T H E C O L L E G E O F A E R O N A U T I C S
C R A N F I E L D
THE EFFECT OF CURVATURE ON THE STRESS CONCENTRATIONS
AROUND HOLES IN SHELLS
by
D. S. Houghton and A. Rothwell
TECHNISCHE HOGESCHOOL DELFTVLIEGTUIGBOUWKUNDE
BI3LI0IHEEK A 6 nov. 6 2
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
T h e Effect of C u r v a t u r e on the S t r e s s C o n c e n t r a t i o n s A r o u n d H o l e s in S h e l l s b y -D . S. Houghton*, M . S c . ( E n g . ) , A. M . I . M e c h . E . , A . F . R . A e . S . , and A. R o t h w e l l , B S c , M . S . , D . C . A e . SUMMARY An e x p e r i m e n t a l i n v e s t i g a t i o n h a s b e e n c a r r i e d out t o i n v e s t i g a t e t h e effect of c u r v a t u r e on the s t r e s s c o n c e n t r a t i o n s a r o u n d h o l e s in s h e l l s t r u c t u r e s . Two m e t h o d s h a v e b e e n e m p l o y e d : -(1) A r a l d i t e c y l i n d e r s , c o n t a i n i n g h o l e s of v a r i o u s s h a p e s , s u b j e c t e d t o a x i a l t e n s i o n , i n t e r n a l p r e s s u r e and t o r s i o n w e r e e x a m i n e d b y the p h o t o e l a s t i c f r o z e n s t r e s s t e c h n i q u e . (2) A l u m i n i u m a l l o y c u r v e d p a n e l s and h e m i s p h e r e s w e r e u s e d in conjunction with m i n i a t u r e e l e c t r i c a l s t r a i n g a u g e s .T h e r e s u l t s a r e c o m p a r e d with the t h e o r e t i c a l s o l u t i o n s and s u g g e s t that the c u r v a t u r e effect can be s i g n i f i c a n t , p a r t i c u l a r l y for t h e c a s e of s h e a r o r b i a x i a l l o a d i n g .
* P a p e r p r e s e n t e d to the Second I n t e r n a t i o n a l C o n f e r e n c e on S t r e s s A n a l y s i s , P a r i s , A p r i l 1962.
Notation
Introduction 1 Experimental Procedure 2
2 . 1 . The Photoelastic Investigations 2 2 . 2 . The Experiments on Curved Panels 4 2 . 3 . The Experiments on the
Hemi-spherical Shell 4 Theoretical Considerations 5 3 . 1 . Flat Plate Theory 5 3 . 2 . Circular Hole in Cylindrical SheU 6
3 . 3 . Circular Hole in Spherical Shell 6
Discussion of Results 7
Conclusions 8 References 9 Figures
X y
f , f» radial and tangential direct s t r e s s e s fr> ' e radial and tangential bending s t r e s s e s q applied shear s t r e s s
6 angle on c i r c l e or e l l i p s e
a radius of c i r c u l a r hole, or s e m i - m a j o r axis of elliptical hole b s e m i - m i n o r a x i s of elliptical hole
R radius of curvature of s h e l l t s h e e t t h i c k n e s s
V P o i s s o n ' s ratio
p Internal p r e s s u r e
a angle on the unit c i r c l e
1. Introduction
In the analysis of the aircraft p r e s s u r e cabin and the atomic reactor s t r u c t u r e , a precise knowledge of the s t r e s s distribution around the various shaped openings is required. These may form windows or access doors, in which case their dimensions might be conaparable with the overall dimensions of the shell, and clearly the determination of the s t r e s s concentrations around these openings is important.
It is generally considered that the factors which influence the magnitude of the s t r e s s concentration a r e the shape of the hole and the dimensions of the reinforcement, and provided that the hole size is small compared to the radius of the curvature of the shell, the geometry of the shell itself is thought to be of secondary importance.
In all the recent investigations the assumption has been made that shell curvature has little effect on the s t r e s s concentration around the hole, and to simplify the
theoretical analysis the problem is considered to be identical to that of a hole in an infinite plane sheet.
Little experimental work has been published to justify this assumption; in fact most of the experimental work which is a v a i l a b l e ' ^ ' ^ ' has been carried out on flat plates, owing to the simplification of the n e c e s s a r y apparatus.
