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The use of nonlinear dynamic analysis provides significant uncertainties on the seismic demand, especially when recorded ground motions are used. As these uncertainties strongly depend on ground motion selection and modification (GMSM) methods, a spectrum-compatibility criterion and a method based on the minimization of the scaling factor are compared in this work. The variability of a representative engineering demand parameter (EDP), obtained by subjecting ten reinforced concrete structures to different groups of records, is investigated through a sensitivity study based on the "Tornado diagram analysis." The results show that the variability of the structural demand produced by the variation of the ground motion profile amplifies significantly with the increase in complexity and irregularity of the structures. More specifically, for regular structures, the selected GMSM criteria provide very similar variability while with the increase of irregularities, the spectrum-compatibility criterion produces a minimization of the demand uncertainty.

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... A number of scientific studies focusing on the influence of ground motion records on the structural response of buildings have been performed in the past [1][2][3][4][5]. Among the various findings, the structural response estimates were seen to be highly dependent on the type of ground motions, i.e. artificial, real or simulated, as well as on the methods utilized for selection and scaling of the records [6][7][8]. Current codes implicitly recognize the variability of the seismic response of buildings introduced by input ground-motions by setting a minimum number of records, which varies from code to code, to be selected and scaled with the aim of providing realistic estimates of mean seismic demands [9][10][11]. This uncertainty is usually biased when using code-based record selection methods which is because of the fact that the uncertainty in the ground motion intensity is not properly addressed when selection and scaling are performed to reduce the mismatch in relation to a target spectrum, and thus cannot be used to capture the dispersion in the structural responses [12,13]. ...

... It has been shown that the accuracy of the structural demand estimates is of crucial importance in ensuring reliable application of demand-based safety assessment procedures. Several possible sources of bias in the estimation of structural demands have already been identified in previous works [6,18], such as the scaling of ground motion records, the pairs of M-R considered in the record selection or even the record-to-record variability within a given group, although no consensus has yet been achieved on how these parameters may actually influence the estimation of both local and global deformation demands. These various issues will be discussed in the next sections of the paper with the aim of clarifying some open questions. ...

... Hence, it can be concluded that the consideration of the control of the mismatch at the record level may introduce bias in both local and global deformation demand estimates when using the ASCE scaling method. Similarly to what was observed by Cantagallo et al. [6], the influence of the mean scale factors on the variability of both local and global structural demands, estimated with analysis conducted with code-based scaled records, seems to be characterized by a coefficient of correlation of about 0.4 to 0.5 (Figure 10 (B)). ...

The recent concerns regarding the seismic safety of the existing building stock have highlighted the need for an improvement of current seismic assessment procedures. Alongside with the development of more advanced commercial software tools and computational capacities, nonlinear dynamic analysis is progressively becoming a common and preferable procedure in the seismic assessment of buildings. Besides the complexity associated with the formulation of the mathematical model, major issues arise related with the definition of the seismic action, which can lead to different levels of uncertainty in terms of local and global building response. Aiming to address this issue, a comparative study of different code-based record selection methods proposed by Eurocode 8, ASCE41-13 and NZS1170.5:2004 is presented herein. The various methods are employed in the seismic assessment of four steel buildings, designed according to different criteria, and the obtained results are compared and discussed. Special attention is devoted to the influence of the number of real ground motion records selected on the estimation of the mean seismic response and, importantly, to the efficiency that is achieved when an additional selection criteria, based on the control of the spectral mismatch of each individual record with respect to the reference response spectrum, is adopted. The sufficiency of the methods with respect to the pairs of M–R of the selected group of records and the robustness of the scaling procedure are also examined. The paper closes with a study which demonstrates the suitability of a simplified probability-based approach recently proposed for estimating mean seismic demands. Copyright © 2015 John Wiley & Sons, Ltd.

... While nearly all the current application of sensitivity analysis using Tornado diagram is limited to material and modeling RVs with some limited contributions from loading uncertainty, [7] developed Tornado diagram for 10 different 3D RC building portfolio using three ground motion selection and scaling techniques. They used a force-based fiber frame model using the commercial software Midas. ...

