Finite Element Analysis of Continuous Plates Using a High-Performance Programming Language (MATLAB)

Excel Obumneme Amaefule, Owen Chukwuebuka Abuka, Nnaemeka Chukwudi Nwachukwu, Chukwuebuka Oscar Nnadi, Chibueze Nelson Ogbonna, Habeeb Keji Adeyemo, Meshack Ibeamaka Ogundu, Sunday Okechukwu, Don-Ugbaga Chinomso, Gregory Ezeokpube

Abstract

This paper uses MATLAB, a finite element software program to compare the results of several finite analysis methods for continuous plates and check the degree of correlation with the exact values obtained by Timoshenko (1959) and Cheung (1996). The results showed little or no significant difference between plates in finite elements. Two different finite Numerical techniques are used. The Finite Strip and Exact methods and their results are compared to the results from the MATLAB program. Finite element analysis (FEA) workflow using MATLAB includes generation of meshes, geometry creation, defining physics of load, initial conditions and boundary problems, calculation, and results from visualization. FEA is a very general approach for solving Equations in science and engineering. This work offers solutions to the increasing errors associated with several other numerical methods when solving any equations of plates (сontinuous). It makes it easier to calculate and design larger structures through geometry discretization of plates and plains into more minor elements.



Keywords


Finite Element Analysis; Finite Element Method; MATLAB; Continuous Plates; Finite Strip Method; Exact Method

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References


Belounar, L., & Guenfoud, M. (2005). A new rectangular finite element based on the strain approach for plate bending. Thin-Walled Structures, 43(1), 47–63. doi: 10.1016/j.tws.2004.08.003

Bernoulli, J. (1789). Essai theorique sur les vibrations de plaques elastiques rectangularies et libers. Novi Commentari Acad Petropolit, 5, 197–219

Cauchy, A. (2009). Oeuvres complétes: Sur l'équilibre et le mouvement d'une plaque solide. doi: 10.1017/CBO9780511702679.014

Cheung, M. S., Chidiac, S. E., & Li, W. (1996). Finite Strip Analysis of Bridges. London: CRC Press.

Clebsch, A., Flamant, A. A., & de Saint-Venant, A. C. (1883). Théorie de l'élasticité des corps solides. Paris: Dunod.

Clough, R. W. (1960). The Finite Element Method in Plane Stress Analysis. Retrieved from https://www.semanticscholar.org/paper/The-Finite-Element-Method-in-Plane-Stress-Analysis-Clough/035536cf1b0157b3cc7a6a19ed1b66638b388553

Cole, V. (2006, January 25). A Critical History of Computer Graphics and Animation. Retrieved from https://www.engadget.com/2006-01-25-a-critical-history-of-computer-graphics-and-animation.html

Cook, R. D. (2002). Concepts and applications of finite element analysis. New York: Wiley.

Courant, R. (1943). Variational methods for the solution of problems of equilibrium and vibrations. Bulletin of the American Mathematical Society, 49, 1–23.

Dawe, D. J. (1984). Matrix and finite element displacement analysis of structures. Oxford: Claredon Press.

Düster, A., & Rank, E. (2001). The p-version of the finite element method compared to an adaptive h-version for the deformation theory of plasticity. Computer Methods in Applied Mechanics and Engineering, 190(15–17), 1925–1935. doi: 10.1016/s0045-7825(00)00215-2

Euler, L. (1776). De motu vibratorio tympanorum. Retrieved from https://scholarlycommons.pacific.edu/cgi/viewcontent.cgi?article=1301&context=euler-works

Germain, S. (1826). Remarques sur la nature, les bornes et l'étendue de la question des surfaces élastiques et équation générale de ces surfaces. Paris.

Han, J.-G., Ren, W.-X., & Huang, Y. (2007). A wavelet-based stochastic finite element method of thin plate bending. Applied Mathematical Modelling, 31(2), 181–193. doi: 10.1016/j.apm.2005.08.020

Hastings, J. K., Judes, M. A., & Brauer, J. R. (1985). Accuracy and Economy of Finite Element Magnetic Analysis. 33rd Annual National Relay Conference.

Hrennikoff, A. (1941). Solution of Problems of Elasticity by the Framework Method. Journal of Applied Mechanics, 8(4), A169–A175. doi: 10.1115/1.4009129

Kelvin, W. T., & Tait, P. G. (2013). Treatise on Natural Philosophy (Vol. 1). Oxford: Clarendon Press.

