Proceedings of the ASME 27th International Conference on Offshore Mechanics and Arctic Engineering OMAE2008 June 15-20, 2008, Estoril, Portugal

Proceedings of the ASME 27th International Conference on Offshore Mechanics and Arctic Engineering OMAE2008 June 15-20, 2008, Estoril, Portugal OMAE 2008-57836 STRESS CONCENTRATION FACTORS OF LONGITUDINAL AND TRANSVERSE PLAIN DENTS ON STEEL PIPELINES Bianca C. Pinheiro COPPE/UFRJ - Ocean Engineering Dept. Rio de Janeiro, RJ, BRAZIL Ilson P. Pasqualino COPPE/UFRJ - Ocean Engineering Dept. Rio de Janeiro, RJ, BRAZIL ABSTRACT The objective of this work is to evaluate the stress concentration induced by longitudinal and transverse plain dents on steel pipelines under cyclic internal pressure. This work is within a study to propose a new methodology to assess the fatigue life of dented steel pipelines based on the current high cycle fatigue theory. This methodology employs stress concentration factors induced by plain dents, which are used to modify material S-N curves of metallic structures under high cycle fatigue loadings. The proposed assessment methodology was validated according to small-scale fatigue test results of steel pipe models with spherical dents under cyclic internal pressure. Here, stress concentration factors induced by longitudinal and transverse plain dents on steel pipes under internal pressure are obtained from a previously developed finite element model. Several finite element analyses are carried out in a parametric study. Analytical expressions are developed to estimate stress concentration factors for these two different dent geometries as function of pipe and dent geometric parameters. With the inclusion of these expressions, the proposed assessment methodology is improved and is now able to deal with three different plain dent geometries: spherical, longitudinal and transverse dents. INTRODUCTION One of the possible failure modes of oil and gas pipelines is the high cycle fatigue due to stress concentration in dented sections [1]. Cyclic loadings may be generated by fluid pressure and temperature changes and, in the case of offshore pipelines, also by the action of streams. This type of defect can be generated by the impact of heavy equipment or objects and may comprise different types and geometry, accumulating high levels of plastic strain and residual stresses. In view of the possibility of high cycle fatigue failure, the assurance of a safe pipeline operation is obtained with a consistent assessment of the stress field in the mechanical damages. The assessment of the structural integrity of pipes subjected to cyclic loadings can avoid flow interruptions caused by unnecessary repairs or leaks. The induced stress concentration factors (SCFs) have been the main focus of previous studies on the fatigue behavior of dented steel pipelines [2-9]. It was found that the associated stress concentration factors are mainly function of the pipe and dent dimensions, the material properties and the boundary conditions [2-8]. These studies also involved the proposal of methods to assess the fatigue life of dented pipelines based on empirical formulations and numerical simulations [5-8]. Some of these methods have been discussed by Cosham and Hopkins []. The numerical models based on finite element (FE) analysis, for instance, can precisely evaluate SCFs for even complex structures, but it requires an experienced professional and several hours of computer processing. Direct formulations derived from FE results can be of practical application to obtain the desired SCFs and carry out the fatigue analysis through the more suitable theoretical approach. The aim of this work is to evaluate the stress concentration induced by longitudinal and transverse plain dents on steel pipelines under cyclic internal pressure. This work is part of a study to propose a new methodology to assess the fatigue life of dented steel pipelines based on the current high cycle fatigue theory. This methodology employs stress concentration factors induced by plain dents, which are used to modify material S-N curves of metallic structures under high cycle fatigue loadings. 1 Copyright 2008 by ASME

