1

1

1950

2

x y yz = zx = 0

yz = zx = 0

{ }T= x y z xy { } (1)

{} T

{ }T

= x y z xy{ } =uxx

uyy

uzz

uxy

+uyx

(2)

ux uy x y

2

x

y

z

xy

=E(1 )

(1+ )(1 2 )

11 1

0

11

10

1 11 0

0 0 01 22(1 )

x

y

z

xy

(3)

E

x y z z = 0

z = 0 (3) z —

x

y

xy

=E

1 2

1 0

1 0

0 0 (1 ) / 2

x

y

xy

(4)

x y z z = 0

z

{ x , y ,

xy } { x , y , xy } (3)

x

y

xy

=E(1 )

(1+ )(1 2 )

1 /(1 ) 0

/(1 ) 1 0

0 0 (1 2 ) /[2(1 )]

x

y

xy

(5)

(4) (5)

{ } = D[ ]{ } (6)

x

x+

xy

y+ Xx = 0 (7)

xy

x+

y

y+Xy = 0 (8)

Xx Xy x y

( )

3

D S

St St Su

St {t}T = {tx , t y}

St

x xy

xy y

cos

sin

=

t xt y

[ ]{n}= {t} St (9)

x

{n}T={cos , sin } {t}= 0

(9) (Cauchy)

{n}{t}

t xt y

dx

dy

dSt

y

x

xy

xy

[ 1] (9) 1

x y

xdy+ xydx = t xdSt

xydy+ ydx = t ydSt

cos = dy / dSt sin = dx / dSt (9)

4

x cos + xy sin = t x

xy cos + y sin = t y [ ]

Su

ux = ux uy = uy Su (10)

(2) (7, 8) (6)

(9, 10)

3

h x y xy{ }x

y

xy

dxdy

D

= h ux uy{ }Xx

X y

dxdy

D+ h ux uy{ }

t xt y

StdSt (11)

h { }T{ }dxdy

D= h u{ }

TX{ }dxdy

D+ h u{ }

Tt{ }

StdSt (12)

h z 1 ux uy Suux = 0 uy = 0 x y xy ux uy

x =( ux )x

, y =( uy )

y, xy =

( ux )y

+( uy )

x (13)

1 h

5

(11) u{ }

(7, 8)

h x

x+

xy

y+ Xx

ux +

xy

x+

y

y+ Xy

uy

dxdy

D= 0 (a)

x x u x + xy uy( ) +y xy ux + y u y( )

=x

xux +

xy

xuy +

xy

yux +

y

yuy

+ x

ux( )x

+ xy

uy( )x

+ xy

ux( )y

+ y

uy( )y

= x

x+

xy

y

ux +

xy

x+

y

y

uy + x y xy{ }

x

y

xy

(b)

(13) (b) (a)

h x

x+

xy

y+ Xx

ux +

xy

x+

y

y+ Xy

uy

dxdy

D

= h x

x+

xy

y

ux +

xy

x+

y

y

uy

dxdy

D

+h x y xy{ }x

y

xy

dxdy

D

h x y xy{ }x

y

xy

dxdy+

Dh X x ux + X y uy[ ]dxdy

D

= hx x ux + xy uy( ) +

y xy ux + y uy( )

dxdy

D

h { }T{ }dxdy+

Dh X{ }

Tu{ }dxdy

D= 0 (c)

(c) ( ) 2

(c) = h nx x ux + xy u y( ) + ny xy ux + y uy( )[ ]dSS

2

6

= h nx ny{ }x xy

xy y

uxuy

dS

S= h n{ }

T[ ] u{ }dS

S (d)

nx ny (d) n{ }T[ ] = t{ }

T

(9) Su ux = uy = 0 u{ }

Su (d) S St

S = St + Su St

(d)

h n{ }T[ ] u{ }dS

S= h t{ }

Tu{ }dSt

St (e)

(e) (c)

h t{ }T

u{ }dStSt

h { }T{ }dxdy +

Dh X{ }

Tu{ }dxdy

D= 0 (d)

(12)

