iDesign
Enabling Seismic Design Decision-Making under Uncertainty
The primary objective is the development of a simple yet accurate method for performance-based design of structures in seismic areas. In essence, we seek to revolutionize the standard process that every professional structural engineer undertakes to design a structure subject to seismic forces. The reason is that recent earthquakes have shown that buildings reflecting current design approaches may reduce the rate of fatalities, but often result to staggering monetary losses and disruption of functionality. Thus, earthquakes can still financially cripple entire cities, or even countries.
To holistically quantify such effects, the concept of “seismic performance” is employed. This characterizes the behavior of a given structure under seismic loads. Ideally this is based on the use of metrics that are of immediate use to engineers, e.g., story forces and deformations, but also to stakeholders, such as monetary losses, human casualties and time-to-repair (or replace). At a simpler level, one may quantify damage by using simpler engineering metrics such as global ductility or maximum interstory drift. “Performance-based” earthquake engineering is concerned with tackling the dual problems of assessment and design. Assessment is the direct process of estimating the performance of a given (existing) structure. Design is the inverse problem, whereby a (new or rehabilitated) structure, its members and properties are sought to assure a desired performance under a given seismic hazard. As typically befitting such dualities, the direct path of assessment is by far the simpler of the two. Designing a structure to achieve a desired level of performance is an indirect process that is currently solvable only through arduous iterations: It is simply not applicable in practice. To make things worse, the considerable uncertainty inherent in earthquakes (i.e., when, where, how intense) and structures (what has been constructed versus what was designed on paper, what are the material properties, issues of corrosion, aging etc.) make this problem even more difficult.
To offer a practicable path for performance-based design, three important innovations have been introduced. First, a computationally efficient approach is proposed for rapidly establishing the effect of model uncertainties on the seismic performance (Figure 1). This allows the quantification of the consequences of uncertainties and their inclusion in subsequent analyses. Second, a simplified, yet accurate formula is offered for evaluating the seismic performance in terms of the mean annual frequency of damage occurring in the building. Thus, intuition is gained on the structural parameters that influence the seismic behavior of the building, while an analytical estimation of the building performance becomes possible. Finally, the concept of Yield Frequency Spectra (Figure 2) is developed, whereby an engineer can directly determine the required strength and stiffness of the structure given the seismic hazard at the building site and the owner’s requirements on how frequently it sustains low or hign levels of damage. Both analytical approximations and accurate numerical solutions are available, encoded in open-source software that can estimate Yield Frequency Spectra within seconds to provide a reliable design basis for any building and any user requirements.
Project Funding
EU Research Executive Agency
Marie Curie Actions, Continuing Integration Grant
FP7-PEOPLE-2011-CIG
Collaborators
D. Vamvatsikos, NTU Athens
A.K. Kazantzi, NTU Athens
M. Fragiadakis, NTU Athens
D. Giannopoulos, NTU Athens
M. Aschheim, Santa Clara University, USA
Time Period
Sep 2011 – Aug 2015
Relevant Publications
Journal Papers
Kazantzi A.K., Vamvatsikos D. (2015). Intensity measure selection for vulnerability studies of building classes. Earthquake Engineering and Structural Dynamics, 44(15): 2677-2694
Summary | The selection of a scalar Intensity Measure (IM) for performing analytical vulnerability (loss) assessment across a building class is addressed. We investigate the ability of several IM choices to downgrade the effect of seismological parameters (sufficiency) as well as reduce the record‐to‐record variability (efficiency) for both highrise and lowrise sets of ‘index’ buildings. These characteristics are explored in unprecedented detail, employing comparisons and statistical significance testing at given levels of local engineering demand parameters (story drift ratios and peak floor accelerations) that relate to losses, instead of global variables such as the maximum interstory drift. Thus, a detailed limit‐state‐specific view is offered for the suitability of different scalar IMs for loss assessment. As expected, typical single‐period spectral values are found to introduce unwanted bias at high levels of scaling, both for a single as well as a class of buildings. On the other hand, the geometric mean of the spectral acceleration values estimated at several periods between the class‐average second‐mode and an elongated class‐average first‐mode period offers a practical choice that significantly reduces the spectral‐shape bias without requiring the development of new ground motion prediction equations. Given that record selection remains a site‐ and building‐specific process, such an improved IM can help achieve reliable estimates for building portfolios, as well as single structures, at no additional cost. Copyright © 2015 John Wiley & Sons, Ltd.
