未來研究方向

由於本文的元素切線剛度矩陣為一近似剛度矩陣，這也許會影響平衡迭 代的收斂速度，因此以後研究可以考慮以非線性理論推導元素的切線剛度 矩陣，並探討其對平衡迭代的影響。因本研究使用的三角平面元素具面內 旋轉自由度及高精度，如果將此三角平面元素與三角板元素疊加成三角殼 元素，應可提高薄殼結構之幾何非線性分析的精度。

34

參考文獻

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[2] B. M. Fraejis de Veubeke, “Displacement and equilibrium models in the finite element method”, Stress Analysis (O. C. Zienkiewicz, G. Holister, eds.), 145-197, John Wiley & Sons, 1965.

[3] C. A. Felippa, “Refined finite element analysis of linear and nonlinear two-dimensional structures”, University of California, Ph. D. Dissertation, 1966.

[4] C. A. Felippa, “A study of optimal membrane triangles with drilling freedoms”, Computer Methods in Applied Mechanics and Engineering, 192, 2125-2168, 2003.

[5] C. A. Felippa, C. Militello, “Developments in variational methods for high performance plate and shell elements”, Analytical and Computational Models for Shells, The American Society of Mechanical Engineers, 3, 191-216, 1989.

[6] D. J. Allman, “A Compatible Triangular Element Including Vertex Rotations for Plane Elasticity Analysis”, Computers and Structures, 19, 1-8, 1984.

[7] P. G. Bergan, C. A. Felippa, “A triangular membrane element with rotational degrees of freedom”, Computer Methods in Applied Mechanics and Engineering, 50, 25-69, 1985.

[8] D. J. Allman, “The constant strain triangle with drilling rotations : a simple prospect for shell analysis”, The Mathematics of Finite Elements and Applications, 6, 233-240, 1987.

[9] D. J. Allman, “Evaluation of the constant strain triangle with drilling rotations”, International Journal of Numerical Methods in Engineering, 26, 2645-2655, 1988.

[10] D. J. Allman, “Variational validation of a membrane finite element with drilling rotations”, Communications in Numerical Methods in Engineering, 9, 345-351, 1993.

[11] C. A. Felippa, “The extended free formulation of finite elements in linear elasticity”, Journal of Applied Mechanics, 56, 609-616, 1989.

[12] K. Alvin, H. M. de la Fuente, B. Haugen, C. A. Felippa, “Membrane triangles with corner drilling freedoms : I. The EFF element’, Finite Elements in Analysis and Design, 12, 163-187, 1992.

[13] C. A. Felippa, C. Militello, “Membrane triangles with corner drilling freedoms : II. The ANDES element”, Finite Elements Analysis and Design, 12, 189-201, 1992.

[14] C. A. Felippa, S. Alexander, “Membrane triangles with corner drilling freedoms : III. Implementation and performance evaluation”, Finite Elements Analysis Design, 12, 203-239, 1992.

[15] A. B. Sabir, “A rectangular and triangular plane elasticity element with drilling degrees of freedom”, Proceedings of the second International Conference on Variational Methods in Engineering, 17-25, 1985.

[16] R. D. Cook, “A plane hybrid element with rotational d.o.f. and adjustable stiffness”, International Journal of Numerical Methods in Engineering, 1, 24, 1499-1508, 1987.

[17] R. D. Cook, “Modified formulations for nine-dof plane triangles that include vertex rotations”, International Journal of Numerical Methods in Engineering, 31, 825-835, 1991.

[18] R. Piltner, R. L. Taylor, “Triangular finite elements with rotational degrees of freedom and enhanced strain modes”, Computers and Structures, 75, 361-368, 2000.

36

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

[20] P. Fajman, “New triangular plane element with drilling degrees of freedom”, Journal of Engineering Mechanics, 128, 413-418, 2002.

[21] R. D. Cook, “Improved Two-Dimensional Finite Element”, Journal of the Structural Division, 100, 1851-1863, 1974.

