journal+of+composite+materials 1988 shi 694 716

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Journal of Composite

http://jcm.sagepub.com/content/22/8/694The online version of this article can be found at:

DOI: 10.1177/002199838802200801

1988 22: 694Journal of Composite MaterialsGuimin Shi and Gerard Bezine

Bending ProblemsA General Boundary Integral Formulation for the Anisotropic Plate

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A General Boundary IntegralFormulation for theAnisotropic Plate

Bending Problems

GUIMIN SHIAND GERARD BEZINE

Laboratoire de Mcanique des Solides (UA CNRS 861)40, avenue du Recteur Pineau

86022 Poitiers, FRANCE

(Received January 17, 1987)(Revised June 26, 1987)

ABSTRACT

In this paper, a new direct boundary integral element method is presented for the analy-sis of Kirchhoffs anisotropic plate bending problems. The two boundary integral equationsare derived from the generalized Rayleigh-Green identity after introducing the fundamen-

tal singular solution ofan

infinite plate corresponding to the problem of interest. Bya sim-

ple discretization procedure with straight elements for the boundary, and constant assump-tion for the unknown boundary functions, two boundary integral equations are obtained inthe matrix form. Several computational examples concerning orthotropic plate bendingproblems are presented. The numerical results obtained by our method as compared withsome analytical results show that the present numerical scheme is a versatile tool whichgives a satisfactory accuracy.Key words: boundary integral equations, anisotropic plate bending, Kirchhoffs theory,

DBEM.

INTRODUCTION

I NRECENT YEARS, the boundary element methods applied to the flexure of thinelastic plates according to the Kirchhoffls theory have achieved some satisfac-

tory results. The pioneering work ofthe subject established the boundary integralequations mostly by means of the so-called indirect methods, which have certainshortcomings from the viewpoint of general-purpose and numerical analysis.During the past few years, however, Bezine [1], Stern [2], and Du, Yao and Song[3] et al. independently proposed the boundary integral equations in terms of theboundary variables having definite physical meanings from the different startingpoints for the analysis of isotropic thin plate bending problems. This method isusually known as the direct method. Recent developments and some interestingextensions of the direct boundary element methods applied to the flexure of iso-tropic plate were investigated by Shi [4] and Bezine [5].

Journal of COMPOSITE MATERIALS, Vol. 22 -August 1988

0021-9983/88/08 0694-23 $4.50/0@ 1988 Technomic Pubhshmg Co., Inc




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However, for the analysis of thin anisotropic plate bending problems, very fewattempts have been made up until now by the direct boundary element methods.To the knowledge of the authors, the BEM appears to have been used only twice.First, Wu andAltiero [6] proposed a numerical solution procedure for the bend-ing problems of anisotropic plates based on an indirect method, and they com-puted some example problems. Later, Kamiya and Sawaki [7] investigated asimplified direct boundary element method for the flexure of an orthotropic plate

by analogywith the method devised for the

isotropic plate bending problems,and

gave a single numerical example of an orthotopic clamped circular plate.In the present paper, we propose a new Direct Boundary Element Method

(DBEM) for the analysis of Kirchhof~s anisotropic plate bending problems. Thisformulation derives from the well-known generalized Rayleigh-Green identity inthe case of anisotropic plate bending theory [8]. Since the boundary integralequations are established in terms of the principal physical quantities, namely,the deflection, the normal slope, the bending moment, the Kirchhoffls equivalentshear force and thejumps of twisting moment, which appear in the conventionalboundary conditions, this method is applicable to anisotropic plate bending prob-lems under arbitrary boundary conditions, and particularly when the domain hasa non-smooth boundary. For this aim, we propose an original approach in orderto eliminate the twisting moment at the corners on the boundary of the plate.Andfor the first time we give the expressions of all the kernels involved in the two

boundary integral equations. Some numerical examples illustrate the precision ofthe current method for the practical applications.

Kirchhofrs Theory ofAnisotropic Plate

Suppose an anisotropic plate in the cartesian coordinates system o-xyz (Figure1), occupying the domain Sl with a boundary r. This plate is loaded by a transverse

Figure 1. Definition of the plate.




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load intensity p(x,y).According to the Kirchhoffs theory of anisotropic plateflexure [10,11], the fundamental equations of the problem can be written as fol-lows :

1. Governing equation

where w(x,y) is the lateral deflection, Dll, D,2, D22, D66, D16, D26 are theflexural rigidities of the anisotropic plate.

2. Boundary conditions

where ii is the outward unit normal vector to theboundary r,

and Mn, Vnare respectively the normal bending moment and the Kirchhoi~s equivalentshear force on the boundary r.

