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- Pages: 17
b) Resolver la estructura por el método matricial con el mínimo número de nudos
posibles y calcular; Reacciones y Valores de; Cortantes, Momentos, Función de Momentos y Elástica. Graficar todos los resultados.
6[tn / m ]
4[tn ]
4[tn / m]
65�
2
2
2
2[tn ]
4.5
[m ]
CONFIGURACIÓN DE NUDOS
q3
q1 A
B
1
q2
q5
q4
C
2
q6
ANÁLISIS DE CARGAS BARRA 1
9.750 [ tn]
6[tn / m]
2.250 [ tn ]
B
A
5.50 [ tn � m]
2.5 [ tn � m]
2
2
MB = -
a
b
q� a3 � a� 4-3 � � 12 � L� L�
M B = -2.5 [ tn � m]
q[tn / m] A MA
q� a2 12
� a� a� � 6- � 8-3 � � � L� � L� � M A = 5.500 [ tn � m] MA =
B MB
[m ]
� q� a � a2 � a � RA = 2- 2 � 2- � � � 2 � L � L� � R A =9.750 [ tn ] q� a3 2� L2
� a� 2- � � � L� R B =2.250 [ tn ] RB =
BARRA 2 5.641[ tn ]
12.359 [ tn ]
C
B
8.987 [ tn � m]
12.821[ tn � m]
2
4.5
q[tn / m] C MA
B MB
a
b
[m ]
q� a2 MC = 12
� a� a� � 6- � 8-3 � � � L� � L� � M C = 5.500 [ tn � m] MB = -
q� a3 � a� 4-3 � � 12 � L� L�
M B = -2.5 [ tn � m] RC =
q� a � a2 2- 2 2 � � L
R C =9.750 [ tn ]
� � a� 2- � � � � L� �
q� a3 � a � RB = 2- � � 2� L2 � L � R B =2.250 [ tn ] 1.548 [ tn ]
0.452 [ tn ]
B
C
1.917 [ tn � m]
2[tn ]
2
p �� a b2 L2 M B = 1.917 [ tn � m]
0.852 [ tn � m ]B = M
4.5
p� a2 � b 2 L = -0.852 [ tn � m]
MC = MC C
B
MB
q[tn]
MC
b2 � b� RA = p 2 � 3-2 � L� L � R A = 1.548 [ tn ] a2 L2
a� � 3-2 � � L� � R C = 0.452 [ tn ] RC = p
a
b
4.093 [ tn ]
4[tn / m]
11.907 [ tn ]
C
B
7.070 [ tn � m]
11.969 [ tn � m]
2[tn]
2
4.5 RESUMEN DE LAS CARGAS
9.750 [ tn ]
9.968[ tn ]
11.907 [ tn ]
5.500 [ tn � m] A
MA
1
VA
B
VB
4.570 [ tn � m]
2
LOS MATRICES DE LAS RIGIDEZ � 12 6L -12 6L � EI � 6L 4L2 -6L 2L2 � � K= 3 � L �-12 -6L 12 -6L � � 2 2� � 6L 2L -6L 4L �
BARRA 1 24 -12 24 � �12 �24 64 -24 32 � 9000 � � K= 64 � -12 -24 12 -24 � � � �24 32 -24 64 �
2 4 3 1 3375 -1687.5 3375 � �1687.5 �3375 9000 -3375 4500 � � � � -1687.5 -3375 1687.5 -3375 � � � 4500 -3375 9000 � �3375
C
VC
11.969 [ tn � m]
=
BARRA 2 39 -12 39 � �12 �39 169 -39 84.5 � 9000 � � = K= -12 -39 12 -39 � 274.625 � � � �39 84.5 -39 169 �
3 4 5 6 �393.263541 1278.10651 -393.263541 1278.10651 � �1278.10651 5538.46154 -1278.10651 2769.23077 � � � � -393.263541 -1278.10651 393.263541 -1278.10651 � � � 2769.23077 -1278.10651 5538.46154 � �1278.10651
MATRIZ GLOBAL DE LA RIGIDEZ 6 5 3 1 2 4 3375.000 -1687.500 3375.000 0 0 VA -9.75 � q1 =0 � � �� 1 �1687.500 � � � � � � 2 2 3375.000 9000.000 -3375.000 4500.000 0 0 M A -5.5 q =0 � � � �� � 3� -1687.500 -3375.000 2080.764 -2096.893 -393.264 1278.107 �� VB -9.968 � q 3 =0 � � �4 �= � �� � 4 � 3375.000 450 0.000 2096.893 14538.4 62 1491.124 2769.231 -4.57 � q =? � � � �� 5� 0 0 -393.264 -1278.107 393.264 -1278.107 �� VC -11.907 � q 5 =0 � � � � � � � � 6 0 1278.107 2769.231 -1278.107 5538.462 �� 11.969 � q 6 =? � � � 0 CALCULO DE INCÓGNITAS (q4 Y q6) 14538.462 2769.231�� q 4 � �-4.57 � � �6 �= � �2769.231 5538.462 �� � 11.969� q � � � �� q 4 = -0.00080239 q 6 = 0.00256226 CALCULO DE REACCIONES Y MOMENTO VA = 7.042 [ tn ]
M B = 1.889 [ tn � m] VB = 14.925 [ tn ]
VC = 9.658 [ tn ] FORMULAS DE APLICACIÓN PARA EL CÁLCULO DE MOMENTOS, CORTANTE, ROTACIÓN ANGULAR Y LA FECHA
ξ=
2 � (x-xn )-1 xm -xn
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 DONDE:
n = El desplazamiento vertical θ= La rotacion M= El momento Q= El cortante Toda estas formulas mencionados seran aplicados en la tabla