A previously developed finite element model [11,12] is modified to generate stress concentration factors for plain longitudinal and transverse dents on steel pipes under internal pressure. The SCFs obtained from the numerical model are validated according to small-scale fatigue test results of steel pipe models with spherical dents under cyclic internal pressure. For each dent geometry several finite element analyses are carried out in a parametric study to evaluate SCFs for different pipe and dent dimensions. The numerical analyses comprise an elastic-plastic simulation of the denting process followed by an elastic determination of the stress concentration factor. With the aid of the Buckingham s theorem, the most relevant geometric parameters are used to develop analytical expressions to estimate SCFs for the three different dent geometries studied as function of pipe and dent geometric parameters. Additionally, an expression for short longitudinal dents is also proposed. Finally, as a result of an extensive numerical study about stress concentrations on dented steel pipes, the proposed methodology comprises simple expressions to estimate stress concentration factors induced by three different plain dent geometries: spherical, longitudinal and transverse dents. This methodology can then be used to assess the fatigue life of dented steel pipelines based on the current high cycle fatigue theory. NUMERICAL MODEL A numerical procedure based on the FE method was developed to estimate stress concentration factors induced by plain dents on pipes under internal pressure. It comprises a nonlinear three-dimensional shell type elastic-plastic model, developed using the mainframe ABAQUS, release 6.5 [13]. This model was initially developed to generate SCFs for spherical dents [11,12], as shown in Fig. 1. FIGURE 1: FE MESH AND THE ANALYTICAL SURFACE SIMULATING THE DENTING TOOL TO PRODUCE SPHERICAL DENTS. The denting tools were simulated by analytical rigid surfaces. The cylindrical denting tool is positioned parallel or perpendicularly in the case of longitudinal or transverse dents, respectively (Figs. 2 and 3). FIGURE 2: FE MODEL WITH DENTING TOOL TO PRODUCE LONGITUDINAL DENTS. FIGURE 3: FE MODEL WITH DENTING TOOL TO PRODUCE TRANSVERSE DENTS. Constitutive Model In order to accurately model the pipe response to denting, spring back and cyclic pressurization, a plastic constitutive behavior was adopted within the potential flow rule, assuming the von Mises yield function under combined isotropic and kinematic hardening. As the rerounding effect during the first cycle of pressurization confers reverse plastic deformation to the pipe, the Bauschinger effect must be taken into account. Since cyclic uniaxial tensile tests were not carried out, an approximation routine using only half cycle [13] was adopted to estimate this effect within the kinematic hardening theoretical approach. Model Geometry The FE model is comprised of the pipe and the denting tool. The pipe is defined by the diameter of the middle surface and the wall thickness ( t). To produce longitudinal and transverse dents, a pipe longitudinal length corresponding to 12D was adopted. According to preliminary FE analyses for these new dent geometries, it was found necessary to increase this initial length to sufficiently minimize any interaction between the stress field in the dented region and the pipe edges. To minimize computational time in the numerical analyses, a quarter-symmetry model was used, assuming planes of symmetry in longitudinal and transverse directions, planes 1-2 and 2-3, respectively (Figs. 1-3). Finite Element Mesh Since shell elements may be adopted whenever the diameter to thickness ratio is 20 or higher, solid continuum elements were not required. The model FE mesh was generated 2 Copyright 2008 by ASME

