x y x-y σ x + τ xy + X σ y B = + τ xy + Y B = S x = σ x l + τ xy m S y = σ y m + τ xy l σ x σ y τ xy X B Y B S x S y l m δu δv [ ( σx δu + τ )

1 8 6 No-tension 1. 1 1.1................................ 1 1............................................ 5.1 - [B].................................. 5................................. 6.3.......................................... 7.4........................................ 7 3. No-tension 8 3.1.......................................... 8 3....................................... 11 3.3............................................ 1 4. 13 4.1 subroutine................................... 13 4. FEM subroutine function....................... 14 15.1.............................................. 15. -........................................ 16.3................................ 16.4........................................ 16

1. 1.1 x y x-y σ x + τ xy + X σ y B = + τ xy + Y B = S x = σ x l + τ xy m S y = σ y m + τ xy l σ x σ y τ xy X B Y B S x S y l m δu δv [ ( σx δu + τ ) ( xy + X σy B + δv + τ )] xy + Y B d = (1) (1) f g = (f g) f g () [ (δu σx ) = + (δu τ xy) + (δv σ y) + (δv τ ] xy) d [ (δu) σ x + (δu) τ xy + (δv) σ y + (δv) ] τ xy d + (δu X B + δv Y B ) d () () 1 Green - D ( P + Q ) ( d = P C n + Q ) ds n [ (δu σx ) + (δu τ xy) + (δv σ y) + (δv τ ] xy) d ( = δu σ x s n + δu τ xy n + δv σ y n + δv τ xy ) ds n = [δu(σ x l + τ xy m) + δv(σ y m + τ xy l)]ds s = (δu S x + δv S y )ds (3) s u = ɛ x v = ɛ y v + u = τ xy () [ (δu) σ x + (δu) τ xy + (δv) σ y + (δv) ] τ xy d = (δɛ x σ x + δɛ y σ y + δγ xy τ xy )d (4) (δɛ x σ x + δɛ y σ y + δγ xy τ xy )d = (δus x + δvs y )ds + (δux B + δvy B )d (5) s 1

(5) {δɛ} T {σ}d = s {δu} T {S}ds + {δu} T {F }d (6) (6) V s {δɛ} T {σ}dv = {δu} T {S}d + V V {δu} T {F }dv (7) = t z t ( ) {δɛ} T {σ}dv = {δɛ} T {σ}d dz = t {δɛ} T {σ}d V (6) t 1. t (1) [N] {u} {ɛ} {u nd } {u} =[N]{u nd } (8) {ɛ} =[B]{u nd } (9) (6) {u nd } {δu nd } [B] T {σ}d = {δu nd } [N] T {S}ds + {δu nd } [N] T {F }d (1) s [B] T {σ}d = s [N] T {S}ds + [N] T {F }d (11) - [D e ] t {f nd } =[k]{u nd } (1) [k] = t [B] T [D e ][B]d {f nd } = t [N] T {S}ds + t [N] T {F }d s {ɛ} = [B]{u nd } {σ} = [D e ]{ɛ} = [D e ][B]{u nd } {f nd } [k] {u nd } [B] - t {ɛ} {σ} [D e ] - [N] {S} {F }

{S} (t C [N]T {S}ds) {F } = {F Bx, F By } T {w} [N] {F } = [N]{w} (13) {F B } {F B } = t [N] T {F }d = t [N] T [N]d {w} (14) () {ɛ } {σ} = [D e ]{ɛ ɛ } = [D e ][B]{u nd } [D e ]{ɛ } (15) {T nd } {T nd } = t [B] T [D e ]{ɛ }d (16) {f nd } = [k]{u nd } {T nd } (17) {u nd } = [k] 1 {f nd + T nd } (18) (15) (3) - - E ν α T Hooke ɛ x αt = 1 E [σ x ν(σ y + σ z )] ɛ y αt = 1 E [σ y ν(σ z + σ x )] ɛ z αt = 1 E [σ z ν(σ x + σ y )] γ xy = (1 + ν) E τ (1 + ν) xy γ yz = E τ yz γ zx = x y σ z = τ yz = τ zx = ɛ x αt = 1 E (σ x νσ y ) ɛ y αt = 1 E (σ y νσ x ) γ xy = 1 ν σ x = σ y = E ν 1 ɛ x αt 1 ν τ 1 ν ɛ y αt xy γ xy (1 + ν) E τ zx (1 + ν) E τ xy 3

