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 and Historical Review ............................................................................. 1
1.1
Bree Diagram ..................................................................................................... 2
1.2
Linear Temperature Difference .......................................................................... 4
2. Theory .......................................................................................................................... 5
2.1
Discussion .......................................................................................................... 5
2.2
Conduction in a Hollow Cylinder ...................................................................... 6
2.3
Forced Convection Inside a Hollow Cylinder .................................................... 7
2.4
ASME Requirements.......................................................................................... 8
2.5
2.4.1
Thermal Ratcheting ASME Code Requirements ................................... 8
2.4.2
Linear Regression Calculation ............................................................... 8
Numerical FEA Methods ................................................................................... 9
3. Results and Discussion .............................................................................................. 11
3.1
ABAQUS Analysis Inputs ............................................................................... 11
3.1.1
Boundary Conditions............................................................................ 11
3.2
Calculation of Convective Heat Transfer Coefficient ...................................... 19
3.3
Thermal Analysis Results................................................................................. 22
3.4
Stress Analysis Results..................................................................................... 26
4. Conclusions ................................................................................................................ 30
5. References .................................................................................................................. 32
ii
6. Appendix A, Program Files ....................................................................................... 33
7. Appendix B, ABAQUS Card Definitions .................................................................. 34
7.1
Discussion ........................................................................................................ 34
7.2
Thermal Analysis ABAQUS File ..................................................................... 34
7.3
7.2.1
Node Section ........................................................................................ 34
7.2.2
Elements Section .................................................................................. 34
7.2.3
Analysis Information Section ............................................................... 35
Stress Analysis ABAQUS File ......................................................................... 38
7.3.1
Analysis Information Section ............................................................... 38
iii
LIST OF TABLES
Table 1: Pipe Size Dimensions from Table A-6 of [8] ................................................................. 11
Table 2: Heat Transfer Analysis Material Properties for Alloy N06600 [1] ................................ 14
Table 3: Structural Analysis Material Properties for Alloy N06600 [1] ...................................... 15
Table 4: Water Properties from Table A-3 of [8] ......................................................................... 16
Table 5: Thermal Transient Temperature vs Time ....................................................................... 17
Table 6: Tabular Calculation of h, Hot Flow ................................................................................ 19
Table 7: Tabular Calculation of h, Cold Flow .............................................................................. 20
Table 8: Calculation of Maximum Negative ∆T1 ......................................................................... 23
Table 9: Calculation of Maximum Positive ∆T1 ........................................................................... 24
Table 10: Program Files ................................................................................................................ 33
iv
LIST OF FIGURES
Figure 1: Bree’s Shakedown Diagram [3], [4]................................................................................ 2
Figure 2: Illustration of Temperature Profile 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 ...................................................................................................... 13
Figure 5: T vs time for one cycle .................................................................................................. 17
Figure 6: T vs time for 20 cycles .................................................................................................. 18
Figure 7: h vs T for 500 gpm Hot Flow ........................................................................................ 20
Figure 8: h vs T for 500 gpm Cold Flow ...................................................................................... 21
Figure 9: Thermal Analysis Line .................................................................................................. 22
Figure 10: ∆T1 vs time .................................................................................................................. 25
Figure 11: Hoop Stress vs Hoop Strain for 1000, 2000, and 3000 psi ......................................... 26
Figure 12: Hoop Stress vs Displacement for 1000, 2000, and 3000 psi ....................................... 27
Figure 13: Plastic Hoop Strain vs time, 1000 to 3000 psi in 1000 psi Increments ....................... 28
Figure 14: Plastic Hoop Strain vs time, 1000 to 2000 psi in 100 psi Increments ......................... 29
Figure 15: Pressure for Onset of Thermal Ratcheting .................................................................. 30
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
D
Mean diameter
in
di
Inner diameter
in
Do
Outer diameter
in
∆T1
Linear through-wall temperature difference
°F
E
Young’s Modulus
psi
h
Convective heat transfer coefficient
BTU/in2/s/°F
k
Thermal conductivity
BTU/in/s/°F
L
Length from flow entry region
in
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
Pressure stress
psi
σt
Thermal stress
psi
σy
Yield strength
psi

Kinematic viscosity
ft2/s
Poisson’s ratio
none
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, popularized by the
work of Bree, occurs due to severe pressure and thermal stresses and is a low-cycle failure mode.
Bree used simple tube geometry in his initial study of thermal ratcheting.
This project extends Bree’s work by analyzing thermal ratcheting for more complex geometry.
The results of this project predict the onset of thermal ratcheting for valve nozzle geometry with
the use of FEA methods.
This report documents a numerical FEA method for predicting the onset of ratcheting for more
complex geometry with the use of ABAQUS and compares the results to current analytical
methods used in the ASME commercial code.
