Thin Rectangular Plates Research Articles

Anti-Buckling Design of Rectangular Thin Plate Based on Local Surface Nanocrystallization

In this paper, a novel local surface nanocrystallization treatment is introduced to design the anti-buckling rectangular plate. The mechanical properties and critical buckling loads of the plates are greatly improved by the surface nanocrystallization technology. Several local nanocrystallization layouts, including the horizonal stripes distribution, the vertical stripes distribution and the spaced latticed blocks distribution, are designed and numerical simulations are carried out to evaluate the stability of the plates. Results show that the critical buckling load was significantly improved by the local nanocrystallization treatment. Among all the designs, the critical buckling loads for the vertical nanocrystallization layouts is the optimal one. And the technology can also be extended to the anti-buckling design of other structures.

Analytical Bending Solutions of Orthotropic Rectangular Thin Plates with Two Adjacent Edges Free and the Others Clamped or Simply Supported Using Finite Integral Transform Method

For the first time, the finite integral transform method is introduced to explore the accurate bending analysis of orthotropic rectangular thin plates with two adjacent edges free and the others clamped or simply supported. Previous solutions mostly focused on plates with simply supported and clamped edges, but the existence of free corner makes the solution procedure much complex to solve by conventional inverse/semi‐inverse methods. Compared with the conventional methods, the employed method eliminates the need to preselect the deflection function, which makes it more reasonable and theoretical for calculating the mechanical responses of the plates. Moreover, the approach used can also analyze static problems of moderately thick plates and thick plates with the same boundary conditions investigated in this article. Finally, comprehensive analytical results obtained in this paper illuminate the validity of the proposed approach by comparing with the previous literature and finite element method by using (ABAQUS) software.

Analysis of the Behavior of a Square Plate in Free Vibration by FEM in Ansys

In the realization of mechanical structures, achieving stability and balance is a problem commonly encountered by engineers in the field of civil engineering, mechanics, aeronautics, biomechanics and many others. The study of plate behavior is a very sensitive subject because it is part of the structural elements. The study of the dynamic behavior of free vibration structures is done by modal analysis in order to calculate natural frequencies and modal deformations. In this paper, we present the modal analysis of a thin rectangular plate simply supported. The analytical solution of the differential equation is obtained by applying the method of separating the variables. We are talking about the exact solution of the problem to the limit values. However, numerical methods such as the finite element method allow us to approximate these functions with greater accuracy. It is one of the most powerful computational methods for predicting dynamic response in a complex structure subject to arbitrary boundary conditions. The results obtained by MEF through Ansys 15.0 are then compared with those obtained by the analytical method.

Analysis of the free vibration of thin rectangular plates with cut-outs using the discrete singular convolution method

Abstract In this paper, free vibration of thin rectangular plates with cut-outs for different boundary conditions is analyzed by using the discrete singular convolution (DSC) method. First, the entire plate configuration is divided into appropriate sub-domains. The edges involved with cut-outs in each sub-domain are treated as free boundary conditions and the Taylor series expansion method is adopted for solving. Then, the convergence of the DSC method is tested and the frequency parameters were validated by comparing with the exact solutions by Leissa. At last, the influence of rectangular and U-shaped cut-outs structure parameters on the natural frequency is studied.

Transverse Vibration of Thin Rectangular Orthotropic Plates on Translational and Rotational Elastic Edge Supports: A Semi-analytical Approach

The free vibration analysis of thin orthotropic plates with elastically constrained edges is presented using the energy-based Rayleigh–Ritz (RR) method. Various edge conditions are modeled with rotational and translational linear springs. The complete set of admissible functions, which is a combination of (1) trigonometric functions, (2) the unit function, (3) the linear function, and (4) the lowest order polynomial, has been used in the RR method. It has been demonstrated that the use of a combination of the lowest order polynomial and the cosine series results in a rapid convergence of the solution, without any ill-conditioning of the admissible functions upon expansion. In particular, this work proposes a simple guideline for determining a set of trial functions that can be universally utilized for the vibration analysis of plates with non-classical boundary conditions. The convergence and exactness of this approach have been demonstrated through several examples. The results indicate that the elastically restrained stiffness and the plate aspect ratio impact the frequency parameters considerably.

