Various Combinations Of Boundary Conditions Research Articles

Steady-state response of a cantilever plate subjected to harmonic displacement excitation at the base

An interest in the dynamic steady-state response of cantilever plates to harmonic lateral and rotational displacement imposed along the clamped edge has arisen in connection with the projecting of lifetimes of electronic components mounted on the plate lateral surface. Analytical type solutions to the problem are obtained by exploiting the superposition method, a method which has previously been successfully exploited to obtain accurate solutions for free vibration problems involving rectangular plates with various combinations of boundary conditions, point supports, etc. This newly developed approach to free vibration problem analysis has been modified here to handle forced vibration problems, in particular, to calculate response of cantilever plates of two different geometries to a range of base excitation frequencies. The theoretical work has been supported by a careful parallel experimental program. Very good agreement between theory and experiment has been encountered. It is expected that the theoretical approach described will provide a powerful analytical means for obtaining accurate solutions to various other rectangular plate forced vibration problems.

SHEAR BUCKLING OF THIN PLATES USING THE SPLINE COLLOCATION METHOD

This paper presents a highly accurate method for analyzing the critical shear buckling load of thin elastic rectangular plates. The solutions are approximated by the extended spline collocation method (SCM). Using the quintic table in place of the complex quintic B-spline functions, one can easily formulate the field equation of shear buckling loads for a thin elastic rectangular plate. Through the generalized eigenvalue analysis, the shear buckling loads and mode shapes for the plate can be determined precisely. Numerical examples are given for the critical shear buckling load of plates with various combinations of boundary conditions, aspect ratios, and uni- and bi-directional compressive/tensile loadings. The solutions obtained by the SCM are compared with those by the finite element method, the Lagrangian multiplier method, and the extended Kantorovich method under several types of boundary conditions. Compared with the other methods, the proposed SCM is not only more accurate, but also easier for computation.

Stochastic nonlinear free vibration of laminated composite plates resting on elastic foundation in thermal environments

This paper presents the nonlinear free vibration analysis of laminated composite plates resting on elastic foundation with random system properties in thermal environments. System parameters are modeled as basic random variables for accurate prediction of system behavior. A C0 nonlinear finite element based on HSDT in von Karman sense is used to descretize the laminate. A direct iterative method in conjunction with first-order perturbation technique is outlined and applied to solve the stochastic nonlinear generalized eigenvalue problem. The developed stochastic procedure is successfully used for thermally induced nonlinear free vibration problem with a reasonable accuracy. Numerical results for various combinations of boundary conditions, geometric parameters, amplitude ratios, foundation parameters and thermal loading have been compared with those available in literature and an independent MCS. Some new results are also presented which clearly demonstrate the importance of the randomness in the system parameters on the response of the structures.

Elastoplastic buckling analyses of rectangular plates under biaxial loadings by the differential qudrature method

The paper investigates the elastoplastic buckling behavior of thin rectangular plates under biaxial loadings by using the differential quadrature (DQ) method for the first time. Both J 2 ′ flow and deformation plasticity theories are adopted and iteration processes are involved due to the material non-linearity. The methodology and procedures are worked out in detail. The buckling behavior of thin rectangular plates with various combinations of boundary conditions and under various loadings is studied. It is found that the DQ results are compared well with existing analytical solutions for plates with two boundaries simply supported and the others either simply supported or clamped. Buckling loads for plates with other combinations of boundary conditions, no analytical solutions are available in such cases, are also presented. The existence of an optimal loading path for the deformation theory model, firstly reported by Durban and Zuckerman, is also observed in such cases. The possible reason is given to explain the large discrepancy in predictions for thicker plates by the J 2 ′ flow theory and deformation plasticity theory.

Exact vibration analysis of variable thickness thick annular isotropic and FGM plates

Annular plates are used in many engineering structures. In many cases variable thickness is used in order to save weight and improve structural characteristics. In recent years functionally graded materials (FGM) are used in many engineering applications. A FGM plate is an inhomogeneous composite made of two constituents (usually ceramic and metal), with both the composition and the material properties varying smoothly through the thickness of the plate. An optimal distribution of material properties may be obtained. The plate vibrations will have a strong bending–stretching coupling effect. The equations of motion including the effect of shear deformations using the first-order shear deformation theory are derived and solved exactly for various combinations of boundary conditions. The solution is obtained by using the exact element method. Exact vibration frequencies and modes are given for several examples for the first time.

