Politecnico di Milano
School of Industrial and Information Engineering
MSc in Mechanical Engineering
Advanced Manufacturing Processes
2021-2022 Academic Year
FEM simulation of tube bending processes
Francesco Flamini 995700
Claudia Ghidoni Pastorelli 976445
Matteo Lamberti 995030
Massimiliano Moreni 977593
1
ABSTRACT
Tube bending is a metal forming process (usually a cold one) highly used in lots of
industrial fields, requiring precision and skilled knowledge. The following work is
mainly focused on the application of Finite Element Method (through many different
softwares, depending on the applications needed) in order to create a simplified model
of a particular structure, to prevent the most common defects and observe the parts’ (or
assemblies’) behaviour. All the most important bending processes will be introduced
and, for each of them, a numerical model will be associated, based on the specific
conditions we are taking into account.
2
INDEX
LIST OF FIGURES…………………………………………………………………………………4
INTRODUCTION…………………………………………………………………………………..8
BENDING PROCESSES AND FEM SIMULATION…………………………………………...10
•
•
•
•
•
•
•
•
Rotary draw bending………………………………………………………………………..11
Roll bending………………………………………………………………………………...24
Heat induction bending ……………………………………………………………………..31
Laser bending……………………………………………………………………………….36
Shear bending……………………………………………………………………………….41
Push bending………………………………………………………………………………..49
Press bending………………………………………………………………………………..66
Hydro bending………………………………………………………………………………68
REFERENCES…………………………………………………………………………………….71
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LIST OF FIGURES
Fig. 1. Tube bending process [taken from Google images]
Fig. 2. Tube bending process using heat [taken from Google images]
Fig. 3. FEM analysis of bent tubes [taken from Google images]
Fig. 4. FEM analysis of bent tubes [taken from Google images]
Fig. 5. Rotary draw bending
Fig. 6. Sketch of NC bending process of tube
Fig. 7. Eight-node hexahedral isoparametric element in natural and Cartesian coordinate systems
Fig. 8. A FEM model of tube bending
Fig. 9. Deformed meshes when bending angles are (a) 0.0; (b) 0.3; (c) 0.6;(d) 0.9; (e) 1.2; (f) 1.57 rad
Fig. 10. Stress distribution along the bending direction of tube with bending angle 0.3 rad
Fig. 11. Relationship between maximal wall thickness changing ratio and bending angle
Fig. 12. Elastic-plastic finite element model of springback of thin-walled tube NC bending
Fig. 13. Schematic diagram of mandrel simplified reasonably
Fig. 14. Curves of bending velocity and mandrel drawing velocity vs time
Fig. 15. Constraint condition of springback computing
Fig. 16. Simulation and experiment results of springback angle
Fig. 17. Curve of bending angle vs springback angle of stainless steel tube bending
Fig. 18. Curves of bending angle vs springback angle of stainless steel tube bending with different relative
bending radius
Fig. 19. Curves of bending angle vs springback angle of stainless steel tube bending with different relative
tube diameter
Fig. 20. Curves of ending angle vs springback angle of tube bending with different elastic modulus
Fig. 21. Curves of bending angle vs springback angle of tube bending with different yield stress
Fig. 22. Curves of bending angle vs springback angle of tube bending with different strength factor
Fig. 23. Curves of bending angle vs springback angle of tube bending with different hardenability value
Fig. 24. Forming principle of rotary draw bending method and sketch of standard mandrel
Fig. 25. FEM model for NC bending process
Fig. 26. Geometry model of mandrel with three balls
Fig. 27. Comparison between the experimental and simulation results cencerning the cross-section distorsion vs
bending angle
Fig. 28. Tools and axis of the three-roll-push-bending machine
Fig. 29. Three main steps of the three-roll-push-bending process
Fig. 30. Tube section
Fig. 31. Tool set-up for the Three-Roll-Push-Bending-Twisting with a description of the kinematic of the tools
4
Fig. 32. Final setup of the machine. The roller B is constrained to A by a spherical-joint element, which allows
the centre of B to rotate about that of A, keeping the same relative distance
Fig. 33. Tube mock-up ‘S’: Comparison between measured and numerical profiles. The red points represent
the edges of the final meshed, whereas the measured points have been plotted with colour scaled in relation to
their deviation (or distance) from the mesh-surfaces. 3D visual of the whole tube (a), view of portion
undergoing TRPBTon the ZX- (b) and YZ-plane (c); curvatures obtained via RDB (d) (f)
Fig. 34. S-shaped tube mock-up. (f) and (g) residual stresses distribution in terms of Von Misses equivalent
stress while (c) and (d) accumulated plastic strain in the zones bent via RDB. (a), (c), (f) bottom view, (b), (d),
(g) top view. In (b) the welding seam is highlighted
Fig. 35. Variation of residual stresses and plastic strain along path-1 (f) and -2 (g) are plotted after the
completion of the forming process. Path-2 follows the welding seam. e stress tensor orientation, S11 is aligned
with the longitudinal direction
Fig. 36. Compressive test with numerical results and experimental ones: (a) reaction force, (b) ux displacement,
(c) uy displacement vs z-displacement along the loading-axis; in the model FEM-stress the residual stress and
accumulated strain from the previous process phases are considered, on the contrary, these aspects are not
taken in account in the model FEM-virgin
Fig. 37. Induction bending machine
Fig. 38. FEM meshed model
Fig. 39. Pipe shape after loading
Fig. 40. Pipe shape after unloading
Fig. 41. Temperature distribution in local heating zone
Fig. 42. Circumferential equivalent plastic strain
Fig. 43. Wall thickness and thickening
Fig. 44. Ovality/spring back angle with bending angle (a/b)
Fig. 45. Ovality with load step
Fig. 46. Circumferential scanning scheme
Fig. 47. Factors of influence
Fig. 48. Finite element model
Fig. 49. Sketch of laser scanning process
Fig. 50. Temperature with time
Fig. 51. Axial stress with time
Fig. 52. Axial strain with time
Fig. 53. Displacement with time
Fig. 54. Axial plastic strain/time
Fig. 55. Bending angle with n of scans
Fig. 56. Schematic sketch of shear bending equipment
Fig. 57. Schematic illustration of shear bending process
Fig. 58. Typical products by various pushing forces: (a) splitting; (b) successful; (c) wrinkling; (d) rupturing
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Fig. 59. Finite element meshes after the deformation
Fig. 60. Distribution of tube effective strain
Fig. 61. Strain distributions across tube width in the steady state region (line TB)- maximum strains have
occurred around π = 90°
Fig. 62. Distribution of thickness strain along routes T and B
Fig. 63. Cross section configurations of the deformed tubes with different initial thickness obtained by
simulation and experiment: (a) simulation, (b) experiment and(c) experiment (t0=3.0 mm)
Fig. 64. Effect of the axial pushing pressure on the cross section ovality of the deformed tube
Fig. 65. Configuration of the deformed tube
Fig. 66. Effect of the initial thickness on the thickness strain of the tube-sheared part obtained by simulation
and experiment.
Fig. 67. Relation between the pushing and shearing strokes
Fig. 68. Effect of the tube initial thickness on the as a function of the tube initial thickness obtained by
simulation shearing load obtained by simulation
Fig. 69. Effect of the initial thickness and pushing pressure on the distribution of thickness strain along the
hoop direction of the tube-sheared part
Fig. 70. Tube bending stress and strain analysis
Fig. 71. FE modelling of tube free bending
Fig. 72. Model assembly position
Fig. 73. Bending simulation under condition
Fig. 74. Bending simulation under condition
Fig. 75. Bending simulation under condition
Fig. 76. Bending simulation under condition
Fig. 77. Experiments of 3D free bending under different conditions
Fig. 78. Schematic of an elbow granular-media-based push-bending process: (a) before bending, (b) bending
Fig. 79. FEM model of tubular blank and DEM model of granular filler
Fig. 80. Stress-strain curve of 1Cr18Ni9Ti
Fig. 81. Distributions of stress in granular filler and tube: (a) granular filler, (b) tube
Fig. 82. Forming force of the granular-media-based push-bending process: simulation and experimental results
Fig. 83. Effects of particle sizes of granular fillers on wrinkling of thin-wall elbow tube: (a) particle size 0.98
mm, (b) particle size 1.58 mm, (c) particle size 2.08 mm
Fig. 84. The contacts’ distribution on tube wall (unfolded into a plane): (a) particle size 0.98 mm, (b) particle
size 1.58 mm, (c) particle size 2.08 mm
Fig. 85. The tube bending process treated
Fig. 86. Meshed geometries of die, tube and rod
Fig. 87. Urethane rod penetrating the tube
Fig. 88. Predicted tube geometry after bending
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Fig. 89. Von Mises stress distribution in the formed tube
Fig. 90. Von Mises strain distribution in the formed tube
Fig. 91. MOS bending technique
Fig. 92. Influence factors on free bending technology
Fig. 93. Bending profile experimentally used
Fig. 94. Measured geometry of EN AW-5182 tubes
Fig. 95. Standard deviation and density function of experimental bending
Fig. 96. Strong wrinkling at radius 1
Fig. 97. Weak wrinkling at radius 4
Fig. 98. Decrease of wrinkling by decrease of lubricant (a to c)
Fig. 99. Simulation result of test geometry (max EPS): a) batch C4 b) batch C1, C3
Fig. 100. Technique used to measure radius and bending angle
Fig. 101. Correlation between experiment and simulation varying the weld seam position
Fig. 102. (a)ANOVA in terms of EPS inner. (b) Scatter plot
Fig. 103. (a)ANOVA in terms of bending angle. (b) of diameter and thickness in relation to EPS inner ANOVA
in terms of bending radii
Fig. 104. Press bending process (1. upper roll; 2-3 lower rolls; 4 – pipe (Semi-manufactured); 2θ - bending
angle; h – the distance covered by the upper roll during the bending; R – the bending radius)
Fig. 105. FEM of press bending
Fig. 106. The final position of press bending
Fig. 107. Stress-strain relation reached through tube tensile test
Fig. 108. Distribution of wall thickness of angle 2Ζ=25°
Fig. 109. Distribution of wall thickness of angle 2Ζ=90°
Fig. 110. Wall thickness trend
Fig. 111. Diagram of double-layered tube specimen
Fig. 112. Dimensions of ultra-thin-walled elbow (number 1-1 to 5-5 are the measuring locations of section
flattening after bending)
Fig. 113. Tooling of the double layered tube hydro bending
Fig.114. Bending results under different levels of internal pressure: (a) defect-free double-layered elbow
formed under Py; (b) section view of double-layered elbow under pressure Py; (c) wrinkling under pressure
0.1Py; (d) rupture under pressure 1.2Py; (e) ultra-thin-walled elbow formed under pressure 0.8Py; and (f) ultrathin-walled elbow formed under pressure Py
Fig. 115. Finite element meshes for double-layered tube hydro bending: (a) shell type outer tube (b) solid type
outer tube
7
INTRODUCTION
As one kind of key lightweight components for “bleeding” transforming and loading carrying with enormous
quantities and diversities, metallic bent tubular parts have attracted increasing applications in many hightechnological industries such as aviation, aerospace, shipbuilding, automobile, energy and health care since
they satisfy the current needs for products with lightweight, high strength and high performance from both
materials and structure aspects. To bend the tubular materials with certain bending radius, bending angle and
shapes, the tube bending, one important branch of the tubular plastic forming fields, has been a vital
manufacturing technology for lightweight products. Many tube bending approaches have been developed in
response to the different demands of tube specification, shapes, materials and forming tolerance.
Fig. 1. Tube bending process [taken from Google images]
Fig. 2. Tube bending process using heat [taken from Google images]
According to the forming conditions, there are cold bending at room temperature and hot (heat) bending with
elevated temperature. From the viewpoint of loading conditions, there are pure bending, compression bending,
stretch bending, roll bending, rotary draw bending and laser bending. There are stainless steel tube bending,
aluminium alloy tube bending, copper tube bending, magnesium alloy tube bending and titanium alloy tube
bending in materials point of views. From aspects of tubular shapes, there are round tube bending, rectangular
tube bending and other irregular section tube bending. There are seamless tube bending and welded tube
bending from the tube fabrication respect. For any bending process, upon bending deformation, the complex
uneven tension and compression stress distributions are induced at the extrados and intrados of bending tube
respectively, which may cause multiple defects or instabilities such as wrinkling, over thinning (cracking),
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cross-section distortion and springback. The accurate prediction and efficient controlling of these physical
phenomena are the ever existing knots in forming manufacturing .Up to now, regarding different tube bending
processes with various loading conditions, great efforts have been conducted on investigations of the multiple
defects/instabilities, selection/optimization of the forming parameters and tooling to promote the development
of tube bending science and technology by using analytical, physical experimental and numerical methods.
For example using a FEM analysis we’re able to optimize the process non only in terms of reducing the defects
of the tubes but even from an economical point of view.
Fig. 3 and Fig. 4. FEM analysis of bent tubes [taken from Google images]
The optimization strategies associated to a particular type of process require enough knowledge about the basic
physical stages determining the achievement of the desired results. Furthermore, to obtain a correct
optimization, the quantitative results reached through a numerical analysis are of high importance too. Even if
the analytical approach is limited (mathematical equations describing the process too much complicated), the
finite element method (able to consider numerous dependencies on physical constants as well as to describe
complicated geometries and kinematics) allows us to describe coupled simulations of plastic and thermal
processes.
