Post on 13-Aug-2020
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