An Investigation into the use of FEA methods
For the Prediction of Thermal Stress Ratcheting
by
Stephen Charles Huse
A Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Engineering Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
December, 2014
i
CONTENTS
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
LIST OF SYMBOLS ........................................................................................................ vi
ACKNOWLEDGMENT.................................................................................................. vii
KEYWORDS .................................................................................................................. viii
ABSTRACT ...................................................................................................................... ix
1. Introduction .................................................................................................................. 1
2. Historical Review......................................................................................................... 2
2.1
Bree Diagram ..................................................................................................... 2
2.2
Linear Thermal Discontinuity ............................................................................ 4
3. Theory .......................................................................................................................... 6
3.1
Discussion .......................................................................................................... 6
3.2
Conduction in a Hollow Cylinder ...................................................................... 7
3.3
Forced Convection Inside a Hollow Cylinder .................................................... 8
3.4
ASME Requirements.......................................................................................... 9
3.5
3.4.1
Thermal Ratcheting ASME Code Requirements ................................... 9
3.4.2
Linear Regression Calculation ............................................................... 9
Numerical FEA Methods ................................................................................. 10
4. Method of Procedure.................................................................................................. 12
4.1
Discussion ........................................................................................................ 12
4.2
Thermal Analysis ABAQUS File ..................................................................... 12
4.3
4.2.1
Node Section ........................................................................................ 12
4.2.2
Elements Section .................................................................................. 13
4.2.3
Analysis Information Section ............................................................... 13
Stress Analysis ABAQUS File ......................................................................... 16
ii
4.3.1
4.4
Analysis Information Section ............................................................... 16
ABAQUS Analysis Inputs ............................................................................... 18
4.4.1
Boundary Conditions............................................................................ 18
5. Results ........................................................................................................................ 25
5.1
Calculation of Convective Heat Transfer Coefficient ...................................... 25
5.2
Thermal Analysis Results................................................................................. 28
5.3
Stress Analysis Results..................................................................................... 32
6. Discussion and Conclusions ...................................................................................... 37
7. References .................................................................................................................. 39
Appendix A, Program Files ............................................................................................. 40
iii
LIST OF TABLES
Table 1: Pipe Size Dimensions from Table A-6 of [8] ................................................................. 18
Table 2: Material Properties for Alloy N06600 from [1] ............................................................. 21
Table 3: Water Properties from Table A-3 of [8] ......................................................................... 22
Table 4: Thermal Transient Temperature vs Time ....................................................................... 23
Table 5: Tabular Calculation of h, Hot Flow ................................................................................ 25
Table 6: Tabular Calculation of h, Cold Flow .............................................................................. 26
Table 7: Calculation of Minimum ∆T1 ......................................................................................... 29
Table 8: Calculation of Maximum ∆T1......................................................................................... 30
Table 9: Program Files .................................................................................................................. 40
iv
LIST OF FIGURES
Figure 1: Bree’s Shakedown Diagram [3], [4]................................................................................ 3
Figure 2: Illustration of Temperature Gradients from Figure NB-3653.2(b)-1 of [1] .................... 4
Figure 3: Stress vs time from page 2 of [6] .................................................................................. 10
Figure 4: Valve Nozzle Model ...................................................................................................... 20
Figure 5: T vs time for one cycle .................................................................................................. 23
Figure 6: T vs time for 20 cycles .................................................................................................. 24
Figure 7: h vs T for 500 gpm Hot Flow ........................................................................................ 26
Figure 8: h vs T for 500 gpm Cold Flow ...................................................................................... 27
Figure 9: Thermal Analysis Line .................................................................................................. 28
Figure 10: ∆T1 vs time .................................................................................................................. 31
Figure 11: Hoop Stress vs Hoop Strain for 1000, 2000, and 3000 psi ......................................... 33
Figure 12: Hoop Stress vs Displacement for 1000, 2000, and 3000 psi ....................................... 34
Figure 13: Plastic Hoop Strain vs time ......................................................................................... 35
Figure 14: Plastic Hoop Strain vs time, 1000 to 2000 psi in 100 psi Increments ......................... 36
Figure 15: Pressure for Onset of Thermal Ratcheting .................................................................. 37
v
LIST OF SYMBOLS
Symbol
Description
Units
A
Surface area
in2
α
Mean coefficient of thermal expansion
in/in/°F
cp
Specific heat
BTU/lb
Distance from flow entry region in diameter
lengths
diameters
D
Mean diameter
in
di
Inner diameter
in
Do
Outer diameter
in
∆T1
Linear through-wall temperature gradient
°F
∆T2
Surface temperature gradient
°F
E
Young’s Modulus
psi
h
Convective heat transfer coefficient
BTU/in2/s/°F
k
Thermal conductivity
BTU/in/s/°F
Nusselt number
none
Pressure
psi
Prandtl number
none
Radius
in
Reynold’s number
none
ρ
Density
lb/in3
T
Temperature
°F
t
Time
s
tw
Wall thickness
in
σp
Primary stress
psi
σt
Thermal secondary stress
psi
σy
Yield strength
psi

Kinematic viscosity
ft2/s
Poisson’s ratio
none
D/L
Nu
P
Pr
r
Re

vi
ACKNOWLEDGMENT
I would like to thank my wife, Sarah, for being supportive and helpful during the long hours
spent on this project. Thanks also to my fellow workers at Electric Boat for guidance and thanks
to Ernesto for being a great advisor.
vii
KEYWORDS

