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TECHNICAL PAPERS

# Null-Field Integral Equation Approach for Plate Problems With Circular Boundaries

[+] Author and Article Information
Jeng-Tzong Chen

Department of Harbor and River Engineering, National Taiwan Ocean University, No. 2, Pei-Ning Road, Keelung 20224, Taiwanjtchen@mail.ntou.edu.tw

Chia-Chun Hsiao

Department of Harbor and River Engineering, National Taiwan Ocean University, No. 2, Pei-Ning Road, Keelung 20224, Taiwan

Shyue-Yuh Leu

Hydraulic Engineering Department, Sinotech Engineering Consultants, 171 Nanking E. Road, Sec. 5, Taipei 10570, Taiwan

J. Appl. Mech 73(4), 679-693 (Oct 18, 2005) (15 pages) doi:10.1115/1.2165239 History: Received August 05, 2005; Revised October 18, 2005

## Abstract

In this paper, a semi-analytical approach for circular plate problems with multiple circular holes is presented. Null-field integral equation is employed to solve the plate problems while the kernel functions in the null-field integral equation are expanded to degenerate kernels based on the separation of field and source points in the fundamental solution. The unknown boundary densities of the circular plates are expressed in terms of Fourier series. It is noted that all the improper integrals are transformed to series sum and are easily calculated when the degenerate kernels and Fourier series are used. By matching the boundary conditions at the collocation points, a linear algebraic system is obtained. After determining the unknown Fourier coefficients, the displacement, slope, normal moment, and effective shear force of the plate can be obtained by using the boundary integral equations. Finally, two numerical examples are proposed to demonstrate the validity of the present method and the results are compared with the available exact solution, the finite element solution using ABAQUS software and the data of Bird and Steele.

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## Figures

Figure 1

The perforated Kirchhoff plate subject to the essential boundary conditions

Figure 2

Degenerate kernel for U(s,x)

Figure 3

Adaptive observer system when integrating the corresponding circular boundaries

Figure 4

Vector decomposition (collocation on x and integration on Bj)

Figure 5

Collocation point and boundary contour integration in the null-field integral equation

Figure 6

Boundary integral equation for the domain point

Figure 7

Flowchart of the present method

Figure 8

An annular plate subject to the essential boundary conditions

Figure 9

The contour plot of displacement for the annular plate subject to the essential boundary conditions by using a different method

Figure 10

Error estimation of the moment and shear force on the boundaries for the concentric circular domain

Figure 11

A circular plate containing three circular holes subject to the essential boundary conditions

Figure 12

The contour plots of displacement for the plate containing three circular holes subject to the essential boundary conditions by using different methods

Figure 13

Parseval sum versus terms of Fourier series

## Errata

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