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DISEÑO DE LOSAS ALIGERADAS EN DOS DIRECCIONES - METODO COEFICIENTES DEL ACI 1. APLICACIÓN DEL METODO - VIGAS RIGIDAS Se debe determinar si la viga de apoyo es rígida o no para saber si utilizamos el método. α Para esto utilizamos el coeficiente

6.00

Viga 3

Viga 2

Viga 1 4.25

Losa

�=(𝐸_𝑐𝑏.𝐼_ 𝑏)/ Viga 4 (�_��.�_� ) α1 = α2 = α3 = α4 = αprom =

Asumiendo Espesor de Losa Ecb = Ecs Viga 1 27.00 40.00 cm Viga 2 27.00 40.00 cm Viga 3 27.00 40.00 cm Viga 4 27.00 40.00 cm Ib1 = Ib2 = Ib3 = Ib4 =

144000.00 144000.00 144000.00 144000.00

cm4 cm4 cm4 cm4

20.00

cm

B= B= B= B=

3.00 2.13 4.25 3.00

Is1 = Is2 = Is3 = Is4 =

200000.00 cm4 141666.67 cm4 283333.33 cm4 200000.00 cm4

0.72 1.02 0.51 0.72 0.74

Ok, se puede considerar que la viga es muy rígida con respecto a la losa y que no se deforma al aplicar la carga

2. VERIFICACION DEL ESPESOR DE LA LOSA αprom = 0.74

fy = 4200.0 Kg/cm2 fy = 411.76 Mpa

Formula 9.6.3.3 NTE E.060 h = Espesor minimo: h = 16 Ok No debe ser menor que 12.5 cm

m m m m

ANALISIS Y DISEÑO DE LOSA ALIGERADA EN DOS DIRECCIONES

CASO Continuo Libre

DATOS Sobrecarga (S/C) = 300 Kg/m2 Recubrimiento = 2.50 cm Tab. Repartida = 0 Kg/m2 f'c = 210 Kg/cm2 fy = 4200 Kg/cm2 Alig. En dos direc = 360 Kg/m2 Peso especifico C° 2400 Kg/m3

4.250 m

6.000 m 1. DETERMINACION DEL ESPESOR DE LA LOSA.

h=

20.00 m

2. METRADO DE CARGAS CARGA MUERTA (Cm) Peso de Alig. 20 360 Kg/m2 Peso Muerto = 180 Kg/m2 Peso Tab. Equiv. 0 Kg/m2 Cm = 540 Kg/m2

CARGA ULTIMAS

CARGA VIVA (Cv) Sobrecarga (S/C) 300 Kg/m2 Cv = 300 Kg/m2 3. ANALISIS Restricciones de los apoyos Libre ,

, , , Libre , , ,

Wu cm =

756 Kg/m2

Wu cv  1.7 * Cv

Wu cv =

510 Kg/m2

Wu  1.4 * Cm 1.7 * Cv

Wu = 1266 Kg/m2

Para b = 1 m

Wu =

CASO :

1266 Kg/m

CASO 1

, , , , , ,

,

,

Wu cm  1.4 * Cm

Libre

A m  B

A= B=

4.250 6.000

m=

0.708

, Libre

cm 3.1 Determinacion de los Momentos Negativos cv De la Tabla 13.1 de la NORMA E.060: Coeficientes del ACI:

Ca =

Cb =

0.70 0.708 0.75

Ver Tabla #VALUE! Ver Tabla

0.70 0.708 0.75

Ver Tabla #VALUE! Ver Tabla

297 Kg/m2 297.088532383

Ma neg  Ca wu A 2 Ma (-) =

#VALUE!

Mb neg  Cb wu B 2 Mb (-) =

#VALUE!

3.2 Determinacion de los Momentos Positivos De la Tabla 13.2 y 13.3 de la NORMA E.060: Coeficientes del ACI: COEFICIENTES POR CARGA MUERTA 0.70

0.068

Ma posd  Ca wud A2

Ca =

Cb =

0.708 0.75

0.067 0.061

0.70 0.708 0.75

0.016 0.017 0.019

Ma posd  Ca wud A2 Ma Pos =

913 Kg-m

Mb posd  Cb wud B2 Mb Pos =

449 Kg-m

COEFICIENTES POR CARGA VIVA

Ca =

Cb =

0.70 0.708 0.75

0.068 0.067 0.061

0.70 0.708 0.75

0.016 0.017 0.019

Ma pos l  Ca wul A2 Ma Pos =

616 Kg-m

Mb pos l  Cb wul B2 Mb Pos =

303 Kg-m

Momentos Positivos Ma  Ma posd  Ma posl

Ma (+) =

1528 Kg-m

Mb  Mb pos d  Mb posl

Mb (+) =

752 Kg-m

FORMA DE GRAFICAR MOMENTOS EN LOSAS MB (-)

MB(+)/3

MB (+)

B

A

MA (-)

MA (+)

LOSA EN DOS DIRECCIONES

MA(+)/3

4. DISEÑO DE REFUERZO POR FLEXION Diseño para un ancho b = 100.0 cm 4.1 REFUERZO NEGATIVO Datos bw = f'c = fy = d=

25.00 cm 210 Kg/cm2 4200 Kg/cm2 17.02 cm

Para Mu-: a= As = As Vigueta =

#VALUE! #VALUE! #VALUE! #VALUE! 1 Φ 1/2 + 1 Φ 3/8

Peralte Efectivo d = hn - Rec.- (3/8")/2 d= 17.02 cm

a 

As . fy 0 . 85 . f ' c .b

2.00 cm²

As 

#VALUE!

