TECHNISCHE HOGESCHOOL DELFT •ÜCHTVAART- EN RlilMTEVAABTTÏCHWtElf
BIBLIOTHEEK KJuyverweg 1 - OFLPT
THE COLLEGE OF AERONAUTICS
CRANFIELD
THE INFLUENCE OF FRAME PITCH AND STIFFNESS
ON THE STRESS DISTRIBUTION IN PRESSURISED
CYLINDERS
by
r
Nate No. 79
F e b r u a r y , 1958.
T H E C O L L E G E O F A E R O N A U T I C S C R A N F I E L D
The I n f l u e n c e of Frame P i t c h and S t i f f n e s s on t h e S t r e s s D i s t r i b u t i o n i n P r e s s u r i s e d C y l i n d e r s . b y -D. S . Houghton, M . S c . ( E n g ) , A . F , R , A e . S . , A.M.I.Meoh.E. SUMMARY An a n a l y s i s i s made of t h e s t r e s s e s o c c u r r i n g i n s t r i n g e r r e i n f o r c e d c y l i n d e r s due -^-,0 t h e r e s t r a i n i n g a c t i o n of t h e f r a m e s , Graphs a r e p r e s e n t e d showing t h e e f f e c t of v a r i a t i o n i n frame p i t c h and s t i f f n e s s , on t h e b e n d i n g moment a r d s h e a r f o r c e i n t h e s k i n s , and t h e hoop s t r e s s i n t h e s k i n s between f r a m e s . The r e s u l t s a r e u s e d t o show how t h e optimum s t r u c t u r a l geometry can be chosen f o r any given s t r e s s r a t i o s .
N O T A T I O N
x,y,z, are co-ordinates: 'x' is measured along the
length of the element, 'y' along the arc and 'z'
perpendicular to the arc.
u,v,w. Displacements in x,y, z, directions
p Pressure differential
R Radius of cylinder
Ar,
Cross sectional area of frame
A Cross sectional area of stringer
s
t Skin thickness
t = _s Equivalent thickness of stringers
^ b
t = "^f
y 7 — Equivalent thickness of frames
^f '
•&- Frame p i t c h
b , S t r i n g e r p i t c h
I Moment of I n e r t i a of s t r i n g e r
Ê Young's Modulus
V Poisson's Ratio
e e , S t r a i n s in d i r e c t i o n s of x and y
X y -^ c c L o n g i t u d i n a l and hoop s t r e s s i n s k i n <T cr S t r i n g e r ajid frame s t r e s s X y ^ G Bending momentN'
'/unit width
Sheax force/ ..
-r-Afu
parallel to 'z' axis
T^ Longitudinal load/^.^ ^.^^^
To Circumferential load/ .. ^ .,
2 /\init length
2
iJR
^ 1 n
2[tjl -y^) + t]
1
-1, Introduction
It is well known that in aircraft pressure cabins, the presence of structural discontinuities such as cutouts or reinforcements, tend to disrupt the simple membrane theory, and substantially effect the local stress distribution,
In this note, the case of a circular cylinder having longitudinal
and transverse stiffeners in the form of stringers and frames is considered, and the effects of varying the frame geometry on the stress distribution are shovvn.
Considering the effect of crack propagation in pressure cabins, the present trend is to relate the critical crack length to the maximiJm hoop stress in the skin, and to design the frame pitch to be less than the critical crack length. No evidence is available to shov; the effect of a non-uniform stress distribution such as experienced in an aircraft pressure cabin, but it would seem reasonable to expect the crack
propagation rate to reduce in the region of the frames, since here the hoop stress will be substantially less than the maximum hoop stress. For this reason results are presented showing the influence of frame
geometry on this hoop stress ratio, in addition to the shear and bending stresses occuring between frames, since only by including these latter effects can a correct analysis of the crack behaviour be carried out.
An assessment is made in which the frame geometry is varied in such a way that the condi';ion of constant frame -weight is maintained. The effect of having light frames of approximately half the frame pitch of present pressurised aircrafu;, is to substantially reduce the maximum hoop stress ±n the skin, giving a critical crack length of two or three times the frame pitch. The influence of frame geometry on the frrjne stress is considered.
