Influence of Gravity-Induced Axial Force on the Natural Frequencies of an Inclined Cantilever Beam
DOI:
10.29303/jpm.v21i2.11634Published:
2026-04-07Downloads
Abstract
The limitations of conventional horizontal cantilever models in capturing gravity-induced axial effects in inclined structures that influence stiffness and natural frequencies. This study investigates the influence of gravity on the dynamic behavior of an inclined cantilever beam, with particular emphasis on a 10-m steel beam configuration. Unlike conventional horizontal cantilever models, structural inclination generates a spatially varying axial compressive force that increases from zero at the free end to a maximum value at the fixed support. The objective of this study is to analyze how this gravity-induced axial force distribution affects the natural frequencies and effective stiffness of the beam. To achieve this, a high-fidelity numerical model was developed in MATLAB, where the beam was discretized into 100 Euler–Bernoulli elements to capture the gradual variation of axial load and the resulting geometric softening effect. The analytical formulation is based on the Rayleigh–Ritz method, in which the spatial axial force distribution is incorporated into the potential energy expression. The predicted natural frequencies were validated through frequency-domain analysis of the transient response obtained from numerical simulation. Both analytical and numerical results indicate that, for an inclination angle of 60°, the first and second natural frequencies are approximately 1.84 Hz and 11.61 Hz, respectively. These results demonstrate that beam inclination significantly reduces the effective structural stiffness due to gravity-induced geometric softening. The findings highlight the importance of incorporating spatially varying geometric stiffness in the dynamic analysis, stability assessment, and vibration-sensitive design of large-scale inclined cantilever structures.
Keywords:
Geometric Stiffness; Inclined Cantilever Beam; Natural Frequency; Self-weight; Rayleigh–Ritz MethodReferences
R. Ali, M. Beh, A. Y. Al-Dweik, B. S. Yilbas, and M. Sunar, “Closed-form solution of nonlinear oscillation of a cantilever beam using λ-symmetry linearization criteria,” Heliyon, vol. 8, no. 11, p. e11673, 2022, doi: 10.1016/j.heliyon.2022.e11673.
A. Janchikoski and J. F. Meireles, “Dynamic analysis of a free vibrating cantilever beam,” Revista Científica Multidisciplinar Núcleo do Conhecimento, pp. 123–146, 2023.
P. J. Waste et al., “Determination of natural frequencies and mode shapes of a cantilever beam using finite element method,” Int. J. Sci. Res. Eng. Manag., vol. 9, no. 7, pp. 9–11, 2025, doi: 10.55041/IJSREM51691.
M. R. Chiarelli, “Dynamic response of beams under random loads,” Mathematics, vol. 13, 2025.
A. Moghadam, H. G. Melhem, and A. Esmaeily, “A proof-of-concept study on a proposed ambient-vibration-based approach to extract pseudo-free-vibration response,” Eng. Struct., vol. 212, p. 110517, 2020, doi: 10.1016/j.engstruct.2020.110517.
Y. Madhuranthakam and S. K. Chakrapani, “Analytical study of nonlinear flexural vibration of a beam with geometric, material and combined nonlinearities,” Vibration, vol. 7, pp. 464–478, 2024.
G. Shukla and Rajbahadur, “Numerical and analytical investigations of vibration and buckling in composite cantilever beams,” J. Inf. Syst. Eng. Manag., vol. 10, 2025.
L. Li, “Natural vibration characteristics of a cantilever beam parametrically excited by pitch motion,” Int. J. Struct. Stab. Dyn., vol. 24, no. 12, pp. 1–41, 2024, doi: 10.1142/S021945542550124X.
Z. Shi, C. Wang, and G. Yao, “Stability and nonlinear vibrations of an inclined axially moving beam considering self-weight,” Commun. Nonlinear Sci. Numer. Simul., vol. 133, p. 107966, 2024, doi: 10.1016/j.cnsns.2024.107966.
X. Fu, S. Wei, and H. Ding, “Weight effects on vertical transverse vibration of a beam with a nonlinear energy sink,” Appl. Sci., vol. 15, 2025.
D. Gritsenko, J. Xu, and R. Paoli, “Transverse vibrations of cantilever beams: Analytical solutions with general steady-state forcing,” Appl. Eng. Sci., vol. 3, 2020, doi: 10.1016/j.apples.2020.100017.
G. Wang, H. Ding, and L. Chen, “Dynamic effect of internal resonance caused by gravity on the nonlinear vibration of vertical cantilever beams,” J. Sound Vib., vol. 474, p. 115265, 2020, doi: 10.1016/j.jsv.2020.115265.
C. Iandiorio and P. Salvini, “Large displacements of slender beams in plane: Analytical solution by means of a new hypergeometric function,” Int. J. Solids Struct., vol. 185–186, pp. 467–484, 2020, doi: 10.1016/j.ijsolstr.2019.09.006.
V. Ondra and B. Titurus, “Free vibration and stability analysis of a cantilever beam axially loaded by an intermittently attached tendon,” Mech. Syst. Signal Process., vol. 158, p. 107739, 2021, doi: 10.1016/j.ymssp.2021.107739.
G. Wang, H. Ding, and L. Chen, “Gravitational effects and mode interactions of vertical cantilever beams,” Int. J. Nonlinear Mech., vol. 123, p. 103493, 2020, doi: 10.1016/j.ijnonlinmec.2020.103493.
K. Kumar and S. S. Gupta, “Planar nonlinear dynamics of a hanging geometrically exact beam with a moving mass,” Int. J. Solids Struct., vol. 325, p. 113704, 2026, doi: 10.1016/j.ijsolstr.2025.113704.
