Orthotropic Plates Research Articles

Estimation of the stress-strain state of orthotropic plates on elastic foundation using the bimoment theory

The study is devoted to the development of the bimoment theory and the method for calculating thick plates in the framework of the three-dimensional theory of elasticity. The basic relations and equations of motion of the plate presented are constructed with respect to the forces, moments and bimoments arising from the nonlinearity of the law of distribution of displacements and stresses over the plate thickness. Bending and vibrations of isotropic and orthotropic plates are considered an example of calculations. The resulting solution showed the efficiency and accuracy of the proposed bimoment theory in assessing the stress-strain state of thick plates.

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.

STUDY OF THE MATERIAL PROPERTIES INFLUENCE AND CONTACT CONDITIONS ON THE STRESS-STRAIN STATE DURING THE INTERACTION OF A STAMP AND A PLATE

The article examines the results obtained when applying the perturbation method to the solution of individual contact problems.Attention is paid to the influence of non-ideal properties of materials of interaction bodies. The influence of the complex geometry of the contact zone is also taken into account. In the article, mathematical models of problems on the interaction of an elastic orthotropic semi-infinite plate with cylindrical anisotropy in the form of a curvilinear sector are constructed and considered. In the process of interaction between the stamp and the plate, there are areas of sliding and adhesion. The original problems of the theory of elasticity, difficult in their initial formulation, in the process of applying the perturbation method are reduced to the sequential solution of simpler problems of the theory of functions of a complex variable. Analytical solutions to the problem of the interaction of an elastic anisotropic semi-infinite plate in the form of a curvilinear sector and a rigid stamp were obtained, taking into account the presence of areas of sliding and adhesion in the contact zone. As significant results obtained after solving the formulated contact problems, it is possible to indicate the found regularities of stress distribution under the stamp, as well as the dependence of this distribution on the dimensions of the contact area. An analysis of the dependence of the size of the coupling zone on the stiffness characteristics of the sector was also carried out, taking into account different materials (for the sector), the opening angle of the sector and the dimensions of the stamp. In order to further develop the method and prove the convenience of applying the perturbation method in complex mixed boundary value problems, a study of the stress-strain state of an elastic anisotropic plate for an analogue of the Galin problem was carried out, while cases for different plate materials were considered. Corresponding analytical solutions were also obtained. The dependence of the size of the adhesion area on the dimensions of the stamp and the opening angle of the sector, as well as on the physical properties of the material of the sector, was analyzed. Possible boundary transitions are performed in the work and a comparison with known scientific results is made. The approach proposed by the authors makes it possible to obtain analytical solutions to practically important problems of contact interaction, to carry out possible assessments of the stress-strain state of thick-walled structures with reinforcing elements, stamps, overlays, etc. The results may also be useful. when designing mining works. At the same time, such important issues as the peculiarities of the physical properties of the materials of the interacting bodies, the complex geometry of the contact zone, the occurrence of areas of sliding and adhesion in the contact zone are taken into account.

Saint-Venant and Karman equations for orthotropic prestretched plate when exposed to temperature

Saint-Venant and Karman equations for orthotropic prestretched plate when exposed to temperature

Estimation of seismic resistance of multi-storey buildings based on a continuum model

A dynamic continuum plate model of a multi-story building is proposed in the form of a cantilevered orthotropic plate, developed in the framework of Timoshenko’s theory of plates, which describes seismic vibrations of buildings. Formulas are given for determining the reduced moduli of elasticity, shear, and density of a plate model of a building. Transverse natural and forced oscillations of a nine-story building under nine-point seismic impacts are considered. Numerical results are obtained.

