Study and development of FEM-models used in expansion analyses of pipelines. Cristina Lindholm

Transkript

1 Study and development of FEM-models used in expansion analyses of pipelines Master of Scince Thesis Stockholm 7

2 Study and development of FEM-models used in expansion analyses of pipelines by Master of Science Thesis MMK 7:17 MME 794 KTH Machine element SE-1 44 STOCKHOLM i

3 Master of science thesis MMK 7:17 MME 794 Study and development of FEM models used in expansion analyses of pipelines Approved Examiner Sören Andersson Commissioner Tore Søreide,REIERTSE AS Supervisor Ulf Sellgren Contact person Sigurd Trier Abstract REIERTSE AS performs expansion and buckling analyses of pipelines using ASYS, a finite element modelling and analysis tool. In an expansion analysis the pipeline is modelled with thin walled pipe elements called PIPE which allows plastic deformation. However, in a recent analysis the results retrieved from the PIPE element were incorrect for strains about 1 percent. One main issue was that the PIPE element overestimated the strains. The purpose with this Master Thesis was to become familiar with the theories and analysis methods used in pipeline design. By using the knowledge gained a short examination of the weakness in the PIPE element was carried out and a first development of a new model for expansion analysis in ASYS was made. First, a detailed initial study was made. The analytically derived functions were examined and a model was created in ASYS using PIPE elements with linear elastic material model. This model was, when compared to the analytically expected results, very accurate and could thereby be used in further analysis. A short study of the PIPE element was carried out for a linear elastic perfectly plastic material model. The results from PIPE were compared to analytically derived results and to results from Pipeline Analysis System (PAS a D analysis tool developed at REIERTSE. It was seen that results from calculations involving the PIPE element differs from the expected results. When the rotation is small and the cross section is almost completely elastic the expected value and PIPE element output is practically identical but with increasing rotation and plasticity the accuracy is lessen this due to the few number of integration points. When more than 3 percent of the cross section is plastic the element output is utterly incorrect. The conclusion is that the PIPE element is not suitable in applications where plastic deformation of a major part of the cross section is expected. One solution is to create a model with a beam shell assembly. The PIPE elements in the model where inaccuracies have been found can be replaced by shell elements. A first development of such a model was made. The beam shell assembly was exposed to the same loads as a model with only PIPE elements. The strains where compared and was found to be smaller in the beam shell assembly than in the PIPE model, indicating that a model with a beam shell assembly can be used. Even though more development and testing are needed the first results are satisfying. i

4 Examensarbete MMK 7:17 MME 794 FEM modeller för expansionsanalys av piplines Approved Examinator Sören Andersson Uppdragsgivare Tore Søreide,REIERTSE AS Handledare Ulf Sellgren Kontaktperson Sigurd Trier Sammanfattning REIERTSE AS utför expansions och bucklingsanalyser på pipelines genom att använda ASYS. I dessa analyser modelleras rören med hjälp av ett tunnväggigt rörelement kallat PIPE. Detta element tillåter plastisk deformation. I en analys nyligen utförd på REIERTSE med töjningar på ca 1 till 1.5 procent var resultaten från PIPEelementet felaktiga i jämförelse med resultat från andra applikationer. Ett huvudproblem var att PIPEelemntet överskattar töjningarna. Syftet med detta examensarbete är att sätta sig in i teorier och analysmetoder som ligger bakom konstruktionsarbetet för pipelines. Genom utnyttjande av den kunskap som erhållits gjordes först en kort undersökning av PIPEelementets brister. Efter detta gjordes en första utveckling en ny expansionsmodell. En noggrann förstudie utfördes, de analytiskt härledda formlerna studerades och en modell med användande av PIPE element gjordes där en linjär elastisk material modell nyttjades. PIPEmodellen var, i jämförelse med de teoretiskt förväntade värdena, mycket noggrann och kunde därför användas i fortsatt analys. En liten modell bestående av ett PIPEelement med plastiska egenskaper undersöktes. Förväntat moment vid olika rotationsförskjutningar jämfördes med utdata från PIPE. Det fastslogs att utdata från PIPEelementet skilde sig från det förväntade värdet av två anledningar. är rotationen är så liten att nästan hela tvärsnittarean är elastisk är det förväntade värdet och PIPEmodellens utdata så gott som identiska men med ökad rotationslast så minskar noggrannheten. Detta på grund av de få integrationspunkterna definierade i PIPEelementet. Ökas rotationen ytterligare så mer än 3 procent av tvärsnittsarean är plastiskt deformerat är utdata från PIPEelementet helt felaktiga. Slutsatsen är att PIPE inte är att rekommendera i analyser där en stor del av tvärsnittsarean förväntas bli plastisk deformerat. Ett förslag till lösning är att skapa en ny modell där delar av modellen består av skalelement. Genom att byta ut de delar av modellen där problem med noggrannhet uppstår och ersätta dessa med skalelement kan en så kallad beam shell assembly vara en förnuftigare modell. Ett första försök till utveckling av en sådan modell har gjorts. Den nya modellen och en modell bestående av enbart PIPEelement utsattes för samma analys och töjningar jämfördes. Det visade sig att töjningar i beam shell modellen var lägre än de funna i modellen med endast PIPEelement. Detta är en indikation på att en modell med både PIPEelement och skalelement kan användas i expansionsanalyser. Det är nödvändigt med vidare utveckling och testning av modellen, men de första resultaten är tillfredsställande. ii

