Thin Plate Finite Element Research Articles

Uticaj temperaturskog gradijenta na savijanje tankih ploča

Within the theory of thermo-elasticity, the temperature field of thin plates is commonly defined via two parameters: temperature in the mid-plane and linear temperature gradient normal to the mid-plane. First, the paper analytically proves the justification of that assumption in machine structures. Then, in an analytical closed form, applying the integral transformation method, the thin plate deflection caused by a constant temperature gradient is defined. It is shown that, in that case, the plate deflection does not depend on its thickness but only on the plate dimensions in the mid-plane. Analytically defined values are compared to corresponding values obtained by applying the thin plate finite element, where the temperature field is described using the two mentioned parameters. This finite element is defined and programmed within the Komips program package. The influence of the temperature gradient on the behavior of constructions mostly depends on the type of material. That is why the behavior of some structural elements made of brass, steel, and concrete is analyzed in this paper.

Derivation of Elastic Constants for Orthotropic Plates

The paper presents a procedure for modeling an orthotropic thin plate finite element so to have an equivalent structural response as the steel stiffened plate of same overall dimensions. New, equivalent element, homogenizes the structural properties of plates and stiffeners in element’s stiffness matrix. By use of such elements, a significant reduction in overall size of a global finite element model of large stiffened panel structures could be achieved. The algorithm of proposed procedure is fully explained and, as an example, applied to a standard global model of steel ship hull structure. Structural response comparison is made between reduced model made of newly defined thin finite elements and conventional ones. Advantages are emphasized and remedies of some recognized shortcomings are suggested.

A new component mode synthesis for dynamic mixed thin plate finite element models

This paper presents a methodology for the reduction in dynamic mixed finite element models (DM-FEMs) based on the use of a sub-structuring primal methods adapted to such models. We implement a DM-FEM for Kirchhoff–Love thin plates using the Hellinger–Reissner variational mixed formulation adapted to dynamic, and give a quick insight of its convergence. This model uses both displacement and generalized stress fields within the plate, obtained as a primary result, but the numerical size of the model is bigger than with a primal displacement model. Thus, we choose to offset this complication by reducing the model with a totally new sub-structuring reduction method, especially adapted to DM-FEMs. The aim of our method is to adapt sub-structuring reduction methods commonly used for primal displacement FEM only (such as Craig and Bampton method) and split the reduction in the two fields. With these displacement methods, the whole structure is splitted into few smaller ones, and each of them is condensed with eigenmodes of the substructure and static connections between them. The principle of our method is to build, for each substructure, a reduced basis for the displacements according to the existing method, and a projection of the primal basis for the stresses. A new reduced basis for the whole mixed model is then built up exclusively with modes taken from the primal model. That method reduces significantly the number of degrees of freedom and keeps the properties and advantages of the mixed formulation.

Elastic response of an orthogonally reinforced plate

Abstract This paper develops a three-dimensional fully elastic analytical model of a solid plate that has two sets of embedded, equally spaced stiffeners that are orthogonal to each other. The dynamics of the solid plate are based on the Navier–Cauchy equations of motion of an elastic body. This equation is solved with unknown wave propagation coefficients at two locations, one solution for the volume above the stiffeners and the second solution for the volume below the stiffeners. The forces that the stiffeners exert on the solid body are derived using beam and bar equations of motion. Stress and continuity equations are then written at the boundaries and these include the stiffener forces acting on the solid. A two-dimensional orthognalization procedure is developed and this produces an infinite number of double indexed algebraic equations. These are all written together as a global system matrix. This matrix can be truncated and solved resulting in a solution to the wave propagation coefficients which allows the systems displacements to be determined. The model is verified by comparison to thin plate theory and finite element analysis. An example problem is formulated. Convergence of the series solution is discussed. The frequency limitations of the model are examined.

