Submodel stress analysis of pressure vessel closure
hardware
This example demonstrates the use of surface-based submodeling as
a technique to obtain more accurate solutions than those obtained using
node-based submodeling.
The particular scenarios studied include
cases where:
the submodel displacement field is expected to differ from the global
model displacement field by a rigid translation and
the geometry of the submodel differs from the global model in a region
whose response is primarily load controlled.
This example examines the stress behavior of closure head standpipe
structures in a nuclear reactor vessel closure assembly. The vessel assembly
forms the pressure boundary surrounding the fuel core. This example considers
the following loading conditions:
pre-tension load in the stud bolts,
constant internal pressure, and
loading due to the control rod drive mechanism
(CDM) plug.
The loading conditions cover the most basic structural operation of a
reactor vessel. The International System of units
(SI) is used in the following sections to
describe the model. The analysis itself is performed in English units. The
model and analysis are derived from details of the Shippingport pressurized
water reactor (1958).
Geometry
The problem domain comprises a cylindrical vessel shell, a hemispherical
bottom head, a dome-shaped closure head, and the closure and seal assembly, as
shown in
Figure 1.
The overall height of the vessel shell including the bottom head is 7650 mm
(301 in). The bottom head has an inner radius of 1410 mm (55.5 in) and a
thickness of 157 mm (6.18 in). The inlet nozzles on the bottom head are not
considered in this example. The inner radius of the vessel shell is 1380 mm
(54.5 in), and the thickness is 213 mm (8.40 in). The closure head has a height
of 2330 mm (91.8 in), an inner radius of 1310 mm (51.5 in), and a thickness of
210 mm (8.25 in).
The closure head includes eight standpipes with
CDM plugs inserted in each. The standpipes
have an inside diameter of 472 mm (12 in) and an outside diameter of 630 mm (16
in) and extend roughly 1000 mm (25 in) above the closure head. The
CDM plugs have an outside diameter of 465 mm
(11.8 in) and a flange diameter of 630 mm (16 in) and are 748 mm (19 in) tall.
Each CDM plug sits in a closure head standpipe
on a 404 mm (10.25 in) diameter ledge.
The closure head is attached to the vessel shell by a seal and closure
assembly. The assembly includes 40 stud bolts passing through the bolting
flanges of the closure head and the vessel shell, each of which is restrained
by two cap nuts (one on each end). Each stud bolt is 2290 mm (90 in) in length
and has a diameter of 146 mm (5.75 in). The closure nuts are 304 mm (12 in)
long with a thickness of 28.6 mm (1.13 in). To complete the closure assembly,
an Omega seal is welded to the under surface of the closure head and top
surface of the vessel shell.
Materials
All components are constructed of high-strength steel.
Boundary conditions and loading
A pre-tension load of 2200 kN (5 × 106 lbf) is applied to each
bolt. The inner surfaces of the head and the vessel shell are subject to a
constant pressure of 1.38 × 107 Pa (2000 psi) from the water.
Interactions
Contact occurs between
the reactor vessel and closure head,
the lower nuts and the reactor vessel bolting flange,
the upper nuts and the closure head bolting flange, and
the closure head standpipe and CDM
plug.
Model terminology
This example illustrates the use of the submodeling technique in ways that
are generalizations of the
Abaqus
user interface concept of a global model driving the response of a submodel.
Specifically, some submodel analyses described in this section represent
material domains that are adjacent to, rather than lying within, the domain
considered in the initial analysis. To more clearly describe the models in this
example, the term “source model” is used instead of “global model.” A source
model is a model that provides solution results to a subsequent submodel
analysis.
Abaqus modeling approaches and simulation techniques
The objective of the analyses in this example is an understanding of
stresses in the region of the closure head standpipe.
Summary of analysis cases
Case 1Reactor closure analysis:
reference solution
The stress distribution in the closure
head is determined from a single, finely meshed model that includes closure
head standpipe and CDM plug details.
Case 2Submodeling of the closure
head standpipe region
A defeatured source model of the vessel
assembly is analyzed first. Features excluded are the closure head standpipes
and the CDM plugs. This model then drives a
submodel with a more detailed representation of the closure head standpipe
region.
Case 3Submodel application of
CDM hardware loading
The CDM
plug is analyzed separately as a source model. Boundary conditions are
introduced on the CDM plug where it interacts
with the closure head standpipe seating ledge to determine the surface traction
characteristics at this interface. This model is then used to drive a submodel
of the remaining vessel assembly.
In the two submodel analysis cases the node-based submodeling technique, in
which the submodel is driven with displacements, is compared to the
surface-based submodeling technique, in which the submodel is driven with
stresses. The three cases are discussed in more detail below.
