A probabilistic approach for estimating the seismic response of elasto‐plastic SDOF systems

The seismic response of elasto‐plastic structures to both recorded and generated accelerograms is characterized by a large scattering of the results, even for accelerograms with similar peak ground acceleration values and frequency content. According to current code recommendations a design value of the seismic response of an elasto‐plastic structure can be computed as the mean of the responses to a certain number of spectrum‐fitting generated accelerograms. A more effective probabilistic approach is presented herein. It allows the analyst to calculate a design value of the seismic response characterized by a predefined non‐exceedance probability using a limited number of generated accelerograms. The results of the performed analyses are presented in diagrams that can be used for structural design applications. The applicability of the proposed method is demonstrated in the case of an elasto‐plastic structural system and the results are compared with those obtained applying current code recommendations. Copyright © 2005 John Wiley & Sons, Ltd.     

Seismic performance and new design procedure for chevron‐braced frames

The paper is concerned with the seismic design of steel‐braced frames in which the braces are configured in a chevron pattern. According to EuroCode 8 (EC8), the behaviour factor q, which allows for the trade‐off between the strength and ductility, is set at 2.5 for chevron‐braced frames, while 6.5 is assigned for most ductile steel moment‐resisting frames. Strength deterioration in post‐buckling regime varies with the brace's slenderness, but EC8 adopts a unique q value irrespective of the brace slenderness. The study focuses on reevaluation of the q value adequate for the seismic design of chevron‐braced frames. The present EC8 method for the calculation of brace strength supplies significantly different elastic stiffnesses and actual strengths for different values of brace slenderness. A new method to estimate the strength of a chevron brace pair is proposed, in which the yield strength (for the brace in tension) and the post‐buckling strength (for the brace in compression) are considered. The new method ensures an identical elastic stiffness and a similar strength regardless of the brace slenderness. The advantage of the proposed method over the conventional EC8 method is demonstrated for the capacity of the proposed method to control the maximum inter‐storey drift. The q values adequate for the chevron‐braced frames are examined in reference to the maximum inter‐storey drifts sustained by most ductile moment‐resisting frames. When the proposed method is employed for strength calculation, the q value of 3.5 is found to be reasonable. It is notable that the proposed method does not require larger cross‐sections for the braces compared to the cross‐sections required for the present EC8 method. Copyright © 2005 John Wiley & Sons, Ltd.     

Equivalent viscous damping for displacement-based seismic design of hysteretic damped braces for retrofitting framed buildings

A displacement-based design (DBD) procedure aiming to proportion hysteretic damped braces (HYDBs) in order to attain, for a specific level of seismic intensity, a designated performance level of a structure is proposed for the retrofitting of framed buildings. A key step for the reliability of the DBD procedure is the selection of the equivalent viscous damping in order to account for the energy dissipated by the damped braced frame. In this paper, expressions of the equivalent damping are obtained considering the energy dissipated by the HYDBs and the framed structure. To this end, dynamic analyses of an equivalent single degree of freedom system, whose response is idealized by a trilinear model, are carried out considering real accelerograms matching, on the average, Eurocode 8 (EC8) response spectrum for a medium subsoil class. Then, a three-storey reinforced concrete (r.c.) framed structure of a school building, designed in a medium-risk seismic region according to the Italian code in force in 1975, is supposed as retrofitted as if in a high-risk seismic region of the current seismic code (NTC08) by the insertion of HYDBs. Nonlinear static analyses are carried out to evaluate the vulnerability of the primary structure, characterized by the lack of interior girders along the floor slab direction, and to select optimal properties of the HYDBs. The effectiveness of the retrofitting solutions is checked referring to nonlinear dynamic analyses, considering artificially generated accelerograms whose response spectra match those adopted by NTC08 for the earthquake design levels corresponding to the serviceability and ultimate limit states.     

