Showing papers on "Plate theory published in 1968"
TL;DR: In this paper, a recently developed triangular equilibrium element was applied to finite element analysis of some plate bending problems, and demonstrated to have a straightforward and satisfactory application and to possess advantages over the conventional triangular displacement element.
Abstract: Further details are given of a recently developed triangular equilibrium element which is then applied, in conjunction with the complementary energy principle, to the finite element analysis of some plate bending problems. The element is demonstrated to have a straightforward and satisfactory application and to possess advantages over the conventional triangular displacement element.
329 citations
TL;DR: Theoretical study of plate thickness effect on bending stress distribution around slotted cracks is presented in this article, where the authors study the effect of the thickness of the plate on the stress distribution.
Abstract: Theoretical study of plate thickness effect on bending stress distribution around slotted cracks
157 citations
106 citations
55 citations
TL;DR: In this article, the governing equations on bending of multilayered orthotropic (both core and facings) sandwich plates are developed through the use of variational methods, where in-plane stresses in the cores are neglected as is the variation of stresses across the facings.
Abstract: The governing equations on bending of multilayered orthotropic (both core and facings) sandwich plates are developed through the use of variational methods. The plate is assumed to consist of j facings and (j — 1) cores, having different thicknesses and possessing different orthotropic elastic properties. In-plane stresses in the cores are neglected as is the variation of stresses across the facings. The developed theory is an extension of that given in Ref. 7. Nomenclature a, b = length of rectangular plate in the x- and y-directions, respectively Dxz, Dyz — transverse shear rigidities of plate DXj Dy = bending rigidities of plate in x- and ^-directions, respectively DXy = twisting rigidity of plate Eix, Eiy = Young's moduli of the ith facing in the x- and ^/-directions, respectively Gixz, Giyz = shear moduli of the ith core hi = thickness of the ith core Lr(r = 1-8) = Lagrange multipliers MX, My, MXy = stress couples per unit length of plate NX, Ny, NXy = normal and in-plane shear stress resultants per unit length of plate pz = load function per unit area applied normal to the plate QX) Qy = transverse shear forces per unit length of plate
36 citations
TL;DR: In this article, a method of developing a stiffness matrix for a rectangular, moderately thick plate element is presented, thus enabling some of the advantages of finite element treatments (variable plate thickness, anisotropy, etc.) to be applied to certain problems in which thin plate theory might be considered inadequate.
33 citations
TL;DR: In this article, the duality between the basic equations and the dependent variables in problems of stretching and of bending of plates is applied to the finite element method, and a displacement method is a stress function method in the bending problem and vice versa.
Abstract: The duality between the basic equations and the dependent variables in problems of stretching and of bending of plates is applied to the finite element method. A displacement method in the stretching problem is a stress function method in the bending problem and vice versa. The stress function method has the same properties of accuracy and convergence as the well established dual displacement method for plate stretching, using only two equations per node. A computer program, originally written for the analysis of plane stress and plane strain problems by the displacement method, is used to solve plate bending problems. Results show that a high degree of accuracy may be achieved for the stress couples. The determination of the deflection and slopes is made from the curvatures and involves no loss of accuracy. Comparisons with results of the fully compatible displacement method are presented.
25 citations
01 May 1968
TL;DR: In this paper, the authors analyzed the symmetrical bending of laterally loaded circular micropolar plates and found that the results of the new plate theory are dependent on Poisson's ratio and two new parameters which are related to the material constants.
Abstract: This paper analyses the symmetrical bending of laterally loaded circular micropolar plates. The linear theory of micropolar plates has been used to investigate the effect of microstructure of the medium on bending of circular micropolar plates. It is found that the results of the new plate theory are dependent on Poisson's ratio and two new parameters which are related to the material constants of the micropolar plate theory.
22 citations
TL;DR: In this paper, an integral equation method is developed for the analysis of thin elastic plates based on a Green's formula and adapted for the solution of the biharmonic problem of plate analysis.
Abstract: An integral equation method is developed for the analysis of thin elastic plates. The integral equations are based on a Green's formula and are adapted for the solution of the biharmonic problem of plate analysis. By means of this method, a numerical solution is obtained for a simply-supported, uniformly loaded corner plate. Deflections and bending moments along the diagonal and shear forces along the edges are found. Values of these quantities previously calculated by means of a finite difference technique for a right-angled plate are confirmed by this new method and the analysis is extended to corner plates with different angles at the corner. Furthermore, for a right-angled corner plate, detailed contour plots of bending and twisting moments inside the plate are presented. With a digital computer these results are obtained without undue difficulty, and it appears that the integral equation method is well suited to this type of problem.
