In Section 3.2 second-order methods for computing the reliability index were discussed. Mostly, these methods can only be used when the limit state function is explicitly available, except for the FORM method that uses Newton-Raphson recursive procedure. Another way of calculating the probability of failure and reliability index is using simulation methods. Simulation methods can be used for both explicit and implicit limit state functions. In simulation methods many random samples for each of the basic random variables in the problem are generated in a way that they can represent their true probability distributions and stochastic characteristics. Using each realisation of the random variables in the problem and analysing the problem determin- istically will lead to a realisation of the problem itself. Each of these deterministic analyses is referred to as a simulation trial. A large number of simulation trials give the overall stochastic
characteristics of the problem, especially when the number of simulation trials tends to infin- ity [19]. For instance, if the limit state function for a beam with respect to its moment-bearing capacity is formulated where the limit state is in terms of two random variables, By using a simulation method, it is possible to obtain several random samples for each one the random variables according to their probability distributions, and deterministically evaluate the limit state function for each realisation of both of the random variables. This will lead to finding the stochastic characteristics of the limit state itself which can be used to calculate the probability of failure and the reliability index (refer to Section 3.3.6).
Simulation methods are suitable for calculations by a computer [3], and available software products include special subroutines that make it easy to apply simulation methods in a com- puter environment [22]. As a result, simulation methods provide an appropriate tool to verify the results obtained through other reliability methods (such as FORM) or confirm the validity of the results obtained by a new method.
The Monte Carlo simulation is one of the most common methods of simulation that follows the concepts explained above. It’s based on generating random samples according to the probability distributions of the basic random variables which are treated as inputs. These inputs are used for the deterministic analyses of the limit sate function. The obtained results are assessed to get the probability of failure and reliability index. This procedure is explained in detail below. In Sections 3.3.1 to 3.3.5 different sampling methods are discussed. In Section 3.3.6 the calculation of failure probability and reliability index is explained (the approach to evaluate the obtained results for the limit state). Finally, in Section 3.3.7 the method to check the accuracy of the obtained results is shown.
3.3.1 Generation of random values (inverse transform sampling)
The generation of random numbers is possible if a random number generator which can generate uniformly distributed random numbers between 0 and 1 (the probability of the occurrence of all the random numbers between 0 and 1 is equal) is available. Computer programs like Matlab have the ability to generate uniformly distributed random numbers between 0 and 1. Also, in the available literature, tables are provided that give random numbers in this interval. When a random number ui is generated, this number can be equated to the cumulative distribution function of the random variable as shown in Equation 3.27.
ui = FX(xi) → xi = F
−1
X (ui) (3.27)
It can be demonstrated as a two-step procedure:
This algorithm is very general, and given the accessible functions in a programming language like Matlab can be easily implemented for a Monte Carlo simulation. Throughout this thesis, when the inverse transform sampling is used for generating random samples for the basic random variables, the Monte Carlo method is referred to as the direct Monte Carlo simulation (DMCS).
3.3.2 Latin Hypercube Sampling (LHCS)
In Latin hypercube sampling the probability density function of each random variable is divided into different partitions, and then from each of these “division” a representative value is selected. The representative values are combined in a way that each representative value is used once and only once in the simulation. The following steps can be applied for evaluation of the limit state function (G) [37]:
G = G(X1, X2, ..., Xj) (3.28)
1. Divide the probability density function of each of the random variables into (n) intervals in the considered range (usually [−∞, +∞]). Partitioning should be performed in a way that the probability of Xi being in the interval is1/n.
2. Choose a representative value from each interval. This representative value has to be randomly selected; nevertheless, if the number of intervals is large the middle value of the interval can be selected as well.
3. After steps 1 and 2 are performed there will be (n) representative values for each of the random variables in the limit state function. The objective is to create (n) combinations so that each representative value is only used once (it should be noted that there would be nj combinations available if each value was not supposed to be used only once). For instance, if there are two random variables X1 and X2 together with n intervals, then the vectors of representative values will be as below:
V1 = {x11, x12, ..., x1n}
V2 = {x21, x22, ..., x2n}
Where xjn in vector Vj is the nth representative value of the jth random variable.
