# C 2 Stress Indices for Back-to-Back Welded Pipe Bends

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1 C 2 Stress Indices for Back-to-Back Welded Pipe Bends M. Irfan Haq 1, Mani Aggarwal Engineering Mechanics Department, Ontario Power Generation Inc., 889 Brock Road, Pickering, ON, L1W 3J2, Canada. 1 ABSTRACT In ASME Boiler and Pressure Vessel Code the C 2 stress index for back-to-back bends welded together is taken as a product of the C 2 index of girth butt weld and C 2 index of the elbow. The Code considers that the secondary stresses from the weld and mismatch are negligible for a thick pipe but it recommends a relation to calculate C 2 for weld in a pipe of thickness less than inch. In our previous publication, a methodology using Finite Element Method (FEM) was presented to calculate the girth butt weld C 2 stress index. This paper presents C 2 stress indices for different configurations of back-to-back welded pipe bends using FEM. One bend is rotated relative to the other from 0 o to 180 o. Both amplitude and location of highest stress are investigated as a function of the angle between bends and bend angle. Different bend angles from 30 to 90 degrees have been analyzed. INTRODUCTION In a typical piping configuration there are several girth butt welds in straight pipe segments, between straight pipes and bends and between back-to-back pipe bends. This study focuses on back-to-back welded pipe bends. In the design rules of NB-3600 [1] Equation 10, Equation 11 and Equation 12 evaluation, nominal bending stress is multiplied by C 2 stress index. As per NB (a) the C 2 stress index for back-to-back welded bends is calculated as follows; for curved pipe or butt welding elbows welded together or joined by a piece of straight pipe less than one pipe diameter long, the stress indices shall be taken as the product of the indices for the elbow or curved pipe and the indices for the girth butt weld except for B 1 and C 3 which are exempted Thus C 2(butt welded bends) = C 2(bend) * C 2(weld) (1) The C 2 index for elbow provided in Table NB-3681 (a)-1 is derived from theoretical analyses using in-plane moment loading. The formula given in the ASME Code, NB for calculating C 2 stress index for bends is 1.95 C 2 = but not < 1.50 (2) 2/3 h where tr h = (3) 2 r m t = nominal wall thickness R = nominal bend radius and r m = mean pipe radius (D o -t)/2; D o = outside diameter The C 2 index for girth butt weld is given by C 2 = /t but not > 2.1 (4) For t inch. From the above equations it is evident that ASME Code assumes C 2 indices for back-to-back bends are not a function of bend angle or angle between bends. From our work, we found that the C 2 stress indices for back-to-back bends vary with the 1

2 bend angle as well as angle between bends. The purpose of this work is to present C 2 indices calculations for different backto-back welded bend configurations using finite element method. In our previous publication [2] C 2 indices for two configurations of back-to-back bends were presented. These were for 90 o bends with 0 o and 90 o angle between the bends. The methodology used for calculating the indices is presented in detail in Reference [2]. In this paper a wide range of bend angles is covered. These include 30 o, 45 o, 60 o, 75 o and 90 o bends. The angle between bends is varied from 0 o to 180 o with increments of 30 o. Figure 1 shows the 90 o bends with different angles between bends analyzed. In addition to the weld cap a central-line offset is added to simulate the maximum fabrication tolerance. Figure 2 shows a schematic of two back-to-back bends with central line mismatch. Figures 3 to 5 show some of the configurations of back-toback welded bends used in this paper. FINITE ELEMENT MODELS A parametric finite element model using the ANSYS [3] Advanced Parametric Design Language APDL was developed. This parametric model consists of two back-to-back pipe bends that can lie in different planes. These bends can have different bend angles ranging from 30 o to 90 o. A straight pipe approximately 5 pipe diameters long is added at the end of each bend. For all finite element models, the pipe thickness was set to 0.2 inch, outside diameter to 2 inch and the elbow radius 1.5 pipe diameter. A typical computational model is shown in Figure 6. These finite element models were developed using 20-node hexahedral brick type elements (SOLID186 in ANSYS) with quadratic displacement formulation. In each of these models, there are 3 layers of elements in the thickness direction and 36 elements along the pipe circumferential direction for each layer. Along the axial direction of bends there are 20 elements for 90 o bends, 17 for 75 o bends, 14 for 60 o bends, 11 for 45 o bends and 8 for 30 o bends. Along straight pipes there are 20 elements with a 2.5:1 ratio between the size of elements at both ends and the size of the elements close to the elbow where higher stress gradients are expected. At both ends of the computational models, the nodes are associated with a pilot node that is surface-to-surface type element in ANSYS program. This is an element with one node whose motion governs the motion of the entire group of nodes to which it is connected. The pilot node provides a convenient means of imposing boundary conditions such as rotations and translations as well as forces and moments. One end of the computational model was fully constrained so that all the translational and rotational degrees of freedom for the pilot node were fixed. At the other end, a constant moment was applied at the pilot node. The weld contour is derived from a real application of back-to-back elbows with a welded joint. The length in axial direction of weld cap is approximately 6.2 mm. The height of weld cap is conservatively chosen as 1 mm and the thickness profile along the axial direction approximated by a spline fit. The meshed structure of the butt welded joint is presented in Figure 7. For presenting the mesh density through the thickness in the butt weld region, one quarter of the computational model is not shown. COMPUTATIONAL ANALYSES STEPS The analysis steps are shown in Figure 8. More details are given below. Calculation of Weld Offset Angle Finite element analyses are employed to find the worst-possible central line mismatch direction. At the girth butt weld location, a 1/32" offset is used for these analyses. A computer macro is used to automate the procedure for creating (preprocessing) the models with the offset, analyzing and post-processing the results. The two regions located before and after the offset plane are connected by bonded contact surface-to-surface elements in ANSYS. The reason for connecting the two parts with contact elements is to create continuous parts meshed with brick type finite elements. One end of the model is fixed while bending moment is applied on the other end. The macro automatically executes the 1/32" offsets in radial directions for angles from 0 o to 360 o, with increments of 10 o for the offset angle. Figure 9 shows that the offset angle is determined counter-clockwise from extrados of bend 1. From experience 10 o is sufficient angular increment of the offset to capture the worst offset direction. For each position, the model is solved and post-processed for maximum linearized stresses at the offset location. The worst possible weld mismatch is obtained by plotting the highest linearized membrane plus bending stresses at the cross section close to the offset location. In order to prevent mathematical singularities in the finite element model created by the offset method, the maximum linearized stresses are summarized at locations 1 mm away from the offset location along the model centre axis. The stresses are linearized on each path through the pipe wall. These paths are created at angles from 0 o to 360 o around the circumference of the wall, with 10 o angles increment. Planes of the linearized membrane plus bending results are created at (n+1) positions along each elbow s centre axis, where n is the number of elements along the axial direction of the elbow. 2