3. Bending and twisting moments, and transverse shear forces inside So




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4. Operators defined on the boundary

where a = (x,ii), R is the curvature radius at a smooth point of the boundaryr, and Tn (.) is the operator of the twisting moment. The constants are definedas

and

and




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Generalized Rayleigh-Green Identity and Fundamental Solution

The boundary integral equations for the bending problems of thin isotropicplates were obtained by several authors in different ways. Now we are going to es-tablish two boundary integral equations of anisotropic plate bending problems byusing the Generalized Rayleigh-Green identity. In the case of anisotropic plate,this identity can be written as follows [8]

where w and u are two functions with sufficient degree of continuity, m is thenumber of corner points of the boundary, [[ ...]].,, is the jump at corner s, alongthe boundary arc corrdinate s, and 00 (.) is defined by

We introduce now the fundamental singular solution r~ which satisfies

where 6(Q,P) is the Dirac delta-function behaving singular at the source pointP(xo,yo). The solution of Equation (10) stands for the deflection at point Q(x,y) ofan infinite anisotropic plate subjected to a concentrated lateral load D22 at pointP (xo~yo).

Lekhnitskii proved that the characteristic equation of Equation (1), i.e.




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could not have real roots in the case of anisotropic homogeneous elastic plates[11]. Let

be the roots of the characteristic Equation (11). Defining a polar coordinatessystem (rO) such as

then the fundamental singular solution wS(r,O) = WS(Q,P) can be expressed as fol-lows [6,8,9]

where

and




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and

with

It should be noted that in the functions Rk(r,6) and ~(r,9), the coefficient a is anarbitrary constant for our method. One can always choose, for example, thata = 1 (In fact the value of a does not modify the numerical solution).

Note also that if the flexural rigidities satisfy

then, the solution given in Equation (14) takes an indeterminate form. For sucha case, the fundamental singular solution can be written as [6,12]

BoundaryIntegral Equations

From the generalized Rayleigh-Green identity [8], ifwe choose w as the deflec-tion of the plate under consideration, and u as the fundamental solution WS(Q,P),and making use of the properties of Dirac 6-function in connection with Equation(1), the following integral representation is obtained:




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This equation gives, in fact, the value of deflection w at a point P (xo, yo) inside0 in terms of the

boundaryknown and unknown variables.

In the same way as for the analysis of istropic plate bending problems, whenthe interior point P (xo, yo) approaches to one point on the boundary r, one ob-tains the first boundary integral equation [5,8] :

For a plate bending problem, there are always two unknowns at any point of the

boundary. Consequentlywe must establish a second

boundary integral equationfor solving the problem of interest. This one is obtained by differentiating Equa-tion (21) with respect to point P (xo, yo) in the direction of the outward unit nor-mal no at P (xo, yo). This new boundary integral equation can be formed as fol-lows

We give inAnnexe the expressions of aw,lan, Mn(wS), Vn(WS), Tn(wS), awlan.,awlanoan, aM&dquo;(w$)lano, av&dquo;(ws)lano and c3T&dquo;(ws)lano which appear in twoboundary integral equations (21) and (22).

Equations (21) and (22) involve four fundamental boundary variables, i.e.,deflection w, normal slope awlan, bending moment M&dquo;(w) and Kirchhoffsequivalent shear force Y&dquo;(w), whose two should be the unknowns of the problem




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and two others are given by the two boundary conditions. However, in the otherhand, the twisting moment T&dquo;(w) is not considered as a fundamental variable inour numerical treatment, and it will be expressed in terms of the fundamentalvariables in the following.

In fact, from the expression ofM,, and Tn associated with w in the formulas (4),one can obtain for Mn(w) and T,,(w) the following forms:

where the coefficients are defined as

From Equation (23) we can then easily express T&dquo;(w) as follows

After the discretization of the boundary, one can always approximatea2wlanat = alat(awlan) by the nodal values of awlan, and a2wlat2 by those ofw, at several nodes near the corner points of the boundary. So we prove that thetwisting moment T&dquo;(w) can be always expressed in terms of w, awlan and M,,(w).




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In these conditions, one can calculate the two boundary unknowns by applyingEquations (21) and (22).

Numerical Formulation

The boundary r is discretized into n straight elements N, (i = 1, ..., n) witha node C, defined at the middle point of each segment. The boundary variablesw, awlc3n, Mn(w) and V,,(w) are supposed to be constant along each element N,,their values

being those taken by the variables at the node C,. So,on

placingsuc-

cessively the source point P(xo, Yo) at every node C,(i = 1, ..., n), one canfinally write Equations (21) and (22) in the matrix form as follows

where [~i], [B,], [C1], [Dl] and [Az], [B2], [C2], LD2] are eight n by n matriceswhose components can be computed numerically from the boundary integrals inEquations (21) and (22); {w}, lawlan), {Mn} and {Vn} are four nxl vectors whose

componentsare the nodal unknowns or knowns;

