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Research Papers

# Two Simple Triangular Plate Elements Based on the Absolute Nodal Coordinate Formulation

[+] Author and Article Information
Oleg Dmitrochenko

Department of Mechanical Engineering, Lappeenranta University of Technology, Skinnarilankatu 34, 53851 Lappeenranta, Finlandoleg.dmitrochenko

Aki Mikkola

Department of Mechanical Engineering, Lappeenranta University of Technology, Skinnarilankatu 34, 53851 Lappeenranta, Finlandaki.mikkola@lut.f

Here, the cross product $r1′×r2′$ is represented using a concept of the skew-symmetric matrix $r̃1′$ associated with vector $r1′$.

The area element is $dΔ=dp1dp2$; that is why $2Δ$ is the Jacobian $|∂(dp1dp2)/∂(Δi,Δj)|$ when $Δk=1–Δi–Δj$, see Eqs. 1,5.

J. Comput. Nonlinear Dynam 3(4), 041012 (Sep 04, 2008) (8 pages) doi:10.1115/1.2960479 History: Received June 09, 2007; Revised November 26, 2007; Published September 04, 2008

## Abstract

In this paper, two triangular plate elements based on the absolute nodal coordinate formulation (ANCF) are introduced. Triangular elements employ the Kirchhoff plate theory and can, accordingly, be used in thin plate problems. As usual in ANCF, the introduced elements can exactly describe arbitrary rigid body motion when their mass matrices are constant. Previous plate developments in the absolute nodal coordinate formulation have focused on rectangular elements that are difficult to use when arbitrary meshes need to be described. The elements introduced in this study have overcome this problem and represent an important addition to the absolute nodal coordinate formulation. The two elements introduced are based on Specht’s and Morley’s shape functions, previously used in conventional finite element formulations. The numerical solutions of these elements are compared with previously proposed rectangular finite element and analytical results.

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## Figures

Figure 1

Triangular area and its coordinates

Figure 2

Triangle element based on Specht’s shape functions

Figure 3

A simple triangle element based on Morley’s shape functions

Figure 4

Discontinuity of a mesh using Morley’s elements

Figure 5

Discontinuity of two Morley’s elements

Figure 6

Modeling of artificial discontinuity

Figure 7

Mesh used for a test problem

Figure 8

Free-falling plate-shaped pendulum

Figure 9

Displacements of the tip of the plate pendulum

Figure 10

Simulation of Dali’s soft clocks by triangles

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