the effect of material and ground motion uncertainty on the seismic vulnerability of rc structure

7/30/2019 The Effect of Material and Ground Motion Uncertainty on the Seismic Vulnerability of RC Structure

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Engineering Structures 28 (2006) 289303

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The effect of material and ground motion uncertainty on the seismicvulnerability curves of RC structure

Oh-Sung Kwona,, Amr Elnashaib

aUniversity of Illinois, 205 North Mathews, Urbana, IL 61801, United StatesbMid-America Earthquake Center, University of Illinois, 205 North Mathews, Urbana, IL 61801, United States

Received 23 September 2004; received in revised form 17 June 2005; accepted 12 July 2005

Available online 3 October 2005

Abstract

Starting from the premise that vulnerability curves are an indispensable ingredient of earthquake loss assessment, this paper focuses on

establishing the relative effect of strong-motion variability and random structural parameters on the ensuing vulnerability curves. Moreover,

the effect of the selection of statistical models used to present simulation results is studied. A three story ordinary moment resisting reinforced

concrete frame, previously shake-table tested, is used as a basis for the fragility analysis. The analytical environment and the structural model

are verified through comparison with shaking-table test results. The selection of ground motion sets, definition of limit states, statistical

manipulation of simulation results, and the effect of material variability are investigated. No approximations are used to reduce the sample

size or minimize the analytical effort, in order that attention is focused on the parameters under investigation. Notwithstanding the limited

scope of the study, the results presented indicate that the effect of randomness in material response parameters is far less significant than the

effect of strong-motion characteristics. Therefore, the importance of scrupulous selection and scaling of strong-motion and use of appropriate

limit states and statistical models is emphasized.

2005 Elsevier Ltd. All rights reserved.

Keywords: Seismic vulnerability; Ground motion uncertainty; Material uncertainty; Ordinary moment resisting concrete frame

1. Introduction

Vulnerability curves relate strong-motion shaking sever-

ity to the probability of reaching or exceeding a specified

performance limit state. Strong-motionshaking severity may

be expressed by an intensity (I), peak ground parameters

(a, v or d) or spectral ordinates (Sa , Sv or Sd) corresponding

to an important structural period. The number of limit states

used varies between three and five. In this paper, three limitstates considered as the most significant are used; service-

ability, damage control and collapse prevention.

Vulnerability curves play a critical role in regional

seismic risk and loss estimation as they give the probability

of attaining a certain damage state when a structure is

Corresponding author. Tel.: +1 217 265 5497; fax: +1 217 333 3821.E-mail addresses: [email protected] (O.-S. Kwon),

[email protected] (A. Elnashai).

0141-0296/$ - see front matter 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engstruct.2005.07.010

subjected to a specified demand. Such loss estimations are

essential for the important purposes of disaster planning and

formulating risk reduction policies. The driving technical

engines of a regional seismic risk and loss estimation system

are [12]:

Seismic hazard maps (i.e. peak ground parameters or

spectral ordinates).

Vulnerability functions (i.e. relationships of conditional

probability of reaching or exceeding a performance limit

state given the measure of earthquake shaking).

Inventory data (i.e. numbers, location and characteristics

of the exposed system or elements of a system).

Integration and visualization capabilities (i.e. data

management framework, integration or seismic risk and

graphical projection of the results).

The scope of this study is to present vulnerability

curves of a reinforced concrete structure subjected to
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290 O.-S. Kwon, A. Elnashai / Engineering Structures 28 (2006) 289303

Table 1

Categorization of vulnerability curvea

Category Characteristics

Empirical vulnerability curve

Feature Based on post-earthquake survey

Most realistic

Limitation Highly specific to a particular seismo-tectonic, geotechnical and built environment

The observational data used tend to be scarce and highly clustered in the low-damage, low-ground-motion severity range

Include errors in building damage classification

Damage due to multiple earthquakes may be aggregated

Sample ref. [24]

Judgmental vulnerability curve

Feature Based on expert opinion

The curves can be easily made to include all the factors

Limitation The reliability of the curves depends on the individual experience of the experts consulted

A consideration of local structural types, typical configurations, detailing and materials inherent in

the expert vulnerability predictions

Sample Ref. ATC-13 [1]

Analytical vulnerability curve

Feature Based on damage distributions simulated from the analyses

Reduced bias and increased reliability of the vulnerability estimate for different structures

Limitation Substantial computational effort involved and limitations in modeling capabilities

The choices of the analysis method, idealization, seismic hazard, and damage models influence the

derived curves and have been seen to cause significant discrepancies in seismic risk assessments

Sample Ref. [21,26,9]

Hybrid vulnerability curve

Feature Compensate for the scarcity of observational data, subjectivity of judgmental data, and modeling

deficiencies of analytical procedures

Modification of analytical or judgment based relationships with observational data and

experimental results

Limitation The consideration of multiple data sources is necessary for the correct determination of

vulnerability curve reliability

Sample Ref. [19]

a Mainly excerpted from [27].

various ground motion sets and to investigate the effects of

material uncertainties and selected ground motion sets on

the obtained vulnerability curves. In addition, several other

aspects of vulnerability curve derivation are investigated

such as the selection of ground motion duration, definition of

limit states, selection of damping parameter, and statistical

manipulation.

