**The inclusion of nozzle flexibility (directional spring rate) in pipe-stress analysis results in more realistic piping reactions on pressure-vessel nozzles, whereby it will be easier to meet the limited nozzle-load capacity of the nozzles. It will surely contribute to a cost-effective design that ensures structural integrity demands**

Piping systems are characteristic of every process plant. Piping systems connect various process equipment items, such as pressure vessels, pumps, compressors, turbines, heat exchangers and so on. It is common practice that a formal pipe-stress analysis is performed for critical piping systems. The pipe-stress analysis performed should comply with the required design code or standard. This analysis not only requires assessing the stresses in the piping system against the allowable stresses, but also assessing the piping reactions that the piping system exerts on the connecting equipment. The problem that arises here often focuses on the assessment of the piping reactions, which in many cases should adhere to company specifications. The reason why the piping reactions exceed the permissible values according to the company specification often lies in the fact that the pipe-stress engineer has considered the terminal point to the equipment infinitely rigid rather than as an element with finite flexibility in his or her pipe stress analysis.

The assumption of a rigid connection for most pressure vessels in the low- or moderate-pressure regime is very conservative and will not lead to realistic piping reactions. Hence, the magnitude of the nozzle loads from the piping-system stress analysis may be overly conservative, depending on the boundary conditions used in the analysis of the piping system.

While a pressure vessel can be considered relatively stiff, it is not infinitely rigid at the nozzles. Completely constraining the model of the piping system where it connects to the pressure vessel will result in external nozzle loads that may be an order-of-magnitude higher than reality. The solution can be found by inclusion of nozzle flexibility into the pipe-stress analysis. The pressure-vessel engineer can provide values for the stiffness of the pressure vessel at the nozzles locations for the piping stress analyst to use. This inclusion will produce more accurate loads at the nozzles. Most systems would have the piping forces and moments increased by two or three times when the connections are considered rigid, as compared with that calculated with flexible connections.

Among the six degrees of freedom at a vessel connection, the flexibilities in the directions of the two bending moments (*M*_{x} and *M*_{z}) and the direct axial force (*F*_{y}) are considered significant (Figure 1). Flexibilities of torsion and direct shear directions are generally ignored and considered rigid.

## Vessel-piping interface flexibility

As mentioned before, equipment nozzles are normally modeled as rigid piping junctions, which result in nozzle loads that do not adequately reflect the ability of the nozzle to rotate and displace. To better define the loading on the equipment nozzles it becomes apparent that the nozzle stiffness would need to be included in the piping flexibility analysis.

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 endpoint for the piping can be unnecessarily conservative or can also render non-conservative results for piping stresses.

The nozzle stiffness and stress results using different calculation methods, such as WRC-297 [*1*], PD 5500 Annex G [*2*], and finite element analysis (FEA), are quite diverse. The analysis results for FEA gives the most reliable and realistic results compared to the other methods. However, FEA can increase the time and cost parameters associated with the normal design process. However, due to mostly psychological concerns, the inclusion of vessel flexibility in the piping analysis is still not universal. Some vessel engineers worry that the inclusion of shell flexibility will ultimately result in a stiffer piping system that might cause damage to the vessel. This is partly true, but it mostly has an adverse effect on the quality of the plant.

Procedures for calculating stiffness coefficients, taken from Refs. 1 and 2, are described in Refs. 3 and 4.

## Nozzle flexibilities

Consider an example of a vessel with a pad-reinforced nozzle in the cylindrical shell. The data for such a vessel are presented in Table 1. The nozzle flexibilities for the pressure vessel according to Table 1 are shown in Table 2.

Table 2 shows that the nozzle flexibilities calculated with different software programs are almost the same. All of these software programs are based on numerical FEA. Furthermore, the nozzle flexibilities have also been calculated in accordance with WRC 297 and PD 5500 Annex G, as included in the pipe stress analysis software package CAESAR II. The results of this exercise are shown in Table 3. Note that, for both methods, *extrapolation* of the curves is required. However, extrapolation will lead to a stiffer intersection.

The nozzle flexibilities computed with NozzlePRO, FE/Pipe and PV Elite differ significantly from those computed with CAESAR II according to WRC 297 and PD 5500 Annex G.

