BOUNDARY ELEMENT METHOD APPLIED TO THE ANALYSIS OF THIN PLATES

F. PARES and S. DE LEON

E.T.S. Ingenieros Industriales, Avda Reina Mercedes, s/n, 41012~Sevilla, Spain

(Received 27 Muy 1986)

Ahstrsct-The classic Kirchhoff plate theory is formulated through integral representation decomposing the biharrnonic field equation in two coupled harmonic equations. Two domain integrals appear, one inherent to the problem and the other due to the previous decomposition. Both of them are evaluated defining equivalent boundary integrations, using collocation method in several points on the boundary and the domain. These equivalent integrations are the same as those involved in the integral representation of the Poisson equation, therefore no extra integrations are needed.

1. lNTRODUCTlON

The problem of applying the Boundary Element Method to the analysis of thin elastic plates is not completely closed. Referring to the direct formu- lations published to date, ail of them present diffi- culties or limitations.

The formulation presented by Bezine [I], Stern [2] and Costa and Brebbia [3], defining an integral equa- tion through the use of a reciprocity theorem over the biharmonic equation and obtaining a second one by directional derivative of the first, would be impecca- ble, if a divergent integral were not involved. The main difference between the works of the authors referred to, is the way of evaluating this integral. A great effort is being made in this field and in the near future the problem might be solved.

Katayama et af. [4] tried another way making use of an existing analogy between the plates bending problem and the plane stress problem. Simply sup- ported plates cannot be considered with this alterna- tive and the domain integrations which appear are evaluted by dividing the domain into cells.

Anyway, none of the approaches described deal either with the case of a general load not analytically convertible in boundary loading, nor with the case of a load given in a discrete form but using cells.

Paris and de Le6nf5J formulated the problem of simply supported plates, decomposing the bihar- manic equation into two harmonic and, for such plates, uncoupled equations. With this alternative, no divergent integrals appear because the integral representation of the plate bending problem is re- duced to that of two Poisson equations. Due to the uncoupling, the boundary conditions can be directly applied.

In this work, a general treatment for solving any combination of boundary conditions and loading system without employing cells is presented. The same decomposition as in [5] is used, and the domain integrations which appear due to the coupling of both

harmonic equations and to a general load are evalu- ated in an approximate way through equivalent boundary integrations. This is done following the ideas of Brebbia and Nardini [6] dealing with the elastodynamic study of continuum using B.E.M.

In a case where boundary conditions cannot be applied directly, the variables involved in the formu- lation are related to the physical magnitudes of the problem, using the approximation made of this along the boundary.

2. DEFINiWON OF THE PROBLEM. INTEGRAL REPRESENTATION

Considering the case of a plate of general shape and loading as shown in Fig. I.

The plate can be supporied along andfor inside the domain. The field equation corresponding to the Kirchhoff theory is:

DV4 W(X,Y) =P(x,Y)> (1)

where D represents the flexure stiffness, w the trans- versal displacement of the plate and p represents the transversal loading.

Other design variables of the problem are the slopes, the moments and the shear forces. All of them can be related to the deflection:

4: = w.z (2)

MT8 = - D [VW., &, + (1 - v> w.zpl (3)

Q,= - Dw:~,. (4)

The usual situations along an edge, in a local coordinate system are:

simply supported edge A4, = aa; w = @ (5)

clamped edge q;=Q;;w=* (6)

free edge g=R,;M*=lii,, (7)

where &, is the equivalent shear force used in the

225

226 F. PARTS and S. DE LEON

A -x 3. EVALL’ATlON OF THE DOMAIN INTEGRALS

The domain integrals that appear in eqns (I?) and (13) have the general form:

I = IL(P, R)g(R) dA(R);

RED.PED or dD. (14)

I

Fig. 1. The plate

Kirchhoff model.

K.=Q.+f$

Using an intermediate variable, M,

M= 2 (v = Poisson’s ratio), (9) C(J’)5,(P)= i,(Ql%$$d,,QJ I 20

+ the field equation (1) can be alternatively written in the following form:

Supposing that g(R) admits a representation in the form:

g(R)=r,J;(R) j= I,.... L, (15)

where the functions 4 associated to L points J represent a general system of coordinates and CZ, are the defining coefficients of g in such a system.

