Nelinearna statika štapnih konstrukcija. Prednapete gipke konstrukcije od užadi

Velikost: px
Začni prikazovanje s strani:

Download "Nelinearna statika štapnih konstrukcija. Prednapete gipke konstrukcije od užadi"

Transkripcija

1 Nelinearna statika štapnih konstrukcija Prednapete gipke konstrukcije od užadi

2

3

4

5

6 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

7 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

8 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

9 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

10

11 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

12

13 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) x 5 a (x 5)2 + 1

14 L : x Þ b (x + 1) b (x 5) x + 1 a (x + 1) x 5 a (x 5)2 + 1 = 0

15 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))

16 x (k+1) = x (k) 0

17

18 k x (k) f ( x (k)) 0 5,0 0, , , , , , , , , , ,

19 k x (k) f ( x (k)) f 1( x (k)) 0 7,0 1, , , , , , , , , , , , ,0 0,0 5 neodredeno neodredeno

20

21 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

22 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

23 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

24 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

25 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

26 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

27 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

28 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

29 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

30 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

31 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

32 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

33 df ( ) v x (k) x x (k)w + f ( x (k)) = 0 = x (k) 0 dx

34 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, [1, ,267 67] T 0, [1, ,364 79] T 0, [1, ,367 74] T 1, [1, ,367 74] T L = 9, [1,0 1,0] T L = 9, [2,0 2,0] T L = 9,488 59

35 D 0 = [ 2,5; 7,5] ˆ [ 2,5; 7,5] 201 ˆ 201 točaka

36 D 1 = [4,0; 6,0] ˆ [4,0; 6,0] 201 ˆ 201 točaka

37 D 2 = [4,998; 5,002] ˆ [0,998; 0,002] 201 ˆ 201 točaka

38 D 3 = [4, ; 5, ] ˆ [0, ; 0, ] 201 ˆ 201 točaka

39 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

40

41 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

42 L : (x, y) Þ b b b x 2 + y 2 + (x 5) 2 + y 2 + (x 1,425) 2 + (y 5) 2

43 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

44

45 D 0 = [ 2,5; 7,5] ˆ [ 2,5; 7,5]

46 f ( x (k+1) ) ă α f ( x (k) )

47 D 0 = [ 2,5; 7,5] ˆ [ 2,5; 7,5] 201 ˆ 201 točaka

48 D 1 = [4,0; 6,0] ˆ [4,0; 6,0] 201 ˆ 201 točaka

49 D 2 = [4,998; 5,002] ˆ [0,998; 0,002] 201 ˆ 201 točaka

50 D 3 = [4, ; 5, ] ˆ [0, ; 0, ] 201 ˆ 201 točaka

51 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}

52

53

54

55

56

57

58

59 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

60 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}

61 q rubni = 1, q unutarnji = 1

62 q rubni = 5, q unutarnji = 1

63 q rubni = 1, q unutarnji = 5

64 q rubni = 25, q unutarnji = 5

65 q rubni = 5, q unutarnji = 1

66 q {i,j} = {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

67 S {i,j} = {i, j} P B

68 q {i,j} = {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

69 S {i,j} = {i, j} P B

70 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}

71 S {i,j} = {i, j} P B k = 1

72 S {i,j} = {i, j} P B k = 3

73 S {i,j} = {i, j} P B k = 5

74 S {i,j} = {i, j} P B k = 7

75 S {i,j} = {i, j} P B k = 9

76 S {i,j} = {i, j} P B k = 13

77 S {i,j} = {i, j} P B k = 23

78 S {i,j} = {i, j} P B k = 53

79 S {i,j} = {i, j} P B k = 169 S {i,j} {i, j} P B

80 S {i,j} = 10 S {i,j} = 1 ako je {i, j} rubni kabel ako je {i, j} unutarnji kabel

81 S {i,j} = 10 S {i,j} = 1 ako je {i, j} rubni kabel ako je {i, j} unutarnji kabel

82 S {i,j} = 10 S {i,j} = 1 ako je {i, j} rubni kabel ako je {i, j} unutarnji kabel

83 S {i,j} = 10 S {i,j} = 1 ako je {i, j} rubni kabel ako je {i, j} unutarnji kabel

84 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}

85

86

87

88

89

90

91

92