To establish a framework for the study, a brief review of

vulnerability curves and the procedures used in this study are

presented. Details of the structural and analytical models,

alongside a brief description of the analytical platform

and uncertainties in the analysis, are introduced and limit

states are defined for the selected demonstration structure.Sensitivity analyses are conducted and vulnerability curves

are derived. The results are fully investigated to identify

trends and derive conclusions on the relative sensitivity to

input motion and the randomness of material properties.

2. Brief review of analytical vulnerability functions

Vulnerability functions exhibit considerable variability

depending on the approaches used in their derivation.

The factors that influence the vulnerability functions are

input ground motion sets, severity indices of ground

motions, performance limit states, source of structural

damage data, structural modeling method, analysis platform

characteristics, analysis method, consideration of epistemic

uncertainty, etc.

Based on the sources of data, vulnerability curves may

be sub-divided into four categories [27] as summarized in

Table 1. A class of the curves are based on observational

data from post-earthquake surveys (e.g. [24,27]) while

others are based on analytical simulation (e.g. [21,26,

9]). Empirical vulnerability curves should be inherently

more realistic than their analytical counterparts should,

since they are based on the observed damage of actual

structures subjected to real strong motion. They have,

however, limitations in general application since the curves

are derived for a specific seismic region and a sample that

is not necessarily similar to that sought. On the other hand,

analytical vulnerability curves can be derived for general

purposes, but the choice of analytical model, simulation

method, and required computational power pose challenges

for the development of the required relationship.

One of the main criteria for the selection of the method

is the availability of structural damage data; either the

observation of post-earthquake losses or the analytical

simulation. Observational data are realistic, but are often

neither statistically viable nor homogeneous. The data


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O.-S. Kwon, A. Elnashai / Engineering Structures 28 (2006) 289303 291

Fig. 1. Flow chart for the derivation of analytical vulnerability curves.

from simulation, on the other hand, is constrained by

computational power and reliability of analytical tools. With

the expansion of computational power and the development

of reliable analysis tools, the limitations in the analytical

derivation of vulnerability curves are diminishing.

For the analytical derivation of vulnerability curves,

Mosalam et al. [21] used single-degree-of-freedom (SDOF)

systems, representations of the pushover curves of infilled

and bare frames. In HAZUS [23], the variability in seismic

demand is provided without explicit consideration of the

influence of the structural parameters such as damping,

period, and yield strength level. Reinhorn et al. [26] usedconstant yield-reduction-factor inelastic spectra with the

capacity spectrum method to evaluate inelastic response.

The above studies used simplified methods because the

derivation of vulnerability curves required a large amount

of simulation. Thus, the results are approximate as these

methods neglect the effect of higher modes, hysteretic

damping, and limit states based on local failure.

In this study, inelastic dynamic response-history analysis

is adopted for the sample building using ZEUS-NL [13] as

the analysis platform. Multi-threading techniques are used to

deploy ZEUS-NL on large computer clusters, thus reducing

very substantially the computational time.

3. Procedure of vulnerability curve derivation

Four aspects of the derivation process mainly affect

vulnerability curves as shown in Fig. 1. These are structure,

hazard definition, simulation method, and vulnerability

analysis. Each component can be divided into a number

of sub-tasks. By definition, vulnerability analysis is

probabilistic since each of the constituent components is

uncertain. Some of the uncertainties are inherently random

while others are consequences of lack of the knowledge.

In this study, only uncertainties in material properties and

input motion are considered. Those uncertain components

are indicated by the shaded areas in Fig. 1. The remaining

components, such as limit state, analysis method, structural

modeling, etc., also affect the derived vulnerability curves.

The estimation of these uncertainties, however, inevitably

includes subjective judgment. In this study, the latter

variables are assumed deterministic. Hence, the derived

vulnerability curves are conditional on the assumption thatthe input, the analysis methods, and the output from the

analysis reflect reality.

A structure representing an important sub-class in the

Mid-America region is chosen; a medium rise RC frame

structure with no seismic detailing. The ultimate strength

of concrete and yield strength of steel are selected as

random variables. An analytical model of the structure is

constructed and verified through comparison with shaking

table experiments. The limit states of the structure are

defined based on material response and sectional behavior

obtained from pushover analysis. The seismic hazard is

defined by selecting several ground motion sets. Full

combination of ground motion sets and random materialproperties are used for the simulation. Notwithstanding

previous studies that suggested the use of a few records, the

current investigation employs full Monte Carlo simulation

under all earthquake records, scaled at small intervals.

4. Selection of reference structure for simulation

A three-story ordinary moment resisting concrete frame

(OMRCF) is chosen for this benchmark study. It is

appreciated that the curves developed may not be generally

applicable to the loss estimation of all RC buildings. The

sample structure, however, serves ideally the purposes ofthis study, since experimental results under earthquake

loading exist for detailed verification of the analytical model.

Moreover, it is postulated that the curves derived could be

applicable to the sub-class of medium rise RC buildings with

limited ductility and no seismic design provisions; this might

be applicable to many areas in the mid-west of the USA and

Central/Northern Europe.

