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TAIL EQUIVALENT LINEARIZATION METHODS FOR SEISMIC RESPONSE SPECTRUM ANALYSIS
... where a g ðtÞ denotes the base acceleration modeled as follows [48][49][50]: ...
Structural Reliability Analysis in 8 the Presence of Random and 9 Interval Variables 10 This paper presents a novel procedure based on first-order reliability method (FORM) 11 for structural reliability analysis in the presence of random parameters and interval 12 uncertain parameters. In the proposed formulation, the hybrid problem is reduced to 13 standard reliability problems, where the limit state functions are defined only in terms 14 of the random variables. Monte Carlo simulation (MCS) for hybrid reliability analysis 15 (HRA) is presented, and it is shown that it requires a tremendous computational effort; 16 FORM for HRA is more efficient but still demanding. The computational cost is signifi-17 cantly reduced through a simplified procedure, which gives good approximations of 18 the design points, by requiring only three classical FORMs and one interval analysis 19 (IA), developed herein through an optimization procedure. FORM for HRA and its 20 simplified formulation achieve a much improved efficiency than MCS by several orders 21 of magnitude, and it can thus be applied to real-world engineering problems. 22 Representative examples of stochastic dynamic analysis and performance-based engineer-23 ing are presented.
In this paper, a novel method to determine the distribution of a random variable from a sample of data is presented. The approach is called Generalized Kernel Density Maximum Entropy Method (GKDMEM), because it adopts a Kernel Density (KD) representation of the target distribution, while its free parameters are determined through the principle of Maximum Entropy (ME). Here, the ME solution is determined by assuming that the available information is represented from generalized moments, which include as their subsets the power and the fractional ones. The proposed method has several important features: (i) applicable to distributions with any kind of support, (ii) computational efficiency because the ME solution is simply obtained as a set of systems of linear equations, (iii) good trade-off between bias and variance, and (iv) good estimates of the tails of the distribution, in presence of samples of small size. Moreover, the joint application of GKDME with a bootstrap resampling allows to define credible bounds of the target distribution. The method is first benchmarked through an example of stochastic dynamic analysis. Subsequently, it is used to evaluate the seismic fragility functions of a reinforced concrete frame, from the knowledge of a small set of available ground motions.
Three methods of stochastic equivalent linearizations defined in the broad framework of structural reliability analysis are presented. These methods are (1) the Gaussian equivalent linearization method (GELM), here defined for the first time as a linear response surface in terms of normal standard random variables; (2) the tail equivalent linearization method (TELM), here reinterpreted as a stochastic critical excitation method; and (3) a novel equivalent linearization called the tail probability equivalent linearization method (TPELM). The
Gaussian equivalent linear system (GELS) is the equivalent linear system (ELS) obtained by minimizing the difference between the variance of the GELS and the original nonlinear system. The tail equivalent linear system (TELS) is the ELS having the same critical excitation as the original system. The tail probability equivalent linear system (TPELS) is the ELS obtained by minimizing the difference between the tail probability of the equivalent system and the original nonlinear system. The knowledge of the ELS allows the evaluation of engineering
quantities of interest—e.g., first-passage probabilities—through the application of the random vibration analysis to these systems. Shortcomings and advantages of the three methods are presented and illustrated through applications to selected representative nonlinear oscillators. Finally, the methods are applied to an inelastic multi-degree-of-freedom (MDOF) system, showing their scalability to systems of higher complexity.
The usefulness of passive energy dissipation devices to reduce seismic response of structures is now well established. In this paper, a method is presented to obtain the amount and the placement of viscous or visco-elastic damping necessary for having a suitable reduction of the response quantities. It is based on the minimization of the Expected value of the stochastic Dissipated Power (EDP) by the structure. The optimal placement of the dampers is obtained through the solution of a simple optimization problem. The method gives also values of the total capacity damping and the upper bound values of damping to each story. At first, the theory of the EDP is illustrated through simple applications to SDOF and 2DOF systems; then, the effectiveness of the method is shown through its application to a ten-story shear-type building, subjected to stochastic ground motion model, with different values of soil profile.
This paper addresses the study of the seismic response of structures to response-spectrum-compatible accelerograms. The number of methodologies proposed in the last three decades to simulate artificial earthquake ground motion testifies the relevance of this subject in the scientific community. However, the implications of the selection of models and hypothesis adopted and their impact on the structural response have not been thoroughly highlighted yet. This contribution shows for the first time that different ground motion models, having identical response spectrum at 5% damping, peak ground acceleration, strong motion phase and total duration, can lead to significant discrepancies in the structural responses even for proportionally damped linear behaving structures, although all the models are satisfying the response-spectrum compatibility criteria. The results show clearly the weakness in the current response-spectrum-compatible criteria provided by seismic codes and the necessity of more robust conditions for the simulation of artificial earthquake ground motions.
