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Computational approach to analysis and design of hydroforming process for an automobile lower arm
Computers and Structures 80 (2002) 1295–1304 www.elsevier.com/locate/compstruc
Computational approach to analysis and design of hydroforming process for an automobile lower arm Jeong Kim a, Sung-Jong Kang b, Beom-Soo Kang a
a,*
ERC/NSDM, Pusan National University, Pusan 609-735, South Korea b Catholic University of Taegu, Kyungbuk 712-702, South Korea Received 7 June 2001; accepted 15 March 2002
Abstract Tubular hydroforming has attracted increased attention in the automotive industry recently. In this study, a professional finite element program for analysis and design of tube hydroforming processes has been developed, called HydroFORM-3D, which is based on a rigid-plastic model. The friction calculation between die and workpiece has been dealt with carefully by introducing a new scheme in three-dimensional surface integration. With the developed program, HydroFORM-3D, the hydroforming process for an automobile lower arm is analyzed and designed. The manufacturing process for a lower arm consists of tube bending, preforming, and final hydroforming. To accomplish successful hydroforming process design, thorough investigation on proper combination of process parameters such as internal hydraulic pressure, axial feeding, and tool geometry is required. This paper includes the study on the influences of the forming conditions on the hydroforming of a lower arm by using simulation to predict strain and tube shape during bending, preforming, and final hydroforming processes. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Hydroforming process; Preform design; FEM; Lower arm
1. Introduction Due to several advantages of hydroforming process over conventional methods, it has become a new mainstream technology for the manufacturing of automotive structural components. Since tubular hydroforming has attracted increased attention in the automotive industry for making a wide variety of structural components such as sub-frames, engine cradles, and exhaust manifolds, the process design of the hydroforming process is required to improve the product quality and productivity. To manufacture a tube-hydroformed component, a circular tube is inserted into a suitable die, sealed on both sides by two horizontal cylinders, and subsequently
formed to conform to the shape of the die cavity by internal hydraulic pressure. Over the years a great deal of experience has been accumulated in this field mainly through the trial-and-error approach. But the sound design by ‘‘trial-and-error’’ can be very expensive and time consuming. On the other hand, the numerical approach of the finite element method in hydroforming can increase existing knowledge and reduce tool cost and lead-time by providing virtual tryout before tool construction and predicting the formability in advance.
2. Hydroforming of a lower arm *
Corresponding author. Tel.: +82-51-510-2310; fax: +82-51513-3760/3960. E-mail address: [email protected] (B.-S. Kang).
In many cases, the hydroformed components undergo multi-stage processes since the hydroforming frequently requires reduction in wear and scrap levels,
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which results in cost saving and high surface quality. The process sequence for producing the lower arm by hydroforming involves the following steps: • pre-bending of the starting blank tube, • preforming of the pre-bent tube by stamping, • final sizing by hydroforming of the preformed tube. Prior to the hydroforming operation, the tube must be bent to the approximate centerline of the final part to enable the tube to be placed in the die cavity. After then, a preforming operation is carried out to the pre-bent tube. When hydroforming a pre-bent axial component, such as a lower arm, the maximum expansion limit in the pre-bent blank zone is lower than those straight axis components, because there is not enough material feeding in this area [1,2]. Then it is difficult to obtain a sound final product from pre-bent tube in one hydroforming operation, and a preforming process is needed. Preform design in hydroforming refers to the design of an intermediate geometry close to the final shape, and it derives the final product without defects and excessive loss of material. After the pre-bending process, the pre-bent tube has to undergo the preforming process before the hydroforming operation because the diameter of the tube is larger than the smallest dimension of the final product. For hydroforming process, selection of the tube blank size is a critical problem. If the diameter of initial tubular blank greatly exceeds the smallest component diameter, the cross section will be collapsed when the die closes. On the other hand, when the diameter of the blank is too small, it is possible that the component cannot be fully formed due to excessive circumferential elongation. This will cause bursting failure in the premature component. Therefore, a carefully-designed preforming process can contribute significantly to reduce production cost and it is a dominant factor for successful hydroforming [3,8]. In this study, the hydroforming process for an automobile lower arm is analyzed by using three-dimensional finite element program, HydroFORM-3D, based on a rigid-plastic model. The program is developed with modification of the previous rigid-plastic program, addition of three-dimensional die description for tube hydroforming, and suggestion of new frictional force calculation. The goal of this study is to evaluate the program by accomplishing proper design and control of processes for producing hydroformed lower arm component most economically. This study also describes the influences of forming conditions, such as the loading path of the hydraulic pressure, the axial feeding displacement, and the friction condition between tubular workpiece and die wall on the hydroforming process. The numerical simulations by using HydroFORM-3D are carried out on the bending, the preforming of stamping, and the hydroforming.
