Segundo Ejercicio Matricial.docx
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b) Resolver la estructura por el método matricial con el mínimo número de nudos posibles y calcular; Reacciones y Valores de; Cortantes, Momentos, Función de Momentos y Elástica. Graficar todos los resultados.
5[tn / m]
4[tn / m]
3,5[tn � m]
2
2
2[tn ]
4.5
4
CONFIGURACIÓN DE NUDOS q3 q1 A
1
q2
q5
B
q4
q7
C
2
q6
ANÁLISIS DE CARGAS BARRA 1
0
0
0 A
B
2
0
BARRA 2 0.148 [ tn ]
0.889 [ tn � m]
5[tn / m] 7.852 [ tn ]
C
B
2
2[tn]
4
BARRA 3 2.700 [ tn ]
6.300 [ tn ]
[m ]
5.333 [ tn � m]
3
D
q8
4[tn / m] 2.700 [ tn � m]
D
C
4.5
4.050 [ tn � m]
RESUMEN 0.148 [ tn]
0
A
B
1
0
4.389
VA
10.552 [ tn ]
C
2
2.633
VB
6.300 [ tn ]
3
VD
VC
LOS MATRICES DE LAS RIGIDEZ 12 6L -12 6L � � � EI � 6L 4L2 -6L 2L2 � � K= L3 � -12 -6L 12 -6L � � � 6L 2L2 -6L 4L2 � � BARRA 1
12 � � 9000 12 K= � 8 � -12 � 12 �
12 16 -12 8
-12 -12 12 -12
12 � �13500 13500 -13500 13500 � � 8 � 9000 � �= �13500 18000 -13500 � -12 � � -13500 -13500 13500 -13500 � � � � 16 � �13500 9000 -13500 18000 �
BARRA 2
12 � � 9000 36 � K= 216 � -12 � 36 �
36 144 -36 72
-12 -36 12 -36
36 � � 500 � � 72 � � 1500 = -36 � � -500 � � 144 � � 1500
1500 6000 -1500 3000
-500 -1500 500 -1500
D
1500 � 3000 � � -1500 � � 6000 �
4.050
BARRA 3
12 � 9000 � 27 K= 91.125 � � -12 � 27 �
27 81 -27 40.5
-12 -27 12 -27
27 � � 1185.185 � � 40.5� � 2666.667 = -27 � � -1185.185 � � 81 � � 2666.667
2666.667 8000.000 -2666.667 4000.000
-1185.185 -2666.667 1185.185 -2666.667
2666.667 � 4000.000 � � -2666.667 � � 8000.000 �
MATRIZ GLOBAL DE LA RIGIDEZ 1 1� 13500 � 2 13500 � 3� -13500 � 4 13500 � 5� 0 � 6� 0 � 7 0 8� 0 �
2 13500 18000 -13500 9000 0 0 0 0
3 4 -13500 13500
5 0
6 0
7 0
-13500 9000 14000 -12000
0 -500.0
0 1500.0
0 0
-12000 24000
-1500.0
3000.000
0
-500
-1500
1685.185
1166.667
-1185.185
1500
3000.000 1166.667
0
0
-1185.185 -2666.667
1185.185
0
0
2666.667
-2666.667
14000.000 -2666.667 4000.000
CALCULO DE INCÓGNITAS
24000 3000.000 �� -4.389 � � q4 � � � = �6 � � 3000.000 14000 �� 2.633 � q � � �� � � q 4 = -0.000212 q 6 = 0.000234 CALCULO DE REACCIONES Y MOMENTO VA = 2.863[ tn � m] M A = 1.909 [ tn � m] VB = 3.043 [ tn � m] VC = 11.143 [ tn � m] VD = 5.677 [ tn � m] M D = 3.116 [ tn � m]
8 0
VA = q1 � � �� �2 � � � MA = 0 q � � �� 3 �� VB = 0 q � � ��4 � � 0 q � � � �� = � VC = 2666.667 � q 5 � � ��6 � � 4000.000 �� q � � � � V = -2666.667 q 7 � � ��8 � � D MD = 8000.000 �� � q � � �
0
� � 0 � -0.148 � � -4.389 � -10.552 � � 2.633 � -6.30 � � 4.050 �
Xn Le
ξ
H1 -1
H2 1
H3 0
H4 0
q 0
q
0
2
0.00000 0.00000
0
2
-0.50000 0.84375
0.28125 0.15625 -0.09375 0.00000 0.00000
0
2
0.00000
0.50000
0.25000 0.50000 -0.25000 0.00000 0.00000
0
2
0.50000
0.15625
