ingenieria sismica

Halla el analisis estatatico en la direccion "X" de l a siguiente estructura.

♣ C = 0.4 x 0.4 mC VP VP ♣ Vp = 0.4 x 0.60 m

VS 2 VS 2 VS 2

♣ Vs = 0.4 x 0.4 m

5.0 m 5.4 ♣N° pisos= 3m ♣H. de piso a techo

h1= 3.5 h2= 3.5 h3= 3.5 mB VP 1 VP 2 ♣ e.losa= 0.17 m

VS 1 VS 1 VS 1

♣ P losa = 2805.4 ♣ universidad - Lima

5.0 m m ♣ Pacab= 100 kgm2

♣ con azotea= 25%A VP 1 VP 2 ♣ f'c= 210 kg/cm2

♣ fy= 4200 kg/m21 7.4 m 2 7.4 m 3 ♣ P.concre = 2400 kg/m2

♣ demas pisos = 50%7.0 m 7.0 m 0 ♣ S/C = 300

1._ Metrado de cargas

Elemento peso area long repet. 1er piso 2do piso 3er pisoPeso de Losa 1 280 35 - 1 9800 9800 9800Peso de Losa 2 280 35 - 1 9800 9800 9800Peso de Losa 3 280 35 - 1 9800 9800 9800Peso de Losa 4 280 35 - 1 9800 9800 9800

Viga Principal (0.40m*0.65m)viga principal 1 2400 0.24 7.0 3 12096 12096 12096viga principal 2 2400 0.24 7.0 3 12096 12096 12096

Viga Secundaria (0.40m*0.40m)Viga secundaria 1 2400 0.16 5.0 3 5760 5760 5760Viga secundaria 2 2400 0.16 5.0 3 5760 5760 5760

Columna (0.40 x 0.40 m )2400 0.16 3.5 9 120962400 0.16 3.5 9 120962400 0.16 1.8 9 6048

Acabado 1 100 35 - 1 3501 3501 3501Acabado 2 100 35 - 1 3501 3501 3501Acabado3 100 35 - 1 3501 3501 3501Acabado 4 100 35 - 1 3501 3501 3501

50% en los demas pisos , 25% azoteas/c 300 170.24 - 1 25536 25536 12768

Peso por piso 126548 126548 107732 6048Peso total 259144 kg.m

259.144 tn

1 2

3 4

3628812096120961209636288

1.75 Pesos relativos3.5 m

P3 = 107.732 tn3.5 P2 = 126.548 tn

3.5 m P1 = 126.548 tnP1'= 6.048

3.5 pt= 366.876 tn3.5 m

2.- Calculando Fuerzas sismicasZ = 0.4U universidad = 1.5F T = hn = 10.5 = 0.3

ei 35S = 1.2 Tp= 0.6R = 8C = 2.5Tp = 2.5x 0.6 = 5 2.5

T 0.3

Vb = ZUCS x P = 82.55R

PESO Hi Pi Hi x Pi % Vb F Vn3 10.5 107.732 1131.186 46% 82.5 38.0 38.02 7.0 126.548 885.836 36% 82.5 29.7 67.71 3.5 126.548 442.918 18% 82.5 14.9 82.55

2459.94 100% 82.55

38.0

29.7

14.9

𝑍_3𝑆_3

≤

Columana =C = 40 x 40 cm Ic = 213333Vp = 40 x 60 cm Ivp = 720000Vs = 40 x 40 cm Iv = 213333E= 15 x (210)^.5 = 217.371 Tn/cm2

ANALIZANDO EN EL EJE "X"

kv= 973 kv= 972.9740 x 60 40 x 60

kc= 609.52 609.52 609.52

1.60

40x 3.19

40x 1.60

40x

350 a= 0.44

40 0.61

40 0.44

40

5.76 7.98 5.76kv= 973 kv= 972.97

40 x 60 40 x 60kc= 609.52 609.52 609.52

1.60

40x 3.1940x 1.60

40x

350 a= 0.44

40 0.61

40 0.44

40

5.76 7.98 5.76kv= 973 kv= 972.97

40 x 60 40 x 60kc= 609.52 609.52 609.52

350 1.60

40x 3.19

40x 1.60

40x

a= 0.58

40 0.71

40 0.58

40

7.57 9.23 7.57

740 740

5.76 7.98 5.76 = 19.50 x 3 = 58.51

5.76 7.98 5.76 = 19.50 x 3 = 58.51

k=

K=

k=

K=

k=

K=

7.57 9.23 7.57 = 24.36 x 3 = 73.08

981

38.0

m3= 0.11K3= 5.76 K3 = 7.98 K3 = 5.76 K3= 19.50

+ + 29.7

=m2= 0.13

K2= 5.76 K2 = 7.98 K2 = 5.76 K2= 19.50

14.9

m1= 0.13K1 = 7.57 K1 = 9.23 K1 = 7.57 K1= 24.36

nivel w T U3 13.33 0.47 1.952 12.30 0.51 1.521 13.74 0.46 0.61

m= 0.10982

m= 0.128999

m= 0.128999

1.- Calculamos la matriz de rigidez

43.8622 -19.50 0k = -19.50 39.00 -19.50

0 -19.50 19.50

2.- Calculamos la matriz de masa:

