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Note on stresses and displacements in square plates and cylinders with pressurized central circular holes
NUCLEAR ENGINEERING AND DESIGN 5 (1967) 142-149. NORTH-HOLLAND PUBLISHING COMP., AMSTERDAM
NOTE ON STRESSES A N D DISPLACEMENTS IN SQUARE PLATES A N D CYLINDERS W I T H PRESSURIZED CENTRAL CIRCULAR HOLES T. SLOT
Nuclear Materials and Propulsion Operation, General Electric Company Cincinnati, Ohio, USA Received 20 January 1967
Accurate elastic solutions for perforated squares subjected to internal pressure are reported for hole diameter to plate width ratios of 0.5, 0.6, 0.7, 0.8, 0.85 and 0.9. A point-matching procedure is used to generate solutions that satisfy the internal boundary conditions exactly and the external boundary conditions approximately. The degree of approximation is examined by inspection of residual tractions along the outer boundary. Plane-stress and plane-strain conditions are considered, and the superposition by which the solutions for external pressure can be obtained from the solutions for internal pressure is indicated.
1. INTRODUCTION T h i s note is c o n c e r n e d with p l a n e - s t r e s s and p l a n e - s t r a i n solutions for the p r o b l e m defined in fig. 1, w h e r e a hollow s q u a r e is shown s u b j e c t e d to i n t e r n a l p r e s s u r e . It is known f r o m the t h e o r y of e l a s t i c i t y that solutions of this type d e s c r i b e the conditions found in'thin plates and long c y l i n d e r s , r e s p e c t i v e l y . A c c u r a t e solutions that s a t isfy the boundary conditions for the hole exactly and the boundary conditions f o r the s q u a r e edge in c l o s e a p p r o x i m a t i o n a r e r e p o r t e d for D / W r a t i o s of 0.5, 0.6, 0.7, 0.8, 0.85 and 0.9. They w e r e obtained by p o i n t - w i s e m a t c h i n g of the boundary conditions for the outer boundary, in the s a m e m a n n e r that p r e v i o u s solutions w e r e obtained for hollow hexagons under i n t e r n a l p r e s s u r e [1]. By e m p l o y in g the p r i n c i p l e of s u p e r p o s i t i o n , the solutions for e x t e r n a l p r e s s u r e a r e a l s o r e a d i l y d e t e r m i n e d f r o m the solutions for i n t e r n a l p r e s s u r e . T h e p r o b l e m of the hollow s q u a r e with unif o r m loading applied to the inner or o u t e r bounda r y was c o n s i d e r e d in s e v e r a l other t h e o r e t i c a l i n v e s t i g a t i o n s [2-6]. P o i n t - m a t c h i n g p r o c e d u r e s r e s e m b l i n g the one used for the solutions in this note w e r e r e p o r t e d by Sekiya [2] and by Roy [3]. Sekiya obtained the s t r e s s e s at the hole for p r e s s u r i z e d hollow s q u a r e s with D / W r a t i o s of 0.25 and 0.5 and showed that the r e s u l t s for the second c a s e w e r e in good a g r e e m e n t with those r e p o r t e d by Kawaguchi [4], who solved this p r o b -
y D
X
o
J D
~
w
Fig. 1. Square plate with central hole subjected to internal pressure. l e m by a c o n f o r m a l mapping method. Roy obtained the s t r e s s e s for the i n n e r and outer boundary of p r e s s u r i z e d hollow s q u a r e s with D / W r a t i o s r a n g i n g f r o m 0.1 to 0.9. S t r e s s e s for the inner boundary w e r e given in t ab u l ar and g r a p h i cal f o r m , s t r e s s e s at the outer boundary only in g r a p h i c a l f o r m . Hengst [5] d ev el o p ed a l e a s t s q u a r e s a v e r a g i n g p r o c e d u r e to a p p r o x i m a t e conditions at the outer boundary. S t r e s s e s at the inner boundary w e r e d e t e r m i n e d for D / W r a t i o s
STRESSES AND DISPLACEMENTS IN SQUARE PLATES AND CYLINDERS from 0.1 to 0.5 and :[or two conditions of loading, n a m e l y s y m m e t r i c and a n t i s y m m e t r i c u n i f o r m loading of the outer boundary. A l e a s t - s q u a r e s a v e r a g i n g p r o c e d u r e was also used by Schlack and Little [6], who r e p o r t e d the s t r e s s e s at the hole and along the line of s y m m e t r y 0 =0 (see fig. 1) for D / W r a t i o s from 0.1 to 0.9. R e s u l t s tabulated for both s y m m e t r i c and a n t i s y m m e t r i c loading of the outer boundary were shown to be in g e n e r a l a g r e e m e n t with c o r r e s p o n d i n g r e s u l t s from the p a p e r s by Hengst and Roy. In c o m p a r i s o n with the i n v e s t i g a t i o n s cited above, the r e s u l t s p r e s e n t e d in the sequel a r e the most c o m p r e h e n s i v e for the loading condition and D / W r a t i o s c o n s i d e r e d . S t r e s s e s and d i s p l a c e m e n t s a r e tabulated for a l a r g e n u m b e r of locations along the i n n e r boundary, the outer b o u n d a r y and the axes of s y m m e t r y . A detailed a n a l y s i s is included of r e s i d u a l t r a c t i o n s at the outer boundary, which p e r m i t s an a s s e s s m e n t of the a c c u r a c y of the p o i n t - m a t c h i n g p r o c e d u r e employed. The effect of P o i s s o n ' s r a t i o on the values of the d i s p l a c e m e n t s is also considered. S t r e s s e s in s q u a r e plates with p r e s s u r i z e d c e n t r a l holes w e r e also m e a s u r e d on p h o t o e l a s tic models. E a r l y e x p e r i m e n t s by D u r e l l i and B a r r i a g e [7] were followed by an i m p r o v e d a n a l y s i s by Riley et al. [8]. Subsequently, North and Mantle [9] obtained s i m i l a r photoelastic s o l u tions by a different e x p e r i m e n t a l technique. The r e s u l t s in the last two p a p e r s appear to be in g e n e r a l a g r e e m e n t with the t h e o r e t i c a l solutions, except for high D / W r a t i o s , where s i g n i f i c a n t d i f f e r e n c e s were noted. A m o r e f a v o r a b l e o v e r all c o m p a r i s o n was o b s e r v e d between n u m e r i c a l and photoelastie r e s u l t s for hollow hexagons
[10,11].
2. ANALYSIS A digital c o m p u t e r p r o g r a m s p e c i a l l y d e v e l oped for the a n a l y s i s of plane p e r f o r a t e d s t r u c t u r e s was used to g e n e r a t e the n u m e r i c a l s o l u tions of i n t e r e s t . A d e s c r i p t i o n of this p r o g r a m and v a r i o u s solutions obtained with it were the s u b j e c t of an e a r l i e r publication in this j o u r nal [1]. In e s s e n c e , the p r o g r a m p e r m i t s the u s e r to c o n s t r u c t a s u i t a b l e s t r e s s function by s e l e c t i o n of a finite n u m b e r of t e r m s f r o m the following b i h a r m o n i c s e r i e s in p o l a r c o o r d i nates: F = a o log r + bo r2 + :Dn(anrn+2 + bnr n + Cnr -n + dnr_n+2) ( s i n nO~ \COS nO/ '
143
where ao, bo, an, bn, cn and dn a r e a r b i t r a r y coefficients, and n = 2, 3,4, . . . . It can be shown that the t e r m s f r o m this s e r i e s f u r n i s h solutions t h a t s a t i s f y the usual r e q u i r e m e n t s of e q u i l i b r i u m , compatibility and s i n g l e - v a l u e d d i s p l a c e m e n t s ; f u r t h e r m o r e , that boundary conditions placed on s t r e s s e s or d i s p l a c e m e n t at a point r e s u l t in l i n e a r a l g e b r a i c equations in the coefficients. The coefficients may t h e r e f o r e be d e t e r m i n e d from a set of b o u n d a r y conditions that m a t c h e s the n u m b e r of t e r m s in the chosen s t r e s s function. A c c o r d i n g ly, the computer p r o g r a m allows s t r e s s e s and d i s p l a c e m e n t s in any d i r e c t i o n to be specified for a r b i t r a r y points on the boundary. Once the coefficients a r e known, s t r e s s e s and d i s p l a c e m e n t s can be evaluated for any point of the s t r u c t u r e . The c o m p u t e r p e r f o r m s all the n e c e s s a r y calculations. To satisfy the conditions of s y m m e t r y in the p r o b l e m under c o n s i d e r a t i o n , the choice of the v a l u e s of n was limited to m u l t i p l e s of 4. Specifically, the solutions for D / W r a t i o s of 0.5-0.85 w e r e obtained by choosing the t e r m s connected with coefficients a o and bo, and the cosine t e r m s connected with n - v a l u e s of 4 , 8 , . . . ,24, for a total of 26 t e r m s in the s t r e s s function. To obtain s a t i s f a c t o r y a c c u r a c y for the case D / W = 0.9, the n u m b e r of t e r m s was i n c r e a s e d to 42 by the addition of t e r m s connected with n - v a l u e s of 28, 32, 36, 40. For the sector of the s q u a r e between lines of s y m m e t r y OB and OD in fig. 1, the boundary conditions along the outer edge BD a r e ax = 0 and Txy = O. T h e s e conditions were specified for a n u m b e r of d i s c r e t e points on BD. In the f i r s t five p r o b l e m s (D/W = 0.5-0.85) the points w e r e chosen at Y/YD = O, 0.2, 0.4, 0.6, 0.8, 0.9 and 1; in the sixth p r o b l e m (D/W= 0.9) additional points w e r e chosen at Y/YD = 0.1, 0.3, 0.5 and 0.7. It may be v e r i f i e d that for N values of n, t h e s e boundary conditions r e s u l t in 2N + 1 r e l a t i o n s between 4N + 2 coefficients, where N = 6 for the f i r s t five p r o b l e m s and N= 10 for the sixth p r o b lem. With r e f e r e n c e to the paper by Slot and Yalch [1], the additional 2 N + 1 r e l a t i o n s needed for the evaluation of the coefficients a r e p r o vided by the boundary conditions for the hole. It is shown t h e r e that by introducing s p e c i a l r e l a tions between the coefficients, ~ r = - p i and ~r0 =0 along the e n t i r e edge of the hole. R e g a r d i n g the p l a n e - s t r e s s d i s p l a c e m e n t s at the hole, they can be e x p r e s s e d in the following form:
144
T. SLOT Here, subscripts ~ and e are used to distinguish between plane-stress and plane-strain displacements, respectively.