We demonstrate the effect of nozzle flexibility using a piping configuration in which the piping connects to a nozzle in the middle of a curved head of a vertical pressure vessel on the one hand and to the cylindrical shell of a horizontal pressure vessel on the other. It is important to mention that the configuration is not aimed at a stress-wise optimal result, but is randomly chosen to show the effects of nozzle flexibility.

## An example

For the configuration shown in Figure 2, the piping reactions were determined with the CAESAR II Pipe Stress Program. This concerns an NPS 6-in. (NB 150) schedule 40 pipe with Class 150 connection flanges on the vessels. Both pressure vessels have an outside diameter of 1,000 mm and a wall thickness of 10 mm. The system is designed for an internal pressure of 7 bars (0.7 MPa) and a temperature of 350°C. Nozzle 1 (top nozzle) is located in the middle of the top head of the vertical vessel and Nozzle 2 (lowest nozzle) is located in the middle of the cylindrical shell of the horizontal vessel with a length of 4,000 mm between the tangent lines. The horizontal shell rests on two saddles of which the right saddle is a fixed point and the left saddle is a sliding point. The distance between the symmetrically placed saddles is 3,000 mm. The vertical pressure vessel is supported by a skirt and attached to a concrete foundation by means of anchor bolts. The contents of the pipe system contain a gaseous mixture with a fluid density of 0 kg/m ^{3}. Insulation of the piping has been omitted. Two situations have been analyzed:

** Situation #1.** Without taking the nozzle flexibility into account for the two nozzle connections, which means that the connection points are assumed to be completely rigid.

** Situation #2.** The nozzle flexibility in the CAESAR II run has been taken into account for both Nozzle 1 and Nozzle 2. Flexibility of both nozzles have been computed with the numerical FE/Pipe software package. The results of the CAESAR II pipe stress analysis are shown in Table 4.

## Results of piping reactions

To summarize, the following situations are distinguished:

** Situation #1.** Nozzle 1 and Nozzle 2 without nozzle flexibility (rigid)

** Situation #2.** Nozzle 1 and Nozzle 2 both with nozzle flexibility

Note that nozzle flexibility is synonymous with directional spring rate.

The following is extracted from the CAESAR II Pipe Stress Analysis Software:

Load case definition key

Case 1 (OPE): *W* + *T1* + *P1*

Case 2 (SUS): *W* + *P1*

Case 3 (EXP): *L3* = *L1* – *L2*

Where:

OPE = operating load case

SUS = sustained load case

EXP = expansion load case

W = Dead weight (pipe weight, insulation weight, refractory weight, cladding weight, fluid weight, rigid weight

*T1*= Thermal Set 1 (Temperature #1)

*P1* = Pressure set 1 (Pressure #1)

*L3* = *L1*– *L2*(a combination case that combines the displacements, forces, and stresses)

Table 4 gives an overview of the results of the calculations for the piping reactions for Situations #1 and #2. It should be noted that the shear forces are normally left out of consideration, since they have a minor effect on the stress levels in the vicinity of the nozzle intersection.

Table 5 gives a summary of directional spring rate (nozzle flexibility) for Nozzle 1 on top of the head, and for Nozzle 2 on the cylindrical shell.

## Assessment interface between vessel and piping

As another exercise, we perform the nozzle-load analysis for the same two situations as above according to EN 13445-3, using VES – Software. The software combines the effects of simultaneously acting pressure, axial load and bending moment where the following conditions must be met:

|Φ _{P} | ≤ 1.0 (Individual load ratio)

|Φ _{B} | ≤ 1.0 (Individual load ratio)

|Φ _{I} | ≤ 1.0 (Load interaction ratio)

Note that the load ratios for shear loads have been ignored because the effect is negligible.

The results of the calculations are presented in Table 6.

The load ratios marked in red exceed the permitted unity ratios (1.0) according to EN 13445-3 [*5*]. This means that the nozzle load calculated with CAESAR II for the nozzle on the horizontal pressure vessel is too high.

It turns out that if we perform a calculation in accordance with both WRC 107 [*6*] and WRC 297 [*1*] to evaluate the stresses for Nozzle 2, assuming that the intersection of the nozzles is completely rigid, then the nozzle loads are even not permissible. However, when the flexibility of the nozzle is taken into account, it appears that for Nozzle 2 the nozzle loads according to EN 13445-3 are amply permissible. This also applies if the nozzle loads are evaluated according to WRC 297 and FE/Pipe (FEA). We can therefore conclude that the results where the nozzle is considered flexible are more favorable in terms of nozzle loading. An evaluation of the flange loads according to different methods resulted in overloaded flanges class 150 for both situations, which means that a higher flange class (class 300) must be chosen.

