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Finite element analysis of hydro-forming process of a toroidal shell
International Journal of Machine Tools & Manufacture 39 (1999) 1439–1450
Finite element analysis of hydro-forming process of a toroidal shell Shijian Yuana,*, Z.R. Wanga, Qing Heb a
Division of Metal Forming, #435, Harbin Institute of Technology, Harbin 150006, People’s Republic of China b Engineering Technology Associates, Inc., 1133 E. Maple Road, Suite 200, Troy, MI 48038, USA Received 1 September 1998
Abstract A hydro-forming technology of toroidal shells was proposed for manufacturing large diameter elbow pipes. The deformation process was simulated using an explicit dynamic finite element code ETA/DYNAFORM. The stress/strain distribution during hydro-forming and thinning in the deformed shell were analyzed, and the reason for observed wrinkling was discussed. The deformed shape of the toroidal shell and wrinkle configuration were compared with those from experiment. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Hydro-forming; Hydro-bulging; Elbow pipe; Toroidal shell; Finite element analysis
1. Introduction The integrally hydro-bulging forming (IHBF) technology was proposed in 1985 to overcome the difficulties in manufacturing spherical vessels with the conventional method, in which petals are formed using dies and a press and then assembled and welded into a spherical shell [1]. Since then, numerous spherical vessels of mild steel, stainless steel and aluminum have been commercially manufactured using IHBF, for water storage tanks, ornamental objects and spherical digesters, amongst which the maximum diameter and wall thickness have reached 9.4 m and 24 mm, respectively [2]. In 1993, IHBF was successfully extended to form ellipsoidal shells [3]. Elbow pipes are one of the key parts in a pipe system used in petrochemical, chemical, mechanical industries, etc., to connect two pipes in different directions and adjust thermal stress caused * Corresponding author. 0890-6955/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 9 8 ) 0 0 0 9 6 - 0
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by temperature change. Conventional technologies, such as the wind- bending method and the push-bending method, can only be applied to the manufacture of medium and small size standard series elbow pipes [4]. The maximum diameter of elbow pipes obtained by these conventional methods is not more than 1 m. With the development of modern industry, pipe systems have a tendency toward larger diameters. Therefore, it is necessary to develop a new method for manufacturing large diameter elbow pipes. Based on IHBF technology of spherical and ellipsoidal shells, a new method for manufacturing large elbow pipes was proposed by which a toroidal shell is first hydro-formed and then the hydro-formed toroidal shell is cut into several elbow pipes with the desired angle. Some experimental research has been conducted indicating the feasibility of this process [5]. This paper further discusses the feasibility of IHBF technology and the ability of FE simulation to analyze and predict this forming process.
2. Hydro-forming process of a toroidal shell The main process of manufacturing elbow pipes using hydro-forming of a toroidal shell can be briefly described as follows: a toroidal shell with a polygonal cross-section, afterwards called a polyhedral toroidal shell, is first assembled and welded, then the shell is filled with liquid medium (usually water) and then pressurized by means of a pump; with increasing internal pressure, a plastic deformation occurs, and the polygonal cross-section gradually becomes a circular one; as a result, a toroidal shell with circular cross-section is finally obtained, as shown in Fig. 1. According to requirements, the deformed toroidal shell can be cut into four 90° elbow pipes, six 60° elbow pipes or other angle elbow pipes. Compared with conventional methods, this method has advantages such as low production cost, short production cycle and the ability to manufacture large diameter elbow pipes on site without pipes, special equipment and dies.
Fig. 1. Hydroforming process of a toroidal shell. (1) pressure source; (2) one way valve; (3) pressure meter; (4) pipe; (5) toroidal shell.
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Fig. 2. Structure of the toroidal shell used in analysis.
