Plastic buckling in gas transmission line-pipes, cold
formed from thermo-mechanically-controlled rolling
of low-alloy steel plates
by
Vaibhaw Vishal
B.Tech., Mechanical Engineering,
Indian Institute of Technology, Bombay, India (2000)
S.M, Mechanical Engineering,
Massachusetts Institute of Technology, Cambridge, USA (2003)
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2007
© Massachusetts Institute of Technology 2007. All rights reserved.
Author .......................................................
artment of Mechanical Engineering
) rch 5, 2007
Certified by ......................
David M. Parks
Professor of Mechanical Engineering
Thesis Supervisor
Accepted by............................
MASSACHUSETTS IN __
OFTECHNOLOGY
JUL 18 2007
LIBRARIES
E
Lallit Anand
Chairman, Department Committee on Graduate Students
Plastic buckling in gas transmission line-pipes, cold formed from
thermo-mechanically-controlled rolling of low-alloy steel plates
by
Vaibhaw Vishal
Submitted to the Department of Mechanical Engineering
on March 5, 2007, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Mechanical Engineering
Abstract
The need for energy infrastructure has led to transportation of gases over long distances.
The strength-grade of pipeline steels used for transportation of gases has been increasing
to reduce the cost of the overall pipeline system. Along with higher strength, adequate
fracture toughness and resistance to plastic buckling are required of pipes installed in
earthquake- or frost-prone regions. To get higher strength with adequate deformability in low-alloy pipeline steels, plates for pipes are typically made today by thermomechanically-controlled rolling processes, which introduce strong crystallographic texture
and anisotropy in the pipes. The plates are then cold-formed into pipes, which introduces
further anisotropy and residual stresses in the pipe.
In the current work, effects of various steps of the pipe manufacturing process, such
as rolling, cold forming, etc., on residual stress, hardening moduli, plastic anisotropy, and
eventually, to the buckling resistance of the pipe, are studied. Effects of various types
of geometric perturbation on plastic buckling response of pipes are also studied. Due to
the crystallographic texture and cold-forming, crystal plasticity-based constitutive models
instead of Mises plasticity-based constitutive models may be better suited to model the
pipe. In the current work, crystal plasticity-based material models are used to predict the
buckling response of pipes. Results show that the buckling strain in uniaxial compression,
predicted using a crystal plasticity-based model, is
-
20% less than the one predicted using
an "equivalent" Mises plasticity-based model, for a pipe with d/t ratio of 51. Further
results show that variation in material properties and residual stresses caused by cold
forming reduces the buckling strain by ~ 30%, for a pipe with d/t ratio of 51.
Thesis Supervisor: David M. Parks
Title: Professor of Mechanical Engineering
2
Acknowledgments
I would like to thank my advisor, Prof. David Parks, for his constant support, guidance,
and advice throughout this work. I have learnt a lot from him and sincerely thank him for
all the insightful discussions we had. I would also like to thank my committee members,
Prof. Raul Radovitzky and Prof. Tomasz Wierzbicki, for their invaluable suggestions
during this work.
I would like to thank all my fellow graduate students in the mechanics and materials
group, especially to Dora, Deepti, and Suvrat for their help and encouraging remarks during my stay at MIT. I also thank Ray Hardin for handling all the administrative works
efficiently.
I am especially grateful to my parents and my sister for their constant encouragement
and support. I am also very thankful to Linika, Maanas, Shweta and Sachin for creating
a homely atmosphere here in Cambridge.
3
Contents
6
List of Figures
List of Tables
18
Nomenclature
19
1
22
Motivation & Introduction
1.1
Plastic buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.2
Pipe manufacturing processes
1.3
Outline of the thesis
. . . . . . . . . . . . . . . . . . . . . . . . . 31
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
43
2 Mechanical Modeling
2.1
2.2
2.3
Modeling of manufacturing steps
. . . . . . . . . . . . . . . . . . . . . . . 43
2.1.1
Modeling of UO forming . . . . . . . . . . . . . . . . . . . . . . . . 43
2.1.2
Modeling of mechanical expansion or E-step . . . . . . . . . . . . . 44
Material modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.1
Crystal plasticity model . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.2
Mises plasticity model . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.2.3
Modeling of strain aging . . . . . . . . . . . . . . . . . . . . . . . . 66
Modeling of compression of pipe . . . . . . . . . . . . . . . . . . . . . . . . 67
4
3 Results
70
3.1
Effects of anisotropy of flow rule . . . . . . . . . . . . . . . . . . . . . . . . 76
3.2
Effects of UO forming
3.3
Effects of strain aging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4
Effects of expansion strain . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.5
Effects of operating pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 80
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 Summary and Future Directions
4.1
Future directions
107
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
A Buckling Simulations with Single-Pass vs. Multi-Pass E-step
112
B Effects of Using All Grains in the Taylor Model
114
C Perturbation Study
118
C.1 Perturbation study with Mises plasticity model
C.2 Perturbation study with Crystal plasticity model
. . . . . . . . . . . . . . . 118
. . . . . . . . . . . . . . 119
D Pipe Buckling with Higher Mesh Density
154
Bibliography
157
5
List of Figures
1-1
Relation between buckling strain
(6b)
in axial compression and pipe di-
ameter to thickness ratio (d/t) for X-80 grade conventional pipeline steels
[1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5
1-2
Examples of engineering tensile stress vs. engineering tensile strain curves
for high-strength pipeline steels [2]. Triangles denote onset of necking near
the respective tensile strengths.
. . . . . . . . . . . . . . . . . . . . . . . . 25
1-3
Large deformation caused by fault movement [3].
1-4
Large deformation caused by frost heave to a pipe [3]. . . . . . . . . . . . . 27
1-5
Buckling damage to a buried pipeline [4]. . . . . . . . . . . . . . . . . . . . 27
1-6
Schematic showing how a lower value of Y/T results in higher values of
. . . . . . . . . . . . . . 26
tangent m odulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1-7
Optical micrograph of high deformability pipeline steel showing ferritebainite microstructure [1]; bainite is the dark phase in the micrograph.
1-8
. . 31
Schematic of Round-house type versus Luders elongation type of tensile
stress-strain curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1-9
Schematic representation of the pipe manufacturing processes. . . . . . . . 33
1-10 Longitudinal tensile stress-strain curve of TMCP developed line-pipe before
and after the polymer coating [2]. . . . . . . . . . . . . . . . . . . . . . . . 35
6
1-11 Tensile engineering stress-strain curves of TMCP developed ferrite-bainite
(F+B) dual-phase line-pipes and conventional hot-rolled line-pipes [2]. . . . 36
1-12 Axial compression test results of X-80 grade pipes, showing relation between buckling strain and d/t ratio of TMCP-developed pipes and conventionally hot-rolled pipes [5].
. . . . . . . . . . . . . . . . . . . . . . . . . . 37
1-13 Pole figures showing crystallographic texture in X-100 grade pipeline steel
[6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1-14 Experimental true stress vs.
true strain curve obtained from standard
ASTM tension test for an X100 steel [6].
. . . . . . . . . . . . . . . . . . . 39
1-15 Experimental true stress vs. true strain curve obtained from standard
ASTM compression test for an X100 steel [6].
2-1
. . . . . . . . . . . . . . . . 40
Schematic showing the modeling of UO forming, only a small strip indicated
by ABCDEFGH is modeled in the finite element framework. Here, RD
stands for rolling direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2-2
Schematic showing the modeling of mechanical expansion. Only a small
circumferential strip shown as the shaded part here is modeled in the finite
element framework. The cylinder indicated by dotted lines is the state of
pipe at the end of UO-forming step and the cylinder indicated by solid lines
is the state of pipe at the end of mechanical expansion step.
. . . . . . . . 46
2-3
Schematic of the F = F*FP decomposition. . . . . . . . . . . . . . . . . . 49
2-4
Cumulative volume fraction plot as a function of number of grains, based
on full 860 orientations of initial popLA dataset. . . . . . . . . . . . . . . . 57
2-5
Longitudinal true stress-strain curves in RD, obtained from experiments
and from finite element simulations. . . . . . . . . . . . . . . . . . . . . . . 60
7
2-6
Longitudinal true stress-strain curves in TD, obtained from experiments
and from finite element simulations. Due to the expansion strain of E-step,
there is an abrupt change in tangent modulus on tensile yielding and a
gradual change in compression.
2-7
. . . . . . . . . . . . . . . . . . . . . . . . 61
Evolution of area contraction in simple tension test on a cylindrical specimen in RD direction. Results from both experimental measurements and
simulations are displayed [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2-8
Algorithm that shows how the implementation of crystal plasticity with UO
is perform ed.
2-9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Algorithm that shows how the implementation of crystal plasticity without
UO is perform ed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2-10 Schematic showing how the modeling of compression is performed in pipes
to determine its buckling resistance. Only a small circumferential strip,
subtending an angle of 0.5730 on the center line, is modeled in the finite
element framework for computational efficiency. . . . . . . . . . . . . . . . 68
2-11 Geometry and boundary conditions of the pipeline subjected to axial compression load shown on an axisymmetric plane. . . . . . . . . . . . . . . . . 69
3-1
A small geometric perturbation in pipe wall, axisymmetric section of the
pipe is shown . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . .
71
3-2
Pipe mesh shown on an axisymmetric plane. . . . . . . . . . . . . . . . . . 72
3-3
Shape of pipe-section at various stages of shortening. (a) Pipe-shortening
= 0.5%, (b) pipe-shortening = 1.0%, (c) pipe-shortening = 1.5%, (d) pipe-
shortening = 2.0% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3-4
Nominal stress vs. shortening curve obtained with the implementation of
crystal plasticity with UO material model.
8
. . . . . . . . . . . . . . . . . . 75
3-5
Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material models of the Mises plasticity without UO and
crystal plasticity without UO.
3-6
. . . . . . . . . . . . . . . . . . . . . . . ... . 81
Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material models of the Mises plasticity without UO and
crystal plasticity without UO.
3-7
. . . . . . . . . . . . . . . . . . . . . . . . . 82
Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material models of the Mises plasticity without UO and
crystal plasticity without UO.
3-8
. . . . . . . . . . . . . . . . . . . . . . . . . 83
Effects of UO forming on buckling strain and peak nominal stress obtained
with the implementation of crystal plasticity with UO. . . . . . . . . . . . . 84
3-9
Effects of UO forming on buckling strain and peak nominal stress obtained
with the implementation of crystal plasticity with UO. . . . . . . . . . . . . 85
3-10 Effects of UO forming on buckling strain and peak nominal stress obtained
with the implementation of crystal plasticity with UO. . . . . . . . . . . . . 86
3-11 Effects of strain aging on nominal stress vs. shortening strain curve obtained
with crystal plasticity-based material models.
. . . . . . . . . . . . . . . . 87
3-12 Effects of strain aging on nominal stress vs. shortening strain curve obtained
with crystal plasticity-based material models.
. . . . . . . . . . . . . . . . 88
3-13 Effects of strain aging on nominal stress vs. shortening strain curve obtained
with crystal plasticity-based material models.
. . . . . . . . . . . . . . . . 89
3-14 Effects of strain aging on buckling strain, crystal plasticity-based material
models are used for the simulations. . . . . . . . . . . . . . . . . . . . . . . 90
3-15 Effects of expansion strain on nominal stress vs. shortening strain curve
obtained with the implementation of crystal plasticity with UO.
9
. . . . . . 91
3-16 Effects of expansion strain on nominal stress vs. shortening strain curve
obtained with the implementation of crystal plasticity with UO.
. . . . . . 92
3-17 Effects of expansion strain on nominal stress vs. shortening strain curve
obtained with the implementation of crystal plasticity with UO.
. . . . . . 93
3-18 Effects of expansion strain on buckling strain of pipe obtained with the
implementation of crystal plasticity with UO. . . . . . . . . . . . . . . . . . 94
3-19 Variation of circumferential residual stresses across the pipe wall with varying level of expansion strains obtained with the implementation of crystal
plasticity with UO.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3-20 Variation of axial residual stresses across the pipe wall with varying level
of expansion strains obtained with the implementation of crystal plasticity
with U O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3-21 Variation of circumferential residual stresses across the pipe wall with varying level of expansion strains obtained with the implementation of crystal
plasticity with UO.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3-22 Variation of axial residual stresses across the pipe wall with varying level
of expansion strains obtained with the implementation of crystal plasticity
with U O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3-23 Variation of circumferential residual stresses across the pipe wall with varying level of expansion strains obtained with the implementation of crystal
plasticity with UO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3-24 Variation of axial residual stresses across the pipe wall with varying level
of expansion strains obtained with the implementation of crystal plasticity
w ith U O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
10
3-25 Deformation of an axial strip when it is cut out from the pipe. The three
figures are for all the three d/t ratio considered. . . . . . . . . . . . . . . . 101
3-26 Deformation of an axial strip when it is cut from a pipe. An expansion
strain of 1.5% results in the largest radius of curvature. . . . . . . . . . . . 102
3-27 Effects of change in internal pressure on nominal stress vs. shortening curve
obtained with the implementation of crystal plasticity with UO.
. . . . . . 103
3-28 Effects of change in internal pressure on nominal stress vs. shortening curve
obtained with the implementation of crystal plasticity with UO.
. . . . . . 104
3-29 Effects of change in internal pressure on nominal stress vs. shortening curve
obtained with the implementation of crystal plasticity with UO.
. . . . . . 105
3-30 Effects of change in internal pressure on buckling strain of the pipe, obtained with the implementation of crystal plasticity with UO. . . . . . . . . 106
4-1
Buckling strain vs. diameter to thickness ratio plots for all the three implementations of material model considered. . . . . . . . . . . . . . . . . . 109
A-1 Comparison of nominal stress vs. shortening curves obtained with singlepass and multi-pass mechanical expansion models. . . . . . . . . . . . . . . 113
B-1 Comparison of tensile stress-strain curve in RD and TD obtained with the
crystallographic texture of 125 and 860 weighted grains. . . . . . . . . . . . 115
B-2 Comparison of compressive stress-strain curve in RD and TD obtained with
the crystallographic texture of 125 and 860 weighted grains.
. . . . . . . . 116
B-3 Nominal stress vs. shortening curves obtained with the crystallographic
texture of 125 and 860 weighted grains. . . . . . . . . . . . . . . . . . . . . 117
C-1 Schematic showing perturbation in inner and outer diameter. . . . . . . . . 121
11
C-2 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of the Mises plasticity without UO. Magnitude perturbations in inner diameter, with a fixed wavelength A = 1.5 x t,
are performed in these simulations.
. . . . . . . . . . . . . . . . . . . . . . 122
C-3 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of the Mises plasticity without UO. Magnitude perturbations in inner diameter, with a fixed wavelength A = 1.5 x t,
are performed in these simulations.
. . . . . . . . . . . . . . . . . . . . . . 123
C-4 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of the Mises plasticity without UO. Magnitude perturbations in inner diameter, with a fixed wavelength A = 1.5 x t,
are performed in these simulations. . . . . . . . . . . . . . . . . . . . . . . 124
C-5 Effect of change of perturbation magnitude (a) in inner diameter of pipe
on its buckling strain with a fixed wavelength (A = 1.5 x t) of perturbation.
Mises plasticity without UO material models are used in the simulations.
.
125
C-6 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of the Mises plasticity without UO. Magnitude perturbations in outer diameter, with a fixed wavelength A = 1.5 x t,
are performed in these simulations.
. . . . . . . . . . . . . . . . . . . . . . 126
C-7 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of the Mises plasticity without UO. Magnitude perturbations in outer diameter, with a fixed wavelength A = 1.5 x t,
are performed in these simulations.
12
. . . . . . . . . . . . . . . . . . . . . . 127
C-8 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of the Mises plasticity without UO. Magnitude perturbations in outer diameter, with a fixed wavelength A = 1.5 x t,
are performed in these simulations.
