Abstract
Hightemperature pressurized pipelines design requires special attention, as restrained thermal stresses are high. Due consideration should be given to thermal expansion, as stresses in bends of expansion loops are significant. Also, the modelling of the soilpipe interaction using soil characteristics, especially friction and lateral resistance, is important when analyzing hightemperature pipelines. This paper describes a numerical procedure for the analysis of global and local buckling behavior of high temperature pressurized buried pipelines. Results of finite element calculations are presented and discussed.
Pipeline; buckling analysis; thermal analysis; soilstructure interaction
Buckling analysis of hightemperature pressurized pipelines with soilstructure interaction
R. A. Einsfeld^{I}; D. W. Murray^{II}; N. YoosefGhodsi^{III}
^{I}Instituto Politécnico IPRJ, Universidade do Estado do Rio de Janeiro, Caixa Postal 97282, 28601970 Nova Friburgo, RJ. Brazil. einsfeld@iprj.uerj.br
^{II}Department of Civil & Environmental Engineering, University of Alberta, Edmonton, AB, Canada T6G 2G7. dwm@civil.ualberta.ca
^{III}Centre for Engineering Research CFER, 250 Karl Clark Road, Edmonton, AB, Canada, T6N 1E4. N.Yoosef@cfertech.com
ABSTRACT
Hightemperature pressurized pipelines design requires special attention, as restrained thermal stresses are high. Due consideration should be given to thermal expansion, as stresses in bends of expansion loops are significant. Also, the modelling of the soilpipe interaction using soil characteristics, especially friction and lateral resistance, is important when analyzing hightemperature pipelines. This paper describes a numerical procedure for the analysis of global and local buckling behavior of high temperature pressurized buried pipelines. Results of finite element calculations are presented and discussed.
Keywords: Pipeline, buckling analysis, thermal analysis, soilstructure interaction
Introduction
When a pipeline is operated at high internal pressure and temperature, it will attempt to expand and contract for differential temperature changes. Normally the line is not free to move because of the plane strain constraints in the longitudinal direction and soil friction effects. For positive differential temperature it will be subjected to an axial compressive load and when this load reaches some critical value the pipe may experience vertical (upheaval buckling) or lateral (snaking buckling) movements that can jeopardize the structural integrity of the pipeline. In these circumstances, an evaluation of the pipeline behavior should be performed in order to ensure the pipeline structural integrity during operation in such demanding loading conditions. Performing such analysis, the correct mitigation measures for thermal buckling can be taken into account either by acceptance of bar buckling but preventing the development of excessive bending moment, or by preventing any occurrence of bending.
In the early eighties, pipeline technology started to account for inservice buckling. A series of papers, e.g. Hobbs (1984) and Taylor and Ben Gan (1986), proposed analytical tools to predict the occurrence and the consequence of inservice buckling. In the late eighties and early nineties analytical tools were superseded in many circumstances due to the request to consider the plastic capacity of the pipe section. When the stresses exceed the elastic limit, a more sophisticated nonlinear analysis is required to account for the nonlinear state of stress. As the problems increased in their complexity, numerical models took the place of the analytical tools in the preference of engineers. Nevertheless, analytical models are still useful in the prediction of the buckling behavior as will be seen in the sequence of this paper.
The buckling mode expected in a specific case depends, among other factors, on the boundary conditions, magnitude and shape of the initial pipe imperfection, the type of axial constraints and soil reaction. The main effect of the nonlinear stressstrain relationship is to allow the prediction of the formation of a plastic hinge in the most loaded pipe cross section in the middle of the buckle. This paper describes a numerical procedure developed by the PipeTec Group of the Department of Civil Engineering of the University of Alberta for the analysis of global and local buckling behavior of high temperature pressurized pipelines. This study aims to supply the engineering staff and pipeline operators with a tool that allows them to evaluate the actual conditions of the pipe in different load conditions.
Nomenclature
A = crosssection area of the pipe
E = elastic modulus
F_{m} = maximum axial friction force
I = second moment of area of the pipe?
