PARCIAL 1-3 “Parrillas” – ANÁLISIS MATRICIAL AVANZADO
Nicolás Ortiz Álvarez – 1094580633
Nota: Las unidades de los resultados mostrados a continuación están expresados en las siguientes
unidades:
-
Distancia: metros (m)
Fuerza: kilo Newtons (kN)
Momento: kilo Newtons × metro (kN.m)
Desarrollo
a)
Debido a que se tiene un ángulo α de 35°, se determinan entonces las distancias “a” y “b” de la
siguiente manera:
- a:
tan(35 − 15) =
𝑎
→ 𝑎 = 10 × tan(20) → 𝑎 = 3.640 𝑚
10
Pág. 1/12
-
b:
tan(35) =
1.1 Nodos
- Número de nodos
=
𝑏
→ 𝑏 = 10 × tan(35) → 𝑏 = 7.000 𝑚
10
4
Nodo
Coord X
Coord Y
Rest Giro X
Rest Giro Y
Rest Desp. Z
Carga Mx Global
1
0
0
1
1
1
0
Carga My
Global
Z Global
Asent X
Asent Y
Giro
2
8
0
0
0
0
0
3
18
-3.63970234
1
0
1
0
4
18
7.00
1
0
1
0
0
0
0
0
0
0
0
0
-140
0
0
0
0
0
0
0
0
0
0
0
Nota: Para el sistema de coordenadas, se toma como coordenada x = 0m, y = 0m el nodo 1.
1.2 Secciones
- Número de secciones
=
Sección
b
h
Área
Inercia
3
1
0.70
0.60
4.20E-01
1.26E-02
2
0.50
0.60
3.00E-01
9.00E-03
3
0.80
0.60
4.80E-01
1.44E-02
2
2
3
2
1
0.5
0.6
-20.0
0.2
8.33E+06
0.01240
3
2
4
3
1
0.8
0.6
35.0
0.2
8.33E+06
0.03110
1.3 Materiales
- Número de materiales =
Material
Módulo de elasticidad, E
1
1
2.00E+07
1.4 Elementos
- Número de elementos =
Elemento
Nudo i
Nudo j
Sec
Mat
b
h
α
v
G
J
3
1
1
2
1
1
0.7
0.6
0.0
0.2
8.33E+06
0.02441
2 ANÁLISIS
Grados de libertad (GL) = 12
Número de restricciones (Nrest) = 7
Grado de Indeterminación cinemática (GIC) = 5
Pág. 2/12
2.1 Grados de Libertad de los Nodos
θx
θy
z
Nodo 1
6
7
8
Nodo 2
1
2
3
Nodo 3
9
4
10
Nodo 4
11
5
12
- Fuerzas fijas en los elementos
Elemento
1
2
3
L
8.000
10.642
12.208
w
18.000
12.000
15.000
a
4.000
P
70.000
Mfy ij
-166.00
-113.25
-186.29
Mfy ji
166.00
113.25
186.29
zf ij
107.00
63.85
91.56
zf ji
107.00
63.85
91.56
2.2 Matriz de Fuerzas Fijas
GL
[FF]
1
4
68.12
-93.01
262.41
106.42
5
152.60
6
0.00
-166.00
107.00
38.73
63.85
-106.85
91.56
2
3
7
8
9
10
11
12
[FFN]
[FFA]
Esta matriz de fuerzas fijas se obtuvo aplicando la matriz de transformación (en la sección 2.4 se
muestran dichas matrices) a cada una de las matrices de fuerzas fijas de los elementos en
coordenadas globales, se la siguiente manera:
2.2.1 Vector de Fuerzas Fijas por elemento (Coordenadas locales)
Nodo i
Nodo j
Elemento 1
1
2
GL
Mfx ij
Mfy ij
zf ij
Mfx ji
Mfy ji
zf ji
6
7
8
1
2
3
Nodo i
Nodo j
Elemento 2
2
3
GL
0.00
-166.00
107.00
0.00
166.00
107.00
1
2
3
9
4
10
Pág. 3/12
Nodo i
Nodo j
Elemento 3
2
4
GL
0.00
-113.25
63.85
0.00
113.25
63.85
1
2
3
11
5
12
0.00
-186.29
91.56
0.00
186.29
91.56
2.2.2 Vector de Fuerzas Fijas por elemento (Coordenadas Globales)
Elemento 1
Elemento 1
Nodo i
Nodo j
Elemento 2
Elemento 2
Nodo i
Nodo j
1
2
GL
Mfx ij
Mfy ij
zf ij
Mfx ji
Mfy ji
zf ji
2
3
Elemento 3
Elemento 3
Nodo i
Nodo j
GL
0.00
-166.00
107.00
0.00
166.00
107.00
6
7
8
1
2
3
2
4
GL
-38.73
-106.42
63.85
38.73
106.42
63.85
1
2
3
9
4
10
1
2
3
11
5
12
106.85
-152.60
91.56
-106.85
152.60
91.56
2.3 Matriz de Fuerzas Externas y Desplazamientos/Giros (Inicial)
GL
[F]
[U]
1
4
0
0
-140
0
0
0
0
0
5
0
0
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
3
7
8
9
10
11
12
[FN]
[FA]
[UN]
[UA]
Para la futura verificación de reacciones en los apoyos, se hallan las fuerzas totales aplicadas a la
parrilla:
Total de fuerzas fijas =
Total de fuerzas externas =
524.82
140.00
664.82
2.4 Matrices de rigidez por elemento
Elemento 1
Propiedades del elemento
J = 2.44E-02
cos(αij) = 1.0
I = 1.26E-02
E = 2.0E+07
G = 8.33E+06
12EI/L³
