EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS Earthquake Engng Struct. Dyn. 2009; 38:951–972 Published online 2 December 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.876 Alternative non-linear demand estimation methods for probability-based seismic assessments F. Jalayer ∗, † and C. A. Cornell Department of Civil and Environmental Engineering, Stanford University, U.S.A. SUMMARY Alternative non-linear dynamic analysis procedures, using real ground motion records, can be used to make probability-based seismic assessments. These procedures can be used both to obtain parameter estimates for specific probabilistic assessment criteria such as demand and capacity factored design and also to make direct probabilistic performance assessments using numerical methods. Multiple-stripe analysis is a non-linear dynamic analysis method that can be used for performance-based assessments for a wide range of ground motion intensities and multiple performance objectives from onset of damage through global collapse. Alternatively, the amount of analysis effort needed in the performance assessments can be reduced by performing the structural analyses and estimating the main parameters in the region of ground motion intensity levels of interest. In particular, single-stripe and double-stripe analysis can provide local probabilistic demand assessments using minimal number of structural analyses (around 20 to 40). As a case study, the displacement-based seismic performance of an older reinforced concrete frame structure, which is known to have suffered shear failure in its columns during the 1994 Northridge Earthquake, is evaluated. Copyright 2008 John Wiley & Sons, Ltd. Received 3 June 2008; Revised 6 October 2008; Accepted 6 October 2008 KEY WORDS: probabilistic seismic assessments; non-linear dynamic analysis; demand and capacity factored design; global instability; multiple-stripe analysis; performance-based design; earthquake engineering 1. INTRODUCTION One of the main attributes distinguishing performance-based earthquake engineering from tradi- tional earthquake engineering is the definition of quantifiable performance objectives that balance desirable structural performance and life cycle costs [1]. A performance objective can be expressed by the probability of exceeding a specified limit state or performance level defined in terms of ∗ Correspondence to: F. Jalayer, Department of Structural Engineering, University of Naples ‘Federico II’, Via Claudio 21, Naples, Italy. † E-mail: [email protected]Contract/grant sponsor: U.S. National Science Foundation; contract/grant number: EEC-9701568 Copyright 2008 John Wiley & Sons, Ltd.
Transcript
EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2009; 38:951–972Published online 2 December 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.876
Alternative non-linear demand estimation methods forprobability-based seismic assessments
F. Jalayer∗,† and C. A. Cornell
Department of Civil and Environmental Engineering, Stanford University, U.S.A.
SUMMARY
Alternative non-linear dynamic analysis procedures, using real ground motion records, can be used to makeprobability-based seismic assessments. These procedures can be used both to obtain parameter estimatesfor specific probabilistic assessment criteria such as demand and capacity factored design and also tomake direct probabilistic performance assessments using numerical methods. Multiple-stripe analysis isa non-linear dynamic analysis method that can be used for performance-based assessments for a widerange of ground motion intensities and multiple performance objectives from onset of damage throughglobal collapse. Alternatively, the amount of analysis effort needed in the performance assessments can bereduced by performing the structural analyses and estimating the main parameters in the region of groundmotion intensity levels of interest. In particular, single-stripe and double-stripe analysis can provide localprobabilistic demand assessments using minimal number of structural analyses (around 20 to 40). As acase study, the displacement-based seismic performance of an older reinforced concrete frame structure,which is known to have suffered shear failure in its columns during the 1994 Northridge Earthquake, isevaluated. Copyright q 2008 John Wiley & Sons, Ltd.
Received 3 June 2008; Revised 6 October 2008; Accepted 6 October 2008
KEY WORDS: probabilistic seismic assessments; non-linear dynamic analysis; demand and capacityfactored design; global instability; multiple-stripe analysis; performance-based design;earthquake engineering
1. INTRODUCTION
One of the main attributes distinguishing performance-based earthquake engineering from tradi-tional earthquake engineering is the definition of quantifiable performance objectives that balancedesirable structural performance and life cycle costs [1]. A performance objective can be expressedby the probability of exceeding a specified limit state or performance level defined in terms of
∗Correspondence to: F. Jalayer, Department of Structural Engineering, University of Naples ‘Federico II’, Via Claudio21, Naples, Italy.
Contract/grant sponsor: U.S. National Science Foundation; contract/grant number: EEC-9701568
Copyright q 2008 John Wiley & Sons, Ltd.
952 F. JALAYER AND C. A. CORNELL
life cycle cost and/or some other structural performance parameter. Probabilistic performanceobjectives need to take into consideration the uncertainty in the ground motion and structuralmodeling parameters.
Demand and capacity factor design (DCFD), named because of its similarity to the LRFDprocedures, is a probability-based format for performance-based seismic design and assessmentof structures [2]. The DCFD probabilistic performance objective is represented by an analyticclosed-form expression after making certain simplifying assumptions about the main componentsof the DCFD format [3]. Implementation of DCFD format’s closed-form representation for seismicdesign or assessment involves obtaining parameter estimates. In addition to relevant seismic hazardinformation, a suite of non-linear dynamic analyses can be organized to obtain such parameterestimates. Alternatively, the integral expression for the probability of exceeding a specified perfor-mance level can be calculated directly using numerical methods. The results can be used as abasis for evaluating the DCFD estimations when the assumptions underlying the derivation of itsclosed-form are not satisfied.
This paper discusses how to implement non-linear dynamic analyses in probabilisticperformance-based assessment of structures. The probabilistic performance objective for theassessments is defined by taking into account the uncertainty in the ground motion. The non-lineardynamic procedures introduced in this paper can be carried out both for the full range of groundmotion intensities of interest and also for a limited range of ground motion intensities (or ‘hazard’)levels. In particular, multiple-stripe analysis (MSA) is a non-linear dynamic analysis proceduresuitable for probability-based assessments for multiple performance levels. Alternatively, for thepurpose of minimizing the analysis effort, non-linear dynamic methods such as single-stripeanalysis or double-stripe analysis can be used in order to make seismic assessments concentratingon a single limit state. These methods have already been implemented by researchers in seismicassessment of structures (e.g. [4–9]). The seismic probabilistic assessments discussed in this paperare performed for both the ‘collapse’ limit state and intermediate damage levels such as the onsetof damage limit state in the structure.
In some cases, the displacement response is either too large to be meaningful or it is simplyunavailable owing to numerical non-convergence in the structural analysis software. Both situations,which may signal an over-all lack of stability in the structure, may be treated in a similar manner byassuming that the displacement response in the structure is arbitrarily large. This paper addressessuch cases by presenting a methodology for structural performance assessment in cases wheresome of the records’ displacement responses are too large to be included among the body of thedisplacements computed.
The maximum inter-story drift ratio response in the structure and over the entire ground motiontime-history has been chosen as the the engineering demand parameter in the design/assessmentprocess. This is a particularly suitable choice for moment-resisting frame structures, since it relatesthe global response of the structure to joint rotations where most of the inelastic behavior inthe moment-resisting frames is concentrated. The spectral acceleration at the first-mode periodand denoted by Sa(T1) or simply Sa has been adopted as the intensity measure variable. Thischoice is supported by studies by Shome and Cornell [10]. They demonstrate that for first-modedominated moment-resisting frame structures with first-mode periods lying within the moderaterange (e.g. around T =1.0s), the spectral acceleration of the first mode is sufficient for relayingthe primary ground motion characteristics to the structural response. Recent studies have foundthat vector-valued intensity measures prove to be more effective [11, 12]. Nevertheless, the focusof this work is on how to organize a series of non-linear dynamic analyses once a suitable choice
ALTERNATIVE NON-LINEAR DEMAND ESTIMATION METHODS 953
for an intensity measure is being made. In what follows, we shall simply refer to the case specificmeasures: spectral acceleration and maximum inter-story drift.
