SAGE Profile V6.3.2 User Manual - Volume 3

SAGE Profile Version 6.3.2 User Manual

VOLUME 3

THEORY MANUAL Prepared By Fugro Engineers SA/NV Document Ref.: SAGE Profile V6.3.2 User Manual - Volume 3.doc Revision: 04 Date: 17/03/2005

Document Title:

SAGE Profile V6.3.2 - User Manual - Volume 3 - 04 Theory Manual

Document Ref.:

SAGE Profile V6.3.2 User Manual - Volume 3.doc

REVISION STATUS Rev

Status

Compiled

Reviewed

by

Date

by

Date

00

Issued for Comments

MMA

July 2003

JFW

July 2003

01

Issued for V6.3

MMA

Oct 2003

DCA

Nov 2003

02

Issued for V6.3.1

HFA

May 2004

JWI

May 2004

03

Issued for V6.3.1

MMA

Sept 2004

04

Issued for V6.3.2

RDE

Feb 2005

HFA

Mar 2005

Signatory Le Legend: DCA MMA HFA JWI

Approved by

Date

JWI

Nov 2003

JWI

Mar. 2005

David Ca Cathie Matthi thieu Ma Mallié Hendrik Falepin Jean-François Win Wintgens

Copyright © 1995 - 2005 Fugro Engineers SA/NV No warranty, expressed or implied is offered as to the accuracy of results from this program. The program should not be used for design unless caution is exercised in interpreting the results, and independent independent calculations are available to verify the general correctness of the results. Fugro Engineers SA/NV accept no responsibility for the results of the program and will not be deemed responsible for any liability arising from use of the program.

Document Title:

SAGE Profile V6.3.2 - User Manual - Volume 3 - 04 Theory Manual

Document Ref.:

SAGE Profile V6.3.2 User Manual - Volume 3.doc

REVISION STATUS Rev

Status

Compiled

Reviewed

by

Date

by

Date

00

Issued for Comments

MMA

July 2003

JFW

July 2003

01

Issued for V6.3

MMA

Oct 2003

DCA

Nov 2003

02

Issued for V6.3.1

HFA

May 2004

JWI

May 2004

03

Issued for V6.3.1

MMA

Sept 2004

04

Issued for V6.3.2

RDE

Feb 2005

HFA

Mar 2005

Signatory Le Legend: DCA MMA HFA JWI

Approved by

Date

JWI

Nov 2003

JWI

Mar. 2005

David Ca Cathie Matthi thieu Ma Mallié Hendrik Falepin Jean-François Win Wintgens

Copyright © 1995 - 2005 Fugro Engineers SA/NV No warranty, expressed or implied is offered as to the accuracy of results from this program. The program should not be used for design unless caution is exercised in interpreting the results, and independent independent calculations are available to verify the general correctness of the results. Fugro Engineers SA/NV accept no responsibility for the results of the program and will not be deemed responsible for any liability arising from use of the program.

CONTENT

Content 1. 1.1. 1. 1. 1.2. 1. 2.

GENERAL DESCRIPTION ........................ ............................. ............................ ............... 5 Intr In trod oduc ucti tion on........................... ........................... ............................ ............................ ............................. ....... 5 Overvi Over view ew.......................... .......................... ............................. ............................ ............................ ............. 6 1.2. 1. 2.1. 1. Gene Ge nera ral l ............................ ............................ ............................ ............................ ............................. .. 5 1.2.2. 1.2 .2.

Non-l No n-line ineari aritie ties s ............................ ............................ ............................. .................... 6

1.2. 1. 2.3. 3.

Capa Ca pabi bili liti ties es ........................... ............................ ............................. ......................... 7

1.2.4. 1.2.4. Pipeli Pip eline ne Ana Analys lysis is Fea Featur tures es ............................ ............................. ............................ 8 1.2.5. 1.2 .5. Main Mai n Ass Assum umpti ption ons s .......................... ............................. ............................ ............... 8 1.3.. Pip 1.3 Pipeli eline ne Mod Modell elling ing.......................... .......................... ............................ ............................. ....................... 10

2.

1.3. 1. 3.1. 1.

Mesh Me shin ing g th the e pi pipe pe ............................ ............................ ............................. ............. 10

1.3. 1. 3.2. 2.

Lay La y Do Down wn......................... ......................... ............................ ............................. ............................ 10

1.3.3. 1.3.3. 1.3.4. 1.3 .4.

Residual Residu al Lay Ten Tensio sion n .......................... ............................. ............................ ........ 11 Soil-P Soi l-Pipe ipe Int Intera eracti ction on........................ ........................ ............................. ............................ ............. 11

1.3.5. 1.3.5. 1.3. 1. 3.6. 6.

Modelling Modellin g Thre Three-Di e-Dimens mensiona ionall Prob Problems lems ........................ ............................. .......... 12 Buri Bu ried ed Pi Pipe pe ........................... ............................ ............................. ....................... 13

1.3.7. 1.3 .7.

Curre Cu rrent nt and Wav Waves es......................... ......................... ............................. ............................ ............. 14

FINITE ELEMENT ............................ ............................ ............................ ........................ 16 2.1. 2.1. 2.2.. 2.2

Elementt Des Elemen Descri cripti ption on............................ ............................ ............................ ............................. .................. 16 Elemen Ele mentt Loa Loadin ding g............................. ............................ ............................ ........................ 17 2.2.1. 2.2.1. 2.2.2.. 2.2.2

Generall Lo Genera Loadi ading ng ........................ ........................ ............................. ............................ .................. 17 Internal Inter nal and exter external nal pres pressure sures s ........................... ............................ ..................... 17

2.2. 2. 2.3. 3.

Temp Te mper erat atur ure e ......................... ............................ ............................. ....................... 19

2.2.4. 2.2.4. Effect Eff ective ive axi axial al for force ce......................... ......................... ............................. ............................ ............. 19 2.3.. Mat 2.3 Materi erial al Pro Proper pertie ties s ......................... ............................ ............................. ....................... 20

2.4. 2. 4.

2.3.1. 2.3. 1. 2.3. 2. 3.2. 2.

Pipe Pl Pipe Plas asti tici city ty ............................ ............................ ............................ ............................. .................. 20 Oval Ov alis isat atio ion n ............................ ............................ ............................ ........................ 26

2.3. 2. 3.3. 3.

Soil So il Pl Plas asti tici city ty ......................... ......................... ............................ ............................. ....................... 29

Cont Co ntac actt ............................ ............................. ............................ ............................ ........... 34 2.4.1. 2.4. 1. 2.4.2.. 2.4.2

Introd Intr oduc ucti tion on........................... ........................... ............................ ............................. ....................... 34 Synchron Sync hronous ous Cont Contact act Algo Algorith rithm m .......................... ............................ ..................... 34

2.4.3. 2.4.3. Asynchro Asyn chronous nous Conta Contact ct Algo Algorithm rithm ............................. ............................ ................ 37 2.5. Geom Geometric etrical al Non-L Non-Linear inearity ity............................ ............................ ............................. ............................ ........ 38 2.6.. 2.6

Soluti Sol ution on Tec Techniq hniques ues ........................... ............................ ............................. .................. 39

2.6.1. 2.6.1. Incremen Incre mental/It tal/Iterati erative ve Proce Process ss ......................... ............................ .......................... 40 2.7.. Sig 2.7 Sign n Con Conven ventio tion n ............................. ............................ ............................ ........................ 45 2.8. 2.8. 2.9. 2. 9.

Flow Ch Flow Char arts ts........................... ........................... ............................ ............................ ............................. ..... 48 Glos Gl ossa sary ry .......................... ............................. ............................ ............................ ........... 50

SAGE Profile V6.3.2 - Theory Manual

3

GENERAL DESCRIPTION

3.