An approximate method for nonstationary solution of nonlinear systems under random excitation is presented. The nonlinearities in the restoring force are polynomial type. The excitation is either shot noise or filtered shot noise. The solution is based on a Markov-vector approach. The unsteady Fokker-Planck-Kolmogorov equation for the nonlinear system is solved approximately by a Galerkin method where a time-dependent Hermite-series expansion is used and the equation is reduced to a system of first-order ordinary differential equations. The method is illustrated by numerical examples on a damped Duffing oscillator and a Ramsberg-Osgood yielding system. Comparison of results obtained herein with exact stationary solution and Monte Carlo results indicates that the proposed method is powerful and efficient for study of nonlinear systems.
A performance-based earthquake engineering (PBEE) methodology was developed at the Pacific Earthquake Engineering Research (PEER) Center. The method is based on explicit determination of performance, e.g., monetary losses, in a probabilistic manner where uncertainties in earthquake ground motion, structural response, damage, and losses are explicitly considered. There is an increasing trend towards use of probabilistic performance-based design (PPBD) methods in practice. Therefore, the International Federation for Structural Concrete (fib) initiated a task group to disseminate PPBD methods. This article is a contribution to this task group summarizing and demonstrating the PEER PBEE methodology in a useful manner to practicing engineers.
The subject of this paper is the simulation of one-dimensional, uni-variate, stationary, Gaussian stochastic processes using the spectral representation method. Following this methodology, sample functions of the stochastic process can be generated with great computational efficiency using a cosine series formula. These sample functions accurately reflect the prescribed probabilistic characteristics of the stochastic process when the number N of the terms in the cosine series is large. The ensemble-averaged power spectral density or autocorrelation function approaches the corresponding target function as the sample size increases. In addition, the generated sample functions possess ergodic characteristics in the sense that the temporally-averaged mean value and the autocorrelation function are identical with the corresponding targets, when the averaging takes place over the fundamental period of the cosine series. The most important property of the simulated stochastic process is that it is asymptotically Gaussian as N -> oo. Another attractive feature of the method is that the cosine series formula can be numerically computed efficiently using the Fast Fourier Transform technique. The main area of application of this method is the Monte Carlo solution of stochastic problems in engineering mechanics and structural engineering. Specifically, the method has been applied to problems involving random loading (random vibration theory) and random material and geometric properties (response, variability due to system stochasticity).
Performance-based earthquake engineering aims to quantify performance of facilities using metrics that are of immediate use to both engineers and stakeholders. A rigorous yet practical implementation of a performance-based earthquake engineering methodology is developed and demonstrated for an idealized building. The methodology considers seismic hazard, structural response, resulting damage, and repair costs associated with restoring the building to its original condition, using a fully consistent, probabilistic analysis of the associated parts of the problem. The methodology can be generalized to consider other performance measures such as casualties and down time, though these have not been pursued. The proposed procedure is consistent with common building design, construction, and analysis practices such that it can be readily adopted in earthquake engineering practice today.
The well-known modal superposition method for the evaluation of seismic response by the complete quadratic modal combination rule (CQC) is revisited. The most widely used version of the CQC rule utilizes correlation coefficients derived for white-noise excitation and neglects the influence of peak factor variation on the response. Here a simplified procedure for evaluation of correlation coefficients and peak factors consistent with the power spectral density of seismic excitation is proposed. The procedure is based on an approximate analytic expression for direct evaluation of the power spectral density of the excitation consistent with any prefixed response spectrum, and the evaluation of the consistent correlation coefficients and peak factors by using analytical expressions. The ranges of system dynamic parameters for which the correlation coefficients derived for white-noise excitation are not adequate are pointed out. Then the influence of the assumptions for CQC rule derivation on the evaluation of nodal response parameters is investigated, and the role played by the correlation coefficients and peak factors is pointed out.
An extension of the Tail-Equivalent Linearization Method (TELM) to the frequency domain is presented. The extension defines the Tail-Equivalent Linear System in terms of its frequency-response function. This function is obtained by matching the design point of the nonlinear response with that of the linearized response, thus guaranteeing the equivalence of the tail probability of the latter and the first-order approximation of the tail probability of the nonlinear response. The proposed approach is particularly suitable when the input and response processes are stationary, as is usually the case in the analysis of marine structures. When linear waves are considered, the Tail-Equivalent Linear System possesses a number of important properties, such as the capability to account for multi-support excitations and invariance with respect to scaling of the excitation. The latter property significantly enhances the computational efficiency of TELM for analysis with variable sea states. Additionally, the frequency-response function of the Tail-Equivalent Linear System offers insights into the geometry of random vibrations discretized in the frequency domain and into the physical nature of the response process. The proposed approach is applied to the analysis of point-in-time and first-passage statistics of the random sway displacement of a simplified jack-up rig model.
