Modal Espectral Ejemplo

CURSO Ingeniería Sismo Resistente II Análisis Sísmico por Superposición Modal Espectral

Ing. Omart Tello Malpartida

Introducción  25 0  M   0 0  0

k1  50000

0

0

25 0

0

0

0 20 0 0

0 20

0

0

0

  0  0  0   15  0

k2  40000

0 0  k1  k2 k2  0  k2 k2  k3 k3 k3 k3  k4 k4 K   0  0 k4 k4  k5 0  k5 0 0  0

Ing. Sismo Resistente II

k3  40000

  0  0  k5   k5  0

k4  30000

k5  20000

0.00 0.00   90000.00 40000.00 0.00   0.00 0.00 40000.00 80000.00 40000.00   40000.00 70000.00 30000.00 0.00  K   0.00  0.00 0.00 30000.00 50000.00 20000.00   0.00 0.00 20000.00 20000.00   0.00

Ing. Omart Tello Malpartida

Calculo de Frecuencias Angulares y Periodos de vibración K1  M

0.00  0.00  3600.00 1600.00 0.00  1600.00 3200.00 1600.00 0.00  0.00   2000.00 3500.00 1500.00 0.00  K1   0.00  0.00 1500.00 2500.00 1000.00 0.00   1333.33 1333.33  0.00 0.00  0.00

1

K

evp  eigenvals ( K1)

 6016.82  4227.54   evp   175.66   1093.43    2619.89

5to 4to 1ro 2do 3ro

Ing. Sismo Resistente II

Freq  sort ( evp )

 175.66   1093.43   Freq   2619.89  4227.54    6016.82

1ro 2do 3ro 4to 5to

Ing. Omart Tello Malpartida

Calculo de Frecuencias Angulares y Periodos de vibración i  1  5 i  Freq

i

T  i

2  i

1  13.25

T  0.47

2  33.07

T  0.19

3  51.18

T  0.12

4  65.02

T  0.10

5  77.57

T  0.08

Ing. Sismo Resistente II

1 2 3 4 5

Ing. Omart Tello Malpartida

Calculo de modos de vibración v  eigenvecs ( K1) 5to

 0.387  0.585  v   0.642  0.298   0.085

Ing. Sismo Resistente II

4to

1ro

2do

3ro

  0.265  0.337  0.551   0.571 

0.613 0.141 0.328 0.433 0.240 0.302 0.514 0.458 0.430 0.349 0.543 0.550 0.126 0.250 0.634 0.700

Ing. Omart Tello Malpartida

Normalizando los modos de vibración

V  1

V  4

v v

v v

3

1 3

2

1 2

 1.000   2.140    V   3.045  1  3.896     4.487 

 1.000   0.392    V   0.748  4  0.886     0.408 

Ing. Sismo Resistente II

V  2

V  5

v v

1 4

v v

4

1

1 1

 1.000   1.567    V   1.063  2  0.384     2.134 

V  3

v v

5

1 5

 1.000   0.613    V   0.778  3  1.273     1.319 

 1.000   1.511    V   1.659  5  0.770     0.219 

Ing. Omart Tello Malpartida

Matriz de modos de vibración Normalizados   V1 1 1   V   1 2  1      V1 3 1    V1 4  1    V1 5 1 

 1.000  2.140     3.045  3.896   4.487 Ing. Sismo Resistente II

V21  1 V22  1 V23  1 V24  1 V25  1

V31  1 V32  1 V33  1 V34  1 V35  1

V41  1 V42  1 V43  1 V44  1 V45  1

V51  1  V52  1  V53  1   V54  1   V  55  1 

1.000

1.000

1.000

1.567

0.613 0.392

  1.511  1.659  0.770   0.219 

1.063 0.778 0.748 0.384 1.273 0.886 2.134 1.319 0.408

1.000

Ing. Omart Tello Malpartida

Calculo de Me y Ke

T

Me    M  

T

Ke    K 

Ing. Sismo Resistente II

 930.53  0.00  Me   0.00  0.00   0.00

0.00

0.00

180.20 0.00 0.00

0.00 0.00

105.02 0.00

0.00

0.00

58.23

0.00

0.00

0.00

  0.00  0.00  0.00   149.69 0.00

0.00 0.00 0.00   163461.99 0.00  0.00 197033.12 0.00  0.00 0.00   Ke   0.00 0.00 275138.89 0.00 0.00   0.00 0.00 0.00 246170.59 0.00    0.00 0.00  0.00 0.00 900633.61  

