In order to facilitate the broad application of the reliability method, a gen- eralized reliability problem is required to be defined, derived from the load- resistance case presented earlier. However, in most engineering applications,
R and S will not comprise single variables but will be a function of a number
of basic variables which contribute the limit-state function. All basic variables can be represented by the vector X. Now, by expressing the generalized limit-
state function as G(X), the failure probability for the joint probability density
function fX(x)can be expressed as:
Pf = P [G(X) ≤ 0] = Z ... Z G(X)≤0 fX(x)dx (2.7)
In most cases of evaluating the generalized failure probability, the integration of the probability density functions cannot be performed analytically, and must be approximated using appropriate methods; of which the two leading approaches are transformation methods and simulation methods. Using simulation meth- ods, such as Monte Carlo methods, the multi-dimensional integral can be eval- uated. Conversely, transformation methods are used when bypassing the inte- gration is desirable, and the joint probability density function is transformed to a multi-normal probability density function which can be described by its mo- ments. Although often seen to be competing methods, the belief held by some researchers is that these methods should be seen as complementary; as one method may be more appropriate for a specific problem over another (Bjerager 1990).
It is possible to evaluate the failure probability through direct integration, but only in a limited number of instances; specifically where the limit-state function is linear and all random variables are normally or lognormally distributed. For this reason, it is largely considered an impractical method to solve for the failure probability.
2.3 Second Moment Transformation and Simulation Methods
2.3.1
Problem Formulation
The First-Order Second Moment (FOSM) method was developed to linearise the nonlinear limit-state function using a Taylor series expansion about a lin- earisation point. The location of this point is best chosen to be the design point, being the point of maximum likelihood. However, it has previously been lin- earised at the mean values of the random variables, giving rise to the name Mean Value First-Order Second Moment (MVFOSM) method. Moreover, the benefit of this method is that it is easier to locate this point than that of the de- sign point, but does not offer as good an approximation. However, linearising the surface at the mean leads to an invariance problem, where the analysis of equivalent limit-state functions will result in a disagreement of the reliability indices. To correct this invariance problem, the first-order reliability method was developed.
2.3.2
First-Order Reliability Method
First-order reliability methods (FORM) involve transforming non-Normal ran- dom variables into comparable Normal random variables that can be described using their first-order moments. This can be achieved using methods such as the Rosenblatt (Rosenblatt 1952) or the approximate Nataf transformations. However, by transforming the random variables, the limit-state function is also transformed, and is usually now represented as a nonlinear function. In order to compute the reliability index, FORM requires a linearisation of the limit-state surface at a point that provides a better approximation than seen with MV- FOSM. The linearisation is achieved through a Taylor series expansion about a point on the limit-state surface, optimally chosen to be the design point u∗.
The prominent computational demand of FORM is through the location of u∗,
and methods to locate this point are discussed later. A general algorithm is developed based on the location methods, which is repeated until the solution converges to a point where u∗ and β stabilise in terms of value.
FORM addresses the invariance problem present using MVFOSM by approxi- mating the limit-state surface at a point as opposed to the mean value of the random variables. But, as can be seen, using an expansion method to linearise the limit-state surface becomes less accurate as the level of curvature of this surface increases, and as such, use of FORM becomes less desirable in these
2.3.3
Second-Order Reliability Method
Second-order reliability method (SORM) is an extension of FORM, but without the need to linearise the limit-state surface. Instead, a hyperparabolic surface is fitted the limit-state surface at the location of the design point. Due to this, SORM is capable of dealing with problems of a higher degree of complexity than FORM, as the method can be extended to highly curved limit-state surfaces. A number of methods have been proposed to evaluate the failure probability using SORM (Der Kiureghian et al. 1987, Hohenbichler and Rackwitz 1988, Tvedt 1990, Der Kiureghian and Stefano 1991), but the simplest implementa- tion of the method involved asymptotic approximations (Breitung 1984) and multiplied the FORM result by a correction factor:
Pf ≈ Φ(−β) n−1 Y i=1 1 √ 1 + βki (2.8) As can be seen, the correction factor is a function of the limit-state curvatures
ki at the design point. Thus, the problem reduces to one of determining the
curvatures of the limit-state surface.
Figure 2.3: FORM linearisation and SORM approximation in standard normal space
2.3 Second Moment Transformation and Simulation Methods
2.3.4
Monte Carlo Simulation for Reliability Analysis
A direct method to evaluate the probability density integral for the limit state function is through simulation methods. The use of simulation methods was proposed as an alternative to the high computational demand required for solv- ing through direct integration, and was specifically helpful in solving for non- linear limit-state functions. Monte Carlo simulation involves artificially running a large number of experiments based on the numerical model, with the output being a function of the number of experimental failures observed. When ap- plied to structural reliability, the failure probability is calculated as being the number of the instances the limit-state function was violated across the total number of experiments run N.
Pf ≈
n[G(xi) ≤ 0]
N (2.9)
To evaluate the accuracy of a Monte Carlo simulation, the coefficient of variance of the failure probability δpf should be checked, and is defined as:
δpf = σpf µpf = s 1 − pf N pf (2.10) Values of 2–5% for δpf are typically deemed to be acceptable. Knowing the ac-
ceptable levels of CoV, and having a target reliability index/failure probability, the number of samples required for an acceptable Monte Carlo simulation can be found from the following formula:
N = 1 δ2 pf 1 − pf pf ! (2.11) It should be noted that the Monte Carlo simulation method is only a practical alternative method when the number of simulations is less than the number of integration points required for a numerical integration. Additionally, the Monte Carlo simulation can be optimized by sampling in the area of the design point. This greatly improves the efficiency of the method and is referred to as Importance Sampling. However, a transformation method, such as FORM, must be conducted in order to locate the design point.
2.3.5
Practical Implementation Aspects
As the computational difficulties from the past have been rectified through de- velopment of robust methods and commercially available software applications which implement them, there remains little reason to hinder the widespread application of the structural reliability method. Remaining difficulties include the sufficient education of engineers to be able to fully utilize the method and work with the output. However, as with any advanced numerical method and associated software application, the quality of the input data is of primary im- portance as small input errors can manifest in large output errors. To effectively model the input variables, the most appropriate probability distribution must be chosen, of which will be highlighted in the following sections.