Calculation of the spherical elements of non-uniform thickness for structures with holes based on the variational RVR-method
DOI:
https://doi.org/10.15587/1729-4061.2020.217091Keywords:
orthotropic shell of inhomogeneous thickness with holes, Reissner principle, R-function theoryAbstract
This paper proposes a theoretically substantiated and universal new method to calculate the three-dimensional stressed-strained state of the statically loaded multi-link orthotropic shell of arbitrary thickness, made of heterogeneous material (a composite). The numerical-analytical RVR method used in this work is based on the Reissner principle, Vekua method, the R-function theory, as well as the algorithm of two-way assessment of the accuracy of approximate solutions to variational problems. In contrast to the classical principles by Lagrange and Castigliano, the application of the mixed variational Reissner principle yields an increase in the accuracy of solving boundary-value problems due to the independent variation of the displacement vector and the stress tensor. Vekua method makes it possible, as a result of expanding the desired functions into a Fourier series based on Legendre polynomials, to replace a solution to the three-dimensional problem with a regular sequence of solutions to the two-dimensional problems in the process of refining the models of shells. The R-function theory that takes into consideration, at the analytical level, the geometric information on boundary-value problems for multi-relationship regions is necessary to build the structures of solutions that accurately meet different boundary conditions. When studying spatial boundary-value problems, the constructed algorithm for a two-way integrated assessment of the accuracy of approximate solutions makes it possible to automate the search for such a number of approximations at which the process of solutions’ convergence becomes persistent. For an orthotropic spherical shell made from the material of non-uniform thickness and weakened by the pole holes, the RVR-method capabilities are shown on the numerical examples of solving the relevant boundary-value problems. The results of the reported research have been discussed, as well as the features typical of the new method, which could be effectively applied when designing responsible shell-type elements of structures in the different sectors of modern industry
References
- Nagle, A., Wowk, D., Marsden, C. (2020). Three-dimensional modelling of interlaminar normal stresses in curved laminate components. Composite Structures, 242, 112165. doi: https://doi.org/10.1016/j.compstruct.2020.112165
- Ahmadi, I. (2018). Three-dimensional stress analysis in torsion of laminated composite bar with general layer stacking. European Journal of Mechanics - A/Solids, 72, 252–267. doi: https://doi.org/10.1016/j.euromechsol.2018.05.003
- Bohlooly, M., Kulikov, G. M., Plotnikova, S. V., Kouchakzadeh, M. A. (2020). Three-dimensional stress analysis of structures in instability conditions using nonlinear displacement-based and hybrid-mixed quadrilaterals based on SaS formulation. International Journal of Non-Linear Mechanics, 126, 103540. doi: https://doi.org/10.1016/j.ijnonlinmec.2020.103540
- Huang, S., Qiao, P. (2020). A new semi-analytical method for nonlinear stability analysis of stiffened laminated composite doubly-curved shallow shells. Composite Structures, 251, 112526. doi: https://doi.org/10.1016/j.compstruct.2020.112526
- Washizu, K. (1982). Variational methods in elasticity and plasticity. New York, 542.
- Li, H., Pang, F., Gao, C., Huo, R. (2020). A Jacobi-Ritz method for dynamic analysis of laminated composite shallow shells with general elastic restraints. Composite Structures, 242, 112091. doi: https://doi.org/10.1016/j.compstruct.2020.112091
- Li, H., Cong, G., Li, L., Pang, F., Lang, J. (2019). A semi analytical solution for free vibration analysis of combined spherical and cylindrical shells with non-uniform thickness based on Ritz method. Thin-Walled Structures, 145, 106443. doi: https://doi.org/10.1016/j.tws.2019.106443
- Wang, Y., Gu, Y., Liu, J. (2020). A domain-decomposition generalized finite difference method for stress analysis in three-dimensional composite materials. Applied Mathematics Letters, 104, 106226. doi: https://doi.org/10.1016/j.aml.2020.106226
- Hii, A. K. W., Minera, S., Groh, R. M. J., Pirrera, A., Kawashita, L. F. (2019). Three-dimensional stress analyses of complex laminated shells with a variable-kinematics continuum shell element. Composite Structures, 229, 111405. doi: https://doi.org/10.1016/j.compstruct.2019.111405
- Zhgenti, V. S. (1991). Study of the stress state of isotropic thick-walled shells of nonuniform structure. Soviet Applied Mechanics, 27, 459–465. doi: https://doi.org/10.1007/BF00887769
- Khoma, I. Y. (1996). Stressed state of an inhomogeneous transversely isotropic spherical shell with a circular strip and given nonlinearly varying tangential stress. Int. Appl. Mech., 32, 955–963. doi: https://doi.org/10.1007/BF02086480
- Vekua, I. N. (1965). Teoriya tonkih pologih obolochek peremennoy tolshchiny. Vol. 30. Tbilisi, 3–103.
