xyz,, uvw,, Bernoulli-Euler u c c c v, w θ x c c c dv ( x) dw uxyz (,, ) = u( x) y z + ω( yz, ) φ dx dx c vxyz (,, ) = v( x) zθ x ( x) c wxyz (,, ) =
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1 ,, uvw,, Bernoull-Euler u v, w θ dv ( ) dw u (,, ) u( ) ω(, ) φ d d v (,, ) v( ) θ ( ) w (,, ) w( ) θ ( ) (11.1) ω φ φ dθ / dφ v v θ u w u w 11.1 θ θ θ 11. vw, (11.1) u du d v d w ε d d d u v ω γ φ w u ω γ φ ε ε γ 0 ( ) ( ) ( ) (11.) 66
2 σ Eε τ Gγ τ Gγ σ σ τ 0 (11.3) e 1 du d v d w 1 ω ω V E ddd Gφ ddd d d d (11.4) (, ) (11.5) d d d 1 l e du d v d w V EA EI EI 0 GKφ d l A I dd, I dd dd ω ω K dd (11.6) (11.5) dd 0, dd 0, dd 0 (11.7) (11.1) u v, w θ u ( ) 1 u u l l j v v l v l l l l l l l l l l ( ) 1 3 θ 3 j θj w l θ 3 wj l θj w ( ) 1 3 l l l l l l l l l θ( ) 1 θ θj l l θ dv / d, θ dw / d (11.8)(11.5) (11.8) 67
3 e 1 e T e V { U } [ k]{ U } (11.9) T { U e } { u v w θ θ θ uj vj wj θj θj θj} (11.10) [ k] k11 0 k 0 0 k k k 0 k 0 k k Sm k k 77 0 k k86 0 k k93 0 k k k k k113 0 k k119 0 k k k 0 k k EA 1EI 1EI GK k11, k, k 3 33, k 3 44 l 4EI 4EI k k k k l k k, k k, k k, k k, k k 53, 55, 6, k k, k k, k k, k k, k k k113 k53, k115 k55 /, k119 k53, k1 111 k55 k k, k k /, k k, k k (11.11) (11.1) (11.11){ U e } { U e } (3.) e e { U } Tg { U } (11.13) T { U e } { u v w θ θ θ uj vj wj θj θj θj} (11.14) T { U e } { u v w θ θ θ uj vj wj θj θj θj} (11.15) 68
4 [ T ] [ T ] T g, [ T] [ T ] [ T ] (11.16) (11.13)(11.9) k (11.11) [ k] T g k Tg (11.17) (11.17)[ T ] l m n [ T] l m n (11.18) l m n [ k] k11 Sm. k1 k k31 k3 k33 k41 k4 k43 k44 k51 k5 k53 k54 k55 k61 k6 k63 k64 k65 k66 k71 k7 k73 k74 k75 k76 k77 k81 k8 k83 k84 k85 k86 k87 k88 k91 k9 k93 k94 k95 k96 k97 k98 k99 k101 k10 k103 k104 k105 k106 k107 k108 k109 k1010 k111 k11 k113 k114 k1 15 k116 k117 k118 k119 k1110 k1111 k k k k k k k k k k k k (11.19) k k l k l k l , k k lm k lm k lm k k m k m k m , , k k l n k l n k l n k k m n k m n k m n k k n k n k n ( ) , , n k6ln, k44 k44l k55l k66l k k k l l k k l m k l m k k l ( ) k k l m k l m, k k k m m, k k m n k m n , k k l m k l m k l m k k m k m k m ( ) k k l n k l n, k k m n k m n, k k k n n k k l n k l n k l n, k k m n k55mn k66mn, k66 k44n k55n k66n 69
5 k k, k k, k k, k k, k k, k k, k k k k, k k, k k, k k, k k, k k, k k, k k k k, k k, k k, k k, k k, k k, k k, k k 83, k 99 k k k, k k, k k ( ), ( ) ( ) k104 k44l k55l k66l k105 k44l m k55l m k66l m k106 k44l n k55l n k66l n k k, k k, k k, k k k k, k k, k k ( ) ( ) 1 1 k114 k105, k115 k44m k55m k66m k116 k44m n k55m n k66m n k k, k k, k k, k k, k k k k, k k, k k ( ) 1 k14 k106, k15 k116k16 k44n k55n k66n k k, k k, k k, k k, k k, k k (11.0) (11.18) (,, ) (,, ) (,, ),( j, j, j) (1) l ( )/, l m ( )/, l n ( )/ l (11.1) j j j l ( ) ( ) ( ) j j j (),, (a) P P P θ 70
