Este documento presenta la resolución de un ejercicio utilizando el método matricial para analizar una estructura compuesta por vigas y columnas sometida a cargas. Se describe el proceso de 1) numerar los nodos y barras, 2) obtener las fuerzas actuantes en cada nodo, 3) calcular las matrices de rigidez locales de cada barra, y 4) transformar estas matrices a un sistema global para obtener la matriz de rigidez total de la estructura.

1. EJERCICIO 2: RESUELTO MÉTODO MATRICIAL REGIDECES EN SISTEMA GLOBAL
Tutorial del ejercicio resuelto - Canal Youtube: Jhon Muchica Sillo
Analizar completamente la estructura mostrada considerando que: reacciones
axiales
b t fc= 210 kg/cm2 esfuerzo cortante
0.3 0.4 momento flector
0.4 0.4 ELACTICIDAD: kg/cm2
0.3 0.3
6000 kg 4000 kg/m
m
500 kg
3000 kg 3000 kg/m 2000 kg/m
m
4 m 3 m
W
L
P
W
L
L/2
VIGA
COL. 1
COL. 2
COL
2:
0.30
x
0.30
VIGA: 0.30 x 0.40
VIGA: 0.30 x 0.40 VIGA: 0.30 x 0.40
3.5
3.5
COL
2:
0.30
x
0.30
COL
2:
0.30
x
0.30
COL
1:
0.40
x
0.40
217370.6512
COL
1:
0.40
x
0.40
A B
A B

2. EJERCICIO 2: RESUELTO MÉTODO MATRICIAL REGIDECES EN SISTEMA GLOBAL
Tutorial del ejercicio resuelto - Canal Youtube: Jhon Muchica Sillo
1. Numeración de Nodos y barras. La numeración de nodos y barras es como sigue a continuación.
2. Fuerzas y momentos transmitidos a los nodos (sistema local).
-8000 -8000
6000 kg -5333 4000 kg/m 5333
m
500 kg
-6250 -6250 -3000 -3000
3000 kg -4250 3000 kg/m 4250 -1500 2000 kg/m 1500
m
4 m 3 m
2.1. Matriz de fuerzas actuantes (sistema local): en nodos extremos de cada barra es la siguiente
barras 6, 7 y 8
f4x = 3000 kg
f4y = -6250 kg
M4 = -425000 kg-cm
f5x = 0 kg
f5y = -9250 kg
M5 = 275000 kg-cm
f6x = = 0 kg
f6y = -3000 kg
M6 = 150000 kg-cm
f7x = 6000 kg
f7y = -8000 kg
M7 = -533333 kg-cm
f8x = 0 kg
f8y = -8000 kg
M8 = 533333 kg-cm
3.5
3.5
1
1
2 3
4 5 6
7 8
2 3
4 5
6 7
8

3. EJERCICIO 2: RESUELTO MÉTODO MATRICIAL REGIDECES EN SISTEMA GLOBAL
Tutorial del ejercicio resuelto - Canal Youtube: Jhon Muchica Sillo
3. Matriz de Transformada de cada barra. Se considera que la dirección global x de las está dirigido desde el nodo numeral
menor hasta el nodo numeral mayor. Luego aplicando la fórmula de la matriz de transformación.
Formula de la Matriz de Transformación
cos Ө -senӨ 0 0 0 0 cos Ө senӨ 0 0 0 0
senӨ cosӨ 0 0 0 0 -senӨ cos Ө 0 0 0 0
0 0 1 0 0 0 0 0 1 0 0 0
0 0 0 cosӨ -senӨ 0 0 0 0 cos Ө senӨ 0
0 0 0 senӨ cosӨ 0 0 0 0 -senӨ cos Ө 0
0 0 0 0 0 1 0 0 0 0 0 1
Para columnas: la matriz de Transformación y su respectiva transpuesta tiene el valor de (Ө = 90 °)
6E-17 -1 0 0 0 0 6E-17 1 0 0 0 0
1 6E-17 0 0 0 0 -1 6E-17 0 0 0 0
0 0 1 0 0 0 0 0 1 0 0 0
0 0 0 6E-17 -1 0 0 0 0 6E-17 1 0
0 0 0 1 6E-17 0 0 0 0 -1 6E-17 0
0 0 0 0 0 1 0 0 0 0 0 1
… Para VIGAS: la matriz de Transformación y su respectiva transpuesta tiene el valor de (Ө = 0 °)
1 0 0 0 0 0 1 0 0 0 0 0