4

2

1

2

3

45

6

7 8

1

2

3

x

y

Uxe3

Uye3

Uxe2

Uye2

7

x y

Uxe, Uy

e Fxe , Fy

e

{U e}T = {Uxe1 , Uy

e1, Uxe2 , Uy

e2, , U xen , Uy

en} (14)

{F e }T = {Fxe1 , Fy

e1 , Fxe2 , Fy

e2 , , Fxen , Fy

en} (15)

n

N I(xJ , y J , zJ ) =1 for I = J

0 for I J

(I, J = 1 ~ n) (16)

{U} {u}T = {ux , uy}

uxuy

=N1 0 N 2 0 . . . N n 0

0 N1 0 N 2 . . . 0 N n

Uxe1

Uye1

Uxe 2

Uye 2

Uxen

Uyen

(17)

{u}= [N]{U e} (18)

ux uy

8

x

y

xy

=

uxxuyy

uxy

+uyx

=

N 1

x0

N 2

x0 . . .

N n

x0

0N 1

y0

N 2

y. . . 0

N n

yN 1

y

N 1

x

N 2

y

N 2

x. . .

N n

y

N n

x

Uxe1

Uye1

Uxe 2

Uye 2

Uxen

Uyen

(19)

{ } = [B]{U e} (20)

(3.17) (3.19) {U e} {u} { }

{u} { } 2

[N] [B]

N[ ] B[ ]

ux = A+ Bx +Cy (21)

uy = D + Ex +Fy (22)

A F (21) x I y I ( I =1, 2, 3 I )

UxeI Uy

eI A F

Uxe1= A +Bx1 +Cy1 (23)

Uxe2= A +Bx 2 +Cy2 (24)

Uxe3= A+ Bx3 +Cy3 (25)

Uye1= D +Ex1 + Fy1 (26)

Uye2= D +Ex 2 + Fy2 (27)

Uye3= D+ Ex3 + Fy3 (28)

A F (21, 22)

9

uxuy

=

N1 0 N2 0 N3 0

0 N1 0 N 2 0 N 3

Uxe1

Uye1

Uxe2

Uye2

Uxe3

Uye3

= N[ ] U e{ } (29)

N1=12{(x 2y3 x 3y 2 ) + (y 2 y3 )x + (x 3 x 2 )y} (30)

N2=12{(x 3y1 x1y3 ) + (y 3 y1 )x + (x1 x3 )y} (31)

N3=12{(x1y2 x 2y1 )+ (y1 y2 )x + (x 2 x1 )y} (32)

2 = x1y2 + x 2y 3 + x 3y1 x1y3 x 2y1 x 3y2

(29) (32)

x

y

xy

=

uxxuyy

uxy

+uyx

=

N1

x0

N2

x0

N3

x0

0N1

y0

N2

y0

N3

yN1

yN1

xN2

yN2

xN3

yN3

x

Uxe1

Uye1

Uxe2

Uye2

Uxe3

Uye3

=12

y2 y 3 0 y3 y1 0 y1 y 2 0

0 x 3 x 2 0 x1 x 3 0 x 2 x1

x3 x 2 y 2 y3 x 1 x 3 y3 y1 x 2 x1 y1 y 2

Uxe1

Uye1

Uxe2

Uye2

Uxe3

Uye3

= [B]{U e} (33)

B[ ]

10

h { }T{ }dxdy

D= h u{ }

TX{ }dxdy

D+ h u{ }

Tt{ }

StdSt (34)

{ } = D[ ]{ } { } = [B]{U e}

{ }= [B]{ U e}

h { Ue}T B[ ]T D[ ] B[ ]{U e}dxdy

D

= h Ue{ }TN{ }

T X{ }dxdyD

+ h Ue{ }TN{ }

T t{ }St

dSt (35)

{ U e} {U e}

h U e{ }T

B[ ]T D[ ] B[ ]dxdy

D( ) U e{ }

= h Ue{ }T

N{ }T X{ }dxdy

D+ h U e{ }

TN{ }

T t{ }St

dSt (36)

