95 2 x y yz = zx = yz = zx = { } T = { x y z xy } () {} T { } T = { x y z xy } = u u x y u z u x x y z y + u y (2) x u x u y x y
x y z xy E( ) = ( + )( 2) 2 2( ) x y z xy (3) E x y z z = z = (3) z x y xy E = 2 ( )/2 x y xy (4) x y z z = z { x, y, xy }{ x, y, xy }(3) x y xy /( ) E( ) = /( ) ( + )( 2) ( 2)/[2( )] x y xy (5) (4)(5) { }= [ ] {} (6) x x + xy y + X x = (7) xy x + y y + X y = (8) X x X y x y ( ) 2
S S t S t S u S t {t} T = {t x, t y } S t x xy xy y cos sin = t x t y [ ]{n} = {t} S t (9) x {n} T ={cos, sin}{t} = (9)(Cauchy) ds t {n} {t} x dy t x t y xy dx xy y [ ] (9) x y x dy + xy dx = t x ds t xy dy + y dx = t y ds t cos = dy /ds t sin = dx /ds t (9) 3
x cos + xy sin = t x xy cos + y sin = t y [] S u u x = u x u y = u y S u () (2)(7, 8)(6) (9, ) 3 { } x y h x y xy xy dxdy = h { u x u y } X x dxdy + h u X x u y y { } t x ds S t t t () y h { } T { }dxdy = h { u} T { X }dxdy + h { u} T {} t ds t (2) S t h z u x u y S u u x = u y = x y xy u x u y x = (u x) x, y = (u y ) y, xy = (u x ) + (u ) y y x (3) h 4
(){ u} (7, 8) h x x + xy y + X x u x + xy x + y y + X y u y dxdy = (a) ( x xu x + xy u y ) + ( y xyu x + y u y ) = x x u x + xy x u y + xy y u x + y y u y u ( x ) + x x ( ) u ( x ) + xy y + u ( ) y y y + u y xy x = x x + xy u x + xy y x + y u y + { x y xy } x y (b) y xy (3)(b)(a) h x x + xy y + X x u x + xy x + y y + X y u y dxdy = h x x + xy u y x + xy x + y u y y dxdy { } x y +h x y xy xy dxdy h { x y xy } x y dxdy + h [ X x u x + X y u y ]dxdy xy = h ( x xu x + xy u y ) + y ( xyu x + y u y ) dxdy h { } T { }dxdy + h { X} T { u }dxdy = (c) (c)( ) 2 (c) = h [ n x ( x u x + xy u y )+ n y ( xy u x + y u y )]ds S 2 5
= h { n x n y } x xy u x xy y u y ds = h { n} T [ ] { u }ds (d) S S n x n y (d){} n T [ ]= {} t T (9)S u u x = u y = { u} S u (d)s S t S = S t + S u S t (d) h { n} T [ ] { u }ds = h {} t T { u }ds (e) t S (e)(c) S t h {} t T { u}ds h { } T { }dxdy + h { X} T { u }dxdy = (d) t S t (2) 4 2 5 4 3 3 6 7 8 2 y x 2 6
x y U e x, U e y F e x, F e y {U e } T = {U x, U y, U x, U y,, U en x, U en y } (4) {F e } T = {F x, F y, F x, F y,, F en x, F en y } (5) n N I (x J, y J, z J ) = for I = J for I J (I, J =~n) (6) {U}{u} T ={u x, u y } u x u y = N N 2... N n N N 2... N n en en (7) {u} = [N]{U e } (8) u x u y 7
x y xy = u x u x x u y y y + u y x N x = N y N y N x N 2 x N 2 y... N 2 y N 2 x N n x...... N n y N n y N n x en en (9) {} = [B]{U e } (2) (3.7)(3.9){U e }{u}{} {u}{} 2 [N][B] [ N][ B] u x = A+ Bx + Cy (2) u y = + Ex + Fy (22) A F(2) x I y I (I =, 2, 3I ) U ei x U ei y A F U x = A + Bx + Cy (23) U x = A + Bx 2 + Cy 2 (24) U x = A+ Bx 3 + Cy 3 (25) U y = + Ex + Fy (26) U y = + Ex 2 + Fy 2 (27) U y = + Ex 3 + Fy 3 (28) A F(2, 22) 8
