Thin Rectangular Plates Research Articles

A ragged porous hollow tubular carbon nitride towards boosting visible-light photocatalytic hydrogen production in water and seawater

Photocatalytic water splitting to produce hydrogen, especially seawater, shows great economic value and development prospects to deal with the energy crisis. In this paper, urea and melamine were employed as base precursor materials to obtain four different morphologies of tubular g-C3N4: stacked rectangular thin plates with flocculent structure, ragged porous hollow tubes, porous hollow tubes and tetragonal hollow prismatic tubes by simply regulating the heating rate. Among them, the ragged porous hollow tubular g-C3N4 (TCN-1.5) exhibit an amazing hydrogen evolution rate of 8683 μmol h−1 g−1 from water, which is 19.3 times that of the massive g-C3N4 (449 μmol h−1 g−1), and still maintain a high hydrogen evolution rate (6782 μmol h−1g−1) from sea water. The analysis results show that by controlling the change of the heating rate, the ammonia overflow rate can be effectively controlled to produce abundant voids and pits to form volume defects, which can exhibit higher specific surface area and serve as a highly active center for the catalytic reaction and effectively improve the photocatalytic performance. This study provides a simple design idea to construct a novel type of ragged porous hollow tubular g-C3N4 for efficient photocatalytic hydrogen production from water and seawater.

Simplified Closed-Form Method to Large Deflection Stability Analysis of Thin Rectangular Plate

At the moment, the analyses of postbuckling behavior of thin rectangular plates of constant thickness are generally based on von Karman’s large deflection equations which make use of stress functions and trigonometric variables. These equations are coupled, nonlinear partial differential equations of fourth order each. Getting a closed-form solution of these equations is near impossible and tedious. The present work presents a simplified closed-form general equation using a variational approach for large deflection analysis. The approach adopted here is devoid of Airy’s stress functions. A new strain-displacement equation is formulated, and the total potential energy equation is minimized. The resulting compatibility equation was solved to obtain the general governing stability equation under large deflection. This governing equation was applied to a plate simply supported all-around using polynomial displacement function and numerical results were obtained. To validate the numerical results obtained, they were compared with the values obtained by Levy whose results are generally acclaimed as exact and with two others. It was observed that the minimum and maximum percentage differences were 0% and 41.56% at stress parameters 3.66 and 21.45 respectively. Also, for deflection to thickness ratio (w/t), the present results showed a close agreement with those of Levy with a minimum and maximum percentage difference of 0.07% and 41.56% at w/t of 0 and 3.376 respectively. Importantly, the present result lies between two other research results. We, therefore, conclude that the present work is adequate and a new simple closed-form approach to understanding and predicting the postbuckling strength of thin rectangular plates.

On the acoustic radiation of all edges free rectangular thin plate

In this study, transverse vibrations and acoustic radiation of an all-edges-free baffled rectangular thin plate are examined by Rayleigh's formulation. While the Rayleigh-Ritz method is applicable for a wide variety of boundary condition combinations, in this study, the method is used for evaluating the modal characteristics of an all edges free plate. Acoustic pressure levels are calculated by Rayleigh's acoustic formula for a free field. This study provides a quick formula for those who are interested in calculating the sound pressure of a baffled plate on a specific point for a free field while only requiring the vibration of the surface to be expressed in terms of 2D power series. The results obtained from the formula are compared with the finite element analysis results.

Derivation of the equations of motion and boundary conditions of a thin plate via the variational method

Small deflections in both a thin rectangular plate and a thin circular plate are studied via the variational method. In order to apply Hamilton’s principle to this system, the potential energy is expressed in terms of strain and stress tensors. Quantities such as the gradient displacement tensor and the traction vector are reviewed. It is showed the advantage of the variational method as a technique which allows to obtain the equations of motion and the boundary conditions simultaneously.