ABAQUS

Convection

Elastic Plastic

Fatigue

FEA

Heat Transfer

Piping

Ratcheting

Valve
viii
ABSTRACT
The prediction of the onset of thermal ratcheting is a necessary step in the design of nuclear
piping and pressure vessels. The failure mechanism of thermal ratcheting occurs due to severe
pressure and thermal stresses and is a low-cycle failure mode.
This report documents a numerical FEA method for predicting the onset of ratcheting with the
use of ABAQUS [2] and compares the results to the current analytical methods used in the
ASME commercial code [1].
FEA calculation methodology in the computer program ABAQUS [2] was applied to 3” NPS
Schedule 80 piping connected to typical valve nozzle geometry. The thermal ratcheting analysis
involved the creation of two ABAQUS analyses, a heat transfer analysis and a structural elasticplastic analysis which imports the heat transfer analysis output.
ix
1. Introduction
Nuclear power plants, in particular, are susceptible to high thermal ratcheting strains due to rapid
increases and decreases in the temperature of the water flowing through the piping and pressure
vessels. When cold water from outside of the plant quickly flows through hot piping, the inside
of the pipe thermally contracts while the outside circumference 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 back through the same piping
resulting in the inside of the pipe thermally expanding while the outside remains cold creating a
compressive thermal stress on the inside of the pipe.
Related to the local through-wall temperature gradient is the gross thermal expansion and
contraction of the piping arrangement due to changes in the mean temperature of the piping.
Constrained expansion results in secondary moments which bend the piping and create stress.
The arrangement of the pipes and support structure greatly influences this expansion moment.
For this report, however, the effects of mean thermal expansion of the piping system are not
included in the secondary stress. The secondary stress from through-wall temperature gradients
will be focused on as the linear temperature gradient is the largest factor in thermal ratcheting as
seen in the ASME requirements [1].
The previously discussed loads combined with large primary stresses due to high pressures result
in plastic strain and thermal ratcheting. This report documents a method for predicting the onset
of thermal ratcheting by the use of the FEA software, ABAQUS [2].
1
2. Historical Review
Thermal ratcheting failure was popularized by the work of J. Bree [3], [4]. In his articles, 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 predicted the stress combinations
necessary for plastic strains to accumulate in piping and pressure vessels.
2.1 Bree Diagram
Bree analyzed a condition in which pressure builds up in nuclear fuel cans due to off gassing of
fission materials. Combined with the pressure was a thermal gradient that was present when the
reactor was operating, but not present when the reactor was cold. This cyclic thermal load
causes yielding of the material, maintaining stress at the yield strength [3]. When the plant cools
down, the residual stress may cause further plastic strains. Therefore, both cooldown and heatup
can result in plastic deformation that accumulates until fatigue failure occurs. The prevention of
this fatigue failure is the purpose for thermal ratcheting requirements in the ASME commercial
code.
2
Figure 1: Bree’s Shakedown Diagram [3], [4]
Figure 1 is Bree’s shakedown diagram, Figure 3 of [3], for non-work hardening material and
constant yield strength σy with respect to temperature. The different regions of Figure 1 are as
follows: E is the purely 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 accumulate to
3
failure, and lastly, R1 and R2 are the ratcheting regions where the combination of primary and
secondary stresses are sufficient to result in eventual failure of the structure.
The X axis of Figure 1 is equal to the primary stress over the yield strength. For primary stress
due to internal pressure in a cylinder, the stress can be calculated with a thin-walled
approximation resulting in  p 
PD
PD
which leads to X _ axis 
where the material
2t w y
2t w
yield strength is taken at the average bulk fluid temperature of the thermal transient.
The Y axis of Figure 1 is equal to the maximum secondary stress range due to a linear thermal
gradient over the yield strength. The stress resulting from a linear through wall temperature
gradient is  t 
ET1
ET1
Y
_
axis