One previous attempt to study the s t r e s s e s around circular and elliptical holes in cylinders was carried out by Richards'"^' on Xylonite models. The s t r e s s e s were measured by standard electrical strain gauge techniques in conjunction with an automatic strain gauge r e c o r d e r . The r e s u l t s indicated that significant bending s t r e s s e s a r i s e in the vicinity of the hole, and Richards concluded that the 'flat plate' theory is inadequate to completely describe the s t r e s s distributions around the hole.
A closer examination of these papers suggests that for an unreinforced hole in a shell subjected to internal p r e s s u r e this discrepancy is not unreasonable, since much depends upon whether the t r a n s v e r s e p r e s s u r e loading a c r o s s the hole is allowed to be reacted at the edge of the hole, or whether this is reacted externally. The flat plate theory when applied to unreinforced holes does not make allowance for the effect of out of plane forces. F o r the reinforced hole, it is generally considered that provided the hole is neutral, or nearly so, and the t r a n s v e r s e loading is applied in a prescribed m a n n e r " ' , then close agreement should be obtained with the flat plate solution. In other words, the s t r e s s concentrations arising in the vicinity of an unreinforced hole in a curved shell will be in excess of those given by flat plate theory, but if the hole is suitably reinforced it is considered that the discrepancies should become insignificant.
The theoretical analysis of the s t r e s s distribution around holes in circular cylinders is restricted to very small unreinforced circular holes. The solution for the biaxial loading case is given by L u r i e ' ^ ' , and this solution has been discussed by Penney^''. L u r i e ' s solution has been extended by Withum'"' to the
case of a cylinder subjected to torsion, and again the solution is restricted to very small holes. The solution for a circular hole in a spherical shell under
subject to certain conditions on the size of the hole. An alternative solution is given by Penney^^'.
(4)
The r e s u l t s of a number of experiments which were undertaken to investigate the significance of the effect of curvature on the s t r e s s concentration around c i r c u l a r and elliptical holes a r e reported. A s e r i e s of photoelastic experiments using the frozen s t r e s s method has been conducted on unreinforced circular and elliptical holes in Araldite cylinders, subjected to various loading conditions. A further s e r i e s of t e s t s was carried out to find the effect of curvature on the s t r e s s concentrations around unreinforced circular holes in curved panels and in a hemispherical shell, using electrical strain gauges.
The hole dimensions a r e comparable with the large openings which occur in many p r e s s u r e vessel designs, and in consequence are in excess of those given by the limits of the Lurie-Withum solution for cylindrical shells.
The experimental r e s u l t s a r e compared with the various theoretical solutions and also with the corresponding flat plate solution.
2. Experimental Procedure
In o r d e r to study the effect of shell curvature on the s t r e s s concentrations around unreinforced holes, the following s e r i e s of experiments were conducted:
(a) Photoelastic investigations using Araldite cylinders which contained either circular or elliptical holes.
(b) Experiments on curved aluminium alloy panels containing a c i r c u l a r hole. (c) Experiments on a hemispherical shell having a circular hole.
2 . 1 . The Photoelastic Investigations The Range of the Experinnents
The cylinders under test were cast from Araldite B and Hardener 901, a s supplied by Aero Research Ltd. , in the ratio of four to one, and machined to the dimensions given in Fig. 1. Three sizes of circular hole having diameters 0.8 in. , 1.0 in. and 1.2 in. , were examined. In addition an elliptical hole having a minor axis 0.85 in. and major axis 1.2 in. (with its minor axis parallel to the longitudinal axis of the cylinder) was l a t e r examined. The apparatus shown in Fig. 2 enabled the cylinders to be loaded by axial tension, torsion or internal p r e s s u r e .
The frozen s t r e s s technique was used to study the s t r e s s concentrations in the vicinity of the holes. This method was found convenient since only small externally applied forces were required. Also, in the case of internal p r e s s u r e loading, the presence of the sealing plug covering the hole made it n e c e s s a r y to remove the loading before analysing the s t r e s s e s .