## Evaluation methodologies

It should be recognized that the main purpose of piping-stress analysis is to ensure the structural integrity of the piping and to maintain the operability of the system. The latter function is mainly to ensure that the piping forces and moments applied to connecting equipment are not excessive. Excessive piping loads may hinder the proper functionality of the equipment. The function of maintaining system operability requires the investigation of the interface effects with connecting equipment. It is therefore quite crucial to evaluate the acceptability of the loads exerted by the piping on the nozzle flanges and the local stresses in the vicinity of the nozzle-vessel intersection. The detailed evaluations are as follows, using the nomenclature defined here:

Nomenclature:

*D*_{BC} = bolt circle diameter

*F* = axial tensile force on flange

*F*_{M} = moment factor (according Table UG-44-1)

*G* = diameter of effective gasket reaction

*K*_{V} = “Koves” factor (according paragraph 2 of D 0701) is moment correction factor

*M*= bending moment on flange

*P*_{D} = design pressure

*P*_{eq} = equivalent pressure (for checking against flange rating)

*P*_{R} = pressure rating = 0.84 MPa

The following applies to Nozzle 1 on top head of Figure 2:

*F* = *F*_{Y}

*M*= ( *M*_{X}^{2} + *M*_{Z}^{2}) ^{1/2}

*M*_{T} = *M*_{Y} (torsional moment)

The following applies to Nozzle 2 on cylindrical shell of Figure 2:

*F* = *F*_{X}

*M*_{T} = *M*_{X}

*M*_{L} = *M*_{Y} (longitudinal moment)

*M*_{C} = *M*_{Z} (circumferential moment)

The following are flange rating evaluation methods for flanges conforming to ASME B16.5 and ASME B16.47 subjected to external loads, using data from Table 5.

To evaluate the maximum flange load for Nozzle 2 on the cylindrical shell Situation #1 (rigid nozzle),

*M* = (*M*_{C}^{2} + *M*_{L}^{2})^{1/2} = (4,308^{2} + 459^{2}) ^{1/2} =

4,332.383 Nm = 4,332,383 Nmm

*F* is a compressive force, so can be entered at a value of 0 N.

Equation (1) is the Kellogg method for determining the equivalent pressure:

(**1**)

Using the value for *M* determined above, Equation (1) gives:

*P*_{eq} = 16 × 4,332,383/π194.6 ^{3} + 0 + 0.7 = 3.694 MPa

This is much greater than the pressure rating of 0.84 MPa.

Equation (2) is evaluation of *P*_{eq} according to Paragraph 4.3 Chapter D 0701 (Rules for pressure vessels):

(**2**)

Using the value for *M* determined above, Equation (2) gives:

*P*_{eq} = 16 × 4,332,383/(π194.6^{2} × 241.3 × 2.7) + 0 + 0.7 = 1.5943 MPa

This is also greater than the pressure rating of 0.84 MPa.

Another method for determining *P _{eq} is*the UG-44 ASME VIII-1 method, given by Equation (3):

(**3**)

Using the value for *M* determined above, the left side of the inequality of Equation (3) gives:

*P*_{eq} = 16 × 4,332,383/(π194.6^{3}) + 0 + 0.7 = 3.694 MPa, which is greater that the expression on the right of the inequality:

(1 + 1.2)0.84 = 1.848 MPa.

In fact, it is important to note that *none* of the conditions are met for the most heavily loaded nozzle flange (Nozzle 2, Situation #1)! However, when we enter the bending moment that applies to Situation #2, it appears that the flange load according to UG-44 can be tolerated for the class 150 flange, because Equation (1) in this case is:

*M* = (*M*_{X}^{2} + *M*_{Z}^{2})^{1/2} = (1,619^{2} + 224^{2}) ^{1/2} =

1,634.423 Nm = 1,634,423 Nmm

In this case, the left side of the inequality of Equation (3) gives:

*P*_{eq} = 16 × 1,634,423/(π194.6 ^{3}) + 0 + 0.7 = 1.8296 MPa

which is less than the right side of the inequality of Equation (3),

(1 + 1.2)0.84 =1.848 MPa.

