Third-order Shear Deformation Beam Theory Research Articles

Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams

• Nonlinear behaviour is of a hardening type. • Both symmetric and asymmetric modes are present. • Larger gradient index results in shift in resonance. This paper aims at investigating the nonlinear vibration characteristics of axially functionally graded (AFG) shear deformable tapered beams subjected to external harmonic excitations. A coupled nonlinear longitudinal-transverse-rotational set of equations governing the motion of the AFG system is derived utilising the third-order shear deformation beam theory via Hamilton's energy principle. The beam under consideration is tapered; i.e. the width of the beam varies along the length. The tapered geometry along with the nonuniform material properties arising from the AFG nature of the beam increases the complexity in the modelling and numerical simulations. The expressions for the kinetic and potential energies of the AFG shear-deformable tapered beam together with the expressions for the work of damping and external excitation are derived and implemented in Hamilton's principle. The nonlinear partial differential equations are discretised making use of the Galerkin technique and solved with the aid of a continuation scheme. The effect of different parameters such as the gradient index and the tapered ratios on the force- and frequency-amplitude diagrams of the AFG system is examined.

Geometrically nonlinear analysis of sandwich beams under low velocity impact: analytical and experimental investigation

Nonlinear low velocity impact response of sandwich beam with laminated composite face sheets and soft core is studied based on Extended High Order Sandwich Panel Theory (EHSAPT). The face sheets follow the Third order shear deformation beam theory (TSDT) that has hitherto not reported in conventional EHSAPT. Besides, the two dimensional elasticity is used for the core. The nonlinear Von Karman type relations for strains of face sheets and the core are adopted. Contact force between the impactor and the beam is obtained using the modified Hertz law. The field equations are derived via the Ritz based applied to the total energy of the system. The solution is obtained in the time domain by implementing the well-known Runge-Kutta method. The effects of boundary conditions, core-to-face sheet thickness ratio, initial velocity of the impactor, the impactor mass and position of the impactor are studied in detail. It is found that each of these parameters have significant effect on the impact characteristics which should be considered. Finally, some low velocity impact tests have been carried out by Drop Hammer Testing Machine. The contact force histories predicted by EHSAPT are in good agreement with that obtained by experimental results.

Nonlocal strain gradient beam model for postbuckling and associated vibrational response of lipid supramolecular protein micro/nano-tubules

As a supramolecular construction, lipid protein micro/nano-tubules can be utilized in a variety of sustained biological delivery system. The high slenderness ratio of lipid tubules makes their hierarchical assembly into a desired architecture difficult. Therefore, an accurate prediction of mechanical behavior of lipid tubular is essential. The objective of this study is to capture size dependency in the postbuckling and vibrational response of the postbuckled lipid micro/nano-tubules more comprehensively. To this purpose, the nonlocal strain gradient elasticity theory is incorporated to the third-order shear deformation beam theory to develop an unconventional beam model. Hamilton's principle is put to use to establish the size-dependent governing differential equations of motion. After that, an improved perturbation technique in conjunction with Galerkin method is employed to obtain the nonlocal strain gradient load-frequency response and postbuckling stability curves of lipid micro/nano-tubules. It is revealed that by taking the nonlocal size effect into consideration, the influence of the type (geometrical parameters) of an axially compressed lipid micro/nano-tubule on its natural frequency in order decreases and increases within the prebuckling and postbuckling regimes. While the strain gradient size dependency plays an opposite role which causes that the influence of the type of lipid micro/nano-tubule on its natural frequency corresponding to the prebuckling and postbuckling domains increases and decreases, respectively.

