bending moment capacity of pipes

8/3/2019 Bending Moment Capacity of Pipes

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Offshore Mechanical and Arctic Engineering, July 11-16, 1999

BENDING MOMENT CAPACITY OF PIPES

Sren Hauch and Yong Bai

American Bureau of Shipping

Offshore Technology Department

Houston, Texas

USA

ABSTRACT

In most modern pipeline design, the required minimum wall

thickness is determined based on a maximum allowable hoopstress under design pressure. This is an efficient way to come up

with an initial wall thickness design, based on the assumption thatpressure will be the governing load. However, a pipeline may be

subjected to additional loads due to installation, seabed contours,impacts and high-pressure/high-temperature operating conditions

for which the bending moment capacity is often the limitingparameter. If in-place analyses for the optimal route predict thatthe maximum allowable moment to a pipeline is going to beexceeded, it will be necessary to either increase the wall thickness

or, more conventionally, to perform seabed intervention to reducethe bending of the pipe.

In this paper the bending moment capacity for metallic pipes has

been investigated with the intention of optimising the costeffectiveness in the seabed intervention design withoutcompromising the safety of the pipe. The focus has been on thederivation of an analytical solution for the ultimate load carrying

capacity of pipes subjected to combined pressure, longitudinalforce and bending. The derived analytical solution has beenthoroughly compared against results obtained by the finite elementmethod.

The result of the study is a set of equations for calculating the

maximum allowable bending moment including proposed safetyfactors for different target safety levels. The maximum allowable

moment is given as a function of initial out-of-roundness, truelongitudinal force and internal/external overpressure. The

equations can be used for materials with isotropic as well as an-isotropic stress/strain characteristics in the longitudinal and hoopdirection. The analytical approach given herein may also be usedfor risers and piping if safety factors are calibrated in accordance

with appropriate target safety levels.

Keywords: Local buckling, Collapse, Capacity, Bending,Pressure, Longitudinal force, Metallic pipelines and risers.

NOMENCLATURE

A Area

D Average diameter

E Youngs modulus

F True longitudinal force

Fl Ultimate true longitudinal force

f0 Initial out-of-roundness

M Moment

MC Bending moment capacity

Mp Ultimate (plastic) moment

p Pressure

pc Characteristic collapse pressure

pe External pressurepel Elastic collapse pressure

pi Internal pressure

pl Ultimate pressure

pp Plastic collapse pressure

py Yield pressure

r Average pipe radius

SMTS Specified Minimum Tensile Strength

SMYS Specified Minimum Yield Strength

t Nominal wall thickness

Strength anisotropy factor

y Distance to cross sectional mass centre

C Condition load factor

R Strength utilisation factor Curvature

Poissons ratio

h Hoop stress

hl Limit hoop stress for pure pressure

l Longitudinal stress

ll Limit longitudinal stress for pure longitudinal force

Angle from bending plane to plastic neutral axis

OMAE99, PL-99-5033 Hauch & Bai 1


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INTRODUCTION

Nowadays design of risers and offshore pipelines is often based ona Limit State design approach. In a Limit State design, allforeseeable failure scenarios are considered and the system isdesigned against the failure mode that is most critical to structural

safety. A pipe must sustain installation loads and operationalloads. In addition external loads such as those induced by waves,

current, uneven seabed, trawl-board impact, pullover, expansiondue to temperature changes etc need to be considered. Experience

has shown that the main load effect on offshore pipes is bendingcombined with longitudinal force while subjected to externalhydrostatic pressure during installation and internal pressure whilein operation. A pipe subjected to increased bending may fail due

to local buckling/collapse or fracture, but it is the localbuckling/collapse Limit State that commonly dictates the design.The local buckling and collapse strength of metallic pipes has

been the main subject for many studies in offshore and civil

engineering and this paper should be seen as a supplement to theongoing debate. See Murphey & Langner (1985), Winter et al

(1985), Ellinas (1986), Mohareb et al (1994), Bai et al (1993,1997) etc.

BENDING MOMENT CAPACITY

The pipe cross sectional bending moment is directly proportionalto the pipe curvature, see Figure 1. The example illustrates aninitial straight pipe with low D/t (


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importance in the design of pipelines, but the main parameters

will generally be those that are studied in this paper.

FAILURE MODES

As pointed out in the previous section the ultimate momentcapacity is highly dependent on the amount of longitudinal forceand pressure loads and for cases with high external pressure also

initial out-of-roundness. To clarify the approach used in thedevelopment of the analytical equations and to give a betterunderstanding of the obtained results, characteristics of theultimate strength for pipes subjected to single loads and combinedloads are discussed below.

The cross sectional deformations just before failure of pipessubjected to single loads are shown in Figure 2.

