Bending Of Elastic Plates Research Articles

Comparison of approximation methods for bending of finite and infinite elastic plates: The traction boundary value problem

A generalized Fourier series method is designed to approximate the solution of the Neumann problem in a finite domain and in an infinite domain with a hole, for the mathematical model of bending of elastic plates with transverse shear deformation. The results in the two cases are compared and contrasted, and the analysis is illustrated by examples with specific boundary conditions.

Analysis of Berger nonlinear elastic static plate bending of rectangular plates

This paper deals with a nonlinear static plate model based on Berger theory, which is a specific case of a generalization of the Woinowsky-Krieger mathematical model of beam bending. It is considered a plate bending with forces acting in the middle plane of the plate and a contact problem with an elastic foundation, where the normal compliance condition is employed. A variational equation of the problem and a functional of the total potential energy corresponding to the variational equation are derived. Under additional assumptions on the data (e.g., clamped plate), the existence and uniqueness of the solution are proved. A numerical solution is based on the Galerkin method and Courant approximation. The theory is illustrated by a numerical example.

Space-time generalized finite difference method for solving the thin elastic plate bending under dynamic loading

In this paper, the space-time generalized finite difference method (ST-GFDM) with supplementary nodes is applied to solve the thin elastic plate bending under dynamic loading. The proposed method treats the temporal dimension as another spatial dimension, thus the d-dimensional problem in space can be viewed as a new (d+1)-dimensional problem in the space-time domain. This method effectively reduces the complexity of numerical solution of the problem by simultaneously discretizing the temporal and spatial dimensions in the space-time domain. It avoids the temporal-difference measurement while maintaining all the advantages of the GFDM. The ST-GFDM leads a sparse linear system due to its local meshless feature, which is suitable for solving large-scale and long-time bending problems. Moreover, the supplementary nodes on the boundary are introduced to make the system well-determined. Several numerical examples are illustrated to demonstrate the accuracy and stability of the proposed method.

Dilation-Invariant Bending of Elastic Plates, and Broken Symmetry in Shells

We propose bending energies for isotropic elastic plates and shells. For a plate, we define and employ a surface tensor that symmetrically couples stretch and curvature such that any elastic energy density constructed from its invariants is invariant under spatial dilations. This kinematic measure and its corresponding isotropic quadratic energy resolve outstanding issues in thin structure elasticity, including the natural extension of primitive bending strains for straight rods to plates, the assurance of a moment linear in the bending measure, and the avoidance of induced mid-plane strains in response to pure moments as found in some commonly used analytical plate models. Our analysis also reveals that some other commonly used numerical models have the right invariance properties, although they lack full generality at quadratic order in stretch. We further extend our result to naturally-curved rods and shells, for which the pure stretching of a curved rest configuration breaks dilation invariance; the new shell bending measure we provide contrasts with previous \emph{ad hoc} postulated forms. The concept that unifies these theories is not dilation invariance, but rather through-thickness uniformity of strain as a definition of pure stretching deformations. Our results provide a clean basis for simple models of low-dimensional elastic systems, and should enable more accurate analytical probing of the structure of singularities in sheets and membranes.

Comparison of structural functions method approximationsof the solution of a linear elastic layered plate bending problem

Comparison of structural functions method approximationsof the solution of a linear elastic layered plate bending problem

A GFDM with supplementary nodes for thin elastic plate bending analysis under dynamic loading

This work proposes a meshless generalized finite difference method (GFDM) with supplementary nodes for the solution of thin elastic plate bending under dynamic loading. The first- and second-order time derivatives in equilibrium equation are firstly discretized with the Houbolt method. The numerical solution of the resulting spatial problem at each time step is then calculated by the GFDM. The supplementary nodes on the boundary of the problem domain are introduced in the GFDM to yield the well-determined linear system of equations. Numerical example is provided to verify the accuracy and the stability of the developed approach, and these preliminary results reveal the potential of the method.

