1
Predimensionamiento De
Columnas, Vigas Y Zapatas Datos: Niveles = 6
Cv = 500 kg/m2
Cm = 300 kg/m2
Cvaz = 100 kg/m2
Cp del suelo = 1.03 kg/cm2
σ del suelo = 1903 kg/m3
S/C = 500 kg/ m2
E. piso = 0.15 m
F’c = 210 kg/cm2
F’y = 4200 kg/cm2
Φ de columna central y lateral = ¾”
Φ de columna esquina = 5/8”
1. PREDIMENSIONAMIENTO DE COLUMNAS. COLUMNA 1:
At1 = 5x4.25=21.25 m2
PGp=1.2(300)+1.6(500) =1160kg/m2
PGaz = 1.2 (300)+1.6 (100) = 520 kg/m2
Pt = (5x1160x21.25)+ (520x1x21.25)=134300kg
1º piso:
Asc=(1.1)(134300)
(0.25)(210)= 2813.90 cm2= (53.05)2= 55x55 cm
2
2º piso:
Asc=(1.1)(109650)
(0.25)(210)= 2297.43 cm2= (47.93)2= 50x50 cm
3º piso:
Asc=(1.1)(85000)(0.25)(210)
= 1780.95 cm2=(42.20)2 = 45x45 cm
4º piso:
Asc=(1.1)(60350)
(0.3)(210)= 1053.73 cm2= (32.46)2= 35x35 cm
5º piso:
Asc=(1.1)(35700)
(0.3)(210)= 623.33cm2= (24.97)2= 25x25 cm
6º piso:
Asc=(1.1)(35700)
(0.3)(210)= 623.33cm2= (24.97)2= 25x25 cm
COLUMNA 2:
At =5x2.25=11.25 m2
PGp =1.2(300)+1.6(500 )=1160kg/m2
PGaz = 1.2 (300)+1.6 (100) = 520 kg/m2
Pt = (5x1160x11.25)+ (520x11.25) =71100kg
1º piso:
Asc=(1.25)(71100)
(0.25)(210)= 1692.86 cm2= (41.14)2= 45x45 cm
2º piso:
Asc=(1.25)(58050)
(0.25)(210)= 1382.14 cm2= (37.18)2= 40x40 cm
3
3º piso:
Asc=(1.8)(45000)(0.25)(210)
= 1071.43 cm2= (32.71)2= 35x35 cm
4º piso:
Asc=(1.25)(31950)
(0.25)(210)= 760.71 cm2= (27.58)2= 30x30 cm
5º piso:
Asc=(1.25)(18900)
(0.25)(210)= 450cm2= (21.21)2= 25x25 cm
6º piso:
Asc=(1.25)(18900)
(0.25)(210)= 450cm2= (21.21)2= 25x25 cm
COLUMNA 3:
At=4.25x2.5 =10.625 m2
PGp=1.2(300)+1.6(500) =1160kg/m2
PGaz= 1.2 (300)+1.6 (100) = 520 kg/m2
Pt= (5x1160x10.625)+ (520x10.625) = 67150kg