E U r = -(1 - v)PiR - 2 a o R - 1 + 4Zn(anRn+l +dnR_n+l) ( s i n nO~ \cos n0/
E U0 = 4Zn(anR n+ l - d n R - n + l )
(
~ cOS
'
n0)
\ s i n nO/ "
These expressions were derived by eliminat i o n of c o e f f i c i e n t s bo, b n a n d c n f r o m t h e g e n eral expressions for the displacements and the above mentioned special relations between the coefficients required to satisfy the boundary c o n d i t i o n s a t t h e h o l e . It i s n o t e d t h a t f o r t w o s o l u t i o n s w i t h d i f f e r e n t P o i s s o n ' s r a t i o s , s a y v1 a n d v2, t h e d i s p l a c e m e n t s a t t h e h o l e a r e r e l a t e d as follows:
(Ur)v=v2 : (Ur)u:Vl + (v 2 - V l ) P i R / E , (Uo)v=v2 = (Uo)v=Vl • It is e v i d e n t f r o m t h e s e r e l a t i o n s t h a t in t h e c a s e of e x t e r n a l l o a d i n g , i . e . , Pi = 0, t h e d e f o r m a t i o n of t h e h o l e i s n o t d e p e n d e n t o n t h e v a l u e of P o i s s o n ' s r a t i o . ( T h i s o b s e r v a t i o n m a y b e generalized to include thin plates with multiple perforations that are loaded only along the ext e r n a l b o u n d a r y [1].) B y t h e r e c i p r o c i t y t h e o r e m , i t f o l l o w s t h a t i n t h e c a s e of i n t e r n a l p r e s s u r e t h e d e f o r m a t i o n of t h e s q u a r e b o u n d a r y i s n o t d e p e n d e n t on t h e v a l u e of P o i s s o n ' s r a t i o . F r o m t h e a n a l o g y [12] t h a t e x i s t s b e t w e e n plane-stress a n d p l a n e - s t r a i n p r o b l e m s it f o l lows that the in-plane stresses are the same for both. The axial stresses present under planestrain conditions may be obtained from Hooke's law w i t h t h e a s s u m p t i o n of z e r o a x i a l s t r a i n , o r a z = v((Tr + ~0) = V(ax + Cry). In a c c o r d a n c e w i t h the transition rules provided by the analogy, the displacements for plane strain may be obtained from the displacements for plane stress by rep l a c i n g E w i t h E / ( I - v 2) a n d v b y v / ( l - v ) . T h e following relations result for the displacements at the inner and outer boundary:
3. R E S U L T S The computer solutions acquired for hollow squares under internal pressure loading are s u m m a r i z e d in t a b l e s 1 - 9 . In p a r t i c u l a r , t a b l e s I-6 give the stresses and displacements at vario u s p o i n t s a l o n g t h e b o u n d a r i e s a n d t h e a x e s of s y m m e t r y f o r t h e s i x v a l u e s of D / W c o n s i d e r e d . A graphical representation of t h e b o u n d a r y s t r e s s e s is s h o w n in f i g s . 2 a n d 3. T h e h i g h e s t stresses is f o u n d o n t h e i n n e r b o u n d a r y f o r t h l o w e r v a l u e s of D / W a n d on t h e o u t e r b o u n d a r y f o r t h e h i g h e r v a l u e s of D / W . O n t h e o u t e r b o u n d a r y t h e m a x i m u m s t r e s s o c c u r s a t 0 = 0, w h e r e a s on t h e i n n e r b o u n d a r y t h e l o c a t i o n of m a x i m u m s t r e s s v a r i e s f r o m 0 = 45 ° f o r t h e t h r e e l o w e r D / W r a t i o s to s m a l l e r v a l u e s of 0 for the three higher D/W ratios. Table 1 Values of (~0/Pi along inner boundary (AC)
D/w
60 0 5 10 15 20 25 30 35 40 45
0.5
0.6
0.7
0.8
0.85
0.9
1.412 1.423 1.454 1.501 1.557 1.615 1.667 1.710 1.736 1.746
1.599 1.625 1.697 1.803 1.926 2.048 2.154 2.235 2.285 2.302
1.834 1.899 2.077 2.326 2.594 2.835 3.021 3.144 3.212 3.233
2.107 2.306 2.831 3.500 4.122 4.570 4.811 4.888 4.883 4.871
2.258 2.658 3.674 4.878 5.851 6.394 6.534 6.422 6.245 6.165
2.352 3.412 5.824 8.135 9.497 9.867 9.534 8.862 8.235 7.982
Table 2 Values of (~y/Pi along outer boundary (BD)
D/W Y /YD
0.5
0.6
0.7
0.8
0.85
0.9
Inner boundary:
( Ur) e = (1 - v2)(Ur)a + (1 + v ) v 2 p i R / E , (uo)~ = ( 1 - v 2 ) ( v o ) a
;
Outer boundary:
( Ux)~ = (1 - v 2 ) ( U x ) a , (Uy)e = (1 - v2)(Uy)~ .
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1.018 0.973 0.848 0.674 0.481 0.297 0.142 0.032 -0.022 -0.022 0
1.753 3.106 6.064 1.663 2.913 5.566 1.423 2.406 4.292 1.101 1.755 2.765 0.748 1.085 1.363 0.434 0.536 0.375 0.184 0.141 -0.232 0.018 -0.086 -0.502 -0;053 -0.145 -0.436 -0.039 -0.075 -0.164 0 0 0
9.174 8.258 5.962 3.361 1.222 -0.061 -0.750 -1.019 -0.793 -0.254 0
15.75 13.40 8.266 3.597 0.453 -1.258 -1.963 -1.997 -1.382 -0.386 0
STRESSES AND DISPLACEMENTS IN S QUARE P L A T E S AND CYLINDERS Table 3 V a l u e s of (YO/Pi and (Yr/Pi along r a d i a l line 0 = 0 (AB)
D/W
p* 0.5
0.6
0.7
Table 4 V a l u e s of (YO/Pi and (~r/Pi along r a d i a l line 0 = ~
0.85
0.9
0.5
0.6
0.7
1.412 1,134 0.950 0.865 0.879 1.018
1.599 1.511 1.426 1.412 1.505 1.753
1.834 2.044 2.179 2.353 2.636 3.106
0 0,2 0,4 0,6 0,8 1
-1 -0.594 -0.324 -0.148 -0.040 0
-1 -0.664 -0.389 -0.184 -0.051 0
-1 -0.726 -0.452 -0.222 -0.062 0
0.8
0.85
0.9
6.165 3.467 1.395 0.092 -0.268 0
7.982 4.331 1.429 -0.218 -0.463 0
-1 -0.228 0.174 0.276 0.129 0
-1 -0.043 0.473 0.510 0.210 0
gO/Pi
~0/Pi 0 0.2 0.4 0,6 0.8 1
(CD)
D/W
p *
0.8
145
2.107 2.931 3.609 4.285 5.069 6.064
2.258 3.716 4.995 6.249 7.602 9.174
2.352 5.141 7.711 10.25 12.87 15.75
0 0.2 0.4 0.6 0.8 1
1,746 0.991 0.617 0.306 0.055 0
2.302 1.386 0.841 0.378 0,046 0
3.233 1.950 1.083 0.398 -0.001 0
-1 -0.814 -0.552 -0.291 -0.089 0
-1 -0.841 -0.581 -0.306 -0.089 0
0 0.2 0.4 0.6 0,8 1
-1 -0.432 -0.200 -0.067 0.002 0
-1 -0.463 -0.215 -0.058 0.013 0
-1 -0.447 -0.173 -0.009 0.035 0
* p = (r- rA)/(r B- rA) .
*p
(r- rc)/(rD- rc).