Relaxation properties of the ice cover

The aim of this work was to generalize the known data and conduct experiments to determine the relaxation properties of ice in the ice cover under short-term (no more than 1 min.) loading. The problem lies in the fact that when one is solving applied problems of ice engineering, ice is often considered as an elastic isotropic material, and its stress-strain state (SSS) is studied in terms of the theory of bending of elastic plates. This does not allow performing theoretical calculations when resonant flexural gravity waves (IGW) are excited by moving loads, because under these conditions, the deflections of the ice increase to infinity and the known solutions become unusable. In fact, ice clearly manifests the properties of a quasi-isotropic medium, and the relationship between stresses and deformations is of a viscoelastic nature. It is noted in the work that, depending on the mode in which external loads act on the ice cover, its inelastic properties affect the nature of its behavior in different ways, while the viscoelastic properties of the ice cover are well described by the linear models of Maxwell or Kelvin-Voigt inelastic continuous media. The experimental material is duly processed and analysis is carried out of the results of experimental studies performed in the field by loading the ice cover with balanced loads using a specially made loading device, which was a frame with three supports. The design of the device made it possible to load the ice cover with balanced loads, which made it possible to exclude the influence of false elasticity of water on the results of experiments. For the rheological models of ice behavior indicated, the most probable ranges of changes in the relaxation times of stresses and deformations of the ice cover in the ice conditions considered are given. The results obtained can be used in theoretical studies of ice engineering problems.

On the Boundary Value Problems of Bending of Thin Elastic Plates With Surface Effects

Abstract We modify classical thin plate theory by incorporating surface effects via the Gurtin–Murdoch surface model to accommodate the mechanical behavior of thin plates at the nanoscale. We formulate the corresponding Dirichlet and Neumann boundary value problems and establish uniqueness results in the appropriate function spaces. In addition, we obtain the fundamental solution of the governing system of equations, which is central to further studies concerning well-posedness analysis of the model by the boundary integral equation method. Finally, we validate our model by comparison with results in the existing literature.

Stiffness of Monolithic Reinforced-Concrete Slab Structures

It is shown that the numerical analyses of slab reinforced concrete structures are, as a rule, based on the classical theory of bending of elastic plates. According to the results of theoretical analysis performed by using the finite-element method and shell theory, it was established that the flexural stiffnesses of slab elements with tubular inserts in both directions are different. Simulating the operation of slabs with inserts of square, rectangular, and circular cross sections and applying the finite-element method, we determine the values of flexural stiffness for the plates with geometric parameters typical of actual slab structures.

A passive wireless surface acoustic wave sensor for pillar load measurement

This paper presents a passive wireless pillar load measurement method based on surface acoustic wave (SAW) technology. A strain transfer mechanical structure was designed, and the strain distribution of an elastic substrate was analyzed using elastic thin plate bending theory. In addition, the relationship between the strain and frequency shift of the SAW resonator (SAWR) was analyzed, and a strain sensitivity numerical analysis using different bonding directions was performed. Results corroborated that the differential structure of resonators, one of which is placed at the edge along the direction of a diameter and the other is placed at the center with an angle of 45° to the diameter, is a suitable form of the SAW load sensors. Moreover, the SAWR parameters were optimized to obtain a high-performance resonator. Furthermore, The SAWRs and strain transfer mechanical structures were manufactured, the pillar load measurement system was set, and the proposed SAW load measurement method was evaluated. Experimental results affirmed that the load sensitivity is 8.64 kHz/kN. The SAW pillar load method is feasible for practical application.

