Showing papers on "Plate theory published in 1971"

Effect of Environment on the Elastic Response of Layered Composite Plates

Abstract: A generalized Duhamel-Neumann form of Hooke's law is used to develop laminated plate equations which include the effect of expansional strains. Such strains are induced in composite materials by temperature, absorption by a polymeric matrix material of a swelling agent such as water vapor, and by sudden expansion of absorbed gases in the matrix. Solutions to specific boundary value problems are presented for both symmetric and nonsymmetric laminates. Numerical results indicate that in addition to inducing residual stresses, expansional strains can substantially affect the gross response characteristics of a composite material.

On the Propagation of Flexural Waves in an Orthotropic Laminated Plate and Its Response to an Impulsive Load

Abstract: The dynamic equations of orthotropic laminated plates are derived from the concepts of Timoshenko's beam theory to include the effects of transverse shear and rotatory inertia. The propagation of flexural waves is discussed. The transient response of a rectangular plate to a normal impact is investigated. We also consider briefly the influence of internal friction related to the damping on the response of the plate.

A review of the three-dimensional stress problem for a cracked plate

Abstract: An analytical study has been carried out to examine the influence of plate thickness on the stress distribution around the crack. The qualitative feature of the three-dimensional solution is first determined by an asymptotic expansion of the stresses and displacements in terms of the cylindrical polar coordinates r, θ, z for small values of r which is referenced from the border of a semi-infinite crack. It is found that the stresses σrr, σθθ, σ zz , and σ rθ are singular of the order r $$r{\text{ }}^{{\text{ - }}\tfrac{1}{2}} $$ , but the transverse shear stresses σ rz and σθz , are bounded for plates under stretching and bending. The intensity of crack-border stress field becomes a function of the thickness coordinate z. Knowing that the problem prohibits any exact analytical solutions of a quantitative nature, the three-dimensional equations of elasticity will be approximated by appealing to minimum principles in the calculus of variations. Guided by the results obtained from the asymptotic expansions, each one of the six stress components is assumed to be the product of two functions, one being assigned to describe the stress distribution in the plane of the plate and the other across the thickness. The z-distribution of the stresses may either be pre-assigned arbitrarily or determined from the plane strain condition ahead of the crack. On the basis of the principle of minimum complementary energy, a system of three simultaneous differential equation in two variables is obtained and solved for the problem of an infinite plate containing a through crack by means of integral representations. Determined in closed elementary form are the detailed structure of the three-dimensional crack-edge stress field. The stress-intensity factor, which varies in the thickness direction, is shown to be a function of the ratio of plate thickness to crack length and is found to increase rapidly in magnitude as the plate thickness is perturbed slightly from zero. The present analysis suggests a method by which the effect of a finite plate thickness can be incorporated into an examination of the fracture toughness of cracked sheet specimens.

Vibration analysis of rotating cantilever plates

Abstract: A finite element technique is used to determine the natural frequencies and the mode shapes of a cantilever plate mounted on the periphery of a rotating disc. The plane of the plate is assumed to make any arbitrary angle with the plane of rotation of the disc. The distributed centrifugal force is resolved into two components—one acting in the plane of the plate and the other normal to the plate. The stresses produced in the middle surface of the plate due to the in-plane forces are first determined. The increase in the bending stiffness of the plate elements due to these in-plane stresses is obtained in a manner similar to that used in the stability analysis of plates. The component of the distributed centrifugal force normal to the plate surface is added to the inertia force. From the results of computations carried out for various values of the aspect ratio, the speed of rotation, the disc radius and the setting angle, empirical formulae are derived giving the effect of these parameters on the natural frequencies. These empirical formulae are observed to be in agreement with the corresponding known formulae for rotating cantilever beams, when the aspect ratio is high.

Further study of composite laminates under cylindrical bending.

Abstract: In a series of three papers [1,2,3], the range of applicability of classical laminated plate theory (CPT) in describing the response of composite laminates under static bending has been examined. Briefly, exact solutions within the framework of linear elasticity theory were developed and compared to the respective solutions governed by CPT [4,5,6]. Numerical data calculated based on simple harmonic load distributions have indicated rather wide discrepancy between the two solutions for laminates having low span-to-depth ratios. At high aspect ratios however, the CPT solution is in good agreement with the elasticity solution.

