structural analysis of a hexagonal tube or ring …
TRANSCRIPT
STRUCTURAL ANALYSIS OF A HEXAGONAL TUBE OR RING UNDER INTERNAL PRESSURE
C. Do Flowers / ' /
JUNE, 1966
BAlTELLE MEMORlAL INSTITUTE / PACIFIC NORTHWEST WOFUfORY
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I t PACIFIC NORTHWEST LABORATORY I
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operafed. by I 0AlTEl.G MEMORIAL INSTITUTE I
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I
I 1 MIII(IEP~V/FOUTIIE U. S. ATOMIC EW EAGY COMMISJION
BNWL-263
UC-80, Reactor Technology
STRUCTURAL ANALYSIS OF A HEXAGONAL TUBE
OR RING UNDER INTERNAL PRESSURE
BY C. D. Flowers
Development and Technology Engineering Sect ion FFTF Project
June, 1966
PACIFIC NORTHWEST LABORATORY RICHLAND, WASHINGTON
T A B L E O F CONTENTS Page
N u m b e r
. . . . . . . . . . . . . . . . . . . . . . . . . . . . ABSTRACT !
I N T R O D U C T I O N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUMMARY AND C O N C L U S I O N S
A N A L Y S I S
. . . . . . . . . . . . . . . . . . . . . . . A n a l y s i s M o d e l
. . . . . . . . . . . . . . . . . . Stresses A n d D e f l e c t i o n s
PROBLEM FORMULA T I ON
C a l c u l a t i o n O f Mo . . . . . . . . . . . . . . . . . . . . . Stresses . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . M a x i m u m C o r n e r Stresses I n H e x T u b e
. . . . . . . . . . . . . . . . . . . . . . . . . D e f l e c t i o n s
G R A P H I C A L R E S I L T S F O R HEXAGONAL T U B E . . . . . . . . . . . . . . D I S C U S S I O N . . . . . . . . . . . . . . . . . . . . . . . . . . . NOMENCLATURE' . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . . . . . . . . A P P E N D I X A . D E R I V A T I O N O F E Q U A T I O N S . . . . . . . . . . . . . . A P P E N D I X B . V E R I F I C A T I O N O F A S S U M P T I O N S . . . . . . . . . . . . R E F E R E N C E S . . . . . . . . . . . . . . . . . . . . . . . . . . .
BNWL- 2 6 3
P r i n t e d i n USA. P r i c e $ 3 . 0 0 . A v a i l a b l e f rom t h e C l e a r i n g h o u s e f o r F e d e r a l S c i e n t i f i c a n d T e c h n i c a l I n f o r m a t i o n ,
N a t i o n a l Bu reau o f S t a n d a r d s , U . S . Depa r tmen t o f Commerce, S p r i n g f i e l d , V i r g i n i a
STRUCTURAL ANALYSIS CF A HEXAGONAL TUBE OR R I N G UNDER INTERNAL PRESSUFB
ABSTRACT
The maximum s t r e s s i n hexagonal r e a c t o r tubes occurs a t t h e i n s i d e
sur face of t h e corner where t h e f i b e r s a r e i n tens ion . This maximum
s t r e s s c o n s i s t s p r imar i ly of t h e bending s t r e s s e s caused by t h e r i g i d i t y
of t h e corner . This corner r i g i d i t y induces bending of t h e wal l s , t hus
g iv ing maximum d e f l e c t i o n a t t h e middle of t he f l a t s i d e ,
An e l a s t i c s t r e s s and d e f l e c t i o n a n a l y s i s was made f o r a hexagonal
tube o r a hexagonal r f n g submitted t o a cons tant i n t e r n a l pressure load.
The a n a l y s i s assumes a homogeneous and i s o t r o p i c medium, and uses l i n e a r
small d e f l e c t i o n theory. A hyperbolic s t r e s s d i s t r i b u t i o n was assumed
f o r analyzing t h e corners of t h e tube o r r i n g .
A l l r e s u l t s a r e presented i n equat ion form. Also, f o r t he hexagonal
tube, t h e r e a r e p l o t s of maximum corner primary membrane p lus bending
s t r e s s , and maximum wal l d e f l e c t i o n ac ross t h e f l a t s . These graphs a r e
dimensionless, and a r e app l i cab le t o a use fu l range of hexagonal tube
s f zes .
INTRODUCTION
I n t e r e s t has been generated a t Battelle-Northwest f n hexagonal-shaped
tubes mainly because of t h e i r a p p l i c a t i o n t o f a s t f l u x , tube-type r e a c t o r s .
The p resen t conceptual core des ign of t h e Fas t Flux Tes t F a c f l f t y (FFTF)
fncorpora tes a hexagonal l a t t i c e which i s made up of skewed hexagonal
r e a c t o r tubes. These tubes have a hexagonal shape i n t h e core zone and
a c y l i n d ~ i c a l shape elsewhere. The r e a c t o r i s sodium cooled, and t h e
tubes func t ion a s high v e l o c i t y sodium flow condui t s which a r e exposed
t o i n t e r n a l p re s su re loads . The skewed tube concept i s shown i n Figure 1
and t h e hexagonal core l a t t i c e i s shown i n Figure 2.
The use of skewed hexagonal r e a c t o r tubes provides a c l o s e l y packed
a r r a y which minimizes sodium volume between tubes i n t h e core zone, and
a l s o provides a good f u e l packing e f f i c i e n c y wi th in each tube . This
compact core i s a n a i d t o achieving t h e high f l u x t h a t i s des i r ed i n t h e
F'FTF. The low sodium coo lan t p re s su res t h a t a r e r equ i red a l s o make t h e
hex-shaped r e a c t o r tube f e a s i b l e . Since the r e a c t o r t ubes a r e submitted
t o t h i s i n t e r n a l pressure , a n e l a s t i c s t r u c t u r a l a n a l y s i s of t h e tubes
i s requi red a s a p a r t o f t h e conceptual i n v e s t i g a t i o n s o f core design.
Both pressure-induced s t r e s s e s and d e f l e c t i o n s a r e o f i n t e r e s t i n a
complete mechanical a n a l y s i s o f t h e tube. For the hexagonal tube under
i n t e r n a l pressure , t h e t o t a l primary s t r e s s c o n s i s t s of t h e gene ra l
membrane s t r e s s and t h e bending s t r e s s . The d i f f i c u l t y i n analyzing the
hexagonal tube comes from t h e presence of t he bending s t r e s s e s which a r e
caused by i n t e r n a l bending moments. These i n t e r n a l moments a l s o c o n t r i b u t e
t o t h e t o t a l outward d e f l e c t i o n of t h e tube wa l l s .
An e l a s t i c s t r e s s and d e f l e c t i o n a n a l y s i s of a hexagonal box under
i n t e r n a l p re s su re has been c a r r i e d o u t previous ly . (2) Resu l t s of t h i s
p r i o r a n a l y s i s a r e a v a i l a b l e only i n equat ion form, and a r e not gene ra l
enough f o r convenient a p p l i c a t i o n t o t h e p resen t problem. Also, t h e
a n a l y s i s r e f e r r e d t o does not inc lude t h e p l a t e e f f e c t t h a t e x i s t s i n
t h e tube wal l s . The tube wa l l s a r e a c t u a l l y long p l a t e s and a r igo rous
a n a l y s i s should inc lude p l a t e theo ry a s wel l .