... Sensitivity of type and complexity of the RC buildings; adapted from[7] ...

Sensitivity analysis is a crucial step in computational mechanics and earthquake engineering. Sensitivity analysis of a model (either numerical or physical) aims at quantifying the relative importance of each input parameter, their potential interaction, and their effects on the model response. Sensitivity analysis has a long-term application in structural engineering more specifically on reinforced concrete RC structures. Many researchers benefited from the results of sensitivity analysis to reduce the uncertainty domain to those variables which are very important. This further helps in uncertainty quantification by accelerating the entire process. This state-of-the-art technical report provides a comprehensive review of classical sensitivity analysis techniques, followed by an in-depth review of all the related documents that implemented a full sensitivity analysis or partially adopted it for uncertainty quantification-related discussions. This review report on sensitivity analysis of reinforced concrete structures is a valuable contribution to the field of computational mechanics and earthquake engineering. This review highlights the importance of selecting an appropriate sensitivity analysis technique to achieve reliable results in structural analysis and design. By providing insights into the advantages and limitations of various sensitivity analysis techniques, this report can guide researchers, practitioners, and decision-makers in selecting the most appropriate technique for their specific applications. It is observed that the outcome of a sensitivity analysis depends heavily on the applied technique to perform the sensitivity assessment which eventually may cause a significant bias during the decision-making. This report paves the road for better selection of a sensitivity analysis technique in problems related to structural and earthquake engineering. The findings of this review have significant implications for improving the accuracy and reliability of structural analysis, ultimately leading to safer and more resilient structures.

... Many research has focused on the ground motion records' influence on the structural response of the buildings (Bommer and Acevedo 2004;Iervolino and Cornell 2005;Iervolino et al. 2008;Ay and Akkar 2012;Cantagallo et al. 2014). The estimation of structural responses is highly dependent on the method used in ground motion selection as well as on the type of ground motions (i.e., artificial, real or simulated). ...

... The estimation of structural responses is highly dependent on the method used in ground motion selection as well as on the type of ground motions (i.e., artificial, real or simulated). The record to record variability effect on structural response should be minimized by the record selection process because the design capacity should increase with higher uncertainties (Cantagallo et al. 2014). Therefore, it is essential to define a selection method that produces lower variation in structural demand. ...

Proper selection of ground motion records plays an important role in time history analysis of structural systems. Consistency of average response spectrum of a record set with the target spectrum is the only criterion, that normally is considered in code-based frameworks. However, a spectrum curve cannot represent all the characteristics of a seismic event. To improve the code-based ground motion selection method, the compatibility of three scalar frequency content intensity measures (Np, T0, and Tavg) is integrated in the proposed selection method of this paper. The harmony search (HS) optimization algorithm is utilized to find the best ground motion set from a database and their corresponding scale factors. In order to investigate the accuracy and efficiency of the proposed method on seismic demand estimation of structures, four steel moment resisting frames are used in the analysis (4, 6, 9 and 12-story). It is concluded that combinations of T0 and Tavg with the code-based method are effective for low-rise and mid-rise buildings, respectively, since these methods give reliable responses with less error dissipation.

... Cantagallo et al. assessed the demand sensitivity of ten 3D regular and irregular RC structures to GM scaling methods and showed that as structural irregularity increases, spectrum-compatible records maintain the same precision for the estimated demand. In addition, they showed that EDP variation is consistent with records' deviation from the target spectrum [144]. Huang et al. compared the four different scaling methods and concluded that distribution scaling (i.e., CS matching to median and dispersion of a target spectral ordinate distribution) produces unbiased estimates of mean response and reasonably (yet sometimes conservative) capture response dispersion. ...