A unified approach for the design of high ductility steel frames with concentric braces in the framework of Eurocode 8

Eurocode 8 (EC8) stipulates design methods for frames with diagonal braces and for chevron braced frames, which differ as regards the numerical model adopted, the value of the behavior factor q and the estimation of the lateral strength provided by braces. Instead, in this paper, the use of the same design method is suggested for both types of concentrically braced frames. The design method is a generalization of the one proposed for chevron braced frames in a previous study. A numerical investigation is conducted to assess the reliability of this design method. A set of concentrically braced frames is designed according to the EC8 and proposed design methods. The seismic response of these frames is determined by nonlinear dynamic analysis. Finally, it is demonstrated that the proposed design method is equivalent to those provided by EC8, because it can ensure the same level of structural safety which would be expected when using EC8. Copyright © 2013 John Wiley & Sons, Ltd.     

Shaking table test and numerical simulation of a 1/2‐scale self‐centering reinforced concrete frame

Self‐centering reinforced concrete frames are developed as an alternative of traditional seismic force‐resisting systems with better seismic performance and re‐centering capability. This paper presents an experimental and computational study on the seismic performance of self‐centering reinforced concrete frames. A 1/2‐scale model of a two‐story self‐centering reinforced concrete frame model was designed and tested on the shaking table in State Key Laboratory of Disaster Reduction in Civil Engineering at Tongji University to evaluate the seismic behavior of the structure. A structural analysis model, including detailed modeling of beam–column joints, column–base joints, and prestressed tendons, was constructed in the nonlinear dynamic modeling software OpenSEES. Agreements between test results and numerical solutions indicate that the designed reinforced concrete frame has satisfactory seismic performance and self‐centering capacity subjected to earthquakes; the self‐centering structures can undergo large rocking with minor residual displacement after the earthquake excitations; the proposed analysis procedure can be applied in simulating the seismic performance of self‐centering reinforced concrete frames. To achieve a more comprehensive evaluation on the performance of self‐centering structures, research on energy dissipation devices in the system is expected. Copyright © 2015 John Wiley & Sons, Ltd.     

Alternative closed‐form solutions for the mean rate of exceedance of structural limit states

Two new closed‐form expressions representing the mean rate of exceedance of a given limit state are presented herein. These proposals overcome limitations that were identified with the original formulation of the well‐known SAC/FEMA approach. The new expressions involve new parametric functions for the modeling of the seismic hazard data and for the demand evolution for increasing values of the earthquake intensity measure. Given the carefully selected parametric form of these functions, mathematical tractability is able to be maintained to establish two new closed‐form solutions representing the mean rate of exceedance of a given limit state. The function proposed for the hazard exhibits nonlinear behavior in log‐log space and is able to represent the actual hazard data over a wider range of earthquake intensity levels. The function proposed for the demand evolution addresses issues related to the inadequate performance of the SAC/FEMA approach when force‐based demand parameters such as the shear force are considered. To illustrate the applicability of the new closed‐form solutions, the probability of occurrence of several limit states is determined for a reinforced concrete structure. Copyright © 2013 John Wiley & Sons, Ltd.     

Influence of the spatial distribution of energy-dissipating bracing elements on the seismic response of multistorey frames

Several reinforced concrete frames with different dissipator distributions, and a conventional moment-resisting frame, are compared in order to select the best dissipator distribution from the point of view of seismic response and structural design. The structures with dissipators are designed according to a criterion proposed in the present paper. Each frame is excited with a set of eleven simulated accelerograms. The choice of the best dissipation distribution is based mainly on the differences between the mean of the maximum overturning moments developed at the base of the frames and between the weights of steel reinforcement and concrete resulting from the structural design of each frame. A comparison of initial construction costs of a building with dissipators and a conventional building shows that the former is 3·5 per cent more expensive.     

