THE INFLUENCE OF TENSION, PRESSURE AND WEIGHT ON PIPE AND RISER DEFORMATIONS AND STRESSES

c.P. SPARKS

Institut Fra~i' du Ntrole Rueil·Malmaison

France

ABSTRACT

The influence of tension, pressure, and weight on different aspects of pipe and riser behavior has al­ready been the subject of many articles (see references). In spite of this it has frequently been misunderstood, sometimes with serious consequences. The object of this paper is to explain the subject clearly and the approach is therefore deliberately very elementary.

The principal problems of bending, buckling, yield­ing and strains, usually treated in separated papers, are here treated homogeneously. The widely used con­cepts of "effective tension" and "effective stress" are given interpretations, easy to visualise, which are related to analogous concepts in other engineering fields.

Finally, eight diverse particular examples of the influence of tension, pressure and weight on pipe and riser behavior, are discussed.

NOMENCLATURE

a Lateral deflection

E Young's modulus of elasticity

F e Effective Compressive force (F e = - Te)

I Inertia (flexural)

L Length

M Moment

Pe External pressure

Pi Internal pressure

P Axial load

PE Euler buckling load

r External radius e r i Internal radius

S External Section (= 2 for cylinder) e ... r e S1 Internal Section (= ... r? for cylinder)

~

\

External Section of joint

Internal Section of joint 2 = 'I'.r

"Effective tension" (calculated from considerations of apparent weight of pipe/riser plus contents)

Ttw

True wall tension

U Upthrust

Shear force

Apparent linear weight of pipe/riser plus contents

True linear weight (in air) of pipe/riser (without contents)

Wtrue True weight (in air)

E Axial strain

v Poisson's ratio

Pe

Density of external fluid

Pi Density of internal fluid

"c

"f

Circumferential stress

Bending stress

"I.e "Effective stress" (see Eq. 6)

° "End effect stress" (see Eq. 7) P

Radial stress Ttw

0tw True wall axial stress (= g-:s-) e i

0vm Von Mises'equivalent stress

~ An in-wall stress (see Eq. 8)

1. INTRODUCTION

The effects of tension, pressure and weight on pipe behavior have been studied for more than a century. Various aspects of the problem have been treated in numerous articles and text books, between which there are no fundamental contradictions. This wealth of

literature, however, has not sufficed to remove con­fusion and in the past even eminent engineers have publicly disputed certain aspects of the problem. ~ore recently risers have, as a result, at times been lncorrectly tensioned and errors in the calculation of limit stresses have been frequent. More usually the consequences of the confusion are less serious and simply result in long hours of discussion.

There are many ways of deriving the effects of tension, pressure and weight on pipe deformations and stresses-- almost as many ways as there are articles on the subject. The ones included in this paper have been chosen for their simplicity but are not claimed to be therefore more valid than any other proofs that the reader may prefer.

Some authors take up strong positions for or against the use of such concepts as "effective tension" and "effective stress". This author feels that a clear understanding of these concepts is important, and that engineers should be able to calculate correctly in terms of true wall forces and stresses or in terms of effec­tive ones. Examples of the use of both are included at the end of the paper.

2. BASIC PRINCIPLES

It may seem excessively elementary but, before con­sidering the effects of tension, pressure and weight on pipe or riser behavior, it is wise to take a close look at Archimedes' famous law, shown in Fig. 1.

--

U = upthrust = weight of fluid displaced FIG. 1 ARCHlMEDES'LAW

Concerning the law, it is important to realize:

- it can only be applied to pressure fields that are completely closed,

- it says nothing about internal forces (such as those acting on the dashed line in Fig. 1), it says nothing about stresses.

Mathematical proof of the law, for the general case, is not obvious (for non-mathematicians) but proof from phYsical considerations is very simple-- if the law were not true, the fluid within an enclosed pres-

2

sure field would either rise or fall (see Fig. 2)

0\ 1-

" /' = ..-+ , ,

I " '

Thus:

FIG. 2 PRESSURE AND WEIGHT ACTING IN A FLUID. EQUIVALENCE RESULTING FROM RESOLUTION OF FORCES,

IN ANY DIRECTION (OR MOMENTS, ABOUT ANY POINT)

Seldom mentioned in engineering paJers is an im­portant corollary to the law, which is very relevant to pipe and riser problems. Just as the combined ef­fects of the enclOSing pressure field and the self weight of the enclosed fluid can produce no resultant force {in any direction}, so their combined effects can produce no resultant moment anywhere in the fluid {which would be incapable of reSisting the associated stresses}. This implies that the equivalence given in Fig. 2 is true for moments as well as forces. Thus the moment induced anywhere in a submerged body, by a closed pressure field, is exactly the same as that in~ duced by the distributed upthrust.

