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Facultad de ingeniería civil - Análisis Matricial
UNIVERSIDAD NACIONAL DE SAN ANTONIO ABAD DEL CUSCO
FACULTAD DE INGENIERIA CIVIL
ASIGNATURA: ANALISIS MATRICIAL DE ESRUCTURAS
ESTUDIANTE:
TORRES APAZA DIEGO ARMANDO 111845
HUYHUA MONTES HERIXS Página 1
EJERCICIOS RESUELTOS
Facultad de ingeniería civil - Análisis Matricial
1. Resolver por el método de rigidez, para el siguiente pórtico que se muestra en la figura:
Considere I=500in^4, A=10in^2, E=29(10^3)ksi
AE/L 0 0 -AE/L 0 0
0 12EI/L^3 6EI/L^2 0 -12EI/L^3 6EI/L^2
K= 0 6EI/L^2 4EI/L 0 -6EI/L^2 2EI/L
-AE/L 0 0 AE/L 0 0
0 -12EI/L^3 -6EI/L^2 0 12EI/L^3 -6EI/L^2
0 6EI/L^2 2EI/L 0 -6EI/L^2 4EI/L
0 0 0 0
0 0 0 0
T= 0 0 1 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0 0 1
[K ¿¿CG]=[T ]T [K ] [T ]¿
Solución
1.1. Matriz de rigidez en coordenadas locales para la barra 1
HUYHUA MONTES HERIXS Página 2
λ xλ x
−λ y−λ y
λ x −λ yλ x−λ y
Facultad de ingeniería civil - Análisis Matricial
E(ksi) A(in^2) I(in^4) L(in)
29000 10 500 240
AE/L 12EI/L^3 6EI/L^2 4EI/L 2EI/L
1208.33333 12.5868056 1510.41667 241666.667 120833.333
Nodos X Y
1 0 0 (240-0)/240 (0-0)/240
2 240 0 1 0
4 6 5 1 2 3
1208.33 0 0 -1208.33 0 0 4 1 0 0 0 0 0
0 12.5868 1510.42 0 -12.5868 1510.42 6 0 1 0 0 0 0
K1= 0 1510.42 241667 0 -1510.42 120833 5 T= 0 0 1 0 0 0
-1208.33 0 0 1208.33 0 0 1 0 0 0 1 0 0
0 -12.5868 -1510.42 0 12.5868 -1510.42 2 0 0 0 0 1 0
0 1510.42 120833 0 -1510.42 241667 3 0 0 0 0 0 1
4 6 5 1 2 3
1 0 0 0 0 0 1208.3 0 0 -1208 0 0 4
0 1 0 0 0 0 0 12.587 1510.4 0 -12.59 1510.4 6
[T ]T= 0 0 1 0 0 0 [KCG]1=¿ 0 1510.4 241667 0 -1510 120833 5
0 0 0 1 0 0 -1208 0 0 1208.3 0 0 1
0 0 0 0 1 0 0 -12.59 -1510 0 12.587 -1510 2
0 0 0 0 0 1 0 1510.4 120833 0 -1510 241667 3
1.2. MATRIZ DE RIGIDEZ EN COORDENADAS LOCALES PARA LA BARRA 2
1 2 3 7 8 9
1208.33 0 0 -1208.33 0 0 1 0 -1 0 0 0 0
0 12.5868 1510.42 0 -12.5868 1510.42 2 1 0 0 0 0 0
K= 0 1510.42 241667 0 -1510.42 120833 3 T= 0 0 1 0 0 0
-1208.33 0 0 1208.33 0 0 7 0 0 0 0 -1 0
0 -12.5868 -1510.42 0 12.5868 -1510.42 8 0 0 0 1 0 0
0 1510.42 120833 0 -1510.42 241667 9 0 0 0 0 0 1
HUYHUA MONTES HERIXS Página 3
λ x λ y
Facultad de ingeniería civil - Análisis Matricial
1 2 3 7 8 9
0 1 0 0 0 0 12.587 0 1510.4 -12.59 0 1510.4 1
-1 0 0 0 0 0 0 1208.3 0 0 -1208 0 2
[T ]T=¿ 0 0 1 0 0 0 [KCG]1=¿1510.4 0 241667 -1510 0 120833 3
0 0 0 0 1 0 -12.59 0 -1510 12.587 0 -1510 7
0 0 0 -1 0 0 0 -1208 0 0 1208.3 0 8
0 0 0 0 0 1 1510.4 0 120833 -1510 0 241667 9
1.3. MATRIZ DE RIGIDEZ EN COORDENADAS GLOBALES
1 2 3 4 5 6 7 8 9
1220.92014 0 1510.41667 -1208.3333 0 0 -12.586806 0 1510.41667 1
0 1220.92014 -1510.4167 0 -1510.4167 -12.586806 0 -1208.3333 0 2
1510.41667 -1510.4167 483333.333 0 120833.333 1510.41667 -1510.4167 0 120833.333 3
KCG= -1208.3333 0 0 1208.33333 0 0 0 0 0 4
0 -1510.4167 120833.333 0 241666.667 1510.41667 0 0 0 5
0 -12.586806 1510.41667 0 1510.41667 12.5868056 0 0 0 6