Table 5 V a l u e s of EUr/PiR and EUo/PiR along i n n e r b o u n d a r y
Table 6 V a l u e s of EUx/PiR and EUy/piR along o u t e r b o u n d a r y (BD)
(~r/Pi -1 -0.784 -0.516 -0.264 -0.077 0
ar/P~
(AC)
0.5
0.6
0.7
0.8
0.85
0.9
Y/YD 0.5
EUr/PiR - plane s t r e s s (v=0.3) 0 5 10 15 20 25 30 35 40 45
1.951 1.947 1.935 1.916 1.894 1.870 1.848 1.831 1.819 1.815
2.436 2.426 2.395 2.349 2.295 2.238 2.186 2.145 2.118 2.109
3.347 3.318 3.237 3.118 2.979 2.840 2.717 2.623 2.564 2.544
5.459 5.364 5.103 4.732 4.325 3.942 3.627 3.400 3.265 3.221
7.860 7.659 7.114 6.370 5.591 4.900 4.362 3.995 3,786 3.719
0 -0.020 -0.038 -0.051 -0.058 -0.058 -0.050 -0,037 -0.020 0
0 -0.046 -0.085 -0.114 -0.128 -0.126 -0.109 -0.080 -0.042 0
0 -0.103 -0,190 -0.250 -0,275 -0,266 -0,226 -0,163 -0,085 0
0 -0.258 -0.467 -0.595 -0.630 -0.583 -0.476 -0.332 -0.169 0
0 -0.445 -0,792 -0.982 -1.006 -0.899 -0.710 -0.480 -0.240 0
13.19 12.65 11.26 9.498 7.812 6.441 5.455 4.828 4.494 4.391
0 -0.872 -1.495 -1.762 -1.712 -1.454 -1.093 -0.710 -0.344 0
Radial d i s p l a c e m e n t s f o r v=v 2 f r o m r a d i a l d i s p l a c e m e n t s for V=Vl=0.3:
(Ur)v=v2 = (Ur)v=Vl + (v2 - V'I)PiR/E . D i s p l a c e m e n t s f o r plane s t r a i n f r o m :
(vr) e = (1 - v 2 ) l V r ) a + (1 + v ) ~ 2 p i R / E , (U0) e = ( 1 - v2)(U0)(r. ( S u b s c r i p t s (Y and E denote plane s t r e s s strain, respectively.)
0.6
0.7
0.8
0.85
0.9
EUx/PiR - plane s t r e s s (all P)
EUo/PiR - plane s t r e s s (all lJ) 0 5 10 15 20 25 30 35 40 45
-1 -0.341 -0.004 0.134 0.083 0
D/W
D/W
00
4.871 2.824 1.310 0.273 -0.135 0
and plane
0 0.2 0.4 0.6 0.8 1
1.338 1.256 1.070 0.889 0.794 0.783
1.902 1.768 1.472 1.203 1.072 1.058
2.880 2.623 2.087 1.643 1.453 1.439
5.053 4.416 3,206 2.328 2.013 2.005
7.482 6.294 4.212 2.854 2.409 2.410
12.84 10.08 5.927 3.609 2.935 2.953
EUy/PiR - plane s t r e s s (all p) 0 0.2 0.4 0.6 0.8 1
0 0.384 0.652 0.773 0.790 0.783
0 0.547 0.910 1.058 1.070 1.058
0 0.818 1,316 1.476 1.460 1.439
0 1.360 2.052 2.161 2.050 2.005
0 1.889 2.693 2.701 2.481 2.410
0 2.876 3.726 3.483 3.064 2.953
D i s p l a c e m e n t s f o r plane s t r a i n f r o m :
(Vx)¢ = (1 - v2)(Ux)(X ;
(Uy)E= (1 - V2)(Vy)0, ,
( S u b s c r i p t s (~ and ¢ denote plane :~train, r e s p e c t i v e l y . )
stress
and plane
146
T. SLOT
V a l u e s of
Y/YD
gx/Pi
Table 7 a l o n g o u t e r b o u n d a r y (BD) *
V a l u e s of
Table 8 o u t e r b o u n d a r y (BD) *
7xy/P i a l o n g
D/W 0.5
0.6
0.7
D/W
0.8
0.85
0,95
0 0 0 0 0 0 0 0,05 0.1 0 0 0 0.001 0.004 0.021 0.15 O.2 0 0 0 0 0 0 0.25 0,3 0 0 -0.001 - 0 . 0 0 3 -0.010 - 0 , 0 5 4 0.35 0.4 0 0 0 0 0 0 0,45 0.5 0.001 0,001 0.002 0.004 0.012 0.057 0.55 0.6 0 0 0 0 0 0 0.65 0.7 -0.001 -0.002 -0.003 -0.006 -0,009 -0.025 0.75 0.8 0 0 0 0 0 0 0,85 0.001 0.001 0.002 0.003 0.001 - 0 . 0 0 3 0.9 0 0 0 0 0 0 0,95 - 0 . 0 0 2 - 0 . 0 0 4 - 0 . 0 0 6 - 0 . 0 1 1 0.004 0,013 1 0 0 0 0 0 0 * V a l u e s of
lax/Pi[
Y/YD
0.95 0 -0.002 0 0.00( 0 -0.010 0 -0.014 0 0,006 0 -0.005 0 J ~04 0 -0.003 0 -0,005 0 0,048 0
s m a l l e r than 0.0005 e n t e r e d a s 0.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 O.5 0.55 0.6 0,65 0.7 0.75 0.8 0.85 0.9 0.95 1
0.5
0.6
0.7
0.8
0.85
0.9 5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.001
0,004
0.015
0.043
0.165
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0,005
0.024
0.131
0 0 0 0.001 0
0 0 0 0,001 0
0 0 0 0 0
0 -0.001 0 -0.001
0 -0,020 0 -0.019
0 -0.034 0 -0.026
* V a l u e s of ]
7xy/Pil
-0,008
-0.007
,
(~x/Pi and "rxy/Pi a l o n g
0.5
0.6
0.7
o u t e r b o u n d a r y (BD) *
0.8
0185
0.9 t
0.9
0.013 -0.034 0.036 -0.016 0.003
0 0 0 -0.002 -0.001
gx/Pi "ffx/Pi = f~J yj _ y~ dy 0 0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8 1
0
1
0 0 0 -0.001 0
0 0 0.001 -0.001 -0.001
0
0
0 -0.001 0.001 -0,002 -0.002
0.001 -0,002 0.003 -0.004 -0,003
0.003 -0.006 0.007 -0.006 -0.003
-0.001
-0.001
-0.001
v,,Jt, i 0 0.2 0.4 0.6 0.8
0,2 0.4 0.6 0.8 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0
1
0
0
0
*
V a l u e s of
[ffx/Pi]
"~ 26 t e r m s in s t r e s s functiofi (N = 6). $ 42 t e r m s in s t r e s s f u n c t i o n (N = 10).
and
IYxy/Pil
r~i j
0
0
-0,001
TxY/Pi dy Yj-Yi
-0,005 0,004 -0,004 0.003 0
-0.136
-0.146
0
0 -0.00~ 0 0.002 0 -0.010 0 -0.014 0 0.006 0 -0.005 0 0.004 0 -0,003 0 -0,005 0 -0.048 0
s m a l l e r t h a n 0.0005 e n t e r e d a s 0.
D/W
Yi/YD Yj/YD
-0.029
0
Table 9
V a l u e s of
-0.029
0.9 $
-0.017 0.018 -0.019 0.015 -0.001
-0.084 0.089 -0.093 0,081 -0.003
0.004 -0.013 0 0 0.014
-0.001
-0,002
0,001
s m a l l e r t h a n 0.0005 e n t e r e d a s 0.
STRESSES AND DISPLACEMENTS IN SQUARE PLATES AND CYLINDERS 10
I
I
I
~
I
16
I
147
Jl
I
I
14 12
0.9
10 •
o
®
4
a
6
~
4
0.7
Q
3
2
2
C D/W
0
= 0.5
= 0.5
1 _ 0
D/W
--2 I
f
I
I
I
I
1
I
5
10
15
20
25
30
35
40
-~ 45
0
I
I
I
I
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Distance ratio Y/YD
Angle e, degrees
Fig. 2. Stress distribution along inner boundary (AC).
Fig. 3. Stress distribution along outer boundary (BD).
