Nelinearna statika štapnih konstrukcija Prednapete gipke konstrukcije od užadi
ÿ j P N i S {i,j} te i,j tı = 0 ÿ j P N i S {i,j} te i,j tj = 0 ÿ j P N i S {i,j} te i,j tk = 0 te i,j l {i,j} = = x j x i l {i,j} tı + y j y i l {i,j} tj + z j z i l {i,j} b (x j x i ) 2 + (y j y i ) 2 + (z j z i ) 2 t k
ÿ j P N i S {i,j} ÿ j P N i S {i,j} ÿ j P N i S {i,j} x j x i a (xj x i ) 2 + (y j y i ) 2 + (z j z i ) 2 = 0 y j y i a (xj x i ) 2 + (y j y i ) 2 + (z j z i ) 2 = 0 z j z i a (xj x i ) 2 + (y j y i ) 2 + (z j z i ) 2 = 0
ÿ S x j x i = 0 S ÿ x j x i = 0 l j P N {i,j} l i j P N {i,j} i ÿ S y j y i = 0 S ÿ y j y i = 0 l j P N {i,j} l i j P N {i,j} i ÿ S z j z i = 0 S ÿ z j z i = 0 l j P N {i,j} l i j P N {i,j} i
L ( ) ÿ {x k, y k, z k } k P Nf = {i,j} P B = ÿ {i,j} P B l {i,j} b (x j x i ) 2 + (y j y i ) 2 + (z j z i ) 2 BL({x k, y k, z k } k P Nf ) = ÿ x i x j = 0 Bx i l j P N {i,j} i BL({x k, y k, z k } k P Nf ) = ÿ y i y j = 0 By i l j P N {i,j} i BL({x k, y k, z k } k P Nf ) = ÿ z i z j = 0 Bz i l j P N {i,j} i
L : (x, y) Þ b x 2 + y 2 + b (x 5) 2 + y 2 + b (x 1,425) 2 + (y 5) 2 x a x2 + y + x 5 a 2 (x 5)2 + y + x 1,425 a 2 (x 1,425)2 + (y 5) = 0 2 y a x2 + y 2 + y a (x 5)2 + y 2 + y 5 a (x 1,425)2 + (y 5) 2 = 0
L : x Þ b (x x q ) 2 + y 2 q + b (x x v ) 2 + y 2 v L 1 (x) : x Þ x + 1 a (x + 1)2 + 16 + x 5 a (x 5)2 + 1
L : x Þ b (x + 1) 2 + 4 2 + b (x 5) 2 + 1 2 x + 1 a (x + 1)2 + 16 + x 5 a (x 5)2 + 1 = 0
f : x Þ a x + b f (k) : x Þ f 1( x (k)) x + [ f ( x (k)) f 1( x (k)) x (k)] f 1( x (k)) x + [ f ( x (k)) f 1( x (k)) x (k)] = 0 f 1( x (k)) v x x (k)w + f ( x (k)) = 0 x (k) 0 = x (k) f( x (k) ) f 1( x (k))
x (k+1) = x (k) 0
k x (k) f ( x (k)) 0 5,0 0,832 050 294 337 843 7 1 4,201 999 773 524 741 0,168 995 782 368 261 5 2 3,885 621 945 428 617 0,029 479 992 136 465 56 3 3,804 066 684 172 385 0,001 336 731 305 122 907 4 3,800 009 152 526 629 3,001 708 215 433 219 10 6 5 3,800 000 000 046 348 1,520 039 649 705 041 10 11
k x (k) f ( x (k)) f 1( x (k)) 0 7,0 1,788 854 381 999 832 0,111 803 398 874 989 5 1 8,999 999 999 999 995 1,891 885 890 830 651 0,022 722 339 927 220 84 2 74,261 050 441 561 94 1,998 486 398 580 294 0,000 040 382 998 518 193 74 3 49 414,050 167 456 65 1,999 999 996 518 797 1,409 011 728 364 766 10 13 4 1,419 434 593 264 463 10 13 2,0 0,0 5 neodredeno neodredeno
L : (x, y) Þ b x 2 + y 2 + b (x 5) 2 + y 2 + b (x 1,425) 2 + (y 5) 2 x a x2 + y + x 5 a 2 (x 5)2 + y + x 1,425 a 2 (x 1,425)2 + (y 5) = 0 2 y a x2 + y 2 + y a (x 5)2 + y 2 + y 5 a (x 1,425)2 + (y 5) 2 = 0
L : (x, y) Þ b x 2 + y 2 + b (x 5) 2 + y 2 + b (x 1,425) 2 + (y 5) 2 x a x2 + y 2 + x 5 a (x 5)2 + y 2 + x 1,425 a (x 1,425)2 + (y 5) 2 = 0 y a x2 + y 2 + y a (x 5)2 + y 2 + y 5 a (x 1,425)2 + (y 5) 2 = 0
