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Content
Acknowledgements....................................................................................................................... 4
1
Introduction .......................................................................................................................... 5
1.1
Work outline.................................................................................................................. 5
1.2
Objective ....................................................................................................................... 5
1.3
Straightening in a 7-roll multi staggered roll straightening machine (327). ......................... 7
1.4
The tensile test curve ..................................................................................................... 9
1.4.1
2
3
4
Anelasticity............................................................................................................11
1.5
Modulus of elasticity .....................................................................................................11
1.6
Determination of proof strength, Rp0.2. .........................................................................12
1.7
Residual stress ..............................................................................................................12
1.8
The Bauschinger effect ..................................................................................................13
1.9
Process chain ................................................................................................................14
1.10
Cyclic stress-strain behaviour .........................................................................................14
Test material ........................................................................................................................15
2.1
Material A.....................................................................................................................15
2.2
Material B.....................................................................................................................15
Means and methods .............................................................................................................16
3.1
Simulation and calculations of adequate strain ...............................................................16
3.2
Tensile test ...................................................................................................................20
3.3
Low cycle fatigue test or simulated straightening ............................................................20
3.4
Authentic tests..............................................................................................................20
3.5
Collection of specimens .................................................................................................20
3.7
Tensile test procedure ...................................................................................................25
3.8
Low cycle fatigue test procedure....................................................................................27
Results .................................................................................................................................34
4.1
A volume element´s way through the straightener ..........................................................34
4.1
Calculation model .........................................................................................................35
4.2
FEM-simulation .............................................................................................................40
4.3
Modulus of elasticity .....................................................................................................42
4.4
Tensile tests..................................................................................................................44
-1-
4.4.1
Material A .............................................................................................................44
4.4.2
Material B .............................................................................................................47
4.5
5
Low cycle fatigue test ....................................................................................................49
4.5.1
Material A dynamic & true proof strength curve ......................................................50
4.5.2
Material B dynamic & true proof strength curve ......................................................55
4.6
Hardness test................................................................................................................58
4.7
Micro structure .............................................................................................................60
Discussion ............................................................................................................................63
5.1
Tensile testing procedure ..............................................................................................63
5.2
Tensile test results ........................................................................................................64
5.2.1
In pilgering direction ..............................................................................................64
5.2.2
Transverse to pilgering direction.............................................................................64
5.3
Modulus of elasticity .....................................................................................................64
5.4 Analytical calculation and FEM-simulation............................................................................65
6
Conclusions ..........................................................................................................................66
7
Proposed measures for short time quality improvement.........................................................66
8
Proposed further investigations.............................................................................................67
9
References ...........................................................................................................................68
Appendix 1- Derivation of analytical beam model ..........................................................................69
A1
A1.1
Elementary case analysis ...............................................................................................69
Forces of reaction,
displacement and support angles in beam part 1: ........................69
Displacement and support angles: .........................................................................................70
Originating from point load: ..................................................................................................70
Originating from bending moment: .......................................................................................71
A1.2
Forces of reaction,
displacement and support angles in beam part 2: ........................71
Displacement and support angles: .........................................................................................73
Originating from point load: ..................................................................................................73
Originating from bending moment: .......................................................................................73
A1.3
Forces of reaction,
displacement and support angles in beam part 3: ........................74
Displacement and support angles: .........................................................................................75
Originating from point load: ..................................................................................................75
Originating from bending moment: .......................................................................................75
-2-
Appendix 2-Stiffness matrix ..........................................................................................................77
Appendix 3-Matlab code for analytical beam model.......................................................................80
Appendix 4-Matlab code for the limited FEM-model ......................................................................83
Appendix 5-Additional lcf proof strength diagrams ........................................................................91
-3-
Acknowledgements
I want to direct a special and personal thanks to my instructors Marie-Louise Nyman and
Stanislav Riljak. Marie-Louise, you have been most supporting throughout the entire project.
I´m glad that I could share your long and grounded experience in the cold working processes.
It has also been very valuable that you have introduced me to the right people. Stanislav has
been very important in the work with building the analytical model, and I thank you for our
brain storming in the beginning and the more definite rules for the strength analysis.
Stefan Heino introduced me in his analytical model of the straightening. He also gave me a
very thorough literature survey. Gou-Cai has explained a lot of the in depth mechanisms
involved in the cold working and straightening processes. I will also thank Olle Eriksson who
has made the most efficient and time preserving MATLAB program, for analysing raw-data
from the low cycle fatigue tests.
Rolf Gustavsson has performed the FEM-simulation and been most helpful on the work with
the analytical model.
Fredrik Meurling has made a parallel investigation with much the same aim, our discussions
and interchange of experience between the two projects have been a big help. (1)
I also want to thank Stefan and Annika in the work shop, which succeeded in getting the
specimens ready on time, Birger who has contributed with the hardness measurement and
microstructure and Gunnar, Håkan and Dan who have performed the mechanical tests.
Last but not least I want to thank Carl-Filip Lindahl, who took from his time to read through
this work and gave me very valuable advices to improve the disposition.
-4-
1
Introduction
1.1
Work outline
An investigation of the materials deformation history prior to the straightening process will be
made. The deformation history and further investigations together will hopefully make it
possible to state how much the material properties are changed during the straightening
process.
Authentic field studies will be carried out on tubes taken from the pilgering machine and after
the 7- roll multi staggered roll straightening machine, RSM. From these tubes specimens will
be manufactured and tested to show the different process steps effect on the proof strength,
Rp0.2 value.
Parallel to the above specified procedure there will be a simulated cold working and
straightening process. To simulate the process there will be specimens manufactured. The
specimens will then be exposed to pre strain and or low cycle fatigue tests, LCF. When the
specimens originate from hollows, the specimens will be pre strained to suitable levels
depending on the material. In the simulated process it will be easier to isolate different
essential parameters and to see the specific effect on the proof strength of those. The
simulation process of the straightening will be carried out such as an LCF test with varying
strain levels.
An analytical beam model will be helpful during the LCF tests. The analytical model can
predict strain levels which will be input into the LCF simulation.
Because of the analytical models limitation to predict only elastic strain there will also be a
FEM simulation made. With help from the FEM simulation we hope to get a qualitative
estimation of how to interpolate the analytical model to plastic deformation.
1.2
Objective
Traditionally the straightening machines were developed to prevent strain hardening during
the process. The major part of tubes that have been straightened has also traditionally been
annealed. For some decades some steel manufactures have learned to benefit the austenitic
and duplex steels inherent possibilities to strain harden. Cold working induces strain
hardening and that lead to considerably higher yield strength. The experience from
straightening of heavily cold worked steel grades is therefore limited depending on the above
described history. Straightening has mostly been a concern of how to get the tubes straight
enough, with little thought of that the process itself may affect the material properties.
In Sandvik Materials Technology, SMT, there are some grades especially situated in oil and
gas applications that are of different structures. Two of them are Material A and Material B.
Their strength are increased considerably through pilgering at room temperature and followed
-5-
by straightening without any annealing in between. SMT need to know more about these two
materials behavior depending on straightening.
SMT has therefore decided to make a deeper investigation of the straightening process of the
above steel grades which are straightened in the machine 327. The machine 327 is a RSM,
just as described above.
Some investigations have already been made, but the results are ambiguous.
With help from an analytical model, LCF tests and a FEM-simulation, the increase of know
how should also give better possibilities to set right process parameters in the straightening
machine 327. With more specific process parameters one should be able to predict and
achieve desired outcome.
Because of the large amount of specimens extracted from just one tube, it will also give better
statistics on variations of proof strength throughout the tube.
-6-
1.3
Figure 1:
Straightening in a 77-roll multi staggered roll straightening machine
(327).
Photo of the multi staggered cross roll straightening machine 327.
There are two kinds of continuous straightening processes, reeling and cross roll
straightening, see fig.1. The reeling machine is mostly used for making bars straight and the
latter for tubes. The cross roll straightener has only concave rolls and the reeling machine has
got concave pressure rolls and convex working rolls. (2) The major difference between
reeling and cross roll straightening is that in a reeling machine a bending moment is built up
even in the specific roll pair, and therefore more of the tube ends can also be straightened. In a
cross roll straightener the tube will pass through 5, 6, 7, 9, 10 or as in the most recent
machines 12 rolls.
To make possibilities to considerable straightening, bending is used. Straightening by bending
is made by three point bending, that is the rolls are displaced from the centre line.
Also ovalization, application of compressive load between the pair of rolls, contributes to
straightening of the tubes, see figure 2. In the right picture the arrows to the right show that
through the tube wall the stress changes from compression on the inner wall, to tensile on the
outer face. Of course the picture is exaggerated to be more descriptive. In the upper part there
will be tensile stress induced even on the inner face, but not in magnitudes that will induce
plastic strain.
-7-
The ovalization will make the tubes more cylindrical in cross section, it will also result in
relaxation and equalization of residual stresses, in that sense the ovalization will also make the
tube more straight, but not in the magnitude as bending.
Figure 2:
Depiction of ovalization. The picture to the right shows that the stress changes from compressive to
tensile from the inside to the outside during simultaneous outer point loads.
-8-
1.4
The tensile test curve
In cold worked strain hardening steels there are some differences in the stress strain diagram
compared to an ordinary carbon steel diagram, see figure 3. In a carbon steel there is often a
first well defined straight elastic part.
In a cold worked, not annealed, austenitic or duplex stainless steel on the other hand the look
of the initial part is often nothing like the previous at all, see fig 4. The initial part does in fact
consist of two parts even though that it is hard to recognize. The first elastic part is sometimes
very hard to find or determine, because it is so small, for deeper information, see the part
about modulus of elasticity below. The part that follows the first small part is then curved due
to anelasticity, see passage 1.4.1 below. As the modulus of elasticity should be established in
the first small linear elastic part, it is quite obvious that it is not as straightforward as in the
earlier mentioned material group.
The very first part is also affected by the tensile testing machines inner friction and could
therefore not be used for determination of the modulus of elasticity. To get a reliable result of
the modulus of elasticity and later the proof strength another approach to determine the proof
strength was therefore developed, described under the passage 1.6
Figure 3:
Depiction of ordinary carbon steel, with a first well defined straight elastic part, followed with
Lüders strain and then the strain hardening part before yielding sets in. Source: SS-EN standard
10002-1, pp. 21
-9-
Material A, strain 0.8 %
stress-strain diagram
900
800
Stress [MPa]
700
600
500
400
2-3-12_1
300
200
100
0
0
1
2
3
4
5
Strain [%]
Figure 4:
Sample of a cold worked material A steel exposed to a few load changes of a total strain of 0,8 %,
note that prior to yielding there is a slight change of curvature basically all the way, it is definitely
not straight.
In SMT there is also a mathematical algorithm in the evaluation software that works with the
tensile testing machine. It should however be mentioned that this algorithm is made mainly
for the above described steels with a well defined first straight part.
It should be emphasized that a lower value of the modulus of elasticity will give a higher
proof strength, since the slope of the curve will turn the curve to the right. When the modulus
of elasticity turns to right the values will then meet the curve further up in the diagram.
- 10 -
1.4.1
Anelasticity
The phenomenon anelasticity itself is divided into two modes. namely the after effect and the
internal friction mode. (3) When a constant stress is applied to a specimen, the specimen will
instantaneously respond with an elastic strain. The unrelaxed modulus is defined at that zero
time. When time elapses the strain will gradually relax and at infinite time the modulus will
describe the relaxed modulus. In figure 5 unloading of a plastically strained specimen is
depicted. If the unloading is very fast it will describe the part from A to e2 and with time the
remaining strain will approach to point e3. If the tensile testing machine is set to one specific
strain rate, the unloading part will look like the right side part in figure 6.
In materials that experiences cyclic loading the anelasticity will make the amplitude of
vibrations decrease and that will result in energy loss by internal friction. (5)
Figure 5: Depiction showing the aftereffect.