P u r e p r e s s u r eP u r e l o n g i t u dP u r e b e n d i n g

Figure 2: Pipe cross sectional deformation of pipes subjected tosingle loads.

PUREBENDING

A pipe subjected to increasing pure bending will fail as a result of

increased ovalisation of the cross section and reduced slope in thestress-strain curve. Up to a certain level of ovalisation, thedecrease in moment of inertia will be counterbalanced by

increased pipe wall stresses due to strain hardening. When the lossin moment of inertia can no longer be compensated for by thestrain hardening, the moment capacity has been reached andcatastrophic cross sectional collapse will occur if additional

bending is applied. For low D/t, the failure will be initiated on thetensile side of the pipe due to stresses at the outer fibres exceedingthe limiting longitudinal stress. For D/t higher than approximately30-35, the hoop strength of the pipe will be so low compared to

the tensile strength that the failure mode will be an inwardbuckling on the compressive side of the pipe. The geometrical

imperfections (excluding corrosion) that are normally allowed in pipeline design will not significantly influence the moment

capacity for pure bending, and the capacity can be calculated as,SUPERB (1996):

tDSMYSt

DMp

= 20015.005.1 ( 0 )

where D is the average pipe diameter, t the wall thickness and

SMYS the Specified Minimum Yield Strength.

( ) SMYStD /0015.005.1 represents the average

longitudinal cross sectional stress at failure as a function of the

diameter over wall thickness ratio. The average pipe diameter isconservatively used in here while SUPERB used the outer

diameter.

PUREEXTERNAL PRESSURE

Theoretically, a circular pipe without imperfections should

continue being circular when subjected to increasing uniformexternal pressure. However, due to material and/or geometricalimperfections, there will always be a flattening of the pipe, whichwith increased external pressure will end with a total collapse ofthe cross section. The change in out-of-roundness, caused by the

external pressure, introduces circumferential bending stresses,where the highest stresses occur respectively at the top/bottom and

two sides of the flattened cross-section. For low D/t ratios,material softening will occur at these points and the points will

behave as a kind of hinge at collapse. The average hoop stress atfailure due to external pressure changes with the D/t ratio. For

small D/t ratios, the failure is governed by yielding of the crosssection, while for larger D/t ratios it is governed by elastic

buckling. By elastic buckling is meant that the collapse occursbefore the average hoop stress over the cross section has reached

the yield stress. At D/t ratios in-between, the failure is acombination of yielding and elastic collapse.

Several formulations have been proposed for estimating the

external collapse pressure, but in this paper, only Timoshenkosand Haagsmas equations are described. Timoshenkos equation,which gives the pressure at beginning yield in the extreme fibres,will in general represent a lower bound, while Haagsmas

equation, using a fully plastic yielding condition, will represent anupper bound for the collapse pressure. The collapse pressure of

pipes is very dependent on geometrical imperfections and here inspecial initial out-of-roundness. Both Timoshenkos and

Haagsmas collapse equation account for initial out-of-roundnessinside the range that is normally allowed in pipeline design.

Timoshenkos equation giving the pressure causing yield at theextreme pipe fibre:

05.11 02 =+

++ elpcelpc ppppt

Dfpp

( 0 )

where:

pel =

3

2 )1(

2

D

tE

( 0 )

pp =D

tSMYS 2 ( 0 )

and:pc = Characteristic collapse pressure

f0 = Initial out-of-roundness, (Dmax-Dmin)/DD = Average diameter

t = Wall thicknessSMYS = Specified Minimum Yield Strength, hoop directionE = Youngs Module

= Poissons ratio



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It should be noted that the pressure pc determined in accordanceto Eq. (2) is lower than the actual collapse pressure of the pipe and

it becomes equal to the latter only in the case of a perfectly roundpipe. Hence, by using pc calculated from Eq. (2) as the ultimate

value of pressure, the results will normally be on the safe side(Timoshenko and Gere, 1961).

Haagsmas equation giving the pressure at which fully plastic

yielding over the wall thickness occurs can be expressed as:

020223 =+

+ pelcpelpcelc pppt

Dfpppppp

( 0 )

and represent the theoretical upper bound for the collapsepressure. For low D/t, the collapse pressure will be closer to thecollapse pressure calculated by Haagsmas equation than that

calculated by Timoshenkos equation (Haagsma and Schaap,1981).

The use of Timoshenkos and Haagsmas equations relatesspecifically to pipes with initially linear elastic material propertieswhere the elastic collapse pressure can be derived from classicalanalysis. This would be appropriate for seamless pipes or for pipesthat have been subjected to an annealing process. However, for

pipes fabricated using the UO, TRB or UOE method there aresignificant non-linearitys in the material properties in the hoop

direction, due to residual strains and the Bauschinger effect. Theseeffects may be accounted for by introducing a strength reduction

factor to the plastic collapse pressure term given by Eq. (4). In thisstudy no attempt has been given to this reduction factor, but

according to DNV 2000 the plastic collapse pressure is to bereduced with 7% for UO and TRB pipes and with 15% for UOE

pipes.

