結論 …

本文以共旋轉(co-rotational formulation)有限元素推導法及增量疊代法來探討薄殼結構在位移負荷作用下的幾何非線性行為。由本文分析之數值例題的結果，可得以下的結論

(1)

本文所使用的數值計算方法可以分析結構受位移負荷作用時的行為，並有正確的結果。

(2)

在分析過程中亦發現本文在增量平衡迭代時將幾何剛度矩陣加入元素切線剛度矩陣中可以有效地改善收斂速度。

(3)

由本文例題結果可以知道結構受多個位移負荷作用與受多個力負荷作用時，結構的行為有很大的差異。

(4)

本文所使用的元素切線剛度矩陣僅為一近似剛度矩陣，故不能使用系統切線剛度的行列式值偵測平衡路徑的分歧點。

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附錄 A DKT 元素的形狀函數

在(2.29)式裡面的H_x與H_y分別有 9 個分量，其表示式為[32]

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

)

附錄 B CST 元素的剛度矩陣

附錄 C 分佈反力之節點值與節點反力的關係

) Y ( W

R1 R₂

1 2 j j+1

1 2 3

Rj₊

RN₊

1 N+

L 2

j j+1 N+1 )

j ( W

∆L

圖 C.1 分佈反力W_j(j=1~ N+1)之示意圖

θ

y 3

θ

h 1

圖 2.1 三角殼元素的示意圖及節點自由度

圖2.2 總體座標與元素座標 X1

I − 1

x

3 v

u

ion configurat Current

ion configurat Initial

1 nt displaceme

nodal Membrane

} v u 0 u 0 0 { u

:

=

₂ ₃ ₃

圖2.5 CST元素在元素座標上的變形位移

θ

x

n

圖 2.6 變形前板元素中心面之單位法向量受旋轉向量作用的

示意圖

n θ

1 3

x′

β

β x

O

Φ

∆

Φ

∆ n′

n

x′

j n′

圖2.8 決定板元素節點變形轉角的第3個步驟的示意圖

R

圖 4.7 A點、B點、C點之反力–負荷參數曲線圖(例題一, Case 4)

圖 4.8 A點、B點、F點之反力–負荷參數曲線圖(例題一, Case 5)

0 100 200 300

0 20 40 60 80 100 120 140

Re ac ti o n ( k N )

Loading parameter λ (mm) R _A

R _B R _F

R _A +R _B +R _F

A B

圖 4.10 E點之反力–負荷參數曲線圖(例題二, Case 1)

0 10 20 30

0 1 2 3 4

Present Hsiao[33]

R eaction ( k N)

Loading parameter λ (mm)

圖 4.11 A點之位移–負荷參數曲線圖(例題二, Case 1)

0 10 20 30

-30 -20 -10 0 10

Disp lacem en t ( m m )

Loading parameter λ (mm) U _A X 100

V _A X 100

W _A

圖 4.12 B點與E點之反力–負荷參數曲線圖(例題二, Case 2)

0 10 20 30 40

-20 -10 0 10 20

30 R _E

R _B

R _E +R _B

Re ac ti o n ( k N )

Loading parameter λ (mm)

圖 4.13 B點與 E點之位移–負荷參數曲線圖(例題二, Case 2)

0 10 20 30 40

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

D is p la ce me n t ( mm)

Loading parameter λ (mm) U _E

V _E

U _B

V _B

圖 4.14 B點與 E點之反力–負荷參數曲線圖(例題二, Case 3)

0 10 20 30

0 2 4 6

R eaction P ( k N)

Loading parameter λ (mm) R _E

R _B

R _E +R _B

0 10 20 30 0

2 4 6

圖 4.15 A 點、B點、C點之反力–負荷參數曲線圖

(例題二, Case 4)

8 Loading parameter λ (mm)

Re ac ti o n ( k N )

R _A R _B R _C

R _A +R _B +R _C

圖 4.16 A 點、B點、F點之反力–負荷參數曲線圖 (例題二, Case 5)

7 0 10 20 30

-1 0 1 2 3 4 5 6

R eactio n ( k N)

Loading parameter λ (mm) R _A

R _B R _F

R _A +R _B +R _F

圖 4.17 B點、E點之位移–負荷參數曲線圖(例題二, Case 5)

0 10 20 30

-0.1 0.0 0.1 0.2

U _B V _B U _E V _E

Disp lacem ent ( m m )

Loading parameter λ (mm)

4.18 (a)例題三之Hinged cylindrical shell結構示意圖

4.18 (c)

W Z ,

R R

θ θ ^{X ,} ^U

圖位移負荷之前視圖 (d)力負荷之俯視圖 (e)力負荷之前視圖

) (e

λ

) (c

A a b

L

U X , V

Y , R sin θ λ

a b

sin θ R )

(d

B C

E D

I J

F

K

L

W Z ,

R R

θ θ ^{X ,} ^U

λ

–負荷參數

圖 4.19 A 點之位移W_A λ_D曲線圖

(例題三, Case 1)

0 3 6 9 12 15 18

0 10 20 30 40 50

5 X 5 10 X 10 15 X 15 Displacem ent W A (m m )

λ _D (mm)

0.0 0.5 1.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Mesh λ _R (N/mm) 10 X 10 2.515 20 X 20 2.511 30 X 30 2.510 100 X 100 2.509 Wp / λ R

Y/L

在不同Mesh的數值比較圖 (例題三 Case 1 , 位移控制)

0 10 20 30 40 50 60 70 80 90 100 1.5

2.0 2.5 3.0 3.5 4.0

W p(0)

N

(例題三 Case 4 , mesh 10×10)

–位移負荷參數

圖 4.30 A點之位移W_A λ_D曲線圖

(例題三, 位移負荷)

0 5 10 15

0 10 20 30 40 50

Case 1 2 3 4 W A ( mm)

λ _D (mm)

圖 4.31 F點之位移W –位移負荷參數_F λ_D曲線圖

Case 1 2 3 4 W A (m m )

U _avg (mm)

圖 4.34 A點之位移W_A–力邊界上X軸方向位移平均數

Uavg

0.0 1.0 0.

