1. 藉由本方法未來可研究識別已成形加工之纏繞式複合材料空間椼 架之管殼結構的各項彈性常數之可行性。
2. 可研究識別已成形加工且具有彈性邊界之複合材料積層樑結構的 各項彈性常數及彈性支撐之參數識別的可行性。
3. 未來可研究識別經特殊處理已成形加工的合金材料之各種結構體 的各項彈性常數。
4. 運用此方法識別已成形加工之三明治複合材料積層板及樑結構的 各項彈性常數識別的可行性。
5. 運用此方法發展研究奈米材料之性質研究。
參 考 文 獻
1. R. M. Jones, 1975, “Mechanics of Composite Materials”, McGraw-Hill book company.
2. S. W. Tsai, H. T. Hahn, 1980, “Introduction to Composite Materials”, Techomic publishing Co., Inc.
3. J. R. Vinson, R. L. Sierakowski, 1986, “The Behavior of Structures Composed of Composite Materials”, Martinus Nijhoff publishers.
4. S. R. Swanson, 1997, “Introduction to Design and Analysis with Advanced Composite Materials”, Prentice-Hall International, Inc.
5. J. A. Snyman, L. P. Fatti , 1987, “A Multi-Start Global Minimization Algorithm with Dynamic Search Trajectories”, Journal of Optimization Theory and Applications, 54(1), pp. 121-141.
6. T. Y. Kam, J. A. Snyman , 1991, “Optimal Design of Laminated Composite Structures Using a Global Optimization Technique”, Journal of Composite Structures, 19, pp. 351-370.
7. 朱高弘, 1992, “受挫屈複合材料板之最佳設計”,國立交通大學機械 工程研究所碩士論文。
8. 陳大智, 1992, “複合材料板之多層次最佳設計”,國立交通大學機械 工程研究所碩士論文。
9. 賴峰民, 1993, “複合材料積層板設計及製造”, 國立交通大學機械工 程研究所碩士論文.
10. 廖紹助, 1994, “複合材料積層板之輕量設計”,國立交通大學機械工 程研究所碩士論文。
11. 賴峰民, 1997, “複合材料結構首層破壞之最佳化設計” ,國立交通大 學機械工程研究所博士論文.
12. 林志宏, 1999, “複合材料彈性常數之識別”,國立交通大學機械工程 研究所碩士論文。
13. 王文庭, 2000, “複合材料結構彈性常數識別” ,國立交通大學機械工 程研究所博士論文.
14. G. N. Vanderplaats, 1984, “Numerical Optimization Techniques for Engineering Design with Applications”, McGraw Hill Inc., New York.
15. 李清榮, 2006, “彈性支撐複合材料板結構系統參數之識別” ,國立交 通大學機械工程研究所博士論文.
16. E. M. Salchenberger Cinar, N. A. Lash, 1992, “Neural Networks: A New Tool for Predicting Thrift Failures,” Decision Sciences, 23, pp.
889-916.
17. R. S. Sexton, N. D. Jatinder, Gupta, 2000, “Comparative Evaluation of Genetic Algorithm and Backpropagation for Training Neural Networks”, Information Sciences, 129, pp. 45-59.
18. R. S. Sexton, R. E. Dorsey, J. D. Johnson, 1998, “Toward Global Optimization of Neural Networks: A Comparison of the Genetic Algorithm and Backrogation”, Decision Support System, 22, pp.
171-186.
19. C. C. Huang, Y. F. Huang, 1991, “Bounds on the Numbers of Hidden Neuron in Multiplayer Perceptions”, IEEE Trans. Neural Networks, 2(1), pp. 47-55.
20. C. Y. Lin, C. Fleury, 1992, “Genetic Algorithms in Optimization Problems with Discrete and Integer Design Variables”, Engineering Optimization, 19, pp. 309-327.
21. D. E. Goldberg, C. H. Kuo, 1987, “Genetic Algorithms in Pipeline Optimization”, Journal of Computing in Civil Engineering, 1, pp.
128-141.
22. T. Y. Kam, T. Y. Lee, 1994, “Crack Size Identification Using an Expanded Mode Method”, International Journal of Solids and Structures, 31, pp. 925-940.
23. T. Y. Lee, T. Y. Kam, 1993, “Detection of Crack Location via a Global Minimization Approach”, International Journal of Engineering Optimization, 21, pp. 147-159.
24. T. Y. Kam, T. Y. Lee, 1992, “Detection of Cracks from Modal Test Data”, International Journal of Engineering Fracture Mechanics, 42 (2), pp. 381-387.
25. T. Y. Kam, T. Y. Lee, 1994, “Identification of Crack Size via an Energy Approach”, International Journal of Nondestructive Evaluation, 13(1), pp. 1-11.