Mu . fy.(d  a / 2)

Longitud de desarrollo 1.427 2.02

4.2 REFUERZO POSITIVO Datos bw = f'c = fy = d=

100.00 cm 210 Kg/cm2 4200 Kg/cm2 17.02 cm As 

Para Mu+: a= As = As Vigueta =

Datos bw = f'c = fy = d= Para Mu+: a= As = As Vigueta =

1528 Kg-m 0.57 cm 2.42 cm² 0.97 cm² 1 Φ 1/2

a 

Mu . fy.(d  a / 2) As . fy 0 . 85 . f ' c .b

Para Mu+/3: a= As = As Vigueta =

Ok

1.29 cm²

752 Kg-m 0.28 cm 1.18 cm² 0.47 cm² 1 Φ 3/8

5.1 ACERO MINIMO Cuantia Minima: 1.028 cm2 0.411 cm2 1 Φ 3/8

5.1 ACERO MAXIMO Cuantia Maxima As max = As max/vig =

Mu . fy.(d  a / 2)

a 

As . fy 0 . 85 . f ' c .b

0.71 cm²

100.00 cm 210 Kg/cm2 4200 Kg/cm2 17.02 cm As 

Mu . fy.(d  a / 2)

a 

As . fy 0 . 85 . f ' c .b

Para Mu+/3: a= As = As Vigueta =

Ok

0.71 cm²

5. ACERO MAXIMO Y MINIMO

As min = As min/vig =

509 Kg-m 0.19 cm 0.80 cm² 0.32 cm² 1 Φ 3/8

As 

6.923 cm2 2.769 cm2 2 Φ 1/2

0.0024

As min   min .b .d

0.71 cm²

Ok

0.75*ρb

= ρb =

0.0162675 0.02169

As max   max .b .d

2.58 cm²

As Insuficiente

251 Kg-m 0.09 cm 0.39 cm² 0.16 cm² 1 Φ 3/8

0.71 cm²

CASO Coef. CASO 1 1.00 Ca Ver Tabla Cb Ver Tabla 0.95 Ca Ver Tabla Cb Ver Tabla 0.90 Ca Ver Tabla Cb Ver Tabla 0.85 Ca Ver Tabla Cb Ver Tabla 0.80 Ca Ver Tabla Cb Ver Tabla 0.75 Ca Ver Tabla Cb Ver Tabla 0.70 Ca Ver Tabla Cb Ver Tabla 0.65 Ca Ver Tabla Cb Ver Tabla 0.60 Ca Ver Tabla Cb Ver Tabla 0.55 Ca Ver Tabla Cb Ver Tabla 0.50 Ca Ver Tabla Cb Ver Tabla

CASO 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50

Coef. CASO 1 Ca 0.036 Cb 0.036 Ca 0.040 Cb 0.033 Ca 0.045 Cb 0.029 Ca 0.050 Cb 0.026 Ca 0.056 Cb 0.023 Ca 0.061 Cb 0.019 Ca 0.068 Cb 0.016 Ca 0.074 Cb 0.013 Ca 0.081 Cb 0.010 Ca 0.088 Cb 0.008 Ca 0.095 Cb 0.006

CASO 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50

Coef. Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb

CASO 1 0.036 0.036 0.040 0.033 0.045 0.029 0.050 0.026 0.056 0.023 0.061 0.019 0.068 0.016 0.074 0.013 0.081 0.010 0.088 0.008 0.095 0.006

As 

Mu . fy.(d  a / 2)

a 

As . fy 0 . 85 . f ' c .b

Ok

Ok

ANALISIS Y DISEÑO DE LOSA ALIGERADA EN DOS DIRECCIONES

DATOS Sobrecarga (S/C) = 300 Kg/m2 Recubrimiento = 2.50 cm Tab. Repartida = 0 Kg/m2 f'c = 210 Kg/cm2 fy = 4200 Kg/cm2 Alig. En dos direc = 360 Kg/m2 Peso especifico C° 2400 Kg/m3

4.250 m

6.000 m 1. DETERMINACION DEL ESPESOR DE LA LOSA.

h=

20.00 m

2. METRADO DE CARGAS CARGA MUERTA (Cm) Peso de Alig. 20 1000 Kg/m2 Peso Muerto = 180 Kg/m2 Peso Tab. Equiv. = 0 Kg/m2 Cm = 1180 Kg/m2

CARGA ULTIMAS

CARGA VIVA (Cv) Sobrecarga (S/C) = 300 Kg/m2 Cv = 200 Kg/m2 3. ANALISIS Restricciones de los apoyos Continuo .