5 Table of Contents Abstract...i Sammanfattning... ii Table of Contents... iii Table of Symbols and Abbreviations...v 1 Introduction... 1 Theoretical models....1 Effective axial force and submerged weight How the pipeline will buckle Vertical buckling ects Lateral buckling ects Buckling behaviour on trigger berm Horizontally straight pipeline Buckling behaviour on trigger berm Horizontally curved pipeline Conclusion Feed in length and maximum allowable moment Feed in length and strain Feed in length and deflection Combining feed in length in the case of lateral buckling, even seabed 15.4 How the axial ective force is built up over time Verification of analytical model using ASYS The model The ASYS analysis Vertical buckling Lateral buckling on vertical trigger berm Lateral buckling on even seabed The parameters and variables in the cases Results Vertical buckling Lateral buckling on vertical trigger berm Perfect straight pipe on perfect even seabed Lateral buckling on even seabed Conclusion Plasticity, Ansys and PAS Model of short plastic beam Theory and analytical model PIPE and PAS model Loading PIPE with moment The importance of integration points a short study iii

6 4..1 Analytical vs. numerical Analytical vs. PAS Conslusion Development and verification of a new model Building a shell Modelling a beam shell assembly MPC BEAM Study of the connections Modelling the Seabed and the contact Forces due to pressure pipeline initially created on flat seabed Modelling Link elements Lowering the pipeline Establishing contact with an un even seabed Forces due to pressure pipeline laid down flat seabed Buckling of pipeline lying on trigger berm Vertical buckling Lateral buckling Conclusion of the beam shell assembly and further development References Appendix...i 7.1 Appendix. Vertical buckling on even seabed...i 7. Appendix. Lateral buckling on even seabed... iii 7.3 Appendix. Combined buckling...vi 7.4 Appendix. Parameters and variables... vii 7.5 Appendix. ASYS input files for elastic PIPE...viii 7.6 Appendix. Graphs of vertical and lateral deflection... xx 7.7 Appendix. Moments in partly plastic cross section....xxiv 7.8 Appendix. Input files for short beam in ASYS and PAS...xxv 7.9 Appendix. Matlab input file for study of integration points...xxviii 7.1 Appendix. Input file for beam shell assembly. MPC BEAM4...xxix iv

7 Table of Symbols and Abbreviations Latin symbols Index A Area Initial value C Length between supports 1 Part one/principle 1 D Diameter Part two/principle E Young allow Allowable F Force ansys Value from ASYS output g Gravity As laid Value as laid I Moment of inertia buckle Of buckle k Soil stiffness c Coating L Length calc Calculated value M Moment e Elastic Axial force e External n Force per meter length Effective p Pressure feed in Feed in q Weigh per meter i Internal R Radius lat Lateral r Mean radius of pipeline lay From lay phase T Temperature lift When lift off t Thickness oper In operation u Elongation p Plastic V Shear force press Value when exposed to pressure v Lateral displacement p+t Pressure and temperature w Vertical displacement s Steal soil Soil Greek symbols sub Submerged α Heat coicient true True δ Deflection vert Vertical ε Strain w Water θ Angel x Axial µ Coicient of friction y Yield υ ρ σ Poisons ratio Density Stress DV MPC PAS Det norske veritas Multi points constraints Pipeline Analysis System v

8 1 Introduction REIERTSE AS is a orwegian main contractor supplying multidiscipline process facilities to the oil and gas industry offshore and onshore. They take on full scale analysis and design of pipelines based on the offshore standard developed by Det orske Veritas (DV []. Submerged steel pipelines are used for transporting oil and gas from oil fields at the sea bottom to land or a platform. The difference in temperature and pressure in the hot fluids inside the pipe and the surrounding sea water will lead to axial expansion, rendering an elongation of the pipeline. If the expansion is restrained an axial compression force will develop in the pipeline. A large amount of inner forces in a pipeline is unwanted and numerous analyses have been made on how to lower the large forces in the pipeline. Due to the friction between the pipeline and the seabed, the large axial forces and moments that are built up in the pipe will cause deflection. This is called global buckling. It has been observed that when the pipeline buckles the ective axial force in the pipeline is lowered. This has lead to a discipline called designing for buckling. In order to do this accurate expansion and buckling analyses has to be performed. REIERTSE AS performs expansion and buckling analyses on pipelines using ASYS, a finite element modelling and analysis tool. In an expansion analysis the pipeline is modelled with thin walled pipe elements called PIPE which allows plastic deformation. However in a recent analysis the results received from PIPE has been proven incorrect for strains about 1 percent when compared to results from other applications. One issue is that PIPE overestimates the strains. This master thesis is divided into four chapters First the theoretical models used are derived. Second a model using linear elastic material model for PIPE elements is developed in ASYS and verified against the theories. Third the weakness found when using PIPE is investigate Forth and last a new model is developed. By using a beam shell assembly, the PIPE elements where errors have been found can be replaced by shell elements. And by this create a higher accuracy in the model. 1