. . . . . . . . . . . . . . . . . . . . . . 128
C-9 Effect of change of perturbation magnitude (a) in outer diameter of pipe
on its buckling strain with a fixed wavelength (A = 1.5 x t) of perturbation.
Mises plasticity without UO material models are used in the simulations.
.
129
C-10 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of the Mises plasticity without UO. Wavelength perturbations in inner diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these simulations. . . . . . . . . . . . . 130
C-11 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of the Mises plasticity without UO. Wavelength perturbations in inner diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these simulations. . . . . . . . . . . . . 131
C-12 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of the Mises plasticity without UO. Wavelength perturbations in inner diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these simulations. . . . . . . . . . . . . 132
C-13 Effect of change of perturbation wavelength (A) in inner diameter of pipe
on its buckling strain with a fixed magnitude (a = 4% of t) of perturbation.
Mises plasticity without UO material models are used in the simulations.
13
.
133
C-14 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of the Mises plasticity without UO. Wavelength perturbations in outer diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these simulations. . . . . . . . . . . . . 134
C-15 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of the Mises plasticity without UO. Wavelength perturbations in outer diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these simulations. . . . . . . . . . . . . 135
C-16 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of the Mises plasticity without UO. Wavelength perturbations in outer diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these simulations. . . . . . . . . . . . . 136
C-17 Effect of change of perturbation wavelength (A) in outer diameter of pipe
on its buckling strain with a fixed magnitude (a = 4% of t) of perturbation.
Mises plasticity without UO material models are used in the simulations.
.
137
C-18 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of crystal plasticity with UO. Magnitude
perturbations in inner diameter, with a fixed wavelength A = 1.5 x t, are
performed in these simulations.
. . . . . . . . . . . . . . . . . . . . . . . . 138
C-19 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of crystal plasticity with UO. Magnitude
perturbations in inner diameter, with a fixed wavelength A = 1.5 x t, are
performed in these simulations.
. . . . . . . . . . . . . . . . . . . . . . . . 139
14
C-20 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of crystal plasticity with UO. Magnitude
perturbations in inner diameter, with a fixed wavelength A = 1.5 x t, are
performed in these simulations.
. . . . . . . . . . . . . . . . . . . . . . . . 140
C-21 Effect of change of perturbation magnitude (a) in inner diameter of pipe
on its buckling strain with a fixed wavelength (A = 1.5 x t) of perturbation.
Crystal plasticity with UO material models are used in the simulations.
. .
141
C-22 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of crystal plasticity with UO. Magnitude
perturbations in outer diameter, with a fixed wavelength A = 1.5 x t, are
performed in these simulations.
. . . . . . . . . . . . . . . . . . . . . . . . 142
C-23 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of crystal plasticity with UO. Magnitude
perturbations in outer diameter, with a fixed wavelength A = 1.5 x t, are
performed in these simulations.
. . . . . . . . . . . . . . . . . . . . . . . . 143
C-24 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of crystal plasticity with UO. Magnitude
perturbations in outer diameter, with a fixed wavelength A = 1.5 x t, are
performed in these simulations.
. . . . . . . . . . . . . . . . . . . . . . . . 144
C-25 Effect of change of perturbation magnitude (a) in outer diameter of pipe
on its buckling strain with a fixed wavelength (A = 1.5 x t) of perturbation.
Crystal plasticity with UO material models are used in the simulations.
15
. . 145
C-26 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of crystal plasticity with UO. Wavelength
perturbations in inner diameter, with a fixed perturbation magnitude a =
4% of t, are performed in these simulations.
. . . . . . . . . . . . . . . . . 146
C-27 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of crystal plasticity with UO. Wavelength
perturbations in inner diameter, with a fixed perturbation magnitude a =
4% of t, are performed in these simulations.
. . . . . . . . . . . . . . . . . 147
C-28 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of crystal plasticity with UO. Wavelength
perturbations in inner diameter, with a fixed perturbation magnitude a =
4% of t, are performed in these simulations.
. . . . . . . . . . . . . . . . . 148
C-29 Effect of change of perturbation wavelength (A) in inner diameter of pipe
on its buckling strain with a fixed magnitude (a = 4% of t) of perturbation.
Crystal plasticity with UO material models are used in the simulations.
. . 149
C-30 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of crystal plasticity with UO. Wavelength
perturbations in outer diameter, with a fixed perturbation magnitude a =
4% of t, are performed in these simulations.
. . . . . . . . . . . . . . . . . 150
C-31 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of crystal plasticity with UO. Wavelength
perturbations in outer diameter, with a fixed perturbation magnitude a =
4% of t, are performed in these simulations.
16
. . . . . . . . . . . . . . . . . 151
C-32 Nominal stress vs. shortening curve obtained from pipe buckling simulations with the material model of crystal plasticity with UO. Wavelength
perturbations in outer diameter, with a fixed perturbation magnitude a =
4% of t, are performed in these simulations.
. . . . . . . . . . . . . . . . . 152
C-33 Effect of change of perturbation wavelength (A) in outer diameter of pipe
on its buckling strain with a fixed magnitude (a = 4% of t) of perturbation.
Crystal plasticity with UO material models are used in the simulations.
. . 153
D-1 Mesh details for the coarse and the fine mesh, the mesh is shown on an
axisymmetric plane. Coarse mesh is the default mesh, which is used in the
rest of the thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
D-2 Comparison of nominal stress vs. shortening curves obtained with the
default mesh and a much finer mesh. . . . . . . . . . . . . . . . . . . . . . 156
17
List of Tables
2.1
Material properties for the crystal plasticity model. . . . . . . . . . . . . . 59
3.1
Dimensions and operating pressure for the three modeled pipes. . . . . . . 71
18
Nomenclature
D
:
De
: Elastic rate of stretching tensor.
DP
:Plastic rate of stretching tensor.
Rate of stretching tensor.
E
Young's modulus.
Et:
Tangent modulus.
F
Deformation gradient.
Fe
Elastic part of deformation gradient.
FP
:Plastic part of deformation gradient.
L
Fourth-order anisotropic elasticity tensor.
L
Length of a pipe segment.
L
Velocity gradient.
Le
:Elastic part of velocity gradient.
LP
:Plastic part of velocity gradient.
P
Axial compressive load on pipe.
T
Tensile strength of steel used.
19
Y
:
Yield strength of steel used.
d
:
Outer diameter of the pipe.
di
:Inner diameter of the pipe.
ga
:Slip resistance of the slip system a.
Initial slip resistance.
go
gs
:Saturation slip resistance.
h
Self hardening rate.
h',3
a component (ath row and
ho
Initial hardening rate.
hs
:Saturation hardening rate.
m
Rate sensitivity parameter.
thcolumn) of the matrix of hardening moduli.
Moa
:Slip plane normal for the a slip system in the reference frame.
ma
:Slip plane normal for the a slip system in the current frame.
p
Internal pressure in pipe.
q
Latent hardening ratio.
so
:
Slip direction for the a slip system in the reference frame.
so
:
Slip direction for the a slip system in the current frame.
t
Thickness of the pipe wall.
a
Index of the ath slip system.
Eb
o
-n
:Buckling strain.
:Reference shear rate.
:Rate of shear on slip system a.
20
"Y
:
Total accumulated slip at a material point.
A
:
Applied axial displacement on pipe segment.
Q
:Rate of spin tensor.
e
:
Elastic rate of spin tensor.
O7A
:
Plastic rate of spin tensor.
o
:
Cauchy stress tensor.
T
:
Resolved shear stress on slip system a.
r1
:
Kirchhoff stress tensor
r
e
Jaumann rate of Kirchhoff stress based on elastic spin.
21
Chapter 1
Motivation & Introduction
Natural gas is the world's third largest source of primary energy behind coal and oil. The
demand for natural gas has been ever-increasing because of its "less unfavorable" impact
on the environment. Since gas reserves are often far away from the main market, its economic transportation becomes an important part of the business. Long-distance pipelines
are safe and economic for transporting natural gas from production site to supplier, while
the use of LNG tankers is less economical and often burdened with safety and/or environmental questions.
In the last few decades, the American pipeline industry has used X50- to X65-grade
1 steel
for the construction of pipelines. Many studies have indicated that using a low-
alloy higher strength grade pipeline steel, such as X80-, X100-, or X120-grade steel, to
transport natural gas may lead to cost savings of the order of 5 - 15% [7, 8, 9]. The
savings are achieved by lowering cost in many areas, such as material cost, compression
cost, construction cost, etc., as described below.
'In the pipeline industry a strength-grade of "Xnn" means that the minimum specified yield strength
of the steel used is "nn" ksi (x 7 x nn MPa). A strength grade of X50 corresponds to a minimum yield
strength of 50 ksi (~ 350 MPa) for the steel used.
22
Replacement of lower-strength steel with low-alloy high-strength steel allows for using a pipe with reduced wall-thickness, resulting in fewer tons of steel required for the
pipeline, thereby reducing the material cost. Although cost per ton of steel increases
with its strength grade, the savings in overall weight can more than offset any increase
in steel cost [10].
Using thinner pipes leads to secondary cost savings, such as lower
pipe transportation costs, and because of lower weight, construction costs are lower since
thinner-wall pipe requires less time and material to weld.
Another substantial advantage of using a higher-strength grade steel is that higher
operating pressure can be used. At higher operating pressures, an equivalent amount of
natural gas can be transported with a smaller-diameter pipe. With moderate increase
in operating pressure, smaller diameter, high-strength steel still has a thinner wall than
large-diameter, conventional steel pipelines operated at a lower pressure [10]. This results
in further reduction of material cost, construction cost, and transportation cost. The
in-situ anti-corrosion polymer coating cost also goes down as diameter reduces.
A further advantage of operating the pipe at higher pressure is the possibility of reducing the number of intermediate compressor stations. Although the horsepower required
at the initiating station is higher, the horsepower required at each intermediate station
and the spacing between compressors increases as the operating pressure is increased [10],
leading to overall cost saving.
Therefore, to reduce the overall cost of the pipeline system, the strength grade of
pipeline steels used for transportation of natural gas has been increasing. Accordingly,
23
X100-grade pipeline steel was put to practical use for the first time in 2002 [11].
Since gas reserves are often far from the market, the transportation pipelines may
have to go through environmentally severe regions such as, seismic regions, cold regions,
deep-water, and sour gas environment. For pipes laid in such regions, various other material properties such as, toughness, deformability, sour resistance, etc., are required of the
pipelines. One of the most challenging regions that the pipeline may have to go through
is thought to be the earthquake and frost prone regions where large plastic deformation is
expected to be introduced to buried pipelines. New pipeline design methodology, so called
"strain based design", for seismic and frost prone regions has been developed [12, 13], and
high strain line-pipe applications have been enabled through this new design concept.
This design methodology says that in the the presence of crack-like defects, the pipe
should deform plastically to a specified macroscopic strain before failure instead of failing
in a brittle manner. Further, higher resistance against the larger compressive and tensile
strains are required of pipelines. A critical failure event in compressive deformation of
pipe is buckling, and the pipe needs to have enough resistance to buckling. Fig. 1-1 shows
the relation between buckling strain (in axial compression) and pipe diameter to thickness ratio (d/t) for conventional pipelines. For these pipelines, buckling strain decreases
with increasing d/t ratio, and thicker pipes are needed in seismic or frost-prone regions
for sufficient buckling resistance. Furthermore, as shown in Fig. 1-2, it is a general trend
that higher-strength material has lower uniform elongation until necking and lower plastic
buckling resistance. Using a thicker pipe to address buckling concerns clearly offsets any
material cost advantage obtained by using a higher strength grade steel. Therefore, to
reduce the overall cost of the pipeline system, along with higher strength, higher strain
to necking of the material is required to prevent plastic buckling in pipelines which are
24
1.5
Uni-axial compression
Conventional pipes
1.0
-
:'
G
C-
Improved'
material ,
0.5
05
E =b 35 t/d(%)
0 '
30
40
50
60
70
d/t
Figure 1-1: Relation between buckling strain (sb) in axial compression and pipe diameter
to thickness ratio (d/t) for X-80 grade conventional pipeline steels [1].
1000
yX100
CU
800
y
X80
600
X65
400
C)
200
0
0
5
10
15
Strain (%)
Figure 1-2: Examples of engineering tensile stress vs. engineering tensile strain curves
for high-strength pipeline steels [2]. Triangles denote onset of necking near the respective
tensile strengths.
25
installed in permafrost or earthquake-prone regions. Fig. 1-3 shows large deformation
induced by a seismic event, and Fig. 1-4 shows large deformation induced by frost heave
Fau11t movement
Figure 1-3: Large deformation caused by fault movement [3].
to a pipe, and Fig. 1-5 shows damage caused by plastic buckling to an actual buried pipe.
Plastic buckling of the pipelines is a very critical failure mode as this may lead to
fracture of pipe, especially during load reversal. For example, in frost prone regions, if
frost causes plastic buckling, subsequent thawing of land may lead to low cycle fatigue
and cause fracture. To design pipes with higher buckling resistance, it is important to
understand and quantify buckling resistance of the pipes. In this work, quantification of
plastic buckling resistance for high-strength pipeline steel is performed. Effects of various
manufacturing steps on plastic buckling resistance are also studied in this work. Important aspects of plastic buckling in mechanical structures are presented next.
26
Figure 1-4: Large deformation caused by frost heave to a pipe [3].
4"Local Buckling
Figure 1-5: Buckling damage to a buried pipeline [4].
27
1.1
Plastic buckling
Plastic buckling/localization occurs in a wide variety of circumstances; it is a more-orless abrupt change from a smoothly varying deformation pattern to a pattern involving
one or more regions of highly localized deformation. Classical examples of localization
include necking of the round bar tensile specimen, shear band localization in structural
metals, rocks, etc. As shown in [14], the final collapse mode of an axially compressed steel
plate strip involves one buckle rather than a periodic pattern. Tvergaard and Needleman
[15] pointed out that the basic mechanism for buckling localization is associated with
a bifurcation in the vicinity of maximum load point. This is in close analogy with the
Consid re's [16] treatment of necking of a tensile bar, where necking instability occurs in
the bar at maximum load.
Buckling of cylindrical shells, plates, bars, etc. has been studied extensively by many
researchers; classical treatments of the subject are available in [17, 18, 19, 20]. Most of the
solutions that are available in textbooks deal with linear-elastic buckling. But, the type
of buckling that occurs in these high-strength gas pipelines is plastic buckling because
the d/t ratio (diameter to thickness ratio) of the pipe is typically in the range of 50 - 70,
which is not large enough for linear-elastic wall buckling to be an issue. For a steel pipe
with a d/t ratio of 70, the nominal stress required in pipe walls for local elastic buckling
under axial compression loading is exceedingly large, as given by the classical solution
[19],
O-c,. =
1.2lE t
E ~13.6 GPa.
d
Here o-c,. is nominal axial compressive stress in the pipe wall and E = 207 GPa is the
Young's modulus of the steel. Material would yield and plastic buckling would occur at
much lower nominal stress values than 3.6 GPa, as the yield strength and tensile strength
28
of pipeline steels is at most, of the order of 1 GPa (~
140 ksi). Currently, in the United
States, the strength-grade of steels used in pipelines are largely in the range of X50 to
X65. Higher-strength grade X80 steel pipelines are being used internationally. Furthermore, even higher strength-grade steels, such as X100, are also being used internationally
in pipelines at very few places.
Fundamentally, the (bifurcation) buckling load in the rate-independent plastic range
is proportional to the tangent modulus, Et (= g), of the material [21, 22, 23, 24]. Higher
values of tangent modulus over the first few percent of strain, where plastic buckling is
critical, are required to achieve higher buckling resistance, as measured by strain to buckling, or strain to peak stress, etc. Higher values of tangent modulus can be achieved by
lowering the ratio of yield strength (Y) to tensile strength (T), or by increasing the work
hardening exponent of the steel [13]. Fig. 1-6 schematically shows how a lower value of
Y/T (at a fixed T) results in higher values of tangent modulus. It should be noted that
in the pipeline industry, the tensile yield strength is traditionally defined as the value of
stress at 0.5% total strain, instead of the stress corresponding to a prescribed (e.g., 0.2%)
offset plastic strain.
