Indi an Journaillf Engineering & Materials Sciences Vlll.6. Junc 1l)l)9. pp . 125- 134

Interaction effect of rectangular hole and arbitrarily oriented elliptical hole or crack in infinite plate subjected to uniform tensile loading at infinity

Vijay G Ukadgaonkcr & Pradeep J Awasare

Department of Mechanica l Engineering, Indian Institute of Technology, Powai, Mumbai 400076. India Received 27 Mar 1998: accepted 4 January 1999

The problem of interaction of stresses due to the presence of a crack or elliptical hole near a rectangular opening in. an infinite plate subjected to uniform stresses at infinity is studied in th is paper. This problem is of importance in a passenger aircraft when a crack is found in the vicinity of the door. The stress concentration factors near the hole and the stress intensity factors near the crack tip are evaluated for various orientations of crack and various lengths and distances of the crack from the hole. In this analysis, the two complex stress function s are found out using Schwarz Alternating Method.

The problem of interaction effect of two elliptical holes is important in the pressure vessel industries and nuclear industries. Similarly, the problems of interaction effect of rectangular hole and the elliptical hole have importance in pressure vesse l industries and the aerospace industries. Most of the airplanes have the openin gs like doors and windows, which are of the rectangul ar shape, or the windows are modified to the elliptical shape to reduce the stress concentration due to corners of the rectangular. opening. It is also observed that the cracks are originating from the high stress concentration area at the corner points of the rectangular open ing.

The interacti on effect of the two rectangular openings fo r in-plane loading is analysed by U kadgaonker el al.I to determ ine stress concentration around the ho les. Some of the authors had simplifi ed the problem of two rectangular holes to two elliptical holes, such that one of the ellipse of larger dimension corresponds to the door and smaller one to window. But actual shape of the opening have to be taken into consideration while analysi ng the problem. The est imati on of the SIF va lues of corner cracks at the doors and windows of rectangular shape is the most practical problem . The need for stress analysis of such situations moti vated to solve the problem of interaction effect of the rectangular hole and arbitrarily oriented elliptical hole.

Statement of the Problem The two dimensiona l elasto-static problem of

infinite plate with rectangular hole and arbitrarily

oriented elliptical hole subjected to uniform load ing at infinity is considered as shown in Fig. I . Let the rectangular hole have side length dimensions a and b with rounded corners of radius r c, and the elliptical hole with semi-major axis length g and semi-m inor ax is h. The semi-major ax is of the e lliptica l hole is arbitrari Iy oriented at angle f3 with respect to the rectangular hole axis. The plate is subj ected to uniform stretch p at an angle a with rectangul ar side a . Both the hole geometries are stress free , hence the resultant force on the hole boundaries will be zero.

First, single hole problem for the rectangular hole and elliptical hole in an infinite plate subj ected to uniform loading at infinity is solved, then the stress functions are translated to the center of second hole to determine the boundary conditi on at the ficti ti ous second hole boundary . From th is boundary.cond iti on the corrected stress functions are obtained to' create a stress free hole . Superposing the translated and corrected stress functions , one gets .the fin al so lution valid around the second hole boulldary as exp lained by Ukadgaonker and Pati l2 .

Single hole solution for rectangular hole

The single hole solution is obtained by Savi n's method' and by mapping the infinite region outside of the unit circle on the infinite region outs ide of the rectangular hole boundary as, •

,I, ( / ) - p R \ / al a2 a, a4 l 'f'1 ':>1 -- ':>1 +-+ )+-5 +-7 4 SI S I SI SI

R/ - 2ia p ':>I e

2

126 INDIAN 1. ENG. MATER. Sel.,JUNE 1999

p

" t Yo

" ~., --- - --+--_1, ------1f---

20 I I

I· .\

J

Fig. I-Infinite plate with arbitrarily oriented rectangular hole and elliptical hole subjected to uniform loading

+ pR [ 6·IS18

+ "2St' + " 3S14

+ " 4S12

+ "5 ] 2 s? - 1111 SI7

- 31112S15

- 51113S13

- 7 1114 SI

... (I)

where :?' I=CVI .(SI) IS mapping function for

rectangular hole boundary, such that

(

ml 1112 111 ) ZI = WI ((I) = R (I + - + - ) + 5

(I (I ( I

Only five terms in mapping function will be sufficient to define the rectangular boundary with straight sides and rounded corners, hence the infinite series of mapping function is truncated to first five terms. The 1111; m2, 111 ) , 1114 and R will define the size and corner radius of the rectangular hole.

