Soluc. Examen Final 2014-i Marco 3d

EJEMPLO MARCO 3D

1. COORDENADAS GLOBALES

NUDO SCG NUDO SCG1 372 383 394 405 416 427 438 449 4510 4611 4712 4813 4914 5015 5116 5217 5318 5419 5520 5621 5722 5823 5924 6025 6126 6227 6328 6429 6530 6631 6732 6833 6934 7035 7136 72

5

6

7

8

9

10

11

12

1

2

3

4

ELEMENTO NUDO SCL ELEMENTO NUDO SCL1 492 503 514 525 536 547 558 569 5710 5811 5912 6013 6114 6215 6316 6417 6518 6619 6720 6821 6922 7023 7124 7225 7326 7427 7528 7629 7730 7831 7932 8033 8134 8235 8336 8437 8538 8639 8740 8841 8942 9043 9144 9245 9346 9447 9548 96

10

11

8

11

12

7

8

6

8

9

3

5

4

6

5

7

4

5

1

2

2

3

1

2

2. COORDENADAS LOCALES

ELEMENTO NUDO SCL ELEMENTO NUDO SCL97 14598 14699 147

100 148101 149102 150103 151104 152105 153106 154107 155108 156109 157110 158111 159112 160113 161114 162115 163116 164117 165118 166119 167120 168121 169122 170123 171124 172125 173126 174127 175128 176129 177130 178131 179132 180133 181134 182135 183136 184137 185138 186139 187140 188141 189142 190143 191144 192

12

8

16

9

11 12

11

2

15

3

5 6

10

5

14

6

11 12

9

2

13

3

8 9

3. DETERMINAR LOS ke (S.C.L.)

ke =

4. DETERMINAR LA MATRIZ DE TRANSFORMACIÓN T DE LOS ELEMENTOS

T =

5. DETERMINAR LOS Ke (S.C.G.)

Ke = TT *ke * T

6. DETERMINAR LA MATRIZ DE RIGIDEZ DE LA ESTRUCTURA

MATRIZ DE RIGIDEZ DE LA ESTRUCTURA ( VER ARCHIVO EXCEL )

7. DETERMINAR LAS CARGAS DE FIJACIÓN DE CADA ELEMENTO (QF)

8. DETERMINAR LAS CARGAS EQUIVALENTES DE CADA ELEMENTO (QEQ) - S.C.L.

qeq = - qf

9. DETERMINAR LAS CARGAS EQUIVALENTES DE CADA ELEMENTO (QEQE) - S.C.G

Qeqe = Tt * qeq

10. DETERMINAR LAS CARGAS EQUIVALENTES TOTALESQeq

11. DETERMINAR LAS CARGAS EN LOS NUDOS (Qn) - S.C.G.

• Se forma el vector “Qn” con las cargas puntuales conocidas y desconocidas que actúan en los nudos de la estructura con.

• Si no conocemos la carga puntual que actúa en esa coordenada Global, como es en el caso de los apoyos, se expresa por:

Donde x=coordenada global

VECTOR Qn:

SCG SCG SCG

Qn1 1 8 13 4 25

Qn2 2 0 14 0 26

Qn3 3 0 15 0 27

Qn4 4 0 16 0 28

Qn= Qn5 5 Qn= 0 17 Qn= 0 29

Qn6 6 0 18 0 30

4 7 Qn19 19 8 31

0 8 Qn20 20 0 32

0 9 Qn21 21 0 33

0 10 Qn22 22 0 34

0 11 Qn23 23 0 35

0 12 Qn24 24 0 36

VECTOR Qn:

SCG SCG SCG

Qn37 37 8 49 4 61

Qn38 38 0 50 0 62

Qn39 39 0 51 0 63

Qn40 40 0 52 0 64

Qn= Qn41 41 Qn= 0 53 Qn= 0 65

Qn42 42 0 54 0 66

4 43 Qn55 55 8 67

0 44 Qn56 56 0 68

0 45 Qn57 57 0 69

0 46 Qn58 58 0 70

0 47 Qn59 59 0 71

0 48 Qn60 60 0 72

12. DETERMINAR EL VECTOR DE CARGAS (Q) - S.C.G.

• Se forma el vector ”Q” con la suma del vector “Qeq” del paso 10 y el vector “Qn” del paso 11.