[22] Gouri Dhatt, Gilbert Touzot, The Finite Element Method Displayed, John Wiley & Sons, 1984.

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

[24] Thomas J. R. Hughes, The Finite Element Method : Linear Static and Dynamic Finite Element Analysis, Prentice-Hall , 1987.

[25] O. C. Zienkiewicz, R. L. Taylor, The Finite Element Method : The Basis, Butterworth-Heinemann, 2000.

[26] I. Holand and P. G. Bergan, “Higher order finite element for plane stress”, Discussion, Journal of the Engineering Mechanics Division, 2, 698-702, 1968.

[27] K. M. Hsiao, “Nonlinear analysis of general shell structures by flat triangular shell element”, Computers and Structures, 25, 665-675, 1987.

[28] C. Pacoste, “Co-rotational flat facet triangular element for shell instability”, Computer Methods in Applied Mechanics and Engineering, 156, 75-110, 1998.

[29] K. D. Kim, G. R. Lomboy, S. C. Han, “A co-rotational 8-node assumed strain shell element for postbuckling analysis of laminated composite plates and shells”, Computational Mechanics, 30, 330-342, 2003.

[30] J. M. Battini, C. Pacoste, “On the choice of local element frame for corotational triangular shell elements”, Communications in Numerical Methods in Engineering, 20, 819-825, 2004.

[31] Y. Urthaler, J. N. Reddy, “A corotational finite element formulation for the analysis of planar beams”, Communications in Numerical Methods in Engineering, 21, 553-570, 2005.

[32] K. M. Hsiao, H. J. Horng, Y. R. Chen, “A corotational procedure that handles large rotation of spatial beam structures”, Computers and Structures, 27, 769-781, 1987.

[33] K. M. Hsiao, “Corotational total Lagrangian formulation for three- dimensional beam element”, American Institute of Aeronautics and Astronautics, 30, 797-804, 1992.

[34] K. M. Hsiao, “A co-rotational finite element formulation for buckling and postbuckling analyses of spatial beams”, Computer Methods in Applied Mechanics and Engineering, 188, 567-594, 2000.

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

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

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

[38] M. A. Crisfield, “A fast incremental/iterative solution procedure that handles 'snap-through'”, Computers and Structures, 13, 55-62, 1981.

[39] M. A. Crisfield, Non-linear Finite Element Analysis of Solids and Structures, John Wiley & Sons, 1991.

[40] Herbert Reismann, Peter S. Pawlik, Elasticity Theory and Applications, John Wiley & Sons, 1980.

[41] R. D. Cook, “On the Allman triangle and a related quadrilateral element”, Computers and Structures, 22, 1065-1067, 1986.

[42] A. Ibrahimbergovic, F. Frey, “Membrane quadrilateral finite elements with rotational degrees of freedom”, Engineering Fracture Mechanics, 43, 13-24, 1992.

38

[43] S. L. Sun, M. W. Yuan, and P. Chen, “A practical quadrilateral membrane element with drilling degrees of freedom”, Acta Mechanica Solida Sinica, 10, 179-188, 1997.

表 4.1 懸臂梁受到彎矩作用下端點 C 的側向位移V _C

Mesh subdivisions

Element

Mesh Type BC

Load

Type 1×2 2×2 4×2 8×2 16×2 32×2 ALL-3I [4] - BC12 LNS - 0.39 5.42 38.32 76.48 87.08 CST [4] - BC11 LNS - 1.28 4.82 15.75 36.36 54.05 FF84 [4] - BC12 LNS - 96.27 96.34 96.58 97.17 98.36 LST-Ret [4] - BC12 LNS - 9.46 28.93 59.58 81.04 89.05 OPT [4] - BC12 LNS - 100.07 99.96 99.99 99.99 99.99

