My title https://doi.org/10.14311/APP.2022.36.0001 Acta Polytechnica CTU Proceedings 36:1–5, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague RELIABILITY OF A STRUCTURE STRICKEN BY A TORNADO Milan Holický Czech Technical University in Prague, Klokner Institute, Šolínova 7, 166 08 Prague, Czech Republic correspondence: milan.holicky@cvut.cz Abstract. A tornado may become a very dangerous climatic action for any structure, depending on the tornado intensity and extent of the structure. The origin of a tornado, its range and further activity appear to be unpredictable and completely random. The strike of a tornado to a structure depends on the tornado movement and the structure extent. The tornado intensity is specified by the wind speed, following observed damages and consequences. The probability of a tornado strike on a structure depends on the extent of the structure itself, on the tornado range, and the extent of the considered reference area, for which data concerning tornado appearance are available. The probability of failure of a structure is given by the product of the probability of contact of a tornado with the structure and the probability of exceeding the design speed of the tornado wind. The probability of exceeding the design wind speed depends on the assumed distribution of wind speed and the frequency of tornadoes during the required lifetime of the structure. The target failure probability of common structures presented in available documents is 10−6 per year, and 10−7 per year of structures in power plants. However, it is not clear how these criteria have been derived. Further development of reliability theory of tornado-stricken structures is recommended to be focused on the risk analysis of appropriate systems, of which the considered structures are elements, on the target failure probability, on detailed analysis of the probability distribution of tornado wind speed and their frequency during the required lifetime of the structures. Keywords: Design wind speed, power plant, probability, structure, target failure probability, tornado. 1. Introduction Reliability analysis of a structure stricken by a tornado is treated in several documents and prescriptions [1–3] that are focused primarily on nuclear power plants. Recent document [2] provides detailed recommenda- tions for reliability verification of power plants using probabilistic methods considering random properties of structures and tornado effects. All the available doc- uments include analysis of two inseparable factors of tornado action affecting the reliability of a structure: (1) tornado strike on structure, (2) the intensity of tornado effects. Both factors have random character depending on several local circumstances and conditions that are often assessed based on observed consequences and available experience. The factor (1) depends on the frequencies of the tornado in the reference territory where the tornado actions are observed. Specification of such a territory depends on social and economic aspects, on longtime observation of tornado, and empirical meteorological information. Tornado strike on the considered struc- ture depends surface area of the reference territory. It is a random variable that can be only assessed. The factor (2) the intensity of tornado effects is com- monly described by wind speed in the air twirl. The wind speed is again a random variable having a large variability of unknown parameters that can be assessed based on observed destructive consequences of tornado action. The design value of the wind speed can be estimated assuming usually Weibull distribution and approximate parameters. Because of uncertain param- eters (including the skewness) the estimated design wind speed may be imprecise. The tornado intensity is usually classified using graduated tables for ground wind speeds. Two scales indicated in Table 1 are commonly applied: Fuji and TORRO scales. Fuji scale (having 6 grades) is pri- marily used in USA, TORRO scale (refine Fuji scale having 10 grades) is used in Europe. The lowest grade of both scales starts with the wind speed of 17 m/s, the highest grade from 121 m/s. 2. Probability of structural failure The probability of structural failure pf due to tornado action can be assessed considering probability pA of tornado strike on a given structure, and simultaneous exceedance of wind speed of the design value vd, when the structure may fail with the probability pv . The resulting failure probability pf is therefore given as the product of two probabilities: pf = pA × pv (1) Probability pA of tornado strike depends [2] on area of reference territory A0, for which records of