Simply-supported Boundary Conditions Research Articles

Vibration analysis of defective graphene sheets using nonlocal elasticity theory

Many papers have studied the free vibration of graphene sheets. However, all this papers assumed their atomic structure free of any defects. Nonetheless, they actually contain some defects including single vacancy, double vacancy and Stone-Wales defects. This paper, therefore, investigates the free vibration of defective graphene sheets, rather than pristine graphene sheets, via nonlocal elasticity theory. Governing equations are derived using nonlocal elasticity and the first-order shear deformation theory (FSDT). The influence of structural defects on the vibration of graphene sheets is considered by applying the mechanical properties of defective graphene sheets. Afterwards, these equations solved using generalized differential quadrature method (GDQ). The small-scale effect is applied in the governing equations of motion by nonlocal parameter. The effects of different defect types are inspected for graphene sheets with clamped or simply-supported boundary conditions on all sides. It is shown that the natural frequencies of graphene sheets decrease by introducing defects to the atomic structure. Furthermore, it is found that the number of missing atoms, shapes and distributions of structural defects play a significant role in the vibrational behavior of graphene. The effect of vacancy defect reconstruction is also discussed in this paper.

Dynamic stability of functionally graded nanobeam based on nonlocal Timoshenko theory considering surface effects

Based on nonlocal Timoshenko beam theory, dynamic stability of functionally graded (FG) nanobeam under axial and thermal loading was investigated. Surface stress effects were implemented according to Gurtin-Murdoch continuum theory. Using power law distribution for FGM and von Karman geometric nonlinearity, governing equations were derived based on Hamilton's principle. The developed nonlocal models have the capability of interpreting small scale effects. Pasternak elastic medium was employed to represent the interaction of the FG nanobeam and the surrounding elastic medium. A parametric study was conducted to focus influences of the static load factor, temperature change, gradient index, nonlocal parameter, slenderness ratio, surface effect and springs constants of the elastic medium on the dynamic instability region (DIR) of the FG beam with simply-supported boundary conditions. It was found that differences between DIRs predicted by local and nonlocal beam theories are significant for beams with lower aspect ratio. Moreover, it was observed that in contrast to high temperature environments, at low temperatures, increasing the temperature change moves the origin of the DIR to higher excitation frequency zone and leads to further stability. Considering surface stress effects shifts the DIR of FG beam to higher frequency zone, also increasing the gradient index enhances the frequency of DIR.

Thermo-electro-magneto-mechanical bending behavior of size-dependent sandwich piezomagnetic nanoplates

Thermo-electro-magneto-mechanical bending analysis of a sandwich nanoplate is presented in this paper based on Kirchhoff’s plate theory and nonlocal theory. The sandwich nanoplate includes an elastic nano-core and two piezomagnetic face-sheets actuated by applied electric and magnetic potentials. The governing equations for the electro-magneto-mechanical bending are derived in terms of the displacement components and electric and magnetic potentials. Then, the problem is solved analytically by using Navier’s method. A parametric study is presented to show the effects of the nonlocal parameter, temperature rise, applied electric and magnetic potentials on the bending behaviors of sandwich nanoplates for simply-supported boundary conditions. As a main result of study, it is concluded that the deflection decreases as applied electric potential increases and applied magnetic potential decreases. In addition, the increase of nonlocal parameter leads to increase of deflection and maximum electric potential through the thickness direction.

Thermoelastic beam in modified couple stress thermoelasticity induced by laser pulse

In this study, the thermoelastic beam in modified couple stress theory due to laser source and heat flux is investigated. The beam are heated by a non-Guassian laser pulse and heat flux. The Euler Bernoulli beam theory and the Laplace transform technique are applied to solve the basic equations for coupled thermoelasticity. The simply-supported and isothermal boundary conditions are assumed for both ends of the beam. A general algorithm of the inverse Laplace transform is developed. The analytical results have been numerically analyzed with the help of MATLAB software. The numerically computed results for lateral deflection, thermal moment and axial stress due to laser source and heat flux have been presented graphically. Some comparisons have been shown in figures to estimate the effects of couple stress on the physical quantities. A particular case of interest is also derived. The study of laser-pulse find many applications in the field of biomedical, imaging processing, material processing and medicine with regard to diagnostics and therapy.

