P. R. Heyliger and J. N. Reddy, A higher order beams finite element for bending and vibration problems, Journal of sound and vibration, vol.126, issue.2, pp.309-326, 1988.

P. O. Friberg, Coupled vibrations of beams-an exact dynamic element stiffness matrix, International Journal for numerical methods in engineering, vol.19, issue.4, pp.479-493, 1983.

E. Dokumaci, An exact solution for coupled bending and torsion vibrations of uniform beams having single cross-sectional symmetry, Journal of Sound and Vibration, vol.119, issue.3, pp.443-449, 1987.

J. M. Gere and Y. K. Lin, Coupled vibrations of thin-walled beams of open cross section, Journal of Applied Mechanics, vol.25, issue.3, pp.373-378, 1958.

V. Z. Vlasov, Thin-walled Elastic Beams. Moscow, 2 nd edition, translated from Russian and published for the national Science foundation, 1961.

S. P. Timoshenko, D. H. Young, W. Jr, and W. , Vibration problems in engineering, 1974.

R. E. Bishop, S. M. Cannon, and S. Miao, On coupled bending and torsional vibration of uniform beams, Journal of sound and vibration, vol.131, issue.3, pp.457-464, 1989.

N. S. Trahair, Flexural-torsional buckling of structures. E and FN Spon, 1993.

F. Bleich, Buckling strength of metal structures, 1952.

A. Chajes, Principles of structural stability theory, 1974.

M. Petyt, Introduction to finite element vibration analysis, 2010.

J. R. Banerjee, Coupled bending-torsional dynamic stiffness matrix for beam elements, International journal for numerical methods in engineering, vol.28, issue.6, pp.1283-1298, 1989.

J. R. Banerjee, Explicit frequency equation and mode shapes of a cantilever beam coupled in bending and torsion, Journal of Sound and Vibration, vol.224, issue.2, pp.267-281, 1999.

C. Mei, Coupled vibrations of thin-walled beams of open section using the finite element method, International Journal of Mechanical Sciences, vol.12, issue.10, pp.883-891, 1970.

R. S. Barsoum, Finite element method applied to the problem of stability of a non-conservative system, International Journal for Numerical Methods in Engineering, vol.3, issue.1, pp.63-87, 1971.

P. O. Friberg, Beam element matrices derived from Vlasov's theory of open thin-walled elastic beams, International Journal for Numerical Methods in Engineering, vol.21, issue.7, pp.1205-1228, 1985.

M. Ohga, H. Takao, and T. Hara, Natural frequencies and mode shapes of thinwalled members, Computers and structures, vol.55, issue.6, pp.971-978, 1995.

J. Lee and K. S. , Flexural-torsional coupled vibration of thin-walled composite beams with channel sections, Computers and structures, vol.80, issue.2, pp.133-144, 2002.

J. Lee and K. S. , Free vibration of thin-walled composite beams with Ishaped cross-sections, Composite Structures, vol.55, issue.2, pp.205-215, 2002.

E. C. Pestel and F. A. Leckie, Matrix methods in Elastomechanics, 1963.

A. M. Ebner and D. P. Billington, Steady state vibration of damped Timoshenko beams, Journal of the Structural Division, vol.94, issue.3, pp.737-760, 1968.

H. H. Chen and K. M. Hsiao, Quadruply coupled linear free vibrations of thinwalled beams with a generic open section. Engineering Structures, vol.30, pp.1319-1334, 2008.

F. De-borbon, A. Mirasso, and A. D. , A beam element for coupled torsional-flexural vibration of doubly unsymmetrical thin walled beams axially loaded, Computers and Structures, vol.89, pp.1406-1416, 2011.

M. Tanaka and A. N. Bercin, Free vibration solution for uniform beams of nonsymmetrical cross section using Mathematica. Computers and structures, vol.71, pp.1-8, 1999.

A. Arpaci and B. E. , On free vibration analysis of thin-walled beams with nonsymmetrical open cross-sections. Computers and structures, vol.80, pp.691-695, 2002.

C. N. Chen, Dynamic equilibrium equations of non-prismatic beams defined on an arbitrarily selected co-ordinate system, Journal of sound and vibration, vol.230, issue.2, pp.241-260, 2000.