ɛ z = σ z = ν(σ x + σ y ) EαT τ yz = τ zx = ɛ x αt = 1 E [(1 ν )σ x ν(1 + ν)σ y ] ɛ y αt = 1 E [(1 ν )σ y ν(1 + ν)σ x ] γ xy = 1 ν ν σ x = σ y = E ν 1 ν ɛ x (1 + ν)αt (1 + ν)(1 ν) 1 ν ɛ y (1 + ν)αt τ xy γ xy (1 + ν) E τ xy {σ} = σ x σ y τ xy αt {ɛ } = αt {ɛ} = ɛ x ɛ y γ xy (1 + ν)αt {ɛ } = (1 + ν)αt - [D e ] = [D e ] = E 1 ν 1 ν ν 1 E 1 ν ν 1 ν ν ν 1 ν 1 ν E (1 + ν)(1 ν) 4

..1 - [B] u,v (a,b) [ 1,1] u i,j,k,l v i,j,k,l u =N 1 (a, b) u i + N (a, b) u j + N 3 (a, b) u k + N 4 (a, b) u l v =N 1 (a, b) v i + N (a, b) v j + N 3 (a, b) v k + N 4 (a, b) v l {u} {u nd } {u} = { } [ ] u N1 N = N 3 N 4 v N 1 N N 3 N 4 u i v i u j v j u k v k u l v l = [N]{u nd } (19) N 1 = 1 4 (1 a)(1 b) N = 1 (1 + a)(1 b) 4 N 3 = 1 4 (1 + a)(1 + b) N 4 = 1 (1 a)(1 + b) () 4 N 1 a = 1 4 (1 b) N a = +1 4 (1 b) N 3 a = +1 4 (1 + b) N 4 a = 1 (1 + b) 4 N 1 b = 1 4 (1 a) N b = 1 4 (1 + a) N 3 b = +1 4 (1 + a) N 4 b = +1 (1 a) (1) 4 {ɛ} {u nd } u v {ɛ} = = u + v N 1 N 1 N 1 N 1 N N N N N 3 N 3 N 3 N 3 N 4 N 4 N 4 N 4 N i Jacobi [J] N i a N i b [J] = a b [J] 1 = i=1 u i v i u j v j u k v k N i N i N i = [J] N i N = [J] 1 a i N i b ( ) 4 Ni 4 ( ) a a x Ni i a y i [ ] = i=1 ( ) i=1 4 Ni 4 ( ) b b x Ni i b y = J11 J 1 J 1 J i [ ] 1 J J 1 det(j) J 1 J 11 i=1 u l v l = [B]{u nd } () (3) (4) det(j) = J 11 J J 1 J 1 (5) 5

()(4) { } N i = 1 N i J det(j) a J N i 1 b { } N i = 1 N i J 1 det(j) a + J N i 11 b (6) (4) (x i,y i ) (1)(4) [J] [J] (1)(5)(6) () - [B] [B] a b [J] x i,j,k,l y i,j,k,l J 11 = N 1 a x i + N a x j + N 3 a x k + N 4 a x l J 1 = N 1 a y i + N a y j + N 3 a y k + N 4 a y l J 1 = N 1 b x i + N b x j + N 3 b x k + N 4 b x l J = N 1 b y i + N b y j + N 3 b y k + N 4 b y l. 4 [k] t - [D e ] 1 [k] = t 1 1 1 [B] T [D e ][B] det(j) da db (7) {ɛ } {T nd } {σ} {R nd } 1 {T nd } = t {R nd } = t 1 1 1 1 1 1 1 [B] T [D e ]{ɛ } det(j) da db (8) [B] T {σ} det(j) da db (9) αt p α (N 1 φ i + N φ j + N 3 φ k + N 4 φ l ) {ɛ } = αt p = α (N 1 φ i + N φ j + N 3 φ k + N 4 φ l ) (1 + ν)αt p (1 + ν) α (N 1 φ i + N φ j + N 3 φ k + N 4 φ l ) {ɛ } = (1 + ν)αt p = (1 + ν) α (N 1 φ i + N φ j + N 3 φ k + N 4 φ l ) α ν T p {φ i φ j φ k φ l } T [N 1 N N 3 N 4] [N] Gauss 1 1 f(a, b) da db 1 1 i=1 j=1 n n H i H j f(a i, b j ) (3) 6

