未來研究方向

w

本文採用 DKT 板元素的位移場，該元素沒有定義元素內部的側向位移

，本文在數值模擬時元素節點內力僅取到線性項且元素切線剛度矩陣僅 為近似的切線剛度矩陣，這些也許都會影響平衡迭代的收斂速度和偵測平 衡路徑上分歧點及挫屈模態的準確性，因此以後研究可以考慮把 DKT 板元 素的位移場替換成具有側向位移場 的三角板元素以及考慮元素節點內力 的二次項和完整的元素切線剛度矩陣。

w

參考文獻

[1] K. J. Bathe, L. W. Ho, “A simple and effective element for analysis of general shell structures”, Computer and Structure, vol. 13, pp. 673–681, 1981.

[2] C. Pacoste, “Co-rotational flat facet triangular element for shell instability”, Computer Methods in Applied Mechanics and Engineering, vol. 156, pp.

75–110, 1998.

[3] P. Khosravi, R. Ganesan, R. Sedaghati, “An efficient facet shell element for corotational nonlinear analysis of thin and moderately thick laminated composite structures”, Computer and Structure, vol. 86, pp. 850–858, 2008.

[4] K. M. Hsiao, “Nonlinear analysis of general shell structures by flat triangular shell element”, Computer and Structure, vol. 25, pp. 665–674, 1987.

[5] J. M. Battini, C. Pacoste, “On the choice of the linear element for corotational triangular shells”, Computer Methods in Applied Mechanics and Engineering, vol. 195, pp. 6362–6377, 2006.

[6] 楊禮龍, “薄殼結構在位移負荷作用之下的幾何非線性分析＂, 交通大 學機械工程學系碩士論文, 台灣, 新竹, 2006.

[7] 楊水勝, 拘限彈性薄板受側向位移負荷的行為研究, 交通大學機械工 程學系碩士論文, 台灣, 新竹, 2007.

[8] 林寬政, “平面三角形殼元素之改善研究＂, 交通大學機械工程學系碩 士論文, 台灣, 新竹, 2010.

[9] 盧致群,“高階平面三角形殼元素之研究＂, 交通大學機械工程學系碩 士論文, 台灣, 新竹, 2011.

[10] Z. X. Li, L. Vu-Quoc, “An efficient co-rotational formulation for curved triangular shell element”, International Journal for Numerical Methods in Engineering, vol. 72, pp. 1029–1062, 2007.

[11] P. Khosravi, R. Ganesan, R. Sedaghati, “Corotational non-linear analysis of thin plates and shells using a new shell element”, International Journal for Numerical Methods in Engineering, vol. 69, pp. 859–885, 2007.

[12] L. Kang, Q. Zhang, Z. Wang, “Linear and geometrically nonlinear analysis of novel flat shell elements with rotational degrees of freedom”, Finite Elements in Analysis and Design, vol. 45, pp. 386–392, 2009.

[13] N. Carpenter, H. Stolarski, T. Belytschko, “A flat triangular shell element with improved membrane interpolation”, Communications in Applied Numerical Methods, vol. 1, pp. 161–168, 1985.

[14] W. Chen, Y. K. Cheung, “Refined non-conforming triangular elements for analysis of shell structures”, International Journal for Numerical Methods in Engineering, vol. 46, pp. 433–455, 1999.

[15] J. G. Kim, J. K. Lee, Y. K. Park, “A new 3-node triangular flat shell element”, Communications in Numerical Methods in Engineering, vol. 18, pp. 153–159, 2002.

[16] J. M. Battini, “A modified corotational framework for triangular shell elements”, Computer Methods in Applied Mechanics and Engineering, vol.

196, pp. 1905–1914, 2007.

[17] 林育丞, “具旋轉自由度之三角平面元素的共旋轉推導法＂, 交通大學 機械工程學系碩士論文, 台灣, 新竹, 2008.

[18] J. L. Batoz, K. J. Bathe, L. W. Ho, “A study of three-node triangular plate bending elements”, International Journal for Numerical Methods in Engineer, vol. 15, pp. 1771–1812, 1980.

[19] A. Boudaoud, P. Patrício, Y. Couder, M. B. Amar, “Dynamics of singularities in a constrained elastic plate”, Nature, vol. 407, pp. 718–720, 2000.