3. Theoretical background In many cases of hydroforming, the plastic strains are so large that the elastic strains are negligible by comparison. Thus the rigid-plastic finite element method is adopted in this study because of its effective algorithm for numerical simulation of the forming process. The general description of this numerical method can be found in the previous publications [4,5,7]. For the completion of the investigation, a brief summary of the approach based on infinitesimal deformation theory is included. Consider a body of volume V bounded by a surface S with the traction ti prescribed on a part of the surface SF and the velocity ui enforced on the remainder of the surface SU . If the inertia effects and body forces are assumed to be ignored, the deformation of the control volume V is characterized by the following relations: rij;j ¼ 0 on V rij nj ¼ ti on SF ui ¼ ui on SU
ð1Þ
where r is the Cauchy stress tensor and n is the unit outward normal to the workpiece. From the constitutive equation the ratios of the plastic strain rate are defined by e_ij ¼
of ð r; Y Þ _ k orij
ð2Þ
in which k_ is the plastic multiplier that can be calculated after the condition f ¼ 0, where f is the plastic potential, identified as the scalar function that defines the elastic limit surface and is described by the von Mises yield criterion f ð r; Y Þ ¼ r Y ¼ 0
ð3Þ
Y is the yield stress in simple tension, and depends on the evolution of the isotropic work-hardening, which is obeyed by the n-power law r ¼ Ken
ð4Þ
where K, n are plastic coefficient and strain-hardening exponent, respectively. The equivalent plastic strain is given by the following at time t Z t e ¼ e_ dt ð5Þ 0
The finite element formulation is to begin with a variational principle related to total potential energy. It requires that among admissible velocities ui that satisfies the conditions of compatibility and incompressibility as well as the velocity boundary conditions:
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Z
Z
Z K ti ui dS ¼ 0 ð6Þ ðe_v e_v Þ2 dV v v 2 SF qffiffi qffiffi where r ¼ 32ðr0ij r0ij Þ1=2 , e_ ¼ 23ðe_ij e_ij Þ1=2 , e_v ¼ e_ii , and r0ij , K are the deviatoric stress tensor and the penalty constant with a large positive value, respectively. The first order variation of the functional equation (6) can be written as Z Z Z ti dui dS ¼ 0 r de_ dV þ K e_v de_v dV ð7Þ dp ¼ p¼
re_ dV þ
v
v
SF
Eq. (7) is the basic equation for the finite element discretization and can be converted to non-linear algebraic equations by utilizing the discretization procedure. Here, the workpiece is discretized with eight node hexahedral elements for three-dimensional deformation. In general cases, the implicit procedure uses an automatic incremental strategy based on the success rate of a full Newton iterative solution method. For a linearized form of Eq. (7), the Taylor expansion near an assumed solution point u ¼ uði1Þ gives at the ði 1Þth increment, 2 op op ðiÞ þ Du ¼ 0 ð8Þ ouI u¼uði1Þ ouI ouJ u¼uði1Þ I ðiÞ
where DuI is the first-order correction of the velocity vector uði1Þ and the second term is the current tangent stiffness matrix. Once the solution of Eq. (8) for the velocity correction term is obtained, the assumed velocity uðiÞ is updated according to the following uðiÞ ¼ uði1Þ þ aDuðiÞ
ð9Þ
where a is deceleration coefficient 0 < a 6 1, which is used to avoid divergence during iterations. A small value a, say 0.1 or 0.2, is usually used at the beginning of iteration and then it is gradually increased towards 0.6 so as to reach a fast convergence rate. Two convergence criteria can be used for determining the termination of the iteration. The first convergence criterion is satisfied if the error norm of the velocity becomes very small, namely, kDuk=kuk 6 n
ð10Þ
where Euclidean vector norm is defined as kDuk ¼ ðuT uÞ1=2 , and n is a given tolerance, with recommended value of order 104 . The second convergence criterion is determined by measuring the norm of the force-residual, also called function norm, kop=ouk 6 d
ð11Þ
where d is a small value varying with the magnitude of yield stress. A converged solution can be found only when either of the convergence criteria, Eq. (10) or Eq. (11) is satisfied.
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4. Die and contact description in three-dimensional hydroforming For the analysis of the hydroforming process and the modeling of arbitrarily curved die surfaces encountered in actual industrial components, one of the most important and difficult problems is the treatment of contact between die surfaces and the deformed workpiece. Hydroforming processes are non-stationary and the geometry of the workpiece progressively changes. Therefore, to obtain reasonable and stable solutions, the contact between the nodal points and die surfaces must be carefully checked and modified progressively as well. Some discussions about the contact problem are presented and a simple but effective contact algorithm is proposed in this study. 4.1. Description of the die surface The description scheme of the die surface can have a significant effect on computational time. Moreover, since a discontinuous change in the sliding direction of nodes during forming analysis causes a computational difficulty in convergence, the die surface must be described accurately and smoothly to obtain a stable numerical solution for a large displacement and large deformation analysis of hydroforming processes. In this study, the die surface is described by a set of four-nodequadrilateral C 0 patches whose normal directions indicate the interior of the die, and compared with the larger deformation of the workpiece, the die is treated as a rigid body. Such a representation of the die geometry makes it easy to model. It could be seen next that the contact algorithm for this approach is very simple and efficient. 4.2. Contact analysis Typical contact analysis in current use can be classified into direct method, penalty method, and Lagrange multiplier method. In this study, the direct method is used for contact analysis. For every incremental step, the minimum time is calculated for the free node to contact with the die or for the contacted node to touch the other die. If this minimum time is smaller than the desired incremental time, let the incremental time during this step be equal to the minimum time and these nodes are declared to be in contact with their corresponding die. This scheme will avoid the node entering into the die. For the contacted nodes, it is important to have a correct evaluation of the die in normal direction as well as a correct local frame basis definition for an accurate setting of the nodal boundary condition. The contact search includes a global search and a local search. The global search intends to find candidate die patches that may be in contact with the node and its main objective is
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to increase the speed of the contact analysis. The local search intends to select one patch for contact checking with the node, among all candidate die patches. Every die surface patch has limited boundaries in the x, y, and z directions as shown in Fig. 1. By using the boundary concept, the global search finds the candidate die patches which may be in contact with the node. After the global search, one patch in contact with the node must be selected among all of the candidate patches found in the global search. The local search is carried out to finish this using the area comparison scheme shown in Fig. 2. In the figure, the quadrilateral ABCD is one of the candidate die patches and O is the location of the node projection on this patch. The area comparison scheme is expressed as follows:
Here, the Ap is the area of the patch, A1 ; A2 ; A3 , and A4 are the areas of the triangles which consist of the die patch described with four nodes. g is a small positive tolerance for calculation, usually taken as about 0.001. In numerical programming, to eliminate the equations expressing geometrical relationships for the nodes in contact with dies, the coordinate system is transformed from global to local at the nodes. After the local search, a reference system of local axes is set with respect to the contact die patch. The normal direction vector of the contact patch defines one of the local coordinate axes, while the other two are arbitrary and the only condition required is that they must be orthogonal on the rectangular element. 4.3. Friction effects
jA1 þ A2 þ A3 þ A4 Ap j 6g : Ap
contact
jA1 þ A2 þ A3 þ A4 Ap j >g: Ap
not contact
ð12aÞ
ð12bÞ
In this approach, Coulomb condition for friction along the die-workpiece interface is considered for industrial hydroforming operations. Along the die-workpiece interface there exists a neutral point where the velocity of the deforming material relative to the die becomes zero. In order to evaluate the stiffness at a neutral point, where the sudden change of direction of the frictional force would make the simulation unstable, the approximation of the velocity-dependent frictional force vector is given as the following [9]: ur fr ffi lN qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 urs þ u2rt þ u20
ð13Þ
where N is the normal force, l the frictional factor, the relative velocity vector ur is represented by urs and urt components in the direction of the local coordinate axis s and t, which are defined over the surface of the die, and u0 is a small value for 102 –103 . Then the friction term can be added to Eq. (7), Fig. 1. The schematic representation of boundary concept for the global search.