• Si no conocemos la suma en la coordenada global, como es en el caso de los apoyos porque no se conocen las reacciones, se expresa por:

Donde x=coordenada global


SCG Q

SCG Q

SCG Q

1 Qn1

13 8

25 4 2 Qn2

14 0

26 0

3 Qn3

15 -67

27 -23 4 Qn4

16 -27.0833333

28 -10.4166667

5 Qn5

17 -14

29 10 6 Qn6

18 0

30 0

7 4

19 Qn19

31 8 8 0

20 Qn20

32 0

9 -23

21 Qn21

33 -61 10 -10.4166667

22 Qn22

34 -27.0833333

11 10

23 Qn23

35 34 12 0

24 Qn24

36 0


SCG Q

SCG Q

SCG Q

37 Qn37

49 8

61 4 38 Qn38

50 0

62 0

39 Qn39

51 -61

63 -23 40 Qn40

52 27.0833333

64 10.4166667

41 Qn41

53 -14

65 10 42 Qn42

54 0

66 0

43 4

55 Qn55

67 8 44 0

56 Qn56

68 0

45 -23

57 Qn57

69 -55 46 10.4166667

58 Qn58

70 27.0833333

47 10

59 Qn59

71 34 48 0

60 Qn60

72 0

13. DETERMINAR EL VECTOR DE DESPLAZAMIENTOS (D) - S.C.G.

• Se determina el vector “D” de desplazamientos que tiene cada coordenada Global.

• Si no conocemos el desplazamiento en la coordenada global, como es en el caso de los nudos libres, se expresa por:

Donde: x=coordenada global

VECTOR D:

SCG SCG SCG

0 1 D13 13 D25 25

0 2 D14 14 D26 26

-0.025 3 D15 15 D27 27

0 4 D16 16 D28 28

D = 0 5 D = D17 17 D = D29 29

0 6 D18 18 D30 30

D7 7 0 19 D31 31

D8 8 0 20 D32 32

D9 9 0 21 D33 33

D10 10 0 22 D34 34

D11 11 0 23 D35 35

D12 12 0 24 D36 36

VECTOR D:

SCG SCG SCG

0 37 D49 49 D61 61

0 38 D50 50 D62 62

0 39 D51 51 D63 63

0 40 D52 52 D64 64

D = 0 41 D = D53 53 D = D65 65

0 42 D54 54 D66 66

D43 43 0 55 D67 67

D44 44 0 56 D68 68

D45 45 -0.025 57 D69 69

D46 46 0 58 D70 70

D47 47 0 59 D71 71

D48 48 0 60 D72 72

14. CÁLCULO DE Do y Qx

Para calcular los vectores “Do” y “Qx” se procede a ordenar la matriz ensamblada; formando sub matrices, de acuerdo al vector de fuerzas globales, separando las fuerza externas de las reacciones; es decir, lo conocido de lo desconocido, obteniendo así la matriz reordenada “K”.


Ordenándolo de la siguiente forma:

Entonces para hallar “Do” y “Qx” despejamos de:Donde:


Obteniéndose así:


SCG Do

SCG Do

SCG Do

SCG Do 7 0.01428397

25 0.01415233

43 0.00898497

61 0.00885333

8 0.00023597

26 -0.00039678

44 0.0003481

62 -0.00028464 9 -0.02590882

27 -0.00086994

45 -0.0012723

63 -0.02538393

10 0.00258593

28 -0.0032951

46 0.00331944

64 -0.00256159 11 0.00571281 29 0.0049938

47 0.00112248

65 0.00040348

12 0.00053525 30 0.00053525

48 0.00053525

66 0.00053525 13 0.03161609 31 0.03172569

49 0.0123621

67 0.01247169

14 -0.00697642 32 0.00694323

50 -0.00709522

68 0.00682443 15 -0.02653189 33 -0.00142452

51 -0.00197438

69 -0.02574921

16 -0.0006959 34 -0.00818396

52 0.0082285

70 0.00074044 17 0.00330447 35 0.00734066

53 -0.00302309

71 0.0010131

18 0.0007363 36 0.0007363

54 0.0007363

72 0.0007363


SCG Qx

SCG Qx 1 -6.2908996

37 -14.8339267

2 -2.97549277

38 -3.84437157 3 88.8978542

39 124.45273

4 4.05387938

40 5.25351788 5 -29.3460465

41 -32.9617868

6 -0.44898743

42 -0.44898743 19 -9.16607326

55 -17.7091004

20 3.84437157

56 2.97549277 21 85.0950041

57 37.5544118

22 -5.27137201

58 -4.07173351 23 -32.9864611

59 -36.6022014

24 -0.44898743

60 -0.44898743

15. CÁLCULO DE LAS FUERZAS Q

Donde:


SCG Q

SCG Q

SCG Q

1 -6.2908996

13 8

25 4 2 -2.97549277

14 1.1369E-13

26 -1.4566E-13

3 88.8978542

15 -67

27 -23 4 4.05387938

16 -27.0833333

28 -10.4166667

5 -29.3460465

17 -14

29 10 6 -0.44898743

18 5.6621E-15

30 -6.6613E-15

7 4

19 -9.16607326

31 8 8 -2.4869E-14

20 3.84437157

32 -3.4106E-13

9 -23

21 85.0950041

33 -61 10 -10.4166667

22 -5.27137201

34 -27.0833333

11 10

23 -32.9864611

35 34 12 -8.6597E-15

24 -0.44898743

36 -9.881E-15


SCG Q

SCG Q

SCG Q

37 -14.8339267

49 8

61 4 38 -3.84437157

50 -2.6257E-13

62 1.4344E-13

39 124.45273

51 -61

63 -23 40 5.25351788

52 27.0833333

64 10.4166667

41 -32.9617868

53 -14

65 10 42 -0.44898743

54 7.3275E-15

66 -2.3315E-15

43 4

55 -17.7091004

67 8 44 2.6812E-14

56 2.97549277

68 4.9194E-13

45 -23

57 37.5544118

69 -55 46 10.4166667

58 -4.07173351

70 27.0833333

47 10

59 -36.6022014

71 34 48 -4.4409E-15

60 -0.44898743

72 -4.4409E-15

16. CÁLCULOS DE LAS DEFORMACIONES EN LOS ELEMENTOS

d1

-0.02500000

-0.025908820.014283970.000235970.000535250.002585930.00571281

d2

-0.025908820.014283970.000235970.000535250.002585930.00571281-0.026531890.03161609-0.006976420.0007363-0.00069590.00330447

d3

000000

-0.000869940.01415233-0.000396780.00053525-0.00329510.0049938

d4

-0.000869940.01415233-0.000396780.00053525-0.00329510.0049938

-0.001424520.031725690.006943230.0007363

-0.008183960.00734066

d5

000000

-0.00127230.008984970.0003481

0.000535250.003319440.00112248

d6

-0.00127230.008984970.0003481

0.000535250.003319440.00112248-0.001974380.0123621

-0.007095220.00073630.0082285

-0.00302309

d7

-0.02500000

-0.025383930.00885333-0.000284640.00053525-0.002561590.00040348

d8

-0.025383930.00885333-0.000284640.00053525-0.002561590.00040348-0.025749210.012471690.006824430.0007363

0.000740440.0010131

d9

0.00023597-0.025908820.014283970.005712810.000535250.002585930.0003481-0.00127230.008984970.001122480.000535250.00331944

d10

-0.00039678-0.000869940.014152330.0049938

0.00053525-0.0032951

-0.00028464-0.025383930.008853330.000403480.00053525-0.00256159

d11

-0.01428397-0.025908820.00023597-0.002585930.000535250.00571281-0.01415233-0.00086994-0.000396780.0032951

0.000535250.0049938

d12

-0.00898497-0.00127230.0003481

-0.003319440.000535250.00112248-0.00885333-0.02538393-0.000284640.002561590.000535250.00040348

d13

-0.00697642-0.026531890.031616090.003304470.0007363-0.0006959

-0.00709522-0.001974380.0123621

-0.003023090.00073630.0082285

d14

0.00694323-0.001424520.031725690.007340660.0007363

-0.008183960.00682443-0.025749210.012471690.00101310.0007363

0.00074044

d15

-0.03161609-0.02653189-0.006976420.00069590.0007363

0.00330447-0.03172569-0.001424520.006943230.008183960.0007363

0.00734066

d16

-0.0123621-0.00197438-0.00709522-0.00822850.0007363

-0.00302309-0.01247169-0.025749210.00682443-0.000740440.00073630.0010131

de=TeD

17. CÁLCULO DE LAS FUERZAS qe

q 1

88.8978542-6.2908996

-2.97549277-0.448987434.053879378-29.3460465-88.89785426.2908996012.9754927650.4489874287.8480916824.182448142

