Showing papers on "Timoshenko beam theory published in 2015"

A new simple shear and normal deformations theory for functionally graded beams

Abstract: In the present work, a simple and refined trigonometric higher-order beam theory is developed for bending and vibration of functionally graded beams The beauty of this theory is that, in addition to modeling the displacement field with only 3 unknowns as in Timoshenko beam theory, the thickness stretching effect (ez ≠ 0) is also included in the present theory Thus, the present refined beam theory has fewer number of unknowns and equations of motion than the other shear and normal deformations theories, and it considers also the transverse shear deformation effects without requiring shear correction factors The neutral surface position for such beams in which the material properties vary in the thickness direction is determined Based on the present refined trigonometric higher-order beam theory and the neutral surface concept, the equations of motion are derived from Hamilton's principle Numerical results of the present theory are compared with other theories to show the effect of the inclusion of transverse normal strain on the deflections and stresses

Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory: Chebyshev collocation method

Abstract: Flexural vibration analysis of beams made of functionally graded materials (FGMs) with various boundary conditions is considered in this paper. Due to technical problems during FGM fabrication, porosities and micro-voids can be created inside FGM samples which may lead to the reduction in density and strength of materials. In this investigation, the FGM beams are assumed to have even and uneven distributions of porosities over the beam cross-section. The modified rule of mixture is used to approximate material properties of the FGM beams including the porosity volume fraction. In order to cover the effects of shear deformation, axial and rotary inertia, the Timoshenko beam theory is used to form the coupled equations of motion for describing dynamic behavior of the beams. To solve such a problem, Chebyshev collocation method is employed to find natural frequencies of the beams supported by different end conditions. Based on numerical results, it is revealed that FGM beams with even distribution of porosities have more significant impact on natural frequencies than FGM beams with uneven porosity distribution.

A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory

Abstract: This paper presents a nonlocal shear deformation beam theory for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The developed theory account for higher-order variation of transverse shear strain through the depth of the nanobeam, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the nanobeam. A shear correction factor, therefore, is not required. In addition, this nonlocal nanobeam model incorporates the length scale parameter which can capture the small scale effect and it has strong similarities with Euler–Bernoulli beam model in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived from Hamilton\'s principle. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.

An exact solution for the nonlinear forced vibration of functionally graded nanobeams in thermal environment based on surface elasticity theory

Abstract: In the present investigation, an exact solution is proposed for the nonlinear forced vibration analysis of nanobeams made of functionally graded materials (FGMs) subjected to thermal environment including the effect of surface stress. The material properties of functionally graded (FG) nanobeams vary through the thickness direction on the basis of a simple power law. The geometrically nonlinear beam model, taking into account the surface stress effect, is developed by implementing the Gurtin–Murdoch elasticity theory together with the classical Euler–Bernoulli beam theory and using a variational approach. Hamilton’s principle is utilized to obtain the nonlinear governing partial differential equation and corresponding boundary conditions. After that, the Galerkin technique is employed in order to convert the nonlinear partial differential equation into a set of nonlinear ordinary differential equations. This new set is then solved analytically based on the method of multiple scales which results in the frequency–response curves of FG nanobeams in the presence of surface stress effect. It is revealed that by increasing the beam thickness, the surface stress effect diminishes and the maximum amplitude of the stable response is shifted to the higher excitation frequencies.

Size-dependent thermo-electrical buckling analysis of functionally graded piezoelectric nanobeams

Abstract: In the present study, thermo-electrical buckling characteristics of functionally graded piezoelectric (FGP) Timoshenko nanobeams subjected to in-plane thermal loads and applied electric voltage are carried out by presenting a Navier type solution for the first time. Three kinds of thermal loading, namely, uniform, linear and nonlinear temperature rises through the thickness direction are considered. Thermo-electro-mechanical properties of FGP nanobeam are supposed to vary smoothly and continuously throughout the thickness based on power-law model. Eringen's nonlocal elasticity theory is exploited to describe the size dependency of nanobeam. Using Hamilton's principle, the nonlocal governing equations together with corresponding boundary conditions based on Timoshenko beam theory are obtained for the thermal buckling analysis of graded piezoelectric nanobeams including size effect and they are solved applying analytical solution. According to the numerical results, it is revealed that the proposed modeling can provide accurate critical buckling temperature results of the FG nanobeams as compared some cases in the literature. In following a parametric study is accompanied to examine the effects of the several parameters such as various temperature distributions, external electric voltage, power-law index, nonlocal parameter and aspect ratio on the critical buckling temperature difference of the size-dependent FGP nanobeams in detail. It is found that the small scale effect and electrical loading have a significant effect on buckling temperatures of FGP nanobeams.

Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method

Abstract: This study presents free vibration analysis of rotating functionally graded Timoshenko beam made of porous material using the semi-analytical differential transform method.The material properties are supposed to vary along the thickness direction of the beam according to the rule of mixture, which is modified to approximate the material properties with the porosity phases. The frequency equation is obtained using Hamilton’s principle. It is demonstrated that the DTM has high precision and computational efficiency in the vibration analysis of porous FG rotating beams. The good agreement between the results of this article and those available in literature validated the presented approach. Detailed mathematical derivations are presented and numerical investigations are performed, while emphasis is placed on investigating the effect of the several parameters such as porosity, functionally graded microstructure, thickness ratio, rotational speed and hub radius on the normalized natural frequencies of porous FG rotating beams in detail.