Present M11 BC11 LNS 100.09 99.95 100.54 101.32 102.49 104.14 M11 BC11 M1S 100.39 100.55 100.90 101.52 102.65 104.37 M12 BC11 LNS 100.09 100.12 100.49 101.10 102.18 103.81 M12 BC11 M1S 100.45 100.52 100.70 101.24 102.33 104.01 M11 BC12 LNS 100.02 100.04 100.03 100.12 100.47 101.16 M11 BC12 M1S 100.37 100.64 100.39 100.33 100.64 101.39 M12 BC12 LNS 100.03 100.06 100.04 100.11 100.39 100.97 M12 BC12 M1S 100.43 100.45 100.25 100.25 100.53 101.17 M11 BC13 LNS 100.00 100.00 100.00 100.00 100.00 100.00 M11 BC13 M1S 100.33 100.60 100.36 100.21 100.17 100.23 M12 BC13 LNS 100.00 100.00 100.00 100.00 100.00 100.00 M12 BC13 M1S 100.38 100.41 100.21 100.14 100.15 100.21 M11 BC14 LNS 100.00 99.99 99.99 100.02 100.09 100.22 M11 BC14 M1S 100.33 100.58 100.35 100.23 100.26 100.44 M12 BC14 LNS 100.00 99.99 99.99 100.03 100.12 100.29 M12 BC14 M1S 100.38 100.40 100.20 100.17 100.27 100.49 BC11 U =0 at X =0 and V =0 at X = Y =0

BC12 U =θ =0 at X =0 and V =0 at X = Y =0 BC13 U =γxy =θ =0 at X =0 and V =0 at X = Y =0

BC14 U ⁼θ ⁼⁰ at X =0, 0γxy = at (X,Y) (= 0,±h 2), and V =0 at

=0

= Y X

LNS linear distributed normal stresses at free end, see figure 4.1(d).

M1S concentrated moment applied at mid-point of the side, see figure 4.1(e).

ALL-3I [9] Allman triangle element integrated by 3-point interior rule, published in 1988.

CST [1] constant strain triangle element.

FF84 [7] free formulation triangle element, published in 1984.

LST-Ret [4] retrofitting linear strain triangle element.

OPT [4] optimal membrane triangle element.

40

表 4.2 懸臂梁受到剪力作用下端點 C 的側向位移V_C

Mesh subdivisions

Element BC 2×1 2×2 4×1 8×2 16×4 32×8 4×1^D ALL-3I [4] BC21^* - 0.1514 - 0.3432 0.3510 0.3546 - CST [4] BC21 - 0.0635 - 0.1961 0.2940 0.3379 - FF84 [4] BC21 - 0.3179 - 0.3530 0.3550 0.3555 - LST-Reta [4] BC21^* - 0.2019 - 0.2523 0.3243 0.3485 - OPT [4] BC21 - 0.3284 - 0.3620 0.3571 0.3561 -

Present BC21 0.3574^† 0.3561 0.3729^† 0.4073 0.3804 0.3641 0.3709^† BC23 0.3331^† 0.3325 0.3474^† 0.3529 0.3549 0.3556 0.3459^† BC24 0.3331^† 0.3350 0.3474^† 0.3572 0.3558 0.3557 0.3459^†

AQ [42] BC21 - - 0.3283 - - - 0.3379 MQ3 [43] BC22 - - 0.3491 0.3524 0.3545 - 0.3212 SD6 [44] BC22 - - 0.3514 - - - 0.3454 BC21 U = V =0 at X =0

BC22 U = V =⁰ at (^X,Y) (= ^{0 h,}− ²) and U =0 at (^X,Y) (= ^{0 h, 2})

BC23 U =V =εy =γxy =θ =0 at X =0

BC24 U =V =εy =θ =0 at X =0 and γxy =0 at (X,Y) (= 0,±h 2)

ALL-3I [9] Allman triangle element integrated by 3-point interior rule, published in 1988.

CST [1] constant strain triangle element.

FF84 [7] free formulation triangle element, published in 1984.

LST-Ret [4] retrofitting linear strain triangle element.

OPT [4] optimal membrane triangle element.

AQ [42] Allman-type quadrilateral obtained from an eight-node quadrilateral.

MQ3 [43] quadrilateral element with cubic displacement interpolation.