Single hole solution for elliptical hole

Similarly, the single hole solution for the infinite plate with elliptical hole subjected to uniform loading at infinity at an angle ( a - (3) with the major axis of the ellipse is also obtained by mapping the outside

region of the elliptical hole on outside of the unit circle as,

p S [ (2 e2i

(a - /l) - n )] ¢2 «(2) = 4 (2 + (2

(I"" ) _ _ p S [ I"" e-2/(a-fJ)

'1'2 ., 2 - 2 ., 2

+ ~ _ (1 + n si ) (2 e2i(a-

p) - n)]

S2 S2 (si - n)

Let NI = (2 e2i(u - fJ) - n) and N2 = (e2i

( a -P 1 - n)

which are complex constants, hence <P2(l;2) and \jhCS2) will become

p S [ Nil ¢2((2) = 4 ( 2 + Z and

[

(2 e-2i(a - /l) -I _ pS 2 i

'fI2«(2 ) - - 2 + _I _ (I + n ~2) N2 J" ... (2)

(2 (2 «(2 -n)

UKADGAONKER & AWASARE: INTERACTION OF STRESSES IN AN INFINITE PLATE 127

where Z2 = (02(S2) is a mapping function for the ellip­tical hole boundary, such that

z2 =W2(S2)=S(S2 +!!....), S=g+h and S2 2

g-h n=--

g+h

Two hole solution around elliptical hole The single hole solutions of the individual holes

and first approximation solution do not take into consideration the interaction effect of the neighbouring hole. The effect of neighbouring hole on the stress distribution can be accounted for by Schwarz's Alternating Method . The stress functions given by Eq. (I) are transferred to the center of the second hole by rotation and translation of coordinates system as shown in Fig. 2.

tPI2«(2)=tPl ((2 +co)eip

)e-iP

and 1f112«(2) = IfII ((2 + co) eiP )eip + Co tPt2«(2) where Co is root of the equation Zo = (OI (co) such that

• r-- ~l

"'/ "

Z"ZIZ.

Fig. 2-Translation and rotation of coordinates in z-plane and 1;,­plane for rectangular hole and elliptical hole

I col> 1.0 and there will be only one root, which is

outside of the unit circle. The translated stress func­tions at the center of elliptical hole are

a3 a4

+ BP e6iP + BP 7 e8iP

p R BP e - 2i(a - Pl

1f112«(2) =- 2

pR +--

2

&( e8iP Bp8 + &2 e6iP Bp6 + &3 e4iP BP4

+ &4 e2iP BP2 + &S

Bp9 e8iP -:- m( BP7 e6iP _ 3m2 BpS e4iP

3 2P - 5m3 BP e I - 7 m4 BP

+ Co tPt2 «(2) ., .(3)

where BP = S2 + Co From these stress functions (Eq . (3» , the boundary condition on fictitious elliptical hole boundary is ob­tained as,

r ( ) ,/, () (O2('2),/,' () -(-) (4) JI2 12 = Y'1 2 12 +, Y'12 '2 + '1'12 12 (02 (/2 )

To obtain the stress free boundary at elliptical hole, a problem of an infinite plate with elliptical hole is solved in which the elliptical hole boundary is sub­jected to negative loading given by Eq. (4) and there is no loading at infinity. This will give the corrected stress functions valid around elliptical hole as,

tP22(t;2) = 121 + 122 + 123 + 124 + 125 + 131 + 132 + 141

+ 152 + IS3 + 1~4 + 155

'l'22(t;2) = 161 + 162 + 1(J3 + 164 + 165 + 171 + 172 ... (5)

Superposition of equations (3) and (5) gives the solu­tions valid around the elliptical hole, for the problem of interaction of elliptical hole with rectangular hole in an infinite plate.

tP2 «(2) = tPI 2 «(2) + tP12 «(2)

1f12«(2) = 1f112«(2) + IfI n «(2) .. . (6)

Two hole solution around rectangular hole

The stress fUllctions given by Eq . (2) are transferred to the center of the rectangular hole by rotation and translation of coordinate system. 111

reverse order of the elliptical hole as shown 111

Fig. (2).

