International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Influence on structural reliability of uncertainty in estimated extreme values of load-effects Reid, Stuart G.; Naess, Arvid Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 Influence on Structural Reliability of Uncertainty in Estimated Extreme Values of Load-Effects Stuart G. Reid Professor, School of Civil Engineering, The University of Sydney, Sydney, Australia Arvid Naess, Professor, CeSOS and Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway ABSTRACT: Extreme values of time-varying loads are often estimated to serve as design loads for the purposes of structural design. Uncertainty in the estimation of the design loads inevitably leads to uncertainty in the resultant levels of structural reliability. Uncertainty is assessed for estimates of extreme wind loads calculated using statistical methods based on the average conditional exceedance rate (ACER), fitting of a Gumbel distribution and Peaks-over-Threshold (POT). The ACER method gave the best results, but all the methods gave results which would normally be considered to be sufficiently accurate for engineering applications. However, for structures designed on the basis of the estimated values of V100, the uncertainty in the estimated design loads produced very uncertain probabilities of failure with a significant increase in their expected value. It is concluded that the uncertain distribution of the probabilities of failure must be taken into account when evaluating structural safety and a ‘fiducial confidence function’ is proposed for this purpose. 1. INTRODUCTION Structures are often designed to resist extreme time-varying load-effects that are estimated on the basis of loading data obtained by sampling over short periods of time. Statistical techniques can be used to estimate extreme values corresponding to the sample data, but estimated extreme values are inevitably influenced by sampling variability and uncertainty concerning the nature of the extreme value distribution. For the purposes of structural design to resist time-varying loads (e.g. from wind, snow and earthquakes) Standards often specify characteristic extreme loads with return periods (or annual exceedance probabilities) that depend on the importance and design life of a structure. The specified design loads are intended to ensure appropriate levels of structural reliability, but uncertainty in the estimation of the characteristic design loads inevitably leads to uncertainty in the resultant levels of structural reliability, thereby reducing the dependability of the design process. In this paper, uncertainty in the estimation of characteristic design loads is evaluated for a representative example of wind loading based on synthetic wind speed data obtained by sampling from a stochastic process model that has been shown to give realistic results (Naess & Clausen, 2001). Synthetic data are used to estimate a characteristic wind speed V100 corresponding to a return period of 100 years, using statistical methods based on the average conditional exceedance rate (ACER), fitting of a Gumbel distribution, and Peaks-over-Threshold (POT). Considering estimated values of V100 obtained from 100 sets of synthetic data, the variable results obtained by the different methods are compared with each other, and the accuracy of the results is assessed in relation to the theoretically exact value of V100 given by the underlying (known) stochastic process model. Furthermore, the significance of uncertainty in the estimation of V100 is assessed with regard to the dependability of design procedures based on V100. Structural reliability is assessed for a 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 typical design requirement based on V100 and the variable results based on 100 sets of synthetic data are compared for design loads based on the ACER, Gumbel and POT methods. Finally, the variability of the resultant structural reliability is assessed with regard to its impact on the dependability of the design process.  The term ‘fiducial confidence’ is used to refer to the level of confidence that a decision-making process is dependable. The concept of fiducial confidence is discussed and a relevant ‘fiducial confidence function’ is used to assess the level of confidence that a structure is ‘safe’ with regard to the uncertain probability of failure. 