The Stresses Around Some Unreinforced Cutouts Under Various Loading Conditions

TEC

Mkhiel CoA Report No. 146

THE COLLEGE OF A E R O N A U T I C S

CRANFIELD

STRESSES AROUND SOME UNREINFORCED CUTOUTS

UNDER VARIOUS LOADING CONDITIONS

by

EEPOPT NO. 146 March, 1961.

T H E C O L L E G E OF A E R O N A U T I C S

C R A N F I E L D

The Stresses Around Some Unreinforced Cutouts Under Various Loading Conditions

b y -D. S. Houghton, M.Sc.(Eng.), A . M . I . M e c h . E . , A . F . R . A e . S . and A. Rothwell, B.Sc.CEng.), M . S . , D . C . A e . SUMMARY

A number of experimental results are given for unreinforced circular, elliptical and square cut-outs with rounded c o r n e r s , under a variety of loading conditions. These results are then compared with the infinite flat plate solution, using the complex s t r e s s function and the method of conformal transformation.

It is generally considered sufficiently accurate for a plane s t r e s s solution to be applied to problems of cut-outs in cylindrical shells, provided that the cut-out dimensions are small compared with the radius of curvature of the shell. In order to investigate this effect of shell curvature, a number of experiments were carried out on cut-outs in two pressurised cylinders, and to obtain a wider range of loading conditions, a further s e r i e s of experiments was conducted using a plane loading frame.

Two aluminium alloy cylinders, of 44 in. and 60 in. diameter, were pressurised using a i r , and the s t r e s s conditions around the cut-outs examined using electrical resistance strain gauge technique?.

The plane loading frame enabled any combination of bi-axial tension and shear to be applied to a plane sheet containing the cut-out under

examination. In this way the effects of body forces, which might a r i s e in such applications as aircraft and nuclear r e a c t o r s , could be simulated.

The experimental results which are given generally show a close agreement v/ith the theoretical plane s t r e s s solution.

CONTENTS P a g e S u m m a r y Notation 1, Introduction 1 2 . E x p e r i m e n t a l Work 2 3 . T h e o r y 4 4 , D i s c u s s i o n of R e s u l t s 9 5< Conclusions 11 6 . R e f e r e n c e s 12

Appendix 1. - The method of evaluating the function (f> (S) and the edge s t r e s s

d i s t r i b u t i o n i s shown for an e l l i p t i c a l and a s q u a r e cut-out in a plane s h e e t subjected

to 2:1 b i - a x i a l t e n s i o n 14

N O T A T I O N

P,Q polar co-ordinates

U Airy stress function

cr radial stress P Cg tangential stress r . shear stress pd z , S complex v a r i a b l e s 96, x,^^ complex s t r e s s functions $ , * d e r i v a t i v e s of complex s t r e s s functions C coefficients in t r a n s f o r m a t i o n function n

cr complex c o - o r d i n a t e on the unit c i r c l e

co(^) t r a n s f o r m a t i o n function

R s c a l e factor in t r a n s f o r m a t i o n function

r^ ,r^ c o n s t a n t s depending upon s t r e s s e s at infinity

N^, Ng p r i n c i p a l s t r e s s e s at infinity

a d i r e c t i o n of p r i n c i p a l s t r e s s

a , b coefficients in s e r i e s f o r complex s t r e s s functions

n n '^ Q c r i t i c a l s t r e s s combination

\ stress concentration factor

m parameter for eccentricity of ellipse

a semi-major axis of ellipse, or semi-width of square

b semi-minor axis of ellipse

k eccentricity of ellipse

fi angle in plane of ellipse or square

1

-1. Introduction

In the design of most shell s t r u c t u r e s , cut-outs have to be provided for access doors and windows, or to accommodate som^e intersecting s t r u c t u r e . It is well known that their presence causes perturbations in the original s t r e s s system, so that in the region of the cut-out the local s t r e s s level in the shell may be increased many t i m e s . These s t r e s s concentrations can of course be considerably reduced, and in some cases entirely eliminated by the use of a suitably chosen shape of cut-out, and by the use of a suitable reinforcing member around the edge. It is the analysis of these cut-outs that constitutes one of the major problems in the design of shell s t r u c t u r e s .

In the cabin structure of a pressurised aircraft, a long fatigue life must be achieved for minimum structural weight, and the effect of any s t r e s s concentrations which may arise in the doors and windows must be closely examined. A similar situation exists in nuclear reactor design, where the true s t r e s s conditions at all points in the structure must be accurately determined.

The factors which influence the magnitude of these s t r e s s con-centrations are the shape of the cut-out and the dimensions of the reinforcement. The geometry of the shell itself is generally of only secondary importance, provided that the size of the cut-out is small compared with the radius of curvature of the shell. If this condition is fulfilled, the coupling between the radial displacement and the s t r e s s e s in the middle plane of the shell is small, so that the problem may be reduced to that of a cut-out in an infinite plane sheet, which is subjected to the appropriate s t r e s s conditions at infinity. In all of the recent theoretical investigations this assumption is made, and according to Gurney^ is justified for the case of a cut-out in a circular cylinder provided that the ratio of cylinder diameter to cut-out diameter is not l e s s than 4 : 1 .

2 3 4 More recently, Mansfield , Hicks , and Wittrick have made similar assumptions and have developed the use of both real s t r e s s functions and complex variable methods to predict the s t r e s s

con-centration arojnd unreinforced and reinforced cut-outs of various shapes.