ELECTROMAGNETIC RESPONSE OF LAYERED MAGNETO-ELECTRO-ELASTIC THIN RECTANGULAR PLATE UNDER MODERATELY LARGE DEFLECTION

The analytical solution of a layered thin magneto-electro-elastic rectangular plate is presented. The governing equation is based on classical laminate plate theory and von Karman's stress function; capturing the effects of moderately large deflection. Electromagnetic fields are determined in terms of mechanical unknowns by solving Maxwell's equations of electrostatics and magnetostatics. The condensation of electric and magnetic state into plate kinematics, coupled with stress function definition, derives the governing non-linear partial differential equation of motion. Solution for both simply supported and clamped transverse boundary condition is obtained using Galerkin method, which reduces the system into an ordinary differential equation of cubic and quadratic nonlinearity. Numerical results of laminated plate with constituent piezoelectric barium titanite and piezomagnetic cobalt ferrite is produced utilizing electromagnetic boundary and continuity conditions. The effect of transverse elastic boundary condition on through the thickness variation of electric and magnetic potential under linear and moderately large deflection is produced.

Numerical study of nonlinear wave flow past vertically submerged plate over uneven bottom

This present study investigates the horizontal wave force and overturning moment on a submerged vertically thin rectangular plate on a free surface (z = 0 m) over a flat and uneven bottom structure under various wave steepness conditions and relative openings in the numerical wave tank (NWT) modelled using ANSYS Fluent software. The interface of the fluids is tracked using the volume of fluid (VOF) method. At the inlet of the NWT, the inflow velocity method is followed to produce the desired waves and at the end, numerical damping is used to minimize the wave reflection. Further, the effects of wave characteristics, relative openings, and uneven bottom conditions on the horizontal wave force and overturning moment are analyzed in the intermediate water depth. The horizontal wave force (Fx) and overturning moment (My) on the vertical plate are registered higher with case 3 (α4 = 37.5%) for different wave steepness attributed to the higher slope of the bottom structure. The present work also incorporates the development of a plate type wave energy converter (WEC) to harvest renewable wave energy from the ocean.

Impulse deformation of triangular plates based on the classical theory

This article discusses the impulse effects of various loads on triangular, isosceles, elastic, isotropic plates. Analytical solutions of the direct problem of determining the internal moments and deflections of the plate, as well as the numerical results of calculations of specific loading case are presented. Goal. The goal is to develop a method for solving direct problems of determining internal moments and deflections in rectangular triangular, isosceles, elastic, thin, isotropic plates. Methodology. To solve the direct problem, the Navier method, the classical theory of modeling vibrations of thin plates and the Laplace transform are used. Results. A technique has been obtained that allows one to obtain numerical and analytical dependences for calculating the internal moments and deflections in a triangular plate. Originality. For the first time, a technique was developed for solving direct non-stationary problems to determine the internal moments and deflections in rectangular triangular, isosceles, elastic, thin, isotropic plates based on the classical theory. Practical value. The obtained analytical dependences can be used to simulate impulse vibrations of square and isosceles rectangular triangular thin isotropic elastic plates, which can be critical structural elements.

Free-field sound radiation measurement with multiple synchronous cameras

This paper investigates the possibility of using photogrammetry to obtain the sound radiation into free-field from flexural vibrations of distributed thin structures. A simple model problem is considered, where multiple (i.e. two or more) cameras are used to track at a regular grid of target points the flexural vibration of a baffled thin rectangular flat plate structure. The sound radiation into free-field is then derived from the Rayleigh integral, which is approximated into a finite sum assuming the plate is discretised into a regular mesh of elements whose centres correspond to the grid of points where the flexural vibrations are measured with the cameras. The paper shows that the vibration measurement errors produced by the image pixelisation in the cameras compensate over the grid of target points so that the estimate of the radiated sound field with the Rayleigh integral formulation is comparatively more accurate.

