5to EXAMEN PARCIAL.TEMA: MÉTODO ITERATIVO DE CROSS 1.3 PROCESO ITERATIVO
EJERCICIO 3ro 1 4to 2En el pórtico cargado como se muestra, por el método de Cross determinar: 0.242 0.758 0.758 0.242
1.062 -15.224 30 55 15.2243.430 10.732 5.366-0.042 -7.802 -15.604 -4.9871.900 5.944 2.972 -0.099-0.050 -1.089 -2.177 -0.696
3.3 T 3.3 T 0.276 0.863 0.432 0.011-0.012 -0.168 -0.335 -0.107
3.3 T/m 0.044 0.136 30 30 30 30 0.068 0.013-0.003 -0.031 -0.062 -0.020
12 T 0.008 0.025 0.013 0.004
-0.001 -0.006 -0.013 -0.004 325 cm0.002 0.005 0.003 0.0010.000 -0.001 -0.003 -0.0010.000 0.001 0.001 0.000
4.8 T 4.8 T 0.000 0.000 -0.001 0.0000.000 0.000 0.000 0.000
3.7 T 3.7 T 3.25 m 0.000 0.0004.8 T/m
3.7 T/m 6.614 -6.614 5.884 -5.884
3 1ro 4 5to 5 2do0.096 0.079 0.825 0.442 0.196 0.051 0.310 0.923 0.077
18 T -22.144 40 70 22.144 -17.069 35 65 17.0692.124 1.753 18.267 9.134 -2.493 -7.877 -15.753 -1.3161.715 -0.849 -1.698 -0.754 -0.197 -1.189 -0.595-0.083 -0.069 -0.714 -0.357 -0.348 0.274 0.549 0.0460.950 0.095 0.190 0.085 0.022 0.133 0.067-0.100 -0.083 -0.862 -0.431 -0.054 -0.031 -0.062 -0.005
5.25 m 0.138 0.114 0.228 0.101 0.027 0.160 0.08040 30 -0.024 -0.020 -0.208 40 50 -0.104 -0.010 -0.037 40 28 -0.074 -0.006
0.022 0.033 0.067 0.030 0.008 0.047 0.023-0.005 -0.004 -0.045 -0.023 -0.002 -0.011 -0.022 -0.0020.004 0.008 0.016 0.007 0.002 0.011 0.005-0.001 -0.001 -0.010 -0.005 0.000 -0.003 -0.005 0.0000.001 0.002 0.003 0.002 0.000 0.002 0.001
0.000 0.000 -0.002 -0.001 0.000 -0.001 -0.001 0.000 525 cm0.000 0.000 0.001 0.000 0.000 0.001 0.0000.000 0.000 0.000 0.000 0.000 0.000 0.0000.000 0.000 0.000 0.000 0.000 0.000 0.0000.000 0.000 0.000 0.000 0.000
1. PÓRTICO SIN DESPLAZAMIENTO 4.739 1.576 -6.315 29.164 -0.530 -3.046 -25.588 1.284 -1.284
3.3 T 3.3 T
3.3 T/m
12 T
0.788 -0.265 0.000
4.8 T 4.8 T6 7 8
3.7 T 3.7 T4.8 T/m
3.7 T/m
640 cm 640 cm
18 T
1.4 CORTANTES EN LAS COLUMNAS
12 R23.493 Tn 2.748 Tn
6.614 Tn.m 5.884 Tn.m
3.493 Tn 2.748 Tn