Stability of transient solution of moderately thick plate by finite-difference method

Abstract: : The condition of stability of the finite-difference equation for the transient response of a 'moderately thick' plate is derived. The effect of transverse shear and rotatory inertia on the stability condition are examined. For the same size of space increment the stability condition for a moderately thick plate requires a much smaller time increment than the corresponding one for a 'thin' plate in order to insure numerical stability. It is found that when either the effect of transverse shear or rotatory inertia is included, the sufficient stability condition is identical to that for the thin plate. (Author)

Dynamic Stability of Plates by Finite Elements

Abstract: Convergent finite element equations for dynamic stability of plates dependent on vibration and buckling modes

Triangular Plate Bending Elements with Enforced Compatibility

Abstract: A technique for establishing complete normal slope compatibility at the interfaces of triangular plate bending elements is presented. The nodal connections result in a deficiency of one parameter per common boundary for providing automatic conformity. Each deficiency is compensated by a constraint, imposed by the Lagrange multiplier method, which forces compatibility at the midside. The element displacement is described by a complete cubic interpolation polynomial. Nodal freedoms are the displacement tv, slopes wX9 Wy at each corner and w at the centroid. Derivation of the stiffness and other matrices required for the basic element is straightforward. Results presented, including deflections, critical loads and vibration frequencies are in general more accurate than from other available elements of equivalent complexity.

An improved estimate for the error in the classical, linear theory of plate bending

Abstract: The relative mean square error in the three-dimensional stress field predicted by classical plate theory is shown to be OQi/L^)2, where h is the plate thickness and Lj. is a mean square measure of the wavelength of the midplane deformation pattern. This improves a recent result of Nordgren who obtained a relative error estimate of 0(h/L*). The improved error estimate, which, like Nordgren's, is based on the PragerSynge hypercircle theorem in elasticity, is obtained by constructing a kinematically admissible three-dimensional displacement field that depends on the solution of the classical plate equations but which yields an accurate, nonzero distribution of the transverse shearing strain through the thickness.

Large‐Deflection Solution of the Coaxial‐Ring‐Circular‐Glass‐Plate Flexure Problem

Abstract: Nonlinear deflections and radial surface stresses in thin elastic circular plates laterally deformed into symmetrical concave shapes are analyzed. The deformations are induced by loading each plate with a small center ring while the plate is resting on a ring of nearly the same diameter as the plate. The rings and plate are coaxial, i.e. concentric. Center deflections up to 3 and 4.5 times the plate thickness were predicted for ring-diameter ratios of 0.5 and 0.2, respectively. The predicted deflection profiles at various loads agreed quite well with those which were determined experimentally on a chemically strengthened glass plate. This analysis provides a new criterion for using the coaxial-ring loading method for flexural strength testing of brittle materials.

A bound on the error in plate theory

Abstract: Introduction. A solution to a boundary value problem in the classical two-dimensional theory of plates is generally accepted as an approximate solution to a corresponding boundary value problem in the three-dimensional theory of elasticity provided that the plate is sufficiently thin. This conclusion is supported by several exact solutions for plates in the theory of elasticity [1] and by the fact that the equations of plate theory can be obtained from the equations of elasticity theory as the leading terms in parametric expansions [2], [3], Further, Morgenstern [4] has shown that the stresses and strains obtained from a solution in plate theory converge in a mean-square sense to a solution in elasticity theory as the plate thickness approaches zero. Related theorems on meansquare convergence of parametric expansions for a problem in beam theory are stated by Babuska and Prager [5]. In the present paper we derive an explicit expression for the mean-square error in the components of stress obtained from a solution in plate theory with respect to the exact solution of a corresponding problem in the theory of elasticity. In addition, a precise bound is given for the relative mean-square error. The derivation employs the hypersphere theorems of Prager and Synge [6] in the theory of elasticity. In the course of the derivation the equations of plate theory are obtained in two ways by minimization of portions of both the potential energy and the complementary energy. The general expression obtained for the error contains only quantities which are available from a solution in plate theory. Our results and the previous investigations of convergence [4], [5] show that the relative mean-square error in plate theory is proportional to the thickness of the plate in general. This is somewhat surprising since the exact solutions for plates in elasticity theory [1] give a relative error proportional to the square of the thickness. This form for relative error also is indicated by the parametric expansions [2], The discrepancy in our result can be attributed to the expression obtained for the components of transverse shear stress, which differs from the classical expression. We have been unable to derive the classical expression by the present method. (See note added in proof at the end of this paper.) I. Function space concepts in elasticity. We consider a three-dimensional elastic body R bounded by a closed surface S. With reference to a system of rectangular Cartesian coordinates z,(i = 1, 2, 3), the field equations of the linear theory of elasticity read as follows [1]: equilibrium (in the absence of body force)