9
BENDING PROCESSES AND FEM SIMULATION
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Rotary draw bending
It is the most common process used in the bending industry. This type of bending has a great variety of tooling
options and offers the best quality and accuracy in terms of bending results, particularly for tight radii and thin
wall tubes. The characteristic of this bending method is that the centre bending die, which is used to form the
angle of tube parts, rotates with the workpiece together, and the die set sometimes is equipped with a wiper
and a mandrel, depending on the size and shape of workplace. A die set up for a rotary draw bending is shown
in Fig. 5.
Fig. 5 Rotary draw bending
Rotary draw bending is called in this way because the tube is being drawn into the bending area past the tangent
point. At one end, the tube is tightly pressed between a centre forming die and a clamp die that holds the tube.
The pressure die restrains the free end of the workpiece and allows it to move in a straight line, avoiding the
part to kick out. As the workpiece is being drawn and rotating around the centre die, the pressure die, which is
either static or boosted, transfers the workpiece to the centre die at the tangent point, so as to get the desired
angle and radius.[1]
In the following pages some examples of rotary draw bending technique applied to different processes are
presented in addition of FE simulations.
Using a three-dimensional rigid-plastic FE simulation system called TBS-3D (tube bending simulation by 3D
FEM) a Finite Element simulation system of an NC bending process of a thin-walled tube (D/t ≥ 20, where D
is the external diameter and t is the wall thickness) has been simulated neglecting the effect of springback;
moreover deformed meshes under different bending stages, the stress distribution along the bending direction,
and the relationship between the maximal wall thickness changing ratio and the bending angle have been
studied.
Thin-walled tube forming processes are widely used in the aerospace and automobile fields since the products
obtained are much lighter in weight and with enough strength.
For this manufacturing process NC bending machine instead of a traditional bending methods could be used
since the first can provide higher precision and efficiency; this also means that the bending process will be
affected by an higher number of process parameters and the design phase will be more complicated with
respect to the conventional trial-and-error approach.
The NC tube bending process can be represented as follows:
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Fig. 6 Sketch of NC bending process of tube.
The final desired shape is obtained through the use of the pressure die, the clamp die and the bending die.
Wrinkling and ovalizing of the cross-section are smoothed by the usage of a mandrel and wiper die, meanwhile
the pressure die can also offer a pushing force along the longitudinal direction to avoid an extreme reduction
of wall thickness in the outer tensile area.
The traditional tube bending process has been rebuilt using a FE simulation through which it is possible to
perform a “virtual bending” on a computer, helping not only in design phase but also in the analysis and
optimization of the process and the die parameters.
As a result numerous and sophisticated “trial-and-error” experiments can be avoided, reducing cost and time
without losing in quality and efficiency.
The eight-node hexahedral isoparametric element is adopted to simulate the tube bending process. The
following figures show this element in natural and Cartesian coordinate systems.
Fig. 7 Eight-node hexahedral isoparametric element in natural and Cartesian coordinate systems.
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The velocity and displacement vector of a node inside the element is:
where π₯π , π¦π , π§π are the coordinates, π’π₯π , π’π¦π , π’π§π are the velocities of some one-node of this element, ππ is the
shape function of the eight-node hexahedral isoparametric element with ππ , ππ , ππ natural coordinates of the
node (i is the number of the node of the element).
An important parameter that needs to be set up during the NC bending process of tube is the friction, so an
arctangent friction model has been introduced into the FE simulation using the following equation:
Where m is the friction factor, k the yield shear stress, π’π is the velocity vector of the workpiece relative to the
die, π’0 a verysmall positive number as compared to π’π , and t the unit vector in the direction of π’π .
For what concerns the FE model triangular elements have been chosen to simulate the pressure die, the bending
die and the wiper die, the tube has been discretized into some eight-node hexahedral isoparametric elements.
The following picture is showing the element choice:
Fig. 8. A FEM model of tube bending.
Computation conditions:
•
•
•
•
The
material
used
is
304
stainless
steel and the flow stress model is π =
1356(0.001 + π)0.549 πππ.
Tube dimensions: outer diameter of 28 mm, length 228 mm, and wall thickness of 1 mm.
Process parameters: the rotation speed of the bending die is 0.05 rad/s , the bending radius is 56 mm,
and the pressure die remains stationary.
Die parameters: the length of the wiper die, the pressure die and the mandrel are 140 mm.
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•
Friction condition: the friction factor between the pressure die and tube is 0.01 and the friction
factor between the wiper die and tube is 0.0.
As a conclusion to this bending process some pictures collecting the obtained results have been reported below:
Fig. 9. Deformed meshes when bending angles are (a) 0.0; (b) 0.3; (c) 0.6;(d) 0.9; (e) 1.2; (f) 1.57 rad.
Fig. 9 shows that mashes deform differently according to the different deformation stages when the bending
angles are 0.0 ,0.3 ,0.6 ,0.9 ,1.2 and 1.57 rad.
The deformation is mostly focused on the bending area, the higher is the distance from the bending area, the
smaller is the deformation. In part (a), L presents the length to be bent into a 90°(1.57 rad) bent tube; (f) shows
the shape of the tube after 90°bending and dL presents the elongation of the tube itself after the bending.
Fig. 10. Stress distribution along the bending direction of tube with bending angle 0.3 rad.
From this figure that shows the stress distribution along the bending direction with a bending angle of 0.3 rad,
it can be understood that the outer area is undergoing tensile stress , while the inner area is undergoing
compressive stress and the stress neutral layer moves closer to the inner area.
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Fig. 11. Relationship between maximal wall thickness changing ratio and bending angle.
Figure 11 shows the link between the maximal wall thickness changing ratio and the bending angle: the
maximal wall thickness reducing ratio in the outer tensile area doesn’t change a lot with increase of the
bending angle and the maximal wall thickness increasing ratio in the inner compression area increases linearly
with the bending angle.
As a conclusion of this FE simulation system for the NC precision bending process of thin-walled tube it can
be stated that: this process makes the tube elongates to some degree and the outer area is undergoing tensile
stress while the inner area is undergoing a compressive stress meanwhile the neutral layer moves closer to
the inner area. As reported above the maximal wall thickness reduction ratio in the outer tensile area
changes a bit with an increase of the bending angle and the maximal wall thickness increase ratio
in the inner compression area increases linearly with bending angle. [2]
Effect of the springback on the forming quality of rotary draw bending
In the previous pages the springback effect has been neglected, but it is a key factor that affects the forming
quality of tube bending since after the bending process the elastic deformation releases and the bent-tube is
sprung back. [3]
The elastic-plastic finite element method can be used to analyse the springback back process of thin-walled
tube NC precision bending and the entire process is solved through the use of two algorithms: the dynamic
explicit algorithm and the static implicit algorithm. This numerical simulation method can be useful also to
study the effect of geometry and material parameters on the springback phenomenon to effectively control it
and the factors that affect the forming quality.
If the value of the springback of NC tube bending is beyond the error range, the accuracy of tube bending
can’t satisfy the requirement of applying, so that in the past many researchers performed lot of studies on it
making reference to the springback theory of beam bending and considering as parameters the action centre of
the pressure die and the friction force between mandrel and tube. The results were improved using an
approximate expression in trigonometric form for the displacement field and as constitutive relation the total
deformation theory was employed; moreover without increasing the accuracy of the required hardware, the
method of bend-rebend was used in addition of process control.
Nowadays by numerical simulation method, combined with theoretical analysis and experiments, the
springback rule of thin-walled tube NC precision bending and the effect of other parameters have been carried
out.
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The whole process includes three phases: tube bending, mandrel drawing and tube springback; the processes
of tube bending and mandrel drawing are dynamic, this leads to the usage of the dynamic explicit algorithm,
while the process of tube springback is static, so the static implicit algorithm is used to solve it. During
simulation tube bending process can be divided into two stages: tube bending and mandrel drawing are
performed at the first stage whose operations are controlled through the loading of dies, in the second stage
the process of tube springback is carried out.
Independelty from the fact that a dynamic explicit algorithm or a static implicit algorithm is used in FEM, it is
required to solve the nonlinear equilibrium equation:
π = ππ’Μ + πΉ − π = 0
Where M is the system mass matrix, π’Μ is the vector of accel-eration, F is the vector of internal force and P is
the vector of external force [4].
Since the aim of the simulation is to predict with a certain accuracy the springback phenomenon, the process
of retracting mandrel must be taken into account and, in this case, the method for predicting the springback
angle available in literature could not be used. The springback process is the result of a stress self-balancing
of tube which depends both on the bending process and on the retracting mandrel: the total springback angle
considering retracting mandrel is much smaller than that not considering it and the maximum difference
between them on average is 107%.[4]
The software chosen to establish the finite element model is DYNAFORM (LS-DYNA) since it can do both
dynamic explicit and static implicit computing and it can model a seamless transition between the two
algorithms.
Figure 12 shows the elastic-plastic FEM of spring back.
Fig. 12 .Elastic-plastic finite element model of springback of thin-walled tube NC bending
Establishment of finite element model
To reduce the computational cost it has been taken advantage of the symmetrical structure so only half of the
model has been established, then a symmetry constraint is assigned. Firstly the geometry models of dies are
required, only after the ones of the contact surfaces between dies and tube; for the mandrel structure a
simplified model shown below has been used:
Fig. 13. Schematic diagram of mandrel simplified reasonably
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The mandrel body and mandrel heads are linked to each other with CONSTRAINED-JOINT-SPHERICM.
Firstly the mesh of the tube with a refinement in the field of big deformation has been computed, then the one
of the dies is obtained by offsetting the mesh of the tube. For the simulation fully-integrated-shell-elements
and five integral points along the thickness are used and for the material model a transversely anisotropic
material model is chosen followed by some experimental results.
The contact condition is treated by using a penalty function method [4] in which a contact force (Pc)
proportional to the penetration deepness is assigned to the slave nodes to avoid the situation in which slave
nodes could penetrate master surfaces; the mathematical expression for the contac force is ππ = πΎπ × πΏ, πΏ ≥
0 where πΎπ = 0.1 is the contact interface stiffness.
For what concerns the friction, Coulomb friction model in which the friction force fc is proportional to the
contact force ππ ( ππ = π × ππ where π is the friction coefficient) is used.
In the loading phase, during tube bending, the clamp and the bending dies rotate, the pressure die pushes
ahead, and the mandrel keeps stationary. When the bending process is finished (π‘0 ), the mandrel begins to
be drawn back till the process is finished (π‘1 ).
Bending velocity and mandrel drawing velocity vs time are shown below:
Fig.14. Curves of bending velocity and mandrel drawing velocity vs time
For what concerns the constraint condition shown in figure 15 the translation and rotary of bent-tube must be
constrained in order to unchange the energy of spring back computing; the constraints are imposed at the end
of the bent-tube since it deforms hardly.
Fig. 15. Constraint condition of springback computing
The spring back time needs to be reasonably estimated, so that a comparison between different spring back
times applied to the model and the computing results has been performed, when the effect of the increasing
time on the computing precision is smaller than the requirement an optimal spring back time is found.
To control the whole springback process it has been divided into small several steps (artificial stabilization
with a scale factor =0.005 [4]), this is necessary to accelerate the convergence of springback computing and
increase the accuracy.
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At each iteration the matrix is updated.
The experiment equipment W27YPC-63 full-automatic hydraulic pressure tube bender controlled by
programmable logic controller is used to validate the finite element model of thin-walled tube NC bending;
since the experimental results and the ones from the simulation are in agreement with an error ratio that
doesn’t exceed 18.57%, the finite element model is considered reasonable.
Fig.16. Simulation and experiment results of springback angle
Following the validated FEM, the springback rules of 1Crl8Ni9Ti stainless steel tube and LF2M aluminium
alloy tube NC bending with thickness 1 mm, diameter 38mm and bending radius 57mm have been studied.
It is possible to see that when the bending angle is small, the trend of springback angle with bending angle
is nonlinear, the reason is that the whole bent-tube process considers both the curved part and the transition
part and the springback of the transition part is more significant. On the contrary, when the bending angle
is big, the springback angle changes linearly with the bending angle and this is because the transition part
kept stable after the bending process is developed to steady forming.
The trend changing of bending angle vs springback angle is displaced in the picture below.
Fig.17. Curve of bending angle vs springback angle of stainless steel tube bending
The effects of relative bending radius and relative tube diameter on the springback have been studied.
When the bending angle is big: the bigger the bending radius is, the bigger the springback is; the reason
is that as the length increases more material took place in the springback. On the contrary, when the bending
angle is small: the bigger the bending radius is, the smaller the springback is and the more the nonlinear
changing trend of springback angle with bending angle is; this is because the transition part of benttube is shorter and the process of tube bending is developed to steady forming early.
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Fig. 18. Curves of bending angle vs springback angle of stainless steel tube bending with different relative bending
radius
When the bending angle is big: the bigger the tube diameter is, the bigger the springback is. The reason of
that is the same of the previous case. When the bending angle is small: the bigger the tube diameter is, the
smaller the springback is and more the nonlinear changing trend of springback angle with bending angle
is; this is because the rate of the elastic deformation in the same whole deformation is smaller
Fig. 19. Curves of bending angle vs springback angle of stainless steel tube bending with different relative tube
diameter
If now the effect of the retracting mandrel is accounted for the total springback angle is much smaller than that
not considering it and the maximum difference between them in the paper is 107.34% so to predict accurately
the springback, the retracting of the mandrel needs to be considered. [4]
Also the effect of material parameters on the springback rule has been carried out and they have been set on
the values of lCrl8Ni9Ti stainless steel tube.
The parameters changed are the elastic modulus E, yield stress ππ , strength factor K and the hardenability
value n.
