Master’s Project Proposal:
An Investigation into the use of FEA methods for the
prediction of Thermal Stress Ratcheting
By: Huse, Stephen
08/30/2014
Abstract:
The selected topic for my master’s project is the analysis of thermal ratcheting through
the use of numerical FEA methods. Thermal ratcheting results from a combination of
severe pressure, moment, and thermal stresses. This project will compare the current
analytical methods used in the ASME commercial code for thermal ratcheting to the
FEA results of this project, with the goal being to be able to more accurately predict
accumulated ratcheting strain for complex geometry.
The first part of the project will focus on development of the FEA calculation methods
using the computer program ABAQUS, Reference (a) for simple pipe geometry. This
FEA calculation will require two models, one for heat transfer analysis and the other for
elastic-plastic analysis (which uses the results from the heat transfer analysis). The
models will include pipe cross section, thermal properties, mechanical properties, and
load conditions. The second part of the project will broaden the focus to include thermal
ratcheting analysis for a tee fitting’s more complex geometry.
Background:
The failure mode of thermal ratcheting was initially considered in the work of Bree,
reference (c). In his article, he proposed what is now known as the Bree diagram or
shakedown diagram, as shown in Figure (1). The Bree diagram was created from
analyses of thin walled tubing in nuclear applications where thermal gradient stresses
can be very high. The diagram analytically predicted the conditions necessary for
plastic strains to accumulate in piping and pressure vessels.
Figure 1: Bree’s Shakedown Diagram from Reference (b) figure 3.1,
for axisymmetric shells under constant internal pressure
and cyclical thermal gradient stress
For figure (1), the X axis is primary stress over yield stress. For primary stress due to
internal pressure the stress can be calculated with a thin walled approximation resulting
PDo
PDo
in
which leads to X 
where P is pressure, Do is outer diameter, t is pipe
2tS y
2t
wall thickness, and Sy is yield strength at the average fluid temperature of the transient.
The Y axis is half of the maximum secondary stress range over yield stress Y 
SE
.
2S y
Primary stress is loads such as deadweight and pressure that do not reduce when
strain occurs, but will continue until ductile failure occurs. Secondary stress is loads
such as thermal expansion moments and thermal gradient stress that will reduce when
strain occurs.
The labeled regions in Figure (1) are as follows: E is the pure elastic region where no
plastic strain occurs, S1 and S2 are the plastic shakedown regions where initially,
plastic strain accumulates but then tapers off as the pipe settles into a purely elastic
response, P is the plastic stability region where plastic strain will cycle between the
maximum and minimum stresses, but will not continue to failure, and lastly, R1 and R2
are the ratcheting regions where the combination of primary and secondary stresses
result in eventual failure of the structure.
The thermal discontinuity that Bree considered was a linearized temperature gradient
through the wall of the piping. Temperature gradients, as illustrated in Figure (2), are
the sum of the mean temperature, T, the linearized temperature gradient, V (more
commonly written as ∆T1), and the surface temperature gradient, ∆T2.
Figure 2: Illustration of temperature gradients from
Reference (e), Figure NB-3653.2(b)-1
Problem Description:
Nuclear power plants in particular are susceptible to high thermal ratcheting strains due
to rapid increases and decreases in the temperature of the bulk water flowing through
piping. When cold water from outside the power plant quickly flows through piping that
was previously hot, the inside of the pipe thermally contracts while the outside diameter
remains hot, causing a through wall temperature gradient resulting in tensile stress.
After the piping cools down, hot water from inside the plant can quickly flow through the
piping resulting in the inside of the pipe thermally expanding while the outside
temporarily remains cold. This temperature inequality or gradient creates a
compressive thermal stress.
Related to the local through wall gradients and stresses is the gross thermal expansion
and contraction of the piping system resulting in potentially high secondary moments
which will bend the piping creating stress. For this project, the response of external
factors such as secondary moment will be assumed.
The previously discussed loads combined with large primary stresses due to high
pressures result in plastic strain and thermal ratcheting. This project will attempt to
predict the amount of accumulated plastic strain due to thermal ratcheting for a given
load set by the use of the FEA software, Abaqus, Reference (a).
Methodology:
Thermal ratcheting strain will be calculated using the current requirements of the ASME
Boiler and pressure vessel code, Reference (e), Section III, Division 1 – NB-3653.7. As
input, this code requires that the linear through wall gradient of temperature, ∆T1, be
calculated.
1   T 
T
 kr
  c p
r r  r 
t
where r is radius, k is thermal conductivity of the cylinder, T is temperature (time and
location dependent), t is time, ρ is density, and cp is specific heat.
