JHON AYALA CARDOZO

TRABAJO DE INGENIERIA ANTISISMICA CALCULO DE RIGIDEZ LATERAL V1

V2

C2

V3

C2

C2

V1

V2

C2

C2 V2

C1

C1

5

Luces: Nombre:

Jhon

Apellido Paterno: Apellido Materno:

V2: h = L/12 = = b= h/2 V3: h = L/12 = b= h/2 =

7

Ayala

= =

4 5

Cardozo

=

7

0.33 m = 0.18 m =

Medidas Constructivas 0 .3 5 0 .2 5

m m (ancho minimo E.060)

0.42 m = 0.23 m =

Medidas Constructivas 0.45 0 .2 5

m m (ancho minimo E.060)

0.58 m = 0.3 m =

Medidas Constructivas 0 .6 0 .3

m m

Columna:

b= h=

0.3 0.4

m m

Datos:

f'c = Ec=

210 15000√f'c

=

217370.65 Kg/cm2 2173706.5119

Tn/m2

3

C1

Vigas:

V1: h = L/12 = = b= h/2

C2 V3

C1

4 Predimensionamiento

3

V3

C2 V1

C2

3.5

Calculo de Inercias:

I

bh 

3

12

I colum= bxh3/12 = I v1= bxh3/12 = I v2= bxh3/12 =

160000.00 89322.92 189843.75

cm4 cm4 cm4

= = =

I v3= bxh3/12 =

540000.00

cm4

=

0.0016 m4 0.00089323 m5 0.00189844 m6 0.0054

m7

A. METODO DE CROSS a1) Calculo de Rigideces:

EI L

K=

Como "E" es constante solo calculamos la relacion I/L En vigas: En columnas: I cm3 I Kv1= = 223.31 Kc1= L L

Kv2=

I L

=

379.69

Kv3=

I L

=

771.43

cm3

Kc2=

I L

=

457.14

=

533.33

cm3

a1) Calculo de los F actores de Distribución:

NUDOS V2

V1 4

V3

8 C2

C2 V1

12 C2

16 C2

V2

3

V3

7 C2

C2 V1

11 C2

15 C2

V2

V3

2

6 C1

10 C1

14 C1

1

5

9

13

C1

NUDO 2

FD2-1 =

FD2-3 =

-

=

457.14

457.14

+

457.14 533.33 +

223.31

+

533.33 533.33 +

223.31

223.31

=

0.3766

=

0 .4 3 9 4

=

cm3

cm3


I

bh 

3

12

I colum= bxh3/12 = I v1= bxh3/12 = I v2= bxh3/12 =

160000.00 89322.92 189843.75

cm4 cm4 cm4

= = =

I v3= bxh3/12 =

540000.00

cm4

=

0.0016 m4 0.00089323 m5 0.00189844 m6 0.0054

m7

A. METODO DE CROSS a1) Calculo de Rigideces:

EI L

K=

Como "E" es constante solo calculamos la relacion I/L En vigas: En columnas: I cm3 I Kv1= = 223.31 Kc1= L L

Kv2=

I L

=

379.69

Kv3=

I L

=

771.43

cm3

Kc2=

I L

=

457.14

=

533.33

cm3

a1) Calculo de los F actores de Distribución:

NUDOS V2

V1 4

V3

8 C2

C2 V1

12 C2

16 C2

V2

3

V3

7 C2

C2 V1

11 C2

15 C2

V2

V3

2

6 C1

10 C1

14 C1

1

5

9

13

C1

NUDO 2

FD2-1 =

FD2-3 =

-

=

457.14

457.14

+

457.14 533.33 +

223.31

+

533.33 533.33 +

223.31

223.31

=

0.3766

=

0 .4 3 9 4

=

cm3

cm3

-

457.14

+

533.33 +

223.31

+

533.33 533.33 +

223.31

+

533.33 533.33 +

223.31

+

223.31 533.33 +

223.31

.

