Reddy Beam Theory Research Articles

Vibration analysis of embedded size dependent FG nanobeams based on third-order shear deformation beam theory

In this paper, free vibration characteristics of functionally graded (FG) nanobeams embedded on elastic medium are investigated based on third order shear deformation (Reddy) beam theory by presenting a Navier type solution for the first time. The material properties of FG nanobeam are assumed to vary gradually along the thickness and are estimated through the power-law and Mori-Tanaka models. A two parameters elastic foundation including the linear Winkler springs along with the Pasternak shear layer is in contact with beam. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived based on third order shear deformation beam theory through Hamilton's principle and they are solved applying analytical solution. According to the numerical results, it is revealed that the proposed modeling can provide accurate frequency results of the FG nanobeams as compared to some cases in the literature. The obtained results are presented for the vibration analysis of the FG nanobeams such as the influences of foundation parameters, gradient index, nonlocal parameter and slenderness ratio in detail.

Buckling analysis of CNT-reinforced beams with arbitrary boundary conditions

Axial buckling characteristics of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) are investigated. Various types of beam theories namely as Euler–Bernoulli beam theory, Timoshenko beam theory and Reddy beam theory are used to analyze the buckling behavior of carbon nanotube-reinforced composite beams. Generalized differential quadrature (GDQ) method is utilized to discretize the governing differential equations along with four commonly used boundary conditions. The material properties of the nanocomposite beams are obtained using molecular dynamic (MD) simulation corresponding to both of short-(10,10) SWCNT and long-(10,10) SWCNT composites which are embedded by amorphous polyethylene matrix. Then the results obtained directly from MD simulations are matched with those calculated by the rule of mixture to extract appropriate values of carbon nanotube efficiency parameters accounting for the scale-dependent material properties. Selected numerical results are presented to indicate the influences of nanotube volume fraction and end supports on the critical axial buckling loads of nanoconposite beams relevant to long- and short-nanotube composites.

Thermal effect on dynamics of thin and thick composite laminated microbeams by modified couple stress theory for different boundary conditions

In this article, thermal effect on free vibration behavior of composite laminated microbeams based on the modified couple stress theory is presented. The proposed anisotropic model is developed by using a variational formulation. The governing equations and boundary conditions are obtained based on a modified couple stress theory and using the principle of minimum potential energy and considering different beam theories, i.e., Euler–Bernoulli, Timoshenko and Reddy beam theories. Unlike the classical beam theories, this model contains a material length scale parameter and can capture the size effect. Free vibration of a simply supported beam is solved by utilizing Fourier series. In addition, the fundamental frequency is achieved by using the generalized differential quadrature method for four types of cross-ply laminations with clamped–clamped, clamped–hinged and hinged–hinged boundary conditions for different beam theories. For investigating different parameters including temperature changes, material length scale parameter, beam thickness, some numerical results on different cross-ply laminated beams are presented. The fundamental frequency of different thin and thick beam theories is investigated by increasing slenderness ratio and thermal loads. The results prove that the modified couple stress theory increases the natural frequency under the thermal effects for free vibration of composite laminated microbeams.

A unified beam theory with strain gradient effect and the von Kármán nonlinearity

In this paper, the governing equations and finite element formulations for a microstructure‐dependent unified beam theory with the von Kármán nonlinearity are developed. The unified beam theory includes the three familiar beam theories (namely, Euler‐Bernoulli beam theory, Timoshenko beam theory, and third‐order Reddy beam theory) as special cases. The unified beam formulation can be used to facilitate the development of general finite element codes for different beam theories. Nonlocal size‐dependent properties are introduced through classical strain gradient theories. The von Kármán nonlinearity which accounts for the coupling between extensional and bending responses in beams with moderately large rotations but small strains is included. Equations for each beam theory can be deduced by setting the values of certain parameters. Newton's iterative scheme is used to solve the resulting nonlinear set of finite element equations. The numerical results show that both the strain gradient theory and the von Kármán nonlinearity have a stiffening effect, and therefore, reduce the displacements. The influence is more prominent in thin beams when compared to thick beams.

