未來研究方向

本文推測 D-cones 的移動和結構變形可能與結構邊界狀態有關，

因文獻上未提到實驗時的夾持方式，故無法得知文獻實驗中真正的結 構初始邊界狀態，以及實驗中是否因夾具摩擦力不足，產生邊界某些 位置有滑動位移，未來應以實驗確定夾持的邊界條件。本文有限元素 分析中所採用的元素假設薄膜變形與彎曲變形之間無耦合作用，而元 素切線剛度矩陣為一近似剛度矩陣，這可能影響分析結果，因此往後 可以考慮採用薄膜變形與彎曲變形之間有耦合作用的元素，並以精確 的切線剛度矩陣來偵測分歧點及挫曲模態。

參 考 文 獻

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圖 1.1 文獻[23]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)

α α X d

Y Z

mm d 165

20

=

=

^°

α

圖2.1 柱狀薄板結構示意圖

X₃G

X₂G

X₁G

x₁E

x₃E

x₂E

h v3

u3

w3 3

θ

z

3

θ

y

3

θ

x

3 2

1

圖2.2 三角元素的示意圖及節點自由度

u

2

v

3

u

3

x

₂E

x

₁E

3

1 2

ion configurat Current

ion configurat Initial

圖2.3 CST 元素在元素座標上的變形位移

s

s

s α

12

α

13

α

²³

n

12

n

13

n

23

x

₁E

x

₂E

5

1

3

6

4

2

圖 2.4 DKT 元素的節點及其三邊上的局部座標示意圖

n

d

n

θ

θ

x₂E

x₁E

圖2.5 變形前板元素中心面之單位法向量 n 受旋轉向量θ 作用的示意圖

O

α α

x

₃E ^I

x

₃^E

x

₂E

E I

x

₂

E I

x

₁

x

₁E

(a)

β

x

₂E

x

₂

′

E

x

₁E

x

₁

′

E

β

E

E

x

x

₃

, ′

₃

O (b)

圖2.6 元素座標的剛體旋轉(a)面外旋轉(out-of plane rotation) (b)

j

x₁

′

E

Φtj

∆ Φ

tj

∆ n′

dj

x₃

′

E

dj In′

n

′

u

x₂

′

E

圖 2.7 決定板元素節點變形轉角的第 3 個步驟之示意圖

n

R′

R

φ

(a)

λ

θ

− λ

_θ

Z

X

λ

d

λ

E

Z

X E

(b)

圖3.1 (a)第一階段位移負荷圖 (b)第二階段位移負荷圖

α α

d

L W

h

(b) (a)

(c)

L=35cm W=17.5cm d=16.5cm h=0.35mm α=20°

E=3.8×10

⁹

Nm

^-2

ν=0.4

E D

A B

C G

F

I

H X

Y

Z

X X

X Z Z

α α

F H

圖4.1 (a)圓柱薄板結構示意圖 (b)俯視圖 (c)前視圖

L=254mm R=2540mm h=6.35mm θ=0.1rad

E=3102.75MPa ν=0.3

L

L

R

h

θ θ

E D

A

C

B

(a)

W Z

,

U X

, λ

E

E

θ θ

R R

(b)

圖4.2 (a)結構示意圖 (b)位移負荷圖

0 5 10 15 20 25 30 -0.4

-0.2 0.0 0.2 0.4 0.6 0.8

Reaction RE (kN)

Loading parameter

λ_E

(mm) [2]

Present (a) Present (b)

圖4.3 結構中點的反力-位移負荷參數曲線圖

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0

5 10 15 20 25 30

Displacement wE (mm)

Loading parameter λ_θ (rad)

24x48 30x60 40x80

mesh

圖4.4 第一階段 E 點之位移-負荷參數圖

0 5 10 15 20

0 5 10 15 20

Reaction (N)

Loading parameter λ_E (mm)