2. A COMPREHENSIVE ASSESSMENT EXAMPLE: AN OLDER REINFORCEDCONCRETE FRAME
An older reinforced concrete frame structure in Los Angeles is selected as a comprehensivecase study in probabilistic assessment of an earthquake-damaged existing structure. This structurehas served as a test-bed for the activities of the Pacific Earthquake Engineering (PEER) Center(http://www.peertestbeds.net). For demonstrating the probabilistic assessment procedure, the accu-racy of the mathematical modeling of this particular building was not imperative. Nevertheless, ananalysis software is chosen that is capable of modeling cyclic stiffness and strength degradationbehavior in reinforced concrete. To this effect, DRAIN2D-UW, a modified version of DRAIN2D,which was produced by Professor Jose Pincheira’s research team in the University of Wisconsin(see [13, 14]), is chosen as the analysis software. One of the transverse frames in the case-studystructure is modeled using DRAIN2D-UW. Figure 1(a) illustrates the schematic model of theframe and centerline dimensions. The members are modeled using a beam–column element. Thedegrading behavior is concentrated in two rotational springs at the two ends (flexure) and a transla-tion spring in the middle of the element (shear). The hysteresis behavior of a typical beam–columnelement is also shown in the figure (Figure 1(b)). The figure also illustrates the static pushovercurve for the structure (Figure 1(c)). Note that it is plotted with respect not to the roof drift, as itis customary, but to the response measure used here, maximum (over all stories) of the inter-storydrift ratio. The onset of significant structural damage (defined here as an inter-story drift of 0.007)in the frame is marked on the curve. The small-amplitude natural frequencies of the first twomodes are computed to be 1.25 and 3.66Hz, respectively.
3. RECORD SELECTION
For this exercise, a set of 30 ground motion records was selected from the PEER-NGA Database[15] for California sites. These records were all California events recorded on stiff soil (Geomatrixsoil types C and D : deep soil [15]) and were selected from a moment magnitude and (closest)source-to-site distance range of:
6.5 � M�7
15 � R�32(1)
Addressing the current issues in ground motion selection is not the focus of this study. It isgenerally presumed that one should select records representative of the events that dominate theprobabilistic hazard for the ground motion levels of interest, as determined informally or formally,e.g. by de-aggregation [16, 17]. The magnitude range, 6.5�M�7, is presumed to be representativeof the events likely to cause severe ground motions at this site. With the exception of potential near-source and directivity-influenced records, the most important record characteristics (e.g. spectralshape), other than general amplitude level, are comparatively insensitive to distance. In most casesin this study, the records will be scaled relative to their recorded values; therefore, the recorded
Figure 1. (a) Transverse frame of an existing RC frame structure (dimensions are in meters); (b) exampleof degrading hysteretic behavior; and (c) the static pushover curve.
amplitude is not directly relevant. Directivity issues (e.g. pulse-like records) are beyond the scopeof this particular study (see, e.g. [11, 18]). Issues of nonlinear response sensitivity to magnitudeand distance and record scaling are discussed by Shome et al. [10]. Nonetheless, the subject ofsite-specific record selection and modification for non-linear demand estimation deserves, and isthe subject of, further research (e.g. [12, 19, 20]).
4. PROBABILISTIC FRAMEWORK FOR DESIGN AND ASSESSMENT OF STRUCTURES
A probability-based statement for a particular performance objective can be expressed in terms ofthe mean annual frequency of exceeding a limit state �LS. The mean annual frequency of structuraldemand variable �max exceeding the limit state threshold variable (capacity) CLS can be calculated
ALTERNATIVE NON-LINEAR DEMAND ESTIMATION METHODS 955
by expanding it with respect to all the possible values y of the demand variable:
�LS=�(�max>CLS)=∫yFCLS(y) ·|d��max(y)| (2)
FCLS—also known as the fragility function—is the cumulative probability distribution function(CDF) for capacity variable, CLS, and ��max(y)—also known as the drift hazard—is the (mean)annual frequency of exceeding the demand variable, �max. In the procedures described herein, thelimit states are assumed to be predicted by a critical maximum inter-story displacement and tohave a capacity represented by a probability distribution in such displacement terms.
In a similar manner, the expression for mean annual frequency of exceeding the demand variable��max(y) (also known as the drift hazard [2]) can be expanded with respect to all the possiblevalues x of the intensity measure, here spectral acceleration Sa:
��max(y)=�(�max>y)=∫xG�max|Sa(y|x) ·|d�Sa(x)| (3)
where G�max|Sa(y|x) is the conditional probability of exceeding a specified level �max= y of themaximum inter-story drift ratio for a given intensity measure level, Sa= x , which is also referredto as the conditional complementary cumulative distribution function (CCDF) of demand, �max,for a given Sa value. �Sa(x) is the mean annual frequency of exceeding spectral acceleration, alsoknown as the spectral acceleration hazard function. Detailed discussion of Equations (2) and (3)can be found in a report by the authors [3].4.1. Probabilistic demand assessments in the range of large displacements
There are situations in which the calculated displacement response of the structure is not credibleor, in effect, not finite; these cases may demonstrate themselves as numerical non-convergencein the structural analysis software or excessively large displacements (e.g. drift ratios more than10%) beyond the validity of the mechanical model. In still other cases to be discussed below,we shall define collapse to have occurred based on the excessive rate of displacement growth perunit increase in ground motion intensity. This section discusses probabilistic demand assessmentswhen a subset of the (dynamic) displacement-based responses of the structure is of these types. Forsimplicity, we refer to these cases of extreme dynamic displacement response in the structure as‘collapse cases’; with the awareness that these global collapse cases constitute only one potentialtype of structural collapse. The expression for conditional CCDF of demand �max= y for a givenspectral acceleration, Sa= x can be expanded as (see [21]):
where PNC |Sa(x)=1−PC |Sa(x) is the probability that there are no cases of collapse observed undera ground motion with a given spectral acceleration, x , or more briefly, the conditional probability ofnon-collapse given spectral acceleration. Given collapse, the conditional probability of exceedingany finite drift demand, G�max|C,Sa(.), is assumed to be equal to 1:
G�max|C,Sa(y|x)= P[�max>y|collapse, Sa= x]=1 (5)
Using the above assumption, Equation (4) can be re-written as follows:
Replacing the above expression for G�max|Sa(.) in Equation (3), we obtain an expression for drifthazard in the region of global dynamic instability or collapse in the structure:
Equation (7) provides an alternative expression for drift hazard in which the ‘collapse’ cases areexplicitly addressed. It will be demonstrated later in this paper how non-linear dynamic analysisprocedures can be employed to make demand assessments in the range of large displacementsbased on the alternative expression for drift hazard in Equation (7). It can be noted that for small yvalues Equation (7) reduces to Equation (3) and that for large y it is equal to the mean annualfrequency of the collapse limit state.