PUREINTERNAL PRESSURE

For Pure internal pressure, the failure mode will be bursting of thecross-section. Due to the pressure, the pipe cross section expands

and the pipe wall thickness decreases. The decrease in pipe wallthickness is compensated for by an increase in the hoop stress. At

a certain pressure, the material strain hardening can no longercompensate for the pipe wall thinning and the maximum internal

pressure has been reached. The bursting pressure can inaccordance with API (1998) be given as:

( )D

tSMTSSMYSp

burst

+=2

5.0 ( 0 )

where ( )SMTSSMYS +5.0 is the hoop stress at failure.

PURETENSION

For pure tension, the failure of the pipe, as for bursting, will be aresult of pipe wall thinning. When the longitudinal tensile force isincreased, the pipe cross section will narrow down and the pipewall thickness decrease. At a certain tensile force, the cross

sectional area of the pipe will be reduced so much that themaximum tensile stress for the pipe material is reached. An

additional increase in tensile force will now cause the pipe to fail.

The ultimate tensile force can be calculated as:

( ) ASMTSSMYSFl += 5.0 ( 0 )

where A is the cross sectional area and

( )SMTSSMYS +5.0 the longitudinal tensile stress at

failure.

PURECOMPRESSION

A pipe subjected to increasing compressive force will be subjectedto Euler buckling. If the compressive force is further increased,the pipe will finally fail due to local buckling. If the pipe isrestrained except for in the longitudinal direction, the maximum

compressive force may be taken as:

( ) ASMTSSMYSFl += 5.0 ( 0 )

where A is the cross sectional area and

( )SMTSSMYS +5.0 the longitudinal compressive stressat failure.

COMBINED LOADS

For pipes subjected to single loads, the failure is, as describedabove, dominated by either longitudinal or hoop stresses. Thisinteraction can, neglecting the radial stress component and theshear stress components, be described as:

122

2

2

2

=+

hl

h

hlll

hl

ll

l

( 0 )

where l is the applied longitudinal stress, h the applied hoop

stress and ll and hl the limit stress in their respective direction.

The limit stress may differ depending on whether the applied load

is compressive or tensile. is a strength anisotropy factordepending on the ratio between the limit stress in the longitudinal

and hoop direction respectively. The following definition for thestrength anisotropy factor has been suggested by the authors ofthis paper for external and internal overpressure respectively:

l

c

F

pD

=

4

2

( 0 )

l

b

F

pD

=

4

2

( 0 )

For pipes under combined pressure and longitudinal force, Eq. (9)

may be used to find the pipe strength capacity. Alternatives to Eq.(9) are Von Mises, Trescas, Hills and Tsai-Hills yield condition.

Experimental tests have been performed by e.g. Corona andKyriakides (1988). For combined pressure and longitudinal force,

the failure mode will be similar to the ones for single loads.

In general, the ultimate strength interaction between longitudinalforce and bending may be expressed by the fully plasticinteraction curve for tubular cross-sections. However, if D/t ishigher than 35, local buckling may occur at the compressive side,

leading to a failure slightly inside the fully plastic interactioncurve, Chen and Sohal (1988). When tension is dominating, the



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pipe capacity will be higher than the fully plastic condition due to

tensile and strain-hardening effects.

As indicated in Figure 2, pressure and bending both lead to a crosssectional failure. Bending will always lead to ovalisation and

finally collapse, while pipes fails in different modes for externaland internal overpressure. When bending is combined with

external overpressure, both loads will tend to increase theovalisation, which leads to a rapid decrease in capacity. For

bending combined with internal overpressure, the two failuremodes work against each other and thereby strengthen the pipe.For high internal overpressure, the collapse will always beinitiated on the tensile side of the pipe due to stresses at the outer

fibres exceeding the material limit tensile stress. On thecompressive side of the pipe, the high internal pressure will tendto initiate an outward buckle, which will increase the pipediameter locally and thereby increase the moment of inertia and

the bending moment capacity of the pipe. The moment capacitywill therefore be expected to be higher for internal overpressurecompared with a corresponding external pressure.

ADDITIONAL FAILUREMODE

In addition to the failure modes described above, fracture is a possible failure mode for all the described load conditions. Inparticular for the combination of tension, high internal pressureand bending, it is important to check against fracture because of

the high tensile stress level at the limit bending moment. Thefracture criteria are not included in this paper, but shall beaddressed in design.

