Showing papers on "Bending of plates published in 1982"
TL;DR: In this article, a 4-node, 12-degree-of-freedom quadrilateral element for thin plate bending is presented, which is based on a generalization of the efficient and reliable triangular element DKT.
Abstract: A review of 4-node, 12 degrees-of-freedom quadrilateral elements for thin plates is presented. A new element called DKQ is discussed. The formulation is based on a generalization of the efficient and reliable triangular element DKT presented in References 1 and 2 and on the rectangular element QC presented in Reference 3. These elements are derived using the so-called discrete Krichhoff technique. A detailed numerical evaluation of the behaviour of the DKQ element for the computation of displacements and stresses for thin plate bending problems is presented and discussed. The DKQ element appears to be a simple and reliable 12 degrees-of-freedom thin plate bending element.
330 citations
TL;DR: In this paper, the bending behavior of a rectangular plate is analyzed with the help of a refined higher-order theory, based on a higher order displacement model and the three-dimensional Hooke's laws for plate material, giving rise to a more realistic quadratic variation of the transverse shearing strains and linear variation of transverse normal strain through the plate thickness.
204 citations
TL;DR: In this article, an approach which is concerned with the formulation of incompatible elements for solid continuum and for plate bending problems by the Hellinger-Reissner principle is presented.
Abstract: An element stiffness matrix can be derived by the conventional potential energy principle and, indirectly, also by generalized variational principles, such as the Hu-Washizu principle and the Hellinger-Reissner principle. The present investigation has the objective to show an approach which is concerned with the formulation of incompatible elements for solid continuum and for plate bending problems by the Hellinger-Reissner principle. It is found that the resulting scheme is equivalent to that considered by Tong (1982) for the construction of hybrid stress elements. In Tong's scheme the inversion of a large flexibility matrix can be avoided. It is concluded that the introduction of additional internal displacement modes in mixed finite element formulations by the Hellinger-Reissner principle and the Hu-Washizu principle can lead to element stiffness matrices which are equivalent to the assumed stress hybrid method.
181 citations
161 citations
TL;DR: In this paper, a general finite element formulation for plate bending problem based on a higher-order displacement model and a three-dimensional state of stress and strain is attempted, which incorporates linear and quadratic variations of transverse normal strain and transverse shearing strains and stresses respectively through the thickness of the plate.
151 citations
TL;DR: In this article, the influence of axial forces on the elastic-plastic bending and springback of a beam is examined and the relationship between the curvature of the beam and the load for the case of pure bending is obtained.
88 citations
TL;DR: In this article, two new finite elements are developed for the Mindlin theory plate bending problem based on the modified Hellinger-Reissner principle with independent transverse shear strains.
Abstract: Two new finite elements are developed for the Mindlin theory plate bending problem. The formulation is based on the modified Hellinger-Reissner principle with independent transverse shear strains. Numerical examples indicate that, with properly assumed transverse shear strains, these new elements designated as PLAT8 and PLAT8H do not exhibit locking effect even for very thin plates.
83 citations
TL;DR: In this paper, the authors examined the earthquakes within the oceanic lithosphere and are associated with the bending of plates before subduction and found that the pattern of stresses within the bending oceanic seafloor lithosphere revealed by the depths and focal mechanisms of these intraplate earthquakes is one of horizontal, deviatoric tension down to a depth of about 25 km.
82 citations
TL;DR: In this article, a massless flexible circular plate with a rigid core supported on a layered viscoelastic half-space and subjected to harmonic vertical and rocking excitation is studied.
Abstract: The dynamic response of a massless flexible circular plate with a rigid core supported on a layered viscoelastic half-space and subjected to harmonic vertical and rocking excitation is studied. The mixed boundary-value problem for the case of relaxed contact conditions between the plate and the half-space is reduced to Fredholm integral equations of the second kind which are solved numerically. The effects of flexibility of the plate on the force-displacement relationship, on the motion of different points on the plate, and on the distribution of contact stresses beneath the plate are studied numerically. In general, it has been found that all of these aspects of the response of the plate are highly dependent on the flexibility of the plate relative to that of the supporting half-space.
