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HIGHER ORDER BEAM THEORY IN LINEAR ANALYSIS OF BEAMS – AXIAL MODES OF ARBITRARY CROSS SECTIONS

Authors:
ICOVP 2019
HIGHER ORDER BEAM THEORY IN LINEAR ANALYSIS OF BEAMS –
AXIAL MODES OF ARBITRARY CROSS SECTIONS
Amalia Argyridi, Zinon Chatzopoulos*, Evangelos Sapountzakis
Institute of Structural Analysis and Antiseismic Research, School of Civil Engineering
National Technical University of Athens
Athens, 15780, Greece
a.argyridi@gmail.com, zinon.hatz@hotmail.com, cvsapoun@central.ntua.gr
ABSTRACT
In most cases in the analysis of beam-like structures, Euler – Bernoulli beam theory assumptions are
adopted, while in the case of non-negligible shear deformation effect, these assumptions are relaxed by using
Timoshenko beam theory. However, both theories maintain the assumptions that plane cross sections remain
plane (no out-of-plane deformation) and that their shape does not change after deformation (no in-plane
deformation). In order to take into account shear lag effects in the context of a beam theory, the inclusion of
non-uniform warping is necessary, relaxing the assumption of plane cross section. The shear flow associated
with non-uniform warping leads also to in plane deformation of the cross section, relaxing the assumption
that the cross section shape does not change after deformation. For this purpose the so-called higher order
beam theories have been developed taking into account shear lag [1] and distortional (in-plane deformation)
effects [2]. In this paper, a higher order beam theory is employed for linear static analysis of beams of
arbitrarily shaped, homogeneous cross-section, taking into account warping and distortional phenomena due
to axial [3], shear, flexural, and torsional behavior [4]. The beam is subjected to arbitrary axial, transverse
and/or torsional concentrated or distributed load, while its edges are restrained by the most general linear
boundary conditions. The analysis consists of two stages. In the first stage, where the Boundary Element
Method is employed, a cross sectional analysis is performed based on the so-called sequential equilibrium
scheme establishing the possible in-plane (distortion) and out-of-plane (warping) deformation patterns (axial,
flexural and torsional modes) of the cross section. In the second stage, where the Finite Element Method is
employed, the extracted deformation patterns are included in the linear static analysis multiplied by
respective independent parameters expressing their contribution to the beam deformation. The four rigid
body displacements of the cross section together with the aforementioned independent parameters consist of
the degrees of freedom of the beam. The finite element equations are formulated with respect to the
displacement and the independent warping and distortional parameters. In the present paper, the higher order
beam theory developed in [3] is employed in order to examine axial warping and distortional modes. In
addition, numerical examples with practical interest are presented in order to highlight the importance of
axial modes in linear static analysis of beams.
References
[1] Reissner, E. (1946), Analysis of shear lag in box beams by the principle of minimum potential energy.
Quarterly of Applied Mathematics, 4 (3), 268 – 278.
[2] Schardt, R. (1994b), Lateral Torsional and Distortional Buckling of Channel- and Hat-Sections. Journal of
Constructional Steel Research, 31 (2-3), 243-265.
[3] Argyridi, A.K and Sapountzakis, E.J. (2019), Advanced Analysis of Arbitrarily Shaped Axially Loaded
Beams Including Axial Warping and Distortion. Thin-Walled Structures, 134, 127-147.
[4] Dikaros, I.C. and Sapountzakis, E.J. (2017), Distortional Analysis of Beams of Arbitrary Cross Section by
BEM. Journal of Engineering Mechanics, ASCE, 143 (10): 04017118,
DOI:10.1061/(ASCE)EM.1943-7889.0001340.
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Article
In this paper, a higher order beam theory is developed for the analysis of beams of homogeneous cross-section, taking into account warping and distortional phenomena due to axial, shear, flexural and torsional behavior. The beam can be subjected to arbitrary axial, transverse and/or torsional concentrated or distributed load, while its edges are restrained by the most general linear boundary conditions. The analysis consists of two stages. In the first stage, where the Boundary Element Method is employed, a cross sectional analysis is performed based on the so-called sequential equilibrium scheme establishing the possible in-plane (distortion) and out-of-plane (warping) deformation patterns of the cross-section. In the second stage, where the Finite Element Method is employed, the extracted deformation patterns are included in the beam analysis multiplied by respective independent parameters expressing their contribution to the beam deformation. The four rigid body displacements of the cross-section together with the aforementioned independent parameters consist the degrees of freedom of the beam. The finite element equations are formulated with respect to the displacements and the independent warping and distortional parameters. Numerical examples of axially loaded beams are solved to emphasize the importance of axial mode. In addition, numerical examples of various loading combinations are presented to demonstrate the range of application of the proposed method.
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This paper presents a general formulation for the distortional analysis of beams of arbitrary cross section under arbitrary external loading and general boundary conditions. The nonuniform distortional/warping distributions along the beam length are taken into account by employing independent parameters multiplying suitable deformation modes accounting for in-plane and out-of-plane cross-sectional deformation (distortional/warping functions). The paper proposes a novel procedure for cross-sectional analysis which results in the solution of separate boundary value problems for each resisting mechanism (flexure, torsion) on the cross-sectional domain instead of relying on eigenvalue analysis procedures encountered in the literature. These distortional and warping functions are computed employing a boundary element method (BEM) procedure. Subsequently, sixteen boundary value problems are formulated with respect to displacement and rotation components as well as to independent distortional/warping parameters along the beam length and solved using the analog equation method (AEM), a BEM-based technique. After the establishment of kinematical components, stress components on any arbitrary point of each cross section of the beam can be evaluated, yielding a prediction in good agreement with three-dimensional finite-element method (FEM) solutions, in contrast to conventional beam models.
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For thin-walled sections, lateral and lateral-torsional buckling are often affected by distortion of the section and this can severely reduce the critical stress. A calculation method for the critical stress which takes account of distortional effects, based on Generalised Beam Theory (GBT), is presented in this paper. The limiting slenderness at which distortional effects begin to take effect is evaluated for channel- and hat-sections on the basis of parametric studies. Approximate formulae for the limiting slenderness are then given.
Analysis of shear lag in box beams by the principle of minimum potential energy
  • E Reissner
Reissner, E. (1946), Analysis of shear lag in box beams by the principle of minimum potential energy. Quarterly of Applied Mathematics, 4 (3), 268 -278.