本文之目的在於運用彈塑性增量理論結合有限元素法，模擬分析金屬板材之U型彎曲加工製程中所產生之翹曲現象。本文遵循Prandtl-Reuss塑流法則與von Mises之降伏條件，結合有限變形理論及updated Lagrangian formulation (ULF)之觀念建立一增量型彈塑性大變形三維有限元素分析程式。並利用以四邊形四節點之退化殼元素(degenerated shell element)所推導之形狀函數耦合入剛性矩陣中，組成三維有限元素之分析模式，並且使用廣義 法則處理金屬板材在U型彎曲加工分析中，包含元素之降伏判斷、最大容許應變增量、最大容許旋轉增量、料片與模具間節點之接觸與分離判斷等問題。
藉由數值模擬，可以獲得U型彎曲加工製程之全部變形履歷，例如沖頭負荷與位移關係、料片厚度分佈、應力與應變分佈及製品之最終外形等。本文並設計一組模具進行實驗，經由實驗數據與模擬分析之數據比較顯示其結果相當符合，更加證明以本有限元素分析程式模擬U型彎曲加工製程之可行性及可靠性。

The aims of this thesis is to create an elasto-plastic incremental finite element computer code to simulate the warping behavior of the U- bending process of a sheet metal. A methodology for formulating an elasto-plastic three-dimensional finite element model, which is based on Prandtl-Reuss flow rule and von Mises yield criterion respectively, associated with an updated Lagrangian formulation, is developed to simulate sheet metal forming process. The shape function derived from a four-node quadrilateral degenerated shell element is associated into the stiffness matrix to constitute the finite element model. An extended algorithm is proposed to formulate the boundary condition, such as nodal penetration and separation, strain increment and rotation increment, and altered elasto-plastic state of material.
The whole deformation history, such as the distribution of thickness, stress and strain during the bending process was obtained. The experimental results of U-bending process were made in comparing with the simulation results. The results of both after compensation were accurate.

1. Y. Yamada, N. Yoshimura and T. Sakurai, “Plastic Stress Strain Matrix and Its Application for the solution of Elastic-Plastic Problems by Finite Element Method”, Int. J. Sci, Vol.10, pp.343-354 (1968).
2. K. Iwata, K. Osakada and S. Fujino, “Analysis of Hydroststic Extrusion by the Finite Element Method”, Trans. ASME, J. of Engng. for Ind. , Vol.94, pp.697-703 (1972).
3. E. I. Odell, “A Study of Wall Ironing by The Finite Element Technique”, Trans. ASME, Journal of Engineering for industry, Vol.100, pp.31-36 (1978).
4. C. H. Lee and Kobayashi, “Elastoplastic Analysis of Plane Strain and Axisymmetric Flat Punch Indentation by the Finite Element Method”, Int. J. Mech. Sci., Vol.12, pp.349-370 (1970).
5. C. H. Lee and Kobayashi, “New Solutions to Rigid-Plastic Deformation Problems Using a Matrix Method”, Trans. ASME, Journal of Engineering for industry, Vol.95, pp.865-873 (1973).
6. S. I. Oh and S. Kobayashi, “Finite Element Analysis of Plane Strain Sheet Bending”, Int. J. Mesh. Sci.,Vol.22, pp.583-594 (1979).
7. C. C. Chen and S. Kobayashi, “Rigid-Plastic Finite Element Analysis of Ring Compression”, Appl. Num. Method to Forming Process, AMD-28, pp.163-174 (1978).
8. H. D. Hibbit, P. V. Marcal and J. R. Rice, “A Finite Element Formulation for Problem of Large Strain and Large Displacement”, Int. J. Solids Struct., Vol.6, pp.1069-1086(1970).
9. R. M. McMeeking and J. R. Rice, “Finite-Element Formulation for Problems of Large Elastic-Plastic Deformation”, Int. J. Solids Struct.,Vol.11, pp.601-616 (1975).
10. A. S. Wifi, “An Incremental Complete Solution of the Stretch Forming and Deep Drawing of a Circular Blank Using a Hemisphrical Punch”, Int. J. Mech. Sci.,Vol.10, pp.23-31 (1976).
11. A. Makinouchi, H. Ogawa and Y. Tozawa, “Simulation of Sheet Bending Processes by Elastic-Plastic Finite Element Method”, CIRP, Vol.38, pp.279-282 (1989).
12. J. C. Nagtegaal, D. M. Park, and J. R. Rice, “On Numerically Accurate Finite Element Solutions in the Fully Plastic Range”, Computer Method in Applied Mechanics and Engineering, Vol.4, pp.153-177 (1974).
13. E. D. Pugh, E. Hinton, and O. C. Zienkiewicz, “A Study of Quadrilateral Plate Bending Element with Reduced Integration”, International Journal for Numerical Methods in Engineering, Vol.12, pp.1059-1079 (1978).
14. T. J. R. Hughes, “Generalization of Selective Integration Procedures to Anisotropic and Nonlinear Media”, International Journal for Numerical Methods in Engineering, Vol.15, pp.1413-1418(1980).
15. J. C. Nagtegaal and N. Rebelo, “On the Development of A General Purpose Finite Element Program for Analysis of Forming Process“, Int. J. Num. Meth. Engng. , Vol.25, pp.113-131 (1988).
16. A. Makinouchi and S. Yoshida, “Finite Element Simulation of the Draw-Bending Process of Sheet Metals”, J. of JSTP, Vol.28, No.313, pp.135-142 (1987).
17. Y. M. Huang, H. Takizawa, A. Makinouchi, T. Nakagawa, “Elastic-Plastic Analysis of V-Bending Process”, Spring Proceeding of Plastic Working, Cho-Fu, Tokyo, pp.275-278 (1989).
18. Y. Hayashi and M. Takagi, “Control of Side Wall Curl in Draw-Bending of High Strength Steel Sheets“, Proceeding of Advance Technology of Plasticity, Vol.1, pp.735-740 (1984).
19. A. Makinouchi, “Finite Element Modeling of Draw-Bending Process of Sheet Metal”, Proceeding of the NUMIFORM’86 Conference, Gothenburg, Vol.25-29, August, pp.327-332 (1986).
20. J. K. Kim and P. F. Thomson, “Springback and Side-Wall Curl of Galvanized and Galvalume Steel Sheet”, J. of Mechanical Working Technology, Vol.19, pp.223-238 (1989).
21. Y. M. Huang and D. K. Leu, “An Elasto-Plastic Finite Analysis of Sheet Metal U-Bending Process”, J. of Material Processing Technology, Vol.48, pp.151-157 (1995).
22. 陳聰嘉， “金屬板材V形彎曲成形製程馬鞍現象之研究”，台灣科技大學博士論文，台北(民國93年)。