Kazantzi A.K., Vamvatsikos D., Lignos D.G. (2014). Seismic performance of a steel moment-resisting frame subject to strength and ductility uncertainty. Engineering Structures, 78: 69-77
Abstract | The reliable estimation of the seismic performance of structures requires quantifying the aleatory and epistemic uncertainties of the system parameters. This is efficiently achieved for a case study of a four-story steel moment-resisting frame through several important advances. First, a state-of-the-art numerical model is formed with full spatial parameterization of its strength and plastic deformation properties. Empirical relationships derived from experimental data are used to model the cyclic behavior of steel sections using probabilistically distributed parameters that include intra- and inter-component correlation. Finally, incremental dynamic analysis and Monte Carlo simulation are employed to accurately assess the seismic performance of the model under the influence of uncertainties. Of interest is the extent to which model parameter uncertainties may trigger negative demand-capacity correlation in structural fragility evaluation, where, for example, a lower ductility capacity for a component may decrease the threshold for local failure while at the same time raising the local demand estimate from an uncertainty-aware model. With respect to the examined steel moment-resisting frame and considering three construction quality levels (i.e. very good, average, low) as per FEMA P-58, it is shown that, despite the good agreement of the evaluated structural demands obtained with and without consideration of the model parameter uncertainties for well-designed modern buildings, the potential demand-capacity correlation is likely to give rise to unconservative estimates of fragility for local damage-states, especially in cases where substandard quality control is exercised during construction.
Vamvatsikos D. (2014). Seismic Performance Uncertainty Estimation via IDA with Progressive Accelerogram-wise Latin Hypercube Sampling. ASCE Journal of Structural Engineering, 140(8): A4014015
Abstract | An algorithm is proposed for the rapid estimation of the influence of model parameter uncertainties on the seismic performance of structures using incremental dynamic analysis (IDA) and Monte Carlo simulation with Latin hypercube sampling. It builds upon existing methods that quantify the uncertainty for structural models with nondeterministic parameters by performing IDA with multiple ground motion records on each model realization out of a predetermined sample. However, their practical application is restricted due to (1) the inability to determine a priori the required number of samples and (2) the disproportionate increase of the number of analyses in realistic multiparameter models. To address these issues, two fundamental changes are incorporated. First, Latin hypercube sampling is applied progressively by starting with a small sample that is doubled successively until the desired accuracy is achieved. Second, parameter sampling is performed on a record-by-record basis rather than maintaining the same model over an entire record suite, thus expanding the model sample size without increasing the number of nonlinear dynamic analyses. Using strong-column and weak-column models of a steel moment-resisting frame, the algorithm is shown to possess excellent scalability, extending the original methodology to be applicable to large-scale models with hundreds of random variables.
Vamvatsikos D. (2014). Accurate application and second-order improvement of the SAC/FEMA probabilistic formats for seismic performance assessment. ASCE Journal of Structural Engineering, 140(2): 04013058
Abstract | The SEAONC/ATC/CUREE (SAC/FEMA) probabilistic framework is based on a closed-form expression to estimate analytically the value of the risk integral convolving seismic hazard and structural response. Despite its practicality, implementation has been hindered by reduced accuracy due to a number of approximations that are needed to achieve a desirable form, the most significant being the power-law fitting of the seismic hazard curve. To mitigate this problem, two approaches are given: (1) selecting an appropriately biased power-law fit, and (2) offering a novel closed-form expression involving a second-order approximation. Where blind application of the original format could involve errors in excess of 100% for the predicted mean annual frequency of limit-state (LS) exceedance, biased fitting reduces it to less than 50% for many practical cases, whereas the new closed-form expression consistently brings it below 10%. While other sources of error still remain, the robustness achieved opens new avenues of application for this popular format.