FEA calculation methodology in the computer program was applied to 3” NPS Schedule 80
piping connected to typical valve nozzle geometry. The thermal ratcheting analysis involved the
creation of two analyses, a heat transfer analysis and a structural elastic-plastic analysis which
imports the heat transfer analysis output.
The heat transfer analysis calculated temperature versus time at each node of the ABAQUS
model based on the thermal conductivity, specific heat, and density. The structural analysis then
calculated the stress due to the unequal temperature distribution and thermal expansion coupled
with an internal pressure. The structural models used an elastic, perfectly plastic material
assumption where the material does not experience strain hardening.
ix
1. Introduction and Historical Review
Nuclear power plants 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 difference 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.
The secondary stress due to through-wall temperature differences, specifically the difference
assuming an equivalent linear temperature distribution, is the focus of this report. This is
supported by the ASME requirement [1] which solely uses the equivalent linear temperature
difference as the major factor for predicting thermal ratcheting. Mean thermal expansion and
contraction of the piping result in moments which bend the piping and create secondary stress,
however, these effects are not considered for this report.
The previously discussed loads combined with large pressure stresses 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].
Thermal ratcheting failure in nuclear systems 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 stresses can be very high. The diagram predicted the stress
combinations necessary for plastic strains to accumulate in piping and pressure vessels.
Bree analyzed a condition in which pressure builds up in nuclear fuel cans due to off gassing of
fission materials. Combined with the pressure is thermal stress due to through-wall temperature
differences which are present during reactor operation, but not present when the reactor is cold.
1
This cyclic thermal load causes yielding of the cladding material and plastic strain, maintaining
stress at the yield strength [3]. Residual stresses may cause more plastic strain when the plant
cools back down. Therefore, both cooldown and heatup can result in plastic strain accumulation
to fatigue failure.
The prevention of fatigue failure is the purpose for thermal ratcheting
requirements in the ASME commercial code.
1.1 Bree Diagram
Figure 1: Bree’s Shakedown Diagram [3], [4]
2
Figure 1 is Bree’s shakedown diagram, Figure 3 of [3], for non-work hardening material and
constant yield strength  y with respect to temperature. The diagram is a plot of pressure stress
versus thermal stress, normalized to the yield strength. The following paragraphs describe the
different regions of material behavior.
E is the purely elastic region where no plastic strain occurs. This is bounded by the sum of
pressure and thermal stress set equal to the yield strength. S1 and S2 are the plastic shakedown
regions where plastic strain initially occurs but then the pipe settles into a purely elastic response.
It is seen that for pressure less than half of yield, the shakedown region is defined by a thermal
stress less than twice of the yield strength.
P is the plastic stability region where plastic strain will cycle between the maximum and
minimum stresses, but will not continue to accumulate to failure, and lastly, R1 and R2 are the
ratcheting regions where the combination of pressure and thermal stresses are sufficient to result
in eventual failure of the structure.
The X axis of Figure 1 is equal to the pressure stress over the yield strength. For hoop stress due
to internal pressure in a cylinder, the stress can be calculated with a thin-walled approximation
resulting in  p 
PD
which is divided by the material yield strength  y at the average bulk
2t w
fluid temperature of the thermal transient.
The Y axis of Figure 1 is equal to the maximum thermal stress range due to a through-wall
temperature difference over the yield strength. The stress resulting from a linear through-wall
temperature difference is  t 
ET1
[1], [3] where v is Poisson’s ratio.  t is divided by the
21  v 
material yield strength  y , taken at the average bulk fluid temperature of the thermal transient.
3
1.2 Linear Temperature Difference
The thermal discontinuity that Bree considered was a linear temperature difference through the
wall of the piping. The profile of temperature, as illustrated in Figure 2, is the sum of the mean
temperature T, the linear temperature difference V, and the surface temperature difference. V is
equal to ∆T1 in the thermal stress equation and is defined as the range of the temperature
difference between the inside and outside surface of the pipe assuming an equivalent linear
temperature distribution [1].
Figure 2: Illustration of Temperature Profile from Figure NB-3653.2(b)-1 of [1]
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 temperature difference creates
thermal bending stresses that lead to ratcheting failure. The surface temperature difference
creates surface stresses which results in crack initiation and fatigue crack failure.
4
2. Theory
2.1 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].
Pressures and severe temperature differences are limited such that the structure does not enter the
ratcheting regime.
Pressure is a primary stress that does not reduce when strain occurs, but will advance to ductile
failure. Thermal stresses due to through-wall temperature differences are secondary stresses that
do 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. Hardening is modeled in ABAQUS with isotropic hardening by default. Yielding is
governed by the Von Mises yield surface in ABAQUS.
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.
5
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
linear through-wall difference of temperature, ∆T1, be known. The following sections will
describe the calculation of ∆T1.