Unfortunately little experimental evidence has been recorded in the literature to support the use of the flat plate theory, and the

2

-The work which was carried out by Richards on Xylonite model cylinders suggested that the flat plate theory is inadequate to define the complete s t r e s s distribution, since significant bending effects were observed. However Ref. 6 obtained quite a good correlation between a nvimber of photoelastic r e s u l t s , which were obtained on cylinders, and the flat plate theory.

The only theoretical work known to the authors v/hich includes the effect of shell curvature is by Liirie , and this is restricted to small unreinforced circular cut-outs. A closer examination of these papers suggests that for the unreinforced cut-out this discrepaincy is not

unreasonable, since much depends upon whether the t r a n s v e r s e loading across the cut-out is allowed to be reacted at the boundary. The flat plate theory when applied to unreinforced cut-outs does not make

allowance for the effect of out of plane forces. For the reinforced cut-out however, it is generally considered that provided the cut-out is neutral, or nearly so, and that the shear distribution around the boundary is in a predescribed manner, then a close agreement should result with the flat plate theory.

In the following paper, a number of experimental results a r e given for circular, elliptical and square unreinforced cut-outs. Some of the initial experiments were carried out on cylinders which were pressurised using a i r . These were used to confirm the use of the flat plate theory, and further experiments were subsequently carried out which made use of a plane loading frame. This had the advantage that loading other than p r e s s u r e loading could be more easily exanained.

The analytical work which is presented makes use of the complex s t r e s s function and conformal transformation technique of Muskhelishvili

9 1 0 1 1 1 9

which has been used previously by Wittrick ' and by the authors-^^» •^''. 2. Experimental Work

In o r i e r to investigate the validity of the theoretical work, a number of experimental investigations have been undertaken.

Two cylinders were used for the initial experiments, and these measured 44 in. and 60 in. in diameter and were constructed from

18 s . w . g . (0.048 in.) L.72 aluminium alloy sheet. The cylinders were pressurised by using air which was controlled to give a nominal hoop s t r e s s of the order of 10,000 Ib/in^, and were mounted on a specially designed trolley which enabled a free longitudinal expansion of the

3

-cylinder under test to take place (Fig. 14). Investigations were carried out on circular, elliptical and square cut-outs with rounded corners in these cylinders, having dimensions ranging from 6 in. to 10 in.

The air p r e s s u r e was contained by using a plate which was rolled to the contour of the cylinder and shaped so as to fit accurately into the cut-out (Fig. 15). This was supported by an external structure in such a way that the t r a n s v e r s e p r e s s u r e loading was not reacted at the boundary of the cut-out. This procedure eliminated the substantial bending s t r e s s e s which would otherwise result, and which for these unreinforced cut-outs would be inconsistent with the flat plate solution. These experiments enabled the effects of shell curvature on the 'in plane' s t r e s s e s to be examined separately.

Strain gauge readings were taken using Tinsley 6H and 6K electrical resistance strain gauges, in conjunction with a Savage and Parsons

r e c o r d e r . Typical strain gauge positions are shown in Fig. 19. These preliminary experiments gave a good agreement with the theoretically predicted s t r e s s concentration factors using the flat plate theory. This suggested that for the range of cut-outs under examination, any effects of shell curvature on the s t r e s s concentration factor were small and that the use of the flat plate theory was justified.

In view of this, and because of the considerable experimental difficulties arising, it was decided to conduct a more ambitious s e r i e s of experiments using a plane loading frame (Figs. 11, 12). This loading frame enabled bi-axial s t r e s s ratios other than the 2:1 biaxial p r e s s u r e s t r e s s e s to be studied, and in addition, cut-outs which are subjected to both shear and direct loading could be examined.

The aluminium alloy panels which were used in the plane loading frame were 28 in. square and were made from 16 s . w . g . (.034 in.) L. 72 aluminium alloy sheet. Tinsley 6H and 6K electrical resistance strain gauges were again used with the Savage and Parsons r e c o r d e r . It was found that provided the maximum panel cut-out dimension did not exceed 5 i n . , any panel boundary effects were negligible, and in this way reasonable comparison could be obtained with the infinite flat plate theory.

The s t r e s s concentrations around circular, elliptical and square cut-outs with rounded corners were investigated in the plane loading frame.

4

-The panel was loaded through heavy edge members by means of a linkage system, which was arranged so that if required the effects of direct s t r e s s combined with shear s t r e s s could be examined (Fig. 13). The loads weie produced by two 3000 lb. turnbuckles used in conjunction with calibrated dynamometers. While tensile loading gave edge s t r e s s e s of up to 1000 Ib/in^, the shear loading was restricted by panel buckling considerations to 700 Ib/in^. The strain gauge locations are shown in F i g s . 16, 17, 18. The gauges in the region of the cut-out were

Tinsley 6H, and the backing paper was trimmed so that strain m e a s u r e -ments could be obtained as close as possible to the edge. It was found that measurement of the tangential s t r e s s could effectively be made at a distance of 0.1 in. from the edge of the cut-out,

3. Theory

The s t r e s s e s in the region of circular and elliptic cut-outs in an infinite plane sheet, have been obtained by Inglis-^^ using the real s t r e s s function with an elliptical co-ordinate system.. This method has been developed by Wello^'^ and Hicks"^ for certain other cut-out problems.