0 1 0 0 0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 0 1 0 0 0
0 0 0 1 0 0 0 0 0 1 0 0
0 0 0 0 1 0 0 0 0 0 1 0
0 0 0 0 0 1 0 0 0 0 0 1
4. Matriz de Rigidez de las barras en (sistema local). LA MATRIZ DE RIGIDEZ de cada barra se obtiene aplicando la
siguiente expresión:
0 0
0
0 0
0
0 0
0
0
0
0
0
0
0
0
T
T
(K)
T

4. EJERCICIO 2: RESUELTO MÉTODO MATRICIAL REGIDECES EN SISTEMA GLOBAL
Tutorial del ejercicio resuelto - Canal Youtube: Jhon Muchica Sillo
4.1. Para facilitar la construcción de la MATRIZ RIGIDEZ LOCAL de cada barra, se ará uso de la siguiente tabla
inicio 1 final 4
BARRA base cm) alto (cm) L(cm) A I E
1 30 30 350 900 67500 217370.65
AE/L 4EI/L 2EI/L
558953.103 4106.59423 718653.9897 167685931 83842965.46
558953.10 0 0 -558953.10 0 0
0 4106.59 718653.99 0 -4106.59 718653.99
0 718653.99 167685931 0 -718653.99 83842965.5
-558953.103 0 0 558953.10 0 0
0 -4106.594227 -718653.99 0 4106.59 -718653.99
0 718653.990 83842965.5 0 -718653.99 167685931
… inicio 2 final 5
BARRA base cm) alto (cm) L(cm) A I E
2 40 40 350 1600 213333.3333 217370.65
AE/L 4EI/L 2EI/L
993694.405 12978.8657 2271301.498 529970350 264985174.8
993694.41 0 0 -993694.41 0 0
0 12978.87 2271301.5 0 -12978.87 2271301.5
0 2271301.50 529970350 0 -2271301.50 264985175
-993694.405 0 0 993694.41 0 0
0 -12978.8657 -2271301.5 0 12978.87 -2271301.5
0 2271301.498 264985175 0 -2271301.50 529970350
inicio 3 final 6
BARRA base cm) alto (cm) L(cm) A I E
3 30 30 350 900 67500 217370.65
AE/L 4EI/L 2EI/L
558953.103 4106.59423 718653.9897 167685931 83842965.46
558953.10 0 0 -558953.10 0 0
0 4106.59 718653.99 0 -4106.59 718653.99
0 718653.99 167685931 0 -718653.99 83842965.5
-558953.103 0 0 558953.10 0 0
0 -4106.594227 -718653.99 0 4106.59 -718653.99
0 718653.990 83842965.5 0 -718653.99 167685931
inicio 4 final 7
BARRA base cm) alto (cm) L(cm) A I E
4 30 30 350 900 67500 217370.65
AE/L 4EI/L 2EI/L
558953.103 4106.59423 718653.9897 167685931 83842965.46
558953.10 0 0 -558953.10 0 0
0 4106.59 718653.99 0 -4106.59 718653.99
0 718653.99 167685931 0 -718653.99 83842965.5
-558953.103 0 0 558953.10 0 0
0 -4106.594227 -718653.99 0 4106.59 -718653.99
0 718653.990 83842965.5 0 -718653.99 167685931

5. EJERCICIO 2: RESUELTO MÉTODO MATRICIAL REGIDECES EN SISTEMA GLOBAL
Tutorial del ejercicio resuelto - Canal Youtube: Jhon Muchica Sillo
inicio 5 final 8
BARRA base cm) alto (cm) L(cm) A I E
5 40 40 350 1600 213333.3333 217370.65
AE/L 4EI/L 2EI/L
993694.405 12978.8657 2271301.498 529970350 264985174.8
993694.41 0 0 -993694.41 0 0
0 12978.87 2271301.5 0 -12978.87 2271301.5
0 2271301.50 529970350 0 -2271301.50 264985175
-993694.405 0 0 993694.41 0 0
0 -12978.8657 -2271301.5 0 12978.87 -2271301.5
0 2271301.498 264985175 0 -2271301.50 529970350
inicio 4 final 5
BARRA base cm) alto (cm) L(cm) A I E
6 30 40 400 1200 160000 217370.65
AE/L 4EI/L 2EI/L
652111.954 6521.11954 1304223.907 347793042 173896521