2. EXTREME VALUE PREDICTION We consider 20 years of synthetic wind speed data, amounting to 2000 data points, which is not much for detailed statistics. However, this case may represent a real situation when nothing but a limited data sample is available. In this case it is crucial to provide extreme value estimates utilizing all data available.  The extreme value statistics will be analyzed by application to synthetic data for which the exact extreme values can be calculated (Naess and Clausen, 2001). In particular, it is assumed that the underlying (normalized) stochastic process Z(t) is stationary and Gaussian with mean value zero and standard deviation equal to one. It is also assumed that the mean zero up-crossing rate ν+(0)  is such that the product ν+(0)T=103 where T=1 year, which seems to be typical for the wind speed process. Using the Poisson assumption, the distribution of the yearly extreme value of Z(t) is then calculated by the formula  𝐹? ™ = exp −𝜈? 𝜂 𝑇 =                                exp −10? exp −𝜂?/2   (1)   where T=1 year, ν+(η) is the mean up-crossing rate per year, and η is the scaled wind speed. The 100-year return period value η100yr  is then calculated from the relation F1yr (η100yr ) = 1−1/100 , which gives η100yr = 4.80 . The Monte Carlo simulated data to be used for the example are generated based on the observation that the peak events extracted from measurements of the wind speed process, are usually separated by 3-4 days. This is done to obtain approximately independent data, as required by the POT method. In accordance with this, peak event data are generated from the extreme value distribution  𝐹?? 𝜂 = exp −𝑞  exp −𝜂?/2                    (2) where q=ν+(0)T=10 , which corresponds to T=3.65 days, and F1yr (η) = F 3d (η)( )100  which makes η100yr =4.80  an exact value for the simulated data. Since the data points (i.e. T=3.65 days maxima) are independent, εk(η) is independent of k. Therefore we put k=1. Since we have 100 data from one year, the data amounts to 2000 data points. The ACER function ε1(η) for the chosen range of η-values, and the 95% confidence interval, are estimated following Naess et al. (2013). From the estimated ACER function ˆε1(η)the predicted 100 year return level is obtained  from ˆε1(η100yr ) =10−4 and a nonparametric boot-strapping method was also used to estimate a 95% confidence interval based on 1000 resamples of size 2000. The POT prediction of the 100 year return level was based on using maximum likelihood estimates (MLE) of the parameters in the Generalized Pareto (GP) distribution for a specific choice of threshold. The 95% confidence interval was obtained from the parametrically bootstrapped PDF of the POT estimate for the given threshold. A sample of 1000 data sets was used. One of the unfortunate features of the POT method is that the estimated 100 year value may vary significantly with the choice of threshold. So also for the synthetic data. We have followed the standard recommended procedures for identifying a suitable threshold (Coles, 2001). 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3 Note that in spite of the fact that the true asymptotic distribution of exceedances is the exponential distribution, the POT method used here is based on adopting the GP distribution. The reason is simply that this is the recommended procedure (Coles, 2001), which is somewhat unfortunate but understandable. The reason being that the GP distribution provides greater flexibility in terms of curve fitting. If the correct asymptotic distribution of exceedances had been used on this example, poor results for the estimated return period values would be obtained. The price to pay for using the GP distribution is that the estimated parameters may easily lead to an asymptotically inconsistent extreme value distribution.  The 100 year return level predicted by the Gumbel method was based on using the method of moments for parameter estimation on the sample of 20 yearly extremes. This choice of estimation method is due to the small sample of extreme values. The 95% confidence interval was obtained from the parametrically bootstrapped PDF of the Gumbel prediction. This was based on a sample of size 10,000 data sets of 20 yearly extremes. The results obtained by the method of moments were compared with the corresponding results obtained by using the maximum likelihood method. While there were individual differences, the overall picture was one of very good agreement.  