The following pictures show the trends changing the parameters above:
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Fig. 20. Curves of ending angle vs springback angle of
tube bending with different elastic modulus
Fig. 22. Curves of bending angle vs springback angle
of tube bending with different strength factor
fig. 21. Curves of bending angle vs springback
angle of tube bending with different yield stress
fig. 23. Curves of bending angle vs springback angle
of tube bending with different hardenability value
The significance of effect of these four parameters on the springback of thin-walled tube NC bending become
weaker and weaker changing the rate of springback angle when the bending angle computed was 180°.
Effect of the mandrel on the quality and bending limit in rotary draw bending
A quite important element in the rotary draw bending process to increase the quality of the product and the
bending limit is the mandrel, so that an analytical model of it including mandrel shank and balls has been
performed; moreover some mandrel parameters such as mandrel diameter d, mandrel extension e, number of
balls n, thickness of balls k, space length between balls p and nose radius r need to be selected preliminarly
and so some reference formulas was required to do this. To validate the analytical model an experiment has
been carried out and a 3D elastic–plastic FE model of the NC bending process is established using the dynamic
explicit FEM code ABAQUS/Explicit. From this analysis becomes straightforward to see the influence of
mandrel stress distribution during the bending process and its impotance in the NC precision bending process
such as wrinkling prevention.
The following figure shows the tools required in the thin-walled tube NC precision rotary draw beding process:
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Fig. 24. Forming principle of rotary draw bending method and sketch of standard mandrel.
During this bending process the tube is clamped against the bend die, it is drawn by the bend die and the clamp
die then it rotates along the groove and after it has passed the tangent point it rotates along the groove of the
bend die to obtain the desired degree of bending radius. Since the forming process has been completed the
mandrel is withdrawn and the tube is unloaded. During this operation the tube needs to substain rigid contact
forces from the tooling system made of the bend die, the clamp die, the pressure die, the wiper die and the
mandrel, thus precison and strict control of parameters are required to obtain the wanted final shape.
To provide a rigid support during the bending process the ball-and-socket-type flexible mandrel, including
balls, is positioned inside the hollow tube; to select preliminarly the mandrel parametrs some formulas have
been deduced after the stating of important assumpions:
1) No interference is present between the mandrel shank and the balls, this means an ideal contact
condition.
2) The balls rotate about Z-axis and should point to the bending center O.
3) The bending deformation starts from the early stable stage to stable one, and only the materials in the
local regions near the tangent line deform largely so that in these regions wrinkling, over-thinning and
section ovalization may occur.
The maximum extension length
The maximum mandrel extension length ππππ₯ can be computed by the following equation:
and it shows that the mandrel extension length largely depends on the ratio D/t , the bending radius R and the
mandrel diameter d. If this maximum value is exceeded the tube will interfere with the mandrel shank and over
thins or then cracks. There exists also a minimum value of e, ππππ , that it is equal to r, if it is further reduced
the mandrel and wiper die would not cooperate well and the wrinkling may happen.
The thickness of balls
The thickness of the balls k should be a reasonable value, this means that if it too low the mechanical connection
strength may be compromised, whereas if it is too high the clearance between tube and point C on balls is too
big so that the contact is no more rigid. The next formula shows the link between some parameters and k:
where ππππ₯ is the maximum value of clearance.
21
The number of flexible balls
This vaule can be computed using the following formula:
Where the cross-section degree with maximum ovalization is α and πΌπ is the effective arc of ovalization.
The space length between balls
The space length between balls p is obtained by looking at the geometry shown in the above picture and the
formula:
In order to verify the above proposed analytical modeling of mandrel, some experiments have been carried
out and the resulting parametrs can reasonably justify the analytical model.
FEM model
If compared the static implicit algorithm, the dynamic explicit FE algorithm shows important advantages such
as little solution costs, few difficulties in simulating complex contact and large deformation process and it is
also able to predict imporant problems as wrinkling and cross-section distortion phenomena without
performing several experiments.
The following pictures illustrate the FE model of the process and the geometry model of the mandrel shank
and flexible balls:
Fig. 25. FEM model for NC bending process.
Fig. 26. Geometry model of mandrel with three balls
For what concerns the choice of element-type the four-node doubly curved thin shell S4R had been adopted to
describe three-dimensional deformable tube, moreover to describe with sufficient accuracy the bending
process five integration points have been selected across the thickness. Two parts can be identified axially in
the tube: a clamped portion and a to be bent portion, this last is the one that will actually undergo the bending
deformation. In addition to what already said the external and internal rigid tools are modeled as rigid bodies
using 4-node 3-D bilinear quadrilateral rigid element R3D4 and total number of meshing element is 5889.
For what concerns the material properties, the uniaxial tension test applied to stainless steel is used to obtain
them and the traditional Coulomb model has been chosen to describe the interfaces’friction conditions; 4 values
of friction coefficient have been chosen according to the different friction situatins present.
Some boundary conditions need to be set in the FE model:
1) ‘‘Surface-to-surface contact’’ was chosen to represent the contact interfaces between the tube and the
die so that sliding is allowed between the 2 surfaces.
2) ‘‘Kinematic constraints’’ is used to describe the mechanical constraints in addition with ‘‘Penalty
method’’ to improve the computation efficiency and accuracy.
22
3) The ‘‘Displacement/rotates’’ and ‘‘Velocity/angular’’ were used to apply the same boundary
constraints and loadings as in the true bending process.
4) Since the relative movement between mandrel shank and flexible balls is quite complex,
the‘‘Connector element’’ is employed to define the contact conditions between mandrel shank and
floating balls.
At the end, following the same forming rule and dies structure as the NC bending machine, the previously
described elastic–plastic FE model has been validated by using PLC controlled bender W27YPC-63NC.
The following figure shows the comparison of FE cross-section distortion degrees with the experimental
ones; it is easy to see that the maximum distortion degree increases linearly with the bending angle and the
error between the simulation and the experimental results is lower than 5%, this means that the model is
considered correct.
Fig. 27. Comparison between the experimental and simulation results cencerning the cross-section distorsion vs bending
angle
The effect of mandrel parameters has been invistigated using the established explicit FE model and it shows
the following results:
1) The larger is the mandrel size the more the neutral axial will be moved outward and this will increase
the minimum wrinkling energy and anti-wrinkling ability, but it makes outside tube over-thinning.
2) When the mandrel extension length increases, the neutral axial will be moved outward but the ability
in restraining the wrinkles is much less than the one of the mandrel size, moreover the excessive
extension will cause the tube over-thin or even crack.
3) The significance of ball number’s effect on the neutral axial position is between the ones of the
mandrel size and mandrel extension length: increasing the ball numbers improves the cross-section
distortion degrees in an efficient way, while the role for controlling the distortion is limited when the
ball number exceeds the one computed through the formulas and increasing the ball number will lead
to the over-thinning of the tube and at an higher manufacturing cost. [5]
23
Roll bending
Operating principle
A set of tools for the three-roll-push-bending process is shown in figure 28. The process, based on two holding
rolls, one bending roll and one setting roll, is driven by axes which are numerical controlled. Among these the
C-axis designates the feed of the tube. The setting roll is moved in the bending plane by either rotating around
the centre of the bending roll (Y-axis) or translating in radial direction (P-axis). Although three rolls are
sufficient to perform the bending operation, a rear holding roll is used to stabilize the tube during the bending
operation.[6]
Fig. 28. Tools and axis of the three-roll-push-bending machine.
The three-roll-push-bending process can be described by three steps as displayed in figure 29. The set-up stage
includes the clamping of the tube and the closing of the tool. After this the starting step follows where the
setting roll is moved to the end position with a simultaneous feeding of the tube, which is performed by the Caxis. When the desired position of the setting roll is reached, a constant radius R is bent by constant feeding of
the tube.
1. Set up
2. Starting
3. Bending
Fig. 29. Three main steps of the three-roll-push-bending process.
An arc is formed depending on the position of the setting roll defined by P-axis and Y-axis and the feeding
rate of the C-Axis. As the setting roll can be moved while the tube is being pushed forward and since the
machine has an additional A-axis allowing for rotation of the tube around the C-axis during the bending
process, arbitrary 3D geometries can be bent.
In article [7] is shown a numerical simulation of three-roll push bending plus twisting (TRPBT) of rectangular
tubes. For this purpose, tubes are subjected to different combinations of these forming processes and then
measured using a Coordinate Measuring Machine (CMM). The results are used to validate the FE simulation
of the forming methods.
24
Fig.30. Tube section.
Set up of the process for the numerical simulation
The scheme of the configuration used for this process is shown in Fig. 31, consists of a fixed main (A) and a
mobile slave roller (B) constrained to the former. The roller B, whose rotation axis is parallel to those of A,
and the roller C are fixed on an arm, rotating about the same axis of A. An angular actuator, operating under
displacement control, produces the angular displacement θ of B. The rear side of the tube is constrained to a
booster, which pushes the tube forwards.
C
C
Fig. 31. Tool set-up for the Three-Roll-Push-Bending-Twisting with a description of the kinematic of the tools.
The tube is forced to pass through a system of rollers that are free to rotate about their centres. In the setting
phase, the tube is approaching the rollers. Two holding rollers support the upstream part of the tube preventing
undesired deformations in the zone that has not been deformed yet. After the tube has passed through the two
rollers, the roller B moves about the rotation axis of A according to an angular displacement θ, causing the
curvature of the tube. Up to this point, the tube keeps going forward while the rollers remain fixed. The value
of θ determines the curvature on the plane x-z, illustrated in Fig. 31. From the knowledge of θ, it is possible to
evaluate the theoretical curvature Rc that the tool set-up should impose. The analytical procedure is based on
finding the radius of the circle tangent to both rollers A and B having the centre lying on the x-axis. However,
25
the springback induces effectively a greater curvature Rc,eff with respect to the theoretical one Rc, as shown
in Fig. 31.
Successively, the tube is deformed by the twisting roller, inducing a curvature Rtws on the plane y-z. The
twisting can be set up by adjusting the position HS of the twisting rollers. For HS = 0 the twisting roller is
tangential to the tube profile and twisting is not produced. The shapes of the samples obtained by experimental
tests are acquired using a CMM. The measured profiles of the tubes will be compared with those estimated by
the numerical analysis.
FE model of three-roll-push-bending plus twisting
Below is explained a numerical model able to predict shape and mechanical behaviour of formed tubes with
satisfactory accuracy, in a reasonable time and using computational resources within reach of most companies
working in the tube-forming branch. Acceptable and realistic accuracy target was set at 10%. The workpiece
is approximated as shell elements. This approximation is reasonable due to the small ratio between the
thickness of the tube and the remaining dimensions. Thanks to this approximation computation time decreases.
Infact, the exploitation of brick elements would imply an increment in computational heaviness, accompanied
by more numerical instabilities. Obviously, shell elements cannot capture the stress concentration and local
deformation occurring at the junctions between the tube walls. Nevertheless, they provide a satisfactory
representation of the macroscopic geometry and the global deformation process of the tube which is the
required output of the model. The optimal shell size, obtained experimentally, is equal to 1x5 mm and the
longest side of each element is parallel to the tube axis. The shell element thickness is imposed to be entirely
in the interior of the tube and, thus, the nodes are representative of the external surface. Material constitutive
properties are implemented considering the results of the tensile tests. The material model is formulated based
on the Hill yielding criterion with an isotropic hardening rule. The mechanical properties of the welding seam
are inferred from microhardness tests and implemented in the FE model. The rollers as well are modelled with
shell elements but are regarded as rigid bodies (with infinite stiffness in order to reduce the computational cost)
because their compliance is very low in comparison with that of the workpiece. For computational lightness,
it has assumed that the cylinders can freely rotate about their centre and their contact with the workpiece is
assumed frictionless but in a real machine tool, the cylinders are not completely free to rotate so this is an
assumption that can lead to oversimplify the model. Furthermore, the high contact pressure tool-workpiece
may induce tangential frictions forces may affect the final shape. To account it a Coulombian model wherein
the friction coefficient along the tangential direction is taken equal to 0.25 has been used.
Fig. 32. Final setup of the machine. The roller B is constrained to A by a spherical-joint element, which allows the
centre of B to rotate about that of A, keeping the same relative distance.
Model of the S-shaped tubular mock-up
The FE model of the above-described forming processes is now used to simulate the fabrication of the Sshaped mock-up. Thanks to the FE model we’re able to obtain not only the mesh but also all the nodal
information in terms of hardening, plastic strain, residual stress, mass, displacement, velocity. Both the final
26
geometry and its material properties (hardening, plastic strain, residual stresses) are recovered in a successive
simulation to reproduce the compression test.
Fig. 33 compares numerically predicted and measured profiles of the S-shaped mock-up. The following table
reports the curvature radii estimated both from measurements and numerical model.
Measured and estimated curvature radii from mock-up.
Fig. 33. Tube mock-up ‘S’: Comparison between measured and numerical profiles. The red points represent the edges of
the final meshed, whereas the measured points have been plotted with colour scaled in relation to their deviation (or
distance) from the mesh-surfaces. 3D visual of the whole tube (a), view of portion undergoing TRPBTon the ZX- (b) and
YZ-plane (c); curvatures obtained via RDB (d) (f).
27
Once again, the validity of the FE model is confirmed, as 5 out 6 curvature radius values are well below the
accuracy threshold (10% from the real value). Until now, the comparison has shown only the capability of the
FE analysis to predict correctly the final geometry of the workpiece but as we will see the model is also able
to compute the distribution of residual stresses, strain hardening and plastic deformations.
