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On the Reliability of Steel Frames Exposed to Snow Load: Considering the Effect of Epistemic Uncertainty

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Extreme snow falls are often leading to severe damage and even collapse of structures, causing significant economic losses and claiming lives; e.g. recently 2005/06 Central Europe, 2010/11 North-Eastern USA. The causes are diverse, but some studies
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  NOTE:  The reference period for snow model and reliability analyses in the study is 1 year. It is not stated in the main text, and Fig.6 displays the 50year target reliability: 3.8, which should be 4.7 for 1 year. 2014.09.29. Árpád Rózsás  EUROSTEEL 2014, September 10-12, 2014, Naples, Italy ON THE RELIABILITY OF STEEL FRAMES EXPOSED TO SNOW LOAD Considering the Effect of Epistemic Uncertainty Árpád Rózsás, László Gergely Vigh Budapest University of Technology and Economics, Deptartment of Structural Engineering, Hungary rozsas.bme@gmail.com, geri@vbt.bme.hu KEYWORDS: snow, steel frame, firefly optimization, Bayesian, reliability, Eurocode. ABSTRACT This paper examines the reliability level of the Eurocode’s partial factor design (PFD) method for steel frames subjected to snow load. This is motivated by the frequent damage and collapse of structures under heavy snowfall [1, 2]. We present a general approach which can help to comprehensively investigate the compliance of current Eurocode partial factor provisions and to assess whether the collapses most frequently observed at steel structures can be attributed to the deficiency of standardized provisions. To alleviate the limitations of previous studies, the general framework is intended to efficiently perform parametric reliability analysis using state-of-the-art numerical methods, has the potential to identify the crucial parameters, the sufficient modelling level and to justify the assumptions made in previous studies. Additionally the effect of epistemic uncertainties through Bayesian statistical analysis is also taken into account. The main components and workflow of the proposed algorithm are illustrated in Fig. 1 . Optimization  partial factor design StatisticalanalysisObservationsReliabilityanalysisFEM structuralanalysis  ANSYS  MATLAB MATLAB + FERUM  + (SUMO) MATLAB R + OPENBUGS    Fig. 1 . General framework of the reliability assessment The presented approach is illustrated through a simple steel frame, which is analysed considering various load-ratios. First, actual snow measurements are evaluated using classical, maximum likelihood based and Bayesian methods. Lognormal, Generalized Extreme Value (GEV) and Gumbel distributions are considered. The results of the statistical analyses are summarized in Table 1 . Table 1.  Ground snow loads for 20, 50 and 100 years return periods [kN/m 2 ] Maximum likelihood (MLE) Bayesian posterior (BP) Bayesian predictive posterior (BPP) Gumbel GEV Lognormal Gumbel GEV Lognormal Gumbel GEV Lognormal q  20  0.880 0.950 1.252 0.905 1.018 1.287 0.915 1.023 1.364 q  50  1.065 1.213 1.760 1.097 1.319 1.822 1.126 1.355 1.979 q  100  1.203 1.429 2.209 1.243 1.571 2.297 1.259 1.665 2.562   Using these results partial factor based optimization of the frames is performed. The firefly metaheuristic algorithm is chosen to carry out the optimization [3]. This optimization is used to get fully utilized structures per PFD which is indispensable for reliability assessment of the method. Cross-sectional level limit state function is considered taking into account the global stability of the frame approximately. The objective function is the overall weight of the structure; the constraints are enforced through penalty functions. After the PFD, the safety level of the structures is analyzed. The reliability analysis is performed using FORM, SORM and importance sampling Monte Carlo (ISMC) methods. The results are presented in Fig. 2 . Fig. 2.   