V. H. Cortínez, M. T. Piovan, and S. P. Machado, DQM for vibration analysis of composite thin-walled curved beams, 2001.

R. D. Ambrosini, J. D. Riera, and R. F. Danesi, A modified Vlasov theory for dynamic analysis of thin-walled and variable open section beams, Engineering Structures, vol.22, issue.8, pp.890-900, 2000.

B. Rozmarynowski and C. Szymczak, Non-linear free torsional vibrations of thin-walled beams with bisymmetric cross-section, Journal of Sound and Vibration, vol.97, issue.1, pp.145-152, 1984.

D. Egidio, A. Luongo, A. Vestroni, and F. , A non-linear model for the dynamics of open cross-section thin-walled beams Part I: formulation, International Journal of non-linear Mechanics, vol.38, issue.7, pp.1067-1081, 2003.
URL : https://hal.archives-ouvertes.fr/hal-00798001

Y. Yaman, Vibrations of open-section channels: a coupled flexural and torsional wave analysis, Journal of Sound and Vibration, vol.204, issue.1, pp.131-158, 1997.

A. S. Gendy and A. F. Saleeb, Vibration analysis of coupled extensional/flexural/torsional modes of curved beams with arbitrary thin-walled sections, Journal of Sound and Vibration, vol.174, issue.2, pp.261-274, 1994.

A. Proki?, On triply coupled vibrations of thin-walled beams with arbitrary cross-section, Journal of Sound and Vibration, vol.279, issue.3-5, pp.723-737, 2005.

A. Proki?, On fivefold coupled vibrations of Timoshenko thin-walled beams. Engineering Structures, vol.28, pp.54-62, 2006.

E. Carrera, G. Giunta, and P. M. , Beam structures: classical and advanced theories, 2011.

E. Carrera, M. Cinefra, M. Petrolo, and Z. E. , Finite element analysis of structures through unified formulation, 2014.

E. Carrera and A. Varello, Dynamic response of thin-walled structures by variable kinematic one-dimensional models, Journal of Sound and Vibration, vol.331, issue.24, pp.5268-5282, 2012.

E. Carrera and A. Pagani, Free vibration analysis of civil engineering structures by component-wise models, Journal of Sound and Vibration, vol.333, issue.19, pp.4597-4620, 2014.

A. Pagani, R. Augello, and E. Carrera, Frequency and mode change in the large deflection and post-buckling of compact and thin-walled beams, Journal of Sound and Vibration, vol.432, pp.88-104, 2018.

E. Carrera and M. Filippi, A refined one-dimensional rotordynamics model with three-dimensional capabilities, Journal of Sound and Vibration, vol.366, pp.343-356, 2016.

F. Mohri, L. Azrar, and M. Potier-ferry, Vibration analysis of buckled thinwalled beams with open sections, Journal of Sound Vibration, vol.275, pp.434-446, 2004.

A. Arpaci, S. E. Bozdag, and E. Sunbuloglu, Triply coupled vibrations of thinwalled open cross-section beams including rotary inertia effects, Journal of Sound and Vibration, vol.260, issue.5, pp.889-900, 2003.

J. R. Banerjee, S. Guo, and W. P. Howson, Exact dynamic stiffness matrix of a bending-torsion coupled beam including warping, Computers and structures, vol.59, issue.4, pp.613-621, 1996.

S. S. Rao, Mechanical vibrations, vol.5, 2004.

S. H. Eslimy, J. R. Banerjee, and A. J. Sobey, Response of a bending-torsion coupled beam to deterministic and random loads, Journal of Sound and Vibration, vol.195, issue.2, pp.267-283, 1996.

S. M. Hashemi and M. J. Richard, A Dynamic Finite Element (DFE) method for free vibrations of bending-torsion coupled beams, Aerospace Science and Technology, vol.4, issue.1, pp.41-55, 2000.

L. Jun, S. Rongying, H. Hongxing, and J. Xianding, Coupled bending and torsional vibration of axially loaded Bernoulli-Euler beams including warping effects, Applied Acoustics, vol.65, issue.2, pp.153-170, 2004.

L. Jun, L. Wanyou, S. Rongying, and H. Hongxing, Coupled bending and torsional vibration of nonsymmetrical axially loaded thin-walled Bernoulli-Euler beams, Mechanics Research Communications, issue.6, pp.697-711, 2004.

