matriz de rigidez

1

Ing. Diego J. Barahona R., M.E.

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2


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Determinar la matriz de rigidez para el análisis sísmico de cada uno de los pórticos de la estructura

que se muestra en la figura. Considere:

f´c= 210 Kg/cm²

fy= 4200 Kg/cm²

Iv= 0.50 Igv

Ic= 0.80 Igv

Vigas de 30x35 cm

Figura N°1: Descripción de la estructura

1.) Análisis sísmico pórtico 1

Procedimiento tipo:

a) Seleccionamos el modelo matemático y definimos los grados de libertad

Se utilizo el modelo II, donde se considera que todos los elementos de la estructura son axialmente

rígidos A∞ para definir los grados de libertad que en este caso son:

𝑁𝐺𝐿 = 3(9) − 3(3) − 1(10)

𝑵𝑮𝑳 = 𝟖

3


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Figura N°2: Grados de libertad y numeración de elementos

b.) Determinar la matriz de rigidez de miembro para vigas y columnas

Cálculos tipo:

Viga 30x35cm

𝐸 = 15000 ∗ √𝐹´𝑐 ∗ 10

𝐸 = 15000 ∗ √210 ∗ 10

𝑬 = 𝟐𝟏𝟕𝟑𝟕𝟎𝟔. 𝟓𝟏𝟏𝟗 𝑻/𝒎²

𝑰𝒈𝒗 =𝑏 ∗ ℎ³

12

𝑰𝒈𝒗 =0.3 ∗ 0.35³

12= 0.0011

𝐼𝑣 = 0.5 ∗ 𝐼𝑔𝑣

𝐼𝑣 = 0.5 ∗ 0.001 = 𝟎. 𝟎𝟎𝟎𝟓

𝑘 =4𝐸𝐼𝑣

𝐿

𝑘 =4 ∗ (2173706.5119 ) ∗ 0.0005

4

𝒌 = 𝟏𝟏𝟔𝟒. 𝟗𝟕𝟎𝟖

𝑎 =2𝐸𝐼𝑣

𝐿

𝑎 =2 ∗ (2173706.5119 ) ∗ 0.0005

4

𝒂 = 𝟓𝟖𝟐. 𝟒𝟖𝟓

𝐾𝑣 = K a

a k

4


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𝐾𝑣4𝑚 = 1164.971 582.485

582.485 1164.971

𝐾𝑣5𝑚 = 931.977 465.988

465.988 931.977

Columna 30x3Ocm

𝑰𝒈𝒄 =𝑏 ∗ ℎ³

12

𝑰𝒈𝒄 =0.3 ∗ 0.30³

12= 0.000675

𝐼𝑐 = 0.8 ∗ 𝐼𝑔𝑐

𝐼𝑐 = 0.8 ∗ 0.0016 = 𝟎. 𝟎𝟎𝟎𝟓𝟒

𝑡 =12𝐸𝐼𝑐

𝐿³

𝑡 =12 ∗ (2173706.5119 ) ∗ 0.00054

3³

𝒕 = 𝟓𝟐𝟏. 𝟔𝟗𝟎

𝑏 =6𝐸𝐼𝑐

𝐿²

𝑏 =6 ∗ (2173706.5119 ) ∗ 0.0005

3²

𝒃 = 𝟕𝟖𝟐. 𝟓𝟑𝟒

𝑘 =4𝐸𝐼𝑐

𝐿

𝑘 =4 ∗ (2173706.5119 ) ∗ 0.00054

3

𝒌 = 𝟏𝟓𝟔𝟓. 𝟎𝟔𝟗

𝑎 =2𝐸𝐼𝑐

𝐿

𝑎 =2 ∗ (2173706.5119 ) ∗ 0.00054

3

𝒂 = 𝟕𝟖𝟐. 𝟓𝟑𝟒

𝑲𝒄 =

t -b -t -b

-b k b a

-t b t b

-b a b k

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𝑲𝒄𝟑𝒎 = 521.690 -782.534 -521.690 -782.534

-782.534 1565.069 782.534 782.534

-521.690 782.534 521.690 782.534

-782.534 782.534 782.534 1565.069

Nota: El procedimiento anterior se utiliza para determinar las matrices de rigidez para cada

elemento del pórtico tanto en vigas como en columnas de diferente sección y longitud.

