Problem Of Rectangular Plate Research Articles

Transient Thermoelastic Analysis of Rectangular Plates with Time-Dependent Convection and Radiation Boundaries

Approximate analytical solutions are presented for the transient thermoelastic problem of rectangular plates with time-dependent convection and radiation boundaries. To include the nonlinear radiation boundary, the whole heating process is divided into several time steps, and a linearized approximation is used to simplify the radiation term for each step. The one-dimensional transient temperature along the thickness direction is solved using the technique of the separation of variables. The displacement and stress solutions are obtained by applying the state-space method to the three-dimensional thermoelasticity equations. The accuracy of the present solutions is demonstrated by comparison with the reported results in the open literature and the finite element solutions. In the numerical examples, two kinds of thermal boundaries, namely, time-independent convection boundaries and time-dependent convection and radiation boundaries, are considered to show the availability of the present solutions.

Unified solution of some problems of rectangular plates with four free edges based on symplectic superposition method

PurposeA unified framework for solving the bending, buckling and vibration problems of rectangular thin plates (RTPs) with four free edges (FFFF), including isotropic RTPs, orthotropic rectangular thin plates (ORTPs) and nano-rectangular plates, is established by using the symplectic superposition method (SSM).Design/methodology/approachThe original fourth-order partial differential equation is first rewritten into Hamiltonian system. The class of boundary value problems of the original equation is decomposed into three subproblems, and each subproblem is given the corresponding symplectic eigenvalues and symplectic eigenvectors by using the separation variable method in Hamiltonian system. The symplectic orthogonality and completeness of symplectic eigen-vectors are proved. Then, the symplectic eigenvector expansion method is applied to solve the each subproblem. Then, the symplectic superposition solution of the boundary value problem of the original fourth-order partial differential equation is given through superposing analytical solutions of three foundation plates.FindingsThe bending, vibration and buckling problems of the rectangular nano-plate/isotropic rectangular thin plate/orthotropic rectangular thin plate with FFFF can be solved by the unified symplectic superposition solution respectively.Originality/valueThe symplectic superposition solution obtained is a reference solution to verify the feasibility of other methods. At the same time, it can be used for parameter analysis to deeply understand the mechanical behavior of related RTPs. The advantages of this method are as follows: (1) It provides a systematic framework for solving the boundary value problem of a class of fourth-order partial differential equations. It is expected to solve more complicated boundary value problems of partial differential equations. (2) SSM uses series expansion of symplectic eigenvectors to accurately describe the solution. Moreover, symplectic eigenvectors are orthogonal and directly reflect the orthogonal relationship of vibration modes. (3) The SSM can be carried to bending, buckling and free vibration problems of the same plate with other boundary conditions.

Computation of Limit Loads for Bending Plates

The purpose of this paper is to present a method for calculating the upper bound limit loads of plate bending using a conforming Hsieh-Clough-Tocher (HCT) element. These limit loads can be obtained from Koiter’s kinematic shakedown theorem for the case of one load vertex instead of using the kinematic limit theorem. When combining this theorem with the approximated displacement field, the limit analysis turns into an optimization problem and can be effectively solved by Second-Order Cone Programming (SOCP). Several benchmark plate problems such as square, rectangular, and L-shape plates are investigated to illustrate the effectiveness of the proposed solution. The results of the proposed method show good agreement with the results of previous studies. The maximum error is only 2.91% for the fully clamped rectangular plate problem.

Analytical solutions of free vibration for rectangular thin plate and right-angle triangle plate on the Winkler elastic foundation based on the symplectic superposition method

Abstract Based on the symplectic superposition method, the free vibration models of rectangular and right-angle triangle plates on the Winkler elastic foundation are established in the present paper, and the modes and frequencies are studied. In addition, the theoretical calculation model and finite element analysis model of rectangular thin plate and right-angle triangle plate on elastic foundation are established by using Mathematica software and ABAQUS software. It proves that the symplectic superposition method converges very fast and has a good consistency with the finite element simulation results. Analytical results show that foundation stiffness, aspect ratio, and boundary condition have great influences on vibration frequency and mode shape for structures. This paper solved the free vibration problem of rectangular plate and right-angle triangle plate on elastic foundation by using symplectic superposition method. Compared with the inverse or semi-inverse method, this method avoids the process of assuming the form about the solution, hence the result of this method is completely rational.