ur dpf ¼ fr dur ¼ lN qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dur 2 urs þ u2rt þ u20
ð14Þ
Then the derivatives of Eq. (4) can be evaluated as opf ursi ffi ¼ lNi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oursi 2 u þ u2 þ u2
ð15aÞ
opf urti ffi ¼ lNi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ourti u2 þ u2 þ u2
ð15bÞ
rsi
rsi
0
rti
rti
0
0
1
u2rsi o pf 1 B C ¼ lNi @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 A 2 oursi 2 2 2 2 2 2 ursi þ urti þ u0 ursi þ urti þ u0 2
Fig. 2. The schematic representation of area comparison scheme for the local search.
ð15cÞ
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0
1299
1
u2rti o2 pf 1 C B q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ lN @ i q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 A ou2rti 2 2 2 u2rsi þ u2rti þ u20 ursi þ urti þ u0 ð15dÞ o2 pf ursi urti ¼ lNi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 oursi ourti u2rsi þ u2rti þ u20
ð15eÞ
For any node in contact with the die surface, the frictional force is calculated. 4.4. The implementation of hydraulic pressure According to the governing equation, the pressure force applied to the nodal force can be expressed as Z opSF ¼ p/i dS ð16Þ ovn SF where, p is the pressure force during the hydroforming process, and /i is the shape function of the ith node. The pressure force does not have effect on the stiffness matrix. The integration of the hydraulic pressure term in Eq. (16) is carried out by 5 5 Gauss quadrature on the pressure surface. The direction of the pressure force coincides with the normal direction of the element surface where the hydraulic pressure is applied.
Fig. 3. (a) A conventional stamping-welded lower arm assembly and (b) solid model to be hydroformed.
5. Preform analysis of hydroforming of a lower arm 5.1. Problem description A lower arm is one of the suspension parts, which is placed at the front of passenger room and supports a cross member and a knuckle component. It plays important roles not only in performing the rotational center of a tire which affects wheel alignment but also in reducing the vibration while driving. When the automotive component, a lower arm, is produced by hydroforming process, it can substitute for the conventional stampedwelding lower arm with the same function, but with high rigidity and low cost. In Fig. 3, the assembly of conventional stamped-welded lower arm and the one from hydroforming are shown respectively. Fig. 4 shows the geometry and several different cross sections along the axis of the hydroformed lower arm. The tubular blank is assumed to be a rigid-plastic material obeying the n-power law in Eq. (4), of which stress–strain relation is listed in Table 1. The Coulomb friction model is used and the material is assumed as isotropic. The friction coefficient used in preforming operation was set at 0.1. The initial outer diameter of the tube is 63.5 mm and the initial thickness is 2.6 mm. The original length before the initial pre-bending is 600 mm.
Fig. 4. Schematic view of the lower arm.
Due to the geometric symmetry of the final part, only a quarter of the tube is molded using brick elements.
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Table 1 Tensile properties of the material used STKM-11A n
K-value ð r ¼ Ke Þ (MPa) Work-hardening exponent, n Yield stress (MPa)
468.3 0.206 215.1
5.2. Analysis results of pre-bending process by FEM simulation To produce a lower arm by hydroforming process, first a pre-bending process is carried out so that the tube fits into the die cavity. In certain cases where the overall part geometry is not relatively complex, the bending may be achieved as the final die is closed. In most cases a bending process is done separately. During the bending operation, in general, the tube material thins at the outside of the bend and thickens at the inside. Prebending simulation of the tube is performed using a conventional rotary draw bending machine and the bending angle is 73. Rotary draw bending is the most popular, cost-effective bending method for thin walled tubes. The pre-bending model using this method needs a tubular blank and a die set which includes a bent die, a pressure die, a clamp die and a wiper die as shown in Fig. 5. A wiper die is used to avoid the wrinkle [6]. The tube bending process has been simulated using the rigidplastic FEM program, HydroFORM-3D. Fig. 6 is the deformed shape of the tube after pre-bending. From the results, the maximum effective plastic strain is 0.226 and maximum effective stress is 368.0 MPa. The minimum thickness is 2.41 mm in tensile area while the original thickness is 2.6 mm. This prediction of 7.3% thinning is close to the actual measurement of 10.0% thinning.
Fig. 5. Initial finite element model of the bending tool.
Fig. 6. Deformed shape after pre-bending operation.
5.3. Analysis results of preforming process by FEM simulation After the pre-bending process, the pre-bent tubes have to undergo a preforming process before the hydroforming operation because the diameter of the tube is larger than the smallest dimension of the final product. The preforming simulation model is composed of a prebent tube, which results from previous pre-bending process, and a split die for forming to reduce the outer diameter of tube to the height of final product. Fig. 7 is the schematic view of the lower stamping and hydroforming die created from CAD software. Fig. 8 shows a pre-bent tubular blank and a set of upper and lower stamping dies. To get the final preformed shape, a stamping process is required. Since the minimum dimension of the lower arm is 15.2 mm as shown in Fig. 4,
Fig. 7. Schematic view of the lower stamping and hydroforming die.
J. Kim et al. / Computers and Structures 80 (2002) 1295–1304
Fig. 8. Finite element model of stamping die and pre-bent blank.
1301
Fig. 10. Finite element model of hydroforming process.
6. Analysis of a lower arm hydroforming process the stroke of 24.15 mm is needed for stamping process when a tubular blank with the outer diameter of 63.5 mm is used. At first the pre-bent tubular blank is inserted to the hydroforming dies and then the stamping process is performed as the dies are closed. Fig. 9 shows the deformed geometry and several cross sections after stamping operation. The cross sections are not circular any longer unlike those ones after pre-bending. The maximum effective plastic strain is 0.414 and the maximum effective stress 379.3 MPa. The minimum thickness is 2.38 mm, which implies that the maximum thinning is increased slightly by 9.2% from 2.6 mm original tube thickness.
Fig. 9. Deformed shape and cross sections after preforming operation.