SD6 [44] a practical quadrilateral membrane element.

* Requires one drilling freedom to be fixed, else stiffness is singular.

† VC =(VA +VB) 2

表 4.3 Cook 題目端點 C 的垂直位移V_C

Mesh subdivisions

Element BC 1×1 2×2 4×4 8×8 16×16 32×32 64×64 ALL-3I [4] BC31^* - 21.61 23.00 23.66 23.88 23.94 - CST [4] BC31 - 11.99 18.28 22.02 23.41 - - FF84 [4] BC31 - 20.36 22.42 23.41 23.79 23.91 - LST-Reta [4] BC31^* - 19.82 22.62 23.58 23.86 23.94 - OPT [4] BC31 - 20.56 22.45 23.43 23.80 23.91 23.95

Present BC31 32.93^† 25.65 24.86 24.29 24.08 24.01 23.98 BC32 23.00^† 23.59 24.00 23.99 23.97 23.97 23.97 BC33 23.00^† 23.08 23.78 23.89 23.93 23.95 23.96 BC31 U = V =0 at X =0

BC32 U =V =εy =γxy =θ =0 at X =0

BC33 U =V =εy =θ =0 at X =0 and γxy =0 at (^X,Y) ( ) (= ^{0,0 , 0,44}). ALL-3I [9] Allman triangle element integrated by 3-point interior rule, published in 1988.

CST [1] constant strain triangle element.

FF84 [7] free formulation triangle element, published in 1984.

LST-Ret [4] retrofitting linear strain triangle element.

OPT [4] optimal membrane triangle element.

* Requires one drilling freedom to be fixed, else stiffness is singular.

† VC =(VD+VE) 2

42

表 4.4 Cook 題目 A 點的最大主應力σ_A(max)與 B 點的最小主應力σ_B(min)

(max)

σA σ_B(min)

Element BC 2×2 4×4 8×8 2×2 4×4 8×8 CST [4] BC31 0.0760 0.1498 0.1999 -0.0360 -0.1002 -0.1567 FF84 [4] BC3 0.1700 0.2129 0.2309 -0.1804 -0.1706 -0.1902

Present BC32 0.2302 0.2380 0.2386 -0.1724 -0.2047 -0.2039

BC33 0.2301 0.2331 0.2363 -0.1861 -0.2066 -0.2041 BC34 0.2077 0.2322 0.2364 -0.1914 -0.2067 -0.2040

SD6 [44] BC31 0.1955 0.2308 0.2360 -0.1856 -0.1930 -0.1998 BC31 U = V =0 at X =0

BC32 U =V =εy =γxy =θ =0 at X =0

BC33 U =V =εy =θ =0 at X =0 and γxy =0 at (X,Y) ( ) (= 0,0 , 0,44). CST [1] constant strain triangle element.

FF84 [7] free formulation triangle element, published in 1984.

SD6 [44] a practical quadrilateral membrane element.

表 4.5 簡支梁受到兩端彎矩作用下的中心位移與端點轉角

Element Mesh Load Center displacement End rotation

M41 CM 1.5000 0.6000

M42 CM 1.2625 0.6172

M41 M4V 1.5003 0.7056

MQ3 [43]

M42 M4V 1.4160 0.6843

M41 CM 1.5000 0.6000

M42 CM 1.4473 0.5879

M41 M4V 1.5104 0.6519

SD6 [44]

M42 M4V 1.4556 0.6595

M41 CM 1.5036 0.6301

M42 CM 1.5078 0.6342

M43 CM 1.5098 0.6117

M41 M4V 1.5343 0.7451

M42 M4V 1.5435 0.7651

M43 M4V 1.5710 0.6518

M41 LNS 1.5000 0.6000

M42 LNS 1.5000 0.6000

Present^†

M43 LNS 1.5000 0.6000

CM couple moment applied at two ends of beam, see figure 4.7(c).

M4V concentrated moment applied at the four vertices, see figure 4.7(c).

LNS linear distributed normal stress, see figure 4.7(c).

