SISMORESISTENTE

MODOS DE VIBRACIΓ“N AnΓ‘lisis del edificio en Y. Datos: 𝐸𝑐 = 200 𝑇𝑛/π‘π‘š2 Calcular: ο‚· ο‚· ο‚· ο‚· Matriz de masa [π‘š] =? Matriz de

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MODOS DE VIBRACIΓ“N AnΓ‘lisis del edificio en Y. Datos: 𝐸𝑐 = 200 𝑇𝑛/π‘π‘š2 Calcular: ο‚· ο‚· ο‚· ο‚·

Matriz de masa [π‘š] =? Matriz de rigidez [π‘˜] =? Matriz de frecuencia [𝑀 2 ] =? Grafica de los modos normalizados [Φ𝑛 ] =?

2

SOLUCIΓ“N:

Inercias del pΓ³rtico 1x-4x 𝒃 βˆ— π’‰πŸ‘ 𝑰= 𝟏𝟐 INERCIA 1 𝐼1 =

40 βˆ— (40)3 = 213333.33 π‘π‘š4 12

INERCIA 2 𝐼2 =

30 βˆ— (40)3 = 160000 π‘π‘š4 12

INERCIA 3 𝐼3 =

30 βˆ— (30)3 = 67500 π‘π‘š4 12

Rigideces del edificio 𝐾= 𝐾1 =

12𝐸𝐼 𝐻3

12 βˆ— (200 𝑇𝑛/π‘π‘š2 )(213333.33π‘π‘š4 ) (400 π‘π‘š)3 𝐾1 = 7.999 𝑇𝑛/π‘π‘š

𝐾2 = 𝐾2 =

12 𝐸𝐼 𝐻3

12 βˆ— (200 𝑇𝑛/π‘π‘š2 )(160000π‘π‘š4 ) (300π‘π‘š)3 𝐾2 = 14.22 𝑇𝑛 /π‘π‘š K2=k3=k4

𝐾= 𝐾5 =

12𝐸𝐼 𝐻3

12 βˆ— (200 𝑇𝑛/π‘π‘š2 )(67500 π‘π‘š4 ) (300 π‘π‘š)3 𝐾5 = 6.00 𝑇𝑛/π‘π‘š K5=K6

Resultados SENTIDO X

Columnas

PΓ³rtico 1X – 4X

PΓ³rtico 2X-3X

TOTAL

K1 K2 K3 K4 K5 K6

63,992 113,776 113,776 113.776 36.00 24.00

156.248 296.296 296.296 113.776 85.36 56.888

220.24 410.072 410.072 227.552 121.36 80.88

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DEFORMADA

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MATRIZ DE K (RIGIDEZ) π‘˜1 + π‘˜2 βˆ’π‘˜2 0 βˆ’π‘˜2 π‘˜2 + π‘˜3 βˆ’π‘˜3 0 βˆ’π‘˜3 π‘˜3 + π‘˜4 [π‘˜] = 0 0 βˆ’π‘˜4 0 0 0 [ 0 0 0

8π‘˜1 + 8π‘˜2 βˆ’8π‘˜2 0 βˆ’8π‘˜2 8π‘˜2 + 8π‘˜3 βˆ’8π‘˜3 0 βˆ’8π‘˜3 8π‘˜3 + 8π‘˜4 [π‘˜] = 0 0 βˆ’8π‘˜4 0 0 0 [ 0 0 0

0 0 0 0 0 0 βˆ’π‘˜4 0 0 π‘˜4 + π‘˜5 βˆ’π‘˜5 0 π‘˜5 + π‘˜6 βˆ’π‘˜6 βˆ’π‘˜5 βˆ’π‘˜6 π‘˜6 ] 0

0 0 0 0 0 0 βˆ’8π‘˜4 0 0 8π‘˜4 + 8π‘˜5 βˆ’6π‘˜5 0 6π‘˜5 + 4π‘˜6 βˆ’4π‘˜6 βˆ’6π‘˜5 βˆ’4π‘˜6 4π‘˜6 ] 0

K1 (MATRIZ DE RIGIDEZ) 177.768 βˆ’113.776 0 0 0 0 βˆ’113.776 227.552 βˆ’113.776 0 0 0 0 βˆ’113.776 βˆ’113.776 0 0 227.552 [π‘˜] = 𝑇𝑛/π‘π‘š2 0 0 0 βˆ’113.776 149.776 βˆ’36 0 0 βˆ’36 60 βˆ’24 0 [ 0 0 0 0 βˆ’24 24 ]

Inercia del pΓ³rtico 2x-3x INERCIA 1 50 βˆ— (50)3 𝐼1 = = 520833.33 π‘π‘š4 12 INERCIA 2 40 βˆ— (50)3 𝐼2 = = 416666.67 π‘π‘š4 12 INERCIA 3 𝐼3 =

30 βˆ— (40)3 = 160000 π‘π‘š4 12

8

Rigideces del edificio 𝐾= 𝐾1 =

12𝐸𝐼 𝐻3

12 βˆ— (200 𝑇𝑛/π‘π‘š2 )(520833.33π‘π‘š4 ) (400 π‘π‘š)3 𝐾1 = 19.531 𝑇𝑛/π‘π‘š

𝐾2 = 𝐾2 =

12 𝐸𝐼 𝐻3

12 βˆ— (200 𝑇𝑛/π‘π‘š2 )(416666.67π‘π‘š4 ) (300π‘π‘š)3 𝐾2 = 37.037 𝑇𝑛 /π‘π‘š K2=K3

