Showing papers on "Bending of plates published in 1972"

The Part-Through Surface Crack in an Elastic Plate

Abstract: Elastic plate with part-through surface crack, determining stress intensity factor for remote tensile and bending loads

Stress Analysis of Thick Laminated Composite and Sandwich Plates

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...

Two hybrid elements for analysis of thick, thin and sandwich plates

Abstract: Two general quadrilateral elements for plate bending are developed. Each has 12 degrees of freedom and accounts for transverse shear deformation effects. Each is built from four triangular elements whose properties are derived by the assumed-stress hybrid approach. Matrices needed to generate the stiffness properties of the triangular elements are explicitly stated so as to facilitate their use. Numerical test cases, in which shear deformation effects range from negligible to very important, are used to illustrate the behaviour of the elements. It is concluded that the simpler of the two quadrilaterals is one of the best plate elements currently available.

Finite element analysis of plates with curved edges

Abstract: The application of the high precision triangular plate bending element to problems with curved boundaries is considered. Appropriate edge conditions for nodal points on these boundaries are derived. The error inherent in representing the shape of a curved boundary by a series of straight segments is found to be the limiting factor on accuracy, while the effect of approximations in the actual boundary conditions is minor. To overcome the first type of error, the high precision element is modified to include one curved edge. Substantial improvements in accuracy are obtained, as demonstrated in example calculations for circular and elliptical plates.

Large Deflection of Rectangular Orthotropic Plates

Abstract: A theoretical analysis is presented for the large deflection elastic behavior of clamped, uniformly loaded, rectangular, orthotropic plates. The governing differential equations including the nonlinear terms in the sense of von Karman are solved by the method of perturbation. This results in the predictions of the load deflection relations, and bending stresses are presented in the graphical form for glass-epoxy, boron-epoxy and graphite-epoxy plates having various aspect ratios. In the case of isotropic plates, the present solution reduces to two of the existing solutions.

On the Analysis of Anisotropic Rectangular Plates

Abstract: : Extensive use of energy methods in conjunction with classical beam mode functions has been used to obtain approximate solutions to homogeneous, or symmetrically laminated, anisotropic plate problems. Because of the existence of cross- elasticity bending stiffness terms, the beam functions do not satisfy the natural boundary conditions. As a result, bending moments and stresses, which are of practical interest, may converge to the wrong solution or may not converge at all. Furthermore, bending deflections, buckling loads, and fundamental vibration frequencies converge very slowly for highly anisotropic materials. This report shows that improved results can be obtained for anisotropic plates which contain strong cross-elasticity effects by using a classical Fourier analysis which satisfies both the geometric and natural boundary conditions. Numerical results are presented for bending under transverse load, buckling under biaxial compression and pure shear, and natural frequencies of flexural vibrations. Both homogeneous and laminated plates are discussed.

Strongly Coupled Stress Waves in Heterogeneous Plates

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.

The collapse load of reinforced concrete plate

Abstract: Numerical calculation of the collapse load of a reinforced concrete plate is considered. Mathematical programming formulations are derived by using finite element representation. The plate is assumed to obey the square yield criteria. It is shown that good lower bound solutions can be obtained on a systematic basis. Upper bound solutions are approximated result of yield-line theory. An example of the square built-in plate loaded uniformly is given.

Turbulence‐Induced Plate Vibrations: Some Effects of Fluid Loading on Finite and Infinite Plates

Abstract: The effects of a fluid environment on the flow‐induced vibration of finite and infinite plates are theoretically examined. Qualitative effects of fluid loading are deduced from the mathematical models. Examples of computed flow‐induced vibration statistics for identical plates, with and without water loading, illustrate the quantitative effects of fluid loading. The fluid environment exerts forces on the plate in temporal phase with the plate inertial and resistive forces. For heavy fluids (i.e., water), the inertial component of the fluid loading is several orders of magnitude greater than the resistive component. Comparison of fluid‐loaded finite and infinite plate statistics reveals that the infinite plate velocity spectral density provides a good approximation to the level and general shape of the finite plate velocity spectral density. However, the cross‐spectral characteristics of the infinite and finite plate vibrations are substantially different.

Mixed finite element method for axisymmetric shells

Abstract: A mixed variational formulation is used as basis for developing a mixed finite element method for axisymmetric shell. The independent unknowns of the method are the axial and radial displacement components, the rotation of the normal to the middle surface and the meridional bending stress couple. The basic element is a frustrum of curved meridian. General advantages of the mixed method are presented, one of which is the possibility of using piece-wise linear functions of the meridional arclength to represent the basic unknowns. Test results are presented for plate bending, transverse shear deformation, membrane behaviour, edge-zone bending, bending near the junction of two shells, convergence of the method and accuracy of middle surface curvature interpolation. Shell geometries considered are cylindrical, conical, spherical and ellipsoidal. Good results are obtained which should increase interest in the relatively less known and less tested mixed method as compared to the stiffness method.