H. Gökda? and O. Kopmaz, Coupled bending and torsional vibration of a beam with in-span and tip attachments, Journal of Sound and Vibration, vol.287, issue.3, pp.591-610, 2005.

A. Proki? and D. Luki?, Dynamic analysis of thin-walled closed-section beams, Journal of Sound and Vibration, vol.302, issue.4-5, pp.962-980, 2007.

A. Burlon, G. Failla, and F. Arena, Coupled bending and torsional free vibrations of beams with in-span supports and attached masses, European Journal of Mechanics-A/Solids, vol.66, pp.387-411, 2017.

A. Burlon, G. Failla, and F. Arena, Coupled bending-torsional frequency response of beams with attachments: exact solutions including warping effects, Acta Mechanica, vol.229, issue.6, pp.2445-2475, 2018.

L. Librescu and O. Song, Thin-walled composite beams: theory and application, vol.131, 2005.

M. J. Klausbruckner and R. J. Pryputniewicz, Theoretical and experimental study of coupled vibrations of channel beams, Journal of Sound and Vibration, vol.183, issue.2, pp.239-252, 1995.

D. Ambrosini, On free vibration of nonsymmetrical thin-walled beams. Thin-Walled Structures, vol.47, pp.629-636, 2009.

D. Ambrosini, Experimental validation of free vibrations from nonsymmetrical thin walled beams. Engineering Structures, vol.32, pp.1324-1332, 2010.

P. Ribeiro and R. Carneiro, Experimental detection of modal interaction in the nonlinear vibration of a hinged-hinged beam, Journal of sound and vibration, vol.277, issue.4-5, pp.943-954, 2004.

G. Piana, E. Lofrano, A. Manuello, and G. Ruta, Natural frequencies and buckling of compressed non-symmetric thin-walled beams, Thin-Walled Structures, vol.111, pp.189-196, 2017.

J. Zhou, S. Wen, F. Li, and Y. Zhu, Coupled bending and torsional vibrations of non-uniform thin-walled beams by the transfer differential transform method and experiments. Thin-Walled Structures, vol.127, pp.373-388, 2018.

Z. P. Bazant, E. Nimeiri, and M. , Large-deflection spatial buckling of thinwalled beams and frames, ASCE J Eng Mech Div, vol.99, issue.EM6, pp.1259-1281, 1973.

Y. B. Yang and W. Mcguire, A procedure for analysing space frames with partial warping restraint, International Journal for Numerical Methods in Engineering, vol.20, issue.8, pp.1377-1398, 1984.

P. J. Morrell, J. R. Riddington, F. A. Ali, and H. A. Hamid, Influence of joint detail on the flexural/torsional interaction of thin-walled structures, Thin-walled structures, vol.24, issue.2, pp.97-111, 1996.

C. Basaglia, D. Camotim, and N. Silvestre, Torsion warping transmission at thin-walled frame joints: kinematics, modelling and structural response, Journal of Constructional Steel Research, vol.69, issue.1, pp.39-53, 2012.

H. D. Hibbitt, B. I. Karlsson, and P. Sorensen, ABAQUS theory manual, version 6.3, 2006.

P. Mikell and . Groover, Fundamentals of Modern Manufacturing, 2010.

J. Zhang, R. J. Perez, and E. J. Lavernia, Documentation of damping capacity of metallic, ceramic and metal-matrix composite materials, Journal of Materials Science, vol.28, issue.9, pp.2395-2404, 1993.

C. T. Sun and Y. P. Lu, Vibration damping of structural elements, 1995.

L. Cremer and M. Heckl, Structure-borne sound: structural vibrations and sound radiation at audio frequencies, 2013.

F. Mohri, A. Brouki, and J. C. Roth, Theoretical and numerical stability analyses of unrestrained, mono-symmetric thin-walled beams, Journal of Constructional Steel Research, vol.59, issue.1, pp.63-90, 2003.

F. Mohri, N. Damil, and M. P. Ferry, Large torsion finite element model for thin-walled beams, Computers and Structures, vol.86, issue.7-8, pp.671-683, 2008.
URL : https://hal.archives-ouvertes.fr/hal-01513868

F. Mohri, C. Bouzerira, and M. Potier-ferry, Lateral buckling of thin-walled beam-column elements under combined axial and bending loads, Thin-Walled Structures, vol.46, issue.3, pp.290-302, 2008.