3. Finite element model The structure of the polyhedral toroidal shell prior to hydro-forming used in the experiment [5] is shown in Fig. 2. The medium radius of the elbow pipe R0 ⫽ 150 mm, the radius of the pipe r0 ⫽ 50 mm, and the wall thickness t ⫽ 1.5 mm. The cross-section of the polyhedral toroidal shell is a hexagon. The polyhedral toroidal shell was composed of four kinds of sub-shells: subshell A—a cylindrical shell subjected to external pressure; sub-shell B—a conical shell subjected to external pressure; sub-shell C—a conical shell subjected to internal shell; and sub-shell D—a cylindrical shell subjected to internal pressure. According to structural symmetry, half of a quarter of the polyhedral toroidal shell was taken as a finite element model. The shell material is Q235A—a mild steel commonly used in China for manufacturing pressure vessels and pipes. Its mechanical properties are shown in Table 1. The code used for this finite element analysis was eat/DYNAFORM. It was specifically developed for sheet metal forming simulation and is an integrated pre and post processor and finite element solver, LS/DYNA. LS/DYNA is an explicit, dynamic finite element code which has been applied in many sheet metal forming problems. Examples include the stamping of automobile components [6], conventional deep drawing [7], tube hydro-forming [8] and hydro-forming of spherical vessels [9]. 4. Results and discussion 4.1. Variation of cross-sectional profile Because the toroidal shell is an axisymmetrical structure, a variation of any section can represent the deformation tendency of the whole shell. Thus, only the deformation of a cross-section Table 1 Mechanical properties of the shell material Q235A Elastic modulus E (MPa)
Poisson’s ratio
Strength coefficient K in ⫽ K⑀n (MPa)
Strain hardening exponent n
2.08 ⫻ 105
0.28
648.0
0.20
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is discussed here. Fig. 3 is variation of a cross-sectional profile during the hydro-forming process. It can be seen from the figure that the cross-section of the polyhedral toroidal shell is deformed gradually from a hexagon to a circle with increasing internal pressure, which is consistent with the results of the experiment. A slight difference between the FEA simulation and the experiment is that the weld seams joining sub-shell A(C) and sub-shell B(D) move inwardly during simulation, whereas they do not move in experiment. The main reason for this is that during the experiment the real strength and thickness of the weld seams are higher than that of the sheet metal so that they have strong stiffness and almost do not move; while in simulation, the strength and thickness of weld seams are set to be equal to that of sheet metal so that movement and larger deformation occur under the effect of bending moment. For the weld seam between subshell A and sub-shell B, the total values of the radial movement are 1.88 mm; for the weld seam between sub-shell C and sub-shell D, the total value is 1.65 mm. Fig. 4 is a comparison of displacements at the central point of various sub-shells between FEA simulation and experiment. The displacements calculated in the FEA simulation at the central points of the conical shells (sub-shell B and sub-shell C) are in good agreement with the experimental data. If the inward movement values of the weld seams are considered, the displacements at the central points of the cylindrical shells (sub-shell A and sub-shell D) are also in good agreement with the experimental data.
Fig. 3. Profile variation of section. (a) Initial stage, p ⫽ 2.0 MPa; (b) middle stage, p ⫽ 5.0 MPa; (c) final stage, p ⫽ 8.0 MPa.
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Fig. 4. Comparison of displacement between FE simulation and experiment. (a) Central point of sub-shell A; (b) central point of sub-shell B; (c) central point of sub-shell C; (d) central point of sub-shell D.
4.2. Stress and strain distribution Distributions of the global stress x and y in initial and final hydro-forming stage are shown in Figs. 5 and 6, respectively. It is necessary to point out that the stress and strain components used in post processing in the present version of DYNAFORM are in the global Cartesian coordinate system. However, it is usually more convenient to use the cylindrical coordinate system for representing and analyzing stresses and strains in a rotation shell. For an axisymmetric rotation shell such as a toroidal shell, the stress and strain components in meridional and circumferential direction are the same in every cross-section. But in the global Cartesian coordinate system, they look different in every cross-section due to the different projection angle. In order to make clear this problem, the global stress and strain components in the area near the X–Z plane are used in the following analysis. The global stress/strain in the X-axis direction corresponds the meridional
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Fig. 5. Distribution of global stress x (unit: MPa). (a) Initial stage, p ⫽ 2.0 MPa; (b) final stage, p ⫽ 8.0 MPa.
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Fig. 6. Distribution of global stress y (unit: MPa). (a) Initial stage, p ⫽ 2.0 MPa; (b) final stage, p ⫽ 8.0 MPa.