New Straightforward Benchmark Solutions for Bending and Free Vibration of Clamped Anisotropic Rectangular Thin Plates

Abstract This study presents new straightforward benchmark solutions for bending and free vibration of clamped anisotropic rectangular thin plates by a double finite integral transform method. Being different from the previous studies that took pure trigonometric functions as the integral kernels, the exponential functions are adopted, and the unknowns to be determined are constituted after the integral transform, which overcomes the difficulty in solving the governing higher-order partial differential equations with odd derivatives with respect to both the in-plane coordinate variables, thus goes beyond the limit of conventional finite integral transforms that are only applicable to isotropic or orthotropic plates. The present study provides an easy-to-implement approach for similar complex problems, extending the scope of finite integral transforms with applications to plate problems. The validity of the method and accuracy of the new solutions that can serve as benchmarks are well confirmed by satisfactory comparison with the numerical solutions.

Modified Fourier-Ritz Method to Analyze Transverse Vibration of Rectangular Plates with Additional Mass

In practical engineering, rectangular thin plate structures with concentrated mass points or equivalent to concentrated mass points have a wide range of applications in mechanical engineering, electronic engineering, and vehicle engineering, such as supporting workbenches, ship decks, PCB boards and so on. The modified Fourier Ritz method is used to establish the numerical analysis model of rectangular thin plate with lumped mass under general boundary conditions, which can avoid the problems of non-derivability or discontinuity at the boundary of thin plates in traditional methods. In addition, the Fourier expansion in the form of cosine function plus polynomial has better convergence than that in the form of sine function. The calculation method of mass matrix and stiffness matrix of rectangular thin plate with concentrated mass is given, the parameters of different boundary constraints are analyzed, and the influence of the size, position and quantity of concentrated mass on the modal of rectangular plate is discussed. The method and its results can be applied to the vibration analysis and vibration control of rectangular thin plates.

Postbuckling behaviour of isotropic thin rectangular plates with one free edge

This work aims to evaluate the individual stiffness of six plates’ boundary conditions with one free edge to obtain the specific mathematical models for predicting the postbuckling loads of the six plates’ boundary conditions under consideration. The shape profiles of each of the six plates were differentiated and substituted into the individual stiffness integrals to obtain the numerical values of the individual stiffness. The individual stiffness values were then substituted into the total bending and membrane stiffness expressions and evaluated. The resulting total and membrane stiffness expressions were thereafter substituted into the general postbuckling equation and evaluated to obtain specific mathematical models for the six plate types to predict the postbuckling loads of each plate. The newly formulated mathematical models were validated by carrying out numerical predictions of the postbuckling loads of each plate. The critical load obtained was compared with those in the literature and was found adequate. Additionally, the results showed a gradual increase in the strength of plate beyond the initial yield point which is in line with the behavior of plates. Based on these observations it was concluded that the newly formulated mathematical models for predicting the postbuckling strength of thin isotropic rectangular plates considered here were adequate and that the models will provide an easy means of analyzing thin plates for postbuckling loads.

On transmission Zeros of piezoelectric structures

The evaluation of transmission zeros is of great importance for the control engineering applications. The structures equipped with piezoelectric patches are complex to model and usually require finite element approaches supplemented by model reduction. This study rigorously investigates the influence of mesh size, model reduction, boundary conditions (free and clamped), and sensor/actuator configuration (collocated and non-collocated) on the evaluation of transmission zeros of the piezoelectric structures. The numerical illustrations are presented for a thin rectangular plate equipped with a single pair of piezoelectric voltage sensor/ voltage actuator. Through the examples considered in this study, a link is presented between the static response (or static deflected shape) and the transmission zeros of the piezoelectric structures. This interesting observation forms the basis of: (i) a local mesh refinement strategy for computationally efficient estimation of the transmission zeros and (ii) a physical interpretation of the pole-zero pattern in the case of piezoelectric structures. The physical interpretation developed in this study helps in qualitatively explaining the pole-zero patterns observed for different configurations. It is also shown that this understanding of the relation between the static deformed shape and the transmission zeros can be used by the practitioners to modify the pole-zero pattern through a careful selection of the orientation and the size of the piezoelectric patches.