652111.95 0 0 -652111.95 0 0
0 6521.12 1304223.91 0 -6521.12 1304223.91
0 1304223.91 347793042 0 -1304223.91 173896521
-652111.954 0 0 652111.95 0 0
0 -6521.119536 -1304223.91 0 6521.12 -1304223.91
0 1304223.907 173896521 0 -1304223.91 347793042
inicio 5 final 6
BARRA base cm) alto (cm) L(cm) A I E
7 30 40 300 1200 160000 217370.65
AE/L 4EI/L 2EI/L
869482.605 15457.4685 2318620.279 463724056 231862027.9
869482.60 0 0 -869482.60 0 0
0 15457.47 2318620.28 0 -15457.47 2318620.28
0 2318620.28 463724056 0 -2318620.28 231862028
-869482.605 0 0 869482.60 0 0
0 -15457.46853 -2318620.28 0 15457.47 -2318620.28
0 2318620.279 231862028 0 -2318620.28 463724056
inicio 7 final 8
BARRA base cm) alto (cm) L(cm) A I E
8 30 40 400 1200 160000 217370.65
AE/L 4EI/L 2EI/L
652111.954 6521.11954 1304223.907 347793042 173896521
652111.95 0 0 -652111.95 0 0
0 6521.12 1304223.91 0 -6521.12 1304223.91
0 1304223.91 347793042 0 -1304223.91 173896521
-652111.954 0 0 652111.95 0 0
0 -6521.119536 -1304223.91 0 6521.12 -1304223.91
0 1304223.907 173896521 0 -1304223.91 347793042

6. EJERCICIO 2: RESUELTO MÉTODO MATRICIAL REGIDECES EN SISTEMA GLOBAL
Tutorial del ejercicio resuelto - Canal Youtube: Jhon Muchica Sillo
5 Y 6Matriz de Rigidez en Sistema Global. Para obtener la MATRIZ DE RIGIDEZ en el sistema global, se usa la expresión
que transforma dicha matriz del sistema LOCAL al GLOBAL. De esta manera se obtienen las matrices de Rigidez
1X 1Y R1 4X 4Y R4
… 3E-11 -4107 -7E+05 -3E-11 4106.6 -7E+05 4106.6 0.0 -718654.0 -4106.6 0.0 -718654.0 1X
558953 3E-13 4E-11 -6E+05 -3E-13 4E-11 0.0 558953.1 0.0 0.0 -558953.1 0.0 1Y
0 718654 2E+08 0 -7E+05 8E+07 -718654.0 0.0 167685930.9 718654.0 0.0 83842965.5 R1
-3E-11 4106.6 718654 3E-11 -4107 718654 -4106.6 0.0 718654.0 4106.6 0.0 718654.0 4X
-6E+05 -3E-13 -4E-11 558953 3E-13 -4E-11 0.0 -558953.1 0.0 0.0 558953.1 0.0 4Y
0 718654 8E+07 0 -7E+05 2E+08 -718654.0 0.0 83842965.5 718654.0 0.0 167685930.9 R4
2X 2Y R2 5X 5Y R5
…. 6E-11 -12979 -2E+06 -6E-11 12979 -2E+06 12978.9 0.0 -2271301.5 -12978.9 0.0 -2271301.5 2X
993694 8E-13 1E-10 -1E+06 -8E-13 1E-10 0.0 993694.4 0.0 0.0 -993694.4 0.0 2Y
0 2E+06 5E+08 0 -2E+06 3E+08 -2271301.5 0.0 529970349.6 2271301.5 0.0 264985174.8 R2
-6E-11 12979 2E+06 6E-11 -12979 2E+06 -12978.9 0.0 2271301.5 12978.9 0.0 2271301.5 5X
-1E+06 -8E-13 -1E-10 993694 8E-13 -1E-10 0.0 -993694.4 0.0 0.0 993694.4 0.0 5Y
0 2E+06 3E+08 0 -2E+06 5E+08 -2271301.5 0.0 264985174.8 2271301.5 0.0 529970349.6 R5
3X 3Y R3 6X 6Y R6
3E-11 -4107 -7E+05 -3E-11 4106.6 -7E+05 4106.6 0.0 -718654.0 -4106.6 0.0 -718654.0 3X
558953 3E-13 4E-11 -6E+05 -3E-13 4E-11 0.0 558953.1 0.0 0.0 -558953.1 0.0 3Y
0 718654 2E+08 0 -7E+05 8E+07 -718654.0 0.0 167685930.9 718654.0 0.0 83842965.5 R3
-3E-11 4106.6 718654 3E-11 -4107 718654 -4106.6 0.0 718654.0 4106.6 0.0 718654.0 6X
