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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)
qf
qf
qf
qf
8. DETERMINAR LAS CARGAS EQUIVALENTES DE CADA ELEMENTO (QEQ) - S.C.L.
qeq = - qf
qeq
qeq
qeq
qeq
9. DETERMINAR LAS CARGAS EQUIVALENTES DE CADA ELEMENTO (QEQE) - S.C.G
Qeqe = Tt * qeq
Qeqe
Qeqe
Qeqe
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
12. DETERMINAR EL VECTOR DE CARGAS (Q) - S.C.G.
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
12. DETERMINAR EL VECTOR DE CARGAS (Q) - S.C.G.
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”.
14. CÁLCULO DE Do y Qx
Ordenándolo de la siguiente forma:
Entonces para hallar “Do” y “Qx” despejamos de:Donde:
14. CÁLCULO DE Do y Qx
Obteniéndose así:
14. CÁLCULO DE Do y Qx
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
14. CÁLCULO DE Do y Qx
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:
15. CÁLCULO DE LAS FUERZAS Q
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
15. CÁLCULO DE LAS FUERZAS Q
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)
DIAGRAMAS DE FUERZAS AXIALES(plano 1-2)
DIAGRAMA DE FUERZAS CORTANTES 2-2 (plano1-2)
DIAGRAMA DE FUERZAS CORTANTES 2-2 (plano1-2)
DIAGRAMA DE FUERZAS CORTANTES 3-3 (plano1-3)
DIAGRAMA DE FUERZAS CORTANTES 3-3 (plano1-3)
DIAGRAMA DE MOMENTOS FLECTORES 2-2 (plano 1-3)
DIAGRAMA DE MOMENTOS FLECTORES 2-2 (plano 1-3)
DIAGRAMA DE MOMENTOS FLECTORES 3-3 (plano 1-2)
DIAGRAMA DE MOMENTOS FLECTORES 3-3 (plano 1-2)
DIAGRAMA DE MOMENTOS TORSORES (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