Thermal flexure of an infinite plate with a doubly periodic array of holes

Nuclear Engineering and Design 53 (1979) 71-78 © North-Holland Publishing Company

THERMAL FLEXURE OF AN INFINITE PLATE WITH A DOUBLY PERIODIC ARRAY OF HOLES K. YAMADA Tokushima University, Tokushima, Japan and Y. TAKEUTI Department of Mechanical Engineering, University of Osaka Prefecture, Mozu, Sakai 591, Japan Received 23 September 1978

A theoretical method is presented for bending problems of perforated plates. The method is illustrated by giving the solution for an infinite plate with a doubly periodic set of circular holes having a square or triangular pattern under a uniform temperature difference between upper and lower faces of the plate within the framework of the Poisson-Kirchhoff theory of thin plates. Numerical results are given for the bending moment around holes and curves of stress concentration factors. The results show the power and flexibility of the technique. The solution obtained here can be used, just as it is, for a plate with holes of arbitrary shape and array. Also the extension of the method to a plate under a class of loads other than thermal load, e.g. uniform bending or twisting moment, is quite straightforward.

1. Introduction

intended to obtain the elastic constants of a fictitious solid plate which behaves in the same manner as perforated plate, and then plate bending theory can be used to calculate the overall deflection of this fictitious plate. Therefore, in this paper, the bending problem is solved using the Poisson-Kirchhoff theory of plates for a plate completely perforated with circular holes forming a square or triangular pattern under a uniform temperature gradient through thickness. The method of analysis is based on the fictitious load method in earlier work [3] on the plane stress in the plate under consideration. The only difference between the plane stress and bending problems lies in the fact that the fictitious loads, to be applied inside the hole and to be determined so as to satisfy the boundary conditions, are acting in the plane of a plate for plane stress and are acting perpendicularly to it for bending problems. In this paper, an improved and more suitable form for the load distr~ution is used to the bending problems. The technique, together with the least squares method, is effective, as in the previous work [3], for holes of arbitrary shape and array. By using a Fourier

The problem of strength and deformations for an infinite plate with a doubly periodic array of holes is of great importance to the design of structural elements in nuclear power plants or heat exchangers and also to the mathematical theory of elasticity. Thus many investigations related to the problem have been carried out by several authors so far. Among examples solving the problem with the two-dimensional theory of elasticity, the solutions of Saito [1 ] and Bailey and Hicks [2] are typical ones, and recently the present authors [3] also attacked the problem anew. Contrary to our knowledge, the solutions of bending for such plate, based on the bending theory of plates, are not to be found in the literature. As for the theoretical investigations related intensely to the problem, the only solutions that have been obtained are by Tamate [4] for an infinite row of holes and by Yamashida [5] for two infinite parallel rows of holes. The usual approach presented so far to this problem, as represented by the works of Malkin [6] and Horvay [7], has been only the approximate analytical one 71

K. Yamada, Y. Takeuti / Thermal flexure o f an infinite plate

72

expansion of the fictitious load having a form compatible with the symmetry of the given applied load, the method can be applied straight-forwardly to a plate under the twisting moment or shearing force acting on the edges of the plate.

2. Analysis

As shown in figs. l(a) and (b), let us consider a perforated plate having a square or triangular pattern of

circular holes of radius R with periods 2a and 2b in the x and y directions, respectively. Furthermore, it is assumed that the positive direction and notations of the bending and twisting moments, and shearing force per unit length of arc in the plate follow from those in the text of Timoshenko [8]. In the Poisson-Kirchhoff theory of thin plates, the transverse deflection w o under a linear temperature gradient through thickness is determined as a particular solution of the equation V 2 [V2Wo + (I + v)ar] = 0,

( 7 2 = ~}2/~X2 + 02/~)y2),

(1)