T a b l e s 7-9 f u r n i s h insight into the a c c u r a c y of the s o l u t i o n s by allowing inspection of the d e g r e e of a p p r o x i m a t i o n a c h i e v e d for the boundary conditions along edge BD. In t a b l e s 7 and 8, v a l ues of ax and Txy a r e shown for the points at which boundary conditions w e r e s p e c i f i e d , as w e l l as f o r the points located midway the f o r m e r . T a b l e 9 g i v e s the a v e r a g e v a l u e s of t h e s e s t r e s s e s in f i v e equal i n t e r v a l s along BD and a l s o for the e n t i r e length of BD. It is a p p a r ent f r o m t h e s e t a b l e s that the actually obtained boundary conditions c l o s e l y a p p r o x i m a t e the t r u e boundary conditions f o r the s q u a r e edge of the plate. F o r the p a r t i c u l a r c a s e of D / W = 0.9, it was m en t i o n ed b e f o r e that the n u m b e r of t e r m s was i n c r e a s e d f r o m 26 to 42 in o r d e r to i m p r o v e the a c c u r a c y of the solution. To i l l u s t r a t e the i m p r o v e m e n t obtained, the boundary conditions applicable to both solutions a r e r e p o r t e d in the tables. C o r r e s p o n d i n g solutions for e x t e r n a l p r e s s u r e may be obtained by s u b t r a c t i n g the solution for i n t e r n a l p r e s s u r e f r o m the solution for p r e s s u r e on both b o u n d a r i e s . T h e l a t t e r c a s e of l o a d ing p r o d u c e s a u n i f o r m e q u i - b i a x i a l s t r e s s field of i n t e n s i t y -p, w h e r e p is the applied p r e s s u r e . S t r e s s e s and d i s p l a c e m e n t s for this c a s e a r e t h e r e f o r e as follows:
Plane
stress:
Plane
strain:
Inner boundary:
a r / P = ao/P = -1 E Ur/PR = - (1 - ,) Vo =O
Zr/P = e o / P =
-1;
%/p = -2 v E U r / p R = -(1+ v)(1-2 ~)
u0=0
Outer boundary:
~ x / P = ~ y / P = -1 s V x / P n : -(1- v)(W/D)
a x / P = e y / P : -1, az/p = -2v EVx/pR = -(1+ v)(1-2 v)(W/D)
Ev/
n= -(1 - v ) ( W / D ) ( y / y D )
EUy/pR
=
-(1+,)(1-2,)(W/D)(y/yD)
4. CONCLUSION A s u m m a r y has been p r e s e n t e d of an a p p r o x i m a t e a n a l y s i s of the s t r e s s e s and d i s p l a c e m e n t s in s q u a r e p e r f o r a t e d p l a t e s and c y l i n d e r s due to i n t e r n a l or e x t e r n a l p r e s s u r e loading. It has been shown that a c c u r a t e n u m e r i c a l a n s w e r s f o r
148
T. SLOT Table 10 Comparison with r e s u l t s f r o m other investigations
D/W
r
e°
Hengst [5]
Kawaguchi [4]
Sekiya [2]
Schlack and Little [6]
Present analysis
1.365 1.438 1.711 1.810
1.408 1.667 1.746
1.412 1.501 1.667 1.746
Roy [3]
~e/Pi 0.5 0.5 0.5 0.5
R R R R
0 15 30 45
1.416 1.503 1.667 1.745
1.424 1.511 1.676 1.756
1.385 1.486 1.688 1.723
0.6 0.7 0.8 0.9
R R R R
0 0 0 0
1.522 1.754 2.133 2.920
1.595 1.839 2.110 2.415
1.599 1.834 2.107 2.352
0.6 0.7 0.8 0.9
R R R R
45 45 45 45
2.412 3.360 4.940 8.316
2.304 3.240 4.866 7.990
2.302 3.233 4.871 7.982
0.5 0.6 0.7 0.8 0.9
½W ½W ½W ½W ½W
1.037 1.781 3.107 6.074 15.60
1.018 1.753 3.106 6.064 15.75
(~y/Pi 0 0 0 0 0
1.220
t h e s e p r o b l e m s of e n g i n e e r i n g i n t e r e s t c o u l d b e obtained by generating solutions for the biharmonic field equation that satisfy the boundary conditions for the hole entirely and the boundary conditions for the square edge in a point-wise m a n n e r . In t h e a b s e n c e of r i g o r o u s p r o o f f o r convergence of t h e n u m e r i c a l p r o c e d u r e inv o l v e d , t h e a c c u r a c y of t h e a c q u i r e d s o l u t i o n s h a s b e e n e x a m i n e d b y c o m p u t a t i o n of r e s i d u a l tractions along the square boundary. O t h e r t h e o r e t i c a l i n v e s t i g a t i o n s of t h e p r o b lem treated here have been cited from the litera t u r e . In c o m m o n w i t h t h e p r e s e n t w o r k , t h e s e investigations also involved series solutions that satisfied the internal boundary conditions exactly and the external boundary conditions in approxim a t i o n . A l i m i t e d c o m p a r i s o n of r e s u l t s b y d i f f e r e n t m e t h o d s i s p r o v i d e d i n t a b l e 10. ( N u m e r i cal results attributed to Hengst and Sekiya were computed by the author from equations provided by the investigators.) Although there are some differences, it would hardly be appropriate to j u d g e t h e p o t e n t i a l a c c u r a c y of t h e v a r i o u s m e t h o d s o n t h e b a s i s of t h i s c o m p a r i s o n , f o r two r e a s o n s : 1) t h e n u m b e r of t e r m s u s e d t o g e n e r a t e the solutions varies considerably, a n d 2) n o t enough information is given on the stresses at t h e o u t e r b o u n d a r y . A d e t a i l e d c o m p a r i s o n on t h e
0.986
1.15 1.94 3.32 6.14 14.73
b a s i s of a n e q u a l n u m b e r of t e r m s i n t h e s t r e s s f u n c t i o n w o u l d b e of i n t e r e s t , b u t i s b e y o n d t h e s c o p e of t h i s n o t e . In c o n c l u s i o n , a t t e n t i o n is c a l l e d t o r e c e n t p u b l i c a t i o n s b y G r o s s et al. [13], K o b a y a s h i e t a l . [14], W i l s o n et al. [15], H u l b e r t a n d N i e d e n f u h r [16] a n d B a i l e y and: F i d l e r [17], w h i c h a r e concerned with solutions for complex structural p r o b l e m s o b t a i n e d w i t h t h e a i d of p o i n t m a t c h i n g of b o u n d a r y c o n d i t i o n s . ( A d d i t i o n a l r e f e r e n c e s may be found in the papers by Hulbert and Nied e n f u h r [16] a n d Slot a n d Y a l c h [1].)
5. N O T A T I O N
Co, b o = c o e f f i c i e n t s in s t r e s s f u n c t i o n an, bn, Cn, d n = c o e f f i c i e n t s in s t r e s s f u n c t i o n , n = 2,3,4,... E, v = Young's modulus, Poisson's ratio F = biharmonic stress function P, Pi = pressure loading parameters R, D = r a d i u s , d i a m e t e r of h o l e Ur, U 0 =displacements in polar coordinates Ux, Uy = d i s p l a c e m e n t s in r e c t a n g u l a r c o ordinates W = w i d t h a c r o s s f i a t s of s q u a r e
S:FRESSES AND DISPLACEMENTS IN SQUARE PLATES AND CYLINDERS
r~ °r, ~0, TrO
= summation symbol = s t r e s s c o m p o n e n t s in p o l a r c o o r dinates = s t r e s , s components in r e c t a n g u l a r coordinates
ACKNOWLEDGEME].~ T h e w o r k r e p o r t e d h e r e i n o r i g i n a t e d in p a r t w i t h d e v e l o p m e n t p r o j e c t s s p o n s o r e d b y t h e U.S. A t o m i c E n e r g y C o m m i s s i o n u n d e r C o n t r a c t AT (40-1)-2847. P e r m i s s i o n to publish this note is gratefully acknowledged.
REFERENCES
[1] T. Slot and J . P . Y a l c h , S t r e s s analysis of plane p e r f o r a t e d s t r u c t u r e s by point-wise matching of boundary conditions, Nucl. Eng. Design 4 (1966) 163. [2] Tsuyoshi Sekiya, An approximate solution in the p r o b l e m s of elastic plates with an a r b i t r a r y e x t e r nal f o r m and a c i r c u l a r hole, P r o c . Fifth Japan Natl. Congr. Appl. Mech. (1955) p. 95. [3] S.K. Roy, On the b i - h a r m o n i c analysis of s t r e s s fields for hydrostatic loading in relation to analytical relaxational ~nd photoelastic r e s u l t s for openings in s t r u c t u r e s , P r o e . 3rd Congr. Theor. Appl. Mech., Bangalore, India (1957) p. 71. [4] Tatuji Kawaguchi, S t r e s s distribution of a thick tube, subjecting to an internal p r e s s u r e , which has a c r o s s section of regular polygon and a c i r cular hole in its center, Trans. Japan Soc. Mech. Eng. 18 (1952) 36 (in Japanese). [5] H.Hengst, Beitrag zur Beurteilung des Spannungszustandes einer gelochten Scheibe, ZAMM 18 (1938) 44.
149
[6] A. L. Schlack and R.W. Little, Elastostatic p r o blem of a p e r f o r a t e d square plate, J. Eng. Mech. Div., P r o c . ASCE 90 (1964) 171. [7] A. J. Durelli and J. Barriage, S t r e s s distribution in square plates with hydrostatically loaded central c i r c u l a r holes, J. Appl. Mech., Trans. ASME 77 (1955) 539. [8] W. F. Riley, A. J. Durelli and P. S. Theocaris, F u r ther s t r e s s studies on a square plate with a p r e s surized central c i r c u l a r hole, P r o c . 4th Midwest Conf. Solid Mech., Univ. of Texas, Austin, Texas, (1959). [9] W. P. T. North and J. B. Mantle, Photoelastic study of s t r e s s e s in hydrostatically loaded cylinders with noncircular external boundaries, Exp. Mech. 2 (1962) 91. [10] P. D. Flynn, S t r e s s e s in hollow hexagons under external p r e s s u r e , Exp. Mech. 2 (1961) 148. [11] T. Slot, On the accuraey of two-dimensional photoelastic e x p e r i m e n t s in which uniform edge loading is applied by means of the inflatable tubing t e c h nique, Exp. Mech. 7 (1967). [12] R.D. Mindlin and M.G. Salvadori, Analogies, in: Handbook for experimental s t r e s s analysis, ed. M. Hetenyi (Wiley, New York, 1950) chapter 16. [13] B. Gross, J . E . Strawley and W. F. Brown, S t r e s s intensity factors for a single-edge-notch tension specimen by boundary collocation of a s t r e s s function, NASA TN D-2395 (1964). [14] A. S. Kobayashi, R.D. Cherepy and W. C. Kinsel, A numerical procedure for estimating the s t r e s s intensity factor of a crack in a finite plate, J. Basic Eng., Trans. ASME 86 (1964) 681. [15] H. B. Wilson J r . , J. L. Hill and J. G. Goree, Mathematical studies of composite m a t e r i a l s II (U), Spec. Rept. No. S-50, Rohm and Haas Company, Redstone Arsenal R e s e a r c h Division, Huntsville, Alabama (1965). [16] L . E . Hulbert and F.W. Niedenfuhr, Accurate calculation of s t r e s s distributions in multiholed plates, J. Eng. Ind., Trans. ASME 87 (1965) 331. [17] R.W. Bailey and R. Fidler, S t r e s s analysis of plates and shells containing p a t t e r n s of r e i n f o r c e d holes, Nucl. Eng. Design 3 (1966) 41.