Xn 0 0 0 0 0 0 0 0 0 4 4 4 4 4 4 4 4 4 4 4 4 4
Le ξ H1 H2 H3 H4 q q q 4 -1 1 0 0 0 0 0 0 4 -0.75 0.95703 0.19141 0.04297 -0.02734 0 0 0 4 -0.5 0.84375 0.28125 0.15625 -0.09375 0 0 0 4 -0.25 0.68359 0.29297 0.31641 -0.17578 0 0 0 4 0 0.5 0.25 0.5 -0.25 0 0 0 4 0.25 0.31641 0.17578 0.68359 -0.29297 0 0 0 4 0.5 0.15625 0.09375 0.84375 -0.28125 0 0 0 4 0.75 0.04297 0.02734 0.95703 -0.19141 0 0 0 4 1 0 0 1 0 0 0 0 6.5 -0.84615 0.98316 0.13109 0.01684 -0.01092 0 -0.00080 0 n 6.5 -0.69231 0.93628 0.22030 0.06372 -0.04005 0 -0.00080 0 6.5 -0.53846 0.86482 0.27310 0.13518 -0.08193 0 -0.00080 0 6.5 -0.38462 0.77424 0.29495 0.22576 -0.13109 0 -0.00080 0 6.5 -0.23077 0.67000 0.29131 0.33000 -0.18207 0 -0.00080 0 6.5 -0.07692 0.55758 0.26764 0.44242 -0.22940 0 -0.00080 0 6.5 0.07692 0.44242 0.22940 0.55758 -0.26764 0 -0.00080 0 6.5 0.23077 0.33000 0.18207 0.67000 -0.29131 0 -0.00080 0 6.5 0.38462 0.22576 0.13109 0.77424 -0.29495 0 -0.00080 0 6.5 0.53846 0.13518 0.08193 0.86482 -0.27310 0 -0.00080 0 6.5 0.69231 0.06372 0.04005 0.93628 -0.22030 0 -0.00080 0 6.5 0.84615 0.01684 0.01092 0.98316 -0.13109 0 -0.00080 0 6.5 1 0 0 1 0 0 -0.00080 0
q M Q -0.00080 3.61075 -2.70806 -0.00080 2.25672 -2.70806 -0.00080 0.90269 -2.70806 -0.00080 -0.45134 -2.70806 -0.00080 -1.80538 -2.70806 -0.00080 -3.15941 -2.70806 -0.00080 -4.51344 -2.70806 -0.00080 -5.86747 -2.70806 -0.00080 -7.22150 -2.70806 0.00256 -1.52685 3.11707 0.00256 -0.40219 3.11707 0.00256 0.72246 3.11707 0.00256 1.84712 3.11707 0.00256 2.97177 3.11707 0.00256 4.09642 3.11707 0.00256 5.22108 3.11707 0.00256 6.34573 3.11707 0.00256 7.47038 3.11707 0.00256 8.59504 3.11707 0.00256 9.71969 3.11707 0.00256 10.84435 3.11707 0.00256 11.96900 3.11707
CALCULO DE MOMENTOS, CORTANTE, ANGULO DE ROTACIÓN Y LA FLECHA
ϴ
n
0 0.00016 0.00025 0.00026 0.00020 0.00006 -0.00015 -0.00044 -0.00080 -0.00092 -0.00097 -0.00096 -0.00089 -0.00076 -0.00056 -0.00030 0.00002 0.00040 0.00085 0.00136 0.00193 0.00256
GRAFICAS DE MOMENTO, CORTANTE, ANGULO Y FLECHA
3.625 [ tn ]
6[tn / m]
1.690 [ tn ]
1.889 [ tn � m]
2 VA = 7.042 [tn]
2
2
4[tn / m]
2[tn ]
4.5
VB = 14.925 [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
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
Q=
[m ] VC = 9.658 [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
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
n =
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 A
q5
q3
q9
E
F
D
C
B
q7
q2
G
q8
q6
q4
q13
q11
ANÁLISIS DE LAS BARRAS q A MA
MB
L
6 [ tn ] 2 [ tn � m]
[m ]
6[tn / m]
A
q� L2 12 M A = 2 [ tn � m]
q� L 2 R A = 6 [ tn ]
MA =
B
6 [ tn ] B
RA =
q� L2 12 M B = 2 [ tn � m]
q� L 2 R B =6 [ tn ]
MB =
RB =
2 [ tn � m]
2
C MC
BB MB
L
0
L
[m ]
0
0
B B
C C
0
0
0
0
C
D
1
1
D
D MD
C MC
E
E
F
0
q15 H
q10
q12
q16 14
MD
ME
ME
L
L
0
0
0
D
E
MF
0
0
0
0
E
F
0
4[tn / m] FF
MF
L
q� L2 12 M A = 2.083 [ tn � m]
G MG
[m ] 5 [ tn ]
RA =
q� L2 MB = 12 M B = 2.083[ tn � m]
5 [ tn ] 4[tn / m ]
q� L 2 R B =5 [ tn ]
RB =
2.083 [ tn � m]
G
F
2.083
q� L 2 R A = 5 [ tn ]
MA =
2
G
MG
1.333
L
H MH
4 [ tn]
4 [ tn ]
G
H
q� L2 12 M A = 1.333 [ tn � m]
q� L 2 R A = 4 [ tn ]
q� L2 12 M B = 1.333 [ tn � m]
q� L 2 R B =4 [ tn ]
MA =
RA =
MB =
RB =
1.333[ tn � m]
2.5
RESUMEN
q1
q3
q5
q7
q9
q11
q13
q15
6
6
3.625 2.083
2
2 A
D
C