Seismic loss estimation of non‐ductile reinforced concrete buildings

Non‐ductile reinforced concrete buildings represent a prevalent construction type found in many parts of the world. Due to the seismic vulnerability of such buildings, in areas of high seismic activity non‐ductile reinforced concrete buildings pose a significant threat to the safety of the occupants and damage to such structures can result in large financial losses. This paper introduces advanced analytical models that can be used to simulate the nonlinear dynamic response of these structural systems, including collapse. The state‐of‐the‐art loss simulation procedure developed for new buildings is extended to estimate the expected losses of existing non‐ductile concrete buildings considering their vulnerability to collapse. Three criteria for collapse, namely first component failure, side‐sway collapse, and gravity‐load collapse, are considered in determining the probability of collapse and the assessment of financial losses. A detailed example is presented using a seven‐story non‐ductile reinforced concrete frame building located in the Los Angeles, California. Copyright © 2012 John Wiley & Sons, Ltd.     

Improved risk‐targeted performance‐based seismic design of reinforced concrete frame structures

This paper presents a procedure for seismic design of reinforced concrete structures, in which performance objectives are formulated in terms of maximum accepted mean annual frequency (MAF) of exceedance, for multiple limit states. The procedure is explicitly probabilistic and uses Cornell's like closed‐form equations for the MAFs. A gradient‐based constrained optimization technique is used for obtaining values of structural design variables (members' section size and reinforcement) satisfying multiple objectives in terms of risk levels. The method is practically feasible even for real‐sized structures thanks to the adoption of adaptive equivalent linear models where element‐by‐element stiffness reduction is performed (2 linear analyses per intensity level). General geometric and capacity design constraints are duly accounted for. The procedure is applied to a 15‐storey plane frame building, and validation is conducted against results in terms of drift profiles and MAF of exceedance, obtained by multiple‐stripe analysis with records selected to match conditional spectra. Results show that the method is suitable for performance‐based seismic design of RC structures with explicit targets in terms of desired risk levels.     

Optimum seismic design of reinforced concrete frames according to Eurocode 8 and fib Model Code 2010

Traditional seismic design, like the one adopted in Eurocode 8 (EC8), is force‐based and examining a single level of seismic action. In order to provide improved control of structural damage for different levels of seismic action, the new fib Model Code 2010 (MC2010) includes a fully fledged displacement‐based and performance‐based seismic design methodology. However, the level of complexity and computational effort of the MC2010 methodology is significantly increased. Hence, the use of automated optimization techniques for obtaining cost‐effective design solutions becomes appealing if not necessary. This study employs genetic algorithms to derive and compare optimum seismic design solutions of reinforced concrete frames according to EC8 and MC2010. This is important because MC2010 is meant to serve as a basis for future seismic design codes. It is found that MC2010 drives to more cost‐effective solutions than EC8 for regions of low seismicity and better or similar costs for regions of moderate seismicity. For high‐seismicity regions, MC2010 may yield similar or increased structural costs. This depends strongly on the provisions adopted for selecting the set of ground motions. In all cases, MC2010 provides enhanced control of structural damage. Copyright © 2016 John Wiley & Sons, Ltd.     

Approximate seismic risk assessment of building structures with explicit consideration of uncertainties

An approximate seismic risk assessment procedure for building structures, which involves pushover analysis that is performed utilizing a deterministic structural model and uncertainty analysis at the level of the equivalent SDOF model, is introduced. Such an approach is computationally significantly less demanding in comparison with procedures based on uncertainty analysis at the level of the entire structure, but still allows for explicit consideration of the effect of record‐to‐record variability and modelling uncertainties. A new feature of the proposed pushover‐based method is the so‐called probabilistic SDOF model. Herein, the proposed methodology is illustrated only for reinforced concrete (RC) frames, although it could be implemented in the case of any building structure, provided that an appropriate probabilistic SDOF model is available. An extensive parametric analysis has been performed within the scope of this study in order to develop a probabilistic SDOF model, which could be used for the seismic risk assessment of both code‐conforming and old, that is, non code‐conforming RC frames. Based on the results of risk analysis for the four selected examples, it is shown that the proposed procedure can provide conservative estimates of seismic risk with reasonable accuracy, in spite of the employed simplifications and the relatively small number of Monte Carlo simulations with Latin hypercube sampling, which are performed at the level of the SDOF model. An indication of the possible default values of dispersion measures for limit‐state intensities in the case of low to medium‐height RC frames is also presented. Copyright © 2014 John Wiley & Sons, Ltd.     