26. T. Y. Kam, F. M. Lai, and T. M. Chao, 1999, “Optimum Design of Foam-Filled Laminated Composite Sandwich Plates”, International Journal of Solids & Structures, 36(19), pp. 2865-2889.
27. T. Y. Kam, F. M. Lai, 1999, “Experimental and theoretical predictions of first-ply failure strength of laminated composite plates”, International Journal of Solids and Structures, 36(16), 2379-2395.
28. T. Y. Kam, F. M. Lai, S. C. Liao, 1996, “Minimum Weight Design of Laminated Composite Plates Subject to Strength Constraint”, Journal of AIAA , 34(8), pp.1699-1708.
29. T. Y. Kam, F. M. Lai, 1995, “Design of Laminated Composite Plates for Optimal Dynamic Characteristics Using a Constrained Global Optimization Technique”, International Journal of Computer Methods in Applied Mechanics & Engineering, 120, pp. 389-402.
30. T. Y. Kam, F. M. Lai, H. F. Sher, 1995, “Optimal Parameters for Curing Gr/Ep Composite Laminates”, International Journal of Materials Processing Technology, 48, pp. 357-363.
31. T. Y. Kam, F. M. Lai, 1995, “Maximum Stiffness Design of Laminated Composite Plates via a Constrained Global Optimization Approach”, International Journal of Composite Structures, 32(1) 391-398.
32. T. Y. Kam, M. D. Lai, 1989, “Multilevel Optimal Design of Laminated Composite Plate Structures”, Computers and Structures, 31(2), pp.
197-202.
33. T. Y. Kam, C. K. Liu, 1998, “Stiffness identification of laminated composite shafts”, International Journal of Mechanical Sciences, 4(9),927-936.
34. W. T. Wang, T. Y. Kam, 2001, “Elastic constants identification of shear deformable laminated composite plates”, ASCE, Journal of Engineering Mechanics, 127(11), pp. 1117-1123.
35. W. T. Wang, T. Y. Kam, 2000, “Material characterization of laminated composite plates via static testing”, International Journal of Composite Structures, 50(4), pp. 347-352.
36. T. Y. Kam, C. H. Lin, W. T. Wang, 2000, “Identification of material constants of composite laminated pressure vessels using measured strains”, ASME, Journal of Engineering Materials and Technology, 122(4), pp. 425-427.
37. S. V. Hoa, S. Z. Sheng, P. Ouellette, 2003, “Determination of elastic proerties of triax composite materials”, Composites Science and Technology, 63, pp.437-443.
38. A. K. Bledzki, A. Kessler, R. Rikards, A. Chate, 1999, “Determination of elastic constants of glass/epoxy unidirectional laminates by the vibration testing of plates”, Composites Science and Technology, 59, pp.
2015-2024.
39. J. C. Marín, J. Cañas, F. París, J. Morton, 2002, “Determination of G12
by means of the off-axis tension test. Part І: review of gripping systems and correction factors”, Composites: Part A (33), pp. 87-100.
40. J. C. Marín, J. Cañas, F. París, J. Morton, 2002, “Determination of G12 by means of the off-axis tension test. Part II: a self-consistent approach to the application of correction factors”, Composites: Part A, 33, pp.
101-111.
41. M. J. Pindera, C. T. Herakovich, 1986, “Shear characterization of unidirectional composites with the off-axis tension test”, Experimental Mechanics, pp. 103-112.
42. N. J. Pagano, J. C. Halpin, 1968, “Influence of end constraint in the
testing of anisotropic bodies”, Journal of Composite Mechanicals, 2(1), pp. 18-31.
43. C. T. Sun, I. Chung, 1993, “An oblique end-tab design for testing off-axis composite specimens”, Composites, 24(8), pp. 619-623.
44. M. Grédiac, E. Toussaint, F. Pierron, 2002, “Special virtual fields for the direct determination of material parameters with the virtual fields method. 1 – Principle and definition”, International Journal of Solids and Structures, 39, pp. 2691-2705.
45. M. Grédiac, E. Toussaint, F. Pierron, 2002, “Special virtual fields for the direct determination of material parameters with the virtual fields method. 2 – Application to in-plane properties”, International Journal of Solids and Structures, 39, pp. 2707-2730.
46. M. Grédiac, E. Toussaint, F. Pierron, 2003, “Special virtual fields for the direct determination of material parameters with the virtual fields method. 3- Application to the bending rigidities of anisotropic plates”, International Journal of Solids and Structures, 40, pp. 2401-2419.
47. S. F. Hwang, C. S. Chang, 2000, “Determination of elastic constants of materials by vibration testing”, Composite Structures, 49, pp. 183-190.
