Alb Portico Montenegro.pdf

“AÑO DEL DIALOGO Y LA RECONCILIACIÓN NACIONAL”

UNIVERSIDAD NACIONAL “SANTIAGO ANTÚNEZ DE MAYOLO’’

FACULTAD DE INGENIERÍA CIVIL DETERMINACION MATRIZ DE RIGIDEZ LATERAL DEL EDIFICIO CURSO: ALBAÑILERIA ESTRUCTURAL

DOCENTE: ING. OLAZA HENOSTROS Hugo

ALUMNO: MONTENEGRO TORRES Carlos Santiago

Huaraz – Perú Marzo 2019

PLANO DE PLANTA, EJES A ANALIZAR RESALTADOS TANTO EN "X", COMO EN "Y".

ELEMENTO

RIGIDEZ DE MUROS- SECCION TRANSFORMADA 1. Se calculara la Rigidez de los muros Eje "D" y Eje "1" el plano se encuetra en el primer trabajo. 2. Calculo de rigidez de cada uno de los muros que contemplan el eje tanto en eje "x" e "y". 3. Luego ensamblaremos la estructura hasta 2 niveles.

CARACTERÍSTICAS MECÁNICAS DE LOS MATERIALES a. Albañilería f'b = 145.00 kg/cm2 f'm = 65.00 kg/cm2 v'm = 8.10 kg/cm2 Em = 325,000.00 T/m2 (Em = 500*f'm) Gm = 130,000.00 T/m2 (Gm = 0.40*Em) A= AREA - SECCION TRANSFORMADA b. Concreto f'c = Ec =

210.00 kg/cm2 218,819.79 kg/cm2

Factor de relación:

𝒏= n=

𝑬𝒄 𝑬𝒎 0.673 DE LA ESTRUCTURACION- SECCION TRANSFORMADA DIRECCION X-X MURO X1, X4 REVISAR LA HOJA 15 DEL MANUAL DE DISEÑO EN ALBAÑILERIA REF. RNE E070 ART. 24.6 ANALISIS ESTRUCTURAL

CALCULO DEL CENTROIDE DE LA SECCION TRANSFORMADA 𝑋𝑖 𝐴 ∗ 𝑋𝑖 ELEMENTO ANCHO LARGO AREA 1 0.72 0.12 0.09 0.06 0.005184 2 1.347 0.12 0.16 0.06 0.0096984 3 0.12 2.91 0.35 1.575 0.54999 4 1.347 0.12 0.16 3.09 0.4994676 5 0.72 0.12 0.09 3.09 0.266976 = = 0.85 1.331316

𝑋 𝑋= Ac=

𝐴 ∗ 𝑋𝑖 = 𝐴 0.378

1.575

ELEMENTO 1 2 3 4 5

b 0.72 1.347 0.12 1.347 0.72

MOMENTO DE INERCIA RESPECTO AL EJE YY 𝑑 𝐼𝑌𝑌 + 𝑑2 ∗ 𝐴 𝐼𝑌𝑌 h 0.12 0.00010368 1.515 0.371103849 0.12 0.00019397 1.515 0.801686538 2.91 0.24642171 0 0.24642171 0.12 0.00019397 -1.515 0.198501408 0.12 0.00010368 -1.515 1.940211468 𝐼𝑌𝑌 = 3.557924973 𝑚4 MURO X2,X3 REVISAR LA HOJA 15 DEL MANUAL DE DISEÑO EN ALBAÑILERIA REF. RNE E070 ART. 24.6 ANALISIS ESTRUCTURAL

ELEMENTO 1 2 3 4

0.808 0.12 1.347 0.72

0.2 2.68 0.12 0.12 =

ELEMENTO 1 2 3 4

𝑋

CALCULO DEL CENTROIDE DE LA SECCION TRANSFORMADA 𝑋𝑖 𝐴 ∗ 𝑋𝑖 ANCHO LARGO AREA 0.16 0.32 0.16 0.09 0.73

0.1 1.54 2.94 2.94 =

0.01616 0.495264 0.4752216 0.254016 1.2406616

𝑋=

MOMENTO DE INERCIA RESPECTO AL EJE YY 𝐼𝑌𝑌 𝐼𝑌𝑌 + 𝑑2 ∗ 𝐴 𝑑 b h 0.808 0.2 0.00053867 1.59665 0.412506425 0.12 2.68 0.19248832 0.15665 0.200380582 1.347 0.12 0.00019397 -1.2433 0.250074562 0.72 0.12 0.00010368 -1.2433 0.133670145 𝐼𝑌𝑌 = 0.996631714 𝑚4

Ac=

𝐴 ∗ 𝑋𝑖 = 𝐴 0.36

1.69665445

DIRECCION Y-Y MURO Y1,Y8. REVISAR HOJA 15 DEL MANUAL DE DISEÑO EN ALBAÑILERIA REF. RNE E070 ART. 24.6 ANALISIS ESTRUCTURAL

𝑌

ELEMENTO 1 2 3 4 5

CALCULO DEL CENTROIDE DE LA SECCION TRANSFORMADA 𝑌𝑖 𝐴 ∗ 𝑌𝑖 ANCHO LARGO AREA 0.808 0.2 0.16 0.1 0.01616 0.12 2.1 0.25 1.25 0.315 0.808 0.2 0.16 2.4 0.38784 0.728 0.12 0.09 2.44 0.2131584 0 0 0.00 0 0 = = 0.66 0.9321584

ELEMENTO 1 2 3 4

MOMENTO DE INERCIA RESPECTO AL EJE XX 𝐼𝑥𝑥 𝐼𝑥𝑥 + 𝑑2 ∗ 𝐴 𝑑 b h 0.808 0.2 0.00053867 1.30690413 0.27655121 0.12 2.1 0.09261000 0.15690413 0.09881396 0.808 0.2 0.00053867 -0.99309587 0.15991496 0.728 0.12 0.00010483 -1.03309587 0.09334303 𝐼𝑥𝑥 = 0.62862316 𝑚4

𝑌=

𝐴 ∗ 𝑌𝑖 = 𝐴

1.40690413

3. PROPIEDADES DE SECCION DE LAS VIGAS.

ELEMENTO 1 2

CALCULO DEL CENTROIDE DE LA SECCION TRANSFORMADA ANCHO LARGO AREA 𝑌𝑖 𝐴 ∗ 𝑌𝑖 0.12 0.3 0.04 0.15 0.0054 0.6 0.15 0.09 0.225 0.02025 = = 0.13 0.02565

ELEMENTO 1 2

𝑌 𝑌=

𝐴 ∗ 𝑌𝑖 = 𝐴

MOMENTO DE INERCIA RESPECTO AL EJE XX 𝐼𝑥𝑥 𝐼𝑥𝑥 + 𝑑2 ∗ 𝐴 𝑑 b h 0.12 0.3 0.00027000 0.05357143 0.00037332 0.6 0.15 0.00016875 -0.02142857 0.00021008 𝐼𝑥𝑥 = 0.00058339 𝑚 4