9 Theoretical models Figure 1 demonstrates a pipeline at seabed with the coordinate system used in this thesis. z, w y, v x, u Figure 1. Demonstrating a pipeline at seabed with the coordinate system used in the thesis. To be able to predict the behaviour of the pipeline during operation an analytical tool is needed. The construction of this tool is divided into three steps as described below. 1 Description of the forces acting on a small section of a pipeline o This will give the submerged vertical weight q sub and the ective axial force o. Both necessary tools to describe and understand the actions of pipelines. Due to pipe/soil interaction a pipeline is restrained. This will cause an unpredicted expansion and buckling of the pipeline. By designing for buckling [3] The expansion problem is solved The ective axial force can be limited One can prevent buckling in unfavourable regions Description of how a pipeline has a tendency to buckle. This is divided into studies of vertical, lateral and combined buckling behaviour. o Study of vertical buckling on even seabed This gives expressions of lift off length, Lbuckle, maximum deflection o δ vert and the needed axial force in the pipeline for lift off from seabed,. lift Study of lateral buckling on even seabed with uniform friction. There are several lateral buckling modes depending upon a number of factors for example imperfections of the pipeline and the pipe/soil interaction. Here the first and second mode will be described. Expressions for describing the length of the deformed pipeline, deflection at centre δ and the needed force to initiate lateral sliding lat are presented. vert

10 o lat lift It is found that < on even seabed. Indicating that if there are no imperfections on the pipeline lateral sliding will occur before vertical buckling. Buckling on trigger berm It is observed in the vertical case that the necessary ective lift off force is inversely proportional to the initial deflection at the centre. To be able to control the vertical buckling imperfections has to be built into the system. For example using trigger berms. Buckling behaviour on trigger berm/ un even seabed with a horizontally straight pipeline. This shows that the needed ective lateral buckling force is about 5% of the ective lift off force. Leading to the conclusion that lateral buckling will occur as soon as the pipeline lifts off the ground/trigger berm. This ect causes the pipeline to snap through. By building in lateral as well as vertical imperfections this un wanted behaviour can be controlled. Buckling behaviour on trigger berm/un even seabed of pipeline with initial lateral imperfections. A pipeline with both vertical and lateral imperfections lowers the necessary ective axial force to initiate buckling. This gives a more controlled deflection development which is desired. Where the elongation of the pipeline is a slow process. 3 Description of how much a pipeline can be allowed to buckle. o The buckling can be controlled by building in imperfections. But with continuant buckling the pipeline can be damaged. This can be controlled by limiting the elongation. o The limit is the maximum allowable bending moment M allow in the pipeline which is a second order function of the deflection. o The total elongation is the feed in length u feed i The feed in length is equal to the axial expansion ( ε Lbuckle of the pipe. The change in strain ε can be found by using Hooke s law. ii The feed in length can also be described as a second order function of the deflection. o Using the second connection for u feed it is found that the allowable feed in displacement is highly dependent on the allowable moment limit M allow. o By combining i and ii the allowable pipe distance between axially fixed points C can be determined and the elongation is controlled. allow 3

11 true p e ρ s A s g p i.1 Effective axial force and submerged weight The understanding of the ective axial force and its ects is fundamental since dominates how steel pipelines responses to loading. To illustrate the concept first consider a small section of a pipeline, Figure. The forces acting on the pipeline being Internal pressure, p i External pressure, p e Weight of steel pipe ρ s As Figure. Forces acting on a submerged pipeline section in equilibrium true, the true axial force acting on the steel pipe found by integrating the This introduces steel stress over the steel cross section area [4]. fundamental, and true can cause but the difference is true is the force acting only on the steel pipe. depends of all axial forces, this is further explained in the following. If the section is assumed to be a closed surface the ects of the external pressure can be understood using the law of Archimedes [4] The ect of the water pressure on a submerged body is an upward directed force equal in size of the weight of the water displaced by the body This means that the external pressure p e can be replaced by a vertical weight and axial force as displayed in Figure 3. p e p e p e A e ρ w A e g Figure 3. External forces acting on a pipeline section in equilibrium. The top pipe section is the real forces. This is equal to the two pipes in the middle. The external pressure acting on the closed pipe section (midright can be described as the bottom pipe section. This means that the only parts needed to describe external pressure of a pipe section is the forces at the pipe ends and the buoyancy A similar assumption can be made for the internal pressure p i replacing it with the weight of the content acting downward integrated over the surface and an axial force, Figure 4. 4

12 p i p A i i p i ρ i A i g Figure 4. Internal forces acting on a pipeline section in equilibrium. The only parts needed to describe internal pressure of a pipe section are the forces at the pipe ends and the weight of the contents These assumptions give an equivalent system to Figure. ρ w A e g true p A e e p A i i ρ s ρ i A i A s g g q sub Figure 5. Resulting forces acting on a small section of a pipeline The resultant of the vertical weights is the submerged weight: qsub = ρ s As g + ρi Ai g ρ w Ae g [/m] (1 In case of coating an extra parameter qcoat = ρ c Ac g adds to the weight. The axial forces resultant is the ective axial force = true + pe Ae pi Ai [] ( When < a pipeline locked axially is in compression which could cause buckling. By observing ( it is understood that external pressure stabilizes the buckling ect while internal pressure destabilizes. An unrestrained pipeline is free to move axially. In that case the ective axial force is zero. This can be understood if the unrestrained pipe is seen with an end cap as in Figure 6. The forces acting on the end cap being internal and external pressure. This gives the true axial force as 5

