Variable-node plate and shell elements with assumed natural strain and smoothed integration methods for nonmatching meshes

This chapter shows, through the example of the addition of a plate and shell element to freeware FEM-object, an object-oriented (C++) finite element program, how object-oriented approaches, as opposed to procedural approaches, make finite element codes more compact, more modular, and versatile but mainly more easily expandable, in order to improve the continuity and the compatibility between software of research and industrial software. The fundamental traits of object-oriented programming are first briefly reviewed, and it is shown how such an approach simplifies the coding process. Then, the isotropic shell and orthotropic plate formulations used are given and the discretized equations developed. Finally, the necessary additions to the FEM-object code are reviewed. Numerical examples using the newly created plate membrane plate element are shown.

Refined shell elements for the analysis of functionally graded structures

Geometrically nonlinear analysis of laminate composite plates and shells using the eight-node hexahedral element with one-point integration

A finite element formulation is developed including the fully coupled electrical-structural constitutive relations. An effort has been made to develop the most general formulation applicable to laminated composite plates with inclusion of piezoceramic sensors/actuators. To this end, the von Karman large deflection theory was included, due to the results obtained from the experiments conducted. Solution of the large deflection problem lead to an electrical-structural coupled tangent stiffness, which is believed to be the first time reported in the literature. paper presents comparison of the finite element simulation to experimental results for both static and dynamic piezoceramic sensors in addition to dynamic piezoceramic actuation. INTRODUCTION Piezoelectric material maintains a complex and intricate relationship between several physical sciences such as materials, thermal, electric fields, and elasticity. Evidence of the piezoelectric's complexity can be demonstrated by the seemingly continuous research conducted since its discovery in 1880 by Pierre and Jaques Curie. Much of the research set out to determine analytical models describing the piezoelectric phenomenon and lead to discrete devices such as extremely sharp crystal filters and stable oscillators. Mindlin developed classical methods describing the analysis on waves and vibrations in isotropic elastic plates concurrently with high frequency vibration of crystal plates. Allik and Hughes developed a solid, or three dimensional, finite element to model transducers and crystal oscillators. However, given that smart structures typically consist of thin piezoelectric layers attached to structures that are several orders of magnitude greater in thickness, the solid finite element formulation leads to an inherently inefficient process in which to model the complete physical structure. Subsequently Tzou and Tseng * Graduate Student, Student member AIAA. t Professor, Associate Fellow AIAA. This paper is declared a work of the U.S. Government and is not subject to copy right protection in the United States. formulated a new thin piezoelectric solid finite element in the form of shell and plate elements. When considering piezoelectrics in conjunction with structural members, linear piezoelectric theory simplifies the piezoelectric effect. Linear piezoelectric theory assumes an isothermal adiabatic process and quasi-static electric field equations. research presents a comparison of the fully coupled electricalstructural finite element analysis of an isotropic panel with a surface mounted piezoelectric patch to experimental results. PIEZOELECTRIC RELATIONS Piezoelectric material inherently possesses coupling between electrostatics and structural dynamics. Linear piezoelectric theory yields an intrinsically coupled pair of piezoelectric constitutive equations. One describes the direct piezoelectric effect where the state of strain gives rise to an electric polarization, and the other describes the converse effect where an applied electrical field develops a state of strain. For a thin piezoelectric layer, the following constitutive relations result from linear piezoelectric theory (1)

In this investigation, a non-incremental solution procedure for the finite rotationand large deformation analysis of plates is presented. The method, whichis based on the absolute nodal coordinate formulation, leads to plateelements capable of representing exact rigid body motion. In thismethod, continuity conditions on all the displacement gradients areimposed. Therefore, non-smoothness of the plate mid-surface at the nodalpoints is avoided. Unlike other existing finite element formulationsthat lead to a highly nonlinear inertial forces for three-dimensionalelements, the proposed formulation leads to a constant mass matrix, andas a result, the centrifugal and Coriolis inertia forces are identicallyequal to zero. Furthermore, the method relaxes some of the assumptionsused in the classical and Mindlin plate models and automatically satisfiesthe objectivity requirements. By using a generalcontinuum mechanics approach, a relatively simple expression for theelastic forces is obtained. Generalization of the formulation to thecase of shell elements is discussed. As examples of the implementationof the proposed method, two different plate elements are presented; oneplate element does not guarantee the continuity of the displacementgradients between the nodal points, while the other plate elementguarantees this continuity. Numerical results are presented in order todemonstrate the use of the proposed method in the large rotation anddeformation analysis of plates and shells. The numerical results, whichare compared with the results obtained using existing incrementalprocedures, show that the solution obtained using the proposed methodsatisfies the principle of work and energy. These results are obtainedusing explicit numerical integration method. Potential applications ofthe proposed method include high-speed metal forming, vehiclecrashworthiness, rotor blades, and tires.