For other shapes of cut-out, for example square and triangular cut-outs with rounded c o r n e r s , the method of conformal transformation which has been developed by Muskhelishvili^ and used extensively by Savin^^'l^ can be employed. This method can be used for the solution of an unreinforced cut-out of any shape provided that the transformation function can be expressed as a simple polynomial function.

The method and notation of Muskhelishvili have been followed closely h e r e ,

The s t r e s s components in an elastic plane s t r e s s system may be expressed in t e r m s of a single real s t r e s s function (the Airy s t r e s s function)

U ( p , 6) ,

satisfying, in the absence of body forces, the biharmonic equation V^ U = 0 ,

JL+1

9 , 1 9^

ap2 p ' dp p^ ' dQ^ '

5

-The s t r e s s components in the p l a n e , r e f e r r e d to a p o l a r co-o r d i n a t e s y s t e m (P , 9), a r e given by 9^U cr P

'^e

and '•pe Using the c z the b i h a r m o n i c 92

^

T' OF ~'

9^U 9P^ ' 9 ,1 9 U . " 9p \ ' dd' • omplex v a r i a b l e 16 = p e equation b e c o m e s 9^U _ 2 ^ > '? d7 96^ '

and the r e a l s t r e s s function m a y be e x p r e s s e d in complex f o r m , in t e r m s of the two com.plex functions,

<P (z) and x(z) ,

by _

2U = Z9i(z) + z 0(z) + x(z) + x(z) , w h e r e a b a r d e n o t e s the com^plex conjugate.

If (p (z) and X (z) a r e h o l o m o r p h i c functions, e v e r y e x p r e s s i o n of t h i s f o r m r e p r e s e n t s a b i h a r m o n i c function.

The s t r e s s components in the p o l a r c o - o r d i n a t e s y s t e m a r e then given b y and w h e r e cr +0-Q = 2 [ $ ( z ) + $(z) ] cr^ -cr +2iTpQ = 2 r z $ ' ( z ) +<P(z) 1 e^^® V^(z) = x' (z) , $ ( z ) = 9i'(z) , >^(z) = f'iz) ,

6

-The region 2 in the S -plane may be transformed into the region S in the z-plane by a conformal transformation function

z = a)(S) ,

where co (S) is a single valued analytic function. Transformation functions of the form

(Z) = R S + -7- + — + . . .

have been obtained, which transform an infinite plane sheet containing an elliptic or approximate square cut-out in the z-plane into the region outside the unit circle

16 cr = e

in the ^-plane. Similar transformation functions may be obtained for cut-outs of other shapes .

If the functions previously written

<p{z) ^ (z) i ( z ) *(z)

are now denoted by

Uz) f,^(z) $,Jz) *,(z)

then in the new notation

and

^(^) = ^,,(z) = i^Ji'^iO],

i'(s)

w ' ( ^ )

In the transformed plane the s t r e s s components, denoted by cr , cr„ and r in the curvilinear co-ordinate system corresponc

r 6 re

7 -P ö

[$(S)

and <^ +_{p pö} 217- „ + $ ( ^ ) 2^' = 4 Re

•

h^^]

'

( g ) . $ ' ( S ) + a)'(^)*(S) P^ Ui'U) ^ w h e r e Re d e n o t e s the r e a l p a r t . At the edge of an u n r e i n f o r c e d c u t - o u t p pe

so that to find the s t r e s s at the edge of the cut-out the function 'i'(^) i s not r e q u i r e d , and the p r o b l e m i s r e d u c e d to that of finding the function * (S) satisfying the b o u n d a r y conditions of the p r o b l e m .

The e x p r e s s i o n for the edge s t r e s s then b e c o m e s

cTg = 4 R e . [ * <S) ] .

T h e b o u n d a r y condition on the edge of the cut-out ( s i n c e t h e r e a r e no e x t e r n a l f o r c e s applied to the edge of the cut-out) m a y be w r i t t e n

w h e r e cr 0 (cr) + i 6 CO (cr)

(717)

. 0^' (cr) + .^(cr) = 0 , e on the unit c i r c l e . and

The complex s t r e s s functions ^ iZ) and ^(Z) m a y b e e x p r e s s e d

0 ( ^ ) = 0^(2;) + <p^ (^)

^ ( 5 ) = f^(0 + f^ (S) ,

w h e r e the functions 0 (S) and f {^) r e p r e s e n t the s t a t e of s t r e s s at infinity, and (}> (X^, if- (S) a r e the p e r t u r b a t i o n functions due to the p r e s e n c e of the c u t - o u t .

Hence 9^^ (^) and f (t) a r e given by

0^ (^) = Rr^s

and

8

-w h e r e r = ^ ( N , + N, ) ,

1 / ^ . ^T V - 2 i a

and N^ , N^ a r e the v a l u e s of the p r i n c i p a l s t r e s s e s at infinity, a i s the angle m a d e by the d i r e c t i o n of N with the a x i s 6 = 0, and R i s the s c a l e f a c t o r in the t r a n s f o r m a t i o n function.