-6E+05 -3E-13 -4E-11 558953 3E-13 -4E-11 0.0 -558953.1 0.0 0.0 558953.1 0.0 6Y
0 718654 8E+07 0 -7E+05 2E+08 -718654.0 0.0 83842965.5 718654.0 0.0 167685930.9 R6
4X 4Y R4 7X 7Y R7
3E-11 -4107 -7E+05 -3E-11 4106.6 -7E+05 4106.6 0.0 -718654.0 -4106.6 0.0 -718654.0 4X
558953 3E-13 4E-11 -6E+05 -3E-13 4E-11 0.0 558953.1 0.0 0.0 -558953.1 0.0 4Y
0 718654 2E+08 0 -7E+05 8E+07 -718654.0 0.0 167685930.9 718654.0 0.0 83842965.5 R4
-3E-11 4106.6 718654 3E-11 -4107 718654 -4106.6 0.0 718654.0 4106.6 0.0 718654.0 7X
-6E+05 -3E-13 -4E-11 558953 3E-13 -4E-11 0.0 -558953.1 0.0 0.0 558953.1 0.0 7Y
0 718654 8E+07 0 -7E+05 2E+08 -718654.0 0.0 83842965.5 718654.0 0.0 167685930.9 R7
5X 5Y R5 8X 8Y R8
6E-11 -12979 -2E+06 -6E-11 12979 -2E+06 12978.9 0.0 -2271301.5 -12978.9 0.0 -2271301.5 5X
993694 8E-13 1E-10 -1E+06 -8E-13 1E-10 0.0 993694.4 0.0 0.0 -993694.4 0.0 5Y
0 2E+06 5E+08 0 -2E+06 3E+08 -2271301.5 0.0 529970349.6 2271301.5 0.0 264985174.8 R5
-6E-11 12979 2E+06 6E-11 -12979 2E+06 -12978.9 0.0 2271301.5 12978.9 0.0 2271301.5 8X
-1E+06 -8E-13 -1E-10 993694 8E-13 -1E-10 0.0 -993694.4 0.0 0.0 993694.4 0.0 8Y
0 2E+06 3E+08 0 -2E+06 5E+08 -2271301.5 0.0 264985174.8 2271301.5 0.0 529970349.6 R8
4X 4Y R4 5X 5Y R5
… 652112 0 0 -7E+05 0 0 652112.0 0.0 0.0 -652112.0 0.0 0.0 4X
0 6521.1 1E+06 0 -6521 1E+06 0.0 6521.1 1304223.9 0.0 -6521.1 1304223.9 4Y
0 1E+06 3E+08 0 -1E+06 2E+08 0.0 1304223.9 347793041.9 0.0 -1304223.9 173896521.0 R4
-7E+05 0 0 652112 0 0 -652112.0 0.0 0.0 652112.0 0.0 0.0 5X
0 -6521 -1E+06 0 6521.1 -1E+06 0.0 -6521.1 -1304223.9 0.0 6521.1 -1304223.9 5Y
0 1E+06 2E+08 0 -1E+06 3E+08 0.0 1304223.9 173896521.0 0.0 -1304223.9 347793041.9 R5
5X 5Y R5 6X 6Y R6
869483 0 0 -9E+05 0 0 869482.6 0.0 0.0 -869482.6 0.0 0.0 5X
0 15457 2E+06 0 -15457 2E+06 0.0 15457.5 2318620.3 0.0 -15457.5 2318620.3 5Y
0 2E+06 5E+08 0 -2E+06 2E+08 0.0 2318620.3 463724055.9 0.0 -2318620.3 231862027.9 R5
-9E+05 0 0 869483 0 0 -869482.6 0.0 0.0 869482.6 0.0 0.0 6X
0 -15457 -2E+06 0 15457 -2E+06 0.0 -15457.5 -2318620.3 0.0 15457.5 -2318620.3 6Y
0 2E+06 2E+08 0 -2E+06 5E+08 0.0 2318620.3 231862027.9 0.0 -2318620.3 463724055.9 R6
7X 7Y R7 8X 8Y R8
652112 0 0 -7E+05 0 0 652112.0 0.0 0.0 -652112.0 0.0 0.0 7X
0 6521.1 1E+06 0 -6521 1E+06 0.0 6521.1 1304223.9 0.0 -6521.1 1304223.9 7Y
0 1E+06 3E+08 0 -1E+06 2E+08 0.0 1304223.9 347793041.9 0.0 -1304223.9 173896521.0 R7
-7E+05 0 0 652112 0 0 -652112.0 0.0 0.0 652112.0 0.0 0.0 8X
0 -6521 -1E+06 0 6521.1 -1E+06 0.0 -6521.1 -1304223.9 0.0 6521.1 -1304223.9 8Y
0 1E+06 2E+08 0 -1E+06 3E+08 0.0 1304223.9 173896521.0 0.0 -1304223.9 347793041.9 R8

7. EJERCICIO 2: RESUELTO MÉTODO MATRICIAL REGIDECES EN SISTEMA GLOBAL
Tutorial del ejercicio resuelto - Canal Youtube: Jhon Muchica Sillo
7.
Matriz
ensanblada
de
Rigideces
(sistema
glocal).
Al
aplicar
procedimiento
de
emsamblaje
de
la
Matriz
de
Rigidez,
se
tiene
la
Matriz
de
Rifgidez
de
toda
la
estructura.