K = elastic stiffness for the axial friction
L = pipe length
N_{x} = tensile axial force at position x
N_{0} = residual laytension
u_{L} = relative axial displacement at the outer end point
u_{m} = mobilization displacement that corresponds to F_{m}/k
P = compressive reactive end force at the wrinkle
S = buckle length
= buckle length for a large f
x, y = Cartesian coordinates
= maximum buckle amplitude
w = selfweight per unit length
Greek Symbols
DT = differential temperature change
e_{T} = temperature axial strain
e_{p} = Poisson's ratio internal pressure axial strain
f = soilpipe coefficient of friction
Numerical Tools for Analysis of Buckling Pipelines
Some numerical tools for pipeline analysis are well known by researchers and engineers and have been tested in different situations in the last twenty years. This is the case of PIPLINIII (Structural Software Development, 1981), PlusOne (Andrew Palmer and Associates, 1995), PIPSOL (Nixon, 1994) and ABP (Zhou and Murray, 1994). All of these programs are based on pipe beam elements and elasticplastic soil springs. However, they differ significantly in the formulation of their pipe beam element. For the PIPSOL pipe element, the material is modelled as nonlinear elastic. Consequently, the PIPSOL solution runs are faster than the runs of the other three programs. The pipe elements in PIPLIN, PlusOne and ABP can model the elasticplastic behavior of the pipe material. All the programs assume that the pipe crosssection remains circular. Therefore, crosssectional deformation, including ovalization and local buckling, are not simulated. The recent version of program ABAQUS (Hibbitt et al., 2000) also incorporates pipeline beam elements, soilpipe interaction and large displacements in a way that one can model a considerable length of pipeline and predict the overall behavior of the structure for different load conditions.
ABP is a program for analysis of buried pipelines as well as aboveground pipelines. It was developed at the Department of Civil Engineering of the University of Alberta for research purposes. The program performs nonlinear analysis of twodimensional pipesoil systems subject to internal pressure, temperature differential, gravity and overburden loads, and differential soil settlement. The pipe element in ABP has an elasticplastic material model. The pipe is modeled by conventional elasticplastic beam elements. A number of integration points are placed on one crosssection and there are several integration crosssections in one element (YoosefGhodsi and Murray, 2002). The soil surrounding the pipeline is simulated by three types of elasticperfectlyplastic soil springs, i.e. bearing springs under the pipeline, uplift springs above the pipeline and longitudinal springs along the pipeline. The ultimate strength and the stiffness can be individually specified for each type of soil spring at each element. A classical model for soilstructure interaction is shown in Fig. 1 (Nyman, 1983).
The crosssection of the pipe in ABP is assumed to remain circular. As a result, the effects of crosssectional deformation, such as ovalization and local buckling, are not simulated. A special material model has been derived for this procedure (Zhou, and Murray, 1993), which has the longitudinal stress and strain as the active stress and strain, and the strains in the circumferential and radial directions as reactive strains. The stresses in the radial and circumferential directions remain constant in order to balance the internal pressure. The strains are determined from the active stress and strain by using the von Mises yield criterion, the associated flow rule and the condition of constant stress in both the circumferential and radial directions.
As crosssectional deformation, including ovalization and local buckling, is not simulated in ABP, the evaluation for local buckling is being performed by finite element (FE) analysis with shell elements. Since local buckling of pipes involves severe displacements and rotations, a large displacementlarge rotation formulation needs to be adopted. Standard finite element structural codes available on the market, e.g. ABAQUS, ADINA and ANSYS, have been introduced to analyze the effects of nonlinear material and large displacements. In this research, the commercial FE analysis package ABAQUS was used to predict pipe behavior. It includes the S4RF element type that is an efficient large displacement, large rotation and finite membrane strain shell element.
The development of local buckling and wrinkling that characterize the behavior of pipeline structures has been under investigation in the Department of Civil Engineering at the University of Alberta since the beginning of the last decade. Tests were performed in fullsized pipes subjected to combined axial load, internal pressure and bending moments. The tests demonstrate that the experimental and analytical studies of pipe behavior are consistent with the mechanism by which wrinkling develops in the postbuckling behavior of structures (Murray, 1997).