If c, is a general function related to L through

(8) VZ5,=1;. (16)

then, applying again the second Green formula to eqn

(t6),

V2M = -p

These expressions admit an integral representation

(10)

using the second Green theorem:

(11) v2w = _;.

C(P) M(P) + s

M(Q) “(” ‘) h(Q) - L1D dn

= $QW',QMQ)

C(P),,‘(P) + s w(Q)a"P' ‘) ds(Q) ?D an

= I ~D~(QNW'~Q)d~(Q)

+; M(B)I//(P,fGdAW, (13)

where P and Q are points belonging to the boundary, whereas B belongs to the domain. I// is the funda- mental solution of Laplace equation and C(P) is a term whose value depends on the local shape of the boundary at point P.

If the values of w or M are required inside the domain, eqns (12) and (I 3) can be used, taking

found, then it is possible to evaluate eqn (14) through boundary integrals

- I ~(R)$U’, R)dA(R). (17) D

And, if a particular solution of eqn (16) can be

Usually it is not possible to define zj so that eqn (15) could be satisfied at every point. Therefore, a criterion must be established, in order to determine the values of CL, to approximate g with known 1; functions, generally polynomials. In all the examples presented in this work,

I;= I -f(J,Z) (19)

5,=ti?(J,Z)-~i’(J,Z), (20) : being the distance between the point J, which the function L is associated to, and a general point Z. This distance has been taken in a dimensionless form, dividing it by a characteristic length of the plate, in order to avoid numerical problems.

When g(R) has a very simple expression (only possible in eqn (12): punctual, constant or ramp load), a more direct system to evaluate the integral can be used [7].

C(P) = I.

Variables such as Q, and M,R also admit an integral 4. GENERATION OF THE SYSTEM OF EQUATIONS

representation [5]. In this way, the values of these Independently of the way to evaluate the domain variable inside the domain can be computed once the integration, an approximation on the variables ‘it’, values of the variables along the boundary have been &/dn, M and aMIan which appear in the boundary found, solving eqns (12) and (13). integrals of eqns (12) and (13) must be defined.

Boundary Element Method in the analysis oi thin plates 217

Assuming the boundary as a set of N rectilinear elements, a linear evolution of the variables using its values at the extreme points is defined over the elements. Details of the process can be found in [8].

In this way, the integral expressions (12) and (13) give rise to the next linear system of equations:

IHI fW -PI W’t = (01 (21)

WI lw) -IGl tq”I = {El, (22)

where {w}, {qw}, {M} and {q”‘} are the values of the variables at the N points of the boundary used in the approximation. [H] and [G] are matrices of known coefficients, once the integrations defined in eqns (12) and (I 3) have been performed.

To evaluate the domain integration vectors ff)) and {E}, the expression (18) is going to be used. Because of the similarities between the boundary integrals that appear in eqn (18) and those existing in eqns (12) and (13), fo) and {E) can be expressed in the following way:

to) = -([HIEI-[Gl{q’f)I~~ (23)

{El = -(~~1~5~-~~11~‘~)~8~~ (24)

where {<} and fqtj are (N x L) matrices, L being again the number of points where approximating functions 4 are defined.

The coefficients of these matrices are therefore immediately obtained particularizing the L, tj, and corresponding (I;‘, functions for the N points on the boundary.

The L coefficients of the vectors fz) and (p) are obtained by collocation, fixing the values of the load p and intermediate variable M in L points.

b~=vI{~~ (25)

tM*I = Fl is 1. 6351

In order to simplify the computations the same L points have of course been used in both cases. However, whereas (cc) can be directly determined as the load is known, it is not possible to evaluate {/?I because M is an intermediate and unknown variable. Besides, it is necessary to place some of the L points inside the domain so as to obtain a correct represent- ation of the function M. These Lipoints represent LZ extra unknowns. The remaining LB( = L - LI) points do not supply new unknowns because they belong to the boundary. So as to provide the values of M in these LI points, M’, integral equations like eqn (12) [with C(P) = I] can be applied in each one of these points. After the approximation, the follow- ing set of equations appear:

WI {MI -WI iq”‘) + {M’I

= -(i~7rr1-~c~l14q)Ir}I (27)

where, simitarly to eqns (21) and (22), [Hq and [G’] are matrices of [LI x N] known coefficients, once the integrations defined in eqn (12) have been per- formed, in this case from the tl points.

