Showing papers on "Bending of plates published in 1988"
TL;DR: In this paper, a stiffness method has been used to study dispersive wave propagation in a laminated anisotropic plate with arbitrary number of laminae, where the axis of isotropy of each lamina lies in the plane of the lamina.
Abstract: A stiffness method has been used in this article to study dispersive wave propagation in a laminated anisotropic plate. The advantage of this method is in its usefulness in obtaining numerical results for the dispersion characteristics of waves propagating in a plate with an arbitrary number of arbitrarily anisotropic laminae. This method has been applied here, as a way of illustration, to a plate made up of transversely isotropic laminae with the axis of isotropy of each lamina lying in the plane of the lamina. Results thus obtained are shown to agree well with the exact solutions for isotropic and transversely isotropic single layered plates. Numerical results are presented for cross‐ply (0°/90°/0°) laminated composite plates and show that the frequency spectrum in this case differs considerably from that for a single layered (0°) plate.
177 citations
TL;DR: In this paper, a method for calculating the locations and sizes of delaminations which occur in a rectangular, fiber reinforced composite plate subjected to non-penetrating (low velocity) impact of a solid object is presented.
Abstract: A method is presented for calculating the locations and sizes of delaminations which occur in a rectangular, fiber reinforced composite plate subjected to nonpenetrating (low velocity) impact of a solid object. The plate may be simply supported or clamped along its edges. In-plane loads or in-plane strains may be imposed on the plate during the impact. The method includes two steps. First, the stresses and strains in the plate are calculated by a three-dimensional, transient finite element method using 8-node brick elements with incompatible modes. Second, the locations, lengths, and widths of delaminations inside the plate are predicted by means of a proposed failure criterion, which is based on the concept of dimensional analysis. The finite element method and the failure criterion were implemented by a computer code which can be used to calculate the impactor position and velocity, the displacements of the plate, the stresses and strains inside the plate during the impact, and the locations and dimensions of the delaminations after the impact. Parametric studies were performed to illustrate the information which can be generated by the computer code.
126 citations
TL;DR: A polynomial displacement basis for the three-node plate bending element (Zienkiewicz-triangle) is developed from a relaxed C1-continuity requirement called the interpolation test, which provides a general convergence criterion for non-conforming shape functions and a practical guideline to select a proper displacement basis.
Abstract: A polynomial displacement basis for the three-node plate bending element (Zienkiewicz-triangle) is developed from a relaxed C1-continuity requirement called the interpolation test. The test provides a general convergence criterion for non-conforming shape functions and a practical guideline to select a proper displacement basis. The resulting simple displacement type element passes the patch test. Several reduced numerical integration schemes are discussed and numerical testing provides a comparison with the standard element formulated by Zienkiewicz.
116 citations
TL;DR: In this article, a finite element formulation for flexure of a symmetrically laminated plate based on a higher-order displacement model and a three-dimensional state of stress and strain is presented.
Abstract: A finite element formulation for flexure of a symmetrically laminated plate based on a higher-order displacement model and a three-dimensional state of stress and strain is presented here. The present higher-order theory incorporates linear variation of transverse normal strains and parabolic variation of transverse shear strains through the plate thickness, and as a result it does not require shear correction coefficients. A nine-noded Lagrangian parabolic isoparametric plate bending element is described. The applications of the element to bending of laminated plates with various loading, boundary conditions, and lamination types are discussed. The numerical evaluations also include the convergence study of the element used. The present solutions for deflections and stresses are compared with those obtained using the three-dimensional elasticity theory, closed-form solutions with another high-order shear deformation theory, and the Mindlin's theory. In addition, numerical results for a number of new problems, not available in the literature, are presented for future reference.
113 citations
TL;DR: In this paper, a new triangular plate element is presented based on independent interpolations for slopes, displacement and shear forces, and it is shown that it does not suffer from any defect common to other Mindlin plate elements.
Abstract: A new triangular plate element is presented. This new element is based on independent interpolations for slopes, displacement and shear forces, and it is shown that it does not suffer from any defect common to other Mindlin plate elements. Several examples are presented to illustrate the behaviour of this new element.