4.2. Demand and capacity factored design format
The probabilistic design criterion corresponding to a particular limit state, LS, can be expressedin terms of the likelihood of not exceeding the threshold of acceptable probability level:
�LS�Po (8)
where Po is the acceptable level associated with exceeding limit state LS. Examples of limit statesmight be fracturing of connections in a steel moment frame or yielding of beam reinforcement ina reinforced concrete frame. The choice of a value for acceptable or tolerable rate of exceedingthe limit state, Po, depends on the particular objectives sought in the design of the structure. Aftermaking a set of assumptions and some re-arranging, involving taking the demand-related terms toone side and the capacity-related ones to the other, the analytic form for a DCFD format can bederived as illustrated in Equation (9) below. The reader can refer to a report by the authors [3] fora complete description of the derivation
FC � FD(Po)
FC = �CLS·exp(− 1
2 (k/b)�2CLS
)
FD = ��max|Po Sa ·exp( 12 (k/b)�2�max|Sa)
(9)
where PoSa is the spectral acceleration value with a mean annual frequency of being exceededequal to Po, ��max|Po Sa is the median �max at a spectral acceleration equal to PoSa, ��max|Sa is thefractional standard deviation (i.e. standard deviation of the natural logarithm) of demand given thespectral acceleration level Sa, and b is the slope of the (logarithm of ) median displacement-demandversus spectral acceleration curve if it is approximated by a power-law function of the form a ·Sab.Also, �CLS
is the median capacity and �CLSis the fractional standard deviation of capacity. k is the
slope of the (logarithm of) spectral acceleration hazard curve if it is approximated by a power-lawfunction of the form k0 ·x−k . Therefore, k/b is a sensitivity factor reflecting the change in theprobability with respect to the change in the displacement-based demand.
In order to derive Equation (9), it has been assumed that the demand given spectral acceler-ation and the capacity can both be described by lognormal probability distributions denoted byLN(��max|Po Sa,��max|Sa) and LN(�CLS
,�CLS), respectively. It is also assumed that the spectral accel-
eration hazard curve and median demand-spectral acceleration relationship can be represented, atleast over a local region sufficiently wide to yield reasonable numerical accuracy, by power-law-type expressions in the form �Sa(x)=k0 ·x−k and ��max|Sa =a ·Sab, respectively. It is also assumed
ALTERNATIVE NON-LINEAR DEMAND ESTIMATION METHODS 957
implicitly that the probability of global collapse is negligible in the range of ground motion anddisplacements under consideration; this is equivalent to assuming that PNC |Sa(x)=1−PC |Sa(x)=1for all spectral acceleration values. Under these conditions all or virtually all the dynamic analysisoutput for drift demand given different spectral acceleration levels converge and are reliable.
Equation (9) states the DCFD design criterion, which corresponds to deciding whether thefactored demand, denoted by FD(Po) at tolerable probability Po is less than or equal to factoredcapacity, denoted by FC. Equality is achieved when the mean annual frequency �LS of exceedingthe limit state LS is equal to the allowable probability, Po. DCFD design format in Equation (9)can be viewed as an alternative representation of the probabilistic design criterion in Equation (8),where the design criterion is expressed in terms of structural response parameters instead ofprobabilities or annual frequencies. This quality facilitates both incorporating the DCFD designcriteria within common structural engineering guidelines and the analyses needed to conduct aprobability-based assessment.
4.3. Drift hazard and factored demand
Based on the same set of simplifying assumptions outlined in the derivation of the DCFD designcriterion, an analytic closed-form solution for the drift hazard in Equation (3) can be derived(see [3] for the details of its derivation):
where Sxa denotes the spectral acceleration that ‘corresponds’ to a drift equal to x . More precisely, Sxais the inverse of x=��max|Sxa =a ·(Sxa )b, i.e. Sxa =�−1
�max|Sxa =(x/a)1/b. This is the spectral accelerationvalue that has (approximately) a 50% chance of causing a drift response equal to or greater than x .�Sa(S
xa ) is the spectral acceleration hazard value for (i.e. the mean annual frequency of exceeding)
Sxa . It can be shown that (see [3]) factored demand at allowable probability Po is equal to themaximum inter-story drift value with mean annual frequency of being exceeded (drift hazard)equal to, Po, namely:
Po=��max(FD(Po)) (11)
Exploiting the relationship between factored demand and drift hazard in Equation (10), theprobabilistic demand PD(Po) at allowable probability Po is defined as follows:
PD(Po)=�−1�max
(Po) (12)
where the probabilistic demand at allowable probability Po is the demand value that will beexceeded with a mean annual frequency of Po. In the special case where the drift hazard isexpressed by the analytic form in Equation (10), the probabilistic demand at Po will be equal tothe factored demand at Po. The term PD(Po) is introduced as a parallel to factored demand. Beingdefined in terms of ��max , the probabilistic demand is not necessarily confined by the assumptionsthat underlie the derivation of factored demand. For example, if the structural displacement demandis so large that it is close to the onset of global stability in the structure, its probability distributiongiven spectral acceleration—most likely—is not going to be described by a lognormal probabilitydistribution and the definitions proposed in Equation (9) are not going to be sufficient for obtainingfactored demand and factored capacity. Probabilistic demand provides us a bench-mark againstwhich the factored demand calculated based on the DCFD formulation can be compared.
The hazard value �Sa(x) at spectral acceleration Sa= x is defined as the mean annual frequencythat the intensity of the future ground motion events occurring at the site are greater than or equalto a specific value. As proposed by the development underlying Department of Energy Standard1020 (see [22, 23]), one can fit a power-law type of expression to the hazard curve:
�Sa ≈ P(Sa�x)=k0 ·x−k (13)
where k0 and k are the parameters defining the power-law approximation. In fact, one of theassumptions underlying the derivation of the DCFD closed-form is that �Sa(x) be represented bythe power-law form in Equation (13). Hence, in order to obtain k0 and k estimates for the purposeof DCFD performance assessments, one needs to obtain a proper linear approximation of the givenspectral acceleration hazard curve. This can be achieved by fitting a line to the (logarithm of) thehazard curve in the ‘region of interest’; the region of interest can be determined based on the limitstate LS for which the performance assessments are done. For instance, if the region of interestcorresponds to (rare) earthquake ground-motions with a rate of being exceeded of about 2% in 50years, one needs to fit a line to the hazard curve around Sa value that corresponds to a hazard valueof 2% in 50 years. Figure 2 shows (on a two-way logarithmic scale) a site-specific hazard curvecalculated for a site located in Van Nuys, CA, at T=0.85s, T being (close to) the first naturalperiod of the structure. The figure also illustrates a linear approximation at annual frequency ofabout 2
1000 (i.e. 10% in 50 years). The (absolute value of the) slope of the fitted line is the k valuein the region of interest; here, it is equal to 2.6.
In the framework of DCFD performance assessments, the hazard for spectral acceleration isalso needed to obtain the spectral acceleration value PoSa corresponding to a specified tolerableexceedance rate Po. Figure 2 also shows PoSa values at two different tolerable probability levels of
10–1
10–4
10–3
10–2
10–1
100
100 101
Sa(T=0.85 sec) [g]
Mea
n an
nual
freq
uenc
y of
exc
eeda
nce
Sa
k=2.6Po=0.03
Po=0.0084
Figure 2. A site-specific hazard Curve, Van Nuys, CA. The hazard curve is approximated by a line (onthe two-way logarithmic paper) in the region of interest.