68 citations
TL;DR: In this paper, the general plane problem for an infinite strip containing multiple cracks perpendicular to its boundaries is reduced to a system of singular integral equations and two specific problems of practical interest are then studied in detail.
Abstract: The general plane problem for an infinite strip containing multiple cracks perpendicular to its boundaries is considered. The problem is reduced to a system of singular integral equations. Two specific problems of practical interest are then studied in detail. The first problem explores the interaction effect of multiple edge cracks in a plate or beam under tension or bending. The second problem is that of a rectangular plate containing an arbitrarily oriented crack in the plane of symmetry. Particular emphasis is placed on the problem of a plate containing an edge crack and subjected to concentrated forces.
56 citations
TL;DR: In this article, expressions for the vibration of and sound radiation from a fluid-loaded elastic plate which is stiffened by a finite number of parallel beams are obtained for point and line excitation of a plate with equally spaced beams compared with those for a corresponding periodically stiffened plate.
TL;DR: In this paper, a study of the flutter and divergence instabilities of a rectangular plate with two independent loading parameters is made, where the plate is subjected to the combined action of a tangential follower force and a unidirectional axial force along one edge.
TL;DR: In this paper, the dynamic response of a plate bearing on an elastic half-space and subjected to harmonic forces is analyzed, where both the flexibility and three-dimensionalality of the plate are taken into account.
Abstract: Analytical results are presented for the dynamic response of a plate bearing on an elastic half-space and subjected to harmonic forces. The present work represents a departure from existing analyses in that herein both the flexibility and three-dimensionality of the plate are taken into account. Displacements and contact stresses are presented for square plates having a practical range of flexural stiffness. The harmonic analysis is conducted within the framework of a global stiffness solution, in which the plate and subgrade impedance matrices are formulated independently in accordance with a prescribed discretization pattern. Then compatibility of displacements and equilibrium of forces are enforced at the plate-subgrade interface. Solutions are presented for massless square plates subjected to harmonic point, uniform pressure, and moment loadings. The effect of the plate's mass on its response is studied by starting with a perfectly flexible massive plate and subsequently increasing the stiffness toward the limit of a completely rigid plate.
TL;DR: In this article, a previously developed analysis of the flexural vibration of isotropic rectangular plates is extended to include the presence of a membrane stress system, which consists of biaxial direct stress plus inplane shearing stress and is uniform throughout the plate.
TL;DR: In this paper, the authors proposed two families of isodynes related to two characteristic directions of a plane stress field and additional redundant information, which can be applied to determine all three stress components in photoelastic models and in original machine or structural parts using isodyne coatings.
Abstract: The term "isodynes" has been proposed by Pindera and Mazurkiewicz to denote a new family of characteristic lines of plane stress fields. These lines carry information on total normal forces acting on related cross sections and yield the distribution and values of related normal and shear stress components. Two families of isodynes related to two characteristic directions yield the values of all three components of a plane stress field and additional redundant information. Isodyne photoelasticity methods can be applied to determine all three stress components in photoelastic models and in original machine or structural parts using isodyne coatings. The term "gradient photoelasticity" has been proposed by Pindera and Hecker to denote a new method of photoelasticity which utilizes relationships between the curvature of light paths in a photoelastic object and the gradients of symmetrical and distortional parts of stress/strain tensors. Utilizing a basic mathematical model of photoelastic effect presented by Ramachandran and Ramaseshan, gradient photoelasticity yields the momentary values of absolute and relative photoelastic coefficients. Both methods can be applied to determine the values of stress intensity factors for arbitrary cracks and all stress components in composite structures.