tornado actions are available for n years, on extend of structure Ak and also on the total tornado extent Ator. Above 1 https://doi.org/10.14311/APP.2022.36.0001 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en Milan Holický Acta Polytechnica CTU Proceedings Fuji TORRO Damage Wind Detailed description of damage F0 T0T1 Slight damage 17-24 m/s 25-32 m/s Accidentally demolished chimneys and wooden fences, minor damage to the roofing, damaged advertisements and road signs next to roads, broken tree branches, sporadically uprooted trees with shallow roots, traces of a tornado are visible in the field. F1 T2T3 Moderate damage 33-41 m/s 42-51 m/s Partially destroyed roofing, moving cars are pushed off the roads, mobile homes are shifted from the foundations, overturned or heavily damaged, flimsier buildings and shelters are badly damaged to destroyed, larger trees with stronger roots are sporadically uprooted and broken F2 T4T5 Moderate damage 52-61 m/s 62-72 m/s The roofs of less rigidly built buildings are completely torn down, mobile homes and flimsier structures are completely destroyed, in brick and stronger wooden houses, the side and front walls are not yet seriously damaged. Light cars float, small and light debris become projectiles, most isolated large trees are uprooted or broken. F3 T6T7 Significant damage 73-83 m/s 84-95 m/s Roofs and some walls are completely torn from the structure even in well-built buildings, heavier cars are buoyed, trains and locomotives are overturned, most trees in continuous forests are uprooted and broken, standing trees and tree stumps are partially debarked by flying debris. F4 T8T9 Severe damage 96-107 m/s 108-120 m/s Reinforced concrete buildings are significantly dama- ged, brick (brick) and stone buildings are severely (mostly irreparably) damaged, less solid buildings are completely levelled, the ruins of flimsy buildings are scattered at considerable distances from their foundations, cars are drifted by the air (just above the ground). ) or towed over long distances, large and heavy "projectiles" are formed from flying debris, tree stumps are completely stripped of bark. F5 T10 Completedestruction > 121 m/s Reinforced concrete buildings are badly damaged, other buildings are completely destroyed, cars are carried through the air like projectiles over consi- derable distances, fields are completely devoid of vegetation, crops are mostly torn out even with roots. Table 1. Tornado intensity scales of Fuji and Torro. mentioned variables A0, Ator and Ak are random vari- ables of considerable variability that require statistical and probabilistic analysis. In addition, tornado extent Ator may be assessed as product of the length L and width w (Ator ≈ L × w) [2]. Assuming that the tornado actions occur within the reference territory A0 uniformly, then the probability pA for one year can be estimated from the mean values of involved variables Ak, Ator and A0 using the formula: pA ≈ (Ak + Ator)/A0/n, (2) where n denotes the number of years when torna- dos are observed in the reference territory A0. More detailed probabilistic analysis concerning the specific location of a structure can be provided on the basis of the local conditions within the reference territory A0. An example. The probability pA of a tornado strike per year is strongly dependent on the extent of reference area A0. In South Moravian region A0 = 7188 km2 and area Ak + Ator = Ak + L × w = 1 + 4 × 0, 5 = 3 km2. Then the probability pA determined from one observation is about ≈ 3/7188 ≈ 4 × 10−4. However, when A0 is considered as a circle with the centre at the structure and the radius of 100 km, then A0 ≈ 31400 km2, and pA ≈ 3/31400 ≈ 1 × 10−4. 2 vol. 36/2022 Reliability of a structure stricken by a tornado Design values vd [m/s] for pv Parameters vinf [m/s] µ [m/s] σ [m/s] α 10−5 10−6 10−7 WEIBMIN 30 [m/s] 42 10 1,5 127 142 157 WEIBMIN 7 [m/s] 42 10 1,0 108 118 127 WEIBMIN 21 [m/s] 42 10 0,5 92 98 104 WEIBMIN 10 [m/s] 42 10 0 80 84 87 NORMAL ± INF 42 10 0 85 90 94 Table 2. Design values of wind speed vd for given skewness α and exceedance probability pv . 