Ing. Omart Tello Malpartida

Calculo de Vector de masas T L   .M .I participantes

1   1   I   1  1   1

Ing. Sismo Resistente II

T

L    M  I

 284.63  45.73    L   19.08   11.83     8.31 

Ing. Omart Tello Malpartida

Calculo de Masas Efectivas L( 1  1 ) Le  1 1 Me

Le

2 1





( 2  1 ) 

Me

Le

Le

4 1

5 1





Le

1 1

Le

2 1

L( 4  1 )

Le

3 1

 3.47

Masa efectiva total

Le

4 1

 2.4

Mtotal  105.00

2

( 5  5)

Ing. Sismo Resistente II

 5    Mtotal  Le i 1   i  1 



2

( 4  4)

L( 5  1 ) Me

 11.6

Li 2  i 1 M i n

2

( 3  3)

Me

 87.06

2

( 2  2)

L( 3  1 ) Le  3 1 Me

2

( 1  1)

L

Li 2 Lei  Mi

Le

5 1

 0.46

Ing. Omart Tello Malpartida

Factor de Participación Modal (FPMi)

 T .M .I Li FPM i  i  T   .M . M i

L  1 

( 1  1)

Me

 1  0.31

( 1  1)

L  2 

( 2  1)

Me

 2  0.25

Sumatoria de participacion de modos

( 2  2)

5

L  3 

( 3  1)

Me

 3  0.18

( 3  3)



 i  1.00

i1

L  4 

( 4  1)

Me

 4  0.2

( 4  4)

L  5 

( 5  1)

Me

 5  0.06

( 5  5)

Ing. Sismo Resistente II

Ing. Omart Tello Malpartida

Factor de Participación de Masas P

1

Le  100  1  1  Mtotal

P 



Le

2 1



Mtotal

P

Le  100  3  1 

P

P

3

4

5

P  82.92 1

> 90 %

 100

2

 Li 2  M  Pi   i  Mtotal

P  11.05 2

P  3.30

Sumatoria de participacion efectiva

Le  100  4  1 

P  2.29

Le  100  5  1 

  5  Suma  P i  i  1 

P  0.44

Mtotal

Mtotal

Mtotal

Ing. Sismo Resistente II

3

4



Suma  100.00

5

Ing. Omart Tello Malpartida

ANALISIS MODAL ESPECTRAL

Espectro NTE-030 Z U S R

= = = =

0.4 1.3 1.2 6.0

Lima Centro Comercia S2 Muro estruct.

g Tp

= =

980 0.6

ZUS.g R

=

101.92

T

C = 2.5( Tp/T)

C

0.08 0.10 0.12 0.19 0.47

18.75 15.00 12.50 7.89 3.19

2.5 2.5 2.5 2.5 2.5

Ing. Sismo Resistente II

Sa ZUS.g.C R 254.8 254.8 254.8 254.8 254.8

T

C = 2.5( Tp/T)

C

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

15.00 7.50 5.00 3.75 3.00 2.50 2.14 1.88 1.67 1.50 1.36 1.25 1.15 1.07 1.00 0.94 0.88 0.83 0.79 0.75 0.71 0.68 0.65 0.63 0.60 0.58 0.56 0.54 0.52 0.50

2.50 2.50 2.50 2.50 2.50 2.50 2.14 1.88 1.67 1.50 1.36 1.25 1.15 1.07 1.00 0.94 0.88 0.83 0.79 0.75 0.71 0.68 0.65 0.63 0.60 0.58 0.56 0.54 0.52 0.50