B
VA
q2
E
q4
G
F
q6
4
0.75
q8 VD
LOS MATRICES DE LAS RIGIDEZ � 12 6L -12 6L � EI � 6L 4L2 -6L 2L2 � � K= 3 � L �-12 -6L 12 -6L � � 2 2� � 6L 2L -6L 4L � BARRA 1 �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 2 �12 6 -12 6 � �108000 54000 -108000 54000 � � � 4 -6 2� 36000 -54000 18000 � 9000 �6 �= �54000 � 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 � � � 4 -6 2� 36000 -54000 18000 � 9000 �6 �= �54000 � K= 1 � -12 -6 12 -6 � � -108000 -54000 108000 -54000 � � � � � 4 � �54000 18000 -54000 36000 � �6 2 -6
BARRA 4
9
3
H
q10
q12
q16 1.333 q14
VH
�12 6 -12 6 � �108000 54000 -108000 54000 � � �54000 4 -6 2� 36000 -54000 18000 � 9000 �6 � � � = K= -12 -6 12 -6 � � -108000 -54000 108000 -54000 � 1 � � � � � 4 � �54000 18000 -54000 36000 � �6 2 -6 BARRA 5 �12 6 -12 6 � �108000 54000 -108000 54000 � � �54000 4 -6 2� 36000 -54000 18000 � 9000 �6 � � � = K= -12 -6 12 -6� � -108000 -54000 108000 -54000 � 1 � � � � � 4 � �54000 18000 -54000 36000 � �6 2 -6 BARRA 6 -12 15 � �6912 8640 -6912 8640 � �12 15 � � � 9000 15 25 -15 12.5� K= � �= �8640 14400 -8640 7200 � 15.625 � -12 -15 12 -15 � � -6912 -8640 6912 -8640 � � � � � �15 12.5 -15 25 � �8640 7200 -8640 14400 � BARRA 7 �12 12 -12 12� �13500 13500 -13500 13500 � � �13500 18000 -13500 8� 9000 � 9000 �12 16 -12 � � � = K= � � � � -12 -12 12 -12 -13500 -13500 13500 -13500 8 � � � � 8 -12 16� �13500 9000 -13500 18000 � �12
MATRIZ GLOBAL DE LA RIGIDEZ
1
2
1 �13500 � 2 �13500 -13500 3 � � 4 �13500 5 � 0 � 6 � 0 7 � 0 � 8 � 0 9 � 0 � 10 � 0 11 � 0 � 12 � 0 � 13 � 0 14 � 0 � 15 � 0 16 � � 0
3
4
5
6
7
8
9
10
11
12
13
14
15
16
13500
-13500
13500
0
0
0
0
0
0
0
0
0
0
0
18000
-13500
9000
0
0
0
0
0
0
0
0
0
0
0
-13500
121500
40500 -108000 54000
0
0
0
0
0
0
0
0
0
40500
54000 -54000
0
9000
18000
0
0
0
0
0
0
0
0
0
-108000
54000
0
0
0
0
0
0
0
72000
-54000
18000
0
0
0
0
0
0
0
-108000
54000
0
0
0
0
0
0
-108000 -54000 216000
0
54000
0
0
0
-108000 -54000
0
0
0
54000
18000
0
0
0
0
0
-108000 -54000 216000
0
0
0
0
0
54000
18000
0
0
0
0
0
0
0
-108000 -54000
0
0
0
0
0
0
0
54000
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
18000
0
216000 0
0
72000 -54000 0
18000
0
0
0
0
0
0
-108000
54000
0
0
0
18000
0
0
0
72000 -54000
114912 -45360 -6912
8640
0
50400 -8640
7200
0
-6912
-8640
20412
4860
-13500
0
8640
7200
4860
32400 -13500
0
0
0
0
-13500 -13500
13500
0
0
0
0
13500
-13500
18000 -45360
0 0 0 0 �121500.0 40500.0 -108000.0 54000.0 �40500.0 54000.0 -54000.0 18000.0 0 0 0 0 � � -108000.0 -54000.0 216000.0 0.0 54000 0 0 0 � 18000.0 0.0 72000.0 18000 0 0 0 �54000.0 � 0 0 54000 18000 72000 -54000 18000 0 � 0 0 0 -54000 216000 0 -108000 � 0 � 0 0 0 0 18000 0 72000 -54000 � 0 0 0 0 -108000 -54000 114912 � 0 � 0 0 0 0 0 54000 18000 -45360 � � 0 0 0 0 0 0 0 -6912 � 0 0 0 0 0 0 8640 � 0 � 0 0 0 0 0 0 0 � 0
0 0 0 0 0 54000 18000 -45360 50400 -8640 7200 0
9000
0 0 0 0 0 0 0 -6912 -8640 20412 4860 13500
� q1 � � �2 � q � � 0 � � �3 � q 0 � � � � � q4 � 0 � � q5 � 0 � � � � q6 0 � � � � q7 � 0 � � � � � q8 � 0 ���9 � = q 0 � �10 � � � q � 0 � �11 � � q 0 � �12 � � q � 0 � �13 � � q 13500 � �14 � � q � 9000 � �15 � � � q � -13500 � �16 � q � � 18000 � �
0
VA � � MA � � � � � � � � V �D � � � � � � � � � � � VH � � �
0 0 � �-6 � � � 0 0 � � �2 � 0 0 � �0 � �� � 0 0 � �0 � 0 0 � �0 � �� � 0 0 � �0 � = 0 0 � �0 � �� � 8640 0 �� -3.000 � 7200 0 �� -2.083 � �� � 4860 13500 � � -9.000 � �� � 32400 9000 � � 0.750 � 9000 18000 � 1.333 � �� � �
-6
� -2 � � -6 � � 2 � � 0 � 0 � -3.625� � 0 � � 0 � 0 � � -3 � -2.083� � -9 � 0.75 � � -4 � 1.333 � �
CALCULO DE LAS INCÓGNITAS q 3 = 0.000179 q 4 = 0.000256 q 5 = 0.000354 q 6 = 0.000002 q 8 = -0.000802 q 9 = -0.001225 q10 = -0.001530 q11 = -0.002825 q12 = -0.001553 q13 = -0.003990 q14 = 0.001009 q16 = 0.002562 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
Xm Xn 2 0 2 0 2 0 2 0 2 0 3 2 3 2 4 3 4 3 5 4 5 4 6 5 6 5 8.5 6 8.5 6 8.5 6 8.5 6 8.5 6 10.5 8.5 10.5 8.5 10.5 8.5 10.5 8.5
Le ξ H1 2 -1 1 2 -0.5 0.84375 2 0 0.5 2 0.5 0.15625 2 1 0 1 0 0.50000 1 1 0 1 0 0.50000 1 1 0 1 0.00000 0.50000 1 1.00000 0.00000 1 0.00000 0.50000 1 1.00000 0.00000 2.5 -0.60000 0.89600 2.5 -0.20000 0.64800 2.5 0.20000 0.35200 2.5 0.60000 0.10400 2.5 1.00000 0.00000 2 -0.50000 0.84375 2 0.00000 0.50000 2 0.50000 0.15625 2 1 0
H2
H3
H4
q
q
0 0.28125 0.25 0.09375 0 0.25000 0 0.25000 0 0.25000 0.00000 0.25000 0.00000 0.25600 0.28800 0.19200 0.06400 0.00000 0.28125 0.25000 0.09375 0
0 0.15625 0.5 0.84375 1 0.50000 1 0.50000 1 0.50000 1.00000 0.50000 1.00000 0.10400 0.35200 0.64800 0.89600 1.00000 0.15625 0.50000 0.84375 1
0 -0.09375 -0.25 -0.28125 0 -0.25000 0 -0.25000 0 -0.25000 0.00000 -0.25000 0.00000 -0.06400 -0.19200 -0.28800 -0.25600 0.00000 -0.09375 -0.25000 -0.28125 0
0 0 0 0 0 0.00018 0.00018 0.00035 0.00035 0 0 -0.00122 -0.00122 -0.00283 -0.00283 -0.00283 -0.00283 -0.00283 -0.00399 -0.00399 -0.00399 -0.00399
0 0 0 0 0 0.00026 0.00026 0 0 -0.00080 -0.00080 -0.00153 -0.00153 -0.00155 -0.00155 -0.00155 -0.00155 -0.00155 0.00101 0.00101 0.00101 0.00101
q q 0.00018 0.00026 0.00018 0.00026 0.00018 0.00026 0.00018 0.00026 0.00018 0.00026 0.00035 2.3E-06 0.00035 2.3E-06 0 -0.00080 0 -0.00080 -0.00122 -0.00153 -0.00122 -0.00153 -0.00283 -1.55E-03 -0.00283 -1.55E-03 -0.00399 0.00101 -0.00399 0.00101 -0.00399 0.00101 -0.00399 0.00101 -0.00399 0.00101 0 0.00256 0 0.00256 0 0.00256 0 0.00256
M 0.11049 0.63156 1.15262 1.67369 2.19475 -2.28418 -4.76312 -7.24205 -9.72099 -6.55014 -3.37930 -0.20845 2.96239 6.71624 8.38708 10.05793 11.72877 13.39962 9.82046 6.99131 4.16215 1.33300
Q 1.04213 1.04213 1.04213 1.04213 1.04213 -4.95787 -4.95787 -4.95787 -4.95787 6.34169 6.34169 6.34169 6.34169 11.39393 11.39393 11.39393 11.39393 11.39393 -12.46582 -12.46582 -12.46582 -12.46582
ϴ 0 2.06E-05 7.02E-05 1.49E-04 2.56E-04 1.98E-04 2.34E-06 -3.31E-04 -8.02E-04 -1.25E-03 -1.53E-03 -1.63E-03 -1.55E-03 -1.23E-03 -8.07E-04 -2.95E-04 3.11E-04 1.01E-03 1.63E-03 2.10E-03 2.41E-03 2.56E-03
0 3.95E-06 2.54E-05 7.89E-05 1.79E-04 2.98E-04 3.54E-04 2.78E-04 0.00E+00 -5.22E-04 -1.22E-03 -2.02E-03 -2.83E-03 -3.52E-03 -4.04E-03 -4.32E-03 -4.32E-03 -3.99E-03 -3.32E-03 -2.38E-03 -1.25E-03 0
GRAFICAS DE MOMENTO, CORTANTE, ANGULO Y FLECHA
3.625 [ tn ]
6[tn / m]
1.690 [ tn ]
1.889 [ tn � m]
2 VA = 7.042 [tn]
2
2
4[tn / m]
2[tn ]
4.5
VB = 14.925 [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
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
[m ]
VC = 9.658 [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
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
posibles y calcular; Reacciones y Valores de; Cortantes, Momentos, Función de Momentos y Elástica. Graficar todos los resultados.