[1], [3] which leads to
.
21  v  y
21  v 
2.2 Linear Thermal Discontinuity
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 (also known as ∆T1), and the surface
temperature gradient, ∆T2.
Figure 2: Illustration of Temperature Gradients from Figure NB-3653.2(b)-1 of [1]
4
Changes in mean temperature do not cause local stresses to occur, but do cause thermal
expansion moments in a constrained run of piping. The linearized or average temperature
difference creates thermal stresses that lead to ratcheting failure.
The surface temperature
gradient creates surface stresses which results in crack initiation and fatigue crack failure.
5
3. Theory
3.1 Discussion
Thermal ratcheting is a low cycle fatigue mechanism that accumulates plastic strain with each
stress cycle [5]. 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. Current ASME analysis
requirements in Section III NB-3653.7 are designed to prevent ratcheting from starting [1].
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 [5].
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 after
yielding, occurs in many materials and continues as loading increases until the ultimate tensile
strength is reached at which point the material experiences ductile failure. An isotropic 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 [6]. For this report, an elastic perfectly plastic material model is
assumed.
The stress history is not always well known and can affect the analysis. The earlier that larger
stress cycles are applied the earlier that failure of the material will occur. However, because
cyclic history is usually unknown, the worst case loading history is assumed for design analyses.
Thermal ratcheting strain is calculated using the current requirements of the ASME Boiler and
pressure vessel code [1] Section III, Division 1 – NB-3653.7. As input, the code requires that the
6
linear through-wall gradient of temperature, ∆T1, be known.
The following sections will
describe the calculation of ∆T1.
3.2 Conduction in a Hollow Cylinder
The general heat transfer partial differential equation for a hollow cylinder is
1   T 
T
 kr
  c p
r r  r 
t
[1]
where temperature, T, is time and location dependent and material properties are for the cylinder.
For steady-state conditions, the right hand side of Equation [1] goes to zero and simplifies to
1   T 
 kr
  0 . Multiplying by r, dividing by k (independent of r for isotropic materials) and
r r  r 
integrating gives r
T
 A , where A is the first integration constant. Dividing by r gives
r
T A
 , which integrates to T r   A ln r   B . Boundary conditions are then used to solve for
r r
A and B.
For non steady state conditions, such as when temperature varies with time, the easiest way to
solve Equation [1] for ∆T1 is by numerical methods. Also, a common and conservative analysis
assumption is that the outside of the pipe is perfectly insulated, having convective heat loss of
zero resulting in a slightly higher ∆T1. This simplifying assumption is reasonable based on the
heat transfer rate for free convection between metal and air versus the rate for forced convection
between water and metal, and the rate of thermal conduction in metals. The result of this
comparison is that heat transfer for metal conduction and forced convection is much faster than
metal to air heat transfer in free convection. Additionally, much of the hot piping in proximity to
manned areas is insulated for safety, further reducing heat loss to the environment.
7
The initial temperature of the pipe and the temperature versus time of the inside radius are
needed for solving Equation [1]. The temperature of the inside of the cylinder depends on the
energy transferred due to forced convection from the fluid flowing inside of the cylinder.
3.3 Forced Convection Inside a Hollow Cylinder
The convective heat transfer coefficient, h, for turbulent flow inside a cylinder is calculated with
the Dittus-Boelter equation which is given in Equation (3.2.99) of [7].
Nu  0.023 Re 0.8 Pr n
where Nu 
[2]
vdi
hd i
, Re 
, n is 0.4 for the fluid cooling the pipe and 0.3 for the fluid heating
k

the pipe, k is for the fluid, and v in the numerator of the equation for the Reynold’s number is
velocity. All properties are at bulk fluid temperature. The qualifications for Equation [2] is that
0.7 ≤ Pr ≤ 160, Re > 10000, and D/L>10. By inspection, the water properties from Table 3
satisfy the requirement for Pr. Re is satisfied based on the problem parameters. D/L is the
measure of lengths in diameters from the entry region. It is assumed that the location of analysis
is more than 10 diameters from the entry region.
Knowing the fluid temperature versus time and fluid flow rate versus time, the convective heat
transfer coefficient, h, can be calculated. The convective heat transfer coefficient is then used to
calculate the heat transferred through convection to the piping, Q  hAT where A is the area
of heat transfer and ∆T is the temperature difference between the bulk fluid temperature and the
inside surface of the cylinder. Heat transferred by convection is based on the surface area, the
difference in temperature between the bulk fluid and inside surface of the pipe, and the
convective heat transfer coefficient, h.
8
3.4 ASME Requirements
3.4.1 Thermal Ratcheting ASME Code Requirements
ASME Section III, Division 1 – NB-3653.7 [1] requirements for thermal ratcheting is that the
range of ∆T1 between any two transients is
T1 
y ' y
0.7 E
C4
[3]
where C4 is an equation constant (equal to 1.0 for NiCrFe material), E and α are taken at the
ambient temperature of 70 °F, σy is at the average fluid temperature of the transients, and y’=1/X
for 0 < X < 0.5 and y’=4*(1-X) for 0.5 < X < 1.0 from ASME NB-3222.5, where X is the Bree
diagram axis, X _ axis 
PD
.
2t w y
3.4.2 Linear Regression Calculation
ΔT1 at each time increment is found by calculating the linear regression of the temperatures
through the wall. Once the linear regression fit is known, ∆T1 is the temperature difference from
the inside of the pipe to the outside. The line equation is in the form T  A  Bx where x is the x
coordinate, A is A  T  B x , B is
 x  x T  T 
 x  x 
n
B
i 1
i
i
2
n
i 1
i
[4]
, and a horizontal bar over a variable denotes the average of the variable through the wall. The
temperature difference from the inside of the pipe to the outside is then B times the wall
thickness or T1  Bt w .
9
3.5 Numerical FEA Methods
ABAQUS accepts the convective heat transfer coefficient and bulk fluid temperature as input to
calculate the heat transferred between the fluid and the piping. ABAQUS also calculates the
temperatures necessary to calculate ∆T1 through the numerical analysis of Equation [1]. To
model cyclic thermal cycles, the analysis temperatures are increased and decreased repeatedly.
The stress analysis ABAQUS file then imports the varying temperatures at each node and applies
a constant pressure. The pressure is applied to the inside of the pipe at the nominal value and at
the ends of the pipe due to end effects. The end effect pressure is equal to the nominal pressure
times
Pend
the
ratio
of
cross
sectional
area
of
the
fluid
over
the
metal.
PnomDi2
Pnom Di2

 2
.
4 Do2 / 4  Di2 / 4
Do  Di2

 

