Huse FinalReport

8/17/2019 Huse FinalReport

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A Finite Element Model for the Prediction of

Thermal Ratcheting in a Pipe to Valve Nozzle Connection

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 InstituteHartford, Connecticut

December, 2014


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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

2.4.1 Thermal Ratcheting ASME Code Requirements ................................... 8

2.4.2 Linear Regression Calculation ............................................................... 8

2.5 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


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6. Appendix A, Program Files ....................................................................................... 33

7. Appendix B, ABAQUS Card Definitions .................................................................. 34

7.1 Discussion ........................................................................................................ 34

7.2 Thermal Analysis ABAQUS File ..................................................................... 34

7.2.1 Node Section ........................................................................................ 34

7.2.2 Elements Section .................................................................................. 34

7.2.3 Analysis Information Section ............................................................... 35

7.3 Stress Analysis ABAQUS File ......................................................................... 38

7.3.1 Analysis Information Section ............................................................... 38


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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


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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 at 1000, 1600, and 2000 psi ............................................ 29

Figure 15: Difference in Final Cumulated Plastic Strain vs Pressure ........................................... 30


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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

Nu Nusselt number none

PPressure psi

Pr Prandtl number none

r Radius in

Re 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


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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.


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KEYWORDS

ABAQUS

Convection

Elastic Plastic

Fatigue

FEA

Heat Transfer

Piping

Ratcheting

Valve


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ABSTRACT

This project investigates the thermal ratcheting problem in a complex geometry consisting of 3”

piping connected to typical valve nozzle geometry. The prediction of the onset of thermal

ratcheting is a necessary step in the design of nuclear piping and pressure vessels since failurecan occur by low-cycle fatigue due to severe pressure and thermal stresses. A thermo-

mechanical finite element model was created using ABAQUS for the prediction of the onset of

thermal ratcheting. The results of the finite element analysis were validated by comparison to

those using current analytical methods. 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.


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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 pipethermally 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.


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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]


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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 inw

pt

PD2

which is divided by the material yield strength y at the average bulk

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 v

T E t

12

1 [1], [3] where v is Poisson’s ratio.

t is divided by the

material yield strength y , taken at the average bulk fluid temperature of the thermal transient.


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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 isequal 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.


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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 thatdo 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.


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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,

t

T c

r

T kr

r r p

1 [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

01

r

T kr

r r . Multiplying by r, dividing by k (independent of r for isotropic materials) and

integrating gives Ar

T r

, where A is the first integration constant. Dividing by r gives

r

A

r

T

, which integrates to Br Ar T ln . Boundary conditions are then used to solve for

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.


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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].

n Nu Pr Re023.0 8.0 [2]

wherek hd Nu i ,

ivd Re , Pr is the Prandtl number [8], n is 0.4 for the fluid cooling the pipe

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, T hAQ 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.


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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

417.0

'C

E

yT

y

[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 Bx AT where x is the distance through the

wall, A is x BT A , and B is


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n

i i

n

i ii

x x

T T x x B

1

2

1

[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 w Bt T 1 .

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.

222

22

2

4/4/4 io

inom

io

inomend

D D

D P

D D

D P P

.

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.


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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.


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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 symm etry 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 3 NPS, Schedule 80

Outer Diameter, Do 3.5 in

Thickness, tw 0.3 in

Inner Diameter, di 2.9 in

Mean Diameter, D 3.2 in

Length 10.0 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


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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.


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Figure 4: Valve Nozzle Model

Tangent Length = 0.5”

Length = 2.0”

Length = 4.0”

Pipe Length = 10.0”

Line of

Symmetry

Restrained Axially

Remains parallel to x axis,

simulating attaching pipe


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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 c p=k/TD/ρ where TD is thermal diffusivity from Table

TCD, and ρ is converted to units of lb/ft3 = 0.3*12

3=518.4

Table 2: Heat Transfer Analysis Material Properties for Alloy N06600 [1]

TemperatureT (°F)

Conductivityk

(10-3

BTU/s/in/°F)

Specific Heatcp (BTU/lb)

Densityρ (lb/in.

3)

70 0.199 0.108

0.30

100 0.201 0.109

150 0.206 0.111

200 0.211 0.113

250 0.215 0.114

300 0.222 0.116

350 0.227 0.116

400 0.234 0.118

450 0.238 0.118500 0.245 0.120

550 0.250 0.121

600 0.257 0.122

650 0.262 0.123

700 0.269 0.125

750 0.273 0.126

800 0.280 0.128

850 0.287 0.130

900 0.292 0.131


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Table 3: Structural Analysis Material Properties for Alloy N06600 [1]

TemperatureT (°F)

Densityρ (lb/in.

