Effects of variable transformations on errors in FORM results

doi:10.1016/j.ress.2004.11.018

Reliability Engineering & System Safety

Volume 91, Issue 1, January 2006, Pages 112-118

https://doi.org/10.1016/j.ress.2004.11.018 Get rights and content

Abstract

On the basis of studies on second partial derivatives of the variable transformation functions for nine different non-normal variables the paper comprehensively discusses the effects of the transformation on FORM results and shows that senses and values of the errors in FORM results depend on distributions of the basic variables, whether resistances or actions basic variables represent, and the design point locations in the standard normal space. The transformations of the exponential or Gamma resistance variables can generate +24% errors in the FORM failure probability, and the transformation of Frechet action variables could generate −31% errors.

Introduction

The basic variables of environmental actions and structural resistances for most of structural engineering problems are not normally distributed. The ideas behind the first order reliability method (FORM), a basic method for structural reliability, are first to transform all non-normal basic variables into independent standard normal variables, to linearize the limit-state function at the design point in the standard normal space, and then to evaluate the approximate failure probability corresponding to the distance of the origin to a tangent plan of the limit-state surface at the design point in the standard normal space as the reliability index. The errors in FORM reliability index come from a linear approximation of the curve safety margin surface in the standard normal space.

A variable transformation function T(x) transforms an input non-normal basic variable vector x into an independent standard normal basic variable vector u $u = T (x)$ A limit-state function in the input basic variable space $G (x) = 0$ then is transformed into $g (u) = G (T^{- 1} (u)) = 0$

The nonlinearity of g(u) comes from the two sources: nonlinearity of the limit-state function G(x) in the physical space and nonlinearity of the transformation function T. Second partial derivatives of the transformation functions certainly are an additional source which induces nonlinearity of the safety margin in the equivalent standard normal space. Even though G(x) is linear, a nonlinear T can certainly make g(u) nonlinear and can produce errors in FORM results. Errors in FORM reliability index has been concerned and studied by many authors since 1970s. As a result of these studies, different SORM algorithms are recommended. The effects of linearization of the safety margin in the standard normal space are studied in Refs. [1], [2], [3] and others. Fiessler et al. [1] studied the error in FORM reliability induced by combined nonlinear G(x) with two lognormal variables, a Gumbel variable together with four normal variables as an example. However, when the FORM accuracy is studied in the standard normal space, effects of the two causes on the error are mixed together and effects of only transformation of each of different non-normal variables on FORM reliability cannot be evaluated. Hohenbichler and Rackwitz. [4] studied the error in FORM reliability induced by only transformations of two exponential distributions. Comprehensively evaluating errors in FORM reliability induced by transformations of different non-normal variables is meaningful for improving reliability evaluation.

In addition, structural reliabilities for serviceability limit states have been studied for years. While the ultimate limit state is usually corresponding to a low failure probability, the serviceability limit state is usually corresponding to a quite high failure probability. ISO 2394 [5] recommends a target safety index of zero for the reversible serviceability limit state and a target safety index of 1.5 for the irreversible serviceability limit state. A target safety index of zero is corresponding to a failure probability of 0.5, target safety index of 1.5 is corresponding to a failure probability of 0.07. For reliabilities for the ultimate limit state asymptotic characteristics of FORM reliability error when the safety index goes to positive infinity can be used to evaluate reliability algorithms, as some authors did [2]. However, for reliabilities for the serviceability limit state asymptotic characteristics of the reliability error are not important and the case where the safety index has a small value even near by zero should be studied. Therefore, the accuracy of FORM results for non-normal input basic variables needs more study.

This paper comprehensively studies nonlinearity of the transformation functions for nine non-normal variables, discusses characteristics of the nonlinearity when values of the input basic variables are in different regions and quantitatively gives the effects of different non-normal distributions on errors in the FORM results.

Section snippets

Errors in FORM results generated by nonlinearity of transformations

Whether positive or negative the errors in FORM failure probabilities are depends on the second derivatives of the safety margin g(u) to the independent standard normal variables u_i, i=1,2,… at the design point. If the safety domain is concave, that is $\partial^{2} G (x) / \partial u_{i}^{2} > 0$ , i=1,2,…, FORM overestimates the failure probability, while in the opposite case, FORM underestimates the failure probability. The second order reliability method (SORM) proposed to reduce errors in FORM results induced by nonlinear

Characteristics of nonlinearity of transformation functions

In order to simplify the discussion, only one-to-one transformations, for instance, Nataf's distribution, are discussed in the paper. Denoting a marginal probability distribution function of one input variables by F_i(x_i) and a marginal distribution function of an independent standard normal variable produced by an one-to-one transformation by Φ(u_i), a marginal transformation function is $x_{i} = T_{i}^{- 1} (u_{i}) = F_{i}^{- 1} [Φ (u_{i})]$

Its second derivative is $\frac{d^{2} x_{i}}{d u_{i}^{2}} = \frac{d^{2} {F_{i}^{} [Φ (u_{i})]}}{d u_{i}^{2}}$

This paper considers characteristics

Example 1

Consider a linear safety margin $G (x) = x_{1} - x_{2} + \sqrt{2}$ where x₁ and x₂ are independent basic variables with means μ_x1 and μ_x2 of 1 and standard deviation σ_x1 and σ_x2 of 1, respectively.

If x₁ and x₂ are normally distributed, FORM analytically gives an accurate reliability index and an accurate failure probability $β = \frac{μ_{G}}{σ_{G}} = \frac{μ_{1} - μ_{2} + 1.4142}{\sqrt{σ_{1}^{2} + σ_{2}^{2}}} = 1.0, P_{f 0} = Φ^{- 1} (- 1) = 0.1587$ with a design point $u_{1}^{*} = - 0.707, u_{2}^{*} = 0.707$ in the standard normal space. Obviously, x₁ is a resistance variable and x₂ is an action variable. Such a

Conclusions

When basic variables are not normally distributed, transformations of input basic variables into equivalent normal variables can produce additional nonlinearity of safety margins in the equivalent normal space, even for linear safety margins in the input variable space. Such additional nonlinearity of safety margins in the equivalent normal space certainly influences the accuracy of the FORM failure probability. Effects of such transformations on FORM failure probabilities depend on

Acknowledgements

The paper is a result of a work financially supported by the China Ministry of Science and Technology under a National Key Fundamental Development Programming Project ‘Fundamental Research on Safety of Key Civil Engineering Structures in Hazardous Environment’ numbered 2002CB4127009.

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R. Hohenbichler et al.
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There are more references available in the full text version of this article.

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Technical NoteEffects of variable transformations on errors in FORM results

Abstract

Introduction

Section snippets

Errors in FORM results generated by nonlinearity of transformations

Characteristics of nonlinearity of transformation functions

Example 1

Conclusions

Acknowledgements

Quadratic limit states in structural reliability

J Eng Mech ASCE

Asymptotic approximations for multinormal integrals

J Eng Mech ASCE

Improvement of second-order reliability estimates by importance sampling

J Eng Mech ASCE

Technical Note
Effects of variable transformations on errors in FORM results