3. INTERNAL FORCES IN A SUBMERGED BODY

The forces acting on a segment of a submerged body (such as the lower part of the body shown in Fig. 1) include a pressure field that is not closed. In order to apply Archimedes'law, it is necessary to add in the "missing pressure," to close the pressure field, and to deduct the force {Pe,Se} to which such an addition

gives rise. This procedure is shown in Fig. 3, in which the force/pressure system acting on the segment is first decomposed into two parts. The pressure field, once closed, is replaced by the Archimedian up­thrust. But the sum of this upthrust plus the true weight (Wtrue ) is plainly equal to the apparent weight

(Wa ) of the segment.

The force(Pe,Se) equal and opposite to that created

artifiCially by the addition of the "missing pressure" has to be added to the true tension (Ttrue ) acting

normal to the section. The sum of these two forces, as shown in Fig. 3, is the "effective tension" (Te) :

(l)

Figure 3 shows that the effective tension (T ) can be found by resolution of forces normal to the s~ction, and that it only depends on the apparent weight (Wa ) of the segment. Eq. (1) then provides the simplest way of deducing the true tension (T

true,.

The moment (M) and shear force (V) compatible with the true tension (Ttrue )' true weight (Wtrue ) and hydro-

static pressure are identical to those compatible with

(ktapparent weight (Wa ) and the corresponding effec­t.WLo tension (Te ).

4. PIPE/RISER CURVATURE A,..D DEFLECTIONS

The procedure shown in Fig. 3 can be extended and applied to the analysis of forces acting on a segment of pipe or riser. The resultant figure is then slightly more complicated since internal pressure effects must be considered, as well as external ones. This procedure has already been demonstrated in Ref. 17, Fig. 1. There, the total force/pressure system was first decomposed into three parts. The external and internal pressure fields were separately closed by the addition of "missing pressures" so that they could be replaced by the Archimedean upthrust on the segment (plus contents) and the internal fluid weight, to which they are respectively equivalent. The sum of this upthrust, internal fluid weight and the true weiqht (Wt.8U is plainly equal to the apparent weight of the pipe segment plus contents.

In order not to modify the total force system, it was necessary to add, to the true wall tension, forces equal and opposite to those created artificially by the addition of the "missing pressures" mentioned above. The sum of these forces is the "effective tension" (Te) :

(2 )

From the corollary of section 2. the equivalent force systems must produce identical bending effects. They therefore give rise to identical curvatures and deflections.

These results can be obtained even more directly by considering Fig.4, which showns three distinct force/pressure systems each of which is in static equilibrium. The three systems always fit together like pieces of a jig-saw puzzle, no matter what the shape of the segment. By superimposing them as shown, lateral pressures are made to cancel out. The resultant axial force is the effective tension given by Equ.2. The resultant weight term is the sum of the true weight of the pipe wall (Wt. SL), plus the true weight of the contained fluid (Wc• SU, less the true weight of the displaced fluid (Wf. Sl). But this is the definition of the apparent weight (Wa• Sl) of the segment plus contents. (This definition remains valid even when the fluids are compressible).

In the fluid columns, shears are zero and moments are negligible. Thus shears and moments in the equivalent system are the same as those in the pipe wall. They have been omitted from Fig.4 for clarity.

-5.-

'0

It is simplest, for all cases more complicated than that of a uniform cylinder, to calculate pipe/riser curvature and deflections from

the apparent weight (wa) of the pipe/riser plus contents and the corresponding effective tension (Te). This is true of all bending related phenomena, including buckling.

Those who prefer to calculate bending effects directly from the true wall tension (Ttw ) and true wall weight (Wt) and hydrostatic pressures, may do so and will obtain the same results. They must however be parti­curlarly careful to take into account correctly all the effects of the hydrostatic pressures, which is not entirely straightforward for complicated riser geometries.

Eq. 2 is valid for pipes of any cross section, made from any material. Axial resolution shows that, in the static case, the evolution of the effective tension ( c5Te / c5U depends only on the apparent weight of the pipe/riser plus contents. For a riser, therefore, the simplest procedure is to calculate the effective tension (Te) at any point, from the top tension and the apparent weight (Wa) of the intervening length of the riser (plus contents). The true wall tension (Ttw) can then be found from Eq. (2).

- Upthrust

M ~\ Tc......- (Effective tension) r ......- .

