Response Surface with random factors for seismic fragility of reinforced concrete frames N. Buratti, B. Ferracuti, M. Savoia * DISTART-Structural Engineering, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy article info Article history: Received 25 February 2008 Received in revised form 20 April 2009 Accepted 24 June 2009 Available online 25 July 2009 Keywords: Seismic reliability Response Surface Fragility curves Practical reliability methods abstract Seismic fragility curves for reinforced concrete (RC) frame structures are evaluated considering the uncer- tainties in both structural parameters and seismic excitation. Response Surface (RS) models with random block effects are used to solve the problem in an approximate way with good computational efficiency. The RS models are calibrated through numerical data obtained by non-linear incremental dynamic anal- yses performed using different sets of ground-motions, strength distributions in frame elements, and val- ues of the random variables adopted to describe the uncertainties in the structural behaviour. The present work is mainly focused on the problem of obtaining a reasonable compromise between result soundness and computational effort. With reference to a three storey frame structure, a series of numerical tests is presented. Different simulation plans, defined following the theory of Design of Experiments (DOE), and simplified polynomial RS models are employed. The fragility curves obtained by different methods are compared, using the results from full Monte Carlo simulation as the reference solution. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction The assessment of structural reliability against earthquakes is presently one of the most interesting topics in seismic engineering, from both research and application point of view [1]. In seismic reliability analysis, the fragility functions of the structures, to- gether with the seismic hazard of the considered site, are the fun- damental information required for the risk assessment. Hence, a great research effort has been devoted to the development of effi- cient procedures for the evaluation of the fragility functions. Re- views of recent studies can be found in Pinto et al. [2] and Der Kiureghian [3]. The main difficulties when performing a complete structural reliability analysis depend on (i) the large number of random variables required to describe the uncertainties affecting the structural behaviour (ii) the complex modelling of the uncer- tainty related with the definition of the ground-motion. The fragility functions can be generated by using [4]: (i) obser- vation of damages caused by earthquakes, (ii) experimental data (usually at the component level), (iii) detailed analytical models, (iv) simplified analytical models, and (v) design information and engineering judgement. The different procedures listed above have different degrees of accuracy. In principle, data collected from actual earthquakes or from experimental tests provide for information with the highest confidence. They can be used to develop a fragility model for some categories of recurrent structures. In other cases, they can give use- ful data to verify the predictions of analytical models. Simplified or detailed analytical models may be used to predict the structural behaviour under earthquakes: analytical models can take random- ness and uncertainties in earthquake and structure into account. Since the mid-seventies, simplified analytical models (named ‘‘practical reliability methods”) specialized for seismic problems have been developed mainly with reference to nuclear and other important installations [2]. Recently, these approaches have been set in a clear theoretical framework and have been applied also for more ordinary types of constructions. Several applications of this kind of methods can be found in the literature, see for exam- ple: Hwang and Jaw [4]; Hwang and Huo [5]; Singhal and Kiremidj- ian [6], [7]; Shinozuka et al. [8]; Cornell et al. [9]; Rosowsky and Ellingwood [10]; Wen et al. [11]; Ramamoorthy et al. [12]. The accuracy of the approximate methods with respect to the rigorous formulations needs to be well understood in order for these methods to be used properly. Finally, the use of design information and engineering judge- ment is an economical way to obtain fragility data. However, the result may be very sensitive to the subjective judgement, espe- cially if it is based on the opinions of a few engineers [4]. In the present study, a detailed analytical model has been used for the reliability analysis of RC structures under seismic actions. In this case, due to the non-linear behaviour, the response of RC struc- tures subjected to severe ground-motions must be obtained by computationally demanding non-linear Finite Element analysis. Then, in order to reduce the computational effort, an approach 0167-4730/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.strusafe.2009.06.003 * Corresponding author. Tel.: +39 0512093254; fax: +39 0512093236. E-mail address: marco.savoia@unibo.it (M. Savoia). Structural Safety 32 (2010) 42–51 Contents lists available at ScienceDirect Structural Safety journal homepage: www.elsevier.com/locate/strusafe