0.11 0.00 0.00m = 0.00 0.13 0.00

0.00 0.00 0.13

k =43.86 -19.50 0.00-19.50 39.00 -19.500.00 -19.50 19.50

-0.0018 w^6 1.559 w^4

ORDENANDO

0.0018 w^6 -1.559 w^4

w1 =

[𝐾]−𝑊^2 [𝑀]=0[𝑲]− 𝒘^𝟐 [𝑴]=𝟎

w2 =w3 =

3.- Calculamos la matriz de aceleracion:

Z = 0.4U = 1.5

s 1.2R = 8

Tp = 0.6g = 981

C=

Para t1= 0.710

C= 2.11379914053796 Utilizamos

Para t2= 0.259

C= 5.8013687728792 Utilizamos

Para t3= 0.184C= 8.13528569154862 Utilizamos

A1 =A2 =A3 =

4.- Caculo de modos de vibracion:

k =43.86 -19.501717 Utilizamos-19.50 0.00 0.000.00 0.18 0.00

MODO 1 :w1^2 = 78.3933

35.25311 -19.5017 0.00-19.5017 28.8908 -19.500.0000 -19.502 9.39

11 =∅

[𝐾]−𝑊^2 [𝑀]∅=0

2.5*𝑇𝑝/𝑡 ≤ 2.5

21 =∅31 =∅

MODO 2:w2^2 = 590.49

-20.98 -19.50 0.00-19.50 -37.17 -19.500.00 -19.50 -56.67

11 =∅21 =∅31 =∅

MODO 3:w3^2 = 1161.17

-83.66 -19.50 0.00-19.50 -110.79 -19.500.00 -19.50 -130.29

11 =∅21 =∅31 =∅

MATRIZ MODO DE VIBRACION :

=∅

5.- Calculamos la matriz modal y espectral

MODO 1 :

MODO 2 :

MODO 3:

6.- Caculamos la matriz de desplazamiento

FACTOR DE PARTICIPACION : 0.58

[𝑃]=[Φ]^𝑇∗[𝑀]∗1

P = 0.03-0.23

MATRIZ DIAGONAL DE EIGENVALORES :


1.1550 0.0237U = 2.0879 -0.0255

4.3367 0.0088

7.- Calculamos la matriz de frecuencia lateral

43.86 -19.50 0.00F = -19.50 39.00 -19.50

0.00 -19.50 19.50

9.944 1.537F = -25.663 -1.629

43.856 0.669

8.- Calculo 8.- CALCULO DE LA CORTANTE BASAL

V = 28.137 0.578

Vb = 28.15 Tn

43.937 m= 0.1098Tn

k3 = 19.50 43.937

[𝑈]=[Φ]∗[𝑃]∗[𝐴]∗[Ω^2 ]^(−1)

[𝐹]=[𝐾]∗[𝑈]

[𝑉] 〖=([𝐹]^𝑇∗1)〗^𝑇

26.265 m= 0.1290Tn

k2 = 19.50 70.202

10.627 m= 0.1290Tn

k1 = 24.36 80.829

Tn*seg2/cm Z =U =

sk3 = 19.50 Tn/cm R =

Tp =g =

Tn*seg2/cm

Para t1= 0.710k2 = 19.50 Tn/cm

C= 2.1137991

Tn*seg2/cm Para t2= 0.259

C= 5.8013688k1 = 24.36 Tn/cm

Para t3= 0.184C= 8.1352857

Tn/ cm

Tn*seg2/cm

0.11 0.00 0.00-W^2 * 0.00 0.13 0.00 = 0

0.00 0.00 0.13

-323.738 w^2 9264.687 = 0

ORDENANDO

323.738 w^2 -9264.687 = 0

8.854 RAD/S T1 = 0.710 S

[𝐾]−𝑊^2 [𝑀]=0

24.300 RAD/S T2 = 0.259 S34.076 RAD/S T3 = 0.184 S

2.5

2.5

2.5

220.725 220.725 0 0220.725 A = 0 220.725 0220.725 0 0 220.725

0.11 0.00 0.00-W^2 * 0.00 0.13 0.00

0.00 0.00 0.13

*11∅21∅ = 031∅

1.000 cm

1.808 cm3.755 cm

*11∅21∅ = 031∅

1.000 cm-1.076 cm0.370 cm

*11∅21∅ = 031∅

1.000 cm-4.290 cm0.642 cm

1.000 1.00 1.0001.808 -1.08 -4.2903.755 0.37 0.642

1.420 0.704 1.838 0.625Φ = 1.273 -1.978 -2.683

2.644 0.681 0.4020.544

0.704 1.273 2.6441.599 Φ ^T = 1.838 -1.978 0.681

0.625 -2.683 0.402

0.583 0.000 0.000

[𝑃]=[Φ]^𝑇∗[𝑀]∗1

P = 0.000 0.035 0.0000.000 0.000 -0.226

78.39 0.00 0.00 0.01276 0.00000Ω^2 = 0.00 590.49 0.00 Ω^-2 = 0.00000 0.00169

0.00 0.00 1161.17 0.00000 0.00000

220.725 0.000 0.000A = 0.000 220.725 0.000

0.000 0.000 220.725

-0.0268 U1 = 1.16 cm0.1151 U2 = 2.09 cm-0.0172 U3 = 4.34 cm

*1.162.094.34

-3.421 F1 = 10.627 Tn 15.348 F2 = 26.265 Tn 1-2.580 F3 = 43.937 Tn 1

-0.654

[𝐹]=[𝐾]∗[𝑈]