0.09375 0.84375 -0.28125 0.00000 0.00000
0
2
1.00000
0.00000
0.00000 1.00000 0.00000 0.00000 0.00000
2
6
-0.83333 0.98032
0.14005 0.01968 -0.01273 0.00000 -0.00021
2
6
-0.66667 0.92593
0.23148 0.07407 -0.04630 0.00000 -0.00021
2
6
-0.50000 0.84375
0.28125 0.15625 -0.09375 0.00000 -0.00021
2
6
-0.33333 0.74074
0.29630 0.25926 -0.14815 0.00000 -0.00021
2
6
-0.16667 0.62384
0.28356 0.37616 -0.20255 0.00000 -0.00021
2
6
0.00000
0.50000
0.25000 0.50000 -0.25000 0.00000 -0.00021
2
6
0.16667
0.37616
0.20255 0.62384 -0.28356 0.00000 -0.00021
2
6
0.33333
0.25926
0.14815 0.74074 -0.29630 0.00000 -0.00021
2
6
0.50000
0.15625
0.09375 0.84375 -0.28125 0.00000 -0.00021
2
6
0.66667
0.07407
0.04630 0.92593 -0.23148 0.00000 -0.00021
2
6
0.83333
0.01968
0.01273 0.98032 -0.14005 0.00000 -0.00021
2
6
1.00000
0.00000
0.00000 1.00000 0.00000 0.00000 -0.00021
8
4.5
-0.77778 0.96571
0.17558 0.03429 -0.02195 0.00000 0.00023
8
4.5
-0.55556 0.87380
0.26886 0.12620 -0.07682 0.00000 0.00023
8
4.5
-0.33333 0.74074
0.29630 0.25926 -0.14815 0.00000 0.00023
8
4.5
-0.11111 0.58299
0.27435 0.41701 -0.21948 0.00000 0.00023
8
4.5
0.11111
0.41701
0.21948 0.58299 -0.27435 0.00000 0.00023
8
4.5
0.33333
0.25926
0.14815 0.74074 -0.29630 0.00000 0.00023
q 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0
q -0.00021 -0.00021 -0.00021 -0.00021 -0.00021 0.00023 0.00023 0.00023 0.00023 0.00023 0.00023 0.00023 0.00023 0.00023 0.00023 0.00023 0.00023 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
M
Q
ϴ
n
1.90858 -2.86287
0.00000
0.47714 -2.86287
0.00007
-0.95429 -2.86287
0.00005
-2.38572 -2.86287
-0.00004
-3.81716 -2.86287
-0.00021
0.58793
0.03217
-0.00018
0.60402
0.03217
-0.00015
0.62011
0.03217
-0.00011
0.63619
0.03217
-0.00008
0.65228
0.03217
-0.00004
0.66837
0.03217
-0.00001
0.68445
0.03217
0.00003
0.70054
0.03217
0.00007
0.71663
0.03217
0.00011
0.73272
0.03217
0.00015
0.74880
0.03217
0.00019
0.76489
0.03217
0.00023
-1.55676 0.62270
0.00014
-1.24541 0.62270
0.00006
-0.93406 0.62270
0.00000
-0.62270 0.62270
-0.00004
-0.31135 0.62270
-0.00007
0.00000
-0.00008
0.62270
8
4.5
0.55556
0.12620
0.07682 0.87380 -0.26886 0.00000 0.00023
8
4.5
0.77778
0.03429
0.02195 0.96571 -0.17558 0.00000 0.00023
8
4.5
1
0
0.00000 1.00000 0.00000 0.00000 0.00023
0.0000 0 0.0000 0 0.0000 0
0.00000 0.00000 0.00000
CALCULO DE MOMENTOS, CORTANTE, ANGULO DE ROTACIÓN Y LA FLECHA
0.31135
0.62270
-0.00007
0.62270
0.62270
-4.32E-05
0.93406
0.62270
0.00E+00
5[tn / m]
4[tn / m]
3,5[tn � m]
3.116 [ tn � m]
1.909 [ tn � m]
2
2
2[tn ]
VA = 2.863 [ tn ] VB = 3.043 [ tn ]
4
4.5 VC = 11.14 [ tn ]
[m ] VD = 5.667 [ tn ]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
n =
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