Identification of critical ground motions for seismic performance assessment of structures

A method is established to identify critical earthquake ground motions that are to be used in physical testing or subsequent advanced computational studies to enable seismic performance to be assessed. The ground motion identification procedure consists of: choosing a suitable suite of ground motions and an appropriate intensity measure; selecting a computational tool and modelling the structure accordingly; performing Incremental Dynamic Analysis on a non‐linear model of the structure; interpreting these results into 50th (median) and 90th percentile performance bounds; and identifying the critical ground motions that are close to these defining probabilistic curves at ground motion intensities corresponding to the design basis earthquake and the maximum considered earthquake. An illustrative example of the procedure is given for a reinforced concrete highway bridge pier designed to New Zealand specifications. Pseudodynamic tests and finite element based time history analyses are performed on the pier using three earthquake ground motions identified as: (i) a Design Basis Earthquake (10% probability in 50 years) with 90 percent confidence of non‐exceedance; (ii) a Maximum Considered Event (2% probability in 50 years) representing a median response; and (iii) a Maximum Considered Event representing 90 percent confidence of non‐exceedance. Copyright © 2006 John Wiley & Sons, Ltd.     

Seismic vulnerability evaluation of RC moment frame buildings in moderate seismic zones

A multi‐level seismic vulnerability assessment of reinforced concrete moment frame buildings located in moderate seismic zones (0.25g) is performed on a set of ductile versions of low‐ to mid‐rise two‐dimensional moment frames. The study is illustrated through application to comparative trial designs of two (4‐ and 8‐story) buildings adopting both space‐ and perimeter‐framed approaches. All frames are dimensioned as per the emerging version of the seismic design code in Egypt. These new seismic provisions are in line with current European norms for seismic design of buildings. Code‐compliant designs (CCD), as well as a proposed modified code design relaxing design drift demands for the investigated buildings, are examined to test their effectiveness and reliability. Applying nonlinear inelastic incremental dynamic analyses, fragility curves (FC) for the frames are developed corresponding to various code‐specified performance levels. Code preset lower and upper bounds on design acceleration and drift, respectively, are also addressed along with their implications, if imposed, on the frames seismic performance and vulnerability. Annual spectral acceleration hazard curves for the case study frames are also generated. Estimates for mean annual frequency (MAF) of exceeding various performance levels are then computed through an integration process of the data resulting from the FC with the site hazard curves. The study demonstrates that the proposed design procedure relaxing design drift demands delivers more economic building designs relative to CCDs, yet without risking the global safety of the structure. The relaxed design technique suggested herein, even though scoring higher, as expected by intuition, MAF of exceeding various code‐limiting performance levels expressed in terms of interstory drift ratios, still guarantees a reasonably acceptable actual margin against violating code limits for such levels. Copyright © 2010 John Wiley & Sons, Ltd.     

