sa(T1,5%) 3.0 m/s2
Slight 98.28 97.56 97.58 98.08 98.00 97.98 98.08 97.65 97.72Moderate 74.18 74.09 70.98 79.02 70.10 77.94 77.90 74.13 74.13Extensive 10.52 8.16 9.81 6.79 6.45 10.69 6.40 7.17 7.51Complete 3.15 3.25 4.16 3.74 3.11 3.21 4.18 2.47 2.36
Sa(T1,5%) 6.0 m/s2
Slight 99.95 99.91 99.91 99.94 99.93 99.93 99.94 99.92 99.92Moderate 96.46 96.44 95.64 97.53 95.40 97.31 97.30 96.45 96.45Extensive 46.23 40.64 44.64 36.93 35.92 46.60 35.76 38.00 38.93Complete 24.16 24.60 28.28 26.61 23.95 24.40 28.36 20.98 20.41
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requiring only 30 recordincident angle pairs. The
new procedure is compared with three implementa-
tions and with the same incident angle. The different
implementations have been employed in order to
perform probabilistic safety analysis of two 3D mid-
rise RC buildings. Both buildings have been designed
to fulfil the provisions of Eurocodes 2 and 8 (CEN2002, 2003).
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