ALTERNATIVE NON-LINEAR DEMAND ESTIMATION METHODS 959
Po=0.0084 and 0.03. The probability levels are marked on the hazard axis and their correspondingspectral acceleration values, PoSa= 0.70g and 0.40g, respectively, are calculated by finding the Savalue that corresponds to a hazard level, Po, via the hazard curve.
6. ALTERNATIVE METHODS FOR DEMAND ESTIMATION USING NON-LINEARDYNAMIC PROCEDURES
The non-linear dynamic procedures discussed in this section are classified into two groups. Thefirst group, nominated as wide-range, is suitable for making probabilistic assessments over a widerange of tolerable probability levels. MSA and incremental dynamic analysis (IDA) are two suchmethods. Multiple-stripe analysis is suitable for making probability-based assessments for a (wide)range of spectral acceleration values and/or limit states and IDA can be implemented in order tolocate the onset of global dynamic instability (collapse capacity) in the structure. The second group,nominated as narrow-range, is suitable for making probabilistic assessments for a tight intervalof tolerable probability values. Single-stripe analysis and double-stripe analysis are examples ofthe narrow-range methods discussed in this section; they can be useful for making performanceassessments when limiting the number of structural analyses is a priority.
6.1. Multiple-stripe analysis (MSA)
MSA, as suggested by its name, refers to a group of ‘stripe’ analyses performed at multiple spectralacceleration levels, where a stripe analysis consists of structural analyses for a suite of groundmotion records that are scaled to a common spectral acceleration. The suite of ground motionrecords used for performing each stripe analysis should ideally be representative of the seismicthreat at the corresponding spectral acceleration; however, it is common but not necessarily alwayswell justified (e.g. [8, 12, 21]) to use the same suite of records for all the spectral accelerationlevels.
Figure 3 illustrates the MSA results for the suite of ground motion records used in this study.The ‘collapsing’ cases (full circles) are distinguished from the ‘non-collapsing’ ones, where theonset of ‘collapse’ for a ground motion record is identified as the point where maximum inter-storydrift response increases ‘drastically’ when the spectral acceleration of the record is increased bya ‘small’ amount. Later in this paper, in the section discussing IDA, the onset of collapse for agiven ground motion record will be defined in more precise terms.
6.2. Using MSA results to calculate the drift hazard
The results of MSA can be used to calculate the drift hazard directly from Equation (3). Morespecifically, the CCDF for demand for a given spectral acceleration, G�max|Sa(y|x) in Equation (3) isestimated by the fraction of the stripe response that exceeds the value, y, for a spectral accelerationequal to x at each stripe (this fraction is also known as the Empirical Distribution, see [24]). Thespectral acceleration hazard function �Sa is estimated by the hazard curve in Figure 2. We havecalculated the drift hazard from Equation (3) using numerical integration for multiple values ofdrift, y, and have plotted the resulting drift hazard curve in Figure 4 (thick line). The numericalintegration can be conducted in any number of ways; we simply interpolated linearly the hazardcurve for spectral acceleration and the CCDF for demand given spectral acceleration. In calculating
Figure 3. Multiple-stripe analysis for the existing RC frame using the suite of 30 ground-motion recordingsin Table I; ‘collapse’ points are identified.
the drift hazard in this section, we ignored the distinction between ‘collapsing’ points and ‘non-collapsing’ ones and included all the data points. This was possible since we did not have anynon-convergence cases in which the structural response is simply not available.
We have used the resulting drift hazard curve in Figure 4 to obtain the probabilistic demandvalues at two tolerable annual annual frequency levels equal to Po=0.0084 and Po=0.03. Thesmaller probability level,‡ Po=0.0084, is a probability level that we believe is associated withthe structure being on the verge of collapse (this particular probability level has been chosenmerely for demonstration purposes). Therefore, it poses a severe test to the analytical expressionfor DCFD format (Equation (9)), permitting us to demonstrate its potential limitations. The largerPo level is associated with the onset of damage in structural member. At this level, the analyticalapproximations are expected to be quite robust. The probabilistic demand associated with Po=0.03is equal to 0.7%, which is close to a global ductility of one with respect to the onset of significantdamage in the structure, as judged by Figure 1. For a tolerable probability level of Po=0.0084,the probabilistic demand equal to, 2%, which corresponds to a ductility of 3 relative to the onsetof significant structural damage.
The probabilistic demand values obtained from the drift hazard curve in this section are goingto be used to benchmark the factored demand estimates obtained later in this paper. We are goingto refer to the results here as the ‘best estimate’ for demand because they are obtained based ona less restrictive set of assumptions compared with the factored demand estimates. It should bekept in mind that even our ‘best estimate’ is subject to various sources of uncertainty not dealtwith explicitly here, including those associated with structural modeling, limited sample size, andspectral acceleration hazard estimation.
‡We have sometimes used the term ‘probability’ in order to refer to the mean annual frequency of exceedance, since,for the type of rare events studied, the corresponding numerical values are very close (and small).
ALTERNATIVE NON-LINEAR DEMAND ESTIMATION METHODS 961
Table I. Ground motion records for 6.5�M�7.0 and 15�r�30km selected from PEER Database;soil type: C, D (Geo-Matrix); rclosest distance to fault rupture; M moment magnitude; SS: strike slip;
RN: reverse thrust; RO: reverse-oblique.