~7i}, {p2}are two known col-

umns calculated from the surface integrals, and ft,], {t2} are ruel vectors takinginto account the jumps in Equations (21) and (22).By using Equation (25), the vectors {tl} and {t2} can be expressed in function

of M, fawlani and IM,,]. Consequently Equations (26) and (Z7) can be rewrittenin the following forms

where [A{], [Bl], [Cnand [~2], [B2 ] , [C21, and { p i } , { p2are the modified ma-

trices and vectors after having added the jumps at the corners of the boundaryinto the corresponding matrices and vectors in Equations (26) and (27).For a discretization of the boundary into n constant elements, we have, a priori,

2 unknowns at each node C, and hence 2n unknowns on the boundary. In fact,Equations (28) and (29) contain 4n fundamental variables, i.e., [w], ~w/3~,{M&dquo;} and I Vnwhose 2n are given by the boundary conditions [2], accordingly itremains 2n variables which are the unknowns ofthe problem. On the other hand,we have exactly 2n equations given by (28) and (29). Consequently we can finallyobtain a system of 2n linear equations for solving 2n unknowns




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where {Xis a column vector of the 2n unknowns, [E] is a 2nx2n matrix, and ( f]is a known 2nxvector.When the linear system (30) is solved, and all of the variables on boundary are

known, then it is possible to compute the deflection w at an arbitrary point insidethe domain So by using the integral representation (20).

Numerical Results

The first applicationof our Direct

BoundaryElement

Method (DBEM) isa

square orthotropic plate simply-supported on its four edges. The flexural rigid-ities can be written as

and

where we set Ex = 2.068 x 10&dquo; Nlm2, E~ = E)15, VX = 0.3, Gxy = 6.055 x101 Nlm2 and h = 0.01 m. The boundary of the plate is discretized into 40straight elements, each edge having 10 equal length segments.For such a square orthotropic plate bending problem, an anlytical solution is

given in [10,11]. Moreover, Wu andAltiero have studied this plate subjected to

uniformly distributed lateral load, and given some computational results [6]. Inour practical computation, we have obtained the numerical results for the normalslope and the Kirchhoffls equivalent shear force on the boundary, and for thedeflection at some points inside the domain.We have treated two cases:

~ an uniform load per unit area p~ a concentrated load F at the centre of the plate

In Table 1 we compare the numerical results obtained by the present DBEM withthe results given by the analytical solution [11] for deflection values at the centreof the plate in these two different loads cases.




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Table 1. Deflection at the centre of a simply-supported orthotropic plate.

Where a is the length of edges of the plate. We see that the numerical resultscomputed by our DBEM are in very good agreement with the exact solution,since the error is only about 0.5%.Also, results for the deflection along the two axes of symmetry of the plate

under an uniform load or a concentrated force are shown in Figure 2 and Figure3 respectively.Once again, we compared our numerical results with the analytical solution

and, for the case of an uniform load, with the results given by Wu andAltiero [6].These results show that our method has a very good precision.

In the end we present the numerical boundary results for the normal slope inFigure 4 and Figure 5, and for the Kirchhoffs equivalent shear force in Figure 6and

Figure7

alongthe two

demi-edgesof the

platefor the two load

types respec-tively.All these results are in good agreement with those available from analytical

solution or other numerical methods. They prove the accuracy of our before men-tioned numerical technique.Our second example problem consists of a square orthotropic plate whose all

Figure 2. Deflection along two demi-axes of symmetry of a simply-supported orthotropicplate with an uniform load.




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Figure 3. Deflection along two demi-axes of symmetry of a simply-supported orthotropicplate with a concentrated force at its centre.

Figure 4. Normal slope along two demi-edges of a simply-supported orthotropic plate withuniform load.




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Figure 5. Normal slope along two demi-edges of a simply-supported orthotropic plate witha concentrated force at its centre.

Figure 6. Kirchhoffs shear force along two demi-edges of a simply-supported orthotroppicplate with an uniform load.




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Figure 7. Kirchhoffs shear force along two demi-edges of a simply-supported orthotropicplate with a concentrated force at its centre.

Figure 8. Deflection along the axis of symmetry of a cantilever orthotropic plate with a con-centrated force at its centre.




