Timoshenko beam theory

About: Timoshenko beam theory is a research topic. Over the lifetime, 9426 publications have been published within this topic receiving 200570 citations.

Impact analysis of fiber reinforced polymer honeycomb composite sandwich beams

Abstract: Large scale fiber reinforced polymer (FRP) composite structures have been used in highway bridge and building construction. Recent applications have demonstrated that FRP honeycomb sandwich panels can be effectively and economically applied for both new construction and rehabilitation and replacement of existing structures. This paper is concerned with impact analysis of an as-manufactured FRP honeycomb sandwich system with sinusoidal core geometry in the plane and extending vertically between face laminates. The analyses of the honeycomb structure and components including: (1) constituent materials and ply properties, (2) face laminates and core wall engineering properties, and (3) equivalent core material properties, are first introduced, and these properties for the face laminates and equivalent core are later used in dynamic analysis of sandwich beams. A higher-order impact sandwich beam theory by the authors [Yang MJ, Qiao P. Higher-order impact modeling of sandwich beams with flexible core. Int J Solids Struct 2005;42(20):5460–90] is adopted to carry out the free vibration and impact analyses of the FRP honeycomb sandwich system, from which the full elastic field (e.g., deformation and stress) under impact is predicted. The higher order vibration analysis of the FRP sandwich beams is conducted, and its accuracy is validated with the finite element Eigenvalue analysis using ABAQUS; while the predicted impact responses (e.g., contact force and central deflection) are compared with the finite element simulations by LS-DYNA. A parametric study with respect to projectile mass and velocity is performed, and the similar prediction trends with the linear solution are observed. Furthermore, the predicted stress fields are compared with the available strength data to predict the impact damage in the FRP sandwich system. The present impact analysis demonstrates the accuracy and capability of the higher order impact sandwich beam theory, and it can be used effectively in analysis, design applications and optimization of efficient FRP honeycomb composite sandwich structures for impact protection and mitigation.

A new locking-free formulation for planar, shear deformable, linear and quadratic beam finite elements based on the absolute nodal coordinate formulation

Abstract: Many widely used beam finite element formulations are based either on Reiss- ner's classical nonlinear rod theory or the absolute nodal coordinate formulation (ANCF). Advantages of the second method have been pointed out by several authors; among the ben- efits are the constant mass matrix of ANCF elements, the isoparametric approach and the existence of a consistent displacement field along the whole cross section. Consistency of the displacement field allows simpler, alternative formulations for contact problems or in- elastic materials. Despite conceptional differences of the two formulations, the two models are unified in the present paper. In many applications, a nonlinear large deformation beam element with bending, ax- ial and shear deformation properties is needed. In the present paper, linear and quadratic ANCF shear deformable beam finite elements are presented. A new locking-free continuum mechanics based formulation is compared to the classical Simo and Vu-Quoc formulation based on Reissner's virtual work of internal forces. Additionally, the introduced linear and quadratic ANCF elements are compared to a fully parameterized ANCF element from the literature. The performance of the respective elements is evaluated through analysis of con- ventional static and dynamic example problems. The investigation shows that the obtained linear and quadratic ANCF elements are advantageous compared to the original fully para- meterized ANCF element.

Sound wave propagation in single-walled carbon nanotubes with initial axial stress

Abstract: This paper studies the vibrational characteristics of single-walled carbon nanotubes (SWNTs) with initial axial loading based on the theory of nonlocal elasticity. The consistent equations of motion for the nonlocal Euler-Bernoulli and Timoshenko beam models are provided taking into account the initial axial stress. The small scale effect on CNT wave propagation dispersion relation is explicitly revealed for different CNT wave numbers and diameters by theoretical analyses and numerical simulations. In addition, the applicability of the two beam models is explored by numerical simulations. The research work reveals the significance of the effects of small scale, transverse shear deformation and rotary inertia on wave propagation in short SWCNTs with initial axial loading.

An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams

Abstract: The theoretical results relevant to the vibration modes of Timoshenko beams are here used as benchmarks for assessing the correctness of the numerical values provided by several finite element models, based on either the traditional Lagrangian interpolation or on the recently developed isogeometric approach. Comparison of results is performed on both spectrum error (in terms of the detected natural frequencies) and on the l2 relative error (in terms of the computed eigenmodes): this double check allows detecting for each finite element model, and for a discretization based on the same number of degrees-of-freedom, N, the frequency threshold above which some prescribed accuracy level is lost, and results become more and more unreliable. Hence a quantitative way of measuring the finite element performance in modeling a Timoshenko beam is proposed. The use of Fast Fourier Transform is finally employed, for a selected set of vibration modes, to explain the reasons of the accuracy decay, mostly linked to a poor separation of the natural frequencies in the spectrum, which is responsible of some aliasing of modes.

Dynamic properties of AFM cantilevers and the calibration of their spring constants

Abstract: A hybrid method is introduced for the calibration of the spring constants of atomic force microscopy cantilevers. It is based on the minimization of the difference between the modelled and experimentally determined full-field displacement maps of the surface of the cantilever in motion at several resonant frequencies. The dynamic mechanical response of the cantilever to periodic motion is measured in a vacuum by means of a scanning vibrometer. Given the dimensions of the cantilever, the obtained surface displacements together with analytical or numerical models are used to resolve the physical unknowns of the probe. These are the elastic properties of the cantilever, and the residual stress state built up during the deposition of the reflective coating on the backside of the cantilever. The scanning vibrometry experiment allows the precise determination of the first ten resonant frequencies and the modes associated. After optimization of the elastic properties and the surface stress, the relative agreement between all resonant frequencies is better than 1% with the finite element model and 2% with the Timoshenko beam equation. The agreement between surface displacements is also excellent when the damping constant of the system has been determined, except for the first lateral mode, which exhibits strong coupling to a reflection of the first torsional mode. Because all the displacements at resonance are known, it is possible to decouple these modes, and the result is shown to compare well with the model. The cantilever being fully characterized (geometry, materials, residual stress state and boundary conditions), it is straightforward to deduce all its spring constants, in the linear and nonlinear elastic regimes.