The arrangement of the polariscope to measure the mean s t r e s s e s in the cylinder is shown in Fig. 3. A simple crossed diffuse light polariscope was used
with monochromatic sodium light. The p o l a r i s e r was set inside the cylinder, with the hole under examination between it and the analyser. By this arrangement, s t r e s s e s in the far wall of the cylinder did not interfere with those being measured. The Specimen Loading Mechanism
Since the specimen wasl loaded at temperature the available oven size
dictated the overall dimension of the loading mechanism. The oven was a Towers thermostatically controlled electric oven, and the loading rig consisted essentially of two p a r t s :
(a) A frame for holding the specimen, together with the n e c e s s a r y nnechanism for applying tension and torsion.
(b) The pressurlsatlon equipment.
The tensile load was applied by weights applied at the end of a 4:1 lever a r m attached by a flexible cable to the centre of the top plate of the cylinder. Torsion was applied by weights attached by cables around a 5.4 in. dia. disc which was screwed to the top plug. Arrangements were made to counterbalance the weight of this disc during the experiment.
The p r e s s u r l s a t l o n equipment was designed to give a constant p r e s s u r e , and to compensate for variations in the a i r volume in the system due to temperature changes. This was achieved by using a balloon as an air r e s e r v o i r to which p r e s s u r e was applied by a constant head of water. The p r e s s u r e was measured on a water manometer and could be varied by altering the height of the constant head. It was found that this method could control the internal p r e s s u r e of 1 lb/in2 to within 0.5 per cent during the time the specimen was cooling from 130OC.
During pressurlsatlon of the cylinder, the holes were sealed by a metal plug which was covered inside the cylinder by thin polythene sheeting held in position by a thin film of silicone oil. The polythene had the advantage of softening at temperature and taking up the contour of the cylinder wall. Initially this plug was supported externally (method 'A') so that the p r e s s u r e loading on the hole would not be carried by the cylinder. This method was intended to minimise bending s t r e s s e s at the edge of the hole. The t e s t s were subsequently repeated using a very thin, flexible metal plug which was supported by the edge of the hole (method 'B') so that the p r e s s u r e loading on the hole would be reacted at the edge. This method is more consistent with the theoretical solutions.
The Frozen S t r e s s Technique
The frozen s t r e s s method was found to be satisfactory for this s e r i e s of experiments. The identification of the fringes proved quite easy in most c a s e s . Because of the limited space inside the cylinders, the Senarmont method of determining the fractional fringe o r d e r was employed since it did not require a quarter wave plate between the polariser and the specimen.
The main disadvantage of the frozen s t r e s s method was the presence of compressive edge effects which appeared with time, but this could be overcome
by analysing the cylinder as soon as possible after the specimen had been removed from the oven. By examining the cylinder without slicing, the mean value of the s t r e s s was obtained. The method of testing does not allow bending s t r e s s e s in the cylinder to be measured.
2 . 2 . The Experiments on Curved Panels
A s e r i e s of experiments were conducted using four L. 72 aluminium alloy curved panels (Fig. 4) which were 20 in. long and 10 in. developed width, containing a 2 in. diameter circular hole. The panel dimensions were chosen to give an adequate variation of p a r a m e t e r _ ^ .
Rt
The dimensions of the panels used in this s e r i e s of experiments were:
Panel No. Radius of Thickness Curvature (in.) (in.) 1 6 .028 2 6 .022 3 12 .022 4 12 .036 The panels were loaded in tension in a standard tensile testing machine. A
number of special end plates were designed to minimise bending effects in the panel. A preliminary s e r i e s of experiments indicated that the panel dimensions were such that panel end effects did not disturb the s t r e s s distribution in the vicinity of the hole.
Strains were measured by means of Tinsley 6H electrical resistance strain gauges with a conventional strain gauge r e c o r d e r . The strain gauge positions a r e shown in Fig. 5.
2 . 3 . The Experiments on the Hemispherical Shell
Only one specimen was tested for this s e r i e s of experiments. This consisted of a 16 in. diameter hemispherical shell containing a 3 in. diameter hole, made by spinning a sheet of 16 s . w . g . (0.064 in.) L.59 aluminium alloy sheet (Fig. 6). This method of manufacture has proved satisfactory for making h e m i s p h e r e s , although some variation in thickness around the contour of the hemisphere was to be expected. It was found that in the region of the hole the thickness was substantially constant at 0.062 in.