6[tn / m ]
4[tn ]
4[tn / m]
65�
2
2
2
2[tn ]
4.5
[m ]
CONFIGURACIÓN DE NUDOS
q3
q1 A
B
1
q2
q5
q4
C
2
q6
ANÁLISIS DE CARGAS BARRA 1
9.750 [ tn]
6[tn / m]
2.250 [ tn ]
B
A
5.50 [ tn � m]
2.5 [ tn � m]
2
2
MB = -
a
b
q� a3 � a� 4-3 � � 12 � L� L�
M B = -2.5 [ tn � m]
q[tn / m] A MA
q� a2 12
� a� a� � 6- � 8-3 � � � L� � L� � M A = 5.500 [ tn � m] MA =
B MB
[m ]
� q� a � a2 � a � RA = 2- 2 � 2- � � � 2 � L � L� � R A =9.750 [ tn ] q� a3 2� L2
� a� 2- � � � L� R B =2.250 [ tn ] RB =
BARRA 2 5.641[ tn ]
12.359 [ tn ]
C
B
8.987 [ tn � m]
12.821[ tn � m]
2
4.5
q[tn / m] C MA
B MB
a
b
[m ]
q� a2 MC = 12
� a� a� � 6- � 8-3 � � � L� � L� � M C = 5.500 [ tn � m] MB = -
q� a3 � a� 4-3 � � 12 � L� L�
M B = -2.5 [ tn � m] RC =
q� a � a2 2- 2 2 � � L
R C =9.750 [ tn ]
� � a� 2- � � � � L� �
q� a3 � a � RB = 2- � � 2� L2 � L � R B =2.250 [ tn ] 1.548 [ tn ]
0.452 [ tn ]
B
C
1.917 [ tn � m]
2[tn ]
2
p �� a b2 L2 M B = 1.917 [ tn � m]
0.852 [ tn � m ]B = M
4.5
p� a2 � b 2 L = -0.852 [ tn � m]
MC = MC C
B
MB
q[tn]
MC
b2 � b� RA = p 2 � 3-2 � L� L � R A = 1.548 [ tn ] a2 L2
a� � 3-2 � � L� � R C = 0.452 [ tn ] RC = p
a
b
4.093 [ tn ]
4[tn / m]
11.907 [ tn ]
C
B
7.070 [ tn � m]
11.969 [ tn � m]
2[tn]
2
4.5 RESUMEN DE LAS CARGAS
9.750 [ tn ]
9.968[ tn ]
11.907 [ tn ]
5.500 [ tn � m] A
MA
1
VA
B
VB
4.570 [ tn � m]
2
LOS MATRICES DE LAS RIGIDEZ � 12 6L -12 6L � EI � 6L 4L2 -6L 2L2 � � K= 3 � L �-12 -6L 12 -6L � � 2 2� � 6L 2L -6L 4L �
BARRA 1 24 -12 24 � �12 �24 64 -24 32 � 9000 � � K= 64 � -12 -24 12 -24 � � � �24 32 -24 64 �
2 4 3 1 3375 -1687.5 3375 � �1687.5 �3375 9000 -3375 4500 � � � � -1687.5 -3375 1687.5 -3375 � � � 4500 -3375 9000 � �3375
C
VC
11.969 [ tn � m]
=
BARRA 2 39 -12 39 � �12 �39 169 -39 84.5 � 9000 � � = K= -12 -39 12 -39 � 274.625 � � � �39 84.5 -39 169 �
3 4 5 6 �393.263541 1278.10651 -393.263541 1278.10651 � �1278.10651 5538.46154 -1278.10651 2769.23077 � � � � -393.263541 -1278.10651 393.263541 -1278.10651 � � � 2769.23077 -1278.10651 5538.46154 � �1278.10651
MATRIZ GLOBAL DE LA RIGIDEZ 6 5 3 1 2 4 3375.000 -1687.500 3375.000 0 0 VA -9.75 � q1 =0 � � �� 1 �1687.500 � � � � � � 2 2 3375.000 9000.000 -3375.000 4500.000 0 0 M A -5.5 q =0 � � � �� � 3� -1687.500 -3375.000 2080.764 -2096.893 -393.264 1278.107 �� VB -9.968 � q 3 =0 � � �4 �= � �� � 4 � 3375.000 450 0.000 2096.893 14538.4 62 1491.124 2769.231 -4.57 � q =? � � � �� 5� 0 0 -393.264 -1278.107 393.264 -1278.107 �� VC -11.907 � q 5 =0 � � � � � � � � 6 0 1278.107 2769.231 -1278.107 5538.462 �� 11.969 � q 6 =? � � � 0 CALCULO DE INCÓGNITAS (q4 Y q6) 14538.462 2769.231�� q 4 � �-4.57 � � �6 �= � �2769.231 5538.462 �� � 11.969� q � � � �� q 4 = -0.00080239 q 6 = 0.00256226 CALCULO DE REACCIONES Y MOMENTO VA = 7.042 [ tn ]
M B = 1.889 [ tn � m] VB = 14.925 [ tn ]
VC = 9.658 [ tn ] FORMULAS DE APLICACIÓN PARA EL CÁLCULO DE MOMENTOS, CORTANTE, ROTACIÓN ANGULAR Y LA FECHA
ξ=
2 � (x-xn )-1 xm -xn
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 DONDE:
n = El desplazamiento vertical θ= La rotacion M= El momento Q= El cortante Toda estas formulas mencionados seran aplicados en la tabla
Xn 0 0 0 0 0 0 0 0 0 4 4 4 4 4 4 4 4 4 4 4 4 4