1.5
Modulus of elasticity
One could certainly have opinions about why a materials scientist should read about modulus
of elasticity, ME, in an introduction to a diploma work concerning steel, which should be one
of our deepest rooted knowledge.
However there are several different phenomena that can affect the determination and values of
the modulus of elasticity in a heavily cold worked strain hardening stainless steel.
Dislocations will rearrange compared to a normal annealed steel structure. That itself may
lower the ME, up to in special cases 20 %. We often say that ME could not change because it
is a parameter directly connected to inter forces from one atom to the other. What is forgotten
in that discussion is that the material properties in commercial steel alloys is set by the
dislocation structure. Since ME is a material property that property can change if the
dislocation structure changes.
On the other hand other investigations say that anelasticity will affect the determination of
ME with traditional tensile testing a lot in cold worked strain hardening steels and should
- 11 -
therefore not be recommended. If the ME is determined with resonance frequency testing
instead the ME is still constant. (6)
1.6
Determination of proof strength, Rp0.2.
Because of the problems concerning the determination of the modulus of elasticity it is very
important to establish a standard procedure, to minimize management errors. Then it is
possible to compare results from different tests with a minimum of uncertainty. If the tensile
test had been carried out theoretically perfect, the modulus of elasticity would be the
derivative of the curve when the stress is zero. But when the stress is zero errors tend to blow
up. That is why the tensile test curve often has to be extrapolated backwards in that region,
due to loss of data.
In the standard SS-EN 10002-1, 13.1, a special procedure is recommended with a hysteresis
loop, see fig. 6. That specific procedure is especially suitable to determine proof strength in
these hard evaluated materials showing anelastic behavior.
Figure 6:
1.7
Depiction of determination of proof strength with a hysteresis loop. Source: SS-EN standard 100021, pp. 21.
Residual stress
Residual stresses are internal stresses that are not necessary to keep a body in equilibrium
with its surrounding. (7) Yet they almost always exist in atomic, granular or macro scale.
Residual stresses originate from miss fittings between different spatial areas. As indicated
there are three types of residual stresses. Type I or macro stresses work over ranges that can
be discerned with your eyes. Type II or inter granular stresses work over granular ranges and
type III work on atomic scale.
On macro scale, plastic strain bending of a beam is an illustrating example. The outer fibre of
the beam has been stretched the most, and the inner fibre the least. After unloading, that will
result in that the outer and inner fibres will compete to remain in their respective state. That
will give rise to residual stresses on a macro scale.
- 12 -
Type II residual stresses equilibrate typically over ranges of 3-10 granular dimensions. Inter
phase thermal stresses in a metal matrix composite is an example of type II.
Type III residual stresses exist on atomic scale and equilibrate within a granule. Dislocations
and point defects will result in type III residual stresses.
1.8
The Bauschinger
Bauschinger effect
The Bauschinger effect was first discovered in 1886 and it is named after the finder. The
effect describes what happens in a material, exposed to tensile or compression strain, when it
experiences reversed loading, see fig. 7a. The first half cycle A-D is called forward loading,
tensile segment in fig. 7b, and it is not necessary that it starts in tensile direction. When the
load changes sign in point D the reversed loading starts. D-F is the compression loading
portion. The material in reversed loading will yield at a lower stress, tensile or compressive,
than in the previous load direction, that is the Bauschinger effect. In fig. 7b a measure of the
Bauschinger effect can be read off as the difference between the tensile loading and the
amount of the mirrored compression loading portion.
What cause the Bauschinger effect are changes in the dislocation substructure that occur in
load reversing and also changes in internal stress systems.
Figure 7:
Depiction of the Bauschinger effect. Source: Fatigue of materials, S.Suresh, p. 98.
The Bauschinger effect can also be used to see what effect different dislocation mechanisms
have on strain hardening.
- 13 -
1.9
Process chain
The tubes have gone through several steps in the process chain. The molten steel is first casted
in the continuous casting machine. The cast billets are later hot rolled to bars. The bars are
then peeled and cut into short billets. Machine turning adds a radius in one of the billets ends,
to prevent cracks in the hot extrusion. The prepared billets are heated up to 1200 °C before
put into the extrusion press. In the extrusion press the billets are transformed in to a tube that
looks a lot like the final result, except for the dimensions. The tubes have again passed a hot
step and now they have to be straightened and pickled. The following two steps are the ones
that will set the tubes final mechanical properties. In the pilgering the steel is heavily strained
plastically, which gives strain hardening and increased strength.
In oil and gas applications that high strength is particular desirable, when the tubes from the
platform down to the bottom can build up to several thousand of meters and have to uphold
their own weight.
Unfortunately strain hardened steel is sensitive to altered strain directions, the steel
experiences Bauschinger effect and if put to several load changing cycles also cyclic
softening. That is why the last step, cross roll straightening, has to be performed with extreme
precautions.
1.10
Cyclic stressstress-strain behaviour
Metals are meta stable when put out to cyclic changing loads. How the material will respond
to stress-strain can change drastically when put out to just cyclical strain. If the material has
been annealed or cold deformed prior to these cyclic strain changes is of major importance.
“When well annealed FCC single crystal ,suitably oriented for single slip, are subjected to
cyclic strains under fully reversed loading, rapid hardening is noticed even in the initial few
cycles” After that relatively short period hardening ceases to increase and a state called
“saturation” sets in. (8)
While a cold deformed material immediately will show cyclic softening after the introductory
Bauschinger effect. (9)
- 14 -
2
Test material
There are two grades of material included in this investigation, Material A and Material B.
2.1
Material A
The raw material was manufactured from a round bar with a diameter of 264 mm. The round
bar was then extruded to 197 x 20 in kst 927. The hollow was then pilgered to our final
dimension 156, 3 x 12, 6 mm.
2.2
Material B
The raw material was manufactured from a round bar with a diameter of 264 mm. The round
bar was then extruded to 162, 7x21 in kst 927. The hollow was then pilgered to our final
dimension 127, 5 x 15, 8 mm.
- 15 -
3
Means and methods
The analytical beam model is very simplified. There was therefore a FEM-simulation made
with the software MSC Marc. The purpose with the FEM-simulation was to verify if the
analytical calculations are in the right magnitude. The RSM has got crossed rolls, which
makes the analysis quite complex. First the tube is cylindrical, and then the rolls have got a
hyperbolic shape. That together will make the stress state between the ovalizing rolls hard to
analyze in any other way than with FEM-simulation.
The physical test methods that were carried out in this investigation were tensile test, low
cycle fatigue test, hardness test and microstructure evaluation. Also an analytical model was
made to decide which strain levels should be used in the low cycle fatigue test,
see appendix 1.
3.1
Simulation and calculations
calculations of adequate strain
The goal with the simulation was not to make a perfect simulation. It was to verify the old and
the new analytical beam models. Therefore the rolls were simplified and the mesh was set
quite coarse.
To be able to simulate the deformation with finite element analysis there is need for a true
stress-strain curve. From the tensile test, one curve was modified to true stress-strain. The
tensile test curve of specimen 4-2-12_1 was used. The starting length of the strain gauge is
12.5 mm in the tensile test machine. From the fracture point of strain 14.6193 %, the force
was read from the log of specimen 4-2-12_1, 5255 N. The local fracture neck was measured
to 2,025 mm. The fracture stress could then be calculated.
5255 · 4
·4
1632 · · 2,025
The stress-strain curve of specimen 4-2-12_1, see figure 8, was changed with a completing
line from the above calculated stress-strain point to the tangent of where the elastic strain
stops, see figure 9.
- 16 -
Sanicro 28 4-2-12_1
Stress-Strain curve
900
800
700
Stress [MPa]
600
500
400
4-2-12_1
300
200
100
0
0
Figure 8:
1
2
3
4
5
6
7
8
9
Strain [%]
10
11
12
13
14
15
The look of the technological stress-strain curve from the tensile testing machine.
Sanicro 28 4-2-12_1
True stress-strain curve
1400
Stress [Mpa]
1200
1000
800
600
4-2-12_1
400
200
0
0
1
2
3
4
5
6
7
8
9
10
11
Strain [%]
Figure 9:
The modified stress-strain curve to true stress-strain curve.
- 17 -
12
13
14
15
The meaning with the low cycle fatigue test was to be able to use it as a simulation method for
the real straightening process in the cross roll straightening machine 327. It is then important
to know which strain levels the tubes go through in reality and adopt them in the tests that will
be performed. Therefore it has been a little beam FEM model made(10), see figure 11. The
problem has also been analyzed with a traditional beam model, see figure 10. The two
separate calculations can then verify the correctness of the other.
Before this investigation started there was a similar beam model to that in figure 10. The
major difference between the two is that in the prior model the supports B and C were
represented as pinned connections. When that model was developed the focus was to get the
answer of which displacement could be needed at the most to get a certain strain. When this
investigation first started it was with the purpose to get the answer of what strain a volume
element could possibly have experienced. Together they will give some kind of mean value
theorem. In between the two calculated values the true value must be found. In a cross roll
straightener the contact in between the roll and the tube will give a line contact. That should
induce a distributed load instead of a point load and a line contact should also induce more of
fixed support than pin connection, that’s why fixed support has been chosen here. If further
investigations will be executed it would be preferable to install load cells to get the
opportunity to read the force from respective roll. It would then be possible to decide more
precisely what character the supports are of.
Figure 10: Load case configuration.
Figure 11: Displacement designations for this specific load case configuration.
- 18 -
Table 1:
Scheme of how the separate element matrices will affect the global stiffness matrix. GM means
global matrix and EM means element matrix.
As shown in table 1, 5 loads are known and 3 displacements. The matrix then has to be
rearranged before it can be solved. The 3 unknown forces are moved to the left side, and the
displacements were moved to the right side, see appendix 2. The resulting equation system
was then solved in MATLAB. To be able to sketch the displacement, transverse force
diagram and moment diagram, there was also need for an elementary case analysis. That is
because the FEM-model only delivers the resulting forces and moments and to analyze the
separate displacements in respective beam part one has to know the separate contributions to
the part.
- 19 -
3.2
Tensi
Tensile
sile test
Tensile tests were carried out on both pilgered and straightened material. The tensile tests
were performed in pilgering direction, PD, and transverse to PD, TTPD. It was then possible
to see if there was any anisotropy in the material, from the tensile tests.
3.3
Low cycle fatigue test or simulated straightening
straightening
The low cycle fatigue tests have been carried out on material that has only passed through the
pilgering process. These tests were only performed in PD. The procedure was much the same
as in a low cycle fatigue test, with changing stress states from tensile to compression and
tensile again, and so on.
When a tube passes through a cross roll straightener the stress varies from tensile to
compressive, just as described above. One of the main purposes with this investigation was to
clarify whether the low cycle fatigue test could be used as a simulated straightening process.
3.4
Authentic tests
There were specimens collected from one specific tube throughout the last two steps in the
process chain. The specimens were collected as described in the pictures 12 to 15. The
collected specimens were later put through tensile testing in PD and TTPD.
3.5
Collection of specimens
The investigated tube was cut into seven smaller tube sections, see figure 12. After pilgering
there were two pieces cut off for further testing, number 1 and 2. The length of tube section 1
and 2 was 500 mm. The remaining tube was cut into five tube sections after straightening.
Number 3 and 5 were 700 mm long and number 4 was 500 mm long.
To always be able to trace from where the specific sample was collected, the original tube was
arbitrary marked with one original circumferential 12-direction, by analogy with the watch.
That original 12-direction was later transferred to the tube sections gradually as they were cut
off. The end that was oriented towards the end of the original tube was also marked to make
the tube sections direction fully constrained.
Tube section 1 and 2 were marked in the inner ends, section 3 and 5 were marked in the outer
ends and section 4 was marked in the end directed towards 1.
Each specimen was marked due to the code X_XX_XXX, where X stands for the lengthwise
number 1-5 in figure 12, XX for the lengthwise number 1-7 in figure 13 and XXX for the
circumferential number 12, 3, 6 or 9 in figure 13.