The functions <p i^) and ^ ( g ) , which a r e h o m o m o r p h i c outside the unit c i r c l e including the point at infinity, m a y be expanded into s e r i e s of the form a a a ^ ' ' ^ b b b „ ^ ^ ( S ) = — + - 7 + - r + . . . = s b s . o ^ ^ 2 ^ 3 n On the unit c i r c l e and the t e r m 1 0) (o-) w'(cr) m a y be expanded into a s e r i e s of a s c e n d i n g p o w e r s of cr ,

By substituting t h i s , t o g e t h e r with the s e r i e s expansions for <p(Z) and f {^), into the b o u n d a r y condition on the edge of the cut-out the coefficients a m a y be d e t e r m i n e d by equating coefficients of p o w e r s of cr in the b o u n d a r y condition.

Since the function ^ (S) alone i s sufficient to obtain the s t r e s s at the edge of the c u t - o u t , it i s not n e c e s s a r y to d e t e r m i n e the coefficients b .

n

In t h e appendix, the function (p ( g ) and the s t r e s s d i s t r i b u t i o n cr at the edge of the cut-out have been evaluated for two p a r t i c u l a r c a s e s of an e l l i p t i c a l and a s q u a r e c u t - o u t , to i l l u s t r a t e the method of solution.

9

-The s t r e s s concentration factor ^ is based on the 'critical s t r e s s 17

combination' which is given by

Q = cr^ + cr^ - cr cr + 3r , r e r e rö and the s t r e s s concentration factor is then given by

Q 4. Discussion of Results

The s t r e s s concentration factors obtained from the infinite flat plate theory, for unreinforced circular and elliptical cut-outs under various systems of bi-axial tension and pure shear, are shown in F i g s . 1, 3 and 4. .

For the circular and elliptical cut-outs under bi-axial tension, the maximum s t r e s s concentration occurs at the edge of the cut-out, where the tangent is in the direction of the maximum applied s t r e s s . The maximum value of f^ is dependent on the ratio n:l of the applied s t r e s s e s . An increase in n above 2 leads to a relatively small increase in the s t r e s s concentration factor. F o r a circular cut-out subjected

to 2:1 bi-axial tensions, the s t r e s s concentration factor is approximately 3. For an elliptical cut-out the s t r e s s concentration factor also depends on the eccentricity b / a of the ellipse, and a smaller value of X is

produced when the major axis of the ellipse is in the direction of the maximum applied s t r e s s . There is an optimum value of b / a for the minimum s t r e s s concentration, which is dependent on the ratio of the applied s t r e s s e s .

Under pure shear, the s t r e s s concentration factor for an elliptical cut-out is again dependent on the eccentricity b / a and the direction of the axes. F o r a circular cut-out under pure shear a symmetrical s t r e s s distribution is obtained and the maximum value of "k is approximately 2.25.

The theoretical s t r e s s concentration factors for unreinforced square cut-outs having various corner radii, and under various systems of

bi-axial tension and pure shear, a r e shown in F i g s . 7 and 8. By retaining the first few t e r m s only in the transformation function for the square, a limited number of different corner radii may be obtained. The profiles of the approximate squares obtained by retaining two and three t e r m s only in the transformation function are shown in Fig. 6.

10

-By suitable modification of the coefficients, Wittrick has obtained a more general transformation function for a square having any corner radius. This method may also be applied to triangular cut-outs and other shapes.

F o r a square cut-out the maximum s t r e s s concentration, under both bi-axial tension and pure shear, occurs at the corner of the

square, and the maximum value of X is dependent on the corner radius r / a . Very high s t r e s s concentrations are obtained when the corner radius is small. Under bi-axial tension, the maximum s t r e s s concentration depends on the ratio n:l of the applied s t r e s s e s , and is actually decreased as the value of n is increased.

The s t r e s s concentrations for loading cases other than those shown, may be obtained by superposition of the curves. In this way the effect of biaxial tensions v/hich are not parallel to the axes of sjmimetry of the cut-out, and the effect of combined tensile and shear loading may be deduced.

A number of preliminary t e s t s were carried out on circular and elliptic cut-outs in the p r e s s u r e cylinders to investigate the importance of the effect of curvature of the shell. It was deduced from these that, for the range of cut-outs investigated, the effect of curvature was small and could conveniently be neglected. For this reason a large part of the subsequent experimental work was carried out on the plane

loading frame.

The experimental results shown in Fig. 10 were obtained on the 60 in. diameter cylinder subjected to internal p r e s s u r e , and those shown in F i g s . 2, 5 and 9 were obtained using the plane loading frame.

In order to examine the strain distribution across the panel in the plane loading frame, some preliminary tests were carried out on an uncut panel (Fig. 12). These results indicated that under tensile loading, the strains along the horizontal and vertical axes were between -12% and

-6% of the nominal values. Under shear loading, the strain was

approximately constant across the panel width. There was a tendency for the shear s t r e s s to peak near the c o r n e r s , where the strain was 5% higher than the nominal strain.

The qualitative agreement between the experimental and theoretical results is good. The experimental values are generally lower than the theoretical, and this is to be expected since the strain gauge readings are

1 1

-made at a distance of 0.1 in. from the edge of the cut-out in most c a s e s , owing to the physical dimensions of the strain gauges. Extrapolation of the strain gauge readings could have been carried out in order to eliminate this effect, but the preliminary attempts showed that large e r r o r s could occur in the extrapolation owing to rapid changes in strain away from the edge of the cut-out.

In the experiments on the square cut-out, when both the p r e s s u r e cylinder and the plane loading frame were used, the maximum s t r e s s concentration at the corner of the square was difficult to m e a s u r e , owing to the rapid increase in the s t r e s s concentration in this region and the size of the strain gauge.