0.41.51.28

0.6981

C=Utilizamos 2.5

Utilizamos 2.5

Utilizamos 2.5

0.000000.000000.00086

Columana =C = 40 x 40 cm Ic = 213333Vp = 40 x 60 cm Ivp = 720000Vs = 40 x 40 cm Iv = 213333E= 15 x (210)^.5 = 217.371 Tn/cm2

ANALIZANDO EN EL EJE "X"

kv= 427 kv= 42740 x 40 40 x 40

kc= 609.52 609.52 609.52

0.70

40x 1.40

40x 0.70

40x

350 a= 0.26

40 0.41

40 0.26

40

3.36 5.34 3.36kv= 427 kv= 427

40 x 40 40 x 40kc= 609.52 609.52 609.52

0.70

40x 1.40

40x 0.70

40x

350 a= 0.26

40 0.41

40 0.26

40

3.36 5.34 3.36kv= 427 kv= 427

40 x 40 40 x 40kc= 609.52 609.52 609.52

350 0.70

40x 1.40

40x 0.70

40x

a= 0.44

40 0.56

40 0.44

40

5.77 7.25 5.77

500 500

3.36 5.34 3.36 = 12.07 x 3 = 36.22

3.36 5.34 3.36 = 12.07 x 3 = 36.22

k=

K=

k=

K=

k=

K=

5.77 7.25 5.77 = 18.79 x 3 = 56.37

981

38.0

m3= 0.11K3= 3.36 K3 = 5.34 K3 = 3.36 K3= 12.07

+ + 29.7

=m2= 0.13

K2= 3.36 K2 = 5.34 K2 = 3.36 K2= 12.07

14.9

m1= 0.13K1 = 5.77 K1 = 7.25 K1 = 5.77 K1= 18.79

nivel w T U3 10.49 0.60 3.142 9.67 0.65 2.461 12.07 0.52 0.79

m= 0.10982

m= 0.128999

m= 0.128999

1.- Calculamos la matriz de rigidez

30.8637 -12.07 0k = -12.07 24.15 -12.07

0 -12.07 12.07


0.11 0.00 0.00m = 0.00 0.13 0.00

0.00 0.00 0.13

k =30.86 -12.07 0.00-12.07 24.15 -12.070.00 -12.07 12.07

-0.0018 w^6 1.027 w^4

ORDENANDO

0.0018 w^6 -1.027 w^4

w1 =

[𝐾]−𝑊^2 [𝑀]=0[𝑲]− 𝒘^𝟐 [𝑴]=𝟎

w2 =w3 =


Z = 0.4U = 1.5

s 1.2R = 8

Tp = 0.6g = 981

C=

Para t1= 0.710

C= 2.11379914053796 Utilizamos

Para t2= 0.259

C= 5.8013687728792 Utilizamos


A1 =A2 =A3 =


k =30.86 -12.074021 Utilizamos-12.07 0.00 0.000.00 0.18 0.00

MODO 1 :w1^2 = 78.3933

22.25465 -12.0740 0.00-12.0740 14.0354 -12.070.0000 -12.074 1.96

11 =∅

[𝐾]−𝑊^2 [𝑀]∅=0

2.5*𝑇𝑝/𝑡 ≤ 2.5

21 =∅31 =∅

MODO 2:w2^2 = 590.49

-33.98 -12.07 0.00-12.07 -52.02 -12.070.00 -12.07 -64.10

11 =∅21 =∅31 =∅

MODO 3:w3^2 = 1161.17

-96.65 -12.07 0.00-12.07 -125.64 -12.070.00 -12.07 -137.72

11 =∅21 =∅31 =∅


=∅


MODO 1 :

MODO 2 :

MODO 3:



[𝑃]=[Φ]^𝑇∗[𝑀]∗1

P = -0.17-0.29



0.3488 -0.0582U = 0.6429 0.1638

3.9576 -0.0309


30.86 -12.07 0.00F = -12.07 24.15 -12.07

0.00 -12.07 12.07

3.003 -3.775F = -36.471 5.032

40.022 -2.351


V = 6.554 -1.094

Vb = 6.65 Tn

40.139 m= 0.1098Tn

k3 = 12.07 40.139

[𝑈]=[Φ]∗[𝑃]∗[𝐴]∗[Ω^2 ]^(−1)

[𝐹]=[𝐾]∗[𝑈]