13 and the ective axial force true = p A p A i = pi Ai pe Ae + pe Ae pi Ai i e e = End cap p i true p e Figure 6. The forces acting on an unrestrained pipeline can be found if the pipeline is seen with end caps. The true axial force is given only by the internal and external pressure. Consider a pipeline lying on the seabed free to move axially. If there were no soil/pipeinteraction buckling would not be a problem. However, this is not the case since friction act as virtual anchors building up the ective axial force in the pipeline, Figure 7. If a pipeline is allowed to buckle, the development of ective force is modified as pipe feeds in to the buckle. The force in the buckle drops as the buckle develops [5], Figure 7. Virtual anchor Virtual anchor Straight pipeline Restrained pipeline Unrestrained pipeline x Buckle Figure 7. The soil/pipe interaction creates virtual anchors which build up the ective axial force. This is the force that will make the pipeline deflect. With increasing buckling the ective axial force decreases. This means that designing the pipeline to buckle solves the expansion problem and is a way to limiting the ective axial force. And since buckling will appear due to soil/pipe interaction built in buckling will prevent buckling in unfavourable areas.. How the pipeline will buckle It is concluded that one way to lower the ective axial force in the pipeline during operation is allowing it to deflect. To be able to control the buckling of the pipeline it is essential to understand how the pipeline has a tendency to move...1 Vertical buckling ects The most basic case of buckling, vertical buckling of straight pipeline on idealized even hard seabed is displayed in Figure 8. The submerged weight is acting downwards on every part of the pipeline. The ective axial force is negative in compression. x 6

14 q sub [ / m ] w x w = dw = dx M = L L Figure 8. Vertical buckling behaviour on even, hard seabed An infinite small section of the pipeline can be displayed as in Figure 9. q sub + d M + dm M V w V + dv w + w dx dx Figure 9. A small section of a pipeline. Equilibrium from Figure 9 gives the differential equation for vertical buckling on even seabed 4 d w d w EI + = q 4 sub dx dx With the general solution [6] (symmetry gives even function qsub w( x = A + B cos kx x k = EI The boundary condition w ( L =, w ( L = gives qsub L L cos kl L cos kx x w( x = + k sin kl k sin kl The third boundary condition M ( L = EIw ( L = gives tan( kl = kl kl 4.5 The expression now reading q sub EI EI x w( x = cos x EI The buckling length is inversely proportional to the ective force L = 4. 5 EI (3 (4 7

15 Maximum deflection δ, vert at x =, and lift off ective force by q sub, and EI qsubei lift off qsubei w( = δ vert = 15.7, 3.96 = δ vert More details are given in appendix 7.1. is negative when the pipe buckles. The true axial force when the pipe deflects then is lift true q subei = p + i Ai p 3.96 e Ae δ vert lift off can now be expressed (5 (6 For a normal pipe scenario the internal pressure will be dominating giving q subei p 3.96 i Ai > p + > e Ae true δ vert This means that the steel pipe will be in tension when buckling occurs. This is an essential observation since it is against traditional engineering assumptions... Lateral buckling ects When describing lateral buckling the ects of pipe/soil interactions must be considered. If the pipeline is considered ideally straight and lying on ideally even seabed the lateral restraint of the seabed will give the buckling mode. The lateral restraint is given by the lateral soil stiffness k soil which can be seen as a spring resisting the pipeline to move. If the spring is too weak, as for very soft soil, there is hardly any resistance when the pipeline starts to move. The lateral (horizontal dynamic stiffness k soil is defined as k soil = F L / δ L [/m/m], where F L is the incremental horizontal force between pipe and soil per unit length of pipe, and δ L is the associated incremental horizontal displacement of the pipe. Free spanning pipelines paragraph [7] If the seabed is considered stiff to hard the buckling mode will be similar to the vertical buckling case. In the same way as the ground prevents the pipeline to dig down in the vertical case the soil will be able to resist the shear forces and act as a fixed support, Figure 1. The friction is considered uniform. 8

16 µ q sub [ / m ] v x v = dv = dx M = L Figure 1. Lateral buckling behaviour on hard seabed. First buckling mode will be similar to the vertical case with the difference of the friction coicient µ. lat µ qsubei = 3.96 δ (7 lat If the friction coicient is less than one, lateral sliding will occur before vertical on hard even seabed. The shear force that the ground needs to be able to withstand is found by V ( L 5 L EI = qsub L = qsub 4. (8 In the case of second mode buckling, Figure 11, the soil is not strong enough to withstand the shear forces at L 1 giving a three buckle shape. µ q sub v x 1 v 1 = v = v = v v = v v = v v = v = M = µ q sub µ q sub L 1 L Figure 11. Lateral buckling behaviour on seabed. Second buckling mode, the seabed cannot withhold the shear force at L 1 Equilibrium as in Figure 9 gives the characteristic equation 4 d v d v EI + q sign( v 4 = subµ dx dx With the general solutions for section 1 and qsub q v ( x = A + A1 cos kx x, v ( x = B + B1 x + B sin kx + B3 cos kx + The five first boundary conditions at L 1 given by continuity v L = v (, v ( L = v (, v ( L = v (, v ( L = v ( sub 1 x 1 ( =, 9

