Stress analysis of pipelines with small misalignment under internal pressure using fast modelling system ABAP

. This paper presents a numerical finite element analysis of pipelines with small misalignment under internal pressure using a developed fast pipeline modelling system ABAP using Python development environment of ABAQUS. The effect of small misalignment is shown to affect at most 10% of stress levels. Extensive parametric analyses have been conducted on the pipe length, pipe thickness. Pipe length has trival effect on non-uniform stress distribution when length/diameter ratio is greater than 1. For thinner pipe, misalignment amplifies the circumferential non-uniformity more significantly than thicker pipe. This paper serves to enhance the understanding of pipes with small installation misalignment defects under pure internal pressure.


Introduction
Pipelines under internal pressure may experience burst failure and lead to economic loss and threats to safety of humans. The assessment of integrity of pipelines with various kinds of defects is pivotal to maintain safe operation. There are various researches on these aspects. For example, Chen et al. [1] presented an analytical method to calculate the burst pressure of long corroded pipelines under internal pressure. The existence of corrosion flaws reduces the burst pressure. Furthermore Jin et al. [2] conducted integrity assessment of the pipelines containing an isolated corrosion pit by the similar analytical method. Ye et al. [3] innovatively presented an analytical large deformation collapse model of pipelines under excessive external pressure by including the effect of plastic hardening of materials. Similarly Yan et al. [4] studied the collapse failure and equal potential energy load of shallow cylindrical shells. A beam-model was developed by Yan et al. [5] to investigate the corrosion-induced thickness loss on the critical collapse pressure through the tangent modulus method to account for plasticity effect. Netto et al. [6] also studied the burst pressure of pipelines when initial thickness loss was considered. Finally Ye et al. [7] and Yan et al. [8] presented analytical investigations of corroded pipes and shallow cylindrical shells. The thickness non-uniformity was shown to affect significantly the collapse pressure. From above, the thickness-loss defect's effect on critical pressure has been well understood. But in some occasional case, especially during installation, pipes may be welded imprecisely and it leads to some misalignment defect as illustrated in Fig.1. The assessment of the severity of the misalignment defect is critical for safety maintenance of the entire piping system. Thus this paper presents some finite element analyses of pipelines with misalignment defect under internal pressure based on our previously developed fast piping modelling system ABAP. Extensive parametric analyses(based on linear perturbation analysis in ABAQUS) are conducted to illustrate the effect of thickness, misalignment and pipe lengths on the stress distribution. To connect pipes with misalignment, ABAP system(see section 4) automatically detects the misalignment defect and generats a connecting instance based on orphan mesh method in ABAQUS. The Dofs are fully coupled between connecting instance and pipes. Internal pressure is imposed on the internal surfaces of this assembly with end cap effect considered. The end cap effect is modelled by binding the nodes of two end crosssections to reference points at centers through BEAM type constraints in ABAQUS and imposing forces normal to the bound cross-sections at ends of magnitudes equal to . Note that in this paper 1 By linear perturbation analysis, without loss of generality, internal pressure is fixed as1MPa . After calculation, we use linear stress linearization functionality in ABAQUS to use 64(64 elements are uniformly assigned in circumferential direction) pairs of points circumferentially to extract stress information along each line linked by each pair(see Fig.3 and Fig.5 for circumferential indices). Fig.4 shows the Mises contour when 1 The mises stress level is higher at index 32 and lower at index 0. Fig. 5 shows the Mises stress contour when pipe is thicker with 1 2 0.1m t t   and similarly the convex part has lower stress level.  Fig. 6 shows the Tresca stress and Mises stress of linearized stress along each pair of points. By stress linearization the averaged stress along thickness is called membrane stress and the linearized stress at one end of thickness is called membrane+bending1(the other is called membrane+bending2 and in this analysis membrane+bending1 stress is always greater than membrane+bending2).

Parametric analysis on circumferential stress distribution
Tresca stress of membrane+bending1 is largest and Tresca stress of membrane is lower. By definition Tresca stress is always larger than Mises stress. The abscissa from 0 to 63 means circumferential indices as in Fig.5. As index increases from 0 to 32, all stresses increase and by symmetry as index increases from 32 to 63, all stress decreases.   Fig.7, the Tresca stress(membrane) and Tresca stress(membrane+bending1) are presented for various misalignment parameter e. As e increases, the effect of circumferential index is increasing. For example, at index 0, for e=0.02, the Tresca(membrane) stress is about 4.35MPa and at index 32, the corresponding stress is about 4.47MPa. However for e=0.07, the stress increases from about 4.22MPa to about 4.63MPa. The trend is similar for membrane+bending1 case. This analysis shows the significant effect of misalignment parameter e.  Fig. 8 shows the Tresca stress(membrane) for various thicknesses. The Tresca stress ratio is defined as the Tresca stress(membrane) divided by maximum Tresca stress(membrane) for all circumferential indices. As thickness decreases the non-uniformity of stress distribution increases significantly. For example when 1 2 0.05m t t   , ratio has minimumvalue about 0.88 and when 1 2 0.1m t t   the ratio has minimum value about 0.91. This shows that thinner pipe is more significantly affected by misalignment defect. Fig. 9 presents the effect of pipe lengths on Tresca stress distribution by studying circumferential distribution of Tresca stress(membrane) for 1 2 1,2,3,4,5,6m L L   and fixed 2 / 0.07 e L  , 1 2 0.05,0.1m. t t   All lines coincide indicating that pipe length has trivial effect on stress distribution.

Brief introduction of ABAP system
ABAP(namely ABAqus & Python) system was developed by Python by the first author and the kernel is based on secondary python development environment of ABAQUS software. The developed GUI interface is able to input the node locations, pipe connectivity information and geometric parameters. The kernel is responsible to accept the GUI-generated command text, interpret the text, build geometric pipe model, mesh the pipes by structural grids, assembly the pipes and impose the load to submit the job. All parametric analyses in this paper are based on this system quite conveniently without the tedious process of repeatedly building finite element models manually.

Conclusions
(1) Under internal pressure the pipe misalignment defect slightly affects the circumferential stress distribution. For example, a 4-degree misalignment(corresponding to e/L2=0.07) leads to about 10% variation of stress circumferentially.
(2) The misalignment's effect is more significant when pipe is thinner.
(3) The pipe length has trivial effect on stress distribution circunferentially. (4) All parametric analyses can be conveniently conducted by ABAP system without manually repeatedly building FEA models. This shows the effectiveness of ABAP system in enhancing the analysis efficiency.