Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-07T07:46:40.371Z Has data issue: false hasContentIssue false
  • English
  • Français

SERIES SOLUTION OF LARGE DEFORMATION OF A BEAM WITH ARBITRARY VARIABLE CROSS SECTION UNDER AN AXIAL LOAD

Part of: Equations and inequalities involving nonlinear operators

Published online by Cambridge University Press:  09 March 2010

SHIJUN LIAO
SHIJUN LIAO*
Affiliation:
State Key Lab of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China (email: sjliao@sjtu.edu.cn)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A general analytic approach is proposed for nonlinear eigenvalue problems governed by nonlinear differential equations with variable coefficients. This approach is based on the homotopy analysis method for strongly nonlinear problems. As an example, a beam with arbitrary variable cross section acted on by a compressive axial load is used to show its validity and effectiveness. This approach provides us with great freedom to transfer the original nonlinear buckling equation with variable coefficients into an infinite number of linear differential equations with constant coefficients that are much easier to solve. More importantly, it provides us with a convenient way to guarantee the convergence of solution series. As an example, the beam displacement and the critical buckling load can be obtained for arbitrary variable cross sections. The influence of nonuniformity of moment of inertia is investigated in detail and the optimal distributions of moment of inertia are studied. It is found that the critical buckling load of a beam with the optimal distribution of moment of inertia can be approximately 21–22% larger than that of a uniform beam with the same average moment of inertia. Mathematically, this approach is rather general and thus can be used to solve many other linear/nonlinear differential equations with variable coefficients.

Keywords

eigenvaluenonlinearityvariable coefficientHAMseriesarbitrary beam

MSC classification

Secondary: 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems
Type
Research Article
Information
The ANZIAM Journal , Volume 51 , Issue 1 , July 2009 , pp. 10 - 33
Copyright
Copyright © Australian Mathematical Society 2010