Structural analysis and full-scale compression test
In order to obtain a correct prediction of the structural response of the mock-up under in-service loading
conditions we need the estimation of the strain hardening and residual stresses distribution after the
manufacturing process. For this purpose, the residual stress and the plastic strain field are extracted from the
FE model and shown in Figs. 34 and 35.
Fig. 34. S-shaped tube mock-up. (f) and (g) residual stresses distribution in terms of Von Misses equivalent stress while
(c) and (d) accumulated plastic strain in the zones bent via RDB. (a), (c), (f) bottom view, (b), (d), (g) top view. In (b) the
welding seam is highlighted.
28
Fig. 35 illustrates residual stresses and plastic strain after the manufacturing process measured along two paths
(path-1 and path-2) running along the two opposite tube walls of minor width.
Fig. 35.Variation of residual stresses and plastic strain along path-1 (f) and -2 (g) are plotted after the completion of the
forming process. Path-2 follows the welding seam. e stress tensor orientation, S11 is aligned with the longitudinal
direction.
Such walls have been selected because are expected to undergo larger strains than the lateral walls. Path 2 runs
along the welding seam.
29
It is reasonable to argue that the in-service mechanical response of the component is influenced by the stressstrain history experienced during the forming process. To further investigate this issue, the mock up is
subjected after manufacturing to an instrumented destructive compression test. The outcomes of the
experimental test are interpreted by comparison with an FE simulation of the compression test (Fig. 36). For
this purpose, the numerical model of the tube is recovered and exploited for the simulation, reproducing the
same loading conditions. To highlight the effects of the loading history experienced by the mock-up during
the forming process, two different FE analyses have been performed: the former (FEM stress) takes into
account the residual stresses, plastic strain and hardening accumulated during the fabrication process, the latter
(FEM virgin) assumes the material as virgin.
Fig.36. Compressive test with numerical results and experimental ones: (a) reaction force, (b) ux displacement, (c) uy
displacement vs z-displacement along the loading-axis; in the model FEM-stress the residual stress and accumulated
strain from the previous process phases are considered, on the contrary, these aspects are not taken in account in the model
FEM-virgin.
Results
It is interesting to observe that the FE simulation of the compression test is able to satisfactorily reproduce the
buckling phenomenon and the overall displacement field if the actual material conditions are implemented in
the numerical model. Conversely, if the numerical model does not incorporate the forming load history (FEM
virgin in Fig. 36), the load-bearing capability of the structure is significantly underestimated, and the tube
displacements differ to a larger extent from the experimental ones. Considering the load history experienced
by the tube during forming improves significantly the accuracy of the FE structural analysis. In this way, the
maximal magnitude of compressive load, which is crucial to assess the mechanical resistance, is accurately
estimated with a relative error of 0.3%. However the FE model has difficulty in predict the evolution of the
bucking phenomenon.
30
Heat induction bending
Pipeline transmission is the dominant way for long distance transportation of oil and gas, because it allows
high efficiency, high safety and low costs. So there is a persistent development of pipeline projects, in which
bend pipes are important fittings for changing direction of oil and gas: about 30-40% of the pipes are bent
during the construction.[8]
Today, there is a very high demand for transporting oil and gas by pipelines with high pressure in order to
increase the capacity: this service condition requires high strength.
Since it can be performed in the field cold bending is preferred, but when the bend radius is small with a thickwall pipe it is necessary a hot bending process (Induction bending), which applies localized induction heating
and after a subsequent water cooling, allowing the microstructure to be very changed and so the mechanical
properties, such as the yield strength. So, through this thermal treatment an increase of hardness is obtained,
but it is always followed by a higher brittleness: in order to guarantee the safety, subsequent tempering heat
treatment is required to obtain a good toughness, relaxing also the microstructure from stresses.[9]
Local induction heating is an advanced technique with important advantages: high efficiency, higher
production quality and lower costs; however there some problems like springback and cross section ovality:
during bending, the wall of the bending outside is thinned and the wall of the bending inside is thick-ened, the
cross-section of the bend pipe becomes oval and bending angle suffers some spring-back caused by elastic
deformation after unloading.
In engineering design, the thinning of the pipe wall is not allowed to exceed 7% and the ovality is not allowed
to exceed about 12%, so the bending radius is generally greater than 3.5 times the outside diameter of
the pipe when using conventional pipe-bending techniques with local induction heating.
Usually, the widely used materials are carbon steels, stainless steels (Duplex, Super), special steels such as
Inconel and also Aluminium, Titanium, Monel, Copper, Clad Pipe; as we had already underlined the most
classic application is for oil and gas transportation, but also important applications in petrol-chemical sector,
power plants, industrial equipment, building constructions and also ship building.
Induction Bending principle [8] [9] [10]
Fig. 37. Induction bending machine
The machine is composed of some mainly important components: pipe, pusher, coils for local induction
heating, bending arm with a clamp, water cooling device and guide rollers, as shown in figure above.
31
During the pipe is bended, it is contemporary heated up by a local electro-induction process; the hot deformed
zone is controlled then by water wall. Usually, induction heating coils and cooling coils are mounted to ensure
uniform heating in the range of 850-1100 °C: the induction coil can be adjusted with a 3-plane movement.
By adjusting the radius arm and front clamp, the required bend radius can be fixed. There is one pointer to
display a correct degree of turning and arc lengths are marked on the pipe, which can be moved slowly, even
if the bending force is applied by a fixed radius arm arrangement.
Once everything is set as required, hydraulic pressure, water level, and switches are inspected and then the
induction bending operation is started. Upon reaching the required temperature range, the pipe is pushed
forward slowly at a speed of 10-40 mm/min, and the operation is stopped when the specified bend angle and
arc length are reached. Just beyond the induction coil the heated material is quenched using a water spray.
During the process of bending pipe, deformed metal is generally subjected to tension stress and the wall of
pipe thins on the bending outside, while it is subjected to compression stress and the wall of pipe thickens on
the bending inside. Moreover, the cross-section geometry of the bent pipe is oval.
Process Parameters
The important parameters that affect the induction bending process are:
•
•
•
•
•
•
•
Pipe Diameter
Surface Contamination
Process Parameters like Temperature, Speed, Cooling rate, etc
Bend Radius
Bend Angle
Process Interruptions
Hardenability of the Pipe Material, etc.
Induction vs cold bending
Induction Bending
Must
Slow Process
Better control
Highly efficient complex machineries are used for
hot bending.
Hot bending Process is efficient for larger pipe
diameters
High costs
Requires lower force
Cold Bending
No
Fast Process as no heating & cooling involved
Less control
Simple machines
Not suitable for large pipe diameters. Cold bending
is limited to smaller diameter piping only.
Not so expensive
Higher force
FEM finite element model [8]
As we have understood before it’s very important to control the thickness of the pipe to ensure its performance.
Thickness variation is one of the most important design criteria and, also, quality control variables to ensure
mechanical properties. To obtain a required uniform thickness, pipe pushing load is needed to control with
revers moment.
Large deformation elastic-plastic finite element method (FEM) is used to develop a coupling thermomechanical simulation system: the whole deformation process of bending pipe has been realized by the
computer. By FEM simulation, stress variation, strain and temperature fields and cross-section geometry with
bending pipe process can be obtained. In addition, the spring-back of bending pipe after unloading can be
predicted. All of them are very important for guiding practically the process.
32
FEM involves multiple non-linear factors (material and geometry), and contacts during the de- formation
process of bending pipe. In addition, local induction heating technique makes it more difficult to obtain the
model and solution. For the simulation, elastic-plastic large deformation FEM is used to develop 3D coupling
thermo-mechanical simulation system for the solution.
In this particular investigation, we consider as the stock' s material the pipeline steel X70: it is assumed
isotropic and the Von Mises yield criterion for yield behaviour. To model the material behaviour the following
equation is used, given by a constitutive model: the flow stress at which the material starts to flow is function
of the equivalent strain, the equivalent strain rate, and temperature.
Induction bending is a thermo-mechanical process analysed through solving the heat conduction equation
coupled with deformation analysis. During the bending process, the temperature distribution of the pipe can
be calculated using the partial differential equation:
(where k, is thermal conductivity; p is density; c is specific heat; qv is heat generation term representing the
heat released due to plastic work; x, y are position coordinate; t is time)
In this evaluation we have to consider the water-cooling process and so find also the related heat loss, by
considering a convection heat transfer process.
Coupling thermo and mechanical issues, a FE model for hot elbowing pipe has been built in this study.
According to the symmetry of the pipe bending structure about the x, y plane, only a half-part of +z is analysed
in order to reduce the number of element. Guide rollers are defined as rigid along with ignoring the influence
of friction between rollers and pipe on bending pipe to reduce simulation time. Bending arm and clamp are
replaced with large rigid elastic elements.
The effect of these simplifications on bending pipe is weak and fits the practical process. The main calculating
conditions are the following: pipe original diameter (Do), 1 000 mm; wall thickness (b), 20 mm; bending radius
(R), 6 000 mm; pushing speed (v), 0. 5 mm/s and so on. The FEM meshed model is shown in the following
Figure, in which 2858 3D solid elements are involved.
Fig. 38. FEM meshed model
33
Analysis and discussion
After using the simulation system previously developed and loaded, the whole forming process of pipe bending
was simulated. The pipe shape after loading is shown in figure 39 with a bending angle 94", while 40 gives
the pipe shape after unloading. It is evident that a greater springback of pipe-bending occurs.
Fig. 39. Pipe shape after loading
Fig. 40. Pipe shape after unloading
The temperature distribution in the local heating zone is shown in the subsequent figure, during the bending
process. First, the node temperature varies from room temperature 20 °C to 900 °C, and then drops quickly
after a while due to the heat radiation and water cooling. The circumferential equivalent plastic strain at a
certain cross-section of the deformation area is shown in 42, when bending angle is about 45°.
Fig. 41. Temperature distribution in local heating zone
Fig. 42. Circumferential equivalent plastic strain
The equivalent plastic strain value decreases at the former half stage and increases at the later stage from
bending outside to bending inside. Furthermore, strain of bending inside is more than outside. So, the
distribution rule agrees with that of pipe-bending theory. The relationship between wall thinning or thickening
ratio of the pipe wall thickness and bending angle for the outside, middle and inside of bending is shown in
Fig. 43. Wall variation at the frontend of pipe is relatively greater than any other. The main reason is that the
deformation is difficult at the beginning of bending. According to the engineering quality specification of the
thin-wall and large-diameter pipe, the thinning ratio of the wall thickness on the bending outside is not allowed
to exceed 7%. Generally, the change ratio of the wall thickness is expressed in the form of the following
equation:
∅=
π0 − π1
π0
34
where ∅ is change ratio of the wall thickness; b0, is initial wall thickness; and b1, is formed wall thickness. In
addition, it can be seen from Fig. 43 that the wall thickness increases at the middle. The neutral layer of pipebending process leans to the bending outside. Based on the bending theory of material mechanics, this
phenomenon will occur as deformation cross-section is subjected to axial force besides shear force and torque
force. So, the simulation results are in accord with that of theory. Also we must consider quality requirements
for engineering design for the ovality (10-12%) of the bent pipe, calculated by:
πΏ=
ππππ₯ − ππππ
(ππππ₯ + ππππ ) β 2
Fig. 43. Wall thickness and thickening
The relationship between the ovality and the bending angle is demonstrated in the figure 44. With an increase
in bending angle the ovality firstly increases, and then decreases near to the end of bent pipe as shown in figure
44 (a). Moreover, it can be also seen that the ovality of the pipe after unloading is less than that prior to
unloading. The relationship between spring-back angle and bending angle after unloading is illustrated in
figure 44 (b). The spring-back angle curve shows a gradual increase with the bending angle. Figure 45, instead,
presents the relationship between the ovality of certain cross-section near the leading end and load step. At the
beginning of bending, the ovality increases gradually with an increase in load steps, but reaching a high value,
it remains at a steady value.
Fig. 44. Ovality/spring back angle with bending angle (a/b)
Fig. 45. Ovality with load step
35
Laser bending
Laser bending is a noncontact method that used laser energy in order to obtain 2D bending, 3D shaping, and
precision alignment of metallic and non-metallic components: sheets and tubes can be formed depending upon
the non-uniform thermal stresses induced by laser heating. Compared to laser bending of sheet metals, the laser
tube bending is more complex because much more process parameters are involved. It is also become one of
the basic parts of laser metal forming because it overcomes the troubles that characterize the tube bending
technique. Moreover, tube bending is an attractive application in many high-technological industries such as
aerospace, aviation, automobile, shipbuilding, energy and health care.
The bending process is based on the non-uniform distribution of temperatures on specimen surface, which
generate different thermal stress between the irradiated area and the cold side of the tube. The mechanism used
in laser tube bending is Buckling mechanism where the diameter of the laser beam is chosen to be much bigger
than the tube thickness as seen in Fig. 46. Hence the inner side of the tube which faced the laser beam is
homogenized heated in thickness direction causing material expansion. The expansion is restricted by unheated
material which leads to plastic compression. Due to the cooling, the material suffers shrinkage and shortening
in axial direction forcing the tube to bend towards of laser beam. [11] [12]
Laser Bending principle [11]
Fig. 46. Circumferential scanning scheme
Laser tube bending is similar to the heat treatment process and the surface temperature of the material has to
be less than the melting point. During the process there is a lot of nonlinear phenomena like the temperature,
microstructure, and stress field changes: all are significantly related to each other. In this case we can describe
a tube with circumferential scanning shown in the figure above: the tube rotates on its axial axis around 180°,
where the laser light is defocused on the tube surface with laser beam diameter much bigger than tube
thickness. The heated area suffers wall thickening and compressive plastic deformation because of the thermal
expansion limitation by the unheated material. When the laser source is turning off a rapid cooling occur, with
material that shrinks on the heated surface. As a result, the shortening of the material in the axial direction
forced the tube to bend towards of laser source. During bending process heat is generated because of strain
energy, but it is very small compared to input laser beam energy so it can be neglected.