24X48 30X60 40X80

mesh

圖4.5 第二階段 E 點之反力-負荷參數圖

0 5 10 15 20 0

5 10 15 20

Reaction (N)

Loading parameter λ_E (mm) [23]Numerical

[23]Experiment Present

圖4.6 E 點之反力-負荷參數曲線圖

0 25 50 75 100 125 150 175 0

5 10 15 20 25 30

Z (mm)

X (mm)

0.001 0.013 0.0931 0.3491

(rad) λ

θ

圖 4.7 第一階段不同位移負荷λ_θ下 AB 線段在 Z 方向的變形圖

0 50 100 150 200 250 300 350

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04

Displacement w-wE (mm)

Y (mm) 0.001 0.013 0.0931 0.3491 (rad) λ

θ

圖 4.8 第一階段不同位移負荷λ_θ下 CD 線段在 Z 方向對 E 點的 相對位移圖

0 50 100 150 200 250 300 350 -0.2

-0.1 0.0 0.1 0.2 0.3

Reaction R X (N)

Y (mm)

0.001 0.013 0.0931 0.3491

θ

(mm) λ

圖4.9 第一階段邊界 HI 在不同位移負荷λ_θ下X方向的 反力分布圖

0 50 100 150 200 250 300 350 -0.15

-0.10 -0.05 0.00 0.05 0.10 0.15

Reaction RY (N)

Y (mm)

0.001 0.013 0.0931 0.3491

θ

(mm) λ

圖4.10 第一階段邊界HI 在不同位移負荷λ_θ下Y方向的 反力分布圖

0 50 100 150 200 250 300 350 -0.08

-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10

Reaction R Z (N)

Y (mm)

0.001 0.013 0.0931 0.3491

θ

(mm) λ

圖4.11 第一階段邊界 HI在不同位移負荷λ_θ下Z方向的 反力分布圖

0 50 100 150 200 250 300 350 -2

-1 0 1 2 3 4

Reaction R X (N)

Y (mm)

4.339 10.05 11.32 12.47 13.67 15.08

E

(mm) λ

圖4.12 第二階段邊界HI在不同位移負荷λ_E下X方向的 反力分布圖

0 50 100 150 200 250 300 350 -1.0

-0.5 0.0 0.5 1.0 1.5

Reaction RY (N)

Y (mm) 4.339

10.05 11.32 12.47 13.67 15.08

E

(mm) λ

圖4.13 第二階段邊界HI在不同位移負荷λ_E下Y方向的 反力分布圖

0 50 100 150 200 250 300 350 -0.4

-0.3 -0.2 -0.1 0.0 0.1

Reaction R Z (N)

Y (mm)

4.339 10.05 11.32 12.47 13.67 15.08

E

(mm) λ

圖4.14 第二階段邊界HI 在不同位移負荷λ_E下Z方向的 反力分布圖

0 0.5L L 0

10 15 20 25 30

Z (mm)

E

= 15.2 λ

13.67

E

= λ

12.47

E

= λ 10.05

E

= λ

4.339

E

= λ

E

= 0 λ

7.023

E

= λ

圖4.15 第二階段不同位移負荷下CD線段在Z方向的變 形圖

0 5 10 15 20 25 30

Z (mm)

d 0 0.5d

E

= 15.2 λ

E

= 0 4.339 λ

E

=

λ λ

_E

= 7.023

13.67

E

= λ

10.05

E

= λ

12.47

E

= λ

圖4.16 第二階段不同位移負荷下 AB線段在Z方向的 變形圖

0 50 100 150 200 250 300 350 0

2 4 6

Displacement u (10-10

Y (mm) 4.339 10.05 11.32 12.47 13.67 15.08

E

(mm) λ

圖 4.17 第二階段不同位移負荷λ_E下CD線段之 位移 u分布圖

mm)

0 25 50 75 100 125 150 175 -5

-4 -3 -2 -1 0 1 2 3

Displacement v (10-5 mm)

