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Masters Theses Student Theses and Dissertations

1949

Analytical investigation of thin and moderately thick-walled tubing Analytical investigation of thin and moderately thick-walled tubing

under peripheral and biaxial loading under peripheral and biaxial loading

Frank Joseph Cizek

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Recommended Citation Recommended Citation Cizek, Frank Joseph, "Analytical investigation of thin and moderately thick-walled tubing under peripheral and biaxial loading" (1949). Masters Theses. 4949. https://scholarsmine.mst.edu/masters_theses/4949

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ANALYTICAL INVESTIGATION OF THIN AND MODERATELY

THICK-WALLED TUBING UNDER PERIPHERAL

AND BIAXIAL L~ADING

BY:

FBANK J. CIZEK

A

THESIS

submitted to the faculty of the

SCHOOL OF MINES AND METALLURGY OF THE UNIVERSITY OF MISSOURI

in partial fulfillment of the work required for the

Degree of

MASTER OF SCIENCE IN MECHANICAL ENGINEERING

Bo lle, Missour i

1949

------

i

Approved bY~Professor of

ACKNOWLEDGEMENT

The author is grateful to Professor A. J. Miles under

whose gUidance this work was done. He also wishes to thank

him for his re.ading of the manuscript.

The author is also thankful to Assistant Professor

N. Costakos for his valuable assistance.

i1

ili

CONTENTS

Page

Aeknow le dgement s •......•.......................•...... 1 i

List of illustr ations .••.•••••••••••••••••••..••••..•. liii

Intr oduction 1

Beview of literature .................................. 4

Part I: The case of elastic collapse of tubes byan external pressure acting alone............ 6

Notations for Part II ......... ........................ 12

Part II: The case of elastic collapse of tubes byan external pressure acting simultaneouslywith an axial end load •••••••••••••••••••••• 13

Part III: The case of plastic collapse of tubes byan external pressure acting alone •••••••••• 27

Part IV: Yielding of tubes under biaxial loading ••••• 34

Conclusions •••....•............•......•..•....•.••..•.• 40

Summar y .............•....................••........... 42

Bibliography •.......................................•• 43

Vita..... .. .. 45

Fig.

LIST OF ILLUSTRATIONS

Cross section view of cylinder showingmean arc before aQd after deformation ••••••••••

Elastic curves for collapsing pressureand cr itica1 stress of steel •••••.•••••••••••••

iiii

Page

7

11

Element of thin tube dx " ds.Also longitudinal axis of t~be ................. 14

5.

6.

7.

8.

Various projections of element infigure no. 3 after strain ••••••••••••.••••••.••

Stress-strain relationship forhy pothe t 1cal steel s •••.•••.•••.•.•.••••...•••••

Critical stress at collapse forhypothetical steels ••••••••••••••••••••••••••••

Collapsing pr essur as for hypothet 1calsteels .

Effect of biaxial loading 1n tubingon the yield stress ••••••••••••••••••.•••••••••

Effect of an end tension on the externalcollapse pressure and ciroumferentialstress of tubing ............................•.•

15

29

31

32

38

39

1

INTRODUCTION

The behavior under loads of thin and moderately thick

walled tUbing with various combinations of loadings has been

investigated by' many researchers. The importance of such 8

problem is indicated by the number of individuals who have

studied tubing.

The decision to choose this subject was made after it

was realized that certain industr ies had tubing failures

Which were in many ways unexplainable. The oil industry and

its casing program is an example of such an industry. Their

casings were sufficiently strong in the early years of oil

production to meet requirements, but with the advent of

longer casing strings required in deeper holes, failures

became more frequent. Of course, it was simple enough to

increase the thickness of casing to a point where it was

strong enough to withstand the pressures and loads associated

With deeper holes, but such a method led to over weight cas­

ing and a subsequent increase in cost. The more costly this

procedure became, the more desirable it became to spend

money for r esear ch.

One of the reasons that casing failures became more

frequent with greater drilling depths was that the casing

underwent 8 greater tensile loading due to the weight of the

casing. This loading on the c8sing was accompanied by an

increased external fluid pressure on the outside surface of

2

the casing from the fluid pressures in the various strata

below the earth's surface. The casing is usually surrounded

by cement on the outside, but in some cases the cement does

not always fill the entire annular space. Such being the

case, the casing is sUbjected to the full effect of the ex­

ternal fluid pressure.

The problem then, resolves itself into a study of the

behavior of thin-walled and moderately thick-walled cylinders

under the effects of an external fluid pressure acting alone,

or the e ffe ct s of an external fluid pr e ssure acting simul­

taneously with an aXial loading. In this thesis a theoret­

ical or analytical ap!I"oaeh will be used. Some application

and reference w1ll be made to oil-well casings in particular.

but the solutions w11l not be restricted to oil-well casings

alone.

It must be recognized that the tubing problem involves

the problem of unstability before the proportional limit of

the metal is exceeded and a similar problem after the propor­

tional limit is exceeded. This breaks down the investigation

into four main divisions. The first is a diseussion of the

effect of external fluid pressure acting alone on tubes that

fail by becoming unstable before the proportional limit of

the met al is reached. The second part 1s similar to the

first but involves the effects produced by an axial load act­

ing in conjunction with the external pressure. These two

cases complete the stUdy of failure due to unstabllity at

stresses below the elastic limit or within what is called the

:3

elastic range.

The next two parts or the thesis are concerned with the

effects of the types of loadings mentioned before, that is,

external pressure alone or with external pressure and aXial

loading combined, but in a stress range above the propor­

tional limit of the metal. This range, which 1s for moder­

ately thick tubes, is known as the plastic range. The

limits of this range are the proportional limit and the

yield point of the tubing or casing material.

