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b) Resolver la estructura por el método matricial con un mínimo de 8 nudos y calcular; Reacciones y Valores de; Cortantes, Momentos, Función de Momentos y Elástica. Graficar todos los resultados.
q1
q5
q3
A
D
C
B
q2
C B MC
BA MB L
0
q7
q9
E
F
0
A
B
L
0
0
0
0
B
C
1
D ME
L
0 0
E
D ME
0
0 0
1
E
0
F
1
q
MF
L
0
E
D
0
0
q15 H
q10
MD
1
C MD
q8
C
B MC
[m ]
0
G
q6
q4
q13
q11
q12
q14 16
E ME
F
L
0.75 [ tn ]
q� L2 30 M E = 0.33[ tn � m]
3 �� q L 20 R E = 0.75 [ tn ]
ME =
2.5[tn / m] 1.75 [ tn ]
RE =
q� L2 20 M F = 0.5 [ tn � m]
7 �� q L 20 R F = 1.75 [ tn ]
MF =
F
E
0.333[ tn � m]
MF
0.5 [ tn � m]
2 q2
3.25 [ tn ]
q1
F
G
1.166 [ tn � m]
RF =
5 [ tn / m ] 2.5 [ tn / m ]
F
4.25 [ tn ]
G
1.333[ tn � m]
2 q
E ME
F
L
2.7 [ tn ]
q� L2 30 M G = 2.7 [ tn � m]
3 �� q L 20 R G = 2.7 [ tn ]
MG =
6.30 [ tn ]
4[tn / m]
RG =
q� L2 20 M H = 4.05 [ tn � m]
7 �� q L 20 R H = 6.30 [ tn ]
MH =
F
E
2.7 [ tn � m]
MF
4.050 [ tn � m]
2
RH =
RESUMEN
q1
q3
q5 0
0
0
q7
A
B
q11 1.25 0.333 0.666
0
0
0
q9 0
D
C
q2 LOS MATRICES DE LAS RIGIDEZ VA � 12 6L -12 6L � 2 2� EI � K= 3 � 6L 4L -6L 2L � L �-12 -6L 12 -6L � � 2 2� � 6L 2L -6L 4L �
E
q4
F 3.5 q6
VC
q13
6.95
5
q15
1.317 G
q8
H
q10
q12
6.30
q16 4.050 q14 VGH
BARRA 1 �12 6 -12 6 ��108000 54000 -108000 54000 � � 2 �� 54000 36000 -54000 18000 � 9000 �6 4 -6 �� � � K= � �� � -12 -6 12 -6 -108000 -54000 108000 -54000 1 � �� � 4 ��54000 18000 -54000 36000 � �6 2 -6 BARRA 2
�12 6 -12 6 ��108000 54000 -108000 54000 � � 2 �� 54000 36000 -54000 18000 � 9000 �6 4 -6 �� � � K= � �� � -12 -6 12 -6 -108000 -54000 108000 -54000 1 � �� � 4 ��54000 18000 -54000 36000 � �6 2 -6 BARRA 3
�12 6 -12 6 ��108000 54000 -108000 54000 � � 2 �� 54000 36000 -54000 18000 � 9000 �6 4 -6 �� � � K= � �� � -12 -6 12 -6 -108000 -54000 108000 -54000 1 � �� � 4 ��54000 18000 -54000 36000 � �6 2 -6 BARRA 4
�12 6 -12 6 ��108000 54000 -108000 54000 � 9000 �6 4 -6 2 �� 54000 36000 -54000 18000 � � �� � K= � 1 � -12 -6 12 -6 �� -108000 -54000 108000 -54000 � � �� � 4 ��54000 18000 -54000 36000 � �6 2 -6
BARRA 5
�12 12 -12 12 ��13500 13500 -13500 13500 � � ��13500 18000 -13500 9000 � 9000 �12 16 -12 8 �� � � K= � �� � -12 -12 12 -12 -13500 -13500 13500 -13500 8 � �� � �12 8 -12 16 ��13500 9000 -13500 18000 �
BARRA 6
�12 12 -12 12 ��13500 13500 -13500 13500 � 9000 �12 16 -12 8 ��13500 18000 -13500 9000 � � �� � K= � 8 � -12 -12 12 -12�� -13500 -13500 13500 -13500 � � �� � �12 8 -12 16 ��13500 9000 -13500 18000 � BARRA 7
27 -12 27 ��1185.185 2666.667 -1185.185 2666.667 � �12 9000 �27 81.0 -27 40.5��2666.667 8000.000 -2666.667 4000.000 � K= �� � � 91.125 � � -12 -27 12 -27 �� -1185.185 -2666.667 1185.185 -2666.667 � � �� � �27 40.5 -27 81.0 ��2666.667 4000.000 -2666.667 8000.000 � FORMULAS DE APLICACIÓN PARA EL CÁLCULO DE MOMENTOS, CORTANTE, ROTACIÓN ANGULAR Y LA FECHA 2 � (x-xn )-1 xm -xn
ξ=
M=
EI [ 6ξq1 +(3ξ-1)Lq 2 -6q3ξ+(3ξ+1)Lq 4 ] L2
Q=
6EI [ 2q1 +Lq 2 -2q3 +Lq 4 ] L3
θ=
3q 3q1 2 q q (ξ -1)+ 2 (ξ-1)(3ξ+1)+ 3 (1-ξ 2 )+ 4 (3ξ-1)(ξ+1) 2Le 4 2L e 4
L L n (ξ) =H1q1 + q 2 H 2 +H3 q 3 + q 4 H 4 2 2 1 H1 = (2-3ξ+ξ 3 ) 4 1 H 2 = (1-ξ-ξ 2 +ξ 3 ) 4 1 H 3 = (2+3ξ-ξ 3 ) 4 1 H 4 = (-1-ξ+ξ 2 +ξ 3 ) 4
MATRIZ GLOBAL DE LA RIGIDEZ 108000 � � 54000 � � -108000 � 54000 � � 0 � 0 � � 0 � 0 � � 0 � � 0 � 0 � � 0 � 0 � � 0 � 0 � � 0 �
VA 0 q1 � � � �� �2 � � � � MA 0 36000 -54000 18000 0 0 0 0 0 0 0 0 0 0 0 0 q � � � �� � �� 0 -54000 216000 0 -108000 54000 0 0 0 0 0 0 0 0 0 0 q3 � � � ��4 � � 0 18000 0 72000 -54000 18000 0 0 0 0 0 0 0 0 0 0 q � � � �� � �� VC 0 0 -108000 -54000 216000 0 -108000 54000 0 0 0 0 0 0 0 0 q5 � � � ��6 � � -3.5 � 0 54000 18000 0 72000 -54000 18000 0 0 0 0 0 0 0 0 q � � �� � �� 0 0 0 0 -108000 -54000 216000 0 -108000 54000 0 0 0 0 0 0 q7 � � � ��8 � � 0 0 0 0 54000 18000 0 72000 -54000 18000 0 0 0 0 0 0 q � � � �� �9 �= � � �� 1.25 0 0 0 0 0 -108000 -54000 121500 -40500 -13500 13500 0 0 0 0 q � �� � � -0.333 � 0 0 0 0 0 54000 18000 -40500 54000 -13500 9000 0 0 0 0 �� q10 � � � ��11 � � -5 0 0 0 0 0 0 0 -13500 -13500 27000 0 -13500 13500 0 0 q � � � �� �� -0.666 � 0 0 0 0 0 0 0 13500 9000 0 36000 -13500 9000 0 0 q12 � � � ��13 � � VG -6.95 � 0 0 0 0 0 0 0 0 0 -13500 -13500 14685 -10833 -1185 2667 �� q � � -1.367 � 0 0 0 0 0 0 0 0 0 13500 9000 -10833 26000 -2667 4000 �� q14 � � � ��15 � � VH -6.30 � 0 0 0 0 0 0 0 0 0 0 0 -1185 -2667 1185 -2667 �� q � � �16 � � M H 4.050 � 0 0 0 0 0 0 0 0 0 0 0 2 667 4000 -2667 8000 � q � � �� � 54000
-108000 54000
216000.0 � � 0.0 � � 54000.0 � 0.0 � � 0.0 � 0.0 � � 0.0 � 0.0 � � 0.0 � � 0.0 �
0
0.0 72000.0 18000.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0
54000.0 18000.0 72000.0 -54000.0 18000.0 0.0 0.0 0.0 0.0 0.0
0
0.0 0.0 -54000.0 216000.0 0.0 -108000.0 54000.0 0.0 0.0 0.0
0
0
0.0 0.0 18000.0 0.0 72000.0 -54000.0 18000.0 0.0 0.0 0.0
0
0.0 0.0 0.0 -108000.0 -54000.0 121500.0 -40500.0 -13500.0 13500.0 0.0
0
0.0 0.0 0.0 54000.0 18000.0 -40500.0 54000.0 -13500.0 9000.0 0.0
0
0
0.0 0.0 0.0 0.0 0.0 -13500.0 -13500.0 27000.0 0.0 13500.0
0
0.0 0.0 0.0 0.0 0.0 13500.0 9000.0 0.0 36000.0 9000.0
0
0
0 0 � � � � � � 0 0 � � � � � 0 -3.5 � � � � 0 0 � � � � � � 0 0 �= � � 0 1.25 � � � � � 0 -0.3333� � � � 13500 � � -5 � � � 9000 -0.666 � � � � 26000 � -1.367 � � � � �
CALCULO DE INCÓGNITAS q 3 = 5.30172E-05 q 4 = 5.30172E-05 q 6 = -0.000212069 q 7 = -0.000226351 q 8 = -0.000237292 q 9 = -0.000467899 q10 = -0.000242462 q11 = -0.000657123 q12 = 0.000159199 q14 = 0.000233514
FORMULAS DE APLICACIÓN PARA EL CÁLCULO DE MOMENTOS, CORTANTE, ROTACIÓN ANGULAR Y LA FECHA 2 � (x-xn )-1 xm -xn
ξ=
M=
EI [ 6ξq1 +(3ξ-1)Lq 2 -6q3ξ+(3ξ+1)Lq 4 ] L2
Q=
6EI [ 2q1 +Lq 2 -2q3 +Lq 4 ] L3
θ=
3q 3q1 2 q q (ξ -1)+ 2 (ξ-1)(3ξ+1)+ 3 (1-ξ 2 )+ 4 (3ξ-1)(ξ+1) 2Le 4 2L e 4
L L n (ξ) =H1q1 + q 2 H 2 +H3 q 3 + q 4 H 4 2 2 1 H1 = (2-3ξ+ξ 3 ) 4 1 H 2 = (1-ξ-ξ 2 +ξ 3 ) 4 1 H 3 = (2+3ξ-ξ 3 ) 4 1 H 4 = (-1-ξ+ξ 2 +ξ 3 ) 4
CALCULO DE MOMENTOS, CORTANTE, ANGULO DE ROTACIÓN Y LA FLECHA X 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Xm Xn 1 0 1 0 1 0 2 0 2 0 3 2 3 2 4 3 4 3 6 4 6 4 6 4 6 4 8 6 8 6 8 6 8 6 12.5 8 12.5 8 12.5 8 12.5 8 12.5 8 12.5 8 12.5 8 12.5 8 12.5 8
Le 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5
ξ
H1
H2
H3
H4
-1 0 1 0.5 1 0 1 0 1 -0.500 0.000 0.500 1.000 -0.500 0.000 0.500 1.000 -0.778 -0.556 -0.333 -0.111 0.111 0.333 0.556 0.778 1.000
1 0.50000 0 0.15625 0 0.50000 0 0.50000 0 0.84375 0.50000 0.15625 0.00000 0.84375 0.50000 0.15625 0.00000 0.96571 0.87380 0.74074 0.58299 0.41701 0.25926 0.12620 0.03429 0
0 0.25000 0 0.09375 0 0.25000 0 0.25000 0 0.28125 0.25000 0.09375 0.00000 0.28125 0.25000 0.09375 0.00000 0.17558 0.26886 0.29630 0.27435 0.21948 0.14815 0.07682 0.02195 0
0 0.50000 1 0.84375 1 0.50000 1 0.50000 1 0.15625 0.50000 0.84375 1.00000 0.15625 0.50000 0.84375 1.00000 0.03429 0.12620 0.25926 0.41701 0.58299 0.74074 0.87380 0.96571 1