Current fully probabilistic approaches to performance-based earthquake engineering describe structures' behavior under a wide range of seismic hazard levels. These approaches require a detailed representation of ground motion (GM) uncertainty at all considered hazard levels, yet different GM selection methods lead to different estimations of structural performance. This paper presents a holistic review of the current practices in GM representation and selection for structural demand analysis through a performance-based lens. The multidisciplinary nature of GM selection, ranging from earth science to engineering seismology and statistics, has created a preponderance of literature to find the best practice for probabilistic assessment of structures in terms of computational efficiency and statistical accuracy. Many of these studies focus individually on GM selection or structural analysis, and the relatively scarce review papers either focus on code-based GM selection or do not specifically address risk-based evaluations by overlooking the interaction between GM selection and structural analysis. This paper aims to aid researchers in selecting appropriate GMs as part of a statistically valid and robust probabilistic demand analysis without performing an exhaustive literature review. Discussion on the available computational tools and their trade-offs for risk-based assessment of single structures is provided. While the problem-specific nature of GM selection means that no pre-selected set of GM/IM is applicable to all cases, the comprehensive narrative of this paper is expected to aid analysts in reaching a more informed decision.

... In addition, recent studies showed that it is possible to obtain different code-compatible ground motion record sets by selecting and scaling from hundreds of ground motion records available in digital databases (Iervolino et al. 2008;Kayhan et al. 2011;Kayhan 2016). Hence, estimated seismic demands representing the structural responses to seismic excitation vary significantly and could be accepted as random variables that change according to code-compatible record sets used for nonlinear time history analyses (Cantagallo et al. 2014;Macedo and Castro 2017). Recently, various studies were carried out to investigate the efficiency of the selecting and scaling of ground motion records according to various seismic codes. ...

Over the past 20 years, significant socio-economic losses have been encountered in Turkey due to several moderate to large earthquakes. The studies published after the earthquakes concurringly emphasized that multistory reinforced concrete (RC) buildings, mostly 3–7 story ones, collapsed or were heavily damaged as a result of inadequate seismic performance. Global drift ratio demands are mostly used as a representative quantity for determining the behavior of structures when subjected to earthquakes. In this study, three representative mid-rise RC buildings are analyzed by nonlinear time history analysis using code-compatible real ground motion record sets and the calculated global drift ratio demands of these buildings are statistically evaluated. Ground motion record sets compatible with the design spectrum defined for local soil classes in the Turkish Earthquake Code (TEC-2007) are used for the analyses. In order to evaluate the effect of the number of ground motions on drift ratio demands, five different ground motion record sets with 7, 11 and 15 ground motion records are used separately for each local soil class. Results of this study indicate that (1) the dispersion of global drift ratio demands calculated for individual ground motion records in record sets is high, (2) local soil class has no significant effect on dispersion. However, dispersion increases in a direct proportion to the number of ground motion records in a record set, (3) the mean of global drift ratio demands calculated for different ground motion record sets may differ although they are compatible with the same design spectrum, (4) the mean of the drift demands obtained from different ground motion record sets compatible with a particular design spectrum can be accepted as simply random samples of the same population at 95% confidence level.

... Another controversy about the spectral matching is about the most significant range of periods to check. In fact, while usually the fundamental period characterizing the dynamic response of the structure is assumed to be the one corresponding to the first vibrational mode, alternative assumptions can be made [12], referring to the "nonlinear period" of the structure, defined as the one corresponding to the initial branch of the bilinear idealized capacity curve obtained from the non-linear static (pushover) analysis, according to Eurocode 8, which, in turn, depends on the distribution of lateral loads. ...

... Another controversy about the spectral matching is about the most significant range of periods to check. In fact, while usually the fundamental period characterizing the dynamic response of the structure is assumed to be the one corresponding to the first vibrational mode, alternative assumptions can be made [12], referring to the "nonlinear period" of the structure, defined as the one corresponding to the initial branch of the bilinear idealized capacity curve obtained from the non-linear static (pushover) analysis, according to Eurocode 8, which, in turn, depends on the distribution of lateral loads. ...

