Abstract
An analytical solution of the nonlinear problem of cantilever bending by a vertical force is presented, presented in elementary functions with calculation formulas for direct determination of the main parameters of a bent cantilever depending on the given value of the force load (module), such as coordinates of the cantilever outline, bending angles and curvature along the length of the cantilever, moments of forces and internal bending energy, as well as simplified formulas for finding the coordinates of the free end of the cantilever. Comparison of the obtained calculated values with graphical and tabular data of known numerical (exact) solutions gave a fairly high convergence of results (<1–2%), calculation examples are given. The obtained results can also be used to determine (inversely) the rigidity of the cantilever rods of arbitrary cross section, or the elastic modulus of the cantilever material at known sections, including when designing protective structures against dangerous slope geophysical processes.
Similar content being viewed by others
REFERENCES
E. P. Popov, Theory and Calculation of Flexible Elastic Rods (Nauka, Moscow, 1986) [in Russian].
Y. V. Zakharov and K.G. Okhotkin, “Nonlinear bending of thin elastic rods,” J. Appl. Mech. Tech. Phys. 43, 739–744 (2002). https://doi.org/10.1023/A:1019800205519
Yu. V. Zakharov and A. A. Zakharenko, “Dynamic instability in the nonlinear problem of a cantilever,” Vychisl. Tekhnol. 4 (1), 48–54 (1999).
K. N. Anakhaev, “A contribution to calculation of the mathematical pendulum,” Dokl. Phys. 59, 528–533 (2014). https://doi.org/10.1134/S1028335814110081
L. Miln-Tomson, “Elliptic integrals,” in Handbook on Special Functions, Ed. by M. Abramowitz and I. A. Stegun (Nauka, Moscow, 1977), pp. 401–441.
L. Miln-Tomson, “Elliptic Jacobi functions and theta functions,” in Handbook on Special Functions, Ed. by M. Abramowitz and I. A. Stegun (Nauka, Moscow, 1977), pp. 380–400.
K. N. Anakhaev, “On the improvement of hydromechanical methods for calculating potential (filtration) flows,” in Engineering Systems - 2009 (RUDN, Moscow, 2009), Vol. 2, pp. 588–595.
K. N. Anakhaev, “On definition of Jacobi elliptic functions,” Vestn. RUDN, Ser. Mat. Inform. Fiz., No. 2, 90–95 (2009).
K. N. Anakhaev, “Complete elliptic integrals of the third kind in problems of mechanics,” Dokl. Phys. 62, 133–135 (2017). https://doi.org/10.1134/S1028335817030053
K. N. Anakhaev, “Elliptic integrals in nonlinear problems of mechanics,” Dokl. Phys. 65, 142–146 (2020). https://doi.org/10.1134/S1028335820040011
Ya. M. Parkhomovskii, “Approximate formulas for elliptic integrals and examples of their application to two problems of nonlinear statics of elastic beams,” Uch. Zap. TsAGI 9 (4), 75–86 (1978).
N. S. Astapov, “Approximate formulas for deflection of compressed flexible bars,” J. Appl. Mech. Tech. Phys. 37, 573–576 (1996). https://doi.org/10.1007/BF02369735
A. V. Anfilof’ev, “Mid–span deflection and end–shortening of a rod after buckling,” J. Appl. Mech. Tech. Phys. 42, 352–357 (2001). https://doi.org/10.1023/A:1018852625202
D. M. Zuev, “Sagging deflection of cantiliever loaded with a transversal load. Approximate formulas for modification of linear theory,” Akt. Probl. Aviats. Kosmonavt. 1 (14), 294–296 (2018).
D. M. Zuev and K. G. Okhotkin, “Modified formulas for maximum deflection of a cantilever under transverse loading,” Kosm. App. Tekhnol. 4 (1), 28–35 (2020). https://doi.org/10.26732/j.st.2020.1.04
Y. V. Zakharov, K. G. Okhotkin, and A. Y. Vlasov, “Approximate formulas for sagging deflection of an elastic rod under transverse loading,” J. Appl. Mech. Tech. Phys. 43, 745–747 (2002). https://doi.org/10.1023/A:1019852222357
K. N. Anakhaev, “Calculation of potential flows,” Dokl. Phys. 50, 154–157 (2005). https://doi.org/10.1134/1.1897992
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by I. K. Katuev
About this article
Cite this article
Anakhaev, K.N. The Problem of Nonlinear Cantilever Bending in Elementary Functions. Mech. Solids 57, 997–1005 (2022). https://doi.org/10.3103/S0025654422050028
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0025654422050028