/......-V 'W ! I (Apparent weight)

FIG. 3 EQUIVALENT FORCE SYSTEMS ACTING ON PART OF A SUBMERGED BODY

5. EFFECTIVE TENSION - A PHYSICAL INTERPRETATION

Effective tension (Te) is a force with a physical interpretation that can be easily visualized. Once it has been recognized, the correct formulation of the ef­fective tension equation becomes straightforward, even for very complicated examples. To understand it, it is first necessary to consider briefly the method of anal­ysis of composite columns.

Engineers familiar with this subject know that,

I

Wt·bl- , ~

I ~I Ttw

Pip, wall

f

Pj.Sj P •. S.

Internal fluid column Di,placed fluid column

W a. 0 l (AHarlllt ...... of iii ... ,au, Cllltllltl,

T e (Effectiv, tln,iln)

-s--

nen calculating bending effects in axially loaded composite columns, it is necessary to consider:

i) the "total axial force" acting in the column section;

ii) the "total bending moment" developed in the section; (or the "total flexural rigidity" (EI) in the case of a column of elastic materials).

These principles are true no matter how many ele­ments of different materials (not necessarily elastic) make up the section, no matter what the adhesion (if any) between them, and no matter what their relative bending stiffnesses. It makes no difference how the "total axial force" is distributed between the various elements, or whether it is concentrated principally or entirely in only one of them. The same is true for the distribution of the "total bending moment." In reality, interaction forces develop naturally between the ele­ments, and assure that they all experience the same ,transverse deformation. These interaction forces do not have to be considered, if the column global behavior only is being analysed.

For those who prefer mathematical explanations, the above principles can be explained in terms of the standard Euler fourth order differential equation (see Eq. (44) of REF. 19), for the simple case of a uniform axially loaded composite beam of elastic materials. If the interaction forces are included, it is possible to write down the differential equation separately for each element, in terms of its rigidity (EI), relative to the beam axis, and its "true axial force". Summing all the equations leads to the cancelling out of all the interaction forces. This procedure then yields the sum of all the rigidities (EI), as coefficient of the fourth order term, and the sum of all the aXial forces, as the coefficient of the second order term. This is in accordance with the principles i) and ii) above. The same argument can still be used even if an element of the section has zero rigidity (EI), as in the case of a cable or fluid column.

.)

FIG. 5 COMPOSITE COLUMN SECTIONS THREE EXAMPLES IN REINFORCED CONCRETE

c)

Fig. 5 shows thre~ examples of composite columns taken from the field of concrete engineering. The pipe/riser problem most closely resembles the column section of Fig. 5b), but with the concrete infill re­placed by the internal fluid. Since such a column is a composite column, according to the description given above, it is interesting to see whether the pipe engi­neer's.concept of "effective tension" corresponds to the concrete engineer's concept of "total axial force". A glance at Eq. (2) shows that the "total axial (ten­sile) force" in the pipe/riser column section (includ­ing contained fluid) is indeed equal to the first two terms (Ttw - Pi.Si) of the equation. Thus, in the absence of external pressure, the two concepts corre­spond.

When external pressure is present, the last term (Pe'Se) of Eq. (2) represents the (compressive) force

in the displaced fluid column, which of course has zero bending resistance. Convention requires that ten­sion be positive. The (tensile) force (-Pe,Se) in the displaced fluid column is the datum tension, which must be deducted from the total column tension (Ttw-p· .Si), to give the "effective" component, as far as ben~ing is concerned. Indeed the same deduction would be neces­sary in the case of any other composite column subject­ed to external pressure. Eq. (2) should logically be rewritten in the form of Eq. (2A). The above reasoning then leads to the physical interpretation of effective tension given below (in italics).

(2A)

. "Effective tension" Is the total force in the pipe/riser column, Including contained fluid(s). less the force in the displaced fluid column (tension positive).

This definition still applies when external axial forces act on a pipe or riser. It also applies when the internal fluid(s) are in a state of flow. Wall friction does modify the internal distribution of forces between the pipe walles) and the internal fluid(s) but does not change the total column force, thus does not change the effective tension.

Once the definition has been understood, the ef­fective tension equation, analogous to Eq. (2), can be written down immediately for even complicated examples. One could imagine, for example, a riser consisting of irregular shaped tubes within tubes, made of different non-elastic materials, filled with compressible fluids of variable density at different pressures! From the definition (in italics), the corresponding effective tension, at any point, would be equal to the sum of the (tensile) forces in all the pipe walls and all the internal fluids less the (tensile) force in the dis­placed fluid column. This result could also be ob­tained by drawing the figure analogous to Fig. 4.