BNWL
PiR r Sodium
Closed Loor, _
(Tubes Ilavt, Hrxagonal
\ \ C r o s s c c t i o n Through C o r r Rcgion)
Reac tor Zone
High P r e s s u r e T u b e s h e e t s
- 2 6 3
Flow
Sodium 1 low In
FIGURE 1. FFTF Skewed Tube S c h e m a t i c
4 .5 in.
6 in. C losed Loop-Fue l ( 2 )
Open Loop-Mater ia l (8) k Neutron S o u r c e (1 )
4. 5 i n Loop -
T e s t :y (1)
5 in.
6 in. Open ~ o & ~ - ~ u e l (2 ) \ Safety Rod with M a t e r i a l s T e s t (3 )
0 Q D r i v e r F u e l T u b e s
Q with Zoned F u e l Loading
FIGURE 2. FFTF Hexagonal Core Lattice
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I n t h e present ana lys i s t h e s o l u t i o n i s more s t ra ight forward; t h e
maximum s t r e s s and d e f l e c t i o n equations a r e w r i t t e n i n a genera l form so
a s t o apply t o any hex-shaped tube o r r ing . Also, s ince t h e primary
i n t e r e s t i s i n t h e hex tube, dimensfonless s t r e s s and d e f l e c t i o n graphs
a r e presented f o r t h e convenience of the designer and include a range
of hexagonal tube s i z e s . The graphs a r e app l i cab le , wi th in t h e e l a s t i c
l i m i t , f o r any pressure and temperature loads. A p p r o p ~ i a t e p l a t e theory
terms a r e a l s o included i n t h e tube ana lys i s .
SUMMARY AND CONCLUSIONS
The maximum s t r e s s i n t h e hexagonal tube o r r i n g was found t o occur
a t t h e i n s i d e sur face of t h e corner where the f i b e r s a r e i n tens ion .
This maximum s t r e s s c o n s i s t s p r imar i ly of t h e bendfng s t r e s s e s t h a t
a r i s e due t o t h e r i g i d i t y of t h e corner . The maximum d e f l e c t i o n occurs
a t t he middle of t h e f l a t s ide and i s due p r imar i ly t o t h e bendfng of
t h e wa l l s s ince t h e corney i s f a i r l y r i g i d .
For the hexagonal r i n g t h e r e s u l t s a r e given only i n equation form.
However, s ince our primary i n t e r e s t i s i n tubes , t h e r e s u l t s f o r hexagonal
shaped tubes a r e presented i n g raph ica l form. Graphs f o r both maximum
corner s t r e s s ( ~ i ~ u r e 7) and ~naximum tube d e f l e c t i o n ac ross the f l a t s
( ~ i ~ u r e 8) a r e included. I t i s ~ecommended t h a t ~ / t r a t i o s 2 2 be used
when applying these graphs i n order t o e l iminate high s t r e s s c o n c e n t ~ a -
t f o n s i n t h e corners . The maximum s t r e s s graph can a l s o be used con-
venient ly t o determfne the necessary wal l th ickness and corner r a d i u s f o r
a tube submitted t o given pressure and t e m p e ~ a t u r e loads ,
Since the FFTF r e a c t o r tubes experience a wide v a r i e t y of loading
condi t ions (temperature bowing, tubesheet r eac t fons , middle support
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- 6 -
r e a c t i o n s , e t c .) , f w t h e r s tudy i s requi red before t h e f e a s i b i l i t y of t h e
hexagonal r e a c t o r tube can be determined.
ANALYSIS
Analysis Model
The model used i n t h i s a n a l y s i s i s shown i n Figure 3. The model
i s assumed t o be i n f i n i t e i n l eng th , homogeneous and i s o t r o p i c , and
t o have a uniform pressure (p) throughout t h e l eng th , Since t h e
a x i a l p re s su re drop i n a r e a l tube i s gradual , a cons tant p re s su re
assumption i s v a l i d . The tube c r o s s s e c t i o n can be broken i n t o two
primary elements--the f l a t wa l l s and the rounded corners , and,due t o
symmetry, h a l f of one wal l and h a l f of one corner w i l l be used f o r
a n a l y s i s , A free-body diagram of the model i s shown i n Figure 4.
A f o r c e and moment balance must f i rst be made on t h e model before
t h e s t r e s s e s and d e f l e c t i o n s can be determined. The only unknown term
i s t h e moment Moo The f o r c e s Px and P a r e e a s i l y determined from Y
equi l ibr ium, and t h e moment equat ions f o r t h e s t r a i g h t p o r t i o n and
f o r t h e curved por t fon can be w r i t t e n f n terms of M, which i s unknown.
Since sma l l - s t r a in l i n e a r e l a s t f c theo ry i s used, t h e moment M,
i s found by combining t h e s t r a i n energy o f bending of t h e two elements
and us ing C a s t i g l i a n o b theorem t o f ind t h e s lope a t x = 0 . By apply-
ing t h e proper boundary condition--that t he s lope a t x = 0 i s equal t o
zero--Mo can be obtained. Once Mo i s known, t h e s t r e s s and d e f l e c t i o n
a n a l y s i s can be made,
S t r e s s e s And Def lec t ions
F l a t Side: The s t r a i g h t p o r t i o n of t h e tube has been assumed a s
a f l a t p l a t e and t h e d i f f e r e n t i a l equat ion f o r t h e c y l i n d r i c a l bending
SIGN CONVENTION
Outside o f tube
%kg@ Inside o f tube
F I G U R E 3 . Analytical Model
FIGURE 4 . Free Body Diagram
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of p l a t e s a p p l i e s . The equa t ion is:
( ~ o m e n c l a t u r e l i s t on p. 30 ) (Ref. 6, p. 5 )
where D = f l e x u r a l r i g i d i t y of f l a t p l a t e .
The d e f l e c t i o n s a r e assumed t o be small i n comparison with t h e
th i ckness (y<t /2) , and s ince supe rpos i t i on can be used i n l l n e a r
e l a s t i c theory , d i r e c t s t r e s s P,/A i s combined with bending s t r e s s $0
g e t t h e t o t a l wa l l s t r e s s . Def lec t ions a r e obta ined by d i r e c t i n t e -
g r a t i o n of t h e gene ra l d i f f e r e n t i a l equat ion and a p p l i c a t i o n of t h e
necessary boundary cond i t i ons . The corner d e f l e c t i o n must f i rs t be
found and t h e second boundary cond i t i on (g iven below) must be app l i ed
t o determine t h e i n t e g r a t i o n cons tan t and t h e t o t a l d e f l e c t l o n
equat ion.
S ince we have a second o rde r d i f f e r e n t i a l equat ion, t h e necessary
boundary cond i t i ons f o r our f l a t p l a t e model can be s t a t e d .
Boundary Condit ions
1. A t x = 0 , s lope = 0.
2. Def lec t ion of s t r a i g h t p o r t i o n a t X = L is equal t o d e f l e c t i o n
of corner a t 8 = C .
