Orthotropic Rectangular Plates Research Articles

Buckling Delamination of a Rectangular Orthotropic Thick Plate with an Embedded Rectangular Crack Subjected to an Axial Compressive Force

A 3D buckling delamination problem for a simply supported rectangular orthotropic composite thick plate containing a rectangular crack and subjected to an axial compressive force is studied. In the initial state, crack faces have an infinitesimal imperfection. The development of this imperfection caused by an external compressive loading along the crack is investigated within the framework of the 3D geometrically nonlinear elasticity theory of anisotropic bodies. The corresponding boundary-value problems are solved numerically by the 3D FEM. The influence of material and geometrical parameters of the plate on its critical buckling force is evaluated and discussed.

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.

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.

New analytical free vibration solutions of orthotropic rectangular thin plates using generalized integral transformation

A first endeavor is made to obtain the analytical free vibration solutions of orthotropic rectangular thin plates utilizing the generalized integral transformation technique. Owing to the nature of the problem, it is very hard to get the exact solution of the title problem by common inverse/semi-inverse method. In solution procedure, the vibrating beam function is selected as the integral core to form the generalized integral transformation pair. Then, the high order partial differential equation under specific boundary conditions is converted to linear algebraic equations and the exact solution is achieved in a straightforward way. One of the advantages of the proposed method is its simplicity and versatility and does not require pre-determining the deflection function which makes the solving procedure more reasonable. The method has wide applications range and can handle other elastic plate problems, such as shear buckling, buckling, and bending. The present results are validated by comparing with the existing analytical solutions which show satisfactory agreement.

Analytical buckling loads for corner supported rectangular orthotropic and symmetrically laminated plates

AbstractIn this research we present a new analytical solution for finding the buckling loads of thin isotropic and orthotropic rectangular plates in which all four corners are supported. This new solution is also capable of solving all cases where few or all the edges are supported. The methods that are 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. Using this new method, exact buckling loads and buckling modes of many new cases of classical boundary conditions are found. Results are given for several stiffness ratios in both directions of the plate, and for uni‐directional and bi‐directional loading.

Calculation of Non-Homogeneous Anisotropic Rectangular Plates with Arbitrary Fixation on the Contur

The developing of the effective methodic of elastic orthotropic plates’ calculation and the research on the base of their state under different boundary conditions are of great importance nowadays. The representation of the received results in the form, convenient for practical use, is also important. For practical applications in engineering are important tables for determining deflections and internal forces of structures. Such tables for the isotropic case under various conditions of plate support on the contour are given in many works. As for the anisotropic plates, there are no such tables, with the exception of one Huber table compiled for a freely supported rectangular orthotropic plate, depending on the relationship between the stiffness values. Here is a method of calculating the non-homogeneous anisotropic rectangular plates with arbitrary fixation on the contour is set forth, which is reduced to a boundary value problem. The main idea of a calculated general methodic of linear marginal differential tasks calculation is based on underlying of the main part of a solution. Such approach is proved by means of development and some generalization of common positions of a variational method of marginal tasks of mathematical physics of self-conjugated tasks solution. To solve a system of equations in terms of displacements using finite difference method (FDM) in combination with different variations of analytical solutions. It is advisable to construct a numerical solution of the problem so that in difficult cases the support fixing and uploading solution sought, not directly, but in the form of amendments to the known solution for simple cases of reference to consolidate and uploading at finding the solutions which the analytical methods or the FDM with sparse mesh may be used. Given as examples are the results of calculation for a series of square orthotropic plates with a fixed boundary under the action of uniformly distributed load.

Dynamic stiffness formulation and free vibration analysis of specially orthotropic Mindlin plates with arbitrary boundary conditions

A novel dynamic stiffness formulation which includes the effects of shear deformation and rotatory inertia is proposed to carry out the free vibration analysis of thick rectangular orthotropic plates i.e. the formulation is based on an extension of the Mindlin theory to orthotropic plates in a dynamic stiffness context. The modified trigonometric basis is used to construct the general solution for the free vibration problem of the plate in series form, permitting the derivation of an infinite system of linear algebraic equations which connect the Fourier coefficients of the kinematic boundary conditions of its four edges. The boundary conditions are essentially the deflections and angles of rotations constituting the displacement vector and shear forces and bending moments constituting the force vector. The force-displacement relationship is then obtained by relating the two vectors via the dynamic stiffness matrix. Essentially the ensuing infinite system forms the fundamental basis of the paper, which defines the dynamic stiffness matrix for an orthotropic Mindlin plate in an exact sense. Numerical results are obtained by using the Wittrick-Williams algorithm as solution technique. The theory is first validated using published results and then further illustrated by a series of numerical examples. The paper concludes with significant conclusions.

