Simply-supported Boundary Conditions Research Articles

Mechanics of shear and normal deformable doubly-curved delaminated sandwich shells with soft core

In this work the first- and second-order shear and normal deformation as well as the third-order shear deformation theories are applied to delaminated sandwich shells having constant radii of curvatures. The normal deformation of the first- and second-order models was captured by quadratic functions in the core, while it was ignored in the facesheets. The third-order model was created using cubic displacement functions in the terms of through-thickness coordinate in the core, the facesheets were still captured by second-order shear deformable shell theory. The governing equations were deduced through the virtual work principle, eventually three possible scenarios were considered: core-core, face-core and face-face delamination, respectively. To prove the applicability of the developed shell models several examples were solved including the former three scenarios combined with elliptic and hyperbolic doubly-curved sandwich shell geometries with different facesheet/core material pairs. The traditional simply-supported boundary conditions were assumed involving the Lévy-type formulation. Moreover, the state space method was applied to solve the examples. The mechanical fields calculated were compared to results obtained through spatial finite element models. The J-integral was also determined and separated into mode-II and mode-III components making it possible to perform the comparison to energy release rates computed by the virtual crack closure method. The results indicate that the proposed models capture the mechanical fields with high accuracy in the vicinity of points lying along the delamination front. The J-integrals are also very well predicted by the proposed shell models.

A comprehensive electro-magneto-elastic buckling and bending analyses of three-layered doubly curved nanoshell, based on nonlocal three-dimensional theory

The present paper studies the three-dimensional magneto-electro-elastic bending and buckling analyses of three-layered doubly curved nanoshells based on nonlocal elasticity theory. The kinematic relations are developed based on two-variable sinusoidal transverse shear and thickness deformation theory. The transverse deflection is decomposed into bending, shear and thickness deformation components by using a two-variable sinusoidal shear deformation theory. The nanoshell is subjected to in-plane mechanical, electrical and magnetic loads in which these forces are accounted as external works in constitutive relations for buckling analysis. This introduces a variable transverse displacement along the thickness direction. The principle of virtual work is used to derive the governing equations of bending and buckling. Numerical results are obtained based on Navier’s technique for simply-supported boundary conditions. A parametric analysis is performed to investigate the influence of the nonlocal parameter, the applied electric and magnetic potentials on the magneto-electro-elastic bending and buckling responses of doubly curved nanoshells.

Thermoelastic Damping and Frequency Shift in Kirchhoff Plate Resonators Based on Modified Couple Stress Theory With Dual-Phase-Lag Model

The present investigation deals with study of thermoelastic damping and frequency shift of Kirchhoff plate resonators by using generalized thermoelasticity theory of dual-phase-lag model. The basic equations of motion and heat conduction equation are written with the help of Kirchhoff-Love plate theory and dual phase lag model. The analytical expressions for thermoelastic damping and frequency shift of modified couple stress dual-phase-lag thermoelastic plate have been obtained. A computer algorithm has been constructed to obtain the numerical results. Influences of modified couple stress dual-phase-lag thermoelastic plate, dual- phase-lag thermoelastic plate and Lord-Shulman (L-S, 1967) thermoelastic plate with few vibration modes on the thermoelastic damping and frequency shift are examined. The thermoelastic damping and frequency shift with varying values of length and thickness are shown graphically for clamped-clamped and simply-supported boundary conditions. It is observed from the results that the damping factor and frequency shift have noticed larger value in the presence of couple stress for varying values of length but opposite effect are shown for varying values of thickness in case of both vibration modes and boundary conditions.

A comparative study on the vibration of functionally graded Magneto-Electro-Elastic rectangular plate rested on a Pasternak foundation based on exponential and first-order shear deformation theories

In this paper, the in-plane and out-of-plane free vibrations of the functionally graded magneto-electro-elastic (FG-MEE) rectangular plate on a pasternak foundation were evaluated based on exponential shear deformation theory (ESDT) and first order shear deformation theory (FSDT). The material properties of FG-MEE varied along the thickness according to a power-law distribution model. It was assumed that the FG-MEE plate is subjected to initial external electrical and magnetic potentials; mreover, simply-supported boundary conditions were considered for all the edges of the FG-MEE plate. Firstly, the corresponding partial differential equations (PDEs) were derived using Hamilton’s principle, then, the natural frequencies were determined by solving the obtained equations through Navier’s approach according to the assumed boundary conditions. The results revealed that the natural frequencies of the plate decrease/increase with the increase of the electric/magnetic potentials. Moreover, the results showed a 0.03%, difference between the natural frequencies of the plate with a thickness-to-length ratio of 0.1 based on FSDT and ESDT; when the aspect ratio was increased to 0.2 and 0.3 this difference rose to 0.2% and 0.5%, respectively.