0.20357143

PORTICO EJE "D" DIRECCION X -X

CÁLCULO DE RIGIDEZ PARA LOS MUROS

PROPIEDADES MECANICAS PARA LA ALBAÑILERIA: tonf

f'm ≔ 650 ―― 2

h ≔ 2.4 m

m

tonf

E ≔ 500 ⋅ f'm = 325000 ―― 2 m tonf G ≔ 0.4 ⋅ E = 130000 ―― m2

Calculo de K1, K4, K5 y K8 : I ≔ 3.558

Ac ≔ 0.378

A ≔ 0.85

12 ⋅ E ⋅ I ϕ ≔ ―――― = 49.024 G ⋅ Ac ⋅ h 2

⎡ 12 ⋅ E ⋅ I ⎤ 6⋅E⋅I 0 ―――― ⎥ ⎢ ―――― 3 2 ( ) ( ) h ⋅ (1 + ϕ) ⎥ ⎢ h ⋅ (1 + ϕ) ⎡ 0 24078.84 ⎤ ⎢ ⎥ ⎢ 20065.7 A ⎥ E ⋅ ―― 0 0 K1 ≔ ⎢ 0 115104.167 0 ⎥=⎢ ⎥ ( ) (h) ⎢ ⎥ ⎣ 24078.84 0 510707.108 ⎦ E ⋅ I ⋅ ((4 + ϕ)) ⎥ ⎢ 6⋅E⋅I 0 ――――― ⎢ ―――― ⎥ 2 h ⋅ ((1 + ϕ)) ⎦ ⎣ h ⋅ ((1 + ϕ)) ⎡ 20065.7 0 24078.84 ⎤ ⎥ K4 ≔ K1 = ⎢ 0 115104.167 0 ⎢ ⎥ 0 510707.108 ⎦ ⎣ 24078.84 ⎡ 12 ⋅ E ⋅ I -6 ⋅ E ⋅ I -12 ⋅ E ⋅ I -6 ⋅ E ⋅ I ⎤ 0 0 ―――― ―――― ―――― ⎥ ⎢ ―――― 3 h 2 ⋅ ((1 + ϕ)) h 3 ⋅ ((1 + ϕ)) h 2 ⋅ ((1 + ϕ)) ⎥ ⎢ h ⋅ ((1 + ϕ)) ⎢ ⎥ A A E ⋅ ―― -E ⋅ ―― 0 0 0 0 ⎢ ⎥ ( ) ( ) h h ) ) ( ( ⎢ ⎥ E ⋅ I ⋅ ((4 + ϕ)) 6⋅E⋅I E ⋅ I ⋅ ((2 - ϕ)) ⎥ ⎢ -6 ⋅ E ⋅ I 0 0 ――――――――― ――――― ⎢ ―――― ⎥ h ⋅ ((1 + ϕ)) h ⋅ ((1 + ϕ)) ⎥ h 2 ⋅ ((1 + ϕ)) h 2 ⋅ ((1 + ϕ)) K5 ≔ ⎢ 6⋅E⋅I 12 ⋅ E ⋅ I 6⋅E⋅I ⎢ -12 ⋅ E ⋅ I ⎥

⎡ 20065.7 0 24078.84 ⎤ ⎥ K4 ≔ K1 = ⎢ 0 115104.167 0 ⎢ ⎥ 0 510707.108 ⎦ ⎣ 24078.84 ⎡ 12 ⋅ E ⋅ I -6 ⋅ E ⋅ I -12 ⋅ E ⋅ I -6 ⋅ E ⋅ I ⎤ 0 0 ―――― ―――― ―――― ⎥ ⎢ ―――― 3 2 3 2 ( ) ( ) ( ) h 1 + ϕ h 1 + ϕ h 1 + ϕ h ⋅ ⋅ ⋅ ⋅ ((1 + ϕ)) ⎥ ) ) ) ( ( ( ⎢ ⎢ ⎥ A A E ⋅ ―― -E ⋅ ―― 0 0 0 0 ⎢ ⎥ ( ) ( ) (h) (h) ⎢ ⎥ E ⋅ I ⋅ ((4 + ϕ)) 6⋅E⋅I E ⋅ I ⋅ ((2 - ϕ)) ⎥ ⎢ -6 ⋅ E ⋅ I 0 0 ―――― ――――― ―――― ――――― ⎢ 2 ⎥ h ⋅ ((1 + ϕ)) h ⋅ ((1 + ϕ)) ⎥ h ⋅ ((1 + ϕ)) h 2 ⋅ ((1 + ϕ)) K5 ≔ ⎢ 6⋅E⋅I 12 ⋅ E ⋅ I 6⋅E⋅I ⎢ -12 ⋅ E ⋅ I ⎥ 0 0 ―――― ―――― ―――― ⎥ ⎢ ―――― 3 2 3 2 ( ) ( ) ( ) ( ) h ⋅ 1 + ϕ h ⋅ 1 + ϕ h ⋅ 1 + ϕ h ⋅ 1 + ϕ ) ) ) ) ( ( ( ( ⎢ ⎥ A A ⎢ ⎥ -E ⋅ ―― E ⋅ ―― 0 0 0 0 ⎢ ⎥ ((h)) ((h)) ⎢ ⎥ E ⋅ I ⋅ ((2 - ϕ)) 6⋅E⋅I E ⋅ I ⋅ ((4 + ϕ)) ⎥ ⎢ -6 ⋅ E ⋅ I 0 0 ―――― ――――― ―――― ――――― ⎢ 2 ( h ⋅ ((1 + ϕ)) h ⋅ ((1 + ϕ)) ⎥⎦ h 2 ⋅ ((1 + ϕ)) ⎣ h ⋅ (1 + ϕ))

⎡ 20065.7 0 -24078.84 -20065.7 0 -24078.84 ⎤ ⎢ ⎥ 0 115104.167 0 0 -115104.167 0 ⎢ ⎥ -24078.84 0 510707.108 24078.84 0 -452917.892 ⎥ K5 = ⎢ 0 24078.84 20065.7 0 24078.84 ⎥ ⎢ -20065.7 ⎢ ⎥ 0 -115104.167 0 0 115104.167 0 ⎢⎣ -24078.84 0 -452917.892 24078.84 0 510707.108 ⎥⎦ ⎡ 20065.7 0 -24078.84 -20065.7 0 -24078.84 ⎤ ⎢ ⎥ 0 115104.167 0 0 -115104.167 0 ⎢ ⎥ -24078.84 0 510707.108 24078.84 0 -452917.892 ⎥ K8 ≔ K5 = ⎢ 0 24078.84 20065.7 0 24078.84 ⎥ ⎢ -20065.7 ⎢ ⎥ 0 -115104.167 0 0 115104.167 0 ⎢⎣ -24078.84 0 -452917.892 24078.84 0 510707.108 ⎥⎦