4

REVIEW OF LITERATURE

A survey of the field of thin-walled tubing shows that

a great deal of literature has been written on the subject.

The various authors have shown originality and resourceful­

ness in their work in studying each phase of tubing design.

Heretofore however, there have been few who combIned under

one writing so complete an analysis such as has been under­

taken in this thesis. Most of the work done from a theoret_

ical approach has been done piecemeal. i.e., only parts of

the whole problem have been considered at one writing. The

work of Holmquist and Nadai have come closest to the most

comprehensive work on this SUbject. This thesis enlarges

upon their work in that it provides a rational expression

for the design of short thin-walled cylinders within the

elastic range under either tension or compression axially and

under external pr e ssur e. It also shows mor e concisely that

in the case of casing design for the oil fields that the

effect of end loads is negligible.

Prescott must be given much of the credit for having

developed the mathematical derivation of the expression for

thin elastic tu~s under biaxial stresses but did not apply

his work, as has been done here, to oil casing.rThe work on collapsing pressures acting alone on thin

tubes in the elastic range has been investigated as long ago

as 1850. In this thesis it is applied to casing and is used

here not as something original but used to make this invest­

igataon more complete and understandable.

The practical experiments conducted elsewhere for

checking the theory given in the work of this thesis are

very limited and suggest a large field for future invest­

igation.

5

so

6

PART I: THE CASE OF ELASTIC COLLAPSE OF TUBES BY AN

EXTERNAL PRESSURE ACTING ALONE

First the derivation of a formula tor the buckling of

thin tUbes by external pressure alone in the elastic range

will be considered. Referring to Figure 1 and those

notations alongside the figure. the tube in the bent position

wi 11 be treated as in equlli br ium in a slightly defor med

shape. Considering the axes oA and XOY as axes of symmetry,

the compressive force at X and Y 1s

Q =: p(Y.., - V'o ) -::. f X6

The bending moment at any cross section C is

M: Mo + Q XF - (' XZ · g__ 2-

=Mo -t- r XO· XF - f nz.t KO -\ -lJ:: M. +P X • XF - ~ Xi • (1)

From the law of cosines

1'0 \:; Xz"& + XO1._ 2~ • XO • COC;~~

but co 8 Il. '= ,Xl=-i!

Fe) -= ~"'-+ XO~- 2. XF XO,and

SUbstituting in equation (1).

Defor'medMe(l~ arc.

7

Q

~«/" ._-"-

e

p

/\~ Mean QYC/

t

Q

Cl'"O:S~ SeetlOrl '/lew of Cylinder Showing Mean Arc Befol"'e a.ndA tel'" oefor'rnotlon.

r m - mean r~diu8

D - outside cip.meter f cylinderP - rreasur per unl~ lRn[thr - cOffipressi e stress at X and Y

Q - com~res iv r ,ree At X and Y

w - r~d:'~l Gisplec811.ent 9t :.:u,y roirt

~o - radial disrl~cement at roint X1~ - befl<iir<~ l!,oment Be .inc It cross a8ctlrns X ar d Y

With this expression for the bending moment and from

figur e 1,

8

then

...J ¥O == r", - W

(2)

The differential equation for the deflection curve of a( .

long circular tube as given by Timoshenko 1) is

(1) S. Timoshenko, Theory of Elastic Stability, N.Y.,McGrawHill. 1936. p. 2Q7

Ewhere E' = L - V\

{) :; Poi sson' s ratio

E :: Young's modUlus of elasticity

I -.: Moment of inertia

substituting the moment expression (2) into the above

equation the differential equation that follows is

~~ +W" -W-: [ M. +pI'... W- We)] ~

and

Let1.1. '31 = '4o ~~ ,

E'I

'- .~

_ ..- Mo r\'"n + raJ r""" WQ",-

E'I(3)

(4)

9

then the solution of equation (3) becomes:

. ~ ,w..~ ASIt1f8 -+ e COS f9 +- Me lin t f rm W..

. E'l -+ PI""":The derivative of w with respect to e is then

1ft ~ ·fA co~ fa - f B S iii f9

(5)

and also from symmetry

(dWl ? 0 • ( aw2 -= o..d9: S'::D ./ ai'i:Ez

With these two conditions it is seen then that

A=o E SUi f!:: CI Z. •

Then 'f must be an even integer and the smallest root

different from zero in this equat 10n 1s when fie 2. This

means that a critical situation occurs at some pressure, Pcr.

This value 1s found when f: 2 is substituted in equation (4).

-U1~... - r

M' (6)

(7)

~3

1. - t.... -11.~ ::.D-t qt'kl

2equation (6) becomes

_2.E.' _ 21: j

~r - (DIt -I)' - (I-:~j .(1)4-1)3

From

If this 1s a thin tube, that is, it D/t is large, a

linear distribution of circumferential stress may be assumed

and therefore the circumferential stress is given by

r=~ _ (7a)Lt

10

and the stress at the critical pressure is

e Dt'l.( t - v'l )(0 - t )\

(8)

The assumption that AO and XC are axes of symmetry is(2)

consistent with tests made by Sturm.- These equations

(2) sturm, R. G., A Study of the Collapsing Pressure ofThin Walled Cylinders .. Doctorate thesis. Univ. of Ill,

are developed for the elastic range and are not necessarily

true where the elastic limit of the material 1s exceeded.

A typical plot of equations (7) and (8) is shown in Figure 2.