2
-2. The Stress Distribution in a Pressurised Circular Cylinder. Considering the d.ement shown in Fig. 1, the following equations result from equilibrium conditions
dN dx dG
R
dx and T + N + p = 0, = 0, 2* (T is assumed independent of 'x'), EI G = , 2 d w dx E x p r e s s i n g T and Tp i n t e r m s of s t r a i n components, TOiere and As t h e n du dx ' w R . ( t a n g e n t i a l d i s p l a c e m e n t i s z e r o bjr syiimetry). T - ^ t ^ U - i^^) + t ( 1 . 0 1 ) ( 1 . 0 2 ) ( 1 . 0 3 )3
and Tp can be expressed as :
-T^ = E fiyf + pR (1 - ^ ) , (1.04)
, ., t
v^
t^
where t' = - ,
-I- y 2 ^ (^ „j^2)2 ^ t^^ ^j^2) Xand
(p
= 1 - 1^ t
2 [ t ^ d - y^)+ t ] •
From equation (l,Ol), putting
k-ii
= — 5 - ,
IR
^ + A- A = H ^ . (1.05)
dx
The s o l u t i o n i s
w = C, c o s ti X cosh ^ x + Cp s i n jux s i n h jux + C, c o s iJx s i n h /ix + C, s i n lix cosh )ux + w ,
I ^
°0.06)
where w = -^— M
o *
Et'
which reduces to
v/ = C. cos
lix
cosh
lux
+ Co sin
^x
sinh /JX + w ,
° ( 1 . 0 7 )
( s i n c e w i s symmetric i n x ) ,
and -r- = Ai(Co - C.) s i n l^x c o s h /Jx + H (C. + C„)cos Mx s i n h lix,
^ '^ "^
1 «^ (>,.08)
Using the condition of zero slope at the frames
X - - f
2
^> II. *^ and p u t t i n g f _ Q 2 - ' xy o " t a n 6 - t a n h 6 / o n \
*^^^ °2 = - t i ^ e ^ tanh e S = ' ' ^ v ^'-^^^
At t h e frame, from c o n d i t i o n s of e q u i l i b r i u m , o r ^2 As d w E I b a3O
X d/w d ^ •f ^ f 2 ^ ^ f 2R2 = w A„b f 2 2IR fCw)
X=• t
ƒ. • f 2 ( 1 . 1 0 ) dx' 2= 2 /i^ (Cp cos /i X c o s h jux - C s i n /^x s i n h /jx),
and —w _ 2 / i ' j (Cp-C ) c o s i^x s i n h ;jx - (C + C p ) s i n ^x cosh /Jx [ , dx '' ' t h e n u s i n g t h e c o n d i t i o n s i n ( l . l O ) g i v e s 2 M^!(Cp-C ) c o s e s i n h e -(C + C ) s i n 6 c o s h e ] A^b p ) u^ c o s o cosn o + u^ s m o s m n o + w^ | , 2IR I C c o s e cosh 6 + Cp s i n 6 s i n h 6 + w I and s u b s t i t u t i o n of e q u a t i o n ('1.09) g i v e s - w 0 . = jt — - f t c o s e c o s h 6 + p s i n 6 s i n h 6 + —r- f ( l + p ) s i n ö c o s h 6 + ( l - p ) c o s 6 s i n h el
By substituting
2 tan e „
tan e -, tanh 0 ^°'' ^ + ^ '
and 2 tanh 6 ^ ^
for 1 - p ,
tan 0 + tanh 6
5
-the equations for 0 and Gp can be more conveniently expressed as
o w (sin 6 cosh 6 + cos 6 sinh 6 ) 1
^ (sin2 6 + sinh2 ö) + - ^ (sin 0 + sinh 6 )
Or» c^ = - w^ y^ , (1.11)
and Cg = •* w^ y 2 » (1.12)
, ^ 2 sin 0 cosh 0 - cos 0 sinh 0
vmere = = p , V
' 1 sin 0 cosh 0 + cos 0 sinh 0
Finally the bending moment, shear force and hoop force equations can be expressed as : -(1) The Bending M o m e n t y ^ . ^ ^ . ^ ^ ^ : 2 EI d w EI / \ G = T—- —K = 2 /i -r— (Go c o s n x cosh a x - C s i n /j x s i n h /i x ) , '^ dx"^ * "^^ ^
"é = G' = y s i n /ix s i n h Hx - y p c o s jJx cosh A'x. ( 1 . 1 3 )
(2) The Shear F o r c e / ., . , , , ; ^ ' / u n i t v/idth
N ^ - S 1 ^ = - 2 / i ' | i r (Co - C.) cos ^x s i n h ^x -(C +C ) s i n /ix
or '^ / = N' = ( Yo ~ y^)°°^ ^ ^ ^^"* A' X - ( y. + y 2 ) s i n /i x
9^ P cosh /i X ,
I L v x i I. •.i.j\^'. li^ r i'_^V/l...»'••-•! >'_• ^ ' 1 . VÜEGTU:GBOüWiCU^^iDC _ 6 - Kanaablraat 10 - DL-LFT
(5) The Hoop F o r c e ^ ^ . , ^ •^^^^^:
As ^ 2 = ^ + p R ( l ~ 0 ) ,
T
then _2 ^ -i - 0 ( y^ cos /i x cosh /i x + yg s i n ^x sinh / i x ) ,
T
7 ~
3. Discussion of Results.
Using equations 1.13, 1.14 and 1.15, F i g . 3 shows the values of the bending moment, shear force and hoop force parameters G' N ' and T' occ\jrring at the frame positions for a practical range of frame pitch and stiffness. F i g s . 4-, 5 and 6 show the same parameters at positions between frames. In this case, the frame stiffness parameter 2t«
jjA^ h a s been vaiied between 0,5 and 20 and the frame pitch parameter between ,60 and 3.0. A s a n aid to simplifying this work F i g . 2 w a s constructed to show the variation of the parameter <p with the skin thickness (t) and the ratio of stringer area t o stringer pitch (t ) , It w a s found that for many aircraft pressure cabins having typical structural geometry, a constant value of ^ = ,90 could be used with reasonable accuracy.
Using this appro:dmate value for 0, F i g , ( 7a) shows the ratio of maximum hoop stress in the skin to the nominal hoop stress (as
predicted b y simple 'boiler t h e o r y ' ) , against t h e frame pitch parameter
^ ^ f 2t'
—T— and the frame stiffness paxameter -rr~' T h e maximum hoop stress ii'C
is seen to equal the nominal hoop stress at a value of f __ n -zfi
2t' ^ Ii6
for a l l values of —— , This means that at this unique value of f ,
^^f 2 the frame size ceases to influence the maximum hoop stress in the skins.
The rea.son for this is seen b y examination of equations 1,15 a n d 1,11. Since the maximum hoop stress occurs midway between frames
J 2 (when X = o), then equation 1.15 reduces to -r (l p ) = y ..
— — (that is _2 ) approaches 1,0, then y approaches zero giving
2' pR 2 f a solution to equation (l.ll) which is independent of the parameter r-r—
of f
— p — = 2,3é. In a similar manner Fig. 7b shows the hoop stress in the skin at the position of the frames.
- 8 «
Fig. (8) shows the ratio of the direct stress in the frames cr
• pR' ^
to the nominal stress. The parameter cr is given as -'T— where R'
y
^
is the mean radius of the frame. For relatively shallow frames this
can be approximated by the nominal hoop stress in the skin •t^,
The ratio of maximum hoop stress in the skin to the frame direct
stress is given in Fig. 9(a) for various geometries. In Fig. 9('b)
2
this ratio is plotted against che frame pitch parameter .- f and tho
parameter T — . If the stringer area remains constant, which is reasonable
since the stringer geometry is determined by considerations other than
pressure, then for any given skin thickness t, the lines of constant
•T— are lines of constant weight.
•^ The use of these results can best be illustrated by examples.
CASE 1.
(a) Skin thiokness
Diameter
Frame pitch
Frame area
Stringer pitch
Stringer area
t
2R b^n
t '
t
/i 22 t '
t'
t • = rr = = =: = =: = = =!,048 in.
10 f t .
27 i n ..345
in?
6.25 i n .
.129 in?