48. L. Marin, L. Elliott, D. B. Ingham, D. Lesnic, 2004, “parameter identification in isotropic linear elasticity using the boundary element methob”, Endineering Analysis with Boundary Element, pp. 221-233.
49. K. Genovese, L. Llamberti and C. Pappalettere, 2004, “A new hybrid technique for in-plane characterization of orthotropic materials”, Society for Experimental Mechanics, pp. 58-592
50. L. R. Deobald, R. F. Gibson, 1988, “Determination of elastic constants
of orthotropic plates by a model analysis/Rayleigh-Ritz technique”, Journal of Sound and Vibration, pp. 269-283
51. P. S. Frederiksen, 1997, “Experimental procedure and results for the identification of elastic constants of thick orthotropic plates”, journal of composite materials, 31(4), pp. 360-382.
52. F. Moussu, M. Nivoit, 1993, “Determination of elastic constants of orthotropic plates by a modal analysis/method of superposition”, Journal of Sound and Vibration, pp. 149-163.
53. K. E. Fallstrom, 1991, “Determining material properties in anisotropic plates using Rayleigh’s method”, Polymer Composites, pp. 360-314.
54. M. Taktak, F. Dammak, S. Abid, M. Haddar, 2005, “A mixed-hybrid finite element for three-dimensional isotropic helical beam analysis”, International Journal of Mechanical Sciences, 47, pp. 209-229
55. C. M. Mota Soares, M. Moreira de Freitas, A. L. Araújo and P.
Pederson, 1993, “Identification of material properties of composite plate specimens”, Composite Structures, 25, pp. 277-285.
56. H. Sol, H. Hua, J. De Visscher, J. Vantomme and W. P. De Wilde, 1997, “A mixed numerical/experimental technique for the nondestructive identification of the stiffness properties of fibre reinforced composite materials”, NDT & E International, 30(2), pp.
85-91.
57. W. P. Wilde, H. Sol, 1987, “Anisotropic material identification using measured resonant frequencies of rectangular composite plates”, Composite Structures, 4(2), pp.2317-2324.
58. R. Rikards, A. Chate, W. Steinchen, A. Kessler, A. K. Bledzki, 1999,
“Method for identification of elastic properties of laminates based on experiment design”, Composites, 30B, pp. 279-289.
59. K. H. Ip, P. C. Tse, T. C. Lai, 1998, “Material characterization for orthotropic shells using modal analysis and Rayleigh-Ritz models”, Composites, 29B, pp. 397-409.
60. A. L. Araújo, C. M. Mota Soares, M. Moreira de Freitas, 1996,
“Characterization of material parameters of composite plate specimens using optimization and experimental vibrational data”, Composites, 27B, pp. 185-191.
61. ASTM, 1990, “Standards and Literature References for Composite Materials”, 2nd Ed., West Conshohocken, Pa.
62. E. C. Cojocaru, H. Irschik, H. Gattringer, 2004, “Dynamic response of an elastic bridge due to a moving elastic beam”, Composite and Structures, 82, pp. 931-934.
63. A. B. Lipen, A. V. Chigarev, 1998, “The displacements in an elastic half-space when a load moves along a beam lying on its surface”, Journal Appl. Maths Mechs,62(5), pp.791-796.
64. H. S. Zibdeh, R. Rackwitz, “Response moments of an elastic beam subjected to poissonian moving loads”, Journal of Sound and Vibration, 188(4), pp.479-495.
65. B. Nadler, M. B. Rubin, 2003, “Determination of hourglass coefficients in the theory of a Cosserat point for nonlinear elastic beams”, International Journal of Solids and Structures, 40, pp. 6163-6188.
66. IMSL, 1994, “User’s manual”, Version 3.0, IMSL Inc.
67. S. W. Tsai, 1987, “Composite design”, Think Composites, Dayton.
68. F. Mujika, A. Valea, P. Ganan, I. Mondragon, 2005, “Off-axis flexure test: A new method for obtaining in-plane shear properties”, Journal of Composite Material, pp. 953-980.
69. F. Mujika, L. Berglund, J. Varna, I. Mondragon, 2002, “45∘flexure test for measurement of in-plane shear modulus”, Journal of Composite Material, 36, pp. 2313-2338.
70. F. Mujika, I. Mondragon, 2003, “On the displacement field for unidirectional off-axis composites in 3 point flexure. Part 1: analytical approach”, Journal of Composite Material, 37, pp.1041-1066.
71. F. Mujika, A. de Benito, I. Mondragon, 2003, “On the displacement field for unidirectional off-axis composites in 3 point flexure. Part 2:
numerical and experimental results”, Journal of Composite Material, 37, pp. 1191-1217.
72. R. J. Benjamin, C. A. Cornell, 1970, “Probability, statistics, and decision for civil engineers”, McGraw-Hill, New York.