1º piso:
Asc=(1.25)(67150)
(0.25)(210)= 1598.81 cm2= (39.99)2= 40x40 cm
2º piso:
Asc=(1.25)(54825)
(0.25)(210)= 1305.36 cm2= (36.13)2= 40x40 cm
3º piso:
Asc=(1.25)(42500)
(0.25)(210)= 1011.91 cm2= (31.81)2= 35x35 cm
4
4º piso:
Asc=(1.25)(30175)
(0.25)(210)= 718.452 cm2= (26.8)2= 30x30 cm
5º piso:
Asc=(1.25)(17850)
(0.25)(210)= 425cm2= (20.6)2= 25x25 cm
6º piso:
Asc=(1.25)(17850)
(0.25)(210)= 425cm2= (20.6)2= 25x25 cm
COLUMNA 4:
At=2.25x2.5=5.625 m2
PGp=1.2(300)+1.6(500)=1160kg/m2
PGaz= 1.2 (300)+1.6 (100)= 520 kg/m2
Pt= (5x1160x5.625)+ (520x5.625)=35550kg
1º piso:
Asc=(1.5)(35550)(0.20)(210)
= 1269.64 cm2= (35.6)2= 40x40 cm
2º piso:
Asc=(1.5)(29025)(0.20)(210)
= 1036.61 cm2= (32.19)2= 35x35 cm
3º piso:
Asc=(1.5)(22500)(0.20)(210)
= 803.57 cm2= (28.3)2= 30x30 cm
4º piso:
Asc=(1.5)(15975)(0.20)(210)
= 570.54 cm2= (23.9)2= 25x25 cm
5º piso:
Asc=(1.5)(9450)(0.20)(210)
= 337.5 cm2= (18.37)2= 25x25 cm
5
6º piso:
Asc=(1.5)(2925)(0.20)(210)
= 104.46 cm2= (10.22)2= 25x25 cm
Resumen de datos
6
2. PREDIMENSIONAMIENTO DE VIGAS:
Principales:
P=(𝐿)
(10)=
(5)(10)
= 0.5
b=(𝐿)
(20)=
(5)(20)
=0.25
Vp = 50cm x 25cm
Secundaria:
P=(𝐿)
(12)=
(4.5)(12)
= 0.375= 40
b=(𝐿)
(20)=
(4.5)(20)
=0.225= 25
Vs =40 cm x 25cm
3. PREDIMENSIONAMIENTO DE ZAPATAS: Zapata N° 1
CVT = (5000(5)+100) (21.25) = 55250Kg = 55.25Tn
CMT = (300) (21.25) (6) =38250Kg = 38.25Tn
Peralte efectivo
d = 0.08 * 𝜙Ac * 𝐹ʹ𝑦√𝐹ʹ𝑐
≥ 0.004 * ∅A * Fʹy
d = 0.08 * 1.905 * 4200√210
≥ 0.004 * 1.905 * 4200
d = 44.17 ≥ 32
Altura de la zapata será:
hz = d + r + ∅b = 44.17 + 10 +1.588 = 55.758cm = 60cm = 0.6m
7
El peralte efectivo de diseño será:
dʹ = 60 – 10 – 1.588 = 48.41cm = 0.4841m
Esfuerzo neto del terreno
𝜎𝑛 = 1.03 ∗ (1002) − 1903(. 6) − 500 − 24008(.15)
𝜎𝑛 = 6858.2 𝑘𝑔𝑚2 = 6.858𝑇𝑛/𝑚2
Área de zapata
Az = 𝑷𝒆𝒔𝒕.𝜎n
Az = (𝟓𝟓.𝟐𝟓+𝟑𝟖.𝟐𝟓)𝑻𝒏𝟔.𝟖𝟓𝟖𝑻𝒏/𝒎𝟐
= 13.634𝑚2
Lv1 = Lv2
T = S = √13.634 ± (.𝟓𝟓−.𝟓𝟓𝟐
) = 3.692𝑚 ≅ 3.7𝑚
S = 3.7m T = 3.7m
LV1 = LV2 = 𝟑.𝟕−𝟎.𝟓𝟓
𝟐= 1.575𝑚
Reacción ultima
Wnu = (𝟑𝟖.𝟐𝟓)(𝟏.𝟐)+𝟓𝟓.𝟐𝟓(𝟏.𝟔)
(𝟑.𝟕)𝟐= 𝟗.𝟖𝟏𝑻𝒏
𝒎𝟐 = 9810𝑘𝑔/𝑚2
verificación de cortante
∅vc > Vu
∅(0.53)√𝐹ʹ𝑐 (𝑆)dʹ > Wnu(S)(LV − dʹ)
0.75(0.53)√210(10)(3.70)(0.4841) > 9.81(3.7)(1.575 − 0.4841)
103.18 > 39.596
8
corte por punzonamiento
m = dʹ + t1 = 0.4841+0.55 = 1.034m
n = dʹ + t2 = 0.4841+0.55 = 1.034m
Vu < ∅Vc
Vu = Pdin – Wnu (m) (n)
Vu = 134.3 – 9.81 (1.034)2 = 123.812Tn
bo = 2m + 2n = 4(1.034) = 4.136m
𝛽𝑐 = 0.550.55
= 1
∅(0.27)(2 + 4𝛽𝑐
)√𝐹ʹ𝑐 𝑏𝑜𝑑
∅Vc ∅(0.27)(∝𝑠𝑑𝑏𝑜
+ 2)√𝐹ʹ𝑐 𝑏𝑜𝑑
∅ ∗ 1.1√𝐹ʹ𝑐 𝑏𝑜𝑑
0.75(0.27) �2 + 41�√210 (10)(4.136)(0.4841) = 352.53𝑇𝑛
∅Vc 0.75(0.27) �(40)(.4841)4.136
+ 2� √210 (10)(4.136)(0.4841) = 392.595𝑇𝑛
0.75 ∗ 1.1 ∗ √210 (10)(4.136)(0.4841) = 239.375𝑇𝑛
Acero
Mu = 𝑊𝑛𝑢2
(𝐿𝑣1)2 ∗ 𝑆
Mu = 9.812
(1.575)2 ∗ (3.7) = 45.02 𝑇𝑛𝑚
= 45.02 ∗ 105𝑘𝑔/𝑐𝑚
9
Área del acero
As = 𝑀𝑢
∅𝐹ʹ𝑦(𝑑ʹ−𝑎2) = As=
(𝑎)(0.85)(𝐹ʹ𝑐)(𝑆)𝐹ʹ𝑦
45.02∗105
0.9(4200)(48.41− 𝑎2) =
𝑎(0.85)(210)(370)4200
50.02∗105
6.6045∗104= 48.41𝑎 − 0.5𝑎2
0.5𝑎2 − 48.41𝑎 + 75.736 = 0
𝑎 =48.41 ± �48.412 − 4(0.5)(75.736)
1= 48.41 ± 46.81𝑎
𝒂 = 𝟏.𝟓𝟗𝟏𝒄𝒎
As1= 45.0∗105
0.9(4200)(48.41−1.5912 )
= 𝟐𝟓.𝟎𝟏𝟑𝒄𝒎𝟐
As2= 1.591(0.85)(210)(370)4200
= 𝟐𝟓.𝟎𝟏𝟖𝒄𝒎𝟐
As=25.018 𝒄𝒎𝟐
Verificación de acero mínimo
As> As𝑚𝑖𝑛𝑖𝑚𝑜
25.0188>0.0018 (48.41) (370)
25.018>32.241
As = 32.241𝒄𝒎𝟐
Nv⅝ = 32.241
2= 16.12 ≅ 17
S = 370−2(2.75)−1.588
16= 22.09 ≅ 0.22𝑚
