Extreme Winds by Direction Sector
The peaks-over-threshold approach can be applied to winds separated by direction sector. This is done in Figure 2.4 which shows a wind speed versus return period for a number of wind directions using combined gust data from Melbourne Airport, Australia, for synoptic wind events:
- • Gust wind speeds separately analysed for 11 direction sectors of 22.5° width. For the remaining 5 sectors (ENE to SSE), there was insufficient data for a meaningful analysis to be carried out.
- • Combined data from all direction sectors, analysed as a single data set.
- • A combined distribution obtained by combining distributions for directional sectors, according to Equation (2.8).
A peaks-over-threshold approach was used, with the shape factor, k, fixed at 0.1 (see Appendix G). This resulted in distributions forming lines, curving slightly on the wind speed (linear) versus return period (logarithmic) graphs.
There is good agreement for the ‘all-directions’ line, and the ‘combined directions’ line, indicating the independence of the data from the various direction sectors, and the validity of Equation (2.8)
Figure 2.4 Probability distributions fitted to gust data for direction sectors for Melbourne Airport, 1970-2008.
Bootstrapping and Confidence Limits
As far back as the 1950s, Gumbel (1958) emphasized the importance of calculating confidence limits when making predictions for long average recurrence intervals from limited data sets. A convenient way of doing this is by use of a simulation or ‘bootstrapping’ technique, in which a large number of samples of random data are generated using random numbers representative of the cumulative distribution function, with parameters determined from the actual data (e.g. Naess and Clausen, 2001).
Figure 2.5 shows the 10% and 90% percentile limits obtained by simulating 50 samples of 50years of annual maxima (2,500 in total), and fitting each sample, using the Method of Moments (Section 2.3.3) to the Gumbel Distribution of Equation (2.11), with и equal to 29.2m/s and a equal to 3.0 m/s, It can be seen that the predictions of V_{1>000} can be up to 4 m/s from the correct value of 50 m/s. Even the V_{so} predictions can depart from the correct value of 41 m/s by up to 2.5 m/s.
The standard deviations of extreme wind predictions can also be obtained using the bootstrapping approach. This contributes to uncertainty estimates for wind loading required for structural reliability studies (see Table 2.6). For the example in Figure 2.5, the standard deviation of the 50-year ARI (V_{J0}) estimates (based on 50-year samples) is 1.62 m/s. For V, ooo» the standard deviation is 2.82 m/s. Note that these are uncertainties determined assuming that the underlying windspeed measurements are accurate, and that the correct probability distribution has been chosen, and
Figure 2.5 Confidence limits obtained by ‘bootstrapping’, for predictions using the Gumbel Distribution.
other sources of uncertainty need to be included in reliability studies and load factor determination.
Prediction of Extreme Winds from Tornados
The prediction of extreme wind speeds from small, rare extreme events like tornadoes is very difficult, as they seldom strike anemometers. However, damage surveys after such events can often be used to estimate the area of their ‘footprints’ with wind speeds exceeding defined values (e.g. Fujita, 1971).
The probability of a tornado strike on the site of a structure, with a wind speed exceeding a value U„ in a time period, T, years, can be written as Equation (2.20) (Wen and Chu, 1973).
where v is the average rate of occurrence of tornados, per square kilometre, per year, in the region where the structure is located, and a_{v} is the average ‘footprint’ area of tornadoes with wind speed greater than 17,. v can also be written as nlA_{R}, where n is the average number of tornados in a region (such as a state, county or shire, or a 1° square) of area A_{R} km^{2}.
Equation (2.20) applies to a ‘point target’. A transmission line represents a ‘line target’ and the tornado-risk model for a complete transmission line is discussed in Section 13.3.1.
Parent Wind Distributions
For some design applications, it is necessary to have information on the distribution of the complete population of wind speeds at a site. An example is the estimation of fatigue damage for which account must be taken of damage accumulation over a range of windstorms (see Section 5.6). The population of wind speeds produced by synoptic windstorms at a site is usually fitted with a distribution of the Weibull type:
Equation (2.21) represents the probability density function for mean wind speeds produced by synoptic events. There are two parameters: a scale factor, c, which has units of wind speed, and a shape factor, w, which is dimensionless (see also Appendix C3.4). The probability of exceedance of any given wind speed is given by Equation (2.22):
Figure 2.6 Example of a Weibull distribution fit to parent population of synoptic winds.
Typical values of c are 3 - 10 m/s, and w usually falls in the range of 1.3 - 2.0. An example of a Weibull fit to recorded meteorological data is shown in Figure 2.6.
Several attempts have been made to predict extreme winds from knowledge of the parent distribution of wind speeds, and thus make predictions from quite short records of wind speed at a site (e.g. Gomes and Vickery, 1977b). The ‘asymptotic’ extreme value distribution for a Weibull parent distribution is the Type I, or Gumbel, distribution. However, for extremes drawn from a finite sample (e.g. annual maxima), the ‘penultimate’ Type I, as discussed in Section 2.3.5, is the more appropriate extreme value distribution. However, it should be noted that the Weibull Distribution, the Type I Extreme Value Distribution, and the ‘penultimate’ distribution will all give unlimited wind speeds with reducing probability of exceedance.