Calculo de K2, K3, K6 y K7 : I ≔ 0.9966

A ≔ 0.73124

Ac ≔ 0.36

12 ⋅ E ⋅ I = 14.418 ϕ ≔ ―――― G ⋅ Ac ⋅ h 2 ⎡ 12 ⋅ E ⋅ I ⎤ 6⋅E⋅I 0 ―――― ⎢ ―――― ⎥ 3 2 h ⋅ ((1 + ϕ)) ⎥ ⎢ h ⋅ ((1 + ϕ)) ⎡ 18235.278 0 21882.333 ⎤ A ⎢ ⎥ ⎢ ⎥ E ⋅ ―― 0 99022.083 0 K2 ≔ ⎢ 0 0 = ⎥ ⎢ ⎥ ((h)) 0 161215.05 ⎦ ⎢ ⎥ ⎣ 21882.333 ( ) 6 ⋅ E ⋅ I E ⋅ I ⋅ 4 + ϕ ) ( ⎢ ―――― 0 ―――― ⎥ ⎢⎣ h 2 ⋅ ((1 + ϕ)) h ⋅ ((1 + ϕ)) ⎥⎦ ⎡ 18235.278 0 21882.333 ⎤ ⎥ K3 ≔ K2 = ⎢ 0 99022.083 0 ⎢ ⎥ 0 161215.05 ⎦ ⎣ 21882.333 ⎡ 12 ⋅ E ⋅ I -6 ⋅ E ⋅ I -12 ⋅ E ⋅ I -6 ⋅ E ⋅ I ⎤ 0 0 ―――― ―――― ―――― ⎥ ⎢ ―――― 3 2 3 2 ( ) ( ) ( ) h 1 + ϕ h 1 + ϕ h 1 + ϕ h ⋅ ⋅ ⋅ ⋅ ((1 + ϕ)) ⎥ ) ) ) ( ( ( ⎢ ⎢ ⎥ A A 0 0 0 0 E ⋅ ―― -E ⋅ ―― ⎢ ⎥ ((h)) ((h)) ⎢ ⎥ ( ) ( ) E ⋅ I ⋅ (4 + ϕ) 6⋅E⋅I E ⋅ I ⋅ (2 - ϕ) ⎥ ⎢ -6 ⋅ E ⋅ I 0 0 ―――― ――――― ―――― ――――― ⎢ 2 ( h ⋅ ((1 + ϕ)) h ⋅ ((1 + ϕ)) ⎥ h ⋅ (1 + ϕ)) h 2 ⋅ ((1 + ϕ)) ⎥ K6 ≔ ⎢ -12 ⋅ E ⋅ I 6⋅E⋅I 12 ⋅ E ⋅ I 6⋅E⋅I ⎢ ―――― 0 0 ―――― ―――― ―――― ⎥ ⎢ h 3 ⋅ (1 + ϕ) h 2 ⋅ ((1 + ϕ)) h 3 ⋅ ((1 + ϕ)) h 2 ⋅ ((1 + ϕ)) ⎥ ) ( ⎢ ⎥ A A ⎢ ⎥ 0 0 0 0 -E ⋅ ―― E ⋅ ―― ⎢ ⎥ ((h)) ((h)) ⎢ -6 ⋅ E ⋅ I E ⋅ I ⋅ ((2 - ϕ)) 6⋅E⋅I E ⋅ I ⋅ ((4 + ϕ)) ⎥ ⎢ ―――― ⎥ 0 0 ――――― ―――― ――――― 2 h ⋅ ((1 + ϕ)) h ⋅ ((1 + ϕ)) ⎦⎥ h 2 ⋅ ((1 + ϕ)) ⎣⎢ h ⋅ ((1 + ϕ))

( ) ( ) ⎢ ⎥ E ⋅ I ⋅ ((4 + ϕ)) 6⋅E⋅I E ⋅ I ⋅ ((2 - ϕ)) ⎥ ⎢ -6 ⋅ E ⋅ I 0 0 ―――― ――――― ―――― ――――― ⎢ 2 ( h ⋅ ((1 + ϕ)) h ⋅ ((1 + ϕ)) ⎥ h ⋅ (1 + ϕ)) h 2 ⋅ ((1 + ϕ)) ⎥ K6 ≔ ⎢ -12 ⋅ E ⋅ I 6⋅E⋅I 12 ⋅ E ⋅ I 6⋅E⋅I ⎢ ―――― 0 0 ―――― ―――― ―――― ⎥ ⎢ h 3 ⋅ (1 + ϕ) h 2 ⋅ ((1 + ϕ)) h 3 ⋅ ((1 + ϕ)) h 2 ⋅ ((1 + ϕ)) ⎥ ) ( ⎢ ⎥ A A ⎢ ⎥ 0 0 0 0 -E ⋅ ―― E ⋅ ―― ⎢ ⎥ ((h)) ((h)) ⎢ -6 ⋅ E ⋅ I E ⋅ I ⋅ ((2 - ϕ)) 6⋅E⋅I E ⋅ I ⋅ ((4 + ϕ)) ⎥ ⎢ ―――― ⎥ 0 0 ――――― ―――― ――――― ⎢⎣ h 2 ⋅ ((1 + ϕ)) h ⋅ ((1 + ϕ)) h ⋅ ((1 + ϕ)) ⎥⎦ h 2 ⋅ ((1 + ϕ))

⎡ 18235.278 0 -21882.333 -18235.278 0 -21882.333 ⎤ ⎢ ⎥ 0 99022.083 0 0 -99022.083 0 ⎢ ⎥ -21882.333 0 161215.05 21882.333 0 -108697.45 ⎥ K6 = ⎢ 0 21882.333 18235.278 0 21882.333 ⎥ ⎢ -18235.278 ⎢ ⎥ 0 -99022.083 0 0 99022.083 0 ⎢⎣ -21882.333 0 -108697.45 21882.333 0 161215.05 ⎥⎦ ⎡ 18235.278 0 -21882.333 -18235.278 0 -21882.333 ⎤ ⎢ ⎥ 0 99022.083 0 0 -99022.083 0 ⎢ ⎥ -21882.333 0 161215.05 21882.333 0 -108697.45 ⎥ K7 ≔ K6 = ⎢ 0 21882.333 18235.278 0 21882.333 ⎥ ⎢ -18235.278 ⎢ ⎥ 0 -99022.083 0 0 99022.083 0 ⎢⎣ -21882.333 0 -108697.45 21882.333 0 161215.05 ⎥⎦ CÁLCULO DE RIGIDEZ PARA LAS VIGAS

