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Experimental investigation of the nonlinear behaviour of shallow spherical shells
NUCLEAR STRUCTURAL ENGINEERING 1 (1965) 285-294. NORTH-HOLLAND PUBLISHING COMP., AMSTERDAM
E ~ N T A L BEHAVIOUR
INVESTIGATION OF THE NONLINEAR OF SHALLOW SPHERICAL SHELLS *
J. D. W. HOSSACK Department of Mechanical Engineering, University of Strathclyde, Glasgow, Scotland
Received 16 January 1965
This paper presents a brief review of some recent contributions in the field of instability of shallow spherical shells. The causes of the apparent inconclusive nature of earlier experimental investigations are examined. This is followed by a report of work carried out in such a manner as to eliminate, as far as possible, some of the factors responsible for experimental error. The results of an experimental investigation of the non-linear stress and deformation characteristics of shallow shell behaviour are presented. Good experimental agreement with theory is shown to be dependent on the preparation of uniform stress free specimens of accurate geometry and on the requirement that the shell be supported in a manner consistent with the assumptions of relevant theory. 1. INTRODUCTION
2. REVIEW
One of the m a n y p r o b l e m s which face d e s i g n e r s of shell s t r u c t u r e s in the field of n u c l e a r e n g i n e e r i n g , i s that of i n s t a b i l i t y . Though e x t e n sive t h e o r e t i c a l a n d 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 s have been c a r r i e d out, the p r o b l e m is still u n r e s o l v e d . The p r e s e n t state of knowledge can do little to help d e s i g n e r s of e n g i n e e r i n g s h e l l s which a r e f r e q u e n t l y subject to a v a r i e t y of load conditions. Such s h e l l s may have p e n e t r a t i o n s and r e i n f o r c i n g pads which, together with i m p e r f e c t i o n s f r o m t r u e s p h e r i c a l f o r m , make the s t a b i l i t y a n a l y s i s of the shell as a whole e x t r e m e l y difficult. Before such a n a l y s e s can be c a r r i e d out, it i s f i r s t n e c e s s a r y to e s t a b l i s h with some c e r t a i n t y , the behaviour of i d e a l i s e d s h e l l s . Though much work has b e e n done with t h i s object in view, the m e a s u r e of a g r e e m e n t of e x p e r i m e n t a l data with r e l e v a n t t h e o r y is s t i l l disappointing. The p r e s e n t study is d i r e c t e d t o ward an a n a l y s i s of the f a c t o r s which influence e x p e r i m e n t a l work. Much of the p r e v i o u s e x p e r i m e n t a l work was p e r f o r m e d on c l a m p e d s h e l l s and the r e s u l t s of t h i s i n v e s t i g a t i o n suggest that some of t h i s work m a y have b e e n affected by u n c e r t a i n t i e s s u r r o u n d i n g methods adopted to obt a i n this b o u n d a r y condition.
In the study of the i n s t a b i l i t y of shell s t r u c t u r e s , the s p h e r i c a l shell has been the subject of many t h e o r e t i c a l and e x p e r i m e n t a l i n v e s t i g a tions. E a r l y t h e o r e t i c a l a n a l y s i s [1-3] of the s t a b i l i t y of the complete s p h e r i c a l shell under e x t e r n a l p r e s s u r e , were b a s e d on the l i n e a r , c l a s s i c a l concept of buckling. However the exp e r i m e n t a l r e s u l t s [4] indicated collapse p r e s s u r e s which were a f r a c t i o n of the t h e o r e t i c a l l y d e t e r m i n e d values. The apparent f a i l u r e of the c l a s s i c a l r e a s o n i n g to apply to p r a c t i c a l s h e l l s , led w o r k e r s to e x a m i n e the behaviour of a s h a l low s p h e r i c a l cap which was a s s u m e d to be c l a m p e d at the boundary. This a s s u m p t i o n i n volved a b a s i c change in the a n a l y s i s of the p r o b l e m . The e s s e n t i a l difference between the p r o b l e m of the complete sphere and that of the cap, was that the s p h e r e was a s s u m e d to be in a state of m e m b r a n e c o n t r a c t i o n p r i o r to buckling, while the a s s u m e d support conditions of the shallow cap caused this p r o b l e m to be one of continuous d e f o r m a t i o n f r o m the f i r s t application of the load. P r e s e n t e d in t h i s f o r m , the buckling of the shallow cap was c l e a r l y a l a r g e deflection p h e n o m e n o n with the r e s u l t that the governing diff e r e n t i a l equations b e c a m e n o n - l i n e a r . In this r e s p e c t , the equations d e r i v e d by Biezeno [5], R e i s s n e r [6] and Kaplan and Fung [7] were e s s e n t i a l l y s i m i l a r . A l a r g e volume of the published l i t e r a t u r e has dealt with solutions of t h e s e
* Accepted by T. A. Jaeger. Adv.: A. S. Tooth.
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equations, but s u c c e s s was not a t t a i n e d until Budiansky [8] and, independently, Weinitschke [9] obtained n u m e r i c a l solutions for the buckling p r e s s u r e s of a c l a m p e d shallow cap over a wide r a n g e of r i s e to t h i c k n e s s ratio. A l e s s e r volume of the l i t e r a t u r e has dealt with e x p e r i m e n t a l work. In 1954, Kaplan and Fung [7] m a n u f a c t u r e d s h e l l s by spinning f r o m hot m a g n e s i u m alloy. The s p e c i m e n s were t e s t e d u n d e r both oil and a i r p r e s s u r e c o r r e s p o n d i n g to constant volume and constant p r e s s u r e c o n d i tions. They came to the c o n c l u s i o n that the loading m e d i u m had no significant effect on the buckling load. The e x p e r i m e n t a l values of the c o l lapse p r e s s u r e showed an a p p r e c i a b l e s c a t t e r . In 1959, Homewood, B r i n e and Johnson [10] r e ported f u r t h e r e x p e r i m e n t a l work on p r e s s u r e loaded c l a m p e d shells. The r e s u l t s showed a s i m i l a r degree of s c a t t e r . In 1962, E v a n - I w a n o w s k i , Cheng and Loo [11] r e p o r t e d on e x p e r i m e n t a l work c a r r i e d out on the point loading of both f r e e l y supported and c l a m p e d shells. Both m e t a l l i c and n o n - m e t a l l i c s p e c i m e n s were tested. The m e t a l l i c s p e c i m e n s in copper, a l u m i n i u m and steel w e r e f o r m e d by the h y d r o f o r m p r o c e s s , with s p h e r i c a l r a d i i v a r y i n g f r o m 5 in. to 11.2 in. and b a s e r a d i i v a r y i n g f r o m 3.812 in. to 0.697 in, They noticed that c e r t a i n shells exhibited d i s c o n t i n u o u s l o a d deflection behaviour. Though this phenomenon would appear to be c o n s i s t e n t with a change in the deflection mode shape, the a u t h o r s did not c o m m e n t on this aspect but c l a i m e d that, in most c a s e s , r o t a t i o n a l s y m m e t r y was p r e s e r v e d . T h e i r r e s u l t s showed a significant s c a t t e r which might be a t t r i b u t e d to t h e i r e x p e r i m e n t a l t e c h nique and the difficulty of d e t e r m i n i n g the g e o m e t r y of some of t h e i r s p e c i m e n s which, in some c a s e s , were v e r y s m a l l . It would also appear that some of t h e i r c a l c u l a t e d v a l u e s for the shell p a r a m e t e r were i n c o n s i s t e n t with the published m e a s u r e m e n t s . The lack of e x p e r i m e n t a l a g r e e ment with e s t a b l i s h e d t h e o r e t i c a l work b e c a m e m o r e s e r i o u s in the r a n g e of s h e l l s showing the d i s c o n t i n u o u s behaviour and for the s h e l l s which were r e c o g n i s a b l y n o n - s y m m e t r i c in d e f o r m a tion. L a t e r in the s a m e year Evan-Iwanowski p r e s e n t e d f u r t h e r r e s u l t s on the combined point and p r e s s u r e loading of shells. In this i n v e s t i g a tion he found that the c r i t i c a l c o m b i n a t i o n of point and p r e s s u r e load was independent of the o r d e r in which they w e r e applied - a not u n e x pected r e s u l t . He also r e p o r t e d that for some positions of an e c c e n t r i c a l l y placed point load, the value of the c r i t i c a l c o m b i n a t i o n of loading