The validation of ABP has been made through comparison with results produced by programs exhaustively used, and considered as a standard by the industry, as PIPLIN. Some comparisons are presented in the work of YoosefGhodsi and Murray, 2002. The numerical procedure described in this paper, with the use of ABP and ABAQUS, was applied in many situations that occurred in the field. Nevertheless, the results of these investigations are property of the industry companies that contracted the analysis and, consequently, the publication of these results is quite limited. An example of industrial consultation that was published, concerning the case of NPS12 Norman Wells Pipeline, can be seen in the paper of YossefGhodsi et al., 2000.
Pipeline Anchor Length and Far Field Conditions
If a long straight pipe is subjected to an increased temperature, the thermal expansion is totally constrained and there is no longitudinal displacement at any section. In this case, the pipe develops selfequilibrating longitudinal compressive stress due to the constraint of the free expansion. Nevertheless, for a certain magnitude of the generated compressive force, global and/or local buckling can occur in a location of relative weakness, such as at an imperfection, in which the pipe cannot sustain the prewrinkling stress. This process allows the pipeline to form a wrinkle at which displacements localize because of slip movements relative to the soil toward the wrinkle location.
At points in the pipe, remote from the wrinkle, there is no motion of the pipe relative to the soil. The first point that one encounters transversing along the pipe in the direction away from the wrinkle where no relative movement occurs, is called a 'soilanchor' point. For a pipeline of length L moving from this anchorpoint in the direction of the wrinkle, as shown in Fig. 2a, the end displacement u_{L} is obtained solving the differential equation of equilibrium that arises from the free body diagram of Fig. 2b, where P is the compressive reactive end force at the wrinkle and N_{x} is the tensile axial force at position x.
If the right end of pipeline model represented in Fig. 2a is free to move, the reactive force P is equal to zero. In this case, the interaction between the pipe length and the end displacement is shown in Fig. 3. The most interesting aspect of this interaction is that the maximum end displacement reaches a limit beyond which it no longer increases as the embedded length increases. This means that there is no point in modelling a long pipeline using an embedded length longer than the one corresponding to the anchor length L_{a} and, therefore, much computational effort can be saved in this way.
Nevertheless, if far field conditions are imposed at the outer ends of the model, only a small portion of the pipe, shorter than the anchor length, has to be modelled. This portion is usually called the characteristic part of the pipe. The pipe sections adjacent to this characteristic part are assumed to be horizontal and straight. The far field conditions are obtained by solving the differential equations that govern the elasticplastic pipesoil interaction shown in Fig. 2b. This solution gives the axial force Nx that should be applied at the inner ends of the model in order to simulate the behavior of the pipe sections within the characteristic portion of the pipe. Solving the differential equations (Andrew Palmer and Associates, 1995), one can obtain:
where N_{0} is the residual laytension, E is the elastic modulus, A is the crosssection area of the pipe, e_{T} is the temperature axial strain, e_{p} is the Poisson's ratio internal pressure axial strain, u_{L} is the relative axial displacement at the outer end point, F_{m} is the maximum axial friction force for the elasticperfectly plastic interactive pipesoil shearslip relationship, and u_{m} is the mobilization displacement that corresponds to F_{m}/k, where k is the elastic stiffness for the axial friction.
Local Buckling Evaluation
As mentioned before, the pipe crosssectional deformation, including ovalization and local buckling, is not simulated in ABP. Because of this, for problems involving pipeslip, a technique was developed that combines local buckling results obtained from FE analysis with pipesoilslip results obtained from differential equations solutions. It is assumed that a FE model segment that contains a local wrinkle is compressed by thermal expansion of two anchor length segments. The interaction between the anchor length segments of the pipe and the FE model segment is represented in Fig.4.