Vamvatsikos D. (2013). Derivation of new SAC/FEMA performance evaluation solutions with second-order hazard approximation. Earthquake Engineering and Structural Dynamics, 42(8): 1171-1188
Summary | A novel set of SAC/FEMA‐style closed‐form expressions is presented to accurately assess structural safety under seismic action. Such solutions allow the practical evaluation of the risk integral convolving seismic hazard and structural response by using a number of idealizations to achieve a simple analytical form. The most heavily criticized approximation of the SAC/FEMA formats is the first‐order power‐law fit of the hazard curve. It results to unacceptable errors whenever the curvature of the hazard function becomes significant. Adopting a second‐order fit, instead, allows capturing the hazard curvature at the cost of necessitating new analytic forms. The new set of equations is a complete replacement of the original, enabling (a) accurate estimation of the mean annual frequency of limit‐state exceedance and (b) safety checking for specified performance objectives in a code‐compatible format. More importantly, the flexibility of higher‐order fitting guarantees a wider‐range validity of the local hazard approximation. Thus, it enables the inversion of the formulas for practically estimating the allowable demand or the required capacity to fulfill any design objective. Copyright © 2012 John Wiley & Sons, Ltd.
Conference Papers
Giannopoulos D.G., Vamvatsikos D. (2015). Influence of rotated ground motion components on the response distribution of inelastic oscillators. Proceedings of the COMPDYN2015 Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Crete, Greece
Abstract | Recent research has shown that structures may experience quite different response when the components of any given ground motion record are rotated in the horizontal plane. On the other hand, significant differences are also associated with using different ground motion records. This paper examines whether it is preferable to use a small set of records, each rotated to multiple incidence angles, or a larger set with fewer (or no) rotations, in the case we wish to perform a fixed number of nonlinear dynamic analyses and we have little site information. To this purpose we collected a relatively large, non-pulsive set of records and applied it to an elastoplastic single-degree-of-freedom (SDOF) system. Finally, bootstrapping was performed to investigate the effect of the number of rotation angles studied versus the number of records on the statistics of response. The results indicate that, in all cases, and especially when a small number of nonlinear analyses is allocated, the effect of record-torecord variability clearly outweighs the incidence angle influence. Using a small set of records can lead to unreliable results by both inaccurate estimation of the central value and severe underestimation of the dispersion.
Kazantzi A.K., Vamvatsikos D. (2015). A next generation scalar intensity measure for analytical vulnerability studies. Proceedings of the COMPDYN2015 Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Crete, Greece
Abstract | To assess the seismic performance of a structural system within an analytical context, we need, among others, to specify a ground motion Intensity Measure (IM). The wary IM selection is undoubtedly an important step towards the successful implementation of a risk assessment, since insufficient and/or inefficient IMs can induce unwanted bias and variance in the vulnerability estimates. Supplementary issues related to practicality, necessitate the use of IMs for which ground motion prediction relationships exist, such as the elastic response spectral values (i.e. acceleration, velocity and displacement). Several past studies suggested as an improvement the use of IMs defined as the geometric mean of spectral acceleration values computed over a period range. The latter range may span between periods that are below, at or above the fundamental one. Some of these choices were proven to significantly improve both efficiency and sufficiency of the IM compared to more commonly used counterparts.