Modelling Method of Dynamic Characteristics of Marine Thin-Walled Structure

Abstract Thin-walled structures are very popular in industries, especially in the field of shipbuilding. There are many types of equipment and structures of ships, which are made up of thin-walled structures such as hull, deck and superstructure. Therefore, the analysis and understanding of the static and dynamic characteristics of a thin-walled structure are very important. In this article, we focus on vibration analysis of a typical thin-walled structure-rectangular plate, a basic structure of the hull. Vibration analysis of a rectangular thin plate is conducted by two methods: numerical modelling method of the finite element on Patran-Nastran software platform and experimental method implemented in the laboratory of Gdynia Maritime University. Thin rectangular plate is fixed one end by four clamping plates and is modelled with finite elements and different meshing densities. The numerical model of thin rectangular plate is divided into four cases. Case 1, thin rectangular plate, and clamping plates are modelled with two-dimensional elements. Case 2, the rectangular thin plate is modelled with two-dimensional elements; the clamping plates are modelled with three-dimensional elements. Case 3, both the rectangular thin plate and clamping plates are modelled with three-dimensional elements. Case 4, the rectangular thin plate, and clamping plates are modelled with three-dimensional elements with larger mesh density to increase the accuracy of the calculation results. After that, the results of vibration analysis according to the numerical modelling method on Patran-Nastran software platform for these cases were compared with the measurement results. From there, assess the accuracy of analysis results of selected numerical model methods and the ability to widely apply this numerical model method to other marine structures.

New Analytical Solutions of Buckling Problem of Rotationally-Restrained Rectangular Thin Plates

A double finite sine integral transform method is employed to analyze the buckling problem of rectangular thin plate with rotationally-restrained boundary condition. The method provides more reasonable and theoretical procedure than conventional inverse/semi-inverse methods through eliminating the need to preselect the deflection function. Unlike the methods based on Fourier series, the finite integral transform directly solves the governing equation, which automatically involves the boundary conditions. In the solution procedure, after performing integral transformation the title problem is converted to solve a fully regular infinite system of linear algebraic equations with the unknowns determined by satisfying associated boundary conditions. Then, through some mathematical manipulation the analytical buckling solution is elegantly achieved in a straightforward procedure. Various edge flexibilities are investigated through selecting the rotational fixity factor, including simply supported and clamped edges as limiting situations. Finally, comprehensive analytical results obtained in this paper illuminate the validity of the proposed method by comparing with the existing literature as well as the finite element method using (ABAQUS) software.

Aspect ratio dependence of the effective electromechanical coupling coefficient Ke of the transverse vibration mode for a piezoelectric rectangular thin plate

In this paper, the apparent elasticity method is used to derive the transverse effective electromechanical coupling coefficient Ke of a piezoelectric rectangular thin plate with an arbitrary aspect ratio G (defined as the ratio of the two transverse dimensions). A unified formula Ke with regard to G is obtained, and the dependence of Ke on G is presented and discussed. The results show that, at G = 1 (square thin plate), Ke drops to 0 at the low-frequency branch, whereas it reaches its maximum value at the high-frequency branch. It is proposed that the maximum value of Ke corresponds to the planar vibration mode. In addition, thin plates made of the PZT-4 ceramic with different aspect ratios were fabricated and measured to check the validity of the above method. For the low-frequency branch of Ke, the experimental results were consistent with the trend of the theoretical values. For the high-frequency branch of Ke within the range of 0.2 ≤ G ≤ 0.55 and 1.82 ≤ G ≤ 5, however, the experimental results exhibited large deviations compared with the theoretical values owing to the negative influence of disturbing resonances. In contrast, the experimental results within the range of 0.6 ≤ G ≤ 1.67 were in good agreement with the theoretical values and reached a maximum value at G = 1. It is suggested that the piezoelectric plates with 0.6 ≤ G ≤ 1.67 should be preferred for practical applications due to their undisturbed high-frequency resonances with high values of Ke.