A closedform differential equation relationship is established to determine the relative slip that will occur at the end of the anchor length segments due to a temperature change. This slip is dependent upon the internal resisting force within the buckle (that can be determined using the FE model). Superimposing the plot of end force vs. end slip displacement onto the plot of resisting force vs. shortening of the FE model segment permits the analyst to determine the deformation in the buckle for a compatible axial force and displacement by determining the point of intersection of the two curves, thus yielding a solution for the mechanism of wrinkle formation with slip.
This strategy is exemplified in Fig. 5, where the soilslip solutions for the anchor length segments for different temperature changes are represented by the dashed inclined curves, and the resisting force vs. shortening of the FE model segment is represented by the continuous curve. In this latter curve, the peak load corresponds to the limit point for the initiation of the local buckling. It can be seen that the wrinkle will not form for a differential temperature change DT_{1}, as the soilslip solution intercepts the curve obtained from the FE model segment in the ascending part (Point A) where the resisting force of the FE model segment exceeds the applied force arising from the expansion of the anchor segments. On the other hand, the wrinkle is already developed for a differential temperature change DT_{2}, as the intersection happens to be in the descending part of the FE model curve (Point B) after traversing the region where expansion forces exceed the FE resisting force.
Predictive Analytical Models for Global Buckling
The deformed configuration in global buckling analysis is triggered by the initial outofstraightness (IOS) imposed on the numerical model. The positioning of the IOS can be predicted by the use of analytical models that consider the elastic behavior of the pipe. Good results can be obtained for the global buckling length using this formulation (Hobbs, 1984). Nevertheless, due to the elasticplastic constitutive law applied in the numerical model, in contrast with the elastic formulation of the analytical model, great differences are encountered in the results obtained for the buckling amplitude. These considerations will be exemplified in the following section.
According to Hobbs (1984), the linear differential equation governing the deflected shape of the pipeline has the form (see Fig. 6):
where S is the buckle length, n^{2}=P/EI, P is the axial load at the buckle, E is the elastic modulus, I is the second moment of area of the pipe, m=w/EI for upheaval buckling (vertical mode), w is the selfweight per unit length, and f is the coefficient of friction between the pipe and subgrade.
Solving Eq.(3) and applying a compatibility equation that accounts for the reduction in the axial force in the buckle, the following result for the maximum amplitude of the buckle (see Fig. 6) is obtained for upheaval buckling:
The buckle length obtained for the case of a very large coefficient of friction is:
where A is the crosssectional area of the pipe.
For snaking buckling (lateral mode) shown in Fig. 7, the differential equation governing the deflection is the same as Eq.(3), except that m=fw/EI. For snaking buckling the following results are obtained:
In this analysis it is assumed that the cross section of the pipe remains circular. This is true at least in the initial stages of buckling, although the global buckling responses discussed here may lead to local buckling and failure of the pipe by yielding and ovalization. More details about this derivation can be seen in Hobbs (1984). In this work, only the derived equations for buckle length and amplitude are presented as they are of interest to predict the pipe global buckling configuration.
Example of Snaking Buckling Analysis
The following study was performed using the ABP program for the overall buckling analysis, in combination with the ABAQUS program for the local buckling analysis. The pipe under consideration was an API X65 grade line pipe with an outside diameter D equal to 12.75 in. (323.85 mm) and a wall thickness t of 0.25 in. (6.35 mm). Consequently, the diameter to thickness relationship is D/t = 51.
The model shown in Fig. 8 was used for the analysis performed with the ABP program. Only a characteristic portion of the pipe was modelled as the far field condition was imposed at the outer pipe ends. The pipe model was equally divided in 64 threenode beam elements. The analysis was performed for the snaking behavior considering a differential temperature up to 60º C. The IOS input in the model had a maximum amplitude equal to 50 mm, as shown in Fig. 8. Note that the vertical scale has been exaggerated in order to make the initial imperfection visible. Besides the temperature change, the pipe was also subjected to an internal pressure equal to 3.0 MPa. The distance between the IOSs (32 m) was initially estimated using Eq.5. This equation gives S = 31.2 m for fw = 1.0 N/mm and E = 200,000 MPa.