In order to get an idea about the performance of the ACER, POT and Gumbel methods, 100 independent 20 yr MC simulations as discussed above were done. The values over the 100 simulated cases are presented as average 100 year value and average 95% confidence interval: 1) ACER – 4.81 and (4.41, 5.09) fitted, (4.48, 5.18) bootstrapped; 2) Gumbel – 4.84 and (4.37, 5.40); 3) POT – 4.72 and (4.27, 5.23). It is seen that the average of the 100 predicted 100 year return levels is slightly better for the ACER method than for both the POT and the Gumbel methods. Moreover, the range of predicted 100 year return levels by the ACER method is 4.34-5.36, whilst the range for the POT method is 4.19-5.87, and for the Gumbel method 4.41-5.71. Hence, by these measures, the ACER method performs consistently better than the other methods. It is also observed from the estimated 95% confidence intervals that the ACER method, as implemented in this paper, provides slightly higher accuracy than the other two methods. As a final comparison, the 100 bootstrapped confidence intervals obtained for the ACER and Gumbel methods missed the target value three times, whilst for the POT method this number was 18. The relative accuracy and consistency of the results can also be assessed with regard to the mean, standard deviation (s.d.) and Coefficient of Variation (CoV) of the estimated values of the 100-year wind speed V100, as shown in Table 1.    Table 1: Statistics of V100 estimates  ACER Gumbel POT mean 4.81 4.84 4.72 s.d. 0.202 0.252 0.280 CoV 0.042 0.052 0.059  For all methods the mean error (cf. V100=4.80) is less than 2% and the CoV is 4-6%. For engineering applications, such results would normally be considered to be of good accuracy and consistency. However, for structural designs based on V100, the uncertainty in the estimation of V100 has a significant effect on the structural reliability, as discussed below.  3. USE OF EXTREME CHARACTERISTIC VALUES IN DESIGN In many design Standards, the minimum design load for a natural hazard (e.g., wind, snow or earthquake) is specified in relation to the annual probability of exceedance of the design load, dependent on the importance and design life of the structure. For example, for structures in Australia and New Zealand, the annual probability of exceedance of the design load is specified in the loading Standard (AS/NZS 1170.0) and values given for ultimate limit state design wind events are shown in Table 2. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 Table 2: Annual probabilities of exceedance of ultimate limit state design wind events for Australia and New Zealand (AS/NZS 1170.0). Design life (L)   Importance Annual exceedance probability Australia NZ < 6 mths normal   1/100  major  1/250 5 yrs normal  1/50 1/250  major 1/100 1/500 25 yrs normal  1/200 1/250  major 1/500 1/500 50 yrs normal  1/500 1/500  major 1/1000 1/1000 100 yrs normal  1/1000 1/1000  major 1/2500 1/2500  For Australian structures, the Standard also states that for a design working life (L) between 5 and 100 years, the annual probability of exceedance for wind and earthquake events may be taken as r/L, where the lifetime risk (r) is as shown in Table 3.   Table 3: Lifetime risk (r) for ultimate limit state wind and earthquake design events for Australia (AS/NZS 1170.0) Importance Risk of exceedance of design event (r) Minor 0.20 to 0.25 Normal 0.10 to 0.125 Major 0.04 to 0.05 post-disaster 0.020 to 0.025  Therefore, in accordance with AS1170.0, the 100-year wind load can be used for ultimate limit state design for a normal structure with a design life not exceeding 12.5 years. The relevant design wind load-effect is  𝑊? = 𝐵?𝑉™??    (3) where Bn is the nominal (estimated) coefficient (load-effect factor) that relates the wind load-effect to the square of the wind speed. Considering the effects of wind loading alone, the basic design requirement is ∅𝑅? =𝑊? = 𝐵?𝑉™??     (4) where φ is the resistance factor, and Rn is the nominal (calculated) resistance. 4. STRUCTURAL RELIABILITY The lifetime reliability implicit in the basic design requirement above has been evaluated for a structure with a design life of 12.5 years.  The reliability has been evaluated for a structure with a resistance R that is lognormally distributed with a coefficient of variation of 0.2 and a mean value Rmean dependent on the design resistance, assuming ∅𝑅? = 𝑂. 65  𝑅™??    (5) Structural failure will occur if the resistance R is exceeded by the peak wind load-effect W which is a function of the load-effect factor B and the peak wind speed V: 𝑊 = 𝐵𝑉?    (6) Assuming that the uncertainty in the estimation of Bn is negligible, and normalizing for Bn=B=1, the peak wind speeds described in Section 2 (above) give peak wind load effects W (over a period of T years) with a cumulative distribution function 𝐹? 𝑤 = 𝑒𝑥𝑝[−  1000. 𝑇. 𝑒𝑥𝑝 −𝑤/2   ]   (7) (where FW(w) is a Gumbel distribution function). In accordance with the above, if designs were based on an accurate assessment of V100 the nominal probability of failure would be 0.00913, corresponding to a reliability index of 2.36. This is a very low reliability index (high probability of failure) but it is consistent with a previous finding that adequate reliability is apparently achieved in practice through the unintentional use of very conservative estimates of the nominal load-effect factor Bn (Reid, 2012) For designs based on the estimates of V100 calculated for 100 samples of synthetic wind speed data using the ACER, Gumbel and POT methods of analysis, the mean, standard deviation (s.d.) and Coefficient of Variation 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 (CoV) of the nominal probabilities of failure are shown in Table 4.    Table 4: Statistics of the nominal probability of failure.  ACER Gumbel POT mean 0.0118 0.0120 0.0211 s.d. 0.0112 0.0108 0.0213 CoV 0.95 0.90 1.01  The results show that uncertainty in the estimates of V100 increases the expected probability of failure by about 30% for ACER and Gumbel estimates, and 130% for POT estimates. These increases are significant. Furthermore, the uncertainty in the estimates of V100 leads to very much larger uncertainties in the resultant probabilities of failure, with resultant CoVs ranging from 0.90-1.0. These very large uncertainties have a significant impact on the dependability of the results and the level of confidence placed in the design process. 5. CONFIDENCE Statistical measures of confidence are based on the probability that a random value will fall within some limits. For example, a characteristic fractile of a distribution of the probability of failure pf is associated with a corresponding level of statistical confidence that pf will not exceed that value. Although this is the accepted meaning of ‘confidence’ in the context of statistics, in the context of risk-based decision-making ‘confidence’ has a different meaning and is based on a different concept of ‘fiducial confidence’. In risk-based decision-making, fiducial confidence is related to the concern that unreliable risk estimates may mislead decision-makers, resulting in bad decisions that could have unacceptable consequences. 5.1. Fiducial confidence in structural safety In the assessment of structural safety, reliance on uncertain risk estimates raises concerns because uncertainty about the true probability of failure may lead to acceptance of an engineering design solution that is less safe than it should be (in relation to a target value of the probability of failure pft), but the main concern is that the uncertainty could lead to the design of a structure that would be unsafe (with pf>pft) and would subsequently fail.  Therefore the probability that a structure is unsafe (with pf>pft) and will fail is a measure of the ‘lack of confidence’ in the safety of the structure. The probabilistic measure 𝑃? ™  of the ‘lack of confidence’ that a structure is safe with regard to a target probability of failure pft can be determined from the probability distribution of the probability of failure 𝑓??(𝑝), giving:  𝑃? ™ =    (?? ™ 𝑓?? 𝑝 . 𝑝)𝑑𝑝    (8) A complementary measure C of ‘proba-bilistic confidence’ that a structure is safe (in relation to pft) is then given by: 𝐶? ™ = 1− 𝑃? ™ = 1− (?? ™ 𝑓?? 𝑝 . 𝑝)𝑑𝑝   (9) The probability  𝑃? ™ can be interpreted as the probability of failure attributable to ‘bad’ designs for which pf exceeds pft, whilst the complementary probability 𝐶? ™ is a measure of fiducial reliability (in relation to the avoidance of failures attributable to ‘bad’ designs). More generally, in the absence of a specific target probability of failure pft, the fiducial confidence associated with a decision based on a selected characteristic value of the probability of failure (pfc) is a decreasing function of the probability of failure attributable to outcomes with pf exceeding pfc, and a related ‘fiducial confidence function’ GPf(pfc) is defined as the proportion of the total probability of failure Pf that is attributable to outcomes where pf does not exceed the selected characteristic value pfc. Accordingly, 𝐺?? 𝑝™ = (𝑓??? ™? 𝑝 . 𝑝)𝑑𝑝/𝑃?  (10) where 𝑃? = (𝑓???? 𝑝 . 