Abstract
The paper presents a closed-form analytical solution for the finite displacement analysis of cantilever beams end-loaded, having an unvarying section. Extension to supported beams loaded at the middle point is obvious. The solution accounts of slender beams, so that shear deformation effect is neglected. Material elasticity holds. Concentrated end loads are considered within the solution. Several former approaches to this classical problem can be found in the scientific literature. Typical methods provides an integral form, solved by means of incomplete elliptic functions. Alternatively, a massive number of numerical integrations techniques have been proposed, such as “Non linear shooting method”, “Automatic Taylor expansion technique (ATET)”, “Runge-Kutta method”, etc.
In the present paper, the solution is gained through a parametric representation of the average line of the bending beam, after an expansion based on Gaussian hypergeometric functions. One of the main advantage is that an end-to-end analysis is possible, thus avoiding the integration throughout the full length of the beam. The method considers straight lines as well as circular shaped lines; the way to shift between the two cases is suggested within the paper. The reference system to get an easier solution of the integrals is taken so that it is aligned with the resulting load applied. Graphic presentations of some examples compare the analytic solutions here provided with numerical solutions, finite element method results and an experimental evidence.
The original version of this chapter was revised: First author name has been updated. The correction to this chapter is available at https://doi.org/10.1007/978-981-13-2273-0_33
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18 November 2018
In the original version of the book, the following belated correction has been incorporated:
References
Hodges, D.H., Orniston, R.A.: Stability of elastic bending and torsion of uniform cantilever rotor blades in hover with variable structural coupling. National Aeronautics and space administration (NASA), Washington D.C (1976)
Truesdell, C., Euler, L.: The Rational Mechanics of Flexible or Elastic Bodies 1638-1788. Birkhäuser, Basel (1960)
Barten, H.J.: On the deflection of a cantilever beam. Q. Appl. Math. 2, 168–171 (1944)
Bisshop, K.E., Drucker, D.C.: Large deflection of cantilever beams. Q. Appl. Math. 3, 272–275 (1945)
Banerjee, A., Bhattacharya, B., Mallik, A.K.: Large deflection of a cantilever beams with geometric non-linearity: analytical and numerical approaches. Int. J. Non-Linear Mech. 43, 366–376 (2008)
Tari, H.: On a parametric large deflection study of Euler-Bernoulli cantilever beams subjected to a combined tip point loading. Int. J. Non-Linear Mech. 49, 90–99 (2013)
Shvartsman, B.S.: Analysis of large deflections of a curved cantilever subjected to a tip-concentrated follower force. Int. J. Non-Linear Mech. 50, 75–80 (2013)
Nallathambi, A.K., Lakshmana Rao, C., Srinivasan, S.M.: Large deflection of a constant curvature cantilever beam under follower load. Int. J. Mech. Sci. 52, 440–445 (2010)
Reissner, E.: On one dimensional finite strain beam theory: the plane problem. J. Appl. Math. Phys. 23, 795–804 (1972)
Carlson, B.C.: Some series and bounds for incomplete elliptic integrals. J. Math. Phys. 40, 125–134 (1961)
Landau, L.: Theory of elasticity, translated from Russian by J.B. Sykes and W.H. Elsevier Reid, Pergamon Press, London (1959)
Frish-Fay, R.: Flexible Bars. Butterworths, London (1962)
Belluzzi, O.: Scienza delle costruzioni, vol. 4. Zanichelli, Bologna (1998)
Timoshenko, S., Gere, J.M.: Theory of Elastic Stability. McGraw Hill, New York (1963)
Jairazbhoy, V.A., Petukhov, P., Qu, J.: Large deflection of thin plates in convex or concave cylindrical bending. J. Eng. Mech. 138, 230–234 (2011)
Bona, F., Zelenika, S.: A generalized elastica-type approach to the analysis of large displacements of spring-strips. J. Mech. Eng. Sci. 211, 509–517 (1997)
Visner, J.C.: Analytical and experimental analysis of the large deflection of a cantilever beam subjected to a constant, concentrated force, with constant angle, applied at the free end. A thesis presented to the graduate faculty of the university of akron (2007)
Howell, P.D., Kozyreff, G., Ockendon, R.: Applied Solid Mechanics. Cambridge University Press, New York (2009)
Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Trascendental Functions, vol. 1. McGraw-Hill, New York (1953)
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Iandiorio, C., Salvini, P. (2019). An Analytical Solution for Large Displacements of End-Loaded Beams. In: Abdel Wahab, M. (eds) Proceedings of the 1st International Conference on Numerical Modelling in Engineering . NME 2018. Lecture Notes in Mechanical Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-13-2273-0_25
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DOI: https://doi.org/10.1007/978-981-13-2273-0_25
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