[20] J. H. Argyris, I. Fried, D. W. Scharpf, “The tuba family of plate elements for the matrix displacement method”, The Aeronautical Journal of the Aeronautical Society, vol. 72, pp. 701–709, 1968.

[21] D. J. Allman, “A compatible triangular element including vertex rotations for plane elasticity analysis”, Computers and Structures, vol. 19, pp. 1–8, 1984.

[22] L. Damkilde, M. Gronne, “An improved triangular element with drilling rotations”, Proceedings of the 15th Nordic Seminar on Computational Mechanics, pp. 135–138, 2002.

[23] C. A. Felippa, “A study of optimal membrane triangles with drilling freedoms”, Computer Methods in Applied Mechanics and Engineering, vol.

192, pp. 2125–2168, 2003.

[24] G. P. Bazeley, Y. K. Cheung, B. M. Irons, O. C. Zienkiewicz, “Triangular elements in plate bending—conforming and nonconforming solutions”, Proceeding First Conference on Matrix in Structural Mechanics, Air Force Institute of Technology, Dayton, Ohio, pp. 66–80, 1966.

[25] R. W. Clough, J. L. Tocher, “Finite element stiffness matrices for analysis of plate bending”, Proc. Conf. on Matrix Methods in Structural Mechanics, pp. 515–545, 1965.

[26] J. A. Stricklin, W. E. Haisler, P. R. Tisdale, R. Gunderson, “A rapidly converging triangular plate element”, AIAA Journal, vol. 7, pp. 180–181, 1969.

[27] D. J. Allman, “Triangular finite element bending with constant and linearly varying bending elements”, High Speed Comput. Elastic Struct. Tom 1, pp.

105–107, 1971.

[28] J. L. Batoz, “An explicit formulation for an efficient triangular plate-bending element”, International Journal for Numerical Methods in Engineering, vol. 18, pp. 1077–1089, 1982.

[29] G. R. Cowper, E. Kosko, G. M. Lindberg, M. D. Olson, “Static and dynamic applications of a high-precision triangular plate bending element”, AIAA Journal, vol. 7, pp. 1957–1965, 1969.

[30] K. Y. Sze, X. H. Liu, S. H. Lo, “Popular benchmark problems for geometric nonlinear analysis of shells”, Finite Elements in Analysis and Design, vol. 40, pp. 1551–1569, 2004.

[31] E. H. Boutyour, H. Zahrouni, M. Potier-Ferry, M. Boudi, “Bifurcation points and bifurcated branches by an asymptotic numerical method and Pade approximants”, International Journal for Numerical Methods in Engineering, vol. 60, pp. 1987–2012, 2004.

[32] V. V. Kuznetsov, S. V. Levyakov, “Phenomenological invariant-based finite-element model for geometrically nonlinear analysis of thin shells”, Computer Methods in Applied Mechanics and Engineering, vol. 196, pp.

4952–4965, 2007.

[33] E. H. Boutyour, L. Azrar, M. Potier-Ferry, “Vibration of buckled elastic structures with large rotations by an asymptotic numerical method”, Computers and Structures, vol. 84, pp. 93–101, 2006.

[34] A. Eriksson, C. Pacoste, “Element formulation and numerical techniques for stability problems in shells”, Computer Methods in Applied Mechanics and Engineering, vol. 191, pp. 3775–3810, 2002.

[35] P. Patrício, W. Krauth, “Numerical solutions of the Von Karman equations for a thin plate”, Journal of Modern Physics, vol. 8, pp. 427–434, 1997.

[36] K. Wisniewski, “A shell theory with independent rotations for relaxed Biot stress and right stretch strain”, Computational Mechanics, vol. 21, pp.

101–122, 1998.

[37] C. Sansour, H. Bufler, “An exact finite rotation shell theory, its mixed variational formulation and its finite element implementation”, International journal for numerical methods in engineering, vol. 34, pp.

73–115, 1992.

[38] K. Wisniewski, E. Turska, “Kinematics of finite rotation shells with in-plane twist parameter”, Computer Methods in Applied Mechanics and Engineering, vol. 190, pp. 1117–1135, 2000.

[39] M. Li, F. Zhan, “The finite deformation theory for beam, plate and shell.

