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Flexibility analysis of the vessel-piping interface
International Journal of Pressure Vessels and Piping 81 (2004) 181–189 www.elsevier.com/locate/ijpvp
Flexibility analysis of the vessel-piping interface Martin M. Schwarz TUV Austria, Department for Pressure Equipment, Krugerstr. 16, A-1015 Vienna, Austria
Abstract The connections of a pressure vessel with the interconnecting piping system are the boundaries for the computer model built for the analysis of the vessel. The stiffness and the imposed displacements at these points significantly influence the stresses in the piping as well as forces and moments imposed on the connection to the adjacent pressure equipment. This paper summarizes the results of numerous FEM analyses to establish stiffness coefficients for nozzles in spherical and cylindrical pressure vessels. q 2003 Elsevier Ltd. All rights reserved. Keywords: Piping; Stiffness coefficient; Shells
1. Introduction When building a model for computerized analysis of piping, the connecting points to vessels are usually modeled as fixed points. Most computer programs allow defining imposed displacements and stiffness coefficient at the nozzle-shell intersection. Some programs also provide tools for calculating the stiffness coefficients. The procedures used are usually derived from two publications: (a) BS PD 5500 Appendix G [1] and (b) Welding Research Council Bulletin 297 [2] BS PD 5500 gives procedures for both spherical and cylindrical vessels, whereas the WRC 297 bulletin deals only with nozzles in cylindrical shells. The stiffness of the nozzle heavily influences the stresses in the piping system and also the forces and moments acting on the nozzle itself. Defining the nozzle as a fixed end point for the piping can be unnecessarily conservative or can also render non-conservative results for piping stresses. Both publications do not cover the complete range of vessel –nozzle geometries that are of interest for piping analyses. A comparison of stiffness coefficients calculated following the different procedures given in the two above
mentioned publications shows significant deviations. These two reasons were the main motive for studying the stiffness of nozzles in cylindrical and spherical shells by means of FEM analyses.
2. General First step of an analysis is the set up of an appropriate parametric model. First, the relevant parameters describing the model have to be identified. These are: R r T t L E F ML MC w g
At the present state of this study, the following restrictions apply: –
E-mail address: [email protected] (M.M. Schwarz). 0308-0161/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2003.11.008
mean radius of the shell or sphere mean radius of the nozzle wall thickness of the shell or sphere wall thickness of the nozzle length of the shell modulus of elasticity force acting on the nozzle longitudinal moment acting on the nozzle circumferential moment acting on the nozzle nozzle deflection caused by radial force nozzle rotation caused by external moment
Nozzle is perpendicular to the surface of the shell/ sphere
182
– –
M.M. Schwarz / International Journal of Pressure Vessels and Piping 81 (2004) 181–189
Nozzle is in equal distance to the ends of the cylindrical shell No reinforcement pads
Three different types of models were used to calculate the stiffness coefficients. Model 1: Quarter of a sphere with a nozzle for calculating the stiffness for a nozzle force acting in the direction of the nozzle centerline. Model 2: Quarter of a cylinder (longitudinal section) with a nozzle to calculate the stiffness against a nozzle force acting in the direction of the nozzle centerline and also for calculating the stiffness for a longitudinal moment. Model 3: Quarter of a cylinder (normal section) with a nozzle to calculate the stiffness for a circumferential moment. For all three models 8 node structural linear elastic shell elements were used. Since in most cases overestimating the stiffness of a nozzle leads to conservative results, the boundary conditions were defined to represent the worst (i.e. stiffest) possible case.
Deflection parameter : def ¼ w
ET 2 FR
ð2Þ
The results in BS PD 5500 are based on the Bijlaard method [3], in which the nozzle is represented by a rigid cylindrical block fixed to the spherical shell. Modeling the nozzle as a cylindrical shell requires one additional parameter for the ratio of wall thicknesses. The ratio T=t is the parameter for the family of curves in the diagrams illustrating the results of the present investigation. The results are provided in Fig. 1. The agreement with the results given in BS PD 5500 is very good. For very small ratios of the wall thickness, i.e. thick walled nozzles in a thin walled sphere, the deflection parameter approaches the results of BS5500 for the completely rigid nozzle. For thin walled nozzles the decrease in stiffness compared to the rigid nozzle can become a significant benefit.
