Influence of in-Plane Deformation in Higher Order Beam Theories

Sapountzakis, Evangelos; Argyridi, Amalia

doi:10.2478/scjme-2018-0028

Abstract

Comparing Euler-Bernoulli or Tismoshenko beam theory to higher order beam theories, an essential difference can be depicted: the additional degrees of freedom accounting for out-of plane (warping) and in-plane (distortional) phenomena leading to the appearance of respective higher order geometric constants. In this paper, after briefly overviewing literature of the major beam theories taking account warping and distortional deformation, the influence of distortion in the response of beams evaluated by higher order beam theories is examined via a numerical example of buckling drawn from the literature.

References

[1] J. Murin, M. Aminbaghai, V. Goga, V. Kutiš , J. Paulech 1 , J. Hrabovský. Effect of nonuniform torsion on elastostatics of a frame of hollow rectangular cross-section. Journal of Mechanical Engineering - Strojnícky časopis 2018 (68), No. 2, 35 - 52.
Search in Google Scholar Back to article
[2] R. Schardt. Lateral torsional, and distortional buckling of channel-, and hat-sections. Journal of Constructional Steel Research 1994 (31), No. 2 - 3, 243 - 265.
Search in Google Scholar Back to article
[3] E. Reissner. Analysis of shear lag in box beams by the principle of minimum potential energy. Quarterly of Applied Mathematics 1946 (4), No. 3, 268 - 278.
Search in Google Scholar Back to article
[4] D. J. Malcolm, R.G. Redwood. Shear lag in stiffened box-girders. J. Struct. Div. ASCE 1970 (96), No. 7, 1403 - 1415.
Search in Google Scholar Back to article
[5] K. R. Moffatt, P. J. Dowling. Shear lag in steel box-girder bridges. Struct. Engineer 1975 (53), No. 10, 439 - 448.
Search in Google Scholar Back to article
[6] N.W. Murray. Introduction to the theory of thin-walled structures. Oxford University Press, 1986.
Search in Google Scholar Back to article
[7] Eurocode 3: Design of Steel Structures - Part 1.5: Plated Structural Elements, European Committee for Standardization, 2004, prEN 1993-1-5.
Search in Google Scholar Back to article
[8] Eurocode 4: Design of Composite Steel and Concrete Structures - Part 1.1: General Rules and Rules for Buildings, European Committee for Standardization, 2004, prEN 1994-1-1.
Search in Google Scholar Back to article
[9] Eurocode 4: Design of Composite Steel and Concrete Structures - Part 2: General Rules and Rules for Bridges, European Committee for Standardization, 2004, prEN 1994-2.
Search in Google Scholar Back to article
[10] Eurocode 2: Design of concrete structures - Part 2: Concrete bridges - Design and detailing rules, European Committee for Standardization, 2005, EN 1992-2.
Search in Google Scholar Back to article
[11] C. A., Ie, J. B. Kosmatka. On the analysis of prismatic beams using first-order warping functions. International Journal of Solids, and Structures 1992 (29), No. 7, 879 - 891.
Search in Google Scholar Back to article
[12] Eurocode 9: Design of aluminium structures - Part 1.3: Structures susceptible to fatigue, European Committee for Standardization, 2007, EN 1999-1-3.
Search in Google Scholar Back to article
[13] N.H., Park, N.H., Lim, Y.J. Kang. A consideration on intermediate diaphragm spacing in steel box girder bridges with a doubly symmetric section. Engineering Structures 2003 (25), No. 13, 1665 - 1674.
Search in Google Scholar Back to article
[14] V. Vlasov. Thin-walled elastic beams. Israel Program for Scientific Translations, Jerusalem, 1963.
Search in Google Scholar Back to article
[15] R.N. Wright, S.R. Abdel-Samad, A.R. Robinson. BEF analogy for analysis of box girders. J. Struct. Div., ASCE 1968 (94), No. 7, 1719 - 1743.
Search in Google Scholar Back to article
[16] L.T. Stavridis. Structural Systems: Behaviour, and design. Thomas Telford Limited, 40 Marsh Wall, London, 2010.