REFERENCES ............................ ............................ ............................ ............................. 54


4

GENERAL DESCRIPTION

1. GENERAL DESCRIPTION DESCRIPTION 1.1

Introduction SAGE Profile is a suite of programs for pipeline profile analysis developed by Fugro Engineers SA/NV (Formerly THALES GeoSolutions). Operating with a standard Graphical User Interface (GUI) under Microsoft Windows, the suite enables the full range of pipeline profile analysis tasks to be performed efficiently. Volume 3 (this volume) of the set of documentation for SAGE Profile is concerned with the theory on which SAGE Profile is based and its finite element engine, PipeNet. The series of SAGE Profile documentation comprises: comprises:

•

SAGE Profile User Manual Volume 1: SAGE Profile Interface

•

SAGE Profile User Manual Volume 2: PipeNet

•

SAGE Profile User Manual Volume 3 (this document)

•

SAGE Profile Validation Reports

1.1.1 General PipeNet is a finite element program for two- and three-dimensional pipeline stress analysis. Its capabilities include non-linear pipe bending, non-linear soil response (bearing capacity, and axial and lateral frictional resistance), large deformation analysis and buckling. General forms of loading include: self weight (including piggy-back lines), lay tension, point loads (e.g. anodes), distributed loads (e.g. current), prescribed displacements (e.g. lifting during trenching) as well as internal and external pressures and general temperature distributions. A schematic diagram depicting some of the programs 2D capabilities is shown in Figure 1.


5

GENERAL DESCRIPTION

1.2

Overview

Figure 1: Schematic showing some features of pipe2D. Analyses are divided into loading stages corresponding to different stages in the life of the pipeline. Within each stage the loading is applied incrementally to follow the geometric and material non-linearities which may ensue. Thus, a typical pipeline analysis is defined by a laydown stage, followed by (for example) hydrotest loading in a second stage. A third operational stage could follow. In fact, an unlimited number of loading stages are possible with loading applied in any sequence following laydown.

1.2.1 Non-linearities Non-linearities in SAGE Profile arises from:

•

Material non-linearity: pipe plasticity and ovalisation (see Sections 2.3.1 and 2.3.4)

• •

Soil non-linearity: soil plasticity and friction (see Section 2.3.5)

•

Geometric non-linearity: large displacements (including large rotations) and buckling (see Section 2.5). Contact non-linearity: touch-down/lift-off phenomena (see Section 2.4).

The principal mechanisms of non-linear pipe bending and buckling are illustrated in Figure 2.


6

GENERAL DESCRIPTION

Figure 2: Pipe Bending Non-linearities.

1.2.2 Capabilities

• • • • • • • • • • • • • •

Two- and three-dimensional analysis Euler-Bernoulli beam-column elements Variable pipe diameter and wall thickness General specification of vertical, lateral, axial distributed loads (e.g. weight of pipe, current loads) Point loads at arbitrary spacing and location Prescribed displacements (local and global coordinates) Arbitrarily or uniformly spaced field joints with reduced stiffness Two non-linear pipe moment-curvature models (Moment-curvature defined by the Ramberg-Osgood curve or based on a uniaxial stress-strain curve) General seabed profile including a cross-slope (3D) General pipeline profile in plan (3D) Non-linear vertical, axial and lateral soil support Modelling of column buckling (upheaval and snaking) Pipe air filled or fluid filled, submerged or in air Internal and external fluid pressures


7

GENERAL DESCRIPTION

• • • • • •

General variation of temperature along pipe axis General variation of initial tension along pipe axis End fixity: free, rigid, pinned Calculation of hoop stresses and true wall stresses Restart capability Output control and plot file

1.2.3 Pipeline Analysis Features Stress analysis of a pipeline in a finite element model requires special handling of the circular pipe cross-section if simple beam-column elements are used as in PipeNet. While the beam-column elements model satisfactorily the overall pipeline configuration, the analysis must handle the effects of internal and external fluid pressure on the stresses in the pipe wall as well as the thermal strains that can develop due to temperature effects. All of the most important features of circular pipe behaviour are included in PipeNet:

• Effect of internal and external pressure on the ends of the pipe • Hoop stress developed by internal and external pressure • Axial force developed in the pipe wall due to the Poisson’s effect of the hoop stress

• • • •

Thermal strains and axial force in pipe wall due to temperature changes Output of true wall stress and hoop stress for equivalent stress calculation Lay tension may be specifically included Sag tensions are calculated automatically as a result of modelling geometric changes (large deformations)

• Moment-curvature relationship based on longitudinal stresses in the pipe wall and the uniaxial stress strain behaviour, accounting for the biaxial stress conditions in the pipe wall

1.2.4 Main Assumptions All finite element modelling makes certain simplifying assumptions regarding reality. The principal assumptions existing in the present version of SAGE Profile are noted in this section.

• Plane sections normal to the pipe axis remain plane. • Bending moment can be obtained by integrating for σx around the pipe wall as shown in Equation .

• Axial non-linearity arising from pure axial stresses/strains is assumed to be negligible.

• The pipe section is thin-walled (i.e. the wall thickness is less than about one tenth of the radius). Therefore: The hoop stress is considered to be constant over the pipe wall (i.e. no o variation with the radial co-ordinate). Biaxial stress state prevails in the pipe wall (effect of radial stresses/strains is negligible).

o

• Material stress-strain curve can be approximated by the Ramberg-Osgood form.


8

GENERAL DESCRIPTION

• Non-linearities in the vertical soil support curves and in the explicit momentcurvature method are non-linear rather than plastic. Unloading results in the same non-linear relationship being followed. For axial soil springs and for the stress-strain based moment curvature model, unloading following plasticity is elastic.

• The pipe is assumed to have zero diameter for seabed contact - the soil friction and support acts on the centreline of the pipe.

• Soil supports are "lumped" at each node and act in translation only. Soil rotational stiffness is ignored.

• For 3D analysis, the non-linearity in the pipe (Ramberg-Osgood model) is not history dependent. The single moment-curvature relationship is assumed to apply on the plane of maximum bending moment, irrespective of previous loading history (i.e. bending on other planes).

• External water pressures are calculated based on the seabed elevation. This avoids continual updating of the loading at each loading increment when most depth variations are very small for profile analysis.


9

GENERAL DESCRIPTION

1.3

Pipeline Modelling

1.3.1 Meshing the pipe The pipe mesh is based on the discrete seabed profile. The seabed is defined by a series of KP’s versus Elevation points, called KP points. To each defined KP point of the seabed, there will be a corresponding pipe node. In other words, the coordinates of ith pipe node will be equal to the coordinates of the i th KP point:

x = KPi ∀i  i yi = Elevationi

Equation 1 With

• xi,yi • KPi • Elevationi

horizontal and vertical coordinates of pipe node i horizontal coordinates of the i th KP point vertical coordinate of the i th KP point

If the *HORIZONTAL_PIPE option is activated, the pipe is set horizontal and touching the highest point of the seabed. Therefore, Equation 2

x = KPi ∀i  i yi = Max(Elevationk , ∀k )

In this case, the initial pipe length, on which the weight loading will be computed, is equal to the “KP length”. The “KP length” corresponds to the length of the seabed profile projection on the horizontal axis. The mesh can be refined either by increasing the number of KP points either by directly interpolating the seabed or by specifying the number of pipe elements between two seabed KP’s (see control parameter 13 or the “FE Tune” window of the interface). Important note: SAGE Profile interface will always start the analyses using the *HORIZONTAL_PIPE option and then lay the pipeline on the seabed nodes (see Section 1.3.2).

1.3.2 Lay Down Pipelay is simulated in a simplified manner by "lowering" the pipeline in 2D to the frictionless seabed in a series of increments (typically 100 or more) selected by the program such that no more than one node touches down per increment. The standard boundary conditions for lay down consist in fixing axially the left-hand extremity. In order to stabilise the solution, very soft springs are added at both ends. When a point touches down the soil stiffness is invoked; conversely, if a point is lifting off the soil stiffness is removed. Two different contact algorithms can be used:

•

the Synchronous Touch-Down (see Section 2.4.2)

•

the Asynchronous Touch-Down (see Section 2.4.3)

The pipelay process is thus one of the non-linear problems that the program solves. Note that usually the axial friction is not activated during the lay down phase to avoid any axial constraint. In 3D the lay down is done also in 2D. Once the pipeline is on the seabed the model is then deformed into 3D (see Section 1.3.5).


