# Prediction of Design Wind Speeds and Structural Safety

## Introduction and Historical Background

The establishment of appropriate design wind speeds is a critical first step towards the calculation of design wind loads for structures. It is also usually the most uncertain part of the design process for wind loads, and requires the statistical analysis of historical data on recorded wind speeds.

In the 1930s, the *Gaussian* distribution (Appendix C3.1), with a symmetrical bell-shaped probability density, was used to represent extreme wind speeds, and for the prediction of long-term design wind speeds. However, this failed to take note of the earlier theoretical work of Fisher and Tippett (1928), establishing the limiting forms of the distribution of the largest (or smallest) value in a fixed sample, that depends on the form of the tail of the parent distribution.

The use of extreme value analysis for design wind speeds lagged behind the application to flood analysis. E. J. Gumbel (1954) promoted the use of the simpler Type I extreme value distribution for such analysis. However, von Mises (1936) and Jenkinson (1955) showed that the three asymptotic distributions of Fisher and Tippett could be represented as a single *Generalized Extreme Value (GEV) Distribution*; this is discussed in one of the following sections. By the 1950s and the early 1960s, several countries had applied extreme value analyses to predict design wind speeds. Mainly, Type I (by now also known as the ‘Gumbel Distribution’), was used for these analyses. The concept of *return period,* the reciprocal of the probability of exceedance of an epoch (usually annual) extreme was defined originally by Gumbel (1941) and became widely adopted in the following twenty years.

The use of probability and statistics as the basis for the modern approach to wind loads was, to a large extent, a result of the work of A.G. Davenport in the 1960s. This was recorded in many papers (e.g. Davenport 1961), although there were others, such as Shellard (1958, 1963) in the United Kingdom, and Whittingham (1964) in Australia, who also applied GumbePs methods to make predictions of extreme wind speeds.

In the 1970s and 1980s, the enthusiasm was tempered for the then standard ‘Gumbel analysis’ due to events such as Cyclone ‘Tracy’ in Darwin, Australia (1974) and severe gales in Europe (1987). In those events, the previous design wind speeds that had been determined by a Gumbel fitting procedure were exceeded by quite an extent. This highlighted the importance of the following factors:

- • sampling errors inherent in the recorded data base, usually less than 50 years, and
- • the need for separation of data originating from different storm types.

The need to separate the recorded data by storm type was recognized in the 1970s by Gomes and Vickery (1977a).

The development of probabilistic methods in structural design was parallel with their use in wind engineering, followed by pioneering work by Freudenthal (1947, 1956) and Pugsley (1966). This field of research and development is known as ‘structural reliability’ theory. Limit-states design, based on probabilistic concepts, was steadily introduced into design practice from the 1970s onwards.

This chapter discusses modern approaches to the use of extreme value analysis for prediction of extreme wind speeds for the design of structures. Related aspects of structural design and safety are discussed in Section 2.9.

## Principles of Extreme Value Analysis

The theory of extreme value analysis of wind speeds, or other geophysical variables, such as flood heights, or earthquake ground accelerations, is based on the application of one or more of the three asymptotic extreme value distributions identified by Fisher and Tippett (1928). These are discussed in the following section. They are asymptotic in the sense that they are the correct distributions for the largest of an *infinite* population of independent random variables of known probability distribution. In practice, of course, there will be a finite number in a population, but in order to make predictions, the asymptotic extreme value distributions are still used as empirical fits to the extreme data. Out of the three variables, which one is theoretically ‘correct’, depends on the form of the tail of the underlying parent distribution. However, unfortunately this form is not usually known with certainty due to lack of data. Physical reasoning has sometimes been used to justify the use of one or the other of asymptotic extreme value distributions.

Gumbel (1954, 1958) covered the theory of extremes and the state of the art in the 1950s; although dated, this work is still widely referenced for its methods of fitting the Type I extreme value distribution (see Section 2.3.1

following). A useful review of the various methodologies available for the prediction of extreme wind speeds, including those discussed in this chapter, was given by Palutikof *et al.* (1999). Coles (2001) discussed the principles of extreme value analysis from the point of view of a statistician. Torrielli *etal.* (2013) reviewed many known methods of extreme value analysis, including all of those discussed in the following sections, and tested them with a simulated data series representing more than 12,000years of wind speeds.