The hydroforming simulation model is composed of the preformed tube, which is from the result of preforming operation, and the same dies for stamping process. Fig. 10 shows the preformed blank and a set of upper, and lower dies, and an axial feeding ram. 6.1. Influence of internal hydraulic pressure on formability To evaluate the formability of the lower arm hydroforming with respect to internal hydraulic pressure, numerical simulation with three loading paths shown in Fig. 11 is performed. The maximum axial feeding and the friction coefficient between the preformed tube and the die wall was set at 50 mm and 0.05 respectively while
Fig. 11. Description of loading paths.
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Fig. 13. The variations of (a) the die clamping forces and (b) the axial feeding forces during hydroforming process.
6.2. Influence of axial feeding on formability
Fig. 12. Deformed shape of the workpiece for different loading paths: (a) Path 1, (b) Path 2.
the internal hydraulic pressures varied. From the analysis, the simulation results show that for Paths 1 and 2 the material could not fill the inner cavity of the die shown in Fig. 12. Fig. 12(a) shows the deformed shape at the pressure level B on Path 1 shown in Figs. 11 and 12(b) at the pressure level A on Path 2. From these results, to get the completely filled hydroformed part, the internal hydraulic pressure should be started at over 30 MPa and the final hydraulic pressure should be higher than 40 MPa. The numerical analysis predicts a minimum thickness of 1.73 mm in the final part (33.5% thinning) at the outside of the bend under the loading Path 3. Fig. 13 shows the variations of clamping forces and axial feeding forces during the hydroforming process. For Path 3, a clamping force of 3403 kN and the axial force of 162 kN for 50 mm axial feeding were required to complete soundly hydroforming process.
To evaluate the formability of the lower arm hydroforming with respect to axial feeding displacement, hydroforming simulations with the three axial feeding cases, no feeding, 50 and 100 mm, are carried out. The load condition was Path 2 shown in Fig. 11, and the friction coefficient between the preformed tube and the die wall was fixed at 0.05 while the axial feeding varied. It is noted that the case of no axial feeding leads to a high possibility of bursting failure at the outside of the bend due to the maximum thinning. As the axial feeding displacement increases, the formability becomes higher, when the three cases are considered. This is because more material is supplied into the potential bursting failure site by increasing the axial feeding. But excessive material feeding not only causes the weight increment on the final part but also may lead to buckling or wrinkling failure. Thus, the axial feeding must be appropriate to manufacture the required sound part. The maximum thinning is predicted to 42.7% (the minimum thickness of 1.49 mm) at the outside of the bend with no feeding condition. Table 2 shows the maximum clamping forces and maximum axial feeding forces for these cases. For
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Table 2 Maximum clamping forces and axial feeding forces for different axial feeding displacements Axial feeding displacements
No feeding
50 mm
100 mm
Clamping force of hydroforming press [kN] Axial feeding force [kN]
3512
3403
3520
N/A
162
307
the axial feeding displacement of 100 mm, 3520 kN for clamping and 307 kN for axial feeding were needed. 6.3. Influence of friction coefficient on formability To evaluate the formability of the lower arm hydroforming with respect to the friction coefficient between the preformed tube and the die wall, hydroforming process with four different friction coefficient values, 0.0, 0.05, 0.1, and 0.2, is analyzed. The loading condition was Path 2 shown in Fig. 11 and the axial feeding was fed up to 50 mm for each case. From simulation results, the formability increases while the friction coefficient decreases. It implies that the better lubrication condition makes the material fed into possible failure region easily. And for the friction condition of over 0.1, the high possibility of local wrinkling or bursting is predicted. The whole deformed product and some cross sections predicted by the numerical simulation with friction coefficient of 0.05 is shown in Fig. 14. The cross section shapes are formed well to conform to the die cavity. The maximum thinning is predicted 52.3% (the minimum thickness of 1.24 mm) at the outside of the bend under the friction condition of 0.2. For the friction condition of 0.2, the compression force reaches the maximum value of 211 kN. In that case, the maximum die clamping force for sealing needs 3158 kN. Fig. 15 shows the distribution of the equivalent stress predicted by the numerical simulation with friction coefficient of 0.05. The maximum effective plastic
Fig. 14. Cross sections after finishing hydroforming process.
Fig. 15. Distribution of the equivalent stress of the hydroformed lower arm.
strain is 0.738 and the maximum effective stress 475.2 MPa.
7. Conclusion For analysis and design of tube hydroforming, a FEM program of rigid-plastic approach is presented. A simple concept for die contact and non-contact to workpiece and a friction calculation algorithm in tube hydroforming are implemented newly in the developed program, HydroFORM-3D. To manufacture a sound automobile component, lower arm, a systematic approach on the analysis and design of the hydroforming process was suggested by using HydroFORM-3D based on the rigid-plastic finite element method. The simulation results present that the blank used in the hydroforming process should undergo pre-bending and preforming process at first. An optimum process trial is proposed through the numerical simulation to select a suitable internal hydraulic pressure level, axial feeding displacement path and friction coefficient between workpiece and die wall. This work shows that the algorithm of the rigidplastic finite element program, HydroFORM-3D, can provide valuable information regarding the tube hydroforming process of an automobile lower arm and also dramatically improve the analysis and design capabilities on the hydroforming process. Through the FEM-based computer-aided engineering approach proposed in this study, the designer can improve the design efficiency and avoid expensive and time-consuming trial and error and extensive process design experience.
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Acknowledgements This work has been completed by the support of the ERC/NSDM, Pusan National University, Korea, and the authors thank for this support. Also the last author would like to acknowledge the support of the BK21 program.