New analytic free vibration solutions of orthotropic rectangular plates by a novel symplectic approach

This paper aims at analytically solving the free vibration problems of orthotropic rectangular plates, which have found broad applications as a class of crucial aerostructures. A novel symplectic superposition method is presented to give analytic solutions of plates without two opposite edges simply supported (i.e., non-Levy-type plates). Since these solutions are very difficult to obtain within the classical Lagrangian system framework, the new approach proceeds in the Hamiltonian system, in physics, and in the symplectic space, in mathematics. The validity of separation of variables and symplectic eigenexpansion guarantees the realization of new analytic solutions in a rigorous step-by-step way, which cannot be achieved by some classical analytic methods where the predetermination of solution forms is usually inevitable. Representative challenging problems, including those incorporating two adjacent free edges (e.g., cantilever and free plates), are successfully solved and validated by the refined finite element modeling. Comprehensive natural frequency and mode shape results show the accuracy of the present solutions. The generality of the symplectic superposition method enables one to pursue more analytic solutions of similar problems that are governed by higher-order partial differential equations.

Investigation of energy release rate under mode II loading in a rectangular orthotropic plate with a band crack using 3D finite element analysis

A rectangular plate containing a band crack and made of orthotropic elastic material is considered and the energy release rate (ERR) component in mode II at this crack front is investigated. It is assumed that the crack’s edge planes are parallel to the upper and lower face planes of the plate and the edge planes of the crack are loaded with uniformly distributed sliding forces. The corresponding boundary value problem is formulated within the scope of the three-dimensional linear theory of elasticity and the solution to this problem is found by employing 3D finite element method (3D FEM) modeling. Numerical results on the ERR for mode II and on the influence of the mechanical-orthotropic properties of the plate material are presented and discussed. In particular, it is established that the influence of the shear modulus in a plane which is simultaneously perpendicular to the crack’s front and crack’s plane on the ERR is more significant than that caused by the change of another modulus of the orthotropic material.

Dynamic calculation of nonlinear oscillations of viscoelastic orthotropic plate with a concentrated mass

Plates, panels and shells made of composite material with fixed objects in the form of an additional mass have found a wide use due to their viscoelastic and strength properties. An analysis of their dynamic behavior indicates a significant effect of inhomogeneity of an associated mass type on their strength. The problem of oscillations of a viscoelastic orthotropic rectangular plate with an associated mass is considered according to the Kirchhoff-Love hypothesis in a geometrically nonlinear statement. This problem is reduced to solving the systems of nonlinear integro-differential equations with singular relaxation kernels, solved by the Bubnov-Galerkin method in combination with a numerical method based on the use of quadrature formulas. The numerical values of the approximate solution have been calculated in the Delphi programming environment. At wide range of changes in physicomechanical and geometrical parameters, the behavior of the plate has been studied. The effect of viscoelastic and inhomogeneous material properties, concentrated mass and their location on the oscillatory process of a rectangular plate is shown.

Two-dimensional generalized finite integral transform method for new analytic bending solutions of orthotropic rectangular thin foundation plates

In this paper, a two-dimensional generalized finite integral transform method is developed for new analytic bending solutions of orthotropic rectangular thin foundation plates. The vibrating beam functions are adopted as the integral kernels to construct the integral transform pairs. By imposing the transform to the governing equation, utilizing some inherent properties of the beam functions, the title problem is converted to that of solving a system of linear algebraic equations, by which the new analytic solutions are elegantly obtained in a straightforward way. Numerical examples validate the present method as well as the solutions yielded by satisfactory agreement with the literature and finite element analysis.

Free Transverse Vibration of Rectangular Orthotropic Plates with Two Opposite Edges Rotationally Restrained and Remaining Others Free

Many types of engineering structures can be effectively modelled as orthotropic plates with opposite free edges such as bridge decks. The other two edges, however, are usually treated as simply supported or fully clamped in current design practice, although the practical boundary conditions are intermediate between these two limiting cases. Frequent applications of orthotropic plates in structures have generated the need for a better understanding of the dynamic behaviour of orthotropic plates with non-classical boundary conditions. In the present study, the transverse vibration of rectangular orthotropic plates with two opposite edges rotationally restrained with the remaining others free was studied by applying the method of finite integral transforms. A new alternative formulation was developed for vibration analysis, which provides much easier solutions. Exact series solutions were derived, and the excellent accuracy and efficiency of the method are demonstrated through considerable numerical studies and comparisons with existing results. Some new results have been presented. In addition, the effect of different degrees of rotational restraints on the mode shapes was also demonstrated. The present analytical method is straightforward and systematic, and the derived characteristic equation for eigenvalues can be easily adapted for broad applications.