Lv =370 – 2(7.5) = 355 = 3.55𝑚
10
17 ∅ ⅝ @ 0.22
Zapata N° 2
CVT = (500(5)+100) (11.25) = 29250Kg = 29.25Tn
CMT = (300) (11.25) (6) =20250Kg = 20.25Tn
Peralte efectivo
d = 0.08 * 𝜙Ac * 𝐹ʹ𝑦√𝐹ʹ𝑐
≥ 0.004 * ∅A * Fʹy
d = 0.08 * 1.905 * 4200√210
≥ 0.004 * 1.905 * 4200
d = 44.17 ≥ 32
Altura de la zapata será:
hz = d + r + ∅b= 44.17 + 10 +1.588 = 55.758cm = 60cm = 0.6m
El peralte efectivo de diseño será:
dʹ = 60 – 10 – 1.588 = 48.41cm = 0.4841m
Esfuerzo neto del terreno
𝜎𝑛 = 1.03 ∗ (1002) − 1903(. 6) − 500 − 24008(.15) 𝜎𝑛 = 6858.2 𝑘𝑔
𝑚2 = 6.858𝑇𝑛/𝑚2
Área de zapata
Az = 𝑷𝒆𝒔𝒕.𝜎n
Az = (𝟐𝟗.𝟐𝟓+𝟐𝟎.𝟐𝟓)𝑻𝒏𝟔.𝟖𝟓𝟖𝑻𝒏/𝒎𝟐
= 7.22𝑚2
11
Lv1 = Lv2
S= Lv2 + t2 Lv2 = S- t2
T= 2 Lv1 + t1 Lv1= 𝑇−𝑡12
2AZ=T2-T (t1-2t2)
2(7.22)= T2-T (0.45-2(0.45))
14.44= T2 + 0.45T
0= T2 + 0.45T -14.44
𝑇 =−0.45 ± �0.452 − 4(1)(14.44)
2 = 3.585 𝑚
2S2=AZ-S (t1-2t2)
2S2=7.22-S (0.45-2(0.45))
S = 0.45±�0.452−4(2)(7.22)4
= 2.015 𝑚
Lv2 = 2.015-0.45 = 1.565
Lv1 =𝟑.𝟓𝟖𝟓−𝟎.𝟒𝟓
𝟐 =1.567
Lv2=Lv1=1.60
S=1.6+ (0.45)=2.05 m
T=2(1.6)+ (0.45)= 3.65 m
Reacción ultima
Wnu = (𝟐𝟎.𝟐𝟓)(𝟏.𝟐)+𝟐𝟗.𝟓𝟓(𝟏.𝟔)
(𝟐.𝟎𝟓)(𝟑.𝟔𝟓)= 𝟗.𝟓𝟕𝑻𝒏
𝒎𝟐 = 9570𝑘𝑔/𝑚2
m = dʹ + t1 = 0.4841+0.45 = 0.934m
n = dʹ + t2 = 0.4841+0.45 = 0.934m
12
verificación de cortante
EJE “x”
∅vc > Vu
∅(0.75)√𝐹ʹ𝑐 (𝑆)dʹ > Wnu(S)(LV − dʹ)
0.75(0.53)√210(10)(3.65)(0.4841) > 9.57(3.65)(1.6 − 0.4841)
101.783 > 38.979
EJE “y”
∅(0.75)√𝐹ʹ𝑐 (𝑆)dʹ > Wnu(S)(LV − dʹ)
0.75(0.53)√210(10)(2.05)(0.4841) > 9.57(2.05)(1.6 − 0.4841)
57.166 > 21.8923
corte por punzonamiento
Vu < ∅Vc
Vu = Pdin – Wnu (m) (n)
Vu = 71.58 – 9.57 (0.934)2 = 63.23 Tn
bo = 2m + 2n = 4(0.934) = 3.736m
𝛽𝑐 = 0.450.45
= 1
∅(0.27)(2 + 4𝛽𝑐
)√𝐹ʹ𝑐 𝑏𝑜𝑑
∅Vc ∅(0.27)(∝𝑠𝑑𝑏𝑜