was r a i s e d a s c o m p a r e d with the r e s u l t when the point load was applied at the apex. The e x p e r i m e n t a l work so f a r r e v i e w e d is c h a r a c t e r i s e d by the s c a t t e r of r e s u l t s , a f a c t o r which i n e v i t a b l y r e d u c e s its value a s a t e s t of r e l e v a n t theory. The w o r k e r s on the t h e o r e t i c a l side of the p r o b l e m , have examined the v a r i o u s c a u s e s which might account for t h e s e effects. In this r e s p e c t , Budiansky [8] found that the effect of i n i t i a l i m p e r f e c t i o n of an a s s u m e d s y m m e t r i c f o r m could not in itself account for the d i s p a r i t y between the t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s . He t h e r e f o r e concluded that an explanation m u s t lie in supposed n o n - s y m m e t r i c behaviour. Seve r a l p a p e r s have been published p r e s e n t i n g a n a l y s e s of n o n - s y m m e t r i c deformation. The work of P a r m e r t e r [12], Huang [13] and T h u r s t o n [14] a r e in s u b s t a n t i a l a g r e e m e n t while that of W e i n i t s c h ke [15] p r e d i c t e d a different t r e n d for the c r i t i c a l p r e s s u r e s of c l a m p e d s h e l l s . Though the l a t t e r theory was in some a g r e e m e n t with e a r l y e x p e r i m e n t a l work, the a g r e e m e n t would appear to be to some extent f o r t u i t o u s in the light of r e c e n t 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 s [12]. The f a c t o r s which c r i t i c a l l y influence e x p e r i m e n t a l work, have only been a p p r e c i a t e d r e l a t i v e l y r e c e n t l y . Since i n s t a b i l i t y is total s t r e s s dependent, the influence of i n i t i a l s t r e s s e s in the s p e c i m e n , whether they o r i g i n a t e in m a n u f a c t u r e or in the test set up, is not easy to evaluate. T h i s is p a r t i c u l a r l y t r u e since the governing d i f f e r e n t i a l equations a r e n o n - l i n e a r . The m a n u f a c t u r e of s p e c i m e n s by s p i n n i n g i n t r o d u c e s both r e s i d u a l s t r e s s effects and v a r i a t i o n s in t h i c k n e s s . U n l e s s c a r e is taken to s t r e s s r e l i e v e s p e c i m e n s , i n i t i a l s t r e s s effects may also be p r e s e n t in other t e c h n i q u e s of m a n u f a c t u r e . It thus b e c o m e s c l e a r , that for valid e x p e r i m e n t a l data to be obtained, close attention must be d i r e c t e d to producing a s t r e s s f r e e , u n i f o r m shell whose g e o m e t r y can be a c c u r a t e l y d e t e r m i n e d . Much of the work on c l a m p e d s h e l l s has been c a r r i e d out by a t t e m p t i n g to achieve t h i s bounda r y condition by mounting the s p e c i m e n between r i n g s . All the available evidence i n d i c a t e s that this method p r o v i d e s only a poor a p p r o x i m a t i o n to the fully fixed condition and, in addition, may induce r a n d o m f o r c e and m o m e n t a c t i o n s which may cause p r e m a t u r e collapse of the shell. At best, the method may introduce an u n a c c e p t a b l e degree of s c a t t e r into the e x p e r i m e n t a l r e s u l t s . The degree of fixity is of c r i t i c a l i m p o r t a n c e in the p r o b l e m of i n s t a b i l i t y and it is doubtful whether a fully fixed support condition can e v e r be achieved even in the l a b o r a t o r y . While the ex-
NON-LINEAR BEHAVIOUR OF SHALLOW SPHERICAL SHELLS p e r i m e n t a l work of P a r m e n t e r [12] shows an e n c o u r a g i n g m e a s u r e of a g r e e m e n t with t h e o r e t i c a l a n a l y s i s , the s c a t t e r p r e s e n t in his r e s u l t i n d i c a t e s that p e r h a p s even m o r e r e f i n e d e x p e r i m e n t a l t e c h n i q u e s m a y be n e c e s s a r y though it is r e c o g n i s e d that additional p r o b l e m s a r e i n t r o duced when n o n - s y m m e t r i c behaviour is e x pected. In view of the p r a c t i c a l difficulties a s s o c i ated with d i s p l a c e m e n t dependent b o u n d a r y c o n ditions, the p r e s e n t work avoids t h e s e p r o b l e m s by adoption of force dependent conditions. The s i m p l e s t of t h e s e is, of c o u r s e , the fully f r e e support w h e r e both the m e r i d i o n a l m e m b r a n e and bending a c t i o n s a r e z e r o on the boundary. The e x p e r i m e n t a l work was a i m e d at the d e t e r m i u a t i o n of the full e q u i l i b r i u m path for both point and p r e s s u r e loaded shells. In addition to t h i s , the work included a t h e o r e t i c a l and e x p e r i m e n t a l a n a l y s i s of the v a r i a t i o n and s t r e s s and d e f o r m a t i o n along the e q u i l i b r i u m path. T o s i m plify the e x p e r i m e n t a l a n a l y s i s , the p r o b l e m was r e s t r i c t e d to s h e l l s for which s y m m e t r i c a l b e h a v i o u r had b e e n e s t a b l i s h e d .
3. E X P E R I M E N T A L
WORK
3.1. T e s t s p e c i m e n s In view of the s t r i c t g e o m e t r i c a l r e q u i r e m e n t s and the n e c e s s a r y s t r e s s f r e e c h a r a c t e r of the s h e l l s , it was decided to f o r m t h e s e f r o m sheet a l u m i n i u m alloy. The alloy chosen c o n t a i n e d 2% m a g n e s i u m and 0.25% m a n g a n e s e . T h i s m a t e r i a l had good l i n e a r s t r e s s - s t r a i n p r o p e r t i e s in the a n n e a l e d state. 3.1.1. D e t e r m i n a t i o n of e l a s t i c c o n s t a n t s a. Young' s m o d u l u s An attempt to obtain Young's m o d u l u s f r o m suitably s t r a i n - g a u g e d t e n s i l e s p e c i m e n s yielded v a r i a t i o n s in E of about 6%. To avoid the u s e of s t r a i n gauges, a m o r e f u n d a m e n t a l approach was adopted. A s t r i p of the sheet m a t e r i a l was cut and set up as a s i m p l y supported b e a m . The app l i c a t i o n of equal loads to the overhung ends of the b e a m produced c o n s t a n t bending m o m e n t b e tween the supports. Thus the b e a m bent to f o r m a c i r c u l a r a r c over the r e g i o n of c o n s t a n t b e n d ing m o m e n t . By m e a s u r i n g the c e n t r a l deflection of the b e a m r e l a t i v e to the s u p p o r t s , the r a d i u s of c u r v a t u r e of the c e n t r a l p o r t i o n of the b e a m could be e x p r e s s e d in t e r m s of the deflection and the span. F r o m the s i m p l e bending r e l a t i o n , the value of Young' s modulu~ was obtained as a f u n c tion of the r a d i u s of c u r v a t u r e , the bending m o -
287
ment and the second m o m e n t of a r e a of the beam. A graph of the bending m o m e n t against the c e n t r a l deflection was drawn. The slope of t h i s line, which was p r o p o r t i o n a l to the value of Young's modulus, was d e t e r m i n e d by m i n i m i s i n g the l e a s t s q u a r e s deviation of the m e a s u r e d points. The value of Young's m o d u l u s so obtained was within l e s s than 2% of the accepted value of 10 x 106 l b / i n . 2. b. Poisson's ratio Since the effect of slight error in the value of Poisson's ratio w a s of less importance, the value w a s obtained f r o m the ratio of transverse to axial strain gauge m e a s u r e m e n t in a tensile test.
3.1.2. M a n u f a c t u r e of t e s t s p e c i m e n s Two sets of dies of 10 in. base d i a m e t e r were m a c h i n e d on a profile following lathe to r a d i i of c u r v a t u r e of 80 in. and 100 in. The o r d i n a t e s of the die p r o f i l e s w e r e checked for a c c u r a c y to within 0.0002 in. D i s c s 10 in. d i a m e t e r were cut f r o m alloy sheet with a t o l e r a n c e of 0.001 in. in the r a d i u s . A set of dies with the disc between was then heated in an oven to a p p r o x i m a t e l y 360oc and m a i n t a i n e d t h e r e for four hours. The oven and contents w e r e then ailowed to cool so that the f o r m i n g p r o c e s s took about 24 h o u r s to c o m plete. By this p r o c e s s , the r e s i d u a l f a b r i c a t i o n s t r e s s e s w e r e r e d u c e d to a m i n i m u m . 3.1.3. M e a s u r e m e n t of s p h e r i c a l r a d i u s On r e m o v a l f r o m the dies, the s p e c i m e n was m e a s u r e d at 21 points a c r o s s a d i a m e t e r so that a complete p r o f i l e of the shell could be d e t e r mined. The r a d i u s of c u r v a t u r e of the shell was t a k e n as the r a d i u s of the c i r c l e which had the m i n i m u m l e a s t s q u a r e s deviation f r o m the m e a s u r e d o r d i n a t e s . The equation of a c i r c l e , tangent to the X - a x i s at the o r i g i n is x2 + y2 + 2Ry = 0 . In g e n e r a l the m e a s u r e d points will not s a t i s fy t h i s equation exactly. The s q u a r e of the e r r o r which is a s s u m e d to be a function of the r a d i u s R is given by f(R) = ~(x 2 +y2 + 2Ry)2 . F o r m i n i m u m deviation a f = 0 = 2ZRy 2 + Zx2y + Zy3 aR Hence R
~x2y + ~y3 2~y 2
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It was found that the value of the s p h e r i c a l r a d i u s so obtained a g r e e d c l o s e l y with the n o m i nal r a d i u s of c u r v a t u r e of the dies. This i n d i cated no m e a s u r a b l e " s p r i n g - b a c k " of the s p e c i men on r e m o v a l f r o m the dies and hence this provided another check on the s t r e s s f r e e c h a r a c t e r of the shell. F u r t h e r m e a s u r e m e n t of other typical d i a m e t e r s showed good s y m m e t r y and consistent curvature. The d i m e n s i o n s of the shells t e s t e d under point and p r e s s u r e loading were: Spherical Base Thickness Shell r a d i u s (in.) r a d i u s (in.) (in.) parameter 80 5 0.036 75.35 80 5 0.048 42.39 80 5 0.064 23.84 I00 5 0.036 48.23 100 5 0.048 27.13 100 5 0.064 15.26 3.2. Test apparatus In o r d e r to obtain the e q u i l i b r i u m path through both stable and unstable r a n g e s , a "rigid" loading device was used. A typical e q u i l i b r i u m path is shown in fig. 1.
