Triangular Element for Multilayer Sandwich Plates

MITC technique extended to variable kinematic multilayered plate elements

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.

This article discusses the problem of symmetric vibrations of a three-layer plate pivotally supported along the edges in the longitudinal direction is solved. The materials of the plate layers are assumed to be elastic and isotropic. It is believed that the plate is not symmetrical in thickness, i.e. different thicknesses of layers. A brief review of scientific papers on theories of calculation of three-layer and multilayer plates is given. The equations of vibration of a three-layer plate, previously developed by the authors, are accepted as resolving equations. The stress and displacement components are expressed, as well as the oscillation equations, through the main parts of the longitudinal and transverse displacements of the points of the “intermediate” plane of the middle layer, which is at some distance from the middle plane of the plate. The conditions of the articulated plumage are formulated with respect to the displacements of the points of the middle layer. The initial conditions are assumed to be zero. The formulated problem is solved by an analytical-numerical method using a software package. The obtained numerical results are presented in the form of graphs of the dependences on time and the longitudinal coordinate of the displacements and voltages of the points, both the middle and the bearing layers of the plate. The calculations were performed for steel materials of the bearing layers, and for the middle layer the physic-mechanical parameters of the polymer material were adopted. The corresponding conclusions are made according to the results of numerical calculations. It was established that slight transverse displacements of the plate points appear due to the action of longitudinal external loads. Moreover, due to the insignificance of these displacements with symmetric vibrations of a three-layer plate, transverse displacements can be neglected. With symmetrical vibrations of the plate, longitudinal normal voltages are also excited despite the fact that the edges of the plate are free of external loads. These voltages are generated by the action of external tangential and normal voltages on the boundary surfaces.

The edge-based smoothing discrete shear gap method (ES-DSG3) using three-node triangular elements is combined with a C0-type higher-order shear deformation theory (HSDT) to give a new linear triangular plate element for static, free vibration, and buckling analyses of laminated composite plates. In the ES-DSG3, only the linear approximation is necessary, and the discrete shear gap method (DSG) for triangular plate elements is used to avoid the shear locking and spurious zero energy modes. In addition, the stiffness matrices are calculated relying on smoothing domains associated with the edges of the triangular elements through an edge-based strain smoothing technique. Using the C0-type HSDT, the shear correction factors in the original ES-DSG3 can be removed and replaced by two additional degrees of freedom at each node. The numerical examples demonstrated that the ES-DSG3 show remarkably excellent performance compared to several other published elements in the literature.

C0 triangular elements based on the Refined Zigzag Theory for multilayer composite and sandwich plates

Large amplitude free flexural vibrations of thin plates of arbitrary shape

Vibration of arbitrarily-shaped triangular plates with elastically restrained edges

Hybrid stress-based finite elements with side displacement fields have been used to generate equilibrium models having the property of equilibrium in a strong form. This paper establishes the static and kinematic characteristics of a flat triangular hybrid equilibrium element with both membrane and plate bending actions of general polynomial degree p. The principal characteristics concern the existence of hyperstatic stress fields and spurious kinematic modes. The former are shown to exist for p>3, and their significance to finite element analysis is reviewed. Knowledge of the latter is crucial to the determination of the stability of a mesh of triangular elements, and to the choice of procedure adopted for the solution of the system of equations. Both types of characteristic are dependent on p, and are established as regards their numbers and general algebraic forms. Graphical illustrations of these forms are included in the paper. Copyright © 2005 John Wiley & Sons, Ltd.

Bending, vibration and buckling of simply supported thick multilayered orthotropic plates: An evaluation of a new displacement model

This paper proposes a simple method for deriving the geometric stiffness matrix (GSM) of a three‐node triangular plate element (TPE). It is found that when the GSM of the element is combined into the global one of the structure, this structural stiffness matrix becomes symmetric and satisfies both the rigid body rule and incremental force and moment equilibrium (IFE) conditions, which are basically two fundamental conditions for analysis of mechanics. The former condition has been widely used in the community of mechanics; while the latter one, to our best knowledge, has never been considered. Advantages with the GSM derived are that derivations only need simple matrix operations without cumbersome non‐linear virtual strain energy derivations and tedious numerical integrations and more appealingly, this derived GSM can be explicitly given for applications. In addition, based on IFE and the rigid body rule conditions, a reasonable GSM for the three‐node TPE must be asymmetric; however, an asymmetric matrix usually gives rise to tedious numerical calculation especially in geometrically nonlinear problems and further, greatly influences computation efficiency. Fortunately, the skew‐symmetric parts of the derived GSM can be canceled out once they are merged into the global stiffness matrix of the structure. In this regard, this structural stiffness matrix becomes a symmetric one and thus enhances its effectiveness. Finally, several examples are provided for validating the robustness of the derived GSM.

In the paper a problem of the control vibrations of the smart aluminum plate with SFSF (Supported-Free-Supported-Free) boundary conditions is considered. In order to obtain effectively control system the plate is equipped in two piezo-patch actuators and two piezo-patch sensors oriented in two perpendicular directions X and Y, respectively. Such locations and orientations of the piezo-elements allow to determine the mathematical models separately for odd and even modes. By using Dynamical Signal Analyzer (DSA) the aforementioned models have been obtained on the laboratory stand. In results of experimental identification, both models Godd(s) and Geven(s) described vibrations of the plate in the selected frequency range from 10 Hz to 250 Hz are obtained. In the next stage the process of design optimal control law is carried out for such obtained models into two steps. In the first step the local controller LQRodd is designed to control only the lowest three odd modes of the plate. On the other hand, the controller LQReven is designed to control lowest two even modes. Results of simulation investigations of both closed-loop systems were verified on the laboratory stand with using Dspace controller 1005.

The vibration and dynamic stability problems of an orthotropic rectangular plate subjected to a non-conservative follower force are studied. The governing differential equation derived by Brunelle and Oyibs in affine space is used. The Levy-type solution method is used to solve the vibration problems. The optimization package ADS is employed to find the dynamic stability loading. The effect of generalized rigidity ratio D* on the stability boundaries is presented.

A simplified geometric stiffness in stability analysis of thin-walled structures by the finite element method

Coupled analysis of 3D structural–acoustic problems using the edge-based smoothed finite element method/finite element method

A semi-analytical postprocessing method, termed the equilibrium/compatibility method, is employed for computation of hitherto unavailable through-thickness variation of interlaminar shear stresses in the neighborhood of the bilayer interface circumferential reentrant corner line of an internal (part-through) elliptical cylindrical hole weakening an edge-loaded laminated composite plate. A C 0 -type triangular composite plate element, based on the assumptions of transverse inextensibility and layerwise constant shear-angle theory, is used to first compute the in-plane stresses and layerwise through-thickness average interlaminar shear stresses, which serve as the starting point for computation of through-thickness distribution of interlaminar shear stresses in the vicinity of the bilayer interface circumferential reentrant corner line of the internal hole. The errors in the same stresses computed by the conventional equilibrium method are more severe in the presence of the bilayer interface circumferential reentrant corner line singularity arising out of the internal (part-through) elliptical cylindrical hole than their circular counterparts, and are, as before, found to violate the interfacial compatibility condition. The computed interlaminar shear stresses can vary from negative to positive through the thickness of a cross-ply plate in the neighborhood of this kind of stress singularity.