PROPIEDADES MECANICAS DEL CONCRETO: f'c ≔ 210

kgf ―― cm 2

h ≔ 2.4 m

tonf E ≔ 150000 ⋅ ‾‾‾ f'c = 2173706.512 ―― 2 m

para la viga K9, K12 a ≔ 1.58

b ≔ 1.70

L≔1

⎡ 12 ⋅ E ⋅ I 6⋅E⋅I 12 ⋅ E ⋅ I ⋅a ―――― ―――+ ―――― ⎢ L3 L2 L3 ⎢ ⎢ 6⋅E⋅I 12 ⋅ E ⋅ I 4⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I ⋅a ⋅ a2 ⎢ ―――+ ―――― ―――+ ―――⋅ ((2 ⋅ a)) + ―――― L ⎢ L2 L3 L2 L3 K≔⎢ -12 ⋅ E ⋅ I -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I ⎢ - ―――― ⋅a ―――― ―――― ⎢ L3 L2 L3 ⎢ 6⋅E⋅I 12 ⋅ E ⋅ I 2⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I ⎢ ―――+ ―――― ⋅ b ―――+ ―――⋅ ((a + b)) + ―――― ⋅a⋅b ⎢⎣ L L2 L3 L2 L3

I ≔ 0.000583392857142857 ⎤ -12 ⋅ E ⋅ I 6⋅E⋅I 12 ⋅ E ⋅ I ⋅b ―――― ―――+ ―――― ⎥ L3 L2 L3 ⎥ ⎥ -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I 2⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I - ―――― ⋅ a ―――+ ―――⋅ ((a + b)) + ―――― ⋅a⋅b⎥ ―――― L ⎥ L2 L3 L2 L3 ⎥ 12 ⋅ E ⋅ I -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I ⎥ - ―――― ⋅b ―――― ―――― 3 2 3 ⎥ L L L ⎥ -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I 4⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I 2 ⎥ ( ) - ―――― ⋅b ⋅b ―――― ―――+ ―――⋅ (2 b) + ―――― ⎥⎦ L L2 L3 L2 L3

⎡ 15217.498 31652.396 -15217.498 33478.496 ⎤ ⎢ 31652.396 67105.109 -31652.396 68367.147 ⎥ K9 ≔ K = ⎢ ⎥ ⎢ -15217.498 -31652.396 15217.498 -33478.496 ⎥ ⎣ 33478.496 68367.147 -33478.496 74920.816 ⎦ ⎡ 15217.498 31652.396 -15217.498 33478.496 ⎤ ⎢ 31652.396 67105.109 -31652.396 68367.147 ⎥ K12 ≔ K = ⎢ ⎥ ⎢ -15217.498 -31652.396 15217.498 -33478.496 ⎥ ⎣ 33478.496 68367.147 -33478.496 74920.816 ⎦

para la viga 11 y 14 a ≔ 1.3

b ≔ 1.58

L ≔ 1.0

⎡ 12 ⋅ E ⋅ I 6⋅E⋅I 12 ⋅ E ⋅ I ⋅a ―――― ―――+ ―――― ⎢ L3 L2 L3 ⎢ ⎢ 6⋅E⋅I 12 ⋅ E ⋅ I 4⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I ⋅a ⋅ a2 ⎢ ―――+ ―――― ―――+ ―――⋅ ((2 ⋅ a)) + ―――― L ⎢ L2 L3 L2 L3 K≔⎢ -12 ⋅ E ⋅ I -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I ⎢ - ―――― ⋅a ―――― ―――― ⎢ L3 L2 L3 ⎢ 6⋅E⋅I 12 ⋅ E ⋅ I 2⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I ⎢ ―――+ ―――― ⋅ b ―――+ ―――⋅ ((a + b)) + ―――― ⋅a⋅b 2 3 2 3 ⎢ L

⎤ -12 ⋅ E ⋅ I 6⋅E⋅I 12 ⋅ E ⋅ I ⋅b ―――― ―――+ ―――― ⎥ L3 L2 L3 ⎥ ⎥ -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I 2⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I - ―――― ⋅ a ―――+ ―――⋅ ((a + b)) + ―――― ⋅a⋅b⎥ ―――― 2 3 2 3 L ⎥ L L L L ⎥ 12 ⋅ E ⋅ I -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I ⎥ - ―――― ⋅b ―――― ―――― ⎥ L3 L2 L3 ⎥ -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I 4⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I 2 ⎥ ( ) - ―――― ⋅b ⋅b ―――― ―――+ ―――⋅ (2 b) + ―――― 2 3 2 3 ⎥ L

⎡ 12 ⋅ E ⋅ I 6⋅E⋅I 12 ⋅ E ⋅ I ⋅a ―――― ―――+ ―――― ⎢ L3 L2 L3 ⎢ ⎢ 6⋅E⋅I 12 ⋅ E ⋅ I 4⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I ⋅a ⋅ a2 ⎢ ―――+ ―――― ―――+ ―――⋅ ((2 ⋅ a)) + ―――― L ⎢ L2 L3 L2 L3 K≔⎢ -12 ⋅ E ⋅ I -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I ⎢ - ―――― ⋅a ―――― ―――― ⎢ L3 L2 L3 ⎢ 6⋅E⋅I 12 ⋅ E ⋅ I 2⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I ⎢ ―――+ ―――― ⋅ b ―――+ ―――⋅ ((a + b)) + ―――― ⋅a⋅b ⎢⎣ L L2 L3 L2 L3

⎤ -12 ⋅ E ⋅ I 6⋅E⋅I 12 ⋅ E ⋅ I ⋅b ―――― ―――+ ―――― ⎥ L3 L2 L3 ⎥ ⎥ -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I 2⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I - ―――― ⋅ a ―――+ ―――⋅ ((a + b)) + ―――― ⋅a⋅b⎥ ―――― L ⎥ L2 L3 L2 L3 ⎥ 12 ⋅ E ⋅ I -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I ⎥ - ―――― ⋅b ―――― ―――― 3 2 3 ⎥ L L L ⎥ -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I 4⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I - ―――― ⋅b ⋅ b2 ⎥ ―――― ―――+ ―――⋅ ((2 b)) + ―――― 2 3 2 3 ⎥⎦ L L L L L

⎡ 15217.498 27391.497 -15217.498 31652.396 ⎤ ⎢ 27391.497 50572.819 -27391.497 55706.189 ⎥ K11 ≔ K = ⎢ ⎥ ⎢ -15217.498 -27391.497 15217.498 -31652.396 ⎥ ⎣ 31652.396 55706.189 -31652.396 67105.109 ⎦

