The experimental and analytical study of geometrically nonlinear bending of a cantilever beam under a distributed gravity load | Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika – Tomsk State University Journal of Mathematics and Mechanics. 2022. № 78. DOI: 10.17223/19988621/78/8

The experimental and analytical study of geometrically nonlinear bending of a cantilever beam under a distributed gravity load

This paper describes an approximate analytical solution for the geometrically nonlinear bending of a thin elastic cantilever beam under a uniformly distributed gravity load. The solution is based on the linearized Euler-Bernoulli equation of mechanics of materials. Traditionally, such a linear approach is used for small (geometrically linear) deflections. The authors have modified the original equation with an arc-length preservation condition. The modified solution allows one to obtain bending shapes, deflection, and axial displacement in the range of loads corresponding to geometrically nonlinear bending of a beam (large deflections). An experimental study is conducted to verify the proposed solution. A thin steel band bent by gravity is used as a sample. Changes in the length of the bent sample part allow one to obtain various dimensionless load parameters. The deflections and axial displacements averaged on experimental statistics are determined. Bending shapes are obtained by the least square method of 5^th order. Experimental and theoretical data are shown to be in good agreement. This fact confirms that the approximate analytical solution can be applied to solve large deflection problems in a wider range of loads than normally considered in the original linear theory.

Download file

Counter downloads: 59

Keywords

cantilever, geometrically nonlinear bending, large deflections, distributed load, experiment

Authors

Name	Organization	E-mail
Zuev Dmitriy M.	Reshetnev Siberian State University of Science and Technology	zuevdmitriy93@yandex.ru
Makarov Dmitriy D.	Reshetnev Siberian State University of Science and Technology	dima-makarov-98@mail.ru
Okhotkin Kirill G.	Reshetnev Siberian State University of Science and Technology; JSC Information Satellite Systems Reshetnev	okg2000@mail.ru

Всего: 3

References

Лопатин А.В. и др. Геометрически нелинейная модель трансформируемого обода боль шой космической антенны с гибкими композитными элементами // Вестник Сибирского государственного аэрокосмического университета имени академика М.Ф. Решетнева. 2012. № 5 (45). С. 75-80.

Li M., Tang H.X., Roukes M.L. Ultra-sensitive NEMS-based cantilevers for sensing, scanned probe and very high-frequency applications // Nat. Nanotechnol. 2007. V. 2, No. 2. P. 114120. doi: 10.1038/nnano.2006.208

Li X., Bhushan B., Takashime K., Baek C., Kim Y. Mechanical characterization of micro / nanoscale structures for MEMS / NEMS applications using nanoindentation techniques // Ultramicroscopy. 2003. V. 97, No. 1-4. P. 481-494. doi: 10.1016/S0304-3991(03)00077-9

Zhang A., Chen G. A comprehensive elliptic integral solution to the large deflection problems of thin beams in compliant mechanisms //j. Mech. Robot. 2013. V. 5, No. 2. P. 1-10. doi: 10.1115/1.4023558

Rohde F. V. Large deflections of a cantilever beam with uniformly distributed load // Q. Appl. Math. 1952. No. 2. P. 337-338.

Frisch-Fay R. The analysis of a vertical and a horizontal cantilever under a uniformly distributed load //j. Franklin Inst. 1961. V. 271, No. 3. P. 192-199. doi: 10.1016/0016-0032(61)90148-X

Schmidt R., DaDeppo D.A. Large deflections of heavy cantilever beams and columns // Q. Appl. Math. 1970. V. 28, No. 3. P. 441-444. doi: 10.1090/qam/99779

Wang C.Y. A critical review of the heavy elastica // Int. J. Mech. Sci. 1986. V. 28, No. 8. P. 549-559. doi: 10.1016/0020-7403(86)90052-4

Scarpello G.M., Ritelli D. Exact Solutions of Nonlinear Equation of Rod Deflections Involving the Lauricella Hypergeometric Functions // International Journal of Mathematics and Mathematical Sciences. 2011. V. 2011. Art. 838924, 22 pages. doi: 10.1155/2011/838924

Chen L. An integral approach for large deflection cantilever beams // Int. J. Non. Linear. Mech. Elsevier. 2010. V. 45, No. 3. P. 301-305. doi: 10.1016/j.ijnonlinmec.2009.12.004

Belendez T., Neipp C., Belendez A. Numerical and Experimental Analysis of a Cantilever Beam: a Laboratory Project to Introduce Geometric Nonlinearity in Mechanics of Materials // Int. J. Eng. Educ. 2003. V. 19, No. 6. P. 885-892.

Barbieri E. Analytical solution of the cantilevered elastica subjected to a normal uniformly distributed follower load // Int. J. Solids Struct. 2020. V. 202. P. 486-494. doi: 10.1016/j.ijsolstr.2020.06.031

Gosar Z., Kosel F. Large deflection states of euler-bernoulli slender cantilever beam subjected to combined loading // SYLWAN. 2014. V. 158, No. 5. P. 489-499.

Lee H.C., Durelli A.J., Parks V.J. Stresses in largely deflected cantilever beams subjected to gravity //j. Appl. Mech. Trans. ASME. 1964. V. 36, No. 2. P. 323-325. doi: 10.1115/1.3564633

Bahari A.R., Yunus M.A., Abdul Rani M.N., Ayub M.A., Nalisa A. Numerical and Experimental Investigations of Nonlinearity Behaviour in A Slender Cantilever Beam // MATEC Web Conf. 2018. V. 217. P. 1-6. doi: 10.1051/matecconf/201821702008

Brojan M., Cebron M., Kosel F. Large deflections of non-prismatic nonlinearly elastic cantilever beams subjected to non-uniform continuous load and a concentrated load at the free end // Acta Mech. Sin. Xuebao. 2012. V. 28, No. 3. P. 863-869. doi: 10.1007/s10409-012-0053-3

Зуев Д.М., Охоткин К.Г. Модифицированные выражения для стрелы прогиба консоли в случае поперечной нагрузки // Космические аппараты и технологии. 2020. Т. 4, № 1. С. 28-35. doi: 10.26732/j. st.2020.1.04

Тимошенко С.П., Гере Д. Механика материалов : учебник для вузов. 2-е, стер. изд. СПб. : Лань, 2002. 672 с.

Варданян Г.С., Андреев В.И., Атаров Н.М., Горшков А.А. Сопротивление материалов с основами теории упругости теории и пластичности. М. : АСВ, 1995. 572 с.