## Comment

If the evaluation according to the above methodologies fails, there is an option to evaluate the flange connection according to ASME BPVC Section VIII – Division 1; Appendix 2 [*7*] in which the external loads are converted into an equivalent pressure and added to the design pressure that must be successively entered in the flange calculation. For the relevant NPS 6-in. Class 150 flange, this means that the flange complies. Such a check was also carried out in accordance with EN 13445-3; clause 11 and it was found that the flange rating was satisfactory.

## Natural frequency

Natural frequency and mode shapes are dynamic properties of the structure. They are controlled by the mass and stiffness of the system. The natural frequency and mode shapes describe the tendency of the structure to vibrate when subjected to dynamic loading. Natural frequencies and mode shapes of a structure are determined by modal analysis. They are computed starting from the mode with the lowest frequency. The lowest natural frequency is called the fundamental natural frequency. Since only the modes with lower frequencies get a significant response to the source of excitation, only modes with lower frequencies are calculated for the analysis. Assigning nozzle flexibility influences the dynamic behavior of the pipework. For the system in question, the differences in natural frequencies are relatively small (Table 7). If the nozzle flexibility is taken into account, this leads to a lower natural frequency of the pipework, which can be seen as a general trend in piping systems. Furthermore, the magnitude of the system stresses and the stress distribution are influenced by considering the flexibility of the nozzle. The intensity of the system stresses decreases if the nozzle flexibility is taken into account in the pipe stress analysis.

## Concluding remarks

From this study, it can be concluded that for relatively thin pressure vessels, it is very attractive to take the nozzle flexibility into account in the pipe-stress analysis. This not only results in more realistic nozzle loads compared to when the piping-vessel interface is considered completely rigid, but can also lead to advantages for the piping layout (narrower footprint). In addition, you prevent the pressure-vessel nozzle from requiring additional reinforcement on top of that required for internal pressure. Another advantage is that in many cases a higher flange rating of the connecting flange with the equipment in particular can be avoided. Anyway, inclusion of the nozzle flexibility shows a decreasing trend in the magnitude of nozzle loads. Overall, it is plausible that incorporating nozzle flexibility into the pipe-stress analysis provides benefits with respect to lower exerted loads on pressure-vessel nozzles, more realistic system stresses and stress distribution without compromising the structural integrity.

*Edited by Gerald Ondrey*

## Acknowledgement

The author is grateful to Mamdouh Abdel Alim for performing the CAESAR II pipe stress analyzes together with the constructive consultations, and Farzad Gardaneh for running the FE/Pipe and WRC 297 analyses.

## References

1. WRC 297 “Local Stresses In Cylindrical Shells Due To External Loadings On Nozzles”— Supplement to WRC Bulletin No. 107 (Revision I); Mershon, J. L., Mokhtarian, K., Ranjan, G.V. and Rodabaugh, E. C., Bulletin Circular by Welding Research Council, Inc., 1984.

2. PD 5500:2021+A2:2022, “Specification for Unfired Pressure Vessels.”

3. Schwarz, Martin M., Flexibility Analysis of the Vessel – Piping Interface, International Journal of Pressure Vessels and Piping, 81, pp. 181–189, 2004.

4. E. Weiss, W. and Joost, H., Local and global flexibility of nozzle-to-vessel intersections under local loads as boundary conditions for piping system design, International Journal of Pressure Vessels and Piping, 73, pp. 241–247, 1997.

5. EN 13445-3:2021: “Unfired Pressure Vessels Design.”

7. WRC 107 “Local Stresses In Spherical And Cylindrical Shells Due To External Loadings”— Wichman, K.R., Hopper, A.G. and Mershon, J.L., Bulletin Circular by Welding Research Council, Inc.,1965.

8. ASME BPVC-VIII-1: July 1, 2023; Section VIII — Rules for Construction of Pressure Vessels, Division 1.

## Author

Walther Stikvoort ([email protected]) is a renowned authority in the field of mechanical and structural integrity of static pressure equipment. He has more than 50 years of experience in pressure vessel and piping design and has developed numerous technical standards and practices to improve the asset integrity of leading operating companies. He is the author of numerous peer-reviewed international journal articles in the field of mechanical and structural integrity. During his career he was regularly active in developing and teaching courses and training to mechanical engineers in his area of expertise and he was a member of various expertise committees. He is currently active as a consultant on static pressure equipment integrity for the engineering community on request.