Corner: For determining d e f l e c t i o n s , t h e corner i s assumed f i x e d
a s shown i n Figure 3 I n assuming t h e corner i s f i x e d , we assume t h a t
t h e ex tens ion of t h e f l a t s i d e due t o t h e membrane t e n s l o n i s approxi-
mately enough t o al low t h e f l a t s i d e t o d e f l e c t outward and assume t h e
proper cu rva tu re , Also, t h e change i n l e n g t h of t h e s i d e due t o
d i r e c t membrane t e n s i o n i s q u i t e small i n comparison with t h e tube
wa l l de f l ec f ion . A sample c a l c u l a t i o n i n Appendix B shows t h i s
assumption t o be reasonable.
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- 10 -
The d e f l e c t i o n of t h e corner i s c a l c a l a t e d by us ing C a s t i g l i a n o 3 s
theorem tind a modi f ica t ion of t h e t o t a l s t r s i n energy equat ion f o r 3
t h i c k r i n g s (Ref. 4, p. 84 ) . We have included a f a c t o r of (I-'&) i n
t h e r i n g equat ion t o make it app l i cab le t o curved p l a t e s . This y i e l d s
t h e fo l lowinp t o t a l s t r a i n energy equa-tion:
2 2 t-- - . -
2AE AEI 2AG d s
The d e f l e c t i o n s due t o each of t h e l o a d s on t h e curved p o r t i o n
Py, Px, and ko shown i n Figure 5 a r e c a l c u l a t e d by us ing C a s t i g l i a n o ' s
theorem.
The t o t a l c c r a e r de f ' l e c t io r ; ,A , i s t h e n found by superpor:l+,b3n.
F ipo l ly , by apply ing t h e second boundary cond i t i oc , t h e i n t e g r a t i o n
constar,% and t h e t o t 2 1 deflection eciurt.icn f c r ti?? vwtili can be obtoS fie?.
To o b t a i n " Le cc rne r s i r e s e e s , c-.,rvcd F' - : te t heo ry (which i s
assumed t o be t h e same a s curved b a r . thecry f c r t h i s c s s e ) xi.1.l be
used. There a r e two methods of s c i ; r t i o n t h a t can be used 50 o b t a i n
t h e co rne r s t r e s s e s : (1) t h e hyperbol ic s t r e s s d i s t r i b u t i o n method
( R e f . 4, p. 65 ) , and (2) t h e exac t s t r e s s d i s t . r i b u t i o n method ( ~ e f . 5,
p . 61) . I n t h e f i rs t method it i s assumed t h a t t r a n s v e r s e c ros s s e c t i o n s
remain p lane and normal t o t h e c e n t e r l i n e and t h e t r a d i a l s t r e s s e s
a r e equa l t o zero. With t h e s e assumptions t h e d i s t r i b u t i o n of t h e
normal s t r e s s e s , re, over any c r o s s s e c t i o n fo l lows a hyperbol ic law.
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- 12 - The second method assumes a s t r e s s func t ion of t h e form
= A i n r + ~r~ i n r + cr2 t D' (Ref. 5 9 p. 58)
which a p p l i e s t o a l l problems with axisymmetrical s t r e s s d i s t r i b u -
t i o n and no body fo rces . I n t h i s method t h e r a d i a l s t r e s s e s a r e not
neglected--which makes it t h e more exac t of the two methods.
However, according t o Timoshenko ( ~ e f . 5, p. 64) , t h e hyperbolic
s o l u t i o n g i v e s very accura te r e s u l t s f o r b / ~ ~ r a t i o s l e s s than
t h r e e o r fou r . Using b/Ri<4, t h e hyperbolic s o l u t i o n w i l l be
assumed t o be accura t e f o r ~ / t > 5 / 0 ,
I n most p r a c t i c a l a p p l i c a t i o n s , ~ / t r a t i o s of l e s s than 5/6 w i l l
no t be encountered a s t h i s w i l l letis t o high s t r e s s concent ra t ions
which a r e undas i rab le , FOP these reasons t h e hyperbolic s t r e s s d i s -
t r i b u t i o n method has been chosen t o analyze t h e co rne r s of t h e hexa-
gonal tube. From Timoshenko, Ref. 5, p. 64, it i s es t imated t h a t t h e
e r r o r introduced i n t h i s a n a l y s i s by using the hyperbolic s t r e s s d i s -
t r i b u t i o n method ( i n s t e a d of t he exac t method) should not exceed
f i v e percent . To show t h a t t h e r a d i a l s t r e s s e s a r e small i n compari-
son with t h e Q-s t resses , a sample c a l c u l a t i o n using t h e exac t method
has been made i n Appendix B.
PROBLEM FORMlJLlZ_T_=-:
Ca lcu la t ion Of Mo
By equi l ibr ium we g e t t h e f o r c e s per u n i t l eng th of tube
Px = p (LC + Ri) and
Py = p L
The moment.% i s unknown. However, t h e moment equat ions f o r t he
s t r a i g h t s e c t i o n and t h e curved s e c t i o n can be w r i t t e n i n terms of &be
- 1 3 - BNWL- 203
The moment equat ion f o r t h e s t r a i g h t s e c t i o n pe r u n i t l eng th of tube
The moment equat ion f o r t h e curved s e c t i o n i s t h e sum of t h e moments
imposed by t h e loadings shown i n Figure 5. The p re s su re loading
causes no moment, so Me w i l l c o n s i s t o f moments induced by bo, Px-Pp,
and Py where n
Px-Pp = p ( L C + Ri)-pRi = p~n9 and Py = pL
From t h i r , t h e moment equat ion pe r u n i t l eng th o f tube f o r t he curved
s e c t i o n i s n
The moment, &, i n Equations (1) and (2) cannot be determined from
t h e condi t ions o f s t a t i c equilibrium, and t h e r e f o r e a n a d d i t i o n a l con-
d i t i o n must be used. The moment, &, can be determined by us ing s t r a i n
energy methods and t h e cond i t i on t h a t t h e s lope o r r o t a t i o n a t
x = 0 ( ~ o i n t o f a p p l i c a t i o n of M,) i s equal t o zero. C a s t i g l i a n o ' s
theorem f o r t h e s lope o r r o t a t i o n a t t h e p o i n t o f a p p l i c a t i o n o f M, i s
where )bo i s t h e s lope o r r o t a t i o n a t t h e s ec t ion , & i s t h e moment
a t t h e s e c t i o n and U i s t h e t o t a l bending s t r a i n energy of t h e member.
For t h e tube wal l , t h e s t r a i n energy equat ion f o r t h e bending o f f l a t
p l a t e s t o a c y l i n d r i c a l su r f ace i s
where M i s t h e moment a t any s e c t i o n and D i s t h e f l e x u r a l r i g i d i t y
of a f l a t p l a t e .