Case 1 Reactor closure analysis: reference solution
This reference case determines the stress response of the reactor vessel
assembly when subjected to boltup and pressure loading using a single analysis.
By comparison, the other modeling cases make use of submodeling. The reactor
vessel assembly is cyclic-symmetric with respect to the axis of the cylindrical
vessel body and only one-quarter of the whole assembly is modeled. The global
geometry is shown in
Figure 2.
Analysis types
A static stress analysis is performed.
Mesh design
The vessel is meshed with C3D20R elements, and the closure head is meshed with C3D10M elements. All other parts including the head, bolts, and Omega
seal are meshed with C3D8R elements. The mesh is shown in
Figure 3.
Materials
The elastic model is used in all models with a Young’s modulus of 2.07 ×
1011 N/m2 (3.0 × 107 lbf/in2) and a
Poisson’s ratio of 0.29.
Boundary conditions
Symmetric boundary conditions are applied to the two side surfaces of the
vessel quarter. The nodes on the centerline are constrained separately and are
free to move only along the vessel central axis. The node located at the center
of the vessel bottom surface is pinned to give the model a statically
determinant condition.
Loads
A pre-tension load of 2200 kN (5 × 106 lbf) is applied to each
stud bolt in the model. The inner surfaces of the head, the vessel body, and
the nozzle are subject to a constant pressure of 1.38 × 107 Pa (2000
psi) from the water.
Bolting of the CDM plug to the closure head
standpipe, defined as the CDM assembly, is
simulated in this analysis by the application of a pair of concentrated loads
acting through distributing coupling constraints. Refer to
Figure 4
for identification of the loaded regions of each
CDM assembly. For each
CDM plug one coupling constraint acts on the
top surface of the plug (region B). For each standpipe one coupling constraint
acts across the top surface (region C). The reference node for each coupling
constraint is positioned along the center axis of the
CDM assembly so that vertical concentrated
loads can be applied without generating overturning moments. A bolting force of
106 kN (2.4 × 105 lbf) for each CDM
assembly is chosen as adequate to overcome the liftoff force due to the vessel
internal pressure. This force is applied vertically in the up direction to the
standpipe coupling constraint. A downward force is applied to the accompanying
coupling constraint on the CDM plug, but this
force is lessened by the amount of the pressure-generating liftoff force due to
the operating pressure acting on region A (shown in
Figure 4),
since the capping of this region is not considered explicitly in the model.
Based on the diameter of region A, this pressure liftoff force equals 99 kN
(2.26 × 105 lbf).
Constraints
The Omega seal is tied to the flange surfaces of the vessel head and the
vessel body. As mentioned above, distributing coupling constraints are applied
to the CDM plugs and the closure head
standpipes.
Interactions
Small-sliding contact definitions are prescribed between
the reactor vessel and closure head,
the lower nuts and the reactor vessel bolting flange,
the upper nuts and the closure head bolting flange, and
the CDM plug and the accompanying
seating ledge in the closure head.
Analysis steps
The analysis is performed in a single static step with automatic
stabilization to help establish the contact between the stud bolts, head, seal,
and vessel. Results show that the static dissipation energy is minimal compared
to the strain energy; therefore, its effect on the response can be neglected.
Output requests
Default field and history output requests are specified in the step.
Results and discussion
This case is provided as a reference. Submodel analysis results are compared
to these results in the discussion of Case 2 and Case 3 below.
Case 2 Submodel analysis of the closure head standpipe region
This case is representative of the most common submodel analysis approach: a
global analysis of a coarse source model followed by a detailed submodel
analysis representing a smaller region of the source model. Here, the coarse
source model excludes details of the CDM plug
and closure head standpipe as an illustration of a source model with
significant defeaturing—a common motivation for subsequent submodel analysis.
The submodel comprises a portion of the closure head and two
CDM plugs, using a finer mesh and with feature
details included. The relation between the source model and submodel is shown
in
Figure 5.
Analysis types
A static stress analysis is performed.
Mesh design
In the source model the vessel and closure head are meshed with C3D20R elements; all the other parts including the head, bolts, and
Omega seal are meshed with C3D8R elements. The source model mesh is shown in
Figure 6.
For the submodel the CDM plug is meshed
with C3D8R elements and the closure head is meshed with C3D10M elements. The submodel mesh is shown in
Figure 7.
Materials
The material model is the same as in Case 1.
Boundary conditions
The source model boundary conditions reflect those applied in Case 1.