D E F'l..c'C'rl O N
Fig. 1. Typical load-deflection curve for a shallow shell. (Symmetrical behaviour.) If the load is applied i n c r e m e n t a l l y , the shell will follow a stable e q u i l i b r i u m path up to the t u r n i n g point on the path at A. Any f u r t h e r i n c r e a s e in the load c a u s e s the shell to ' s n a p ' suddenly f r o m A to B on the second stable p o r t i o n of the path. D e c r e a s e of the load f r o m B in the s e c ond stable r a n g e , c a u s e s the shell to follow the path down to C. F u r t h e r d e c r e a s e in the load p r o d u c e s a second ' s n a p ' to D on the f i r s t stable portion of the path. Thus by v a r y i n g the load, only stable e q u i l i b r i u m s t a t e s can be achieved. At any i n t e r m e d i a t e load between the snapping conditions, t h e r e will be, in g e n e r a l , t h r e e e q u i l i b r i u m d i s p l a c e m e n t s , two of which will be s t a -
Fig. 2. Test equipment.
NON-LINEAR BEHAVIOUR OF SHALLOW SPHERICAL SHELI~ ble and one unstable. It is t h e r e f o r e obvious that if the c o m p l e t e e q u i l i b r i u m path i s to be o bta in ed , it is n e c e s s a r y to s p e c i f y the d e f l e c t i o n r a t h e r than the load. M e r e l y by s p e c i f y i n g the d e f l e c t i o n and s i m u l t a n e o u s l y m e a s u r i n g the load, the c o m p l e t e e q u i l i b r i u m path can be t r a v e r s e d . Since a x i a l s y m m e t r y is a s s u m e d in o r de r to s i m p l i f y the s t r e s s a n a l y s i s , the d e f l e c tion at the c e n t r e of t h e s h e l l is a convenient p a r a m e t e r to u s e to d e t e r m i n e the e q u i l i b r i u m path. The t e s t a p p a r a t u s is shown in fig. 2. The she l l is p l a c e d in a support r i n g a t t a c h e d to the top of the p r e s s u r e c h a m b e r . A b e a m ABCD is supp o r t ed in b a l l - b e a r i n g s at B in such a m a n n e r that the length AB e q u a l s the length BC. A t h r e a d e d wh eel f i t s c l o s e l y into a slot in the b e a m at C. By t u r n i n g the wheel, a t h r e a d e d r o d i s a d v a n c e d a x i a l l y th r o u g h th e b e a m . A dial gauge l o c a t e d above the r o d r e c o r d s the a x i a l movement. 3.2.1. Point load The s h e l l was p l a c e d c o n v e x side up f o r t h e s e t e s t s . The load wa s t r a n s m i t t e d to the s h e ll by a
/
2,
0
0"8
t.0
t.S
ball-bearing located Since the b e a m w as f o r c e in the r o d w as at A. The load w as c r e m e n t s of 0.01 in.
289
at the l o w e r end of the rod. f r e e l y supported at B, the r e c o r d e d on a p r o v i n g - r i n g m e a s u r e d at s u c c e s s i v e i n of the c e n t r a l deflection.
3.2.2. P r e s s u r e load In the c a s e of the p r e s s u r e load t e s t s , the sh el l w as p e n e t r a t e d at the apex and a t t a c h e d to the r o d so that the c o n c a v e side was u p p e r m o s t . In t h e s e t e s t s , the p r o v i n g r i n g w as r e m o v e d and a d j u s t a b l e stops w e r e set at D to r e s t r i c t the f r e e m o v e m e n t of the b e a m to 0.001 in. The p r e s s u r e was c o n t r o l l e d by a r e d u c i n g v a l v e and a n e e d l e v a l v e . The p r e s s u r e was m e a s u r e d on a w a t e r m a n o m e t e r with a m a x i m u m e f f e c t i v e height of 5 f t . To c a r r y out the t e s t , the d e f l e c tion was g i v en to the sh el l by set t i n g the wheel at C. T h i s p r o d u c e d a f o r c e in the r o d co n n ect ed to the sh el l at the apex. The p r e s s u r e w as then g r a d u a l l y i n c r e a s e d in the p r e s s u r e c h a m b e r until the b e a m just ' f l o a t e d ' between the p r e s e t l i m i t s . At t h i s p r e s s u r e , the sh el l was in e q u i l i b r i u m at the g i v en d e f l e c t i o n with no f o r c e in the r o d at C. In t h i s way the stable and unstable
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N O N - L I N E A R BEHAVIOUR OF SHALLOW S P H E R I C A L SHELLS
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292
J . D . W . HOSSACK
r a n g e s of the e q u i l i b r i u m path were t r a v e r s e d . Some leakage of the air took place at the support but this was f a i r l y constant and, to some extent, facilitated the p r o b l e m of c o n t r o l l i n g the p r e s sure. In o r d e r to a s s e s s the effect of the hole at the apex of the shell, the stable r a n g e s of the e q u i l i b r i u m path were e x a m i n e d on s h e l l s with no such p e n e t r a t i o n s . No a p p r e c i a b l e difference was r e c o g n i s e d . The t e s t s on both point and p r e s s u r e loaded s h e l l s were r e p e a t e d on f u r t h e r s p e c i m e n s to a s s e s s the degree of c o n s i s t e n c y of r e s u l t s . It was found that the c r i t i c a l load could be r e p r o duced to within about 3-4%. 3.3. Strain m e a s u r e m e n t Six shells were s t r a i n gauged u s i n g 36, ~ in. l i n e a r , 70 ohm foil s t r a i n gauges attached to both s u r f a c e s of the shell along a m e r i d i o n a l line in the c i r c u m f e r e n t i a l and r a d i a l d i r e c t i o n s . Thus the state of s t r a i n could be d e t e r m i n e d at nine points on both top and bottom s u r f a c e s of the shell. The s t r a i n r e a d i n g s were r e c o r d e d u s i n g ' S o l a r t r o n Data Logging E q u i p m e n t ' . This equipment r e a d voltage changes c o r r e s p o n d i n g to s t r a i n s on a digital v ~ l t m e t e r and p r i n t e d out the r e a d i n g s at speeds up to 10 c h a n n e l s per second. All the s u r f a c e s t r a i n s were m e a s u r e d at each
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4. THEORETICAL ANALYSIS A t h e o r e t i c a l a n a l y s i s was c a r r i e d out for both point and p r e s s u r e loading following the method proposed by Biezeno [5]. The governing d i f f e r e n t i a l equations d e r i v e d by Biezeno a r e e s s e n t i a l l y s i m i l a r to the finite deflection equat i o n s obtained by R e i s s n e r [6]. The complete s o lution is p r e s e n t e d e l s e w h e r e [16] but an outline of the method i s included here. The solution of the shallow shell equations is a s s u m e d to be r e l a t e d to that for the i n e x t e n s i o n al bending of a flat plate by an u n d e t e r m i n e d multiple with the addition of a l i n e a r t e r m . The a s s u m e d solution c o n t a i n s two independent cons t a n t s C 1 and C2, and is then substituted into the shallow shell equations. By equating the r o t a t i o n at the boundary and the c e n t r a l deflection c o m puted f r o m the shell equations with the c o r r e sponding v a l u e s of r o t a t i o n and deflection f r o m the a s s u m e d f o r m , two s i m u l t a n e o u s a l g e b r a i c cubic equations r e s u l t . By s u b t r a c t i o n , the cubic equations may be r e d u c e d to a single cubic which m a y then be solved by s t a n d a r d n u m e r i c a l p r o -
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Fig. 8. Critical pressure loads for freely supported shells.
