Calculation of a Lower Bound Ratchet Limit Part 2 - Application to a Pipe Intersection with
Dissimilar Material Join
James Ure
a
, Haofeng Chen
a
, David Tipping
b
a
Dept of Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow, Scotland, G1 1XJ
b
Central Engineering Support, EDF Energy Nuclear Generation Ltd., Barnwood, Gloucester, GL4 3RS
Abstract: In an accompanying paper in this issue a lower bound method based on Melan's theorem
was derived and implemented into the Linear Matching Method ratchet analysis procedure. This
paper presents a ratchet analysis of a pipe intersection subject to cyclic thermo-mechanical loading
using the proposed numerical technique. This work is intended to demonstrate the applicability of
the lower bound method to a structure commonly seen in industry and also to better understand
the behaviour of this component when subjected to cyclic loading. The pipe intersection considered
here has multiple materials with temperature dependent properties. Verification of the results is
given via full elastic-plastic analysis in Abaqus.
Keywords: Shakedown, Ratchet limit, Lower bound, Linear Matching Method, Pipe Intersection
1. Introduction
Pipe intersections and branch connections are common in piping systems, and their assessment is an
important consideration in ensuring the structural integrity of existing plant systems. The
widespread use of pipe intersections has led to many studies of their behaviour over a number of
years from early elastic analyses (Lekkerkerker, 1971) through to more recent analyses considering
creep effects including the effects of welded regions (Han et al, 2010). In particular, studies to
calculate limit and plastic collapse loads are common for both internal pressure and bending modes,
for example the semi-theoretical solutions proposed by Kim et al (2006). These equations use
several geometric parameters to predict whether failure will occur in the nozzle, the main pipe or
the intersection of the two for a monotonically applied bending moment or pressure.
Despite their common use in piping systems, very few studies have been published regarding this
geometry subject to cyclic loading. A study by Nadarajah et al (1996) used the Elastic Compensation
Method to conduct a parametric study of this geometry subject to internal pressure and cyclic
bending moments. Systematically altering the mean radii and shell thicknesses of the pipe and
nozzle allowed a variety of thin walled (radius to thickness ratio between 50 and 200) to be
analysed. Both the limit load surface and elastic shakedown limit was calculated for each geometry
using an elastic-perfectly plastic material model. The parametric studies found a strong interaction
between the loads, with small nozzles (having a radius of less than one fifth of that of the main pipe)
having a near linear interaction curve. Larger nozzles (i.e. with a radius of two fifths of the main pipe)
still displayed a strong interaction between the loads, but had a circular interaction curve which is
more akin to the conventional Bree like shakedown boundary.
In addition to this study, the shakedown response of a single thick walled pipe intersection was
analysed in the EPERC Design by Analysis manual (1999). This geometry consisted of two materials
and also contained the weld detail at the join between the nozzle and main pipe. Several analysis
methods were used including the Elastic Compensation Method, direct analysis using elastic-plastic
FEA and stress categorisation methods. The Elastic Compensation Results presented for this case
show little interaction between the bending moment and internal pressure. This is attributed to the
thick shells, which serve to isolate the loadings from each other. Again, internal pressure and cyclic
in-plane moments were considered. Apart from these studies, no further shakedown examples of
this geometry could be found.
Within the UK nuclear industry assessment code R5 (Ainsworth, 2003), components are allowed to
operate beyond traditional elastic shakedown limits into the reverse plasticity region (also known as
global shakedown). However, because R5 is based on simplified assessment procedures, at times it
may prove difficult to show that a component is in shakedown status (whether elastic or global). To
alleviate this, EDF Energy has incorporated the Linear Matching Method (Chen 2010a, 2010b; Chen
and Ponter, 2010) into their R5 research programme. The Linear Matching Method is a direct
method for assessing shakedown and ratchet limits which avoids the need for full nonlinear finite
element analysis. Instead the bounding theorems of Koiter (1960) and Melan (1936) are used to
calculate the shakedown and ratchet limits of the component.
In an accompanying paper elsewhere in this issue (Chen et al, 2012) the theoretical basis is given for
the extension of Melan's theorem to the assessment of the ratchet limit, which shows the boundary
between global shakedown and ratcheting behaviour. The numerical implementation of this
theorem is also described and demonstrated through the benchmark of a plate with a central hole
subject to cyclic thermal and mechanical loading. The purpose of the current paper is to apply this
method to a pipe intersection with a dissimilar material join. Firstly, this will increase the knowledge
of the behaviour of this component when subject to cyclic thermal and mechanical loading.
Secondly, this will verify the applicability of the latest LMM procedure to larger and more complex
problems than the typical benchmarks seen. The remainder of this paper briefly summarises the
Linear Matching Method before going on to explain the geometry and finite element model used
here. Two load cases are considered: a constant internal pressure and 1) cyclic thermal loading and
2) a cyclic in-plane bending moment applied to the branch pipe. Discussion of the results is provided
and verification is given through a full nonlinear elastic-plastic step-by-step analysis in Abaqus.