NOTE ON STRESSES A N D DISPLACEMENTS IN SQUARE PLATES A N D CYLINDERS W I T H PRESSURIZED CENTRAL CIRCULAR HOLES T. SLOT
Nuclear Materials and Propulsion Operation, General Electric Company Cincinnati, Ohio, USA Received 20 January 1967
Accurate elastic solutions for perforated squares subjected to internal pressure are reported for hole diameter to plate width ratios of 0.5, 0.6, 0.7, 0.8, 0.85 and 0.9. A point-matching procedure is used to generate solutions that satisfy the internal boundary conditions exactly and the external boundary conditions approximately. The degree of approximation is examined by inspection of residual tractions along the outer boundary. Plane-stress and plane-strain conditions are considered, and the superposition by which the solutions for external pressure can be obtained from the solutions for internal pressure is indicated.
1. INTRODUCTION T h i s note is c o n c e r n e d with p l a n e - s t r e s s and p l a n e - s t r a i n solutions for the p r o b l e m defined in fig. 1, w h e r e a hollow s q u a r e is shown s u b j e c t e d to i n t e r n a l p r e s s u r e . It is known f r o m the t h e o r y of e l a s t i c i t y that solutions of this type d e s c r i b e the conditions found in'thin plates and long c y l i n d e r s , r e s p e c t i v e l y . A c c u r a t e solutions that s a t isfy the boundary conditions for the hole exactly and the boundary conditions f o r the s q u a r e edge in c l o s e a p p r o x i m a t i o n a r e r e p o r t e d for D / W r a t i o s of 0.5, 0.6, 0.7, 0.8, 0.85 and 0.9. They w e r e obtained by p o i n t - w i s e m a t c h i n g of the boundary conditions for the outer boundary, in the s a m e m a n n e r that p r e v i o u s solutions w e r e obtained for hollow hexagons under i n t e r n a l p r e s s u r e [1]. By e m p l o y in g the p r i n c i p l e of s u p e r p o s i t i o n , the solutions for e x t e r n a l p r e s s u r e a r e a l s o r e a d i l y d e t e r m i n e d f r o m the solutions for i n t e r n a l p r e s s u r e . T h e p r o b l e m of the hollow s q u a r e with unif o r m loading applied to the inner or o u t e r bounda r y was c o n s i d e r e d in s e v e r a l other t h e o r e t i c a l i n v e s t i g a t i o n s [2-6]. P o i n t - m a t c h i n g p r o c e d u r e s r e s e m b l i n g the one used for the solutions in this note w e r e r e p o r t e d by Sekiya [2] and by Roy [3]. Sekiya obtained the s t r e s s e s at the hole for p r e s s u r i z e d hollow s q u a r e s with D / W r a t i o s of 0.25 and 0.5 and showed that the r e s u l t s for the second c a s e w e r e in good a g r e e m e n t with those r e p o r t e d by Kawaguchi [4], who solved this p r o b -
y D
X
o
J D
~
w
Fig. 1. Square plate with central hole subjected to internal pressure. l e m by a c o n f o r m a l mapping method. Roy obtained the s t r e s s e s for the i n n e r and outer boundary of p r e s s u r i z e d hollow s q u a r e s with D / W r a t i o s r a n g i n g f r o m 0.1 to 0.9. S t r e s s e s for the inner boundary w e r e given in t ab u l ar and g r a p h i cal f o r m , s t r e s s e s at the outer boundary only in g r a p h i c a l f o r m . Hengst [5] d ev el o p ed a l e a s t s q u a r e s a v e r a g i n g p r o c e d u r e to a p p r o x i m a t e conditions at the outer boundary. S t r e s s e s at the inner boundary w e r e d e t e r m i n e d for D / W r a t i o s
STRESSES AND DISPLACEMENTS IN SQUARE PLATES AND CYLINDERS from 0.1 to 0.5 and :[or two conditions of loading, n a m e l y s y m m e t r i c and a n t i s y m m e t r i c u n i f o r m loading of the outer boundary. A l e a s t - s q u a r e s a v e r a g i n g p r o c e d u r e was also used by Schlack and Little [6], who r e p o r t e d the s t r e s s e s at the hole and along the line of s y m m e t r y 0 =0 (see fig. 1) for D / W r a t i o s from 0.1 to 0.9. R e s u l t s tabulated for both s y m m e t r i c and a n t i s y m m e t r i c loading of the outer boundary were shown to be in g e n e r a l a g r e e m e n t with c o r r e s p o n d i n g r e s u l t s from the p a p e r s by Hengst and Roy. In c o m p a r i s o n with the i n v e s t i g a t i o n s cited above, the r e s u l t s p r e s e n t e d in the sequel a r e the most c o m p r e h e n s i v e for the loading condition and D / W r a t i o s c o n s i d e r e d . S t r e s s e s and d i s p l a c e m e n t s a r e tabulated for a l a r g e n u m b e r of locations along the i n n e r boundary, the outer b o u n d a r y and the axes of s y m m e t r y . A detailed a n a l y s i s is included of r e s i d u a l t r a c t i o n s at the outer boundary, which p e r m i t s an a s s e s s m e n t of the a c c u r a c y of the p o i n t - m a t c h i n g p r o c e d u r e employed. The effect of P o i s s o n ' s r a t i o on the values of the d i s p l a c e m e n t s is also considered. S t r e s s e s in s q u a r e plates with p r e s s u r i z e d c e n t r a l holes w e r e also m e a s u r e d on p h o t o e l a s tic models. E a r l y e x p e r i m e n t s by D u r e l l i and B a r r i a g e [7] were followed by an i m p r o v e d a n a l y s i s by Riley et al. [8]. Subsequently, North and Mantle [9] obtained s i m i l a r photoelastic s o l u tions by a different e x p e r i m e n t a l technique. The r e s u l t s in the last two p a p e r s appear to be in g e n e r a l a g r e e m e n t with the t h e o r e t i c a l solutions, except for high D / W r a t i o s , where s i g n i f i c a n t d i f f e r e n c e s were noted. A m o r e f a v o r a b l e o v e r all c o m p a r i s o n was o b s e r v e d between n u m e r i c a l and photoelastie r e s u l t s for hollow hexagons
[10,11].
2. ANALYSIS A digital c o m p u t e r p r o g r a m s p e c i a l l y d e v e l oped for the a n a l y s i s of plane p e r f o r a t e d s t r u c t u r e s was used to g e n e r a t e the n u m e r i c a l s o l u tions of i n t e r e s t . A d e s c r i p t i o n of this p r o g r a m and v a r i o u s solutions obtained with it were the s u b j e c t of an e a r l i e r publication in this j o u r nal [1]. In e s s e n c e , the p r o g r a m p e r m i t s the u s e r to c o n s t r u c t a s u i t a b l e s t r e s s function by s e l e c t i o n of a finite n u m b e r of t e r m s f r o m the following b i h a r m o n i c s e r i e s in p o l a r c o o r d i nates: F = a o log r + bo r2 + :Dn(anrn+2 + bnr n + Cnr -n + dnr_n+2) ( s i n nO~ \COS nO/ '
143
where ao, bo, an, bn, cn and dn a r e a r b i t r a r y coefficients, and n = 2, 3,4, . . . . It can be shown that the t e r m s f r o m this s e r i e s f u r n i s h solutions t h a t s a t i s f y the usual r e q u i r e m e n t s of e q u i l i b r i u m , compatibility and s i n g l e - v a l u e d d i s p l a c e m e n t s ; f u r t h e r m o r e , that boundary conditions placed on s t r e s s e s or d i s p l a c e m e n t at a point r e s u l t in l i n e a r a l g e b r a i c equations in the coefficients. The coefficients may t h e r e f o r e be d e t e r m i n e d from a set of b o u n d a r y conditions that m a t c h e s the n u m b e r of t e r m s in the chosen s t r e s s function. A c c o r d i n g ly, the computer p r o g r a m allows s t r e s s e s and d i s p l a c e m e n t s in any d i r e c t i o n to be specified for a r b i t r a r y points on the boundary. Once the coefficients a r e known, s t r e s s e s and d i s p l a c e m e n t s can be evaluated for any point of the s t r u c t u r e . The c o m p u t e r p e r f o r m s all the n e c e s s a r y calculations. To satisfy the conditions of s y m m e t r y in the p r o b l e m under c o n s i d e r a t i o n , the choice of the v a l u e s of n was limited to m u l t i p l e s of 4. Specifically, the solutions for D / W r a t i o s of 0.5-0.85 w e r e obtained by choosing the t e r m s connected with coefficients a o and bo, and the cosine t e r m s connected with n - v a l u e s of 4 , 8 , . . . ,24, for a total of 26 t e r m s in the s t r e s s function. To obtain s a t i s f a c t o r y a c c u r a c y for the case D / W = 0.9, the n u m b e r of t e r m s was i n c r e a s e d to 42 by the addition of t e r m s connected with n - v a l u e s of 28, 32, 36, 40. For the sector of the s q u a r e between lines of s y m m e t r y OB and OD in fig. 1, the boundary conditions along the outer edge BD a r e ax = 0 and Txy = O. T h e s e conditions were specified for a n u m b e r of d i s c r e t e points on BD. In the f i r s t five p r o b l e m s (D/W = 0.5-0.85) the points w e r e chosen at Y/YD = O, 0.2, 0.4, 0.6, 0.8, 0.9 and 1; in the sixth p r o b l e m (D/W= 0.9) additional points w e r e chosen at Y/YD = 0.1, 0.3, 0.5 and 0.7. It may be v e r i f i e d that for N values of n, t h e s e boundary conditions r e s u l t in 2N + 1 r e l a t i o n s between 4N + 2 coefficients, where N = 6 for the f i r s t five p r o b l e m s and N= 10 for the sixth p r o b lem. With r e f e r e n c e to the paper by Slot and Yalch [1], the additional 2 N + 1 r e l a t i o n s needed for the evaluation of the coefficients a r e p r o vided by the boundary conditions for the hole. It is shown t h e r e that by introducing s p e c i a l r e l a tions between the coefficients, ~ r = - p i and ~r0 =0 along the e n t i r e edge of the hole. R e g a r d i n g the p l a n e - s t r e s s d i s p l a c e m e n t s at the hole, they can be e x p r e s s e d in the following form:
144
T. SLOT Here, subscripts ~ and e are used to distinguish between plane-stress and plane-strain displacements, respectively.
