Four-node Plate Element Research Articles

Dynamic stability of the Mindlin-Reissner plate using a time-modulated axial force

A time-periodic axial force acting on a continuous structure can be destabilized by parametric excitation. Moreover, it is capable of increasing the equivalent damping and shortening the transient vibrations. In this work, this concept is extended, and dynamic stability of a Mindlin-Reissner plate is analyzed. The FE formulation for the four-node plate element is derived by employing the variational approach including the action of an axial force. The stability of the time-periodic system is evaluated both numerically and analytically, using Floquet theory and first-order approximation based on the averaging method, respectively. The influence of plate length, thickness, and aspect ratio on dynamic stability is highlighted. It is demonstrated that the instability tongue at a parametric combination resonance frequency of the summation type is shifted toward higher axial force amplitudes by increasing the plate width and decreasing the plate length. Alternatively, the axial force acting in the vicinity of a parametric combination frequency of the difference type shows a parametric antiresonance which is maximized by exactly the opposite tuning of the plate design. This guideline is important when plate design under the action of a harmonic axial force is considered.

Modeling of partially delaminated composite plates resting on two-parameter elastic foundation with improved layerwise theory

In this work, we aim to build the mathematical model for delaminated composite plates resting on elastic foundation by means of improved layerwise theory and finite element implementation for free vibration analysis. The proposed model takes into account the transverse normal and shear deformation of the elastic foundation so as to investigate their effect on the dynamic characteristics of delaminated plates. The four-node plate element is used for the finite element formulations with Langrage interpolation functions for in-plane structural unknowns and Hermite cubic interpolation functions for out-of-plane structural unknowns. The delamination is simulated by means of Heaviside unit step functions allowing for the discontinuity of displacement fields. The effects of both longitudinal locations and interfacial locations of delamination are investigated. The results show that when the delamination located closer to the clamped end and at the middle plane, it has much more severe influence on its natural frequencies. The foundation parameters also have significant effect on the natural frequencies and frequency shift due to delamination. Different layup stacking sequences are also considered in this study.

Assumed strain finite element for natural frequencies of bending plates

PurposeThis paper aims to describe the formulation of a new finite element by assuming the strain field rather than the displacement field and by using the Reissner–Mindlin plate theory for the free vibration analysis of bending plates. This quadrilateral element consists of four-nodes and twelve degrees of freedom. The suggested element is based on assumed functions of the strain field that satisfy the compatibility equation.Design/methodology/approachAfter the proposition of the new element, several numerical tests for plates with regular and distorted meshes are presented to assess the performance of the new element. In addition, a parametric study is carried out to analyze the effects of biaxial loads on the natural frequencies of square plates with various boundary conditions. Detailed discussions are proposed after each benchmark problem.FindingsThe formulated element has verified the shear locking test and passes the patch test. The obtained results from the developed element show an excellent accuracy and fast convergence, and the natural frequencies are in excellent agreement when compared with analytical and other available numerical solutions.Originality/valueThe present element is simple in its formulation and has been proven to be applicable to thin or thick plate situations with sufficient accuracy. This element with full integration is free from shear locking, however, the numerical results provided by the standard four-node plate element R4 element show locking phenomena in thin plates. In addition to these features, the imposition of the compatibility conditions and the rigid body modes allow obtaining a finite element with higher-order terms for displacements field, which can increase the performance of the finite elements.

A mixed finite element model with enhanced zigzag kinematics for the non-linear analysis of multilayer plates

In the present work, a non-linear finite element model for the analysis of composite and sandwich plates is proposed. The underlying variational formulation is based on the kinematics of the refined zigzag theory (RZT) as well as on a modified version of the two-field Hellinger–Reissner (HR) functional, where the displacements and the transverse shear stresses are involved as independent field variables. In fact, a consistent expression for each shear stress function is obtained by integrating the local equilibrium equations written in terms of derivated strain components. Regardless of the number of material layers, the proposed four-node plate element, which is labeled HR–RZT, exhibits only seven nodal degrees of freedom. The additionally introduced variables are effectively eliminated on the element level. One key advantage of the HR–RZT formulation is the fact that interlayer-continuity of the transverse shear stresses is automatically obtained without resorting to any post-processing procedures. The same applies to the zero stress conditions at the outer surfaces of the laminate. Further, due to the enhanced kinematics, the layerwise distortion of the laminate’s cross-section can also be accurately captured. The performance of the novel element formulation is studied by several numerical applications, where the solutions of the two-dimensional HR–RZT elements are verified by comparison with fully three-dimensional finite element models.

Finite element analysis of laminated composite plates using zeroth-order shear deformation theory

In this paper a generalized finite element model is developed for static and dynamic analyses of laminated composite plates using zeroth-order shear deformation theory (ZSDT). The theory ensures the parabolic distribution of transverse shear stresses across the plate thickness. A four-noded plate element is considered in this model and the generalized nodal variables are expressed using Lagrangian linear interpolation functions and Hermitian cubic interpolation functions. The solutions of the finite element model have been compared with the existing solutions for symmetric and antisymmetric laminated composite plates. The comparison confirms that the ZSDT can be efficiently used for finite element analysis of both thin and thick plates with high accuracy.

Transient analysis of biocomposite laminates with delamination.