Kirchhoff, G. (1985). Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal), 40, 51–88. doi: 10.1515/crll.1850.40.51

Kreyszig, E. (2011). Advanced Engineering Mathematics (10th ed.). Retrieved from https://www.bau.edu.jo/UserPortal/UserProfile/PostsAttach/59003_3812_1.pdf

Krishnamoorthy, C. S. (2011). Finite element analysis theory and programming (2nd ed.). New Delhi: Tata McGraw Hill Education Private Limited.

Levy, M. (1877). Mémoire sur la théorie des plaques élastiques planes. Journal de mathématiques pures et appliquées, 3(3), 219–306.

Lim, G. T., & Reddy, J. N. (2003). On canonical bending relationships for plates. International Journal of Solids and Structures, 40(12), 3039–3067. doi: 10.1016/s0020-7683(03)00084-2

Mathew, J., Ma, L., Tan, A., Weijnen, M., & Lee, J. (Eds.). (2012). Engineering Asset Management and Infrastructure Sustainability. doi: 10.1007/978-0-85729-493-7

Melosh, R. J. (1963). Basis for derivation of matrices for the direct stiffness method. AIAA Journal, 1(7), 1631–1637. doi: 10.2514/3.1869

Nwagozie, I. L. (2008, January). Finite Element Modelling Of Engineering Systems. Retrieved from https://www.researchgate.net/publication/297739827_FINITE_ELEMENT_MODELLING_OF_ENGINEERING_SYSTEMS_With_Emphasis_in_Water_Resources

Poisson, S. D. (1828). Mémoire sur l'équilibre et le mouvement des corps élastiques. N. d.

Reddy, J. N. (2006). Theory and Analysis of Elastic Plates and Shells. doi: 10.1201/9780849384165

Reissner, E., & Stein, M. (1951). Torsion and transverse bending of cantilever plates. Retrieved from https://ntrs.nasa.gov/citations/19930090894

Rockey, K. C. (1974). The Finite Element method: A Basic Introduction. London: Collins professional and Technical Book.

Ross, C. (1996). Finite Element Techniques in Structural Mechanics. London: Woodhead Publishing.

Saibel, E., & Tadjbakhsh, I. (1960). Large deflections of circular plates under uniform and concentrated central loads. Zeitschrift Für Angewandte Mathematik Und Physik ZAMP, 11(6), 496–503. doi: 10.1007/bf01595401

Segerlind, L. (1985). Applied Finite Element Analysis (2nd ed.). New York: Wiley.

Semie, A. G. (2010, June). Numerical Modelling of Thin Plates using the Finite Element Method. Retrieved from http://etd.aau.edu.et/bitstream/handle/123456789/7199/Addisu%20Gezahegn.pdf?sequence=1&isAllowed=y

Shames, I. H., & Dym, C. L. (2017). Energy and Finite Element Methods in Structural Mechanics. doi: 10.1201/9780203757567

Shudhir, N. (2012). Plate Bending Analaysis Using Finite Element Method. Retrieved from http://ethesis.nitrkl.ac.in/3303/1/108ME015.pdf

Speare, P. R. S., & Kemp, K. O. (1977). A simplified reissner theory for plate bending. International Journal of Solids and Structures, 13(11), 1073–1079. doi: 10.1016/0020-7683(77)90077-4

Strang, G., & Fix, G. (2008). An Analysis of the Finite Element Method (2nd ed.). Cambridge: Wellesley-Cambridge Press.

Szabó, B., & Actis, R. (2011). Simulation governance: New technical requirements for software tools in computational solid mechanics. Retrieved from https://www3.nd.edu/~powers/vv.presentations/szabo.pdf

Thompson, E. G. (2004). Introduction to the Finite Element Method: Theory, Programming and Applications. New York: Wiley.

Timoshenko, S. P., & Woinowsky-Kerieger, S. (1959). Theory of Plates and Shells (2nd ed.). New York: McGraw-Hill Book Co.

Topping, B. H. V. (Ed.). (2003). Developments in Structural Engineering. doi: 10.1201/9781482298567

Turner, M. J., Clough, R. W., Martin, H. C., & Topp, L. J. (1956). Stiffness and Deflection Analysis of Complex Structures. Journal of the Aeronautical Sciences, 23(9), 805–823. doi: 10.2514/8.3664

Ventsel, E., & Krauthammer, T. (2001). Thin Plates and Shells. doi: 10.1201/9780203908723

Williams, M. S., & Todd. J. D. (2000). Structures: Theory and analysis. London: Red Globe Press.

Zienkiewicz, O. C., Taylor, R. L., & Zhu, J. Z. (2013). The finite element method: its basis and fundamentals (7th ed.). Oxford: Butterworth-Heinemann.


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