Moment-based methods

“Moment-based methods” benefit from Taylor expansion series that are used to linearize the limit-state function G(x) in the point of interest. Depending on the order of the Taylor expansion (first or second-order) these methods are referred as First Order Second Moment (FOSM) or Second Order Second Moment (SORM) respectively, where the probability distributions of the limit-state function are approximated by their first and second moments (µG and σG) to determine the reliability index β. Other method based on the approximation of the limit-state function in a particular point is the First Order Reliability Method (FORM), whose development can be derived from the FOSM. The FORM is based on the concept of the Hasofer and Lind Reliability-Index (Hasofer and Lind [83]), which determines the reliability of the structure through an optimization problem.

2.2.2.1 First Order Second Moment Method (FOSM)

This method, also referred as Mean Value First-Order Second-Moment Method (MV- FOSM) attempts to simplify the integration process of Equation 2.2 through the ap- proximation of the limit-state function G(x) through a first-order Taylor series ex- pansion in the vicinity of the mean values of the random variables for estimating its reliability. Assuming that the random variables x are statistically independent, the limit-state function can be approximated as follows:

˜

where µx is a vector with the mean values of the random variables x. Through this expansion the first and second-order statistical moments (mean and standard devia- tion) of the function can be easily obtained. The mean value and standard deviation of the approximate limit-state function ˜G(x) are:

µ_G˜ ≈ E[G(µx)] = G(µx) (2.9a) σ_G˜ =  n X i=1  ∂G(µ_x) ∂xi 2 · σ2 xi 1₂ (2.9b)

According to Equation 2.6, the reliability index β can be expressed as:

β = µG˜ σ_G˜ = G(µx)  Pn i=1  ∂G(µx) ∂xi 2 · σ2 xi 1₂ (2.10)

The reliability index β provides the exact probability of failure when all the random variables are statistically independent, normally distributed and the limit-state func- tion is linear. In other cases the probability of failure can not be obtained directly with Equation 2.10, although can be approximated following other methods.

One of the main drawbacks of the FOSM is that for limit-state functions with high nonlinearities or large coefficients of variation cv = σ_µ, this linearization is not precise enough (Choi et al [39]). Moreover the method provides different results of the relia- bility index β for equivalent formulations of the same limit-state function. Therefore the FOSM is not invariant since the result of β varies depending on how the limit-state function is expressed.

Although the implementation of the FOSM is simple, these drawbacks advise against its use. Moreover the accuracy of the method is not acceptable for low probabilities of failure (Pf < 10−5) or non linear responses (AIR5080 [4]). The same can be stated for the Second Order Second Moment (SOSM) method, where the accuracy is barely improved despite the additional computational cost derived from introducing a second- order term in the Taylor expansion.

2.2.2.2 Hasofer and Lind (HL) Reliability-index. First Order Reliability Method (FORM)

Hasofer and Lind (Hasofer and Lind [83]) proposed in 1974 a methodology to obtain the structural reliability that considers the invariance of the β index and that was developed for uncorrelated and normally distributed random variables x. This method

requires to convert x to their standard Normal distribution N (0, 1), becoming standard normalized random variables u through the well-known transformation:

u = x − µx σx

(2.11)

where u is the vector of the uncorrelated and normalized random variables. Therefore the limit-state function in the new standard and independent u-space is denoted as G(u) = 0.

In this new space the structural reliability index β is defined by Hasofer and Lind as the minimum distance from the origin of the u-space to the failure surface G(u) = 0 and thus is expressed as β = min kuk. Figure 2.3 shows the transformation of the failure surface from the x-space to the u-space and the representation of β in the u- space for a two-variable case. It also shows the first-order Taylor approximation of the limit-state function ˜G in the vicinity of the MPP.

x1 x2 G(x) = 0 Failure region G(x) < 0 Safe region G(x) > 0 x-space µx u1 u2 G(u) = 0 Safe region G(u) > 0

Failure region G(u) < 0

u-space β O MPP ˜ G

Figure 2.3: Mapping of Failure Surface from x-space to u-space and representation of β.

Therefore the obtention of the reliability index β can be expresed as an optimization problem that is presented in Equation2.12:

β = min kuk (2.12a)

subject to:

The solution of this problem (u∗) is denoted as Most Probable Failure Point (MPP) and represents the most probable values of the random variables when the limit-state of the structure G = 0 is reached, thus the MPP is the point of maximum likehood in a failure situation. The reliability index β, which allows to obtain the probability of failure through Pf = Φ(−β), is the distance from the origin of the u-space to the MPP u∗ and is expressed as:

β = G(u ∗_{) −}Pn i=1 ∂G(u∗) ∂ui u ∗ i s Pn i=1  ∂G(u∗₎ ∂ui 2 (2.13)

where n is the dimension of vector u, namely the number of random variables. In the case of non-Gaussian random variables, the evaluation of the structural reliabil- ity β through Equation 2.13 is inappropiate. In such cases the transformation of the random variables to the uncorrelated normalized space of Equation 2.11 becomes nonlinear and is denoted as u = T (x), being the reverse transformation denoted as x = T−1(u). Some approachs to perform this nonlinear tranformation are presented in Hohenbichler and Rackwitz [88], Rossemblatt [175], Der Kiureghian and Liu [51], Rackwitz and Fiessler [169] or Box and Cox [26]. However, when the random variables x are not normally distributed, one of the most common approaches is to convert the non-Gaussian variables into equivalent normal variables. One way to achieve this is through the “equivalent normal distribution”, which lies in matching the CDF and the PDF of the non-Gaussian random distribution and the equivalent normal distribution at the MPP (Rossemblatt [175]), as exposed Equation 2.14.

Fxi(x ∗ i) = Fx0_i(x ∗ i) = Φ( x∗_i − µ_x0 i σ_x0 i ) (2.14a) fxi(x ∗ i) = fx0_i(x ∗ i) = 1 σ_x0 i φ(x ∗ i − µx0_i σ_x0 i ) (2.14b)

where xiis the original non-Gaussian random variable and x

0

iis the equivalent normally distributed random variable. From these equations the mean and standard deviation values of the equivalent standard distribution µ_x0

i and σx 0 i can be derived: µ_x0 i = x ∗ i − Φ −1 [Fxi(x ∗ i)]σx0_i (2.15a) σ_x0 i = φ(Φ−1[Fxi(x ∗ i)]) fxi(x ∗ i) (2.15b)

This methodology is known as the Hasofer Lind - Rackwitz Fiessler method (HL-RF). The steps of the HL and HL-RF method as exposed below:

1. If the random variables are non-Gaussian, transform them to equivalent Gaus- sian random variables through Equation 2.14. Define the mean and standard deviation of the Normal distribution.

2. Perform the tranformation of Equation 2.11from the x-space to the u-space. In the first iteration k = 0 the design points are the mean values of the random variables (x0 = µx).

3. Obtain the linear approximation of the limit-state function in the vicinity of the design point ˜G(uk). This requires the evaluation of G(uk) and ∇G(uk), which is usually performed through FE analysis or simulation codes.

4. Compute αk = −

∇G(uk)

k∇G(uk)k

5. Compute β as exposed in Equation 2.13. 6. Compute new design point as uk+1 = β · αk.

7. Repeat steps from 3 to 6 until the convergence of β. In the HL-RF algorithm, compute (µ_x0, σ

x0) with Equation 2.15 before going to step 3.

A flowchart of the HL/HL-RF algorithm is presented in Figure 2.4.