73. T. K. Hwang, C. S. Hong, C. G. Kim, 2003, “Probabilistic deformation and strength prediction for a filament wound pressure vessel”, composites: Part B 34, pp. 481-497.
74. M. Drechsler, 1998, “Sensitivity analysis of complex models”, Biological Conservation, 86, pp. 401-412.
75. A. Chan, X. L. Liu, W. K. Chiu, 2004, “Sensitivity analysis of potential tests for determining the interlaminar shear modulus of fibre reinforced
composites”, Composite Structures, 66, pp. 109-114.
附錄 程式流程圖
Multi-start sampling
n=n+1
Strain analysis
Subroutine 1
n=1
二.副程式 1 流程圖(擴增拉格蘭吉乘子法流程圖)
Subroutine 2
ζ
三. 副程式 2 流程圖(軌跡收尋流程圖)
Set i=0,k=0 Start new trajectory
k=k+1
Satisfy termination condition?
{ }
trajectory
Downhill?
i=0
i=i+1
Yes i=0 i=0
表3-1. Gr/ep 複合材料積層樑單一階段識別法之靈敏度
Sensitivity (103) C.O.V.
Fiber angle [(θ°/-θ°)6]s
Elastic constant
ε ε strains
E1(GPa) -2.204 -0.1095 -0.629 5.70% 0.27% 0.38% 5.72%
E1(GPa) -2.660 -2.701 -12.093 35.12% 25.26% 5.32% 43.59%
E2(GPa) 7.071 7.095 69.74 148.55% 105.56% 48.85% 188.67%
G12(GPa) -1.034 1.036 0.0078 2.92% 2.08% 0.001% 3.59%
45°
ν 12 -1.259 0.080 -8.682 8.11% 0.37% 1.86% 8.33%
E1(GPa) -2.046 -4.857 -11.179 50.89% 34.89% 4.64% 61.87%
E2(GPa) 0.492 1.449 6.811 18.03% 15.34% 4.16% 24.04%
G12(GPa) -2.992 -4.242 -23.224 15.32% 6.27% 1.98% 16.67%
60°
ν 12 -5.617 -20.345 -51.213 63.62% 66.55% 9.67% 92.58%
表3-2. Gl/ep 複合材料積層樑單一階段識別法之靈敏度
Sensitivity (103) C.O.V.
Fiber angle [(θ°/-θ°)6]s
Elastic constant
ε ε strains
E1(GPa) -4.646 -0.154 -2.136 5.41% 0.08% 0.34% 5.42%
E2(GPa) 0.6017 3.7398 -4.4371 3.31% 8.95% 3.31% 10.10%
G12(GPa) 0.05571 -0.4611 37.9622 0.06% 0.22% 5.58% 5.58%
ν 12 11.544 -12.348 162.96 30.72% 21.12% 42.24% 56.34%
E1(GPa) -5.408 -5.663 -28.818 16.81% 9.05% 3.88% 19.48%
表3-3. Gr/ep 複合材料積層板兩階段識別法之靈敏度 Sensitivity (103)
Level Fiber angle [(θ°/-θ°)2]s
Elastic constant
ε ε
∂
∂
x
xi
ε ε
∂
∂
y
xi
Var[xi] C.O.V.
E1(GPa) -0.74 -0.79 0.0011 25.42%
E2(GPa) -0.11 -0.14 0.000028 2.83%
G12(GPa) -0.56 0.59 0.000625 3.66%
First 45°
ν12 -0.0084 0.02 0.0000002 0.14%
E1(GPa) -1.12 -0.04 0.00006 5.30%
15°
E2(GPa) 0.14 1.33 0.000082 9.82%
E1(GPa) -0.96 -0.18 0.00021 9.90%
30°
E2(GPa) -0.34 1.055 0.0004 21.83%
E1(GPa) -0.51 -3.205 0.0046 46.22%
Second
60°
E2(GPa) -0.204 0.054 0.00017 14.14%
表3-4. Gl/ep 複合材料積層板兩階段識別法之靈敏度 Sensitivity (103)
Level Fiber angle [(θ°/-θ°)2]s
Elastic constant
ε ε
∂
∂
x
xi
ε ε
∂
∂
y
xi
Var[xi] C.O.V.