0 -0.25000 0 -0.28125 0 -0.25000 0 -0.25000 0 -0.09375 -0.25000 -0.28125 0.00000 -0.09375 -0.25000 -0.28125 0.00000 -0.02195 -0.07682 -0.14815 -0.21948 -0.27435 -0.29630 -0.26886 -0.17558 0
q 0 0 0 5.30E-05 5.30E-05 0.00E+00 0.00E+00 -2.26E-04 -2.26E-04 -4.68E-04 -4.68E-04 -4.68E-04 -4.68E-04 -6.57E-04 -6.57E-04 -6.57E-04 -6.57E-04 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
q 0 0 0 5.30E-05 5.30E-05 -2.12E-04 -2.12E-04 -2.37E-04 -2.37E-04 -2.42E-04 -2.42E-04 -2.42E-04 -2.42E-04 1.59E-04 1.59E-04 1.59E-04 1.59E-04 2.34E-04 2.34E-04 2.34E-04 2.34E-04 2.34E-04 2.34E-04 2.34E-04 2.34E-04 2.34E-04
q
q 0.00005 0.00005 0.00005 0.00005 0.00005 0.00005 0.00000 -0.00021 0.00000 -0.00021 -0.00023 -2.4E-04 -0.00023 -2.4E-04 -0.000468 -0.00024 -0.000468 -0.00024 -0.00066 0.00016 -0.00066 0.00016 -0.00066 0.00016 -0.00066 0.00016 0.00000 0.00023 0.00000 0.00023 0.00000 0.00023 0.00000 0.00023 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
M 1.90862 0.47715 -0.95431 -3.10151 -3.81724 -0.22700 -0.13677 -0.04653 0.04370 1.09224 1.80747 2.52271 3.23795 2.11918 0.33442 -1.45035 -3.23511 -1.55676 -1.24541 -0.93406 -0.62270 -0.31135 0.00000 0.31135 0.62270 0.93406
Q ϴ -2.86293 0 -2.86293 6.63E-05 -2.86293 5.30E-05 -2.86293 -1.16E-04 -2.86293 -2.12E-04 0.18047 -2.27E-04 0.18047 -2.37E-04 0.18047 -2.42E-04 0.18047 -2.42E-04 1.43047 -2.02E-04 1.43047 -1.21E-04 1.43047 -8.19E-07 1.43047 1.59E-04 -3.56953 3.27E-04 -3.56953 3.95E-04 -3.56953 3.64E-04 -3.56953 2.34E-04 0.62270 1.38E-04 0.62270 6.05E-05 0.62270 0.00E+00 0.62270 -4.32E-05 0.62270 -6.92E-05 0.62270 -7.78E-05 0.62270 -6.92E-05 0.62270 -4.32E-05 0.62270 0
n 0 1.99E-05 5.30E-05 4.06E-05 0.00E+00 -1.10E-04 -2.26E-04 -3.46E-04 -4.68E-04 -5.81E-04 -6.63E-04 -6.95E-04 -6.57E-04 -5.32E-04 -3.47E-04 -1.53E-04 0.00E+00 9.23E-05 1.41E-04 1.56E-04 1.44E-04 1.15E-04 7.78E-05 4.04E-05 1.15E-05 0
5[tn / m]
4[tn / m]
3,5[tn � m]
3.116 [ tn � m]
1.909 [ tn � m]
2
2
2[tn ]
VA = 2.863 [ tn ] VB = 3.043 [ tn ]
4
4.5 VC = 11.14 [ tn ]
[m ] VD = 5.667 [ tn ]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
n =
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 1010.51111.51212.5
q=