These curves are called the elastic curves.

l

11

-+-+-++-++-++++-+++ L-t-4 -f- -. 1-- -. - t-+-t-+-++-+++-t-t--+-+--t-H:-+-r-+--t-+-t-+-+++++++--+- -j

1-+--1-++-+-+-+++++++1

---+-c--+--11----+_+-Ftr- --~_I--- '- ~

+ -1-++++-++- - --t - -j- -- f- - 1--" - - -

-I-- - -- - .= -}"-:-r= -+ t- j-~.:: ,..:. :1- --. t--: f- - f-f_ - - ++-- -- -t-=-1'--

H--jC-+--f -l f- - - -H- I--- -- -j + ~- -1- i +- -- - t-· -rt-, -=l-t-+-++++-+--+- --+ - ~- ---- -t -I- t- f---- -- -" f- --It "iT' -- 1-- .. +-t- - ,,- --I-h"

: f---- -:= +:' +-;: - .." r FI-- +" - -lr-' - - -- t ~ ~ T . - H:1l-1- .-- '-r-, --t t 1 -" - +1 --1- +1-'- f-++t:

I-- ~ 1'::1-11;~tt- =t.J '.', r'~;. ·'iJj':l _I~ "l"fl:~~T- - 'i---LI:~ tJ+~'.,:" .. ~"t '::"-~f·--t-,-- '-liiJr I'H---+-H-i t- ", Tj- + -I' ), ,- .. , -1 t ' .. r +-t----- + 1'--1 . l-+f·

elTJfT -~,l, Iitll•• -i ~"h~· '''; I-- .. - .~ +- . - .. - I - --iofl Ir. " -+ - "1.-," - -f-' ----I-+-l-+i~' I'"~.,.,l - :tJ tt

- -- - ---. ~ UII":~t!l-p Q'I""'" - . :at _jL.~:]~~ ~- :.. ' .1~= -tr+-- -- +:-'"t.>- -f- -- ,- -- -I- - - .-.t-- 1- t-· j -- -H-- ++ t -. - - ---.1--- -- - - -HT~-H+r .1--1""

-- 1- . -. - -tt~+ - . f--- - -- - il± --- -- ---t1-j--HT~-- ;-=t- '-

sx _

r m

p

T

f

'J'

Y

At

NOTATIONS FOR PART II

distance along mean cir cumterence

distance along the axis of the cylinder

mean r adius of cylinder

angle of polar coordinate

stresses along longitudinal axis

stresses along mean eir cumfer ence

displacements along x-axis

radial displacement

displacement of angle e

unit external pressure

bending moments par unit length

shear str ess

t.or que

shearing forces per unit length

Young's modulus of elasticity

shearing modulus of elasticity

radius of curvature

curvatur e

Poisson t S rat 10

shearing strain

angle of twist per unit length

thickness of cylinder wall

angle of inclination of stresses fi'+If)(

angle of element after strain

unit strain in the x - direction

unit strain in the direction e

12

13

PART II: THE CASE OF ELASTIC COLLAPSE OF TUBES .BY AN

EXTERNAL PRESSURE ACTING SIMULTANEOUSLY WITH

AN AXIAL END LOAD

For the case of a oylinder with an external pressure p,

and a simultaneous axial load of either tension or compres­

sion, a general set of equations must be found. The general

set of equations found in the following work will be for the

case with a compressive axial load. Then with these equations

and certain end conditions depending upon the application, a

general expression for the collapsing pressure with the effect

of an end load can be deVised.

Referring to an element of oylinder in the unstrained

state, an example of which is found in figure 3, a particle

of the middle surface is at a position defined by x, r m, Q.

After strain the particle will be at a position x+ u l '

r m'" wl ' and e +'1, _her e ul' wI' and'\, sr e displaoements of

the axial distance, radial distance, and the polar angle e

respectively. Let the element have the dimensions dX.ds.

~he element of middle surface is considered to be in equllib-

rlum by the following forces, moments, and stresses aoting

as shown in figure 4:

~ +<G - compressive stress in the middle surface

in an axial direction

G+G$ - compressive stress in the middle surface

in a peripheral direction

14

15

~~J\ I

F.+~dJcI ~X

(e:t)Top View of clemcl"ltAfter Strout) ShOW'f16Torque ; Moments 0'-'\"

---- - --_.... -... - •...•... -. __._-

(d) Left Side View of­E.1~Mel'\t After StrQln

(c) Frol'\t- V...wof element Atter S'ttttM

( 1)

16

Wl bending moment per unit length acts along

the same section as does ~ 1"" q-x

M2 bending mome~t per unit length acts along

t.he same section as does <1"2. +<Ij.

~ shearing stress in the middle surface

T t.orque in the middle surface

Fl shearing force per unit length acting on

the same face as V1

F2 shear ing tor oe per unit length act ing on

the same tace as M2p ext.ernal pressure

r angle which Ci+q'}( makes with the x - axis

d~ - 1ncluded angle of element atter strain

It i8 to be noted that J "1=..0 .... " and u1=uo+u,

where wo~ constant. and uo=linear tunction ot x. (This w1l1

be shown later). CODsider also that .0 1s produced solely by

t.he stress Ii and that tJ; produces uo•

Noting tigure 4b which is a view from the convex side of

the element., t.he equation of equilibrium in an unstrained

direction along the x-axis, considering the stresses acting

in the middle surface always and neglecting quantities of

higher order. ia

This can be written, s1nc.~ is small relatively speaking, as

~ .,.. Dz: d" a7.u,~lC ~ - v~ as" 0. 0 •

(2)

(3)