The prototype structure was originally designed for the

purpose of an experimental study [6]. The building has three

and four bays in EW and NS directions, respectively. The

story height is 3.7m (12 ft) and the bay width is 5.5 m (18 ft).

The total building height is 11 m (36 ft). It is designed for

gravity loads since wind loads seldom govern for low-risebuildings, and is non-seismically detailed. The provisions

of ACI 318-89 code, with Grade 40 steel [ fy = 276 MPa

(40 ksi)] and ordinary Portland cement [ fc = 24 MPa (3.5

ksi)], was employed. The plan and elevation layouts of the

structure are given in Fig. 2. The analyzed frame is shaded

in the figure. For detailed design information, reference is

made to Bracci et al. [6].

5. Selection of analysis method and environment

Several analysis methods have been proposed to

determine the seismic demand of structures subjected to


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Fig. 2. Plan and elevation of prototype structure [6], (a) plan, (b) elevation.

earthquake loading. Static pushover analysis, conventional,

modal, or adaptive, yields the capacity and collapse

mechanism of a structure. For seismic response assessment,

though, a seismic demand procedure is required, where

effective damping or equivalent ductility is accounted

for. Such equivalence may introduce approximations in

the analysis results. In addition, the more irregular the

structure is and more peculiar the strong-motion records

are, the less representative are pushover results of dynamic

response. This limitation lends weight to the use of dynamic

analysis which deals with the coupled demandcapacity

problem [11], especially for an irregular structure.Since structures not designed to resist seismic loads

usually fail in localized modes, their response is not likely to

be well estimated by static methods. Moreover, in order to

focus attention on other approximations in the vulnerability

functions derivation, it was decided to deploy the most

accurate and generally applicable method available for

seismic demand and supply evaluation; inelastic dynamic

response-history analysis. Use is made of the Mid-America

Earthquake Center analysis environment ZEUS-NL [13].

Elements capable of modeling material and geometric

nonlinearity are available in the program. The sectional

stressstrain state is obtained through the integration of the

inelastic material response of the individual fibers describing

the section. The Eulerian approach towards geometric

nonlinearity is employed on the element level. Therefore,

full account is taken of the spread of inelasticity along the

member length and across the section depth as well as the

effect of large member deformations. Since the sectional

response is calculated at each loading step from inelasticmaterial models that account for stiffness and strength

degradation, there is no need for sweeping assumptions

on the momentcurvature relationships required by other

analysis approaches. In ZEUS-NL, conventional pushover,

adaptive pushover, Eigen analysis, and dynamic analyses are

available and have been tested on the member and structural

levels ([14,15,7,20,25] amongst others). Recently, ZEUS-

NL was used to steer a full scale 3D RC frame testing

campaign, and the a priori predictions were shown to be

accurate and representative of the subsequently undertaken

pseudo-dynamic tests [17,18].

In the following section, the verification of the analysis

model and environment through comparison with shakingtable experiments are undertaken.

6. Verification of analytical model

The structural model and analysis environment are ver-

ified through comparison of response history analysis with

shake table test by Bracci et al. [6]. In response history anal-

ysis, damping, mass and stiffness are key parameters that

affect the assessment result. The verification is undertaken

in terms of structural periods and global displacement time

histories since local stressstrain measurements are not

available in the published literature.

6.1. Structural period

Columns and beams are divided into six and seven

elements, respectively, in the numerical model. Mass is

deposited at the beam and column connections, as shown

in Fig. 3. Material properties are taken from the reported

test result of the experimental model. The elastic structural

periods from eigenvalue analysis are 0.898, 0.305, and

0.200 s for the first, second, and third modes, respectively.

Bracci et al. [6] conducted a snap-back test before running

shaking table experiments to estimate the natural periodsof the 1/3 scale specimen and found that the periods, after

conversion to full scale using similitude laws, were 0.932,

0.307, and 0.206 s. The experimental values under small

amplitude testing are just 3%4% longer than the analytical

values which might have resulted from minor cracking in the

test specimen. These values give credence to the analytical

model.

6.2. Displacement response history verification

Fig. 4 depicts the comparison of the 3rd story displace-

ments of the 1/3 scale specimen and analysis using a 1/3


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Fig. 3. Analytical model configuration.

scale analytical model. For the analysis, Rayleigh damp-

ing is used for small amplitude ground excitation with

PGA of 0.05g, in which the damping ratio was taken from

the snap-back test, [6]. For moderate and severe ground

motions of 0.20g and 0.30g, damping other than hysteretic

is not included. Fig. 4(a) does not show good agreement

whilst Fig. 4(b) and (c) show very good agreement between

experimental and analytical results. This is attributed to thefact that at low level ground motion, it is difficult to accu-

rately estimate the level of damping and to model initial

cracks due to curing and experimental set-up. On the con-

trary, at medium to high earthquake motion, the inelastic

response from the fully cracked section mainly governs the

behavior, thus reducing the effect of cracking and small

amplitude damping. Good agreement in the moderate-to-

large amplitude shaking verifies that the analytical model

represents the experimented structure well. It is also ob-

served that assuming the same level of damping from the low

amplitude to the collapse level ground motion could result

in non-conservative vulnerability curves at medium-to-highground motion intensities. In this study, it is assumed that

there is no source of damping other than hysteretic. Thus,

for low level ground motions, the vulnerability curves might

be on the conservative side. Assuming no viscous damping

may cause spurious higher mode oscillation in a structure

without any other sources of energy dissipation. In the stud-

ied structure, however, this is not a significant issue because,

(a) the shaking table experiment confirmed that the numer-

ical model is of good accuracy, (b) the height of the struc-

ture is short, and with no irregularity, hence there is limited

possibility for spurious modes, and (c) the inelastic concrete

material used in the analysis shows hysteretic damping even

under very small magnitude of vibrations, thus it damps out

very short period spurious modes.