Part IV. The FE formulation of Mindlin plate and shell based on Green-Lagrangian strain”, Computer Methods in Applied Mechanics and Engineering, vol. 182, pp. 187–203, 2000.

[40] C. Sansour, H. Bednarczyk, “The Cosserat surface as a shell model, theory and finite-element formulation”, Computer Methods in Applied Mechanics and Engineering, vol. 120, pp. 1–32, 1995.

[41] L. E. Malvern, Introduction to the Mechanics of a Continuous Medium, Prentice-Hall Englewood Cliffs, N. J. ,1969

[42] Th. v. Kármán, Festigkeistprobleme im Maschinenbau.,

Encyklopädie der Mathematischen Wissenschaften, 4/4, 311-385, 1910.

[43] P. G. Ciarlet, “A Justification of the von Karman Equations”, Archive for Rational Mechanics and Analysis, vol. 73, pp. 349–389, 1980.

[44] J. M. Battini, C. Pacoste, “On the choice of local element frame for corotational triangular shell elements”, Communications in Numerical Method in Engineering, vol.20, pp. 819–825, 2004

[45] H. Goldstein, Classical Mechanics, Addision-Wesley Publishing Company, 1980.

[46] K. M. Hsiao, “Corotational Total Lagrangian Formulation for Three-Dimensional Beam Element”, AIAA Journal, 30, No. 3, pp. 797–804, 1992.

[47] D. J. Dawe, Matrix and Finite Element Displacement Analysis of Structures, Clarendon Press, 1984.

[48] T. J. Chung, Continuum Mechanics, Prentice-Hall, N. J.,1988

[49] R. D. Cook, Concepts and Applications of Finite Element Analysis, John Wiley & Sons, 1981

[50] O. C. Zienkiewicz, The Finite Element Method in Engineering Science, CENTRAL BOOK CO., 1971.

[51] A. P. Boresi, K. P. Chong, Elasticity In Engineering Mechanics, John Wiley & Sons, 1999.

[52] M. A. Crisfield, “A fast incremental/iterative solution procedure that handles snap-through”, Computers and Structures, vol. 13, pp.55–62, 1981 [53] B. Nour-Omid, C. C. Rankin, “Finite rotation analysis and consistent

linearization using projectors”, Computer Methods in Applied Mechanics and Engineering, vol. 93, pp. 353–384, 1991.

[54] J. M. Battini, “A non-linear corotational 4-node plane element”, Mechanics Research Communications, vol. 35, pp. 408–413, 2008.

[55] K. Wisniewski, E. Turska, “Four-node mixed Hu-Washizu shell element with drilling rotation”, International Journal for Numerical Methods in Engineering, vol. 90, pp. 506–536, 2012

[56] M. Li, F. Zhan, “The finite deformation theory for beam, plate and shell.

Part V. The shell element with drilling degree of freedom based on Biot strain”, Computer Methods in Applied Mechanics and Engineering, vol.

189, pp. 743–759, 2000.

表 4.1 圓柱殼片段受到單點集中力作用的挫屈負荷(例題 4.3，Mesh 10×10)



Tx Type of Kg Buckling load

(a) (0) 528.949

(a) (1) 528.946

(a) (2) 528.726

(b) (0) 529.166

(b) (1) 529.163

(b) (2) 529.093

表 4.2 槽型斷面梁的挫屈負荷 (例題 4.4)

Mesh T_x Type of Kg Buckling load

Present (2+2+2)×14 (a) (0) 3022.17 (2+2+2)×14 (a) (1) 2999.39 (2+2+2)×14 (a) (2) 3005.15 (2+2+2)×14 (b) (0) 3137.76 (2+2+2)×14 (b) (1) 3112.96 (2+2+2)×14 (b) (2) 3119.37 (4+4+4)×60 (a) (0) 2969.00 (4+4+4)×60 (a) (1) 2957.80 (4+4+4)×60 (a) (2) 2972.12 (4+4+4)×60 (b) (0) 2995.84 (4+4+4)×60 (b) (1) 2984.41 (4+4+4)×60 (b) (2) 2999.13 ref.[5] (2+2+2)×15 - - 2618.26

(4+4+4)×60 - - 2529.29

表 4.3 受壓之簡支承板的挫屈負荷 (例題 4.5)

