Isoparametric Quadrilateral Finite Element Research Articles

Free vibration behavior in functionally graded material plates with dual efficient quadrilateral finite elements

Functionally graded materials represent advanced composites with typically two materials that exhibit gradual variations in their mechanical properties throughout the structure’s thickness in general, as determined by a presumed mathematical model. The current investigation focuses on analyzing free vibrations of functionally graded material plate structures with two efficient isoparametric quadrilateral finite elements (Lagrange and Serendipity) together for the first time and different material combinations based on first-order shear deformation theory. The analysis was carried out on relatively thin as well as medium thick functionally graded plates with varying power law indices and at varied boundary conditions. The framework equation was formed with Hamilton’s principle, and solutions were done with finite element method, considering a shear correction factor for the behavioral study of varying material properties. The findings align well with the information presented in prior literature. After validating the present model, parametric analysis was conveyed to show the impact of length-to-thickness ratio, aspect ratio, volume fractions, and skew angles for two novel materials on the fundamental frequencies and mode shapes. The results thus obtained with the two quadrilateral elements have a significant effect on the changes in these parameters. Also, the natural frequencies of different functionally graded plates, including two novel materials, were compared and discussed.

FE sloshing modelling in bidimensional cavity using wave equation

Tuned liquid column damper (TLCD) is a kind of passive damper that use water sloshing motion to control structural vibration in tall buildings, wind turbines and ships. This paper presents finite element analysis of sloshing in 2D water tank using a pressure based Eulerian approach. The fluid domain is discretised by an isoparametric quadrilateral finite element. The cavity was modelled with rigid contours and free surface in order to study the decoupled liquid reservoir. Next, the cavity was coupled to a flexible structure with free vibration and forced vibration. The numerical results obtained by the created program were validated using experimental results [1] presenting acceptable errors with less than 5k elements.

An Enhanced Tire Model for Dynamic Simulation based on Geometrically Exact Shells

Abstract In the present work, a tire model is derived based on geometrically exact shells. The discretization is done with the help of isoparametric quadrilateral finite elements. The interpolation is performed with bilinear Lagrangian polynomials for the mid-surface as well as for the director field. As time stepping method for the resulting differential algebraic equation a backward differentiation formula is chosen. A multilayer material model for geometrically exact shells is introduced, to describe the anisotropic behavior of the tire material. To handle the interaction with a rigid road surface, a unilateral frictional contact formulation is introduced. Therein a special surface to surface contact element is developed, which rebuilds the shape of the tire.

Damage Detection of A T-Shaped Panel by Wave Propagation Analysis in the Plane Stress / Wykrywanie Uszkodzen W Tarczy Typu T Z Uzyciem Analizy Propagacji Fal W Płaskim Stanie Naprezenia

Abstract A computational approach to analysis of wave propagation in plane stress problems is presented. The initial-boundary value problem is spatially approximated by the multi-node C0 displacement-based isoparametric quadrilateral finite elements. To integrate the element matrices the multi-node Gauss-Legendre-Lobatto quadrature rule is employed. The temporal discretization is carried out by the Newmark type algorithm reformulated to accommodate the structure of local element matrices. Numerical simulations are conducted for a T-shaped steel panel for different cases of initial excitation. For diagnostic purposes, the uniformly distributed loads subjected to an edge of the T-joint are found to be the most appropriate for design of ultrasonic devices for monitoring the structural element integrity

A rotation-free quadrilateral thin shell subdivision finite element

We present a rotation-free, subdivision-based, isoparametric quadrilateral finite element for the analysis of doubly-curved thin shells. The element is a generalization of spline-based C1 finite elements to curved domains, and general mesh connectivity. It uses semi-local basis functions that span the two-neighbourhood of a node. The traditional difficulty with this class of elements comes from the fact that these basis functions depend on the valence of the nodes of the element. For elements with 4-valence nodes the basis functions are bicubic polynomials, but there are no simple closed-form expressions for them when the nodes have general connectivity patterns. In the paper, we show how the basis functions for the general case may be efficiently evaluated as linear combinations of translation and scaling transformations of the bicubic polynomials. The performance of the resulting element is illustrated on a hemispherical shell model problem. Copyright © 2005 John Wiley & Sons, Ltd.