5.91E+03
4EI/L
1.26E+05
2EI/L
6.30E+04
Nodo i = 1
xi = 0.00
yi = 0.00
α = 0.0°
Nodo j = 2
xj = 8.00
yj = 0.00
sen(αij) = 0.0
Long. = 8.0
Dist. x = 8.00
Dist. y = 0.00
GJ/L
2.54E+04
6EI/L²
2.36E+04
Pág. 4/12
K Local Elemento 1
ul i
ul i
vl i
θl i
ul j
vl j
θl j
vl i
θl i
ul j
vl j
θl j
2.54E+04
0.00E+00
0.00E+00
-2.54E+04
0.00E+00
0.00E+00
0.00E+00
1.26E+05
-2.36E+04
0.00E+00
6.30E+04
2.36E+04
6
7
8
1
2
3
1.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.00E+00
-2.36E+04
5.91E+03
0.00E+00
-2.36E+04
-5.91E+03
-2.54E+04
0.00E+00
0.00E+00
2.54E+04
0.00E+00
0.00E+00
0.00E+00
6.30E+04
-2.36E+04
0.00E+00
1.26E+05
2.36E+04
0.00E+00
2.36E+04
-5.91E+03
0.00E+00
2.36E+04
5.91E+03
Transformada Elemento 1
K Global Elemento 1
GL
6
7
8
1
2
3
6
25425.2
0.0
0.0
-25425.2
0.0
0.0
0.0
126000.0
-23625.0
0.0
63000.0
23625.0
0.0
-23625.0
5906.3
0.0
-23625.0
-5906.3
-25425.2
0.0
0.0
25425.2
0.0
0.0
0.0
63000.0
-23625.0
0.0
126000.0
23625.0
0.0
23625.0
-5906.3
0.0
23625.0
5906.3
7
8
1
2
3
Elemento 2
Propiedades del elemento
J = 1.24E-02
cos(αij) = 0.9
I = 9.00E-03
E = 2.0E+07
G = 8.33E+06
12EI/L³
1.79E+03
4EI/L
6.77E+04
2EI/L
3.38E+04
Nodo i = 2
xi = 8.00
yi = 0.00
-α = 20.0°
Nodo j = 3
xj = 18.00
-yj = 3.64
-sen(αij) = 0.3
Long. = 10.6
Dist. x = 10.00
-Dist. y = 3.64
GJ/L
9.71E+03
6EI/L²
9.54E+03
K Local Elemento 2
ul i
ul i
vl i
θl i
ul j
vl j
θl j
9.71E+03
0.00E+00
0.00E+00
-9.71E+03
0.00E+00
0.00E+00
vl i
0.00E+00
6.77E+04
-9.54E+03
0.00E+00
3.38E+04
9.54E+03
θl i
0.00E+00
-9.54E+03
1.79E+03
0.00E+00
-9.54E+03
-1.79E+03
Pág. 5/12
ul j
-9.71E+03
0.00E+00
0.00E+00
9.71E+03
0.00E+00
0.00E+00
vl j
0.00E+00
3.38E+04
-9.54E+03
0.00E+00
6.77E+04
9.54E+03
θl j
0.00E+00
9.54E+03
-1.79E+03
0.00E+00
9.54E+03
1.79E+03
Transformada Elemento 2
1
2
3
9
4
10
0.940
-0.342
0.000
0.000
0.000
0.000
0.342
0.940
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
0.940
-0.342
0.000
0.000
0.000
0.000
0.342
0.940
0.000
0.000
0.000
0.000
0.000
0.000
1.000
K Global Elemento 2
GL
1
2
3
9
4
10
1
16490.5
18623.4
-3261.7
-4618.8
13993.8
3261.7
18623.4
60879.5
-8961.5
13993.8
28735.6
8961.5
-3261.7
-8961.5
1792.3
-3261.7
-8961.5
-1792.3
-4618.8
13993.8
-3261.7
16490.5
18623.4
3261.7
13993.8
28735.6
-8961.5
18623.4
60879.5
8961.5
3261.7
8961.5
-1792.3
3261.7
8961.5
1792.3
2
3
9
4
10
Elemento 3
Propiedades del elemento
J = 3.11E-02
cos(αij) = 0.8
I = 1.44E-02
E = 2.0E+07
G = 8.33E+06
12EI/L³
1.90E+03
4EI/L
9.44E+04
2EI/L
4.72E+04
Nodo i = 2
xi = 8.00
yi = 0.00
α = 35.0°
Nodo j = 4
xj = 18.00
yj = 7.00
sen(αij) = 0.6
Long. = 12.2
Dist. x = 10.00
Dist. y = 7.00
GJ/L
2.12E+04
6EI/L²
1.16E+04
K Local Elemento 3
ul i
ul i
vl i
θl i
ul j
vl j
θl j
vl i
θl i
ul j
vl j
θl j
2.12E+04
0.00E+00
0.00E+00
-2.12E+04
0.00E+00
0.00E+00
0.00E+00
9.44E+04
-1.16E+04
0.00E+00
4.72E+04
1.16E+04
1
2
3
11
5
12
0.819
0.574
0.000
0.000
0.000
0.000
-0.574
0.819
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
0.819
0.574
0.000
0.000
0.000
0.000
-0.574
0.819
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.00E+00
-1.16E+04
1.90E+03
0.00E+00
-1.16E+04
-1.90E+03
-2.12E+04
0.00E+00
0.00E+00
2.12E+04
0.00E+00
0.00E+00
0.00E+00
4.72E+04
-1.16E+04
0.00E+00
9.44E+04
1.16E+04
0.00E+00
1.16E+04
-1.90E+03
0.00E+00
1.16E+04
1.90E+03
Transformada Elemento 3
Pág. 6/12
K Global Elemento 3
GL
1
2
3
11
5
12
1
45291.6
-34362.5
6650.6
1276.7
-32144.0
-6650.6
-34362.5
70305.5
-9498.1
-32144.0
24675.7
9498.1
6650.6
-9498.1
1899.6
6650.6
-9498.1
-1899.6
1276.7
-32144.0
6650.6
45291.6
-34362.5