E1(GPa) -0.267 1.400 0.000105 32.41%
E2(GPa) 0.335 -5.864 0.001636 31.35%
G12(GPa) -1.13 0.73 0.00025 3.82%
First 45°
ν12 -0.008 -0.35 0.000006 0.94%
E1(GPa) -0.41 0.005 0.000004 5.21%
15°
E2(GPa) 1.34 8.05 0.00035 22.51%
E1(GPa) -0.41 0.02 0.00001 8.03%
30°
E2(GPa) -2.03 9.52 0.00245 59.77%
E1(GPa) -0.11 -2.23 0.000125 28.97%
Second
60°
E2(GPa) -0.50 0.02 0.00008 11.00%
表3-5. Gr/ep 複合材料積層板四應變一階段識別之靈敏度 Sensitivity (103)
Fiber angle [(θ°/-θ°)2]s
Elastic constant
ε ε
∂
∂
45 x
xi
ε ε
∂
∂
45 y
xi
ε ε
∂
∂
30 x
xi
ε ε
∂
∂
30 x
xi
Var[xi] C.O.V.
E1(GPa) -0.169 0.237 0.218 -0.173 0.0000822 6.19%
E2(GPa) -0.921 -1.276 1.077 1.066 0.00237 52.77%
G12(GPa) -0.562 0.533 0.034 -0.035 0.000583 3.53%
45°+30°
ν12 -18.823 -14.021 21.16 0.027 0.671 272.78%
表3-6. Gr/ep 複合材料積層樑兩階段識別法之靈敏度 Sensitivity (103)
Level Fiber angle [(θ°/-θ°)2]s
Elastic constant
ε ε
∂
∂
x
xi
ε ε
∂
∂
y
xi
Var[xi] C.O.V.
E1(GPa) -1.39 -1.40 0.0001752 10.21%
E2(GPa) -0.23 -0.26 0.0000052 1.22%
G12(GPa) -1.03 1.09 0.0001000 1.46%
First 45°
ν12 -0.08 -0.07 0.0000005 0.24%
E1(GPa) -2.21 -0.11 0.0000103 2.20%
15°
E2(GPa) 0.26 2.49 0.0000134 3.97%
E1(GPa) -1.77 -0.33 0.0000338 3.97%
30°
E2(GPa) -0.66 1.99 0.0000679 8.94%
E1(GPa) -0.98 -5.39 0.0006553 17.40%
Second
60°
E2(GPa) -0.38 0.10 0.0000272 5.66%
表3-7. Gl/ep 複合材料積層樑兩階段識別法之靈敏度 Sensitivity (103)
Level Fiber angle [(θ°/-θ°)2]s
Elastic constant
ε ε
∂
∂
x
xi
ε ε
∂
∂
y
xi
Var[xi] C.O.V.
E1(GPa) -0.26 -0.27 0.0000020 4.58%
E2(GPa) -1.08 -1.33 0.0000349 4.62%
G12(GPa) -0.83 0.81 0.0000200 1.08%
First 45°
ν12 -0.17 -0.08 0.0000007 0.32%
E1(GPa) -0.46 -0.01 0.0000007 2.16%
15°
E2(GPa) 1.53 9.37 0.0000620 9.52%
E1(GPa) -0.45 -2.35 0.0000016 3.27%
30°
E2(GPa) -2.35 12.17 0.0005125 27.38%
E1(GPa) -0.11 -2.40 0.0000189 11.25%
Second
60°
E2(GPa) -0.56 0.03 0.0000134 4.43%
表4-1 Graphite/Epoxy 之複材板結構之各項彈性常數值 Material constant
Material type
E1(GPa) E2(GPa) G12(GPa) ν12 Graphite/Epoxy 146.5 9.22 6.84 0.3
C.O.V. 0.7% 1.2% 3.2% 0.19%
(Source:本表由金大仁實驗室依 ASTM[61]規範量測而得)
表 5-1. F= 1kN之Gr/ep複材板結構[(θ°/-θ°)4/θ°]之理論應變值 Strain
Fiber angle θ
εx (10-4) εy (10-4) γxy(10-4)
15° 2.479 -2.395 -0.5088 30° 5.228 -6.554 -0.4371 45° 12.67 -8.985 -0.3747 60° 22.68 -6.554 -0.3352
表 5-2. 以F= 1kN理論應變值識別Gr/ep 複材板結構[(θ°/-θ°)4/θ°] 之各 項彈性常數值
Identified material constant Fiber angle θ
E1(GPa) E2 (GPa) G12 (GPa) ν12 15° 146.54
(0.03%)†
9.25 (0.3%)
6.84 (0%)
0.30 (0%) 30° 146.58
(0.05%)
9.25 (0.3%)
6.84 (0%)
0.30 (0%) 45° 146.71
(0.1%)
9.17 (0.5%)
6.84 (0%)
0.30 (0%) 60° 146.07
(0.3%)
9.23 (0.1%)
6.84 (0%)
0.30 (0%)
†Value in parentheses denotes percentage difference between identified and actual data.