Coupling of two- and three-dimensional composite shell elements in linear and non-linear applications

A numerical model for the non-linear analysis of laminates is presented. The developed element is based on piecewise polynomial interpolation in thickness direction. Although the total number of degrees of freedom is comparable to a discretization with 3D brick elements this approach provides several advantages, e.g. a simple mesh generation due to a 2D-type data structure and a better bending behavior. The developed isoparametric quadrilateral finite element allows prediction of the complete stress state. Furthermore, a transition element is presented which is used to couple the developed element with 5-parameter shell elements. Several linear and non-linear examples are presented to illustrate the performance of the developed numerical model.

Large deflection inelastic pseudo-transient analysis of laminated composite plates

A pseudo-transient (static) linear, geometric, material and combined geometric and material non-linear analyses of composite laminates are presented. A nine-noded isoparametric quadrilateral finite element belonging to the Lagrangiar family is used in space discretization. An explicit time marching scheme is employed for time integration of the resulting discrete ordinary differential equations with the special forms of diagonal fictitious mass and/or damping matrices. Elasto-plastic material behaviour is incorporated using the flow theory of plasticity. In particular, a modified version of Hill's initial yield criterion is used in which anisotropy parameters of plasticity are introduced. The shear deformation is accounted for by assuming a constant transverse shear strain across the thickness of the laminate and the geometric non-linearity is considered in the sense of von Karman strains. The layered element approach is adopted for the treatment of plastic behaviour through the thickness. A wide range of numerical examples is presented to demonstrate the validity and efficiency of the present approach. The results for combined non-linearity are also presented. The variety of results presented here, which are based on realistic material properties of often-used advanced laminated composite plates, and especially those for combined non-linear analysis, should serve as a reference for future investigations.

On the numerical analysis of local effects in composite structures

A numerical model for the nonlinear analysis of thick laminates is presented. Using a multiplicative decomposition of the displacement vector in shell space, the 3D problem is reduced to a 2D problem. Although the total number of degrees of freedom is comparable to 3D brick elements this approach provides several advantages, e.g. a simplified mesh generation due to a 2D-type data structure and a better bending behavior. The developed isoparametric quadrilateral finite element is capable of predicting interlaminar stresses and local effects. Several linear and nonlinear examples are presented to illustrate the performance of the developed numerical model.

Effects of element distortions on the performance of isoparametric elements

AbstractWe discuss the effects of element distortions on the performance of displacement‐based isoparametric quadrilateral finite elements. Suitable example problems for both the Lagrangian and the serendipity types of elements are used to show numerically the effects of element distortions. The effects of angular and curved‐edge distortions are evaluated analytically in terms of the abilities of the elements to represent exactly various polynomial displacement fields. We then solve a plane stress problem adaptively to demonstrate the effect of element distortions on the accuracy of the stress distribution. The conclusion is that the Lagrangian elements are not affected by angular distortions and are therefore the more reliable elements for general use.

Fourier continuum finite elements for large deformation problems

A procedure to analyze axisymmetric solids undergoing large nonaxisymmetric deformation is presented. The plane of axisymmetry is modeled by isoparametric quadrilateral finite elements with dependence in the circumferential direction determined by a continuous Fourier series. Although the Fourier harmonics are coupled for such analyses, the size of the resulting model can be much smaller than it would be if brick elements were to be used. This model reduction can significantly decrease the amount of computational effort needed for analyzing this class of problems. Different schemes for solving the coupled nonlinear equations are evaluated by analyzing two structures subjected to different loadings and exhibiting various amounts of geometric nonlinearity. The coupled systems of linear simultaneous equations occurring at each Newton-Raphson iteration are solved by both direct and indirect solution schemes. Several types of Gauss-Seidel and conjugate gradient iterations are shown to be quite efficient and robust. The effectiveness of these indirect solution techniques are a result of using the uncoupled stiffness matrices to precondition the coupled systems of linear simultaneous equations. The continuous Fourier series approach also provides a very simple, efficient, and reliable way to conduct adaptive refinement in the circumferential direction.