[𝑉] 〖=([𝐹]^𝑇∗1)〗^𝑇

37.033 m= 0.1290Tn

k2 = 12.07 77.172

5.381 m= 0.1290Tn

k1 = 18.79 82.553

Tn*seg2/cm Z =U =

sk3 = 12.07 Tn/cm R =

Tp =g =

Tn*seg2/cm

Para t1= 0.710k2 = 12.07 Tn/cm

C= 2.1137991


C= 5.8013688k1 = 18.79 Tn/cm

Para t3= 0.184C= 8.1352857

Tn/ cm

Tn*seg2/cm

0.11 0.00 0.00-W^2 * 0.00 0.13 0.00 = 0

0.00 0.00 0.13

-141.418 w^2 2739.195 = 0

ORDENANDO

141.418 w^2 -2739.195 = 0

8.854 RAD/S T1 = 0.710 S

[𝐾]−𝑊^2 [𝑀]=0

24.300 RAD/S T2 = 0.259 S34.076 RAD/S T3 = 0.184 S

2.5

2.5

2.5

220.725 220.725 0 0220.725 A = 0 220.725 0220.725 0 0 220.725

0.11 0.00 0.00-W^2 * 0.00 0.13 0.00

0.00 0.00 0.13

*11∅21∅ = 031∅

1.000 cm

1.843 cm11.347 cm

*11∅21∅ = 031∅

1.000 cm-2.815 cm0.530 cm

*11∅21∅ = 031∅

1.000 cm-8.005 cm0.702 cm

1.000 1.00 1.0001.843 -2.81 -8.005

11.347 0.53 0.702

3.824 0.262 0.918 0.344Φ = 0.482 -2.583 -2.752

2.967 0.487 0.2411.090

0.262 0.482 2.9672.908 Φ ^T = 0.918 -2.583 0.487

0.344 -2.752 0.241

0.474 0.000 0.000

[𝑃]=[Φ]^𝑇∗[𝑀]∗1

P = 0.000 -0.170 0.0000.000 0.000 -0.286

78.39 0.00 0.00 0.01276 0.00000Ω^2 = 0.00 590.49 0.00 Ω^-2 = 0.00000 0.00169

0.00 0.00 1161.17 0.00000 0.00000

220.725 0.000 0.000A = 0.000 220.725 0.000

0.000 0.000 220.725

-0.0187 U1 = 0.35 cm0.1497 U2 = 0.66 cm-0.0131 U3 = 3.96 cm

*0.350.663.96

-2.385 F1 = 5.381 Tn 14.000 F2 = 37.033 Tn 1-1.966 F3 = 40.139 Tn 1

-0.351

[𝐹]=[𝐾]∗[𝑈]

0.41.51.28

0.6981

C=Utilizamos 2.5

Utilizamos 2.5

Utilizamos 2.5

0.000000.000000.00086

Portico A Portico B Portico C

10 11 12 3

7 8 9 2

4 5 6 1

Ec = 15000 x (210)^.5 x 10 = 2173707 Tn /m2

C

B

A

10 11 12 3

7 8 9 2

6 EIc 6 EIc 6 EIc

12 EIc 12 EIc 12 EIc

6 EIc 6 EIc 6 EIc

4 5 6 1

6 EIc 6 EIc 6 EIc

12 EIc 12 EIc

12 EIc

6 EIc 6 EIc 6 EIc

K11 = 12 EIc + 12 EIc + 12 12 EIc + 12 EIc + 12 EIc = 72 EIch23 h13 h23 h13 h23 h13 h13

K21 = - 12 EIc - 12 - 12 EIc = -36 EIc K31 = 0h23 h23 h23 h23

K41 = 6 EIc - 6 EIc = 0 K51 = 6 EIc - 6 EIc = 0h12 h22 h12 h22

K61 = 6 EIc - 6 EIc = 0K71 = 6 EIc

h12 h22 h22

K81 = 6 EIc K91 = 6 EIch22 h22

K101 = 0 K111 = 0 K121= 0

h22 h22 h22

h23 h23 h23

h22 h22 h22

h12 h12 h12

h13 h13

h13

h12 h12 h12

10 11 12 3

6 EIc 6 EIc 6 EIc


6 EIc 6 EIc 6 EIc

7 8 9 2

6 EIc 6 EIc 6 EIc


6 EIc 6 EIc 6 EIc

4 5 6 1

K12 = - 12 EIc - 12 EIc - 12 EIc = -36 EIch23 h3 h3 h3

K22 = 12 EIc + 12 EIc + 12 EIc + 12 EIc + 12 EIc + 12 EIc = 72 EIch33 h23 h3 h3 h3 h3 h3