References

[1]Abbas, Z., Wang, Y., Hayat, T. and Oberlack, M., “Hydromagnetic flow in a viscoelstic fluid due to the oscillatory stretching surface”, Int. Nonlinear Mech. 43 (2008) 783–793.CrossRefGoogle Scholar
[2]Abbasbandy, S., “The application of the homotopy analysis method to nonlinear equations arising in heat transfer”, Phys. Lett. A 360 (2006) 109–113.CrossRefGoogle Scholar
[3]Abbasbandy, S., “Solitary wave equations to the Kuramoto–Sivashinsky equation by means of the homotopy analysis method”, Nonlinear Dynam. 52 (2008) 35–40.CrossRefGoogle Scholar
[4]Abbasbandy, S. and Parkes, E. J., “Solitary smooth hump solutions of the Camassa–Holm equation by means of the homotopy analysis method”, Chaos Solitons Fractals 36 (2008) 581–591.CrossRefGoogle Scholar
[5]Akyildiz, F. T. and Vajravelu, K., “Magnetohydrodynamic flow of a viscoelastic fluid”, Phys. Lett. A 372 (2008) 3380–3384.CrossRefGoogle Scholar
[6]Alizadeh-Pahlavan, A., Aliakbar, V., Vakili-Farahani, F. and Sadeghy, K., “MHD flows of UCM fluids above porous stretching sheets using two-auxiliary-parameter homotopy analysis method”, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 473–488.CrossRefGoogle Scholar
[7]Allan, F. M., “Derivation of the Adomian decomposition method using the homotopy analysis method”, Appl. Math. Comput. 190 (2007) 6–14.Google Scholar
[8]Boley, A. B., “On the accuracy of the Bernoulli–Euler theory for beams of variable section”, J. Appl. Mech. ASME 30 (1963) 373–378.CrossRefGoogle Scholar
[9]Chang, D. and Popplewell, N., “A non-uniform, axially loaded Euler–Bernoulli beam having complex ends”, Quart. J. Mech. Appl. Math. 49 (1996) 353–371.CrossRefGoogle Scholar
[10]Clausen, T., “Über die Form architektonischer Säulen”, Bull. Phys.-Math. Acad. Imp. Sci. Saint Pétersbourg 9 (1851) 368–379.Google Scholar
[11]Dugush, Y. A. and Eisenberger, M., “Vibrations of non-uniform continuous beams under moving loads”, J. Sound Vibration 254 (2002) 911–926.CrossRefGoogle Scholar
[12]Elishakoff, I. and Rollot, O., “New closed-form solutions for buckling of a variable stiffness column by Mathematica”, J. Sound Vibration 224 (1999) 172–182.CrossRefGoogle Scholar
[13]Fang, X. Q., Hu, C. and Du, S. Y., “Strain energy density of a circular cavity buried in semi-infinite functionally graded materials subjected to shear waves”, Theor. Appl. Fract. Mech. 46 (2006) 166–174.CrossRefGoogle Scholar
[14]Farjoun, Y. and Neu, J., “The tallest column—a dynamical system approach using a symmetry solution”, Stud. Appl. Math. 115 (2005) 319–337.CrossRefGoogle Scholar
[15]Farjoun, Y. and Neu, J., “The optimal shape of a javelin”, Stud. Appl. Math. 115 (2005) 339–354.CrossRefGoogle Scholar
[16]Firouz-Abadi, R. D., Haddadpour, H. and Novinzadeh, A. B., “An asymptotic solution to transverse free vibrations of variable-section beams”, J. Sound Vibration 304 (2007) 530–540.CrossRefGoogle Scholar
[17]Ganapathi, M., “Dynamic stability characteristics of functionally graded materials shallow spherical shells”, Compos. Struct. 79 (2007) 338–343.CrossRefGoogle Scholar
[18]Gao, X. W., Zhang, C., Sladek, J. and Sladek, V., “Fracture analysis of functionally graded materials by a BEM”, Compos. Sci. Technol. 68 (2008) 1209–1215.CrossRefGoogle Scholar
[19]Hilton, P. J., An introduction to homotopy theory (Cambridge University Press, Cambridge, 1953).CrossRefGoogle Scholar
[20]Huang, X. L. and Shen, H. S., “Nonlinear vibration and dynamic response of functionally graded plates in thermal environments”, Internat. J. Solids Structures 41 (2004) 2403–2427.CrossRefGoogle Scholar
[21]Katsikadelis, J. T. and Tsiatas, G. C., “Non-linear dynamic analysis of beams with variable stiffness”, J. Sound Vibration 270 (2004) 847–863.CrossRefGoogle Scholar
[22]Katsikadelis, J. T. and Tsiatas, G. C., “Buckling load optimization of beams”, Arch. Appl. Mech. 74 (2005) 790–799.CrossRefGoogle Scholar
[23]Keller, J. B., “The shape of the strongest column”, Arch. Ration. Mech. Anal. 5 (1960) 275–285.CrossRefGoogle Scholar
[24]Lee, B. K., Wilson, J. F. and Oh, S. J., “Elastica of cantilevered beams with variable cross sections”, Internat. J. Non-Linear Mech. 28 (1993) 579–589.Google Scholar
[25]Lee, S. K., Mace, B. R. and Brennan, M. J., “Wave propagation, reflection and transmission in nonuniform one-dimensional waveguides”, J. Sound Vibration 304 (2007) 31–49.CrossRefGoogle Scholar
[26]Liang, S. X. and Jeffrey, D. J., “Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation”, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 4057–4064.CrossRefGoogle Scholar