4. Spherical shell under bending moment The geometry parameter u is the same as above. The rotation parameter is
3. Spherical shell under radial force
gr ET 2 pffiffiffiffiffi M R=T
For better comparability of the results, the same dimensionless parameters as in BS PD 5500 have been used:
rot ¼
1:82r Geometry parameter : u ¼ pffiffiffiffi RT
Similar to the case with external force the results, reproduced in the diagrams, show very good agreement with those of BS PD 5500 (Fig. 2).
ð1Þ
Fig. 1. Nozzle in spherical shell—radial force.
ð3Þ
M.M. Schwarz / International Journal of Pressure Vessels and Piping 81 (2004) 181–189
183
Fig. 2. Nozzle in spherical shell—bending moment.
5. Cylindrical shell under radial load The geometry of the model is defined by the following dimensionless parameters:
The results are presented in two steps. In the first step a diagram or a set of diagrams is used from which the stiffness coefficient for a radial nozzle force KC 0F ¼
T=R L=R r=R T=t
thickness over radius of shell length over radius of shell nozzle radius over shell radius wall thickness of shell over w.t. of nozzle
F ETw
ð4Þ
depending of T=R; L=R and r=R for a configuration with equal wall thickness in shell and nozzle, i.e. T=t ¼ 1; can be derived.
Fig. 3. Nozzle in cylindrical shell—radial force, r=R ¼ 0:05 –0.2
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M.M. Schwarz / International Journal of Pressure Vessels and Piping 81 (2004) 181–189
Fig. 4. Nozzle in cylindrical shell—radial force, r=R ¼ 0:3–0.5.
In the second step a second diagram is used, showing the influence of T=t on the stiffness coefficient. The stiffness coefficient for a configuration with T=t different to 1 can then be calculated by KCF ¼ KC 0F KTF
ð5Þ
KC 0F can be found in Figs. 3 and 4 and KTF in Fig. 5. For different values of r=R interpolation between diagrams is necessary.
Afterwards the influence of a wall thickness ratio other then 1 on the stiffness coefficient is shown separately.
6. Cylindrical shell—external moment The procedure for external longitudinal or circumferential moment is the same as described above.
Fig. 5. Nozzle in cylindrical shell—radial force, influence of wall thickness ratio.
M.M. Schwarz / International Journal of Pressure Vessels and Piping 81 (2004) 181–189
185
Fig. 6. Nozzle in cylindrical shell—longitudinal moment.
KCMU ¼ KC 0MU KTMU
Longitudinal external moment ML for T=t ¼ 1 ET 3 g ¼ KC 0ML KTML
KC 0Ml ¼
ð6Þ
KCML
ð7Þ
MU ¼ for T=t ¼ 1 ET 3 g
KC 0MU can be found in Figs. 8 and 9 and KTML in Fig. 10.
7. Comparsion
KC 0Ml can be found in Fig. 6 and KTML in Fig. 7. Circumferential external moment KC 0MU
ð9Þ
ð8Þ
The comparison of the FEM results with calculations according to BS PD 5500 Appendix G can be split up into three parts.
Fig. 7. Nozzle in cylindrical shell—longitudinal moment, influence of wall thickness ratio.
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M.M. Schwarz / International Journal of Pressure Vessels and Piping 81 (2004) 181–189
Fig. 8. Nozzle in cylindrical shell—circumferential moment, r=R ¼ 0:05 –0.2.