Vibration of shear deformable plates with variable thickness — first-order and higher-order analyses

This work presents accurate numerical calculations of the natural frequencies for elastic rectangular plates of variable thickness with various combinations of boundary conditions. The thickness variation in one or two directions of the plate is taken in polynomial form. The first-order shear deformation plate theory of Mindlin and the higher-order shear deformation plate theory of Reddy have been applied to the plate analysis. The governing equations and the boundary conditions are derived using the dynamic version of the principle of minimum of the Lagrangian function. The solution is obtained by the extended Kantorovich method. This approach is combined with the exact element method for the vibration analysis of members with variable flexural rigidity, which provides for the derivation of the exact dynamic stiffness matrix of varying cross-sections strips. The large number of numerical examples demonstrates the applicability and versatility of the present method. The results obtained by both shear deformation theories are compared with those obtained by the classical thin plate theory and with published results.

Vibration and stability of ring-stiffened Euler–Bernoulli tie-bars

Tie-bars are frequently used in structural and mechanical engineering applications. To satisfy requirements like weight reduction, stability improvement, etc., the tie-bars are stiffened with rings. In this paper a method is developed to calculate the natural frequencies, buckling axial force, etc., of the ring-stiffened tie-bars. The dynamics of the ringed and the unringed portions of the beam are treated separately. The mode shape of the first portion was expressed as the superposition of two functions multiplied by constants. Consideration of continuity of deflection and of slope and compatibility of bending moment and shearing force at the first step enabled the mode shape of the second portion to be expressed as the superposition of two functions but multiplied by the same constants as in the first portion. This procedure was recursively carried out up to the last portion. The frequency equation was then derived from the boundary conditions at the end. Buckling of the tie-bar was considered as the case when one of the natural frequencies is zero. The first three frequency parameters and the first two buckling dimensionless axial forces are tabulated for tie-bars stiffened with various number of rings and for various combinations of boundary conditions. The calculation procedure can handle any number and any type of ring-stiffeners.

Buckling of intermediate ring supported cylindrical shells under axial compression

This paper is concerned with the elastic buckling of axially compressed, circular cylindrical shells with intermediate ring supports. The simple Timoshenko thin shell theory and the more sophisticated Flügge thin shell theory have been adopted in the modeling of the cylindrical shells. We used these two representative theories to examine the sensitivity of the buckling solutions to the different degree of approximations made in shell theories. By dividing the shell into segments at the locations of the ring supports, the state-space technique is employed to derive the solutions for each shell segment and the domain decomposition method utilized to impose the equilibrium and compatibility conditions at the interfaces of the shell segments. First-known exact buckling factors are obtained for cylindrical shells of one and multiple intermediate ring supports and various combinations of boundary conditions. Comparison studies are carried out against benchmark solutions and independent numerical results from ANSYS and p-Ritz analyses. The influence of the locations of the ring supports on the buckling behaviour of the shells is examined.

Stability of variable thickness shear deformable plates—first order and high order analyses

This work presents analysis of the buckling loads for thick elastic rectangular plates with variable thickness and various combinations of boundary conditions. Both the first order and high order shear deformation plate theories have been applied to the plate's analysis. The effects of higher order non-linear strain terms (curvature terms) are considered as well. The governing equations and the boundary conditions are derived using the principle of minimum of potential energy. The solution is obtained by the extended Kantorovich method. This approach is combined with the exact element method for the stability analysis of compressed members with variable cross-section, which provides for the derivation of the exact stiffness matrix of tapered strips including the effect of in-plane forces. The results from the two shear deformation theories are compared with those obtained by the classical thin plate's theory and with published results.

Stability and vibration of shear deformable plates––first order and higher order analyses

This work presents the highly accurate numerical calculation of the natural frequencies and buckling loads for thick elastic rectangular plates with various combinations of boundary conditions. The Reissener–Mindlin first order shear deformation plate theory and the higher order shear deformation plate theory of Reddy have been applied to the plate’s analysis. The governing equations and the boundary conditions are derived using the dynamic version of the principle of minimum of the total energy. The solution is obtained by the extended Kantorovich method. This approach is combined with the exact element method for the vibration and stability analysis of compressed members, which provides for the derivation of the exact dynamic stiffness matrix including the effect of in-plane and inertia forces. The large number of numerical examples demonstrates the applicability and versatility of the present method. The results obtained by both shear deformation theories are compared with those obtained by the classical thin plate’s theory and with published results. Many new results are given too.