This work is aimed at investigating the variation of the seismic response of RC structures subjected to different ensembles of ground motions in the nonlinear dynamic analysis for the seismic assessment of existing RC buildings. At the current time indeed, all the main International Seismic Codes provide a soil classification which is based on the shear wave velocity, the soil morphology and the assumed distance from the fault source. Depending on the soil properties, a suitable elastic spectrum is provided, defined on the basis of average properties assumed for the soil. To perform a nonlinear dynamic analysis, an ensemble of ground motions compatible to the elastic spectrum must be selected. The ensemble can be made by ground motions compatible to the Code spectrum for the assumed soil-type, or, alternatively, by assuming the ground motions to be consistent with the bed rock Code spectrum and by filtering them according with the specific stratigraphy of the site soil. In this work a comparison among these different approaches, all compatible to the European (Eurocode 8, EC8) and Italian (NTC 2008) Code provisions, has been made on a case-study, i.e. a real irregular RC Italian building. The comparison has been made in terms of seismic response, i.e. by checking the maximum displacement and interstory drift induced by the assumed ensembles of ground motions.

The non-structural components play a vital role in industrial buildings. The past study indicates that the non-structural components damage loss surpass the structural damage loss. As a result, after an earthquake in critical, essential, and lifeline buildings, non-structural components that must be in operational conditions are essential. Industrial buildings accommodate a wide range of processes and provide workers with the necessary equipment for efficient industrial operations. The centre of mass and the centre of stiffness or rigidity in the building do not align because of the irregular distribution of floor mass, inducing torsion in the building. According to Indian Standard (IS 1893-part 4:2015), linear time history analysis with design basis earthquake is required to generate floor response spectra for use in non-structural component design. Thus, linear time history analysis was initially performed to generate floor response spectra, and then work was extended with nonlinear time history analysis. The primary goal of this present study is to focus on the impact of torsional irregularity on the floor response continuum with tri-directional earthquake time histories. Regular and torsional irregular buildings with G + 2 storeys are considered for this. The building location is considered in the highest seismic zone in India and positioned on medium soil. The study concluded that torsional irregularity in the structure resulted in floor response spectra in the two orthogonal horizontal and vertical directions with significant design forces for each direction of earthquake excitation. Thus, the structural designer must employ these three direction response spectra for precise and protective non-structural support design.

The need for industrial buildings has increased multifold, with a wide range in the irregular distribution of floor mass and stiffness. The uneven distribution of floor mass causes mass eccentricity in the floor, inducing torsion in the building. The aim of the present study is to highlight the influence of torsional irregularity due to mass eccentricity on the multi-directional floor response spectra. IS 1893 (Part 4): 2015 suggested time-history analysis of the design basis earthquake to develop floor response spectra, adopting linear analysis technique. Eleven sets of real orthogonal records were chosen to perform time-history analysis to obtain a mean response. The present study observed that the building’s with regular geometry and without stiffness variation in it, a slight mass eccentricity significantly increases floor response spectra. Moreover, actual industrial buildings have significant variations in mass, stiffness, and strength, resulting in further intensifying floor response spectra. Thus, the structural designer needs to use tri-directional floor response spectra for precise and protective non-structural component support design for appropriate response. IS 1893 (Part 1): 2016 suggests reduced stiffness for beam and column for earthquake analysis. Floor response spectra with reduced stiffness were also studied here, and the response was compared with non-linear analysis. Non-structural components lateral design forces as per floor response spectra approach and Indian Standard provisions are also included in the study.

Os terremotos são responsáveis pelo colapso e danificação de edificações e grandes perdas de vidas ao redor do mundo. O Brasil, apesar de localizado no interior de uma placa tectônica, possui um considerável histórico de atividade sísmica. Dentre os diversos tipos de edificações que podem ser afetadas por terremotos, aquelas que apresentam irregularidades como, por exemplo, vigas de transição, pé-direito duplo ou assimetrias têm se mostrado as mais vulneráveis. Funções de fragilidade se enquadram nas ferramentas estatísticas que vêm sido desenvolvidas nos últimos anos e amplamente utilizadas para avaliar o desempenho sísmico de estruturas, entretanto existe uma falta de estudos deste tipo no Brasil. Diante disso, este presente trabalho tem como objetivo desenvolver funções de fragilidade para estruturas de concreto armado com irregularidades estruturais e dimensionadas de acordo com as normas brasileiras de projeto. Inicialmente, tipologias estruturais com irregularidades são definidas e modelos numéricos de elementos finitos são gerados utilizando o software OpenSees. Análises dinâmicas não-lineares são então conduzidas nesses modelos para estimar a demanda sísmica, que é combinada com a capacidade estrutural para gerar as funções de fragilidade. Os resultados demonstraram o efeito prejudicial da presença de um maior pé-direito e também da distribuição irregular das alvenarias de vedações (pilotis) na fragilidade. Essas funções são posteriormente combinadas com curvas de ameaça sísmica disponíveis na literatura para estimar a probabilidade de falha das tipologias analisadas em um período de 50 anos para uma capital estadual localizada em uma região com atividade sísmica, a cidade de Natal no Rio Grande do Norte. Os resultados desta análise indicaram que os pórticos analisados apresentam probabilidade de colapso inadequada na cidade analisada, de acordo com critérios de aceitabilidade encontrados na literatura técnica. http://repositorio.unicamp.br/jspui/handle/REPOSIP/358300