. , , Libre , , ,

Wu cm = 1652 Kg/m2

Wu cv  1.7 * Cv

Wu cv =

Wu = 1992 Kg/m2

Para b = 1 m

Wu =

.

Libre

A m  B

.

0.70 0.708 0.75

0.086 0.086 0.085

0.70 0.708 0.75

0 0.000 0

Ma neg  Ca wu A 2 Ma (-) =

3088 Kg-m

Mb neg  Cb wu B 2 Mb (-) =

0 Kg-m

3.2 Determinacion de los Momentos Positivos De la Tabla 13.2 y 13.3 de la NORMA E.060: Coeficientes del ACI: COEFICIENTES POR CARGA MUERTA 0.70

0.035

A= B= m=

3.1 Determinacion de los Momentos Negativos De la Tabla 13.1 de la NORMA E.060: Coeficientes del ACI:

Cb =

1992 Kg/m

CASO 5

Continuo

Ca =

340 Kg/m2

Wu  1.4 * Cm 1.7 * Cv

CASO :

, , , , ,

.

.

Wu cm  1.4 * Cm

Ma posd  Ca wud A2

Ca =

Cb =

0.708 0.75

0.035 0.033

0.70 0.708 0.75

0.005 0.005 0.007

Ma posd  Ca wud A2 Ma Pos =

1034 Kg-m

Mb posd  Cb wud B2 Mb Pos =

317 Kg-m

COEFICIENTES POR CARGA VIVA

Ca =

Cb =

0.70 0.708 0.75

0.051 0.050 0.047

0.70 0.708 0.75

0.011 0.011 0.013

Ma pos l  Ca wul A2 Ma Pos =

309 Kg-m

Mb pos l  Cb wul B2 Mb Pos =

139 Kg-m

Momentos Positivos Ma  Ma posd  Ma posl

Ma (+) =

1344 Kg-m

Mb  Mb pos d  Mb pos l

Mb (+) =

456 Kg-m

FORMA DE GRAFICAR MOMENTOS EN LOSAS MB (-)

MB(+)/3

MB (+)

B

A

MA (-)

MA (+)

LOSA EN DOS DIRECCIONES

MA(+)/3

4. DISEÑO DE REFUERZO POR FLEXION Diseño para un ancho b = 100.0 cm 4.1 REFUERZO NEGATIVO Datos bw = f'c = fy = d= Para Mu-: a= As = As Vigueta =

25.00 cm 210 Kg/cm2 4200 Kg/cm2 17.02 cm 3088 Kg-m 5.36 cm 5.70 cm² 2.28 cm² 1 Φ 1/2

Peralte Efectivo d = hn - Rec.- (3/8")/2 d= 17.02 cm

a 

As . fy 0 . 85 . f ' c .b

1.29 cm²

As 

Mu . fy.(d  a / 2)

As Insuficiente

Longitud de desarrollo 1.427 2.02

4.2 REFUERZO POSITIVO Datos bw = f'c = fy = d=

100.00 cm 210 Kg/cm2 4200 Kg/cm2 17.02 cm As 

Para Mu+: a= As = As Vigueta =

1344 Kg-m 0.50 cm 2.12 cm² 0.85 cm² 1 Φ 3/8

Mu . fy.(d  a / 2)

Para Mu+/3: a= As . fy a  As = 0 . 85 . f ' c .b As Vigueta = As Insuficiente 0.71 cm²

448 Kg-m 0.16 cm 0.70 cm² 0.28 cm² 1 Φ 3/8

Datos bw = f'c = fy = d=

100.00 cm 210 Kg/cm2 4200 Kg/cm2 17.02 cm

Para Mu+: a= As = As Vigueta =

456 Kg-m 0.17 cm 0.71 cm² 0.28 cm² 1 Φ 3/8

As 

Mu . fy.(d  a / 2)

a 

As . fy 0 . 85 . f ' c .b

Para Mu+/3: a= As = As Vigueta =

0.71 cm²

Ok

0.0024

As min   min .b .d

0.71 cm²

Ok

5. ACERO MAXIMO Y MINIMO 5.1 ACERO MINIMO Cuantia Minima: As min = As min/vig =

1.028 cm2 0.411 cm2 1 Φ 3/8

5.1 ACERO MAXIMO Cuantia Maxima As max = As max/vig =

6.923 cm2 2.769 cm2 2 Φ 1/2

0.75*ρb

= ρb =

0.0162675 0.02169

As max   max .b .d

2.58 cm²

As Insuficiente

152 Kg-m 0.06 cm 0.24 cm² 0.09 cm² 1 Φ 3/8

Continuo Libre

4.250 6.000 0.708

CASO Coef. CASO 5 1.00 Ca 0.075 Cb 0.000 0.95 Ca 0.079 Cb 0.000 0.90 Ca 0.080 Cb 0.000 0.85 Ca 0.082 Cb 0.000 0.80 Ca 0.083 Cb 0.000 0.75 Ca 0.085 Cb 0.000 0.70 Ca 0.086 Cb 0.000 0.65 Ca 0.087 Cb 0.000 0.60 Ca 0.088 Cb 0.000 0.55 Ca 0.089 Cb 0.000 0.50 Ca 0.090 Cb 0.000