E U r = -(1 - v)PiR - 2 a o R - 1 + 4Zn(anRn+l +dnR_n+l) ( s i n nO~ \cos n0/
E U0 = 4Zn(anR n+ l - d n R - n + l )
(
~ cOS
'
n0)
\ s i n nO/ "
These expressions were derived by eliminat i o n of c o e f f i c i e n t s bo, b n a n d c n f r o m t h e g e n eral expressions for the displacements and the above mentioned special relations between the coefficients required to satisfy the boundary c o n d i t i o n s a t t h e h o l e . It i s n o t e d t h a t f o r t w o s o l u t i o n s w i t h d i f f e r e n t P o i s s o n ' s r a t i o s , s a y v1 a n d v2, t h e d i s p l a c e m e n t s a t t h e h o l e a r e r e l a t e d as follows:
(Ur)v=v2 : (Ur)u:Vl + (v 2 - V l ) P i R / E , (Uo)v=v2 = (Uo)v=Vl • It is e v i d e n t f r o m t h e s e r e l a t i o n s t h a t in t h e c a s e of e x t e r n a l l o a d i n g , i . e . , Pi = 0, t h e d e f o r m a t i o n of t h e h o l e i s n o t d e p e n d e n t o n t h e v a l u e of P o i s s o n ' s r a t i o . ( T h i s o b s e r v a t i o n m a y b e generalized to include thin plates with multiple perforations that are loaded only along the ext e r n a l b o u n d a r y [1].) B y t h e r e c i p r o c i t y t h e o r e m , i t f o l l o w s t h a t i n t h e c a s e of i n t e r n a l p r e s s u r e t h e d e f o r m a t i o n of t h e s q u a r e b o u n d a r y i s n o t d e p e n d e n t on t h e v a l u e of P o i s s o n ' s r a t i o . F r o m t h e a n a l o g y [12] t h a t e x i s t s b e t w e e n plane-stress a n d p l a n e - s t r a i n p r o b l e m s it f o l lows that the in-plane stresses are the same for both. The axial stresses present under planestrain conditions may be obtained from Hooke's law w i t h t h e a s s u m p t i o n of z e r o a x i a l s t r a i n , o r a z = v((Tr + ~0) = V(ax + Cry). In a c c o r d a n c e w i t h the transition rules provided by the analogy, the displacements for plane strain may be obtained from the displacements for plane stress by rep l a c i n g E w i t h E / ( I - v 2) a n d v b y v / ( l - v ) . T h e following relations result for the displacements at the inner and outer boundary:
3. R E S U L T S The computer solutions acquired for hollow squares under internal pressure loading are s u m m a r i z e d in t a b l e s 1 - 9 . In p a r t i c u l a r , t a b l e s I-6 give the stresses and displacements at vario u s p o i n t s a l o n g t h e b o u n d a r i e s a n d t h e a x e s of s y m m e t r y f o r t h e s i x v a l u e s of D / W c o n s i d e r e d . A graphical representation of t h e b o u n d a r y s t r e s s e s is s h o w n in f i g s . 2 a n d 3. T h e h i g h e s t stresses is f o u n d o n t h e i n n e r b o u n d a r y f o r t h l o w e r v a l u e s of D / W a n d on t h e o u t e r b o u n d a r y f o r t h e h i g h e r v a l u e s of D / W . O n t h e o u t e r b o u n d a r y t h e m a x i m u m s t r e s s o c c u r s a t 0 = 0, w h e r e a s on t h e i n n e r b o u n d a r y t h e l o c a t i o n of m a x i m u m s t r e s s v a r i e s f r o m 0 = 45 ° f o r t h e t h r e e l o w e r D / W r a t i o s to s m a l l e r v a l u e s of 0 for the three higher D/W ratios. Table 1 Values of (~0/Pi along inner boundary (AC)
D/w
60 0 5 10 15 20 25 30 35 40 45
0.5
0.6
0.7
0.8
0.85
0.9
1.412 1.423 1.454 1.501 1.557 1.615 1.667 1.710 1.736 1.746
1.599 1.625 1.697 1.803 1.926 2.048 2.154 2.235 2.285 2.302
1.834 1.899 2.077 2.326 2.594 2.835 3.021 3.144 3.212 3.233
2.107 2.306 2.831 3.500 4.122 4.570 4.811 4.888 4.883 4.871
2.258 2.658 3.674 4.878 5.851 6.394 6.534 6.422 6.245 6.165
2.352 3.412 5.824 8.135 9.497 9.867 9.534 8.862 8.235 7.982
Table 2 Values of (~y/Pi along outer boundary (BD)
D/W Y /YD
0.5
0.6
0.7
0.8
0.85
0.9
Inner boundary:
( Ur) e = (1 - v2)(Ur)a + (1 + v ) v 2 p i R / E , (uo)~ = ( 1 - v 2 ) ( v o ) a
;
Outer boundary:
( Ux)~ = (1 - v 2 ) ( U x ) a , (Uy)e = (1 - v2)(Uy)~ .
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1.018 0.973 0.848 0.674 0.481 0.297 0.142 0.032 -0.022 -0.022 0
1.753 3.106 6.064 1.663 2.913 5.566 1.423 2.406 4.292 1.101 1.755 2.765 0.748 1.085 1.363 0.434 0.536 0.375 0.184 0.141 -0.232 0.018 -0.086 -0.502 -0;053 -0.145 -0.436 -0.039 -0.075 -0.164 0 0 0
9.174 8.258 5.962 3.361 1.222 -0.061 -0.750 -1.019 -0.793 -0.254 0
15.75 13.40 8.266 3.597 0.453 -1.258 -1.963 -1.997 -1.382 -0.386 0
STRESSES AND DISPLACEMENTS IN S QUARE P L A T E S AND CYLINDERS Table 3 V a l u e s of (YO/Pi and (Yr/Pi along r a d i a l line 0 = 0 (AB)
D/W
p* 0.5
0.6
0.7
Table 4 V a l u e s of (YO/Pi and (~r/Pi along r a d i a l line 0 = ~
0.85
0.9
0.5
0.6
0.7
1.412 1,134 0.950 0.865 0.879 1.018
1.599 1.511 1.426 1.412 1.505 1.753
1.834 2.044 2.179 2.353 2.636 3.106
0 0,2 0,4 0,6 0,8 1
-1 -0.594 -0.324 -0.148 -0.040 0
-1 -0.664 -0.389 -0.184 -0.051 0
-1 -0.726 -0.452 -0.222 -0.062 0
0.8
0.85
0.9
6.165 3.467 1.395 0.092 -0.268 0
7.982 4.331 1.429 -0.218 -0.463 0
-1 -0.228 0.174 0.276 0.129 0
-1 -0.043 0.473 0.510 0.210 0
gO/Pi
~0/Pi 0 0.2 0.4 0,6 0.8 1
(CD)
D/W
p *
0.8
145
2.107 2.931 3.609 4.285 5.069 6.064
2.258 3.716 4.995 6.249 7.602 9.174
2.352 5.141 7.711 10.25 12.87 15.75
0 0.2 0.4 0.6 0.8 1
1,746 0.991 0.617 0.306 0.055 0
2.302 1.386 0.841 0.378 0,046 0
3.233 1.950 1.083 0.398 -0.001 0
-1 -0.814 -0.552 -0.291 -0.089 0
-1 -0.841 -0.581 -0.306 -0.089 0
0 0.2 0.4 0.6 0,8 1
-1 -0.432 -0.200 -0.067 0.002 0
-1 -0.463 -0.215 -0.058 0.013 0
-1 -0.447 -0.173 -0.009 0.035 0
* p = (r- rA)/(r B- rA) .
*p
(r- rc)/(rD- rc).