4. In order to create n combinations from the representative values, j random permutations of p = 1, 2, ..., n can be created, and then components can be called form each vector of rep- resentative values to create the combinations. For instance, consider a limit state function G = X1− X2, if it is intended to have 5 random samples for each of the random variables, 5 intervals are considered in a way that they divide the probability density function of each of the random variables into 5 partitions so that the probability of random variable Xi occurring in each interval is 1/5 . Then from each of the intervals a representative
value is randomly selected. The vectors of representative values can be shown as below. V1 = [x11, x12, x13, x14, x15]
V1 = [x21, x22, x23, x24, x25]
Next, two random permutation of p = 1, 2, ..., 5 are created which could be P1= [3, 5, 1, 2, 4] and P2 = [1, 2, 3, 4, 5].Therefore, the first combination that is used to evaluate G is com- posed of the third component of the representative vector V1 and the first component of the representative vector V2. This way all of the n combinations can be created. In most of the computer software packages there are ways of randomly permuting numbers (such as “randperm” in Matlab).
5. Eventually, the limit state function G can be evaluated for each of the combinations of random variables. As a result, an approximation of failure probability will be obtained (refer to Section 3.3.7).
The advantage of Latin Hypercube sampling is that it creates a more representative sampling of the distribution since it ensures a good spread of the samples of each random variable even though the number of generated samples is low [30]. In fact, Latin Hypercube sampling will prevent clustering of non-representative sampled values, and it ensures sampling form the tails of the probabilistic distribution of random variables (for instance, if 1000 samples are to be obtained, one sample is taken from the interval [−∞, 0.001] and one sample is taken from the interval [0.999, +∞]. Therefore, upper tail and lower tail are included in the sampling (whereas in direct Monte Carlo sampling none, one or many samples maybe drawn). In general, Latin Hypercube sampling gives more accurate results, and the estimates of failure probability will have a lower coefficient of variation [30]. This matter is demonstrated in Chapter 5 where the application of this sampling method for the component-level reliability evaluation of truss structures is investigated.
3.3.3 Systematic sampling
If the limit state function is in terms of (k) random variables: G = G(X1, X2, ..., Xk)
In order to get (n) sample values for each one of the variables a matrix P can be formed where each column of the matrix includes an independent random permutation of the values i = 1, 2, 3..., n. Components of matrix P can be denoted as Pjk which corresponds to xjk, and xjk is jth simulated value of the kth random variable in the limit state function. Accordingly, xjk can be obtained as below [32]
Using the inverse function of F we get: xjk = Fk−1 2pjk − 1 2N (3.30)
In Equations 3.29 and 3.30 Fkis the cumulative distribution function of the kthrandom variable.
3.3.4 Updated system sampling
Updated system sampling can be seen as an improvement or modification to system sampling [16]. In system sampling when the matrix of random samples is formed there is a correlation between different columns of the sample matrix. The very existence of this correlation can undermine the efficiency of sampling. In updated system sampling an effort is made in order to reduce the correlation among the columns of the random sample matrix.