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
M=
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Q=
q1
q5
q3
A
D
C
B
q2
C B MC
BA MB L
0
q7
q9
E
F
0
A
B
L
0
0
0
0
B
C
1
D ME
L
0 0
E
D ME
0
0 0
1
E
0
F
1
q
MF
L
0
E
D
0
0
q15 H
q10
MD
1
C MD
q8
C
B MC
[m ]
0
G
q6
q4
q13
q11
q12
q14 16
E ME
F
L
0.75 [ tn ]
q� L2 30 M E = 0.33[ tn � m]
3 �� q L 20 R E = 0.75 [ tn ]
ME =
2.5[tn / m] 1.75 [ tn ]
RE =
q� L2 20 M F = 0.5 [ tn � m]
7 �� q L 20 R F = 1.75 [ tn ]
MF =
F
E
0.333[ tn � m]
MF
0.5 [ tn � m]
2 q2
3.25 [ tn ]
q1
F
G
1.166 [ tn � m]
RF =
5 [ tn / m ] 2.5 [ tn / m ]
F
4.25 [ tn ]
G
1.333[ tn � m]
2 q
E ME
F
L
2.7 [ tn ]
q� L2 30 M G = 2.7 [ tn � m]
3 �� q L 20 R G = 2.7 [ tn ]
MG =
6.30 [ tn ]
4[tn / m]
RG =
q� L2 20 M H = 4.05 [ tn � m]
7 �� q L 20 R H = 6.30 [ tn ]
MH =
F
E
2.7 [ tn � m]
MF
4.050 [ tn � m]
2
RH =
RESUMEN
q1
q3
q5 0
0
0
q7
A
B
q11 1.25 0.333 0.666
0
0
0
q9 0
D
C
q2 LOS MATRICES DE LAS RIGIDEZ VA � 12 6L -12 6L � 2 2� EI � K= 3 � 6L 4L -6L 2L � L �-12 -6L 12 -6L � � 2 2� � 6L 2L -6L 4L �
E
q4
F 3.5 q6
VC
q13
6.95
5
q15
1.317 G
q8
H
q10
q12
6.30
q16 4.050 q14 VGH
BARRA 1 �12 6 -12 6 ��108000 54000 -108000 54000 � � 2 �� 54000 36000 -54000 18000 � 9000 �6 4 -6 �� � � K= � �� � -12 -6 12 -6 -108000 -54000 108000 -54000 1 � �� � 4 ��54000 18000 -54000 36000 � �6 2 -6 BARRA 2
�12 6 -12 6 ��108000 54000 -108000 54000 � � 2 �� 54000 36000 -54000 18000 � 9000 �6 4 -6 �� � � K= � �� � -12 -6 12 -6 -108000 -54000 108000 -54000 1 � �� � 4 ��54000 18000 -54000 36000 � �6 2 -6 BARRA 3
�12 6 -12 6 ��108000 54000 -108000 54000 � � 2 �� 54000 36000 -54000 18000 � 9000 �6 4 -6 �� � � K= � �� � -12 -6 12 -6 -108000 -54000 108000 -54000 1 � �� � 4 ��54000 18000 -54000 36000 � �6 2 -6 BARRA 4
�12 6 -12 6 ��108000 54000 -108000 54000 � 9000 �6 4 -6 2 �� 54000 36000 -54000 18000 � � �� � K= � 1 � -12 -6 12 -6 �� -108000 -54000 108000 -54000 � � �� � 4 ��54000 18000 -54000 36000 � �6 2 -6
BARRA 5
�12 12 -12 12 ��13500 13500 -13500 13500 � � ��13500 18000 -13500 9000 � 9000 �12 16 -12 8 �� � � K= � �� � -12 -12 12 -12 -13500 -13500 13500 -13500 8 � �� � �12 8 -12 16 ��13500 9000 -13500 18000 �
BARRA 6
�12 12 -12 12 ��13500 13500 -13500 13500 � 9000 �12 16 -12 8 ��13500 18000 -13500 9000 � � �� � K= � 8 � -12 -12 12 -12�� -13500 -13500 13500 -13500 � � �� � �12 8 -12 16 ��13500 9000 -13500 18000 � BARRA 7
