Shock and Vibration

Volume 2015, Article ID 574846, 12 pages

http://dx.doi.org/10.1155/2015/574846

## Reliability Analysis of Damaged Beam Spectral Element with Parameter Uncertainties

Department of Computational Mechanics, University of Campinas (UNICAMP), 13083-970 Campinas, SP, Brazil

Received 12 December 2014; Accepted 24 May 2015

Academic Editor: Gang Li

Copyright © 2015 M. R. Machado and J. M. C. Dos Santos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper examines the influence of uncertainty parameters on the wave propagation responses at high frequencies for a damaged beam structure in the structural reliability context. The reliability analyses were performed using the perturbation method, First-Order Reliability Method (FORM), and response surface method (RSM) which were compared with Monte Carlo simulation (MCS) under the spectral element method environment. The simulated results were performed to investigate the effects of material property and geometric uncertainties on the response at high frequency modes, such as the computational efficiency of reliability methods. For the first time, the spectral element method is used in the context of reliability analysis at medium and high frequency bands applied to damage detection. It has shown the effects of parameters uncertainty on the dynamic beam response due on an impulsive load and the robustness of each method. Numerical examples in a bending vibrating beam with random parameters are performed to verify the computational efficiency of the present study.

#### 1. Introduction

At medium and high frequency bands approaches like the finite element method (FEM) create large numerical models, which can require very high computational times. Statistical Energy Analysis (SEA) produces small models but without spacial variation in subsystems [1]. An alternative to these problems is the spectral element method (SEM) [2–5]. It consists in the analytical solution of the displacement wave equation in the frequency domain written in the form of a finite element. Then, a spectral element is equivalent to an infinite number of finite elements. This characteristic and the spectral domain make SEM more suitable to solve high frequency band problems. Its wave propagation formulation is also more adequate to solve damage detection problems. In the last decade, researches about damage detection concentrate on methods that use elastic wave propagation at medium and high frequencies to detect and quantify structural damage [6–9]. They use the evidence that material discontinuities, such as a crack, generate changes in the elastic waves propagating into the structure [10]. Some particular advantages of elastic wave-based damage detection methods include their capacity to propagate over significant distances and their high sensitivity to discontinuities near the wave propagation path. The presence of a structural damage introduces a local flexibility change that modifies its vibration response [11–13]. Therefore, vibration energy can be used to investigate the damaged condition of a structure. The treatment of uncertainties using spectral element method is recent [14, 15], and very few were made related with detection and assessment of the damage. Recently, the authors [16–18] and some other researchers [19–21] have presented works in damage detection using wave propagation in the context of uncertainty quantification and stochastic SEM model.

Engineering analysis consists in verifying and certifying that the system complies with many performance criteria, safety, and durability under different kinds of solicitation. It is well known that there are many uncertainty sources in external loads and structural parameters. Consequently, the necessity to incorporate this information in the system design is increasing. With the aim to include uncertainty in system design, reliability methods have been developed in the last few decades. First-order second-moment (FOSM), First-Order Reliability Method (FORM), and Second-Order Reliability Method (SORM) [22–24] are methods based on Hasofer and Lind transformation [25], which include uncertainty in the reliability analyses. Frequently, in the reliability analysis of complex structures, the limit state function cannot be expressed in a closed form. Typically, it needs to be evaluated implicitly through an approximated solution, such as a finite element method. The reliability analysis can also be carried out using Monte Carlo (MC) simulation [26]. However, MC requires a large number of realizations to converge, which could be an expensive technique and for large structures it may be infeasible. In order to reduce the computation time the response surface method (RSM) has been applied [27–32]. Usually, in this case, the actual limit state function is approximated by a polynomial function. Melchers and Choi [33, 34] present the techniques for reliability analysis of engineering structures using probability theory. In the context of damage detection and crack propagation, reliability methods were employed using approximated techniques such as boundary element method (BEM) [35], finite element method [22, 23, 33, 36], and finite element model updating with wave propagation [37]. In this paper a new contribution to the structural damage detection of a nonpropagating crack in the context of reliability methods using the spectral element method is presented.

In this study the structural damage detection problem using the the spectral element method together with reliability algorithms is presented. The damaged beam spectral element [6] is extended to include uncertainty in material property and geometric parameters. A straightforward procedure to estimate the parameter randomness is the Monte Carlo simulation [26]. It consists in obtaining a large number of deterministic analyses with different realizations of random variables. For complex structures with implicit performance function the evaluation of each realization is computationally expensive and the reliability analysis could be impracticable. Therefore, MC simulation becomes computationally infeasible to estimate the failure probability for large structures. Approximated methods like First-Order Reliability Method (FORM) and response surface method (RSM) [28] provide more appropriated tools to estimate the structural failure in these cases.

To consider probabilistic damage detection problems and random crack parameters, it is required to couple the reliability procedures with the spectral element model. This coupling can be performed using either direct coupling or response surface method. In the direct coupling the limit state function derivatives are calculated directly based on the numerical response of the SEM model [36]. In this approach, the limit state function remains implicit and is defined by the SEM model responses. The basic procedure consists in directly coupling the reliability model, FORM, with the numerical beam model based on SEM. In the reliability analysis the RSM is used to approximate the beam response at the vicinity of the most probable failure point in terms of the random variables (material properties, geometries, etc.). The beam response is estimated by a local polynomial approximations in an iterative process, as the search of the design point is evaluated [35]. Numerical tests are presented in order to compare the coupling of the numerical model based on SEM with reliability methods and the computational efficiency of each approach.

#### 2. Spectral Element Method

The spectral element method is similar in style to the finite element method. However, there are two important differences between these methods. The first one is that SEM is a wave propagation formulation written in the frequency domain. The second one is that the element interpolation function is the exact analytical solution of the differential equation. Based on these characteristics the number of elements required for a spectral model will coincide with the number of discontinuities in the structure (Figure 1).