Mesh Tx Type of Kg Buckling load

Present 4×6 (a) (0) 60.9536

4×6 (a) (1) 60.9551

4×6 (a) (2) 60.7836

4×6 (b) (0) 63.0364

4×6 (b) (1) 63.0380

4×6 (b) (2) 62.8510

4×10 (a) (0) 59.3858

4×10 (a) (1) 59.3903

4×10 (a) (2) 59.2272

4×10 (b) (0) 61.7979

4×10 (b) (1) 61.8026

4×10 (b) (2) 61.6275

6×12 (a) (0) 58.5703

6×12 (a) (1) 58.5704

6×12 (a) (2) 58.4311

6×12 (b) (0) 59.5753

6×12 (b) (1) 59.5754

6×12 (b) (2) 59.4167

8×20 (a) (0) 58.1172

8×20 (a) (1) 58.1191

8×20 (a) (2) 57.9698

8×20 (b) (0) 58.5838

8×20 (b) (1) 58.5882

8×20 (b) (2) 58.4232

ref.[5] 4×10 - - 59.1703

8×20 - - 58.4762

表 4.4 T 型斷面梁的挫屈負荷 (例題 4.6)

Mesh T_x Type of Kg Buckling load Present (2+2+4)×20 (a) (0) 2993.08

(2+2+4)×20 (a) (1) 2992.78 (2+2+4)×20 (a) (2) 2990.83 (2+2+4)×20 (b) (0) 3098.05 (2+2+4)×20 (b) (1) 3097.52 (2+2+4)×20 (b) (2) 3095.79 (3+3+5)×30 (a) (0) 2942.77 (3+3+5)×30 (a) (1) 2942.56 (3+3+5)×30 (a) (2) 2940.45 (3+3+5)×30 (b) (0) 3005.97 (3+3+5)×30 (b) (1) 3005.67 (3+3+5)×30 (b) (2) 3003.68 (5+5+8)×50 (a) (0) 2863.83 (5+5+8)×50 (a) (1) 2863.71 (5+5+8)×50 (a) (2) 2861.54 (5+5+8)×50 (b) (0) 2890.41 (5+5+8)×50 (b) (1) 2890.27 (5+5+8)×50 (b) (2) 2888.11 ref.[5] (2+2+4)×20 - - 3103.86

(5+5+8)×50 - - 2890

表 4.5 槽型斷面梁的極限點 (例題 4.7)

Mesh T_x W_J^L Limit point Present (1+2+1)×20 (a) 19.9372 1111.40

(1+2+1)×20 (b) 19.6097 1159.65 (2+4+2)×40 (a) 20.4243 1112.90 (2+4+2)×40 (b) 21.1579 1128.17 ref.[5] (1+2+1)×20 - - 1011.92

(3+8+3)×56 - - 1080.86

*W_J^L為 J 點在極限點的側向位移

表 4.6 圓柱殼受四個徑向集中力作用的挫屈負荷 (例題 4.12)

Mesh T_x Kg B₁ B₂ B₃

 ^W_A^/R  ^W_A^/R  ^W_A^/R

Present 8×48 (a) (0) 6.037 0.0673 10.537 0.1202 11.738 0.1370 8×48 (a) (1) 6.033 0.0673 10.537 0.1202 11.738 0.1370 8×48 (a) (2) 6.033 0.0673 10.537 0.1202 11.738 0.1370 8×48 (b) (0) 6.626 0.0766 10.987 0.1244 11.839 0.1410 8×48 (b) (1) 6.626 0.0766 10.987 0.1244 11.841 0.1410 8×48 (b) (2) 6.619 0.0765 10.986 0.1244 11.816 0.1409

12×72 (a) (0) 6.219 0.0708 10.382 0.1193 11.539 0.1363 12×72 (a) (1) 6.219 0.0708 10.382 0.1193 11.539 0.1363 12×72 (a) (2) 6.218 0.0709 10.382 0.1193 11.538 0.1363

12×72 (b) (0) 6.516 0.0754 10.489 0.1202 11.575 0.1371 12×72 (b) (1) 6.515 0.0754 10.489 0.1202 11.575 0.1371 12×72 (b) (2) 6.508 0.0752 10.489 0.1202 11.574 0.1371

ref.[32] 8×48 - - 6.590 0.0767 10.210 0.1179 11.295 0.134 12×72 - - 6.397 0.0738 10.251 0.1183 11.351 0.135

