C. Allin Cornell

Earthquakes, Records, and Nonlinear Responses

The estimation of MDOF nonlinear structural response given an earth-quake of magnitude M at distance R is studied with respect to issues such as the benefits and harms of (1) first scaling the records, (2) selecting records from the “wrong” magnitude, (3) alternative choices for how to scale the records, and (4) scaling records to a significantly higher or lower intensity, etc. We find that properly chosen scaling can reduce the necessity of the number of nonlinear analyses by a factor of about four, and that proper scaling does not introduce any bias. Several global and local nonlinear damage measures are considered. A five-DOF model of a steel structure is used; other cases are under study. The paper finishes with a demonstration of the use of such results in the estimation of the annual probability of exceeding a specified interstory ductility (drift) or other damage measures.

STRUCTURE-SPECIFIC SCALAR INTENSITY MEASURES FOR NEAR SOURCE AND ORDINARY EARTHQUAKE GROUND MOTIONS

Introduced in this paper are several alternative ground-motion intensity measures (IMs) that are intended for use in assessing the seismic performance of a structure at a site susceptible to near-source and/or ordinary ground motions. A comparison of such IMs is facilitated by defining the “efficiency” and “sufficiency” of an IM, both of which are criteria necessary for ensuring the accuracy of the structural performance assessment. The efficiency and sufficiency of each alternative IM, which are quantified via (i) nonlinear dynamic analyses of the structure under a suite of earthquake records and (ii) linear regression analysis, are demonstrated for the drift response of three different moderate- to long-period buildings subjected to suites of ordinary and of near-source earthquake records. One of the alternative IMs in particular is found to be relatively efficient and sufficient for the range of buildings considered and for both the near-source and ordinary ground motions.

APPLIED INCREMENTAL DYNAMIC ANALYSIS

We are presenting a practical and detailed example of how to perform incremental dynamic analysis (IDA), interpret the results and apply them to performance-based earthquake engineering. IDA is an emerging analysis method that offers thorough seismic demand and capacity prediction capability by using a series of nonlinear dynamic analyses under a multiply scaled suite of ground motion records. Realization of its opportunities requires several steps and the use of innovative techniques at each one of them. Using a nine-story steel moment-resisting frame with fracturing connections as a test bed, the reader is guided through each step of IDA: (1) choosing suitable ground motion intensity measures and representative damage measures, (2) using appropriate algorithms to select the record scaling, (3) employing proper interpolation and (4) summarization techniques for multiple records to estimate the probability distribution of the structural demand given the seismic intensity, and (5) defining limit-states, such as the dynamic global system instability, to calculate the corresponding capacities. Finally, (6) the results can be used to gain intuition for the structural behavior, highlighting the connection between the static pushover (SPO) and the dynamic response, or (7) they can be integrated with conventional probabilistic seismic hazard analysis (PSHA) to estimate mean annual frequencies of limit-state exceedance. Building upon this detailed example based on the nine-story structure, a complete commentary is provided, discussing the choices that are available to the user, and showing their implications for each step of the IDA.

Record Selection for Nonlinear Seismic Analysis of Structures

This study addresses the question of selection and amplitude scaling of accelerograms for predicting the nonlinear seismic response of structures. Despite the current practices of record selection according to a specific magnitude-distance scenario and scaling to a common level, neither aspect of this process has received significant research attention to ascertain the benefits or effects of these practices on the conclusions. This paper hypothesizes that neither these usual principal seismological characteristics nor scaling of records matters to the nonlinear response of structures. It then investigates under what conditions this hypothesis may not be sustainable. Two classes of records sets are compared in several case studies: one class is carefully chosen to represent a specific magnitude and distance scenario, the other is chosen randomly from a large catalog. Results of time-history analyses are formally compared by a simple statistical hypothesis test to assess the difference, if any, between nonlinear demands of the two classes of records. The effect of the degree of scaling (by first-mode spectral acceleration level) is investigated in the same way. Results here show (1) little evidence to support the need for a careful site-specific process of record selection by magnitude and distance, and (2) that concern over scenario-to-scenario record scaling, at least within the limits tested, may not be justified.