[k] t {T nd } t {R nd } t n i=1 j=1 n i=1 j=1 n i=1 j=1 n H i H j [B(a i, b j )] T [D e ][B(a i, b j )] det(j(a i, b j )) (31) n H i H j [B(a i, b j )] T [D e ]{ɛ } det(j(a i, b j )) (3) n H i H j [B(a i, b j )] T {σ} det(j(a i, b j )) (33) 4 (n = ) a b H a b 4.3 i j a b H 1 1.5773569.5773569 1. 1 +.5773569.5773569 1. 1 +.5773569 +.5773569 1..5773569 +.5773569 1. {σ} {u} {σ(a i, b j )} = [D e ][B(a i, b j )]{u nd } (34) - [B] a b {σ} a b 4 4.4 {F } = {F Bx, F By } T {w} [N] {F } = [N]{w} (35) [ N1 N [N] = N 3 N 4 N 1 N N 3 ] N 4 (36) {w} = γ {k h k v k h k v k h k v k h k v } T (37) γ k h k v {F B } {F B } = t [N] T {F }d = t γ [N] T [N]d {w } (38) {w} γ {w } = { k h k v k h k v k h k v k h k v } T (39) 7

3. No-tension 3.1 Zienkiewicz No-tension No-tension No-tension No-tension 1 No-tension 1 3 [k] { f} {f} {f m } {u} { f} = {f} + {f m } {u} = {} { f} [k] { u} { u} = [k] 1 { f} 4 {u} = {u} + { u} {ɛ} = [B]{u} 5 {σ} = [D e ]{ɛ ɛ } 6 7 8 σ 1 σ σ ts {σ} {σ t } {σ} = {σ} {σ t } {R} {R} = t [B] T {σ}d { f} { f} = {f} + {f m } {R} 9 { f} { u} 3 8

Setting of [k] { f} = {f} + {f m } {u} = {} i= i=i+1 { u} = [k] 1 { f} {u} = {u} + { u} {ɛ} = [B]{u} {σ} = [D]{ɛ ɛ } σ 1, σ < σ ts Yes No {σ} = {σ} {σ t } {R} = t [B] T {σ}d { f} = {f} + {f m } {R} No f < δ End Yes 9

i 1 σ i 1 f i 1 = f R i 1 f i 1 u i u i u i ɛ i σ i ɛ i σ i = σ i σ t σ t σ i σ i R i f i = f R i i [K] {f} { u i } [K] { fi 1} [K] { fi} [K] {σ i } = [D]{ɛ i ɛ } {σ i } = {σ i } {σ t } {Ri 1} {Ri} 1 {u i } 1

3. No-tension {σ x, σ y, τ xy } T {σ x, σ y, τ xy } T (1) x y θ (I ) () σ 1 = σ x + σ y + 1 (σ x σ y ) + 4τ xy (4) σ = σ x + σ y 1 (σ x σ y ) + 4τ xy (41) θ = 1 ( ) τxy tan 1 (4) σ x σ y σ x > σ y τ xy θ = θ σ x > σ y τ xy < σ x < σ y σ x = σ y τ xy > σ x = σ y τ xy < θ = θ + 18 θ = θ + 9 θ = 45 θ = 135 σ x = σ y τ xy = (θ = ) x y φ σ x = σ x + σ y σ y = σ x + σ y τ xy = σ x σ y x-y + σ x σ y σ x σ y σ x = σ 1 + σ σ y = σ 1 + σ τ xy = σ 1 σ cos φ + τ xy sin φ (43) cos φ τ xy sin φ (44) sin φ + τ xy cos φ (45) + σ 1 σ σ 1 σ σ 1 σ σ x = σ 1 (1 + cos φ) σ 1 σ y = σ 1 (1 cos φ) τ xy = σ 1 sin φ σ x = σ (1 cos φ) σ σ y = σ (1 + cos φ) τ xy = σ sin φ 11 cos φ (46) cos φ (47) sin φ (48) (49) (5)

No-tension {σ x,σ y,τ xy } σ 1 > ( ) σ 1 σ σx c1 = σ x σ 1 (1 + cos φ) σ 1 σy c1 = σ y σ 1 (1 cos φ) τxy c1 = τ xy + σ 1 sin φ σx c = σx c1 σ (1 cos φ) σ σy c = σx c1 σ (1 + cos φ) τxy c = τxy c1 σ sin φ φ = θ σ 1 σ 1 {R} {σ c1 x, σ c1 y, τ c1 xy} T {σ c x, σ c y, τ c xy} T 3.3 ( u i ) n {( ui ) n } < 1 1 6 u i { u i } i n m mm 1 3 mm (51) (5) 1