17

Next, resolving the forces in the direction of the normal

through the middle element (see figures 4c and 4d) and

neglecting fx in compar ison with Qj , another equation of

equlli br ium is

t a's di~ dl< - dx~ ds - ds~t dx - C dJ(((J;.T~)d9-r pd5 dx =- 0

which in turn i8

~'w ;>t;. dFi dfJ-t 0; ~"t - ~ - ~ -t-(Q;.t-(JS)(i{S + P = 0 _

~ is not negleoted as befor e because of the term ~

which may be lar gee

The equation of equilibrium in the unstrained direction

ds, atter neglecting unimportant terms, is, from figures

4c and 4d

t;- uf; + tr.:. ~ O)~t - t e£ -E ~ =: 0"'gS WI I QJx.... ;;J)( z.,S

Referring to figure 4a the summation of moments about the

edge dx is

(4)

The summation of moments about the edge ds is

~l 4- C)T + F == 0 (5)d~ a~ I •

The expression ~ is the curvature of the strained

circumferential element which will be shown to be

Writing the above equation apprOXimately then

~ _ ,I _ _ _l ,r .~l,W)ds - ~.... ""'~o r","l.lW+}il: •

(6)

(7)

18

Neglecting quantities ot higher order, the following ex­

pressions using the above equations can be written

(8)

Since ~~is the circumferential stress while the tube is

a circular cylinder of (r m+..o )' that Is, while -, Fl , '2'

are all zero, and since equation (2) must remain true for

this particular condition of the tube, it follows from that

equation and equation (8) that

(10)

Saying that r m=(rm+-wo ) approximately, equation (2) be­

comes with the aid ot (8) and (10)

Also from equation (9) equation (3) becomes

(12)

Solving for '2 in equation (4) and Fl in equation (5)

and substituting them in equation (11) and (12) the

fo llowlng expr esaions ar e der Ived :

19

The important equations ot equilibrium are then

equations (13) and (14) and (1) written as follows

(15)

With the equations of equilibrium now decided upon,

expressions tor the displacements will be written next.

Remembering that ~ and fzare assumed to be solely respon­

sible for the displacements Uo and Wo respectively and that

these stresses are constant, then

(16 )

(17)

This shows that Uo is a linear function ot x. and Wo is a

constant as stated before in the opening paragraphs of Part

II. The longitudinal strain of the middle surface in the

x - direction 1s

~u.o + e .dX lX /

( 18)

20

in the direction e it is

The shearing strain is

v_ r: ~'" '~L«..0- M~+~»8'

since Uo is not a funotion of e.(20)

From the general formula for the curvature of El curve at

polar coordinates (r,9) and by negleoting the squares of

differential coeffioients of r the general formula beoomes,

where f' is the radius of curvature,

I I r. I oP.r)P= rl l - r O(ii •

In the particular problem at hand r::rm+-wl after strain.

Therefore the curvature after strain along the circumference

where M2 acts 1s

1 _ I [, - I .....(Y...... ..,,)17i- rM +\\', ~+~ oe"t

- I [I d~W, ]- fW\ +"'1 '?J /) 7-YM+-W.

l [1- ~ ~,.~1- - ---r""" r~ ~ aD"'" • (21)

Equation (21) 1s found by approximations and has been used

before in the form or equation (6). The general formulas

for the curYature in cartesian coordinates are

1 ~X1.-= L' + (%)J~and

21

appr oXimate ly.

The curvature along the tube in the direction of the x-axis

where Ml acts, atter strain 1s

(22)

The change in curvat.ure cl with Ml positive along the X-axis

isc"WC,= c>"W

C> )("\. - 0 - .- - (23)oX""

from (22). Also froll (21) the change 1n curvature along the

circumferential direction with M2 positive is

C", = J.- - -'-r. (\ - '!J., _.L ()"'W, )Yr\I\ rlI\ rWI r...., 08'"

(24)

The equation for the angle of twist per unit length

according to Prescott (3) is

(3) J. Prescott, Applied Elasticity. London. Longmans,Green and Co .• 1924 p. 547

~. (25)

22

The relations between stresses and st.rains are

-E:~ : <r; + ~ - -V (q-~ +-<1";)

- E ~ = <f"l.. +~ - .,) (cr, -+-~ )

E. s y= ?;

Combining these equations with (16) and (17) then the above

equations become

E(~ + ;~) -:: a-~ -vq;

Es(r~ ~ + -t ~):: 1

It can be shown from the theory of thin plates that

M, -= E'~ (C I + V("t. )

M'l. :. E.'l: ( C"t. ~ vel)

also that

(26)

(27)

(28)

(29)

( 31)

Equations 26. 27. 28, 29, 30, and 31 are expressions for the

displacements.

Now with the equation of displacements and equilibrium

found, it remains t.o solve these equations. According to

23

the accepted mathematical theory, '4~ all the displacements

(4) Prescott, Ope cit., P. 552

(32)

when the tube begins to buckle.

be expre ssed by

U =A Cos n I) CDS

rM \::. ~ S u'\ 'fl () SIr'\

W=c CDS nB ~1V\

are functions of the coordinate x and the polar angle Q.

The displacements can then

kX-r......kx~

Kx....-.-~

The constant k depends on the end conditions of the tube.

