Showing papers on "Timoshenko beam theory published in 2019"

Nonlinear free vibrations of bi-directional functionally graded micro/nano-beams including nonlocal stress and microstructural strain gradient size effects

Abstract: Recently, advanced materials whose properties vary within a continuous pattern have been put to use to design and manufacture modern structures. In the current investigation, size dependencies are captured in the nonlinear free vibration characteristics of micro/nano-beams made of bi-directional functionally graded materials (2D-FGM). With the aid of the nonlocal strain gradient elasticity theory and the variational principle, the size-dependent nonlinear differential equations of motion are derived within the framework of the refined hyperbolic shear deformation beam theory. It is supposed that the material properties are distributed exponentially along longitudinal direction, and vary based on the power law function in lateral direction. Moreover, the deviation of the associated physical neutral plane from the mid-plane counterpart is taken into consideration. By employing a numerical solution methodology on the basis of the generalized differential quadrature method (GDQM) together with Galerkin technique and pseudo arc-length continuation method, the nonlocal strain gradient frequency-deflection responses of 2D-FGM micro/nano-beam are obtained corresponding to various values of longitudinal and lateral material property indexes and small scale parameters. It is revealed that the increment made by the strain gradient size dependency in the value of the nonlinear frequency is more than the reduction caused by the nonlocality, especially for the lower maximum deflection imposed to the 2D-FGM micro/nano-beam. Also, it is indicated that for lower values of the material property gradient indexes, the reduction in the nonlinear frequency caused by the lateral functionally graded pattern in the absence of the axial functionally graded pattern is more than that made by the vice versa case. However, for higher values of the material property gradient indexes, an opposite observation is seen.

Nonlocal strain gradient nonlinear resonance of bi-directional functionally graded composite micro/nano-beams under periodic soft excitation

Abstract: With the aid of advanced design techniques, functionally graded materials as promising new materials can be fabricated into various micro/nano-structures to acquire stronger mechanical performance. In this work, the size-dependent nonlinear primary resonance of periodic soft excited micro/nano-beams made of bi-directional functionally graded materials (2D-FGMs) is studied. To accomplish this end, the nonlocal strain gradient theory of elasticity is utilized within the framework of the refined hyperbolic shear deformation beam theory to construct a size-dependent beam model. On the basis of the variational approach using the principle of Hamilton, the non-classical differential equations of motion are achieved. Thereafter, a discretization scheme based numerical solving process via generalized differential quadrature method (GDQM) together with the pseudo-arclength continuation technique, the nonlocal strain gradient frequency response and amplitude response associated with the nonlinear primary resonance of 2D-FGM micro/nano-beams with different boundary conditions are obtained. It is displayed that the nonlocal size dependency makes a reduction in the oscillation amplitudes associated with both of the bifurcation points, but the strain gradient size effect causes to increase them. These patterns are more significant for the second bifurcation point and for the simply supported-simply supported boundary conditions in comparison with the clamped-clamped ones. Also, it is observed that by increasing the value of both of the axial and lateral material property gradient indexes, the peak of the oscillation amplitude and its associated excitation frequency increase.

Free vibration of imperfect sigmoid and power law functionally graded beams

Abstract: In the present work, free vibration of beams made of imperfect functionally graded materials (FGMs) including porosities is investigated. Because of faults during process of manufacture, micro voids or porosities may arise in the FGMs, and this situation causes imperfection in the structure. Therefore, material properties of the beams are assumed to vary continuously through the thickness direction according to the volume fraction of constituents described with the modified rule of mixture including porosity volume fraction which covers two types of porosity distribution over the cross section, i.e., even and uneven distributions. The governing equations of power law FGM (P-FGM) and sigmoid law FGM (S-FGM) beams are derived within the frame works of classical beam theory (CBT) and first order shear deformation beam theory (FSDBT). The resulting equations are solved using separation of variables technique and assuming FG beams are simply supported at both ends. To validate the results numerous comparisons are carried out with available results of open literature. The effects of types of volume fraction function, beam theory and porosity volume fraction, as well as the variations of volume fraction index, span to depth ratio and porosity volume fraction, on the first three non-dimensional frequencies are examined in detail.