Table 5 V a l u e s of EUr/PiR and EUo/PiR along i n n e r b o u n d a r y
Table 6 V a l u e s of EUx/PiR and EUy/piR along o u t e r b o u n d a r y (BD)
(~r/Pi -1 -0.784 -0.516 -0.264 -0.077 0
ar/P~
(AC)
0.5
0.6
0.7
0.8
0.85
0.9
Y/YD 0.5
EUr/PiR - plane s t r e s s (v=0.3) 0 5 10 15 20 25 30 35 40 45
1.951 1.947 1.935 1.916 1.894 1.870 1.848 1.831 1.819 1.815
2.436 2.426 2.395 2.349 2.295 2.238 2.186 2.145 2.118 2.109
3.347 3.318 3.237 3.118 2.979 2.840 2.717 2.623 2.564 2.544
5.459 5.364 5.103 4.732 4.325 3.942 3.627 3.400 3.265 3.221
7.860 7.659 7.114 6.370 5.591 4.900 4.362 3.995 3,786 3.719
0 -0.020 -0.038 -0.051 -0.058 -0.058 -0.050 -0,037 -0.020 0
0 -0.046 -0.085 -0.114 -0.128 -0.126 -0.109 -0.080 -0.042 0
0 -0.103 -0,190 -0.250 -0,275 -0,266 -0,226 -0,163 -0,085 0
0 -0.258 -0.467 -0.595 -0.630 -0.583 -0.476 -0.332 -0.169 0
0 -0.445 -0,792 -0.982 -1.006 -0.899 -0.710 -0.480 -0.240 0
13.19 12.65 11.26 9.498 7.812 6.441 5.455 4.828 4.494 4.391
0 -0.872 -1.495 -1.762 -1.712 -1.454 -1.093 -0.710 -0.344 0
Radial d i s p l a c e m e n t s f o r v=v 2 f r o m r a d i a l d i s p l a c e m e n t s for V=Vl=0.3:
(Ur)v=v2 = (Ur)v=Vl + (v2 - V'I)PiR/E . D i s p l a c e m e n t s f o r plane s t r a i n f r o m :
(vr) e = (1 - v 2 ) l V r ) a + (1 + v ) ~ 2 p i R / E , (U0) e = ( 1 - v2)(U0)(r. ( S u b s c r i p t s (Y and E denote plane s t r e s s strain, respectively.)
0.6
0.7
0.8
0.85
0.9
EUx/PiR - plane s t r e s s (all P)
EUo/PiR - plane s t r e s s (all lJ) 0 5 10 15 20 25 30 35 40 45
-1 -0.341 -0.004 0.134 0.083 0
D/W
D/W
00
4.871 2.824 1.310 0.273 -0.135 0
and plane
0 0.2 0.4 0.6 0.8 1
1.338 1.256 1.070 0.889 0.794 0.783
1.902 1.768 1.472 1.203 1.072 1.058
2.880 2.623 2.087 1.643 1.453 1.439
5.053 4.416 3,206 2.328 2.013 2.005
7.482 6.294 4.212 2.854 2.409 2.410
12.84 10.08 5.927 3.609 2.935 2.953
EUy/PiR - plane s t r e s s (all p) 0 0.2 0.4 0.6 0.8 1
0 0.384 0.652 0.773 0.790 0.783
0 0.547 0.910 1.058 1.070 1.058
0 0.818 1,316 1.476 1.460 1.439
0 1.360 2.052 2.161 2.050 2.005
0 1.889 2.693 2.701 2.481 2.410
0 2.876 3.726 3.483 3.064 2.953
D i s p l a c e m e n t s f o r plane s t r a i n f r o m :
(Vx)¢ = (1 - v2)(Ux)(X ;
(Uy)E= (1 - V2)(Vy)0, ,
( S u b s c r i p t s (~ and ¢ denote plane :~train, r e s p e c t i v e l y . )
stress
and plane
146
T. SLOT
V a l u e s of
Y/YD
gx/Pi
Table 7 a l o n g o u t e r b o u n d a r y (BD) *
V a l u e s of
Table 8 o u t e r b o u n d a r y (BD) *
7xy/P i a l o n g
D/W 0.5
0.6
0.7
D/W
0.8
0.85
0,95
0 0 0 0 0 0 0 0,05 0.1 0 0 0 0.001 0.004 0.021 0.15 O.2 0 0 0 0 0 0 0.25 0,3 0 0 -0.001 - 0 . 0 0 3 -0.010 - 0 , 0 5 4 0.35 0.4 0 0 0 0 0 0 0,45 0.5 0.001 0,001 0.002 0.004 0.012 0.057 0.55 0.6 0 0 0 0 0 0 0.65 0.7 -0.001 -0.002 -0.003 -0.006 -0,009 -0.025 0.75 0.8 0 0 0 0 0 0 0,85 0.001 0.001 0.002 0.003 0.001 - 0 . 0 0 3 0.9 0 0 0 0 0 0 0,95 - 0 . 0 0 2 - 0 . 0 0 4 - 0 . 0 0 6 - 0 . 0 1 1 0.004 0,013 1 0 0 0 0 0 0 * V a l u e s of
lax/Pi[
Y/YD
0.95 0 -0.002 0 0.00( 0 -0.010 0 -0.014 0 0,006 0 -0.005 0 J ~04 0 -0.003 0 -0,005 0 0,048 0
s m a l l e r than 0.0005 e n t e r e d a s 0.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 O.5 0.55 0.6 0,65 0.7 0.75 0.8 0.85 0.9 0.95 1
0.5
0.6
0.7
0.8
0.85
0.9 5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.001
0,004
0.015
0.043
0.165
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0,005
0.024
0.131
0 0 0 0.001 0
0 0 0 0,001 0
0 0 0 0 0
0 -0.001 0 -0.001
0 -0,020 0 -0.019
0 -0.034 0 -0.026
* V a l u e s of ]
7xy/Pil
-0,008
-0.007
,
(~x/Pi and "rxy/Pi a l o n g
0.5
0.6
0.7
o u t e r b o u n d a r y (BD) *
0.8
0185
0.9 t
0.9
0.013 -0.034 0.036 -0.016 0.003
0 0 0 -0.002 -0.001
gx/Pi "ffx/Pi = f~J yj _ y~ dy 0 0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8 1
0
1
0 0 0 -0.001 0
0 0 0.001 -0.001 -0.001
0
0
0 -0.001 0.001 -0,002 -0.002
0.001 -0,002 0.003 -0.004 -0,003
0.003 -0.006 0.007 -0.006 -0.003
-0.001
-0.001
-0.001
v,,Jt, i 0 0.2 0.4 0.6 0.8
0,2 0.4 0.6 0.8 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0
1
0
0
0
*
V a l u e s of
[ffx/Pi]
"~ 26 t e r m s in s t r e s s functiofi (N = 6). $ 42 t e r m s in s t r e s s f u n c t i o n (N = 10).
and
IYxy/Pil
r~i j
0
0
-0,001
TxY/Pi dy Yj-Yi
-0,005 0,004 -0,004 0.003 0
-0.136
-0.146
0
0 -0.00~ 0 0.002 0 -0.010 0 -0.014 0 0.006 0 -0.005 0 0.004 0 -0,003 0 -0,005 0 -0.048 0
s m a l l e r t h a n 0.0005 e n t e r e d a s 0.
D/W
Yi/YD Yj/YD
-0.029
0
Table 9
V a l u e s of
-0.029
0.9 $
-0.017 0.018 -0.019 0.015 -0.001
-0.084 0.089 -0.093 0,081 -0.003
0.004 -0.013 0 0 0.014
-0.001
-0,002
0,001
s m a l l e r t h a n 0.0005 e n t e r e d a s 0.
STRESSES AND DISPLACEMENTS IN SQUARE PLATES AND CYLINDERS 10
I
I
I
~
I
16
I
147
Jl
I
I
14 12
0.9
10 •
o
®
4
a
6
~
4
0.7
Q
3
2
2
C D/W
0
= 0.5
= 0.5
1 _ 0
D/W
--2 I
f
I
I
I
I
1
I
5
10
15
20
25
30
35
40
-~ 45
0
I
I
I
I
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Distance ratio Y/YD
Angle e, degrees
Fig. 2. Stress distribution along inner boundary (AC).
Fig. 3. Stress distribution along outer boundary (BD).