⎡ 15217.498 27391.497 -15217.498 31652.396 ⎤ ⎢ 27391.497 50572.819 -27391.497 55706.189 ⎥ K14 ≔ K = ⎢ ⎥ ⎢ -15217.498 -27391.497 15217.498 -31652.396 ⎥ ⎣ 31652.396 55706.189 -31652.396 67105.109 ⎦

para la viga 10 y 13 a ≔ 1.30

b ≔ 1.70

L ≔ 2.45

⎡ 12 ⋅ E ⋅ I 6⋅E⋅I 12 ⋅ E ⋅ I ⋅a ―――― ―――+ ―――― ⎢ L3 L2 L3 ⎢ ⎢ 6⋅E⋅I 12 ⋅ E ⋅ I 4⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I ⋅a ⋅ a2 ⎢ ―――+ ―――― ―――+ ―――⋅ ((2 ⋅ a)) + ―――― L ⎢ L2 L3 L2 L3 K≔⎢ -12 ⋅ E ⋅ I -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I ⎢ - ―――― ⋅a ―――― ―――― ⎢ L3 L2 L3 ⎢ 6⋅E⋅I 12 ⋅ E ⋅ I 2⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I ⎢ ―――+ ―――― ⋅ b ―――+ ―――⋅ ((a + b)) + ―――― ⋅a⋅b ⎢⎣ L L2 L3 L2 L3

⎤ -12 ⋅ E ⋅ I 6⋅E⋅I 12 ⋅ E ⋅ I ⋅b ―――― ―――+ ―――― ⎥ L3 L2 L3 ⎥ ⎥ -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I 2⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I - ―――― ⋅ a ―――+ ―――⋅ ((a + b)) + ―――― ⋅a⋅b⎥ ―――― L ⎥ L2 L3 L2 L3 ⎥ 12 ⋅ E ⋅ I -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I ⎥ - ―――― ⋅b ―――― ―――― ⎥ L3 L2 L3 ⎥ -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I 4⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I - ―――― ⋅b ⋅ b2 ⎥ ―――― ―――+ ―――⋅ ((2 b)) + ―――― 2 3 2 3 ⎥⎦ L L L L L

⎡ 1034.773 2612.801 -1034.773 3026.71 ⎤ ⎢ 2612.801 7114.925 -2612.801 7124.842 ⎥ K10 ≔ K = ⎢ ⎥ ⎢ -1034.773 -2612.801 1034.773 -3026.71 ⎥ 7124.842 -3026.71 9370.73 ⎦ ⎣ 3026.71 ⎡ 1034.773 2612.801 -1034.773 3026.71 ⎤ ⎢ 2612.801 7114.925 -2612.801 7124.842 ⎥ K13 ≔ K = ⎢ ⎥ ⎢ -1034.773 -2612.801 1034.773 -3026.71 ⎥ 7124.842 -3026.71 9370.73 ⎦ ⎣ 3026.71

CÁLCULO DE LA MATRIZ A ⎡1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0⎤ A1 ≔ ⎢ 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎣0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0⎦ ⎡1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0⎤ A2 ≔ ⎢ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎣0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0⎦ ⎡1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0⎤ A3 ≔ ⎢ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎣0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0⎦ ⎡1 A4 ≔ ⎢ 0 ⎢ ⎣0 ⎡1 ⎢0 ⎢ 0 A5 ≔ ⎢ ⎢0 ⎢0 ⎢⎣ 0

0 0 0 0 0 0 1 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0⎤ 0⎥ ⎥ 0⎦ 0⎤ 0⎥ ⎥ 0⎥ 0⎥ 0⎥ 0 ⎥⎦

⎡1 A4 ≔ ⎢ 0 ⎢ ⎣0 ⎡1 ⎢0 ⎢ 0 A5 ≔ ⎢ ⎢0 ⎢0 ⎢⎣ 0

0 0 0 0 0 0 1 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0⎤ 0⎥ ⎥ 0⎦ 0⎤ 0⎥ ⎥ 0⎥ 0⎥ 0⎥ 0 ⎥⎦

⎡1 ⎢0 ⎢ 0 A6 ≔ ⎢ ⎢0 ⎢0 ⎢⎣ 0

0 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 1 0

0 0 0 0 0 1

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0⎤ 0⎥ ⎥ 0⎥ 0⎥ 0⎥ 0 ⎥⎦

⎡1 ⎢0 ⎢ 0 A7 ≔ ⎢ ⎢0 ⎢0 ⎢⎣ 0

0 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 1 0

0 0 0 0 0 1

0 0 0 0 0 0

0⎤ 0⎥ ⎥ 0⎥ 0⎥ 0⎥ 0 ⎥⎦

⎡1 ⎢0 ⎢ 0 A8 ≔ ⎢ ⎢0 ⎢0 ⎢⎣ 0

0 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 1 0

0⎤ 0⎥ ⎥ 0⎥ 0⎥ 0⎥ 1 ⎥⎦

⎡0 ⎢0 A9 ≔ ⎢ ⎢0 ⎣0

0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0⎤ 0⎥ ⎥ 0⎥ 0⎦

⎡0 ⎢0 A10 ≔ ⎢ ⎢0 ⎣0

0 0 0 0

0 0 0 0

0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0⎤ 0⎥ ⎥ 0⎥ 0⎦

⎡0 ⎢0 A11 ≔ ⎢ ⎢0 ⎣0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0⎤ 0⎥ ⎥ 0⎥ 0⎦

⎡0 ⎢0 A12 ≔ ⎢ ⎢0 ⎣0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

0⎤ 0⎥ ⎥ 0⎥ 0⎦

⎡0 ⎢0 A13 ≔ ⎢ ⎢0 ⎣0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

0⎤ 0⎥ ⎥ 0⎥ 0⎦

⎡0 ⎢0 A14 ≔ ⎢ ⎢0 ⎣0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

0⎤ 0⎥ ⎥ 0⎥ 1⎦

CÁLCULO DE LA MATRIZ DEL PORTICO EN EL EJE X-X

CÁLCULO DE LA MATRIZ DEL PORTICO EN EL EJE X-X Ki ≔ 1 i ≔ 1 14 T K ((portico)) ≔ ∑ Ai ⋅ Ki ⋅ Ai i

KG ≔ A1 T ⋅ K1 ⋅ A1 + A2 T ⋅ K2 ⋅ A2 + A3 T ⋅ K3 ⋅ A3 + A4 T ⋅ K4 ⋅ A4 + A5 T ⋅ K5 ⋅ A5 + A6 T ⋅ K6 ⋅ A6 + A7 T ⋅ K7 ⋅ A7 + A8 T ⋅ K8 ⋅ A8 + A9 T ⋅ K9 ⋅ A9 + A10 T ⋅ K10 ⋅ A10 + A11 T ⋅ K11 ⋅ A11 + A12 T ⋅ K12 ⋅ A12 + A

⎡ 153203.912 ⎢ -76601.956 ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ 0 KG = ⎢ 0 ⎢ ⎢ 0 ⎢ -24078.84 ⎢ 0 ⎢ ⎢ -21882.333 ⎢ 0 ⎢ ⎢ -21882.333 ⎢ 0 ⎢ -24078.84 ⎣

-76601.956 76601.956 0 24078.84 0 21882.333 0 21882.333 0 24078.84 0 24078.84 0 21882.333 0 21882.333 0 24078.84

0 0 245425.832 31652.396 -15217.498 33478.496 0 0 0 0 -115104.167 0 0 0 0 0 0 0

0 24078.84 31652.396 1088519.326 -31652.396 68367.147 0 0 0 0 0 -452917.892 0 0 0 0 0 0