Wind Loads and Structural Safety
The development of structural reliability concepts - i.e. the application of probabilistic methods to the structural design process - has accelerated the adoption of such methods into wind engineering since the 1970s. The assessment of wind loads is only one part of the total structural design process, which also includes the determination of other loads and the resistance of structural materials. The structural engineer must proportion the structure so that collapse or overturning has a very low risk of occurring, and defined serviceability limits on deflection, acceleration, etc. are not exceeded very often.
Limit-States Design
Limit-states design is a rational approach to the design of structures, which has now become accepted around the world. Explicitly defining the ultimate and serviceability limit states for design, the method takes a more rational approach to structural safety by defining ‘partial’ load factors (‘gamma’ factors) for each type of loading, and a separate resistance factor (‘phi’ factor) for the resistance. The application of the limit states design method is not, in itself, a probabilistic process, but probability is usually used to derive the load and resistance factors.
A typical ultimate limit-states design relationship involving wind loads, is as follows:
where
(p is a resistance factor,
R is the nominal structural resistance, y_{D} is the dead load factor,
D is the nominal dead load, is the wind load factor,
Wis the nominal wind load.
In this relationship, the partial factors,
y_{D}, and y_{w} are adjusted separately to take account of the variability and uncertainty in the resistance, dead load and wind load. The values used also depend on what particular nominal values have been selected. Often a final calibration of a proposed design formula is carried out by evaluating a ‘safety’, or ‘reliability’, index as discussed in the following section, for a range of design situations, e.g. various combinations of nominal dead and wind loads.
Probability of Failure and the Safety Index
A quantitative measure of the safety of the structural design process, the safety index, or reliability index, is used as a method of calibration of existing and future design methods for structures. As will be explained
Figure 2.1 Probability densities for load effects and resistance.
in this section, there is a one-to-one relationship between this index and a probability of failure, based on the exceedance of a design resistance by an applied load (but not including failures by human errors and other accidental causes).
The structural design process is shown in its simplest form in Figure 2.7. The process consists of comparing a structural load effect, S, with the corresponding resistance, R. In the case of limit states associated with structural strength or collapse, the load effect could be an axial force in a member or a bending moment, or the corresponding stresses. In the case of serviceability limit states, S and R may be deflections, accelerations or crack widths, or their acceptable limits.
The probability density functions f_{s}(S) and /_{K}(R) for a load effect, S, and the corresponding structural resistance, R are shown in Figure 2.7. (Probability density is defined in Section C2.1 in Appendix C.) Clearly, S and R must have the same units. The dispersion or ‘width’ of the two distributions represents the uncertainty in 5 and R.
Failure (or unserviceability) occurs when the resistance of the structure is less than the load effect. The probability of failure will now be determined, assuming S and R are statistically independent.
The probability of failure occurring at a load effect between S and S+5S = [probability of load effect lying between S and S + 5S] x [probability of resistance, R, being less than S]
where F_{r}(R) is the cumulative probability distribution of R, and,
The terms in the product in Equation (2.24) are the areas shown in Figure 2.7.
The total probability of failure is obtained by summing, or integrating, Equation (2.24) over all possible values of S (between and +°°):
Substituting for F_{r}(S) from Equation (2.25) into Equation (2.26),
where f(S, R) is the joint probability density of S, R.
The acceptable values of the probability of failure in practice, computed from Equation (2.27) are normally very small numbers, typically lxlO-^{2 }to 1 x 10-^{5}.
The safety, or reliability index is defined according to Equation (2.28), and normally takes values in the range of 2 -5.
where Ф-'() is the inverse cumulative probability distribution of unitary normal (Gaussian) variate, i.e. a normal variate with a mean of zero and a standard deviation of one.
The relationship between the safety index, /1, and the probability of failure, p_{f}, according to Equation (2.28) is shown plotted in Figure 2.8.
Equations (2.26) and (2.27) can be evaluated exactly when S and R are assumed to have Gaussian (normal) or lognormal (Appendix C3.2) probability distributions. However, in other cases, (which includes those involving wind loading), numerical methods must be used. Numerical methods must also be used when, as is usually the case, the load effect, S, and resistance, R, are treated as combinations (sums and products) of separate random variables, with separate probabilistic characteristics.
Details of structural reliability theory and practice can be found in a number of texts on the subject (e.g. Blockley (1980), Melchers (1987), Ang and Tang (1990)). Reliability concepts as applied to wind loading were addressed by Rojiani and Wen (1981), Davenport (1983), Pham etal. (1983), and Kasperski and Geurts (2005).
Nominal Return Periods for Design Wind Speeds
The return periods (or annual recurrence intervals) for the nominal design wind speeds in various wind loading codes and standards are discussed in
Figure 2.8 Relationship between safety index and probability of failure.
