THE LIMITS OF TAYLOR’S EXPANSION SERIES FOR THE STUDY OF ELASTIC CURVES IN ISOTROPIC BEAMS, STATIC DETERMINATE, WITH A CONSTANT MOMENT OF INERTIA

Adrian-Ioan BOTEAN

Abstract


This paper proposes a detailed study on the evaluation of deflection and rotation for two distinct types of supports, aiming to analyze structural behavior under various loading conditions. To conduct this analysis, the Taylor series expansion method is employed, a mathematical tool that allows for the approximation of nonlinear functions by incorporating higher-order terms. Thus, in the context of deflection and rotation analysis, various loading scenarios are investigated, each with its specific characteristics, and the Taylor series expansion is applied to approximate structural behavior at points of interest, considering initial conditions and the characteristics of the support type. This methodology facilitates obtaining approximate solutions for deflection and rotation, particularly in regions where exact solutions are difficult to determine due to the complexity of geometry or load distribution. A central aspect of the study is the identification of the limitations of the Taylor series expansion method concerning the analyzed cases.


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Gere, J.M., Goodno, B.J., 2013, Mechanics of Materials, Eighth Edition, Cengage Learning, p. 1,130.

Hibbeler, R.C., 2005, Mechanics of Materials, Sixth Edition, Pearson Education International, p. 873.

Beer, F.P., Johnston, E.R., 1985, Mechanics of Materials, McGraw-Hill, Singapore, p. 610.

Gere, J.M., Timoshenko, S.P., 1997, Mechanics of Materials, PWS, USA, p. 1,130.

Popov, E.P., 1990, Engineering Mechanics of Solids, Prentice-Hall, USA, p. 727.

Botean, A.I., 2019, Rezistenţa materialelor. Solicitări simple, Ediţia a II-a revizuită şi adăugită, Editura UTPRESS, Cluj-Napoca, p. 454.

Ruiyu, B., Nan, Y., Bo, L., Guimin, C., 2024, Nonlinear strain energy formulation of spatially strip flexures, Mechanism and Machine Theory, 195 (105594), p. 18.

Botean, A.I., 2024, Approximation of the elastic curve differential equation by trigonometric series for an isotropic beam with a constant moment of inertia, complexly loaded, according to the Euler – Bernoulli theory, Acta Technica Napocensis, Series: Applied Mathematics, Mechanics, and Engineering, Vol.67, Issue I, p. 55-68.

Botean, A.I., 2023, The Use of Trigonometric Series for the Study of Isotropic Beam Deflection, Mathematics, 11(6), 1426, p. 16.

Ike, C.C., 2022, Fourier series method for finding displacements and stress fields in hyperbolic shear deformable thick beams subjected to distributed transverse loads, Journal of Computational Applied Mechanics, 53(1), p. 126-141.

Tang, Y., Qing, H., 2024, Finite element formulation for higher-order shear deformation beams using two-phase local/nonlocal integral model, Arch Appl Mech 94, p. 1243–1262.

Patuelli, C., Cestino, E., Frulla, G., 2023, A Beam Finite Element for Static and Dynamic Analysis of Composite and Stiffened Structures withBending-Torsion Coupling, Aerospace, 10, 142, p. 28.

Helmi, K., Taylor, T., Zarafshan, A., Ansari, F., 2015, Reference free method for real time monitoring of bridge deflections, Engineering Structures, 103, p. 116-124.

Ke, W., Gang, Z., 2022, Solutions to large beam – deflection problems by Taylor series and Pade approximant for compliant mechanism, Mechanism and Machine Theory, 177 (105033, p. 25.

Tat, J.T., I-Ming, C., Guilin, Y., Wei, L., 2010, A generic approximation model for analyzing large nonlinear deflection of beam – based flexure joints, Precision Engineering 34, p. 607-618.

Hatami, M., Vahdani, S., Ganji, D.D., 2014, Deflection prediction of a cantilever beam subjected to static co-planar loading by analytical methods, HBRC Journal, 10, p. 191-197.

Sladek, J., Sladek, V., Hosseini, S.M., 2021, Analysis of a curved Timoshenko nano-beam with flexoelectricity, Acta Mech, 232, p. 1563-1581.

Fariba, F., 2015, A new method for analysis of beam deflection using Taylor's expansion series, Computational Research Progress in Applied Science & Engineering, Volume 03(06), p. 178-185.