c.) Matrices de rigidez de miembros y vectores de colocación para el pórtico 1

0 0 1 3

Vc1=

521.690 -782.534 -521.690 -782.534 0

-782.534 1565.069 782.534 782.534 0

-521.690 782.534 521.690 782.534 1

-782.534 782.534 782.534 1565.069 3

0 0 1 4

Vc2=

3220.306 -4830.459 -3220.30594 -4830.459 0

-4830.459 9660.918 4830.459 4830.459 0

-3220.306 4830.459 3220.306 4830.459 1

-4830.459 4830.459 4830.459 9660.918 4

0 0 1 5

Vc3=

1236.597 -1854.896 -1236.597 -1854.896 0

-1854.896 3709.792 1854.896 1854.896 0

-1236.597 1854.896 1236.597 1854.896 1

-1854.896 1854.896 1854.896 3709.792 5

1 3 2 6

Vc4=

901.480 -1126.849 -901.480 -1126.849 1

-1126.849 1878.082 1126.849 939.041 3

-901.480 1126.849 901.480 1126.849 2

-1126.849 939.041 1126.849 1878.082 6

6


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1 4 2 7

Vc5=

5564.689 -6955.861 -5564.689 -6955.861 1

-6955.861 11593.101 6955.861 5796.551 4

-5564.689 6955.861 5564.689 6955.861 2

-6955.861 5796.551 6955.861 11593.101 7

1 5 2 8

Vc6

2136.840 -2671.051 -2136.840 -2671.051 1

-2671.051 4451.751 2671.051 2225.875 5

-2136.840 2671.051 2136.840 2671.051 2

-2671.051 2225.875 2671.051 4451.751 8

3 4

Vc7 =

1164.971 582.485 3

582.485 1164.971 4

4 5

Vc8 =

931.977 465.988 4

465.988 931.977 5

6 7

Vc9 =

1164.971 582.485 6

582.485 1164.971 7

7 8

Vc10 =

931.977 465.988 7

465.988 931.977 8

7


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d.) Ensamblaje directo de la matriz de rigidez de la estructura

