An analysis of an unstiffened cylindrical shell subjected to internal pressure and axial loading

Michie'

..GL DELFT

'/KUNDE

^veq 10 - DELFT C o A N o t e N o . 114

10

THE COLLEGE OF AERONAUTICS

CRANFIELD

AN ANALYSIS OF AN UNSTIFFENED CYLINDRICAL

SHELL SUBJECTED TO INTERNAL PRESSURE AND

AXIAL LOADING

by

NOTE N^.114 March, 1961.

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

An Analysis of an Unstiffened Cylindrical Shell Subjected to Internal P r e s s u r e and /ixial Loading

b y

-D. S. Houghton, M.Sc.(Eng.), A . M . I . M e c h . E . , A . F . R . A e . S . and

D, J . Johns, M.Sc.(Eng.), M . I . A e . S .

SUMMARY

General equations are obtained for the deflections and s t r e s s e s in long thin unreinforced cylinders, which are subjected to an axial

load and internal p r e s s u r e . By making suitable simplifying assumptions, results are presented which show the variation of the structural weight parameter with the structural axial loading index, for both pressurised and unpressurised shells. An allowance is m.ade for the effects of shell initial eccentricities on the buckling s t r e s s coefficient K, in accordance with R . A e . S . data sheet 04.01.01.

Extreme cases are considered, in which the shell is assumed to be either fully effective (K = 0.6). or completely ineffective (K = 0), in resisting axial compressive loads. F o r this latter case, complete p r e s s u r e stabilisation of the shell is considered, and it is shown that the weight penalty involved in using this design philosoj^hy, is negligible for a certain range of the structural loading index.

A simple modification to the analysis for this case, i . e . K = 0,

CONTENTS

Summary Notation Introduction Theory

Discussion of the parameter K Results

4 . 1 . The unpressurised shell 4 . 2 . The pressurised shell

The effect of an external longitudinal bending moment

References Figures 1 and 2

NOTATION A , Ag constants in equation (1)

D flexural rigidity E Young's modulus

P axial compressive force

f allowable stress

a

f buckling stress K buckling coefficient

(, head of liquid in vertical shell

M longitudinal bending moment n longitudinal acceleration (in g's) p internal pressure

p stabilising pressure s

R radius of cylindrical shell T^ longitudinal force/in. T circumferential force/in.

2 '

t s h e l l t h i c k n e s s

w r a d i a l d i s p l a c e m e n t

X axial c o - o r d i n a t e along length of shell m e a s u r e d from the l o w e r support point

a weight p a r a m e t e r defined in equation (14) cr d e n s i t y of s h e l l m a t e r i a l

V Poisson's ratio

f

Notation (Continued)

structural index, defined after equation (12) 2M

E E , a n d \ir' = f +

ïT R E

initial shell irregularity Et

hoop s t r e s s f then the condition for f i s a a Ew pRi^ R "^ 2t vF 27rRt (7)

It can b e shown that when x i s s m a l l equations (2) and (3) m a y be simplified, with v e r y l i t t l e e r r o r , to give r e s p e c t i v e l y

and w w nR^ Et nR^ Et V + p^\ ( 1 -<p),. p' + P^ (1 - e ) . (8) (9) Substitution of equations (8) and (9) into (7) y i e l d s

!f-PMl -t)^f

_{' - ! > ]}

for clamped edge conditions, and a s i m i l a r e x p r e s s i o n for the s i m p l y supported edge with 0 r e p l a c e d by 6.

When equation (6) a p p l i e s , i . e . the s h e l l i s just s t a b i l i s e d by i n t e r n a l p r e s s u r e , p = p^ in equation (10) which b e c o m e s , hi nR . ^ F T

^^^ ^Rt

4

T h e s e r e s u l t s a r e for a long s h e l l , but an investigation of the effects of finite length of s h e l l s u g g e s t s that the m o s t c o n s e r v a t i v e f a c t o r i s about 1.09. T h i s f a c t o r will b e applied in the subsequent

a n a l y s i s , and will be a s s u m e d to apply for e i t h e r fully claïnped o r s i m p l y supported e d g e s . Hence i f v = j equation (11) can be w r i t t e n a s

^ = 1.09 | ( r ) - 2 . 1 5 K ( ^ ) . (12) ~np-g-E" w h e r e f index in c o m p r e s s i o n . + _E , and X F TTR , i s the s t r u c t u r a l

4

-3. Discussion of the parameter K

The results which a r e presented in the R . A e . S . Data Sheet 04.01.01 may be used to determine the buckling s t r e s s coefficient K for thin

walled unstiffened circular cylinders under combined axial coinpression and internal p r e s s u r e . It is generally considered that the depth of

initial irregularity of the shell wall (6) plays a dominant part in predicting the buckling s t r e s s of circular cylinders, and this fact is considered in the R . A e . S . Data Sheet 04.01.01, where K is plotted against — for various values of the parameter ^, (•—) . The results

t L t

of a large number of experiments have been correlated in determining these curves, which show the dependence of 6/t on R/t.