27 -12 27 ��1185.185 2666.667 -1185.185 2666.667 � �12 9000 �27 81.0 -27 40.5��2666.667 8000.000 -2666.667 4000.000 � K= �� � � 91.125 � � -12 -27 12 -27 �� -1185.185 -2666.667 1185.185 -2666.667 � � �� � �27 40.5 -27 81.0 ��2666.667 4000.000 -2666.667 8000.000 � FORMULAS DE APLICACIÓN PARA EL CÁLCULO DE MOMENTOS, CORTANTE, ROTACIÓN ANGULAR Y LA FECHA 2 � (x-xn )-1 xm -xn
ξ=
M=
EI [ 6ξq1 +(3ξ-1)Lq 2 -6q3ξ+(3ξ+1)Lq 4 ] L2
Q=
6EI [ 2q1 +Lq 2 -2q3 +Lq 4 ] L3
θ=
3q 3q1 2 q q (ξ -1)+ 2 (ξ-1)(3ξ+1)+ 3 (1-ξ 2 )+ 4 (3ξ-1)(ξ+1) 2Le 4 2L e 4
L L n (ξ) =H1q1 + q 2 H 2 +H3 q 3 + q 4 H 4 2 2 1 H1 = (2-3ξ+ξ 3 ) 4 1 H 2 = (1-ξ-ξ 2 +ξ 3 ) 4 1 H 3 = (2+3ξ-ξ 3 ) 4 1 H 4 = (-1-ξ+ξ 2 +ξ 3 ) 4
MATRIZ GLOBAL DE LA RIGIDEZ 108000 � � 54000 � � -108000 � 54000 � � 0 � 0 � � 0 � 0 � � 0 � � 0 � 0 � � 0 � 0 � � 0 � 0 � � 0 �
VA 0 q1 � � � �� �2 � � � � MA 0 36000 -54000 18000 0 0 0 0 0 0 0 0 0 0 0 0 q � � � �� � �� 0 -54000 216000 0 -108000 54000 0 0 0 0 0 0 0 0 0 0 q3 � � � ��4 � � 0 18000 0 72000 -54000 18000 0 0 0 0 0 0 0 0 0 0 q � � � �� � �� VC 0 0 -108000 -54000 216000 0 -108000 54000 0 0 0 0 0 0 0 0 q5 � � � ��6 � � -3.5 � 0 54000 18000 0 72000 -54000 18000 0 0 0 0 0 0 0 0 q � � �� � �� 0 0 0 0 -108000 -54000 216000 0 -108000 54000 0 0 0 0 0 0 q7 � � � ��8 � � 0 0 0 0 54000 18000 0 72000 -54000 18000 0 0 0 0 0 0 q � � � �� �9 �= � � �� 1.25 0 0 0 0 0 -108000 -54000 121500 -40500 -13500 13500 0 0 0 0 q � �� � � -0.333 � 0 0 0 0 0 54000 18000 -40500 54000 -13500 9000 0 0 0 0 �� q10 � � � ��11 � � -5 0 0 0 0 0 0 0 -13500 -13500 27000 0 -13500 13500 0 0 q � � � �� �� -0.666 � 0 0 0 0 0 0 0 13500 9000 0 36000 -13500 9000 0 0 q12 � � � ��13 � � VG -6.95 � 0 0 0 0 0 0 0 0 0 -13500 -13500 14685 -10833 -1185 2667 �� q � � -1.367 � 0 0 0 0 0 0 0 0 0 13500 9000 -10833 26000 -2667 4000 �� q14 � � � ��15 � � VH -6.30 � 0 0 0 0 0 0 0 0 0 0 0 -1185 -2667 1185 -2667 �� q � � �16 � � M H 4.050 � 0 0 0 0 0 0 0 0 0 0 0 2 667 4000 -2667 8000 � q � � �� � 54000
-108000 54000
216000.0 � � 0.0 � � 54000.0 � 0.0 � � 0.0 � 0.0 � � 0.0 � 0.0 � � 0.0 � � 0.0 �
0
0.0 72000.0 18000.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0
54000.0 18000.0 72000.0 -54000.0 18000.0 0.0 0.0 0.0 0.0 0.0