The constant pressure and varying thermal cycles result in a stress load set similar to Figure 3
where the first curve is primary stress versus time and the second curve is secondary stress
versus time.
Figure 3: Stress vs time from page 2 of [6]
10
The geometry, material properties, and pressure films for the analysis files were created in the
ABAQUS pre-processor software, HYPERMESH. Load conditions are added by direct editing
of the .inp file as described in Section 4. The ABAQUS stress analysis can use nonlinear FEA
methods for calculating large plastic strains; however this restricts the output of the total strain.
11
4. Method of Procedure
4.1 Discussion
This section describes the analysis files and the inputs to the thermal and stress analyses which
are provided in the Appendices. The student version of ABAQUS limits the user to 1000 nodes
per model. In order to conserve the number of nodes, modeling is done axisymmetrically.
ABAQUS axisymmetric analysis, by default, defines the Y axis as the axis of symmetry equating
R,Z,θ with X,Y,Z respectively. Bending moments were not calculated for this analysis since a
three-dimensional model would be needed, requiring the full version of ABAQUS. The slight
disadvantage to three-dimensional modeling is the increased computational times whereas an
axisymmetric model may take seconds, a complex three-dimensional model could take minutes
or hours to complete.
Section 4.2 details the thermal analysis model. Section 4.3 details the changes from the thermal
model for the stress analysis. Section 4.4 details the inputs to ABAQUS.
4.2 Thermal Analysis ABAQUS File
The ABAQUS file is separated into three main sections which are nodes, elements, and analysis
information. The majority of manual editing is done in the analysis information section of the
ABAQUS input file. ** is a delimiter in the files that tells ABAQUS to ignore the line, which is
useful for commenting or blank space.
4.2.1 Node Section
The first section defines node locations. *NODE, NSET=ALL denotes the start of the node
section. *NODE tells ABAQUS that the following lines will have a node number then node
coordinates based on analysis type. Since the analysis is 2D axisymmetric, two coordinates are
given: radial (X) and longitudinal (Y). NSET=ALL creates a set of node numbers. Appending
the *NODE card with NSET=ALL places all nodes into the set ALL which is then used for
assigning the initial temperature of all the nodes.
12
4.2.2 Elements Section
The second section is initiated with the card *ELEMENT, TYPE=DCAX8, ELSET=Pipe.
*ELEMENT tells ABAQUS that the following lines will have an element number then nodes
defining the element. These are automatically created by HYPERMESH in the correct order.
TYPE=DCAX8 defines the element type as D for diffusive heat transfer, C for non-twisting, AX
for axisymmetric, and 8 for 8-noded quadratic second order element. ELSET=Pipe creates a set
of elements under the name “Pipe”. Appending the *ELEMENT card with ELSET places all
elements defined in the card into the set which is then used for assigning material properties to
the elements.
4.2.3 Analysis Information Section
The third section is where most editing of ABAQUS input files occurs. While it is laborious to
manually enter node and element information, the analysis section is much faster to manually
edit rather than navigating through a user interface that was designed to run every type of
analysis that ABAQUS is capable of.
The following is one of the many ways to order and build the analysis section.
4.2.3.1 Material Definitions
*MATERIAL, NAME=N06600 tells ABAQUS that the following material property cards apply
to the material named N06600.
*CONDUCTIVITY, TYPE=ISO tells ABAQUS that the following lines will have thermal
conductivity in BTU/s/in/°F then the temperature in °F at which each applies. ISO denotes the
property applies equally in all directions.
*SPECIFIC HEAT tells ABAQUS that the following lines will have specific heat in BTU/lb then
the temperature in °F at which each applies.
13
*DENSITY tells ABAQUS that the following line will have density in lb/in3 at 70 °F. For
material property cards with only one line, the property is applied to all temperatures.
*ELASTIC, TYPE = ISOTROPIC tells ABAQUS that the following lines contain Young’s
modulus in psi then Poisson’s ratio then the temperature in °F at which each applies.
ISOTROPIC denotes the property applies equally in all directions.
*EXPANSION, ZERO = 70.0, TYPE = ISO tells ABAQUS that the following lines contain the
mean coefficient of thermal expansion in in/in/°F then the temperature in °F at which each
applies. ZERO defines the ambient temperature at which no thermal expansion occurs. ISO
denotes similar properties in all directions.
*PLASTIC tells ABAQUS that the following lines will have stress in psi then plastic strain then
the temperature in °F at which each applies. A plastic strain of 0.0 denotes the yield strength at
which plastic deformation begins. Entering plastic strain of 0.0 at each temperature creates an
elastic perfectly plastic material definition.
*SOLID SECTION, ELSET=Pipe, MATERIAL=N06600 places the material properties labeled
N06600 onto the named set of elements. The line following this card is the attribute line, for
which 1.0 is for default attributes.
4.2.3.2 Transient Information
*ELSET, ELSET=P2 creates a set of elements from the following lines and labels the set as P2.
This is used to define a set of elements that border the inside edge and have the second edge of
the element at the inside of the piping. An easy way to find this set of elements is by defining a
pressure on the inside of the model in HYPERMESH.
*INITIAL CONDITIONS, TYPE=TEMPERATURE tells ABAQUS the initial temperature of