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stress/strain, and the global stress/strain in the Y-axis direction corresponds the circumferential stress/strain in the cylindrical coordinate system. It can be seen from Figs. 5 and 6 that the meridional stress (global stress x near the X–Z plane) on various sub-shells is in tensile state during the entire hydro-forming process; while the circumferential stress (global stress y near the X–Z plane) is in compressive state on sub-shell A and sub-shell B and in the area adjacent to the weld seam between sub-shell C and sub-shell D at the initial hydro-forming stage. The compressive circumferential stress becomes tensile stress at the final hydro-forming stage as the polyhedral toroidal shell is finally formed into an ideal toroidal shell. This compressive circumferential stress is liable to result in wrinkling in sub-shell A and sub-shell B which are subjected to external pressure, the details will be discussed later. Fig. 7 shows the strain distribution at the final hydro-forming stage. From this figure we can see that the circumferential strain in the central area of sub-shell A and sub-shell B and the area near the weld seam between sub-shell C and sub-shell D is still in compressive state; the meridional strain is in tensile state in most areas of the shell except the weld seam between sub-shell B and sub-shell C where the meridional strain is compressive due to the effect of bending. 4.3. Thickness distribution Fig. 8 is the thickness distribution in the final deformed toroidal shell. It can be seen from Fig. 8 that the maximum thinning occurs at the weld seams. This phenomenon is not consistent with the experiment in which the weld seams have almost no thinning because their strength is higher than that of the sheet metal and the thickness of weld seams is larger than that of the sheet metal. As discussed earlier, in simulation the strength and thickness of weld seams are set to be equal to that of sheet metal so that they move and large plastic deformation occurs under the effect of bending moment. This plastic deformation results in larger thinning in the weld seam areas than other areas of the shell. It is reasonable to exclude thinning occurring at the weld seams in analysis. Thus, there is a maximum thinning in the central area of sub-shell C and sub-shell D. The value of the maximum thinning rate or the difference between the maximum thickness and the minimum thickness is so small that the thickness can be considered to be uniform in the deformed toroidal shell. Moreover, the thickness near the central line of sub-shell A is increased little due to the effect of the compressive circumferential stress. 4.4. Wrinkling Some tiny wrinkles at sub-shell A and sub-shell B were found in the experiment, as shown in Fig. 9(a). The wrinkles obtained in the simulation are shown in Fig. 9(b), which match very well with ones shown in the experiment. The occurrence of the wrinkles is caused by the compressive circumferential stress existing on sub-shell A and sub-shell B, as mentioned above. For a cylindrical shell being subjected to external pressure, the circumferential stress is expressed by
⫽ ⫺ pr/t where p is the external stress, r is the radius of the cylindrical shell, and t is the thickness of the cylindrical shell. When the value of the circumferential stress reaches a critical stress value
S. Yuan et al. / International Journal of Machine Tools & Manufacture 39 (1999) 1439–1450
Fig. 7. Strain distribution at final stage. (a) Global strain ⑀x; (b) global strain ⑀y.
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Fig. 8.
Thickness distribution in the deformed shell (unit: mm).
Cr, wrinkling occurs. This is consistent to a conical shell being subjected to external pressure. Although tiny wrinkles would not have serious influence on the use of the elbow pipe in some cases, it is necessary to do further research to avoid occurrence of the wrinkles on sub-shells being subjected to external pressure by redesigning the structure of the polyhedral toroidal shell prior to hydro-forming and/or adapting other technological measures.
5. Conclusions The hydro-forming process of a polyhedral toroidal shell was simulated by the explicit dynamic finite element code eta/DYNAFORM. The results obtained from this FEA simulation, especially the deformed shape and wrinkle distribution, have a good agreement with those from the experiment. The thickness of the deformed toroidal shell is very even. FEA results further show that the method of manufacturing elbow pipes by hydro-forming toroidal shells is feasible. Wrinkles occur because of the compressive circumferential stress existing in the sub-shells being subjected to external pressure, which can be prevented by redesigning the structure of the polyhedral toroidal shell prior to hydro-forming and/or adapting other technological measures. Further research into this problem will be done in the near future.
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Fig. 9. Comparison of wrinkles observed in FE simulation and experiment. (a) Experiment; (b) simulation.
