02 estimación del periodo de la estructura15 o

ESTIMACIÓN DEL PERIODO

DE LA ESTRUCTURA

Fórmulas de Wilbur.

Determinación de Rigideces

Diseño Sísmico de Edificios (Bazan-Meli)

𝑀𝑜𝑑𝑒𝑙𝑜 𝑒𝑛 𝐶𝑜𝑚𝑝𝑢𝑡𝑎𝑑𝑜𝑟𝑎.

𝑃𝑙𝑎𝑛𝑡𝑎 𝐵𝑎𝑗𝑎

𝑁𝑖𝑣𝑒𝑙 1







𝑋

z

𝑦

vigas: 90𝑥45 𝑐𝑚

𝐸 = 11000 𝑓´𝑐

𝑓´𝑐 = 250𝑘𝑔

𝑐𝑚2

𝑐𝑜𝑙𝑢𝑚𝑛𝑎𝑠: 80𝑥80𝑐𝑚

𝑘0

𝑘1

𝑘2

𝑘3

𝑘4

𝑘5

𝑘6

𝑘7𝐻

𝐻

𝐻

𝐻

𝐻

𝐻

𝐻

𝐻

𝐻 = 4.3 𝑚

𝐼𝑐 𝐼𝑐𝐼𝑐 𝐼𝑐 𝐼𝑐 𝐼𝑐𝐼𝑐








𝐼𝑣𝐼𝑣 𝐼𝑣 𝐼𝑣 𝐼𝑣 𝐼𝑣








𝐼𝑐=2733750 𝑐𝑚4

𝐼𝑣 = 3413333.33 𝑐𝑚4









𝑋

z

𝑦


𝐸 = 11000 𝑓´𝑐

𝑓´𝑐 = 250𝑘𝑔

𝑐𝑚2


𝑘0

𝑘1

𝑘2

𝑘3

𝑘4

𝑘5

𝑘6

𝑘7𝐻

𝐻

𝐻

𝐻

𝐻

𝐻

𝐻

𝐻

𝐻 = 4.3 𝑚

















𝐼𝑐=2733750 𝑐𝑚4

𝐼𝑣 = 3413333.33 𝑐𝑚4

𝐼𝑣 𝐼𝑣

𝐼𝑐𝐼𝑐 𝐼𝑐𝐻

𝐼𝑚 = 71458333𝑐𝑚4









𝑌

Z

𝑋


𝐸 = 11000 𝑓´𝑐

𝑓´𝑐 = 250𝑘𝑔

𝑐𝑚2


𝑘0

𝑘1

𝑘2

𝑘3

𝑘4

𝑘5

𝑘6

𝑘7𝐻

𝐻

𝐻

𝐻

𝐻

𝐻

𝐻

𝐻

𝐻 = 4.3 𝑚

𝐼𝑐 𝐼𝑐𝐼𝑐 𝐼𝑐








𝐼𝑣 𝐼𝑣 𝐼𝑣








𝐼𝑐=2733750 𝑐𝑚4

𝐼𝑣 = 3413333.33 𝑐𝑚4









𝑌

Z

𝑋


𝐸 = 11000 𝑓´𝑐

𝑓´𝑐 = 250𝑘𝑔

𝑐𝑚2


𝑘0

𝑘1

𝑘2

𝑘3

𝑘4

𝑘5

𝑘6

𝑘7𝐻

𝐻

𝐻

𝐻

𝐻

𝐻

𝐻

𝐻

𝐻 = 4.3 𝑚

















𝐼𝑐=2733750 𝑐𝑚4

𝐼𝑣 = 3413333.33 𝑐𝑚4

𝐼𝑣

𝐼𝑐𝐼𝑐

𝐻 𝑁𝑖𝑣𝑒𝑙 8

𝐼𝑚 = 71458333𝑐𝑚4

RIGIDECES EN DIRECCION X

A B C D SUMA (Ton/m)