The tube sections were later divided into smaller rings. Those smaller rings were then cut into
circumferential sectors. From those sectors lengthwise tensile-, transverse tensile- and
- 20 -
Bauschinger specimens could be machine turned. No further machining was executed after
turning.
The final machine turned specimens were collected in groups of three pieces, see fig. 13.
From the position marked X-1-12 three transverse tensile specimens were collected, from X2-12 three lengthwise tensile specimens and from X-3-12 three Bauschinger specimens.
However in grade Material B specimens were only collected from position 12 and 6, 7
specimens from each position. The tube was too small to allow collection from other
positions.
The cross roll straightening machine 327, which is installed in tube mill 63 in Sandviken has a
distance of 900 mm between the working rolls. In a cross roll straightener the ends are not
properly straightened in about half the distance between the working rolls, that is about
450mm in our case. But they have been adequately ovalized. That is why equal sets of
specimens have been collected from both ends of section 3 and 5 in figure 12.
Figure 12: Depiction of the tested tube sections location and numbering along the original tube. The
unnumbered sections have been withdrawn from further testing.
- 21 -
Figure 13: Depiction of the location, on the particular tube section, from where the specimens have been
collected. The specimens were collected due to this configuration for the sections marked 1 and 2, in
figure 12. These tube sections were cut off after pilgering. Also note the lengthwise and
circumfential marking code. Position 6 and 7 marks the mirrored positions of position 1 and 2.
Figure 14: Depiction of the location, on the particular tube section, from where the specimens have been
collected. The specimens have been collected due to this configuration for the second and the sixth
tube section, in figure 1. These tube sections were cut out after straighten.
- 22 -
Figure 15: Depiction of the location, on the particular tube section, from where the specimens have been
extracted. The specimens have been collected due to this configuration for the fourth tube section, in
figure 2. These tube sections were cut out after straightening.
- 23 -
LCF (front and back)
2
3
1
3
1
4
4
4
1
1
3
3
2
12
12
36
Total No. of samples
24
24
72
120
Tensile test ⊥ (centre piece)
Tensile test // (centre piece)
3
6
2
3
1
1
4
4
4
4
1
1
3
1
3
3
24
24
12
12
Total No. of samples
- 24 -
Total
Subtotal
Tensile test // (front and back)
4C30
4C30
4C30
4C30
Number/
position
Tensile test ⊥ (front and back)
No. of
positions
(clockwise)
Tube pieces/tube
No. of
positions
(lengthwise)
Specimens collected from tubes after straightening
Kind of
specimen
Table 3:
Total
Subtotal
Tensile test // (front and back)
4C30
4C30
Bauschinger
Number/
position
Tensile test ⊥ (front and back)
No. of
positions
(clockwise)
Tube pieces/tube
No. of
positions
(lengthwise)
Specimens collected from tubes after pilgering.
Kind of
specimen
Table 2:
48
48
12
12
120
3.7
Tensile test procedure
The tensile tests were performed in a 100 kN RKM servo hydraulic machine, see figure 16.
Figure 16: Photo over the tensile test machine.
Figure 17: Photo over the tensile test specimens called 4C30.
The machine worked test specimens were tested according to ISO standard 6892-1 fig. 6. That
specific procedure, with a hysteresis loop, is well suited for materials with considerable
anelastic strain, see introduction chapter about determination of proof strength.
- 25 -
The procedure is that the specimen is installed in a tensile test machine and drawn according
to practice. When the strain is well over 0.2 %, see figure 18 , the specimen is unloaded to
about 10 % of current stress. The specimen is drawn again and this time until fracture occurs.
In materials showing anelastic strain, the stress-strain curve will now describe a hysteresis
loop with a look like a section of a lens. Modulus of elasticity is then evaluated as the slope of
the line connecting the peripheries of the lens. A parallel line to the previous one is
constructed tangent to the initial part of the stress-strain curve. That is it will sometimes cut
the abscissa in negative values. The line is then displaced parallel with a 0.2 strain, in usual
manner. It is now possible to read Rp0.2 as the intersection between the constructed sloped
line and the stress-strain curve. In this investigation the hysteresis loop was introduced at a
total strain of 1.5 %.
Figure 18: Description of how the evaluation of Rp0.2 with hysteresis loop is made.
- 26 -
3.8
Low
Low cycle fatigue test procedure
The low cycle fatigue, LCF, tests were performed in a 100kN Instron servo hydraulic fatigue
machine, see figure 19.
Figure 19: Photo over the machine where the low cycle fatigue tests were performed.
- 27 -
In table 4 a test plan over the LCF tests for Material A is represented.
Number of
specimens tested
Number of cycles
Strain in %
Number of
specimens tested
Number of cycles
Strain in %
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
Strain in %
1-3-12
1-3-3
1-3-6
1-3-9
1-4-12
1-4-3
1-4-6
1-4-9
1-5-12
1-5-3
1-5-6
1-5-9
2-3-12
2-3-3
2-3-6
2-3-9
2-4-12
2-4-3
2-4-6
2-4-9
2-5-12
2-5-3
2-5-6
2-5-9
Number of cycles
Specimen
Number of
specimens tested
Low cycle fatigue test plan for Material A.
Number of
specimens
Table 4:
1
1
15
15
0,4
0,55
1
1
15
15
0,8
0,6
1
1
15
15
1,2
0,65
1
1
1
15
15
15
0,4
0,4
0,4
1
1
1
15
15
15
0,8
0,8
0,8
1
1
1
15
15
15
1,2
1,2
1,2
2
1
0,8
1
2
0,8
2
1
0,8
1
5
0,8
2
2
0,8
1
7
0,8
2
3
0,8
1
10
0,8
- 28 -
In table 5 a test plan over the LCF tests for Material B is represented.
Number of
specimens tested
Number of cycles
Strain in %
2
2
1
1
1
1
7
1
5
1
0,8
0,8
0,6
0,8
1,2
1
1
1
1
1
3
10
10
10
5
0,8
0,8
0,6
0,8
1,2
Strain in %
Strain in %
4
3
2
2
3
Number of cycles
Number of cycles
6-3-12
6-3-12
6-3-6
6-3-6
6-3-6
Number of
specimens tested
Specimen
Number of
specimens tested
Low cycle fatigue test plan for Material B.
Number
Table 5:
1
5
0,8
1
10
1,2
The machine worked Bauschinger test specimens, see figure 20, were tested with a changing
load, from tensile to compression and so on until reaching determined number of cycles. The
established strain was equal in positive and negative mode. The test data values were
collected in one long file, including all cycles from respective specimen, and are note very
manageable. That file is of the format *.raw and could be imported to excel, see figure 21. To
get a quantitaive comparison of the proof strength with or without cyclic loadchanging a
method to determine proof strength of each tensile half cycle from the long data collection
above was neded. A MATLAB program was therefore used to first create useful outputs, in
the shape of one *.raw file for each half cycle.
Figure 20: Photo over the Bauschinger test specimen.
- 29 -
Stress [MPa]
Sanicro 28, strain 0,8 %
LCF curve for specimen 2-5-12
-0,9
-0,7
-0,5
-0,3
900
800
700
600
500
400
300
200
100
0
-100
-0,1
-200
-300
-400
-500
-600
-700
-800
-900
Cycle 1
0,1
0,3
0,5
0,7
0,9
Cycle 2
Cycle 3
Strain [%]
Figure 21: The look of the raw data when it is in one long file and has not been divided into half cycles.
To determine proof strength in tensile tests or fatigue tests, a software called Cyclic EDC is
used at SMT. Cyclic EDC is programmed according to ISO standard 6892-1, but not the
procedure with the hysteresis loop. According to the procedure in ISO 6892-1 the program
tries to find the straightest part in the steepest first part of the tensile curve. If strains lower
than proof strength should be used it is not possible to use the hysteresis loop, since it should
be introduced after the proof strength. Since the first part isn´t especially straight in these
materials at all, the determination with Cyclic EDC is not exact. Even though this procedure is
more suited to materials that have not been cold worked the errors tend to equalize each other,
which have been investigated with comparison between proof strength determined with
hysteresis loop method and with Cyclic EDC. Proof strength determined with hysteresis loop
in general often lies about 20 MPa higher than determined with Cyclic EDC. Though Cyclic
EDC has been used to determine proof strength in the half cycles from the origin long output
file in this investigation.
A tensile half cycle represented with a *.raw file, prepared with the MATLAB program, could
then be imported to Cyclic EDC, see figure 22. In Cyclic EDC the proof strength for that
particular half cycle can be determined.
- 30 -
Figure 22: Screen dump from Cyclic EDC, determination of proof strength, Rp0.2, after the *.raw file has been
divided in to half cycle parts
It has also been noted that the software Cyclic EDC has got an upper limit of ME set to 210
GPa. Therefore all values above 210 GPa should be considered with precautions, they could
be higher. A lower value of ME will influence the proof strength value in a positive way that
is a lower ME will give a higher proof strength value.
The proof strength values from respective tensile half cycle were then noted in an excel sheet
and represented as curves in figure 23. It is called dynamic because the test was running
continuous and the proof strength values here could therfore still remain some anelastic rest.
- 31 -
Stress [Mpa]
Material A, strain 0,8%
Rp0.2 dynamic
860
840
820
800
780
760
740
720
700
680
660
640
620
600
580
560
540
520
500
837
797
795
2-3-12
657
637
617
0
1
640
619
603
2
2-4-12
632
606
594
3
611
596
583
4
605
578
5
600
572
6
2-5-12
592
7
588
8
581
9
10
Cycles [No.]
Figure 23: A diagram from Excel showing the dynamic proof strength values determined with EDC Cyclic.
To secure respective dynamic proof strength value also a last tensile test was performed, see
figure 24. That proof strength has been called true here.
Stress [Mpa]
Material A, strain 0,8%
Rp0.2 true
860
840
820
800
780
760
740
720
700
680
660
640
620
600
580
560
540
520
500
837
797
795
2-3-12
673
660
647
2-4-12
623
2-5-12
591
583
0
1
2
3
4
5
567
6
7
8
9
10
Cycles [No.]
Figure 24: Diagram over the true proof strength value from the final tensile test, when the specimens were
drawn until fracture occurred.
- 32 -
The dynamic proof strength and the true were then brought together in one diagram beginning
with the dynamic part and of course ending with the true, see figure 25.
Stress [MPa]
Material A, strain 0,8%
Rp0.2 dynamic & true
860
840
820
800
780
760
740
720
700
680
660
640
620
600
580
560
540
520
500
837
797
795
2-3-12
657
637
617
0
1
640
619
603
2
2-4-12
632
606
594
3
611
596
583
4
605
583
578
5
600
572
6
2-5-12
592
591
7
588
8
581
9
567
10
Cycles [No.]
Figure 25: The resulting diagram from the dynamic and true proof strength brought together.
- 33 -
4
Results
4.1
A volume element´s
element´s way through the straightener
Before the proof strength results and the LCF results are shown, there is an analysis
concerning a volume element´s way through the straightener. The analysis has been made
from two perspectives, ovalization and bending. The analysis is only qualitative.
Let us start with the bending analysis, because it is the most critical, regarding strength, see
figure 26. The point will enter from right and the first point is determined to be perfectly
synchronized with the first maximum of bending moment, see figure 26. The point will alter
position with 211° from one roller set to the next. By the ovalization roll pair the point is still
quite well in phase with the maximum bending moment, by the third and the fifth roll the
point is about as much out of phase for both. By the fourth roll it is completely out of phase.
With reference to this knowledge, the point could theoretically speaking change stress sign
five times, but it is more likely that it happens twice or three times.
Figure 26: Depiction over a separate point’s way through the cross roll straightener. Here the focus is on
bending.
