×

zbMATH — the first resource for mathematics

Nonlinear flexural-torsional dynamic analysis of beams of variable doubly symmetric cross section – application to wind turbine towers. (English) Zbl 1281.74019
Summary: In this paper, a boundary element solution is developed for the nonlinear flexural-torsional dynamic analysis of beams of arbitrary doubly symmetric variable cross section, undergoing moderate large displacements, and twisting rotations under general boundary conditions, taking into account the effect of rotary and warping inertia. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading in both directions and to twisting and/or axial loading. Four boundary-value problems are formulated with respect to the transverse displacements, to the axial displacement, and to the angle of twist and solved using the analog equation method, a boundary element method (BEM) based technique. Application of the boundary element technique yields a system of nonlinear coupled differential-algebraic equations (DAE) of motion, which is solved iteratively using the Petzold-Gear backward differentiation formula (BDF), a linear multistep method for differential equations coupled with algebraic equations. Numerical examples of great practical interest including wind turbine towers are worked out, while the influence of the nonlinear effects to the response of beams of variable cross section is investigated.

MSC:
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74H45 Vibrations in dynamical problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
74S15 Boundary element methods applied to problems in solid mechanics
Software:
NX; FEMAP
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Mohri, F.; Azrar, L.; Potier-Ferry, M., Vibration analysis of buckled thin-walled beams with open sections, J. Sound Vib., 275, 434-446, (2004)
[2] Machado, S.P.; Cortinez, V.H., Free vibration of thin-walled composite beams with static initial stresses and deformations, Eng. Struct., 29, 372-382, (2007)
[3] Lopes Alonso, R.; Ribeiro, P., Flexural and torsional non-linear free vibrations of beams using a p-version finite element, Comput. Struct., 86, 189-1197, (2008)
[4] Sapountzakis, E.J.; Dourakopoulos, J.A., Nonlinear dynamic analysis of Timoshenko beams by BEM. part I: theory and numerical implementation, Nonlinear Dyn., 58, 295-306, (2009) · Zbl 1183.74341
[5] Sapountzakis, E.J.; Dourakopoulos, J.A., Nonlinear dynamic analysis of Timoshenko beams by BEM. part II: applications and validation, Nonlinear Dyn., 58, 307-318, (2009) · Zbl 1183.74342
[6] Sapountzakis, E.J.; Tsipiras, V.J., Nonlinear nonuniform torsional vibrations of bars by the boundary element method, J. Sound Vib., 329, 1853-1874, (2010)
[7] Sapountzakis, E.J.; Dikaros, I.C., Non-linear flexural-torsional dynamic analysis of beams of arbitrary cross section by BEM, Int. J. Non-Linear Mech., 46, 782-794, (2011)
[8] Kitipornchai, S.; Trahair, N.S., Elastic behaviour of tapered monosymmetric I-beams, J. Struct. Div., 101, 1661-1678, (1975)
[9] Kameswara Rao, C.; Mirza, S., Free torsional vibrations of tapered cantilever I-beams, J. Sound Vib., 124, 489-496, (1988)
[10] Bradford, M.A., Elastic buckling of tapered monosymmetric I-beams, J. Struct. Eng., 114, 977-996, (1988)
[11] Eisenberger, M.; Reich, Y., Static, vibration and stability analysis of non-uniform beams, Comput. Struct., 31, 567-573, (1989) · Zbl 1236.34089
[12] Eisenberger, M., Exact solution for general variable cross-section members, Comput. Struct., 41, 765-772, (1991) · Zbl 0850.73306
[13] Ronagh, H.R.; Bradford, M.A.; Attard, M.M., Nonlinear analysis of thin-walled members of variable cross-section. part I: theory, Comput. Struct., 77, 285-299, (2000)
[14] Ronagh, H.R.; Bradford, M.A.; Attard, M.M., Nonlinear analysis of thin-walled members of variable cross-section. part II: application, Comput. Struct., 77, 301-313, (2000)
[15] Sapountzakis, E.J.; Mokos, V.G., Nonuniform torsion of composite bars of variable thickness by BEM, Int. J. Solids Struct., 41, 1753-1771, (2004) · Zbl 1045.74610
[16] Abdel-Jaber, M.S.; Al-Qaisia, A.A.; Abdel-Jaber, M.; Beale, R.G., Nonlinear natural frequencies of an elastically restrained tapered beam, J. Sound Vib., 313, 772-783, (2008)