5. Conclusions

The results are given of a number of tests carried out on un-reinforced cut-outs of various shapes in pressurised cylinders and in a plane loading frame.

These results are compared with the theory, and the general agreement is found to be good in most c a s e s .

Having now established the technique, it is hoped to continue

with a further programme of experimental work for reinforced cut-outs of various shapes, under a variety of loading conditions.

12 -6. References 1. 2. 3. 4. 5. 6 . 7. 8. 9. Gurney, C. Mansfield, E . H . Hicks, R. Wittrick, W.H. Richards, T.H. Houghton, D.S. Lurie, A . I . Muskhelishvili, N . I . Wittrick, W.H.

Analysis of the s t r e s s e s in a flat plate with a reinforced circular hole under edge forces.

A.R. C. , R & M. 1834, 1938. Neutral holes in a plane sheet. A . R . C . , R & M . 2 8 1 5 , 1955.

Reinforced elliptical holes in stressed plates.

Journal of the Royal Aeronautical Society, v o l . 6 1 , 1957, pp 688-693. Stresses around reinforced elliptical holes, with applications to p r e s s u r e cabin windows.

Aeronautical Quarterly, vol.10, Nov. 1959. Stress distribution in pressurised cabins: An experimental study by means of

Xylonite models.

A. R. C. 19,360. Strut. 1999, 1957. Stress concentrations around cut-outs in a cylinder. (To be published by Royal Aeronautical Society),

Statics of thin walled elastic shells. Ogiz. Moscow. 1947.

Some basic problems of the mathematical theory of elasticity.

Noordhof f, Groningen, Holland, 1953. Some simple transformation functions for square and triangular holes with rounded c o r n e r s .

Aeronautical Quarterly, v o l . 1 1 , May 1960, pp 195-199.

13 -References (Continued) 10. Wittrick, W.H. 1 1 . 12. Houghton, D . S . Hillel, A. , Rothwell, A . , Arthurs, T . D . Analysis of s t r e s s concentrations at reinforced holes in infinite sheets. Aeronautical Quarterly, v o l . 1 1 , Aug. 1960.

Nuclear reactor containment buildings and p r e s s u r e v e s s e l s .

Butterv/orths, London, 1960, pp 191-220. Unpublished thesis work conducted at the College of Aeronautics, Cranfield.

1 3 . Inglis, C.E. S t r e s s e s in a plate due to the presence of

cracks and sharp c o r n e r s .

T r a n s , of the Inst. Naval Arch, vol.55, 1913, p 219.

14.

1 5 .

Wells, A. A.

Savin, G.N.

On the plane s t r e s s distribution in an infinite plane with rim-stiffened elliptical opening.

Quarterly Journal of Mechanics and Applied Mathematics, v o l . 3 , 1950, p 23. Stress concentrations around holes. Veb. Verlag Technik, Berlin, 1956. (German translation). 16. 17. Godfrey, D . E . R . Timoshenko,S. , Goodier, J . N .

RepoT-t on Savin's 'Stress concentration around holes'.

A. R.C.18460, Struct. 1895, 1956. Theory of elasticity.

14

-APPENDIX 1

The method of evaluating the function 0 ( ?) and the edge s t r e s s d i s t r i b u t i o n i s shown for an e l l i p t i c a l and a s q u a r e cut-out in a plane s h e e t subjected to 2:1 b i - a x i a l t e n s i o n .

(a) E l l i p t i c a l Cut-out ( F i g . 20a)

T h e r e g i o n outside an e l l i p s e in the z - p l a n e m a y be t r a n s f o r m e d

on to the r e g i o n outside the unit c i r c l e in the S -plane by the t r a n s f o r n a a t i o n function

z = u (S) = RU + f - ) . H > 0 0< m <1

The c i r c l e

kl = 1

c o r r e s p o n d s to an e l l i p s e with c e n t r e at the origin and s e m i - a x e s

a = b = and t h e r e f o r e m = w h e r e k = R(l + m) , R(l - m) , 1 - k 1 + k ' b a

Differentiation of the t r a n s f o r m a t i o n function gives

a ) ' ( Ü = R ( l - | ^ ) . On the unit c i r c l e i e S = cr = e and . "^ ~- ^ • T h e r e f o r e " (cr) = cr + —

15

-o r expanding t h i s equati-on a s an a s c e n d i n g p -o w e r s e r i e s

J i L k l = 2 L + ( i + „ i 2 ) c ^ + m ( l + m ' ) c r ' + . . . . cr

The complex s t r e s s functions m a y be w r i t t e n

<^iÜ = 0 (S) + 0 (S) ,

o 1 o 1

and the functions cj) (^) and i^ (S) a r e obtained from the conditions at infinity. ^ ^ In t h i s c a s e N , = 2 , N^ = 1 , a •== 0 . T h e r e f o r e V = ^

(N

+ N ) = I 1 4 1 2 4 1 /TVT TVT \ " 2 i « 1 r ^ = - 2<Ni - N j e = " 2 ' and ^ (^) = J RS , ^ ( g ) = - ^ R S . 1 'S

The p e r t u r b a t i o n functions m a y be expanded in the f o r m

°° - n . ^ ( ^ ) = E b ^ S , and differentiation gives

°° - n - 1

16 -T h e r e f o r e 0 0 - _ n + l . 3 ^ i n 4 and V'(cr) = ? \ ^ - 2 ' ? •

T h e b o u n d a r y condition to be satisfied on the unit c i r c l e i s w(cr)

i> (cr) + - ^ ^ ^ . (f>' (cr) + f (cr) = 0

w ' ( a )

Substitution in the b o u n d a r y condition gives

^ Ro- + Z a cr"" 4 1 n + [ ^ + (1 + m2)(T + m ( l + m^)cr' + . . . J x r 3 _, " - n+1 1 [ ^ R - Z na^ cr J 1 R ~ r- n - - . - + 2 b cr 2 cr i n = 0 .