- 34 -
The analysis of the ovalization is easier. Also here the point will enter from right and the point
is determined to hit the second roller set perfectly, the ovalizing roll pair, see figure 27. By the
second ovalizing roll pair, the point will not experience any ovalization. The depiction shows
that while a volume element travels through the straightener it will experience only one
ovalization cycle for sure, but it is possible that it won´t be ovalized at all, if the rolls are set at
improper angles.
Figure 27: Depiction over a separate point’s way through the cross roll straightener. Here the focus is on
ovalization.
4.1
Calculation model
The undetermined beam was solved with a little beam FEM-model and an elementary case
analysis, see figure 10 and 11. In figure 28 the plot of the tube deflection is represented. The
plot itself is a good check that the calculations are made correctly. The boundary conditions
were as said in passage 3.1 fixed in the middle supports and pinned in both ends. The
difference in boundary conditions in the outer tube sections will make the maximum
deflection occur on the half nearest to the pinned support.
- 35 -
Figure 28: Plot of deflection from MATLAB.
In figure 29 the transverse force diagram is plotted. Because the forces are represented as
point loads, the transverse force diagram will show only constant force levels.
- 36 -
Figure 29: Plot of transverse forces from MATLAB.
In figure 30 the moment diagram is represented. Because the transverse force diagram only
shows constant levels the moment diagram will show linear moment changes.
- 37 -
Figure 30: Plot of moments from MATLAB.
The resulting strain from the beam calculation analysis is represented in figure 31. The
maximum strain is achieved by the left ovalizing roll pair, as the deflection was set to the
highest value there. A value of about 1.5 % is read from the diagram. According to the other
model the strain is 0.6 %. The strain should then lie in between 0.6 and 1.5 % when looking at
isolated bending.
- 38 -
Figure 31: Plot of strains from MATLAB.
- 39 -
4.2
FEMFEM-simulation
The FEM-simulation was made parallel to the analytical calculations. As can be seen in figure
32 the sharp edges of the rolls will make considerable indentation in the tube. This is because
the model of the rolls was simplified. The simplification was made to generate a more quick
computer calculation. Of course this is not acceptable at all and the radius should have been
reintroduced. It is not worth to talk about the strain levels in this stage.
Figure 32: Shows the point contact that comes from the deletion of the outer radius.
In figure 33 we can follow how the indentation propagates throughout the entire simulation.
Also here the lack of the radius makes the most of the strain. Therefore the strain levels are
wrong here also.
- 40 -
Figure 33: Shows that the lack of the outer radius will propagate throughout the whole simulation.
The mesh was as mentioned earlier set quite coarse. In figure 34 we can see the effect of the
coarse mesh. The deformation runs like a caterpillar around the tube. If the mesh had been
perfect it would have rendered in a smooth tape instead of a caterpillar. As previous pictures
the strain levels should be ignored.
- 41 -
Figure 34: Shows the caterpillar pattern due to the quite coarse mesh.
On this stage we can´t draw a lot of quantitative conclusions. We can say one thing from these
pictures. If the angle of the rolls aren´t set correctly to the present tube diameter we will end
up with a tube that consists of very inhomogeneous mechanical properties.
4.3
4. 3
Modulus of elasticity
ME had to be evaluated with the same methods as used for determination of proof strength.
To get the right proof strength values one also needs correct input from ME. That’s why two
diagrams over the ME have been inserted here. In figure 35 a diagram over ME in Material A
is represented. It starts on 210 GPa and immediately drops to 192 during the first cycle. (11)
At the 10th cycle the ME is down at 179 GPa.
- 42 -
Material A, modulus of elasticity
Bauschinger specimen 0.8 %
220
Modulus of elasticity [GPa]
210
200
192
188
186
180
184
184
182
181
181
180
179
160
1-3-12
140
120
100
0
1
2
3
4
5
6
7
8
9
10
Cycles [No.]
Figure 35: Diagram over modulus of elasticity for a Material A specimen.
In figure 36 the ME diagram over Material B is represented
Material B, modulus of elasticity
Bauschinger specimen 0.8 %
Modulus of elasticity [GPa]
220
200
186
180
165
160
163
161
160
158
158
157
157
155
152
6-3-6
140
120
100
0
1
2
3
4
5
6
7
8
9
10
Cycles [No.]
Figure 36: Diagram over modulus of elasticity for a Material B specimen. It is not likely to get values that go
below 170.
- 43 -
4.4
4. 4
Tensile tests
4.4.1
Material A
Figure 37 shows the tensile proof strength of Material A TTPD before straightening. Notice
the low variance in the sections and from one section to the other. Section 1 is front end and
section 2 is rear end. The diagram shows results from 24 specimens in total. The median is
about 710 MPa.
Material A, after pilgering
Rp0.2 TTPD
860
840
800
780
Tube Section 1
700
698
684
680
0
1-1-12
720
1-1-9
1-1-6
740
707
701
2-1-3
760
1-1-3
Rp0.2 [MPa]
820
719
707
702
706
702
Tube Section 2
4
727
724
722
710
721
712
702
8
Position
Figure 37: Tensile proof strength transverse to pilgering direction, before straightening. Tube section 1 is front
end and section 2 is the rear end.
Figure 38 shows the tensile strength of Material A in PD before straightening for front and
rear end. Also here we see quite uniform properties from front to rear end and in circumfential
direction. The median is about 810 MPa, 100 MPa higher than TTPD, that means that the
materail is showing considerable anisotropic properties.
- 44 -
Material A, after pilgering
Rp0.2 PD
Rp0.2 [MPa]
808
799
800
802
792
814
800
2-2-3
1-2-12
820
1-2-6
1-2-3
840
1-2-9
860
824
813
801
807
813
803
819
813
814
808
780
760
Tube section 1
740
Tube section 2
720
700
680
0
4
Position
8
Figure 38: Tensile strength in pilgering direction before straightening. Tube section 1 is front end and section 2
is the rear end.
In figure 39 the tensile strength of Material A TTPD, after straightening, is shown. Compared
to figure 37 the level is far from the uniform level before straightening. The median is about
760 MPa. The raise of the proof strength TTPD was therefore about 50 MPa. However the
level is far from uniform. It is the ovalization that affects the tensile strength TTPD, see figure
2.
- 45 -
Material A, after straightening
Rp0.2 TTPD
860
840
760
Tube Section 3
Tube Section 3X
5-7-3
780
5-1-3
3-1-3
3-1-6
3-1-9
3-1-12
Rp0.2 [MPa]
800
4-1-3
3-7-3
820
Tube Section 4
740
Tube Section 5X
720
Tube Section 5
700
680
0
4
8
12
16
20
Position
Figure 39: Tensile strength transverse to pilgering direction after straightening. Tube section 1 is front end and
section 2 is the rear end.
- 46 -
In figure 40 the result after straightening in PD is represented. In the ends the level from
before straightening has not been affected at all. In the passage about how the cross roll
straightener works it was mentioned that the ends will not be straightened through bending,
just through ovalization and therefore there will be no proof strength loss in PD in the ends. In
the other end of the same tube section the specimens were collected so far from the ends that
the effect from bending started to become considerable. The lower values represent a proof
strength of about 690 MPa. The drop in PD was therefore about 120 MPa at the most during
the straightening process.
Material A, after straightening
Rp0.2 PD
800
5-6-3
Tube section 3
780
Tube section 3 X
3-6-3
Rp0.2 [MPa]
820
5-2-3
840
4-2-3
3-2-3
3-2-6
3-2-9
3-2-12
860
760
Tube section 4
740
Tube section 5
720
Tube section 5 X
700
680
0
4
8
12
16
20
Position
Figure 40: The proof strength after straightening in PD. Note the considerable drop much likely depending on
the Bauschinger effect from combined bending and load change.
4.4.2
Material B
Unfortunately we could only get tensile tests before straightening for the Material B, see
figure 41 and 42. The anisotropy TTPD was considerable in this material, which was not the
case in Material A.
- 47 -
1
6-1-12
6-1-9
0
Tube Section 6
6-1-6
1090
1070
1050
1030
1010
990
970
950
930
910
890
870
850
830
810
790
770
750
6-1-3
Rp0.2 [MPa]
Material B, before straightening
Rp0.2 TTPD
2
Position
3
4
Figure 41: Proof strength TTPD for Material B. Note the difference in levels, a difference of 250MPa.
Rp0.2 [MPa]
6-2-12
6-2-9
6-2-3
1090
1070
1050
1030
1010
990
970
950
930
910
890
870
850
830
810
790
770
750
6-2-6
Material B, before straightening
Rp0.2 PD
Tube section 6
0
1
2
Position
3
Figure 42: Proof strength in PD for Material B. Smooth level of about 1010MPa.
- 48 -
4
4.5
4. 5
Low cycle fatigue test
In figure 43 a plot of a specimen that has only been strained to an elastic level of 0.4 % is
plotted. Even though the specimen was only elastically loaded the stress-strain diagram is far
from a straight line just as discussed in the passage about anelasticity.
Material A, strain 0.4 %
elastic load change
700
500
300
100
1-4-3
-0,43
-0,33
-0,23
-0,13
-0,03
-100
0,07
0,17
0,27
0,37
-300
-500
-700
Figure 43: Graph over 15 load change cycles with a strain of 0.4 %. Note that the 15 curves almost run in the
same trace and therefore show elastic strain, without showing a straight line.
In table 4 the specimens that were tested with low cycle fatigue test are shown. The specimens
did not show any plastic strain and could therefore not give any proof strength values for
dynamic analysis. One of those specimens is shown in figure 43. In table 6 the proof strength
after 15 cycles for 3 of the specimens are shown.
Specimen
Number of
specimens tested
Number of cycles
Strain in %
Proof strength
[MPa]
Proof strength for the 3 specimens that were exposed to
cyclic strain that was elastic.
Number
Table 6:
1-4-3
1-4-6
1-4-9
3
3
3
1
1
1
15
15
15
0,4
0,4
0,4
801
804
801
- 49 -
The specimens that were strained to a level of 1.2 % were not possible to determine the proof
strength of, because suddenly the EM showed irrelevant values from the software. It could
possibly have been determined by hand.
4.5
4.5.1
Material A dynamic & true proof strength curve
The background of the determination of the LCF diagram was derived in the passage 3.4.
From the initial level the drop was about the same as in the tensile tests. The proof strength
drops about 200 MPa immediately. Since the sign of the load, for sure, has changed this is the
Bauschinger effect. The drop is then quite uniform from cycle to cycle and that is much likely
due to cyclic softening. In the end the last tensile test pops up a bit. The test has rested for a
week after the first cycling before the final tensile test to fracture was performed. It could be
because of that the strain velocity was too high in the test and that not all the anelastic effect
had disappeared in the dynamic test.
Stress [MPa]
Material A, 0.8%
Rp0.2, dynamic & true
860
840
820
800
780
760
740
720
700
680
660
640
620
600
580
560
540
520
500
801
1-3-12
596
0
1
2
584 575
568 559
567
554 550 547
542 540 535 532
531 529
3
4
5
6
7
8
9
Cycles [No.]
10
11
12
13
14
15
Figure 44: The LCF proof strength diagram for specimen 1-3-12 in Material A. Note the initial drop due to the
Bauschinger effect, the continuous cyclic softening and finally maybe an anelastic effect.
- 50 -
Stress [MPa]
Material A, strain 0.8%
Rp0.2, dynamic & true
860
840
820
800
780
760
740
720
700
680
660
640
620
600
580
560
540
520
500
787
1-4-3
627
0
1
2
612
3
599
4
590
588 579
574 567
563 558 554
550 547 545 542
5
6
7
8
9
10
11
12
13
14
15
Cycles [No.]
Figure 45: The LCF proof strength diagram for specimen 1-4-3 in Material A.
Stress [MPa]
Material A, strain 0.8%
Rp0.2, dynamic & true
860
840
820
800
780
760
740
720
700
680
660
640
620
600
580
560
540
520
500
791
1-4-6
589
0
1
2
578
3
566 560
554 547 544
539 535 532 530 528
526 524
4
5
6
7
8
9
Cycles [No.]