NUDO 3

FD3-2 =

FD3-4 =

FD3-7 =

533.33

533.33

533.33

=

0 .4 1 3 4

=

0 .4 1 3 4

=

0.1731

NUDO 4

FD4-3=

533.33 533.33 + 223.31

=

0.7049

FD4-8=

223.31 533.33 + 223.31

=

0.2951

NUDO 6

FD6-2 =

FD6-7 =

FD6-10 =

FD6-5 =

223.31

223.31

223.31

223.31

+

223.31 533.33 + 379.69 +

457.14

+

533.33 533.33 + 379.69 +

457.14

+

379.69 533.33 + 379.69 +

457.14

+

457.14 533.33 + 379.69 +

457.14

+

223.31 533.33 + 379.69 +

533.33

+

533.33 533.33 + 379.69 +

533.33

+

379.69 533.33 + 379.69 +

533.33

+

533.33 533.33 + 379.69 +

533.33

=

0.1401

=

0.3347

=

0.2383

=

0.2869

=

0.1337

=

0.3194

=

0.2274

=

0.3194

NUDO 7

FD7-3 =

FD7-8 =

FD7-11 =

FD7-6 =

223.31

223.31

223.31

223.31

NUDO 8

FD8-4 =

FD8-12 =

FD8-7 =

223.31

223.31

223.31

+

223.31 379.69 +

533.33

+

379.69 379.69 +

533.33

+

533.33 379.69 +

533.33

+

379.69 533.33 + 771.43 +

457.14

+

533.33 533.33 + 771.43 +

457.14

=

0.1965

=

0.3341

=

0.4693

NUDO 10

FD10-6 =

FD10-11=

FD10-14 =

FD10-9 =

379.69

379.69

771.43 379.69

+

533.33 +

771.43 +

457.14

457.14 379.69

+

533.33 +

771.43 +

457.14

+

379.69 533.33 + 771.43 +

533.33

+

533.33 533.33 + 771.43 +

533.33

+

771.43 533.33 + 771.43 +

533.33

+

533.33 533.33 + 771.43 +

533.33

+

379.69 771.43 +

533.33

+

771.43 771.43 +

533.33

+

533.33 771.43 +

533.33

=

0.1773

=

0.2490

=

0.3602

=

0.2135

=

0.1712

=

0.2405

=

0.3478

=

0.2405

NUDO 11

FD11-7 =

FD11-12=

FD11-15 =

FD11-10 =

379.69

379.69

379.69

379.69

NUDO 12

FD12-8 =

FD12-16 =

FD12-11 =

379.69

379.69

379.69

=

0.2254

=

0.4580

=

0.3166

NUDO 14

FD14-10 =

FD14-15=

FD14-13=

771.43

771.43

771.43

+

771.43 533.33 +

457.14

+

533.33 533.33 +

457.14

+

457.14 533.33 +

457.14

+

771.43 533.33 +

533.33

+

533.33 533.33 +

533.33

+

533.33 533.33 +

533.33

=

0.4378

=

0.3027

=

0.2595

=

0.4197

=

0.2902

=

0.2902

NUDO 15

FD15-11 =

FD15-16=

FD15-14=

771.43

771.43

771.43

NUDO 16

FD16-12=

771.43 771.43 + 533.33

=

0.5912

FD16-15=

533.33 771.43 + 533.33

=

0.4088

a3)PORTICO A RESOLVER CON FUERZAS HORIZONTALES 3

4 2

1

8

12

16

7

11

15

6

10

14

5

9

13

3

2

1 Panteo de la Ecuación: -2.3003 X1

+ 15.6005 X2

+

-19.369 X3

=

1

15.8678 X1

+

-1.3123 X2

+

-7.7999 X3

=

2

-4.4567 X1

+

13.9508 X2

+

-11.128 X3

=

3

Resolviendo: X1=

0.440

X2=

0.763

X3=

0.511

CALCULO DE LOS MOMENTOS FINALES en TN.m NUDO MIEMBRO

1 1--2

2 2--6

2--1

2--3

∆1 (X1)

-4.45 -4.58 -0.11 4.70

∆2(X2)

1.34

∆3(X3)

-0.20 -0.21 -0.16 0.37

NUDO

6

MIEMBRO

6--2

2.63

6--5

3--2

3 3--7

4 3--4

4--3

3.43 -1.36 -2.07 -0.32

1.72 -4.35 -5.68 0.08 1.34

5.60

4--8

0.32 -4.45

2.70 -2.70 0.90

0.94 -2.28 -1.59 1.59 -0.12

7 6--10

6--7

7--3

5 5--6

7--6

7--11

7--8

8--4

8 8--7

9 8--12 9--10

∆1 (X1)