0 0 -15217.498 -31652.396 214296.438 -30865.695 -1034.773 3026.71 0 0 0 0 -99022.083 0 0 0 0 0

0 21882.333 33478.496 68367.147 -30865.695 404465.841 -2612.801 7124.842 0 0 0 0 0 -108697.45 0 0 0 0

0 0 0 0 -1034.773 -2612.801 214296.438 24364.786 -15217.498 31652.396 0 0 0 0 -99022.083 0 0 0

0 21882.333 0 0 3026.71 7124.842 24364.786 382373.649 -27391.497 55706.189 0 0 0 0 0 -108697.45 0 0

0 0 0 0 0 0 -15217.498 -27391.497 245425.832 -31652.396 0 0 0 0 0 0 -115104.167 0

0 24078.84 0 0 0 0 31652.396 55706.189 -31652.396 1088519.326 0 0 0 0 0 0 0 -452917.892

0 0 -115104.167 0 0 0 0 0 0 0 130321.665 31652.396 -15217.498 33478.496 0 0 0 0

-24078.84 24078.84 0 -452917.892 0 0 0 0 0 0 31652.396 577812.218 -31652.396 68367.147 0 0 0 0

0 0 0 0 -99022.083 0 0 0 0 0 -15217.498 -31652.396 115274.354 -30865.695 -1034.773 3026.71 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ … ⎥⎦

CÁLCULO DE LA MATRIZ LATERAL DEL PORTICO EN EL EJE X-X KL ≔ KLL - KLO ⋅ KOO -1 ⋅ KOL ⎡ 153203.912 -76601.956 ⎤ KLL ≔ ⎢ ⎣ -76601.956 76601.956 ⎥⎦ ⎡0 0 0 0 0 0 0 0 0 -24078.84 0 -21882.333 0 -21882.333 0 -24078.84 ⎤ KLO ≔ ⎢ ⎣ 0 24078.84 0 21882.333 0 21882.333 0 24078.84 0 24078.84 0 21882.333 0 21882.333 0 24078.84 ⎥⎦ ⎡ 0 ⎢ 0 ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ 0 ⎢ 0 ⎢ KOL ≔ ⎢ 0 ⎢ -24078.84 ⎢ 0 ⎢ ⎢ -21882.333 ⎢ 0 ⎢ -21882.333 ⎢ 0 ⎢ ⎢⎣ -24078.84 ⎡ 245425.832 ⎢ 31652.396 ⎢ ⎢ -15217.498 ⎢ 33478.496 ⎢ 0 ⎢ 0 ⎢ 0 ⎢ ⎢ 0 KOO ≔ ⎢ ⎢ -115104.167 0 ⎢ ⎢ 0 ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ 0 ⎣

⎤ 0 24078.84 ⎥ ⎥ 0 ⎥ 21882.333 ⎥ ⎥ 0 21882.333 ⎥ ⎥ 0 ⎥ 24078.84 ⎥ ⎥ 0 24078.84 ⎥ ⎥ 0 ⎥ 21882.333 ⎥ ⎥ 0 21882.333 ⎥ ⎥ 0 ⎥ 24078.84 ⎥⎦

31652.396 1088519.326 -31652.396 68367.147 0 0 0 0 0 -452917.892 0 0 0 0 0 0

-15217.498 -31652.396 214296.438 -30865.695 -1034.773 3026.71 0 0 0 0 -99022.083 0 0 0 0 0

33478.496 68367.147 -30865.695 404465.841 -2612.801 7124.842 0 0 0 0 0 -108697.45 0 0 0 0

0 0 -1034.773 -2612.801 214296.438 24364.786 -15217.498 31652.396 0 0 0 0 -99022.083 0 0 0

0 0 3026.71 7124.842 24364.786 382373.649 -27391.497 55706.189 0 0 0 0 0 -108697.45 0 0

0 0 0 0 -15217.498 -27391.497 245425.832 -31652.396 0 0 0 0 0 0 -115104.167 0

0 0 0 0 31652.396 55706.189 -31652.396 1088519.326 0 0 0 0 0 0 0 -452917.892

-115104.167 0 0 0 0 0 0 0 130321.665 31652.396 -15217.498 33478.496 0 0 0 0

⎡ 146013.117 -67033.577 ⎤ -1 KL ≔ KLL - KLO ⋅ KOO ⋅ KOL = ⎢ ⎥ ⎣ -67033.577 60549.83 ⎦

0 -452917.892 0 0 0 0 0 0 31652.396 577812.218 -31652.396 68367.147 0 0 0 0

0 0 -99022.083 0 0 0 0 0 -15217.498 -31652.396 115274.354 -30865.695 -1034.773 3026.71 0 0

0 0 0 -108697.45 0 0 0 0 33478.496 68367.147 -30865.695 243250.791 -2612.801 7124.842 0 0

0 0 0 0 -99022.083 0 0 0 0 0 -1034.773 -2612.801 115274.354 24364.786 -15217.498 31652.396

0 0 0 0 0 -108697.45 0 0 0 0 3026.71 7124.842 24364.786 221158.599 -27391.497 55706.189