Nonlinear resonance responses of geometrically imperfect shear deformable nanobeams including surface stress effects

In this work, a thorough investigation is presented into the nonlinear resonant dynamics of geometrically imperfect shear deformable nanobeams subjected to harmonic external excitation force in the transverse direction. To this end, the Gurtin–Murdoch surface elasticity theory together with Reddy’s third-order shear deformation beam theory is utilized to take into account the size-dependent behavior of nanobeams and the effects of transverse shear deformation and rotary inertia, respectively. The kinematic nonlinearity is considered using the von Kármán kinematic hypothesis. The geometric imperfection as a slight curvature is assumed as the mode shape associated with the first vibration mode. The weak form of geometrically nonlinear governing equations of motion is derived using the variational differential quadrature (VDQ) technique and Lagrange equations. Then, a multistep numerical scheme is employed to solve the obtained governing equations in order to study the nonlinear frequency–response and force–response curves of nanobeams. Comprehensive studies into the effects of initial imperfection and boundary condition as well as geometric parameters on the nonlinear dynamic characteristics of imperfect shear deformable nanobeams are carried out through numerical results. Finally, the importance of incorporating the surface stress effects via the Gurtin–Murdoch elasticity theory, is emphasized by comparing the nonlinear dynamic responses of the nanobeams with different thicknesses.

A new nonlocal elasticity theory with graded nonlocality for thermo-mechanical vibration of FG nanobeams via a nonlocal third-order shear deformation theory

ABSTRACTIn the present research, free vibration study of functionally graded (FG) nanobeams with graded nonlocality in thermal environments is performed according to the third-order shear deformation beam theory. The present nanobeam is subjected to uniform and nonlinear temperature distributions. Thermo-elastic coefficients and nonlocal parameter of the FG nanobeam are graded in the thickness direction according to power-law form. The scale coefficient is taken into consideration implementing nonlocal elasticity of Eringen. The governing equations are derived through Hamilton's principle and are solved analytically. The frequency response is compared with those of nonlocal Euler–Bernoulli and Timoshenko beam models, and it is revealed that the proposed modeling can accurately predict the vibration frequencies of the FG nanobeams. The obtained results are presented for the thermo-mechanical vibrations of the FG nanobeams to investigate the effects of material graduation, nonlocal parameter, mode number, slenderness ratio, and thermal loading in detail. The present study is associated to aerospace, mechanical, and nuclear engineering structures that are under thermal loads.

A dynamic stiffness method for analysis of thermal effect on vibration and buckling of a laminated composite beam

Thermal effects on vibration and buckling behaviors of generally layered composite beams with arbitrary boundary conditions are dealt with in this paper. The composite beam is modeled using third-order shear deformation beam theory in which the Poisson effect is incorporated. A constant temperature change through the beam thickness is assumed. An exact dynamic stiffness matrix is formulated by directly solving the differential equations of motion governing the natural vibration of the composite beams subjected to uniform temperature changes along the beam thickness. Application of the derived dynamic stiffness matrix together with the Wittrick–Williams algorithm to compute the natural frequencies and buckling temperature changes of two particular composite beams is discussed. The correctness and accuracy of the derived dynamic stiffness matrix is evaluated by comparing the present results with the available solutions in literature. The influences of Poisson effect, boundary condition, temperature change, thermal expansion coefficient and material anisotropy on the natural frequencies of the composite beams are studied.

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.

Electro-magnetic effects on nonlocal dynamic behavior of embedded piezoelectric nanoscale beams

This article investigates vibration behavior of magneto-electro-elastic functionally graded nanobeams embedded in two-parameter elastic foundation using a third-order parabolic shear deformation beam theory. Material properties of magneto-electro-elastic functionally graded nanobeam are supposed to be variable throughout the thickness based on power-law model. Based on Eringen’s nonlocal elasticity theory which captures the small size effects and using Hamilton’s principle, the nonlocal governing equations of motions are derived and then solved analytically. Then, the influences of elastic foundation, magnetic potential, external electric voltage, nonlocal parameter, power-law index, and slenderness ratio on the frequencies of the embedded magneto-electro-elastic functionally graded nanobeams are studied.

