Master’s Project Proposal:
An Investigation into the use of FEA methods for the
prediction of Thermal Stress Ratcheting
By: Huse, Stephen
09/03/14
Abstract:
The selected topic for my master’s project is the prediction for the onset of thermal
ratcheting through the use of numerical FEA methods. Thermal ratcheting results from
a combination of severe pressure and thermal stresses and results in fatigue failure at
low cycles. 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 the onset of thermal ratcheting 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 input of geometry, thermal properties, mechanical properties, and
load conditions. The second part of the project will broaden the focus to include thermal
ratcheting analysis for a valve nozzle.
Background:
Thermal ratcheting is a low cycle fatigue mechanism that accumulates plastic strain with
each stress cycle. Structures such as nuclear piping systems are subjected to the type
of low cycle, high stress conditions that result in plastic strain and thermal ratcheting.
The current ASME analysis requirements for thermal ratcheting are designed to prevent
ratcheting from starting. Primary and secondary stresses are limited such that the
structure does not enter the ratcheting regime. Primary stresses are loads such as
deadweight and pressure that do not reduce when strain occurs, but will continue until
ductile failure occurs. Secondary stresses are loads such as thermal expansion
moments and thermal gradient stress that will reduce when strain occurs. In the design
of piping systems, it is important to give special attention to locations prone to stress
concentrations such as welds or geometry discontinuities.
Accurate modeling of accumulated plastic strain due to ratcheting is hindered by many
complex and hard to model factors. Material hardening and cyclic stress history are two
of the major factors that are difficult to accurately model. Kinematic hardening, the
increase in strength past yield, occurs in many materials and continues as loading
increases until the ultimate tensile strength at which point the material experiences
ductile failure. A linear kinematic hardening model will tend to under predict thermal
ratcheting accumulated strains while a nonlinear kinematic hardening model will tend to
either over predict ratcheting strains or predict elastic shakedown. For this project, a
simplifying assumption will be the use of elastic, perfectly plastic material properties.
The history of stress cycles is not always well known and can affect the analysis. The
earlier that large stress cycles are applied the earlier that failure of the material will
occur. However, because cyclic history may not be well known, the worst case loading
history is usually assumed for analyses.
The failure mode of thermal ratcheting was made popular by 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 fuel applications where thermal gradient
stresses can be very high. The diagram analytically predicted the stress combinations
necessary for plastic strains to accumulate in piping and pressure vessels.
Bree analyzed a condition in which pressure built up in nuclear fuel cans due to off
gassing of fission materials. Combined with this pressure was a thermal gradient
present when the reactor was operating, but zero when the plant was cold. This cyclic
thermal load can cause yielding of the material to maintain stress at the yield strength.
When the plant cools down, the residual stress may cause further plastic strains.
Therefore, both cooldown and heatup can produce plastic strains that accumulate until
fatigue failure occurs. The prevention of this fatigue failure is the reason for thermal
ratcheting checks in commercial code.
Figure 1: Bree’s Shakedown Diagram from Reference (c) figure 3 for non-work
hardening material with yield stress Sy unchanged by changes in temperature
For figure (1), the X axis is primary stress over yield stress. For primary stress due to
internal pressure in a cylinder, the stress can be calculated with a thin walled
PDo
PDo
approximation resulting in
which leads to X _ axis 
where P is pressure, Do
2t y
2t
is outer diameter, t is pipe wall thickness, and σy is yield strength at the average fluid
temperature of the transient.
The Y axis is half of the maximum secondary stress range due to a linear thermal
gradient over yield stress. The stress resulting from a linear through wall temperature
gradient is
t 
ET1
where σt is the maximum elastic thermal stress, E is Young’s
21  v 
modulus, α is the mean thermal coefficient of linear expansion, ∆T1 is the linear
temperature gradient across the wall, and v is Poisson’s ratio. This leads to
Y_axis =
ET1
where the terms are the same as before.
21  v  y
The different 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
The mean temperature causes no local stresses to occur, but will cause thermal
expansion moments in the constrained run of piping. The surface temperature gradient
will cause surface stresses that may lead to crack initiation and fatigue crack failure.
The linearized or average temperature difference will cause stresses that lead to
thermal ratcheting failure.
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 and pressure vessels. When cold water from outside the 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 on the inside of the pipe. 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 on the inside of
the pipe.
Related to the local through wall gradients and stresses is the gross thermal expansion
and contraction of the piping system from changes in the mean temperature of the
piping resulting in potentially high secondary moments which bend the piping and create
stress. However, for this project the secondary stress will be limited to stress due to
through-wall temperature gradients.
The previously discussed loads combined with large primary stresses due to high
pressures can result in plastic strain and thermal ratcheting. This project will attempt to
predict the onset of thermal ratcheting by the use of the FEA software, ABAQUS,
Reference (a).
The student version of ABAQUS limits the user to 1000 nodes per model. In order to
conserve the number of nodes, modeling will be done axisymmetrically. For bending
moments, a three-dimensional half-symmetry model would be created, but this involves
many nodes and the full version of ABAQUS. The slight disadvantage to threedimensional modeling is increased computational times whereas an axisymmetric
model may take seconds, a complex three-dimensional model could take minutes or
hours to complete.
