Piping and pressure vessel engineers face a complex challenge when designing safe, reliable systems: accurately evaluating stresses, deformations, and fatigue risks in intricate components like nozzles, elbows, saddles, and pipe shoes.
Finite element programs like NozzlePRO and FEPipe offer powerful tools to simplify and automate much of this process, providing outputs such as stress intensification factors (SIFs), flexibilities, stiffnesses, and allowable loads. However, understanding how these values are derived—and how to apply them—is critical for effective use.
This following blog walks through the key concepts behind SIFs, stiffness, and allowable loads, explains how NozzlePRO and FEPipe compute them, and offers practical guidance for engineers performing pipe stress analysis or component design.
SIFs, stiffnesses (k), and allowable loads are provided for nearly any component available in NozzlePRO shell models as part of the software output, apart from some structural attachments.
In NozzlePRO, these reports can be found in the results tables and are included by default in the reports users can export. FEPipe also automatically generates SIFs and flexibilities for individual nozzles in the Nozzles, Plates and Shells template or in the standard intersection templates. In addition to the automatic generation of SIFs and Flexibilities, the user can also easily generate SIFs and Flexibilities for a variety of user defined model types. SIFs and flexibilities have been generated for mitered elbows, Wye intersections, jacketed pipe junctions, and conic sections.
These results are readily available in the software’s text and table outputs, simplifying the post-processing required for stress analysts. However, some structural attachments may require manual evaluation.
The basis for the allowable loads in the provided reports is based on the separation of secondary and primary allowable loads and stresses, where secondary failures are assumed to be principally due to membrane and bending stresses, where primary failures (due to high local loads) are principally due to membrane only stresses. The allowable for secondary stresses is SPS (3S or 2Sy), and the allowable for primary loads is SPL, (1.5Sh or Syh). For both primary and secondary allowable loads, the loads are reported in sets of maximum individually occurring, conservative simultaneous, and realistic simultaneous loads. The maximum individually occurring loads indicate the highest load or pressure the model can handle in a single direction. A single load from this column applied to the load screen input will result in stress output near 99% of the allowable stress for the respective primary or secondary case.
NozzlePRO and FEPipe provide three sets of allowable load combinations:
For the set of conservative simultaneous loads, this set is generated using the pressure entered in the input by user, and the resulting set of external loads in this column will result in about 2/3rds or 67% of the allowable stress when applied together in the loads screen. For the example below, 100 psi was entered in the input for design pressure, and this is used to determine the set of loads that can be applied together that will result in maximum stresses at about 67% of the allowable stress. For the realistic simultaneous column, the same is done as conservative simultaneous but stresses with this set of loads will be near 99% of the allowable stress.
The user can verify this by entering the loads appropriately into the input again and submitting the model for analysis. Note: the output for allowable loads uses in-lb and N-mm units, and the input screen for loads uses ft-lb and N-m units.
The Stress Intensification Factor (SIF) quantifies the localized peak stress relative to a nominal stress under bending, pressure, or axial loads. It is calculated as:
The nominal stress in the part for a piping component subject to bending loads is M/Z where M is the moment applied on the component, and Z is the section modulus of the matching pipe welded to the part being analyzed. For pressure loads, the nominal stress is typically taken as the longitudinal pressure stress in the header or vessel, found from PD/4T (although the circumferential nominal pressure stress, PD/2T, could be used as well).
The nominal stress definition must clearly be stated when the SIF is used. For axial loads, the nominal stress is found in F/A, where A is the axial cross-sectional area. When making piping calculations, all piping programs compute the bending moment in the pipe, i.e. the M, and the section modulus of the matching pipe, the Z. The nominal stress is easily found from M/Z. The actual stress in the component under study (say a welding tee) is found by multiplying the M/Z found in the piping program by the SIF.
The B31 piping codes, however, use a slightly different basic equation for the SIF. This definition was first used by Markl in his papers on Piping Flexibility in the 1950’s. Markl’s definition of the SIF is the ratio of the actual stress in the part due to a moment M, divided by the nominal stress in a girth butt weld due to M.
This adjustment to the calculation of the stress intensification factor definition has caused confusion because essentially a second “SIF” has been introduced that does not produce the true peak stress in the part when multiplied by the nominal stress, called the fatigue strength reduction factor (FSRF). Girth butt welds have FSRFs of between 1.7 and 2.0 and are material dependent; stainless having a smaller value, closer to 1.0. Thus, the true peak stress in a girth butt weld due to a moment M, can be found from:
Where M is the moment in the pipe with the butt weld, and Z is the section modulus of the pipe with the butt weld. Also, since the definition of peak stress reported in ASME reports has changed to a range stress since 2021, it’s important to note that here, both peak and nominal stress are defined as amplitude stresses. In terms of the nominal stress in a straight pipe without a girth butt weld, the B31 SIF can be expressed:
This last definition is used in FEPipe. To compute the SIF for models where it is not automatically generated, the following procedure is used:
This procedure is illustrated in the vessel head example below; this complete example may be provided upon request:
Figure: Geometry for SIF Generation
Fatigue cracks due to cyclic external loads and pressure typically occur at the locations shown above. The highest stress will be in the nozzle if the nozzle thickness is smaller than the header thickness, and the highest will tend to occur in the vessel or header if the vessel or header thickness is smaller.