References [1] Z.R. Wang, T. Wang, D.C. Kang, The technology of the hydro-bulging of whole spherical vessels and experimental analysis, Journal of Mechanical Working Technology 18 (1989) 85–94. [2] S.J. Yuan, Z.R. Wang, Safety analysis of 200 m3 PLG tank manufactured by the dieless hydrobulging technology, Journal of Materials Processing Technology 70 (1997) 215–219. [3] ZENG Yuansong, S.J. Yuan, Z.R. Wang, Research on the hydrobulge forming of ellipsoidal shells, Journal of Materials Processing Technology 72 (1997) 28–31. [4] Lu Xiaoyang, Xu Binye, Forming process analysis of medium frequency induction heated pushing bending of tubes, Proceedings of 2nd Pacific-Asia Conference on Engineering Plasticity and Application, 1994, pp. 647–652.
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[5] S.J. Yuan, Z.R. Xu Zhang, HAIWANG Wang, The integrally hydro-forming process of pipe elbows, International Journal of Pressure Vessel & Piping 75 (1998) 7–9. [6] R.G. Whirly, B.E. Englmann, R.W. Logan, Some aspects of sheet forming simulation using explicit finite element technique, Numerical Methods for Simulation of Industrial Metal Forming Processes, ASME, 1992, CEDVol.5/AMD, Vol. 156, pp. 77–83. [7] E. Harpell, C. Lamontagne, M.J. Worswick, P. Martin, M. Finn, C. Galbrainth, Deep drawing and stretching of AL 5754 sheet: LS-DYNA3D simulations and validation, Proceedings of the 3rd International Conference Mumisheet’ 96, Dearborn, Michigan, September 29–October 3, 1996, pp. 386–389. [8] H.J. Kim, B.H. Jcon, H.Y. Kim, J.J. Kim, Finite element analysis of the liquid bulge forming process, Advanced Technology of Plasticity 1993—Proceedings of 4th International Conference on Technology of Plasticity, September 1993, Beijing, pp. 545–550. [9] S.H. Zhang, J. Danckert, S.J. Yuan, Z.R. Wang, W.J. Hang, Spherical and spherical steel structure products made by using integral hydro-bulge forming technology, Journal of Construction Steel Research 4A (1998) 1–3.
Finite element analysis of hydro-forming process of a toroidal shell Shijian Yuana,*, Z.R. Wanga, Qing Heb a
Division of Metal Forming, #435, Harbin Institute of Technology, Harbin 150006, People’s Republic of China b Engineering Technology Associates, Inc., 1133 E. Maple Road, Suite 200, Troy, MI 48038, USA Received 1 September 1998
Abstract A hydro-forming technology of toroidal shells was proposed for manufacturing large diameter elbow pipes. The deformation process was simulated using an explicit dynamic finite element code ETA/DYNAFORM. The stress/strain distribution during hydro-forming and thinning in the deformed shell were analyzed, and the reason for observed wrinkling was discussed. The deformed shape of the toroidal shell and wrinkle configuration were compared with those from experiment. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Hydro-forming; Hydro-bulging; Elbow pipe; Toroidal shell; Finite element analysis
1. Introduction The integrally hydro-bulging forming (IHBF) technology was proposed in 1985 to overcome the difficulties in manufacturing spherical vessels with the conventional method, in which petals are formed using dies and a press and then assembled and welded into a spherical shell [1]. Since then, numerous spherical vessels of mild steel, stainless steel and aluminum have been commercially manufactured using IHBF, for water storage tanks, ornamental objects and spherical digesters, amongst which the maximum diameter and wall thickness have reached 9.4 m and 24 mm, respectively [2]. In 1993, IHBF was successfully extended to form ellipsoidal shells [3]. Elbow pipes are one of the key parts in a pipe system used in petrochemical, chemical, mechanical industries, etc., to connect two pipes in different directions and adjust thermal stress caused * Corresponding author. 0890-6955/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 9 8 ) 0 0 0 9 6 - 0
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by temperature change. Conventional technologies, such as the wind- bending method and the push-bending method, can only be applied to the manufacture of medium and small size standard series elbow pipes [4]. The maximum diameter of elbow pipes obtained by these conventional methods is not more than 1 m. With the development of modern industry, pipe systems have a tendency toward larger diameters. Therefore, it is necessary to develop a new method for manufacturing large diameter elbow pipes. Based on IHBF technology of spherical and ellipsoidal shells, a new method for manufacturing large elbow pipes was proposed by which a toroidal shell is first hydro-formed and then the hydro-formed toroidal shell is cut into several elbow pipes with the desired angle. Some experimental research has been conducted indicating the feasibility of this process [5]. This paper further discusses the feasibility of IHBF technology and the ability of FE simulation to analyze and predict this forming process.