ID Earthquake Station & Comp M r Mech Sa(T1)
1 Loma Prieta 10/18/89 Agnews State Hospital, 090 6.9 28.2 RO 0.232 Northridge 01/17/94 LA, Baldwin Hill, 090 6.7 31.3 RN 0.253 Imperial Valley 10/15/79 Compuertas, 285 6.5 32.6 SS 0.0814 Imperial Valley 10/15/79 Plaster City, 135 6.5 31.7 SS 0.065 Loma Prieta 10/18/89 Hollister Diff Array, 255 6.9 25.8 RO 0.676 San Fernando 02/09/71 LA, Hollywood Stor Lot, 180 6.6 21.2 RN 0.147 Loma Prieta 10/18/89 Anderson Dam (Downst), 270 6.9 21.4 RO 0.298 Loma Prieta 10/18/89 Coyote Lake Dam (Downst), 285 6.9 22.3 RO 0.299 Imperial Valley 10/15/79 El Centro Array #12, 140 6.5 18.2 SS 0.1810 Imperial Valley 10/15/79 Cucapah, 085 6.5 23.6 SS 0.4011 Northridge 01/17/94 LA, Hollywood Stor FF, 360 6.7 25.5 RN 0.6112 Loma Prieta 10/18/89 Sunnyvale, Colton Ave, 270 6.9 28.8 RO 0.3613 Loma Prieta 10/18/89 Anderson Dam (Downst), 360 6.9 21.4 RO 0.3114 Imperial Valley 10/15/79 Chihuahua, 012 6.5 28.7 SS 0.5115 Imperial Valley 10/15/79 El Centro Array #13, 140 6.5 21.9 SS 0.1316 Imperial Valley 10/15/79 Westmorland Fire Station, 090 6.5 15.1 SS 0.1017 Loma Prieta 10/18/89 Hollister South & Pine, 000 6.9 28.8 RO 1.0218 Loma Prieta 10/18/89 Sunnyvale, Colton Ave., 360 6.9 28.8 RO 0.2519 Superstition Hills(B) 11/24/87 Wildlife Liquefaction Array, 090 6.7 24.4 SS 0.2620 Imperial Valley 10/15/79 Chihuahua, 282 6.5 28.7 SS 0.6321 Imperial Valley 10/15/79 El Centro Array #13, 230 6.5 21.9 SS 0.1122 Imperial Valley 10/15/79 Westmorland Fire Station, 180 6.5 15.1 SS 0.1323 Loma Prieta 10/18/89 Halls Valley, 090 6.9 31.6 RO 0.2224 Loma Prieta 10/18/89 Waho, 000 6.9 16.9 RO 0.8025 Superstition Hills 11/24/87 Wildlife Liquefaction Array, 360 6.7 24.4 SS 0.5326 Imperial Valley 10/15/79 Compuertas, 015 6.5 32.6 SS 0.1627 Imperial Valley 10/15/79 Plaster City, 045 6.5 31.7 SS 0.0328 Loma Prieta 10/18/89 Hollister Diff Array, 165 6.9 25.8 RO 0.6729 San Fernando 02/09/71 LA, Hollywood Stor Lot, 090 6.6 21.2 RN 0.3030 Loma Prieta 10/18/89 Waho, 090 6.9 16.9 RO 0.72
6.2.1. Using MSA results to calculate drift hazard in the range of global dynamic instability. Justas with G�max|Sa(y|x), the conditioned-on-non-collapse CCDF, G�max|NC,Sa(y|x) can be estimatedby fitting an Empirical Distribution to MSA data. The only difference is that the samples areconditioned not only on spectral acceleration but also on non-collapse, meaning that for each‘stripe’ analysis level in Figure 3, we discard the full circles and consider only the remainingdata points as our samples. The probability of non-collapse, PNC|Sa(x), can also be estimatedempirically by the fraction of the non-collapse data points (i.e. thin circles) at each stripe . Oncewe have estimated G�max|NC,Sa(y|x) and PNC|Sa(x), we can calculate the drift hazard numericallyfrom Equation (7). The resulting drift hazard is plotted (dashed line) in Figure 4 together withthe drift hazard obtained above without treating the collapse data separately. It can be observedthat the two curves are identical for the low drift values where the probability of collapse givenspectral acceleration is small. This was expected because the drift hazard at small values of driftdemand converges to the drift hazard given no-collapse (as noted before). However, the two curvesstart to diverge as the drift demand values become larger and the drift hazard considering the
Figure 4. Drift hazard curve calculated by numerical integration of Equations (3) (thick line) and(7) (dashed line) considering explicitly for ‘collapse’ cases. The probabilistic demand estimates at
Po from both curves are also shown on the plot.
collapse information approaches its lower limit, which is the mean annual frequency of exceedingthe collapse limit state, �LS (as also noted before). The figure also illustrates the probabilisticdemand obtained for the two tolerable probability levels Po=0.0084 and 0.03 from the drift hazardcurve. We can observe that the allowable probability level of Po=0.0084 is very close to (but stillgreater than) the mean annual frequency of collapse (i.e. the lower limit indicating global dynamicinstability). This is also indicated by the significant difference in the demand estimates (2 versus2.7% when ‘collapse’ information is taken into account). The two curves yield identical resultsfor probabilistic demand at Po=0.03, which is consistent with the presumption that the structurehas not yet undergone significant damage at this tolerable probability level.
6.3. Using MSA results to estimate factored demand
The results of MSA can also be used to calculate the factored demand from Equation (9); thisentails estimating the parameters ��max|Sa,��max|Sa , and b at each spectral acceleration level. Wehave based our calculations on the ‘counted’ statistics. In order to obtain the ‘counted’ statisticalparameters of a data set, the data are first sorted in the ascending order. The counted median ofthe data is the drift value such that 50% of the points lie below it. The counted fractional standarddeviation is estimated here by the average of ln(�84th/�50th) and ln(�50th/�16th), where the symbols�16th and �84th denote the drift values corresponding to 16 and 84% percentiles of the ordereddata, respectively. The b value can be estimated as the local tangent or slope of the median curvefor a given spectral acceleration value plotted in the logarithmic scale. The counted 16th, 50th and84th percentiles of the response for each spectral acceleration level are shown in Figure 5.
The multiple-stripe method provides a picture of how both the general trend (median) and thedispersion (fractional standard deviation) of the response evolve under incrementally increasingground motion levels. Between about 0.80g and 1.10g, the median curve starts to ‘soften’, indicatinga more rapid increase in response for a given increase of spectral acceleration value. This is
ALTERNATIVE NON-LINEAR DEMAND ESTIMATION METHODS 963
100
Maximum interstory drift ratio, θmax
Sa(
T1)
[g]
= 0.70g
PoSa
PoSa
= 0.40gb = 2.70
b = 1.60
ηθmax|Sa = 0.0183
ηθmax|Sa = 0.007
Figure 5. MSA: estimates for, ��max|Sa and b values at Po Sa=0.40g and 0.70g.
accompanied by an increase in fractional standard deviation. The large response dispersion forspectral acceleration values beyond 1.0g implies that the median is determined with decliningaccuracy.
Figure 5 illustrates the application of MSA in obtaining local estimates for ��max|Sa,��max|Sa , andb-value at PoSa=0.40g and 0.70g, which correspond to Po=0.0084 and 0.03, respectively. Thefactored demand at Po=0.0084 and 0.03 can be calculated from Equation (9) based on the localparameter estimates, ��max|Sa,��max|Sa and b-value:
FD(PoSa=0.40) = ��max|Po Sa · exp(1
2
k
b�2�max|Sa
)=0.007×exp
(1
2
2.6
1.6(0.35)2
)
= 0.0073=0.73% (14)
and
FD(PoSa=0.70)=0.0183×exp( 122.62.7 (0.49)
2)=0.0205=2.05% (15)
These estimates are very close to the probabilistic demand values PD(0.03)=0.7% andPD(0.0084)=2%, respectively; which were obtained using numerical integration of Equation (3)earlier in the paper as the ‘best-estimates’ for demand.
6.4. Single- and double-stripe analysis
The single-stripe analysis, as its name suggests, involves structural dynamic analyses for a setof records scaled to a common spectral acceleration value as described before. The output ofsingle-stripe analysis is referred to here as the ‘stripe response’. Similarly, the common spectralacceleration value is referred to as the spectral acceleration of the stripe. The stripe response canbe used to estimate the median and (fractional) standard deviation for the inter-story drift demandconditioned on the spectral acceleration of the stripe. The statistical parameter estimations provided
Figure 6. ‘Stripe’ response at Po Sa=0.70. The lognormal distribution fitted to the ‘stripe’ response basedon its (counted) median and fractional standard deviation are also plotted.
by the stripe analysis depend on the choice of the spectral acceleration to which the records arescaled. One possibility is to choose the ground motion intensity level that has a probability ofbeing exceeded equal to the allowable probability level Po.
With this strategy, the spectral acceleration of the stripe is chosen as the spectral acceler-ation that corresponds to Po=0.0084 (the probability level corresponding to severe damage)from the hazard curve (Figure 2) and is equal to PoSa=0.70. This choice is driven by the fact thatthe median demand at PoSa appears—as the main parameter affecting the factored demand—inthe expression for factored demand FD(Po) in Equation (9). In order to obtain ‘the stripe’response, the set of ground motion records are scaled to PoSa=0.70 and are applied to the modelstructure. The maximum inter-story drift ratio response to the set of scaled records is plottedtogether with the underlying lognormal distribution model fitted to data, in Figure 6.
