Elsevier

Engineering Structures

Volume 33, Issue 12, December 2011, Pages 3392-3401
Engineering Structures

Reliability analysis and inspection updating by stochastic response surface of fatigue cracks in mixed mode

https://doi.org/10.1016/j.engstruct.2011.07.003Get rights and content

Abstract

The analysis of engineering structures under fatigue crack growth aims at ensuring an appropriate reliability level over the entire operational lifetime. This paper deals with a new approach, namely the Stochastic Response Surface, to couple finite element analysis and reliability methods. The stochastic collocation method provides an explicit expression of the limit state function related to fatigue failure. This expression is used in first and second order reliability methods in order to compute the failure probability at a given structural age. When inspection is carried out, the structural reliability can be easily updated in terms of the observed crack length. Two numerical applications dealing with fatigue crack growth are presented to illustrate the proposed method, showing its performance in terms of numerical efficiency and accuracy.

Highlights

► Non-intrusive approach based on stochastic collocation and reliability methods. ► The structural reliability is updated in terms of the observed crack length. ► Two applications dealing with fatigue crack growth show its performance.

Introduction

Structural integrity is mainly concerned by predicting the safe remaining life which is governed by Fatigue Crack Growth (FCG). Due to the complexity of this phenomenon, the high level of uncertainty and randomness makes the predictive models very poor in providing effective decision-making tools. To model the FCG, the use of time consuming finite element models is obligatory in order to simulate structures and mechanical parts where the crack propagation path is very often nonlinear, involving bifurcation and mixed tension-shear mode.

Since FCG is a slow cumulative process, this failure can be avoided by means of appropriate scheduling of inspections. Each inspection adds new information about the cumulated damage (i.e. crack length). This additional information can be used to update the structural reliability and therefore to allow us to make better decisions about the rescheduling of next inspections and the optimization of maintenance operations. As material, geometry and loading uncertainties should be considered for structural integrity assessment, the approach should be probabilistic.

The probabilistic crack growth has been largely considered in the literature, in which restrictive assumptions are often introduced to simplify the model complexity. Macias et al. [1] have applied the concepts of probabilistic fracture mechanics to study the integrity of plate containing a central crack. Under crack failure in mode I, their approach combines the Finite Element Analysis (FEA) with First Order Reliability Method (FORM), using quadratic response surface. Mohamed et al. [2] have developed a general reliability method coupling FORM with any general-purpose finite element software. They extended the methodology to the case of correlated and compound random variables (i.e. variables with embedded random statistical parameters). The proposed approach is applied to evaluate the reliability of cracked membrane subjected to thermal loading. Pendola et al. [3] have proposed a probabilistic method to perform nonlinear fracture analysis, using either the direct coupling method or the quadratic response surface method, in order to combine reliability and finite element models. They have demonstrated that the quadratic response surface method becomes more attractive for incremental finite element analyses. Zhao et al. [4], [5] have devoted considerable efforts to study the fatigue reliability of steel-bridge components. In a first paper [4], they have proposed a model based on Linear Elastic Fracture Mechanics (LEFM) to evaluate the structural integrity of welded connections in steel bridges. They integrated a damage function derived from Paris’ FCG law to quantify the cumulated damage in terms of crack length. The first-order second moment reliability method has been used to estimate the probability of failure. They have shown that, compared to the most commonly used AASHTO (American Association of State Highway and Transportation Officials) approach, which is based on the S–N curve, the LEFM-based approach gives almost the same results in terms of reliability index, in addition to providing useful information about crack length. In a second paper [5], the LEFM-based approach has been extended to incorporate informations collected through non-destructive inspections. These informations are modeled by different margin events and the failure probability is updated using the Bayesian approach. In the same context, Crémona [6] proposed a methodology to update the reliability of cracked welded joints, using FORM analysis combined with Bayesian updating. Chung et al. [7] have also used the LEFM-based approach to optimize the inspection scheduling for steel bridges. Rajasankar et al. [8] have studied the integrity of offshore tubular joints, by Monte-Carlo simulations and FORM analysis. The proposed approach has been extended to evaluate the reliability of joints containing two cracks, which are modeled as a series system. The results of periodic inspections have been used to update the failure probability of the joints. Kulkurani and Achenbach [9] have developed an analytical probabilistic method to optimize the inspection schedule for surface-breaking cracks subject to fatigue loading based on the minimization of the total cost. In their study, only the crack length is taken as a random variable and the crack growth is governed by Paris’ law. Under these simplified assumptions, the probability distribution of the crack length at any inspection time can be easily derived. The welded joints in marine structures and bridges have gained a large amount of interest in reliability assessment under FCG. Darshuk [10] has proposed a probabilistic methodology to evaluate the integrity of welded steel bridges, taking into account the irregularity of loads induced by the traffic. His study is concentrated on bridge details, such as welded T-joints containing semi-elliptical defects. The failure probability is calculated by FORM and updated using periodic inspection results. Righiniotis [11] has studied the reliability of cover plate joints subjected to random loading spectrum. The proposed approach is based on two-stage crack propagation law and the failure probability is computed by Monte-Carlo simulations.