TL;DR: In this article, the free vibration of a simply-supported rectangular plate having a straight notch simulating a through-crack in the plate is analyzed to obtain its eigenvalues and the dynamic stress concentration in the front of the notch.
Abstract: Free vibration of a simply-supported rectangular plate having a straight notch simulating a through-crack in the plate is analyzed to obtain its eigenvalues and the dynamic stress concentration in the front of the notch. That is, the plate is divided into two parts along the notch and then, Fredholm integral equations of the first kind are derived for the internal moment and shearing force, using Green functions satisfying the boundary conditions of each part and continuity conditions of deflection and deflection angle of the original plate. The integral equations are transformed into the algebraic equations by the numerical integration and subdomain method, to calculate the eigenvalues and the moment and shearing force distributions. They are numerically calculated with regard to the lower four modes and the effects of the aspect ratio of the plate and the length and location of the notch on them are discussed in detail.
TL;DR: In this article, the effects of various parameters on thermal buckling loads were studied and the resulting equations were linearized and integrated through the thickness of the plate to obtain the thermal elastic plate equations.
Abstract: Equations of motion for a transversely isotropic thick plate with thermal effects in a general state of nonuniform initial stress where the effects of transverse shear and rotary inertia are included are derived. The method is to perturb the nonlinear equations of elasticity by an incremental deformation. The resulting equations are linearized and integrated through the thickness of the plate to obtain the thermal elastic plate equations. A reduced set of equations for a thick plate with thermal effects is also given. Finally, the thermal buckling problems are solved for a simply supported rectangular plate in a state of uniform compressive (or tensile) initial stress plus initial bending stress combined with uniform thermal compressive stress plus thermal bending stress. The effects of various parameters on thermal buckling loads are studied.
TL;DR: In this article, free axisymmetric vibrations of an isotropic, elastic, non-homogeneous circular plate of linearly varying thickness have been studied on the classical theory of plates.
TL;DR: In this article, an effective approximate nonlinear deflection analysis of heated flat plates using the boundary element method is presented, based on both the simplified Berger equation governing nonlinear plate bending and the weighted residual formulation technique for the boundary value problem.
TL;DR: In this paper, the drilling freedom is defined in the triangular natural coordinate system and the method of nested interpolations is then used to derive a nine-freedom membrane element with u, v, φ at each vertex.
TL;DR: In this paper, a hybrid stress finite element procedure for the solution of bending stress intensity factors of a plate with a through-the-thickness crack is presented, where the dominant singular crack tip stress field is embedded in the crack tip singular elements and only regular polynomial functions are assumed in the far field elements.
Abstract: A hybrid stress finite element procedure for the solution of bending stress intensity factors of a plate with a through-the-thickness crack is presented. Reissner's sixth-order plate theory including the effects of transverse shear deformation is used. The dominant singular crack tip stress field is embedded in the crack tip singular elements and only regular polynomial functions are assumed in the far field elements. The stress intensity factors can be calculated directly from the crack tip singular stress solution functions. The effects of the plate thickness, the ratio between the crack size and the inplane dimension of the plate, and the singular element size on the stress intensity factor solution are investigated.
The effects of the explicit enforcement of traction-free conditions along crack surfaces, which are the natural boundary conditions in the present hybrid stress finite element model, are also investigated. The numerical results of bending of a plate with a straight central crack compare favourably with analytical solutions. It is also found that the explicit enforcement of traction-free conditions along crack surfaces is mandatory to obtain meaningful results for the Mode I type of bending stress intensity factor.
Book•
01 Aug 1982
TL;DR: In this paper, a plate bending kinematics-rectangular coordinates-polar coordinates-radial symmetry has been studied in the context of geometry of surfaces and principal directions.