the nodes. In the following lines is the node set, ALL, then the initial temperature, 70 °F.
14
*AMPLITUDE, NAME=TEMPAMP1, VALUE=ABSOLUTE tells ABAQUS that the following
lines have time in seconds then the temperature in °F, repeating up to 4 times per line. This
inputs the temperature versus time curve for use in the calculation of energy transferred in
convection. Multiple curves were used to define the full transient in order to minimize run time
of the stress analysis.
*AMPLITUDE, NAME=FILMAMP1, VALUE=ABSOLUTE is the same card type as for the
temperature curves but is instead inputting the convective heat transfer coefficient versus time.
Similar to the temperature curve, the film curve is divided into multiple curves.
*INCLUDE,INPUT=5cycles.th.inp tells ABAQUS to insert the lines found in the 5cycle.th file.
This card is used to reduce the repetition of lines in the main file by running 5 thermal cycles
with one line of code.
4.2.3.3 Step Definition in 5cycles.th
In order to reduce the repetition of multiple lines in the main ABAQUS thermal analysis file,
lines were added in a separate file. After properties and thermal inputs are defined in the main
file, the analysis steps are imported from this file.
*STEP, INC=5000 initiates a step with up to 5000 discrete analysis increments. The cards
between this and the following *END STEP card will define a step of the analysis. Multiple
steps are entered to reduce run times of the analysis.
*HEAT TRANSFER, DELTMX=20.0 tells ABAQUS that the following line defines the initial
time increment, the length of time to run the step for, the minimum time step size, the maximum
time step size, and steady state option where 0.0 denotes no steady state analysis. DELTMX
defines the maximum difference in temperature allowed between adjacent nodes. The ABAQUS
program will use the DELTMX control to automatically increase or decrease the time of each
increment.
15
*FILM, AMPLITUDE=TEMPAMP1, FILM AMPLITUDE=FILMAMP1 tells ABAQUS that
the following lines apply the time versus temperature and time versus heat transfer coefficient
curves to the elements by element set, edge of element, temperature (dummy value since
AMPLITUDE=TEMPAMP is appending the card), and film coefficient (dummy value since
FILM AMPLITUDE=FILMAMP is appending the card).
The lines *NODE FILE, FREQUENCY=1 | NT | *EL FILE | COORD, TEMP | *EL
FILE,POSITION=NODES, FREQUENCY=1 | TEMP create a binary data file of temperatures at
each time step which are then imported into the stress analysis later.
*END STEP defines the completion of the analysis step. The lines from *STEP to *END STEP
are then repeated to define the full transient and to create five thermal cycles.
4.3 Stress Analysis ABAQUS File
The stress analysis file has the same geometry and material properties as the thermal file, but the
analysis information and element type are different. The element type is CAX8 for structural
analysis instead of DCAX8.
4.3.1 Analysis Information Section
Other than the material property cards, the analysis information section for the stress analysis is
different from the thermal analysis section as detailed below.
*BOUNDARY tells ABAQUS that the following lines will have a node then degree of freedom
(2 is Y) then prescribed displacement where 0.0 is no deflection, essentially anchoring the node
in the selected degree of freedom.
*EQUATION tells ABAQUS that the following lines will have the number of variables for an
equation for node displacements multiplied by a factor and equal to zero, alternating with the
next line which inputs the variable information. The variable information is given as the first
16
node, degree of freedom, multiplication factor, second node, degree of freedom, and
multiplication factor. This card is used to tell ABAQUS that the nodes on the free end of the
pipe can move in the Y direction but must all have the same Y displacements.
*AMPLITUDE, NAME=PRESS,VALUE=ABSOLUTE defines the time versus pressure curve
in the following lines. This value controls the pressure on the model and is iterated to induce
ratcheting. PRESS is defining the pressure on the inside of the pipe. PRESE is defining the
longitudinal pressure due to end effects on the pipe.
*ELSET, ELSET=P1E creates a set of elements from the following lines and labels the set as
P1E. This is used to define a set of elements that border the top edge of the pipe and has the first
edge of the element at the end. This set will have the PRESE amplitude pressure applied.
*STEP initiates the load set.
When INC is not included, the default number of analysis
increments allowed is up to 100.
*STATIC, DIRECT tells ABAQUS to discretize the stress analysis by the input in the following
line which gives the time of each increment and the total time.
*TEMPERATURE, FILE=valve.th, BSTEP=1, BINC=1,ESTEP=2,EINC=1 tells ABAQUS to
import temperatures from the thermal file from step 1, increment 1 to step 2, increment 1 up to
the amount of time requested, 10 seconds. Therefore the next analysis step does not duplicate an
analysis time. Modifying the thermal file usually requires modifying this card as well.
*DLOAD, AMPLITUDE=PRESS tells ABAQUS that the following lines have the following
information: element, edge of element, and dummy value for pressure as the appended amplitude
card for PRESS overwrites these values. *DLOAD, AMPLITUDE=PRESE is the same card
except that it applies the end pressure effects.
17
The analysis steps are repeated until all thermal analysis steps are used. The use of many time
steps allows for the varying of time increments to speed up the run time of the total analysis.
4.4 ABAQUS Analysis Inputs
This section presents the information that is entered into the ABAQUS input files. Table 1
details the geometry of the piping which is connected to the valve nozzle.