The best characterizer of this stress is the stress in either the nozzle or the head element immediately adjacent to the high stress point. The stresses in the tapered weld elements have not been correlated with the different failure modes and so are not used (they theoretically should not correlate as well, but little work has been done with these).
Note that it is the peak stress at the toe of these fillet welds that is sought for the SIF calculation since the SIF is the multiplier used to compute the tendency to fail due to fatigue-type loadings.
The SIF is, however, composed of two components, one for the discontinuity, and one for the fillet toe concentration. For this reason, the SIF is also currently used for sustained, (or non-cyclic) load evaluation as well as for cyclic load evaluation.
Figure: FEPipe Model to Accurately Compute Stresses at Failure Points
The model shown above has “weld elements” placed between the nodes 25 and 30, and between the nodes 30 and 35. It is suggested that welded intersection models are made up of header and nozzle weld sections so that the connection of the “nominal” section of the nozzle to the weld is done in a planar manner. This tends to avoid the arbitrary inplane or “boring” degree of freedom problem associated with 5 degree-of-freedom shell element formulations.
Figure: Loading Ring Model
The elbow and attached pipe at the top of the tower could have been modeled, but for SIF and flexibility calculations only a portion of the nozzle is typically modeled, (Uses should be careful with this practice, especially where large, lateral openings are involved, and where an accurate pressure SIF is desired. Cutting short or large D/T, large d/D nozzle models that do not have stiffening flange pairs may introduce stiffness and strength adjacent to the nozzle penetration that does not exist.)
Boundary effects through pipe are considered significant over a distance roughly 2D away from the part discontinuity (these are boundary effects due to ovalization, not due to local meridional direction discontinuities).
In the figure above, the elbow is within the 2D distance, and the elbow inplane and outplane stiffness will theoretically provide for different SIFs and flexibilities in the inplane and outplane directions because the elbows flex differently in these directions. In cases like this, where the sensitivity of the result is to be evaluated, the user encourages to “play” with the lengths framing into the intersection to see if changing these values has much effect on the calculated SIF or flexibility. If the incoming nozzle length is changed in the above problem to simulate the effect of the elbow, and the SIF doesn’t change very much (5%), then the result is not sensitive to this value and can be used without concern.
If a flange exists at the end of a nozzle that is being studied, then the flange will usually insist that plane sections remain plane at that intersection, and rigid loading rings can be used at the flange location, and the model can be comfortably stopped there. (The exception to this rule of course is large diameter, short, thin nozzles.
These usually have a small design pressure, but may have a large hydro-test pressure, and the flanges on the end of these nozzles may rotate and bend due to the hydrotest displacements.
These flange connections can be modeled using combinations of intersection and string models.
Flexibility calculations are automatically performed for each standard model described above when SIFs are generated. It is often interesting to see how the stiffnesses vary from technique-to-technique, and from the header to the branch. The variation in bending stiffness along pressure vessels with large openings should be particularly interesting to the pressure vessel analyst.
The SAM processor in FEPipe computes nozzle in cylinder stiffnesses according to WRC 297, and according to NB3685. The General SAM screen is shown below, and the results for a 24x12x0.5 intersection are included.
When a flexibility calculation is performed, the user typically wants to know how a local part of a component deflects when loaded with pipe-type loads, for example, a vessel head. When the pipe leaving a head is loaded, the vessel wall surrounding the connection undergoes shell-type distortions. If these shell-type distortions are omitted from a piping thermal analysis, excessively large forces and moments may be shown to exist at the pipes connection to the vessel.
These large forces and moments do not really exist and so can both complicate and increase the cost of the design. By including the magnitude of these local shell-type distortions in the piping analysis the reduction on forces and moments that they provide can be evaluated.
E. Rodabaugh in WRC 329 showed how, for pad reinforced intersections, not including the proper intersection stiffness could cause an error to be made in the design.
In the example, a pad was placed on an intersection that a piping analysis showed to be overstressed. The piping analyst did not consider the increased stiffness at the junction due to the pad--as most piping programs, and most analysts do not.
In the example, the reduction in stress due to the pad was less than the increase in load due to the greater stiffness. The net effect was to increase the stress at the pad reinforced intersection by including the pad. This is certainly not the desired result.
FEPipe calculates the stiffnesses at nozzles in standard cylinders using a quite complicated internal algorithm that includes the flexibility of the surrounding header and nozzle. A simple example like below can illustrate it.