-12.586806 0 -1510.4167 0 0 0 12.5868056 0 -1510.4167 7
0 -1208.3333 0 0 0 0 0 1208.33333 0 8
1510.41667 0 120833.333 0 0 0 -1510.4167 0 241666.667 9
1.4. CALCULO DE LOS DESPLAZAMIENTOS DESCONOCIDOS
5 1220.92014 0 1510.41667 -1208.3333 0 D1
0 0 1220.92014 -1510.4167 0 -1510.4167 D2
0 = 1510.41667 -1510.4167 483333.333 0 120833.333 D3
0 -1208.3333 0 0 1208.33333 0 D4
0 0 -1510.4167 120833.333 0 241666.667 D5
D1 0.69575393
D2 -0.00155071
D3 = -0.0024876
D4 0.69575393
D5 , 0.00123411
1.5. CALCULO DE LAS REACCIONES DESCONOCIDAS
HUYHUA MONTES HERIXS Página 4
Facultad de ingeniería civil - Análisis Matricial
Q6 -1.87378009 K
Q7 = -5 K
Q81.87378009
1 K
Q9750.292778
1 K.in
1.6. CALCULO DE FUERZAS PARA LA BARRA 1
q=[k ] [T ][D ]
q1 K T D
N4 1208.33 0 0 -1208.33 0 0 1 0 0 0 0 0 0.695754
N6 0 12.5868 1510.42 0 -12.5868 1510.42 0 1 0 0 0 0 0
N5 = 0 1510.42 241667 0 -1510.42 120833 0 0 1 0 0 0 0.001234
F1 -1208.33 0 0 1208.33 0 0 0 0 0 1 0 0 0.695754
F2 0 -12.5868 -1510.42 0 12.5868 -1510.42 0 0 0 0 1 0 -0.00155
F3 0 1510.42 120833 0 -1510.42 241667 0 0 0 0 0 1 -0.00249
q1
N4 0
N6 -1.8737801
N5 = 5.6843E-14
F1 0
F2 1.87378009
F3 -449.70722
1.7. CALCULO DE FUERZAS PARA LA BARRA 2
q2 K T D
N1 1208.33 0 0 -1208.33 0 0 0-1 0 0 0 0 0.695754
N2 0 12.5868 1510.42 0 -12.5868 1510.42 1 0 0 0 0 0 -0.00155
N3 = 0 1510.42 241667 0 -1510.42 120833 0 0 1 0 0 0 -0.00249
F7 -1208.33 0 0 1208.33 0 0 0 0 0 0-1 0 0
F8 0 -12.5868 -1510.4 0 12.5868 -1510.42 0 0 0 1 0 0 0
F9 0 1510.42 120833 0 -1510.42 241667 0 0 0 0 0 1 0
HUYHUA MONTES HERIXS Página 5
Facultad de ingeniería civil - Análisis Matricial
q2
N1 1.87378
N2 5
N3 = 449.707
F7 -1.8738
F8 -5
F9 750.293
1.8. GRAFICA DE LA SOLUCION
2. ANALICE EL PORTICO DE LA FIGURA POR EL METODO MATRICIAL DE LOS DESPLAZAMIENTOS.
Dimensiones bxh (mm) Viga 300x500
HUYHUA MONTES HERIXS Página 6
Facultad de ingeniería civil - Análisis Matricial
Columna 300x300E=19KN/mm^2
Solución
2.1. Matriz de rigidez en coordenadas locales para la barra 1
HUYHUA MONTES HERIXS Página 7
Facultad de ingeniería civil - Análisis Matricial
E(KN/m2) A(m2) I(m^4) L(m)
19000000 0.15 0.003125 6
AE/L(KN/m) 12EI/L^3(KN/m) 6EI/L^2(KN) 4EI/L(KN.m) 2EI/L(KN.m)
475000 3298.611111 9895.833333 39583.33333 19791.66667
Nodos X Y
1 0 3 (6-0)/6 (3-3)/6
2 6 3 1 0
1 2 3 4 5 6
475000 0 0 -475000 0 0 1 0 0 0 0 0
0 3298.61 9895.83 0 -3298.6 9895.83 0 1 0 0 0 0
[K]= 0 9895.83 39583.3 0 -9895.8 19791.7 [T]= 0 0 1 0 0 0
-475000 0 0 475000 0 0 0 0 0 1 0 0
0 -3298.6 -9895.8 0 3298.61 -9895.8 0 0 0 0 1 0
0 9895.83 19791.7 0 -9895.8 39583.3 0 0 0 0 0 1
1 2 3 4 5 6
1 0 0 0 0 0 475000 0 0 -475000 0 0 1
0 1 0 0 0 0 0 3298.6111 9895.8333 0 -3298.611 9895.8333 2[T]t
= 0 0 1 0 0 0 [KCG]= 0 9895.8333 39583.333 0 -9895.833 19791.667 3
0 0 0 1 0 0 -475000 0 0 475000 0 0 4
0 0 0 0 1 0 0 -3298.611 -9895.833 0 3298.6111 -9895.833 5
0 0 0 0 0 1 0 9895.8333 19791.667 0 -9895.833 39583.333 6