0
0.0 0.0 -54000.0 216000.0 0.0 -108000.0 54000.0 0.0 0.0 0.0
0
0
0.0 0.0 18000.0 0.0 72000.0 -54000.0 18000.0 0.0 0.0 0.0
0
0.0 0.0 0.0 -108000.0 -54000.0 121500.0 -40500.0 -13500.0 13500.0 0.0
0
0.0 0.0 0.0 54000.0 18000.0 -40500.0 54000.0 -13500.0 9000.0 0.0
0
0
0.0 0.0 0.0 0.0 0.0 -13500.0 -13500.0 27000.0 0.0 13500.0
0
0.0 0.0 0.0 0.0 0.0 13500.0 9000.0 0.0 36000.0 9000.0
0
0
0 0 � � � � � � 0 0 � � � � � 0 -3.5 � � � � 0 0 � � � � � � 0 0 �= � � 0 1.25 � � � � � 0 -0.3333� � � � 13500 � � -5 � � � 9000 -0.666 � � � � 26000 � -1.367 � � � � �
CALCULO DE INCÓGNITAS q 3 = 5.30172E-05 q 4 = 5.30172E-05 q 6 = -0.000212069 q 7 = -0.000226351 q 8 = -0.000237292 q 9 = -0.000467899 q10 = -0.000242462 q11 = -0.000657123 q12 = 0.000159199 q14 = 0.000233514
FORMULAS DE APLICACIÓN PARA EL CÁLCULO DE MOMENTOS, CORTANTE, ROTACIÓN ANGULAR Y LA FECHA 2 � (x-xn )-1 xm -xn
ξ=
M=
EI [ 6ξq1 +(3ξ-1)Lq 2 -6q3ξ+(3ξ+1)Lq 4 ] L2
Q=
6EI [ 2q1 +Lq 2 -2q3 +Lq 4 ] L3
θ=
3q 3q1 2 q q (ξ -1)+ 2 (ξ-1)(3ξ+1)+ 3 (1-ξ 2 )+ 4 (3ξ-1)(ξ+1) 2Le 4 2L e 4
L L n (ξ) =H1q1 + q 2 H 2 +H3 q 3 + q 4 H 4 2 2 1 H1 = (2-3ξ+ξ 3 ) 4 1 H 2 = (1-ξ-ξ 2 +ξ 3 ) 4 1 H 3 = (2+3ξ-ξ 3 ) 4 1 H 4 = (-1-ξ+ξ 2 +ξ 3 ) 4
CALCULO DE MOMENTOS, CORTANTE, ANGULO DE ROTACIÓN Y LA FLECHA X 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Xm Xn 1 0 1 0 1 0 2 0 2 0 3 2 3 2 4 3 4 3 6 4 6 4 6 4 6 4 8 6 8 6 8 6 8 6 12.5 8 12.5 8 12.5 8 12.5 8 12.5 8 12.5 8 12.5 8 12.5 8 12.5 8
Le 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5
ξ
H1
H2
H3
H4
-1 0 1 0.5 1 0 1 0 1 -0.500 0.000 0.500 1.000 -0.500 0.000 0.500 1.000 -0.778 -0.556 -0.333 -0.111 0.111 0.333 0.556 0.778 1.000
1 0.50000 0 0.15625 0 0.50000 0 0.50000 0 0.84375 0.50000 0.15625 0.00000 0.84375 0.50000 0.15625 0.00000 0.96571 0.87380 0.74074 0.58299 0.41701 0.25926 0.12620 0.03429 0
0 0.25000 0 0.09375 0 0.25000 0 0.25000 0 0.28125 0.25000 0.09375 0.00000 0.28125 0.25000 0.09375 0.00000 0.17558 0.26886 0.29630 0.27435 0.21948 0.14815 0.07682 0.02195 0
0 0.50000 1 0.84375 1 0.50000 1 0.50000 1 0.15625 0.50000 0.84375 1.00000 0.15625 0.50000 0.84375 1.00000 0.03429 0.12620 0.25926 0.41701 0.58299 0.74074 0.87380 0.96571 1