6. PIPE/RISER WALL STRESSES

6.1. General

The axial tension (Ttw ) in the pipe/riser wall, given by Eq. (2), is independent of the pipe material. Likewise the pressure induced circumferential force does not depend on the material. However, the result­ant distribution of stresses (axial, circumferential and radial) within the wall, will depend on the par­ticular characteristics of the material. For example, flexible pipes and those made from fiber reinforced resins, are usually provided with different reinforced layers to resist the different forces. If there is interaction between the layers, this must be considered when calculating the performance of such pipes.

6.2. Thick Walled Elastic Pipes

The formulae for the pressure induced circumferen­tial and radial stresses in a thick walled elastic pipe, which can be found in standard works of reference (see ref. 1 and 2), were first presented by LAME in 1831. LAME showed that the mean of the two stresses is the same at all points within the wall section

a +0 cr· (-2- = constant). This implies that the resulting

axial strain (the Poisson effect) is the same at all points in the cross section. Since plane transverse sections therefore remain plane, the wall axial stress

/(atw) is also constant across the section. This stress can be found by dividing Eq. (2) by the riser wall sec­tion (Se-Si). The triaxial stresses are finally:

• axial

LAME jCirCUmferential

equations} radial

where:

• "effective strese"

• "end effect stress"

a - T P

T e

s:s.-e 1. Pi·SCPe·Se

S -So e 1.

(3)

(4)

(5)

(6)

(7)

The "effective stress" (a e) is so called since it is equal to the effective ten~1.on (Te) divided by the wall section (Se-Si). The "end effect stress" (ap ) is so called since it is equal to the axial stress that would be induced in the pipe wall, by internal and ex­ternal pressures, if the pipe were cut and closed at the section under conSideration.

FIG. 6 EQUIVALENT IN-WALL STRESS SYSTEMS ACTING IN THICK WALLED ELASTIC TUBES

(induced by pressure and tension)

It is interesting to note that the "end effect stress" (ap ) appears in all three equations (3, 4, 5). This seems surprising, but the appendix of this paper shows that it is not just an extraordinary coincidence.

Resolution of stresses (or the Mohr stress circle) leads to two simple ways of representing the wall stresses. These are shown on the elemental stress cubes of Fig. 6. The stresses on the two cubes are completely equivalent. Neither way of representing them is more valid than the other. The right hand cube shows that at all points within the wall section, there are the following:

a constant hydrostatic stress (a p )' equal to the "end effect stress". This stress may be com­pressive or tensile;

- a distortional shear stress (Tl acting on planes at 45° to radial directions. This stress varies across the wall section and has its greatest va­lue on the inner surface (where Sr = Silo

- a constant axial "effective stress" (aLe).

7. EFFECTIVE STRESS - A PHYSICAL INTERPRETATION

It is a fact that effective stress (aLe)' as given in Section 6, is equal to the effective tension divided by the pipe/riser wall section, but that is not the justification for its use. It has a real physical significance which can be easily visualised, as in Fig. 6.

The effectilJt! stress (a 2e) is the amount by which the (l)Cia/ wall

.trt!ss (a tw) exceeds the in-wall hydrostatic strt!ss (a p).

Such a stress is familiar to soil mechanics engi­neers used to performing triaxial tests on soil samples. An important parameter in such a test (see Fig. 7) is the deviator stress, which is the difference between the vertical stress and the hydrostatic pressure acting on the sample. This deviator stress is analogous to the effective stress in the pipe wall.

"'~==============~I

FIG. 7 TRIAXIAL APPARATUS FOR TESTING SOIL SAMPLES

It is true that effective stress, which can only be applied to elastic cylindrical tubes, is of more limited ~nterest than effective tension, which can be universally apptied. If used with care, however, it can sometimes considerably simplify otherwise obscure problems (see Section 10).

8. LIMIT STRESSES

, The differential pressure and corresponding wall stresses, that provoke the onset of yielding, have been the subject of research for more than a century. The Tresca criteria (1864) is still oft~n used, but the von Mises' criteria (1913) has found more general accept­ance.

It usually considered that the threshold of yield­ing has been reached once the von Mises'equivalent stress (avm) reaches the yield stress at some point in the pipe wall. This stress, for a triaxial system, is given by:

2 a2 vm

2 2 2 (a l -a2 ) + (a

2-a

3) + (a

3-a

l) +

2 2 2 6('12 + '23 + '31) (9)

The stresses shown on the two cubes of Fig. 6 are precisely equivalent and give the fOflowing equations for the von Mises'stress:

2 a2 = (a _a)2 + (a -a )2 + (a -at

)2 (10) vm tw c c r r w

or

222 avm = a I.e + 3., (11)

It seems remarkable at first that it is the effec­tive stress (ate) that appears in Eq. (11), and not the true axial stress (atw)' However, a glance at Fig. 6 shows this must ~e so, since a hydrostatic stress (ap ) can have no influence on the von Mises'stress.