2. Hydro-forming process of a toroidal shell The main process of manufacturing elbow pipes using hydro-forming of a toroidal shell can be briefly described as follows: a toroidal shell with a polygonal cross-section, afterwards called a polyhedral toroidal shell, is first assembled and welded, then the shell is filled with liquid medium (usually water) and then pressurized by means of a pump; with increasing internal pressure, a plastic deformation occurs, and the polygonal cross-section gradually becomes a circular one; as a result, a toroidal shell with circular cross-section is finally obtained, as shown in Fig. 1. According to requirements, the deformed toroidal shell can be cut into four 90° elbow pipes, six 60° elbow pipes or other angle elbow pipes. Compared with conventional methods, this method has advantages such as low production cost, short production cycle and the ability to manufacture large diameter elbow pipes on site without pipes, special equipment and dies.
Fig. 1. Hydroforming process of a toroidal shell. (1) pressure source; (2) one way valve; (3) pressure meter; (4) pipe; (5) toroidal shell.
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Fig. 2. Structure of the toroidal shell used in analysis.
3. Finite element model The structure of the polyhedral toroidal shell prior to hydro-forming used in the experiment [5] is shown in Fig. 2. The medium radius of the elbow pipe R0 ⫽ 150 mm, the radius of the pipe r0 ⫽ 50 mm, and the wall thickness t ⫽ 1.5 mm. The cross-section of the polyhedral toroidal shell is a hexagon. The polyhedral toroidal shell was composed of four kinds of sub-shells: subshell A—a cylindrical shell subjected to external pressure; sub-shell B—a conical shell subjected to external pressure; sub-shell C—a conical shell subjected to internal shell; and sub-shell D—a cylindrical shell subjected to internal pressure. According to structural symmetry, half of a quarter of the polyhedral toroidal shell was taken as a finite element model. The shell material is Q235A—a mild steel commonly used in China for manufacturing pressure vessels and pipes. Its mechanical properties are shown in Table 1. The code used for this finite element analysis was eat/DYNAFORM. It was specifically developed for sheet metal forming simulation and is an integrated pre and post processor and finite element solver, LS/DYNA. LS/DYNA is an explicit, dynamic finite element code which has been applied in many sheet metal forming problems. Examples include the stamping of automobile components [6], conventional deep drawing [7], tube hydro-forming [8] and hydro-forming of spherical vessels [9]. 4. Results and discussion 4.1. Variation of cross-sectional profile Because the toroidal shell is an axisymmetrical structure, a variation of any section can represent the deformation tendency of the whole shell. Thus, only the deformation of a cross-section Table 1 Mechanical properties of the shell material Q235A Elastic modulus E (MPa)
Poisson’s ratio
Strength coefficient K in ⫽ K⑀n (MPa)
Strain hardening exponent n
2.08 ⫻ 105
0.28
648.0
0.20
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is discussed here. Fig. 3 is variation of a cross-sectional profile during the hydro-forming process. It can be seen from the figure that the cross-section of the polyhedral toroidal shell is deformed gradually from a hexagon to a circle with increasing internal pressure, which is consistent with the results of the experiment. A slight difference between the FEA simulation and the experiment is that the weld seams joining sub-shell A(C) and sub-shell B(D) move inwardly during simulation, whereas they do not move in experiment. The main reason for this is that during the experiment the real strength and thickness of the weld seams are higher than that of the sheet metal so that they have strong stiffness and almost do not move; while in simulation, the strength and thickness of weld seams are set to be equal to that of sheet metal so that movement and larger deformation occur under the effect of bending moment. For the weld seam between subshell A and sub-shell B, the total values of the radial movement are 1.88 mm; for the weld seam between sub-shell C and sub-shell D, the total value is 1.65 mm. Fig. 4 is a comparison of displacements at the central point of various sub-shells between FEA simulation and experiment. The displacements calculated in the FEA simulation at the central points of the conical shells (sub-shell B and sub-shell C) are in good agreement with the experimental data. If the inward movement values of the weld seams are considered, the displacements at the central points of the cylindrical shells (sub-shell A and sub-shell D) are also in good agreement with the experimental data.