Table 1: Pipe Size Dimensions from Table A-6 of [8]
Description
Value
Units
Geometry
Outer Diameter, Do
Thickness, tw
Inner Diameter, di
Mean Diameter, D
Length
3 NPS, Schedule 80
3.5
0.3
2.9
3.2
10.0
in
in
in
in
in
4.4.1 Boundary Conditions
The geometry for the valve nozzle is detailed in
Figure 4. The valve end is anchored axially while the pipe end is allowed to thermally grow,
simulating a flexible piping system. The pipe end is constrained to axially displace equally at all
points along the radius, simulating the attaching pipe.
If a piping system is arranged as a straight run from anchor to anchor, then the boundary
conditions would be modeled as axially constrained at both ends. However, this would produce
enormous compressive stress, and so is avoided in practice. Common practice is to introduce
flexibility into the arrangement with bends and stress loops in order to allow the piping to
thermally grow.
18
Remains parallel to x axis,
simulating attaching pipe
Line of
Symmetry
Pipe Length = 10.0”
Tangent Length = 0.5”
Length = 2.0”
Length = 4.0”
Restrained Axially
19
Figure 4: Valve Nozzle Model
Table 2 details the material properties entered into ABAQUS for the piping and the valve nozzle.
The material is assumed to be a Nickel Chromium Iron composition, commonly known as
Inconel. The specific material properties taken are for NiCrFe, Alloy N06600 seamless pipe and
tube, Spec SB-167 for sizes ≤ 5 inches from Reference [1], Section II, Part D, Material
Properties, Tables Y-1, TE-4, TCD, TM-4, and PRD. Conductivity was converted from units of
BTU/hr/ft/°F by dividing by (3600*12). Also, 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
20
Table 2: Material Properties for Alloy N06600 from [1]
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
Mean Coefficient of
Thermal Expansion
α (10-6 in./in./°F)
0.31
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
31.0
30.3
29.9
29.4
0.30
29.0
28.6
28.1
27.6
27.1
21
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
Table 3 details the water properties used to calculate the convective heat transfer
coefficient that is input into ABAQUS. The results of this calculation are provided in
Section 5.
Table 3: Water Properties from Table A-3 of [8]
Temperature
T (°F)
32
40
50
60
70
80
90
100
150
200
250
300
350
400
450
500
550
600
Conductivity
Kinetic Viscosity
Density
Prandtl
K (BTU/hr/ft/°F)
-5
ρ (lb/ft )
Number
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
2
v x 10 (ft /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
3
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
22
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
Table 4 provides the assumed temperature versus time data used for the thermal
transient. This transient is then repeated twenty times in order to calculate if ratcheting
is occurring as seen in Figure 6. Figure 5 graphs the information entered in Table 4.
Table 4: Thermal Transient Temperature vs Time
t
(s)
0
5
50
55
100
T
(°F)
70
600
600
70
70
T vs time
700
Temperature (°F)
600
500
400
300
T (°F)
200
100
0
-10
0
10
20
30
40
time (s)
Figure 5: T vs time for one cycle
23
50
60
T vs time
700
Temperature (°F)
600
500
400
300
T (°F)
200
100
0
0
500
1000
1500
time (s)
Figure 6: T vs time for 20 cycles
24
2000
5. Results
Section 5 details the results of the thermal and stress analysis as well as the calculation
of the convective heat transfer coefficient.
5.1 Calculation of Convective Heat Transfer Coefficient
Table 5 and Table 6 provide the calculated values for the convective heat transfer
coefficient with an assumed flow rate of 500 gallons per minute, gpm. Flow rate was
converted from gpm to in/s using the conversions 231 in3 = 1 gallon, 60 sec = 1 min, and
by dividing by the cross-sectional area, πdi2/4=6.605 in2. Table 5 is graphed in Figure 7
and Table 6 is graphed in Figure 8.
Table 5: Tabular Calculation of h, Hot Flow
T
(°F)
70
100
150
200
250
300
350
400
450
500
550
600
Flow
(gpm)
500
Flow
(in/s)
Re
291.44
553700
793137.8
1230444
1721179
2181866
2667827
3105407
3452482
3786593
4047738
4222460
4284102
Pr
6.82
4.52
2.74
1.88
1.45
1.18
1.02
0.927
0.876
0.87
0.93
1.09
25
Nu
1609
1896
2318
2708
3028
3344
3614
3823
4046
4259
4495
4769
h
(BTU/in2/s/°F)
0.00446
0.00551
0.0071
0.00852
0.00957
0.01054
0.01128
0.01163
0.01185
0.01187
0.01166
0.01112
Table 6: Tabular Calculation of h, Cold Flow
T
(°F)
Flow
(gpm)
600
550
500
450
400
350
300
250
200
150
100
70
500
Flow
(in/s)
Re
291.44
4284102
4222460
4047738
3786593
3452482
3105407
2667827
2181866
1721179
1230444
793137.8
553700
Pr
1.09
0.93
0.87
0.876
0.927
1.02
1.18
1.45
1.88
2.74
4.52
6.82
Nu
h
(BTU/in2/s/°F)
4810
4462
4201
3993
3794
3621
3399
3143
2884
2564
2204
1949
0.011212
0.011576
0.011702
0.011698
0.011537
0.011302
0.010718
0.009935
0.009071
0.007858
0.006404
0.005399
It is seen in Figure 7 and Figure 8 that the coefficient is not well represented by only the
start and end points; therefore, each data point is entered into ABAQUS for the
amplitude card containing the curve of film coefficient versus time.
h vs T for 500 gpm Hot Flow
0.012
0.011
h (BTU/in^2/s/°F)
0.01
0.009
0.008
0.007
0.006
h (BTU/in^2/s/°F)
0.005
0.004
0
100
200
300
T (°F)
400
Figure 7: h vs T for 500 gpm Hot Flow
26
500
600
h vs T for 500 gpm Cold Flow
0.012
0.01
0.009
0.008
0.007
0.006
h (BTU/in^2/s/°F)
0.005
0.004
600
500
400
300
200
100
T (°F)
Figure 8: h vs T for 500 gpm Cold Flow
27
0
h (BTU/in^2/s/°F)
0.011
5.2 Thermal Analysis Results
The results of the thermal analysis file was exported into MS Excel for an analysis slice
in order to calculate the linear temperature gradient. The line of analysis that ∆T1 is
calculated from, is at the transition from the pipe outer diameter to the 30° slope as
shown in Figure 9.
Line of ∆T1
analysis
Node 145
Figure 9: Thermal Analysis Line
28