13 citations
TL;DR: In this article, a one-dimensional approximate theory was derived that accounts for the first eight modes of propagation of extensional waves in uniform, isotropic, linear elastic bars of rectangular cross section.
Abstract: A one‐dimensional approximate theory is derived that accounts for the first eight modes of propagation of extensional waves in uniform, isotropic, linear elastic bars of rectangular cross section. The theory provides an optimal quadratic approximation to the lateral spatial dependence of the three‐dimensional bar response. Applicable to motions in bars of arbitrary rectangular section, it is possible to predict bar responses that are neither plane stress nor plane strain, but are transitional between the two. The relation of bar to plate theory is established, and an identification of bar and plate modes is proposed. Discussion of dispersion of harmonic waves is limited to bars of square section. Typical spectra are given, and their variation with Poisson's ratio is established.
12 citations
TL;DR: In this article, two analytical methods are presented for use in bridge analysis, slope deflection and finite differences, using IBM 1620 and IBM 7094, respectively, and compared.
Abstract: Two analytical methods are presented for use in bridge analysis. The inclusion of orthotropic plate theory permits consideration of any type of deck configuration. Structural continuity between all supporting members and their flexibilities may be considered. Complete computer programs have been developed in order to apply these techniques. One method, slope deflections utilizes the IBM 1620; the other technique, finite differences, uses the IBM 7094. All of the developed equations have been applied to problems consisting of isotropic plates. Complete agreement from both methods and previously analyzed structures occurs for these test cases. An example orthotropic bridge structure is initially designed and then analyzed, when subjected to a truck loading. The resulting deflection and moment data, throughout the structure, are obtained by both analytical methods. These data are compared, resulting in excellent agreement between methods.
TL;DR: In this paper, a closed-form solution for the elastic bending of an infinite plate with a circular hole was obtained based on a three-dimensional thick plate theory developed earlier by Lee and Donnell (7).
Abstract: A closed-form solution is obtained for the elastic bending of an infinite plate with a circular hole, the solution is based on a three-dimensional thick plate theory developed earlier by Lee and Donnell (7). Numerical results of the stress concentration factor obtained here are compared to those based on Reissner and classical thin plate theories. The three-dimensional series solution given by Alblas is discussed. Twisting load is handled as usual by superposition of bending loads. A simple formula relating the stress concentration factors of bending and twisting is reached, and numerical values of twisting stress concentration factor are included.
TL;DR: In this article, the transverse shearing stresses in a thin, simply-supported, rectangular plate are determined by means of the plate equations developed by Eric Reissner, and the load distribution may be defined by any function which can be represented by a double cosine series.
Abstract: By means of the plate equations developed by Eric Reissner, the transverse shearing stresses in a thin, simply-supported, rectangular plate are determined. The load distribution may be defined by any function which can be represented by a double cosine series. The solution for uniform loading is discussed in detail. Experimental results obtained by the frozen-stress technique of photoelasticity are compared with the predictions of the Reissner theory and the classical theory. The boundary conditions used are the vanishing of deflection, bending moments, and twisting moments. The results show a marked difference between the two theories near the corner zones of the plate.
TL;DR: In this article, it was shown that the structures of the moment resultant singularities in the couple-stress theory depend upon the elastic constants of the plate and that the stress intensity factor is always larger than that predicted in the Reissner theory for this problem.
TL;DR: Close form solutions for the slope of the linearly varying thickness mirrors were obtained for fixed and simply supported boundary conditions along the central hole and maximum deflections were obtained by numerical integration and compared with the results for comparable constant thickness mirrors.
Abstract: Most cassegrainian mirrors supported along the central hole are designed for deflection tolerances using the theory for solid, constant thickness plates. Where tolerances are critical, the mirror is usually made thicker, thereby reducing the deflection, but also increasing the weight of the mirror. Weight can be reduced by using a honeycomb design; however, manufacturing problems result because of the inherent complexity. To circumvent the disadvantages of excessive weight in the solid, constant thickness design and the complexity of the honeycomb design, a lightweight, yet simple design would be desirable. A possible lightweight, yet simple design would be a solid mirror of linearly varying thickness, decreasing in thickness from the center to the outer edge. As mirrors of linearly varying thickness may provide the best solution under combined deflection and weight restraints, a design basis is required and found in small deflection plate theory. The work of H. Conway was extended to account for pressure loading proportional to mirror density for the case when Poisson's ratio is ?. Closed form solutions for the slope of the linearly varying thickness mirrors were obtained for fixed and simply supported boundary conditions along the central hole. Maximum deflections were obtained by numerical integration and compared with the results for comparable constant thickness mirrors.