Time domain aero-thermo-elastic instability of two-dimensional non-linear curved panels with the effect of in-plane load considered

This study presents aero-thermo-elastic Instability of two-dimensional Non-linear Curved Panels. Aero-thermo-elasticity plays an important role in the design and optimization of supersonic aircrafts. Furthermore, the transient and nonlinear effects of the thermal and aerodynamic environment encompassing a curved surface cannot be ignored. Accordingly, a homogenous curved plate with a high length-to-width ratio and simply-supported boundary conditions is assumed. The effect of large deflection is included in the equations through von Karman non-linear strain–displacement relations. The thermal load is assumed to be a steady-state temperature non-uniform distribution. Structural properties such as modulus of elasticity and thermal expansion coefficient are assumed to be temperature-dependent. The novelty is incorporating first- and third-order piston theory for the non-linear curved panel flutter analysis under the effects of in-plane and thermal loads. Hamilton’s principle is used and partial differential equations are derived. The semi-analytical weighted residual method for the nonlinear curved panel is utilized. The fourth- and fifth-order Runge–Kutta iterative method are deployed to obtain the non-linear aero-thermo-mechanical deflections. Non-linear frequency analysis of cambered panel with the combined effects of aerodynamics, thermal and in-plane loads is investigated for the first time. The increase in panel curvature leads to a complicated behavior in the non-linear structural frequency variations. With increasing in-plane compressive load, complicated oscillating behavior is observed. More critical instability boundary for cambered panel is detected through the use of third-order piston theory. In addition, with an increase in panel curvature from 0 to 3, the panel displacement increases and for higher camber ratio, it decreases.

Nonlinear vibration of axially moving simply-supported circular cylindrical shell

This paper investigates the nonlinear free and forced vibration of axially moving simply-supported thin circular cylindrical shells in the sub-harmonic region, considering the effect of viscous structure damping. Motion equations of axially moving circular cylindrical shells in cylindrical coordinates are derived with the aid of the Hamilton principle and employing the Donnell–Mushtari nonlinear shells’ theory. Three nonhomogeneous nonlinear partial differential equilibrium equations concerning displacements across the three directions of cylindrical coordinate are reduced to two equations as a result of using the definition of the airy function. The particular and private solution of airy stress function is obtained utilizing the assumption of the form of the displacement in the radial direction based on the simply-supported boundary conditions and combination of axisymmetric and asymmetric driven and companion modes. By gaining the airy function, the third motion equation is discretized using the Galerkin method into the set of coupled nonlinear nonhomogeneous ordinary differential equations. The perturbation method and Runge–Kutta 4th order are employed as a semi-analytical and a numerical solution method, which in practice leads to the prediction of the nonlinear frequencies and amplitudes of various modes of vibration at different velocities and external excitation fluctuation domain and frequency. The bifurcation analysis of the velocity and external excitation parameters in sub-harmonic regions shows the changes in the instability condition of the system. It causes the appearance of the pitchfork bifurcation and the Neimark–Sacker bifurcation points. The accuracy of both, the direct normal form and numerical method results are compared against each other and validated in the particular case in the absence of the velocity with available data. The system would be more stable at a higher speed near 1:1 external resonance due to the activation of more asymmetric vibration modes and the growth of the nonlinear softening characteristic in the absence of companion vibration modes.

On static bending of multilayered carbon nanotube-reinforced composite plates

In this paper, the bending behavior of single-walled carbon nanotube-reinforced composite (CNTRC) laminated plates is studied using various shear deformation plate theories. Several types of reinforcement material distributions, a uniform distribution (UD) and three functionally graded distributions (FG), are inspected. A generalized higher-order deformation plate theory is utilized to derive the field equations of the CNTRC laminated plates where an analytical technique based on Navier\'s series is utilized to solve the static problem for simply-supported boundary conditions. A detailed numerical analysis is carried out to examine the influence of carbon nanotube volume fraction, laminated composite structure, side-to-thickness, and aspect ratios on stresses and deflection of the CNTRC laminated plates.