3)

Young’sModulus

E (106 psi)

Poisson’sRatio

v

Mean Coefficient ofThermal Expansion

α (10-6

in./in./°F)

Yield Stressσy (ksi)

70

0.30

31.0

0.31

6.8 30.0

100 6.9 30.0150 7.0 29.2

200 30.3 7.1 28.6

250 7.2 28.0

300 29.9 7.3 27.4

350 7.4 26.8

400 29.4 7.5 26.2

450 7.6 25.7

500 29.0 7.6 25.2

550 7.7 24.7

600 28.6 7.8 24.3

650 7.9 23.9700 28.1 7.9 23.5

750 8.0 23.2

800 27.6 8.0 22.9

850 8.1 22.6

900 27.1 8.2 22.3


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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)

Conductivity

K (BTU/hr/ft/°F)

Kinetic Viscosity

v x 10-5

(ft2 /s)

Density

ρ (lb/ft3)

Prandtl

Number

32 0.319 1.93 62.4 13.7

40 0.325 1.67 62.4 11.6

50 0.332 1.4 62.4 9.55

60 0.34 1.22 62.3 8.03

70 0.347 1.06 62.3 6.82

80 0.353 0.93 62.2 5.89

90 0.359 0.825 62.1 5.13

100 0.364 0.74 62 4.52150 0.384 0.477 61.2 2.74

200 0.394 0.341 60.1 1.88

250 0.396 0.269 58.8 1.45

300 0.395 0.22 57.3 1.18

350 0.391 0.189 55.6 1.02

400 0.381 0.17 53.6 0.927

450 0.367 0.155 51.6 0.876

500 0.349 0.145 49 0.87

550 0.325 0.139 45.9 0.93

600 0.292 0.137 42.4 1.09


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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)

T(°F)

0 70

5 600

50 600

55 70

100 70

Figure 5: T vs time for one cycle

0

100

200

300

400

500

600

700

-10 0 10 20 30 40 50 60

T e m p

e r a t u r e ( ° F )

time (s)

T vs time

T (°F)


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Figure 6: T vs time for 20 cycles

0

100

200

300

400

500

600

700

0 500 1000 1500 2000

T e m p e r a t u r e ( ° F )

time (s)

T vs time

T (°F)


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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 in

2. 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)

Flow

(gpm)

Velocity

(in/s) Re Pr Nu

h

(BTU/in2 /s/°F)

70

500 291.44

553700 6.82 1609 0.00446

100 793137.8 4.52 1896 0.00551

150 1230444 2.74 2318 0.0071

200 1721179 1.88 2708 0.00852

250 2181866 1.45 3028 0.00957

300 2667827 1.18 3344 0.01054

350 3105407 1.02 3614 0.01128

400 3452482 0.927 3823 0.01163

450 3786593 0.876 4046 0.01185

500 4047738 0.87 4259 0.01187550 4222460 0.93 4495 0.01166

600 4284102 1.09 4769 0.01112


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Table 7: Tabular Calculation of h, Cold Flow

T(°F)

Flow(gpm)

Velocity(in/s)

Re Pr Nuh

(BTU/in2 /s/°F)

600

500 291.44

4284102 1.09 4810 0.011212

550 4222460 0.93 4462 0.011576500 4047738 0.87 4201 0.011702

450 3786593 0.876 3993 0.011698

400 3452482 0.927 3794 0.011537

350 3105407 1.02 3621 0.011302

300 2667827 1.18 3399 0.010718

250 2181866 1.45 3143 0.009935

200 1721179 1.88 2884 0.009071

150 1230444 2.74 2564 0.007858

100 793137.8 4.52 2204 0.006404

70 553700 6.82 1949 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.

Figure 7: h vs T for 500 gpm Hot Flow

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.011

0.012

0 100 200 300 400 500 600

h ( B T U / i n ^ 2 / s / ° F )

T (°F)

h vs T for 500 gpm Hot Flow

h (BTU/in^2/s/°F)


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Figure 8: h vs T for 500 gpm Cold Flow

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.011

0.012

0100200300400500600

h ( B T U / i n ^ 2 / s / ° F )

T (°F)

h vs T for 500 gpm Cold Flow

h (BTU/in^2/s/°F)


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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 differencewhich 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.

Figure 9: Thermal Analysis Line

Line of ∆T1

analysis

Node 145


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23

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) nodeT

(°F)x

(in) (Ti-Tm)(xi-xm) (xi-xm)2

5.31

129 544.94 1.45 -37.85 0.0225

130 499.58 1.47 -27.16 0.0172

131 455.87 1.49 -18.36 0.0127

132 414.51 1.51 -11.43 0.0088

133 376.14 1.53 -6.26 0.0056

134 340.82 1.54 -2.71 0.0032

135 308.79 1.56 -0.61 0.0014

136 280.15 1.58 0.23 0.0004

137 254.77 1.60 0.00 0.0000

138 232.65 1.62 -1.12 0.0004139 213.78 1.64 -2.96 0.0014

140 198.01 1.66 -5.32 0.0032

141 185.22 1.68 -8.06 0.0056

142 175.28 1.69 -11.00 0.0088

143 168.17 1.71 -14.00 0.0127

144 163.86 1.73 -16.90 0.0172

145 162.38 1.75 -19.54 0.0225

Average 292.64 1.60

Sum -183.05 0.14

B*tw -183.05/0.14*0.3= -382.84


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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)node

