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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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