using ABAQUS S8R5 second-order quadrilateral thin shell elements [13], with five degrees of freedom per node (three translations and two in-surface rotations). The formulation of this element type assumes large rotations and infinitesimal plasticity. The change in thickness with deformation is ignored in this element, which converges to thin shell theory as the thickness decreases (the Love-Kirchhoff hypothesis is satisfied numerically). Figures 1-3 show the FE mesh and the analytical surface simulating the denting tool for each dent geometry, i.e., spherical, longitudinal and transverse dents. The pipe mesh is more refined at the damaged region, near the middle section, and presents a smooth transition to a coarser mesh close to the edge. A mesh sensitivity study was accomplished to define a proper FE refinement and to avoid time consuming analyses. Based on this study, the refinement in the middle region was done with one element at each 5 degrees in hoop direction, keeping an element aspect ratio of approximately 1 to 1. Boundary Conditions and Loading The model simulates the following load steps: plastic denting, removal of the denting tool (spring back), and two cycles of internal pressure. Initially, the denting tool punches the pipe external surface generating stress levels greater than the yield stress of the material. When the denting tool is removed, the pipe wall returns elastically, but the geometry remains partially deformed according to the denting depth and the level of plasticity attained. Finally, two equal cycles of internal pressure are applied with further plastic straining of the damaged region in the first cycle. Two different boundary conditions can be imposed at the pipe edge: axially constrained edge and closed edge. The last condition simulates a pressure vessel with the application of an axial load equivalent to an end cap. This load is directly applied at a reference node, placed in the center of the pipe section, and transmitted to the pipe edge through a kinematic coupling constraint. Model Validation The FE model was validated from experimental results of strain history during denting simulation followed by application of cyclic internal pressure on small-scale steel pipe models [11]. In these experiments, only spherical dents were induced on the pipe models. To reproduce the experimental tests, the FE model was set with the average dimensions, material properties and boundary conditions of the small-scale specimens [11]. In spite of some observed discrepancies, graphs of numerical and experimental strain results along the denting process showed good agreement. Also, the experimental and numerical strain behaviors of the dented models under internal pressure were compared [11]. It was showed that the numerical model can reproduce the experiments within engineering accuracy and precisely determine stress concentration factors [11]. PARAMETRIC STUDY The numerical model was used to carry out several FE analyses in a parametric study to determine stress concentration factors for different pipe and dent dimensions. A parametric study was previously carried out to generate SCFs for spherical dents [11,12]. Here, the parametric study comprised longitudinal and transversal dents. From the obtained results, analytical expressions were developed to estimate stress concentration factors for the three plain dent geometries, spherical, longitudinal and transverse dents. FE Analyses Based on experimental and numerical evidences, dented steel pipes under cyclic internal pressure might be considered to deform elastically after the first cycle (rerounding), if the maximum internal pressure is not increased [11]. Therefore, the residual stresses could be neglected and stress concentration factors may be obtained from simple linear elastic analyses. In this case, the SCFs are only related to the dent geometry and the condition at which the dent was induced is not taken into account. Consequently, each numerical analysis of the parametric study comprised a nonlinear elastic-plastic analysis, to simulate the denting process and to generate the deformed pipe geometry, and a subsequent elastic analysis, to determine the