Boundary integral formulation of plate bending problems

MECHANICS RESEARCH COMMUNICATIONS

0093-6413/84 $3.00 + .00

Vol. 11(4),245-251, 1984. Printed in the USA

Copyright (c) 1984 Pergamon Press Ltd.

BOUNDARY INTEGRAL FORMULATION OF PLATE BENDING PROBLEMS N. Kamiya and Y. Sawaki Department of Mechanical Engineering, Mie University Kamihamacho, Tsu 514 Japan

(Received 26 March 1984; accepted for p~nt 20 April 1984) Introduction The field equation governing deflection of thin elastic plates of constant thickness under the conventional assumption after Kirchhoff and Love is represented by the two-dimensional inhomogeneous biharmonic differential equation. The inhomogeneous term occurs from specified distribution of lateral load. The above-mentioned biharmonic differential equation together with the associated boundary conditions can be formulated as integral equations which are conveniently employed to analyze numerically the elastic bending of arbitrary boundary conditions using the so-called Boundary Element Method. The boundary element method or more generally the integral equation method, the authors believe, is highly convenient and elegant when the formulation can be achieved on the boundary alone. In general, such smart formulation is possible only for linear, homogeneous differential equations, and therefore the integral equation ruling the plate bending problem involves one domain integral of the inhomogeneous load term. The present study concerns an attempt at transformation of the domain integral into the corresponding boundary integral. The method relies on the well-known integral theorems due to Gauss or Green and is mathematically rigorous. Distributed lateral loads expressed in terms of a power series of the coordinates including constant, linear and quadratic variations are illustrated. Since higher order variations can be treated similarly, problems with arbitrary load distribution are formulated as the perfect "boundary" integral equation. Consequently, discretization and numerical calculation over the domain are wiped out thoroughly and utility of the boundary element method for the plate bending analysis is emphasized in the extreme. 245

246

N. KAMIYA and Y, SAWAKI

Integral Equation Consider

Formulation

a thin elastic plate of arbitrary profile

by a smooth curve ?S

(FIG. 1).

surrounded

S and n denote the domain

occupied by the plate and the unit outward normal on the boundary. Let the plate be supported appropriately rigid body displacements. of thin elastic plates, following biharmonic

to prevent

trivial

Obeying the Kirchhoff-Love

assumption

deflection w of the plate satisfies

differential

the

equation:

V4w = p/D

(l)

where p and D stand for lateral bending rigidity of the plate. the inhomogeneous

load distribution

and constant

The right hand side of Eq.

term in the differential

equation,

(I) is

which is

denoted by ¢ in what follows. The boundary-value equation

problem constructed by the differential

(i) and related boundary conditions

the following

two integral

equations

is transformed

for the smooth boundary

1 I ~S {K[v(X, Y)]w(Y) 1 w(X) = I klJ(x)v(x, X)dS - ~2 ~ S . . . . . ~V

Unit Outward Normal: 8

-_

oi

x2'Y2

~ k

//~ D°main: S / ~ B o u n d a r y

Boundary:

0

~S

m- Xl , Yl

FIG. 1 Notations

Point: X

into [1-3]

BOUNDARY INTEGRALS FOR PLATE BENDING

~ v(x, X)dS - ~ ~xfskb(x ) i f s{3 x[v(X, Y)]w(Y)

~-q(X) =

1

~

~

shear

represented

~nxBn (X, Y)M[w(Y)]

~

w h e r e q, M a n d K a r e effective

247

slope,

force

(X, Y)K[w(Y)]}ds

normal bending

on t h e b o u n d a r y

moment and K i r c h h o f f

respectively,

which are

as follows;

q = ~3w

(4) r-232~. _2 3 2

M : - D [ V 2 - ( I - ~ ) ~n2T~- ~

32

2nln2~2)]

nz~-~-

(n~

~ ~- -~-~) ~ K = D{(1-V)~--~ [nln2(~-~-

In the above equations

~ denotes

term related to discontinuity to Eqs.

(3)

(2) and

-

(5)

n~)~_~._, ox,ox22]

Poisson ratio.

v~± 3n }

(6)

Each additional

of twisting moment must be added

(3) for the nonsmooth boundary

function v(x, y) appearing

_

in Eqs.

(2) and

[2, 3].

The

(3) is the fundamental

solution to the original biharmonic equation

v(~,

y)

1 r21og r

= - 8-9

where r represents

(~ ar21og r, a =

(7)

1

- ~-9 )

the distance between two arbitrary points x

and

It has been known that, are formulated

since the integral equations

(2) and

(3)

in terms of the four boundary quantities w, q, M

and K, two of them are specified by the boundary condition and the remaining

t~o are unknown, most bending problems

solved systematically One may mention,

using Eqs.

however,

(2) and

can be

(3).

that besides boundary discretization

the domain S occupied by the plate must be discretized

into

248

N. KAMIYA and Y. SAWAKI

a set of appropriate

internal

domain integrals sides of Eqs.

appearing

(2) and

on the boundary discretization efficient Complete

If the such integrals

integrals,

numerical

any more and becomes

Boundary

Integral

easier,

can be

computation based

element method does not require

from the practical

The following

in order to compute the two

in each first term of the right hand

(3).

replaced by the boundary

cells

tedius

domain

simpler and more

point of view.

Formulation

two domain integral

remain

in Eqs.