a)  Reliability indices for Gumbel maximum likelihood case; b)  summary of reliability indices corresponding to Gumbel and GEV assumptions CONCLUSIONS Based on the completed analyses, the main conclusions are the following: ( i ) The safety level of steel frames subjected to snow load is fairly sensitive to the distribution type used to describe the dominant variable load. This assertion is valid irrespectively of the load-ratio. ( ii ) Frames designed  per partial factor method in accordance with Eurocode, i.e., using Gumbel distribution assumption and frequentist statistics (MLE) to derive characteristic loads, may not have adequate safety level in the high load-ratio region (  >0.6) with the same assumptions made in the reliability analysis. If GEV distribution is used in the reliability analysis the safety level significantly decreases, by ~14% compared to Gumbel model, which means a 7 times increase in the probability of failure. ( iii ) In case of GEV distribution the Bayesian posterior model gives about 80% higher probability of failure than the GEV-MLE. ( iv ) The epistemic uncertainty has a significant effect on the reliability of steel frames, in case of Gumbel distribution the incorporation of epistemic uncertainty into the reliability analysis yields to ~35% increase in the probability of failure. For the GEV distribution the increase is about 400-500%. ( v ) The reliability level substantially varies with the load-ratio, this cannot be captured by one single partial factor of the snow load. It should be emphasized that the statistical analysis is based on a 50-year long observation and the numerical analysis is limited to few examples. The investigation is being extended to further design cases and to the sensitivity analysis of measurements data, to able to draw general conclusions. REFERENCES [1] Geis, J, Strobel, K, Liel, A, “Snow-Induced Building Failures”, Journal of Performance of Constructed Facilities, Vol. 26, pp. 377-388, 2011. [2] Holicky, M, Sykora, M, “Failures of Roofs under Snow Load: Causes and Reliability Analysis” in Forensic Engineering 2009 , American Society of Civil Engineers, pp. 444-453, 2009. [3] Yang, X-S,  Engineering Optimization: An Introduction with Metaheuristic Applications , John Wiley & Sons, Hoboken, New Jersey, 2010. a) 3.43.63.84.04.20.2 0.4 0.6 0.8 Load − ratio, χ  [ − ]    R  e   l   i  a   b   i   l   i   t  y   i  n   d  e  x ,        β    [     −    ] FORM SORM ISMC b =3.72  b) 3.03.54.00.2 0.4 0.6 0.8 Load − ratio, χ  [ − ]    R  e   l   i  a   b   i   l   i   t  y   i  n   d  e  x ,        β    [     −    ] distribution statistics GUM GEV MLE BP BPP b =3.72    EUROSTEEL 2014, September 10-12, 2014, Naples, Italy ON THE RELIABILITY OF STEEL FRAMES EXPOSED TO SNOW LOAD Considering the Effect of Epistemic Uncertainty Árpád Rózsás, László Gergely Vigh Budapest University of Technology and Economics, Deptartment of Structural Engineering, Hungary rozsas.bme@gmail.com, geri@vbt.bme.hu INTRODUCTION Extreme snow falls are often leading to severe damage and even collapse of structures, causing significant economic losses and claiming lives [1]; e.g. recently 2005/06 Central Europe, 2010/11  North-Eastern USA. The causes are diverse, but some studies have identified the lack of adequate safety level provided by standards [3, 4]. Although modern structural standards such as Eurocode are based on probabilistic principles, the safety and combination factors are based on limited and approximate calculations and alignment to historically proven practice [5]. Previous researches typically apply – often questionable – simplifications in the reliability assessment of structures subjected to snow load, such as: a) linear response assumption, b) analysis is limited to cross-section resistance (global structural analysis is omitted), c) limited to certain limit states, d) limited number of examples, e) distribution type of snow load, f) epistemic uncertainties neglected, etc. These restrictions make cumbersome to draw general conclusions. The aim of this paper is to present a general approach which can help to comprehensively investigate the compliance of current Eurocode partial factor provisions and to assess whether the collapses most frequently observed at steel structures can be attributed to the deficiency of standardized provisions. To alleviate the limitations of previous studies, the framework is intended to efficiently perform parametric reliability analysis using state-of-the-art numerical methods, designed to be easily extendable by additional limit states and design considerations, has the  potential to identify the crucial parameters, the sufficient modelling level and to justify the assumptions made in previous studies, as well as invokes Bayesian statistics to examine the effect of epistemic uncertainty of snow load models on the safety level. The paper briefly discusses the proposed general framework and the algorithm is illustrated through an example with special emphasis on the epistemic uncertainty and distribution type of snow loads. 