Forced vibration of sinusoidal FG nanobeams resting on hybrid Kerr foundation in hygro-thermal environments

ABSTRACTSize-dependent forced vibration behavior of functionally graded (FG) nanobeams subjected to an in-plane hygro-thermal loading and lateral concentrated and uniform dynamic loads is investigated via a higher-order refined beam theory, which captures shear deformation influences needless of any shear correction factor. The nanobeam is in contact with a three-parameter Kerr foundation consisting of upper and lower spring layers as well as a shear layer. Hygro-thermo-elastic material properties of the nanobeam are described via power-law distribution considering exact position of the neutral axis. Through nonlocal elasticity theory of Eringen and Hamilton's principle, the governing equations of higher-order FG nanobeams on Kerr foundation under dynamic loading are derived. These equations are solved for simply-supported and clamped-clamped boundary conditions. A detailed parametric study is performed to show the importance of moisture concentration rise, temperature rise, material composition, nonlocality, Kerr foundation parameters, and boundary conditions on forced vibration characteristics and resonance frequencies of FG nanobeams. As a consequence, Kerr foundation parameters lead to a significant delay in the occurrence of resonance frequencies.

Diagonal piezoelectric sensors on cylindrical shells

Piezoelectric sensors are effective for distributed health monitoring and sensing of structures. The signals of piezoelectric sensors are related to the orientation of the sensors. In this study, a diagonal piezoelectric sensor is proposed for cylindrical shells. The sensor is made of a rectangular piezoelectric patch and diagonally attached on the shell surface; and piezoelectric actuators are used for excitation. An analytical model of the sensor is derived based on thin shell assumption with simply-supported boundary conditions. The orientation angle of the piezoelectric sensor is introduced as an independent variable. The proposed model consists of an integral term over the electrode area, which is divided into three regions for calculation. The sensing signal is decomposed into six components to evaluate the contributions of the strain components. Case studies on signals with respect to the orientation and aspect ratios are accomplished. The cylindrical shell with piezoelectric actuators and diagonal sensors is fabricated and tested under laboratory condition. Comparison of theoretical results with experimental data is conducted, and the model of the diagonal sensors is validated. The errors between the predictions and experimental results are less than 10% for all evaluated modes.

A semi-analytical method for vibration analysis of functionally graded carbon nanotube reinforced composite doubly-curved panels and shells of revolution

The object of this paper is to present a novel semi-analytical method and its associated applications for linear vibration analyses of functionally graded carbon nanotube reinforced composite (FG-CNTRC) doubly-curved panels and shells of revolution on with arbitrary boundary conditions. Distribution of the carbon nanotubes through the thickness of the structures may be uniform or functionally graded and four types of the CNTs distribution are considered in this paper. Properties of the composite media are determined by a refined rule of mixtures approach which contains the efficiency parameters. The translation and rotation displacements of the doubly-curved structures are uniformly expressed as the superposition of a standard cosine Fourier series and several auxiliary functions introduced to eliminate all potential discontinuities of the original displacement function and its derivatives at the edges. Based on that, the energy expression of the FG-CNTRC doubly-curved panels and shells of revolution is examined where the first-order shear deformation elasticity theory is considered. Lastly, to solve the natural frequencies as well as the associated mode shapes by means of the Ritz-variational energy method. Unlike other existing methods, the proposed method is capable of handling various combinations of boundary constraints in a unified fashion, including free, simply-supported, clamped and elastic-supported boundary conditions. Comprehensive studies on the convergence, accuracy, stability and efficiency of the method are derived via the comparison with existing results reported in publications. The parametric studies concerning the influence of the geometrical parameters, CNTs distributions, volume fraction of CNTs as well as boundary restraint parameters on free vibration of FG-CNTRC doubly-curved panels and shells of revolution is also investigated in detail.

