Research Article
Some closed-form solutions for buckling of straight beams with varying cross-section by Variational Iteration Method with Generalized Lagrange Multipliers
Abstract
This
study aims to derive approximate closed-form solutions for critical loads of
straight beams with variable cross-section. The governing equations are derived
for purely flexible beam for small displacements and rotation and turned into
non-dimensional form. Approximate solutions to the set of equations for stability
problems are searched by Variational Iteration Method with Generalized Lagrange
Multipliers. It turns out that highly accurate approximate buckling loads of cantilever
beams with variable section can be obtained in closed-form. Many novel
closed-form solutions for critical load of such structures, which may serve as
benchmark solutions, are presented.
References
- [1] Elishakoff, I., Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions, CRC Press, Boca Raton, 2005.[2] Antman, S.S., The theory of rods, pp.641-703 of Linear Theories of Elasticity and Thermoelasticity, Truesdell C. (ed.), Berlin, Springer, 1973.[3] Love, A.E.H., A Treatise on the Mathematical Theory of Elasticity, Cambridge, at the University press, 4th edition, 1927.[4] Timoshenko, S., History of Strength of Materials, McGraw-Hill, New York, 1953.[5] Oldfather, W.A., Ellis C.A., Brown D.M., Leonhard Euler’s Elastic Curves. Isis, 20(1), 72-160, 1933.[6] Euler, L., Sur la force des callones. Memories de L’Academie des Sciences et Belles-Letteres (Berlin) 13, 252–282, 1759. (in French)[7] Engesser, F., Ueber Krickfestigkeit gerader Staebe. Zeitschrift Architekten und Ingineure in Hannover 35, 455 (1899). (in German).[8] Dinnik, A.N., Design of columns of varying cross-section, Transactions of the ASME, Applied Mechanics, 1929.[9] Duncan, N.J., Galerkin’s method in mechanics and differential equations. Aeronautical Research Committee Reports and Memoranda, No. 1798, 1937.[10] Elishakoff, I., Pellegrini, F., Exact and effective approximate solutions of some divergent type non-conservative problems. Journal of Sound and Vibration, 114, 144-148, 1987.[11] Elishakoff, I., Pellegrini, F., Application of Bessel and Lommel functions and undetermined multiplier Galerkin method version for instability of non-uniform column. Journal of Sound and Vibration, 115, 182-186, 1987 [12] Elishakoff, I., Pellegrini, F., Exact solution for buckling of some divergence type non-conservative systems in terms of Bessel and Lommel functions. Computer Methods in Applied Mechanics and Engineering, 66, 107-119, 1988[13] Elishakoff, I., Inverse buckling problem for inhomogeneous columns. International Journal of Solids and Structures, 38(3), 457–464, 2001.[14] Elishakoff, I., Eisenberger, M., Delmas, A., Buckling and vibration of functionally graded columns sharing Duncan’s mode shape, and new cases. Structures, 5, 170–174, 2016.[15] Suresh S, Mortensen A. Fundamentals of functionally graded materials. London, UK: IOM Communications Limited, 1998.[16] Mahamood, R.S., Akinlabi, E.T., Functionally Graded Materials, Springer, 2017.[17] Ruta, G.C., Varano, V., Pignataro, M., Rizzi N.L., A beam model for the flexural-torsional buckling of thin-walled memberse with some applications. Thin-Walled Structures, 46, 816,822, 2008.[18] Ruta, G., Pignataro, M., Rizzi, N., A direct one-dimensional beam model for the flexural-torsional buckling of thin-walled beams. Journal of Mechanics of Materials and Structures, 1(8), 1479-1496, 2006.[19] A. Tatone, A., Rizzi, N., A one-dimensional model for thin-walled beams, pp. 312–320 in Trends in applications of mathematics to mechanics, edited by W. ed. Schneider et al., Longman, Avon, 1991.[20] Gupta, R.K., Gunda, J.B., Janardhan, G.R., Rao, G.V., Post-buckling analysis of composite beams: Simple and accurate closed-form expressions. Composite Structures, 92, 1947-1956, 2010.[21] Mercan, K., Civalek, O., Comparison of Small Scale Effect Theories for Buckling Analysis of Nanobeam. International Journal of Engineering and Applied Sciences, 9(3), 87-97, 2016.[22] Abbondanza, D., Battista, D., Morabito, F., Pallante, C., Barretta, R., Luciano, R., de Sciarra, F.M., Ruta, G., Modulated linear dynamics of nanobeams accounting for higher gradient effects. International Journal of Engineering and Applied Sciences, 8(2), 1-20, 2016.