In the author's experience more errors are commit­ted using Eq. (10) than Eq. (11). However, since the von Mises'stress (avm) is of such importance, it is perhaps advisable to calculate it both ways (see example 10.6).

When a pipe is subjected to bending, the resultant bending stresses (af) must be added to the axial stresses (atw and ate)' The von Mises'stress (avm) must then be checked on the inner surface, where, is a maximum, and on the outer surface, where the bending stress is a maximum (see example 10.6).

9. AXIAL STRAIN

It has already been pointed out (see section 6.2) that, in an elastic tube, the mean of the circumferen­tial and radial stresses is constant and equal to the "end effect stress" (a p.) at all points in the wall section. The axial strain (£) can therefore be ex­pressed in terms of the true wall stress (atw ) as:

(12)

or in terms of the effective stress (ate) as:

0" 0 £ : ~ + ':£(1-2,,)

E E (13)

The in wall shear stress (,) can plainly produce no axial strain (see Fig. 6) and therefore does not appear in Eq. (13).

10. PARTICULAR EXAMPLES

The following are a selection of particular exam­ples that have provoked much discussion in the past, in the author's and other's experience. They include buckling problems, in which compressive forces are considered positive (in contrast with the rest of the paper) •

10.1. Buckling of Pressurised Tubes

Two identical tubular columns, articulated and free to slide at extremities, are shown in Fig. 8. If one tube is subjected to internal pressure (Pi), while the other is not, which of the two columns can withstand the higher axial load (P) without buckling?

I) wilh ialeraal ~ure ~) Willlaul inlernal pressure

FIG. 8 TWO AXIALLY LOADED TUBULAR COLUMNS

The two columns, as shown (Fig. 8) are subjected to the same effective compressive force (Fe:P). Since it is the effective force that controls all bending and buckling phenomena (see section 4), the two columns buckle for the same value of P -- namely for P equal to the Euler buckling load (PE):

·l.EI 7

(14)

The true wall tension (Ttw ) does depend on the in­ternal pressure (see Eq. 2) but, since that is only part of the total force in the column section, it does not control bending and buckling.

10.2. Buckling of a Suspended Drill String

A vertical drill string being lowered into very deep water is shown in Fig. 9a. The question has often been asked whether the large upthrust (U) acting at the lower end can eventually cause the string to buckle.

Provided that the apparent weight of the string is positive, the effective tension is everywhere positive, therefore the string does not buckle, no matter how large the upthrust (U). Indeed a cable (see Fig. 9b) without rigidity, does not buckle either.

Those, who find this question troublesome, may find it helpful to reflect on the reason why the upthrust (U), acting at the lower end, does not even cause the displaced fluid column to buckle (see Fig. 9c). This column, which is equivalent to the limiting case of a

drill string without rigidity and with zero apparent weight, is nevertheless perfectly capable of resisting the axial thrust (U).

t U t U a) Drill stnng b) Cable

II 1\ II II II II II II II II II II II II II II U

t U c) Ditplaced fluid column

FIG. 9 UPTHRUST (U) ACTING AT THE LOWER ENDS OF VERTICAL COLUMNS

10.3. Tension in Connector Bolts

It is sometimes asked whether the force in riser connector bolts (or dogs) is equal to the effective tension (Te) or the wall tension (Ttw ). Such a pro­duction riser connector is shown schematically in Fig. 10.

Slcti,. Sl

s"

.) Bolted .... ICIOf ') VertICIl I .....

FIG: 10 RISER CONNECTOR BOLT TENSION (Tbolts)

The tension in the bolts (Tbolts) is not neces­sarily equal to either of the mentioned tensions. It depends on the Joint seal interior and exterior areas (Sji and Sje), since these determine the forces in the internal, and the displaced, fluid columns, at the connector level. It also depends, of course, on the seal prestress. Ignoring the latter, and taking SL to be the (arbitrary) outer sectional area of the

enclosing cylinder represented by the dashed line (see Fig. 10), the bolt force can be calculated from the true wall tension (Ttw ) or from the effective tension (Te ). In the following, the apparent weights of the connector flanges and bolts are neglected:

a) ~=!~~_!~~~_~~_~~=!~~ (Ttw )

Resolving vertically across the dashed lines of Fig. lOb):

Thus: ,

(Ttw - Pi·Si + Pe·Se) + (Pi·Sji - Pe·Sje)

From Eq. (2) therefore:

(15)

b) ~sing~!~!ive tension

The effective tension in the connector is the same as that in the riser section immediately above or be­low it. Therefore (see Eq. 2):

(16)

It should be noted that the bolts are also subject­ed to the external hydrostatic pressure.