3 Figures 10 to 14 present the variation of linearized stress with offset angle for sample analyses. For example, in Figure 10 the highest linearized stress ( MPa) occurs at offset angle of 10 o for 30 o back-to-back bends with 180 o angle between the bends. Similarly, in Figure 12 maximum linearized stress is MPa occurring at 30 o for 60 o bends with 60 o angle between the bends. Figure 15 presents the variation of worst angle of offset for each bend type with the angle between the bends. Calculation of C 2 Stress Index The worst offset angle corresponding to the highest linearized membrane plus bending stress (as calculated in the previous section) is implemented into the finite element models. The weld cap as presented in Figure 7 is added on the outside surface of the back-to-back elbows. The loading used is the same as for finding the worst offset direction. An ANSYS macro is used to implement the offset direction and weld cap, mesh, analyze and post-process the results. Maximum linearized membrane plus bending stress through the wall are determined for each bend and weld region. membrane plus bending stress is divided by the nominal stress (in a straight pipe) to determine the C 2 index. The nominal stress is calculated using σ nom = (MD o )/(2I) (5) where M is the applied moment D o is the outer diameter and I is the moment of inertia. The ASME Code (NB-3682 (a)) general definition of C 2 is C 2 = σ/σ nom (6) The same nominal stress is used for calculating the indices for bends and weld, although the weld does not have uniform thickness. The comparison between the C 2 stress indices from this paper and ASME Code methodology is presented in Table 1 for few sample calculations. In column A, the bend type is given and in column B the angle between the bends is shown. Column C presents the maximum P m +P b stresses in the bends while column D gives the maximum P m +P b in the weld region. Nominal stress is given in column E and the C 2 indices (from this paper) for the bend and weld are given in column F & G respectively. The C 2(butt welded bends) (which is 3.50 for the thickness and bend radius used in this study) calculated as per the ASME Code methodology is in column H. Column I provides the ratio between the C 2 stress indices as predicted by linear elastic finite element analyses presented in column F and C 2 stress indices calculated as per ASME Code. Finally, column J is the ratio between the C 2 stress indices as calculated by finite element analyses summarized in column G and C 2 stress indices as per ASME Code. Figure 16 shows the C 2 indices for bend calculated using finite element method in this paper plotted as a function of angle between bends for each bend angle analyzed. Similarly, Figure 17 is C 2 indices for weld calculated using finite element method in this paper plotted as a function of angle between bends for each bend angle analyzed. The curves in the Figures 16 and 17 are linear trend line fit for the data with the R 2 -squared value (coefficient of determination) above RESULTS AND CONCLUSIONS A finite element method for calculating the C 2 stress index for back-to-back welded bends including the centre line offset and girth butt weld cap is presented in this paper. A linear elastic finite element approach is used. A methodology to predict the worst offset direction for the central line mismatch by trying incrementally varying offset angle is also presented. The results are presented in the form of C 2(bend) for five back-to-back bend types; 30 o, 45 o, 60 o, 75 o and 90 o with angle between the bends from 0 o to 180 o. The ASME Code equations used to calculate the C 2 indices are not function of bend angle, but from our work it can be seen that the C 2(bend) is a function of bend angle as well as the angle between the bends. Similarly the C 2(weld) for the five back-to-back bend types is also a function of the bend angle and angle between the bends. From the analysis results it can be concluded that; methodology to calculate the worst offset central line mismatch direction presented in this paper depends on the geometrical configuration of the back-to-back bends. The worst offset direction is a function of the bend angle and angle between the bends. 3

