Classical Thin Plate Theory Research Articles

Band gaps and dynamics of locally resonant meta-plate with stiffness adjustable low frequency resonator

This paper presents the band gaps and dynamics of locally resonant meta-plate with stiffness micro-adjustable high static and low dynamic resonator, in which a Euler-buckled beam provides negative stiffness. The resonator is periodically installed on a thin plate through a threaded linkage. By changing the axial displacement of the buckled beam, the total stiffness of the resonator is adjusted. Through the static analysis, the critical axial compression required for the buckled beam to provide negative stiffness is discussed. With the help of classical thin plate theory, the dynamic equation of the system is obtained. The band gaps of the locally resonant meta-plate are calculated by employing the plane wave expansion method. Considering the simply supported boundary condition, dynamic responses of the finite size locally resonant plate are shown with the aid of numerical simulation. The effects of the axial compression of the buckled beam, the mass of resonator, the damping of oscillators and plates, and spring stiffness of the system on the band gap are analyzed in detail.

Free Vibration Analysis of Functionally Graded Annular Circular Plates Using Classical Thin Plate Theory Based on Physical Neutral Surface

Free Vibration Analysis of Functionally Graded Annular Circular Plates Using Classical Thin Plate Theory Based on Physical Neutral Surface

Modeling the deformation of an elastic element of a small spacecraft in its plane during temperature shock

A basic mathematical model of the deformation of a large elastic element of a small spacecraft in its plane is constructed. Deformations are caused by a temperature shock after a small spacecraft leaves the Earth’s shadow on the solar portion of the orbit. The model is used to conduct a computational experiment with the aim of assessing perturbations acting on a small spacecraft due to temperature shock. The temperature distribution during thermal shock is described by a one-dimensional model of thermal conductivity. The classical theory of thin plates is used to determine the deformations. The results of the estimation of disturbing factors are obtained as a result of a computational experiment for a model small spacecraft. These results indicate the need to compensate for the impact of temperature shock for small technological spacecraft. The data obtained can be used in the design of small space-craft for technological purposes.

Effect of temperature on sound transmission loss of laminated composite plate

In this investigation, by an analytical approach, the influence of several key parameters, especially the temperature on the sound isolation capacity of the symmetrically finite orthotropic laminated composite plate is studied. The plate is modeled with classic thin-plate theory and is assumed to be simply supported on all four sides. The incident acoustic pressure is modeled as a harmonic plane wave impinging on the plate at an arbitrary angle. The sound transmission loss is calculated from the ratio of incident to transmitted acoustic powers

Vibroacoustic behavior of a rectangular composite plate in thermal environment

This paper is a study of the vibroacoustic behavior of orthotropic laminated composite rectangular plate under a sound wave excitation in thermal environments. An improved analytical procedure has been developed that allows for an efficient solution of the finite composite plate sound transmission problem. A symmetrically orthotropic laminated composite plate is considered. The plate is modeled with classic thin-plate theory and is assumed to be clamped on all four sides. The incident acoustic pressure is modeled as a harmonic plane wave impinging on the plate at an arbitrary angle. The sound transmission loss is calculated from the ratio of incident to transmitted acoustic powers.

Wave Mode Control of Cantilever Slab Structure of T-Beam Bridge with Large Aspect Ratio

The cantilever plate structure in a T-beam bridge with a large aspect ratio will cause vibration under the influence of environmental disturbance and self-stress, resulting in fatigue damage of the plate structure. Wave control based on elastic wave theory is an effective method to suppress the vibration of the cantilever plate structure in a beam bridge. Based on the classical thin plate theory and the wave control method, the active vibration control of the T-shaped cantilever plate with a large aspect ratio in the beam bridge is studied in this paper. The wave mode control strategy of structural vibration is analyzed and studied, the controller is designed, the vibration mode function of the cantilever plate is established, and the control force/sensor feedback wave control is implemented for the structure. The dynamic response of the cantilever plate before and after applying wave control force is analyzed through numerical examples. The results show that the response of the structure is intense before control, but after wave control, the structure increases damping, absorbs the energy carried by the elastic wave in the structure, weakens the sharp response, and changes the natural frequency of the structure to a certain extent.

