ABSTRACT
Gas transmission pipe resistance to external damage is a subject of great attention at Gaz de France and in Europe. Existing results cover part of the necessary criteria for the residual life of damaged pipelines, but more knowledge is needed on defect creation. We propose to complement existing experimental work which is limited to the explored range of parameters by validated numerical models. The first, simple static denting model aims at optimizing the conditions for calculating the residual stress distribution needed to assess the fatigue life of dents and dents and gouges. The second, more complex dynamic puncture model calculates both the puncture force and the puncture energy for a given pipe, excavator and tooth geometry. These models can contribute to enhance the external damage prevention policies of transmission pipeline operators.
1. INTRODUCTION
Gas transmission pipeline failure is mainly due to external damage : the failure percentage related to external mechanical damage ranges from 55 % in the U.S.A. to around 70 % in Europe. The european transmission networks being more recent than american ones, they are less subject to corrosion. Hence, pipeline resistance to external damage like dents, gouges, dents and gouges, and puncture is currently a subject of investigation in Europe. Two aspects have to be tackled in this context :
- defect creation (comparing pipe resistance to the machinery damaging capacity)
- defect stability (whether the defect evolves to a leak or a break)
The second aspect is generally quite well understood, especially due to the power industry's needs (Miller, 1988), and has also led to specific developments for transmission linepipe (see Jones, 1982, Kiefner, 1969, Kiefner et al., 1973, Maxey, 1987, Muntinga and Koning, 1990). Nevertheless, subjects like dent and gouge fatigue strength are not yet solved. The first aspect is less known, except : some experimental evidence of part-wall defects created in a laboratory by Battelle (Columbus), without relating pipe resistance to the damaging action, as well as a simplified pipe denting model by Spieckhout et al. (1986). Semi-empirical criteria for pipeline resistance to external damage already exist for some cases, like denting. Puncture resistance is one of the subjects that are still evolving. Experimental data is available, but it does neither give complete insight into the phenomena, nor cover the whole range of pipes.
2. SCOPE OF WORK
Gaz de France developed two specific 3D elasto-plastic finite element modeling approaches in order to treat dent and gouge fatigue strength and puncture resistance, which are both related to denting. Denting means deforming the pipe with an aggression tool (for instance, an excavator tooth). This phenomenon is considered from different points of view according to the expected result :
- when studying dent and gouge fatigue strength, the variation of the residual stress
distribution between extreme pressures is sought
- for puncture resistance, local material failure is the relevant criterion
We describe here the main features of these models, namely the handling of the various non-linearities, their validation by comparing them to experimental results, and their range of applications.
3. PROBLEM DEFINITION AND MODEL CHOICE
The mechanical problem treated here is modeled by an impactor forced vertically on the upper generatrix of a pressurized closed-ended pipe resting on a V-shaped support (see Fig.1). The largest dimension of the impactor is aligned with the pipe generatrix. In order to limit the stiffening effect of the end caps, the pipe is between 4 to 10 times longer than its diameter. The support is shorter than the pipe, so the vertical motion of the end caps is not restrained. The external damage process is either static - simulated by a hydraulic jack imposed displacement - , or dynamic, - simulated by the free fall of an extended excavator arm from a given height. The range of velocities is : about 0.005 m/s in the static case, and 3 to 5 m/s in the dynamic case. This velocity range would indicate quasi-static modelling in both cases. But the ways in which the loads are applied : imposed displacement in the static case, and imposed initial velocity and kinetic energy in the dynamic case, lead to different modelling approaches. The static case can be modeled as such, but the so-called "dynamic" case has to be treated by a technique which does not impose force and displacement, the two latter being results of the calculation. "Explicit" finite element codes solve this problem by taking into account inertia terms - here, those related to the excavator arm. The main common feature of these models is that they take into account several non-linearities which are critical in order to correctly describe denting : large displacements (and strains in the dynamic case), elasto-plastic behaviour law with strain hardening, and contact.