17 The three boundary conditions at L are caused by the soils lateral stiffness. The elastic length, l e, is the length for the moment at L to be damped out and highly dependent on k soil as 4EI l e = 4 k soil Here the soil stiffness is assumed large enough to withhold the shear forces at L which will give a short elastic length. And hereby can the moment and the deflection at L can be considered zero. v ( L =, v ( L =, v ( L = This is an adequate assumption for most soil types [8]. k soil The boundary conditions give L 1 and as L L1 =, L = k k And v1 and v qsub EI v1 ( x = µ cos kx. 5x ( ( x.86sin kx cos kx +. q EI v ( x = µ x sub 5 See appendix 7. for more detailes (9 (1 The deflection in the centre is δ lat lat µ q = sub sub δ lat EI eeded ective axial force to initiate sliding µ q EI = 3.45 lat is lower than before buckling vertical. (11 (1 lift indicating that a pipeline lying on stiff even seabed will slide lateral..3 Buckling behaviour on trigger berm-horizontally straight pipeline As seen in eq.(6 the lift off force is inversely proportional to the initial maximum deflection. On an ideally even seabed the force needed for lift off is infinitive. Since there always are imperfections on the seabed or the pipeline, an unknown small initial deflection will cause a large unpredictable lift off force. One solution to predict and control lift is placing triggerberms, Figure 1, at the sea bottom to create initial imperfections. A trigger berm is normally created by dumping stones at the seabed, this form a small hill for the pipeline to lie on. 1

18 q sub [ / m ] Post buckled δ vert Trigger berm δ, vert L Figure 1. Vertical buckling behaviour of pipeline lying on a trigger-berm When lying on the trigger berm the pipeline is locked laterally preventing lateral buckling. When lift off occurs the contact pressure is reduced and the lateral friction is equal to zero. This can be seen as a fixed fixed system giving the lateral buckling force according to Euler. 4π EI π EI lat = (L Half the buckling length is given by eq.(4 giving the ective lateral buckling force as a function of ective lift off force lat π EI π EI π lift = = = L (4.5/ k.5 lat lift =. 49 (13 The conclusion is that lateral buckling appears as soon as lift off occurs. This means that when the pipeline is lifted from the trigger berm it will rapidly deflect lateral causing snapthrough Figure 13. To avoid critical snap through ects lateral imperfection can be build in. (Compression = L L Vertical buckling Lateral buckling Lift off Combined lat/vert buckling Snap trough δ,vert lat =. 49 δ, vert δ lat lift Figure 13. The needed force for lift-off is depended of the vertical initial deflection. When lifted the friction that is keeping the pipeline at its lateral position is reduced to zero which makes the pipeline snap lateral causing so-called snap-through ect...4 Buckling behaviour on trigger-berm -Horizontally curved pipeline As seen in Figure 13 a trigger berm lower the ective lift off force but since the needed force for lateral buckling is lower than the needed force for continued vertical buckling the pipeline snaps lateral when it is lifted from the seabed 11

19 Built in lateral imperfections, Figure 14, can lower the snap through ect z, w y,v δ,lat δ,vert x,u Figure 14. Curved pipeline lying on trigger berm gives a pipeline initially defected in both vertical as lateral direction To find the needed force to initiate lateral sliding for the combined case it is seen that the lift lateral friction resistance is zero when lift off occurs ( = [8]. The modified lateral friction resistance is then * q = sub qsub 1 lift lift Where is given by eq. (7 lift qsubei = b, b = 3.96 δ vert The criterion to initiate lateral sliding is from eq.(1 and with the modified lateral friction resistance * lat µ qsubei = a, a = 3.45 δ lat This gives the ective compression at start lateral sliding as (appendix 7.3 lat lift δ vert a 1 1 b δ lat = + + µ (14 δ 4 lat b a µδ vert Figure 15 demonstrates how the needed force for lateral sliding decreases with increasing lateral imperfection. 1

20 Effective axial force at start sliding [M meter triggheight eeded axial ective for lift off 5.7 M Coicent of friction, Initial lateral imperfection [m] Figure 15. A pipeline lying on a one meter trigger-berm, with increasing lateral imperfection the needed force to initiate sliding is decreased. Lateral imperfections on un even seabed give a lower ective axial force to initiate lateral buckling. This means that the snap through ect is eliminated and the deflection of the pipeline is more controlled. This is demonstrated in Figure 16 Small initial lateral imperfection Large initial lateral imperfection time Figure 16. With a large lateral imperfection the needed axial force to initiate sliding is lowered and the pipeline slowly deflects instead of having snap through. This is caused since the ective axial force is slowly built up over time...5 Conclusion When snap through occurs it indicates that the pipeline contains large amount of inner forces. To avoid the snap through ects it is possible to create pipelines with initial defections. This would lower the ective axial when loaded with internal pressure and temperature rise. It is proven that lateral buckling will occur before vertical. By combining vertical and lateral initial imperfection the buckling behaviour can be considered satisfying. If the imperfections are large enough the pipeline will slowly expand and the snap through ect is eliminated. 13