For the pipe model, the following inputs will be used:
Table 1: Pipe Geometry from Reference (i), Table A-6
Description
Geometry
Outer Diameter
Thickness
Inner Diameter
Length
Value
3 NPS, Schedule 80
3.5
0.3
2.9
10.0
Units
inches
inches
inches
inches
Table 2: Material Properties for NiCrFe, seamless pipe and tube, Spec SB-167, Alloy N06600, size ≤ 5 inches
from Reference (e), Section II, Part D, Material Properties, Tables Y-1, TE-4, TCD, TM-4, and PRD
Temperature
T (°F)
70
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
Conductivity
k
(10-3 BTU/s/in/°F)
0.199
0.201
0.206
0.211
0.215
0.222
0.227
0.234
0.238
0.245
0.250
0.257
0.262
0.269
0.273
0.280
0.287
0.292
Specific Heat
Cp (BTU/lb)
0.108
0.109
0.111
0.113
0.114
0.116
0.116
0.118
0.118
0.120
0.121
0.122
0.123
0.125
0.126
0.128
0.130
0.131
Density, ρ
(lb/in.3)
Young’s
Modulus, E (106
psi)
Poisson’s
Ratio, v
31.0
30.3
29.9
29.4
0.30
29.0
28.6
28.1
27.6
27.1
0.31
Mean Coefficient of
Thermal Expansion
α (10-6 in./in./°F)
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.6
7.7
7.8
7.9
7.9
8.0
8.0
8.1
8.2
Yield
Stress
σy (ksi)
30.0
30.0
29.2
28.6
28.0
27.4
26.8
26.2
25.7
25.2
24.7
24.3
23.9
23.5
23.2
22.9
22.6
22.3
Note: Conductivity was converted from units of BTU/hr/ft/°F by dividing by (3600*12)
Specific heat was calculated from the equation cp=k/TD/ρ where TD is thermal diffusivity from Table TCD, and ρ is
converted to units of lb/ft3 = 0.3*123=518.4
Table 3: Water Properties from Reference (i), Table A-3
Temperature
T (°F)
32
40
50
60
70
80
90
100
150
200
250
300
350
400
450
500
550
600
Conductivity
K (BTU/hr/ft/°F)
0.319
0.325
0.332
0.34
0.347
0.353
0.359
0.364
0.384
0.394
0.396
0.395
0.391
0.381
0.367
0.349
0.325
0.292
Kinetic Viscosity
v (ft2/s)
1.93
1.67
1.4
1.22
1.06
0.93
0.825
0.74
0.477
0.341
0.269
0.22
0.189
0.17
0.155
0.145
0.139
0.137
Density
ρ (lb/ft3)
62.4
62.4
62.4
62.3
62.3
62.2
62.1
62
61.2
60.1
58.8
57.3
55.6
53.6
51.6
49
45.9
42.4
Prandtl Number
13.7
11.6
9.55
8.03
6.82
5.89
5.13
4.52
2.74
1.88
1.45
1.18
1.02
0.927
0.876
0.87
0.93
1.09
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 is dependent on time is by
numerical methods. Also, a common analysis assumption is that the outside of the pipe
is perfectly insulated, having a convective heat loss of zero resulting in a slightly higher
∆T1 which is conservative. This simplifying assumption is good based on comparing
heat transfer rates of convection between water and metal, the conductivity of metal,
and the convection between metal and air. The result of the comparison is that the
metal conduction and fluid to metal forced convection is over an order of magnitude
greater than metal to air under free convection. Additionally, much of the hot piping in
proximity to manned areas have insulation for safety. If the piping has insulation on its
outside then the assumption of no heat loss is further reinforced.
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
hd
k
where Nu is the Nusselt number equal to
Re is the Reynolds number equal to
vd

,
,
Pr is the Prandtl number,
n is 0.4 for the fluid cooling the pipe and 0.3 for the fluid heating the pipe,
h is the convective 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 kinematic viscosity,
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 analysis step is repeated. The second ABAQUS model
will then have constant pressure applied and will read the varying thermal cycles
resulting in a stress loadset as shown in figure 3 where the top is primary stress and the
bottom is secondary stress. For the project, the loading conditions will be iterated to
initiate ratcheting.
Figure 3: Stress versus time from Reference (h), Pg 2
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 be elastic-perfectly plastic.
The effects of reduced integration elements and convergence studies may 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. The student version of
ABAQUS is limited to 1000 node models, which limits the modeling to axisymmetric
models.
Expected Outcomes / Objectives:
Calculate onset of thermal ratcheting 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 valve nozzle
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).
g) Bari, S. (2001). Constitutive Modeling for Cyclic Plasticity and Ratcheting.
h) Cailletaud, G. (2003). UTMIS Course 2003 – Stress Calculations for Fatigue - 6.
Ratcheting. Ecole des Mines de Paris: Centre des Materiaux.
i) Kreith, F. (1965). Principles of heat transfer. Second edition. Scranton, Pa.:
International Textbook.