100 citations
TL;DR: In this article, a direct boundary element formulation for Reissner's plate bending theory is reviewed and found to be also applicable to external problems in infinite plates, which bears close resemblance with the standard plane strain boundary element implementation producing singular integrals of the same order.
69 citations
TL;DR: In this paper, a finite element formulation for flexure of a generally orthotropic plate based on a higher-order displacement model and a three-dimensional state of stress and strain is presented.
62 citations
TL;DR: This paper proceeds from the field-consistency paradigm to understand why the original element and even the element modified by the 2 ∗ 2 Gaussian rule cannot achieve, consistently, the true shear strain constraints in the penalty limit of thin plate behaviour, and derives the optimal shear strains definitions that leave the element free of all problems in the rectangular form.
51 citations
TL;DR: In this paper, the bending of a Winkler foundation is solved by using an original boundary integral equation method involving the fundamental solution for plate flexure problems, and an integral representation for the second member of the equation is given.
49 citations
TL;DR: In this paper, the authors examined thermal contraction stresses due to temperature changes in a thin, semi-infinite rectangular plate bounded by the spreading center axis, and calculated the state of thermal contraction stress calculated from this model is characterized by large tensile stresses at the boundaries of the plate and relatively low stresses in the plate interior.
Abstract: Nonuniform contraction of the oceanic lithosphere as it cools and thickens following its formation at the axis of a spreading center results in a complex three-dimensional state of deviatoric stress which can be separated into two parts: a thermal bending stress due to changes in the vertical temperature distribution and a thermal contraction stress due to lateral variations in the vertically averaged temperature. We examine thermal contraction stresses due to temperature changes in a thin, semi-infinite rectangular plate bounded by the spreading center axis. Two stress-free boundaries, representing transforms or fracture zones, define the plate width or ridge segment length L. With the bottom of the elastic plate defined by a prescribed temperature, the plate thickens as the square root of age, as expected for a thermal boundary layer due to vertical conductive cooling. Above this prescribed temperature, elastic stresses are assumed to relax quickly. Stresses are obtained by properly accounting for the rate of accumulation of the vertically averaged stress as initially stress-free material is added to bottom of the cooling, thickening plate. The state of thermal contraction stress calculated from this model is characterized by large tensile stresses at the boundaries of the plate and relatively low stresses in the plate interior. At the ridge axis, ridge-parallel tensile stresses are about 300 MPa, the same as if the plate were not allowed to contract in this direction. Along the transform boundary, the maximum transform-parallel tensile stress occurs at a distance L/2 from the ridge-transform intersection, where its magnitude is comparable to the stress at the ridge axis. The tangential stresses at the plate boundary decrease rapidly with distance from the boundary; at a distance of L/4 from the ridge axis the ridge axis parallel stresses are one tenth of their ridge axis magnitude. The stress magnitudes are independent of both spreading rate and ridge segment length. A large transform-parallel tensile stress may control the length of transform offsets. Thermal bending moments are influenced by the large thermal contraction stresses near the ridge axis. However, a short distance from the ridge these moments attain their free horizontal contraction values which previous studies have shown to cause observable bending of the plate and a geoid anomaly at fracture zones. Flexure due to thermal bending moments will concentrate bending stresses at a distance from the fracture zone determined by the flexural length of the plate, thus providing a natural length scale controlling the spacing of transforms.
49 citations
TL;DR: It is shown that aspect ratio stiffening in membrane elements is partially due to the same modelling error that produces shear locking, and rules-of-thumb are suggested by the same analysis that will insure the absence of errors due to shearlocking at the cost of additional degrees of freedom.
Abstract: The causes of shear locking and other discretization errors are analysed using a physically interpretable notation. This analysis provides insights that allow the errors due to shear locking to be removed either directly or indirectly. St. Venant's principle is incorporated into the stiffness matrix to directly eliminate shear locking from bending elements. Rules-of-thumb are suggested by the same analysis that will insure the absence of errors due to shear locking at the cost of additional degrees of freedom. It is also shown that aspect ratio stiffening in membrane elements is partially due to the same modelling error that produces shear locking. The source of parasitic shear is also identified and a direct procedure for eliminating it is given. A two node Timoshenko beam element and a four node membrane element are fully developed in symbolic form. The procedure is directly applicable to plate bending elements.
