Bars under Torsional Loading: A Generalized Beam Theory Approach

Abstract

In this paper both the static and dynamic analyses of the geometrically linear or nonlinear, elastic or elastoplastic nonuniform torsion problems of bars of constant or variable arbitrary cross section are presented together with the corresponding research efforts and the conclusions drawn from examined cases with great practical interest. In the presented analyses, the bar is subjected to arbitrarily distributed or concentrated twisting and warping moments along its length, while its edges are supported by the most general torsional boundary conditions. For the dynamic problems, a distributed mass model system is employed taking into account the warping inertia. The analysis of the aforementioned problems is complete by presenting the evaluation of the torsion and warping constants of the bar, its displacement field, its stress resultants together with the torsional shear stresses and the warping normal and shear stresses at any internal point of the bar. Moreover, the construction of the stiffness matrix and the corresponding nodal load vector of a bar of arbitrary cross section taking into account warping effects are presented for the development of a beam element for static and dynamic analyses. Having in mind the disadvantages of the 3D FEM solutions, the importance of the presented beamlike analyses becomes more evident.

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

“…Following the standard assumptions in beam theory [1], we consider the stress tensor throughout the beam structure to be of the following particular form:…”

“…Next, we choose a virtual velocity fieldvðxÞ ¼v x ðxÞe x þ v y ðxÞe y þv z ðxÞe z , withv x ðxÞ;v y ðxÞ, andv z ðxÞ denoting the components ofvðxÞ with respect to an orthonormal base with vectors e x ; e y , and e z (see Figure 2); and with x ¼ xe x þ ye y þ ze z denoting the location vector, whereby x is the coordinate in direction of the beam axis, and coordinates y and z are measured from the geometrical center (abbreviated as GC) of the cross section. Focusing on slender, torsioncompliant beams, we extend the classical Euler-Bernoulli kinematics by modes of torsional deformation in line with the review paper of Sapountzakis [1], mathematically expressed by the following virtual velocity components:…”

“…This situation motivates the development of realistic "nonclassical" beam theories; and according to a recent review paper on the topic [1], the pertinent state of the art involves combining classical equilibrium conditions for torsional beams-where the axial gradient of the torsional moment is equal to the distributed torsional loading, see Eq. (33) in Ref.…”

“…(33) in Ref. [1]-with extended deformation characteristics related to the cross section of the beam, including primary as well as secondary torsional warping modes. In this context, it remains unclear whether (and if yes, how) stress resultants (such as the classical torsional moment), and the equilibrium conditions they are assigned to, relate to the underlying 3D continuum on which the stress and strain tensors are defined.…”

Angle of Twist of a Double-Bent Leaf Flexure under Torsion

“…Torsional theory with warping has been presented in many studies. Examples include a generalized beam theory of bars under torsional loading presented in [21], and a theory of torsional analysis for an open thin-walled beam developed in [22] and [23]. However, the case of partially restrained warping was not considered in these studies.…”