表4.7開口型半球殼 B點位移之線性解(例題4.13，P = 1)

100

V_B

4×4 8×8 12×12 16×16 24×24 64×64 Present 8.3123 8.9851 9.2743 9.3503 ref.[55] 9.2866 9.3824 9.3839 9.3714

表4.8開口型半球殼 B點位移的非線性解(例題4.13，P = 400) Mesh ^T_x V_B

Present 12×12 (a) 7.934

12×12 (b) 7.790

16×16 (a) 8.045

16×16 (b) 7.954

24×24 (a) 8.111

24×24 (b) 8.076

ref.[30] 12×12 - 8.178 16×16 - 8.178 ref.[39] 16×16 - 8.112

24×24 - 8.148 ref.[56] 16×16 - 7.629

24×24 - 7.714

表4.9 聚酯圓柱薄殼受兩階段負荷作用的挫屈負荷 (例題4.15)

Mesh T _x Type of Kg B ₁ W_E B ₂ W_E 24×48 (a) (0) 9.5614 9.3320 18.2032 13.5716 24×48 (a) (1) 9.5674 9.3358 18.1976 13.5712 24×48 (a) (2) 9.5613 9.3313 18.2121 13.5710 24×48 (b) (0) 10.8078 9.9395 16.4032 13.4939 24×48 (b) (1) 10.7919 9.9314 16.3995 13.4940 24×48 (b) (2) 10.7903 9.9301 16.3948 13.4941 30×60 (a) (0) 9.4440 9.3173 18.2274 13.6123 30×60 (a) (1) 9.4440 9.3173 18.2255 13.6123 30×60 (a) (2) 9.4299 9.3098 18.2310 13.6123 30×60 (b) (0) 10.0587 9.6315 16.5281 13.5306 30×60 (b) (1) 10.0579 9.6311 17.9329 13.5969 30×60 (b) (2) 10.0555 9.6299 17.9362 13.5967

*使用Case(b)分析過程中，T_x僅對剛度作用

圖 1.1 文獻[19]實驗所觀察到四種變形轉換(a-d)及 對應於a-c 圖結構的上視圖(e-g)

圖 2.1 旋轉向量 a

b



b

圖 2.2 薄殼中 P、Q 點之位移以及元素座標與中平面座標之關係圖 ,n

3

xS

Q

x₂S dx x₁S

x₁E

x₂E

x₃E

P Q

u

v w P

3 3 0x^S,e

x₁S 0

x₂S

0 dX

圖 2.3 元素節點 j 中心面之x 軸受旋轉向量_ij^B θ_nj作用的示意圖

θ

nj



Bj

x

₁

Bj

x

₂

Bj

x

₃

Bj E

x x

₃

,

₃^

θ

nj

j

Bj

x

₁ ^

Bj

x

₁ ^

0

Bj

x

₂^

Bj

x

₂^

0

j



3

j



3

圖 2.4 元素節點 j 中心面之⁰x₁^B_j^軸受旋轉向量_{3j 3}e^E作用的示意圖

E Bj Bj

x x x

_{3 3 3}

0 ^

,

^

,

j

(a)

(b) (c)

圖 4.1 半圓環受到單點集中力作用 (a)結構尺寸示意圖 (b)網格 18×1 示意圖 (c)網格 18×2 示意圖

F 2

A

1 d

20 R

 V X₂^G,

U X₁^G, 25 . 0 , 10⁷ 

 v E

A A

0 5 10 15 20 25 30 0

2 4 6 8 10 12

Load F (10

3

)

Displacement -V

_A

ref.[54]

(a) (b)

20x1 40x2

圖4.2 半圓環受到單點集中力作用之負荷－位移曲線圖

(a)

(b)

圖4.3 直角構架受到端點剪力作用 (a)結構尺寸示意圖 (b)網格M21與網格M22示意圖

F A

9

1 1

9 U X₁^G, V

X₂^G,



1 , 3 . 0 , 10

3 ⁷  

 t

E 

M21 M22

0 1 2 3 4 5 6 7 0

1 2 3 4

Load F (10

4

)

Displacement U

_A

ref.[54]

(a) (b)