Ground motion selection is arguably the most crucial step in performance-based seismic design and risk assessment, as it bears the highest level of uncertainty that needs to be propagated through the calculations. The presently typical approach to capture this uncertainty in predicted structural responses is to use a suite of ground motions that match a uniform-hazard response spectrum. This study examines the downstream effects of ground motion selection made through a variety of conditioning criteria based on one, two, or average of more-than-two intensity measures, and also with respect to different earthquake characteristics. Ductile and nonductile reinforced concrete moment frames with varying heights are used as model problems in detailed parametric studies. A risk-based approach is adopted to develop a point-of-comparison reference demand. Seismic responses obtained using different ground motion suites compiled using three different conditioning criteria are then compared against this reference to determine the extent to which the selected ground motion suites can capture various critical structural responses.

The seismic damage evaluated through Non Linear Time-History Analyses is significantly affected by the response quantity chosen to represent the seismic responses. Starting from the theory of tolerance regions, a generic upper limit of the seismic responses is proposed. The method is applied to a reinforced concrete structure subjected to different record combinations. For each considered damage index and record combination, the upper limit damage is compared with the average value suggested by seismic codes. The proposed method yields a higher seismic damage than the average response and an increase of the damage indices as the number of records decreases.

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.

Probabilistic seismic hazard analysis (PSHA) integrates over all potential earthquake occurrences and ground motions to estimate the mean frequency of exceedance of any given spectral acceleration at the site. For improved communication and insights, it is becoming common practice to display the relative contributions to that hazard from the range of values of magnitude, M, distance, R, and epsilon, ɛ, the number of standard deviations from the median ground motion as predicted by an attenuation equation.
The proposed disaggregation procedures, while conceptually similar, differ in several important points that are often not reported by the researchers and not appreciated by the users. We discuss here such issues, for example, definition of the probability distribution to be disaggregated, different disaggregation techniques, disaggregation of R versus ln R, and the effects of different binning strategies on the results. Misconception of these details may lead to unintended interpretations of the relative contributions to hazard.
Finally, we propose to improve the disaggregation process by displaying hazard contributions in terms of not R, but latitude, longitude, as well as M and ɛ. This permits a display directly on a typical map of the faults of the surrounding area and hence enables one to identify hazard-dominating scenario events and to associate them with one or more specific faults, rather than a given distance. This information makes it possible to account for other seismic source characteristics, such as rupture mechanism and near-source effects, during selection of scenario-based ground-motion time histories for structural analysis.

Among all the possible options to define the seismic input for structural analysis, natural recordings are emerging as the most attractive. Easily accessible waveform databases are available and evidence shows that only a relatively limited number of criteria has to be considered in selection and scaling to get an unbiased estimation of seismic demand. Like many codes worldwide, Eurocode 8 (EC8) allows the use of real ground-motion records for the seismic assessment of structures. The main condition to be satisfied by the chosen set is that the average elastic spectrum does not underestimate the code spectrum, with a 10% tolerance, in a broad range of periods depending on the structure's dynamic properties. The EC8 prescriptions seem to favour the use of spectrum-matching records, obtained either by simulation or manipulation of real records. The study presented herein investigates the European Strong-Motion Database with the purpose of assessing whether it is possible to find real accelerogram sets complying with the EC8 spectra, while accounting for additional constraints believed to matter in the seismic assessment of buildings, as suggested by the current best practice. Original (un-scaled) accelerogram sets matching EC8 criteria were found, for the case of one-component (P-type) and spatial sets (S-type), for the spectra anchored to the Italian peak acceleration values. The average spectra for these sets tend to be as close as possible to the code spectrum. Other sets, requiring scaling, have been found to match the non dimensional (country-independent) EC8 spectral shape. These sets have also the benefit of reducing, in respect to the un-scaled sets, the record-to-record variability of spectra. Combinations referring to soft soil, stiff soil, and rock are presented here and are available on the internet at http://www.reluis.it/