4. STATIC DENTING MODEL
4.1. General model description
This model was developed to represent the mechanical state of a dented and gouged pipe, and only the denting modelling is presented here (Toumbas and Meziere, 1995). Actual defects can be quite long, so we made the simplifying assumption that the total length of the dent is created at once. This allows for applying the load as an imposed displacement, and for a simple treatement of the contact non-linearity, which is explained in more detail below. The SYSTUS finite element code was used for this static model. Symmetries of the pipe and the loads allowed to study only one quarter of the structure, leading to a reasonable mesh size (2800 nodes, 5400 elements). The mesh was refined in the impactor-pipe contact area in order to allow for accurate tracking of the steep strain gradients. The minimum element size was in the order of magnitude of the pipe thickness. Thin triangular shell elements with linear interpolation for the shell's in-plane displacement and cubic interpolation for bending were used. The shell element includes 9 stations in its thickness. The updated-Lagrangian resolution method accounts for the large displacements non-linearity and small strains. The steel behaviour is represented by a non-linear stress-strain curve with isotropic hardening and the Von Mises elasto-plastic criterium.
4.2. Contact modelling
We looked for simplicity when treating this case, especially regarding the treatment of the contact non-linearity by imposed displacements. This is a realistic assumption as long as the impactor can be considered as very rigid. In previous studies, the denting process may have been simulated by a displacement imposed directly to the pipe nodes. This loading permanently "sticks" the denting tool onto the pipe, and thus does not reflect the physical reality of the contact zone that changes during the denting process. In order to overcome this limitation, some surface contact elements have to be used. The SYSTUS code having no such elements for three-dimensional shell structures, we created a nodal contact by using non-linear spring elements. In order to model the pipe/impactor reactions, one defines a fictitious one-side compression spring with normal stiffness K between the denting tool and the pipe. The spring stiffness K is zero in tension and is given on the compression side by :
with : p : penalty coefficient , E : Young's modulus, e: pipe thickness.
The penalty coefficient p of these springs is an adjustable parameter. It represents the hardness of the pipe material when it comes into contact with the impactor. Its value can vary from 0,001 up to 1, but its influence on the results does not exceed 10 %. We found that there is an adequate value for K for each material and for which physical contact phenomena are well represented. For the cases studied here, the value of p was adjusted by comparison with experimental values, and then kept constant for all the cases. In order to insure a good description of the changing extension of the contact zone, the springs are positioned at all the pipe nodes which could get in contact with the impactor. Its shape is given by the initial position of the springs, and the displacement is imposed to all springs at the same time. The compressed springs are in actual contact, the others are not in contact. The total denting force is obtained by summing up the reactions of all contact springs being activated during the vertical displacement of the impactor.
4.3. Static model validation
We validated this model on several pipe diameters, either with internal pressure up to total dents of 10 % of the diameter, or without pressure, up to total dents of 15 %. Experimental data from static denting tests was used for model validation, where the impactor is a new excavator tooth. We illustrate this validation procedure by an example of a pressurized (load factor about 0.8) 914 mm diameter pipe dented by a 120 mm long CASE -POCLAIN CAL 60 tooth. Fig.2 represents the evolution of the denting force versus impactor displacement, which describes the global structure response. The maximum deviation between computed and experimental results is about 9 % up to a total dent of 5.5 % of the diameter. Simulated elastic-plastic springback (not presented here) compares also well with test observations.
4.4 Static model range of applications
This model gives satisfactory results for significant dents and was validated for narrow impactor-pipe contact areas which are aligned with the pipe axis. In this case, the impactor length is the only geometrical parameter, so it is simple and does not require precise impactor surface meshing. This model is used for several purposes :
- determine the dent depth created by a given force (i.e. a given machinery)
- when combined with springback calculations, it allows to relate total and residual
dent depths more accurately than available techniques
- it gives the residual stress variation between extreme pressure values in the dented
zone which leads to the remaining life-time under internal pressure
variation fatigue loading of the dent
- when combined with a local model, it is used to determine the remaining
life-time under fatigue loading of a dent and gouge
5. DYNAMIC MODEL
5.1. General model description
Modelling the denting until puncture needs to add some failure criterion, as described in details below. In this case, the local phenomena play an important role, so the tooth geometry and the contact model have to be accurate. If in addition, the damaging process is due to the rapid loading imposed by the large inertia of earth moving equipment (several tons), special explicit finite element techniques must be used. The short time scale associated with the puncture process - about 10 ms - indicate that wave phenomena might appear, confirming the need for explicit techniques. These criteria led us to use the ABAQUS-Explicit FE code in order to develop our model (Zarea and Toumbas, 1996).