NON-LINEAR BEHAVIOUR OF SHALLOW c e d u r e s such as C a r d a n ' s method. T h i s r e s u l t s in the d e t e r m i n a t i o n of the c o n s t a n t s C 1 and C 2. The e q u i l i b r i u m path, s t r e s s r e s u l t a n t s and s u r face s t r a i n s m a y then be computed. 5. E X P E R I M E N T A L
RESULTS
In the analysis of the experimental strains, it was found that when the stress resultants were computed, good agreement with theory w a s obtained for the bending actions. The corresponding results for the m e m b r a n e actions were less satisfactory by comparison. This w a s due to the fact that the behaviour of the shallow shell w a s predominantly a bending phenomenon with relatively small m e m b r a n e action. For this reason, it was decided to adopt a m o r e reasonable approach and to compare actual surface strains directly. The results of the experimental investigation are presented graphically and compared with theory which is shown as a full line. The experimental and theoretical equilibrium paths are shown in fig. 3a. The changing deflected form is illustrated in fig. 3b where the ratio of the deflection at any radius to the deflection at the centre is plotted along a meridian. Fig. 5 shows plots of the surface strains at approximately equal intervals along the equilibrium path. In the case of the point load, the occurrence of local yield w a s reflected in the surface strains near the apex. The effect was m o r e noticeable in the case of the circumferential strains. Similar graphs are presented for the pressure loaded shells (figs. 4,6). The strain gauges in the vicinity of the hole at the apex do not appear to have been seriously affected by any stress concentration effects. In general, the experimental results for the pressure loaded shells are in better agreement with theory than those of the point loaded shells. It is probable that this is due to the occurrence of local yield in the neighbourhood of the concentrated load. The buckling loads are compared with theoretical results for both points and pressure loaded shells in fig. 7 and fig. 8, respectively. For both types of load, shells with ~ less than about 20 do not buckle but deform in a non-linear manner. 6. CONCLUSION The r e s u l t s of t h i s i n v e s t i g a t i o n indicate that when the g e n e r a l g o v e r n i n g d i f f e r e n t i a l equations a r e solved for a set of b o u n d a r y conditions which
S P H E R I C A L SHELLS
2~
c a n be r e p r o d u c e d with some d e g r e e of c e r t a i n ty, good n u m e r i c a l a g r e e m e n t with e x p e r i m e n t a l work can be achieved. The m e a s u r e of a g r e e ment of e x p e r i m e n t a l work with r e l e v a n t t h e o r y is also dependent on the c a r e f u l p r e p a r a t i o n of u n i f o r m , s t r e s s f r e e s p e c i m e n s of r e g u l a r and definite g e o m e t r y . It is c l e a r f r o m the p r e s e n t work and that of p r e v i o u s a u t h o r s , that the t h e o r e t i c a l a n a l y s i s of the buckling p r o b l e m is only of r e a l value when the b a s i c a s s u m p t i o n s of the a n a l y s i s a r e f u l filled in p r a c t i c a l applications. The p r o b l e m i s such that even a l i m i t e d d e p a r t u r e f r o m t h e s e a s s u m p t i o n s m a y w e l l produce effects which c a n not be a c c u r a t e l y a s s e s s e d in the light of p r e s e n t knowledge. The behaviour of shallow shells can only be adequately p r e d i c t e d for a somewhat l i m i t e d r a n g e of p r o b l e m s . The design of p r a c t i c a l shells is u s u a l l y d e pendent on f a c t o r s other than c o n s i d e r a t i o n s of shell stability. The p r o b l e m is, t h e r e f o r e , u s u ally one of checking e x i s t i n g d e s i g n s for p o s s i b l e buckling. F o r this p u r p o s e , the types of a n a l y s e s at p r e s e n t available a r e r e s t r i c t e d to r a t i o n a l l y s y m m e t r i c c a s e s . The author f e e l s that in view of the c r i t i c a l influence of such f a c t o r s as i n i t i a l s t r e s s , i m p e r f e c t i o n and u n c e r t a i n t y of support r e s t r a i n t , it would be unwise to a s s u m e that c r i t i c a l loads, deduced f r o m a n a l y s e s which p r e dict snapping, can be r e l i e d upon to justify i n c r e a s e d c a r r y i n g capacity. It would s i m i l a r l y be unwise to a s s u m e that any useful m e a s u r e of r e s t r a i n t a t t h e support may be r e l i e d upon in p r a c t i c e , though a d e s i g n b a s e d on an a s s u m e d f r e e support would undoubtedly be c o n s e r v a t i v e in the a b s e n c e of other effects. It may, however, be acceptable to p e r m i t l a r g e n o n l i n e a r deflections provided that t h e s e a r e r e c o g n i s e d as such and that buckling does not occur. F o r this, the shell p a r a m e t e r ~ = a 4 / R 2 t 2 should be l i m i t e d to about 15 for both point and p r e s s u r e load conditions. In most e n g i n e e r i n g applications, the b e h a v iour of a shallow cap would p r o b a b l y a p p r o x i m a t e to that of a f r e e l y supported shell even when some m e a s u r e of r e s t r a i n t is provided. The p r o b l e m of m o r e extensive s h e l l s would most p r o b a b l y be approached by i n t r o d u c i n g s t i f f e n e r s a r r a n g e d in such a m a n n e r as to divide the shell into a n u m b e r of shallow caps of l i m i t e d shell p a r a m e t e r . In such a s t r u c t u r e , the conditions of continuity of the shell over the s t i f f e n e r s would p r o b a b l y produce some degree of r e s t r a i n t at the cap boundary. T h u s the a s s u m p t i o n of a f r e e support would be c o n s e r v a t i v e , though this m u s t be b a l a n c e d a g a i n s t the effect of i n i t i a l s t r e s s and i m p e r f e c t i o n s .
294
J.D.W. HOSSACK REFERENCES
7. NOTAT~N R a t h w
5 =w/h 50 = a4(Rt)-2 E v
PR/Et 3 PRa2 / E t 3
s p h e r i c a l r a d i u s of s h e l l b a s e r a d i u s of s h e l l t h i c k n e s s of s h e ll r i s e at c e n t r e of s h e l l horizontal radius d e f l e c t i o n of s h e l l dimensionless deflection d i m e n s i o n l e s s d e f l e c t i o n at c e n t r e of s h e ll shell parameter Young' s m o d u lu s Poisson' s ratio d i m e n s i o n l e s s point load p a r a m e ter d i m e n s i o n l e s s p r e s s u r e load p a rameter
ACKNOWLEDGEMENT The author w i s h e s to thank P r o f e s s o r A. S. T. T h o m s o n f o r the f a c i l i t i e s of the D e p a r t m e n t of M e c h a n i c a l E n g i n e e r i n g of the U n i v e r s i t y of Strathclyde. Thanks a r e a l s o due to P r o f e s s o r R. M. Kenedi and Dr. A.S. Tooth f o r t h e i r i n t e r e s t and guidance in the p r e p a r a t i o n of t h i s p a p e r .
[1] R. Zoelly, Thesis, Zurich University (1915). [2] E. Schwerin, Z. Angew. Math. Mech. 2 (1922) 81. [3] A. Van der Neut, Thesis, Delft University (1932). [4] T.V. Karman and H.S. Tsien, J. Aero. Sci. 7 (1939) 43. [5] C.B. Biezeno, Z..Angew. Physik 15 (1935) 10; reproduced in C.B.Biezeno and R.Grammel, Teehnische Dynamik (Springer, Berlin, 1939). [6] E. Reissner, Proc. Symp. Appl. Math., vol. 3 (McGraw-Hill, New York, 1950)p. 27. [7] A. Kaplan and Y. C. Fung, A non-linear theory of bending and buckling of thin elastic shallow spherical shells, Note N.A.C.A. 3212 (1954). [8] B. Budiansky, Proc. IUTAM Symp. on the Theory of Thin Elastic Shells, Delft, 1959 (North-Holland Publishing Comp., Amsterdam, 1960) p. 64. [9] H. Weinitsehke, J. Math. Phys. 38(1960) 209. [10] R. H. Homewood, A.C. Brine and A.E. Johnson, Buckling instability of monocoque shells (Society for Experimental Stress Analysis, 1959). [11] R. M. Evan-Iwanowski, H.S. Cheng and. T.C. Loo, Proc.'Appl. Mech., Fourth U.S. Natl. Congr., vol. 1 (1962). [12] R. R. Parmerter, The buckling of clamped shallow spherical shells under uniform pressure, A.F.O. S.R. report 5367 (1963). [13] N. C. Huang, A.I.A.A.J. 1 (1963) 945; also Harvard University Technical Report No. 15 (1963). [14] G. A. Thurston, A.I.A.A.J. 2 (1964) [15] H.J.Weinitschke, Asymmetric buckling of clamped shallow spherical shells, N.A.S.A., TN D-1510 (1962) p. 481. [16] J.D.W.Hossack, Thesis in preparation, University of Strathelyde, Glasgow.