Filippi, M., Pagani, A., Petrolo, M., Colonna, G., Carrera, E., 2015, Static and free vibration analysis of laminated beams by refined theory based on Chebyshev polynomials, Composite Structures, 132, p. 1248-1259.

Timoshenko, S.P., Goodier, J.N., 1970, Theory of Elasticity, McGraw-Hill, p. 608.

Kreyszig, E., 2011, Advanced Engineering Mathematics, John Wiley & Sons, p. 1,280.

Boyce, W.E., DiPrima, R.C., 2017, Elementary Differential Equations and Boundary Value Problems, Wiley, p. 624.

Fung, Y.C., 2001, Classical and Computational Solid Mechanics, World Scientific Publishing, p. 952.

Tari, H., 2013, On the parametric large deflection study of Euler-Barnoulli cantilever beams subjected to combined tip point loading, International Journal of Non-Linear Machanics, 49, p. 90-99.

Fouaidi, M., Jamal, M., Belouaggadia, N., 2020, Nonlinear bending analysis of functionally graded porous beams using the multiquadric radial basis functions and a Taylor series-based continuation procedure, Composite Structures, 252 (112593), p. 12.

Boresi, A.P., Schmidt, R.J., Sidebottom, O.M., 1993, Advanced Mechanics of Materials, Wiley, p. 811.

Bathe, K.J., 1996, Finite Element Procedures, Prentice Hall, p. 736.

Zienkiewicz, O.C., Taylor, R.L., 2005, The Finite Element Method for Solid and Structural Mechanics, Elsevier, p. 736.

Reddy, J.N., 2014, An Introduction to the Finite Element Method, McGraw-Hill Education, p. 816.

Cook, R.D., Malkus, D.S., Plesha, M.E., 2002, Concepts and Applications of Finite Element Analysis, Wiley, p. 733.

Ling, M., Yuan, L., Zhang, X., 2024, Geometrically nonlinear analysis of compliant mechanisms using a dynamic beam constraint model (DBCM), Mechanism and Machine Theory, 191 (105489), p. 25.

Zhang, P., Fu, Y., 2013, A higher-order beam model for tubes, European Journal of Mechanics A/Solids, 38, p. 12-19.

Peng, X.-L., Zhang, L., Yang, Z.-X., Feng, Z.-Y., Zhao, B., Li, X.-F., 2020, Effect of the gradient on the deflection of functionally graded microcantilever beams with surface stress, Acta Mech, p. 4185-4198.

Hughes, T.J.R., 2000, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover Publications, p. 704.

Gere, J.M., Timoshenko, S.P., 1999, Mechanics of Materials, PWS Publishing, p. 913.

Logan, D.L., 2016, A First Course in the Finite Element Method, Cengage Learning p. 992.

Crandall, S.H., Dahl, N.C., Lardner, T.J., 1999, An Introduction to the Mechanics of Solids, McGraw-Hill, p. 628.

Chandrupatla, T.R., Belegundu, A.D., 2011, Introduction to Finite Elements in Engineering, Pearson, p. 448.

Love, A.E.H., 1944, A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, p. 643.

Smith, T.A., Brown, D.E., 2022, Modern Approaches in Beam Theory, Springer, p. 350.

Bazant, Z.P., Cedolin, L., 2010, Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories, World Scientific Publishing p. 1,040.

Budiansky, B., 1974, Theory of Buckling and Post-Buckling Behavior of Elastic Structures, Advances in Applied Mechanics, (14), p. 65.

Timoshenko, S.P., Gere, J.M., 1961, Theory of Elastic Stability, McGraw-Hill, p. 541.

Thompson, J.M.T., Hunt, G.W., 1984, Elastic Instability Phenomena, Wiley, p. 209.

Nash, W.A., Sen, I.C., 1998, Strength of Materials: Theory and Problems, Schaum's Outline Series, McGraw-Hill, p. 432.

Megson, T.H.G., 2019, Aircraft Structures for Engineering Students, Butterworth-Heinemann, p. 1,040.

Clough, R.W., Penzien, J., 2003, Dynamics of Structures, McGraw-Hill Education, p. 738.

Aliabadi, M.H., 2002, The Boundary Element Method: Applications in Solids and Structures, Wiley, p. 598.

Reddy, J.N., 2013, Energy Principles and Variational Methods in Applied Mechanics, Wiley, p. 768.


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