0 -0.25000 0 -0.28125 0 -0.25000 0 -0.25000 0 -0.09375 -0.25000 -0.28125 0.00000 -0.09375 -0.25000 -0.28125 0.00000 -0.02195 -0.07682 -0.14815 -0.21948 -0.27435 -0.29630 -0.26886 -0.17558 0
q 0 0 0 5.30E-05 5.30E-05 0.00E+00 0.00E+00 -2.26E-04 -2.26E-04 -4.68E-04 -4.68E-04 -4.68E-04 -4.68E-04 -6.57E-04 -6.57E-04 -6.57E-04 -6.57E-04 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
q 0 0 0 5.30E-05 5.30E-05 -2.12E-04 -2.12E-04 -2.37E-04 -2.37E-04 -2.42E-04 -2.42E-04 -2.42E-04 -2.42E-04 1.59E-04 1.59E-04 1.59E-04 1.59E-04 2.34E-04 2.34E-04 2.34E-04 2.34E-04 2.34E-04 2.34E-04 2.34E-04 2.34E-04 2.34E-04
q
q 0.00005 0.00005 0.00005 0.00005 0.00005 0.00005 0.00000 -0.00021 0.00000 -0.00021 -0.00023 -2.4E-04 -0.00023 -2.4E-04 -0.000468 -0.00024 -0.000468 -0.00024 -0.00066 0.00016 -0.00066 0.00016 -0.00066 0.00016 -0.00066 0.00016 0.00000 0.00023 0.00000 0.00023 0.00000 0.00023 0.00000 0.00023 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
M 1.90862 0.47715 -0.95431 -3.10151 -3.81724 -0.22700 -0.13677 -0.04653 0.04370 1.09224 1.80747 2.52271 3.23795 2.11918 0.33442 -1.45035 -3.23511 -1.55676 -1.24541 -0.93406 -0.62270 -0.31135 0.00000 0.31135 0.62270 0.93406
Q ϴ -2.86293 0 -2.86293 6.63E-05 -2.86293 5.30E-05 -2.86293 -1.16E-04 -2.86293 -2.12E-04 0.18047 -2.27E-04 0.18047 -2.37E-04 0.18047 -2.42E-04 0.18047 -2.42E-04 1.43047 -2.02E-04 1.43047 -1.21E-04 1.43047 -8.19E-07 1.43047 1.59E-04 -3.56953 3.27E-04 -3.56953 3.95E-04 -3.56953 3.64E-04 -3.56953 2.34E-04 0.62270 1.38E-04 0.62270 6.05E-05 0.62270 0.00E+00 0.62270 -4.32E-05 0.62270 -6.92E-05 0.62270 -7.78E-05 0.62270 -6.92E-05 0.62270 -4.32E-05 0.62270 0
n 0 1.99E-05 5.30E-05 4.06E-05 0.00E+00 -1.10E-04 -2.26E-04 -3.46E-04 -4.68E-04 -5.81E-04 -6.63E-04 -6.95E-04 -6.57E-04 -5.32E-04 -3.47E-04 -1.53E-04 0.00E+00 9.23E-05 1.41E-04 1.56E-04 1.44E-04 1.15E-04 7.78E-05 4.04E-05 1.15E-05 0
5[tn / m]
4[tn / m]
3,5[tn � m]
3.116 [ tn � m]
1.909 [ tn � m]
2
2
2[tn ]
VA = 2.863 [ tn ] VB = 3.043 [ tn ]
4
4.5 VC = 11.14 [ tn ]
[m ] VD = 5.667 [ tn ]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
n =
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 1010.51111.51212.5
q=
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
M=
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
Q=
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