E ~ N T A L BEHAVIOUR
INVESTIGATION OF THE NONLINEAR OF SHALLOW SPHERICAL SHELLS *
J. D. W. HOSSACK Department of Mechanical Engineering, University of Strathclyde, Glasgow, Scotland
Received 16 January 1965
This paper presents a brief review of some recent contributions in the field of instability of shallow spherical shells. The causes of the apparent inconclusive nature of earlier experimental investigations are examined. This is followed by a report of work carried out in such a manner as to eliminate, as far as possible, some of the factors responsible for experimental error. The results of an experimental investigation of the non-linear stress and deformation characteristics of shallow shell behaviour are presented. Good experimental agreement with theory is shown to be dependent on the preparation of uniform stress free specimens of accurate geometry and on the requirement that the shell be supported in a manner consistent with the assumptions of relevant theory. 1. INTRODUCTION
2. REVIEW
One of the m a n y p r o b l e m s which face d e s i g n e r s of shell s t r u c t u r e s in the field of n u c l e a r e n g i n e e r i n g , i s that of i n s t a b i l i t y . Though e x t e n sive t h e o r e t i c a l a n d 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 s have been c a r r i e d out, the p r o b l e m is still u n r e s o l v e d . The p r e s e n t state of knowledge can do little to help d e s i g n e r s of e n g i n e e r i n g s h e l l s which a r e f r e q u e n t l y subject to a v a r i e t y of load conditions. Such s h e l l s may have p e n e t r a t i o n s and r e i n f o r c i n g pads which, together with i m p e r f e c t i o n s f r o m t r u e s p h e r i c a l f o r m , make the s t a b i l i t y a n a l y s i s of the shell as a whole e x t r e m e l y difficult. Before such a n a l y s e s can be c a r r i e d out, it i s f i r s t n e c e s s a r y to e s t a b l i s h with some c e r t a i n t y , the behaviour of i d e a l i s e d s h e l l s . Though much work has b e e n done with t h i s object in view, the m e a s u r e of a g r e e m e n t of e x p e r i m e n t a l data with r e l e v a n t t h e o r y is s t i l l disappointing. The p r e s e n t study is d i r e c t e d t o ward an a n a l y s i s of the f a c t o r s which influence e x p e r i m e n t a l work. Much of the p r e v i o u s e x p e r i m e n t a l work was p e r f o r m e d on c l a m p e d s h e l l s and the r e s u l t s of t h i s i n v e s t i g a t i o n suggest that some of t h i s work m a y have b e e n affected by u n c e r t a i n t i e s s u r r o u n d i n g methods adopted to obt a i n this b o u n d a r y condition.
In the study of the i n s t a b i l i t y of shell s t r u c t u r e s , the s p h e r i c a l shell has been the subject of many t h e o r e t i c a l and e x p e r i m e n t a l i n v e s t i g a tions. E a r l y t h e o r e t i c a l a n a l y s i s [1-3] of the s t a b i l i t y of the complete s p h e r i c a l shell under e x t e r n a l p r e s s u r e , were b a s e d on the l i n e a r , c l a s s i c a l concept of buckling. However the exp e r i m e n t a l r e s u l t s [4] indicated collapse p r e s s u r e s which were a f r a c t i o n of the t h e o r e t i c a l l y d e t e r m i n e d values. The apparent f a i l u r e of the c l a s s i c a l r e a s o n i n g to apply to p r a c t i c a l s h e l l s , led w o r k e r s to e x a m i n e the behaviour of a s h a l low s p h e r i c a l cap which was a s s u m e d to be c l a m p e d at the boundary. This a s s u m p t i o n i n volved a b a s i c change in the a n a l y s i s of the p r o b l e m . The e s s e n t i a l difference between the p r o b l e m of the complete sphere and that of the cap, was that the s p h e r e was a s s u m e d to be in a state of m e m b r a n e c o n t r a c t i o n p r i o r to buckling, while the a s s u m e d support conditions of the shallow cap caused this p r o b l e m to be one of continuous d e f o r m a t i o n f r o m the f i r s t application of the load. P r e s e n t e d in t h i s f o r m , the buckling of the shallow cap was c l e a r l y a l a r g e deflection p h e n o m e n o n with the r e s u l t that the governing diff e r e n t i a l equations b e c a m e n o n - l i n e a r . In this r e s p e c t , the equations d e r i v e d by Biezeno [5], R e i s s n e r [6] and Kaplan and Fung [7] were e s s e n t i a l l y s i m i l a r . A l a r g e volume of the published l i t e r a t u r e has dealt with solutions of t h e s e
* Accepted by T. A. Jaeger. Adv.: A. S. Tooth.
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equations, but s u c c e s s was not a t t a i n e d until Budiansky [8] and, independently, Weinitschke [9] obtained n u m e r i c a l solutions for the buckling p r e s s u r e s of a c l a m p e d shallow cap over a wide r a n g e of r i s e to t h i c k n e s s ratio. A l e s s e r volume of the l i t e r a t u r e has dealt with e x p e r i m e n t a l work. In 1954, Kaplan and Fung [7] m a n u f a c t u r e d s h e l l s by spinning f r o m hot m a g n e s i u m alloy. The s p e c i m e n s were t e s t e d u n d e r both oil and a i r p r e s s u r e c o r r e s p o n d i n g to constant volume and constant p r e s s u r e c o n d i tions. They came to the c o n c l u s i o n that the loading m e d i u m had no significant effect on the buckling load. The e x p e r i m e n t a l values of the c o l lapse p r e s s u r e showed an a p p r e c i a b l e s c a t t e r . In 1959, Homewood, B r i n e and Johnson [10] r e ported f u r t h e r e x p e r i m e n t a l work on p r e s s u r e loaded c l a m p e d shells. The r e s u l t s showed a s i m i l a r degree of s c a t t e r . In 1962, E v a n - I w a n o w s k i , Cheng and Loo [11] r e p o r t e d on e x p e r i m e n t a l work c a r r i e d out on the point loading of both f r e e l y supported and c l a m p e d shells. Both m e t a l l i c and n o n - m e t a l l i c s p e c i m e n s were tested. The m e t a l l i c s p e c i m e n s in copper, a l u m i n i u m and steel w e r e f o r m e d by the h y d r o f o r m p r o c e s s , with s p h e r i c a l r a d i i v a r y i n g f r o m 5 in. to 11.2 in. and b a s e r a d i i v a r y i n g f r o m 3.812 in. to 0.697 in, They noticed that c e r t a i n shells exhibited d i s c o n t i n u o u s l o a d deflection behaviour. Though this phenomenon would appear to be c o n s i s t e n t with a change in the deflection mode shape, the a u t h o r s did not c o m m e n t on this aspect but c l a i m e d that, in most c a s e s , r o t a t i o n a l s y m m e t r y was p r e s e r v e d . T h e i r r e s u l t s showed a significant s c a t t e r which might be a t t r i b u t e d to t h e i r e x p e r i m e n t a l t e c h nique and the difficulty of d e t e r m i n i n g the g e o m e t r y of some of t h e i r s p e c i m e n s which, in some c a s e s , were v e r y s m a l l . It would also appear that some of t h e i r c a l c u l a t e d v a l u e s for the shell p a r a m e t e r were i n c o n s i s t e n t with the published m e a s u r e m e n t s . The lack of e x p e r i m e n t a l a g r e e ment with e s t a b l i s h e d t h e o r e t i c a l work b e c a m e m o r e s e r i o u s in the r a n g e of s h e l l s showing the d i s c o n t i n u o u s behaviour and for the s h e l l s which were r e c o g n i s a b l y n o n - s y m m e t r i c in d e f o r m a tion. L a t e r in the s a m e year Evan-Iwanowski p r e s e n t e d f u r t h e r r e s u l t s on the combined point and p r e s s u r e loading of shells. In this i n v e s t i g a tion he found that the c r i t i c a l c o m b i n a t i o n of point and p r e s s u r e load was independent of the o r d e r in which they w e r e applied - a not u n e x pected r e s u l t . He also r e p o r t e d that for some positions of an e c c e n t r i c a l l y placed point load, the value of the c r i t i c a l c o m b i n a t i o n of loading