Assume that there are (k) random variables in the limit state function and the objective is to produce (n) random samples of each random variable. Using systematic sampling matrix R with n rows and (k) columns is formed which contains the random samples. And matrix P is the permutation matrix that contains (n) permutations of i = 1, 2, 3, .., n in each of its k columns. The correlation between the columns of matrix P can be obtained using the so-called Spearman parameter [16]: Tij = 6P n (Rni− Rnj)2 N (N − 1)(N + 1) (3.31)
Tij is known as the Spearman parameter between the two columns i and j, and its value is in the range [-1,1] since it’s stating the correlation. By finding all the Tij values, matrix T can be formed. Matrix T is symmetric and if all the columns of matrix R are independent matrix T is equal to unit matrix. In order to modify the matrix of random variables, first the Cholesky partition of matrix T should be obtained. Cholesky partitioning can be shown as below:
T = QQT (3.32)
Here Q is the lower triangular matrix. If S is shown as the inverse of Q:
S = Q−1 (3.33)
Using S, it is possible to perform a modification to matrix R utilising Equation 3.33:
Rs= R × ST (3.34)
If Spearman matrix Ts is formed for Rs, it can be observed that Ts is closer to unit matrix compared to T . In other words, the correlation between the columns of the random sample matrix is decreased. As the number of modifications increases, the columns get closer to becom-
ing thoroughly independent. This method is used in Chapter 5 together with Latin Hypercube sampling method. It is hence provided with more details in that chapter.
3.3.5 Hybrid sampling methods
Any combination of the sampling methods mentioned in the above sections can be used to ameliorate the efficiency of the Monte Carlo simulation [28]. For instance, it is feasible to use a combination of Latin hypercube sampling and updated system sampling and so forth.
3.3.6 Calculation of probability of failure using simulation methods
Monte Carlo simulation sample values (random realisations) can be drawn for each of the random variables involved in the limit state function through any of the sampling methods specified earlier. Each of these sample values (realisations) can be substituted in the limit state function, and the limit state function can be evaluated deterministically using these sample values. Failure is when the value of the limit state function is smaller than zero. Therefore, a binary vector can be formed for each evaluation of the performance function. If the computed value of the limit state function is smaller than 0 (failure) the corresponding value in the binary vector is 1 and 0 otherwise. This is shown in Equation 3.35 where (n) is the number of simulations. The summation of components of the binary vector will yield the total number of failures for the limit state function. Accordingly, If Nf is the number of failures (total number of times when g < 0 due to substitution of sample values into the limit state function), and n is the total number of simulations, an approximation of the failure probability can be calculated as is shown in Equation 3.36. I(i) = ( 0 G ≥ 0 1 G < 0 i = 1, 2, 3..., n (3.35) Pf = Nf n (3.36)
Another method of calculating the failure probability from Monte Carlo simulation is to cal- culate the mean value and standard deviation of the limit state function using the acquired evaluation values of the limit state function in each simulation. Then, by using Equation 2.17 the reliability index can be calculated. Having obtained the reliability index, it is possible to calculate the probability of failure:
Pf = Φ(−β) (3.37)
Equations 3.38 and 3.39 can be employed to obtain an unbiased estimation of mean and vari- ance [22]
s2 = 1 n − 1 n X i=1 (xi− m)2 (3.39)
The second method assumes that the limit state function follows a normal distribution. Ob- viously, this assumption is not true in many cases where the basic random variables are not normally distributed. However, in cases of high reliability index values (low probability of fail- ure) where the first method leads to zero probability of failure, this method can be used to get an estimation of the failure probability and the reliability index.
3.3.7 Accuracy and efficiency of probability estimation with MCS
It should be mentioned that despite the simplicity of direct Monte Carlo simulation, it can be subject to considerable error if the number of simulations is not large enough. There are some methods to determine the accuracy of a crude Monte Carlo simulation or to determine the required number of simulations to reach certain accuracy.
One method to determine the accuracy of a Monte Carlo simulation is to approximate a bino- mial probability distribution with a normal distribution, and make an estimation about 95% confidence interval of the calculated failure probability. It will lead to Equation 3.40 [19]:
e% = v u u t 1 − PfT N × PfT × 200% (3.40)
In Equation 3.40, pTf is the true probability of failure and N is the number of simulations. Off course, probability of failure is not known a-priori, but if the objective is to confirm the results of other reliability assessment methods such as FORM, this formula can be useful in determining the number of necessary simulations, since already an estimation of the failure probability is available. Usually in engineering problems, it is recommended to have k million numbers of simulation with k being the number of basic random variables [19].