Le ξ H1 H2 H3 H4 q q q 4 -1 1 0 0 0 0 0 0 4 -0.75 0.95703 0.19141 0.04297 -0.02734 0 0 0 4 -0.5 0.84375 0.28125 0.15625 -0.09375 0 0 0 4 -0.25 0.68359 0.29297 0.31641 -0.17578 0 0 0 4 0 0.5 0.25 0.5 -0.25 0 0 0 4 0.25 0.31641 0.17578 0.68359 -0.29297 0 0 0 4 0.5 0.15625 0.09375 0.84375 -0.28125 0 0 0 4 0.75 0.04297 0.02734 0.95703 -0.19141 0 0 0 4 1 0 0 1 0 0 0 0 6.5 -0.84615 0.98316 0.13109 0.01684 -0.01092 0 -0.00080 0 n 6.5 -0.69231 0.93628 0.22030 0.06372 -0.04005 0 -0.00080 0 6.5 -0.53846 0.86482 0.27310 0.13518 -0.08193 0 -0.00080 0 6.5 -0.38462 0.77424 0.29495 0.22576 -0.13109 0 -0.00080 0 6.5 -0.23077 0.67000 0.29131 0.33000 -0.18207 0 -0.00080 0 6.5 -0.07692 0.55758 0.26764 0.44242 -0.22940 0 -0.00080 0 6.5 0.07692 0.44242 0.22940 0.55758 -0.26764 0 -0.00080 0 6.5 0.23077 0.33000 0.18207 0.67000 -0.29131 0 -0.00080 0 6.5 0.38462 0.22576 0.13109 0.77424 -0.29495 0 -0.00080 0 6.5 0.53846 0.13518 0.08193 0.86482 -0.27310 0 -0.00080 0 6.5 0.69231 0.06372 0.04005 0.93628 -0.22030 0 -0.00080 0 6.5 0.84615 0.01684 0.01092 0.98316 -0.13109 0 -0.00080 0 6.5 1 0 0 1 0 0 -0.00080 0
q M Q -0.00080 3.61075 -2.70806 -0.00080 2.25672 -2.70806 -0.00080 0.90269 -2.70806 -0.00080 -0.45134 -2.70806 -0.00080 -1.80538 -2.70806 -0.00080 -3.15941 -2.70806 -0.00080 -4.51344 -2.70806 -0.00080 -5.86747 -2.70806 -0.00080 -7.22150 -2.70806 0.00256 -1.52685 3.11707 0.00256 -0.40219 3.11707 0.00256 0.72246 3.11707 0.00256 1.84712 3.11707 0.00256 2.97177 3.11707 0.00256 4.09642 3.11707 0.00256 5.22108 3.11707 0.00256 6.34573 3.11707 0.00256 7.47038 3.11707 0.00256 8.59504 3.11707 0.00256 9.71969 3.11707 0.00256 10.84435 3.11707 0.00256 11.96900 3.11707
CALCULO DE MOMENTOS, CORTANTE, ANGULO DE ROTACIÓN Y LA FLECHA
ϴ
n
0 0.00016 0.00025 0.00026 0.00020 0.00006 -0.00015 -0.00044 -0.00080 -0.00092 -0.00097 -0.00096 -0.00089 -0.00076 -0.00056 -0.00030 0.00002 0.00040 0.00085 0.00136 0.00193 0.00256
GRAFICAS DE MOMENTO, CORTANTE, ANGULO Y FLECHA
3.625 [ tn ]
6[tn / m]
1.690 [ tn ]
1.889 [ tn � m]
2 VA = 7.042 [tn]
2
2
4[tn / m]
2[tn ]
4.5
VB = 14.925 [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
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
Q=
[m ] VC = 9.658 [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
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
n =
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 A
q5
q3
q9
E
F
D
C
B
q7
q2
G
q8
q6
q4
q13
q11
ANÁLISIS DE LAS BARRAS q A MA
MB
L
6 [ tn ] 2 [ tn � m]
[m ]
6[tn / m]
A
q� L2 12 M A = 2 [ tn � m]
q� L 2 R A = 6 [ tn ]
MA =
B
6 [ tn ] B
RA =
q� L2 12 M B = 2 [ tn � m]
q� L 2 R B =6 [ tn ]
MB =
RB =
2 [ tn � m]
2
C MC
BB MB
L
0
L
[m ]
0
0
B B
C C
0
0
0
0
C
D
1
1
D
D MD
C MC
E
E
F
0
q15 H
q10
q12
q16 14
MD
ME
ME
L
L
0
0
0
D
E
MF
0
0
0
0
E
F
0
4[tn / m] FF
MF
L
q� L2 12 M A = 2.083 [ tn � m]
G MG
[m ] 5 [ tn ]
RA =
q� L2 MB = 12 M B = 2.083[ tn � m]
5 [ tn ] 4[tn / m ]
q� L 2 R B =5 [ tn ]
RB =
2.083 [ tn � m]
G
F
2.083
q� L 2 R A = 5 [ tn ]
MA =
2
G
MG
1.333
L
H MH
4 [ tn]
4 [ tn ]
G
H
q� L2 12 M A = 1.333 [ tn � m]
q� L 2 R A = 4 [ tn ]
q� L2 12 M B = 1.333 [ tn � m]
q� L 2 R B =4 [ tn ]
MA =
RA =
MB =
RB =
1.333[ tn � m]
2.5
RESUMEN
q1
q3
q5
q7
q9
q11
q13
q15
6
6
3.625 2.083
2
2 A
D
C
B
VA
q2
E
q4
G
F
q6
4
0.75
q8 VD
