Orthotropic Rectangular Plates Research Articles

Nonlinear transient coupled thermo-electromechanical dynamic responses of the piezoelectric structure with the high-order temperature-dependent material properties

This work aims to study the nonlinear transient coupled thermo-electromechanical dynamic responses of the piezoelectric structure with the high-order temperature-dependent material properties. The multi-field coupling nonlinear model is established based on the non-Fourier thermoelasticity theories by introducing the function of high-order temperature dependency of the arbitrary material property. The virtual work principle and the nonlinear finite element approach are proposed to investigate the two-dimensional case of the orthotropic rectangular piezoelectric plate of the crystal class mm2 subjected to the zonal transient heating loads. The influences of the high-order temperature-dependent material parameters on the structural responses and wave propagations are discussed.

Symplectic superposition method for vibration problems of orthotropic rectangular thin plates with four corners point-supported

Symplectic superposition method for vibration problems of orthotropic rectangular thin plates with four corners point-supported

A method to estimate the rectangular orthotropic plate elastic constants using least-squares and Chladni patterns

A method to retrieve the elastic constants of rectangular wooden plates is presented, relying on the measurement of a set of eigenfrequencies and the identification of the corresponding mode shapes, and belonging to the more general category of non-destructive inverse parameter estimation methods. Compared to previous work, the current method is effective with any choice of boundary conditions. Furthermore, the error function is linear in the elastic constants, which may be computed easily via a matrix inversion. This framework lends itself naturally to a physical interpretation of the results in terms of linear combinations of eigenmodes, yielding new sets of modes and associated combined mode shapes in which the elastic constants are completely uncoupled. Several numerical benchmark tests and experimental cases are treated in detail, highlighting the reliability of the proposed methodology in cases of interest in acoustics and musical acoustics.

On the Dynamic Characteristics of Orthotropic Rectangular Plates under the Influence of Moving Distributed Masses and Resting on a Variable Elastic Pasternak Foundation

This work studies the dynamic characteristics of orthotropic rectangular plates under the influence of moving distributed masses and resting on a variable elastic Pasternak foundation. The governing equation is a fourth order partial differential equation with variable and singular coefficients. The solutions to the problem are obtained by transforming the fourth order partial differential equation for the problem to a set of coupled second order ordinary differential equations using the technique of Shadnam et al, then simplified using asymptotic method of Struble [11]. The closed form solution is analyzed, resonance conditions are obtained and the results are depicted graphically for both cases of moving distributed mass and moving distributed force

Hamiltonian system-based analytic thermal buckling solutions of orthotropic rectangular plates

New analytic solutions for the thermal buckling of orthotropic rectangular plates are presented using the symplectic superposition method (SSM). The SSM deals with higher-order partial differential equations within the Hamiltonian framework, distinguishing it from conventional methods in the Lagrangian framework. The solution procedure involves dividing the original problem into two subproblems, utilizing the techniques in the symplectic space, like separating variables and symplectic eigen expansion, to achieve analytic solutions, and using superposition to obtain the final solutions. One notable advantage of the SSM is the absence of predetermined assumptions, which is a departure from conventional semi-inverse methods. The new analytic solutions under challenging non-Lévy-type boundary conditions (BCs) were rarely addressed in prior research due to their complex mathematical characteristics, which reveals the main novelty. Comprehensive thermal buckling results are provided for isotropic/orthotropic plates under various BCs, validated by those from the finite element method with a maximum difference being 2.7374% and literature with a maximum difference being 5.9201%. Parameter analyses quantify the effects of BCs, aspect ratio, and thickness-to-length ratio on critical buckling temperatures and mode shapes.