Similarly, the submodel has symmetric boundary conditions applied to the two
side surfaces of the closure head. In the node-based submodel analysis,
submodel boundary conditions are applied to the submodel boundary. In the
surface-based submodel analysis, a boundary condition is applied in the
2-direction on the coupling constraint for each
CDM plug in the closure head to suppress the
rigid body mode.
Loads
The source model loads reflect those applied in Case 1 except for the
bolting of the CDM plugs to the closure head,
which is introduced in the submodel.
In the surface-based submodel analysis, submodel distributed loads are
applied to the submodel boundary surface.
Constraints
Distributing coupling constraints are applied to the
CDM plugs and the closure head standpipes, as
in Case 1.
Interactions
Contact interactions are the same as in Case 1.
Analysis steps
The global analysis of the source model is performed in a single static
step with automatic stabilization to help establish the contact between the
stud bolts, head, seal, and vessel. The submodel analysis is performed in a
single static step.
Output requests
Default field and history output requests are specified in the step.
Results and discussion
Von Mises stress results are compared along paths in two regions in the closure
head:
the first comparison is made along a ligament through the closure head
shell, as shown in
Figure 8;
and
the second comparison is made along a circular path in the vicinity of
the CDM hardware seating ledge, as shown in
Figure 9.
By reviewing the stress distribution comparisons (discussed below), you can
see that in this case, surface-based submodeling is superior to node-based
submodeling for results lying within the main closure head shell. In the upper
region of the standpipe, neither method provides adequate results indicating
that the level of defeaturing in the source model is too great for an accurate
submodel analysis of the standpipe region.
Closure head shell ligament
Figure 10
compares the von Mises stress distribution on the path shown in
Figure 8
for the reference model, the node-based submodel solution, and the
surface-based submodel solution. These results show that the surface-based
submodel solution provides a more accurate stress distribution than the
node-based submodel technique in this region. This result is consistent with
the guidelines documented in
Surface-Based Submodeling,
namely that a surface-based solution is more accurate in cases where the
environment is load controlled—the vessel pressurization dominates the closure
head response in the shell region—and the submodel geometry differs from the
source model geometry—the source model does not include the standpipe detail.
In practice, the classification of an analysis according to these
guidelines, particularly the classification of load-controlled vs.
displacement-controlled, is often not obvious nor is the reference solution
available for comparison. Therefore, you should always compare measures of
interest between the source model and the submodel on or near the submodel
driven boundary and confirm that they show reasonable agreement. In this case
the von Mises stress is compared along a path, shown in
Figure 7,
cutting across the submodel driven boundary. In
Figure 11
the results comparison shows that the surface-based submodel solution provides
a stress distribution on the submodel boundary that more closely matches that
for the global solution of the source model. This plot also shows the reference
solution, which shows better agreement with the surface-based submodel solution
at the outer edge of the shell and better agreement with the node-based
submodel solution at the inner edge of the shell. Hence, the agreement between
global model and submodel stress distributions, while necessary, is not
sufficient to confirm an adequate submodel solution at all locations. You must
also use judgment as to whether geometric differences are too great between the
source model and submodel.
Standpipe seating ledge
Figure 12
compares the von Mises stress distribution on the path shown in
Figure 9
for the reference model, the node-based submodel solution, and the
surface-based submodel solution. The stress results near the seating ledge show
that neither submodeling technique is clearly superior or provides adequate
accuracy. This follows from the fact that the standpipe and seating ledge
region did not appear at all in the source model analysis; the defeaturing was
too severe in this case for an adequate submodel solution in this region.
This seating ledge stress comparison makes it clear that although a
favorable comparison of results on the submodel boundary, as was done in
Figure 11,
is necessary, it is not sufficient to ensure adequate submodel results in all
locations in the model. In this case the seating ledge region was absent
entirely from the defeatured source model, and you should not expect accurate
results in this region.
Case 3 Submodel application of CDM plug
loading
This case represents an atypical use of submodeling in which the source
model is associated with a small part of the structure and the submodel
comprises most of the overall structure. Here, the source model focuses on the
CDM plugs to predict how each of the
CDM plugs loads the closure head.
The subsequent submodel analysis uses results from the source analysis for
loading the remainder of the structure. The regions considered for the source
model and submodel are shown in
Figure 13.
Mesh design
In the source model the CDM plug hardware
is meshed with C3D8R elements. The source model mesh is shown in
Figure 13.
The geometry for the remaining structure is also shown in this figure to
illustrate the plug positioning relative to the overall reactor assembly.
The submodel mesh is nearly identical to that shown in
Figure 3
for Case 1. The only difference is that the
CDM plugs are excluded from the model in Case
3.