Elastic Shear Buckling Coefficients for Oblique Plates

Oblique plates are utilized in a variety of applications in steel structures such as panel zones. As shear stresses generally act upon panel zones, it is important to evaluate the shear buckling resistance of panel zones, especially as thinner steel plates have been used in recent years. In contrast to the stability problem of rectangular plates, which has been extensively studied, the elastic buckling strengths of oblique plates are not entirely clear, even under basic stress conditions such as uniform shear stress. This study investigated the elastic shear buckling stress of oblique plates. The main goals were to examine the buckling behavior of simply and clamped supported oblique plates and propose convenient and reliable design equations to predict the elastic shear buckling coefficients. By adopting the energy method based on Timoshenko’s plate buckling theory, the influence of the geometry of the oblique plates and the direction of the shear stress were investigated. Furthermore, highly convenient design equations for elastic shear buckling coefficients were proposed. The accuracy of the proposed equation was assessed using extensive parametric studies based on the finite element method using MSC Marc software.

A review of the analytical solution methods for the eigenvalue problems of rectangular plates

This paper presents a review of typical analytical solution methods for the eigenvalue problems of rectangular plates. Apart from the exact Levy and Navier methods, this paper mainly focuses on non Levy-type of solution methods, including the Kantorovich-Krylov method, the separation-of-variable methods, the series expansion-based methods, the symplectic eigenfunction expansion method and the dynamic stiffness method. For each method, the motivation factors, development history and relevant works are introduced first, then a rectangular thin plate vibration problem is taken as an example to describe solution procedures. All methods are compared in terms of mode function forms, the principles based on, features and application scopes. Numerical results showed that all methods can obtain highly-accurate frequencies and mode functions.

Comparative Analysis of Two-Dimensional Steady State Heat Flow Problem of a Rectangular Plate by Analytical and Numerical Approach

Heat transfer problems play a very vital role in many fields of engineering. This present analysis work focuses on analytical and numerical approaches to resolve a two-dimensional steady-state heat conduction problem of a rectangular plate. The temperatures at the inside points of the rectangular plate are calculated analytically and the results are verified numerically. For numerical solution of the problems programming is done in using METLAB software. The material of the rectangular plate used is steel. It will be seen from the literature survey that rigorous analytical solutions are available only for a very few simple boundary conditions and these conditions does not seem to be favourable for complex boundaries. However, majority of engineering problems with simple as well as complex boundary conditions will be solved with the help of numerical approaches. In this analysis work, an attempt is made to obtain solutions using numerical method for a two-dimensional steady state heat conduction problems of a steel rectangular plate. The mathematical formulation of problems is done using Laplace method. The results obtained from the two different methods that is analytical and numerical are then compared with each other. The agreement between the analytical and numerical results is a sign of the accuracy of solution method.

MLS-Based Numerical Manifold Method for Modeling the Cracked Rock Considering the Contact of the Crack Surface

To simulate the moving boundary problems, the moving least square–based numerical manifold method, abbreviated as MLS-based NMM, was proposed. The MLS-based NMM has been applied successfully to open crack problems, which exhibits the high accuracy and strong robustness. In this study, we extend the MLS-based NMM to simulate the cracked rock considering the contact of the crack surface. Simultaneously, in order to simulate the progressive failure of the cracked rock, an improved strength-based criterion is proposed. The criterion is based on the Mohr–Coulomb criterion and maximum tensile stress criterion. Because rock can be regarded as a quasi-brittle material, a characteristic distance is used to calculate the crack tip stress and correct the crack propagation direction which avoids the phenomenon of “Zig-zag” for the crack propagation path based on the fracture mechanics criterion. The proposed strength-based criterion can acquire the crack tip stress and propagation direction and also realize the automatic determination of the crack propagation length in each step of the crack growth. A Brazilian disc problem and a rectangular plate problem are adopted to verify the numerical model. At last, the numerical model is applied to study the progressive failure process of the rock slope. The results indicate that the proposed method can deal with the crack propagation in the rock and the opening/sliding of rock blocks along discontinuities in a natural way.

Effect of extreme in-plane and transverse stiffness ratios on frequencies and load transfer between face sheets and core of rectangular sandwich plates

Challenging issues for sandwich structures include accurately finding their frequencies and how the load transfers from face sheets to the core when their material stiffness ratio (FCSR) in either the axial or the transverse directions varies from 1 to 107. Here we use both the equivalent single layer and the layer wise shear and normal deformable plate theories to analyze problems for rectangular plates of different thickness/length ratios. The Ritz method with Jacobi polynomials as basis functions is employed to numerically solve three-dimensional linear elasticity theory equations. Transverse stresses are calculated using a one-step stress recovery scheme. It is found that the FCSR in the transverse direction determines the minimum degree of complete polynomials in the thickness coordinate needed to accurately compute the lowest six frequencies and the transverse stresses. New results include the following: (i) the FCSR in the transverse direction has a greater influence than that in the axial direction on the computed frequencies and transverse stresses, and (ii) whereas both transverse normal and shear stresses in the core contribute to transmitting loads between the two face sheets for FCSR < 103, however, only the former is effective for FCSR ≥ 105.