1X
1Y
R1
2X
2Y
R2
3X
3Y
R3
4X
4Y
R4
5X
5Y
R5
6X
6Y
R6
7X
7Y
R7
8X
8Y
R8
4106.6
0.0
-718654.0
0
0
0
0
0
0
-4106.6
0.0
-718654.0
0
0
0
0
0
0
0
0
0
0
0
0
1X
0.0
558953.1
0.0
0
0
0
0
0
0
0.0
-558953.1
0.0
0
0
0
0
0
0
0
0
0
0
0
0
1Y
-718654.0
0.0
167685930.9
0
0
0
0
0
0
718654.0
0.0
83842965.5
0
0
0
0
0
0
0
0
0
0
0
0
R1
0
0
0
12978.9
0.0
-2271301.5
0
0
0
0
0
0
-12978.9
0.0
-2271301.5
0
0
0
0
0
0
0
0
0
2X
0
0
0
0.0
993694.4
0.0
0
0
0
0
0
0
0.0
-993694.4
0.0
0
0
0
0
0
0
0
0
0
2Y
0
0
0
-2271301.5
0.0
529970349.6
0
0
0
0
0
0
2271301.5
0.0
264985174.8
0
0
0
0
0
0
0
0
0
R2
0
0
0
0
0
0
4106.6
0.0
-718654.0
0
0
0
0
0
0
-4106.6
0.0
-718654.0
0
0
0
0
0
0
3X
0
0
0
0
0
0
0.0
558953.1
0.0
0
0
0
0
0
0
0.0
-558953.1
0.0
0
0
0
0
0
0
3Y
0
0
0
0
0
0
-718654.0
0.0
167685930.9
0
0
0
0
0
0
718654.0
0.0
83842965.5
0
0
0
0
0
0
R3
-4106.6
0.0
718654.0
0
0
0
0
0
0
660325.1
0.0
0.0
-652112.0
0.0
0.0
0
0
0
-4106.6
0.0
-718654.0
0
0
0
4X
0.0
-558953.1
0.0
0
0
0
0
0
0
0.0
1124427.3
1304223.9
0.0
-6521.1
1304223.9
0
0
0
0.0
-558953.1
0.0
0
0
0
4Y
-718654.0
0.0
83842965.5
0
0
0
0
0
0
0.0
1304223.9
683164903.7
0.0
-1304223.9
173896521.0
0
0
0
718654.0
0.0
83842965.5
0
0
0
R4
0
0
0
-12978.9
0.0
2271301.5
0
0
0
-652112.0
0.0
0.0
1547552.3
0.0
0.0
-869482.6
0.0
0.0
0
0
0
-12978.9
0.0
-2271301.5
5X
0
0
0
0.0
-993694.4
0.0
0
0
0
0.0
-6521.1
-1304223.9
0.0
2009367.4
1014396.4
0.0
-15457.5
2318620.3
0
0
0
0.0
-993694.4
0.0
5Y
0
0
0
-2271301.5
0.0
264985174.8
0
0
0
0.0
1304223.9
173896521.0
0.0
1014396.4
1871457796.9
0.0
-2318620.3
231862027.9
0
0
0
2271301.5
0.0
264985174.8
R5
0
0
0
0
0
0
-4106.6
0.0
718654.0
0
0
0
-869482.6
0.0
0.0
873589.2
0.0
718654.0
0
0
0
0
0
0
6X
0
0
0
0
0
0
0.0
-558953.1
0.0
0
0
0
0.0
-15457.5
-2318620.3
0.0
574410.6
-2318620.3
0
0
0
0
0
0
6Y
0
0
0
0
0
0
-718654.0
0.0
83842965.5
0
0
0
0.0
2318620.3
231862027.9
718654.0
-2318620.3
631409986.8
0
0
0
0
0
0
R6
0
0
0
0
0
0
0
0
0
-4106.6
0.0
718654.0
0
0
0
0
0
0
656218.5
0.0
718654.0
-652112.0
0.0
0.0
7X
0
0
0
0
0
0
0
0
0
0.0
-558953.1
0.0
0
0
0
0
0
0
0.0
565474.2
1304223.9
0.0
-6521.1
1304223.9
7Y
0
0
0
0
0
0
0
0
0
-718654.0
0.0
83842965.5
0
0
0
0
0
0
718654.0
1304223.9
515478972.8
0.0
-1304223.9
173896521.0
R7
0
0
0
0
0
0
0
0
0
0
0
0
-12978.9
0.0
2271301.5
0
0
0
-652112.0
0.0
0.0
665090.8
0.0
2271301.5
8X
0
0
0
0
0
0
0
0
0
0
0
0
0.0
-993694.4
0.0
0
0
0
0.0
-6521.1
-1304223.9
0.0
1000215.5
-1304223.9
8Y
0
0
0
0
0
0
0
0
0
0
0
0
-2271301.5
0.0
264985174.8
0
0
0
0.0
1304223.9
173896521.0
2271301.5
-1304223.9
877763391.5
R8
[K]

8. EJERCICIO 2: RESUELTO MÉTODO MATRICIAL REGIDECES EN SISTEMA GLOBAL
Tutorial del ejercicio resuelto - Canal Youtube: Jhon Muchica Sillo
7.1
Inversa
de
la
Matriz
Reducida.