-0.13 -4.62 -0.23 4.98 -1.14 4.30 -1.53 -1.63 0.09 -0.01 -0.08 -4.45

∆2(X2)

1.59

∆3(X3)

-0.17 -0.15 -0.18 0.50

NUDO MIEMBRO

10 11 12 13 10--6 10--11 10--14 10--9 11--7 11--12 11--1511--10 12--8 12--16 12--1113--14

∆1 (X1)

-0.11 5.21 -0.48 -4.63 -0.96 -1.08 -2.71 4.74

∆2(X2)

1.25 -6.09 3.55

1.92

2.09 -5.60 -0.13 -7.10 -0.35 7.58 -2.18 4.63 -2.45 0.60 0.93

1.47

1.29 -0.14 6.98

1.40 -3.81

1.40 -3.14 1.74 -0.07

0.06

0.37 -0.43 -4.46

0.09 -6.93 -2.07 -4.02 6.08

0.81

-0.12 0.34 -0.20 -0.02 0.81 -3.79 2.08

∆3(X3)

NUDO

0.90

1.35

2.40 -3.74 -0.10

14 15 16 14--1014--15 14--1315--1115--16 15--1416--1216--15 -0.44 5.06 -4.62 -2.99 -1.38 4.37 0.47 -0.47

MIEMBRO ∆1 (X1) ∆2(X2)

3.88 -5.59 1.70

∆3(X3)