Buckling Analysis of Symmetrically Laminated Rectangular Thin Plates under Biaxial Compression

In this parametric study, the buckling analysis of symmetrically laminated rectangular thin plates assumed to be subjected to biaxial compression was presented. The simply supported boundary condition was considered at the edges of the symmetrically laminated quasi-isotropic, cross-ply and angle-ply plates. The Rayleigh-Ritz Method was used to specify the critical buckling load of the plates based on the Classical Laminated Plate Theory (CLPT). A convergence study was achieved by increasing the number of parameters of assumed shape function. Validation of isotropic case was verified. The effects of the lamination types, plate aspect ratios (a/b, b/a) and thickness on the critical buckling load of the laminated plates under bi-axial compression were then investigated. The results were compared with Finite Element Method (FEM) solutions performed by ANSYS software package and fairly good agreement was obtained. Non-dimensional results were tabulated and presented for practical use for designers.

On the Solution to Pre- Buckling of a Thin Rectangular Plate Subjected to a Thick Strip in-plane Loading

The current paper deals with the problem of the simply supported thin rectangular plate subjected to the intermediate strip in-plane loading. Based on the strain energy method (Fourier ansatz), the critical (minimum value) of buckling stress occurrence was determined in a general form dependent only on the strip thickness, strip location, plate width and stress magnitude. Compatible with the classical columns Euler method it was found that the plate stability is decreased with the increasing of the plate width due to larger induced stresses. Also, strip location relative to the support region was found to influence the buckling (same analogy to the Euler buckling theory; consider the strip as a both sides pressed rod). Additionally, the strip width parameter increase is likely to cause larger buckling stress. Moreover, expressions that includes both axial and transverse loads for different extended cases configurations were also derived and examined based on the strain energy method alongside explanation for possible applications (thin aluminum plate welding). In a general view, it was found that the cases of combined axial and perpendicular loading action are less stabilized than cases where only one kind of loading configuration is participated. Finally, the buckling stress was found to agree qualitatively with the cited literature.

Synergistic effect of memory-size-microstructure on thermoelastic damping of a micro-plate

The design of high-performance micro-mechanical resonators requires sufficiently accurate analysis of their thermoelastic damping. In this paper, a micro-resonator is modeled as a thin rectangular plate in vibration, and its thermoelastic damping is characterized in terms of the inverse quality factor Q−1. The Kirchhoff plate theory containing the memory, nonlocal, and size effects is presented, where the memory effect of heat flow is described by a convolution integral for time, its nonlocal effect is captured by the Guyer–Krumhansl model for space and the size effect for microstructures is characterized by Hadjesfandiari and Dargush’s consistent couple stress theory. The temperature field related to the deflection is solved first, and the governing equation of the transverse motion of a vibrating plate is obtained later. Two methods including the complex frequency approach and energy dissipation approach are employed to derive the analytical expression for the inverse quality factor. Emphasis is focused on the synergistic effect of non-locality, memory, and microstructure on the thermoelastic dissipation in micro-plates. Additionally, the influences of boundary conditions, vibration modes, and structural dimension on the thermoelastic damping are also analyzed. The temperature dependence of the thermoelastic damping is predicted and anelastic internal friction is elucidated.

An extended separation-of-variable method for free vibrations of orthotropic rectangular thin plate assemblies

Rectangular plate assemblies are widely used in various engineering fields, but there are not closed-form solutions for them with arbitrary homogeneous boundary conditions. Hence, this work is to develop an extended separation-of-variable method to achieve closed-form analytical solutions for the vibrating problems of orthotropic rectangular thin plate assemblies. In this method, the mode function of each plate is in an explicit separation-of-variable form, and the frequencies corresponding to two-direction eigenfunctions are mathematically independent of each other. This method can deal with all homogeneous boundary conditions. The two-direction eigenvalue equations can be solved simultaneously, and the obtained closed-form eigensolutions satisfy the Rayleigh’s principle exactly. Numerical experiments validate the accuracy of the present solutions, and parametric study is conducted.