Free vibration of thick laminated circular and annular plates using three-dimensional finite element analysis

This paper presents a numerical analysis of free vibration of thick laminated circular plates, having free, clamped as well as simply-supported boundary conditions at outer edges. The finite element method that works on the basis of three-dimensional theory of elasticity is employed to evaluate the first ten natural frequencies of uniform circular plate. The effect of thickness ratios, fiber orientation angle (angle-ply and cross-ply), stacking sequences (symmetric and antisymmetric), radius ratios and boundary conditions on frequency parameter are discussed in detail. The first five natural modes of flexural vibrations for different boundary conditions are presented in pictorial forms. Verification of the accuracy of these results is verified by appropriate convergence study and checked with the results available in the literature.

Bending and stability analysis of size-dependent compositionally graded Timoshenko nanobeams with porosities

In this article, static deflection and buckling of functionally graded (FG) nanoscale beams made of porous material are carried out based on the nonlocal Timoshenko beam model which captures the small scale influences. The exact position of neutral axis is fixed, to eliminate the stretching and bending coupling due to the unsymmetrical material change along the FG nanobeams thickness. The material properties of FG beam are graded through the thickness on the basis of the power-law form, which is modified to approximate the material properties with two models of porosity phases. By employing Hamilton's principle, the nonlocal governing equations of FG nanobeams are obtained and solved analytically for simply-supported boundary conditions via the Navier-type procedure. Numerical results for deflection and buckling of FG nanoscale beams are presented and validated with those existing in the literature. The influences of small scale parameter, power law index, porosity distribution and slenderness ratio on the static and stability responses of the FG nanobeams are all explored.

Exact three-dimensional static analysis of single- and multi-layered plates and shells

Abstract This new work proposes an exact three-dimensional static analysis of plates and shells. One-layered and multilayered isotropic, orthotropic, sandwich and composite structures are investigated in terms of displacements and in-plane and out-of-plane stresses through the thickness direction. Proposed structures are completely simply-supported and a transverse normal load is applied. The proposed method is based on the 3D equilibrium equations written using general orthogonal curvilinear coordinates which are valid for spherical shells. Cylindrical shell, cylinder and plate results are obtained as particular cases of 3D spherical shell equations. All the considered structures are analyzed without any geometrical approximation. The exact solution is possible because of simply-supported boundary conditions and harmonic form for applied loads. The shell solution is based on a layer-wise approach and the second order differential equations are solved using the redouble of variables and the exponential matrix method. A preliminary validation of the model is made using reference results in the literature. Thereafter, the proposed exact 3D shell solution is employed with confidence to provide results for one-layered and multilayered plates, cylinders, cylindrical shell panels and spherical shell panels. All these geometries are analyzed via a unified and general solution, and the obtained results can be used to validate future numerical methods proposed for plates and shells (e.g., the finite element method or the differential quadrature method). Proposed results allow to remark substantial features about the thickness of the structures, their geometry, the zigzag effects of displacements, the interlaminar continuity of displacements and transverse stresses, and boundary loading conditions for stresses.

Postbuckling of piezolaminated cylindrical shells with eccentrically/concentrically stiffeners surrounded by nonlinear elastic foundations

A fully analytical approach on thermal and mechanical buckling and postbuckling of cylindrical shell surrounded on nonlinear elastic foundation is presented. The shell is composed of composite material, piezoelectric actuator, and eccentrically/concentrically isotropic stringers and rings. The equilibrium and compatibility equations of shell are derived based on Kirchhoff assumptions taking into account von Karman nonlinearity. Two types of simply-supported boundary conditions are considered as freely movable and immovable edges. The equations are solved by definitions of stress function and applying Galerkin method. Numerical examples are well verified with available data in the literature. Several parametric investigations are conducted to examine the effects of voltage, different stiffeners, lay-up configuration, and nonlinear elastic foundations on equilibrium paths.

A simple analytical model for free vibration of orthotropic and functionally graded rectangular plates

With the widespread application of composite structures in engineering, the static and dynamic analyses of these structures have become important in the past years. Composite and functionally graded plates have been frequently being used and are important parts of many engineering structures. Numerous plate theories and solution procedures have been presented to predict the linear dynamic response of composite and functionally graded plates. This study aims to develop a simple analytical model to predict the free vibration response of orthotropic and functionally graded rectangular plates with clamped and simply-supported boundary conditions. To this end, Mindlin’s shear deformation plate theory has been used to obtain the equations of motion in terms of transverse displacement and rotations of mid-plane of the plate. After expressing the rotation parameters in terms of the transverse displacement, they have been substituted into equation of transverse motion of the plate. Then, for each boundary condition, an appropriate function has been assumed for the transverse displacement of the plate. Using the orthogonality of these functions, the natural frequencies of the plate have been determined. Some examples have been provided to validate the proposed model.