-6650.6
-32144.0
24675.7
-9498.1
-34362.5
70305.5
9498.1
-6650.6
9498.1
-1899.6
-6650.6
9498.1
1899.6
2
3
11
5
12
2.5 Matriz de Rigidez
KNN
KNA
KAN
KAA
3
4
2.5.1. Matriz de rigidez ensamblada
GL
1
2
1
87207.315
-15739.04194
3388.926797
13993.81446
-32144.0405
-25425.21866
0
0
-4618.754901
3261.722977
1276.710383
-6650.649775
-15739.04194
257184.9638
5165.377552
28735.60242
24675.65828
0
63000
-23625
13993.81446
8961.510228
-32144.0405
9498.112219
2
3
4
5
6
7
8
9
10
11
12
5
6
3388.926797 13993.81446 -32144.0405 -25425.21866
5165.377552 28735.60242 24675.65828
0
9598.17449 -8961.510228 -9498.112219
0
-8961.510228 60879.49986
0
0
-9498.112219
0
70305.46391
0
0
0
0
25425.21866
23625
0
0
0
-5906.25
0
0
0
-3261.722977 18623.41531
0
0
-1792.302046 8961.510228
0
0
6650.649775
0
-34362.45725
0
-1899.622444
0
9498.112219
0
7
8
9
0
63000
23625
0
0
0
126000
-23625
0
0
0
0
0
-23625
-5906.25
0
0
0
-23625
5906.25
0
0
0
0
-4618.754901
13993.81446
-3261.722977
18623.41531
0
0
0
0
16490.45566
3261.722977
0
0
10
11
12
3261.722977 1276.710383 -6650.649775
8961.510228 -32144.0405 9498.112219
-1792.302046 6650.649775 -1899.622444
8961.510228
0
0
0
-34362.45725 9498.112219
0
0
0
0
0
0
0
0
0
3261.722977
0
0
1792.302046
0
0
0
45291.64068 -6650.649775
0
-6650.649775 1899.622444
2.5.2. Matriz de rotación
GL
1
2
3
4
5
6
7
8
9
10
11
12
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0.940
0
0
0
0
0.342
0
0
0
0
0
0
0
0.819
0
0
0
0
0
-0.574
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
-0.342
0
0
0
0
0.940
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0.574
0
0
0
0
0
0.819
0
0
0
0
0
0
0
0
0
0
0
0
1
10
11
2
3
4
5
6
7
8
9
10
11
12
2.5.3. Matriz de rigidez global rotada
GL
1
2
1
87207.315
-15739.04194
3388.926797
11570.17697
-27063.14748
-25425.21866
0
0
-9126.376326
3261.722977
-17391.24428
-6650.649775
-15739.04194
257184.9638
5165.377552
31788.79998
38650.18013
0
63000
-23625
3321.729329
8961.510228
-12177.48034
9498.112219
2
3
4
5
6
7
8
9
10
11
12
3
4
2.5.4. Rotación de Fuerzas fijas y Fuerzas externas
Matriz de fuerzas fijas rotada
GL
[FF]
1
2
3
4
5
6
7
8
9
10
11
12
5
6
3388.926797 11570.17697 -27063.14748 -25425.21866
5165.377552 31788.79998 38650.18013
0
9598.17449 -9536.639993 -11595.05404
0
-9536.639993 67657.8687
0
0
-11595.05404
0
94366.3155
0
0
0
0
25425.21866
23625
0
0
0
-5906.25
0
0
0
-4.54747E-13 -3.63798E-12
0
0
-1792.302046 9536.639993
0
0
0
0
0
0
-1899.622444
0
11595.05404
0
68.12
-93.01
262.41
113.25
186.29
0.00
-166.00
107.00
0.00
63.85
0.00
91.56
7
8
9
0
63000
23625
0
0
0
126000
-23625
0
0
0
0
0
-23625
-5906.25
0
0
0
-23625
5906.25
0
0
0
0
-9126.376326
3321.729329
-4.54747E-13
-2.72848E-12
0
0
0
0
9712.086829
4.54747E-13
0
0
Matriz de fuerzas externas rotada
GL
[F]
1
2
3
[FFN]
4
5
6
7
8
[FFA]
9
10
11
12
Pág. 7/12
0
0
-140
0
0
0
0
0
0
0
0
0
[FN]
[FA]
12
3261.722977 -17391.24428 -6650.649775
8961.510228 -12177.48034 9498.112219
-1792.302046
0
-1899.622444
9536.639993
0
0
0
0
11595.05404
0
0
0
0
0
0
0
0
0
4.54747E-13
0
0
1792.302046
0
0
0
21230.78908
0
0
0
1899.622444
2.6 Determinación de desplazamientos desconocidos
[KNN]-1( [FN] - [FFN] - [KNA] [UA] )
[UN] =
GL
[U]
1
0.00115
0.00596
-0.07746
-0.01559
-0.01360
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
2
3
4
5
6
7
8
9
10
11
12
Giro en coordenadas
globales
0.0040
Nodo 4
0.00000
-0.01559
0.0000
Coordenadas globales
-0.00533
4x
-0.01465
4y
[UN]
[UA]
Nodo 5
0.00000
-0.01360
0.0000
0.00780
-0.01114
5x
5y