3. Design wind speed The design value vd of the wind speed v is defined for the specified exceedance probability pv when the structural failure may occur, so: pv ≈ P (v > vd), (3) Weibull distribution of wind speed v is considered in available documents [1–3]. It is the lower bound limited distribution characterized by three parame- ters: the mean µ, standard deviation σ a and skewness α. In the professional literature and software prod- ucts (MATHCAD, EXCEL, EasyFit) another three parameters are also used: for example c1 (also α or b), c2 (also b or a) and the lower bound vinf (for standardized variables uinf ). The parameter c1 depends solely on the skewness α, the parameter c2 depends on the parameter c1 (skewness α) and standard deviation σ, the lower bound of the standardized variable uinf depends only on the parameter c1 (on skewness α), the lower bound of the real variable vinf depends on three parameters: on the mean µ, standard deviation σ and also on the skewness α (on parameter c1) [4]. The skewness α affects therefore the lower bound vinf and shape of the distribution for a given mean and standard deviation σ, and significantly affects result- ing probability of tornado effect. Appendix provides a copy of MATHCAD sheet enabling application of the Weibull distribution. An example of determining the design values of wind speed vd for given skewness α and exceedance probability pv is indicated in Table 2. It follows from Table 1 and Figure 1 that the rel- atively high skewness α = 1, 5 leads to high design values vd of wind speed, by 50 % greater than for the skewness α = 0 (normal distribution). However, it is not clear how the actual skewness α can be determined from available data. It may be also recommended to utilize experience from the distribution of usual wind speed or the information concerning lower bound vinf indicated in Table 1. 4. Number of tornado strikes During the required lifetime of a structure several tornados, in general, say N , may occur [3]. If the distribution function of a singular tornado is Φ(x, 1), then the overall distribution function of all N tornados is given by power function Φ(x, 1)N [4, 5]: Figure 1. Weibull and normal distribution for µ = 42 m/s and σ = 10 m/s. Φ(x, N ) = Φ(x, 1)N (4) Then the design value (upper fractile) vd,N of the distribution function Φ(x, 1)N may be greater the design value determined from a single distribution Φ(x, 1). The occurrence number N may be in the case of less frequent tornados equal to the number of years n used in Equation 2. Variation of the design value vd,N of tornados for given exceedance probabilities with the occurrence number N is indicated in Table 2. Probability density functions and partly distribution functions for occur- rence numbers N = 1 and 10 are shown in Figure 3. An example. Table 3 demonstrates that the result- ing design values vd,N for given number of tornados N > 1 are greater than the values for one tornado N = 1. With increasing N the design value increases up to 35 %. Consequently, when designing a structure the number of tornados N during the assumed lifetime is necessary to consider. When during the assumed lifetime of a structure the number of tornados is N , then the probability pv,N , that the wind speed vN exceeds the design values vd with the probability: pv,N = P(vN > vd) (5) 3 Milan Holický Acta Polytechnica CTU Proceedings Occurrence number of tornados Lower bound Mean value Standard deviation Skewness Design value vd,N for exceedance probability N vinf [m/s] µ [m/s] σ [m/s] α 10−5 10−6 10−7 1 30 42 10 1,5 127 142 157 10 30 61 10 1 142 157 172 100 30 80 10 1 157 172 187 1000 30 98 10 1 172 187 201 Table 3. Variation of the design value vd,N of tornados with the occurrence number N . Figure 2. Distributions of tornado wind speed for N = 1 and 10. Table 4 indicates the probability P(vN > vd) for design values vd determined for a single tornado (N = 1) when structural failure may occur. Design values vd are determined as before for the probabilities 10−5, 10−6 a 10−7. The probability P(vN > vd) for the design value vd determined from one observation (N = 1) is therefore increased by the order of tornado occurrence N . 5. Failure probability Failure probability of a structure due to tornado strike is generally given by Equation1 1 as the product of the probability pA, of tornado strike, and the prob- ability pv , that wind speed exceeds design value vd when structural failure may occur. In the case of one isolated tornado the failure probability pf,1 is given as: pf,1 = pA × pv = P(A0, Ak, Ator) × P(v > vd) (6) In the case of N tornados during the required life- time the failure probability pf,N is given by the prod- uct: pf,N = pA × pv,N = P(A0, Ak, Ator) × P(vN > vd) (7) It is obvious that when several tornados (N > 1) are assumed, the failure probability is greater than when only one tornado (N = 1) is considered. An example. Consider as before the reference area A0 = 7188 km2 and extent Ak + Ator = 2 + 1 = 3 km2. Then the probability pA was determined from one observation ≈ 3/7188 ≈ 4 × 10−4. When the structure is design assuming the probability P(v > vd) = 10−5, then the expected probability is ≈ pf,1 ≈ pA × pv = 4 × 10−9. However, when during the assumed lifetime number of tornados is N = 10, then the failure proba- bility is about ≈ pf,N = pA × pv,N ≈ 4 × 10−8. Thus, it is the failure probability ten times greater. 