0 0 0

2bI

0 a

2a

where v is Poisson's ratio, ct is the coefficient of linear thermal expansion and r is linear temperature gradient through thickness. The temperature distribution in a thin plate is expressible in the form

T(x, y, z) = To(X, y) + zr(x, y),

where the first term To(x, y) relates to a plane stress problem and the second term zr(x, y) relates to a bending problem. If the deflection is small, the two problems are separable. Therefore, it can be permissible to take only the second term here. Now, if it is assumed that the temperature gradient through the thickness is uniform in the plane of the plate, i.e.

r(x, y) = ro = const.,

(3)

then as a particular solution of eq. (1), in which eq, (3) is substituted, the following form is taken, as is usually done: w0 = 0 .

l

(2)

(4)

Then the bending moment, twisting moment and shearing force in the plate are

gxo =My 0 = - D ( 1 + v)Otro , Mxyo = Qxo = Qyo = 0 ,

Fig. 1. Infinite plate with a doubly periodic array of holes. (a) Rectangular array; (b) triangular array.

(5)

where D = Eh3/12(1 - v 2) is the flexual rigidity, E is Young's modulus and h is the plate thickness. The particular solution, eq. (5), may not satisfy the boundary conditions. To satisfy the boundary conditions, we may use the same technique as in our previous paper [3] on plane stress in the plate under consideration; first we would apply surface load p with

K. Yamada, Y. Takeuti / Thermalfiexure of an infinite plate m=l

2

3

h

X= 1 , 2 , 3 .... ;

j

o t.-o+u+u+,,--I -V

n=l

+

+ v + v + v

2 3

t, v ._L

-

+

--

-

+

-

+

+

--

+

--

+

-

--

+

X

73

/~= 1 , 2 , 3 . . . . .

where M

4v ~

N

~j

M

4Um~__1

Bu =--

7ra

16

Pmn

[m•nu\

.=,'"T

)tan

XTru

,

N

e m pmn z(nlarro I t a n -IdnO

~

= n=t M

/a

\ 2b ]

2b '

N

Ck/a=~._m~_ 1 n=l ~ Pmn Z ----~-

Xrru z(nlaTrv~ #rrv × tan-2a-a \ 2b ! tan 2---b-'

in which the following abridged notations have been introduced:

Y Fig, 2. Geometry of areas of load.

='~0 unknown intensity on areas set up appropriately inside the holes and seek the bending moment, twisting moment and shearing force due to the surface load. Then, we superpose these solutions on the particular solutions, eq. (5), and finally determine the intensity of the load p so as to satisfy the boundary conditions along holes. The surface load p to be applied on the area inside the holes, under the symmetric condition in the state given by eq. (5), must be symmetric with respect to the x and y axes, i.e.

p(x,y)=p(-x,y)=p(x, -y)=p(-x, -y).

p=

oo

Xnx

= oo

Axcos--

a

+~B

u=l

#Try

ucos-

h~rx

/arty cos-b

(1

m o r n : even, m or n: o d d ,

(sin2(') Z(")

m or n: even,

= l(cos2(')

m or n: o d d .

In the case of the triangular array, the Fourier series o f the surface load Pmn can be obtained in the following way. Referring to fig. l(b), first, if the surface load •Pmn is applied only in the holes with periods 2a and 2 b in the x - y coordinate system,p has, from eq. (7), the following form in a Fourier series: oo

Pl =

= oo

+

oo

A h cos

Xrrx a

+~

/.t=l

/any Bu cosb

oo

Z)

h=l ~=1

X= 1 , 2 , 3 .... ;

hrrx

cos--

t/

#Try

cos-

b '

ta = 1,2, 3, ....