T

(°F)x

(in) (Ti-Tm)(xi-xm) (xi-xm)

2

55.35

129 162.38 1.45 19.54 0.0225

130 205.924 1.47 11.38 0.0172

131 246.177 1.49 5.23 0.0127

132 282.868 1.51 0.92 0.0088

133 315.847 1.53 -1.74 0.0056

134 345.366 1.54 -2.97 0.0032

135 371.56 1.56 -2.96 0.0014

136 394.543 1.58 -1.91 0.0004

137 414.549 1.60 0.00 0.0000

138 431.814 1.62 2.61 0.0004

139 446.487 1.64 5.77 0.0014

140 458.7 1.66 9.34 0.0032

141 468.542 1.68 13.19 0.0056

142 476.129 1.69 17.20 0.0088

143 481.525 1.71 21.25 0.0127

144 484.768 1.73 25.22 0.0172

145 485.863 1.75 28.98 0.0225

Average 380.77 1.60

Sum 151.05 0.14B*tw 151.05/0.14*0.3= 315.92


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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.

Figure 10: ∆T1 vs time

5.3, -383

55.4, 316

-500

-400

-300

-200

-100

0

100

200

300

400

0 10 20 30 40 50 60 70

T e m p e r a t u

r e ( ° F )

time (s)

∆T1 vs time

∆T1 (F)

∆T1 vs time

∆T1 (°F)


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26

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


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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


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28

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.

Figure 13: Plastic Hoop Strain vs time, 1000 to 3000 psi in 1000 psi Increments

0

0.002

0.004

0.006

0.008

0.01

0.012

0 500 1000 1500 2000 2500

S t r a i n

time (s)

Cumulative Plastic Strain vs time

1000 psi

2000 psi

3000 psi


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29

The analysis was run for different iterations of pressure between 1000 and 2000 psi, in

increments of 100 psi. Figure 14 plots the plastic hoop strain versus time for 1000,

1600, and 2000 psi. Below about 1600 psi, the graph settles into a cyclic plastic strain

response. Around 1600 psi, the plastic strain levels off. Above about 1600 psi, the

plastic strain is seen to accumulate.

Figure 14: Plastic Hoop Strain vs time at 1000, 1600, and 2000 psi

0

0.0005

0.001

0.0015

0.002

0.0025

0 500 1000 1500 2000

S t r a i n

time (s)

Plastic Hoop Strain vs time

1000 psi

1600 psi

2000 psi


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30

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 atwhich 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 of final accumulated plastic strain versus pressure for each

100 psi increment in pressure from 1000 to 2000 psi. This slope of the curve is around

5*10-7

(1/psi) for pressures below about 1600 psi. However, at the onset of ratcheting,

the slope begins to increase rapidly to about 40*10-7

(1/psi) at 2000 psi. From Figure 15,

it is seen that ABAQUS predicts a pressure for the onset of ratcheting somewhere

between 1500 and 1600 psi.

Figure 15: Difference in Final Cumulated Plastic Strain vs Pressure

1600

0.00E+00

5.00E-05

1.00E-04

1.50E-04

2.00E-04

2.50E-04

3.00E-04

3.50E-04

4.00E-04

4.50E-04

1000 1500 2000

D i f f e r e n c e i n S t r a i n

Pressure (psi)

Difference in

Plastic Hoop Strain


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31

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.

This results in X= 26980*3.*2

2.3*2000

2 ywt

PD

0.40 and for X


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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: DassaultSystèmes Simulia Corp.

[3] Bree, J. (1967). Elastic-plastic behaviour of thin tubes subject to internalpressure and intermittent high-heat fluxes with application to fast nuclearreactor fuel elements. Journal of Strain Analysis, 2(3), 226-238.

[4] Bree, J. (1989). Plastic deformation of a closed tube due to interaction ofpressure stresses and cyclic thermal stresses. International Journal ofMechanical Sciences, 31(11/12), pp. 865-892.

[5] Bari, S. (2001). Constitutive Modeling for Cyclic Plasticity and Ratcheting .PhD 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.


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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



Valve13.st.inp ABAQUS Standard structural analysis, 1300 psi iterationValve14.st.inp ABAQUS Standard structural analysis, 1400 psi iteration







Valve30.st.inp ABAQUS Standard structural analysis, 3000 psi iterationReportCalculations.xls

MS Excel Workbook for calculating h, ∆T1, and plottingresults

1st.pyPython program for sequentially running ABAQUS files usingthe command “abaqus python 1st.py”


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7. Appendix B, ABAQUS Card Definitions

7.1 Discussion

This section describes the analysis file structure used in ABAQUS. The student versionof 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 thenode 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


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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.


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*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


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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.


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*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.


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*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, 02,21,1 DOF n DOF n uu

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


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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.

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