This study investigates the efficiency and sufficiency of a newly developed scalar IM that combines the geometric mean IM concept with the significant duration of the ground motions. Improving the geometric mean IMs via including the significant duration of the ground motions, was driven by recent findings suggesting there is a strong tie between the collapse capacity of a structure and the ground motion duration. Hence, the performance of the proposed next generation IM is addressed in detail by means of comparisons and statistical significance tests. The testing is performed at specific levels of local engineering demand parameters that are closely related to losses, using a testbed capacity-designed steel moment-resisting frame. It was demonstrated that ground motion duration is closely related to the collapse capacity whereas its effect at lower demand levels is insignificant. Hence, the proposed IM may be employed to improve the estimates in collapse assessment studies. Nevertheless, at least for steel moment-resisting frame buildings that exhibit moderate cyclic degradation rates and sustain most losses prior to the global collapse state, the significant duration is anticipated to only minimally affect the evaluated vulnerability and consequently may be disregarded.
Vamvatsikos D., Katsanos E.I., Aschheim M.A. (2015). A case study in performance-based design using yield frequency spectra. Proceedings of the SECED 2015 Conference, Cambridge, UK
Abstract | An analytical formulation is offered to allow performance-based seismic design to be achieved following a direct code-compatible procedure. The approach builds upon the use of the yield displacement as a robust system characteristic. A new format for displaying seismic demands known as Yield Frequency Spectra is introduced to quantitatively link performance objectives with the base shear seismic coefficient for a fixed value of yield displacement. Analytical expressions allow estimating the design base shear strength to satisfy any number of performance requirements, foregoing the need for a behaviour factor. The effect of uncertainties is naturally introduced to inject the proper conservatism for, e.g., the natural randomness in the ground motion or lack of knowledge in modelling and analysis. Finally, an 8-story reinforced concrete frame is designed, showing that EN1998 may not achieve the stated performance targets, while the proposed approach can match them with a single iteration.
Vamvatsikos D. (2015). A view of seismic robustness based on uncertainty. Proceedings of the 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12, Vancouver, Canada
Abstract | A simplified view of robustness and redundancy is presented for the seismic assessment and design of structures. It is argued that the topological simplicity of the seismic load, i.e., its highly correlated nature of application for practically every component in all but the ultra-long structures, means that simpler formulations can be devised compared to blast, wave or wind hazards. In that sense, robustness may be considered to express the influence of structural uncertainties on the seismic performance of the structure, in essence showing the available margin of safety subject to material variability. Quantitatively, a pertinent robustness/redundancy index is defined as the ratio of two different estimates of the mean annual frequency (MAF) of exceeding a limit-state of interest, such as global collapse: On the denominator lies the MAF estimate that incorporates all sources of variability while on the nominator is the estimate that neglects structural uncertainties. It is shown that a simple closed form solution is available that directly relates robustness to the dispersion of response due to model parameter uncertainty. As an example, a steel frame where only beams are allowed to yield is shown to be more robust compared to another version where columns become the sacrificial element.
Vamvatsikos D., Aschheim M.A. (2014). Direct performance-based seismic design of structures using Yield Frequency Spectra. Proceedings of the 10th U.S. National Conference on Earthquake Engineering, Anchorage, AK, USA
Abstract | Yield Frequency Spectra (YFS) are employed to enable the direct design of a structure subject to a set of performance objectives. YFS offer a unique view of the entire solution space for structural performance. This is measured in terms of the mean annual frequency (MAF) of exceeding arbitrary ductility (or displacement) thresholds, versus the base shear strength of a structural system with given yield displacement and backbone capacity curve. Using publicly available software tools or closed-form solutions, YFS can be nearly instantaneously computed for any system whose performance is characterized by response quantities that can be satisfactorily approximated by an equivalent nonlinear single-degree-of-freedom oscillator. Thus, stated performance objectives can be directly related to the strength and stiffness of the structure. The combination of ductility (or displacement) demand and its mean annual frequency of exceedance that governs the design is readily determined, allowing a satisfactory design to be realized in a single step.