The r e l a t i o n f o r D i s
(Ref. 6 , p. 5 )
Equation(&) a l s o holds f o r curved p l a t e s i f R / t >2 (Ref. 3,
p. 171) , b u t f o r R/t<2, a more e x a c t r e l a t i o n f o r t h e bending s t r a i n
energy i n a curved p l a t e i s
This equat ion i s found by us ing Hookefs Law and assuming t h a t
t h e l o n g i t u d i n a l s t r e s s i s equal t o zero i n t h e curved p l a t e . By
doing t h i s h e (1 - ) term appears f n the energy equat ion f o r t h e
curved p l a t e .
It i s shown i n Appendix B t h a t t h e e r r o r introduced i n calcu-
l a t i n g M, by us ing Equat ion(4)for t h e s t r a i n energy of t h e corner
in s t ead of t h e curved p l a t e energy equat ion i s l e s s t h a n one percent ,
A value of Z / t = P was used f o r t h e comparison and t h e discrepancy
was approximasely 0.7 percent . Based on t h i s , we w i l l use Equation(&)
f o r t h e s t r a i n energy due t o bending i n both t h e f l a t p l a t e and t h e
curved p l e t e .
Applying Cas t ig l i ano s t h e o ~ 6 . i (Eq. 3) t o Equation (4) over t h e
e n t i r e s t r u c t u r e , we g e t A ,'J
1 M - & X + L J M&%~ Jb- D bM 0 D
wal l aM0 corner
For t h e f l a t p l a t e t h e moment Mx i s
2 Mx = JZ- - Mo
2
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and t h e limits of i n t e g r a t i o n a r e from x = 0, L.
For t h e curved p l a t e t h e moment % i s
and t h e limits of i n t e g r a t i o n a r e from 0 = 0, v/6. Also, f o r t h e
curved por t ion d s = R d 8. For both ~ q u a t i o n s ( 1 ) and (2), @ = -1.
From these condi t ions , and from t h e condi t ion t h a t we can super-
impose t h e energy of t h e two -ec t ions , t h e r e l a t i o n f o r Mo becomes
( ~ e r i v s t i o n i n Appendix A)
This equat ion can be expressed i n t h e fol lowing dimensionless form:
A s an a i d i n using t h e s t r e s s and d e f l e c t i o n equat ions derived i n
Mo t h i s a n a l y s i s , -2 i s p l o t t e d a s a func t ion of (L/R) i n Figure 6 on PL
page 16 . Having completed t h e s t a t i c equi l ibr ium balance, we can now
solve f o r s t r e s s e s and d e f l e c t i o n s . S t r e s s e s
Wall S t r e s s e s : The combined bending s t r e s s and d i r e c t s t r e s s f o r
t h e c y l i n d r i c a l bending of p l a t e s i s
~ q u a t i o n ( 6 ) i s genera l and a p p l i e s t o a hexagonal r i n g a s we l l a s
t o a hexagonal tube ,
For t h e hexagonal r i n g
Px = Load
Ax = Area of c r o s s sec t ion .
Mx = Bending moment a t any s e c t i o n of f l a t s ide .
y = Distance from c e n t r o i d a l a x i s t o f i b e r s where s t r e s s i s
des i red .
I = Moment of i n e r t i a of c ros s sec t ion .
For t h e s t r e s s per u n i t l eng th of t h e hexagonal tube, t h e terms
a r e a s fol lows
y = Same a s above.
Corner S t r e s s e s : The t o t a l corner s t r e s s c o n s i s t s of bending and
d i r e c t s t r e s s e s combined. The equat ion f o r pure bending s t r e s s i n a
curved p l a t e f s
-My 1
Cb = ~ e ( r t jl) ( R e f . 4, p . 67)
where y ' i s measured p o s f t i v e l y outward from the n e u t r a l a x i s . Since
i t i s more convenlent t o measure t h e f i b e r s from the een t ro fda l a x i s ,
t he fol lowing equat ions have been modified so t h a t y i s t h e d i s t ance
from t h e e e n t r o i d a l a x i s , measured p o s i t i v e outward.
For a rec tangular c ross sec t ion , t h e bending s t r e s s i n t h e corner
becomes
where M = MQ and
Equation (9a) i s derived i n Appendix A .
The d i r e c t s t r e s s can be expressed by t h e fol lowing no ta t ion
The normal f o r c e Pn can b e s t be obtained by r e f e r r i n g t o Figure 5.
From Figure 5.
Pn = ( P ~ - P ~ ) ~ o s €J + Pp + P s i n @,or Y
Pn = pL O c o s 8 + pRi + pL s i n 8
and pL 0 cos 8 + pRj + pL s i n @
0 = 8, A
The t o t a l c o m e r s t r e s s can now be expressed a s the sum of t h e
bending and d i r e c t s t r e s s e s .
- -- pL fl cos 8 + pL s i n 8 + pRi c ~ T o t . A + Y8 AR [-1
Equation (9) f o r t h e corner s t r e s s e s i s genera l and w i l l apply t o a
hexagonal r i n g with r ec t angu la r c ross s e c t i o n a s wel l a s t o a
hexagonal tube. For s t r e s s e s per u n i t l eng th of t h e hexagonal tube ,
t h e a rea term ( A ) w i l l equal t h e th i ckness it). I t must be remem-
bered t h a t t h e s t r e s s equat ion, (7a), a p p l i e s only t o a hexagonal r i n g
when t h e p ing has a r ec t angu la r c r o s s sec t ion .
Maximum Corner S t r e s s e s I n Hexagonal Tube: The h ighes t s t r e s s e s
t h a t occur i n t h e hexagonal tube w i l l occur i n t h e corners and w i l l
be a t t h e i n s i d e and ou t s ide su r faces where t h e f i b e r s a r e i n t e n s i o n
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- 19 - and compression, respect ively . By maximizing Equation (9), the maximum
s t r e s s e s a re found t o occur a t t h e ins ide surface a t 8 = v/6, o r a t
t h e middle of t h e corner.
Maximum s t r e s s occurs a t
Q = V/6
Y = - - ( ~ t the corner, y i s measured pos i t ive outward from 2
t h e cent roidal axis . )
with A = t
therefore ,
Using a f i r s t approximation f o r Z (Z =(1/12t/R) , the maximum
s t r e s s equation can be wr i t t en i n the following dimensionless form
Equation(l0) i s derived i n Appendix A.
qua ti on (lo), using the f i r s t approximation f o r Z, i s appl icable
only f o r R/t>2 since the remaining terms i n t h e Z-series become
negl ig ib le . For ~ / t < _ 2 the s e r i e s f o r Z should be used, employing
a s many terms a s a r e necessary f o r the required accuracy. Usually,
t h e f i rs t two o r th ree terms of the s e r i e s a r e a l l t h a t a re needed,
A graph of omax/p versus L/R with R / t a s a parameter has been p lo t t ed
i n Figure 7.
