vigas-y-losas-y-columnas-de-concreto-armado

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I.S.T.P "JOSÉ CARLOS MARIATEGUI" CONCRETO ARMADO VI SEMESTRE
DISEÑOS DE :
• VS EN EL SEGUNDO NIVEL • COLUMNA DEL CUARTO PISO
• VP EN ELQUINTO PISO • COLUMNA Y ZAPATA EN EL SEGUNDO PISO
• LOSA EN EL PRIMER PISO • COLUMNA Y ZAPATA EN EL TERCER PISO
UBICACIÓN :ANCASH =
=
=
=
=
H
SENTID O DEL ALIGERADO
COL. EN EL 2º PISO
G VS
ZAPATA
F
E
COLUMNA EN EL 3º PISO
D
ZAPATA
C
B VS EN 2º PISO
COLUM 4º PISO
A
1 2 3 4 5 6
VS
L
O
S
A
 E
N
 E
L
 1
º 
P
IS
O
V
P
V
P
V
P
V
P
4
.7
0
V
P
4
.5
0
V
P
VS
7
.6
0
VS
5
.2
0
V
P
VS
P
L
A
C
A
7
.5
0
V
P
5
.5
0
V
P
VS
6
.5
0
V
P
PLANO A DESARROLLAR
Ancho Tributario
5.90
VS VS VS
3 er piso 3.60
V
P
4 er piso 3.40
5 er piso 3.20
VS VS
V
P
 
E
N
 E
L
 5
º 
P
IS
O
5.60
DATOS GENERALES
USO : OFICINAS
Nº DE PISO : 5
1 er piso 4.00
2 er piso 3.80
P
L
A
C
A
5.307.30 4.50 5.60
4
.6
0
DOCENTE : ING. JUAN LAJO RODRIGUEZ ALUMNO :ROBERTO FLORES RAMIREZ
I.S.T.P "JOSÉ CARLOS MARIATEGUI" CONCRETO ARMADO VI SEMESTRE
DISEÑO DE VS EN EL SEGUNDO NIVEL
DATOS :
S/C = Kg/m2 salas de archivo OFICINAS
PISO T = Kg/m2
F'c = Kg/m2 h d
Fy = Kg/m2
ESPESOR DE LOSA = cm
VIGA SECUNDARIA = x m
RECUBRIMIENTO = 6 cm
Pp de losa de 20cm = Kg/m2
γ del concreto = Kg/m
3
= ton/m3
C
B VS EN 2º PISO
A
1 2 3 4 5 6
METRADO DE CARGAS 
Se realiza por ancho tributario AT = -
AT =
CARGA MUERTA : CM
Pp losa = ton/m2 x = ton/m
Pp viga = x x = ton/m
Pp piso T = x = ton/m
CM = ton/m
CARGA VIVA : CV Se realiza con el ancho tributario sin descontar la viga
CV = S/C x AT
x = ton/m
CV = ton/m
CALCULO DEL WU
RNC
Wu= 1.5 (CM)+1.8(CV)
Wu= ton/m
PARA EL CALCULO DE " q "
KU = Ø q F'c ( 1-0.59 q )
KU = q ( 1-0.59 q )
KU = q - q
2
q
2 - q + KU = 0
q
2 q
para tomar el menor
0.3
4
.5
0
V
P
5
0
30
20
0.5
44
300
b
VS
P
L
A
C
A
P
L
A
C
A
4
.7
0
V
P
500
100
210
4200
A
n
ch
o
 t
ri
b
u
ta
ri
o
28.30
4.60 0.30
4.30
4
.6
0
VS
5.60 5.30 5.60
4.30 0.43
0.300 4.30 1.29
2400 2.4
7.30 4.50
2.08
0.50 4.60 2.30
2.30
0.30 0.50 2.4 0.36
0.10
= 0
111.51
q =
1.695 - 1.695 
2 
-
1.695 q +
KU
= 0
111.5
7.26
111.5 111.5
q2 -
+
KU