Vamvatsikos D., Aschheim M.A., Kazantzi A.K. (2014). Direct performance-based seismic design: Avant-garde and code-compatible approaches. Proceedings of the 9th European Conference on Structural Dynamics (EURODYN 2014), Porto, Portugal
Abstract | Current force-based seismic design codes use design spectra and system-specific behavior factors to satisfy two pre-defined structural limit-states: Serviceability and Life Safety. Instead, performance-based seismic design aims to design a structure to fulfill any number of target performance objectives, defined as user-prescribed levels of structural response, loss or casualties to be exceeded at a maximum rate less than a given mean annual frequency (MAF). Even at its simplest structural response basis, the inverse probabilistic nature of the requirements has not yet allowed a satisfying solution without cumbersome cycles of re-design and re-analysis. An alternative approach is proposed, relying on a new format for visualizing seismic performance, termed Yield Frequency Spectra (YFS). YFS offer a unique view of the entire solution space for structural performance, as indexed by the MAF of exceeding arbitrary ductility (or displacement) thresholds, versus the base shear strength of a structural system with given yield displacement and backbone capacity curve. YFS can be instantly computed for any system that is satisfactorily approximated by a single-degree-of-freedom oscillator, as in any nonlinear static procedure. Thus, stated performance objectives are directly related to the strength and stiffness of the structure while fully incorporating aleatory and epistemic sources of uncertainty, as needed to achieve any required level of confidence. The combination of ductility (or displacement) demand and its exceedance MAF is readily determined, allowing a satisfactory initial design to be realized in a single step. Using a simple safety factor approach given the period and the hazard curve slope, the benefits can also apply to contemporary seismic codes, essentially introducing true performance-based capabilities in a traditional format.
Giaralis A., Vamvatsikos D. (2014). Local wavelet-based spectral “epsilon” modification of ground motions in support of incremental dynamic analysis. Proceedings of the 2nd International Conference on Vulnerability and Risk Analysis and Management (ICVRAM2014), Liverpool, UK
Abstract | A novel scaling algorithm for ground motion accelerograms (GMs) is proposed in support of incremental dynamic analysis used to establish dependable statistical relationships between scalable intensity measures (IMs) and engineering demand parameters (EDPs) within a performance based earthquake engineering framework. Specifically, an iterative harmonic wavelet based scheme is employed to accomplish “surgical” changes to the spectral shape of suites of GMs to span various pre-defined levels of the spectral acceleration at the structural fundamental natural period, the most widely adopted IM. Since the target IM values may not be accomplished precisely by the local spectral modifications, a second step involving global uniform GM scaling is further considered. The proposed algorithm requires significantly smaller global (amplitude) scaling factors compared to the currently used scaling approach in which no initial local GM modification is undertaken. A numerical application of the proposed algorithm to elastoplastic structural systems shows the extent to which spectral shape may influence the displacement response of yielding structures and explains the conservative bias introduced by uniform global scaling.
Vamvatsikos D., Aschheim M.A. (2014). A code-compatible application of yield frequency spectra for direct performance-based design. Proceedings of the 2nd European Conference on Earthquake Engineering and Seismology (2ECEES), Istanbul, Turkey
Abstract | An analytical formulation is offered for performance-based seismic design following a direct procedure based on code-like requirements. The approach builds upon the use of the yield displacement as a robust system characteristic. A new format for displaying seismic demands known as Yield Frequency Spectra is introduced to quantitatively link performance objectives with the base shear seismic coefficient, for a fixed value of yield displacement. The closed-form solutions developed in the SAC/FEMA project are inverted to provide analytical solutions for estimating the design base shear strength implied by typical code requirements, involving both strength and displacement limitation checks. It is shown that the consideration of aleatory and epistemic sources of uncertainty result in an increase of 30% or more in the strength required to achieve the stated performance objectives, depending on the site and structure characteristics. While the blanket safety factors embodied in the seismic code may cover this difference, they do so inconsistently. Instead, the simple expressions derived can offer uniform safety at no additional complexity.