T a b l e s 7-9 f u r n i s h insight into the a c c u r a c y of the s o l u t i o n s by allowing inspection of the d e g r e e of a p p r o x i m a t i o n a c h i e v e d for the boundary conditions along edge BD. In t a b l e s 7 and 8, v a l ues of ax and Txy a r e shown for the points at which boundary conditions w e r e s p e c i f i e d , as w e l l as f o r the points located midway the f o r m e r . T a b l e 9 g i v e s the a v e r a g e v a l u e s of t h e s e s t r e s s e s in f i v e equal i n t e r v a l s along BD and a l s o for the e n t i r e length of BD. It is a p p a r ent f r o m t h e s e t a b l e s that the actually obtained boundary conditions c l o s e l y a p p r o x i m a t e the t r u e boundary conditions f o r the s q u a r e edge of the plate. F o r the p a r t i c u l a r c a s e of D / W = 0.9, it was m en t i o n ed b e f o r e that the n u m b e r of t e r m s was i n c r e a s e d f r o m 26 to 42 in o r d e r to i m p r o v e the a c c u r a c y of the solution. To i l l u s t r a t e the i m p r o v e m e n t obtained, the boundary conditions applicable to both solutions a r e r e p o r t e d in the tables. C o r r e s p o n d i n g solutions for e x t e r n a l p r e s s u r e may be obtained by s u b t r a c t i n g the solution for i n t e r n a l p r e s s u r e f r o m the solution for p r e s s u r e on both b o u n d a r i e s . T h e l a t t e r c a s e of l o a d ing p r o d u c e s a u n i f o r m e q u i - b i a x i a l s t r e s s field of i n t e n s i t y -p, w h e r e p is the applied p r e s s u r e . S t r e s s e s and d i s p l a c e m e n t s for this c a s e a r e t h e r e f o r e as follows:
Plane
stress:
Plane
strain:
Inner boundary:
a r / P = ao/P = -1 E Ur/PR = - (1 - ,) Vo =O
Zr/P = e o / P =
-1;
%/p = -2 v E U r / p R = -(1+ v)(1-2 ~)
u0=0
Outer boundary:
~ x / P = ~ y / P = -1 s V x / P n : -(1- v)(W/D)
a x / P = e y / P : -1, az/p = -2v EVx/pR = -(1+ v)(1-2 v)(W/D)
Ev/
n= -(1 - v ) ( W / D ) ( y / y D )
EUy/pR
=
-(1+,)(1-2,)(W/D)(y/yD)
4. CONCLUSION A s u m m a r y has been p r e s e n t e d of an a p p r o x i m a t e a n a l y s i s of the s t r e s s e s and d i s p l a c e m e n t s in s q u a r e p e r f o r a t e d p l a t e s and c y l i n d e r s due to i n t e r n a l or e x t e r n a l p r e s s u r e loading. It has been shown that a c c u r a t e n u m e r i c a l a n s w e r s f o r
148
T. SLOT Table 10 Comparison with r e s u l t s f r o m other investigations
D/W
r
e°
Hengst [5]
Kawaguchi [4]
Sekiya [2]
Schlack and Little [6]
Present analysis
1.365 1.438 1.711 1.810
1.408 1.667 1.746
1.412 1.501 1.667 1.746
Roy [3]
~e/Pi 0.5 0.5 0.5 0.5
R R R R
0 15 30 45
1.416 1.503 1.667 1.745
1.424 1.511 1.676 1.756
1.385 1.486 1.688 1.723
0.6 0.7 0.8 0.9
R R R R
0 0 0 0
1.522 1.754 2.133 2.920
1.595 1.839 2.110 2.415
1.599 1.834 2.107 2.352
0.6 0.7 0.8 0.9
R R R R
45 45 45 45
2.412 3.360 4.940 8.316
2.304 3.240 4.866 7.990
2.302 3.233 4.871 7.982
0.5 0.6 0.7 0.8 0.9
½W ½W ½W ½W ½W
1.037 1.781 3.107 6.074 15.60
1.018 1.753 3.106 6.064 15.75
(~y/Pi 0 0 0 0 0
1.220
t h e s e p r o b l e m s of e n g i n e e r i n g i n t e r e s t c o u l d b e obtained by generating solutions for the biharmonic field equation that satisfy the boundary conditions for the hole entirely and the boundary conditions for the square edge in a point-wise m a n n e r . In t h e a b s e n c e of r i g o r o u s p r o o f f o r convergence of t h e n u m e r i c a l p r o c e d u r e inv o l v e d , t h e a c c u r a c y of t h e a c q u i r e d s o l u t i o n s h a s b e e n e x a m i n e d b y c o m p u t a t i o n of r e s i d u a l tractions along the square boundary. O t h e r t h e o r e t i c a l i n v e s t i g a t i o n s of t h e p r o b lem treated here have been cited from the litera t u r e . In c o m m o n w i t h t h e p r e s e n t w o r k , t h e s e investigations also involved series solutions that satisfied the internal boundary conditions exactly and the external boundary conditions in approxim a t i o n . A l i m i t e d c o m p a r i s o n of r e s u l t s b y d i f f e r e n t m e t h o d s i s p r o v i d e d i n t a b l e 10. ( N u m e r i cal results attributed to Hengst and Sekiya were computed by the author from equations provided by the investigators.) Although there are some differences, it would hardly be appropriate to j u d g e t h e p o t e n t i a l a c c u r a c y of t h e v a r i o u s m e t h o d s o n t h e b a s i s of t h i s c o m p a r i s o n , f o r two r e a s o n s : 1) t h e n u m b e r of t e r m s u s e d t o g e n e r a t e the solutions varies considerably, a n d 2) n o t enough information is given on the stresses at t h e o u t e r b o u n d a r y . A d e t a i l e d c o m p a r i s o n on t h e
0.986
1.15 1.94 3.32 6.14 14.73
b a s i s of a n e q u a l n u m b e r of t e r m s i n t h e s t r e s s f u n c t i o n w o u l d b e of i n t e r e s t , b u t i s b e y o n d t h e s c o p e of t h i s n o t e . In c o n c l u s i o n , a t t e n t i o n is c a l l e d t o r e c e n t p u b l i c a t i o n s b y G r o s s et al. [13], K o b a y a s h i e t a l . [14], W i l s o n et al. [15], H u l b e r t a n d N i e d e n f u h r [16] a n d B a i l e y and: F i d l e r [17], w h i c h a r e concerned with solutions for complex structural p r o b l e m s o b t a i n e d w i t h t h e a i d of p o i n t m a t c h i n g of b o u n d a r y c o n d i t i o n s . ( A d d i t i o n a l r e f e r e n c e s may be found in the papers by Hulbert and Nied e n f u h r [16] a n d Slot a n d Y a l c h [1].)
5. N O T A T I O N
Co, b o = c o e f f i c i e n t s in s t r e s s f u n c t i o n an, bn, Cn, d n = c o e f f i c i e n t s in s t r e s s f u n c t i o n , n = 2,3,4,... E, v = Young's modulus, Poisson's ratio F = biharmonic stress function P, Pi = pressure loading parameters R, D = r a d i u s , d i a m e t e r of h o l e Ur, U 0 =displacements in polar coordinates Ux, Uy = d i s p l a c e m e n t s in r e c t a n g u l a r c o ordinates W = w i d t h a c r o s s f i a t s of s q u a r e
S:FRESSES AND DISPLACEMENTS IN SQUARE PLATES AND CYLINDERS
r~ °r, ~0, TrO
= summation symbol = s t r e s s c o m p o n e n t s in p o l a r c o o r dinates = s t r e s , s components in r e c t a n g u l a r coordinates
ACKNOWLEDGEME].~ T h e w o r k r e p o r t e d h e r e i n o r i g i n a t e d in p a r t w i t h d e v e l o p m e n t p r o j e c t s s p o n s o r e d b y t h e U.S. A t o m i c E n e r g y C o m m i s s i o n u n d e r C o n t r a c t AT (40-1)-2847. P e r m i s s i o n to publish this note is gratefully acknowledged.
REFERENCES
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149
[6] A. L. Schlack and R.W. Little, Elastostatic p r o blem of a p e r f o r a t e d square plate, J. Eng. Mech. Div., P r o c . ASCE 90 (1964) 171. [7] A. J. Durelli and J. Barriage, S t r e s s distribution in square plates with hydrostatically loaded central c i r c u l a r holes, J. Appl. Mech., Trans. ASME 77 (1955) 539. [8] W. F. Riley, A. J. Durelli and P. S. Theocaris, F u r ther s t r e s s studies on a square plate with a p r e s surized central c i r c u l a r hole, P r o c . 4th Midwest Conf. Solid Mech., Univ. of Texas, Austin, Texas, (1959). [9] W. P. T. North and J. B. Mantle, Photoelastic study of s t r e s s e s in hydrostatically loaded cylinders with noncircular external boundaries, Exp. Mech. 2 (1962) 91. [10] P. D. Flynn, S t r e s s e s in hollow hexagons under external p r e s s u r e , Exp. Mech. 2 (1961) 148. [11] T. Slot, On the accuraey of two-dimensional photoelastic e x p e r i m e n t s in which uniform edge loading is applied by means of the inflatable tubing t e c h nique, Exp. Mech. 7 (1967). [12] R.D. Mindlin and M.G. Salvadori, Analogies, in: Handbook for experimental s t r e s s analysis, ed. M. Hetenyi (Wiley, New York, 1950) chapter 16. [13] B. Gross, J . E . Strawley and W. F. Brown, S t r e s s intensity factors for a single-edge-notch tension specimen by boundary collocation of a s t r e s s function, NASA TN D-2395 (1964). [14] A. S. Kobayashi, R.D. Cherepy and W. C. Kinsel, A numerical procedure for estimating the s t r e s s intensity factor of a crack in a finite plate, J. Basic Eng., Trans. ASME 86 (1964) 681. [15] H. B. Wilson J r . , J. L. Hill and J. G. Goree, Mathematical studies of composite m a t e r i a l s II (U), Spec. Rept. No. S-50, Rohm and Haas Company, Redstone Arsenal R e s e a r c h Division, Huntsville, Alabama (1965). [16] L . E . Hulbert and F.W. Niedenfuhr, Accurate calculation of s t r e s s distributions in multiholed plates, J. Eng. Ind., Trans. ASME 87 (1965) 331. [17] R.W. Bailey and R. Fidler, S t r e s s analysis of plates and shells containing p a t t e r n s of r e i n f o r c e d holes, Nucl. Eng. Design 3 (1966) 41.