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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: [email protected]

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.

78 N.D. Lagaros


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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

Structure and Infrastructure Engineering 79


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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.

80 N.D. Lagaros


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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).

82 N.D. Lagaros


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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.

84 N.D. Lagaros


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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

86 N.D. Lagaros


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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).



13/19

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).

88 N.D. Lagaros


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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.



15/19

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.

90 N.D. Lagaros


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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


Sa(T1,5%) 6.0 m/s2


92 N.D. Lagaros


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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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