2.2. Matriz de rigidez en coordenadas locales para la barra 2
E(KN/m2) A(m2) I(m^4) L(m)
19000000 0.09 0.000675 3
AE/L(KN/m) 12EI/L^3(KN/m) 6EI/L^2(KN) 4EI/L(KN.m) 2EI/L(KN.m)
570000 5700 8550 17100 8550
Nodos X Y
HUYHUA MONTES HERIXS Página 8
λ x λ y
λ x λ y
Facultad de ingeniería civil - Análisis Matricial
3 0 0 (0-0)/3 (3-0)/3
1 0 3 0 1
570000 0 0 -570000 0 0 0 1 0 0 0 0
0 5700 8550 0 -5700 8550 -1 0 0 0 0 0
[K]= 0 8550 17100 0 -8550 8550 [T]= 0 0 1 0 0 0
-570000 0 0 570000 0 0 0 0 0 0 1 0
0 -5700 -8550 0 5700 -8550 0 0 0 -1 0 0
0 8550 8550 0 -8550 17100 0 0 0 0 0 1
7 8 9 1 2 3
0 -1 0 0 0 0 5700 0 -8550 -5700 0 -8550 7
1 0 0 0 0 0 0 570000 0 0 -570000 0 8
[T]t= 0 0 1 0 0 0 [KCG]= -8550 0 17100 8550 0 8550 9
0 0 0 0 -1 0 -5700 0 8550 5700 0 8550 1
0 0 0 1 0 0 0 -570000 0 0 570000 0 2
0 0 0 0 0 1 -8550 0 8550 8550 0 17100 3
2.3. Matriz de rigidez en coordenadas locales para la barra 3
E(KN/m2) A(m2) I(m^4) L(m)
19000000 0.09 0.000675 3
AE/L(KN/m) 12EI/L^3(KN/m) 6EI/L^2(KN) 4EI/L(KN.m) 2EI/L(KN.m)
570000 5700 8550 17100 8550
Nodos X Y
2 6 3 (6-6)/3 (0-3)/3
4 6 0 0 -1
570000 0 0 -570000 0 0 0 -1 0 0 0 0
0 5700 8550 0 -5700 8550 1 0 0 0 0 0
[K]= 0 8550 17100 0 -8550 8550 [T]= 0 0 1 0 0 0
-570000 0 0 570000 0 0 0 0 0 0 -1 0
0 -5700 -8550 0 5700 -8550 0 0 0 1 0 0
HUYHUA MONTES HERIXS Página 9
λ x λ y
Facultad de ingeniería civil - Análisis Matricial
0 8550 8550 0 -8550 17100 0 0 0 0 0 1
4 5 6 10 11 12
0 1 0 0 0 0 5700 0 8550 -5700 0 8550 4
-1 0 0 0 0 0 0 570000 0 0 -570000 0 5
[T]t= 0 0 1 0 0 0 [KCG]= 8550 0 17100 -8550 0 8550 6
0 0 0 0 1 0 -5700 0 -8550 5700 0 -8550 10
0 0 0 -1 0 0 0 -570000 0 0 570000 0 11
0 0 0 0 0 1 8550 0 8550 -8550 0 17100 12
2.4. MATRIZ DE RIGIDEZ EN COORDENADAS GLOBALES
1 2 3 4 5 6 7 8 9 10 11 12
480700 0 8550 -475000 0 0 -5700 0 8550 0 0 0 1
0 573298.6 9895.833 0 -3298.61 9895.833 0 -570000 0 0 0 0 2
8550 9895.833 56683.33 0 -9895.83 19791.67 -8550 0 8550 0 0 0 3
-475000 0 0 480700 0 8550 0 0 0 -5700 0 8550 4
0 -3298.61 -9895.83 0 573298.6 -9895.83 0 0 0 0 -570000 0 5[KCG]= 0 9895.833 19791.67 8550 -9895.83 56683.33 0 0 0 -8550 0 8550 6
-5700 0 -8550 0 0 0 5700 0 -8550 0 0 0 7
0 -570000 0 0 0 0 0 570000 0 0 0 0 8
8550 0 8550 0 0 0 -8550 0 17100 0 0 0 9
0 0 0 -5700 0 -8550 0 0 0 5700 0 -8550 10
0 0 0 0 -570000 0 0 0 0 0 570000 0 11
0 0 0 8550 0 8550 0 0 0 -8550 0 17100 12
2.5. CALCULO DE LOS DESPLAZAMIENTOS DESCONOCIDOS
0 480700 0 8550 -475000 0 0 D1
-75 0 573298.6 9895.833 0 -3298.611 9895.833 D2
-75 = 8550 9895.833 56683.33 0 -9895.833 19791.67 D3
0 -475000 0 0 480700 0 8550 D4
-75 0 -3298.611 -9895.833 0 573298.6 -9895.833 D5
75 0 9895.833 19791.67 8550 -9895.833 56683.33 D6
D1 1.8225E-05 m
D2 -0.0001316 m
HUYHUA MONTES HERIXS Página 10
Facultad de ingeniería civil - Análisis Matricial
D3 = -0.0020372 rad
D4 -1.823E-05 m
D5 -0.0001316 m
D6 0.0020372 rad
2.6. CALCULO DE LAS REACCIONES DESCONOCIDAS
Q7 -5700 0 -8550 0 0 0 1.8225E-05 17.314 KN