A wooden plug with a standard 1/8 in. diameter 'O' ring was used to seal the specimen (Fig. 7) which was p r e s s u r i s e d using air at 18 Ib/in^. The s t r e s s e s were measured by Tinsley 6H and 6K electrical strain gauges located as shown in Fig. 8, together with a Savage and P a r s o n s strain gauge r e c o r d e r .
3. Theoretical Considerations 3 . 1 . Flat Plate Theory (Fig. 16)
The solution for the s t r e s s e s around an unreinforced elliptical hole in an infinite plane sheet may be obtained by the use of the contiplex s t r e s s function and the method of conformal transformation'^'.
The transformation function
z - A ( S + f )
t r a n s f o r m s an ellipse In the z-plane on to the unit circle in the ^-plane, where a - b
m a + b
and a , b a r e the s e m i - m a j o r and s e m i - m i n o r axes respectively. Complex s t r e s s functions a r e found which satisfy the boundary conditions at the edge of the hole and at infinity.
F o r an elliptical hole in a plate subjected to uniform biaxial tensions f^ and X f , the s t r e s s fa at the edge of the hole is given by
(f + f ) (1 - m*) + 2(f - f ) (m - cos 2a) f^ = _2E 1 X 2^
1 + m - 2m cos 2 a
and when the plate is subjected to uniform s h e a r q the edge s t r e s s is . 4q sin 2«
1 + m - 2m cos 2a
The angle a is the angle on the unit c i r c l e , and the corresponding angle 6
on the ellipse is given by i_
tan 6 = — tan a
a
F o r a c i r c u l a r hole m = 0 ,
and the expression for the edge s t r e s s under biaxial tension reduces to
f ^ = (f + f ) - 2(f - f ) cos 26 e X y X y
and under uniform s h e a r
fg = 4q sin 2 e ,
3 . 2 . Circular Hole in Cylindrical Shell (Fig. 17)
The solution for a small circular hole in a cylindrical shell under uniform biaxial tension has been given by L u r i e ' ^ ' . The solution is r e s t r i c t e d to v e r y small holes for which
Rt ^^ ^ • where a is the radius of the hole,
R is the radius of curvature of the cylindrical shell
and t is the sheet thickness.
When the shell is subj ected to biaxial tensions f and f , the direct
J X y
s t r e s s fg at the edge of the hole is given by ffi = (f + f ) - 2(f - f ) cos 26
0 X y X y
+ V3(i -v'). X- I T [2 f - ( f - 3 f ) cos 2 e l .
4 Rt L y X y J Lurie also obtains expressions for the bending s t r e s s in the shell at the edge of the hole.
(6) The corresponding solution for the case of uniform s h e a r is given by Withum , with the same restriction on hole size. In this case the s t r e s s at the edge of the hole is given by
fg = q r 4 + V3(l -v"). I . ^ J sin 26
3 . 3 . Circular Hole in Spherical ShellA solution for the s t r e s s concentration around a circular hole in a spherical shell may be obtained from the general equations for spherical shells given by Novozhilov'^'. The solution is not applicable to very small values of the curvature p a r a m e t e r ( a ' / R t j: 1. approximately) where R is the spherical radius.
The direct s t r e s s f Q at the edge of the hole in a spherical shell subjected to internal p r e s s u r e p is given by
r
^6 2t 1 +
nëö-T^)./^] .
If the ratio a/R is small (a/R ^ \, approximately) and there a r e no bending s t r e s s e s at the edge of the hole according to this solution.
(8)
An alternative solution has been given by Penney , for which expressions for the edge s t r e s s may be written approximately:
In the r a n g e ^ * ^l ^ < 0.5 Rt f E 5 e 2t 2 + 2.6
af_
Rt and f o r ' ^ > 0.5 Rt f ER e 2t 1.2 + 2.50 a ^ Rt( T a k i n g v = ^) and in t h i s c a s e the bending s t r e s s e s at the edge of the hole a r e not z e r o .
NOTE
In the t h e o r e t i c a l s o l u t i o n s f o r a s e a l e d hole in a c y l i n d r i c a l o r s p h e r i c a l s h e l l u n d e r i n t e r n a l p r e s s u r e , it i s a s s u m e d that the p r e s s u r e loading on the hole i s r e a c t e d by a s h e a r f o r c e at the edge of the h o l e , so that t h i s p a r t of the l o a d i n g i s c a r r i e d by the s h e l l itself.