K0 37226.326 37226.326 37226.326 37226.326 148905.3038

K1 25247.2523 25247.2523 25247.2523 25247.2523 100989.0091

K2 24311.799 24311.799 24311.799 24311.799 97247.19591

K3 24311.799 24311.799 24311.799 24311.799 97247.19591

K4 24311.799 24311.799 24311.799 24311.799 97247.19591

K5 24311.799 24311.799 24311.799 24311.799 97247.19591

K6 24311.799 24311.799 24311.799 24311.799 97247.19591

K7 24311.799 24311.799 24311.799 24311.799 97247.19591

K8 13381.0151 13381.0151 26762.03024

RIGIDECES EN DIRECCION Y

1 2 3 4 5 6 7 SUMA (Ton/m)

K0 20227.2746 20227.2746 20227.2746 20227.27459 20227.2746 20227.2746 20227.2746 141590.9221

K1 13345.9107 13345.9107 13345.9107 13345.91073 13345.9107 13345.9107 13345.9107 93421.37513

K2 12762.543 12762.543 12762.543 12762.54302 12762.543 12762.543 12762.543 89337.80116

K3 12762.543 12762.543 12762.543 12762.54302 12762.543 12762.543 12762.543 89337.80116

K4 12762.543 12762.543 12762.543 12762.54302 12762.543 12762.543 12762.543 89337.80116

K5 12762.543 12762.543 12762.543 12762.54302 12762.543 12762.543 12762.543 89337.80116

K6 12762.543 12762.543 12762.543 12762.54302 12762.543 12762.543 12762.543 89337.80116

K7 12762.543 12762.543 12762.543 12762.54302 12762.543 12762.543 12762.543 89337.80116

K8 7620.93707 7620.937068 7620.93707 22862.81121

Análisis Estático

DATOS DIRECCION (X,Y)

Q 2

c 0.45

F.I. 1.5

a´=(c/Q)*F.I. 0.3375

𝑭𝒊=𝒂´

𝒘𝒊 𝒘𝒊∗𝒉𝒊

∗𝒘𝒊𝒉𝒊

Calculo de Fuerzas (Fi).

𝐶𝑂𝑀𝑃𝑅𝑂𝐵𝐴𝐶𝐼Ó𝑁𝑉0𝑊𝑇

= 0.3375

Nivel wi (Ton) hi (m) wi*hi Fi (Ton) Vi(Ton)

8 151.2 38.7 5851.9 102 102

7 1150.6 34.4 39581.5 689 790

6 1311.8 30.1 39486.1 687 1478

5 1311.8 25.8 33845.2 589 2066

4 1311.8 21.5 28204.3 491 2557

3 1311.8 17.2 22563.5 393 2950

2 1311.8 12.9 16922.6 294 3244

1 1311.8 8.6 11281.7 196 3440

PB. 1311.8 4.3 5640.9 98 3539

10484.6 203377.6

Determinación del periodo fundamental de

la estructura, en dos direcciones.

𝑻 = 𝟐𝝅 𝑾𝒊𝑿𝒊

𝟐

𝒈 𝑭𝒊𝑿𝒊

𝑇𝑋 = 1.205 𝑠𝑒𝑔.

𝑇𝑌 = 1.252 𝑠𝑒𝑔.

