2.4 Probability distributions of random variables
2.4.2 Some important types of random variable distribution
In this section some of the most important distributions that a random variable can have are discussed, and the fundamental parameters of each of these distributions will be presented. Knowing the basic parameters of these random variables is of paramount importance in struc- tural reliability evaluation. It will be demonstrated in the following chapters that in each case of reliability evaluation the fundamental parameters of a certain distribution (such as how load effect is distributed or how resistance is distributed) play a significant role in the resulting value of the reliability index.
2.4.2.1 Uniform random variables
In case of uniform random variables, the probability density function will have a constant value over the range where it is defined. It is in the form of a function such as f (x) = c. It means that all the values of random variable have the same likelihood of emergence. If it is defined in the range [a, b], the function will have the value as below:
ϕX = 1
a − b (2.24)
If [X] is not in the range [a,b], the value of the function will be zero. The shape of this type of random variable is shown in Figure 2.6 below.
2.4.2.2 Normal random variables
One of the common distributions that can be used for a random variable is a normal distribution. In fact, normal distribution is the most important type of probability distribution in structural reliability [37]. Its probability density function follows the form:
ϕX(x) = 1 σX√2πexp " −0.5 x − µ X σX 2# (2.25)
Figure 2.6: CDF and PDf of a uniform distribution shown respectively [22]
There is no closed-form of the cumulative distribution function available for normal random variables. As a result, there are usually some tables available that give the probability for the standardised realisations of a normal distribution. Also programs like Excel or Matlab have some built-in functions that can be used for this purpose. Throughout this thesis these two programs are used for the relevant calculations. A random variable can be standardised using Equation 2.8. In this case the standardised random variable will have a mean value of zero (µU = 0) and standard deviation of one (σU = 1).
The normal distribution is often used as a theoretical model in structural reliability. It is used to model some load variables like self-weight, mechanical properties such as strength, and geometrical properties [22]. The normal distribution is defined in the interval [−∞, +∞], and its shape is symmetric, as is apparent form Figure 2.7. Consequently, the skewness for a normal distribution will be zero.
2.4.2.3 Lognormal random variables
If a random variable (X) is considered, then it is stated that X possesses a lognormal distribution when Y = Ln(X) has a normal distribution. Lognormal distribution can only be defined for positive values (or absolute values). In order to calculate the probability of a realisation of a lognormal random variable, the fact that Y has a normal distribution is utilised. Thus, it is possible to calculate the probability for a realisation of Y by just using the cumulative distribution function of a standardized normal random variable:
P (Y ≤ y) = ΦY(y − µY
σY
) (2.26)
In order to use Equation 2.26 the values of µY and σY are necessary which are the mean value and standard deviation of the lognormal random variable Y respectively. It is known that y = ln(x), hence µY = µ
ln(x) and σY = σln(x). It is suggested that Equations 2.27 and 2.28 be used to get
these parameters [37]: σln(x)2 = ln(wX2 + 1) (2.27) µ ln(x) = ln(µx) − 0.5σ 2 ln(x) (2.28)
By obtaining the values of mean and variance using these two equations, it is possible to obtain the probability density function of a lognormal distribution which is shown by ϕ(x) [37]:
ϕ(x) = d dXΦ ln(x) − µ ln(X) σln(x) ! = 1 xσln(X)φ ln(x) − µ ln(X) σln(x) ! (2.29)
The Lognormal distribution is commonly used in structural reliability. This is especially the case for modelling the resistance properties of different materials like steel, concrete, timber and so forth [14, 22]. Sometimes for some methods of structural reliability analysis, it is needed to calculate the equivalent normal distribution parameters of a lognormal distribution. This matter is discussed comprehensively in Chapter 3.
2.4.2.4 Extreme value distributions
Extreme value distributions, as is evident form the name, are used to model the extreme values of a specific phenomenon over time. They are useful in modelling wind load or seismic load and other types of phenomena where the distribution of maximum or minimum values is important [37]. These distributions often follow an exponential function. There are three types of extreme value distribution: extreme Type I (Gumbel distribution), extreme Type II, and extreme Type III (Weibull distribution).
The first type of extreme value is mostly referred to as Gumbel distribution. It is sometimes used to develop a probabilistic model for wind loads. Mathematically, its CDF and PDF are
Figure 2.7: Comparison between normal, lognormal and extreme Type I distribution
expressed respectively as below:
ΦX(X) = exp(− exp(−α(x − u)) (2.30)
u ≈ µ − 0.45σ (2.31)
The two parameters (α) and (u) are distribution parameters, and they can be calculated in terms of the mean value (µ) and the standard deviation (σ). The approximate equation given underneath can be used to calculate these parameters [37]:
α ≈ 1.2825
σ (2.32)
u ≈ µ − 0.45σ (2.33)
Extreme type II distribution is addressed as Frechet distribution. It is sometimes used to model the maximum seismic load values that are applied to a structure [37]. Extreme type III is the so-called Weibull distribution. This type of extreme value distribution is beneficial when it is intended to model the minimal values of a certain phenomenon, say material properties such as strength.
Chapter
3
Methods of Structural Reliability Analysis
3.1
Introduction to structural reliability
In Chapter 2 the concepts of probability of failure and the reliability index were discussed. In this chapter the methods of reliability analysis will be discussed. There are different methods to evaluate the reliability of a component when its limit state function (safety margin) is known. Each of these methods has its own strengths and weaknesses. Some of the most useful and important methods will be discussed. At the end, a case will be made about what method to choose and why to choose a specific reliability method. An algorithm will be developed to do a computerized component-level reliability analysis of a structure. The focus will be on simulation methods like Monte Carlo Simulation and first order reliability methods like FORM. These methods are discussed in the following sections below.
3.2
First Order Reliability Methods (FORM)
First order reliability methods are one of the common and effective ways of reliability analysis [22]. These methods were developed based on the so-called “second moment” methods that were mainly developed by Cornell [9]. Second moment methods as the name suggests use the information on first and second moments of the random variables to calculate the reliability index. In this section, firstly the first-order second moment method (FOSM) is discussed (Section 3.2.1). Next, the extensions to this method are discussed in Sections 3.2.2 and 3.2.3.