10

GENERAL DESCRIPTION

1.3.3 Residual Lay Tension Before the laying process the pipe lay tension may be modelled by specifying the appropriate nodal load at the right extremity of the pipeline.

1.3.4 Soil-Pipe Interaction The soil is modelled as a series of lumped springs distributed along the pipeline route as sketched in Figure 3. The soil springs models are described in Section 2.3.5. The pipe-seabed contact algorithms are described in section 2.4. SpringForce

SpringForce ux

y (Posit ive Upwards)

Frictional Spring uy Vertical Spring

KP

PipeLine

Seabed

Figure 3: Soil-Pipeline Interaction


11

GENERAL DESCRIPTION

1.3.5 Modelling Three-Dimensional Problems Three-dimensional analyses in PipeNet require additional information to 2D analyses. Some of this information is indicated schematically on Figure 4

Figure 4: Additional features for 3D pipe model.

In particular, the route co-ordinates and 3D boundary conditions must be supplied along with a definition of the lateral soil resistance for the pipe. Although PipeNet is a general analysis program, a recommended method for performing 3D pipeline analysis has been developed. This consists in performing the pipe laydown in the normal 2D way - equivalent to ignoring the plan pipe curvature effects during the lay process. Equilibrium in the vertical plane (including pipe nonlinearity) is thus found. The second step is to "switch" into 3D mode using the appropriate keyword. This process involves the calculation of the out-of-balance moments due to the plan curvature of the pipe and the application of these loads to t he laterally restrained pipeline (by soil resistance) in full 3D mode. A true equilibrium position is found in which the soil resistance and the internal pipe bending moments are in equilibrium. Some lateral movement of the pipeline is observed at this stage.


12

GENERAL DESCRIPTION

Once the equilibrium positions on the seabed in both the vertical and seabed plane have been found the analysis can proceed in the normal way.

1.3.6 Buried Pipe A risk of upheaval buckling exists for subsea buried pipelines subject to a large temperature increase and axial restraint (due to friction with the sea bed for example). This tendency of the pipe to lift is restricted by the weight of soil above the pipe and by the shear resistance generated along the failure surface in the backfill soil. SAGE Profile interface allows the user to add an extra uniformly distributed load (UDL) due to soil cover (i.e. backfill). Basically this udl is equal to the weight of soil cover. However, it can also incorporate the equivalent shear resistance of the backfill by using the two following models: Schaminee’s formula

 Z  FCover = γ 'De Z1 + f  D e  

Equation 3 With ·

Fcover

Uplift resistance (equivalent UDL)

·

γ'

Soil submerged unit weight (kN/m)

·

De

Pipe diameter (m)

·

Z

Cover depth (m)

·

f

Uplift coefficient

Pedersen’s formula Equation 4

  D   Z  1 + D e  FCover = γ 'D e Z1 + 0.1 e  + f   2Z   Z D   e   


   

2

13

GENERAL DESCRIPTION

1.3.7 Current and Waves Currents and waves action can be taken into account as an equivalent UDL using the Morison’s equations. DnV RP E 30.5 is used to calculate the forces on the pipeline due to current and wave as described below. Forces acting in the x- and z-directions:

•

Drag force (kN/m):

f d

Equation 5

•

1

= ρ w D e C d (U s cos βi + U c )2 2

Inertia force (kN/m)

Equation 6

ΠD e2

 2ΠUS   sin βi γ w C m   TU 

f i

=

f l

= γ w D e Cl (U s cos βi + U c )2

4

Forces acting in the y-direction:

•

Lift force (kN/m)

Equation 7

1

2

With

• • • • • • • • •

γw

Sea water unit weight (kN/m³)

De

Pipe diameter (m)

Us

Significant near bottom velocity amplitude (m/s)

Uc

Current velocity (m/s)

Tu

Mean zero up crossing period (s)

βi

Incidence angle (°)

Cd

Drag coefficient

Cl

Lift coefficient

Cm

Inertia coefficient

The approach angle of the current/waves, β (measured relative to north, anticlockwise being positive) must be entered for 3D analysis only. In 2D analysis, the angle of incidence does not have to be entered. The lift force fl is always computed assuming β = 0°. The drag and inertia forces, f d and f i respectively, are computed for all values of the maxima are used as input in the analysis.


β. Only

14

GENERAL DESCRIPTION

In the y-direction: Equation 8

UDL = f l (βi

1

= 0) = ρ w D e C l (U s + U c )2 2

In the x-direction: Equation 9

UDL − X = − sin(βi ) max(f d

+ f i )

UDL − Z = − cos(βi ) max(f d

+ f i )

In the z-direction: Equation 10


15

FINITE ELEMENT

2. FINITE ELEMENT 2.1

Element Description Modelling of the pipeline is performed using a finite element approach with standard six degree of freedom in 2D and twelve degree of freedom in 3D elastic beam-column elements (axial, lateral, and rotational deformations). Elastic behaviour of the pipeline is defined by the elastic modulus (E), the moment of inertia (I) and the cross-sectional area (A s) of the pipe wall.

uy1

DIEGREESOFFREEDOM

uy2

θz1 ux1 A, E, I Node 1

ux2 Node 2

θz2 ELEMENTFORCES M1

M2

N1 T1

T2

N2

Figure 5: Bernoulli Beam 2D Element.

The effect of axial load on the element stiffness is included (Cook et al, 1989) using a stability function method (Smith and Griffiths, 1988).


16

FINITE ELEMENT

2.2

Element Loading

2.2.1 General Loading Point loads and distributed loads are computed in the normal way (see Cook et al, 1989). Loadings may be specified as distributed by defining a KP range or point by point defining a single KP value. Distributed loading applies to pipe weight, weight of contents, pressure, temperature, current and waves, etc. Point definitions are useful f or defining pipe lay tension, anode weight, clump weights, buoyancy attachments, etc.

2.2.2 Internal and external pressures The effects of external water pressure and internal fluid pressure on pipelines and risers has been elaborated by Sparks (1980, 1984) and Chakrabati and Frampton (1982). These works form the basis of the implementation in SAGE Profile. While the beam-column elements model satisfactorily the overall pipeline configuration, the analysis must handle the effects of internal and external fluid pressure on the stresses in the pipe wall as well as the thermal strains that can develop due to temperature effects. The most important features of circular pipe behaviour are:

•

Effect of internal and external pressure on the ends of the pipe

•

Hoop stress developed by internal and external pressure

•

Axial force developed in the pipe wall due to the Poisson’s effect of the hoop stress

•

Thermal strains and axial force in pipe wall due to temperature changes (see section 2.2.3)

Note that the formulae presented in this section are based on the assumption of thinwalled pipe. A pipe can be considered as thin-walled if its wall thickness is less than about on e tenth of its radius (see Roark and Young, 1975). This section provides a summary of the implementation following the work of Hoskins (1982). An axially restrained pipe is subject to both axial and hoop stresses as a result of the external and internal pressure loading on the pipe wall and over the cross-section of the pipe. These may be quantified approximately (Hoskins, 1982): Hoop stress: Equation 11

σh =

( pi Di - pe De ) 2t

Axial wall stress: Equation 12

σa =

ν ( pi Di - pe De ) 2t

Resulting in an axial tensile force due to the Poisson effect (F p1) which may be approximated as:


17

FINITE ELEMENT

Equation 13

F p1 =

π ν ( pi Di2 - pe De2 ) 2

The axial force induced by fluid pressure (F p2) over the cross-section is compressive if pi > pe, thus yielding Equation 14

F p2 =

- π ( pi Di2 - pe De2 ) 4

Thus a resultant axial force over the whole cross-section for a fully restrained pipe (often called the effective axial force) of F p = Fp1 + Fp2 is obtained: Equation 15

F p = -

π 4

( pi Di - pe De ) ( 1 - 2 ν ) 2

2

Note that if pi > pe then σa > 0 (tensile) while F p < 0 (compressive), resulting in a tendency of the pipe to buckle despite the tensile wall stress. The force Fp is introduced into the finite element analysis as a fixed end force ("body force") and causes deformation in the pipe according to the degree of restraint provided by the boundary conditions and soil resistance. Clearly, the computed effective axial force in the beam/column will be different from this full-restrained value. In order to calculate the axial stress in the wall of the pipeline for stress checks, it is necessary to obtain the true wall axial force from the calculated axial force by removing the end effects of the fluid pressure (Sparks, 1984): Equation 16

Ftw = F - Fi + Fe

Where F is the calculated effective tension or compression (tension positive but with water pressure considered positive in compression), and F i and Fe are the internal and external fluid pressure forces on the section. Equation 17

Fi = - pi Ai Fe = - pe Ae

Alternatively, the true wall stress (σtw) can be defined according to: Equation 18

σtw =

Ftw As

Note that the External water pressures are calculated based on the seabed elevations. This avoids continual updating of the loading at each increment when most depth variations are very small for pipeline profile analysis.