stress concentration factor. This methodology was adopted to filter strain hardening effects on SCFs. Although residual stresses can change SCFs, preliminary analyses have shown that its influence is small. Other parameter that could be mentioned is the stress stiffening generated by the internal pressure. Different conditions of average internal pressure lead to varying pipe stiffness, affecting the resulting SCFs. The effect of an average internal pressure on the SCFs may be subject of future analyses. The elastic-plastic analysis comprised the load steps of plastic denting and removal of the denting tool (spring back). The deformed pipe geometry obtained from this analysis was then assumed as the initial configuration for the elastic analysis. It comprised a single load step analysis, corresponding to a small internal pressure, sufficient to generate the elastic response and calculate the stress concentration factor. A Young modulus of 205,000 MPa and a Poison ratio of 0.3 were adopted as elastic properties. The FE analyses comprising spherical dents covered D/t ratios of 20, 30, 40, 50 and 60, spherical denting tool diameters of 0.25, 0.50, 0.75 and 1D and dent depths ( d/d) of approximately 2, 4, 6, 8, and 12%. In this case, both boundary conditions were investigated, axially constrained edge and closed edge. Figure 4 shows typical FE results of von Mises stresses on the pipe external surface after the introduction of a spherical dent. For the FE analyses concerning longitudinal and transverse dents, the same D/t ratio and dent depth ( d/d) variations were assumed, but only cylindrical denting tool diameters of 0.25 and 1D were considered. In the case of longitudinal dents, the length of the cylindrical analytical surface assumed values corresponding to 1, 2 and 3D. Additionally, for the denting tool 3 Copyright 2008 by ASME

diameter of 1D, a length of 0.5D was adopted to produce short longitudinal dents. In these analyses, only the closed edge boundary condition was investigated. Typical FE results of von Mises stresses on the pipe external surface after the introduction of longitudinal and transverse dents are shown in Figs. 5 and 6, respectively. Theoretical stress concentration factors (K t ) where obtained from the elastic analyses, referred to the von Mises equivalent stress and calculated as and from 3.4 to 11.6 for D/t equal to 60. Unsurprisingly, slender pipes under cyclic internal pressure tend to present higher SCFs. Although the closed edge condition has presented higher SCFs, the boundary condition does not pose significant effect on SCFs. The differences vary from 4% to almost zero, being higher for thicker pipes (D/t = 20). Kt σmax σnom (1) where max is the peak (maximum) stress and nom is the nominal stress [14] for an undamaged thin walled circular cylindrical shell. All the stress values are acquired at the centroid of the elements. The use of the von Mises equivalent stress allows direct reading from S-N curves. Through the concept of notch sensitivity factor ( q), the theoretical stress concentration factor can be used to calculate the fatigue stress concentration factor (K f ) [15]: K 1 1 K q (2) f t which relates the fatigue limit of an unnotched part to the one with a stress raiser [15]. FIGURE 5: TYPICAL FE RESULTS OF VON MISES STRESSES ON THE PIPE EXTERNAL SURFACE AFTER THE INTRODUCTION OF A LONGITUDINAL DENT. FIGURE 6: TYPICAL FE RESULTS OF VON MISES STRESSES ON THE PIPE EXTERNAL SURFACE AFTER THE INTRODUCTION OF A TRANSVERSE DENT. FIGURE 4: TYPICAL FE RESULTS OF VON MISES STRESSES ON THE PIPE EXTERNAL SURFACE AFTER THE INTRODUCTION OF A SPHERICAL DENT. The theoretical stress concentration factor depends exclusively on the part geometry, showing no dependency to the material properties [14]. Therefore, a careful characterization of the dent and pipe geometry is of great importance. In addition to diameter to thickness ratio ( D/t) and dent depth ( d), others geometric parameters analyzed are the dent length (l) and width (w), both evaluated at d/2 (Fig. 7). In the case of spherical dents, the stress concentrations factors vary approximately from 2.0 to 5.3 for D/t equal to 20 Analytical Expressions Although sophisticated FE models can precisely evaluate SCFs, a direct approach through a simple equation is obviously more practical, provided the precision of the result is assured. For this reason, an analytical methodology was proposed hereafter. The peak stress on spherical dents can be assumed to depend on the primary involved variables [11], that is σmax f σnom,d,l,w,d,t (3) 4 Copyright 2008 by ASME