There are many influencing factors on laser tube forming, such as forming process parameters, material’s
properties, optic parameters, and geometrical parameters. Figure 47 gives factors of influence on laser tube
bending.
36
Fig. 47. Factors of influence
The melting point and yield strength are basic conditions to restrict to the forming process. For a given forming
objective, material’s properties and geometrical parameters of the tube are already decided. The aim of the
laser tube bending is to acquire the desired bending deformation by the reasonable match of the process
parameters. As a high nonlinear forming process, it is hard to express the relationship of bending angle and
process parameters by using a general formula; so the finite element method is a powerful tool to study the
process. Using finite element method (FEM) process simulation, can be not only the final bending angle be
acquired, but also kinds of fields, such as the stress, strain, displacement, and temperature fields.
The FEM process simulation is an ideal researching method and very complex: a series of treatment techniques
are necessary according to the characteristics of the laser tube bending. In this paper, it is adopted a
thermomechanical FEM model considering the correlation between material’s properties and temperature. To
some extent, the FEM process simulation can realize the optimization of forming processes, too. However, it
is hard to obtain the optimum process parameters only through FEM simulation.
FEM simulation
It’s considered a typical coupled thermal-mechanical problem, where the temperature and displacement fields
affect each other. The finite element method based on the 3D thermal elastic–plastic was established to realize
the simulation of the laser bending of tubes and so to reduce analytical errors significantly. The fields of
thermal stress, temperature, and displacement can be obtained simultaneously by using the coupled model.
In laser bending process, the temperature and deformation of the heated zone are characterised by sharp
variations during a very short time. Steep temperature and stress gradients occur. Higher precision can be
acquired using brick element. Thus, brick elements are selected when the sheet is discretized during simulation.
The brick element is divided into full integral element and reduced integral element according to numerical
integral way. For bending problem of the thin sheet, we can obtain unreasonable results because of larger
precision loss, while selecting full integral element. Using reduced integral way, not only the calculation time
of integration can be decreased, but also analytical precision can be improved because of eliminating the
influence of imperfect high-order term. Meanwhile, a fine mesh is necessary around the laser beam path due
to steep temperature and stress gradients in the heated zone. It is essential to have at least eight integration
points in the thickness direction; in order to cut the total number of elements in the simulation, a coarse mesh
is used outside the heating zone.
37
1. Boundary and initial conditions
The natural cooling-down method is adopted in the paper: in other word, the tube is cooled down
naturally in the air. Heat convection and radiation exists between the tube and the surroundings, but
therefore, the irradiation boundary condition is nonlinear.
The displacements at one end of the tube are restricted, and the other surfaces are displacement free.
It is assumed that the tube is initially stress free and strain free. Initial temperature of the tube is 20
°C.
2. Simulation
A specimen of Φ 10×80×1 mm (diameter × length × thickness) is analysed. Twenty-node brick
elements are used. The total element number is 10,400. The finite element mesh is shown in figure 48;
in 49 instead the scanning process.
Fig. 48. Finite element model
Fig. 49. Sketch of laser scanning process
The process parameters are:
• Laser output power: P=1 kW
• Scanning velocity: v=1.5 m/min
• Laser spot diameter: d=5 mm
• Initial temperature of sheet metal: T0=20 °C
• Scanning wrap angle: 180°
The right figure shows the sketch of the laser scanning process and cross-section of the tube in the
heated zone. The central zones of laser scanning side and non-scanning side are selected to study the
characteristics of the laser tube bending.
38
•
Single scan process
Fig. 50. Temperature with time
Fig. 51. Axial stress with time
The left figure shows the variation of temperature with time. The laser spot gets to the zone I at 0.294
s. The irradiation lasts for 0.196 s. The temperature reaches its peak value at 1,099.25 °C at 0.49 s.
Subsequently, the temperature drops down sharply until it reaches to the room temperature. The stress
and strain vary with the change of temperature greatly.
Right figure, instead, gives the variation of axial stress with time. At the beginning, the stress state is
tensile: when the laser spot irradiates the zone I, the thermal expansion of materials due to temperature
rise goes up, but the surrounding materials restrict the expansion and then the stress of zone I become
too compressive. At the beginning of the cooling stage, the stress in axial direction recovers from
compressive to tensile because of the shrinkage. Subsequently, the tensile stress diminishes gradually.
Finally, the stress state is in compressive due to the more shrinkage of material.
Zone II is not irradiated directly by laser beam. The temperature rises slowly due to the heat
conduction. The temperature reaches its peak value at 111.64 °C at 5.80 s. There is a great gradient
between the laser scanning and non-scanning side. At the beginning, the stress state in zone II is
compressive. The stress keeps a lower value all the time. Thus, the plastic deformation in zone II is
not induced.
Fig. 52. Axial strain with time
Fig. 53. Displacement with time
In figure 52 is showed the variations of the thermal strain, total strain, and plastic strain with time. The
change of thermal strain is similar to that of temperature. The thermal strain increases from zero to its
peak value 0.0143, and then decreases to zero finally. This corresponds with the movement of laser
spot. The axial plastic strain is dependent upon the axial stress. At the beginning of the laser irradiation,
the material in the heated zone expands due to the temperature rise. When the temperature continues
to go up, the axial stress of in the heated zone becomes too compressive because of the restriction of
39
the surrounding materials. The axial compressive stress rises greatly with the increase of the
temperature. In the meanwhile, the yield strength decreases with the temperature rise. The compressive
plastic deformation is brought out in the heated zone.
Obviously, the gradient and development of the temperature between the laser scanning side and the
nonscanning side leads to the changing complexity of the stress and strain. The final strain of the laser
scanning side resulting by the shrinkage of the heated zone is −0.0084 in axial direction. The length
of the laser scanning side becomes shorter than that of non-scanning side after cooling. The length
difference between both sides makes the tube produce the bending angle.
•
Multi-scan process
Fig. 54. Axial plastic strain/time
Fig. 55. Bending angle with n of scans
A bending angle acquired in a scan is small. In order to obtain the desired bending angle in practical
engineering, the tube has to be scanned repeatedly along the same scanning path. Figure 54 gives the
variation of the axial plastic strain of the irradiated side with time. The axial plastic strain of the
irradiated zone becomes larger and larger with the increase of the number of scans. Meanwhile, the
axial plastic strain of the non-irradiated side is almost zero. The different step of the axial plastic strain
difference is generated between two successive scans. Thus, the axial plastic strain difference between
the irradiated side and non-irradiated side augments in the multi-scan laser bending process.
Consequently, the bending angle of the tube increases.
Figure 55 instead, gives the influence of the number of scans on bending angle, it is seen that the
relationship between the number of scans and the bending angle keeps linear; the bending angle
acquired in the first scan is largest.
•
Conclusions
-
The gradient and development of the temperature cause the changing complexity of the stress and
strain.
Consequently, the length of the laser scanning side is shorter than non-scanning after cooling.
The length difference between both sides makes the tube produce the bending angle.
The relationship between the number of scans and the bending angle is about in direct ratio.
-
40
Shear bending
Tube shear bending is a suitable bending technology to realize a considerable small bending radii in order to
minimize the space of bent tubular parts. The forming process can also be studied through an elasto-plastic 3D
finite element simulation and the results on the deformation behaviour and strain distribution demonstrate that
the forming process is a combination of shearing and bending deformations.
Nowadays light weights, low cost and compact products are the most requested on the market moreover tubular
components, especially, with small bending radii need to fit into tighter spaces to be adopted in hydraulics,
fuel piping and in many devices used in narrow spaces such as automotive engine parts and air conditioning
appliances. Another important challenge in case of tubes is to produce one-piece tubular form to replace casting
or multiple welding operations.
With conventional bending methods the minimum bending radius that can be obtained is almost more than 1–
2 times of the tube original diameter, with shear bending of tubes on the other hand very small radii, compact
and unified elbow tubular parts can be manufactured.
A typical frame of a shear bending equipment and a schematic illustration of the process are shown below in
the pictures 56 and 57:
Fig. 56. Schematic sketch of shear bending equipment
41
Fig. 57. Schematic illustration of shear bending process
The tube (#4, Fig.57) is inserted in the hole of dies (#5,#7) and the mandrels (#6, #8) are placed inside where
the mandrel of moving die is fixed to the moving die itself. Then using a hydraulic cylinder (#11) the
moving die slides and applies a shearing force on tube; at the same time, another hydraulic cylinder (#1)
pushes the end face of tube and material is supplied into dies during the process. Thanks to this configuration
the tube is undergoing a shearing deformation continuously. Two load cells (#2, #10) measure the values of
pushing and shearing forces and the hydraulic cylinders are connected to a hydraulic power pack and controlled
independently. The final shape of the tube is a crank shape product and ππ and ππ radii (die corner radius and
mandrel head radius, respectively) determine the inside and outside bending radii of the bent tube.
Experimental data
Process parameters used:
-
A1050, aluminium tube
diameter π·0 =30mm
wall thickness π‘0 = 1.5 mm
die corner radius 3 mm
radial clearance of die-tube and mandrel-tube of 0.1 mm
During the shear bending process, to overcome the friction force and to avoid the extreme thinning outside, a
pushing force is applied on the tube-end side; the resulting product obtained by applying different values of
the force is shown in the following picture:
Fig. 58. Typical products by various pushing forces: (a) splitting; (b) successful; (c) wrinkling; (d) rupturing
42
Only in the case (b) a correct shear bending product is obtained, if the pushing pressure on the tube is not
enough to send material into the dies a tube splitting happens (case(a)) and, on the contrary, if the pushing
pressure exceeds a threshold a wrinkling in the tube outbreaks (case(c)), finally, in the worst case of too high
pressure applied there is the rupture of the workpiece (case(d)); therefore the amount of pushing force should
be accurately measured.
Finite element simulation settings:
-
ELFEN finite element code
By exploiting the symmetrical shape, only half of the model is considered
Dies and mandrels are assumed to be rigid
Tube is modelled using solid elements
The number of tube elements in the thickness are 4 and in circumferential direction are 20
A mesh refinement has been done in the axial direction where there is a severe deformation
The bending angle is 90°
The bending die has been moved by 100 mm
The used flow stress-strain relation of the tube material is π = π(π)π
The isotropic hardening rule is assumed
Von Mises yield criterion is used
The following table collects the simulation parameters:
During the experiment many values of pushing pressure have been used, only for P=30 MPa no failure
mechanisms have occurred in tube.
On the base of the deformation of the meshes around different routs (T=Top, L=lateral, B=bottom), the amount
of deformation can be stated: the elements on the tube lateral side (route L) have sustained more shearing
deformation if compared to the top and bottom sides, in fact approaching to routes T and B the degree of
deformation reduces and elements in these regions have been subjected to significantly less shear distortion.
The figure number 59 shows what said:
43
Fig. 59. Finite element meshes after the deformation
In a conventional pure bending process the tube suffers only a pure bending moment since the inside and
outside bending radii are concentric and there is a single bending centre; on the contrary in shear bending, on
the base of the die geometry, each element of tube feels an additional shearing force as well as a bending
moment. The amount of deformation shown by each element is related to its position in the peripheral
direction: elements around routes T and B are subjected to a high bending deformation due to die geometry
whereas the maximum shearing occurs on route L.
FEM results
The effective strain distribution after the bending process is shown below:
Fig. 60. Distribution of tube effective strain
It can be seen clearly that in the vertical part of the tube the effective strain along tube longitudinal direction
is uniform whereas it varies in tube width direction.
44
When π = 90° the maximum effective strain (ε≈1) is reached in the region across tube width and the result is
in good agreement with the corresponding theoretical value for a 2D ideal shearing process. It can be seen that
moving towards the centre (route L), the bending component reduces as the shear components increase until
the shear strain becomes uniform and reaches its maximum value. Moving to route B or T, since the shearing
deformation decreases, the effective and normal strains reduce.
Figure 61 shows the effective strain distribution of tube outer layer in the steady state region across tube width
(lines TB) and the x and y strain components:
Fig. 61. Strain distributions across tube width in the steady state region (line TB)- maximum strains have occurred
around π = 90°
For the value of the tube thickness strain along routes T and B a comparison between simulation and
experimental results is shown in Fig. 62.
Fig. 62. Distribution of thickness strain along routes T and B
In this experiment a pushing pressure of 55 MPa has been applied on the tube. Point C on route B (Fig.62)
and point F on route T are under compression in the longitudinal direction and thickness strain in these regions
is positive; on the contrary point D on route T and point E on route B are under tension and therefore
thickness strains in these points are negative. [13]
45
Effects of initial thickness in shear bending process
In the previous pages the effects of initial thickness on the deformation characteristics during shear bending
process wasn’t discussed, in the following paper this aspect has been investigated both by experiments and
numerical simulation through the use of FEM (ELFEN code). The results show that, under the effect of same
pushing pressure, increasing the initial thickness the tube thinning and cross section deformation increase.
The ability of a tube to be bent is strictly correlated to geometrical factors such as the bending radius, the
diameter and the thickness of the tube. Experiments and numerical simulations show that during shear bending
processes no failure occurs only if the pushing pressure is inside a suitable range and performing some
experiments it has been demonstrated that the range of pushing pressure for a successful forming enlarges if
the tube initial thickness is increased.
