Random Eigenvibrations of Internally Supported Plates by the Boundary Element Method
Michał Guminiak1, Marcin Kamiński2,*
1 Department of Structural Mechanics, Institute of Structural Analysis, Poznań University of Technology, Piotrowo 5 Street, Poznań, 60-965, Poland
2 Department of Structural Mechanics, Faculty of Civil Engineering, Architecture & Environmental Engineering, Łódź University of Technology, Al. Politechniki 6, Łódź, 90-924, Poland
* Corresponding Author: Marcin Kamiński. Email: 
(This article belongs to the Special Issue: Selected Papers from the 26th International Conference on Computer Methods in Mechanics (CMM2025))
Computer Modeling in Engineering & Sciences https://doi.org/10.32604/cmes.2025.071887
Received 14 August 2025; Accepted 22 October 2025; Published online 03 December 2025
Abstract
The analysis of the dynamics of surface girders is of great importance in the design of engineering structures such as steel welded bridge plane girders or concrete plate-column structures. This work is an extension of the classical deterministic problem of free vibrations of thin (Kirchhoff) plates. The main aim of this work is the study of stochastic eigenvibrations of thin (Kirchhoff) elastic plates resting on internal continuous and column supports by the Boundary Element Method (BEM). This work is a continuation of previous research related to the random approach in plate analysis using the BEM. The static fundamental solution (Green’s function) is applied, coupled with a non-singular formulation of the boundary and domain integral equations. These are derived using a modified and simplified formulation of the boundary conditions, in which there is no need to introduce the Kirchhoff forces on a plate boundary. The role of the Kirchhoff corner forces is played by the boundary elements placed close to a single corner. Internal column or linear continuous supports are introduced using the Bèzine technique, where the additional collocation points are introduced inside a plate domain. This allows for significant simplification of the BEM computational algorithm. An application of the polynomial approximations in the Least Squares Method (LSM) recovery of the structural response is done. The probabilistic analysis will employ three independent computational approaches: semi-analytical method (SAM), stochastic perturbation technique (SPT), and Monte-Carlo simulations. Numerical investigations include the fundamental eigenfrequencies of an elastic, thin, homogeneous, and isotropic plate.
Keywords
Plate natural vibrations; stochastic boundary element method; semi-analytical probabilistic method; stochastic perturbation technique; Monte-Carlo simulation
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