18

FINITE ELEMENT

Note also that SAGE Profile includes the internal fluid pressure head in the calculation of the internal pressure pi. We have

p i = p io

Equation 19

+ ρfluidgh

where

•

pio

Hydrotest or operational pressure defined at the mean sea level.

•

ρfluid

Internal fluid density

2.2.3 Temperature For an axially restrained pipe, an increase in temperature creates a compressive force in the pipe wall:

Ft = - As E α th

Equation 20

∆T

With

•

As

Cross-sectional area of steel (m 2)

•

αth

Coefficient of thermal expansion (1/°C)

•

E

Elastic modulus of steel

This force is introduced as a "body force" in the same way as pressure induced loads.

2.2.4 Effective axial force As explained in section 2.2.2, the effective axial force F eff is defined combining Equation 16 and Equation 17 (Sparks, 1984):

Feff = Ftw − pi A i + p e A e Where

•

Ftw

True-wall axial force (kN)

•

pi

Internal pressure (kPa), sum of the test/operational pressure defined at the mean sea level and the pressure induced by the internal fluid.

•

pe

External pressure (kPa)

•

Ai, Ae

Internal and external cross-section (m²)

Ftw is the axial force in the pipe generated by the combination of all applied loads. It is important to notice that the external pressure is applied at the lay down phase during which the pipe is axially unrestrained (see section 1.3.2). Therefore, since the pipe is free to deform axially, the external pressure won’t generate any axial stress, hence no axial load. This definition is in accordance with the DnV recommended formulae for the effective axial force for fully restrained pipe (see section 5C210 znd 5C211, DnV 2000).


19

FINITE ELEMENT

2.3

Material Properties

2.3.1 Pipe Plasticity Two methods of modelling pipe plasticity (bending) are available in SAGE Profile. These are: 1.

The explicit moment-curvature method in which the moment-curvature relationship is expressed in the form of a single Ramberg-Osgood curve.

2.

A stress-strain based moment-curvature relationship in which the momentcurvature relationship is continually updated according to the stress state in the pipe wall.

2.3.2 Moment – Curvature Relationship The spread of plasticity across a pipe section is characterised by a reduction in the stiffness and results in a non-linear moment-curvature relationship. This may be defined in terms of the Ramberg-Osgood equation (Murphey and Langer, 1985; Ramberg and Osgood, 1943):

κ M  M  = + A   κ0 M 0  M0 

Equation 21

B

With

•

κo and Mo

nominal curvature and moment

•

A and B

Ramberg-Osgood coefficients


20

FINITE ELEMENT

B=2 3

A

5 10

1.0

∞

El 1.0 1+A

Figure 6: Ramberg-Osgood Curve

The parameters (A, B, Mo and κo) are chosen to fit the moment-curvature relationship (see Figure 6) obtained by integrating across the section for a given curvature: Equation 22


M = ∫ A σ x y dA

21

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Figure 7: Stress-Strain Curve with σ y at 0.2% strain

The Ramberg-Osgood parameters A, B and M o depend on steel yield stress and section properties D and t. However, if the nominal moment, M o, is nondimensionalised via the yield moment M y, the parameters can be found for a particular steel grade. Myield = 2 σyield I / De

Equation 23

Typically, σy may be defined as the 0.2% proof stress, or as the nominal yield stress corresponding to a nominal strain e.g. 0.5% (see Figure 7). For example, for X65 steel, the integration yields: A = 0.49

B = 9.04

M0 / Myield = 1.175


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2.3.3 Stress-Strain Relationship The stress-strain based moment-curvature model deals with the bending of a circular pipe section under combined axial loading and bending. In this model, the stress state at various points around the circumference forms the basis of the allowable longitudinal stress state, which in turn defines the moment that can be supported by the section. Axial and hoop stress/strain effects are combined with bending effects to define the longitudinal stress and strain state around the pipe wall. Bending in both planes is considered and bending strains are related to curvatures from simple bending theory. The maximum longitudinal stresses are governed by the Von Mises yield criterion and strain hardening properties of the material. Given any particular stress/strain state that satisfies equilibrium and the material stress-strain relations, the incremental stresses for a known increment of strain can be found. Thus, by an incremental procedure the stress at the new strain state can be found. The moment supported by the pipe may be found by integrating the longitudinal stress around the section.

Figure 8: Pipe Bending The formulation of bending in a single plane is presented below. Bending in both planes is obtained by analogy.


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FINITE ELEMENT

The strain in the longitudinal direction, Figure 8) is given by

εx , at a distance y from the pipe centroid (see

Equation 24

εx = ε - εi + (κezz + κ pzz) y

With

•

ε

•

εI

initial axial strain arising from thermal loading and from the Poisson effect of hoop strains/stresses caused by pressure loads.

•

kzze, kzzp

deformation strain calculated from the change in axial length

elastic and plastic curvatures about the z-axis

For the case of elastic behaviour, the associated stresses in the longitudinal direction can be calculated from:

σx = σtw + BF E κezz y

Equation 25 Where

•

σtw

true wall stress, defined as the mean axial stress in pipe wall

•

BF

Bending factor (required by some codes practice)

•

E

elastic modulus

For elastic-plastic behaviour, it is necessary to use appropriate stress-strain relationships involving plastic deformation. It is assumed that the stresses in the pipe wall may be approximated as biaxial; this means that radial stress is assumed negligible. Therefore biaxial stress-strain relations may be developed from standard elasticity and plasticity theory (e.g. Valliappan, 1981). The equivalent stress,

σeq, is given by:

σeq =

Equation 26

(σ2x + σ2h - σx σh + 3 τ2)

With

•

σh

hoop stress

•

τ

shear stress (i.e.

Yielding will occur when

τxh)

σeq becomes equal to the yield stress in uni-axial tension σy.


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FINITE ELEMENT

The equivalent plastic strain εpeq is normally derived from the component plastic strains in the principal directions, which are not computed in the simplified pipe bending model. However, as shown by Klever et al (1994), it can be approximated by basing it on the longitudinal plastic strain εpx and the ratio of the hoop stress to the yield stress using the formula:

ε px

ε = p eq

Equation 27

1-

3 4

m

2

m = σx / σh

where

Since this approximate formula does not account for reductions in the hoop stress, in the program the maximum value calculated during a series of load stages is output. For complex loading cases, is it recommended to base engineering decisions on the longitudinal plastic strain rather than the equivalent strain. From laboratory tests, the relation between uniaxial stress and strain may be obtained for a particular material. In the formulation used in SAGE Profile, a stress-strain curve in the Ramberg-Osgood form has been adopted β

 σ  ε σ  = + α   σ εy yield  σyield 

Equation 28 With

•

εy

nominal yield strain (typically at 0.5%)

•

σyield

nominal yield stress

•

a and b

Ramberg-Osgood parameters.