was r a i s e d a s c o m p a r e d with the r e s u l t when the point load was applied at the apex. The e x p e r i m e n t a l work so f a r r e v i e w e d is c h a r a c t e r i s e d by the s c a t t e r of r e s u l t s , a f a c t o r which i n e v i t a b l y r e d u c e s its value a s a t e s t of r e l e v a n t theory. The w o r k e r s on the t h e o r e t i c a l side of the p r o b l e m , have examined the v a r i o u s c a u s e s which might account for t h e s e effects. In this r e s p e c t , Budiansky [8] found that the effect of i n i t i a l i m p e r f e c t i o n of an a s s u m e d s y m m e t r i c f o r m could not in itself account for the d i s p a r i t y between the t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s . He t h e r e f o r e concluded that an explanation m u s t lie in supposed n o n - s y m m e t r i c behaviour. Seve r a l p a p e r s have been published p r e s e n t i n g a n a l y s e s of n o n - s y m m e t r i c deformation. The work of P a r m e r t e r [12], Huang [13] and T h u r s t o n [14] a r e in s u b s t a n t i a l a g r e e m e n t while that of W e i n i t s c h ke [15] p r e d i c t e d a different t r e n d for the c r i t i c a l p r e s s u r e s of c l a m p e d s h e l l s . Though the l a t t e r theory was in some a g r e e m e n t with e a r l y e x p e r i m e n t a l work, the a g r e e m e n t would appear to be to some extent f o r t u i t o u s in the light of r e c e n t 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 s [12]. The f a c t o r s which c r i t i c a l l y influence e x p e r i m e n t a l work, have only been a p p r e c i a t e d r e l a t i v e l y r e c e n t l y . Since i n s t a b i l i t y is total s t r e s s dependent, the influence of i n i t i a l s t r e s s e s in the s p e c i m e n , whether they o r i g i n a t e in m a n u f a c t u r e or in the test set up, is not easy to evaluate. T h i s is p a r t i c u l a r l y t r u e since the governing d i f f e r e n t i a l equations a r e n o n - l i n e a r . The m a n u f a c t u r e of s p e c i m e n s by s p i n n i n g i n t r o d u c e s both r e s i d u a l s t r e s s effects and v a r i a t i o n s in t h i c k n e s s . U n l e s s c a r e is taken to s t r e s s r e l i e v e s p e c i m e n s , i n i t i a l s t r e s s effects may also be p r e s e n t in other t e c h n i q u e s of m a n u f a c t u r e . It thus b e c o m e s c l e a r , that for valid e x p e r i m e n t a l data to be obtained, close attention must be d i r e c t e d to producing a s t r e s s f r e e , u n i f o r m shell whose g e o m e t r y can be a c c u r a t e l y d e t e r m i n e d . Much of the work on c l a m p e d s h e l l s has been c a r r i e d out by a t t e m p t i n g to achieve t h i s bounda r y condition by mounting the s p e c i m e n between r i n g s . All the available evidence i n d i c a t e s that this method p r o v i d e s only a poor a p p r o x i m a t i o n to the fully fixed condition and, in addition, may induce r a n d o m f o r c e and m o m e n t a c t i o n s which may cause p r e m a t u r e collapse of the shell. At best, the method may introduce an u n a c c e p t a b l e degree of s c a t t e r into the e x p e r i m e n t a l r e s u l t s . The degree of fixity is of c r i t i c a l i m p o r t a n c e in the p r o b l e m of i n s t a b i l i t y and it is doubtful whether a fully fixed support condition can e v e r be achieved even in the l a b o r a t o r y . While the ex-
NON-LINEAR BEHAVIOUR OF SHALLOW SPHERICAL SHELLS p e r i m e n t a l work of P a r m e n t e r [12] shows an e n c o u r a g i n g m e a s u r e of a g r e e m e n t with t h e o r e t i c a l a n a l y s i s , the s c a t t e r p r e s e n t in his r e s u l t i n d i c a t e s that p e r h a p s even m o r e r e f i n e d e x p e r i m e n t a l t e c h n i q u e s m a y be n e c e s s a r y though it is r e c o g n i s e d that additional p r o b l e m s a r e i n t r o duced when n o n - s y m m e t r i c behaviour is e x pected. In view of the p r a c t i c a l difficulties a s s o c i ated with d i s p l a c e m e n t dependent b o u n d a r y c o n ditions, the p r e s e n t work avoids t h e s e p r o b l e m s by adoption of force dependent conditions. The s i m p l e s t of t h e s e is, of c o u r s e , the fully f r e e support w h e r e both the m e r i d i o n a l m e m b r a n e and bending a c t i o n s a r e z e r o on the boundary. The e x p e r i m e n t a l work was a i m e d at the d e t e r m i u a t i o n of the full e q u i l i b r i u m path for both point and p r e s s u r e loaded shells. In addition to t h i s , the work included a t h e o r e t i c a l and e x p e r i m e n t a l a n a l y s i s of the v a r i a t i o n and s t r e s s and d e f o r m a t i o n along the e q u i l i b r i u m path. T o s i m plify the e x p e r i m e n t a l a n a l y s i s , the p r o b l e m was r e s t r i c t e d to s h e l l s for which s y m m e t r i c a l b e h a v i o u r had b e e n e s t a b l i s h e d .
3. E X P E R I M E N T A L
WORK
3.1. T e s t s p e c i m e n s In view of the s t r i c t g e o m e t r i c a l r e q u i r e m e n t s and the n e c e s s a r y s t r e s s f r e e c h a r a c t e r of the s h e l l s , it was decided to f o r m t h e s e f r o m sheet a l u m i n i u m alloy. The alloy chosen c o n t a i n e d 2% m a g n e s i u m and 0.25% m a n g a n e s e . T h i s m a t e r i a l had good l i n e a r s t r e s s - s t r a i n p r o p e r t i e s in the a n n e a l e d state. 3.1.1. D e t e r m i n a t i o n of e l a s t i c c o n s t a n t s a. Young' s m o d u l u s An attempt to obtain Young's m o d u l u s f r o m suitably s t r a i n - g a u g e d t e n s i l e s p e c i m e n s yielded v a r i a t i o n s in E of about 6%. To avoid the u s e of s t r a i n gauges, a m o r e f u n d a m e n t a l approach was adopted. A s t r i p of the sheet m a t e r i a l was cut and set up as a s i m p l y supported b e a m . The app l i c a t i o n of equal loads to the overhung ends of the b e a m produced c o n s t a n t bending m o m e n t b e tween the supports. Thus the b e a m bent to f o r m a c i r c u l a r a r c over the r e g i o n of c o n s t a n t b e n d ing m o m e n t . By m e a s u r i n g the c e n t r a l deflection of the b e a m r e l a t i v e to the s u p p o r t s , the r a d i u s of c u r v a t u r e of the c e n t r a l p o r t i o n of the b e a m could be e x p r e s s e d in t e r m s of the deflection and the span. F r o m the s i m p l e bending r e l a t i o n , the value of Young' s modulu~ was obtained as a f u n c tion of the r a d i u s of c u r v a t u r e , the bending m o -
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ment and the second m o m e n t of a r e a of the beam. A graph of the bending m o m e n t against the c e n t r a l deflection was drawn. The slope of t h i s line, which was p r o p o r t i o n a l to the value of Young's modulus, was d e t e r m i n e d by m i n i m i s i n g the l e a s t s q u a r e s deviation of the m e a s u r e d points. The value of Young's m o d u l u s so obtained was within l e s s than 2% of the accepted value of 10 x 106 l b / i n . 2. b. Poisson's ratio Since the effect of slight error in the value of Poisson's ratio w a s of less importance, the value w a s obtained f r o m the ratio of transverse to axial strain gauge m e a s u r e m e n t in a tensile test.
3.1.2. M a n u f a c t u r e of t e s t s p e c i m e n s Two sets of dies of 10 in. base d i a m e t e r were m a c h i n e d on a profile following lathe to r a d i i of c u r v a t u r e of 80 in. and 100 in. The o r d i n a t e s of the die p r o f i l e s w e r e checked for a c c u r a c y to within 0.0002 in. D i s c s 10 in. d i a m e t e r were cut f r o m alloy sheet with a t o l e r a n c e of 0.001 in. in the r a d i u s . A set of dies with the disc between was then heated in an oven to a p p r o x i m a t e l y 360oc and m a i n t a i n e d t h e r e for four hours. The oven and contents w e r e then ailowed to cool so that the f o r m i n g p r o c e s s took about 24 h o u r s to c o m plete. By this p r o c e s s , the r e s i d u a l f a b r i c a t i o n s t r e s s e s w e r e r e d u c e d to a m i n i m u m . 3.1.3. M e a s u r e m e n t of s p h e r i c a l r a d i u s On r e m o v a l f r o m the dies, the s p e c i m e n was m e a s u r e d at 21 points a c r o s s a d i a m e t e r so that a complete p r o f i l e of the shell could be d e t e r mined. The r a d i u s of c u r v a t u r e of the shell was t a k e n as the r a d i u s of the c i r c l e which had the m i n i m u m l e a s t s q u a r e s deviation f r o m the m e a s u r e d o r d i n a t e s . The equation of a c i r c l e , tangent to the X - a x i s at the o r i g i n is x2 + y2 + 2Ry = 0 . In g e n e r a l the m e a s u r e d points will not s a t i s fy t h i s equation exactly. The s q u a r e of the e r r o r which is a s s u m e d to be a function of the r a d i u s R is given by f(R) = ~(x 2 +y2 + 2Ry)2 . F o r m i n i m u m deviation a f = 0 = 2ZRy 2 + Zx2y + Zy3 aR Hence R
~x2y + ~y3 2~y 2
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It was found that the value of the s p h e r i c a l r a d i u s so obtained a g r e e d c l o s e l y with the n o m i nal r a d i u s of c u r v a t u r e of the dies. This i n d i cated no m e a s u r a b l e " s p r i n g - b a c k " of the s p e c i men on r e m o v a l f r o m the dies and hence this provided another check on the s t r e s s f r e e c h a r a c t e r of the shell. F u r t h e r m e a s u r e m e n t of other typical d i a m e t e r s showed good s y m m e t r y and consistent curvature. The d i m e n s i o n s of the shells t e s t e d under point and p r e s s u r e loading were: Spherical Base Thickness Shell r a d i u s (in.) r a d i u s (in.) (in.) parameter 80 5 0.036 75.35 80 5 0.048 42.39 80 5 0.064 23.84 I00 5 0.036 48.23 100 5 0.048 27.13 100 5 0.064 15.26 3.2. Test apparatus In o r d e r to obtain the e q u i l i b r i u m path through both stable and unstable r a n g e s , a "rigid" loading device was used. A typical e q u i l i b r i u m path is shown in fig. 1.