Typically, the Y/T ratio tends to increase with increasing strength, and it becomes
difficult to balance higher strength and lower Y/T ratios. Therefore, to achieve specific
microstructures in low-alloy steels which result in both higher strength and higher deformability, plates for pipelines are typically made by a thermo-mechanically controlled
rolling process (TMCP) [5, 1, 2]. The manufacturing process involves controlled rolling
followed by accelerated cooling, giving rise to a dual-phase microstructure of ferrite and
a mixture of bainite and martensite, as shown in the optical micrograph of Fig. 1-7.
29
h.
T I--------------------------- - - - ---Et1
y, .~~---
Et2
C,,
Y2
U)Y
2
T
Et2
0.5%
YI
T
> Etj
Strain
Figure 1-6: Schematic showing how a lower value of Y/T results in higher values of tangent
modulus.
30
The presence of elongated bainite in the microstructure results in a "round-house" type
N
020
m
Rolling direction (RD)""""
Figure 1-7: Optical micrograph of high deformability pipeline steel showing ferrite-bainite
microstructure [1]; bainite is the dark phase in the micrograph.
stress-strain curve, as shown in Fig. 1-8, rather than a Luders elongation type curve in
which -the hardening is delayed after onset of yield, typical of ferrite-pearlite microstructures obtained from a conventional hot-rolling process [1]. In the Luders elongation type
stress-strain curve, tangent modulus drops to nearly zero at the onset of yield, leading to
immediate plastic buckling at the yield point. Therefore, a round-house type stress-strain
curve is much preferred, as it improves the buckling resistance of the pipeline.
1.2
Pipe manufacturing processes
Schematically, the major pipe manufacturing steps are shown in Fig. 1-9.
In the first
step, thermo-mechanically-controlled rolling of plate followed by accelerated cooling is
performed in several passes to obtain the desired thickness, microstructure, and strength.
In this figure, RD stands for rolling direction, TD stands for transverse direction, and ZD
stands for thickness direction. The thermo-mechanically-controlled rolling is performed to
obtain the appropriate microstructure in the plate, which results in both higher strength
31
CI)
U)
U)
2
1
Round-house type
Luders elongation type
Strain
Figure 1-8: Schematic of Round-house type versus Luders elongation type of tensile stressstrain curves.
32
Accelerat(
cooling
Tab Welding
Thermo-mechanically controlled Rolling
Crimping
Edge milling and beveling
I
U-forming
Inside welding
O-forming
F
I
Tack welding
Corrosion Resistant
Polymer Coating
Outside welding
Mechanical expansion
Figure 1-9: Schematic representation of the pipe manufacturing processes.
33
and higher hardening. In the next step tabs are welded to the corners of the plate for
better handleability during the rest of the manufacturing processes. Edge milling and
beveling is performed next to even out the edges of the plate. Edge crimping is performed
next so that when the plate is formed into a pipe the two ends match properly. To form
pipes from these rolled plates, a so-called "UOE" forming process is utilized. In the Ustep, plate is plastically bent by tooling to resemble the shape of a 'U', whereupon the
resulting stock is subsequently deformed into the general shape of an '0' and longitudinal
seam welding is performed. The composite UO step together produces maximum bending
strains of the order of t/d (ratio of wall thickness to diameter of the pipe) in the pipewall. Next, in an expansion, or E-step, an essentially uniform radial expansion of about
1% is applied. This cold expansion step is performed to get good roundness in the pipe
and to reduce the residual stresses, both in the seam-welding region and throughout the
wall, due to prior elastic-plastic UO bending. Finally, a thin external polymer coat is
applied to the pipe in the field, just before the pipe is laid at an actual field site, in order
to improve its environmental corrosion resistance. Since the pipe is heated slightly, to
~ 230'C, to apply the polymer coat, a microstructural process called strain aging can
occur in the pre-deformed pipe, resulting in a slight increase of the flow strength and
Y/T ratio, as shown in Fig. 1-10. The increase in flow strength during the strain aging
process is typically of the order of 30 MPa - 50 MPa. Further details of the strain aging
process are given in [2]. Stress-strain curves of the TMCP developed ferrite-bainite dual
phase line-pipes are compared to conventionally developed line-pipes in Fig. 1-11. All
TMCP developed line-pipes have lower Y/T ratio and higher strain to necking leading
to higher plastic buckling resistance. The improvement in buckling resistance of these
TMCP-developed pipes compared to the conventional hot-rolled pipes is experimentally
validated [5]. As shown in Fig. 1-12, strains to buckling of TMCP-developed X-80 grade
34
800
After coating
7001I
I.
,600
C,,
Co
. 500
-I
As UOE
400
300
0
1
2
3
4
5
Strain (%)
Figure 1-10: Longitudinal tensile stress-strain curve of TMCP developed line-pipe before
and after the polymer coating [2].
35
1000
- -
X100
800
p600
--
--------- ----..-
X65
2400
-W
Developed (F+B)
-
200
Conventional (B)
-----
0
0
5
10
15
Strain (%)
Figure 1-11: Tensile engineering stress-strain curves of TMCP developed ferrite-bainite
(F+B) dual-phase line-pipes and conventional hot-rolled line-pipes [2].
36
pipes in axial compression tests are
-
1.5 times those of conventionally hot-rolled X-80
grade pipes.
2.0
O Conventional Hot-rolled
*
TMCP-developed
0 1.5
1.0
~0.5
I3
0'
3)
40
50
0
60
70
d/t
Figure 1-12: Axial compression test results of X-80 grade pipes, showing relation between
buckling strain and d/t ratio of TMCP-developed pipes and conventionally hot-rolled
pipes [5].
Thermo-mechanically controlled rolling process introduces strong crystallographic texture in the pipe wall, which may lead to anisotropic mechanical behavior. Fig. 1-13 shows
stereographic pole figures
2
indicating strong crystallographic rolling texture in X-100
grade pipeline steel. Furthermore, the UO-step introduces linearly-varying level of bending strain and the E-step introduces uniform expansion strain in the pipe wall. The strains
introduced during the UOE forming steps can be as large as 2 - 3%, and may result in
2
To find out more about how to read pole figures, please refer to [25].
37
varying levels of strain hardening and residual stresses in the pipe wall.
(002)
(011)
ND
RD
w
Figure 1-13: Pole figures showing crystallographic texture in X-100 grade pipeline steel
[6].
Owing to the textured microstructure imposed by the controlled rolling and nonuniform hardening introduced by the UOE forming processes, the mechanical properties of
the pipeline steel can be highly anisotropic and spatially heterogeneous through-thickness.
Fig. 1-14 shows true stress vs. true strain curves for standard ASTM tension tests, and
Fig. 1-15 shows true stress vs. true strain curves for standard ASTM compression tests
on an X100 steel. It is evident from the figures that the material has strong anisotropy,
since the stress-strain curves in the different directions are different from one another.
The anisotropy in yield, flow, and work hardening behavior can potentially affect the
structural integrity of the pipelines, including their plastic buckling and collapse resistance. To make a conservative estimate of buckling strain, material properties in the
material direction of the minimum strain hardening and the highest Y/T ratio are often
used in isotropic plasticity models [26]. This simplification can overly under-estimate
38
1000
900
800
0n
700
U)
600
_U
500
0
400
300
D
--
enio
200
100
I
0
0
0.5
1.0
1.5
I
I
I-
2.0
2.5
3.0
3.5
True T ensile S train (%)
-
TD Tension
4.0
4.5
5.0
Figure 1-14: Experimental true stress vs. true strain curve obtained from standard ASTM
tension test for an X100 steel [6].
39
1000
.. .. .. ... .. . .
900
Cu
0~
800
U)
U)
700
0
Co
0
600
U)
U)
500
.
.
..
.
.
.
..
..
..
.
.
.
. .
.
.
.
.
..
. .
.
.
.
.
.
.
.
-m
- . . . . . . ..
I.0.i
.
.
. .. . . ... . . .. . . . .... . . .. . . . ... . . . . . . .
-................
0
..
D oprsso
-
I-
a. 400
E
TD Copeso
..
0
0
0
300
I-
I- 200
100
- R D Compression
0
0.0
0.5
1.0
I
I
1.5
2.0
2.5
3.0
3.5
4.0
True Compressive S train (%)
4.5
5.0
Figure 1-15: Experimental true stress vs. true strain curve obtained from standard ASTM
compression test for an X100 steel [6].
40
the load-carrying capability of the pipes [26]. Therefore, more representative models are
needed to predict buckling resistance of these pipelines. Furthermore, residual stresses in
the pipe wall may also affect the buckling resistance of the pipeline as indicated in [27].
Mises plasticity with isotropic hardening is one of the most simple constitutive models
that can be used to predict the plastic buckling resistance of these pipes. Other classical
plasticity models such as Hill's anisotropic model or Mises plasticity with kinematic hardening can also be utilized. Some numerical. results using both Hill's anisotropic model
and Mises plasticity with kinematic and isotropic hardening are presented in [28]. Since
the material used in the pipeline steel has a strong crystallographic texture, polycrystal
plasticity-based material models, instead of Mises plasticity or Hill's anisotropy-based
material models may be better-suited to simulate the buckling behavior of the pipes.
Further, most of the models available in literature for predicting buckling strains of highstrength pipeline steels do not account for residual stresses incurred during manufacturing
processes. In the present work, a crystal plasticity-based material model in which X100
strength grade pipeline steel is modeled as a BCC polycrystal will be used to simulate
buckling in the pipes. Also, residual stresses will be included in the model for better
prediction of buckling strains.
1.3
Outline of the thesis
Details of finite element modeling of buckling of pipes are presented in Chapter 2. Constitutive model of poly-crystal plasticity along with the procedures to determine the material
properties for crystal plasticity model are also described in this chapter.
Results obtained with the models of Chapter 2 are presented in Chapter 3. In partic41
ular, effects of thermo-mechanical rolling, UOE forming, strain aging, etc. are presented
in this chapter.
In Chapter 4 the modeling framework is reviewed, results are discussed, and possible
future directions are presented.
Since plastic buckling in pipelines can be very sensitive to small perturbations in the
geometry of the pipe, a perturbation study is performed and presented in Appendix C.
42
Chapter 2
Mechanical Modeling
2.1
Modeling of manufacturing steps
As seen in the last chapter, the pipe has gone through a number of cold forming steps
after thermo-mechanical rolling. These forming steps may introduce variation in material
properties and residual stresses across the pipe wall, which may eventually affect the
buckling resistance of the pipe. Therefore, it is important to carefully model the pipe
manufacturing steps in the finite element framework for a better prediction of the final
buckling resistance. The details of how the major manufacturing steps are modeled in a
finite element framework are presented here.
2.1.1
Modeling of UO forming
As seen in Fig. 1-9, the U-forming, 0-forming, and seam-welding operations are performed
to deform a flat plate into a pipe. The strain induced during this UO-forming operation
may be both circumferentially and longitudinally non-uniform, but for modeling purposes
this deformation is assumed to be uniform, i.e., the strains are assumed to not depend
on the circumferential or axial location. This assumption leads to significant savings in
43
computational effort, as instead of modeling the whole plate only a small section of the
plate can be modeled. Fig. 2-1 schematically shows how the modeling of UO-forming is
performed. Only a small strip denoted by ABCDEFGH in Fig. 2-1 is modeled in the
finite element framework, as all such strips are assumed to have identical deformations.
Pure bending about RD (rolling direction) is applied on face AEHD and on face BFGC to
simulate the forming of plate into a pipe. Generalized plane strain conditions are enforced
in the rolling direction. Although the deformation of this strip during UO forming is
axially uniform, a long strip with 200 elements in axial direction is modeled, since it will
be needed in modeling the buckling of pipe, as explained in Section 2.3.
2.1.2
Modeling of mechanical expansion or E-step
Mechanical expansion of pipes are performed in several passes/bites. Typically pipesegments are ~ 10 m long, and the length of the expansion tool is
-
1m: therefore, in
one pass/bite a 1 m long segment of pipe is expanded. The procedure for expanding the
pipe is as follows [3].
1. Expansion tool is advanced axially with an overlap of ~ 20 - 30% of the length of
the tool between two passes.
2. Expansion tool is moved radially to desired expansion strain.
3. Expansion tool is retracted radially.
4. The above procedure is repeated until half the length of the pipe has been expanded.
5. Expansion tool is taken to the other end and expansion process as described above
in items 1 to 3 is repeated until the whole pipe is expanded.
44
A
H
G
A
F
B
RD
UO forming
H
DG
A
RD
Figure 2-1: Schematic showing the modeling of UO forming, only a small strip indicated
by ABCDEFGH is modeled in the finite element framework. Here, RD stands for rolling
direction.
45
Modeling of this complex expansion process in so many steps proves to be computationally expensive. Therefore, a much simpler version in which the whole pipe is expanded
in just one pass is utilized, as shown in Fig. 2-2. Again, since the pipe and deformations
H
-
G
Figure 2-2: Schematic showing the modeling of mechanical expansion. Only a small
circumferential strip shown as the shaded part here is modeled in the finite element
framework. The cylinder indicated by dotted lines is the state of pipe at the end of UOforming step and the cylinder indicated by solid lines is the state of pipe at the end of
mechanical expansion step.
are assumed to be axisymmetric, only a small circumferential segment of the pipe, shown
as the shaded part (strip ABCDEFGH) in Fig. 2-2, is modeled in the finite element
framework. To enforce axisymmetric deformation, the nodes that are on plane ADH are
constrained to move within the ADH plane, and the nodes that are on plane BCG are
constrained to move within the BCG plane. Also, generalized plane strain conditions are
46
enforced in the axial direction. The results obtained with the resulting simple "one-pass
expansion" model is compared to results obtained with a multi-pass expansion model in
Appendix A. It is shown in Appendix A that the buckling strain obtained with these two
models of mechanical expansions are essentially identical.
2.2
Material modeling
As seen in the last chapter, due to controlled thermo-mechanical rolling the pipe line
steel has developed a strong crystallographic texture, which results in anisotropy in yield
and flow behavior of material. For this reason, a crystal plasticity based material model
is utilized to simulate plastic buckling in the pipes. For comparison purposes, a Mises
plasticity based model is also utilized in the simulations. The crystal plasticity model is
presented next.
2.2.1
Crystal plasticity model
The physics of single-crystal plasticity was established during the early 20th century. The
notable works are those of, Ewing, et al. [29], Bragg [30], Taylor, et al. [31, 32, 33, 34],
Schmid [35], and others. Their experiments indicated that at low homologous temperatures the major source of plastic deformation in metals is the dislocation movement in
the crystal lattice. The dislocation movements occur on certain crystallographic planes
in certain crystallographic directions, and crystal structure of the metal is not altered
by the plastic flow. The preferred crystallographic planes are called slip planes and they
are typically those planes on which the area atom packing density is the highest. The
preferred crystallographic direction in which the dislocation movement occurs on a slip
plane is called the slip direction and is typically a direction on the slip plane in which the
linear atom packing density is the highest. The slip plane and the slip direction together
47
are called a slip system. Anisotropic constitutive models for large elastic-plastic deformation of single crystals of ductile materials have been formulated by various researchers
[36, 37, 38, 39, 40]. In these models plastic deformation is assumed to occur by crystallographic slip alone; other deformation producing mechanisms such as, twinning, phase
transformation, diffusion, or grain boundary sliding are not considered.
In the current formulation of crystal plasticity, the deformation of a crystal is "taken"
as resulting from two independent mechanisms, an overall elastic distortion of the lattice
and a plastic deformation that does not disturb the lattice orientation. Consequently
the overall deformation gradient, F, can be multiplicatively decomposed into two parts,
namely F' and FP, where F' corresponds to the elastic part of the deformation and
FP corresponds to the plastic part. This decomposition, often termed the Kroner-Lee
decomposition [41, 42], is schematically shown in Fig. 2-3. The intermediate configuration,
known as a relaxed lattice configuration, is an imaginary stress-free configuration [43]. All
the stress due to deformation gradient F comes from lattice distortion during the 'F"
part of the deformation. In equation form, this decomposition can be written as,
F = FFP,
(2.1)
with det FP = 1
(indicates plastic incompressibility).