表 5-3. 以不同之Nx及Ny 的Gr/ep複材板結構[(θ°/-θ°)4/θ°]之理論應變 值
Stress resultant Strain Nx (kN/m) Ny (kN/m)
Fiber angle θ
εx (10-5) εy (10-5) γxy(10-6)
15° 3.001 5.349 -8.591 30° 5.877 -3.027 -7.562 45° 16.32 -9.675 -6.745 5 1
60° 32.05 -8.262 -6.339 15° 0.1265 41.11 -12.42 30° -1.988 24.19 -11.58 45° 5.534 5.534 -11.24 5 5
60° 24.19 -1.988 -11.58 15° 4.377 -12.53 -6.674 30° 9.809 -16.63 -5.551 45° 21.71 -17.28 -4.497 5 -1
60° 35.99 -11.40 -3.717 15° 7.312 -48.30 -2.839 30° 17.67 -43.85 -1.528 45° 32.49 -32.49 0 5 -5
60° 43.85 -17.67 1.528
表 5-4. 以 不 同 之 Nx 及 Ny的 理 論 應 變 值 識 別Gr/ep 複 材 板 結 構 [(θ°/-θ°)4/θ°]之各項彈性常數值
Stress resultant Identified material constant Nx
(kN/m)
Ny (kN/m)
Fiber angle
θ E1(GPa) E2 (GPa) G12 (GPa) ν12
†Value in parentheses denotes percentage difference between identified and actual data.
表 5-5. 以不同之Nx及Ny 的Gl/ep複材板結構[(θ°/-θ°)4/θ°]之理論應變 值
Stress resultant Strain Nx (kN/m) Ny (kN/m)
Fiber angle θ
εx (10-5) εy (10-5) γxy(10-5)
15° 12.05 4.861 -1.642 30° 17.79 -3.422 -1.908 45° 31.80 -11.15 -1.635 5 1
60° 45.97 -9.059 -1.062 15° 7.441 47.34 -1.832 30° 7.275 35.46 -2.475 45° 17.21 17.21 -2.725 5 5
60° 35.46 7.275 -2.475 15° 14.35 -16.38 -1.547 30° 23.05 -22.86 -1.625 45° 39.09 -25.32 -1.090 5 -1
60° 51.23 -17.23 -0.3551 15° 18.96 -58.85 -1.357 30° 33.56 -61.74 -1.058 45° 53.68 -53.68 0 5 -5
60° 61.74 -33.56 1.058
表 5-6. 以 不 同 之 Nx 及 Ny的 理 論 應 變 值 識 別Gl/ep 複 材 板 結 構 [(θ°/-θ°)4/θ°]之各項彈性常數值
Stress resultant Identified material constant Nx
(kN/m)
Ny (kN/m)
Fiber angle θ
E1(GPa) E2 (GPa) G12 (GPa) ν12
†Value in parentheses denotes percentage difference between identified and actual data.
表5-7. F= 0.5kN Gr/ep複材板結構[θ°] 9之理論應變值 Strain
Material
type Layup
ε*x ε*y γ*xy
[15°]9 2.268E-4 -5.517E-5 -4.624E-4 [30°]9 5.520E-4 -1.048E-4 -7.380E-4 [45°]9 9.533E-4 -1.297E-4 -7.528E-4 Gr/Ep
[60°]9 1.305E-3 -1.048E-4 -5.660E-4
表5-8.以F= 0.5kN的理論應變值識別Gr/ep 複材板結構[θ°] 9之各項彈 性常數值
Identified material constant Material
type Layup
†Value in parentheses denotes percentage difference between identified and actual data.
表5-9. F= 0.1kN Gl/ep複材板結構[θ°] 9之理論應變值 Strain
Material
type Layup
ε*x ε*y γ*xy
[15°]9 1.107E-4 -3.500E-5 -1.225E-4 [30°]9 1.923E-4 -6.510E-5 -1.740E-4 [45°]9 2.777E-4 -8.014E-5 -1.408E-4 Gl/Ep
[60°]9 3.330E-4 -6.510E-5 -6.978E-5
表5-10.以F= 0.1kN的理論應變值識別G1/ep 複材板結構[θ°] 9之各項彈 性常數值
Identified material constant Material
type Layup
†Value in parentheses denotes percentage difference between identified and actual data.
表 5-11. 以Nx= 16.667kN/m及Ny= 0 的Gr/ep複材板結構 [(30°/-30°)4/30°]之實驗應變值
Measured strain Derived strains Specimen
No.
