?T CoA Note No, 105
THE COLLEGE OF AERONAUTICS
CRANFIELD
i
THERMAL BUCKLING OF A FREE CIRCULAR PLATE
by
NOTE NO. 105 August 1960
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
Thermal Buckling of a F r e e Circular Plate
b y -J . P.H.Webber, D . C . A e . , and D.S.Houghton, M.Sc.(Eng.), A . F . R . A e . S . , A . M . I . M e c h . E . SUMMARY
The buckling of a free circular plate subjected to a temperature field, which varies only in the radial direction, is considered. The problem is first investigated experimentally and the mode of deflection and form of the temperature distribution are measured. Expressions a r e developed for the thermal s t r e s s e s and the deflection mode of the plate, which are used for a theoretical small deflection energy analysis. This solution is found to compare favourably with the experimental r e s u l t s .
* This work was begun by the first author, under the supervision of the second, in part fulfilment for the award of the Diploma of the College of Aeronautics.
CONTENTS
Summary
Notation
1. Introduction
2. The experimental investigation 2 . 1 . Specimen
2 . 2 . The experimental apparatus 2 . 3 . Instrumentation 3. Experimental results 4. Theoretical analysis 4 . 1 . General r e m a r k s 4 . 2 . Assumptions 4 . 3 . Solution
4 . 4 . Comparison with experiment
5. References
NOTATION
A constant in deflection mode of plate b r a d i u s of p l a t e D f l e x u r a l rigidity E Young's modulus h p l a t e t h i c k n e s s M r M m o m e n t and t o r q u e r e s u l t a n t s
re
n index Q s h e a r r e s u l t a n t r r a d i a l c o - o r d i n a t e d s e l e m e n t al s u r f a c e a r e a of plate T t e m p e r a t u r e r i s e above ambient T t e m p e r a t u r e at c e n t r e of p l a t eAT t e m p e r a t u r e difference between c e n t r e and edge of p l a t e AT value of AT t o p r o d u c e buckling
t t i m e
w l a t e r a l d i s p l a c e m e n t of m e d i a n s u r f a c e of plate
a coefficient of t h e r m a l expansion
/? , /?2 c o n s t a n t s in deflection mode of plate
Z, coefficient » E a
r € the r a t i o
Notation (Continued)
''re
6 <t> V^ v ^ membrane shear s t r e s s angular co-ordinate s t r e s s function g2 1 Laplace's operator » -2— + — 9r^ r operator » [v^J^ 9^ 3021
-1. Introduction
A free circular plate may buckle if temperature gradients are set up in the material.
The case considered here is when the temperature field varies only in the radial direction, giving rise to compressive s t r e s s e s which may be large enough to cause buckling. This problem has been
investigated by C o t t e r e l r ^ ' who found a large discrepancy between his experimental and theoretical r e s u l t s . The difference was not explained and it was stated that further experimental work should be carried out.
(2)
It has been suggested by Hoff , that the accepted methods of
stability analysis might not be applicable to the case of thermal s t r e s s e s because these s t r e s s e s are influenced by the deformations which the body undergoes when it buckles. To clarify his point, he investigated both theoretically and experimentally, the behaviour of a column subjected to a uniform temperature rise while placed between two fixed platens'^'. He found that at the moment of buckling, first order small displacements caused second order small changes in the thermal s t r e s s e s , and that the results of a classical analysis agreed with those obtained by experiment. It was then concluded that the thermal buckling of plates and shells could be predicted using the accepted methods of analysis, but that further studies of their reliability should be undertaken.
2. The experimental investigation
A photograph of the circular plate specimen, with the experimental apparatus is shown in Fig. 1.
