Existence and continuous dependence of solutions for equilibrium configurations of cantilever beam

Apassara Suechoei; Parinya Sa Ngiamsunthorn; Waraporn Chatanin; Somchai Chucheepsakul; Chainarong Athisakul; Danuruj Songsanga; Nuttanon Songsuwan; Apassara Suechoei; Parinya Sa Ngiamsunthorn; Waraporn Chatanin; Somchai Chucheepsakul; Chainarong Athisakul; Danuruj Songsanga; Nuttanon Songsuwan

doi:10.3934/mbe.2022572

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 12: 12279-12302. doi: 10.3934/mbe.2022572

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Existence and continuous dependence of solutions for equilibrium configurations of cantilever beam

1.
Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, Bangkok 10140, Thailand
2.
Department of Civil Engineering, Faculty of Engineering, King Mongkut's University of Technology Thonburi, Bangkok 10140, Thailand
3.
Learning Institute, King Mongkut's University of Technology Thonburi, Bangkok 10140, Thailand

Academic Editor: Jesús Martín Vaquero

Received: 09 June 2022 Revised: 07 August 2022 Accepted: 11 August 2022 Published: 22 August 2022

This article explores the equilibrium configurations of a cantilever beam described by the minimizer of a generalized total energy functional. We reformulate the problem as a boundary value problem using the Euler-Lagrange condition and investigate the existence and uniqueness of minimizers. Furthermore, we discuss the dependence of solutions on the parameters of the boundary value problems. In addition, the Adomian decomposition method is derived for approximating the solution in terms of series. Finally, numerical results for the equilibrium configurations of cantilever beams are presented to support our theoretical analysis.
- equilibrium configuration,
- existence and uniqueness of solution,
- Euler-Lagrange theorem,
- Adomian decomposition method
Citation: Apassara Suechoei, Parinya Sa Ngiamsunthorn, Waraporn Chatanin, Somchai Chucheepsakul, Chainarong Athisakul, Danuruj Songsanga, Nuttanon Songsuwan. Existence and continuous dependence of solutions for equilibrium configurations of cantilever beam[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 12279-12302. doi: 10.3934/mbe.2022572

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Abstract

This article explores the equilibrium configurations of a cantilever beam described by the minimizer of a generalized total energy functional. We reformulate the problem as a boundary value problem using the Euler-Lagrange condition and investigate the existence and uniqueness of minimizers. Furthermore, we discuss the dependence of solutions on the parameters of the boundary value problems. In addition, the Adomian decomposition method is derived for approximating the solution in terms of series. Finally, numerical results for the equilibrium configurations of cantilever beams are presented to support our theoretical analysis.