Sa ZUS.g.C R 254.8 254.8 254.8 254.8 254.8 254.8 218.4 191.1 169.9 152.9 139.0 127.4 117.6 109.2 101.9 95.6 89.9 84.9 80.5 76.4 72.8 69.5 66.5 63.7 61.2 58.8 56.6 54.6 52.7 51.0

Ing. Omart Tello Malpartida

ANALISIS MODAL ESPECTRAL Espectro NTE-030 Z U S R

= = = =

0.4 1.3 1.2 6.0

Lima Centro Comercia S2 Muro estruct.

g Tp

= =

980 0.6

ZUS.g R

=

101.92

Espectro NTE-030 300.0

Sa (m/seg2)

250.0 200.0 Sa

150.0 100.0

T

C = 2.5( Tp/T)

C

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 21

15.00 7.50 5.00 3.75 3.00 2.50 2.14 1.88 1.67 1.50 1.36 1.25 1.15 1.07 1.00 0.94 0.88 0.83 0.79 0.75 0 71

2.50 2.50 2.50 2.50 2.50 2.50 2.14 1.88 1.67 1.50 1.36 1.25 1.15 1.07 1.00 0.94 0.88 0.83 0.79 0.75 0 71

T

C = 2.5( Tp/T)

C

0.08 0.10 0.12 0.19 0.47

18.75 15.00 12.50 7.89 3.19

2.5 2.5 2.5 2.5 2.5

50.0 0.0 T (seg)

Ing. Sismo Resistente II

Sa ZUS.g.C R 254.8 254.8 254.8 254.8 254.8 254.8 218.4 191.1 169.9 152.9 139.0 127.4 117.6 109.2 101.9 95.6 89.9 84.9 80.5 76.4 72 Sa8 ZUS.g.C R 254.8 254.8 254.8 254.8 254.8

Ing. Omart Tello Malpartida

ANALISIS MODAL ESPECTRAL Calculo de Desplazamientos Máximos 1er Modo:

 1 1  

2do Modo:

1  13.25

T  0.47

Del espectro :

Sa1  254.8

 1.00   2.14    1   3.05   3.90     4.49 

 2 2  

 1.00     1.57  2   1.06   0.38     2.13 

Ing. Sismo Resistente II

Sd1 

Umax2  2 Sd2  2

Sa

2

Sa1

  1

2

Sd1  1.45

 2  0.25

2

Sa2  254.8





 0.444   0.950    Umax1   1.351   1.728     1.991 

T  0.19

Del espectro :

Sv

 1  0.31

1

Umax1  1 Sd1  1

2  33.07

Sd 

Sd2 

Sa2

2

2

Sd2  0.23

 0.059     0.093  Umax2   0.063   0.023     0.126 

Ing. Omart Tello Malpartida

ANALISIS MODAL ESPECTRAL Calculo de Desplazamientos Máximos 3er Modo:

3  51.18

Del espectro :

 3 3  

 1.00    0.61   3   0.78   1.27     1.32 

4to Modo:

Ing. Sismo Resistente II

Sd3 

Sa3  254.8

4  65.02

 1.00    0.39   4   0.75   0.89     0.41 

 3  0.18

3

Sa3

3

T  0.10

Sa4  254.8

Umax4  4 Sd4  4

Sd3  0.10

 4  0.20

4

Sd4 

2

 0.018    0.011   Umax3   0.014   0.023     0.023 

Umax3  3 Sd3  3

Del espectro :

 4 4  

T  0.12

Sa4

4

2

Sd4  0.06

 0.012    0.005   Umax4   0.009   0.011     0.005 

Ing. Omart Tello Malpartida

ANALISIS MODAL ESPECTRAL Calculo de Desplazamientos Máximos

5  77.57

5to Modo:

Del espectro :

