Tehran International Congress on Manufacturing Engineering (TICME2007)
December 10-13, 2007, Tehran, Iran
Application of Neural Network and FEM to optimize
load path of T-shape tube hydroforming
A. Jaamialahmadi1, M. Kadkhodayan2, E. Masumi3
Department of Mechanical Engineering
Ferdowsi University of Mashhad,
91775-1111, Mashhad, Iran
Corresponding Author E-mail: kadkhoda@um.ac.ir
Abstract
During tube hydroforming (THF) process, failure modes such as buckling, necking
and bursting may occur because axial feeding and internal pressure are imposed
simultaneously. As load path has great influence on THF, prediction of required
specificationsproperties of final product is difficult and time consuming work. In this
study, Neural Network algorithm and ANSYS LS-DYNA and ANSYS Program
Language Design (APDL), was used to predict final product specifications properties
such as bulging height and thinning (thickness reduction) of T- shape branch
workpiece by using stress based FLD. FE model and Neural Network were verified
using experimental result for a determined load path. Finally direct search method
has been used to obtain optimum load path for higher formability.
Keywords: Tube hydroforming, Bursting failure, Load path, Neural Network, Direct
search pattern
1. Introduction
There has been an accelerating growth in production of hydroformed parts in recent
years. Regarding process features and various blanks, hydroforming (HF) could be
classified into sheet HF and tube HF [1, 2]. Tube hydroforming (THF) offers
advantages as compared to conventional method via stamping and welding process.
It could produce lightweight parts and reduce the costs of part because tooling costs
are lower than that of other methods.
Likewise, part consolidation could be attained by decreasing the number of
processes and parts for the final product [3]. However, failures such as buckling,
necking, bursting and so on could occur in the THF process, because loading
conditions, internal fluid pressure and axial feeding, are complex and imposed
simultaneously.
Since bursting in hydroforming processes is a consequence of necking, which is a
condition of local instability under excessive tensile stresses, prediction of necking
initiation is an important issue before designing the details of processes [4]. The
well-known diffuse necking criterion for a close-end tube can be deduced from Hill’s
uniqueness principle, as it was done by Yamada and Aoki [5]. Nefussi and
Combescure [6] discussed Swift’s two criteria for sheet forming and for tube
hydroforming where necking and buckling are considered simultaneously. It is found
1
2
3
- Lecturer
- Associated Professor
- BSc
1
Tehran International Congress on Manufacturing Engineering (TICME2007)
December 10-13, 2007, Tehran, Iran
in the literature that path-independent stress-based FLD (FLSD) can be used well for
all forming processes without concern for path effects [7]. In addition, it is reported
that the path-dependent nature of the strain-based FLD causes the formability
prediction method to become ineffective in the analysis of complex forming process,
especially hydroforming [8].
In this paper, the stress based hydroforming limit was used to determine bursting
failure with (APDL). Neural network was used to predict final properties of T-branch
workpiece. After experimental verification of FE model and Neural Network with a
determined load path, using direct search method, optimum load path for higher
formability obtained.
2. Determination of analytical forming limit curve for bursting
failure
2.1. Incremental theory of plasticity
In order to predict onset of necking condition, Swift's criterion for plastic instability
and Hill’s general theory for the uniqueness to the boundary value problem was
used. Consider a thin-walled, close-end tube with internal pressure P and axial load
F, which are applied independently. Von-Misses yield criterion for isotropic material
is,
2  ( x   y ) 2  ( z   y ) 2  ( x   z ) 2 ,
(1)
for plane stress condition, effective stress can be written as,
   12   22   1 2
,
(2)
where σ1 and σ2 are the principal stress.
The plastic strain increment is given as follows,
d 
4(d 12  d 22  d 1 d 2 )
,
3
(3)
where dε1 and dε2 are the principal strain increment.
The plastic strain increments are given from Levi-Misses flow rule and
incompressibility condition as the following,
1 d
(2 1   2 )
2
1 d
d 2 
(2 2   1 )
2
d 3  (d 1  d 2 )
d 1 
(4)
2.2. Plastic instability criterion and forming limit stress
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Tehran International Congress on Manufacturing Engineering (TICME2007)
December 10-13, 2007, Tehran, Iran
It is assumed that principal stresses ratio α and principal strain ratio β are maintained
as constants: α = σ2/σ1 and β =dε2/dε1.
2.2.1. Hill’s local necking criterion for sheet metal
Hill’s local necking criterion predicts onset of plastic instability when following
conditions are satisfied simultaneously [9],
d 1  1 ( d1  d 2 )
(5)
d 2   2 ( d1  d 2 )
Then the local necking criterion in terms of sub-tangent ZL, is obtained based on
Equations (3) and (4),
1
1 d
(1   )


.
Z L  d 2(1     2 ) 3 / 2
(6)
2.2.2. Diffuse necking criterion
It may be assumed that for the long tubes the stress distribution to be uniform and
that the instability occurs when the following simultaneous conditions are satisfied
[10]:
d 1   2 d 2   1d 1 ,
d 2   2 (2 d 2  d 1 ) .
(7)
The variation of the principal stress components in terms of the principal hoop stress
can be derived as:
d
d 1 
( 2    1) 22 ,

(8)
3 d 2
d 2 
2 ,
2 
thus the plastic instability condition in terms of sub-tangent Z for tube
hydroforming is obtained as:
1 1 d 1  (2  1) 2  2(2   )(1   ) A
.



