Bending of clamped orthotropic thin plates: polynomial solution

Goloskokov, Dmitriy P.; Matrosov, Alexander V.

doi:10.1177/10812865221075280

Cited by 3 publications

(1 citation statement)

References 18 publications

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“…Tey attained the free vibration parameters of a clamped circular plate. Goloskokov and Matrosov [39] analysed an anisotropic body for the threedimensional elastodynamic problem. Based on relationships developed with MIFs, the bending of an orthotropic plate is considered.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Analysis of Isotropic Homogeneous Beams Using the Method of Initial Functions and Comparison with Classical Beam Theories and FEM

Namdeo,

Dubey,

Dorji

2023

Complexity

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The main aim of this study is the dynamic analysis of isotropic homogeneous beams using the method of initial functions (MIFs) and comparison with classical beam theories and FEM. Also, this research employs the state space methodology, with a special emphasis on isotropy, to analyse simply supported beam systems. A mathematical model for the dynamic response of beams is constructed using the method of initial functions. The novelty of this study lies in its approach to dynamic analysis, where isotropic homogeneous beams are explored without making assumptions, thus ensuring increased precision using the method of initial functions. Importantly, the approach remains free from restrictive assumptions and relies solely on mathematical formulations, yielding results superior to classical beam theories such as Euler–Bernoulli, Timoshenko, and Rayleigh beam theories. In this work, the application of MIFs of various orders (4th, 6th, 8th, and 10th) to calculate natural frequencies is explored, enabling a thorough examination of the beam’s dynamic characteristics. In addition, parameters such as normal stresses, shear stresses, and deflections in different directions are considered to provide a comprehensive understanding of beam behaviour. To validate the findings, a detailed comparison with a finite element method (FEM) is conducted, achieving excellent agreement between the analytical results and FEM solutions. Furthermore, the influence of Poisson’s ratio (μ) on natural frequencies is investigated by varying its value from 0.18 to 0.30. The research also explores the deviation of plane stress values of the beam section from those estimated using the FEM for the corresponding components.

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Section: Introductionmentioning

confidence: 99%

Dynamic Analysis of Isotropic Homogeneous Beams Using the Method of Initial Functions and Comparison with Classical Beam Theories and FEM

Namdeo,

Dubey,

Dorji

2023

Complexity

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Sector Plate Bending: Using Computer Algebra Systems

Goloskokov

Matrosov

2023

Proceedings of Petersburg Transport University

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Purpose: To investigate the stress-strain state of a thin homogeneous isotropic plate in the form of a sector by numerical-analytical method. To consider the possibility of using computer algebra systems (CAS) to calculate sectoral plates operating under bending due to a transverse load. To demonstrate the effectiveness of applying one of these CASs through the example of the Maple system for calculations using the Ritz method — performing analytical transformations when calculating the integral that determines the total potential energy functional, forming and solving the main resolving system of linear algebraic equations with respect to unknown numerical coefficients in the formula approximating the deflection of the plate, visualization of the obtained solution. Methods: A direct method is used to solve the variational problem of minimizing the functional of the total potential energy of deformation of a thin homogeneous isotropic plate in the form of a sector — the Ritz method. The solution is constructed in the form of a series in terms of basis functions. As basis functions, polynomial functions are chosen that exactly satisfy all boundary conditions. Results: An approximate numerical-analytical solution has been obtained for the problem of bending a sectoral plate in the form of a quarter of a circle, clamped along the contour and loaded with a uniformly distributed load. The effectiveness of using the Maple analytical computing system for solving the problem of bending a sectoral plate by the variational Ritz method is demonstrated. It is shown that the resulting solution quickly converges both for deflection and for bending moments and stresses. Practical significance: The algorithm proposed in the article for solving the problems of bending sectoral plates can be recommended for practical use in determining the stress-strain state of such plates.

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An Exact In-Plane Equilibrium Equation for Transversely Loaded Large Deflection Membranes and Its Application to the Föppl-Hencky Membrane Problem

Sun,

Wu,

et al. 2023

Mathematics

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In the existing literature, there are only two in-plane equilibrium equations for membrane problems; one does not take into account the contribution of deflection to in-plane equilibrium at all, and the other only partly takes it into account. In this paper, a new and exact in-plane equilibrium equation is established by fully taking into account the contribution of deflection to in-plane equilibrium, and it is used for the analytical solution to the well-known Föppl-Hencky membrane problem. The power series solutions of the problem are given, but in the form of the Taylor series, so as to overcome the difficulty in convergence. The superiority of using Taylor series expansion over using Maclaurin series expansion is numerically demonstrated. Under the same conditions, the newly established in-plane equilibrium equation is compared numerically with the existing two in-plane equilibrium equations, showing that the new in-plane equilibrium equation has obvious superiority over the existing two. A new finding is obtained from this study, namely, that the power series method of using Taylor series expansion is essentially different from that of using Maclaurin series expansion; therefore, the recurrence formulas for power series coefficients of using Maclaurin series expansion cannot be derived directly from that of using Taylor series expansion.

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Bending of clamped orthotropic thin plates: polynomial solution

Cited by 3 publications

References 18 publications

Dynamic Analysis of Isotropic Homogeneous Beams Using the Method of Initial Functions and Comparison with Classical Beam Theories and FEM

Dynamic Analysis of Isotropic Homogeneous Beams Using the Method of Initial Functions and Comparison with Classical Beam Theories and FEM

Sector Plate Bending: Using Computer Algebra Systems

An Exact In-Plane Equilibrium Equation for Transversely Loaded Large Deflection Membranes and Its Application to the Föppl-Hencky Membrane Problem

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