New bi-cubic spline wavelet based thin plate finite elements

New bi-cubic spline wavelet based thin plate finite elements

Comparison between ANCF and B-spline surfaces

This paper compares the descriptions of surfaces used in computational geometry (CG) methods and the finite element (FE) kinematics. Such a description is necessary for the successful Integration of Computer Aided Design and Analysis (I-CAD-A). B-spline surface geometry is defined using rigid recurrence formulas that employ the concept of the knot multiplicity to define the degree of continuity at the breakpoints. The knot multiplicity concept as well as the rigid recurrence formulas do not offer the flexibility provided by FE formulations. Furthermore, the concept of degrees of freedom is not considered in developing CG methods. Consequently, problems can be encountered in using the recurrence formulas to model certain types of joints. It is, therefore, important to adopt a FE method that is consistent with CG methods in order to be able to establish an efficient interface between CAD systems and analysis tools. The FE formulation used in this investigation for the surface description is the Absolute Nodal Coordinate Formulation (ANCF). It is shown in this paper that B-spline surfaces can be converted to ANCF thin plate finite elements without any geometric distortion. To this end, a linear transformation that defines the relationship between the B-spline surface control points and the ANCF position and gradient vectors is developed. The resulting ANCF thin plate finite elements define a unique displacement and rotation field, and their geometry is invariant under an arbitrary rigid body rotations. The analysis presented in this paper clearly shows that all B-spline surfaces can be converted to ANCF meshes that have the same original CAD geometry. The converse is not true; that is, not all ANCF thin plate finite elements can be converted to B-spline surfaces. This sheds light on the flexibility and computational advantage that can be gained by the use of the FE method in the analysis as compared to the use of CG methods. The paper also shows how a B-spline surface can be converted systematically to a volume representation (full parameterization) using ANCF geometry.

New rectangular plate elements based on twist-Kirchhoff theory

We introduce a new framework for the development of thin plate finite elements, the “twist-Kirchhoff theory”. A family of rectangular plate elements is derived that takes advantage of the special structure of this new theory. Particular attention is focused on the lowest-order member of the family, an eight degree-of-freedom, four-node element with mid-side rotations whose stiffness matrix is exactly computed with one-point Gaussian quadrature. We prove a convergence theorem for it and various error estimates. These are also generalized to the higher-order elements in the family. Numerical tests corroborate the theoretical results.

Multiscale analysis of nanoscale thin film considering surface effects: Thermomechanical properties

The classical model for a thin film, based on continuum theory, is independent of size, and ignores surface effects. But the surface-to-bulk ratio becomes very large in small-scale structures such as nanofilms, nanowires, and nanobeams, and surface effects play an important role. Molecular dynamics simulation has been a conventional way to analyze these ultrathin structures, but structures in the range between submicro and micro are difficult to analyze by molecular dynamics simulation due to the restrictions of computing resources and time. In the present study, a continuum-based, size-dependent model is developed for predicting the thermomechanical properties of nanoscale structures, especially thin films. The proposed continuum-based thin plate finite element is efficient and reliable for the prediction of

Global and local behaviour based composite damping studies on thin-walled box structure

Abstract The local (plate)/global (beam) vibration and damping behavior of composite thin-walled box member subjected to vibratory environment are studied and presented in this paper. This investigation is carried out by the use of 3D beam and thin plate finite elements and the corresponding modal frequencies/damping values of composite box beam are predicted by modal strain energy method. Application examples illustrate the ability of the 3D beam and thin plate element to predict both local and global modal frequencies/damping of hollow box sections. The influence of L / h ratio, b / h ratio and ply angle on the frequency and loss factor of composite box beam is investigated. In addition, an attempt has also been made to investigate the effect of temperature on the composite damping characteristics of rectangular box type section.

Thin plate neotectonic models of the Australian plate

Thin plate finite element models of the neotectonic deformation of the Australian plate have been calculated in order to estimate the stress and strain rate within the plate, specifically concentrating on the Australian continent. The model includes plate‐bounding faults, an anelastic brittle‐ductile layered rheology and the option of laterally varying elevation and heat flow. The results of the models are compared to (1) the velocity of geodetic benchmarks on the Australian plate, (2) the spreading rate of the mid‐oceanic ridges along the Australian plate's margins, (3) the direction of the maximum horizontal principal stress, (4) the stress regime within the plate, and (5) the crustal thickness estimated from the depth to the base of Mohorovicic discontinuity's transition zone. A variety of models are tested with a wide range of input parameters. The model with the smallest misfit with observations predicts that the strain rate for most of the Australian continent is approximately 10−17 s−1. This model has a slightly lower strain rate in the central Australia and is higher off the northern coast of Australia than for the rest of the continent. Strain rates of this magnitude would be difficult to observe from geodetic or geologic data for most parts of Australia but would be enough to generate much of the seismicity that has been observed over the last century.

Improved Secant Matrices for the Postbuckling Analysis of Thin Composite Plates

To study the postbuckling behavior of imperfect laminated composite plates, improved incremental or secant matrices are presented in this paper using what is called additional displacement formulation (ADF). These secant matrices are derived using the Marguerre's shell theory and they can be used in combination with any thin plate finite element. The advantage of the present formulation is that it involves no numerical approximation in forming the initial imperfection matrices as opposed to earlier secant matrices published in the literature using total displacement formulation. With the addition of shear stiffness matrix and little modification, the present incremental matrices could be extended to model postbuckling behavior of plates using the first-order shear deformation theory. The secant matrices presented in this study are shown to be very accurate in tracing the postbuckling behavior of thin isotropic and laminated composite plates with general initial imperfections.

