scispace - formally typeset
Search or ask a question

Showing papers on "Plate theory published in 1972"


Journal ArticleDOI
James R. Rice1, N. Levy1
TL;DR: In this paper, an elastic plate with part-through surface crack, determining stress intensity factor for remote tensile and bending loads was used to calculate the stress intensity for bending loads.
Abstract: Elastic plate with part-through surface crack, determining stress intensity factor for remote tensile and bending loads

485 citations


Journal ArticleDOI
TL;DR: In this paper, the authors consider the response of multi-ply laminates with a view toward examining the generality of previous conclusions regarding the range of validity of the approximate theory normally used in the analysis of these bodies, namely, classical laminated plate theory.
Abstract: SEVERAL recent papers1–5 have addressed the problem of defining the exact (elastic) response of composite laminates under static bending. However, all of these studies have treated laminates consisting of only a few layers, while in practical applications, composite structures may consist of many layers, in some cases, 100 or more. It is therefore appropriate that we consider the response of multi-ply laminates with a view toward examining the generality of previous conclusions regarding the range of validity of the approximate theory normally used in the analysis of these bodies, namely, classical laminated plate theory (CPT).6 Open image in new window Fig. 1 Normal stress distribution.

442 citations


Journal ArticleDOI
TL;DR: In this article, a simple differential equation is derived to describe constrained-layer damping in nonsymmetric sandwich plates and beams composed of isotropic and homogeneous layers, and the natural boundary conditions related to this equation are determined.
Abstract: A simple differential equation is derived to describe constrained-layer damping in nonsymmetric sandwich plates and beams composed of isotropic and homogeneous layers. The natural boundary conditions related to this equation are determined and some typical numerical results obtained by this equation are given. The equation is valid within the linear theories of elasticity and viscoelasticity in the absence of any constraints on thicknesses, positions, symmetries, and densities of the layers.

215 citations


Journal ArticleDOI
TL;DR: In this article, the authors show that laminated plate theory based on the Kirchhoff hypothesis becomes inaccurate for determining gross plate reformation due to the relatively soft interlaminar shear modulus in high per-formance composites.
Abstract: Because of the relatively soft interlaminar shear modulus in high per formance composites, laminated plate theory based on the Kirchhoff hypothesis becomes inaccurate for determining gross plate re...

189 citations



James R. Rice1
01 Jan 1972
TL;DR: In this paper, a model for the analysis of long part-through surface cracks in the walls of plate or shell structures is presented, with the part-cracked section represented as a line-spring in the middle surface.
Abstract: A model is discussed for the analysis of long part-through surface cracks in the walls of plate or shell structures. Such problems are formulated within the context of two dimensional plate and shell theory with the part-cracked section represented as a line-spring in the middle surface. The spring allows relative separations and rotations of the middle surface, and constitutive laws relating these discontinuities to the prevailing force and moment per unit length at any point are taken from the plane strain solution for a strip in combined tension and bending, which contains an edge crack of a corresponding depth. Prior work is reviewed and further line spring constitutive laws are discussed as appropriate to elastic analysis with thermal or residual stresses and to elastic-plastic analysis, with yielding in the ligament between the crack front and far wall in the latter case.

70 citations


Journal ArticleDOI
TL;DR: In this paper, the authors established a mathematically consistent formulation for the dynamic plate equation, utilizing Hamilton's Principle in conjunction with the three dimensional theory of elasticity, and proved that for a variable Young's modulus and a constant Poisson's ratio the resulting formulations for plates and beams are the same as those for the corresponding homogeneous problems, if a modified flexural ridigity is used.
Abstract: In the past, the analyses of floating ice plates subjected to static or dynamic loads were based on the theory of a thin homogeneous plate, although in actual floating ice plates Young's modulus may vary strongly with depth. Recently,A. Assur concluded, on the basis of a heuristic argument, that the solutions obtained for homogeneous plates may be used for floating ice plates, if a modified flexural rigidity is used. The purpose of the present paper is to study this question, by establishing a mathematically consistent formulation for the dynamic plate equation, utilizing Hamilton's Principle in conjunction with the three dimensional theory of elasticity. It is proven that for a variable Young's modulus and a constant Poisson's ratio the resulting formulations for plates and beams are the same as those for the corresponding homogeneous problems, if a modified flexural ridigity is used; thus confirmingAssur's conclusion. It is shown that the stress distribution is not linear and that the stress formula\(\sigma _{\max } = M{{z_0 } \mathord{\left/ {\vphantom {{z_0 } I}} \right. \kern- ulldelimiterspace} I}\) used by a number of investigators for the determination of the carrying capacity of a floating ice plate, as well as for the computation of failure stresses from tests on floating ice beams, is not applicable. Correct formulas are derived, corresponding stress distributions are presented and the consequences of the findings discussed.