Dynamic stiffness vibration analysis using a high‐order beam model

AbstractThis work presents the derivation of the exact dynamic stiffness matrix for a high‐order beam element. The terms are found directly from the solutions of the differential equations that describe the deformations of the cross‐section according to the high‐order theory, which include cubic variation of the axial displacements over the cross‐section of the beam. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments. Using the dynamic stiffness matrix exact vibration frequencies for beams with various combinations of boundary conditions are tabulated and compared with results from the Bernoulli–Euler and Timoshenko beam models. Copyright © 2003 John Wiley & Sons, Ltd.

Buckling loads of variable thickness thin isotropic plates

This work presents accurate solutions for bifurcation buckling loads of rectangular thin plates with thickness that varies in the directions parallel to the two sides. The plates are subjected to biaxial compression and various combinations of boundary conditions are considered. The calculation of the critical loads was carried out by using the extended Kantorovich method. For the resulting ordinary differential equation, an exact method for the stability analysis of compressed members with variable flexural rigidity is used. The buckling load is found as the inplane load that makes the determinant of the stiffness matrix equal zero. New, exact results are given for many cases of uni-directional and bi-directional variation in thickness. The results are very accurate and exact for cases where an exact solution is available.

Effects of heterogeneities on capillary pressure–saturation–relative permeability relationships

In theories of multiphase flow through porous media, capillary pressure–saturation and relative permeability–saturation curves are assumed to be intrinsic properties of the medium. Moreover, relative permeability is assumed to be a scalar property. However, numerous theoretical and experimental works have shown that these basic assumptions may not be valid. For example, relative permeability is known to be affected by the flow velocity (or pressure gradient) at which the measurements are carried out. In this article, it is suggested that the nonuniqueness of capillary pressure–relative permeability–saturation relationships is due to the presence of microheterogeneities within a laboratory sample. In order to investigate this hypothesis, a large number of “numerical experiments” are carried out. A numerical multiphase flow model is used to simulate the procedures that are commonly used in the laboratory for the measurement of capillary pressure and relative permeability curves. The dimensions of the simulation domain are similar to those of a typical laboratory sample (a few centimeters in each direction). Various combinations of boundary conditions and soil heterogeneity are simulated and average capillary pressure, saturation, and relative permeability for the “soil sample” are obtained. It is found that the irreducible water saturation is a function of the capillary number; the smaller the capillary number, the larger the irreducible water saturation. Both drainage and imbibition capillary pressure curves are found to be strongly affected by heterogeneities and boundary conditions. Relative permeability is also found to be affected by the boundary conditions; this is especially true about the nonaqueous phase permeability. Our results reveal that there is much need for laboratory experiments aimed at investigating the interplay of boundary conditions and microheterogeneities and their effect on capillary pressure and relative permeability.

Stiffness of Toyoura Sand from Dilatometer Tests

A series of dilatometer test profiles were carried out in a calibration chamber on Toyoura sand using a standard blade and a special research dilatometer. Pluviated sand samples were prepared using various combinations of boundary conditions, relative density, and stress history. The research dilatometer probe consists of a standard blade modified internally to allow monitoring of the complete pressure-displacement soil response during a test. Precise performance of small unload-reload cycles can also be performed to evaluate elastic deformation properties. The pressure-displacement response during the expansion tests clearly shows the influence of stress history and density. The stiffness of the Toyoura sand from both dilatometers were also compared to that measured in laboratory monotonic shear tests and to various empirical relationships. The results suggest that the moduli measured from the unload-reload cycles during the research dilatometer tests reflect the elastic properties of the soil and can be correlated to results from resonant column tests and monotonic torsional shear tests. Modulus values measured with the standard dilatometer are more representative of an elasto-plastic response.

Differential cubature method: A solution technique for Kirchhoff plates of arbitrary shape

A treatment for bending analysis of Kirchhoff plates using the differential cubature (DC) method is presented. To the authors' knowledge, this represents the first known successful application of the DC method to this problem. The plates treated are subjected to various combinations of boundary conditions and of arbitrary plate geometry. A discussion on the implementation issues is presented and several numerical test examples are included to demonstrate the applicability, efficiency and simplicity of the approximate procedures. The accuracy of the procedures is verified by direct comparing the present numerical results of different plate shapes with the existing exact analytical solutions [1] and other sources of numerical values.

