Engineering Transactions, 62, 1, pp. 33-59, 2014
10.24423/engtrans.42.2014

An Alternative Approach of Initial Stability Analysis of Kirchhoff Plates by the Boundary Element Method

Michał GUMINIAK
Poznan University of Technology Piotrowo 5, 60-965 Poznań
Poland

An initial stability of Kirchhoff plates is analysed in the paper. Proposed approach avoids Kirchhoff forces at the plate corner and equivalent shear forces at a plate boundary. Two unknown variables are considered at the boundary element node. The governing integral equations are derived using Betti theorem. The integral equations have the form of boundary and domain integral equations. The constant type of boundary element are used. The singular and non-singular formulation of the boundary-domain integral equations with one and two collocation points associated with a single boundary element located at a plate edge are presented. To establish a plate curvature by double differentiation of basic boundary-domain integral equation, a plate domain is divided into rectangular sub-domains associated with suitable collocation points. A plate curvature can also be establish by considering three collocation points located in close proximity to each other along line parallel to one of the two axes of global coordinate system and establishment of appropriate differential operators.
Keywords: the boundary element method, Kirchhoff plates, initial stability, fundamental solution.
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Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).

References

Burczyński T., The Boundary Element Method in Mechanics (in Polish), Technical-Scientific Publishing house, Warszawa, 1995.

Altiero N.J., Sikarskie D.L., A boundary integral method applied to plates of arbitrary plane form, Computers and Structures, 9, 163–168, 1978.

Bezine G., Gamby D.A., A new integral equations formulation for plate bending problems, Advances in Boundary Element Method, Pentech Press, London, 1978.

Stern M., A general boundary integral formulation for the numerical solution of plate bending problems, International Journal od Solids and Structures, 15, 769–782, 1978.

Hartmann F., Zotemantel R., The direct boundary element method in plate bending, International Journal of Numerical Method in Engineering, 23, 2049–2069, 1986.

Debbih M., Boundary element method versus finite element method for the stress analysis of plates in bending, MSc Thesis, Cranfield Institute of Technology, Bedford, 1987.

Debbih M., Boundary element stress analysis of thin and thick plates, PhD Thesis, Cranfield Institute of Technology, Bedford, 1989.

Abdel-Akher A., Hartley G.A., Evaluation of boundary integrals for plate bending, International Journal of Numerical Method in Engineering, 28, 75–93, 1989.

Hartley G.A., Development of plate bending elements for frame analysis, Engineering Analysis with Boundary Element, 17, 2, 93–104, 1997.

Beskos D.E., Dynamic analysis of plates by boundary elements, Applied Mechanics Review, 7, 26, 213–236, 1999.

Wen P.H., Aliabadi M.H., Young A., A boundary element method for dynamic plate bending problems, International Journal of Solids and Structures, 37, 5177–5188, 2000.

Wrobel L.C., Aliabadi M.H., The Boundary Element Methods in Engineering, McGraw-Hill College, ISBN 0–07–707769–5, 2002.

Katsikadelis J.T., A boundary element solution to the vibration problem of plates, Journal of Sound and Vibration, 141, 2, 313–322, 1990.

Katsikadelis J.T., A boundary element solution to the vibration problem of plates, International Journal of Solids and Structures, 27, 15, 1867–1878, 1991.

Katsikadelis J.T., Yotis A.J., A New boundary element solution of thick plates modelled by Reissner’s theory, Engineering Analysis with Boundary Elements, 12, 1, 65–74, 1999.

Katsikadelis J.T., Sapountzakis E.J., Zorba E.G., A BEM Approach to Static and Dynamic Analysis with Internal Supports, Computational Mechanics, 7, 1, 31–40, 1990.

Katsikadelis J.T., Kandilas C.B., A flexibility matrix solution of the vibration problem of plates based on the Boundary Element Method, Acta Mechanica, 83, 1–2, 51–60, 1990.

Katsikadelis J.T., Sapountzakis E.J., A BEM Solution to dynamic analysis of plates with variable thickness, Computational Mechanics, 7, 5–6, 369–379, 1991.

Shi G., Flexural vibration and buckling analysis of orthotropic plates by the boundary element method, Int. J. Solids Structures, 12, 26, 1351–1370, 1990.

Guminiak M., Okupniak B., Sygulski R., Analysis of plate bending by boundary element method, [in:] Proceedings of European Conference on Computational Mechanics ECCM-2001, Abstarcts, 1, 176–177, Z. Waszczyszyn, J. Pamin [Eds.], Krakow 2001.