La transformación se hace multiplicando los giros obtenidos por la matriz de transformación, para
el Nodo 4, por la matriz del Elemento 2, y para el Nodo 5, por la matriz del Elemento 3, ya que a su
giro correspondiente le corresponde su matriz de transformación respectiva del elemento. A
continuación, se muestras dichas sub-matrices:
-
Submatriz (resaltada en verde) de transformación para transformar los giros del Nodo 4:
Transformada Elemento 2
-
1
2
3
9
4
10
0.940
-0.342
0.000
0.000
0.000
0.000
0.342
0.940
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
0.940
-0.342
0.000
0.000
0.000
0.000
0.342
0.940
0.000
0.000
0.000
0.000
0.000
0.000
1.000
Submatriz (resaltada en verde) de transformación para transformar los giros del Nodo 5:
Transformada Elemento 3
1
2
3
11
5
12
0.819
0.574
0.000
0.000
0.000
0.000
-0.574
0.819
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
0.819
0.574
0.000
0.000
0.000
0.000
-0.574
0.819
0.000
0.000
0.000
0.000
0.000
0.000
1.000
Pág. 8/12
2.7 Determinación de Fuerzas desconocidas
[FA] =
[FFA] - [FA] + [KAN] [UN] + [KAA] [UA]
GL
[FA]
6
-29.278
-1620.581
423.720
48.017
111.174
-199.440
129.924
7
8
9
10
11
12
2.8. Determinación de las fuerzas internas de los elementos
Elemento 1
[Fe]
Mfx ij
Mfy ij
zf ij
Mfx ji
Mfy ji
zf ji
-29.28
-1620.58
423.72
29.28
-913.18
-209.72
=
[Fef]
0.000
-166.000
107.000
0.000
166.000
107.000
2.54E+04
0.00E+00
0.00E+00
-2.54E+04
0.00E+00
0.00E+00
+
0.00E+00
1.26E+05
-2.36E+04
0.00E+00
6.30E+04
2.36E+04
K Global Elemento 1
0.00E+00 -2.54E+04
-2.36E+04 0.00E+00
5.91E+03 0.00E+00
0.00E+00 2.54E+04
-2.36E+04 0.00E+00
-5.91E+03 0.00E+00
0.00E+00
6.30E+04
-2.36E+04
0.00E+00
1.26E+05
2.36E+04
0.00E+00
2.36E+04
-5.91E+03
0.00E+00
2.36E+04
5.91E+03
1.40E+04
2.87E+04
-8.96E+03
1.86E+04
6.09E+04
8.96E+03
3.26E+03
8.96E+03
-1.79E+03
3.26E+03
8.96E+03
1.79E+03
x
[Ue]
0.00000
0.00000
0.00000
0.00115
0.00596
-0.07746
x
[Ue]
0.00115
0.00596
-0.07746
-0.00533
-0.01465
0.00000
[FLe]
GL
Mfx ij
Mfy ij
zf ij
Mfx ji
Mfy ji
zf ji
6
7
8
1
2
3
-29.28
-1620.58
423.72
29.28
-913.18
-209.72
Elemento 2
[Fe]
Mfx ij
Mfy ij
zf ij
Mfx ji
Mfy ji
zf ji
163.52
476.41
16.53
8.72
-3.18
111.17
=
[Fef]
-38.733
-106.418
63.851
38.733
106.418
63.851
1.65E+04
1.86E+04
-3.26E+03
-4.62E+03
1.40E+04
3.26E+03
+
1.86E+04
6.09E+04
-8.96E+03
1.40E+04
2.87E+04
8.96E+03
K Global Elemento 2
-3.26E+03 -4.62E+03
-8.96E+03 1.40E+04
1.79E+03 -3.26E+03
-3.26E+03 1.65E+04
-8.96E+03 1.86E+04
-1.79E+03 3.26E+03
[FLe]
GL
1
2
3
9
4
10
Mfx ij
Mfy ij
zf ij
Mfx ji
Mfy ji
zf ji
Pág. 9/12
-9.28
503.60
16.53
9.28
0.00
111.17
Elemento 3
[Fe]
Mfx ij
Mfy ij
zf ij
Mfx ji
Mfy ji
zf ji
-192.80
436.77
53.19
-75.85
-53.11
129.92
=
[Fef]
106.849
-152.597
91.558
-106.849
152.597
91.558
4.53E+04
-3.44E+04
6.65E+03
1.28E+03
-3.21E+04
-6.65E+03
+
-3.44E+04
7.03E+04
-9.50E+03
-3.21E+04
2.47E+04
9.50E+03
K Global Elemento 3
6.65E+03 1.28E+03
-9.50E+03 -3.21E+04
1.90E+03 6.65E+03
6.65E+03 4.53E+04
-9.50E+03 -3.44E+04
-1.90E+03 -6.65E+03
2
3
11
5
12
-6.65E+03
9.50E+03
-1.90E+03
-6.65E+03
9.50E+03
1.90E+03
x
[Ue]
0.00115
0.00596
-0.07746
0.00780
-0.01114
0.00000
[FLe]
GL
1
-3.21E+04
2.47E+04
-9.50E+03
-3.44E+04
7.03E+04
9.50E+03
Mfx ij
Mfy ij
zf ij
Mfx ji
Mfy ji
zf ji
92.59
468.36
53.19
-92.59
0.00
129.92
Los valores resaltados en Amarillo corresponden a las reacciones en los apoyos de la parrilla, al
sumarlos vemos que nos da un valor de 664.82 kN, los correspondientes a la totalidad de fuerzas
aplicadas mostradas en la sección 2.3. Es decir, la condición de equilibrio CUMPLE.