(36) { U e}

h B[ ]T D[ ] B[ ]dxdy

D( ) U e{ } = h N{ }T X{ }dxdy

D+ h N{ }

T t{ }St

dSt (37)

[K e ] U e{ } = Fe{ } (38)

[K e ]= h B[ ]T D[ ] B[ ]dxdy

D (39)

F e{ } = h N{ }T X{ }dxdy

D+ h N{ }

T t{ }St

dSt (40)

[K e ] F e{ }

(39) D[ ]

B[ ] h dxdyD

= h

[K e ]= h B[ ]T D[ ] B[ ] (41)

11

K11e K12

e K13e K14

e K15e K16

e

K21e K22

e K23e K24

e K25e K26

e

K31e K32

e K33e K34

e K35e K36

e

K41e K42

e K43e K44

e K45e K46

e

K51e K52

e K53e K54

e K55e K56

e

K61e K62

e K63e K64

e K65e K66

e

Uxe1

Uye1

Uxe2

Uye2

Uxe3

Uye3

=

Fxe1

Fye1

Fxe 2

Fye 2

Fxe3

Fye3

(42)

M N

2N 2N

K11 K12 K13 K14 . . . K1,2N

K21 K22 K23 K24 . . . K2,2N

K31 K32 K33 K34 . . . K3,2N

K41 K42 K43 K44 . . . K4,2N

. . .

K2N ,1 K2N ,2 K2N,3 K2N,4 . . . K2N,2N

Ux1

Uy1

Ux2

Uy2

UyN

=

Fx1

Fy1

Fx2

Fy2

FyN

(43)

(42)

1 2 3 i j k

1

12

1

[K e ]

K[ ]

1 2i 1 2 2i 3 2 j 1

4 2 j

5 2k 1 6 2k

(43)

K[ ]

0

K 0

0 0 1 0

0

Ux1

Uy1

Uxi

UyN

=

Fx1 K1,2i 1Ux

i

Fy1 K2,2i 1Ux

i

Ux

i

FyN K2N ,2i 1U x

i

(44)

Uxi=Ux

i i x Uxi

Uxi Ux

i

0 2i 1

Uxi=Ux

i

i x Uxi= 0

U{ }

13

CG

U{ }

U e{ } 1

U e{ }

{ } = B[ ] U e{ } , { } = D[ ]{ } (45)

3

14

5 FEM

(ADINA)

ADINA

ADINA PC

b)

100mm 10mm 1mm 70000 MPa

0.25 100N

5.714mm

4 4(a)

5.733mm

4(b)

2.105mm

1 2 3 4 5 6 7 8 9 10 11

12 13 14 15 16 17 18 19 20 21 22

23 24 25 26 27 28 29 30 31 32 33

STRESS-YY

RST CALC

TIME 1.000

180.0

120.0

60.0

0.0

-60.0

-120.0

-180.0

MAXIMUM217.9

MINIMUM-213.9

1 2 3 4 5 6 7 89

1011

12 13 14 15 16 17 18 1920

2122

23 24 25 26 27 28 29 3031

3233

34 35 36 37 38 39 4041

4243

44 45 46 47 48 49 5051

5253

54 55 56 57 58 59 6061

6263

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

STRESS-YY

RST CALC

TIME 1.000

540.0

360.0

180.0

0.0

-180.0

-360.0

-540.0

MAXIMUM611.0

MINIMUM-611.0

(a)８節点要素を用いた場合

(b)定ひずみ三角形要素を用いた場合

図13.4　片持ち梁の解析例

15

5.1 4

8

c)

5

1/4

5.2 5

STRESS-ZZ

RST CALC

TIME 1.000

3250.

2750.

2250.

1750.

1250.

750.

250.

MAXIMUM3445.

MINIMUM-110.1

1 2 3 4 5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26 27 28 29

30

31

32

3334

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66 6768

69

70 7172

73

74 75 7677

78 79 80 81

82 83 84 85

8687

88

89

90 9192

93

94 9596

97

98 99 100 101

102 103 104 105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

図13.5　穴開き板の引張の解析例5

16

17

8

11