[4]
[5] [6]
[7]
References [1] Dohmann F, Hartl CH. Tube hydroforming––research and practical application. J Mat Proc Technol 1997;71:174–86. [2] Schmoeckel D, Hielscher C. Metal forming of tubes and sheet with liquid and other flexible media. Ann CIRP 1999;48(2):497–513. [3] Liu J, Hahn D. Optimization of process and tool development for tube hydroforming using computer simulation. In:
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Innovations in Tube Hydroforming Technology International Conference, Michigan, ERC/NSM, 13–14 June, 2000. Marques JMB, Martins PAF. Three-dimensional finite element contact algorithm for metal forming. Int J Num Meth Eng 1990;30:1341–54. Kobayahsi S, Oh SI, Altan T. Metal forming and the finite element method. New York: Oxford University Press; 1989. Chen CC, Kobayashi S. New solutions to rigid plastic deformation problems using a matrix method. Trans ASME J Eng Ind 1978;95:865–73. Yang JB, Jeon BH, Oh SI. Tube bending technology of hydroforming process for an automotive part. J Mat Proc Technol 2001;111:175–81. Kim J, Lei LP, Kang BS. Preform design in hydroforming of automobile lower arm by FEM. In: IMCC2000 International Manufacturing Conference in China, Hong Kong, 16–17 August, 2000. Lei LP, Kim J, Kang BS. Analysis and design of hydroforming process for automobile real axle housing by FEM. Int J Machine Tools Manufact 2000;40:1691–708.
Computational approach to analysis and design of hydroforming process for an automobile lower arm Jeong Kim a, Sung-Jong Kang b, Beom-Soo Kang a
a,*
ERC/NSDM, Pusan National University, Pusan 609-735, South Korea b Catholic University of Taegu, Kyungbuk 712-702, South Korea Received 7 June 2001; accepted 15 March 2002
Abstract Tubular hydroforming has attracted increased attention in the automotive industry recently. In this study, a professional finite element program for analysis and design of tube hydroforming processes has been developed, called HydroFORM-3D, which is based on a rigid-plastic model. The friction calculation between die and workpiece has been dealt with carefully by introducing a new scheme in three-dimensional surface integration. With the developed program, HydroFORM-3D, the hydroforming process for an automobile lower arm is analyzed and designed. The manufacturing process for a lower arm consists of tube bending, preforming, and final hydroforming. To accomplish successful hydroforming process design, thorough investigation on proper combination of process parameters such as internal hydraulic pressure, axial feeding, and tool geometry is required. This paper includes the study on the influences of the forming conditions on the hydroforming of a lower arm by using simulation to predict strain and tube shape during bending, preforming, and final hydroforming processes. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Hydroforming process; Preform design; FEM; Lower arm
1. Introduction Due to several advantages of hydroforming process over conventional methods, it has become a new mainstream technology for the manufacturing of automotive structural components. Since tubular hydroforming has attracted increased attention in the automotive industry for making a wide variety of structural components such as sub-frames, engine cradles, and exhaust manifolds, the process design of the hydroforming process is required to improve the product quality and productivity. To manufacture a tube-hydroformed component, a circular tube is inserted into a suitable die, sealed on both sides by two horizontal cylinders, and subsequently
formed to conform to the shape of the die cavity by internal hydraulic pressure. Over the years a great deal of experience has been accumulated in this field mainly through the trial-and-error approach. But the sound design by ‘‘trial-and-error’’ can be very expensive and time consuming. On the other hand, the numerical approach of the finite element method in hydroforming can increase existing knowledge and reduce tool cost and lead-time by providing virtual tryout before tool construction and predicting the formability in advance.
2. Hydroforming of a lower arm *
Corresponding author. Tel.: +82-51-510-2310; fax: +82-51513-3760/3960. E-mail address: [email protected] (B.-S. Kang).
In many cases, the hydroformed components undergo multi-stage processes since the hydroforming frequently requires reduction in wear and scrap levels,
0045-7949/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 2 ) 0 0 0 8 1 - 0
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which results in cost saving and high surface quality. The process sequence for producing the lower arm by hydroforming involves the following steps: • pre-bending of the starting blank tube, • preforming of the pre-bent tube by stamping, • final sizing by hydroforming of the preformed tube. Prior to the hydroforming operation, the tube must be bent to the approximate centerline of the final part to enable the tube to be placed in the die cavity. After then, a preforming operation is carried out to the pre-bent tube. When hydroforming a pre-bent axial component, such as a lower arm, the maximum expansion limit in the pre-bent blank zone is lower than those straight axis components, because there is not enough material feeding in this area [1,2]. Then it is difficult to obtain a sound final product from pre-bent tube in one hydroforming operation, and a preforming process is needed. Preform design in hydroforming refers to the design of an intermediate geometry close to the final shape, and it derives the final product without defects and excessive loss of material. After the pre-bending process, the pre-bent tube has to undergo the preforming process before the hydroforming operation because the diameter of the tube is larger than the smallest dimension of the final product. For hydroforming process, selection of the tube blank size is a critical problem. If the diameter of initial tubular blank greatly exceeds the smallest component diameter, the cross section will be collapsed when the die closes. On the other hand, when the diameter of the blank is too small, it is possible that the component cannot be fully formed due to excessive circumferential elongation. This will cause bursting failure in the premature component. Therefore, a carefully-designed preforming process can contribute significantly to reduce production cost and it is a dominant factor for successful hydroforming [3,8]. In this study, the hydroforming process for an automobile lower arm is analyzed by using three-dimensional finite element program, HydroFORM-3D, based on a rigid-plastic model. The program is developed with modification of the previous rigid-plastic program, addition of three-dimensional die description for tube hydroforming, and suggestion of new frictional force calculation. The goal of this study is to evaluate the program by accomplishing proper design and control of processes for producing hydroformed lower arm component most economically. This study also describes the influences of forming conditions, such as the loading path of the hydraulic pressure, the axial feeding displacement, and the friction condition between tubular workpiece and die wall on the hydroforming process. The numerical simulations by using HydroFORM-3D are carried out on the bending, the preforming of stamping, and the hydroforming.