SOLVING THE VIBRATION OF THICK SYMMETRIC LAMINATES BY REISSNER/MINDLIN PLATE THEORY AND THE p-RITZ METHOD

A global p-Ritz method is employed for vibration analysis of thick rectangular laminates with various combinations of boundary conditions. In the method a set of boundary beam characteristic orthogonal polynomials is used as admissible functions in the Ritz minimization procedures. To incorporate the effects of transverse shear deformation and rotary inertia, first-order Reissner/Mindlin plate theory is employed. Several example problems are presented for symmetric laminates of different boundary conditions. Numerical results are carefully established through convergence studies. Comparisons are made with the known published values to verify the accuracy of the p-Ritz method. Finally results in terms of non-dimensional frequency parameters for various boundary conditions, aspect ratios and relative thickness ratios are presented.

Optimal‐order convergence rates for Eulerian‐Lagrangian localized adjoint methods for reactive transport and contamination in groundwater

AbstractEulerian–Langrangian localized adjoint methods (ELLAM) were developed to solve convection–diffusion–reaction equations governing contaminant transport in groundwater flowing through a porous medium, subject to various combinations of boundary conditions. In this article, we prove optimal‐order error estimates and some superconvergence results for the ELLAM schemes. In contrast to many existing estimates for a variety of numerical methods, which often contain the temporal derivatives of the exact solution, our error estimates contain the total derivatives of the exact solution but do not involve any temporal derivatives of the exact solution. © 1995 John Wiley & Sons, Inc.

Simulation of Compressible Convection: A Comparative Study of Boundary Conditions

Numerical simulation of compressible hydrodynamics in the context of astrophysical convection is necessarily restricted to a limited volume of an entire star: however, the physical parameters inside that volume are determined by what goes on throughout the star as a whole. The aim of boundary conditions (BCs) is to model the contact (both mechanical and thermal) between the computational domain and the rest of the star. Different investigators have used various combinations of BCs. Here we explore how the choice of BC affects certain aspects of the solutions of three-dimensional compressible convection. As regards mechanical BCs, we examine both closed and open domains

The equivalence between the wave propagation method and Bolotin's method in the asymptotic estimation of eigenfrequencies of a rectangular plate

The Kirchhoff thin plate equation is an important mathematical model in structural dynamics. Because this partial differential equation has order four, analysis of vibration eigenfrequencies is much more complicated than that of a vibrating membrane, whose governing PDE has order two. As a matter of fact, for the simplest rectangular geometry the plate boundary value problems do not even allow the separation of the two space variables, thus one must resort to alternative approaches. Bolotin's method, published in 1961, offers such an asymptotic alternative. More recently, the three authors generalized Keller-Rubinow's wave method to the thin plate equation, constituting a different approach. In this paper, we establish that for various combinations of boundary conditions (clamped, hinged, roller-supported and free), after some conversion the wave method and Bolotin's method yield an identical set of transcendental equations for asymptotic values of the eigenfrequencies of a rectangular plate, thus confirming the equivalence of these two methods.

Buckling and postbuckling behavior of delaminated sandwich beams

An investigation was performed to study the buckling and postbuckling behavior of sandwich beams containing lengthwise and depthwise through-the-width delaminations. An analytical beam model was developed to predict the buckling load of the beam and to describe its postbuckling response for arbitrarily situated delaminations and various combinations of boundary conditions. Griffith's energy release rate model was employed to predict the stability of delamination propagation under external loading and to determine the direction of delamination growth. Parametric studies over a wide range of beam geometries, damage sizes and locations, composite facings and beam boundary conditions were carried out to study their effects on the overall behavior of the sandwich structure, as well as its damage tolerance. The results demonstrated that a sandwich construction is very ‘sensitive’ to the presence of delaminations situated at the core-faceplate interface. Premature buckling failure occurs at external loads which are significantly lower than the buckling load for a ‘perfect’ sandwich beam; in ‘imperfect’ beams with composite faceplates, the layup sequence affects significantly the load-carrying capacity of the beam; varying either the boundary conditions in a sandwich beam or the lengthwise location of a delamination has a small effect on the postbuckling behavior of the beam. Delaminations located within composite faceplates have less pronounced influence, and as the defect is moved outwards the limit load may reach the buckling load corresponding to that of the ‘perfect’ beam. The proposed model is capable of analyzing the postbuckling behavior of both sandwich and composite laminated beams for arbitrary locations of the delamination, and various combinations of boundary conditions.