10
11
12
Figure 46: The LCF proof strength diagram for specimen 1-4-6 in Material A.
- 51 -
13
14
15
582
Stress [MPa]
Material A, strain 0.8%
Rp0.2, dynamic & true
860
840
820
800
780
760
740
720
700
680
660
640
620
600
580
560
540
520
500
787
652
0
1
2
635
3
1-4-9
621
4
609 603
594 589
591
583 580 574
570 566 564 561
5
6
7
8
9
10
11
12
13
14
15
Cycles [No.]
Figure 47: The LCF proof strength diagram for specimen 1-4-9 in Material A.
Stress [Mpa]
Material A, strain 0,8%
Rp0.2 dynamic
860
840
820
800
780
760
740
720
700
680
660
640
620
600
580
560
540
520
500
797
786
779
2-3-12_1_1
673
2-3-12_1_2
637
619
0
1
606
2
3
2-3-12_8_5
596
4
583
5
Cycles [No.]
Figure 48: The LCF proof strength diagram for the specimens 2-3-12 in Material A. They were tested 1, 2 and
5 cycles.
- 52 -
Stress [Mpa]
Material A, strain 0,8%
Rp0.2 dynamic & true
860
840
820
800
780
760
740
720
700
680
660
640
620
600
580
560
540
520
500
810
795
755
2-4-12_7_1
668
638
617
0
1
647
646
603
2
2-4-12_2_2
2-4-12_2_1
594
3
583
591
578
4
5
572
6
7
8
Cycles [No.]
Figure 49: The LCF proof strength diagram for the specimens 2-4-12 in Material A. They were tested 2 and 7
cycles.
Stress [MPa]
Material A, strain 0,8%
Rp0.2 dynamic
860
840
820
800
780
760
740
720
700
680
660
640
620
600
580
560
540
520
500
841
837
818
2-5-12_10
657
650
639
640
601
0
1
632
623
614
588
2
3
4
2-5-12_3_2
611
605
5
600
6
2-5-12_3_1
592
7
588
8
581
9
567
10
Cycles [No.]
Figure 50: The LCF proof strength diagram for the specimens 2-5-12 in Material A. They were tested 3 and 10
cycles.
- 53 -
Stress [MPa]
Material A, strain 0,8%
Rp0.2 dynamic & true
860
840
820
800
780
760
740
720
700
680
660
640
620
600
580
560
540
520
500
837
797
795
2-3-12
657
637
617
0
1
640
619
603
2
632
606
594
3
2-4-12
611
596
583
605
583
578
4
5
6
Cycles [No.]
600
572
2-5-12
592
591
7
588
8
581
9
567
10
Figure 51: The LCF proof strength diagram for the specimens 2-3-12, 2-4-12 and 2-5-12 in Material A. They
were tested 5, 7 and 10 cycles.
- 54 -
4.5
4.5.2
Material B dynamic & true proof strength curve
The procedure for the LCF-test for Material B was the same as for Material A. In figure 52 a
proof strength droop of about 150 MPa. It is not as much as in Material A on a relative basis.
The drop is even though of such a high level that there is no process way that could
compensate for this kind of proof strength loss.
A difference from Material A was that a more frequent drop was noted in the last fracture
tensile test.
Stress [Mpa]
Material B, strain 0.6%
Rp0.05 dynamic & true
1100
1050
1000
950
900
850
800
750
700
650
600
550
500
450
924
900
775
755
742
6-3-6_10_1
721
699
682
669
647
6-3-6_1_1
642
634
499
0
1
2
3
4
5
6
7
8
9
10
Cycles [No.]
Figure 52: The LCF proof strength diagram for the specimens 6-3-6 in Material B. They were tested 1 and 10
cycles. Note that the proof strain is only 0.05 due to low plastic strain.
- 55 -
Stress [Mpa]
Material B, strain 0.8%
Rp0.2 dynamic & true
1100
1050
1000
950
900
850
800
750
700
650
600
550
500
450
1085
1051
948
926
837
804
907
895
782
766
917
753
741
730
723
717
742
6-3-6_5_1
6-3-6_10_1
0
1
2
3
4
5
6
Cycles [No.]
7
8
9
10
Figure 53: The LCF proof strength diagram for the specimens 6-3-6 in Material B. They were tested 5 and 10
cycles.
Stress [Mpa]
Material B, strain 0,8%
Rp0.2 dynamic & true
1100
1050
1000
950
900
850
800
750
700
650
600
550
500
450
6-3-12_10_1
6-3-12_7_1
6-3-12_7_2
6-3-12_5_1
6-3-12_3_1
6-3-12_1_2
6-3-12_1_1
0
1
2
3
4
5
6
7
8
9
10
Cycles [No.]
Figure 54: The LCF proof strength diagram for the specimens 6-3-12 in Material B. They were tested 1, 3, 5, 7
and 10 cycles.
- 56 -
Stress [Mpa]
Material B, strain 1,2%
Rp0.2 dynamic & true
1100
1050
1000
950
900
850
800
750
700
650
600
550
500
450
6-3-6_10_1
6-3-6_5_1
6-3-6_1_1
0
2
4
6
8
10
Cycles [No.]
Figure 55: The LCF proof strength diagram for the specimens 6-3-6 in Material B. They were tested 1, 5 and
10 cycles.
- 57 -
4.6
4. 6
Hardness test
In the Material A there were hardness tests preformed TTPD and in PD. As expected the
hardness was highest at outer surface, because the strain from rolling was the highest there.
The hardness raise a bit at the inner surface, and the strain should have been higher there as
well in the pilgering operation. There were no major differences between the ends of the tube,
therefore the conclusion can be made that ovalization affects the hardness values the most,
since it works along all the tube.
Material A, 3 kp
hardness after pilgering, TTPD
340
330
Hardness [HV]
320
310
300
1-12T
290
3M-12T
3-12T
280
270
260
0
1
2
3
4
5
6
7
8
9
Position across section [mm]
10
11
12
Figure 56: Hardness transverse to pilgering direction, 0 is at outer surface and 1 is at the inner side of the
very ends, 3 is at the end towards 1 and 3M is at the end away from 1 on the same tube section as 3.
There were some differences in PD compared to TTPD. There were no differences in the
same graph, only anisotropy. The middle part in PD falls linearly towards the lowest level at
inner surface. These properties were very uniform throughout the entire tube, even though
showing on some anisotropic behavior.
- 58 -
Material A, 3kp
hardness after pilgering in PD
340
330
Hardness [HV]
320
310
300
1-12
290
3-12
3M-12
280
270
260
0
1
2
3
4
5
6
7
8
9
10
11
12
Position across section[mm]
Figure 57: Hardness parallel to pilgering direction, 0 is at outer surface and 1 is at the inner side of the very
ends, 3 is at the end towards 1 and 3M is at the end away from 1 on the same tube section as 3.
Material B, 3kp
hardness after pilgering, TTPD and in PD
380
Hardness [HV]
370
360
350
6-3-12_TPD
6-3-12_PD
340
330
320
0
1
2
3
4
5 6 7 8 9 10 11 12 13 14 15
Position across section [mm]
Figure 58: Hardness transverse to and parallel to pilgering direction. 0 is at outer surface.
- 59 -
In the Material B the hardness TTPD and in PD has been plotted in the same diagram. There
are no major differences between the two directions, although more specimens should be
evaluated before any big conclusions can be made.
4.7
4. 7
Micro structure
Prior to this study there was a microstructure analyze made on the extruded material on this
specific working order. The micro structure was approved. The microstructures in fig. 59-62
shows the outer and inner surfaces, in pilgering direction and transverse to pilgering direction,
of pilgered and pilgered and straightened material. No foreign phases were observed. In fig.
59-61 structures of material A are shown. In fig. 62structures of material B are shown. A great
lot of working twins can be observed in material A structures. In the upper figs, taken in
pilgering direction, it can be recognized that the grains have been stretched during the
working operation. It is even easier to notice the strain in pilgering direction in the material B
structures, see the upper pictures in fig. 62.
Figure 59: Microstructure, from position 1-12, Material A.
- 60 -
Figure 60: Microstructure, from position 3-12, Material A.
Figure 61: Microstructure, from position 3-12M, Material A
- 61 -
Figure 62: Microstructure, from position 6-12, Material B.
- 62 -
5
Discussion
5.1
Tensile testing procedure
In tensile testing with hysteresis loop, the loop was introduced at a total strain of 1.6 %. It
would have been possible to introduce it as early as 0.65 %. The consequence of that has not
been further analysed. The residual stresses are more normalised the more strain that is
induced. That would plead for a late loop introduction as above. On the other hand higher
strain level will strengthen specific texture components. Though 1.6 % is a fairly low strain
level. The strain hardening will also continue, but the slope in that region is almost zero. In
the standard the recommendation is to unload to only 10% of stress when the hysteresis loop
is introduced. In this investigation the unload level is about 30 % of the hysteresis loops
introduction level. That itself will make the value of ME to increase. A higher value of ME in
its turn will result in a lower proof strength value.
This entire investigation rendered in a quite thorough sub investigation about determination of
proof strength in materials showing considerable anelastic behaviour. It has already been
mentioned that the proof strength earlier was determined only with the EDC Cyclic software.
On an early stage it was clear that the ME could vary considerably from one specimen to
another. Further the specimens also showed ME a lot below 170GPa. Something was odd
with this behaviour. It is recognized that ME could vary as declared in the introduction, but
when it does it almost always do it in a continuous way as what is recognized in the cycles
after the first. However from the first cycle to the other there was a big drop of the ME from
sometimes 210GPa to 190GPa, see figures 35 and 36. As also described in the introduction,
ME can look like it has changed a lot even though what sometimes really has happened is that
ME has been evaluated with tensile test procedure that is not appropriate for these steels with
anelastic behaviour. A determination with resonance frequency will then give more correct
values (6). With the background with the above discussion one could definitely wonder what
is the proof strength then, if the ME is not correct determined and the proof strength must be
determined with an offset from the ME. A way to evaluate if the hysteresis loop procedure
presents the right proof strength could be to first determine the proof strength with the Cyclic
software and then know a preliminary value. Then prepare some specimens and make a new
tensile test up to that preliminary level and then unload and look if the load drop will meet the
abscissa at a strain of 0.2 %. If that is not the case the procedure could be adjusted and redone
until the load drop hits 0.2 remaining strain, then the unloading was introduced at the proof
strength. This could also be valuable in another question. In the hysteresis loop procedure the
hysteresis loop ME is translated to make the tangent to the initial part of the tensile curve. The
tangent is then extrapolated backwards to the abscissa. That point often lies on the negative
side of the ordinate, even though the tensile test started from the origin. The above procedure
could show if that is to conservative thinking.
- 63 -
Even though a new procedure with a hysteresis loop was introduced to the tensile testing it
could unfortunately not be used in the LCF testing and the Cyclic software had to be used.
This is exemplified with the figures 35 and 36.
5.2
Tensile test results
5.2.1
In pilgering direction
As described in passage 4.4.1 the proof strength drops about 120 MPa in pilgering direction at
the most. One could almost see the pitch from the simulation in figure 34. The middle section
shows the same behavior of the values as the inner ends which is just as one could predict
with the background of how the machine works. This is particularly serious because it means
that the final mechanical properties will be set in the straightening process, not in the
pilgering process. Compensation for this major drop in the pilgering means that the reduction
must increase. With increased reduction comes a problem to fulfill requirements of fracture
elongation. The Bauschinger effect might affect the elongation in a positive sense. These
thoughts have to be investigated further to get a more quantitative model.