Here FP accounts for accumulated plastic deformation by plastic slip on a set of slip
systems (soa, mo). Here so' is the slip direction and mo
is the slip plane normal for
the 'a' slip system in the reference and relaxed lattice configurations. The slip direction
48
CL
Deformed configuration
F = FeFP
Fe
FP
Reference configuration
Relaxed lattice configuration
Figure 2-3: Schematic of the F = F'FPdecomposition.
49
and slip plane normal at the current state of deformation are given by,
S
=
Fso, and
(2.2)
m
=
F
(2.3)
TmOQ
respectively. It should be noted that transforming s" and ma in this manner preserves
their orthogonality. The velocity gradient tensor is given by,
L - grad v = D +0,
(2.4)
where, D = sym L and 0 = skew L are rate of stretching and spin tensor, respectively.
L
=PF-'
=
eFe- + Fe.FP" FPe-1
L+
LP.
Here,
F6
L e=
Fe~
=
sym L' + skew L'
=
De+fe, and
LP =
=
FePFPlFe-1
sym LP + skew LP
= DP + aP,
where D' and fO* are elastic rate of stretching and spin tensor, respectively, and DP and
50
OP are plastic rate of stretching and spin tensor, respectively. Further, LP is given by the
flow rule,
LP
=
Z
asa 0 ma,
(2.5)
where a is the index of the slip system and a is the rate of shear on slip system a. The
shear rate on slip system a is assumed to be given by the power law form [44, 45],
a
Osgn(Ta)
T a 1/m
,
(2.6)
ga
where m is the rate sensitivity parameter, with m -+ 0 denoting the rate independent
limit, O is a reference shear rate, r is the resolved shear stress on slip system a, and ga
is the slip resistance of the slip system a. The resolved shear stress on slip system a is
given by,
Ta
=
(7-a(sa 0
(2.7)
ma),
where a is the Cauchy stress tensor. The slip resistance, ga, is taken to evolve according
to
ia=Zh/ ,4I 3 ,
(2.8)
where h'1 is a matrix of hardening moduli that describes the rate of increase of deformation resistance on slip system a due to shearing on slip system
#;
it describes both
self-hardening and latent-hardening of the slip systems. Several simple forms for the hardening matrix have been proposed in the past, and are reviewed in [46, 47, 48]. For the
current study, the following simple form of the hardening moduli shall be used:
haO = qh+ (1 - q)haf,
(2.9)
where h denotes the self-hardening rate, the parameter q represents the latent hardening
51
ratio, and
6
is the Kronecker delta symbol. The self-hardening rate, h, takes the form
[39, 49]
h = h, + (ho - h,) sech 2
ho
,
(2.10)
where ho is initial hardening rate, h, is saturation hardening rate, go is initial slip resistance, g, is saturation slip resistance, and - (total accumulated slip at a material point)
is given by
/=j
a dt.
(2.11)
Assuming no effect of slip on crystal elasticity, the elastic constitutive relation is given by,
rVe = ID,
(2.12)
where rve is the Jaumann rate of Kirchhoff stress based on elastic spin 0', and C is the
fourth-order anisotropic elasticity tensor. Here, the Kirchhoff stress tensor is given by,
r
=
(det F)o,
(2.13)
and its Jaumann rate is given by,
.Ve =
-Tff'+ -+T .
(2.14)
For a cubic crystal such as BCC iron, the elasticity tensor may be specified in terms of
three stiffness parameters, c1 1 , c12 , and c4 4 [50].
This completes the structure of a single crystal constitutive model. For polycrystalline
materials, a material point is visualized as an aggregate of many single crystals, and the
constitutive response at the material point is taken as a suitable weighted average of the
52
constitutive responses of the individual crystals comprising this representative aggregate.
Various averaging procedures, such as the Sachs-type models [51], the Taylor-type models
[34], relaxed constraint models [52], and self-consistent models [53, 54], have been proposed in the literature to make this transition from the micro-response of the individual
grains to the macro response of the polycrystalline aggregate.
In the Sachs model, the stress field in each individual grain comprising the aggregate
representative volume element (RVE) is taken to be uniform and equal to the macroscopic
stress at the material point in the polycrystalline material. Consequently, equilibrium is
satisfied throughout the aggregate, and while compatibility is satisfied within the individual grains, it is usually violated between the grains in the aggregate.
In contrast, the Taylor model assumes that the deformation gradient in each grain
of the polycrystalline aggregate is uniform and equal to the macroscopic deformation
gradient at the material point.
Consequently, compatibility is automatically satisfied
throughout the aggregate, and equilibrium is satisfied within the individual grains, but
not across grain boundaries.
Both the Sachs and Taylor models discussed above are fully constrained models, since
they impose constraints on all components of either stress or deformation gradients in the
individual grains. In the relaxed constraint models, the full constraints of the Taylor-type
models are relaxed as deformation progresses in individual grains. However, the problem
with this model is that it is difficult to prescribe a generalized criterion for relaxation.
The self consistent models attempts to take into account the influence of grain interac-
53
tions, but these models require significantly more computational effort when compared to
the other models discussed earlier. The results obtained with these self-consistent models may not be more accurate than the ones obtained with much simpler Taylor model [55].
Taylor model has been successfully adapted to many problems dealing with polycrystal plasticity of cubic metals [56, 57, 58]. For the current study, the Taylor model will
be used for averaging. Although the rate-independent, rigid-plastic, small strain version
of the Taylor model was proposed almost seventy years ago [34], its generalization to a
large deformation, elasto-viscoplastic form has been relatively recent [59]. In the Taylor
model, the macroscopic Cauchy stress (&) at a material point in crystalline aggregate is
given by the volume average of Cauchy stress over all representative individual grains,
N
Z v(k)(.(k)
E
(2.15)
k=1
where
(k)
denotes the stress in grain k, V(k) denotes the volume fraction (or weight
fraction) of grain k, and N is the total number of grains which are representing the
macroscopic material point. Note that,
N
E V(k)
1.
k=1
In the current work X100 steel is modeled as a BCC polycrystal, each individual crystal
having a different set of orientation angles with respect to the global coordinate frame.
BCC crystals are different from FCC crystals, which at low homologous temperatures
have twelve well-defined slip systems (four slip planes with three slip direction on each
slip plane). Corresponding BCC crystals can have up to 48 active slip systems. The large
number of slip systems becomes computationally demanding; therefore, as a simplification,
54
a so-called "pencil glide" type of slip is often taken to occur in BCC crystals [60]. In pencil
glide, the four < 111 > -type slip directions, so', are well-defined, but slip planes can
be any plane containing the slip direction. In practice the activated plane is chosen to
be the one on which the resolved shear stress is a maximum; algorithms for slip-plane
determination are explained in [60]. The resolved shear stress is expressed in terms of the
slip vectors in the reference configuration as
7a
moa. (F"e-laFeso).
=
(2.16)
The reference normal, mo
0 , of the slip plane containing the slip direction on which the
resolved shear stress -r is a maximum is given by
0o a-
tasoa
ta._ (a.
(soa. tcl)2
/ta-
(2.17)
where,
V = F"-lcF'so.
(2.18)
The above polycrystalline constitutive model is used as a FORTRAN SUBROUTINE/UMAT
with a commercially available software package, ABAQUS [61], to perform finite element
simulations. The material properties for the crystal plasticity model are determined as
follows.
Determination of Material Properties for Crystal Plasticity Model
The elastic stiffness properties for the X-100 steel, modeled as an aggregate of BCC
crystals, are taken to be the same as those of iron, and are given by,
cl = 237 GPa, c12 = 141 GPa, and c44 = 116 GPa.
55
To capture the effects of UOE forming on buckling resistance of the pipes, material properties of the plate need to be determined. To obtain the crystal hardening parameters of
the steel at plate stage, a trial and error-based curve-fitting procedure is used to match
results of ASTM compression and tension tests (in the rolling and transverse direction)
with results of corresponding finite element simulations. Experimental results of ASTM
simple compression tests of X100 pipeline steel are shown in Fig. 1-15. The rolling direction (RD) compression experiments were performed on a cylindrical specimen of diameter
6.3 mm taken from the center of a pipe-wall of thickness
-
15 mm; the axis of the cylin-
drical specimen is parallel to the axis of the pipe, which is also the rolling direction of the
plate stock. The pipe-stock, from which the specimens are cut, has gone through thermomechanical rolling and UOE forming processes. The crystallographic texture, shown in
Fig. 1-13, is also obtained from this pipe-stock. The experimental pole figure data were
processed by using the texture conversion program popLA [62], and a weight file containing 860 weighted grain orientations was obtained. Here, in the framework of the Taylor
model, the texture of material point is taken as an average of the 125 highest-weighted
grains, instead of all the 860 weighted grains given by popLA, to reduce computational
time. When using only 125 grains in the Taylor model, the volume fraction of each of
these 125 grain is scaled up by a common factor so that the total volume fraction represented by these 125 grains becomes 100% again. The cumulative distribution function as
a function of number of highest weighted grains are plotted in Fig. 2-4. As seen from this
figure, 125 grains capture about 60% of the volume fraction of the material. Although,
60% volume fraction seems a relatively small fraction, the buckling predictions obtained
with these 125 grains representing 60% volume fraction is very close to the one obtained
with all the 860 grains, as presented in Appendix B.
56
1
- -
0.9
- - -
-
-
- -
-
-1-1-
.
- -
--
--
0.8
0
0
0.7
-.
(U
E
. . . -.-.-... -.-.-.-
0.6
-.
. .. . .. . .. .
--
0- 0.5
-
--
-
-.
...
.
-..
......- -
0.4
0.3
0.2
0.1
0
0
100
200
300
400
500
600
700
800
900
Number of grains
Figure 2-4: Cumulative volume fraction plot as a function of number of grains, based on
full 860 orientations of initial popLA dataset.
57
To obtain the hardening parameters, finite element simulations on one 8-noded element with a single integration point are performed. It should be noted that the crystallographic texture is measured on a specimen taken from formed pipe, as opposed to from an
un-formed plate, but the crystallographic texture before and after UOE forming process
should essentially remain unchanged, as the amount of strain incurred during the UOE
step is less than ~ 3% at any location. Accordingly, local radian measures of crystallographic rotations associated with forming should differ from those of the plate stock by
angles of similar magnitude. Since the compression experiment is performed on a small
cylindrical specimen centered on the pipe-wall mid-thickness, the effective history of strain
"seen" by this specimen during pipe forming is dominated by the 1% expansion strain in
circumferential direction incurred during the E-step. Therefore, in the one-element simulation, a tensile strain of 1% is applied in TD (transverse direction), and then the load is
removed to get an updated reference configuration for the subsequent compression
and tension simulations. The compression and tension simulations were then performed
and matched with the corresponding experimental compression and tension tests. The
algorithm for determining the crystal hardening parameters is given below.
1. A set of crystal hardening parameters are assumed at the plate stage. The crystallographic texture at this stage is assumed to be identical to the experimentally
measured crystallographic texture at the pipe stage (after UOE forming). These
properties are given to an 8-noded cubic element.
2. A 1% expansion strain in TD is applied and the load is subsequently removed to
simulate the mechanical expansion step. The resulting material state corresponds to
the material state at an excised specimen removed from the center of the pipe-wall
after UOE-forming.
3. Now the tension and compression simulations in TD and RD are performed to match
58
them with experimental ASTM compression and tension tests.
4. The above three steps are repeated with a new guess for the crystal hardening
properties at step-1 until a reasonable match between experiments and simulation
is achieved at step-3.
Following the above algorithm, crystal hardening parameters are obtained. Table 2.1 summarizes the values of crystal hardening parameters. As shown in Fig. 2-5, the agreement
Initial slip resistance, go
Saturation slip resistance, g,
Initial hardening rate, ho
Saturation hardening rate, h,
Rate sensitivity parameter, m
Latent hardening ratio, q
Reference shear rate, ,o
253 MPa
296 MPa
600 MPa
50 MPa
0.005
1.0
0.001 sa
Table 2.1: Material properties for the crystal plasticity model.
between tension and compression experiments in rolling direction (RD) and corresponding
finite element simulations is very good with these hardening parameters. The agreement
between simulations and test data for TD loading is reasonable, as shown in Fig. 2-6 [6].
The values of crystal yield/hardening parameters, given in Table 2.1, are taken to
be effective in all crystals after the TMCP rolling process. This rolled state, with the
above-mentioned hardening parameters, is taken as a reference, and further mechanical
processing such as UOE-forming can be performed starting from this material state.
Although the steel is dual-phase, consisting primarily of a softer ferrite-phase and a
harder bainite-martensite-phase, in the present work it is modeled as an effective singlephase BCC poly-crystal. Authors in [6] have used this assumption and shown that this
59
1000
900
800
700
600
500
RD-ten-EX.
-RD-ten-SIM.
RD-com-EX.
RD-com-SIM.
400
300
200
100
0
0
'
0.04
0.02
0.06
0.08
True Strain
Figure 2-5: Longitudinal true stress-strain curves in RD, obtained from experiments and
from finite element simulations.
60
1000
900 800 -A'-'o-EX
-~
A
700f
'As
S0
600
A
500
TD-com-EX.
TD-com-SIM.
TD-ten-EX.
TD-ten-SIM.
~400
+
-------
300
200
100
0
0
0.02
0.04
0.06
0.08
'I'
0.1
0.12
True Strain
Figure 2-6: Longitudinal true stress-strain curves in TD, obtained from experiments and
from finite element simulations. Due to the expansion strain of E-step, there is an abrupt
change in tangent modulus on tensile yielding and a gradual change in compression.
61
modeling captures the experimentally-observed anisotropy of the macroscopic flow rule
very well, as demonstrated by their simulation of ovalization of a solid cylinder under
tension. Fig. 2-7 shows the ovalization of cross-section in simple tension test on a cylindrical specimen in RD direction. Due to the anisotropy of flow rule, diametral contraction
in the thickness direction is more (in magnitude) than the diametral contraction in the
transverse direction.
UO-forming operations produce different levels of strain across the pipe-wall, which
result in varying material properties and varying residual stresses across the pipe-wall. To
determine if the residual stresses and variation in material properties affect the buckling
resistance, two different implementations of the crystal plasticity model are performed.
In the first implementation the residual stresses and variation in material properties are
accounted for, while they are not accounted in the second implementation.
The first implementation is shown schematically in Fig. 2-8. In this implementation,
crystallographic texture and material properties of Table 2.1 are given to the as-rolled
plate and an idealized UOE forming is performed, as explained in Section 2.1. The resulting material state which has all the information about through-thickness variation
of residual stresses and material properties serves to provide initial conditions for subsequent buckling simulations. The modeling details of the buckling simulations are given
in Section 2.3. Since the UO steps are explicitly simulated in this implementation, it will
be referred to as the crystal plasticity with UO.
The second implementation is shown schematically in Fig. 2-9. In this implementation
the material properties, as given in Table 2.1, are taken and assigned to a cubic element. A
62
L
RD
0
ZD-SIM.
--
TD - SIM.
ZD-EXP.
TD - EXP.
x
-0.1
*
-0.2
0
u/L = 0.10
L-
*
x
-0.3
0
U
x
u/L = 0.14
-0.4
x
-0.5
u/L = 0. 16
IP&
1 1 1 1 1 1 1 1 1 1 I
0.05
0.1
I
I
I
I
0.15
I
0.2
Normalized Displacement in RD, u/L
Figure 2-7: Evolution of area contraction in simple tension test on a cylindrical specimen
in RD direction. Results from both experimental measurements and simulations are
displayed [6].