3.493 Tn 2.748 Tn 4.739 Tn.m
3.046 Tn.m1.1 CÁLCULO DE MOMENTOS DE EMPOTRAMIENTO
3.493 Tn 2.748 Tn 18 R1
ME12 = -15.224 Tnm ME21 = 15.224 Tnm 0.450 Tn ME34 = -22.144 Tnm ME43 = 22.144 Tnm 0.151 Tn 0.244 TnME45 = -17.069 Tnm ME54 = 17.069 Tnm
1.576 Tn.m 0.530 Tn.m 1.284 Tn.m
0.450 Tn 0.151 Tn 0.244 Tn1.2 CÁLCULO DE FACTORES DE DISTRIBUCIÓN y
NUDO 1 k12 649.902 FD12 0.758I12 415937.5 cm4 k13 207.692 FD13 0.242I13 67500 cm4 suma 857.595
NUDO 2I21 415937.5 K21 649.902 F21 0.758I24 67500 K24 207.692 F24 0.242 0.450 Tn 0.151 Tn 0.244 Tn
suma 857.595 0.0000.788 Tn.m 0.265 Tn.m
NUDO 3 K31 207.692 F31 0.096I31 67500 K34 1786.458 F34 0.825I34 1143333.33 K36 171.429 F36 0.079I36 90000 suma 2165.579
NUDO 4 K43 1786.458 F43 0.442 R2 = -12.746 TnI42 67500 K42 207.692 F42 0.051 R1 = -17.309 TnI43 1143333.33 K45 1251.546 F45 0.310I47 416666.667 K47 793.651 F47 0.196 0.19647994I45 800989.583 suma 4039.348
NUDO 5 K54 1251.546 F54 0.923I54 800989.583 K58 104.533 apoyo movilF58 0.077I58 73173.3333 suma 1356.080
a) Los momentos en las barrasb) Las reacciones en los apoyosc) }los diagramas de N ,V y Md) Las deflexiones en los centros del as vigas
|
2. PÓRTICO CON DESPLAZAMIENTO EN EL NIVEL 3-4-5
1
3 Δ
6
2.1 MOMENTOS DE EMPOTRAMIENTO
ME36 = ME63
ME47 = ME74
ME58 = ME85
EΔ
2.2 PROCESO ITERATIVO
4to0.242 0.758-0.218 0.053 0.165-0.349 -8.7972.213 6.933-0.115 -1.6310.423 1.324-0.026 -0.3080.081 0.253 -0.006 -0.0630.017 0.052-0.001 -0.0130.003 0.0110.000 -0.0030.001 0.0020.000 -0.0010.000 0.001
2.075 -2.075
2do0.096 0.079 0.825
-195.918 200.454
-0.435 -0.358 -3.7410.026 7.250-0.699 -0.575 -6.0031.107 1.297
-0.231 -0.190 -1.9830.211 0.334-0.052 -0.043 -0.4500.040 0.078-0.011 -0.009 -0.0980.008 0.017-0.002 -0.002 -0.0210.002 0.004-0.001 0.000 -0.0050.000 0.0010.000 0.000 -0.001
-0.036 -197.096 197.133
-0.589-195.918-196.507
2.3 CORTANTES EN LAS COLUMNAS
0.627 Tn
2.075 Tn.m
0.627 Tn
0.627 Tn-0.036 Tn.m
|
IER 0.627 Tn
74.972 Tn
197.096 Tn.m