5.2.2
Transverse to pilgering direction
The ovalization is more predictable than bending, because of no load changes, just as
described in the introduction. Therefore it is possible to affect the anisotropy with various
ovalization levels. In the tensile testing TTPD it was showed that the proof strength increased
from about 710MPa to about 760MPa. It was clear that the proof strength didn´t drop in a
single point. When the proof strength in PD was determined after straightening the ovalization
could be adjusted to get a more corresponding value of Rp0.2 in the transverse direction and
therefore minimize anisotropy. However the pitch must be adjustable in order to get in to
phase with the diameter of the tube. In short time period it might be possible to run a tube
twice and put the second helical in between the first one. A double straightening together with
a possibility to change the angle of the working rolls slightly could give a tube with quite
homogeneous properties.
5.3
Modulus of elasticity
As told in passage 4.3 ME started on 210 GPa and immediately fell to 192 during the first
cycle. At the 10th cycle the ME was down at 179 GPa. In a heavily cold worked strain
hardening steel showing anelastic behavior that is not strange. There are reported cases were
ME drops up to 20 %, in our case that was 42 units and means a lower value of 168 GPa, from
that point of view these values could be possible. The fact should though be dealt with caution
because in these materials determination of ME with tensile testing is not recommended and
there are other investigations that say that ME is fairly constant (6)(12).
It should be mentioned that even though the software for evaluation of proof strength
produces an answer, there is no guaranties that the answer is correct. There is a great
possibility that there is an anelastic contribution that affects the result from the software.
- 64 -
There is also an indication from the fluctuations of the ME that strengthens that theory. It
should also be mentioned that there is a built in, in the software, upper limit by 210 GPa.
When the Rp0.2 value turns up in the LCF tests it is a sign that ME in the previous cycle
might have been wrong evaluated.
In the material that was only strained 0.4% ME was kept in quite reasonable levels compared
to the specimens that were only strained 0.8 and 1.2%. The levels of values below 170 GPa
should be considered with great scepticisms.
To read out from the discontinuous first drop, something happens when load direction
changes, and that happens immediately and only once. So far despite level of plastic strain,
only if there are some plastic strain that is enough. Afterwards the stress drop is continuous,
just like in investigations about anelasticity and change of ME says. Three mechanisms might
be involved. The most possible explanation is that the first drop is due to Bauschinger effect
and the continuous drop thereafter is due to cyclic softening.
5.4 Analytical calculation and FEMFEM-simulation
As described in passage 3 the problem is quite complex from an analytical calculation view.
The analysis was therefore simplified and a complementary FEM-model would be
encouraging to have. The deflections were set to 7.5, 1.3 and 3.2 mm. Both the analytical
beam model and the little FEM-model gave strain levels of about 1.5 %. The old beam model
gave about 0.6 %. The true strain level should be somewhere in between depending on
boundary conditions. With thought of the LCF-tests it seems quite right. The drop in the LCFtest for Material A was between 120-200 MPa, in the first cycle. In the authentic tests the drop
was at the most about 120 MPa. In the LCF-test the whole specimen was strained too the
same level. In the authentic test the outer fiber of the tube will have the strain level printed
above. That means that the strain level will decrease towards the middle of the tube. To this
discussion the very coarse adjustment of the straightening machine should be added. With this
background the correlation with the analytical model plus the LCF-test and the authentic tests
should be considered very good.
When it comes to the FEM- and analytical model the correlation is very bad. On this level the
FEM-model can´t be used for any quantitative analysis. Compressive stresses were
exaggerated, especially on the outer surface and right beneath it. The rolls were simplified,
see passage 4.2. The outer fillets were removed in order to make the simulation to run
quicker. One run lasted for three days anyway. Also the mesh was quite coarse. The mesh is
always made from straight lines even though they are connected into a circle. A coarse mesh
will then result in sharp edges. The sharp edges will then give stress concentrations depending
on the coarse mesh. That can be seen as caterpillars in the helical around the tube.
At last I want to refer to a line in the passage 1.2, objective. “The main purpose of this
investigation is to clarify when specific grades soften or harden during the straightening
- 65 -
process.” The mission was to answer the question whether the proof strength increased or
decreased during the straightening process. That must be considered as fulfilled.
6
Conclusions
The strength decreases drastically, about 120 MPa, and immediately in pilgering direction
when the load changes sign and the material is deformed plastically.
The strength increases, about 50 MPa, transverse to pilgering direction when the material is
deformed plastically in circumferential direction by ovalization.
The strength varies a lot more after straightening then after pilgering, because of not proper
rolls for the tube diameter. That also gives a tube with strongly inhomogene properties,
compared to after pilgering.
The absolute ends of the straightened tube are not affected concerning straightening by
bending, therefore it is not likely to find strength values representative for the whole tube
there.
The hardness is not affected much under normal straightening parameters.
The hardness is higher at outer surface.
The correlation between the analytical beam model and the FEM-simulation is at the moment
not god at all; therefore more job remains on the FEM-model to see if the load case on the
whole is possible to break down with this approach.
Some kind of adjustment of the straightener is desired to be able to adjust the angle from the
tube to the rolls that gives the right pitch, and therefore gives more concise material
properties.
7
Proposed measures for short time quality improvement
Let the tubes pass the straightener two times with lengthwise adjustment, so that the bending
moments will overlap from the first lap to the second, and then give more homogeneous
properties.
The displacement should be considered as a process parameter, therefore the displacement
sensors should be fool proof and the displacement configuration should be noted on respective
working order.
- 66 -
8
Proposed further investigations
Evaluate modulus of elasticity with resonance frequency equipment.
Compare the pressurized and the tensile effected material during bending and straightening
press straightening to cross roll straightened material.
Map a tube more seriously to check against the route through the straightening machine.
Investigation about how the stress states vary from pilgering to straightening.
Evaluate what ovalization is needed to make a particular increase of the transverse tensile
strength, whit the thought of creating better anisotropic properties.
- 67 -
9
References
1. Meurling, Fredrik. Results of the 4 1/2" and 5 1/2" trial of the Sandvik Material C. 2011.
2. Cross-Roll Straighteners and their Performance. Talkuder, N K, Singh, A N och W
Johnson. 1990, Journal of Materials Processing Technology, ss. 101-109.
3. Bocquet, J L, Brebec, G och Limoge, Y. Diffusion in Metals and Alloys. [bokförf.]
Robert W Cahn och Peter Haasen. Physical Metalurgy 1. u.o. : Elsevier Science B. V., 1996.
4. Smallman, R E och Bishop, R J. Modern Physical Metallurgy and Materials Engineering.
u.o. : Butterwort Heinemann, 1999.
5. Youngs modulus measured in different ways on different steels subjected to various
pretretments. Engberg, G. 1995.
6. Residual Stress Overiview Part 1-2. Withers, P. J. och Bhadeshia, H. K. D. H. u.o. :
Science and Technology, 2000.
7. Laird, Campbell. Fatigue. [bokförf.] Robert W. Cahn. Physical Metallurgy 3. u.o. :
Elsevier Science, 1996.
8. Suresh, S. Fatigue of Materials. u.o. : Cambridge Univ Press, 1992.
9. Hutton, David V. Fundamentals of finite element analysis. u.o. : McGraw-Hill , 2004.
10. Low Temperature Elastic Constants of Deformed Polycrystalline Copper. Ledbetter, H
M och Kim, S A. u.o. : MateriaLs Science and Engineering A, 1988, Vol. 101.
11. Physical Metallugy. Cahn, Robert W och Haasen, Peter. 4, u.o. : Elsevier Science B.
V., Vol. I.
12. Determination of the Anelastic Modulus for several Metals. Alexopoulos, P. S. u.o. : Acta
Metallurgica, 1980, Vol. 29.
- 68 -
Appendix 11- Derivation of analytical beam model
A1
Elementary case analysis
A1.1
Forces of reaction,
displacement and support angles in beam part 1:
Figure 63: Depiction of beam part 1, point load elementary case.
Force balance in figure 64 and symmetry induces:
!
"!_! $%
1
Figure 64: Depiction of beam part 1, moment elementary case.
Force and momentum balance in figure 65 induces:
&: ( "!_ 0 2
) *++,-. /: 0"! ( 1 · (3) induces:
1·
2
0 3
30"!
- 69 -
3
4
45%
6
(1) and (4) induces:
R 8 R 8! ( R 8 9%
3
5
:;%
<
(4) in (2) induces:
"!_ 3
45%
6
6
(1) and (6) induces:
"! "!_! ( "!_ $%
(
45%
6
7
Displacement and support angles:
Originating from point load:
> [email protected] · 6 AB DEF · AG1 3 A @ 3 @ H I
6
!
!
$6 C
$% 6 C
· · J?1 · K 3 KB · 3 LM
$% 6 C
· · ? K · 3 LB
$% 6 C
··L
· N 8
KL EF !
DEF
DEF
DEF
!
!
K
!
!
H
!
!
6C
!
$% 6 2
· K ?1 · ( B
$% 6 2
·K·
DEF
!
!
P"!_! ?Q A B DEF
!
!
!
$% 6 2
DEF
· QA1 ( Q
!
H
- 70 -
· N 9
KL EF !
H
62
Originating from bending moment:
> [email protected] , 0 0B DEF · G0 [email protected] 3 [email protected] ( @ H ( 0" @ 3 @ H I
!
62
DEF · 0" ? · K 3 LB
62
!
K
DEF · 0" L 62
!
· 0 10
KL EF "
H
H
62
P"!_ 0 0 4S ·6
DEF
(
45 ·6
HEF
!
H
· EF 0" 11
6
(8) - (11) induces the following equation system:
T
· N
KL EF !
!
H
KL
A1.2
6C
· EF N!
62
3 KL · EF 0"
H
6
HEF
62
0"
7.4 · 10VH Y
0
X
W
12
Forces of reaction,
displacement and support angles in beam part 2:
Figure 65: Depiction of beam part 2, point load elementary case.
Force balance in figure 66 and symmetry induces:
"_! Z!_! $2
13
- 71 -
Figure 66: Depiction of beam part 2, moment elementary case.
Force and momentum balance in figure 67 induces:
(14) induces:
) *++,-. /: 30" 3 1 · Z!_ 0 14
) *++,-. [: 0Z! ( 1 · "_ 0 15
Z!_ 3
452
16
"_ 3
4\%
17
6
(15) induces:
6
(13) and (17) induces:
" "_! ( "_ $2
3
4\%
18
$2
3
452
19
6
(13) and (16) induces:
Z! Z!_! ( Z!_ 6
(7) and (18) induces:
" "! ( "
$%
(
!
N! ( N 3 0Z! 3 0"! 20
45%
6
(
$2
3
!
4\%
6
6
- 72 -
Displacement and support angles:
Originating from point load:
> [email protected] AB ·
KL
!
!
21
$2 6 C
EF
P"_! PZ!_! ?Q A B !
·
KL
H
$2 6 2
EF
22
Originating from bending moment:
> [email protected] B DEF · G0 [email protected] 3 [email protected] ( @ H ( 0" @ 3 @ H I
!
62
! ! H
! H
· 0 ]2 · 3 3 · ?B ( ?B ^ ( 0" ] 3 ?B ^
DEF
62
62
DEF
!
!
· J0 ? · K 3 K · ( LB ( 0" ? · K 3 LBM
K
H
!
!
K
!
DEF · J0 ?L 3 L ( LB ( 0" ?L 3 LBM
62
62
DEF
L
D
!
· JL 0 ( L 0" M
H
H
K
23
P"_ 4S ·6
(
45 ·6
24
PZ!_ 4S ·6
(
45 ·6
25
HEF
DEF
DEF
HEF
!
(21) to (22) induces the following equation system:
· N
KL EF !
6C
T H · 62 N
KL EF · N
KL EF H
62
3 KL · EF 0
H
62
3 H · EF 0
!
6
3 D · EF 0
!
6
3 KL · EF 0"
H
62
3 D · EF 0"
!
6
3 H · EF 0"
!
6
- 73 -
7.4 · 10VH Y
_
_
0
0
X
_
_
W
26
A1.3
A1.3
Forces of reaction,
displacement and support angles in beam part 3:
Figure 67: Depiction of beam part 3, point load elementary case.