The relationship between the temperature change and the maximum amplitude of an equilibrium configuration is shown in Fig. 9. The early peak in the curve corresponds to the critical thermal loading condition of the overall buckling of the pipe. At this point (57º C), the pipe develops an "S" shape as the buckling amplitude jumps to the next equilibrium configuration, i.e. it snaps through as shown in Fig. 9.
Figure 10 shows the deformed configuration of the pipe in the postbuckling regime for a temperature change of 57º C. The buckled configuration shows a buckling halfwave length of approximately 34 m and a buckling amplitude of 1.7 m. The buckling amplitude calculated using Eq. 4 was 0.62 m. This difference was expected as the analytical model considers the material elastic behavior and the numerical model considers both the plastic behavior for the pipe material and for the pipesoil interaction.
The next step in the snaking analysis is to evaluate whether wrinkling will be triggered by the snaking buckle. This is done by applying a loading sequence that develops the wrinkling mechanism in a FE model that represents a segment of the pipeline in the region of greatest curvature of the snaking mode. The segment considered in this example has a length of 1.83 meters, that corresponds to approximately 6 times the diameter of the pipe. The end axial load vs. end axial displacement curve for the beam segment model is then obtained from the FE analysis. This diagram is plotted against the soilpipe interaction diagram related to a selected temperature change, as shown in Fig. 11. As shown previously in Fig. 5, the point of intersection between the two curves indicates whether the pipe will buckle locally during the global snaking buckling.
The loading sequence that develops the buckling mechanism considers an initial curved configuration followed by an axial displacement of the end points towards each other. This latter displacement simulates the pipe anchor length expansion due to the change in temperature. The initial curved configuration may be supposed to be formed during the pipe laying operation. In the worst scenario, a stress corresponding to the yield stress is assumed to occur in the extreme fibers of the pipe, as it is unlikely that the contractor and owner would knowingly permit plastic deformation to occur during the laying operation.
In the curve related to the ABAQUS FE model presented in Fig. 11, the peak load corresponds to the limit point for the initiation of the local buckling. For the temperature of 57º C, which corresponds to the snapthrough behavior shown in Fig. 9, the intersection occurs at the ascending part of the diagram and, therefore, no local buckling will occur at this stage. Nevertheless, for a temperature of 100º C, the positions of the two curves indicate that the expansive driving force exceeds the resisting force of the wrinkle and therefore the wrinkle will develop a large amplitude in the descending part of the diagram. In this case, a severe wrinkle is expected to develop and threatens the pipe integrity at that point. From Fig. 11 one can say that a temperature over 90º C should be avoided in order not to compromise the pipeline safety requirements.
The deformed configuration for an axial displacement of 60 mm is shown in Fig. 12 and it is possible to see how severe the wrinkle configuration for this local buckling behavior is.
Concluding Remarks
A numerical procedure for the analysis of global and local buckling of pipelines has been presented. The numerical technique considers the use of a pipesoil interaction formulation for the determination of the global buckling configuration, and the use of a FE commercial package (ABAQUS) for the local buckling evaluation. The IOS was predicted with the use of an elastic analytical model. It was shown that the buckling length and the expected deformed postbuckling pipeline configuration were obtained with a good approximation using this model.
This study is intended to supply engineering staff and pipeline operators with a tool that permits them to evaluate the susceptibility to buckling for different load conditions on the pipe. In this way, the correct mitigation measures for thermal buckling can be taken into account either by acceptance of global buckling but prevention of the development of excessive bending moment, or by prevention of any occurrence of bending.
Acknowledgements
The first author is thankful to the Brazilian agency Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for the grant that permitted him to spend a sabbatical year at the University of Alberta, Canada.
Paper accepted May, 2003
Technical Editor: Atila P. Silva Freire
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Publication Dates

Publication in this collection
18 Mar 2004 
Date of issue
Apr 2003
History

Accepted
May 2003