Fig. 3. Profile variation of section. (a) Initial stage, p ⫽ 2.0 MPa; (b) middle stage, p ⫽ 5.0 MPa; (c) final stage, p ⫽ 8.0 MPa.
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Fig. 4. Comparison of displacement between FE simulation and experiment. (a) Central point of sub-shell A; (b) central point of sub-shell B; (c) central point of sub-shell C; (d) central point of sub-shell D.
4.2. Stress and strain distribution Distributions of the global stress x and y in initial and final hydro-forming stage are shown in Figs. 5 and 6, respectively. It is necessary to point out that the stress and strain components used in post processing in the present version of DYNAFORM are in the global Cartesian coordinate system. However, it is usually more convenient to use the cylindrical coordinate system for representing and analyzing stresses and strains in a rotation shell. For an axisymmetric rotation shell such as a toroidal shell, the stress and strain components in meridional and circumferential direction are the same in every cross-section. But in the global Cartesian coordinate system, they look different in every cross-section due to the different projection angle. In order to make clear this problem, the global stress and strain components in the area near the X–Z plane are used in the following analysis. The global stress/strain in the X-axis direction corresponds the meridional
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Fig. 5. Distribution of global stress x (unit: MPa). (a) Initial stage, p ⫽ 2.0 MPa; (b) final stage, p ⫽ 8.0 MPa.
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Fig. 6. Distribution of global stress y (unit: MPa). (a) Initial stage, p ⫽ 2.0 MPa; (b) final stage, p ⫽ 8.0 MPa.
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stress/strain, and the global stress/strain in the Y-axis direction corresponds the circumferential stress/strain in the cylindrical coordinate system. It can be seen from Figs. 5 and 6 that the meridional stress (global stress x near the X–Z plane) on various sub-shells is in tensile state during the entire hydro-forming process; while the circumferential stress (global stress y near the X–Z plane) is in compressive state on sub-shell A and sub-shell B and in the area adjacent to the weld seam between sub-shell C and sub-shell D at the initial hydro-forming stage. The compressive circumferential stress becomes tensile stress at the final hydro-forming stage as the polyhedral toroidal shell is finally formed into an ideal toroidal shell. This compressive circumferential stress is liable to result in wrinkling in sub-shell A and sub-shell B which are subjected to external pressure, the details will be discussed later. Fig. 7 shows the strain distribution at the final hydro-forming stage. From this figure we can see that the circumferential strain in the central area of sub-shell A and sub-shell B and the area near the weld seam between sub-shell C and sub-shell D is still in compressive state; the meridional strain is in tensile state in most areas of the shell except the weld seam between sub-shell B and sub-shell C where the meridional strain is compressive due to the effect of bending. 4.3. Thickness distribution Fig. 8 is the thickness distribution in the final deformed toroidal shell. It can be seen from Fig. 8 that the maximum thinning occurs at the weld seams. This phenomenon is not consistent with the experiment in which the weld seams have almost no thinning because their strength is higher than that of the sheet metal and the thickness of weld seams is larger than that of the sheet metal. As discussed earlier, in simulation the strength and thickness of weld seams are set to be equal to that of sheet metal so that they move and large plastic deformation occurs under the effect of bending moment. This plastic deformation results in larger thinning in the weld seam areas than other areas of the shell. It is reasonable to exclude thinning occurring at the weld seams in analysis. Thus, there is a maximum thinning in the central area of sub-shell C and sub-shell D. The value of the maximum thinning rate or the difference between the maximum thickness and the minimum thickness is so small that the thickness can be considered to be uniform in the deformed toroidal shell. Moreover, the thickness near the central line of sub-shell A is increased little due to the effect of the compressive circumferential stress. 4.4. Wrinkling Some tiny wrinkles at sub-shell A and sub-shell B were found in the experiment, as shown in Fig. 9(a). The wrinkles obtained in the simulation are shown in Fig. 9(b), which match very well with ones shown in the experiment. The occurrence of the wrinkles is caused by the compressive circumferential stress existing on sub-shell A and sub-shell B, as mentioned above. For a cylindrical shell being subjected to external pressure, the circumferential stress is expressed by
⫽ ⫺ pr/t where p is the external stress, r is the radius of the cylindrical shell, and t is the thickness of the cylindrical shell. When the value of the circumferential stress reaches a critical stress value
S. Yuan et al. / International Journal of Machine Tools & Manufacture 39 (1999) 1439–1450
Fig. 7. Strain distribution at final stage. (a) Global strain ⑀x; (b) global strain ⑀y.