Buckling and postbuckling of composite beams in hygrothermal environments

This paper presents an investigation into the buckling and postbuckling of composite beams in hydrothermal environments. The free expansion due to both temperature variation and moisture absorption has been taken into consideration. In the meantime, the material properties are considered to be temperature- and moisture-dependent. A micromechanics-based model is proposed to consider the effect of the fiber volume fraction, temperature and moisture on the mechanical and hygrothermal material properties. The equilibrium equations are derived from the principle of virtual work where the small strain, moderate rotation assumption has been adopted. The classical Euler–Bernoulli beam (EBB) theory and the higher-order shear-deformation Reddy beam (RB) theory are presented. The critical buckling loads and the postbuckling amplitude are calculated at varying temperature, moisture, and fiber volume fraction. The model has been validated versus results published in the literature. The temperature variation has been found to significantly reduce the buckling load and increase the postbuckling amplitude. However, the contribution of the moisture concentration on the critical buckling loads has been found insignificant according to the assumptions made in this study. However, moisture has a noticeable effect on the postbuckling response.

Size dependent pull-in behavior of functionally graded sandwich nanobridges using higher order shear deformation theory

In this study the behavior of beam-type micro/nano electromechanical systems (MEMSs/NEMSs) made of functionally graded sandwich materials subjected to electrostatic actuation effect and intermolecular Casimir forces is investigated. Euler–Bernoulli, Timoshenko and higher order shear deformation (Reddy) beam theories are used to model the system. The modified couple stress theory is used to investigate size dependency in such systems. The three cases of softcore, hardcore, and functionally graded core are considered for the sandwich structure. The principle of minimum potential energy and variational approach are used to derive the governing nonlinear equations and boundary conditions of the system. Considering the nonlinearity of equations in the two force and geometric cases, the efficient generalized differential quadrature method is used to discretize the equations and boundary conditions, and the discretized equations are solved numerically afterwards. In the end, shear deformation effect and size effect on the system behavior are examined.

An investigation of stress wave propagation in a shear deformable nanobeam based on modified couple stress theory

This paper presents a new investigation for propagation of stress wave in a nanobeam based on modified couple stress theory. Using Euler–Bernoulli beam theory, Timoshenko beam theory, and Reddy beam theory, the effect of shear deformation is investigated. This nonclassical model contains a material length scale parameter to capture the size effect and the Poisson effect is incorporated in the current model. Governing equations of motion are obtained by Hamilton’s principle and solved explicitly. This solution leads to obtain two phase velocities for shear deformable beams in different directions. Effects of shear deformation, material length scale parameter, and Poisson’s ratio on the behavior of these phase velocities are investigated and discussed. The results also show a dual behavior for phase velocities against Poisson’s ratio.

Nonlocal Reddy beam model for free vibration analysis of multilayer nanoribbons incorporating interlayer shear effect

Abstract The aim of this study is to investigate the interlayer shear effect on nonlocal free vibration of multilayer graphene nanoribbons (MLGNRs) based on Reddy beam theory. The major novelty of the study is using the higher order shear deformation theory of Reddy to take the van der Waals’ interaction of the layers into account and calibrating the nonlocal parameter with the aids of molecular dynamics (MD) simulations to match the best results. Furthermore, the shear correction factor of nonlocal Timoshenko theory is calculated for modeling the MLGNRs. MLGNRs have broad applications in mechanical devices, such as resonators, sensors and actuators. Accordingly the present results can be used as a reference for future works in which the interlayer shear effects has substantial influences on the mechanical behavior of MLGNRs. The present study outstands for its modeling simplicity and low computational time and cost as well as excellent accuracy.

Flapwise vibration of rotating composite beams

In the present study, vibration of rotating composite beams is studied. Different beam theories are used in the formulation including Euler–Bernoulli, Timoshenko and Reddy beam theories. Ritz method is used in the solution of the problem. Simple polynomials are chosen for the displacement field. The continuity of transverse stresses is satisfied among the layers. Results are obtained for different orthotropy ratios, rotation speed, hub ratio, length to thickness ratio of the rotating composite beam and different boundary conditions.

Thermo-mechanical vibration analysis of a single-walled carbon nanotube embedded in an elastic medium based on higher-order shear deformation beam theory

In this study, the thermal effect on the free vibration characteristics of embedded Single-walled carbon nanotubes (SWCNTs) based on the size-dependent Reddy higher order shear deformation beam theory subjected to in-plane thermal loading is investigated by presenting a Navier-type solution and employing a semi-analytical Differential transform method (DTM) for the first time. In addition, the exact nonlocal Reddy beam theory solution presented here should be useful to engineers designing nanoelectromechanical devices. The smallscale effect is considered based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton’s principle, and they are solved by applying DTM. Numerical results reveal that the proposed modeling and semi-analytical approach can provide more accurate frequency results of the SWCNTs compared to analytical results and some cases in the literature. The detailed mathematical derivations are presented, and numerical investigations are performed, whereas emphasis is placed on investigating the effect of several parameters such as small-scale effects, boundary conditions, mode number, thickness ratio, temperature change, and Winkler spring modulus on the natural frequencies of the SWCNTs in detail. The vibration behavior of SWCNTs is significantly influenced by these effects. Results indicate that the inclusion of size effect results in a decrease in nanobeam stiffness and leads to a decrease in natural frequency. Numerical results are presented to serve as benchmarks for future analyses of SWCNTs.