q 2

67.71865296-11.49107194.75752982-0.17093996-4.46114306-11.7834704-67.718653

11.49107188-4.757529820.170939961-9.8114464

-22.6897452

q 3

85.09500415-9.166073263.844371567-0.44898743-5.27137201-32.9864611-85.09500419.166073257-3.844371570.448987428-10.1061143-3.67783193

q 4

60.274758492.80267212510.73657005-0.17093996-12.1197512-1.10994666-60.2747585-2.80267213

-10.736570.170939961-20.08995899.517963035

q 5

124.4527298-14.8339267-3.84437157-0.448987435.253517882-32.9617868-124.45273

14.833926743.8443715670.44898742810.12396839-26.3739202

q 6

76.3056083-18.8026721

-10.73657-0.1709399612.10327986-18.8172696-76.305608318.8026721310.736570050.17093996120.10643029-37.5907468

q 7

37.55441182-17.70910042.975492765-0.44898743-4.07173351-36.6022014-37.554411817.7091004-2.975492770.448987428-7.83023755-34.2342002

q 8

39.70098025-4.50892812-4.75752982-0.170939964.444671688-8.14374581-39.70098024.5089281194.75752982

0.1709399619.827917772-5.38303855q 9

-7.31261053-6.438359380.6157134482.341718572-1.53928362-17.09241537.3126105346.438359377-0.61571345-2.34171857-1.53928362-15.0993816

q 10

-7.312610536.4378062750.6157134482.341718572-1.5392836215.097998837.312610534-6.43780627-0.61571345-2.34171857-1.5392836217.09103254

q 11

-8.584458834.617560619-0.42041205-3.288800011.26123615315.2593037

8.584458831-4.617560620.4204120513.2888000071.26123615312.44606002

q 12

-8.5844588318.70876215-0.42041205-3.288800011.26123615357.532908298.584458831-18.70876210.4204120513.2888000071.26123615354.71966461

q 13

7.747049935-3.733990563.6558001223.22796054-9.13950031-21.4593714-7.747049933.73399056-3.65580012-3.22796054-9.139500312.78941859

q 14

7.7470499353.7274020113.6558001223.22796054-9.13950031-2.80588996-7.74704993-3.72740201-3.65580012-3.22796054-9.1395003121.44290002

q 15

7.1468720034.452643519-2.98952011-4.187484468.9685603455.46178469-7.146872

-4.452643522.9895201154.1874844568.96856034521.25407643

q 16

7.14687200311.57161774-2.98952011-4.187484468.96856034526.81870735

-7.146872-11.57161772.9895201154.1874844568.96856034542.61099909

qe=kede

18. CÁLCULO DE LAS FUERZAS LINEALES qFe qFe=qf+qe

qfinal 1

94.8978542 AXIAL-6.2908996 C2

-2.97549277 C3-0.44898743 M TORSOR4.05387938 M2-29.3460465 M3-82.8978542 AXIAL6.2908996 C2

2.97549277 C30.44898743 M TORSOR7.84809168 M24.18244814 M3

qfinal 2

72.218653 AXIAL-11.4910719 C24.75752982 C3-0.17093996 M TORSOR-4.46114306 M2-11.7834704 M3-63.218653 AXIAL11.4910719 C2-4.75752982 C30.17093996 M TORSOR-9.8114464 M2

-22.6897452 M3

qfinal 3

91.0950041 AXIAL-9.16607326 C23.84437157 C3-0.44898743 M TORSOR-5.27137201 M2-32.9864611 M3-79.0950041 AXIAL9.16607326 C2-3.84437157 C30.44898743 M TORSOR-10.1061143 M2-3.67783193 M3

qfinal 4

64.7747585 AXIAL2.80267213 C2

10.73657 C3-0.17093996 M TORSOR-12.1197512 M2-1.10994666 M3-55.7747585 AXIAL-2.80267213 C2

-10.73657 C30.17093996 M TORSOR-20.0899589 M29.51796303 M3

qfinal 5

130.45273 AXIAL-14.8339267 C2-3.84437157 C3-0.44898743 M TORSOR5.25351788 M2-32.9617868 M3-118.45273 AXIAL14.8339267 C23.84437157 C30.44898743 M TORSOR10.1239684 M2-26.3739202 M3