K32 = - 12 EIc - 12 EIc - 12 EIc = -36 EIc K42 = -6 EIch33 h3 h3 h3 h22

K52 = -6 EIc K62 =-6 EIc

K72 = 0h2 h2

K82 = 0 K92 = 0 K102 = 6 EIch32

K112 = 6 EIc K122 = 6 EIch2 h2

h32 h2 h2

h33 h3 h3

h32 h2 h2

h22 h2 h2

h23 h3 h3

h22 h2 h2

10 11 12 3

6 EIc 6 EIc 6 EIc


6 EIc 6 EIc 6 EIc

7 8 9 2

4 5 6 1

K13 = 0

K23 = - 12 EIc - 12 EIc - 12 EIc = -36 EIch3 h3 h3 h3

K33 = 12 EIc + 12 EIc + 12 EIc = 36 EIch3 h3 h3 h3

K43 = 0 K53 = 0 K63 = 0

K73 = -6 EIc K83 = -6 EIc K93 = -6 EIch2 h2 h2

K103 = -6 EIc K113 = -6 EIc K123 = -6 EIch2 h2 h2

h2 h2 h2

h3 h3 h3

h2 h2 h2

10 11 12 3

7 8 9 2

2 EIc

h2 6 EIc

4 EIc 2 EIv

h2 L1

4 5 6 1

4 EIc 4 EIv

h1 L1

6 EIc

2 EIc

h1

K14 = 6 EIc - 6 EIc = 0 K24 = -6 EIc K34 = 0h12 h22 h22

K44 = 4 EIc + 4 EIc + 4 EIv = 8 EIc + 4 EIv K54 = 2 EIvh1 h2 L1 h2 L1 L1

K64 = 0 K74 = 2 EIv K84 = 0h2

K94 = 0 K104 = 0 K114 = 0 K124 = 0

h22

h12

10 11 12 3

7 8 9 2

2 EIch 6 EIc

h2

4 EIv 4 EIc 2 EIvL h L

4 5 6 1

2 EIv 4 EIc 4 EIvL h L

6 EIc2 EIc h2h

K15 = 6 EIc - 6 EIc = 0 K25 = - 6 EIc K35 = 0h2 h2 h2

K45 = 2 EIv K55 = 4 EIc + 4 EIc + 4 EIv + 4 EIv = 8 EIc + 8 EIvL h h L L h L

K65 = 2 EIv K75 = 0 K85 = 2 EIc K95 = 0L h

K105 = 0 K115 = 0 K125 = 0

10 11 12 3

7 8 9 2

2 EIc6 EIc hh2

4 EIv 4 EIcL h

4 5 6

2 EIv 4 EIcL h

6 EIch2 2 EIc

h

K16 = 0 K26 = - 6 EIc K36 = 0h2

K46 = 0 K56 = 2 EIvL

K66 = 4 EIc + 4 EIc + 4 EIc = 8 EIc + 4 EIv K76 = 0h h L h L

K86 = 0 K96 = 2 EIc K106 = 0h

K116 = 0 K126 = 0

10 11 12 3

2 EIch 6 EIc

h24 EIc 2 EIvh L

7 8 9 2

4 EIc 4 EIvh L

6 EIc

h22 EIch

4 5 6 1

K17 = 6 EIc K27 = 0 K37 = - 6 EIch2 h2

K47 = 2 EIc K57 = 0 K67 = 0h

K77 = 4 EIc + 4 EIc + 4 EIv = 8 EIc + 4 EIv K87 = 2 EIvh h L h L L

K97 = 0K107 = - 2 EIc K117 =

0K127 =

0h

10 11 12

2 EIc 3

h 6 EIc

h24 EIc

4 EIv h 2 EIvL L

7 8 9 2

2 EIv 4 EIc 4 EIvL h 6 EIc L

h2

2 EIch 1

4 5 6

K18 = 6 EIc K28 = 6 EIc - 6 EIc = 0 K38 = - 6 EIch2 h2 h2 h2

K48 = 0 K58 =2 EIc

K68 = 0h

K78 = 2 EIv K88 = 4 EIv + 4 EIv + 4 EIc + 4 EIc = 8 EIv + 8 EIcL L L h h L h

K98 = 2 EIv K108 = 0 K118 = 2 EIc K128 = 0L h

10 11 12 3

2 EIc

6 EIc h

h24 EIv 4 EIc

L h7 8 9 2

2 EIv 4 EIcL 6 EIc h

h22 EIch

4 5 6 1

K19 = 6 EIc K29 = 6 EIc - 6 EIc = 0 K39 = - 6 EIch2 h2 h2 h2

K49 = 0 K59 = 0 K69 = 2 EIch

K79 = 0 K89 = 2 EIvL

K99 = 4 EIc + 4 EIc + 4 EIv = 8 EIc + 4 EIv K109 = 0h h L h L

K119 = 0 K129 = 2 EIch

2 EIvL

10 11 12 3

4 EIc 4 EIvh L

6 EIc2 EIc h2

h

7 8 9 2

4 5 6 1

K110 = 0 K210 = 6 EIc K310 = - 6 EIch2 h2

K410 = 0 K510 = 0 K610 = 0

K710 = 2 EIc K810 = 0 K910 = 0h

K1010 = 4 EIc + 4 EIv K1110 = 2 EIv K1210 = 0h L L

4 EIv 2 EIvL L

10 11 12

2 EIv 4 EIc 4 EIv 3L h L

6 EIc

2 EIc h2h

7 8 9 2

14 5 6

K111 = 0 K211 = 6 EIc K311 = - 6 EIch2 h2

K411 = 0 K511 = 0 K611 = 0

K711 = 0 K811 = 2 EIc K911 = 0h

K1011 = 2 EIv K1111 = 4 EIv + 4 EIv + 4 EIc = 8 EIv + 4 EIcL L L h L h

K1211 = 2 EIvL

4 EIvL

10 11 12 3

2 EIv 4 EIcL 6 EIc h

h22 EIch

7 8 9 2

4 5 6 1

K112 = 0 K212 = 6 EIc K312 = - 6 EIch2 h2

K412 = 0 K512 = 0 K612 = 0

K712 = 0 K812 = 0 K912 = 2 EIch

K1012 = 0 K1112 = 2 EIv K1212 = 4 EIv + 4 EIcL L h

1 2 3 4 5 6 7 8 9 10 11 12

1 72 EIc -36 EIc0 EIc 0 EIc 0 EIc 0 EIc

6 EIc 6 EIc 6 EIc0 EIc 0 EIc 0 EIc

h13 h3 h2 h2 h2

2 -36 EIc 72 EIc -36 EIc -6 EIc -6 EIc -6 EIc0 EIc 0 EIc 0 EIc

6 EIc 6 EIc 6 EIch23 h3 h3 h22 h2 h2 h2 h2 h2

30 EIc

-36 EIc 36 EIc0 EIc 0 EIc

0 EIc -6 EIc -6 EIc -6 EIc -6 EIc -6 EIc -6 EIch3 h3 h2 h2 h2 h2 h2 h2

40 EIc

-6 EIc0 EIc

8 EIc+ 4 EIv 2 EIv 0 EIc

2 EIc0 EIc 0 EIc 0 EIc 0 EIc 0 EIc

h22 h2 L1 L h

K =

50 EIc