Vibrational analysis of orthotropic rectangular plate having combination of circular thickness and parabolic temperature

The objective of this study is to investigate the natural vibration of a rectangular plate made of orthotropic material with circular thickness (two dimensions) and temperature variation on the plate is parabolic (two dimensions) in nature. The solution to the problem is obtained by utilizing the Rayleigh-Ritz technique and the first four frequency modes are obtained under clamped edge conditions. The study aims to provide numerical data that demonstrate how circular variation in tapering parameters of plate can effectively control and optimized vibrational frequencies of the plate. Orthotropic rectangular plate, thermal gradient, circular tapering, aspect ratio.

Influence of plate thickness on the results of residual stresses determination by through hole drilling in orthotropic composites of different fiber orientation

The main objective of this study is to characterize residual stress in composite orthotropic plates using hole drilling method. The values of hole diameter increment in the principal anisotropic directions, which were obtained from electronic speckle-pattern interferometry on the external surface of the specimen, serve as the initial experimental information. The theoretical foundation of the approach is based on S. G. Lekhnitsky's analytical solution, which describes a stress concentration along the edge of the central open hole in a rectangular orthotropic plate under tension in principal anisotropy directions. The transitional model, that is essential link for deriving residual stress components from initial experimental data, provides the unequivocally solution to the properly posed inverse problem. Firstly, the validation of the transitional model regarding prescription of residual stress values for specific plies within a specimen’s thickness is based on calibration experiments. These experiments yield data for unidirectional and cross-ply composite plates of three different thicknesses. The results demonstrate that the values of principal residual stress components for cross-ply composite material are independent of plate thickness. This means that the transition model, based on the measurements of deformation responses to through hole drilling on the external plate face, can be reliably implemented to characterize the maximal residual stress values present in the middle plane of the composite plate. An investigation into the effect of hole diamter values on the determination of residual stress is conducted for cross-ply and angle-cross-ply composites. It is shown that a hole of diameter of 3.2 mm provides the most reliable results.

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.

Symplectic superposition solutions for free in-plane vibration of orthotropic rectangular plates with general boundary conditions

This work reports new analytic free in-plane vibration solutions for orthotropic non-Lévy-type rectangular plates, i.e., those without two opposite edges simply supported, by the symplectic superposition method (SSM), which has never been applied to in-plane elasticity problems in any existing works. Such analytic solutions are not accessible through conventional analytic methods as seeking analytic solutions that meet both the governing partial differential equations and various non-Lévy-type boundary conditions is an acknowledged challenge in mechanical analysis of plates. The clamped and free plates are considered as two most representative cases of non-Lévy-type plates. The SSM is implemented in the Hamiltonian system-based symplectic space, where the separation of variables and the symplectic eigen expansion prove to be well-grounded. These two mathematical treatments are adopted to first gain the analytic solutions of two elementary problems. The final analytic free in-plane vibration solutions are obtained by superposition of the two elementary problems. Comprehensive new natural frequencies and vibration modes are studied and validated by reference solutions from the finite element method or other approaches. The rigorous solution procedure, fast convergence, and highly accurate results render the present framework capable of serving as benchmarks for future comparison and applicable to analytic investigation of more plate problems.

