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Finite element analysis of plate and beam models
Thesis (PhD (Mathematics and Applied Mathematics))--University of Pretoria, 2006.
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University of Pretoria
2013
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| _version_ | 1867613666508013568 |
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| access_status_str | Open Access |
| author2 | Van der Merwe, A.J. |
| author_browse | Van der Merwe, A.J. |
| author_facet | Van der Merwe, A.J. |
| collection | Thesis |
| dc_rights_str_mv | © 2006, University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. |
| description | Thesis (PhD (Mathematics and Applied Mathematics))--University of Pretoria, 2006. |
| format | Thesis |
| id | oai:repository.up.ac.za:2263/30240 |
| institution | University of Pretoria (South Africa) |
| last_indexed | 2026-06-10T12:39:46.144Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UPSpace — University of Pretoria Institutional Repository |
| publishDate | 2013 |
| publishDateRange | 2013 |
| publishDateSort | 2013 |
| publisher | University of Pretoria |
| publisherStr | University of Pretoria |
| record_format | dspace |
| source_str | UPSpace — University of Pretoria Institutional Repository |
| spelling | oai:repository.up.ac.za:2263/30240 Finite element analysis of plate and beam models Van der Merwe, A.J. Van Rensburg, N.F.J. anneke.labuschagne@up.ac.za Labuschagne, Anneke Numerical solutions Differential equations Numerical analysis Finite element method Mathematical models UCTD Thesis (PhD (Mathematics and Applied Mathematics))--University of Pretoria, 2006. We consider linear mathematical models for elastic plates and beams. To be specific, we consider the Euler-Bernoulli, Rayleigh and Timoshenko theories for beams and the Kirchhoff and Reissner-Mindlin theories for plates. The theories mentioned above refer to the partial differential equations that model a beam or plate. The contact with other objects also need to be modelled. The equations that result are referred to as “interface conditions". We consider three problems concerning interface conditions for plates and beams: A vertical slender structure on a resilient seating, the built in end of a beam and a plate-beam system. The vertical structure may be modelled as a vertically mounted beam. How- ever, the dynamics of the seating must be included in the model and this increases the complexity of a finite element analysis considerably. We show that the interface conditions and additional equations can be accommodated in the variational form and that the finite element method yields excellent results. Although the Timoshenko model is considered to be better than the Euler- Bernoulli model, some authors do not agree that it is an improvement for the case of a cantilever beam. In a modal analysis of a two-dimensional beam model, we show that the Timoshenko model is not only better, but it provides good results when the beam is so short that one is reluctant to use beam theory at all. In applications, structures consisting of linked systems of beams and plates are encountered. We consider a rectangular plate connected to two beams. Combining the Reissner-Mindlin plate model and the Timoshenko beam model can be seen as a first step towards a better model while still avoiding the complexity of a fully three-dimensional model. However, the modelling of the plate-beam system is more complex than in the case of the classical theory and the mathematical analysis and numerical analysis present additional difficulties. A weak variational form is derived for all the model problems. This is necessary to apply general existence and uniqueness results. It is also necessary to apply general convergence results and derive error bounds. The setting for the weak variational forms are product spaces. This is due to the complex nature of the model problems. Mathematics and Applied Mathematics unrestricted 2013-09-07T18:22:20Z 2006-12-08 2013-09-07T18:22:20Z 2006-09-07 2006-12-08 2006-12-08 Thesis Labuschagne, A 2006, Finite element analysis of plate and beam models, PhD thesis, University of Pretoria, Pretoria, viewed yymmdd < http://hdl.handle.net/2263/30240 > http://hdl.handle.net/2263/30240 http://upetd.up.ac.za/thesis/available/etd-12082006-135946/ © 2006, University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. application/pdf application/pdf application/pdf application/pdf application/pdf application/pdf application/pdf application/pdf University of Pretoria |
| spellingShingle | Numerical solutions Differential equations Numerical analysis Finite element method Mathematical models UCTD Finite element analysis of plate and beam models |
| title | Finite element analysis of plate and beam models |
| title_full | Finite element analysis of plate and beam models |
| title_fullStr | Finite element analysis of plate and beam models |
| title_full_unstemmed | Finite element analysis of plate and beam models |
| title_short | Finite element analysis of plate and beam models |
| title_sort | finite element analysis of plate and beam models |
| topic | Numerical solutions Differential equations Numerical analysis Finite element method Mathematical models UCTD |
| url | http://hdl.handle.net/2263/30240 http://upetd.up.ac.za/thesis/available/etd-12082006-135946/ |