A two variable refined plate theory for the bending analysis of functionally graded plates

Bending analysis of functionally graded plates using the two variable refined plate theory is presented in this paper.The number of unknown functions involved is reduced to merely four,as against five in other shear deformation theories. The variationally consistent theory presented here has, in many respects,strong similarity to the classical plate theory. It does not require shear correction factors,and gives rise to such transverse shear stress variation that the transverse shear stresses vary parabolically across the thickness and satisfy shear stress free surface conditions.Material properties of the plate are assumed to be graded in the thickness direction with their distributions following a simple power-law in terms of the volume fractions of the constituents.Governing equations are derived from the principle of virtual displacements, and a closed-form solution is found for a simply supported rectangular plate subjected to sinusoidal loading by using the Navier method.Numerical results obtained by the present theory are compared with available solutions,from which it can be concluded that the proposed theory is accurate and simple in analyzing the static bending behavior of functionally graded plates.

Control Mechanism of Surface Subsidence and Overburden Movement in Backfilling Mining based on Laminated Plate Theory

The backfilling mining technology is a type of high-efficiency coal mining technology that is used to address the environmental issues caused by the caving mining technology.In this paper,the mechanical model of symmetrical laminated plate representing the overburden movement caused by the backfilling mining technology is established,and the governing differential equation of the motion of the overburden is derived.The boundary conditions of the mechanical model are put forward,and the analytical solution of the overburden movement and surface subsidence is obtained.The numerical model of the overburden movement and surface subsidence,under mining with backfilling,is established by means of the FLAC3D numerical software,which aims to systematically study the influence of backfilling compactness,mining thickness,and mining depth on the overburden movement and surface subsidence in backfilling mining.When the compactnessηis less than 70%,the overburden movement and surface subsidence is greater,while whenηis greater than 70%,the overburden movement and surface subsidence is reduced significantly.On this basis,the control mechanism of surface subsidence and overburden movement in backfilling mining is obtained.The suitable backfilling compactness is the key to controlling surface subsidence and overburden movement in backfilling mining.

KRMN-TYPE EQUATIONS FOR A HIGHER-ORDER SHEARDEFORMATION PLATE THEORY AND ITS USE IN THE THERMAL POSTBUCKLING ANALYSIS

arman-type nonlinear large deflection equations are derived occnrding to theReddy's higher-order shear deformation plate theory and used in the thermal postbuckling analysis The effects of initial geometric imperfections of the plate areincluded in the present study which also includes th thermal effects.Simply supported,symmetric cross-ply laminated plates subjected to uniform or nomuniform parabolictemperature distribution are considered. The analysis uses a mixed GalerkinGolerkinperlurbation technique to determine thermal buckling louds and postbucklingequilibrium paths.The effects played by transverse shear deformation plate aspeclraio, total number of plies thermal load ratio and initial geometric imperfections arealso studied.

Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

Based on the nonlocal theory and Mindlin plate theory,the governing equations(i.e.,a system of partial differential equations(PDEs)for bending problem)of magnetoelectroelastic(MEE)nanoplates resting on the Pasternak elastic foundation are first derived by the variational principle.The polynomial particular solutions corresponding to the established model are then obtained and further employed as basis functions with the method of particular solutions(MPS)to solve the governing equations numerically.It is confirmed that for the present bending model,the new solution strategy possesses more general applicability and superior flexibility in the selection of collocation points.The effects of different boundary conditions,applied loads,and geometrical shapes on the bending properties of MEE nanoplates are evaluated by using the developed method.Some important conclusions are drawn,which should be helpful for the design and applications of electromagnetic nanoplate structures.

Temperature distribution effects on buckling behavior of smart heterogeneous nanosize plates based on nonlocal four-variable refined plate theory

This work presents a theoretical study for thermo-mechanical buckling of size-dependent magneto-electro-thermo-elastic func-tionally graded(METE-FG)nanoplates in thermal environments based on a refined trigonometric plate theory.Temperature field has uniform,linear,and nonlinear distributions across the thick-ness.Nonlinear thermal loadings are considered as heat conduc-tion(HC)and sinusoidal temperature rise(STR).A power law function is applied to govern the gradation of material properties through the nanoplate thickness.Considering coupling impacts between magneto,electro,thermo-mechanical loadings,the equa-tions of motion,and distribution of magneto-electrical field across the thickness direction of the METE-FG nanoplate are derived.The exact solutions for critical buckling temperatures of METE-FG nanoplates are introduced implementing Navier’s method.Moreover,the accuracy of the present formulation is examined by comparing the obtained results with published ones.Furthermore,the effects played by the magneto-electrical field,various temperature rises,nonlocality,power law index,side-to-thickness ratio,and aspect ratio on the critical buckling tempera-ture response are all investigated and reported.