Modal identification experimental investigations and calculation method for biaxial tensioned membranes

Abstract The utilization of membrane structures in space engineering has attracted considerable attention in recent decades due to ultra-light weight characteristics. The measurement and calibration of dynamic performance on ground is challenging due to air effects and equipment limitation. In this paper, a novel vacuum-level vibration-testing device is developed to realize an environment of adjustable air pressure. An improved membrane tension system is utilized to guarantee simply-supported boundary conditions and a uniform stress field, which is more consistent with theoretical and simulation requests. The modal identification method is modified based on only four measuring points because of limited space of the vacuum tank, which is a combination of FFT (Fast Fourier Transform) and SSI (Stochastic Subspace Identification) methods. The effects of non-uniform stress field and air pressure on membrane modal are investigated based on the proposed testing device and method. The effect of the non-uniform stress field on membrane frequency is analyzed by theoretical, numerical and experimental methods, and the results show that the non-uniform stress field resulting from uneven tensions has little effect on the frequency of biaxial tensioned membranes. Meanwhile, an empirical formula is proposed to predict the first three structural frequency of a specific biaxial tensioned membrane under different air pressures.

Non-local approach to free vibrations and buckling problems for cylindrical nano-structures

Non-local higher order free vibration and buckling analysis of a cylindrical nano-shell is studied in this paper based on non-local 3D elasticity theory and third order shear deformation theory proposed by Prof. J.N. Reddy. One can conclude that combination of important theories such as Reddy's shear deformation theory and non-local Tiersten-Mindlin’s theory to a more complicated structure such as cylindrical nano-structures results in an important and novel work in context of structural mechanics. The governing equations of non-local elastic bending are reduced to one linear homogeneous differential equation for simply-supported boundary conditions. In the strain or stress non-local problems different physical formulations have different effects on the searched values, understood in the sense of weakening or enhancement the buckling/vibration results in comparison with the classical global approach.

Nonlocal dynamic stability analysis of a Timoshenko nanobeam subjected to a sequence of moving nanoparticles considering surface effects

In this study, the dynamic stability of a Timoshenko nanobeam under the intermittent movement of nanoparticles is investigated considering surface effects. Simply-supported boundary conditions are assumed for a rectangular cross-section beam. The elastic foundation is modeled as a Pasternak elastic foundation. Also, the role of the nanoparticle inertia is considered. The nanoparticles move over the beam continuously with a constant velocity, and the friction between particles, and the beam is ignored. Governing equations are derived by Hamilton's principle in conjunction with nonlocal and Gurtin-Murdoch theories. Dynamic stability analysis of the nanobeam under induced excitation of moving nanoparticles is carried out, and the stability and instability regions are derived by incremental harmonic balance method (IHBM). The stability of the system is described in the plane (mass-velocity). A detailed study is conducted to examine the effects of various parameters such as the small scale parameter, surface constants, Pasternak foundation coefficients, and nanoparticles inertia on the dynamic instability region (DIR) of the nanobeam. The results show that considering surface theory has a negligible effect on the DIR of the nanobeam under the intermittent passage of nanoparticles. Pasternak and Winkler spring constants increase the stiffness and stability of the nanobeam. Also, considering moving mass inertia and small scale parameter shifts the DIR to the lower frequency zone.

A procedure of the method of reverberation ray matrix for the buckling analysis of a thin multi-span plate

A procedure of the method of reverberation ray matrix (MRRM) is developed to perform the buckling analysis of thin multi-span rectangular plates having internal line supports or stiffeners. A computation algorithm for the reverberation ray matrix in the MRRM is derived to determine the buckling loading. Specifically, the analytical solutions are presented for the buckling of the structure having two opposite simply-supported or clamped-supported edges with spans, while the constraint condition of two remaining edges may be in any combination of free, simply-supported, and clamped boundary conditions. Furthermore, based on the analysis of matrices relating to the unknown coefficients in the solution form for the deflection in terms of buckling modal functions, some recursive equations (REs) for the MRRM are introduced to generate a reduced reverberation ray matrix with unchanged dimension when the number of spans increases, which promotes the computation efficiency. Several numerical examples are given, and the present results are compared with the known solutions to illustrate the validity and accurateness of the MRRM for the buckling analysis.