It follows from the Buckingham s theorem [16] that this can be reduced to a relationship between non-dimensional variables, such as: σmax D d l t F,,, (4) σnom t D w w or, from Eq. (2), D d l t K t F,,, (5) t D w w where the four non-dimensional terms, assumed from length dimension variables, should characterize the stress concentration factors induced by spherical dents. As the available results of SCFs differ slightly between the two boundary conditions studied, in the case of spherical dents, only the data for closed edges were used to adjust the mentioned parameters, since this condition produced higher SCFs. Finally, the resulting expression was obtained for plain spherical dents: 0.72 0.59 1.41 0.13 D d l t K t 11.3 (9) t D w w In Fig. 8, this expression is compared against the available FE results. The line fitting of this expression presents a mean deviation of 2.66%. It can be verified that the proposed expression for spherical dents gives accurate approximations for K t < 5, although higher K t values can be obtained within a reasonable engineering precision. The same procedure was carried out for longitudinal and transverse dents. Additionally, an expression for the case of short longitudinal dents was also proposed. The obtained expression for longitudinal dents is: 1.50 1. 0.30 0.70 D d l t K t 1 3.1218 () t D w w In the case of short longitudinal dents, the proposed expression is: Kt FIGURE 7: DENT DIMENSIONS AT HALF DEPTH. The Eq. (5) can be expressed as the following series [16]: α1 D An t n0 α2 d D α3 l w n α4 t w (6) 4.66 1. 2.28 3.53 D d w t K t 1 0.2047 (11) t D l w Finally, for transverse dents the following expression was obtained: 0.44 0.57 0.54 0.39 D d l w K t 11.5788 (12) t D w t Since K t 1, it can be assumed that for n = 0, A 0 = 1. Then, a first order expression was proposed to fit the numerical data, neglecting terms with powers n > 1: Kt 1 A1 B (7) where A 1 is the angular coefficient of the linear equation and the non-dimensional geometric parameter B in given by In Figs. 9-11, the expressions given by equations ( )-(12) are compared against the associated available FE results. The line fitting of the expression concerning longitudinal dents presented a mean deviation of 3.02%. A mean deviation of 5.30% was obtained from the line fitting of the expression for short longitudinal dents. Finally, the mean deviation corresponding to the line fitting for transverse dents was equal to 2.78%. α1 α2 α3 α D d l t 4 B (8) t D w w The parameters A 1 1, 2, 3 and 4 were determined in order to achieve an accurate correlation between Eq. (7 ) and the available FE results. 5 Copyright 2008 by ASME

13 14 11 12 K t 9 7 5 K t 8 6 4 3 2 1 0 1 2 3 4 5 6 7 8 9 (D/t) 0.72 (d/d) 0.59 (l/w) 1.41 (t/w) 0.13 0 0 20 30 40 50 60 (D/t) 4.66 (d/d) 1. (w/l) 2.28 (t/w) 3.53 FIGURE 8. STRAIGHT LINE FITTING OF THE FE RESULTS FOR SPHERICAL DENTS (CLOSED EDGE CONDITION). 35 FIGURE. STRAIGHT LINE FITTING OF THE FE RESULTS FOR SHORT LONGITUDINAL DENTS. 14 30 12 25 20 8 K t K t 15 6 4 5 2 0 0 1 2 3 4 5 6 7 8 9 (D/t) 1.50 (d/d) 1. (l/w) 0.30 (t/w) 0.70 FIGURE 9. STRAIGHT LINE FITTING OF THE FE RESULTS FOR LONGITUDINAL DENTS. 0 0 1 2 3 4 5 6 7 8 (D/t) 0.44 (d/d) 0.57 (l/w) 0.54 (w/t) 0.39 FIGURE 11. STRAIGHT LINE FITTING OF THE FE RESULTS FOR TRANSVERSE DENTS. 6 Copyright 2008 by ASME