4X
4Y
R4
5X
5Y
R5
6X
6Y
R6
7X
7Y
R7
8X
8Y
R8
5.995E-05
3.615E-07
-5.794E-08
5.871E-05
6.395E-08
-7.843E-08
5.847E-05
-4.752E-07
-3.973E-08
7.422E-05
4.083E-07
-6.059E-09
7.424E-05
3.762E-08
-1.585E-08
4X
3.615E-07
1.782E-06
-3.962E-09
3.607E-07
8.334E-09
-2.220E-09
3.587E-07
-7.328E-09
3.496E-10
1.561E-06
1.780E-06
-4.073E-09
1.560E-06
9.025E-09
-4.257E-09
4Y
-5.794E-08
-3.962E-09
1.675E-09
-5.741E-08
2.084E-09
-1.368E-11
-5.720E-08
2.568E-10
6.341E-11
-1.602E-07
-4.120E-09
-2.222E-10
-1.593E-07
2.173E-09
3.210E-10
R4
5.871E-05
3.607E-07
-5.741E-08
5.899E-05
6.504E-08
-7.837E-08
5.875E-05
-4.764E-07
-4.008E-08
7.395E-05
4.081E-07
-8.102E-09
7.397E-05
3.840E-08
-1.404E-08
5X
6.395E-08
8.334E-09
2.084E-09
6.504E-08
9.951E-07
-1.155E-10
6.773E-08
1.159E-08
-3.646E-09
-3.987E-07
9.750E-09
2.024E-09
-3.970E-07
9.943E-07
2.293E-09
5Y
-7.843E-08
-2.220E-09
-1.368E-11
-7.837E-08
-1.155E-10
7.962E-10
-7.784E-08
2.426E-09
-1.944E-10
-1.944E-07
-2.640E-09
1.595E-10
-1.950E-07
1.203E-10
3.398E-11
R5
5.847E-05
3.587E-07
-5.720E-08
5.875E-05
6.773E-08
-7.784E-08
5.965E-05
-4.792E-07
-4.132E-08
7.361E-05
4.058E-07
-8.019E-09
7.364E-05
4.126E-08
-1.397E-08
6X
-4.752E-07
-7.328E-09
2.568E-10
-4.764E-07
1.159E-08
2.426E-09
-4.792E-07
1.776E-06
6.133E-09
-8.523E-07
-8.617E-09
4.757E-10
-8.539E-07
1.232E-08
1.814E-10
6Y
-3.973E-08
3.496E-10
6.341E-11
-4.008E-08
-3.646E-09
-1.944E-10
-4.132E-08
6.133E-09
1.738E-09
-1.405E-08
4.400E-10
-5.512E-11
-1.388E-08
-3.697E-09
-4.323E-12
R6
7.422E-05
1.561E-06
-1.602E-07
7.395E-05
-3.987E-07
-1.944E-07
7.361E-05
-8.523E-07
-1.405E-08
2.044E-04
2.317E-06
-7.348E-08
2.034E-04
-8.241E-07
-2.664E-07
7X
4.083E-07
1.780E-06
-4.120E-09
4.081E-07
9.750E-09
-2.640E-09
4.058E-07
-8.617E-09
4.400E-10
2.317E-06
3.566E-06
-8.364E-09
2.316E-06
1.191E-08
-7.762E-09
7Y
-6.059E-09
-4.073E-09
-2.222E-10
-8.102E-09
2.024E-09
1.595E-10
-8.019E-09
4.757E-10
-5.512E-11
-7.348E-08
-8.364E-09
2.204E-09
-7.172E-08
4.437E-09
-3.012E-10
R7
7.424E-05
1.560E-06
-1.593E-07
7.397E-05
-3.970E-07
-1.950E-07
7.364E-05
-8.539E-07
-1.388E-08
2.034E-04
2.316E-06
-7.172E-08
2.040E-04
-8.222E-07
-2.679E-07
8X
3.762E-08
9.025E-09
2.173E-09
3.840E-08
9.943E-07
1.203E-10
4.126E-08
1.232E-08
-3.697E-09
-8.241E-07
1.191E-08
4.437E-09
-8.222E-07
1.999E-06
4.264E-09
8Y
-1.585E-08
-4.257E-09
3.210E-10
-1.404E-08
2.293E-09
3.398E-11
-1.397E-08
1.814E-10
-4.323E-12
-2.664E-07
-7.762E-09
-3.012E-10
-2.679E-07
4.264E-09
1.864E-09
R8

9. EJERCICIO 2: RESUELTO MÉTODO MATRICIAL REGIDECES EN SISTEMA GLOBAL
Tutorial del ejercicio resuelto - Canal Youtube: Jhon Muchica Sillo
8. Cálculo de desplazamientos ( Sistema global) para aplicar la ECUACIÓN FUNDAMENTAL
se tiene que (F) presentA la siguiente expresión:
f4x = 3000 kg
f4y = -6250 kg
M4 = -425000 kg-cm
f5x = 0 kg
f5y = -9250 kg
M5 = 275000 kg-cm
f6x = 0 kg
f6y = = -3000 kg
M6 = 150000 kg-cm
f7x = 6000 kg
f7y = -8000 kg
M7 = -533333.3 kg-cm
f8x = 0 kg
f8y = -8000 kg
M8 = 533333.3 kg-cm
Reemplazando las matrices respectivas U7x = 1.343065 U8x = 1.334985
U7y = -0.02327 U8Y = -0.031748
U4x = 0.612056 cm U7 = -0.001628 U8 = -0.000586
U4y = -0.014026 cm
U4 = -0.001531 rad
U5x = 0.608501 cm
U5y = -0.020846 cm
U5 = -0.001245 rad
U6x = 0.605605 cm
U6y = = -0.010637 cm U4x = 0.612056 U5x = 0.608501 U6x = 0.605605
U6 = 4.31E-05 rad U4y = -0.014026 U5y = -0.020846 U6y = -0.010637
U7x = 1.343065 cm U4 = -0.001531 U5 = -0.001245 U6 = 4.31E-05
U7y = -0.02327 cm
U7 = -0.001628 rad
U8x = 1.334985 cm
U8Y = -0.031748 cm
U8 = -0.000586 rad
{F}
{F}=
{U}=
4 5 6
7 8

10. EJERCICIO 2: RESUELTO MÉTODO MATRICIAL REGIDECES EN SISTEMA GLOBAL
Tutorial del ejercicio resuelto - Canal Youtube: Jhon Muchica Sillo
9. Conversión de desplazamientos globaes obtenidos al sistema local. Los 10. Cálculo de esfuerzos en barras (Sistema Local). Seguidamente, se calcula los esfuerzos internos
desplazamiento globales se transforman al sistema de coordenadas locales en cada barra aplicando la siguiente ecuación de equilibrio.