63
# Take experimental crystallographic texture.
# Take material properties, as given in Table 2.1.
# Give the above properties to a plate.
# Perform UOE forming on plate to deform it into a pipe.
- Note: Residual stresses incurred during UOE forming
are accounted for.
- Note: Variation of material properties across the pipewall is accounted for.
# Perform subsequent buckling simulations.
Figure 2-8: Algorithm that shows how the implementation of crystal plasticity with UO
is performed.
64
# Take experimental crystallographic texture.
# Take material properties, as given in Table 2.1.
# Give the above properties to an 8-noded cubic element.
I
# Perform 1% expansion in TD and unload.
# This gives a new set of material properties.
# Give the above obtained material properties to entire pipe.
- Note: No macroscopic residual stresses.
- Note: Spatially homogeneous material properties.
I
# Perform subsequent buckling simulations.
Figure 2-9: Algorithm that shows how the implementation of crystal plasticity without
UO is performed.
65
1% tensile strain followed by unloading are applied in TD to get an updated, macroscopically unstressed material state. These updated material properties are given as initial
conditions to the pipe and subsequent buckling simulations are performed, as explained
in Section 2.3. It should be noted that the E-step in this implementation is effectively
performed on the plate, not on the pipe, resulting in zero residual stress and spatiallyconstant values of slip strengths and slip hardening moduli across the wall-thickness of
the pipe at the beginning of the buckling simulation. Since no UO-steps are explicitly
simulated in this implementation, it will be referred to as crystal plasticity without
Uo.
2.2.2
Mises plasticity model
For comparison, an implementation of Mises plasticity (isotropic hardening) without UO
is done.
The reference true stress-strain curve for the implementation of the Mises
plasticity without UO is taken to be the same as the experimental true stress vs. true
strain curve of the standard ASTM compression test in RD, as shown in Fig. 2-5.
2.2.3
Modeling of strain aging
Strain aging increases the yield strength of the material slightly. In the framework of
crystal plasticity models, modeling of strain aging is done by increasing the slip strength
of all slip systems by a (appropriate) constant amount. The magnitude of slip-strength
increase is determined by iteration until the desired increase in yield strength is achieved
for subsequent compression simulation on an 8-noded element in RD.
66
2.3
Modeling of compression of pipe
Longitudinal compression simulations of pipes are performed to calculate their buckling
resistance, as measured by strain to peak load. Many experimental data reported in
literature suggest that plastic buckling of relatively thick pipes under axial compression
occurs in an axisymmetric mode [63, 64, 65], particularly so for pipes which are pressurized
internally [66]. For the present work the diameter to thickness ratios of the pipes are in
the range of 50-70 and the pipes are pressurized internally with a pressure that causes
circumferential stress of
-
40% of the yield strength of the steel used.
Under these
conditions, longitudinal compression of full 3D cylinder with an isotropic material model
also predicts an axisymmetric plastic buckling mode, as shown in Fig. 2-10. Therefore, for
predicting buckling strain using crystal plasticity models a "3D axisymmetric" model is
utilized in the finite element framework, as shown by the radial section in Fig. 2-10. Again,
to enforce axisymmetric deformation, the nodes that are on plane ADH are constrained
to move within the ADH plane, and the nodes that are on plane BCG are constrained to
move within the BCG plane. Before the application of any compressive displacements, an
internal pressure, p, is applied in the pipeline to simulate the operating conditions. No
shear traction along with displacement boundary conditions are then applied to both ends
of the pipe, to simulate compression leading to buckling. Pipe geometry and boundary
conditions are shown more clearly in Fig. 2-11, which shows a 2D view of plane BCG.
The results obtained with these mechanical models are presented and compared in the
next chapter.
67
'e
Figure 2-10: Schematic showing how the modeling of compression is performed in pipes to
determine its buckling resistance. Only a small circumferential strip, subtending an angle
of 0.5730 on the center line, is modeled in the finite element framework for computational
efficiency.
68
A = applied displacement
L
Load, P
G
C
AI
I
~~ti
B
F
d/2
Internal pressure, p
Figure 2-11: Geometry and boundary conditions of the pipeline subjected to axial compression load shown on an axisymmetric plane.
69
Chapter 3
Results
Results obtained with the mechanical models outlined in Chapter 2 are presented in this
chapter. Three different diameter to thickness ratio (d/t) of pipes ranging from 51 to 72
are modeled. The dimensions and operating pressures of the pipe segments are given in
Table 3.1. For each pipe, operating pressure is chosen so that it results in 300 MPa (=
43% of yield strength of X100 steel) of circumferential stress in the pipe wall, which is
typical of the pipeline industry. To initiate the buckling, a slight perturbation in the wall
thickness of the pipe is made, as shown in Fig. 3-1. The perturbation magnitude, a, and
the perturbation wavelength, A, for each of the three pipe segments are also given in Table 3.1. Since plastic buckling response may be highly sensitive to any kind of geometric
inhomogeneity in the pipeline, a geometric perturbation study on these three pipelines is
performed and presented in Appendix C.
In the implicit finite element model, 8-noded brick elements with one integration point,
along with hourglass stabilization, are used to mesh the circumferential strip. Number of
element in r-direction is taken to be 5, in 9-direction is taken to be 1, and in z-direction
is taken to be 200. The mesh on an axisymmetric plane is shown in Fig 3-2. Results
70
d/t
d (mm)
t (mm)
L (mm)
pd/2t (MPa)
a
A
Pipe-1
51
1020.0
20.00
3000.0
300.0
4% of t = 0.80 mm
1.50 x t = 30 mm
Pipe-2
64
1219.2
19.05
3000.0
300.0
4% of t = 0.76 mm
1.57 x t = 30 mm
Pipe-3
72
914.4
12.70
3000.0
300.0
4% of t = 0.51 mm
2.36 x t = 30 mm
Table 3.1: Dimensions and operating pressure for the three modeled pipes.
t
L
4-
a
d/2
Figure 3-1: A small geometric perturbation in pipe wall, axisymmetric section of the pipe
is shown.
71
obtained with a denser mesh are compared with the results obtained with the current
mesh in Appendix D. The results indicate that the current mesh is sufficiently dense, as
further increasing the mesh density does not alter the buckling response of the pipe.
*
L
d/2
Figure 3-2: Pipe mesh shown on an axisymmetric plane.
Fig. 3-3 shows the shape of a pipe-section of length 600 mm at various stages of
shortening strain, obtained with the implementation of crystal plasticity with UO. Mises
stress contours are plotted at each stage of shortening. Fig. 3-4 shows the corresponding
72
S, Mises
(Ave. Crit.: 75%-)
+9.560e+08
+8.885e+08
+8.210e+08
+7. 535e+08
+6.860e+08
+6.185e+08
+5.510e+08
+4.835e+08
+4.160e+08
+3 .485e+08
+2.135e+08
+1. 460e+08
I I
z
r
(a)
(b)
(c)
(d)
Figure 3-3: Shape of pipe-section at various stages of shortening. (a) Pipe-shortening
= 0.5%, (b) pipe-shortening = 1.0%, (c) pipe-shortening = 1.5%, (d) pipe-shortening
=
2.0%.
73
nominal compressive stress vs. shortening curve for the pipe. In this figure,
nominal compressive stress =
4P
7r(d2-
d?)'
(3.1)
where di is the inner diameter of the pipe and P is the total compressive load on the pipe.
Shortening strain is given by,
shortening strain =
,
(3.2)
where L is the initial length of the pipe and A is the relative axial displacement of one
end of the cylinder to the other. The stress contour in Fig. 3-3(a) is plotted at a pipeshortening of 0.5%, where the compressive loads are still increasing; the pipe has not
buckled yet. Fig. 3-3(b) shows stress contours and buckled shape just after the max
load is passed, at a pipe-shortening value of 1.0%. Further, Fig. 3-3(c) and (d) show the
buckled shape at higher pipe-shortening values of 1.5% and 2.0%, respectively. The corresponding (a), (b), (c), and (d) points are also shown on the nominal stress vs. shortening
curve in Fig. 3-4.
To quantify the buckling resistance of the pipe, buckling strain Eb, shall be defined as,
Ebw= spr
L
a
where Pmax is the maximum value of compressive load achieved during compression.
74
(33)
700
(b)
600
500
CL
0.
0
400
300
E
0
pd/2t = 300 MPa
d/t = 51
L = 3.0m
d= 1020.0 mm
t = 20.0 mm
200
E
z
100
A'
0.0
..
............
0.25
0.50
... ...............
0.75
1.00
1.25
Shortening (%)
1.50
1.75
2. 00
Figure 3-4: Nominal stress vs. shortening curve obtained with the implementation of
crystal plasticity with UO material model.
75
3.1
Effects of anisotropy of flow rule
To quantify the effects of anisotropy of flow rule nominal stress vs. pipe shortening curves
obtained with the material model of the Mises plasticity without UO and crystal plasticity
without UO are compared for all the three pipes given in Table 3.1. Fig. 3-5 shows the
curves for the pipe with d/t = 51, Fig. 3-6 shows the curves for the pipe with d/t = 64,
and Fig. 3-7 shows the curves for the pipe with d/t = 72. The isotropic Mises plasticity
material model predicts ~ 5% lower peak stress for all the three d/t ratio considered, and
10 - 20% lower buckling strain compared to the one predicted by the anisotropic material
model of crystal plasticity without UO. These results indicate that anisotropy in yield,
flow, and hardening affects the pre-buckling and buckling responses of the pipes.
3.2
Effects of UO forming
To quantify the effects of UO forming on compression and buckling of pipes, compression
simulations of pipes are performed with the material model of crystal plasticity with UO
and crystal plasticity without UO. The nominal stress vs. shortening curves obtained for
these material models are compared for all the three pipes. Fig. 3-8 shows the curves
for the pipe with d/t = 51, Fig. 3-9 shows the curves for the pipe with d/t = 64, and
Fig. 3-10 shows the curves for the pipe with d/t = 72. The peak nominal stress achieved
with these two material model implementations are almost identical to each other for all
the three d/t ratio considered, but the buckling strain obtained with the implementation
of crystal plasticity without UO is ~ 20 - 30% higher than buckling strain predicted with
the implementation of crystal plasticity with UO. This indicates that the variation of
crystal hardening parameters and residual stresses across the pipe wall, incurred during
76
the UO-step in the implementation of crystal plasticity with UO model, deteriorates the
buckling resistance of the pipes significantly.
3.3
Effects of strain aging
To study the effects of strain aging, buckling simulations for all the three pipes are performed. The material model of crystal plasticity with UO are used for these simulations.
Strain aging results in an increase of the effective yield strength of the material by 30-50
MPa. Its implementation is performed after the UOE step, as explained in Section 2.2.3.
The nominal stress vs. shortening curve obtained for the d/t ratio of 51, 64, and 72, are
given in Fig. 3-11, Fig. 3-12, and Fig. 3-13, respectively. Fig. 3-14 shows a plot of buckling strain as a function of yield strength increase during strain aging process. It can be
seen from these figures that strain aging of 30-50 MPa increase the peak nominal stress
by ~ 10% as compared to the case when no strain aging is performed, but its effect on
buckling strain is minimal, as less than 3% change in buckling strain occurs in all the
cases considered.
3.4
Effects of expansion strain
Circumferential expansion of pipe is performed to get better roundness in pipes, as explained in Section 2.1.2. Currently the industry uses a total of 1% expansion strain during
this mechanical expansion process [3]. A study to determine whether a different expansion
strain would result in better buckling resistance of pipes is performed and presented in
this section.
Fig. 3-15 shows nominal stress vs. shortening curves obtained for the pipe with d/t =
77
51, with varying level of mechanical expansion strain applied during the E-step. As seen
from this figure, a buckling strain of 0.91% is obtained with 0.0% expansion strain; when
the expansion strain is increased to 0.5%, buckling strain reduces to 0.88%, and when
the expansion strain is increased to 1.0%, buckling strain increases to 0.98%. When
the expansion strain is increased to 1.5% buckling strain further increases to 1.01%, and
when the expansion strain is increased to 2.0% buckling strain reduces to 0.98%. The
highest buckling strain value is obtained for an expansion strain of 1.5%. Nominal stress
vs. shortening curves for d/t ratio 64 and 72 gives similar results, as shown in Fig. 316 and Fig. 3-17, respectively. All three pipes give the highest buckling strain when an
expansion of 1.5% is performed during the E-step. These findings are summarized in
Fig. 3-18, which plots buckling strain of pipes with varying levels of expansion strain for
all the three pipe geometries considered.
A further advantage of performing mechanical expansion is that it reduces the residual
stresses in the pipe wall. Varying the expansion strain during the E-step changes the residual stress pattern in the pipe wall. This might be the primary cause of changes in buckling
strain with a change in expansion strain. The circumferential and axial residual stress
pattern for the pipe with d/t = 51 are plotted in Fig. 3-19 and Fig. 3-20, respectively.
As seen from Fig. 3-19, no expansion strain results in very high values of circumferential
residual stresses (~
+900 MPa) in the pipe wall. An expansion strain of 0.5% reduces
the circumferential residual stress to a more reasonable
-
+300 MPa. Another increase
in expansion strain to 1.0% further reduces the residual stresses
-
±100 MPa. As can
be seen from this figure, the lowest "overall" value of circumferential residual stresses are
reached for an expansion strain of 1.5%, which also results in the highest buckling strain,
as seen previously.
78
Due to generalized plane strain conditions in axial direction during UOE forming,
residual stresses in the axial direction are also present.
These residual stresses (as a
function of expansion strain) are shown in Fig. 3-20. Again, no expansion results in the
highest value of residual stresses (~
±350 MPa.) in the axial direction. Increasing the
expansion strain to 0.5% reduces the residual stresses in the axial direction as well, but
the fractional drop in axial residual stresses is not as much as in the case of circumferential
residual stresses. Again, an expansion strain of 1.5% results in the lowest axial residual
stresses.
Residual stress plots for the other two pipes with d/t ratio of 64 and 72 are shown in
Fig. 3-21, Fig. 3-22, Fig. 3-23, and Fig. 3-24. Again, an expansion strain of 1.5% during
the E-step results in the lowest value of residual stresses in circumferential and axial direction for both of these pipes. The buckling strain for both of these pipes are also the
highest with a 1.5% expansion strain.
The axial residual stresses causes the modeled strip to curl out if it is cut from the
pipe, i.e., if the axisymmetry boundary conditions are removed after the UOE forming
the strip deforms into a circular arc. Fig. 3-25 shows the deformed shape of the strip for
all the three pipes considered. An expansion strain of 1.0% is utilized in these simulations.
The radius of curvature for the pipe with a d/t ratio of 51 is obtained to be 13.4 m, for a
pipe with d/t ratio of 64 is obtained to be 11.5 m, and for a pipe with d/t ratio of 72 is
obtained to be 8.3 m. Further, as seen from Fig. 3-26, for a pipe with d/t ratio of 51, an
expansion strain of 1.5% results in the smallest curvature of the strip. This is expected
as a 1.5% expansion strain results in the smallest overall values of axial residual stresses,
79
as seen before.
3.5
Effects of operating pressure
A study for the effects of changes in operating pressure of the pipe on its buckling resistance is performed. The nominal stress vs. shortening curves obtained with the implementation of crystal plasticity with UO are given in Fig. 3-27, Fig. 3-28, and Fig. 3-29 for
pipes with d/t = 51, d/t = 64, and d/t = 72, respectively. Fig. 3-30 shows the buckling
strain vs. circumferential stress (pd/2t) plots for the three pipes.
From these figures it can be seen that, as the internal pressure is increased, the buckling strain increases but the peak nominal stress decreases. For relatively thicker pipe
the increase in buckling strain with an increase in pressure is significant (when pd/2t is
increased from 250 MPa to 400 MPa for the pipe with d/t = 51, buckling strain increases
by 24%). But, for thinner pipes the increase in buckling strain is not so significant (when
pd/2t is increased from 250 MPa to 400 MPa for the pipe with d/t = 72, buckling strain
increases by only 11%). Since buckling in gas pipelines are largely governed by imposed
strains (ground movement, earthquake, etc.), and not by stresses, it would be desirable
to use higher internal pressures so that the buckling strain is larger. However, at higher
internal pressures, circumferential stresses becomes larger, and fracture of pipe or seam
weld becomes an important issue to consider.