Plane Stress Analysis of Simply Supported Rectangular Plates by The Nodal Line Finite Difference Method.(Dept.C)

The nodal line finite difference method developed earlier by the author has been successfully applied to bending analysis of rectangular plates. In the present work, the method is extended to solve plane stress problems of rectangular plates. The analysis deals with the in-plane displacements and requires the solution of two simultaneous second-order partial differential equations. These equations are transformed into two simultaneous ordinary differential equations which in turns are cast in two simultaneous nodal line difference equations. Numerical results obtained by the finite strip method and this indicates the applicability and the power of the present technique.

A simple finite element procedure for free vibration of rectangular thin and thick plates

A simple finite element procedure is introduced and developed to solve the free vibration problem of rectangular plates. The proposed method first reduces the original plate problem to two simple beam problems. Each beam problem is then discretized separately with the help of one-dimensional finite element method (FEM). Since only the one-dimensional FEMs are dealt with in the proposed approach, the implementation of the proposed approach is much easier than the case where the two-dimensional FEM is applied to the problem. The reliability and applicability of the proposed method are demonstrated herein through the solution of some illustrative problems, including the vibration problems of thin and thick rectangular plates. The simulations have been done for plates with different boundary conditions including the simply supported, clamped, and free boundary conditions. The results generated by the proposed method are compared with analytical and numerical results available in the literature and excellent agreement is achieved.

Generalization of a sandwich structure model: Analytical studies of bending and buckling problems of rectangular plates

The paper is devoted to generalization of the analytical model of sandwich structures. The continuous variation of mechanical properties in thickness direction of the structure wall is proposed. The individual nonlinear theory of deformation of the straight line normal to the neutral surface is developed. This analytical model of sandwich structures is presented in detail for the example rectangular plate. Based on the principle of stationary potential energy two differential equations of equilibrium of the plate are obtained. The system of equations is analytically solved. The maximum deflection and critical loads of the example plates are derived. Results of these analytical studies are presented in Figures and Tables.

Analysis of bending plates with mixed boundary conditions using generalized equations of finite difference method

Currently, plates with characteristics of high thermal insulation, sound insulation and strength are sufficiently and widely used in structural engineering. To describing the connect of the plates with other elements, it is possible to give many calculation schemes on the different views. In this paper, it is considered the problem of rectangular plates, each side of which are fixed with hinged support or fixed support. This paper deals with the discontinuous points of boundary conditions with the use generalized equations of finite difference method. The obtained results of the analysis of stress state in discontinuous area of boundary conditions are compared with the results of work of Smirnov V.A.

Stability of SCSC Rectangular Plates under Intermediate and End Uniaxial Loads

AbstractThis paper is concerned with a new buckling problem of rectangular plate subjected to uniaxial loads. The considered plate has two clamped edges parallel to x-axis while the other edges parallel to y-axis are simply supported (SCSC) subjected to combined uniaxial loads. The analysis of the plate under study is performed using minimum potential energy techniques andthe results are examined using finite element method. The energy method depends on the assumed deflection function, which satisfies exactly the end conditions and capable of representing the deflected plate surface. The analysis furnishes an approximate stability solutions; presented in curves describe the relation between the ratio of the intermediate load to the end load both, aspect ratio and the location of the intermediate load. These results can be used simply to design plates or walls that have to support intermediate floors/loads.Comparisons between the results of the energy technique and the finite element methods show very good agreement.

Hencky Bar-Net Model for Vibration of Rectangular Plates with Mixed Boundary Conditions and Point Supports

This paper is concerned with the development of the Hencky bar-net model (HBM) for free vibration analyses of rectangular plates with mixed boundary conditions and point supports. The HBM is a two-dimensional discrete net system composed of rigid segments connected by frictionless hinges and rotational springs. In the model, bending is accommodated by rotational springs at each joint while the twisting by a diagonal spring system in each grid cell. The total mass of the plate is distributed as lumped mass at each joint and the continuous boundary stiffness of plate is simulated by springs located at the edge joints. Owing to the discrete property of HBM, it is able to readily handle any boundary conditions of plates including mixed boundary conditions and point supports. The HBM is herein used to solve some vibration problems of rectangular plates with mixed boundary conditions and point supports to demonstrate its accuracy and convenience for plate analyses.