It will be noticed that ., the radial displacement, will have

the same value at e=o and e=211". Therefore it is apparent

that n must be an even integer. Substitution is now made

into equations 26 through 31 which yield the following

24

With these values and the equations of equilibrium 13,

14, and 15 the following three equations can be written~

{ 1'11. f,Hc:1. + ~ (1-11) ril.} A-! (H·v)l'Ik~-VkC",O (33)

-vk A t ~ z(l-v)nk.... I~"'~ Tl'I \ e - ~ k1.~, -I-(I\"':.~~-I\ C

-l- ; ~"''''1. ~ (1\"'+ k....)...- rj1. -" 1<:1.) C. :. 0 (34)

-~(1+ -0) n k.A f ~ - k1.~ +,,'" -+ ~ (I-V)(I + ~S.:)\(' B

+ nC + ;:l'"~ (n (1'(1._ 1) -I- VI k..... \ C ,,0 (35)

From the last three simultaneous equations, A, B, and C can

be eliminated. This is done by determinants and gives the

following relationship. This relationship is the result

after neglecting squares and products of the extremely small~ t1.

quantities -= ,~ , and -E' E.' ''2.r~

--i(, -V) k."'~, ~ ( Vl'L + k:.l.)"'" +n1.. +-.L Kl. + Z. V1<.1."\

-i(1 -v)~, ~ (l'\1.-I)(n'\.+ 'i......t-rl'-l<.... ')

4o..L (i -v)..:L. \ r~'1.+ K\)4+ VlJ.+ ~ n'\. lit- -t 2(' -1» k4l'2.. rl..rM't. 1"-1 J

'+.L ~ v' !"Z._ r_2rl- _7.,..4 k"l. - (1 +V - z.v'" ) ..ilot<.""- V1<.l.1. 'Z. \.' - I "1.r~ l J

( 36)

25

Treating k as a small fraction, as it would be in most

practical cases, and arbitrarily selecting the important

terms of equation 36, the resulting equation may be

considered to be correct

(37)

Equation 37 can be modified to include the value of the

external pressure by assuming a linear distribution of stress

~l on a longitudinal section which is given by

<l: :: p(t>-ttt. 2..~

which was shown before by equation 10. Then equation 37 be-

comes, solVing for pressure

It is noticed that the eXistence of an end thrust <J'; causes

the tube to Collapse at a smaller value of pressure than if

~ were zero. Likewise an end load representing a tension

(C\substituted as a negative quantity) will increase the

pressure for collapse.

The preceding equation and the results therefrom can

only be realized if k is a reasonably large value, and if

the Dft ratio is large. The value of k as stated before is

a quantity which depends on the end conditions of the tube.

If it is possible" at the ends of the tube 1n question, to

restrain the ends so that no buckling occurs at these points,

then " .. 0 at t.hese points. With ":::0, k becomes froll

26

equation 32

k =1r'r'~L

where L 1s the length of the tube. In the specific case of

oil casing, the end restraints would be imposed by the

couplings at the joints.

The value n is an even integer as shown before and is

to be found by substituting even numbers starting with 2 in

equation 38. The value of n is the value of the even number

that yields the smallest pressure.

For oil casing where most of the lengths encountered

are in the magnitude of 30 ft. or more, k would be small

enough so that equation 38 oan be reduced to

p:: E'~(D-t)

.,. _ b--r(~- -

L

n=2 for long tubes

or

This equation is the same as was derived before for 8. thin

tube acted upon by an external pressure only. This then

leads to the important oonclusion that the end loads have

little effect upon the collapsing pressure for long tubes in

the elastic range.

27

PART III: THE CASE OF PLASTIC COLLAPSE OF TUBES BY Jill

EXTERNAL PRESSURE ACTING ALONE

Since many of the commercial tubes do not tall in the

range of elastic collapse, the case of plastic collapse ~

an exter ne I pr e saur e act tng a lone w111 now be consider ed.

The tubes within this range are those which D/t ratios

small enough to place them in a range of stress beyond the

proportional limit. This is indicated in figure 2.

It has been shown that for the case of buckling of

columns, that the substitution of a reduced modulus of

elast i c1 ty 1n the buck.ling for mUla will pr edict accur ately

the buck.ling stress in the plastic range. On this basis

and since the instability formulas for both columns and

tubes run closely parallel, it is seen that the similarity

can be extended to substituting a reduced modulus in the

collapsing formulas. With this in mind then, equations 7

and 8 of Part I can then be written as follows

(3)

(2)

(I)

<J;..-= (,--V"\.)(D-tY~

4 E. e'E-=-- Lr [YE +~J

2E.... ,r :: " -~--~<:.y I_V1) (D/t -I)~

Eo,.. Dt"t.

where

28

Er :. reduced modulUs

E = Young's modulus

E' = slope of stress.strain curve

at a point on the curve

All other symbols the same as before

From the preceding it is apparent that the critical

pressure and stress will depend upon the nature of the stress

-strain curve beyond the proportional limit. The exact

variation of this curve in the plastic range for oil field

casing would have to be known if ~ were to be calculated.

For purposes of clarity two examples in the use of equations

1 and 2 will be shown. For these examples it will be pro-

posed that the steels behave in accordance with stress.

strain curves as shown in figure 5. Steel no. 1 has a

par aholtc str e ss distr i but ion beyond the pr oport ional limit

and steel no. 2 has a straight horizontal line relationship

beyond the same point.

Both steels are to have a proportional limit stress of

40 x 103 psi and steel no. I is to have a yield strength of

60 x 103 psi as defined by a O.~ permanent set. From fig. 5

and the stress-strain curve for steel no. 1 the portion of

the curve from the proportional limit to the defined yield

strength can be represented by the following equation.