Parametric stability and complex dynamical behavior of functionally graded rectangular thin plates subjected to in-plane inertial disturbance

In this study, the impact of in-plane inertial disturbance from environment on the dynamical behavior of a functionally graded (FG) rectangular plate is discussed. The inertial disturbance is modeled as a parametric excitation. The dynamical model of FG plate subjected to an uniaxial inertia force of an in-plane disturbance are developed according to von Kármán’s large deflection theory. A reduced low-dimensional nonlinear ordinary differential equations of considered FG plate are then obtained by employing Galerkin’s procedure. Both multiple scales method and Floquet’s theory are applied to analyze the parametric instability regions for small-deflection vibration of FG plate. The effects of nonlinearity are also discussed through numerical integration. It is found that the instability regions vary greatly with the variation of frequency ratio and it cannot come to a clear rule or regular pattern, but three regions, i.e. two principle instability areas and a combination instability area, always exist. The damping affects the parametric stability in a very complicated way. And the nonlinear nonzero response for combination resonance displays complex dynamical behaviors induced by bifurcations.

New Analytic Shear Buckling Solution of Clamped Rectangular Plates by a Two-Dimensional Generalized Finite Integral Transform Method

A first attempt is made in this paper to explore new analytic shear buckling solution of clamped rectangular thin plates by a two-dimensional generalized finite integral transform method. The problem is classical but challenging due to the mathematical difficulty in handling the complex boundary value problem of the governing higher-order partial differential equation (PDE). Taking the vibrating beam functions as the integral kernels, and imposing the double transform, the problem comes down to solving a system of linear algebraic equations, thereby the analytic solution is obtained in a straightforward way. The present method is confirmed to be highly accurate with fast convergence, which agrees very well with both the finite element method (FEM) and energy method from the literature. The new analytic solution obtained may serve as a benchmark for validating other numerical and approximate methods.

Free vibration analysis of simply supported rectangular plates

In this paper, numerical analysis for free vibration of simply supported thin rectangular plates has been simulated using Ansys. The vibration of plate follows Kirchhoff plate theory. The natural frequency calculated using Ansys has been compared with Levy type solution results available in the literature. It is observed that the natural frequencies of the simply supported rectangular plate is in close agreement with exact solution results as reported in literature.

Bending of orthotropic rectangular thin plates with two opposite edges clamped

This work presents integral transform solutions of the bending problem of orthotropic rectangular thin plates with constant thickness, subject to five sets of boundary conditions: (a) fully clamped; (b) three edges clamped and one edge simply supported; (c) three edges clamped and one edge free; (d) two opposite edges clamped, one edge simply supported, and one edge free; and (e) two opposite edges clamped and two edges free. By adopting eigenfunctions of Euler–Bernoulli beams with corresponding boundary conditions for each direction of the plate, the governing fourth-order partial differential equation is integral transformed into a system of linear algebraic equations. Boundary conditions at the free edges are treated exactly by carrying out integral transform in the boundary formulations, which are incorporated in the transformed governing equations by integration by parts. The numerical difficulties with the high-order beam functions are overcome by using modified exponential forms, thus limiting the eigenfunctions to the range between −2 and 2. Analytical integration forms are used for the integrals of the coefficients of the transformed equations, further avoiding numerical difficulties with large high-order eigenvalues. The accuracy and convergence of the solutions are shown through numerical examples in comparison with available solutions in the literature and with finite element solutions obtained by using Abaqus program.

Simplification through regression analysis on the dynamic response of plates with arbitrary boundary conditions excited by moving inertia load

Dynamic response of a thin rectangular plate traversed by a moving inertia load with arbitrary boundary condition is investigated through this paper. The inertia effect of mass is considered and relevant formulation is established based on the full-term of acceleration, employing the method of Boundary Characteristic Orthogonal Polynomials, BCOP. To acquire the complete solution of partial differential equations governing on the plate, the Galerkin method is used to separate the temporal function from the spatial one. The problem is formulated in the state space and applying the numerical method of Matrix Exponential the complete solution would be achieved. In the numerical studies, a comprehensive parametric study is performed for both cases of loading when inertia effect is included or neglected. Several mass and aspect ratios for the plate with major types of boundary conditions CCCC, SSSS, CFCF and SFSF are accounted for presenting the results. Dynamic amplification factor against velocity parameter is scrutinized within many graphs alongside with a time history analysis of dynamic deflection for the plate's mid-span. Investigating on the dynamic response concludes to the critical boundary condition upon moving mass. By introducing a conversion factor, the margin of inertia and the critical velocity where happened would be achieved, then through a regression analysis a curve fitting model of polynomials is proposed. Corresponding coefficients testify the goodness of fit for such regression which are reported within tables. Referring to this simplified model of conversion factor pertaining to the specific boundary condition, it would be possible to handle the problem in moving load case without undertaking the complexities arisen from inertia contribution into the formulation. Having derived the factor from simplified model which has been calculated for a specific mass and velocity ratio, then multiplying into the moving load response, the complete solution for moving mass would be achieved.