Vibration of the rotating rectangular orthotropic Mindlin plates with an arbitrary stagger angle

In this article, the vibration analysis of rotating moderately thick cantilever orthotropic plate is analytically investigated. Based on the first order shear deformation plate theory, the partial differential equations of motion are derived using Hamilton’s principle. The centrifugal inertia forces and Coriolis effects due to the rotation are all considered. The analytical approaches, both extended Kantorovich method and extended Galerkin method, are employed to obtain the solution of the problem. Results obtained by these two methods are compared with those available in the open literature and good agreements are observed. The effects of various parameters, individually or in combination, on the vibrational behaviors are analyzed in detail. From the studies, it is found that in the rotating plates, when the stiffness ratio increases, the crossing/veering phenomenon occurs in the lower mode orders and/or rotation speeds. The results show that for each stiffness ratio, the effect of hub radius ratio is more significant on the out-of-plane mode frequencies than on the in-plane ones.

Double scaling and master curve to predict Kts for elliptically notched orthotropic plates from Kts in circularly notched isotropic plates

The key role played by the so-called stress concentration factors (symbolically usually referred to as Kts) while performing either analysis or design in both mechanical engineering and structural engineering is a well-proven fact; hence, the corresponding accuracy and ease of determination in their estimation have critical implications related to engineering matters. In a previous work, we showed that starting with the Kts for rectangular isotropic plates with circular holes the Kts for rectangular orthotropic plates with elliptic holes can be easily and accurately predicted, using a double scaling procedure (a geometric scaling and a material scaling) together with a basic Kt curve employed as a master curve. In the present work we investigate, in an engineering heuristic manner, whether the stated double scaling procedure exhibits a more general structure so that it can accurately be applied to different plate stress raiser geometries. Specifically, we examine if starting with the known Kts for rectangular isotropic plates with circular notches, used as a master curve, the Kts for orthotropic plates with elliptic notches can be easily and accurately predicted as well. Using a simple hand calculator, we show that the resulting proposed set of predicting mathematical expressions makes virtually unnecessary the use of complex numerical programs (FEM based for example) to determine the corresponding Kts, producing at the same time accurate results with a maximum average recorded error of 2.59%. Additionally, given that a Kt predictive procedure for holed plates was presented in our previous work, and for notched plates in this work, we found necessary first to examine the hypothesis that suggests using Kts from holed plates as good approximations for Kts for notched ones and vice versa. We found this hypothesis not to be valid for either isotropic or orthotropic plates. In the course of this work it was found, rather significantly, that the well-known double square root material contribution to Kts is valid for both holed plates as well as notched ones.

Thermal effect on vibration and buckling analysis of thin isotropic/orthotropic rectangular plates with crack defects

The present research is concerned with the vibration analysis of thin isotropic and orthotropic rectangular plates with crack defects under thermal environmental conditions. In the literature, there are only few studies reported in this direction. Based on the classical plate theory, the governing equations of the isotropic and orthotropic cracked rectangular plates can be derived, in which a surface crack located at the plate center is formulated based on a line-spring model. Since the dynamic behavior of structural elements is significantly affected by thermal effects, a thermal buckling analysis for isotropic and orthotropic plates is also conducted. A uniform heating load on the cracked rectangular plates is considered and the critical buckling temperature of the plates with or without cracks is investigated. The discrete singular convolution (DSC) method is then applied to formulate the eigenvalue equations for the cracked rectangular plates under various thermal conditions. The DSC technique is an ingenious method in stability and dynamic analysis of plates, not only it is a flexible local method to handle complex geometries and boundary conditions, but also it performs as a global approach with a high degree of accuracy. To go beyond the limitation of the DSC method, the use of Taylor’s series expansion method is incorporated for the treatment of free boundary conditions. In addition, this is the first attempt to explore its application on the analysis of cracked rectangular plates under thermal effects. In this work, the vibration of isotropic and orthotropic cracked rectangular plates with various combinations of boundary conditions is studied. A special restrained manner of simply supported conditions that are permissible for in-plane movements is also analyzed. The obtained solutions herein are compared with the existing results to verify the accuracy and reliability. Besides, accurate first-known solutions are also presented.