EXPRESSION FOR ULTIMATE MOMENT CAPACITY

In the following section, an analytical solution to the ultimatemoment capacity for pipes subjected to combined loads is derived.

To keep the complexity of the equations on a reasonable level, thefollowing assumptions have been made:

The pipe is geometrically perfect except for initial out-of-roundness

The cross sectional geometry does not change before the

ultimate moment is reached

The cross sectional stress distribution at failure can beidealised in accordance with Figure 3.

The interaction between limit longitudinal and hoop stress

can be described in accordance with Eq. (9)

FAILURELIMITSTRESS

The pipe wall stress condition for the bending moment Limit Statecan be considered as that of a material under bi-axial loads. It is in

here assumed that the interaction between average cross sectionallongitudinal and hoop stress at pipe failure can be described by

Eq. (12). The failure limit stresses are here, neglecting the radial

stress component and the shear stress components, described as afunction of the longitudinal stress l, the hoop stress h and

the failure limit stresses under uni-axial load ll and hl in

their respective direction. The absolute value of the uni-axial limitstresses, which should not mistakenly be taken as the yield stress,are to be used, while the actual stresses are to be taken as positivewhen in tension and negative when in compression.

122

2

2

2

=+

hl

h

hlll

hl

ll

l

( 0 )

where is a strength anisotropy factor depending on the hl/ llratio.

Solving the second-degree equation for the longitudinal stress l gives:

( )2

211

=

hl

h

ll

hl

h

lll

( 0 )

comp is now defined as the limit longitudinal compressive stress

in the pipe wall and thereby equal to l as determined above withthe negative sign before the square root. The limit tensile stress

tens is accordingly equal to l with the positive sign in front of

the square root.

( )2

211

=

hl

h

llhl

h

llcomp

( 0 )

( )2

211

+=

hl

h

ll

hl

h

lltens

( 0 )

THEBENDINGMOMENT

The bending moment capacity of a pipe can by idealising the crosssectional stress distribution at failure in accordance with Figure 3.,

be calculated as:

( ) tenstenstenscompcompcompC yAyAM hl +=,( 0 )

Where Acomp and Atens are respectively the cross sectional area in

compression and tension, y their mass centres distance to the

pipe mass centre and the idealised stress level.

A t e n s

A c o m p

P l a n o f b e n d i n g

r a v

t

t e n s

y t e

y c o m p

c o m p

P l a s t i cn e u t r a la x e s

Figure 3: Pipe cross section with stress distribution diagram(dashed line) and idealised stress diagram for plastified cross

section (full line).



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For a geometrical perfect circular pipe, the area in compressionand tension can approximately be calculated as:

trAcomp

2= ( 0 )

( ) trAtens

=2 ( 0 )

The distance from the mass centre to the pipe cross section centrecan be taken as:

( )

sinrycomp = ( 0 )

( )

=

sinry tens ( 0 )

where ris the average pipe wall radius and the angle from the

bending plan to the plastic neutral axis. The plastic neutral axis isdefined as the axis at which the longitudinal pipe wall stresses

change from tensile to compressive, see Figure 3.

Inserting Eq. (17) to (20) in Eq. (16) gives the bending momentcapacity as:

( ) ( ) ( ) tenscompC trtrMhl

sin2sin2 22,

+=

( 0 )

LOCATIONOFFULLY PLASTICNEUTRAL AXIS

The angle to the fully plastic neutral axis from the plane ofbending can be deduced from the following simplified expressionfor the true longitudinal pipe wall force:

tenstenscompcomp AAF += ( 0 )

where the area in compression Acomp is calculated as:

trAcomp

2= ( 0 )

and the area in tension Atens as;

( ) trAtens = 2 ( 0 )

Giving:

( )(tenscomp

trF += 2 ( 0 )

Solving Eq. (25) for gives:

( )tenscomptens

tr

trF

=2

2( 0 )

or

( )l

t enscomp

tensltrF

2, =

= ( 0 )

FINAL EXPRESSIONFOR MOMENTCAPACITY

Substituting the expression for the plastic neutral axis, Eq. (27),into the equation for the moment capacity, Eq. (21) gives:

( )

( ) comp

tenscomp

tensl

C trtrM hl

+

= sin2sin2 22,

( 0 )and substituting the expression for tensile and compressive stress,

Eq. (14) and (15) into Eq. (28) gives the final expression for thebending moment capacity:

( ) ( )

(

=

2

22

,

11

2cos114

ll

l

hl

h

llC trM hl

( 0 )or alternatively and more useful in design situations:

( ) ( )

( )

=

2

2

2

,

11

2cos11

l

l

ppFC

p

p

p

F

F

p

pMM

( 0 )where

MC = Ultimate bending moment capacityMp = Plastic moment

p = Pressure acting on the pipepl = Ultimate pressure capacity

F = True longitudinal force acting on the pipeFl = True longitudinal ultimate force

When the uni-axial limit stress in the circumferential and

longitudinal direction are taken as the material yield stress and

set to , Eq. (29) and (30) specialises to that presented by among

others Winter et al (1985) and Mohareb et al (1994).