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ …⎦

PORTICO EJE "1" DIRECCION Y -Y

CÁLCULO DE RIGIDEZ PARA LOS MUROS

PROPIEDADES MECANICAS PARA LA ALBAÑILERIA: tonf

f'm ≔ 650 ―― 2

h ≔ 2.4 m

m

tonf

E ≔ 500 ⋅ f'm = 325000 ―― 2 m tonf G ≔ 0.4 ⋅ E = 130000 ―― m2

Calculo de K1, K2, K3 y K4 : I ≔ 0.628623

Ac ≔ 0.3

A ≔ 0.66

12 ⋅ E ⋅ I ϕ ≔ ―――― = 10.914 G ⋅ Ac ⋅ h 2

⎡ 12 ⋅ E ⋅ I ⎤ 6⋅E⋅I 0 ―――― ⎥ ⎢ ―――― 3 2 ( ) ( ) h ⋅ (1 + ϕ) ⎥ ⎢ h ⋅ (1 + ϕ) ⎡ 0 17863.214 ⎤ ⎢ ⎥ ⎢ 14886.012 A ⎥ E ⋅ ―― 0 0 K1 ≔ ⎢ 0 89375 0 ⎥=⎢ ⎥ ( ) (h) ⎢ ⎥ ⎣ 17863.214 0 106561.888 ⎦ E ⋅ I ⋅ ((4 + ϕ)) ⎥ ⎢ 6⋅E⋅I 0 ――――― ⎢ ―――― ⎥ 2 h ⋅ ((1 + ϕ)) ⎦ ⎣ h ⋅ ((1 + ϕ)) ⎡ 14886.012 0 17863.214 ⎤ ⎥ K2 ≔ K1 = ⎢ 0 89375 0 ⎢ ⎥ 0 106561.888 ⎦ ⎣ 17863.214 ⎡ 12 ⋅ E ⋅ I -6 ⋅ E ⋅ I -12 ⋅ E ⋅ I -6 ⋅ E ⋅ I ⎤ 0 0 ―――― ―――― ―――― ⎥ ⎢ ―――― 3 h 2 ⋅ ((1 + ϕ)) h 3 ⋅ ((1 + ϕ)) h 2 ⋅ ((1 + ϕ)) ⎥ ⎢ h ⋅ ((1 + ϕ)) ⎢ ⎥ A A E ⋅ ―― -E ⋅ ―― 0 0 0 0 ⎢ ⎥ ( ) ( ) h h ) ) ( ( ⎢ ⎥ E ⋅ I ⋅ ((4 + ϕ)) 6⋅E⋅I E ⋅ I ⋅ ((2 - ϕ)) ⎥ ⎢ -6 ⋅ E ⋅ I 0 0 ――――――――― ――――― ⎢ ―――― ⎥ h ⋅ ((1 + ϕ)) h ⋅ ((1 + ϕ)) ⎥ h 2 ⋅ ((1 + ϕ)) h 2 ⋅ ((1 + ϕ)) K3 ≔ ⎢ 6⋅E⋅I 12 ⋅ E ⋅ I 6⋅E⋅I ⎢ -12 ⋅ E ⋅ I ⎥

⎡ 14886.012 0 17863.214 ⎤ ⎥ K2 ≔ K1 = ⎢ 0 89375 0 ⎢ ⎥ 0 106561.888 ⎦ ⎣ 17863.214 ⎡ 12 ⋅ E ⋅ I -6 ⋅ E ⋅ I -12 ⋅ E ⋅ I -6 ⋅ E ⋅ I ⎤ 0 0 ―――― ―――― ―――― ⎥ ⎢ ―――― 3 2 3 2 ( ) ( ) ( ) h 1 + ϕ h 1 + ϕ h 1 + ϕ h ⋅ ⋅ ⋅ ⋅ ((1 + ϕ)) ⎥ ) ) ) ( ( ( ⎢ ⎢ ⎥ A A E ⋅ ―― -E ⋅ ―― 0 0 0 0 ⎢ ⎥ ( ) ( ) (h) (h) ⎢ ⎥ E ⋅ I ⋅ ((4 + ϕ)) 6⋅E⋅I E ⋅ I ⋅ ((2 - ϕ)) ⎥ ⎢ -6 ⋅ E ⋅ I 0 0 ―――― ――――― ―――― ――――― ⎢ 2 ⎥ h ⋅ ((1 + ϕ)) h ⋅ ((1 + ϕ)) ⎥ h ⋅ ((1 + ϕ)) h 2 ⋅ ((1 + ϕ)) K3 ≔ ⎢ 6⋅E⋅I 12 ⋅ E ⋅ I 6⋅E⋅I ⎢ -12 ⋅ E ⋅ I ⎥ 0 0 ―――― ―――― ―――― ⎥ ⎢ ―――― 3 2 3 2 ( ) ( ) ( ) ( ) h ⋅ 1 + ϕ h ⋅ 1 + ϕ h ⋅ 1 + ϕ h ⋅ 1 + ϕ ) ) ) ) ( ( ( ( ⎢ ⎥ A A ⎢ ⎥ -E ⋅ ―― E ⋅ ―― 0 0 0 0 ⎢ ⎥ ((h)) ((h)) ⎢ ⎥ E ⋅ I ⋅ ((2 - ϕ)) 6⋅E⋅I E ⋅ I ⋅ ((4 + ϕ)) ⎥ ⎢ -6 ⋅ E ⋅ I 0 0 ―――― ――――― ―――― ――――― ⎢ 2 ( h ⋅ ((1 + ϕ)) h ⋅ ((1 + ϕ)) ⎥⎦ h 2 ⋅ ((1 + ϕ)) ⎣ h ⋅ (1 + ϕ))

⎡ 14886.012 0 -17863.214 -14886.012 0 -17863.214 ⎤ ⎢ ⎥ 0 89375 0 0 -89375 0 ⎢ ⎥ -17863.214 0 106561.888 17863.214 0 -63690.174 ⎥ K3 = ⎢ 0 17863.214 14886.012 0 17863.214 ⎥ ⎢ -14886.012 ⎢ ⎥ 0 -89375 0 0 89375 0 ⎢⎣ -17863.214 0 -63690.174 17863.214 0 106561.888 ⎥⎦ ⎡ 14886.012 0 -17863.214 -14886.012 0 -17863.214 ⎤ ⎢ ⎥ 0 89375 0 0 -89375 0 ⎢ ⎥ -17863.214 0 106561.888 17863.214 0 -63690.174 ⎥ K4 ≔ K3 = ⎢ 0 17863.214 14886.012 0 17863.214 ⎥ ⎢ -14886.012 ⎢ ⎥ 0 -89375 0 0 89375 0 ⎢⎣ -17863.214 0 -63690.174 17863.214 0 106561.888 ⎥⎦ CÁLCULO DE RIGIDEZ PARA LAS VIGAS

PROPIEDADES MECANICAS DEL CONCRETO: f'c ≔ 210

kgf ―― cm 2

h ≔ 2.4 m

tonf E ≔ 150000 ⋅ ‾‾‾ f'c = 2173706.512 ―― 2 m

para la viga K5, K6 a ≔ 1.10

b ≔ 1.10

L ≔ 3.15

⎡ 12 ⋅ E ⋅ I 6⋅E⋅I 12 ⋅ E ⋅ I ⋅a ―――― ―――+ ―――― ⎢ L3 L2 L3 ⎢ ⎢ 6⋅E⋅I 12 ⋅ E ⋅ I 4⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I ⋅a ⋅ a2 ⎢ ―――+ ―――― ―――+ ―――⋅ ((2 ⋅ a)) + ―――― L ⎢ L2 L3 L2 L3 K≔⎢ -12 ⋅ E ⋅ I -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I ⎢ - ―――― ⋅a ―――― ―――― ⎢ L3 L2 L3 ⎢ 6⋅E⋅I 12 ⋅ E ⋅ I 2⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I ⎢ ―――+ ―――― ⋅ b ―――+ ―――⋅ ((a + b)) + ―――― ⋅a⋅b ⎢⎣ L L2 L3 L2 L3

⎤ -12 ⋅ E ⋅ I 6⋅E⋅I 12 ⋅ E ⋅ I ⋅b ―――― ―――+ ―――― ⎥ L3 L2 L3 ⎥ ⎥ -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I 2⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I - ―――― ⋅ a ―――+ ―――⋅ ((a + b)) + ―――― ⋅a⋅b⎥ ―――― L ⎥ L2 L3 L2 L3 ⎥ 12 ⋅ E ⋅ I -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I ⎥ - ―――― ⋅b ―――― ―――― ⎥ L3 L2 L3 ⎥ -6 ⋅ E ⋅ I 12 ⋅ E ⋅ I 4⋅E⋅I 6⋅E⋅I 12 ⋅ E ⋅ I - ―――― ⋅b ⋅ b2 ⎥ ―――― ―――+ ―――⋅ ((2 b)) + ―――― ⎥⎦ L L2 L3 L2 L3