[23] He, J.H., A new approach to non-linear partial differential equations. Communications in Nonlinear Science and Numerical Simulation, 2, 230–235, 1997. [24] He, J.H., Variational iteration method for delay differential equations, Communications in Nonlinear Science and Numerical Simulation, 2, 235–236, 1997. [25] He, J.H., Variational iteration method a kind of non-linear analytical technique: some examples. International Journal of Nonlinear Mechanics, 34, 699–708, 1999.[26] He, J.H., Variational iteration method some recent results and new interpretations. Journal of Computational and Applied Mathematics, 207, 3–17, 2007. [27] He, J.H., Wu, X.H., Variational iteration method: new development and applications. Computers and Mathematics with Applications, 54,881–894, 2007.[28] Turkyilmazoglu, M., An optimal variational iteration method. Applied Mathematics Letters, 24(5), 762–765, 2011.[29] Yilmaz, E., Inc, M., Numerical simulation of the squeezing flow between two infinite plates by means of the modified variational iteration method with an auxiliary parameter. Nonlinear Science Letters A, 1, 297–306, 2010.[30] Hosseini, M.M., Mohyud-Din, S.T., Ghaneai H., Usman, M., Auxiliary parameter in the variational iteration method and its optimal determination. International Journal of Nonlinear Sciences and Numerical Simulation, 11(7), 495–502, 2010.[31] Herişanu, N., Marinca, V., A modified variational iteration method for strongly nonlinear problems. Nonlinear Science Letters A, 1, 183–192, 2010.[32] Altintan, D., Ugur, O., Generalisation of the Lagrange multipliers for variational iterations applied to systems of differential equations. Mathematical and Computer Modelling. 54, 2040-2050, 2011.[33] He, J.H., Notes on the optimal variational iteration method. Applied Mathematics Letters. 25(10), 1579-1581, 2012.[34] Coskun, S.B., Atay, M.T., Determination of critical buckling load for elastic columns of constant and variable cross-sections using variational iteration method. Computers and Mathematics with Applications, 58, 2260,2266, 2009.[35] Chen, Y., Zhang, J., Zhang, Z., Flapwise bending vibration of rotating tapered beams using variational iteration method. Journal of Vibration and Control, 22(15), 3384-3395, 2016.[36] Eroglu, U., Large deflection analysis of planar curved beams made of functionally graded materials using variational iterational method. Composite Structures, 136, 204–216, 2016.[37] Yun-dong, L., Yi-ren, Y., Vibration analysis of conveying fluid pipe via He’s variational iteration method. Applied Mathematical Modelling, 43, 409,420.[38] Eroglu, U., Tufekci, E., Small-Amplitude free vibrations of straight beams subjected to large displacements and rotation. Applied Mathematical Modelling, 53, 223-241, 2018.[39] Budiansky, B., Theory of buckling and postbuckling behavior of elastic structures, pp. 1–65 in Advances in applied mechanics,14, Academic Press, New York, 1974.[40] Timoshenko, S.P., Gere, J.M., Theory of Elastic Stability. McGraw-Hill, New York, 1961.[41] Pease, M.C., Methods of Matrix Algebra. Academic Press, New York, 1965.[42] Tufekci, E., Arpaci, A., Exact solution of in-plane vibrations of circular arches with account taken of axial extension, transverse shear and rotatory inertia effects. Journal of Sound and Vibration 209 (5):845–56, 1998.[43] Tufekci, E., Arpacı, A., Analytical solutions of in-plane static problems for non-uniform curved beams including axial and shear deformations. Structural Engineering and Mechanics, 22 (2):131–50, 2006.[44] Yildirim, V., Several Stress Resultant and Deflection Formulas for Euler-Bernoulli Beams under Concentrated and Generalized Power/Sinusoidal Distributed Loads, International Journal of Engineering and Applied Sciences, 10(2), 35-63, 2018.[45] Wei, D.J., Xani S.X., Zhang, Z.P., Li, X.F., Critical load for buckling of non-prismatic columns under self-weight and tip force. Mechanics Research Communications, 37, 554-558- 2010.[46] Wang, C.M., Wang, C.Y., Reddy, J.N., Exact Solutions for Buckling of Structural Members. CRC Press, Boca Raton, 2005.