Analytical approach for vertical floor acceleration of regular RC frames under earthquake excitation

Vertical floor acceleration (VFA) analysis arises as an important issue for seismic design of floor-attached nonstructural components (e.g., ceilings). However, to date, estimating the VFA is hardly accessible in the literature. In this paper, an analytical method is proposed for the VFA of regular RC frames. Here “regular” means the frame has uniform spans along each principal direction, so the structure can be represented by a single-span analogy. Firstly, the fundamental frequency, modal mass, and modal participation factor of a clamped floor slab are analyzed based on the classic thin-plate theory. Then, the dynamic properties of the clamped floor slab are adjusted to account for the effect of the flexible beam restraints (non-clamped). According to the adjusted dynamic parameters, the beam-restrained slab is substituted by a single-degree-of-freedom system, and is assembled with the column-beam system to form the global dynamic model for the structure. Based on the assembled model, the analytical process for the VFA is presented. The feasibility of the model is validated using the finite element modelling (FEM) method. The vertical dynamic properties and the peak VFA of a single-, 3-, 6-, and 9-story frame are computed using the FEM and the proposed analytical approach. Results show that the VFA obtained from the analytical approach is close to the FEM outcomes. Furthermore, important features of VFA are revealed by the computation. It is shown that the VFA could be 10 times as large as the peak vertical ground acceleration. A larger VFA appears at the higher stories, yet the relationship between the VFA and the height is nonlinear because of the effects of the higher-order modes.

Vibrations of axially traveling plates partially coupled with a viscous, orthogonally flowing liquid

AbstractA semi‐analytical method is presented to study the free vibration of an axially traveling plate partially submersed in viscous and orthogonally flowing liquid. The method is based on the classical thin plate theory and the finite element theory. The Navier‐Stokes equation, Bernoulli's equation and the velocity potential function are used to express the liquid. The formulation of the dynamical liquid‐induced force is derived by kinematic boundary conditions of the plate‐liquid interfaces. The governing equations of the partially immersed traveling plate are obtained via the Hamilton's principle. Numerical calculations show that as the plate thickness increases, the liquid viscosity effect on natural frequencies of the plate becomes weaker and weaker. Besides, the magnitude of the liquid viscosity effect is found to be positively related to the distance between the plate and rigid wall, and the width of liquid domain. The plate is easier to reach the critical speed when the orthogonal liquid velocity is larger. Moreover, liquid viscosity plays important role on the vibrational behavior of the immersed traveling plate when the liquid velocity is relatively large. Several verifications have been conducted to test the effectiveness and accuracy of the developed method.

Fundamental frequency maximization of composite rectangular plates by sequential permutation search algorithm

Due to the classical thin plate theory based governing equations of composite rectangular plates for dynamic problems are linear homogeneous equations on bending stiffness coefficients, a general sequential permutation search (SPS) algorithm is presented to optimize stacking sequences of composite laminates by exploiting convexity of lamination parameters, sensitivity of bending stiffness, and bending-twisting coupling stiffness feature. Numerical examples are conducted for 8-layer composite rectangular plates with different boundaries to maximize the fundamental frequency evaluated by the Rayleigh–Ritz method, and the results are compared with those of layerwise optimization approach and genetic algorithm, demonstrating robustness and efficiency of SPS.

Semi-analytical static analysis of nonlocal strain gradient laminated composite nanoplates in hygrothermal environment

In this work, the bending behavior of nanoplates subjected to both sinusoidal and uniform loads in hygrothermal environment is investigated. The present plate theory is based on the classical laminated thin plate theory with strain gradient effect to take into account the nonlocality present in the nanostructures. The equilibrium equations have been carried out by using the principle of virtual works and a system of partial differential equations of the sixth order has been carried out, in contrast to the classical thin plate theory system of the fourth order. The solution has been obtained using a trigonometric expansion (e.g., Navier method) which is applicable to simply supported boundary conditions and limited lamination schemes. The solution is exact for sinusoidal loads; nevertheless, convergence has to be proved for other load types such as the uniform one. Both the effect of the hygrothermal loads and lamination schemes (cross-ply and angle-ply nanoplates) on the bending behavior of thin nanoplates are studied. Results are reported in dimensionless form and validity of the present methodology has been proven, when possible, by comparing the results to the ones from the literature (available only for cross-ply laminates). Novel applications are shown both for cross- and angle-ply laminated which can be considered for further developments in the same topic.