APPLICABLERANGEFOR MOMENTCAPACITYEQUATION

To avoid complex solutions when solving Eq. (30), theexpressions under the square root must be positive, which givesthe theoretical range for the pressure to:

22 1

1

1

1

lp

p( 0 )

where the ultimate pressure pl depends on the load condition and

on the ratio between the limit force and the limit pressure.

Since the wall thickness design is based on the operating pressureof the pipeline, this range should not give any problems in the

design.



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Given the physical limitation that the angle to the plastic neutral

axis must be between 0 and 180 degrees, the equation is valid forthe following range of longitudinal force:

( ) ( )

2

2

2

1111

+

llll p

p

F

F

p

p

p

p

( 0 )

where the ultimate loads Fl and pl depend on the load condition

and on the ratio between the ultimate true longitudinal force Fland the ultimate pressure pl.

For the design of pipelines, this range is normally not going to

give any problems, but again, the range may be reduced due to thequestion of fracture.

FINITE ELEMENT MODEL

This section describes how a pipe section is modelled using the

finite element method. The finite element method is a method

where a physical system, such as an engineering component orstructure, is divided into small sub regions/elements. Each elementis an essential simple unit in space for which the behaviour can be

calculated by a shape function interpolated from the nodal valuesof the element. This in such a way that inter-element continuitytends to be maintained in the assemblage. Connecting the shapefunctions for each element now forms an approximating function

for the entire physical system. In the finite element formulation,the principles of virtual work together with the established shapefunctions are used to transform the differential equations ofequilibrium into algebraic equations. In a few words, the finite

element method can be defined as a Rayleigh-Ritz method inwhich the approximating field is interpolated in piece wise fashionfrom the degree of freedom that are nodal values of the field. Themodelled pipe section is subject to pressure, longitudinal force and

bending with the purpose of provoking structural failure of the pipe. The deformation pattern at failure will introduce both

geometrical and material non-linearity. The non-linearity of the buckling/collapse phenomenon makes finite element analyses

superior to analytical expressions for estimating the strengthcapacity.

In order to get a reliable finite element prediction of the

buckling/collapse deformation behaviour the following factorsmust be taken into account:

A proper representation of the constitutive law of the pipematerial

A proper representation of the boundary conditions

A proper application of the load sequence The ability to address large deformations, large rotations, and

finite strains

The ability to model/describe all relevant failure modes

The material definition included in the finite element model is ofhigh importance, since the model is subjected to deformations

long into the elasto-plastic range. In the post-buckling phase,strain levels between 10% and 20% are usual and the material

definition should therefore at least be governing up to this level. In

the present analyses, a Ramberg-Osgood stress-strain relationshiphas been used. For this, two points on the stress-strain curve are

required along with the material Youngs modules. The two pointscan be anywhere along the curve, and for the present model,

Specified Minimum Yield Strength (SMYS) associated with astrain of 0.5% and the Specified Minimum Tensile Strength

(SMTS) corresponding to approximately 20% strain has beenused. The material yield limit has been defined as approximately

80% of SMYS.

The advantage in using SMYS and SMTS instead of a stress-straincurve obtained from a specific test is that the statistical uncertainty

in the material stress-strain relation is accounted for. It is therebyensured that the stress-strain curve used in a finite elementanalysis in general will be more conservative than that from aspecific laboratory test.

To reduce computing time, symmetry of the problem has beenused to reduce the finite element model to one-quarter of a pipesection, see Figure 4. The length of the model is two times the

pipe diameter, which in general will be sufficient to catch all

buckling/collapse failure modes.

The general-purpose shell element used in the present model

accounts for finite membrane strains and allows for changes in shellthickness, which makes it suitable for large-strain analysis. The

element definition allows for transverse shear deformation and usesthick shell theory when the shell thickness increases and discreteKirchoff thin shell theory as the thickness decreases.

Figure 4 shows an example of a buckled/collapsed finite elementmodel representing an initial perfect pipe subjected to pure bending.

Figure 4: Model example of buckled/collapsed pipe section.

For a further discussion and verification of the used finite elementmodel, see Bai et al (1993), Mohareb et al (1994), Bruschi et al(1995) and Hauch & Bai (1998).

ANALYTICAL SOLUTION VERSUS FINITE ELEMENT

RESULTS

In the following, the above-presented equations are compared

with results obtained from finite element analyses. First are thecapacity equations for pipes subjected to single loads compared

with finite element results for a D/t ratio from 10 to 60. Secondlythe moment capacity equations for combined longitudinal force,

pressure and bending are compared against finite element results.