The current and previous verification of the modeling

approach and analysis platform lend weight to the confident

use of the analytical tools to investigate the effects of

parameter variation within the fragility analysis presented in

this paper.

7. Uncertainties in capacity and demand

In the derivation of vulnerability functions, a probabilistic

approach is adopted owing to uncertainties in the hazard

(demand) as well as structural supply (capacity). Some of

those uncertainties stem from factors that are inherently

random (referred to as aleatoric uncertainty), or from lack of

knowledge (referred to as epistemic uncertainty) [30]. In this

paper, the effects of aleatory uncertainties from material and

ground motion on the vulnerability curves are investigated.

Epistemic uncertainty considerations are beyond the scope

of this study.

7.1. Material uncertainty

7.1.1. Concrete strength

Barlett and MacGregor [3] investigated the relationship

between the strength of cast-in-place concrete and specified

concrete strength. When concrete is 1 year old, the ratio of

the average in-place strength to the specified strength was

1.33 and 1.44 for short and long elements, respectively, with

a coefficient of variation of 18.6%. The variation of strength

throughout the structure for a given mean in-place strength


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Fig. 4. Comparison of dynamic analysis: (a) 3rd story displacement Taft 0.05g, (b) 3rd story displacement Taft 0.20g, (c) 3rd story displacement Taft

0.30g.

depends on the number of members, number of batches, and

type of construction. In this study, it is assumed that there

is no variability of concrete strength since the structure is

a low-rise building of limited volume that would have been

constructed in a relatively short period. Thus, a coefficient

of variation of 18.6% is adopted. The specified concrete

strength (or design strength) of the considered structure

was 24 MPa. In-place concrete strength is assumed 1.40

times larger than the specified strength (33.6 MPa). Normal

distribution assumption is adopted for the concrete strength.

7.1.2. Steel strength

Mirza and MacGregor [22] reported results of about

4000 tests on Grade 40 and 60 bars. The mean

values and coefficient of variation of the yield strength

were 337 MPa (48.8 ksi) and 10.7%, respectively. The

probability distribution of modulus of elasticity of Grade

40 reinforcing steel followed a normal distribution with a

mean value 201,327 MPa (29,200 ksi) and a coefficient

of variation of 3.3%. Due to the low level of variability

observed, the modulus of elasticity is assumed deterministic

(201,327 MPa) in this study. The structure was designed

with grade 40 steel, thus the mean steel strength is assumed

337 MPa. The steel strength is assumed to follow a normal

distribution.

7.2. Input motion uncertainty

7.2.1. Selection of ground motion

In this study, nine sets of ground motions are used forthe derivation of vulnerability curves. Derived vulnerability

curves for each set are compared to gain insight into the

effect of ground motion variation on fragility analysis.

The first three sets of ground motions are based on the

ratio of peak ground acceleration to peak ground velocity

(a/v). Zhu et al. [31] discussed these three categories

of earthquake ground motions and their engineering and

seismological significance. The a/v ratio implicitly accounts

for many seismo-tectonic features and site characteristics

of earthquake ground motion records. Sawada et al. [28]

concluded that low a/v ratios signify earthquakes with

low predominant frequencies, broader response spectra,


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Table 2

Properties of the selected ground motions based on a/v ratio

a/v ratio Earthquake event/Location ML Date Soil type D (km) A max (m/s2) a/v ratio (g/ms1)

Low

Bucharest/Romania 6.40 3/4/1977 Rock 4 1.906 0.275

Erzincan/Turkey Unknown 3/13/1992 Stiff soil 13 3.816 0.382

Aftershock of Montenegro/Yugoslavia 6.20 5/24/1979 Alluvium 8 1.173 0.634

Kalamata/Greece 5.50 9/13/1986 Stiff soil 9 2.109 0.657Kocaeli/Turkey Unknown 8/17/1999 Unknown 101 3.039 0.750