Based on this literature review, we can notice that in almost all the proposed fatigue reliability methods, even for engineering structures such as bridges and offshore joints, the FCG is always assumed to follow the opening fracture mode (i.e. mode I). This highly simplified modeling of the crack growth process is adopted to provide analytical formulation for the fracture problem. However, this assumption is far from being acceptable for real structures with complex geometry, irregular loading, imperfect boundary conditions, and random defect shapes and orientations. As a result, the cracks mostly propagate in mixed mode, and consequently the crack path is curved, which can only be modeled by numerical implicit models, such as FEA. Consequently, numerical difficulties arise as many analysis runs become necessary for reliability assessment, leading to prohibitive computation effort.

Basically, the failure probability can be computed by using Monte-Carlo simulations [12], but the extremely high number of FEA runs makes it impossible to deal with engineering structures. An appropriate approximation is provided by the second moment reliability methods such as First and Second Order Reliability Methods, known as FORM/SORM [13]. These methods allow us to couple reliability methods with FEA, using an algorithmic scheme known as the direct coupling method. However, the implicit response provided by FEA leads to a significant number of FEA calls and errors due to numerical differentiation of the mechanical response, especially when nonlinear phenomena are involved. The Response Surface Method (RSM) [14] appeared as a good alternative to overcome the problem of implicit mechanical response. This approach is founded on second order polynomial approximation of the mechanical response, through a limited number of FEA calls. Many approaches have been proposed to improve the efficiency of this method (e.g. [15], [16], [17]), such as updating, resizing and re-using the experiment design points. As the approximation is only valid in a localized region of the design space, the accuracy has a direct impact on the computed failure probability.

In the present work, an alternative to the above approaches is proposed on the basis of a Stochastic Response Surface Method (SRSM). This approach is based on the stochastic collocation method to construct an approximation of the performance function in the standard random space. The specificity of the proposed approach is that the explicit function is constructed by taking into account all the available statistical information of the uncertain input parameters. Consequently, the approximation is valid, not only in the vicinity of the most likely failure point, but also in the whole random space. The proposed approach has been applied to evaluate the reliability of structures subjected to mixed mode FCG. In the following, the reliability method is developed and applied to the fatigue crack growth in mixed mode. The updating procedure is described according to the inspection results concerning the observed crack length. The performance of the proposed approach is evaluated through two numerical applications dealing with FCG.

Section snippets

Stochastic response surface for fatigue crack growth

The reliability analysis aims at computing the failure probability Pf with respect to a given failure criterion. In FCG, this criterion can be described by the limit state function corresponding to the critical crack length. As the mixed mode FCG is evaluated through implicit FEA response, we propose herein to perform the reliability analysis by the Stochastic Response Surface based on the stochastic collocation method [18]. The proposed method allows us to compute the statistical moments and

Reliability analysis and sensitivity measures

For the limit state function G(X), the failure probability Pf of the structure is given by: Pf=Pr[G(X)0]=G(X)0fX(x)dx where fX(X) is the joint pdf of the vector of random variables X={X1,X2,,Xn}T. In the standard Gaussian space, this probability of failure Pf is written: Pf=Pr[H(U)0]=H(U)0ϕn(u)du where ϕn is the n-dimensional Gaussian pdf and H(U) is the image of G(X), in the Gaussian space. In order to avoid integration of the above formula, the First Order Reliability Method (FORM)

Reliability updating through inspections

Structures experiencing FCG require periodic inspections in order to avoid the risk of failure and consequently to ensure safety for human lives and equipment. When inspection does not involve crack length measurements, two types of result could be reported, regarding whether the crack is detected or not. These two events can be written as: a<ad: crack is not detectedaad: crack is detected where ad is the detectable crack length which characterizes the detectability threshold of the inspection

Implementation

The general concepts of reliability analysis using the Stochastic Response Surface Method can be divided into the following steps (Fig. 1):

  • 1.

    Define the probabilistic model for the uncertain input variables X.

  • 2.

    Perform the probabilistic transformation T from the physical space to the standard Gaussian space (Eq. (7)).

  • 3.

    Construct the stochastic response surface (Eq. (11)), by using calls to the FEA modeling the complete crack propagation in mixed mode, under fatigue cycles.

  • 4.

    Compute the reliability index

Numerical applications

Two numerical examples are presented hereafter. The first one aims at explaining the applied methodology and validating the results through convergence and sensitivity analyses. The second example concerns a mechanical component under mixed mode propagation, solved by finite element analysis; in this example, inspection results are introduced to update the structural reliability. In these applications, the vector X of uncertain input parameters involved in the reliability problem is assumed to

Conclusion

The coupling between mechanical analysis and reliability methods represents a powerful tool to assess the integrity of a wide range of structures and mechanical systems. As the mechanical response is always implicit for real structures, the combination between reliability and finite element analysis implies prohibitive computation costs. To overcome this difficulty, the Stochastic Response Surface Method (SRSM) has been developed in the framework of fatigue crack growth in mixed mode. The

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