ε*x (10-4) ε*y (10-4) ε*45 (10-4) γ*xy(10-4) 1 2.687
(+2.8%)†
-3.361 (+2.6%)
-0.4481 (+1.7%)
-0.2222 (+1.7%) 2 2.772
(+6%)
-3.467 (+5.8%)
-0.4658 (+5.7%)
-0.2366 (+8.3%) 3 2.713
(+3.8%)
-3.431 (+4.7%)
-0.4747 (+7.7%)
-0.2314 (+5.9%) average 2.724
(+4.2%)
-3.420 (+4.7%)
-0.4629 (+5.0%)
-0.2301 (+5.3%) C.O.V. 1.6% 1.6% 2.9% 3.2%
†Value in parentheses denotes the percentage difference between actual and measured strains.
表 5-12. 以Nx= 16.667 kN/m 及 Ny= 0 的Gr/ep複材板結構[(30°/-30°)4/30°]之 實驗應變值識別之各項彈性常數值
Measured strain Identified material constant Case
†Value in parentheses denotes percentage difference between identified and actual data.
§This case uses average measured strains for identification
表 5-13. 以Nx= 10kN/m及 Ny= 0 的Gr/ep複材板結構[(45°/-45°)4/45°]之 實驗應變值
Measured strain Derived strain Specimen
No.
ε*x (10-4) ε*y (10-4) ε*45 (10-4) γ*xy(10-4) 1 4.095
(+7.7%)†
-2.971 (+10.2%)
0.5045 (+1.4%)
-0.1150 (+2.3%) 2 4.084
(+7.4%)
-2.921 (+8.4%)
0.5233 (+5.0%)
-0.1184 (+5.3%) 3 4.028
(+5.9%)
-2.888 (+7.2%)
0.5121 (+3.0%)
-0.1158 (+3%) average 4.069
(+7.0%)
-2.927 (+8.6%)
0.5130 (+3.2%)
-0.1164 (+3.6%) C.O.V. 0.9% 1.4% 1.7% 1.5%
†Value in parentheses denotes the percentage difference between actual and measured strains.
表 5-14. 以Nx= 10kN/m 及 Ny= 0 的Gr/ep複材板結構[(45°/-45°)4/45°]之實 驗應變值識別之各項彈性常數值
Measured strain Identified material constant Case
†Value in parentheses denotes percentage difference between identified and actual data.
§This case uses average measured strains for identification
表 5-15. F= 3N Gr/ep複材樑結構[(θ°/-θ°)6]s之理論應變值 Strain
Fiber angle θ
Identified elastic constant Fiber angle θ
†Value in parentheses denotes percentage difference between identified and actual data.
表 5-17. 理 論 應 變 值 以 隨 機 多 起 始 點 識 別 Gr/ep 複 材 樑 結 構 [(30°/-30°)6]s之各項彈性常數值
Starting Point Elastic constant point
No.
stage E1(GPa) E2 (GPa) G12 (GPa) ν12
No. of iterations
Initial 130.37 3.22 13.36 0.26 1
Final 146.56 9.23 6.84 0.30
5
Initial 159.88 16.88 7.54 0.49 2
Final 146.56 9.23 6.84 0.30
5
Initial 226.93 4.76 4.54 0.38 3
Final 146.56 9.23 6.84 0.30
8
Initial 265.99 2.29 16.45 0.39 4
Final 146.56 9.23 6.84 0.30
10
Initial 304.55 5.65 8.02 0.103 5
Final 146.56 9.23 6.84 0.30
10
Global minimum 146.56 (0.04%)†
9.23 (0.1%)
6.84 (0%)
0.30 (0%)
Probability 0.997835
†Value in parentheses denotes percentage difference between identified and actual data.
表 5-18. F= 1N Gl/ep複材樑結構[(θ°/-θ°)6]s之理論應變值
Identified elastic constant Fiber angle θ
†Value in parentheses denotes percentage difference between identified and actual data.
表 5-20. F= 3N Gr/ep複材樑結構[(45°/-45°)6]s之實驗應變值 Measured strains
Specimen
No. ε*x (10-4) ε*y (10-4) γ*xy(10-5) 1 3.932
(+1.6%)†
-2.775 (+1.3%)
-1.321 (+2.4%) 2 3.997
(+3.3%)
-2.821 (+3.0%)
-1.350 (+4.7%) 3 4.017
(+3.8%)
-2.841 (+3.7%)
-1.341 (+4.0%) average 3.982
(+2.9%)
-2.812 (+2.6%)
-1.337 (+3.6%)
C.O.V. 1.1% 1.2% 1.1%
†Value in parentheses denotes the percentage difference between actual and measured strains.
表 5-21. 實驗應變值識別Gr/ep 複材樑結構[(45°/-45°)6]s之各項彈性常 數值
Measured strain Identified elastic constant Specimen
No. ε*x Average
strain
3.982 -2.812 -1.337 141.38 (3.5%)
†Value in parentheses denotes percentage difference between identified and actual data.