Vibration suppression of advanced plates embedded magnetostrictive layers via various theories

Vibration suppression analysis of a laminated composite plate embedded magnetostrictive layers is presented with/without the transverse shear and normal strains effects are taken into account in this study. For purpose of vibration suppression, the velocity feedback control with constant gain distributed is applied. The formulation of problem is written to give five theories are Euler-Bernoulli’s classical plate theory, the Timoshenko’s first-order and Reddy’s third-order shear deformation plate theories, simple and refined sinusoidal shear deformation plate theories and other theories. The governing equations of motion are obtained using the Hamilton’s principle. Navier’s method is applied to discuss the solution of vibration problem at the simply-supported boundary conditions. Some effects are extensively studied and discussed such the impact of material properties, modes, thickness and number of the magnetostrictive layers, lamination schemes, magnitude of the feedback coefficient and location of the magnetostrictive layers on vibration suppression of the system. Numerical results are reported and illustrated, and various conclusions are formulated.

A comparison of the Bolotin and incremental harmonic balance methods in the dynamic stability analysis of an Euler–Bernoulli nanobeam based on the nonlocal strain gradient theory and surface effects

This study addresses the dynamic stability of an Euler–Bernoulli nanobeam under time-dependent axial loading based on the nonlocal strain gradient theory (NSGT) and considering the surface stress effects. The studied nanobeam cross-section was rectangular, and simply-supported boundary conditions were assumed. Moreover, a uniform thermal gradient was applied to the nanobeam. The elastic medium was modeled based on the Pasternak theory. The strain–displacement relations were derived using the Von Kármán equations. The governing equations were obtained by the energy method and applying the Hamilton's principle. Furthermore, the Bolotin and Incremental Harmonic Balance (IHB) methods were used to solve the differential equations. This study investigates the impact of such parameters as the small-scale parameter, the material length scale, surface effects, elastic medium parameters, temperature variations, geometry, and the static loading factor on the Dynamic Instability Region (DIR). The results are suggestive of the shift of the DIR to lower frequency zone by increasing the small-scale Eringen's nonlocal theory parameter, whereas an increase in the material length scale from the strain gradient theory moves the region to higher frequencies. In case the said parameters are equal, the result conforms to the classical beam theory. In addition, assuming a Pasternak medium and taking into account the effects of surface stress (Young's modulus and the residual stress of the surface) shifts the DIR to higher frequencies, whereas applying a compressive static load moves the region to lower frequencies. Moreover, depending on the thermal expansion coefficient of the medium, temperature variations can also displace the DIR.

Three-dimensional thermo-electro-mechanical buckling analysis of functionally graded piezoelectric micro/nano-shells based on modified couple stress theory considering thickness stretching effect

The present work uses a new three-dimensional framework for thermo-electro-mechanical buckling analysis of functionally graded piezoelectric cylindrical nano/micro-shells subjected to axial mechanical compression, an external applied voltage and uniform temperature rising. To account thickness stretching effect, three-dimensional shear and normal deformation theory is employed. Size dependency is modeled based on modified couple stress theory. Principle of virtual work is used to derive governing equations. The solution procedure is developed based on Navier’s technique for simply-supported boundary conditions. The importance of the present new framework is justified through comparison between present results and corresponding previous lower-order analysis. The full numerical results are presented in terms of significant parameters of the problem such as some dimensionless geometric parameters, temperature change, dimensionless length scale parameter and applied voltages.

Size dependent free vibration analysis of functionally graded piezoelectric micro/nano shell based on modified couple stress theory with considering thickness stretching effect

Higher-order shear and normal deformation theory is used in this paper to account thickness stretching effect for free vibration analysis of the cylindrical micro/nano shell subjected to an applied voltage and uniform temperature rising. Size dependency is included in governing equations based on the modified couple stress theory. Hamilton’s principle is used to derive governing equations of the cylindrical micro/nano shell. Solution procedure is developed using Navier technique for simply-supported boundary conditions. The numerical results are presented to investigate the effect of significant parameters such as some dimensionless geometric parameters, material properties, applied voltages and temperature rising on the free vibration responses.

Natural frequency, damping and forced responses of sandwich plates with viscoelastic core and graphene nanoplatelets reinforced face sheets

The free and forced vibration analysis of a sandwich plate with the viscoelastic core and face layers reinforced functionally with multilayered graphene nanoplatelets is presented. Different graphene nanoplatelet distributions are considered through the thickness, and the effective properties of the graphene reinforced nanocomposite are obtained by the rule of mixture. The equations of motion are extracted using Hamilton’s principle and assuming the classical thin plate theory for face layers and the first-order shear deformation theory for the thick viscoelastic core. Assuming the simply-supported boundary condition for all edges, the displacement components are proposed by Fourier series and the complex eigenvalue problem is solved to obtain the natural frequencies as well as the loss factors. The results are validated with available investigations, and effects of some important parameters on the free and forced responses of the sandwich plate are studied.