80
800
- -
Mises plasticity without UO
Crystal plasticity without UO
700-
I..
- -- .....
-
.-..................
6 00 -
S500-
0.
pd/2t = 300 MPa
E 0
01
L = 3.0 m
= 1020.0 mmt= 20.0 mm
-E 200 -d
o
0.0
0.25
0.50
0.75
1.00
1.25
Shortening (%)
1.50
1.75
2.00
Figure 3-5: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material models of the Mises plasticity without UO and crystal plasticity without
Uo.
81
800
- -
Mises plasticity without UO
Crystal plasticity without UO
700
600
-4
500
-
-M
404
0
400
414
pd/2t =300 MPa
d/t =64
L =3.0:m
d= 1219.2 mm
t= 19.1 mm
0
z 300
200
.
100
.
-
C
0.0
0.25
0.50
0.75
1.00
1.25
Shortening (%)
1.50
1.75
2.00
Figure 3-6: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material models of the Mises plasticity without UO and crystal plasticity without
Uo.
82
800
-- -
Mises plasticity without UO
Crystal plasticity without UO
700 F
(.
600
500
U)
400
E
1z
0
0
300
pd/2t =300 MPa
d/t = 72
.
L =3.0 m
d= 914.4 mm
t= 12.7mm
200
100
n
0.0
-.
0.25
...........
0.50
0.75
1.00
1.25
Shortening (%)
......... .......
1.50
1.75
2.00
Figure 3-7: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material models of the Mises plasticity without UO and crystal plasticity without
Uo.
83
800
-- -
Crystal plasticity with UO
Crystal plasticity without UO
700 I
CD
0
0)
U)
600
500
400
300
= 300 MPa
d/t = 51
L = 3.Om
d = 1020.0 mm
. .pd/2t
200 -
..
t = 20.0 mm
100 .
0
0. 0
.
. .
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Shortening (%)
Figure 3-8: Effects of UO forming on buckling strain and peak nominal stress obtained
with the implementation of crystal plasticity with UO.
84
800
-
700
SCrystal plasticity with UO
Crystal plasticity without UO
rsa patct-wtotU
-
600
-M
500
-
0
U)
0
U)
400
pd/2t =.300 MPa
d/t = 64
L = 3.0m
d = 1219.2 mm
t= 19.1 mm
300
200 -
..
100 - . .
0
0.0
0.25
0.50
1.25
0.75
1.00
Shortening (%)
1.50
1.75
2.00
Figure 3-9: Effects of UO forming on buckling strain and peak nominal stress obtained
with the implementation of crystal plasticity with UO.
85
800
-- -
Crystal plasticity with UO
Crystal plasticity without UO
700
0
600
0
500
400
..
300
E
-
...... ................... ................ ... ...... ... . .
-
pd/2t = 300 MPa
d/t = 72
L =3.O m
d =914.4 mm
t = 12.7 mm
-
200
z
100
n
0.0
0.25
0.50
0.75
1.00
1.25
Shortening (%)
1.50
1.75
2.00
Figure 3-10: Effects of UO forming on buckling strain and peak nominal stress obtained
with the implementation of crystal plasticity with UO.
86
800
-- -
700 CL
CD
600
No Strain Ageing
Strain Ageing = 30 MPA
Strain Ageing = 50 MPA
-'
-
500
(A
CL
0
4001
300
pd/2t = 300 MPa
-
d/t= 51
L =3.O m
0
..
200 .......-
.. .
.......
I ....
d= 1020.0
mm
.... .............
t=20.0 mm
z
100
0
0.0
0.25
0.50
1.25
Shortening (%)
0.75
1.00
1.50
1.75
2.00
Figure 3-11: Effects of strain aging on nominal stress vs. shortening strain curve obtained
with crystal plasticity-based material models.
87
800
-
700 - .
.
-
.Strain
.
No Strain Ageing
Strain Ageing = 30 MPA
Ageing = 50 MPA
-
-0
(U
600
EA
U)
500
a)
400
E
(0
U)
pd/2t = 300 MPa
300
d/t = 64
L =3.0 m
d= 1219.2 mm
t= 19.1 mm
200
-
100
0
0.0
h
0.25
.
0.50
0.75
1.00
.
1.25
-
.
1.50
1.75
2.00
Shortening (%)
Figure 3-12: Effects of strain aging on nominal stress vs. shortening strain curve obtained
with crystal plasticity-based material models.
88
800
-- -
(U
700 F
No Strain Ageing
Strain Ageing = 30 MPA
Strain Ageing = 50 MPA
0. 600
500
-30
0
0
400
0
300
pd/2t
=
300 MPa
d/t = 72
L = 3.0 m
d = 914. mm
t =12.7 mm
200
z
-
100
n
0.0
0.25
0.50
0.75
1.00
1.25
Shortening (%)
1.50
1.75
2.00
Figure 3-13: Effects of strain aging on nominal stress vs. shortening strain curve obtained
with crystal plasticity-based material models.
89
1.1
- d/t ratio = 51
d/t ratio =64
'.'
d/t ratio =72
-
1 -
.
.
. .
.
. .
.
.......
.......
-...... .... .. ... .....
0.8
-..... ...... ....... ..... ............. .. ......... ..
C 0.7
-. . . . . . . .
...... ...... ..... . . .. ... ....... . .. . ...
0.6 E
' . ... ... ..... ... ... .. .... ....... .... ... .... .
Ar
0
10
30
20
40
50
Strain ageing (MPa)
Figure 3-14: Effects of strain aging on buckling strain, crystal plasticity-based material
models are used for the simulations.
90
800
Expansion strain = 0.0%
7.. -
- -
Expansion strain
--
Expansion strain = 1.0%
=
Expansion strain = 1.5%
Expansion strain = 2.0%
-
600 CO
500 -
u' 400-0
0.5%
pd/2t ='300 MPa
d/t = 51
L = 3.&m
d =1020.0mm
t=20.0 mm
S 300 -.....
- 200 0
z
1000
0.0
0.25
0.50
0.75
1.00
1.25
Shortening (%)
1.50
1.75
2.00
Figure 3-15: Effects of expansion strain on nominal stress vs. shortening strain curve
obtained with the implementation of crystal plasticity with UO.
91
800
Expansion strain = 0.0%
700
.
- -
Expansion strain = 0.5%
-
Expansion strain
-'
2
=
1.0%
Expansion strain = 1.5%
Expansion strain = 2.0%
600500
-
.
1. 400
E
pd2t =300
d/t = 64
L = 3.0 m
d = 1219.2 mm
t =19.1 MM
300To
-200 z
0
1000
0.0
0.25
0.50
0.75
1.00
1.25
Shortening (%)
1.50
1.75
2.00
Figure 3-16: Effects of expansion strain on nominal stress vs. shortening strain curve
obtained with the implementation of crystal plasticity with UO.
92
800
Expansion strain = 0.0%
- -
--
-
7100
Expansion strain = 0.5%
Expansion strain = 1.0%
" Expansion strain = 1.5%
Expansion strain
,
Cn
=
2.0%
600
(n 500 (I)
-
400
0.
E
o 300-
pd/2t =300 MPa
d/t = 72
.C
E 200-
L = 3.0 m
d = 914.4 mm
t= 12.7 mm
z
100-
0
0.0
0.25
0.50
0.75
1.00
1.25
Shortening (%)
1.50
1.75
2.00
Figure 3-17: Effects of expansion strain on nominal stress vs. shortening strain curve
obtained with the implementation of crystal plasticity with UO.
93
1.1
FF
S0.9
--
-
.8---
.....
.. . .. . .. . . *.
.. ..
d/t ratio = 51
d/t ratio = 64
-*
0.6
.. . .. ..
+'6"d/t
0.0
0.5
1.0
1.5
ratio = 72
2.0
Expansic n S train (%)
Figure 3-18: Effects of expansion strain on buckling strain of pipe obtained with the
implementation of crystal plasticity with UO.
94
1000
800
-
600--
400
Po5%
200 -1.10%
0.-
- I.
0-
. ..............
-..
02.0%
-
-200 -
-400--
-600-
-800--
-1000
0.0
1
0.2
0.4
0.6
0.8
1.0
Figure 3-19: Variation of circumferential residual stresses across the pipe wall with varying
level of expansion strains obtained with the implementation of crystal plasticity with UO.
95
400
300.- .0.5%
200
1.0%
100-..-.
1.5%
2.0%
0-
-100.-
-200-
.-
-300
-400'
0.0
-~
0.2
0.4
0.6
(r-r.)/(r -r.)
0.8
1.0
Figure 3-20: Variation of axial residual stresses across the pipe wall with varying level of
expansion strains obtained with the implementation of crystal plasticity with UO.
96
1000
I
I
I
I
I
800 F
600 F
400 F
0.5%
200
-s 1.0%
00-
- - 1.5%
oF.
2.0%
0.-200 -- 000 ..
..
..
...
...
..
.......
...
-800 F
O. 0
...............
.....
...
...
......
..
..
..
..
..
..
..
.
-600k
-1000'
............ .... .........
...
....
...
...
...
...
......
-400 H
0.2
..... ..... ............... .... -
I
III
-
0.4
0.6
(r-r 1)/(r 0 -r.)
I
0.8
1.0
Figure 3-21: Variation of circumferential residual stresses across the pipe wall with varying
level of expansion strains obtained with the implementation of crystal plasticity with UO.
97
400
300
0.5%
200
1.0%
100
FI
1.5%
2.0%
0.
0
m
--
-100
..... ...
....
...........
-200 .
. 0 0
......
-
-300
-400'
0. 0
II
II
0.2
0.4
0.6
(r-r.)/(r -r)
0.8
1.0
Figure 3-22: Variation of axial residual stresses across the pipe wall with varying level of
expansion strains obtained with the implementation of crystal plasticity with UO.
98
1 00C I
I
I
0.0%
800 F
600 F
400 -
0.5%
200
/
/
(U
'M I an
1.0%
0
1.5%
4t
-200
...........
....... ...........
.
.. 2.0%
.....
........
-
-400 k
.
..
...
..
........
........... ...................-
-600 F
...
.........
..
.......
.....
-800 F
....
....
........
..
...
............. ..................-
-10000.0
0.2
0.4
................... ...... -
0.6
0.8
1.0
Figure 3-23: Variation of circumferential residual stresses across the pipe wall with varying
level of expansion strains obtained with the implementation of crystal plasticity with UO.
99
400
0.0%
300
0.5%
-
200 - .
1.0%
100
.
1.5%
.
2.0%
-
0
-100
-
...
-200-
- 300 -
-400'
0.0
.... ....
0.2
...............
0.4
0.6
(r-r.)/(r -r.)
-..
...... ..
0.8
1.0
Figure 3-24: Variation of axial residual stresses across the pipe wall with varying level of
expansion strains obtained with the implementation of crystal plasticity with UO.
100
d/t = 51
d/t = 64
d/t = 72
Figure 3-25: Deformation of an axial strip when it is cut out from the pipe. The three
figures are for all the three d/t ratio considered.
101
1.0% expansion strain
2.0% expansion strain
1.5% expansion strain
Figure 3-26: Deformation of an axial strip when it is cut from a pipe. An expansion strain
of 1.5% results in the largest radius of curvature.
102
800
- -
700 -...
(U
not
pd/2t = 250
pd/2t = 300
pd/2t = 350
pd/2t = 400
MPa
M Pa
MPa
M Pa
-
600
500
400
0
1..
0
d/t =51
L =3.O m
300
E
200
d=1020.0 mm
t=20.0 mm
-
100 -I
10
0.0
-
0.25
0.50
. .
0.75
1.00
1.25
Shortening (%)
1.50
1.75
I
2.00
Figure 3-27: Effects of change in internal pressure on nominal stress vs. shortening curve
obtained with the implementation of crystal plasticity with UO.
103
800 -
pd/2t = 250 MPa
-'
700 600
pd/2t = 300 M Pa
pd/2t = 350 MPa
pd/2t = 400 M Pa
--
-
500
E
400
E
L.
0
0
300
d/t =64
200
L = 3.0 m
d = 1219.2 mm
t= 19.1 mm
100
0
0.0
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Shortening (%)
Figure 3-28: Effects of change in internal pressure on nominal stress vs. shortening curve
obtained with the implementation of crystal plasticity with UO.
104
800
--
" pd/2t = 350 MPa
pd/2t =400 M Pa
700
600
0
pd/2t = 250 MPa
pd/2t = 300 MPa
--
-
500
400
0
I-
300
200
d/t = 72
L = 3.0 m
d = 9 14 .4 m m . .....
........... ...
t= 12.7 mm
100
I
0
0.0
0.25
0.50
0.75
1.00
I
-
1.25
1.50
1.75
2.00
Shortening (%)
Figure 3-29: Effects of change in internal pressure on nominal stress vs. shortening curve
obtained with the implementation of crystal plasticity with UO.
105
-e-
d/t ratio = 51
d/t ratio = 64
"
d/tratio=72
1.1
-.0 - - --..
0
0.6
250
.
350
300
-0-
400
pd2t (MPa)
Figure 3-30: Effects of change in internal pressure on buckling strain of the pipe, obtained
with the implementation of crystal plasticity with UO.
106
Chapter 4
Summary and Future Directions
In the present thesis, effects of material anisotropy in yield and flow on buckling behavior
of high-strength pipelines are studied. Results obtained with isotropic (Mises plasticity)
and anisotropic (crystal plasticity) material models are compared in the previous chapter.
The results indicate that due to anisotropy in yield and flow behavior of the high-strength
steel, the predicted buckling strain is higher (by 10 - 20% depending on the d/t ratio)
compared to the buckling strain predicted by an equivalent isotropic material model.
Effects of (cold) UOE forming on buckling response of the pipe are also studied in this
work. Results obtained, with the implementation of crystal plasticity based model, indicated that residual stresses and variation in material properties get introduced during the
UOE forming process. These residual stresses and variation in material properties reduces
the buckling strain predictions (by 20 - 30% depending on the d/t ratio) for the pipes, as
compared to the predictions given by an equivalent crystal plasticity based model which
does not take residual stresses and variation in material properties into account.
Furthermore, it was observed that a 1.0 - 1.5% expansion strain during the E-step
107
results in the highest buckling strain predictions for the pipe, which is in-line with the
current industry practice of applying a 1% plastic expansion strain during the E-step. It
was also seen that for a pipe with d/t ratio of 51, a 1% expansion strain during E-step
effectively reduces the residual stresses in the circumferential direction from
to
t100 MPa, but the axial residual stresses reduces only from
±
-
t900 MPa
+
+350 MPa to
±120
±
MPa.
As shown in Fig. 4-1, buckling strain predictions with the simple material model of
Mises plasticity, which neither accounts for material anisotropy nor accounts for residual stresses and variations in material properties incurred during UO forming process,
is very close to the predictions obtained with a much more complex model of crystal
plasticity with UO which takes into account both the material anisotropy and the UOE
forming operations. The reasons for the two predictions to be close are that while material
anisotropy tends to increase the buckling strain prediction, the effects of residual stresses
and variation in material properties incurred during UO forming tends to reduce the buckling strain prediction by similar amounts, bringing the two predictions close to each other.
The effects of strain aging on buckling strain of the pipes are also studied in this work.
It is shown in the previous chapter that modern strain aging has a minimal effect on the
buckling strain of pipes, as the highest change in buckling strain in any pipe obtained from
the present model is less than 3% for any increase in yield strength in the range 30-50 MPa.
108
1.4
--
1.3
Crystal plasticity without UO
Mises plasticity without UO
+ Crystal plasticity with UO
.4
1.2
.4
r 1.1
6,,
.