69 citations


Journal ArticleDOI
TL;DR: In this article, the problem of placing an inclined crack centrally placed in a generally orthotropic rectangular plate under tension was investigated and the modified mapping-collocation technique introduced by Bowie was used.
Abstract: The problem of an inclined crack centrally placed in a generally orthotropic rectangular plate under tension is investigated. The modified mapping-collocation technique introduced by Bowie ...

63 citations


Journal ArticleDOI
TL;DR: Stiffened rectangular plates parametric instability under in-plane sinusoidal dynamic forces, using mathematical model with stiffeners as discrete elements as discussed by the authors, using stiffener as discrete element.
Abstract: Stiffened rectangular plates parametric instability under in-plane sinusoidal dynamic forces, using mathematical model with stiffeners as discrete elements

42 citations



01 Jan 1972
TL;DR: In this article, a simple structural model of an aircraft wing is used to show the effects of strength (stress) and flutter requirements on the design of minimum-weight aircraft-wing structures.
Abstract: A simple structural model of an aircraft wing is used to show the effects of strength (stress) and flutter requirements on the design of minimum-weight aircraft-wing structures. The wing is idealized as an isotropic sandwich plate with a variable cover thickness distribution and a variable depth between covers. Plate theory is used for the structural analysis, and piston theory is used for the unsteady aerodynamics in the flutter analysis. Mathematical programming techniques are used to find the minimum-weight cover thickness distribution which satisfies flutter, strength, and minimum-gage constraints. The method of solution, some sample results, and the computer program used to obtain these results are presented. The results indicate that the cover thickness distribution obtained when designing for the strength requirement alone may be quite different from the cover thickness distribution obtained when designing for either the flutter requirement alone or for both the strength and flutter requirements concurrently. This conclusion emphasizes the need for designing for both flutter and strength from the outset.

Journal ArticleDOI
TL;DR: In this paper, a modified approach based on the orthotropic plate theory for computing the natural frequencies of bridge slabs is presented through a set of empirical relationships between the plate parameters, and the results from the present investigation are compared with those obtained by other methods; good agreements are obtained.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the case of coupled stress waves generated by an impulsive load applied at one end of a semi-infinite plate and derived a hyperbolic system of equations in which a strong coupling in the second derivatives exists.
Abstract: Consideration of coupled stress waves generated by an impulsive load applied at one end of a semiinfinite plate. For the field equations governing the one-dimensional coupled waves a hyperbolic system of equations is obtained in which a strong coupling in the second derivatives exists. The method of characteristics described by Chou and Mortimer (1967) is extended to cover the case of strong coupling, and a study is made of the transient stress waves in a semiinfinite plate subjected to an initial step input. Coupled discontinuity fronts are found to propagate at different velocities. The normal plate stress and the bending moment at different time regimes are illustrated by graphs.

Journal ArticleDOI
TL;DR: In this article, a triangular and a rectangular finite element, both based on the hybrid formulation, are presented for the analysis of plate bending problems of arbitrary plan-form, which demonstrate generally the superiority of the hybrid elements over the equivalent displacement elements.

Journal ArticleDOI
TL;DR: In this paper, a lumped-parameter model of a rectangular plate is developed by assuming fundamental mode solutions and using Hamilton's Principle and the Euler equations to set up the differential equation of motion for the system.

Journal ArticleDOI
TL;DR: In this article, the assumed displacement fields and the assumed-stress hybrid principle were used to derive rectangular and general quadrilateral plate elements with 12 degrees of freedom (DOF).
Abstract: Rectangular and general quadrilateral plate elements having 12 degrees-of-freedom are treated in this paper. New elements are derived, two based upon assumed displacement fields and others upon the assumed-stress hybrid principle. Some of these elements account for the effects of transverse shear deformation. The hybrid elements admit the possibility of continuously variable element stiffness. The new elements, as well as closely related elements previously derived, are subjected to numerical tests in order to evaluate their relative merits. These tests include computation of displacements and moments in homogeneous and sandwich plates.