4
KU
111.5
2
0.9 210
189 111.5
111.5 189
111.5
-
189
DOCENTE : ING. JUAN LAJO RODRIGUEZ ALUMNO :ROBERTO FLORES RAMIREZ
I.S.T.P "JOSÉ CARLOS MARIATEGUI" CONCRETO ARMADO VI SEMESTRE
DIAGRAMA A ANALIZAR
m m m m m
Luces
Promedio
Mu =
KU = Mu/bd2
q =
ρ = (q x f`c )
fy
AS= ρ x b x d
6 Ø de 3/4 7 Ø de 3/4 4 Ø de 3/4 5 Ø de 3/4 5 Ø de 3/4 4 Ø de 3/4
4 Ø de 1'' 2 Ø de 3/4 2 Ø de 1'' 3 Ø de 3/4 2 Ø de 1''
uniformizando
2 Ø de 3/4 3 Ø de 3/4 4 Ø de 3/4 1 Ø de 3/4 1 Ø de 3/4
4 Ø de 1''
5.60
20.30
18.13 11.25 13.39 14.97 9.33
5.83 9.33 8.28 10.83
0.00820
0.00852 0.01014 0.01134 0.00707
0.141
0.141 0.125 0.164
5.60
47.58 15.82
28.98
24.50
33.75
21.95
37.13
28.00
7.30 5.90 5.05 5.45 5.45
7.30 4.500 5.60 5.30
0.227
0.01538 0.00442 0.00707 0.00627
0.01301 0.01374
0.260
0.308 0.088
17.18
0.275 0.170 0.203
5.60
7.26
1/11 1/11 1/10
5.6
1/10
7.3 4.50 5.30
 1/16
25.27
1/14
24.50
24.18
43.51
16.26
1/14 1/16 1/16 1/16
14.23
 1/16
41.63
16.83
14.23
19.60
12.75
21.56
27.63 9.19
DOCENTE : ING. JUAN LAJO RODRIGUEZ ALUMNO :ROBERTO FLORES RAMIREZ
I.S.T.P "JOSÉ CARLOS MARIATEGUI" CONCRETO ARMADO VI SEMESTRE
DOCENTE : ING. JUAN LAJO RODRIGUEZ ALUMNO :ROBERTO FLORES RAMIREZ
I.S.T.P "JOSÉ CARLOS MARIATEGUI" CONCRETO ARMADO VI SEMESTRE
DOCENTE : ING. JUAN LAJO RODRIGUEZ ALUMNO :ROBERTO FLORES RAMIREZ
I.S.T.P "JOSÉ CARLOS MARIATEGUI" CONCRETO ARMADO VI SEMESTRE
4 Ø de 3/4
DOCENTE : ING. JUAN LAJO RODRIGUEZ ALUMNO :ROBERTO FLORES RAMIREZ
I.S.T.P "JOSÉ CARLOS MARIATEGUI"
CONCRETO ARMADO VI SEMESTRE
DISEÑO DE VP EN EL QUINTO NIVEL
DATOS :
S/C = Kg/m
2
salas de archivo OFICINAS
PISO T = Kg/m
2
F'c = Kg/m
2 h d
Fy = Kg/m
2
ESPESOR DE LOSA = cm
VIGA PRINCIPAL = x m
RECUBRIMIENTO = 6 cm
Pp de losa de 20cm = Kg/m
2
γ del concreto = Kg/m
3
= ton/m3
A B C D E F G H
4
5 PLACA VP EN EL 5º PISO
6
METRADO DE CARGAS 
Se realiza por ancho tributario AT = -
AT = CARGA VIVA : CV
CARGA MUERTA : CM Se realiza con el ancho tributario sin descontar la viga
Pp losa = ton/m2 x = ton/m CV = S/C x AT
Pp viga = x x = ton/m
Pp piso T = x = ton/m x = ton/m
CM = ton/m CV = ton/m
20
0.7 0.35
0.588
0.10 5.10
500
64
100
7
0
210
4200