M21 M22

圖4.4 直角構架受到端點剪力作用之負荷－位移曲線圖

(a)

(b) (c)

圖4.5 圓柱殼片段受到單點集中力作用 (a)結構尺寸示意圖 (b)力負荷圖 (c)網格10×10 示意圖

L=254mm R=2540mm h=6.35mm θ=0.1rad

E=3102.75MPa ν=0.3

L

L

R

h

 

E D

A

C

B

R R

 

E U

X₁^G, W

X₃^G,

F

A B

C D

0 5 10 15 20 25 30 -400

-200 0 200 400 600 800

Load F (N)

Displacement -W

_E

(mm)

ref.[31]

(a) (b)

Primary path Secondary path

圖4.6 圓柱殼片段受到單點集中力作用之負荷－位移曲線圖

(a)

(b)

圖4.7 槽型斷面梁之側向扭轉挫屈(例題4.4) (a)結構尺寸示意圖

150

L E 21000

10

b  0.3

10 h

1 t

A B

C D

E F

G I

t

W X₃^G,

V X₂^G,

U X₁^G,

 I

G

F E

D

C B

A

b

h L

(a)

100

L b50 t 2

100

E  0.3

(b)

圖4.8 受壓之簡支承板 (a)結構尺寸示意圖 (b)網格 4×6示意圖

A B

C D

A B

C D

 

U X₁^G, V X₂^G,

L

b

(a)

(b)

圖4.9 T型斷面梁 (a)結構尺寸示意圖 (b)網格(2+2+3)×4示意圖

A C B

E G F

D I

C G P

W X₃^G,

V X₂^G,

U X₁^G,

t I

G F

E

D

C B

A

J

L

b

h

450

L E 70960

38

b  0.321

65 h

1 t

(a)

(b)

圖4.10 槽型斷面梁(例題4.7) (a)結構尺寸示意圖 (b)網格(1+2+1)×2示意圖

A B

C D

E F

G I

1100

L E 2.110⁶

75

b  0.3

25 h

h

C B X₁^G,U V

X₂^G,

W X₃^G,

D A

E

F G

I

K J

M

 

P

P

L t b

0 5 10 15 20 25 30 35 40 45 0

200 400 600 800 1000 1200

L o ad factor 

Displacement -W

_J

ref.[16]

(a) (b)

(1+2+1) x 20 (2+4+2) x 40

圖 4.11槽型斷面梁之負荷－位移曲線圖(例題4.7)

(a)

(b)

圖4.12 直角梁受到單點集中力作用(例題4.8) (a)結構尺寸示意圖 (b)網格(2+3)×2 示意圖

1400

L E 193300 75

.

47

b  0.3 75

.

72 h

5 .

2 t

A

B

C

D E

F

U X₁^G, W X₃^G,

A

B

C

D

F

E P

V X₂^G,

L t

b

h

0 20 40 60 80 100 0

2 4 6 8 10

Load P (10

3

)

Displacement U

_A

ref.[5] OPT ref.[5] ALL (a) (b)

(2+2) x 25 (4+6) x 60

圖4.13 直角梁受到單點集中力作用之負荷－位移曲線圖(例題4.8)

(a)

(b)

圖4.14 懸臂圓柱殼受到單點集中力作用(例題4.9) (a)結構尺寸示意圖 (b)網格16×16示意圖

016 .

1

R E2.068510⁷ 048

.

3

L  0.3 03

.

0 t

A

B

C D

t

F

F

L U X₁^G, V

X₂^G,

W X₃^G,

end Free

end Clamped B

C D

A

R

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0

200 400 600 800

Load F

Displacement -W

_A

/R ref.[16]

ref.[30]

(a) (b)

16x16 32x32 40x40

圖4.15 懸臂圓柱殼受到單點集中力作用之負荷－位移曲線圖(例題4.9)

(a)

10

R t0.5 104



E  0.2

(b)

圖 4.16 半球殼受到單點集中力作用(例題4.10) (a)結構尺寸示意圖 (b)網格 12×12示意圖

Bj

x₃ x₂^B_j

Bj

x₁ R

W X₃^G,

V X₂^G, U

X₁^G,

2F

B A

t

F F 2

2

F 2

C

B A

C

圖 4.17 半球殼受到單點集中力作用之負荷－位移曲線圖(例題 4.10)

0 1 2 3 4 5 6 7 8 9

0 5 10 15 20 25 30 35 40 45

Load F

Displacement

ref.[16]

(a) (b)

12x12

UB V_A

(a)

953 .