The horizontal deflection of the header shown in the figure above should not be included in the axial stiffness calculation for the nozzle. This deflection will be modeled by the beam elements framing into the intersection. The actual desired deflection is the local deflection of the shell that will not be considered by a standard beam model of the intersection.
For the standard intersection models FEPipe builds internally standard beam models of the intersection and loads them in the standard stiffness directions of axial, inplane, outplane and torsion, and then compares the movement of the beam-only model to the movement of the shell model. When generating stiffnesses by hand using finite element results, the user must be careful how much of the displacement that is included as a “Stiffness” deflection is actually part of a beam-type deflection. For the standard cylindrical intersections, FEPipe automatically subtracts the beam-type deflections of the model. For the Nozzles, Plates & Shells template stiffness calculations, FEPipe cannot effectively subtract out beam model performance characteristics, and so finds stiffnesses using the standard approaches.
Kax = Fax/disp(ax)
Kin = Min/Rot(in)
Kout = Mout/Rot(out)
Ktor = Mtor/Rot(tor))
The user must be sure in these cases that the flexibility or displacement in the finite element model, i.e. the disp(ax), Rot(in), and so on, represent local flexibilities, and not beam displacements.
For example, let’s look at the rotational stiffness of a nozzle in a vessel head. The rotational stiffness of the vessel head is defined as a point rotational spring at the surface of the vessel at the nozzle centerline penetration whose stiffness is calculated by:
Pipe-type rotations are those where beam assumptions are valid, i.e. where plane sections remain plane. In some cases, the rotation of a vessel head can be affected by the local shell stiffness of the nozzle leaving it. Some portion of the nozzle must always be included in the head analysis.
Flanges adjacent to the head can also affect the locally calculated stiffness of the head. A rule-of-thumb is to get at least two branch diameters away from the locally flexible area being studied before applying boundary conditions, (unless there are flanges closer), and to apply these boundary conditions through a “rigid loading ring,” to make sure that plane sections remain plane at the point of application of the load.
Note: This may not be a conservative assumption, but this is for the user to resolve. If a flange on the end of a nozzle is not radially stiff, then the “flexible” flange will not reinforce, or stiffen the local head connection, and large stresses due to the same moment may be the result.
User defined geometries, (i.e. string models of piping components), should be loaded with a moment, preferably through a loading ring at a point sufficiently removed from the discontinuity being studied. The rotation of the loading ring due to the moment is read from output plots of the results and used to compute the local flexibility of the head.
It is important not to include excessive flexibility of components other than the head in the flexibility calculation. The thin-walled model below for example, can be seen to include piping rotations as well as vessel rotations. The stiffness calculated from the analysis below is not the stiffness sought since the bending of the nozzle shown in the figure below will be included in the piping analysis.
A calculated flexibility (or stiffness) should not include flexibilities that will be included in a standard piping model, only flexibilities that won’t, (and that can be defined adequately using point springs).
The plot below, on the other hand, shows a system where the head, and not the pipe contributes the majority of the displacement due to moment. This is the type of deformed shape plot that is sought when the user is looking for good stiffnesses to use back in a pipe stress program.
To find rotations due to external moments applied, the user should first deform the model to make sure that the load case used is the one intended. For example, in the following plotted result, load case 3 was used to draw the deflected shape.
As can be seen from the axis, this rotation is definitely due to our 160,000 in.lb. moment of the Z axis. We want to enable contour mapping, and this is found under the graphics menu as shown below.
Now, when the displacements are plotted, the exact value of the rotation on the end of the model can be picked off using the “find_contour_value” option. The plot showing the rotation about Z and the picked value is shown below:
The rotational stiffness is calculated by: M/rot = (160,000) in.lb. / (0.0421) radians = 3,800,475 in.lb./radian, and since most pipe stress programs accept rotational stiffnesses in in.lb./degree, this value must be multiplied: (3,800,475)(pi/180) = 66,330 in.lb./deg. For the geometry shown above, FEPipe makes this calculation automatically, (i.e. in Nozzles, Plates and Shells.) The technique is demonstrated so that the user can generate his own stiffnesses using other FEPipe templates where automatic calculations have not been installed yet.
In the above figure, note the unusual Z rotations on the side of the nozzle. Remember that shell elements are only 5 degree of freedom elements, and the Z or boring degrees of freedom normal to the shell are not defined. The elements on the nozzle that shoe the “red” RZ rotations are showing their RZ components that are not defined for the element and so are meaningless. This is the reason for always using the contour value pointer when getting rotational values from the ends of models. The scale may be distorted by undefined DOFs, and the contour_value pointer always displays the exact value at a particular location. This is true with any shell/plate finite element program.
FEPipe automates these calculations in the Nozzles, Plates & Shells module, but users can replicate the method manually for advanced modeling situations.