.4
.4
1.0
.4
.4
.4
.4
0 0.9
.4
.4
.4
.4
0.8k
.4
.4
0.7
50
I
55
I
60
65
70
75
d/t
Figure 4-1: Buckling strain vs. diameter to thickness ratio plots for all the three implementations of material model considered.
109
4.1
Future directions
In the current work, X100 steel is modeled as a single phase BCC polycrystal. In reality,
the steel is closer to a two phase material consisting of a softer ferrite-phase and a harder
bainite-martensite-phase, as shown in Fig. 1-7. Morphological texture leads to elongated
grain shapes in both phases. The steel could be modeled as a dual phase material, given
access to relevant experimental data. Also more hardening parameters for both the phases
would be needed. The Bauschinger effect would become more prominent if the steel is
modeled as a dual phase material, as the two phases have very different strength-levels [5].
In the current work, longitudinal compression simulations are performed to quantify
buckling resistance of the pipeline. Other types of loading conditions, such as bending
may be imposed on the pipeline to quantify its buckling resistance. Implementation of the
3D poly-crystal plasticity model, as described in this work, may prove computationally
expensive, as half the pipe instead of a thin radial segment would need to be modeled.
To reduce computational effort, implementation of poly-crystal plane stress shell model
may be desirable.
In the present work, the strain induced during UO forming is assumed to be independent of axial and circumferential coordinates. In reality, UO-forming is performed using
mechanical tools, and there would inevitably be non-uniformity in the UO-forming process, which would lead to circumferential and axial variation in bending strains, stresses,
hardening moduli, etc. More representative UO forming operations can be performed,
but experimental data and more details of the UO-forming operations would be needed.
In the present work, geometric perturbation studies are performed and presented in
110
Appendix C. The type of perturbations considered in the present work are limited to
inner diameter and outer diameter perturbations, which locally change the thickness of
the pipe, keeping it axisymmetric. Other types of geometric perturbations, such as out
of roundness variation of pipe may be considered. The out of roundness variation would
break the axisymmetry of the pipe and a full 3D model or shell-based model of the pipe
would be needed. Given relevant experimental data, thickness variation as a function of
axial and circumferential coordinates may also be accounted for in the pipe geometric
model.
The (cold) UOE formed high-strength pipelines are also used for deep water applications. In deep water applications, collapse bucking due to external pressure is a more
dominant mode of failure. Deep water pipelines are much thicker with the d/t ratio in the
range of 10 - 15. With such a low value of d/t ratio, the bending strains introduced during
UOE forming are more severe and may lead to higher circumferential residual stresses and
material anisotropy in the pipeline. An analogous collapse buckling study on such deep
water pipelines may be performed.
111
Appendix A
Buckling Simulations with
Single-Pass vs. Multi-Pass E-step
In this appendix, buckling response of pipe-1 (dimensions given in Table 3.1) obtained
with the single-pass E-step model is compared with the buckling response obtained with
the multi-pass E-step model, as described in Section 2.1.2. The nominal compressive
stress vs. shortening curves obtained with the two models of mechanical expansion are
compared in Figure A-1. As seen from the figure, the two models predict almost identical
peak stress and buckling strain. Therefore, for the purpose of modeling the E-step, a
single-pass mechanical expansion is utilized in the thesis, as it saves computational effort.
112
800
- -
Single pass Estep
Multi pass Estep
700 H
600
0 500
0)
U)
400
300
pd/2t = 300 MPa
d/t = 51
L =3.6m
d 1020.0 mm
t=20.0 mm
-.
200
100 -
0.0
.... ...
0.25
0.50
....... ... ... . .......... -..
0.75
1.00
1.25
Shortening (%)
1.50
1.75
2.00
Figure A-1: Comparison of nominal stress vs. shortening curves obtained with single-pass
and multi-pass mechanical expansion models.
113
Appendix B
Effects of Using All Grains in the
Taylor Model
In the framework of crystal plasticity based model, a polycrystalline material is visualized
as an aggregate of many single crystals. In this work, the Taylor model is utilized to determine the average polycrystal constitutive response of the material. In this thesis, the
texture of a material point is taken as an average of 125 highest-weighted grains, instead
of all the 860 weighted grains given by the experiments. The effects of this simplification
on material and buckling response are studied in this appendix.
Tensile and compressive material stress-strain curves obtained with the material model
of crystal plasticity with UO are shown in Fig. B-1 and Fig. B-2, respectively. The curve
marked as "860 weighted grains" utilizes all the 860 weighted grains obtained from the
experiments and the curve marked as "125 weighted grains" utilizes only the 125 highestweighted grains for the crystallographic texture in the Taylor averaging scheme. It can
be seen from the figure that 860 weighted grains and 125 weighted grains predict almost
identical material response in both tension and compression for loading in both rolling
114
and transverse direction. This result implies that material anisotropy is captured well
with only 125 highest-weighted grains, and there is no need to use all the 860 weighted
grains to represent the crystallographic texture of the material.
1000
900800
700....
Rolling
direction
6000
500-
0
400 -
I-
2
3000200
-
- -
100j
- -
U
0.0
0.5
1.0
1.5
RD I ension Simulation
TD Tension Simulation
RD Tension Simulation
TD Tension Simulation
2.0
2.5
3.0
3.5
True T ensile S train (%)
*~
k 1 grains)
(125 grains)
(860 grains)
(860 grains)
4.0
4.5
-~-
5.0
Figure B-1: Comparison of tensile stress-strain curve in RD and TD obtained with the
crystallographic texture of 125 and 860 weighted grains.
Fig. B-3 shows the nominal stress vs. shortening curve obtained for pipe buckling
simulation on pipe-1, whose dimensions are given is Table 3.1. The material model of
Crystalplasticity with UO is utilized in the simulations. From this figure it can be seen that
if the crystallographic texture is assumed to be consisting of 125 highest-weighted grains,
115
1000
900
0.
800
U)
700
600
Rolling
direction
CL
0
500
400
300
200
- -
100
-'(V
0.0
0.5
1.0
RD Compression
TD Compression
RD Compression
TD Compression
Simulation
Simulation
Simulation
Simulation
(125 grains)
(125 grains)
(860 grains)
(860 grains)
1.5
2.0
4.0
2.5
3.0
3.5
True Compressive S train (%)
4.5
5.0
Figure B-2: Comparison of compressive stress-strain curve in RD and TD obtained with
the crystallographic texture of 125 and 860 weighted grains.
116
the obtained nominal stress vs. shortening curve is identical to the one obtained with
assuming the crystallographic texture to be consisting of 860 weighted-grains. Therefore,
for computational efficiency the crystallographic texture of the material is assumed to be
made up of only 125 highest-weighted grains in the rest of this thesis.
800
Crystal plasticity with UO (125 weighted grains
-
Crystal plasticity with UO (860 weighted grains)
700 F
600
500
400
0A
U)
E
pd/2t = 300 MPa
d/t = 51
L =3.0 m
d =1020.0 mm
t=20.0 mm
0 300
200
100
0
0.0
0.25
0.50
0.75
1.00
1.25
Shortening (%)
1.50
1.75
2.00
Figure B-3: Nominal stress vs. shortening curves obtained with the crystallographic
texture of 125 and 860 weighted grains.
117
Appendix C
Perturbation Study
Plastic buckling resistance may be highly sensitive to any geometric imperfections present
in the pipes. To quantify the effects of geometric inhomogeneity a study is performed and
presented here.
Two types of perturbations, namely perturbation in inner diameter and perturbation in outer diameter are considered, as shown schematically in Fig. C-1. The variable
parameters are perturbation magnitude, a, and its wavelength, A.
C.1
Perturbation study with Mises plasticity model
Fig. C-2, C-3, and C-4 shows the nominal stress vs. shortening curve obtained with the
material model of Mises plasticity without UO for pipes of d/t ratio 51, 64, and 72, respectively. Magnitude perturbations in inner diameter, with a fixed wavelength A = 1.5 x t, are
performed in these simulations. As can be seen from theses figures, buckling strain reduces
as the perturbation magnitude in inner diameter is increased. Fig. C-5 plots the buckling
strains as a function of perturbation magnitude for all the three d/t ratios considered. As
seen from this figure, an increase in the perturbation magnitude of inner diameter from
118
2% (of t) to 8% (of t) reduces the buckling strain by ~ 25% for a pipe with d/t ratio of 51.
Similar magnitude perturbations in outer diameter are also performed. The nominal
stress vs. shortening curves obtained with these perturbations are shown in Fig. C-6, C-7,
and C-8 for pipes of d/t ratio 51, 64, and 72, respectively. Fig. C-9 plots the buckling
strains as a function of perturbation magnitude in outer diameter for all the three d/t
ratios considered. As can be seen from Fig. C-9, an increase in magnitude perturbation
in outer diameter from 2% (of t) to 8% (of t) reduces the buckling strain by ~ 14% for a
pipe with d/t ratio of 51.
From Fig. C-5 and Fig. C-9, it can be concluded that a perturbation in inner diameter
changes the buckling strain by a larger amount compared to when a similar perturbation
in outer diameter is performed.
Wavelength perturbation studies on outer and inner diameter are also performed. The
results obtained with these variations are shown in Fig. C-10 to Fig. C-17. As seen from
Fig. C-13 and Fig. C-17, a four-fold increase in inner diameter perturbation wavelength
reduces the buckling strain by ~ 30% but the same four fold increase in outer diameter
perturbation wavelength reduces the buckling strain by only
-
20%, for a pipe with a d/t
ratio of 51.
C.2
Perturbation study with Crystal plasticity model
Magnitude and wavelength perturbation studies with the material model of Crystal plasticity with UO are also performed. The results obtained with crystal plasticity based
119
material model are shown in Fig. C-18 to Fig. C-33. As was the case with the Mises
plasticity model, with the crystal plasticity-based model, the difference between buckling
strain predictions are much higher when perturbations in inner diameter are performed
compared to when perturbations in outer diameter are performed. For example, for the
pipe with a d/t ratio of 51, a perturbation in the magnitude of inner diameter from 2%
(of t) to 8% (of t) reduces the buckling strain by
-
30%, as seen in Fig. C-21, where as a
perturbation in the magnitude of outer diameter from from 2% (of t) to 8% (of t) reduces
the buckling strain by only ~ 17%, as seen in Fig. C-25.
Similarly, a wavelength perturbation in inner diameter from 0.75 x t to 3.00 x t results
in a reduction of ~ 26% in buckling strain predictions for the pipe with a d/t ratio of 51,
as seen in Fig. C-29. A similar perturbation in the wavelength of outer diameter reduces
the buckling strain by only
-
17% for the pipe with a d/t ratio of 51, as seen in Fig. C-33.
Again, it can be concluded that any type of perturbation in inner diameter changes
the buckling strain by a larger amount compared to when a similar perturbation in outer
diameter is performed.
120
t
t
L
L
I
HI
d/2
d/2
$I
I
A ZO A
(a) Perturbation in inner diameter
(b) Perturbation in outer diameter
Figure C-1: Schematic showing perturbation in inner and outer diameter.
121
800
i
n
I
,
-Perturbation magnitude, a = 0.02t
- - Perturbation magnitude, a = 0.04t
-
Perturbation magnitude, a = 0.06t
700 k
Perturbation magnitude, a = 0.08t
600
0
0)
U)
500
400
pd/2t = 300 MPa
300
d/t = 51
L = 3.0m
020.0 mm
d
t=20.0 mm
200
100
n
0.0
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Shortening (%)
Figure C-2: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of the Mises plasticity without UO. Magnitude perturbations in
inner diameter, with a fixed wavelength A = 1.5 x t, are performed in these simulations.
122
800- -
700-
Pe-tPIrurbation
Perturbation
" Perturbation
Perturbation
magnitude,
magnitude,
magnitude,
magnitude,
a = 0.02t
a = 0.04t
a = 0.06t
a = 0.08t
6004' '500--
400I-
E
00
S300
-d/t
. ..
200--E 00
z
pd/2t =300 MPa
= 64
L = 3.0 m
d = 1219.2 mm
t = 19.1 mm
1000
0.0
0.25
0.50
0.75
1.00
1.25
Shortening (%)
1.50
1.75
2.00
Figure C-3: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of the Mises plasticity without UO. Magnitude perturbations in
inner diameter, with a fixed wavelength A = 1.5 x t, are performed in these simulations.
123
800
-Perturbation
Perturbation
Perturbation
. Perturbation
---
.
.
700
magnitude,
magnitude,
magnitude,
magnitude,
a=
a=
a=
a=
0.02t
0.04t
0.06t
0.08t
600
(U
500
E
400
E
0.
300
pd/2t-= 300 MP
d/t = 72
L = 3.0 m
d = 914.4 mm
t= 12.7 mm
200
zU
100
-
0
0.0
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Shortening (
Figure C-4: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of the Mises plasticity without UO. Magnitude perturbations in
inner diameter, with a fixed wavelength A = 1.5 x t, are performed in these simulations.
124
1.3
--
d/t ratio = 51
d/t ratio = 64
'..
d/t ratio = 72
-
1. 2
1.1
....
....
...
.....
...
...
. ..... .... .
1
C
(0 . 9
f
f
-
....... ...........
ff
~Im ~
0.8
*
11,
0.7 F
II ,
~
w
.4
III
,II
III
111111
1
I
,-I.
I1.I
/
.
IIII
1
.,.0i10..
.
-
0.6F
0.5
2
4
6
8
Perturbation magnitude (% of pipe wall thickness)
Figure C-5: Effect of change of perturbation magnitude (a) in inner diameter of pipe on
its buckling strain with a fixed wavelength (A = 1.5 x t) of perturbation. Mises plasticity
without UO material models are used in the simulations.
125
800
i
-- Perturbation
Perturbation
. ......
.
Perturbation
Perturbation
700
0
magnitude, a = 0.02t
magnitude, a = 0.04t
magnitude, a = 0.06tmagnitude, a = 0.08t
600
500
400
-U)
0
pd/2t = 300 MPa
d/t = 51
L. 300
L = 3.Om
200
mm
-...
.d=1020.0
-
t=20.0 mm
100
0
0.0
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Shortening (%)
Figure C-6: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of the Mises plasticity without UO. Magnitude perturbations in
outer diameter, with a fixed wavelength A = 1.5 x t, are performed in these simulations.
126
800P
-
-Perturbation
magnitude, a = 0.02t
700- Perturbation magnitude, a = 0.04t
700 -
Perturbation magnitude, a = 0.06tPerturbation magnitude, a = 0.08t
....
600
500
4
400-
E
0 300-
= 300 MPa
.pd/2t
d/t =64
- 200
L = 3.0 m
d = 1219.2 mm
z
t= 19.1 mm
100
0
0.0
0.25
0.50
0.75
1.00
1.25
Shortening (%)
1.50
1.75
2.00
Figure C-7: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of the Mises plasticity without UO. Magnitude perturbations in
outer diameter, with a fixed wavelength A = 1.5 x t, are performed in these simulations.
127
800
Perturbation magnitude,
-- Perturbation magnitude,
'I " Perturbation magnitude,
Perturbation magnitude,
- -
700-
a=
a=
a=
a=
0.02t
0.04t
0.06t
0.08t
600 -
0
500
400
0
1..
pd/2t =300 MPa
d/t = 72
300
L = 3.0 m
200
n
-
-
100
0.0
-...
d=914.4 mm
t =12.7 mm
-
0.25
0.50
1.25
0.75
1.00
Shortening (%)
1.50
1.75
2.00
Figure C-8: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of the Mises plasticity without UO. Magnitude perturbations in
outer diameter, with a fixed wavelength A = 1.5 x t, are performed in these simulations.
128
1.4
d/t ratio = 51
d/t ratio = 64
'
d/t.ratio
= 72
.. ....... .........
.. ....
. -e-
................
..
1.3
1.2
0
1.1F
CO
In
1~ ~
0.9 F
a,..
5
~
........................
m m
-
......
0
0.8
..............
. ... ... . . ..
. ......
..