4 . D i s c u s s i o n of R e s u l t s
T h e r e s u l t s of the p h o t o e l a s t i c i n v e s t i g a t i o n s on c i r c u l a r and e l l i p t i c a l h o l e s in t h e A r a l d i t e c y l i n d e r s , u n d e r v a r i o u s l o a d i n g c o n d i t i o n s , a r e shown in F i g s . 9, 10 and 1 1 .
F o r the t o r s i o n l o a d i n g c a s e it i s found t h a t s h e l l c u r v a t u r e h a s a significant effect on the s t r e s s c o n c e n t r a t i o n a r o u n d the h o l e . F o r the c i r c u l a r hole the m a x i m u m s t r e s s c o n c e n t r a t i o n i n c r e a s e s with hole d i a m e t e r , and for the l a r g e s t hole t e s t e d (1.2 i n . d i a . ) the m a x i m u m s t r e s s c o n c e n t r a t i o n e x c e e d s the flat p l a t e t h e o r y by about 40 p e r c e n t . It i s a l s o found t h a t the p o s i t i o n of maiximum s t r e s s c h a n g e s with i n c r e a s e in hole d i a m e t e r . Although only one s i z e of e l l i p t i c a l hole w a s t e s t e d , it would a p p e a r that the effects of c u r v a t u r e a r e s i m i l a r .
When the c y l i n d e r i s 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 , l i t t l e i n c r e a s e in the s t r e s s c o n c e n t r a t i o n due to c u r v a t u r e i s obtained if the p r e s s u r e l o a d i n g on the h o l e i s r e a c t e d e x t e r n a l l y . T h i s i s p r o b a b l y due to an effective r e d u c t i o n in the l o c a l p r e s s u r e l o a d i n g in the v i c i n i t y of the h o l e . H o w e v e r , if the p r e s s u r e on t h e hole i s r e a c t e d at the edge of the hole s o t h a t t h i s p a r t of the l o a d i n g i s s u p p o r t e d by the s h e l l itself, a significant i n c r e a s e in the s t r e s s c o n c e n t r a t i o n i s o b t a i n e d .
When t h e c y l i n d e r i s s u b j e c t e d to a x i a l t e n s i o n no i n c r e a s e in s t r e s s c o n c e n t r a t i o n i s found.
In F i g . 1 1 , the v a r i a t i o n of t h e m a x i m u m s t r e s s c o n c e n t r a t i o n with t h e c u r v a t u r e p a r a m e t e r >'"a*/Rt i s plotted for the c i r c u l a r h o l e s , and a c o m p a r i s o n i s m a d e with the t h e o r e t i c a l s o l u t i o n s of L u r i e and Withum, which a r e r e s t r i c t e d to v a l u e s of the c u r v a t u r e p a r a m e t e r a / R t « 1.
In p r a c t i c a l a p p l i c a t i o n s the v a l u e of the c u r v a t u r e p a r a m e t e r a / R t m a y g r e a t l y e x c e e d unity, and to i n v e s t i g a t e the effect of c u r v a t u r e for l a r g e r v a l u e s
In o r d e r to show the effect of curvature on the s t r e s s concentrations around holes in spherical shells under internal p r e s s u r e , a test was carried out on a hemispherical shell containing a c i r c u l a r hole. Fig. 13 shows the variation of the theoretical maximum s t r e s s concentration (obtained from Ref. 8) with the p a r a m e t e r a / R t . F o r one particular diameter of hole (* a*/Rt = 2.09) the experimental r e s u l t s of the variation of the radial and tangential direct s t r e s s e s
with the distance from the edge of the hole a r e given in Fig. 14, and the corresponding r e s u l t s for the radial and tangential bending s t r e s s e s in the shell a r e given in
F i g . 15. These r e s u l t s a r e compared with the theoretical solution derived from Ref. 9. Good general agreement is obtained, although the theoretical r e s u l t s a r e somewhat in excess of the experimental. The maximum s t r e s s concentration is found to be considerably g r e a t e r than that predicted by flat plate theory, and it i s seen that the bending s t r e s s e s away from the edge of the hole a r e significant. 5. Conclusions
The effect of curvature on the s t r e s s concentrations around holes in shells may be considerable in many c a s e s , particularly for holes in a cylinder under torsion or internal p r e s s u r e , and for holes in a spherical shell under internal p r e s s u r e .