The following picture illustrates both the experimental and simulated results related to the cross section
deformation using tubes with different thickness value but with the same amount of pushing pressure (P= 40
MPa) applied to understand the effect of the initial thickness on the deformation behaviour.
Fig. 63. Cross section configurations of the deformed tubes with different initial thickness obtained by simulation and
experiment: (a) simulation, (b) experiment and(c) experiment (t0=3.0 mm)
The thicker the initial tube is, the greater its cross section deformation is, especially around the inner part of
the bending and this result is completely against the traditional behaviour in conventional tube bending; the
reason is that increasing the initial thickness the bending rigidity of the tube rises and the bending becomes
harder as a direct consequence. Therefore, the tube material cannot follow the die shape well and the tube cross
section needs to sustain more deformation. What stated is shown in Fig. 63 (c), in fact a gap is formed
between the tube and die wall due to a shrinkage registered along the diameter.
However, increasing the pushing pressure ovality of the cross section can be reduced; the next figure shows
the experimental and simulation results of the relation between the pushing pressure and the cross section
ovality of the deformed tube.
46
Fig. 64. Effect of the axial pushing pressure on the cross section ovality of the deformed tube
The thinning of tube wall during shearing deformation increases the lower the pushing pressure is since during
the process the tube sustains elongation and its cross section deforms; if the pushing pressure is enough high
the ovality is prevented, always assuring not to have wrinkling as a drawback of an excessive pressure applied.
π‘
π‘0
Another aspect is considered: the effect of the initial thickness on the thickness strain ππ‘ = ln ( ).
Experiments and simulation state that thickness strain decreases with an increasing of the initial thickness.
Fig. 65. Configuration of the deformed tube.
Fig. 66. Effect of the initial thickness on the thickness
strain of the tube-sheared part obtained by
simulation and experiment.
In order to explain the reason of this behaviour, the relation between the pushing and shearing strokes of the
moving die (Figure 67) as a function of the tube initial thickness and the effect of the tube initial thickness on
shearing load (Figure 68) are plotted.
47
Fig. 67. Relation between the pushing and shearing strokes
as a function of the tube initial thickness obtained by simulation
Fig. 68. Effect of the tube initial thickness on the
shearing load obtained by simulation
It can be seen that in the graph shown in figure 67 the slope of the curves decreases by increasing the tube
initial thickness, this means that a lower amount of material is supplied into the dies increasing the tube
initial thickness.
So under the action of the same pressure, the tube shows more elongation and the shearing load increases by
increasing the tube initial thickness, as displaced in figure 68.
Moreover by applying an higher pushing force on the tube it is possible to understand that thinning can be
reduced; this is confirmed by the results shown in the following figure:
Fig. 69. Effect of the initial thickness and pushing pressure on the distribution of thickness strain along the hoop
direction of the tube-sheared part
These curves demonstrate that applying an higher pushing pressure on the thick tubes can prevent the tube
wall from thinning. [14]
In conclusion, to summarize what has been said previously:
-
Increasing the tube thickness, the bending rigidity increases too, as a consequence the tube cannot
follow the die profile and the cross section ovality increases accordingly.
Raising the applied pushing pressure the cross section high ovality can be prevented.
A thick wall tube has more tendency to thinning and rupture with respect to a thin tube but to reduce
these drawbacks the applied pushing pressure must be as high as possible within a range in which
wrinkle does not occur.
48
Push bending
3D free bending process is an advanced metal forming technology, characterised by high flexibility and
efficiency. It can be easily found that copper tubes filled with low melting point alloys are able to induce the
most uniform stress distribution, associated to better formability and moderate thickness changing rates. Tube
bending process is one of the main used manufacturing technologies for lightweight production and 3D free
bending technology has the big advantage of being suitable for most tubes and profiles due to its geometrical
flexibility and efficiency. It can be used to produce complex hollow tubes with asymmetric cross-sections and
continuous varying radii. Also, the bending of different R values and angles can be achieved without changing
the die, which greatly extends the flexibility of the bending machine. On the other hand, during the tube
bending process, wall thickness and cross section changes are unavoidable. However, due to the axial force,
wall thickness and cross section changes are restrained during 3D free bending technology. 3D free bending is
a dieless forming technology, making it hard to keep stable and restricted springback during the bending
process. However, several materials could be used inside the tubes for supporting roles. These materials are
not only used for restraining thickness changes and springback of the tubes, but also to further improve
formability. [15]
Figure 70 shows 3D bending process, during which the internal side of the tube is subjected to compressive
stress while the outside is associated to tensile stress. For this reason a thickening of the inner side of the tube
and reduction of its outer portion is obtained. In the meantime, the tube was subjected to feeding force, so the
neutral layer offset outwards to restrain reduction of the wall thickness and reduce the deformation of the cross
section of the bent tube. The tube bending stress and strain analysis can be represented always in Fig.70, where
ππ and ππ are bending radius of inside and outside wall of tube, respectively, t is the thickness of tube at any
point and t0 is the initial thickness of tube, N is the interaction between filling materials and tube, and P is the
feeding force.
Fig.70. Tube bending stress and strain analysis
FEM simulation:
ABAQUS/explicit is applied to numerically simulate the forming process and the model used for this process
is shown in Fig.71, including a tube, a bend die, a spherical bearing, a guider, a pressing and different filling
materials. Each die model created during the geometry modelling was oriented in its own coordinate system
and was independent of the other geometry models, then the dies and tube were assembled in the global
coordinate system. These data were put in CATIA software and then imported to ABAQUS/explicit FE code.
The tube was made of copper and filling materials were steel balls, PU rubber and low melting point alloys,
respectively. The interaction among the parts is set to general contact and the analysis step is adjusted to
dynamic explicit. Since the lubrication unit is designed near the bend die, the coefficient of friction between
the dies and the tube is assumed as a very low value. Thus, based on a great deal of simulation results, the
coefficient of friction between the dies and the copper tubes were identified as 0.02. Meanwhile, the pressing
and guider are set to encase. The tube specifically built for a specified movement speed in the Z direction and
the bearing is set having two movements respectively in the X and Y direction, while the remaining degrees
of freedom are all fixed.
49
Fig.71 FE modelling of tube free bending
To simulate the free bending process using different materials, simulations for four copper tubes are carried
out based on the conditions summarized in the table. The filled materials included steel balls, PU rubbers, and
low melting point alloys. The relative position sizes are shown in Fig. 72.
Fig.72 Model assembly position
Influences of different filling materials in 3D free bending can be investigated through a simulation without
filling materials (1). The simulation results are represented in Fig.73.
Fig.73. Bending simulation under condition (1)
The simulation results under condition (2) are presented in Fig. 74. The simulation data of bending process
using low melting point alloys are represented in Fig.75. The low melting point alloy establishes a complete
contact with tube wall after solidification and the stress distribution is uniform during bending process. Ideal
forming results are indicated in Fig.75. The simulation results under condition (4) are shown in Fig. 76. Due
to the lowest elastic modulus, PU rubber was unable to provide internal pressure. If compared to condition (1)
(Fig. 73), during the forming process, the tube bears less stress and the stress distribution is more uniform
when the filling materials were replaced with PU rubber.
50
Fig.74. Bending simulation under condition (2)
Fig.75. Bending simulation under condition (3)
Fig.76. Bending simulation under condition (4)
The experimental process and results under different conditions are shown in Fig. 77.
Fig.77. Experiments of 3D free bending under different conditions
51
Results:
•
•
•
•
The interaction N between filling materials and tube has a significant effect on formability. The more
the uniform distribution of N is, the better the formability and moderate change in thickness attained
along the radial direction.
Compared to the flexible filling, copper tubes filled with solid filling (such as low melting point alloys
and steel balls) could contribute to better formability and less change in thickness.
On the other hand, filling tube with steel balls led to bad formability and great change in thickness
than unfilled tubes. Meanwhile, large numbers of spherical indentation are developed on the inner
surface results in degradation of the inner surface quality.
The ellipticity decreased regardless of the filling material and filling with low melting point alloys
reduced ellipticity to 1.467%.
A 3D FEM-DEM coupling numerical model can be usually developed to investigate the interaction between
thin-wall elbow and granular filler for a granular-media-based thin-wall elbow push-bending process. [16]
This procedure is about filling a tube through granular media and pushing it into a die to bend a tubular blank
into an elbow shape. Depending on the mechanical properties of granular filler an elbow tube having t/D<0.01
(wall thickness/outer diameter) and R/D<1.5 (ratio of bending radius/outer diameter) can be produced. To use
in a correct way the 3D FEM-DEM model it should consider both the tubular blank deformation behaviour
(since it is continuum, FEM is applied) and mechanical properties of granular filler (based on discrete media,
so a discrete element method DEM will be applied).
Referring to traditional normal-diameter normal-wall elbow tubes push-bending process, it consists of pushing
a tube between a die and a mandrel to bend a tubular blank obtaining a final elbow shape. Unfortunately, there
exist many problems using the traditional push-bending process to form large diameter thin-wall elbow tubes,
especially wall thinning-fracture and wrinkling. For that reason, in recent years granular-media-based forming
process has been developed and it’s now commonly used due to its flexibility, resistance to high forming
temperature and good characteristics associated to pressure-transmission. While on one side, in traditional
push-bending process, forming force of punch is immediately imposed on tubular blank with filler providing
passive supporting in the tube, on the other side, in the new tube bending process, forming force of punch is
applied to granular media which, with their flow in mould cavity, transform the tubular blank into a curvature.
After a 3D FEM-DEM is developed, using this coupling model, some key forming parameters, such as forming
force, wall thickness distribution, wrinkling are simulated and compared to experimental results.
About the mechanical process, a tubular blank with predetermined dimensions and filled with granular media
is inserted into the die and then deformed according to the shape of the die. The tubular blank is bended under
forming force of punch, constraint force of die, and interaction of granular filler. The forming force of punch
and constraint force of die can be analysed and studied using FEM simulation. On the contrary the granular
filler cannot be simulated through FEM, since it’s a discrete system, not a continuum one.
52
Fig. 78. Schematic of an elbow granular-media-based push-bending process: (a) before bending, (b) bending
In the granular-media-based push-bending process, the deformation behaviour of tubular blank can be
determined by minimum principle of potential energy:
(1)
where {δ}^e is displacement vector of nodes, [k] e is element stiffness matrix, {F}e is equivalent nodal forces
which can be expressed as:
(2)
where {P} collects forming force of punch and constraint force of die, {N} is interaction between tubular blank
and granular filler. Considering Eq. (2), Eq. (1) can be rewritten as:
(3)
In the granular-media-based push-bending process, the mechanical characteristics and motions of granular
filler can be determined by minimum principle of generalized complementary energy:
(4)
where {ζ}e is displacement vector of discrete elements in granular filler, [D(α)]e is tangent stiffness matrix of
granular media, {N} is interaction between tubular blank and granular filler, {T} collects external forces except
{N} and α ≥ 0 denotes granular media only produces compressive contact forces. The third and fourth
equations can be numerically solved with FEM (finite element method) and DEM (discrete element method)
to
analyse
the
granular-media-based
push-bending
process.
Talking about FEM-DEM coupling method, DEM simulation involves following the motion of every particle
in the flow and modelling each collision or contact between particles and between particles and boundary. The
time evolution of the particles is advanced using an explicit finite difference scheme. A damping is imposed
on the dynamic equations of motion to dissipate kinetic energy and obtain static equilibrium. The granular
material considered cast iron and all the properties are listed in the first table below. In addition, the friction
coefficient is set to 0.1 in DEM. The main geometry conditions associated to tubular blank are reported in the
53
second table. The size considered for a single FEM element of tube is 1mm x 1mm to obtain the best
compromise in terms of accuracy and stability. The die and punch were modelled in FEM to be rigid. Tubular
blank is set to 1Cr18Ni9Ti stainless steel, the material parameters are listed in the third table, and the stressstrain curve of 1Cr18Ni9Ti is shown in Fig. 3. The friction coefficient is set to 0.02 in FEM. The forming
process simulation is based on the following passages: at first a downward displacement is applied on the
punch and it produces a forming force on the granular fillet. The fillet confined in tubular blank is compressed,
applying contact forces on tube inner walls. The contact forces between granular filler and tube drive the
tubular blank into a curvature. The contact forces between granular filler and tube are calculated in DEM, and
the boundary determined by the geometry of deformed tube is provided by FEM. Simultaneously, at the end,
the tube deformation is simulated through FEM while the forming force (due to contact forces between the
granular filler and the tube) is given by the application of DEM. To couple FEM and DEM models, a particular
module for data-exchanging needs to be set. Applying it contact forces on the rigid wall elements in DEM and
corresponding deformable tube elements in FEM can be easily associated. The contact forces on a wall element
in DEM are equally imposed on the nodes of the corresponding tube element in FEM in a format of
concentrated force. The nodes’ coordinates on a tube element in FEM are used to create the corresponding
wall element in DEM. At this point the total time consumption of 3D FEM-DEM analysis can be split in three
parts: time-consuming of DEM simulation for granular media π‘π·πΈπ , time consuming of FEM simulation for
tubular blank π‘πΉπΈπ , and time-consuming of data transmission π‘πππ‘π−π‘ππππ . The total consumption of time is
finally expressed as:
π‘πππ’πππππ = π‘π·πΈπ + π‘πΉπΈπ + π‘πππ‘π−π‘ππππ
For the granular-media-based push-bending process, π‘πππ‘π−π‘ππππ mainly depends on read/write of files
estimated by data volume, π‘π·πΈπ and π‘πΉπΈπ are mainly based on total number of elements, which can be
determined knowing element size and dimension of tubular blank. Wrinkling and bending phenomena can be
observed imposing the element size of FEM almost equal to the particle size used in DEM. That means that,
in conclusion, the total time-consuming π‘πππ’πππππ depends mostly on particle size of granular media and tubular
blank dimensions. Generally, in 3D coupling analysis for granular-media-based push-bending process, the
most time-consuming is about 200 h.