Improved performance‐based seismic assessment of buildings by utilizing Bayesian statistics

This paper addresses two important issues of concern to practicing engineers and researchers alike in application of performance‐based seismic assessment (PBSA) methodology on buildings: (i) the number of ground motion records required to exercise PBSA—current practice (FEMA P‐58‐1) requires eleven or more pairs of motions for this purpose, and (ii) the time and effort associated with performing the number of nonlinear response history analyses required to exercise PBSA. We present a method for exercising of PBSA that employs classical linear modal analysis to develop a first estimate (i.e., a priori) of probability distribution of loss, followed by utilizing Bayesian statistics to update this estimate using estimates of loss obtained by utilizing a small number of nonlinear response history analyses of a detailed model of the building (i.e., posterior). The proposed technique is used to assess the distribution of monetary loss of two case studies, a 4‐story reinforced concrete moment‐resisting frame building and a 20‐story steel moment‐resisting frame building, both located in Los Angeles, for a ground motion hazard with 10% probability of exceedance in 50 years. The efficiency of the proposed PBSA method is demonstrated by showing the similarity between the distribution of monetary loss at each story of case study buildings obtained from the traditional/sophisticated PBSA methodology and the proposed PBSA method in this study. Copyright © 2015 John Wiley & Sons, Ltd.     

Risk‐based seismic design for collapse safety

Risk‐based seismic design, as introduced in this paper, involves the use of different types of analysis in order to satisfy a risk‐based performance objective with a reasonable utilization rate and sufficient reliability. Differentiation of the reliability of design can be achieved by defining different design algorithms depending on the importance of a structure. In general, the proposed design is iterative, where the adjustment of a structure during iterations is the most challenging task. Rather than using automated design algorithms, an attempt has been made to introduce three simple guidelines for adjusting reinforced concrete frames in order to increase their strength and deformation capacity. It is shown that an engineer can design a reinforced concrete frame in a few iterations, for example, by adjusting the structure on the basis of pushover analysis and checking the final design by means of nonlinear dynamic analysis. A possible variant of the risk‐based design algorithm for the collapse safety of reinforced concrete frame buildings is proposed, and its application is demonstrated by means of an example of an eight‐storey reinforced concrete building. Four iterations were required in order to achieve the risk‐based performance objective with a reasonable utilization rate. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonlinear dynamic response of r.c. framed structures subjected to near-fault ground motions

The nonlinear dynamic response of reinforced concrete (r.c.) framed buildings subjected to near-fault ground motions is studied to check the effectiveness of current code provisions with reference to study cases. Three-, six- and twelve-storey r.c. plane frames, representative of symmetric framed buildings, are designed according to the European seismic code (EC8), assuming medium and high ductility classes and stratigraphic profiles A (rock) and D (soft soil) in a high-risk seismic region. The nonlinear seismic analysis is performed using a step-by-step procedure; a bilinear model idealizes the behaviour of the r.c. frame members. Artificially generated motions (matching EC8 response spectra for subsoil classes A and D) and horizontal motions (recorded on rock- and soft soil-site at near-fault areas) are considered. The results indicate that near-fault ground motions may require a special consideration in the code, in particular when designing r.c. framed structures placed on a soft soil-site; particular attention should be paid to the design of the frame members of the lower storeys.     

Seismic design of RC structures: A critical assessment in the framework of multi‐objective optimization

The assessment of seismic design codes has been the subject of intensive research work in an effort to reveal weak points that originated from the limitations in predicting with acceptable precision the response of the structures under moderate or severe earthquakes. The objective of this work is to evaluate the European seismic design code, i.e. the Eurocode 8 (EC8), when used for the design of 3D reinforced concrete buildings, versus a performance‐based design (PBD) procedure, in the framework of a multi‐objective optimization concept. The initial construction cost and the maximum interstorey drift for the 10/50 hazard level are the two objectives considered for the formulation of the multi‐objective optimization problem. The solution of such optimization problems is represented by the Pareto front curve which is the geometric locus of all Pareto optimum solutions. Limit‐state fragility curves for selected designs, taken from the Pareto front curves of the EC8 and PBD formulations, are developed for assessing the two seismic design procedures. Through this comparison it was found that a linear analysis in conjunction with the behaviour factor q of EC8 cannot capture the nonlinear behaviour of an RC structure. Consequently the corrected EC8 Pareto front curve, using the nonlinear static procedure, differs significantly with regard to the corresponding Pareto front obtained according to EC8. Furthermore, similar designs, with respect to the initial construction cost, obtained through the EC8 and PBD formulations were found to exhibit different maximum interstorey drift and limit‐state fragility curves. Copyright © 2007 John Wiley & Sons, Ltd.     