Kx(Ton/m) Vi/K(m) xi(cm) wi*xi^2(ton-cm2) Fi*xi(ton-cm) Tx

26762.0 0.0038 0.2 5.8 20.0

97247.2 0.0081 0.2 42.6 132.5

97247.2 0.0152 0.2 44.5 126.6

97247.2 0.0212 0.2 37.5 99.6

97247.2 0.0263 0.1 28.7 72.5

97247.2 0.0303 0.1 19.4 47.7

97247.2 0.0334 0.1 10.9 26.8

100989.0 0.0341 0.1 4.4 11.4

148905.0 0.0238 0.0 0.7 2.3 1.205

194.5 539.4

Ky(Ton/m) Vi/K(m) xi(cm) wi*xi^2(ton-cm2) Fi*xi(ton-cm) Ty

22863.0 0.0 0.2 6.8 21.7

89338.0 0.0 0.2 49.9 143.4

89338.0 0.0 0.2 52.2 137.0

89338.0 0.0 0.2 43.9 107.7

89338.0 0.0 0.2 33.5 78.4

89338.0 0.0 0.1 22.6 51.5

89338.0 0.0 0.1 12.6 28.9

93421.0 0.0 0.1 5.0 12.1

141591.0 0.0 0.0 0.8 2.5 1.2525

227.3 583.2

Análisis Dinámico (Modal Espectral)

RIGIDEZ DE ENTREPISO EN DIRECCION X (Ton/m)

K0 148905 K1 100989 K2 97247 K3 97247 K4 97247 K5 97247 K6 97247 K7 97247 K8 26762

Matriz de rigidez lateral

Klat

K0 K1

K1

0

0

0

0

0

0

0

K1

K1 K2

K2

0

0

0

0

0

0

0

K2

K2 K3

K3

0

0

0

0

0

0

0

K3

K3 K4

K4

0

0

0

0

0

0

0

K4

K4 K5

K5

0

0

0

0

0

0

0

K5

K5 K6

K6

0

0

0

0

0

0

0

K6

K6 K7

K7

0

0

0

0

0

0

0

K7

K7 K8

K8

0

0

0

0

0

0

0

K8

K8

PROPIEDADES

Pesos en Toneladas Altura de entrepiso en m Modulo de elasticidad Ton/m2

w0 1311.8 H 4.30

w1 1311.8 E 14000 2500

w2 1311.8

w3 1311.8 E 7 105

w4 1311.8

w5 1311.8

w6 1311.8

w7 1150.6

w8 151.2

Matriz de masas (Ton*seg2)/m

M

w0

g

0

0

0

0

0

0

0

0

0

w1

g

0

0

0

0

0

0

0

0

0

w2

g

0

0

0

0

0

0

0

0

0

w3

g

0

0

0

0

0

0

0

0

0

w4

g

0

0

0

0

0

0

0

0

0

w5

g

0

0

0

0

0

0

0

0

0

w6

g

0

0

0

0

0

0

0

0

0

w7

g

0

0

0

0

0

0

0

0

0

w8

g

M

133.721

0

0

0

0

0

0

0

0

0

133.721

0

0

0

0

0

0

0

0

0

133.721

0

0

0

0

0

0

0

0

0

133.721

0

0

0

0

0

0

0

0

0

133.721

0

0

0

0

0

0

0

0

0

133.721

0

0

0

0

0

0

0

0

0

133.721

0

0

0

0

0

0

0

0

0

117.288

0

0

0

0

0

0

0

0

0

15.413

Matriz de Masas.

Frecuencias y Periodos.

Frecuencias

freq genvals Klat M( )

W2

2.656 103

1.594 103

1.994 103

2.337 103

2.853 103

1.101 103

617.229

236.81

27.181

Periodos

T 2

W

T

0.122

0.157

0.141

0.13

0.118

0.189

0.253

0.408

1.205

MODOS

modos genvecs Klat M( )