74.972 Tn
74.972 Tn
196.507 Tn.m
2.1 CORTANTE EN LAS COLUMNAS
R2´ =V1´ =
"0.992Tn + 15.416 Tn "-0.922-58.394-15.416-222.494-12.660
2. PÓRTICO CON DESPLAZAMIENTO EN EL NIVEL 3-4-5
2
4 Δ 5
Δ
7 8
2.1 MOMENTOS DE EMPOTRAMIENTO
= -195.918 Tn.m 0
= -907.029 Tn.m
= -159.289 Tn.m
= 100
2.2 PROCESO ITERATIVO
5to0.758 0.242
23.1290.083
-17.595 -5.6173.467 0.837-3.262 -1.0410.662 0.150-0.615 -0.196
0.126 0.038-0.125 -0.0400.026 0.009-0.026 -0.0080.005 0.002-0.006 -0.0020.001 0.000 -0.001 0.0000.000 0.000
-17.260 17.260 -400.907029
1ro0.442 0.197 0.051 0.310 0.923
-907.029 400.907 178.685 46.259 281.179 140.590-1.871 -2.809 -28.126 -56.25214.500 6.463 1.673 10.170 5.085-3.002 -0.521 -2.347 -4.6932.594 1.156 0.299 1.819 0.910
(−6𝐸𝐼∆)/𝐿^2 (−6𝐸𝐼∆)/𝐿^2 (−6𝐸𝐼∆)/𝐿^2
-0.992 -0.098 -0.420 -0.840 0.667 0.297 0.077 0.468 0.234
-0.225 -0.020 -0.108 -0.2160.156 0.069 0.018 0.109 0.055-0.049 -0.004 -0.025 -0.0500.035 0.015 0.004 0.024 0.012-0.011 -0.001 -0.006 -0.011
0.008 0.003 0.001 0.005 0.003-0.002 0.000 -0.001 -0.0020.002 0.001 0.000 0.001 0.0010.000 0.000 0.000 -0.001
412.717 -720.339 44.878 262.743 84.822
93.345-907.029-813.684
1
2.3 CORTANTES EN LAS COLUMNAS
R´219.119 Tn
17.260 Tn.m
19.119 Tn
19.119 Tn 44.878 Tn.m
19.119 Tn V1´
292.195 Tn 16.157
720.339 Tn.m 84.822
292.195 Tn 16.157
292.195 Tn 16.1570.000
813.684 Tn.m
2.1 CORTANTE EN LAS COLUMNAS
= -19.747= -403.070 Tn= 403.373
"0.992Tn + 15.416 Tn "-0.922-58.394-15.416-222.494-12.660
Δ
3ro0.077-159.289
79.644-4.693
-0.392
-0.070
-0.018
-0.004
-0.001
0.000
0.000
-84.822
-159.289159.289
0.000
Tn
Tn.m
Tn
Tn
3 PÓRTICO CON DESPLAZAMIENTO EN NIVEL 1-2
1 Δ
3
6
3.1. MOMENTOS DE EMPOTRAMIENTO
ME13 = ME3 =ME24 = ME4 =
EΔ =
3.2. PROCESO ITERATIVO
2do0.242 0.758
-383.432 145.321
57.623 180.48817.022 -36.0464.604 14.420-2.134 -3.1531.279 4.008-0.492 -0.840 0.322 1.010-0.096 -0.2080.074 0.230-0.020 -0.0470.016 0.051-0.004 -0.0100.004 0.011-0.001 -0.0020.001 0.0020.000 0.0000.000 0.0010.000 0.0000.000 0.000