Vibration analysis of FG nanobeams based on third-order shear deformation theory under various boundary conditions

In this study, free vibration of functionally graded (FG) micro/nanobeams based on nonlocal third-order shear deformation theory and under different boundary conditions is investigated by applying the differential quadrature method. Third-order shear deformation theory can consider the both small-scale effects and quadratic variation of shear strain and hence shear stress along the FG nanobeam thickness. The governing equations are obtained by using the Hamilton's principle, based on third-order shear deformation beam theory. The differential quadrature (DQ) method is used to discretize the model and attain the natural frequencies and mode shapes. The properties of FG micro/nanobeam are assumed to be chanfged along the thickness direction based on the simple power law distribution. The effects of various parameters such as the nonlocal parameter, gradient index, boundary conditions and mode number on the vibration characteristics of FG micro/nanobeams are discussed in detail.

Dynamic modeling of smart shear-deformable heterogeneous piezoelectric nanobeams resting on Winkler–Pasternak foundation

Free vibration analysis is presented for a simply supported, functionally graded piezoelectric (FGP) nanobeam embedded on elastic foundation in the framework of third-order parabolic shear deformation beam theory. Effective electro-mechanical properties of FGP nanobeam are supposed to be variable throughout the thickness based on power-law model. To incorporate the small size effects into the local model, Eringen’s nonlocal elasticity theory is adopted. Analytical solution is implemented to solve the size-dependent buckling analysis of FGP nanobeams based upon a higher-order shear deformation beam theory where coupled equations obtained using Hamilton’s principle exist for such beams. Some numerical results for natural frequencies of the FGP nanobeams are prepared, which include the influences of elastic coefficients of foundation, electric voltage, material and geometrical parameters and mode number. This study is motivated by the absence of articles in the technical literature and provides beneficial results for accurate FGP structures design.

Thermal Buckling Analysis of Size-Dependent FG Nanobeams Based on the Third-Order Shear Deformation Beam Theory

In this paper, the thermal effects on the buckling of functionally graded (FG) nanobeams subjected to various types of thermal loading including uniform, linear and non-linear temperature changes are investigated based on the nonlocal third-order shear deformation beam theory. The material properties of FG nanobeam are supposed to vary gradually along the thickness direction according to the power-law form. The governing equations are derived through Hamilton’s principle and solved analytically. Comparison examples are performed to verify the present results. Obtained results are presented for thermal buckling analysis of FG nanobeams such as the effects of the power-law index, nonlocal parameter, slenderness ratio and thermal loading in detail.

Analytical solution for nonlocal buckling characteristics of higher-order inhomogeneous nanosize beams embedded in elastic medium

In this paper, buckling characteristics of nonhomogeneous functionally graded (FG) nanobeams embedded on elastic foundations are investigated based on third order shear deformation (Reddy) without using shear correction factors. Third-order shear deformation beam theory accounts for shear deformation effects by a parabolic variation of all displacements through the thickness, and verifies the stress-free boundary conditions on the top and bottom surfaces of the FG nanobeam. A two parameters elastic foundation including the linear Winkler springs along with the Pasternak shear layer is in contact with beam in deformation, which acts in tension as well as in compression. The material properties of FG nanobeam are supposed to vary gradually along the thickness and are estimated through the power-law and Mori-Tanaka models. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. Nonlocal equations of motion are derived through Hamilton's principle and they are solved applying analytical solution. Comparison between results of the present work and those available in literature shows the accuracy of this method. The obtained results are presented for the buckling analysis of the FG nanobeams such as the effects of foundation parameters, gradient index, nonlocal parameter and slenderness ratio in detail.

Free Vibration Analysis of Functionally Graded Beams Resting on Elastic Foundation in Thermal Environment

The free vibration of functionally graded (FG) beams with various boundary conditions resting on a two-parameter elastic foundation in the thermal environment is studied using the third-order shear deformation beam theory. The material properties are temperature-dependent and vary continuously through the thickness direction of the beam, based on a power-law distribution in terms of the volume fraction of the material constituents. In order to discretize the governing equations, the differential quadrature method (DQM) in conjunction with the Hamilton’s principle is adopted. The convergence of the method is demonstrated. In order to validate the results, comparisons are made with solutions available for the isotropic and FG beams. Through a comprehensive parametric study, the effect of various parameters involved on the FG beam was studied. It is concluded that the uniform temperature rise has more significant effect on the frequency parameters than the nonuniform case.