For the spherical shell the results agree very good with the results according to BS PD 5500. The results given in this paper also cover hollow nozzles, which BS PD 5500 does not. The decrease in stiffness can be significant for thin walled nozzles. For forces on nozzles in cylindrical vessels the results according to the different approaches agree quite well, but only for a wall thickness ratio of T=t ¼ 0:5:
At this value the stiffness of a cylindrical vessel under a patch load, as calculated by BS PD 5500 Appendix G, is nearly the same as the stiffness of a nozzle. For bending moments acting on nozzles in cylindrical vessels the procedure in BS5500 gives values for the stiffness coefficients which are by far too small. According to the BS PD 5500 procedure, the stiffness coefficients are calculated by substituting the real nozzle by
Fig. 9. Nozzle in cylindrical shell—circumferential moment, r=R ¼ 0:3–0.5
M.M. Schwarz / International Journal of Pressure Vessels and Piping 81 (2004) 181–189
187
Fig. 10. Nozzle in cylindrical shell—circumferential moment, influence of wall thickness ratio.
two patch loads. The deflection at these patches and their distance gives a rotation. The deflection at each patch is treated as completely independent of the deflection at the other patch. This may be the main reason for the great deviation in the results of this work and WRC 297 on one hand and BS PD 5500 on the other hand.
In WRC Bulletin 297, the stiffness coefficients for moment loading are shown in Fig. 60 of the bulletin. In this paper reproductions of this figure, with some of the FEM results of this work included, are shown (Figs. 11 and 12). The agreement of the result is good within a certain range of the parameters. The deviations in the results for parameters outside of this range are caused by
Fig. 11. Comparison of the results with WRC 297 Bulletin Fig. 60 Longitudinal Moment; Filled symbols denote FEM results.
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M.M. Schwarz / International Journal of Pressure Vessels and Piping 81 (2004) 181–189
Fig. 12. Comparison of the results with WRC 297 Bulletin Fig. 60 Circumferential Moment; Filled symbols denote FEM results.
the differences in the boundary conditions applied in the models. These differences have minor influence in models with small r=R values and larger influence for models with r=R does to 0.5, or even larger. The deformation of the cylindrical shell is global to the cross section and not localized around the nozzle.
The last two figures show a comparison with the results of finite element calculations of pipe branches. Though pipe branches are similar to the vessel-piping intersection, the results presented by Xue, Widera and Wei in Ref. [4] should be comparable. Fig. 13 shows results for cylinder– cylinder intersections with an in plane moment. These results were
Fig. 13. Comparison with Xue, Widera an Weis results (dashed lines) for a in plane moment (i.e. longitudinal moment).
M.M. Schwarz / International Journal of Pressure Vessels and Piping 81 (2004) 181–189
189
Fig. 14. Comparison with Xue, Widera and Weis results (dashed lines) for a out-of-plane moment (i.e. circumferential moment).
obtained with t=T ¼ 1 and L=R ¼ 1: The agreement is very good, especially for large nozzles. Fig. 14 shows results for cylinder–cylinder intersections with an out-of-plane moment. The agreement is not as good as for the in plane moment, which is most likely caused by the different boundary conditions which have great influence on the stiffness of the cylinder – cylinder intersection.
8. Conclusion More than 1000 finite element models have been analyzed to establish the values for the diagrams shown. Although a large piece of the parameter space is covered, there are still a lot of vessel – nozzle configurations not covered by this work. Due to the restrictions of the parametric model which gives reasonable results only within a certain range of parameters, the following limits were applied:
0:005 $ r=R $ 0:5; 1 $ L=R $ 5; 2 $ T=t $ 0:5
ð10Þ
Future work to extend these limits will follow. The diagrams can be useful for piping analysts who have to model the vessel-piping interface.
References [1] Specification for unfired fusion welded pressure vessels. BS PD 5500, British Standards Institution, London. [2] Local Stresses in Cylindrical Shells Due to External Loadings on Nozzles—Supplement to WRC Bulletin 107—(Revision 1), by Mershon JL, Mokhtarian K, Ranjan GV, Rodabaugh EC. August 1984, revised September; 1987. p. 88. [3] On the stresses from local loads on spherical pressure vessels and pressure vessel heads by Bijlaard PP; WRC Bulletin No. 34; New York, USA: The Welding Research Council; 1957. [4] Flexibility factors and stress intensification indices for pipe branch connections subjected to in plane and out-of plane moments by Liping X, Widera GEO, Zhizhong W, Proceedings ICPVT-10; 2003. Vienna, Austria.