Search in Google Scholar Back to article
[17] L.F. Boswell, S.H. Zhang. A box beam finite element for the elastic analysis of thinwalled structures. Thin-Walled Structures 1983 (1), No. 4, 353 - 383.
Search in Google Scholar Back to article
[18] L.F. Boswell, S.H. Zhang. The effect of distortion in thin-walled box-spine beams. International Journal of Solids and Structures 1984 (20), No. 9 - 10, 842 - 862.
Search in Google Scholar Back to article
[19] S.H. Zhang, L.P.R. Lyons. A thin-walled box beam finite element for curved bridge analysis. Computers and Structures 1984 (18), No. 6, 1035 - 1046.
Search in Google Scholar Back to article
[20] S.H. Zhang, L.P.R. Lyons. The application of the thin-walled box beam element to multibox bridge analysis. Computers and Structures 1984 (18), No. 5, 795 - 802.
Search in Google Scholar Back to article
[21] B. Kermani, P. Waldron. Analysis of continuous box girder bridges including the effects of distortion. Computers and Structures 1993 (47), No. 3, 427 - 440.
Search in Google Scholar Back to article
[22] N.I. Kim, M.Y. Kim. Exact dynamic/static stiffness matrices of non-symmetric thinwalled beams considering coupled shear deformation effects. Thin-Walled Structures 2005 (43), No. 5, 701 - 734.
Search in Google Scholar Back to article
[23] N.H. Park, S. Choi, Y.J. Kang. Exact distortional behavior, and practical distortional analysis of multicell box girders using an expanded method. Computers and Structures 2005 (83), No. 19 - 20, 1607 - 1626.
Search in Google Scholar Back to article
[24] L.F. Boswell, Q. Li. Consideration of the relationships between torsion, distortion, and warping of thin-walled beams. Thin-Walled Structures 1995 (21), No. 2, 147 - 161.
Search in Google Scholar Back to article
[25] R. Schardt. Verallgemeinerte technicsche biegetheory. Germany: Springler - Verlag, 1989.
Search in Google Scholar Back to article
[26] R. Schardt. Generalised Beam Theory - An adequate method for coupled stability problems. Thin-Walled Structures 1994 (19), No. 2 - 4, 161 - 180.
Search in Google Scholar Back to article
[27] J.M. Davies, P. Leach. First-order Generalized Beam Theory. Journal of Constructional Steel Research 1994 (31), No. 2 - 3, 187 - 221.
Search in Google Scholar Back to article
[28] J.M. Davies, P. Leach, D. Heinz. Second-order Generalized Beam Theory. Journal of Constructional Steel Research 1994 (31), No. 2 - 3, 221 - 241.
Search in Google Scholar Back to article
[29] P. Leach. The calculation of modal cross-section properties for use in the Generalized Beam Theory. Thin-Walled Structures 1994 (19), No. 1, 61 - 79.
Search in Google Scholar Back to article
[30] P. Leach, J. M. Davies. An experimental verification of the Generalized Beam Theory applied to interactive buckling problems. Thin-Walled Structures 1996 (25), No. 1, 61 -79.
Search in Google Scholar Back to article
[31] N. Silvestre, D. Camotim. First-order Generalised Beam Theory for arbitrary orthotropic materials. Thin-Walled Structures 2002 (40), No. 9, 755 - 789.
Search in Google Scholar Back to article
[32] R. Gonçalves, D. Camotim. GBT local, and global buckling analysis of aluminium, and stainless steel columns. Computers and Structures 2004 (82), No. 17 - 19, 1473 - 1484.
Search in Google Scholar Back to article
[33] R. Gonçalves, D. Camotim. Thin-walled member plastic bifurcation analysis using Generalised Beam Theory. Advances in Engineering Software 2007 (38), No. 8 - 9, 637 - 646.
Search in Google Scholar Back to article
[34] R. Gonçalves, D. Camotim. Generalised Beam Theory-based finite elements for elastoplastic thin-walled metal members. Thin-Walled Structures 2011 (49), No. 10, 1237 - 1245.
