**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 in-service 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 in-service 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 In the late eighties and early nineties analytical tools were superseded in many circumstances due to the request to consider the plastic capacity of the pipe section.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 stress-strain 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* = cross-section 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*= tensile axial force at position

_{x}*x*

*N*= residual lay-tension

_{0}*u*= relative axial displacement at the outer end point

_{L}*u*= mobilization displacement that corresponds to

_{m}*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**

D*T *= differential temperature change

e* _{T}* = temperature axial strain

e

*= Poisson’s ratio internal pressure axial strain*

_{p}f = soil-pipe 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 PIPLIN-III (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 elastic-plastic soil springs. However, they differ significantly in the formulation of their pipe beam element. For the PIPSOL pipe element, the material is modelled as non-linear 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 elastic-plastic behavior of the pipe material. All the programs assume that the pipe cross-section remains circular. Therefore, cross-sectional deformation, including ovalization and local buckling, are not simulated. The recent version of program ABAQUS (Hibbitt et al., 2000) also incorporates pipeline beam elements, soil-pipe 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 above-ground pipelines. It was developed at the Department of Civil Engineering of the University of Alberta for research purposes. The program performs non-linear analysis of two-dimensional pipe-soil systems subject to internal pressure, temperature differential, gravity and overburden loads, and differential soil settlement. The pipe element in ABP has an elastic-plastic material model. The pipe is modeled by conventional elastic-plastic beam elements. A number of integration points are placed on one cross-section and there are several integration cross-sections in one element (Yoosef-Ghodsi and Murray, 2002). The soil surrounding the pipeline is simulated by three types of elastic-perfectly-plastic 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 soil-structure interaction is shown in Fig. 1 (Nyman, 1983).

The cross-section of the pipe in ABP is assumed to remain circular. As a result, the effects of cross-sectional 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 cross-sectional 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 displacement-large 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 full-sized 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 post-buckling 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 Yoosef-Ghodsi 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 Yossef-Ghodsi 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 self-equilibrating 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 pre-wrinkling 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 ‘soil-anchor’ point. For a pipeline of length *L* moving from this anchor-point 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*is the tensile axial force at position

_{x}*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 elastic-plastic pipe-soil 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 lay-tension,

*E*is the elastic modulus,

*A*is the cross-section area of the pipe, e

*is the temperature axial strain, e*

_{T}*is the Poisson’s ratio internal pressure axial strain,*

_{p}*u*is the relative axial displacement at the outer end point,

_{L}*F*is the maximum axial friction force for the elastic-perfectly plastic interactive pipe-soil shear-slip relationship, and

_{m}*u*is the mobilization displacement that corresponds to

_{m}*F*, where

_{m}/k*k*is the elastic stiffness for the axial friction.

**Local Buckling Evaluation**

As mentioned before, the pipe cross-sectional deformation, including ovalization and local buckling, is not simulated in ABP. Because of this, for problems involving pipe-slip, a technique was developed that combines local buckling results obtained from FE analysis with pipe-soil-slip 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 closed-form 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 soil-slip 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 D*T _{1}*, as the soil-slip 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 D

*T*, as the intersection happens to be in the descending part of the FE model curve (Point

_{2}*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 out-of-straightness (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 elastic-plastic 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 cross-sectional 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=*f*w/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 three-node 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 f*w* = 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 post-buckling regime for a temperature change of 57º C. The buckled configuration shows a buckling half-wave 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 pipe-soil 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 soil-pipe 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 snap-through 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 pipe-soil 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 post-buckling 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.

**References**

Andrew Palmer and Associates, 1995, “PlusOne Version 4.1 – Modelling of Upheavel Buckling of Buried/Trenched Pipelines – Theoretical Manual”, SAIC Ltd., Artillery House, Artillery Row, London, England. [ Links ]

Hibbitt, H.D., Karlsson, B.I., and Sorensen, P., 2000, “ABAQUS Theoretical Manual”, Version 4.5, March 1984. [ Links ]

Hobbs, R.E., 1984, “In-Service Buckling of Heated Pipeline”, Journal of Transportation Engineering, V.110, N.2, March 1984, pp.175-189. [ Links ]

Murray, D.W., 1997, “Local Buckling, Strain Localization, Wrinkling and Post-Buckling Response of Line Pipe”, Engineering Structures, V.19, N.5, pp. 360-371. [ Links ]

Nyman, K.J., 1983, “Thaw Settlement Analysis for Buried Pipelines in Permafrost”, Pipelines in Adverse Environments – II, ASCE Special Publication, Edited by Mark B. Pickell, pp. 300-325. [ Links ]

Nixon, J.F., 1994, PIPSOL, “Stress Analysis Program for Pipes Subjected to Ground Displacements”, Nixon Geotech Limited, Calgary, Alberta, Canada. [ Links ]

Structural Software Development Inc., 1981, “PIPLIN-III Computer Program for Stress and Deformation Analysis of Pipelines”, Structural Software Development Inc., Berkeley, California. [ Links ]

Taylor, N., and Ben Gan, A., 1986, “Submarine Pipeline Buckling – Imperfection Studies”, Thin-Walled Structures, V.4, pp.295-323. [ Links ]

Yoosef-Ghodsi, N., Cheng, J.J.R., Murray, D.W., Doblanko, R., and Wilkie, S., 2000, “Analytical Simulation and Field Measurements for a Wrinkle on the Norman Wells Pipeline”, Proc. of the Inter. Pipeline Conference IPC 2000, ASME (OMAE Division), Oct. 1-5, 2000, Calgary, Alberta, Canada, pp. 931-938. [ Links ]

Yoosef-Ghodsi, N., and Murray, D.W., 2002, “Analysis of Buried Pipelines with Thermal Applications”, Structural Engineering Report No. 246, Dept. of Civil and Environmental Engineering, University of Alberta, Edmonton, Alberta, Canada. [ Links ]

Zhou, Z. J., and Murray, D.W., 1993, “Behaviour of Buried Pipelines Subjected to Imposed Deformations”, 12^{th}Int. Conference on Offshore and Arctic Engineering, ASCE, V.II, pp.5-122. [ Links ]

Zhou, Z.J. and Murray, D.W., 1994, “Analysis of Buried Pipelines (ABP) – Users Manual”, Dept. of Civil Engineering, University of Alberta, Edmonton, Alberta, Canada. [ Links ]

Source : Pipeline Buckling