An experimental study of the dynamic plastic behavior of rectangular plates

Abstract: An experimental investigation has been undertaken in order to study the behavior of fully clamped rectangular plates when subjected to uniformly distributed impulsive velocities. The total energy of the dynamic loads was sufficiently large to cause plastic flow of the plate material and maximum permanent transverse deflections from 0.3 to nearly 10 times the corresponding plate thickness. The rectangular plates had aspect ratios (BETA) of 1/4, 1/2, 3/4 and 1 and were made from either hot rolled mild steel, which is a strain-rate sensitive material, or aluminum 6061 T6, which is almost strain-rate insensitive. A rigid, perfectly plastic method, which included the influence of geometry changes, gave theoretical predictions which agreed favorably with the corresponding experimental results recorded on aluminum 6061 T6 rectangular plates.

Some Elements for Analysis of Plate Bending

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.

Finite Deflections of Uniformly Loaded, Clamped, Rectangular, Anisotropic Plates

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

Comparison of Finite Elements for Plate Bending

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.

Cellular Structures of Arbitrary Plan Geometry

Abstract: A method of analysis and computer program are presented for cellular structures of constant depth with arbitrary geometry in plan view. The development is based on the finite element method. Special elements are developed to capture both the membrane and plate bending behavior of the deck and web components. The well established direct stiffness method is used for the element assembly. The structure may be subjected to a variety of force and displacement boundary conditions such as distributed dead and live loads in addition to concentrated nodal loads and prescribed nodal displacements. After solving for the unknown nodal displacements, the reactions are computed in addition to the internal forces which are output at points selected by the user.

Stresses in Circular Plates on Elastic Foundations

Abstract: The von Karmen equations with the Winkler foundation terms added are solved for a uniform load by assuming an infinite series expansion for the plate deflections. The spatial functions chosen include both regular and modified Bessel functions and are chosen to be the eigenfunctions for the linear biharmonic operator. A coupled set of nonlinear algebraic equations governing the plate deflections are derived using the Galerkin averageing techniques. Dimensionless plots of plate deflections and stresses are shown for both fixed and simply supported plates with both radially movable and immovable edges for various values of the dimensionless stiffness. The limitations of the approximate stress equations commonly used for large amplitude plate deflections are presented.

Large deflection of elastic plates under patch loading

Abstract: Two methods based on: (1) the direct finite difference approach; and (2) the dynamic relaxation method are presented for the treatment of the elastic large deflection behavior of plates under transverse loading. The merits of the methods in respect of formulation, accuracy, storage and computing time requirements are pointed out together with the main aspects of the finite element method for the analysis of the same problem. The large deflection behavior of square plates under a central patch loading (a concentrated load distributed over a finite area) is investigated. Numerical solutions are offered in a general and condensed form for plates with simply supported and clamped boundary conditions, and for a range of patch sizes. The solutions are shown to be of particular value in the assessment of stresses under a patch loading in practical plate problems. The solutions can serve as a useful guide line for the advanced design of the plate components of plated structures.

Elasto-plastic bending of cracked plates, including the effects of crack closure

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

Rectangular Finite Element for Plate Bending Analysis Based on Hellinger-Reissner's Variational Principle

Abstract: Based on Hellinger-Reissner's variational principle, a rectangular finite element for plate bending is derived with bilinearly varying deflection and incompletely linear moments. This method is an example of the mixed method first proposed by Herrmann, where both displacement and stress are adopted simultaneously as fundamental variables. Because continuity conditions required on the fundamental variables are much relaxed compared with compatible models, suitable shape functions can be easily derived and calculation of the element matrix can be performed by hand. The explicit expression of the element matrix for general orthotropic plates is presented in this paper. A few simple test problems are computed in order to demonstrate the accuracy of the present model, and it is shown that it gives improved results compared with Herrmann's triangular element. Convergence is also assured when suitably small mesh sizes are adopted, and this element can be utilized in general practical problems.

Anisotropic tapered elements using displacment models

Abstract: Two FORTRAN IV subroutines that generate the stiffness, mass and geometric matrices for plate bending and plane stress elements using displacement models are described. The elements can be of either rectangular or triangular shape with anisotropic properties together with a linear variation in thickness. Although such a formulation is straightforward theoretically, emphasis has been placed on the programming technique where the amount of algebra involved in the matrix manipulations is avoided. Exact integration is automatically performed since a closed-form solution con be obtained.