Data mining has been performed on the result,; of an aerodynamic design optimization of a two-stage-to-orbit reusable launch vehicle flyback-booster wing. Three data mining techniques were compared, including selforganizing map, functional analysis of variance, and the rough set theory. The optimization problem had four aerodynamic objective functions and 71 wing shape design variables. The hypothetical design database resulting from the optimization contained a total of 302 solutions which included 102 nondominated solutions. Consequently, the acquired knowledge of the design space consisted of general design characteristics, correlation between objective functions, and the effects of these design variables on the objective functions, for both nondominated as well as all solutions. The comparison also revealed the similarities and differences among the three data mining techniques used in this study. Even though all three techniques discovered detailed design knowledge and the results produced by the combination of all three methods compensated disadvantages of each method when applied individually, it was discovered that the self-organizing map produced the overall best results. Moreover, this study has also shown that the knowledge acquired from both nondominated solutions and from all solutions found was consistent despite the differences between the design spaces. Furthermore, it was shown that data mining is essential for visualizing results of an evolutionary multi-objective optimization problem and extracting useful design knowledge from these results.

This paper presents a method to compute consistent response sensitivities of force-based finite element models of structural frame systems to both material constitutive and discrete loading parameters. It has been shown that force-based frame elements are superior to classical displacement-based elements in the sense that they enable, at no significant additional costs, a drastic reduction in the number of elements required for a given level of accuracy in the computed response of the finite element model. This advantage of force-based elements is of even more interest in structural reliability analysis, which requires accurate and efficient computation of structural response and structural response sensitivities. This paper focuses on material non-linearities in the context of both static and dynamic response analysis. The formulation presented herein assumes the use of a general-purpose non-linear finite element analysis program based on the direct stiffness method. It is based on the general so-called direct differentiation method (DDM) for computing response sensitivities. The complete analytical formulation is presented at the element level and details are provided about its implementation in a general-purpose finite element analysis program. The new formulation and its implementation are validated through some application examples, in which analytical response sensitivities are compared with their counterparts obtained using forward finite difference (FFD) analysis. The force-based finite element methodology augmented with the developed procedure for analytical response sensitivity computation offers a powerful general tool for structural response sensitivity analysis. Copyright © 2004 John Wiley & Sons, Ltd.

Ground-motion prediction equations (GMPE) for horizontal peaks of acceleration and velocity, and for horizontal response spectral ordinates, have employed a variety of definitions for the horizontal component of motion based on different treatments of the two horizontal traces from each accelerogram. New definitions have also recently been introduced and some of these will be used in future GMPEs. When equations using different horizontal-component definitions are combined in a logic-tree framework for seismic-hazard analysis, adjustments need to be made to both the median values of the predicted ground-motion parameter and to the associated aleatory variability to achieve compatibility among the equations. Because there is additional aleatory variability in the empirical ratios between the median values for different components, this uncertainty also needs to be propagated into the transformed logarithmic standard deviation of the adjusted equations. This study provides ratios of both medians and standard deviations for all existing component definitions with respect to the geometric mean of the two horizontal components, which is currently the most widely used in prediction equations. The standard deviations on the ratios of the medians are also reported. This article also discusses the issue of the ratios of different horizontal component definitions in relation to the specification of seismic input for dynamic structural analyses, highlighting the importance of consistency between the component definition used to derive the elastic design-response spectrum and the way that biaxial dynamic loading input is prepared.