New analysis model for rotor-bearing systems based on plate theory

The purpose of this work is to develop a new analysis model for angular-contact,ball-bearing systems on the basis of plate theory instead of commonly known approaches that utilize spring elements.Axial and radial stiffness on an annular plate are developed based on plate,Timoshenko beam,and plasticity theories.The model is developed using theoretical and inductive methods and validated through a numerical simulation with the finite element method.The new analysis model is suitable for static and modal analyses of rotor-bearing systems.Numerical examples are presented to reveal the effectiveness and applicability of the proposed approach.

EQUIVALENT BOUNDARY INTEGRAL EQUATIONS WITH INDIRECT UNKNOWNS FOR THIN ELASTIC PLATE BENDING THEORY

Equivalent Boundary Integral Equations (EBIE) with indirect unknowns for thin elastic plate bending theory, which is equivalent to the original boundary value problem, is established rigorously by mathematical technique of non-analytic continuation and is fully proved by means of the variational principle. The previous three kinds of boundary integral equations with indirect unknowns are discussed thoroughly and it is shown that all previous results are not EBIE.

THE FIRST-ORDER APPROXIMATION OF NON-KIRCHHOFF-LOVE THEORY FOR ELASTIC CIRCULAR PLATE WITH FIXED BOUNDARY UNDER UNIFORM SURFACE LOADING (Ⅲ)──NUMERICAL RESULTS

ased upon the differential equations and their related boundary conditions givenin the previous papers[1, 2], using a global interpolation method, this paper presents anumerical solution to the axisymmetric bending problem of non-Kirchhoff-Love theoryfor circular plate with fixed boundary under uniform surface loading. All the numericalresults obtained in this paper are compared with that of Kirchhoff-Love classicaltheory[3] and E. Reissner's modified theory[4]

Preliminary Report on the Theory of Elastic Circular Plate with no Kirchhoff-Love Assumptions

In this paper, the theory of elastic circular plate with no classical Kirchhoff-Love assumptions is established on the basis of a previous paper. In this theory, no classical Kirchhoff-Love assumptions are pre-assumed and the axial symmetrical analytic solution of fixed circular plate under the action of uniform pressure is obtained. Comparison of this solution and the known classical solution shows that this new solution agrees better than classical solution with the experiment measurement.This gives also the quantitative effect of the thickness on the deflection of circular plate with moderate thickness.

ANALYTICAL RELATIONS BETWEEN EIGENVALUES OF CIRCULAR PLATE BASED ON VARIOUS PLATE THEORIES

Based on the mathematical similarity of the axisymmetric eigenvalue problems of a circular plate between the classical plate theory（CPT）, the first-order shear deformation plate theory（FPT） and the Reddy＇s third-order shear deformation plate theory（RPT）, analytical relations between the eigenvalues of circular plate based on various plate theories are investigated. In the present paper, the eigenvalue problem is transformed to solve an algebra equation. Analytical relationships that are expressed explicitly between various theories are presented. Therefore, from these relationships one can easily obtain the exact RPT and FPT solutions of critical buckling load and natural frequency for a circular plate with CPT solutions. The relationships are useful for engineering application, and can be used to check the validity, convergence and accuracy of numerical results for the eigenvalue problem of plates.

Comparison of Linear Level I Green-Naghdi Theory with Linear Wave Theory for Prediction of Hydroelastic Responses of VLFS

Very Large Floating Structures (VLFS) have drawn considerable attention recently due to their potential significance in the exploitation of ocean resources and in the utilization of ocean space. Efficient and accurate estimation of their hydroelastic responses to waves is very important for the design. Recently, an efficient numerical algorithm was developed by Ertekin and Kim (1999). However, in their analysis, the linear Level I Green-Naghdi (GN) theory is employed to describe fluid dynamics instead of the conventional linear wave (LW) theory of finite water depth. They claimed that this linear level I GN theory provided better predictions of the hydroelastic responses of VLFS than the linear wave theory. In this paper, a detailed derivation is given in the conventional linear wave theory framework with the same quantity as used in the linear level I GN theory framework. This allows a critical comparison between the linear wave theory and the linear level I GN theory. It is found that the linear level I GN theory can be regarded as an approximation to the linear wave theory of finite water depth. The consequences of the differences between these two theories in the predicted hydroelastic responses are studied quantitatively. And it is found that the linear level I GN theory is not superior to the linear wave theory. Finally, various factors affecting the hydroelastic response of VLFS are studied with the implemented algorithm.

