Crack parameter estimation in structures using finite element modeling

https://doi.org/10.1016/S0020-7683(03)00286-5Get rights and content

Abstract

This paper addresses the problem of linear crack quantification, crack depth estimation and localization, in structures. An optimization technique based on a finite element model for cracked structural elements is employed in the estimation of crack parameters for beam, truss and two-dimensional frame structures. The modal data for the cracked structures are obtained by solving the corresponding eigenvalue problem. The error in the modal data is simulated by an additive noise that follows the normal distribution. The simulated reduced modal data is expanded using the eigenvector projection method. Numerical examples showed that this technique gives good results for cracks with high depth ratio. The accuracy of the estimated crack parameters depends on (1) the number of modes used, (2) the error level in the cracked structure modal data and (3) the number of measured degrees of freedom in the case of reduced modal data.

Introduction

In the past few years the problem of health monitoring and fault detection of structures has received considerable consideration. It was noted that fault cause changes in the dynamic response of the structure. The changes can be considered as an indication of the health of the structure. Consequently, these methods of fault detection are based on the comparison of the dynamic response of the healthy structure with the dynamic response of the defected structure.

The comparison is carried out through some algorithm, which employs the modal data of the healthy and defected structure. Therefore, the fault detection problem is dependant on the modal data for the healthy structure, the modal data for the defected structure and the algorithm that uses these data and provides information about the state of the structure. Each of these items has its own aspects and associated problems that affect the results of the fault detection.

In fault detection literature, four different levels for fault detection have been identified. The first level is the detection of a fault in which the algorithm just indicates that a fault exists in the structure. This level is the simplest level and a simple comparison of the modal data for the healthy and defected structures can achieve this task. In level two, the algorithm locates the defect in the structure. Level three in fault detection gives an estimate of the severity of the defect in addition to the detection and localization (levels one and two). The last level (level four) is that which provides an estimate of the remaining operational life of the structure.

Several algorithms for fault detection were invented and developed during the past few years. The differences between these algorithms are the type of dynamic data that is used and the level of detection. A full literature survey for the fault detection and health monitoring of structures, is presented by Scott et al. (1996) and Mottershead and Friswell (1993). The methods that are based on the frequency change are classified as either forward (Fox, 1992; Friswell et al., 1994) or inverse problems (Dado, 1997; Dado and Abuzid, in press). Such methods have a significant limitation because of the low sensitivity of the natural frequencies for cracks with small depth. Therefore, high level of defect or measurements with high accuracy is required for the detection of cracks with small depth. These methods are considered as level one (detection).

The second set of algorithms is based on the change in mode shapes. The modal assurance criteria (MAC) as an indication about the location of a defect, was first introduced by West (1984). Since then several different measures for the fault detection such as the partial modal assurance (PMAC) and coordinate modal assurance (COMAC) criterion (Kim and Bartkowicz, 1993) and the structural translational and rotational error check (Mays, 1992), were developed. These algorithms do not need a model for the defected structure, which is considered as being their main advantage. A similar set of algorithms uses the change in the mode shape curvature or the strain mode shapes as an alternative for the mode shapes. These methods have the same advantage that is they do not need a model for the structure.

Another class of fault detection algorithms are based on the dynamically measured flexibility matrix of the defected structure, similar to these methods that are based on the change on mode shapes, different criterion were developed to provide and indication about the defect. The direct comparisons of the flexibility of the heathery and delectated structure (Pandy et al., 1991), the unity check method (Lin, 1990) and the stiffness error matrix (Park et al., 1988) were used throughout the fault detection literature.

Matrix update methods (Simth, 1992; Linder and Goff, 1993; Kaouk and Zimmerman, 1994) are based on modifying the structure matrices (mass, stiffness and damping matrices) such that the modal data for the defected structure is reproduced. The differences between these methods are the objective function, the constraints definition and the numerical schemes employed in the optimization.