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- 21 -
I t should be noted here t h a t t h e express ion P,/A f o r t h e d i r e c t
s t r e s s e s i s not e n t i r e l y c o r r e c t s ince t h i s r e l a t i o n i s only v a l i d
f o r a s t r a i g h t member. S ince we have a curved member, t h e s t r e s s on
t h e concave s i d e of t h e corner w i l l be somewhat l a r g e r than P/A. A
c o r r e c t i o n f a c t o r (K) f o r t h i s term i s suggested i n Reference 3 on
page 152, and i f t h e c o r r e c t i o n f a c t o r were included, Equation (10)
would be a s fo l lows
The c o r r e c t i o n f a c t o r (K) suggested i n Reference 3 i s a f u n c t i o n
of R / t and inc reases wi th decreasing R / t . The maximum value of K f o r
our range of R / t would be about 1 .7 ( ~ / t = 0.85) . The maximum e r r o r
introduced by l e t t i n g K = 1 (rnax. a t R / t = 0.85 and L/R = 1 f o r our
range) i s about -20 percent . That f s, f o r K = 1, t h e s t r e s s e s a t
most a r e 20 percent low. However, a s R / t and L/R become l a r g e r , t he
e r r o r goes t o zero very quickly. Since t h e e r r o r introduced by not
us ing t h e suggested c o r r e c t i o n f a c t o r i s of s i g n i f i c a n t magnitude
on ly f o r a small range of R / t and L/R, we have assumed K t o be equal
t o 1. The shaded r eg ion on t h e graph i n Figure 7 i n d i c a t e s t h e a rea
where t h e s t r e s s e s (K 1) a r e 10-20 percent l e s s than the s t r e s s e s
would be i f t h e suggested c o r r e c t i o n f a c t o r were used. The dashed
l i n e i n d i c a t e s t h e approximate 10-percent e r r o r . A l l o t h e r s t r e s s e s
ou t s ide of t h i s r eg ion can be assumed t o have no apprec iable e r r o r .
Def lec t ions
The maximum d e f l e c t i o n of t h e hexagonal tube w i l l be a t t h e
middle of t h e f l a t s i d e o f t h e tube o r a t x = 0 on our model. We s t a r t
with t h e genera l equat ion f o r c y l i n d r i c a l bending of t h e tube wa l l
which i s a f l a t p l a t e . The gene ra l equat ion i s
11 M = D dy/dx = D y (11)
Rewriting Equat ion(l l ) and inc luding t h e moment equat ion f o r t h e
s i d e g ives
I n t e g r a t i o n of Equation(12) t o g e t t h e s lope y i e l d s
Using t h e boundary condition--at x = 0, t he s lope y s equals zero, we
g e t C1 = 0.
Now, i n t e g r a t i n g t o g e t t h e d e f l e c t i o n , we g e t
To eva lua te C2 t h e d e f l e c t i o n of t h e f l a t s i d e a t x = L must be
equated t o t h e d e f l e c t i o n of t h e curved p o r t i o n a t 8 = 0 .
To o b t a i n t h e corner d e f l e c t i o n s t r a i n , energy methods and
Cas t ig l i ano* theorem w i l l be used. The t o t a l s t r a i n energy equat ion
f o r a curved p l a t e i s
2 2 11-Q)M ~2 ( ~ - V ~ ) M N dv2
=l[Z,eR += - AER + -1 ds
(Ref. 4, p a 84)
The (I-$) term i s included t o account f o r t h e p l a t e e f f e c t . For a
2 - r i n g a n a l y s i s (1-3 ) - 1.
To o b t a i n de f l ec t ions , C a s t i g l i a n o s s theorem w i l l be used: t h i s
g ives the d e f l e c t i o n due t o t h e load F. The d e f l e c t i o n s due t o each
of t h e loads shown i n Figure 4 w i l l be determined and added t o o b t a i n
t h e t o t a l d e f l e c t i o n of t h e corner .
The pressure load on t h e curved p o r t i o n causes no r e l a t i v e de-
f l e c t i o n of t h e end of t h e corner . Therefore, s ince t h e corner has
been assumed f ixed a t t h e middle, t h e p res su re on t h e curved member
causes no d e f l e c t i o n a t t h e end. The d e r i v a t i o n s of a l l fol lowing
equat ions can be found i n Appendix A .
Deflec t ion due t o Py:
For t h e load Py we g e t
Mpy = PyR s i n Q
N ~ Y = Py Sin P = pL
Y
Since P l i e s along t h e l i n e of des i red d e f l e c t i o n , no dummy load i s Y
required. S u b s t i t u t i o n i n t o Equation (14) y i e l d s
Deflect ion due t o Px:
To ob ta in t h e d e f l e c t i o n i n t h e des i r ed d i r e c t i o n f o r t h i s load,
a dummy load Q i s needed a t Q = 0 i n t h e Py d i r e c t i o n .
For Px, inc luding t h e d m y load , we g e t
Mpx = -PxR(l-cos €4) + QR s i n €3
Npx = Px cos 8 + Q s i n 8 P, = pL \17
vpx = -P, s i n 8 + Q cos Q
S u b s t i t u t i o n i n t o Equation (14) y i e l d s
Def l ec t ion due t o moment Q . 0
Again, a dummy load F i s needed t o g e t t h e d e f l e c t i o n i n t h e
des i r ed d i r e c t i o n .
For t h e moment & , inc luding t h e dummy load , we g e t 0
M = Meo + FR s i n 'J
N = F s i n Q
V = F c o s Q
S u b s t i t u t i o n i n t o ~ ~ u a t i o n ( l 4 ) ~ i e l d s
Equations (151, (16), and (17) a r e app l i cab le t o a hexagonal r i n g pro-
vided t h e c o r r e c t l oads Fy, Px and % a r e used i n t h e equat ions . 0
To g e t t h e t o t a l d e f l e c t i o n of t h e corner (A) , Equations (151, (16),
and (17) a r e added. This y i e l d s t h e fol lowing equat ion f o r t h e t o t a l
corner d e f l e c t i o n .
where R
(Ref. A, p o 70) t
F i n a l l y , we must go back t o ~ ~ u a t i o n ( l 3 ) and equate the d e f l e c t i o n s
o f t h e corner and t h e wa l l a t x = L t o o b t a i n t h e i n t e g r a t i o n con-
s tant ,C2. Applying t h e boundary condi t ion we can say
y = A a t x = L
This gives the following equation f o r C2
Now, the t o t a l d e f l e c t i o n equation f o r the tube wall can be
wr i t t en a s
Since the maximun def lec t ion i s known t o occur a t x = 0, the
maximum def lec t ion equation i s
Equation(20) i s applicable t o e f t h e r a hexagonal r i n g o r t o a hexa-
gonal tube. FOP the r ing , the f l e x u r a l r i g i d i t y D ( f o r p la tes ) i s
replaced by EI, The maximum def lec t ion equation f o r the tube wall (per
u n i t length of tube) can be w r i t t e n i n a dimensionless form s imi la r t o
the dimensionless s t r e s s equation. To obta in t h i s equation, l e t
A = t
3
With the proper subs t i tu t ions , Equation (20) y i e l d s the following
dimensionless equation
where ~ / e = Same a s before, and UE/G = 2 (I+ v) . This gives the maximum def lec t ion i n the funct ional form
Equation (21), der ived i n Appendix A, i s app l i cab le t o any s i z e
hexagonal tube made of any l i n e a r l y e l a s t i c , i s o t r o p i c and homogeneous
ma te r i a l .
