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# Discussion: “Isotropic Clamped-Free Thin Annular Circular Plate Subjected to a Concentrated Load” (, 2006, ASME J. Appl. Mech., 73, pp. 658–663)OPEN ACCESS

[+] Author and Article Information
J. T. Chen

Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan, R.O.C.jtchen@mail.ntou.edu.tw

W. M. Lee

Department of Mechanical Engineering, China Institute of Technology, Taipei 11581, Taiwan, R.O.C.wmlee@cc.chit.edu.tw

H. Z. Liao

Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan, R.O.C.

J. Appl. Mech 76(1), 015501 (Nov 12, 2008) (4 pages) doi:10.1115/1.2937145 History: Received July 25, 2007; Revised March 25, 2008; Published November 12, 2008

## Introduction

In this interesting paper (1), a concentrated load was applied to the clamped-free annular plate. The problem domain was divided into two parts by the cylindrical section where a concentrated load was applied. The author used the Trefftz method (2) to construct the homogeneous solutionDisplay Formula

$u=∑m=0∞Rm(r)cosmθ$
(1)
in each part. By substituting Eq. 1 into the governing equation, the author could determine $Rm(r)$. Mathematically speaking, the series in Eq. 1 can be seen as the summation of Trefftz bases. To simulate the concentrated force, a circularly distributed force using the Fourier series is used. Then, the author utilized two boundary conditions (BCs) in each part, two continuity, and two equilibrium conditions on the interface to determine the eight unknown coefficients. Variation of deflection coefficients, radial moment coefficients, and shear coefficients along radial positions and angles was presented. However, some results are misleading. To investigate these inconsistencies, both null-field integral formulation and finite element method (FEM) using the ABAQUS are adopted to revisit this problem. In addition, two unclear issues in Ref. 1 are discussed. One is the simulation of concentrated load and the other is the operator of shear force.

In Adewale’s paper (1), the author expanded the concentrated load to the Fourier seriesDisplay Formula

$P≈P[12+∑k=1∞2sin(2k−1)π2(2k−1)πcos(2k−1)θ],0≤θ≤π2$
(2)
By summing up the series of Eq. 2, the result converges to 1 as shown in Fig. 1, which does not show the behavior of the Dirac-delta function. The Dirac-delta function $δ(x)$ should satisfy the identity as follows:Display Formula
$∫−∞∞δ(x)dx=1$
(3)
Equation 2 cannot satisfy Eq. 3 such that the strength of the concentrated loading is 1. The author seems to improperly transform the concentrated load to a circularly distributed one. If this load is distributed along an angle from 0 to $π∕2$, the results of the deflection coefficient in Fig. 5 of Ref. 1 would be untrue.

## Definition of Shear Force

For the clamped-free annular plate problems as shown in Fig. 2, the shear force on the inner circle is zero for the free boundary. Therefore, the author obtained the shear forceDisplay Formula

$∣(∂3∂r3−1r2∂∂r+1r∂2∂r2−m2r2∂∂r)Rm(r)∣r=a=0shearforcefree$
(4)
According to the displacement of Eq. 1 and the definition of shear force operator in Szilard’s book (3), the shear force can be derived asDisplay Formula
$∂3Rm(r)∂r3−1r2∂Rm(r)∂r+1r∂2Rm(r)∂r2+2m2r3Rm(r)−m2r2∂Rm(r)∂r+(1−ν)[m2r3Rm(r)−m2r2∂Rm(r)∂r]forshearforce$
(5)
where $ν$ is the Poisson ratio. Equation 4 is unreasonable since it does not involve the Poisson ratio. In literature, many articles had reported the definition of shear force operator, e.g., Refs. 1-5. We summarized the shear force operators in Table 1. After careful comparison, Adewale’s shear force operator differs from the others and consequently, this difference may cause inconsistent results.

## Alternative Derivation of the Analytical Solution Using the Null-Field Integral Formulation

The first boundary integral equations for the domain point can be derived from the Rayleigh–Green identity as follows (5-6):Display Formula