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four edges are clamped. For solving this problem we have used the same dis-cretization for the boundary as in the first example. Numerical results are thedeflection inside the domain, the bending moment and the Kirchhoi~s equivalentshear force on boundary. The computational results show that the maximaldeflection value occurs at the centre of the plate:

for an uniformly distributed load p.And

for a concentrated force F applied at the centre of the plate. In Table 2 we givethe numerical results on boundary for the bending moment Mn and the Kir-chhof~s equivalent shear force Vn in the dimensionless forms.To test the performance of our DBEM for various boundary conditions, we

have treated as a last numerical example a square cantilever orthotropic plateunder a concentrated loadF at its centre. The discretization ofthe boundary is thesame as in the previous examples. In the practical computation, we have obtained

for this problem the numerical results for the deflectionw

along the axis of sym-metry (Figure 8), the deflection w and the normal slope awlan along the freeedges (Figure 9 and 10). For the bending moment M,, and the KirchhofTs equiva-

Table 2. Boundary numerical results obtained by our DBEMfor a clamped orthotropic plate.




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Figure 9. Deflection along the free edges of a cantilever orthotropic plate with a concen-trated force at its centre.

Figure 10. Normal slope along the free edges of a cantilever orthotropic plate with a con-centrated force at Its centre.




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lent shear force Vn along the clamped edge, the numerical results show that themaximum values

and

occur at the centre of the clamped edge for both M~ and Vn.

CONCLUSION

A new direct boundary integral equation-boundary element method for thesolution of Kirchhofps anisotropic plate bending problems has been presented.Owing to the characteristics ofthe two proposed boundary integral equations andto the new numerical treatment for the twisting moment on boundary, thismethod can be used to analyse the flexure of anisotropic plates with any planforms under arbitrary conventional boundary conditions, particularly for theplates with a boundary having some corner points. Numerical results of the prac-tical computation of some example problems show that this direct boundary ele-ment

method hasa

satisfactory accuracy.And one

ofthe most

interestingfeatures

of the present numerical technique is the considerable reduction in the data prep-aration to run a practical anisotropic plate bending problem.

ANNEXE

The kernel functions involved in the boundary integral Equations (21) and (22)can be obtained by using the boundary operators (4) with respect to the funda-mental solution w$(r,9) given in (14). In other words, one can give the expressionsof the kernels as follows:




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where o = (x, no), and the constants have been defined in (5), (6) and (7).The derivatives of the fundamental solution in (33) and (34) can be expressed

by those of the functions R,(r,9), S,(r,6) (i = 1,2) given in (15) and (16). For ex-ample :

So we give only the derivatives of R,(rO) and S,(r,9) in the following.Derivatives of R,(r,9) (i = 1,2):




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Derivatives of S, (r, 8) (I = 1,2):




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REFERENCES

1.Bezine, G "Boundary Integral Formulation for Plate Flexure withArbitrary Boundary Condi-tions," Mechanics Research Commumcations, 5(4):197-206 (1978).

2. Stern, M. "A General Boundary Integral Formulation for the Numerical Solution of Plate Bend-ing Problems, Int. J. Solids and Structures, 15 769-782 (1979).

3. Du, Q., Z. Yao and G. Song. "Solutions of Some Plate Bending Problems Using the BoundaryElement Method,"Appl. Math. Modelling, 8(1):15-22 (1984).




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4. Shi, G. "On Boundary Element Methods for Thin Plate Flexure withArbitrary Boundary Con-ditions," some Preliminary Research Reports on Boundary Element Method, No. 84-1, TsinghuaUniversity, Beijing, China (1984).

5. Bezine, G. "Application de la Mthode des quations Intgrales la Rsolution des ProblmesStatiques et Dynamiques Stationnaires ou Transitoires de Flexion de Plaques Minces. Extension des Problmes de Mcanique des Fluides Pour des coulements Visqueux en Rgime deStokes," Thse de Docteur s Sciences Physquies, Poitiers, France (1982).

6. Wu, B. C. and N. I.Altiero. "A New Numerical Method for theAnalysis ofAnisotropic Thin-Plate Bending Problems,"

Comp.Math.

Appl.Mech.

Engg. ,25:343-353 (1981).

7. Kamiya, N. and Y. Sawaki. "A General Boundary Element Method for BendingAnalysis of Or-thotropic Elastic Plates," Res. Mechanica, 5:329-334 (1982).

8. Shi, G. "Etude de la Flexion des PlaquesAmsotropes par la Mthode des quations Intgralesde Frontire," Rapport de DEA, Poitiers, France (1986).

9. Suchar, M. "On Singular Solutions in the Theory of Anisotropic Plates," Bulletin de lAcadmiePolonaise des Sciences. Srie des Sciences Techniques, 12:29-38 (1964).

10. Timoshenko, S. and S. Woinowsky-Krieger. "Theory of Plates and Shells," New York :McGraw-Hill (1959).

11. Lekhnitski, S. G. "Anisotropic Plates," New York:Gordon and Breach (1968).12. Lukasiewicz, S. "Local Loads in Plates and Shells," The Netherland:Sijthoffand Noordhoff In-

ternational Pub. (1979).13. Zienkiewicz, O. C. "The Finite Element Method," 3rd Edition, England:McGraw-Hill (1977).



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