2 . 1 . Specimen
The circular plate was cut from a nominal 16 s . w . g . rolled sheet of commercial mild steel, and measured 10.78 in.dia. x .060 in. thick. Tensile tests on the material gave a value of Young's modulus of
29.7 X 10^ Ib/in^. , a yield s t r e s s of 22,300 Ib/in^. and a 0.1 per cent proof s t r e s s of 31,000 Ib/in^.
In order to sixpport the specimen it was found necessary to drill a ^/8 in. diameter hole in the centre of the plate which was then screwed to a tripod stand. This method of attachment allows dial gauge measurements to be taken, and is assumed to have a negligible coi-straining effect on the plate.
2
-2. -2. The experimental apparatus
A photograph of the experimental apparatus is shown in Fig. 1(a). A Calor gas burning ring was used as a heating source, and an overhead Dexion structure gave support to the gas supply pipe which was connected to a four way junction piece, giving four inlets to the copper burning ring.
Provision was made for a shield to be lowered between the edge of the plate and burning ring to dissipate the flames.
It was hoped to induce radial temperature gradients in the plate by applying a uniform heat input to the edge of the plate at a high rate of heating. Although a high rate could easily be obtained it was not possible, at the same time, to provide uniformity of heating around the edge of the plate. Recourse was made to a very slow heating r a t e , in which the plate was brought to a near uniform temperature r i s e , and a radial temperature gradient was induced by cooling the plate in the centre by compressed a i r . This method was found to give satisfactory r e s u l t s ,
2 . 3 . Instrunaentation
The buckling of the plate was investigated under transient cooling conditions, and in order to obtain a time history of the temperature distribution and deflection in the plate, recording equipment had to be used. This equipment consisted of a 12-channel m i r r o r galvonometer trace recording set, made by New Electronics Products Ltd. , Type
1000, and a Shackman Auto c a m e r a .
40 s . w . g . Chrome-Alumel thermocouples, of equal r e s i s t a n c e s , were used to obtain the overall temperature distribution in the plate. These were connected to the recording set which had been previously calibrated using a representative thermocouple.
The buckling point and mode of deflection were measured by taking photographs of the dial gauges during cooling of the plate, with a
Shackman Auto c a m e r a . Since the speed of the t r a c e recording and c a m e r a are knov/n, the temperature distribution and mode of deflection could be collated at any instant. The buckling point was obtained by examining the dial gauge readings in order to establish the time at which large plate deflections began.
3
-3. Experimental results
Fig. 2 shows the measured temperature distribution in the plate at different time intervals during the t e s t , and in particular the critical radial temperature distribution. This i s , in fact, a mean distribution corresponding to the temperature mid-way through the plate thickness.
The mode of deflection is shown in Figs. 3(a) and 3(b) where the actual experimental points are indicated. These are seen to agree well with the theoretical mode used below in paragraph 4.
F o r the form of temperature distribution shown in Fig. 2 the critical temperature differential between the centre and edge of the plate is seen to be 39°C. having an estimated accuracy of t 3,0° C.
4. Theoretical analysis
4 . 1 . General r e m a r k s
The results obtained from experiment enable expressions to be derived representing the form of temperature distribution in the plate and the mode of deformation. The radial and hoop s t r e s s e s are
determined, and the work done by the internal forces are calculated. The strain energy of bending of the plate is determined and on equating these two expressions, the point of neutral equilibrium is found directly; a variational procedure is not required.
4 . 2 . Assumptions
The following assumptions are made :
-(i) The plate is considered to be free.
(ii) The mechanical properties of the material are independent of t e m p e r a t u r e .
(iii) The plate material is isotropic.
(iv) The temperature distribution is axi-symmetrical. (v) The thermal stressing problem may be considered
as one of plane s t r e s s .
The following r e m a r k s concerning the above assumptions may be made.
4
-(i) The presence of a / 8 in. diameter attachment hole at the centre of the plate is assumed to
affect only the s t r e s s distribution in the immediate locality of the hole, leaving the s t r e s s e s over the outer region (which are the cause of buckling), identical to the case of the free plate.
(ii) The maximum uniform temperature r i s e attained in the plate was of the order of 150°C. and since the material used was steel, it is felt that this assumption is not unreasonable.