Pseudo-Transient Analysis of Composite Shells Including Geometric and Material Non-Linearities

An unified approach is presented for the pseudo-transient (static) linear, geometric, material and combined geometric and material non-linear analyses of lami nated composite shells. A nine-noded isoparametric quadrilateral finite element belonging to the Lagrangian family is used in space discretization. An explicit time marching scheme is employed for time integration of the resulting discrete ordinary differential equations with the special forms of diagonal fictitious mass and/or damping matrices. The elasto plastic material behaviour is incorporated using a flow theory of plasticity. In particular, the modified version of Hill's initial yield criterion is used in which anisotropy parameters of plasticity are introduced. The shear deformation is accounted for using an extension of Sander's shell theory, and the geometric non-linearity is considered in the sense of von Karman strains. The layered element approach is adopted for the treatment of elasto plastic behaviour through the thickness. A wide range of numerical examples are pre sented to demonstrate the validity and efficiency of the present approach. The results for combined non-linearity are also presented. The variety of results presented here is based on realistic material properties of more commonly used advanced composite materials. The results of the combined non-linear analysis should serve as reference for future in vestigations.

Thermal buckling of laminated plates using a shear flexible finite element

The buckling behavior of laminated composite plates subjected to a uniform temperature field is studied. Transverse shear flexibility is accounted for in the analysis using the thermo-elastic version of the first-order shear deformation theory. The prebuckling behavior of the plate under a uniform temperature field is studied prior to predicting the initial thermal buckling load. Numerical results are presented for thermal buckling problems using a nine-noded isoparametric quadrilateral finite element. The influence of boundary conditions, ply orientation, and plate geometries on the critical buckling temperature is examined.

A shear deformable finite element for the analysis of general shells of revolution

A 64-dof isoparametric quadrilateral finite element is presented for the analysis of generally laminated shells of revolution. The effects of shear deformation and rotary inertia are accounted for by using shear deformation theory that employs the parabolic shear strain variation across the thickness. The classical thin shell theory is the special case of shear deformation theory used in the present study. Thus, the thin shell element also can be obtained from the present thick shell element by simply having the displacement parameters ( u 1 and v 1,) associated with the shear rotations as zeros. The numerical results presented illustrate the performance of the element and the effects of shear deformation.

Kinematics of small and large deformations of continua

The kinematics of small and large deformations (displacements, rotations and strains) is described by use of the engineering strain, the logarithmic strain, the Seth‐Hill class of strains and the rate‐type strains derived using the Lagrangian and the ‘Relative’ descriptions. The displacement gradient is computed for two and three dimensions and the error associated with use of the small rotation approximation is plotted. The components of the rotation tensor are derived for a four‐noded isoparametric quadrilateral finite element for determining the error due to small displacement and rotation approximations. Finally, the various strain measures are computed and plotted for representative problems.

Stress, buckling and vibration of hybrid bodies of revolution

The analysis is applicable to bodies of revolution composed of thin shell segments, thick segments and discrete rings. The thin shell segments are discretized by the finite difference energy method and the thick or solid segments are treated as assemblages of 8-node isoparametric quadrilateral finite elements of revolution. Suitable compatibility conditions are formulated through which these dissimilar segments are joined without introduction of large spurious discontinuity stresses. Plasticity and primary or secondary creep are included. Axisymmetric prebuckling displacements may be moderately large. The nonlinear axisymmetric problem is solved in two nested iteration loops at each load level or time step. In the inner loop the simultaneous nonlinear equations corresponding to a given tangent stiffness are solved by the Newton-Raphson method. In the outer loop the plastic and creep strains and tangent stiffness are calculated by a subincremental procedure. The linear response to nonaxisymmetric loading is obtained by superposition of Fourier harmonics. Many examples are given to demonstrate the scope of the computer program, BOSOR6, derived from the analysis and to illustrate certain stress concentration effects in shell-type structures which cannot adequately be treated with use of thin shell theory.

A finite element procedure for the large deformation dynamic response of axisymmetric solids

A computational procedure for the large deformation dynamic response of solids is presented. The underlying mechanics, the constitutive theories of interest, the spatial discretization, and the time integration scheme are each discussed. The mechanics is carried out in the current configuration described by a fixed spatial coordinate system and using the Cauchy stress. Elastic, elastoplastic, viscoelastic and curshable foam constitutive theories are examined. A bilinear isoparametric quadrilateral finite element is employed for the spatial discretization. An explicit central difference time integration scheme and artificial viscosity are used to compute the response. The results of five computations are presented and compared with either experimental results or exact answers.