PSI - Issue 83

Amal Lahrizi et al. / Procedia Structural Integrity 83 (2026) 162–170

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1. Introduction In modern industry, monitoring the condition of structures and detecting damage at an early stage are crucial to ensuring the safety and performance of systems. Structures in service are subject to varied and cyclical stresses, which can lead to structural defects, such as cracks, that can cause catastrophic failures. The ability to predict and prevent these failures is essential to maintaining structural integrity. Non-destructive inspection methods, particularly those based on vibrations, have emerged as effective and economical approaches for monitoring changes in the properties of structures. Cracks, by altering the dynamic stiffness of components, have an impact on vibration characteristics, particularly natural frequencies and eigenmodes. This leads to crack detection being approached as an inverse problem, where modal data from testing is used to identify crack characteristics. Among the key structural elements, beams play an essential role in the distribution of loads. However, cracks are proving to be a worrying threat to their integrity. While traditional non-destructive testing methods have their limitations for complex structures, vibration based methods, which take into account variations in natural frequencies, are gaining in relevance. These frequency variations are more accessible to measurement and offer crucial information for detecting and predicting the location and extent of cracks [1]. Over the past few decades, research efforts have been prominently directed towards utilizing vibrations and variations in natural frequencies for the identification of structural damage. Numerous research papers have played a pivotal role in advancing this field. In the study by Outassafte et al. [2], the focus was on detecting cracks in circular arches through an examination of natural frequency variations, employing the Firefly hybrid algorithm. Y. El Khouddar et al. [3] delved into the influence of hygrothermal effects on the free and forced vibrations of functional gradient piezoelectric beams, presenting potential applications in challenging environments. A parallel approach was pursued by Outassafte et al. in their investigation [4], where the authors analyzed the linear and geometrically non linear plane vibrations of a damaged circular arc, thereby laying the foundational groundwork for a promising methodology in predicting structural damage through natural frequency variations. Recent research has expanded the application of vibration-based damage detection techniques to functionally graded beams (FGMs). Within the context of FGM structures, Banerjee et al. [5] introduced a novel approach to crack modeling and detection. They utilized frequency contour analysis and a response surface model with a genetic algorithm (GA) to successfully detect cracks in Timoshenko fiberglass beams subjected to transverse vibrations. Another noteworthy contribution to the condition monitoring of FGM structures comes from Lahrizi et al. [6,7]. Their work applied an optimization algorithm based on the transit search technique to predict and identify cracks in functional gradient beam structures, showcasing a highly efficient detection method. While not directly focused on crack detection, the paper [8] imparts valuable insights into the forced vibrational behavior of complex beams, contributing to a more profound understanding of vibrational responses in functional gradient structures. Finally, in [9], the authors conducted a non-linear analysis of forced vibrations of piezoelectric functional gradient beams in a thermal environment, concentrating on non-linear responses and shedding light on the vibrational behavior under various environmental conditions, which could have implications for crack detection strategies. Collectively, these research papers provide a comprehensive foundation for the study of modeling and detecting cracks in piezoelectric functional gradient beams subjected to transverse vibrations. The present paper places a specific emphasis on the innovative application of the bee algorithm to address the intricate challenge of crack detection in beams. The bee algorithm emerges as a promising solution, capitalizing on natural frequency changes for precise crack prediction. This study signifies a substantial contribution to the enhancement of predictive maintenance practices and the overall durability of structures exposed to diverse stresses.

Nomenclature 11 A

Tensile stiffness coefficient

11 B

Flexural-tensile coupling stiffness coefficient

( , ) E z T Young's modulus z M Bending moment x N Force axiale n

Volume fraction exponent