FREE VIBRATION ANALYSIS OF RECTANGULAR PLATES WITH VARIABLE THICKNESS

FREE-VIBRATION ANALYSIS OF RIGHT TRIANGULAR PLATES WITH VARIABLE THICKNESS

This investigation is concerned with the notion of concentrated loads in classical elastostatics and related issues. Following a limit treatment of problems involving concentrated internal and surface loads, the orders of the ensuing displacements and stress singularities, as well as the stress resultants of the latter, are determined. These conclusions are taken as a basis for an alternative direct formulation of concentrated-load problems, the completeness of which is established through an appropriate uniqueness theorem. In addition, the present work supplies a reciprocal theorem and an integral representation-theorem applicable to singular problems of the type under consideration. Finally, in the course of the analysis presented here, the theory of Green's functions in elastostatics is extended.

Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory

Exact results for the stress intensity factor are presented for an external circular crack with oppositely directed concentrated loads applied to the crack surfaces. This result is specialized to the case of a semi-infinite crack in an infinite body with concentrated loads on the crack. A procedure is then suggested by which one can obtain from the corresponding plane result the approximate three-dimensional Green's function (concentrated load result) for any straight crack in an infinite elastic body. This procedure is used to determine the Green's functions for a finite-length crack in an infinite body, and is then used in conjunction with a suggested “slicing” procedure to obtain approximate three-dimensional Green's function for plates of finite thickness and infinite extent, containing finite length cracks. Previously existing solutions for crack problems are compared with results obtained by application to plate tension and bending problems of the three-dimensional Green's functions. The results indicate that the procedure yields satisfactory results when stress gradients through the plate thickness are not excessive. However, an accurate assessment of the validity of the slicing procedure awaits further progress in three-dimensional crack analysis.

FREE VIBRATION OF ORTHOTROPIC SQUARE PLATES WITH A SQUARE HOLE

The frequencies and modes of free vibration, coupled in bending and twisting, of a free-free, sweptback, uniform beam are obtained from its differential equations of motion. The modes of vibration of a nonuniform, sweptback wing are then determined by the Ritz method. The discussion of the methods and various practical aspects of the analysis are presented from an engineering, rather than a mathematical, point of view.

Free vibration analysis of beams and rectangular plates with free edges by the discrete singular convolution

Free vibration analysis of continuous rectangular plates

This paper presents an experimental and numerical study of the effect of residual thermal stresses on the mechanical behavior of diagonally reinforced plates. The study focuses on the analysis of carbon/epoxy square plates onto which diagonal reinforcements were glued. These reinforcements were glued using two different methodologies: the first method was to glue the reinforcements at operating temperature (22 °C), while in the second methodology reinforcements were cured in an autoclave at 177 °C. Mechanical behavior assessment was based on the stiffness, free vibration, and buckling tests. For the study of the stiffness of the plates, an optical technique to determine the transversal displacement caused by the presence of a static load was employed. For the vibration tests, the natural frequencies associated with the first four modes of free vibration were determined by using a dynamic signal analyzer. For the linear and non-linear buckling, compression tests on a universal testing machine were performed, determining the displacements produced during the test using the digital image processing method. All experimental results were compared with results obtained from numerical approximations made with commercial software. The results show the effect of residual thermal stresses caused during the manufacturing process on the mechanical performance of diagonally reinforced plates.

Expressions for Green's function of the free rectangular plate and structures consisting of beams and plates are derived by analytical and semianalytical methods to be used in a distributed element program. Thereby, the boundary-value problem for the forced vibrations of the free plate excited harmonically by a concentrated load at an arbitrary point is divided into three appropriate plate problems, with solutions of the Levy-type. Then, Green's functions for structures being more complex are gained using the substructure technique, whereby Green's functions of the free substructures are taken as basis. Using the superposition principle, they can be reduced to Green's functions for simpler boundary conditions. Green's function for the whole structure then yields the receptance matrices for arbitrary connecting points.