10.4. Failure of Pressurized Tubes with and with­out "End Effect"

If two identical sections of tube (elastic mate­rial) are tested in air, and if one is subjected to "end effect", while the other is not (see Fig. 11), it is interesting to know which tube will withstand the

,higher internal pressure.

It seems surpriSing at first, but it is the tube subjected to "end effect", which withstands the higher pressure. This is immediately clear if effective stresses are considered:

Tube with "end effect" I Tube without "end effect" 1------------There is no exterior force I The true wall axial force acting on the tube. Thus I is zero. Therefore: the total force (effective I tension) is zero. There- I fore: I

T e = 0

a U= 0

I I I I I

CJvm

./3. T (Eq.ll)1

Ttw o

a I.e= - ap (see Eq.3)

CJ = 1a2 + 3l (Eq.ll) vm p

The tube with "end effect" is plainly subjected to a lower von Mises-stress and can therefore resist a higher pressure.

The same conclusion can be reached if true wall stresses are used and substituted into Eq. (10). The true wall force and stresses, for the two cases, are:

" " cr

" + T C P

" - T r p

vm = .13. T (see Eq.lO)

T = 0 tw

" = 0 tw

ol c

cr r

" vm

= cr + T P

= cr - T p

=10 2+3T2 p

., ro. wit_I "etod effect"

FIG. 11 TWO SECTIONS OF TUBE SUBJECTED TO INTERNAL PRESSURE (Pi)

(Eq.10)

Note that the effective force in the tube without "end effect" is not zero. It is compressive and equal to Pi ,Si' the total force in the "column" section.

10.5. Von Mises'stress in Axially Loaded Pressurized Tubes

It is sometimes asked whether the von Mises'stress can be reduced by applying an axial force P (tensile or compressive) to a pressurized tube (see Fig. 12) •

~CC============~============:J

IIIternal pm$UI1 Pi

..!.. <c:::===:===:=' =====:::J

P --

FIG. 12 IDENTICAL PRESSURIZED TUBES SUBJECTED TO TENSILE AND COMPRESSIVE AXIAL LOADS (P)

The two tubes of Fig. 12 are subjected to equal and opposite effective tensions (Te=± P) and therefore to equal and opposite effective stresses. Eq. (11) shows that they are thus subjected to the same von Mises' stress. This stress is a minimum for P = 0 (cf. sec­tion 10.4). An external (effective) axial force (P) whether tensile or compressive, increases the von Mises' stress.

10.6. Calculation of the Von Mises'Stress in a

~

The von IUses' stress (0 vm) can be calculated from true wall triaxial stresses (Eq. 10) or from the ef­fective stress (Eq.1l). Consider a riser (lS 5/S" O.D 1/2" W. T) subjected to the following:

- Effective tension (Te) - Moment (101)

Internal pressure (Pi) - External pressure (Pe)

1,500 KN 150 KN-m

27,000 KN/m2 15,500 KN/m2

In the following, all stresses are in KN/m2.

~!~~~_~~~~~~!~ri~!!~ ~~!!!m0!2_~~~~~

2r. = 0,4471 m Pi·SCPe,Se

79,526 " Se-Si ~ p

2r 0,4734 (Pi-Pe)Se

106,526 m T. e ~ S -So e ~

S. 0,157 2 (PCPe)Si 95,026 m T S -So ~ e e ~

S = 0,176 2 ± M =S7,065 m "n l·ri e

I 0,00050 4 ± M ~71,OlO = m "fe l·re

Using true wall triaxial -----streSSeS---- I ~~ng_=!!=~!j.v=~tr~~~

Tt =T +(~~~-p S )=3011 KN II T = 1,500 KN we ~~ee e T

Ttw I " = e 0tw=g::s.- = 158,474 II I.e Se-Si

e l.

"c • ("p + 't)

cr - (cr - 't) r p

a (Eq. 10) VI

I Internal External I surface surface I -------- -------- I 225,539 229,484 I ("te + "r) 186,052 174,552 I

I T -27,000 -15,000 I 235,293 222,659 I ".1 (Eq. 11)

78,947

Internal External surface surface

146,012 149,957

lU6,526 95,026

235,293 222,659

Note, in this example, the effective tension (Te) was given initially. This would have been found by resolving· forces, knowing the top tension and the ap­parent weight of the intervening length of riser (in­cluding contents). Eq. (2) was then used to calculate the true wall tension (Ttw ). If instead the true wall tension (Ttw ) had been measured, with strain gauges, then Eq. (2) would have been used to give the effective tension (Te ). The calculation of the von Mises'stress, using true wall and effective stresses, would have been the same as shown above.

