A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams

This paper presents a nonlocal sinusoidal shear deformation beam theory for the bending, buckling, and vibration of nanobeams. The present model is capable of capturing both small scale effect and transverse shear deformation effects of nanobeams, and does not require shear correction factors. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion as well as the boundary conditions of the beam are derived using Hamilton’s principle. Analytical solutions for the deflection, buckling load, and natural frequency are presented for a simply supported beam, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory. The comparison firmly establishes that the present beam theory can accurately predict the bending, buckling, and vibration responses of short nanobeams where the small scale and transverse shear deformation effects are significant

Modulated linear dynamics of nanobeams accounting for higher gradient effects

We present some numerical results for the linear dynamics of nanobeams modulated by an axial force, basing on a recent proposal of literature that encompasses both the standard nonlocal elasticity, according to Eringen, and second-order strain elasticity. Three different possibilities for the elastic potential energy provide different responses that highlight the contributions of nonlocality and strain gradient, plus their combination. An axial force affects the linear stationary dynamics of such nanobeams, inducing suitable variation of the natural angular frequencies for benchmark cases, until static buckling occurs when the natural angular frequency vanishes. Effects of the various elastic potentials on this modulation are investigated and thoroughly commented

Free Vibration of Statically Deformed Bi-directionally Graded Timoshenko Beam on Elastic Foundation

In the present work, geometric nonlinearity is taken into consideration to study the free vibration response of nonlinearly deflected Timoshenko beam with bi-directional gradation, resting on an elastic foundation of Winkler type. Exponential variation of material properties is considered along the thickness and length direction. In order to carry out the present numerical study, the entire physical domain of analysis is converted into a normalized domain and suitable number of computational points are created on it to conduct a whole domain analysis. The adopted mathematical formulation is displacement based and semi-analytical. A unique methodology is used for the formulation, in which the static analysis is taken first under large amplitude static force, then a subsequent eigenvalue analysis is performed on the known statically deformed beam profile. Appropriate energy principles are utilized for both analyses to derive the equation of motion in matrix form, which are solved with the help of the algorithm of direct substitution method. Generated results are validated with the results of previously published paper. New sets of result are also presented as benchmark results to represent the impact of bi-directional gradation and elastic foundation on nonlinear vibration response

A modified couple stress elasticity for non-uniform composite laminated beams based on the Ritz formulation

none4siDue to the large application of tapered beams in smart devices, such as scanning tunneling microscopes (STM), nano/micro electromechanical systems (NEMS/MEMS), atomic force microscopes (AFM), as well as in military aircraft applications, this study deals with the vibration behavior of laminated composite non-uniform nanobeams subjected to different boundary conditions. The micro-structural size-dependent free vibration response of composite laminated Euler–Bernoulli beams is here analyzed based on a modified couple stress elasticity, which accounts for the presence of a length scale parameter. The governing equations and boundary conditions of the problem are developed using the Hamilton’s principle, and solved by means of the Rayleigh–Ritz method. The accuracy and stability of the proposed formulation is checked through a convergence and comparative study with respect to the available literature. A large parametric study is conducted to investigate the effect of the length-scale parameter, non-uniformity parameter, size dimension and boundary conditions on the natural frequencies of laminated composite tapered beams, as useful for design and optimization purposes of small-scale devices, due to their structural tailoring capabilities, damage tolerance, and their potential for creating reduction in weight.openJouneghani F.Z.; Babamoradi H.; Dimitri R.; Tornabene F.Jouneghani, F. Z.; Babamoradi, H.; Dimitri, R.; Tornabene, F

A simple shear deformation theory for nonlocal beams

In this paper, a simple beam theory accounting for shear deformation effects with one unknown is proposed for static bending and free vibration analysis of isotropic nanobeams. The size-dependent behaviour is captured by using the nonlocal differential constitutive relations of Eringen. The governing equation of the present beam theory is obtained by using equilibrium equations of elasticity theory. The present theory has strong similarities with nonlocal Euler–Bernoulli beam theory in terms of the governing equation and boundary conditions. Analytical solutions for static bending and free vibration are derived for nonlocal beams with various types of boundary conditions. Verification studies indicate that the present theory is not only more accurate than Euler–Bernoulli beam theory, but also comparable with Timoshenko beam theory