Pz =

=

Ax cos

Xlrs a

+ ~ B u cos /~=1

'

(7)

/arrt b

0o

Xzrs /arrt +~ ~ C~ cos-- cos--, X=l u=l a b

b

(9)

Next, if the surface load Prnn is applied only in the holes with periods 2a and 2b in the s - t coordinate system, it must have the following form in a Fourier series:

oo

oo

+ ~ ~ Cxu c o s - h=l U=I a

~m , Cn

(6)

Regarding the form of distribution of the surface load p, we may assume the following: as shown in fig. 2, inside the holes we set up rectangular areas the side lengths of which are u and o, and apply uniform pressure Pmn on each of the areas alternately in direction from area to area. Taking account of the symmetric condition eq. (6), and taking the number o f areas of load to be M X N (m = 1,2 ..... M; n = 1,2, ...,N), the surface load Pmn can be, in the case of the rectangular array shown in fig. l(a), expanded into a double Fourier series in the following form: oo

(8)

X = 1 , 2 , 3 .... ;

#=1,2,3 .....

0o)

K. Yamada, Y. Takeuti / Thermal flexure of an infinite plate

74

where, A ~, B u and Cxu are given by eq. (8). On the other hand, since s =x - a, t =y - b, eq. (10) assumes the following form:

a•

P2 =

=

(-1)XAx cos

go

X~rx a

Substituting eq. (15)into eq. (14), the bending and twisting moments, and shearing force, can be obtained in the following form: 16b 2 ~ , ,

grry b

+ ~ (-1)UB u cos - ~u=l

N

7r4 ~'L~=I~ =

Pmnmx(x, y),

n=l

oo

go

Xnx

~ (-1)~'+ucxu cos h=~ ~=~ a

+ ~

gTry

cos-

(11)

b

zrv~

1

mx(x,y) = e. ~ h = l

(mMru I

Xnu

Mrx

/kO-~--~Z \ - 2- a ] tan-~-a cos -a

go

,, -,, v

Therefore, finally the Fourier series for p assumes the following form:

+ em -4 a co

P =P~ +P2

x =2

go

+2 ~

Ax cos-

+2

a

~ =2

X=l u=l

tan/mY cos

..y

2b

(O2+ug 2) ( m ~ u ) .

b

Xnu [ngzrv, tan-~--z[-~)

b

gnu Mrx gTry × t a n ~ - cos cos-a b '

Mrx /any cos-a b '

X = 2 , 4 , 6 .... ; g = 2 , 4 , 6 ....

\ 2b ]

= u =, Xg(p2 + g 2 ) 2 Z - -

E B u c o s"+Y --

go

~ Cxu cos

oo

+x~=l=

e ,o

=2 E

1 u=l ~ ~

for single series,

p = hb/a,

X = 1,2, 3 .... ;

g = 1,2, 3 .... (and so on), (16)

(X = 1,3,5 .... ) + ( X = 2 , 4 , 6 .... ) for double series . g = 1,3,5, ~g=2,4,6,

02)

M

M y - 1662 ~ if4

The deflection ~, when a surface load p is acting, is governed by the equation

my(x,y) 7 2 V 2~ =p/D.

=

~ro ~-~ 1 [mXcu k Mru Z.J w5-~_ 2 Z I\- - 2a // tan ~ en-~-~u x=lkp l z("g"'q

+ em 4 a u=l ~5 oo

~rrv

knx

axOy '

~

~cos

mxy (x, y)=

a

+ .=~g

COS

b

M ~)x-

i(Xb/a) 2 ; g2}2

X= 1,2, 3, ...;

0 = 1 , 2 , 3 .....