This equation can be written alternatively as:

 σ  σ   ε = 1 + α    E  σ yield  

   

β−1

Equation 29

For multi-axial stresses and strains, this equation is generalized in terms of equivalent stress and strain. Thus, for any given strain state, the corresponding equivalent stress can be found. In pipe bending analysis the hoop stress and shear stress is assumed to be constant during a load increment. Thus the allowable longitudinal stress σx can be established. This simplified approach enables the effect of hoop stress changes to "harden" or "soften" the moment-curvature relationship, as well as incorporating the "softening" effects of increases in axial stress. It also permits kinematic hardening behaviour to be incorporated whereas the plasticity formulations of Franzen and Stokey (1973) and Klever et al (1994) assume isotropic loading and do not address the effects of changing hoop stress.


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2.3.4 Ovalisation Ovalisation refers to the reduction in circular cross-section diameter, which occurs when a pipe is subject to increasing bending moment. This causes a reduction of the bending stiffness leading to a greater curvature and ovalisation. This rapid loss in bending resistance can eventually lead to bifurcation buckling. At the same time, the strain hardening of the steel walls permits to gradually increase the resisting moment of the pipe. The combined effects of ovalisation and plastic straining therefore result in a flatter moment-curvature relationship (see Figure 9).

Figure 9: Modified Ramberg-Osgood due to Ovalisation Taking into account pipe ovalisation is done by selecting appropriate Ramberg-Osgood coefficients to defined the M-K relation, based on empirical relationship (Murphey, 1985), in order to combine the ovalisation effects with plasticity. Therefore, as implemented in SAGE Profile, ovalisation and plasticity may be treated together when defined by a M-K relation. Equations are derived here below. The critical bending moment M b, corresponding to the critical curvature K b, is the bending moment at which bifurcation buckling occurs or at which the ovalisation effect exceeds the strain hardening effects.


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Figure 10: Ovalisation – Critical Point This limit is dependent on the D/t ratio. At greater curvatures, for D/t ratios < 35, ovalisation results in a gradual fall-off in the moment resistance, whilst for D/t ratios > 50, bifurcation buckling is postulated with a rapid reduction in the moment resistance (see Figure 11).

Figure 11: Ovalisation – Bifurcation Point In practice, the curve defined by the Ramberg-Osgood relationship is considered valid only up to the value of κb. Curvatures beyond κb represent strain states far in excess of those permitted.


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It has been observed on the basis of experiment (Murphey, 1985; Gellin, 1980) that M b is approximately equal to the fully plastic moment, M p. The relationship being given empirically by: Equation 30

M b = M p ( 1.0 - 0.002

De t

)

Corresponding to the bending stress (at the outer fibre) Equation 31

σ b = -

E De κ b 2

=-

M b De 2I

The critical bending strain, εb, for a homogeneous pipe material with strain-hardening properties is reached when the curvature becomes approximately (Murphey and Langer, 1985):

κ b =

Equation 32

t 2

De

The Ramberg-Osgood parameter A is then chosen so that Equation 33

M o = M b,

And

A=

Equation 34

κ b - 1 κ0

Where

κ0 = M0 / E I

Equation 35

The second Ramberg-Osgood parameter B is: Equation 36

B = 16 - 0.07 De / t

The pipe ovalisation, defined as the relative change in diameter, is computed as: Equation 37

ζ=

∆D e De

 D e2   = ζ 0 + γ  κ t  

2

Where ζ 0 is the initial out-of-roundness and γ defined as: Equation 38

D  γ = 0.015   1+ e   120 t 

A typical moment-curvature curve is shown in Figure 9. Note that the non-linear pipe model is based on ovalisation and plasticity due to bending only. Axial force and hydrostatic pressure effects on the moment-curvature relation are not considered. However, these effects may be included on a case-by-case basis by providing SAGE Profile with an appropriate set of A, B, and M o parameters, thus defining a Ramberg-Osgood curve that does include these effects.


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2.3.5 Soil Plasticity Soil Bearing Capacity Soil reaction (soil bearing capacity) may be modelled as piecewise linear spring supporting the pipe, or as a simple elastic-perfectly plastic, or bilinear elastic-plastic curve (see Figure 12).

Figure 12: Typical forms of vertical soil support curves.


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In SAGE Profile, the vertical soil support curves act non-linearly rather than plastically e.g. unloading results in the same non-linear relationship being followed. When the pipe is above the seabed level the springs have zero stiffness. Note that the soil spring curves calculated in SAGE Profile interface take into account the circular shape of the pipe and the increase in bearing area with pipe penetration.

In SAGE Profile, the soil spring curves can be either calculated by different methods or defined manually by the user. The following consists in a brief description of the methods that have been implemented in SAGE Profile.

The two first methods are issued from the DNV standards and apply for both cohesive and cohesionless material. The two others methods presented apply only for cohesive soil. Note that in SAGE Profile, rocks are modelled by high strength clay.

2.3.6 DNV CN30.4 The DNV CN30.4 suggests the following equations for the computations of the vertical soil reaction: Equation 39

For Sands

N q

Equation 40

d q

Equation 42

For Clays

+ γ ' yN q d q ) B

π φ  = e π tan φ tan 2   +   4 2 

N γ

Equation 41

Equation 43

Qu ( y ) = (0.5γ ' BN γ

= 2 1 + N q ) tan φ

= 1 + 1.2 tan(φ )(1 − sin(φ )) 2

  y   Qu ( y ) = 5.14Cu 1 + 0.3 tan( )  + yγ ' B B    

With

•

y

Bearing depth

•

Qu

Vertical soil reaction per unit length at bearing depth y

•

B

Bearing width

•

γ'

Soil submerged unit weight

•

Nq, Ng

Bearing capacity factors (function of the friction angle Φ)

•

Cu

Undrained shear strength


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2.3.7 DNV RP F105 According to the DNV RP F105, the computation of the vertical soil reaction is as follow:

Qu ( y ) = (0.5γ ' BN γ

For Sands

Equation 44

N q

Equation 45

π φ  = e π tan φ tan 2   +   4 2 

N γ

Equation 46

For Clays

Equation 47

+ γ ' yN q ) B

= 1.5 N q − 1) tan φ Qu ( y ) = (5.14Cu + γ ' y ) B

With the same notations as above.

2.3.8 Verley and Lund method The model presented herein is based on a dimensional analysis and back calculation of pipelines with external diameter from 0.2 to 1m. It has been developed to assess the vertical soil reaction of a pipeline lying on cohesive materials under its own weight. The equations are presented hereafter:

y

Equation 48

D

= 0.0071( SG 0.3 ) 3.2 + 0.062( SG 0.3 ) 0.7 Qu

Equation 49

S =

Equation 50

G =

DCu Cu Dγ

With:

•

y

Bearing depth

•

Qu


•

D

Pipe diameter

•

Cu


•

γ

Soil unit weight


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The following range of application is recommended:

• • • • • •

Cu: 0.8 – 70kPa D: 0.2 – 1m y/D: 0 – 0.35 S: 0.05 – 7.5 G: 0.02 – 5 Specific weight of pipe: 1.06 – 2.5

2.3.9 Buoyancy method This method should be used only with very soft clay. The buoyancy method assumes that the soil behaves like a liquid and that the soil-induced buoyancy of the pipeline is equal to the vertical soil reaction. The equation used in this model is presented hereafter:

Qu ( y ) =

Equation 51

y 6 B

(3 y 2

+ 4 B 2 )γ '

With:

•

y

Bearing depth

•

Qu


•

B

Bearing width

•

γ'


2.3.10 Rigid Seabed A "rigid" model (i.e. a very stiff spring with a stiffness of 107 kN/m/m) is available if no soil response curves are specified or the rigid seabed option is selected in the SAGE Profile interface. Soil Friction Axial and lateral soil resistance is modelled as an elastic-perfectly plastic shaped curve as shown in Figure 13.


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FINITE ELEMENT

Figure 13: Typical forms of axial and lateral soil resistance curves The peak resistance (F max) is defined in general terms as the sum of a frictional and an adhesive (cohesive) component. If V is the vertical force (per unit length), then: Equation 52

Fmax = Fa + µ V

Where µ is the friction factor and F a is the adhesion per unit length (either µ or F a may be zero according to the soil resistance model required, or both may be used together). Axial and lateral resistance use the same form but different values of Fa and µ. When unloading occurs the axial soil resistance behaves as an elastic-plastic material and unloads/reloads elastically when below the yield resistance.