RAMIFICATION OF EQUILIBRIA OF A COMPRESSED ELASTIC ORTHOTROPIC PLATE WITH INTERNAL STRESSES

The problem of loss of stability and post-critical behavior of a compressed orthotropic rectangular plate on a nonlinearly elastic base containing continuously distributed fields of dislocations and disclinations and under the influence of a small normal load is considered. The compressive force components are evenly distributed along the edges and act parallel to the main directions of elasticity. The problem is formulated as an analogue of a system of nonlinear Karman equations for an orthotropic plate containing a function called the incompatibility measure, which is expressed in terms of the densities of edge dislocations and wedge disclinations. The system of equations takes into account the small transverse pressure and the reaction of the elastic base in the form of a polynomial of the third degree of deflection. The following boundary conditions are considered: the edges of the plate are freely pinched or movably pivotally supported; two opposite edges of the plate are freely pinched or movably pivotally supported, and the other two are free from loads. The stress function is sought in the form of two components: the stress function caused by the presence of internal sources, determined from the linear boundary value problem, and the stress function caused by the external impact of compressive loads and a nonlinear elastic base, which is determined from the nonlinear boundary value problem. The nonlinear boundary value problem is investigated by the Lyapunov – Schmidt method. To solve the linearized equation from which the critical value of the compressive load is determined, the variational method is used in combination with the difference method. A system of branching equations of the Lyapunov – Schmidt method is constructed, which is investigated by numerical methods. The post-critical behavior of the plate is investigated and asymptotic formulas for new equilibria in the vicinity of the critical load are derived. For various values of compressive loads, the orthotropy parameters of the plate and the intensity parameter of internal stresses, the relations between the values of the base parameters are established, at which its bearing capacity is preserved in the vicinity of the classical value of the critical load.

Modeling of temperature processes in orthotropic boards of radio-electronic devices

The design of electronic equipment that is resistant to external influences is a topical problem. The arrangement of individual circuit parts is modeled in the form of an orthotropic plate. The paper describes the solutions of homogeneous and inhomogeneous thermal conductivity equation for an orthotropic rectangular plate, obtained by the recurrence-operator method. Numerical results are given for two cases of temperature distribution, satisfying zero boundary conditions, during cooling and heating and given initial conditions.

The analytical bending solutions of orthotropic rectangular plates with four clamped edges by the symplectic superposition method

The analytical bending solutions of orthotropic rectangular plates with four clamped edges by the symplectic superposition method

An Accurate Computational Method for Buckling of Orthotropic Composite Plate with Non-Classical Boundary Restraints

New accurate buckling analysis for rectangular orthotropic thin plates with complicated non-classical boundary restraints are conducted through adopting the finite Fourier integral transform approach. Non-classical boundaries such as rotationally restrained edges increase the mathematical difficulty in processing problems of plates, which leads to rare analytical results for benchmark use. The proposed approach is implemented in the framework of integral transform theory, in which trial function for the deflection is not necessary, and offers uniform solution procedures for problems of plates with various boundaries via adopting different integral kernels. The main merits of the approach employed is to enable one to change the complicated title problem into dealing with linear algebraic equations easily solved. Via altering the rotational spring factors introduced, buckling behaviors of plates with Levy-type boundaries and non-Levy-type boundaries can also be studied. Finally, all the given results including critical load factor and mode shape match the FEM analysis exactly, which illustrate the accuracy and validity of the method.

Critical Loads of Uniformly Compressed Orthotropic Rectangular Plate on an Elastic Base

Introduction. The problem of critical loads of a compressed orthotropic rectangular plate on an elastic base was considered. The following orthotropy parameters were set for the plate: Poisson coefficients, Young's modules for the main directions, and the shear modulus of the plate material. The components of the compressive load were uniformly distributed along two opposite edges of the plate and acted parallel to the coordinate axes. The edges of the plate were loosely pinched or pivotally supported. Cases were also considered when two parallel edges of the plate were free from loads, and the other two were freely pinched or pivotally supported.Materials and Methods. The problem was studied on the basis of a system of nonlinear Kármán-type equilibrium equations. The critical values of the load parameter were determined from a linearized problem based on a trivial solution. At the same time, the variational method in combination with the finite difference method was used to solve the boundary eigenvalue problem.Results. The problem was reduced to solving a parametric linear boundary eigenvalue problem. In case of boundary conditions of a movable hinge support, exact formulas of eigenvalues and eigenfunctions were given. While in case of free edge pinching, a variational method was used in combination with a finite-difference method, and a computer program for solving the problem was built. It was established that one or two eigenfunctions expressing the deflection of the plate could correspond to the critical value of the compressive load parameter at which the stability of the compressed plate was lost. The results of numerical calculations of the critical values of the compressive load at different values of the orthotropy parameters were presented, and graphs of the corresponding equilibrium forms were constructed. For the case of a long orthotropic plate on an elastic base, it was established that the main term of the asymptotic expansion of the solution to the linear eigenvalue problem was determined from the problem of critical loads of a compressed beam on an elastic base with an elastic modulus that coincides with the elastic modulus of the plate in the longitudinal direction.Discussion and Conclusions. The problem of critical loads of an orthotropic plate compressed in two directions lying on an elastic base was investigated. As the compressive load component increased along one direction, the critical value of the load compressing the plate along the other direction decreased. If an orthotropic plate was compressed by a load along a direction that corresponded to a greater bending stiffness, then the critical value of the loss of stability was greater than the critical value of the compressive load acting along the direction of a lesser bending stiffness. The presence of an elastic foundation increased the bearing capacity of the compressed plate.