Guminiak M., Analysis of thin plates by the boundary element method using modified formulation of boundary condition [in Polish], Doctoral dissertation, Poznan University of Technology, Faculty of Civil Engineering, Architecture and Environmental Engineering, 2004.

Guminiak M., Sygulski R., Vibrations of system of plates immersed in fluid by BEM, Proceedings of IIIrd European Conference on Computational Mechanics, Solids, Structures and Coupled Problems in Engineering ECCM-2006, pp. 211, C.A. Mota Soares, J.A.C. Rodrigues, J.A.C. Ambrósio, C.A.B. Pina, C.M. Mota Soares, E.B.R. Pereira and

J. Folgado [Eds.], CD enclosed, June 5–9, 2006, Lisbon, Portugal.

Guminiak M., Sygulski R., Initial stability of Kirchhoff plates by the boundary element method using a modified formulation of a boundary condition, Foundations of Civil and Environmental Engineering, 7, 171–186, 2006.

Guminiak M., Sygulski R., The analysis of internally supported thin plates by the Boundary Element Method. Part 1 – Static analysis, Foundation of Civil and Environmental Engineering, 9, 17–41, 2007.

Guminiak M., Sygulski R., The analysis of internally supported thin plates by the Boundary Element Method. Part 2 – Free vibration analysis, Foundation of Civil and Environmental Engineering, 9, 43–74, 2007.

Myślecki K., Approximate fundamental solutions of equilibrium equations for thin plates on an elastic foundation, Archives of Civil and Mechanical Engineering, 1, 4, 2004.

Myślecki K., Metoda elementów brzegowych w statyce dźwigarów powierzchniowych [in Polish], Oficyna Wydawnicza Politechniki Wrocławskiej, Wrocław 2004.

Myślecki K., Oleńkiewicz J., Analiza częstości drgań własnych płyty cienkiej metodą elementów brzegowych [in Polish], Problemy naukowo-badawcze budownictwa, Wydawnictwo Politechniki Białostockiej, Białystok, 2, 511–516, 2007.

Oleńkiewicz J., Analiza drgań wybranych dźwigarów powierzchniowych metodą elementów brzegowych, Rozprawa doktorska, Politechnika Wrocławska, Instytut Inżynierii Lądowej, 2011.

Litewka B., Analysis of Reissner plates by the boundary element method with interaction with liquid, Doctoral dissertation [in Polish], Poznan University of Technology, Faculty of Civil and Environmental Engineering, 2007.

Litewka B., Sygulski R., Application of the fundamental solutions by Ganowicz in a static analysis of Reissner’s plates by the boundary element method, Engineering Analysis with Boundary Elements, 34, 1072–1081, 2010.

Ganowicz R., Selected problems of theory of Reissner and three layer plates [in Polish], Theoretical and Applied Mechanics, 3–4, 55–95, 1966.

Katsikadelis J.T., ynopiaka toixeia, Toμo& II: A , 2 Eo,

EM 2010 (Katsikadelis J.T., Boundary Elements: Vol. II, Analysis of Plates, Second Edition, NTUA, Athens, 260, 2010).

Katsikadelis J.T., The analog equation method – A powerful BEM-based solution technique for solving linear and nonlinear engineering problems, [in:] Boundary Element Method XVI: 167–182, Brebbia C.A. [Ed.], Computational Mechanics Publications, Southampton, 1994.

Nerantzaki M.S., Katsikadelis J.T., Buckling of plates with variable thickness – An Analog Equation Solution, Engineering Analysis with Boundary Elements, 18, 2, 149–154, 1996.

Chinnaboon B., Chucheepsakul S., Katsikadelis J.T., A BEM–based Meshless Method for Buckling Analysis of Elastic Plates with Various Boundary Conditions, International Journal of Structural Stability and Dynamics, 7, 1, 81–89, 2007.

Babouskos N., Katsikadelis J.T., Flutter instability of damped plates under combined conservative and nonconservative loads, Archive of Applied Mechanics, 79, 541–556, 2009.

Katsikadelis J.T., Babouskos N.G., Nonlinear flutter instability of thin damped plates: A solution by the analog equation method, Journal of Mechanics of Materials and structures, 4, 7–8, 1395–1414, 2009.

Guminiak M., Litewka B., Selected problems of thin and thick plates theory in terms of BEM, Foundations of Civil and Environmental Engineering, 15, 41–90, 2012.

Girkmann K., Plane Girders [in Polish], Warszawa, Arkady, 1957.

Timoshenko S., Woinowsky-Krieger S., Theory of plates and shells, Warszawa, Arkady, 1962.




DOI: 10.24423/engtrans.42.2014