A semi-analytical solution for in-plane free waves analysis of rectangular thin plates with general elastic support boundary conditions

A semi-analytical solution for in-plane free waves analysis of rectangular thin plates with general elastic support boundary conditions (BCs) is developed. Based on the classical thin plate theory, the governing differential equations of the rectangular thin plate are introduced. The semi-analytical solution is obtained by superposition of in-plane waves of different patterns and different propagation directions. And it is validated by comparisons of the present results with those given in literatures. Eigensolutions for general BCs and the specific expressions of characteristic equations for various classical BCs are demonstrated. Characteristic curves of the dimensionless wavenumbers of the in-plane longitudinal wave and the in-plane shear wave of the rectangular thin plate with various BCs are presented and discussed. Perfect orthogonality of the in-plane longitudinal wave and the in-plane shear wave of different mode numbers is presented in the characteristic curves of the rectangular thin plate with two opposite sides simply supported. The characteristic curves corresponding to the rest of the classical BCs and all of the non-classical BCs show non-orthogonality of different extents, which reveals the formation mechanism of the variety of the mode shapes of the rectangular thin plate with various BCs. The strengthening and weakening process of the orthogonality and the formation of the intertwinement or independence of the characteristic curves of different mode numbers are elaborated through analysis on elastic support BCs intermediate between two classical BCs, which shows the influence of the BCs on the intrinsic characteristics of in-plane waves of the rectangular thin plate.

Strain‐based displacement field reconstruction method for thin rectangular plate through orthogonal deflection curves bridging

In the existing studies of strain-based displacement field detection for thin rectangular plate, either the boundary conditions are too difficult to handle or the sensors layout and algorithm computing process are too complex to satisfy the requirement of rapid reconstruction of nonlinear vibration shape. In this paper, a strain-displacement conversion method for reconstructing the deformed shape of rectangular plate with typical boundary conditions is presented. This method overcomes the above shortcomings for small deflection nonlinear vibration in linear elastic range. Strains measuring points are uniformly and regularly distributed in the middle area of the plate. Unmeasurable boundary strains will be estimated using the new derived formula according to quadratic interpolation of strains and boundary conditions. The principle of the proposed method is to bridge the orthogonal deflection curves by translating and rotating them. In this way, framework of displacement field can be formed. First, curvature integral method is employed to shape the orthogonal deflection curves of deformed plate. Then, the rotating angles and translating displacements are deduced according to boundary conditions. For some special boundary plates, it even needs to solve nonlinear optimization problem to get the rotating angles. The displacement field reconstructed by the proposed method shows great agreement with the finite element analyses results, which verifies the effectiveness of the methodology.

Benchmark Vibration Frequencies of Square Thin Plates with all Possible Combinations of Classical Boundary Conditions

In this work, a new method is used for the exact vibration analysis of plates with classical boundary conditions. Four classical edge conditions are included: C — clamped, S — Simply supported, F — free, and G — guided. For square plates, all the possibilities add up to 55 cases. The solutions for the natural frequencies of the plates are found in this paper using static analysis. Starting from the equations of motion of an isotropic rectangular thin plate supported on Winkler elastic foundation, with a positive or negative value, the solution for the vibration frequencies of the plate is equivalent to finding the values of the negative elastic foundation that will yield infinite deflection under a point load on the plate. The solution is composed of three parts, the sum of which satisfies exactly both the field equation and the boundary conditions. For zero force, the vibration frequencies are found up to the desired accuracy. Benchmark results of the first six normalized natural frequencies, of isotropic square plates, for all possible 55 combinations of classical boundary conditions are given, many for the first time.