X (mm)

4.339 10.05 11.32 12.47 13.67 15.08

E

(mm) λ

圖 4.18 第二階段不同位移負荷λ_E下AB線段之 位移 v分布圖

0 5 10 15 20 0.0

0.5 1.0 1.5 2.0 2.5

Reaction RX(10-9 N)

Loading parameter (mm) λ

_E

圖4.19 第二階段E點X 向反力-負荷參數圖

0 5 10 15 20 -0.04

-0.02 0.00 0.02 0.04 0.06 0.08

Reaction RY (N)

Loading parameter

λ_E

(mm)

圖4.20 第二階段E點Y 向反力-負荷參數圖

圖4.21(a) 第二階段下λ_E =4.339(mm)的結構變形圖

圖4.21(b) 第二階段下λ_E =10.05(mm)的結構變形圖

圖4.21(c) 第二階段下λ_E =13.67(mm)的結構變形圖

圖4.21(d) 第二階段下λ_E =15.08(mm)的結構變形圖

(a) (b)

50 100 150

0 50 100 150 200 250 300 350

50 100 150

0 50 100 150 200 250 300 350

(c) (d)

50 100 150

0 50 100 150 200 250 300 350

50 100 150

0 50 100 150 200 250 300 350

圖4.22結構變形圖上視圖(a)λ_E =4.339mm (b)λ_E =10.05mm (c)λ_E =13.67mm (d)λ_E =15.08mm

E D

A B

C G

F

I

H X

Y

E1

E E₂ E₃

mm 1 . 0 mm 05 . 0

mm 01 . 0

(a)

0 5 10 15

0 5 10 15 20

Reaction (N)

Loading parameter

λ_E'

(mm) 0

0.1 0.05 0.01

(mm) d

_R

(b)

圖4.23 (a)位移負荷偏移不同距離後的位置圖 (b)位移負荷向右 偏移不同距離下中點的 Z向反力-位移負荷參數曲線圖

0 2 4 6 8 10 12 14 -0.8

-0.6 -0.4 -0.2 0.0 0.2

Reaction RX (N)

Loading parameter

λ_E'

(mm) 0

0.1 0.05 0.01

(mm) d

_R

圖4.24 位移負荷向右偏移不同距離下中點的 X 向反力-位移負 荷參數曲線圖

0 2 4 6 8 10 12 14 -0.05

-0.04 -0.03 -0.02 -0.01 0.00 0.01

Reaction RY (N)

Loading parameter

λ_E'

(mm) 0

0.1 0.05 0.01

(mm) d

_R

圖4.25 位移負荷向右偏移不同距離下點的 Y 向反力-位移負 荷參數曲線圖

50 100 150 0

50 100 150 200 250 300 350

50 100 150

0 50 100 150 200 250 300 350

50 100 150

0 50 100 150 200 250 300 350

(a) (b) (c)

圖4.26 位移負荷向右偏移 0.1mm 的結構變形上視圖(a)λ_E′=4.339mm (b)λ =10.05mm (c)λ =13.06mm

0 25 50 75 100 125 150 175 -2.0

-1.5 -1.0 -0.5 0.0 0.5

Displacement v(10-5 mm)

X (mm)

4.339 10.05 11.32 12.37 13.06

E′

(mm) λ

圖 4.27 E 點向右偏移 0.1mm 時在不同位移負荷λ_E′下 AB 線段之位移 v 分布圖

0 50 100 150 200 250 300 350 -0.002

-0.001 0.000 0.001 0.002 0.003 0.004 0.005

Displacement u(mm)

Y (mm) 4.339 10.05 11.32 12.37 13.06

E′

(mm) λ

圖4.28 E 點向右偏移 0.1mm 時在不同位移負荷λ_E′下

(a)

(a)