MQ3 [43] quadrilateral element with cubic displacement interpolation.

SD6 [44] shear wall element of six degrees of freedom.

† Center displacment =(VA+VB) 2,Endrotation =-θ_C.

44

(a)

(b)

(c)

圖 1.1 (a)CST 元素節點位置與節點參數 (b)LST 元素節點位置與節點參數 (c)QST-10/20C 元素節點位置與節點參數

1

圖 2.1 固定總體座標、元素座標與節點基礎座標

46

圖 2.2 元素變形示意圖

u , x , x₁^E ₁^E

0

v , x , x₂^E ₂^E

0

u2

u3

v3

1

01

, ⁰2 2

x₁₂B 0

x₁₂B 0

x₁₂B

2

α

B 2

0

α

B

B

θ

2 03

3

Initial

Current

(a)

(b)

圖 4.1 網格示意圖 (a)4×2×2 網格 (b) 4×1×2 網格

48

(a)

25 . 0 , 768 ,

1 , 2 ,

32 = = = =

= h t E ν

L

(b)

(c)

(d) (e)

圖 4.2 懸臂梁受到彎矩作用 (a)結構尺寸示意圖 (b) M11 網格 8×2 示意圖 (c)M12 網格 8×2 示意圖 (d)負荷 LNS 示意圖 (e)負荷 M1S 示意圖

2 h

C M

C 2 h

0 2

6 th

= M σ

h U

,

X₁^G C

V , X₂^G

t

L θ M

2 L

(a)

25 . 0 , 30000 ,

1 , 12 ,

48 = = = =

= h t E ν

L

(b)

(c)

圖 4.3 懸臂梁受到剪力作用 (a)結構尺寸示意圖 (b)網格 8×2 示意圖 (c)扭曲網格 4×1^D示意圖 4

L L 4 L 4

6 L 3

L

12 L 2 L

U ,

X₁^G C P

θ

t V

, X₂^G

A

B L

h

50

(a)

-6 -3 0 3 6

-60 -30 0 30 60

σx

Y Exact 16×4 32×8

(b)

-6 -3 0 3 6

-1.0 -0.5 0.0 0.5 1.0

σy

Y Exact 16×4 32×8

(c)

-6 -3 0 3 6

0 1 2 3 4 5

τxy

Y Exact 16×4 32×8

圖 4.4 懸臂梁受到剪力作用不同網格下在X =12處的應力分佈 (a)

σ

_x (b)σ_y (c)τ_xy

(a)

-6 -3 0 3 6

-40 -20 0 20 40

σx

Y Exact 16×4 32×8

(b)

-6 -3 0 3 6

-1.0 -0.5 0.0 0.5 1.0

σy

Y Exact 16×4 32×8

(c)

-6 -3 0 3 6

0 1 2 3 4 5

τxy

Y Exact 16×4 32×8

圖 4.5 懸臂梁受到剪力作用不同網格下在X = 24處的應力分佈 (a)

σ

_x (b)σ_y (c)τ_xy

52

(a)

-6 -3 0 3 6

-20 -10 0 10 20

σx

Y Exact 16×4 32×8

(b)

-6 -3 0 3 6

-1.0 -0.5 0.0 0.5 1.0

σy

Y Exact 16×4 32×8

(c)

-6 -3 0 3 6

0 1 2 3 4 5

τxy

Y Exact 16×4 32×8

圖 4.6 懸臂梁受到剪力作用不同網格下在X =36處的應力分佈 (a)

σ

_x (b)σ_y (c)τ_xy

(a)

(b)

圖 4.7 Cook 題目 (a)結構尺寸示意圖 (b)2×2 網格示意圖 16

44

24 24

P B C

A

d distribute uniformly

1 load shear

total P =

U X₁^G, V

X₂^G,

θ

D

E

1/3 ,

1 ,

1 = =

= E ν

t

54

(a)

0 , 10 ,

1 , 10 ,

1000 = = = ⁵ =

= h t E ν

L andν =0.25

(b)

(c)

(d)