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2.4

Contact

2.4.1 Introduction PipeNet uses two pipe-seabed contact algorithms:

•

Simple touch down algorithm with or without scale-back (“Synchronous TouchDown”): The assumption adopted in the simple touch down algorithm is that all contact between the pipe and the seabed is restricted to a pipe node touching down on a seabed “node” with the same KP value. Contact occurs whether the height of a pipe node is above or below the corresponding seabed node (see Figure 14). Note that this determines the finite element discretisation as the seabed and the pipeline meshes must have the same number of nodes. Automatic load increment size (“scale-back”) can be activated to ensure only one node touches down per load increment and in order to avoid excessive soil reaction forces. This algorithm becomes increasingly approximate as the unevenness of the seabed profile increases.

•

Contact search algorithm (“Asynchronous Touch-Down”): The synchronous touchdown is appropriate for most pipeline problems. However, if the pipe is laid on a steep slope, the pipe nodes (KP points) and the seabed x-co-ordinates (KP points) drift apart (see Figure 16). This leads to testing pipe touchdown against seabed points that are not directly below the pipe nodes

Both algorithms are explained hereafter.

2.4.2 Synchronous Contact Algorithm Unlike the classical two-mesh contact algorithms, which comprise both contactor and target meshes, SAGE Profile synchronous contact algorithm is based on a single mesh, namely that of the pipe. The seabed is represented as a set of nodes forming a boundary in space.

Nodes in Contact y (Positive Upwards)

KP

PipeLine

Seabed

Figure 14:

Synchronous Touchdown

The fundamental unknowns in the problem are the pipe displacements. The relative displacements of the soil-pipe interface elements do not appear as independent variables in the solution, as the non-linearities associated with the soil forces at the interface are introduced as residual forces in the solution.


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FINITE ELEMENT

If any pipe node attempts to cross the seabed boundary, a soil-pipe interface element is introduced into the solution, and a soil force mobilized. The advantages of this method are as follows:

• The classical algorithms require the concept of master and slave nodes/surfaces on order to determine potential contact pairs,

•

Only those spring elements at nodes touching the seabed are introduced into the solution, thus saving considerable computational effort,

•

Pipelines undergo a sequence of touchdown/lift-off states during installation, whereby the final position of the pipe may differ from that of first contact. A more realistic solution is thus obtained by redefining and re-orientating the soil-pipe interface at each touchdown.

Contact state decisions are based on displacement considerations. This has distinct advantages over decisions based on contact forces. This is especially important when dealing with stiff soils, where very large contact forces may develop. The constraint condition is based on the location of the pipe nodes relative to the seabed. This is equivalent to the node-to-node contact adopted in multi-mesh configurations. This leads to a simple, yet highly efficient contact search algorithm. The position of the pipe nodes relative to the seabed determines whether or not the pipe has touched-down. We identify the contact states as follows: 1.

Open State: where the pipe is above the seabed (see Figure 15). The displacement condition is

u p + u sp

Equation 53

> ug

With

•

up

pipe displacement relative the initial position

•

usp

new relative seabed-pipe displacement

•

u g

initial gap between the pipe and the seabed

In this case the contact force, between the pipe and the soil, V is zero, as the soil-pipe interface element has not been mobilized.


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FINITE ELEMENT

ADHESIONSTATE

OPENSTATE y (Positive Upwards)

y (Positive Upwards) Initial PipeLine Position

Initial PipeLine Position KP

KP

up

Current PipeLine Posit ion

ug

ug up

usp

Seabed

Seabed usp

Current PipeLine Posit ion

Figure 15: Contact Search: Open and Adhesion States Definition

2.

Adhesion State: where the pipe is in contact with the seabed and no sliding takes place. Here, the term contact implies penetration into the seabed. The displacement and force conditions are:

u p + u sp

Equation 54

= ug

And the traction force is smaller than the maximum friction resistance (see Figure 15):

Ftraction

Equation 55

< Fa + µV

With

3.

•

Ftraction

Traction Force

•

Fa

Adhesion

•

m

coefficient of friction between pipe and soil

•

V

normal force

Sliding State: as above, but with the addition of sliding, according to Coulomb’s friction law. The displacement condition is as above, and the force condition is now:

Equation 56


Ftraction

= Fa + µV

36

FINITE ELEMENT

An important part of a solution scheme involving contact is the search algorithm, which attempts to determine which elements are in contact. We use the straightforward criteria that an interface element is mobilized if the pipe node elevation is below that of the original seabed elevation. Contacts may appear and disappear, as the pipe nodes touchdown/lift-off the seabed, with the interface elements being subsequently introduced/removed from the solution. Note that during the lay down phase, seabed friction is generally switched off in order to allow free axial movement of the pipe.

2.4.3 Asynchronous Contact Algorithm The asynchronous contact algorithm will allow the pipe to touch down on the seabed at its current KP location, and not at the corresponding seabed KP as in the synchronous touchdown explained above. In order to achieve this an uncoupling of the pipe and seabed nodes is required. y (Positive Upwards)

KP

Underformed PipeLine

Nodes in Contact

Derformed PipeLine

Seabed

Figure 16:

Asynchronous Touchdown

In order to test the pipe-seabed contact, the seabed elevation at current pipe KP is used. This value is linearly interpolated from the seabed elevation.


37

FINITE ELEMENT

2.5

Geometrical Non-Linearity Since lateral deformations in pipelines may be relatively large, particularly when large compressive loads are induced and buckling is approached, the geometric updating of the pipe configuration is performed each increment. Upheaval buckling will occur if the axial forces are sufficiently high and the downward restraining loads are insufficient to maintain contact of the pipe and soil. In a 3D analysis, pipeline snaking may be observed if sufficient out-of-straightness in plan is defined in the pipeline route coordinates. The use of the term “geometrical non-linearity” implies that deformations significantly alter the location or distribution of loads, so that equilibrium equations must be written with respect to the deformed geometry, which is not know in advance. Therefore a displacement state is sought in which the deformed structure is in equilibrium with load applied to it. Here, the term “displacements” refers to both rotation and translation. Typically, pipe snaking or upheaval buckling fall into the category of “geometrically nonlinear problems”. Geometric non-linear behaviour is characterized by a non-linear relationship between load and displacement as the axial force in the member increases. Under compression, a reduction in stiffness occurs as the critical load is approached. Under tension the member tends to stiffen as the tensile force is increased (i.e. membrane effect). A co-rotational formulation has been adopted (Mattiasson et al. 1985) in which a local Cartesian co-ordinate system is “attached” to each element, which continuously translates and rotates as an element deforms. In the co-rotational formulation, each element has 3 degrees of freedom: the displacement of end 2 relative to end 1 along the chord, and the moment inducing rotations at each end relative to the chord. Thus, all rigid body translations and rotations are effectively removed. The global or column buckling effects are accounted for within the element stiffness formulation via the use of stability functions (Smith and Griffiths, 1988). The stiffness matrix (bending components only) is modified dependant on the value of the axial force, F. The accuracy of the approximation depends on the value of the ratio F/F E, where FE is the Euler buckling. This formulation also accounts for stiffening effects due to tensile axial forces.


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2.6

Solution Techniques

2.6.1 Introduction Non-linearities in the problem require an incremental and/or iterative solution technique. PipeNet supports the following non-linearities: 

Pipeline plasticity (material non-linearity)



Soil non-linearity with friction (material non-linearity)



Large displacements (geometric non-linearity)



Buckling (geometric non-linearity)



Soil-pipe contact (boundary non-linearity)

In PipeNet, these non-linearities are treated in parallel using an incremental solution technique, with the option of refining the solution within each increment by iteration. This approach is known as a tangent stiffness method with initial stress iteration (Smith and Griffiths, 1988; Owen and Hinton, 1980). This procedure is also known as the modified Newton-Raphson method.