128 INDIAN J. ENG. MATER. SCI., JUNE 1999

¢-21(SI) = rh (SI -CI)e-iP )eiP

'f21(SI) = 'f2 (SI _CI)e-iP )e-iP -CI ¢ZI(SI)

where CI is root of the equation . Zo = ffi2(CI) such that

I ell > 1.0; there are two roots, one o~ them will be

outside unit circle. The translated stress functions at the center of rectangular hole will be

"',«(,) = P4S [(, _ G, + (;: ~2:)]

11',,«(,) = - p2S [«(, -G,) e-

2ia + «(, ~ G,J

p s [ I + n( I - CI )2 e -2iP 2 , ]_ - do' (I") (") +- + 2 - 2ifJ ) CI'f'21 ':>1 ... I

2 (sl-el)(sl-cl) e -n

The translated stress functions (Eq. (7» at the rectan­gular hole boundary give the boundary condition.

. WI(tl)~( ) -(-) (8) hl(tl )=¢2I(tI)+, 'f'21/1 +'f21 /1 .. . WI (II)

Similar to elliptical hole problem, by applying nega­tive of boundary condition (8) at rectangular hole boundary, the corrected stress functions valid around rectangular hole are obtained as,

¢II (SI) = J 21 + J 22 + J 31 + J 32 + J 4 + J 52

'fIICr;,) =J61 + J 62 +J71 + I n - J IO ... (9)

Superposition of Eqs (7) and (9) gives the solution valid around the rectangular hole, which takes into account the interaction effect of the elliptical hole in an infinite plate.

¢ICSI)= ¢2I Cr;,) + ¢IICSI) VI,(S,) = Vl21(S,) + VI, ,(S,) ... (10)

The stress functions given by Eqs (6) and (10) are valid around the elliptical hole and rectangular hole boundary respectively and satisfY the stress free boundary condition exactly at respective boundaries. By setting different values of m), 1112, m ), m4 aspect ratio and corner radius of the hole can be changed to the required shape. A particular set of values of m),

I1h ... are chosen, as mentioned below, to give the ra­ti~ of sides bla = 0.4133 and corner radius rcla = 0.15, to suit the door and window geometry of the passen­ger aircraft airbus A300B. 1111 = 0.38, 1112 = - 0.1248, m ) = -().O 168, m4 = -0.0002, and R will be given by

R = a

The shape of the rectangular hole opening is shown in Fig. I .

The stress components around the hole are derived by substituting the corresponding stress functions into Eq. (11) as given below:

a + a = a + a = 4 Re [¢'(S)_] x y r 0 w'er;)

CFy -CFr+2ifrLr 2 rw(S)¢"(S)+VI'(O ] ./ w(S)~ ... ( II)

CFO - CFr + 2i fr'(J = 22s2 [w(s) ¢"(S) + VI'(S)] S (V'(S)

The solutions are checked for the correctness condition, i.e., if two holes are separated by very large ligament distance it should give the results corresponding to single hole solution. It is found that SCF at boundary of elliptical hole are R/S times more or less, depending upon the size of the opening. This is due to the stress functions valid around the elliptical hole, which are solely dependent on R and while calculating the stress values the mapping function for the elliptical hole is used . Hence SCF and SIF values are computed by multiplying the respect ive equations with SIR. Similar observation is also made in the case of the stress functions valid around the rectangu lar hole.