[17] Hoseini, S.H.; Pirbodaghi, T.; Ahmadian, M.T.; Farrahi, G.H., On the large amplitude free vibrations of tapered beams: an analytical approach, Mech. Res. Commun., 36, 892-897, (2009) · Zbl 1173.35646
[18] Prathap, G.; Varadan, T.K., Non-linear vibrations of tapered cantilevers, J. Sound Vib., 55, l-8, (1977) · Zbl 0366.73058
[19] Prathap, G.; Varadan, T.K., The large amplitude vibration of tapered clamped beams, J. Sound Vib., 58, 87-94, (1978) · Zbl 0371.70006
[20] Nageswara Rao, B.; Venkateswara Rao, G., Large amplitude vibrations of a tapered cantilever beam, J. Sound Vib., 127, 173-178, (1988)
[21] Nageswara Rao, B.; Venkateswara Rao, G., Large-amplitude vibrations of free-free tapered beams, J. Sound Vib., 141, 511-515, (1990)
[22] Kanaka Raju, K.; Shastry, B.P.; Venkateswara Rao, G., A finite element formulation for the large amplitude vibrations of tapered beams, J. Sound Vib., 47, 595-598, (1976)
[23] Shavezipur, M.; Hashemi, S.M., Free vibration of triply coupled centrifugally stiffened nonuniform beams, using a refined dynamic finite element method, Aerosp. Sci. Technol., 13, 59-70, (2009)
[24] Liao, M.; Zhong, H., Nonlinear vibration analysis of tapered Timoshenko beams, Chaos Solitons Fractals, 36, 1267-1272, (2008) · Zbl 1135.74021
[25] Verma, M.K.; Krishna Murthy, A.V., Non-linear vibrations of non-uniform beams with concentrated masses, J. Sound Vib., 33, 1-12, (1974) · Zbl 0277.73054
[26] Bazoune, A.; Khulief, Y.A.; Stephen, N.G.; Mohiuddin, M.A., Dynamic response of spinning tapered Timoshenko beams using modal reduction, Finite Elem. Anal. Des., 37, 199-219, (2001) · Zbl 1015.74066
[27] Katsikadelis, J.T.; Tsiatas, G.C., Non-linear dynamic analysis of beams with variable stiffness, J. Sound Vib., 270, 847-863, (2004)
[28] Katsikadelis, J.T., The analog equation method. A boundary-only integral equation method for nonlinear static and dynamic problems in general bodies, Theor. Appl. Mech., 27, 13-38, (2002) · Zbl 1051.74052
[29] Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. North-Holland, Amsterdam (1989) · Zbl 0699.65057
[30] Ramm, E., Hofmann, T.J.: In: Mehlhorn, G. (ed.) Stabtragwerke, Der Ingenieurbau (Beam Structures in Civil Engineering), Band Baustatik/Baudynamik. Ernst & Sohn, Berlin (1995)
[31] Rothert, H., Gensichen, V.: Nichtlineare Stabstatik (Nonlinear Frame Analysis). Springer, Berlin (1987) · Zbl 0628.73068
[32] Sapountzakis, E.J.; Mokos, V.G., Warping shear stresses in nonuniform torsion by BEM, Comput. Mech., 30, 131-142, (2003) · Zbl 1128.74344
[33] Sapountzakis, E.J., Nonuniform torsion of multi-material composite bars by the boundary element method, Comput. Struct., 79, 2805-2816, (2001)
[34] Katsikadelis, J.T.: Boundary Elements: Theory and Applications. Elsevier, Amsterdam-London (2002) · Zbl 1051.74052
[35] Brigham, E.: Fast Fourier Transform and Its Applications. Prentice Hall, Englewood Cliffs (1988)
[36] Lewandowski, R., Free vibration of structures with cubic non-linearity-remarks on amplitude equation and Rayleigh quotient, Comput. Methods Appl. Mech. Eng., 192, 1681-1709, (2003) · Zbl 1033.74018
[37] FEMAP for Windows: Finite element modeling and post-processing software. Help System Index. Version 10 (2008) · Zbl 0701.65014
[38] Siemens PLM Software Inc: NX Nastran User’s Guide (2008) · Zbl 1183.74341
[39] Quiligan, A.; O’Connor, A.; Pakrashi, V., Fragility analysis of steel and concrete wind turbine towers, Eng. Struct., 36, 270-282, (2012)
[40] Hansen, M.O.L.: Aerodynamics of Wind Turbines. Earthscan, London (2008)
[41] Jonkman, J.M.: Dynamics modeling and loads analysis of an offshore floating wind turbine. Technical Report NREL/TP-500-41958 (2007)
[42] CEN/TC250: Eurocode 1: actions on structures-general actions—part 1-4: wind actions. prEN 1991-1-4 (2004) · Zbl 1293.74207
[43] Deodatis, G., Simulation of ergodic multivariate stochastic processes, J. Eng. Mech., 122, 778-787, (1996)
[44] Vassilopoulou, I.; Gantes, C.; Gkimousis, I., Response of cable networks under wind loading, Volos · Zbl 1293.74207
[45] Fornberg, B., Generation of finite difference formulas on arbitrarily spaced grids, Math. Comput., 50, 669-706, (1988) · Zbl 0701.65014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.