The transient analysis of smart biocomposite laminates with delamination at ply interfaces is investigated, by using an electro-mechanical coupled improved layerwise theory. The piezoelectric coupling effect was modeled using higher order electric potential. Four-node plate elements were used for finite element implementation. Linear Lagrange interpolation functions were used for in-plane structural unknowns and electric unknowns, and a Hermite cubic interpolation function was used for out-of-plane structural unknowns. Single delamination was seeded in the biocomposite laminates at three different locations, to study the effects of delamination. Numerical results were obtained by using the Newmark-beta algorithm. The results showed that the time history of nodal displacement and sensor output didn't give enough information of delamination. However, the power spectral density of the time history provided clear information of the delamination effects. From the results, it is expected that the proposed theory has the potential to predict the dynamic characteristics and delamination effects in smart biocomposite laminates.

Frequency response analysis of a delaminated smart composite plate

A frequency analysis of smart composite plate with delamination at ply interface was investigated in this article. The modeling was based on an electro-mechanical coupled improved layerwise theory, with implementing finite element method. Four-node plate elements with Lagrange and Hermite cubic interpolation functions were used for in-plane structural unknowns, electric unknowns, and out-of-plane structural unknowns. The general modal reduction method was applied to solve the second-order differential equation. Numerical results showed significant shift of natural frequencies in the frequency response of tip displacement and three sensor outputs due to the presence of delamination. It is found that the delamination locations also influence the natural frequencies of smart composite structure. Thus, the proposed methodology could be a useful tool to develop system identification and structural health monitoring techniques of smart composite structure.

Analysis and optimal design of smart skin structures for buckling and free vibration

Analysis and optimal design of the smart skin, i.e., the conformal load-bearing antenna structure for buckling and free vibration are studied. The smart skin is modeled as an asymmetric composite sandwich structure which consists of different materials. In order to act as antennas or radars, a dielectric layer is embedded on the middle surface of the sandwich structure. The formulation of the smart skin model is based on the first-order shear deformation plate theory, and numerical results are obtained by the finite element method using a four-node plate element with five degrees of freedom at each node. The buckling and vibration behaviors of the smart skin are investigated with the variation of the aspect ratio, Honeycomb core thickness, area of dielectric layer and fiber angle of face sheet. Furthermore, to maximize the buckling load and the natural frequency of the smart skin, the optimal design is performed using a simple genetic algorithm.

Free vibration analysis of plates by using a four-node finite element formulated with assumed natural transverse shear strain

Abstract A study on the free vibration analysis of plates is described in this paper. In order to investigate vibrational characteristics of plates, a four-node plate element is developed by using the assumed natural strains on the basis of Reissner–Mindlin (RM) assumptions which allows us to consider the shear deformation and rotatory inertia effect. All terms related to the plate finite element formulation are consistently defined in the natural domain. Assumed natural strains are derived to alleviate the locking phenomena inherited in the RM plate elements. In particular, the explicit expression of assumed natural transverse shear strain is described in this paper. The natural constitutive equation is used in conjunction with the natural strain terms. Several numerical examples are carried out and their results are then compared with the existing reference solutions.

An eight noded composite plate element for the dynamic analysis of spatial mechanism systems

The development of a shear-deformable laminated plate element, based on the Mindlin plate theory, for use in large reference displacement analysis is presented. The element is sufficiently general to accept an arbitrary number of layers and an arbitrary number of orthotrophic material property sets. Coordinate mapping is utilized so that non-rectangular elements may be modeled. The Gauss quadrature method of numerical integration is utilized to evaluate volume integrals. A comparative study is done on the use of full Gauss quadrature, reduced Gauss quadrature, mixed Gauss quadrature, and closed form integration techniques for the element. Dynamic analysis is performed on the RSSR (Revolute-Spherical-Spherical-Revolute) mechanism, with the coupler modeled as a flexible plate. The results indicate the differences in the dynamic response of the transverse shear deformable eight-noded element as compared to a four-noded plate element. Dynamically induced stresses are examined, with the results indicating that the primary deformation mode of the eight-noded Mindlin plate model being bending.

Derivation of stabilization matrices for mindlin plates from the combined mixed functional and their mathematical characteristics

A new mixed functional is proposed for the analysis of Mindlin plates. The functional is constructed by combining the Hellinger—Reissner mixed functional and the total potential energy linearly. Existence and uniqueness of the solution of the proposed mixed model are proven and finite element equations are derived. The equivalence theorem for mixed elements and reduced/selective integration elements is applied and the stabilization matrix of Belytschko is obtained for the four-node plate element. Using the present method, stabilization matrices which have strong mathematical properties can be obtained for higher-order elements and triangular elements without any difficulty.

A variational justification of the assumed natural strain formulation of finite elements—II. The C0 four-node plate element

Abstract In Part II the four-node C0 plate bending element is used to explore some of the possibilities opened by the theory presented in Part I [C. Militello and C. A. Felippa, Comput. Struct.34, 431–438 (1990)]. This element is chosen because the version presented by Bathe and Dvorkin [Int. J. Numer. Meth. Engng21, 367–383 (1985)], MITC4, can be considered the simplest assumed natural strain element that allows several possibilities to be studied in a straightforward manner. Attention is focused on the governing functionals R and H presented in Part I, assuming independent strain fields only for the transverse shear strains. Besides MITC4, three formulations (two mixed and one hybrid) are considered that collectively represent a variational justification for the assumed strain technique. In addition, reduced and selective-integration elements are examined to compare their behavior with that of the present assumed strain elements.

Transverse shear oscillations in four-node quadrilateral plate elements

Abstract Numerical results show that the transverse shears are free of oscillation for regular or quasi-regular meshes of four-node plate elements based on Mindlin-Reissner theory. However, even small perturbations of these meshes can introduce severe oscillations throughout the mesh. These findings apply to the underintegrated elements with and without stabilization and to the fully integrated elements which use assumed shear strains. A filtering procedure capable of recovering improved shears is presented.