0
0.7'2
4
6
8
Perturbation magnitude (% of pipe wall thickness)
Figure C-9: Effect of change of perturbation magnitude (a) in outer diameter of pipe on
its buckling strain with a fixed wavelength (A = 1.5 x t) of perturbation. Mises plasticity
without UO material models are used in the simulations.
129
800
---
700
CL
Perturbation
Perturbation
Perturbation
Perturbation
wavelength,
wavelength,
wavelength,
wavelength,
X=
X=
? =
X=
0.75
1.50
2.25
3.00
xt
xt
xt
xt
600
0
500
-
------
---
U)
400
0)
a)
pd/2t = 300 MPa
d/t =51
L = 3.0 m
d= 1020.0 mm
t=20.0 mm
300
200
100
-
0
0.0
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Shortening (%)
Figure C-10: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of the Mises plasticity without UO. Wavelength perturbations in
inner diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these
simulations.
130
800
- -
700-
-
CU
Perturbation wavelength, X = 0.75 x t
Perturbation wavelength, X = 1.50 x t
Perturbation wavelength, ? = 2.25 x t -
Perturbation wavelength, X = 3.00 x t
600 -
0
500
400
0
I-
300
200
100
n
0.0
-.. .d=
pd/2t =300 MPa
d/t = 64
L=3.0 m
1219.2 mm
1=19.1 mm
.
-.
0.25
0.50
1.25
Shortening (%)
0.75
1.00
-..
...
1.50
1.75
2.00
Figure C-11: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of the Mises plasticity without UO. Wavelength perturbations in
inner diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these
simulations.
131
-
800
-
L
I
I
Perturbation wavelength, X = 0.75 x t
-- Perturbation wavelength, X = 1.50 x t
- -
700-
-Perturbation wavelength, X = 2.25 x t Perturbation wavelength, X = 3.00 x t
600#A
500
$ 400 C-
E
0
pd/2t =M300 MPa
d/t =72
3.0 m
u 300 --
- 200 -
d =91.4.4 mm
t =12.7 mm
0
Z
1000
0.0
0.25
0.50
0.75
1.00
1.25
Shortening (%)
1.50
1.75
2.00
Figure C-12: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of the Mises plasticity without UO. Wavelength perturbations in
inner diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these
simulations.
132
1.3
--- d/t ratio = 51
-e-
1
d/tratio=64
d/t ratio = 72
1.1...- .
0.70.60.5
0.75
1.50
2.25
Perturbation wavelen th x t)
3.00
Figure C-13: Effect of change of perturbation wavelength (A) in inner diameter of pipe on
its buckling strain with a fixed magnitude (a = 4% of t) of perturbation. Mises plasticity
without UO material models are used in the simulations.
133
800
SPerturbation
Perturbation
Perturbation
Perturbation
- .-
700
wavelength, = 0.75
wavelength, X = 1.50
wavelength,
2.25
wavelength, X == 1.25
xt
xt
x
x tt
Perturbation wavelength, X= 3.00 x t
C.
600
0
-7
500
U)
400
E
0
pd/2t = 300 MPa
d/t = 51
L = 3.Om
300
-d = 1020.0 mm
200
-
t=20.0 mm
z
100
n
0.0
-
.........
0.25
0.50
....
0.75
1.00
1.25
1.50
1.75
2.00
Shortening (%)
Figure C-14: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of the Mises plasticity without UO. Wavelength perturbations in
outer diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these
simulations.
134
800SPerturbation wavelength,
- Perturbation wavelength, X =
Perturbation wavelength, X =
Perturbation wavelength, X=
700IL
0.75 x
0.50 x
2.25 x
3.00 x
t
t
t -
t
600-
0
0
0
U)
500
400
pd/2t = 300 MPa
300
d/t = 64
L = 3.0 m
d= 1219.2 mm
t= 19.1 mm
200 - ...
100
- ......
0
0.0
0.25
..
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Shortening (%)
Figure C-15: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of the Mises plasticity without UO. Wavelength perturbations in
outer diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these
simulations.
135
800
-- Perturbation
-- Perturbation
-Perturbation
Perturbation
700 CU
wavelength,
wavelength,
wavelength,
wavelength,
X=
X=
k=
X=
0.75 x
1.50 x
2.25 x
3.00 x
t
t
t t
600 500
0
0
0
400
pd/2t =300MP
d/t = 72
300
E
-
L =3.O m
200
d 914.4 mm
t= 12.7 mm
100
-.
n
0.0
____
0.25
0.50
-------
0.75
1.00
1.25
1.50
1.75
2.00
Shortening (%)
Figure C-16: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of the Mises plasticity without UO. Wavelength perturbations in
outer diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these
simulations.
136
1.4
-e-o'
1 .3 -.
d/t ratio =51
d/t ratio = 64
d/t ratio = 2
.. .
-..
~1.1CI)
-
0.9
-
0.8-
0. 7
0.75
1.50
2.25
Perturbation wavelenith x t)
3.00
Figure C-17: Effect of change of perturbation wavelength (A) in outer diameter of pipe on
its buckling strain with a fixed magnitude (a = 4% of t) of perturbation. Mises plasticity
without UO material models are used in the simulations.
137
800
- -
700 -
Perturbation magnitude, a = 0.02t
Perturbation magnitude, a = 0.04t
Perturbation magnitude, a = 0.06t
Perturbation magnitude, a = O.08t
IL
600
0
500
400
0
1z
pd/2t = 300 MPa
300
d/t = 51
L = 3.0 m
d=1020.0 mm
t=20.0 mm
200
100 - ... ..
n
0.0
0.25
0.50
0.75
1.00
1.25
Shortening (%)
1.50
1.75
2.00
Figure C-18: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of crystal plasticity with UO. Magnitude perturbations in inner
diameter, with a fixed wavelength A = 1.5 x t, are performed in these simulations.
138
800
Perturbation magnitude, a = 0.02t
---
700 F
.
Perturbation magnitude, a = 0.04t
Perturbation magnitude, a = 0.06t
Perturbation magnitude, a = 0.08t
600
500
0
-
-~
400
0
U)
pd/2t=300 MPa
d/t = 64
L = 3.Om
d = 1219.2 mm
t= 19.1 mm
300
200
100
-
0.0
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Shortening (%)
Figure C-19: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of crystal plasticity with UO. Magnitude perturbations in inner
diameter, with a fixed wavelength A = 1.5 x t, are performed in these simulations.
139
800
- -
-'-
700 k
Perturbation magnitude, a = 0.02t
Perturbation magnitude, a = 0.04t
" Perturbation magnitude, a = 0.06t
Perturbation magnitude, a = 0.08t
600
-%
L-.
CD
500
CL
0
400
0
E..
300
pd/2t 300 MPa"'%.,
200
d/t = 72
L = 3.0m
d = 914.4 mm
t= 12.7mm
0.
-
z
100 - . . .
0
0.0
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Shortening (%)
Figure C-20: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of crystal plasticity with UO. Magnitude perturbations in inner
diameter, with a fixed wavelength A = 1.5 x t, are performed in these simulations.
140
1.3
-o-' d/t ratio = 51
-o- d/t ratio = 64
1.2
.
. .
.
.
.
.
.
. . . . . . . .
.. . . .. .. . ..
. .. .. .....
. .
.
.
/..a.o.7
0
.......
1
C,)
0)
0.9
U
0.8
.=
.. ...
r.... . -
I
m
0.7
....
. . . . . .......
. .......
--
- .
... . . . .
les
..i
.i
. ..
i
. . ... ...
... ....... .....
0.6
0.5
2
4
6
8
Perturbation magnitude (% of pipe wall thickness)
Figure C-21: Effect of change of perturbation magnitude (a) in inner diameter of pipe on
its buckling strain with a fixed wavelength (A = 1.5 x t) of perturbation. Crystal plasticity
with UO material models are used in the simulations.
141
800
---
I
I
Perturbation magnitude, a
'
Perturbation magnitude, a = 0.04t
=
0.02t
Perturbation magnitude, a = 0.06t
... Perturbation magnitude, a = 0.08t
700 -
600
CL
500
U)
0 400
0)
4)
pd/2t
300
= 300 MPa
d/t =51
L = 3.&m
E
d=1020.0 mm
200
t=20.0 mm
100
(n
0.0
-1
-f
0.25
0.50
0.75
1.00
1.25
Shortening (%)
1.50
1.75
2.00
Figure C-22: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of crystal plasticity with UO. Magnitude perturbations in outer
diameter, with a fixed wavelength A = 1.5 x t, are performed in these simulations.
142
800
-
--- 7000.
Perrbation magnitude,
-Perturbation
magnitude,
Perturbation magnitude,
Perturbation magnitude,
a = 0.02t
a = 0.04t
a = 0.06t
a = 0.08t
600500-
. .
.
.
....
.. . . .
4001L
0S 3 0 0 ------ ......
= 300 MPa
pd/2t
....
d/t = 64
L = 3.0 m
-
d = 1219.2 mm
t 19.1 mm
- 2000
1000
0.0
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Shortening (%)
Figure C-23: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of crystal plasticity with UO. Magnitude perturbations in outer
diameter, with a fixed wavelength A = 1.5 x t, are performed in these simulations.
143
800
- -
700 F
I-
Perturbation
Perturbation
" Perturbation
Perturbation
magnitude,
magnitude,
magnitude,
magnitude,
a=
a=
a=
a=
0.02t
0.04t
0.06t
0.08t
600
500
400
'-.
E
pd/2t =300 MPa
d/t = 72
L = 3.0 m
d 914.4 mm
t= 12.7mm
0 300
0
1z
200
100 - .
.
0
0.0
0.25
.
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Shortening (%)
Figure C-24: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of crystal plasticity with UO. Magnitude perturbations in outer
diameter, with a fixed wavelength A = 1.5 x t, are performed in these simulations.
144
-e
1.2 - ..
0.6
.
2
....
'
.....
4
6
dit ratio = 51
d/t ratio = 64
d/t ratio = 72
Perturbation magnitude (% of pipe wall thickness)
8
Figure C-25: Effect of change of perturbation magnitude (a) in outer diameter of pipe on
its buckling strain with a fixed wavelength (A = 1.5 x t) of perturbation. Crystal plasticity
with UO material models are used in the simulations.
145
800
- -
700
(U
-
Perturbation wavelength, X = 0.75
Perturbation wavelength, X = 1.50
Perturbation wavelength, X = 2.25
Perturbation wavelength, X = 3.00
xt
xt
xt xt
600
E~ 500
0
0
0
0
0
400
pd/2t = 300 MPa
300
d/t =51
L = 3.fm
d=1020.0 mm
t=20.0 mm
200
100 - .
0
0.0
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Shortening (%)
Figure C-26: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of crystal plasticity with UO. Wavelength perturbations in inner
diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these
simulations.
146
800
- -
-00-%
700 F
Perturbation wavelength, X = 0.75 x t
Perturbation wavelength, X = 1.50 x t
-'Perturbation wavelength, X = 2.25 x t Perturbation wavelength, X = 3.00 x t
600
500
CL
400
E
z.
- pd/2t.= 300 MPa
0 300
dit = 64
L = 3.0 m
d = 1219.2 mm
t= 19.1 mm
0 200
100
0
0.0
0.25
0.50
0.75
1.00
1. 25
Shortening (%)
1.50
1.75
2.00
Figure C-27: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of crystal plasticity with UO. Wavelength perturbations in inner
diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these
simulations.
147
800
- -
--
700
-.....-
'Perturbation
wavelength, X = 2.25 x t
Perturbation wavelength, X = 3.00 x t
CL
D
Lo
-
Perturbation wavelength, X = 0.75 x t
Perturbation wavelength, X = 1.50 x t
600
500
400
0 300
pd/2t= 300 MP"d/t = 72
L =3.0m
d =914.4 mm
t = 12.7mm
0
E
1z
200
.......-. . .
100
n
0.0
0.25
0.50
0.75
1.00
1.25
Shortening (%)
1.50
1.75
2.00
Figure C-28: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of crystal plasticity with UO. Wavelength perturbations in inner
diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these
simulations.
148
1.2
-'
d/t ratio = 51
d/t ratio = 64
d/t ratio=72
1.1
0.9c
=0.80.7
0.6 --
0.5
0.75
---
1.50
2.25
Perturbation wavelenith x t)
3.00
Figure C-29: Effect of change of perturbation wavelength (A) in inner diameter of pipe
on its buckling strain with a fixed magnitude (a = 4% of t) of perturbation. Crystal
plasticity with UO material models are used in the simulations.
149
800
CL
SPerturbation
Perturbation
Perturbation
Perturbation
700
wavelength, = 0.75
wavelength, X = 0.50
wavelength, X = 2.25
wavelength, k = 3.00
x t
xt
xt
xt
600
Perturbation~
waeegh X
.0
500
400
U..
E
pd/2t = 300 MPa
0 300
d/t = 51
L = 3.0m
d = 1020.0 mm
t=20.0 mm
0 200
100
0
0.0
-TO
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Shortening (%)
Figure C-30: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of crystal plasticity with UO. Wavelength perturbations in outer
diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these
simulations.
150
800
700
SPerturbation
--Perturbation
Perturbation
Perturbation
-. ...-.
wavelength, = 0.75
wavelength, X = 1.50
wavelength, X = 2.25
wavelength, k = 3.00
x t
xt
xt
xt
600
0
-
500
-.
400
0
I-
pd/2t = 300 MPa
300
d/t =64
L = 3.0 m
d 1219.2 mm
t= 19.1 mm
200
100
-
0
0.0
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Shortening (%)
Figure C-31: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of crystal plasticity with UO. Wavelength perturbations in outer
diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these
simulations.
151
800
0.75 x t
SPerturbation wavelength,
Perturbation wavelength, X = 1.50 x t
Perturbation wavelength, X = 2.25 x t
Perturbation wavelength, X = 3.00 x t
700 -..
C-
600
A
500
0
0
400
E
0
0
300
pd/2t= 300 MPad/t =72
L = 3.0 m
d =914.4 mm
t= 12.7 mm
-
200
z
0Ov"',
100
-
0
0.0
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Shortening (%)
Figure C-32: Nominal stress vs. shortening curve obtained from pipe buckling simulations
with the material model of crystal plasticity with UO. Wavelength perturbations in outer
diameter, with a fixed perturbation magnitude a = 4% of t, are performed in these
simulations.
152
1.3
d/t ratio = 51
d/t ratio = 64
-'
1.2
d/t ratio= 72
.1
%m
0.
0.6
0.75
1.50
2.25
Perturbation wavelenith x t)
3.0
Figure C-33: Effect of change of perturbation wavelength (A) in outer diameter of pipe
on its buckling strain with a fixed magnitude (a = 4% of t) of perturbation. Crystal
plasticity with UO material models are used in the simulations.
153
Appendix D
Pipe Buckling with Higher Mesh
Density
A pipe buckling simulation with much finer pipe mesh is performed, and results are
compared to the results obtained with the standard mesh density given in Chapter 3.
The two meshes are shown in Fig. D-1. Nominal stress vs. shortening curves obtained for
pipe-1, whose details are given in Table 3.1, are plotted in Fig. D-2. This figure shows
identical buckling response for the two meshes; therefore, the coarser mesh is used in the
rest of the thesis for computational efficiency.
154
a
a
a
a
L
L
(b
Fi
s Im
(a) Coarse mesh
Figure D-1: Mesh details for the coarse and the fine mesh, the mesh is shown on an
axisymmetric plane. Coarse mesh is the default mesh, which is used in the rest of the
thesis.
155
800
- -
Finer mesh
Default mesh
700
(U
600
.A
.............
........ ..
500
0
0
400
0
z.
pd/2t = 300 MPa
d/t = 51
300
L = 3.Qm
d=1020.0 mm
t =20.0 mm
200
.
100
. .
-
-
.
C
0.0
0.25
0.50
0.75
1.00
1.25
Shortening (%)
1.50
1.75
2.00
Figure D-2: Comparison of nominal stress vs. shortening curves obtained with the default
mesh and a much finer mesh.
156
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