13581.602 -8603.009 -344.315 -2125.402 -816.154 -1126.849 -6955.861 -2671.051

-8603.009 8603.009 1126.849 6955.861 2671.051 1126.849 6955.861 2671.051

-344.315 1126.849 4608.122 582.485 0.000 939.041 0.000 0.000

𝑲 = -2125.402 6955.861 582.485 23350.967 465.988 0.000 5796.551 0.000

-816.154 2671.051 0.000 465.988 9093.520 0.000 0.000 2225.875

-1126.849 1126.849 939.041 0.000 0.000 3043.053 582.485 0.000

-6955.861 6955.861 0.000 5796.551 0.000 582.485 14311.367 465.988

-2671.051 2671.051 0.000 0.000 2225.875 0.000 465.988 5383.728

13581.602 -8603.009

𝑲𝑨𝑨 = -8603.009 8603.009

𝑲𝑨𝑩 = -344.3151115 -2125.401923 -816.1543383 -1126.849456 -6955.860838 -2671.050562

1126.849456 6955.860838 2671.050562 1126.849456 6955.860838 2671.050562

4608.121949 582.4854169 0 939.0412132 0 0

582.4854169 23350.96673 465.9883335 0 5796.550698 0

0 465.9883335 9093.52005 0 0 2225.875468

𝑲𝑩𝑩 = 939.0412132 0 0 3043.05326 582.4854169 0

0 5796.550698 0 582.4854169 14311.36667 465.9883335

0 0 2225.875468 0 465.9883335 5383.727603

0.000232775 -7.11081E-06 8.12533E-07 -7.29617E-05 5.90751E-06 -1.06521E-06

-7.11081E-06 4.67309E-05 -4.7108E-06 5.90751E-06 -1.93988E-05 4.25504E-06

𝑲𝑩𝑩−𝟏 =

8.12533E-07 -4.7108E-06 0.000113038 -1.06521E-06 4.25504E-06 -4.24499E-05

-7.29617E-05 5.90751E-06 -1.06521E-06 0.000354378 -1.69567E-05 2.58789E-06

5.90751E-06 -1.93988E-05 4.25504E-06 -1.69567E-05 7.90623E-05 -1.18025E-05

-1.06521E-06 4.25504E-06 -4.24499E-05 2.58789E-06 -1.18025E-05 0.000183788

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-344.3151115 1126.849456

-2125.401923 6955.860838

𝑲𝑩𝑨 = -816.1543383 2671.050562

-1126.849456 1126.849456

-6955.860838 6955.860838

-2671.050562 2671.050562

e.) Condensación estática de la matriz de rigidez

𝐾𝐿 = 𝐾𝐴𝐴 − 𝐾𝐴𝐵 − 𝐾𝐵𝐵−1 ∗ 𝐾𝐵𝐴

𝑲𝑳 (𝑷ó𝒓𝒕𝒊𝒄𝒐 𝟏) = 9062.125 -4186.473

-4186.473 2803.378



a.) Matrices de rigidez de miembros y vectores de colocación para el pórtico 2

0 0 1 3

Vc1=

1854.896 -2782.344 -1854.896 -2782.344 0

-2782.344 5564.689 2782.344 2782.344 0

-1854.896 2782.344 1854.896 2782.344 1

-2782.344 2782.344 2782.344 5564.689 3

9


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0 0 1 4

Vc2=

5893.562 -8840.344 -5893.56242 -8840.344 0

-8840.344 17680.687 8840.344 8840.344 0

-5893.562 8840.344 5893.562 8840.344 1

-8840.344 8840.344 8840.344 17680.687 4

0 0 1 5

Vc3=

2347.603 -3521.405 -2347.603 -3521.405 0

-3521.405 7042.809 3521.405 3521.405 0

-2347.603 3521.405 2347.603 3521.405 1

-3521.405 3521.405 3521.405 7042.809 5

1 3 2 6

Vc4=

3205.261 -4006.576 -3205.261 -4006.576 1

-4006.576 6677.626 4006.576 3338.813 3

-3205.261 4006.576 3205.261 4006.576 2

-4006.576 3338.813 4006.576 6677.626 6

1 4 2 7

Vc5=

10184.076 -12730.095 -10184.076 -12730.095 1

-12730.095 21216.825 12730.095 10608.412 4

-10184.076 12730.095 10184.076 12730.095 2

-12730.095 10608.412 12730.095 21216.825 7

1 5 2 8

Vc6=

4056.658 -5070.823 -4056.658 -5070.823 1

-5070.823 8451.371 5070.823 4225.685 5

-4056.658 5070.823 4056.658 5070.823 2

-5070.823 4225.685 5070.823 8451.371 8

10


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3 4

Vc7 =

1164.971 582.485 3

582.485 1164.971 4

4 5

Vc8=

931.977 465.988 4

465.988 931.977 5

6 7

Vc9=

1164.971 582.485 6

582.485 1164.971 7

7 8

Vc10=

931.977 465.988 7

465.988 931.977 8

b.) Ensamblaje directo de la matriz de rigidez de la estructura

27542.056 -17445.995 -1224.232 -3889.751 -1549.418 -4006.576 -12730.095 -5070.823

-17445.995 17445.995 4006.576 12730.095 5070.823 4006.576 12730.095 5070.823

-1224.232 4006.576 13407.286 582.485 0.000 3338.813 0.000 0.000

𝑲 = -3889.751 12730.095 582.485 40994.459 465.988 0.000 10608.412 0.000

-1549.418 5070.823 0.000 465.988 16426.157 0.000 0.000 4225.685

-4006.576 4006.576 3338.813 0.000 0.000 7842.597 582.485 0.000

-12730.095 12730.095 0.000 10608.412 0.000 582.485 23313.772 465.988

-5070.823 5070.823 0.000 0.000 4225.685 0.000 465.988 9383.348

27542.056 -17445.995

𝑲𝑨𝑨 = -17445.995 17445.995

𝑲𝑨𝑩 = -1224.231508 -3889.751194 -1549.418002 -4006.575843 -12730.09482 -5070.822551

4006.575843 12730.09482 5070.822551 4006.575843 12730.09482 5070.822551

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13407.28591 582.4854169 0 3338.813202 0 0