3. Theoretical background In many cases of hydroforming, the plastic strains are so large that the elastic strains are negligible by comparison. Thus the rigid-plastic finite element method is adopted in this study because of its effective algorithm for numerical simulation of the forming process. The general description of this numerical method can be found in the previous publications [4,5,7]. For the completion of the investigation, a brief summary of the approach based on infinitesimal deformation theory is included. Consider a body of volume V bounded by a surface S with the traction ti prescribed on a part of the surface SF and the velocity ui enforced on the remainder of the surface SU . If the inertia effects and body forces are assumed to be ignored, the deformation of the control volume V is characterized by the following relations: rij;j ¼ 0 on V rij nj ¼ ti on SF ui ¼ ui on SU
ð1Þ
where r is the Cauchy stress tensor and n is the unit outward normal to the workpiece. From the constitutive equation the ratios of the plastic strain rate are defined by e_ij ¼
of ð r; Y Þ _ k orij
ð2Þ
in which k_ is the plastic multiplier that can be calculated after the condition f ¼ 0, where f is the plastic potential, identified as the scalar function that defines the elastic limit surface and is described by the von Mises yield criterion f ð r; Y Þ ¼ r Y ¼ 0
ð3Þ
Y is the yield stress in simple tension, and depends on the evolution of the isotropic work-hardening, which is obeyed by the n-power law r ¼ Ken
ð4Þ
where K, n are plastic coefficient and strain-hardening exponent, respectively. The equivalent plastic strain is given by the following at time t Z t e ¼ e_ dt ð5Þ 0
The finite element formulation is to begin with a variational principle related to total potential energy. It requires that among admissible velocities ui that satisfies the conditions of compatibility and incompressibility as well as the velocity boundary conditions:
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Z
Z
Z K ti ui dS ¼ 0 ð6Þ ðe_v e_v Þ2 dV v v 2 SF qffiffi qffiffi where r ¼ 32ðr0ij r0ij Þ1=2 , e_ ¼ 23ðe_ij e_ij Þ1=2 , e_v ¼ e_ii , and r0ij , K are the deviatoric stress tensor and the penalty constant with a large positive value, respectively. The first order variation of the functional equation (6) can be written as Z Z Z ti dui dS ¼ 0 r de_ dV þ K e_v de_v dV ð7Þ dp ¼ p¼
re_ dV þ
v
v
SF
Eq. (7) is the basic equation for the finite element discretization and can be converted to non-linear algebraic equations by utilizing the discretization procedure. Here, the workpiece is discretized with eight node hexahedral elements for three-dimensional deformation. In general cases, the implicit procedure uses an automatic incremental strategy based on the success rate of a full Newton iterative solution method. For a linearized form of Eq. (7), the Taylor expansion near an assumed solution point u ¼ uði1Þ gives at the ði 1Þth increment, 2 op op ðiÞ þ Du ¼ 0 ð8Þ ouI u¼uði1Þ ouI ouJ u¼uði1Þ I ðiÞ
where DuI is the first-order correction of the velocity vector uði1Þ and the second term is the current tangent stiffness matrix. Once the solution of Eq. (8) for the velocity correction term is obtained, the assumed velocity uðiÞ is updated according to the following uðiÞ ¼ uði1Þ þ aDuðiÞ
ð9Þ
where a is deceleration coefficient 0 < a 6 1, which is used to avoid divergence during iterations. A small value a, say 0.1 or 0.2, is usually used at the beginning of iteration and then it is gradually increased towards 0.6 so as to reach a fast convergence rate. Two convergence criteria can be used for determining the termination of the iteration. The first convergence criterion is satisfied if the error norm of the velocity becomes very small, namely, kDuk=kuk 6 n
ð10Þ
where Euclidean vector norm is defined as kDuk ¼ ðuT uÞ1=2 , and n is a given tolerance, with recommended value of order 104 . The second convergence criterion is determined by measuring the norm of the force-residual, also called function norm, kop=ouk 6 d
ð11Þ
where d is a small value varying with the magnitude of yield stress. A converged solution can be found only when either of the convergence criteria, Eq. (10) or Eq. (11) is satisfied.
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4. Die and contact description in three-dimensional hydroforming For the analysis of the hydroforming process and the modeling of arbitrarily curved die surfaces encountered in actual industrial components, one of the most important and difficult problems is the treatment of contact between die surfaces and the deformed workpiece. Hydroforming processes are non-stationary and the geometry of the workpiece progressively changes. Therefore, to obtain reasonable and stable solutions, the contact between the nodal points and die surfaces must be carefully checked and modified progressively as well. Some discussions about the contact problem are presented and a simple but effective contact algorithm is proposed in this study. 4.1. Description of the die surface The description scheme of the die surface can have a significant effect on computational time. Moreover, since a discontinuous change in the sliding direction of nodes during forming analysis causes a computational difficulty in convergence, the die surface must be described accurately and smoothly to obtain a stable numerical solution for a large displacement and large deformation analysis of hydroforming processes. In this study, the die surface is described by a set of four-nodequadrilateral C 0 patches whose normal directions indicate the interior of the die, and compared with the larger deformation of the workpiece, the die is treated as a rigid body. Such a representation of the die geometry makes it easy to model. It could be seen next that the contact algorithm for this approach is very simple and efficient. 4.2. Contact analysis Typical contact analysis in current use can be classified into direct method, penalty method, and Lagrange multiplier method. In this study, the direct method is used for contact analysis. For every incremental step, the minimum time is calculated for the free node to contact with the die or for the contacted node to touch the other die. If this minimum time is smaller than the desired incremental time, let the incremental time during this step be equal to the minimum time and these nodes are declared to be in contact with their corresponding die. This scheme will avoid the node entering into the die. For the contacted nodes, it is important to have a correct evaluation of the die in normal direction as well as a correct local frame basis definition for an accurate setting of the nodal boundary condition. The contact search includes a global search and a local search. The global search intends to find candidate die patches that may be in contact with the node and its main objective is
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to increase the speed of the contact analysis. The local search intends to select one patch for contact checking with the node, among all candidate die patches. Every die surface patch has limited boundaries in the x, y, and z directions as shown in Fig. 1. By using the boundary concept, the global search finds the candidate die patches which may be in contact with the node. After the global search, one patch in contact with the node must be selected among all of the candidate patches found in the global search. The local search is carried out to finish this using the area comparison scheme shown in Fig. 2. In the figure, the quadrilateral ABCD is one of the candidate die patches and O is the location of the node projection on this patch. The area comparison scheme is expressed as follows:
Here, the Ap is the area of the patch, A1 ; A2 ; A3 , and A4 are the areas of the triangles which consist of the die patch described with four nodes. g is a small positive tolerance for calculation, usually taken as about 0.001. In numerical programming, to eliminate the equations expressing geometrical relationships for the nodes in contact with dies, the coordinate system is transformed from global to local at the nodes. After the local search, a reference system of local axes is set with respect to the contact die patch. The normal direction vector of the contact patch defines one of the local coordinate axes, while the other two are arbitrary and the only condition required is that they must be orthogonal on the rectangular element. 4.3. Friction effects
jA1 þ A2 þ A3 þ A4 Ap j 6g : Ap
contact
jA1 þ A2 þ A3 þ A4 Ap j >g: Ap
not contact
ð12aÞ
ð12bÞ
In this approach, Coulomb condition for friction along the die-workpiece interface is considered for industrial hydroforming operations. Along the die-workpiece interface there exists a neutral point where the velocity of the deforming material relative to the die becomes zero. In order to evaluate the stiffness at a neutral point, where the sudden change of direction of the frictional force would make the simulation unstable, the approximation of the velocity-dependent frictional force vector is given as the following [9]: ur fr ffi lN qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 urs þ u2rt þ u20
ð13Þ
where N is the normal force, l the frictional factor, the relative velocity vector ur is represented by urs and urt components in the direction of the local coordinate axis s and t, which are defined over the surface of the die, and u0 is a small value for 102 –103 . Then the friction term can be added to Eq. (7), Fig. 1. The schematic representation of boundary concept for the global search.