L : (x, y) Þ b x 2 + y 2 + b (x 5) 2 + y 2 + b (x 1,425) 2 + (y 5) 2 BL Bx : (x, y) Þ x a x2 + y + x 5 a 2 (x 5)2 + y + x 1,425 a 2 (x 1,425)2 + (y 5) 2 BL By : (x, y) Þ y a x2 + y 2 + y a (x 5)2 + y 2 + y 5 a (x 1,425)2 + (y 5) 2
f 1 : (x, y) Þ x a x2 + y 2 + x 5 a (x 5)2 + y 2 + x 1,425 a (x 1,425)2 + (y 5) 2
f 2 : (x, y) Þ y a x2 + y 2 + y a (x 5)2 + y 2 + y 5 a (x 1,425)2 + (y 5) 2
x a x2 + y + x 5 a 2 (x 5)2 + y + x 1,425 a 2 (x 1,425)2 + (y 5) = 0 2 y a x2 + y 2 + y a (x 5)2 + y 2 + y 5 a (x 1,425)2 + (y 5) 2 = 0
x a x2 + y + x 5 a 2 (x 5)2 + y + x 1,425 a 2 (x 1,425)2 + (y 5) = 0 2 y a x2 + y 2 + y a (x 5)2 + y 2 + y 5 a (x 1,425)2 + (y 5) 2 = 0
f (k) 1 : (x, y) Þ Bf 1( x (k), y (k)) v x x (k)w + Bf 1( x (k), y (k)) v y y (k)w ( + f 1 x (k), y (k)) Bx By
f (k) 2 : (x, y) Þ Bf 2( x (k), y (k)) v x x (k)w + Bf 2( x (k), y (k)) v y y (k)w ( + f 2 x (k), y (k)) Bx By
Bf 1 ( x (k), y (k)) v x x (k)w + Bf 1( x (k), y (k)) v y y (k)w ( + f 1 x (k), y (k)) = 0 Bx By Bf 2 Bx ( x (k), y (k)) v x x (k)w + Bf 2( x (k), y (k)) v y y (k)w ( + f 2 x (k), y (k)) = 0 By
Bf 1 ( x (k), y (k)) v x x (k)w + Bf 1( x (k), y (k)) v y y (k)w ( + f 1 x (k), y (k)) = 0 Bx By Bf 2 Bx ( x (k), y (k)) v x x (k)w + Bf 2( x (k), y (k)) v y y (k)w ( + f 2 x (k), y (k)) = 0 By
Bf 1 ( x (k), y (k)) v x x (k)w + Bf 1( x (k), y (k)) v y y (k)w ( + f 1 x (k), y (k)) = 0 Bx By Bf 2 Bx ( x (k), y (k)) v x x (k)w + Bf 2( x (k), y (k)) v y y (k)w ( + f 2 x (k), y (k)) = 0 By Bf 1 Bx Bf 2 Bx Bf 1 By Bf 2 By ([ [ ] [ [ ]([ ]) x (k) x x (k) f1 x (k) y ])$ % (k) y y ],/- + (k) f 2 y (k) = [ ] 0 0 df ( ) v x (k) x x (k)w + f ( x (k)) = 0 dx f 1( x (k)) v x x (k)w + f ( x (k)) = 0, f 1 = df dx
df ( ) v x (k) x x (k)w + f ( x (k)) = 0 = x (k) 0 dx
x (k+1) = x (k) 0 = df ( ) v x (k+1) x x (k+1)w + f ( x (k+1)) = 0 dx k x (k) ( ) f x (k) 0 [2,0 2,0] T 0,279 684 1 [1,905 36 1,267 67] T 0,058 836 1 2 [1,839 83 1,364 79] T 0,001 515 27 3 [1,843 51 1,367 74] T 1,745 56 10 6 [1,843 51 1,367 74] T L = 9,391 85 [1,0 1,0] T L = 9,559 83 [2,0 2,0] T L = 9,488 59
D 0 = [ 2,5; 7,5] ˆ [ 2,5; 7,5] 201 ˆ 201 točaka
D 1 = [4,0; 6,0] ˆ [4,0; 6,0] 201 ˆ 201 točaka
D 2 = [4,998; 5,002] ˆ [0,998; 0,002] 201 ˆ 201 točaka
D 3 = [4,999 998; 5,000 002] ˆ [0,999 998; 0,000 002] 201 ˆ 201 točaka
f 1 : (x, y) Þ x a x2 + y 2 + x 5 a (x 5)2 + y 2 + x 1,425 a (x 1,425)2 + (y 5) 2
f 2 : (x, y) Þ y a x2 + y + y a 2 (x 5)2 + y + y 5 a 2 (x 1,425)2 + (y 5) 2
L : (x, y) Þ b b b x 2 + y 2 + (x 5) 2 + y 2 + (x 1,425) 2 + (y 5) 2
df ( ) v x (k) x x (k)w + f ( x (k)) = 0 dx df dx = [ Bf Bx ] Bf By = Bf 1 Bx Bf 2 Bx Bf 1 By Bf 2 By
D 0 = [ 2,5; 7,5] ˆ [ 2,5; 7,5]