TL;DR: In this article, the equilibrium configurations of a thin circular plate supported on an elastic foundation of the Winkler type that reacts in compression only are investigated, where the plate is assumed to be subjected to eccentric concentrated load and moment as well as a uniformly distributed load.
Abstract: The equilibrium configurations of a thin circular plate supported on an elastic foundation of Winkler type that reacts in compression only are investigated. The plate is assumed to be subjected to eccentric concentrated load and moment as well as a uniformly distributed load. The solution is accomplished by minimizing the total potential energy of the system. As the coordinate functions for the displacement function of the plate, the free vibration mode shapes of the completely free plate are used by including a rigid translation and a rigid rotation. It is found out that the plate will lift-off when the foundation stiffness is low. The results for the plate on a conventional and tensionless Winkler foundations are given in figures and compared.
TL;DR: In this paper, a new direct boundary integral element method is presented for the analy sis of Kirchhoff's anisotropic plate bending problems, and two boundary integral equations are derived from the generalized Rayleigh-Green identity after introducing the fundamen tal singular solution of an infinite plate corresponding to the problem of interest.
Abstract: In this paper, a new direct boundary integral element method is presented for the analy sis of Kirchhoff's anisotropic plate bending problems. The two boundary integral equations are derived from the generalized Rayleigh-Green identity after introducing the fundamen tal singular solution of an infinite plate corresponding to the problem of interest. By a sim ple discretization procedure with straight elements for the boundary, and constant assump tion for the unknown boundary functions, two boundary integral equations are obtained in the matrix form. Several computational examples concerning orthotropic plate bending problems are presented. The numerical results obtained by our method as compared with some analytical results show that the present numerical scheme is a versatile tool which gives a satisfactory accuracy.
TL;DR: Numerical experiments show that this synthesis produces an element that should be identical to other recent formulations of this element based on tensorial transformations or on shear constraint condensation on the edges, but now given a more complete and formal logical basis.
Abstract: In this paper, we demonstrate the use of two conceptual13; principles, the field-consistency requirement and the edge-13; consistency requirement as the basis for deriving a 4-noded13; quadrilateral plate bending element based on Mindlin plate13; theory using Jacobean transformations only. The derivation is now free of the use of such devices as strain-interpolation points and Hrennikoff strain reference lines etc., which have been the basis for many recent formulations of this element. The shear strain constraints are now consistently defined within the element dornain, and 'tangential' shear strains are consistently matched at element boundaries so that there is no locking even13; under extreme distortion - e.g. even when two nodes are collapsed so that the quadrilateral becomes a triangle. Numerical experiments show that this synthesis produces an element that should be identical to other recent formulations of this element based on tensorial transformations or on shear constraint condensation on the edges, but now given a more complete and formal logical basis.
TL;DR: The solution of a nonlinear truss-spring, the analysis of plate bending problems with large deflections under transverse loading, and the calculation of the ultimate load of plates under an in-plane load, using DRM with personal computers are described.
TL;DR: In this paper, the authors developed fast converging series solutions for rectangular, parallelogram and triangular plate bending elements with arbitrary boundary conditions and arbitrary shapes, and subjected to generalized normal loading.
TL;DR: In this article, a combined boundary element-transfer matrix method is proposed for plate bending problems, in which a transfer matrix is obtained, from the sytem of equations derived by the procedure based on the boundary element method.
TL;DR: In this article, the authors extended the analytical strip method of solution for bending of orthotropic rectangular plates to stiffened rectangular plates and derived the behavior of the system by imposing the edge and continuity conditions on the closed form solution of the individual plate strips and beam elements.
TL;DR: In this paper, two classes of exact solutions are derived for the equations of three dimensional linear orthotropic elasticity theory governing flat (plate) bodies in plane strain or axisymmetric deformations.