Thermal buckling of functionally graded piezomagnetic micro- and nanobeams presenting the flexomagnetic effect

Galerkin weighted residual method (GWRM) is applied and implemented to address the axial stability and bifurcation point of a functionally graded piezomagnetic structure containing flexomagneticity in a thermal environment. The continuum specimen involves an exponential mass distributed in a heterogeneous media with a constant square cross section. The physical neutral plane is investigated to postulate functionally graded material (FGM) close to reality. Mathematical formulations concern the Timoshenko shear deformation theory. Small scale and atomic interactions are shaped as maintained by the nonlocal strain gradient elasticity approach. Since there is no bifurcation point for FGMs, whenever both boundary conditions are rotational and the neutral surface does not match the mid-plane, the clamp configuration is examined only. The fourth-order ordinary differential stability equations will be converted into the sets of algebraic ones utilizing the GWRM whose accuracy was proved before. After that, by simply solving the achieved polynomial constitutive relation, the parametric study can be started due to various predominant and overriding factors. It was found that the flexomagneticity is further visible if the ferric nanobeam is constructed by FGM technology. In addition to this, shear deformations are also efficacious to make the FM detectable

On wave propagation in two-dimensional functionally graded porous rotating nano-beams using a general nonlocal higher-order beam model

This paper studies the wave propagation of two-dimensional functionally graded (2D-FG) porous rotating nano-beams for the first time. The rotating nano-beams are made of two different materials, and the material properties of the nano-beams alter both in the thickness and length directions. The general nonlocal theory (GNT) in conjunction with Reddy's beam model are employed to formulate the size-dependent model. The GNT efficiently models the dispersions of acoustic waves when two independent nonlocal fields are modelled for the longitudinal and transverse acoustic waves. The governing equations of motion for the 2D-FG porous rotating nano-beams are established using Hamilton's principle as a function of the axial force due to centrifugal stiffening and displacement. The analytic solution is applied to obtain the results and solve the governing equations. The effect of the features of different parameters such as functionally graded power indexes, porosity, angular velocity, and material variation on the wave propagation characteristics of the rotating nano-beams are discussed in detail

Numerical modeling of bending, buckling, and vibration of functionally graded beams by using a higher-order shear deformation theory

The objective of this work is to analyze the behavior beams functionally graded, simply supported, under different conditions such as bending, buckling, and vibration and this by use shear deformation theories a two-dimensional (2D) and quasi-three-dimensional (quasi-3D). The proposed theories take into account a new field of displacement which includes indeterminate whole terms and contains fewer unknowns, compared to other theories of the literature; by taking account of the effects of the transverse shears and the thickness stretching. In this theory, the distribution of the transverse shear stress is hyperbolic and satisfies the boundary conditions on the upper and lower surfaces of the beam without the need for a shear correction factor. In this type of beam the properties of the materials vary according to a distribution of the volume fraction, the Hamilton principle is used to calculate the equations of motion, and in order to check the accuracy of the theory used comparison is made with the studies existing in the literature

Nonlocal analysis of the flexural–torsional stability for FG tapered thin-walled beam-columns

none5siThis paper addresses the flexural–torsional stability of functionally graded (FG) nonlocal thin-walled beam-columns with a tapered I-section. The material composition is assumed to vary continuously in the longitudinal direction based on a power-law distribution. Possible small-scale effects are included within the formulation according to the Eringen nonlocal elasticity assump-tions. The stability equations of the problem and the associated boundary conditions are derived based on the Vlasov thin-walled beam theory and energy method, accounting for the coupled interaction between axial and bending forces. The coupled equilibrium equations are solved numer-ically by means of the differential quadrature method (DQM) to determine the flexural–torsional buckling loads associated to the selected structural system. A parametric study is performed to check for the influence of some meaningful input parameters, such as the power-law index, the nonlocal parameter, the axial load eccentricity, the mode number and the tapering ratio, on the flexural–torsional buckling load of tapered thin-walled FG nanobeam-columns, whose results could be used as valid benchmarks for further computational validations of similar nanosystems.openSoltani M.; Atoufi F.; Mohri F.; Dimitri R.; Tornabene F.Soltani, M.; Atoufi, F.; Mohri, F.; Dimitri, R.; Tornabene, F