Size-dependent thermal buckling analysis of micro composite laminated beams using modified couple stress theory

The thermal effect on size-dependent buckling analysis of micro composite laminated beams is studied based on the modified couple stress theory. The governing equations and boundary conditions are obtained by using the principle of minimum potential energy. The effect of shear deformation is investigated by considering the Euler–Bernoulli, Timoshenko and Reddy beam theories. In addition, the effect of boundary condition by using the hinged–hinged, clamped–hinged and clamped–clamped boundary conditions and the effect of lamination by applying four kinds of cross ply laminate i.e. [0, 0, 0], [0, 90, 0], [90, 0, 90] and [90, 90, 90] are investigated. Using the Fourier series expansions, the governing equations and boundary conditions are solved analytically just for hinged–hinged boundary condition. Also by using the generalized differential quadrature (GDQ) method, the governing equations and boundary conditions are solved numerically for all of the boundary conditions. Comparison between results obtained by numerical solution, analytical solution and those obtained by literature reveals the accuracy of GDQ method.

Flexural vibration and buckling analysis of single-walled carbon nanotubes using different gradient elasticity theories based on Reddy and Huu-Tai formulations

The aim of the present work is to analyze free flexural vibration and buckling of single-walled carbon nanotubes (SWCNT) under compressive axial loading based on different constitutive equations and beam theories. The models contain a material length scale parameter that can capture the size effect, unlike the classical Euler-Bernoulli or Reddy beam theory. The equations of motion of the Reddy and the Huu-Tai beam theories are reformulated using different gradient elasticity theories, including stress, strain and combined strain/inertia. The equations of motion are derived from Hamilton's principle in terms of the generalized displacements. Analytical solutions of free vibration and buckling are presented to bring out the effect of the nonlocal behavior on natural frequencies and buckling loads. The presented theoretical analysis is illustrated by a numerical example, and the results are qualitatively compared by another results.

Nonlinear Flow-Induced Flutter Instability of Double CNTs Using Reddy Beam Theory

In this study, nonlocal nonlinear instability and the vibration of a double carbon nanotube (CNT) system have been investigated. The Visco-Pasternak model is used to simulate the elastic medium between nanotubes, on which the effect of the spring, shear and damping of the elastic medium is considered. Both of the CNTs convey a viscose fluid and a uniform longitudinal magnetic field is applied to them. The fluid velocity is modified by small-size effects on the bulk viscosity and the slip boundary conditions of nano flow through the Knudsen number (Kn). Using von Karman geometric nonlinearity, Hamilton’s principle and considering longitudinal magnetic field, the nonlinear higher order governing equations for Reddy beam (RB) theory are derived. The differential quadrature method (DQM) is used to obtain the nonlinear frequency and critical fluid velocity (CFV) of the fluid conveying a coupled system. A detailed parametric study is conducted, focusing on the effects of parameters such as magnetic field strength, Knudsen number, aspect ratio, small scale and elastic foundation on the in-phase and out-of-phase vibration of the nanotube. The results indicate that the natural frequency and the critical fluid velocity of double bonded Reddy beams increase with an increase in the longitudinal magnetic field and elastic medium module. Furthermore, the results of this study can be useful for designing and manufacturing micro/nano- double-mechanical systems in advanced mechanics applications by controlling nonlinear frequency with an applied magnetic field.

AN INVESTIGATION ON THE CHARACTERISTICS OF BENDING, BUCKLING AND VIBRATION OF NANOBEAMS VIA NONLOCAL BEAM THEORY

In the present study, a nonlocal hyperbolic shear deformation theory is developed for the static flexure, buckling and free vibration analysis of nanobeams using the nonlocal differential constitutive relations of Eringen. The theory, which does not require shear correction factor, accounts for both small scale effects and hyperbolic variation of shear strains and consequently shear stresses through the thickness of the nanobeam. The equations of motion are derived from Hamilton's principle. Analytical solutions for the deflection, buckling load and natural frequency are presented for a simply supported nanobeam, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory and Reddy beam theories. Present solutions can be used for the static and dynamic analyses of single-walled carbon nanotubes.