Fig. 79. FEM model of tubular blank and DEM model of granular filler
54
Fig. 80. Stress-strain curve of 1Cr18Ni9Ti
In order to fully analyse the fundamental mechanism of the granular-media-based push-bending process the
distributions of stress in granular filler tube can be represented in Figure 81. From there it can be seen how an
important stress concentration occurs at the area A in granular filler corresponding to the major deformation
area B in tube. The starting point of local buckling and cross-section deformation are positioned in the area B
of tube. So, in conclusion, the stress concentration occurring at the area A in granular filler prevents the
presence of wrinkling and cross-section deformation.
Fig. 81. Distributions of stress in granular filler and tube: (a) granular filler, (b) tube
Forming force is an important process parameter determining choice and design of forming equipment. It is
hard to calculate forming force of granular-media-based push-bending process applying current traditional
models and formulas, because of the deformation characteristics of granular media and interaction between
tube and granular filler. At this point through 3D FEM-DEM approach, the forming force of the process can
be calculated and compared to experimental data obtained, which are graphically represented in Figure 82.
55
Fig. 82. Forming force of the granular-media-based push-bending process: simulation and experimental results
Experimental data shows slower and slower increase because lubricating oil between tube and die is gradually
squeezed out leading to friction changes larger. The coupling model cannot reflect this working condition.
Wrinkling is one of the most important defects in thin-wall elbow tube bending. Because of additional axial
compressive force introduced by ‘push’-bending, the wrinkling at inner bending arc becomes more severe. As
it can be noticed looking at Figure 81. stress concentration of granular filler present at the deformation area
prevents the thin-wall tube wrinkling. It must be noticed that not all the granular filler can be correctly used to
face wrinkling phenomenon. Through Figure 83 the effects of particle size on wrinkling of thin-wall elbow
tube during granular-media-based push-bending process are represented. Contacts between granular filler and
tubular blank provide supporting forces for inner bending arc of thin-wall elbow. The contacts’ distribution on
tube wall is unfolded into a plane and shown in Figure 84.
Fig. 83. Effects of particle sizes of granular fillers on wrinkling of thin-wall elbow tube: (a) particle size 0.98 mm, (b)
particle size 1.58 mm, (c) particle size 2.08 mm
56
Fig. 84. The contacts’ distribution on tube wall (unfolded into a plane): (a) particle size 0.98 mm, (b) particle
size 1.58 mm, (c) particle size 2.08 mm
A novel bending method, based on the principles of a push bending process, has been developed to create an
elbow section starting from a straight tube, making advantage of an inner pressure and a rigid die. [17]
The die guides the initial tube into the elbow, then it is pushed by a punch, while internal pressure is used to
avoid wrinkling and buckling phenomena. Going on a FEM simulation will be applied and two different cases
will be compared and analysed: using internal hydraulic pressure and using a urethane rod to supply the internal
pressure.
Fig. 85. The tube bending process treated
A great number of bending processes for transforming a straight tube into an elbow section already exist, but,
at the same time, they present many disadvantages. The new process developed here uses a urethane rod inside
the tube. This tube is then set inside the die and a plunger located at the edge end of the tube and rod exerts an
axial loading pressure on both. In addition a counterpunch is used to control tube and rode at the other
extremity. The final result will give the tube and rod pushed into the die, creating an internal pressure in the
tube. At this point a finite element model can be developed to describe the tube deformation and the
stress/strain distributions inside the tube. The FEM/FEA (or finite element analysis) method applied in this
case is LSDYNA, a non-linear software.
The method of solution used by LSDYNA 3D provides fast solutions for large deformation dynamics and
complex contact/impact problems. Using this integrated product, the problem is pre-processed in ANSYS, the
explicit dynamic solution is obtained using LSDYNA 3D and results are manipulated using the standard
57
ANSYS post-processing tools. Taking advantage of symmetry a half of the problem is modelled using 3D
solid brick elements. Traditionally, shell elements have been favoured for use in simulating bulge forming and
pipe bending operations, but in this case the aim was just to give a proper representation of the deformation of
the urethane rod. Solid elements are actually used to model the tube and the die, to avoid difficulties related to
the mixing of different element types. The tube material behaviour can be described only through a non-linear
elastic-plastic material model, since it is plastically deformed.
In LSDYNA there are two strain rate independent plasticity models available: the classical bilinear kinematic
hardening model and the classical bilinear isotropic hardening model. Both models use two slopes, the elastic
modulus and the tangent modulus to represent the stress–strain behaviour of the material and can be used
satisfactorily for most engineering metals. The only difference between the two models is the hardening
assumption, where the kinematic hardening model assumes secondary yield to occur at 2σy while isotropic
hardening occurs at 2σmax. The urethane rod behaves in a non-linear elastic manner and the LSDYNA
software provides three options for modelling non-linear elastic materials: the Blatz-Ko rubber model is used
for compressible foam type materials such as polyurethane rubbers, the Mooney–Rivlin model is used to model
incompressible rubber materials and the viscoelastic material model is used to model the behaviour of glass
and glass-like materials. In this case the Blatz-Ko model was chosen as the most appropriate material model.
Poisson’s ration is automatically set to 0.463 when using this model and material response is defined through
the strain energy density function:
with πΌ2 and πΌ3 strain invariants. The rod shear modulus was experimentally determined.
Starting from the ANSYS pre-processing tools, simulations are processed to produce a 90° bend in a straight
tube of inner diameter 37.6 mm, outer diameter 40 mm and length 140 mm. The die had an inner radius of 40
mm and a die bend radius of 70 mm. The outer diameter of the urethane rod was 37.6 mm and its length was
140 mm.
Fig. 86. Meshed geometries of die, tube and rod
During the analysis an algorithm is developed to guarantee the correct surface contact between die and tube
and between tube and urethane (rod). The die is modelled as a rigid body and a loading curve defining
displacement over time is applied to the nodes at the free end of the tube and rod to model the punch force. An
elastic Coulomb friction law is used adopting a coefficient of friction equal to 0.1 between die and tube and
0.15 between tube and rod.
58
Problem:
About the simulation results, the first thing noticed is that the meshes associated to tube and rod needed a
redesign at the area nearest the die bend. This is because it is the first deformed location with significant contact
problems. To solve those problems the mesh density was distorted to increase then nearer the die bend. Despite
that approach the redesigned mesh is still subjected to problems related to the contact algorithm. A clear
example is shown in figure 87, where the rod has penetrated the tube and the die.
Fig.87. Urethane rod penetrating the tube
Solutions:
1. Using a first analysis the problem is considered as associated to the fact that the end nodes of the rod,
tube and die are coincident, causing problems for contact algorithm. When trying to establish contact
the algorithm searches for probable contact nodes in an area nearby the node under examination. At
the tube and rod end it is thought that there may be not enough nodes to properly establish contact. To
investigate this case the rod’s length model is reduced so that the end nodes of the rod and tube are not
at the same z-location. At the end the simulation with reduced rod length doesn’t show any significant
improvement.
2. Here is provided a new analysis to find out if a different material model would more accurately predict
the behaviour of the rod, applying a two parameter Mooney- Rivlin material model to represent the
urethane material. Experimental tests are conducted on the urethane material and the Mooney–Rivlin
constants are determined from experimental stress–strain curves. The constants used are C01 =
0.152655E − 03 and C10 = 0.133499E − 02.
It’s found that this model would give better results, but a new issue is now observed in the urethane
model. It is possible to remove the hourglass behaviour and improve the contact algorithm behaviour
changing solution control parameters as advised in the LSDYNA user documentation. There are
however still significant problems in the urethane rod mesh—it seems that the model is having a great
deal of trouble accurately describing the compression of the urethane rod. To solve this problem a hole
was introduced into the urethane rod. The idea of this hole is to allow space for the elements in the
centre of the rod to compress and deform. After this initial deformation the rod deforms normally and
as expected afterwards. It should be noted that this hole would not be present in the actual forming
process but is simply a method adopted to allow the finite element solution of the problem. Results
obtained using Mooney-Rivlin model at the end of the forming process are shown in Fig.88:
59
Fig.88. Predicted tube geometry after bending
In the following figures (89 and 90 respectively) show the distribution of Von Mises stresses and
strains in the deformed tube. It can be clearly seen how the stress distribution is non-homogeneous
and is higher on the back of the tube than on the front portion.
Fig. 89. Von Mises stress distribution in the formed tube
Fig. 90. Von Mises strain distribution in the formed tube
MOS bending process technique
Among all the possible free bending techniques (and in particular push bending techniques) one of the latest
and most significant one is MOS bending process, associated to 6 DOF and mainly used by BMW to produce
semi-finished products for the tube hydroforming process. [18]
For this particular application the bending process is analysed to have a better understanding of the whole
process. In the first experimental bending series, the process precision is measured and evaluated by standard
deviation and process capability. In the second bending series, the analysis focuses on the weld seam position
and the wrinkling behaviour of the tube. The correlations shown between process parameters and the resulting
bending geometry are the basis for the validation of a process simulation model developed in LS-Dyna. The
finite element model is applied to provide a numerical based sensitivity analysis of the bending process,
considering the process and the tube parameters.
60
Generally speaking, the previously mentioned MOS technique is used to create 3D bent tubes as the preforming
step into the hydroforming process. In Fig. 91 it is shown the bending principle on which MOS technique is
based:
Fig. 91. MOS bending technique
The main factors influencing MOS technique are schematized in figure 92, ordered in terms of their effects on
the bending result:
Fig. 92. Influence factors on free bending technology
The process deviation is obtained bending tubes to a 3D part, as it is shown in figure 93. The represented
profile is made of a union of 4 bents characterized by different radii and bending directions.
Fig. 93. Bending profile experimentally used
The tubes used for the bending series are obtained starting from a coil through a roll-forming process and
welded longitudinally using the high-frequency method. The manufacturing process just mentioned is based
on two parts welded together using an electromagnetic field, to have at the end a very strong join. The material
used is an EN AW-5182 aluminium alloy. An inhomogeneity of tube dimensions can be easily noticed, looking
at Figure 94, for three different batches. This inhomogeneity regards, in particular, diameter and sheet
thickness.
61
Fig.94. Measured geometry of EN AW-5182 tubes
The process deviation is rated from the bending line of 26 tubes from the batch C4. The bending line is
reconstructed by means of iterative cylinder fitting through the measured tube surface. To calculate the
deviation between the single bends, the bending line is oriented as follows: 1. The origin of the bending line
is located at the start position (Figure 93) and the local x-axis points tangentially along curve. 2. The z axis is
fixed at the tube end position. 3. After the coordinate transformation to global Cartesian coordinates, the
bending line was interpolated equidistant. The measurement of standard deviation (shown in Figure 95) is
performed along the bending line from start to endpoint. It shows that the deviation increases with increasing
distance from the orientation point. Therefore, the development of the course is nearly linear and can be
expressed as a deviation of 9.2 mm per 2,650 mm of bending line (or 3.54 mm/m bending line). Regarding the
process capability indices, it is now possible to calculate the necessary specification limits for subsequent
forming or joining operations:
(1)
A stable process chain is required, so the process capability index Cp must be greater than 2. Since the tube
profile is considered symmetric from the middle of the bending line the maximum distance in both directions
is 1,325 mm. Therefore, the standard deviation is calculated by π = 1.325 m x 3.54 mm/m = 4.64 mm. The
minimum total specification limit at the end of the tube has to be at least (USL-LSL) = 55.68 mm (calculated
by the previous relation defining Cp (1)). These bounds results in a preforming tool with movable segments to
meet the demands of the bending deviation.
Fig. 95. Standard deviation and density function of experimental bending
FEM model is developed to study numerically the based parameters. The simulation is built in LS-Dyna and
the tools (mandrel, guide and die) are defined as rigid shells, without considering thickness. The stress and
strain distribution from the roll-forming process is simulated and mapped to the closed tube mesh before the
beginning of bending simulation. During the bending process, the tube is pushed with a 10-fold increase in
pushing velocity. Standard penalty-forming contacts were used for all contact zones. The coefficients of
friction are 0.08 for the contact between guide/tube and die/tube and 0.15 for mandrel/tube. The weld line
strength of the tube (e.g., increased yield strength) is constituted by a hardness increase during roll-forming
simulation in the area of the strip edges and also mapped on the tube mesh. The definition of the weld line
thickness differs from the overall sheet thickness (Figure 94 border of thickness distribution) with an increase
to 2.3 mm. Therefore, a single element row is selected along the weld line, and a new thickness is defined for
62
this area. The influence of weld seam position on the bending result and tube wrinkling tendency is analysed
in order to provide an overview of the precision of the simulation model.
While testing 26 tubes of batch C4, wrinkling phenomenon is observed in 10 cases. Strong wrinkling occur on
the profiles only in radius 1 and 4: wrinkling in radius 1 (Figure 96), weak or no wrinkling in radius 4 (figure
97). When the same profile was bent with tubes of batch C1 and C3 (10 tubes each), no wrinkling was observed.