Year 2018,
Volume: 10 Issue: 3, 159 - 175, 04.11.2018
Abstract
References
- [1] Elishakoff, I., Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions, CRC Press, Boca Raton, 2005.[2] Antman, S.S., The theory of rods, pp.641-703 of Linear Theories of Elasticity and Thermoelasticity, Truesdell C. (ed.), Berlin, Springer, 1973.[3] Love, A.E.H., A Treatise on the Mathematical Theory of Elasticity, Cambridge, at the University press, 4th edition, 1927.[4] Timoshenko, S., History of Strength of Materials, McGraw-Hill, New York, 1953.[5] Oldfather, W.A., Ellis C.A., Brown D.M., Leonhard Euler’s Elastic Curves. Isis, 20(1), 72-160, 1933.[6] Euler, L., Sur la force des callones. Memories de L’Academie des Sciences et Belles-Letteres (Berlin) 13, 252–282, 1759. (in French)[7] Engesser, F., Ueber Krickfestigkeit gerader Staebe. Zeitschrift Architekten und Ingineure in Hannover 35, 455 (1899). (in German).[8] Dinnik, A.N., Design of columns of varying cross-section, Transactions of the ASME, Applied Mechanics, 1929.[9] Duncan, N.J., Galerkin’s method in mechanics and differential equations. Aeronautical Research Committee Reports and Memoranda, No. 1798, 1937.[10] Elishakoff, I., Pellegrini, F., Exact and effective approximate solutions of some divergent type non-conservative problems. Journal of Sound and Vibration, 114, 144-148, 1987.[11] Elishakoff, I., Pellegrini, F., Application of Bessel and Lommel functions and undetermined multiplier Galerkin method version for instability of non-uniform column. Journal of Sound and Vibration, 115, 182-186, 1987 [12] Elishakoff, I., Pellegrini, F., Exact solution for buckling of some divergence type non-conservative systems in terms of Bessel and Lommel functions. Computer Methods in Applied Mechanics and Engineering, 66, 107-119, 1988[13] Elishakoff, I., Inverse buckling problem for inhomogeneous columns. International Journal of Solids and Structures, 38(3), 457–464, 2001.[14] Elishakoff, I., Eisenberger, M., Delmas, A., Buckling and vibration of functionally graded columns sharing Duncan’s mode shape, and new cases. Structures, 5, 170–174, 2016.[15] Suresh S, Mortensen A. Fundamentals of functionally graded materials. London, UK: IOM Communications Limited, 1998.[16] Mahamood, R.S., Akinlabi, E.T., Functionally Graded Materials, Springer, 2017.[17] Ruta, G.C., Varano, V., Pignataro, M., Rizzi N.L., A beam model for the flexural-torsional buckling of thin-walled memberse with some applications. Thin-Walled Structures, 46, 816,822, 2008.[18] Ruta, G., Pignataro, M., Rizzi, N., A direct one-dimensional beam model for the flexural-torsional buckling of thin-walled beams. Journal of Mechanics of Materials and Structures, 1(8), 1479-1496, 2006.[19] A. Tatone, A., Rizzi, N., A one-dimensional model for thin-walled beams, pp. 312–320 in Trends in applications of mathematics to mechanics, edited by W. ed. Schneider et al., Longman, Avon, 1991.[20] Gupta, R.K., Gunda, J.B., Janardhan, G.R., Rao, G.V., Post-buckling analysis of composite beams: Simple and accurate closed-form expressions. Composite Structures, 92, 1947-1956, 2010.[21] Mercan, K., Civalek, O., Comparison of Small Scale Effect Theories for Buckling Analysis of Nanobeam. International Journal of Engineering and Applied Sciences, 9(3), 87-97, 2016.[22] Abbondanza, D., Battista, D., Morabito, F., Pallante, C., Barretta, R., Luciano, R., de Sciarra, F.M., Ruta, G., Modulated linear dynamics of nanobeams accounting for higher gradient effects. International Journal of Engineering and Applied Sciences, 8(2), 1-20, 2016.[23] He, J.H., A new approach to non-linear partial differential equations. Communications in Nonlinear Science and Numerical Simulation, 2, 230–235, 1997. [24] He, J.H., Variational iteration method for delay differential equations, Communications in Nonlinear Science and Numerical Simulation, 2, 235–236, 1997. [25] He, J.H., Variational iteration method a kind of non-linear analytical technique: some examples. International Journal of Nonlinear Mechanics, 34, 699–708, 1999.