Dynamic snap-through of shallow spherical shells subjected to thermal shock

Present study deals with the dynamic snap-through phenomenon of an isotropic shallow spherical cap under transient type of thermal loading. The inner surface of the shell is subjected to sudden temperature elevation, whereas the outer surface is kept at reference temperature. Transient thermal shock is applied uniformly and shell thickness is assumed to be thin enough. Therefore, transient heat conduction equation is analytically solved across the thickness direction (one dimensional). Immovable simply-supported boundary conditions are assumed for the shell. Since the boundary conditions and the applied thermal shock are axisymmetric, the governing motion equations of the shell are restricted to the case of axisymmetric. First order shear deformation theory of shells is utilized to approximate the displacement field. The von Kármán type of geometrical non-linearity is used in strain-displacement relations. With the establishment of the associated Hamilton principle, differential equations of motion are extracted. Motion equations are discretized within the shell domain by means of the harmonic differential quadrature method (HDQM). The Newmark time marching scheme based on the constant average acceleration method is applied to turn the motion differential equations into a system of algebraic equations at each time step. Highly coupled equations are solved implementing the well-known Newton-Raphson iterative technique. By means of the Budiansky criterion, critical thermal shock parameters are distinguished. Critical dynamic snap-through temperatures are also verified using the phase-plane presentation. Comparison study is performed to validate the formulation and solution method of the present research with simple cases. Also, parametric investigations are performed to demonstrate the geometrical effects on the dynamic critical buckling temperatures of the shallow spherical shells subjected to thermal shock.

Free and forced vibration of sandwich plates with electrorheological core and functionally graded face layers

The free and forced vibration of a sandwich plate with functionally graded face layers and electrorheological fluid core is presented in this article. Assuming classical thin plate theory for face sheets and first order shear theory for the core layer, the equations of motions are derived using Hamilton’s principle. The simply-supported boundary condition is considered for all edges and the generalized Fourier series are employed to represent the displacements of the sandwich plate. The natural frequencies and loss factors are obtained by solving the complex eigenvalue problem. The results of free and forced responses are compared and validated with the previous articles and effects of electric field, geometric aspect ratio, functionally graded component ratios and core layer thickness on the dynamic behavior of the sandwich plate are studied.

Thermal Buckling Analysis of Circular Bilayer Graphene Sheets Resting on an Elastic Matrix Based on Nonlocal Continuum Mechanics

In this article, the thermal buckling behavior of orthotropic circular bilayer graphene sheets embedded in the Winkler–Pasternak elastic medium is scrutinized. Using the nonlocal elasticity theory, the bilayer graphene sheets are modeled as a nonlocal double–layered plate that contains small scale effects and van der Waals (vdW) interaction forces. The vdW interaction forces between the layers are simulated as a set of linear springs using the Lennard–Jones potential model. Using the principle of virtual work, the set of equilibrium equations are obtained based on the first-order shear deformation theory (FSDT) and nonlocal differential constitutive relation of Eringen. Differential quadrature method (DQM) is employed to solve the governing equations for simply-supported and clamped boundary conditions. Finally, the effects of the small scale parameter, vdW forces, aspect ratio, elastic foundation, and boundary conditions are considered in detail.

Static analysis of functionally graded anisotropic nanoplates using nonlocal strain gradient theory

We study functionally graded nanoplates made of (hexagonal) beryllium crystals. Therefore, a five-variable refined plate theory in conjunction with the nonlocal strain gradient theory is developed. From the best knowledge of authors, it is the first time that mentioned theories are developed for hexagonal materials. The governing equations are obtained using Hamilton’s principle where an analytical technique based on Navier’s series is utilized to solve the static problem for simply-supported boundary conditions. To simplify the equations, the number of unknowns and governing equations are reduced by dividing the transverse displacement into bending, shear, and thickness stretching parts. The obtained results of the displacements are compared with those predicted by other 2D and quasi-3D plate theories available in the literature. We show that the bending characteristics of FG anisotropic nanoplates are influenced by the nonlocal parameter, strain gradient parameter, length-to-thickness ratio, length-to-width ratio and exponential factor. This study offers benchmark results for the static analysis of functionally graded anisotropic nanoplates which could for instance be used for other computational approaches. We also quantify the accuracy of replacing an anisotropic model with an isotropic one and show that the differences in the stresses can grow up to 10% in some conditions.