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Analysis of nonlinear Timoshenko–Ehrenfest beam problems with von Kármán nonlinearity using the Theory of Functional Connections

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  • Yassopoulos, Christopher
  • Reddy, J.N.
  • Mortari, Daniele

Abstract

In this paper, the Theory of Functional Connections (TFC) is used to analyze static beams, accounting for the von Kármán nonlinearity and using the Timoshenko–Ehrenfest beam theory. The authors extend their earlier framework on linear beam bending problems to nonlinear bending problems using TFC. The TFC results and performance parameters are then compared to those of the Finite Element Method (FEM) to both validate the TFC solutions and compare computational efficiencies. In addition, this paper also focuses on the benefits of using TFC over FEM for stress analysis of beam bending problems. Also, a TFC methodology to solve buckling and free vibration problems for the linearized Timoshenko–Ehrenfest beam equations is introduced and validated. The results within this paper suggest that for most static beam bending problems, TFC provides more accurate solutions in terms of the residuals of the differential equations and a faster solution time when compared to the FEM using linear or quadratic approximations. Also, TFC has the added benefit of calculating stress fields that are continuous and smooth everywhere within the domain of the beam while FEM is limited to a piece-wise continuous stress field.

Suggested Citation

  • Yassopoulos, Christopher & Reddy, J.N. & Mortari, Daniele, 2023. "Analysis of nonlinear Timoshenko–Ehrenfest beam problems with von Kármán nonlinearity using the Theory of Functional Connections," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 709-744.
    • Handle: RePEc:eee:matcom:v:205:y:2023:i:c:p:709-744
    DOI: 10.1016/j.matcom.2022.10.015
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    References listed on IDEAS

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    1. Daniele Mortari, 2022. "Theory of Functional Connections Subject to Shear-Type and Mixed Derivatives," Mathematics, MDPI, vol. 10(24), pages 1-16, December.
    2. Enrico Schiassi & Mario De Florio & Andrea D’Ambrosio & Daniele Mortari & Roberto Furfaro, 2021. "Physics-Informed Neural Networks and Functional Interpolation for Data-Driven Parameters Discovery of Epidemiological Compartmental Models," Mathematics, MDPI, vol. 9(17), pages 1-17, August.
    3. Daniele Mortari, 2017. "The Theory of Connections: Connecting Points," Mathematics, MDPI, vol. 5(4), pages 1-15, November.
    4. Hunter Johnston & Carl Leake & Daniele Mortari, 2020. "Least-Squares Solutions of Eighth-Order Boundary Value Problems Using the Theory of Functional Connections," Mathematics, MDPI, vol. 8(3), pages 1-17, March.
    5. Nguyen, Vinh Phu & Anitescu, Cosmin & Bordas, Stéphane P.A. & Rabczuk, Timon, 2015. "Isogeometric analysis: An overview and computer implementation aspects," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 117(C), pages 89-116.
    6. Carl Leake & Hunter Johnston & Daniele Mortari, 2020. "The Multivariate Theory of Functional Connections: Theory, Proofs, and Application in Partial Differential Equations," Mathematics, MDPI, vol. 8(8), pages 1-30, August.
    7. Andrea D’Ambrosio & Enrico Schiassi & Fabio Curti & Roberto Furfaro, 2021. "Pontryagin Neural Networks with Functional Interpolation for Optimal Intercept Problems," Mathematics, MDPI, vol. 9(9), pages 1-23, April.
    8. Daniele Mortari, 2017. "Least-Squares Solution of Linear Differential Equations," Mathematics, MDPI, vol. 5(4), pages 1-18, October.
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    Cited by:

    1. Stojanović, Vladimir & Deng, Jian & Milić, Dunja & Petković, Marko D., 2024. "Vibrational analysis of a coupled damaged Timoshenko beam-arch mechanical system with von Kármán nonlinearities and layer discontinuity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 218(C), pages 334-356.

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