Improving an accurate thin plate element

The accurate 9 df thin plate finite element of Mohr was originally reported to experience ‘numerical precision difficulty’ in some cases [1], resulting in slight loss in accuracy (1 or 2%). The element is thoroughly tested here with 8 dp computation and found to give excellent results without difficulty. Further improvement in results is then obtained using a penalty factor approach to enforce the constraint ∂Φ x ∂y = ∂Φ y ∂x and by using approximate consistent loads.

On two equivalent thin plate finite elements

The 9 df thin plate element of Mohr and Mohr uses cubic interpolation to obtain values of at the third points of the element sides, in turn interpolating from these and the vertex values within the element. Recently this element has been modified and successfully applied to 'potential' problems. Subsequently it was found that the interpolations of the element of Bazeley et al. (1965, 1968) gave identical results for potential problems. In the present paper it is shown that this is because the interpolations of the two elements are exactly equivalent.

Supersonic flutter of laminated thin plates with thermal effects

A 48 degree-of-freedom rectangular laminated thin plate finite element including the effects of thermal and aerodynamic loadings is formulated to study the buckling and supersonic flutter characteristics of thin plate structures. Interactive effects between the critical temperature difference and critical aerodynamic pressure for the plates are also studied. The element formulation is based on the classical lamination theory. The aerodynamic pressure due to supersonic potential flow is described by a two-dimensional steady supersonic theory. The element formulation and solution procedure are evaluated by comparing results of three examples with existing alternative solutions. The practical applicability is demonstrated by performing buckling and supersonic flutter analyses of laminated thin plates under various types of temperature distributions. Based on the numerical results, the effects of aspect ratio, ratio of thermal expansion coefficients, fiber orientation, type of temperature distribution, and flow angularity on these examples are discussed. HIN plates are a popular and useful form of structural components with significant applications in aerospace ve- hicles, such as high-speed aircraft, rocket, and spacecraft, which are subjected to thermal loads due to aerodynamic and/ or solar radiation heating. This results in a temperature dis- tribution over the surface and thermal gradient through the thickness of the plate. Due to boundary constraints, com- pressive stresses are induced and may cause buckling. There- fore, thermal buckling is a significant failure mode governing the design of thin plates used in aerospace structures. For obvious advantages such as the high-strength-to-weight ratio and high-stiffness-to-weight ratio, etc., fiber-reinforced laminated composite materials have been increasingly used in the design and fabrication of thin plate structures. It is of interest to briefly review the works done in thermal buckling of laminated composite plates. For example, Whit- ney and Ashton1 studied the thermal buckling of symmetric, angle-ply, layered composite plates with simply supported edges using energy formulation. The critical temperatures for plates with various types of composite were presented. Chen and Chen2 analyzed the thermal buckling of laminated cylin- drical panels using Galerkin's method. The critical temper- atures for plates with various structural and material param- eters, and boundary conditions were presented. Tauchert and Huang3 investigated thermal buckling of simply supported symmetric angle-ply laminated plates under uniform temper- ature change using the Rayleigh-Ritz method. The critical temperature and associated mode shape for plates with var- ious fiber orientations, numbers of layers, and aspect ratios were presented. Thangaratnam et al. 4 studied the thermal buckling of composite laminated plates using semiloof finite elements. The critical temperatures for plates under various types of temperature distribution, lamination parameters, and boundary conditions were presented. Chen and Chen5 ana- lyzed the thermal buckling of laminated cylindrical plates sub- jected to a nonuniform temperature field using the finite ele- ment method. The critical temperatures for the plates with various lamination angle, modulus ratio, number of layers, plate aspect ratio, and boundary condition were presented.