The t o t a l maximum tube d e f l e c t i o n ( A T ) ac ross t h e f l a t s i s o f
prime i n t e r e s t and i s merely twice t h e d e f l e c t i o n of one wal l ( E ~ . 21 ) ,
To o b t a i n a genera l and use fu l graph f o r t o t a l maximum tube
d e f l e c t i o n ( s i m i l a r t o Figure 7 f o r t he s t r e s s e s ) , we have assumed a
value of 0.3 f o r PoissonP s r a t i o (v ) . The r e s u l t i n g graph (Figure 8
on page 27) g ives A ~ E / ~ L versus L/% with ~ / t a s a parameter.
GRAPHICAL RESULTS FOR HEXAGONAL TUBE UNDER INTERNAL PRESSURE
Figure 6: M,/~L' Versus L/it f o r Hex Tube
Figure 9 ! Maximum Corner S t r e s s i n Hex Tube
Figure 8: Maximum Hex Tube Deflect ion Across F l a t s
DISCUSSION
The gene ra l equat ions derived i n t h i s r e p o r t a r e app l i cab le t o
e i t h e r a hexagonal tube o r a hexagonal r i n g submitted t o a cons tant ,
uniform i n t e r n a l pressure load. However, t h e equat ion f o r t h e corner
bending s t r e s s ( ~ q . 79) w i l l on ly apply t o t h e tube and t o a hexagonal
r i n g with a r ec t angu la r c ross sec t ion . A s o l u t i o n f o r corner s t r e s s e s i n
r i n g s with o t h e r c r o s s s e c t i o n s can be found i n Reference 3, Also, s ince
ou r primary i n t e r e s t i s i n hexagonal tubes , t he maximum s t r e s s and de-
f l e c t i o n equat ions and corresponding graphs apply only t o hexagonal tubes .
Equations t h a t apply t o hexagonal r i n g s a r e noted throughout t h e a n a l y s i s .
FIGURE 8. Maximum I ~ C X ; I ~ O I I ~ ~ T u h ~ f l ~ c t ion Across F l a t s
2 For t h e r i n g app l i ca t ion , the (1-v ) term i s equal t o one, A i s the
c ross-sec t ional a rea of t h e r i n g , and EI r ep laces t h e f l e x u r a l r i g i d i t y
f o r p l a t e s , D, Expected Accuracy
Varying accuracy can be expected f o r t h e d i f f e r e n t graphs and
t h e i r corresponding equat ions . These w i l l be d iscussed sepa ra t e ly .
2 The graph i n Figure 6 , bIo/pL versus (L/R), a s explained i n
Appendix B, i s very accura te and no more than 1 o r 2 percent e r r o r
can be expected.
The accuracy of t h e maximum s t r e s s e s p l o t t e d i n Figure 7 v a r i e s
i n d i f f e r e n t reg ions of t h e graph. A s was explained be fo re , Refer-
ence 3 claims t h a t t he e r r o r i n t h e shaded r eg ion of t h e graph
exceeds 10 percent--with t h e maximum e r r o r approximately equa l t o
20 percent . However, ou t s ide of t h e shaded region , i n Figure 7,
t h e accuracy should be w i t h i n 5 percent , with t h e r e s u l t s f o r t he
l a r g e r R / t and L/R r a t i o s being t h e most accura te . O u r s t r e s s re-
s a l t s a l s o compare very we l l wi th t h e r e s u l t s obtained i n Reference 2.
The accuracy of t he maximum d e f l e c t i o n equat ions and t h e cor-
responding graph i n Figure 8 i s hard t o p i n down. It 3s shown i n
Appendix B t h a t t h e d e f l e c t i o n s i n Figure 8 a r e approximately 5
percent low s i n c e the change i n l e n g t h of t h e tube wal l s due t o t h e
d i r e c t membrane s t r e s s was not included. The main source of e r r o r
i s encountered i n t h e corner s ince any d e f l e c t i o n o r r o t a t i o n of t h e
corner w i l l become magnified a t t h e c e n t e r of t h e wa l l and t h u s can
change t h e t o t a l maxirflum d e f l e c t i o n of t he wa l l by a n apprec iable
amount. Also, s ince it i s impossible t o a c t u a l l y make a p e r f e c t
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- 29 -
corner , t h e maximum tube wa l l d e f l e c t i o n may vary g r e a t l y f o r d i f f e r -
e n t tubes. For a w e l l cons t ruc ted tube wi th t i g h t corner to l e rances ,
it i s est imated t h a t t h e maximum d e f l e c t i o n p l o t i n Figure 8 should
be accura te t o approximately 20 percent o r l e s s .
NOMENCLATURE
A,Ax = Cross-sect ional a rea of a hexagonal r i n g o r of a u n i t length
of t h e hexagonal tube
b = Outside r a d i u s a t corner
D = Flexura l r i g i d i t y of p l a t e
E = Young's Modulus
e = Distance between c e n t r o i d a l and n e u t r a l a x i s a t t h e corner
G = Shear Modulus = ~ / 2 ( 1 +v I = Moment of i n e r t i a of r i n g c ross s e c t i o n
L = One-half t h e l eng th of f l a t s i d e of tube
M = Bending moment
N = Membrane load i n wa l l and corner
P = I n t e r n a l pressure
P = Force on a s e c t i o n of t h e tube wal l o r corner
r , R = Corner r ad ius t o n e u t r a l a x i s , and mean corner r ad ius
Ri = I n s i d e r a d i u s a t t he corner
t = Tube wal l t h i ckness
U = S t r a i n energy
V = Shear f o r c e on tube wall a t corner
0 = Angular coordinate a t t h e corner
Y = Slope of s e c t i o n of tube wall
Y , y , b , A = Deflec t ion
u = S t r e s s
v = Poisson ' s r a t i o
a = Nonuniform shear s t r e s s d i s t r i b u t i o n f a c t o r , a = 1.2 f o r a r ec t angu la r c r o s s sec t ion . )
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ACKNOWLEDGEMENT
The author wishes t o thank M. T . Jakub f o r h i s t echn ica l a s s i s t -
ance i n the formulation of t h i s analys is .
- 32 -
APPENDIX A
DERIVATION OF EQUATIONS
M- Equation U
For t h e wall
M = g - M i n t e g r a t e from x = o , L 2 0 '
For t h e corner
M - - M~ + ~ L R ( s i n o + JS cos o - 6 ) , i n t e g r a t e from - 2
71 0 = o , g , with dx = RdO.
a M - - For both moment equations - - 1 a M 0
Boundary Condition -- Slope = 0 a t point of app l i ca t ion of Mo.
6 cos 0 - 611 RdO
In tegra t ion y ie lds
Solving f o r Mo
Corner S t r e s s
The equation f o r pure bending s t r e s s i n a curved p l a t e i s
For a rec tangular cross section,which i s what we have
and
e = R - r .
I f we l e t
Then we have
It i s more convenient t o measure from t h e cen t ro ida l a x i s when loca t ing
f i b e r s , so t h e d is tance from t h i s a x i s w i l l be c a l l e d y and w i l l be meas-
ured (+) outward.