CASO 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50

Coef. CASO 5 Ca 0.027 Cb 0.018 Ca 0.028 Cb 0.015 Ca 0.029 Cb 0.013 Ca 0.031 Cb 0.011 Ca 0.032 Cb 0.009 Ca 0.033 Cb 0.007 Ca 0.035 Cb 0.005 Ca 0.036 Cb 0.004 Ca 0.037 Cb 0.003 Ca 0.038 Cb 0.002 Ca 0.039 Cb 0.001

CASO 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50

Coef. Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb

CASO 5 0.032 0.027 0.034 0.024 0.037 0.021 0.041 0.019 0.044 0.010 0.047 0.013 0.051 0.011 0.055 0.009 0.059 0.007 0.063 0.005 0.067 0.004

gitud de desarrollo

As 

Mu . fy.(d  a / 2)

a 

As . fy 0 . 85 . f ' c .b

0.71 cm²

Ok

0.71 cm²

Ok

ANALISIS Y DISEÑO DE LOSA ALIGERADA EN DOS DIRECCIONES

CASO Continuo Libre

DATOS Sobrecarga (S/C) = 250 Kg/m2 Recubrimiento = 2.50 cm Tab. Repartida = 0 Kg/m2 f'c = 210 Kg/cm2 fy = 4200 Kg/cm2 Alig. En dos direc = 360 Kg/m2 Peso especifico C° 2400 Kg/m3

4.250 m

6.000 m 1. DETERMINACION DEL ESPESOR DE LA LOSA.

h=

20.00 m

2. METRADO DE CARGAS CARGA MUERTA (Cm) Peso de Alig. 20 360 Kg/m2 Peso Muerto = 180 Kg/m2 Peso Tab. Equiv. 0 Kg/m2 Cm = 540 Kg/m2

CARGA ULTIMAS

CARGA VIVA (Cv) Sobrecarga (S/C) 250 Kg/m2 Cv = 200 Kg/m2 3. ANALISIS Restricciones de los apoyos Continuo .

. , , Libre , , ,

Wu cm =

756 Kg/m2

Wu cv  1.7 * Cv

Wu cv =

340 Kg/m2

Wu  1.4 * Cm 1.7 * Cv

Wu = 1096 Kg/m2

Para b = 1 m

Wu =

CASO :

CASO 6

Libre

A m  B

, Libre

3.1 Determinacion de los Momentos Negativos De la Tabla 13.1 de la NORMA E.060: Coeficientes del ACI:

Ca =

Cb =

0.70 0.708 0.75

0.091 0.091 0.088

0.70 0.708 0.75

0 0.000 0

Ma neg  Ca wu A 2 Ma (-) =

1792 Kg-m

Mb neg  Cb wu B 2 Mb (-) =

0 Kg-m

3.2 Determinacion de los Momentos Positivos De la Tabla 13.2 y 13.3 de la NORMA E.060: Coeficientes del ACI: COEFICIENTES POR CARGA MUERTA 0.70

1096 Kg/m

. , , , , ,

,

,

Wu cm  1.4 * Cm

0.051

Ma posd  Ca wud A2

A= B=

4.250 6.000

m=

0.708

Ca =

Cb =

0.708 0.75

0.051 0.048

0.70 0.708 0.75

0.009 0.010 0.012

Ma posd  Ca wud A2 Ma Pos =

690 Kg-m

Mb posd  Cb wud B2 Mb Pos =

259 Kg-m

COEFICIENTES POR CARGA VIVA

Ca =

Cb =

0.70 0.708 0.75

0.06 0.059 0.055

0.70 0.708 0.75

0.013 0.014 0.016

Ma pos l  Ca wul A2 Ma Pos =

363 Kg-m

Mb pos l  Cb wul B2 Mb Pos =

165 Kg-m

Momentos Positivos Ma  Ma posd  Ma posl

Ma (+) =

1053 Kg-m

Mb  Mb pos d  Mb posl

Mb (+) =

424 Kg-m

FORMA DE GRAFICAR MOMENTOS EN LOSAS MB (-)

MB(+)/3

MB (+)

B

A

MA (-)

MA (+)

LOSA EN DOS DIRECCIONES

MA(+)/3

4. DISEÑO DE REFUERZO POR FLEXION Diseño para un ancho b = 100.0 cm 4.1 REFUERZO NEGATIVO Datos bw = f'c = fy = d= Para Mu-: a= As = As Vigueta =

25.00 cm 210 Kg/cm2 4200 Kg/cm2 17.02 cm 1792 Kg-m 2.86 cm 3.04 cm² 1.22 cm² 1 Φ 1/2

Peralte Efectivo d = hn - Rec.- (3/8")/2 d= 17.02 cm

a 

As . fy 0 . 85 . f ' c .b

1.29 cm²

Ok

As 

Mu . fy.(d  a / 2)

