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Maestria en Ingenieria Estructural MIS-507 "Ingenieria Sismica"
ID: 1068251
ANALISIS ESPECTRAL MODAL
1. Propiedades de la estructura
Gravedad g ≔ 981
Pisos P ≔[[ 1 2 3 4 5 6 7 8 9]] P ≔
T P
Altura h ≔ 4 ⋅[[ 1 1 1 1 1 1 1 1 1]]
Rigidez k ≔ 303 ⋅[[ 1 1 1 1 1 1 1 1 1]]
Pesos W ≔816.44 [[ 1 1 1 1 1 1 1 1 1]]
2. Ensamblaje de la matriz de masa.
i ≔ 1 ‥cols (( W )) M (^) i , i ≔――
W (^) 1 , i
g
M =
832.2528 ⋅ 10 −3 0 0 0 0 0 0 0 832.2528 ⋅ 10 −3 0 0 0 0 0 0 0 832.2528 ⋅ 10 −3 0 0 0 0 0 0 0 832.2528 ⋅ 10 −3 0 0 0 0 0 0 0 832.2528 ⋅ 10 −3 0 0 0 0 0 0 0 832.2528 ⋅ 10 −3 0 0 0 0 0 0 0 832.2528 ⋅ 10 − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 …
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎢⎣
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎥⎦
3. Ensamblaje de la matriz de rigidez.
K ≔‖ ‖
‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖
‖ ‖
for i ∊ 1 ‥cols (( k )) ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ if
else
i ≤cols (( k )) − 1 ‖ ‖ ‖ ‖ ‖ ‖ K (^) i , i ← k (^) 1 , i + k (^) 1 , i + 1 K (^) i , i + 1 ←−1 ⋅ k (^) 1 , i + 1 K (^) i + 1 , i ← K (^) i , i + 1
‖ ‖‖ K^ i^ , i^ ← k^^1 , i K
K =
606 −303 0 0 0 0 0 0 0 −303 606 −303 0 0 0 0 0 0 0 −303 606 −303 0 0 0 0 0 0 0 −303 606 −303 0 0 0 0 0 0 0 −303 606 −303 0 0 0 0 0 0 0 −303 606 −303 0 0 0 0 0 0 0 −303 606 −303 0 0 0 0 0 0 0 −303 606 − 0 0 0 0 0 0 0 −303 303
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Maestria en Ingenieria Estructural MIS-507 "Ingenieria Sismica"
ID: 1068251
4. Calculo de formas modales y periodos de vibración.
A ≔ M −1^ ⋅ K ω2 ≔sort ((eigenvals (( A ))))
Frecuencia circular ω ≔ ‾ ω2 ‾‾=
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎢ ⎣
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎥ ⎦
Periodo natural T ≔―^2 ⋅ π ―= ω
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎢ ⎣
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎥ ⎦
Formas modales
ϕ ≔‖ ‖
‖ ‖ ‖‖
for i ∊ 1 ‥rows (( T )) ‖ ‖ ‖ ϕ^ ←
⟨⟨ i ⟩⟩ eigenvec ⎛ ⎝
A , ω2 (^) i , 1 ⎞ ⎠ ϕ
ϕ =
0.0755 0.2184 −0.3376 −0.4202 −0.4573 0.4448 −0.3841 −0.2818 0. 0.149 0.3841 −0.4573 −0.3376 −0.0755 −0.2184 0.4202 0.4448 −0. 0.2184 0.4573 −0.2818 0.149 0.4448 −0.3376 −0.0755 −0.4202 0. 0.2818 0.4202 0.0755 0.4573 0.149 0.3841 −0.3376 0.2184 −0. 0.3376 0.2818 0.3841 0.2184 −0.4202 0.149 0.4448 0.0755 0. 0.3841 0.0755 0.4448 −0.2818 −0.2184 −0.4573 −0.149 −0.3376 −0. 0.4202 −0.149 0.2184 −0.4448 0.3841 0.0755 −0.2818 0.4573 0. 0.4448 −0.3376 −0.149 −0.0755 0.2818 0.4202 0.4573 −0.3841 −0. 0.4573 −0.4448 −0.4202 0.3841 −0.3376 −0.2818 −0.2184 0.149 0.