4

R E 10.510⁶ 35

.

10

L  0.3125 094

.

0 t

(b)

圖4.18 圓柱殼受一對集中力作用(例題4.11) (a)結構尺寸示意圖 (b)網格 16×24示意圖

U X₁^G,

V X₂^G, W

X₃^G,

t R

P P

L A

B C

D

圖 4.19 圓柱殼受一對集中力作用之負荷－位移曲線圖(例題4.11)

0 1 2 3 4 5

0 1 2 3 4

P (10 4 )

Displacement ref.[30]

(a) (b)

16x24 24x36

WA

UB



UB C 

U

UC



(a)

m

R0.1 E 210¹¹ N /m m

L0.1  0 m h0.001

(b)

圖 4.20 圓柱殼受四個徑向單點集中力作用(例題4.12) (a)結構尺寸示意圖 (b)網格8×48 示意圖

P P

P P

h R

L U

X₁^G, V X₂^G, W X₃^G,

A

B

C D

圖4.21 圓柱殼受四個徑向單點集中力作用之負荷－位移曲線圖(例題 4.12)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0 4 8 12 16 20



Displacement -W

_A

/R

ref.[32]

(a) (b)

8x48 12x72

path Primary

path Secondary

(a)

10

R E 6.82510⁷ 04

.

0

t  0.3

(b)

圖4.22 開口型半球殼受集中力作用(例題4.13) (a)結構尺寸示意圖 (b)網格16×16示意圖

U

X₁^G, X₂^G,V W X₃^G, R

P P P

P t

A B

18

圖4.23 開口型半球殼受集中力作用之負荷－位移曲線圖(例題4.13)

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4

P (10

2

)

Displacement

ref.[30]

ref.[39]

(a) (b)

12x12 16x16 24x24

UA

VB



(a)

6

Ri E 2110⁶

10

Ro  0 03

.

0

t

(b)

圖4.24 裂縫環形板受均勻力負荷作用(例題4.14) (a)結構尺寸示意圖 (b)網格6×30 示意圖

U

X₁^G, X₂^G,V W

X₃^G,

Ri

Ro

P

0 2 4 6 8 10 12 14 16 18 20 0.0

0.2 0.4 0.6 0.8

P

Displacement ref.[30]

(a) (b)

6x30 10x80

圖4.25 裂縫環形板受均勻力負荷作用之負荷－位移曲線圖(例題 4.14) WB

WA

(a)

(b)

(c)

圖4.26圓柱薄殼(例題 4.15) (a)結構示意圖(b)俯視圖(c)前視圖

E D

A B

C G

F

I

H X

Y

35

L

cm

5 .

17

W

cm

5 .

16

d

cm

35 .

0

h

mm

20

 

2 9Nm 10 8 .

3  ^

 E

4 .

0



 

d

L W

h

Z

X X

X Z Z

 

F H

0.0 0.1 0.2 0.3 0.4 0

10 20 30

Displacement W

E

(mm)

Loading parameter  rad 

ref[7]

30x60

圖4.27圓柱薄殼第一階段 E 點之位移－負荷參數曲線圖(例題 4.15)

0 5 10 15 20 25 0

5 10 15 20 25

Reaction (N)

Loading parameter 

_E

(mm)

ref.[19] numerical ref.[19] experiment

ref.[7]

30 x 60

圖 4.28 圓柱薄殼第二階段 E 點之反力－負荷參數曲線圖(例題4.15)

附錄 A 元素座標系統的決定

由(A.2)、(A.3)和(A.4)式可得：

)

current

initial

C 

x₁E^

x₂E^{ 0}x₂^E,x₂^E

E E x x_{1 1}

0 ,

圖 A.1 元素座標示意圖

附錄 B 切線剛度矩陣

k

fixed

j



附錄 C 面積座標(area coordinates)

C.1 面積座標的定義

如圖C.1所示，x、y為三角形中任一點 在直角座標中的座標值，將 點與三角形的三個頂點作連線，就形成了三個小三角形，三個頂點1、2、3 相對應的三個小三角形的面積分別為 、 、 ，令面積座標