en su forma matricial se tiene la siguiente expresión
cosӨ senӨ 0 0 0 0
-senӨ cosӨ 0 0 0 0
0 0 1 0 0 0 *
= 0 0 0 cosӨ senӨ 0
0 0 0 -senӨ cosӨ 0
0 0 0 0 0 1
para columna 1 para columna 1
0 1 0 0 0 0 0 0 N1 7840.13 0 N1 7840.126 kg 7.84 ton
-1 0 0 0 0 0 0 0 V1 1413.14 0 V1 1413.144 kg 1.41 ton
0 0 1 0 0 0 * 0 = 0 M1 311486 0 M1 311485.6 kg-cm 3.11 ton-m
= 0 0 0 0 1 0 0.61206 -0.01403 N4 = -7840.1 - 0 = N4 = -7840.13 kg -7.84 ton
0 0 0 -1 0 0 -0.014 -0.61206 V4 -1413.1 0 V4 -1413.14 kg -1.41 ton
0 0 0 0 0 1 -0.0015 -0.00153 M4 183115 0 M4 183114.7 kg-cm 1.83 ton-m
para columna 2 para columna 2
0 1 0 0 0 0 0 0 N2 20714.3 0 N2 20714.34 kg 20.71 ton
-1 0 0 0 0 0 0 0 V2 5068.9 0 V2 5068.902 kg 5.07 ton
0 0 1 0 0 0 * 0 = 0 M2 1052068 0 M2 1052068 kg-cm 10.52 ton-m
= 0 0 0 0 1 0 0.6085 -0.02085 N5 = -20714 - 0 = N5 = -20714.3 kg -20.71 ton
0 0 0 -1 0 0 -0.0208 -0.6085 V5 -5068.9 0 V5 -5068.9 kg -5.07 ton
0 0 0 0 0 1 -0.0012 -0.00125 M5 722047 0 M5 722047.4 kg-cm 7.22 ton-m
para columna 3 para columna 3
0 1 0 0 0 0 0 0 N3 5945.54 0 N3 5945.538 kg 5.95 ton
-1 0 0 0 0 0 0 0 V3 2517.95 0 V3 2517.954 kg 2.52 ton
0 0 1 0 0 0 * 0 = 0 M3 438835 0 M3 438834.8 kg-cm 4.39 ton-m
= 0 0 0 0 1 0 0.60561 -0.01064 N6 = -5945.5 - 0 = N6 = -5945.54 kg -5.95 ton
0 0 0 -1 0 0 -0.0106 -0.60561 V6 -2518 0 V6 -2517.95 kg -2.52 ton
0 0 0 0 0 1 4.3E-05 4.3E-05 M6 442449 0 M6 442449.1 kg-cm 4.42 ton-m
para columna 4 para columna 4
0 1 0 0 0 0 0.61206 -0.01403 N4 5166.86 0 N4 5166.859 kg 5.17 ton
-1 0 0 0 0 0 -0.01403 -0.61206 V4 731.367 0 V4 731.3666 kg 0.73 ton
0 0 1 0 0 0 * -0.00153 = -0.00153 M4 132069 0 M4 132069.3 kg-cm 1.32 ton-m
= 0 0 0 0 1 0 1.34306 -0.02327 N7 = -5166.9 - 0 = N7 = -5166.86 kg -5.17 ton
0 0 0 -1 0 0 -0.0233 -1.34306 V7 -731.37 0 V7 -731.367 kg -0.73 ton
0 0 0 0 0 1 -0.0016 -0.00163 M7 123909 0 M7 123909 kg-cm 1.24 ton-m
para columna 5 para columna 5
0 1 0 0 0 0 0.6085 -0.02085 N5 10833.1 0 N5 10833.14 kg 10.83 ton
-1 0 0 0 0 0 -0.0208 -0.6085 V5 5268.63 0 V5 5268.633 kg 5.27 ton
0 0 1 0 0 0 * -0.0012 = -0.00125 M5 834674 0 M5 834674.5 kg-cm 8.35 ton-m
= 0 0 0 0 1 0 1.33499 -0.03175 N8 = -10833 - 0 = N8 = -10833.1 kg -10.83 ton
0 0 0 -1 0 0 -0.0317 -1.33499 V8 -5268.6 0 V8 -5268.63 kg -5.27 ton
0 0 0 0 0 1 -0.0006 -0.00059 M8 1009347 0 M8 1009347 kg-cm 10.09 ton-m
para viga 6 para viga 6
1 0 0 0 0 0 0.61206 0.61206 N4 2318.22 3000 N4 -681.777 kg -0.68 ton
0 1 0 0 0 0 -0.014 -0.01403 V4 -3576.7 -6250 V4 2673.266 kg 2.67 ton