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
0
9
0 0.045 0.090.135 0.180.2250.270.3150.360.4050.450.
ϕ 1
⟨⟨ 1 ⟩⟩
P 1
0
9
-0.45-0.36 -0.27-0.18 -0.0900.090.180.270.360.450.
ϕ 1
⟨⟨ 2 ⟩⟩
P 1
0
9
-0.54-0.45 -0.36-0.27-0.18 -0.09 0 0.090.180.270.360.
ϕ 1
⟨⟨ 3 ⟩⟩
P 1
0
9
-0.45-0.36 -0.27-0.18 -0.0900.090.180.270.360.450.
ϕ 1
⟨⟨ 4 ⟩⟩
P 1
0
9
-0.54-0.45 -0.36-0.27-0.18 -0.09 0 0.090.180.270.360.
ϕ 1
⟨⟨ 5 ⟩⟩
P 1
1 ° Modo 2 ° Modo 3 ° Modo 4 ° Modo 5 ° Modo
Maestria en Ingenieria Estructural MIS-507 "Ingenieria Sismica"
ID: 1068251
0
9
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.
Γϕ 1
⟨⟨ 1 ⟩⟩
Γϕ 1
⟨⟨ 2 ⟩⟩
Γϕ 1
⟨⟨ 3 ⟩⟩
Γϕ 1
⟨⟨ 4 ⟩⟩
Γϕ 1
⟨⟨ 5 ⟩⟩
P 1
6. Análisis Espectral Modal.
Data ≔READEXCEL (( “.\Espectro_3Asignación.xlsx” “Sa!A1:B150”, ))
t ≔ Data
⟨⟨ 1 ⟩⟩ Sa ≔ Data
⟨⟨ 2 ⟩⟩
85
110
135
160
185
210
235
260
35
60
285
0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3
2
t
Sa
Espectro de aceleraciones
Factor de moficación de respuesta R ≔ 1
Factor de importancia Ie ≔ 1
Factor C1 (FEMA 440) C 1 ≔ 1
Maestria en Ingenieria Estructural MIS-507 "Ingenieria Sismica"
ID: 1068251
Coeficientes sismicos para cada periodo de vibración.
Csm ≔‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖
for i ∊ 1 ‥rows (( T )) ‖ ‖ ‖ ‖ ‖
Csm (^) i , 1 ←―――――――
linterp ⎛ ⎝
t , Sa , T (^) i , 1 ⎞ ⎠
g ⋅⎛ ⎜ ⎝
―
R Ie
⎞ ⎟ ⎠
Csm
Csm =
Calculo de fuerzas modales laterales fmodal ＝ Γn ⋅ ϕjn ⋅ mj ⋅ Cj
fmodal ≔‖ ‖ ‖ ‖ ‖ ‖ ‖
for i ∊ 1 ‥ 5 ‖ ‖ ‖ ‖ ‖‖
for ii ∊ 1 ‥rows (( ϕ )) ‖ ‖‖
fmodal (^) ii , i ← Γ (^) i , 1 ⋅ ϕ (^) ii , i ⋅ W (^) 1 , ii ⋅ Csm (^) i , 1
fmodal
fmodal =
Cortante basal.