[27]Liao, S. J., “On the relationship between the homotopy analysis method and Euler transform”, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1421–1431.CrossRefGoogle Scholar
[28]Liao, S. J., “The proposed homotopy analysis technique for the solution of nonlinear problems”, Ph. D. Thesis, Shanghai Jiao Tong University, 1992.Google Scholar
[29]Liao, S. J., “A kind of approximate solution technique which does not depend upon small parameters (ii): an application in fluid mechanics”, Internat. J. Non-Linear Mech. 32 (1997) 815–822.CrossRefGoogle Scholar
[30]Liao, S. J., “An explicit, totally analytic approximation of Blasius viscous flow problems”, Internat. J. Non-Linear Mech. 34 (1999) 759–778.CrossRefGoogle Scholar
[31]Liao, S. J., Beyond perturbation—introduction to the homotopy analysis method (Chapman & Hall/CRC Press, Boca Raton, FL, 2003).Google Scholar
[32]Liao, S. J., “A new branch of solutions of boundary-layer flows over an impermeable stretched plate”, Int. Heat Mass Transfer 48 (2005) 2529–2539.CrossRefGoogle Scholar
[33]Liao, S. J., “Notes on the homotopy analysis method: some definitions and theorems”, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 983–997.CrossRefGoogle Scholar
[34]Liao, S. J. and Magyari, E., “Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones”, Z. Angew. Math. Phys. 57 (2006) 777–792.CrossRefGoogle Scholar
[35]Liao, S. J. and Tan, Y., “A general approach to obtain series solutions of nonlinear differential equations”, Stud. Appl. Math. 119 (2007) 297–355.CrossRefGoogle Scholar
[36]Marin, L., “Numerical solution of the Cauchy problem for steady-state heat transfer in two-dimensional functionally graded materials”, Internat. J. Solids Structures 42 (2005) 4338–4351.CrossRefGoogle Scholar
[37]Molabahrami, A. and Khani, F., “The homotopy analysis method to solve the Burgers–Huxley equation”, Nonlinear Anal. B 10 (2009) 589–600.CrossRefGoogle Scholar
[38]Paranicolaou, V. G., “The periodic Euler–Bernoulli equation”, Trans. Amer. Math. Soc. 355 (2003) 3727–3759.CrossRefGoogle Scholar
[39]Poincaré, H., “Second complément à l’analysis situs”, Proc. London Math. Soc. 32 (1900) 277–308.CrossRefGoogle Scholar
[40]Sajid, M. and Hayat, T., “Comparison of HAM and HPM methods for nonlinear heat conduction and convection equations”, Nonlinear Anal. B 9 (2008) 2296–2301.CrossRefGoogle Scholar
[41]Sajid, M., Hayat, T. and Asghar, S., “Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt”, Nonlinear Dynam. 50 (2007) 27–35.CrossRefGoogle Scholar
[42]Sapountzakis, E. J. and Tsiatas, G. C., “Elastic flexural buckling analysis of composite beams of variable cross-section by BEM”, Eng. Struct. 29 (2007) 675–681.CrossRefGoogle Scholar
[43]Shen, H. S., “Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments”, Int. J. Mech. Sci. 44 (2002) 561–584.CrossRefGoogle Scholar
[44]Song, H. and Tao, L., “Homotopy analysis of 1D unsteady, nonlinear groundwater flow through porous media”, J. Coastal Res. 50 (2007) 292–295.Google Scholar
[45]Wang, B. L. and Mai, Y. W., “A periodic array of cracks in functional graded materials subjected to thermo-mechanical loading”, Internat. J. Engrg. Sci. 43 (2005) 432–446.CrossRefGoogle Scholar
[46]Wu, Y. Y., “Analytic solutions for nonlinear long wave propagation”, Ph. D. Thesis, University of Hawaii at Manoa, 2009.Google Scholar
[47]Wu, Y. Y. and Cheung, K. F., “Explicit solution to the exact Riemann problems and application in nonlinear shallow water equations”, Internat. J. Numer. Methods Fluids 57 (2008) 1649–1668.CrossRefGoogle Scholar
[48]Wu, Y. Y. and Cheung, K. F., “Homotopy solution for nonlinear differential equations in wave propagation problems”, Wave Motion 46 (2009) 1–14.CrossRefGoogle Scholar
[49]Yabushita, K., Yamashita, M. and Tsuboi, K., “An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method”, J. Phys. A 40 (2007) 8403–8416.CrossRefGoogle Scholar
[50]Yang, J. and Shen, H. S., “Non-linear analysis of functionally graded plates under transverse and in-plane loads”, Internat. J. Non-Linear Mech. 38 (2003) 467–482.CrossRefGoogle Scholar
[51]Zhu, S.-P., “A closed-form analytical solution for the valuation of convertible bonds with constant dividend yield”, ANZIAM J. 47 (2006) 477–494.CrossRefGoogle Scholar
[52]Zou, L., Zong, Z., Wang, Z. and He, L., “Solving the discrete KdV equation with homotopy analysis method”, Phys. Lett. A 370 (2007) 287–294.CrossRefGoogle Scholar