Literature survey - Hydrostatic press ure

3.4 Hydrostatic press ure

3.4.1 Literature survey

In 1952, Hol t [l1J developed a procedure for computing the allowable ollt-of- roundness of a circular cylindrical shell s ubjected to external pressure. In this paper it was reported that the equation for comput ing the strength of an out-of- round cylinder is

(3.7)

where

i s the numb er of lobes in the initial out-of-roundness pattern , which can be seen in figure 3-1. He lt also developed an equation for the maximum ampl itude which is applicable to cond ition s with elevated temperatures but, this will not be explored in this study.

Along with Holt's equation , another simpler equation was proposed for the reduced buckling load in [261, where the reduced buckling load is found to be

and is shown in figure 3-16.

Su!

qcr

= -~.-.':"c-,="

1 +

^1.5u(I^-^O^.4j)

sol •

(3.8)

Galletly and Bart [13J derived an equation for the reduction in buckling pressure for the case of external hydrostat ic pressure in 1956. It was assumed that the initial imperfection in the cylinders were similar to one of the modes of buckling.

But, actual shells never satisfy this condition. The equation they derived which expressed the initial pressure was

~_(1_q~)(-3+J9 4(1-13 +132) H)

t

^qd

4(1 - 13 + f3')K ^(3.9)

where

900 800 700

v.

- -

.

. / .

600 Q 500 400

X ..

t-- .l(' . _. _...:; ^. . .

/. V ^. ^.

/ '

- - -

300 200 100

o

SOO ¹⁰⁰⁰ ¹⁵⁰⁰ 2000 2500

• • u~9J!!!-

. _~

. --- .

. .

_U=l%

.

u",2"/,

3000 3500 4000

Figure 3-16: Plot showing the re l ationships in Holt's equation .

H 3 _ (2t5

^y ^{) '}

Tqcr

fJ

K

Yield strength of the material.

In the derivation of this

expressIOn,

it was also

aSSllITlf'd

that. the initial imperfection is small (wit.h an ordN of magnitude of the shell wall thickrlf'ss) , wh ich is limited in the context of t his st1ldy.

When Donnell [14 ] initially explo red the effects of impcrf('ctions of cylinders

s1lbjected to external hydrostatic pressure in 1956, he assumed the same

unevenness factor as in his pn"violls collaborated st udy [10] of cyli nders s u bjected

to axial compress ion in 1950 with Wan, outlined in chapter (3.2) on page 67. But this stu dy was inconclusive in respect of the fact that

reduction formula

was

not found.

Lunchick and Short [l5J focllssed their research on the buckling of imperfect shells with an 'accordion' initial deflection in 1957. This nature of imperfection occurs in stiffened cylinders and is caused by shrinkage of frame weld-joints. The imperfection equation t hey used was

Wo = ~e(x ^_ ~2) ^(3.10)

This expression for the initial imperfect ion does not hold true for the present study, as the im perfection is considered to be axisymmetric.

Abdelmoula et al [51} studied the buckling of imperfect cy li nders due to external hydrostatic loading in great detail in 1992. Di strihu ted imperfections, localised imperfections and combinations thereof where considered a nd reduction formulae for each case obtained. They even considered the separate cases of a local ised imperfection and distributed imperfections being in phase and out of phase. This study

was

limited to applications where the Batdorf parameter {Z)

was

large

(Z > 500). The reduction formula they calculated for the case of distributed

imperfections is

q~ ⁼ ^{1 _} ^4.7511 ^?'1 ^{- ",} V(A)'

qcl ifZ t (3.11)

and is shown in fi gure 3-17. Thus the reduction formulae for both axial load ing and hydrostatic pressure are proportional to (1)2/3 ^using the same method.

Yamada and Croll analysed imperfection sensitivity in cylindrical panels in 1989 [46J and cylinders in 1993 [54J. In bot h cases the equations of equilibrium were formulated using the principle of total potential energy. The imperfection distribution was represented by the harmonic exp ressio n

ny .

7rX

Wo = Acos-slD-

I (3.12)

0.8

0.6

~ q.

0 .•

0.2

<".~

0 1 2 3 4 5 6 7

-A t

Figure 3-17: Graph showing Abdelmoula et al [511 relation for the case of distributed imperfections.

where the origin of the co-ord inate system was taken to be on the surface of the cylinder at mid- length. The 1989 study concluded that the complexity of the buckling behaviour displayed by this shap e of shell was show n by the use of numerical experiments using various levels of geometric imperfection.