In such a method, a model for the healthy and defected structures is required. The finite element method is usually used in modeling of the healthy and defected structures. It is usually assumed that the defect affects the stiffness matrix of the defected element but not the mass matrix. The stiffness matrix is represented as a certain fraction of the healthy element stiffness matrix (Abdalla et al., 2000; Kaouk and Zimmerman, 1994) or a localized reduction in the modulus of elasticity. This method is considered as a level three fault detection method.

Many other researches have addressed different issues in fault detection like the effect of noise in the measured modal data, the number of modes used in fault detection, using reduced modal data and applying different schemes in the optimization (Abdalla et al., 2000; Kaouk et al., 1994; Friswell et al., 1997, Friswell et al., 1998, Friswell et al., 1998). It is noted that the fault detection algorithms generally localize the defect by identifying the defected element and the fault severity is indicated as a reduction in the element stiffness. The fourth level of fault detection that provides an estimate of the remaining operational life, needs more information about the defect parameters and the information provided by these fault detection algorithms are not sufficient. Naturally, acquiring more specific information about the defect, which can be used in the fourth level of fault detection, needs a model for the defect that relates the defect parameters to the change in structure dynamic properties (mass, stiffness and damping matrices). Such a model, if available, can be employed in fault detection.

Structures are generally subjected to different forms of faults such as failure of joints, cracks and buckling or complete loss of elements. These defects are generally nonlinear and difficult to model, which can be considered as the main reason for trying to develop fault detection algorithms that do not need a model for the fault such as MAC and the PMAC. However, cracks are the most common type of defects that occur in structures. Much has been done to model the effect of cracks in simple structural elements such as axially loaded members and beams. It was found that analytical modeling of cracks is very difficult especially for practical structures with multiple cracks.

Shpli and Dado (submitted for publication) described a procedure for derivation of modeling finite structural elements with linear (open) cracks in terms of the crack parameters. They derived stiffness matrices for rode, beam and two-dimensional frame finite elements. The derived models are function in the crack depth ratio and the crack location (crack parameters). These models were used in estimating the natural frequencies for beam, truss and two-dimensional frame structures and verified by comparing their results with published experimental and analytical data.

This study employs these models in crack parameter estimation for structures that contain defected elements. The effect of the noise in the modal data, number of modes used and using reduced modal data will be studied for beam, three-dimensional trusses and two-dimensional frame structures. The modal data for the defected structures are obtained by solving the corresponding eigenvalue problem for the cracked structures and the noise is simulated by an additive white noise.

Section snippets

Cracked structure model

A crack, when presented in a structural element causes a reduction in the stiffness matrix of the element. Therefore, the stiffness matrix of a cracked element is expected to be a function of the crack depth and the crack location. Shpli and Dado (submitted for publication) have presented a general procedure for modeling cracked finite structural element. In their model the crack was represented by a localized compliance and the stiffness matrix of the cracked element is expressed in terms of

Objective function

The crack parameter estimation is the process of finding the set of crack parameters (crack depth and crack location) that reproduce the modal data (natural frequencies and mode shapes) of the defected structure. Many objective functions have been suggested to formulate this problem. One objective function (Ruotolo and Surace, 1997) is based on minimizing the difference between the measured and estimated modal data. Mathematically this objective function is given byminimizei∥ωid−ωih∥+∥Uid−Uih

Numerical examples

In this section, three examples for the application of the crack parameter estimation problem for different structures are presented. These examples are a beam with two cracks, a three-dimensional truss with two cracked elements and a two-dimensional frame with one cracked element.

Since no experimental data is available a simulated experimental data is obtained by solving the corresponding eigenvalue problem of the cracked structure. The error in the simulation modal data is emulated by a white

Conclusions

This study showed that the cracked finite element models are efficient in estimating the crack depth ratio and location in complex structures. The results are affected by the noise level, the number of modes used and the reduced modal data. Cracks with small depth ratio are extremely affected by these factors and may be completely lost while cracks with high depth ratio are quantified accurately. It is also noted that using error free modal data will result in accurate estimation of the crack

References (22)

  • M.I. Friswell et al.

    Parameter subset selection in damage location

    Inverse Problems in Engineering

    (1997)
  • Cited by (0)

    View full text