6 θ P 11.3 θ (b) θ θ 11.4 θ (a),(b) (a) e e e e e 0 l l e 0,e m,e m (11.) 1 n n 71
7 * m ± l l, m, n 0 (11.3) l m l m e e e l e m e e n (11.4) l m n n m, m n l l n, n l m m l (11.5) ( ) os e,e n > 0 (11.6) n (11.5),(11.3) n ± l m (11.7) n > 0, m l l, m, n 0 l m l m nl mn l, m, n l m l m l m (11.8) e e e,e θ e e osθ e snθ (11.9) e e snθ e osθ l l l e m m osθ m snθ n n n l l l e m m snθ m osθ n n n (b) l 0 l n e m e e 1 m 0 n 0 n 0 (11.30) (11.30) (11.31) 7
8 l n 0 e m 0 osθ 1snθ n 0 0 (11.3) l n 0 e m 0 snθ 1osθ n , Q, Q P,, P σ dd (11.33) σ dd (11.34) σ dd (11.35) ω ω τ τdd (11.36) Q d d (11.37) 73
9 d Q (11.38) d σ, τ, τ du ( ) d v ( ) d w ( ) σ Eε E d d d v 1 1u θ E E 3 3 l l u j l l vj θ j w θ E 3 3 l l wj θ j u v ω ω 1 1θ τ Gγ G G φ G l l θ j w u ω ω 1 1θ τ Gγ G G φ G l l θ j (11.39)(11.33)(11.38) EA u P [ 1 1] l u j w θ EI 3 3 l l wj θ j v θ EI 3 3 l l v j θ j Q GK θ [ 1 1] l θ j v θ EI θ j 3 3 l v j (11.39) (11.40) (11.41) (11.4) (11.43) (11.44) 74
10 w θ Q EI 3 3 l wj θ j (11.45) A I, dd 0, dd 0 I, K A I dd, I dd dd ω ω K dd (11.46) (11.41), (11.4) w θ EI θ j l wj w 6 6 4θ EI θ j j l wj v θ EI θ j l v j v 6 6 4θ EI θ j j l v j (11.47) (11.48) 75
11 EA EA l l u 1EI 1EI l v 1EI 1EI w θ P Q l Q GK GK θ l l θ 4EI EI u j j l v j EI 4 EI wj l l j l l θ 4EI EI θ l θ EI 4EI l (11.49) (11.13) (11.13)(11.49) { S } [ ]{ e f G U } (11.50) { e } U { S },[ ] { S } T f { P Q Q j j} f G (11.51) [ G] G11 G1 G G17 G18 G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G (11.5) EA EA EA G l, G m, G n, G G, G G, G G EI 1EI 1EI G1 l 3, G m, 3 G3 n 3 G4 l, G5 m, G6 n G G, G G, G G, G G, G G, G G
12 1EI 1EI 1EI G31 l 3, G3 m, 3 G33 n 3 G34 l, G35 m, G36 n G G, G G, G G, G G, G G, G G GK GK GK G l, G m, G n, G G, G G, G G G51 l, G5 m, G53 n 4EI 4EI 4EI G54 l, G55 m, G56 n G G, G G, G G, G G /, G G /, G G / G G, G G, G G, G G, G G, G G G G, G G, G G, G G, G G, G G G71 l, G7 m, G73 n 4EI 4EI 4EI G74 l, G75 m, G76 n G G, G G, G G, G G /, G G /, G G / G G, G G, G G, G G, G G, G G G G, G G, G G, G G, G G, G G
変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy,
変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, z + dz) Q! (x + d x + u + du, y + dy + v + dv, z +
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5,, Euclid.,..,... Euclid,.,.,, e i (i =,, ). 6 x a x e e e x.:,,. a,,. a a = a e + a e + a e = {e, e, e } a (.) = a i e i = a i e i (.) i= {a,a,a } T ( T ),.,,,,. (.),.,...,,. a 0 0 a = a 0 + a + a 0
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1 1 007 1 e iθ = cos θ + isin θ 1) θ = π e iπ + 1 = 0 1 ) 3 11 f 0 r 1 1 ) k f k = 1 + r) k f 0 f k k = 01) f k+1 = 1 + r)f k ) f k+1 f k = rf k 3) 1 ) ) ) 1+r/)f 0 1 1 + r/) f 0 = 1 + r + r /4)f 0 1 f
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