Since errors are frequent, in this calculation, it is as well, as in this example, to evaluate the von Mises'stresses from the true wall triaxial stresses (Eq. 10), as well as from the effective stress (Eq.ll). Calculations using thin walled approximate stress· for­mulae are even more prone to error.

10.7. Pressure Induced Buckling of a Tube with Sliding Expansion Joint

Such a tube, in air, articulated at extremities, is shown in Fig. 13. It is interesting to know what internal pressure will cause the tube to buckle and thus the joint to fail.

As in the first case discussed (section 10.1) the tube will buckle when the effective force (Fe) reaches the Euler buckling load (PE), given by Eq. (14). The pressure required to produce this force depends on the details of the interior of the sliding joinc. As shown in Fig. 13, the fluid can penetrate the joint, which is of section Sj' The effective compressive force (Fe) in

the joint, and therefore in the rest of the tube, is:

F e

(17)

Sliding joint

"\.{ I) Articulated tube with expansion iDint

f b) Detail 01 expansion jOint

FIG. 13 PRESSURE INDUCED BUCKLING OF TUBE WITH EXPANSION JOINT

Buckling of the tube, and thus failure of the joint, takes place for Pi,Sj = PE•

10.S. Pressure Induced Buckling of a Tube without Expansion Joint

Pressure can cause such a tube to buckle, if axial elongation is prevented. Fig. 14 shows a tube, artic­ulated at extremities, and subjected to internal and external pressures.

~ f ... ----L------tf ~ a) Prt-buckling - uial stmn constant (mu)

bl Pust·Backling - effecbwe force conshnt (PEl

FIG. 14 PRESSURE INDUCED BUCKLING OF TUBE WITHOUT EXPANSION JOINT

As the internal pressure is increased the effec­tive compressive force (Fe) in the tube will increase. Once this reaches the Euler load (PE)' buckling takes place. Pre-buckling and post-buckling behavior are quite different.

Up until buckling takes place the axial strain re­mains equal to zero. From this condition and from the internal and external pressures, it is possible to calculate the true wall tension (Ttw ) of the tube. The effective compressive force (Fe) in the tube is then (see Eq. 2):

(18)

Buckling takes place when the effective force (Fe) reaches the Euler force (PEl given by Eq.(14).

For an elastic tube the procedure can be simplified. Eq.(13) gives the effective (tensile) stress (ale) directly, since the strain is zero:

ft

ale a £=0=£+ :(1-2v) (19)

Buckling therefore takes place for:

(20)

From Eq.s (19) and (20) the stress ~ ) required to produce buckling can be found. The coFresponding internal pressure can then be calculated from Eq.(7).

b) ~!-b~cklin~_~~~avi~£ (see Fig. 14)

Once the tube begins to buckle, the strain (E) no longer remains equal to zero. On the other hand, after buckling, the effective force (Fe) remains constant, equal to the Euler buckling force (PE), given by Eq. (14) •

For a non-elastic tube, the true axial tension (Ttw) can be found from Eq. (18) and therefore the axial strain (e) can be deduced.

For an elastic tube, the effective stress (a1

) remains constant, as given by Eq. (20). The axial strain (E) can then be deduced from Eq. (13). It in­creases with 0p'

For a sinusoidal buckle, it is simple to show that the lateral deflection (a) is related to the strain (E) by the following (see Fig. 14):

Thus, the lateral nal pressure (Pi) can The maximum moment in

11. CONCLUSIONS

(21)

deflection (a) induced by inter­be found from Eq.s (13) and (21). the buckled tube is then PE.a.

The concept of "effective tension" can be used to simplify considerably the calculation of bending (and buckling) problems. Likewise, many problems related to elastic tubes can be simplified by using the concept of "effective stress". In ordE't" to avoid errors however, it is helpful to have a clear understanding of the physical significance of these two concepts. In sections 5 and 7 of this paper, they are given physical interpretations, which have not been mentioned in pre­vious papers.

Those who use such concepts should do so with care and should always be capable of visual ising the complete stress system acting within the pipe wall. Equivalent ways of representing the stresses, induced by tension and pressure, are shown in Fig. 6 of this paper.

The examples of section 10 show how these concepts can be applied to a number of different pipe/riser problems.

12. REFERENCES

1. Timoshenko S.- "Strength of Materials" 3rd Ed. Pt II, Kriegas, Huntington, New-York, 1976.

2. Hill R.- "Plasticity". Oxford University Press, 1964, pp. 106-109, 119.

3. Hawk1.ns M.F. and Lamont N.- "The Analysis of Axial Stresses in Drill Stems". Drill. and Proc. Prac., API, 1949, pp. 358-369.