582.4854169 40994.45944 465.9883335 0 10608.41235 0

0 465.9883335 16426.15668 0 0 4225.685459

𝑲𝑩𝑩 = 3338.813202 0 0 7842.597238 582.4854169 0

0 10608.41235 0 582.4854169 23313.77219 465.9883335

0 0 4225.685459 0 465.9883335 9383.347585

8.35435E-05 -1.60866E-06 7.5109E-08 -3.56876E-05 1.62592E-06 -1.14569E-07

-1.60866E-06 2.77074E-05 -1.0721E-06 1.62592E-06 -1.26705E-05 1.11204E-06


7.5109E-08 -1.0721E-06 6.8906E-05 -1.14569E-07 1.11204E-06 -3.10863E-05

-3.56876E-05 1.62592E-06 -1.14569E-07 0.000143023 -4.31852E-06 2.66058E-07

1.62592E-06 -1.26705E-05 1.11204E-06 -4.31852E-06 4.88249E-05 -2.9255E-06

-1.14569E-07 1.11204E-06 -3.10863E-05 2.66058E-07 -2.9255E-06 0.000120716

-1224.231508 4006.575843

-3889.751194 12730.09482

𝑲𝑩𝑨 = -1549.418002 5070.822551

-4006.575843 4006.575843

-12730.09482 12730.09482

-5070.822551 5070.822551

c.) Condensación estática de la matriz de rigidez


𝑲𝑳 (𝑷ó𝒓𝒕𝒊𝒄𝒐 𝟐) = 16262.465 -6783.843

-6783.843 3879.115

12


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a.) Matrices de rigidez de miembros y vectores de colocación para el pórtico 3

0 0 1 3

Vc1=

1104.565 -1656.847 -1104.565 -1656.847 0

-1656.847 3313.695 1656.847 1656.847 0

-1104.565 1656.847 1104.565 1656.847 1

-1656.847 1656.847 1656.847 3313.695 3

0 0 1 4

Vc2=

2934.504 -4401.756 -2934.50379 -4401.756 0

-4401.756 8803.511 4401.756 4401.756 0

-2934.504 4401.756 2934.504 4401.756 1

-4401.756 4401.756 4401.756 8803.511 4

0 0 1 5

Vc3=

2641.053 -3961.580 -2641.053 -3961.580 0

-3961.580 7923.160 3961.580 3961.580 0

-2641.053 3961.580 2641.053 3961.580 1

-3961.580 3961.580 3961.580 7923.160 5

13


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1 3 2 6

Vc4=

1908.688 -2385.860 -1908.688 -2385.860 1

-2385.860 3976.434 2385.860 1988.217 3

-1908.688 2385.860 1908.688 2385.860 2

-2385.860 1988.217 2385.860 3976.434 6

1 4 2 7

Vc5=

5070.823 -6338.528 -5070.823 -6338.528 1

-6338.528 10564.214 6338.528 5282.107 4

-5070.823 6338.528 5070.823 6338.528 2

-6338.528 5282.107 6338.528 10564.214 7

1 5 2 8

Vc6=

4563.740 -5704.675 -4563.740 -5704.675 1

-5704.675 9507.792 5704.675 4753.896 5

-4563.740 5704.675 4563.740 5704.675 2

-5704.675 4753.896 5704.675 9507.792 8

3 4

Vc7 =

1164.971 582.485 3

582.485 1164.971 4

4 5

Vc8=

931.977 465.988 4

465.988 931.977 5

6 7

Vc9=

1164.971 582.485 6

582.485 1164.971 7

7 8

Vc10=

931.977 465.988 7

465.988 931.977 8

14


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18223.373 -11543.251 -729.013 -1936.773 -1743.095 -2385.860 -6338.528 -5704.675