b) Determinación de los diagramas de Fuerza cortante, Momento Flector y Momento Torsor
-
Fuerza cortante:
o
Elemento 1
(+)
o
Elemento 2
(+)
(-)
Pág. 10/12
o
Elemento 3
(+)
(-)
-
Momento Flector:
o
Elemento 1
o
Elemento 2
El momento máximo se presenta en donde el diagrama de cortante intersecta con el eje del
elemento.
Pág. 11/12
o
Elemento 3
El momento máximo se presenta en donde el diagrama de cortante intersecta con el eje del
elemento.
-
Momento Torsor:
o
Elemento 1
(+)
o
Elemento 2
(+)
o
Elemento 3
(-)
Pág. 12/12
ANEXO - Cálculos en Excel
1 ENTRADA DE DATOS
1.1 Nodos
- Número de nodos
=
4
Nodo
Coord X
Coord Y
Rest Giro X
Rest Giro Y
Rest Desp. Z
Carga Mx Global
1
0
0
1
1
1
0
2
8
0
0
0
0
0
3
18
-3.63970234
1
0
1
0
4
18
7.00
1
0
1
0
Carga My Global
0
0
0
0
Z Global
0
0
0
0
-140
0
0
0
0
0
0
0
0
0
0
0
Asent X
Asent Y
Giro
1.2 Secciones
- Número de secciones
=
3
1
0.70
0.60
4.20E-01
1.26E-02
Sección
b
h
Área
Inercia
2
0.50
0.60
3.00E-01
9.00E-03
3
0.80
0.60
4.80E-01
1.44E-02
2
2
3
2
1
0.5
0.6
-20.0
0.2
8.33E+06
0.01240
3
2
4
3
1
0.8
0.6
35.0
0.2
8.33E+06
0.03110
1.3 Materiales
- Número de materiales
=
1
1
Material
2.00E+07
Módulo de elasticidad, E
1.4 Elementos
- Número de elementos
Elemento
Nudo i
Nudo j
Sec
Mat
b
h
α
v
G
J
=
3
1
1
2
1
1
0.7
0.6
0.0
0.2
8.33E+06
0.02441
2 ANÁLISIS
Grados de libertad (GL) = 12
Número de restricciones (Nrest) = 7
Grado de Indeterminación cinemática (GIC) = 5
2.1 Grados de Libertad de los Nodos
θx
θy
z
Nodo 1
6
7
8
Nodo 2
1
2
3
Nodo 3
9
4
10
2.2 Matriz de Fuerzas Fijas
- Fuerzas fijas en los elementos
[FF]
1
4
68.12
-93.01
262.41
106.42
5
152.60
6
0.00
-166.00
107.00
38.73
63.85
-106.85
91.56
3
7
8
9
10
11
12
2.3 Matriz de Fuerzas Externas y Desplazamientos/Giros (Inicial)
¡ NO OLVIDAR SUMAR LAS FUERZAS DE TODOS LOS ELEMENTOS !