2.6.2 Soil-pipe contact and soil material non-linearities Laydown is simulated in an unconditionally stable algorithm if the total load is divided into increments such that no more than one node touches down or lifts off at a time. This is performed automatically in the program by predicting the pipe behaviour for a preliminary load increment and then scaling back the solution and load increment such that the one-node criterion is satisfied. This technique works for all types of seabed. Since the incremental solution is required for touch-down/lift-off, it is convenient to iterate for soil and pipe non-linearity within these increments using an "initial stress" method (Smith and Griffiths, 1988). Thus, during each increment of load, the element forces or moments are checked with the correct values based on the non-linear deformation relationship. Excess forces or moments are redistributed to other adjacent elements until convergence of the iterative solution is achieved. Forces in soil springs are handled in the same way. When an excess moment is identified at a node, the "correcting" loads are calculated by applying an equal and opposite moment, plus a couple which creates zero resultant force on the element. Experience using the program has demonstrated that an unconditionally stable iterative solution for the combined soil non-linearity coupled with the lift-off/touch-down is difficult to achieve. Therefore, by default PipeNet uses a tangent stiffness method (i.e. the Euler incremental algorithm) to handle the pipe-soil interaction to avoid problems of non-convergence. As no attempt is made to correct the solution, the number of increments needs to be selected for each problem but typically about 100 - 200 is sufficient.

2.6.3 Pipe material non-linearity Pipe bending non-linearity in the moment-curvature relationship is solved using the incremental-iterative algorithm (i.e. modified Newton-Raphson). In the finite element analysis, the displacement-based formulation requires that the moment is derived from the element curvature. During plastic deformation the plastic curvature is adopted as the measure of plastic hardening. Because the Ramberg-Osgood formula is implicit for the moment, an iterative algorithm is used to determine the moment for a given curvature.


39

FINITE ELEMENT

If the stress-strain based moment-curvature model is used, the moment supported at a given curvature is defined by equilibrium of a section by equation:

Mzz = ∫ A σx y dA

Equation 57

The solution proceeds in an incremental manner, determining the variation of σx around the pipe and numerically integrating this stress to determine the associated moment. Axial and hoop stresses and strains at the end of the previous load increment are assumed to remain constant during the current increment. During the first iteration for the new load increment, a set of incremental curvatures is defined. These curvatures yield incremental longitudinal strains from Equation 24 at the integration points. These strains are then used along with Equation 26 and Equation 29 as described above, to compute an effective modulus for the increment (Chen and Han, 1985) and the stiffness matrix for the current stress state. The incremental moment derives from this tangent stiffness and the applied incremental curvatures. When axial stresses induce non-symmetric bending - the normal case for plastic bending in pipeline analysis (i.e. the neutral axis is not coincident with the centroid), a further level of iteration is required to find the strain at the centroid of the pipe due to plastic bending and the total moment that can be supported. If the derived moment is less than the current moment, the "excess" is accumulated as a body force and the iterative solution proceeds until convergence is obtained.

2.6.4 Incremental/Iterative Process This section explains, based on Cook and Malkus (1989), the principle of the incremental-iterative process using the modified Newton-Raphson method. The non-linear problem can be written (Cook and Malkus, 1989): Equation 58

K(U) U = F(U)

With

•

U

Displacements (unknown)

•

F(U)

External Forces (known)

•

K(U)

Stiffness Matrix (function of the displacements)

The stiffness matrix K is Equation 59

K(U) = K o + K N(U)

Where Ko and KN are, respectively, the displacement independent and dependent part of the stiffness matrix. Consider that the solution is known for a given displacement U A. The purpose is to find UB corresponding to FB = F(UB), which is the exact solution of Equation . Assuming that the solution is in the neighbourhood of UA a small increment considered:


∆U is

40

FINITE ELEMENT

F ( U B )

∆F  ≈ F(U A ) +    ∆U  ∆U  A UB

Equation 60

= U A + ∆U

Using a truncated Taylor expansion, we have

K t

∆F  =     ∆U  A

Equation 61 The tangent stiffness matrix Kt is defined by Equation 62

Equation 63

K t

∆K U  = K 0 +   N   ∆U  A

Obviously, it is also function of the displacements. In 1-D, it represents to the tangent to the force-displacement curve at point A. Isolating ∆U in Equation 61, we have Equation 64

∆U A = K t−1 ∆R

With Equation 65

∆R = F(U B ) − F(U A )

Where ∆R is called the force imbalance. The words “out-of-balance force” or “residual” are also used. Summarising: 1.

F(U A) is know from the solution to Equation for a given displacement states U A.

2.

The load is increased to FB = F(UB), for which the corresponding displacements UB are not know.


41

FINITE ELEMENT

3.

Since the tangent stiffness matrix is defined by the physics of the problem, Equation 64 allows to compute a first displacement increment ∆U1.

4.

Using Equation 60, a first approximation of the unknown displacements U B,1 can be found.

5.

Using Equation , the corresponding force F B,1 = F(UB,1) is computed.

If the exact solution has been found, the difference between F B and FB,1 must be zero.


42

FINITE ELEMENT

Force F

FULLNEWTON-RAPHSON

b1 FB Fb2

Kt1

FB- Fb1

B

b2

2 FB- FA

Fb1

1 KtA A

FA

∆u1 uA

Displacement ub1

UB

ub2

Figure 17: Full Newton-Raphson Therefore the full Newton-Raphson iteration involves repeated solution of Equation 64, where the tangent stiffness matrix K t and out-of-balance force ∆R are updated after each cycle i. For a given load increment, the solution process seeks to reduce the force imbalance, and consequently ∆Ui, to zero. The modified Newton-Raphson differs from the full Newton-Raphson only in that the tangent stiffness either is not updated or is updated infrequently (i.e. every x iterations). This avoids the expensive repetitions of forming and reducing the tangent stiffness matrix Kt at each iteration. However it requires more iteration to reach a prescribed accuracy. The convergence tolerance in SAGE Profile is based on the load and can be written as:

FA

Equation 66

− FA

FB

≤

Tolerance

With

•

F A

Force at previous iteration

•

FB

Force at current iteration

•

Tolerance

Convergence threshold


43

FINITE ELEMENT

Figure 18: Modified Newton-Raphson

2.6.5 Incremental/iterative Method The discussion in the previous section deals only with locating a single point on the force-displacement curve. Since the full curve is required. An incremental approach is required to which the iterative process is applied. This is the so-called “incrementaliterative process” (Cook and Malkus, 1989). This procedure is also called the “tangentstiffness method with initial stress iteration”. The stiffness matrix is formed every load increment but it is not updated during the iteration process. If the iterative part of the procedure is ignored, the method reduces to the classical “Forward-Euler” method. Here also the stiffness matrix is formed every load increment but the iterative refinement is not performed. It is also referred as the “tangent stiffness” method. It should be noted that this incremental-iterative scheme will successfully handle nonlinear problems provided that the slope of the non-linear load-displacement curve does not change in sign. Therefore, situation such as material softening or snap-though buckling behaviour will require more sophisticated numerical algorithm. This explains why “SAGE Profile cannot handle post-buckling behaviour”.


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2.7

Sign Convention

2.7.1 Co-ordinate System A right hand co-ordinate system with the x-axis along the KP of the pipe (for 2D analyses) and the y-axis upwards is used in the program (see Figure 19).

Figure 19: Static sign convention for displacements and internal actions. Positive directions are indicated. A static sign convention, as shown, is used in PipeNet for deformations and equilibrium forces/moments. x, y, z , α, β, θ , are used in pipe output files. Seabed elevation is, therefore, the y-coordinate and unless otherwise defined will be measured as the depth below mean sea level (It will have anegative sign). For 3D problems, the pipe route is defined by the plan co-ordinates in the X-Z plane ( Figure 4). The KP value defines the plan length of the pipe. Associated with each KP, x, z set is the seabed elevation at that location. During load specification and internal computation a static sign convention is operated (Figure 19). Curvatures are positive when the beam is sagging (centre of curvature in the positive axis direction).

2.7.2 Shear Forces and Moments A deformation sign convention has been adopted in PipeNet output for consistency with normal engineering design practice (Gere and Timoshenko, 1985):

•

Shear force is positive when the algebraic sum of the normal forces to the right of the section is upwards (positive y direction), and

•

Bending moment is positive when the algebraic sum of the moments to the right of the section is anti-clockwise (causing sagging and compression in the upper fibres of the beam).