-11543.251 11543.251 2385.860 6338.528 5704.675 2385.860 6338.528 5704.675

-729.013 2385.860 8455.099 582.485 0.000 1988.217 0.000 0.000

𝑲 = -1936.773 6338.528 582.485 21464.673 465.988 0.000 5282.107 0.000

-1743.095 5704.675 0.000 465.988 18362.929 0.000 0.000 4753.896

-2385.860 2385.860 1988.217 0.000 0.000 5141.405 582.485 0.000

-6338.528 6338.528 0.000 5282.107 0.000 582.485 12661.161 465.988

-5704.675 5704.675 0.000 0.000 4753.896 0.000 465.988 10439.769

𝑲𝑨𝑨 = 18223.373 -11543.251

-11543.251 11543.251

𝑲𝑨𝑩 = -729.0128595 -1936.772502 -1743.095252 -2385.860267 -6338.528189 -5704.67537

2385.860267 6338.528189 5704.67537 2385.860267 6338.528189 5704.67537

8455.099429 582.4854169 0 1988.21689 0 0

582.4854169 21464.67252 465.9883335 0 5282.106824 0

𝑲𝑩𝑩 = 0 465.9883335 18362.92919 0 0 4753.896142

1988.21689 0 0 5141.404613 582.4854169 0

0 5282.106824 0 582.4854169 12661.161148 465.9883335

0 0 4753.896142 0 465.9883335 10439.76895

0.000130576 -4.59937E-06 1.88308E-07 -5.09791E-05 4.27431E-06 -2.76536E-07

-4.59937E-06 5.21728E-05 -1.78948E-06 4.27431E-06 -2.20288E-05 1.79814E-06


1.88308E-07 -1.78948E-06 6.18104E-05 -2.76536E-07 1.79814E-06 -2.82265E-05

-5.09791E-05 4.27431E-06 -2.76536E-07 0.000215541 -1.17232E-05 6.49201E-07

4.27431E-06 -2.20288E-05 1.79814E-06 -1.17232E-05 8.88874E-05 -4.78637E-06

-2.76536E-07 1.79814E-06 -2.82265E-05 6.49201E-07 -4.78637E-06 0.000108855

15


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-729.0128595 2385.860267

-1936.772502 6338.528189

𝑲𝑩𝑨 = -1743.095252 5704.67537

-2385.860267 2385.860267

-6338.528189 6338.528189

-5704.67537 5704.67537



𝑲𝑳 (𝑷ó𝒓𝒕𝒊𝒄𝒐 𝟑) = 11263.151 -4888.780

-4888.780 2982.174711

4.) Análisis sísmico pórtico A


16


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a.) Matrices de rigidez de miembros y vectores de colocación para el pórtico A

0 0 1 3

Vc1=

521.690 -782.534 -521.690 -782.534 0

-782.534 1565.069 782.534 782.534 0

-521.690 782.534 521.690 782.534 1

-782.534 782.534 782.534 1565.069 3

0 0 1 4

Vc2=

2347.603 -3521.405 -2347.603 -3521.405 0

-3521.405 7042.809 3521.405 3521.405 0

-2347.603 3521.405 2347.603 3521.405 1

-3521.405 3521.405 3521.405 7042.809 4

0 0 1 5

Vc3=

1442.697 -2164.046 -1442.697 -2164.046 0

-2164.046 4328.091 2164.046 2164.046 0

-1442.697 2164.046 1442.697 2164.046 1

-2164.046 2164.046 2164.046 4328.091 5

1 3 2 6

Vc4=

901.480 -1126.849 -901.480 -1126.849 1

-1126.849 1878.082 1126.849 939.041 3

-901.480 1126.849 901.480 1126.849 2

-1126.849 939.041 1126.849 1878.082 6

1 4 2 7

Vc5=

4056.658 -5070.823 -4056.658 -5070.823 1

-5070.823 8451.371 5070.823 4225.685 4

-4056.658 5070.823 4056.658 5070.823 2

-5070.823 4225.685 5070.823 8451.371 7

1 5 2 8

Vc6=

2492.981 -3116.226 -2492.981 -3116.226 1

-3116.226 5193.709 3116.226 2596.855 5

-2492.981 3116.226 2492.981 3116.226 2

-3116.226 2596.855 3116.226 5193.709 8

17


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3 4

Vc7=

1553.294 776.647 3

776.647 1553.294 4

4 5

Vc8=

1164.971 582.485 4

582.485 1164.971 5

6 7

Vc9=

1553.294 776.647 6

776.647 1553.294 7

7 8

Vc10=

1164.971 582.485 7

582.485 1164.971 8


11763.1078 -7451.1181 -344.3151 -1549.4180 -952.1801 -1126.8495 -5070.8226 -3116.2257