Abstract: 1. Preliminaries.- 1.0 Motivation.- 1.1 Vectors-algebra.- 1.2 Vectors-calculus.- 1.3 Matrices.- 1.4 Statics-equilibrium.- 1.5 Summation convention and index notation.- 1.6 Elements of beam theory.- 1.7 Conclusions.- 2. Statics and Kinematics of Plate Bending.- 2.0 Introduction.- 2.1 The stress resultants.- 2.2 Principal values.- 2.3 The moment circle.- 2.4 Equilibrium equations-rectangular coordinates.- 2.5 Plate bending kinematics-rectangular coordinates.- 2.6 Equilibrium equations-polar coordinates-radial symmetry.- 2.7 Plate bending kinematics-polar coordinates-radial symmetry.- 2.8 Conclusions.- 3. Elastic Plates.- 3.0 Introduction.- 3.1 Elastic theory of plate bending-moment/curvature relations.- 3.2 Elastic theory of plate bending-governing equation.- 3.3 Circular plates-radial symmetry.- 3.4 Some simple solutions for circular plates.- 3.5 Simple solutions for problems in rectangular coordinates.- 3.6 Further separation of variable features-rectangular plates.- 3.7 Solution by finite differences.- 3.8 Some other aspects of plate theory.- 3.9 Stability of plates.- 3.10 Conclusions.- 4. Plastic Plates.- 4.0 Introduction.- A. Solid metal plates.- 4.1 Yield criteria.- 4.2 The bound theorems.- 4.3 The normality rule.- 4.4 Circular plates-square yield locus.- 4.5 Circular plates-Tresca yield locus.- 4.6 Plates of other shapes-square and regular shapes.- B. Reinforced concrete plates.- 4.7 Yield line theory-I. Fundamentals.- 4.8 Yield line theory-II. Further isotropic examples.- 4.9 Yield line theory-III. Orthotropic problems.- 4.10 Hillerborg strip theory.- 4.11 Conclusions.- 5. Optimal Plates.- 5.0 Introduction.- 5.1 Problem formulation.- 5.2 Constant curvature surfaces and principal directions.- 5.3 Basic results-corners.- 5.4 Some complete results.- 5.5 Moment volumes.- 5.6 Some theory.- 5.7 Conclusions.- 5.8 Exercises.- 6. Bibliography and Exercises.- 6.0 Bibliography.- 6.1 Exercises.- Appendix Geometry of Surfaces.- A.0 The need for geometry.- A.1 Geometry of a plane curve-curvature.- A.2 Length measurement on a surface-first fundamental form.- A.3 The normal to a surface.- A.4 Normal curvature-second fundamental form.- A.5 The derivatives of n-the Weingarten equations.- A.6 Directions on a surface.- A.7 The principal curvatures.- A.8 Principal directions.- A.9 Curvature and twist along the coordinate lines.- A.10 The curvature matrix.- A.11 The curvature circle.- A.12 Continuity requirements.- A.13 Special surfaces.- A.14 Summary-the geometrical quantities required for the construction of a plate theory.
TL;DR: In this article, a finite element procedure for determining the plate bending stiffnesses of a circular voided slab is presented and the results are compared with those obtained from tests on model elastic voided plates.
Abstract: Finite element procedures for determining the plate bending stiffnesses of a circular voided slab are presented and the results are compared with those obtained from tests on model elastic voided plates. Preliminary investigations showed that a simplified plane-strain formulation, rather than a more complex general plane-strain formulation, could be used to determine the transverse flexural stiffness. Torsional stiffnesses were obtained from an analysis using the Prandtl stress-function formulation of the torsion problems. Tests were carried out on four epoxy-resin model plates having different void sizes. Good agreement was obtained between the finite element and experimental results. For design purposes, charts are presented which enable a designer of a concrete voided slab, having a Poisson's ratio of 0.2, to determine the values of the plate bending stiffnesses required for a thin plate analysis of such a slab. It is also shown that simple calculations based upon replacing a circular void with an equivalent square void are often adequate for design purposes.