+ 2)√𝐹ʹ𝑐 𝑏𝑜𝑑
∅ ∗ 1.1√𝐹ʹ𝑐 𝑏𝑜𝑑
13
0.75(0.27) �2 + 41�√210 (10)(3.736)(0.4841) = 318.44 𝑇𝑛
∅Vc 0.75(0.27) �(30)(0.4841)3.736
+ 2� √210 (10)(3.736)(0.4841) = 312.46𝑇𝑛
0.75 ∗ 1.1 ∗ √210 (10)(3.736)(0.4841) = 216.225 𝑇𝑛
Acero
Mu = 𝑊𝑛𝑢2
(𝐿𝑣1)2 ∗ 𝑆
Mu = 9.572
(1.6)2 ∗ (3.65) = 44.71 𝑡𝑛.𝑚 = 44.71 ∗ 105𝑘𝑔/𝑐𝑚
Área del acero
As = 𝑀𝑢
∅𝐹ʹ𝑦(𝑑ʹ−𝑎2) = As=
(𝑎)(0.85)(𝐹ʹ𝑐)(𝑆)𝐹ʹ𝑦
44.71∗105
0.9(0.85)(210)(365) = 48.41 a – 0.5 a2
0.5𝑎2 − 48.41𝑎 + 76.25 = 0
𝑎 =48.41 ± �48.412 − 4(0.5)(76.25)
1 = 48.41 ± 46.81𝑎
𝒂 = 𝟏.𝟔 𝒄𝒎
As1= 44.71∗105
0.9(4200)(48.41−(0.5∗1.6))= 𝟐𝟒.𝟖𝟒 𝒄𝒎𝟐
As2= 1.6(0.85)(210)(365)4200
= 𝟐𝟒.𝟖𝟐 𝒄𝒎𝟐
As=24.84 𝒄𝒎𝟐
Verificación de acero mínimo
As> As𝑚𝑖𝑛𝑖𝑚𝑜
24.84>0.0018 (48.41) (365)
14
24.84> 31.811
As =31.81 𝒄𝒎𝟐
Nv⅝ = 31.812
= 15.91 ≅ 16
S = 365−2(7.5)−1.588
15= 23.22 ≅ 24 𝑐𝑚 ≅ 0.24 𝑚
Lv =205 – 2(7.5) = 190 cm = 1.9 𝑚
16 ∅ ⅝ @ 0.24
Ast = (𝐴𝑠)(𝑆)
𝑇=
(31.81)(205)365
= 17.87 cm2
Nºv⅝ = (17.87)
2= 8.935≅9
Se= (205)−(2∗7.5)−(1.588)
8=23.55 cm=0.24 m
Lv=365-(2*7.5)=350=3.5m
9 ∅ ⅝ @ 0.24
Zapata N° 3
CVT = (5000(5)+100) = 29.25Tn
CMT = (300) (6) (11.25) = 20.25Tn
Peralte efectivo
d = 0.08 * 𝜙Ac * 𝐹ʹ𝑦√𝐹ʹ𝑐
≥ 0.004 * ∅A * Fʹy
d = 0.08 * 1.905 * 4200√210
≥ 0.004 * 1.905 * 4200
d = 44.17 ≥ 32
Altura de la zapata será:
hz = d + r + ∅b= 44.17 + 10 +1.588 = 55.758cm = 60cm = 0.6m
15
El peralte efectivo de diseño será:
dʹ = 60 – 10 – 1.588 = 48.41cm = 0.4841m
Esfuerzo neto del terreno
𝜎𝑛 = 1.03 ∗ (1002) − 1903(. 6) − 500 − 24008(.15)
𝜎𝑛 = 6858.2 𝑘𝑔𝑚2 = 6.858𝑇𝑛/𝑚2
Área de zapata
Az = 𝑷𝒆𝒔𝒕.𝜎n
Az = (𝟐𝟕.𝟔𝟓+𝟏𝟗.𝟏𝟐𝟓)𝑻𝒏
𝟔.𝟖𝟓𝟖𝑻𝒏/𝒎𝟐= 6.817𝑚2
Lv1 = Lv2
S = Lv2 + t2 Lv = S – t2
T = 2Lv1 + t1 Lv1 = 𝑇−𝑡12
2Az = T2 – T(t1-2t2)