Solid mechanics models in application to ophthalmology

The review presents the models on a change in the stress-strain state of the eye shell after vision correction operations and a change in intraocular pressure with the introduction of drugs into the vitreous body. All mathematical models were developed in collaboration with ophthalmologists. Mathematical models that describe the process of determining the true intraocular pressure (IOP) by applanation methods are discussed. New results in solid mechanics were obtained while studying ocular biomechanics problems. For example, the problem of the stability of a spherical shell subjected to a concentrated force and internal pressure was solved and the stability of the axisymmetric equilibrium state of annular inhomogeneous orthotropic plates under normal pressure was studied. The stress-strain states of transversely isotropic spherical and cylindrical layers subjected to internal and external pressures are discussed and the solutions obtained in the framework of the three-dimensional theory are compared with the solutions obtained by non-classical shell theories. This comparison made it possible to assess the accuracy of approximate theories.

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

Strength design of prefabricated corrugated steel plate shear walls under combined compression and shear loads

The prefabricated corrugated steel plate shear wall (PCSPSW) embedded within the frames is composed of one corrugated steel plate (CSP) and two vertical boundary elements (VBEs). Both axial compression and shear loads are inevitable in designing PCSPSWs with vertical corrugation in practical engineering. However, almost no research has been conducted on the strength design of PCSPSWs under combined axial compression and shear loads. This paper investigates the load-bearing mechanism and strength design formulas of PCSPSWs under combined axial compression and shear loads. Firstly, the elastic buckling load of the CSP under axial compression is theoretically derived using the orthotropic plate theory. Then the ultimate load-bearing strength factor of the CSP, φs, only under axial compression is established by introducing the revised normalized height-to-thickness ratio of the CSP, λns, which is related to the compressive elastic buckling load. Thus the compressive load-bearing capacity of the PCSPSW could be obtained by superimposing the contribution of the CSP and VBEs. Subsequently, a parametric study is performed to establish the ultimate load-bearing capacity of the PCSPSW under combined compression and shear loads through the correlation curve of N/Nu-V/Vu. The obtained results show that the normalized slenderness ratio λsc of the CSP could significantly affect the correlation curve. Finally, the weaker constraints provided by fishplates are involved in the analysis, and accordingly the prediction formulas for considering the weaker constraints are proposed. Then this complete design method can provide valuable fundamentals for predicting the load-bearing capacity of PCSPSWs in practical engineering applications.

CONSIDERATION OF THE PHYSICAL NONLINEARITY OF ORTHOTROPIC PLATE IN THE STATIC CALCULATION OF AN INFINITE REGULAR SYSTEM OF PLATES ON AN ELASTIC LAYER

An infinite regular system of orthotropic plates on an elastic layer rigidly connected to a non-deformable base is considered. Nonlinear calculation of an infinite regular system of plates on an isotropic basis is performed by the variational-difference method (VRM), which is characterized by the replacement of differential equations by finite-difference approximations. The numerical solution of the resulting system of equations is performed using an iterative algorithm.
 When finding the variable stiffness of the plate on the elastic layer, the "stiffness – curvature" dependence on the Straw in the directions of the axes of inertia is used. According to the found cylindrical bending stiffness from the ratio of S.P. Timo-shenko, the torsion stiffness of the plate is determined for each iteration. The energy of deformation of the elastic base is replaced by the work of reactive pressures in the contact zone of the structure on the basis of the law of conservation of energy.
 The analysis of the results of elastic and nonlinear calculations (3rd iteration) was carried out for the values of the orthotropic plate sediment and contact stresses in the zone of interaction of the plate with the elastic base.

To the calculation of orthotropic plates taking into account bimoments

The work is devoted to the development of the bimoment theory and the method for calculating thick plates in the framework of the three-dimensional theory of elasticity. The basic relations and equations of motion of the plate are given, constructed with respect to forces, moments and bimoments, arising from the nonlinearity of the law of distribution of displacements and stresses over its thickness. Bending and vibrations of isotropic and orthotropic plates are considered as an example of calculation. The resulting solution showed the efficiency and accuracy of the proposed bimoment theory in assessing the stress-strain state of thick plates. Figs.: Refs.: 11 titles.
 Keywords: thick orthotropic plate, three-dimensional theory of elasticity, bi-moment theory, bending and vibrations, stress-strain state.