q =
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
M=
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Q =
5[tn / m]
4[tn / m]
3,5[tn � m]
2
2
2[tn ]
4.5
4
CONFIGURACIÓN DE NUDOS q3 q1 A
1
q2
q5
B
q4
q7
C
2
q6
ANÁLISIS DE CARGAS BARRA 1
0
0
0 A
B
2
0
BARRA 2 0.148 [ tn ]
0.889 [ tn � m]
5[tn / m] 7.852 [ tn ]
C
B
2
2[tn]
4
BARRA 3 2.700 [ tn ]
6.300 [ tn ]
[m ]
5.333 [ tn � m]
3
D
q8
4[tn / m] 2.700 [ tn � m]
D
C
4.5
4.050 [ tn � m]
RESUMEN 0.148 [ tn]
0
A
B
1
0
4.389
VA
10.552 [ tn ]
C
2
2.633
VB
6.300 [ tn ]
3
VD
VC
LOS MATRICES DE LAS RIGIDEZ 12 6L -12 6L � � � EI � 6L 4L2 -6L 2L2 � � K= L3 � -12 -6L 12 -6L � � � 6L 2L2 -6L 4L2 � � BARRA 1
12 � � 9000 12 K= � 8 � -12 � 12 �
12 16 -12 8
-12 -12 12 -12
12 � �13500 13500 -13500 13500 � � 8 � 9000 � �= �13500 18000 -13500 � -12 � � -13500 -13500 13500 -13500 � � � � 16 � �13500 9000 -13500 18000 �
BARRA 2
12 � � 9000 36 � K= 216 � -12 � 36 �
36 144 -36 72
-12 -36 12 -36
36 � � 500 � � 72 � � 1500 = -36 � � -500 � � 144 � � 1500
1500 6000 -1500 3000
-500 -1500 500 -1500
D
1500 � 3000 � � -1500 � � 6000 �
4.050
BARRA 3
12 � 9000 � 27 K= 91.125 � � -12 � 27 �
27 81 -27 40.5
-12 -27 12 -27
27 � � 1185.185 � � 40.5� � 2666.667 = -27 � � -1185.185 � � 81 � � 2666.667
2666.667 8000.000 -2666.667 4000.000
-1185.185 -2666.667 1185.185 -2666.667
2666.667 � 4000.000 � � -2666.667 � � 8000.000 �
MATRIZ GLOBAL DE LA RIGIDEZ 1 1� 13500 � 2 13500 � 3� -13500 � 4 13500 � 5� 0 � 6� 0 � 7 0 8� 0 �
2 13500 18000 -13500 9000 0 0 0 0
3 4 -13500 13500
5 0
6 0
7 0
-13500 9000 14000 -12000
0 -500.0
0 1500.0
0 0
-12000 24000
-1500.0
3000.000
0
-500
-1500
1685.185
1166.667
-1185.185
1500
3000.000 1166.667
0
0
-1185.185 -2666.667
1185.185
0
0
2666.667
-2666.667
14000.000 -2666.667 4000.000
CALCULO DE INCÓGNITAS
24000 3000.000 �� -4.389 � � q4 � � � = �6 � � 3000.000 14000 �� 2.633 � q � � �� � � q 4 = -0.000212 q 6 = 0.000234 CALCULO DE REACCIONES Y MOMENTO VA = 2.863[ tn � m] M A = 1.909 [ tn � m] VB = 3.043 [ tn � m] VC = 11.143 [ tn � m] VD = 5.677 [ tn � m] M D = 3.116 [ tn � m]
8 0
VA = q1 � � �� �2 � � � MA = 0 q � � �� 3 �� VB = 0 q � � ��4 � � 0 q � � � �� = � VC = 2666.667 � q 5 � � ��6 � � 4000.000 �� q � � � � V = -2666.667 q 7 � � ��8 � � D MD = 8000.000 �� � q � � �
0
� � 0 � -0.148 � � -4.389 � -10.552 � � 2.633 � -6.30 � � 4.050 �
Xn Le
ξ
H1 -1
H2 1
H3 0
H4 0
q 0
q
0
2
0.00000 0.00000
0
2
-0.50000 0.84375
0.28125 0.15625 -0.09375 0.00000 0.00000
0
2
0.00000
0.50000
0.25000 0.50000 -0.25000 0.00000 0.00000
0
2
0.50000
0.15625
0.09375 0.84375 -0.28125 0.00000 0.00000
0
2
1.00000
0.00000
0.00000 1.00000 0.00000 0.00000 0.00000
2
6
-0.83333 0.98032
0.14005 0.01968 -0.01273 0.00000 -0.00021
2
6
-0.66667 0.92593