求解弹性地基上自由矩形中厚板弯曲问题的辛-叠加方法

Based on the symplectic superposition method proposed in recent years, the bending problems of free-edge rectangular thick plates resting on elastic foundations were analytically solved. The original problem was split into 3 subproblems corresponding to the bending problems of rectangular thick plates with 2 opposite edges slidingly clamped and resting on elastic foundations, which were solved with the symplectic geometry method. The analytic solution of the original problem was then obtained through superposition. Compared to the conventional analytic approaches such as the semi-inverse method, the symplectic superposition method has the advantages of both rationality of the symplectic method and regularity of the superposition method. The solution procedure starts from the basic equations of elasticity, and a rigorous derivation yields the analytic solutions, thus extending the scope of problems to be solved. The present method can serve as an effective analytic approach to complex boundary value problems of high-order partial differential equations in elasticity, as represented by the rectangular plate problems.

Hencky bar-net model for buckling analysis of plates under non-uniform stress distribution

The Hencky bar-net model (HBM) has been shown to be a physical structural representation of the finite difference plate model (FDM). This paper extends the HBM to tackle buckling problems of rectangular plates under in-plane non-uniform loads. This class of problems requires a pre-buckling stress analysis involving plane-stress elasticity equations in order to establish the non-uniform stress distributions in the plate domain for input loads to the HBM. The converged results given by HBM compare well with existing reference solutions; thereby verifying the model. The HBM is also used to generate some new buckling solutions for rectangular plates under various point loads and partially distributed loads.

Exact solutions for the free in-plane vibrations of rectangular plates with arbitrary boundary conditions

Though the free in-plane vibration problem of rectangular plates with classical boundary conditions have been studied using various numerical methods like direct separation of variables, the elastic boundary restraint is more general in the practical structural applications compared with the perfect classical boundary condition. Thus, in this paper, an innovative exact solution on the basis of the method of reverberation ray matrix (MRRM) and golden section search (GSS) algorithm is presented to study the exact free in-plane vibration characteristics of rectangular plates with arbitrary boundary conditions. In the present method, the exact natural frequencies and mode shapes can be very easily obtained when two opposite plate edges have either type of the simply supported boundary conditions and the resting plate edges have the arbitrary boundary conditions including the classical case and general elastic restraints. By the comparison with other published exact solutions, it is shown that the present exact solutions have some obvious advantages which are performed in the excellent accuracy, reliability and efficiency. Performing the study, some of the first-known exact results for the free in-plane vibrations of rectangular plates with elastic boundary restraints are also obtained, which can be worked as the benchmark data for the future approximate methods.

Discrete singular convolution and Taylor series expansion method for free vibration analysis of beams and rectangular plates with free boundaries

Since the excellent performance of the discrete singular convolution (DSC) for high order frequency analysis has not been demonstrated for rectangular plates with free edges and corners thus far, this paper proposes a novel approach in applying the DSC for vibration analysis of beams and rectangular plates with arbitrary boundary conditions. The Taylor series expansion method is used to apply the boundary conditions. The existing difficulty encountered at plate corner points is overcome by the proposed method. The essential idea of the novel method is that two Taylor series expansions with different orders are used to eliminate the fictitious points outside the physical domain. Various beam and rectangular plate problems are solved by the proposed DSC and the results are compared with exact solutions, finite element data and results obtained by the differential quadrature method (DQM) using much higher number of grid points. It is demonstrated that the Taylor series expansion method to apply the boundary conditions is general and accurate. It is also found that the performance of the proposed DSC is excellent and independent of boundary conditions.

Application of the method of weighted residuals for determining the distribution of temperatures and radiation creep strains in plates

The paper is devoted to the description of the Weighted Residuals Method in Galerkin form for solution of the non-stationary heat transfer and irradiation creep problems in rectangular plates. The basic methods and dependencies were used for the solution of these problems are considered. The non-stationary heat problem is derived to sequential decisions of the systems of linear algebraic equations for the coefficients in expansions of temperature function in series on the basis functions for the different time moments. The problem of rectangular plate’s bending with specified temperature drops in its sides is preliminary solved in order to obtain the solution of irradiation creep problem. The stress tensor components were found in the plate’s points, are used in associated flow rule for the obtaining the components of creep irradiation strain tensor. The non-stationary heat transfer problem in quadratic area with boundary conditions of Newton and Diriclet types is considered as well as the calculated temperature distributions on co-ordinates in plane and time are presented. The irradiation creep of rectangular plate made from high resistant steel 15Х1М1Ф is considered. Obtained distributions of irradiation creep strains for three regarded values of temperature drops allow us to evaluate the plate deformation and service ability at different time moments.