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2 Visual Basic でわかるやさしい有限要素法の基礎サンプルページこの本の定価判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行当時のものです.

3 URL ( ) ( FAX )

4 (FEM Finite Element Method) FEM FEM FEM FEM FEM FEM Visual Basic FEM FEM FEM FEM FEM FEM ( ) FEM

5 ii Visual Basic ( ) ( ) FEM ( ) (SI ) kgf 1kgf=9.8N 1kgf/mm 2 = N/m 2 =9.8MPa 21,000kgf/mm 2 = N/mm 2 =206GPa

6 Gauss maketk, bound, gauss output deform

7 vii FEM inputdata maketk band, bound bandmatrix (3 ) ( ) FEM fmtk FEM Castigliano

8 viii FEM inputdata fmtk output

9 ix FEM Gauss fmtk =\ FEM FEM

10 2 (plane truss) FEM 4 FEM 2.1 ( (pin joint) ) FEM ( k) f u f = ku (2.1) (Hooke s law) f u (elastic body) 2.3 (element) 1, 2 ( (node) ) f 1,f 2

11 u 1,u 2 (2.1) 1 f 1 = k(u 1 u 2 ) (2.2) 2 f 2 = k(u 2 u 1 ) (2.3) f 2 = k(u 1 u 2 ) (2.2) (2.3) { } [ ]{ } f1 k k u1 = k k f 2 u 2 (2.4) } { f1 f 2 {f}, [ ] k k [K], k k { u1 u 2 } {u} {f} (nodal force vector) {u} (nodal displacement vector) [K] (stiffness matrix) (2.4) {f} =[K]{u} (2.5) (2.5) (2.1) (2.4) (2.5) u 1,u 2 f 1,f 2 f 1,f 2 u 1,u 2 [K] ( [K] (singular matrix) ) f 1,f 2 u 1,u [K]

12 (x, y, z) x ( x, ȳ, z) 4.2 ȳ, z x (4.1) x y z x l 1 m 1 n 1 ȳ l 2 m 2 n 2 (4.1) z l 3 m 3 n 3 x x θ x x l 1,,n 3 l 1 =cosθ x x,,n 3 =cosθ z z (4.2)

13 i, j, (u i,v i,w i ) (u j,v j,w j ) (f xi,f yi,f zi ) (f xj,f yj,f zj ) f xi k 0 0 k 0 0 ū i f yi v i f zi = w i (4.3) f xj k 0 0 k 0 0 ū j f yj v j f zj w j { f} =[ K]{ū} (4.4) f xi,, fzj f xi,,f zj (4.1) f xi l 1 m 1 n f yi l 2 m 2 n f zi = l 3 m 3 n f xj l 1 m 1 n 1 f yj l 2 m 2 n 2 f zj l 3 m 3 n 3 f xi f yi f zi f xj f yj f zj (4.5) { f} =[T ]{f} (4.6) [T ] (transformation matrix) {ū} =[T ]{u} (4.7)

14 {u} = u i v i w i u j v j w j t, {ū} = ū i v i w i ū j v j w j t (4.6) (4.7) (4.4) {f} {f} =[K]{u} (4.8) [K] =[T ] t [ K][T ] (4.9) [T ] 1 =[T ] t (4.10) (4.9) l 1 l, m 1 m, n 1 n l 2 lm ln l 2 lm ln lm m 2 mn lm m 2 mn [K] =k ln mn n 2 ln mn n 2 l 2 lm ln l 2 (4.11) lm ln lm m 2 mn lm m 2 mn ln mn n 2 ln mn n 2 (4.11) l, m, n 4.3 l x,l y,l z L l = l x /L, m = l y /L, n = l z /L (4.12) (4.9) ȳ, z ȳ, z 4.3

15 60 4 (4.8) 4.1 (4.10) (4.9) (4.11) α, β, γ i, j 4.4 l =cosα, m =cosβ, n =cosγ (4.13) k (= AE/L L A E ) j (4.8) f xj = kl 2 (u j u i )+klm(v j v i )+kln(w j w i ) f yj = klm(u j u i )+km 2 (v j v i )+kmn(w j w i ) (4.14) f zj = kln(u j u i )+kmn(v j v i )+kn 2 (w j w i ) S ij S ij = f xj l + f yj m + f zj n (4.15) (4.14) (4.15) l 2 + m 2 + n 2 =1 S ij = kl(u j u i )+km(v j v i )+kn(w j w i ) (4.16)

16 7 ( ) ( 7.1 ) 1 ( (5.12)) 2 3 (finite difference method) 7.1

17 (i, j, k) x, y u i f xi {δ} = v i u j v j, {f} = f yi f xj f yj (7.1) u k v k f xk f yk 7.2 (x, y) x y u(x, y) v(x, y) x, y 1 u = α 1 + α 2 x + α 3 y, v = α 4 + α 5 x + α 6 y (7.2) α 1,,α 6 (7.2) x, y 1 (displacement function) 7.2

18 144 7 (7.2) (7.2) (7.2) 3 i, j, k u i = α 1 + α 2 x i + α 3 y i, v i = α 4 + α 5 x i + α 6 y i, u j = α 1 + α 2 x j + α 3 y j, v j = α 4 + α 5 x j + α 6 y j, (7.3) u k = α 1 + α 2 x k + α 3 y k, v k = α 4 + α 5 x k + α 6 y k (u i,v i ), (u j,v j ), (u k,v k ) x, y (7.3) u i 1 x i y i α 1 v i 1 x i y i α 4 u j = 1 x j y j α 2, v j = 1 x j y j α 5 (7.4) u k 1 x k y k α 3 v k 1 x k y k α 6 1 α 1 1 x i y i u i α 2 = 1 x j y j u j α 3 1 x k y k u k = 1 x j y k x k y j x k y i x i y k x i y j x j y i 2A y j y k y k y i y i y j x k x j x i x k x j x i α 4 α 5 = 1 x j y k x k y j x k y i x i y k x i y j x j y i 2A y j y k y k y i y i y j α 6 x k x j x i x k x j x i u i u j u k v i v j v k (7.5) (7.6) A = 1 1 x i y i 2 1 x j y j = 1 (xj x i )(y k y i ) (x k x i )(y j y i ) 2{ } (7.7) 1 x k y k (7.7) 7.2 (i, j, k) A 7.3 i A A = 1 2 r 1r 2 sin θ θ = θ 2 θ 1 A

19 Gauss 32 Gauss Castigliano CALCOMP , , , ( ) (4 ) 206 ( )

20 , (Bernoulli-Euler) Polygon ,

21 2008 JCLS Printed in Japan ISBN

<4D F736F F D B B BB2D834A836F815B82D082C88C60202D B2E646F63>

<4D F736F F D B B BB2D834A836F815B82D082C88C60202D B2E646F63> 例題で学ぶはじめての塑性力学サンプルページこの本の定価判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/066721 このサンプルページの内容は, 初版 1 刷発行当時のものです. http://www.morikita.co.jp/support/ 03 3817 5670 FAX 03 3815 8199 i 1