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A Review and Study on Ritz Method Admissible Functions with Emphasis on Buckling and Free Vibration of Isotropic and Anisotropic Beams and Plates

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Abstract

The first goal of this work is to present a literature review regarding the use of several sets of admissible functions in the Ritz method. The papers reviewed deal mainly with the analysis of buckling and free vibration of isotropic and anisotropic beams and plates. Theoretically, in order to obtain a correct solution, the set of admissible functions must not violate the essential or geometric boundary conditions and should also be linearly independent and complete. However, in practice, some of the sets of functions proposed in the literature present a bad numerical behavior, namely in terms of convergence, computational time and stability. Thus, a second goal of the present work is to compare the performance of several sets of functions in terms of these three features. To achieve this objective, the free vibration analysis of a fully clamped rectangular plate is carried out using six different sets of functions, along with the study of the convergence of natural frequencies and mode shapes, the computational time and the numerical stability.

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Acknowledgements

The authors greatly appreciate the financial support of FCOMP-01-0124-FEDER-010236 through Project Ref. FCT—Portugal PTDC/EME-PME/102095/2008 and Junta de Andalucía—Spain (Project of Excellence No. P11-TEP-7093). This work was also supported by FCT—Portugal, through IDMEC, under LAETA, project UID/EMS/50022/2013.

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Authors and Affiliations

  1. Departamento de Ingeniería Mecánica y Diseño Industrial, Escuela Superior de Ingeniería, Universidad de Cádiz, Cádiz, Spain

    Pablo Moreno-García

  2. IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    José V. Araújo dos Santos

  3. DEM, ISEP, Instituto Politécnico do Porto, Porto, Portugal

    Hernani Lopes

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  1. Pablo Moreno-García
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Appendix

Appendix

This Appendix presents the plots of admissible functions with \(m=1, 2, 10\) and 15 (Figs. 13, 14, 15, 16, 17 and 18). We see that the shapes of the functions with \(m=1\) and 2 are very similar (with the exception of the Non-Orthogonal Polynomials), but for \(m=10\) and 15 the shape varies substantially. It is also clear a breakdown of the Characteristic Function with \(m=15\) for large values of x (Fig. 13d). The symmetry of the Orthogonal Polynomials is lost for values of \(m>7\).

Fig. 13
figure 13

Characteristic functions: (a) \(m = 1\), (b) \(m=2\), (c) \(m=10\), and (d) \(m=15\)

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Fig. 14
figure 14

Modified characteristic functions: (a) \(m = 1\), (b) \(m=2\), (c) \(m=10\), and (d) \(m=15\)

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Fig. 15
figure 15

Orthogonal polynomials: (a) \(m = 1\), (b) \(m=2\), (c) \(m=10\), and (d) \(m=15\)

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Fig. 16
figure 16

Non-orthogonal polynomials: (a) \(m = 1\), (b) \(m=2\), (c) \(m=10\), and (d) \(m=15\)

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Fig. 17
figure 17

Product of trigonometric functions: (a) \(m = 1\), (b) \(m=2\), (c) \(m=10\), and (d) \(m=15\)

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Fig. 18
figure 18

Static beam functions: (a) \(m = 1\), (b) \(m=2\), (c) \(m=10\), and (d) \(m=15\)

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Moreno-García, P., dos Santos, J.V.A. & Lopes, H. A Review and Study on Ritz Method Admissible Functions with Emphasis on Buckling and Free Vibration of Isotropic and Anisotropic Beams and Plates. Arch Computat Methods Eng 25, 785–815 (2018). https://doi.org/10.1007/s11831-017-9214-7

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