LOS MATRICES DE LAS RIGIDEZ � 12 6L -12 6L � EI � 6L 4L2 -6L 2L2 � � K= 3 � L �-12 -6L 12 -6L � � 2 2� � 6L 2L -6L 4L � BARRA 1 �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 2 �12 6 -12 6 � �108000 54000 -108000 54000 � � � 4 -6 2� 36000 -54000 18000 � 9000 �6 �= �54000 � 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 � � � 4 -6 2� 36000 -54000 18000 � 9000 �6 �= �54000 � K= 1 � -12 -6 12 -6 � � -108000 -54000 108000 -54000 � � � � � 4 � �54000 18000 -54000 36000 � �6 2 -6
BARRA 4
9
3
H
q10
q12
q16 1.333 q14
VH
�12 6 -12 6 � �108000 54000 -108000 54000 � � �54000 4 -6 2� 36000 -54000 18000 � 9000 �6 � � � = K= -12 -6 12 -6 � � -108000 -54000 108000 -54000 � 1 � � � � � 4 � �54000 18000 -54000 36000 � �6 2 -6 BARRA 5 �12 6 -12 6 � �108000 54000 -108000 54000 � � �54000 4 -6 2� 36000 -54000 18000 � 9000 �6 � � � = K= -12 -6 12 -6� � -108000 -54000 108000 -54000 � 1 � � � � � 4 � �54000 18000 -54000 36000 � �6 2 -6 BARRA 6 -12 15 � �6912 8640 -6912 8640 � �12 15 � � � 9000 15 25 -15 12.5� K= � �= �8640 14400 -8640 7200 � 15.625 � -12 -15 12 -15 � � -6912 -8640 6912 -8640 � � � � � �15 12.5 -15 25 � �8640 7200 -8640 14400 � BARRA 7 �12 12 -12 12� �13500 13500 -13500 13500 � � �13500 18000 -13500 8� 9000 � 9000 �12 16 -12 � � � = K= � � � � -12 -12 12 -12 -13500 -13500 13500 -13500 8 � � � � 8 -12 16� �13500 9000 -13500 18000 � �12
MATRIZ GLOBAL DE LA RIGIDEZ
1
2
1 �13500 � 2 �13500 -13500 3 � � 4 �13500 5 � 0 � 6 � 0 7 � 0 � 8 � 0 9 � 0 � 10 � 0 11 � 0 � 12 � 0 � 13 � 0 14 � 0 � 15 � 0 16 � � 0
3
4
5
6
7
8
9
10
11
12
13
14
15
16
13500
-13500
13500
0
0
0
0
0
0
0
0
0
0
0
18000
-13500
9000
0
0
0
0
0
0
0
0
0
0
0
-13500
121500
40500 -108000 54000
0
0
0
0
0
0
0
0
0
40500
54000 -54000
0
9000
18000
0
0
0
0
0
0
0
0
0
-108000
54000
0
0
0
0
0
0
0
72000
-54000
18000
0
0
0
0
0
0
0
-108000
54000
0
0
0
0
0
0
-108000 -54000 216000
0
54000
0
0
0
-108000 -54000
0
0
0
54000
18000
0
0
0
0
0
-108000 -54000 216000
0
0
0
0
0
54000
18000
0
0
0
0
0
0
0
-108000 -54000
0
0
0
0
0
0
0
54000
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
18000
0
216000 0
0
72000 -54000 0
18000
0
0
0
0
0
0
-108000
54000
0
0
0
18000
0
0
0
72000 -54000
114912 -45360 -6912
8640
0
50400 -8640
7200
0
-6912
-8640
20412
4860
-13500
0
8640
7200
4860
32400 -13500
0
0
0
0
-13500 -13500
13500
0
0
0
0
13500
-13500
18000 -45360
0 0 0 0 �121500.0 40500.0 -108000.0 54000.0 �40500.0 54000.0 -54000.0 18000.0 0 0 0 0 � � -108000.0 -54000.0 216000.0 0.0 54000 0 0 0 � 18000.0 0.0 72000.0 18000 0 0 0 �54000.0 � 0 0 54000 18000 72000 -54000 18000 0 � 0 0 0 -54000 216000 0 -108000 � 0 � 0 0 0 0 18000 0 72000 -54000 � 0 0 0 0 -108000 -54000 114912 � 0 � 0 0 0 0 0 54000 18000 -45360 � � 0 0 0 0 0 0 0 -6912 � 0 0 0 0 0 0 8640 � 0 � 0 0 0 0 0 0 0 � 0
0 0 0 0 0 54000 18000 -45360 50400 -8640 7200 0
9000
0 0 0 0 0 0 0 -6912 -8640 20412 4860 13500
� q1 � � �2 � q � � 0 � � �3 � q 0 � � � � � q4 � 0 � � q5 � 0 � � � � q6 0 � � � � q7 � 0 � � � � � q8 � 0 ���9 � = q 0 � �10 � � � q � 0 � �11 � � q 0 � �12 � � q � 0 � �13 � � q 13500 � �14 � � q � 9000 � �15 � � � q � -13500 � �16 � q � � 18000 � �
0
VA � � MA � � � � � � � � V �D � � � � � � � � � � � VH � � �
0 0 � �-6 � � � 0 0 � � �2 � 0 0 � �0 � �� � 0 0 � �0 � 0 0 � �0 � �� � 0 0 � �0 � = 0 0 � �0 � �� � 8640 0 �� -3.000 � 7200 0 �� -2.083 � �� � 4860 13500 � � -9.000 � �� � 32400 9000 � � 0.750 � 9000 18000 � 1.333 � �� � �
-6
� -2 � � -6 � � 2 � � 0 � 0 � -3.625� � 0 � � 0 � 0 � � -3 � -2.083� � -9 � 0.75 � � -4 � 1.333 � �
CALCULO DE LAS INCÓGNITAS q 3 = 0.000179 q 4 = 0.000256 q 5 = 0.000354 q 6 = 0.000002 q 8 = -0.000802 q 9 = -0.001225 q10 = -0.001530 q11 = -0.002825 q12 = -0.001553 q13 = -0.003990 q14 = 0.001009 q16 = 0.002562 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
Xm Xn 2 0 2 0 2 0 2 0 2 0 3 2 3 2 4 3 4 3 5 4 5 4 6 5 6 5 8.5 6 8.5 6 8.5 6 8.5 6 8.5 6 10.5 8.5 10.5 8.5 10.5 8.5 10.5 8.5
Le ξ H1 2 -1 1 2 -0.5 0.84375 2 0 0.5 2 0.5 0.15625 2 1 0 1 0 0.50000 1 1 0 1 0 0.50000 1 1 0 1 0.00000 0.50000 1 1.00000 0.00000 1 0.00000 0.50000 1 1.00000 0.00000 2.5 -0.60000 0.89600 2.5 -0.20000 0.64800 2.5 0.20000 0.35200 2.5 0.60000 0.10400 2.5 1.00000 0.00000 2 -0.50000 0.84375 2 0.00000 0.50000 2 0.50000 0.15625 2 1 0
H2
H3
H4
q
q
0 0.28125 0.25 0.09375 0 0.25000 0 0.25000 0 0.25000 0.00000 0.25000 0.00000 0.25600 0.28800 0.19200 0.06400 0.00000 0.28125 0.25000 0.09375 0
0 0.15625 0.5 0.84375 1 0.50000 1 0.50000 1 0.50000 1.00000 0.50000 1.00000 0.10400 0.35200 0.64800 0.89600 1.00000 0.15625 0.50000 0.84375 1
0 -0.09375 -0.25 -0.28125 0 -0.25000 0 -0.25000 0 -0.25000 0.00000 -0.25000 0.00000 -0.06400 -0.19200 -0.28800 -0.25600 0.00000 -0.09375 -0.25000 -0.28125 0
0 0 0 0 0 0.00018 0.00018 0.00035 0.00035 0 0 -0.00122 -0.00122 -0.00283 -0.00283 -0.00283 -0.00283 -0.00283 -0.00399 -0.00399 -0.00399 -0.00399
0 0 0 0 0 0.00026 0.00026 0 0 -0.00080 -0.00080 -0.00153 -0.00153 -0.00155 -0.00155 -0.00155 -0.00155 -0.00155 0.00101 0.00101 0.00101 0.00101
q q 0.00018 0.00026 0.00018 0.00026 0.00018 0.00026 0.00018 0.00026 0.00018 0.00026 0.00035 2.3E-06 0.00035 2.3E-06 0 -0.00080 0 -0.00080 -0.00122 -0.00153 -0.00122 -0.00153 -0.00283 -1.55E-03 -0.00283 -1.55E-03 -0.00399 0.00101 -0.00399 0.00101 -0.00399 0.00101 -0.00399 0.00101 -0.00399 0.00101 0 0.00256 0 0.00256 0 0.00256 0 0.00256
M 0.11049 0.63156 1.15262 1.67369 2.19475 -2.28418 -4.76312 -7.24205 -9.72099 -6.55014 -3.37930 -0.20845 2.96239 6.71624 8.38708 10.05793 11.72877 13.39962 9.82046 6.99131 4.16215 1.33300
Q 1.04213 1.04213 1.04213 1.04213 1.04213 -4.95787 -4.95787 -4.95787 -4.95787 6.34169 6.34169 6.34169 6.34169 11.39393 11.39393 11.39393 11.39393 11.39393 -12.46582 -12.46582 -12.46582 -12.46582
ϴ 0 2.06E-05 7.02E-05 1.49E-04 2.56E-04 1.98E-04 2.34E-06 -3.31E-04 -8.02E-04 -1.25E-03 -1.53E-03 -1.63E-03 -1.55E-03 -1.23E-03 -8.07E-04 -2.95E-04 3.11E-04 1.01E-03 1.63E-03 2.10E-03 2.41E-03 2.56E-03
0 3.95E-06 2.54E-05 7.89E-05 1.79E-04 2.98E-04 3.54E-04 2.78E-04 0.00E+00 -5.22E-04 -1.22E-03 -2.02E-03 -2.83E-03 -3.52E-03 -4.04E-03 -4.32E-03 -4.32E-03 -3.99E-03 -3.32E-03 -2.38E-03 -1.25E-03 0
GRAFICAS DE MOMENTO, CORTANTE, ANGULO Y FLECHA
3.625 [ tn ]
6[tn / m]
1.690 [ tn ]
1.889 [ tn � m]
2 VA = 7.042 [tn]
2
2
4[tn / m]
2[tn ]
4.5
VB = 14.925 [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
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
[m ]
VC = 9.658 [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
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
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