2.2 Conduction in a Hollow Cylinder
The equation governing transient heat transfer through the wall of 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] 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, which makes
this a reasonable assumption.
6
The initial temperature of the pipe and the temperature as a function of time at the inside radius
are needed to solve 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.
2.3 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 [7].
Nu  0.023 Re 0.8 Pr n
where Nu 
[2]
vdi
hd i
, Re 
, Pr is the Prandtl number [8], n is 0.4 for the fluid cooling the pipe
k

and 0.3 for the fluid heating the pipe, k is for the fluid, and v in the numerator of the equation for
the Reynold’s number is the bulk velocity of the fluid inside of the cylinder. All properties are at
bulk fluid temperature. The qualifications for Equation [2] is that 0.7 ≤ Pr ≤ 160, Re > 10000,
and L/D>10. By inspection, the water properties from Table 4 satisfy the requirement for Pr. Re
is satisfied based on the problem parameters. L/D 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 and velocity 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.
7
2.4 ASME Requirements
2.4.1 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 and y’
correspond to the Bree diagram axes x and y, respectively.
2.4.2 Linear Regression Calculation
From the ASME code [1], ΔT1 is defined as
“[The] absolute value of the range of the temperature difference between the temperature
of the outside surface To and the temperature of the inside surface Ti of the piping product
assuming moment generating equivalent linear temperature distribution, °F”
The equivalent linear temperature distribution at each time increment is calculated with a linear
regression of the temperatures through the wall. The ∆T1 temperature difference is then the
difference in temperature from the inside to the outside surface for the linear regression.
A rough approximation for ΔT1 would be to use the difference in temperature of the inside and
outside surfaces, however, this would overestimate ΔT1 by including surface effects of
temperature. The requirements for thermal ratcheting do not include surface effects, therefore it
is appropriate to use the linear regression results.
The linear regression equation is in the form T  A  Bx where x is the distance through the
wall, A is A  T  B x , and B is
8
 x  x T  T 
B
 x  x 
n
i 1
i
i
2
n
i 1
i
[4]
where 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 .
2.5 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. The program then uses the metal
conductivity, density, and specific heat to calculate the temperatures throughout the model.
These temperatures are used to calculate the linear temperature difference ∆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 the ratio of cross sectional area of the fluid over the metal.
Pend
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 pressure stress versus time and the second curve is thermal secondary
stress versus time where the thermal stress is due to the temperature difference through the pipe
wall.
9
Figure 3: Stress vs time from page 2 of [6]
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 Appendix B, ABAQUS Card Definitions. The ABAQUS stress
analysis can use nonlinear FEA methods for calculating large plastic strains; however this
restricts the output of the total strain.
10
3. Results and Discussion
Section 3 details the inputs and results of the thermal and stress analysis as well as the
calculation of the convective heat transfer coefficient.
3.1 ABAQUS Analysis Inputs
This section presents the dimensions and material properties entered into the ABAQUS input
files. For additional information about the ABAQUS input file, see Appendix B, ABAQUS Card
Definitions. The student version of ABAQUS limits the user to 1000 nodes per model. In order
to conserve the number of nodes, modeling is done axi-symmetrically. 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 calculation involving a three-dimensional model could take minutes
or hours to complete.
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
3.1.1 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
11
end is constrained to axially displace equally at all nodes along the radius, simulating the
attaching pipe, by the use of constraints equating the displacements as described in Section 7.3.1.
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.
12
Line of
Symmetry
Remains parallel to x axis,
simulating attaching pipe
Pipe Length = 10.0”
Tangent Length = 0.5”
Length = 2.0”
Length = 4.0”
Restrained Axially
Figure 4: Valve Nozzle Model
13
Table 2 and Table 3 detail 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
Table 2: Heat Transfer Analysis Material Properties for Alloy N06600 [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
14
Density
ρ (lb/in.3)
0.30
Table 3: Structural Analysis Material Properties for Alloy N06600 [1]
Temperature
T (°F)
70
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
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
15
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 4 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 3.
Table 4: 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
16
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 5 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 5.
Table 5: 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
17
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
18
2000
3.2 Calculation of Convective Heat Transfer Coefficient
Table 6 and Table 7 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. The computed values of the
convective heat transfer coefficient for both hot and cold flows are graphed in Figure 7
and Figure 8 respectively.
