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## Stability analysis of thin walled C-profile beam

### Michał Maciej Bielski

#### Abstract

The following thesis is dedicated to the problem of global stability loss caused by local buckling, and its influence on the structure. The strength of beam structure is highly dependent on the proper stress distribution which is disturbed when buckling occurs. This thesis is divided into three main parts, which concern respectively: the analysis of an undesignated profile beam (as an introduction of the problem), the main analysis of a thin-walled C-profile beam, and the numerical analysis. In the first part the stability issue is discussed in a comprehensive manner, on the general example of a beam showing the influence of geometric and load imperfections on the stability loss. Separated calculations are being conducted with use of two different approaches: the equations of static equilibrium method, and the energetic method – determining the minimum of total potential energy. Comparing them allows to draw conclusions regarding their effectiveness and application in terms of problem discussed. Next analysis concerns the work of thin walled C-profile beam, loaded with bending moment – which reflects the usual load of the beam as the structural element. Based on careful observation of series of cardboard models with different dimensions ratios (length of the bottom-wall in comparison to the height of side-walls), several possible types of local deformations were determined, and their mathematical models were created. This approach is called a geometric method of structure analysis. After that, the internal energy for each local state of deflection was specified, and corresponding equations of total bending moment in relation to deflection angle, for several different parameters, were plotted. Considering those plots, calculated results and the minimum of total potential energy principle, the most probable type of deformation was designated, as well as specific bifurcation point – of conversion between before- and after buckling deformation states. Knowing of the critical load for which buckling deformation occurs is crucial in structure designing process. The thesis focuses on subsequent stages of calculations, and approximations used – starting from a general approach where deflection is presented in sharp edges form, succeeding with arcs rounded with constant or linearly changing radius – all based on experimental observation of earlier mentioned cardboard models. To verify some of the conclusions, a numerical analysis using the finite element method and Ansys Workbench 14.0 software was conducted. The subject of this calculations was a beam model with identical dimensions as in the analytical part. First results concerns the linear buckling of perfect geometry beam, after that the study focused on the non-perfect geometry with different sizes of imperfection. Again, the main purpose was to determine possible buckling modes. Additionally, the numerical analysis considers also stress distribution along the beam. Enlarging the size of imperfection clearly shows how stress distribution converts from typical general bending state to a state modified with local deflection and side-walls buckling of C-profile beam. As with every engineering analysis, the purpose is to create a model possibly close to the real structure – in terms of reaction to applied physical conditions. Therefore to end the study, the suggestion of further improvements of the proposed model were made, which may allow to make the results more accurate.
Diploma type
Engineer's / Bachelor of Science
Diploma type
Engineer's thesis
Author
Michał Maciej Bielski (FPAE) Michał Maciej Bielski,, Faculty of Power and Aeronautical Engineering (FPAE)
Title in Polish
Analiza stateczności belki cienkościennej o profilu C-owym
Supervisor
Krzysztof Kozakiewicz (FPAE/IAAM) Krzysztof Kozakiewicz,, The Institute of Aeronautics and Applied Mechanics (FPAE/IAAM)Faculty of Power and Aeronautical Engineering (FPAE)
Certifying unit
Faculty of Power and Aeronautical Engineering (FPAE)
Affiliation unit
The Institute of Aeronautics and Applied Mechanics (FPAE/IAAM)
Study subject / specialization
, Lotnictwo i Kosmonautyka
Language
(en) English
Status
Finished
Defense Date
25-01-2016
Issue date (year)
2016
Pages
115
Internal identifier
PD; MEL-3418
Reviewers
Tomasz Zagrajek (FPAE/IAAM) Tomasz Zagrajek,, The Institute of Aeronautics and Applied Mechanics (FPAE/IAAM)Faculty of Power and Aeronautical Engineering (FPAE) Krzysztof Kozakiewicz (FPAE/IAAM) Krzysztof Kozakiewicz,, The Institute of Aeronautics and Applied Mechanics (FPAE/IAAM)Faculty of Power and Aeronautical Engineering (FPAE)
Keywords in Polish
stateczność, wyboczenie, profil cienkościenny, praca belki zginanej, punkt krytyczny pracy ustroju, metoda geometryczna
Keywords in English
stability, buckling, thin-walled profile, bending load, critical point of load, bifurcation, geometric method
Abstract in Polish
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MBielski_eng_dipl_thesis.pdf
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Identyfikator pracy APD: 8725

Uniform Resource Identifier
https://repo.pw.edu.pl/info/bachelor/WUT746acee9411143f0b4b5364a86d6411f/
URN
urn:pw-repo:WUT746acee9411143f0b4b5364a86d6411f

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