ur dpf ¼ fr dur ¼ lN qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dur 2 urs þ u2rt þ u20
ð14Þ
Then the derivatives of Eq. (4) can be evaluated as opf ursi ffi ¼ lNi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oursi 2 u þ u2 þ u2
ð15aÞ
opf urti ffi ¼ lNi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ourti u2 þ u2 þ u2
ð15bÞ
rsi
rsi
0
rti
rti
0
0
1
u2rsi o pf 1 B C ¼ lNi @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 A 2 oursi 2 2 2 2 2 2 ursi þ urti þ u0 ursi þ urti þ u0 2
Fig. 2. The schematic representation of area comparison scheme for the local search.
ð15cÞ
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0
1299
1
u2rti o2 pf 1 C B q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ lN @ i q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 A ou2rti 2 2 2 u2rsi þ u2rti þ u20 ursi þ urti þ u0 ð15dÞ o2 pf ursi urti ¼ lNi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 oursi ourti u2rsi þ u2rti þ u20
ð15eÞ
For any node in contact with the die surface, the frictional force is calculated. 4.4. The implementation of hydraulic pressure According to the governing equation, the pressure force applied to the nodal force can be expressed as Z opSF ¼ p/i dS ð16Þ ovn SF where, p is the pressure force during the hydroforming process, and /i is the shape function of the ith node. The pressure force does not have effect on the stiffness matrix. The integration of the hydraulic pressure term in Eq. (16) is carried out by 5 5 Gauss quadrature on the pressure surface. The direction of the pressure force coincides with the normal direction of the element surface where the hydraulic pressure is applied.
Fig. 3. (a) A conventional stamping-welded lower arm assembly and (b) solid model to be hydroformed.
5. Preform analysis of hydroforming of a lower arm 5.1. Problem description A lower arm is one of the suspension parts, which is placed at the front of passenger room and supports a cross member and a knuckle component. It plays important roles not only in performing the rotational center of a tire which affects wheel alignment but also in reducing the vibration while driving. When the automotive component, a lower arm, is produced by hydroforming process, it can substitute for the conventional stampedwelding lower arm with the same function, but with high rigidity and low cost. In Fig. 3, the assembly of conventional stamped-welded lower arm and the one from hydroforming are shown respectively. Fig. 4 shows the geometry and several different cross sections along the axis of the hydroformed lower arm. The tubular blank is assumed to be a rigid-plastic material obeying the n-power law in Eq. (4), of which stress–strain relation is listed in Table 1. The Coulomb friction model is used and the material is assumed as isotropic. The friction coefficient used in preforming operation was set at 0.1. The initial outer diameter of the tube is 63.5 mm and the initial thickness is 2.6 mm. The original length before the initial pre-bending is 600 mm.
Fig. 4. Schematic view of the lower arm.
Due to the geometric symmetry of the final part, only a quarter of the tube is molded using brick elements.
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Table 1 Tensile properties of the material used STKM-11A n
K-value ð r ¼ Ke Þ (MPa) Work-hardening exponent, n Yield stress (MPa)
468.3 0.206 215.1
5.2. Analysis results of pre-bending process by FEM simulation To produce a lower arm by hydroforming process, first a pre-bending process is carried out so that the tube fits into the die cavity. In certain cases where the overall part geometry is not relatively complex, the bending may be achieved as the final die is closed. In most cases a bending process is done separately. During the bending operation, in general, the tube material thins at the outside of the bend and thickens at the inside. Prebending simulation of the tube is performed using a conventional rotary draw bending machine and the bending angle is 73. Rotary draw bending is the most popular, cost-effective bending method for thin walled tubes. The pre-bending model using this method needs a tubular blank and a die set which includes a bent die, a pressure die, a clamp die and a wiper die as shown in Fig. 5. A wiper die is used to avoid the wrinkle [6]. The tube bending process has been simulated using the rigidplastic FEM program, HydroFORM-3D. Fig. 6 is the deformed shape of the tube after pre-bending. From the results, the maximum effective plastic strain is 0.226 and maximum effective stress is 368.0 MPa. The minimum thickness is 2.41 mm in tensile area while the original thickness is 2.6 mm. This prediction of 7.3% thinning is close to the actual measurement of 10.0% thinning.
Fig. 5. Initial finite element model of the bending tool.
Fig. 6. Deformed shape after pre-bending operation.
5.3. Analysis results of preforming process by FEM simulation After the pre-bending process, the pre-bent tubes have to undergo a preforming process before the hydroforming operation because the diameter of the tube is larger than the smallest dimension of the final product. The preforming simulation model is composed of a prebent tube, which results from previous pre-bending process, and a split die for forming to reduce the outer diameter of tube to the height of final product. Fig. 7 is the schematic view of the lower stamping and hydroforming die created from CAD software. Fig. 8 shows a pre-bent tubular blank and a set of upper and lower stamping dies. To get the final preformed shape, a stamping process is required. Since the minimum dimension of the lower arm is 15.2 mm as shown in Fig. 4,
Fig. 7. Schematic view of the lower stamping and hydroforming die.
J. Kim et al. / Computers and Structures 80 (2002) 1295–1304
Fig. 8. Finite element model of stamping die and pre-bent blank.
1301
Fig. 10. Finite element model of hydroforming process.