Search in Google Scholar Back to article
[35] N. Silvestre. Generalised Beam Theory to analyse the buckling behaviour of circular cylindrical shells, and Tubes. Thin-Walled Structures 2007 (45), No. 2, 185 - 198.
Search in Google Scholar Back to article
[36] P. B. Dinis, D. Camotim, N. Silvestre. On the local, and global buckling behaviour of angle, t-section, and cruciform thin-walled members. Thin-Walled Structures 2010 (48), No. 10 -11, 786 - 797.
Search in Google Scholar Back to article
[37] R. Gonçalves, M. R. Corrêa, D. Camotim. A large displacement, and finite rotation thinwalled beam formulation including cross-section deformation. Comput. Methods Appl. Mech. Engrg 2010 (199), No. 23 - 24, 1627 - 1643.
Search in Google Scholar Back to article
[38] N. Silvestre, D. Camotim. On the mechanics of distortion in thin-walled open sections. Thin-Walled Structures. 2010 (48), No. 7, 469 - 481.
Search in Google Scholar Back to article
[39] D. Camotim, P. B. Dinis. Coupled instabilities with distortional buckling in cold-formed steel lipped channel columns. Thin-Walled Structures 2011 (49), No. 5, 562 - 575.
Search in Google Scholar Back to article
[40] P.B. Dinis, D. Camotim. Post-buckling behaviour, and strength of cold-formed steel lipped channel columns experiencing distortional/global interaction. Computers and Structures 2011 (89), No. 3 - 4, 422 - 434.
Search in Google Scholar Back to article
[41] M. Abambres, D. Camotim, N. Silvestre. GBT-Based first-order analysis of elastic plastic thin-walled steel members exhibiting strain-hardening. The IES Journal Part A: Civil, and Structural Engineering 2013 (6), No. 2., 119 - 134. DOI: 10.1080/19373260.2012.757209
Search in Google Scholar Back to article
[42] C. D. Balch, C. R. Steele. Asymptotic solutions for warping, and distortion of thin-walled box beams. Journal of Applied Mechanics, ASME 1987 (54), No. 1, 165 - 173.
Search in Google Scholar Back to article
[43] L. Mentrasti. Distorion (and torsion) of rectangular thin-walled beams. Thin-Walled Structures 1990 (10), No. 3, 175 - 193.
Search in Google Scholar Back to article
[44] G. Ranzi, A. Luongo. A new approach for thin-walled member analysis in the framework of GBT, Thin-Walled Structures 2011 (49), No. 11, 1404 - 1414.
Search in Google Scholar Back to article
[45] J. Jönsson, M. J. Andreassen. Distortional eigenmodes, and homogeneous solutions for semi-discretized thin-walled beams. Thin-Walled Structures 2011 (49), No. 6, 691 - 707.
Search in Google Scholar Back to article
[46] M. J. Andreassen, J. Jönsson. Distortional solutions for loaded semi-discretized thinwalled beams. Thin-Walled Structures 2012 (50), No. 1, 116 - 127.
Search in Google Scholar Back to article
[47] M. J. Andreassen, J. Jönsson. Distortional buckling modes of semi-discretized thinwalled columns. Thin-Walled Structures 2012 (51), 53 - 63.
Search in Google Scholar Back to article
[48] M. J. Andreassen, J. Jönsson. A distortional semi-discretized thin-walled beam element. Thin-Walled Structures 2013 (62), 142 - 157.
Search in Google Scholar Back to article
[49] R. F. Vieira, F. B. Virtuoso, E. B. R. Pereira. A higher order model for thin-walled structures with deformable cross-sections. International Journal of Solids and Structures 2014 (51), No. 3 - 4, 575 - 598.
Search in Google Scholar Back to article
[50] B. W. Schafer, S. Ádány. Understanding, and classifying local, distortional, and global buckling in open thin-walled members. Annual Conference Structural Stability Research Council, Montreal, Canada, 2005.