-305.235 305.235
3ro0.096 0.079 0.825 0.000-383.432
28.811 34.044 28.015 292.5622.302 42.157-4.268 -3.512 -36.679
(_6𝐸𝐼∆)/(𝐿.𝐿)(_6𝐸𝐼∆)/(𝐿.𝐿)
0.640 9.612-0.984 -0.810 -8.4580.161 1.845 -0.193 -0.158 -1.6550.037 0.374-0.039 -0.032 -0.3390.008 0.079-0.008 -0.007 -0.0720.002 0.017 -0.002 -0.001 -0.0150.000 0.0040.000 0.000 -0.0030.000 0.0010.000 0.000 -0.0010.000 0.0000.000 0.000 0.000
-322.922 23.493 299.429
11.747
3.3 CORTANTES EN COLUMNAS
193.279 Tn
305.235 Tn.m
193.279 Tn
193.279 Tn322.922 Tn.m
|
193.279 Tn
6.712 Tn
23.493 Tn.m
6.712 Tn
6.712 Tn
11.747 Tn.m
R´´1 =V´´2 =
3 PÓRTICO CON DESPLAZAMIENTO EN NIVEL 1-2
2 Δ
4 5
7 8
3.1. MOMENTOS DE EMPOTRAMIENTO
-383.432 Tn.m-383.432 Tn.m
100
3.2. PROCESO ITERATIVO
1ro0.758 0.242
-383.432290.641 92.79190.244 4.864-72.092 -23.0167.210 1.109-6.306 -2.0132.004 0.213
-1.680 -0.5360.505 0.043-0.415 -0.1330.115 0.009-0.094 -0.0300.025 0.002-0.021 -0.007 0.006 0.000 |-0.004 -0.0010.001 0.000-0.001 0.0000.000 0.0000.000 0.0000.000
310.138 -310.1384to 5to
0.442 0.197 0.051 0.310 0.000 0.000 0.923 0.077 -383.432
146.281 46.395 29.56784.314 37.579 9.729 59.134 -27.290 -2.277-18.339 -11.508 -13.645 6.74119.224 8.568 2.218 13.483 -6.222 -0.519
(_6𝐸𝐼∆)/(𝐿.𝐿)(_6𝐸𝐼∆)/(𝐿.𝐿)
-4.229 -1.007 -3.111 1.2943.689 1.644 0.426 2.587 -1.194 -0.100-0.827 -0.268 -0.597 0.2620.748 0.333 0.086 0.525 -0.242 -0.020-0.169 -0.066 -0.121 0.0550.158 0.070 0.018 0.111 -0.051 -0.004-0.036 -0.015 -0.026 0.0120.034 0.015 0.004 0.024 -0.011 -0.001-0.008 -0.003 -0.005 0.0030.007 0.003 0.001 0.005 -0.002 0.000-0.002 -0.001 -0.001 0.0010.002 0.001 0.000 0.001 -0.001 0.0000.000 0.000 0.000 0.0000.000 0.000 0.000 0.000 0.000 0.0000.000 0.000 0.000 0.0000.000 0.000 0.000 0.000 0.000 0.000
230.846 48.214 -337.423 58.363 2.921 -2.921
24.107
0.0000.000
1 0.000
3.3 CORTANTES EN COLUMNAS V´´2
199.250 Tn
310.138 Tn.m
199.250 Tn
199.250 Tn 337.423 Tn.m
199.250 Tn R´´1
13.775 Tn 0.556 Tn