The geometry of random vibration problems in the space of standard normal random variables obtained from discretization of the input process is investigated. For linear systems subjected to Gaussian excitation, the problems of interest are characterized by simple geometric forms, such as vectors, planes, half spaces, wedges and ellipsoids. For non-Gaussian responses, the problems of interest are generally characterized by non-linear geometric forms. Approximate solutions for such problems are obtained by use of the first- and second-order reliability methods (FORM and SORM). This article offers a new outlook to random vibration problems and an approximate method for their solution. Examples involving response to non-Gaussian excitation and out-crossing of a vector process from a non-linear domain are used to demonstrate the approach.
SUMMARY A state-of-the-art seismic performance assessment is illustrated through application to a reinforced- concrete moment-frame building designed per current (2003) building code provisions. Performance is quantified in terms of economic losses and collapse safety. The assessment includes site-specific seismic hazard analyses, nonlinear dynamic structural response simulations to collapse, damage analyses, and loss estimation. When selecting ground motion records for nonlinear dynamic analyses that are consistent with a target hazard level expressed in terms of a response spectral value at the building's fundamental period, it is important to consider the response spectral shape, especially when considering higher hazard levels. This was done through the parameter commonly denoted by . Neglecting these effects during record selection is shown to lead to a factor of 5-10 overestimation of mean annual collapse rate. Structural response simulations, which properly account for uncertainties in ground motions and structural modelling, indicate a 2-7% probability of collapse for buildings subjected to motions scaled to a hazard level equivalent to a 2% probability of exceedance in 50 years. The probabilities of component damage and the means and coefficients of variation of the repair costs are calculated using fragility functions and repair-cost probability distributions. The calculated expected annual losses for various building design variants range from 0.6 to 1.1% of the replacement value, where the smaller losses are for above-code design variants and the larger losses are for buildings designed with minimum-code compliance. Sensitivity studies highlight the impact of key modelling assumptions on the accurate calculation of damage and the associated repair costs. Copyright q 2007 John Wiley & Sons, Ltd.
A gradient-free method is developed for finding the design point in nonlinear stochastic dynamic analysis, where the input excitation is discretized into a large number of random variables. This point defines the realization of the excitation that is most likely to give rise to a specific response threshold at a given time. The design point is the essential information in the recently developed tail-equivalent linearization method. The proposed approach employs a variant of the model correction factor method developed by O. Ditlevsen, which is further improved by the use of a novel response surface technique. Example applications to single- and multi-degree-of-freedom hysteretic systems demonstrate the efficiency and accuracy of the method.
A new approach for computing seismic fragility curves for nonlinear structures for use in performance-based earthquake engineering analysis is proposed. The approach makes use of a recently developed method for nonlinear stochastic dynamic analysis by tail-equivalent linearization. The ground motion is modeled as a discretized stochastic process with a set of parameters that characterizes its evolutionary intensity and frequency content. For each selected response (seismic demand) threshold, a linear system is defined that has the same tail probability as the nonlinear response in first-order approximation. Simple linear random vibration analysis with the tail-equivalent linear system then yields the fragility curve. The approach has the advantage of avoiding the selection and scaling of recorded accelerograms and repeated time-history analyses, which is the current practice for developing fragility curves. Copyright © 2009 John Wiley & Sons, Ltd.
A method is presented for efficiently computing small failure probabilities encountered in seismic risk problems involving
dynamic analysis. It is based on a procedure recently developed by the writers called Subset Simulation in which the central idea is that
a small failure probability can be expressed as a product of larger conditional failure probabilities, thereby turning the problem of
simulating a rare failure event into several problems that involve the conditional simulation of more frequent events. Markov chain Monte
Carlo simulation is used to efficiently generate the conditional samples, which is otherwise a nontrivial task. The original version of
Subset Simulation is improved by allowing greater flexibility for incorporating prior information about the reliability problem so as to
increase the efficiency of the method. The method is an effective simulation procedure for seismic performance assessment of structures
in the context of modern performance-based design. This application is illustrated by considering the failure of linear and nonlinear
hysteretic structures subjected to uncertain earthquake ground motions. Failure analysis is also carried out using the Markov chain samples
generated during Subset Simulation to yield information about the probable scenarios that may occur when the structure fails.