圖 4.9 簡支梁受到兩端彎矩作用 (a)結構尺寸示意圖 (b)網格 8×1 示意圖 (c)負荷 LNS 示意圖 (d)負荷 M4V 示意圖

2 M

2 M

2 M

2 M

σ

0

σ

₀ = 6M th²

h 2 L

h t

A C

θ

B

M M

V , X₂^G

U , X₁^G

L 2

56

0.0 0.1 0.2 0.3 0.4

0 1 2 3

Displacements V_C /L

-U_B /L

θ_A /2π

Beam 20×1 50×1 100×1

ML/EI

圖 4.10 簡支梁受到兩端彎矩(LNS)作用，不同網格下之 無因次負荷位移曲線圖。

0.0 0.1 0.2 0.3 0.4

0 1 2 3

Displacements θ_A /2π

-U_B /L

V_C /L

ML/EI

Beam LNS M4V

圖 4.11 簡支梁(Mesh 20×1)受到兩端彎矩作用，不同負荷型態下之 無因次負荷位移曲線圖。

圖 4.12 簡支梁(Mesh 20×1)受到兩端彎矩(LNS)作用，不同蒲松比下之 無因次負荷位移曲線圖。

0.0 0.1 0.2 0.3 0.4

0 1 2 3

Displacements V_C /L -U_B /L

θ_A /2π

Beam ν = 0 ν = 0.25

ML/EI

58

(a)

=1000

L andL=10000,h =10 ,t =1 ,E =10⁵,ν =0andν = 0.25

(b)

(c)

(d) (e)

圖 4.13 懸臂梁受到彎矩作用 (a)結構尺寸示意圖 (b)M61 網格 8×1 與 8×2 示意圖 (c)M62 網格 8×1 與 8×2 示意圖 (d)負荷 LNS 示意圖

(e)負荷 M2V 示意圖

h 2 L 2

h 2

B C A

M 2 M 2

h 2 A

2 0 = 6M /th B

σ

C

h 2

L

A

B θ C

M t U h

, X₁^G V

, X₂^G

0.0 0.2 0.4 0.6 0.8 1.0 1.2 0

2 4 6

−U_C /L

ML/EI

Beam

20×1(M61) 20×1(M62) 50×1(M61) 50×1(M62) 100×1(M61) 100×1(M62)

圖 4.14 懸臂梁(L =1000)受到彎矩(LNS)作用，不同單層網格下的 M -U_C曲線圖。

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0 2 4 6

ML/EI

V_C /L

Beam

20×1(M61) 20×1(M62) 50×1(M61) 50×1(M62) 100×1(M61) 100×1(M62)

圖 4.15 懸臂梁(L =1000)受到彎矩(LNS)作用，不同單層網格下的 M -V_C曲線圖。

60

0.0 0.2 0.4 0.6 0.8 1.0

0 2 4 6

ML/EI

θ_C /2π

Beam

20×1(M61) 20×1(M62) 50×1(M61) 50×1(M62) 100×1(M61) 100×1(M62)

圖 4.16 懸臂梁(L =1000)受到彎矩(LNS)作用，不同單層網格下的 M -θ_C曲線圖。

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0 2 4

6 Beam

20×1(M61) 20×2(M61) 40×1(M61) 40×2(M61)

ML/EI

−U_C /L

圖 4.17 懸臂梁(L=1000)受到彎矩(LNS)作用，不同單層及雙層網格下的 M -U_C曲線圖。

0.0 0.2 0.4 0.6 0.8 1.0 1.2 0

2 4 6

Beam

20×1(M61) 20×2(M61) 40×1(M61) 40×2(M61)

ML/EI

V_C /L

圖 4.18 懸臂梁(L=1000)受到彎矩(LNS)作用，不同單層及雙層網格下的 M -V_C曲線圖。

0.0 0.2 0.4 0.6 0.8 1.0

0 2 4

6 Beam

20×1(M61) 20×2(M61) 40×1(M61) 40×2(M61)