Abstract: Two classes of exact solutions are derived for the equations of three dimensional linear orthotropic elasticity theory governing flat (plate) bodies in plane strain or axisymmetric deformations. One of these is the analogue of the Levy solution for plane strain deformations of isotropic plates and is designated as the interior solutions. The other complementary class correspond to the Papkovich-Fadle Eigenfunction solutions for isotropic rectangular strips and is designated as the residual solutions. For sufficiently thin plates, the latter exhibits rapid exponential decay away from the plate edges. A set of first integrals of the elasticity equations is also derived. These first integrals are then transformed into a set of exact necessary conditions for the elastostatic state of the body to be a residual state. The results effectively remove the asymptoticity restriction of rapid exponential decay of the residual state inherent in the corresponding necessary conditions for isotropic plate problems. The requirement of rapid exponential decay effectively limits their applicability to thin plates. The result of the present paper extend the known results to thick plate problems and to orthotropic plate problems. They enable us to formulate the correct edge conditions for twodimensional orthotropic thick plate theories with stress or mixed edge data.
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TL;DR: In this paper, a method termed "fractal patch test" is presented, in which the patch size is maintained constant while the distorted mesh is refined, for four-node quadrilateral elements used in plane stress and strain analysis, and for plate bending elements.
Abstract: Element consistency is generally checked using the patch test on an element patch of finite size. This condition may in certain cases be too restrictive, and disqualifies elements that appear to be convergent. A method termed ‘fractal patch test’ is presented, in which the patch size is maintained constant while the distorted mesh is refined. Examples are given for four-node quadrilateral elements used in plane stress and strain analysis, and for plate bending elements.
TL;DR: An 8 noded, eccentrically stiffened, plate bending element is introduced in this article, which allows for any number of stiffeners arbitrarily orientated within a plate element without disturbing their individual properties and positions.
Abstract: An 8 noded, eccentrically stiffened, plate bending element is introduced. The formulation allows for any number of stiffeners arbitrarily orientated within a plate element without disturbing their individual properties and positions. This is a distinct improvement over conventional lumped stiffener modelling and equivalent orthotropic plate theory and considerably simplifies the modelling of stiffened structures. A technique is also presented which transforms stiffener positions defined in the global cartesian system to the local, isoparametric coordinates of the plate element which contains the given stiffener. Several examples are given which demonstrate the usefulness of the element.
TL;DR: In this paper, the shear-locking phenomenon in discrete bending analysis of Mindlin/Reissner plates is investigated and mixed/hybrid variational principles are introduced which, unlike the rigorous displacement model, allow systematic derivation of locking-free finite elements.
Abstract: The shear-locking phenomenon in discrete bending analysis of Mindlin/Reissner plates is investigated. Mixed/hybrid variational principles are introduced which, unlike the rigorous displacement model, allow systematic derivation of locking-free finite elements. This is achieved by satisfaction of an auxiliary condition, having the clear physical interpretation of shear-force elimination on account of equilibrium. An example, using competitive techniques, demonstrates the applicability of the idea.
18 Apr 1988
TL;DR: A plate bending element based on the generalized laminate plate theory (GLPT) developed by the senior author is described and its accuracy is investigated by comparison with the exact solutions ofthe generalized plate theory and the 3D-elasticity theory as discussed by the authors.
Abstract: .( SUMMARY A plate bending element based on the generalized laminate plate theory (GLPT) developed by the senior author is described and its accuracy is investigated by comparison with the exact solutions ofthe generalized plate theory and the 3D-elasticity theory. The element accounts for transverse shear deformation and layer wise description of the inplane displacements of the laminate. The element has improved description of the inplane as well as the transverse deformation response. A method for the computation of interlaminar (transverse) stresses is also presented. 1. BACKGROUND Laminated composite plates are often modelled using the classical laminate plate theory (CLPT) or the first-order shear deformation plate theory (FSDT). In both cases the laminate is treated as a single-layer plate with equivalent stiffnesses, and the displacements are assumed to vary through the thickness according to a single expression (see Reddy 1 ), not allowing for possible discontinuities in strains at an interface of dissimilar material layers. Recently, Reddy2 presented a general laminate plate theory that allows layer-wise representation of inplane displacements, and an improved response of inplane and transverse shear deformations is predicted. Similar but different theories have appeared in the literature. 3-6 In the generalized laminate plate theory (0LPT) the equations of three-dimensional elasticity are reduced to differential equations in terms of unknown functions in two dimensions by assuming layer-wise approximation of the displacements through the thickness. Consequently, the strains are different in different layers. Exact analytical solutions of the theory were developed by the authors 7 ,8 to evaluate the accuracy ofthe theory compared to the 3D-elasticity theory. The results indicated that the generalized laminate plate theory allows accurate determination ofinterlaminar stresses. The present study deals with the finite-element formulation of the theory and its application to laminated composite plates. In the interest of brevity only the main equations of the theory are reviewed and the major steps of the formulation are presented. The accuracy of the numerical
TL;DR: In this paper, the authors quantify the combined effects of bending and transverse shear loads on delamination initiation from matrix cracks in graphite-epoxy laminates and derive simple equations relating the total strain energy release rate, G, associated with the delamination growth from a 90 deg. ply crack.