2. The Linear Matching Method
The premise of the linear matching method is that the nonlinear elastic plastic behaviour of metallic
materials can be substituted for a linear material model in an iterative solution scheme. A series of
linear elastic solutions is performed and the modulus is reduced in regions where the stress in the
material has exceeded the yield stress. The subsequent iteration uses these modified modulus
values, which allows the stresses to redistribute in the structure in a way which very closely mimics
that of an elastic plastic material.
The Linear Matching Method is divided into two stages. The first stage considers only the cyclic
loading to evaluate the varying residual stress and the associated plastic strain range. A fixed level of
cyclic loading is applied in this stage. The modulus adjustment procedure allows the stresses to
redistribute and the varying residual stress field is developed at each point in the load cycle. The
second stage then calculates the maximum level of additional steady state loading which will not
cause the component to ratchet. Stage two is essentially a traditional shakedown assessment to
calculate the constant residual stress field where the initial elastic cyclic stress field is augmented by
the varying residual stress calculated in stage 1. The convergence of stage two is based on Koiter's
theorem which states that if 1) any kinematically admissible strain rate can be found such that the
strain rate is compatible with the applied displacement and 2) the plastic dissipation within the
structure is less than or equal to the applied work, then shakedown does not occur.
In the accompanying paper (Chen et al, 2012), an extension to Melan's theorem was derived which
made the calculation of the lower bound on the ratchet limit possible. Melan's theorem states that
for a given load set the structure will shakedown if a constant residual stress field can be found such
that the yield condition is not violated for any combination of cyclic elastic and residual stresses.
If the cyclic loading is large enough to cause reverse plasticity, then a varying residual stress field is
also formed in addition to the constant residual stress present. The extension to Melan's theorem
comes from the idea that this varying residual stress field can be combined with the elastic stress at
the corresponding point in the load cycle to create the total varying stress. With the addition of a
steady state stress to this total varying stress, if a constant residual stress field can be found such
that the yield condition is not violated for any point in the load cycle, then it can be said that the
component is not ratcheting.
The LMM is currently implemented in the Abaqus using the UMAT user subroutine (Abaqus, 2009).
This extension to Melan’s theorem for calculation of the lower bound ratchet limit has been added
into stage 2 of the LMM subroutines. The cyclic elastic stresses and associated varying residual
stresses calculated during stage 1 are combined to form the total varying stress. Stage two uses this,
along with the additional steady state stress, to calculate a constant residual stress field which will
mean that yield is satisfied at each point in the load cycle.
3. Pipe Intersection
The pipe intersection analysed in this paper is similar to that analysed in the EPERC example (1999),
and is shown in Figure 1. The small intersecting pipe is welded to the main pipe, with the weld itself
modelled as a chamfer between the two shells (as per the EPERC example). The main pipe is made
from 316 stainless steel, a common material in nuclear plant components. The intersecting pipe was
chosen to be from low alloy steel SA508 and the weld material is Inconel 82/182. Recently the UK
nuclear industry has been analysing the residual stresses present in this dissimilar weld with a view
to investigating stress corrosion cracking (Brust et al, 2010; Smith et al, 2010a, 2010b).
When conducting thermal analyses it is important to consider the effects of temperature dependent
material properties. The temperature dependent yield stresses used for all three materials is shown
in Table 1. During the analysis linear interpolation/extrapolation is used to calculate the yield stress
at intermediate and outlying temperatures. The data for 316 and SA508 were taken from the British
Standard (2007) and a paper by Hurrell et al (2005) respectively. Material data for the Inconel weld
in the as-welded condition is limited, with the micromechanical tests of Kim et al (2009) being the
only elevated temperature tests which could be found. Linear extrapolation is performed to provide
an approximation to the temperature dependency, which will give a more accurate solution than if
temperature independent properties were assumed throughout the weld. An elastic perfectly plastic
material which satisfies the von-Mises yield criterion is assumed throughout this work.
The Linear Matching Method requires an elastic calculation to be performed for each point in the
load cycle. This provides a starting point for the Linear Matching solution procedure. Temperature
dependent yield stress is considered once the linear matching process has begun, but because the
elastic solutions have been performed before this process a single value of thermal expansion must
be assumed. This is fine for a single ratchet analysis at a fixed level of cyclic thermal load, but to
create the Bree diagram a new set of elastic solutions would need to be created for each level of
thermal loading considered. To avoid this, the worst case thermal expansion scenario was chosen.
When the temperature dependency of the thermal expansion coefficients of the materials in Smith
et al (2010b) is examined, it can be seen that the difference between the three values remains
almost constant. The magnitude increases, but does so uniformly for all three materials. Therefore,
with a maximum temperature considered never larger than 600
0
C, the expansion values at this
temperature were used for all thermal analyses conducted. Therefore values of 1.8E-5 for 316, 1.5E-
5 for Inconel 82/182 and 1.4E-5 for SA508 are used. This assumption gives conservative results in all
cases.