Free Vibration Analysis of Nonuniform Microbeams Based on Modified Couple Stress Theory: an Analytical Solution

In this study, analytical solution is presented to calculate the free vibration frequencies of nonuniform microbeams. Scale effects are modelled using modified couple stress theory and the microbeam is assumed to be thin while Poisson's ratio effects are also taken into account. Nonuniformity is presented by exponentially varying width among the microbeam while the thickness remains constant. Results are presented for simply-supported, cantilever and clamped boundary conditions. First five natural frequency parameter are calculated for different scale and nonuniformity parameters and effects of each parameter on the results are discussed. Also, to understand the effects of Poisson's ratio, small scale and nonuniformity on the first frequency of the nonuniform microbeam and resonance domain, a comprehensive parametric study is done. This research is important in understanding the dynamic behavior of microbeams and effective designs using variable cross section in this type of microstructures.

Three-Dimensional Free Vibration Analysis of Thick Laminated Composite Circular Plates with Simply-Supported Boundary Conditions

This paper presents a numerical analysis of free vibration of thick laminated circular plates, with simply supported boundary conditions at the edges of plates. A finite element analysis is made for obtaining the first five natural frequencies using three-dimensional ‘SOLID185’ of ANSYS. The effects of different fibre orientation angle, thickness to radii ratios, on the free vibration responses are discussed in detail. The accuracy and numerical reliability have been verified by appropriate convergence studies and comparisons with the existing results from literature. These results should be a valuable alternative for validating new computational techniques in future, due to the accuracy, simplicity and versatility of the present analysis.

A New Quasi 3D Nonlocal Plate Theory for Vibration and Buckling of FGM Nanoplates

This paper presents the analyses of free vibration and buckling of functionally graded (FG) nanoplates in thermal environment by using a new quasi-3D nonlocal hyperbolic plate theory in which both shear and normal strains are included. The nonlocal equations of motion for the present problem are derived from Hamilton’s principle. For simply-supported boundary conditions, Navier’s approach is utilized to solve the motion equations. Eringen’s nonlocal theory is employed to capture the effect of the nonlocal parameter on natural frequency and buckling of the FGM nanoplates. Numerical results of the present formulation are compared with those predicted by other theories available in the open literature to explain the accuracy of the suggested theory that contains the shear deformation and thickness stretching. Other numerical examples are also presented to show the influences of the nonlocal coefficient, power law index and geometrical parameters on the vibration and buckling load of FGM nanoplates.

Shell elements with through-the-thickness variable kinematics for the analysis of laminated composite and sandwich structures

Abstract Several efforts have been made in the last years to improve the efficiency and the effectiveness of structural models for the analysis of laminated shell structures. Among the others, many recent and past works in the literature have been aimed at formulating theories of structures that maximize the accuracy of analysis meanwhile reducing the computational costs. In this paper, this objective is pursued by implementing advanced shell theories with through-the-thickness variable kinematic capabilities. By employing the Carrera Unified Formulation (CUF), the proposed shell model is obtained by expressing the displacement field as an arbitrary and, eventually, hierarchical expansion of the primary unknowns along the thickness. Thus, Equivalent-Single-Layer (ESL), Layer-Wise (LW) models as well as variable kinematic models which combine ESL and LW approaches within the shell thickness can be obtained in a straightforward and unified manner. After the unified shell model is formulated, the governing equations and the related finite element arrays are obtained by employing the principle of virtual work. A nine node finite element is implemented to approximate the solution field, and the Mixed Interpolation of Tensorial Components (MITC) method is used to contrast the membrane and shear locking phenomena. Some numerical examples are discussed, including three- and ten-layered cross-ply shells under bi-sinusoidal load and simply-supported boundary conditions, a multilayered spherical panel subjected to bi-sinusoidal load and a sandwich cylinder undergoing bi-sinusoidal pressure. Moreover, various thickness and radius-to-thickness ratios are considered. Whenever possible, the results are compared with those from the literature and from exact elasticity solutions. The analysis of the results clearly shows the enhanced capabilities of the present variable-kinematic shell element, which allows the analyst to opportunely reduce the computational costs and enhance the accuracy of the model only in those regions of the thickness domain where an accurate evaluation of the stress/strain field is needed.