Therefore, the vector f/I> adopts the form

M {/j} = [F]-’

I i

. *. , . * . . . . (28)

M’(M, q”)

In this way {fit is related to the existing variables {M) and fq”‘) along the boundary.

So, substituting eqns (23) and (24) into cqns (21) and (22) and taking into consideration eqns (75) (26). (27) and (28) a final system of ZN equations with unknowns just along the boundary can be estab- lished. In the next paragraph the way to apply the boundary conditions to have only 2N unknowns will be dealt with.

Although it is not especially outstanding LB does not need to be equal to N. In most cases it could be less to simplify the calculations.

5. APPLICATION OF BOUNDARY CONIXTIONS

The variables involved in the presented formu- lation are w, q;, M and qf whereas those appearing in the boundary conditions (5). (6) and (7) are w, q;, MS and K,.

There is no problem in the case of rectilinear simply supported (M = M,) and clamped edge. In other cases supplementary equations to relate both sets of variables are needed. With the help of Frenet formulas for the case of a curve boundary, these relations are:

M-M,-D(t -~)lf~+~,~i (29)

dM a* dw 1 -=K,+D(l -v)- -f-W . an I I as2 l?tl p

(30)

In this way, for a point belonging to a free edge the variables involved in the fo~ulation are related to the data En and R, and a reasonable hypothesis can be established to approximate the evolutions along s. For instance, for the case of a rectilinear boundarv and equal length of the elements, A, the preceding expressions at a point P, would remain:

M(P) = i&(P) - L)(l - v)

X w(P+ I)-2w(P)+rr(P- I)

A* (31)

4.” (P) = R”(P) + D(1 - v)

X q~(P+l)-2q~(~)~q~~P-l)

AZ . (32)

Therefore, only two unknowns IV and q; remain associated to a point P belonging to a free edge.

6. NUMERICAL RESL‘LTS

Three examples are going to be considered in order to check the more relevant aspects of the theory developed.

228 F. PARis and .% DE LEON

0 = 2m

n= Oim

E : 2.1 x IO” kg/m2

v = 0.3

q : 5 x IO’ kg/m2

I Y I I /

Fig. 2. Free, clamped and doubly simply supported square plate.

6. I. Un$ormly loaded, square plate with IWO opposite simpIy supported edges, one clamped edge and the last one free (Fig. 2)

This problem is especially suitable to verify the application of boundary conditions and also the assumption made at the corners to avoid the presence of extra unknowns.

Figures 3, 4 and 5 show the evolution of several

Fig. 3. Transversal displacement, W, and shear transversal force, Q,, along the free edge.

variables along the boundary or the domain. The results are compared with the analytica solution of the problem, which can be found in [9] and, when it is possible, those of other numerical alternatives. More numerical results obtained with this method can be found in [lo].

The results obtained for w and Q, present a similar

accuracy except at the corner, although the latter correspond to a variable not directly managed in the solution of the system of equations. Two reasons contribute to the error occurring at the corner. The first is the use of a backward instead of central, finite difference expression. The second is the use of a plane to relate superabundant variables at the corner [S]. The accuracy, in both cases, should improve with a finer discretization just around the corner. This fact can be checked against the improvement reached with the second discretization used.

The shear normal force Q,, shown in Fig. 4, coincides with the equivalent Kirchhoff shear force K,, because IV,, = 0. However, in general, the formu- lation allows for evaluation in a separate form Q, and M,,,. Of course, K, can be also obtained, if it is desired.