4 calculated C 2 indices increase for back-to-back elbows out-of-plane positions as compared to in-plane positions. But in case of angle between the bends of 180 o (in-plane position), the C 2 indices are the highest. indices for butt welded elbows vary with bend angle and angle between bends. calculations presented in this study summarize the results at the junction between the back-to-back bends, maximum stresses are located close to the centre of the two bends due to ovalization of the cross-section work is continuing to derive a general expression for indices as a function of both the bend angle and angle between bends. ACKNOWLEDGEMENTS Authors wish to acknowledge the support and technical advice from members of the Piping Analysis Section, Engineering Mechanics Department at Ontario Power Generation, Inc. REFERENCES 1. The American Society of Mechanical Engineers, ASME Boiler and Pressure Vessel Code, 2004 Edition including 2005 Addenda, Section III, Division Vlaicu D., Aggarwal M., Li M., Determination of C 2 Stress Index of Back-to-Back Welded Piping Bends Using Finite Element Method, Proceedings of PVP2006-ICPVT-11, 2006 ASME Pressure Vessels and Piping Division Conference, July 23-27, 2006, Vancouver, BC, Canada. 3. ANSYS Version 10.0, ANSYS Inc., Canonsburg, PA., USA. No. Table 1: Sample Calculations A B C D E F G H I J Bend Angle Angle Between Bends Highest P m +P b in B1/B2 Highest P m +P b in Weld Nominal stress σ nom C 2(bend) C 2(weld) C 2(butt welded bends) from ASME Code Ratio 1 Bend Ratio 2 Weld (0 o ) (30 o ) (60 o ) (90 o ) (120 o ) (150 o ) (180 o ) Figure 1: 90 o Back-to-Back Bends With Different Angles Between the Bends 4

5 Bend 2 Bend 1 1/32" Offset Figure 2: Schematic of Back-to-Back Bends Figure 3: 30 o Back-to-Back Bends With 0 o Angle Between the Bends (In-Plane) Figure 4: 45 o Back-to-Back Bends With 0 o Angle Between the Bends Figure 5: 60 o Back-to-Back Bends With 60 o Angle Between the Bends Figure 6: Computational Model Showing Mesh Density for 90 o Bends With 90 o Angle Between Bends Figure 7: Computational Model With 1/4 th Model Removed to Show Mesh at the Weld Cap 5

6 Start Back-to-back welded bends computational model No Weld cap Analyze with 10 o offset around the circumference from 0 o to 360 o Incorporate 1/32" offset at the worst offset direction plus the weld cap between the back-to-back bends Determine worst offset direction Determine the maximum linearized stress intensity in each bend and weld and determine the C 2 stress index End Figure 8: Analysis Procedure to Determine C 2 Stress Indices for Back-to-Back Welded Bends 1/32" Offset Angle Bend 1 E1 Bend 2 Figure 9: Offset Angle is Defined by Moving Counter-Clockwise From Extrados of Bend 1 (E1) Figure 10: Stress vs Offset Angle for 30 o Bends With 180 o Angle Between the Bends Figure 11: Stress vs Offset Angle for 45 o Bends With 0 o Angle Between the Bends 6

7 Figure 12: Stress vs Offset Angle for 60 o Bends With 60 o Angle Between the Bends Figure 13: Stress vs Offset Angle for 75 o Bends With 150 o Angle Between the Bends Figure 14: Stress vs Offset Angle for 90 o Bends With 150 o Angle Between the Bends Bend Angle 30 deg 45 deg 60 deg 75 deg 90 deg Angle Between Bends (deg) Figure 15: Variation of Angle of Offset for Each Bend Type With the Angle Between the Bends 7

8 C2(bend) Bend Angle 30 deg 45 deg 60 deg 75 deg 90 deg Angle Between Bends (deg) Figure 16: C 2(bend) Plotted as a Function of Angle Between Bends for Each Bend Angle Analyzed. C 2(butt welded bends) as per ASME Code is C2(weld) Bend Angle 30 deg 45 deg 60 deg 75 deg 90 deg Angle Between Bends (deg) Figure 17: C 2(weld) Plotted as a Function of Angle Between Bends for Each Bend Angle Analyzed. C 2(butt welded bends) as per ASME Code is

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