Z  d 4
B
( 2    1) 3 / 2
(9)
With a Holomon or power law work hardening equation,   k n , Eq .(9) can be
rewritten as:
1 1 d n A

  .
Z  d  B
(10)
Thus the equivalent strain can be described as:
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Tehran International Congress on Manufacturing Engineering (TICME2007)
December 10-13, 2007, Tehran, Iran
 
4 2
( 1   22   32 )  C 2 ,
3
(11)
where,
C
4 2
(     1) .
3
(12)
According to Equations (10) and (11), the limit strains based on the plastic instability
are:
 1c   2c ,
 2c 
(13)
B n
,
AC
where  1c and  2c are axial and hoop principal critical strains.
From Eq. (4) the limit stresses  1c and  2c , when bursting failure is occurred, can be
obtained as:
2
(2 1c   2c ) ,
3
2
 2c 
(2 2c   1c ) .
3
 1c 
(14)
Finally, the forming limit stress diagram to evaluate the bursting failure can be
obtained as Figure 1.
800
FLSD
700
Major Stress (MPa)
600
500
400
300
200
Yield Locus
100
0
-400
-300
-200
-100
0
100
200
300
400
500
Minor Stress (MPa)
Figure1: Theoretical forming limit stress curve for prediction of
bursting failure.
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Tehran International Congress on Manufacturing Engineering (TICME2007)
December 10-13, 2007, Tehran, Iran
3. Determination of load path by using of Neural Network and direct
search method
3.1. FE model
Direct search method and Neural Network may be coupled together to obtain the
loading path of a T-branch part using ANSYS LS-DYNA and ANSYS parametric
design language (APDL). A cylindrical steel tube of 60.5mm (middle diameter),
235mm (length) and 3.91mm (thickness) was used as blank. The diameter of Tbranches was equal to that of the main tube. The forming load is an internal
hydraulic pressure along with an axial compressive load (tube end feed).
By taking advantage of symmetry, it was possible to model one quarter of T-branch
[11]. The die was modelled only for the surface in contact with the tube. It had a
variable radius along the junction i.e. the radius variation was from 15 mm (at the top
junction of the bend of the die) to infinity (at the side junction of the die). The finite
element model for the tube was built with four node 3D thin shell quadratic mapped
mesh element with assuming that the material follows a power law hardening model.
The die was also meshed with 3D thin shell element, with the assumption that the
die material follows a linear elastic law.
An elastic coulomb friction law with the coefficient of friction of 0.1 was assumed
between the tube and die surface. The material properties used for tube obtained by
a tensile test are: the work hardening law   836 0.1757 MPa, Young’s modulus = 204
GPa, Poisson’s ratio = 0.3, Density = 7860 Kg/m3 and ultimate tensile strength = 615
MPa.
To avoid failure due to bursting, principal stresses were limited to safe zone of FLD
diagram in the entire of FE simulation. The wall thinning was limited to 3.42 mm
(0.875 times of initial thickness) and to prevent failure due to wrinkling, change in the
sign of slope of middle and bottom line of tube was limited to some small values.
3.2. Neural Network
Prediction of final properties of the product in THF is too difficult due to dependency
of process to load path. Here Neural Networks were used to predict the final
properties of product under some arbitrary load paths. The load path was divided to
five stages, and the relation between the pressure and end feed in any stage
assumed to be linear. It was also assumed that the previous stages affects on some
properties of workpiece such as maximum stress, mean stress, minimum thickness,
mean thickness, thickness tolerance and maximum bugling height. If the slope of
pressure-end feed and the mentioned data are known at the beginning of each
stage, the properties at the end of that stage can be predicted.
In this study five Neural Networks were coupled to each other and results of each
stage were used as input to next network (except the last one). Finally, the result of
the fifth stage was final properties of the product predicted by Neural Networks.
Two load path types (Figure 2) were used to obtain necessary data for training
Neural Networks. Using APDL, the slope of this line was changed and if there was
no failure due to bursting or wrinkling the required data was saved at the beginning
and the end of stages. Neural Networks had two layers with backpropagation
algorithm and conjugate gradient algorithm for training the network.
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Tehran International Congress on Manufacturing Engineering (TICME2007)
December 10-13, 2007, Tehran, Iran
6
4
Pressure (MPa)
Pressure (MPa)
4.5
3.5
3
2.5
2
1.5
1
5
4
3
2
1
0.5
0
0
0
5
10
15
20
0
25
5
10
15
20
25
Axial Feed (mm)