Therefore
From t h i s a more s p e c i f i c s t r e s s r e l a t i o n can be obtained. Subs t i tu t ing
i n t o Equation ( 7 ) we ge t
Where
M = M0 - - - Mo + p L ~ ( s i n O + 6 Cos 0 - 6 ) 2
Maximum Corner S t r e s s
The maximum corner stress occurs a t
A = t ( f o r t u b e )
Therefore, s u b s t i t u t i n g i n t o t h e genera l Equation ( 9 ) we g e t
p~ 6 cos 2 + p~ s i n + p~~ " t rnax
0 max
0 max 2L + Ri - =
P t
Since
- Ri
- R - - we f i n a l l y g e t 2
To obta in Equation ( l o ) , we s u b s t i t u t e t h e first approximation f o r Z
i n t o t h e above equation.
Corner Deflection
The der ivat ion of t h e de f lec t ion equations f o r t h e loads P y , Px, and
MO follows.
0
Py Deflection
Loads f o r t o t a l s t r a i n energy, Equation (14)
v~ = P cos 0 Y
Y
Subst i tu t ing i n t o energy equation we ge t
- v2) p2 R~ s i n 2 0 p2 s i n 2 O (P R s i n 0 ) (P s i n 0 ) Y + Y - =I" [" 2 AEeR 2 AE
AER
i
I n t h e curved por t ion ds = Rd0. Since t h e corner i s f ixed a t t h e middle
ll we in tegra te from 0 = 0 t o O = 5.
In tegra t ing and rearranging terms we get
Applying Cas t ig l i ano ' s Theorem we ge t t h e de f lec t ion
Px Deflect ion
To obta in t h e de f lec t ion i n t h e y d i r e c t i o n a dummy load Q must be
applied a t O = 0 i n t h e P d i rec t ion . Y
The required loads become
$ = - PxR(l - cos 0) + QR s i n O X
N~ = P cos O + Q s i n O X
X
v~ = - P s i n O + Q cos O X
X
- - aM - R s i n o a Q
_ - :: - s i n o
- = av cos O aa
Where
I n t h i s case , i n order f o r Cast ig l iano 's Theorem t o apply, we must in te -
g r a t e t h e t o t a l energy equation with respect t o t h e dummy load Q. This gives
t h e following equation.
Subst i tu t ing i n t h e proper terms and l e t t i n g Q = %when in tegra t ing you
ge t t h e de f lec t ion equation f o r P . X
MO Deflection 0
Again a dummy load F i n t h e P d i rec t ion i s needed t o obta in t h e Y
desired def lec t ion.
The required loads become
M = Mg + FR s i n O 0
N = F s i n O
V = F cos O
- - aM - R s i n o aF
- - aN - s i n Q a F
a v - - a F
- cos 0
Where
Subst i tu t ing t h e values i n t o t h e t o t a l energy equation, d i f f e r e n t i a t i n g
ll with respect t o F,and in tegra t ing from O = 0 t o Q = c w e ge t
l T
6 - - MO
0
a N a M ' c[" a M N a N M - + N - - + - - - a F aF AE aF AER aF (1 - v 2 )
Which y i e l d s
The t o t a l corner d e f l e c t i o n ( A ) i s t h e sum o f t h e ind iv idua l d e f l e c t i o n s
due t o t h e loads Py , Px, and M, . 0
S u b s t i t u t i n g t h e c o r r e c t va lues f o r P Px, and Me we g e t Y ' 0
Maximum W a l l Def lec t ion
The maximum t u b e wal l d e f l e c t i o n i s
Y max
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Y a E - ~ [ 0 . 0 3 ( ~ ) ( 1 - v2) - 0.292(1 - v2) + 0 . 2 6 2 ( ~ )
max - AE
Per u n i t length of t h e tube we l e t
Subs t i tu t ing these values i n t o t h e above equation we ge t
Y - a E - @! [,.03(!)(1 - v2) - 0.292(1 - v2) + 0.2621T) max ET
Rearranging t h e above equation we ge t
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APPENDIX B
Fixed Corner Assumption
The f l a t s i d e o f t h e tube w i l l undergo a change i n l eng th due t o
t h e d i r e c t membrane s t r e s s e s i n t h e wall , thus al lowing t h e corner t o
move r a d i a l l y outward. P a r t of t h i s change i n l eng th w i l l be taken up
by t h e outward d e f l e c t i o n and cu rva tu re .o f t h e tube wall . If t h e out-
ward d e f l e c t i o n accounts f o r a l l of t h e change i n l eng th of t h e s i d e ,
t h e corner w i l 1 , i n e f fec t , remain f ixed and w i l l not d e f l e c t . A r e l a t i o n
f o r t he change i n l eng th o f t h e f l a t s i d e can be derived a s fol lows:
o x = %/t
where Px = pL 0
o r ox = pL D/t The u n i t s t r a i n of a f l a t p l a t e i s now w r i t t e n f o r t h e wal l (Ref-
erence 6 , P O 5)
Ex = (1-3) ax@
To o b t a i n t h e t o t a l change i n l eng th of t h e f l a t s ide , we mul t ip ly
t h e u n i t s t r a i n t imes t h e length of t h e f l a t s ide .
o r upon s u b s t i t u t i o n
a L = ~ ~ s ~ . ( I - - ? ) ( P L ~ / E ~ )
For corner movement, t h e component o f A L i n t h e d i r e c t i o n o f de-
f l e c t i o n of t he ad jacen t tube wa l l i s of concern and i s
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For t h e e f f e c t on t h e t o t a l tube d e f l e c t i o n ( ac ross t h e f l a t s ) , a s
p l o t t e d i n Figure 8, we must mul t ip ly Equation (B-2) by 2 t o g e t
Neglecting any negat ive e f f e c t due t o t h e curva ture of t h e wa l l s
(which d e f i n i t e l y e x i s t s ) , Equation (B-3) w i l l add t o t h e t o t a l de f l ec -
t i o n of t h e tube (Figure 8 ) .
A sample c a l c u l a t i o n w i l l be made t o compare t h e a d d i t i o n a l def lec-
t i o n given by Equatfon (B-3) with the d e f l e c t i o n i n Figure 8 f o r which
t h e corner was assumed f ixed .
Sample Ca lcu la t ion
A l a r g e , thin-walled tube , wi th small corner r a d i u s and high in-
t e r n a l pressure , w i l l be used s ince t h i s w i l l g ive a r e l a t i v e l y l a r g e
change i n wal l length .
Tube Dimensions, In. Loading Conditions & Mater ia l Prop.
L = 2.5 p = 100 p s i
t = 0.25
R = 0.25 v = 0.3, ( 1 4 ) = 0.91
6 E = 20 x 1 0 p s i
L/R = 10,R/t = 1
From Figure 8 we can g e t t h e t o t a l tube d e f l e c t i o n wfth the corner
assumed f ixed
s A TE - - *I, - 1050, o r
AT = 13.1 x = 0.131 in.
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Using equation (B-3) we can ca lcu la te the def lec t ion due the change
i n length of the walls.
= 0.000683 in.