Correlation of Response Spectral Values for Multicomponent Ground Motions

Ground-motion prediction (attenuation) models predict the probability distributions of spectral acceleration values for a specified earthquake event. These models provide only marginal distributions, however; they do not specify correlations among spectral accelerations with differing periods or orientations. In this article a large number of strong ground motions are used to empirically estimate these cor- relations, and nonlinear regression is used to develop approximate analytical equa- tions for their evaluation. Because the correlations apply to residuals from a ground- motion prediction, they are in principle dependent on the ground-motion prediction model used. The observed correlations do not vary significantly when the underlying model is changed, however, suggesting that the predictions are applicable regardless of the model chosen by the analyst. The analytical correlation predictions improve upon previous predictions of correlations at differing periods in a randomly oriented horizontal ground-motion component. For correlations within a vertical ground mo- tion or across orthogonal components of a ground motion, these results are believed to be the first of their kind. The resulting correlation coefficient predictions are useful for a range of problems related to seismic hazard and the response of structures. Past uses of previous cor- relation predictions are described, and future applications of the new predictions are proposed. These applications will allow analysts to better understand the properties of single- and multicomponent earthquake ground motions.

Which Spectral Acceleration Are You Using

Analysis of the seismic risk to a structure requires assessment of both the rate of occurrence of future earthquake ground motions (hazard) and the effect of these ground motions on the structure (response). These two pieces are often linked using an intensity measure such as spectral acceleration. However, earth scientists typically use the geometric mean of the spectral accelerations of the two horizontal components of ground motion as the intensity measure for hazard analysis, while structural engineers often use spectral acceleration of a single horizontal component as the intensity measure for response analysis. This inconsistency in definitions is typically not recognized when the two assessments are combined, resulting in unconservative conclusions about the seismic risk to the structure. The source and impact of the problem is examined in this paper, and several potential resolutions are proposed. This discussion is directly applicable to probabilistic analyses, but also has implications for deterministic seismic evaluations.

Nonlinear Soil-Site Effects in Probabilistic Seismic-Hazard Analysis

This study presents effective probabilistic procedures for evaluating ground-motion hazard at the free-field surface of a nonlinear soil deposit located at a specific site. Ground motion at the surface, or at any depth of interest within the soil formation (e.g., at the structure foundation level), is defined here in terms either of a suite of oscillator-frequency-dependent hazard curves for spectral acceleration, , or of one or more spectral acceleration uniform-hazard spectra, each associated s S ( f ) a with a given mean return period. It is presumed that similar information is available for the rock-outcrop input. The effects of uncertainty in soil properties are directly included. This methodology incorporates the amplification of the local soil deposit into the framework of probabilistic seismic hazard analysis (PSHA). The soil amplification is characterized by a frequency-dependent amplification function, AF( f ), where f is a generic oscillator frequency. AF( f ) is defined as the ratio of to the spectral s S ( f ) a acceleration at the bedrock level, . The estimates of the statistics of the ampli- s S ( f ) a fication function are obtained by a limited number of nonlinear dynamic analyses of the soil column with uncertain properties, as discussed in a companion article in this issue (Bazzurro and Cornell, 2004). The hazard at the soil surface (or at any desired depth) is computed by convolving the site-specific hazard curve at the bedrock level with the probability distribution of the amplification function. The approach presented here provides more precise surface ground-motion-hazard estimates than those found by means of standard attenuation laws for generic soil conditions. The use of generic ground-motion predictive equations may in fact lead to inaccurate results especially for soft-clay-soil sites, where considerable amplifi- cation is expected at long periods, and for saturated sandy sites, where high-intensity ground shaking may cause loss of shear strength owing to liquefaction or to cyclic mobility. Both such cases are considered in this article. In addition to the proposed procedure, two alternative, easier-to-implement but approximate techniques for obtaining hazard estimates at the soil surface are also briefly discussed. One is based on running a conventional PSHA with a rock- attenuation relationship modified to include the soil response, whereas the other consists of using a simple, analytical, closed-form solution that appropriately mod- ifies the hazard results at the rock level.