-7451.1181 7451.1181 1126.8495 5070.8226 3116.2257 1126.8495 5070.8226 3116.2257

-344.3151 1126.8495 4996.4456 776.6472 0.0000 939.0412 0.0000 0.0000

𝑲 = -1549.4180 5070.8226 776.6472 18212.4453 582.4854 0.0000 4225.6855 0.0000

-952.1801 3116.2257 0.0000 582.4854 10686.771 0.0000 0.0000 2596.8547

-1126.8495 1126.8495 939.0412 0.0000 0.0000 3431.3769 776.6472 0.0000

-5070.8226 5070.8226 0.0000 4225.6855 0.0000 776.6472 11169.6362 582.4854

-3116.2257 3116.2257 0.0000 0.0000 2596.855 0.0000 582.4854 6358.6803

𝑲𝑨𝑨 = 11763.1078 -7451.1181

-7451.1181 7451.1181

18


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𝑲𝑨𝑩 = -344.3151 -1549.4180 -952.1801 -1126.8495 -5070.823 -3116.2257

1126.8495 5070.8226 3116.2257 1126.8495 5070.8226 3116.2257

4996.4456 776.6472 0.0000 939.0412 0.0000 0.0000

776.6472 18212.4453 582.4854 0.0000 4225.6855 0.0000

𝑲𝑩𝑩 = 0.0000 582.4854 10686.7714 0.0000 0.0000 2596.8547

939.0412 0.0000 0.0000 3431.3769 776.6472 0.0000

0.0000 4225.6855 0.0000 776.6472 11169.6362 582.4854

0.0000 0.0000 2596.8547 0.0000 582.4854 6358.6803

0.000213188 -1.10773E-05 8.78843E-07 -6.02518E-05 8.43922E-06 -1.132E-06

-1.10773E-05 6.10605E-05 -4.28522E-06 8.43922E-06 -2.3893E-05 3.93874E-06

8.78843E-07 -4.28522E-06 0.000104239 -1.13199E-06 3.93874E-06 -4.2932E-05

𝑲𝑩𝑩−𝟏 -6.02518E-05 8.43922E-06 -1.13199E-06 0.000313608 -2.5143E-05 2.76549E-06

8.43922E-06 -2.38926E-05 3.93874E-06 -2.51427E-05 0.000100881 -1.085E-05

-1.13199E-06 3.93874E-06 -4.29317E-05 2.76549E-06 -1.085E-05 0.000175792

-344.3151 1126.8495

-1549.4180 5070.8226

𝑲𝑩𝑨 = -952.1801 3116.2257

-1126.8495 1126.8495

-5070.8226 5070.8226

-3116.2257 3116.2257

e.) Condensación estática de la matriz de rigidez


𝑲𝑳 (𝑷ó𝒓𝒕𝒊𝒄𝒐 𝑨) = 7978.947 -3737.243

-3737.243 2555.035

19


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5.) Análisis sísmico pórtico B


a.) Matrices de rigidez de miembros y vectores de colocación para el pórtico B

0 0 1 3

Vc1=

2060.996 -3091.494 -2060.996 -3091.494 0

-3091.494 6182.987 3091.494 3091.494 0

-2060.996 3091.494 2060.996 3091.494 1

-3091.494 3091.494 3091.494 6182.987 3

0 0 1 4

Vc2=

5893.562 -8840.344 -5893.562 -8840.344 0

-8840.344 17680.687 8840.344 8840.344 0

-5893.562 8840.344 5893.562 8840.344 1

-8840.344 8840.344 8840.344 17680.687 4

0 0 1 5

Vc3=

3622.844 -5434.266 -3622.844 -5434.266 0

-5434.266 10868.533 5434.266 5434.266 0

-3622.844 5434.266 3622.844 5434.266 1

-5434.266 5434.266 5434.266 10868.533 5

20


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1 3 2 6

Vc4=

3561.401 -4451.751 -3561.401 -4451.751 1

-4451.751 7419.585 4451.751 3709.792 3

-3561.401 4451.751 3561.401 4451.751 2

-4451.751 3709.792 4451.751 7419.585 6

1 4 2 7

Vc5=

10184.076 -12730.095 -10184.076 -12730.095 1

-12730.095 21216.825 12730.095 10608.412 4

-10184.076 12730.095 10184.076 12730.095 2

-12730.095 10608.412 12730.095 21216.825 7

1 5 2 8

Vc6=

6260.275 -7825.343 -6260.275 -7825.343 1

-7825.343 13042.239 7825.343 6521.120 5

-6260.275 7825.343 6260.275 7825.343 2

-7825.343 6521.120 7825.343 13042.239 8

3 4

Vc7=

1553.294 776.647 3

776.647 1553.294 4

4 5

Vc8=

1164.971 582.485 4

582.485 1164.971 5

6 7

Vc9=

1553.294 776.647 6

776.647 1553.294 7

7 8

Vc10=

1164.971 582.485 7

582.485 1164.971 8

21


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31583.1538 -20005.7514 -1360.2572 -3889.7512 -2391.0772 -4451.7509 -12730.0948 -7825.3434