Results and Discussion

Effect of ligament thickness for collinear holes along x- axis and a = 90°

The SCF variation along the boundary of the elliptical hole is shown in Fig. 3, for bla = 0.4133 , rcla =0.15, glb = 0.5 , and dlb = (zo - g - a)/b = 1.2 and various values of glh = 2.0, 1.0, 0.5. It shows that maximum SCF is not observed at inner tip of ellipse and its location is displaced by some angle. This is due to the stress concentration at the corners of the rectangular hole. The maximum SCF at the corner of the rectangular hole is changed from its single hole solution, from 4.45 to 4.743 for elliptical hole with glh = 2.0. The maximum SCF for th~ recta~gul~r hole corner is 4.762, when it is interactIng with circular ho le (glh = 1.0), and with elliptical ho le having glh =

0.5, it is 4.837. The variation of the maximum SCF at elliptical hole with ligament thickness for glh = ~ . O,

1.0, and 0.5 is shown in Fig. 4 and its angular location is presented in Table 1. It is observed that for glh =

1.0 and glh = 0.5, the interaction etfect of the rectangular hole diminishes for dlb > 6.0 and 4.0 respectively.

UKADGAON KER & A W ASARE: INTERACTION OF STRESSES IN AN INFINITE PLATE 129

9 I h = 2·0 9 Ih = 1·0

9 I h = 0 ·5

Fig. 3-SCF variation around elliptical hole boundary for collin­ear holes along x-axis and a =90"

Table I-Interaction effect of collinear rectangular hole and elliptical hole along X axis and a = 90°

c .f = 2.0 .f = 10 .f = 05

b h h h

Max . SCF 01 Max. SCF 10 30.876 - 17104 24.194 1.2 15.950 - 172.37 11.354 14 10.913 -173 .67 7.206 16 9.236 - 176.23 5.580 18 8.937 - 180.0 4.869 2.0 8.493 - 180.0 4.536 3.0 6.722 - 180.0 3.578 4.0 5.967 -180.0 3.237 6.0 5.240 -180.0 3.0723 8.0 5.122 -\80.0 3.044

10.0 5.075 - 180.0 3.029

The stress fu nct ions <1>2 ( (,2) and \jJ2( (,2) given in Eg . (6), valid around el liptica l hole, could not give the correct results 'for dlb < 1.0 and Co > 1.0, but <1>1«(,1) and \jJI«(,I) given by Eg. (10), valid around rectangular hole, gives correct results in that range of interaction . Hence, the graph shown in Fig. 5 gives variation of maximum SCF with ligament thickness dlb > 0.1, for glh = 2.0, 1.0 and 0.5. The maximum SCF location has not changed, it remains at the same location as that of single hole solution .

Effect of li game nt thickness for collinear holes along y-ax is and (1 = 0°

The stress concentration variat ion along the elliptical hole is affected by the presence of rectangular hole, when two holes are very close to ~ach other. The stress function s (Eg. (6» cou ld give the correct results only when dlb > 1.0 and Co > 1.0. In thi s case, for elliptical hole with hlg = 2.0 there is

01 Max. SCF 01 - 160.0 18.03 1 - 143 .94 - 162.5 8.170 - 143 .94 - 165 .0 5.015 - 147.76 - 167.5 3.700 - 147.76 - 170.0 3.104 - 15181 - 180.0 2.773 - 15608 - 180.0 2.1 86 - 156.08 - 180.0 2088 0.0

0.0 2044 0.0 0.0 2.025 0.0 0.0 2.016 0.0

only one maximum value of SCF at -90°, while for circular hole with hlg = 1.0 and for elliptical hole with hlg = 0.5 , it shows two maximum va lues of SCF for dlh = 1.2. This is so because; major axis of elliptical hole is parallel to the longer side of the rectangular hole . The interaction effect in these two cases diminish for dlb > 4.5. The variation of maximum SCF at the elliptica l hole and rectangular hole with ligament thickness is plotted in Figs 6 and 7 respectively and its angular location is presented in Tab le 2.

Craqli at the corner of rectangular hole In the case of the infinite plate with rectangular

opening subjected to uniform tension, the crack wil l be developed at the corner of the opening, since it is high Iy stressed area . The further development of the crack depends on the stress intensity factors SIF1 and SIF2 at the crack tip .

