This paper transforms combined loads, applied at an arbitrary point of a thin-walled open section beam, to the shear centre of the cross-section of the beam. Therein, a generalized transformation matrix for loads with respect to the shear centre is derived, this accounting for the bimoments that develop due to the way the combined loads are applied. This and the authors’ earlier paper (World Journal of Mechanics 2021, 11, 205-236) provide a full solution to the theory of thin-walled, open-section structures bearing combined loading. The earlier work identified arbitrary loading with the section’s area properties that are necessary to axial and shear stress calculations within the structure’s thin walls. In the previous paper attention is paid to the relevant axes of loading and to the transformations of loading required between axes for stress calculations arising from tension/compression, bending, torsion and shear. The derivation of the general transformation matrix applies to all types of loadings including, axial tensile and compression forces, transverse shear, longitudinal bending. One application, representing all these load cases, is given of a simple channel cantilever with an eccentrically located end load.
References
[1]
Wagner, H. (1936) Torsion and Buckling of Open Sections. US National Advisory Committee for Aeronautics.
[2]
Vlasov, V.Z. (1961) Thin-Walled Elastic Beams. National Science Foundation, Oldbourne Press, Washington DC.
[3]
Martin, H.C. (1965) Introduction to Matrix Methods of Structural Analysis. McGraw-Hill Companies, New York.
[4]
Przemieniecki, J.S. (1968) Discrete-Element Methods for Stability Analysis of Complex Structures. The Aeronautical Journal, 72, 1077-1086.
Krahula, J.L. (1967) Analysis of Bent and Twisted Bars Using the Finite Element Method. AIAA Journal, 5, 1194-1197. https://doi.org/10.2514/3.4163
[8]
Krajcinovic, D. (1969) A Consistent Discrete Elements Technique for Thin-Walled Assemblages. International Journal of Solids and Structures, 5, 639-662.
[9]
https://doi.org/10.1016/0020-7683(69)90085-7
[10]
Barsoum, R.S. and Gallagher, R.H. (1970) Finite Element Analysis of Torsional and Torsional-Flexural Stability Problems. International Journal for Numerical Methods in Engineering, 2, 335-352. https://doi.org/10.1002/nme.1620020304
[11]
Bleich, F. and Bleich, H.H. (1952) Buckling Strength of Metal Structures. McGraw-Hill, New York.
[12]
Rajasekaran, S. (1977) Finite Element Method for Plastic Beam-Columns. In: Atsuta, T., Ed., Theory of Beam Columns: Space Behavior and Design (pp. 539-608), Vol. 2, J. Ross Publishing, Florida.
[13]
Baigent, A.H. and Hancock, G.J. (1982) Structural Analysis of Assemblages of Thin-Walled Members. Engineering Structures, 4, 207-216.
[14]
https://doi.org/10.1016/0141-0296(82)90010-4
[15]
AL-Sheikh, A.M.S. (1985) Behaviour of Thin-Walled Structures under Combined Loads. Ph.D. Thesis, Loughborough University of Technology, Loughborough.
[16]
Chen, B.-Z. and Hu, Y.-R. (1988) The Torsional Stiffness Matrix of a Thin-Walled Beam and Its Application to Beams under Combined Loading. Computers & Structures, 28, 421-431. https://doi.org/10.1016/0045-7949(88)90080-6
[17]
Dvorkin, E.N., Celentano, D., Cuitiño, A. and Gioia, G. (1989) A Vlasov Beam Element. Computers & Structures, 33, 187-196.
[18]
https://doi.org/10.1016/0045-7949(89)90140-5
[19]
Musat, S.D. and Epureanu, B.I. (1999) Study of Warping Torsion of Thin-Walled Beams with Open Cross-Section Using Macro-Elements. International Journal for Numerical Methods in Engineering, 44, 853-868.
Erkmen, R.E. (2006) Finite Element Formulations for Thin-Walled Members. Ph.D. Thesis, Department of Civil Engineering University of Ottawa, Ottawa.
[22]
Wang, Z.-Q., Zhao, J.-C., Zhang, D.-X. and Gong, J.-H. (2012) Restrained Torsion of Open Thin-Walled Beams Including Shear Deformation Effects. Journal of Zhejiang University Science A, 13, 260-273. https://doi.org/10.1631/jzus.A1100149
[23]
Ferradi, M.K. and Cespedes, X. (2014) A New Beam Element with Transversal and Warping Eigenmodes. Computers & Structures, 131, 12-33.
[24]
https://doi.org/10.1016/j.compstruc.2013.10.001
[25]
Choi, I.S., Jang, G.-W., Choi, S., Shin, D. and Kim, Y.Y. (2017) Higher Order Analysis of Thin-Walled Beams with Axially Varying Quadrilateral Cross Sections. Computers & Structures, 179, 127-139.
[26]
https://doi.org/10.1016/j.compstruc.2016.10.025
[27]
Al-Sheikh, A.M.S. and Rees, D.W.A. (2021) General Stiffness Matrix for a Thin-Walled Open-Section Beam Structure. World Journal of Mechanics, 11, 205-234.
[28]
https://doi.org/10.4236/wjm.2021.1111015
[29]
Al-Sheikh, A.M. and Sharman, P.W. (1983) A Thin-Walled Beam Finite Element. Loughborough University, Loughborough.
[30]
Beaufait, F.W., Rowan, W.H. Hoadleey, P.G. and Hackett, R.M. (1970) Computer Methods of Structural Analysis. Prentice-Hall, Hoboken.
[31]
Zienkiewicz, O.C. (1971) The Finite Element Method in Engineering Science. McGraw-Hill, London.
[32]
Weaver, W. and Gere, J.M. (1980) Matrix Analysis of Framed Structures. 2nd Edition, Springer, Cham.