Finally, Fig. 5 shows excellent results inside the domain, as is inherent to the B.E.M., when an integral expression is used, even in the case of coarse discretizations of the boundary.

In all cases the results have been obtained with two

A 16 elements / 9 points

+ 32 elements/49 points

- Analytical

0.x)6

Boundary Element Method in the analysis of thin plates 229

0.50

16 e&n&s I9 points

32 etements 149 point.5

n Costa and Brebbta f3]

-- Moody 11 II

- &notyticot

0 I I

?I

I I 2x 0.25 0.50 073 I.00 Ti

A

Fig. 4. Shear force, Q., along the clamped edge.

di~creti~tions of the boundary (16 and 32 etements) are needed, because this number governs a series of and two numbers of internal points (9 and 49 points). computations whose amount can represent an im- One important question that arises from the formu- portant drawback of the method. Although this lation presented is the number of internal points that number of points depends on the nature of the

Fig. 5. Bending moments along the symmetry axis.

v a

230 F. PAR& and S. DE LE&

Number of internal points used (NIPU 1

-3 -

Fig, 6. Relative error on the transversaf displacement along the free edge.

function to be approximated, as a general rule it can be said that the number and situation of these must avoid any local rigidity of the function, Of course, when the function has a smooth evolution, this rule leads to a uniform distribution of the points accord- ing to the number of nodes placed along the bound- ary. This has been done in the two cases whose results have been presented.

However, with the aim of having a more quan- titative idea of the influence mentioned, two error curves varying the number of internal points that are going to be analysed (Figs 6 and 7). In both cases the error represented adopts the form:

s, - SO” error = ___ x 100, S 0”

where S, corresponds to the found value of the variable and S,, corresponds to the analytical value. Thirty-two elements always have been used to model the boundary.

For the displacements, even when no internal point is included the error is quasi-constant, remaining lower than 2%. Instead, the relative error for the bending moment increases monotonously when the number of points diminishes. This different behav- iour can be understood by considering the influence

Error %

Ly”l -80 !-

of the domain integral to be evaluated, eqn (I 3) in the integral equation of both points. Whereas in the case of the free edge, the boundary integral has a significant value because displacements and slopes are going to appear, in the case of a clamped edge only the domain integration remains significant. This fact justifies the higher sensibility of the results along this edge in relation to the number of internal points used to evaluate the domain integration.

Anyway, the more refevant result of both curves is that an error lower than IO%, in any case, can be reached also with the use of a small number of points (between 9 and 16 in the case shown). This fact proves that only a reduced extra workload, from a computational point of view, needs be done to evaluate the domain integrations with the proposed alternative.

6.2. Circrtlar clamped plate Grh central free hole and ~~i~~rrn~~ loaded

This example is included to take into consideratjon the conditions (29) and (30) where the curvature appears. The problem is defined in Fig. 8 whereas Fig. 9 shows the evolution of bending moments along the radius, Finally, Table 1 gives a summary of the more relevant values of the solution.

M, (x 8 0.25 a) ,

0 I 5 9 16 25 36 49

NIPU

Fig. 7. Relative error on the normal bending moment along the clamped edge.

Boundary Element Method in the analysis of thin plates 231

Fig. 8. Definition of the problem.

6.3. Clomped square pfate with four intemal rigid supports and ioaded with a concentrated load at the ten ter

Internal supports can easily be taken into consid- eration. In this case the second member of eqn (1) remains

P(KY)- T x,, I-1

where X, is the value of the reaction in each of the NA

existing supports. Due to this modification, in the second member of

eqn (12) a new term appears:

- ,?, WA,) j- Ilt (J’, A,) d.U,), _ 4

where D, represents the surface of the support having considered the reaction uniformly distributed over it.

The system of equations must now include NA new unknowns, the NA reactions. But NA new equations can be added, applying eqn (13) to the NA supports whose displacements are nil because they are consid- ered rigid. Fig. IO. Description of the plate with internal supports.

f/O I I I

+ 1.6

Table 1. Maximum values of K, M, and Q,

P4 W,,, = z--j Eh

&a, = Bqb ’ Q,, = ~9

z B ;:

Analytical 0.0575 0.480 0.750 B.E.M. 0.0585 0.490 0.734

Figure IO represents the problem considered with four internal supports. This problem has been solved by Bezine [ 121 with a mixed formulation of boundary and domain.

