ENTIDADES COOPERANTES DEL SECTOR PETROQUIMICO

In this section a level II formulation of the ultimate limit state shall be undertaken. This reliability analysis is used to calculate the safety index, β, which is then com- pared to the target safety index.

The basic assumptions of this method is that the strength of the material is a s.v. denoted R, that can be described by a log-normal distribution. The self-weight of the bridge, denoted G is a s.v. which is normally distributed and the traffic load effect, Q, is described by either the first and second moments which is based on the assumption of a normal distribution or by a normal tail approximation of the actual distribution cf. section 4.6.6. The failure function then becomes:

g(r, g, q) = r− g − q (4.67)

The basic variables are then transformed into the standard normal space.

The transformation of the s.v. R into the standard normal space Y₁ was done using the approximation

R = µRexp(Y1CovR) (4.68)

while the variables G and Q are transformed using the transformations

G = µG+ Y2σG (4.69)

Q = µ_Q+ Y₂σ_Q (4.70)

where µ_Q and σ_Q are purely the mean, µQ, and standard deviation, σQ, of the

traffic load effect in the case of first and second moment approximation. Whereas

µ_Q and σ_Q are the normal tail approximations (see subsection 4.6.6) in the case of non-normal distributions.

Substitution of (4.68)–(4.70) into (4.67) yields the normalised coordinate system for the limit state function as

g(y) = µRexp(y1CovR)− (µG+ σGy2)− (µQ+ σQy3) (4.71)

recognising that the coefficient of variation, Cov, is given by Cov = σ/µ, and dividing throughout by the nominal traffic load effect defined earlier in section 4.9.1,

Qn, (4.71) becomes g(y) = µR Qn exp(y₁CovR)− µG Qn (1 + CovGy2)− µ_Q Qn (1 + Cov_Q y₃) (4.72)

Also by substitution of ν from (4.62) into (4.72) yields

g(y) = µR

Qn

exp(y₁CovR)− ν(1 + CovGy2)−

µ_Q Qn

Differentiation of the above limit state function yields the gradient vector which at the m-th iteration ∇g(y(m)_{) =}  µR Qn

CovRexp(y₁(m)CovR) − νCovG −

µ_Q Qn

Cov_Q



(4.74) The above equations (4.73) and (4.74) can hence be used in the iteration procedure described in subsection 4.6.5 using the normal tail approximation of section 4.6.6 or not, for the normalised load variable, depending on the type of distribution assumed for the traffic load, i.e. normal or non-normal.

4.10.1

Allowance for Increased Allowable Axle Load

In the reliability method a future increase in allowable axle load was accounted for by adjusting the parameters of the distribution used to describe the traffic load effect. The increase was assumed to be linear, shifting the mean value of the distribution by a factor equivalent to the percentage increase in allowable axle load. The increase in variance of the traffic load effect will therefore be the square of the percentage increase. For the GEV and Gumbel distribution this is equivalent to increasing the location and the scale parameters by the percentage increase, the scale parameter being unchanged. If the allowable axle load is increased by a factored κ then the corresponding adjustment to the GEV and the Gumbel parameters will be given by

σ_inc = κσ (4.75)

µ_inc = κµ (4.76)

ξ_inc = ξ (4.77)

Statistical Models and Tools

5.1

General

In this chapter the statistical tools and mathematical models used in this thesis are introduced together with some of their basic concepts and underlying assumptions. The chapter is broken into three main parts, the first of which describes some of the general statistical tools, terminology and definitions used in the thesis. The second and the third parts both deal with extreme value theory. The second part introduces what is often referred to as the classical approach to extreme value theory, describing the three extreme value distributions and their generalised form of the Generalised Extreme Value distribution. It also discusses how they can be applied to the case of modelling the traffic load effect. The final part describes an alternative method to the classical approach, namely the Peaks-Over-Threshold model and the associated Generalised Pareto Distribution.

Extreme value theory, abbreviated EVT has been used in many disciplines of civil en- gineering to describe the loading distribution of variable loads and even the strength of materials. Extreme value theory has been used in oceanography to describe max- imum sea levels (Shim et al., 1993; de Haan, 1993), in wind engineering (Gross et al., 1993), in mining to estimate the occurrence of large diamond and precious stone deposits (Caers et al., 1998) and to predict extreme traffic load effects on road bridges (Cremona and Carracilla, 1998; O’Connor et al., 1998). It has even been used in financial applications to describe and predict high insurance claims and in stock market applications (Embrechts et al., 1997; Emmer et al., 1998; Rootz´en and Tajvidi, 1997).

As an engineer one is often interested in establishing the maximum loading on a structure in a given reference period of time or more importantly the likelihood of a certain load level being exceeded in a given reference period. This reference period is often, but not always, chosen to be one year, or it may be the structures anticipated service life. We often refer to a 50-year return load, in many instances this may even be as much as a 100-year or even 1000-year. This use of the phrase means the loading level that is likely to be exceeded on average once every 50 years (a 50-

year return load) which is true provided the loading can be described as stationary and that the time between events, i.e. the exceedance of the return load, can be approximated to a Poisson process. Because of the long design lives of bridges and other civil engineering structures we are often forced to extrapolate long past the available data in order to make predictions about events with a very low risk of occurrence. This extrapolation from a mathematical point of view is not strictly justifiable (Caers and Maes, 1998). In the preface to (Coles, 2001) the validity of extrapolation is discussed at length and it is admitted in the book that it is easy to criticise extrapolation of this kind. However, it is also noted that extrapolation is demanded and that the extreme value theory provides the only real model with which to do this.

In order to extrapolate in this manner, a sound mathematical model must form the foundation on which to build. This is provided by the models of the extreme value theory, at least asymptotically, as will be discussed in sections 5.3–5.4.

There are several text books on the subject of extreme value theory and their appli- cations. One of the pioneering works can be found in (Gumbel, 1958), while from an engineering applications point of view the book (Castillo, 1987) is very useful in explaining the fundamentals, while the books (Embrechts et al., 1997; Leadbet- ter et al., 1997) provide a mathematically more rigourous account of the subject together with the underlying assumptions. More recent works on the subject can be found in (Reiss and Thomas, 2001; Coles, 2001). The book (Coles, 2001) is, in my opinion, very informative and well structured, providing the background to the distributions without getting too involved in the pure mathematics. It also shows several examples of the use of the distributions and how they can be used for inference purposes.

However, before discussing extreme value theory some of the diagnostic tools to- gether with some general definitions will be discussed.