Longitud de desarrollo 1.427 2.02

4.2 REFUERZO POSITIVO Datos bw = f'c = fy = d=

100.00 cm 210 Kg/cm2 4200 Kg/cm2 17.02 cm As 

Para Mu+: a= As = As Vigueta =

Datos bw = f'c = fy = d= Para Mu+: a= As = As Vigueta =

1053 Kg-m 0.39 cm 1.66 cm² 0.66 cm² 1 Φ 3/8

a 

Mu . fy.(d  a / 2) As . fy 0 . 85 . f ' c .b

Para Mu+/3: a= As = As Vigueta =

Ok

0.71 cm²

424 Kg-m 0.16 cm 0.66 cm² 0.26 cm² 1 Φ 3/8

5.1 ACERO MINIMO Cuantia Minima: 1.028 cm2 0.411 cm2 1 Φ 3/8

5.1 ACERO MAXIMO Cuantia Maxima As max = As max/vig =

Mu . fy.(d  a / 2)

a 

As . fy 0 . 85 . f ' c .b

0.71 cm²

100.00 cm 210 Kg/cm2 4200 Kg/cm2 17.02 cm As 

Mu . fy.(d  a / 2)

a 

As . fy 0 . 85 . f ' c .b

Para Mu+/3: a= As = As Vigueta =

Ok

0.71 cm²

5. ACERO MAXIMO Y MINIMO

As min = As min/vig =

351 Kg-m 0.13 cm 0.55 cm² 0.22 cm² 1 Φ 3/8

As 

6.923 cm2 2.769 cm2 2 Φ 1/2

0.0024

As min   min .b .d

0.71 cm²

Ok

0.75*ρb

= ρb =

0.0162675 0.02169

As max   max .b .d

2.58 cm²

As Insuficiente

141 Kg-m 0.05 cm 0.22 cm² 0.09 cm² 1 Φ 3/8

0.71 cm²

CASO Coef. CASO 6 1.00 Ca 0.071 Cb 0.000 0.95 Ca 0.075 Cb 0.000 0.90 Ca 0.079 Cb 0.000 0.85 Ca 0.083 Cb 0.000 0.80 Ca 0.086 Cb 0.000 0.75 Ca 0.088 Cb 0.000 0.70 Ca 0.091 Cb 0.000 0.65 Ca 0.093 Cb 0.000 0.60 Ca 0.095 Cb 0.000 0.55 Ca 0.096 Cb 0.000 0.50 Ca 0.097 Cb 0.000

CASO 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50

Coef. CASO 6 Ca 0.033 Cb 0.027 Ca 0.036 Cb 0.024 Ca 0.039 Cb 0.021 Ca 0.042 Cb 0.017 Ca 0.045 Cb 0.015 Ca 0.048 Cb 0.012 Ca 0.051 Cb 0.009 Ca 0.054 Cb 0.007 Ca 0.056 Cb 0.006 Ca 0.058 Cb 0.004 Ca 0.061 Cb 0.003

CASO 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50

Coef. Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb

CASO 6 0.035 0.032 0.038 0.029 0.042 0.025 0.046 0.022 0.051 0.019 0.055 0.016 0.060 0.013 0.064 0.010 0.068 0.008 0.073 0.006 0.078 0.005

As 

Mu . fy.(d  a / 2)

a 

As . fy 0 . 85 . f ' c .b

Ok

Ok

ANALISIS Y DISEÑO DE LOSA ALIGERADA EN DOS DIRECCIONES Continuo Libre DATOS Sobrecarga (S/C) = 100 Kg/m2 Recubrimiento = 2.50 cm 4.250 m Tab. Repartida = 0 Kg/m2 f'c = 210 Kg/cm2 fy = 4200 Kg/cm2 Alig. En dos direc = 360 Kg/m2 6.000 m Peso especifi 2400 Kg/m3 1. DETERMINACION DEL ESPESOR DE LA LOSA. h=

20.00 m

2. METRADO DE CARGAS CARGA MUERTA (Cm) Peso de Alig. 20 360 Kg/m2 Peso Muerto = 180 Kg/m2 Peso Tab. Equiv. 0 Kg/m2 Cm = 540 Kg/m2

CARGA ULTIMAS

Wu cm  1.4 * CmWu cm = 756 Kg/m2

Wu cv  1.7 * CvWu cv = 340 Kg/m2 Wu  1.4 * Cm 1.7 * CvWu = 1096 Kg/m2

CARGA VIVA (Cv) Sobrecarga (S/C) 100 Kg/m2 Cv = 200 Kg/m2

Para b = 1 m

3. ANALISIS Restricciones de los apoyos Libre ,

, , , Libre , , ,

CASO 1

, , , , , ,

,

,

CASO :

Wu = 1096 Kg/m

Libre

A m  B

, Libre

3.1 Determinacion de los Momentos Negativos De la Tabla 13.1 de la NORMA E.060: Coeficientes del ACI:

Ca =

Cb =

0.70 0.708 0.75

Ver Tabla 0.000 Ver Tabla

0.70 0.708 0.75

Ver Tabla 0.000 Ver Tabla

Ma neg  Ca wu A 2 Ma (-) =

0 Kg-m

Mb neg  Cb wu B 2 Mb (-) =

0 Kg-m

3.2 Determinacion de los Momentos Positivos De la Tabla 13.2 y 13.3 de la NORMA E.060: Coeficientes del ACI: COEFICIENTES POR CARGA MUERTA 0.70