u x u y = N N 2 N 3 N N 2 N 3 = [ N] { U e } (29) N = 2 {(x2 y 3 x 3 y 2 ) + (y 2 y 3 )x + (x 3 x 2 )y} (3) N 2 = 2 {(x3 y x y 3 ) + (y 3 y )x + (x x 3 )y} (3) N 3 = 2 {(x y 2 x 2 y )+ (y y 2 )x + (x 2 x )y} (32) 2 = x y 2 + x 2 y 3 + x 3 y x y 3 x 2 y x 3 y 2 (29)(32) x y xy = u x u x x u y y y + u y x N x = N y N y N x N 2 x N 2 y N 2 y N 2 x N 3 x N 3 y N 3 y N 3 x = y 2 y 3 y 3 y y y 2 x 3 x 2 x x 3 x 2 x 2 x 3 x 2 y 2 y 3 x x 3 y 3 y x 2 x y y 2 = [B]{U e } (33) [ B] 9
h { } T { }dxdy = h { u} T { X }dxdy + h { u} T {} t ds t (34) { }= [ ] {}{} = [B]{U e } {} = [B]{U e } h {U e } T B [ ] T [ ] B [ ]{U e }dxdy { } T N = h U e { } T { X }dxdy + h U e { } T {} t ds t (35) S t S t { } T N {U e }{U e } { } T B h U e = h U e ( [ ] T [ ][ B]dxdy ) U e { } T N { } { } T { X}dxdy+ h U e { } T N { } T {} t ds t (36) (36){U e } h( [ B] T [ ][ B]dxdy ){ U e }= h { N} T { X }dxdy + h { N} T {} t ds t (37) S t S t [K e ]{ U e }= { F e } (38) [K e ]= h [ B] T [ ][ B]dxdy (39) { F e }= h N { } T { X }dxdy + h { N} T {} t ds t (4) S t [K e ]{ F e } (39)[ ] [ B]h dxdy = h [K e ]= h[ B] T [ ] [ B] (4)
e e e e e K K 2 K 3 K 4 K 5 K 6 e e e e e e e K 2 K 22 K 23 K 24 K 25 K 26 e e e e e e K 3 K 32 K 33 K 34 K 35 K 36 e e e e e e K 4 K 42 K 43 K 44 K 45 K 46 e e e e e e K 5 K 52 K 53 K 54 K 55 K 56 e e e e e e K 6 K 62 K 63 K 64 K 65 K 66 = F x F y F x F y F x F y (42) M N 2 N 2N K K 2 K 3 K 4... K,2N F x K 2 K 22 K 23 K 24... K 2,2N F y K 3 K 32 K 33 K 34... K 3,2N 2 2 F x 2 2 K 4 K 42 K 43 K 44... K 4,2N = F y... K 2N, K 2N,2 K 2N,3 K 2N,4... K N N 2N,2N F y (43) (42) 23i j k
[K e ] [ K] 2i 2 2i 3 2 j 4 2 j 5 2k 6 2k (43) [ K] K i N F i x K,2i F i y K 2,2i = i (44) F N i y K 2N,2i U x i = U i x i x U i x U i i x 2i i = U i x i i x = { U} 2
CG { U} { U e } { U e } {}= [ B] { U e }, { }= [ ] {} (45) 3 3
5 FEM (AINA) AINA AINA PC b) mm mm mm 7 MPa.25 N 5.74mm 4 4(a) 5.733mm 4(b) 2.5mm STRESS-YY RST CALC TIME. 57 54 55 56 25 26 27 65 67 69 7 73 2 3 4 5 47 44 45 46 6 64 66 68 7 72 2 3 4 37 34 35 36 5 58 48 38 28 75 7 74 6 59 49 39 29 77 8 76 7 5 4 6 3 79 9 78 8 5 4 6 3 8 2 8 9 52 42 62 32 83 2 82 53 43 63 33 85 22 84 54. 36. 8.. -8. -36. -54. (a) MAXIMUM 6. MINIMUM -6. STRESS-YY RST CALC TIME. 23 24 25 26 27 28 29 3 3 32 33 2 3 4 5 6 7 8 9 2 2 22 2 3 4 5 6 7 8 9 8. 2. 6.. -6. -2. -8. (b) MAXIMUM 27.9 MINIMUM -23.9 3.4 4
82 2 83 3 84 4 85 5 25 9 3 7 2 78 98 79 99 8 8 45 65 24 8 2 6 2 2 44 6 74 94 75 95 76 96 4 77 97 7 23 5 9 64 9 7 9 43 57 7 9 4 72 92 73 6 6 93 8 4 22 37 5 66 8 86 39 67 53 56 4 87 68 42 88 36 63 7 69 89 3 2 38 52 59 35 49 2 55 6 33 48 34 9 5 32 8 3 47 7 3 6 5 54 58 62 46 26 2 27 3 28 4 29 5 STRESS-ZZ RST CALC TIME. 325. 275. 225. 75. 25. 75. 25. MAXIMUM 3445. MINIMUM -. 3.5 5 5. 4 8 c) 5 /4 5.2 5 5
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