F. Mohri, N. Damil, and M. Potier-ferry, Pre-buckling deflection effects on stability of thin-walled beams with open sections. Steel and Composite Structures, vol.13, pp.71-89, 2012.

F. Jacob and B. Ted, A first course in finite elements, 2007.

G. R. Liu and S. S. Quek, The finite element method: a practical course, 2003.

K. J. Bathe, Finite element procedures. Klaus-Jurgen Bathe, 2006.

S. B. Dong, W. Jr, and J. A. , Effect of transverse shear deformation on vibrations of planar structures composed of beam-type elements, The Journal of the Acoustical Society of America, vol.53, issue.1, pp.120-127, 1973.

F. Mohri, L. Azrar, and M. Potier-ferry, Flexural-torsional post-buckling analysis of thin-walled elements with open sections. Thin-walled structures, vol.39, pp.907-938, 2001.

F. Mohri, A. Eddinari, N. Damil, and M. P. Ferry, A beam finite element for non-linear analyses of thin-walled elements, Thin-Walled Structures, vol.46, issue.7-9, pp.981-990, 2008.

F. Mohri, N. Damil, and M. Potier-ferry, Linear and non-linear stability analyses of thin-walled beams with monosymmetric I sections, Thin-Walled Structures, vol.48, issue.4-5, pp.299-315, 2010.

R. Matlab, The MathWorks Inc, 2015.

R. Adina and . Inc,

A. K. Noor, J. M. Peters, and B. J. Min, Mixed finite element models for free vibrations of thin-walled beams, Finite Elements in Analysis and Design, vol.5, issue.4, pp.291-305, 1989.

T. P. Vo, J. Lee, and K. Lee, On triply coupled vibrations of axially loaded thinwalled composite beams, Computers and structures, vol.88, issue.3-4, pp.144-153, 2010.

C. T. Nguyen, J. Moon, and H. E. Lee, Natural frequency for torsional vibration of simply supported steel I-girders with intermediate bracings, Thin-Walled Structures, vol.49, issue.4, pp.534-542, 2011.

R. S. Barsoum and R. H. Gallagher, Finite element analysis of torsional and torsional-flexural stability problems, International Journal for Numerical Methods in Engineering, vol.2, issue.3, pp.335-352, 1970.

E. N. Dvorkin, D. Celentano, A. Cuitino, and G. Gioia, A Vlasov beam element. Computers and structures, vol.33, pp.187-196, 1989.

G. Dhatt and J. L. Batoz, Modélisation des structures par éléments finis : poutres et plaques, 1990.

S. Bolouri and H. , Modélisation des structures-poutres élasto-plastique (Doctoral dissertation, 1994.

C. Bo-zhen and H. Yu-ren, The torsional stiffness matrix of a thin-walled beam and its application to beams under combined loading, Computers and structures, vol.28, issue.3, pp.421-431, 1988.

A. Conci and M. Gattass, Natural approach for geometric non-linear analysis of thin-walled frames, International Journal for Numerical Methods in Engineering, vol.30, issue.2, pp.207-231, 1990.

C. F. Kolbrunner and K. Basler, Torsion in Structures: An Engineering Approach, 1969.

A. M. Tralli, A simple hybrid model for torsion and flexure of thin-walled beams, 1986.

G. R. Liu and S. S. Quek, The finite element method: a practical course, 2013.

R. W. Clough and J. Penzien, Dynamics of structures, Journal of Structures, 1975.

W. C. Hurty and M. F. Rubinstein, Dynamics of structures. Prentice-Hall Series in Engineering of the Physical Sciences, 1964.

C. J. Thompson, Classical equilibrium statistical mechanics, 1988.

D. Ewins, Modal Testing: Theory, Practice, and Application, 2000.

J. A. Yura, Fundamentals of beam bracing. Engineering Journal. American institute of steel construction, vol.38, pp.11-26, 2001.

A. R. Flint, The influence of restraints on the stability of beams, The Structural Engineer, vol.29, issue.9, pp.235-246, 1951.