Intermediate

Aftershock of Friuli/Italy 6.10 9/15/1976 Soft soil 12 0.811 1.040

Athens/Greece Unknown 9/7/1999 Unknown 24 1.088 1.090

Umbro-Marchigiano/Italy 5.80 9/26/1997 Stiff soil 27 0.992 1.108

Lazio Abruzzo/Italy 5.70 5/7/1984 Rock 31 0.628 1.136

Basso Tirreno/Italy 5.60 4/15/1978 Soft soil 18 0.719 1.183

High

Gulf of Corinth/Greece 4.70 11/4/1993 Stiff soil 10 0.673 1.432

Aftershock of Montenegro/Yugoslavia 6.20 5/24/1979 Rock 32 0.667 1.526

Aftershock of Montenegro/Yugoslavia 6.20 5/24/1979 Alluvium 16 1.709 1.564

Aftershock of Umbro-Marchigiana/Italy 5.00 11/9/1997 Rock 2 0.412 1.902

Friuli/Italy 6.30 5/6/1976 Rock 27 3.500 1.730

Table 3Properties of the artificial ground motions for Memphis, TN

EQ scenario Scenario 1 Scenario 2 Scenario 3

7.5 @ Blytheville, AR 6.5 @ Marked Tree, AR 5.5 @ Memphis, TN

Memphis, TN (Lowlands)

Set ID Set L-1 Set L-2 Set L-3

PGA (g) 0.1427 0.0632 0.0958

PGV (m/s) 0.1520 0.0576 0.0665

PGD (m) 0.0606 0.0202 0.0138

Memphis, TN (Uplands)

Set ID Set U-1 Set U-2 Set U-3

PGA (g) 0.1407 0.0676 0.1030

PGV (m/s) 0.1290 0.0516 0.0609

PGD (m) 0.0537 0.0178 0.0118

longer durations and medium-to-high magnitudes, long

epicentral distances and site periods. Conversely, high a/v

ratios represent high predominant frequencies, narrow band

spectra, short duration and smallmedium magnitudes, short

epicentral distance and site periods. Ground motions were

classified in the following ranges:

Low: a/v < 0.8g/m s1

Intermediate: 0.8g/m s1 a/v 1.2g/m s1

High: 1.2g/m s1 < a/v.

(1)

Based on the above categorization, three sets of groundmotions are selected in this study (Table 2). The average

response spectra of selected ground motion sets (Fig. 5)

show distinctive difference among each ground motion set.

The remaining six sets used in this study are artificial

ground motions. Drosos [10] generated the bedrock motion

for the Mississippi Embayment in the New Madrid

Seismic Zone. Equivalent linear site response analyses were

conducted to evaluate the soil surface ground motions.

During the site response analysis, shear wave velocity

profiles were randomized to account for the uncertainties in

shear wave velocity and layer thickness. Three of the sets, set

L-1, L-2, and L-3, were generated based on a Lowlands soil

Fig. 5. Average response spectrum of selected ground motion sets.

profile in the Memphis area. The other three sets, set U-1,

U-2, and U-3, were generated based on an Uplands soil

profile in the same region. Each of the three sets of

ground motions were based on three scenario earthquakes;

small, medium, and large, at three epicenter distances,

short, medium, and long. Each set contains ten ground

motions. Table 3 shows the ground motion parameters for

Memphis, TN.


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7.3. Random variables sampling

Three different ground motion sets are combined with

different material properties. For ground motion set low,

normal, and high a/v, ten ultimate concrete strengths, fc,

and ten steel yield strengths, Fy , were generated and a

full combination of material strengths are used, resultingin a total of 100 frames. The analysis results are used

to study the effect of material properties on the structural

response. For ground motion set L-1, L-2, and L-3, which

are artificial ground motions based on Lowlands profile,

50 concrete and steel strengths are generated following the

mean and standard deviation given in Section 7.1 resulting

in 50 different frames. For ground motion set U-1, U-2, and

U-3 based on Uplands profile, 100 concrete and steel

strengths are generated. From the analysis result of these

frames, the effect of sample size is investigated.

8. Ground motion duration and scale factors

8.1. Ground motion duration

The duration of the significant part of strong motion

affects the maximum response when a structure undergoes

inelastic deformations. There are many previous studies

aimed at defining the duration of strong ground motion.

Bommer and Martinez-Pereira [4] assembled 30 definitions

of strong-motion duration suggested by previous researchers

and divided them into four categories: bracketed, uniform,

significant, and structural response-based definitions. Since

the selected motions are to be scaled for vulnerability curve

generation, the duration should be defined in a relativemanner as in the significant duration option. Trifunac

and Brady [29] used significant duration concepts based

on the integral of the square of acceleration, velocity and

displacement where the duration is defined as the interval

between the times at which 5% and 95% of the total integral

is attained. The latter range of duration is meaningful in

characterizing ground motions. From a structural analysis

point of view, however, this duration may not be practical.

For instance, if the above-mentioned interval of total

integral is used as shown in the Fig. 6, the ground motion

acceleration could start at very large values, which may

apply an unrealistic pulse to the structure. Moreover, sincethe major part of ground motion energy is skewed to

the early part of the motion, using identical margins for

the start and the end of the duration is not a reasonable

approach. Based on the above argument, for the current

study, the interval between 0.5% and 95% of the integrals is

used.

8.2. Selection of scale factors

The computational demand for derivation of vulnerability

curve is very large. Analysis time for 100 frames

(combination of 10 concrete and 10 steel strengths)

subjected to five ground motions (e.g. intermediate a/v ratio

ground motion set) that are scaled from PGA of 0.050 .5g

at 0.05g interval was 121 h on a fast PC (Pentium IV

2.65 GHz, 1 GB of RAM) at the time of the current study.

Thus, a reasonable range of PGA levels should be selected to

most effectively utilize the computational power, since each

ground motion set has different spectral acceleration at thefundamental period of the structure. For example, for the

low a/v ratio ground motion set, a PGA of 0.50g was too

large because most structures collapsed at a PGA of 0.2g.