Search in Google Scholar Back to article
[51] D. Camotim, N. Silvestre, R. Gonçalves, P. B. Dinis. GBT-based structural analysis of thin-walled members: overview, recent progress and future developments. M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction 2006 (140), 187 - 204.
Search in Google Scholar Back to article
[52] B. W. Schafer, S. Ádány. Buckling analysis of cold-formed steel members using CUFSM: conventional, and constrained Finite Strip Methods. 18th International Specialty Conference on Cold-Formed Steel Structures, Orlando, Florida, 2006.
Search in Google Scholar Back to article
[53] S. Ádány, B. W. Schafer. Buckling mode decomposition of single-branched open crosssection members via Finite Strip Method: Derivation. Thin-Walled Structures, 2006 (44), No. 5, 563 - 584.
Search in Google Scholar Back to article
[54] S. Ádány, B. W. Schafer. Buckling mode decomposition of single-branched open crosssection members via Finite Strip Method: Application, and examples. Thin-Walled Structures 2006 (44), No. 5, 585 - 600.
Search in Google Scholar Back to article
[55] S. Ádány, B. W. Schafer. A full modal decomposition of thin-walled, single-branched open cross-section members via the constrained Finite Strip Method. Journal of Constructional Steel Research 2008 (64), No. 1, 12 - 29.
Search in Google Scholar Back to article
[56] Z. Li, M.T. Hanna, S. Ádány, B. W. Schafer. Impact of basis, orthogonalization, and normalization on the constrained Finite Strip Method for stability solutions of open thinwalled members. Thin-Walled Structures 2011 (49), No. (9), 1108 - 1122.
Search in Google Scholar Back to article
[57] V. J. Tsipiras, E. J. Sapountzakis. Bars under nonuniform torsion - application to steel bars, Assessment of EC3 guidelines. Engineering Structures 2014 (60), 133 - 147.
Search in Google Scholar Back to article
[58] J. B. Kosmatka, S. B. Dong. Saint-Venant solutions for prismatic anisotropic beams. International Journal of Solids and Structures 1991 (28), No. 7, 917 - 938.
Search in Google Scholar Back to article
[59] C.A. Ie, J. B. Kosmatka. Saint-Venant elasticity solutions of a tip-loaded anisotropic cantilevered beam with an elliptical section. Composites Engineering 1993 (3), No. 12, 1149 - 1164.
Search in Google Scholar Back to article
[60] A. E. H. Love. A treatise on the mathematical theory of elasticity, Dover, New York, 1944.
Search in Google Scholar Back to article
[61] X. S. Xu, W. X. Zhong. H. W. Zhang. The Saint-Venant problem, and principle in elasticity. International Journal of Solids and Structures 1997 (34), No. 22, 2815 - 2827.
Search in Google Scholar Back to article
[62] S. W. Reagan, W. D. Pilkey. Constrained torsion of prismatic bars. Finite Elements in Analysis, and Design 2002 (38), No. 10, 909 - 919.
Search in Google Scholar Back to article
[63] P. F. Pai. High-fidelity sectional analysis of warping functions, stiffness values, and wave properties of beams. Engineering Structures 2014 (67), No. 15, 77 - 95.
Search in Google Scholar Back to article
[64] V. Giavotto, M. Borri, P. Mantegazza, G. Ghiringhelli, V. Carmaschi, G. C. Maffioli, F. Mussi. Anisotropic beam theory, and applications. Computers and Structures 1983 (16), No. 1 - 4, 403 - 413.
Search in Google Scholar Back to article
[65] M. Kazic, S. B. Dong. Analysis of restrained torsion. Journal of Engineering Mechanics, 1990 (116), No. 4, 870 - 891.
Search in Google Scholar Back to article
[66] N. Ghazouani, R. El Fatmi. Extension of the non-uniform warping theory to an orthotropic composite beam. Comptes Rendus Mecanique 2010 (338), No. 12, 704 - 711.
Search in Google Scholar Back to article
[67] N. Ghazouani, R. El Fatmi. Higher order composite beam theory built on Saint-Venant’s Solution. Part-II: Built-in effects influence on the behavior of end-loaded cantilever beams. Composite Structures 2011 (93), No. 2, 567 - 581.