Figures 11 and 12 represent the evolution of the transversal displacement and bending moment.

1. CONCLUSIONS

A formulation dealing with the plates bending problem using B.E.M. has been presented. No div-

Y : 0.3 0 = 3.5Ch-n

U’O25rn

‘I 1014

32 elements/ 48 points

Katayama 14 ] 48 elements 1200 cells

AnalytIcal

Fig. 9. Bending moments along the radius.

F. PARis and S. DE LEON

04 c -- Bezlne [IZI

--f-- 2 etements /49 pants

2x/a

Fig. 1 I. Transversal displacement along the horizontal line of symmetry and the diagonal.

ergent integral appears and the discretization of the domain in the sense of using cells has been avoided. However, the use of some internal points to evaluate the domain integration is necessary. The influence of the number and situation of these points is not of primary importance, and therefore an automated random generation of these points could be consid- ered. In any case, it is obvious that the necessity of including these points means a reduction in the effectiveness of the method.

In this sense, it has to be recognised, from a conceptual point of view, that the problem of plates is not entirely adequate to be approachable by a boundary formulation. This is due to the fact that in most cases the load is applied in the domain by the very nature of the plates. This situation could be compared to plasticity and dynamic problems where domain integrations are inherent to the problem. In all these problems some gadgets have to be included to evaluate the domain integration leading to a

penalization of the time computation compared with the alternative domain method. In the formulation presented here, the main advantage of the B.E.M. has at least been maintained, which is the need to dis- cretize only the boundary. Even in the case of a general and complicated load the user has just to give the value of this load in several representative internal points. It is impossible to reduce the data input for such a domain problem.

Finally, the formulation presented is flexible enough to admit the presence of particular external effects such as temperature, elastic foundation and edges supported by beams [I.

REFERENCES

I. G. P. Bezine, Boundary integral formulation for plate flexure with an arbitrary boundary conditions. Mech. Res. comm. 5, 197-206 (1978).

2. M. Stern, A general boundary integral formulation for the numerical solution of plate bending problems. In!. J. Solids Struct. 15, 769-782 (1979).

-Q- 64 etements 149 points

Fig. 12. Bending moment along the edge.

Boundary Element Method in the analysis of thin pfates

3. J. A. Costa and C. A. Brebbia, Plate bending problems usine B.E.M. In VI Intern. Conf. on Boundary Element

M&x& (Edited by C. A. fjrebbia), pp.- 343-363. Springer, Berlin (1984).

4. T. Katayama, T. Sekiya and H. Tai, Bending analysis of perforated plates with unsupported or clamped edges bv B.E.M. In V Intern. Conf. on Boundary Element

Methods (Edited by C. A. B;ebbia, T. Futagami and M. Tanaka), pp. 517-526 (1983).

5. F. Paris and S. de Le6n. Simply supported plates by the boundary integral equations method. Int. 1. Numer. Merh. Engng 23, 173-191 (1986).

6. C. A. Brebbia and D. Nardini, Dynamic analysis in solid mechanics by an alternative boundary element Drocedure. Soil Dynamic Earthquake Engng 3, 228-233

[1983). 7. S. Lebn, A new integral formulation of the plates

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Plates and Shells. McGraw-Hill, New York (1959). F. Paris and S. Le6n, An alternative analysis of thin elastic plates with any boundary conditions. using B.E.M. VII inrernarional Conference on Boundary

Element Methods (Edited by c. A. Brebbia and G. Maier), DD. 4-17-4-28. Surinner, Berlin 119851. __ N. 1. Moody, Moments‘andre&tions for rectangular plates. US Department of Interior, Bureau of Reclama- tion, Engineering Monograph 27 (1960). G. Bezine, A boundary integral equation method for plate flexure with conditions inside the domain. Inr. J. Numer. Merh. Engng 17, 1647-1657 (1981).