G. Winter, Lateral bracing of beams and columns, ASCE Transactions, Paper, issue.3044, 1960.

A. Taylor and M. Ojalvo, Torsional restraint of lateral buckling, J. Struct. Div, vol.92, issue.2, pp.115-130, 1966.

S. Kitipornchai and . Nj-r, Elastic lateral buckling of I-beams with discrete intermediate restraints. Transactions of the Institution of Engineers, Australia. Civil engineering, issue.2, pp.105-111, 1978.

J. A. Yura, Winter's bracing approach revisited. Engineering structures, vol.18, pp.821-825, 1996.

C. T. Nguyen, J. Moon, and H. E. Lee, Lateral-torsional buckling of I-girders with discrete torsional bracings, Journal of Constructional Steel Research, vol.66, issue.2, pp.170-177, 2010.

F. Mccann, L. Gardner, and M. A. Wadee, Design of steel beams with discrete lateral restraints, Journal of Constructional Steel Research, vol.80, pp.82-90, 2013.

A. Agüero, F. J. Pallarés, and L. Pallarés, Equivalent geometric imperfection definition in steel structures sensitive to lateral torsional buckling due to bending moment, Engineering Structures, vol.96, pp.41-55, 2015.

W. F. Zhang, Y. C. Liu, G. L. Hou, K. S. Chen, J. Ji et al., , 2016.

, Lateral-torsional buckling analysis of cantilever beam with tip lateral elastic brace under uniform and concentrated load, International Journal of Steel Structures, vol.16, issue.4, pp.1161-1173

P. Pezeshky, A. Sahraei, and M. Mohareb, Effect of bracing heights on lateral torsional buckling resistance of steel beams, Proc., 6th Int. Conf. on Engineering Mechanics and Materials CSCE, 2017.

N. Olhoff, Minimum stiffness of optimally located supports for maximum value of beam eigenfrequencies, Journal of Sound Vibration, vol.120, pp.457-463, 1988.

C. K. Rao, Frequency analysis of clamped-clamped uniform beams with intermediate elastic support, Journal of Sound and Vibration, vol.133, issue.3, pp.502-509, 1989.

H. Abramovich and O. Hamburger, Vibration of a uniform cantilever Timoshenko beam with translational and rotational springs and with a tip mass, Journal of Sound and Vibration, vol.154, issue.1, pp.67-80, 1992.

J. S. Wu and H. M. Chou, Free vibration analysis of a cantilever beam carrying any number of elastically mounted point masses with the analytical-and-numericalcombined method, Journal of Sound and Vibration, vol.213, issue.2, pp.317-332, 1998.

C. Y. Wang, Minimum stiffness of an internal elastic support to maximize the fundamental frequency of a vibrating beam, Journal of sound and vibration, vol.259, issue.1, pp.229-232, 2003.

D. Wang, J. S. Jiang, and W. H. Zhang, Optimization of support positions to maximize the fundamental frequency of structures, International journal for numerical methods in engineering, issue.10, pp.1584-1602, 2004.

C. M. Albarrac??, L. Zannier, and R. O. Grossi, Some observations in the dynamics of beams with intermediate supports, Journal of Sound and Vibration, vol.1, issue.271, pp.475-480, 2004.

M. J. Maurizi, D. V. Bambill, P. M. Bellés, D. Rosa, and M. A. , Free vibrations of Bernoulli-Euler beams with intermediate elastic support: a concise thematic recension, Journal of Sound Vibration, vol.281, pp.1238-1239, 2005.

D. Wang, M. I. Friswell, and Y. Lei, Maximizing the natural frequency of a beam with an intermediate elastic support, Journal of sound and vibration, vol.291, issue.3-5, pp.1229-1238, 2006.

H. Gokdag and O. Kopmaz, Natural frequencies of a bending-torsion coupled beam supported by in-span linear springs, Structural Engineering and Mechanics, vol.27, issue.2, pp.259-262, 2007.

. Zbirohowski-koscia, Thin-walled beams. First published By Crosby Lockwook and Son L td, 1967.

W. Jrad, F. Mohri, G. Robin, E. M. Daya, and J. Al-hajjar, Analytical and finite element solutions of free and forced vibration of unrestrained and braced thin-walled beams, Journal of Vibration and Control, p.1077546319878901, 2019.