 5 5  

 1.00   1.51    5   1.66   0.77     0.22 

Ing. Sismo Resistente II

T  0.08

 5  0.06

5

Sa5  254.8

Sd5 

Umax5  5 Sd5  5

Sa5

5

2

Sd5  0.04

 0.002   0.004    Umax5   0.004   0.002     0.001 

Ing. Omart Tello Malpartida

COMBINACION POR SUPERPOSICION MODAL U1 = Sumatoria de Valores Absolutos (A BS)

    U1      

Umax1

 Umax2

 Umax3

 Umax4

 Umax5

Umax1

 Umax2

 Umax3

 Umax4

 Umax5

Umax1

 Umax2

 Umax3

 Umax4

 Umax5

Umax1

 Umax2

 Umax3

 Umax4

 Umax5

Umax1

 Umax2

 Umax3

 Umax4

 Umax5

1 1 2 1 3 1 4 1 5 1

1 1 2 1 3 1 4 1 5 1

1 1 2 1 3 1 4 1 5 1

1 1 2 1 3 1 4 1 5 1

1 1 2 1 3 1 4 1 5 1

        

 0.54    1.06   U1   1.44   1.79     2.15  Ing. Sismo Resistente II

Ing. Omart Tello Malpartida

COMBINACION POR SUPERPOSICION MODAL U2= Raiz Cuadrada de la Suma de los Cuadrados ( RCSC)

     U2        

Umax11  12  Umax21  12  Umax31  12  Umax41  12  Umax51  12  Umax12  1  Umax22  1  Umax32  1  Umax43  1 2

2

2

2



Umax13  12  Umax23  12  Umax33  12  Umax43  12  Umax14  12  Umax24  12  Umax34  12  Umax44  12  Umax15  12  Umax25  12  Umax35  12  Umax45  12 

 0.45    0.95   U2   1.35   1.73     1.99  Ing. Sismo Resistente II

 Umax54  1  Umax53  12   2 Umax54  1   2 Umax55  1  2

Ing. Omart Tello Malpartida

COMBINACION POR SUPERPOSICION MODAL Umax

Norma NTE-030: 0.25ABS+0.75SRSS

 0.25 U11  1    0.25 U12  1   Umax  0.25 U13  1   0.25 U1  4 1   0.25 U1  5 1 

0.75 U2

1 1 

 0.75 U2 2 1   0.75 U2 3 1   0.75 U2 4 1   0.75 U2 5 1 

 1 

 0.47   0.98    Umax   1.37   1.74     2.03 

1 1

100

 1  0.0047

Umax  2 

2 1

100

 2  0.0098

Umax  3 

3 1

100

 3  0.0137

Umax  4 

4 1

100

 4  0.0174

Umax  5 

Ing. Sismo Resistente II

5 1

100

 5  0.0203

Ing. Omart Tello Malpartida

Distorsión de Entrepiso

Ing. Sismo Resistente II

Ing. Omart Tello Malpartida

Calculo de Cortantes de Entrepiso

V  234.93

V  k1   1

1

1

V  k2   2   1 2





V  204.45

V  k3   3   2 3





V  157.52

V  k4   4   3





V  110.52





V  57.88

4

V  k5   5   4 5

Ing. Sismo Resistente II

2 3 4 5

Ing. Omart Tello Malpartida

Fuerzas Sísmicas del Análisis Dinámico

F F F F F

1 2 3 4 5

 V  V  V  V  V

1 2 3 4

 V  V  V  V

F  30.48 2

1

F  46.93 3 4 5

5

2

F  47.00 3

F  52.64 4

F  57.88 5

5

V 



i1

Ing. Sismo Resistente II

F

i

V  234.93

Ing. Omart Tello Malpartida

Fuerzas Sísmicas del Análisis Dinámico (Método Alternativo)

Fd  K

Umax 100

 30.48   46.93    Fd   47.00   52.64    57.88   5

V 



i1

Ing. Sismo Resistente II

F

i

V  234.93

Ing. Omart Tello Malpartida

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