-20005.7514 20005.7514 4451.7509 12730.0948 7825.3434 4451.7509 12730.0948 7825.3434

-1360.2572 4451.7509 15155.8668 776.6472 0.0000 3709.7924 0.0000 0.0000

𝑲 = -3889.7512 12730.0948 776.6472 41615.7772 582.4854 0.0000 10608.4123 0.0000

-2391.0772 7825.3434 0.0000 582.4854 25075.742 0.0000 0.0000 6521.1195

-4451.7509 4451.7509 3709.7924 0.0000 0.0000 8972.8793 776.6472 0.0000

-12730.0948 12730.0948 0.0000 10608.4123 0.0000 776.6472 23935.0900 582.4854

-7825.3434 7825.3434 0.0000 0.0000 6521.120 0.0000 582.4854 14207.2099

𝑲𝑨𝑨 = 31583.1538 -20005.7514

-20005.7514 20005.7514

𝑲𝑨𝑩 = -1360.2572 -3889.7512 -2391.0772 -4451.7509 -12730.095 -7825.3434

4451.7509 12730.0948 7825.3434 4451.7509 12730.0948 7825.3434

15155.8668 776.6472 0.0000 3709.7924 0.0000 0.0000

776.6472 41615.7772 582.4854 0.0000 10608.4123 0.0000

𝑲𝑩𝑩 = 0.0000 582.4854 25075.7425 0.0000 0.0000 6521.1195

3709.7924 0.0000 0.0000 8972.8793 776.6472 0.0000

0.0000 10608.4123 0.0000 776.6472 23935.0900 582.4854

0.0000 0.0000 6521.1195 0.0000 582.4854 14207.2099

7.35575E-05 -1.83451E-06 7.02747E-08 -3.05684E-05 1.80756E-06 -1.0636E-07

-1.83451E-06 2.71653E-05 -8.63305E-07 1.80756E-06 -1.2121E-05 8.93189E-07

𝑲𝑩𝑩−𝟏 = 7.02747E-08 -8.63305E-07 4.53182E-05 -1.06365E-07 8.93189E-07 -2.0838E-05

-3.05684E-05 1.80756E-06 -1.06365E-07 0.000124505 -4.8471E-06 2.47549E-07

1.80756E-06 -1.21205E-05 8.93189E-07 -4.8471E-06 4.73662E-05 -2.352E-06

-1.06365E-07 8.93189E-07 -2.08377E-05 2.47549E-07 -2.352E-06 8.00478E-05

-1360.2572 4451.7509

-3889.7512 12730.0948

𝑲𝑩𝑨 = -2391.0772 7825.3434

-4451.7509 4451.7509

-12730.0948 12730.0948

-7825.3434 7825.3434

22


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𝑲𝑳 (𝑷ó𝒓𝒕𝒊𝒄𝒐 𝑩) = 18888.409 -7968.418