Flexibility analysis of the vessel-piping interface Martin M. Schwarz TUV Austria, Department for Pressure Equipment, Krugerstr. 16, A-1015 Vienna, Austria
Abstract The connections of a pressure vessel with the interconnecting piping system are the boundaries for the computer model built for the analysis of the vessel. The stiffness and the imposed displacements at these points significantly influence the stresses in the piping as well as forces and moments imposed on the connection to the adjacent pressure equipment. This paper summarizes the results of numerous FEM analyses to establish stiffness coefficients for nozzles in spherical and cylindrical pressure vessels. q 2003 Elsevier Ltd. All rights reserved. Keywords: Piping; Stiffness coefficient; Shells
1. Introduction When building a model for computerized analysis of piping, the connecting points to vessels are usually modeled as fixed points. Most computer programs allow defining imposed displacements and stiffness coefficient at the nozzle-shell intersection. Some programs also provide tools for calculating the stiffness coefficients. The procedures used are usually derived from two publications: (a) BS PD 5500 Appendix G [1] and (b) Welding Research Council Bulletin 297 [2] BS PD 5500 gives procedures for both spherical and cylindrical vessels, whereas the WRC 297 bulletin deals only with nozzles in cylindrical shells. The stiffness of the nozzle heavily influences the stresses in the piping system and also the forces and moments acting on the nozzle itself. Defining the nozzle as a fixed end point for the piping can be unnecessarily conservative or can also render non-conservative results for piping stresses. Both publications do not cover the complete range of vessel –nozzle geometries that are of interest for piping analyses. A comparison of stiffness coefficients calculated following the different procedures given in the two above
mentioned publications shows significant deviations. These two reasons were the main motive for studying the stiffness of nozzles in cylindrical and spherical shells by means of FEM analyses.
2. General First step of an analysis is the set up of an appropriate parametric model. First, the relevant parameters describing the model have to be identified. These are: R r T t L E F ML MC w g
At the present state of this study, the following restrictions apply: –
E-mail address: [email protected] (M.M. Schwarz). 0308-0161/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2003.11.008
mean radius of the shell or sphere mean radius of the nozzle wall thickness of the shell or sphere wall thickness of the nozzle length of the shell modulus of elasticity force acting on the nozzle longitudinal moment acting on the nozzle circumferential moment acting on the nozzle nozzle deflection caused by radial force nozzle rotation caused by external moment
Nozzle is perpendicular to the surface of the shell/ sphere
182
– –
M.M. Schwarz / International Journal of Pressure Vessels and Piping 81 (2004) 181–189
Nozzle is in equal distance to the ends of the cylindrical shell No reinforcement pads
Three different types of models were used to calculate the stiffness coefficients. Model 1: Quarter of a sphere with a nozzle for calculating the stiffness for a nozzle force acting in the direction of the nozzle centerline. Model 2: Quarter of a cylinder (longitudinal section) with a nozzle to calculate the stiffness against a nozzle force acting in the direction of the nozzle centerline and also for calculating the stiffness for a longitudinal moment. Model 3: Quarter of a cylinder (normal section) with a nozzle to calculate the stiffness for a circumferential moment. For all three models 8 node structural linear elastic shell elements were used. Since in most cases overestimating the stiffness of a nozzle leads to conservative results, the boundary conditions were defined to represent the worst (i.e. stiffest) possible case.
Deflection parameter : def ¼ w
ET 2 FR
ð2Þ
The results in BS PD 5500 are based on the Bijlaard method [3], in which the nozzle is represented by a rigid cylindrical block fixed to the spherical shell. Modeling the nozzle as a cylindrical shell requires one additional parameter for the ratio of wall thicknesses. The ratio T=t is the parameter for the family of curves in the diagrams illustrating the results of the present investigation. The results are provided in Fig. 1. The agreement with the results given in BS PD 5500 is very good. For very small ratios of the wall thickness, i.e. thick walled nozzles in a thin walled sphere, the deflection parameter approaches the results of BS5500 for the completely rigid nozzle. For thin walled nozzles the decrease in stiffness compared to the rigid nozzle can become a significant benefit.