(iii) The value of Young's modulus was found to be the same in the direction of the grain of the material and in the direction across the grain, by experiment. Other mechanical properties may be taken to be the same in both directions.
(iv) This assumption is justified by experimental data.
(v) The theoretical buckling s t r e s s e s are found to be well below the yield s t r e s s of the material.
(See Fig. 6).
(vi) In a recent article by Gatewood , thermal s t r e s s e s in moderately thick plates are investigated, and it is concluded that if the plate is thin, then the temperature gradients must be extremely large to have any effect on the plane s t r e s s solution.
4 . 3 . Solution
In plane s t r e s s , the thermal problem reduces to the solution of the following differential equation subject to certain boundary conditions'^'.
V^0 + V^^T ' 0, (1)
where ^ is a s t r e s s function and the s t r e s s components are given by
and cr r ^Q ^r6 a ae at 1 90 r 9 r •*• a ^ 2 » a r 1 90 2 96 1 -96^ t 1 3^ 0 r 9 r 99 ( 2 a . b . c . )
5 -A t h i n d i s c w i t h a t e m p e r a t u r e d i s t r i b u t i o n w h i c h i s a f u n c t i o n of t h e r a d i a l d i s t a n c e o n l y , i s in a s t a t e of p l a n e s t r e s s w i t h r o t a t i o n a l s y m m e t r y . F o r a f r e e d i s c , t h e s t r e s s c o m p o n e n t s c a n b e s h o w n t o b e T r d r (6)
^{k [ Trdr i
-a n d - T + — r^ T r d r ( 3 a . b ) F o r t h e p r o b l e m u n d e r c o n s i d e r a t i o n , i t i s found t h a t t h e f o r m of t h e t e m p e r a t u r e d i s t r i b u t i o n c a n b e r e p r e s e n t e d by T + A T ? ' n (4) a n d t h e c o r r e s p o n d i n g s t r e s s c o m p o n e n t s t h e n b e c o m e a n d ^ A T n + 2 ^ A T 1 - 5 n n ^fl - n + 2 1 - ? (n + 1) ( 5 a . b ) T h e e n e r g y e q u a t i o n f o r b u c k l i n g i n p o l a r c o - o r d i n a t e s b e c o m e s ^ ' f o r're
29^w
a 2 r 9 r 2 i 9^ ! _ 9 r r 9 V 902 2(1 -v) h 2 9^w L 9r" 9 w 1 9^w 9 r 2 2 r 9 r 9 w 9 r + / 9 w d s . 3 w 90 9 V 9 r 9 0 / ! j ° (6) A s u i t a b l e d i s p l a c e m e n t m o d e m a y b e r e p r e s e n t e d b y t h e e q u a t i o n w A ? ' 1 2 s i n 20 , ( 7 a ) w h e r e ^^ a n d ^^ a r e s u i t a b l y c h o s e n c o n s t a n t s w h i c h s a t i s f y t h e c o n d i t i o n s at t h e f r e e e d g e ' " ' t h a t^ r d 0
Mj, - 0 .
6 -» 0 ,
and ( 7 b . c )
Substitution of equations (5) and (7a) into equation (6) y i e l d s an equation for t h e c r i t i c a l t e m p e r a t u r e d i f f e r e n t i a l . T h i s b e c o m e s for the c a s e when V » .287
, T , . M2LD , <8,
b"" h S F ( n ) w h e r e F(n) i s given by F ( n ) •• "" n + 2 1.25 , 1.25 ^a , 1 . 0 8 4 „ , 2.67 ^ n + 8 2^ • - " 2^ n + 10 1 2 (9) and w h e r e p ^ - .279 and /? » .063. 1 2T h e s e coefficients give an e x p r e s s i o n for w which c o i n c i d e s with the e x p e r i m e n t a l l y d e t e r m i n e d mode of deflection, s e e F i g s . 3a, b .
d( AT )
The condition =— =» 0 gives a value of n » 2.6 and a dn
c o r r e s p o n d i n g value of AT ^^ =• 23.3 C . for the d i m e n s i o n s and m a t e r i a l of the t e s t s p e c i m e n .
