Showing papers on "Bending of plates published in 2009"

Elastic theory of unconstrained non-Euclidean plates

Abstract: Non-Euclidean plates are a subset of the class of elastic bodies having no stress-free configuration. Such bodies exhibit residual stress when relaxed from all external constraints, and may assume complicated equilibrium shapes even in the absence of external forces. In this work we present a mathematical framework for such bodies in terms of a covariant theory of linear elasticity, valid for large displacements. We propose the concept of non-Euclidean plates to approximate many naturally formed thin elastic structures. We derive a thin plate theory, which is a generalization of existing linear plate theories, valid for large displacements but small strains, and arbitrary intrinsic geometry. We study a particular example of a hemispherical plate. We show the occurrence of a spontaneous buckling transition from a stretching dominated configuration to bending dominated configurations, under variation of the plate thickness.

Dynamics of plate bending at the trench and slab-plate coupling

Abstract: [1] The bending strength of subducting lithosphere plays a critical role in the Earth's plate tectonics and mantle convection, modulating the amount of slab pull transmitted to the surface and setting the boundary conditions under which plates move and deform. However, it is the subject of a lively debate how much of the potential energy of the downgoing plate is consumed in bending the plate and how the lithospheric strength is defined during this process. We model the subduction of a viscoelastic lithosphere, driven solely by the downgoing plate's buoyancy, freely sinking in a passive mantle, represented by drag forces. To investigate the dynamics of bending, (1) we vary the density and the viscosity profile within the plate from isoviscous, where strength is distributed, to strongly layered, where strength is concentrated in a thin core, and (2) we map the stress, strain, and dissipation along the downgoing plate. The effective plate strength during bending is not a simple function of average plate viscosity but is affected by rheological layering and plate thinning. Earth-like layered plates allow for the transmission of large fractions of slab pull (∼75–80%) through the bend and yield a net slab pull of FSPnet = 1 to 6 × 1012 N m−1, which varies with the buoyancy of plates. In all models, only a minor portion of the energy is dissipated in the bending. Surprisingly, bending dissipation hardly varies with lithospheric viscosity because in our dynamic system, the plates minimize overall dissipation rate by adjusting their bending curvature. As a result, bending dissipation, ΦB, is mainly controlled by the bending moment work rate exerted by slab pull. We propose a new analytical formulation that includes a viscosity-dependent bending radius, which allows for assessment of the relative bending dissipation in the Earth's subduction zones using parameters from a recent global compilation. This yields estimates of ΦB/ΦTOT < 20%. These results suggest that plates on Earth weakly resist bending, yet are able to propagate a large amount of slab pull.

Buckling analysis of plates using the two variable refined plate theory

Abstract: Buckling analysis of isotropic and orthotropic plates using the two variable refined plate theory is presented in this paper. The theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate, hence it is unnecessary to use shear correction factors. Governing equations are derived from the principle of virtual displacements. The closed-form solution of a simply supported rectangular plate subjected to in-plane loading has been obtained by using the Navier method. Numerical results obtained by the present theory are compared with classical plate theory solutions, first-order shear deformable theory solutions, and available exact solutions in the literature. It can be concluded that the present theory, which does not require shear correction factor, is not only simple but also comparable to the first-order shear deformable theory.

A theoretical analysis of smart moderately thick shear deformable annular functionally graded plate

Abstract: Based on Mindlin's plate theory, free vibration analysis of moderately thick shear deformable annular functionally graded plate coupled with piezoelectric layers is presented in this paper. A consistent formulation that satisfies the Maxwell static electricity equation is presented so that the full coupling effect of the piezoelectric layer on the dynamic characteristics of the annular FGM plate can be estimated based on the free vibration results. The differential equations of motion are solved analytically for various boundary conditions of the plate through the transformation of variable method. The applicability of the proposed model is analyzed by studying the effect of varying the gradient index of FGM plate on the free vibration characteristics of the structure. For some specific cases, obtained results were cross checked with those existing literatures and furthermore, verified by those obtained from three-dimensional finite element (3D FE) analyses.

Water-wave scattering by submerged elastic plates

Abstract: We present a solution to the water-wave interaction with a submerged elastic plate of negligible thickness by the eigenfunction-matching method. The eigenfunction expansion depends on the solution of a special dispersion equation for a submerged elastic plate and this is discussed in detail. We show how the solution can be calculated for the case of normal incidence on a semi-infinite plate in two spatial dimensions and then extend this solution to obliquely incident waves, to a plate of finite length and to a circular finite plate in three dimensions. Numerical calculations showing various properties of the solutions are presented and a near-orthogonality relation for the eigenfunctions is used to derive an energy-balance relation.