Viscoelastic free vibration analysis of in-plane functionally graded orthotropic plates integrated with piezoelectric sensors: Time-dependent 3D analytical solutions

This paper proposes an accurate three-dimensional framework for elastic and viscoelastic free vibration investigation of in-plane functionally graded (IPFG) orthotropic rectangular plates integrated with piezoelectric sensory layers. The developed analytical framework is capable of considering layer-wise unidirectional linear functional gradation in both stiffness and density of the orthotropic composite layers. 3D piezoelasticity-based governing equations of motion are formulated in mixed form by employing Hamilton’s principle, and further solved analytically for Levy-type support conditions using the power-series-based extended Kantorovich method (EKM) jointly with Fourier series. The displacements, stresses, and electrical variables (electric field and electric potential) are solved as the primary variables that ensure the point-wise interlayer continuity and electro-mechanical support conditions. The viscoelastic property of the orthotropic interlayer is defined by employing Biot model, which is similar to the standard linear viscoelastic model. The correctness and efficacy of the present mathematical model are established by comparing the present numerical results with published literature and 3D finite element results, obtained by utilizing user material subroutine in the commercial FE software ABAQUS. An extensive numerical study is performed for various configurations and thickness ratios to investigate the influences of in-plane gradation, viscoelasticity and their coupled effects on the free-vibration response of hybrid laminated plates. It is found that in-plane gradation of stiffness and density remarkably alters the flexural frequencies and corresponding mode shapes of the hybrid intelligent rectangular plates. The flexural frequencies and stresses in the plate can be modified by selecting suitable grading indexes. Another interesting observation is that the in-plane gradation shows a considerably less effect on the electrical response of piezoelectric layers, which can play a vital role in the design of sensors and actuators for dynamic applications. Further, the numerical study demonstrates a potential time-dependent structural behaviour based on the present viscoelastic modelling. The consideration of viscoelasticity could be crucial for analysing the mechanical behaviour of a wide range of polymer composites more realistically and for prospective temporal programming in smart structural systems by exploiting the viscoelastic effect. Although the present analytical solution has been proposed for the free-vibration investigation of smart in-plane functionally graded (IPFG) viscoelastic plates, it can also be utilized directly to analyze the symmetric and asymmetric laminated piezoelectric smart plates with constant properties.

Improvement of Calculation Technique for Flexible Orthotropic Plates on Elastic Base. Part 2. Calculation Results