F o r internal p r e s s u r e loading, the s t r e s s concentration also depends on the method of supporting the p r e s s u r e on the sealed hole.
2. Houghton, D . S . , Rothwell, A. 3. Richards, T . H . 4. 5. L u r i e , A . I . 6. Withum, D. 7. Penney, R.K. 8. Penney, R.K. 9. Novozhilov, V.V.
The analysis of reinforced circular and elliptical cutouts under various loading conditions.
College of Aeronautics Report 151, 1961. S t r e s s distribution in p r e s s u r i s e d cabins: an experimental study by means of Xylonite ndodels.
A . R . C . 19360, Strut. 1999, 1957.
Unpublished thesis work conducted at the College of Aeronautics by Hopkins, H . L . , Gopal, C.S. , Quicke, D . C . , and the authors. Statics of thin walled elastic s h e l l s .
(Russian). Ogiz, Moscow, 1947.
Circular hole in a cylinder under torsion. (German). Ingenieur-Archiv, vol.26, 1958. Discussion of paper presented by Houghton,D.S. Nuclear reactor containment buildings and p r e s s u r e v e s s e l s .
Butterworths, London, 1960, pp. 316-317. S t r e s s concentrations at unreinforced holes in p r e s s u r i s e d spherical s h e l l s .
P r i v a t e communication, 1960. The theory of thin shells.
Noordhoff, Groningen, The Netherlands, 1959. (English translation).
o I» IN
-e
s-^i IN W
i i i i iN^IA^
FIG. 1. DIMENSIONS OF ARALDITE CYLINDERS
OPAL PERSPEX iCREEN CYLINDRICAL SPECIMEN / A N A L Y S E R ' ^ ' P L A T E 4 SODIUM LIGHT SOURCE
FIG. 3 . ARRANGEMENT O F POLARISCOPE
TELESCOPE
O
, -a*
ALL GAUGES TINSLEY S H D I M C N S I O N S I N I N C H E S .
-+-* „ _!-•-+-* - J
D I R E C T I O N o r LOAD
FIG. 5 . STRAIN GAUGE POSITIONS FOR TESTS ON CURVED PANELS
FIG. 6. HEMISPHERICAL SHELL CON-TAINING A CIRCULAR HOLE
C L A M P I N G RING SASE PLATE AIR SUPPLY, MANOMETCR , AND
STRAIN GAUGE CONNCCTIONS
F I G . 7. DIMENSIONS AND M E T H O D O F S E A L I N G O F H E M I S P H E R I C A L S H E L L T I N S L E Y SK STRAIN GAUGES T IN S L EY S H STRAIN GAUGES F I G . 8. STRAIN G A U G E P O S I T I O N S FOR T E S T S ON H E M I S P H E R I C A L S H E L L
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II
ft
til po^5
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DIRECT STRESS CONCENTRATION AT EDGE O F HOLE ^ OIAECT STRESS CONCENTBATION AT EDGE OF HOLE f^
a'jRi « •mox/(,-3 (T+0-43^) <i"/Rt « I 0 2 0 . 4 0 6 0 8 1-0 « nAoius RADIUS SHEET OF OF MOLE CYLINDER THICKNESS o R t
FIG. 11 PHÜCUEI.ASriC TK\S1> MAXIMUM STHE.SS CONCENTRATION FOR CIHCULAR HOLE IN CYLINDER UNDER TORSION. INTERNAL
0 6
EXPERIMENTAL LAT t»LATE THEORY
RADIUS OF HOLE o RA0U5 OF CUevUkTURE. R. SHEET THICKNESS, 't
Ch4 0-8 IS I- » 0 3-a
FIG. 12. TESTS ON CIRCULAR HOLE IN CURVED PANEL U M S a AXIAL TENSION: MAXIMUM STRESS CONCENTRATION
FKJ. 13. THEORETICAL MAXIMUM STRESS CONCENTRATION FOR CIRCULAR HOLE IN SPHERICAL SHELL tnJDEH INTERNAL
W 3 H 2 Ö M 3 w w ^ 5 o » jj) o 3?:8 o » » a O «ü 5 -TN o
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