5.2. Dynamic FE model description
The pipe puncture FE model was built on the following principles :
- a quarter of the total structure is studyied for symmetry reasons
- the impactor is modeled as a rigid body, with a detailed surface meshing
- the impact-load is modeled by assigning to the impactor an equivalent mass and
an initial velocity
- friction is neglected
- the pipe is modeled with elasto-plastic quadrangular shell elements
- end caps are modeled as rigid bodies
- the pipe rests without friction on the v-shaped support (radial displacement of
the pipe-support contact points is blocked)
- large displacements and large strains are taken into account
- the failure criterion is derived from the Gurson-Tvergaard model
The shell elements are under-integrated, i.e. the surface integrals are obtained with a single integration point. Along the thickness, the integrals are obtained with seven points of integration, corresponding virtually to seven layers. A static elasto-plastic behaviour law with isotropic strain-hardening is used. At large strains, the actual law is extrapolated by a power-law. The mesh contains about 7000 elements, roughly 2000 to describe the impactor surface geometry, and 5000 elements are used for the pipe. The mesh is strongly refined in the impact area (see Fig.3), the minimum element size being close to the pipe thickness. The smallest element size determines the time-step of the explicit algorithm : in our case, its value is about 10-7 s. In order to avoid lengthy calculations induced by the oscillations decay-time due to the explicit treatment of the constant initial internal pressure loading, pressure is imposed by an ABAQUS-Implicit calculation. In this way, a factor of four is gained in CPU-time. The equivalent mass assigned to the impactor is deduced, for instance, from the excavator arm rotation inertia.
5.3. Ductile rupture mechanism : material softening
The Von Mises criterium generally used in elasto-plastic calculations is not sufficient for predicting the actual failure of a steel structure under large ductile strain. The examination of ductile rupture surfaces, given sufficient enlargement, shows the presence of voids around inclusions, which play an important role in the ductile rupture mechanism. A ductile rupture evolves in three stages :
- void nucleation around inclusions
- void (or cavities) growth
- coalescence of cavities which initiates a crack
On the macroscopic scale, this amounts to material softening.
The Gurson-Tvergaard (Gurson, 1977 and Tvergaard, 1981) model programmed into the ABAQUS/Explicit code (ABAQUS, 1994) for shell elements is adapted to the representation of a porous material. The function F describing the yield surface of the elasto-plastic domain according to this theory depends, apart from the Von Mises stress q, on yield stress sy, on stress triaxiality p, on the evolution of the void volume fraction f, and on the parameters q1, q3 (q1=q3=1.5) allowing for control of the instant of rupture :
Material softening can be well described by this model. Failure appears when the softening process accelerates strongly. This acceleration phase is triggered when the void fraction reaches a critical value fc and the material loses its resistance.
5.4. Model implementation and failure criterion
We follow the three steps mentioned above for the failure mechanism, which does not involve any numerically introduced cracks.
5.4.1 Void nucleation.
In this study we do not consider the energy necessary for void nucleation. Additionnally, detailed modern steel microscopic observations show that inclusions are rather spherical. Hence, the initial void ratio (or initial porosity f0 with voids assumed to be spherical) is equal to the inclusions volume fraction fvolMnS%. This value is calculated from the chemical composition of the steel by using the Franklin formula (Batisse et al., 1986), given the volume contents of Sulphur (S%) and Manganese (Mn%) :
5.4.2. Cavities growth.
The cavities growth ratio f/f0 is given by the Gurson-Tvergaard approach.