Analytical mathematical solution for vibrational response of postbuckled laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams

Abstract: An inhomogeneous beam model incorporating size effects is formulated based on the nonlocal strain gradient theory of elasticity within the framework of a third-order shear deformation beam theory to analyze the size dependency in vibration behavior of postbuckled laminated functionally graded (FG) micro-/nanobeams made from graphene platelet-reinforced composite (GPLRC). The graphene platelets are randomly dispersed in each individual layer in such a way that the weight fraction of the nanofiller varies layerwise on the basis of different patterns of FG dispersion. Based upon the Halpin–Tsai micromechanical scheme, the effective material properties of laminated FG-GPLRC micro-/nanobeams are extracted. With the aid of the Hamilton’s principle, the nonlocal strain gradient equations of motion are constructed. After that, an improved perturbation technique is put to use to capture the small-scale effects on the fundamental frequencies of laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams. The size-dependent natural frequencies of laminated third-order shear deformable FG-GPLRC micro-/nanobeams are obtained as function of applied axial compressive load within both the prebuckling and postbuckling domains. It is seen that in the prebuckling regime, the softening-stiffness effect of the nonlocality causes to reduce the natural frequencies of laminated FG-GPLRC micro-/nanobeam, but the hardening-stiffness influence of strain gradient size effect leads to increase the natural frequencies. However, by moving to the postbuckling regime, this pattern becomes vice versa, as the nonlocal size dependency leads to increase the natural frequencies of micro-/nanobeam while the strain gradient size effect causes to reduce them.

Bending and vibrational behaviors of piezoelectric nonlocal nanobeam including surface elasticity

Abstract: This manuscript illustrates coupled effects of nonlocal elasticity and surface properties on static and vibration characteristics of piezoelectric nanobeams using thin beam theory. The mechanical a...

Free vibration analysis of viscoelastic nanotubes under longitudinal magnetic field based on nonlocal strain gradient Timoshenko beam model

Abstract: In this paper, the free vibration of viscoelastic nanotube under longitudinal magnetic field is investigated The governing equation is formulated by utilizing Timoshenko beam model and Kelvin-Voigt model based on the nonlocal strain gradient theory The local adaptive differential quadrature method (LADQM) is applied in the analyzing procedure We also investigated the influences of the nonlocal parameter, structural damping coefficient, material length scale parameter and the longitudinal magnetic field on the natural frequencies of the system The results of this research may be helpful for understanding the potential applications of nanotubes in Nano-Electromechanical System

A targeted review on large deformations of planar elastic beams: extensibility, distributed loads, buckling and post-buckling:

Abstract: In this paper, we give a targeted review of the state of the art in the study of planar elastic beams in large deformations, also in the presence of geometric nonlinearities. The main scope of this...

Vibration analysis of different material distributions of functionally graded microbeam

Abstract: In the current research paper, a quasi-3D beam theory is developed for free vibration analysis of functionally graded microbeams. The volume fractions of metal and ceramic are assumed to be distributed through a beam thickness by three functions, power function, symmetric power function and sigmoid law distribution. The modified coupled stress theory is used to incorporate size dependency of micobeam. The equation of motion is derived by using Hamilton\'s principle, however, Navier type solution method is used to obtain frequencies. Numerical results show the effects of the function distribution, power index and material scale parameter on fundamental frequencies of microbeams. This model provides designers with guidance to select the proper distributions and functions.

A modified series solution for free vibration analyses of moderately thick functionally graded porous (FGP) deep curved and straight beams

Abstract: As a novel class of weight-efficient engineering materials, the functionally graded porous (FGP) beam structures have great potential value. However, the current research on it is relatively small. Based on this research status, the aim of this paper is establishing a unified analytical model to study the vibration behavior of moderately thick functionally graded porous deep curved and straight beam with general boundary conditions. The first-order beam theory which considering the influence of shear deformation, inertia rotary and deepness term are adopted in the formulation. The theoretical solution model is obtained by means of modified series solution which core soul is using the modified Fourier series including a standard cosine Fourier series with two auxiliary terms to expand the admissible function. This fact gives the opportunity to derive the exact solution for FGP beam with general boundary conditions by utilizing a reasonable spring stiffness value at both ends. A series of numerical examples show that the current model has superior convergence characteristics, computational accuracy and stability. On this basis, a series of innovative results are also highlighted in the text, which may be providing basic data for other algorithm research in the future.

Exact solutions for the bending of Timoshenko beams using Eringen’s two-phase nonlocal model:

Abstract: Eringen’s nonlocal differential model has been widely used in the literature to predict the size effect in nanostructures. However, this model often gives rise to paradoxes, such as the cantilever ...