Chapter 15. The most common choice is 50years. There should be no confusion between return period, R, and expected lifetime of a structure, L. The return period is just an alternative statement of annual risk of exceedance, e.g. a wind speed with a 50-year return period is one with an expected risk of exceedance of 0.02 (1/50) in any one year. An annual recurrence interval, R_{h} can easily be converted to the equivalent return period, R_{P}, through application of Equation (2.4), although they are essentially equal for values greater than 10years.
Assuming a stationary (unchanging) wind climate, the risk, r, of exceedance of a wind speed over the lifetime, can be determined by assuming that all years are statistically independent of each other. Then,
Equation (2.29) is very similar to Equation (2.9) in which the combined probability of exceedance of a wind speed occurring over a range of wind directions was determined.
Setting both R_{;}> and L as 50 years in Equation (2.28), we arrive at a value of r of 0.636. There is thus a nearly 64% chance that the 50-year return period wind speed will be exceeded at least once during a 50-year lifetime - i.e. a better than even chance that it will occur. Wind loads derived from wind speeds with this level of risk must be factored up when used for ultimate limit states design. Typical values of wind load factor, y_{w}, are in the range of 1.4-1.6. Different values may be required for regions with different wind speed / return period relationships, as discussed in Section 2.9.
The use of ‘lifetime exceedance probability’ (LEP), rather than annual risk, for design, enables a changing climate (Section 1.8) to be incorporated into the design process (Xu et al., 2020). Equation (2.30) is a modification of Equation (2.29) in which the structural lifetime, L, is divided into n discrete periods, in each of which the probability of exceedance (return period) of the specified wind speed is evaluated based on the expected state of the climate at that time.
The use of a return period, or annual recurrence interval, substantially higher than the traditional 50years, for the nominal design wind speed, avoids the need to have different wind load factors in different regions. This was a consideration in the revision of the Australian Standard for Wind Loads in 1989 (Standards Australia, 1989), which in previous editions, required the use of a special ‘Cyclone Factor’ in the regions of northern coastline affected by tropical cyclones. The reason for this factor was the greater rate of change of wind speed with return period in the cyclone regions. A similar ‘hurricane importance factor’ appeared in some editions of the American National Standard (ASCE, 1993), but was later incorporated into the specified basic wind speed (ASCE, 1998).
In AS1170.2-1989, the wind speeds for ultimate limits-states design had a nominal probability of exceedance of 5% in a lifetime of 50years (a return period of 1,000 years, approximately). In later versions of this Standard, a range of annual recurrence intervals are provided, with values of 100-2,000years specified in other documents, depending on the assessed importance level of a structure, and the perceived risk to human life. A similar approach has been adopted in recent editions of ASCE-7, with a range of mean recurrence intervals from 300 to l,700years.
However, a load factor of 1.0 is normally applied to the wind loads derived in this way - and this factor is the same in both cyclonic/hurricane and non-cyclonic/non-hurricane regions.
Uncertainties in Wind Load Specifications
A reliability study of structural design involving wind loads requires an estimation of all the uncertainties involved in the specification of wind loads - wind speeds, multipliers for terrain, height, topography, pressure coefficients, local and area averaging effects, etc. Some examples of this type of study for buildings and communication towers were given by Pham etal. (1983, 1992).
Table 2.5 Variability of wind loading parameters
Parameter |
Mean/nominal |
Coefficient of variation |
Assumed distribution |
Wind speed (50-year maximum) |
1.12 |
0.28 |
Gumbel |
Directionality |
0.9 |
0.05 |
Lognormal |
Exposure |
0.8 |
0.15 |
Lognormal |
Pressure coefficient |
0.8 |
0.15 |
Lognormal |
Local & area reduction effects |
0.85 |
0.10 |
Lognormal |
Source: From Pham etal. (1983).
Table 2.5 shows estimates by Pham et al. (1983) of mean-to-nominal values of various parameters associated with wind loading calculations for regions affected by tropical cyclones from the Australian Standard of that time. It can be seen from Table 2.5 that the greatest assessed contributor to the variability and uncertainty in wind load estimation is the wind speed itself - particularly as it is raised to a power of two (or greater, when dynamic effects are important) when wind loads and effects are calculated. A secondary contributor is the uncertainty in the ‘exposure’ parameter in Table 2.5, which is also squared, and includes uncertainties in the vertical profile of mean and gust speeds as discussed in earlier sections of this chapter.
Kasperski and Geurts (2005) have also estimated uncertainties (mean and coefficients of variation) for various wind loading parameters, including the expected reductions in uncertainty in the estimation of aerodynamic coefficients by the use of wind-tunnel tests. Those values are comparable to those given in Table 2.5, and both sets are applicable to buildings, for which resonant dynamic response is not significant.
The uncertainties associated with wind loading of wind-sensitive structures, such as long-span bridges and communication towers, are somewhat different and should be treated separately.