ПРОГИБ И ОСЕВАЯ СИЛА В СВОБОДНО-ОПЕРТОЙ ГЕОМЕТРИЧЕСКИ-НЕЛИНЕЙНОЙ БАЛКЕ
ПРОГИБ И ОСЕВАЯ СИЛА В СВОБОДНО-ОПЕРТОЙ ГЕОМЕТРИЧЕСКИ-НЕЛИНЕЙНОЙ БАЛКЕ
DOI:
https://doi.org/10.22337/2587-9618-2025-21-1-39-52Ключевые слова:
large deflection, axial force, nonlinear beam theory, finite difference method, convergence of iterative method, ABAQUS, nonlinear differential equations, principal of virtual workАннотация
The problem of beam bending for large deflections is described in general. The nonlinear beam theory is considered for a simply-supported beam subjected to uniform load. The governing equations for displacements of this beam are derived. Numerical method for solving the governing equations is proposed. Convergence of the numerical method is studied. Numerical results are shown in the form of figures and formulas. These results suggest that the deflection predicted by the nonlinear theory at a specific point can be expressed solely as a function of the linear deflection at the same point. It is also shown how the axial force in the beam depends on the nonlinear deflection. Analytical expression for the axial force in the beam for small deflections is derived without solving the differential equation. For larger deflections, another representation for the axial force is obtained in terms of auxiliary functions that are defined only in terms of nonlinear and linear deflections. Comparison of the present results with the ABAQUS results is given. It is shown that the present theory can quite accurately predict the deflection and axial force in the beam for large deflections. However, the product of the axial force by the co-sine of the slope angle at the support rather than the axial force itself will be a more accurate estimate of the horizontal support reaction.
Библиографические ссылки
Timoshenko S., Woinowsky-Krieger S. Theory of plates and shells. Mcgraw-Hill Book Company, 1959, 594 p.
Woinowsky-Krieger, S. The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 1950, 17, pp. 35–36.
Gibshman E.E. Proektirovanie metallicheskikh mostov [Steel bridge design]. Moscow, Transport, 1969, 416 p. (in Russian)
Belenya E.I., Baldin V.A., Vedenikov G.S. Metallicheskie kontstruktsii [Steel structures]. Moscow, Stroyizdat, 1986, 560 p. (in Russian)
Loc V. Tran, Jarkko Niiranen. A geometrically nonlinear Euler-Bernoulli beam model within strain gradient elasticity with isogeometric analysis and lattice structure applications, Mathematics and Mechanics of Complex Systems, 2020, vol. 8, no. 4.
Paavani D., Aswathy M., Arun C.O., Praveen Krishna I.R. Analysis of geometrically nonlinear Euler-Bernoulli beam using EFGM, IOP Conf. Series: Materials Science and Engineering, 2020, 936, 012050.
Worden K., Tomlinson G.R. Nonlinearity in Structural Dynamics (Detection, Identification and Modelling), Institute of Physics Publishing, Bristol and Philadelphia, 2001.
Lukash P.A. Osnovy nelineinoy stroitel’noi mekhaniki [Foundations of Nonlinear Structural Mechanics]. Moscow, Stroyizdat, 1978, 204 p. (in Russian)
Reddy J.N. An Introduction to Nonlinear Finite Element Analysis. Oxford Univeristy Press, 2010.
Jain S., Tiso P., Haller G. Exact nonlinear model reduction for a von Karman beam: Slow-fast decomposition and spectral submanifolds, Journal of Sound and Vibration, 2018, vol. 423, pp. 195-211.
Mehrpouya M.A., Salehi R., Wong P.J.Y. A Fast and Accurate Numerical Method for Solving Nonlinear Fourth-Order Boundary Value Problems in the Beam Theory. Axioms, 2024, 13, 757.
Kiusalaas J. Numerical Methods in Engineering with Python 3. Cambridge University Press, 2013.
Hoffman J.D. Numerical Methods for Engineers and Scientists. McGraw-Hill, Inc., New York, 1992
LeVeque R.J. Finite Difference Methods for Ordinary and Partial Differential Equations. Society for Industrial and Applied Mathematics, Philadelphia, 2007.
Butler J.S. Numerical Methods for Differential Equations with Python. https://johnsbutler.netlify.app.
Загрузки
Опубликован
Выпуск
Раздел
Лицензия
Copyright (c) 2025 International Journal for Computational Civil and Structural Engineering
Это произведение доступно по лицензии Creative Commons «Attribution-NonCommercial-ShareAlike» («Атрибуция — Некоммерческое использование — На тех же условиях») 4.0 Всемирная.