The t o t a l de f lec t i cn including ALy i s then
= 0.0131 + 0.000683
dT = 0.01378
The di f ference between the value i n Figure 8 and the value j u s t cal-
culated i s
0.00068? % Difference = 0.01378 x 100 = 4.95%
If the e f f e c t of the curvature were included, t h i s d i f ference would
be smaller yet .
Since the e r ro r introduced by assuming the corner t o be f ixed i s
small, the r e s u l t s neglect the membrane expansion,as it i s within the
accuracy of de f lec t ion r e s u l t s (see is cuss ion).
Corner Radial S t resses
I n the previous so lu t ion f o r f i b e r s t resses , it was assumed t h a t
plane cross sect ions remained plane and t h a t r a d i a l s t r e s s e s were equal
t o zero. I n r e a l i t y , r a d i a l s t r e s s e s a re present .
The exact solut ion, using s t r e s s functions i n Reference 5, w i l l be
used t o ca lcu la te the r a d i a l s t resses .
The f irst case i s f o r bending moments only.
From Reference 5, p. 63, we g e t
where
M = -PxR (Reference 1, p. 16) 0
r = Any rad ius where s t r e s s i s des i r ed .
The second case i s f o r shear load, P only. Y' D
(3r = ( 2 Ar - 2B t -) s i n 0 (Reference 5, p. 74) P~ r3
where
b and N = R: -b2 t (R: -t b2) l n - R i
The t h i r d case i s f o r a n end load Px and a n end moment M' = PxR.
The moment M' i s r equ i red t o s a t i s f y t h e end condi t ions on t h e curved
beam. Since t h i s a d d i t i o n a l moment i s included here , it was sub t r ac t ed
from Mg t o o b t a i n t h e moment M i n Equation (B-4). When the r a d i a l 0
s t r e s s e s a r e superimposed, we w i l l g e t t h e c o r r e c t t o t a l r a d i a l s t r e s s .
The equat ion f o r t h e r a d i a l s t r e s s e s caused by Px i s der ived i n Refer-
ence 1, i n d e t a i l . Thfs equat ion i s
2B orpi = ( 2 ~ r - - t a) cos 0 r3 r
where t h e cons tan t s a r e
and
The fou r th case i s f o r t h e r a d i a l s t r e s s e s caused by t h e p res su re ( p )
on t h e i n s i d e su r face of t h e corner . 2
- (1 - 5 ) (Reference 5, p. 60) P - b2-Rf
I f we l e t d = b/Ri and m = r /Ri, a l l of t he above r a d i a l s t r e s s equa-
t i o n s can be w r i t t e n a s
where
NM = ~ $ [ ( d ~ - l ) ~ - ld2 ( l n c I )~ ] ,
and Py = pL
A s expla ined i n Reference 1, t h e maximum r a d i a l s t r e s s e s w i l l occur
m 2 - - 6d d2 + 1 + J d 4 + 14dL + 1
f o r Equations (B-8, B-9, B-10).
The maximum pres su re r a d i a l s t r e s s quati ti on B-7) w i l l occur a t
r = R i . A sample c a l c u l a t i o n w i l l now be made t o compare t h e magnitudes
o f t h e r a d i a l s t r e s s e s and t h e memb~ane s t r e s s e s given i n Figure 7.
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Sample Calculat ion
Tube Dimensions
L = 2,5
Load Condf t fons
p = 50 p s i
From Figure 7 , t h e rr~aximwa membrane s t r e s s i s read a s
&ax/p = 275; Ul,lax = >9P950 p s i P
For t h i s partfculap. tube the rna.rfnm ~ a d i a l s t r e s s e s w i l l occur a t
m = 1.72, o r m = 1.311
and the mximm r a d i a l s t r a sses w e w calcula ted a s
crp =-50.2 p s i
o'pY = 755 p s i
or, = 178 p s i
OrPx = -50 p s i
I t i s apparent t h a t the r a d i a l s t r e s s e s a r e q u i t e small i n comparison
with t h e membrane s t r e s s e s . It should be noted t h a t a t the ins ide
surface ( a t r = R ~ ) , t he only r a d i a l s t r e s s i s t h a t caused by pressure,O, P
quati ti on B-7l0 A l l o thar r a d i a l s t r e s s e s a r e equal t o zero.
A t i n s ide surface
Error I n M, Derivat ion
To f i n d t h e e r r o r introduced i n t h e de r iva t ion of Mo, we must f i r s t
reder ive %, using the more exact r e l a t i o n f o r the bending s t r a i n energy
i n a curved p l a t e which i s
We can then compare t h e two equations by using a small R / t t o see
how much e r r o r i s present .
New Derivation: The s t r a i n energy equation f o r a f l a t p l a t e a p p l i e s
t o the wall . The euqation is
U =(iM2/2D) dx
and the slope i s
I n t h e equat ion f o r the bending s t r a i n energy i n a curved p l a t e ,
d s = RdQ and the slope i s
Superimposing t h e s lopes i n Equations (B-11) and (B-12), t h e t o t a l s lope
a t x = 0 due t o Mo becomes
AEe corner
The r e l a t i o n s f o r M and the i n t e g r a t i o n l i m i t s a r e the same a s f o r
t h e previous M, der iva t ion , Now, applying the boundary condi t ion t h a t
t h e slope i s zero a t x = 0 , we g e t
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In tegrat ing, we ge t
For a un i t length of tube, we can say
A = t,
e RZ --
- 1 t z j and
Subst i tu t ing these values i n t o Equation (B-lh), we can obtain the
following dimensionless equation f o r Mo
t s (k)? (t) (Y) + (i$O()(l- %3@] (B-15)
A sample ca lcu la t ion f o r small R / t , using Equation (B-15), w i l l
now be made and the r e s u l t s w i l l be compared with those given i n Figure 6.
Sample Calculat ion
Let R / t = 1, Z = 0.098, L/R = 10, = 0.3
Subs t i tu t ing i n to Equatfon (B-15), w s ge t
This value compares with 0.184 f o r the same tube from the graph i n Figure 6 ,
Since t h i s d i f ference i s so small, we have used the f l a t p l a t e energy
equation f o r the corner i n ca lcula t ing Moo The value chosen f o r ~ / t has
l i t t l e e f f e c t on the r e s u l t s given i n Figure 6,
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REFERENCES
1. L. Goodman. "Structural Analysis of A Square Box Under In te rna l Pressure,It NAA-SR-TDR-7470, Atomics Internat ional , 06-08-62.
2. L. Goodman. "Structural Analysis of A Hexagonal Box Under In te rna l Pressure,I1 NAA-SR-MEMO-8670, Atomics Internat ional , 06-26-63.
3. F. B. Seely and J. 0. Smith. Advanced Mechanics Of Materials, John Wiley & Sons, Inc., New York, 1959.
4. S. Timoshenko. Strength Of Materials, D. Van Nostrand and Co., Princeton, New Jersey, 1941, Par t 11.
5. S. Timoshenko and J. N. Goodier. Theory Of E l a s t i c i t y , McGraw-Hill, New York, 1951.
6. So Timoshenko and S . Woinowsky-Krieger. Theory Of P la tes And Shel ls , McGraw-Hill, New York, 1959.
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