Finite element algorithm for jointed concrete pavements subjected to moving aircraft

An algorithm based on the finite element method is developed to analyze the dynamic response of multiple, jointed concrete pavements to moving aircraft loads. In the finite element idealization, the pavement-subgrade system is idealized by thin plate finite elements resting on Winkler-type viscoelastic foundation represented by a series of distributed springs and dashpots. The dowel bars at the transverse joints are represented by beam elements. It i assumed that the dowel bar is fully embedded into the pavement thus neglecting dowel-pavement interaction effects. The longitudinal keyed or aggregate interlock joints are modeled by vertical spring elements. The dynamic aircraft-pavement interaction effects are considered in the analysis by modeling the aircraft by masses supported by spring-dashpot systems representing the landing gear of the aircraft. It is assumed that the aircraft travels along a straight line with a specific initial velocity and acceleration. The aircraft-pavement interaction takes the form of two sets of coupled equations which result ina non-symmetric stiffness matrix. An approximate mixed iteration-direct elimination scheme is used to solve for the dynamic equations. The accuracy of the computer code is verified by the available experimental and analytical solutions. A parametric study is conducted to investigate the effects of various parameters on the dynamic response of pavements.

Flutter analysis of composite panels using high-precision finite elements

A high-precision 18-degree-of-freedom triangular thin plate finite element procedure is presented for studying the panel flutter of composite thin plates. Classical lamination theory, together with linearized piston theory, is used to study the effects of aerodynamic damping, boundary constraint, flow orientation, composite filament angle, and orthotropic modulus ratio on the flutter characteristics of both isotropic and composite panels. The numerical results indicate that aerodynamic damping, which enlarges the flutter boundary, is generally beneficial. The panels with stronger structural boundary constraint possess a better capacity to resist flutter instability. Alignment of the composite filament and flow orientations results in the best flutter performance for both fully simply supported and fully clamped panels, whereas alternative flutter-divergence type instability, which is not observed in the isotropic cases, takes place as the composite filament angle varies in the cases of cantilevered panels. The orthotropic modulus ratio is shown to be very effective in enlarging the flutter boundary; however, a reverse trend may occur if the instability type is changed from flutter to divergence. Substantial improvement or even complete avoidance of flutter can be attained if the composite panel is properly tailored.

High-order analysis of generally symmetrical laminated plates under transverse loading

The High-Order technique of Lo et al. for full three-dimensional plate bending analysis is furthered to cope with generally symmetrical rectangular laminates subjected to transverse loading. Assessment of the technique is made through normalised central transverse displacement and surface stress results with solutions from exact elasticity and shear flexible finite elements for a series of test cases. This demonstrates that high-order modelling predicts more accurately the effect of shear deformation than the finite element method, and that the classical approach is preferable for span-to-thickness ratios, s, greater than 40. The specific high-order model of a simply supported rectangular laminated plate with central patch load is developed to provide dimension optimisation data for design of a proposed biaxial test. The analysis shows that, providing s is greater than 35, then the maximum contribution to deflections and stresses from shear deformation for multi-layered plates is 10%. Furthermore, for s in excess of 35, investigation of bending in the biaxial test is found to be appropriate by a classical thin plate finite element.

From Kirchhoff to Mindlin plate elements

AbstractA procedure to generalize Kirchhoff thin plate finite elements so that they can be used for solving thick Mindlin plates is presented. Two typical discrete Kirchhoff elements, a triangular one and a quadrilateral one, are modified. Numerical results for a clamped circular plate show that the accuracy of the new Mindlin elements is the same as that of the original Kirchhoff elements and they can be used in the whole thickness range of Mindlin plate theory.

A simple method for including shear deformations in thin plate elements

AbstractA method is described for including the effect of shear deformations in existing thin plate finite elements, and thereby extending their range of application to include moderately thick plates. The method does not add extra degrees of freedom to the final element, so the thick and thin plate elements can be used interchangeably, and the thick plate solution is not appreciably more expensive than the thin plate solution.It is assumed that the shear deformations are constant over the element and, to account for this, two extra internal shear strain variables are added to the element. Various methods for eliminating these internal variables are examined but it is shown to be impossible to simultaneously satisfy both the constant bending moment and constant shear patch tests, except for parallelograms. However, one method gives elements which pass the constant shear patch test and, although failing the constant bending moment patch test for arbitrary geometries, gives errors which are small enough to be neglected in most engineering applications.This method has been applied to a triangular plate element and it is shown that the results obtained with this element converge (for all practical purposes) to the correct thick plate results.

A new thin plate finite element by basis transformation

A simple nine-freedom thin plate finite element is obtained by using cubic Hermitian interpolation to define displacements at the third points on each side. A tenth “local” freedom at the centroid is then approximated using an interpolation devised by Bazeley et al. together with Hermitian interpolation on the sides. The result is a ten freedom “local” element to which the standard cubic Lagrangian areal coordinate interpolation can be applied. The element is of similar accuracy to a number of other well-known elements of comparable complexity and it provides another useful example of the “method of nested interpolations” which, along with penalty factors and Lagrange multipliers, brings us to the state of the art of the finite element method[1].