The proposed paper is a continuation of the author's theoretical work presented earlier in the development of the theory and methodology for calculating flexible orthotropic reinforced concrete plates on an elastic foundation, taking into account the physical nonlinearity of the plate material. The paper presents numerical results of elastic and nonlinear calculations of an isolated rectangular orthotropic plate on an elastic foundation, modeled by an elastic homogeneous isotropic layer rigidly connected to a non-deformable foundation under the action of an external static load, taking into account the own weight of the reinforced concrete plate. The change in its stiffness at the time of cracking and further active opening of cracks has been taken into account in the calculation of the structure under study. The nonlinear calculation of the studied structure takes into account the change in its rigidity at the time of cracking and further active crack opening. The calculation of a flexible orthotropic plate on an elastic foundation in a nonlinear formulation is performed iteratively by the method of B. N. Zhemochkin. A mixed method of structural mechanics has been used to determine the coefficients of canonical equations and free terms. At the first iteration, the reinforced concrete plate is calculated as linearly elastic, homogeneous, and orthotropic; at the subsequent ones – as linearly elastic, inhomogeneous, and orthotropic at each Zhemochkin site. Camber plates with a clamped normal in the primary system of mixed method due to the action of a concentrated force are determined by the Ritz method when the deflections were represented as a power polynomial in a new original expression that has been proposed for the first time in the Part 1 of the paper. The solution algorithm has been implemented using the Wolfram Mathematica 11.3 computer program. Numerical and graphical results of elastic and non-linear calculations of sediment concrete road plate, contact stresses and bending moment diagrams on the plate are presented in the paper.

Dynamic stiffness formulation for isotropic and orthotropic plates with point nodes

In this paper, the dynamic stiffness method for isotropic and orthotropic rectangular plates with point nodes is developed, making it possible to integrate the dynamic stiffness properties for plates with the dynamic stiffness properties of other elements such as bars and beams, but importantly, the advanced theory allows amalgamation of the dynamic stiffness method with the conventional finite element method for the first time. The derivation of the dynamic stiffness matrices for isotropic and orthotropic plates with point nodes has been accomplished by implementing the Fourier coefficients of the boundary values of the amplitudes of forces and displacements of the plate to form the force–displacement relationship at nodal points, including the corners. This innovative objective has been achieved by developing a new form of discrete Fourier transform technique for modified trigonometric functions. Using some carefully chosen illustrative examples, the convergence of results is ascertained by using different number of node points and their locations on the plate edges. The proposed theory has substantial advantages over conventional dynamic stiffness theories for plates, particularly when applying non-classical different boundary conditions on plate edges. The computed numerical results are discussed with significant conclusions drawn.

Behavior of frequencies of orthotropic rectangular plate with circular variations in thickness and density

In this article, authors dealt with general solution of differential equation of orthotropic rectangular plate with clamped boundary conditions under bi-parabolic temperature variations. Rayleigh Ritz technique is adopted to solve the resultant equation and evaluate the frequencies for first four modes of vibration. The effect of circular thickness and density on frequencies of orthotropic plate are analyzed which is not done yet. The objective of the study is to optimize the frequency modes by choosing the appropriate variation in plate parameters. Th findings of the study complete the objective of the article. All the results are provided in tabular form.

Buckling analysis of isotropic and orthotropic square/rectangular plate using CLPT and different HSDT models

Buckling analysis of simply supported isotropic and orthotropic rectangular plate has been performed using Classical plate theory (CLPT) and various higher-order shear deformation theories (HSDT) developed by Reddy, Karama, Shi, and Touratier. The considered HSDT models utilize parabolic, trigonometric, and exponential distribution functions for shear strain along the transverse direction. The total minimum potential energy approach has been applied to find the governing equations using a generalized higher-order displacement field. The non-dimensional stiffness parameters are calculated for all the considered HSDT models employing governing differential equations. Subsequently, the governing differential equations have been solved using the Navier method. The non-dimensional critical buckling load (Ncr) as per different HSDT models has been discussed for various plate thicknesses. The Ncr of the rectangular plate considering HSDT and CLPT models for various plate thicknesses has been plotted to analyze the numerical result. Finally, a detailed discussion has been done to compare the Ncr values obtained using different shear deformation theories for isotropic and orthotropic plates. From the above critical analysis, the authors wanted to investigate the best suitable plate thickness range for CLPT as well as HSDT models for better accuracy in bucking load calculation.

Optimal projecting in the task of bending of rectangular orthotropic piecewise-homogeneous plate

Optimal projecting in the task of bending of rectangular orthotropic piecewise-homogeneous plate