Limit analysis of plates using the EFG method and second‐order cone programming

Abstract: The meshless element-free Galerkin (EFG) method is extended to allow computation of the limit load of plates. A kinematic formulation that involves approximating the displacement field using the moving least-squares technique is developed. Only one displacement variable is required for each EFG node, ensuring that the total number of variables in the resulting optimization problem is kept to a minimum, with far fewer variables being required compared with finite element formulations using compatible elements. A stabilized conforming nodal integration scheme is extended to plastic plate bending problems. The evaluation of integrals at nodal points using curvature smoothing stabilization both keeps the size of the optimization problem small and also results in stable and accurate solutions. Difficulties imposing essential boundary conditions are overcome by enforcing displacements at the nodes directly. The formulation can be expressed as the problem of minimizing a sum of Euclidean norms subject to a set of equality constraints. This non-smooth minimization problem can be transformed into a form suitable for solution using second-order cone programming. The procedure is applied to several benchmark beam and plate problems and is found in practice to generate good upper-bound solutions for benchmark problems.

Shear failure characteristics of steel plate girders

Abstract: A number of full-scale plate girders are modeled and analyzed to determine their shear failure mechanism characteristics. An objective of this numerical nonlinear large deflection elastoplastic finite element study is to clarify how, when, and why plastic hinges that emerge in experimental tests actually form. It is observed that shear-induced plastic hinges only develop in the end panels. These hinges are caused by the shear deformations near supports and not due to bending stresses arising from tension fields. Also, a comparison between the ultimate capacity of various plate girders and different codes and theories is presented. Finally, it is shown that simple shear panels, in the form of detached plates, do not accurately represent the failure mechanism of web plates.

Analytical Modeling and Vibration Analysis of Partially Cracked Rectangular Plates With Different Boundary Conditions and Loading

Abstract: This study proposes an analytical model for vibrations in a cracked rectangular plate as one of the results from a program of research on vibration based damage detection in aircraft panel structures. This particular work considers an isotropic plate, typically made of aluminum, and containing a crack in the form of a continuous line with its center located at the center of the plate and parallel to one edge of the plate. The plate is subjected to a point load on its surface for three different possible boundary conditions, and one examined in detail. Galerkin's method is applied to reformulate the governing equation of the cracked plate into time dependent modal coordinates. Nonlinearity is introduced by appropriate formulations introduced by applying Berger's method. An approximate solution technique-the method of multiple scales-is applied to solve the nonlinear equation of the cracked plate. The results are presented in terms of natural frequency versus crack length and plate thickness, and the nonlinear amplitude response of the plate is calculated for one set of boundary conditions and three different load locations, over a practical range of external excitation frequencies.

Thermoelastic analysis of functionally graded plates using the element-free kp-Ritz method

Abstract: In this paper, the static responses of metal and ceramic functionally graded plates subject to thermal and mechanical loads are investigated. The first-order shear deformation plate theory is adopted, and the displacement field is expressed in terms of a set of mesh-free kernel particle functions. It is assumed that the material property of each plate exponentially varies through the thickness. The governing equations are solved to obtain the plate displacements and axial stresses using the element-free kp-Ritz method. The effects of the volume fraction, material property, boundary conditions and length-to-thickness ratio on the plate deflection and axial stress are discussed in detail. The numerical results generated from the proposed method agree well with those in the literature.

Non-linear analysis of functionally graded circular plates under asymmetric transverse loading

Abstract: Based on the first-order shear deformation plate theory with von Karman non-linearity, the non-linear axisymmetric and asymmetric behavior of functionally graded circular plates under transverse mechanical loading are investigated. Introducing a stress function and a potential function, the governing equations are uncoupled to form equations describing the interior and edge-zone problems of FG plates. This uncoupling is then used to conveniently present an analytical solution for the non-linear asymmetric deformation of an FG circular plate. A perturbation technique, in conjunction with Fourier series method to model the problem asymmetries, is used to obtain the solution for various clamped and simply supported boundary conditions. The material properties are graded through the plate thickness according to a power-law distribution of the volume fraction of the constituents. The results are verified by comparison with the existing results in the literature. The effects of non-linearity, material properties, boundary conditions, and boundary-layer phenomena on various response quantities in a solid circular plate are studied and discussed. It is found that linear analysis is inadequate for analysis of simply supported FG plates which are immovable in radial direction even in the small deflection range. Furthermore, the responses of FG materials under a positive load and a negative load of identical magnitude are not the same. It is observed that the boundary-layer width is approximately equal to the plate thickness with the boundary-layer effect in clamped FG plates being stronger than that in simply supported plates.

