This paper continues the senior author’s previous investigation of the slowly rotating Timoshenko beam in a horizontal plane whose movement is controlled by the angular acceleration of the disk of the driving motor into which the beam is rigidly clamped. It was shown before that this system preserves the total energy. We consider the problem of stability of the system after introducing a particular type of damping. We show that the energy of only part of the system vanishes. We illustrate obtained solution with the critical case of the infinite value of the damping coefficient.

Static bending of a twisted Timoshenko beam subjected to combined transverse and axial loadings is studied. The equilibrium equations are established in the twist coordinates by applying the principle of minimum potential energy. The governing equations are then reduced into solvable algebraic equations using a finite element approach. The effects of the twist angle, thickness-to-width ratio, length-to-thickness ratio, loading and boundary conditions on the static bending characteristics of the twisted beams are investigated. The present parametric analyses will provide engineers a good insight into the influence of various structural aspects of the twisted beam on its response to different static loads.

In the present study, an enriched continuum mechanics framework is employed to study thesurface effects on bending behavior of silver nanowires (NWs) resting on elasticsubstrate. The Timoshenko beam theory and the Laplace-Young equation are employed toinvestigate static behavior of silver NWs lying on Winkler-Pasternak elastic substrate.Three types of boundary conditions are considered as doubly simply supported (S-S), doublyclamped (C-C) and cantilevered (C-F). Analytical solutions are obtained for NWs withsurface crystallographic orientation of [001] subjected to a concentrated external force.By defining different normalized contact stiffness, extensive numerical results arecarried out to study the influence of effective parameters such as substrate, surface,aspect ratio (L/D) and diameter onthe stiffness of NWs. According to the obtained results, the effect of surface and itsrate of variation on stiffness of NWs lying on Winkler and Winkler-Pasternak elasticfoundation models are more significant in (C-F) type of boundary condition compared to theNWs without foundation. By increasing the modulus of elastic substrate, the effect ofshear deformation increases which it is more considerable in (C-C) and (S-S) NWs restingon the Winkler-Pasternak and Winkler substrate models, respectively.

The aim of this paper is to analyze a low order finite element methodfor a stiffened plate. The plate is modeled by Reissner-Mindlinequations and the stiffener by Timoshenko beams equations. Theresulting problem is shown to be well posed. In the case of concentricstiffeners it decouples into two problems, one for the in-plane plate deformation and the other for the bending of the plate. The analysisand discretization of the first one is straightforward. The second oneis shown to have a solution bounded above and below independently of thethickness of the plate. A discretization based on DL3 finite elementscombined with ad-hoc elements for the stiffener is proposed.Optimal order error estimates are proved for displacements, rotationsand shear stresses for the plate and the stiffener. Numerical tests arereported in order to assess the performance of the method. Thesenumerical computations demonstrate that the error estimates areindependent of the thickness, providing a numerical evidence that themethod is locking-free.



In this paper, the stability of a Timoshenko beam with time delaysin the boundary input is studied. The system is fixed at the leftend, and at the other end there are feedback controllers, in which time delays exist. We prove that this closed loop system iswell-posed. By the complete spectral analysis, we show that there isa sequence of eigenvectors and generalized eigenvectors of thesystem operator that forms a Riesz basis for the state Hilbert space. Hence the system satisfies the spectrum determined growth condition. Then we conclude the exponential stability of the system under certain conditions. Finally, we give some simulations to support our results.


We intend to conduct a fairly complete study on Timoshenko beams with pointwise feedback controls and seek to obtain information about the eigenvalues, eigenfunctions, Riesz-Basis-Property, spectrum-determined-growth-condition, energy decay rate and various stabilities for the beams. One major difficulty of the present problem is the non-simplicity of the eigenvalues. In fact, we shall indicate in this paper situations where the multiplicity of the eigenvalues is at least two. We build all the above-mentioned results from an effective asymptotic analysis on both the eigenvalues and the eigenfunctions, and conclude with the Riesz-Basis-Property and the spectrum-determined-growth-condition. Finally, these results are used to examine the stability effects on the system by the location of the pointwise control relative to the length of the whole beam.