1,029
,0439 i n ^
.1485
2 . 01.925
3.875
« q
-From F i g . 7a -From P i g . 7b -From F i g . 8 -From F i g . 9
l l _ 953 .^2(frame) _ i , , ""2 . -..^ ^ - '^^^ £R = . 6 5 1 &^= - 5 ^ — = = 1 . 7 6 5
+• •<- •/ »y
(^)
Reducing the frame pitch to 13.5 in and the frame area
2 t' to .1725 it- , that is keeping the ratio -r— constant gives:
CASE 2 . ( a )
The geometry i s i d e n t i c a l w?.th c a s e 1 except t h a t t h e frame p i t c h i s 20 i n s . Then Hi - ^ = 1.48 9 + ' ^ = 2 , 8 6 -2. A^o 2(frame) _ . _ ! z . , "^2 , „ ^
^ = - ^
I R= '70
;5:^
= .55
-5:-
=
1.530
t t y y10
-(b) Reducing the p i t c h t o 10" and the frame area t o .1725 in
again keeping the r a t i o r— constant gives :
-2
^ ^ ( f r a m e ) ^ ^
^ = .74 ^ = c73 cr' = ,575 cr = 1.285
t t ^ ^
Case 1 shows that by reducing the frame pitch from 27 in. to 13.5 in., at the same time keeping the structure weight constant gives a substantial reduction in maximum hoop stress in the skin, but increases the frame stress and the hoop stress in the skin at the frame positions. The ratio of maximum hoop stress in the skin to frame stress reduces. Case 2 shows the same trend.
The effect of using the frames as a crack stopper.
Using the results of oases 1 and 2, and beaxing in mind the recent work on crack propagation, one wonders Tiriiich is the best structural configuration in the design of aircraft pressiire cabins.
Having widely spaced frames means high maximum hoop stress, but the stresses in the region of the frames are relatively low, and hence such frames become effective crack stoppers, providing of course that the frame pitch is less than the critical crack length.
If the structure weight is kept constant and the frame pitch is decreased, a reduction of 15^ and 12^ occxirs in the maximum hoop stress for cases 1 and 2 respectively.
This reduction in maxim'jm hoop stress means a substantial increase in the critical crack length. Case 2b shows xhat by reducing the
frame pitch from 20 ins to 10 ins, only gives a slight increase in the stresses at the frames. This means that although such frames are less effective as crack stoppers, the orack woxold have to penetrate probably two frames before the critical crack length is reached.
11
-4 . Conclusions.
Results are presented giving the stress ratios in a reinforced
circxiLar cylinder. These results show the effect of structural variation on these stress ratios.
Some advantage seems to be gained by usiiig relatively closely spaced frames in pressure cabin, design, since such frames serve to substantially reduce the maximum hoop stress in the skins, -«tiich could be used either to reduce the skin thickness or to increase the critical crack length.
5. Acknowledgement;
The author wishes t o thank llür. Ingam of the Mathematics Department College of Aeronautics for a s s i s t i n g with the d e t a i l c a l c u l a t i o n s
of t h i s n o t e , and Mr, A.S.L. Chan who a s s i s t e d with some of the p r e l i m i n a r y c a l c u l a t i o n s .
(T,+ |^dx)dy l('N+5Nd,)dy FIG. I. ^ • 2S -SO -75 I Q I . J S hS tx t
I^^'l
2 \ \ \ ~ \ \ \\V
k
\ \V
N
\ ^ ^ ^ 3! '^'•im
^^\ . l 3 =^,1^
lO-O ISO 2 0 0 3 0 0 , ^ H )THE EFFECT OF FRAME PITCH AND FLEXBUTY ON THE HOOP FORCE AT THE FRAME.
H
l
/k
l
l
V
' / ^ , / /ƒ ^
/ / /f
/1
/ ^ • ^ — - '"r
j ^ / /^ ^— '"'''' -~ -~ - - ^ e's 2XG "*» 150 2 0 0 300 ( 4 = FRAME PITCH) | N ' I - 3 / ^ / / /^ / 1 1 ^ ; ^ ^ ^ N'. — ^r
0-5 10 3 04
i
M* i
t >\ l
THE EFFECT OF FRAME PITCH AND FLEXIBILITY ON THE BENDING MOMENT AT THE FRAME
| - « ^ A | ( ^ . F R A M E P I T C H )
THE EFFECT OF FRAME PITCH AND FLEXBUTY
AJAf... G' O O •» « ' l O 0 -X) ' 4 . ^ y '
y
O -S ^ ° < ^ ' O ^ ^ •o ' ^ ^ toÊi^
/ / . ^^ 10 1 f •9G O •o -o 3 0 O G' •10 O 0 • -— o •o ^i).