1   GENERAL FRAMEWORK The underlying idea of the proposed framework is to connect and integrate state-of-the-art programs which have powerful capabilities in their specific fields to make a versatile, flexible and robust algorithm. The main components and workflow of the proposed algorithm are illustrated in Fig. 1 . As a first step, if measurements are available statistical analysis is performed to determine the characteristic load values. These are passed to a module which performs partial factor based optimization with arbitrary objective function to get fully utilized structures per prescribed limit states. Fully utilized structures are required to assess solely the reliability level of the partial factor design (PFD) and no additional safety margin which may be present in practical design is considered. The optimization module calls the finite element program to get the required  parameters, e.g., internal forces, deflections, critical loads, to verify the limit states; in general this means numerous structural analyses. Once the optimal design is achieved (typically minimal weight) the corresponding geometric and loading model is passed to the reliability analysis module. Statistical analysis is used again to ensure the consistency between the partial factor design and reliability analysis, i.e., the characteristic values should be in accordance with the corresponding density functions. The reliability analysis module uses advanced methods such as FORM, SORM, importance sampling (ISMC), directional sampling, response surface methods, etc., to determine   the safety level of the structures. It can call the finite element program to evaluate the limit state function and it is also capable of representing random variables as random fields and to conduct stochastic finite element analysis. The statistical analysis is performed in Matlab [6] and R [7], the latter is mainly used to perform Bayesian analysis with connection to OpenBUGS [8]. The structural analysis is performed in ANSYS [9] and the reliability analysis is based on FERUM [10] and SUMO [11] toolboxes. The algorithm currently can handle only time-invariant reliability  problems. Optimization  partial factor design StatisticalanalysisObservationsReliabilityanalysisFEM structuralanalysis  ANSYS  MATLAB MATLAB + FERUM  + (SUMO) MATLAB R + OPENBUGS    Fig. 1.  General framework of the reliability assessment 2   STATISTICAL ANALYSIS OF SNOW MEASUREMENTS The snow load for the reliability analysis is determined from meteorologist pre-processed data of actual measurements at Budapest, Hungary, ranging from 1961 to 2010 [12]. Gumbel, Generalized Extreme Value (GEV) and Lognormal distributions are considered along the frequentist maximum likelihood (MLE) and Bayesian approaches. The annual maxima are analyzed and the results are presented in Table 1 . The Bayesian posterior (BP) stands for a distribution defined by the mean of the posterior distribution of the parameters, while the Bayesian predictive posterior (BPP) distribution is calculated as the weighted sum of the  parameters’ posterior distribution and the likelihood function. Table 1.  Ground snow loads for 20, 50 and 100 years return periods [kN/m 2 ] Maximum likelihood Bayesian posterior Bayesian predictive posterior Gumbel GEV Lognormal Gumbel GEV Lognormal Gumbel GEV Lognormal q  20  0.880 0.950 1.252 0.905 1.018 1.287 0.915 1.023 1.364 q  50  1.065 1.213 1.760 1.097 1.319 1.822 1.126 1.355 1.979 q  100  1.203 1.429 2.209 1.243 1.571 2.297 1.259 1.665 2.562 As the table confirms, the distribution type and statistical approach has a significant effect on the quantiles. For the 50-year return period the MLE of GEV and Lognormal models compared to Gumbel distribution predict 14% and 65% larger snow load, respectively. For the GEV distribution the MLE prediction for 20, 50 and 100-year return periods are 7%, 10% and 14% smaller than the BPP estimation respectively. The growing difference can be attributed to the scarcity of data in the large return value region, where the epistemic uncertainty is increasingly predominant. The Bayesian parameter estimation and quantiles are found robust and not sensitive to the prior. In respect of the likelihood value the GEV outperforms the Gumbel distribution, but if the number of variables is also taken into account trough Akaike information criteria (AIC) the Gumbel fits  better, although the difference is negligible. The AIC values of the Gumbel, GEV and Lognormal distributions are 465, 466 and 475 respectively. The Lognormal model for this particular site is unduly conservative, which is also confirmed by Fig. 2.  As the diverging curves show, for larger return periods the difference of the models is increasing. The incorporation of the epistemic uncertainty, especially in the large return period region, with few observations, is important.
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