-6 EIc0 EIc 2 EIv 8 EIc

+ 8 EIv 2 EIv 0 EIc2 EIc

0 EIc 0 EIc 0 EIc 0 EIch2 L1 h L L h

60 EIc

-6 EIc0 EIc 0 EIc 2 EIv 8 EIc

+ 4 EIv 0 EIc 0 EIc2 EIc

0 EIc 0 EIc 0 EIch2 L h L h

7 6 EIc0 EIc

-6 EIc 2 EIc 0 EIc0 EIc


2 EIc0 EIc 0 EIc

h22 h2 h2 h L L h

8 6 EIc0 EIc

-6 EIc0 EIc

2 EIc0 EIc 2 EIv 8 EIc

+ 8 EIv 2 EIv 0 EIc2 EIc

0 EIch22 h2 h L h L L h

9 6 EIc0 EIc

-6 EIc0 EIc 0 EIc

2 EIc0 EIc

2 EIv 8 EIc+

4 EIv0 EIc 0 EIc

2 EIch22 h2 h L h L h

100 EIc

6 EIc -6 EIc0 EIc 0 EIc 0 EIc

-2 EIc0 EIc 0 EIc


h32 h2 h h L L

110 EIc

6 EIc -6 EIc0 EIc 0 EIc 0 EIc 0 EIc

2 EIc0 EIc 2 EIv 8 EIv 4 EIc 2 EIv

h2 h2 h L L h L

120 EIc

6 EIc -6 EIc0 EIc 0 EIc 0 EIc 0 EIc 0 EIc

2 EIc0 EIc

2 EIv 4 EIc+

4 EIvh2 h2 h L h L

C = 40 x 40 cm Vp = 40 x 60 cm h1= 350 cmh2= 350 cm

Ic =b*h3

=213333.3333

Iv =b*h3

=720000.0000

h3 = 350 cm12 12 L1 = 700 cm

L2= 700 cm

77.87 Tn/cm -38.94 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 2271.30 Tn/cm 2271.30 Tn/cm 2271.30 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 0.00 Tn/cm-38.94 Tn/cm 77.87 Tn/cm -38.94 Tn/cm -2271.30 Tn/cm -2271.30 Tn/cm -2271.30 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 2271.30 Tn/cm 2271.30 Tn/cm 2271.30 Tn/cm0.00 Tn/cm -38.94 Tn/cm 38.94 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 0.00 Tn/cm -2271.30 Tn/cm -2271.30 Tn/cm -2271.30 Tn/cm -2271.30 Tn/cm -2271.30 Tn/cm -2271.30 Tn/cm0.00 Tn/cm -2271.30 Tn/cm 0.00 Tn/cm 1954265.66 Tn/cm 447162.48 Tn/cm 0.00 Tn/cm 264985.17 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 0.00 Tn/cm0.00 Tn/cm -2271.30 Tn/cm 0.00 Tn/cm 447162.48 Tn/cm 2848590.63 Tn/cm 447162.48 Tn/cm 0.00 Tn/cm 264985.17 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 0.00 Tn/cm

K = 0.00 Tn/cm -2271.30 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 447162.48 Tn/cm 1954265.66 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 264985.17 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 0.00 Tn/cm2271.30 Tn/cm 0.00 Tn/cm -2271.30 Tn/cm 264985.17 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 1954265.66 Tn/cm 447162.48 Tn/cm 0.00 Tn/cm 264985.17 Tn/cm 0.00 Tn/cm 0.00 Tn/cm2271.30 Tn/cm 0.00 Tn/cm -2271.30 Tn/cm 0.00 Tn/cm 264985.17 Tn/cm 0.00 Tn/cm 447162.48 Tn/cm 2848590.63 Tn/cm 447162.48 Tn/cm 0.00 Tn/cm 264985.17 Tn/cm 0.00 Tn/cm2271.30 Tn/cm 0.00 Tn/cm -2271.30 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 264985.17 Tn/cm 0.00 Tn/cm 447162.48 Tn/cm 1954265.66 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 264985.17 Tn/cm

0.00 Tn/cm 2271.30 Tn/cm -2271.30 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 264985.17 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 1424295.31 Tn/cm 447162.48 Tn/cm 0.00 Tn/cm0.00 Tn/cm 2271.30 Tn/cm -2271.30 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 264985.17 Tn/cm 0.00 Tn/cm 447162.48 Tn/cm 2318620.28 Tn/cm 447162.48 Tn/cm0.00 Tn/cm 2271.30 Tn/cm -2271.30 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 0.00 Tn/cm 264985.17 Tn/cm 0.00 Tn/cm 447162.48 Tn/cm 1424295.31 Tn/cm

0.00 Tn/cm -9.09385341357E-08 2.2002224079521E-08 -8.1354772110038E-08 2.323626378073E-08 -8.80762623248E-09 1.723996767409E-08 -6.70225430412E-09 3.742824255011E-09-9.09385341356986E-08 3.836654373156E-07 -9.0938534135699E-08 2.3236263780727E-08 -4.36898707811E-08 2.323626378073E-08 -6.70225430412E-09 7.578283320855E-09 -6.70225430412E-092.2002224079521E-08 -9.09385341357E-08 5.4354028150746E-07 -8.8076262324758E-09 2.323626378073E-08 -8.135477211E-08 3.742824255011E-09 -6.70225430412E-09 1.723996767409E-08