Seismic vibrations of multi-storey buildings within the framework of the plate model

A continual plate-like dynamic model of a multi-storey building is proposed in the form of a cantilevered orthotropic plate developed in the framework of Timoshenko's theory of plates, which describes the seismic vibrations of buildings. Formulas are given for determining the reduced moduli of elasticity, shear, and density of a plate-like model of a building. The transverse vibrations of the building under seismic effects are considered. Numerical results are obtained.

Bending Analysis of Orthotropic Plate using 5th Order Shear Deformation Theory

A composite material is a combination of different material which can be assembled to provide required engineering properties like bending, buckling, stiffness etc. It can give better properties than those of individual components. A lamina is the building block of modern composite laminated structures. A lamina is also known as a ply, or a layer. A laminate has several layers of lamina, or laminae. Since the early years of engineering, various types of laminated composite materials have been used in various engineering fields, such as in aerospace, military and automotive industries that are subjected to high temperatures. Thermal stress caused due to change in temperature, increase in temperature causes the changes in material properties of a plate which in turn complicates the analysis and design and may affect the structure also. Therefore, the analysis of thermal stress in design of structure is very important. A critical review of available literature for the prediction of the behaviour of laminated composites under thermal conditions is carried out and summarized. This work attempts to review the studies carried out from 2000 on laminated composite plates by representing classification based on Classical laminate plate concept, First order shear deformation theory, Higher order shear deformation theory, Layerwise theory and Thermal stress on laminated plates.

Sparse Bayesian learning for complex‐valued rational approximations

AbstractSurrogate models are used to alleviate the computational burden in engineering tasks, which require the repeated evaluation of computationally demanding models of physical systems, such as the efficient propagation of uncertainties. For models that show a strongly non‐linear dependence on their input parameters, standard surrogate techniques, such as polynomial chaos expansion, are not sufficient to obtain an accurate representation of the original model response. It has been shown that for models with discontinuities or rational dependencies, for example, frequency response functions of dynamic systems, the use of a rational (Padé) approximation can significantly improve the approximation accuracy. In order to avoid overfitting issues in previously proposed standard least squares approaches, we introduce a sparse Bayesian learning approach to estimate the coefficients of the rational approximation. Therein the linearity in the numerator polynomial coefficients is exploited and the denominator polynomial coefficients as well as the problem hyperparameters are determined through type‐II‐maximum likelihood estimation. We apply a quasi‐Newton gradient‐descent algorithm to find the optimal denominator coefficients and derive the required gradients through application of ‐calculus. The method is applied to the frequency response functions of an algebraic frame structure model as well as that of an orthotropic plate finite element model.

The adhesive contact problem for a piecewise-homogeneous orthotropic plate with an elastic patch

A piecewise-homogeneous elastic orthotropic plate, reinforced with a finite patch of the wedge-shaped, which meets the interface at a right angle and is loaded with tangential and normal forces is considered. Using methods of the theory of analytic functions, the problem is reduced to the system of singular integro-differential equations (SIDE) with fixed singularity. Under tension-compression of patch using an integral transformation a Riemann problem is obtained, the solution of which is presented in explicit form. The tangential contact stresses along the contact line are determined and their asymptotic behavior in the neighborhood of singular points is established.

On the bending of rectangular atomic monolayers along different directions: an ab initio study

We study the bending of rectangular atomic monolayers along different directions from first principles. Specifically, choosing the phosphorene, GeS, TiS3, and As2S3 monolayers as representative examples, we perform Kohn–Sham density functional theory calculations to determine the variation in transverse flexoelectric coefficient and bending modulus with the direction of bending. We find that while the flexoelectric coefficient is nearly isotropic, there is significant and complex anisotropy in bending modulus that also differs between the monolayers, with extremal values not necessarily occurring along the principal directions. In particular, the commonly adopted orthotropic continuum plate model with uniform thickness fails to describe the observed variations in bending modulus for GeS, TiS3, and As2S3. We determine the direction-dependent effective thickness for use in such continuum models. We also show that the anisotropy in bending modulus is not associated with the rehybridization of atomic orbitals.