4. Klinkenberg. A.- "Neutral Zones in Drill Pipe and Casing and their significance in Relation to Buck­ling and Collapse". Drill. and Prod. Prac., API, 1951, pp. 64-79.

5. Burrows W.R., Michel R., Ranking A.W.- "A Wall Thick­ness Formula for High Pressure, High Temperature Piping". Trans., ASME, April 1954, pp. 427-444.

6. Morrison J.L.M., Crossland B. and Parry J.S.C.-"The Strength of Thick Cylinders Subjected to Repea­ted Internal Pressure". Journal of Engineering for Industry. ASME, May 1960, pp. 143-153.

7. Fischer W. and Ludwig M.- "Design of Floating Vessel Drilling Riser". Journal of Petroleum Technology, March 1966, pp. 272-280.

8. Palmer A.C. and Baldry J.A.S.-"Lateral Buckling of Axially Constrained Pipelines". J.P.T., Nov. 1974, pp. 1283-1284.

9. Tilbe J.R., Van der Horst G.- "Risers: Key cause to N.Sea Down Time". Oil and Gas Journal, March 10th, 1975, pp. 71-73.

10. Lubinski A.- "Influence of Neutral Axial Stress on Yield and Collapse of Pipe". Journal of Engineering for Industry. ASME, May 1975, pp. 400-407.

11. Morgan G.W. and Peret J .W.- "Applied Mechanics of Marine Riser Systems". Petroleum Engineer, Oct.1974, May 1976.

12. "!organ G.W.- "Analysing Top Tension". Ocean Resour­ces En~ineering. Feb. 1977 pp. 40-50. April 1977, pp. 12-24.

13. Lubinski A.- "Necessary Tension in Marine Risers" Revu~ de 1'Institut Fran~ais du Petrole, vol.XXXII n 0 2, :1arch-April 1977, pp. 233-256. Nov.-Dec. 1977 pp. 873-895.

14. Goins W.C.- "Better Understanding Prevents Tubular Buckling Problems". World Oil. Jan. 1980, pp. 101-106. Feb. 1980, pp. 35-40.

15. Pattillo P.D. and Randall B.V.- "Two Unresolved Problems in Well Bore Hydrostatics". Petroleum Engineer International, July 1980, pp. 24-32.

16. Bournazel C.- "Confsequences de l'effet de fond sur l'instabilite des conduites soumises a une pression interne". Revue de l'Institut Franc;;ais du Petrole, Jan.-Feb. 1980, vol XXXV n 0 1, pp. 79-100.

17. Sparks C.P.- "Mechanical Behavior of Marine Risers. Mode of Influence of Principal Parameters". Journal of Energy Resources TechnolOgy. ASME, Dec. 1980, pp. 214-222.

18. Bernitsas M.M. and Papala:nbrosP.-"Riser Design Optimization Under Gener!'.lized StatiC Load", Inter­maritec"80, Hamburg, Germany, Sept 1980, pp.129-139.

19. McIver D.B. and Olson R.J.- "Effective Tension- Now you see it, now you don't!". 37th Mechanical Engi­neering Workshop and Conference Petroleum, Division ASME, Dallas, Texas, Sept. 13-15 1981, pp. 177-187.

20. Bernitsas M.- "Problems of Marine Riser Design". Marine Technology, Vol. 19 nO I, Janv. 1982 , pp.73-82.

21. Chakrabarti S.K. and Frampton R.E.- "Review of Riser Analysis Techniques". Applied Ocean Research, vol. 4, nO 2, April 1982. pp. 73-90.

13. APPENDIX

The concept of "effective stress" is justified and useful because the LAME equations (4, 5) for a thick walled elastic tube show that the mean of the pressure induced Circumferential and radial stresses is constant across the section and equal to the "end effect stress" (see section 6):

'" a p (22)

It is interesting to ask whether this is limited to elastic tubes.

Fig. 15 shows the circumferential and radial stresses acting on an element within any pipe wall (not necessarily elastic). Resolution in the radial direc­tion leads to :

2a .6r + r.6a r r

FIG. 15 CIRCUMFERENTIAL AND RADIAL STRESSES ACTING ON AN ELEMENT WITHIN A PIPE WALL

(23)

No assumptions about the pipe material character·­istics were necessary in deriving Eq. (23). Thus the

a + a c r integral of (----2---) across the wall section of a pipe

is aways equal to the "end effect fOl'ce." The special characteristic of elastic tubes is that the mean of the two stresses is constant at all points of the wall sec­tion. Thus Eq. (22) is only valid for elastic tubes, whereas Eq. (23) is valid for any circular tube.

12..