Search in Google Scholar Back to article
[68] R. El Fatmi, N. Ghazouani. Higher order composite beam theory built on Saint-Venant’s Solution. Part-I: Theoretical developments. Composite Structures 2011 (93), No. 2, 557- 566.
Search in Google Scholar Back to article
[69] E. Petrov, M. Géradin. Finite element theory for curved, and twisted beams based on exact solutions for three-dimensional solids, Part 1: Beam concept, and geometrically exact nonlinear formulation. Computational Methods in Applied Mechanics, and Engineering. 1998 (165), No. 1 - 4, 43 - 92.
Search in Google Scholar Back to article
[70] M. K. Ferradi, X. Cespedes. A new beam element with transversal, and warping eigenmodes. Computers and Structures 2014 (131), 12 - 33.
Search in Google Scholar Back to article
[71] M. K. Ferradi, X. Cespedes, M. Arquier. A higher order beam finite element with warping eigenmodes. Engineering Structures 2013 (46), 748 - 762.
Search in Google Scholar Back to article
[72] A. Genoese, A. Genoese, A. Bilotta, G. Garcea. A generalized model for heterogeneous, and anisotropic beams including section distortions. Thin-Walled Structures 2014 (74), 85 - 103.
Search in Google Scholar Back to article
[73] I. C. Dikaros, E. J. Sapountzakis. Distortional analysis of beams of arbitrary cross section by BEM. Journal of Engineering Mechanics, ASCE 2017 (143), No. 10, 04017118, DOI:10.1061/(ASCE)EM.1943-7889.0001340
Search in Google Scholar Back to article
[74] N. Ghazouani, R. El Fatmi. Higher order composite beam theory built on Saint-Venant’s solution, Part-II: Built-in effects influence on the behavior of end-loaded cantilever beams. Composite Structures 2011 (93), No. 2, 567 - 581.
Search in Google Scholar Back to article
[75] D. Henriques, R. Gonçalves, D. Camotim. GBT-based finite element to assess the buckling behaviour of steel-concrete composite beams. Thin-Walled Structures 2016, (107), 207 - 220.
Search in Google Scholar Back to article
[76] G.-W. Jang, M.-J. Kim, Y. Y. Kim. Analysis of thin-walled straight beams with generally shaped closed sections using numerically determined sectional deformation functions. Journal of Structural Engineering, ASCE 2012 (138), No. 12, 1427 - 1435.
Search in Google Scholar Back to article
[77] E. J. Sapountzakis, V. G. Mokos. Warping shear stresses in nonuniform torsion by BEM. Computational Mechanics 2003 (30), No. 2, 131 - 142.
Search in Google Scholar Back to article
[78] R. Gonçalves, D. Camotim. Elastic buckling of uniformly compressed thin-walled regular polygonal tubes. Thin-Walled Structures 2013 (71), 35 - 45.
Search in Google Scholar Back to article
[79] E. Sapountzakis, L. Tsellos, I. Dikaros. Advanced beam element under longitudinal external loading by BEM. Proc. of the 16th International Conference on Boundary Elements, and Meshless Techniques, 2015, 267 - 272, Valencia, Spain, July 06-08.
Search in Google Scholar Back to article
[80] M. K. Ferradi, A. Lebée, A. Fliscounakis, X. Cespedes, K. Sab. A model reduction technique for beam analysis with the asymptotic expansion method. Computers and Structures 2016, (172), 11 - 28.
Search in Google Scholar Back to article
[81] A. K. Argyridi, E. J. Sapountzakis. Advanced analysis of arbitrarily shaped axially loaded beams including axial warping and distortion. Thin-Walled Structures 2019 (134), 127 -147.
Search in Google Scholar Back to article
[82] A. K. Argyridi, E .J. Sapountzakis. Higher Order Beam Theory for Linear Local Buckling Analysis. Engineering Structures 2018 (177), 770 - 784.