d/d (%) d/d (%) m a (MPa) Failure N (cycles) Nominal Mean Initial Final (MPa) Test Gerber Goodman Soderberg (Y/N) 5 4.60 6.19 3.01 98.85 97.81 4.67 131.46 156.25 7,357 Y 3.71 5.49 1.92 0.19 98.92 6.06 133.57 159.31 8,885 Y 6.12 6.29 5.95 97.64 51.92 55.47 69.49 82.34 88,761 Y 6.01 6.04 5.98 99.04 52.34 56.02 70.39 83,71 164,471 Y 6.03 6.12 5.93 98.25 31.91 34.12 42.80 50.79 560,229 Y 6.24 6.27 6.21 97.94 33.21 35.49 44.49 52.76 1,167,172 Y 11.81 14.39 9.22 86.51 85.27 89.69 9.61 128.90 6,640 Y 12.73 15.16.3 85.49 83.46 87.68 6.93 125.40 7,236 Y 13.84 14.32 13.35 0.40 54.48 58.36 73.40 89.72 57,572 Y.29.46.11 86.50 42.94 45.17 55.20 64.91 191,525 Y.80.66.94 83.99 40.81 42.80 52.03 60.78 208,452 Y.84.87.81 85.01 35.61 37.39 45.55 53.36 287,671 Y.82.84.79 82.79 23.83 24.96 30.26 35.25 1,153,504 N.45.46.44 83.94 28.32 29.70 36. 42.17 1,787,213 N.75.83.67 83.43 23.82 24.97 30.31 35.36 1,931,722 N TABLE 2: FATIGUE TEST RESULTS. FATIGUE TESTS The validity of the stress concentration factors obtained from the finite element model could be verified from smallscale fatigue tests results of steel pipe models with spherical dents under cyclic internal pressure [12]. Fifteen fatigue tests were conducted on low carbon steel (SAE/AI SI 20) pipe models with nominal external diameter ( D) of 73 mm, wall thickness (t) of 3.05 mm and overall length of 730 mm (D). Previously to the fatigue test, each model was submitted to a denting test using a 63.2 mm diameter cylindrical rod with a spherical tip. The dented models were fitted with specially built closing devices and then subjected to cyclic internal pressure. A servohydraulic machine actuated a hydraulic cylinder that imposed to the test model a harmonically oscillating pressure, with frequency ranging from 0.1 to 1 Hz. This experimental set up is presented in Fig. 11. The fatigue tests comprised two individual series, with dent depths around 5% and % of the pipe external diameter, respectively. The stress concentration factors related to these dent depths, obtained from the FE model, are equal to 3.59 and 5.24 for 5% and % dent depths, respectively. The material properties were obtained through uniaxial tensile tests of test coupons from both series. The properties of main interest are reported in Table 1. The stress-strain curve used as an input data to the FE model is shown in Fig. 13 in terms of true stress and logarithmic plastic strain. FIGURE 11: EXPERIMENTAL SET UP FOR THE FATIGUE TESTS. Nominal d/d (%) S y (MPa) S u (MPa) f 5 264.3 386.2 498.6 255.6 389.6 505.5 TABLE 1: MATERIAL PROPERTIES. (MPa) To assure that all models in a series had very similar initial dent depths, previously to the fatigue tests they were submitted to identical denting and then pressurized once to a level identical to the maximum pressure envisioned to the whole 7 Copyright 2008 by ASME

series. In each series, all models were submitted to approximately the same mean pressure and only the alternating pressure was varied. In both series, the pressure levels were determined trying to cover failures from 5000 to 6 cycles. 500 These curves were defined, for each testing series, considering a unitary notch sensitivity factor in Eq (2), i.e., K f = K t. The k a factor was assumed equal to 0.8 [18]. 800 600 400 450 200 (MPa) 400 350 300 250 0 0.05 0.1 0.15 0.2 0.25 p ln FIGURE 13: TRUE STRESS VERSUS LOGARITHMIC PLASTIC STRAIN CURVE USED IN THE FE MODEL. Table 2 presents the experimental results for both series. Failures were not attained for three tests conducted to simulate failures at 6 cycles for models with dent depths around %; these tests were interrupted. It was observed a reduction of the dent depth during the tests, as can be seen from the measured initial and final dent depths. This ratcheting phenomenon was more pronounced for higher levels of alternating stress and might not occur with other materials. The experimental results were compared with theoretical S-N curves inferred from the material tensile properties as early proposed by Cunha et al. [12]. According to the SAE Fatigue Design Handbook [17], the high cycle portion of an S-N curve would cross the one cycle axis at the true stress of failure in a tensile test ( f ). The endurance limit ( S e ), corresponding to 6 cycles, is inferred from the ultimate tensile strength [15]: Se min 0.5 Su ; 700MPa (13) However, considering the fatigue stress concentration factor (K f ) and the surface condition factor (k a ), the endurance limit of the pipe (S e ) is assumed as k S a e Se (14) K f a (MPa) a (MPa) 0 80 60 40 20 Spherical Dents - d/d=5% SN Curve Test Results - Soderberg Test Results - Goodman Test Results - Gerber 1 1E+002 1E+004 1E+006 N FIGURE 14: FATIGUE TEST RESULTS AGAINST THEORETICAL S-N CURVES FOR 5% DENT DEPTHS. 