TL;DR: In this article, an integral formulation for a plate subjected to periodic transverse vibrations of frequency ω and containing a crack of length 2c, is solved for the Kirchhoff bending stresses.
TL;DR: In this paper, the stiffness matrix for rectangular plywood plate bending elements is developed from the orthotropic material properties of the individual lamina, and the Melosh rectangular plate element representation is used.
Abstract: Bending of plywood plates is investigated using the finite element matrix method. The stiffness matrix for rectangular plywood plate bending elements is developed from the orthotropic material properties of the individual lamina. The Melosh rectangular plate element representation is used. A centrally loaded rectangular plywood plate was analyzed using two finite element mesh sizes. The computed stresses and deflections are compared with experimental results for 7-ply Douglas fir plywood plates. It is found that both deflections and stresses can be calculated accurately by the finite element method if the orthotropic material properties for the lamina are available. A high speed digital computer is essential to implementation of the analysis.
TL;DR: In this article, the finite difference technique is used to obtain a numerical solution for stress resultants and displacements of continuous folded plate structures, where the governing differential equations of plate and elasticity theory are written as four second-order equations involving three displacements and a plate bending moment.
Abstract: The finite difference technique is used to obtain a numerical solution for stress resultants and displacements of continuous folded plate structures. The governing differential equations of plate and elasticity theory are written as four second-order equations involving three displacements and a plate bending moment (\Iu\N, \Iv\N, \Iw\N and M\Dx\N). This technique allows the use of a tridiagonal matrix routine for the solution of the resulting difference equations. Experimental results from three aluminum models are compared with the values predicted by the theory.
01 Jan 1968
TL;DR: Flutter analysis of plates with inplane boundary support flexibility exposed to transverse pressure loading or buckled by uniform thermal expansion is presented in this article, where the authors show that these plates are vulnerable to thermal expansion.
Abstract: Flutter analysis of plates with inplane boundary support flexibility exposed to transverse pressure loading or buckled by uniform thermal expansion
TL;DR: In this article, an experimental study was conducted on two aluminum non-prismatic folded plate structural models and the experimental results obtained from the model study were compared to experimental results derived from the theory.
Abstract: An experimental study was conducted on two aluminum nonprismatic folded plate structural models. The stresses, moments, and deflections predicted by a theory for the analysis of long nonprismatic folded plate structures, which is an extension of ordinary folded plate theory, are compared to the experimental results obtained from the model study. Model 1 is made of eight triangular plates with a span to maximum depth ratio of about three. Model 2 is made of eight trapezoidal plates with a span to maximum depth ratio of about four. The predictions by the theory are shown to be in good agreement with the experimental results for both models. It is also shown that elementary beam theory is inaccurate in predicting the behavior of interior plate elements in regions where large differences exist between the depths of adjacent plates.
TL;DR: In this paper, the authors present the solutions to several plate bending problems as governed by a recent theory developed by Green and Naghdi, into which couple-stress is incorporated.
Abstract: Presented in this paper are the solutions to several plate bending problems as governed by a recent theory developed by Green and Naghdi, into which couple-stress is incorporated. Specifically, each problem considered is subjected to boundary conditions emanating from a singular load distribution acting on the free edge of a semi-infinite plate. The method of integral transforms is applied in the solutions. In general, it is found that the singularities in the shear and moment resultants are of the same order as those given in Reissner's plate theory, however the detailed structures of these singular functions are altered. The present theory also suggests that, in most cases, the maximum magnitudes of the moment resultants are diminished as compared to the corresponding results given in Reissner's theory. Also discussed is the exact relationship between the Green-Naghdi theory and Reissner's theory.
TL;DR: In this article, the flexural problem of a supported circular plate with free overhanging edge by concentrated force is considered, where the material and thickness of the plate in the portions separated by the support can be different and the concentrated force may be applied at an arbitrary point in either region.
Abstract: The flexural problem of a supported circular plate with free overhanging edge by concentrated force is considered. The material and the thickness of the plate in the portions separated by the support can be different and the concentrated force may be applied at an arbitrary point in either region. Solutions of two related problems, each according to the region of the applied force, are obtained and deflections of the plate for both problems are presented. The effect of the loading positions on bending moments and shear resultants at the support is illustrated.
TL;DR: In this paper, the problem of bending instability of a rectangular plate subjected to pure bending is analyzed for various types of transverse cross-sections and it is shown that for a plate to be inherently unstable it depends on the degree of transversal camber and the type of cross-section under consideration.