Calculus of Long Rectangular Plates Embedded in Long Borders with Uniform Vertical Load on a Line Parallel to the Long Borders

Daniel Opruţa, Mihai-Sorin Tripa, Luminiţa Codrea, Cristian Boldor, Dan Dumea, Robert Gyorbiro, Cosmin Brisc, Iulia Bărăian, Petre Opriţoiu, Aurel Chereches and Mihaela Suciu ()
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Daniel Opruţa: Department of Mechanical Engineering, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
Mihai-Sorin Tripa: Department of Design Engineering and Robotics, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
Luminiţa Codrea: Department of Mechanical Engineering, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
Cristian Boldor: Department of Mechanical Engineering, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
Dan Dumea: Department of Mechanical Engineering, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
Robert Gyorbiro: Department of Mechanical Engineering, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
Cosmin Brisc: Department of Mechanical Engineering, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
Iulia Bărăian: Department of Automation, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
Petre Opriţoiu: Department of Land Measurements and Cadaster, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
Aurel Chereches: Department of Road Vehicles and Transport, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
Mihaela Suciu: Department of Mechanical Engineering, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania

Mathematics, 2025, vol. 13, issue 6, 1-20

Abstract: This article presents the Transfer Matrix Method as a mathematical approach for the calculus of different structures that can be discretized into elements using an iterative calculus for future applications in the vehicle industry. Plate calculus is important in construction, medicine, orthodontics, and many other fields. This work is original due to the mathematical apparatus used in the calculus of long rectangular plates embedded in both long borders and required by a uniformly distributed force on a line parallel to the long borders. The plate is discretized along its length in unitary beams, which have the width of the rectangular plate. The unitary beam can also be discretized into parts. As applications, the long rectangular plates embedded on the two long borders and charged with a vertical uniform load that acts on a line parallel to the long borders are studied. A state vector is associated with each side. For each of the four cases studied, a matrix relationship was written for each side, based on a transfer matrix, the state vector corresponding to the origin side, and the vector due to the action of external forces acting on the considered side. After, it is possible to calculate all the state vectors for all sides of the unity beam. Now, the efforts, deformations, and stress can be calculated in any section of the beam, respectively, for the long rectangular plate. This calculus will serve as a calculus of resistance for different pieces of the components of vehicles.

Keywords: long rectangle plate; unit beam; transfer matrix; state vector; charge density; Dirac’s function and operators; Heaviside’s function and operators; vehicle industry (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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