0.23148 0.07407 -0.04630 0.00000 -0.00021
2
6
-0.50000 0.84375
0.28125 0.15625 -0.09375 0.00000 -0.00021
2
6
-0.33333 0.74074
0.29630 0.25926 -0.14815 0.00000 -0.00021
2
6
-0.16667 0.62384
0.28356 0.37616 -0.20255 0.00000 -0.00021
2
6
0.00000
0.50000
0.25000 0.50000 -0.25000 0.00000 -0.00021
2
6
0.16667
0.37616
0.20255 0.62384 -0.28356 0.00000 -0.00021
2
6
0.33333
0.25926
0.14815 0.74074 -0.29630 0.00000 -0.00021
2
6
0.50000
0.15625
0.09375 0.84375 -0.28125 0.00000 -0.00021
2
6
0.66667
0.07407
0.04630 0.92593 -0.23148 0.00000 -0.00021
2
6
0.83333
0.01968
0.01273 0.98032 -0.14005 0.00000 -0.00021
2
6
1.00000
0.00000
0.00000 1.00000 0.00000 0.00000 -0.00021
8
4.5
-0.77778 0.96571
0.17558 0.03429 -0.02195 0.00000 0.00023
8
4.5
-0.55556 0.87380
0.26886 0.12620 -0.07682 0.00000 0.00023
8
4.5
-0.33333 0.74074
0.29630 0.25926 -0.14815 0.00000 0.00023
8
4.5
-0.11111 0.58299
0.27435 0.41701 -0.21948 0.00000 0.00023
8
4.5
0.11111
0.41701
0.21948 0.58299 -0.27435 0.00000 0.00023
8
4.5
0.33333
0.25926
0.14815 0.74074 -0.29630 0.00000 0.00023
q 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 0
q -0.00021 -0.00021 -0.00021 -0.00021 -0.00021 0.00023 0.00023 0.00023 0.00023 0.00023 0.00023 0.00023 0.00023 0.00023 0.00023 0.00023 0.00023 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
M
Q
ϴ
n
1.90858 -2.86287
0.00000
0.47714 -2.86287
0.00007
-0.95429 -2.86287
0.00005
-2.38572 -2.86287
-0.00004
-3.81716 -2.86287
-0.00021
0.58793
0.03217
-0.00018
0.60402
0.03217
-0.00015
0.62011
0.03217
-0.00011
0.63619
0.03217
-0.00008
0.65228
0.03217
-0.00004
0.66837
0.03217
-0.00001
0.68445
0.03217
0.00003
0.70054
0.03217
0.00007
0.71663
0.03217
0.00011
0.73272
0.03217
0.00015
0.74880
0.03217
0.00019
0.76489
0.03217
0.00023
-1.55676 0.62270
0.00014
-1.24541 0.62270
0.00006
-0.93406 0.62270
0.00000
-0.62270 0.62270
-0.00004
-0.31135 0.62270
-0.00007
0.00000
-0.00008
0.62270
8
4.5
0.55556
0.12620
0.07682 0.87380 -0.26886 0.00000 0.00023
8
4.5
0.77778
0.03429
0.02195 0.96571 -0.17558 0.00000 0.00023
8
4.5
1
0
0.00000 1.00000 0.00000 0.00000 0.00023
0.0000 0 0.0000 0 0.0000 0
0.00000 0.00000 0.00000
CALCULO DE MOMENTOS, CORTANTE, ANGULO DE ROTACIÓN Y LA FLECHA
0.31135
0.62270
-0.00007
0.62270
0.62270
-4.32E-05
0.93406
0.62270
0.00E+00
5[tn / m]
4[tn / m]
3,5[tn � m]
3.116 [ tn � m]
1.909 [ tn � m]
2
2
2[tn ]
VA = 2.863 [ tn ] VB = 3.043 [ tn ]
4
4.5 VC = 11.14 [ tn ]
[m ] VD = 5.667 [ tn ]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
n =
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
q =
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
M=
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Q =
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