Fluid damper devices in civil structures such as buildings or bridges are commonly used as energy absorbers for seismic protection. The problem in the response analysis of structures with filled dampers mainly consists in the fact that, due to the strongly nonlinear behavior of such equipments, the response spectrum (RS) technique fails. In this paper by using the concept of power spectral density function compatible with the elastic RS and the statistical linearization technique, the equivalent damping ratio is obtained. It is shown that once the system is linearized, results obtained by Monte Carlo simulation and those obtained by stochastic analysis are in good agreement. Copyright © 2008 John Wiley & Sons, Ltd.
A new, non-parametric linearization method for nonlinear random vibration analysis is developed. The method employs a discrete representation of the stochastic excitation and concepts from the first-order reliability method, FORM. For a specified response threshold of the nonlinear system, the equivalent linear system is defined by matching the “design points” of the linear and nonlinear responses in the space of the standard normal random variables obtained from the discretization of the excitation. Due to this definition, the tail probability of the linear system is equal to the first-order approximation of the tail probability of the nonlinear system, this property motivating the name Tail-Equivalent Linearization Method (TELM). It is shown that the equivalent linear system is uniquely determined in terms of its impulse response function in a non-parametric form from the knowledge of the design point. The paper examines the influences of various parameters on the tail-equivalent linear system, presents an algorithm for finding the needed sequence of design points, and describes methods for determining various statistics of the nonlinear response, such as the probability distribution, the mean level-crossing rate and the first-passage probability. Applications to single- and multi-degree-of-freedom, non-degrading hysteretic systems illustrate various features of the method, and comparisons with results obtained by Monte Carlo simulations and by the conventional equivalent linearization method (ELM) demonstrate the superior accuracy of TELM over ELM, particularly for high response thresholds.
An essential step in FORM, SORM and importance sampling reliability methods is the determination of the so-called design point. This point is the solution of a constrained optimization problem in the outcome space of the random variables, which is commonly solved by an iterative, gradient-based search algorithm. In solving this problem in the context of non-linear finite element reliability analysis, two serious impediments are encountered: (a) for certain material models, the constraint function may have a discontinuous gradient, leading to failure of the search algorithm to converge. (b) The search algorithm may generate trial points too far in the failure domain, where the finite element code fails to produce a result due to lack of numerical convergence. In this paper, remedying strategies are developed for both impediments. The first impediment is addressed by using smooth or smoothed material models, including a smoothed bi-linear model, a Bouc–Wen model and a generalized plasticity model. This is complemented by a proof that sudden elastic unloading does not give rise to gradient discontinuities. The second impediment is addressed by modifying or introducing search algorithms that prevent the trial points from overshooting into the failure domain. Numerical examples are used to demonstrate the two impediments and effectiveness of the proposed remedies.
Efficient methods are presented for digital simulation of a general homogeneous process (multidimensional or multivariate or multivariate-multidimensional) as a series of cosine functions with weighted amplitudes, almost evenly spaced frequencies, and random phase angles. The approach is also extended to the simulation of a general non-homogeneous oscillatory process characterized by an evolutionary power spectrum. Generalized forces involved in the modal analysis of linear or non-linear structures can be efficiently simulated as a multivariate process using the cross-spectral density matrix computed from the spectral density function of the multidimensional excitation process. Possible applications include simulation of (i) wind-induced ocean wave elevation, (ii) spatial random variation of material properties, (iii) the fluctuating part of atmospheric wind velocities and (iv) random surface roughness of highways and airport runways.
During the past two decades, probabilistic risk analysis tools have been applied to assess the performance of new and existing building structural systems. Structural design and evaluation of buildings and other facilities with regard to their ability to withstand the effects of earthquakes requires special considerations that are not normally a part of such evaluations for other occupancy, service and environmental loads. This paper reviews some of these special considerations, specifically as they pertain to probability-based codified design and reliability-based condition assessment of existing buildings. Difficulties experienced in implementing probability-based limit states design criteria for earthquake are summarized. Comparisons of predicted and observed building damage highlight the limitations of using current deterministic approaches for post-earthquake building condition assessment. The importance of inherent randomness and modeling uncertainty in forecasting building performance is examined through a building fragility assessment of a steel frame with welded connections that was damaged during the Northridge Earthquake of 1994. The prospects for future improvements in earthquake-resistant design procedures based on a more rational probability-based treatment of uncertainty are examined.
Open system for earthquake engineering simulation
- F Mckenna
- G L Fenves
- M H Scott
F. Mckenna, G.L. Fenves, M.H. Scott, Open system for earthquake engineering simulation. Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA, 2003, http://opensees.berkeley.edu