Q8 0 -570000 0 0 0 0 -0.00013158 75.000 KN
Q9 = 8550 0 8550 0 0 0 -0.0020372 = -17.262 KN.m
Q10 0 0 0 -5700 0 -8550 -1.8225E-05 -17.314 KN
Q11 0 0 0 0 -570000 0 -0.00013158 75.000 KN
Q12 0 0 0 8550 0 8550 0.0020372 17.262 KN.m
2.7. CALCULO DE FUERZAS PARA LA BARRA 1
q1 K T D
N1 475000 0 0 -475000 0 0 1 0 0 0 0 0 1.8225E-05 17.314 KN
N2 0 3298.611 9895.833 0 -3298.611 9895.833 0 1 0 0 0 0 -0.0001316 0 KN
N3 = 0 9895.833 39583.33 0 -9895.833 19791.67 0 0 1 0 0 0 -0.0020372 = -40.32 KN.m
F4 -475000 0 0 475000 0 0 0 0 0 1 0 0 -1.823E-05 -17.314 KN
F5 0 -3298.611 -9895.833 0 3298.611 -9895.833 0 0 0 0 1 0 -0.0001316 0 KN
F6 0 9895.833 19791.67 0 -9895.833 39583.33 0 0 0 0 0 1 0.0020372 40.32 KN.m
2.8. CALCULO DE FUERZAS PARA LA BARRA 2
q2 K T D
N7 570000 0 0 -570000 0 0 0 1 0 0 0 0 0 75 KN
N8 0 5700 8550 0 -5700 8550 -1 0 0 0 0 0 0 -17.314 KN
N9 = 0 8550 17100 0 -8550 8550 0 0 1 0 0 0 0 = -17.262 KN.m
F1 -570000 0 0 570000 0 0 0 0 0 0 1 0 1.8225E-05 -75 KN
F2 0 -5700 -8550 0 5700 -8550 0 0 0 -1 0 0 -0.0001316 17.314 KN
F3 0 8550 8550 0 -8550 17100 0 0 0 0 0 1 -0.0020372 -34.68 KN.m
2.9. CALCULO DE FUERZAS PARA LA BARRA 3
q3 K T D
N4 570000 0 0 -570000 0 0 0 -1 0 0 0 0 -1.823E-05 75 KN
N5 0 5700 8550 0 -5700 8550 1 0 0 0 0 0 -0.0001316 17.314 KN
N6 = 0 8550 17100 0 -8550 8550 0 0 1 0 0 0 0.0020372 = 34.68 KN.m
F10 -570000 0 0 570000 0 0 0 0 0 0 -1 0 0 -75 KN
HUYHUA MONTES HERIXS Página 11
Facultad de ingeniería civil - Análisis Matricial
F11 0 -5700 -8550 0 5700 -8550 0 0 0 1 0 0 0 -17.314 KN
F12 0 8550 8550 0 -8550 17100 0 0 0 0 0 1 0 17.262 KN.m
2.10. GRAFICA DE LA SOLUCION
HUYHUA MONTES HERIXS Página 12
Facultad de ingeniería civil - Análisis Matricial
3. RESOLVER EL SIGUIENTE PORTICO
HUYHUA MONTES HERIXS Página 14
Facultad de ingeniería civil - Análisis Matricial
3.1. Matriz de rigidez en coordenadas locales para la barra 1
1 2 3 4 5 6
0.16667 0 0 -0.1667 0 0 1 0 0 0 0 0
0 0.11111 0.33333 0 -0.1111 0.33333 0 1 0 0 0 0
[K]= 0 0.33333 1.33333 0 -0.3333 0.66667 [T]= 0 0 1 0 0 0
-0.1667 0 0 0.16667 0 0 0 0 0 1 0 0
0 -0.1111 -0.3333 0 0.11111 -0.3333 0 0 0 0 1 0
0 0.33333 0.66667 0 -0.3333 1.33333 0 0 0 0 0 1
HUYHUA MONTES HERIXS Página 17
Facultad de ingeniería civil - Análisis Matricial
1 2 3 4 5 6
1 0 0 0 0 0 0.166667 0 0 -0.166667 0 0 1
0 1 0 0 0 0 0 0.111111 0.333333 0 -0.111111 0.333333 2[T]t
= 0 0 1 0 0 0
[KCG]= 0 0.333333 1.333333 0 -0.333333 0.666667 3
0 0 0 1 0 0 -0.166667 0 0 0.166667 0 0 4
0 0 0 0 1 0 0 -0.111111 -0.333333 0 0.111111 -0.333333 5
0 0 0 0 0 1 0 0.333333 0.666667 0 -0.333333 1.333333 6
3.2. Matriz de rigidez en coordenadas locales para la barra 2
7 8 9 1 2 3
0.33333 0 0 -0.3333 0 0 0 1 0 0 0 0
0 0.44444 0.66667 0 -0.4444 0.66667 -1 0 0 0 0 0
[K]= 0 0.66667 1.33333 0 -0.6667 0.66667 [T]= 0 0 1 0 0 0
-0.3333 0 0 0.33333 0 0 0 0 0 0 1 0
0 -0.4444 -0.6667 0 0.44444 -0.6667 0 0 0 -1 0 0
0 0.66667 0.66667 0 -0.6667 1.33333 0 0 0 0 0 1
7 8 9 1 2 3