Vibration analysis of nonlocal beams made of functionally graded material in thermal environment

In this paper, thermal vibration behavior of functionally graded (FG) nanobeams exposed to various kinds of thermo-mechanical loading including uniform, linear and non-linear temperature rise embedded in a two-parameter elastic foundation are investigated based on third-order shear deformation beam theory which considers the influence of shear deformation without the need to shear correction factors. Material properties of FG nanobeam are supposed to be temperature-dependent and vary gradually along the thickness according to the Mori-Tanaka homogenization scheme. The influence of small scale is captured based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton’s principle and they are solved applying analytical solution. The comparison of the obtained results is conducted with those of nonlocal Euler-Bernoulli beam theory and it is demonstrated that the proposed modeling predicts correctly the vibration responses of FG nanobeams. The influences of some parameters including gradient index, nonlocal parameter, mode number, foundation parameters and thermal loading on the thermo-mechanical vibration characteristics of the FG nanobeams are presented.

A Nonlocal Higher-Order Shear Deformation Beam Theory for Vibration Analysis of Size-Dependent Functionally Graded Nanobeams

In this paper, free vibration characteristics of functionally graded (FG) nanobeams based on third-order shear deformation beam theory are investigated by presenting a Navier-type solution. Material properties of FG nanobeam are supposed to change continuously along the thickness according to the power-law form. The effect of small scale is considered based on nonlocal elasticity theory of Eringen. Through Hamilton’s principle and third-order shear deformation beam theory, the nonlocal governing equations are derived and they are solved applying analytical solution. According to the numerical results, it is revealed that the proposed modeling can provide accurate frequency results for FG nanobeams as compared to some cases in the literature. The numerical investigations are presented to investigate the effect of the several parameters such as material distribution profile, small-scale effects, slenderness ratio and mode number on vibrational response of the FG nanobeams in detail. It is concluded that various factors such as nonlocal parameter and gradient index play notable roles in vibrational response of FG nanobeams.

On the free vibration characteristics of postbuckled third-order shear deformable FGM nanobeams including surface effects

Abstract On the basis of an efficient numerical solution methodology, the free vibration response of third-order shear deformable nanobeams made of functionally graded materials (FGMs) around the postbuckling domain is investigated incorporating the effects of surface free energy. Gurtin–Murdoch elasticity theory in conjunction with von Karman geometric nonlinearity is implemented into the classical third-order shear deformation beam theory. In order to consider balance conditions on the surfaces of nanobeam, it is assumed that the bulk normal stress is distributed cubically through the thickness. The material properties are assumed to be graded in the thickness direction based on power-law distribution. Using Hamilton’s principle, the non-classical nonlinear governing differential equations of motion and associated boundary conditions are derived. Then generalized differential quadrature method is employed to discretize the governing equations and solution domain according to the Chebyshev–Gauss–Lobatto grid points. Subsequently, the pseudo-arc length continuation technique is utilized to solve the nonlinear problem. It is demonstrated that in contrast to the prebuckling domain, in the postbuckling domain, the natural frequency of FGM nanobeam decreases by increasing the value of material property gradient index. Also, it is revealed that the surface effect plays more important role on the vibration characteristics of the buckled FGM nanobeams with lower thicknesses.

Surface effects on the nonlinear forced vibration response of third-order shear deformable nanobeams

Given the high surface to volume ratio, the nonlinear forced vibration behavior of third-order shear deformable nanobeams in the presence of the both effects of surface stress and that of surface inertia is investigated. Gurtin–Murdoch elasticity theory is utilized within the framework of third-order shear deformation beam theory to develop a novel non-classical beam model to incorporate surface effects into the forced vibration analysis of nanobeams. A cubic variation through the thickness of nanobeam is considered for the normal stress component of the bulk in order to satisfy the surface equilibrium equations. Hamilton’s principle is used to derive size-dependent nonlinear governing differential equations of motion. The equations are solved numerically using generalized differential quadrature method with an iterative algorithm on the basis of shifted Chebyshev–Gauss–Lobatto grid points. Subsequently, based on the Galerkin’s technique, the set of nonlinear partial differential equations are reduced into a time-varying set of ordinary differential equations of Duffing type. At the end, the pseudo arc-length method is employed to solve the set of nonlinear equations of the time domain. It is observed that by increasing the beam thickness, surface effects on the nonlinear forced vibration behavior of nanobeam diminish which leads to increasing the deviation from the linear response.