Conversely, for high a/v ratio ground motion set, 0.50g

PGA level was not large enough to derive the vulnerability

curve for the collapse limit state. Thus, for low and high a/v

ratios, additional selective analyses had to be performed to

improve the resolution of the vulnerability curves.

For the determination of the range of reasonable PGA

scaling, the capacity spectrum method was utilized. The

capacity curves were obtained from adaptive pushover

analysis, and demand curves were converted from the

elastic displacement and acceleration spectra of each groundmotion set. For accurate estimation of maximum PGA

scaling at which the structure collapses, elastic demand

should be decreased to consider inelasticity using effective

damping [2,5] or ductility ratio [8]. In this analysis, however,

rough estimation of PGA scales is undertaken using elastic

demand spectra and inelastic capacity spectra.

9. Limit state definition

In ATC 40 [2] and FEMA-273 [16], four limit states are

defined based on global behavior (interstory drift) as well

as element deformation (plastic hinge rotation). Rossettoand Elnashai [27] used five limit states for derivation

of vulnerability curves based on observational data while

Chryssanthopoulos et al. [9] used only two limit states. In

the latter studies, the global limit states are independent

of the specific response of the structure. For example, the

FEMA-273 [16] life safety level limit state of interstory

drift (ISD) for non-ductile moment resisting frame is 1.00%

regardless of gravity force levels or the details of structural

configuration within the sub-class of structure.

For rigorous analysis, it is necessary to define limit

states for each individual structure, since the deformational

capacity could be affected by many other factors such asgravity force level, irregularity, anticipated plastic hinging

mechanism, etc. In this study, three limit states are defined

for the prototype structure based on the first yielding

of steel, attainment of maximum element strength, and

maximum confined concrete strain during the adaptive

pushover analysis. These are termed, serviceability,

damage control, and collapse prevention, limit states,

respectively. In this study, the local damage of individual

structural element, such as beam, column, or beamcolumn

joint, is not accounted for. Only interstory drift is used

as a global measure of damage. The 1st story drift

which corresponds to each limit state, for the prototype


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Fig. 6. Significant duration of a ground motion.

Fig. 7. Definition of limit states: (a) serviceability state, (b) damage control state, (c) collapse state.

structure, is 0.57%, 1.2% and 2.3% for the selected

three limit states, as shown in Fig. 7. In the figure, c01

through c31 represent the bottom element of the 1st

story columns as indicated in Fig. 3. It is assumed that

theses limit states are also applicable to the 2nd and 3rd

stories.


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Fig. 8. Distinction of collapse state and lognormal distribution assumption

(ground motion normal a/v ratio, PGA = 0.40g).

10. Simulation and vulnerability curve derivation

The total simulation analysis of the frame using ground

motion set low, normal, and high a/v ratio required

456 h using a Pentium IV-2.65 GHz PC for a total

of 23,000 response history analyses. For the analysis

of frames using ground motion set L-1 through U-3, a

mass-simulation environment was developed. The mass-

simulation approach deploys 32 processes simultaneously

on the super-computer IBM p690 at the National Center

for Supercomputing Applications (NCSA). Interstory drifts

are retrieved from each run and used for statistical analysis.The maximum interstory drifts are assumed to follow a log-

normal distribution. The normal distribution is avoided since

its abscissa ranges from negative infinity to positive infinity,

which implies that negative maximum interstory drift could

exist.

The analysis platform used for simulation accounts for

geometric nonlinearity as well as material inelasticity. Thus,

when the structure is subjected to large seismic demands,

instability under gravity loading may ensue. Since the drift

of unstable structures depends on convergence criteria of

the analysis tool, it is concluded that interstory drift at

such response states should not be included in the statisticalanalysis. Thus, the structure is in the collapse state if the

maximum interstory drift is larger than 2.3%, as discussed in

Section 9. The value, 2.3% is applicable only to the studied

structure, and possibly the sub-class of non-seismically

designed medium-rise RC frames. To include the statistics

of collapsed frames, the total probability theorem is adopted.

The probability where a frame maximum interstory drift

will be larger than a certain limit state is calculated as

below:

P(ISD > ISDLimit) = P(ISD > ISDLimit | E1) P(E1)

+ P(ISD > ISDLimit | E2) P(E2)

Fig. 9. Derived vulnerability curves using various methods normal a/v

ratio, limit state = ISD 0.57%.

= P(ISD > ISDLimit | E1) P(E1)

+ 1.0 P(E2

) (2)

where P(E1) and P(E2) represents the probability that the

structure is in a stable or in a collapse state, respectively.

As shown in Fig. 8, a lognormal distribution was assumed

for structures with ISD < 2.3%, and the structures

were considered to have collapsed if ISD 2.3%. This

assumption provides conservative results. Fig. 9 shows a

sample vulnerability curve for the 0.57% ISD limit state

using intermediate a/v ratio ground motion sets. Without

considering collapse state ISD, normal and lognormal

distribution assumptions show lower probability of attaining

the limit state at larger PGA level, which is due to the

misleading average and coefficient of variance of unstablestructures.