T h e coefficients a m a y be obtained from t h i s equation by c o m p a r i n g coefficients of negative p o w e r s of cr.

Thus

a, = (2f2L)H ,

and a l l the o t h e r coefficients a a r e z e r o . n

It i s not n e c e s s a r y to d e t e r m i n e the coefficients b since the n

function 0 ( ^ ) alone i s sufficient to obtain the edge s t r e s s d i s t r i b u t i o n . T h e r e f o r e

17 -^ A.,iY\ 3 -^ (2-3m)R a n d i> {K>) = T R ~ •

^

2V

A.Uy\ i ^2 _ 2-3m Now t (^) = M ^ = 4 '^ 2 O)

Ms)

S - na and the edge s t r e s s i s given by

erg = 4 R e [ i ( S ) On the b o u n d a r y

ie

t, = <x = e ,

and the edge s t r e s s i s

3 - 2 c o s 2 e - 3m^ + 2m

er.

e 1 - 2m cos 26 + m 2

T h u s an e x p r e s s i o n i s obtained for the s t r e s s d i s t r i b u t i o n at the edge of an e l l i p t i c a l cut-out of any e c c e n t r i c i t y in a plane sheet subjected to 2:1 b i - a x i a l t e n s i o n .

It should be noted that the angle 6 in the e x p r e s s i o n for the edge s t r e s s r e f e r s to the unit c i r c l e , and the c o r r e s p o n d i n g angle /S on the e l l i p s e i s given by t a n ^ = ktane . In t h i s c a s e , the c r i t i c a l s t r e s s combination

[Q]^

- 3 , and t h e s t r e s s c o n c e n t r a t i o n factor i s X = ^ . / 3 A c i r c u l a r cut-out m a y be r e g a r d e d a s a p a r t i c u l a r c a s e of an e l l i p t i c a l cut-out with m = 0, and in t h i s c a s e the edge s t r e s s b e c o m e s

0-- = 3 - 2 c o s 26

18

-(b) Square Cut-out (Fig. 20b)

The region outside a square in the z-plane may be transformed on to the region outside the unit circle in the ^-plane by the transformation function

.,, J, 1 ^ 1 1 .1 , ^

z = w( S) = Rl S + - + + . . .

^ 6S^ 56 if 176^^' 384^^= ' R > 0 .

By retaining the first few terms only of this expansion, an approximate square with rounded corners is obtained, and by increasing the number of terms the corner radius is reduced.

If the first two terms only are retained in the transformation function

w (S ) = R( Z

-an approximate square is obtained having semi-width

a = | R .

and corner radius

r =0.12a .

The exact profile of the approximate square obtained by this transformation is shown in Fig. 6(a).

Differentiation of the transformation function gives

CO' iO = R ^ l + —

On the unit circle

S = 0- = e , and — 1

19 -T h e r e f o r e 0) ia) ü)' (cr) cr -1 + 6cr' a^ o r e x p a n d i n g t h i s a s a n a s c e n d i n g p o w e r s e r i e s 0) (cr) 1 co'(o-) 60-j . 13 3 ^ 1 2 ^ 24°^ ""'•• A s f o r t h e e l l i p t i c a l c u t - o u t , t h e c o m p l e x s t r e s s f u n c t i o n s m a y b e e x p r e s s e d b y 0 ( S ) = Z a^ S " " + j R S 1 n 4 a n d ^ ( S ) = Z b S 1 n -n

_I-RS

oo ^ ^ v-i_j_ 1 Q 7-7—T- = - Z n a o- + - R , 0'(cr) i n 4 ' Z b cr" 1 n 2 cr 1^ (cr) B y s u b s t i t u t i n g t h e s e e x p r e s s i o n s i n t h e b o u n d a r y c o n d i t i o n 0) (cr) i>{cr) + ?S'(cr) + ^|r{(T) = 0 , CO' (cr) a n e q u a t i o n i s o b t a i n e d t o d e t e r m i n e t h e c o e f f i c i e n t s a , a s - R r + Z 4 1 + a cr n -n 6 cr' ^ 13 3 12 24 - n+1 - R - Z n a or 4 1 n - i ? H- Z b cr" 2 cr _n = 0 .

20

-By c o m p a r i n g coefficients of negative p o w e r s of cr in t h i s equation we find

\

=

TR

.

a, • i n ,

and all the o t h e r coefficients a a r e z e r o . n T h e r e f o r e

0(s)

=

| R S

+ f. 1

^ " ^ ? ? R

**' ..Ms)

K'

and the edge s t r e s s i s given by

CTQ = 4 Re h ( g ) l . On the b o u n d a r y

S = cr = e ,

and the edge s t r e s s i s

63-72 cos 26 ° e 35+28 cos 46

+ i ^

8 • s^ '

3 ^z 3 7 ' ^ " 8

^1

»

for a s q u a r e cut-out having c o r n e r r a d i u s 0 . 1 2 a in a plane s h e e t subjected to 2:1 b i - a x i a l t e n s i o n .