By means of Green's functions, an exact solution has been found for the thin elastic rectangular plate, built-in along one edge and simply supported along the other edges, for various static loading conditions. The results for deflections and moments have been compared with an approximate method by Timoshenko and the finite element method for different loading cases. It has been shown that results given by Timoshenko and finite element methods agree well with the exact solution, presented in this work.

This work shows a three-dimensional (3D) layerwise model for static and free vibration analyses of functionally graded piezomagnetic materials (FGPM) spherical shell structures where magnetic and elastic fields are completely coupled. The 3D magneto-elastic governing equations for spherical shells are made of the three equations of equilibrium in three-dimensional form and the three-dimensional divergence equation for the magnetic induction. Governing equations are written in the orthogonal mixed curvilinear reference system (α, β, z) allowing the analysis of several curved and flat geometries (plates, cylindrical shells and spherical shells) thanks to proper considerations of the radii of curvature. The static cases, actuator and sensor configurations and free vibration investigations are proposed. The resolution method uses the imposition of the Navier’s harmonic forms in the two in-plane directions and the exponential matrix methodology in the transverse normal direction. Single-layered and multilayered simply-supported FGPM structures have been investigated. In order to understand the behavior of FGPM structures, numerical values and trends along the thickness direction for displacements, stresses, magnetic potential, magnetic induction and free vibration modes are proposed. In the results section, a first assessment phase is proposed to demonstrate the validity of the formulation and to fix proper values for the convergence of results. Therefore, a new benchmark section is presented. Different cases are proposed for several material configurations, load boundary conditions and geometries. The possible effects involved in this problem (magneto-elastic coupling and effects related to embedded materials and thickness values of the layers) are discussed in depth for each thickness ratio. The innovative feature proposed in the present paper is the exact 3D study of magneto-elastic coupling effects in FGPM plates and shells for static and free vibration analyses by means of a unique and general formulation.

The superposition method is employed to obtain an accurate analytical type solution for the free vibration frequencies and mode shapes of multi‐span bridge decks. Free edge conditions are imposed on the long edges running in the direction of the deck. Inter‐span support is of the simple (knife‐edge) type. The analysis is valid regardless of the number of spans or their individual lengths. Exact agreement is found when computed results are compared with known eigenvalues for bridge decks with all spans of equal length. Mode shapes and eigenvalues are presented for typical bridge decks of three and four span lengths. In each case torsional and non‐torsional modes are studied.

An approximate method is extended for analyzing the free vibration problem of simply supported orthotropic square plate with a square hole. In this paper, a square plate with a square hole is transformed into an equivalent square plate with non-uniform thickness byconsidering the hole as an extremelythin part of the equivalent plate. Therefore, the dynamic characteristics of a plate with a hole can be obtained by analyzing the equivalent plate. The Green function, which is the discrete solution for the deflection of the equivalent plate, is used to obtain the characteristic equation of the free vibration. The effects of the side to thickness ratio, hole side to plate side ratio and the variation of the thickness on the frequencies are considered. Some numerical analyses are carried out for the simplysupported orthotropic square plate with a square hole. The efficiency and accuracyof the numerical solutions bythe present method are investigated.

This study investigates the free vibration of a moderately thick rectangular plate, which is composed of functionally graded materials and floating on incompressible fluid. Material properties are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fraction of the constituent. The governing equations of the plate are analytically derived based on the first-order shear deformation theory with consideration of rotational inertial effects and transverse shear stresses. Applied pressure on the free surface of the plate is obtained by the velocity potential function together with Bernoulli’s equation. The equation governing on the oscillatory behaviour of the fluid is obtained by solving Laplace equation with satisfying the boundary conditions. The natural frequencies and shape modes of the rectangular plate are determined by decoupling and solving the motion equations system. Then, analyses of the numerical results of free vibrations and the effects of the different parameters such as thickness to length of the plate, boundary conditions, fluid density, index of volume fraction and the height of the fluid on the frequencies are investigated. Finally, the results of this research in limit case is compared and validated with the results of other researchers and finite element model.