Analysis on free vibration and critical buckling load of a FGM porous rectangular plate

The properties of functionally gradient materials (FGM) are closely related to porosity, which has effect on FGM's elastic modulus, Poisson's ratio, density, etc. Based on the classical theory of thin plates and Hamilton principle, the mathematical model of free vibration and buckling of FGM porous rectangular plates with compression on four sides is established. Then the dimensionless form of the governing differential equation is also obtained. The dimensionless governing differential equation and its boundary conditions are transformed by differential transformation method (DTM). After iterative convergence, the dimensionless natural frequencies and critical buckling loads of the FGM porous rectangular plate are obtained. The problem is reduced to the free vibration of FGM rectangular plate with zero porosity and compared with its exact solution. It is found that DTM gives high accuracy result. The validity of the method is verified in solving the free vibration and buckling problems of the porous FGM rectangular plates with compression on four sides. The results show that the elastic modulus of FGM porous rectangular plate decreases with the increase of gradient index and porosity. Furthermore, the effects of gradient index and porosity on dimensionless natural frequencies and critical buckling loads are further analyzed under different boundary conditions with constant aspect ratio, and the effects of aspect ratio and load on dimensionless natural frequencies under different boundary conditions.

Hydroelastic analysis of Very Large Floating Structures in variable bathymetry regions by multi-modal expansions and FEM

A novel frequency domain numerical method for Very Large Floating Structure (VLFS) hydroelasticity is developed. The problem is formulated in the 2D ocean waveguide, featuring a realistic seabed bathymetry and the presence of inhomogeneous, elastic plates of varying thickness and negligible draft. An in vacuo modal expansion for the elastic body deflection, modelled as a structural plate, is employed to decouple the hydrodynamics from structural mechanics. The inhomogeneous plate is considered to undergo cylindrical bending, while depending on the structure slenderness and excitation wavelength the classical thin plate theory and Mindlin’s model, accounting for first order shear deformation effects are implemented. A weighted residual approach is employed to cast the formulated problems into a mixed weak form for which dimensionality reduction is sought. This is achieved by an enhanced vertical representation for the wave potential, able to accurately account for abrupt bathymetric changes, following Athanassoulis and Belibassakis (1999). The reduced two-field, weak problem is solved by means of the Finite Element Method (FEM). Finally, a series of comparisons are carried out against published results for a range of configurations.

Natural frequencies of plates of medium thickness

A comparison is made of the frequencies obtained by the classical theory of thin plates and the refined theory of plates of medium thickness. Considered a rectangular plate with articulation along the contour. To determine the fundamental frequency, the Bubnov-Galerkin variational method is used. It is shown that with a relative plate thickness h/a = 0.2, the calculation refinement is approximately 15-20% for the values of the fundamental frequency of natural vibrations of a rectangular plate.

Influence of the plate thickness on the frequency of its natural vibration

A comparison is made of the frequencies obtained by the classical theory of thin plates and the refined theory of plates of medium thickness. Considered a rectangular plate with articulation along the contour. To determine the fundamental frequency, the Bubnov-Galerkin variational method is used. It is shown that with a relative plate thickness h/a = 0.2, the calculation refinement is approximately 15-20% for the values of the fundamental frequency of natural vibrations of a rectangular plate.

On the Boundary Value Problems of Bending of Thin Elastic Plates With Surface Effects

Abstract We modify classical thin plate theory by incorporating surface effects via the Gurtin–Murdoch surface model to accommodate the mechanical behavior of thin plates at the nanoscale. We formulate the corresponding Dirichlet and Neumann boundary value problems and establish uniqueness results in the appropriate function spaces. In addition, we obtain the fundamental solution of the governing system of equations, which is central to further studies concerning well-posedness analysis of the model by the boundary integral equation method. Finally, we validate our model by comparison with results in the existing literature.

Sound Radiation of Orthogonal Antisymmetric Composite Laminates Embedded with Pre-Strained SMA Wires in Thermal Environment

The interest of this article lies in the sound radiation of shape memory alloy (SMA) composite laminates. Different from the traditional method of avoiding resonance sound radiation of composite laminates by means of structural parameter design, this paper explores the potential of adjusting the modal peak of the resonant acoustic radiation by using material characteristics of shape memory alloys (SMA), and provides a new idea for avoiding resonance sound radiation of composite laminates. For composite laminates embedded with pre-strained SMA, an innovation of vibration-acoustic modeling of SMA composite laminates considering pre-stain of SMA and thermal expansion force of graphite-epoxy resin is proposed. Based on the classical thin plate theory and Hamilton principle, the structural dynamic governing equation and the frequency equation of the laminates subjected to thermal environment are derived. The vibration sound radiation of composite laminates is calculated with Rayleigh integral. Effects of ambient temperature, pre-strain, SMA volume fraction, substrate ratio, and geometrical parameters on the sound radiation were analyzed. New laws of SMA material and pre-strain characteristics on sound radiation of composite laminates under temperature environment are revealed, which have theoretical and engineering functional significance for vibration and sound radiation control of SMA composite laminates.