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STRENGTHCAPACITYOFPIPESSUBJECTEDTO SINGLELOADS

As a verification of the finite element model, the strengthcapacities for single loads obtained from finite element analysesare compared against the verified analytical expressions describedin the previous sections of this paper. The strength capacity has

been compared for a large range of diameter over wall thicknessto demonstrate the finite element models capability to catch the

right failure mode independently of the D/t ratio.

For all analyses presented in this paper, the average pipe diameteris 0.5088m, SMYS = 450 MPa and SMTS = 530 MPa. In Figure 5

the bending moment capacity found from finite element analysishas been compared against the bending moment capacityequation, Eq. (1). In Figure 6 the limit tensile longitudinal forceEq. (7), in Figure 7 the collapse pressure Eq. (2, 5) and in Figure 8

the bursting pressure Eq. (6) are compared against finite elementresults. The good agreement presented in figure 5-8 between finiteelement results and analytical solutions generally accepted by theindustry, gives good reasons to expect that the finite element

model also give reliable predictions for combined loads.

10 20 30 40 50 600

1

2

3

4

5

6

7x 10

6

Diameter Over Wall Thickness

UltimateMomentCapacity

X = FE results___ = Analytical

Figure 5: Moment capacity as a function of diameter over wall

thickness for a pipe subjected to pure bending.

10 20 30 40 50 600.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

7


UltimateTrueLongitudinalForce = FE results

___ = Analytical

Figure 6: Limit longitudinal force as a function of diameter overwall thickness for a pipe subjected to pure tensile force.

10 20 30 40 50 600

1

2

3

4

5

6

7

8

9x 10

7


CollapsePre

ssure

X = FE results___ = Haagsma

- - - = Timoshenko

Figure 7: Collapse pressure as a function of diameter over wallthickness for a pipe subjected to pure external overpressure.

Initial out-of-roundness f0 equal to 1.5%.

10 20 30 40 50 601

2

3

4

5

6

7

8

9

10

x 107


BurstPressure

X = FE results___

= Analytical

Figure 8: Bursting pressure as a function of diameter over wallthickness for a pipe subjected to pure internal overpressure.

STRENGTHCAPACITYFOR COMBINED LOADS

For the results presented in Figures 9-14 the following pipedimensions have been used:

D/t = 35fo = 1.5 %

SMYS = 450 MPaSMTS = 530 MPa

= 1/5 for external overpressure and 2/3 for

internal overpressure

Figures 9 and 10 show the moment capacity surface given by Eq.(31). In Figure 9, the moment capacity surface is seen from the

external pressure, compressive longitudinal force side and inFigure 10 it is seen from above. Figures 5 to 8 have demonstrated

that for single loads, the failure surface agrees well with finiteelement analyses for a large D/t range. To demonstrate that Eq.(31) also agrees with finite element analyses for combined loads,the failure surface has been cut for different fixed values of

longitudinal force and pressure respectively as demonstrated in



9/13


Figure 10 by the full straight lines. The cuts and respective finite

element results are shown in Figures 11 to 14. In Figure 11 themoment capacity is plotted as a function of pressure. The limit

pressure for external overpressure is given by Haagsmas collapseequation Eq. (5) and the limit pressure for internal overpressure by

the bursting pressure Eq. (6). For the non-pressurised pipe, themoment capacity is given by Eq. (1). In Figure 12, the moment

capacity is plotted as a function of longitudinal force. The limitforce has been given by Eq. (7) and (8). For a given water depth,

the external pressure will be approximately constant, while theaxial force may vary along the pipe. Figure 13 shows the momentcapacity as a function of longitudinal force for an externaloverpressure equal to 0.8 times the collapse pressure calculated by

Haagsmas collapse equation Eq. (5). Figure 14 again shows themoment capacity as a function of longitudinal force, but this timefor an internal overpressure equal to 0.9 times the plastic buckling

pressure given by Eq. (4). Based on the results presented in

Figures 11 to 14, it is concluded that the analytically deducedmoment capacity and finite element results are in good agreementfor the entire range of longitudinal force and pressure. However,the equations tend to be a slightly non-conservative for external

pressure very close to the collapse pressure. This is in agreement

with the previous discussion about Timoshenkos and Haagsmascollapse equations.

Figure 9: Limit bending moment surface as a function of pressure

and longitudinal force.

Figure 10: Limit bending moment surface as a function of

pressure and longitudinal force including cross sections for whichcomparison between analytical solution and results from finite

element analyses has been performed.