Hydroelastic Analysis of Very Large Floating Structures Using Plate Green Functions

Great attention has been paid to the development of very large floating structures. Owing to their extreme large size and great flexibility, the coupling between the structural deformation and fluid motion is significant. This is a typical problem of hydroelasticity. Efficient and accurate estimation of the hydroelastic response of very large floating structures in waves is very important for design. In this paper, the plate Green function and fluid Green function are combined to analyze the hydroelastic response of very large floating structures. The plate Green function here is a new one proposed by the authors and it satisfies all boundary conditions for free-free rectangular plates on elastic foundations. The results are compared with some experimental data. It is shown that the method proposed in this paper is efficient and accurate. Finally, various factors affecting the hydroelastic response of very large floating structures are also studied.

A refined finite element method for bending analysis of laminated plates integrated with piezoelectric fiber-reinforced composite actuators

This research presents a finite element formulation based on four-variable refined plate theory for bending analysis of cross-ply and angle-ply laminated composite plates integrated with a piezoelectric fiber-reinforced composite actuator under electromechanical loading. The four-variable refined plate theory is a simple and efficient higher-order shear deformation theory, which predicts parabolic variation of transverse shear stresses across the plate thickness and satisfies zero traction conditions on the plate free surfaces. The weak form of governing equations is derived using the principle of minimum potential energy, and a 4-node non-conforming rectangular plate element with 8 degrees of freedom per node is introduced for discretizing the domain. Several benchmark problems are solved by the developed MATLAB code and the obtained results are compared with those from exact and other numerical solutions, showing good agreement.

Isogeometric nonlocal strain gradient quasi-three-dimensional plate model for thermal postbuckling of porous functionally graded microplates with central cutout with different shapes

This study presents the size-dependent nonlinear thermal postbuckling characteristics of a porous functionally graded material(PFGM) microplate with a central cutout with various shapes using isogeometric numerical technique incorporating nonuniform rational B-splines. To construct the proposed non-classical plate model, the nonlocal strain gradient continuum elasticity is adopted on the basis of a hybrid quasithree-dimensional(3D) plate theory under through-thickness deformation conditions by only four variables. By taking a refined power-law function into account in conjunction with the Touloukian scheme, the temperature-porosity-dependent material properties are extracted. With the aid of the assembled isogeometric-based finite element formulations,nonlocal strain gradient thermal postbuckling curves are acquired for various boundary conditions as well as geometrical and material parameters. It is portrayed that for both size dependency types, by going deeper in the thermal postbuckling domain, gaps among equilibrium curves associated with various small scale parameter values get lower, which indicates that the pronounce of size effects reduces by going deeper in the thermal postbuckling regime. Moreover, we observe that the central cutout effect on the temperature rise associated with the thermal postbuckling behavior in the presence of the effect of strain gradient size and absence of nonlocality is stronger compared with the case including nonlocality in absence of the strain gradient small scale effect.

ANALYTICAL VARIATIONAL METHOD OF SOLUTION FOR STRESS CONCENTRATION FACTORS OF FINITE PLATES WITH A PIN HOLE

First of all, Laurent series expansions of stress and displacement fields satisfying all the governing equations of plane problems in theory of elasticity are derived. Next, the variational method is applied to satisfy the boundary conditions of a plate with a pin hole. Then, the coefficients in Laurent series and the stress concentration factor can be determined. Finally, the convergency tests and systematical results are given for engineering applications.