Surface energy effects on the free vibration characteristics of postbuckled third-order shear deformable nanobeams

The prime aim of the current study is to predict the free vibration behavior of third-order shear deformable nanobeams in the vicinity of postbuckling configuration and in the presence of surface effects which includes surface elasticity, residual surface stress and surface inertia. To accomplish this end, Gurtin–Murdoch elasticity theory within the framework of third-order shear deformation beam theory is employed. In order to satisfy the balance conditions between the bulk and surfaces of nanobeam, a cubic distribution is considered for the normal stress through the thickness. By using Hamilton’s principle, the non-classical governing differential equations of motion including von Karman geometric nonlinearity are derived. After using generalized differential quadrature (GDQ) method to discretize the governing equations on the basis of Chebyshev–Gauss–Lobatto grid points, the pseudo-arc length continuation technique is utilized to solve the eigenvalue problem. The natural frequencies of nanobeam corresponding to the both prebuckling and postbuckling domains are obtained for various buckling mode shapes based on the numerical solution strategy. It is demonstrated that in the prebuckling domain of the first vibration mode shape, increasing of beam thickness leads to lower natural frequency for all types of boundary conditions, but this behavior becomes reverse in the postbuckling domain.

Static and free vibration analyses of small-scale functionally graded beams possessing a variable length scale parameter using different beam theories

Abstract This article puts forward a modified couple stress theory based approach of analysis for small-scale functionally graded beams, that possess a variable length scale parameter. Presented procedures are capable of predicting static and dynamic beam responses according to three different beam theories, namely: Euler–Bernoulli beam theory, Timoshenko beam theory and third-order shear deformation beam theory. A variational method is used in conjunction with the modified couple stress theory to derive the governing partial differential equations. All properties of the small-scale functionally graded beams – including the length scale parameter – are assumed to be functions of the thickness coordinate in the derivations. The governing equations are solved numerically through the use of the differential quadrature method (DQM). Numerical results are generated for small-scale functionally graded beams, that comprise ceramic and metallic materials as constituent phases. Both small-scale beams subjected to static loading and those undergoing free vibrations are considered in the computations. Comparisons of the numerical results to those available in the literature point out that developed techniques lead to results of high accuracy. Further numerical results are provided, which demonstrate the responses of small-scale functionally graded beams estimated by the three different beam theories as well as provide insight into the influences of material parameters upon the static deflections and natural vibration frequencies.

Nonlinear free vibration analysis of functionally graded third-order shear deformable microbeams based on the modified strain gradient elasticity theory

In the present investigation, a numerical analysis is conducted to predict size-dependent nonlinear free vibration characteristics of third-order shear deformable microbeams made of functionally graded materials (FGMs). For this purpose, the modified strain gradient elasticity theory and von Karman geometric nonlinearity are implemented into the classical third-order shear deformation beam theory to develop a nonclassical higher-order beam model including three additional length scale parameters to capture size effect efficiently. It is assumed that the material properties of the FGM microbeams are evaluated by the Mori–Tanaka homogenization technique. On the basis of the Hamilton’s principle, the size-dependent nonlinear governing differential equations of motion and associated boundary conditions are derived and then discretized along various end supports by employing generalized differential quadrature (GDQ) method. A direct iterative process corresponding to both positive and negative deflection cycles is adopted. Secondly, a parametric study is performed to demonstrate the influences of the values of dimensionless length scale parameter, material property gradient index and length to thickness aspect ratio on the linear and nonlinear natural frequencies of FGM microbeams.