The statistical parameters of the ‘stripe’ response can now be used to estimate the medianand fractional standard deviation at the spectral acceleration . We have calculated the ‘countedmedian’ and ‘counted fractional standard deviation’ of the stripe response, denoted by ��max|Sa and��max|Sa , respectively. The counted median and fractional standard deviation of the stripe response atPoSa=0.70 are equal to 0.0183 and 0.49, respectively. These quantities are also shown in Figure 6together with the lognormal distribution fitted to the stripe response. The arrow on the right-mostpart of the graph indicates that there are maximum inter-story drift values larger than 10% (i.e.relative displacement more than 10% of the story height) and hence not shown in the figure.This observation is consistent with the large value obtained for dispersion parameter, ��max|Sa .The median and fractional standard deviation can also be calculated from the sample mean andstandard deviation of the logarithm of the stripe response, instead of using the counted percentiles.However, difficulties may ensue if one encounters dynamic runs in which numerical convergence isnot obtained, and/or if very large, unrealistic displacements are produced by the structural analysisprogram. The single-stripe response does not provide displacement versus spectral accelerationslope information; therefore it is unable to estimate the b parameter appearing in the expression for
ALTERNATIVE NON-LINEAR DEMAND ESTIMATION METHODS 965
100
"Firs
t Mod
e" S
a(T
=0.
80)
[g]’s
b=3.60
Figure 7. Double-stripe analysis, estimation of b-value by performinga second stripe analysis at Po Sa=0.80.
factored demand (Equation (9)). As a preliminary guess, we assume that the b value is equal to 1,implying that the median response given spectral acceleration denoted by ��max|Sa is proportionalto Sa. Elastic behavior and the ‘equal displacement rule’ [25] are special cases of this condition.The factored demand, which is calculated using the estimates of ��max|Sa and ��max|Sa obtainedfrom single-stripe analysis method and a b-value of one, is:
Comparing this result against the probabilistic demand at Po=0.0084, which is equal toPD(0.0084)=2% (Figure 4), we can observe that the single-stripe prediction for factored demandis significantly larger. We speculate that taking the default value of b=1 may have led to sucha conservative estimate of the factored demand. This brings up the question of how much extraanalysis effort is required in order to acquire an estimate of the parameter, b. A minimum numberof two stripe responses are necessary for getting information about the b value; hence, we considerperforming a ‘double-stripe’ method. The double-stripe method consists of two separate single-stripe analyses, namely, the original stripe response plus an additional stripe response ‘sufficientlyclose’ to the original stripe. In this example we place the second stripe at PoSa=0.80. The double-stripe response is plotted in Figure 7. The b value is estimated as the slope of the line on thelog–log plot connecting the medians of the two stripes:
b̂=ln
�̂�max|Sa=0.80
�̂�max|Sa=0.70
ln 0.800.70
=3.6 (17)
It should be noted that the estimates for ��max|Sa and ��max|Sa are the same as those obtainedfor the single-stripe analysis above and the second stripe is used for estimating the b value.
Figure 8. (a) Histogram for the stripe response at Po Sa=0.70 and (b) stripe responsede-aggregated into collapse and non-collapse parts.
The factored demand in this case is equal to:
FD(PoSa=0.70)=0.0183×exp( 122.63.6 (0.49)
2)=0.0183×1.09=0.020=2% (18)
The result is identical to the probabilistic demand at Po=0.0084, which has been calculatedby numerical integration. However, we should note that the position of the second stripe, withrespect to the original one, plays a critical role in estimation of the b value. If the second stripeis too far, the estimated b value may not be representative of the local slope around the originalstripe. If it is too close, the estimated value may fail to represent the general trend in spectralacceleration versus demand curve. Thus, the accuracy of a double-stripe analysis is dependant onthe analyst’s judgment in choosing the spacing between the stripes. A suggestion is to choosethe second Sa stripe to be above the original value by a fraction (of that value) equal to 1
4 or 12
of ��max|Sa (i.e. S2a = S1a +fraction· ��max|Sa ·S1a ). The improvement observed in the double-stripeestimate for factored demand over the single-stripe method, emphasizes the importance of obtaininglocal estimates of the b-value in the region of interest especially when the structural response isclose to the onset of global dynamic instability, where b may differ substantially from unity.
6.4.1. Single-stripe analysis in the region of global dynamic instability. As indicated by arrowsin Figures 6 and 7, a number of stripe response values get quite large. Figure 8(a) illustrates thehistogram for the maximum inter-story drift values at PoSa=0.70g. If the onset of global dynamicinstability in the structure is identified by maximum inter-story drift values more than 5%, we candetect four cases of ‘collapse’ from the histogram. In the previous sections, the stripe responsewas used to estimate only the first two moments of the response for a given spectral acceleration.Assuming that the stripe response is described by a log-normal distribution, these two moments aresufficient for obtaining the full distribution of response for a given spectral acceleration. However,in the presence of too many ‘collapse’ points in the stripe response, the log-normal assumption may
ALTERNATIVE NON-LINEAR DEMAND ESTIMATION METHODS 967
not be very suitable. The probability that the stripe response exceeds a given value can be writtenin a more general manner by distinguishing between ‘collapse’ and ‘no collapse’ response values(Equation (6)). In order to derive Equation (6) it was assumed that the probability of exceeding�max given that collapse has taken place is equal to one. Loosely speaking, this means that we‘lump’ all the collapse �max values at some arbitrarily large value on the maximum inter-story driftaxis, as it is schematically shown in Figure 8(b).
The data provided by the stripe response can be used to estimate the probability distributionfor the demand G�max|Sa(y|x) at PoSa=0.70g using Equation (6). The (counted) median andfractional standard deviation of the non-collapse portion of the data can be calculated as explainedbefore. Assuming that the non-collapse portion of the data is described by a log-normal probabilitydistribution, median and fractional standard deviation are sufficient for calculating G�max|NC,Sa(y|x).Moreover, the probability of non-collapse at PoSa=0.70g can be estimated by the ratio of non-collapse cases to the total number of structural analyses. For example, P̂NC|Sa(0.70) is equal to2630 =0.87.The factored demand given that collapse has not taken place (hereafter referred to as: given no
collapse) denoted by FD(PoSa|NC) can also be calculated using the median and fractional standarddeviation of the non-collapse portion of the stripe response:
It can be noted that the (counted) median and standard deviation of the non-collapse part ofthe stripe response are equal to 1.6% and 0.46; whereas the same (counted) statistics calculatedusing the entire stripe response were equal to 1.8% and 0.49 respectively. Also, the factoreddemand at PoSa=0.70g given no collapse takes place is equal to 2.1%; whereas the factoreddemand calculated without making the collapse/no collapse distinction was calculated to be equalto 2.5%. This is another way of representing factored demand, namely, reporting the estimatedFD(PoSa=0.70|NC)=2.1% together with the estimated probability of non-collapse at PoSa=0.70g, P̂NC|Sa(0.70)=0.87. The probability of non-collapse at the spectral acceleration of thestripe, which is equal to the fraction of non-collapse cases in the stripe response, could also beregarded as an indication of how far the structure is from the regime of global dynamic instability.In other words, the lower the probability of non-collapse for the stripe, the larger the fraction ofcollapsing cases. This could be an indicative of severe non-linearity in the structure.