800 600 400 200 0 80 60 40 20 Spherical Dents - d/d=% SN Curve Test Results - Soderberg Test Results - Goodman Test Results - Gerber Run Out Results 1 1E+002 1E+004 1E+006 N FIGURE 15: FATIGUE TEST RESULTS AGAINST THEORETICAL S-N CURVES FOR % DENT DEPTHS. The fatigue results were compared with the zero mean stress S-N curve with the aid of the Goodman, Gerber and Soderberg criteria [15]. Each individual point was correct by its corresponding mean stress (Table 2). 8 Copyright 2008 by ASME

Figures 14 and 15 show how the experimental results compare to the theoretical S-N curves proposed by Cunha et al. [12]. Both Gerber and Goodman criteria seem to bring the experimental results to the S-N fatigue curve used. It must be pointed out that the true SCFs of the individual fatigue points may be quite different from the nominal values used to generate this curve. It becomes clear that the elastic stress concentration factor can be used to predict the fatigue life of a dented pipeline and that, for this case, the notch sensitivity factor can be assumed to be equal to 1.0. Moreover, this assumption is conservative, since K t K f, as can be verified from Eq. (2). The failure of the vast majority of the models occurred by cracking at the dent shoulders. The cracks initiated on the pipe external surface and propagated predominantly in the longitudinal direction, as can be seen in Fig. 16, where the fatigue cracks developed in one model is highlighted by inspection with liquid dye penetrating. These results are consistent with the numerical results, which indicated, for spherical dents, the external surface at the dent shoulder as the region of maximum stress concentration. Additionally, according to FE results, tensile residual hoop stresses were observed in this region, as consequence of the denting process [11]. FIGURE 16: FATIGUE CRACKS DEVELOPED IN MODEL 7D. CONCLUSIONS As part of an extensive research project, this work evaluates the stress concentration induced by longitudinal and transverse plain dents on steel pipelines under cyclic internal pressure. This project aims to propose a new methodology to assess the fatigue life of dented steel pipelines based on the current high cycle fatigue theory. This methodology employs stress concentration factors induced by plain dents and material S-N curves of metallic structures. A previously developed finite element model is modified to generate stress concentration factors for cylindrical dents. The SCFs obtained from the numerical model are validated according to small-scale fatigue test results of steel pipe models with spherical dents under cyclic internal pressure. The fatigue test results provided ground to validate the proposed methodology for fatigue analysis of damaged steel pipelines under cyclic internal pressure, employing SCFs obtained from analytical expressions, and to demonstrate its application in practical problems. For each dent geometry several finite element analyses are carried out in a parametric study to evaluate SCFs for different pipe and dent dimensions. Analytical expressions are developed to estimate SCFs for the three different dent geometries studied as function of pipe and dent geometric parameters. Additionally, an expression for short longitudinal dents is also proposed. Finally, the proposed methodology includes simple expressions to estimate stress concentration factors induced by three different plain dent geometries (spherical, longitudinal and transverse dents) and can be used to assess the fatigue life of dented steel pipelines based on the current