b
35
300
2400 2.4
7.50
5
.4
5
4.50 4.70 5.20 7.60 6.50 5.50
0.50 5.45 2.73
2.73
5.10
0.300 5.10 1.53
0.35 0.70
5
.3
0
5
.6
0
5.45 0.35
0.51
2.63
2.4
DOCENTE : ING. JUAN LAJO RODRIGUEZ ALUMNO :ROBERTO FLORES RAMIREZ
I.S.T.P "JOSÉ CARLOS MARIATEGUI"
CONCRETO ARMADO VI SEMESTRE
CALCULO DEL WU PARA EL CALCULO DE " q "
RNC
Wu= 1.5 (CM)+1.8(CV) KU = Ø q F'c ( 1-0.59 q )
Wu= ton/m KU = q ( 1-0.59 q )
KU = q - q
2
q
2 - q + KU = 0
q
2 q
para tomar el menor
DIAGRAMA A ANALIZAR
m m m m m m m
Luces
Promedio
Mu =
KU = Mu/bd
2
q =
ρ = (q x f`c )
fy
AS= ρ x b x d
4 Ø de 1/2 4 Ø de 5/8 5 Ø de 5/8 5 Ø de 3/4 7 Ø de 3/4 5 Ø de 3/4 6 Ø de 3/4 5 Ø de 3/4
2 Ø de 3/4 2 Ø de 3/4 1 Ø de 3/4 + 2 Ø de 5/8 5 Ø de 3/4 2 Ø de 1'' 3 Ø de 3/4 6 Ø de 3/4
6.50
1/14 1/16
1/10
5.50
12.80 12.21 14.95 31.94 23.36
11.20 18.72 21.68 32.94 39.97
4.50 4.700 5.20 7.60
1/14 1/16 1/16 1/16 1/16
4.50 4.60 4.95 6.40 7.05
0
111.51 111.51 111.51
111.51
-
189
+
KU
111.51
2
8.847
4.5 4.70 5.2 7.60 6.50 7.50
 1/16 1/10 1/10 1/11 1/11 1/10
q2 - 1.695 q +
KU
= 0
111.51
q =
1.695 - 1.695 
2 
- 4
KU
=
8.847 0.9 210
189 111.51
111.51 189
7.50
5.50
31.10
16.73 35.55
31.85
6.00 6.50
7.50
37.38
16.30
22.22
11.67
26.07
24.79
21.70
 1/16
7.81 13.06
8.93 8.52
15.12
10.43
22.98
22.28
27.88
0.091 0.064 0.143
0.084 0.132 0.163 0.127 0.1510.042
0.049
0.072
0.046 0.057 0.127
0.0064 0.0076 0.0062
0.0023 0.0029 0.0064 0.0046 0.0032 0.0072
14.77
14.28
18.29
0.124
0.0021
0.0024
0.0036 0.0042 0.0066 0.0082
4.75 8.08
5.45 5.19
9.43
6.40 10.21
14.23
7.19
16.97
16.05
13.87
DOCENTE : ING. JUAN LAJO RODRIGUEZ ALUMNO :ROBERTO FLORES RAMIREZ
I.S.T.P "JOSÉ CARLOS MARIATEGUI"
CONCRETO ARMADO VI SEMESTRE
ton/m
A
n
ch
o
 t
ri
b
u
ta
ri
o
DOCENTE : ING. JUAN LAJO RODRIGUEZ ALUMNO :ROBERTO FLORES RAMIREZ
I.S.T.P "JOSÉ CARLOS MARIATEGUI"
CONCRETO ARMADO VI SEMESTRE
para tomar el menor
5 Ø de 3/4
DOCENTE : ING. JUAN LAJO RODRIGUEZ ALUMNO :ROBERTO FLORES RAMIREZ
I.S.T.P "JOSÉ CARLOS MARIATEGUI" CONCRETO ARMADO VI SEMESTRE
ANALISIS DE DISEÑO DE COLUMNAS - ESQUINERA
B
Nº
35 x 70 I
30 x 50 II
40 x 40 III
Kg/m2 IV
Kg/m2 V
Ton/m3
20
PISO T = Kg/m2