GL
2
Nodo 4
11
5
12
Elemento
1
2
3
L
8.000
10.642
12.208
w
18.000
12.000
15.000
GL
[F]
1
4
0
0
-140
0
524.82
5
0
0
140.00
664.82
6
0
0
0
0
0
0
0
[FA]
0
0
0
0
0
0
0
x
[Ue]
0.00000
0.00000
0.00000
0.00115
0.00596
-0.07746
x
[Ue]
0.00115
0.00596
-0.07746
-0.00533
-0.01465
0.00000
x
[Ue]
0.00115
0.00596
-0.07746
0.00780
-0.01114
0.00000
a
4.000
P
70.000
Mfy ij
-166.00
-113.25
-186.29
Mfy ji
166.00
113.25
186.29
zf ij
107.00
63.85
91.56
zf ji
107.00
63.85
91.56
2
3
[FFN]
Total de fuerzas fijas =
Total de fuerzas externas =
7
Para verificación de racciones
8
[FFA]
9
10
11
12
[U]
[FN]
0
0
0
0
2.2.1 Vector de Fuerzas Fijas por elemento (Coordenadas locales)
Elemento 1
1
2
Nodo i
Nodo j
Nodo i
Nodo j
GL
Elemento 2
2
3
Nodo i
Nodo j
GL
0.00
-166.00
107.00
0.00
166.00
107.00
6
Mfx ij
Mfy ij
zf ij
Mfx ji
Mfy ji
zf ji
7
8
1
2
3
1
2
3
9
4
10
Elemento 3
2
4
GL
0.00
-113.25
63.85
0.00
113.25
63.85
1
2
3
11
5
12
0.00
-186.29
91.56
0.00
186.29
91.56
2.2.2 Vector de Fuerzas Fijas por elemento (Coordenadas Globales)
Elemento 1
Elemento 1
Nodo i
Nodo j
1
2
Elemento 2
Elemento 2
Nodo i
Nodo j
GL
0.00
-166.00
107.00
0.00
166.00
107.00
6
Mfx ij
Mfy ij
zf ij
Mfx ji
Mfy ji
zf ji
Elemento 3
Elemento 3
Nodo i
Nodo j
2
3
GL
7
8
1
2
3
1
2
3
9
4
10
2
4
GL
-38.73
-106.42
63.85
38.73
106.42
63.85
1
2
3
11
5
12
106.85
-152.60
91.56
-106.85
152.60
91.56
2.4 Matrices de rigidez por elemento
Elemento 1
Propiedades del elemento
Nodo i = 1
xi = 0.00
yi = 0.00
α = 0.0°
Nodo j = 2
xj = 8.00
yj = 0.00
sen(αij) = 0.0
Long. = 8.0
Dist. x = 8.00
Dist. y = 0.00
cos(αij) = 1.0
K Global Elemento 1
I = 1.26E-02
E = 2.0E+07
G = 8.33E+06
J = 2.44E-02
GL
6
7
8
1
2
3
6
25425.2
0.0
0.0
-25425.2
0.0
0.0
0.0
126000.0
-23625.0
0.0
63000.0
23625.0
0.0
-23625.0
5906.3
0.0
-23625.0
-5906.3
-25425.2
0.0
0.0
25425.2
0.0
0.0
0.0
63000.0
-23625.0
0.0
126000.0
23625.0
0.0
23625.0
-5906.3
0.0
23625.0
5906.3
7
8
1
2
GJ/L
2.54E+04
6EI/L²
12EI/L³
4EI/L
2EI/L
2.36E+04
5.91E+03
1.26E+05
6.30E+04
3
K Local Elemento 1
ul i
vl i
2.54E+04
0.00E+00
0.00E+00
-2.54E+04
0.00E+00
0.00E+00
ul i
vl i
θl i
ul j
vl j
θl j
0.00E+00
1.26E+05
-2.36E+04
0.00E+00
6.30E+04
2.36E+04
Elemento 2
θl i
0.00E+00
-2.36E+04
5.91E+03
0.00E+00
-2.36E+04
-5.91E+03
Transformada Elemento 1
ul j
vl j
-2.54E+04
0.00E+00
0.00E+00
2.54E+04
0.00E+00
0.00E+00
0.00E+00
6.30E+04
-2.36E+04
0.00E+00
1.26E+05
2.36E+04
θl j
0.00E+00
2.36E+04
-5.91E+03
0.00E+00
2.36E+04
5.91E+03
6
7
8
1
2
3
1.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
Propiedades del elemento
yi = 0.00
-α = 20.0°
Nodo j = 3
xj = 18.00
-yj = 3.64
-sen(αij) = 0.3
Long. = 10.6
Nodo i = 2
Dist. x = 10.00
xi = 8.00
-Dist. y = 3.64
cos(αij) = 0.9
K Global Elemento 2
I = 9.00E-03
E = 2.0E+07
G = 8.33E+06
J = 1.24E-02
GL
1
2
3
9
4
10
1
16490.5
18623.4
-3261.7
-4618.8
13993.8
3261.7
18623.4
60879.5
-8961.5
13993.8
28735.6
8961.5
-3261.7
-8961.5
1792.3
-3261.7
-8961.5
-1792.3
-4618.8
13993.8
-3261.7
16490.5
18623.4
3261.7
13993.8
28735.6
-8961.5
18623.4
60879.5
8961.5
3261.7
8961.5
-1792.3
3261.7
8961.5
1792.3
2
3
9
4
GJ/L
9.71E+03
6EI/L²
12EI/L³
4EI/L
2EI/L
9.54E+03
1.79E+03
6.77E+04
3.38E+04
10
K Local Elemento 2
ul i
vl i
9.71E+03
0.00E+00
0.00E+00
-9.71E+03
0.00E+00
0.00E+00
ul i
vl i
θl i
ul j
vl j
θl j
0.00E+00
6.77E+04
-9.54E+03
0.00E+00
3.38E+04
9.54E+03
Elemento 3
θl i
0.00E+00
-9.54E+03
1.79E+03
0.00E+00
-9.54E+03
-1.79E+03
Transformada Elemento 2
ul j
vl j
-9.71E+03
0.00E+00
0.00E+00
9.71E+03
0.00E+00
0.00E+00
0.00E+00
3.38E+04
-9.54E+03
0.00E+00
6.77E+04