Table 7 provides the data taken from the ABAQUS thermal file for the first time of
minimum ∆T1 (-383 °F). Node, time, and temperature are from the ABAQUS output
and the remaining cells are calculated in accordance with Section 3.4.2.
Table 7: Calculation of Minimum ∆T1
time
(s)
5.31
Average
Sum
B*tw
node
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
T
x
(°F)
(in)
544.94
499.58
455.87
414.51
376.14
340.82
308.79
280.15
254.77
232.65
213.78
198.01
185.22
175.28
168.17
163.86
162.38
292.64
(Ti-Tm)(xi-xm)
1.45
1.47
1.49
1.51
1.53
1.54
1.56
1.58
1.60
1.62
1.64
1.66
1.68
1.69
1.71
1.73
1.75
1.60
(xi-xm)2
-37.85
-27.16
-18.36
-11.43
-6.26
-2.71
-0.61
0.23
0.00
-1.12
-2.96
-5.32
-8.06
-11.00
-14.00
-16.90
-19.54
0.0225
0.0172
0.0127
0.0088
0.0056
0.0032
0.0014
0.0004
0.0000
0.0004
0.0014
0.0032
0.0056
0.0088
0.0127
0.0172
0.0225
-183.05
0.14
-183.05/0.14*0.3=
-382.84
29
Table 8 provides the data taken from the ABAQUS thermal file for the first time of
maximum ∆T1 (316 °F). Node, time, and temperature are from the ABAQUS output and
the remaining cells are calculated in accordance with Section 3.4.2.
Table 8: Calculation of Maximum ∆T1
time
(s)
55.35
Average
Sum
B*tw
node
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
T
x
(°F)
(in)
162.38
205.924
246.177
282.868
315.847
345.366
371.56
394.543
414.549
431.814
446.487
458.7
468.542
476.129
481.525
484.768
485.863
380.77
1.45
1.47
1.49
1.51
1.53
1.54
1.56
1.58
1.60
1.62
1.64
1.66
1.68
1.69
1.71
1.73
1.75
1.60
30
(Ti-Tm)(xi-xm)
(xi-xm)2
19.54
11.38
5.23
0.92
-1.74
-2.97
-2.96
-1.91
0.00
2.61
5.77
9.34
13.19
17.20
21.25
25.22
28.98
0.0225
0.0172
0.0127
0.0088
0.0056
0.0032
0.0014
0.0004
0.0000
0.0004
0.0014
0.0032
0.0056
0.0088
0.0127
0.0172
0.0225
151.05
151.05/0.14*0.3=
0.14
315.92
Figure 10 plots the result of the linear regression of temperature through the wall of the
valve nozzle transition versus time for the first thermal cycle. Negative ∆T1 indicates a
decrease in temperature from the inside edge to the outside edge.
∆T1 vs time
400
55.4, 316
300
Temperature (°F)
200
100
0
-100
0
10
20
30
40
50
-200
∆T1 ( F)
-300
5.3, -383
-400
-500
time (s)
Figure 10: ∆T1 vs time
31
60
70
5.3 Stress Analysis Results
For the assumed thermal transient, the structural analysis was iterated to predict the
onset of ratcheting. Ratcheting was analyzed at Node 145, as shown in Figure 9, which
is at the outside of the thermal analysis slice shown in Figure 9. Figure 11 plots the hoop
stress (psi) versus hoop strain for 1000 psi (blue), 2000 psi (green), and 3000 psi
(yellow). Ratcheting is easily seen in the 3000 psi iteration, and is slightly seen in the
2000 psi iteration. In the 1000 psi iteration, it is difficult to judge whether ratcheting is
occurring.
32
Figure 11: Hoop Stress vs Hoop Strain for 1000, 2000, and 3000 psi
Figure 12 plots the hoop stress (psi) versus displacement (in) for iterations of 1000 psi
(blue), 2000 psi (green), and 3000 psi (yellow). All three iterations show shakedown
occurring. Ratcheting is easily seen in the 3000 psi iteration. The 1000 and 2000 psi
iterations are difficult to judge whether ratcheting is occurring from Figure 12.
33
Figure 12: Hoop Stress vs Displacement for 1000, 2000, and 3000 psi
Figure 13 plots the plastic hoop strain versus time for iterations of 1000 psi, 2000 psi,
and 3000 psi. This metric easily shows iterations with accumulating plastic strain. From
Figure 13 it is seen that the pressure for the onset of thermal ratcheting is between 1000
and 2000 psi.
34
Plastic Strain vs time
0.012
0.01
Strain
0.008
1000 psi
0.006
2000 psi
0.004
3000 psi
0.002
0
0
500
1000
1500
2000
time (s)
Figure 13: Plastic Hoop Strain vs time
35
2500
Figure 14 plots the plastic hoop strain versus time in increments of 100 psi between 1000
and 2000 psi. Up to 1600 psi, the plastic strain levels off, with each steady state strain
slightly different due to the increase in pressure. From 1700 psi and up, the plastic strain
is seen to accumulate.
Plastic Hoop Strain vs time
0.0025
0.002
1000 psi
1100 psi
1200 psi
0.0015
Strain
1300 psi
1400 psi
1500 psi
0.001
1600 psi
1700 psi
1800 psi
1900 psi
0.0005
2000 psi
0
0
500
1000
1500
2000
time (s)
Figure 14: Plastic Hoop Strain vs time, 1000 to 2000 psi in 100 psi Increments
36
6. Discussion and Conclusions
The prediction of the onset of thermal ratcheting with the use of ABAQUS is possible
for complex geometry in order to facilitate the design of piping and pressure vessels.
The thermal and structural analysis models successfully calculated a pressure limit at
which plastic strain begins to accumulate. Maintaining design pressures below the
calculated pressure results will prevent the failure mechanism of thermal ratcheting from
occurring. The thermal models also facilitated the calculation of thermal through wall
temperature gradients for comparison to the ASME code limits.
Figure 15 plots the difference in strain at the end time for each increment from the
previous, lower increment, as seen on the right of Figure 14. From Figure 15, it is seen
that ABAQUS predicts a pressure for the onset of accumulated plastic strain of just over
1500 psi.
Difference in Plastic Hoop Strain from
Previous Pressure Iteration after 20
cycles vs Pressure
4.50E-04
4.00E-04
Difference in Strain
3.50E-04
3.00E-04
2.50E-04
2.00E-04
Difference in
Plastic Hoop Strain
1.50E-04
1.00E-04
1600
5.00E-05
0.00E+00
1000
1500
2000
Pressure (psi)
Figure 15: Pressure for Onset of Thermal Ratcheting
37
From the equations in Section 3.4.1 the pressure when ratcheting begins based on ASME
code can be solved for. First, the equation for y’ is selected. The pressure for onset of
yield is likely less than 2000 psi and the yield strength is 26980 psi, linearly interpolated
at the average transient temperature of 335 °F, X=
PD
2000 * 3.2