0.985 0.990 0.995 1.000 1.005 1.010

980 0.5

λ _F (N/mm) U _avg (mm) 3.056 0.366 7.933 1.336 11.56 3.078 14.31 6.169 17.33 15.97 U i / U av g

Y/L

圖 4.35 力邊界上X軸方向之無因次位移分佈圖

(例題三 Case 1 , mesh 10×10)

軸方向之無因次位移分佈圖 (例題三 Case 2 , mesh 10

圖 4.36 力邊界上X

×10)

0.980 0.985 0.990 0.995 1.000 1.005 1.010

0.0 0.5 1.0

λ _F (N/mm) U _avg (mm) 2.906 0.360 7.555 1.313 11.03 3.020 14.75 8.356 16.60 15.61 U i / U av g

Y/L

軸方向之無因次位移分佈圖 (例題三 Case 3 , mesh 10

圖 4.37 力邊界上X

×10)

0.0 0.5 1.0

0.980 0.985 0.990 0.995 1.000 1.005 1.010

U i / U av g

Y/L

λ _F (N/mm) U _avg (mm)

2.668 0.324

6.569 1.160

10.61 3.310

12.01 4.890

14.65 11.46

軸方向之無因次位移分佈圖 (例題三 Case 4 , mesh 10

圖 4.38 力邊界上X

×10)

0.0 0.5 1.0

0.980 0.985 0.990 0.995 1.000 1.005 1.010

U i / U av g

λ _F (N/mm) U _avg (mm) 2.590 0.329 6.276 1.160 9.442 2.731 11.45 4.855 13.99 11.36

Y/L

圖 4.39 受位移負荷之λ_R–λ_D曲線圖及受力負荷之λ_F– 曲線圖(例題三)

0 5 10 15

Uavg

Case λ _R - λ _D λ _F - U _avg

1

2

3

4 λ R , λ F ( N /mm)

λ _D , U _avg (mm)

結構示意圖 (b)位移負荷示意圖

0 1 2 3 4

0 5 10 15 20 25 30

-W

-U

V

-2 -1 0 1 2

Case 2 2a V _A

V _B

D is p la ce me n t ( cm)

Loading parameter λ (cm)

圖 4.45 C點之位移–負荷參數曲線圖(例題四, Case 2)

0 1 2 3 4 5 6 7

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

D is p la ce me n t ( cm)

Loading parameter λ (cm) Case

2 2a W _C

U _C V _C

圖 4.46 A 點與B點之反力–負荷參數曲線圖(例題四, Case 3)

0.9 0 1 2 3 4 5 6 7

-0.3 0.0 0.3 0.6

R eaction ( N )

Loading parameter λ (cm) R _A

R _B

R _A +R _B

圖 4.47 A點與B 點之位移–負荷參數曲線圖(例題四, Case 3)

0 1 2 3 4 5 6 7 8

-0.9 -0.6 -0.3 0.0 0.3

Displacem en t ( cm )

Loading parameter λ (cm) U _A

U _B

V _A

V _B

圖 4.48 C點之位移–負荷參數曲線圖(例題四, Case 3)

0 1 2 3 4 5 6 7 8

-6 -5 -4 -3 -2 -1 0 1

Disp lacem en t ( cm )

Loading parameter λ (cm) W _C

U _C x 10

V _C x 10

在文檔中薄殼結構在位移負荷作用下之幾何非線性分析 (頁 40-110)

(1)

(2)

(3)

(4)

參 考 文 獻

附錄 A DKT 元素的形狀函數

附錄 B CST 元素的剛度矩陣

附錄 C 分佈反力之節點值與節點反力的關係

L 2

θ

θ

θ

x

x

x 3

1

2

ion configurat Initial 3

1 2

x

x

x

ion configurat

m equilibriu th

I − 1

2

3

x x

x

ion configurat Current

n φ

s

s s

n

n

n

α

α

α

x

5 4

3

1 6 2

x

x

2 u

x

3 v

u

ion configurat Current

ion configurat Initial

1

nt displaceme

nodal Membrane

} v u 0 u 0 0 { u

:

=

θ

θ

x

x

n

n

x

x

x

x

α α

O x

x

)

(a

( )

b

x

x

x′

x′

x′

參考文獻

Loading parameter λ (mm) U _A X 100

W _A

Loading parameter λ (mm) R _E

R _B

R _E +R _B

Loading parameter λ (mm) U _E

U _B

Loading parameter λ (mm) R _E

R _B

R _E +R _B

Loading parameter λ (mm) R _A

R _B R _C

R _A +R _B +R _C

Loading parameter λ (mm) R _A

R _B R _F

R _A +R _B +R _F

Loading parameter λ (mm) U _A X 100

V _A X 100

W _A

30 R _E

R _B

R _E +R _B