Exact solutions of boundary value problems in the theory of plate bending in a half-strip: basics of the theory

Using the boundary value problem on the bending of a thin elastic semi-infinite plate in which the long sides are free, while a self-balanced bending moment and a generalized shearing force are specified at its end as an example, we consider the main steps in constructing exact solutions to the boundary value problems of bending of thin elastic rectangular plates. The solutions are constructed in the form of series in Papkovich–Fadle eigenfunctions. The unknown expansion coefficients are determined in the same way as in classical periodic solutions in trigonometric series and have the same structure, i.e., are expressed via the Fourier integrals of the boundary functions specified at the half-strip end. The systems of functions biorthogonal to the Papkovich–Fadle eigenfunctions constructed here are used in this case. The exact solutions possess properties that are not inherent in any of the known solutions in the theory of plate bending. Some of them are discussed in the paper. The final formulas describing the exact solution of the boundary value problem are simple and can be easily used in engineering practice. The results that we have obtained previously when solving the boundary values problems of the plane theory of elasticity in a rectangular region form the basis for our work.

Reduced-order strategy for meshless solution of plate bending problems with the generalized finite difference method

Abstract This paper presents some recent advances on the numerical solution of the classical Germain-Lagrange equation for plate bending of thin elastic plates. A meshless strategy using the Generalized Finite Difference Method (GFDM) is proposed upon substitution of the original fourth-order differential equation by a system composed of two second-order partial differential equations. Mixed boundary conditions, variable nodal density and curved contours are some of the explored aspects. Simulations using very dense clouds and parallel processing scheme for efficient neighbor selection are also presented. Numerical experiments are performed for arbitrary plates and compared with analytical and Finite Element Method solutions. Finally, an overview of the procedure is presented, including a discussion of some future development.

The Robin problem for bending of elastic plates

The solution of the Robin problem in a finite domain for the system of equations modeling the bending of elastic plates with transverse shear deformation is approximated by means of a generalized Fourier series method closely connected to the structure of the boundary integral equation treatment of the problem. The theory is exemplified by numerical computation that shows a high degree of accuracy and efficiency.

Bending of elastic plates with micro-voids

In this paper, we present a systematic development of unified formulation for bending of elastic plates with micro-voids. The homogenization concept is applied with replacing real medium with micro-voids by a homogeneous medium with effective material coefficients. The effective material coefficients are calculated in advance by solving properly selected micro-structural 3D boundary value problems in representative volume elements with specified shape and volume content of micro-voids. The unified formulation for plate bending problems is developed from 3D elasticity with taking into account the assumptions of three plate bending theories such as Kirchhoff-Love theory for bending of thin plates, 1st and 3rd order shear deformation theories for thick plates. Switching among these theories is allowed by proper selection of two key factors introduced in the unified formulation. The influence of the porosity on bending of porous plates subjected to transversal loading is studied via the numerical simulations.

Bending of elastic plates with transverse shear deformation: The Neumann problem

A generalized Fourier series method is constructed to approximate the solution of the Neumann problem in a finite domain for the system of equations governing the bending of elastic plates with transverse shear deformation. The method is illustrated by an example with computation performed by 3 different techniques that are contrasted and compared for efficiency, accuracy, and stability.

Shape derivative of the energy functional for the bending of elastic plates with thin defects

The paper deals with an equilibrium problem for a homogeneous isotropic elastic plate with a thin rigid inclusion and interfacial crack. We provide an explicit formula for the first shape derivative of the energy functional in the direction of a given vector field by means of a volume integral. For specific examples of the vector field, we derive some representations of the formula in terms of path-independent contour integrals.

An elastic plate bending equation of second-order accuracy

A study is carried out of a thin plate of constant thickness made of linearly elastic material which is transversally isotropic and heterogeneous in the thickness direction. Asymptotic expansions in powers of the relative plate thickness are constructed, and the bending equation of second-order accuracy (the SA model) is delivered. The results of the SA model are compared with the Kirchhoff–Love classical model and with the Timoshenko–Reissner (TR) model, as well as with the exact solution. To this end, some problems for a functionally gradient plate bending, and for a multi-layer plate bending and free vibration are solved and analysed. The range of plate heterogeneity, for which the error of the approximate models is small, is established. The TR model and the SA model are proved to yields results close to each other and the exact results for a very broad range of heterogeneity. That is why the generalized TR model for one-layered homogeneous transversely isotropic plate is proposed. Parameters of this model are chosen so that the results are close to the exact results and the results by the SA model. For the Navier boundary conditions, the analytical solution of 3D problems for a rectangular heterogeneous plate is constructed.