References

[1]	M. Bilinska, K. D. Kristensen, M. Dalstra, Cantilevers: Multi-tool in orthodontic treatment, Dent. J., 10 (2022), 135. https://doi.org/10.3390/dj10070135 doi: 10.3390/dj10070135
[2]	J. P. Hollowell, N. Yoganandan, E. C. Benzel, Spinal implant attributes: cantilever beam fixation, in Spine Surgery, Elsevier, 2005 (2005), 1418–1429. https://doi.org/10.1016/B978-0-443-06616-0.50112-7
[3]	J. A. Birdwell, J. H. Solomon, M. Thajchayapong, M. A. Taylor, M. Cheely, R. B. Towal, et al., Biomechanical models for radial distance determination by the rat vibrissal system, J. neurophysiol., 98 (2007), 2439–2455. https://doi.org/10.1152/jn.00707.2006 doi: 10.1152/jn.00707.2006
[4]	S. Gohari, S. Sharifi, Z. Vrcelj, New explicit solution for static shape control of smart laminated cantilever piezo-composite-hybrid plates/beams under thermo-electro-mechanical loads using piezoelectric actuators, Compos. Struct., 145 (2016), 89–112. https://doi.org/10.1016/j.compstruct.2016.02.047 doi: 10.1016/j.compstruct.2016.02.047
[5]	S. Gohari, S. Sharifi, Z. Vrcelj, A novel explicit solution for twisting control of smart laminated cantilever composite plates/beams using inclined piezoelectric actuators, Compos. Struct., 161 (2017), 477–504. https://doi.org/10.1016/j.compstruct.2016.11.063 doi: 10.1016/j.compstruct.2016.11.063
[6]	R. Levien, The Elastica: A Mathematical History, Technical Report UCB/EECS-2008-103, EECS Department, University of California, Berkeley, 2008. Available from: https://www2.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-103.pdf.
[7]	A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, 2013.
[8]	M. Batista, Analytical treatment of equilibrium configurations of cantilever under terminal loads using jacobi elliptical functions, Int. J. Solids Struct., 51 (2014), 2308–2326. https://doi.org/10.1016/j.ijsolstr.2014.02.036 doi: 10.1016/j.ijsolstr.2014.02.036
[9]	A. Della Corte, F. dell?Isola, R. Esposito, M. Pulvirenti, Equilibria of a clamped euler beam (elastica) with distributed load: Large deformations, Math. Models Methods Appl. Sci., 27 (2017), 1391–1421. https://doi.org/10.1142/S0218202517500221 doi: 10.1142/S0218202517500221
[10]	M. Jin, Z. Bao, Sufficient conditions for stability of euler elasticas, Mech. Res. Commun., 35 (2008), 193–200. https://doi.org/10.1016/j.mechrescom.2007.09.001 doi: 10.1016/j.mechrescom.2007.09.001
[11]	S. Timoshenko, Theory of Elastic Stability, Tata McGraw-Hill Education, 1970. Available from: https://www.academia.edu/43072340/Timoshenko.
[12]	D. Baroudi, I. Giorgio, A. Battista, E. Turco, L. A. Igumnov, Nonlinear dynamics of uniformly loaded elastica: Experimental and numerical evidence of motion around curled stable equilibrium configurations, ZAMM, 99 (2019), e201800121. https://doi.org/10.1002/zamm.201800121 doi: 10.1002/zamm.201800121
[13]	M. Batista, F. Kosel, Cantilever beam equilibrium configurations, Int. J. Solids Struct., 42 (2005), 4663–4672. https://doi.org/10.1016/j.ijsolstr.2005.02.008 doi: 10.1016/j.ijsolstr.2005.02.008
[14]	S. Navaee, R. E. Elling, {Equilibrium configurations of cantilever beams subjected to inclined end loads}, J. Appl. Mech., 59 (1992), 572–579. https://doi.org/10.1115/1.2893762 doi: 10.1115/1.2893762
[15]	B. Shvartsman, Large deflections of a cantilever beam subjected to a follower force, J. Sound Vib., 304 (2007), 969–973. https://doi.org/10.1016/j.jsv.2007.03.010 doi: 10.1016/j.jsv.2007.03.010
[16]	M. Batista, A simplified method to investigate the stability of cantilever rod equilibrium forms, Mech. Res. Commun., 67 (2015), 13–17. https://doi.org/10.1016/j.mechrescom.2015.04.009 doi: 10.1016/j.mechrescom.2015.04.009
[17]	B. Dacorogna, Direct Methods in the Calculus of Variations, Springer New York, NY, 78 (2007). https://doi.org/10.1007/978-0-387-55249-1
[18]	L. Tonelli, Opere Scelte: Calcolo Delle Variazioni, Edizioni Cremonese, 2 (1960).
[19]	G. Adomian, Modification of the decomposition approach to the heat equation, J. Math. Anal. Appl., 124 (1987), 290–291. https://doi.org/10.1016/0022-247X(87)90040-0 doi: 10.1016/0022-247X(87)90040-0
[20]	L. Bougoffa, A. Khanfer, Existence and uniqueness theorems of second-order equations with integral boundary conditions, Bull. Korean Math. Soc., 55 (2018), 899–911. https://doi.org/10.4134/BKMS.b170374 doi: 10.4134/BKMS.b170374
[21]	L. Bougoffa, d R. C. Rach, Solving nonlocal initial-boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations by the adomian decomposition method, Appl. Math. Comput., 225 (2013), 50–61. https://doi.org/10.1016/j.amc.2013.09.011 doi: 10.1016/j.amc.2013.09.011
[22]	L. Bougoffa, R. C. Rach, A. Mennouni, An approximate method for solving a class of weakly-singular volterra integro-differential equations, Appl. Math. Comput., 217 (2011), 8907–8913. https://doi.org/10.1016/j.amc.2011.02.102 doi: 10.1016/j.amc.2011.02.102
[23]	K. Bisshopp, D. C. Drucker, Large deflection of cantilever beams, Q. Appl. Math., 3 (1945), 272–275. https://doi.org/10.1090/QAM/13360 doi: 10.1090/QAM/13360
[24]	R. Kumar, L. Ramachandra, D. Roy, A multi-step linearization technique for a class of boundary value problems in non-linear mechanics, Comput. Mech., 39 (2006), 73–81. https://doi.org/10.1007/s00466-005-0009-6 doi: 10.1007/s00466-005-0009-6
[25]	H. Tari, On the parametric large deflection study of euler–bernoulli cantilever beams subjected to combined tip point loading, Int. J. Non-Linear Mech., 49 (2013), 90–99. https://doi.org/10.1016/j.ijnonlinmec.2012.09.004 doi: 10.1016/j.ijnonlinmec.2012.09.004
[26]	M. Brojan, M. Cebron, F. Kosel, 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., 28 (2012), 863–869. https://doi.org/10.1007/s10409-012-0053-3 doi: 10.1007/s10409-012-0053-3
[27]	Z. Girgin, F. E. Aysal, H. Bayrakçeken, Large deflection analysis of prismatic cantilever beam comparatively by using combing method and iterative dqm, J. Polytech., 23 (2020). https://doi.org/10.2339/politeknik.504480
[28]	M. Mutyalarao, D. Bharathi, B. N. Rao, Large deflections of a cantilever beam under an inclined end load, Appl. Math. Comput., 217 (2010), 3607–3613. https://doi.org/10.1016/j.amc.2010.09.021 doi: 10.1016/j.amc.2010.09.021
[29]	D. Singhal, V. Narayanamurthy, Large and small deflection analysis of a cantilever beam, J. Inst. Eng. India Ser. A, 100 (2019), 83–96. https://doi.org/10.1007/s40030-018-0342-3 doi: 10.1007/s40030-018-0342-3
[30]	I. Stakgold, M. J. Holst, Green's Functions and Boundary Value Problems, John Wiley & Sons, 2011. https://doi.org/10.1002/9780470906538
[31]	M. Benchohra, J. Henderson, S. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York, 2006.
[32]	J. Saetang, P. Sa Ngiamsunthorn, W. Chatanin, S. Chucheepsakul, C. Athisakul, A. Suechoei, Post buckling analysis of a cantilever beam subjected to a compressive load, in ICMA-MU 2020 Book on the Conference Proceedings: International Conference in Mathematics and Applications, (2020), 139–146. Available from: https://www.researchgate.net/publication/362539960.

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