- Salo, V., Rakivnenko, V., Nechiporenko, V., Kirichenko, A., Horielyshev, S., Onopreichuk, D., Stefanov, V. (2019). Calculation of stress concentrations in orthotropic cylindrical shells with holes on the basis of a variational method. Eastern-European Journal of Enterprise Technologies, 3 (7 (99)), 11–17. doi: https://doi.org/10.15587/1729-4061.2019.169631
- Guo, W., Zhu, J., Guo, W. (2020). Equivalent thickness-based three dimensional stress fields and fatigue growth of part-through cracks emanating from a circular hole. Engineering Fracture Mechanics, 228, 106927. doi: https://doi.org/10.1016/j.engfracmech.2020.106927
- Salo, V. A. (2003). Kraevye zadachi statiki obolochek s otverstiyami. Kharkiv: NTU «KhPI», 216.
- Salo, V. A. (2000). Dokazatel'stvo dostatochnogo priznaka shodimosti metoda Rittsa dlya smeshannogo variatsionnogo printsipa Reyssnera. Vestnik Har'kovskogo gosudarstvennogo politehnicheskogo universiteta, 95, 70–75.
- Salo, V. A. (2003). O dvustoronney otsenke tochnosti priblizhennyh resheniy zadach teorii obolochek, poluchennyh metodom Rittsa dlya neekstremal'nogo funktsionala Reyssnera. Dopovidi NAN Ukrainy, 1, 53–57.
- Reissner, E. (1950). On a Variational Theorem in Elasticity. Journal of Mathematics and Physics, 29 (1-4), 90–95. doi: https://doi.org/10.1002/sapm195029190
- Pramod, A. L. N., Natarajan, S., Ferreira, A. J. M., Carrera, E., Cinefra, M. (2017). Static and free vibration analysis of cross-ply laminated plates using the Reissner-mixed variational theorem and the cell based smoothed finite element method. European Journal of Mechanics - A/Solids, 62, 14–21. doi: https://doi.org/10.1016/j.euromechsol.2016.10.006
- Faghidian, S. A. (2018). Reissner stationary variational principle for nonlocal strain gradient theory of elasticity. European Journal of Mechanics - A/Solids, 70, 115–126. doi: https://doi.org/10.1016/j.euromechsol.2018.02.009
- Morachkovskii, O. K., Romashov, Y. V., Salo, V. A. (2002). The Method of R-Functions in the Solution of Elastic Problems on the Basis of Reissner's Mixed Variational Principle. International Applied Mechanics 38, 174–180. doi: https://doi.org/10.1023/A:1015760826979
- Nechyporenko, V. М., Salo, V. А., Litovchenko, Р. I., Kovbaska, В. V., Verkhorubov, D. О. (2016). Using of the theory of R-functions for producing а rational interference fit. Zbirnyk naukovykh prats Natsionalnoi akademiyi Natsionalnoi hvardiyi Ukrainy, 2, 72–76.
- Timoshenko, S., Woinowsky-Krieger, S. (1987). Theory of Plates and Shells. New York: McGraw-Hill Book Company, 580.
- Salo, V. A. (2004). O kontsentratsii napryazheniy okolo otverstiya v uprugoy sfericheskoy obolochke. Voprosy proektirovaniya i proizvodstva konstruktsiy letatel'nyh apparatov, 37 (2), 66–72.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2020 Valentin Salo, Vladimir Nechiporenko, Valeriia Rakivnenko, Stanislav Horielyshev, Natalia Gleizer, Alexander Kebko
This work is licensed under a Creative Commons Attribution 4.0 International License.
The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.
A license agreement is a document in which the author warrants that he/she owns all copyright for the work (manuscript, article, etc.).
The authors, signing the License Agreement with TECHNOLOGY CENTER PC, have all rights to the further use of their work, provided that they link to our edition in which the work was published.
According to the terms of the License Agreement, the Publisher TECHNOLOGY CENTER PC does not take away your copyrights and receives permission from the authors to use and dissemination of the publication through the world's scientific resources (own electronic resources, scientometric databases, repositories, libraries, etc.).
In the absence of a signed License Agreement or in the absence of this agreement of identifiers allowing to identify the identity of the author, the editors have no right to work with the manuscript.
It is important to remember that there is another type of agreement between authors and publishers – when copyright is transferred from the authors to the publisher. In this case, the authors lose ownership of their work and may not use it in any way.