(17)

N

pg = xu(p2 + ~2)2

x z(ngrro]

-

X=l . = ~

I.trry

['-~]

1662 E ~ P m n m x y ( X , Y ) , l~xy=(1-v)~ m=l n=l

\ 2b ] t a n

=

\ 2a / t a n ~ - a

cos---b ' M

In the case of the rectangular array, the deflection due to p given by eq. (7), which is substituted in eq. (13), assumes the following form:

..y

oo

X tan , - ~ - cos a

-- v) C32~

6 y = - D ; ( \ 7 2 w ) .(14)

.,o

\ 2b ] tan - ~ " c°s b

X=l = XU(p2+g2)2

= -D~-~--U + v axe_] ,

(fix = - D ~x (~72w),

XTrx

c o s - -a

go

Then the bending moment, twisting moment and shearing force are given by the following relations:

~'[xy =D(1

Pmnmy(x,y),

m=l n=l

(13 )

LOx2 + v by 2 ] ,

N

~

16b ~

~

[rnkTru~ ~nu Z[~)tan -~a

XTrx

gTry

sin--a sin--b '

(18)

N

E Pmnqx(x,Y)

~3- m=ln=l

a (15)

7r v ~ 1 [m~Iru~ X~ru X~rx qx (x, y) = an ~-~x~__l~ Z [ - ~ ) tan -~a sin --a

75

K. Yamada, Y. Takeuti / Thermal flexure o f an infinite plate oo

oo

a

+ X=] E u=l E

p z(mX. 1 X/a(p2 +/a2) IT/tan

z(n_ww]

/a.o. X~x

X ~--~!tan~Sm--a M

(~y_

16b E

wry

cos--b '

f

2a

a-R

1 ztn/arw) #no qy(x'y)=em 4 a ~ = I g-I [ ' - ~ - ] t a n - ~ - sin lany b x=l = X/a(02 +/a2) Z

larry

X tan - ~ - cos a sin - ~

tan ~

\-2-b-/

AIx -32b2 ~ if4

(23)

o

From these equations, Mx and My can be obtained in the following form. For the rectangular array, from eq. (22), M

(20)

x=~ (-1) x z[---ff-)tan [mXrru\

mxo = en = ~

M

4b 2u ~

l~y =My 0

(24)

x~ru 2a '

N

~ Pmnmyo ,

rra am=l n=l

oo

/,,t--I-

IT]tan

(25)

u;;"

For the triangular array, from eq. (23),

N

~ Pmnmx(x,Y)

m=l n=l

(21)

I

[Mx

where mx(x, y) is given by eq. (16) with X and/a given by eq. (12). The other bending moments, twisting moment and shearing force can be written down immediately from the corresponding solutions for the rectangular array; therefore, the results are not shown here. Next, let us consider the boundary conditions. First of all, the condition that the plate is subjected to the uniform bending moments Mxo and My 0 given by eq. (5) has t=obe sati=_sfied.For this purpose, the above obtained M x and My alone are not sufficient, anew one must assume certain uniform bending momentsMx and My around the x andy axes, respectively, from which the above condition will have the following form:

For a rectangular array,

32b' M N

]

° + - - z"5 rn---, ~ n=l~Pmnmx° ,

/14x = 1 -R/----~

mx°=-en 1 - bR1 l-4

b x=2 ~92 Z

tan -2a -

rru k l ZIn_wrvl ~arty #rrR + em -~ a v = -~ ~ - - ~ ] tan ~ sin --ff--(0 2 + v/Jz) z(mMru~ Xrru + X=l ~ U=l ~ h/a2(p2+ /a2)z \ 2a ] t a n -2a X

z(n/aTrv] tan , /auv -~- sin #lrR ~T] b

My

o + ~71,5

b

(26)

'

E Pmnmyo ,

= n=l

e~

=bgxo, o

N

~ Pmnmxo , a?x =gxo _4~_~o m~=l = n=,

Also in the case of the triangular array, the bending and twisting moments, and shearing force due to the surface load, can be obtained in the same way as in the rectangular array or will be written down immediately from the results of the rectangular array, attention being paid to the analogy between the forms of Fourier series for the rectangular and triangular array. For example, the bending moment Mx is M

+£)x=°dy=bMxo.

f

rt u E

Xrrx

(22)

b-R

fo

N

E Pmnqy(x,y), tz~

~afro

= Uro.

For a triangular array,

(19)

7t3 m=l n=l

.

