01. ANÁLISIS MATRICIAL DE PÓRTICOS.pdf

1. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 1
ANÁLISIS MATRICIAL DE
PÓRTICOS
ANÁLISIS MATRICIAL DE PÓRTICOS
Ing. Erly Marvin Enriquez Quispe
ing_erlyenriquez@hotmail.com
1. INTRODUCCIÓN
Los pórticos son empleados en edificios donde no se conoce la distribución de los
espacios durante el cálculo y se desea dar amplia libertad de uso. Los pórticos desarrollan
una buena ductilidad bajo la acción sísmica. Su elevada hiperestaticidad y el
comportamiento más allá del límite elástico, permiten la redistribución de efectos sísmicos
y los hace adecuados para resistir fuerzas laterales en edificios altos. Sin embargo su
comportamiento se ve afectado por elementos no estructurales y sus grandes
deformaciones.
Figura 1. Pórtico de un edificio en concreto armado
El empleo de la notación matricial presenta dos ventajas en el cálculo de estructuras.
Desde el punto de vista teórico, permite utilizar métodos de cálculo en forma compacta,
precisa y, al mismo tiempo, completamente general. Esto facilita el tratamiento de la teoría
de estructuras como unidad, sin que los principios fundamentales se vean oscurecidos por
operaciones de cálculo, por un lado, o diferencias físicas entre estructuras, por otro. Desde
el punto de vista práctico, proporciona un sistema apropiado de análisis de estructuras y
determina una base muy conveniente para el desarrollo de programas de computación. En
contraste con estas ventajas, debe admitirse que los métodos matriciales se caracterizan
por una gran cantidad de cálculo sistemático.

2. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 2
ANÁLISIS MATRICIAL DE
PÓRTICOS
2. NOTACIÓN
𝐴: Á𝑟𝑒𝑎 𝑑𝑒 𝑙𝑎 𝑠𝑒𝑐𝑐𝑖ó𝑛 𝑡𝑟𝑎𝑛𝑠𝑣𝑒𝑟𝑠𝑎𝑙
𝐸: 𝑀ó𝑑𝑢𝑙𝑜 𝑑𝑒 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑑𝑎𝑑
𝐼: 𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑑𝑒 𝑖𝑛𝑒𝑟𝑐𝑖𝑎
𝐿: 𝐿𝑜𝑛𝑔𝑖𝑡𝑢𝑑 𝑑𝑒𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑜
𝑃: 𝐹𝑢𝑒𝑟𝑧𝑎 𝑎𝑥𝑖𝑎𝑙 𝑒𝑛 𝑒𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑜
{𝐹}:𝑉𝑒𝑐𝑡𝑜𝑟 𝑑𝑒 𝑓𝑢𝑒𝑟𝑧𝑎𝑠 𝑒𝑛 𝑙𝑎 𝑒𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎
{𝐹𝑎}:𝑉𝑒𝑐𝑡𝑜𝑟 𝑑𝑒 𝑓𝑢𝑒𝑟𝑧𝑎𝑠 𝑑𝑒 𝑓𝑢𝑒𝑟𝑧𝑎𝑠 𝑎𝑐𝑡𝑢𝑎𝑛𝑡𝑒𝑠
{𝐹′𝑒𝑖}: 𝑉𝑒𝑐𝑡𝑜𝑟 𝑑𝑒 𝑓𝑢𝑒𝑟𝑧𝑎𝑠 𝑑𝑒 𝑒𝑚𝑝𝑜𝑡𝑟𝑎𝑚𝑖𝑒𝑛𝑑𝑜 𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑜 𝑒𝑛 𝑐𝑜𝑜𝑟𝑑𝑒𝑛𝑎𝑑𝑎𝑠 𝑙𝑜𝑐𝑎𝑙𝑒𝑠
{𝐹𝑒𝑖}: 𝑉𝑒𝑐𝑡𝑜𝑟 𝑑𝑒 𝑓𝑢𝑒𝑟𝑧𝑎𝑠 𝑑𝑒 𝑒𝑚𝑝𝑜𝑡𝑟𝑎𝑚𝑖𝑒𝑛𝑑𝑜 𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑜 𝑑𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑜𝑠 𝑒𝑛 𝑐𝑜𝑜𝑟𝑑. 𝑔𝑙𝑜𝑏𝑎𝑙𝑒𝑠
{𝐹𝑒}: 𝑉𝑒𝑐𝑡𝑜𝑟 𝑑𝑒 𝑓𝑢𝑒𝑟𝑧𝑎𝑠 𝑑𝑒 𝑒𝑚𝑝𝑜𝑡𝑟𝑎𝑚𝑖𝑒𝑛𝑑𝑜 𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑜 𝑑𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑜𝑠 𝑒𝑛 𝑐𝑜𝑜𝑟𝑑. 𝑔𝑙𝑜𝑏𝑎𝑙𝑒𝑠
{𝐹1}:𝑉𝑒𝑐𝑡𝑜𝑟 𝑑𝑒 𝑓𝑢𝑒𝑟𝑧𝑎𝑠 𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑎𝑠 𝑒𝑛 𝑙𝑎 𝑒𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎
{𝐹2}:𝑉𝑒𝑐𝑡𝑜𝑟 𝑑𝑒 𝑓𝑢𝑒𝑟𝑧𝑎𝑠 𝑑𝑒𝑠𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑎𝑠 𝑒𝑛 𝑙𝑎 𝑒𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎 (𝑅𝑒𝑎𝑐𝑐𝑖𝑜𝑛𝑒𝑠)
[𝑘]: 𝑀𝑎𝑡𝑟𝑖𝑧 𝑑𝑒 𝑟𝑖𝑔𝑖𝑑𝑒𝑠 𝑑𝑒𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑜 𝑒𝑛 𝑐𝑜𝑜𝑟𝑑𝑒𝑛𝑎𝑑𝑎𝑠 𝑔𝑙𝑜𝑏𝑎𝑙𝑒𝑠
[𝑘’]: 𝑀𝑎𝑡𝑟𝑖𝑧 𝑑𝑒 𝑟𝑖𝑔𝑖𝑑𝑒𝑠 𝑑𝑒𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑜 𝑒𝑛 𝑐𝑜𝑜𝑟𝑑𝑒𝑛𝑎𝑑𝑎𝑠 𝑙𝑜𝑐𝑎𝑙𝑒𝑠
[𝑘𝐸]: 𝑀𝑎𝑡𝑟𝑖𝑧 𝑑𝑒 𝑟𝑖𝑔𝑖𝑑𝑒𝑧 𝑒𝑙á𝑠𝑡𝑖𝑐𝑎 𝑑𝑒𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑜 𝑒𝑛 𝑐𝑜𝑜𝑟𝑑𝑒𝑛𝑎𝑑𝑎𝑠 𝑙𝑜𝑐𝑎𝑙𝑒𝑠
[𝑘𝐺]: 𝑀𝑎𝑡𝑟𝑖𝑧 𝑑𝑒 𝑟𝑖𝑔𝑖𝑑𝑒𝑧 𝑔𝑒𝑜𝑚é𝑡𝑟𝑖𝑐𝑎 𝑑𝑒𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑜 𝑒𝑛 𝑐𝑜𝑜𝑟𝑑𝑒𝑛𝑎𝑑𝑎𝑠 𝑙𝑜𝑐𝑎𝑙𝑒𝑠
[𝑘𝑖]: 𝑀𝑎𝑡𝑟𝑖𝑧 𝑑𝑒 𝑟𝑖𝑔𝑖𝑑𝑒𝑧 𝑑𝑒𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑜 𝑖 𝑒𝑛 𝑐𝑜𝑜𝑟𝑑𝑒𝑛𝑎𝑑𝑎𝑠 𝑙𝑜𝑐𝑎𝑙𝑒𝑠
[𝐾]: 𝑀𝑎𝑡𝑟𝑖𝑧 𝑑𝑒 𝑟𝑖𝑔𝑖𝑑𝑒𝑧 𝑔𝑙𝑜𝑏𝑎𝑙 𝑑𝑒 𝑙𝑎 𝑒𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎
[𝐾11]: 𝑀𝑎𝑡𝑟𝑖𝑧 𝑞𝑢𝑒 𝑟𝑒𝑙𝑎𝑐𝑖𝑜𝑛𝑎 𝑓𝑢𝑒𝑟𝑧𝑎𝑠 𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑎𝑠 𝑐𝑜𝑛 𝑑𝑒𝑠𝑝𝑙𝑎𝑧𝑎𝑚𝑖𝑒𝑛𝑡𝑜𝑠 𝑑𝑒𝑠𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑜𝑠
[𝐾12]: 𝑀𝑎𝑡𝑟𝑖𝑧 𝑞𝑢𝑒 𝑟𝑒𝑙𝑎𝑐𝑖𝑜𝑛𝑎 𝑓𝑢𝑒𝑟𝑧𝑎𝑠 𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑎𝑠 𝑐𝑜𝑛 𝑑𝑒𝑠𝑝𝑙𝑎𝑧𝑎𝑚𝑖𝑒𝑛𝑡𝑜𝑠 𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑜𝑠
[𝐾21]: 𝑀𝑎𝑡𝑟𝑖𝑧 𝑞𝑢𝑒 𝑟𝑒𝑙𝑎𝑐𝑖𝑜𝑛𝑎 𝑓𝑢𝑒𝑟𝑧𝑎𝑠 𝑑𝑒𝑠𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑎𝑠 𝑐𝑜𝑛 𝑑𝑒𝑠𝑝𝑙𝑎𝑧𝑎𝑚𝑖𝑒𝑛𝑡𝑜𝑠 𝑑𝑒𝑠𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑜𝑠
[𝐾22]: 𝑀𝑎𝑡𝑟𝑖𝑧 𝑞𝑢𝑒 𝑟𝑒𝑙𝑎𝑐𝑖𝑜𝑛𝑎 𝑓𝑢𝑒𝑟𝑧𝑎𝑠 𝑑𝑒𝑠𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑎𝑠 𝑐𝑜𝑛 𝑑𝑒𝑠𝑝𝑙𝑎𝑧𝑎𝑚𝑖𝑒𝑛𝑡𝑜𝑠 𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑜𝑠
{𝑆𝑖}: 𝑉𝑒𝑐𝑡𝑜𝑟 𝑑𝑒 𝑓𝑢𝑒𝑟𝑧𝑎𝑠 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑠 𝑑𝑒𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑜 𝑖 𝑒𝑛 𝑐𝑜𝑜𝑟𝑑𝑒𝑛𝑎𝑑𝑎𝑠 𝑙𝑜𝑐𝑎𝑙𝑒𝑠
[𝑇]: 𝑀𝑎𝑡𝑟𝑖𝑧 𝑑𝑒 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛
[𝑇𝑖]: 𝑀𝑎𝑡𝑟𝑖𝑧 𝑑𝑒 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛 𝑑𝑒𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑜 𝑖
{𝑢𝑖}: 𝑉𝑒𝑐𝑡𝑜𝑟 𝑑𝑒 𝑑𝑒𝑠𝑝𝑙𝑎𝑧𝑎𝑚𝑖𝑒𝑛𝑡𝑜 𝑑𝑒𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑜 𝑖 𝑒𝑛 𝑐𝑜𝑜𝑟𝑑𝑒𝑛𝑎𝑑𝑎𝑠 𝑙𝑜𝑐𝑎𝑙𝑒𝑠
{∆}:𝑉𝑒𝑐𝑡𝑜𝑟 𝑑𝑒 𝑑𝑒𝑠𝑝𝑙𝑎𝑧𝑎𝑚𝑖𝑒𝑛𝑡𝑜𝑠 𝑑𝑒 𝑙𝑎 𝑒𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎
{∆1}: 𝑉𝑒𝑐𝑡𝑜𝑟 𝑑𝑒 𝑑𝑒𝑠𝑝𝑙𝑎𝑧𝑎𝑚𝑖𝑒𝑛𝑡𝑜𝑠 𝑑𝑒𝑠𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑜𝑠 𝑑𝑒 𝑙𝑎 𝑒𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎
{∆2}: 𝑉𝑒𝑐𝑡𝑜𝑟 𝑑𝑒 𝑑𝑒𝑠𝑝𝑙𝑎𝑧𝑎𝑚𝑖𝑒𝑛𝑡𝑜𝑠 𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑜𝑠 𝑑𝑒 𝑙𝑎 𝑒𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎
Ø: Á𝑛𝑔𝑢𝑙𝑜 𝑑𝑒𝑙 𝑒𝑗𝑒 𝑑𝑒𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑜 𝑐𝑜𝑛 𝑙𝑎 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙

3. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 3
ANÁLISIS MATRICIAL DE
PÓRTICOS
3. MATRIZ RIGIDEZ DE UN ELEMENTO EN COORDENADAS LOCALES
E, A, I, L
1
4
2
1
6
3
1
{
𝐹1
𝐹2
𝐹3
𝐹4
𝐹5
𝐹6}
=
[
𝑘11 𝑘12 𝑘13
𝑘21 𝑘22 𝑘23
𝑘31 𝑘32 𝑘33
𝑘14 𝑘15 𝑘16
𝑘24 𝑘25 𝑘26
𝑘34 𝑘35 𝑘36
𝑘41 𝑘42 𝑘43
𝑘51 𝑘52 𝑘53
𝑘61 𝑘62 𝑘63
𝑘44 𝑘45 𝑘46
𝑘54 𝑘55 𝑘56
𝑘64 𝑘65 𝑘66]{
∆1
∆2
∆3
∆4
∆5
∆6}
Cálculo de la primera columna de la matriz rigidez:
∆1= 1 ; ∆2= 0 ; ∆3= 0 ; ∆4= 0 ; ∆5= 0 ; ∆6= 0
E, A, I, L
1 2
1
D=1
{
𝐹1
𝐹2
𝐹3
𝐹4
𝐹5
𝐹6}
=
[
𝑘11
𝑘21
𝑘31
𝑘41
𝑘51
𝑘61]
=
[
𝐸𝐴/𝐿
0
0
−𝐸𝐴/𝐿
0
0 ]
Cálculo de la segunda columna de la matriz rigidez:
∆1= 0 ; ∆2= 1 ; ∆3= 0 ; ∆4= 0 ; ∆5= 0 ; ∆6= 0
E, A, I, L
2
1 5
6
2
D=1
1
{
𝐹1
𝐹2
𝐹3
𝐹4
𝐹5
𝐹6}
=
[
𝑘12
𝑘22
𝑘32
𝑘42
𝑘52
𝑘62]
=
[
0
12𝐸𝐼 𝐿3
⁄
6𝐸𝐼 𝐿2
⁄
0
−12𝐸𝐼 𝐿3
⁄
6𝐸𝐼 𝐿2
⁄ ]

4. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 4
ANÁLISIS MATRICIAL DE
PÓRTICOS
Cálculo de la tercera columna de la matriz rigidez:
∆1= 0 ; ∆2= 0 ; ∆3= 1 ; ∆4= 0 ; ∆5= 0 ; ∆6= 0
E, A, I, L
2
1 5
2
D=1
1
{
𝐹1
𝐹2
𝐹3
𝐹4
𝐹5
𝐹6}
=
[
𝑘13
𝑘23
𝑘33
𝑘43
𝑘53
𝑘63]
=
[
0
6𝐸𝐼 𝐿2
⁄
4𝐸𝐼 𝐿
⁄
0
−6𝐸𝐼 𝐿2
⁄
2𝐸𝐼 𝐿
⁄ ]
Cálculo de la cuarta columna de la matriz rigidez:
∆1= 0 ; ∆2= 0 ; ∆3= 0 ; ∆4= 1 ; ∆5= 0 ; ∆6= 0
1
1 D=1
2
{
𝐹1
𝐹2
𝐹3
𝐹4
𝐹5
𝐹6}
=
[
𝑘14
𝑘24
𝑘34
𝑘44
𝑘54
𝑘64]
=
[
−𝐸𝐴/𝐿
0
0
𝐸𝐴/𝐿
0
0 ]
Cálculo de la quinta columna de la matriz rigidez:
∆1= 0 ; ∆2= 0 ; ∆3= 0 ; ∆4= 0 ; ∆5= 1 ; ∆6= 0
E, A, I, L
1
5 D=1
2
1
3
2
{
𝐹1
𝐹2
𝐹3
𝐹4
𝐹5
𝐹6}
=
[
𝑘15
𝑘25
𝑘35
𝑘45
𝑘55
𝑘65]
=
[
0
− 12𝐸𝐼 𝐿3
⁄
−6𝐸𝐼 𝐿2
⁄
0
12𝐸𝐼 𝐿3
⁄
−6𝐸𝐼 𝐿2
⁄ ]

5. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 5
ANÁLISIS MATRICIAL DE
PÓRTICOS
Cálculo de la sexta columna de la matriz rigidez:
∆1= 0 ; ∆2= 0 ; ∆3= 0 ; ∆4= 0 ; ∆5= 0 ; ∆6= 1
2
E, A, I, L
1
2 5
D=1
1
{
𝐹1
𝐹2
𝐹3
𝐹4
𝐹5
𝐹6}
=
[
𝑘16
𝑘26
𝑘36
𝑘46
𝑘56
𝑘66]
=
[
0
6𝐸𝐼 𝐿2
⁄
2𝐸𝐼 𝐿
⁄
0
− 6𝐸𝐼 𝐿2
⁄
4𝐸𝐼 𝐿
⁄ ]
Matriz rigidez en coordenadas locales:
[𝑘]′ =
[
𝐸𝐴
𝐿
0 0 −
𝐸𝐴
𝐿
0 0
0
12𝐸𝐼
𝐿3
6𝐸𝐼
𝐿2
0 −
12𝐸𝐼
𝐿3
6𝐸𝐼
𝐿2
0
6𝐸𝐼
𝐿2
4𝐸𝐼
𝐿
0 −
6𝐸𝐼
𝐿2
2𝐸𝐼
𝐿
−
𝐸𝐴
𝐿
0 0
𝐸𝐴
𝐿
0 0
0 −
12𝐸𝐼
𝐿3
−
6𝐸𝐼
𝐿2
0
12𝐸𝐼
𝐿3
−
6𝐸𝐼
𝐿2
0
6𝐸𝐼
𝐿2
2𝐸𝐼
𝐿
0 −
6𝐸𝐼
𝐿2
4𝐸𝐼
𝐿 ]

6. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 6
ANÁLISIS MATRICIAL DE
PÓRTICOS
4. MATRIZ RIGIDEZ DE UNA BARRA EN COORDENADAS GLOBALES
X'
Y'
x'i
x'j
y'i
y'j
X
Y
xi
xj
yj
yi
Ø
Mi
Mj
𝑥𝑖
′
= 𝑥𝑖𝑐𝑜𝑠Ø + 𝑦𝑖𝑠𝑒𝑛Ø
𝑦𝑖
′
= −𝑥𝑖𝑠𝑒𝑛Ø + 𝑦𝑖𝑐𝑜𝑠Ø
𝑀𝑖
′
= 𝑀𝑖
𝑥𝑗
′
= 𝑥𝑗𝑐𝑜𝑠Ø + 𝑦𝑗𝑠𝑒𝑛Ø
𝑦𝑗
′
= −𝑥𝑗𝑠𝑒𝑛Ø + 𝑦𝑗𝑐𝑜𝑠Ø
𝑀𝑗
′
= 𝑀𝑗
[
𝑥𝑖
′
𝑦𝑖
′
𝑀𝑖
′
𝑥𝑗
′
𝑦𝑗
′
𝑀𝑗
′
]
=
[
𝑐𝑜𝑠Ø
−𝑠𝑒𝑛Ø
0
0
0
0
𝑠𝑒𝑛Ø
𝑐𝑜𝑠Ø
0
0
0
0
0
0
1
0
0
0
0
0
0
𝑐𝑜𝑠Ø
−𝑠𝑒𝑛Ø
0
0
0
0
𝑠𝑒𝑛Ø
𝑐𝑜𝑠Ø
0
0
0
0
0
0
1] [
𝑥𝑖
𝑦𝑖
𝑀𝑖
𝑥𝑗
𝑦𝑗
𝑀𝑗]
[𝑇] =
[
𝑐𝑜𝑠Ø
−𝑠𝑒𝑛Ø
0
0
0
0
𝑠𝑒𝑛Ø
𝑐𝑜𝑠Ø
0
0
0
0
0
0
1
0
0
0
0
0
0
𝑐𝑜𝑠Ø
−𝑠𝑒𝑛Ø
0
0
0
0
𝑠𝑒𝑛Ø
𝑐𝑜𝑠Ø
0
0
0
0
0
0
1]
𝐹′ = 𝑇𝐹 → 𝐹 = 𝑇−1
𝐹′
𝐹 = 𝑘∆→ 𝐹′ = 𝑘′∆′
∆′
= 𝑇∆→ 𝐹 = 𝑇−1
𝑘′𝑇∆
[𝑘] = [𝑇]𝑇[𝑘]′[𝑇]

7. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 7
ANÁLISIS MATRICIAL DE
PÓRTICOS
5. PROCEDIMIENTO DE CÁLCULO
1) Determinar la matriz rigidez [𝑘′𝑖] de los elementos en coordenadas locales.
2) Determinar la matriz de transformación [𝑇𝑖] de los elementos.
3) Determinar la matriz rigidez de los elementos en coordenadas globales.
[𝑘𝑖] = [𝑇𝑖]𝑇[𝑘′𝑖][𝑇𝑖]
4) Ensamblar la matriz rigidez de la estructura [𝐾].
5) Determinar el vector fuerzas de empotramiento perfecto de los elementos {𝐹′𝑒𝑖}.
6) Determinar el vector fuerzas de empotramiento perfecto de la estructura {𝐹𝑒𝑖}.
{𝐹𝑒𝑖} = [𝑇]𝑇{𝐹′𝑒𝑖}
7) Determinar el vector fuerzas en los nudos {𝐹}.
{𝐹} = {𝐹
𝑎} − {𝐹
𝑒}
8) Particionar la matriz de rigidez de la estructura.
{
𝐹1
𝐹2
} = [
𝐾11 𝐾12
𝐾21 𝐾22
] {
∆1
∆2
}
9) Determinar las deformaciones desconocidas de la estructura. Si {∆2} = {0} luego:
{∆1} = [𝐾11]−1{𝐹1}
10) Determinar las Reacciones de la estructura. Si {∆2} = {0} luego:
{𝐹2} = [𝐾21]{∆1}
11) Determinar las fuerzas internas en los elementos.
{𝑆𝑖} = [𝑘′𝑖][𝑇𝑖]{𝑢𝑖} + {𝐹′𝑒𝑖}

8. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 8
ANÁLISIS MATRICIAL DE
PÓRTICOS
6. EJEMPLO DE APLICACIÓN
1.. GEOMETRÍA DEL PÓRTICO
L1: Longitud del primer tramo (m) 5.00
L2: Longitud del segundo tramo (m) 6.00
L3: Longitud del tercer tramo (m) 5.00
H1: Altura del primer entrepiso (m) 4.00
H2: Altura del segundo entrepiso (m) 3.00
H3: Altura del tercer entrepiso (m) 3.00
L1 L2 L3
H1
H2
H3
w
w
w w
w
w w
w
w
L1 L2 L3
1 1
2
3
2 4
5
6
3 7
8
9
4 10
11
12
6 16
17
18
7 19
20
21
8 22
23
24
1234
35
36
1131
32
33
1028
29
30
9 25
26
27
1337
38
39
1440
41
42
1543
44
45
1646
47
48
5 13
14
15
1 2 3 4
8 9 10 11
15 16 17 18
5 6 7
w
12 13 14
19 20 21
H1
H2
H3
w w
w
w
w
w w w

9. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 9
ANÁLISIS MATRICIAL DE
PÓRTICOS
2.. PROPIEDADES DE LOS ELEMENTOS
E: Módulo de Elasticidad (t/m
2
) 2173707
bv: Base de las vigas (m) 0.30
hv: Peralte de las vigas (m) 0.50
bc: Base de las columnas (m) 0.30
hc: Peralte de las columnas (m) 0.40
ELEM. E (t/m
2
) I (m
4
) A (m
2
) L (m) Ø (°) w (t/m)
1 2173707 0.0016 0.1200 4.00 90 0
2 2173707 0.0016 0.1200 4.00 90 0
3 2173707 0.0016 0.1200 4.00 90 0
4 2173707 0.0016 0.1200 4.00 90 0
5 2173707 0.0031 0.1500 5.00 0 5
6 2173707 0.0031 0.1500 6.00 0 5
7 2173707 0.0031 0.1500 5.00 0 5
8 2173707 0.0016 0.1200 3.00 90 0
9 2173707 0.0016 0.1200 3.00 90 0
10 2173707 0.0016 0.1200 3.00 90 0
11 2173707 0.0016 0.1200 3.00 90 0
12 2173707 0.0031 0.1500 5.00 0 5
13 2173707 0.0031 0.1500 6.00 0 5
14 2173707 0.0031 0.1500 5.00 0 5
15 2173707 0.0016 0.1200 3.00 90 0
16 2173707 0.0016 0.1200 3.00 90 0
17 2173707 0.0016 0.1200 3.00 90 0
18 2173707 0.0016 0.1200 3.00 90 0
19 2173707 0.0031 0.1500 5.00 0 5
20 2173707 0.0031 0.1500 6.00 0 5
21 2173707 0.0031 0.1500 5.00 0 5

10. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 10
ANÁLISIS MATRICIAL DE
PÓRTICOS
3.. MATRIZ RIGIDEZ DE LOS ELEMENTOS EN COORDENADAS LOCALES
65211 0 0 -65211 0 0
0 652 1304 0 -652 1304
0 1304 3478 0 -1304 1739
-65211 0 0 65211 0 0
0 -652 -1304 0 652 -1304
0 1304 1739 0 -1304 3478
65211 0 0 -65211 0 0
0 652 1304 0 -652 1304
0 1304 3478 0 -1304 1739
-65211 0 0 65211 0 0
0 -652 -1304 0 652 -1304
0 1304 1739 0 -1304 3478
65211 0 0 -65211 0 0
0 652 1304 0 -652 1304
0 1304 3478 0 -1304 1739
-65211 0 0 65211 0 0
0 -652 -1304 0 652 -1304
0 1304 1739 0 -1304 3478
65211 0 0 -65211 0 0
0 652 1304 0 -652 1304
0 1304 3478 0 -1304 1739
-65211 0 0 65211 0 0
0 -652 -1304 0 652 -1304
0 1304 1739 0 -1304 3478
65211 0 0 -65211 0 0
0 652 1630 0 -652 1630
0 1630 5434 0 -1630 2717
-65211 0 0 65211 0 0
0 -652 -1630 0 652 -1630
0 1630 2717 0 -1630 5434
54343 0 0 -54343 0 0
0 377 1132 0 -377 1132
0 1132 4529 0 -1132 2264
-54343 0 0 54343 0 0
0 -377 -1132 0 377 -1132
0 1132 2264 0 -1132 4529
65211 0 0 -65211 0 0
0 652 1630 0 -652 1630
0 1630 5434 0 -1630 2717
-65211 0 0 65211 0 0
0 -652 -1630 0 652 -1630
0 1630 2717 0 -1630 5434
k'2 =
k'1 =
k'3 =
k'4 =
k'5 =
k'6 =
k'7 =

11. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 11
ANÁLISIS MATRICIAL DE
PÓRTICOS
86948 0 0 -86948 0 0
0 1546 2319 0 -1546 2319
0 2319 4637 0 -2319 2319
-86948 0 0 86948 0 0
0 -1546 -2319 0 1546 -2319
0 2319 2319 0 -2319 4637
86948 0 0 -86948 0 0
0 1546 2319 0 -1546 2319
0 2319 4637 0 -2319 2319
-86948 0 0 86948 0 0
0 -1546 -2319 0 1546 -2319
0 2319 2319 0 -2319 4637
86948 0 0 -86948 0 0
0 1546 2319 0 -1546 2319
0 2319 4637 0 -2319 2319
-86948 0 0 86948 0 0
0 -1546 -2319 0 1546 -2319
0 2319 2319 0 -2319 4637
86948 0 0 -86948 0 0
0 1546 2319 0 -1546 2319
0 2319 4637 0 -2319 2319
-86948 0 0 86948 0 0
0 -1546 -2319 0 1546 -2319
0 2319 2319 0 -2319 4637
65211 0 0 -65211 0 0
0 652 1630 0 -652 1630
0 1630 5434 0 -1630 2717
-65211 0 0 65211 0 0
0 -652 -1630 0 652 -1630
0 1630 2717 0 -1630 5434
54343 0 0 -54343 0 0
0 377 1132 0 -377 1132
0 1132 4529 0 -1132 2264
-54343 0 0 54343 0 0
0 -377 -1132 0 377 -1132
0 1132 2264 0 -1132 4529
65211 0 0 -65211 0 0
0 652 1630 0 -652 1630
0 1630 5434 0 -1630 2717
-65211 0 0 65211 0 0
0 -652 -1630 0 652 -1630
0 1630 2717 0 -1630 5434
k'8 =
k'9 =
k'10 =
k'11 =
k'12 =
k'13 =
k'14 =

12. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 12
ANÁLISIS MATRICIAL DE
PÓRTICOS
86948 0 0 -86948 0 0
0 1546 2319 0 -1546 2319
0 2319 4637 0 -2319 2319
-86948 0 0 86948 0 0
0 -1546 -2319 0 1546 -2319
0 2319 2319 0 -2319 4637
86948 0 0 -86948 0 0
0 1546 2319 0 -1546 2319
0 2319 4637 0 -2319 2319
-86948 0 0 86948 0 0
0 -1546 -2319 0 1546 -2319
0 2319 2319 0 -2319 4637
86948 0 0 -86948 0 0
0 1546 2319 0 -1546 2319
0 2319 4637 0 -2319 2319
-86948 0 0 86948 0 0
0 -1546 -2319 0 1546 -2319
0 2319 2319 0 -2319 4637
86948 0 0 -86948 0 0
0 1546 2319 0 -1546 2319
0 2319 4637 0 -2319 2319
-86948 0 0 86948 0 0
0 -1546 -2319 0 1546 -2319
0 2319 2319 0 -2319 4637
65211 0 0 -65211 0 0
0 652 1630 0 -652 1630
0 1630 5434 0 -1630 2717
-65211 0 0 65211 0 0
0 -652 -1630 0 652 -1630
0 1630 2717 0 -1630 5434
54343 0 0 -54343 0 0
0 377 1132 0 -377 1132
0 1132 4529 0 -1132 2264
-54343 0 0 54343 0 0
0 -377 -1132 0 377 -1132
0 1132 2264 0 -1132 4529
65211 0 0 -65211 0 0
0 652 1630 0 -652 1630
0 1630 5434 0 -1630 2717
-65211 0 0 65211 0 0
0 -652 -1630 0 652 -1630
0 1630 2717 0 -1630 5434
k'17 =
k'18 =
k'19 =
k'20 =
k'21 =
k'15 =
k'16 =

13. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 13
ANÁLISIS MATRICIAL DE
PÓRTICOS
4.. MATRIZ TRANSFORMACIÓN DE LOS ELEMENTOS
0.00 1.00 0 0 0 0
-1.00 0.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 0.00 1.00 0
0 0 0 -1.00 0.00 0
0 0 0 0 0 1.00
0.00 1.00 0 0 0 0
-1.00 0.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 0.00 1.00 0
0 0 0 -1.00 0.00 0
0 0 0 0 0 1.00
0.00 1.00 0 0 0 0
-1.00 0.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 0.00 1.00 0
0 0 0 -1.00 0.00 0
0 0 0 0 0 1.00
0.00 1.00 0 0 0 0
-1.00 0.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 0.00 1.00 0
0 0 0 -1.00 0.00 0
0 0 0 0 0 1.00
1.00 0.00 0 0 0 0
0.00 1.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 1.00 0.00 0
0 0 0 0.00 1.00 0
0 0 0 0 0 1.00
1.00 0.00 0 0 0 0
0.00 1.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 1.00 0.00 0
0 0 0 0.00 1.00 0
0 0 0 0 0 1.00
1.00 0.00 0 0 0 0
0.00 1.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 1.00 0.00 0
0 0 0 0.00 1.00 0
0 0 0 0 0 1.00
T7 =
T2 =
T3 =
T4 =
T5 =
T6 =
T1 =

14. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 14
ANÁLISIS MATRICIAL DE
PÓRTICOS
0.00 1.00 0 0 0 0
-1.00 0.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 0.00 1.00 0
0 0 0 -1.00 0.00 0
0 0 0 0 0 1.00
0.00 1.00 0 0 0 0
-1.00 0.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 0.00 1.00 0
0 0 0 -1.00 0.00 0
0 0 0 0 0 1.00
0.00 1.00 0 0 0 0
-1.00 0.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 0.00 1.00 0
0 0 0 -1.00 0.00 0
0 0 0 0 0 1.00
0.00 1.00 0 0 0 0
-1.00 0.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 0.00 1.00 0
0 0 0 -1.00 0.00 0
0 0 0 0 0 1.00
1.00 0.00 0 0 0 0
0.00 1.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 1.00 0.00 0
0 0 0 0.00 1.00 0
0 0 0 0 0 1.00
1.00 0.00 0 0 0 0
0.00 1.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 1.00 0.00 0
0 0 0 0.00 1.00 0
0 0 0 0 0 1.00
1.00 0.00 0 0 0 0
0.00 1.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 1.00 0.00 0
0 0 0 0.00 1.00 0
0 0 0 0 0 1.00
T8 =
T9 =
T10 =
T11 =
T12 =
T13 =
T14 =

15. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 15
ANÁLISIS MATRICIAL DE
PÓRTICOS
0.00 1.00 0 0 0 0
-1.00 0.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 0.00 1.00 0
0 0 0 -1.00 0.00 0
0 0 0 0 0 1.00
0.00 1.00 0 0 0 0
-1.00 0.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 0.00 1.00 0
0 0 0 -1.00 0.00 0
0 0 0 0 0 1.00
0.00 1.00 0 0 0 0
-1.00 0.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 0.00 1.00 0
0 0 0 -1.00 0.00 0
0 0 0 0 0 1.00
0.00 1.00 0 0 0 0
-1.00 0.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 0.00 1.00 0
0 0 0 -1.00 0.00 0
0 0 0 0 0 1.00
1.00 0.00 0 0 0 0
0.00 1.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 1.00 0.00 0
0 0 0 0.00 1.00 0
0 0 0 0 0 1.00
1.00 0.00 0 0 0 0
0.00 1.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 1.00 0.00 0
0 0 0 0.00 1.00 0
0 0 0 0 0 1.00
1.00 0.00 0 0 0 0
0.00 1.00 0 0 0 0
0 0 1.00 0 0 0
0 0 0 1.00 0.00 0
0 0 0 0.00 1.00 0
0 0 0 0 0 1.00
T20 =
T21 =
T15 =
T16 =
T17 =
T18 =
T19 =

16. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 16
ANÁLISIS MATRICIAL DE
PÓRTICOS
5.. MATRIZ RIGIDEZ DE LOS ELEMENTOS EN COORDENADAS GLOBALES
37 38 39 25 26 27
652 0 -1304 -652 0 -1304 37
0 65211 0 0 -65211 0 38
-1304 0 3478 1304 0 1739 39
-652 0 1304 652 0 1304 25
0 -65211 0 0 65211 0 26
-1304 0 1739 1304 0 3478 27
40 41 42 28 29 30
652 0 -1304 -652 0 -1304 40
0 65211 0 0 -65211 0 41
-1304 0 3478 1304 0 1739 42
-652 0 1304 652 0 1304 28
0 -65211 0 0 65211 0 29
-1304 0 1739 1304 0 3478 30
43 44 45 31 32 33
652 0 -1304 -652 0 -1304 43
0 65211 0 0 -65211 0 44
-1304 0 3478 1304 0 1739 45
-652 0 1304 652 0 1304 31
0 -65211 0 0 65211 0 32
-1304 0 1739 1304 0 3478 33
46 47 48 34 35 36
652 0 -1304 -652 0 -1304 46
0 65211 0 0 -65211 0 47
-1304 0 3478 1304 0 1739 48
-652 0 1304 652 0 1304 34
0 -65211 0 0 65211 0 35
-1304 0 1739 1304 0 3478 36
25 26 27 28 29 30
65211 0 0 -65211 0 0 25
0 652 1630 0 -652 1630 26
0 1630 5434 0 -1630 2717 27
-65211 0 0 65211 0 0 28
0 -652 -1630 0 652 -1630 29
0 1630 2717 0 -1630 5434 30
28 29 30 31 32 33
54343 0 0 -54343 0 0 28
0 377 1132 0 -377 1132 29
0 1132 4529 0 -1132 2264 30
-54343 0 0 54343 0 0 31
0 -377 -1132 0 377 -1132 32
0 1132 2264 0 -1132 4529 33
k3 =
k4 =
k5 =
k6 =
k1 =
k2 =

17. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 17
ANÁLISIS MATRICIAL DE
PÓRTICOS
31 32 33 34 35 36
65211 0 0 -65211 0 0 31
0 652 1630 0 -652 1630 32
0 1630 5434 0 -1630 2717 33
-65211 0 0 65211 0 0 34
0 -652 -1630 0 652 -1630 35
0 1630 2717 0 -1630 5434 36
25 26 27 13 14 15
1546 0 -2319 -1546 0 -2319 25
0 86948 0 0 -86948 0 26
-2319 0 4637 2319 0 2319 27
-1546 0 2319 1546 0 2319 13
0 -86948 0 0 86948 0 14
-2319 0 2319 2319 0 4637 15
28 29 30 16 17 18
1546 0 -2319 -1546 0 -2319 28
0 86948 0 0 -86948 0 29
-2319 0 4637 2319 0 2319 30
-1546 0 2319 1546 0 2319 16
0 -86948 0 0 86948 0 17
-2319 0 2319 2319 0 4637 18
31 32 33 19 20 21
1546 0 -2319 -1546 0 -2319 31
0 86948 0 0 -86948 0 32
-2319 0 4637 2319 0 2319 33
-1546 0 2319 1546 0 2319 19
0 -86948 0 0 86948 0 20
-2319 0 2319 2319 0 4637 21
34 35 36 22 23 24
1546 0 -2319 -1546 0 -2319 34
0 86948 0 0 -86948 0 35
-2319 0 4637 2319 0 2319 36
-1546 0 2319 1546 0 2319 22
0 -86948 0 0 86948 0 23
-2319 0 2319 2319 0 4637 24
13 14 15 16 17 18
65211 0 0 -65211 0 0 13
0 652 1630 0 -652 1630 14
0 1630 5434 0 -1630 2717 15
-65211 0 0 65211 0 0 16
0 -652 -1630 0 652 -1630 17
0 1630 2717 0 -1630 5434 18
k7 =
k11 =
k12 =
k10 =
k8 =
k9 =

18. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 18
ANÁLISIS MATRICIAL DE
PÓRTICOS
16 17 18 19 20 21
54343 0 0 -54343 0 0 16
0 377 1132 0 -377 1132 17
0 1132 4529 0 -1132 2264 18
-54343 0 0 54343 0 0 19
0 -377 -1132 0 377 -1132 20
0 1132 2264 0 -1132 4529 21
19 20 21 22 23 24
65211 0 0 -65211 0 0 19
0 652 1630 0 -652 1630 20
0 1630 5434 0 -1630 2717 21
-65211 0 0 65211 0 0 22
0 -652 -1630 0 652 -1630 23
0 1630 2717 0 -1630 5434 24
13 14 15 1 2 3
1546 0 -2319 -1546 0 -2319 13
0 86948 0 0 -86948 0 14
-2319 0 4637 2319 0 2319 15
-1546 0 2319 1546 0 2319 1
0 -86948 0 0 86948 0 2
-2319 0 2319 2319 0 4637 3
16 17 18 4 5 6
1546 0 -2319 -1546 0 -2319 16
0 86948 0 0 -86948 0 17
-2319 0 4637 2319 0 2319 18
-1546 0 2319 1546 0 2319 4
0 -86948 0 0 86948 0 5
-2319 0 2319 2319 0 4637 6
19 20 21 7 8 9
1546 0 -2319 -1546 0 -2319 19
0 86948 0 0 -86948 0 20
-2319 0 4637 2319 0 2319 21
-1546 0 2319 1546 0 2319 7
0 -86948 0 0 86948 0 8
-2319 0 2319 2319 0 4637 9
22 23 24 10 11 12
1546 0 -2319 -1546 0 -2319 22
0 86948 0 0 -86948 0 23
-2319 0 4637 2319 0 2319 24
-1546 0 2319 1546 0 2319 10
0 -86948 0 0 86948 0 11
-2319 0 2319 2319 0 4637 12
k13 =
k14 =
k15 =
k16 =
k17 =
k18 =

19. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 19
ANÁLISIS MATRICIAL DE
PÓRTICOS
1 2 3 4 5 6
65211 0 0 -65211 0 0 1
0 652 1630 0 -652 1630 2
0 1630 5434 0 -1630 2717 3
-65211 0 0 65211 0 0 4
0 -652 -1630 0 652 -1630 5
0 1630 2717 0 -1630 5434 6
4 5 6 7 8 9
54343 0 0 -54343 0 0 4
0 377 1132 0 -377 1132 5
0 1132 4529 0 -1132 2264 6
-54343 0 0 54343 0 0 7
0 -377 -1132 0 377 -1132 8
0 1132 2264 0 -1132 4529 9
7 8 9 10 11 12
65211 0 0 -65211 0 0 7
0 652 1630 0 -652 1630 8
0 1630 5434 0 -1630 2717 9
-65211 0 0 65211 0 0 10
0 -652 -1630 0 652 -1630 11
0 1630 2717 0 -1630 5434 12
k19 =
k20 =
k21 =

20. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 20
ANÁLISIS MATRICIAL DE
PÓRTICOS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
66757 0 2319 -65211 0 0 0 0 0 0 0 0 -1546 0 2319 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
0 87600 1630 0 -652 1630 0 0 0 0 0 0 0 -86948 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2
2319 1630 10072 0 -1630 2717 0 0 0 0 0 0 -2319 0 2319 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3
-65211 0 0 121100 0 2319 -54343 0 0 0 0 0 0 0 0 -1546 0 2319 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4
0 -652 -1630 0 87978 -498 0 -377 1132 0 0 0 0 0 0 0 -86948 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5
0 1630 2717 2319 -498 14600 0 -1132 2264 0 0 0 0 0 0 -2319 0 2319 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6
0 0 0 -54343 0 0 121100 0 2319 -65211 0 0 0 0 0 0 0 0 -1546 0 2319 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7
0 0 0 0 -377 -1132 0 87978 498 0 -652 1630 0 0 0 0 0 0 0 -86948 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8
0 0 0 0 1132 2264 2319 498 14600 0 -1630 2717 0 0 0 0 0 0 -2319 0 2319 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9
0 0 0 0 0 0 -65211 0 0 66757 0 2319 0 0 0 0 0 0 0 0 0 -1546 0 2319 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10
0 0 0 0 0 0 0 -652 -1630 0 87600 -1630 0 0 0 0 0 0 0 0 0 0 -86948 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11
0 0 0 0 0 0 0 1630 2717 2319 -1630 10072 0 0 0 0 0 0 0 0 0 -2319 0 2319 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12
-1546 0 -2319 0 0 0 0 0 0 0 0 0 68303 0 0 -65211 0 0 0 0 0 0 0 0 -1546 0 2319 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13
0 -86948 0 0 0 0 0 0 0 0 0 0 0 174549 1630 0 -652 1630 0 0 0 0 0 0 0 -86948 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14
2319 0 2319 0 0 0 0 0 0 0 0 0 0 1630 14709 0 -1630 2717 0 0 0 0 0 0 -2319 0 2319 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15
0 0 0 -1546 0 -2319 0 0 0 0 0 0 -65211 0 0 122645 0 0 -54343 0 0 0 0 0 0 0 0 -1546 0 2319 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16
0 0 0 0 -86948 0 0 0 0 0 0 0 0 -652 -1630 0 174926 -498 0 -377 1132 0 0 0 0 0 0 0 -86948 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17
0 0 0 2319 0 2319 0 0 0 0 0 0 0 1630 2717 0 -498 19237 0 -1132 2264 0 0 0 0 0 0 -2319 0 2319 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18
0 0 0 0 0 0 -1546 0 -2319 0 0 0 0 0 0 -54343 0 0 122645 0 0 -65211 0 0 0 0 0 0 0 0 -1546 0 2319 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19
K = 0 0 0 0 0 0 0 -86948 0 0 0 0 0 0 0 0 -377 -1132 0 174926 498 0 -652 1630 0 0 0 0 0 0 0 -86948 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20
0 0 0 0 0 0 2319 0 2319 0 0 0 0 0 0 0 1132 2264 0 498 19237 0 -1630 2717 0 0 0 0 0 0 -2319 0 2319 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21
0 0 0 0 0 0 0 0 0 -1546 0 -2319 0 0 0 0 0 0 -65211 0 0 68303 0 0 0 0 0 0 0 0 0 0 0 -1546 0 2319 0 0 0 0 0 0 0 0 0 0 0 0 22
0 0 0 0 0 0 0 0 0 0 -86948 0 0 0 0 0 0 0 0 -652 -1630 0 174549 -1630 0 0 0 0 0 0 0 0 0 0 -86948 0 0 0 0 0 0 0 0 0 0 0 0 0 23
0 0 0 0 0 0 0 0 0 2319 0 2319 0 0 0 0 0 0 0 1630 2717 0 -1630 14709 0 0 0 0 0 0 0 0 0 -2319 0 2319 0 0 0 0 0 0 0 0 0 0 0 0 24
0 0 0 0 0 0 0 0 0 0 0 0 -1546 0 -2319 0 0 0 0 0 0 0 0 0 67409 0 -1014 -65211 0 0 0 0 0 0 0 0 -652 0 1304 0 0 0 0 0 0 0 0 0 25
0 0 0 0 0 0 0 0 0 0 0 0 0 -86948 0 0 0 0 0 0 0 0 0 0 0 152812 1630 0 -652 1630 0 0 0 0 0 0 0 -65211 0 0 0 0 0 0 0 0 0 0 26
0 0 0 0 0 0 0 0 0 0 0 0 2319 0 2319 0 0 0 0 0 0 0 0 0 -1014 1630 13549 0 -1630 2717 0 0 0 0 0 0 -1304 0 1739 0 0 0 0 0 0 0 0 0 27
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1546 0 -2319 0 0 0 0 0 0 -65211 0 0 121752 0 -1014 -54343 0 0 0 0 0 0 0 0 -652 0 1304 0 0 0 0 0 0 28
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -86948 0 0 0 0 0 0 0 0 -652 -1630 0 153189 -498 0 -377 1132 0 0 0 0 0 0 0 -65211 0 0 0 0 0 0 0 29
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2319 0 2319 0 0 0 0 0 0 0 1630 2717 -1014 -498 18078 0 -1132 2264 0 0 0 0 0 0 -1304 0 1739 0 0 0 0 0 0 30
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1546 0 -2319 0 0 0 0 0 0 -54343 0 0 121752 0 -1014 -65211 0 0 0 0 0 0 0 0 -652 0 1304 0 0 0 31
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -86948 0 0 0 0 0 0 0 0 -377 -1132 0 153189 498 0 -652 1630 0 0 0 0 0 0 0 -65211 0 0 0 0 32
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2319 0 2319 0 0 0 0 0 0 0 1132 2264 -1014 498 18078 0 -1630 2717 0 0 0 0 0 0 -1304 0 1739 0 0 0 33
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1546 0 -2319 0 0 0 0 0 0 -65211 0 0 67409 0 -1014 0 0 0 0 0 0 0 0 0 -652 0 1304 34
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -86948 0 0 0 0 0 0 0 0 -652 -1630 0 152812 -1630 0 0 0 0 0 0 0 0 0 0 -65211 0 35
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2319 0 2319 0 0 0 0 0 0 0 1630 2717 -1014 -1630 13549 0 0 0 0 0 0 0 0 0 -1304 0 1739 36
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -652 0 -1304 0 0 0 0 0 0 0 0 0 652 0 -1304 0 0 0 0 0 0 0 0 0 37
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -65211 0 0 0 0 0 0 0 0 0 0 0 65211 0 0 0 0 0 0 0 0 0 0 38
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1304 0 1739 0 0 0 0 0 0 0 0 0 -1304 0 3478 0 0 0 0 0 0 0 0 0 39
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -652 0 -1304 0 0 0 0 0 0 0 0 0 652 0 -1304 0 0 0 0 0 0 40
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -65211 0 0 0 0 0 0 0 0 0 0 0 65211 0 0 0 0 0 0 0 41
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1304 0 1739 0 0 0 0 0 0 0 0 0 -1304 0 3478 0 0 0 0 0 0 42
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -652 0 -1304 0 0 0 0 0 0 0 0 0 652 0 -1304 0 0 0 43
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -65211 0 0 0 0 0 0 0 0 0 0 0 65211 0 0 0 0 44
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1304 0 1739 0 0 0 0 0 0 0 0 0 -1304 0 3478 0 0 0 45
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -652 0 -1304 0 0 0 0 0 0 0 0 0 652 0 -1304 46
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -65211 0 0 0 0 0 0 0 0 0 0 0 65211 0 47
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1304 0 1739 0 0 0 0 0 0 0 0 0 -1304 0 3478 48

21. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 21
ANÁLISIS MATRICIAL DE
PÓRTICOS
7.. VECTOR FUERZAS DE EMPOTRAMIENTO PERFECTO DE CADA ELEMENTO
0.00 37
0.00 38
0.00 39
0.00 25
0.00 26
0.00 27
0.00 40
0.00 41
0.00 42
0.00 28
0.00 29
0.00 30
0.00 43
0.00 44
0.00 45
0.00 31
0.00 32
0.00 33
0.00 46
0.00 47
0.00 48
0.00 34
0.00 35
0.00 36
0.00 25
12.50 26
10.42 27
0.00 28
12.50 29
-10.42 30
0.00 28
15.00 29
15.00 30
0.00 31
15.00 32
-15.00 33
0.00 31
12.50 32
10.42 33
0.00 34
12.50 35
-10.42 36
F'e4 =
F'e5 =
F'e6 =
F'e3 =
F'e7 =
F'e1 =
F'e2 =

22. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 22
ANÁLISIS MATRICIAL DE
PÓRTICOS
0.00 25
0.00 26
0.00 27
0.00 13
0.00 14
0.00 15
0.00 28
0.00 29
0.00 30
0.00 16
0.00 17
0.00 18
0.00 31
0.00 32
0.00 33
0.00 19
0.00 20
0.00 21
0.00 34
0.00 35
0.00 36
0.00 22
0.00 23
0.00 24
0.00 13
12.50 14
10.42 15
0.00 16
12.50 17
-10.42 18
0.00 16
15.00 17
15.00 18
0.00 19
15.00 20
-15.00 21
0.00 19
12.50 20
10.42 21
0.00 22
12.50 23
-10.42 24
F'e12 =
F'e13 =
F'e14 =
F'e8 =
F'e9 =
F'e10 =
F'e11 =

23. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 23
ANÁLISIS MATRICIAL DE
PÓRTICOS
0.00 13
0.00 14
0.00 15
0.00 1
0.00 2
0.00 3
0.00 16
0.00 17
0.00 18
0.00 4
0.00 5
0.00 6
0.00 19
0.00 20
0.00 21
0.00 7
0.00 8
0.00 9
0.00 22
0.00 23
0.00 24
0.00 10
0.00 11
0.00 12
0.00 1
12.50 2
10.42 3
0.00 4
12.50 5
-10.42 6
0.00 4
15.00 5
15.00 6
0.00 7
15.00 8
-15.00 9
0.00 7
12.50 8
10.42 9
0.00 10
12.50 11
-10.42 12
F'e17 =
F'e18 =
F'e19 =
F'e20 =
F'e21 =
F'e15 =
F'e16 =

24. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 24
ANÁLISIS MATRICIAL DE
PÓRTICOS
8.. VECTOR FUERZAS DE EMPOTRAMIENTO PERFECTO DE LA ESTRUCTURA
0.00 37
0.00 38
0.00 39
0.00 25
0.00 26
0.00 27
0.00 40
0.00 41
0.00 42
0.00 28
0.00 29
0.00 30
0.00 43
0.00 44
0.00 45
0.00 31
0.00 32
0.00 33
0.00 46
0.00 47
0.00 48
0.00 34
0.00 35
0.00 36
0.00 25
12.50 26
10.42 27
0.00 28
12.50 29
-10.42 30
0.00 28
15.00 29
15.00 30
0.00 31
15.00 32
-15.00 33
0.00 31
12.50 32
10.42 33
0.00 34
12.50 35
-10.42 36
Fe5 =
Fe6 =
Fe7 =
Fe3 =
Fe4 =
Fe1 =
Fe2 =

25. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 25
ANÁLISIS MATRICIAL DE
PÓRTICOS
0.00 25
0.00 26
0.00 27
0.00 13
0.00 14
0.00 15
0.00 28
0.00 29
0.00 30
0.00 16
0.00 17
0.00 18
0.00 31
0.00 32
0.00 33
0.00 19
0.00 20
0.00 21
0.00 34
0.00 35
0.00 36
0.00 22
0.00 23
0.00 24
0.00 13
12.50 14
10.42 15
0.00 16
12.50 17
-10.42 18
0.00 16
15.00 17
15.00 18
0.00 19
15.00 20
-15.00 21
0.00 19
12.50 20
10.42 21
0.00 22
12.50 23
-10.42 24
Fe14 =
Fe8 =
Fe9 =
Fe10 =
Fe11 =
Fe12 =
Fe13 =

26. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 26
ANÁLISIS MATRICIAL DE
PÓRTICOS
0.00 13
0.00 14
0.00 15
0.00 1
0.00 2
0.00 3
0.00 16
0.00 17
0.00 18
0.00 4
0.00 5
0.00 6
0.00 19
0.00 20
0.00 21
0.00 7
0.00 8
0.00 9
0.00 22
0.00 23
0.00 24
0.00 10
0.00 11
0.00 12
0.00 1
12.50 2
10.42 3
0.00 4
12.50 5
-10.42 6
0.00 4
15.00 5
15.00 6
0.00 7
15.00 8
-15.00 9
0.00 7
12.50 8
10.42 9
0.00 10
12.50 11
-10.42 12
Fe17 =
Fe18 =
Fe19 =
Fe20 =
Fe21 =
Fe15 =
Fe16 =

27. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 27
ANÁLISIS MATRICIAL DE
PÓRTICOS
0.00 1
12.50 2
10.42 3
0.00 4
27.50 5
4.58 6
0.00 7
27.50 8
-4.58 9
0.00 10
12.50 11
-10.42 12
0.00 13
12.50 14
10.42 15
0.00 16
27.50 17
4.58 18
0.00 19
27.50 20
-4.58 21
0.00 22
12.50 23
-10.42 24
0.00 25
12.50 26
10.42 27
0.00 28
27.50 29
4.58 30
0.00 31
27.50 32
-4.58 33
0.00 34
12.50 35
-10.42 36
0.00 37
0.00 38
0.00 39
0.00 40
0.00 41
0.00 42
0.00 43
0.00 44
0.00 45
0.00 46
0.00 47
0.00 48
Fe =

28. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 28
ANÁLISIS MATRICIAL DE
PÓRTICOS
9.. VECTOR FUERZAS EN LOS NUDOS
0.00 1
-12.50 2
-10.42 3
0.00 4
-27.50 5
-4.58 6
0.00 7
-27.50 8
4.58 9
0.00 10
-12.50 11
10.42 12
0.00 13
-12.50 14
-10.42 15
0.00 16
-27.50 17
-4.58 18
0.00 19
-27.50 20
4.58 21
0.00 22
-12.50 23
10.42 24
0.00 25
-12.50 26
-10.42 27
0.00 28
-27.50 29
-4.58 30
0.00 31
-27.50 32
4.58 33
0.00 34
-12.50 35
10.42 36
R37 37
R38 38
R39 39
R40 40
R41 41
R42 42
R43 43
R44 44
R45 45
R46 46
R47 47
R48 48
F =

29. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 29
ANÁLISIS MATRICIAL DE
PÓRTICOS
10.. VECTOR DESPLAZAMIENTOS EN LOS NUDOS
0.000098 1
-0.000926 2
-0.001085 3
0.000043 4
-0.002294 5
-0.000278 6
-0.000043 7
-0.002294 8
0.000278 9
-0.000098 10
-0.000926 11
0.001085 12
-0.000018 13
-0.000798 14
-0.000531 15
-0.000007 16
-0.001962 17
-0.000240 18
0.000007 19
-0.001962 20
0.000240 21
0.000018 22
-0.000798 23
0.000531 24
-0.000053 25
-0.000531 26
-0.000730 27
-0.000024 28
-0.001309 29
-0.000210 30
0.000024 31
-0.001309 32
0.000210 33
0.000053 34
-0.000531 35
0.000730 36
0.000000 37
0.000000 38
0.000000 39
0.000000 40
0.000000 41
0.000000 42
0.000000 43
0.000000 44
0.000000 45
0.000000 46
0.000000 47
0.000000 48
D =

30. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 30
ANÁLISIS MATRICIAL DE
PÓRTICOS
11.. VECTOR REACCIONES EN LOS APOYOS
0.99 0.00 0.99 37
34.65 0.00 34.65 38
-1.34 0.00 -1.34 39
0.29 0.00 0.29 40
85.35 0.00 85.35 41
-0.40 0.00 -0.40 42
-0.29 + 0.00 = -0.29 43
85.35 0.00 85.35 44
0.40 0.00 0.40 45
-0.99 0.00 -0.99 46
34.65 0.00 34.65 47
1.34 0.00 1.34 48
R =

31. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 31
ANÁLISIS MATRICIAL DE
PÓRTICOS
12.. VECTOR FUERZAS INTERNAS EN LOS ELEMENTOS
0.000000 34.65 0.00 34.65 37
0.000000 -0.99 0.00 -0.99 38
0.000000 -1.34 0.00 -1.34 39
-0.000053 -34.65 0.00 -34.65 25
-0.000531 0.99 0.00 0.99 26
-0.000730 -2.61 0.00 -2.61 27
0.000000 85.35 0.00 85.35 40
0.000000 -0.29 0.00 -0.29 41
0.000000 -0.40 0.00 -0.40 42
-0.000024 -85.35 0.00 -85.35 28
-0.001309 0.29 0.00 0.29 29
-0.000210 -0.76 0.00 -0.76 30
0.000000 85.35 0.00 85.35 43
0.000000 0.29 0.00 0.29 44
0.000000 0.40 0.00 0.40 45
0.000024 -85.35 0.00 -85.35 31
-0.001309 -0.29 0.00 -0.29 32
0.000210 0.76 0.00 0.76 33
0.000000 34.65 0.00 34.65 46
0.000000 0.99 0.00 0.99 47
0.000000 1.34 0.00 1.34 48
0.000053 -34.65 0.00 -34.65 34
-0.000531 -0.99 0.00 -0.99 35
0.000730 2.61 0.00 2.61 36
-0.000053 -1.88 0.00 -1.88 25
-0.000531 -1.03 12.50 11.47 26
-0.000730 -3.27 10.42 7.15 27
-0.000024 1.88 0.00 1.88 28
-0.001309 1.03 12.50 13.53 29
-0.000210 -1.86 -10.42 -12.27 30
-0.000024 -2.61 0.00 -2.61 28
-0.001309 0.00 15.00 15.00 29
-0.000210 -0.48 15.00 14.52 30
0.000024 2.61 0.00 2.61 31
-0.001309 0.00 15.00 15.00 32
0.000210 0.48 -15.00 -14.52 33
0.000024 -1.88 0.00 -1.88 31
-0.001309 1.03 12.50 13.53 32
0.000210 1.86 10.42 12.27 33
0.000053 1.88 0.00 1.88 34
-0.000531 -1.03 12.50 11.47 35
0.000730 3.27 -10.42 -7.15 36
=
S7 = k7 x T7 x + F'7 = +
=
S6 = k6 x T6 x + F'6 = + =
S5 = k5 x T5 x + F'e5 = +
S4 = k4 x T4 x + F'e4 = + =
k3 x T3 x + F'e3 = + =
S3 =
+
=
=
S2 = k2 x T2 x + F'e2
=
=
S1 = k1 x T1 x + F'e1 +

32. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 32
ANÁLISIS MATRICIAL DE
PÓRTICOS
-0.000053 23.17 0.00 23.17 25
-0.000531 -2.87 0.00 -2.87 26
-0.000730 -4.54 0.00 -4.54 27
-0.000018 -23.17 0.00 -23.17 13
-0.000798 2.87 0.00 2.87 14
-0.000531 -4.07 0.00 -4.07 15
-0.000024 56.83 0.00 56.83 28
-0.001309 -1.02 0.00 -1.02 29
-0.000210 -1.49 0.00 -1.49 30
-0.000007 -56.83 0.00 -56.83 16
-0.001962 1.02 0.00 1.02 17
-0.000240 -1.56 0.00 -1.56 18
0.000024 56.83 0.00 56.83 31
-0.001309 1.02 0.00 1.02 32
0.000210 1.49 0.00 1.49 33
0.000007 -56.83 0.00 -56.83 19
-0.001962 -1.02 0.00 -1.02 20
0.000240 1.56 0.00 1.56 21
0.000053 23.17 0.00 23.17 34
-0.000531 2.87 0.00 2.87 35
0.000730 4.54 0.00 4.54 36
0.000018 -23.17 0.00 -23.17 22
-0.000798 -2.87 0.00 -2.87 23
0.000531 4.07 0.00 4.07 24
-0.000018 -0.70 0.00 -0.70 13
-0.000798 -0.50 12.50 12.00 14
-0.000531 -1.64 10.42 8.78 15
-0.000007 0.70 0.00 0.70 16
-0.001962 0.50 12.50 13.00 17
-0.000240 -0.84 -10.42 -11.26 18
-0.000007 -0.80 0.00 -0.80 16
-0.001962 0.00 15.00 15.00 17
-0.000240 -0.54 15.00 14.46 18
0.000007 0.80 0.00 0.80 19
-0.001962 0.00 15.00 15.00 20
0.000240 0.54 -15.00 -14.46 21
0.000007 -0.70 0.00 -0.70 19
-0.001962 0.50 12.50 13.00 20
0.000240 0.84 10.42 11.26 21
0.000018 0.70 0.00 0.70 22
-0.000798 -0.50 12.50 12.00 23
0.000531 1.64 -10.42 -8.78 24
=
S14 = k14 x T14 x + F'14 = + =
S13 = k13 x T13 x + F'13 = +
=
S12 = k12 x T12 x + F'12 = + =
S11 = k11 x T11 x + F'11 = +
=
S10 = k10 x T10 x + F'10 = + =
S9 = k9 x T9 x + F'9 = +
S8 = k8 x T8 x + F'8 = + =

33. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 33
ANÁLISIS MATRICIAL DE
PÓRTICOS
-0.000018 11.17 0.00 11.17 13
-0.000798 -3.57 0.00 -3.57 14
-0.000531 -4.71 0.00 -4.71 15
0.000098 -11.17 0.00 -11.17 1
-0.000926 3.57 0.00 3.57 2
-0.001085 -5.99 0.00 -5.99 3
-0.000007 28.83 0.00 28.83 16
-0.001962 -1.12 0.00 -1.12 17
-0.000240 -1.64 0.00 -1.64 18
0.000043 -28.83 0.00 -28.83 4
-0.002294 1.12 0.00 1.12 5
-0.000278 -1.73 0.00 -1.73 6
0.000007 28.83 0.00 28.83 19
-0.001962 1.12 0.00 1.12 20
0.000240 1.64 0.00 1.64 21
-0.000043 -28.83 0.00 -28.83 7
-0.002294 -1.12 0.00 -1.12 8
0.000278 1.73 0.00 1.73 9
0.000018 11.17 0.00 11.17 22
-0.000798 3.57 0.00 3.57 23
0.000531 4.71 0.00 4.71 24
-0.000098 -11.17 0.00 -11.17 10
-0.000926 -3.57 0.00 -3.57 11
0.001085 5.99 0.00 5.99 12
0.000098 3.57 0.00 3.57 1
-0.000926 -1.33 12.50 11.17 2
-0.001085 -4.42 10.42 5.99 3
0.000043 -3.57 0.00 -3.57 4
-0.002294 1.33 12.50 13.83 5
-0.000278 -2.23 -10.42 -12.65 6
0.000043 4.69 0.00 4.69 4
-0.002294 0.00 15.00 15.00 5
-0.000278 -0.63 15.00 14.37 6
-0.000043 -4.69 0.00 -4.69 7
-0.002294 0.00 15.00 15.00 8
0.000278 0.63 -15.00 -14.37 9
-0.000043 3.57 0.00 3.57 7
-0.002294 1.33 12.50 13.83 8
0.000278 2.23 10.42 12.65 9
-0.000098 -3.57 0.00 -3.57 10
-0.000926 -1.33 12.50 11.17 11
0.001085 4.42 -10.42 -5.99 12
=
S21 = k21 x T21 x + F'21 = +
=
S20 = k20 x T20 x + F'20 = + =
S19 = k19 x T19 x + F'19 = +
=
S18 = k18 x T18 x + F'18 = + =
S17 = k17 x T17 x + F'17 = +
=
S16 = k16 x T16 x + F'16 = + =
S15 = k15 x T15 x + F'15 = +

34. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 34
ANÁLISIS MATRICIAL DE
PÓRTICOS
13.. DIAGRAMA DE FUERZAS AXIALES
14.. DIAGRAMA DE FUERZAS CORTANTES

35. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 35
ANÁLISIS MATRICIAL DE
PÓRTICOS
15.. DIAGRAMA DE MOMENTOS FLECTORES

36. ING. ERLY MARVIN ENRIQUEZ QUISPE Pág. 36
ANÁLISIS MATRICIAL DE
PÓRTICOS
7. CONCLUSIONES
- El método de la matriz rigidez proporciona un sistema apropiado de análisis de
estructuras y determina una base muy conveniente para el desarrollo de programas
de computación.
- El método de la matriz rigidez nos permite obtener las deformaciones de una
estructura para poder comparar con las deformaciones permisibles que se indican
en las normas.
- El método de la matriz rigidez nos permite obtener las fuerzas internas de los
elementos de una estructura para un posterior diseño.
- Se debe ser consciente que sin un modelo adecuado o sin una interpretación final,
el refinamiento en el análisis carece de sentido.
8. BIBLIOGRAFÍA
- Jairo Uribe Escamilla. Análisis de estructuras.
- R. C. Hibbeler. Análisis Estructural.
- Erly Marvin Enriquez Quispe. Análisis Matricial de Estructuras.