21 .3 Feed in length and maximum allowable moment The limiting parameter for a pipeline subjected to loads due to installation, seabed contours and high pressure/high temperature operating conditions is often found to be the bending moment capacity [9]. The maximum allowable bending moment M allow can be found as a second order ect of the deflection. When a pipeline is designed to buckle it is of interest to find how large the elongation can get, how much pipe that can be fed in. The elongation is called the feed in length u feed in. By using u feed in a connection between M allow and the allowable pipe distance between axially fixed points C allow can be found. u feed in can be described in two different ways, either by using Hooke s law or by using geometry..3.1 Feed in length and strain Basic solid mechanics says that the change in strain in a linear elastic material is the change in length divided with the original length. This means that the feed in length can be described as u = εl feed in buckle L buckle is the total length of the deformed parts in the pipeline. The axial strain in a pipeline is given by Hooke s law. If the conditions are such that the internal pressure is much larger than the external pressure the pipeline will be in plane stress [1]. This is the case in most design scenarios. But in deep water environment the external pressure can be large enough to affect the results. Therefore a three dimension stress state model is used. 1 (15 ε axial = [ σ axial ν ( σ hoop + σ radial ] + α T E where the stresses based on thin walled theory are true ( x σ axial = A pi Di pede σ hoop = t p i + pe σ radial σ radial is the stress at mid surface of the pipe wall when a linear stress distribution is assumed. By using the definition of ective axial stress eq.( in eq.(15 the total axial strain change ε = ε ε can be expressed as operation as laid 1 Di 1 ε = + pi Ai pi Asν + Asα T E t (16 The external pressure is considered constant. ote that pi is the internal pressure difference relative to as laid..3. Feed in length and deflection As seen in Figure 17 below a small section of the deformed pipeline is u feed in longer than in the un deformed state. If the section is small enough the deformed pipeline section can be s 14

22 considered straight and the deformed length can be expressed by the original length and the angle as in eq. (17. w θ Deformed pipeline Straight pipeline L Figure 17. The correlation between a straight section length and a deformed pipe L = L + u feed cosθ By using eq.(17 the elongation of the section can be described (1 cosθ u feed = L cosθ Taylor expansion gives 4 θ θ cosθ = ! 4! (17 Since the angle is considered small u feed in can be simplified to θ θ u feed = L /1 = L dw If the small section is infinitesimal dx θ =, L = dx, u feed = du dx 1 dw du feed = dx dx Concluding that the total elongation of the entire deformed pipeline is u feed = 1 L L dw dx dx [ symmetry] L dw dx dx feed ( Combining feed in length in the case of lateral buckling, even seabed Since lateral deflection will occur before vertical buckling, only lateral sliding will be described. For second mode lateral buckling the expression describing the deflection is eq. (1. By using it with eq. (18 the feed in length can be found as (appendix 7. L1 L 3 / ( µ qsub ( EI (19 u = + feed ( v1 ( x dx ( v ( x dx = / And the maximum bending moment, (at x = Figure 18, is µ qsubei M lat = EIv ( = 4.89 ( ( 15

23 µ q sub v 1 u feed µ q sub M lat x L L 1 µ 1 q sub u feed Figure 18. Feed in length and moment at lateral buckled pipeline By using eq. (19 the bending moment can be expressed as 3 / 7 4 / 7 / 7 M lat = 1.38 ( µqsub ( EI u feed The limiting parameter is M allow and it give the allowable feed in displacement as 7 / allow M allow u feed =.3 3 / ( µ q ( EI sub The assumptions made in the beginning of this chapter that the bending moment is a relevant parameter to use as a limit for the pipeline is proven true. As seen in eq. (1 the allowed feed in length is highly dependent on the allowable bending moment. Another important factor is the friction coicient µ. If the friction of the soil is high less elongation is allowed. Eq. (1 with eq. (16 gives the allowable pipe distance between two supports. allow allow L u buckle feed Callow = = ε 1 Di 1 pi Ai pi As + As T E + ν t α Or inserted for allow u feed Callow = ( µ qsubei Di pi Ai pi Asν + EAsα T M allow t When the pipeline is designed for buckling it is necessary that the buckling behaviour is controlled. With the increasing elongation of the pipeline in operation the bending moment keeps rising. Since the maximum allowed moment in the pipeline is a known parameter it is used to determine maximum allowed feed in. To limit the feed in length the maximum distance between two supports for free spanning pipelines is used..4 How the axial ective force is built up over time The ective axial force is increasing with rising internal pressure and temperature until it reaches the needed force for deflection. If a pipeline is ideally straight the ective axial force will keep increasing with increasing pressure and temperature. Consider a fully restrained pipeline. The built up ective axial force at a certain internal pressure and temperature can be found by using eq. (16 where the elongation is zero ( ( (1 (1 lay ε = ε ( p, T,, p ε ( p, T,, p =. oper i e as laid i e (1 16

24 lay is the ective residual lay tension, the ective axial force the pipeline is subjected to when laid down. This is a known parameter caused by the real lay down conditions. The equation now reads 1 lay Di 1 ε = ( + pi Ai pi Asν + Asα T = E t which gives lay Di 1 = pi Ai + pi Asν AsαE T t (3 This is known as the fully constrained axial force [5] If the pipeline can be idealised as thin walled, eq. (3 can be approximated to lay pi Ai ( 1 ν AsαE T (4 The error of the simplification is less than 1% for D e / t larger than 15 [4]. 17

25 3 Verification of analytical model using ASYS A model was built in ASYS and exposed for the scenarios examined in the analytically derived functions in chapter. The results from ASYS were compared with the analytical results. The purpose is to create an accurate model with linear elastic material model that can be converted into a model with a pipe with plastic abilities used in later analysis. The model uses the same theories and procedure as the model used for analysis at REIERTSE. However the model developed in this thesis is a simplified version where three specified analysis are to be carried out. Three different scenarios are analysed with an ASYS generated model 1 Vertical buckling on even seabed. Using a trigger berm to get the pipeline to respond. The pipeline is restrained lateral and a rising temperature and internal pressure will build up to make it buckle vertical. o The height of the trigger berm will be varied to get the berm height/ diagram. Lateral buckling on trigger. The pipeline is laid down as in case one but a small lateral force before the internal pressure and temperature is put on will make it deflect lateral. 3 Lateral buckling on even seabed. An initial imperfection created by forcing part of the pipeline to move will be used as a lateral trigger. As in the vertical case the internal pressure and the temperature rise will make it buckle. o The lateral friction is varied to understand how this affects the needed buckle force. 3.1 The model The ASYS model is build up by four element types. Each one explained below. PIPE LIK1 TARGE17 COTA175 LIK1 PIPE COTA175 ormal TARGE17 Figure 19. The different element types used in the ASYS model to create the buckling cases For the pipeline the PIPE element is used. 18

