,, 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
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 3 3 3 3 ( ) 1 3 θ 3 j θj 3 3 3 3 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
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 k33 0 0 0 k44 0 0 k 0 k 0 k 0 0 0 53 55 k Sm. 6 66 k71 0 0 0 0 0 k 77 0 k8 0 0 0 k86 0 k88 0 0 k93 0 k95 0 0 0 k99 0 0 0 k104 0 0 0 0 0 k1010 0 0 k113 0 k115 0 0 0 k119 0 k1111 0 k 0 0 0 k 0 k 0 0 0 k 1 16 18 11 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, 66 71 11 77 11 8 86 6 88 k k, k k, k k, k k, k k 93 33 95 53 99 33 104 44 1010 44 k113 k53, k115 k55 /, k119 k53, k1 111 k55 k k, k k /, k k, k k 1 6 16 66 18 6 11 66 (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
[ 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 1 13 14 15 16 17 18 19 110 111 11 (11.19) 11 11 33 k k l k l k l 1 11 33, 11 33 k k lm k lm k lm k k m k m k m 31 11 33, 3 11 33, 33 11 33 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 ( ) 41 53 6, 4 53 6, 43 53 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 51 53 6 5 53 6 53 53 6 54 44 55 66, 55 44 55 66 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 61 53 6 6 53 6 63 53 6 k k l n k l n k l n, k k m n 64 44 55 66 65 44 k55mn k66mn, k66 k44n k55n k66n 69
k k, k k, k k, k k, k k, k k, k k 71 11 71 73 31 74 41 75 51 76 61 77 11 k k, k k, k k, k k, k k, k k, k k, k k 81 1 8 83 3 84 4 85 5 86 6 87 7 88 k k, k k, k k, k k, k k, k k, k k, k k 83, k 99 k 33 91 31 9 3 93 33 94 43 95 53 96 63 97 73 98 k k, k k, k k 101 41 10 4 103 43 ( ), ( ) ( ) 1 1 1 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 107 74 108 84 109 94 1010 44 k k, k k, k k 111 51 11 5 113 53 ( ) ( ) 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 117 75 118 85 119 95 1110 54 1111 55 k k, k k, k k 11 61 1 6 13 63 ( ) 1 k14 k106, k15 k116k16 k44n k55n k66n k k, k k, k k, k k, k k, k k 17 76 18 86 19 96 110 64 111 65 11 66 (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
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
* 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
l n 0 e m 0 osθ 1snθ n 0 0 (11.3) l n 0 e m 0 snθ 1osθ n 0 0 0 11.5 11.5 0, 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
d Q (11.38) d σ, τ, τ du ( ) d v ( ) d w ( ) σ Eε E d d d v 1 1u 6 1 4 6 6 1 6 θ E E 3 3 l l u j l l vj θ j w 6 1 4 6 6 1 6 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 6 1 4 6 6 1 6 θ EI 3 3 l l wj θ j v 6 1 4 6 6 1 6 θ EI 3 3 l l v j θ j Q GK θ [ 1 1] l θ j v 1 6 1 6 θ EI θ j 3 3 l v j (11.39) (11.40) (11.41) (11.4) (11.43) (11.44) 74
w 1 6 1 6 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 6 4 6 θ EI θ j l wj w 6 6 4θ EI θ j j l wj v 6 4 6 θ EI θ j l v j v 6 6 4θ EI θ j j l v j (11.47) (11.48) 75
EA EA 0 0 0 0 0 0 0 0 0 0 l l u 1EI 1EI 0 0 0 0 0 0 0 0 3 3 l v 1EI 1EI w 0 0 0 0 0 0 0 0 3 3 θ P Q l Q GK GK θ 0 0 0 0 0 0 0 0 0 0 l l θ 4EI EI 0 0 0 0 0 0 0 0 u j j l v j EI 4 EI 0 0 0 0 0 0 0 0 wj l l j l l θ 4EI EI 0 0 0 0 0 0 0 0 θ l θ EI 4EI 0 0 0 0 0 0 0 0 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 G13 0 0 0 G17 G18 G19 0 0 0 G G G G G G G G G G G G G G G G G G G G G G G G 0 0 0 G G G 0 0 0 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 1 3 4 5 6 7 8 9 10 11 1 31 3 33 34 35 36 37 38 39 310 311 31 44 45 46 410 411 41 51 5 53 54 55 56 57 58 59 510 511 51 61 6 63 64 65 66 67 68 69 610 611 61 71 7 73 74 75 76 77 78 79 710 711 71 81 8 83 84 85 86 87 88 89 810 811 81 (11.5) EA EA EA G l, G m, G n, G G, G G, G G 11 1 13 17 11 18 1 19 13 1EI 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 7 1 8 9 3 10 4 11 5 16 76
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 37 31 38 3 39 33 310 34 311 35 31 36 GK GK GK G l, G m, G n, G G, G G, G G 44 45 46 410 44 411 45 41 46 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 / 57 51 58 5 59 53 510 54 511 55 51 56 G G, G G, G G, G G, G G, G G 61 51 6 5 63 53 64 510 65 511 66 51 G G, G G, G G, G G, G G, G G 67 51 68 5 69 53 610 54 611 55 61 56 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 / 77 71 78 7 79 73 710 74 711 75 71 76 G G, G G, G G, G G, G G, G G 81 71 8 7 83 73 84 710 85 711 86 71 G G, G G, G G, G G, G G, G G 87 71 88 7 89 73 810 74 811 75 81 76 77