4. 4.1 subroutine (1) subroutine XGYG3M(ne,kk,kakom,x,y,xg,yg,NELT,NODT,nod) () subroutine XGYG4M(ne,kk,kakom,x,y,xg,yg,NELT,NODT,nod) IPR=1 (3) subroutine XGYG4I(ne,kk,kakom,x,y,xg,yg,NELT,NODT,nod) Gauss Gauss (4) subroutine DMT PL(NSTRES,Eme,poe,d) (5) real(8) function CL PL3(ne,kakom,x,y,NELT,NODT) (6) subroutine BMT PL3(ne,kakom,x,y,b,NELT,NODT) - (7) subroutine SM PL3(ne,kakom,x,y,NSTRES,Em,po,t,sm,NELT,NODT) (8) subroutine MV PL3(ne,kakom,x,y,t,gamma,gkh,gkv,fevec,NELT,NODT) (9) subroutine CLST PL3TS(ne,kakom,x,y,deltaT,dist,NSTRES,Em,po,t,alpha,ts, strsave,fevec,noten,nelt,nodt,nt) (1) subroutine BMT PL4(ne,kakom,x,y,bm,detJ,a,b,NELT,NODT) - (11) subroutine IPOINT4(kk,a,b) Gauss (1) subroutine SM PL4(ne,kakom,x,y,NSTRES,Em,po,t,sm,NELT,NODT) (13) subroutine MV PL4(ne,kakom,x,y,t,gamma,gkh,gkv,fevec,NELT,NODT) (14) subroutine CLST PL4TS(ne,kakom,x,y,deltaT,dist,NSTRES,Em,po,t,alpha,ts, strsave,fevec,noten,nelt,nodt,nt) 13

(15) subroutine PST PL(sigx,sigy,tauxy,ps1,ps,ang) 4. FEM subroutine function (1) subroutine SVEC(ne,nod,nhen,kakom,fevec,ftvec,NELT,NODT) vector. () subroutine SMT(ne,nod,nhen,kakom,sm,tsm,NELT,NODT) matrix (3) subroutine CHBND1(n1,m1,array,nt) Cholesky Cholesky (4) subroutine CHBND11(n1,m1,array,vector,nt) Cholesky (5) integer function IBND(mm,array,fact,nt) Cholesky ib function mm mm 1. 1 3 fact (6) subroutine BND(mm,ib,array,nt) Cholesky mm mm Cholesky mm ib (7) subroutine del spaces(s) csv 14

.1 α 1 6 u =α 1 + α x + α 3 y (53) v =α 4 + α 5 x + α 6 y (54) i j k (u, v) (x, y) α {α} u i 1 x i y i α 1 v i 1 x i y i α u j = 1 x j y j α 3 v j 1 x j y j α 4 u k 1 x k y k α 5 v k 1 x k y k α 6 (55) {α} = [C]{u nd } (56) a i a j a k [C] = 1 b i b j b k c i c j c k a i a j a k b i b j b k c i c j c k 1 x i y i = 1 x j y j 1 x k y k (57) a i = x j y k x k y j b i = y j y k c i = x k x j a j = x k y i x i y k b j = y k y i c j = x i x k (58) a k = x i y j x j y i b k = y i y j c k = x j x i {u nd } {α} = (x k x j )y i + (x i x k )y j + (x j x i )y k (59) Sarrus 3 3 a 11 a 1 a 13 a 1 a a 3 a 31 a 3 a 33 = a 11a a 33 + a 1 a 3 a 13 + a 31 a 1 a 3 a 13 a a 31 a 3 a 3 a 11 a 33 a 1 a 1 { } [ ] [ ] u 1 x y 1 x y = {α} = [C]{u v 1 x y 1 x y nd } = [N]{u nd } (6) [N] [N] = [ ] [ ] 1 x y Ni N [C] = j N k 1 x y N i N j N k (61) N i = a i + b i x + c i y N i = a j + b j x + c j y N i = a k + b k x + c k y (6) 15