0.068

Ma posd  Ca wud A2

A= B=

4.250 6.000

m=

0.708

CASOCoef. ### Ca Cb ### Ca Cb ### Ca Cb ### Ca Cb ### Ca Cb ### Ca Cb ### Ca Cb ### Ca Cb ### Ca Cb ### Ca Cb ### Ca Cb

Ca =

Cb =

0.708 0.75

0.067 0.061

0.70 0.708 0.75

0.016 0.017 0.019

Ma posd  Ca wud A2 Ma Pos = 913 Kg-m Mb posd  Cb wud B2 Mb Pos = 449 Kg-m

COEFICIENTES POR CARGA VIVA

Ca =

Cb =

0.70 0.708 0.75

0.068 0.067 0.061

0.70 0.708 0.75

0.016 0.017 0.019

Ma pos l  Ca wul A2 Ma Pos = 410 Kg-m Mb pos l  Cb wul B2 Mb Pos = 202 Kg-m

Momentos Positivos Ma  Ma pos d  Ma posl

Ma (+) =

1323 Kg-m

Mb  Mb pos d  Mb pos l

Mb (+) =

651 Kg-m

FORMA DE GRAFICAR MOMENTOS EN LOSAS MB (-)

MB(+)/3

MB (+)

B

A

MA (-)

MA (+)

LOSA EN DOS DIRECCIONES

MA(+)/3

4. DISEÑO DE REFUERZO POR FLEXION Diseño para un ancho b = 100.0 cm 4.1 REFUERZO NEGATIVO Datos bw = f'c = fy = d= Para Mu-: a= As = As Vigueta =

25.00 cm 210 Kg/cm2 4200 Kg/cm2 17.02 cm 0 Kg-m 0.00 cm 0.00 cm² 0.00 cm² 1 Φ 3/8

Peralte Efectivo d = hn - Rec.- (3/8")/2 d = 17.02 cm

a 

As . fy 0 . 85 . f ' c .b

0.71 cm²

Ok

Longitud de desarrollo 1.427 Mu As  2.02 . fy.(d  a / 2)

4.2 REFUERZO POSITIVO Datos bw = f'c = fy = d=

100.00 cm 210 Kg/cm2 4200 Kg/cm2 17.02 cm As 

Para Mu+: a= As = As Vigueta =

Datos bw = f'c = fy = d= Para Mu+: a= As = As Vigueta =

1323 Kg-m 0.49 cm 2.09 cm² 0.83 cm² 1 Φ 1/2

a 

Mu . fy.(d  a / 2) As . fy 0 . 85 . f ' c .b

Ok

1.29 cm²

As 

Para Mu+/3: 441 Kg-m 0.16 cm a= As = 0.69 cm² As Vigueta = 0.28 cm² 1 Φ 3/8

651 Kg-m 0.24 cm 1.02 cm² 0.41 cm² 1 Φ 3/8

5.1 ACERO MINIMO Cuantia Minima: 1.028 cm2 0.411 cm2 1 Φ 3/8

5.1 ACERO MAXIMO Cuantia Maxima As max = As max/vig =

As . fy 0 . 85 . f ' c .b

0.71 cm²

Ok

0.71 cm²

Ok

100.00 cm 210 Kg/cm2 4200 Kg/cm2 17.02 cm As 

Mu . fy.(d  a / 2)

a 

As . fy 0 . 85 . f ' c .b

Ok

0.71 cm²

Para Mu+/3: 217 Kg-m 0.08 cm a= As = 0.34 cm² As Vigueta = 0.14 cm² 1 Φ 3/8

5. ACERO MAXIMO Y MINIMO

As min = As min/vig =

a 

Mu . fy.(d  a / 2)

6.923 cm2 2.769 cm2 2 Φ 1/2

0.0024

As min   min .b .d

0.71 cm²

Ok

0.75*ρb

= ρb =

0.0162675 0.02169

As max   max .b .d

2.58 cm²

As Insuficiente

CASO 1 Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla Ver Tabla

CASO 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50

Coef. CASO 1 Ca 0.036 Cb 0.036 Ca 0.040 Cb 0.033 Ca 0.045 Cb 0.029 Ca 0.050 Cb 0.026 Ca 0.056 Cb 0.023 Ca 0.061 Cb 0.019 Ca 0.068 Cb 0.016 Ca 0.074 Cb 0.013 Ca 0.081 Cb 0.010 Ca 0.088 Cb 0.008 Ca 0.095 Cb 0.006

CASO 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50

Coef. Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb

CASO 1 0.036 0.036 0.040 0.033 0.045 0.029 0.050 0.026 0.056 0.023 0.061 0.019 0.068 0.016 0.074 0.013 0.081 0.010 0.088 0.008 0.095 0.006

TABLA 13.1 COEFICIENTES PARA MOMENTOS NEGATIVOS Ma neg  Ca wu A 2

B

Wu = Carga amplificada total A

Mb neg  Cb wu B 2 Relacion

m 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50

CASO 1 CASO 2

CASO 3 CASO 4 CASO 5 CASO 6 CASO 7

CASO 8

CASO 9

0.033 0.061 0.038 0.056 0.043 0.052 0.049 0.046 0.055 0.041 0.061 0.036 0.068 0.029 0.074 0.024 0.080 0.018 0.085 0.014 0.089 0.010