F o r the purpose of this paper, comparisons a r e drawn between the r e s u l t s obtained using either 6/t = 0 or 1, i . e . independent of R/t, or in accordance with the data sheet for the more critical uppermost line B . B . This line r e p r e s e n t s the limiting maximum values of 6/t for over 80 per cent of all experimental results which were used in the correlation.

4. Results

4 . 1 , The unpressurised shell

Fig. 1 shows the structural efficiency of the unpressurised shell in compression, compared with that of the p r e s s u r e stabilised shell, for a shell material having the properties E = 28 x 10" Ih/in., o- = .273 lb/in? and allowable tensile s t r e s s f = 170,000 l b / i n ^ The curve for the unpressurised shell was obtained by using equation (6) with p = 0, in which case the structural index becomes

s

X = 2KE (^)^ . (13)

•p

By assuming values for -j- in the range 100 to 3000, corresponding values of 6/t and hence K were found from the R . A e . S . Data Sheet 04.01.01 line BB. The structural index was found from equation (13) and the weight parameter a which is plotted in Fig. 1 is defined by

5

-4 . 2 . The pressurised shell

• By specifying the allowable tensile s t r e s s to be f = 170,000 lb/in? equation (12) is presented in Fig. 1, for the p r e s s u r e stabilised shell, with the hydrostatic effect neglected, i . e .

\ - 1.09 S ( | ) . 2 . i 5 K ( L ) .

The full line shown is in accordance with R.Ae.S. Data Sheet 04.01.01, and the analysis is similar to that indicated above for the unpressurised shell. The dotted line in Fig. 1 shows the effect of assuming K = 0, i . e . the p r e s s u r e to be such that no compressive s t r e s s is set up in the shell, in which case

f = 1 . 0 9 ( S ) ( | ) .

Comparison of the solid and dotted curves for the pressurised shell shows that for values of the weight parameter a < 2 x 10" lb/in! i. e R

' ' -- > 1500 or \ < 150, the assumption that K = 0 gives negligible e r r o r to the weight parameter,

F r o m Fig. 1 it may be concluded that for low values of the

structural index (low loads, large dimensions), the p r e s s u r e stabilised structure has a significant weight advantage. This result was previously noted in Ref. 2 where curves which correspond almost exactly to the solid lines in Fig. 1, were derived using Ref. 3. This reference provided some of the experimental results considered in deriving the R . A e . S . Data Sheet 04.01.01. la Ref. 2, it was also shown that for high values of the structural index, conventional stiffening is optional in t e r m s of structural efficiency.

The above results prompted an investigation into the effect of the p a r a m e t e r 6 and hence K, on the structural efficiency, for various

t

values of f . The results are presented in Fig. 2, which was obtained using equation (12), and values of b_ ^ 0 (K = 0 6) and K = 0 (-^ -* «=). The value K = 0 corresponds to the case when there is no compiessive s t r e s s in the shell, in which case equation (12) becomes

6

-*a R

- | = 1 . 0 9 ^ if) . (15) The results indicate that the difference between the corresponding

curves, for the extreme values K = 0 and K = 0.6, decreases as

— i n c r e a s e s . This also implies that the effect of initial i r r e g u l a r i t i e s R

d e c r e a s e s as — i n c r e a s e s . This point is further emphasised by the curve plotted for — = 1.0 and >/' = 2 x 10

5. The effect of an external longitudinal bending moment

If an external longitudinal bending moment M is applied to the shell together with the axial load F , then it may be shown that, for the case when K = 0, equation (15) is still applicable provided that the parameter ( f) is defined as

n p^ X 2M

f = -r^ + — +

E E - R ' E

6. References

1. Timoshenko, S. Theory of plates and shells. McGraw Hill, 1940.

2. Sandorff, P . E . Structures considerations in design for space boosters.

A . R . S . Journal, vol.30, No. 11, November 1960, pp 999,

3. H a r r i s , L. , and others

The stability of thin walled unstiffened circular cylinders under axial

compression including the effects of internal p r e s s u r e .

Journal of the Aeronautical Sciences, vol.24, 1959, pp 587-596.

lO M . - 4 lO r S • ^ 1 — _{r r} UNPRESSURISED SHELLS ^

A

y

y

9

y

y

k-y

FIG. 2. OPTIMUM SHELL ^/t RATIOS FOR GIVEN LOADING 0 / AND ALLOWABLE HOOP STRESS 9a.