4. Spherical shell under bending moment The geometry parameter u is the same as above. The rotation parameter is
3. Spherical shell under radial force
gr ET 2 pffiffiffiffiffi M R=T
For better comparability of the results, the same dimensionless parameters as in BS PD 5500 have been used:
rot ¼
1:82r Geometry parameter : u ¼ pffiffiffiffi RT
Similar to the case with external force the results, reproduced in the diagrams, show very good agreement with those of BS PD 5500 (Fig. 2).
ð1Þ
Fig. 1. Nozzle in spherical shell—radial force.
ð3Þ
M.M. Schwarz / International Journal of Pressure Vessels and Piping 81 (2004) 181–189
183
Fig. 2. Nozzle in spherical shell—bending moment.
5. Cylindrical shell under radial load The geometry of the model is defined by the following dimensionless parameters:
The results are presented in two steps. In the first step a diagram or a set of diagrams is used from which the stiffness coefficient for a radial nozzle force KC 0F ¼
T=R L=R r=R T=t
thickness over radius of shell length over radius of shell nozzle radius over shell radius wall thickness of shell over w.t. of nozzle
F ETw
ð4Þ
depending of T=R; L=R and r=R for a configuration with equal wall thickness in shell and nozzle, i.e. T=t ¼ 1; can be derived.
Fig. 3. Nozzle in cylindrical shell—radial force, r=R ¼ 0:05 –0.2
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M.M. Schwarz / International Journal of Pressure Vessels and Piping 81 (2004) 181–189
Fig. 4. Nozzle in cylindrical shell—radial force, r=R ¼ 0:3–0.5.
In the second step a second diagram is used, showing the influence of T=t on the stiffness coefficient. The stiffness coefficient for a configuration with T=t different to 1 can then be calculated by KCF ¼ KC 0F KTF
ð5Þ
KC 0F can be found in Figs. 3 and 4 and KTF in Fig. 5. For different values of r=R interpolation between diagrams is necessary.
Afterwards the influence of a wall thickness ratio other then 1 on the stiffness coefficient is shown separately.
6. Cylindrical shell—external moment The procedure for external longitudinal or circumferential moment is the same as described above.
Fig. 5. Nozzle in cylindrical shell—radial force, influence of wall thickness ratio.
M.M. Schwarz / International Journal of Pressure Vessels and Piping 81 (2004) 181–189
185
Fig. 6. Nozzle in cylindrical shell—longitudinal moment.
KCMU ¼ KC 0MU KTMU
Longitudinal external moment ML for T=t ¼ 1 ET 3 g ¼ KC 0ML KTML
KC 0Ml ¼
ð6Þ
KCML
ð7Þ
MU ¼ for T=t ¼ 1 ET 3 g
KC 0MU can be found in Figs. 8 and 9 and KTML in Fig. 10.
7. Comparsion
KC 0Ml can be found in Fig. 6 and KTML in Fig. 7. Circumferential external moment KC 0MU
ð9Þ
ð8Þ
The comparison of the FEM results with calculations according to BS PD 5500 Appendix G can be split up into three parts.
Fig. 7. Nozzle in cylindrical shell—longitudinal moment, influence of wall thickness ratio.
186
M.M. Schwarz / International Journal of Pressure Vessels and Piping 81 (2004) 181–189
Fig. 8. Nozzle in cylindrical shell—circumferential moment, r=R ¼ 0:05 –0.2.