-8.13547721100381E-08 2.323626378073E-08 -8.8076262324758E-09 5.6078024918155E-07 -9.76407884398E-08 2.574504833453E-08 -1.15834707458E-07 3.664077238898E-08 -1.62932747425E-082.3236263780727E-08 -4.36898707811E-08 2.3236263780727E-08 -9.7640788439823E-08 3.912437206364E-07 -9.76407884398E-08 3.664077238898E-08 -5.88464374228E-08 3.664077238898E-08

-8.80762623247582E-09 2.323626378073E-08 -8.1354772110038E-08 2.5745048334532E-08 -9.76407884398E-08 5.607802491816E-07 -1.62932747425E-08 3.664077238898E-08 -1.15834707458E-071.72399676740927E-08 -6.70225430412E-09 3.7428242550114E-09 -1.1583470745822E-07 3.664077238898E-08 -1.62932747425E-08 7.752096964239E-07 -1.64220078914E-07 5.458877356452E-08-6.70225430412459E-09 7.578283320855E-09 -6.7022543041246E-09 3.6640772388976E-08 -5.88464374228E-08 3.664077238898E-08 -1.64220078914E-07 5.013583121611E-07 -1.64220078914E-073.74282425501136E-09 -6.70225430412E-09 1.7239967674093E-08 -1.6293274742499E-08 3.664077238898E-08 -1.15834707458E-07 5.458877356452E-08 -1.64220078914E-07 7.752096964239E-07

K11 K12 K21 K22

〖 "[" 𝑲_𝟐𝟐]〗^(−𝟏) =

77.87 Tn/cm -38.94 Tn/cm 0.00 Tn/cm

-6.05505663710207 -0.23456956520676 -5.14432378008157

-38.94 Tn/cm 77.87 Tn/cm -38.94 Tn/cm -0.23 Tn/cm 13.46 Tn/cm -7.41 Tn/cm0.00 Tn/cm -38.94 Tn/cm 38.94 Tn/cm -5.14 Tn/cm -7.41 Tn/cm 11.99 Tn/cm

=71.82 Tn/cm -38.70 Tn/cm 5.14 Tn/cm

-38.70 Tn/cm 64.41 Tn/cm -31.53 Tn/cm5.14 Tn/cm -31.53 Tn/cm 26.94 Tn/cm

KL =

KL

𝑲𝑳=[𝑲_𝟏𝟏 ]−[𝑲_𝟏𝟐]∗〖 "[" 𝑲_𝟐𝟐]〗^(−𝟏)∗[𝑲_𝟐𝟏]

E = 217.3706512

m= 0.10982

m= 0.128999

m= 0.128999

1.- Calculamos la matriz de rigidez lateral

71.8181 -38.7020 5.1443k = -38.7020 64.4108 -31.5292

5.1443 -31.5292 26.9438


0.11 0.00 0.00m = 0.00 0.13 0.00

0.00 0.00 0.13

k =71.82 -38.70 5.14-38.70 64.41 -31.535.14 -31.53 26.94

-0.0018 w^6 2.489 w^4

ORDENANDO

0.0018 w^6 -2.489 w^4

w1 =

[𝐾]−𝑊^2 [𝑀]=0[𝑲]− 𝒘^𝟐 [𝑴]=𝟎

w2 =w3 =


Z = 0.4U = 1.5

s 1.2R = 8

Tp = 0.6g = 981

C=

Para t1= 0.710

C= 2.11379914053796 Utilizamos

Para t2= 0.259

C= 5.8013687728792 Utilizamos


A1 =A2 =A3 =


k =71.82 -38.702028 Utilizamos-38.70 0.00 0.005.14 0.18 0.00

MODO 1 :w1^2 = 78.3933

63.20910 -38.7020 5.14-38.7020 54.2981 -31.535.1443 -31.529 16.83

11 =∅

[𝐾]−𝑊^2 [𝑀]∅=0

2.5*𝑇𝑝/𝑡 ≤ 2.5

21 =∅31 =∅

MODO 2:w2^2 = 590.49

6.97 -38.70 5.14-38.70 -11.76 -31.535.14 -31.53 -49.23

11 =∅21 =∅31 =∅

MODO 3:w3^2 = 1161.17

-55.70 -38.70 5.14-38.70 -85.38 -31.535.14 -31.53 -122.85

11 =∅21 =∅31 =∅


=∅


MODO 1 :

MODO 2 :

MODO 3:



[𝑃]=[Φ]^𝑇∗[𝑀]∗1

P = 0.32-0.04



1.4075 0.3274U = 2.2987 0.0590

4.3061 -0.0378


71.82 -38.70 5.14F = -38.70 64.41 -31.53

5.14 -31.53 26.94

34.269 21.040F = -42.178 -7.683

50.786 -1.193


V = 42.877 12.164

Vb = 44.57 Tn

50.806 m= 0.1098Tn

0 0.00 50.806

[𝑈]=[Φ]∗[𝑃]∗[𝐴]∗[Ω^2 ]^(−1)

[𝐹]=[𝐾]∗[𝑈]