Experimentally and Numerical Investigation on Behavior of Annular RC Slabs under Ring Loading

This research presents the experimentally measured displacement and strain at specified locations of the concrete of tested annular reinforced concrete slabs subjected to lateral load with three different ratios of inner to outer radii and simply supported at the outer circumference. Performed 3D model of annular RC slabs under the axisymmetric ring load applied, as close as at the inner edge, and investigated their stress-strain state in the elastic stage. This study contains different approaches based on classical thin-plate (CTP) theory and performed a 3D finite-element (FE) model to predict the fields of radial and circumferential stresses and deflection of the slab. Experimentally investigated the crack widths and crack pattern of the two groups of slabs-group A- radially reinforced and group M–orthogonally reinforced. added a correction factor to the CTP equations, which used to determine both radial and circumferential stresses. Also, investigated the appearance of the first cracks, deflection, failure mode, and the maximum value of the failure load for both cases of the reinforcement.

An improved model for unsymmetric plates

For an unsymmetric plate, a pure bending (plate curvature) inevitably causes a certain amount of stretching to the geometric mid-plane due to the stretching-bending coupling. According to classical thin plate theory, the geometric mid-plane is assumed to remain unstrained under pure bending. In this study, we demonstrate that the classical thin plate theory based on Kirchhoff–Love hypothesis is not accurate enough to model the structural behavior of unsymmetric plates. To overcome this limitation, we propose an improved theoretical model for unsymmetric plates through taking advantages of neutral plane strains in defining the geometric functions instead of mid-plane strains. Subsequently, the new governing equations and energy expression for the cylindrical bending of unsymmetric plates are derived using a modified constitutive equation. An alternative derivation approach based on the general stress equations is also presented for further validation. A direct consideration of stretching-bending coupling in the constitutive equation can significantly reduce the number of unknown parameters in establishing an accurate analytical model for unsymmetric plates. The pure bending problem of unsymmetric plates with small deformation is first studied, for which the improved model proposed in this paper is shown to capture the out-of-plane deformation of unsymmetric plates, accurately. However, many previous works have to take into account the nonlinear von Kármán strains even in the model of this small deformation problem. For the pure plate bending problem with large deformation, few unknown terms are needed for the improved model to give accurate results compared with the conventional mid-plane strain based method. Later, this improved model is applied to predict the stable configurations, the bifurcation/loss-of-bifurcation and the static snap-through phenomena of bistable cross-ply composite laminates. Furthermore, the application of this improved model for the accurate simulation of the nonlinear dynamics of unsymmetric plates is also demonstrated. Applying this proposed improved model, the model is reduced into an analogous one for isotropic or symmetric plates, therefore, the problem of unsymmetric plates can be solved readily and accurately.

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.

Theoretical and experimental research of thin-walled structures interacting with viscous fluid

The results of studying thin-walled plates and cylindrical shells interacting with the stationary or flowing viscous compressible fluid are presented. The numerical solution of the problem has been carried out using the finite element method. The motion of liquid medium is described by the system of linearized Navier-Stokes equations, the solution of which is sought for in the form of the acoustic approximation in terms of velocity perturbation potential. The relations obtained for liquid and the corresponding boundary conditions are transformed by the Bubnov-Galerkin method. The behavior of a thin-walled structure is described in the framework of the classical theory of thin plates. The mathematical formulation of the dynamic problem of an elastic body is based on the variational principle of virtual displacements. Stability estimation is based on the calculation and analysis of complex eigenvalues of a coupled system of equations. The influence of fluid viscosity and other parameters on the natural vibration frequencies and critical velocities responsible for the loss of structural stability has been analyzed. The natural frequencies and the corresponding decrements of harmonic vibrations of rectangular plates located in the air and on the free liquid surface have been studied with the help of developed experimental setup. It has been found out that the damping coefficient corresponding to one vibration mode (bending or torsional) grows with an increase in the number of nodal lines. It has been demonstrated that this relation can be violated during the interaction of the plates with the liquid.