Accurate free vibration solutions of orthotropic rectangular thin plates by straightforward finite integral transform method

This paper aims to obtain the analytical free vibration solution of an orthotropic rectangular thin plate using the finite integral transformation. Due to the mathematical difficulty of complex boundary value problems, it is very hard to solve the title problem with common analytical methods. By imposing the integral transformation, the high-order partial differential equation with specified boundary conditions is converted into linear algebraic equation and the exact solutions with high precision and fast convergence are obtained elegantly. The main advantage of the proposed method is that it is simple and general, and it does not need to pre-determine the deflection function, which makes the solution procedure more reasonable. The method has a wide range of applications and can deal with other elastic plate problems, such as buckling and bending. The present results are validated by comparison with the existing analytical solutions that showed satisfactory agreement. The new analytical solution obtained can be used as a benchmark for validating other numerical and approximate methods.

Deformation Analysis of Hydrostatic Guideways Based on the Cantilever Plate Bending Calculation Method

In this study, a novel analytical model considering practical oil pressure and a cantilever plate bending calculation method is proposed to calculate the deformation of a vertical hydrostatic guideway. Based on the governing equations of rectangular thin plates, the finite integral transformation theory and the series-by-item differential transformation theory are used to solve the bending deformation of the target guideway to avoid the errors caused by an artificially selected deflection function and Kirchhoff’s hypothesis. The deformation values obtained from finite element simulation and experiments are compared with those calculated from the model. The comparison results show that the proposed model is capable of providing deformation values that are in better agreement with the experimental values than simulation values and its computation time is much less than that of the finite element method. The quantitative relationship between the actual oil film thickness and the performance of the hydrostatic guideways considering the size effect of the oil film thickness is studied.

Buckling analysis of thin rectangular FG-CNTRC plate subjected to arbitrarily distributed partial edge compression loads based on differential quadrature method

This paper investigates the buckling behaviors of thin rectangular functionally graded carbon nanotubes reinforced composite (FG-CNTRC) plate subjected to the arbitrarily distributed partial edge compression loads. Three distribution types of carbon nanotubes (CNTs) are considered and the effective material properties of carbon nanotube reinforced composite are given according to the extended rule of mixture, in which the distribution uncertainty of CNTs is also taken into account. Based on the differential quadrature method (DQM) and work equivalent method, the pre-buckling in-plane stress distribution of thin rectangular FG-CNTRC plate are determined firstly. Then the accurate critical buckling loads and buckling modes of FG-CNTRC plate are consequently obtained. By comparing with the results in published literatures, the feasibility and accuracy of present numerical method are validated. Finally, the systematic parametric studies are carried out to elaborate the effects of distribution type, volume fraction and distribution uncertainty of CNTs as well as the distribution width, position and distribution type of partial edge compression loads on the buckling behaviors of thin rectangular FG-CNTRC plate for the first time.

Analytical Buckling Loads for Rectangular Orthotropic and Symmetrically Laminated Plates

In this research, a new analytical solution for finding the buckling loads of thin orthotropic rectangular plates with different boundary conditions is derived. The methods currently known in the literature for finding the buckling loads of plates are mainly numerical. Although some plates with specific boundary conditions have analytical solutions, a comprehensive analytical method providing analytical solutions that fit all possible combinations of boundary conditions is lacking. The solution method in this study is based on the development of a static solution for a plate. The physical meaning of buckling is the loss of stiffness, and it is found as the value of the in-plane loading intensity at which a zero force on the plate surface will generate infinite displacement. The solution is obtained in series form, and the coefficients are solved to match the edge conditions. The solution presented for this plate will fit different boundary conditions of the deflection, slope, shear force, and bending moment along the common edges of neighboring plates. Using this new method, exact buckling loads and the buckling modes of many new cases of classical boundary conditions are found (cases involving clamped, simply supported, guided, and free boundaries). Results are given for several stiffness ratios in both directions of the plate, as well as for unidirectional and several cases of bidirectional loading.