Force balance in figure 68 and symmetry induces:
Z_! `! $C
27
Figure 68: Depiction of beam part 2, moment elementary case.
Force and momentum balance in figure 69 induces:
&: Z_ ( ` 0 28
) *++,-. [: 30Z 3 1 · ` 0 29
(29) induces:
` 3
4\2
6
30
(30) in (28) induces:
Z_ 3` 4\2
6
31
(27) and (31) induces:
- 74 -
Z Z_! ( Z_ (
$C
4\2
6
32
(19) and (32) induces:
Z Z! ( Z
$2
3
!
N ( NH 3 0" 3 0Z 33
6
452
6
(
$C
(
4\2
6
!
(27) and (30) induces:
` `! ( ` 3
$C
34
4\2
6
Displacement and support angles:
Originating from point load:
> [email protected] AB !
·
KL
!
35
$C 6 C
EF
PZ_! P`! ?Q A B !
Originating from bending moment:
> [email protected] , 0" 0B !
PZ_ 45 ·6
HEF
·
KL
H
$C 6 2
EF
36
· 0
37
KL EF Z_
H
62
38
(35) to (38) induces the following equation system:
·
KL
T
!
H
KL
·
$C 6 C
EF
$C 6 2
EF
3 KL · EF 0Z_
H
62
3 H · EF · 0"
!
6
3.2 · 10VH Y
0
- 75 -
X
W
39
a
0
c 40
b
a d · e 41
(40) and (41) induces:
d·e
e
0
·c
b
0
· c 42
db
- 76 -
Appendix 22-Stiffness matrix
- 77 -
- 78 -
dbh
12
j[· H
H
i
Q
6i
[· Q
fkD flL fm!n fHK ( f!
1
36[ · i ]1 ( ^
Q
f!! f!! g[ !
f! f!
!
f!H !
f!H
!
f fH
!
f
fDD fLL f!n!n fKK ( f
1
4[ · i ]1 ( ^
Q
12
3[ · H
Q
f!K f!K 3[ ·
6i
Q
f!!!! fHH ( f!!
k
f!!! fHK ( f!
k
6i
!
fH 3[ · Q
!
2i
Q
fHK fHK ( f! 6[ · i ]1 3
!
fHk fkl flm fm!! !
f!H
f!!!H f!H 3[ ·
D
1
^
QH
f!!!K f!K [ ·
D
1
^
Q
f!! fKK ( f
k
1
^
Q
12
QH
6i
Q
D
1
4[ · i ]1 ( ^
Q
312[
fHD fkL fl!n fm! f!K 6[ · i
f!!H fH [ ·
D
1
!
fKK fKK ( f 4[ · i ]1 ( ^
Q
f!!K fKk fDl fLm f!n!! fH
36[ · i
D
fK
D
6i
Q
2i
[·
Q
f!H!H fHH [ ·
fKD fDL fL!n f!n! fK
2[ · i
12
QH
f!H!K fHK 3[ ·
D
fkk fll fmm fHH ( f!!
1
12[ · ]1 ( H ^
Q
D
1
^
QH
36[ · i ]1 3
fHH fHH ( f!! 12[ · ]1 (
!
D
12[ · ]1 (
4i
[·
Q
fK fK [ ·
H
f!K!K fKK [ ·
H
D
- 79 -
6i
Q
4i
Q
H
Appendix 3-Matlab code for analytical beam model
Below the MATLAB code for the elementary case analysis is found.
%Balkböjning av rör i riktmaskin 327, med elementarfallsmetoden
clc;
format long
%Konstanter
E=2.1e11;% elasticitetsmodul [GPa][N/m2]
%Variabler
D=156.3e-3;%[m] rörets ytterdiameter
t=12.6e-3;% [m] rörets godstjocklek
l=900e-3;% [m] elementets längd
v2=7.4e-3;% [m] rörets utböjning
v4=1.4e-3;% [m] rörets utböjning
v6=3.2e-3;% [m] rörets utböjning
a=1;% korrigering för annan elementlängd
%Beräknade variabler
d=D-2*t;% [m]
I=pi*(D^4-d^4)/64% [m4]
C=E*I
%Balkdel 1
x1=0:1e-3:450e-3;
xi=x1/l;
E1=[(1/48)*(l^3/C) -(3/48)*(l^2/C)
(3/48)*(l^2/C) -(1/3)*(l/C)]
D1=[v2
0];
L1=inv(E1)*D1
L1(1,1)
delta_P1=(L1(1,1)*l^3)/(12*C)*(3/4*xi-xi.^3);
delta_P1M=0;
for i=1:length(delta_P1)-1
delta_P1M(1,i)=delta_P1(1,length(delta_P1)-i);
end
delta_P1=[delta_P1 delta_P1M];
x=0:1e-3:900e-3;
xi=x/l;
L1(2,1)
delta_M1=l^2/(6*C)*(L1(2,1))*(xi-xi.^3);
- 80 -
delta_1=-delta_P1+delta_M1;
%Balkdel 2
x1=0:1e-3:450e-3;
xi=x1/l;
E2=[(1/48)*(l^3/C) -(3/48)*(l^2/C) (-3/48)*(l^2/C)
(3/48)*(l^2/C) (-1/3)*(l/C) (-1/6)*(l/C)
(3/48)*(l^2/C) (-1/6)*(l/C) (-1/3)*(l/C)];
D2=[v4
0
0];
L2=inv(E2)*D2
delta_P2=(L2(1,1)*l^3)/(12*C)*(3/4*xi-xi.^3);
delta_P2M=0;
for i=1:length(delta_P2)-1
delta_P2M(1,i)=delta_P2(1,length(delta_P2)-i);
end
delta_P2=[delta_P2 delta_P2M];
x=0:1e-3:900e-3;
xi=x/l;
delta_M2=l^2/(6*C)*(L2(2,1)*(2*xi-3*xi.^2+xi.^3)+L2(3,1)*(xi-xi.^3));
delta_2=-delta_P2+delta_M2;
%Balkdel 3
x1=0:1e-3:450e-3;
xi=x1/l;
E3=[(1/48)*(l^3/C) -(3/48)*(l^2/C)
(3/48)*(l^2/C) -(1/3)*(l/C)];
D3=[v6
0];
L3=inv(E3)*D3
delta_P3=(L3(1,1)*l^3)/(12*C)*(3/4*xi-xi.^3);
delta_P3M=0;
for i=1:length(delta_P3)-1
delta_P3M(1,i)=delta_P3(1,length(delta_P3)-i);
end
delta_P3=[delta_P3 delta_P3M];
x=0:1e-3:900e-3;
- 81 -
xi=x/l;
delta_M3=l^2/(6*C)*(L3(2,1)*(2*xi-3*xi.^2+xi.^3));
delta_3=-delta_P3+delta_M3;
plot(x,-delta_P3)
hold on
plot(x,-delta_M3)
hold on
plot(x,delta_3)
%Plotta hela balken
x=0:1e-3:2700e-3;
delta=[delta_1(:,1:900) delta_2(:,1:900) delta_3];
%Beräkna reaktionskrafterna
Ra=L1(1,1)/2-L1(2,1)/l%Balkdel 1
Rb_1_1=L1(1,1)/2
Rb_1_2=L1(2,1)/l
Rb_1=Rb_1_1+Rb_1_2
Rb2_1=L2(1,1)/2%Balkdel 2
Rb2_2=L2(3,1)/l
Rb_2=Rb2_1
Rc_1=L2(1,1)+L2(2,1)/l
Rc_2=L3(1,1)/2-L3(2,1)/l%Balkdel 3
Rd=L3(1,1)/2-L3(2,1)/l;
Rb=Rb_1+Rb_2
Rc=Rc_1+Rc_2
Rd
%Beräkna reaktionsmomenten
Mb=-L1(2,1)+L2(2,1)
Mc=-L2(3,1)+L3(2,1)
plot(x,delta)
title('Plot of tube deflection')
xlabel('Position along tube [m]')
ylabel('Deflection [m]')
%axis([0 2.7 -10e-3 0])
set(gca,'XTick',0:0.45:2.7)
set(gca,'YTick',-20e-3:1e-3:0)
grid on
%Kontroll av reaktionskrafterna
Rest=Ra-L1(1,1)+Rb-L2(1,1)+Rc-L3(1,1)+Rd
- 82 -
Appendix 4-Matlab code for the limited FEMFEM-model
Below the MATLAB code that solves the little FEM-model is found.
clc;
format long
%Konstanter
E=2.1e11;
[GPa][N/m2]
% elasticitetsmodul
%Variabler
D=156.3e-3;
t=12.6e-3;
l=450e-3;
v2=7.4e-3;
kraften P1
v4=1.4e-3;
kraften P2
v6=3.2e-3;
kraften P3
a=1;
elementlängd
r=14;
%
%
%
%
[m]
[m]
[m]
[m]
rörets ytterdiameter
rörets godstjocklek
elementets längd
rörets utböjning vid
% [m] rörets utböjning vid
% [m] rörets utböjning vid
% korrigering för annan
% antalet randvillkor
x1=0:1e-3:450e-3;%Definierar balkens olika element
x2=450e-3:1e-3:900e-3;
x3=900e-3:1e-3:1350e-3;
x4=1350e-3:1e-3:1800e-3;
x5=1800e-3:1e-3:2250e-3;
x6=2250e-3:1e-3:2700e-3;
%Beräknade variabler
d=D-2*t;
I=pi*(D^4-d^4)/64;
C=E*I/l^3;
% [m]
% [m4]
%Korrigerat elements styvhetsmatris
k1=C*[12/a^3 6*l/a^2 -12/a^3 6*l/a^2
0 4*l^2/a -6*l/a^2 2*l^2/a
0 0 12/a^3 -6*l/a^2
0 0 0 4*l^2/a];
%Elementets styvhetsmatris
k2=C*[12 6*l -12 6*l
0 4*l^2 -6*l 2*l^2
0 0 12 -6*l
0 0 0 4*l^2];
K=zeros(14);%Ursprunglig global styvhetsmatris
F=[0;1;1;1;0;0;1;1;0;0;1;1;0;1];%Den ursprungliga förskjutningsvektorns
positioner
- 83 -
for i=1:4
for j=1:4
for m=0:1
K(i+10*m,j+10*m)=k1(i,j);%Läser in elememten från
elementstyvhetsmatrisen k1 i den globala matrisen
end%i är rad och j kolumn, den plockar över första raden från
elementstyvhetsmatrisen till globala styvhetsmatrisen
end%m tar hänsyn till att det är den första och den 6:e
elementstyvhetsmatrisen som kommer från k1, dvs det skiljer 10
elementpositioner dem emellan
end
K;
for i=1:4
for j=1:4
for m=1:4
K(i+2*m,j+2*m)=K(i+2*m,j+2*m)+k2(i,j);%Läser in elememten från
elementstyvhetsmatrisen k2 i den globala matrisen
end%m tar hänsyn till att det är 2, 3, 4 och den 5:e
elementstyvhetsmatrisen som kommer från k2, dvs det skiljer 2
elementpositioner dem emellan
end
end
K;
for i=1:r
for j=1:r
if i~=l
K(j,i)=K(i,j);%speglar den globala matrisen i diagonalen och gör
den symmetrisk
end
end
end
K;
j=0;%Räknar igenom förskjutningsvektorn, registrerar RV som ger bidrag,dess
plats och ser hur många obekanta det finns.