5.4.3. Failure criterion.
To describe material fracture, we use the Beremin (1981) approach based on the knowledge of the critical value for the cavity radius expansion ratio (R/R0)cr. This results in a sudden failure, rather than a progressive one in the Gurson-Tvergaard approach. The ratio (R/R0)cr was obtained by tests on axisymmetrical notched specimens of X52 steel (Réglé, 1995) by using the Rice and Tracey (1969) evolution law for R/R0. This critical value was extrapolated to steels of similar structure. The critical ratio has to be related to the critical void fraction fc of the Gurson-Tvergaard model. Following Rousselier (1986), we consider that the integral giving the evolution of the cavity radius expansion ratio is equivalent to the integral giving the evolution of the void fraction f. Thus, we obtain :
For metallurgically clean modern steels the void fraction ranges between roughly 10-4 and 10-3.
5.5. Dynamic model validation
The dynamic puncture model was validated on several different diameter pressurized pipe puncture experiments, including different tooth geometries. We illustrate this validation by a 406 mm diameter pipe puncture test performed with a Case-Poclain 170 C (32 tons) excavator and a 80 mm long Cal 44 new tooth (see Fig.3 for mesh). The equivalent mass of the tooth is about 2000 kg, and its initial speed is 4.42 m/s.
The structure response in terms of force versus displacement is given in Fig.4 for the dynamic model computations and for the measurements.
On these curves we observe that the pipe puncture experimental values are closely predicted by calculations. The numerical and experimental loading curves follow very similar trends, with slight differences probably due to vibrations of the pipe and of the excavator arm, the latter not being taken into account in the numerical model. Nevertheless, the deformation energy up to puncture is almost the same, close to 10,000 joules. Element failure starts locally at about 75 % of the puncture force, and the total row of elements under the tooth fails at the maximum force, materializing well tooth penetration. Thus, even though the structure response differs after puncture, the puncture instant is well identified (see Fig.4). The assumption of negligible friction before puncture is also validated. The divergence after puncture may be attributed to two factors :
- different failure mechanisms when the tooth enlarges the initial hole
- temperature effects not taken into account, which soften additionnally the
material in the final stage
The residual dent profiles calculated at the end of the impact and after elastic springback are compared to those obtained in the experiments (see Fig.5). Good overall comparison is observed, although experimental uncertainties concerning the bent-over end of the longitudinal profile does not allow a full comparison.
5.6 Dynamic model application range
Parametric studies involving different impact speeds showed that a threshold kinetic energy is needed to puncture the pipe. Once puncture is achieved, the puncture force is quite reproducible for given values of tube diameter, thickness, grade, internal pressure and tooth geometry. This means that puncture criteria can be established or enhanced by means of the present dynamic puncture model for a variety of pipes, excavators, and tooth geometries. It can economically complete existing experimental work. Such criteria can be used in two ways :
- for design purposes, define the pipe characteristics that makes it puncture-proof
- for operational use, define the machinery which can be considered as harmless
(unable to puncture) in the vicinity of a given pipe
Additionnally, such a model can be used for dynamic denting modelling without puncture.
6.CONCLUSIONS
Numerical models for static denting and dynamical puncture were developed and validated.These models take into account large non-linearities :
- large displacements and strains
- elasto-plastic material behaviour law
- contact between pipe and impactor
- ductile steel rupture at puncture (dynamic model only)
The simple static denting model contributes to assess the residual life-time for a fatigue loaded dent and gouge defect, as well as for a plain dent. The more complex dynamic puncture model can be used to develop puncture criteria which may improve the gas transmission pipelines' external damage prevention policies.
Acknowledgements
We hereby kindly acknowledge the contribution of the team in charge of experimental work : Mr. G. Dreyfus, P. Cardin, S. Inderbitzin, J. Halhal, J. Skrzypek, as well as Mr R. Batisse, A. Pineau, Y. Mézière, and R. Champavère for useful discussions.
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