6.5. Incremental dynamic analysis (IDA)
IDA [4] consists of obtaining a suite of IDA response curves for a suite of ground motion records,where an IDA curve demonstrates the changes in a specific structural response parameter when thestructure is subjected to a particular ground motion record scaled to successively increasing levelsof intensity. Having a format similar to the familiar force–displacement curve for the responseof an single degrees of freedom system to a monotonically increasing static load, the IDA curveprovides unique information about the nature of the structural response of an multiple degrees offreedom system to a ground motion record.
Alternative curve-fitting routines such as a ‘spline fit’ can be used to get smooth IDA curves[4]. In lieu of employing curve-fitting routines, one can simply connect the response points, fora given ground motion record, by straight lines in order to obtain a (non-smooth) IDA curve; asis illustrated in Figure 9. These IDA curves correspond to the suite of 30 ground motion recordsused in this paper. It should be noted that the data represented in Figure 9 are the same as the
Figure 9. IDA curves with the corresponding onset of global dynamic instability or the ‘collapse’ points.The lognormal probability distributions fitted to the spectral acceleration and the maximum inter-story
drift ratio of the ‘collapse points’ are also shown on the figure.
multiple-stripe data presented in Figure 3. Therefore, the IDA results would provide the sameestimates for factored demand as those presented for MSA. However, unlike MSA, which could(potentially) be done using different sets of records at different intensities, the IDA results pertainto a fixed set of ground motion records.
6.6. Capacity estimation using the results of IDA
Earlier in this paper, we studied demand estimations in the range of global dynamic instabilityin the structure. We are particularly interested in this behavior range because structural responseestimation close to instability can pose a severe challenge to the assumptions underlying the closed-form and analytic formulation of DCFD and also because the non-converging cases observed in thestructural response calculations are often caused by the global instability in the dynamic response.Global instability is identified by an arbitrarily large increase in displacement response subjectedto a small increment in ground motion intensity. This particular behavior is well captured by anIDA curve. Following the definition in the FEMA/SAC Guidelines (see [26]), we have identifiedthe onset of such behavior by a point where the local slope of the IDA curve decreases to a certainpercentage of the initial slope of the IDA curve in the elastic region (Figure 9). This percentageis a more or less an arbitrary value that represents the point where the IDA curve becomes ‘flat’enough; here, we have chosen it to be equal to 16.6% ( 16 ). We can clearly observe the scatter inthe capacity points for the different ground motion records in Figure 9.
6.6.1. Estimation of factored capacity using the results of IDA. Figure 9 also illustrates the‘collapse’ points on the IDA curves; the ‘collapse’ coordinates for each ground motion recordconsists of maximum inter-story drift ratio and the corresponding spectral acceleration value. Thelog-normal curves fitted to these ‘collapse’ coordinates based on their corresponding medians andfractional standard deviations are also shown in the figure. Similar to factored demand estimation,this information can be used for estimating the factored (displacement) capacity from Equation (9).
ALTERNATIVE NON-LINEAR DEMAND ESTIMATION METHODS 969
We have estimated the b-value to be equal to 4; that is, the local slope of the median MSA curve(Figure 5) in the vicinity of the median maximum inter-story drift capacity (shown in Figure 9):
FC=�C ·exp(1
2
k
b�2C
)=0.0278×exp
(−1
2
2.6
40.412
)=0.0278×0.95=0.026=2.6% (20)
By calculating the factored capacity, we can now make a performance assessment of the structurefor the global dynamic instability limit state based on the DCFD format and at the allowableprobability level, Po. Recalling from previous sections, we arrived at the same numerical value forfactored drift demand, FD(Po=0.0084)=2%, by pursuing alternative strategies; namely, factoreddemand estimation using MSA, double-stripe analysis, and direct numerical integration of Equation(3). We can easily verify that the factored demand at Po=0.0084 satisfies the DCFD designcriterion in Equation (9) for the structural limit state of global dynamic instability.
6.7. Estimation of the mean annual frequency of exceeding a limit state
We can also calculate directly the mean annual frequency of exceeding the global instability limitstate (or mean annual frequency of collapse), �LS, from Equation (2); where, the cumulative distri-bution function (CDF) for limit state capacity, FCLS(y), is estimated by an Empirical Distributionfit to the maximum inter-story drift capacity points in Figure 9. This computation results in anestimated mean annual frequency collapse, �LS, equal to 0.0063. An alternative way for calcu-lating �LS is to obtain the lower limit of the drift hazard from Equation (7), which is derived byexplicitly accounting for the collapse cases. The performance of the structure for the limit state ofglobal dynamic instability can also be checked according to the criterion stated in Equation (8) bycomparing the allowable probability level Po against the mean annual frequency of collapse.
7. SUMMARY AND CONCLUSIONS
Probability-based seismic performance assessment of a structure can be based on the results fromalternative non-linear dynamic analysis procedures. These methods can be used both to obtainparameter estimates for specific probabilistic assessment criteria such as demand and capacityfactored design (DCFD) and also to make direct performance assessments using numerical inte-gration methods. This paper describes how non-linear dynamic analysis procedures can be usedin order to make parameter estimates in the context of the probability-based DCFD criterion forstructural performance assessment for multiple performance levels. Alternatively, the structuralperformance can be evaluated by calculating directly the expression for the mean annual frequencyof exceeding a desirable performance level using numerical methods. The results of such directperformance assessments can be used as a basis for bench-marking evaluations based on the DCFDassessment criteria and when the assumptions underlying the derivation of its closed-form are notsatisfied. The performance of a case-study structure for the limit state of global dynamic instabilityis checked both by directly comparing the allowable probability level to the mean annual frequencyof collapse and also using the DCFD design criterion. These alternatives criteria provide consistentconclusions about the safety of the structure for the limit state of global dynamic instability.
Two types of alternative non-linear dynamic analysis methods are presented in this paper. Themethods classified in the first group (which can be referred to as the wide-range methods) map outthe structural response to a wide range of ground motion intensity and structural response levels.
Table II. Alternative non-linear dynamic analysis methods for probability levels Po=0.03 and Po=0.0084;number of analyses needed to achieve 20% standard error; N is the number of stripes (e.g. N =20).
Method No. Analyses Local estimates Comments
MSA at Po=0.03 3× N ��max|Sa , ��max|Sa and b Most accurateSingle-stripe at Po=0.03 3 ��max|Sa and ��max|Sa Least accurateDouble-stripe at Po=0.03 6 ��max|Sa , ��max|Sa and b Careful selection of the second stripeMSA at Po=0.0084 6× N ��max|Sa , ��max|Sa and b Most accurateSingle-stripe at Po=0.0084 6 ��max|Sa and ��max|Sa Least accurateDouble-stripe at Po=0.0084 12 ��max|Sa , ��max|Sa and b Careful selection of the second stripe
MSA and IDA are examples of wide-range non-linear dynamic analysis methods discussed here.It has been demonstrated how these non-linear dynamic analysis procedures can be implementedfor the performance assessments for two different tolerable probability levels that translate intotwo distinct structural performances levels. The performance assessments include the limiting casein structural behavior when the structural response increases to an arbitrarily large amount due toa small increment in ground motion intensity. Demand and capacity estimations for this limitingcase, which is referred to as the global dynamic instability or the ‘collapse’ limit state, may faceimplications that are discussed in this paper. Similar implications arise when structural responseis non-available due to numerical non-convergence in the structural analysis calculations.