0 -1 0 0 0 0 0.444444 0 -0.666667 -0.444444 0 -0.666667 7
1 0 0 0 0 0 0 0.333333 0 0 -0.333333 0 8
[T]t= 0 0 1 0 0 0
[KCG]= -0.666667 0 1.333333 0.666667 0 0.666667 9
0 0 0 0 -1 0 -0.444444 0 0.666667 0.444444 0 0.666667 1
0 0 0 1 0 0 0 -0.333333 0 0 0.333333 0 2
0 0 0 0 0 1 -0.666667 0 0.666667 0.666667 0 1.333333 3
3.3. Matriz de rigidez en coordenadas locales para la barra 3
10 11 12 4 5 6
0.33333 0 0 -0.3333 0 0 0 1 0 0 0 0
0 0.44444 0.66667 0 -0.4444 0.66667 -1 0 0 0 0 0
[K]= 0 0.66667 1.33333 0 -0.6667 0.66667 [T]= 0 0 1 0 0 0
-0.3333 0 0 0.33333 0 0 0 0 0 0 1 0
0 -0.4444 -0.6667 0 0.44444 -0.6667 0 0 0 -1 0 0
0 0.66667 0.66667 0 -0.6667 1.33333 0 0 0 0 0 1
HUYHUA MONTES HERIXS Página 18
Facultad de ingeniería civil - Análisis Matricial
10 11 12 4 5 6
0 -1 0 0 0 0 0.444444 0 -0.666667 -0.444444 0 -0.666667 10
1 0 0 0 0 0 0 0.333333 0 0 -0.333333 0 11
[T]t= 0 0 1 0 0 0
[KCG]= -0.666667 0 1.333333 0.666667 0 0.666667 12
0 0 0 0 -1 0 -0.444444 0 0.666667 0.444444 0 0.666667 4
0 0 0 1 0 0 0 -0.333333 0 0 0.333333 0 5
0 0 0 0 0 1 -0.666667 0 0.666667 0.666667 0 1.333333 6
3.4. Matriz de rigidez en coordenadas locales para la barra 4
7 8 9 10 11 12
0.16667 0 0 -0.1667 0 0 1 0 0 0 0 0
0 0.11111 0.33333 0 -0.1111 0.33333 0 1 0 0 0 0
[K]= 0 0.33333 1.33333 0 -0.3333 0.66667 [T]= 0 0 1 0 0 0
-0.1667 0 0 0.16667 0 0 0 0 0 1 0 0
0 -0.1111 -0.3333 0 0.11111 -0.3333 0 0 0 0 1 0
0 0.33333 0.66667 0 -0.3333 1.33333 0 0 0 0 0 1
7 8 9 10 11 12
1 0 0 0 0 0 0.166667 0 0 -0.166667 0 0 7
0 1 0 0 0 0 0 0.111111 0.333333 0 -0.111111 0.333333 8
[T]t= 0 0 1 0 0 0
[KCG]= 0 0.333333 1.333333 0 -0.333333 0.666667 9
0 0 0 1 0 0 -0.166667 0 0 0.166667 0 0 10
0 0 0 0 1 0 0 -0.111111 -0.333333 0 0.111111 -0.333333 11
0 0 0 0 0 1 0 0.333333 0.666667 0 -0.333333 1.333333 12
3.5. Matriz de rigidez en coordenadas locales para la barra 5
10 11 12 13 14 15
0.25 0 0 -0.25 0 0 1 0 0 0 0 0
0 0.1875 0.375 0 -0.1875 0.375 0 1 0 0 0 0
[K]= 0 0.375 1 0 -0.375 0.5 [T]= 0 0 1 0 0 0
-0.25 0 0 0.25 0 0 0 0 0 1 0 0
0 -0.1875 -0.375 0 0.1875 -0.375 0 0 0 0 1 0
HUYHUA MONTES HERIXS Página 19
Facultad de ingeniería civil - Análisis Matricial
0 0.375 0.5 0 -0.375 1 0 0 0 0 0 1
10 11 12 13 14 15
1 0 0 0 0 0 0.25 0 0 -0.25 0 0 10
0 1 0 0 0 0 0 0.1875 0.375 0 -0.1875 0.375 11
[T]t= 0 0 1 0 0 0
[KCG]= 0 0.375 1 0 -0.375 0.5 12
0 0 0 1 0 0 -0.25 0 0 0.25 0 0 13
0 0 0 0 1 0 0 -0.1875 -0.375 0 0.1875 -0.375 14
0 0 0 0 0 1 0 0.375 0.5 0 -0.375 1 15
3.6. Matriz de rigidez en coordenadas locales para la barra 6
16 17 18 7 8 9
0.25 0 0 -0.25 0 0 0 1 0 0 0 0
0 0.1875 0.375 0 -0.1875 0.375 -1 0 0 0 0 0
[K]= 0 0.375 1 0 -0.375 0.5 [T]= 0 0 1 0 0 0
-0.25 0 0 0.25 0 0 0 0 0 0 1 0
0 -0.1875 -0.375 0 0.1875 -0.375 0 0 0 -1 0 0
0 0.375 0.5 0 -0.375 1 0 0 0 0 0 1
16 17 18 7 8 9
0 -1 0 0 0 0 0.1875 0 -0.375 -0.1875 0 -0.375 16
1 0 0 0 0 0 0 0.25 0 0 -0.25 0 17
[T]t= 0 0 1 0 0 0
[KCG]= -0.375 0 1 0.375 0 0.5 18
0 0 0 0 -1 0 -0.1875 0 0.375 0.1875 0 0.375 7