P

A2

P

A1 A₃

A A₁

  (C.1)

A A₂

  (C.2)

A A₃

  (C.3)



_{21 31 31 21}

2

1 x y x y

A 



^(C.4)

j i

ij x x

x   (C.5)

j i

ij y y

y   (C.6)

其中A為三角形123 的面積，x_i、 代表三角形頂點y_i i的x和 座標。y 、、

稱為三角形中P點的面積座標，固 A

A A

A₁  ₂  ₃  (C.7)

由(C.1)式至(C.3)式與(C.7)式可以得出

1



 

 (C.8)

因、 、之間不是互相獨立的，因此在本文中僅用 、表示三角形中 任意點的面積座標，如圖C.2。

C.2 面積座標與直角座標的關係

P點之面積座標與直角座標之間的關係可表示成[49]



圖 C.1 面積座標表示方法

圖C.2 面積座標示意圖

P

¹

A A

2

A

3

1

2 y 3

(0,0)

x



1 2

3

(1,0) (0,1)



附錄 D 及其微分

附錄 E DKT 元素的形狀函數

4)

)

圖 E.1 DKT 元素的節點及其三邊上的局部座標示意圖

圖 E.2 元素面積座標的示意圖

s

s

s 

12



13



²³

n

12

n

13

n

23

x

₁E

x

₂E

5

1

3

6

4

2

x y

A

1

A

2

A

3

1

2 3

P

附錄 F Projector matrix

文與文獻[53]的三角形殼元素有相同節點位移和旋轉自由度，所以文獻 3]的 Projector matrix 僅要稍加修改，即可適用於本文所推導的殼元素。

殼元素在當前的變形位置，受到節點擾動位移 作用，則在擾動後的元

中 為考慮 Projector matrix 時的元素節點內力， 為不考慮 Projector

atrix 時的元素節點內力。

文之 Projector matrix 可表示成[53]：

(F.3)



p為考慮 Projector matrix k (spin matrix)

~e

附錄 G 文獻[19]的實驗說明

文獻[19] 驗是探討一聚 圓柱薄殼受位移負荷作用後的非線 為，文中將 片的兩個長邊

端相距一固定距離 ，並與水平面維持一固定角

的實 酯 性行

一薄 以夾鉗固定，再將其彎成柱狀殼，兩邊夾持

cm

d 16.5 度 20^(如圖 G.1)，然後在薄殼中心施加一向下的位移負荷，薄片的尺寸和材料參數為長 度 L35cm 、 寬 度 W 17.5cm 、 厚 度 h0.35mm 、 楊 氏 係 數

2

9 /

10 8 .

3 N m E 

中心附近出現兩個對稱 個變形是中心向下位移

mm 5 . 12

心向下位移15

。在

X、Y mm 5 . 11

成一個對薄殼中心轉了一個角度的菱形 d-cone

其實驗中隨著位移負荷的增加

時，薄殼

出現兩 (圖 G.2 的連線形成一個梯形

d-cone 產生

， 薄殼變成波浪狀的圓柱殼 的長度若不夠長，則無法觀察到菱形及

早產生波浪狀的柱狀殼。

，結

b (圖

移到薄殼自由端的邊界時

構連續產生四個特 mm

4

， 殊的幾何變形，如圖 G.2 所示。第一個變形是中心向下位移

軸的 d-cone (developable cone) (圖 G.2a)。第二 個新的 d-cone，而四個 d-cone 圍 時，

)。第三個變形是中心向下位 G.2c)。第四個變形是中

移 時，四個

mm 時，梯形底邊兩個

一個不連續的變化 使 (圖 G.2d)。文獻[19]提到殼

後來的梯形圖形這兩種型態，會提

圖 G.1 柱狀薄板結構示意圖

35

L

cm

5 . 17

W

cm

5 . 16

d

cm

35 . 0

h

mm









X Y Z

20



2

9 /

10 8 .

3 N m E 

4 .

0 W

h



L

 

d

圖 G.2 文獻[19]實驗所觀察到四種變形轉換(a-d)及 對應於 a-c 圖結構的上視圖(e-g)

在文檔中 平面三角形薄殼元素之共旋轉推導法 (頁 79-144)