Vibration and Stability of Functionally Graded Plates Based on a Higher-order Deformation Theory

Abstract: In this article, the governing equations of motion for a functionally graded material plate (FGP) based on a higher-order deformation theory in a general state of non-uniform initial stress are derived. The properties of FGP are assumed varied continuously along the thickness of the plate, according to a simple power law of volume fractions of constituents. With the derived governing equations, the natural frequencies and buckling loads of ceramic—FGM—metal plates subjected to an initial stress are investigated. The initial stress is taken to be a combination of a uniaxial extensional stress and a pure bending stress. A ceramic—FGM—metal plate can become an all-FGM, all-ceramic plate, or all-metal plate by modifying the value of material parameter and volume fraction index. The effects of various parameters and initial stresses on the natural frequencies and buckling loads of FGPs are studied.

Dynamic stability of a metal foam circular plate

Abstract: The study is devoted to a radial compressed metal foam circular plate. Properties of the plate vary across its thickness. The middle plane of the plate is its symmetry plane. First of all, a displacement field of any cross-section of the plate was defined. Afterwards, the components of strain and stress states were found. The Hamilton principle allowed one to formulate a system of differential equations of dynamic stability of the plate. This basic system of equations was approximately solved. The forms of unknown functions were assumed and the system of equations was reduced to a single ordinary differential equation of motion. The equation was then numerically processed that allowed one to determine critical loads for a family of metal foam plates. The results of studies are shown in figures. They show the effect of porosity of the plate on the critical loads. The results obtained for porous plates were compared to homogeneous circular plates.

Axisymmetric bending analysis of thick functionally graded circular plates using fourth-order shear deformation theory

Abstract: In the present article, axisymmetric bending and stretching of functionally graded (FG) circular plates subjected to uniform transverse loading based on fourth-order shear deformation plate theory (FOST) have been studied. Using a fourth-order shear deformation theory, the solutions for deflection and rotation functions of FG plates are presented in terms of the corresponding quantities for a homogeneous plate using the classical plate theory (CPT), from which solutions one can easily obtain the FOST solutions for axisymmetric bending of FG circular plates. It is assumed that the effective mechanical properties of the functionally graded plates through the thickness are continuous functions of the volume fractions of the constituent parts which are themselves defined by a power-law function. Numerical results for maximum deflection and shear stress are presented for various percentages of ceramic–metal volume fractions. These results are also compared with those obtained from the first-order shear deformation plate theory of Mindlin (FST), the third-order shear deformation plate theory of Reddy (TST) as well as the exact three-dimensional elasticity solution. It is found that although the maximum deflections obtained using FOST and TST are close to each other, the through-thickness shear stress is predicted more accurately by the FOST formulation than by the TST.

Exact Solutions for Free Vibrations of Functionally Graded Thick Plates on Elastic Foundations

Abstract: Free vibration analysis of functionally graded thick plates on elastic foundation is carried out based on three-dimensional elasticity. An isotropic plate is assumed with Young's modulus and mass density varying exponentially through the thickness, and Poisson's ratio remaining constant. The state space method is used to derive an exact solution for a simply supported plate by expanding the state variables in trigonometric dual series. The effects of interaction between the plate surface and a Pasternak foundation are treated as the traction boundary conditions of the plate. The solution procedure is validated by comparing with established results for a homogeneous plate. Finally, the influences of foundation stiffness, foundation support, and gradient index on the natural frequencies are investigated.

The effect of transverse shear and normal deformations on the thermomechanical bending of functionally graded sandwich plates

Abstract: A thermomechanical bending analysis for a simply supported, rectangular, functionally graded material sandwich plate subjected to a transverse mechanical load and a through-the-thickness thermal load is presented using the refined sinusoidal shear deformation plate theory. The present shear deformation theory includes the effect of both shear and normal deformations and it is simplified by enforcing traction-free boundary conditions at the plate faces. Material properties and thermal expansion coefficient of the sandwich plate faces are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic material. The equilibrium equations of different sandwich plates are given based on various plate theories. A number of examples are solved to illustrate the numerical results concern thermo-mechanical bending response of functionally graded rectangular sandwich plates. The influences played by transversal shear and normal deformations, plate aspect ratio, side-to-thickness ratio, volume fraction distributions, and thermal and mechanical loads are investigated.