s/c = Kg/m2 Pp de losa de 20cm A VS
s/c = kg/cm2 salas de archivo OFICINAS 1 2
# de veces = 5 pisos - 4 pisos
# de pisos = 1 pisos
AREA VP = ( - ) X = m
2
AREA VS = ( - ) X = m
2
AREA COL. = X m
2
m
2
X = AT =
- = A LOSA =
CALCULO DE CARGAS
a) CARGAS VIVAS CV
X ( - ) X 1
ton
b) CARGAS MUERTAS CM
PPpt = S/C x (AreaT - AreaCol) x # de piso = X ( - ) X 1 = ton
PP loza = A losa x S/C x # = x x 1 = ton
PP vp = Area vp x hvp x γ c' x # de piso = x x x 1 PP vs = ton
PP vs = Area vs x hvs x γc' # de piso = x x x 1 PP vp = ton
PP col = b x t x h piso x γc' 5º piso = x x x PP col = ton
= ton
CARGAS ULTIMA
1.5 CM + 1.8 CV
x ) + ( X )
PU = ton
PU =
PU =( 1.5 6.221
0.160
1.78
3.65 0.40 0.3 0.98
300
100
500
C4 =
fy = 4200
0.40 0.40
VS = 3.80
3.65
 COLUMNA ESQUINERA : del 4º PISO
Nº PISOS
A vp = (AT y - b col ) x b vp
DATOS h
3.60
3.40
f'c = 210 3.20
γ c = 2.4
S/C x ( AreaT - Area COL ) X # pisocv =
3.65
2.25 0.40 0.35 0.65
METRADOS
losa = cm
AREA TOTAL = 3.65 2.25 8.21
AREA LOSA. = 8.21 1.78 6.430
PP pt 0.805
0.16
cv = 4.026
2
.2
5
2
.2
5
A vs = (ATx - b col)xb vpA col = L X A
cv =
0.40 0.40 3.20
VP = 4.00
0.10 8.21 0.16
0.5 8.21
1.17
CM 6.221
2.40 1.088
0.98 0.50 2.40
2.40 1.229
6.430 0.300 PP L 1.929
0.65 0.70
16.579
1.8 4.026
V
P
4
.5
0
7.30
DOCENTE : ING. JUAN LAJO RODRIGUEZALUMNO :ROBERTO FLORES RAMIREZ
I.S.T.P "JOSÉ CARLOS MARIATEGUI" CONCRETO ARMADO VI SEMESTRE
PARA UNA COLUMNA CON ESTRIBOS
REFUERZO LONGITUDINAL
PU = Ø x 0.80 x ( 0.85 x f'c ( Ag - As ) + As x fy )
= x x ( X x ( - As ) + As x
= x ( x ( - As ) + As )
= x ( - As + As )
= x ( + As )
= + As
+ = As
+ = As
= As
As = cm2
ADOPTAMOS = 4 Ø 5/8" 
REFUERZO TRANSVERSAL
* POR CONFINAMIENTO ( NORMA 2.4.6 )
m
LONGITUD = m @ m
m Ø
* ZONAS NO CONFINADAS ( R.C.N.)
16 Ø L = 16 x = cm
Ø 48 Ø e = 48 x = cm Para diseño tomamos el menor
b = 40 cm
1 @
2 @
2 @
resto @
4 Ø 5/8" 
Ø 3/8"
16.58 0.56
0
.4
0
7.35
0.16 4.200
16.579 0.70 0.80 0.85 0.210 0.16
-0.016 16.58
2.252
7.354
16.58 0.016 2.252
-0.016
15.25
0.179 4.200
16.58 0.56 0.029 4.022
16.58
4.200
0.05
0.10
0.15
0.20
3/8" 0.953 45.74
0.953
0.40
0.10
si h = 3.20 3/8"
t = 0.40