Refined model for rectangular thin plates on two-parameter foundations using an iterative procedure

The main purpose of this study is to investigate the influence of soil heterogeneity on the mechanical responses of rectangular plates resting on elastic foundations and to determine reasonable physical model parameters. The bending problem of rectangular thin plates lying on Gibson elastic foundations is solved using an iterative procedure. First, plates and foundations are discussed as a whole system, based on the principle of minimum potential energy. Second, the governing equations and boundary conditions for plates are established. And the equation for the attenuation parameter in the mathematical model is also derived. Then an iterative procedure is used to iterate the two parameters simultaneously, which solves a key issue for the two-parameter foundation model. Third, a double Fourier series with supplementary functions is adopted, and the solution to the bending problem is obtained. The agreements between the numerical results and the literature results prove that the refined model is practical and achievable. The proposed method exhibits a certain universality in analyzing the interactions between rectangular thin plates and elastic foundations. Furthermore, Gibson soil has an influence on the structural responses of rectangular plates on elastic foundations.

Bending of rectangular orthotropic plates with rotationally restrained and free edges: Generalized integral transform solutions

Generalized integral transform solutions for bending problem of orthotropic rectangular thin plates with rotationally restrained and free edges are presented in this work, including four edges rotationally restrained (RRRR), three-edges rotationally restrained and one-edge free (RRRF), and two opposite edges rotationally restrained and the other two edges free (RFRF). The dimensionless governing equation of orthotropic rectangular plate bending with RRRR, RRRF or RFRF boundary conditions are integral transformed in each spatial direction by adopting Euler–Bernoulli beam eigenfunctions with appropriate boundary conditions (RR, RF or FF). The eigenfunctions for beams with RR and RF boundary conditions in numerically stable exponential function form are presented for the first time, as well as the respective characteristic equations for the determination of the eigenvalues. Free edges of the plate are integral transformed and then incorporated into the transformed governing equations by integration by parts. Transversal deflection of the orthotropic rectangular thin plate under arbitrary transversal load distribution can be readily obtained by solving a system of linear algebraic equations. Numerical results are presented for uniform load, hydrostatic pressure, and concentrated load, with the rotational fixity parameter varying between the limiting cases of simply-supported and clamped edges. Through the comparison with available analytical solutions and finite element solutions by Abaqus, the proposed solutions show the rapid convergence and high accuracy. The effects of the rotational restraints on the deflection of orthotropic rectangular plates with the three combinations of boundary conditions are also investigated.

Thin rectangular plates under axial point loading: Accuracy of the classical single Fourier series solution for stresses

The purpose of this article is to clarify certain ambiguities associated with commonly adopted solution strategies of the problem of thin rectangular plates subjected to equal and opposite colinear point loads. The problem is treated as a plane-stress problem where the boundary conditions are prescribed exclusively in terms of tractions. Both an elaborate exact solution and a simplified approximate Single Fourier Series (SFS) solution are employed, shedding light on the accuracy of the latter. This is accomplished, first, by showing that the error implicit in the sine single-Fourier series solution becomes significant for column plates, and, secondly, by re-evaluating the stress distribution at various cross-sections away from the compressed sides, through the application of a truly exact solution to the problem at hand, in which the free-edge conditions are precisely fulfilled. It is shown that the restrictions usually associated with the SFS technique may not be severe and that a wider range of plate shapes with free vertical sides could be amenable to analysis utilizing this simple approximate scheme.

R-Function and variation method for bending problem of clamped thin plate with complex shape

Solving ordinary thin plate bending problem in engineering, only a few analytical solutions with simple boundary shapes have been proposed. When using numerical methods (e.g. the variational method) to solve the problem, the trial functions can be found only it exhibits a simple boundary shape. The R-functions can be applied to solve the problem with complex boundary shapes. In the paper, the R-function theory is combined with the variational method to study the thin plate bending problem with the complex boundary shape. The paper employs the R-function theory to express the complex area as the implicit function, so it is easily to build the trial function of the complex shape thin plate, which satisfies with the complex boundary conditions. The variational principle and the R-function theory are introduced, and the variational equation of thin plate bending problem is derived. The feasibility and correctness of this method are verified by five numerical examples of rectangular, I-shaped, T-shaped, U-shaped, and L-shaped thin plates, and the results of this method are compared with that of other literatures and ANSYS finite element method (FEM). The results of the method show a good agreement with the calculation results of literatures and FEM.