0.061 0.033 0.065 0.029 0.068 0.025 0.072 0.021 0.075 0.017 0.078 0.014 0.081 0.011 0.083 0.008 0.085 0.006 0.086 0.005 0.088 0.003

A B Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb

0.045 0.045 0.050 0.041 0.055 0.037 0.060 0.031 0.065 0.027 0.069 0.022 0.074 0.017 0.077 0.014 0.081 0.010 0.084 0.007 0.086 0.006

0.076 0.072 0.070 0.065 0.061 0.056 0.050 0.043 0.035 0.028 0.022

0.050 0.050 0.055 0.045 0.060 0.040 0.066 0.034 0.071 0.029 0.076 0.024 0.081 0.019 0.085 0.015 0.089 0.011 0.092 0.008 0.094 0.006

0.075

0.071 0.071

0.079

0.075

0.080

0.079

0.067 0.062 0.082

0.083

0.083

0.086

0.085

0.088

0.086

0.091

0.087

0.093

0.088

0.095

0.089

0.096

0.090

0.097

0.057 0.051 0.044 0.038 0.031 0.024 0.019 0.014

TABLA 13.2 COEFICIENTES PARA MOMENTOS POSITIVOS DEBIDO A LA CARGA MUERTA

Ma pos d  Ca wud A2

A

Mb pos d  Cb wud B 2 Relacion

m 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50

CASO 1

B

Wud = Carga muerta amplificada

CASO 2 CASO 3 CASO 4 CASO 5 CASO 6 CASO 7 CASO 8 CASO 9

A B Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb

0.036 0.036 0.040 0.033 0.045 0.029 0.050 0.026 0.056 0.023 0.061 0.019 0.068 0.016 0.074 0.013 0.081 0.010 0.088 0.008 0.095 0.006

0.018 0.018 0.020 0.016 0.022 0.014 0.024 0.012 0.026 0.011 0.028 0.009 0.030 0.007 0.032 0.006 0.034 0.004 0.035 0.003 0.080 0.007

0.018 0.027 0.021 0.025 0.025 0.024 0.029 0.022 0.034 0.020 0.040 0.018 0.046 0.016 0.054 0.014 0.062 0.011 0.071 0.009 0.080 0.007

0.027 0.027 0.030 0.024 0.033 0.022 0.036 0.019 0.039 0.016 0.043 0.013 0.046 0.011 0.050 0.009 0.053 0.007 0.056 0.005 0.059 0.004

0.027 0.018 0.028 0.015 0.029 0.013 0.031 0.011 0.032 0.009 0.033 0.007 0.035 0.005 0.036 0.004 0.037 0.003 0.038 0.002 0.039 0.001

0.033 0.027 0.036 0.024 0.039 0.021 0.042 0.017 0.045 0.015 0.048 0.012 0.051 0.009 0.054 0.007 0.056 0.006 0.058 0.004 0.061 0.003

0.027 0.033 0.031 0.031 0.035 0.028 0.040 0.025 0.045 0.022 0.051 0.020 0.058 0.017 0.065 0.014 0.073 0.012 0.081 0.009 0.089 0.007

0.020 0.023 0.022 0.021 0.025 0.019 0.029 0.017 0.032 0.015 0.036 0.013 0.040 0.011 0.044 0.009 0.048 0.007 0.052 0.005 0.056 0.004

0.023 0.020 0.024 0.017 0.026 0.015 0.028 0.013 0.029 0.010 0.031 0.007 0.033 0.006 0.034 0.005 0.036 0.004 0.037 0.003 0.038 0.002

TABLA 13.3 COEFICIENTES PARA MOMENTOS POSITIVOS DEBIDO A LA CARGA VIVA

Ma pos l  Ca wul A 2

A

Mb pos l  Cb wul B 2 Relacion

m 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50

B

Wul = Carga viva amplificada

CASO 1

CASO 2

CASO 3

CASO 4

CASO 5

CASO 6

CASO 7

0.036 0.036 0.040 0.033 0.045 0.029 0.050 0.026 0.056 0.023 0.061 0.019 0.068 0.016 0.074 0.013 0.081 0.010 0.088 0.008 0.095 0.006

0.027 0.027 0.030 0.025 0.034 0.022 0.037 0.019 0.041 0.017 0.045 0.014 0.049 0.012 0.053 0.010 0.058 0.007 0.062 0.006 0.066 0.004

0.027 0.032 0.031 0.029 0.035 0.027 0.040 0.024 0.045 0.022 0.051 0.019 0.057 0.016 0.064 0.014 0.071 0.011 0.080 0.009 0.088 0.007

0.032 0.032 0.035 0.029 0.039 0.026 0.043 0.023 0.048 0.020 0.052 0.016 0.057 0.014 0.062 0.011 0.067 0.009 0.072 0.007 0.077 0.005

0.032 0.027 0.034 0.024 0.037 0.021 0.041 0.019 0.044 0.010 0.047 0.013 0.051 0.011 0.055 0.009 0.059 0.007 0.063 0.005 0.067 0.004