h/6 = 0.53
0.953
16.58
2.252
0.029
0.56 0.179
DOCENTE : ING. JUAN LAJO RODRIGUEZ ALUMNO :ROBERTO FLORES RAMIREZ
I.S.T.P "JOSÉ CARLOS MARIATEGUI"
CONCRETO ARMADO VI SEMESTRE
ANALISIS DE DISEÑO DE COLUMNA PERIMETRAL
E
Nº
35 x 70 I
30 x 50 II
40 x 40 III
Kg/m2 IV
Kg/m2 V
Ton/m3
20
PISO T = Kg/m2
s/c = Kg/m2 Pp de losa de 20cm C VS
s/c = kg/cm2 salas de archivo OFICINAS 5 6
# de veces = 5 pisos - 3 pisos
# de pisos = 2 pisos
AREA VP = ( - ) X = m
2
AREA VS = ( - ) X = m
2
AREA COL. = X m
2
m
2
X = AT =
- = A LOSA =
CALCULO DE CARGAS
a) CARGAS VIVAS CV
X ( - ) X 2
ton
b) CARGAS MUERTAS CM
PPpt = S/C x (AreaT - AreaCol) x # de piso = X ( - ) X 2 = ton
PP loza = A losa x S/C x # = x x 2 = ton
PP vp = Area vp x hvp x γ c' x # de piso = x x x 2 PP vs = ton
PP vs = Area vs x hvs x γc' # de piso = x x x 2 PP vp = ton
PP col = b x t x h piso x γc' 4º piso = x x x PP col = ton
5º piso = x x x PP col = ton
CM = ton
CARGAS ULTIMA
1.5 CM + 1.8 CV
x ) + ( X )
PU = ton
2.80
 COLUMNA ESQUINERA : del 3º PISO
Nº PISOS
DATOS h
VP = 4.00 V
P
300
500
4200 3.40
f'c = 210 3.20
γ c = 2.4
losa =
C4 = 3.60
fy =
2.80
cm
100
VS = 3.80
METRADOS
A vp = (AT y - b col ) x b vp A vs = (ATx - b col)xb vp A col = L X A
6.40 0.40 0.35 2.10
2.80 0.40 0.3 0.72
0.40 0.40 0.160
cv = 17.8
2.98
AREA TOTAL = 6.40 2.80
AREA LOSA. = 17.92 2.98
0.50 2.40 1.728
23.83
cv = S/C x ( AreaT - Area COL ) X # piso
cv = 0.5 17.92 0.16
0.40 0.40 3.20 2.40 1.229
2.10 0.70 2.40 7.056
0.72
PU =
PU =( 1.5 23.83 1.8 17.76
67.720
14.940 0.300 PP L 8.964
5
.2
2
.6
0
3
.8
0
0.10 17.92 0.16 PP pt 3.552
5.60
6
.4
0
17.92
14.940
0.40 0.40 3.40 2.40 1.306
D
7
.6
DOCENTE : ING. JUAN LAJO RODRIGUEZ ALUMNO :ROBERTO FLORES RAMIREZ
I.S.T.P "JOSÉ CARLOS MARIATEGUI"
CONCRETO ARMADO VI SEMESTRE
PARA UNA COLUMNA CON ESTRIBOS
REFUERZO LONGITUDINAL
PU = Ø x 0.80 x ( 0.85 x f'c ( Ag - As ) + As x fy )
= x x ( X x ( - As ) + As x
= x ( x ( - As ) + As )
= x ( - As + As )
= x ( + As )
= + As
+ = As
+ = As
= As
As = cm2
ADOPTAMOS = 6 Ø DE 1" 
REFUERZO TRANSVERSAL
* POR CONFINAMIENTO ( NORMA 2.4.6 )
m
LONGITUD = m @ m
m Ø
* ZONAS NO CONFINADAS ( R.C.N.)