for i=1:r
if F(i,1)~=0
j=j+1;
m(j)=i;
end
end
%m
%j
K_temp_rad=zeros(j,r);%Skapar en matris för att lagra bidragande kolumner
for i=1:j
K_temp_rad(i,:)=K(m(i),:);
end
K_temp_rad;
K_red=zeros(j,j);%Definierar den reducerade matrisen
for i=1:j
- 84 -
K_red(:,i)=K_temp_rad(:,m(i));
end
K_red;
L=zeros(j,1);
L=-v2*K_red(:,2);%Beräknar lastvektorn, genom att flytta över de kända
förskjutningarna till lastvektorn
L=L-v4*K_red(:,4);
L=L-v6*K_red(:,6);
L2=[0 -1 0 0 0 0 0 0];%Radmatriser för att kunna flytta de okända lasterna
till förskjutningsvektorn
L4=[0 0 0 -1 0 0 0 0];
L6=[0 0 0 0 0 -1 0 0];
K_red(:,2)=L2';%Byter ut kolonnerna som flyttats till lastvektorn emot de
transponerade radmatriserna ovan
K_red(:,4)=L4';
K_red(:,6)=L6';
%K_red
F_red=inv(K_red)*L;%Beräknar förskjutningsvektorn, som också innehåller de
okända lasterna
F_global=zeros(r,1);%Definierar en global förskjutningsvektor
for i=1:j%Flyttar över de beräknade förskjutningarna från den reducerade
förskjutningsvektorn till den globala sanna förskjutningsvektorn
F_global(m(i),1)=F_red(i,1);
end
F_global(3,1)=v2;%Läser in de ursprungliga förskjutningarna i den globala
förskjutningsvektorn
F_global(7,1)=v4;
F_global(11,1)=v6
%F_global
%Beräknar reaktionskrafter och moment
for i=1:r%Beräknar reaktionskrafterna och momenten
L_global(i,1)=K(i,:)*F_global;
end
L_global
for i=1:r/2
if i==1
L_res=L_global(i,1);
else
L_res=L_res+L_global(2*i-1,1);
end
- 85 -
end
L_res;%Kontrollerar resultantkraften efter summering av reaktionskrafterna
och de yttre krafterna
%Balkdel 1
x1=0:1e-3:450e-3;%Definierar halva intervallet, för att formeln bara gäller
upp till halva intervallet
l=900e-3;
xi=x1/l;
delta_P1=(L_global(3,1)*l^3)/(12*(E*I))*(3/4*xi-xi.^3);%Beräknar
utböjningen av punktlasten
delta_P1M=0;%Speglar utböjningen, eftersom den är helt symmetrisk
for i=1:length(delta_P1)-1
delta_P1M(1,i)=delta_P1(1,length(delta_P1)-i);
end
delta_P1=[delta_P1 delta_P1M];%Lägger ihop de två delarna
x=0:1e-3:900e-3;
xi=x/l;
E1=[(1/48)*(l^3/(E*I)) -(3/48)*(l^2/(E*I))%Ekvationssystem av de
superponerade punktlasterna och momenten
(3/48)*(l^2/(E*I)) -(1/3)*(l/(E*I))];
D1=[v2
0];
L1=inv(E1)*D1;%Förskjutningsvektorn
delta_M1=l^2*(L1(2,1)/(6*E*I))*(xi-xi.^3);%Beräknar utböjningen av det
enskilda momentet, OBS! Den globala förskjutningsvektorn innehåller det
resulterande momenten ifrån de olika balkelementens respektive moment
delta_1=-delta_P1+delta_M1;%Beräknar den superponerade utböjningen
Mb_1=L1(2,1)%Beräknar momentbidrag från balkdel 1
%Balkdel 2
x1=0:1e-3:450e-3;
xi=x1/l;
E2=[(1/48)*(l^3/(E*I)) -(3/48)*(l^2/(E*I)) (-3/48)*(l^2/(E*I))
(3/48)*(l^2/(E*I)) (-1/3)*(l/(E*I)) (-1/6)*(l/(E*I))
(3/48)*(l^2/(E*I)) (-1/6)*(l/(E*I)) (-1/3)*(l/(E*I))];
D2=[v4
0
0];
- 86 -
L2=inv(E2)*D2;
delta_P2=(L_global(7,1)*l^3)/(12*(E*I))*(3/4*xi-xi.^3);
delta_P2M=0;
for i=1:length(delta_P2)-1
delta_P2M(1,i)=delta_P2(1,length(delta_P2)-i);
end
delta_P2=[delta_P2 delta_P2M];
x=0:1e-3:900e-3;
xi=x/l;
delta_M2=l^2/(6*(E*I))*(L2(2,1)*(2*xi-3*xi.^2+xi.^3)+L2(3,1)*(xi-xi.^3));
delta_2=-delta_P2+delta_M2;
Mb_2=-L2(2,1)
Mc_1=-L2(3,1)
%Balkdel 3
xi=x1/l;
E3=[(1/48)*(l^3/(E*I)) -(3/48)*(l^2/(E*I))
(3/48)*(l^2/(E*I)) -(1/3)*(l/(E*I))];
D3=[v6
0];
L3=inv(E3)*D3;
delta_P3=(L_global(11,1)*l^3)/(12*(E*I))*(3/4*xi-xi.^3);
delta_P3M=0;
for i=1:length(delta_P3)-1
delta_P3M(1,i)=delta_P3(1,length(delta_P3)-i);
end
delta_P3=[delta_P3 delta_P3M];
x=0:1e-3:900e-3;
xi=x/l;
delta_M3=l^2/(6*(E*I))*(L3(2,1)*(2*xi-3*xi.^2+xi.^3));
delta_3=-delta_P3+delta_M3;
Mc_2=-L3(2,1)
%Plotta hela balken
x=0:1e-3:2700e-3;
- 87 -
delta=[delta_1(:,1:900) delta_2(:,1:900) delta_3];
%delta
plot(x,delta)
title('Plot of tube deflection')
xlabel('Position along tube [m]')
ylabel('Deflection [m]')
%axis([0 2.7 -10e-3 0])
set(gca,'XTick',0:0.45:2.7)
set(gca,'YTick',-20e-3:1e-3:0)
grid on
figure (2)
%Plot av tvärkraftsdiagram
R1=L_global(1,1);
R_1=[0 R1];
X_1=[0 0];
R_2=[R1 R1];
X_2=[0 l/2];
R3=R1+L_global(3,1);
R_3=[R1 R3];
X_3=[l/2 l/2];
R_4=[R3 R3];
X_4=[l/2 l];
R5=R3+L_global(5,1);
R_5=[R3 R5];
X_5=[l l];
R_6=[R5 R5];
X_6=[l l*3/2];
R7=R5+L_global(7,1);
R_7=[R5 R7];
X_7=[l*3/2 l*3/2];
R_8=[R7 R7];
X_8=[l*3/2 l*2];
R9=R7+L_global(9,1)
R_9=[R7 R9];
X_9=[l*2 l*2];
R_10=[R9 R9];
X_10=[l*2 l*5/2];
R11=R9+L_global(11,1)
R_11=[R9 R11];
X_11=[l*5/2 l*5/2];
- 88 -
R_12=[R11 R11];
X_12=[l*5/2 l*3];
R13=R11+L_global(13,1)
R_13=[R11 R13];
X_13=[l*3 l*3];
R=[R_1 R_2 R_3 R_4 R_5 R_6 R_7 R_8 R_9 R_10 R_11 R_12 R_13];
X=[X_1 X_2 X_3 X_4 X_5 X_6 X_7 X_8 X_9 X_10 X_11 X_12 X_13];
plot(X,R)
hold on
X_led=[0 3*l]
Y_led=[0 0]
plot(X_led,Y_led,'k')
title('Transverse force diagram')
xlabel('Position along tube [m]')
ylabel('Force [N]')
%axis([0 3 -10e-3 0])
set(gca,'XTick',0:0.45:2.7)
set(gca,'YTick',-1.5e6:3e5:3e6)
grid on
figure (3)
%Plot av momentdiagram
%Balkdel 1
M_1=-x1*(L_global(3,1)/2)+x1*(Mb_1/l);
e_1=M_1/(E*I)*(D/2);
M_2=-(l/2-x1)*(L_global(3,1)/2)+x2*(Mb_1/l);
e_2=M_2/(E*I)*(D/2);
%Balkdel 2
M_3=-x1*(L_global(7,1)/2)-x1*(Mb_2/l)-(l-x1)*(Mc_1/l); %Sammanlagda
momentet
e_3=M_3/(E*I)*(D/2);
M_stod_1=[M_2(length(M_2)) M_3(1)];
e_stod_1=M_stod_1/(E*I)*(D/2)
M_4=-(l/2-x1)*(L_global(7,1)/2)-x2*(Mc_1/l)-(l/2-x1)*(Mc_1/l);
e_4=M_4/(E*I)*(D/2);
%Balkdel 3
M_5=-x1*(L_global(11,1)/2)-(l-x1)*(Mc_2/l);
e_5=M_5/(E*I)*(D/2);
- 89 -
M_stod_2=[M_4(length(M_4)) M_5(1)]
e_stod_2=M_stod_2/(E*I)*(D/2);
M_6=-(l/2-x1)*(L_global(11,1)/2)-(l/2-x1)*(Mc_2/l);
e_6=M_6/(E*I)*(D/2);
XP_3=[x1 x2 x3 x4 x5 x6]
MP_3=[M_1 M_2 M_3 M_4 M_5 M_6]
plot(XP_3,MP_3)
hold on
plot(X_led,Y_led,'k')
title('Moment diagram')
xlabel('Position along tube [m]')
ylabel('Moment [Nm]')
set(gca,'XTick',0:0.45:2.7)
set(gca,'YTick',-10e5:1e5:10e5)
grid on
figure (4)
e_3p=[e_2(1,451) e_4(1,1)]
x3_p=[x3(1,1) x3(1,451)]
XP_4=[x1 x2 x3 x4 x5 x6];
ep_4=[e_1 e_2 e_3 e_4 e_5 e_6];
plot(XP_4,ep_4)
hold on
plot(X_led,Y_led,'k')
title('Strain')
xlabel('Position along tube [m]')
ylabel('Strain [ ]')
set(gca,'XTick',0:0.45:2.7)
set(gca,'YTick',-15e-3:2e-3:15e-3)
grid on
- 90 -
Appendix 5- Additional lcf proof strength diagrams
Stress [MPa]
Material B, strain 0,8%
Rp0.2 dynamic & true
1100
1050
1000
950
900
850
800
750
700
650
600
550
500
450
1065
1011
976
878
6-3-12_1_1
6-3-12_1_2
0
1
2
3
4
5
6
Cycles [No.]
7
8
9
10
Stress [MPa]
Material B, strain 0,8%
Rp0.2 dynamic & true
1100
1050
1000
950
900
850
800
750
700
650
600
550
500
450
1049
871
851
850
6-3-12_3_1
0
1
2
3
4
5
6
Cycles [No.]
- 91 -
7
8
9
10
Stress [MPa]
Material B, strain 0,8%
Rp0.2 dynamic & true
1100
1050
1000
950
900
850
800
750
700
650
600
550
500
450
1054
900
877
860
860
847
6-3-12_5_1
0
1
2
3
4
5
6
7
8
9
10
Cycles [No.]
Stress [MPa]
Material B, strain 0,8%
Rp0.2 dynamic & true
1100
1050
1000
950
900
850
800
750
700
650
600
550
500
450
1047
879
868
846
834
824
817
809
6-3-12_7_1
0
1
2
3
4
5
6
Cycles [No.]
- 92 -
7
8
9
10
Stress [MPa]
Material B, strain 0,8%
Rp0.2 dynamic & true
1100
1050
1000
950
900
850
800
750
700
650
600
550
500
450
1034
889
861
843
829
817
830
809
6-3-12_7_2
0
1
2
3
4
5
6
7
8
9
10
Cycles [No.]
Stress [MPa]
Material B, strain 0,8%
Rp0.2 dynamic & true
1100
1050
1000
950
900
850
800
750
700
650
600
550
500
450
1052
893
871
857
844
836
825
818
841
814
807
6-3-12_10
0
1
2
3
4
5
6
Cycles [No.]
- 93 -
7
8
9
10
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