+

o

rrv mY°=en-4-bV

£

1

[mMru\

= h-~Z[T)

Mru

XrtR

tanTa sin-a

K. Yarnada, Y. Takeuti / Thermal flexure of an infinite plate

76

-6m(l--R)TT2

1 Z(?lldTrl)~

~-~-~

Id~O ~b---] tan 2b-

4 a u=2 ~

+~ ~

Qxc°sO+QysinO+\~x _(~_Mx \ ~y

(vp2 +la2) Z ~/m}trtu~ - - ~ a J tan ?'lru

x=l u=l 72~(p2 +/12)2

2a

z(nllIrvl /17to sin XTrR X ~---~--! tan 2--b-- ~ '

(27)

where ~ and/a in eqs. (26) and (27) take the values given in eq. (12). In the foregoing analysis, the bending moment, twisting moment and shearing force, eqs. (16)-(21), and uniform bending moments, eqs. (24)-(27), required for equilibrium, with Pmn as unknown quantities, have been obtained. Thus, finally the unknowns Pmn may be determined so that the sum of these solutions satisfy the boundary conditions along holes. If the peripheries of the circular holes are free, referring to fig. 3, the boundary conditions are

Q

(M,),=R = 0 ,

OMro]

' - S~-/,_-R

= 0,

(28)

where Mr, Mro and Qr are the bending and twisting moments, and shearing force acting on the edges r = const., and the second expression is Kirchhoff's equivalent shearing force. Since the solutions obtained in the foregoing analysis have been derived in Cartesian coordinates, the boundary conditions, eq. (28), may be rewritten in more convenient forms for calculations as follows:

+Mx) cos:'o +

R

O3My)sin0 cos20

_ 1__(3~x +3~x _/14y - My)(COS20 - sin20) R + OAlxy (cos20 - sinZ0) sin 0 0x

~Mxy (cos20 - sin20) cos 0 3y 4=

+-~Mxy sin 0 cos 0 = 0 , (Qr - aMro/rbO = 0),

(29)

which are the boundary conditions to be satisfied along the peripheries of the circular holes. Since the foregoing solutions have the same periods as those of holes the satisfaction of the boundary conditions on any one of the holes is sufficient for all boundary conditions on the other holes. Also, on account of symmetry, only a quadrant of one period need be considered. To satisfy the boundary conditions, eq. (29), the method of least squares was used. In this case, the points were selected amply on the periphery of the hole. The reason why the discrete points on the hole are preferred instead of a line integral along it is that the procedure may be applied to holes of arbitrary shape.

3. Numerical examples

+

-- 2Mxy sin 0 cos 0 = O,

0

~-x / sin20 cOs 0

(Mr = 0 ) ,

.I q

....

x

0

For two plates with circular holes forming a square and an equilateral triangular pattern, the numerical calculations were carried out. In both cases, to obtain the bending moment around holes and curves of stress concentration factors, the following values of pitch to diameter ratios were chosen:

air = 1.2, 1.25, 1.3, 1.35, 1.4, 1.5, 1.6, 1.8, 2.0, 3.0, .,

Y Fig. 3. Bending and twisting moments on periphery of hole.

Poisson's ratio of the plate being 0.3. On the other hand, the areas of load, on which Pmn is acting, may be chosen arbitrarily so long as they exist inside the holes. Thus, in this calculation the 36 (M = 6, N = 6) areas were taken with the configuration shown in fig. 4. To use the least squares method, the

K. }ramada, Y. Takeuti / Thermalflexure of an infinite plate 0.66

1.0

x/R

7

77

I

I

LolR=I.2 6

o

1.2s

5

\X,.