⎡ 486.868 1302.373 -486.868 1302.373 ⎤ ⎢ 1302.373 3886.427 -1302.373 3081.268 ⎥ K5 ≔ K = ⎢ ⎥ 486.868 -1302.373 ⎥ ⎢ -486.868 -1302.373 ⎣ 1302.373 3081.268 -1302.373 3886.427 ⎦ ⎡ 486.868 1302.373 -486.868 1302.373 ⎤ ⎢ 1302.373 3886.427 -1302.373 3081.268 ⎥ K6 ≔ K = ⎢ ⎥ 486.868 -1302.373 ⎥ ⎢ -486.868 -1302.373 ⎣ 1302.373 3081.268 -1302.373 3886.427 ⎦

CÁLCULO DE LA MATRIZ A

I ≔ 0.000583392857142857

⎡ 486.868 1302.373 -486.868 1302.373 ⎤ ⎢ 1302.373 3886.427 -1302.373 3081.268 ⎥ K6 ≔ K = ⎢ ⎥ 486.868 -1302.373 ⎥ ⎢ -486.868 -1302.373 ⎣ 1302.373 3081.268 -1302.373 3886.427 ⎦

CÁLCULO DE LA MATRIZ A ⎡1 0 0 0 0 0 0 0 0 0⎤ A1 ≔ ⎢ 0 0 1 0 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎣0 0 0 1 0 0 0 0 0 0⎦ ⎡1 A2 ≔ ⎢ 0 ⎢ ⎣0 ⎡1 ⎢0 ⎢ 0 A3 ≔ ⎢ ⎢0 ⎢0 ⎢⎣ 0

0 0 0 0 0 0 1 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0

0⎤ 0⎥ ⎥ 0⎦ 0⎤ 0⎥ ⎥ 0⎥ 0⎥ 0⎥ 0 ⎥⎦

⎡1 ⎢0 ⎢ 0 A4 ≔ ⎢ ⎢0 ⎢0 ⎢⎣ 0

0 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 1 0

0⎤ 0⎥ ⎥ 0⎥ 0⎥ 0⎥ 1 ⎥⎦

⎡0 ⎢0 A5 ≔ ⎢ ⎢0 ⎣0

0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

0⎤ 0⎥ ⎥ 0⎥ 0⎦

⎡0 ⎢0 A6 ≔ ⎢ ⎢0 ⎣0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

0⎤ 0⎥ ⎥ 0⎥ 1⎦

CÁLCULO DE LA MATRIZ DEL PORTICO EN EL EJE X-X Ki ≔ 1 i ≔ 1 14 T ( ) K (portico) ≔ ∑ Ai ⋅ Ki ⋅ Ai i

KG ≔ A1 T ⋅ K1 ⋅ A1 + A2 T ⋅ K2 ⋅ A2 + A3 T ⋅ K3 ⋅ A3 + A4 T ⋅ K4 ⋅ A4 + A5 T ⋅ K5 ⋅ A5 + A6 T ⋅ K6 ⋅ A6 ⎡ 59544.048 -29772.024 0 0 0 0 0 -17863.214 0 -17863.214 ⎤ ⎢ -29772.024 29772.024 0 17863.214 0 17863.214 0 17863.214 0 17863.214 ⎥ ⎢ ⎥ 0 0 179236.868 1302.373 1302.373 0 0 0 -486.868 -89375 ⎢ ⎥ -63690.174 0 17863.214 1302.373 217010.204 -1302.373 3081.268 0 0 0 ⎢ ⎥ ⎢ ⎥ -486.868 -1302.373 179236.868 -1302.373 -89375 0 0 0 0 0 KG = ⎢ -63690.174 ⎥ 0 17863.214 1302.373 3081.268 -1302.373 217010.204 0 0 0 ⎢ ⎥ -89375 -486.868 0 0 0 0 0 89861.868 1302.373 1302.373 ⎥ ⎢ ⎢ -17863.214 -63690.174 17863.214 0 0 0 1302.373 110448.315 -1302.373 3081.268 ⎥ ⎢ -89375 -486.868 -1302.373 0 0 0 0 0 89861.868 -1302.373 ⎥ ⎢ ⎥ 17863.214 0 0 0 1302.373 3081.268 -1302.373 110448.315 ⎦ -63690.174 ⎣ -17863.214

CÁLCULO DE LA MATRIZ LATERAL DEL PORTICO EN EL EJE X-X KL ≔ KLL - KLO ⋅ KOO -1 ⋅ KOL ⎡ 59544.048 -29772.024 ⎤ KLL ≔ ⎢ ⎣ -29772.024 29772.024 ⎥⎦

⎡ 59544.048 -29772.024 ⎤ KLL ≔ ⎢ ⎣ -29772.024 29772.024 ⎥⎦ ⎡0 0 0 0 0 -17863.214 0 -17863.214 ⎤ KLO ≔ ⎢ ⎣ 0 17863.214 0 17863.214 0 17863.214 0 17863.214 ⎥⎦ ⎡ ⎤ 0 0 ⎢ 0 17863.214 ⎥ ⎢ ⎥ 0 0 ⎢ ⎥ 0 17863.214 ⎥ ⎢ KOL ≔ ⎢ ⎥ 0 0 ⎢ -17863.214 17863.214 ⎥ ⎢ ⎥ 0 0 ⎢ ⎥ ⎣ -17863.214 17863.214 ⎦ ⎡ 179236.868 1302.373 ⎤ -486.868 1302.373 -89375 0 0 0 ⎢ 1302.373 217010.204 -1302.373 3081.268 ⎥ 0 -63690.174 0 0 ⎢ ⎥ 0 0 -89375 0 ⎢ -486.868 -1302.373 179236.868 -1302.373 ⎥ 1302.373 3081.268 -1302.373 217010.204 0 0 0 -63690.174 ⎥ ⎢ KOO ≔ ⎢ -89375 0 0 0 89861.868 1302.373 -486.868 1302.373 ⎥ ⎢ 0 -63690.174 0 0 1302.373 110448.315 -1302.373 3081.268 ⎥ ⎢ ⎥ 0 0 -89375 0 -486.868 -1302.373 89861.868 -1302.373 ⎥ ⎢ 0 0 0 -63690.174 1302.373 3081.268 -1302.373 110448.315 ⎦ ⎣

⎡ 52819.354 -21098.431 ⎤ -1 KL ≔ KLL - KLO ⋅ KOO ⋅ KOL = ⎢ ⎥ ⎣ -21098.431 15684.085 ⎦