6. Target probability of failure Target probability of failure of a structure during the lifetime t denoted ptar,t can be specified based on probabilistic optimization (maximization) of struc- tural utility (cost) Ctot(t) during its lifetime t. This can be analyzed in a simplified way considering the total benefit of functioning structure Cben(t) during the lifetime t, and the possible losses Cf (t) caused by structural failure (within the relevant system) with the target probability ptar,t. The total utility Ctot(t) can be then symbolically expressed as [4, 5]: Ctot(t) = max{Cben(t) − ptar,t × Cf (t)}. (8) The cost Cben(t) includes all kind of benefits during the life time t. Consequences of structural failure Cf (t) cover all the material, ecological and social losses caused by the structural failure. These losses Cf (t) may be extremely large, and could be very difficult to assess. A detailed analysis is outside this contribution and it is recommended to accept a very small target probability, for example, ptar,t = 10−7 per year as indicated in documents [1–3]. Note that the target probability for common struc- tures is usually accepted as ≈ 1.30 × 10−6 for one year (that is reliability index ≈ 4, 7). This corresponds to ≈ 0, 72 × 10−4 per 50 years (and to the reliability index 3,83), per 100 years the target probability is ≈ 1, 31 × 10−4 and reliability index ≈ 3, 65. Varia- tion of the target probabilities ptar,t with reference time t given in years for selected basic probabilities ptar,1 = 10−5, 10−6 and 10−7 is shown in Figure 3. 4 vol. 36/2022 Reliability of a structure stricken by a tornado Number of tornados Lower bound Mean value Standard deviation Skewness The probability P(vN > vd) for design value vd N vinf [m/s] µ [m/s] σ [m/s] α 127 142 157 1 30 42 10 1,5 1.0×10−5 1.0×10−6 1.1×10−7 10 30 61 10 1 9.4×10−5 1.0×10−5 1.1×10−6 100 30 80 10 1 9.3×10−4 1.0×10−4 1.1×10−5 1000 30 98 10 1 9.3×10−3 1.0×10−3 1.1×10−4 Table 4. Probability P(vN > vd) of exceedance the design values vd. Figure 3. Variation of the target probabilities ptar,t with reference time t in years for basic probabilities for one year - ptar,1 = 10−5, 10−6 and 10−7. 7. Concluding remarks (1) A tornado may be for any structure very danger- ous depending on the extent of the structure and tornado and its intensity. Tornado is unpredictable and of random time behaviour. (2) The probability of a tornado strike can be assessed taking into account the extent of a structure and tornado and the extent of the reference territory where tornados are observed. (3) Tornado intensity expressed by wind speed is de- termined after the tornado strike following observed consequences. (4) Probability structural failure is given by a product of the probability of a tornado strike and the prob- ability of wind speed exceedance of design value. (5) Estimation of wind speed exceedance of design value is strongly dependent on assumed wind speed distribution and the number of tornado occurrences during the lifetime of the structure. (6) Available documents provide for target probability for structural failure value 10-7 per year, however, it is unclear how this criterion has been derived. (7) Further reliability analysis of structures stricken by tornados is to be focused on detailed analysis of tornado consequences, on wind speed distribution and the number of tornado occurrences during the lifetime of the structures. References [1] IAEA SAFETY GUIDES 50-SG-S11A. Extreme Meteorological Events in Nuclear Power Plant Siting Excluding Tropical Cyclones. International Atomic Energy Agency, Vienna, Austria, 1981. [2] Pacific Northwest National Laboratory. Tornado Climatology of the Contiguous United States. NUREG/CR-4461, Rev. 2, U.S. Nuclear Regulatory Commission Office of Nuclear Regulatory Research, Washington, 2007. [3] H. Zhu, J. Chen, F. Li, et al. Tornado Hazard Assessment for a Nuclear Power Plant in China. Energy Procedia 127:20-8, 2017. https://doi.org/10.1016/j.egypro.2017.08.091. [4] M. Holický. Reliability analysis for structural design. SUN PRESS, Stellenbosch (South Africa), 2009. [5] M. Holický. Introduction to Probability and Statistics for Engineers. Springer, Heidelberg, 2013. 5 https://doi.org/10.1016/j.egypro.2017.08.091 Acta Polytechnica CTU Proceedings 36:1–5, 2022 1 Introduction 2 Probability of structural failure 3 Design wind speed 4 Number of tornado strikes 5 Failure probability 6 Target probability of failure 7 Concluding remarks References