H. S. A. Ebrahimi-Mamaghani and M. Safarpour, “Nonlocal study of the vibration and stability response of small-scale axially moving supported beams,” Math. Methods Appl. Sci., 2020, doi: 10.1002/mma.6859.
Y. Liu, “Nonlinear dynamic analysis of an axially moving composite laminated cantilever beam,” J. Vib. Eng. Technol., 2022, doi: 10.1007/s42417-022-00750-2.
X. Liu et al., “An exact dynamic stiffness method for multibody systems consisting of beams and rigid bodies,” Mech. Syst. Signal Process., vol. 150, p. 107264, 2021, doi: 10.1016/j.ymssp.2020.107264.
J. R. Banerjee, “An exact method for free vibration of beams and frameworks using frequency-dependent matrices,” Comput. Struct., vol. 292, p. 107235, 2024, doi: 10.1016/j.compstruc.2023.107235.
I. Christiawan, “Secara umum kompromi,” p. 10, 1998.
G. Nitti, G. Lacidogna, and A. Carpinteri, “An analytical formulation to evaluate natural frequencies and mode shapes of high-rise buildings,” Curved Layer. Struct., vol. 8, pp. 307–317, 2021.
L. Ledda, A. Greco, I. Fiore, and I. Calio, “Closed-form exact solution for free vibration analysis of symmetric functionally graded beams,” Symmetry, vol. 16, 2024.
M. S. Sari and S. Faroughi, “Free vibrations of functionally graded porous hanging and standing cantilever beams,” J. Low Freq. Noise Vib. Act. Control, vol. 43, no. 3, pp. 1082–1096, 2024, doi: 10.1177/14613484241238593.
Z. Shi, C. Wang, and G. Yao, “Stability and nonlinear vibrations of an inclined axially moving beam considering self-weight,” Commun. Nonlinear Sci. Numer. Simul., vol. 133, p. 107966, 2024, doi: 10.1016/j.cnsns.2024.107966.
M. Soltani and B. Asgarian, “New hybrid approach for free vibration and stability analyses of axially functionally graded Euler–Bernoulli beams with variable cross-section resting on Winkler–Pasternak foundation,” Lat. Am. J. Solids Struct., vol. 16, no. 3, pp. 1–25, 2019.
M. Debeurre, “Extreme nonlinear dynamics of cantilever beams: Effect of gravity and slenderness on nonlinear modes,” Nonlinear Dyn., vol. 111, no. 14, pp. 12787–12815, 2023, doi: 10.1007/s11071-023-08637-x.
M. A. Eltaher, A. E. Alshorbagy, and F. F. Mahmoud, “Vibration analysis of Euler–Bernoulli nanobeams using finite element method,” Appl. Math. Model., vol. 37, no. 7, pp. 4787–4797, 2013, doi: 10.1016/j.apm.2012.10.016.
Q. Abbas, R. Nawaz, H. Yaqoob, H. Muhammad, and M. Musaddiq, “Nonlinear vibration analysis of cantilever beams: Analytical, numerical and experimental perspectives,” Partial Differ. Equations Appl. Math., vol. 13, p. 101115, 2025, doi: 10.1016/j.padiff.2025.101115.
Y. Chen, “Error analysis based on Euler–Bernoulli beam theory: An example of a simply supported beam bending model,” in Proc. CONF-FMCE 2025 Symp., 2025, pp. 17–24, doi: 10.54254/2755-2721/155/2025.GL23166.
A. Luo, B. Lossouarn, and A. Erturk, “The effects of shear deformation and rotary inertia on the electrical analogs of beams and plates for multimodal piezoelectric damping,” Int. J. Circuit Theory Appl., vol. 52, pp. 2985–2998, 2024, doi: 10.1002/cta.3899.
G. Kanwal, N. Ahmed, and R. Nawaz, “A comparative analysis of the vibrational behavior of various beam models with different foundation designs,” Heliyon, vol. 10, no. 5, p. e26491, 2024, doi: 10.1016/j.heliyon.2024.e26491.
H. Farokhi, E. Kohtanen, and A. Erturk, “Extreme parametric resonance oscillations of a cantilever: An exact theory and experimental validation,” Mech. Syst. Signal Process., vol. 196, p. 110342, 2023, doi: 10.1016/j.ymssp.2023.110342.
H. Farokhi, Y. Xia, and A. Erturk, “Experimentally validated geometrically exact model for extreme nonlinear motions of cantilevers,” Nonlinear Dyn., vol. 107, no. 1, pp. 457–475, 2022, doi: 10.1007/s11071-021-07023-9.
F. H. Jazi, R. Tikani, and M. S. Taki, “Geometric modeling and dynamic analysis of cantilever beams,” Multidiscip. Model. Mater. Struct., vol. 21, no. 4, pp. 916–941, 2025, doi: 10.1108/MMMS-10-2024-0301.
M. A. Mahmoud, “Natural frequency of axially functionally graded tapered cantilever beams with tip masses,” Eng. Struct., vol. 187, pp. 34–42, 2019, doi: 10.1016/j.engstruct.2019.02.043.
License
Copyright (c) 2026 Ananta Kusuma Yoga Pratama, Sunarko Sunarko
This work is licensed under a Creative Commons Attribution 4.0 International License.
The following terms apply to authors who publish in this journal:
1. Authors retain copyright and grant the journal first publication rights, with the work simultaneously licensed under a Creative Commons Attribution License 4.0 International License (CC-BY License) that allows others to share the work with an acknowledgment of the work's authorship and first publication in this journal.
2. Authors may enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., posting it to an institutional repository or publishing it in a book), acknowledging its initial publication in this journal.
3. Before and during the submission process, authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website), as this can lead to productive exchanges as well as earlier and greater citation of published work (See The Effect of Open Access).
Downloads
How to Cite