0 0 0 1 0 0 0 -0.25 0 0 0.25 0 8
0 0 0 0 0 1 -0.375 0 0.5 0.375 0 1 9
3.7. Matriz de rigidez en coordenadas locales para la barra 7
19 20 21 10 11 12
0.5 0 0 -0.5 0 0 0 1 0 0 0 0
0 1.5 1.5 0 -1.5 1.5 -1 0 0 0 0 0
[K]= 0 1.5 2 0 -1.5 1 [T]= 0 0 1 0 0 0
-0.5 0 0 0.5 0 0 0 0 0 0 1 0
0 -1.5 -1.5 0 1.5 -1.5 0 0 0 -1 0 0
0 1.5 1 0 -1.5 2 0 0 0 0 0 1
HUYHUA MONTES HERIXS Página 20
Facultad de ingeniería civil - Análisis Matricial
19 20 21 10 11 12
0 -1 0 0 0 0 1.5 0 -1.5 -1.5 0 -1.5 19
1 0 0 0 0 0 0 0.5 0 0 -0.5 0 20
[T]t= 0 0 1 0 0 0
[KCG]= -1.5 0 2 1.5 0 1 21
0 0 0 0 -1 0 -1.5 0 1.5 1.5 0 1.5 10
0 0 0 1 0 0 0 -0.5 0 0 0.5 0 11
0 0 0 0 0 1 -1.5 0 1 1.5 0 2 12
3.8. Matriz de rigidez en coordenadas locales para la barra 8
22 23 24 13 14 15
0.33333 0 0 -0.3333 0 0 0 1 0 0 0 0
0 0.44444 0.66667 0 -0.4444 0.66667 -1 0 0 0 0 0
[K]= 0 0.66667 1.33333 0 -0.6667 0.66667 [T]= 0 0 1 0 0 0
-0.3333 0 0 0.33333 0 0 0 0 0 0 1 0
0 -0.4444 -0.6667 0 0.44444 -0.6667 0 0 0 -1 0 0
0 0.66667 0.66667 0 -0.6667 1.33333 0 0 0 0 0 1
22 23 24 13 14 15
0 -1 0 0 0 0 0.444444 0 -0.666667 -0.444444 0 -0.666667 22
1 0 0 0 0 0 0 0.333333 0 0 -0.333333 0 23
[T]t= 0 0 1 0 0 0
[KCG]= -0.666667 0 1.333333 0.666667 0 0.666667 24
0 0 0 0 -1 0 -0.444444 0 0.666667 0.444444 0 0.666667 13
0 0 0 1 0 0 0 -0.333333 0 0 0.333333 0 14
0 0 0 0 0 1 -0.666667 0 0.666667 0.666667 0 1.333333 15
3.9. MATRIZ DE RIGIDEZ EN COORDENADAS GLOBALES
HUYHUA MONTES HERIXS Página 21
Facultad de ingeniería civil - Análisis Matricial
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0.61 0 0.67 -0.2 0 0 -0.4 0 0.67 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
0 0.44 0.33 0 -0.1 0.33 0 -0.3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2
0.67 0.33 2.67 0 -0.3 0.67 -0.7 0 0.67 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3
-0.2 0 0 0.61 0 0.67 0 0 0 -0.4 0 0.67 0 0 0 0 0 0 0 0 0 0 0 0 4
0 -0.1 -0.3 0 0.44 -0.3 0 0 0 0 -0.3 0 0 0 0 0 0 0 0 0 0 0 0 0 5
[KCG]= 0 0.33 0.67 0.67 -0.3 2.67 0 0 0 -0.7 0 0.67 0 0 0 0 0 0 0 0 0 0 0 0 6
-0.4 0 -0.7 0 0 0 0.8 0 -0.3 -0.2 0 0 0 0 0 -0.2 0 0.38 0 0 0 0 0 0 7
0 -0.3 0 0 0 0 0 0.69 0.33 0 -0.1 0.33 0 0 0 0 -0.3 0 0 0 0 0 0 0 8
0.67 0 0.67 0 0 0 -0.3 0.33 3.67 0 -0.3 0.67 0 0 0 -0.4 0 0.5 0 0 0 0 0 0 9
0 0 0 -0.4 0 -0.7 -0.2 0 0 2.36 0 0.83 -0.3 0 0 0 0 0 -1.5 0 1.5 0 0 0 10
0 0 0 0 -0.3 0 0 -0.1 -0.3 0 1.13 0.04 0 -0.2 0.38 0 0 0 0 -0.5 0 0 0 0 11
0 0 0 0.67 0 0.67 0 0.33 0.67 0.83 0.04 5.67 0 -0.4 0.5 0 0 0 -1.5 0 1 0 0 0 12
0 0 0 0 0 0 0 0 0 -0.3 0 0 0.69 0 0.67 0 0 0 0 0 0 -0.4 0 0.67 13
0 0 0 0 0 0 0 0 0 0 -0.2 -0.4 0 0.52 -0.4 0 0 0 0 0 0 0 -0.3 0 14
0 0 0 0 0 0 0 0 0 0 0.38 0.5 0.67 -0.4 2.33 0 0 0 0 0 0 -0.7 0 0.67 15
0 0 0 0 0 0 -0.2 0 -0.4 0 0 0 0 0 0 0.19 0 -0.4 0 0 0 0 0 0 16
0 0 0 0 0 0 0 -0.3 0 0 0 0 0 0 0 0 0.25 0 0 0 0 0 0 0 17
0 0 0 0 0 0 0.38 0 0.5 0 0 0 0 0 0 -0.4 0 1 0 0 0 0 0 0 18
0 0 0 0 0 0 0 0 0 -1.5 0 -1.5 0 0 0 0 0 0 1.5 0 -1.5 0 0 0 19