Two separate load cases are considered for this pipe intersection. The first is a steady state internal
pressure and a cyclic thermal load. The second load case is the combined action of a steady state
internal pressure and a cyclic in plane bending moment.
4 Internal Pressure and a Cyclic Thermal Load
The thermal cycle chosen for this analysis has three load instances. The first point is where the
intersection is at ambient temperature, θ
0
, throughout the entire structure. The second point is
where the inner surface is at an elevated temperature, θ
0
+ Δθ, whilst the outer surface remains at
ambient, θ
0
. This results in a linear temperature gradient through the wall of the pipe with a
temperature difference ∆θ. The differential expansion of the inner and outer surfaces results in a
linear distribution of elastic stress through the wall thickness. Finally, the case where the entire
structure is held at a uniform elevated temperature, θ
0
+ Δθ, is considered. The different thermal
expansion coefficients of the materials create thermal stresses at the material boundaries. In
addition to this cyclic thermal loading, an internal pressure is applied. The closed end condition is
assumed, which is applied in the model as an axial tension to both pipes. In addition, both free ends
of the intersection are constrained to expand in plane, which simulates the expansion of a long pipe.
Due to the symmetry present in the geometry and the applied loads, a quarter model with
appropriate symmetry boundary conditions was used to model the pipe intersection. The mesh is
refined in the region of the weld as all of the structural and material discontinuities are in this
region. This gives the model a total of 4038 elements as shown in Figure 2. Element type DC3D20
elements used for the two heat transfer analyses. These temperature distributions were then
applied as predefined fields in linear elastic structural analyses (using C3D20R elements), giving the
stresses shown in Figure 3.
These elastic stresses are used as the starting point for the LMM analysis. In stage 1 the cyclic
stresses are considered so that the reverse plasticity mechanism and plastic strain range can be
calculated. Once this has been found, the maximum level of constant internal pressure which will
not cause the component to ratchet is calculated. This calculation procedure was repeated for
different levels of cyclic loading so that the ratchet boundary could be created. The LMM shakedown
procedure was also employed to calculate the reverse plasticity limit. This limit, which divides the
regions of strict shakedown and global shakedown, completes the Bree diagram for the component
and demonstrates the ability of the LMM to produce lower and upper bound limits for both
shakedown and ratcheting.
4.1 Results
The Bree diagram for the pipe intersection subject to a constant internal pressure and the thermal
cycle is shown in Figure 4. The applied constant internal pressure, P, is normalised against the initial
applied internal pressure, P
0
, of 10MPa. The applied temperature difference, ∆θ, is normalised
against the initial applied temperature difference, ∆θ
0
, of 100
0
C. Both temperature dependent and
temperature independent results (using the yield stress at 20
0
C in Table 1) are plotted, and
demonstrates that temperature dependant yield stress significantly reduces the ratchet and reverse
plasticity limits of the component.
The point corresponding to zero cyclic load (point A in Figure 4) represents the limit load for internal
pressure. The failure mechanism for this loading predicted by the LMM calculation is at the crotch
corner (Figure 5a), which is typical of this geometry subject to pressure loading. This mechanism
dominates the failure at low levels of cyclic loading. When the cyclic thermal load increases in
magnitude (point C), the material mismatch begins to play a more significant role in the failure. The
difference in the thermal expansions causes a reverse plasticity mechanism at the material joins.
When the internal pressure is applied, this then interacts with the concentration at the crotch corner
to produce a failure mechanism which has a contribution from both loads as shown in Figure 5b.
At large levels of cyclic loading (point E), the material mismatch dominates the failure mechanism.
The concentration due to the mismatch at these levels of thermal loading is such that even the
severe stress raiser at the crotch corner is no longer a factor in the ratchet mechanism (Figure 5c).
Figure 6 shows the convergence for Points B and D respectively. These plots show the convergence
of the upper and lower bound multipliers when temperature dependent material properties are
used. Considering point B first, it can be seen that the upper bound solution converges rapidly to the
least upper bound for this finite element mesh. The convergence of the lower bound at this point is
comparatively slow. As highlighted in the benchmark example in the accompanying paper, this is
typical behaviour for both upper and lower bound solutions. Lower bounds are calculated using the
stresses at each integration point in the model. It may be difficult to satisfy yield at integration
points in these regions, and so the lower bound solution will be dictated by these points. The
solution may require a number of increments to redistribute the stress to surrounding points and
bring all stresses below the yield stress. The upper bounds are based on energy integrals over the
volume of the structure, which dilutes the effects of stress concentrations allowing upper bounds to
converge in fewer increments than lower bounds.
Considering point D, the upper bound solution shows the convergence behaviour alluded to in the
accompanying paper and described by O. Barrera et al (2009) where initial rapid convergence then
slows as it approaches the least upper bound. In the vast majority of cases, convergence of upper
bounds tend to be much quicker than lower bounds, but the complex nature of the solution with
many thousands of integration points and significant levels of cyclic loading can mean that other
convergence behaviours could be seen in addition to those seen by O. Barrera et al (2009). The
presence of a convergent lower bound means that for any upper bound convergence behaviour that