Three-dimensional modelling of laminated glass bending on two-dimensional in-plane mesh

Abstract The article presents an analysis of bending of a laminated glass (LG) plate. The analysis uses a 3D physical model in which only a 2D in-plane mesh is applied for numerical analysis. The LG consists of glass panes bonded by films of polymers, so the LG is layered structure. The approach in which the 3D spatial approximation in each layer is a combination of a 2D in-plane approximation with a 1D transverse approximation (along the plate thickness) is used. In consequence, the final numerical model is based on a 2D in-plane finite element mesh. A special postprocessing procedure, devised specially to be used in this method, has been developed which enables visualization of a full 3D deformation as well as all six stress components. The method is relatively simple, robust and effective, and it gives correct 3D results. The accuracy of the method has been tested with the use of some known solutions as well as by comparison with analytical solution, well established FEM procedures and with experimental results. In the examples the two-, three- and four-paned laminated glass is analysed. In spite of the fact that the numerical model is 2D, the full 3D results are obtained. For the sake of clarity, this paper is limited to elastic plate bending, but the method can be directly applied for LG subjected to thermal or thermo-mechanical load.

Elastic flexure controls magma trajectories and explains the offset of primary volcanic activity upstream of mantle plume axis at la Réunion and Hawaii hotspot islands

Surface volcanism at la Réunion and Hawaii occurs with an offset of 150–180 km upstream to the plume axis with respect to the plate motion. This striking observation raises questions about the forcing of plume-lithosphere thermo-mechanical interactions on melt trajectories beneath these islands. Based on visco-elasto-plastic numerical models handled at kilometric resolution, we propose to explain this offset by the development of compressional stresses at the base of the lithosphere, that result from elastic plate bending above the upward load exerted by the plume head. This horizontal compression adopts a disc shape centered around the plume axis: (i) it is 20 km thick, (ii) it has a 150 km radius, (iii) it lays at the base of the elastic part of the lithosphere, i.e., around ∼50–70 km depth where the temperature varies from ∼600 °C to ∼750 °C, (iv) it lasts for 5 to 10 My in an oceanic plate of age greater than 70 My, and (vi) it is controlled by the visco-elastic relaxation time at ∼50–70 km depth. This period of time exceeds the time during which both the Somalian/East-African and Pacific plates drift over the Reunion and Hawaii plumes, respectively. This indicates that this basal compression is actually a persistent feature. It is inferred that the buoyant melts percolating in the plume head pond below this zone of compression and eventually spread laterally until the most compressive principal elastic stresses reverse to the vertical, i.e., ∼150 km away from the plume head. There, melts propagate through dikes upwards to ∼35 km depth, where the plate curvature reverses and ambient compression diminishes. This 30–35 km depth may thus host a magmatic reservoir where melts transported by dykes pond. Only after further magmatic differentiation can dykes resume their ascension up to the surface and begin forming a volcanic edifice. As the volcano grows because of melt accumulation at the top of the plate, the lithosphere is flexed downwards, inducing extra tensile stress at 30–35 km depth and compression at ∼15 km depth (induced by the edifice load). It implies that now the melts pond at ∼15 km and form another magmatic reservoir lying just underneath the crust. These processes explain the ponding of primary (shield) melts at ∼35 km and ∼15 km depths as recorded below La Reunion, Mauritius or Hawaii volcanoes, all shifted by ∼150 km with respect to the plume axis.

THE NONLINEAR BENDING OF SIMPLY SUPPORTED ELASTIC PLATE

In this article, assumptions in Classical Plate Theory (CPT) are explained; followed by concepts involved in Finite Element discretization for elastic Plate bending in CPT. Computer implementation aspects and Numerical Results of CPT elements are also included for analyzing nonlinear bending of simply supported elastic plates.