ML/EI

θ_C /2π

圖 4.19 懸臂梁(L=1000)受到彎矩(LNS)作用，不同單層及雙層網格下的 M -θ_C曲線圖。

62

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0 2 4 6

Displacements θ_C /2π

V_C /L

-U_C /L

Beam

50×1(M61) 100×1(M61) 200×1(M61)

ML/EI

圖 4.20 懸臂梁(L=10000)受到彎矩(LNS)作用，不同網格下之 無因次負荷位移曲線圖。

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0 2 4 6

Displacements

-U_C /L θ_C /2π

V_C /L

ML/EI

Beam LNS M2V

圖 4.21 懸臂梁(L=1000、M61 網格 100×1)受到彎矩作用，不同負荷型態 下之無因次負荷位移曲線圖。

圖 4.22 懸臂梁(L=1000、M61 網格 100×1)受到彎矩(LNS)作用，不同蒲松 比下之無因次負荷位移曲線圖。

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0 2 4 6

Displacements

-U_C /L θ_C /2π V_C /L

ML/EI

Beam

ν = 0

ν = 0.25

64

(a)

0 , 10 ,

1 , 10 ,

1000 = = = ⁵ =

= h t E ν

L andν =0.25

(b)

(c)

(d)

圖 4.23 懸臂梁受到剪力作用 (a)結構尺寸示意圖 (b)M71 網格 8×1 示意圖 (c)M72 網格 8×1 示意圖 (d)二次分佈負荷示意圖

U , X₁^G

L

A

B θ C

P t

h V

, X₂^G

th P 2 3

0 =

τ

A

B C L 2

0.0 0.2 0.4 0.6 0.8 1.0 0

2 4 6

Displacements θ_C /π

Beam

20×1(M71) 20×1(M72) 50×1(M71) 50×1(M72) 100×1(M71) 100×1(M72) V_C /L

-U_C /L

PL2 /EI

圖 4.24 懸臂梁受到剪力作用，不同網格下之無因次負荷位移曲線圖。

0.0 0.2 0.4 0.6 0.8 1.0

0 2 4 6

Displacements θ_C /π

Beam

ν = 0 ν = 0.25

V_C /L

-U_C /L

PL2 /EI

圖 4.25 懸臂梁(M71 網格 20×1)受到剪力作用，不同蒲松比下之 無因次負荷位移曲線圖。

66

(a)

(b)

圖 4.26 直角構架受到端點剪力作用 (a)結構尺寸示意圖 (b)網格 M81 與網格 M82 示意圖

F A

9

1 1

9 U X₁^G, V

X₂^G, θ

1 , 3 . 0 , 10

3× ⁷ = =

= t

E ν

M81 M82

圖 4.27 直角構架受到端點剪力作用，不同網格下之負荷位移曲線圖。

0 1 2 3 4 5 6 7

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Load F/10000

Horizontal displacement of point A (UA) Battini[35]

M81 M82

68

(a)

(b)

圖 4.28 半圓環受到單點集中力作用 (a)結構尺寸示意圖 (b)網格示意圖 A

F 2

=1 d

=20 R A

U X₁^G, V

X₂^G, θ

25 . 0 , 10⁷ =

= ν E

圖 4.29 半圓環受到單點集中力作用下之負荷位移曲線圖。

0 5 10 15 20 25 30

0 1 2 3 4 5 6 7 8 9 10 11

Load F/1000

Vertical displacement of point A (V_A)

Battini[35]

present

70

附錄 A 面積座標(area coordinates)

A.1 面積座標的定義

如圖 A.1 所示，x、y為三角形中任一點P在直角座標中的座標值，將P 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A₁、A₂、A₃，令面積座標

A A₁

λ = (A.1)

A A₂

ξ = (A.2)

A A₃

η = (A.3)

(

_{21 31 31 21}

)

2

1 x y x y

A= − (A.4)

j i

ij x x

x = − (A.5)

j i

ij y y

y = − (A.6)

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

η稱為三角形中P點的面積座標，固 A

A A

A₁ + ₂ + ₃ = (A.7)