Both studies noted that classical buckling analysis provided an unreliable upper bound to buckling pressures due to imp erfections and that the reduced stiffness analysis provided a close lower bound to cases where imperfection sensitive loads are applied. Although no co nclusive relation between the reduced buckling load , cylinder parameters and imperfect ion parameters was acquired in either study, these papers aided to enhance the understanding of buck ling of cylind ers subjected to imperfection sensitive loading case of external pressure.

eroll [601 later reviewed the developments of the reduced stiffness method for

both ax ial load ing and external pressure, from its early developments onward

in 1995. By comparing numerical experiments and experiments performed by

others, Croll concluded that this method is shown to provide a safe prediction of

buckling loads, without the need of mathematically examining the imperfection

1000 900 900 700

eoo

"'"

- -

'b

³⁰⁰

I

_~ ²⁰⁰

j

J

¹⁰⁰^OD¹¹⁰

..

.. ...

2'

itself.

T he ASME Boiler and Pressure Vessel Code UG-80(b)(l) 138] requires that the

difference between the minimum and maximum diameters does not exceed 1% of the nominal diameter. Also, in code UG-80(b)(2), in order to detect fiat spots

and bu lging caused by fabrication, a second requirement is that the maximum deviation from a true ci r cle does not exceed that given in fig ure 3-18 when

- ^...

r"::::::: ^...

~ i'- :-.. - .

-.. ^r- r--. I-... ^r--- ^~-9,' ~ I-....

r- _r---. ^~

~ ^... '"

r--. _r-. _~'" ~ ^r-...

r- _r-... _~

^Q^s,

_:-.. ^['-... _I-.... ^i'- ^r-.

-' 0

~ ... :-..

... I'-..

-'0

...

~ ^...

r-...

0.10 0.2 0.3 0.4 0.5 0.6 0.8 1.0 2 3 .. 5 6 7 8 9 10

o..igll Length + Outside DI.met:er. LIDo

Figure 3-18: Figure obtained from 1980 ASME code [38J.

measured over twice the arc lengt h given in figure 3-19.

Jawad and Farr [47] identified t hat the curves in figure 3-18 are based on the

-

",0

!

•

~ _~

• ~ •

• :§ =

<3

2000

1000 BOO BOO 500 400 300

200

lOO 80

60 50 40 30

0.01 0.02

formula

I~ o'~o. I J

. {

_{... CI}^~

o~~

_#^Q^' _#⁰

"

,.;

,..

^~

'If-c

/ / 15>

. / 09

_, _{#' -}

Q^O

'.1 _o _' , ,

,..~ ^()' ~~ ^<)0

/ / / ^I

. /

'" ^",^#

~

^J^oCJ^,

^t

⁰

^$.

I~"Q'I

^'l'

I J ^1/

/' V ^.t ~,,,. ~~ ^Q ^' J ^V V ^V

. / ' / .. I ^/ ^..f : .A

fC)'::-1IJ

.f Q '

o J iI

~

~^..^{c. _ •}

.

•

... c.

^--~

,0^~'

&'

^QO

. / _ ,V" - ..

^,

o'"!

.. ~ I

./ 7{

~ 1/ /

L.,. 1/

0.04 0.06 0.10 0.2 0.4 0.6 1.0 2 3 4 5 6 8 10

Figu re :1-19: Figu re obtained from 1980 ASME code [ 38] .

e 0.018

- = ( t ) + 0. 01 5n .

t n -

_2,

^(3. ¹³⁾

This formula is based on an analogy between a cy lind rical press ure vessel and a col umn by conside ring t he cy li nder to be made up of a seri es of columns. Harvey [ 44] then went on to show how thi s formula was deri ved , and showed that the equatio n f or intermed iate lengt h cy linders was

constant -= nU,) .

e (3. 14)

t

It

was

noted that it was found by ex periment th at a certain ecce ntricity has less effect

the collapse pressure of a long cylind er than on a s horter one, of the sam e

thickness and diameter. This means that the allowable out-of-roundness decreases with decreasing length until a certain limit is reached and the eft curve flattens out and continues parallel to the i/ Do axis, where here Do is the nominal diameter.

Likewise, the allowable out-of-roundness increases with increasing length until a critical length is reached and the eft c urve flattens and continues parallel to the

i/ Do axis. Therefore, the basi c equation 3.14 is modified to reflect this behaviour

e a

- =

_{n -} (,)+bn

(3. 15) The values of a and b are then obtained such that t he equation satisfies experimental results.

The control of imperfection , including eccentricity, ovality. bulges. dents and fiat

spots is vital if buckling under external pressure is to be avoided. It has been

shown that collapse finally occurs at the location where the initial deviation from

perfect is the greatest. Therefore, safety codes and regulations have established

allowable deviations in overall out-of-roundness and local eccentricities. It is also

required that these be verified by measurernents taken around the circumference

and at intervals along the length, which is the purpose of figure 3-18.