26 PIPE is a uniaxial element with tension compression, bending, and torsion capabilities. The element has six degrees of freedom at each node: translations in the nodal, x, y, and z directions, and rotations about the nodal x, y, and z axes. The element has plastic capabilities. The element input data include two nodes, the pipe outer diameter and wall thickness, optional stress factors, and the isotropic material properties. Internal pressure and external pressure are input as positive values. Only constant pressures are supported for this element. Temperatures may be input as element body loads at the nodes. Ansys element library. PIPE [1] When the simulation starts the pipeline is above the seabed hold up by LIK1 elements. This is to simulate the most probably as laid mode and to create the initial stresses in the pipeline.lik1 elements are tension only element making it suitable simulating lowering. LIK1 is a 3 D spar element having the unique feature of a bilinear stiffness matrix resulting in a uniaxial tension only (or compression only element. With the tension only option, the stiffness is removed if the element goes into compression (simulating a slack cable or slack chain condition. This feature is useful for static guy wire applications where the entire guy wire is modelled with one element. The element is defined by two nodes, the cross sectional area, an initial strain or gap, and the isotropic material properties. The element x axis is oriented along the length of the element from node I toward node J. LIK1 has three degrees of freedom at each node: translations in the nodal x, y, and z directions. o bending stiffness is included in the tension only (cable. Ansys element library. LIK1 [1] The seabed and the trigger berm are built up by quadrilateral target elements TARGE17. TARGE17 is used to represent various 3 D target surfaces for the associated contact elements. This target surface is discredited by a set of target segment elements (TARGE17 and is paired with its associated contact surface via a shared real constant set. For any target surface definition, the node ordering of the target segment element is critical for proper detection of contact. The nodes must be ordered so that the outward normal to the target surface is defined by the right hand rule Each target segment of a rigid surface is a single element with a specific shape, or segment type. The segment types are defined by several nodes and a target shape code, TSHAP. QUAD is a 4 node quadrilateral. Where 1st 4th nodes are corner points (UX, UY, UZ Ansys element library. TARGE17 [1] To establish contact between pipeline and seabed contact elements COTA175 are created on the pipeline nodes. The contact are initiated by using same real constant set for contact and target elements. The node at the middle of the pipeline share the same real constant set 19

27 as the trigger berm. The rest of the pipeline share same the real constant set as the seabed. In the lateral case where there is no triggerberm there is naturally only one real constant set. COTA175 may be used to represent contact and sliding between two surfaces (or between a node and a surface, or between a line and a surface in D or 3 D. [Here node to surface]. The element is applicable to D or 3 D structural contact analyses. This element is located on the surfaces of solid, beam, and shell elements. Contact occurs when the element surface penetrates one of the target segment elements (TARGE169, TARGE17 on a specified target surface. The element is defined by one node. The contact algorithm used has to be specified. The penalty method uses a contact spring to establish a relationship between the two contact surfaces. The spring stiffness is called the contact stiffness. For the penalty method, normal and tangential contact stiffness are required. ASYS automatically defines default tangential contact stiffness that is proportional to µ and the normal stiffness. The maximum allowable elastic slip parameter is required. It is used to control maximum sliding distance when the tangential contact stiffness is updated each iteration. Ansys element library. COTA175, 11.4.Performing a node to surface analysis [1] The contact algorithm is the penalty method. This means that a spring in the target will keep the contact element up. The spring is represented by the vertical soil stiffness and given to ASYS as the normal contact stiffness. This is similar to the lateral soil stiffness described in previous chapter. In the analytically solution the soil was considered very stiff and penetration was not allowed. The values given to the spring in the ASYS model is the values of a very stiff soil [7] but penetration of some centimetres is allowed. This is more accurate to a real case. Since the trigger has to be able to withstand the entire pipeline at aslaid a higher value of k soil is used for the trigger target contact set. The lateral soil stiffness is generated by the vertical soil stiffness and the mobilisation length, the maximum allowable elastic slip. This value is usually a couple of centimetres. 3. The ASYS analysis An analysis of a non conservative system is path dependent: the actual load response history of the system must be followed closely to obtain accurate results. An analysis can also be path dependent if more than one solution could be valid for a given load level (as in a snapthrough analysis. Path dependent problems usually require that loads be applied slowly (that is, using many substeps to the final load value. Ansys 8.. Basic Information About onlinear Analyses [1] To be able to rely on the results it is necessary that the order of the different steps in the analysis corresponds to the actions of the real pipeline. A pipeline is first laid down which induces stresses in the pipeline due to the residual lay force and the bending of the pipeline caused by the shape of the seabed. This is the as laid condition where the pipeline is exposed