Journal ArticleDOI
C. Y. Chia1
TL;DR: In this paper, the authors considered the large deflection behavior of a rectangular anisotropic plate with clamped edges and applied the classical nonlinear theory of elastic plates to the present problem, where the classical assumptions for displacements, the straindisplacement relations associated with the von Karman assumptions, and the equations of equilibrium are the same as in the theory.
Abstract: HE elastic behavior of a rectangular orthotropic plate has been studied by a few authors making use of the von Karmantype large deflection theory. Yusuff1 has considered the postbuckling of the plate under edge compression using Fourier series for both deflection and stress function. The large deflection of the plate under lateral load has been treated by Basu and Chapman.2 Aalami and Chapman3 have also looked at the problem of the plate under transverse and in-plane loads. In the last two studies the finite-difference technique has been used and the solutions have been restricted to a special class of orthotropic materials. The present investigation is concerned with the large deflection behavior of a rectangular anisotropic plate with clamped edges. The classical nonlinear theory of elastic plates is applied to the present problem. Hence the classical assumptions for displacements, the strain-displacement relations associated with the von Karman assumptions, and the equations of equilibrium are the same as in the theory. The material properties or Hooke's law can be introduced at the final stage of the formulation of the governing differential equations. These equations are then solved by the method of perturbation. 4 Because of lack of available solutions for large deflections of anisotropic plates in literature, the present solution is specified for certain special cases and then compared with existing solutions. Contents Let us consider a rectangular plate of length 2a in the x direction, width 2b in the y direction, and thickness h in the z direction under a uniformly distributed load q per unit area. The origin of the coordinate system is chosen to coincide with the center of the midplane of the undeformed plate. The stressstrain relations for a thin homogeneous anisotropic plate may be written as

Journal ArticleDOI
TL;DR: In this paper, the bending of a uniformly loaded circular plate with mixed conditions on the boundary was investigated and two cases were considered: 1) clamped-simply supported and 2) simply supported-free.
Abstract: This paper treats the bending of a uniformly loaded circular plate with mixed conditions on the boundary. Two cases are considered: 1) clamped—simply supported and 2) simply supported—free. Deflections and bending moments are calculated at the center of the plate and compared to results obtained by other investigators. Excellent agreement is found except when the clamped segment or support segment is short compared with the circumference. An auxiliary function, which permits the stress intensity factor to be determined, is also tabulated for each case.



Journal ArticleDOI
TL;DR: The qualitative analysis and the methods of comparison are also relevant to finite elements for types of problems other than plate bending because they are related to computational effort.
Abstract: The purpose herein is to reconsider the earlier results for plate elements in a form which attempts to relate accuracy to computational effort. However, the qualitative analysis and the methods of comparison are also relevant to finite elements for types of problems other than plate bending.

Journal ArticleDOI
TL;DR: In this paper, an improved theory for orthotropic plates with thickness-shear flexibility and subjected to in-plane loading is presented, which is a generalization of the work of Haringx from columns to inplane-loaded plates.
Abstract: A new, improved theory is presented for orthotropic plates with thickness-shear flexibility and subjected to in-plane loading. The improvement introduced herein is the modification of the slopes at which the in-plane stress resultants are assumed to act. This is a generalization of the work of Haringx from columns to inplane-loaded plates. Comparison with the classical (Reissner) shear-flexible plate theory shows that the improved theory imposes loading anisotropy on the governing differential equations and tends to predict higher buckling loads than the overly conservative Reissner theory. Comparison with experimental results for uniaxially-compressed, simply supported sandwich plates with glass-fiber-reinforced facings and hexagonal-cell honeycomb cores indicates that the improved-theory results agree slightly better than do the Reissner-theory results.

