Scrutiny of non-linear differential equations Euler-Bernoulli beam with large rotational deviation by AGM

Abstract

The kinematic assumptions upon which the Euler-Bernoulli beam theory is founded allow it to be extended to more advanced analysis. Simple superposition allows for three-dimensional transverse loading. Using alternative constitutive equations can allow for viscoelastic or plastic beam deformation. Euler-Bernoulli beam theory can also be extended to the analysis of curved beams, beam buckling, composite beams and geometrically nonlinear beam deflection. In this study, solving the nonlinear differential equation governing the calculation of the large rotation deviation of the beam (or column) has been discussed. Previously to calculate the rotational deviation of the beam, the assumption is made that the angular deviation of the beam is small. By considering the small slope in the linearization of the governing differential equation, the solving is easy. The result of this simplification in some cases will lead to an excessive error. In this paper nonlinear differential equations governing on this system are solved analytically by Akbari-Ganji’s method (AGM). Moreover, in AGM by solving a set of algebraic equations, complicated nonlinear equations can easily be solved and without any mathematical operations such as integration solving. The solution of the problem can be obtained very simply and easily. Furthermore, to enhance the accuracy of the results, the Taylor expansion is not needed in most cases via AGM manner. Also, comparisons are made between AGM and numerical method (Runge-Kutta 4th). The results reveal that this method is very effective and simple, and can be applied for other nonlinear problems.

Key words： AGM; critical load of columns; large deformations of beam; nonlinear differential equation

References

1 Schmidt A P, Sidebottom R J. Advanced Mechanics of Materials. New York: John Wiley & Sons, Inc, 1993

2 Libai A, Simmonds J G. The Nonlinear Theory of Elastic Shells. Cambridge: Cambridge University Press, 1998

3 Parcel J I, Moorman R B B. Analysis of Statically Indeterminate Structures. New York: JohnWiley & Sons, Inc, 1955

4 Rostami A K, Akbari M R, Ganji D D, Investigating Jeffery-Hamel flow with high magnetic field and nanoparticle by HPM and AGM. Central European Journal of Engineering, 2014, 4(4): 357–370

5 Akbari M R, Ganji D D, Majidian A, Ahmadi A R. Solving nonlinear differential equations of Vanderpol, Rayleigh and Duffing by AGM. Frontiers of Mechanical Engineering, 2014, 9(2): 177–190

6 Ganji D D, Akbari M R, Goltabar A R. Dynamic vibration analysis for non-linear partial differential equation of the beam-columns with shear deformation and rotary inertia by AGM. Development and Applications of Oceanic Engineering (DAOE), 2014, 3: 22–31

7 Akbari M R, Ganji D D, Ahmadi A R, Analyzing the nonlinear vibrational wave differential equation for the simplified model of tower cranes by algebraic method. Frontiers of Mechanical Engineering, 2014, 9(1): 58–70

8 Akbari M R, Ganji D D, Nimafar M, et al. Significant progress in solution of nonlinear equations at displacement of structure and heat transfer extended surface by new AGM approach. Frontiers of Mechanical Engineering, 2014in press)

9 Salimi M<?Pub Caret?>. Passive and active control of structures. Dissertation for the Master Degree. Tehran: Tarbiat Modarres University, 2003

10 Yang J N, Danielians A, Liu S C. A seismic hybrid control systems for building structures. Journal of Engineering Mechanics, 1991, 117(4): 836–853

11 Ganji Z Z, Ganji D D, Asgari A. Finding general and explicit solutions of high nonlinear equations by the exp-function method. Computers & Mathematics with Applications (Oxford, England), 2009, 58(11–12): 2124–2130

12 Ganji D D. The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer. Physics Letters [Part A], 2006, 355(4–5): 337–341

13 Ning J G, Wang C, Ma T B. Numerical analysis of the shaped charged jet with large cone angle. International Journal of Nonlinear Science and Numerical Simulation, 2006, 7(1): 71–78

14 Das S, Gupta P. Application of homotopy perturbation method and homotopy analysis method to fractional vibration equation. International Journal of Computer Mathematics, 2011, 88(2): 430–441

15 Ghosh S, Roy A, Roy D. An adaptation of Adomian decomposition for numeric-analytic integration of strongly nonlinear and chaotic oscillators. Computer Methods in Applied Mechanics and Engineering, 2007, 196(4–6): 1133–1153

16 Sheikholeslami M, Ganji D D, Ashorynejad H R, et al. Analytical investigation of Jeffery-Hamel flow with high magnetic field and nanoparticle by Adomian decomposition method. Applied Mathematics and Mechanics, 2012, 33(1): 25–36

17 Sfahani M G, Barari A, Omidvar M, Dynamic response of inextensible beams by improved energy balance method. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 2011, 225(1): 66–73

18 Ganji Z Z, Ganji D D, Bararnia H. Approximate general and explicit solutions of nonlinear BBMB equations exp-function method. Applied Mathematical Modelling, 2009, 33(4): 1836–1841

19 Ren Z F, Liu G Q, Kang Y X, et al. Application of He’s amplitude— application of He’s amplitude frequency formulation to nonlinerar oscillators with discontinuities. Physica Scripta, 2009, 80(4): 45003

20 He J H. Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B, 2006, 20(10): 1141–1199

21 Wu X H, He J H. Solitary solutions, periodic solutions and compacton-like solutions using the exp-function method. Computers & Mathematics with Applications, 2007, 54(7–8): 966–986

22 He J H. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 1999, 178(3–4): 257–262

23 Zhou J K. Differential Transformation and Its Applications for Electrical Circuits. Wuhan: Huazhong University Press, 1986