Axial Feed (mm)
Figure 2: Two load paths used for training the network.
3.3. Optimization
Direct search method was used in order to find the optimum load path to obtain
higher formability. This method only needs function's value at any point and there is
no need to any information about gradient of objective function. Developed program
uses result of Neural Network to obtain value of objective function at any point.
4. Results
After training the Neural Networks, they are used in direct search algorithm to obtain
optimum load path. The results are shown in Figure 3. In practice, in the Toos
payvand Company an internal pressure equal to 95 MPa is being required. The
experimental load path is quite similar to path pattern used in [12].The reduction in
required pressure is about 13 MPa. Furthermore, a better formability and more
uniform thickness distribution with the same axial feed may be obtained.
Table 1
Optimum load path results
Maximum
Maximum end Maximum
pressure
feed (mm)
bulging(mm)
(MPa)
FE model 82
40
45
Neural
Network
82
40
44
6
Thickness
Tolerance (mm)
1.85
1.92
Tehran International Congress on Manufacturing Engineering (TICME2007)
December 10-13, 2007, Tehran, Iran
90
Pressure (MPa)
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
Axial Feed (mm)
Figure 3: Optimum load path.
The Figure 4 displays the variation of thickness in bulging region after applying
obtained load path in a FEM model. Figure 5 clearly shows the thickness variation of
a longitudinal strip of elements on the top of the model after deformation.
Figure 4. Thickness variation in T shape tube after optimized load path .
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Tehran International Congress on Manufacturing Engineering (TICME2007)
December 10-13, 2007, Tehran, Iran
Figure 5: Thickness variation of elements on top of the model in
longitudinal direction.
6. Conclusions
1. Neural Networks have been utilized to predict the final geometrical specifications
quality of THF process. By using this method, geometrical specifications of product
can be predicted with a good accuracy without any need to FE simulation or
experiment.
2. Direct search method has been used to obtain optimum load path.
3. The proposed method provides not only potential improvement in formability of Tshape, but also can be used in other THF processes.
4.For a typical T-shape hydroforming, prescribed bulging height and thickness
reduction without any failure, such as wrinkling and diffuse necking, was achieved by
reduction in internal pressure(at leas 13%)
Acknowledgment
The authors would like to thank Toospeyvand Company for its cooperation to get
access technical data.
References
[1] L.H. Lang, Z.R. Wang, D.C. Kang, S.J. Yuan, S.H. Zhang, J. Danckert, K.B.
Nielsen, "Hydroforming highlights: sheet hydroforming and tube hydroforming" , J.
Mater. Process. Technol. Vol.151 (2004), pp.165–177.
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Tehran International Congress on Manufacturing Engineering (TICME2007)
December 10-13, 2007, Tehran, Iran
[2] Ch. Hartl, "Research and advances in fundamentals and industrial applications of
hydroforming", J. Mater. Process. Technol. Vol.167 (2005), pp. 383–392.
[3] M. Ahmetoglu, T. Altan, "Tube hydroforming: state-of-the-art and future trends", J.
Mater. Process. Technol. Vol. 98 (2000), pp. 25–33.
[4] Kim J, Kim YW, Kang BS, Hwang SM. "Finite element analysis for bursting failure
prediction in bulge forming of a seamed tube". Finite Elements in Analysis and
Design, Vol.40 (2004), pp. 953–966.
[5] Xia ZC. "Failure analysis of tubular hydroforming". Journal of Eng. Mater. Techno.
Vol.123 (2001), pp. 423–429.
[6] Nefussi G, Combescure A. "Coupled buckling and plastic instability for tube
hydroforming". Int. J. Mech. Sci. Vol. 44 (2002), pp. 899–914.
[7] Stoughton TB. "A general forming limit criterion for sheet metal forming". Int. J.
Mec. Sci. Vol. 42 (1999), pp.1–27.
[8] Stoughton TB, Zhu X. "Review of theoretical models of the strain-based FLD and
their relevance to the stressbased FLD". Int. J. Plast.; Vol. 20 (2004), pp.1463–86.
[9] R. Hill, "The Mathematical Theory of Plasticity", Oxford University Press, New
York, 1983.
[10] H.W. Swift, "Plastic instability under plane stress", J. Mech. Phys. Solids, Vol.1
(1952), pp. 1–18.
[11] P. Ray, B.J. Mac Donald," Determination of the optimal load path for tube
hydroforming processes using a fuzzy load control algorithm and finite element
analysis", FE in Analysis and Design; Vol.41 (2004), pp.173–192.
[12] H. Kashani Zadeh ,M. Mosavi Mashhadi,“Finite element simulation and
experiment in tube hydroforming of unequal T shapes”, J Mater Process Technol.
Vol.177 (2006), pp. 684-687.
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