qfinal 6

80.8056083 AXIAL-18.8026721 C2

-10.73657 C3-0.17093996 M TORSOR12.1032799 M2-18.8172696 M3-71.8056083 AXIAL18.8026721 C2

10.73657 C30.17093996 M TORSOR20.1064303 M2-37.5907468 M3

qfinal 7

43.5544118 AXIAL-17.7091004 C22.97549277 C3-0.44898743 M TORSOR-4.07173351 M2-36.6022014 M3-31.5544118 AXIAL17.7091004 C2-2.97549277 C30.44898743 M TORSOR-7.83023755 M2-34.2342002 M3

qfinal 8

44.2009802 AXIAL-4.50892812 C2-4.75752982 C3-0.17093996 M TORSOR4.44467169 M2-8.14374581 M3-35.2009802 AXIAL4.50892812 C24.75752982 C30.17093996 M TORSOR9.82791777 M2-5.38303855 M3

18. CÁLCULO DE LAS FUERZAS LINEALES qFe

qfinal 9

-7.31261053 AXIAL6.06164062 C20.61571345 C3-7.65828143 M TORSOR-1.53928362 M2-6.67574863 M37.31261053 AXIAL18.9383594 C2-0.61571345 C3-12.3417186 M TORSOR-1.53928362 M2-25.5160483 M3

qFe=qf+qe

qfinal 10

-7.31261053 AXIAL18.9378063 C20.61571345 C3-7.65828143 M TORSOR-1.53928362 M225.5146655 M37.31261053 AXIAL6.06219373 C2-0.61571345 C3-12.3417186 M TORSOR-1.53928362 M26.67436587 M3

qfinal 11

-8.58445883 AXIAL4.61756062 C2-0.42041205 C3-3.28880001 M TORSOR1.26123615 M215.2593037 M38.58445883 AXIAL-4.61756062 C20.42041205 C33.28880001 M TORSOR1.26123615 M2

12.44606 M3

qfinal 12

-8.58445883 AXIAL18.7087621 C2-0.42041205 C3-3.28880001 M TORSOR1.26123615 M257.5329083 M38.58445883 AXIAL-18.7087621 C20.42041205 C33.28880001 M TORSOR1.26123615 M254.7196646 M3

qfinal 13

7.74704993 AXIAL31.7660094 C23.65580012 C3-6.77203946 M TORSOR-9.13950031 M25.62396194 M3-7.74704993 AXIAL33.2339906 C2-3.65580012 C3-13.2279605 M TORSOR-9.13950031 M2-24.2939147 M3

qfinal 14

7.74704993 AXIAL39.227402 C2

3.65580012 C3-6.77203946 M TORSOR-9.13950031 M224.2774434 M3-7.74704993 AXIAL25.772598 C2

-3.65580012 C3-13.2279605 M TORSOR-9.13950031 M2-5.64043332 M3

qfinal 15

7.146872 AXIAL31.4526435 C2-2.98952011 C3-4.18748446 M TORSOR8.96856034 M229.4617847 M3-7.146872 AXIAL

16.5473565 C22.98952011 C34.18748446 M TORSOR8.96856034 M2-2.74592357 M3

qfinal 16

7.146872 AXIAL38.5716177 C2-2.98952011 C3-4.18748446 M TORSOR8.96856034 M250.8187074 M3-7.146872 AXIAL

9.42838226 C22.98952011 C34.18748446 M TORSOR8.96856034 M218.6109991 M3

DIAGRAMAS DE FUERZAS AXIALES(plano 1-2)

DIAGRAMA DE FUERZAS CORTANTES 2-2 (plano1-2)

DIAGRAMA DE FUERZAS CORTANTES 3-3 (plano1-3)

DIAGRAMA DE MOMENTOS FLECTORES 2-2 (plano 1-3)

DIAGRAMA DE MOMENTOS FLECTORES 3-3 (plano 1-2)

DIAGRAMA DE MOMENTOS TORSORES (plano 1-2)

19.CÁLCULO DE LAS REACCIONESReacciones en Nudo 1 Reacciones en Nudo 4

Reacciones en Nudo 7

Reacciones en Nudo 10

20

. DEFO

RM

AD

A

Soluc. Examen Final 2014-i Marco 3d

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