Numerical analysis - Combi ned hydrostatic pressure and axial l oading

2.8 Combi ned hydrostatic pressure and axial l oading

2.8.2 Numerical analysis

An alyses were performed on the usual range of cy linders. An axial load was applied to the cyli nder. and the hydrostatic buckling pressure determined. The

• axial load for SASOL's F7101B tank was used which was reported to be 30000kg.

This tank is a 75ft (22.86m) diameter tank , which corresponds to the mid-range of this study. The axial load com prises of the s uperstructure as well as the roof plating. This axial load forms 0.4% of the classical axial buckling load for t he cy linder of dime nsions l= 12m , r= 11.43m t= O.Olrn.

The num erical analysis results compared with equation 2. 126 can be seen in figure 2-31. where, again , at lower values of N and therefore Z the discrepancy

.,

'"

!OD

. . V

: y-

. / /~ V

:....---

1!xx) l!OD

"'IJ~ ~

~ ^r--

• . ~

^\l\eOfe~

^cO''''

. ---

'!OD 3!OD

Figure 2-31: Combined loading buckling results for a perfect cylinder.

is smallest.

.

When these buckling r es ults are co mpared with pure hydrostatic press ure buckling, one can see that the axial load r educes the buckling load by a very s mall margin, as seen in figure 2-32.

If this axial load is t hen varied, as the axial load increases the critical buckling

pressure decreases as expected. Various axial loads were te sted for a particular

cylinder, and the results can be seen in figure 2-33.

a

500 1000 1500 2000 2500 3000 3500 4000

Figure 2-32: Comparison of hydrostatic nad combined loading buckling values.

,,;(XX)

"00l

,= r\ ^I- ^-

^- ^{- -}

^{I -}

^- ^I- ^I- ^{- -}

10000 q~

BOOO

6000

~ ~

,o~o

~

^10'-'8

""- ^{I -} ^-

^{- -}

I-- - _;u.

I-

IOn

~

4000 ^~^- ^- -

lOOO

-

'" ""

^!,in'

700

J

""

^lOO) _r ¹¹⁰⁰ ¹²⁰⁰ ¹³⁰⁰ ¹⁴⁰⁰ ¹⁵⁰⁰

Figure 2-33: Graph showing the decrease of pressure bearing capability with increasing axial

load .

PlottC'd as in Showkati & Ansollrian [ 63] a nd compa ring to hyciroi:i t a tic

pr<'SSllrc

bucklin g anal ysis va lllPs , t hi s rC' iation is clp url y ('vidcnt ill fi g llJ'{-' 2-34.

a

-

~ ~

"

r---.--

"

o

"""

-

"00 "00

-

'. """

., _.~

e--

.-

'"00

, ..

T ^- ^{, -}

^p~:lIl\~

''''

~''4 p->1

"'"

'::: _.'"" _-

- ^l-

- _.

- -

_2<00

r

Figure 2-34: Graph showing the comparison of hydrostatic loading with varius cases of combined loading.

The bu cklin g res ults of the t hree form s of loading (namely lat.eral pressure, hydrost.atic pressure, and combined loadin g) c an be Sf'en in fi gure 2-35.

The C'xp0ctf'd trend of decreasing prf'SSllrC' loading capacity is c1e>ari y C'vidf'nt as

onf' progresses from

plln~

lateral ('xtcrnal preS Sllre, to pure hydrostatic f'xternal

pressnre, through to the c ase of combined loading.

'"

Nume I resuts 70

Figure 2-35: Graph showing a comparison of all loading conditions for perfect cylinders.

Chapter 3 IMPERFECT CYLINDRICAL SHELLS

3.1 Introduction

l'hp. st.udy of buckling of real cylinders must take into accoun t the fact that a ll con structions, however carefully mad e, do have defects su ffi ciently large to initiate bucklin g and affect buckling load. The design of vessels s ubjected to external pr ess ure is differe nt to t he design of vesse ls su bj ected to internal press ure because o f t he buck ling instabili ty extern al pressure produces. This factor is evide nt in a ll research of buckling of cylinders whereby the corre lation of act ua l bucking tests with theory h as been poor with seem ingly similar test spec imens and techniques. These discrepancies were the subj ect of widespread cont roversy but , now it is generally accepted that initi al imperfections are the principle cause o f this disagreement . The imperfec tion , such as out-of-roundness, grows as the external pressure is introduced and inc reased , whereas it is 'cancell ed out' in the case of internal pressure. The effect of small local geometric imperf ect ions i s small in stiff structures such as thick-walled cylinders and it is therefor e not necessary to take them in to accoun t. However in more fl exible structures such as thin-wa lled vessels, these imperfect ions can dramatically reduce the buckling load.

De form at ion c urves and critical stresses of the cylindrical shells, taking into

acco unt initial imperfections, was first explored by Donnell and Wan in 1950[10]

Howeve r, t his an alysis assumed t hat t h e ini t ial im perfection has the sam e sha pe as t he buckling configu ration. Since s ha pe varies duri ng buckling, th is assumpt ion of sim ilar ity in shape is theoretically wrong. T h is was shown in the gener al t heo ry of Koitcr [27] an d [20]. Nevert heless, if it wer e t r ue that in t he case of an impe rfection t hat has t he same shape as t he buckled shape was t he most o ppressive case, t he n t his a pproach could furnis h t he critical buckling load co rrectl y.

One t rend that is o bviously prevalent, as is shown in Ko ll ar and Dulacska's publication [43] whi ch looks at previous resear ch in this fi eld, is th at t he crit ical loads o f im perfect cy linde rs dec rease sh arply with the increasing ini t ial imperfect ion.

In t his cha p ter , avail ab le theory will be reviewed. But , as will be shown, th e nat ure of t he im pe rf ec tion considered thro ughout prev io us resea rch varies greatly.

T he refore, charts will be prese nted based almost exclusively on num erical a nalysis results. Since SASO L's interest lies in ou t-o f -ro und tanks as defi ned in t he tank code AD Merkblatte r B6 , out-of-roundn ess will be ex pressed as a percen tage, and is d efi ned as :

(3. 1)

Fo r t he purp oses of t he nume rical an alysis, t he imperfect cylinders are ass um ed

t o ha ve a n ellipt ical cross-sectio n at mid-length . Fig ure 3-1 s hows th e va riables

used a nd t he characterist ics of t he impe rfection. Therefore, the imperfection has

t he form aro und the circumference of e sin (20 ). For th e num erical a nalyses, the

materia l is assum ed to b e carbo n steel wi t h t he materia l properties shown in

ap pendix B .

180'

0.5

e

-<>.5

-1

e

rmln

e 0' 1<---=- - --/+-- >1

rmax

360

⁰

, , , ,

: 270

⁰

Figure 3-1: Figure shO'Ning the definitions of the variables used in the imperfect cylinder study.

Dalam dokumen Buckling of short, thin-walled cylinders, as applied to storage tanks. (Halaman 79-86)

Numerical analysis