(4)

29

1-1- - --I-I+-i-- 1 f-t-- ---' - - -.J-' -T::J--~--t1-1'- -- -- - - f--- "t-- - -+-I-+--li-f-t!·+ t' -+- r- -- c- - 1- --- '-t- 1-' -- - +~f-++~-f-+-t-i

1-1- ,- l +- -- - t-- - - t' -1" .. H-- -. ~-++-.H--!-f-++1+-H-l+-+--H..---j 't-+_- +- '-. . ., f-. -

H··t-l-t+-H-+-H-' t-. -t·+-t-t'H-t-t-t -t-H-t +--H-t- H+-1-++-++t -t-+'j- t'i '-t-+-+-+-H-T- t- -H--+ -+-++-+ -r

1-+++-+-1'-+-+-+-+-1-+(-1' ,~~ " r. r- f-t-l---t-++-+ ++-++H-+++tI'+l-i-m

H-i-+t-r-t--+-t+ -+-1- -1 W·t+ ' .. Ii . -j- 1__,_ t,-+-+.++-H-+

i--H-H-j'f-j"'1-+--t-+-j,-t-H

H-+-1H-+.'i±tl +,- t.t-.+ -It ,~=:= ,_' - t- --.1= r- t--~. +' 1 t-.. I-

~- - f- J 1 ~+- . -tt +t. -t-+ ,- .+- -- r- _1-t - - .. ~ T- ~,-j--+-~. :.;::: -- 1/.,;.j;~~++-H-H. r- ..i I· , " ,Ii , r

J, t .j-

~+-H-+-H-+t--t-1-+-1-TI'-t -, +-+-i7'~i;lljf'--i'\""bb-+--+-.1-++--t-lf-H-t-t-++lH-i-t-tlH--+--++-+--+---H--t- t--rt, t--t-H--j- -f- t I

--t-- -', I100+-+-+-+-+-1 -,.-t . , -,+-+·-l--+++Ir++~~+-H-++-t+H-t-I'l

f---- -1-' v +-++-I,--t-W-+-H-iJ-H-+-H--H-rH-rH--i

f-i----l--.l--t--I,-++--t--1 +, -+- --r

1', - __ .-f---

-~--L, +- - :t- ·t+ -+H--H--t--f-+1'( . '+ 'I I , " +..... __ -f-I-+'H-.~'H--+-tt-ti t l~- c- -- I

~-++iiijOrt--t'-j--t-t-,,1.. __ '..'+-.' '1-," '~1=:;:::=:--j-t-t '1--+--++-+-41. =h, t 'j- T

f-H-'+_iJ"~'-r-t'_ t- ·t;'- j- .+-+-l---tt-~-1+__-+-+-+-++-+04_+-"':+-+';-''--iT"-:'-+t-~i~t-+H-.-t-_-:::=:-+-+_+-\-I-+1-t_-H-++-r_Ht-t"'I - ... - f- - - -- ,- t- -T +- +~t-++-irt-t-+'+I-++-+-;----;~-t-lr-_+-+·+-+

-- -1= -j- --t . + ++++-i---t-t--t'-. . - t-r-f- " , . .. T - -t-+-' +-t-/t--H- L +-+ ~--I-i-+--.~- -+ r- .. T . -- - ,

1-+--t--irI'"!"fr'tI '1· r- l'- ... Ht' ~ " ,-r/- +1'-. +-.j....._-1t--.+-++-t-+ +--t-+-+--lI+H--+-++-1-t+-+++-+-i

- t-- - T' r:(.t~t1 r. - rc- --- -t- t- H- -+:-+-t-t~~~~-+-++-++-+--H+++++tiT,rrTif- - H i'-r--j--\- .. - it -

t· -t--f-+-H +-+'+-H-f-J-+-t-t-t--t-t-'iJ++-l'jr- - r -4'-+--t-t-t-r--r t-+-+-+I~+'H-+H--HH-H-t-r-tti-t+-r-r+-' iH--+~iit-t-t--H-tt-; ..1--41+'+-+-Ir-I-++-t-t-H+~r-ttiTtt -l--t---; -H-H++-+-+-r-t-n-t-1' +_+ ...,

I-++~flt-+-+-rt-+-t-H-' t-- -- '+ ~t-+-+'-H:-+~-t .....

- ~.. ~~~-- I

1-l--l--t+-J-+++++++JHfrHii-t-rti-ttitt1·· . --T'

H--l---t-t-H-W-t-11-+-++-+J-H-+t+J-+ii-r-Hiit _,t-_++--tf__,~,--t.~t-

~-H-t-+-+--t-'H-ttt 1-- f- -+-+-+-I-++-t-t--t-t--t-t--t-n-t11,i-+-+-+-+ -t-t-+ t

+-t+-cl-+--H·+t+-H-·H-rl"1 r-+-+--+-t- 1-- - t '.~ -- - 'Ii '.+++ ·++-H-++t+H-H-m

1- t- ·-I-H---t-I-++-H-+~++r·TnH-t--tl--H. ''', -t- 0:t-,: ',~ --~t-~f'C ~~+--+-tt--+-,t"--t-:-tt--_-H..+-+--4~-:-_,~r-+""=-+t'H_ t­

ttjjttj~tt1j[tl:rL--1.--..L--tLL-..LJ-'L"Lt-J,.·t--,--.L--,-I--,+_-·t-~-~_·::::r:.L-,-f--L-f.L-L..L '::....-1 +-t-i- +~~ . . _. __

30

The explanation of notation is

. ,f=variable strain

f~strain at the defined yield stress

€p~strain at the proportional limit

4"=variable stress

Gf.-:stress at the proportional limit

<iy~stress at the defined yield strength

The value of Et can now be found from equation 4 by finding

~. (Where the equation of a curve is not known, Et 1s

found by measuring the slope of the stress-strain curve).

Then the reduce modulus is solved for and substituted into

equation 2. The stress curve for steel no. 1 in figure 6

is plotted from various values. Assuming, again, a linear

stfess distribution of the hoop stress and using formula 7a

Part I. a plot of collapsing pressures can be made as shown

in figure 7 for steel no. 1.

The same type of procedure is followed for steel no. 2.