28 to only external pressure and the submerged weight. When the operation starts the pipeline is filled with hot medium giving internal pressure and this will rendering in rising temperature of the steel. To simulate this in ASYS the solution is devided into several timestep. Every timestep is then divided into substeps where the loads are applies in small parts. The order in which the timesteps are given are called the timeorder Vertical buckling The timeorder of the vertical buckling case is described in four steps, Figure 3. At the start of the analyses the link elements are locked at there upper node and the pipeline is locked axially at one end and having a residual tension force in the other, this to create the tensions caused by the real lay down conditions. The pipeline is simulated to hang above the seabed. (1 1 Links locked in all directions Pipe locked axially lay Figure. Start condition for ASYS analysis. The axial force in the last node is the residual tension force. This is caused by the real circumstances when a pipeline is laid down. The pipeline is loaded with external pressure as surface load and submerged weight eq.(1 as a nodal load. The link elements are in the next step lowered until the entire pipeline is resting on the seabed. ( q sub p e Lowering ode forces due to buoyancy and weight of concrete, steel and gas Figure 1. When still hanging the pipeline is loaded with external pressure and submerged weight. To thereafter be lowered down. 1

29 Since the link elements are tension only they will go slack if they are compressed. When the pipeline is at the seabed the link elements are deactivated as they are no longer necessary. The pipeline is restrained laterally and the ends are fixed at there positions. (3 3 Deactivate links Pipe locked laterally. Axially at ends Figure. When the pipeline is locked at the seabed the lowering link elements are killed In the last step, the operation phase, the pipeline is loaded with first internal pressure and then temperature. This starts building up the ective axial force which will make the pipeline deflect. 4 T p i Increasing temperature and pressure makes the pipe expand Figure 3. Deflection of the pipeline is caused by operation conditions of internal pressure and rising temperature. 3.. Lateral buckling on vertical trigger berm In the vertical case with lateral buckling the timeorder is almost similar with the difference of a small lateral force instead of lateral locked. This is to make it deflect lateral. Since the pipeline should not be able to buckle lateral without that small force the lateral restraining in case one should not be necessary but tests showed that without this restriction convergence problems occurred Lateral buckling on even seabed In the case with lateral buckling the pipeline is laid down as in the vertical case but without the trigger berm. It will therefore be completely straight when as laid. It is confirmed that

30 without an initial imperfection the needed buckling force is infinite. keeps rising with increasing pressure and temperature according to eq. (4. To make the pipeline buckle laterally a force, F pre, is applied at the nodes in the midsection around the midnode, as seen in Figure 4. Midnode Pipenodes L F pre Figure 4. To make the pipeline laterally imperfect a force is put on the nodes around the midnode at the pipeline. The timeorder for the lateral case is as in Figure 5. lat δ lat Stable postbuckling 5.5 e 6 Stable prebuckling lay "time" Preforce up Preforce down Internal pressure Temperature Figure 5. Lateral loading case. Lateral deflection and ective axial force as functions of time. F pre First a force,, is applied to create the imperfections on the pipeline but since the model is elastic the internal pressure needs to be applied as F pre is lowered down to zero. This is possible since the pipeline is not expected to buckle until the temperature is put on. The needed ective axial force for buckling is higher than the ective axial force that is reached with full internal pressure. At the last timestep, as the temperature is applied, will build up to make the pipeline deflect. As ASYS uses the ewton Raphson method for solving the problem the postbuckling behaviour cannot be followed. The pipeline will in infinite short time move from a stable prebuckling position to a stable postbuckling position. This means that the expected curvature is not possible to be find but the expected value of the ective axial force can be compared with the ASYS values before and after buckling. 3

31 3..4 The parameters and variables in the cases The scenario used for the analyses is an steel pipeline with a concrete coating transporting hot gas at 13 meters water depth. When the pipeline is as laid the temperature is equal to the sea temperature 5 C and the inner pressure is zero. When the pipeline is in operation the temperature is 1 C and the internal pressure is 3 bar. In ASYS the diameter used is for the steel pipe. The main purpose of the coating is to add weight and this factor is included in the submerged weight. The material model used is linear elastic. D e D s D i Figure 6. The pipeline model used has a concrete coating and transports gas. Table 1. Parameters and variables used, for full input see appendix 7.4 Parameter Value Unit Description D e.9 m External diameter D.8 m Steel diameter s D.75 M Internal diameter i i p 3 MPa Internal pressure in operation p 3.7 MPa External pressure e T 95 C Temperature rise EI Mm Bending stiffness q.197 k/m Submerged weight sub The variables δ vert 1,,3 m Trigger berm height µ.4,.6,.8, 1 Friction co For ASYS input files see appendix Results Vertical buckling First the response of a laterally restrained pipeline lying on a trigger berm is examined. shows the as laid and Figure 8 the deflected pipeline. 4

32 Design and development of FEM models used in expansion analyses Master Thesis MMK 7 Figure 7. Pipeline as laid on one meter trigger berm Figure 8. Pipeline vertical buckled caused by internal pressure and temperature. The analytical results are found using eq. (6 where the ective axial force is a function of trigger height. The analytical and ASYS results for needed lift off force at different height of trigger is found in Table. Table. The ective axial force at lift off. Comparison between analytical and ASYS model. It is seen that the analytical result and the result from ASYS correspond satisfyingly. It is also seen that with more initial deflection the error is lessen. Trigger berm height 1m m 3m lift [M] Analytical lift ASYS [M] Error % 1.4%.7% Figure 9 shows the buckling of the pipeline at a one meter trigger berm where the ective axial force is a function of the deflection. 5