11. Effect of analysis parameters on vulnerability curves

11.1. Effect of ground motion set

Fig. 10 compares vulnerability curves for each limit

state derived from nine ground motion sets. For general

application of vulnerability curves, it is necessary to perform

regression analysis to obtain functional forms that are

readily programmable in loss assessment software. For the

purpose of comparison amongst different ground motionsets, however, actual data with linear interpolation are

plotted in order to insure that differences are not masked

by regression smoothing. Fig. 11 shows the coefficient of

variation of ISD from each ground motion set vs. ground

motion level. In this figure, all response data, including

collapse state, are used. From Figs. 10 and 11 the following

observations are made:

(a) Because of large discrepancies in the coefficient of

variation of the maximum interstory drift, each ground

motion set shows significantly different vulnerability

curves.


11/15


Fig. 10. Vulnerability curves for each ground motion set.

(b) The differences between the vulnerability curves

increase for higher damage limits.

(c) The difference between the vulnerability curves in-

creases with ground motion intensity until the structure

reaches the collapse state.

(d) The vulnerability curves do not indicate monotonic

increase in damage prediction at large PGA levels due

to instability of structure and statistical manipulation.

Consequently, it is concluded that the ground motion

sets should be selected with great care since they affect

very significantly the outcome from the fragility analysis

represented by vulnerability curves.

11.2. Effect of material properties

The effect of material properties is investigated using

analysis results from a/v ratio ground motion sets for which

10 concrete ultimate strengths and 10 steel yield strengths

are combined. It is assumed that ISDmax is a function of

ground motion sets, concrete ultimate strength, and steel

yield strength; i.e,

ISDmax = g(X1, X2, X3) (3)

where

X1: ground motion

X2: concrete strength

X3: steel strength.


12/15


Fig. 11. COV ofISDmax for each ground motion set.

Mean ISDmax can be calculated from the result of full

simulation, thus using 100 frames, or from the simulation

of single frame using mean material properties. Fig. 12compares the means values of ISDmax using 100 frames and

using a single frame with mean material properties. Up to

0.25g, the means from the two methods are almost identical.

The difference becomes large as the ground motion intensity

increases.

Assuming that all uncertain variables, i.e. ground motion,

concrete, and steel properties, are statistically independent

and as a first order approximation, the variance of ISD can

be calculated from Eq. (4)

Var(ISDmax) 2 + c

22Var(X2) + c

23Var(X3) (4)

where is the standard deviation of the error term andci is evaluated at xi = X i . The error term represents

the uncertainty of ground motion and has zero mean and a

standard deviation of , hence the ground motion set adds

uncertainty as expressed in the first term of Eq. (4). The c2and c3 values can be calculated numerically in the vicinity of

the mean values of concrete and steel properties. Variances,

Var(X2) and Var(X3), can be calculated from the coefficient

of variation, COV = 0.186, and COV = 0.107, for concrete

and steel, respectively.

Fig. 13(a), (b), (d), and (e) show ISDmax vs. fc and

Fy at small (0.05g) and large (0.35g) PGA levels. From

Fig. 13(a) and (b), different ground motions exhibit differentslopes, d(ISDmax)/d fc. This is because the elastic modulus

of concrete is affected by the ultimate strength, fc, hence,

the structural period is also affected. As a result, the

relationship between ISDmax and fc does not follow a clear

trend, i.e. it is inappropriate to state that higher concrete

strength reduces structural response since ISDmax and fcare correlated with spectral displacement, which is random

in nature. On the contrary, the ISDmax at low PGA level

is rarely affected by the yield strength of steel (Fig. 13(d)

and (e)), since the elastic modulus of steel is constant

regardless of yield strength. Fig. 13(c) and (f) show the

contribution of concrete and steel strengths, which are the

Fig. 12. Mean ISDmax from mean material properties and full simulation.

second and third terms of Eq. (4), to the variance ofISDmax.

In these figures, the first term of Eq. (4) is zero sincethe plots are developed for each ground motion, i.e. the

input is deterministic. The contribution of concrete to the

variance ofISDmax is generally larger than that of steel since

the former affects stiffness, hence period and amplification

characteristics.

11.3. Effect of sample size

The analysis results of ground motion set U-1 are used

where 100 concrete and steel properties are combined to

assess the effect of the sample size on the variance ofISDmax. From the discussion in the previous section, it

is reasonable to state that material variability has very

little effect on the variability of ISDmax at low PGA

levels. The latter observation, however, is not globally

applicable; highly irregular systems with random spatial

distribution of materials may cause failure mode switches.

In such cases, material variability may exhibit more

prominence in fragility analysis. Notwithstanding, based on

the observations in this study, the variances of ISDmax from

the mean frame, 10, 50, and 100 frames were compared.

Variance of the mean frame, Var1, is calculated as in

Eq. (5).

Var1 =n

i=1

(xi X i )2

n 1(5)

where n is the number of ground motions. If multiple frames

are used, the variance can be calculated as below.

Var2 =

j n

i=1

(xi X i )2

j n 1(6)

where j is the number of frames. From the assumption that

the difference in material has little effect on the outcome of

the result, Eq. (6) is rewritten as below.