[𝑉] 〖=([𝐹]^𝑇∗1)〗^𝑇

42.912 m= 0.1290Tn

0 0.00 93.719

40.247 m= 0.1290Tn

0 0.00 133.966

Tn*seg2/cm Z =U =

sR =

Tp =g =

Tn*seg2/cm

Para t1= 0.710

C= 2.1137991


C= 5.8013688

Para t3= 0.184C= 8.1352857

Tn/ cm

Tn*seg2/cm

0.11 0.00 0.00-W^2 * 0.00 0.13 0.00 = 0

0.00 0.00 0.13

-734.547 w^2 12886.684 = 0

ORDENANDO

734.547 w^2 -12886.684 = 0

8.854 RAD/S T1 = 0.710 S

[𝐾]−𝑊^2 [𝑀]=0

24.300 RAD/S T2 = 0.259 S34.076 RAD/S T3 = 0.184 S

2.5

2.5

2.5

220.725 220.725 0 0220.725 A = 0 220.725 0220.725 0 0 220.725

0.11 0.00 0.00-W^2 * 0.00 0.13 0.00

0.00 0.00 0.13

*11∅21∅ = 031∅

1.000 cm

1.633 cm3.059 cm

*11∅21∅ = 031∅

1.000 cm0.180 cm-0.115 cm

*11∅21∅ = 031∅

1.000 cm-1.439 cm0.369 cm

1.000 1.00 1.0001.633 0.18 -1.4393.059 -0.12 0.369

1.196 0.836 2.723 1.555Φ = 1.365 0.490 -2.237

2.558 -0.314 0.5740.367

0.836 1.365 2.5580.643 Φ ^T = 2.723 0.490 -0.314

1.555 -2.237 0.574

0.598 0.000 0.000

[𝑃]=[Φ]^𝑇∗[𝑀]∗1

P = 0.000 0.322 0.0000.000 0.000 -0.044

78.39 0.00 0.00 0.01276 0.00000Ω^2 = 0.00 590.49 0.00 Ω^-2 = 0.00000 0.00169

0.00 0.00 1161.17 0.00000 0.00000

220.725 0.000 0.000A = 0.000 220.725 0.000

0.000 0.000 220.725

-0.0129 U1 = 1.45 cm0.0186 U2 = 2.30 cm-0.0048 U3 = 4.31 cm

*1.452.304.31

-1.676 F1 = 40.247 Tn 11.852 F2 = 42.912 Tn 1-0.783 F3 = 50.806 Tn 1

-0.607

[𝐹]=[𝐾]∗[𝑈]

0.41.51.28

0.6981

C=Utilizamos 2.5

Utilizamos 2.5

Utilizamos 2.5

0.000000.000000.00086

Raíces de una ecuación cúbica

Una ecuación cúbica general con a ≠ 0 tiene la forma: ax³ + bx² + cx + d = 0a b c d

0.0018 -2.4893 734.547 -12886.684 0.00182746165028074x³ - 2.48927888367399x² + 734.547157193912x - 12886.683712948 = 0

Las raíces de una ecuación son los valores de la variable x que satisfacen la ecuación para que valga 0.Cálculo de las raíces de una ecuación cúbica, siendo números reales los coeficientes a, b, c, d.Cada ecuación cúbica, con coeficientes reales, tiene al menos una solución x entre los números reales.

Se pueden distinguir tres casos usando el discriminante: Δ = 18abcd - 4b³d + b²c² - 4ac³ - 27a²dSi Δ > 0, entonces la ecuación tiene tres raíces reales distintas.Si Δ = 0, entonces la ecuación tiene una raíz múltiple y todas sus raíces son reales.Si Δ < 0, entonces la ecuación tiene una raíz real y dos raíces imaginarias.

M =

N =

P =

(-b-POTENCIA((M+RCUAD(P))/2;1/3)-POTENCIA((M-RCUAD(P))/2;1/3))/(3*a) =

Fórmula para las raíces si Δ > 0

Segunda y tercera raíz

Fórmula para las raíces si Δ < 0: x 1 real; x 2 y x 3 imaginarias

2b3 − 9abc + 27a2d =

b2 − 3ac =

M2-4N3 = (2b3 − 9abc + 27a2d)2 − 4(b2 − 3ac)3 =

x1 =

x1 =

x2 = (-b-x1*a+RCUAD(b^2-4*a*c-2*a*b*x1-3*a^2*x1^2))/(2*a)

x3 = (-b-x1*a-RCUAD(b^2-4*a*c-2*a*b*x1-3*a^2*x1^2))/(2*a)

ax³ + bx² + cx + d = 0

0.00182746165028074x³ - 2.48927888367399x² + 734.547157193912x - 12886.683712948 = 0

Las raíces de una ecuación son los valores de la variable x que satisfacen la ecuación para que valga 0.Cálculo de las raíces de una ecuación cúbica, siendo números reales los coeficientes a, b, c, d.Cada ecuación cúbica, con coeficientes reales, tiene al menos una solución x entre los números reales.

Δ = 18abcd - 4b³d + b²c² - 4ac³ - 27a²d 411281.289555606

Si Δ = 0, entonces la ecuación tiene una raíz múltiple y todas sus raíces son reales.Si Δ < 0, entonces la ecuación tiene una raíz real y dos raíces imaginarias.

-1.93816432990034

2.16943908042138

-37.0850834521363

COMPROBACION DE LAS RAICES

(-b-POTENCIA((M+RCUAD(P))/2;1/3)-POTENCIA((M-RCUAD(P))/2;1/3))/(3*a) =

18.7142658452695 -6.98491930961609E-10

944.477715890602 -7.06495484337211E-09

398.959110896678 -9.53150447458029E-10

imaginarias

0.00182746165028074x³ - 2.48927888367399x² + 734.547157193912x - 12886.683712948 = 0

-6.98491930961609E-10

-7.06495484337211E-09

-9.53150447458029E-10

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