9.54E+03
θl j
0.00E+00
9.54E+03
-1.79E+03
0.00E+00
9.54E+03
1.79E+03
1
2
3
9
4
10
0.940
-0.342
0.000
0.000
0.000
0.000
0.342
0.940
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
0.940
-0.342
0.000
0.000
0.000
0.000
0.342
0.940
0.000
0.000
0.000
0.000
0.000
0.000
1.000
Propiedades del elemento
yi = 0.00
α = 35.0°
Nodo j = 4
xj = 18.00
yj = 7.00
sen(αij) = 0.6
Long. = 12.2
Nodo i = 2
Dist. x = 10.00
xi = 8.00
Dist. y = 7.00
cos(αij) = 0.8
K Global Elemento 3
I = 1.44E-02
E = 2.0E+07
G = 8.33E+06
J = 3.11E-02
GL
1
2
3
11
5
12
1
45291.6
-34362.5
6650.6
1276.7
-32144.0
-6650.6
-34362.5
70305.5
-9498.1
-32144.0
24675.7
9498.1
6650.6
-9498.1
1899.6
6650.6
-9498.1
-1899.6
1276.7
-32144.0
6650.6
45291.6
-34362.5
-6650.6
-32144.0
24675.7
-9498.1
-34362.5
70305.5
9498.1
-6650.6
9498.1
-1899.6
-6650.6
9498.1
1899.6
2
3
11
5
GJ/L
2.12E+04
6EI/L²
12EI/L³
4EI/L
2EI/L
1.16E+04
1.90E+03
9.44E+04
4.72E+04
12
K Local Elemento 3
ul i
vl i
2.12E+04
0.00E+00
0.00E+00
-2.12E+04
0.00E+00
0.00E+00
ul i
vl i
θl i
ul j
vl j
θl j
0.00E+00
9.44E+04
-1.16E+04
0.00E+00
4.72E+04
1.16E+04
θl i
0.00E+00
-1.16E+04
1.90E+03
0.00E+00
-1.16E+04
-1.90E+03
2.5 Matriz de Rigidez
KNN
KAN
2.5.1. Matriz de rigidez ensamblada
Transformada Elemento 3
ul j
vl j
-2.12E+04
0.00E+00
0.00E+00
2.12E+04
0.00E+00
0.00E+00
0.00E+00
4.72E+04
-1.16E+04
0.00E+00
9.44E+04
1.16E+04
θl j
0.00E+00
1.16E+04
-1.90E+03
0.00E+00
1.16E+04
1.90E+03
1
2
3
11
5
12
0.819
0.574
0.000
0.000
0.000
0.000
-0.574
0.819
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
0.819
0.574
0.000
0.000
0.000
0.000
-0.574
0.819
0.000
0.000
0.000
0.000
0.000
0.000
1.000
KNA
KAA
GL
1
2
3
4
5
6
7
8
9
10
11
12
1
87207.315
-15739.04194
3388.926797
13993.81446
-32144.0405
-25425.21866
0
0
-4618.754901
3261.722977
1276.710383
-6650.649775
-15739.04194
257184.9638
5165.377552
28735.60242
24675.65828
0
63000
-23625
13993.81446
8961.510228
-32144.0405
9498.112219
3388.926797
5165.377552
9598.17449
-8961.510228
-9498.112219
0
23625
-5906.25
-3261.722977
-1792.302046
6650.649775
-1899.622444
13993.81446
28735.60242
-8961.510228
60879.49986
0
0
0
0
18623.41531
8961.510228
0
0
-32144.0405
24675.65828
-9498.112219
0
70305.46391
0
0
0
0
0
-34362.45725
9498.112219
-25425.21866
0
0
0
0
25425.21866
0
0
0
0
0
0
0
63000
23625
0
0
0
126000
-23625
0
0
0
0
0
-23625
-5906.25
0
0
0
-23625
5906.25
0
0
0
0
-4618.754901
13993.81446
-3261.722977
18623.41531
0
0
0
0
16490.45566
3261.722977
0
0
3261.722977
8961.510228
-1792.302046
8961.510228
0
0
0
0
3261.722977
1792.302046
0
0
1276.710383
-32144.0405
6650.649775
0
-34362.45725
0
0
0
0
0
45291.64068
-6650.649775
-6650.649775
9498.112219
-1899.622444
0
9498.112219
0
0
0
0
0
-6650.649775
1899.622444
2
3
4
5
6
7
8
9
10
11
12
2.5.2. Matriz de rotación
GL
1
2
3
4
5
6
7
8
9
10
11
12
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0.940
0
0
0
0
0.342
0
0
0
0
0
0
0
0.819
0
0
0
0
0
-0.574
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
-0.342
0
0
0
0
0.940
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0.574
0
0
0
0
0
0.819
0
0
0
0
0
0
0
0
0
0
0
0
1
2
3
4
5
6
7
8
9
10
11
12
2.5.3. Matriz de rigidez global rotada
GL
1
2
3
4
5
6
7
8
9
10
11
12
1
87207.315
-15739.04194
3388.926797
11570.17697
-27063.14748
-25425.21866
0
0
-9126.376326
3261.722977
-17391.24428
-6650.649775
-15739.04194
257184.9638
5165.377552
31788.79998
38650.18013
0
63000
-23625
3321.729329
8961.510228
-12177.48034
9498.112219
3388.926797
5165.377552
9598.17449
-9536.639993
-11595.05404
0
23625
-5906.25