0 = T2 + 0.40T-13.634
T=−0.4±�4(13.634)+0.42
2
T=3.5m
S = 0.4±√4∗2∗6.817+0.42
4
S = 1.95m
Lv2 = 1.949 – 0.4 = 1.549
Lv1 = (3.498 – 0.4)/2 = 1.549
16
Lv2=Lv1=1.55m
S = 1.55 + 0.4 =1.95m
T = 2*1.55 + 0.4 = 3.50m
Reacción ultima
Wnu = (𝟐𝟕.𝟔𝟐𝟓)(𝟏.𝟔)+𝟏𝟗.𝟏𝟐𝟓(𝟏.𝟐)𝟏.𝟗𝟓∗𝟑.𝟓
= 𝟗.𝟖𝟑𝟗𝑻𝒏𝒎𝟐
verificación de cortante
Eje “ X”
∅vc > Vu
∅(0.53)√𝐹ʹ𝑐 (𝑆)dʹ > Wnu(S)(LV − dʹ)
0.75(0.53)√210(10)(3.5)(0.4841) > 9.839(3.5)(1.55 − 0.4841)
97.6 > 36.71
Eje “ Y”
0.75(0.53)√210(10)(1.95)(0.4841) > 9.839(1.95)(1.55 − 0.4841)
54.38 > 20.45
verificación por punzonamiento
m = dʹ + t1 = 0.4841 + 0.4 = 0.884m
n = dʹ + t2 = 0.4841 + 0.4 = 0.884m
V𝜇 = 67.115 - 9.839(0.884)2 = 59.46Tn
bo = 3(0.884) = 2.652m
𝛽 = 0.40.4
= 1
17
0.75(0.27) �2 + 41�√210 (10)(2.652)(. 4841) = 226.04𝑇𝑛
∅Vc 0.75(0.27) �(30)(.4841)2.802
+ 2�√210 (10)(2.652)(. 4841) = 281.66𝑇𝑛
0.75 ∗ 1.1 ∗ √210 (10)(2.652)(. 4841) = 153.49𝑇𝑛
Área del acero
As = 𝑀𝑢
∅𝐹ʹ𝑦(𝑑ʹ−𝑎2) = As=
(𝑎)(0.85)(𝐹ʹ𝑐)(𝑇)𝐹ʹ𝑦
𝑀𝑢
0.9(0.85)𝐹ʹ𝑐(𝑇) =𝑑𝑎 − 0.5𝑎2
41.37∗105
0.9(0.85)(210)(350)= 48.41𝑎 − 0.5𝑎2
0.5𝑎2 − 48.41𝑎 + 73.576 = 0
𝑎 =48.41 ± �48.412 − 4(0.5)(73.576)
1 = 48.41 ± 46.81𝑎
𝒂 = 𝟏.𝟓𝟒𝟒𝒄𝒎
As1= 41.37∗105
0.9(4200)(48.41−0.5∗1.544)= 𝟐𝟐.𝟗𝟕𝟒𝒄𝒎𝟐
As2= 1.544(0.85)(210)(350)4200
= 𝟐𝟐.𝟗𝟔𝟕𝒄𝒎𝟐
As = 22.97𝐜𝐦𝟐
Verificación de acero mínimo
As >Aminimo
22.97 >0.0018 (48.41) (350)
22.97 >30.498
As = 30.498𝒄𝒎𝟐
18
Area del acero longitudinal “x”
Nv⅝ = 30.4982
= 15.249 ≅ 16
Se = 350−2(7.5)−1.588
15= 22.23𝑐𝑚 ≅ 0.23𝑚
Lv = 195 – 2(7.5) = 180cm = 1.8𝑚
16 ∅ ⅝ @ 0.23
Área de acero en la sección transversal ”Y”
AST=𝐴𝑠∗𝑆𝑇
= 31.84∗205365
= 17.87𝑐𝑚2
Nv⅝ = 16.962
= 8.495 ≅ 9
Se = 195−2(7.5)−1.588
8= 22.3𝑐𝑚 ≅ 0.23𝑚