Vmodal ≔‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖
for i ∊ 1 ‥cols ⎛⎝ fmodal ⎞⎠ ‖ ‖ ‖ ‖ ‖ ‖ ‖
a ← 0 for ii ∊ 1 ‥rows ⎛⎝ fmodal ⎞⎠ ‖ ‖ ‖ ‖‖
a ← fmodal + ii , i
a
Vmodal ← 1 , i
a
Vmodal
VSRSS ≔
n = 1
cols ⎛⎝ Vmodal ⎞⎠ ⎛ ⎝
Vmodal (^) 1 , n ⎞ ⎠
2
Vnom ≔――――
VSRSS
n = 1
cols (( W )) W (^) 1 , n
VSRSS =684.5154 Vnom =0. Vmodal =[[ 673.0175 114.1626 43.3334 25.2514 7.7597]]
Maestria en Ingenieria Estructural MIS-507 "Ingenieria Sismica"
ID: 1068251
VEmodal =
VESRSS ≔‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖
for i ∊ 1 ‥rows ⎛⎝ VEmodal ⎞⎠ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ a ← 0 for ii ∊ 1 ‥ 5 ‖ ‖ ‖
a ← VEmodal (^) i , ii^2 + a
VESRSS (^) i , 1 ←^ ‾ a ‾
VESRSS
P =
VESRSS =
0
9
0 70 140 210 280 350 420 490 560 630 700
VESRSS.
T P_
Fuerzas cortantes por entrepiso.
T
2
Maestria en Ingenieria Estructural MIS-507 "Ingenieria Sismica"
ID: 1068251
Calculo de desplazamientos ujn ＝ Γn ⋅ ϕjn ⋅ Cn ⋅ g ⋅――
Tn
2
4 ⋅ π
2
umodal ≔‖ ‖
‖ ‖ ‖ ‖ ‖ ‖ ‖
for i ∊ 1 ‥ 5 ‖ ‖ ‖ ‖ ‖ ‖ ‖ for ii ∊ 1 ‥rows (( ϕ )) ‖ ‖ ‖ ‖‖
umodal (^) ii , i ← Γ (^) i , 1 ⋅ ϕ (^) ii , i ⋅ Csm (^) i , 1 ⋅ g ⋅――⋅ ⋅
Ti (^) , 12
4 ⋅ π^2
R C 1
umodal
uSRSS ≔‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖
for i ∊ 1 ‥rows ⎛⎝ umodal ⎞⎠ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ a ← 0 for ii ∊ 1 ‥ 5 ‖ ‖ ‖
a ← umodal (^) i , ii^2 + a
uSRSS (^) i , 1 ←^ ‾ a ‾
uSRSS
umodal =
uSRSS =
0
9
0 1.5 3 4.5 6 7.5 9 10.5 12 13.
9 13.
uSRSS.
P.
Desplazamientos letarales de cada piso.
Maestria en Ingenieria Estructural MIS-507 "Ingenieria Sismica"
ID: 1068251
Calculo de distorsiones de entrepiso.
DISmodal ≔‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖
for i ∊ 1 ‥ 5 ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖
‖‖
for ii ∊ 1 ‥rows (( ϕ )) ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖
‖‖
if
else if
ii ＝ 1 ‖ ‖ ‖ ‖ ‖
umodal (^) ii , i ← Γ (^) i , 1 ⋅ ϕ (^) ii , i ⋅ Csm (^) i , 1 ⋅ g ⋅――――――⋅ ⋅
T (^) i , 12
4 ⋅ π^2 ⋅⎛ ⎝ h (^) 1 , i ⋅ 100 ⎞ ⎠
R C 1
ii > 1 ‖ ‖ ‖ ‖ ‖
umodal (^) ii , i ← Γ (^) i , 1 ⋅⎛ ⎝ ϕ (^) ii , i − ϕ (^) (( (^) ii − 1 )) (^) , i ⎞ ⎠ ⋅ Csm (^) i , 1 ⋅ g ⋅――――――⋅ ⋅
T (^) i , 12
4 ⋅ π^2 ⋅^ ⎛ ⎝ h (^) 1 , i ⋅ 100 ⎞ ⎠
R C 1
umodal
DISSRSS ≔‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖
for i ∊ 1 ‥rows ⎛⎝ DISmodal ⎞⎠ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ a ← 0 for ii ∊ 1 ‥ 5 ‖ ‖ ‖
a ← DISmodal (^) i , ii^2 + a
DISSRSS (^) i , 1 ←^ ‾ a ‾
DISSRSS
0
9
0 0.0006 0.0011 0.0017 0.0022 0.0028 0.0033 0.0039 0.0044 0.005 0.0055 0.
DISSRSS.
T P_
Distorsiones de entrepiso.