13/15


Fig. 13. Effect of material strength on the maximum ISD (a) fc vs. ISDmax at 0.05g, (b) fc vs. ISDmax at 0.35g, (c) c22Var(X2) vs. PGA, (d) Fy vs. ISDmax

at 0.05g, (e) Fy vs. ISDmax at 0.35g, (f) c23Var(X3) vs. PGA.

Var2 =

j n

i=1

(xi X i )2

j n 1=

n

i=1

j(xi X i )

2

j n 1

=n 1

n 1/j

n

i=1

(xi X i )2

n 1=

n 1

n 1/jVar1. (7)

In other words, the increase in sample size considering

material variability, which does not increase variability of

ISDmax, reduces the variance of ISDmax considering only

ground motion uncertainty. Fig. 14(a) shows the coefficient

of variation from 1, 10, 50, and 100 frames vs. PGA

level. From this figure, it is observed that the coefficientof variation does not vary much even if uncertain material

properties are used. With the stipulation that material

properties have little effect on ISDmax, which is true for

low PGA levels, the correction factor in Eq. (7) is applied

to the variance and the resulting coefficients of variation

are compared in Fig. 14(b). The latter figure shows COV

for the mean, 10, 50, and 100 frames that are closer

than before. Comparison of Figs. 14 and 11 confirms that

there is indeed an effect from material variability, but

that such an effect is significantly smaller, verging on

the negligible, compared to the effect of ground motion

uncertainty.

The above observations confirm that the variability in

materials is much less significant than the effect of ground

motion variation. The conclusion is verified by using SDOF

system analysis as shown in Fig. 15 where the spectral

displacement is plotted vs. period and ductility. The period

range, from 0.85 to 0.95 s, corresponds to the period of

the prototype structure with concrete ultimate strength of

mean one standard deviation. The used range covers

most of the possible concrete strength variation. Since steel

yield has practically no effect on the structural period,

the period axis represents variability in concrete only.The ultimate strength of concrete and yield strength of

steel affect the ductility of structures for a given ground

motion level. Since the ductility cannot be estimated without

analysis, a ductility range from 1 to 2 is plotted for

comparison purposes. For the three ground motion sets,

U-1, U-2, and U-3, the mean spectral displacement is

calculated after normalizing the ground motion to a PGA

of 1g. Fig. 15 clearly shows that the structural response

parameters are very significantly affected by the input

motion set, while the effect of material variability is rather

insignificant.


14/15


Fig. 14. Comparison of COV with mean frame and multiple frames: (a)

variance before applying correction factor, (b) variance after applying

correction factor.

12. Conclusion

In this study, vulnerability curves of a three-story RCstructure are derived for nine sets of ground motions. The

effects of ground motion input and material variability are

the main focus of the study, alongside other supporting

aspects of vulnerability curve derivation such as selection

of representative structure, verification of analytical model

and analysis environment, effect of explicit damping,

scaling of ground motion sets, significant duration of

ground motion, and limit state definition. Even though

each of those parameters affects the derived vulnerability

curves, many previous studies have been performed with

simplified models and sweeping assumptions. In the current

study, focus is placed on the most realistic vulnerability

curve derivation through inelastic response history analysis.Conclusions are drawn in the preceding sections. Hereafter,

some of the findings as well as comments on the methods

adopted in this study are reiterated below:

The concrete ultimate strength affects the structural

response under low ground motion intensities since

the concrete elastic modulus is related to the ultimate

strength.

The yield strength of steel has little effect on the structural

response at low ground motion levels.

At high ground motion levels, material properties

contribute to the variability in structural response, but

Fig. 15. Response surface of spectral displacement against period and

ductility.

the resulting variability is much smaller than that due toground motion variability.

Input motion characteristics have a most significant

effect on vulnerability curves. Therefore, meticulous

consideration is required when ground motions are

selected.

The conclusions above are robust because of the verification

of the analysis platform, the care taken in selecting the

reference structure, the wide range of selected and scaled

input motion and the rigor of the statistical analysis. An

important feature of the study is the extensive simulation

undertaken with minimum assumptions. It is noted though

that for structures with higher levels of irregularity, andwhere material variability is spatially uncorrelated, different

failure modes may be observed where interstory drift may

not be an appropriate measure of the structural damage. In

addition, since the definition of limit state and response of

the structure under earthquake loading vary with structural

configuration, the derived vulnerability curve in this study

may not be generally applicable to all RC structures of

limited ductility. The procedures and discussions in this

study, however, provide an insight into the vulnerability

derivation of low ductility structures.

Acknowledgements

The paper is an outcome of the Mid-America Earthquake

Center project DS-3: Advanced Simulation Tools. The Mid-

America Earthquake Center is an Engineering Research

Center funded by the National Science Foundation under

cooperative agreement reference EEC-9701785. The authors

are grateful to Drs. Tom Prudhomme and Cristina Beldica

for their help in having access to the parallel processing

facilities at the National Center for Supercomputing

Applications at the University of Illinois at Urbana-

Champaign.


15/15


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the effect of material and ground motion uncertainty on the seismic vulnerability of rc structure

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