0.035 0.032 0.038 0.029 0.042 0.025 0.046 0.022 0.051 0.019 0.055 0.016 0.060 0.013 0.064 0.010 0.068 0.008 0.073 0.006 0.078 0.005

0.032 0.035 0.036 0.032 0.040 0.029 0.045 0.026 0.051 0.023 0.056 0.020 0.063 0.017 0.070 0.014 0.077 0.011 0.085 0.009 0.092 0.007

CASO 8 CASO 9

A B Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb Ca Cb

0.028 0.030 0.031 0.027 0.035 0.024 0.040 0.022 0.044 0.019 0.049 0.016 0.054 0.014 0.059 0.011 0.065 0.009 0.070 0.007 0.076 0.005

0.030 0.028 0.032 0.025 0.036 0.022 0.039 0.020 0.042 0.017 0.046 0.013 0.050 0.011 0.054 0.009 0.059 0.007 0.063 0.006 0.067 0.004

COMBINACIONES DE ACEROS COMERCIALES

BARRA N° 3 4 5 6 7 8 9 10 11 12

ACEROS COMERCIALES DIAMETRO Peso pulg cm Kg/m 1 Φ 3/8 0.95 0.559 1 Φ 1/2 1.27 0.993 1 Φ 5/8 1.59 1.552 1 Φ 3/4 1.91 2.235 1 Φ 7/8 2.22 3.042 1Φ1 2.54 3.973 1 Φ 1 1/8 2.86 5.028 1 Φ 1 1/4 3.18 6.207 1 Φ 1 3/8 3.49 7.511 1 Φ 1 1/2 3.81 8.938

Area cm2 0.71 1.29 1.98 2.85 3.88 5.1 6.45 8.19 9.58 11.4

COMBINACIONES RECOMENDADAS 1/2 - 3/8"

1/2 - 5/8"

1/2 - 3/4"

5/8 - 3/4"

5/8 - 1"

5/8 - 3/4"

3/4 - 1" 1 Φ 3/4 + 1 Φ 1 1 Φ 1 + 2 Φ 3/4 2 Φ 1 + 1 Φ 3/4 2 Φ 1+ 2 Φ 3/4

Evitando tener diametros excesivamente diferentes

Area (cm2) b min (cm) 7.950 16.89 10.800 21.34 13.050 21.97 15.900 26.42

1/2 - 3/8" 2 Φ 3/8 1 Φ 1/2 + 1 Φ 3/8 3 Φ 3/8 2 Φ 1/2 1 Φ 1/2 + 2 Φ 3/8 4 Φ 3/8 2 Φ 1/2 + 1 Φ 3/8 3 Φ 1/2 2 Φ 1/2 + 2 Φ 3/8 3 Φ 1/2 + 1 Φ 3/8 4 Φ 1/2

Area (cm2) b min (cm) 1.420 14.34 2.000 14.66 2.130 17.83 2.580 14.98 2.710 18.15 2.840 21.32 3.290 18.47 3.870 18.79 4.000 21.96 4.580 22.28 5.160 22.60

11.64 1/2 - 5/8" 1 Φ 1/2 + 1 Φ 5/8 2 Φ 5/8 2 Φ 1/2 + 1 Φ 5/8 1 Φ 1/2 + 2 Φ 5/8 3 Φ 1/2 + 1 Φ 5/8 3 Φ 5/8 2 Φ 1/2 + 2 Φ 5/8 1 Φ 1/2 + 3 Φ 5/8 4 Φ 5/8

Area (cm2) b min (cm) 3.270 15.30 3.960 15.62 4.560 19.11 5.250 19.43 5.850 22.92 5.940 19.75 6.540 23.24 7.230 23.56 7.920 23.88

Area (cm2) b min (cm) 1/2 - 3/4" 4.140 15.62 1 Φ 1/2 + 1 Φ 3/4 5.430 19.43 2 Φ 1/2 + 1 Φ 3/4 5.700 16.26 2 Φ 3/4 6.990 20.07 1 Φ 1/2 + 2 Φ 3/4 8.550 20.71 3 Φ 3/4 6.456 23.88 2 Φ 1/2 + 2 Φ 3/4 11.400 25.16 4 Φ 3/4 Area (cm2) b min (cm) 5/8 - 3/4" 4.830 15.94 1 Φ 5/8 + 1 Φ 3/4 6.810 20.07 2 Φ 5/8 + 1 Φ 3/4 7.680 20.39 1 Φ 5/8 + 2 Φ 3/4 8.790 24.200 3 Φ 5/8 + 1 Φ 3/4 9.660 24.520 2 Φ 5/8 + 2 Φ 3/4 10.530 24.840 1 Φ 5/8 + 3 Φ 3/4 5/8 - 1" 1 Φ 1 + 1 Φ 5/8 2Φ1 2 Φ 1+ 1 Φ 5/8 2 Φ 1+ 2 Φ 5/8 3Φ1

Area (cm2) b min (cm) 7.080 16.57 10.200 17.52 12.180 21.65 14.160 25.78 15.300 22.60