Application of the Ritz method for the bending and stress analysis of thin rectilinear orthotropic composite sector plates

This paper aims to investigate the linear elastic behavior of composite sector plates with rectilinear orthotropy in terms of displacement and stress fields determination. This variety of plates presents layers reinforced with unidirectional fibers organized along straight directions that make a variable angle with the generic radial direction. The fundamental equations for rectilinear orthotropic composite sector plates are obtained according to the Classical Lamination Theory. The mid-surface deflection of the sector plates is derived on the basis of an original analytical approach that combines the principle of virtual works with the Ritz method. Different boundary conditions in terms of displacement constraints are taken into account and, accordingly, the relative discretized forms of the displacement field are written as a linear combination of expressly obtained approximation functions. Commonly employed symmetrical stacking sequences are considered to demonstrate the accuracy and reliability of the proposed method in the final test cases.

Complex Potentials and Integral Equations for Thermoelasticity Problem of Orthotropic Cracked Plate

In the present article, the plane thermoelasticity problem of an infinite orthotropic cracked plate, is studied. The interaction between thermal and mechanical field, in a plate containing a crack and a heat source is investigated. The plate is submitted to normal stresses at infinity, where a heat flow is also activated. Applying the complex potentials method and the technique of Cauchy integrals, a system of singular integral equations is obtained. These equations, allow the determination of the temperature’s distribution, but also of the stress state at any point of the medium.

Rotational and multimodal local resonators for broadband sound insulation of orthotropic metamaterial plates

Locally resonant metamaterials are employed to increase the sound insulation of a host structure. In orthotropic host structures, the direction-dependent bending stiffness leads to a broad coincidence dip in sound transmission loss, which is difficult to overcome through conventional metamaterial solutions based on single-mode translational resonators. For this reason, in this paper we aim at developing orthotropic metamaterials with rotational resonators, exhibiting a direction-dependent effectiveness, and with multimodal resonators, which combine translational and rotational modes and are effective in a broader frequency band. First, we develop an analytical effective medium model of sound transmission in multimodal orthotropic metamaterials, which exhibit similar accuracy to simulations exploiting a wave and finite element method. We then propose a design methodology for realizable geometrical layouts of multimodal resonators, by exploiting numerical optimization to maximize the broadband sound transmission loss of the metamaterial. For two selected design cases, we show that rotational and multimodal metamaterials can suppress the broad coincidence dip of orthotropic plates, when the frequencies of local resonances are tuned within the coincidence region. The results of the work open up new possibilities for broadband sound insulation through metamaterial solutions.

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.

Refined models for free vibration analysis of elastic plates with part-through surface cracks

In this paper, two refined models are proposed for the vibration analysis of isotropic and orthotropic thin plates with part-through surface cracks. The governing equations of cracked plates are formulated via the refined models. Like in the conventional model, crack terms are incorporated in the governing equations. All-over and finite-length cracks (i.e., central and edge cracks) can be directly handled by the new models and their modified forms, respectively. The matched interface and boundary method is then applied to formulate eigenvalue equations. A series of parametric studies for investigating the effects of crack depth and crack length are presented. Numerical results of the proposed models are compared with the existing and finite element solutions to validate their performance and accuracy. The salient feature of the refined models is to complement the deficiency of the conventional model for the analysis of crack depth problems. Several illustrative examples are provided to demonstrate the superior performance of the refined models over the conventional one. The present models are effective for predicting the natural frequencies of thin plates with part-through surface cracks.