16 Ø L = 16 x = cm
Ø 48 Ø e = 48 x = cm Para diseño tomamos el menor
b = 40 cm
1 @
2 @
2 @
resto @
6 Ø DE 1" 
Ø 3/8"
67.720 0.70 0.80 0.85 0.210 0.16 4.200
67.72 0.56 0.179 0.16 4.200
67.72 0.56 0.029 0.179 4.200
67.72 0.56 0.029 4.022
67.72 0.016 2.252
-0.016 67.72 2.252
-0.016 67.72
2.252
30.06
30.06
t = 0.40
h/6 = 0.53 0.10
si h = 3.20 3/8"
0.20
0.953 15.25
3/8" 0.953 45.74
0
.4
0
0.953
0.05
0.10
0.40 0.15
DOCENTE : ING. JUAN LAJO RODRIGUEZ ALUMNO :ROBERTO FLORES RAMIREZ
ANALISIS DE DISEÑO DE COLUMNA CENTRICA
H
Nº
35 x 70 I
30 x 50 II
Kg/m
2 IV
Kg/m
2
V
γ c = Ton/m3
losa = 20 cm
PISO T = Kg/m2 F VS
s/c = Kg/m
2
Pp de losa de 20cm OFICINAS 2 4
s/c = kg/cm2 salas de archivo
# de veces = 5 pisos - 2 pisos
# de pisos = 3 pisos
AREA VP = ( - ) X = m
2
AREA VS = ( - ) X = m
2
AREA COL. = X m
2
m
2
X = AT = m
2
- = A LOSA = m
2
CALCULO DE CARGAS
a) CARGAS VIVAS CV
X ( - ) X 3
ton
b) CARGAS MUERTAS CM
PPpt = S/C x (AreaT - AreaCol) x # de piso = X ( - ) X 3 = ton
PP loza = A losa x S/C x # = x x 3 = ton
PP vp = Area vp x hvp x γ c' x # de piso = x x x 3 PP vs = ton
PP vs = Area vs x hvs x γc' # de piso = x x x 3 PP vp = ton
PP col = b x t x h piso x γc' 3º piso = x x x PP col = ton
4º piso = x x x PP col = ton
5º piso = x x x PP col = ton
= ton
CARGAS ULTIMA
1.5 CM + 1.8 CV
x ) + ( X )
PU = ton
 COLUMNA ESQUINERA : del 2º PISO
Nº PISOS
DATOS h
300
VP = 4.00
VS = 3.80
fy =
A vs = (ATx - b col)xb vpA col = L X A
6.50 0.40 0.35 2.14
0.40 0.3 1.40
0.40 0.40 0.160
cv = 49.0
3.69
AREA TOTAL = 6.50 5.05
AREA LOSA. = 32.83 3.69
29.135 0.300 PP L 26.222
cv = S/C x ( AreaT - Area COL ) X # piso
cv = 0.5 32.83 0.16
0.40 0.40 3.40 2.40
1.3824
1.3056
2.14 0.70 2.40 10.76
1.40 0.50 2.40 5.022
0.40 0.40 3.20 2.40 1.2288
CM 55.72
PU =
PU =( 1.5 55.72 1.8 48.998
171.776
METRADOS
A vp = (AT y - b col ) x b vp
4200
f'c = 210
2.4
100
500
C4 = 40 x 40
5.05
III 3.60
6
.5
0
3.40
3.20
2
.7
5
9.7995
5.05
3
2.25 2.80V
P
4.5 5.6
0.10 32.83 0.16 PP pt
3
.7
5 7
.5
5.
5
32.83
29.135
0.40 0.40 3.60 2.40
G
PARA UNA COLUMNA CON ESTRIBOS
REFUERZO LONGITUDINAL
PU = Ø x 0.80 x ( 0.85 x f'c ( Ag - As ) + As x fy )
= x x ( X x ( - As ) + As x
= x ( x ( - As ) + As )
= x ( - As + As )
= x ( + As )
= + As
+ = As
+ = As
= As
As = cm2
ADOPTAMOS = 4 Ø 1 1/4" + 8 Ø 1''
REFUERZO TRANSVERSAL
* POR CONFINAMIENTO ( NORMA 2.4.6 )
m
LONGITUD = m @ m
m Ø
* ZONAS NO CONFINADAS ( R.C.N.)
16 Ø L = 16 x = cm
Ø 48 Ø e = 48 x = cm Para diseño tomamos el menor
b = 40 cm
1 @
2 @
2 @
resto @
4 Ø 1 1/4" + 8 Ø 1''
Ø 3/8"
###### 0.70 0.80 0.85 0.210 0.16 4.200
171.78 0.56 0.179 0.16 4.200
171.78 0.56 0.0286 0.179 4.200
171.78 0.56 0.0286 4.022
76.269
76.27
t =
0.016 2.252
-0.016 171.78 2.252
45.744
0.40
h/6 = 0.53 0.10
si h = 3.20 3/8"
0.05
0.10
0.40 0.15
0.20
0.953 15.248
3/8" 0.953
0.953
171.78
-0.016 171.78
2.252
0
.4
0