Solving Equation 2-19 in closed-formed is impossible and difficulty in computing probability has led to the development of numerous analytical and numerical techniques. These techniques can further be classified into either simulation methods such as direct Monte-Carlo (MC) or approximation methods such as FORM (First Order Reliability Method) or FOSM (First-order Second Order Reliability Method). For small problems with few random variables the approximation methods such as FORM and FOSM are efficient (Frangopol and Moses 1994) but for more complex problem with many random variables MC simulation seems to be more reliable (Papadrakakis and Papadopoulos 1995).
Φ Φ β 2-22
Limit state function , 0
Safe
Failure
0
Figure 2-35: Reliability index defined as the shortest distance in the space of reduced variables
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2.7.1 First-order Reliability Method
Among all approximation methods, first-order reliability method is considered to be one of the most reliable computational methods. The most basic type of FORM is the first- order second-moment reliability method (FOSM) which is based on the first-order Taylor series approximation of the performance function using mean values of random variables. This method is also referred as Mean Value First-Order Second Moment method (MVFOSM). A Taylor series of the linear performance function is given as
where the terms (i=1,2,…n) are constants and the terms are uncorrelated random variables. By expressing the limit state in terms of the reduced variables ( ) and finding the shortest distance from the origin in the n-dimensional space of reduced variable as explained in Section 2.6, the reliability index can be determined as:
This method uses only second moment statistics (mean and standard deviation) of random variables to calculate β while ignoring the information regarding their probability distributions. If the random variables are uncorrelated and defined by normal distributions, this method can exactly predict the reliability index and probability of failure. Otherwise it only provides an approximation of β.
Later, Hasofer and Lind (1974) proposed a modified approach called Hasofer-Lind reliability index by evaluating the limit state function at a point known as the “design point” instead of the mean values. The design point is a point on the failure surface of =0. For these methods the detailed information on the distribution type for random variables are not taken into account. The Hasofer-Lind reliability index method can be generally considered as the first-order second-moment mean value method when the limit state function is linear. However, when the limit state function is nonlinear, since the
, , … , 2-23
∑
design point is generally not known a priori, an iteration method is required to determine the reliability index.
The procedures to calculate the reliability index can be improved by considering the probabilistic distributions of all random variables (Rackwitz et al. 1978). The basic idea of this procedure, which is generally called First Order Reliability Method (FORM), is to calculate the “equivalent normal” values of the mean and standard deviation for each non- normal random variable and use them in the iterative analysis. It requires the following steps as set out by Nowak and Collins (2000):
(1) Formulate the limit state function ( ) and determine the probability distributions with appropriate parameters for all random variables 1,2, … , involved.
(2) Acquire an initial design point by assuming values for 1 of the random variables (usually mean values). The limit state equation 0 can then be solved for the remaining random variables. This ensures that the design point is on the failure boundary.
(3) Determine the equivalent normal mean ( ) and standard deviation ( ) for each design point ( ).
(4) Calculate the reduced variates ( ) corresponding to the design point ( ) using
(5) Compute the partial derivatives of the limit state function with respect to reduced variates and define a column vector as the vector with elements of partial derivative multiplied by −1: 2-25 . . . where | 2-26
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(6) Determine an estimate of β using the following equation:
(7) Find a new design point in reduced variates for 1 of the variables from
(8) Calculate a column vector containing the sensitivity factors using
(9) Calculate the values of corresponding design point in original coordinates for the 1 using
(10) Solve for the value of the remaining random variable by the setting the limit state function 0.
(10) Repeat steps 3 to 10 until β and the design point converge.
2.7.2 Monte-Carlo (MC) method
The Monte Carlo (MC) method is useful for reliability prediction when the system complexity makes the use of approximation methods such as FOSM and FORM unreliable. Monte Carlo simulation technique is a tool to solve the probability integral presented in Equation 2-19 over the failure domain. This method relies on random sampling from random variable distributions to obtain the numerical estimate probability of failure. The most basic version of Monte Carlo technique is called “crude” or “direct”
where .. . 2-27 α β 2-28 2-29 2-30
Monte Carlo simulation in which pseudo-random sampling is used. That method is quite simple and easy to implement as there are only a few requirements that makes the method applicable to very difficult integration problem. The expected error is in order of 1⁄ in which is the number of samples and it is fairly independent of the number of random variables. However, if the probability of failure is small the direct MC method demonstrates a poor computational efficiency and needs large number of samples to achieve an estimate of probability of failure with sufficient accuracy. The convergence rate, which means how quickly the error decreases with the number of samples, is proportional to 1⁄ . This means that to halve the error, four times more samples are required. As an alternative to improve the classical Monte Carlo simulation, quasi- random sequences can be used instead of pseudo-random samples which leads to what is known as Quasi-Monte Carlo (QMC) method. These sequences are totally deterministic and used to generate the representative samples from the probability distribution. The quasi-random sequences, also called low-discrepancy sequences, improve the performance of Monte-Carlo simulations, offering shorter computational time and achieving a given accuracy by far fewer samples. The rate of convergence of the Quasi- Monte Carlo method is in order of 1⁄ .