Plotting the collapse pressures in the plastic range

trom equation 7. Part I may not always be absolutely correct

since some steels do not yield homogenously. For absolutely

correct curves experimental data 1s necessary for the par­

ticular ease. Holmquist and Nada! have made some tests which(5)

show the validity of these statements.

(5) J .L.Hol.quist and A. Nadai; A Theoretical andExperimental ApP"oach to the Problem ot Collapse ofCasing. A.P.l. Drilling and Production Practice.1939. p. 403

31

32

33

For Poisson's ratio 1n equation 2, an average value

was used. For elastic strains in steel Poisson's ratio is

0.26, where as for purely plastic strains it is 0.5. The

average value is 0.38. The use of an average value 1s•

tantamount to assuming a straight line Yariatlon of Poisson s

ratio 1n the plastic range. This 1s an assumption which will

do for the assumed problem here, but may lead to serious error

otherwise. The variation of Poisson's ratio in the plastic

range in buckling problems should be determined ~ experl-

mente

34

PART IV: YIELDING OF TUBES UNDER BIAXIAL LOADING

The effect of an axial load upon the collapse pressure

and critical stress at the collapsing pressure in the

plastic range for moderately thick-walled cylinders is ot

prime importance and will be considered under this section.

This is a problem in which the consideration of some theory

of failure must be taken into aocount. Here the yield point

of the material may be lowered or raised by an axial load.

This would oertainly change the value of the collapsing

pressure and stress since it was found before that the

collapsing pressure and stress by an external pressur e

acting alone depended upon the yield strength in the plastic

range.

To find the effect on the hoop or circumferential

stress of an axial stress, oonsider the middle surface ot

the thin tube in figure 3. As before, let the axial stress

be ~ and the hoop stress be ~ ; only this time oonsider

them both tensile stresses. Consider the element in a

state before unstability; and neglect the small effect of

the radial stress.

The strain in the circumferential direction will then

be .~ J<r;€s :. - -,

E: E:.

and the str ain in the axial direction will be

Eoi :: ~ -vd;.-E -E..

E ;. Yd()'-Ung' .. modulus and v= Poisson's rat io.

( 2)

The total strain energy per unit volume is

U = <fa" t.)( + Cfi. €s .'-z.. z..

Substituting (1) and (2) into (3) then

U =<h.. [~ -!§ ] + ~ r~ -:e5J-z. E. E '- L E:.. E.

35

(3)

(4)

The strain energy per unit volume at the yield point of the

. material is

~'t.U =~,y 2E..

where <1y = yield stressin pure tension_(5)

The strain energy theory says that the material starts to

flow when Uy = U.

~1. =1- (~1. -\- ~l. _ -Va;(f~)zE. 2E:

<r:,~ :: <J~"\. + ~"l.. -"VG""; ~L. (6)

The Hencky-von Mises theory says the material begins to

flow when

From test results this seems to be the most acceptable

(7)

theory. Solving for 4l. then-------,{- <r; + \ / r.-"t\Jot.. :: Z - V \Jr - (8)

36

If the pressure on a tube is external, the hoop stress Ci'i 1s

compressive and equation 8 becomes

(9)

a;~now represents the yield stress in a circumferential

direction under combined external pressure and axial tension,

which 1s less than it was when an external pressure was

acting alone. The equation for the case of internal pressure

and an end stress can be determined from equation 8 also.

Similarily, other equations can be found from 8 if the

algelr aic signs are adjusted.

A plot of equation 8 is shown in figure 8 and a plot of

are4't.(f}.

ratios, ther e

equation 9 is shown in figure 9. Note that the curves

plotted with the ratios ~ as abscissae and the ratios~y

as ordinates. Since the plotted values are

are many materials which are represented by these curves,

It has been shown by Edwards and Miller by experiment

that some tubes behave as indicated by formUlas 8 and 9. (6)

(6) S.H.Edwards and C,P,Miller, Discussion on the Effectof Combined LongitUdinal Loading and External Pressure on thestrength of Oil-Well Casing, API Drilling and Production Prac­~ice. pp. 483-502. 1939.

The aforementioned investigators also showed by experi_p

ment that the 8ubstitutoion of the r etto rr for the ordinate~ Cy..1: is a valid assumption, where p 1s the reduced collapsingcry .

pressure due to the effect of an axial stress and Pcr is the

collapsing pressure as defined by equation 1 Part III. The

ratio 1: Multiplied by 100 gives the percent of the collapsePGr

pre.sur e Pcr to be used when an axial load 1s applied.

This percentage is also plotted in figure 9.

39

lH~ p. + ~ ,. ~;t~l -J-p...1-_+-~-++:.. +t-++:-.l--+-H.-t-H-H/

40

CONCLUSIONS

An important conclusion is that tUbing collapse is

similar to the buckling encountered in long columns and

much of the same reasoning can be applied to both pro blems.

It can also be concluded that 1n the elastic case. the

effect. of a compressive or tensile stress acting at the end

of the tube has little effect on long tubes and should be

considered in the case of short tUbes. For short tubes the

effect of axial end tension 1s to raise the collapsing ext.

ernal pressure and the effect of axial end compression is

to lower the collapsing pressure.

In the plastic region it is important to notice that

the yield strength of the tube material 1s lowered or raised

~ an end load acting simultaneously with a fluid pressure

depending upon whether the fluid pressure is external or

internal and whether or not it acts with an end load of

tension or oompression. A spec1f1c example is the case of

an external pressure and an end load in tension. Here the

yield strength 1s lowered along the circumferential direc.

tion (and thus the collapsing pressure) by the end tensile

stress. This is the case of many oil well casing problems.