A study of the buckling of laminated composite plates

Abstract: for short axial wavelengths become parallel to the sector line (Pi = 0) with the least slope. For shells with /i = 1, p = 2, the frequency lines of the third and fourth axisymmetric nontorsional modes become parallel to the frequency line of the Rayleigh waves in the material of the outer shell, prior to becoming parallel to the sector line /32 = 0 (Fig. 9), whereas, in the case of shells with p = 2, p — 1, they become parallel to the frequency line of the Rayleigh waves in the material of the inner shell prior to becoming parallel to the sector line ft = o. The frequency lines of the fourth, fifth and sixth torsional modes and the fifth and sixth nontorsional modes for shells with jit = 1, p = 2, become parallel to the sector line /32 = 0 before, for higher values of f not included in the figures, becoming parallel to the sector line ft. = 0. Analogous behavior is observed for shells with /x = 2, p = 1. The behavior of the frequency lines of the flexural modes (n > 1,) is similar to that of the axisymmetric modes, except that they do not exhibit the tendency of becoming parallel to the sector line A = 0, with the greatest slope. For n = 1, for large axial wavelengths the first mode is essentially a uniform translation of the entire cross section, the second mode involves essentially longitudinal shear motion, whereas the third mode is associated with predominantly radial motion (breathing). The three lowest flexural modes are those contained in the bending shell theories, wherein the radial component of the displacement is assumed constant across the thickness of the shell and the tangential and axial components are assumed to vary linearly with the thickness coordinate.

Exact Natural Frequencies and Modal Functions for a Thick Off-Axis Lamina

Abstract: The three-dimensional elastic field equations that describe the vibration of a class of thick, orthotropic, laminae are solved exactly. The class is characterized by an arbitrary, in-plane, angle of inclination of the principal orthotropic axes to the plate axes. The frequency spectra and modal func tions are presented for two types of materials and five inclination angles. Comparison of these results to classical thin plate theory shows that clas sical flexural plate theory is adequate for vibration with wave lengths greater than 20 times the thickness. Thin extensional plate theory is ade quate also, but shows larger errors in some of the test cases than flexural theory. In other cases, extensional theory is astoundingly accurate

Flutter of Clamped Skew Panels in Supersonic Flow

Abstract: The supersonic panel flutter problem of clamped skew panels with in-plane forces is formulated on the basis of the classical, small deflection, thin plate theory using oblique coordinates. The two-dimensional, static approximation is made use of for the aerodynamic loading. Galerkin method with the flutter mode represented in terms of a double series of beam characteristic functions is employed. Results of numerical calculations made for unstressed panels for different combinations of side ratio, angle of skew, and angle of yaw are presented here. The majority of the calculations were made using 16 terms in the series. Convergence is examined in a few typical cases. The dynamic pressure parameter for flutter is found to increase monotonically with the angle of skew for side ratio 1 and to decrease initially before beginning to increase for side ratio 0.5. The results are also compared with those obtained earlier for simply supported panels.

Vibration and Buckling Analysis of Composite Plates and Shells

Abstract: The magnitude of the previously observed effect associated with laminated composites - ie, that a coupling exists between extension and bending if the plies are not balanced in number and fiber orientation - is investigated for buckling and vibration of doubly curved monocoque plates and shells of positive and negative Gaussian curvature In addition, the effect of stacking sequence is examined Solutions are presented which provide a means of simply and economically assessing the magnitude of the coupling and stacking effects for various composite materials and geometric configurations

The application of a curved, mixed-type shell element.