[26] He, J.H., Variational iteration method some recent results and new interpretations. Journal of Computational and Applied Mathematics, 207, 3–17, 2007. [27] He, J.H., Wu, X.H., Variational iteration method: new development and applications. Computers and Mathematics with Applications, 54,881–894, 2007.[28] Turkyilmazoglu, M., An optimal variational iteration method. Applied Mathematics Letters, 24(5), 762–765, 2011.[29] Yilmaz, E., Inc, M., Numerical simulation of the squeezing flow between two infinite plates by means of the modified variational iteration method with an auxiliary parameter. Nonlinear Science Letters A, 1, 297–306, 2010.[30] Hosseini, M.M., Mohyud-Din, S.T., Ghaneai H., Usman, M., Auxiliary parameter in the variational iteration method and its optimal determination. International Journal of Nonlinear Sciences and Numerical Simulation, 11(7), 495–502, 2010.[31] Herişanu, N., Marinca, V., A modified variational iteration method for strongly nonlinear problems. Nonlinear Science Letters A, 1, 183–192, 2010.[32] Altintan, D., Ugur, O., Generalisation of the Lagrange multipliers for variational iterations applied to systems of differential equations. Mathematical and Computer Modelling. 54, 2040-2050, 2011.[33] He, J.H., Notes on the optimal variational iteration method. Applied Mathematics Letters. 25(10), 1579-1581, 2012.[34] Coskun, S.B., Atay, M.T., Determination of critical buckling load for elastic columns of constant and variable cross-sections using variational iteration method. Computers and Mathematics with Applications, 58, 2260,2266, 2009.[35] Chen, Y., Zhang, J., Zhang, Z., Flapwise bending vibration of rotating tapered beams using variational iteration method. Journal of Vibration and Control, 22(15), 3384-3395, 2016.[36] Eroglu, U., Large deflection analysis of planar curved beams made of functionally graded materials using variational iterational method. Composite Structures, 136, 204–216, 2016.[37] Yun-dong, L., Yi-ren, Y., Vibration analysis of conveying fluid pipe via He’s variational iteration method. Applied Mathematical Modelling, 43, 409,420.[38] Eroglu, U., Tufekci, E., Small-Amplitude free vibrations of straight beams subjected to large displacements and rotation. Applied Mathematical Modelling, 53, 223-241, 2018.[39] Budiansky, B., Theory of buckling and postbuckling behavior of elastic structures, pp. 1–65 in Advances in applied mechanics,14, Academic Press, New York, 1974.[40] Timoshenko, S.P., Gere, J.M., Theory of Elastic Stability. McGraw-Hill, New York, 1961.[41] Pease, M.C., Methods of Matrix Algebra. Academic Press, New York, 1965.[42] Tufekci, E., Arpaci, A., Exact solution of in-plane vibrations of circular arches with account taken of axial extension, transverse shear and rotatory inertia effects. Journal of Sound and Vibration 209 (5):845–56, 1998.[43] Tufekci, E., Arpacı, A., Analytical solutions of in-plane static problems for non-uniform curved beams including axial and shear deformations. Structural Engineering and Mechanics, 22 (2):131–50, 2006.[44] Yildirim, V., Several Stress Resultant and Deflection Formulas for Euler-Bernoulli Beams under Concentrated and Generalized Power/Sinusoidal Distributed Loads, International Journal of Engineering and Applied Sciences, 10(2), 35-63, 2018.[45] Wei, D.J., Xani S.X., Zhang, Z.P., Li, X.F., Critical load for buckling of non-prismatic columns under self-weight and tip force. Mechanics Research Communications, 37, 554-558- 2010.[46] Wang, C.M., Wang, C.Y., Reddy, J.N., Exact Solutions for Buckling of Structural Members. CRC Press, Boca Raton, 2005.
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Details
| Primary Language | English |
|---|---|
| Subjects | Engineering |
| Journal Section | Articles |
| Authors | |
| Publication Date | November 4, 2018 |
| Acceptance Date | October 2, 2018 |
| Published in Issue | Year 2018 Volume: 10 Issue: 3 |