130 INDIAN 1. ENG. MATER. SCI., JUNE 1999

~

E 30-0 u ~ C o

~~

C 101 U C o u

• A

•

a = g/b

.. 10 r 9 /h = 2-0

II 10r g/h = 1-0

• 10r 91 h = 0 ·5

5·0, b/a ::. 0-1.133,

= 0-5

8·0 10·0 12-0 Ratio of hole distance to reet . hole side b, t/b

Fig. 4--Effect of ligament distance for collinear holes along x­axis of a =90° for elliptical hole

~

0

u ~ c

.2 ~ C 101 U c 0 u

~ ~ ~

5-1

4 - 9

4-8

4 -7

4 · 6

0 ' 0

.. ,-_, for g/h = 2-0

*,11-_11 10r g/h = 1-0

"'-_1 10r g/h = 0·5

a = 5· 0 , b fa = 0 ·1.133,

9 f b = 0-5

4·0 8-0 12·0 Ratio r:A hole distanct to recto hole side b, C/b

fig. 5--Effect of-ligament distance for collinear holes along x­ax is of a =90° for rectangular hole

2 ~ c

.2 ~ C .. u c o u

1/1

40-0

30 -0

20-0

1/1 ~ 10-0 Vi

• • • • • • a = 5-0, 9 lb =

2-0 4 -0 6-0

for 91 h = 2-0

for 9 1 h = 1 · 0

for 91 h = 0 -5

b/;a = 0-1.133

0-5

8'0 10·0 12-0 Ratio of heir distance to rtet. hole sidt b, C I b

Fig. 6-Effect of ligament distance for collinear holes along y­axis of a =O° for elliptical hole

~

E u ~ c 0

iii !: c .. u c 0 u

:II .. ~

~

4 · 4

4·0

3·6

3·2

2·8

2 · 4 0 -0

.... -_. for g/h = 2·0 ~JIl ___ " for gl h = 1·0

•• --... forg/h= 0 ·5

a = 5·0, b fa = 0-1.133

9 f b = 0 -5

4-0 8 ' 0 12-0 Ratio of holt distance to rtet. hole side b, c/b

Fig. 7- Effect of ligament distance for collinear holes along y­axis of a =O° fo r rectangular hole

Table 2-lnteraction effect of collinear rectangular hole and elliptical hole along Y axis and a = 0° c f. = 2.0 f. = 1.0 f. = 0.5 b II II II

Max. SCF 81 Max. SCF 81 Max. SCF 81

1.0 36.00 - 90.0 9.725 - 11 7.5 8.548 - 40.89 1.2 17.249 - 90.0 7.364 - 115 .0 6.307 - 136.15 1.4 11 .867 - 90.0 6.1 58 - 11 2.5 5.013 - 133 .00 1.6 9.823 - 90.0 5.379 - 110.0 4.133 - 133 .00 1.8 8.809 - 90.0 4.772 - 72.5 3.531 - 129.64 2.0 8.159 -90.0 4.306 - 72.5 3.106 - 129.64 3.0 6.604 -90.0 3.343 - 85.0 2.256 -136.15 4.0 5.612 -90.0 3.133 - 90.0 2.072 90.00 6.0 5.181 -90.0 3.053 90.0 2.031 90.00 8.0 5.080 -90.0 3.030 90.0 2.016 90.00

10.0 5.044 - 90.0 3.018 90.0 2.008 90.00

UKADGAON KER & A W AS ARE: INTERACflON OF STRESSES IN AN INFINITE PLATE 131

The variation of the normalized SIF, for a = 90° and a = 0°, is plotted against crack tip distance ratio db and the angular location. The normalized SIF is defined as the ratio of SIF at crack tip when crack is interacting with hole to SIF at the crack tip with central crack in infinite plate.