Abstract: The variational condition given by Herrmann (1967) for finite element bending analyses of plates is extended to general thin shell problems. Emphasis is laid upon shell behavior and, in particular, on boundary layers and inextensional bending. A study of the boundary layer problem on a cylindrical shell compares the refined element and the Herrmann-Campbell (1968) element. Some numerical results on cross-tube and mitred-bend problems are given.

Buckling Analysis of Plate Structures by the Strip Elements

Abstract: This paper provides a new approach to the buckling analysis of plate structuers whose section are not varied along the longitudinal axis, for example, stiffened plates with one directional stiffeners, corrugated plates, polygonal tubes, sandwich plates with corrugated cores and so on.In this approach, the prebuckling stresses are assumed to be normal stresses in the longitudinal direction only, however the distribution pattern may be arbitrary.The foundamental concepts of this method are as follows. The plate elements which compose the section of the plate structures are divided into the narrow strip elements and then the stiffeness matrixes of these strip elements are derived by assuming the displacement functions and using the energy theorems. In the subsequent processes, the usual techniques of matrix method can be applied and then bucklig load is computed as the lowest eigen value. Several kinds of plate structures were analized by this method and experiments were also conducted. The coincedence was fairly good.

Circular Viscoelastic Plates Subjected to In-Plane Loads

Abstract: This paper is concerned with circular viscoelastic plates subject to in-plane forces. The plates are assumed to have a small arbitrary initial curvature and the increase in curvature as a function of time is determined. The plate material is assumed to follow a linear viscoelastic stress-strain relation and quasi-static small deflection theory is utilized. The general equation for the deflection of a circular viscoelastic plate is derived from the equation of an elastic plate using the well-known correspondence principle. Special cases are: 1) a circular plate with a fixed edge; 2) a circular plate with a simply supported edge; 3) a circular segment; and 4) an annular plate. Circular symmetry is assumed in cases 1, 2 and 4 but the solution developed in case 3 could readily be applied to the other cases if symmetry did not exist. Numerical examples are given for all cases. The applied load is assumed to be uniformly distributed around the boundary of the plate. For an annular plate the load at the inner boundary may differ from the load at the outer boundary. This leads to a variation of the in-plane forces throughout the plate. The load may vary with time in any arbitrarily prescribed manner. The load is assumed to be compressive but the analysis applies equally well for tensile loads.

Effects of Rotary Inertia on the Supersonic Flutter of Sandwich Panels

Abstract: A theoretical analysis is presented for the supersonic flutter of a simply-supported isotropic sandwich panel. The aerodynamic loading is described by a static approximation valid for a wide range of Mach number. Small deflection theory for sandwich plates which considers transverse shear deformations and rotary inertia in addition to bending is used. The purpose of this study is to demonstrate the effect of the rotary inertia parameter on the critical flutter speed of simply-suppor ted panels. Previous investigators have neglected this effect and results obtained are of an anomalous nature for certain plate parameters. The effect of such plate parameters as rotary inertia, transverse shear, and midplane stress resultants on the critical flutter speed are presented in graphical form.

Prediction of collapse loadings for composite highway bridges

Abstract: A THEORETICAL AND EXPERIMENTAL INVESTIGATION INTO THE COLLAPSE BEHAVIOR OF SINGLE-SPAN, SIMPLY SUPPORTED, SKEW, COMPOSITE STEEL GIRDERS AND DECK SLAB HIGHWAY BRIDGES IS DESCRIBED. A LOWER BOUND SOLUTION IS DEVELOPED USING THIN PLATE, SMALL DEFLECTION, ELASTIC- PLASTIC STIFFENED PLATE THEORY. DYNAMIC RELAXATION AND LOAD INCREMENTATION ARE USED TO SOLVE SETS OF FINITE DIFFERENCE EQUILIBRIUM EQUATIONS. RESULTS ARE COMPARED WITH THOSE OBTAINED FROM AN UPPER BOUND YIELD/LINE ANALYSIS AND FROM TESTS ON TWO ONE-SIXTH SCALE MODELS. /RRL (A)/