Modeling and Characterization of a Piezoelectric Energy Harvester Under Combined Aerodynamic and Base Excitations

Abstract: This paper develops and validates an aero-electromechanical model which captures the nonlinear response behavior of a piezoelectric cantilever-type energy harvester under combined galloping and base excitations. The harvester consists of a thin piezoelectric cantilever beam clamped at one end and rigidly attached to a bluff body at the other end. In addition to the vibratory base excitations, the beam is also subjected to aerodynamic forces resulting from the separation of the incoming airflow on both sides of the bluff body which gives rise to limit-cycle oscillations when the airflow velocity exceeds a critical value. A nonlinear electromechanical distributed-parameter model of the harvester under the combined excitations is derived using the energy approach and by adopting the nonlinear Euler–Bernoulli beam theory, linear constitutive relations for the piezoelectric transduction, and the quasi-steady assumption for the aerodynamic loading. The resulting partial differential equations of motion are discretized and a reduced-order model is obtained. The mathematical model is validated by conducting a series of experiments at different wind speeds and base excitation amplitudes for excitation frequencies around the primary resonance of the harvester. Results from the model and experiment are presented to characterize the response behavior under the combined loading.

Free vibration and buckling analysis of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets

Abstract: This paper investigates the free vibration and elastic buckling of sandwich beams with a stiff core and functionally graded carbon nanotube reinforced composite (FG-CNTRC) face sheets within the framework of Timoshenko beam theory. The material properties of FG-CNTRCs are assumed to vary in the thickness direction, and are estimated through a micromechanical model. The governing equations and boundary conditions are derived by using Hamilton's principle and discretized by employing the differential quadrature (DQ) method to obtain the natural frequency and critical buckling load of the sandwich beam. A detailed parametric study is conducted to study the effects of carbon nanotube volume fraction, core-to-face sheet thickness ratio, slenderness ratio, and end supports on the free vibration characteristics and buckling behavior of sandwich beams with FG-CNTRC face sheets. The vibration behavior of the sandwich beam under an initial axial force is also discussed. Numerical results for sandwich beams with uniformly distributed carbon nanotube-reinforced composite (UD-CNTRC) face sheets are also provided for comparison.

A new Timoshenko beam model incorporating microstructure and surface energy effects

Abstract: A new Timoshenko beam model is developed using a modified couple stress theory and a surface elasticity theory. A variational formulation based on Hamilton’s principle is employed, which leads to the simultaneous determination of the equations of motion and complete boundary conditions for a Timoshenko beam. The new model contains a material length scale parameter accounting for the microstructure effect in the bulk of the beam and three surface elasticity constants describing the mechanical behavior of the beam surface layer. The inclusion of these additional material constants enables the new model to capture the microstructure-and surface energy-dependent size effect. In addition, both bending and axial deformations are considered, and the Poisson effect is incorporated in the current model, unlike existing Timoshenko beam models. The new beam model includes the models considering only the microstructure dependence or the surface energy effect as limiting cases and recovers the Bernoulli–Euler beam model incorporating the two effects as a special case. Also, the current model reduces to the classical Timoshenko beam model when the microstructure dependence, surface energy and Poisson’s effect are all suppressed. To demonstrate the new model, the static bending and free vibration problems of a simply supported beam are analytically solved by directly applying the general formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko beam model. In addition, the differences in both the deflection and rotation predicted by the two models are very large when the beam thickness is small, but they are diminishing with the increase of the beam thickness. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the new model is higher than that given by the classical model, with the difference between them being significantly large for very thin beams. These predicted trends of the size effect in beam bending at the micron scale agree with those observed experimentally.

Kinetostatic Modeling of Fully Compliant Bistable Mechanisms Using Timoshenko Beam Constraint Model

Abstract: Fully compliant bistable mechanisms (FCBMs) have numerous applications in both micro- and macroscale devices, but the nonlinearities associated with the deflections of the flexible members and the kinetostatic behaviors have made it difficult to design. Currently, the design of FCBMs relies heavily on nonlinear finite element modeling. In this paper, an analytical kinetostatic model is developed for FCBMs based on the beam constraint model (BCM) that captures the geometric nonlinearities of beam flexures that undergo relatively small deflections. An improved BCM (i.e., Timoshenko BCM (TBCM)) is derived based on the Timoshenko beam theory in order to include shear effects in the model. The results for three FCBM designs show that the kinetostatic model can successfully identify the bistable behaviors and make reasonable predictions for the locations of the unstable equilibrium points and the stable equilibrium positions. The inclusion of shear effects in the TBCM model significantly improves the prediction accuracy over the BCM model, as compared to the finite element analysis (FEA) results. [DOI: 10.1115/1.4029024]

Size-dependent electromechanical buckling of functionally graded electrostatic nano-bridges

Abstract: This study explored the electromechanical buckling (EMB) of beam-type nanoelectromechanical systems (NEMS) by considering the nonlinear geometric effect and intermolecular forces (Casimir force and van der Walls force) based on modified couple stress theory. To model the system, a slender nanobeam made of functionally graded material (FGM) with clamped-guided boundary conditions, which is under compressive or tensile axial loads as well as symmetric and nonlinear electrostatic and intermolecular transverse loads, is used. Considering the Euler–Bernoulli beam theory and using the principle of minimum potential energy and the variational approach, the governing equation as well as the related boundary conditions is derived. To discretize the equation and its related boundary conditions, and to solve the equations, the generalized differential quadrature method (GDQM) is employed. Finally, after validation of the results, the effects of size, length, power law index, and the distance between the two fixed and movable electrodes on the bucking of the system are discussed and examined.