Abstract
The non-linear bending of slender heavy beam resting on rigid plane and subjected to the end point force is studied. Solution of the problem is derived by using series expansion technique. Approximate closed-form analytical solution for the length of the separated segment of elastica is found. It is shown that obtained simple solution can be used as the approximate lower bound for the evaluation of the force that should be applied to the beam’s end to provide the prescribed length of separation, while the known classical solution of linear theory can be used as the upper bound.
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References
J. M. Gere and S. Timoshenko, Mechanics of Materials (Cole, Pacific Grove, CA, 2001).
R. Frisch-Fay, Flexible Bars (Butterworths, London, 1962).
J. T. Holden, “On the finite deflections of thin beams,” Int. J. Solids Struct. 8, 1051–1055 1972.
A. Banerjee, B. Bhattacharya, and A. K. Mallik “Large deflection of cantilever beams with geometric non-linearity: analytical and numerical approaches,” Int. J. Non-Lin. Mech. 43, 366–376 (2008).
L. F. Campanile and A. Hasse, “A simple and effective solution of the elastica problem approach,” Proc. Inst. Mech. Eng., Part C 222, 2513–2516 2008.
L. Chen, “An integral approach for large deflection cantilever beams,” Int. J. Non-Lin. Mech. 45, 301–305 2010.
A. Humer and H. Irschik, “Large deformation and stability of an extensible elastica with an unknown length,” Int. J. Solids Struct. 48, 1301–1310 2011.
L. F. Campanile, R. Jähne, and A. Hasse, “Exact analysis of the bending of wide beams by a modified elastica approach,” Proc. Inst. Mech. Eng., Part C 255, 2759–2764 2011.
E. Turco, “Discrete is it enough? The revival of Piola-Hencky keynotes to analyze three-dimensional Elastica,” Continuum Mech. Thermodyn., 1–19 (2018).
F. Rohde, “Large deflections of a cantilever beam with uniformly distributed load,” Quart. Appl. Math. 11, 337–338 1953.
C. Y. Wang, “A critical review of the heavy elastica,” Int. J. Mech. Sci. 28, 549–559 1986.
B. W. Kooi and M. Kuipers, “A unilateral contact problem with the heavy elastica,” Int. J. Non-Lin. Mech. 19, 309–321 1984.
B. W. Kooi, “A unilateral contact problem with the heavy elastica solved by use of finite elements,” Comput. Struct. 21, 95–103 1985.
J. S. Chen, H. C. Li, and W. C. Ro, “Slip-through of a heavy elastica on point supports,” Int. J. Solids Struct. 47, 261–268 2010.
A. Kimiaeifar, N. Tolou, A. Barari, and J. L. Herder, “Large deflection analysis of cantilever beam under end point and distributed loads,” J. Chin. Inst. Eng. 37, 438–445 2014.
R. Long, N. Tolou, K. R. Shull, and C. Y. Hui, “Large deformation adhesive contact mechanics of circular membranes with a flat rigid substrate,” J. Mech. Phys. Solids 58, 1225–1242 2010.
A. Srivastava and C. Y. Hui, “Nonlinear viscoelastic contact mechanics of long rectangular membranes,” Proc. R. Soc. London, Ser. A 470 (2171), 20140528 (2014).
A. Patil, A. DasGupta, and A. Eriksson, “Contact mechanics of a circular membrane inflated against a deformable substrate,” Int. J. Solids Struct. 67, 250–262 2015.
F. Essenburg, “On surface constraints in plate problems,” J. Appl. Mech. 29, 340–344 1962.
E. Grigolyuk and V. Tolkachev, Contact Problems in the Theory of Plates and Shells (Imported Pubn., 1987).
V. I. Feodosyev, Selected Problems and Questions in Strength of Materials (Mir, Moscow, 1983).
J. H. Kim, Y. J. Ahn, Y. H. Jang, and J. R. Barber, “Contact problems involving beams,” Int. J. Solids Struct. 51, 4435–4439 2014.
E. Lomakin, L. Rabinskiy, V. Radchenko, Y. Solyaev, S. Zhavoronok, and A. Babaytsev, “Analytical estimates of the contact zone area for a pressurized flat-oval cylindrical shell placed between two parallel rigid plates,” Mecanica 53, 3831–3838 2018.
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The study has been funded by basic program AAAA-A19-119012290177-0 of the Institute of Applied Mechanics of the Russian Academy of Sciences.
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Solyaev, Y., Ustenko, A. & Lykosova, E. Approximate Analytical Solution for a Unilateral Contact Problem with Heavy Elastica. Lobachevskii J Math 40, 1010–1015 (2019). https://doi.org/10.1134/S1995080219070163
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DOI: https://doi.org/10.1134/S1995080219070163