f ( x (k+1) ) ă α f ( x (k) )
D 0 = [ 2,5; 7,5] ˆ [ 2,5; 7,5] 201 ˆ 201 točaka
D 1 = [4,0; 6,0] ˆ [4,0; 6,0] 201 ˆ 201 točaka
D 2 = [4,998; 5,002] ˆ [0,998; 0,002] 201 ˆ 201 točaka
D 3 = [4,999 998; 5,000 002] ˆ [0,999 998; 0,000 002] 201 ˆ 201 točaka
ÿ j P N i S {i,j} x j x i l {i,j} = 0 ÿ j P N i S {i,j} y j y i l {i,j} = 0 ÿ j P N i S {i,j} z j z i l {i,j} = 0 E ( {x k, y k, z k } k P Nf ) = ÿ (i,j) P B S {i,j} l {i,j}
ÿ j P N i S {i,j} x j x i l {i,j} = 0 ÿ j P N i S {i,j} y j y i l {i,j} = 0 ÿ j P N i S {i,j} z j z i l {i,j} = 0 ÿ j P N i S {i,j} l {i,j} (x j x i ) = 0 ÿ j P N i S {i,j} l {i,j} (y j y i ) = 0 ÿ j P N i S {i,j} l {i,j} (z j z i ) = 0
ÿ j P N i S {i,j} l {i,j} (x j x i ) = 0 ÿ j P N i S {i,j} l {i,j} (y j y i ) = 0 ÿ j P N i S {i,j} l {i,j} (z j z i ) = 0 ÿ q {i,j} (x j x i ) = 0 j P N i ÿ q {i,j} (y j y i ) = 0 j P N i ÿ q {i,j} (z j z i ) = 0 j P N i S {i,j} l {i,j} = q {i,j}
q rubni = 1, q unutarnji = 1
q rubni = 5, q unutarnji = 1
q rubni = 1, q unutarnji = 5
q rubni = 25, q unutarnji = 5
q rubni = 5, q unutarnji = 1
q {i,j} = 1 @ {i, j} P B vrijednosti sila u štapovima dijagonalnih kabela: 1,416; 1,427; 1,448; 1,480; 1,522; 1,572; 1,631; 1,697 1,480; 1,448; 1,427; 1,416; 1,416; 1,427; 1,448; 1,480
S {i,j} = 1 @ {i, j} P B
q {i,j} = 1 @ {i, j} P B vrijednosti sila u štapovima kabela sljemena i uvale : 2,964; 2,644; 2,355; 2,135; 2,015; 2,015; 2,135; 2,355; 2,644; 2,964 2,964; 2,644; 2,355; 2,135; 2,015; 2,015; 2,135; 2,355; 2,644; 2,964
S {i,j} = 1 @ {i, j} P B
q {i,j} = S {i,j} l {i,j} = S {i,j} = q {i,j} l {i,j} l (k) {i,j} = l (k 1) {i,j} = S (k) {i,j} S (k 1) {i,j} = q(k) {i,j} q (k 1) {i,j} S (k) {i,j} = S {i,j} = q (k) {i,j} q (k 1) {i,j} = S {i,j} S (k 1) {i,j} q (k) {i,j} = q (k 1) {i,j} S {i,j} S (k 1) {i,j}
S {i,j} = 1 @ {i, j} P B k = 1
S {i,j} = 1 @ {i, j} P B k = 3
S {i,j} = 1 @ {i, j} P B k = 5
S {i,j} = 1 @ {i, j} P B k = 7
S {i,j} = 1 @ {i, j} P B k = 9
S {i,j} = 1 @ {i, j} P B k = 13
S {i,j} = 1 @ {i, j} P B k = 23
S {i,j} = 1 @ {i, j} P B k = 53
S {i,j} = 1 @ {i, j} P B k = 169 S {i,j} «1 @ {i, j} P B
S {i,j} = 10 S {i,j} = 1 ako je {i, j} rubni kabel ako je {i, j} unutarnji kabel
S {i,j} = 10 S {i,j} = 1 ako je {i, j} rubni kabel ako je {i, j} unutarnji kabel
S {i,j} = 10 S {i,j} = 1 ako je {i, j} rubni kabel ako je {i, j} unutarnji kabel
S {i,j} = 10 S {i,j} = 1 ako je {i, j} rubni kabel ako je {i, j} unutarnji kabel
q {i,j} = S {i,j} l {i,j} = l {i,j} = S {i,j} q {i,j} S (k) {i,j} = S (k 1) {i,j} = l (k) {i,j} l (k 1) {i,j} = q(k 1) {i,j} q (k) {i,j} l (k) {i,j} = l {i,j} = q (k) {i,j} q (k 1) {i,j} = l(k 1) {i,j} l{i,j} q (k) {i,j} = q (k 1) {i,j} l (k 1) {i,j} l{i,j}