 0.40.
2t w y 2 * .3 * 26980
For X<0.5, y’=1/X. Substituting for X and y’ in equation [3] gives T1 
Solving for P gives P 
2t w y2
0.7T1 DE

2t w y2
0.7 PDE
.
2 * 0.3 * 26980 2
=1323 psi, less than
0.7 * 699 * 3.2 * 31.0 * 6.8
the 1500 psi result from ABAQUS.
Using the yield strength at 70 °F of 30000 psi gives a higher pressure, 1631 psi, which
is slightly over the ABAQUS results. The results from ABAQUS provide realistic
results which are, as expected, slightly higher than the result given for the ASME code.
38
7. References
[1] 2010 ASME boiler & pressure vessel code an international code. (2010).
New York, NY: American Society of Mechanical Engineers.
[2] ABAQUS (Version 6.13) [Software]. (2013). Providence, RI: Dassault
Systèmes Simulia Corp.
[3] 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.
[4] Bree, J. (1989). Plastic deformation of a closed tube due to interaction of
pressure stresses and cyclic thermal stresses. International Journal of
Mechanical Sciences, 865-892.
[5] Bari, S. (2001). Constitutive Modeling for Cyclic Plasticity and Ratcheting.
[6] Cailletaud, G. (2003). UTMIS Course 2003 – Stress Calculations for Fatigue
- 6. Ratcheting. Ecole des Mines de Paris: Centre des Materiaux.
[7] Kreith, F. (2000). The CRC handbook of thermal engineering. Boca Raton,
Fla.: CRC Press.
[8] Kreith, F. (1965). Principles of heat transfer. Second edition. Scranton, Pa.:
International Textbook.
39
Appendix A, Program Files
Table 9 lists the program files which were used in the creation of this report.
Table 9: Program Files
File
Description
Valve.th.inp
ABAQUS Standard heat transfer analysis
5cycles.th.inp
5 thermal cycles imported into valve.th.inp
Valve10.st.inp
ABAQUS Standard structural analysis, 1000 psi iteration
Valve11.st.inp
ABAQUS Standard structural analysis, 1100 psi iteration
Valve12.st.inp
ABAQUS Standard structural analysis, 1200 psi iteration
Valve13.st.inp
ABAQUS Standard structural analysis, 1300 psi iteration
Valve14.st.inp
ABAQUS Standard structural analysis, 1400 psi iteration
Valve15.st.inp
ABAQUS Standard structural analysis, 1500 psi iteration
Valve16.st.inp
ABAQUS Standard structural analysis, 1600 psi iteration
Valve17.st.inp
ABAQUS Standard structural analysis, 1700 psi iteration
Valve18.st.inp
ABAQUS Standard structural analysis, 1800 psi iteration
Valve19.st.inp
ABAQUS Standard structural analysis, 1900 psi iteration
Valve20.st.inp
ABAQUS Standard structural analysis, 2000 psi iteration
Valve30.st.inp
Report
Calculations.xls
ABAQUS Standard structural analysis, 3000 psi iteration
MS Excel Workbook for calculating h, ∆T1, and plotting
results
Python program for sequentially running ABAQUS files using
the command “abaqus python 1st.py”
1st.py
40