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Fig. 8.
Thickness distribution in the deformed shell (unit: mm).
Cr, wrinkling occurs. This is consistent to a conical shell being subjected to external pressure. Although tiny wrinkles would not have serious influence on the use of the elbow pipe in some cases, it is necessary to do further research to avoid occurrence of the wrinkles on sub-shells being subjected to external pressure by redesigning the structure of the polyhedral toroidal shell prior to hydro-forming and/or adapting other technological measures.
5. Conclusions The hydro-forming process of a polyhedral toroidal shell was simulated by the explicit dynamic finite element code eta/DYNAFORM. The results obtained from this FEA simulation, especially the deformed shape and wrinkle distribution, have a good agreement with those from the experiment. The thickness of the deformed toroidal shell is very even. FEA results further show that the method of manufacturing elbow pipes by hydro-forming toroidal shells is feasible. Wrinkles occur because of the compressive circumferential stress existing in the sub-shells being subjected to external pressure, which can be prevented by redesigning the structure of the polyhedral toroidal shell prior to hydro-forming and/or adapting other technological measures. Further research into this problem will be done in the near future.
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Fig. 9. Comparison of wrinkles observed in FE simulation and experiment. (a) Experiment; (b) simulation.
References [1] Z.R. Wang, T. Wang, D.C. Kang, The technology of the hydro-bulging of whole spherical vessels and experimental analysis, Journal of Mechanical Working Technology 18 (1989) 85–94. [2] S.J. Yuan, Z.R. Wang, Safety analysis of 200 m3 PLG tank manufactured by the dieless hydrobulging technology, Journal of Materials Processing Technology 70 (1997) 215–219. [3] ZENG Yuansong, S.J. Yuan, Z.R. Wang, Research on the hydrobulge forming of ellipsoidal shells, Journal of Materials Processing Technology 72 (1997) 28–31. [4] Lu Xiaoyang, Xu Binye, Forming process analysis of medium frequency induction heated pushing bending of tubes, Proceedings of 2nd Pacific-Asia Conference on Engineering Plasticity and Application, 1994, pp. 647–652.
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[5] S.J. Yuan, Z.R. Xu Zhang, HAIWANG Wang, The integrally hydro-forming process of pipe elbows, International Journal of Pressure Vessel & Piping 75 (1998) 7–9. [6] R.G. Whirly, B.E. Englmann, R.W. Logan, Some aspects of sheet forming simulation using explicit finite element technique, Numerical Methods for Simulation of Industrial Metal Forming Processes, ASME, 1992, CEDVol.5/AMD, Vol. 156, pp. 77–83. [7] E. Harpell, C. Lamontagne, M.J. Worswick, P. Martin, M. Finn, C. Galbrainth, Deep drawing and stretching of AL 5754 sheet: LS-DYNA3D simulations and validation, Proceedings of the 3rd International Conference Mumisheet’ 96, Dearborn, Michigan, September 29–October 3, 1996, pp. 386–389. [8] H.J. Kim, B.H. Jcon, H.Y. Kim, J.J. Kim, Finite element analysis of the liquid bulge forming process, Advanced Technology of Plasticity 1993—Proceedings of 4th International Conference on Technology of Plasticity, September 1993, Beijing, pp. 545–550. [9] S.H. Zhang, J. Danckert, S.J. Yuan, Z.R. Wang, W.J. Hang, Spherical and spherical steel structure products made by using integral hydro-bulge forming technology, Journal of Construction Steel Research 4A (1998) 1–3.