Search in Google Scholar Back to article
[83] J. Bocko, P. Lengvarský, J. Šarloši. Buckling analysis of hetero-junction carbon nanotubes. Journal of Mechanical Engineering - Strojnícky časopis 2018, (68), No. 2, 9- 16.
Search in Google Scholar Back to article
[84] I. S. Sohal, W. F. Chen. Local, and post-buckling behavior of tubular beam-columns. ASCE Journal of Structural Engineering 1988 (114), No. 5, 1073 - 1090.
Search in Google Scholar Back to article
[85] J. Davies, P. Leach, D. Heinz. Second-order generalised beam theory. Journal of Constructional Steel Research 1994 (31), No. 2 - 3, 221 - 241.
Search in Google Scholar Back to article
[86] R. Bebiano, C. Basaglia, D. Camotim, R. Gonçalves. GBT buckling analysis of generally loaded thin-walled members with arbitrary flat-walled cross-sections. Thin-Walled Structures 2018 (123), 11 - 24.
Search in Google Scholar Back to article
[87] R. Gonçalves, D. Camotim. Buckling behaviour of thin-walled regular polygonal tubes subjected to bending or torsion. Thin-Walled Structures 2013 (73), 185- 197.
Search in Google Scholar Back to article
[88] C. Basaglia, D. Camotim, R. Goncalves, A. Graca. GBT-based assessment of the buckling behaviour of cold-formed steel purlins restrained by sheeting. Thin-Walled Structures, 2013 (72), 217 - 229.
Search in Google Scholar Back to article
[89] M. Bradford, R. Johnson. Inelastic buckling of composite bridge girders near internal supports. Proceedings of the Institution of Civil Engineers Part 2-Research, and Theory 1987 (83), 143 - 159.
Search in Google Scholar Back to article
[90] S. A. Karamanos, J. L. Tassoulas. Tubular members. I: Stability analysis, and preliminary results. Journal of Engineering Mechanics 1996 (122), No. 1, 64 - 71.
Search in Google Scholar Back to article
[91] S. A. Karamanos, J. L. Tassoulas. Tubular members II: Local buckling, and experimental verification. Journal of Engineering Mechanics 1996 (122), No. 1, 72 - 78.
Search in Google Scholar Back to article
[92] S. A. Karamanos. Bending instabilities of elastic tubes. International Journal of Solids, and Structures 2002 (39), No. 8, 2059 - 2085.
Search in Google Scholar Back to article
[93] S. Houliara, S. A. Karamanos. Buckling, and post-buckling of long pressurised elastic thin-walled tubes under in-plane bending. Int. J. Nonlinear Mech. 2006 (41), No. (4), 491- 511.
Search in Google Scholar Back to article
[94] T. Aoki, Y. Migita, Y. Fukumoto. Local buckling strength of closed polygon folded section columns. J Constr Steel Res, 1991 20(4), 259-70.
Search in Google Scholar Back to article
[95] W. H. Wittrick. A unified approach to the initial buckling of stiffened panels in compression. Aeronautical Quarterly 1968 (19), No. 3, 265 - 283.
Search in Google Scholar Back to article
[96] C.E. Kurt, R.C. Johnson. Cross sectional imperfections, and columns stability. Journal of the Structural Division, 1978 (104), No. 12, 1869 - 1883.
Search in Google Scholar Back to article
[97] N. Koseko, T. Aoki, Y. Fukumoto. The local buckling strength of the octagonal section steel columns. Proceedings of the Japan Society of Civil Engineers 1983 (330), 27 - 36.
Search in Google Scholar Back to article
[98] A.K. Argyridi, E.J. Sapountzakis. Generalized Warping In Flexural-Torsional Buckling Analysis of Composite Beams. Journal of Applied and Computational Mechanics 2016(2), No. 3, 152 - 173.
Search in Google Scholar Back to article
[99] Q. Wang, W.Y. Li. Buckling of thin-walled compression members with shear lag using spline finite member element method. Computational Mechanics 1996 (18), 139 - 146.
Search in Google Scholar Back to article

Influence of in-Plane Deformation in Higher Order Beam Theories

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