Adaptive finite element analysis for damage detection of non-uniform Euler–Bernoulli beams with multiple cracks based on natural frequencies

Abstract

In this study, an adaptive finite element method (FEM) is developed for structural eigenproblems of cracked Euler-Bernoulli beams via the superconvergent patch recovery displacement technique. This research comprises the numerical algorithm and experimental results for free vibration problems (forward eigenproblems) and damage detection problems (inverse eigenproblems). Design/methodology/approach The weakened properties analogy is used to describe cracks in this model. The adaptive strategy proposed in this paper provides accurate, efficient, and reliable eigensolutions of frequency and mode (i.e. eigenpairs as eigenvalue and eigenfunction) for Euler-Bernoulli beams with multiple cracks. Based on the frequency measurement method for damage detection, utilizing the difference between the actual and computed frequencies of cracked beams, the inverse eigenproblems are solved iteratively for identifying the residuals of locations and sizes of the cracks by the Newton-Raphson iteration technique. In the crack detection, the estimated residuals are added to obtain reliable results, which is an iteration process that will be expedited by more accurate frequency solutions based on the proposed method for free vibration problems. Findings Numerical results are presented for free vibration problems and damage detection problems of representative non-uniform and geometrically stepped Euler-Bernoulli beams with multiple cracks to demonstrate the effectiveness, efficiency, accuracy and reliability of the proposed method. Originality/value The proposed combination of methodologies described in the paper, leads to a very powerful approach for free vibration and damage detection of beams with cracks, introducing the mesh refinement, that can be extended to deal with the damage detection of frame structures.

Numerical analysis of the dynamic evolution of mining-induced stresses and fractures in multilayered rock strata using continuum-based discrete element methods

“…The traditional numerical models, such as those that employ the finite element method (FEM) have various limitations, as the meshes cannot be changed easily after model creation and require further mesh refinements 25 ; therefore, the adaptive finite element method (AFEM) was developed for effective mesh refinement. 26 The extended finite element method (XFEM) was developed to simulate multiple fractures, but the accuracy of the results still depends on the quality of mesh generation. 27 Other numerical techniques that consider the initiation and propagation of fractures are being developed, such as bonded particle models (BMPs), 28 the discontinuous deformation analysis (DDA) method, 29 and the boundary element method (BEM).…”

An h-version adaptive FEM for eigenproblems in system of second order ODEs: vector Sturm-Liouville problems and free vibration of curved beams

“…Although the above criteria are the ultimate aims, they cannot be used as the stop criteria for a computing procedure because exact solutions are not usually available. On the current mesh, the superconvergent solution u * k , which possesses higher accuracy than conventional FE solution, can be used instead of the exact solution to estimate the accuracy of the eigenfunction in energy norm (Wang et al, 2018;Zienkiewicz, 2006). In this case, the superconvergent solution of the eigenvalue is obtained through computing the Rayleigh quotient (Wilkinson, 1965;Wilkinson and Reinsch, 1971) by using the superconvergent solutions of the eigenfunction.…”

“…The superconvergent patch recovery displacement method was developed (Wang et al, 2018;Wiberg et al, 1999a;Wiberg et al, 1999b) to acquire the superconvergent displacements of the FE solutions in static and dynamic problems. The displacements provided by this method can be applied to eigenfunctions.…”

“…For example, if element e is a superconvergent computation element and elements e -1 and e þ 1 can be considered its neighbouring elements, all FE nodes in patched elements e -1, e and e þ 1 are selected for the computation process. Further, the superconvergent displacements for element e can be computed as (Wang et al, 2018):…”

“…Further, high-precision solutions of eigenvalues and eigenfunctions play important roles in research and engineering. For example, the representative dynamic analysis of structures based on high-precision solutions of natural frequency (eigenvalue) or vibration mode (eigenfunction) (Chang and Kim, 2016;Wang et al, 2018) enables the accurate identification of the number, size and location of the damages in cracked structures (Fan and Qiao, 2011;Yang and Yang, 2018). However, solutions that meet user-specified error tolerances are challenging to provide, especially for issues involving the coefficients of variable matrices (Huang et al, 2013), coincident and adjacent approximate eigenvalues (Shen and Shieh, 1999), continuous orders of eigenpairs (Lee, 2000), varying boundary conditions (Jin et al, 2013) and variable cross-sections or curvatures (Raveendranath et al, 2000;Tseng et al, 1997;Zhou and Cheung, 2000).…”