SIF variation at corner for a = 90°

Fig. 8 shows the variation of normalized SIF, at the outer crack tip when the inner crack tip distance db > 0.1. The stress functions will not give correct results, when inner' crack tip distance c/b < 0.1, though it is satisfying the minimum center distance condition while evaluating Cauchy's integrals. It is observed that the normalized SIF, at the outer crack increases when 0.1 < d b < 0.3 , for angular location between 0° to 40°. Similarly normalized SIF, at the inner crack

Fig, 8---Effect of crack tip distance and crack angle for a =90° at outer tip

Fig, 9--EfTect of crack tip distance and crack angle for a=90° at inner tip

tip, in the range 0° to 30°, increases with db when 1.2 < db < 1.9, as observed from plot in Fig. 9. Here the stress functions could not give correct results when inner crack tip distance db < 1.2. When db is more than 5.0, Figs 8 and 9 show no interaction effect of crack with rectangular hole.

SIF variation at corner for a = 0°

The plots in Figs 10 and 11 show the normalized SIF, variation between inner crack tip distance ratio db and crack angle for outer and inner crack tips, for loading in x-direction. The normalized SIF at the outer crack is not varying with crack tip distance for angle more than 40°, and it is found that it is less than or almost equal to 1.0.

Fig. 11 shows instantaneous increase in normalized SIF at an angle of 5° - 15° and for db 1.2 to 1.7. It is observed that the normalized SIF, is less than 1.0 in other region. The SIF, values are found to be more than 1.0 for the loading normal to the longer side of the rectangular opening. The crack initiation possibility is more when the loading is in this direction .

Fig, I O--Effect of crack tip di stance and crack angle lor a =O° at outer tip

~

~~'\.

~;~-~ '.

Fig, II---Effect of crack tip di stance and crack angle for a =O" at inner tip

132 INDIAN 1. ENG. MATER. SCI.. JUNE 1999

Conclusions The stress distribution around the elliptical hole or

crack is disturbed more than the rectangular hole, as the elliptical hole or crack approaches near to the rectangular hole, due to interaction effect of each other. The stress functions gives the single hole solution when the ligament distance is more than four times the smaller side dimension of the rectangular hole or in absence of the second hole. It is also observed from the present study that the stress functions valid around the elliptical hole could not give the correct values of SCF and SIF, when the elliptical hole or crack is very close to the rectangular hole.

The stress fu nctions given by Eqs (6) and (10) satisfy the boundary condition exactly. These equations give the closed form solution for stress functions as well as for SCF and SIF havil ;g finite number of terms.

References I Ukadgaonker V G . Gade S V & Awasare P J. Aeronaulics .1.

(in Press). 2 Ukadgaonker V G & Patil 0 13 . ASM£. 115 (1993) 93 . 3 Savin G N. Stress concentration around holes. (Pergoilloll

Press. New York). 1961.

Appendix

0'1=ml +2(AAI +iBBI}, O'J=mJ +2(AAJ +iBBJ},

O'z = mz + 2 ( AAz + i BBz). 0'4 = m4 - 2 (- m4).

(ml-cos20'+5m Jm4 }(3m 4 - 1)-3mzDI AAI = Z , ,

(5m 4 +D, - 1)(3m4 -1}-3D I-

mz - DIAAI AAz = .

(3 m 4 - I)

AAJ = -(m J - m4 AAI )

BBI = sill 2a(3m4 + I} (5 mi - D, - I) (3 m4 + I) + 3 DI2 '

D,BB, BBz =

(3m4 + I)

BBl = - m4 BBI

D, = mlm4 +mJ.

D, = m~ m4 + mlm J + 3m 2 m4 + m2

fJl =m40'1 ' fJ2 =O'lmJ +3m40' z,

fJJ = m2 0'1 + 3m J0'2 + 5m 40'J,

fJ4 = mlO'I + 3m zO'z + 5m J O'J + 7m 40'4'

/1, =0'1 + 3m10' 2 +5m 20'J +7m J0'4 '

fJ6 = 30'2 + 5m lO'J + 7111 20'4' ~ =5a.J + 7m,a4 , fJ8 = 70'4 '

fJq = fJz + fIl l fJl'

fJ lO = 3f11 2 fJI + fJJ.