𝑝)𝑑𝑝    (11) 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6    Figure 1: Distributions of uncertain probabilities of failure, in relation to probability (CDF) and fiducial confidence.  The fiducial confidence function GPf(pfc) is similar to the cumulative probability distribution function FPf(pfc) which provides a measure of ‘statistical confidence’ that pf will not exceed the characteristic value pfc. However, the fiducial confidence GPf(pfc) accounts not only for the likelihood but also the magnitude of the risk outcomes that exceed pfc, and therefore the fiducial confidence GPf(pfc) is always less than or equal to the statistical confidence FPf(pfc). Statistical and fiducial confidence functions are shown in Figure 1 for designs based on the estimates of V100 calculated using the ACER, Gumbel and POT methods of analysis. The nominal target probability of failure (pf=0.00913, given by an accurate estimate of V100) is also indicated The figure shows that the level of confidence that the target probability of failure would not be exceeded is very low, with a statistical confidence of about 50% for the ACER and Gumbel results and about 35% for the POT results, and the corresponding levels of fiducial confidence are only about 20% for the ACER and Gumbel results and less than 10% for the POT results. Also it is shown that for a statistical confidence of 90% (for example), design decisions would be based on a characteristic probability of failure of 2.4%, 2.7% or 4.8% for ACER, Gumbel or POT results, respectively, and for a fiducial confidence of 90% the corresponding characteristic probabilities of failure would be 3.7%, 4.7% and 8.5%. 6. CONCLUSION It has been shown that uncertainty in the estimation of extreme values of load effects has a very significant and deleterious effect on the 0"0.1"0.2"0.3"0.4"0.5"0.6"0.7"0.8"0.9"1"0" 0.02" 0.04" 0.06" 0.08" 0.1" 0.12" 0.14" 0.16" 0.18" 0.2"CDF$/$Confidence$probability$of$failure$Pf$ACER"CDF"ACER"Confidence"Gumbel"CDF"Gumbel"Confidence"POT"CDF"POT"Confidence"nominal"target"12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7 dependability of structural design procedures and the resultant structural reliability.  Uncertainty in the estimation of characteristic design loads was examined with regard to wind loads based on samples of wind speed data. The data were used to estimate a characteristic wind speed V100 with a 100-year return period, using statistical methods based on the average conditional exceedance rate (ACER), fitting of a Gumbel distribution, and Peaks-over-Threshold (POT). The ACER and Gumbel methods were shown to be more accurate than the POT method, but all the methods gave results which would normally be considered to be sufficiently accurate for engineering applications.  However, the main result of the work presented in this paper is the important finding that for structures designed on the basis of the estimated values of V100, the relatively small uncertainty in the estimated design loads would result in the design of structures with very uncertain probabilities of failure, and the overall probability of failure would be significantly greater than the nominal (target) value. It is concluded that the uncertain distribution of the probabilities of failure must be taken into account when evaluating structural safety. A fiducial confidence function has been used for this purpose.  7. REFERENCES Naess, A. and P. H. Clausen (2001). Combination of peaks-over-threshold and bootstrapping methods for extreme value prediction. Structural Safety 23, 315–330. Naess, A., O. Gaidai and O. Karpa (2013). Estimation of Extreme Values by the Average Conditional Exceedance Rate Method. Journal of Probability and Statistics. Vol. 2013, Article ID797014, http://dx.doi.org/10.1155/2013/797014 Reid, S.G. (2013). “The influence of design life in life-cycle civil engineering”, Life-Cycle and Sustainability of Civil Infrastructure Systems, Strauss, Frangopol & Bergmeister (eds), (Proceedings of the Third International Symposium on Life-Cycle Civil Engineering, IALCCE 2012, Vienna, Austria, October 2012), Taylor & Francis, London, UK, 2298-2305. Reid, S.G. (2015). “Confidence in life-cycle civil engineering decisions based on uncertain information”, Life-Cycle of Structural Systems: Design, Assessment, Maintenance and Management, Furuta, Frangopol & Akiyama (eds), (Proceedings of the Fourth International Symposium on Life-Cycle Civil Engineering, IALCCE 2014, Tokyo, Japan, November 2014), Taylor & Francis, London, UK, 2048-2055. 

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