A

C D

E B

G

F

I

H

H ′ I′

F′

G′

θ

c

mm

1

mm

1

X Y

圖4.31 邊界不完美之結構示意圖

0 5 10 15 20 0

5 10 15 20

Reaction (N)

Loading parameter

λ_E

(mm)

Perfect Imperfect

圖4.32 不完美結構之 E 點反力-位移負荷參數曲線圖

0 50 100 150 200 250 300 350 -2

0 2 4 6 8

Reaction R X (N)

Y (mm)

4.325 10.01 11.2 12.31 13.68 17.15

E

(mm) λ

圖4.33 不完美結構在第二階段不同位移負荷λ_E下邊界H ′′I 在X方向的反力分布圖

0 50 100 150 200 250 300 350 -1.5

-1.0 -0.5 0.0 0.5 1.0 1.5

Reaction RY (N)

Y (mm) 4.325

10.01 11.2 12.31 13.68 17.15

E

(mm) λ

圖4.34 不完美結構在第二階段不同位移負荷λ_E下邊界H ′′I 在Y方向的反力分布圖

0 50 100 150 200 250 300 350 -0.6

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2

Reaction R Z (N)

Y (mm) 4.325 10.01 11.2 12.31 13.68 17.15

E

(mm) λ

圖4.35 不完美結構在第二階段不同位移負荷λ_E下邊界H ′′I 在Z方向的反力分布圖

圖4.36(a) 不完美結構在第二階段下λ_E =4.325(mm)的結構變形圖

圖4.36(b) 不完美結構在第二階段下λ_E =12.31(mm)的結構變形圖

圖4.36(c) 不完美結構在第二階段下λ_E =13.68(mm)的結構變形圖

圖4.36(d) 不完美結構在第二階段下λ_E =17 mm.8( )的結構變形圖

(a) (b)

50 100 150

0 50 100 150 200 250 300 350

50 100 150

0 50 100 150 200 250 300 350

(c) (d)

50 100 150

0 50 100 150 200 250 300 350

50 100 150

0 50 100 150 200 250 300 350

圖4.37 不完美結構變形圖的上視圖(a)λ_E =4.325mm (b)λ =12.31mm(c)λ =13.68mm(d)λ =17.8mm

附錄 A DKT 元素的形狀函數

在(2.28)式裡面的H_x與H_y分別有 9 個分量，其表示式為[29]

H_x1 =1.5(a₆N₆ −a₅N₅) H_x2 =b₅N₅ +b₆N₆ H_x3 = N₁ −c₅N₅ −c₆N₆ H_x4 =1.5(a₄N₄ −a₆N₆) H_x5 =b₆N₆ +b₄N₄ H_x6 = N₂ −c₆N₆ −c₄N₄ H_x7 =1.5(a₅N₅ −a₄N₄) H_x8 =b₄N₄ +b₅N₅ H_x9 = N₃ −c₄N₄ −c₅N₅

H_y1 =1.5(d₆N₆ −d₅N₅) H_y2 =−N₁ +e₅N₅ +e₆N₆ H_y3 =−H_x2

H_y4 =1.5(d₄N₄ −d₆N₆) H_y5 =−N₂ +e₆N₆ +e₄N₄ H_y6 =−H_x5

H_y7 =1.5(d₅N₅ −d₄N₄) H_y8 =−N₃ +e₄N₄ +e₅N₅ H_y9 =−H_x8

其中

₂

ij ij

k l

a −x

=

)

則ξ、η對 x 、 的偏微分別為 y

圖 A.1 元素面積座標的示意圖

x y

A

1

A

2

A

3

1

2 3

P

ξ η

6

5 4

3

1 2 ) 0 , 0

( ,0) (1,0) 2

(1 ) 1 , 0 (

2) ,1 0

( )

2 ,1 2 (1

圖 A.2 殼元素節點的自然座標示意圖

附錄 B CST 元素的剛度矩陣

在文檔中 拘限彈性薄板受側向位移負荷的行為研究 (頁 42-102)