Download - 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 :
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
111
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
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 =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 =
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 =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 =∅
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 : 0.47
[𝑃]=[Φ]^𝑇∗[𝑀]∗1
P = -0.17-0.29
MATRIZ DIAGONAL DE EIGENVALORES :
MATRIZ DIAGONAL DE EIGENVALORES :
0.3488 -0.0582U = 0.6429 0.1638
3.9576 -0.0309
7.- Calculamos la matriz de frecuencia lateral
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
8.- Calculo 8.- CALCULO DE LA CORTANTE BASAL
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
Tn*seg2/cm Para t2= 0.259
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
111
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
12 EIc 12 EIc 12 EIc
6 EIc 6 EIc 6 EIc
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
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
12 EIc 12 EIc 12 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
8 EIc+ 4 EIv 2 EIv 0 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
4 EIc+ 4 EIv 2 EIv 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
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 =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 =
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 =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 =∅
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 : 0.60
[𝑃]=[Φ]^𝑇∗[𝑀]∗1
P = 0.32-0.04
MATRIZ DIAGONAL DE EIGENVALORES :
MATRIZ DIAGONAL DE EIGENVALORES :
1.4075 0.3274U = 2.2987 0.0590
4.3061 -0.0378
7.- Calculamos la matriz de frecuencia lateral
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
8.- Calculo 8.- CALCULO DE LA CORTANTE BASAL
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
Tn*seg2/cm Para t2= 0.259
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
111
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