Table 6: Tabular Calculation of h, Hot Flow
T
(°F)
70
100
150
200
250
300
350
400
450
500
550
600
Flow
(gpm)
500
Velocity
(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
19
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 7: Tabular Calculation of h, Cold Flow
T
(°F)
600
550
500
450
400
350
300
250
200
150
100
70
Flow
(gpm)
500
Velocity
(in/s)
Re
291.44
4284102
4222460
4047738
3786593
3452482
3105407
2667827
2181866
1721179
1230444
793137.8
553700
Pr
h
(BTU/in2/s/°F)
Nu
1.09
0.93
0.87
0.876
0.927
1.02
1.18
1.45
1.88
2.74
4.52
6.82
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 would not be well represented in
ABAQUS by a linear ramp from the starting temperature to the end temperature due to
the quadratic curvature of h vs T; 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
20
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
21
0
h (BTU/in^2/s/°F)
0.011
3.3 Thermal Analysis Results
The temperature versus time of each node along an analysis line from the thermal
analysis file was exported into MS Excel in order to apply a linear regression fit per
Section 2.4.2. The result of the linear regression fit was the ∆T1 temperature difference
which is defined by an assumed linear temperature distribution through the wall. The
line of analysis for calculating ∆T1 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
22
The full range of ∆T1 is the difference between the maximum positive and negative
differences, ∆T1.
The assumed sign convention of ∆T1 is negative for a higher
temperature at the inside surface (hot flow) and positive for a lower temperature at the
inside surface (cold flow).
Table 8 provides the data taken from the ABAQUS thermal file for the first time of
maximum negative ∆T1 (-383 °F). Node, time, and temperature are from the ABAQUS
output and the remaining cells are calculated in accordance with Section 2.4.2.
Table 8: Calculation of Maximum Negative ∆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
23
Table 9 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 2.4.2.
Table 9: Calculation of Maximum Positive ∆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
24
(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 calculation of ∆T1 versus time for the
first thermal cycle. For details of the calculation, see Section 2.4.2.
∆T1
∆T1 vs time
time
400
55.4, 316
300
Temperature (°F)
200
100
0
-100
0
10
20
40
30
50
60
-200
∆T1( (°F)
F)
∆T1
-300
5.3, -383
-400
-500
time (s)
Figure 10: ∆T1 vs time
25
70
3.4 Stress Analysis Results
For the assumed thermal transient, multiple internal pressures were analyzed in order to
predict the pressure at 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.
Figure 11: Hoop Stress vs Hoop Strain for 1000, 2000, and 3000 psi
26
Figure 12 plots the hoop stress (psi) versus displacement (in) for iterations of 1000 psi
(blue), 2000 psi (green), and 3000 psi (yellow). The 1000 and 2000 psi iterations show
an initial large change in displacement, followed by a settling into a mostly elastic
response. The 3000 psi iteration shows an initial large change in displacement followed
by a steady increase per cycle due to the thermal ratcheting. The 1000 and 2000 psi
iterations are difficult to judge whether ratcheting is occurring due to the scale.
Figure 12: Hoop Stress vs Displacement for 1000, 2000, and 3000 psi
27
Figure 13 plots the cumulative 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 ratcheting has begun for the 2000 and 3000 psi
iterations, and that somewhere between 1000 and 2000 psi is the pressure for the onset
of thermal ratcheting.
Cumulative 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
2500
time (s)
Figure 13: Plastic Hoop Strain vs time, 1000 to 3000 psi in 1000 psi Increments
28
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
29
4. 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 ∆T1 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
30
From the equations in Section 2.4.1 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 PDE
.
2t w y2
2 * 0.3 * 26980 2
=1323 psi, less than

0.7T1 DE 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.
31
5. 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.
Unpublished doctoral thesis, North Carolina State University, Raleigh, North
Carolina
[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.
32
6. Appendix A, Program Files
Table 10 lists the program files which were used in the creation of this report.
Table 10: 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
33
7. Appendix B, ABAQUS Card Definitions
7.1 Discussion
This section describes the analysis file structure used in ABAQUS. 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. Section 7.2 details the thermal analysis model. Section 7.3 details the
changes from the thermal model for the stress analysis.
7.2 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.
7.2.1
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.
7.2.2
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
34
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.
7.2.3
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.
7.2.3.1 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.
*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.
35
*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.
7.2.3.2 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.
*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
36
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.
7.2.3.3 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.
37
*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.
7.3 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.
7.3.1
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.
38
*EQUATION tells ABAQUS that the following lines will have the number of variables
for an equation followed in the next line by node, displacement direction, and
multiplication factor, repeating to define all variables and setting them equal to zero. To
equate axial displacement for two nodes, two variables are used in the equation, and a
multiplication factor of -1.0 is applied to one displacement, u n1, DOF1  u n 2, DOF 2  0
The variable information is given as the first node n1, degree of freedom DOF1,
multiplication factor 1.0, second node n2, degree of freedom DOF2, and multiplication
factor -1.0.
This is repeated for all nodes along the pipe end resulting in telling
ABAQUS that the nodes on the free end of the pipe can move in the axial direction but
must all have the same axial 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
39
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.
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.
40