Abstract: A major source of delamination damage in laminated composite materials is from low-velocity impact. In thin composite laminates under point loads, matrix cracks develop first in the plies, and delaminations then grow from these cracks at the ply interfaces. The purpose of this study was to quantify the combined effects of bending and transverse shear loads on delamination initiation from matrix cracks. Graphite-epoxy laminates with 90 deg. plies on the outside were used to provide a two-dimensional simulation of the damage due to low-velocity impact. Three plate bending problems were considered: a 4-point bending, 3-point bending, and an end-clamped center-loaded plate. Under bending, a matrix crack will form on the tension side of the laminate, through the outer 90 deg. plies and parallel to the fibers. Delaminations will then grow in the interface between the cracked 90 deg. ply and the next adjacent ply. Laminate plate theory was used to derive simple equations relating the total strain energy release rate, G, associated with the delamination growth from a 90 deg. ply crack to the applied bending load and laminate stiffness properties. Three different lay-ups were tested and results compared. Test results verified that the delamination always formed at the interface between the cracked 90 deg. ply and the next adjacent ply. Calculated values for total G sub c from the analysis showed good agreement for all configurations. The analysis was able to predict the delamination onset load for the cases considered. The result indicated that the opening mode component (Mode I) for delamination growth from a matrix crack may be much larger than the component due to interlaminar shear (Mode II).
TL;DR: In this article, a complementary energy-based variational principle, using first order stress functions, is developed for plane linear elastic couple-stress problems, analogous to that used in a total potential energy based Mindlin/Reissner thick plate bending analysis and as such is a generalization of the classical analogy between plate stretching and plate bending.
Abstract: A complementary energy based variational principle, using first order stress functions, is developed for plane linear elastic couple-stress problems. The principle is analogous to that used in a total potential energy based Mindlin/Reissner thick plate bending analysis and as such is a generalization of the classical analogy between plate stretching and plate bending. Traction boundary conditions are enforced using a Lagrange multiplier technique. The resulting C0 finite element ‘equilibrium stress model’ is validated by investigating the reduction of the stress concentration factor associated with a small hole in a field of uniform tension.
TL;DR: In this article, a high-order shear-deformation theory of plates is developed, which incorporates a quadratic transverse shearstrain distribution through the plate thickness with zero values at the two free surfaces.
TL;DR: In this article, the buckling coefficient and buckling mode of a rectangular plate of non-uniform thickness subject to in-plane compressive loads are computed for the planar direction transverse to the load.
Abstract: Collocation with quintic splines is used to solve the differential equation of transverse deflection of a rectangular plate of nonuniform thickness subject to in-plane compressive loads. An algorithm is presented for the computation of the buckling coefficient and the buckling mode. For numerical computation, a range of values of plate parameters and different boundary conditions are considered, assuming the first symmetric mode of buckling in the planar direction transverse to the load. Results are presented in the form of graphs.
TL;DR: In this article, the vibration of a plate with an interior cut-out was studied using a finite element method, and the Hamilton's variational principal and potential theory were used to derive the governing equations for a Mindlin plate.
TL;DR: In this article, the Bueckner weight function was used for computing the stress intensity factor using a 2D finite element program, and the results showed that plate length has little or negligible influence on fatigue strength provided that the plate length to thickness ratio is below 20 to 50 depending on the plate thickness.