A new point of view for the parameter variation problem in linear multivariable systems is proposed. The output deviations due to parameter variations for an open-loop realization are related by a sensitivity matrix to the output deviations due to parameter variations for a closed-loop (feedback) realization. Using a time-domain integral of the square of the error as a performance index, frequency-domain criteria involving the sensitivity matrix are derived. The criteria are sufficient for insuring that the feedback realization is less affected by parameter variations than an open-loop realization having the same nominal transfer characteristic. Furthermore, the criteria are independent of the integration interval involved in the performance index. A numerical example shows how the criteria may be used in designing multivariable control systems.

This paper presents a fibre beam–column element for the non-linear static and dynamic analysis of reinforced concrete frames. It is assumed that plane sections remain plane and normal to the longitudinal axis. The effects of shear and bond-slip are, thus, presently neglected. The non-linear hysteretic behaviour of the element derives from the constitutive relations of concrete and reinforcing steel fibres into which each section is divided. The element formulation is flexibility-based and relies on force interpolation functions that strictly satisfy the equilibrium of bending moments and axial force along the element. Since the element does not make use of displacement interpolation functions, an iterative algorithm is needed for the determination of the resisting forces during the element state determination. The proposed algorithm is accurate and stable, even in the presence of strength loss, and is, thus, capable of tracing very well the highly non-linear behaviour of R/C members under cyclic load combinations of bending moment and axial force.

The use of Nonlinear Dynamic Analyses is coupled to the demands produced by earthquakes of different intensities. The uncertainties associated with the demands are highly dependent on the variable adopted as the intensity measure (IM). This generates the need to compare different IMs and in particular the dispersion of the demand measure in relation to each IM. For this purpose correlations between maximum inter-story drift demand of nine reinforced concrete Multiple Degree of Freedom tridimensional structures and a number of widely used ground motion intensity parameters have been investigated. To determine the optimum parameter to be used as the IM, two additional intensity parameters are analyzed, the spectral acceleration corresponding to the cracked and non-linear period of each structure, Sa(Tcrack) and Sa(T∗), respectively. The results showed that these two IMs have the best correlation with the deformation demand and their use produces a lower variability of structural response.

Sensitivity analysis, supported by computer hardware and software, can easily overwhelm an analyst or decision maker with data. However, this data can be organized in a readily understandable way using well-designed graphs. Two graphical techniques, spiderplots and tornado diagrams, are commonly used respectively by engineering economists and decision analysts. Their advantages are complementary. Management scientists should often use both to convey their results to,decision makers succinctly and clearly. The simpler tornado diagram can summarize the total impact of many independent variables. An individual spiderplot displays more information about a smaller number of variables. This includes the limits for each independent variable, the impact of each on the dependent outcome, and the amount of change required to reach a break-even point.

The uncertainty in the seismic demand of a structure (referred to as the engineering demand parameter, EDP) needs to be properly characterized in performance-based earthquake engineering. Uncertainties in the ground motion and in structural properties are responsible for EDP uncertainty. In this study, sensitivity of EDPs to major uncertain variables is investigated using the first-order second-moment method for a case study building. This method is shown to be simple and efficient for estimating the sensitivity of seismic demand. The EDP uncertainty induced by each uncertain variable is used to determine which variables are most significant. Results show that the uncertainties in ground motion are more significant for global EDPs, namely peak roof acceleration and displacement, and maximum inter-storey drift ratio, than those in structural properties. Uncertainty in the intensity measure (IM) of ground motion is the dominant variable for uncertainties in local EDPs such as the curvature demand at critical cross-sections. Conditional sensitivity of global and local EDPs given IM is also estimated. It is observed that the combined effect of uncertainties in structural properties is more significant than uncertainty in ground motion profile at lower IM levels, while the opposite is true at higher IM levels. Copyright © 2005 John Wiley & Sons, Ltd.

An orthogonal set of principal axes is defined for earthquake ground motions along which the component variances have maximum, minimum and intermediate values and the covariances equal zero. Corresponding axes are defined which yield maximum values for the covariances. The orthogonal transformations involved are identical in form to those used in the transformation of stress. Examination of real accelerograms reveals that the major principal axis points in the general direction of the epicentre and the minor principal axis is nearly vertical. It is concluded that artificially generated components of ground motion need not be correlated statistically provided they are directed along a set of principal axes.