Free Vibration of Size-Dependent Functionally Graded Microbeams Based on the Strain Gradient Reddy Beam Theory

The microscale vibration characteristics of microbeams made of functionally graded materials (FGMs) are investigated based on the strain gradient Reddy beam theory capable of capturing the size effect. The non-classical governing differential equations, together with the corresponding boundary conditions, are obtained using Hamilton's principle. Then, the free vibration problem of simply supported FGM microbeams is solved using the Navier solution. The natural frequencies of FGM microbeams are calculated corresponding to a wide range of dimensionless length scale parameters, material property gradient indices, and aspect ratios to illustrate the influences of size effect on the vibrational response of FGM microbeams.

Comparison of various refined nonlocal beam theories for bending, vibration and buckling analysis of nanobeams

In this paper, unified nonlocal shear deformation theory is proposed to study bending, buckling and free vibration of nanobeams. This theory is based on the assumption that the in-plane and transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. In addition, this present model is capable of capturing both small scale effect and transverse shear deformation effects of nanobeams, and does not require shear correction factors. The equations of motion are derived from Hamilton's principle. Analytical solutions for the deflection, buckling load, and natural frequency are presented for a simply supported nanobeam, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory and Reddy beam theories.

Size dependent buckling analysis of microbeams based on modified couple stress theory with high order theories and general boundary conditions

In this research, buckling analysis of three microbeam models are investigated based on modified couple stress theory. Using Euler–Bernoulli beam theory (EBT), Timoshenko beam theory (TBT) and Reddy beam theory (RBT), the effect of shear deformation is presented. To examine the effect of boundary condition, three kinds of boundary conditions i.e. hinged–hinged, clamped–hinged and clamped–clamped boundary conditions, are considered. These nonclassical microbeam models incorporated with Poisson effect, contain a material length scale parameter and can capture the size effect. These models can degenerate into the Classical models if the material length scale parameter and Poisson’s ratio are both taken to be zero. Governing equations and boundary conditions are derived by using principle of minimum potential energy. Generalized differential quadrature (GDQ) method is employed to solve the governing differential equations. Also an analytical solution is applied to determine the critical buckling load of microbeams with hinged–hinged boundary condition. Comparison between the results of GDQ and analytical methods reveals the accuracy of GDQ method. Some numerical results are exhibited to indicate the influences of beam thickness, material length scale parameter and Poisson’s ratio on the critical buckling load of these microbeams.

Thermal Effect on Static Bending, Vibration and Buckling of Reddy Beam Based on Modified Couple Stress Theory

A microstructure-dependent Reddy beam theory (RBT) which contain only one material length scale parameter and can capture the size effect in micro-scale material unlike the classical theory is developed .using the variational principle energy the governing equation of motion is derived based on modified couple stress theory for the simply supported beam. the equations obtained are solved by Fourier series and the influence of the length scale parameter and thermal effect on static bending, vibration and buckling analysis of micro-scale Reddy beam is investigated.

A Nonlinear Finite Element Framework for Viscoelastic Beams Based on the High-Order Reddy Beam Theory

A weak form Galerkin finite element model for the nonlinear quasi-static and fully transient analysis of initially straight viscoelastic beams is developed using the kinematic assumptions of the third-order Reddy beam theory. The formulation assumes linear viscoelastic material properties and is applicable to problems involving small strains and moderate rotations. The viscoelastic constitutive equations are efficiently discretized using the trapezoidal rule in conjunction with a two-point recurrence formula. Locking is avoided through the use of standard low-order reduced integration elements as well through the employment of a family of elements constructed using high-polynomial order Lagrange and Hermite interpolation functions.

A nonlocal beam theory for bending, buckling, and vibration of nanobeams

A nonlocal shear deformation beam theory is proposed for bending, buckling, and vibration of nanobeams using the nonlocal differential constitutive relations of Eringen. The theory, which does not require shear correction factor, accounts for both small scale effects and quadratic variation of shear strains and consequently shear stresses through the thickness of the beam. In addition, it has strong similarities with nonlocal Euler–Bernoulli beam theory in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The equations of motion are derived from Hamilton’s principle. Analytical solutions of deflection, buckling load, and natural frequency are presented for a simply supported beam, and the obtained results compare well with those predicted by the nonlocal Timoshenko and Reddy beam theories.