For the spherical shell the results agree very good with the results according to BS PD 5500. The results given in this paper also cover hollow nozzles, which BS PD 5500 does not. The decrease in stiffness can be significant for thin walled nozzles. For forces on nozzles in cylindrical vessels the results according to the different approaches agree quite well, but only for a wall thickness ratio of T=t ¼ 0:5:
At this value the stiffness of a cylindrical vessel under a patch load, as calculated by BS PD 5500 Appendix G, is nearly the same as the stiffness of a nozzle. For bending moments acting on nozzles in cylindrical vessels the procedure in BS5500 gives values for the stiffness coefficients which are by far too small. According to the BS PD 5500 procedure, the stiffness coefficients are calculated by substituting the real nozzle by
Fig. 9. Nozzle in cylindrical shell—circumferential moment, r=R ¼ 0:3–0.5
M.M. Schwarz / International Journal of Pressure Vessels and Piping 81 (2004) 181–189
187
Fig. 10. Nozzle in cylindrical shell—circumferential moment, influence of wall thickness ratio.
two patch loads. The deflection at these patches and their distance gives a rotation. The deflection at each patch is treated as completely independent of the deflection at the other patch. This may be the main reason for the great deviation in the results of this work and WRC 297 on one hand and BS PD 5500 on the other hand.
In WRC Bulletin 297, the stiffness coefficients for moment loading are shown in Fig. 60 of the bulletin. In this paper reproductions of this figure, with some of the FEM results of this work included, are shown (Figs. 11 and 12). The agreement of the result is good within a certain range of the parameters. The deviations in the results for parameters outside of this range are caused by
Fig. 11. Comparison of the results with WRC 297 Bulletin Fig. 60 Longitudinal Moment; Filled symbols denote FEM results.
188
M.M. Schwarz / International Journal of Pressure Vessels and Piping 81 (2004) 181–189
Fig. 12. Comparison of the results with WRC 297 Bulletin Fig. 60 Circumferential Moment; Filled symbols denote FEM results.
the differences in the boundary conditions applied in the models. These differences have minor influence in models with small r=R values and larger influence for models with r=R does to 0.5, or even larger. The deformation of the cylindrical shell is global to the cross section and not localized around the nozzle.
The last two figures show a comparison with the results of finite element calculations of pipe branches. Though pipe branches are similar to the vessel-piping intersection, the results presented by Xue, Widera and Wei in Ref. [4] should be comparable. Fig. 13 shows results for cylinder– cylinder intersections with an in plane moment. These results were
Fig. 13. Comparison with Xue, Widera an Weis results (dashed lines) for a in plane moment (i.e. longitudinal moment).
M.M. Schwarz / International Journal of Pressure Vessels and Piping 81 (2004) 181–189
189
Fig. 14. Comparison with Xue, Widera and Weis results (dashed lines) for a out-of-plane moment (i.e. circumferential moment).
obtained with t=T ¼ 1 and L=R ¼ 1: The agreement is very good, especially for large nozzles. Fig. 14 shows results for cylinder–cylinder intersections with an out-of-plane moment. The agreement is not as good as for the in plane moment, which is most likely caused by the different boundary conditions which have great influence on the stiffness of the cylinder – cylinder intersection.
8. Conclusion More than 1000 finite element models have been analyzed to establish the values for the diagrams shown. Although a large piece of the parameter space is covered, there are still a lot of vessel – nozzle configurations not covered by this work. Due to the restrictions of the parametric model which gives reasonable results only within a certain range of parameters, the following limits were applied:
0:005 $ r=R $ 0:5; 1 $ L=R $ 5; 2 $ T=t $ 0:5
ð10Þ
Future work to extend these limits will follow. The diagrams can be useful for piping analysts who have to model the vessel-piping interface.
References [1] Specification for unfired fusion welded pressure vessels. BS PD 5500, British Standards Institution, London. [2] Local Stresses in Cylindrical Shells Due to External Loadings on Nozzles—Supplement to WRC Bulletin 107—(Revision 1), by Mershon JL, Mokhtarian K, Ranjan GV, Rodabaugh EC. August 1984, revised September; 1987. p. 88. [3] On the stresses from local loads on spherical pressure vessels and pressure vessel heads by Bijlaard PP; WRC Bulletin No. 34; New York, USA: The Welding Research Council; 1957. [4] Flexibility factors and stress intensification indices for pipe branch connections subjected to in plane and out-of plane moments by Liping X, Widera GEO, Zhizhong W, Proceedings ICPVT-10; 2003. Vienna, Austria.