0 0 1 0 0 0 * -0.0015 = -0.00153 M4 -740184 -425000 M4 -315184 kg-cm -3.15 ton-m
= 0 0 0 1 0 0 0.6085 0.6085 N5 = -2318.2 - 0 = N5 = -2318.22 kg -2.32 ton
0 0 0 0 1 0 -0.0208 -0.02085 V5 3576.73 -6250 V5 9826.734 kg 9.83 ton
0 0 0 0 0 1 -0.0012 -0.00125 M5 -690509 425000 M5 -1115509 kg-cm -11.16 ton-m
para viga 7 para viga 7
1 0 0 0 0 0 0.6085 0.6085 N5 2517.95 0 N5 2517.954 kg 2.52 ton
0 1 0 0 0 0 -0.0208 -0.02085 V5 -2945.5 -3000 V5 54.4619 kg 0.05 ton
0 0 1 0 0 0 * -0.0012 = -0.00125 M5 -591212 -150000 M5 -441212 kg-cm -4.41 ton-m
= 0 0 0 1 0 0 0.60561 0.60561 N6 = -2518 - 0 = N6 = -2517.95 kg -2.52 ton
0 0 0 0 1 0 -0.0106 -0.01064 V6 2945.54 -3000 V6 5945.538 kg 5.95 ton
0 0 0 0 0 1 4.3E-05 4.3E-05 M6 -292449 150000 M6 -442449 kg-cm -4.42 ton-m
para viga 8 para viga 8
1 0 0 0 0 0 1.34306 1.34306 N7 5268.63 0 N7 5268.633 kg 5.27 ton
0 1 0 0 0 0 -0.0233 -0.02327 V7 -2833.1 -8000 V7 5166.859 kg 5.17 ton
0 0 1 0 0 0 * -0.0016 = -0.00163 M7 -657242 -533333 M7 -123909 kg-cm -1.24 ton-m
= 0 0 0 1 0 0 1.33499 1.33499 N8 = -5268.6 - 0 = N8 = -5268.63 kg -5.27 ton
0 0 0 0 1 0 -0.0317 -0.03175 V8 2833.14 -8000 V8 10833.14 kg 10.83 ton
0 0 0 0 0 1 -0.0006 -0.00059 M8 -476014 533333 M8 -1009347 kg-cm -10.09 ton-m
{f}
{f}

11. EJERCICIO 2: RESUELTO MÉTODO MATRICIAL REGIDECES EN SISTEMA GLOBAL
Tutorial del ejercicio resuelto - Canal Youtube: Jhon Muchica Sillo
11.1. A continuación, se representa los diagramas de los resultados obtenidos:
M1= 3.11 ton-m M2= 10.52 ton-m M3= 4.39 ton-m
V1 1.41 ton v2= 5.07 ton V3= 2.52 ton
R2= 7.84 ton R2= 20.71 ton R3= 5.95 ton
11.2. A continuación, se representa Diagrama de esfuerzo axial
N7 = 5.27 ton
=
=
N4 = -0.68 ton N5 = 2.52 ton
=
=
=
N2
ton
5.95
N3
ton
10.83
N5
ton
5.17
N4
ton
20.71
N3
7.84
ton

12. EJERCICIO 2: RESUELTO MÉTODO MATRICIAL REGIDECES EN SISTEMA GLOBAL
Tutorial del ejercicio resuelto - Canal Youtube: Jhon Muchica Sillo
11.3. A continuación, se representa Diagrama de Esfuerzo de corte
V7 5.17 ton
V8= 10.83 ton
=
=
V4 2.67 ton
V5 9.83 ton V6 5.95 ton
=
=
=
11.4. A continuación, se representa Diagrama de momentos flectores
M8 -10.09 ton-m
M7 -1.24 ton
=
=
M7
M8
M5 M6 -4.42 ton-m
=
=
M4
M5
M4 -3.15 ton-m M5 -4.41 ton-m
=
=
=
M4
M5
M6
=
=
=
M1
M2
M3
ton-m
10.09
ton-m
7.22
ton-m
4.39
ton-m
4.42
ton
5.27
V5
-11.16
3.11
ton-m
ton-m
1.83
ton-m
10.52
ton-m
1.32
ton-m
1.24
ton-m
8.35
V1
ton
5.07
V2
ton
2.52
V3
ton
0.73
V4
ton
1.41
=
=
=
=
=
=
=
=
=
=