0 0 0 0 0 0 0 0 0 0 -0.5 0 0 0 0 0 0 0 0 0.5 0 0 0 0 20
0 0 0 0 0 0 0 0 0 1.5 0 1 0 0 0 0 0 0 -1.5 0 2 0 0 0 21
0 0 0 0 0 0 0 0 0 0 0 0 -0.4 0 -0.7 0 0 0 0 0 0 0.44 0 -0.7 22
0 0 0 0 0 0 0 0 0 0 0 0 0 -0.3 0 0 0 0 0 0 0 0 0.33 0 23
0 0 0 0 0 0 0 0 0 0 0 0 0.67 0 0.67 0 0 0 0 0 0 -0.7 0 1.33 24
HUYHUA MONTES HERIXS Página 23
Facultad de ingeniería civil - Análisis Matricial
3.10. CALCULO DE LOS DESPLAZAMIENTOS DESCONOCIDOS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
18 0.611 0 0.667 -0.167 0 0 -0.444 0 0.667 0 0 0 0 0 0 D1
-9 0 0.444 0.333 0 -0.111 0.333 0 -0.333 0 0 0 0 0 0 0 D2
-5 0.667 0.333 2.667 0 -0.333 0.667 -0.667 0 0.667 0 0 0 0 0 0 D3
0 -0.167 0 0 0.611 0 0.667 0 0 0 -0.444 0 0.667 0 0 0 D4
-9 0 -0.111 -0.333 0 0.444 -0.333 0 0 0 0 -0.333 0 0 0 0 D5
9 0 0.333 0.667 0.667 -0.333 2.667 0 0 0 -0.667 0 0.667 0 0 0 D6
9 = -0.444 0 -0.667 0 0 0 0.799 0 -0.292 -0.167 0 0 0 0 0 D7
-6 0 -0.333 0 0 0 0 0 0.694 0.333 0 -0.111 0.333 0 0 0 D8
-6 0.667 0 0.667 0 0 0 -0.292 0.333 3.667 0 -0.333 0.667 0 0 0 D9
0 0 0 0 -0.444 0 -0.667 -0.167 0 0 2.361 0 0.833 -0.25 0 0 D10
-18 0 0 0 0 -0.333 0 0 -0.111 -0.333 0 1.132 0.042 0 -0.188 0.375 D11
12 0 0 0 0.667 0 0.667 0 0.333 0.667 0.833 0.042 5.667 0 -0.375 0.5 D12
-12 0 0 0 0 0 0 0 0 0 -0.25 0 0 0.694 0 0.667 D13
-6 0 0 0 0 0 0 0 0 0 0 -0.188 -0.375 0 0.521 -0.375 D14
-4 0 0 0 0 0 0 0 0 0 0 0.375 0.5 0.667 -0.375 2.333 D15
D1 139.703
D2 -44.08484
D3 -18.98516
D4 63.9448
D5 -100.4017
D6 -10.5326
D7 = 68.39549
D8 -27.830
D9 -20.493
D10 12.990
D11 -62.656
D12 -4.498
D13 -23.499
D14 -29.14286
D15 11.3497
3.11. CALCULO DE LAS REACCIONES DESCONOCIDAS
Q16 0 0 0 0 0 0 -0.2 0 -0.4 0 0 0 0 0 0 139.7 -5.139
HUYHUA MONTES HERIXS Página 24
Facultad de ingeniería civil - Análisis Matricial
Q17 0 0 0 0 0 0 0 -0.3 0 0 0 0 0 0 0 -44.1 6.958Q18 0 0 0 0 0 0 0.38 0 0.5 0 0 0 0 0 0 -19 15.402Q19 0 0 0 0 0 0 0 0 0 -1.5 0 -1.5 0 0 0 63.94 -12.738Q20 = 0 0 0 0 0 0 0 0 0 0 -0.5 0 0 0 0 -100 31.328Q21 0 0 0 0 0 0 0 0 0 1.5 0 1 0 0 0 x -10.5 = 14.987Q22 0 0 0 0 0 0 0 0 0 0 0 0 -0.4 0 -0.7 68.4 2.878Q23 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.3 0 -27.8 9.714Q24 0 0 0 0 0 0 0 0 0 0 0 0 0.67 0 0.67 -20.5 -8.100
12.99
-62.7
-4.5
-23.5
-29.1
11.35
3.12. CALCULO DE LA FUERZAS INTERNAS PARA LAS BARRAS
q1 K T D
N1 0.166667 0 0 -0.166667 0 0 1 0 0 0 0 0 139.703007 12.62637
N2 0 0.111111 0.333333 0 -0.111111 0.333333 0 1 0 0 0 0 -44.084836 -3.581819
N3 = 0 0.333333 1.333333 0 -0.333333 0.666667 0 0 1 0 0 0 -18.985158 = -13.56298
F4 -0.166667 0 0 0.166667 0 0 0 0 0 1 0 0 63.9448004 -12.62637
F5 0 -0.111111 -0.333333 0 0.111111 -0.333333 0 0 0 0 1 0 -100.40174 3.581819
F6 0 0.333333 0.666667 0 -0.333333 1.333333 0 0 0 0 0 1 -10.532598 -7.927935
q2 K T D
N7 0.333333 0 0 -0.333333 0 0 0 1 0 0 0 0 68.3954872 5.418181
N8 0 0.444444 0.666667 0 -0.444444 0.666667 -1 0 0 0 0 0 -27.830291 5.373632
N9 = 0 0.666667 1.333333 0 -0.666667 0.666667 0 0 1 0 0 0 -20.49274 = 7.557921