48.214 Tn.m 2.921 Tn.m
13.775 Tn 0.556 Tn
13.775 Tn 0.556 Tn0.000
24.107 Tn.m
-412.460 tn = -412.460 Tn-392.528 tn = -392.528 Tn
4. CÁLCULO DE LAS CONSTANTES
R2 R´212.746 19.747
R1 V´117.309 403.070
R2 + R´2 A
R1 V´1 A
12.746 + 19.747 A
17.309 + 403.070 A
A =B =
5° CALCULOS DE LOS MOMENTOS FINALES
-6.614 -2.075 305.235 3.102 Tm5.884 -17.260 310.138 15.601 Tm6.614 2.075 -305.235 -3.102 Tm4.739 -0.036 -322.922 -5.563 Tm
-5.884 17.260 -310.138 -15.601 Tm-3.046 44.878 -337.423 -13.350 Tm-6.315 197.133 299.429 5.255 Tm29.164 412.717 230.846 40.754 Tm
-25.588 262.743 58.363 -21.036 Tm
ME12
ME21
ME13
ME31
ME24
ME42
ME34
ME43
ME45
1.284 84.822 2.921 2.246 Tm1.576 -197.096 23.493 0.307 Tm0.788 -196.507 11.747 -0.850 Tm
-0.530 -720.339 48.214 -6.368 Tm0.265 -813.684 24.107 -7.298 Tm
-1.284 -84.822 -2.921 -2.246 Tm0.000 0.000 0 0.000 Tm
ME54
ME36
ME63
ME47
ME74
ME58
ME85
V´´2392.528
R´´1412.460
V´´2 B 0
R´´1 B 0
+ 392.528 B = 0.000
+ 412.460 B = 0.000
0.0100.032
B. CALCULO DE REACCIONES
3.3T 3.3T
3.3T/M3.102 15.601
10.938 16.782
3.102
2.666
2.666
5.563 4.8T 4.8T
4.8T/M5.255 40.754
12.971 27.349
0.307
0.103
0.103
0.850
23.909
15.601
15.601
8.908
10.644 T
13.350 3.7T 3.7T
3.7T/M40.754 21.036
18.476 12.604
6.368
2.603
2.603
7.298
62.061
2.246
12.604
2.246
0.428
1.315 T
12.604
C. LOS DIAGRAMASC.1. De fuerzas normales
C.2. De fuerzas cortantes
C.3. De los momentos
D. LAS DEFLEXIONES EN LOS CENTROS DE LAS VIGASD.1. Deflecion en el centro de la viga 1-2
3.3T Q 3.3T
3.3T/M3.102 T.m 15.601 T.m
1 a c b 2
10.938+Q/2 16.782+Q/2
1.6 3.2 1.6
D.2. Deflexiones en el centro de la viga 3-4
4.8T Q 4.8T
𝑀1𝑎=(10.938+𝑄/2)𝑋−3.102−1.65𝑋^2𝑀𝑎𝑐=(10.938+ /2) −3.102−1.65 ^2𝑄 𝑋 𝑋 −3.3(𝑋−1.6)
𝑀2𝑏=(16.782+ /2) −𝑄 𝑋 15.601−1.65 ^2𝑋
𝑀𝑏𝑐=(16.782+ /2) −15.601−1.65 ^2𝑄 𝑋 𝑋 −3.3(𝑋−1.6))
𝜕𝑀1𝑎/𝜕𝑄=𝑋/2𝜕𝑀𝑎𝑐/
= /2𝜕𝑄 𝑋𝜕𝑀2𝑏/
= /2𝜕𝑄 𝑋𝜕𝑀𝑏𝑐/
= /2𝜕𝑄 𝑋𝛿𝑐=1/𝐸𝐼 ∫_0^1.6▒[(10.938)𝑋−3.102−1.65𝑋^2 ] (𝑋/2)𝑑𝑋+1/𝐸𝐼 ∫_1.6^3.2▒[(10.938)𝑋−3.102−1.65𝑋^2−3.3(𝑋−1.6)] (𝑋/2)𝑑𝑋+1/𝐸𝐼 ∫_0^1.6▒[(16.782)𝑋−15.601−1.65𝑋^2 ] (𝑋/2)𝑑𝑋"+" 1/𝐸𝐼 ∫_1.6^3.2▒[(16.782)𝑋−15.601−1.65𝑋^2−3.3(𝑋−1.6)] (𝑋/2)𝑑𝑋