Again, the angle 9 r e f e r s to the unit c i r c l e , and the c o r r e s p o n d i n g angle /? on the s q u a r e i s given by

sin 6 + 7 - sin 36 tan /? = 6

Ï •

cos© - — cos 36

o

By r e t a i n i n g m o r e t e r m s in the t r a n s f o r m a t i o n function, the s t r e s s d i s t r i b u t i o n around s q u a r e c u t - o u t s having c e r t a i n o t h e r c o r n e r r a d i i m a y be obtained.

•s 3 0

f ,

< l

i..s

o u u) I-O <o u O S n =1 ^ B « 3 2 0 4 0 &0 /3 (DEGREES) — (O) SO ICX> 2 5 2 0

s

5 I'S

5

1 0 o s

m

M 20 40 60 /3 (DEGREES) -00 BO ICX>

FIG. 1. Circular cut-out: Theoretical s t r e s s concentration factor when plate is subjected to biaxial tension or shear

.1,000 p.».I. 2,500 2,000 M 1 0 1,000 soo spoo 6,000 j « . 4 , 0 0 0 «2,CXX) bl (COMPRESSION) -2,0<X5 ipOOps.1. 2 , 0 0 0 p.s.'i. .' L _ ^ . • EXPERIMENTAL

( 0 1 " FROM EDGE OF CUT-CJUT) • THEORETICAL 2 0 4 0 6 0 /3 (DEGREES) Co) 1,000 p.i.l. SO too '3,CXX>p.s.l. '^ \ \ . 0 0-^ \ \ S \ EXPERIMENTAL.

(01" FROM EDGE OF CUT-OUT) THEORETICAL.

M

\ > 20 40 60 80 / 3 (DEGREES) " Cc) too iPOOp.s.i. 6,OCX> S,CX50 < 4 , 0 0 0 w 2,000 u "^'^-. \ • • ^ > v 0 » -^ \ _{s \} \ \ _{\ \} " EXPERIMENTAL

( o r FROM EDGE OF CUT-OOT) THEORETICAL. ^^, 0 20 40 60 / 3 ( D E G R E E S ) Q>) 80 too l,OOOp.s.i. 5,000 4 , 0 0 0 •< 3 , 0 0 0 Ü 2,OCX> >4 o o bl 1 0 0 0 / /, / / f I 11 11 11 I I I I i l il IJ -*•—••- EXPERIMENTAL

(01" FROM EDGE OF CUT-OUT)

// ff f £ f / ' / THEORETICAL. (L*--^ v ^ V , \ ^ k «L\ \ \ ' Y \ \ \ \ \ .\ \ \ Y \ 20 4 0 60 / 3 (DEGREES) — 80 too

FIG. 2. Circular cut-out: A comparison between experimental and theoretical s t r e s s distributions when the plate is subjected to biaxial tension or shear.

4 0

20 40 60

/S (DECWEES) —

O»)

80 too

FIG. 3. V2:l Elliptical cut-out: Theoretical s t r e s s concentration factor when plate i s subjected to biaxial tension or shear

k'O'183

3 0 45 6 0 © (DEGREES)

-(o)

75 8 0

FIG. 4. Elliptical cut-out: Theoretical s t r e s s concentration factor for various values of eccentricity when plate is subjected to 2:1 biaxial tension

i , o o o Pt».l. 3 , 0 0 0 2,500 1 2 , 0 0 0 p.s.1. 2^000 >l .o. in SOO I/X30 S(X3 \ **x \ \ \ N V* S N

V^

^ X ^ . . ^ " ^ f-V2 -• •- EXPERIMENTAL

( 0 1 " FROM EDGE Of Cin--OUT)

20 4 0 6 0 BO ICX) / S ( D E G R E E S ) • Ca) s,ooo 4 , 0 0 0 v 3 3 , 0 0 0 b' ^ 2 , ( X > 0 l,OCX) I.CXX) ^».i.

y

_/ -^—•-" •_ f-V? -•---•- EXPERIMENTAL

( o r FROM EDGE OF CUT OUT)

TIJCi-kOCTI/- Al " • 1 2 0 4 0 60 / a ( D E G R E E S ) — * • so 100 ,i3,OCX>p».i. 7,OCX> 6,OCX> S,(XX} I 4,000 * b * 3 , 0 0 0 bl bl 2 , 0 0 0 1,000 - • - - « - EXPERIMENT

( 0 1 " FROM EDGE O F CUT-OUT) THEORETICAL. 2 0 4 0 6 0 BO /S) ( D E G R E E S ) • (C) f = V2 • - EXPERIMENTAL ( 0 1 " FROM E D G E OF C U T - O U T ) — — THECJRETICAL. 5 , 0 0 0 4 , 0 0 0 ^ 3 , 0 0 0 b l 2,OCX> bl (9 1,000 ^ p.»."-ƒ / ƒ / ƒ / / / / / ƒ /

7/

₁₁ ^ ^''"^\ ' a _{\ \} s \ _{\ \} \ \ \

V

\ >^ \ 2 0 4 0 6 0 / 3 (DEGREES) •• 8 0 KX)