fJll = 5mJfJI + fJ4' fJ l2 = 7m4fJI + fJ5' fJlJ = ml fJ9 + fJlD '

fJI4 =3mzfJq +fJl l,fJlS = 5m JfJ9 +fJI Z'

fJI 6 = 7m4fJ9 + fJ6'

fJI7 = mlfJlJ + fJ14 ' fJI8 = 3m2fJIJ + fJlS •

fJI 9 = 5mJfJIJ + fJ1 6' fJzo = 7 m4fJIJ + fJ7 , fJ20 = 7 m4fl3 + fJ7,

E 1= fJl7 - I, &2 = fJI8 + ml ,

&3 = /319 +3m, &4 = fJ 20 +5m J , &, = fJ8 +7m4•

npR pRalJ ezl/l I ZI =-4S2 : l n= - 4c~ F2;

3pRaze 4//1. 5pRaJe 8//1 .

I ZJ =- 4C0 4 FJ , l z4 = - 4c; F"

7 pRa4e 8//1 p Re -Z/(a - /l)

I z' = - F7 : 1 J I = - = 1 JZ ' . 4c08 2BI

pR [ 1. 1 41 =-2/(00)- /(S2)F

1 , = _ pRaleZi

/1 Co [~ _ si ]:

,- 4. c~ BI2

and

p R a1 e61

/1 [~_ S; ].

1 64 = -----4 co' 131 "

16

, = pRa4e81

/1 [_1 __ S~ 1

. 4 c; BlJ

/71 = pRNNSZ

4BI1

1 =_ p Ra, NN e -2i

/l [- AS2 n+ AS I (2] 72 4 C BI1

1 = _ 3p R 0'2 NN e-41

/1

[ - AS4 n + As, (2 ] 7J 4 C 2 BII

1 =_ 5PRalNNe-61/1[ -AS6 n.+AS5(2 ] 74 4 ( .. 811

1 __ 7p Ra4 NN e 81rl [- ASR 11+ AS7 (2']

7' - 4 ('4 Bn '

where

UKADGAONKER & AWASARE: INTERACDON OF STRESSES IN AN INFINITE PLATE

2A6BI2B3 2A5BI2 AsBli 2nABI) + +-----------

BI ,3 BI13 BI( BI(

A6B/~ 3A~BI381; 3nABI; 3ABI2Bi + --- ---+ --'0---''-BI14 BI,4 . BI,4 BI,4

_ AsBI; _ 4ABI3 B~ + ABI;}

BIt BII~ BI16

F = e {_ ~ + 7 A'6B12 _ 2IAISBI] _ 14nAI4 7 0 BI BI2 BI2 BI2 I I I I

+ 9A12 B1 2Bl? + 611C1IIBI2BI, + 3112 ABII _ AllBlj

BI14 BI14 BI14 8114

3nABI; 7 AIJBI; 12AI2BIJB/~ 4nAIIB/~ --------+ +---'-'-:----"-BI14 Bll

s BI,s BI,s

6AIIBI~BI; 12ABIJBI; 4ABI2BI{ --7-':"'+--~"--

BI,s BIt BIt

3AI2 Bli SAil BI3B1; SnABI; I OAI2 Bli B~ +---- ----+--'-"-~----"-

BIt B/~ BI16 Bit

AI,B/~ 6ABI3Bli A BI;} +---+ +--BI17 . BI17 B/~

&le -SiPe~ +&2 e-6ip eg +&3e -4iPe;

+&4e -2iPeg +C~ /(00) = .

e -siPeci - ml e -6iPC; - 3m2e -4ip eg

-SmJe- 2,P eg -7m

4c

O

cle -slPsiBIS +&2e-6iPs; BI6

+ cJe -4iP S; BI4 + c4e -2ip S~ BI2 + csSs

/(Sl) = e-8iPB19 _mle-6iPs; BI7 -3ml e 4iPS;Bls

- SmJe - liP S~ BI3 - 7 m4S~ BI

81 = 1.0 + COS2 • 'BI, = (cg - n)BI) ,

Bil =(cg -3n -2ncOS2) , BI3 =6n+2neOS1 ) ,

C =c~'-n. Bn=n-S;. NN = 1+ n l

A) = 1O.0+3cgn. A4 = 10.0+egn, As =7.0+Scgn,

A6 =21.0+\Ocgn, A7 =3S.0+lOcgn, As =7.0+cgn,

Ag = 12.0 + cgn, All = 9.0+ 7cgn, AI2 = 12.0+ 7cgn,

Au = 21.0 + Scg n, AI4 = 18.0 + Scg n, AIS = 6.0 + egn.