Transient analysis of FGM and laminated composite structures using a refined 8-node ANS shell element

This paper presents an improved 8-node shell element for the analysis of plates and shells. The finite element, based on a refined first-order shear deformation theory, is further improved by the combined use of assumed natural strain method. We analyze the influence of the shell element with the different patterns of sampling points for interpolating different components of strains. Using the assumed natural strain method with proper interpolation functions, the present shell element generates neither membrane nor shear locking behavior even when full integration is used in the formulation. Further, a refined first-order shear deformation theory, which results in parabolic through-thickness distribution of the transverse shear strains from the formulation based on the third-order shear deformation theory, is proposed. This formulation eliminates the need for shear correction factors in the first-order theory. Numerical examples demonstrate that the present element perform better in comparison with other shell elements.

This paper addresses the problem of calculating sensitivity data by direct methods for isoparametric plate or shell elements. Sensitivity parameters of interest include intrinsic properties such as material modulus and plate thickness, as well as geometry variables which influence the size and shape of a structure. The sensitivity calculation therefore must consider the parametric mapping within an element, as well as the influence of geometric variables on the orientation of an element in space. The methods presented specialize directly to continuum elements, in which the co‐ordinate transformation is omitted, or to simple structural members situated arbitrarily in space. Numerical examples are presented which illustrate the accuracy of the proposed techniques, and the effect of discretization error on computed sensitivities.

An efficient variable-node XFEM for modeling multiple crack growth: A Matlab object-oriented implementation

The paper offers a new flat shell finite element. It results from the combination of a membrane and a bending elements, both founded on the strain based formulation. The used membrane element is the T43_Eq. It is a triangular element with four nodes, three nodes at the vertices and the fourth one at the barycenter. Each node has three degrees of freedom, two translations and one rotation around the normal. The coefficients related to the degrees of freedom at the internal node are subsequently removed from the element stiffness matrix by using the static condensation procedure. The interpolation functions of strain, displacements and stresses fields are developed from equilibrium conditions. Then, the plate element used for the construction of the shell element is a triangular four-node thin plate element based on: the deformation approach, the four fictitious node, the static condensation and the analytic integration. The shell element result of this combination is robust, competitive and efficient. The shell element result of this combination is robust, competitive and efficient.

Dynamic fracture analysis of the linearly uncoupled and coupled physical phenomena by the variable-node multiscale XFEM

Purpose — Analysis of nonprismatic members has received a great deal attention from designers and engineers due to their ability in satisfaction of architectural and aesthetic necessities. Using these structural members in complex structures such as aircrafts, turbine blades and space vehicles, exact static and dynamic analyses of these members become more significant. Based on structural/mechanical principles, the purpose of this paper is to present a new method to evaluate exact structural matrices for nonprismatic Euler‐Bernoulli beam elements. Design/methodology/approach — Through introducing the concept of basic displacement functions (BDFs), it is shown that exact shape functions are derived in terms of BDFs. BDFs and their derivatives have structural interpretations; therefore, they are obtained via application of flexibility method. Unlike the conventional methods, which are almost categorized as displacement‐based methods, the flexibility basis of the method ensures the true satisfaction of equilibrium equations at any interior point of the element. Findings — The exact shape functions and consequently structural matrices are derived for general nonprismatic beam elements. Numerical examples are carried out to determine static deflection and natural frequencies, and the results are highly competent with the other methods in literature. Research limitations/implications — The method can be extended to structural analysis of curved beams, plates and shells as well. Moreover, it is possible to derive exact dynamic shape functions via BDFs by solving the governing equation for transverse vibration of beams. Theoretically, the method faces limitation in analysis of nonprismatic beams that converge to a point where cross‐sectional area and moment of inertia are equal to zero. Practical implications — The development of this idea, i.e. BDFs seems to lead to promotive novel approaches for structural analysis and could be a breaking point for developing new elements for plates and shells as it was shown for beam elements. Originality/value — The paper's introduction of special functions, namely BDFs and their application, in both static and dynamic analyses of structures, could be a breaking point in analysis procedures.

A local–global FE model for nonlinear analysis of column-supported shells of revolution

The use of structural adhesives in automotive structures has been increased recently for their role in noise, vibration and harshness (NVH). Therefore, the dynamic behavior of structures containing bonded joints has become an area with numerous investigations over the past decades. Development of accurate formulations capable of representing adhesively bonded joint dynamics is a step forward in constructing the numerical models for one of the most useful kinds of joints in industry. Analysis of the adhesive layer between the two parts requires special assumptions which leads to using nonlinear and three dimensional models. Obtaining shape functions for an adhesive element by using finite element (F.E.) theory is a complicated and difficult task to do. The complexity is increased when it is assumed that the adhesive element is compatible with the plate element. In this paper, a new finite element formulation is developed for the adhesive layer which does not rely on shape functions and is compatible with the plate element. The accuracy of the proposed element is evaluated by using numerical and experimental results.

pq-Adaptive solid finite elements for three-dimensional plates and shells