No significant differences are found analysing the mechanical tube properties, the tube surface roughness and
the tube homogeneity. The maximum effective plastic strain EPS resulted by the simulation are represented in
figure 99. It can be finally concluded that the tube dimensions have a significant influence on wrinkling
behaviour during the free bending process.
Fig. 96. Strong wrinkling at radius 1
Fig. 97. Weak wrinkling at radius 4 Fig. 98. Decrease of wrinkling
by decrease of lubricant (a to c)
Fig. 99. Simulation result of test geometry (max EPS): a)batch C4 b)batch C1, C3
Figure 98 shows the influence of the amount of lubrication on the wrinkling occurrence for the same bending
profile. It follows, therefore, that friction behaviour also has a strong relation to wrinkling occurrence. With
the assumption that a higher amount of lubricant results in a lower friction coefficient, the simulation model
shows strong wrinkling for friction coefficients around 0.10, medium wrinkling around 0.16 and no wrinkling
starting from 0.18. The distribution of the friction coefficient on the different tool couples also results in
different wrinkling behaviour.
In addition to the wrinkling phenomenon analysis, another bending series is performed to show the effect that
weld seam position has on the bending result. A series of a single, two-dimensional bends is performed
changing the weld seam position from 0° to 180° (0°: weld seam position at the inner portion of the bend). A
good method related to radius measurement is developed and shown in figure 100.
Fig. 100. Technique used to measure radius and bending angle
63
After reconstructing the bending line, the bending plane is transformed into the global Cartesian x-y plane. To
determine the exact centre point of the radius on the bending line, an optimal bend was fitted to the bending
line by means of nonlinear optimization (least-square distance). The bending angle was measured using the
straight lines at the start and end of the fitting profile. For the radius measurement, a group of points around
the determined centre of the bend was used to perform a circle regression on the bending line raw data. It is
necessary to get a minimum number of points (in respect to the measurement interval) in the bending area with
constant curvature, as plotted in Figure 100, to receive a robust radius measurement using the regression
method. Though figure 101 it can be observed the weld seam influence on the bending result, plotting measured
angles and radii.
Fig. 101. Correlation between experiment and simulation varying the weld seam position
At this point, after the simulation model has been defined and its performance demonstrated, a numerical based
sensitivity analysis can be performed to gain information about the influence of the parameters shown above
on the tube wrinkling behaviour and the bending result in terms of bending angle and bending radius.
Therefore, the simulation model in LS-Dyna is integrated in the optimization tool LS-Opt to perform a Design
of Experiments (DOE) analysis. For each design point, the run starts with the initialization of the tube mesh,
followed by the bending simulation and a spring-back operation. In the DOE, the following parameters are
varied: diameter (63.85-64.15 mm), sheet thickness (1.9-2.1 mm), coefficient of friction (COF)
guide/die/mandrel (0.08-0.18), mandrel position (-12 – 0 mm in relation to guide position), weld line thickness
(2.1-2.35 mm), weld line position (0°-180°) and scale factor for the yield curve (0.85-1.15). The statistical
experimental design includes 400 sampling points, distributed by a space-filling algorithm. The DOE is studied
as associated to the wrinkling behaviour of the process. Figure 102 (a) shows the plotted result of the analysis
of variance (ANOVA) in terms of input parameters and EPS inner. Here, the tube dimension (diameter and
thickness) shows the highest sensitivity in terms of the EPS inner. Wrinkling, as previously noticed, is caused
by a change in tube diameter. In the scatter plot shown in Figure 102(b), the diameter and the thickness are
plotted over the EPS inner. Data points are divided into two levels. The experiments with EPS below 0.4 have
no wrinkle occurrence, whereas all points with EPS above 0.4 have wrinkles in the inner bend. When we
observe the diameter-to-thickness relation of the tube, we can see those experiments with a big diameter and a
small thickness (big diameter of neutral layer) have no wrinkles. Conversely, tubes with a small diameter in
the neutral layer show wrinkling in the bending result. Analysing the DOE with refers to the bending result as
bending angle and radius the main contributing factors are the mandrel position, the tube diameter and COF
of the die, as represented in figure 103.
64
Fig. 102. (a)ANOVA in terms of EPS inner. (b) Scatter plot
of diameter and thickness in relation to EPS inner
Fig. 103. (a)ANOVA in terms of bending angle. (b)
ANOVA in terms of bending radii
As final step some points can be listed in order to improve the MOS bending technology performance, reducing
the measured process deviation:
•
•
•
•
Change bending profile and tools design to avoid any wrinkling by variation of tube dimension within
the production tolerances
Increase precision of inner and outer tube lubrication to lower process deviations
Increase precision of tube dimension to limit process deviations
Develop tools allowing an adjustment of the contact gap between tools and tube to adapt their position
regarding to tube dimensions
65
Press bending
Press bending technology is based on a device having 3 main rolls: rolls 2 and 3 are fixed on a table (on which
a press is set), while roll 1 is fixed by the battering ram of the same press, as shown in Figure 104. [19]
The process consists of the following passages: in order for the bending to verify, pipe 4 is placed on rolls 2
and 3, as roll 1 moves vertically on the pipe causing it to deform. The bending radius, as well as the bending
angle, are controlled using quotes x and h. The distance between rolls 2 and 3 can be adjusted moving the rolls
horizontally. The three rolls are placed on a surface with a channel in contact with the pipe. The channel’s
dimensions are proportional to the pipe’s diameter. This procedure is highly advantageous because, through
it, different bending radii can be produced without changing the rolls, because the bending radius depends on
the position of the three rolls, not on the one of the rolls. On the opposite the negative point of that procedure
is the fact that the bending radius is found indirectly using quotes h and x and it is more difficult to modify.
Fig.104 Press bending process (1. upper roll; 2-3 lower rolls; 4 – pipe (Semi-manufactured); 2θ - bending angle; h – the
distance covered by the upper roll during the bending; R – the bending radius)
One of the most common problems associated to tubes production is the wall thickness variation, due to the
fact that on the outside of the bending profile the wall is subjected to tensile stress and the wall becomes thin,
while on the inner part of the bending profile compressive stress appears and the tube wall becomes thick. In
this case, to study the wall thickness variation phenomenon for a press bent tube, a, ABAQUS FEM is applied.
Finite element model:
ABAQUS CAE is the main FEM software used to model the press bending process, as shown in figures 105
and 106.
Fig.105 FEM of press bending
Fig.106 The final position of press bending
The tube is modelled as 3D deformable part of which material behaves elastic-plastic and the tools are
modelled as 3D discrete rigid. Shell elements S4R are used to model the tube geometry. Contact between
various pairs of surfaces: bend die-tube, pressure die-tube, wiper die-tube is defined using
*CONTACT_SURFACE_TO SURFACE contact command, which allows the sliding phenomenon between
these surfaces with a Coulomb friction model. The implement friction coefficient was 0,1. The material
66
considered is laminated steel OLT 35(STAS 8183-87). The mechanical properties of the tube material are
obtained applying a tensile test on a straight tube. The stress is derived from load force and instantaneous
geometry of the tube. The strains are determined from extensometer directly applied on the tube during the
test. Figure 107 shows the Stress-strain relation used to describe the material behaviour during the simulation.
In addition, the elastic modulus considered is E=2.1x105 [N/mm2 ] and Poisson’s ratio is 0,3.
Fig.107 Stress-strain relation reached through tube tensile test
If, at first, an angle 2Ζ = 25[°] and a diameter of 27 [mm] are chosen, a distribution of the wall thickness of
the tube bending made using finite elements, is obtained, as shown in figure 108. On the other hand, in figure
109 the distribution of the wall thickness of the tube bending at angle 2θ = 90[°] and diameter 27 [mm] is
represented.
Fig.108 Distribution of wall thickness of angle 2Ζ=25°
Fig.109 Distribution of wall thickness of angle 2Ζ=90°
Fig.110 Wall thickness trend
67
Hydro bending
The typical difficulties associated to manufacturing ultra-thin-walled elbow tubes made through traditional
bending processes can be solved applying the double-layered tube hydro bending method. [20]
Typically the hydroforming process provides a simple alternative to manufacture such thin-walled tube elbows.
The specimen used for tests about the effect of internal pressure, the wall thickness distribution and the bending
defects consists of an outer tube made of carbon steel and an inner tube made of stainless steel, with the
thickness ratio of outer to inner layers equal to 10. The yield pressure of the double-layered tube is derived
from theoretical analysis. In addition, the mechanism of the wrinkling and stabilizing of the inner tube is
defined through the finite element analysis. It can be shown that the stability of the inner layer during bending
improves with the increase of the internal pressure, while the wall thickness of two layers decreases with
increasing internal pressure. The new double-layered tube hydro bending method for manufacturing the
ultrathin-walled elbow components is based on many different procedures:
(a) wrapping the ultra-thin inner tube with a thicker outer tube.
(b) sealing and pressurizing the double-layered tube with a liquid medium.
(c) double-layered tube bending with internal pressure support.
(d) separating the outer layer from the elbow.
The double-layered tube specimen is prepared by wrapping the ultra-thin-walled tube with a mild steel outer
thicker tube, which will be then removed after the hydro bending process. The double-layered tube is sealed
with end caps by welding at the two ends, as shown in figure 111. The inner tube is made of stainless steel,
while the outer one is made of low carbon steel. The mechanical properties of the inner and outer tubes are
shown in the following table, obtained from uniaxial tensile tests on samples cut along the axial direction.
Through figure 112 it can be represented the dimensions of the ultra-thin-walled elbow.
Fig.111. Diagram of double-layered tube specimen
Fig.112. Dimensions of ultra-thin-walled elbow (number 1-1 to 5-5 are the measuring locations of section flattening
after bending)
68
Experimental procedure:
A hydraulic press is used to conduce this experimental method and is identified in figure 113. The internal
pressure is provided by a pressure pump and servo controller to keep a constant value during the bending
process. After the internal pressure is linearly increased to the target pressure, the upper die moves downward
to bend the tube on the boundary of the die cavity. Four pressure values are chosen in the hydro bending
experiments, as shown in the table below. The slide gage is used to measure the wall thickness of two layers
whereby the cross-section dimensions are measured with an outside lock-joint transfer caliper.
Fig.113. Tooling of the double layered tube hydro bending
Changing the levels of internal pressure new experimental results of double-layered hydro-bending can be
obtained and shown in figure 114.
Fig.114 Bending results under different levels of internal pressure: (a) defect-free double-layered elbow formed under
Py; (b) section view of double-layered elbow under pressure Py; (c) wrinkling under pressure 0.1Py; (d) rupture under
pressure 1.2Py; (e) ultra-thin-walled elbow formed under pressure 0.8Py; and (f) ultra-thin-walled elbow formed under
pressure Py.
It is highly possible that the rupture occurred first at the outer layer, as the total elongation of the outer layer
material is much smaller than the one of the inner layers, as mentioned a previous table. The deformation at
the outside arc exceeds the plastic limit of the outer layer and leads to fracture.
The increasing axial stress at the compression side during bending process causes wrinkling (or bifurcation
instability) of the thin-walled tube if the compressive stress exceeds the critical value. The internal pressure
has an important role in the successful forming of the double-layered elbow. On one side, the higher the internal
pressure, the more a ‘‘tight fit’’ bond would be formed during bending. The bond between the two layers
would increase the critical wrinkling stress of the inner layer greatly. On the other side, a larger additional
69
axial tensile stress could be introduced to the inner layer if a higher internal pressure is applied. The additional
axial tensile stress would decrease the magnitude of the axial stress at the compression side.
Finite Element Model:
ABAQUS/Explicit is the finite element software used to study the wrinkling mechanism associated to lower
pressure values. Just half of the numerical model in the view of geometry and loading symmetries is built up,
as shown in figure 115:
Fig.115 Finite element meshes for double-layered tube hydro bending: (a) shell type outer tube (b) solid type outer tube
Upper and lower dies are considered rigid, while the inner tube is modelled using S4R (four-node quadrilateral
shell with reduced integration) elements, with the mesh size of 6 mm and with five integration points along
the thickness direction. The outer tube is separately modelled with shell (S4R) and solid elements (C3D8R,
eight node three-dimensional (3D) brick element) for comparison purposes. For the lateral case, four-layer
brick elements are created along the thickness direction, and the element dimension is around 2.5 mm 3 5 mm
35 mm. The material model applied for two layers is isotropic, homogeneous and elastic–plastic material
following the Mises yield criterion. The ends of the two layers are tied together to simulate the tensile forces
of the outer layer applied to the inner layer. The main properties of the material near the weld line are
considered to be the same as the base material of the external tube. In addition to that, the internal surface of
the inner tube and the end caps are subjected to the application of a constant pressure.
The bending behaviour associated to the double-layered tube in presence of inner pressure is analysed through
experimental and numerical simulations, focusing on the effects of internal pressure on the section flattening,
on the wall thickness distribution and the wrinkling mechanism of the inner layer. The following results are
found:
1. The section stiffness of the double-layered tube can be increased greatly with the increase of the internal
pressure.
2. The thickness distributions of two layers are very similar, with the maximum thinning points located at an
angle from the symmetrical section. The wall thicknesses of the two layers decrease as the internal pressure
increases.
3. The internal pressure delays the onset of wrinkling and enhances the stability of the inner layer. It is shown
from the simulation that the maximum magnitude of compressive axial stress at the inner arc decreases as the
internal pressure increases. The inner tube is stabilized when the maximum magnitude of compressive axial
stress is less than the critical buckling stress of the inner layer throughout the bending process. This is the
reason for the prevention of the wrinkling under higher levels of internal pressure.
70
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