Effective linear damping and stiffness coefficients of nonlinear systems for design spectrum based analysis

A stochastic approach for obtaining reliable estimates of the peak response of nonlinear systems to excitations specified via a design seismic spectrum is proposed. This is achieved in an efficient manner without resorting to numerical integration of the governing nonlinear equations of motion. First, a numerical scheme is utilized to derive a power spectrum which is compatible in a stochastic sense with a given design spectrum. This power spectrum is then treated as the excitation spectrum to determine effective damping and stiffness coefficients corresponding to an equivalent linear system (ELS) via a statistical linearization scheme. Further, the obtained coefficients are used in conjunction with the (linear) design spectrum to estimate the peak response of the original nonlinear systems. The cases of systems with piecewise linear stiffness nonlinearity, along with bilinear hysteretic systems are considered. The seismic severity is specified by the elastic design spectrum prescribed by the European aseismic code provisions (EC8). Monte Carlo simulations pertaining to an ensemble of nonstationary EC8 design spectrum compatible accelerograms are conducted to confirm that the average peak response of the nonlinear systems compare reasonably well with that of the ELS, within the known level of accuracy furnished by the statistical linearization method. In this manner, the proposed approach yields ELS which can replace the original nonlinear systems in carrying out computationally efficient analyses in the initial stages of the aseismic design of structures under severe seismic excitations specified in terms of a design spectrum.     

On the treatment of uncertainty in seismic vulnerability and portfolio risk assessment

In a related study developed by the authors, building fragility is represented by intensity‐specific distributions of damage exceedance probability of various damage states. The contribution of the latter has been demonstrated in the context of loss estimation of building portfolios, where it is shown that the proposed concept of conditional fragility functions provides the link between seismic intensity and the uncertainty in damage exceedance probabilities. In the present study, this methodology is extended to the definition of building vulnerability, whereby vulnerability functions are characterized by hazard‐consistent distributions of damage ratio per level of primary seismic intensity parameter—Sa(T₁). The latter is further included in a loss assessment framework, in which the impact of variability and spatial correlation of damage ratio in the probabilistic evaluation of seismic loss is accounted for, using test‐bed portfolios of 2, 5, and 8‐story precode reinforced concrete buildings located in the district of Lisbon, Portugal. This methodology is evaluated in comparison with current state‐of‐the‐art methods of vulnerability and loss calculation, highlighting the discrepancies that can arise in loss estimates when the variability and spatial distributions of damage ratio, influenced by ground motion properties other than the considered primary intensity measure, are not taken into account.     

Seismic fragility assessment of existing sub-standard low strength reinforced concrete structures

An analytical seismic fragility assessment framework is presented for the existing low strength reinforced concrete structures more common in the building stock of the developing countries.For realistic modelling of such substandard structures,low strength concrete stress-strain and bond-slip capacity models are included in calibrating material models.Key capacity parameters are generated stochastically to produce building population and cyclic pushover analysis is carried out to capture inelastic behaviour.Secant period values are evaluated corresponding to each displacement step on the capacity curves and used as seismic demand.A modified capacity demand diagram method is adopted for the degrading structures,which is further used to evaluate peak ground acceleration from back analysis considering each point on the capacity curve as performance point.For developing fragility curves,the mean values of peak ground acceleration are evaluated corresponding to each performance point on the series of capacity curves.A suitable probability distribution function is adopted for the secant period scatter at different mean peak ground acceleration values and probability of exceedance of limit states is evaluated.A suitable regression function is used for developing fragility curves and regression coefficients are proposed for different confidence levels.Fragility curves are presented for a low rise pre-seismic code reinforced concrete structure typical of developing countries.