Boundary conditions
The submodel analysis of the CDM plug
source model simulates contact with the seating ledge with a boundary condition
constraint on the plug seating surface.
Loads
The loading follows that for Case 1 with the application of the loads split
between the source model and submodel.
Source model analysis
The bolt load applied to the CDM plug is
simulated through a downward concentrated force applied to a distributing
coupling constraint reference node in each of the
CDM plugs. The magnitude of this force is the
bolting force of 106 kN (2.4 × 105 lbf) less the pressure generating
liftoff force of 99 kN (2.26 × 105 lbf), for the reasons detailed in
the Case 1 description.
Submodel analysis
A pre-tension load of 2200 kN (5 × 106 lbf) is applied to each
stud bolt in the model. The inner surfaces of the head, the vessel body, and
the nozzle are subject to a constant pressure of 1.38 × 107 Pa (2000
psi) from the water.
The bolt load applied to the standpipe is simulated through an upward
concentrated force applied to a distributing coupling constraint reference node
in each of the standpipes. The magnitude of this force is the bolting force of
106 kN (2.4 × 105 lbf) less the pressure generating liftoff force of
99 kN (2.26 × 105 lbf).
Constraints
All constraint definitions are the same as in Case 1.
Interactions
All interaction definitions are the same as in Case 1, except that the
contact interaction between the CDM plug and
the standpipe seating ledge is effected through submodel loads and boundary
conditions.
Run procedure
Run the analyses with the input files listed for Case 3 below.
Results and discussion
Stress results are considered on the same paths defined for comparison of
reference and submodel results in Case 2. The von Mises stress distribution on
these paths is compared in
Figure 14
and
Figure 15
for the two forms of submodeling and the reference solution.
These results show that in both the high-stressed region, shown in the
ligament stress plot, and in the vicinity of the seating ledge, the
surface-based submodeling approach provides a more accurate solution. The poor
results for node-based submodeling follow from the fact that the assembly
model—the submodel in this case—elongates along the vessel main axis. The
CDM assembly region experiences this
elongation as a rigid body translation. The standpipe seating ledge, however,
is constrained in its movement by the submodel boundary conditions. These
boundary conditions follow from the separate analysis of the
CDM plug source model that does not consider
the solution-dependent elongation of the vessel assembly.
Discussion of results and comparison of cases
Case 2 and Case 3 illustrate situations where you may see improved accuracy
when using the surface-based submodeling approach.
The effect of stiffness change on submodel analysis
In cases where the submodel stiffness matches that of the source model, you
can expect, using reasonable modeling practices, that the submodel analysis
will provide adequate results. In cases where the submodel stiffness differs,
such as in Case 2, you must exercise caution in evaluating your submodel
solution. Comparison of stress contours on the common boundary of the source
model and submodel can aid you in determining if your solution is adequate. In
the case of significant defeaturing, you should not rely on the submodeling
analysis technique in any form for detailed stress response in areas absent
from the source model, such as the closure head standpipe.
The effect of displacement discrepancies on submodel analysis
In cases where you expect that the submodel displacement solution will
differ from the corresponding source model solution by only a rigid body
motion, such as in Case 3, you can expect that a node-based submodeling
approach will give incorrect results. In this case you can use the alternative
surface-based submodeling of stresses and obtain improved solution accuracy.
Files
Case 1 Reactor closure analysis: reference solution
Reactor vessel closure assembly submodel analysis with
CDM loading effected through surface-based
submodeling.
References
Naval Reactors Branch, Division of Reactor Development,
United States Atomic Energy Commission,The Shippingport Pressurized Water
Reactor, Reading, Massachusetts: Addison Wesley Publishing
Company, 1958.
Figures
Figure 1. Reactor vessel assembly. Figure 2. Reactor vessel assembly model. Figure 3. Reference analysis mesh. Figure 4. Load application areas on the CDM
plug and closure head standpipe. Figure 5. Case 2 closure head submodel relation to the source model. Figure 6. Case 2 global analysis mesh with defeatured closure head. Figure 7. Case 2 submodel analysis showing the mesh (left) and the path
definition for stress comparison to the source model (right). Figure 8. Through-ligament path definition. Figure 9. Seating ledge path definition. Figure 10. Case 2 stress distribution comparison through the closure head
ligament. Figure 11. Comparison of the closure head through-thickness von Mises stress
distribution at the location of the submodel boundary. Figure 12. Case 2 stress distribution comparison around the seating
ledge. Figure 13. Case 3 CDM plug analysis
mesh. Figure 14. Case 3 stress distribution comparison through the closure head
ligament. Figure 15. Case 3 stress distribution comparison around the seating
ledge.