### The Generalized Extreme Value Distribution

The *GEV Distribution* introduced by von Mises (1936), and later re-discovered by Jenkinson (1955), combines the three extreme value distributions into a single mathematical form:

where *F _{V}(U)* is the cumulative probability distribution function (see Appendix C) of the maximum wind speed in a defined period (e.g. one year).

In Equation (2.1), *k* is a shape factor, *a* is a scale factor, and *и* is a location parameter. When £<0, the GEV is known as the *Type II Extreme Value* (or *Frechet)* Distribution; when £>0, it becomes a *Type III Extreme Value Distribution* (a form of the *Weibull* Distribution). As *k* tends to 0, Equation (2.1) becomes Equation (2.2) in the limit. Equation (2.2) is the *Type I Extreme Value Distribution,* or *Gumbel* Distribution.

The GEV with *k* equal to -0.2, 0 and 0.2 are plotted in Figure 2.1, in a form that the Type I Distribution appears as a straight line. As can be seen in the Figure, the Type III Distribution (£=+0.2) approaches a limiting value; it is therefore appropriate for variables that are ‘bounded’ on the high side. It should be noted that the Type I and Type II predict unlimited values; they are therefore suitable distributions for variables that are ‘unbounded’. Since it would be expected that there is an upper limit to the values that can be produced by the atmosphere, the Type III Distribution may be more appropriate for wind speeds.

### Return Period and Average Recurrence Interval

At this point, it is appropriate to introduce the term *return period, R _{r}.* It is the inverse of the complementary cumulative distribution of the extremes (Gumbel, 1941, 1958).

*Figure 2.1* The generalized extreme value distribution (k=-0.2, 0. +0.2).

Thus, if the annual maximum is being considered, then the return period is measured in years. A 50-year return period wind speed has a probability of exceedance of 0.02 (1/50) in any one year. The probability of wind speed, of given return period, being exceeded in the lifetime of a structure is discussed in Section 2.9.3.

The ‘average (or mean) recurrence interval’, *R _{h}* is the average interval between exceedances of high thresholds, and the reciprocal of the average crossing rate. Average recurrence interval is related to return period through Equation (2.4).

A proof of Equation (2.4) is given in Appendix C, and a graph is provided showing the relationship between *R _{P}* and R,. The values converge at high levels and are virtually identical for values beyond 10years (although interestingly there remains a difference of 0.5years).

They diverge for values approaching unity, and R_{;}> cannot be less than 1.0, as this would correspond to a probability of exceedance, exceeding 1.0 by Equation (2.3). On the other hand, *R,* can take fractional values, with a lower limit of 0.

From Equation (2.3), Equation (2.1) for the GEV can be written as:

Then replacing R_{;}> in Equation (2.5) with R, using Equation (2.4),

where

Equation (2.6) is a simplified form of the GEV that will be adopted in subsequent sections and chapters. Note that C in Equation (2.6) is the maximum value of *U* for any *R*i.e. it is an upper limit. (C-D) is the value of *U *for a value of *R,* of 1 year.

### Separation by Storm Type

In Chapter 1, the various types of windstorm that are capable of generating winds strong enough to be important for structural design, were discussed. These different event types will have different probability distributions, and therefore should be statistically analyzed separately; however, this is quite a difficult task as weather bureaus, or meteorological offices do not always record the necessary information. If anemograph records such as those shown in Figures 1.5 and 1.7 are available for older data, these can be used for identification purposes. Modern automatic weather stations (AWS) can generate wind speed and direction data at short intervals of as low as one minute. These can be used to reconstruct time histories similar to those in Figures 1.5 and 1.7 and assist in identifying storm types. Identification criteria to separate storm types have been proposed (e.g. de Gaetano *etal.,* 2014).

The relationship between the combined return period, *R _{P c}* for a given extreme wind speed due to winds of either type, and for those calculated separately for storm types 1 and 2, (Rp,, and

*R*is:

_{P2})Equation (2.7) relies on the assumption that exceedance of wind speeds from the two different storm types in a given year are independent events. Equation (2.7) also shows the relationship between average recurrence intervals, which follows from Equation (2.4).