Three-dimensional free vibration analysis of thick laminated circular plates

In this communication, a numerical analysis regarding free vibration of thick laminated circular plates, having free, clamped as well as simply-supported boundary conditions at outer edges of plates is presented. The employment of finite element is made in this communication. The finite element methodology operates on the basis of three-dimensional theory of elasticity and was employed to assess the natural frequencies for laminated circular plates of various thickness-to-outer radius ratios. The first five natural modes of flexural vibrations for different boundary conditions are presented in pictorial forms. Verification of the accuracy of the results was made using the necessary convergence analysis and checked using literature results.Keywords: Laminated Composite Circular Plate; Natural Frequency; Mode Shape; Finite Element Method

A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams

This paper investigates buckling characteristics of a curved functionally graded (FG) nanobeam based on nonlocal strain gradient elasticity theory accounting the stress for not only the nonlocal stress field but also the strain gradients stress field. The modeling of nanobeam is carried out via a higher order refined beam theory which captures shear deformation influences needless of any shear correction factor. Power-law model is adopted to describe continuous variation of material properties of curved FG nanobeam. The governing equations of nonlocal strain gradient curved FG nanobeam in the framework of refined hyperbolic beam model are obtained using Hamilton’s principle and solved implementing an analytical solution for simply-supported and clamped boundary conditions. To validate the present model, the results are compared with those of straight FG nanobeams by extending the radius of nanobeam to infinity. The effects of nonlocal parameter, length scale parameter, power-law exponent, boundary conditions and slenderness ratio on the buckling response of curved FG nanobeams are investigated.

Nonlinear modeling and analysis of electrically actuated viscoelastic microbeams based on the modified couple stress theory

A nonclassical nonlinear continuum model of electrically actuated viscoelastic microbeams is presented based on the modified couple stress theory to consider the microstructure effect in the framework of viscoelasticity. The nonlinear integral-differential governing equation and related boundary conditions of are derived based on the extended Hamilton's principle and Euler–Bernoulli hypothesis for viscoelastic microbeams with clamped-free, clamped-clamped, simply-supported boundary conditions. The proposed model accounts for system nonlinearities including the axial residual stress, geometric nonlinearity due to midplane stretching, electrical forcing with fringing effect. The behavior of the microbeam is simulated using generalized Maxwell viscoelastic model. A new generalized differential/integral quadrature method is developed to solve the resulting governing equation. The developed model is verified against elastic behavior and a favorable agreement is obtained. Efficiency of the developed model is demonstrated by analyzing the quasistatic pull-in phenomena of electrically actuated viscoelastic microbeams with different boundaries at various material length scale parameters and axial residual stresses in the framework of linear viscoelasticity.

Size-dependent behaviour of functionally graded microbeams using various shear deformation theories based on the modified couple stress theory

This study investigates the mechanical behaviours of functionally graded (FG) microbeams based on the modified couple stress theory. The material properties of these beams are varied through beam’s depth and calculated by using classical rule of mixture and Mori–Tanaka scheme. The displacement fields are presented by using a unified framework which covers various theories including classical beam theory, first-order beam theory, third-order beam theory, sinusoidal beam theory, and quasi-3D beam theories. The governing equations of bending, vibration and buckling problems are derived using the Hamilton’s principle and then solved by using Navier solutions with simply-supported boundary conditions. A number of numerical examples are conducted to show the validity and accuracy of the proposed approaches. Effects of Poisson’s ratio, material length scale parameter, power-law index, estimation methods of material properties and slenderness ratio on deflections, stresses, natural frequencies and critical buckling loads of FG microbeams are examined.