To prove effectively the theoretical work done here,

tests should be made on the actual casing as to the actual

variation of the reduced modUlus of elasticity and Poisson's

ratio in the plastic range. This would probably enable the

petroleum industry to raise their safety factors on casing.

41

The results of this work show that the collapsing

pressure of the moderately thick-walled casing 1s least

affected by an end tension in the range of higher D/t ratios.

SUMMARY

The following are the important points brought out in

this thesis:

1. An expr ession for the collapsing pressur. and

critical stress for the elastic case of thin

tUbing was derived.

2. An expression for the collapsing pressure and

critical stress with an axial end compression

or end tension of thin tubing was derived.

3. Recognition of the two major cases of behavior

of tubing, namely, the elastic and plastic

case was made.

4. An expression for the collapsing pressure and

critical stress for the plastic case of aoder­

ately thick-walled tubing was found.

5. An expression for use in determining the

change of the yield point in a tube under

fluid pressure in the plastic range by an

axial end load was found •.

6. A curve to illustrate point no. 5 wss plotted.

7. A curve to show the decresse in collapsing

pressure and critical stresses as a result of

an axial end tensile stress was plotted.

42

43

BI BLIOGRAPHY

1. Books:

a. Prescott, J. Applied elasticity. London,Longmans, Green and Co., 1924. pp 530_564

b. Tlmoshenko, S. Theory of elasticity. N.Y.,McGraw-Hill, 1934.pp. 416

c. Timoshenko, S. Theory of elastic stability.N.Y., McGraw-Hill, 1936. PP. 204-221

2. Periodicals:

a. C1inedinst, W. O. Collapse safety factors fortapered casing strings. Oil Weekly. Vol. 118,p. 50 (June 25, 1945)

b. Clinedinst, W. O. Drill pipe yield strengthlowered by severe drilling service. Oil & GasJournal. Vol. 45, pp 76-77 (Nov. 2, 1946)

c. Cooley, H. M. Factors involved in design oflong casing str ings. Oi 1 & Gas J ournal. Vol.42, pp. 25-28 (J an. 20, 1944)

d. Jasper, T. M. Correct setting depth for welleasing depends on compression yield point.Oil & Gas Journal. Vo 1. 34, p. 81 (Nov. 14, 1935)

e. Kemler, E. N. The design of casing strings.Oil Weekly. Vol. 103, p. 21 (Dec. 1, 1941).Vol. 104, p. 36 (Dec. 8, 1944). Vol. 104, p. 33(Dec. 15, 1941). Vol. 104, p. 36 (Dec. 22, 1944)Vol. 104, p. 34 (Dec 29, 1941).

f. O'Donnell, L. & Crake, W. S. Mechanical causes ofcasing failure and practices for their control.Oil & Gas Journal. Vol. 42, P. 46 (Dec. 16, 1943)

g. Porter, L. E. Factors to be considered indetermining proper casing size. Oil Weekly. Vol.102, pp. 81-82 (July 14, 1941)

h. Wais, J. Recent developments in casing standardsand design. Oil Weekly. Vol. 126, PP. 37-38(June 9, 1947)

44

BI BLIOGBAPHY(cont.)

3. Publications of Learned Societies:

&. A.P.I. Information on collapsing pressures andsetting depths tor easing. Division of ProductionBulletin No. 5-C-2. 2nd edition, Mar. (1940)

b. Blaine, B. W•• Dunlop, C. A•• and Kemler. E. N.Setting depths for casing. A.P.I. Drilling &Prod. Practice. Pp. 125-182 (1940)

c. C11nedinst t W.O. A rational expre ssion for thecritical collapsing pressure ot pipe under exter­nal pressure. A.P.I. Drilling & Prod. Practice.PP. 383-391 (1939)

d. Edwards. S. H. and Miller, C. p. Discussion onthe e f fe cto! combined longitudinal loading andexternal pressure on the strength of oil-wellcasing. A.P.I. Drilling & Prod. Practice.PP. 483-502 (1939)

e. Holmquist. J. L. and Nadai. A. A theoretical andexperimental approach to the problem of collapseof deep-well casing. A.P.I. Drilling and Prod.Praetics. pp. 392-420 (1939)

f. Main, W. C. Combining bending and hoop stressesto determine collapsing pressure of oil-countryt.ubular goods. A.P.I. Drilling and Prod. PracticePp. 421-431 (1939)

4. UnpUblished Material:

a. Bobo, R. Developments in casing design. Paperwritten for production department of PhillipsPetroleum Co. (1946)

b. Sturm, R. G. A study of the collapsing pressureot thin-walled cylinders. Doctorate thesis,Un i v. 0 f I 11., Ur bana • Ill. ( 1936 )

45

VITA

The author was born 1n Chicago on November ? 1922.

After his primary and secondary education. he entered Morton

Jr. College of Cicero. I111n01s in September 1940. Graduat_

ing in June 1942 he was later enrolled as a student in the

Mechanical Engineering Department of the Illinois Institute

of Technology. Upon his graduation from I.I.T. in February

1944, he went to work as a tool designer for Thompson

Products Ine. of Cleveland. Ohio. Working in this plant

until July 1944, he left to enter the service of the Unit.ed

Statea~ Naval Reserve, as a member of the Civil Engineer

Cor pa. Appr oximate 1y two year s later after hi a separ ation

from active duty with the navy in August 1946 he went to

work as an engineer for the Master Manufacturing Co. of

Chicago.

The author was then appointed instructor 1n mechanical

engineering in February 1947 at the Jlissourl School of Mines

and Metallurgy where he completed enough work for a Master's

degree in Mechanical Engineering.