Letter to the editor

On the governing equations for vibration and stability of a Timoshenko beam: Hamilton's principle

This paper presents a novel analytical formulation to study the impact of shear deformations on beam-columns and piles by means of the Timoshenko-Engesser’s and Timoshenko-Haringx’s theories. The proposed solution enables the analysis of lateral deformation or buckling, while considering the effect of shear deformations. It offers the flexibility to i) incorporate different boundary conditions at the ends of the element (e.g., semi-rigid connections and lateral transverse springs) and ii) consider an inhomogeneous elastic foundation. When certain variables are disregarded, the proposed GDE can capture particular cases of GDEs found in the literature for beam-columns and piles. Examples are provided to demonstrate the simplicity and practicality of the proposed method, and its accuracy is validated against available analytical or numerical methods. The influence of the shear effects, as computed from Engesser’s and Haringx’s methods, on the lateral and buckling responses of beam-column and pile elements is discussed.

This paper presents an analysis of the stability of Timoshenko beams which uses Eringen's nonlocal elasticity theory. A numerical algorithm based on the exact solution for the free vibration of segmental Timoshenko beams was formulated. The algorithm enables one to calculate, with any degree of accuracy, the critical load levels in the beams on the macro and nanoscale. The beams were subjected to conservative and nonconservative static loads. The levels of critical loads in the beams were analysed assuming a functional dependence of the nonlocal parameters on the vibrational frequency and the state of stress.

This paper deals with natural frequencies of short (or thick) beams taking into account effects of rotary inetria and shear deformation. Three versions of the Bresse–Timoshenko beam theory are considered and contrasted in the context of the problem of beam vibration in presence of the single crack, namely the (a) original Bresse–Timoshenko theory, (b) Elishakoff's truncated version and (c) the proposed slope inertia based version. The original Bresse–Timoshenko equations contain fourth order derivative with respect to time that is not present in the classical Bernoulli–Euler theory whose correction was the declared goal of S.P. Timoshenko who himself did not overestimate the importance of this term. The truncated version of the Bresse–Timoshenko was suggested by Elishakoff with no fourth order time derivative incorporated in the theory. In this paper, variational derivation of the truncated version of the Bresse–Timoshenko beam theory is conducted with attendant additional term containing the fourth derivative with respect to axial coordinate and second derivative with respect to time. It appears imperative to conduct thorough comparison of the above three theories in the context of beams containing a crack which is often present in engineering structures. The applicability of these three versions with attendant extensive numerical calculations is specifically studied for the titled problem. It is demonstrated that the original Bresse–Timoshenko beam theory, as an overcomplicated theory, is not needed for crack analysis. Simpler theories, namely the truncated version of this theory or the one based on slope inertia are sufficient for crack analysis in all treated cases.

The purpose of this article is to investigate the changes in the magnitude of natural frequencies and their associated modal shapes of Timoshenko beam with respect to different system design parameters. This beam includes an intermediate extended eccentric rigid mass mounted on two elastic segments. The equilibrium equations which govern the transverse and rotational motions are derived. The application of the developed system frequency equation is demonstrated by several illustrative examples. Several end and intermediate conditions are considered. The influence of, rotary inertia, shear deformation, axial load, eccentric mass and elastic segments step ratio on the system natural frequencies and mode shapes are conducted. Several sets of new results are presented. Comparison of the present model results with the experimental data for shaft integrated with intermediate rigid mass demonstrates the accuracy of the analysis in practical applications. The present model is valid for several industrial applications, such as mechanical, structural, naval and for wider range of applications.

In this comparative study to estimate the values of critical buckling temperatures of carbon nanotubes, modelled as Timoshenko beam, two new nonlocal elasticity models based on Trefftz and Kounadis approaches are addressed. The previous results obtained from literature are compared with those obtained from two new models under simple boundary conditions. The comparative results reveal that the differences between critical buckling temperatures obtained from three nonlocal elasticity models is more pronounced,especially at the high modes and for the short nanotubes.

This paper deals with the analysis of the natural frequencies, mode shapes of an axially loaded multi-step Timoshenko beam combined system carrying several attachments. The influence of system design and the proposed sub-system non-dimensional parameters on the combined system characteristics are the major part of this investigation. The effect of material properties, rotary inertia and shear deformation of the beam system for each span are included. The end masses are elastically supported against rotation and translation at an offset point from the point of attachment. A sub-system having two degrees of freedom is located at the beam ends and at any of the intermediate stations and acts as a support and/or a suspension. The boundary conditions of the ordinary differential equation governing the lateral deflections and slope due to bending of the beam system including the shear force term, due to the sub-system, have been formulated. Exact global coefficient matrices for the combined modal frequencies, the modal shape and for the discrete sub-system have been derived. Based on these formulae, detailed parametric studies of the combined system are carried out. The applied mathematical model is valid for wide range of applications especially in mechanical, naval and structural engineering fields.