-7968.418 4642.355

6.) Análisis sísmico pórtico C


a.) Matrices de rigidez de miembros y vectores de colocación para el pórtico C

0 0 1 3

Vc1=

695.586 -1043.379 -695.586 -1043.379 0

-1043.379 2086.758 1043.379 1043.379 0

-695.586 1043.379 695.586 1043.379 1

-1043.379 1043.379 1043.379 2086.758 3

23


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0 0 1 4

Vc2=

1854.896 -2782.344 -1854.896 -2782.344 0

-2782.344 5564.689 2782.344 2782.344 0

-1854.896 2782.344 1854.896 2782.344 1

-2782.344 2782.344 2782.344 5564.689 4

0 0 1 5

Vc3=

2641.053 -3961.580 -2641.053 -3961.580 0

-3961.580 7923.160 3961.580 3961.580 0

-2641.053 3961.580 2641.053 3961.580 1

-3961.580 3961.580 3961.580 7923.160 5

1 3 2 6

Vc4=

1201.973 -1502.466 -1201.973 -1502.466 1

-1502.466 2504.110 1502.466 1252.055 3

-1201.973 1502.466 1201.973 1502.466 2

-1502.466 1252.055 1502.466 2504.110 6

1 4 2 7

Vc5=

3205.261 -4006.576 -3205.261 -4006.576 1

-4006.576 6677.626 4006.576 3338.813 4

-3205.261 4006.576 3205.261 4006.576 2

-4006.576 3338.813 4006.576 6677.626 7

1 5 2 8

Vc6=

4563.740 -5704.675 -4563.740 -5704.675 1

-5704.675 9507.792 5704.675 4753.896 5

-4563.740 5704.675 4563.740 5704.675 2

-5704.675 4753.896 5704.675 9507.792 8

3 4

Vc7=

1553.294 776.647 3

776.647 1553.294 4

4 5

Vc8=

1164.971 582.485 4

582.485 1164.971 5

6 7

Vc9=

1553.294 776.647 6

776.647 1553.294 7

24


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7 8

Vc10=

1164.971 582.485 7

582.485 1164.971 8


14162.5094 -8970.9737 -459.0868 -1224.2315 -1743.0953 -1502.4659 -4006.5758 -5704.6754

-8970.9737 8970.9737 1502.4659 4006.5758 5704.6754 1502.4659 4006.5758 5704.6754

-459.0868 1502.4659 6144.1626 776.6472 0.0000 1252.0550 0.0000 0.0000

𝑲 = -1224.2315 4006.5758 776.6472 14960.5804 582.4854 0.0000 3338.8132 0.0000

-1743.0953 5704.6754 0.0000 582.4854 18595.923 0.0000 0.0000 4753.8961

-1502.4659 1502.4659 1252.0550 0.0000 0.0000 4057.4043 776.6472 0.0000

-4006.5758 4006.5758 0.0000 3338.8132 0.0000 776.6472 9395.8917 582.4854

-5704.6754 5704.6754 0.0000 0.0000 4753.896 0.0000 582.4854 10672.7631

𝑲𝑨𝑨 = 14162.5094 -8970.9737

-8970.9737 8970.9737

𝑲𝑨𝑩 = -459.0868 -1224.2315 -1743.0953 -1502.4659 -4006.576 -5704.6754

1502.4659 4006.5758 5704.6754 1502.4659 4006.5758 5704.6754

6144.1626 776.6472 0.0000 1252.0550 0.0000 0.0000

776.6472 14960.5804 582.4854 0.0000 3338.8132 0.0000

𝑲𝑩𝑩 = 0.0000 582.4854 18595.9234 0.0000 0.0000 4753.8961

1252.0550 0.0000 0.0000 4057.4043 776.6472 0.0000

0.0000 3338.8132 0.0000 776.6472 9395.8917 582.4854

0.0000 0.0000 4753.8961 0.0000 582.4854 10672.7631

0.000175525 -1.10478E-05 5.25651E-07 -5.58072E-05 8.5823E-06 -7.0253E-07

-1.10478E-05 7.35651E-05 -3.02592E-06 8.5823E-06 -2.7026E-05 2.82279E-06

𝑲𝑩𝑩−𝟏 = 5.25651E-07 -3.02592E-06 6.08368E-05 -7.02531E-07 2.82279E-06 -2.7252E-05

-5.58072E-05 8.5823E-06 -7.02531E-07 0.000268537 -2.5352E-05 1.69654E-06

8.5823E-06 -2.70256E-05 2.82279E-06 -2.53517E-05 0.000118608 -7.7306E-06

-7.02531E-07 2.82279E-06 -2.72522E-05 1.69654E-06 -7.7306E-06 0.000106257

25


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-459.0868 1502.4659

-1224.2315 4006.5758

𝑲𝑩𝑨 = -1743.0953 5704.6754

-1502.4659 1502.4659

-4006.5758 4006.5758

-5704.6754 5704.6754



𝑲𝑳 (𝑷ó𝒓𝒕𝒊𝒄𝒐 𝑪) = 9265.751 -4226.763

-4226.763 2790.453

Documents

matriz de rigidez