-0.5 0 0.5 1

-1

-0.5

0

0.5

1

Pressure / Plastic Collapse Pressure

Moment/PlasticMoment

X = FE results___= Analytical

Figure 11: Normalised bending moment capacity as a function ofpressure. No longitudinal force is applied.

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

True Longitudinal Force / Ultimate True Longitudinal Force


X = FE results___

= Analytical

Figure 12: Normalised bending moment capacity as a function of

longitudinal force. Pressure equal to zero.

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1



X = FE results___ = Analytical

Figure 13: Normalised bending moment capacity as a function oflongitudinal force. Pressure equal to 0.8 times Haagsmascollapse pressure Eq. (5).



10/13


-0.5 0 0.5 1 1.5-1

-0.5

0

0.5

1



X = FE results___

= Analytical

Figure 14: Normalised bending moment capacity as a function oflongitudinal force. Pressure equal to 0.9 times the plastic

buckling pressure Eq. (4).

USAGE/SAFETY FACTORS

The local buckling check can be separated into a check for loadcontrolled situations (bending moment) and one for displacement

controlled situations (strain level). When no usage/safety factorsare applied in the buckling check calculations, the two checks

ought to result in the same bending capacity. In design though,usage/safety factors are introduced to account for modelling and

input uncertainties. The reduction in bending capacity introduced by the usage factors will not be the same for load anddisplacement controlled situations. Due to the pipe moment versusstrain relationship, a higher allowable strength can be achieved for

a given target safety level by using a strain-based criterion than bya moment criterion. In this paper only the allowable bendingmoment criterion is given. This criterion can be used for both loadand displacement controlled situations, but may as mentioned be

overly conservative for displacement controlled situations.

The usage factor approach presented in this paper is based onshrinking the failure surface shown in Figures 9 and 10. Instead of

representing the bending moment capacity, the surface is scaled torepresent the maximum allowable bending moment associatedwith a given target safety level. The shape of the failure surfacegiven Eq. (30) is dictated by four parameters; the plastic moment

Mp, the limit longitudinal force Fl, the limit pressure Pl and the

strength anisotropy factor . To shrink the failure surface usagefactors are applied to the plastic moment, longitudinal limit force

and the limit pressure respectively. The usage factors are functionsof modelling, geometrical and material uncertainties and will

therefore vary for the three capacity parameters. In general, thevariation will be small and for simplification purposes, the mostconservative usage factor may be applied to all capacity loads.

The strength anisotropy factor is a function of the longitudinallimit force and the limit pressure, but for simplicity, no usagefactor has been applied to this parameter. The modellinguncertainty is highly connected to the use of the equation. In the

SUPERB (1996) project, the use of the moment criteria is dividedinto four unlike scenarios; 1) pipelines resting on uneven seabed,2) pressure test condition, 3) continuous stiff supported pipe and4) all other scenarios. To account for the variation in modelling

uncertainty, a condition load factor C is applied to the plastic

moment and the limit longitudinal force. The pressure, which is a

function of internal pressure and water depth, will not besubjected to the same model uncertainty and the condition load

factor will be close to one and is presently ignored. Based on theabove discussion, the maximum allowable bending moment may

be expressed as:

( ) ( )

=

2

2

,2

cos11lRP

p

c

RMpFAllowable

p

pMM

( 0 )where

MAllowable = Allowable bending moment

C = Condition load factor

R = Strength usage factors

The usage/safety factor methodology used in Eq. (33) ensures that

the safety levels are uniformly maintained for all loadcombinations.

In the following guideline for bending strength calculations, thesuggested condition load factor is in accordance with the results

presented in the SUPERB (1996) report, later used in DNV

(2000). The strength usage factors RM, RF and RP are basedon comparison with existing codes and the engineering experience

of the authors.

GUIDELINE FOR BENDING STRENGTH CALCULATIONS

LOCAL BUCKLING:

For pipelines subjected to combined pressure, longitudinal forceand bending, local buckling may occur. The failure mode may

be yielding of the cross section or buckling on the compressiveside of the pipe. The criteria given in this guideline may be usedto calculate the maximum allowable bending moment for agiven scenario. It shall be noted that the maximum allowable

bending moment given in this guideline does not take fractureinto account and that fracture criteria therefore may reduce the

bending capacity of the pipe. This particularly applies for high-tension / high internal pressure load conditions.

LOAD VERSUSDISPLACEMENTCONTROLLED SITUATIONS:

The local buckling check can be separated into a check for loadcontrolled situations (bending moment) and one for

displacement controlled situations (strain level). Due to therelation between applied bending moment and maximum strain

in pipes, a higher allowable strength for a given target safetylevel can be achieved by using a strain-based criterion rather

than a bending moment criterion. The bending moment criterioncan due to this, conservatively be used for both load anddisplacement controlled situations. In this guideline only the

bending moment criterion is given.