由(A.1)式至(A.3)式與(A.7)式可以得出

=1 + +ξ η

λ (A.8)

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

A.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[37]

⎪⎭

72

圖 A.1 面積座標表示方法

圖 A.2 面積座標示意圖

P

¹

A A

2

A

3

1

2 3

x y

(0,0)

1 2

3

(1,0) (0,1)

ξ

η

附錄 B 不完整三階埃爾米特元素的形狀函數及其微分

在(2.4)式與(2.5)式中的N_i

(

i =1,L,9

)

及其對ξ、η的偏微分，可表示為[22]：

i N_i N_i,_ξ N_i,_η

1 λ²(3−2λ)+2a ⁶λ⁽−¹+λ⁾+^2b 6λ(−1+λ)+2c 2 ξλ² +a/2 λ⁽λ −²ξ⁾+^b/2 −2ξλ +c/2 3 ηλ² +a/2 ⁻²λη ^+b/2 λ(λ −2η)+c/2 4 ξ²(3−2ξ)+ 2a ⁶ξ⁽¹−ξ⁾+ ^2b _2c

5 ξ²(−1+ξ)−a ξ(−2+3ξ)−^b −^c 6 ξ²η +a/2 ²ξη +^b/2 ξ² +c/2 7 η²(3−2η)+ 2a 2b 6η(1−η)+2c 8 ξη² +a/2 η² +b/2 ²ξη +^c/2 9 η²(−1+η)−a −b η(−2+3η)−c 其中

η ξ

λ = 1− − (B.1)

ξηλ

=

a (B.2)

) (λ ξ

η −

=

b (B.3)

) (λ η ξ −

=

c (B.4)

在(2.32)式中的N_u,_x、N_u,_y、N_v,_x、N_v,_y為：

η

ξ η

ξ, , , ,

,_{x x u x u}

u N N

N = + (B.5)

74

附錄 C 等效節點外力(equivalent external nodal force)

76

將(C.1)至(C.4)式代入(C.11)式可得

( )

1.均勻分佈力(uniform distributed surface traction)

(1)邊 AB 受到平行於邊的均勻分佈力，如圖 C.2(a)所示，其合力大小為

將(C.19)式代入(C.17)式並積分可得

將(C.23)式代入(C.18)式並積分可得

F2 2.二次分佈力(quadratic distributed surface traction)

(1)邊 AB 受到平行於邊的二次分佈力，如圖 C.3(a)所示，其合力大小為 fqt，且在 A 點與 B 點的分佈力大小為 0，則第k段所受分佈力可用面積 座標ξ 表示成：

78

將(C.27)式代入(C.17)式並積分可得

3 T F3

將(C.35)式代入(C.18)式並積分可得

3 T F4

{

_{1 2 21 2 21 2 21} 3.純彎分佈力(pure bending distributed surface traction)

如圖 C.4 所示，邊 AB 受到純彎分佈力，其彎矩大小為M ，則第k 段所受

將(C.39)式代入(C.18)式並積分可得

2 T F5

在(C.22)式、(C.26)式、(C.30)式、(C.38)式以及(C.42)式中所計算出的^kf_t與

n

kf 須經(2.57)式座標轉換至固定總體座標與節點基礎座標後，並計算k從 第1段到m段，然後將其總合成負荷外力向量P。

80

(a) (b)

圖 C.1 元素受力示意圖 (a)σ_t (b)σ_n

(a) (b)

圖 C.2 均勻分佈力示意圖 (a) f_ut (b) f_un

(a) (b)

圖 C.3 二次分佈力示意圖 (a) f_qt (b) f_qn

圖 C.4 純彎分佈力示意圖 M

1 2 L m

B A

fqt f_qn

1 2 L m 1 2 L m

A B

B A

fut f_un

1 2 L m 1 2 L m

A B

B A

1 2

E 3 x₂

x₁E

tt

1 2

E 3 x₂

x₁E

tn

在文檔中 具旋轉自由度之三角平面元素的共旋轉推導法 (頁 45-92)