AI6 = 12.0 + cg n, AI7 = 36.0 + cg n

ASI = c~ + n, AS2 = 2co

As) = c~ + 6nc~ + n2, AS4 = 4(c~ + nco),

AS5 = cg + ISnc~ + ISn 2c~ + n2 AS6 = 6cb + 20nc~

+6n 2co, X 6 24 12 4

A.1 7 = Co + 2811Co + 70" Co + 2811 Co + 11

A 8 7 S6 s S6 2) '8 J S8 = Co + nco + n Co + con

J = - pS [~+~+~+~l 21 4 SI7 st SIJ SI

SN -2iP'( K -sK) J __ p le- 1+ CI 2

22 - . 4

J = pSe2ia

J = PS(~+fLJ )1 2S1 32 2 ci JI

J ~_~ +ncle +"1"1+ e SN- {I - 2 21p r (r2 nJI2 2iP)}

4 2 cl(cl2e2iP -n) JI(JI 2e2iP -nSI2)

J = pSc,Nle- I ~+fL - 2P ( 1 J 52 4 C

l2 JI2

K = ~ + m3JI2 + m4JI4 + m3J/12

+ 2m4JI]J/,

I SI SI2 SI2 SI) SI]

n~fi; 3m4fi2fi~ m4fi~ +--+ +--

SI] SI4

st

K -!i_ £Jlo 2 - Jis J/~

JI = 1.0- CISI' J/I = 1+2 CISI J/2 = (ml - CI2 lSI - 2c,

J/) = (C12

- ml ) - 2m,cIS" JI4 = (mlcl

2 + 3ml lSI + 2mlc"

Jis = B Jlo = BI JI- B(6+ 5c1SI )

E = I + mleg + m2e; + m)eg + m4c~ ,

EI = 4 + 3mlc~ + 2m2c; + m/:g ,

B=CI8 - mll

o -3m2c14 -5m)cI

2 - 7m4'

B, = 2m,c,o + 12m2ct + 30m3cl2 + 56m4 ,

p S pSN -21p J

OI =__ J = Ie

4 SI '62 4c,JI

J = pS { OIJSI7

+ Ol4St + OlSSI3

+ Ol6S1 } 71 4 S 6 4 2 SI - mlSI - 3m2S1 - Sm3S1 - 7 m4

133

134 INDIAN 1. ENG. MATER. SCI., JUNE 1999

/ JIJt;I' + 014t;t + oi~t;,) + 016t;1 1\..;, = 8 6 4 Z 7

t;1 - mlt;1 - 3mzt;1 - 5m)t;1 - m4

01 = m) + mlm4. /i2 = 1n2 + 3m2m4. 0) = ntl + 5"'.l nt4'

04 =1+7m~,05 =ml/il +02.06 =3m 201 +0).

0, = 5m)01 + /i4' 08 = 7m401• Oq = ml(Y~ + Cib '

010 = 3m2/i~ +/i,. /ill = 5m) O~ +08 , 012 = 7nt4J~ .

£'I:] = mlOq + 0 10 , 0 14 = 3m209 + 01,

0l~ = 5m/iq + 0 12 , 0 16 = 7 m4/iQ •

FCI = OUCI' + 0'4ci + OI~C: + 016CI .

DFCI = 701Jc~ + 5014C: + 3/iI~CI2 + °16 ,

X (, ~ 7 FC = CI -lIIlel - 31111CI - 51115cI -71114

DFC = 8cI' - 6m lc; - 12mlcj1 - IOm1cI