A new approach for free vibration analysis of a system of elastically interconnected similar rectangular plates

A new procedure is proposed to ease the analyses of the free vibration of an elastically connected identical plates system with respect to Kirchhoff plate theory.A structure of n parallel,elastically connected rectangular plates is of concern,whereby the motion is explained by a set of n coupled partial differential equations.The method involves a new change in variables to uncouple equations and form an equal system of n decoupled plates,while each is assumed to be elastically connected to the ground.The differential quadrature method is adopted to solve the decoupled equations.To unravel the original system,the inverse transform is applied.Decoupling the equations enables one to solve them based on the solution methods available for a single plate system.This also diminishes the computational costs of such problems.By considering different boundary conditions,a case study is run to present the method and to validate the results with its counterparts,for which excellent agreement is observed.Assessing the influence of dimensionless thickness,aspect ratio,and stiffness coefficients on the frequencies reveals the different effects of them at the low order of dimensionless natural frequencies in comparison with high orders and for different boundary conditions.

Time-varying nonlinear dynamics of a deploying piezoelectric laminated composite plate under aerodynamic force

Using Reddy’s high-order shear theory for laminated plates and Hamilton’s principle, a nonlinear partial differential equation for the dynamics of a deploying cantilevered piezoelectric laminated composite plate, under the combined action of aerodynamic load and piezoelectric excitation, is introduced. Two-degree of freedom(DOF)nonlinear dynamic models for the time-varying coefficients describing the transverse vibration of the deploying laminate under the combined actions of a first-order aerodynamic force and piezoelectric excitation were obtained by selecting a suitable time-dependent modal function satisfying the displacement boundary conditions and applying second-order discretization using the Galerkin method. Using a numerical method, the time history curves of the deploying laminate were obtained, and its nonlinear dynamic characteristics,including extension speed and different piezoelectric excitations, were studied. The results suggest that the piezoelectric excitation has a clear effect on the change of the nonlinear dynamic characteristics of such piezoelectric laminated composite plates. The nonlinear vibration of the deploying cantilevered laminate can be effectively suppressed by choosing a suitable voltage and polarity.

Isogeometric analysis for free vibration of bidirectional functionally graded plates in the fluid medium

This paper for first time proposes an isogeometric analysis (IGA) for free vibration response of bi-directional functionally graded (BDFG) rectangular plates in the fluid medium. Material properties of the BDFG plate change in both the thickness and length directions via power-law distributions and Mori-Tanaka model. The governing equation of motion of BDFG plate in the fluid-plate system is formulated basing on Hamilton's principle and the refined quasi three-dimensional (3D) plate theory with improved function f(z). The fluid velocity potential is derived from the boundary conditions of the fluid-plate system and is used to determine the added mass. The discrete system of equations is derived from the Galerkin weak form and numerically analyzed by IGA. The accuracy and reliability of the proposed solutions are verified by comparing the obtained results with those published in the literature. Moreover, the effects of the various parameters such as the interaction boundary condition, geometric parameter, submerged depth of plate, fluid density, fluid level, and the material volume control coefficients on the free vibration behavior of BDFG plate in the fluid medium are investigated in detail. Some major findings regarding the numerical results are withdrawn in conclusions.

Force-Based Quadrilateral Plate Bending Element for Plate Using Large Increment Method

A force-based quadrilateral plate element( 4NQP13) for the analysis of the plate bending problems using large increment method( LIM) was proposed. The LIM, a force-based finite element method( FEM),has been successfully developed for the analysis of truss,beam,frame,and 2D continua problems. In these analyses,LIMcan provide more precise stress results and less computational time consumption compared with displacement-based FEM. The plate element was based on the Mindlin-Reissner plate theory which took into account the transverse shear effects.Numerical examples were presented to study its performance including accuracy and convergence behavior,and the results were compared with the results have been obtained from the displacementbased quadrilateral plate elements and the analytical solutions. The4NQP13 element can analyze the moderately thick plates and the thin plates using LIMand is free from spurious zero energy modes and free from shear locking for thin plate analysis.

New exact solutions for free vibrations of rectangular thin plates by symplectic dual method

The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-form natural mode satisfies the governing equation of the eigenvalue problem of thin plate exactly and is applicable for any types of boundary conditions. With all combinations of simplysupported （S） and clamped （C） boundary conditions applied to the natural mode, the mode shapes are obtained uniquely and two eigenvalue equations are derived with respect to two spatial coordinates, with the aid of which the normal modes and frequencies are solved exactly. It was believed that the exact eigensolutions for cases SSCC, SCCC and CCCC were unable to be obtained, however, they are successfully found in this paper. Comparisons between the present results and the FEM results validate the present exact solutions, which can thus be taken as the benchmark for verifying different approximate approaches.