𝛿=51.69.408/𝐸𝐼
4.8T/M5.255T.m 40.754 T.m
3 a c b 4
12.971 27.349
1.6 3.2 1.6
D.3. Deflexion en el centro de la viga 4-5
3.7T Q 3.7T
3.7T/M21.036 T.m 2.246 T.m
4 a c b 5
𝑀3𝑎=(12.971+𝑄/2)𝑋−5.255−1.65𝑋^2𝑀𝑎𝑐=(12.971+ /𝑄2) −𝑋 5.255−1.65 ^2𝑋 −3.3(𝑋−1.6)𝑀2𝑏=(27.349+ /2) −𝑄 𝑋 40.754−1.65 ^2𝑋
𝑀𝑏𝑐=(27.349+ /2) −𝑄 𝑋 40.754−1.65 ^2𝑋 −3.3(𝑋−1.6))
𝜕𝑀3𝑎/𝜕𝑄=𝑋/2𝜕𝑀𝑎𝑐/
= /2𝜕𝑄 𝑋𝜕𝑀2𝑏/
= /2𝜕𝑄 𝑋𝜕𝑀𝑏𝑐/
= /2𝜕𝑄 𝑋𝛿𝑐=1/𝐸𝐼 ∫_0^1.6▒[(12.971)𝑋−5.255−1.65𝑋^2 ] (𝑋/2)𝑑𝑋+1/𝐸𝐼 ∫_1.6^3.2▒[(12.971)𝑋−5.255−1.65𝑋^2−3.3(𝑋−1.6)] (𝑋/2)𝑑𝑋+1/𝐸𝐼 ∫_0^1.6▒[(27.349)𝑋−40.754−1.65𝑋^2 ] (𝑋/2)𝑑𝑋"+" 1/𝐸𝐼 ∫_1.6^3.2▒[(27.349)𝑋−40.754−1.65𝑋^2−3.3(𝑋−1.6)] (𝑋/2)𝑑𝑋
𝛿=47.90016/𝐸𝐼
18.476+Q/2 12.604+Q/2
1.6 3.2 1.6
𝑀4𝑎=(18.476+𝑄/2)𝑋−21.036−1.65𝑋^2𝑀𝑎𝑐=(18.476+ /𝑄2) −𝑋 21.036−1.65 ^2𝑋 −3.3(𝑋−1.6)
𝑀2𝑏=(12.604+ /2) −𝑄 𝑋 2.246−1.65 ^2𝑋
𝑀𝑏𝑐=(12.604+ /2) −𝑄 𝑋 2.246−1.65 ^2𝑋 −3.3(𝑋−1.6))
𝜕𝑀3𝑎/𝜕𝑄=𝑋/2𝜕𝑀𝑎𝑐/
= /2𝜕𝑄 𝑋𝜕𝑀2𝑏/
= /2𝜕𝑄 𝑋𝜕𝑀𝑏𝑐/
= /2𝜕𝑄 𝑋𝛿𝑐=1/𝐸𝐼 ∫_0^1.6▒[(18.476)𝑋−21.036−1.65𝑋^2 ] (𝑋/2)𝑑𝑋+1/𝐸𝐼 ∫_1.6^3.2▒[(18.476)𝑋−21.036−1.65𝑋^2−3.3(𝑋−1.6)] (𝑋/2)𝑑𝑋+1/𝐸𝐼 ∫_0^1.6▒[(12.604)𝑋−2.246−1.65𝑋^2 ] (𝑋/2)𝑑𝑋"+" 1/𝐸𝐼 ∫_1.6^3.2▒[(12.604)𝑋−2.246−1.65𝑋^2−3.3(𝑋−1.6)] (𝑋/2)𝑑𝑋
𝛿=55.61856/𝐸𝐼
𝛿𝑐=1/𝐸𝐼 ∫_0^1.6▒[(10.938)𝑋−3.102−1.65𝑋^2 ] (𝑋/2)𝑑𝑋+1/𝐸𝐼 ∫_1.6^3.2▒[(10.938)𝑋−3.102−1.65𝑋^2−3.3(𝑋−1.6)] (𝑋/2)𝑑𝑋+1/𝐸𝐼 ∫_0^1.6▒[(16.782)𝑋−15.601−1.65𝑋^2 ] (𝑋/2)𝑑𝑋"+" 1/𝐸𝐼 ∫_1.6^3.2▒[(16.782)𝑋−15.601−1.65𝑋^2−3.3(𝑋−1.6)] (𝑋/2)𝑑𝑋
𝛿𝑐=1/𝐸𝐼 ∫_0^1.6▒[(12.971)𝑋−5.255−1.65𝑋^2 ] (𝑋/2)𝑑𝑋+1/𝐸𝐼 ∫_1.6^3.2▒[(12.971)𝑋−5.255−1.65𝑋^2−3.3(𝑋−1.6)] (𝑋/2)𝑑𝑋+1/𝐸𝐼 ∫_0^1.6▒[(27.349)𝑋−40.754−1.65𝑋^2 ] (𝑋/2)𝑑𝑋"+" 1/𝐸𝐼 ∫_1.6^3.2▒[(27.349)𝑋−40.754−1.65𝑋^2−3.3(𝑋−1.6)] (𝑋/2)𝑑𝑋
𝛿𝑐=1/𝐸𝐼 ∫_0^1.6▒[(18.476)𝑋−21.036−1.65𝑋^2 ] (𝑋/2)𝑑𝑋+1/𝐸𝐼 ∫_1.6^3.2▒[(18.476)𝑋−21.036−1.65𝑋^2−3.3(𝑋−1.6)] (𝑋/2)𝑑𝑋+1/𝐸𝐼 ∫_0^1.6▒[(12.604)𝑋−2.246−1.65𝑋^2 ] (𝑋/2)𝑑𝑋"+" 1/𝐸𝐼 ∫_1.6^3.2▒[(12.604)𝑋−2.246−1.65𝑋^2−3.3(𝑋−1.6)] (𝑋/2)𝑑𝑋
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