TL;DR: In this article, the carrying capacity of square hollow structural section T-joints stiffened by a rectangular flange plate is investigated for both branch bending moment and punching shear.
Abstract: The carrying capacity of square hollow structural section T-joints stiffened by a rectangular flange plate is investigated for both branch bending moment and punching shear. The ultimate moment or load is determined from the simple yield line method of which one of three failure modes is applicable depending on the plate length. A large number of combinations of branch, chord, and plate sizes are analysed to provide a statistical basis for making recommendations of optimum plate lengths and thicknesses for stiffened joints in Vierendeel truss applications.
TL;DR: In this article, the displacement variables are chosen in a hierarchical form so that lower order elements can be determined by straightforward reduction of excess terms in a higher order element, except for the nodal displacements at the vertices.
Abstract: Hybrid stress elements are known to provide accurate results for analyses of plate bending; in particular, for the prediction of moment distribution. The construction of hybrid stiffness matrix requires numerical inversion of a moment matrix, evaluation of some relatively complicated boundary integrals and several matrix transformations. Each of these operations can be time-consuming, and the matrix inversion can result in a loss of numerical accuracy.
This paper devises methods to explicitly invert the moment matrices. We find that for triangular elements the inverse is independent of the element shape and is only inversely proportional to its size. We also use a novel set of displacement variables, which greatly simplifies the boundary integration. The displacement variables are chosen in a hierarchical form so that lower order elements can be determined by straightforward reduction of excess terms in a higher order element. Except for the nodal displacements at the vertices, the present approach involves only variables at the midpoints of the sides of an element.
TL;DR: In this paper, the large deflection elastic-plastic bending of a circular plate subjected to radially outward acting bending moments uniformly distributed around its circumference is analyzed, and computer programs are given to facilitate the determination of the distributions of bending moments, in-plane forces, and displacements during the bending and after unloading or springback.
Abstract: The large deflection elastic-plastic bending of a circular plate subjected to radially outward acting bending moments uniformly distributed around its circumference is analyzed, and computer programs are given to facilitate the determination of the distributions of bending moments, in-plane forces, and displacements during the bending and after unloading or springback. Computed examples are given, and the errors developed by small deflection theory are discussed.
TL;DR: In this article, the free vibration and stability of a variable thickness annular plate subjected to a torque are analyzed by the Ritz method, where the transverse deflection of an annular polygonal plate is written in a series of the deflection functions of a uniform thickness polygon without the action of a torque and the kinetic and strain energies of the plate are evaluated analytically.
01 Oct 1982
TL;DR: In this paper, the implementation of a computer code CONE (for C(1) continuity) based on the p-version of the finite element method is described, and a hierarchic family of triangular finite elements of degree p 5 is used.
Abstract: The implementation of a computer code CONE (for C(1) continuity) based on the p-version of the finite element method is described. A hierarchic family of triangular finite elements of degree p 5 is used. This family enforces C(1)-continuity across interelement boundaries, and the code is applicable to fourth order partial differential equations in two independent variables, in particular to the biharmonic equation. Applications to several benchmark problems in plate bending are presented. Sample results are examined and compared with theoretical predictions. In particular the analysis of the bending of a rhombic plate shows a significant improvement over othr published results.
TL;DR: In this paper, a triangular plate bending hybrid element is constructed according to Reissner's principle, which has 9 degrees of freedom and the shear effect is included, and the C0 formulation of thick plate is directly developed to solve the thin plate by imposing the discrete Kirchhoff constraints.
Abstract: A triangular plate bending hybrid element is constructed according to Reissner's principle. It has 9 degrees-of-freedom and the shear effect is included. The C0 formulation of thick plate is directly developed to solve the thin plate by imposing the discrete Kirchhoff constraints. The matching problem for bending and shear in the hybrid model of thick plate is effectively treated by use of the principle of the energy regulation such that the unified analysis of thick and thin plates is realized. Finally, numerical examples are presented.