$8πu(x)=U(ζ,x)−∫BU(s,x)v(s)dB(s)+∫BΘ(s,x)m(s)dB(s)−∫BM(s,x)θ(s)dB(s)+∫BV(s,x)u(s)dB(s),x∊Ω∪B$
(6)
where $B$ is the boundary of the domain $Ω$; $u(x)$, $θ(x)$, $m(x)$, and $v(x)$ are the displacement, slope, normal moment, and effective shear force; and $s$ and $x$ are the source point and field point, respectively. The kernel function $U(s,x)$ in Eq. 6 is the fundamental solution that satisfiesDisplay Formula
$∇4U(s,x)=8πδ(s−x)$
(7)
Therefore, the fundamental solution can be obtained as follows:Display Formula
$U(s,x)=r2lnr$
(8)
where $r$ is the distance between the source point $s$ and field point $x$. The relationship among $u(x)$, $θ(x)$, $m(x)$, and $v(x)$ is shown as follows:Display Formula
$θ(x)=Kθ,x(u(x))=∂u(x)∂nx$
(9)
Display Formula
$m(x)=Km,x(u(x))=ν∇x2u(x)+(1−ν)∂2u(x)∂2nx$
(10)
Display Formula
$v(x)=Kv,x(u(x))=∂∇x2u(x)∂nx+(1−ν)∂∂tx[∂∂nx(∂u(x)∂tx)]$
(11)
where $Kθ,x(⋅)$, $Km,x(⋅)$, and $Kv,x(⋅)$ are the slope, moment, and shear force operators with respect to the point $x$; $∂∕∂nx$ is the normal derivative with respect to the field point $x$; $∂∕∂tx$ is the tangential derivative with respect to the field point $x$; and $∇x2$ is the Laplacian operator. The first null-field integral equations can be derived by moving the field point $x$ outside the domain as follows:Display Formula
$0=U(ζ,x)−∫BU(s,x)v(s)dB(s)+∫BΘ(s,x)m(s)dB(s)−∫BM(s,x)θ(s)dB(s)+∫BV(s,x)u(s)dB(s),x∊ΩC∪B$
(12)
where $ΩC$ is the complementary domain of $Ω$. For the kernel function $U(s,x)$, it can be expanded in terms of degenerate kernel (2,5-7) in a series form as shown below:Display Formula
$U(s,x)={UI(R,θ;ρ,ϕ)=ρ2(1+lnR)+R2lnR−[Rρ(1+2lnR)+12ρ3R]cos(θ−ϕ)−∑m=2∞[1m(m+1)ρm+2Rm−1m(m−1)ρmRm−2]cos[m(θ−ϕ)],R≥ρUE(R,θ;ρ,ϕ)=R2(1+lnρ)+ρ2lnρ−[ρR(1+2lnρ)+12R3ρ]cos(θ−ϕ)−∑m=2∞[1m(m+1)Rm+2ρm−1m(m−1)Rmρm−2]cos[m(θ−ϕ)],ρ>R}$
(13)
where the superscripts $I$ and $E$ denote the interior and exterior cases of $U(s,x)$ kernel depending on the location of $s$ and $x$. For the annular plate clamped at the outer edge and free at the inner edge, the unknown Fourier coefficients of $m$, $v$ on the outer boundary and $u$, $θ$ on the inner boundary can be expanded toDisplay Formula
$v(s)=a0+∑n=1M(ancosnθ+bnsinnθ),s∊outerboundary$
(14)
Display Formula
$m(s)=a¯0+∑n=1M(a¯ncosnθ+b¯nsinnθ),s∊outerboundary$
(15)
Display Formula
$θ(s)=p0+∑n=1M(pncosnθ+qnsinnθ),s∊innerboundary$
(16)
Display Formula
$u(s)=p¯0+∑n=1M(p¯ncosnθ+q¯nsinnθ),s∊innerboundary$
(17)
where $a0$, $an$, $bn$, $a¯0$, $a¯n$, $b¯n$, $p0$, $pn$, $qn$, $p¯0$, $p¯n$, and $q¯n$ are the Fourier coefficients, and $M$ is the number of Fourier series terms in real computation. By substituting all the Fourier coefficients of boundary densities and boundary conditions, the displacement field can be obtained as shown below:Display Formula
$8πu(x)=U(ζ,x)−∫BU(s,x)[a0+∑n=1M(ancosnθ+bnsinnθ)]dB(s)+∫BΘ(s,x)[a¯0+∑n=1M(a¯ncosnθ+b¯nsinnθ)]dB(s)−∫BM(s,x)[p0+∑n=1M(pncosnθ+qnsinnθ)]dB(s)+∫BV(s,x)[p¯0+∑n=1M(p¯ncosnθ+q¯nsinnθ)]dB(s),x∊Ω∪B$
(18)
where $an$, $bn$, $a¯n$, $b¯n$, $pn$, $qn$, $p¯n$, and $q¯n$$(n=0,1,2,…)$ are solved in Ref. 7.

## Results and Discussions

In order to verify the accuracy of Adewale’s results, two alternatives, null-field approach and FEM using ABAQUS , are employed to revisit the annular problem. A concentrated load was applied at the radial center of the annular plate, as shown in Fig. 2. For the clamped-free boundary condition, Figs.  33 show the displacement contours for the Green’s function by using FEM (ABAQUS ) and the present method, respectively. Good agreement is obtained between our analytical solution and FEM result although Adewale (1) did not provide the displacement contour of his analytical solution. For comparison with the available results in Ref. 1, Fig. 4 shows the variation of deflection coefficients, moment coefficients, and shear force coefficients along radial positions or angles for different inner radii. It is also found that FEM results match well with our solution but deviates from Adewale’s outcome (1).

## Concluding Remarks

To verify the accuracy of Adewale’s results and to examine the response of the clamped-free annular plate subjected to a concentrated load, the null-field integral formulation was employed in solving this problem. The transverse displacement, moment, and shear force along the radial positions and angles for different inner radii were determined by using the present method in comparison with the ABAQUS data. Good agreements between our analytical results and those of ABAQUS were made but deviated from Adewale’s data. The outcome of Adewale’s results may not be correct.

## References

Copyright © 2009 by American Society of Mechanical Engineers
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## Figures

Figure 1

Simulation of a concentrated force by Adewale’s (1)(M=101).

Figure 2

Problem statement of an annular plate

Figure 3

Contour plots of the Green’s function for the annular problem (a=0.4, b=1.0, Rζ=0.7, D=1, ν=0.3). (a) Displacement contour by using the FEM (ABAQUS). (b) Displacement contour by using the present method (M=50).

Figure 4

Responses (b=1.0, Rζ=0.7∼0.85, D=1, ν=0.3, kw=wD∕P, kmr=MrD∕P, ks=MsD∕P)

## Tables

Table 1
The definitions of the shear force (a) Szilard, (b) Leissa, (c) the present operator, and (d) Adewale

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