The second group, referred to as the narrow-range methods, as the name suggests have limited(‘narrow’) range of applicability for probability-based performance assessments. Stripe analysisis an example of narrow-range methods, in which the demand parameters in DCFD format areestimated locally in the vicinity of the allowable probability of interest. It is demonstrated that theaccuracy of the single-stripe method can be improved significantly by performing stripe analysisfor another (sufficiently close) spectral acceleration level; the ensemble of the two stripe analysesis referred to as double-stripe analysis.
The alternative non-linear dynamic analysis methods discussed in the paper are synthesized inTable II for the two limit states considered herein represented by probability levels Po=0.03 andPo=0.0084. The number of analyses needed for yielding a standard error (in the estimation of themean drift demand given spectral acceleration) equal to 20% are outlined in the table. It can beobserved that, for all methods considered, the number of analyses required for the probability levelPo=0.03 (150% in 50 years) is about half of that required for the probability level Po=0.0084(42% in 50 years). Also it can be observed that the MSA analysis is able to provide both localestimates for mean and standard deviation of demand and the b-value for each stripe. However,it needs a large number of structural analyses. On the other hand, the single stripe analsis needsminimal number of analyses but is unable to yield a local estimate of the b-value. Double-stripeanalysis needs twice the number of analyses needed for single-stripe analysis but it is able to providea local estimate for the b-value if the second stripe is spaced close enough to the original one.
ACKNOWLEDGEMENTS
The first author would like to mention the passing away of the second author, the late C. Allin Cornell,in December 2007. This work was almost completed before his passing away with the exception of someeditorial modifications (the paper was originally written in two parts, which are now united in a single
ALTERNATIVE NON-LINEAR DEMAND ESTIMATION METHODS 971
article). He has been a teacher and a mentor to the first author ever since the first author started herPhD in 1998. The contributions of the late C. Allin Cornell to earthquake engineering and seismologyare evident from his research beginning from the 1960s. His profound vision, sharp thinking and distinctapproach to problem-solving will continue to be an inspiration to the first author.
REFERENCES
1. Wen YK. Reliability and performance-based design. Structural Safety 2001; 23:407–428.2. Cornell CA, Jalayer F, Hamburger RO, Foutch DA. The probabilistic basis for the 2000 SAC/FEMA steel
moment frame guidelines. ASCE Journal of Structural Engineering 2002; 128(4):526–533.3. Jalayer F, Cornell CA. A technical framework for probability-based demand and capacity factor design (DCFD)
seismic formats. Pacific Earthquake Engineering Research Center Report, 2003; 2003/08.4. Vamvatsikos D, Cornell CA. Incremental dynamic analysis. Earthquake Engineering and Structural Dynamics
2001; 31(3):491–514.5. Cordova PP, Deierlein GG, Mehanny SSF, Cornell CA. Development of a two-parameter seismic intensity measure
and probabilistic assessment procedure. The Second US–Japan Workshop on Performance-Based EarthquakeEngineering Methodology for Reinforced Concrete Building Structures, Sapporo, Japan, Pacific EarthquakeEngineering Center (PEER), 2000; 2000/10; 195–214.
6. Medina R, Krawinkler H. Seismic Demands for non-deteriorating frame structures and their dependence onground motions. Pacific Earthquake Engineering Center (PEER), 2003; 2003/15.
7. Jalayer F, Franchin P, Pinto PE. A scalar damage measure for seismic reliability analysis of RC frames. EarthquakeEngineering and Structural Dynamics 2007; 36(13):2059–2079.
8. Baker JW. Probabilistic structural response assessment using vector-valued intensity measures. EarthquakeEngineering and Structural Dynamics 2007; 36(13):1861–1883.
9. Tothong P, Cornell CA. Structural performance assessment under near-source pulse-like ground motions usingadvanced ground motion intensity measures. Earthquake Engineering and Structural Dynamics 2008; 37:1013–1037.
10. Shome N, Cornell CA, Bazzurro P, Carballo JE. Earthquakes, records and nonlinear responses. EarthquakeSpectra 1998; 14(3):469–500.
11. Luco N, Cornell CA. Structure-specific scalar intensity measures for near-source and ordinary earthquake groundmotions. Earthquake Spectra 2007; 23(2):357–392.
12. Baker JW, Cornell CA. Spectral shape, epsilon and record selection. Earthquake Engineering and StructuralDynamics 2006; 35(9):1077–1095.
13. Pincheira JA, Dotiwala FS, D’Souza JT. Seismic analysis of older reinforced concrete columns. EarthquakeSpectra 1999; 15(2):245–272.
14. Dotiwala FS, D’Souza JT, Pincheira JA. Drain2D-UW User’s Manual. Department of Civil and EnvironmentalEngineering, University of Wisconsin—Madison, 1998.
15. Pacific Earthquake Engineering Research Center: PEER Strong Motion Database. http://peer.berkeley.edu/smcat/[16 September 2008].
16. McGuire RK. Probabilistic seismic hazard analysis and design earthquakes: closing the loop. Bulletin of SeismicSociety of America 1995; 85(5):1275–1284.
17. Bazzurro P, Cornell CA. On disaggregation of seismic hazard. Bulletin of Seismological Society of America 1999;89(2):501–520.
18. Alavi B, Krawinkler H. Consideration of near-fault ground motion effects in seismic design. Proceedings of 12thWorld Conference on Earthquake Engineering, vol. 2665. New Zealand Society for Earthquake Engineering:Upper Hutt, New Zealand, 2000.
19. Iervolino I, Cornell CA. Record selection for non-linear seismic analysis of structures. Earthquake Spectra 2005;21(3):685–713.
20. Luco N, Bazzurro P. Effects of ground motion scaling on non-linear structural response. PEER Life Line Report,1G00, 31 December 2003.
21. Shome N, Cornell CA. Probabilistic seismic demand analysis of nonlinear structures. RMS Program, Report No.RMS35 (Ph.D. Thesis), Stanford University, May 1999.
22. Kennedy RP, Short SA. Basis for seismic provisions of DOE-STD-1020. Lawrence Livermore National Laboratoryand Brookhaven National Laboratory, Report Numbers UCRL-CR-111478 and BNL-52418, 1994.
23. Luco N, Cornell CA. Seismic drift demands for two SMRF structures with brittle connections. StructuralEngineering World Wide. Elsevier Science Ltd.: Oxford, England, 1998. Paper T158-3.
24. Efron B, Tibshirani RJ. An Introduction to Bootstrap. Chapman & Hall/CRC: London, Boca Raton, FL, 1993.25. Veletsos AS, Newmark NM. Effect of inelastic behavior on the response of simple systems to earthquake motions.
Proceedings of the Second World Conference on Earthquake Engineering, Japan, Tokyo, vol. II, 1960; 895–912.26. Yun SY, Hamburger RO, Cornell CA, Foutch DA. Seismic performance evaluation for steel moment frames.
ASCE Journal of Structural Engineering 2002; 128(4):534–545.