z'^y
f?-o
so -1f
J
' //>V)
0 / / ^ / / Kth
1
u
/ s*-i
>01-FIG. 4 . THE EFFECT OF FRAME PITCH AND FLEXIBILITY ON THE BENDING MOMENT AT
^ /, ^ / ^ Mi( 2 / ^ / ' '•60 / > / • — / / / / / / '
y
^ /V
/ y y-^ / / /y
/ / / y^ ^ • ••"^i
N' yi
1i
/^ /f/
/ / '/ / / ^ MÜ-io / / / / / / / ^ .J / / / / / ^T
/ / / / / •c
/ / ^ / ^ .—1 _ J 2 t ' jJAfA
/ / / l-C t H > Ni1
i
1 / / Ml 2 / ^ / / , ' ' — • ÖÜ / / / / ^ — / / / / — / / / >y^
t ^ — / / / / ^ / / /k
" / / \ / N ' ^é
/ ' ^ y ^ y M.. / ^ /X
^ ^ o / '/ / / / / / / ^1
/ / / / / / /1
1
f
/ /y
-—k
\1
1
A
/ 1 / ^ • , " e ) • ^ : r • — •> ^ 4Ü..3.0 <• ^'y
/é
' \li
ih
/ '//'y
V
c
/ / / , /1
II
1
M
/ / / 2 A / ^(1
i o S5 l ö l ö-FK. S. THE EFFECT OF FBAME PITCH AND FLEXIBILITY ON THE SHEAR fORCE AT
Hl.-60 2 Wf O l o ao .30 -40 -SO -60 70 - a o - s o i-o
©
— • — V I -1 , r t 2 _ _ . -• -—-JUAf •lO Ï O - 3 0 - 4 0 SO - 6 0 -TO BO - 9 01-Kit
S O )f-eo
2 t ' pSf •lO ? 0 3 0 4 0 -SO • o - 7 0 - a o . « o ( o©
— I o ' 2 ^ ^y,
^ / > ^ Mk.2.o / / X / /y
_ / /y
/ / / ^ ^ .— y / ^ ^ ,-^' -^ — >JAf - ^ •10 2 0 SO ' 4 0 SO 6 0 TO • ÊO -«O tFIG. 6. THE EFFECT OF FRAME PITCH AND FLEXIBILITY ON THE HOOP FORCE AT
!•»
k
r
» 5 . MAXIMUM HOOP KTRCU fH SKM -•J.»o ^ >IMAL HCH i l l ^
Vk</
^ ^ / / » t T R f l S V . ^ ^ - ; ^ / t ^ ' •CTWCIN F P A M »FIG. 7 A . EFFECT OF FRAME PITCH I STIFFNESS ON T H E
MAXIMUM HOOP STRESS IN THE SKIN BETWEEN FRAMES
r
> HCWP S . NOMINAL ^ \ mat IN SI HOOP 5TBt z = = = " KIN AT FRAk SS M ric Posmow i l O O n-% lOO r • -T— (fl'm MCAN RADIUS' * OF FRAMED 'OF FRAME) ^X.
' ^
f
'l
1
/1/
/ 1 ^A
A
/ / / / ;: ^^ ^ ^ " ^ = 5 5 ^ ^iz;^ —~^^^r
-~ ^ 3 0 - o l O O I 9 0 l O - O 5 - 0 3 0 0 - » 1 2 FIG. e . THE EFFECT OF FRAME PITCH ( STIFFNESSON THE FRAME STRESS
FIG. 7 8 . THE EFFECT OF FRAME PITCH AND STIFFNESS ON THE HOOP STRESS IN THE SKIN IN THE REGION OF THE FRAMES
> o 2- 9 \ ^ ^ . -/ / / / / / — / / • ^ - ^ • ^ o » 1 • o 2 - 0 »o-o I I - s FIG. 9A. LINES OF CONSTANT^ STRUCTURE WEIGHT ><!> — "~= 0 ).2S / ^ ^ ) 5 / ^ / / ' /
èf
\ 1- 0 / ^y
/ / > 2 0 i -3 0 4 - 0 10-O 2 0 - 0 I-S 2 FIG. 9B. ^THE RATIO OF MAXIMUM HOOP STRESS IN SKIN TO FRAME STRESS POR VARIOUS STRUCTURAL GEOMETRIES
