Structural Imposed Load Analysis of Isotropic Rectangular Plate Carrying a Uniformly Distributed Load Using Refined Shear Plate Theory

This presents the static flexural analysis of a three edge simply supported, one support free (SSFS) rectangular plate under uniformly distributed load using a refined shear deformation plate theory. The shear deformation profile used, is in the form of third order. The governing equations were determined by the method of energy variational calculus, to obtain the deflection and shear deformation along the direction of x and y axis. From the formulated expression, the formulars for determination of the critical lateral imposed load of the plate before deflection reaches the specified maximum specified limit and its corresponding critical lateral imposed load before plate reaches an elastic yield stress is established. The study showed that the critical lateral imposed load decreased as the plates span increases, the critical lateral imposed load increased as the plate thickness increases, as the specified thickness of the plate increased, the value of critical lateral imposed load increased and increase in the value of the allowable deflection value required for the analysis of the plate reduced the chances of failure of a structural member. This approach overcomes the challenges of the conventional practice in the structural analysis and design which involves checking of deflection and shear after design; the process which is proved unreliable and time consuming. It is concluded that the values of critical lateral load obtained by this theory achieve accepted transverse shear stress to the depth of the plate variation in predicting the flexural characteristics for an isotropic rectangular SSFS plate. Numerical comparison was conducted to verify and demonstrate the efficiency of the present theory, and they agreed with previous studies. This proved that the present theory is reliable for the analysis of a rectangular plate. Keywords— Allowable deflection, critical imposed load, energy method, plate theories, shear deformation, SSFS rectangular plate

Chladni Figures Simulation on a Rectangular Plate

An analytical model for creating flat Chladni figures is presented. The equation of a standing wave in the simplest boundary conditions and the Fourier transform are used. Top view images are shown at different frequencies. The practical significance of the results obtained for the further development of the field of creating Chladni figures based on standing waves of different physical nature has been determined.

Investigation of the influence of an additional viscoelastic support mass-inertial characteristics on the rectangular plate non-stationary deformation

Modeling additional supports that affect the non-stationary deformation of lamellar structural elements is associated with a number of idealizations and assumptions. Many sources describe the deformation of supported structural elements using absolutely rigid additional supports or stiffeners. In reality, additional supports have viscoelastic properties (viscous and elastic components). When studying non-stationary vibrations, one should also take into account the mass-inertial properties of additional supports. Goal. The goal of the work is: 1) refinement of the existing mathematical model of an additional viscoelastic support by taking into account the influence of its mass-inertial characteristics; 2) study of the influence of these characteristics on the non-stationary deformation of a rectangular plate. Methodology. The non-stationary deformation of beams or plates is described by systems of partial differential equations. For these objects, good results are given by models based on the hypotheses of S.P. Timoshenko, taking into account the inertia of rotation and shear. Such systems of equations can be solved by expanding the sought functions (displacements and angles of rotation) in the corresponding series and using the direct and inverse integral Laplace transform. The determination of the unknown reaction of the additional viscoelastic support, taking into account its mass-inertial characteristics, is carried out on the basis of solving the Volterra integral equations. Results. In this work, an analytical and numerical solution in a general form is obtained, which makes it possible to determine the dependence of the change in time of reaction between the plate and the additional support for various parameters of the mechanical system. Originality. The solution to this problem is based on the further development by the authors of an approach to modeling additional supports in the form of additional unknown non-stationary loads, which are determined from the analysis of Volterra integral equations. Practical value. Examples of calculations for the considered mechanical system at three different values of mass are given. It is shown that the mass-inertial characteristics of the additional support cause a noticeable effect on the oscillatory process, and the changes concern both amplitude and phase characteristics.

Experimental determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part II

In this study coefficients of the multi-parameter Williams power series expansion for the stress field in the vicinity of the central crack in the rectangular plate and in the semi-circular notched disk under bending are obtained by the use of the finite element analysis. In SIMULIA Abaqus, the finite element analysis software, the numerical solutions for these two cracked geometries are found. The rectangular plate with the central crack has the geometry similar to the geometry used in the digital photoelasticity. Numerical simulations of the same cracked specimen as in the experimental photoelasticity method are performed. The numerical solutions obtained are utilized for the determination of the coefficients of the Williams series expansion. The higher-order coefficients are extracted from the finite element method calculations implemented in Simulia Abaqus software package and the outcomes are compared to experimental values. Determination of the coefficients of the terms of this series is performed using the least squares-based regression technique known as the over-deterministic method, for which stresses data obtained numerically in SIMULIA Abaqus software are taken as inputs. The plate with a small central crack has been considered either. This kind of the cracked specimen has been utilized for comparison of coefficients of the Williams series expansion obtained from the finite element analysis with the coefficients known from the theoretical solution based on the complex variable theory in plane elasticity. It is shown that the coefficients of the Williams series expansion match with good accuracy. The higher-order terms in the Williams series expansion for the semi-circular notch disk are found.

EXTRACTION OF ELASTIC, VISCOUS AND INERTIAL COMPONENTS FROM THE TOTAL REACTION OF AN ADDITIONAL SUPPORT ATTACHED TO A RECTANGULAR PLATE

The paper deals with a mechanical system consisting of a hinged rectangular plate and an additional viscoelastic support with considering its mass-inertia. The impact of the characteristics of additional support on the plate strained state is studied by an original approach of extracting elastic, viscous and inertial components from the total reaction. The plate is assumed to be medium thickness, elastic and isotropic. The Timoshenko hypothesis is used for deformation equations. The external non-stationary force initiates plate vibrations. The impact of the additional support is replaced by the action of three unknown independent non-stationary concentrated forces. The basic formulas for deriving system of three Volterra integral equations are proposed. The system is then solved by numerical and analytical method. By discretizing in time the system of Volterra integral equations is reduced to a system of matrix equations. The system of matrix equations is solved with using generalized Kramer’s algorithm for block matrices and Tikhonov’s regularization method. Note that the approach proposed is applicable for other objects with additional supports, such as beams, plates and shells having various boundary contour and boundary supporting. The results of computing elastic, viscous and inertial components of total reactions on the plate are given. The approach proposed is verified by matching the results of computations by two different methods, namely numerical and analytical for one total reaction and numerical for the total reaction obtained by adding elastic, viscous and inertial components.

Three-Dimensional Solution for the Vibration Analysis of Functionally Graded Rectangular Plate with/without Cutouts Subject to General Boundary Conditions

Functionally graded materials (FGMs) structures are increasingly used in engineering due to their superior mechanical and material properties, and the FGMs plate with cutouts is a common structural form, but research on the vibration characteristics of FGMs plate with cutouts is relatively limited. In this paper, the three-dimensional exact solution for the vibration analysis of FGMs rectangular plate with circular cutouts subjected to general boundary conditions is presented based on the three-dimensional elasticity theory. The displacement field functions are expressed as standard cosine Fourier series plus auxiliary cosine series terms satisfying the boundary conditions in the global coordinate system. The plate with circular cutout is discretized into four curve quadrilateral sub-domains using the p-version method, and then the blending function method is applied to map the closed quadrilateral region to the computational space. The characteristic equation is obtained based on the Lagrangian energy principle and Rayleigh–Ritz method. The efficiency and reliability of proposed method are verified by comparing the present results with those available in the literature and FEM methods. Finally, a parametric study is investigated including the cutout sizes, the cutout positions, and the cutout numbers from the free vibration characteristic analysis and the harmonic analysis. The results can serve as benchmark data for other research on the vibration of FGMs plates with cutouts.