For pipe shoes and saddles, SIF and allowable load generation are available in NozzlePRO as an option in the shoe/saddle design module. The user can request SIFs and allowable loads from the shoe/saddle wizard or from the shoe/saddle input forms. Both paths through NozzlePRO are shown below. The added SIF and Allowable options are highlighted in orange.
There are two SIF options for both shoes and saddles. One option is for loading through the saddle/shoe. This is the most common option. The second option is for loading through the run pipe for shoes, or through the vessel only for saddles. Both options are shown schematically below, and this option is also available for nozzles with a cylindrical shell header.
(Left) SIF for loads through run or vessel, (Right) SIF for loads through saddle or shoe
Pipe shoes have failed due to excessive load, vibration, temperature gradients, poor welding, overheating, cyclic loads and local collapse. Often, support allowable loads are taken from company tables or manufacturers data based on Bijlaard/WRC107 type evaluations. These simplified calculations often ignore:
1) Circumferential, longitudinal and torsional moments in the run pipe due to support loads
2) Stress in the support plates
3) Non-rectangular attachment profiles
4) Non-integral wear plates.
The most common stress related issues associated with support design are believed to most often occur from:
1) Using supports for small D/T pipe on large D/T pipe.
2) Using supports as anchors without considering the effect of combined forces and moments
3) Assuming that all supports can accommodate the total suggested pipe span when high thermal loads exist at the support in the piping system.
Many pipe stress analysts ignore support concentrations all together and simply pass the loads from the piping system at a support location to the structural engineer for supporting steel design. The structural engineer typically does not evaluate the stress in the pipe due to structural load and so the local stress in the pipe is often ignored. When there is not a company standard for how the pipe shoes, stops, or saddles should be fastened to the pipe, it is recommended that some design effort should be used to size and configure welds for both external loads and for the potential of relative thermal expansion.
As with typical piping configurations the SIFs developed for pipe shoes and saddles are large when the piping D/T is large. The SIF analysis for shoes is used most often for any pipe shoe or saddle used as an anchor, limit stop or guide that introduces any significant load to the piping system. The SIFs developed from this analysis can be used at the locations in a piping model. Example 5.15 in the PRG help manual illustrates how this capability is applied in detail and what might be expected. When the D/T is greater than 40 torsional SIFs and flexibilities can become considerable.
Where cyclic thermal or dynamic loads are evaluated, the user is advised to apply this capability carefully. For piping, this analysis generally shows when full encirclement support or rings should be added to the pipe support. For vessels, this capability is used most often to evaluate stresses due to axial loads on the vessel. Also, for pipe shoes with D/T ratios greater than about 40 torsional SIFs can be considerably higher than bending SIFs and indicate to the user when rings, pads, or full encirclement reinforcement should be used.
Models for SIF and flexibility applications are also included in examples 5.2, 5.11.1, and 5.15 of the PRG help manual. Pipe stress program users often assume that in-line pipe anchors are rigid with respect to the pipe, serving as a force and moment firewall, preventing loads from one side of the piping system to influence loads on the other.
Frequently it is found that this assumption is invalid, (increasingly so as D/T gets larger), and so flexibility models are also recommended for use in this case. Pipe shoe and saddle flexibility factors are normalized to the bending flexibility of a single diameter of straight pipe. A flexibility of 30 means that 30 diameters of straight pipe are essentially concentrated at a point spring at the shoe or saddle. Thirty extra diameters of pipe is usually significant in the flexibility of a tight piping system. If the user is concerned about the flexibility of the pipe shoe or saddle, they are encouraged to use the wizard to approximate the geometry and compute flexibility factors. If large flexibility factors are produced, then further evaluation may be warranted. Example 5.15 gives a detailed example where flexibilities of the in-line anchor affect the loads on an adjacent equipment nozzle. A typical guided pipe shoe is shown below:
Stress intensification factors, stiffnesses, and allowable loads form the foundation of safe piping and pressure vessel design. While NozzlePRO and FEPipe automate much of this work, a thorough understanding of:
- The definitions behind the calculations
- How and when to apply these values
- The modeling limitations and sensitivities
…is essential for producing accurate, conservative, and efficient designs.
Accurately incorporating flexibility effects, particularly at vessel connections and supports, can prevent costly overdesign and improve long-term system reliability. The advanced capabilities within NozzlePRO and FE/Pipe, when properly leveraged, provide the necessary tools for engineers to confidently tackle these complex design challenges.
Whether you're validating complex intersections, refining flexibility models, or applying advanced SIF calculations to your most challenging designs, our team can help you fully leverage the capabilities of FEPipe and NozzlePRO.
Schedule a technical consult or training session with our experts to optimize your workflows, ensure code compliance, and apply best-practice modeling strategies across your piping and vessel projects.