FIG. 5. ^2:1 Elliptical cut-out: A comparison between experimental and theoretical s t r e s s distributions when the plate i s subjected to biaxial tension or shear

l O i - 08' 0 08' 6 08' -- 0 -- 4 0 2 -0-2 7a = -l2 R - l PROFILE OF EXACT SQUARE CUT-OUT WITH CORRESPONDING CORNER RADIUS SHOWN BY BROKEN LINE. 0-2 0-4 0-6 O-e I'O (o)

^t

l O --0-8h - 0 - 6 r - 0 - 4 0 2 0 2 -< ^ i ) - R ( ' ^ ~ 6 3 + 5 6 i V 7a = 049 R = l •y") 0-3. 0-4 0-6 0 8 10 (b)

FIG. 6. Square cut-out. Profile of approximate square given by retaining two and three t e r m s in the transformation function

20 40 60

/S (DEGREES) •

80 OO

FIG. 7. Square cut-out (— = .12). Theoretical s t r e s s concentration factor when plate is subjected to biaxial tension or shear

30 4 0 so j3 (DEGREES) -(«D •02 ' 0 4 -06 •08 -to '12 RATIO 7a « Oi) 14 -16 •20

FIG. 8. Square cut-out. Theoretical s t r e s s concentration factor for various c o m e r radii when plate is subjected to 2:1 biaxial tension

»1,000 PLI.L 6,<XX> 5 , 0 0 0 T 4,000 3 , 0 0 0 bl f 2 , 0 0 0 l,OCX> l,CX30 p j l l,0<X> P-i.i

j

, J

— '

l

\ \ . ^ % - -12 _ » _ 0 -s THEORETICAL EXPERIMENTAL (01" FROM E D G E ) • . — — • 2 0 4 0 60 8 0 /S ( D E G R E E S ) • Co) 100 10,000 . 8,0c» 2 6 , 0 0 0 2 4,CXX3 oc 2 , 0 0 0 - 2 , 0 0 0 2,(XX3 Pii-I- 3S^. 7 a - - 1 2 THECJRETICAL - • • - EXPERIMENTAL (01" FROM EDGE) 2 0 4 0 60 fb (DEGREES)-00 80 too 1,000 P^'.!-i,ooor B,CX)0 6 , 0 0 0 4 , 0 0 0 C) 2,0(X> a 2 0 4 0 60 80 100 / © ( D E G R E E S ) • (C) -2,0<X> 20 4 0 60 /b (DEGREES) 1

FIG. 9. Square cut-out (— = .12). A comparison between experimental and theoretical s t r e s s distributions when the plate is subjected to biaxial tension or shear

8 bl O Z 8 V) U) bl o:

r

i

V

w -^ ^ 0 - J » 1 « >

L

y

—f—

ti.

J

•2 " ; - / <^

A

\N ^ r/a > 120 EXPERIMENTAL.

'x

X

"""s--- - - 8 - - - n 10 20 30 40 50 yfl (DEGREES) — (b) 6 0 70 80 90

FIG. 10. Square cut-out. A comparison between experimental and theoretical s t r e s s distributions for various corner radii, when plate is subjected to 2:1 biaxial tension

• .it^-!i.-f.trti,»—.

FIG. 1 1 . P L A N E LOADING FRAME SHOWING UNCUT P A N E L

FIG. 12. TYPICAL CUT-OUT IN PANEL O F PLANE LOADING FRAME

IIOJZ^^

TURNBUCKLE

SHEAR LOADING LINKS

DIAGRAMMATIC ARRANGEMENT OF LOADING R3R PANEL UNDER COMBINED BIAXIAL

TENSION AND SHEAR FIG. 13.

-:J|^=«u^l

TUPNBUCKLE-RVBBER SEiM—-.-c^ WOODEN PLUG SEALING PLATE f EDGE OF TANK

METHOD OF SEALING CUT-OUT IN PRESSURE CYLINDER FIG 15. FIXED END DIAGRAMMATIC ARRANGEMENT OF PRESSURE CYLINDER FIG. 14.

ALL GAUCC» T I N S L I Y « H TYPE WITH tXCEPTlOW O f l « . l 7 . l t . l » J i m

DISTANCES TO CENTRC LiWC OF GAUGES

INNER GAUGES OH FROM . EPOE OF CUT-OUT

|o.rfJp.,-,|p.tf

4 DIAMETER

UNREINFORCED C I R C U L A g CUT-OUT

STRAIN GAUGE POSITIONS ( P L A N E LOADING FRAME)

FIG. 16.

ALL GAUGES TINSLEY 6H TYPE DISTANCES TO I OF GAUGES 7 y O.J-O l " o r CUT-OUT DIMENSIONS MAJOR AXIS 4.24* MINOR AXIS S O '

UNREINFORCED ELLIPTICAL CUT-OUT STRAIN GAUGE POSITIONS (^PLANE LOADING FRAME J

OVERAU. P I M E N t l O N IQ 4 0 ' SQUARE

TIMSLEY SH GAUGES INNER GAUGES 0 1 FROM EDGE OF CUT OUT

STRAIN GAUGES APPROX 1U F»OM

tlXie OF tUT-WT

UNREINFORCED SQUARE CUT-OUT STRAIN GAUGE POSITIONS (PLANE LOADING FRAME)

FIG. 18.

UNREINFORCED SQUARE CUT-OUT STRAIN GAUGE POSITIONS (60IN.DIA. PRESSURE CYLINDER)

FIG. 19.

(a)

FIG. 20. CUT OUT DIMENSIONS AND LOADING USED IN APPENDIX