Ground-Motion Amplification in Nonlinear Soil Sites with Uncertain Properties

This work presents a statistical study on the effect of soil layers with uncertain properties on ground-motion intensity at the soil surface. Surface motion is obtained by applying multiple real rock earthquake records at the base of different characterizations of the soil column, each one generated via Monte Carlo simulation. The effect of the soil is studied in terms of a site-specific, frequency-dependent amplification function, AF( f ), where f is a generic oscillator frequency. The goal here is the identification of ground-motion parameters that allow an efficient prediction of AF( f ). We investigated magnitude, M, source-to-site distance, R, of the input bedrock accelerogram along with bedrock ground-motion parameters such as peak ground acceleration, PGAr, and spectral acceleration values, and , both rr S ( f ) S ( f ) aa sc at the generic frequency f and at the specific initial fundamental frequency of vibra- tion, fsc of the soil column. This work includes two case studies: a saturated sandy site and a saturated soft clayey site. In the former, loss of shear strength owing to cyclic mobility is anticipated for severe levels of ground shaking, while in the latter, significant amplification is expected at long oscillator periods. The results show that of the input record is the single most helpful parameter for the prediction of r S ( f ) a AF( f ) at the same oscillator frequency, f. is more informative than PGAr and/ r S ( f ) a or the pair of M and R values of the event that generated the bedrock motion. A sufficiently accurate estimate of the median AF( f ) can be obtained by using 10 or fewer records, which may be selected without undue attention to the specific scenario events (i.e., M and R pairs) that control the hazard at the site. Finally, the effect of the uncertainty in the soil parameters on the prediction error of AF( f ) is of secondary importance compared to that from record-to-record variability. These findings will be used to estimate the hazard at the soil surface in a companion article in this issue (Bazzurro and Cornell, 2004).

Explicit Directivity-Pulse Inclusion in Probabilistic Seismic Hazard Analysis

Probabilistic seismic hazard analysis (PSHA) is widely used to estimate the ground motion intensity that should be considered when assessing a structures performance. Disaggregation of PSHA is often used to identify representative ground motions in terms of magnitude and distance for structural analysis. Forward directivity–induced velocity pulses, which may occur in near-fault (or near-source) motions, are known to cause relatively severe elastic and inelastic response in structures of certain periods. Here, the principles of PSHA are extended to incorporate the possible occurrence of a velocity pulse in a near-fault ground motion. For each magnitude and site-source geometry, the probability of occurrence of a pulse is considered along with the probability distribution of the pulse period given that a pulse does occur. A near-source “narrowband” attenuation law modification to predict ground motion spectral acceleration ( Sa ) amplitude that takes advantage of this additional pulse period information is utilized. Further, disaggregation results provide the probability that a given level of ground motion intensity is caused by a pulse-like ground motion, as well as the conditional probability distribution of the pulse period associated with that ground motion. These extensions improve the accuracy of PSHA for sites located near faults, as well as provide a rational basis for selecting appropriate near-fault ground motions to be used in the dynamic analyses of a structure.

An Empirical Ground-Motion Attenuation Relation for Inelastic Spectral Displacement

This article presents an empirical ground-motion prediction model (attenuation relation) for inelastic (as opposed to elastic) spectral displacement ( Sdi ) for ground motions without forward directivity effects. It is a function of two earthquake parameters, moment magnitude ( M w) and the closest distance to rupture ( R rup), and two bilinear oscillator parameters, an undamped elastic period ( T ) and a yield displacement ( dy ). The dy is introduced via the predicted median strength-reduction factor (![Graphic][1] ), a proxy for the ratio of elastic spectral displacement ( Sde ) to dy , which is identical with the familiar strength-reduction factor ( R ). The proxy ![Graphic][2] recognizes that R can only be estimated indirectly because it implicitly contains the random variable, Sde , which cannot be known a priori ; therefore, the median estimate or predicted median ( Ŝde ) from a conventional (elastic) ground-motion prediction model is used instead to calculate ![Graphic][3] = Ŝde / dy . For enhanced generality, the inelastic spectral displacement prediction model here is based on a ratio concept, that is, the total model is a (any) conventional elastic prediction model coupled with a new inelastic displacement ratio prediction model, with proper statistical correlation between the two. We empirically consider the dependence of this ratio on source and path effects (i.e., M w and R rup), and find that M w is significant, but R rup is not. The resulting prediction model can easily be added to existing probabilistic seismic-hazard analysis (psha) software packages with only one extra structure-specific parameter, dy of the oscillator. In practical engineering applications, this will likely have been estimated from the conventional static pushover analysis of the multi-degree-of-freedom (mdof) structure under consideration. The resulting psha product is a hazard curve for Sdi , the inelastic spectral displacement of a nonlinear oscillator. Such a curve can provide a more direct hazard- based target displacement for nonlinear static procedures (Federal Emergency Management Agency [fema] 356, 2000) and/or a basic input function for new probabilistic seismic-demand analyses that is based on Sdi (as opposed to Sde ) as an efficient and sufficient intensity measure. This new attenuation relationship will be particularly useful in evaluating the performance of existing structures and specified designs with known lateral strength. In particular, unlike most past studies, it does not pre-fix the ductility level. [1]: /embed/inline-graphic-1.gif [2]: /embed/inline-graphic-2.gif [3]: /embed/inline-graphic-3.gif