The PDE general heat transfer equation for a hollow cylinder is
For steady-state conditions, the right hand side goes to zero and the equation is
1   T 
simplified to
 kr
  0 . Multiplying by r, dividing by k (independent of r for
r r  r 
T
 A , where A is the first integration
isotropic materials) and integrating gives r
r
T A
 , which integrates to T r   A ln r   B .
constant. Dividing by r gives
r r
For non steady state conditions, which covers almost all scenarios, the easiest way to
solve the PDE for the maximum ∆T1 when temperature changes with time is by
numerical methods. Also, a common assumption is that the outside of the pipe is
perfectly insulated, having a convective heat loss of zero. This assumption results in a
slightly higher ∆T1 which is conservative.
Fluid temperature versus time, fluid flow rate versus time, and initial temperature of the
pipe are all needed for solving the PDE. The temperature and flow rate of the fluid are
then used to calculate the heat transferred to the pipe through convection. The heat
transferred by convection is based on the surface area, instantaneous difference in
temperature between the bulk fluid and inside surface of the pipe, and the convective
heat transfer coefficient, h.
The convective heat transfer coefficient, h, for turbulent flow inside a cylinder is
calculated with the Dittus-Boelter equation which is seen in Reference (d), Equation
(3.2.99):
Nu  0.023 Re 0.8 Pr n
where Nu is the Nusselt number equal to
Re is the Reynolds number equal to
Pr is the Prandtl number equal to
,
vd
,

cp
k
hd
k
,
n is 0.4 for the fluid cooling the pipe and 0.3 for the fluid heating the pipe,
h is the heat transfer coefficient,
d is the inner diameter,
k is the thermal conductivity of the fluid,
ρ is density of the fluid,
v is velocity,
 is the dynamic viscosity,
cp is the specific heat of the fluid.
All properties are at bulk fluid temperature, Tb. This equation is then solved for h and
used for the heat transferred to the piping with the equation Q  hATb  Tid  where A is
the surface area and Tid is the temperature of the inner diameter of the pipe.
ABAQUS will accept as input the convective heat transfer coefficient and bulk fluid
temperature to perform numerical analysis of the heat transfer distribution. Then, to
model cyclic thermal cycles, the thermal model is run again. The second ABAQUS
model will then have constant pressure applied and will read the varying thermal cycles
and calculate the accumulative plastic strain.
The geometry and material properties and pressure films for the models will be built in
the ABAQUS pre-processor software, hypermesh. Thermal model conditions will be
added by direct editing of the .inp file. The ABAQUS structural model will invoke
nonlinear FEA methods for calculating large plastic strains. The material properties
applied will elastic-perfectly plastic.
The effects of reduced integration elements and convergence studies should be
considered as proof of model integrity.
Resources Required:
The computing resources required includes the FEA analysis software, ABAQUS,
Reference (a) as well as additional supporting software such as MS word, MS excel,
and HYPERMESH (ABAQUS pre-processor). These softwares are all currently
available for use on two different machines. The input files will be created partly by user
input and partly with input from existing thermal analyses.
Expected Outcomes / Objectives:
Calculate accumulated strain with numerical methods.
Compare the accumulated plastic strains from the numerical method with the analytical
method results (Bree diagram and ASME code).
Provide sufficient description of the ABAQUS input file sections to enable readers to
create a thermal ratcheting input file for ABAQUS.
Milestone List:
Task
Project Proposal
Numerical model of Pipe
First Progress report
Numerical model of Tee
Finish Researching references
Second Progress report
Final Draft
Preliminary final report
Final Report
Deadline
9/12
9/19
9/26
10/3
10/10
10/17
11/7
11/28
12/12
References:
a) ABAQUS (Version 6.13) [Software]. (2013). Providence, RI: Dassault Systèmes
Simulia Corp.
b) Shah, V., Majumdar, S., & Natesan, K. (2003). Review and Assessment of
Codes and Procedures for HTGR Components. Argonne, IL: Argonne National
Laboratory.
c) Bree, J. (1967). Elastic-plastic behaviour of thin tubes subject to internal pressure
and intermittent high-heat fluxes with application to fast nuclear reactor fuel
elements.Journal of Strain Analysis, (2), 226-38.
d) Kreith, F. (2000). The CRC handbook of thermal engineering. Boca Raton, Fla.:
CRC Press.
e) 2010 ASME boiler & pressure vessel code an international code. (2010). New
York, NY: American Society of Mechanical Engineers.
f) Moreton, D., & Ng, H. (1981). The Extension and Verification of the Bree
Diagram. Transactions of the International Conference on Structural Mechanics
in Reactor Technology, L(10/2).