The ‘strength’ of an earthquake ground motion is often quantified by an Intensity Measure (IM), such as peak ground acceleration or spectral acceleration at a given period. This IM is used to predict the response of a structure. In this paper an intensity measure consisting of two parameters, spectral acceleration and epsilon, is considered. The IM is termed a vector-valued IM, as opposed to the single parameter, or scalar, IMs that are traditionally used. Epsilon (defined as a measure of the difference between the spectral acceleration of a record and the mean of a ground motion prediction equation at the given period) is found to have significant ability to predict structural response. It is shown that epsilon is an indicator of spectral shape, explaining why it is related to structural response. By incorporating this vector-valued IM with a vector-valued ground motion hazard, we can predict the mean annual frequency of exceeding a given value of maximum interstory drift ratio, or other such response measure. It is shown that neglecting the effect of epsilon when computing this drift hazard curve leads to conservative estimates of the response of the structure. These observations should perhaps affect record selection in the future. Copyright © 2005 John Wiley & Sons, Ltd.

Sensitivity analysis — calculation of the rate of change of response variables with respect to design variables — is a critical component in the process of re-analysis for improvement of trial designs or in seeking an optimum design. This paper presents necessary theorems and provides details for numerical computation of sensitivity matrices for spatially discretized structural systems subjected to dynamic excitation. General results are presented for nonlinear (hysteretic) structures, and explicit numerical examples illustrate the methodology applied to multi-story shear frames whose force-displacement relationship is bilinear hysteretic.

Fiber reinforced polymer (FRP) lamina have been used widely in the last decade to enhance strength and deformation capacity of deficient reinforced concrete (RC) columns. Seismic assessment and retrofit of existing columns in buildings and bridge piers necessitate accurate prediction of the available deformation capacity. In this study, a new analytical model is proposed to represent potential plastic hinge regions of RC columns prior to and after FRP retrofit. A recently developed variable confined concrete representation is employed within the framework of fiber-discretized frame elements to model the compression zone of the FRP-confined region. Confinement distribution within this region is included through the use of a bond model, whereas the effect of lap splices are considered using an effective steel strain concept. Comparisons of analytical estimates with experimentally measured response show that the proposed model is capable of capturing essential features of the response such as strength degradation due to lap splice slippage, and failure due to FRP rupture. Furthermore, a detailed sensitivity study is conducted to determine the parameters whose uncertainty significantly affects the behavior. It is observed that, in estimating the response of existing deficient columns, parameters such as plastic hinge length, concrete strength and splice length are important sources of uncertainty. While for FRP-retrofitted columns, parameters such as jacket stiffness, dilatation strain at splice failure and yield strength of the reinforcing bars are more important sources of uncertainty.

A general finite element solution method for the dynamic response sensitivity of inelastic structures is developed. Employing a direct differentiation method, the gradient equation of motion is solved without iteration and by taking advantage of the available solution of the response. Special attention is given to sensitivities with respect to inelastic material parameters and detailed derivations are made for theJ2 plasticity model with a linear hardening rule. The method can be applied to any other inelastic material model that has an analytically defined yield function and flow rule. The formulation is easily incorporated in existing finite element codes. Numerical examples demonstrate the accuracy and efficiency of the method.

This paper examines the question of which sources of uncertainty most strongly affect the repair cost of a building in a future earthquake. Uncertainties examined here include spectral acceleration, ground-motion details, mass, damping, structural force-deformation behavior, building-component fragility, contractor costs, and the contractor's overhead and profit. We measure the variation (or swing) of the repair cost when each basic input variable except one is taken at its median value, and the remaining variable is taken at its 10th and at its 90th percentile. We perform this study using a 1960s highrise nonductile reinforced-concrete moment-frame building. Repair costs are estimated using the assembly-based vulnerability (ABV) method. We find that the top three contributors to uncertainty are assembly capacity (the structural response at which a component exceeds some damage state), shaking intensity (measured here in terms of damped elastic spectral acceleration, Sa), and details of the ground motion with a given Sa.