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