D E F'l..c'C'rl O N
Fig. 1. Typical load-deflection curve for a shallow shell. (Symmetrical behaviour.) If the load is applied i n c r e m e n t a l l y , the shell will follow a stable e q u i l i b r i u m path up to the t u r n i n g point on the path at A. Any f u r t h e r i n c r e a s e in the load c a u s e s the shell to ' s n a p ' suddenly f r o m A to B on the second stable p o r t i o n of the path. D e c r e a s e of the load f r o m B in the s e c ond stable r a n g e , c a u s e s the shell to follow the path down to C. F u r t h e r d e c r e a s e in the load p r o d u c e s a second ' s n a p ' to D on the f i r s t stable portion of the path. Thus by v a r y i n g the load, only stable e q u i l i b r i u m s t a t e s can be achieved. At any i n t e r m e d i a t e load between the snapping conditions, t h e r e will be, in g e n e r a l , t h r e e e q u i l i b r i u m d i s p l a c e m e n t s , two of which will be s t a -
Fig. 2. Test equipment.
NON-LINEAR BEHAVIOUR OF SHALLOW SPHERICAL SHELI~ ble and one unstable. It is t h e r e f o r e obvious that if the c o m p l e t e e q u i l i b r i u m path i s to be o bta in ed , it is n e c e s s a r y to s p e c i f y the d e f l e c t i o n r a t h e r than the load. M e r e l y by s p e c i f y i n g the d e f l e c t i o n and s i m u l t a n e o u s l y m e a s u r i n g the load, the c o m p l e t e e q u i l i b r i u m path can be t r a v e r s e d . Since a x i a l s y m m e t r y is a s s u m e d in o r de r to s i m p l i f y the s t r e s s a n a l y s i s , the d e f l e c tion at the c e n t r e of t h e s h e l l is a convenient p a r a m e t e r to u s e to d e t e r m i n e the e q u i l i b r i u m path. The t e s t a p p a r a t u s is shown in fig. 2. The she l l is p l a c e d in a support r i n g a t t a c h e d to the top of the p r e s s u r e c h a m b e r . A b e a m ABCD is supp o r t ed in b a l l - b e a r i n g s at B in such a m a n n e r that the length AB e q u a l s the length BC. A t h r e a d e d wh eel f i t s c l o s e l y into a slot in the b e a m at C. By t u r n i n g the wheel, a t h r e a d e d r o d i s a d v a n c e d a x i a l l y th r o u g h th e b e a m . A dial gauge l o c a t e d above the r o d r e c o r d s the a x i a l movement. 3.2.1. Point load The s h e l l was p l a c e d c o n v e x side up f o r t h e s e t e s t s . The load wa s t r a n s m i t t e d to the s h e ll by a
/
2,
0
0"8
t.0
t.S
ball-bearing located Since the b e a m w as f o r c e in the r o d w as at A. The load w as c r e m e n t s of 0.01 in.
289
at the l o w e r end of the rod. f r e e l y supported at B, the r e c o r d e d on a p r o v i n g - r i n g m e a s u r e d at s u c c e s s i v e i n of the c e n t r a l deflection.
3.2.2. P r e s s u r e load In the c a s e of the p r e s s u r e load t e s t s , the sh el l w as p e n e t r a t e d at the apex and a t t a c h e d to the r o d so that the c o n c a v e side was u p p e r m o s t . In t h e s e t e s t s , the p r o v i n g r i n g w as r e m o v e d and a d j u s t a b l e stops w e r e set at D to r e s t r i c t the f r e e m o v e m e n t of the b e a m to 0.001 in. The p r e s s u r e was c o n t r o l l e d by a r e d u c i n g v a l v e and a n e e d l e v a l v e . The p r e s s u r e was m e a s u r e d on a w a t e r m a n o m e t e r with a m a x i m u m e f f e c t i v e height of 5 f t . To c a r r y out the t e s t , the d e f l e c tion was g i v en to the sh el l by set t i n g the wheel at C. T h i s p r o d u c e d a f o r c e in the r o d co n n ect ed to the sh el l at the apex. The p r e s s u r e w as then g r a d u a l l y i n c r e a s e d in the p r e s s u r e c h a m b e r until the b e a m just ' f l o a t e d ' between the p r e s e t l i m i t s . At t h i s p r e s s u r e , the sh el l was in e q u i l i b r i u m at the g i v en d e f l e c t i o n with no f o r c e in the r o d at C. In t h i s way the stable and unstable
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290
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N O N - L I N E A R BEHAVIOUR OF SHALLOW S P H E R I C A L SHELLS
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292
J . D . W . HOSSACK
r a n g e s of the e q u i l i b r i u m path were t r a v e r s e d . Some leakage of the air took place at the support but this was f a i r l y constant and, to some extent, facilitated the p r o b l e m of c o n t r o l l i n g the p r e s sure. In o r d e r to a s s e s s the effect of the hole at the apex of the shell, the stable r a n g e s of the e q u i l i b r i u m path were e x a m i n e d on s h e l l s with no such p e n e t r a t i o n s . No a p p r e c i a b l e difference was r e c o g n i s e d . The t e s t s on both point and p r e s s u r e loaded s h e l l s were r e p e a t e d on f u r t h e r s p e c i m e n s to a s s e s s the degree of c o n s i s t e n c y of r e s u l t s . It was found that the c r i t i c a l load could be r e p r o duced to within about 3-4%. 3.3. Strain m e a s u r e m e n t Six shells were s t r a i n gauged u s i n g 36, ~ in. l i n e a r , 70 ohm foil s t r a i n gauges attached to both s u r f a c e s of the shell along a m e r i d i o n a l line in the c i r c u m f e r e n t i a l and r a d i a l d i r e c t i o n s . Thus the state of s t r a i n could be d e t e r m i n e d at nine points on both top and bottom s u r f a c e s of the shell. The s t r a i n r e a d i n g s were r e c o r d e d u s i n g ' S o l a r t r o n Data Logging E q u i p m e n t ' . This equipment r e a d voltage changes c o r r e s p o n d i n g to s t r a i n s on a digital v ~ l t m e t e r and p r i n t e d out the r e a d i n g s at speeds up to 10 c h a n n e l s per second. All the s u r f a c e s t r a i n s were m e a s u r e d at each
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i n c r e m e n t of deflection so that in an a v e r a g e t e s t some 1100 s t r a i n r e a d i n g s were r e c o r d e d .
4. THEORETICAL ANALYSIS A t h e o r e t i c a l a n a l y s i s was c a r r i e d out for both point and p r e s s u r e loading following the method proposed by Biezeno [5]. The governing d i f f e r e n t i a l equations d e r i v e d by Biezeno a r e e s s e n t i a l l y s i m i l a r to the finite deflection equat i o n s obtained by R e i s s n e r [6]. The complete s o lution is p r e s e n t e d e l s e w h e r e [16] but an outline of the method i s included here. The solution of the shallow shell equations is a s s u m e d to be r e l a t e d to that for the i n e x t e n s i o n al bending of a flat plate by an u n d e t e r m i n e d multiple with the addition of a l i n e a r t e r m . The a s s u m e d solution c o n t a i n s two independent cons t a n t s C 1 and C2, and is then substituted into the shallow shell equations. By equating the r o t a t i o n at the boundary and the c e n t r a l deflection c o m puted f r o m the shell equations with the c o r r e sponding v a l u e s of r o t a t i o n and deflection f r o m the a s s u m e d f o r m , two s i m u l t a n e o u s a l g e b r a i c cubic equations r e s u l t . By s u b t r a c t i o n , the cubic equations may be r e d u c e d to a single cubic which m a y then be solved by s t a n d a r d n u m e r i c a l p r o -
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Fig. 8. Critical pressure loads for freely supported shells.