01 Oct 1972
TL;DR: In this paper, a capability for solving elasto-plastic plate bending problems using assumptions consistent with Kirchhoff plate theory is developed using assumptions that both bending and extensional modes of deformation are admitted with the two modes becoming coupled as yielding proceeds.
Abstract: A capability for solving elasto-plastic plate bending problems is developed using assumptions consistent with Kirchhoff plate theory Both bending and extensional modes of deformation are admitted with the two modes becoming coupled as yielding proceeds Equilibrium solutions are obtained numerically by determination of the stationary point of a functional which is analogous to the potential strain energy The stationary value of the functional for each load increment is efficiently obtained through use of the conjugate gradient This technique is applied to the problem of a large centrally through cracked plate subject to remote circular bending Comparison is drawn between two cases of the bending problem The first neglects the possibility of crack face interference with bending, and the second includes a kinematic prohibition against the crack face from passing through the symmetry plane Results are reported which isolate the effects of elastoplastic flow and crack closure

Journal ArticleDOI
TL;DR: In this paper, the theory of sound radiation from infinite, plane plates due to the interaction of bending waves with density and stiffness fluctuations in the material of the plate was investigated, and experiments conducted with a large steel plate in air show a large measure of agreement with the theory.

Journal ArticleDOI
TL;DR: In this paper, the convergence proof of plate eigenvalue solution from conforming displacement finite elements is presented based on converting a plate free vibration problem in to a corresponding isoperimetric variational problem.
Abstract: The convergence proof of plate eigenvalue solution from conforming displacement finite elements is presented. The analysis is based on converting a thick plate free vibration problem in to a corresponding isoperimetric variational problem. A conforming thick, plate element is used to illustrate the mathematical development. On the basis of the derived asymptotic rate of convergence of the approximate eigenvalues, the authors propose a practical method of improving the numerical solutions. Extension of the mathematical proof to cover classical thin plate finite elements is briefly discussed.

Journal ArticleDOI
TL;DR: In this paper, a new approach to plate theory is developed for the dynamic analysis of multilayered plates, which provides analytical simplifications as well as refinements of the physical description which includes the skin effect.


Journal ArticleDOI
Changhwi Chi1
TL;DR: In this article, the analytical solutions for the vibrational modes of a thin circular flat plate that is simply supported at three points on the circumference are presented, and the mode shapes and corresponding eigenvalues are obtained.
Abstract: The analytical solutions for the vibrational modes of a thin circular flat plate that is simply supported at three points on the circumference are presented. The mode shapes and corresponding eigenvalues are obtained. Results show that the modes can be grouped into four different types depending on the manner by which they receive the pressure at the supported points. The problem is of the mixed boundary value type in that some portion of the boundary is free while the other portion is simply supported.

Journal ArticleDOI
TL;DR: In this article, the problem of transient axisymmetric vibrations of thin circular elastic plates due to sonic boom excitation is investigated, and the equation of motion for a solid circular plate is solved by applying the modified finite Hankel transform and the Laplace transform.
Abstract: The problem of transient axisymmetric vibrations of thin circular elastic plates due to sonic boom excitation is investigated. The equation of motion for a solid circular plate is solved by applying the modified finite Hankel transform and the Laplace transform, and the numerical results are obtained with the help of the digital computer. From the analysis of the data, obtained for the dynamic deflections of the plates for the boom duration, it is concluded that for a normal flight the boom duration has a significant effect on the vibrations of plates as compared to the overpressure of the boom.

Journal ArticleDOI
01 Oct 1972
TL;DR: In this article, the chromatographic plate theory is applied to the displacement development on ion exchangers of a band containing a mixture of two very closely related species, and a practical determination of the distribution of species in a steady state band, when P, e, and the composition of the mixture are known, is given.
Abstract: The chromatographic plate theory is applied to the displacement development on ion exchangers of a band containing a mixture of two very closely related species. When the number of plates of the band P is small and the separation factor α=1 + e is closie to unity so that P, ≪ 1, and if one of the species of the mixture is of a small percentage, then a linear distribution of the concentrations in the band is obtained. In a more general case the distribution's eqution is given. The band is symmetrical only if its initial composition is 50–50. In every other case the plate corresponding to the initial mixture is not the median plate of the band. We have derived a mathematical expression which gives the place of the median in the band. A practical determination of the distribution of species in a steady-state band, when P, e, and the composition of the mixture are known, is given. A limited experimental verification is given for the case of the separation of boron isotopes.