F i g . 4 shows the v a r i a t i o n of the c r i t i c a l t e m p e r a t u r e differential AT with n, and t h i s solution is g e n e r a l l y applicable provided that the f o r m of t e m p e r a t u r e d i s t r i b u t i o n m a y be r e p r e s e n t e d by a s u i t a b l e choice of n, and that the buckling mode r e m a i n s unchanged for v a r i o u s t e m p e r a t u r e d i s t r i b u t i o n s .
4 . 4 . C o m p a r i s o n with e x p e r i m e n t
F o i the s p e c i m e n u n d e r c o n s i d e r a t i o n h " .06 in. , b =» 5.39 in. , and
a and v a r e taken equal t o 11 x 1 0 " " / ° C . , and .287 r e s p e c t i v e l y .
F i g . 5 shows that a good a p p r o x i m a t i o n to the m e a s u r e d t e m p e r a t u r e d i s t r i b u t i o n , at the point of buckling, m a y be obtained by putting n » 0.6 in equation (4). The c o r r e s p o n d i n g value of AT is 3 8 . 0 ° C . , f r o m F i g . 4. This value c o m p a r e s well with AT » §9 + 3 . 0 ° C . from e x p e r i m e n t .
The t h e o r e t i c a l buckling s t r e s s d i s t r i b u t i o n for t h i s c a s e (equ. 5) ig shown in F i g . 6.
7 -References Cotteren, B. Hoff, N . J . Hoff, N . J . Gatewood, B . E . Gatewood, B . E . Goodier, J . N . Timoshenko, S. Wang, C.
Thermal buckling of a free circular plate.
B . A . E . internal memo Structures 378. November, 1957.
Buckling at high temperature. Journal of the Royal Aeronautical
Society. Vol.61, 1957, pp 756-77.
Effects thermiques dans le calcul de la resistance des structures d'avions et d'engins. Organisation Du Traite de I'Aclantique Nord, Palais de Chaillot, P a r i s 16. Janvier 1956. pp 43.
Thermal s t r e s s e s in moderately thick elastic plates.
Journal of Applied Mechanics, Vol. 26, Series C, 1959. pp 432-436.
Thermal s t r e s s e s .
McGraw-Hill Book Co. I n c . , 1957.
Thermal s t r e s s and deformation.
Journal of Applied Mechanics, Vol.24, 1957.
Theory of elastic stability. McGraw-Hill Book Co. Inc. , F i r s t Edition, 1936.
Applied elasticity.
FIG. 1 (a). PHOTOGRAPH OF THE TEST APPARATUS SHOWING AN EXPLODED VIEW OF THE
SPECIMEN
FIG. 1 (b). CIRCULAR PLATE SPECIMEN WITH TRIPOD MOUNTING
6 ' t = O secs
^ ^ * ( B E G I N N I N G O F C O O L I N G )
t = I I - o SECS
o 1-0 2 0 3 0 4 - 0
5-RAOIAL DISTANCE ( r ) IN INCHES
F I G . 2 . TEMPERATURE D I S T R I B U T I O N DURING C O O L I N G .
FIG 3 o . DEFLECTION MODE AT GIVEN RADIUS.
" = ^ (?)'[—'(t)V°"C?r]/"^
z o Iii lil ' t - 1 0' ^-• •• 1 T 0 TEST POINTS .^ ^^""^ ^^ ^ ^ ' . 0 ^ \ \y
^ - - ' j ^ / o l O 2-o 3 0 4 - 0 5 - 0 RADIAL DISTANCE ( r ) IN INCHESCOMPRESSIVE STRESS L B / I N ^ T E N S I L E O O 1 m n > r m •n f z s M (/I C O z 5 / / z o ^ Ü) ^ \ / / r — I / * / r ' / / i