表 5-22. F= 3N Gr/ep複材樑結構[(60°/-60°)6]s之實驗應變值 Measured strains
Specimen
No. ε*x (10-4) ε*y (10-4) γ*xy(10-5) 1 7.002
(+1.2%)†
-2.016 (+0.9%)
-1.240 (+7.5%) 2 7.071
(+2.2%)
-2.041 (+2.1%)
-1.189 (+3.0%) 3 7.123
(+2.9%)
-2.066 (+3.4%)
-1.207 (+4.6%) average 7.065
(+2.1%)
-2.041 (+2.1%)
-1.212 (+5.0%)
C.O.V. 0.9% 1.2% 2.1%
†Value in parentheses denotes the percentage difference between actual and measured strains.
表 5-23. 實驗應變值識別Gr/ep 複材樑結構[(60°/-60°)6]s之各項彈性常 數值
Measured strain Identified elastic constant Specimen
No. ε*x Average
strain
7.065 -2.056 -1.212 139.2
†Value in parentheses denotes percentage difference between identified and actual data.
表 5-24. F= 0.5kN Gr/ep複材板結構[(θ°/-θ°)2]s之理論應變值 Strain
Fiber angle θ
εx (10-4) εy (10-4)
[(15°/-15°)2]s 1.388 -1.352 [(30°/-30°)2]s 2.930 -3.695 [(45°/-45°)2]s 7.119 -5.064
[(60°/-60°)2]s 12.75 -3.695
表 5-25. 以理論應變值第一階段所識別Gr/ep 複材板結構[(45°/-45°)2]s
之各項彈性常數值
Material constant Starting
point No.
Stage
E1(GPa) E2 (GPa) G12 (GPa) ν12
No. of iterations
Initial 736.71 45.75 13.08 0.36 1
Final 130.22 18.80 6.84 0.30
10
Initial 445.59 3.31 17.87 0.26 2
Final 130.22 18.80 6.84 0.30
8
Initial 529.27 25.58 14.04 0.43 3
Final 130.22 18.80 6.84 0.30
7
Initial 429.96 17.10 14.77 0.28 4
Final 130.22 18.80 6.84 0.30
9
Initial 633.82 32.28 19.87 0.37 5
Final 130.22 18.80 6.84 0.30
10
Global minimum 130.22 (11.1%)†
18.80 (103.9%)
6.84 (0%)
0.30 (0%)
Probability 0.997
†Value in parentheses denotes percentage difference between identified and actual data.
表 5-26. 以理論應變值第一階段所識別Gr/ep 複材板結構 [(θ°/-θ°)2]s 之各項彈性常數值
Identified material constant Fiber angle θ
E1(GPa) E2 (GPa) G12 (GPa) ν12 15° 150.79
(2.9%)†
14.30 (55.1%)
1.95 (71.5%)
0.32 (6.7%) 30° 184.31
(25.8%)
31.14 (237.7%)
3.11 (54.5%)
0.30 (0%) 60° No global minimum
†Value in parentheses denotes percentage difference between identified and actual data.
表 5-27. 以理論應變值隨機多起始點第二階段所識別Gr/ep 複材板結 構 [(15°/-15°)2]s之各項彈性常數值
Material constants Starting
point No.
Stage
E1(GPa) E2 (GPa)
No. of iterations
Initial 422.41 24.73 1
Final 146.45 9.22
7
Initial 387.67 2.75 2
Final 146.45 9.22
8
Initial 413.83 18.4 3
Final 146.45 9.22
7
Initial 131.42 19.00 4
Final 146.45 9.22
3
Initial 171.35 23.65 5
Final 146.45 9.22
10
Global minimum 146.45 (0.03%)†
9.22 (0%)
Probability 0.997
†Value in parentheses denotes percentage difference between identified and actual data.
表5-28. 以理論應變值隨機多起始點第二階段所識別Gr/ep 複材板結 構[(30°/-30°)2]s之各項彈性常數值
Material constants Starting
point No.
Stage
E1(GPa) E2 (GPa)
No. of iterations Initial 419.61 23.24
1 Final 146.52 9.22 10
Initial 258.77 4.71 2 Final 146.52 9.22
8
Initial 449.55 1.97 3 Final 146.52 9.22
9
Initial 312.97 7.35 4 Final 146.52 9.22
6
Initial 346.63 19.04 5 Final 146.52 9.22
6
Global minimum 146.52 (0.01%)†
9.22 (0%)
Probability 0.997
†Value in parentheses denotes percentage difference between identified and actual data.
表5-29. 以理論應變值隨機多起始點第二階段所識別Gr/ep 複材板結 構[(60°/-60°)2]s之各項彈性常數值
表5-29. 以理論應變值隨機多起始點第二階段所識別Gr/ep 複材板結 構[(60°/-60°)2]s之各項彈性常數值