11/13


LOCAL BUCKLINGAND ACCUMULATED OUT-OF-ROUNDNESS:

Increased out-of-roundness due to installation and cyclicoperating loads may aggravate local buckling and is to beconsidered. It is recommended that out-of-roundness, due to

through life loads, be simulated using e.g. finite elementanalysis.

MAXIMUMALLOWABLEBENDINGMOMENT:The allowable bending moment for local buckling under loadcontrolled situations can be expressed as:

( ) ( )

=

2

2

, cos11lRP

p

c

RMpFAllowable

p

pMM

whereMAllowable = Allowable bending momentMp = Plastic moment

pl = Limit pressurep = Pressure acting on the pipe

Fl = Limit longitudinal forceF = Longitudinal force acting on the pipe

= Strength anisotropy factor

C = Condition load factor

R = Strength usage factor

STRENGTHANISOTROPYFACTOR:

l

c

F

pD

=

4

2

for external overpressure

l

b

F

pD

=

4

2

for internal overpressure

If possible, the strength anisotropy factor should be verified byfinite element analyses.

PLASTIC(LIMIT) MOMENT:

The limit moment may be given as:

( ) tDSMYSt

DM

PFC

===2

0,00015.005.1

whereSMYS = Specified Minimum Yield Strength in

longitudinal direction

D = Average diameter

t = Wall thickness

LIMITLONGITUDINAL FORCEFOR COMPRESSIONAND TENSION:

The limit longitudinal force may be estimated as:

( ) ASMTSSMYSFl += 5.0where

A = Cross sectional area, which may be

calculated as D t.

SMYS = Specified Minimum Yield Strength in

longitudinal directionSMTS = Specified Minimum Tensile Strength in

longitudinal direction

LIMITPRESSUREFOR EXTERNAL OVERPRESSURECONDITION:

The limit external pressure pl is to be calculated based on:

020223 =+

+ pellpelplell ppp

t

Dfpppppp

where

pel =

3

2 )1(

2

DtE

pp =D

tSMYSfab

2 1)

f0 = Initial out-of-roundness2), (Dmax-Dmin)/D

SMYS = Specified Minimum Yield Strength in hoop

directionE = Youngs Module

= Poissons ratio

Guidance note:1)

fab is 0.925 for pipes fabricated by the UO precess, 0.85

for pipes fabricated by the UOE process and 1 for seamlessor annealed pipes.

2) Out-of-roundness caused during the construction phase anddue to cyclic loading is to be included, but not flattening due

to external water pressure or bending in as-laid position.

LIMITPRESSUREFOR INTERNAL OVERPRESSURECONDITION:

The limit pressure will be equal to the bursting pressure and

may be taken as:

( ) D

t

SMYSSMTSp l2

5.0 +=where

SMYS = Specified Minimum Yield Strength in hoopdirection

SMTS = Specified Minimum Tensile Strength in hoopdirection

LOADAND USAGEFACTORS:

Load factor C and usage factor Rare listed in Table 1.

Table 1: Load and usage factors.Safety Classes

Safety factors

Low Normal High

C

Uneven seabed 1.07 1.07 1.07

Pressure test 0.93 0.93 0.93

Stiff supported 0.82 0.82 0.82

Otherwise 1.00 1.00 1.00

RP Pressure 0.95 0.93 0.90

RF Longitudinal force 0.90 0.85 0.80

RM Moment 0.80 0.73 0.65



12/13


Guidance notes:

- Load Condition Factors may be combined e.g. LoadCondition Factor for pressure test of pipelines resting on

uneven seabed, 1.07 0.93 = 1.00

- Safety class is low for temporary phases. For the operatingphase, safety class is normal and high for area classified aszone 1 and zone 2 respectively.

CONCLUSIONS

The moment capacity equations in the existing codes are for someload conditions overly conservative and for others non-

conservative. This paper presents a new set of design equationsthat are accurate and simple. The derived analytical equationshave been based on the mechanism of failure modes and have

been extensively compared with finite element results. The use of

safety factors has been simplified compared with existing codesand the target safety levels are in accordance with DNV (2000),ISO (1998) and API (1998). The applied safety factormethodology ensures that the target safety levels are uniformly

maintained for all load combinations. It is the hope of the authorsthat this paper will help engineers in their aim to design safer and

more cost-effective pipes.

It is recommended that the strength anisotropy factor be

investigated in more detail.

ACKNOWLEDGEMENT

The authors acknowledge their earlier employer formerly J PKenny A/S now ABB Pipeline and Riser Section for their supportand understanding without which this paper would not have been

possible.

REFERENCES

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Bruschi, R., Monti, P., Bolzoni, G., Tagliaferri, R. (1995), FiniteElement Method as Numerical Laboratory for Analysing Pipeline

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