A vibro-acoustic quality control approach for the elastic properties characterisation of thin orthotropic plates

Many engineering structures are built by assembly of plates, and in recent years there has been a widespread use of composite plates, and in particular of orthotropic plates, to improve dynamic structural performance and reduce structural weight. Depending on the manufacturing process used, orthotropic plates might be affected by variability of the mechanical properties, which in turn will affect their performance. The four elastic constants of orthotropic plates can be obtained via direct static or dynamic tests, or via inverse methods which combine dynamic experiments with mathematical models. However these approaches may not be appropriate for establishing manufacturing variability across a number of manufactured plates, since they might require expensive testing equipment and they may be time consuming. In this paper an inverse procedure based on the Chladni patterns is investigated to analyse thin orthotropic rectangular plates. A detailed mathematical derivation of the equations needed for estimating the elastic constants based on the knowledge of five natural frequencies is presented. The repeatability and reproducibility of the results obtained with the proposed procedure is first assessed, and the evaluation of the manufacturing variability of an ensemble of “nominally identical” plywood plates with the proposed approach is then investigated.

Semi analytical solution of a rigid pavement under a moving load on a Kerr foundation model

This paper analyzes the dynamic response of assumedly rigid road pavement under a constant velocity of traffic loads moving on its surface. The model of the rigid road pavement is a damped rectangular orthotropic plate which is supported by an elastic Kerr foundation. Semi-analytical solutions of the dynamic deflection of an orthotropic plate, with semi rigid boundary conditions are presented by using governing differential equations. The natural frequencies and mode shapes of the system are then, solved by using the modified Bolotin method considering two transcendental equations as the results of solving the solution of two auxiliaries Levy’s plate type problems. The moving traffic loads modeled by varying the amplitudes of dynamic transverse concentrated loads harmonically. Numerical studies on the soil types, foundation stiffness models, varying constant velocities and loading frequencies are conducted to show the effects of the dynamic response behaviors of the plates. The results show that the dynamic responses of the rigid road pavement influenced significantly by the type foundation stiffness models and velocity of the moving load.

Buckling analysis of orthotropic thin rectangular plates subjected to nonlinear in-plane distributed loads using generalized differential quadrature method

ABSTRACTIn the present work, buckling analysis of orthotropic thin rectangular plates with uniform thickness resting on Pasternak foundation are investigated for eight types of boundary conditions: SSSS, CCCC, SCSC, SSSC, SSCC, CCCF, SSFC, and CFCF. Based on classical plate theory, governing differential equation in buckling are solved numerically using generalized differential quadrature method (GDQM) to obtain critical buckling loads and corresponding modes. The kinds of nonlinear loading are presented in six cases including symmetrical and unsymmetrical distribution. In addition, the effects of aspect ratio, orthotropic moduli ratio and coefficients of foundation on the buckling load are illustrated. The present work is the first attempt to consider the influence of the nonlinearity of distributed in-plane bi-directional loading in determination of buckling load and representation of the corresponding shape modes. Some numerical examples are provided to demonstrate good accuracy of the GDQ method to evaluate the critical buckling load in case of nonlinear distributed bi-directional compressive loads. As shown, profile of distributed in-plane loading plays an important role on buckling behavior of the rectangular plate.

Analytical solutions for flexure of rectangular orthotropic plates with opposite rotationally restrained and free edges

The problem of flexural characteristics of rectangular orthotropic thin plates with opposite rotationally restrained and free edges subjected to arbitrary transverse loads is investigated in present study. By introducing a rotational fixity factor to simulate the elastic rotation restraint at plate edges, the finite integral transform method is employed to obtain exact analytical solution of the problem. Numerical solutions for rectangular plates with various aspect ratios and edge flexibility are investigated and compared to the results reported in previous research. The accuracy and convergence of the present formulation are assessed by comparing with previously reported results. The excellent agreement on the results between the proposed method and previous studies indicates that the proposed method is accurate and effective.

On the free vibration of orthotropic and inhomogeneous with spatial coordinates plates resting on the inhomogeneous viscoelastic foundation

ABSTRACTThe paper developed the closed-form solution for the free vibration problem of inhomogeneous orthotropic rectangular plates (IHORPs) resting on the inhomogeneous viscoelastic foundation (IHVEF). The Young's moduli and density of the orthotropic plate vary continuously with respect to three spatial coordinates, while the characteristics of the viscoelastic foundation vary depending on the in-plane coordinates. The relevant motion equation is obtained using the classical plate theory (CPT) and solved using method of separation of variables. The influences of inhomogeneity of orthotropic materials, inhomogeneity of viscoelastic and elastic foundations on the non-dimensional frequencies (NDFs) of plates are studied in detail.