𝐾= 𝐾4 =

12𝐸𝐼 𝐻3

12 βˆ— (200 𝑇𝑛/π‘π‘š2 )(160000 π‘π‘š4 ) (300 π‘π‘š)3 𝐾4 = 14.222 𝑇𝑛/π‘π‘š K4=K5=K6

8π‘˜1 + 8π‘˜2 βˆ’8π‘˜2 0 βˆ’8π‘˜2 8π‘˜2 + 8π‘˜3 βˆ’8π‘˜3 0 βˆ’8π‘˜3 8π‘˜3 + 8π‘˜4 [π‘˜] = 0 0 βˆ’8π‘˜4 0 0 0 [ 0 0 0

0 0 0 0 0 0 βˆ’8π‘˜4 0 0 8π‘˜4 + 8π‘˜5 βˆ’6π‘˜5 0 6π‘˜5 + 4π‘˜6 βˆ’4π‘˜6 βˆ’6π‘˜5 βˆ’4π‘˜6 4π‘˜6 ] 0

K2 (MATRIZ DE RIGIDEZ) 0 0 0 452.544 βˆ’296.296 0 0 0 0 βˆ’296.296 592.592 βˆ’296.296 βˆ’113.776 0 0 0 βˆ’296.296 410.072 [π‘˜] = 𝑇𝑛/π‘π‘š 0 0 0 βˆ’113.776 199.108 βˆ’85.332 0 0 0 βˆ’85.332 142.220 βˆ’56.88 [ βˆ’56.888 56.888 ] 0 0 0 0

K1+K2 (SUMA DE MATRIZ DE RIGIDEZ)

9

0 0 0 630.312 βˆ’410.072 0 0 0 0 βˆ’410.072 820.144 βˆ’410.072 0 βˆ’410.072 637.624 0 0 βˆ’227.552 [π‘˜] = 𝑇𝑛/π‘π‘š 0 0 βˆ’227.552 348.884 βˆ’121.332 0 0 0 0 βˆ’121.332 202.220 βˆ’80.888 [ 0 0 0 βˆ’80.888 80.888 ] 0

Matriz de masa π‘š1,2,3,4 =

𝑃 270 𝑇𝑛 = = 0.275 𝑇𝑛 𝑠 βˆ’2 /π‘π‘š 𝑔 980 π‘π‘š/𝑠 2 π‘š1 = π‘š2 = π‘š3 = π‘š4

π‘š5 =

𝑃 165 𝑇𝑛 = = 0.168 𝑇𝑛 𝑠 βˆ’2 /π‘π‘š 𝑔 980 π‘π‘š/𝑠 2

π‘š6 =

𝑃 75 𝑇𝑛 = = 0.076 𝑇𝑛 𝑠 βˆ’2 /π‘π‘š 𝑔 980 π‘π‘š/𝑠 2

π‘š1 0 0 π‘š2 0 [π‘š] = 0 0 0 0 0 [ 0 0

0 0 0 0 0 0 0 π‘š3 0 0 π‘š4 0 0 0 π‘š5 0 0 0

0 0 0 0 0 π‘š6]

0.275 0 0 0 0 0 0 0.275 0 0 0 0 0 0 0 0 0 0.275 [π‘š] = 𝑇𝑛𝑠 βˆ’2 /π‘π‘š 0.275 0 0 0 0 0 0 0.168 0 0 0 0 [ 0 0 0 0.076] 0 0

10

MATRIZ DE FRECUENCIA DEL SISTEMA 0 0 0 85.20 0 0 0 0 0 0 457.30 0 0 0 0 0 0 1163.80 [𝑀 2 ] = 0 1966.80 0 0 0 0 0 0 0 0 0 2632.50 [ 0 0 4824.10] 0 0 0

FRECUENCIA (W)= W1= W2= W3= W4= W5= W6=

9.230 rad 21.385 rad 34.115 rad 44.349 rad 51.308 rad 69.456 rad

 Periodo de LiberaciΓ³n: 𝑇1 =

2πœ‹ 2 βˆ— 3.1416 = = 0.681 𝑠𝑔 𝑀1 9.230 π‘ π‘”βˆ’1

𝑇2 =

2πœ‹ 2 βˆ— 3.1416 = = 0.294 𝑠𝑔 𝑀2 21.385 π‘ π‘”βˆ’1

𝑇3 =

2πœ‹ 2 βˆ— 3.1416 = = 0.184 𝑠𝑔 𝑀1 34.115 π‘ π‘”βˆ’1

𝑇4 =

2πœ‹ 2 βˆ— 3.1416 = = 0.142 𝑠𝑔 𝑀1 44.349 π‘ π‘”βˆ’1

𝑇5 =

2πœ‹ 2 βˆ— 3.1416 = = 0.122 𝑠𝑔 𝑀1 51.308 π‘ π‘”βˆ’1

𝑇6 =

2πœ‹ 2 βˆ— 3.1416 = = 0.090 𝑠𝑔 𝑀1 69.456 π‘ π‘”βˆ’1

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 Resultados

Matriz normalizada 1.0957 βˆ’π‘‚. 8397 βˆ’π‘‚. 4226 0.6915 βˆ’0.8449 0.6001 1.4258 βˆ’0.6255 0.8509 βˆ’0.6392 0.1309 βˆ’0.5202 0.7493 βˆ’0.9213 βˆ’0.5109 βˆ’1.1544 βˆ’0.7925 0.0653 [Φ𝑛 ] = 0.2201 βˆ’1.01220 0.1522 1.2431 βˆ’0.4531 0.8888 βˆ’1.2282 βˆ’1.1254 0.1730 1.6752 βˆ’0.5825 βˆ’0.0456 [ βˆ’1.3351 βˆ’1.9730 βˆ’1.8503 βˆ’1.9755 0.3953 0.0129 ]

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