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2.7.3 Strains and Stresses Strains and stresses are positive when t ensile. Bending strains and stresses are output for the top of the pipe only in 2D (in the top half of the pipe in 3D). Thus a positive bending moment produces a positive curvature and a negative strain/stress in the upper fibre. For 3D problems, in which bending in two planes can occur, if the pipe is in tension in the upper half of the pipe a positive stress is assigned. Figure 20 shows, by an example of a fixed ended beam subject to a uniformly distributed load, the result of applying this convention.


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Figure 20: Deformation sign convention used in pipe output.


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2.8

Flow Charts The data input algorithm is described in the form of a flow chart in Figure 21. The essential aspects of the analysis algorithm implemented in PipeNet are described in the flow charts given in Figure 22.

Start

Set array parameters and default values

Open data input file

Start next load step

Null or reset load step arrays

No

Read from data input file until keyword encountered

Keyword "Analyse"?

Yes

EOF

Yes

Stop

Perform a load step analysis

No

Execute keyword to read data or control analysis


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Figure 21: Data input flow chart. Start load step analysis Perform basic error checking Start load increment loop Set nominal load increment size for this increment Form elastic pipe stiffness matrix based on current geometry and axial load Add in ta ngent soil stiffness term s if below g round Add in stiffn ess terms for p rescribed d isplacements Reduce equations Null all temporary vectors Start iteration loop for material non-linearity Form load vector for this increment including scale back Add in exce ss loads from ma terial no n-linearit y from previo us iteratio n Solve for latest prediction of unknown displacements Check for convergence - set convergence flag Check for new nodes touching and calculate fraction of load for first node to touch Scale back all incremental loads and deflections Calculate latest prediction of excess forces from soil non-linearity Calculate latest prediction of excess forces from pipe n on-linearity

Convergence achieved?

No

Yes

Update displacements, actions and total applied load Perform system equllibrium check

No

Has 100% of load increment been applied? Yes Output load step results to file

Continue to read next keyword from input file

Figure 22: Analysis flow chart.


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2.9

Glossary f

Uplift coefficient (backfill resistance)

f d

Drag force due to currents and waves

f i

Inertia force due to currents and waves

f l

Lift force due to currents and waves

pe

External water pressure (compression +ve)

pi

Internal water pressure (compression +ve)

t

Pipe wall thickness

up

pipe vertical displacement

usp

seabed-pipe distance

ug

initial gap between the pipe and the seabed

y

Distance from neutral axis

A, B

Ramberg-Osgood curve fitting parameters for moment-curvature

Ae

Total external cross-sectional area of pipe

Ai

Total internal cross-sectional area of pipe

As

Steel cross-sectional area of pipe

Bfooting

Equivalent footing width

Cd

Drag coefficient

Cl

Lift coefficient

Cu


Cm

Inertia coefficient

De

Pipe external diameter

Di

Pipe internal diameter

E

Elastic modulus of pipe steel

F

Effective axial force

F

Force vector

Fa

Soil adhesion force

Fcover

Uplift resistance due to backfill cover

FE

Euler buckling force

Fe

Axial component of effective force due to external pressure on end of pipe

Fi

Axial component of effective force due to internal pressure on end of pipe

FCurrent

Force at current Newton-Raphson iteration

Fold

Force at previous Newton-Raphson iteration

Ftw

True wall axial force in pipe section

Ft

Axial force in pipe wall (constrained section) due to temperature


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Fmax

Maximum frictional resistance

Fp

Effective axial force

Fp1

Axial force due to Poisson’s effect

Fp2

Axial force induced by fluid pressure (direct effect)

I

Moment of inertia

K

Stiffness matrix

M

Bending moment

Mb

Critical bending moment (Ramberg-Osgood)

Mo

Nominal bending moment

Myield

Yield Moment

Nq, Ng

Bearing capacity factors

Qu

Soil bearing capacity

U

Displacement vector

Uc

Current velocity

Us

Significant near bottom velocity amplitude

Tu

Mean zero up crossing period

V

Vertical load on soil spring

Z

Backfill cover depth

∆T

Temperature increase

α, β

Ramberg-Osgood curve fitting parameters for moment-curvature or uniaxial stress-strain curves

αth

Coefficient of thermal expansion

β

Incidence angle

ε

Strain computed from the axial change in length

εx

Longitudinal strain

εy

Nominal yield strain (e.g. 0.5%)

γ'


γw

Water unit-weight

κ,κ0

Curvature, nominal (normalising) curvature

κzze,kzzp

Elastic and plastic curvatures about the z-axis

κb

Critical curvature (Ramberg-Osgood) corresponding to M b

µ

Friction coefficient of Coulomb

ν

Poisson’s ratio

σb

Bending stress (at the outer fiber -> maximum)

σeq

Equivalent stress


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σtw

Axial stress in pipe wall

σh

Hoop stress in pipe wall

σx

Longitudinal stress in pipe wall

σyield

Nominal yield stress, (e.g. at 0.5% strain)

τ

Shear stress

ζ

Pipe ovalisation

ζ0

Initial out-of-roundness


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REFERENCES

3. REFERENCES Brazier L.G. (1927), On the Flexure of Thin Cylinders, Shells and Other Thin Sections, Proc. Royal Society, Series A, Vol.116, pp.104-114. Chakrabarti, S.K. and R.E.Frampton (1982), Review of Riser Analysis Techniques, Applied Ocean Research, Vol.4, No.2, pp73-90. Chen W.F. and D.J.Han, (1985), Tubular Members in Offshore Structures, Pitman. Cook R.D., D.S.Malkus and M.E.Plesha (1989), Concepts and Applications of Finite Element Analysis, Wiley. DnV (2000), “DNV 2000: Rules for Submarine Pipeline Systems”, Den Norkste Veritas. Franzen W.E. and W.F.Stokey (1973), The Elastic-Plastic Behaviour of Stainless Steel Tubing Subjected to Bending, Pressure and Torsion, 2nd Int. Conf. on Pressure Vessel Technology, Part 1, Design and Analysis, ASME, pp457-467. Gellin S. (1980), The Plastic Buckling of Long Cylindrical Shells Under Pure Bending, Int. J. Solid Struct, Vol.16, pp.397-407. Gere J.M. and S.P.Timoshenko (1985), Mechanics of Materials, 2nd Edition, Brooks/Cole, Monterey. Hoskins E.C. (1982), Sub-sea Pipeline Free Span Vibration Analysis, Institute of Petroleum, Pub. No. IP 82-013. Klever,F.J., Palmer, A.C. and Kyriakides, S. (1994) , Limit State Design of High Temperature Pipelines, Offshore Mechanics in Arctic Engineering, Vol V, Pipeline Technology, pp77-92. Ramberg W. and Osgood W.R. (1943), Description of Stress-Strain Curves by Three Parameters. NACA Tech Note 902, July. Mattiasson, K. Bengtsson, A and Samuelsson, K. (1985), On the Accuracy and Efficiency of Numerical Algorithms for Geometrically Nonlinear Structural Analysis, in "Finite Element Methods for Non-Linear Problems", Ed : Bergan,P.G., Bathe, K.J., and Wunderlich, W., Springer-Verlag, Berlin. Murphey C.E. and Langer C.G. (1985) Ultimate Pipe Strength Under Bending, Collapse and Fatigue, ASME Proc. 4th OMAE Symp. Vol.1, pp.467-477. Owen D.R.J. and E.Hinton (1980), Finite Elements in Plasticity, Pineridge Press, Swansea. Roark, R.J. and W.C.Young (1975), Formulas for Stress and Strain, 5th Edition, McGraw Hill. SAGE (1992), The Effect of Ovalisation on the Moment-Curvature Relationship, Internal memo by R.Wilkins, 3 November 1992. Smith I.M. and D.V. Griffiths (1991), Programming the Finite Element Method, 2nd Edition, Wiley.


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SAGE Profile V6.3.2 User Manual - Volume 3

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