-4.54747E-13
-1792.302046
0
-1899.622444
11570.17697
31788.79998
-9536.639993
67657.8687
0
0
0
0
-3.63798E-12
9536.639993
0
0
-27063.14748
38650.18013
-11595.05404
0
94366.3155
0
0
0
0
0
0
11595.05404
-25425.21866
0
0
0
0
25425.21866
0
0
0
0
0
0
0
63000
23625
0
0
0
126000
-23625
0
0
0
0
0
-23625
-5906.25
0
0
0
-23625
5906.25
0
0
0
0
-9126.376326
3321.729329
-4.54747E-13
-2.72848E-12
0
0
0
0
9712.086829
4.54747E-13
0
0
3261.722977
8961.510228
-1792.302046
9536.639993
0
0
0
0
4.54747E-13
1792.302046
0
0
-17391.24428
-12177.48034
0
0
0
0
0
0
0
0
21230.78908
0
-6650.649775
9498.112219
-1899.622444
0
11595.05404
0
0
0
0
0
0
1899.622444
2
3
4
5
6
7
8
9
10
11
12
2.5.4. Rotación de Fuerzas fijas y Fuerzas externas
Matriz de fuerzas fijas rotada
GL
[FF]
1
2
3
4
5
6
7
8
9
10
11
12
68.12
-93.01
262.41
113.25
186.29
0.00
-166.00
107.00
0.00
63.85
0.00
91.56
Matriz de fuerzas externas rotada
GL
[F]
1
0
0
-140
0
0
0
0
0
0
0
0
0
2
3
[FFN]
4
5
6
7
8
[FFA]
9
10
11
12
[FN]
[FA]
2.6 Determinación de desplazamientos desconocidos
[UN] =
[U]
1
0.00115
0.00596
-0.07746
-0.01559
-0.01360
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
2
3
4
5
6
7
8
9
10
11
12
2.7 Determinación de Fuerzas desconocidas
[FA] =
[KNN]-1( [FN] - [FFN] - [KNA] [UA] )
GL
Giro en coordenadas
globales
0.0040
Nodo 4
0.00000
-0.01559
0.0000
Coordenadas globales
-0.00533
4x
-0.01465
4y
[UN]
[UA]
[FA]
6
12
-29.278
-1620.581
423.720
48.017
111.174
-199.440
129.924
2.54E+04
0.00E+00
0.00E+00
-2.54E+04
0.00E+00
0.00E+00
0.00E+00
1.26E+05
-2.36E+04
0.00E+00
6.30E+04
2.36E+04
K Global Elemento 1
0.00E+00
-2.54E+04
-2.36E+04
0.00E+00
5.91E+03
0.00E+00
0.00E+00
2.54E+04
-2.36E+04
0.00E+00
-5.91E+03
0.00E+00
0.00E+00
6.30E+04
-2.36E+04
0.00E+00
1.26E+05
2.36E+04
0.00E+00
2.36E+04
-5.91E+03
0.00E+00
2.36E+04
5.91E+03
1.65E+04
1.86E+04
-3.26E+03
-4.62E+03
1.40E+04
3.26E+03
1.86E+04
6.09E+04
-8.96E+03
1.40E+04
2.87E+04
8.96E+03
K Global Elemento 2
-3.26E+03
-4.62E+03
-8.96E+03
1.40E+04
1.79E+03
-3.26E+03
-3.26E+03
1.65E+04
-8.96E+03
1.86E+04
-1.79E+03
3.26E+03
1.40E+04
2.87E+04
-8.96E+03
1.86E+04
6.09E+04
8.96E+03
3.26E+03
8.96E+03
-1.79E+03
3.26E+03
8.96E+03
1.79E+03
4.53E+04
-3.44E+04
6.65E+03
1.28E+03
-3.21E+04
-6.65E+03
-3.44E+04
7.03E+04
-9.50E+03
-3.21E+04
2.47E+04
9.50E+03
K Global Elemento 3
6.65E+03
1.28E+03
-9.50E+03
-3.21E+04
1.90E+03
6.65E+03
6.65E+03
4.53E+04
-9.50E+03
-3.44E+04
-1.90E+03
-6.65E+03
-3.21E+04
2.47E+04
-9.50E+03
-3.44E+04
7.03E+04
9.50E+03
-6.65E+03
9.50E+03
-1.90E+03
-6.65E+03
9.50E+03
1.90E+03
7
8
Nodo 5
0.00000
-0.01360
0.0000
0.00780
-0.01114
[FFA] - [FA] + [KAN] [UN] + [KAA] [UA]
GL
9
10
11
5x
5y
2.8 Determinación de las fuerzas internas de los elementos
Elemento 1
[FLe]
GL
6
7
8
1
2
3
Mfx ij
Mfy ij
zf ij
Mfx ji
Mfy ji
zf ji
[Fe]
-29.28
-1620.58
423.72
29.28
-913.18
-209.72
Mfx ij
Mfy ij
zf ij
Mfx ji
Mfy ji
zf ji
-9.28
503.60
16.53
9.28
0.00
111.17
Mfx ij
Mfy ij
zf ij
Mfx ji
Mfy ji
zf ji
92.59
468.36
53.19
-92.59
0.00
129.92
Mfx ij
Mfy ij
zf ij
Mfx ji
Mfy ji
zf ji
-29.28
-1620.58
423.72
29.28
-913.18
-209.72
=
[Fef]
0.000
-166.000
107.000
0.000
166.000
107.000
+
=
[Fef]
-38.733
-106.418
63.851
38.733
106.418
63.851
+
=
[Fef]
106.849
-152.597
91.558
-106.849
152.597
91.558
+
Elemento 2
[FLe]
GL
1
2
3
9
4
10
Mfx ij
Mfy ij
zf ij
Mfx ji
Mfy ji
zf ji
[Fe]
163.52
476.41
16.53
8.72
-3.18
111.17
Elemento 3
[FLe]
GL
1
2
3
11
5
12
Mfx ij
Mfy ij
zf ij
Mfx ji
Mfy ji
zf ji
[Fe]
Suma de reacciones =
664.82
-192.80
436.77
53.19
-75.85
-53.11
129.92
[UN]
[UA]