high cycle fatigue theory. ACKNOWLEDGMENTS The authors would like to thank Dr. Sérgio Barros da Cunha from PETROBRAS/TRANSPETRO, the Brazilian Government (FINEP/CTPETRO -21-04), the Brazilian National Petroleum Agency (ANP) and the Submarine Technology Laboratory of COPPE/UFRJ for sponsoring this research work. NOMENCLATURE B non-dimensional dent geometric parameter D pipe external diameter K f fatigue stress concentration factor K t theoretical (or geometric) stress concentration factor N number of stress cycles to failure S e endurance limit of a standard test sample Se endurance limit of a real structural component S y yield stress S u ultimate tensile strength d dent depth k a surface condition modification factor l dent length w width q notch sensitivity factor t pipe wall thickness von Mises equivalent stress alternating stress a f m max nom true stress of failure in a tensile test mean (or static) stress peak (maximum) stress nominal stress 9 Copyright 2008 by ASME

REFERENCES [1] Lyon, D., 2002, Western European Cross-country Oil Pipelines 30-year Performance Statistics, Report No. 1/02, CONCAWE, Brussels. [2] Seng, O. L., Wing, C. Y., Seet, G., 1989, The Elastic Analysis of a Dent on Pressurized Pipe, International Journal of Pressure Vessels and Piping, 38, pp. 369 383. [3] Beller, M., Mattheck, C., Zimmermann, J., 1991, Stress concentrations in pipelines due to presence of dents, Proceedings of the 1 st International Offshore and Polar Engineering Conference, Edinburgh, UK, II, pp. 421-424. [4] Rinehart, A. J., Keating, P. B., 2002, Length Effects on Fatigue Behavior of Longitudinal Pipeline Dents, IPC2002-27244, Proceedings of the 4 th International Pipeline Conference, Calgary, Canada. [5] Alexander, C. R., and Kiefner, J. F., 1997, Effect of Smooth and Rock Dents on Liquid Petroleum Pipelines, Final Report, API Publication 1156, American Petroleum Institute, Washington. [6] Alexander, C. R., and Kiefner, J. F., 1999, Effect of Smooth and Rock Dents on Liquid Petroleum Pipelines (Phase II), API Publication 1156-Addendum, American Petroleum Institute, Washington. [7] Fowler, J. R., Alexander, C. R., Kovach, P. J., et al., 1994, Cyclic Pressure Fatigue Life of Pipelines with Plain Dents, Dents with Gouges, and Dents with Welds, Reports No. PR-201-927 and PR-201-9324, Pipeline Research Committee, American Gas Association, Houston. [8] Fowler, J. R., 1993, Criteria for Dent Acceptability in Offshore Pipeline, OTC 7311, Proceedings of the 25 th Offshore Technology Conference, Houston, pp. 481-493. [9] Rosenfeld, M. J., 1998, Investigations of Dent Rerounding Behavior, Proceedings of the 2 nd International Pipeline Conference, Calgary, Canada, 1, pp. 299-307. [] Cosham, A., and Hopkins, P., 2004, The Effect of Dents in Pipelines Guidance in the Pipeline Defect Assessment Manual, International Journal of Pressure Vessels and Piping, 81(2), pp. 127-139. [11] Pinheiro, B. C., Pasqualino, I. P., and Cunha, S. B., 2006, Stress Concentration Factors of Dented Pipelines, IPC2006-598, Proceedings of the 6 th International Pipeline Conference, Calgary, Alberta, Canada. [12] Cunha, S. B., Pinheiro, B. C., and Pasqualino, I. P., 2007, High Cycle Fatigue of Pipelines with Plain Dents Simulations, Experiments and Assessment, OMAE2007-29130, Proceedings of the 26 th International Conference on Offshore Mechanics and Arctic Engineering, San Diego, California, USA. [13] ABAQUS, 2006, User s and Theory Manuals, Hibbitt, Karlsson, Sorensen, Inc. [14] Pilkey, W. D., 1997, Peterson s Stress Concentration Factors, John Wiley & Sons, New York. [15] Shigley, J. E., and Mischke, C. R., 2001, Mechanical Engineering Design, McGraw-Hill, 6 th Ed., New York. [16] Buckingham, E., 1914, On Physically Similar Systems, Illustration and the Use of Dimensional Equations, Physical Review, 4(4), pp. 345-376. [17] Graham, J. A., 1968, SAE Fatigue Design Handbook, Society of Automotive Engineers, New York. [18] Shigley, J. E., and Mischke, C. R., 1989, Mechanical Engineering Design, McGraw-Hill, 5 th Ed., New York. Copyright 2008 by ASME