F1 -0.333333 0 0 0.333333 0 0 0 0 0 0 1 0 139.703007 -5.418181
F2 0 -0.444444 -0.666667 0 0.444444 -0.666667 0 0 0 -1 0 0 -44.084836 -5.373632
F3 0 0.666667 0.666667 0 -0.666667 1.333333 0 0 0 0 0 1 -18.985158 8.562976
q3 K T D
N10 0.333333 0 0 -0.333333 0 0 0 1 0 0 0 0 12.9899522 12.58182
N11 0 0.444444 0.666667 0 -0.444444 0.666667 -1 0 0 0 0 0 -62.656281 12.62637
N12 = 0 0.666667 1.333333 0 -0.666667 0.666667 0 0 1 0 0 0 -4.4977495 = 20.95117
F4 -0.333333 0 0 0.333333 0 0 0 0 0 0 1 0 63.9448004 -12.58182
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Facultad de ingeniería civil - Análisis Matricial
F5 0 -0.444444 -0.666667 0 0.444444 -0.666667 0 0 0 -1 0 0 -100.40174 -12.62637
F6 0 0.666667 0.666667 0 -0.666667 1.333333 0 0 0 0 0 1 -10.532598 16.92794
q4 K T D
N7 0.166667 0 0 -0.166667 0 0 1 0 0 0 0 0 68.3954872 9.234256
N8 0 0.111111 0.333333 0 -0.111111 0.333333 0 1 0 0 0 0 -27.830291 -4.460609
N9 = 0 0.333333 1.333333 0 -0.333333 0.666667 0 0 1 0 0 0 -20.49274 = -18.71349
F10 -0.166667 0 0 0.166667 0 0 0 0 0 1 0 0 12.9899522 -9.234256
F11 0 -0.111111 -0.333333 0 0.111111 -0.333333 0 0 0 0 1 0 -62.656281 4.460609
F12 0 0.333333 0.666667 0 -0.333333 1.333333 0 0 0 0 0 1 -4.4977495 -8.050162
q5 K T D
N10 0.25 0 0 -0.25 0 0 1 0 0 0 0 0 12.9899522 9.12232
N11 0 0.1875 0.375 0 -0.1875 0.375 0 1 0 0 0 0 -62.656281 -3.714286
N12 = 0 0.375 1 0 -0.375 0.5 0 0 1 0 0 0 -4.4977495 = -11.39043
F13 -0.25 0 0 0.25 0 0 0 0 0 1 0 0 -23.499326 -9.12232
F14 0 -0.1875 -0.375 0 0.1875 -0.375 0 0 0 0 1 0 -29.142859 3.714286
F15 0 0.375 0.5 0 -0.375 1 0 0 0 0 0 1 11.3496965 -3.466711
q6 K T D
N16 0.25 0 0 -0.25 0 0 0 1 0 0 0 0 0 6.957573
N17 0 0.1875 0.375 0 -0.1875 0.375 -1 0 0 0 0 0 0 5.139376
N18 = 0 0.375 1 0 -0.375 0.5 0 0 1 0 0 0 0 = 15.40194
F7 -0.25 0 0 0.25 0 0 0 0 0 0 1 0 68.3954872 -6.957573
F8 0 -0.1875 -0.375 0 0.1875 -0.375 0 0 0 -1 0 0 -27.830291 -5.139376
F9 0 0.375 0.5 0 -0.375 1 0 0 0 0 0 1 -20.49274 5.155568
q7 K T D
N19 0.5 0 0 -0.5 0 0 0 1 0 0 0 0 0 31.32814
N20 0 1.5 1.5 0 -1.5 1.5 -1 0 0 0 0 0 0 12.7383
N21 = 0 1.5 2 0 -1.5 1 0 0 1 0 0 0 0 = 14.98718
F10 -0.5 0 0 0.5 0 0 0 0 0 0 1 0 12.9899522 -31.32814
F11 0 -1.5 -1.5 0 1.5 -1.5 0 0 0 -1 0 0 -62.656281 -12.7383
F12 0 1.5 1 0 -1.5 2 0 0 0 0 0 1 -4.4977495 10.48943
q8 K T D
N22 0.333333 0 0 -0.333333 0 0 0 1 0 0 0 0 0 9.714286
N23 0 0.444444 0.666667 0 -0.444444 0.666667 -1 0 0 0 0 0 0 -2.87768
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Facultad de ingeniería civil - Análisis Matricial
N24 = 0 0.666667 1.333333 0 -0.666667 0.666667 0 0 1 0 0 0 0 = -8.099753
F13 -0.333333 0 0 0.333333 0 0 0 0 0 0 1 0 -23.499326 -9.714286
F14 0 -0.444444 -0.666667 0 0.444444 -0.666667 0 0 0 -1 0 0 -29.142859 2.87768
F15 0 0.666667 0.666667 0 -0.666667 1.333333 0 0 0 0 0 1 11.3496965 -0.533289
3.13. GRAFICA DE LA SOLUCION
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