PSI - Issue 83

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

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3. Crack identification procedure Evolutionary optimisation (EO) algorithms are powerful tools for solving complex problems using global search approaches inspired by natural evolutionary processes. Among these EAs, the Particle Swarm Optimization (PSO) algorithm stands out as one of the most popular and robust methods. PSO is a swarm-based optimisation method inspired by the collective behaviour of groups of individuals in nature, such as swarms of birds or schools of fish. In the context of optimisation, each possible solution is represented by a particle within a multi-dimensional search space. These particles move through the search space following the movements of the other particles in the group, guided by their own personal experience of the best solution found so far. In sum, PSO offers a promising approach to solving complex optimisation problems, such as the prediction of crack properties in a beam, by exploiting swarm dynamics to efficiently explore the search space and reach the global optimum. 3.1. Crack detection using the PSO algorithm To solve the problem of identifying the location and depth of cracks, the PSO algorithm is used. Fig. 2 illustrates the process of solving the problem of determining the depth and location of the hypothetical crack in a bi-clamped beam.

Fig. 2. Flow chart of the PSO algorithm.

By exploiting the fact that the natural frequencies of a beam vary with the depth and location of cracks, it is possible to approach crack detection as an optimisation problem. With this in mind, a cost function is minimised to accurately determine the location and depth of the crack in the beam. This cost function incorporates four measured frequencies from a beam with a hypothetical crack, as well as four frequencies calculated from a beam with specific crack depth and location parameters. To accomplish this task, a cost function is defined as follows:   2 3 * 1 ( , ) i i i c i c d F w f f L     (8) frequency of the cracked beam, and the ith natural bending frequency of the cracked beam calculated through the regression method, respectively. To account for the involvement of natural frequencies from lower modes in the cost function, weighting coefficients, i w , are introduced into the cost function. The i w coefficients are assumed to be 1/ i [13]. In the given expression, i w , i f and * ( , ) i c c f d L represent the ith weighting factor, the ith measured natural bending