Dalam dokumen Buckling of short, thin-walled cylinders, as applied to storage tanks. (Halaman 99-105)

Literature survey

3.4 Hydrostatic press ure

3.4.1 Literature survey

In 1952, Hol t [l1J developed a procedure for computing the allowable ollt-of- roundness of a circular cylindrical shell s ubjected to external pressure. In this paper it was reported that the equation for comput ing the strength of an out-of- round cylinder is

(3.7)

where

i s the numb er of lobes in the initial out-of-roundness pattern , which can be seen in figure 3-1. He lt also developed an equation for the maximum ampl itude which is applicable to cond ition s with elevated temperatures but, this will not be explored in this study.

Along with Holt's equation , another simpler equation was proposed for the reduced buckling load in [261, where the reduced buckling load is found to be

and is shown in figure 3-16.

Su!

qcr

1 +

sol •

(3.8)

Galletly and Bart [13J derived an equation for the reduction in buckling pressure for the case of external hydrostat ic pressure in 1956. It was assumed that the initial imperfection in the cylinders were similar to one of the modes of buckling.

But, actual shells never satisfy this condition. The equation they derived which expressed the initial pressure was

~_(1_q~)(-3+J9 4(1-13 +132) H)

t

4(1 - 13 + f3')K (3.9)

where

v.

.

. / .

X ..

t-- .l(' . . _...:; . . .

/. V . .

- - -

o

•

• u~9J!!!-

. ~

. --- .

. .

.

.

Figure 3-16: Plot showing the re l ationships in Holt's equation .

H 3 _ (2t5

fJ

K

Yield strength of the material.

In the derivation of this

it was also

that. the initial imperfection is small (wit.h an ordN of magnitude of the shell wall thickrlf'ss) , wh ich is limited in the context of t his st1ldy.

When Donnell [14 ] initially explo red the effects of impcrf('ctions of cylinders

s1lbjected to external hydrostatic pressure in 1956, he assumed the same

unevenness factor as in his pn"violls collaborated st udy [10] of cyli nders s u bjected

to axial compress ion in 1950 with Wan, outlined in chapter (3.2) on page 67. But this stu dy was inconclusive in respect of the fact that

reduction formula

not found.

Lunchick and Short [l5J focllssed their research on the buckling of imperfect shells with an 'accordion' initial deflection in 1957. This nature of imperfection occurs in stiffened cylinders and is caused by shrinkage of frame weld-joints. The imperfection equation t hey used was

Wo = ~e(x _ ~2) (3.10)

This expression for the initial imperfect ion does not hold true for the present study, as the im perfection is considered to be axisymmetric.

limited to applications where the Batdorf parameter {Z)

large

(Z > 500). The reduction formula they calculated for the case of distributed

imperfections is

q~ = 1 _ 4.7511 ?'1 - ", V(A)'

qcl ifZ t (3.11)

and is shown in fi gure 3-17. Thus the reduction formulae for both axial load ing and hydrostatic pressure are proportional to (1)2/3 using the same method.

Yamada and Croll analysed imperfection sensitivity in cylindrical panels in 1989 [46J and cylinders in 1993 [54J. In bot h cases the equations of equilibrium were formulated using the principle of total potential energy. The imperfection distribution was represented by the harmonic exp ressio n

ny .

Wo = Acos-slD-

I (3.12)

eroll [601 later reviewed the developments of the reduced stiffness method for

both ax ial load ing and external pressure, from its early developments onward

in 1995. By comparing numerical experiments and experiments performed by

others, Croll concluded that this method is shown to provide a safe prediction of

buckling loads, without the need of mathematically examining the imperfection

eoo

"'"

- -

'b

I

j

J

..

.. ...

2'

itself.

T he ASME Boiler and Pressure Vessel Code UG-80(b)(l) 138] requires that the

4(1 - 13 + f3')K ^(3.9)

t-- .l(' . _. _...:; ^. . .

/. V ^. ^.

. _~

Wo = ~e(x ^_ ~2) ^(3.10)

q~ ⁼ ^{1 _} ^4.7511 ^?'1 ^{- ",} V(A)'

and is shown in fi gure 3-17. Thus the reduction formulae for both axial load ing and hydrostatic pressure are proportional to (1)2/3 ^using the same method.

- ^...

r"::::::: ^...

-.. ^r- r--. I-... ^r--- ^~-9,' ~ I-....

r- _r---. ^~

~ ^... '"

r--. _r-. _~'" ~ ^r-...

r- _r-... _~

_:-.. ^['-... _I-.... ^i'- ^r-.

~ ^...

^~

_, _{#' -}

'.1 _o _' , ,

/ / / ^I

^t

^$.

I J ^1/

/' V ^.t ~,,,. ~~ ^Q ^' J ^V V ^V

. / ' / .. I ^/ ^..f : .A

^--~

^,

^(3. ¹³⁾