. - {ɛ} (u, v) u N i N j N k v {ɛ} = = N i N j N k u + v {u nd } = [B]{u nd } (63) N i N i N j N j N k N k - [B] = 1 b i b j b k c i c j c k c i b i c j b j c k b k = 1 y j y k y k y i y i y j x k x j x i x k x j x i (64) x k x j y j y k x i x k y k y i x j x i y i y j = 1/ [(x k x j )y i + (x i x k )y j + (x j x i )y k ] (65).3 [k] - [B] - [D e ] [k] = t [B] T [D e ][B]d = t [B] T [D e ][B] (66) {ɛ } {σ} {R nd } {T nd } = t {R nd } = t [B] T [D e ]{ɛ }d = t [B] T [D e ]{ɛ } (67) [B] T {σ}d = t [B] T {σ} (68) αt m α (φ i + φ j + φ k )/3 {ɛ } = αt m = α (φ i + φ j + φ k )/3 (1 + ν)αt m (1 + ν) α (φ i + φ j + φ k )/3 {ɛ } = (1 + ν)αt m = (1 + ν) α (φ i + φ j + φ k )/3 α ν T m {φ i, φ j, φ k } T.4 (1) [N] = [ ] [ ] Ni N j N k ζi ζ = j ζ k N i N j N k ζ i ζ j ζ k 16 (69)

ζ i = i ζ j = j ζ k = k 1 x y i = a i + b i x + c i y = 1 x j y j 1 x k y k 1 x y j = a j + b j x + c j y = 1 x k y k 1 x i y i 1 x y k = a k + b k x + c k y = 1 x i y i 1 x j y j (7) (71) (7) (73) (ζ i, ζ j, ζ k ) ζ i + ζ j + ζ k = 1 ζ r 1 (r = i, j, k) (74) () [N]T [N]d [N] T [N] ζ i ζ i ζ i ζ j ζ k ζ i ζ i [ ] ζ i ζ i ζ j ζ k ζ i [N] T [N] = ζ j ζi ζ j ζ k ζ j = ζ i ζ j ζ j ζ j ζ k ζ i ζ j ζ k ζ i ζ j ζ j ζ j ζ k ζ k ζ k ζ i ζ j ζ k ζ k ζ k ζ k ζ i ζ j ζ k ζ k (75) f(x, y)dxdy = 1 1 ζ i + ζ j + ζ k = 1 (x, y) g(ζ i, ζ j, ζ k ) (ζ i, ζ j ) dζ idζ j (76) x =ζ i x i + ζ j x j + ζ k x k x = ζ i (x i x k ) + ζ j (x j x k ) + x k y =ζ i y i + ζ j y j + ζ k y k y = ζ i (y i y k ) + ζ j (y j y k ) + y k (77) (x, y) (ζ i, ζ j ) = ζ i ζ j = x i x k x j x k y i y k y j y k ζ i ζ j =(x k x j )y i + (x i x k )y j + (x j x i )y k = (78) f(x, y)dxdy = 1 1 (ζ i, ζ j, ζ k ) g(ζ i, ζ j, ζ k )dζ i dζ j (79) 1 1 ζ il ζ j m ζ kn dζ i dζ j = l! m! n! (l + m + n + )! l, m, n (8) l =, m = n = l = 1, m = 1, n = 1 1 1 1 ζ il ζ j m ζ kn dζ i dζ j = 1/1 ζ il ζ j m ζ kn dζ i dζ j = 1/4 17

ζ ζ i d = ζ j d = ζ k d = 1/1 = 1/6 ζ i ζ j d = ζ j ζ k d = ζ k ζ i d = 1/4 = 1/1 1/6 1/1 1/1 1/6 1/1 1/1 [N] T [N]d = 1/1 1/6 1/1 1/1 1/6 1/1 1/1 1/1 1/6 1/1 1/1 1/6 (81) (3) {F B } {w} {F B } = t [N] T {F }d = t [N] T [N]d {w} (8) {w} = γ {k h k v k h k v k h k v } T (83) k h k v γ {F B } {F B } = γ t 3 1/ 1/4 1/4 k h 1/ 1/4 1/4 k v 1/4 1/ 1/4 k h 1/4 1/ 1/4 k v 1/4 1/4 1/ k h 1/4 1/4 1/ t γ k v (84) 18

[1] I-1-B 54 5 [] 55 5 [3] S.P. Timoshenko, J.N. Goodier : Theory of Elasticity Third edition, McGRW-HILL BOOK CM- PNY, 1st printing 1985 [4] 1984 6 [5] H.C. Martin / G.F. Carey / 56 1 [6] O.C.Zienkiewicz / 5 1 [7] 1991.7 [8],,, 55 1 19