\

~E 0

z h

0,66

3 1.0

I

I

I*

olR=I./, j

a

'ix

1.6

ylR

J'$o Fig. 4. Configuration of areas of load. 31 points on a quadrant of hole (0 ° ~< 0 < 90 °) were selected at intervals of three degrees. Finally, the upper limits ~, =/~ = 200 in the Fourier series were taken for the square array and X =/J = 300 (150 in number of terms) for the triangular array. The results for the bending moment around the holes are shown in fig. 5 for the square array and in

6 ~ \

-/ - o l R = 1 . 2

/f

~E

2

oIR=I.

I,

0

i

O"

30"

8

o----,4 x

I

60"

i

i

90"

Fig. 5. Distributions of bending moment on the holes with a sqaure array.

Y y

i

O"

30"

t

O

i

60"

i

i

90"

Fig. 6. Distributions of bending moment on the holes with a triangular array.

fig. 6 for the triangular array. In these figures, all values of moment are divided by M 0 = - D ( 1 - v) Otro, the moment produced in a solid plate when its deflection is perfectly restrained under the uniform temperature gradient re. Curves for stress concentration factors are shown in fig. 7. It should be noticed here that the maximum values o f the bending moment Me agree to two significant figures for both square and triangular arrays and only one curve has therefore been shown in fig. 7. The dashed part of the curve is one obtained by extrapolation between the stress concentration 2.0 for an infinite plate with one hole under all-round bending and that for aiR = 3.0 actually calculated in this paper. The accuracy o f the solution was examined by the degree o f satisfaction o f the boundary conditions, eq. (29), on the periphery of the hole which should vanish from an exact solution. In the most severe case where the pitch to diameter ratio is 1.2, the computed bending moment on the hole was 0.06% o f Me for the square array and 0.15% for the triangular array. The computed effective shearing force on the hole was 0.08% o f 6Mo/h (R/h = 10) for the square array and 0.13% for the triangular array. The 6Mo/h is the

K. Yamada, Y. Takeuti / Thermal flexure of an infinite plate

78

N

M°

\

3 -J.

I

m - w f

J

Y

<

.,

Mo

Mo

I

0 0

0.1

0.2

0.3

~ot t

R/a

0.4

0.5

Mo

0.6

0.7

0.8

0.9

1.0

Fig. 7. Maximum bending moment for holes with a square or triangular array for various values of R/a.

shearing force resulting from the shearing stress distributed uniformly through a thickness, the intensity of which equals the maximum bending stress in a plate under the bending moment M0. To check the effects of the upper limits in the Fourier series on the accuracy, the two limits were taken to be X =/a = 200,300 for the square array and X =/a = 300,400 for the triangular array. In each case of the two hole patterns, both results for the peripheral bending moment agree to two significant figures. The accuracy of solution can become as high as we would wish by increasing the number of areas of load. By determining these, it should be noticed that the number of selected points on the hole is determined so that the distance between the selected points is less than half the side length u or v of area of load.

4. Conclusions

The bending problem was solved on the basis of the bending theory of plates for perforated plates with holes of arbitrary shape and array under a uniform

temperature gradient through thickness. Numerical results for circular holes with a square or triangular pattern were given for the bending moment around holes, and curves for stress concentration factors. The approach can be applied to a class of loads other than thermal. To complete this study, a more extensive study is required for such plates under a simple bending or twisting moment, to be carried out in subsequent papers in this series.

References [1] H. Saito, Z. Angew. Math. Mech. 37 (1957) 111. [21 R. Bailey and R. Hicks, J. Mech. Eng. Sci. 2 (1960) 143. [31 K. Mizoguchi and K. Yamada, Bull. JSME 20 (1977) 1515. [4] O. Tamate, Z. Angew. Math. Mech. 37 (1957)431. [5] T. Yamashida, Trans. Japan Soc. Mech. Eng. 34 (1968) 19. [61 I. Malkin, Trans. ASME 74 (1952) 387. [7] G. Horvay, J. Appl. Mech. 19 (1952) 122. [8] S. Timoshenko, Theory of Plates and Shells (McGraw-Hill, New York, 1940).