Vamvatsikos D., Aschheim M.A., Kazantzi A.K. (2013). Direct performance-based seismic design using yield frequency spectra. Proceedings of the Vienna Congress on Recent Advances in Earthquake Engineering and Structural Dynamics (VEESD 2013), Vienna, Austria
Abstract | Yield Frequency Spectra (YFS) are employed to enable the direct design of a structure subject to a set of performance objectives. YFS offer a unique view of the entire solution space for structural performance. This is measured in terms of the mean annual frequency (MAF) of exceeding arbitrary ductility (or displacement) thresholds, versus the base shear strength of a structural system with given yield displacement and backbone capacity curve. Using publicly available software tools or closed-form solutions, YFS can be nearly instantaneously computed for any system that can be satisfactorily approximated by a single-degree-of-freedom oscillator, as in any nonlinear static procedure application. Thus, stated performance objectives can be directly related to the strength and stiffness of the structure. The combination of ductility (or displacement) demand and its mean annual frequency of exceedance that governs the design is readily determined, allowing a satisfactory design to be realized in a single step.
Kazantzi A.K., Vamvatsikos D., Lignos D.G. (2013). Model parameter uncertainty effects on the seismic performance of a 4-story steel moment-resisting frame. Proceedings of the 10th International Conference on Structural Safety and Reliability (ICOSSAR), New York
Abstract | The reliable estimation of the seismic performance of structures requires quantifying the effect of aleatory and epistemic uncertainties of the system parameters. This is efficiently achieved for a case study of a steel moment-resisting frame through several important advances. First, a state-of-the-art numerical model is formed with full parameterization of its strength properties. Empirical relationships derived from experimental data are used to model the cyclic behavior of steel sections using probabilistically distributed parameters that include intra- and inter-member correlation. Incremental dynamic analysis is employed to accurately assess the seismic performance of the model. Finally, an efficient Monte Carlo simulation algorithm is used, based on record-wise incremental Latin Hypercube Sampling to propagate the uncertainties from the numerous parameters to the actual system demand and capacity. Consequently, this modern steel frame is shown to be only mildly sensitive to parameter uncertainties when quality is tightly controlled.
Vamvatsikos D. (2012). Accurate application and higher-order solutions of the SAC/FEMA probabilistic format for performance assessment. Proceedings of the 15th World Conference on Earthquake Engineering, Lisbon, Portugal
Summary | The SAC/FEMA probabilistic framework is based on a pioneering closed-form expression to analytically estimate the value of the risk integral convolving seismic hazard and structural response. Despite its immense practicality, implementation has been hindered by reduced accuracy due to a number of approximations needed to achieve a simple form, the most significant being the power-law approximation of the seismic hazard curve. To mitigate this problem, two approaches are hereby offered, namely (a) selecting an appropriately-biased power-law fit and (b) offering a novel closed-form expression involving a higher order approximation. Where blind application of the original format could involve error in excess of 100% for the predicted mean annual frequency of limit-state exceedance, biased fitting reduces it to less than 25% while for the new closed-form it remains consistently below 10%. Although other sources of error still remain, the robustness achieved opens new avenues of application for this popular format.
Kazantzi A.K., Vamvatsikos D. (2012). A study on the correlation between dissipated hysteretic energy and seismic performance. Proceedings of the 15th World Conference on Earthquake Engineering, Lisbon, Portugal
Abstract | The hysteretic energy dissipated by systems undergoing quasi-static or dynamic loading is often thought to represent a useful measure of their performance when subjected to earthquake excitation. In general, fuller hysteresis loops mean higher seismic energy removal from the structure, which is logically taken to imply better performance when comparing systems with similar strength. However, such observations are typically based on quasi-static loading tests. Dynamic loading conditions differ as energy input and energy dissipation are intimately related with the details of the system’s hysteresis, in ways that often defy current intuition. Using incremental dynamic analysis on story-level oscillators with varying hysteresis characteristics, we can map this connection in detail. Structural response, as measured in terms of maximum or residual deformation, is shown to have little connection to the energy absorption. Therefore, hysteretic energy dissipation cannot quantitatively measure seismic performance but perhaps only serve as a general indicator.
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