modo

0

0.125

0.305

0.48

0.637

0.771

0.876

0.948

0.984

1

0

0.339

0.732

0.902

0.778

0.401

0.107

0.579

0.864

1

0

0.449

0.744

0.419

0.262

0.72

0.568

0.067

0.645

1

0

0.421

0.429

0.212

0.532

0.046

0.509

0.294

0.366

1

0

0.701

0.914

0.995

1

0.928

0.785

0.583

0.335

0.521

0

0.331

0.205

0.102

0.329

0.297

0.032

0.259

0.346

1

0

0.214

0.036

0.197

0.182

0.062

0.228

0.107

0.149

1

0

0.295

0.107

0.323

0.045

0.332

0.019

0.328

0.082

1

0

0.686

0.715

0.442

0.015

0.417

0.704

0.747

0.53

1

Modos en dirección X

1 0.5 0 0.5 1

2

4

6

8

10

modo 1

1 0.5 0 0.5 1

2

4

6

8

10

modo 2

1 0.5 0 0.5 1

2

4

6

8

10

modo 3

1 0.5 0 0.5 1

2

4

6

8

10

modo 4

1 0.5 0 0.5 1

2

4

6

8

10

modo 5

1 0.5 0 0.5 1

2

4

6

8

10

modo 6

1 0.5 0 0.5 1

2

4

6

8

10

modo 7

1 0.5 0 0.5 1

2

4

6

8

10

modo 8

1 0.5 0 0.5 1

2

4

6

8

10

modo 9

RIGIDEZ DE ENTREPISO EN DIRECCION Y (Ton/m)

K0 141591 K1 93422 K2 89338 K3 89338 K4 89338 K5 89338 K6 89338 K7 89338 K8 22863

Matriz de rigidez lateral

Klat

K0 K1

K1

0

0

0

0

0

0

0

K1

K1 K2

K2

0

0

0

0

0

0

0

K2

K2 K3

K3

0

0

0

0

0

0

0

K3

K3 K4

K4

0

0

0

0

0

0

0

K4

K4 K5

K5

0

0

0

0

0

0

0

K5

K5 K6

K6

0

0

0

0

0

0

0

K6

K6 K7

K7

0

0

0

0

0

0

0

K7

K7 K8

K8

0

0

0

0

0

0

0

K8

K8

Frecuencias

freq genvals Klat M( )

W

51.247

49.41

38.019

42.217

46.101

31.798

23.863

14.795

5.015

Periodos

Ty 2

W

Ty

0.123

0.127

0.165

0.149

0.136

0.198

0.263

0.425

1.253

Modos en dirección Y

1 0.5 0 0.5 1

2

4

6

8

10

modo 1

1 0.5 0 0.5 1

2

4

6

8

10

modo 2

1 0.5 0 0.5 1

2

4

6

8

10

modo 3

1 0.5 0 0.5 1

2

4

6

8

10

modo 4

1 0.5 0 0.5 1

2

4

6

8

10

modo 5

1 0.5 0 0.5 1

2

4

6

8

10

modo 6

1 0.5 0 0.5 1

2

4

6

8

10

modo 7

1 0.5 0 0.5 1

2

4

6

8

10

modo 8

1 0.5 0 0.5 1

2

4

6

8

10

modo 9

PERIODOS

M. Estático

M. Dinámico.

𝑇𝑋 = 1.205 𝑠𝑒𝑔.

𝑇𝑌 = 1.252 𝑠𝑒𝑔.

Tx

0.117

0.121

0.128

0.137

0.155

0.187

0.251

0.405

1.195

MODOS

Ty

0.123

0.127

0.165

0.149

0.136

0.198

0.263

0.425

1.253

Masas y Periodos calculados (3D).

Nivel Peso (ton)Masa Traslacional

(Ton*s2/m)Masa

Rotacional.9 151.2 15.41 3012.7

8 1151 117.33 242789.1

7 1312 133.74 276750

6 1312 133.74 276750

5 1312 133.74 276750

4 1312 133.74 276750

3 1312 133.74 276750

2 1312 133.74 276750

P.B. 1312 133.74 276750

Modo Periodo (seg)1 Transversal 1.1556

2 Longitudinal 1.081

3 Rotacional 1.0415

4 0.3572

5 0.3371

6 0.312

7 0.2015

8 0.1926

9 0.1769

10 0.1378

11 0.1323

12 0.1193

Modo 1

Periodo correspondiente al primer modo (Transversal), T=1.1556 seg.

Modo 2

Periodo correspondiente al segundo modo (Longitudinal), T=1.0981 seg.

Modo 3

Periodo correspondiente al primer modo (Rotacional), T=1.0415 seg.

Education

02 estimación del periodo de la estructura15 o