-0.26 0.35 -0.09 2.22 -3.28 1.06

0.13

6.49 -6.62 -4.93 4.93 2.95 -2.95

4 M4-3=

0.80

8 M8-7=

1.48

12 M12-11=

1.91

16 M16-15=

1.51

3 M3-4=

1.25

M7-8=

2.14

M11-12=

2.11

M15-16=

1.83

M3-2=

-0.90

7 M7-6=

-1.33

11 M11-10=

-1.28

15 M15-14=

-1.19

2 M2-3=

0.71

M6-7=

-0.12

M10-11=

-0.54

M14-15=

-0.17

M2-1=

-2.16

6 M6-5=

-2.86

10 M10-9=

-3.36

14 M14-13=

-3.00

M1-2=

-3.31

M5-6=

-3.67

M9-10=

-3.91

M13-14=

-3.74

1

5

9

13

Calculando las cortantes en Tn. V4-3=V3-4=

0.68113

V8-7=V7-8=

1.20451

V12-11=V11-12=

1.34115

V16-15=V15-16=

1.11437

V3-2=V2-3=

-0.064

V7-6=V6-7=

-0.4814

V11-10=V10-11=

-0.6083

V15-14=V14-15=

-0.4546

V2-1=V1-2=

-1.5633

V6-5=V5-6=

-1.8653

V10-9=V9-10=

-2.0775

V14-13=V13-14=

-1.9276

Calculando las deformaciones M



6 EI L

E= 2173706.51 Tn/m2 I= 0.0016 m4 

∆1-2=

-0.0067989 ∆5-6= -0.0075426 ∆9-10 = -0.0080437 ∆13-14= -0.0076886

+ + + +



ML

2



2

6 EI

∆2-1=

-0.0044429 ∆6-5= -0.0058711 ∆10-9= -0.0068959 ∆14-13= -0.0061731

= = = = ∆PISO 1=

∆2-3=

0.00092136 ∆6-7= -0.0001503 ∆10-11 = -0.0007008 ∆14-15= -0.0002239

+ + + +

∆3-2=

-0.0011697 ∆7-6= -0.0017185 ∆11-10= -0.0016606 ∆15-14= -0.0015406

= = = =

-0.0112418 -0.0134137 -0.0149396 -0.0138618

m m m m

-0.0534568 m -0.0002483 -0.0018688 -0.0023614 -0.0017645

m m m m

∆PISO 2=

∆3-4=

0.00161274 ∆7-8= 0.002764 ∆11-12= 0.00273463 ∆15-16= 0.00236921

+ + + +

∆4-3=

0.00103114 ∆8-7= 0.00191143 ∆12-11= 0.00247122 ∆16-15= 0.00195635

= = = = ∆PISO 2=

-0.006243 m

0.00264387 0.00467543 0.00520585 0.00432555

m m m m

0.01685071 m

1 a2)Calculo de los Momentos: SISTEMA MO

3TN 4

2TN

1TN

8

12

16

7

11

15

6

10

14

5

9

13

3

2

1

NO EXISTEN MOMENTOS DE EMPOTRAMIENTO PERFECTO

SISTEMA M3

M8-7

M4-3 4

8

M12-11 12

M16-15 16

3

M3-4

7

M7-8

M11-12

11

15

2

1

Haciendo M3-4 =

6

10

14

5

9

13

-10 Tn.m

M3-4 = M4-3 = M7-8 = M8-7 = M11-12 = M12-11 = M15-16 = M16-15

=

M2=

-10.00 Tn.m

M15-16

2 Momentos calculados en "crossmovnivel3"

CALCULO DE FUERZAS HORIZONTALES:

Sentido Positivo

4

8

12

F3

16

M4-3=

-3.11

M8-7=

-6.15

M12-11=

-7.33

M16-15=

-5.7

M3-4=

-4.47

M7-8=

-7.46

M11-12=

-7.43

M15-16=

-6.4

3

F3-4=

Sentido Negativo

7

-3

Tn

F7-8=

11

-5

Tn F3=

15

F11-12= 0

Tn

F15-16= -4

Tn

-11.128 Tn

F3

3

7

F2

15

M3-2=

2.63

M7-6=

2.89

M11-10=

1.76

M15-14=

2.08

M2-3=

0.72

M6-7=

0.98

M10-11=

0.66

M14-15=

0.69

2 F3-4=

11

6 1

Tn

F7-8=

1

Tn F2=

10 F11-12= 0

Tn

14 F15-16= 1

Tn

-7.7999 Tn

F3

F2

2

6

14

F1

M2-1=

-0.41

M6-5=

-0.29

M10-9=

-0.04

M14-13=

-0.18

M1-2=

-0.39

M5-6=

-0.23

M9-10=

-0.13

M13-14=

-0.19

1 F3-4=

10

5 -0

Tn

F7-8=

-0

Tn F1=

9 F11-12= 0

-19.369 Tn

Tn

13 F15-16= 0

Tn

3

SISTEMA M2

M4-3 4

M8-7 8

M3-4

M16-15

M12-11 12

16

M11-12

M7-8

M15-16

3

M3-2

7

M2-3

6

M7-6

M11-10

11

15

M15-14

14

M14-15

2

1 Haciendo M2-3 =

M6-7

5

10

M10-11

9

13

-10 Tn.m

M2-3 = M3-2 = M6-7 = M7-6 = M10-11 = M11-10 = M14-15 = M15-14

=

M2=

-10.00 Tn.m

M3-4 = M4-3 = M7-8 = M8-7 = M11-12 = M12-11 = M15-16 = M16-15

=

M2=

10.00

Tn.m



Sentido Positivo

4

8

12

F3

16

M4-3=

3.54

M8-7=

6.07

M12-11=

7.97

M16-15=

6.46

M3-4=

7.34

M7-8=

9.93

M11-12=

9.15

M15-16=

8.50

3

F3-4=

Sentido Negativo

7

4

Tn

F7-8=

11

5

Tn F3=

15

F11-12= 0

Tn

F15-16= 5

Tn

13.9508 Tn

F3

3

7

F2

15

M3-2=

-7.44

M7-6=

-9.31

M11-10=

-9.08

M15-14=

-8.6

M2-3=

-5.70

M6-7=

-7.34

M10-11=

-7.98

M14-15=

-7.3

2 F3-4=

11

6 -4

Tn

F7-8=

-6

Tn F2=

10 F11-12= 0

Tn

14 F15-16= -5

Tn

-1.3123 Tn

F3

F2

2

6

14

F1

M2-1=

3.45

M6-5=

2.51

M10-9=

1.69

M14-13=

2.23

M1-2=

1.75

M5-6=

1.18

M9-10=

0.79

M13-14=

1.06

1 F3-4=

10

5 2

Tn

F7-8=

1

Tn F1=

9 F11-12= 0

15.6005 Tn

Tn

13 F15-16= 0

Tn

5

SISTEMA M1

4 3

8

M3-2

12

M7-6

M11-10

7 2

M2-3 M2-1

Haciendo M1-2 =

15

M10-11 10

M6-5

5

M1-2

M15-14

11

M6-7 6

1

16

M14-15 14

M10-9

9

13

M9-10

M5-6

-10 Tn.m M X Y



M 12 ( EI ) X Y L12

( EI )1 2 L X 

M13-14

2

2 Y

Como EI es constante para las columnas: M

M X



Y



1 2

L X

L 1  2

2

2 

M14-13

Y

M1-2 = M2-1 = M5-6 = M6-5 = M9-10 = M10-9 = M13-14 = M14-13

=

-10.00

Tn.m

M2-3 = M3-2 = M6-7 = M7-6 = M10-11 = M11-10 = M14-15 = M15-14

=

13.61

Tn.m



Sentido Positivo

4

8

12

F3

16

M4-3=

-0.73

M8-7=

-0.03

M12-11=

-0.98

M16-15=

-1.0

M3-4=

-4.70

M7-8=

-3.71

M11-12=

-2.45

M15-16=

-3.1

3

F3-4=

Sentido Negativo

7

-2

Tn

F7-8=

11

-1

Tn F3=

15

F11-12= 0

Tn

F15-16= -1

Tn

-4.4567 Tn

F3

3

7

F2

15

M3-2=

7.80

M7-6=

9.77

M11-10=

10.78

M15-14=

9.93

M2-3=

10.67

M6-7=

11.32

M10-11=

11.84

M14-15=

11.4

2 F3-4=

11

6 6

Tn

F7-8=

7

Tn F2=

10 F11-12= 0

Tn

14 F15-16= 7

Tn

15.8678 Tn

F3

F2

2

6

14

F1

M2-1=

-10.41

M6-5=

-10.51

M10-9=

-10.51

M14-13=

-10.51

M1-2=

-10.10

M5-6=

-10.11

M9-10=

-10.11

M13-14=

-10.11

1 F3-4=

10

5 -7

Tn

F7-8=

-7

Tn F1=

9 F11-12= 0

-2.3003 Tn

Tn

13 F15-16= 0

Tn

13 k3a

k3b

k3c

k4d

k2a

k2b

k2c

k2d

k1a

k1b

k1c

k1d

Calculando las rigideces laterales

V

M 

k1a k1b k1c k1d

= = = =

K 1 PISO =

157.1366 157.1366 157.1366 157.1366



Tn/m Tn/m Tn/m Tn/m

k2a k2b k2c k2d

6285.46344 Kg/cm

k3a k3b k3c k3d

K2PISO

= = = =

K3PISO

174.568996 174.568996 174.568996 174.568996 =

Tn/m Tn/m Tn/m Tn/m

6982.75984 Kg/cm

Calculando Rigidez lateral total: K

1 

1 K 1

K=

= = = =

1 

K 2

1 

2244.58345 Tn/m

K 3

174.568996 174.568996 174.568996 174.568996 =

Tn/m Tn/m Tn/m Tn/m

6982.75984 Kg/cm

14

6

7

8

9

10

11

15

1

2

0 0.00 0.00 0.00 0.00 -0.39 0.00

-4.47

-3.11

0.68 -1.07

0.79 -0.87 -2.15 2.07 7.05

0

-1.73 3.52 4.13 3

0

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0.66

0.00 0.00 0.00 0.2595

0

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3.19 0.00 -0.55 -0.61 0.51

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1 1 4 . 0 7

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0 0

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0.21 0.00 -0.26 0.00 0.00 0.00 0.2135

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9

1 3

0

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2.63

0.16 0.00 -0.46 0.00 0.00 0.00 0.2869 0

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0.4134 0

0.4134 0.00 4.13 0.00 -1.73 -0.45 0.68

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5

0 0.00 0.00 0.00 0.00 -0.23 0.00

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0

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- 0 . 5 1 7

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9

1

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0 0 .0 0

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0.2902 -10.00

2.90 2.04 -1.10 -0.76 0.49 -6.42

4

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-5.70

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10.67

1

-0.24 0.35 1.35 -2.81 -1.59 13.61 0.4394

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0.4134 13.61 -5.63 -0.79 0.70 0.67 -0.77