6. Analysis of a lower arm hydroforming process the stroke of 24.15 mm is needed for stamping process when a tubular blank with the outer diameter of 63.5 mm is used. At first the pre-bent tubular blank is inserted to the hydroforming dies and then the stamping process is performed as the dies are closed. Fig. 9 shows the deformed geometry and several cross sections after stamping operation. The cross sections are not circular any longer unlike those ones after pre-bending. The maximum effective plastic strain is 0.414 and the maximum effective stress 379.3 MPa. The minimum thickness is 2.38 mm, which implies that the maximum thinning is increased slightly by 9.2% from 2.6 mm original tube thickness.
Fig. 9. Deformed shape and cross sections after preforming operation.
The hydroforming simulation model is composed of the preformed tube, which is from the result of preforming operation, and the same dies for stamping process. Fig. 10 shows the preformed blank and a set of upper, and lower dies, and an axial feeding ram. 6.1. Influence of internal hydraulic pressure on formability To evaluate the formability of the lower arm hydroforming with respect to internal hydraulic pressure, numerical simulation with three loading paths shown in Fig. 11 is performed. The maximum axial feeding and the friction coefficient between the preformed tube and the die wall was set at 50 mm and 0.05 respectively while
Fig. 11. Description of loading paths.
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Fig. 13. The variations of (a) the die clamping forces and (b) the axial feeding forces during hydroforming process.
6.2. Influence of axial feeding on formability
Fig. 12. Deformed shape of the workpiece for different loading paths: (a) Path 1, (b) Path 2.
the internal hydraulic pressures varied. From the analysis, the simulation results show that for Paths 1 and 2 the material could not fill the inner cavity of the die shown in Fig. 12. Fig. 12(a) shows the deformed shape at the pressure level B on Path 1 shown in Figs. 11 and 12(b) at the pressure level A on Path 2. From these results, to get the completely filled hydroformed part, the internal hydraulic pressure should be started at over 30 MPa and the final hydraulic pressure should be higher than 40 MPa. The numerical analysis predicts a minimum thickness of 1.73 mm in the final part (33.5% thinning) at the outside of the bend under the loading Path 3. Fig. 13 shows the variations of clamping forces and axial feeding forces during the hydroforming process. For Path 3, a clamping force of 3403 kN and the axial force of 162 kN for 50 mm axial feeding were required to complete soundly hydroforming process.
To evaluate the formability of the lower arm hydroforming with respect to axial feeding displacement, hydroforming simulations with the three axial feeding cases, no feeding, 50 and 100 mm, are carried out. The load condition was Path 2 shown in Fig. 11, and the friction coefficient between the preformed tube and the die wall was fixed at 0.05 while the axial feeding varied. It is noted that the case of no axial feeding leads to a high possibility of bursting failure at the outside of the bend due to the maximum thinning. As the axial feeding displacement increases, the formability becomes higher, when the three cases are considered. This is because more material is supplied into the potential bursting failure site by increasing the axial feeding. But excessive material feeding not only causes the weight increment on the final part but also may lead to buckling or wrinkling failure. Thus, the axial feeding must be appropriate to manufacture the required sound part. The maximum thinning is predicted to 42.7% (the minimum thickness of 1.49 mm) at the outside of the bend with no feeding condition. Table 2 shows the maximum clamping forces and maximum axial feeding forces for these cases. For
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Table 2 Maximum clamping forces and axial feeding forces for different axial feeding displacements Axial feeding displacements
No feeding
50 mm
100 mm
Clamping force of hydroforming press [kN] Axial feeding force [kN]
3512
3403
3520
N/A
162
307
the axial feeding displacement of 100 mm, 3520 kN for clamping and 307 kN for axial feeding were needed. 6.3. Influence of friction coefficient on formability To evaluate the formability of the lower arm hydroforming with respect to the friction coefficient between the preformed tube and the die wall, hydroforming process with four different friction coefficient values, 0.0, 0.05, 0.1, and 0.2, is analyzed. The loading condition was Path 2 shown in Fig. 11 and the axial feeding was fed up to 50 mm for each case. From simulation results, the formability increases while the friction coefficient decreases. It implies that the better lubrication condition makes the material fed into possible failure region easily. And for the friction condition of over 0.1, the high possibility of local wrinkling or bursting is predicted. The whole deformed product and some cross sections predicted by the numerical simulation with friction coefficient of 0.05 is shown in Fig. 14. The cross section shapes are formed well to conform to the die cavity. The maximum thinning is predicted 52.3% (the minimum thickness of 1.24 mm) at the outside of the bend under the friction condition of 0.2. For the friction condition of 0.2, the compression force reaches the maximum value of 211 kN. In that case, the maximum die clamping force for sealing needs 3158 kN. Fig. 15 shows the distribution of the equivalent stress predicted by the numerical simulation with friction coefficient of 0.05. The maximum effective plastic
Fig. 14. Cross sections after finishing hydroforming process.
Fig. 15. Distribution of the equivalent stress of the hydroformed lower arm.
strain is 0.738 and the maximum effective stress 475.2 MPa.
7. Conclusion For analysis and design of tube hydroforming, a FEM program of rigid-plastic approach is presented. A simple concept for die contact and non-contact to workpiece and a friction calculation algorithm in tube hydroforming are implemented newly in the developed program, HydroFORM-3D. To manufacture a sound automobile component, lower arm, a systematic approach on the analysis and design of the hydroforming process was suggested by using HydroFORM-3D based on the rigid-plastic finite element method. The simulation results present that the blank used in the hydroforming process should undergo pre-bending and preforming process at first. An optimum process trial is proposed through the numerical simulation to select a suitable internal hydraulic pressure level, axial feeding displacement path and friction coefficient between workpiece and die wall. This work shows that the algorithm of the rigidplastic finite element program, HydroFORM-3D, can provide valuable information regarding the tube hydroforming process of an automobile lower arm and also dramatically improve the analysis and design capabilities on the hydroforming process. Through the FEM-based computer-aided engineering approach proposed in this study, the designer can improve the design efficiency and avoid expensive and time-consuming trial and error and extensive process design experience.
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Acknowledgements This work has been completed by the support of the ERC/NSDM, Pusan National University, Korea, and the authors thank for this support. Also the last author would like to acknowledge the support of the BK21 program.
[4]
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[7]
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