NON-LINEAR BEHAVIOUR OF SHALLOW c e d u r e s such as C a r d a n ' s method. T h i s r e s u l t s in the d e t e r m i n a t i o n of the c o n s t a n t s C 1 and C 2. The e q u i l i b r i u m path, s t r e s s r e s u l t a n t s and s u r face s t r a i n s m a y then be computed. 5. E X P E R I M E N T A L
RESULTS
In the analysis of the experimental strains, it was found that when the stress resultants were computed, good agreement with theory w a s obtained for the bending actions. The corresponding results for the m e m b r a n e actions were less satisfactory by comparison. This w a s due to the fact that the behaviour of the shallow shell w a s predominantly a bending phenomenon with relatively small m e m b r a n e action. For this reason, it was decided to adopt a m o r e reasonable approach and to compare actual surface strains directly. The results of the experimental investigation are presented graphically and compared with theory which is shown as a full line. The experimental and theoretical equilibrium paths are shown in fig. 3a. The changing deflected form is illustrated in fig. 3b where the ratio of the deflection at any radius to the deflection at the centre is plotted along a meridian. Fig. 5 shows plots of the surface strains at approximately equal intervals along the equilibrium path. In the case of the point load, the occurrence of local yield w a s reflected in the surface strains near the apex. The effect was m o r e noticeable in the case of the circumferential strains. Similar graphs are presented for the pressure loaded shells (figs. 4,6). The strain gauges in the vicinity of the hole at the apex do not appear to have been seriously affected by any stress concentration effects. In general, the experimental results for the pressure loaded shells are in better agreement with theory than those of the point loaded shells. It is probable that this is due to the occurrence of local yield in the neighbourhood of the concentrated load. The buckling loads are compared with theoretical results for both points and pressure loaded shells in fig. 7 and fig. 8, respectively. For both types of load, shells with ~ less than about 20 do not buckle but deform in a non-linear manner. 6. CONCLUSION The r e s u l t s of t h i s i n v e s t i g a t i o n indicate that when the g e n e r a l g o v e r n i n g d i f f e r e n t i a l equations a r e solved for a set of b o u n d a r y conditions which
S P H E R I C A L SHELLS
2~
c a n be r e p r o d u c e d with some d e g r e e of c e r t a i n ty, good n u m e r i c a l a g r e e m e n t with e x p e r i m e n t a l work can be achieved. The m e a s u r e of a g r e e ment of e x p e r i m e n t a l work with r e l e v a n t t h e o r y is also dependent on the c a r e f u l p r e p a r a t i o n of u n i f o r m , s t r e s s f r e e s p e c i m e n s of r e g u l a r and definite g e o m e t r y . It is c l e a r f r o m the p r e s e n t work and that of p r e v i o u s a u t h o r s , that the t h e o r e t i c a l a n a l y s i s of the buckling p r o b l e m is only of r e a l value when the b a s i c a s s u m p t i o n s of the a n a l y s i s a r e f u l filled in p r a c t i c a l applications. The p r o b l e m i s such that even a l i m i t e d d e p a r t u r e f r o m t h e s e a s s u m p t i o n s m a y w e l l produce effects which c a n not be a c c u r a t e l y a s s e s s e d in the light of p r e s e n t knowledge. The behaviour of shallow shells can only be adequately p r e d i c t e d for a somewhat l i m i t e d r a n g e of p r o b l e m s . The design of p r a c t i c a l shells is u s u a l l y d e pendent on f a c t o r s other than c o n s i d e r a t i o n s of shell stability. The p r o b l e m is, t h e r e f o r e , u s u ally one of checking e x i s t i n g d e s i g n s for p o s s i b l e buckling. F o r this p u r p o s e , the types of a n a l y s e s at p r e s e n t available a r e r e s t r i c t e d to r a t i o n a l l y s y m m e t r i c c a s e s . The author f e e l s that in view of the c r i t i c a l influence of such f a c t o r s as i n i t i a l s t r e s s , i m p e r f e c t i o n and u n c e r t a i n t y of support r e s t r a i n t , it would be unwise to a s s u m e that c r i t i c a l loads, deduced f r o m a n a l y s e s which p r e dict snapping, can be r e l i e d upon to justify i n c r e a s e d c a r r y i n g capacity. It would s i m i l a r l y be unwise to a s s u m e that any useful m e a s u r e of r e s t r a i n t a t t h e support may be r e l i e d upon in p r a c t i c e , though a d e s i g n b a s e d on an a s s u m e d f r e e support would undoubtedly be c o n s e r v a t i v e in the a b s e n c e of other effects. It may, however, be acceptable to p e r m i t l a r g e n o n l i n e a r deflections provided that t h e s e a r e r e c o g n i s e d as such and that buckling does not occur. F o r this, the shell p a r a m e t e r ~ = a 4 / R 2 t 2 should be l i m i t e d to about 15 for both point and p r e s s u r e load conditions. In most e n g i n e e r i n g applications, the b e h a v iour of a shallow cap would p r o b a b l y a p p r o x i m a t e to that of a f r e e l y supported shell even when some m e a s u r e of r e s t r a i n t is provided. The p r o b l e m of m o r e extensive s h e l l s would most p r o b a b l y be approached by i n t r o d u c i n g s t i f f e n e r s a r r a n g e d in such a m a n n e r as to divide the shell into a n u m b e r of shallow caps of l i m i t e d shell p a r a m e t e r . In such a s t r u c t u r e , the conditions of continuity of the shell over the s t i f f e n e r s would p r o b a b l y produce some degree of r e s t r a i n t at the cap boundary. T h u s the a s s u m p t i o n of a f r e e support would be c o n s e r v a t i v e , though this m u s t be b a l a n c e d a g a i n s t the effect of i n i t i a l s t r e s s and i m p e r f e c t i o n s .
294
J.D.W. HOSSACK REFERENCES
7. NOTAT~N R a t h w
5 =w/h 50 = a4(Rt)-2 E v
PR/Et 3 PRa2 / E t 3
s p h e r i c a l r a d i u s of s h e l l b a s e r a d i u s of s h e l l t h i c k n e s s of s h e ll r i s e at c e n t r e of s h e l l horizontal radius d e f l e c t i o n of s h e l l dimensionless deflection d i m e n s i o n l e s s d e f l e c t i o n at c e n t r e of s h e ll shell parameter Young' s m o d u lu s Poisson' s ratio d i m e n s i o n l e s s point load p a r a m e ter d i m e n s i o n l e s s p r e s s u r e load p a rameter
ACKNOWLEDGEMENT The author w i s h e s to thank P r o f e s s o r A. S. T. T h o m s o n f o r the f a c i l i t i e s of the D e p a r t m e n t of M e c h a n i c a l E n g i n e e r i n g of the U n i v e r s i t y of Strathclyde. Thanks a r e a l s o due to P r o f e s s o r R. M. Kenedi and Dr. A.S. Tooth f o r t h e i r i n t e r e s t and guidance in the p r e p a r a t i o n of t h i s p a p e r .
[1] R. Zoelly, Thesis, Zurich University (1915). [2] E. Schwerin, Z. Angew. Math. Mech. 2 (1922) 81. [3] A. Van der Neut, Thesis, Delft University (1932). [4] T.V. Karman and H.S. Tsien, J. Aero. Sci. 7 (1939) 43. [5] C.B. Biezeno, Z..Angew. Physik 15 (1935) 10; reproduced in C.B.Biezeno and R.Grammel, Teehnische Dynamik (Springer, Berlin, 1939). [6] E. Reissner, Proc. Symp. Appl. Math., vol. 3 (McGraw-Hill, New York, 1950)p. 27. [7] A. Kaplan and Y. C. Fung, A non-linear theory of bending and buckling of thin elastic shallow spherical shells, Note N.A.C.A. 3212 (1954). [8] B. Budiansky, Proc. IUTAM Symp. on the Theory of Thin Elastic Shells, Delft, 1959 (North-Holland Publishing Comp., Amsterdam, 1960) p. 64. [9] H. Weinitsehke, J. Math. Phys. 38(1960) 209. [10] R. H. Homewood, A.C. Brine and A.E. Johnson, Buckling instability of monocoque shells (Society for Experimental Stress Analysis, 1959). [11] R. M. Evan-Iwanowski, H.S. Cheng and. T.C. Loo, Proc.'Appl. Mech., Fourth U.S. Natl. Congr., vol. 1 (1962). [12] R. R. Parmerter, The buckling of clamped shallow spherical shells under uniform pressure, A.F.O. S.R. report 5367 (1963). [13] N. C. Huang, A.I.A.A.J. 1 (1963) 945; also Harvard University Technical Report No. 15 (1963). [14] G. A. Thurston, A.I.A.A.J. 2 (1964) [15] H.J.Weinitschke, Asymmetric buckling of clamped shallow spherical shells, N.A.S.A., TN D-1510 (1962) p. 481. [16] J.D.W.Hossack, Thesis in preparation, University of Strathelyde, Glasgow.