Sequential 1D/2D Finite Element analyses of tramway rails under bending and restrained torsion, based on the principle of virtual power

Abstract

The increased incidence of fractures in aging tramway rails has motivated the development of a new, efficient numerical tool for studying structural mechanics problems related to such rails. We introduce virtual velocity fields reflecting axial, bending, shear, and torsional deformations, which are then translated, via the principle of virtual power, into corresponding equilibrium conditions. The resulting equations are numerically solved by means of sequential 1D and 2D Finite Element analyses, providing displacement quantities and stress resultants along the beam axis, as well as cross-sectional stress distributions. Finally, the key advantages of this modeling approach, including excellent efficiency and reliability, as well as straightforward applicability to a wide range of cross-sectional shapes and structural settings, are demonstrated via numerical examples. ARTICLE HISTORY

A principle of virtual power-based beam model reveals discontinuities in elastic support as potential sources of stress peaks in tramway rails

“…In line with Sapountzakis [38], who presented a displacement field representative for torsion-compliant beams, we consider the virtual velocity field introduced in [28…”

“…represent the virtual velocities of the geometrical center of the cross section (at the longitudinal coordinate x), whileω x (x) stands for the angular virtual velocity with respect to the twist axis, which is the axis parallel to the beam axis passing through the center of twist (abbreviated as CT), being offset from the geometrical center by y CT and z CT ; determination of y CT and z CT has been demonstrated in [28]. Furthermore, ψ I (y, z) denotes the primary warping function describing the contribution of non-restrained, Saint-Venant-type torsion, whereas ψ II (y, z) denotes the secondary warping function, describing the effect of…”

“…2 is solved numerically, in the form of a procedure comprising four computation steps. This strategy has been elaborated for the first time and in minute detail in [28]. In the following, a concise summary of the key aspects related to this numerical solution strategy is presented, with emphasis on the novelties of this paper.…”

“…In this work, we took a new path towards consideration of elastically supported beams in general, and of tramway rails in particular, tying in with the approach recently published by the present paper's authors [28], where virtual kinematics were introduced reflecting stretching, bending, as well as primary and secondary warping deformations of a beam, and through rigorous application of the principle of virtual power [29] respective equilibrium equations and natural boundary conditions could be derived, including higher-order torsion-related force quantities. Eventually, this mathematical framework was utilized for computing displacement quantities along the longitudinal axis of the beam, and corresponding stress distributions in selected cross sections, both by means of the Finite Element method, see also [30] for further details.…”

“…(48)-(50) into Eq. (2) allows for formulating P int in terms of so-called cross-sectional rigidities, see [28] for all related details. Proceeding as explained above, and considering d 4ω…”