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0.2490 13.61 -0.90 -1.64 0.64 0.47 -0.34

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0.3027 13.61 -1.09 -1.97 0.79 0.42 -0.27 11.49

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SISTEMA DE ECUACIONES LINEALES DE 3x3

A

V

R 1.000

A = Matriz de Coeficientes

- 2.300

15. 600

-19.369

X1

V = Matriz de Variables

15.868

-1.312

-7.800

X2

R = Matriz de Resultados

- 4.457

13. 951

-11.128

X3

-1

A

-0.106 -0.182 -0.186

-1

A xR=

0.083 0.052 0.032

0.127 0.280 0.211

X1=

0.440

X2=

0.763

X3=

0.511

=

2.000 3.000

TRABAJO DE INGENIERIA ANTISISMICA CALCULO DE RIGIDEZ LATERAL V1

V2

C3A

V3

C3B

C3C

V1

V2

C2A

C2C V2

C1A

DATOS Columna: b= 0.3 h= 0.4

C1C

7

Ec=

15000√f'c

=

m m

I colum= bxh3/12 = I v1= bxh3/12 = I v2= bxh3/12 = I v3= bxh3/12 =

I

bh 

3

12

160000.00 89322.92 189843.75 540000.00

cm4 cm4 cm4 cm4

h= h= h=

cm cm cm

= = = =

0.0016 0.00089323 0.00189844 0.0054

m4 m5 m6 m7

Rigidez Lateral

K LE



12 EI h

3

En el Primer Nivel En el Segundo Nivel En el Tercer Nivel

350 300 300

3

C1D

5


C2D V3

C1B

4

3

V3

C2B V1

C3D

KLE1= KLE2= KLE3=

9734.15 15457.47 15457.47

217370.65

Kg/cm2

3.5

Rigidez Relativa Elemento

Inercia(cm4)

Longitud

K(Cm3)

C-1

160000.00

350

457.14

C-2

160000.00

300

533.33

C-3

160000.00

300

533.33

V-1

89322.92

400

223.31

V-2

189843.75

500

379.69

V-3

540000.00

700

771.43

Caso de Elemento Empotrado en un lado y continuo en el

K 

KV1

KC

a



a

K LA

C1A

0.488484701

0.397223539

3866.633516

C1B

1.319051107

0.548063603

5334.933317

C1C

2.518066406

0.667999568

6502.407992

C1D

1.6875

0.593220339

5774.495763

SUMA

21478.471

Kg/cm

2147.847

Tn/m

Caso de Elementos Continuos

K 

en ambos Extremos

a

K V 1

 K V 2  K V 3  K V 4

2K C

KV3

K

En el Segundo Nivel Columna K

K V 2

0.50  K 2





K

aK LE

KV4

KC

2  K

KV1 K L



K C

KV2

K L

En el Primer Nivel Columna K

K V 1

 aK LE

a

K LA

C2A

0.418701172

0.173109922

2675.84143

C2B

1.130615234

0.361147937

5582.433406

C2C

2.158342634

0.519039152

8023.032119

C2D

1.446428571

0.419689119

6487.331969

SUMA

22768.639

Kg/cm

2276.864

Tn/m

KV2

En el Tercer Nivel Columna K

a

K LA

C2A

0.418701172

0.173109922

2675.84143

C2B

1.130615234

0.361147937

5582.433406

C2C

2.158342634

0.519039152

8023.032119

C2D

1.446428571

0.419689119

6487.331969

SUMA

22768.639

Kg/cm

2276.864

Tn/m

Resumen de Rigidez Lateral en cada piso Eje X-X (Pórticos Principales)

Pórtico A B C Total

1 2147.847 2147.847 2147.847 6443.5412

PISOS 2

2276.864 2276.864 2276.864 6830.5917

3 2276.864 2276.864 2276.864 6830.5917

Calculo de la rigidez lateral total K



1 1 K 1

K= 2232.16991 Tn/m



1 K 2



1 K 3

JHON AYALA CARDOZO

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