(2) and

(3):

Bl(X ) = [ ',[J(x)v(x, X) dS

~r B2(X) = ~-~ / ~ ( x ) v ( x , S

B2(X)

(9

(lO

X) dS

~

can be estimated

simply through differentiation

on the specified point X on the boundary, enough to consider only BI(X) plate bending problems, of lateral

load,

that is, concentrated

For a finite number of concentrated expressed

and therefore

for the present.

we conventionally

of BI(X)

As practical

encounter

two types

and distributed

lateral

it is

loads, ~(x)

loads. is

as m

~(x)

=

£ qJ(zJ)6(x j=l

where m concentrated at x = z j.

(11)

- zj)

loads of each intensity ~(z j) are acting

Then Eq.

(9) becomes

m

B~(X) = z

BI(X)

and B2(X)

(12)

vo(zJ)v(z j, x)

j-=1

~

~

can be computed without performing

any integral

calculations. As the second problem, lateral

load which

we consider

is thought

arbitrarily

distributed

to be represented by a power series

BOUNDARY INTEGRALS FOR PLATE BENDING

of the coordinates.

In order

procedure,

an inhomogeneous

we suppose

up to q u a d r a t i c

to explain

249

the t r a n s f o r m a t i o n

term r e p r e s e n t e d

by

terms:

~b(x)~ = ~o + @ k X k + 4~£XkX£

(13)

where 40, 4k,1 4kz2 ( k , £ = 1,2) are constant. For t h e r e p e a t e d i n d i c e s , we obey t h e c o n v e n t i o n a l summation c o n v e n t i o n r u l e within the two-dimensional space. In t h e p r e s e n t c a s e , t h e domain i n t e g r a l BI(X) becomes

(14)

B,(X) = a I [40 + ~ x k + 4~LXkX~)r21°g r ds ~

With

respect

following

S

to the function

identities

hold

r21og

r, we notice

in the t w o - d i m e n s i o n a l

that

the

space:

2r ~ i r21og r = V T6 (log r - ~ ) r ~ = V23~

3•

r~log r = V 2

Applying of Eq.

the second (15),

Green

+

a

frj

identity

[

(log r -

{[40

J 3S r

first

(log r - g1 )

(14)

in c o n s i d e r a t i o n

to give

)V2(4 ° + 4~Xkl + 4~£XkX£)dS

z 'h 2 ~ X ) ~ r~ 1 + 4kXk + ~kZ~k ~ Tff [7g ( l o g r - ~ )]

(log

r -

term on the right

transformed,

to Eq.

we have

BI(X)~ = a

The

(15)

finally,

hand

[40 +

side

of Eq.

(16) (16) can be further

250

N. KAMIYA and Y. SAWAKI

BI(X- ) = f 3S (@0T0 + *kTk 1 1 + *~£T~£) ds

(17)

where 3F1(r) T° = a _ ~n

~Fl(r) Tki = a[Xk

nkF i (r)]

~n

2 = a[XkX ~ Tk£

8FI (r) 3n

(Xznk + Xknz)F i (r) + 6k£

r~ Fl(r)

(18)

~F2 (r) ] ~n

1

= i-6 (log

r - ~ )

(19) _

F2(r)

Equation

(17)

r 6

5

288 (log r - ~- )

is f o r m u l a t e d

as a consequence,

two

as complete

integral

equations

system of s i m u l t a n e o u s

"boundary

this

formulation,

complete

for n u m e r i c a l order

terms

term,

similar

boundary

calculation

are i n c l u d e d

" bo un da ry " (2) and

" integral

procedure

equations.

and, a

Using

discretization

Even though higher

in the e x p r e s s i o n

replacement

(3) become

only b o u n d a r y

is sufficient.

integral

of the i n h o m o g e n e o u s

is available

repeatedly.

Application As an i l l u s t r a t i o n

of the b o u n d a r y

thorough

integral

problem plate

boundary

of thin elastic

subjected

linearly

element

formulation

plates, and

using

for the linear

we consider

to the d i s t r i b u t e d

in the x I d i r e c t i o n

analysis

lateral

is u n i f o r m

a clamped

the

bending elliptic

load w hi ch varies in the x 2

direction. Figure axis

2 shows

distributions

of the ellipse.

of the d e f l e c t i o n

The minor

along

axis does not yi el d

the major deflection

BOUNDARY INTEGRALS FOR PLATE BENDING

in this example.

251

Small circles obtained by 48 piecewise

constant boundary elements agree well with the existing rigorous analytical solution

[4].

We emphasize again that the bending analysis of constant thickness elastic thin plates can be proceeded successfully by only boundary discretization with the help of the boundary element method.

(× lO-3 ) 5 BEM

o

4

Analytical

3 C~

1 I

I

I

I

I

I

0,5

1.0

1,5

Xl/b

FIG. 2 Clamped ellipse ( a/b = 3/2 ) References I. S. Bergman and M. Schiffer, Kernel Functions and Elliptic Differential Equations in Mathematical Physics, Academic Press, New York (1953). 2. G. P. Bezine, Mech. Res. Comm., S, 197 (1978). 3. M. Stern, Int. J. Solids Struct., 15, 769 (1979). 4. S. P. Timoshenko and S. Woinowsky-K-{ieger, Theory of Plates and Shells, 2nd ed., McGraw-Hill, New York, Toronto London (1959).