Lv = 350 – 2(7.5) = 335cm = 3.5𝑚
9 ∅ ⅝ @ 0.23
Zapata N° 4
CVT = (500(5)+100) (5.625) = 14625 kg = 14.625Tn
CMT = (300) (5.625) (6) =10125 Kg = 10.125 Tn
Peralte efectivo
d = 0.08 * 𝜙Ac * 𝐹ʹ𝑦√𝐹ʹ𝑐
≥ 0.004 * ∅A * Fʹy
d = 0.08 * 1.588 * 4200√210
≥ 0.004 * 1.588 * 4200
d = 36.82 ≥ 26.68
Altura de la zapata será:
19
hz = d + r + ∅b= 36.82 + 10 +1.588 = 48.408cm = 50cm = 0.5m
Por condiciones de diseño la altura de la zapata será 60 cm
El peralte efectivo de diseño será:
dʹ = 60 – 10 – 1.588 = 48.41cm = 0.4841m
Esfuerzo neto del terreno
𝜎𝑛 = 1.03 ∗ (1002) − 1903(. 6) − 500 − 24008(.15)
𝜎𝑛 = 6858.2 𝑘𝑔𝑚2 = 6.858𝑇𝑛/𝑚2
Área de zapata
Az = 𝑷𝒆𝒔𝒕.𝜎n
Az = (𝟏𝟒.𝟔𝟐𝟓+𝟏𝟎.𝟏𝟐𝟓)𝑻𝒏
𝟔.𝟖𝟓𝟖𝑻𝒏/𝒎𝟐= 𝟑.𝟔𝟏𝑚P
2
Debe cumplir que:
Lv1 = Lv2 S= Lv2 + t2 Lv2 = S- t2
T= Lv1 + t1 Lv1= 𝑇 − 𝑡1
Az = T2-T (t1-t2)
3.61= T2-T (0.40-0.40)
T=1.9m
Az = S2-S (t1-2t2)
3.61= S2-S (0.40-0.40)
S = 1.9 m
Lv2=1.9-0.4=1.5 m
20
Reacción ultima
Wnu = (𝟏𝟎.𝟏𝟐𝟓)(𝟏.𝟐)+𝟏𝟒.𝟔𝟐𝟓(𝟏.𝟔)
(𝟏.𝟗)𝟐= 𝟗.𝟖𝟓𝑻𝒏
𝒎𝟐 = 9850𝑘𝑔/𝑚2
m = dʹ + t1 = 0.4841+0.40 = 0.884m
n = dʹ + t2 = 0.4841+0.40 = 0.884m
bo= m+n=1.768 Bc=0.40/0.40 =1
verificación de cortante
∅(0.75)√𝐹ʹ𝑐 (𝑆)dʹ > Wnu(S)(LV − dʹ)
0.75(0.53)√210(10)(1.9)(0.4841) > 9.85(1.9)(1.5 − 0.4841)
52.98 > 19.01
Corte por punzonamiento
Vu < ∅Vc
Vu = Pdin – Wnu (m) (n)
Vu = 35.55 – 9.85 (0.884)2 = 27.85 Tn
∅(0.27)(2 + 4𝛽𝑐
)√𝐹ʹ𝑐 𝑏𝑜𝑑
∅Vc ∅(0.27)(∝𝑠𝑑𝑏𝑜
+ 2)√𝐹ʹ𝑐 𝑏𝑜𝑑
∅ ∗ 1.1√𝐹ʹ𝑐 𝑏𝑜𝑑
0.75(0.27)(2 + 4)√210 (10)(1.768)(0.4841) = 150.70 𝑇𝑛
∅Vc 0.75(0.27) �(20)(0.4841)1.768
+ 2� √210 (10)(1.768)(0.4841) = 187.77𝑇𝑛
0.75 ∗ 1.1 ∗ √210 (10)(1.768)(0.4841) = 102.325 𝑇𝑛
21
Acero
Mu = 𝑊𝑛𝑢2
(𝐿𝑣1)2 ∗ 𝑆
Mu = 9.852
(1.5)2 ∗ (1.9) = 21.05 𝑡𝑛.𝑚 = 21.05 ∗ 105𝑘𝑔/𝑐𝑚
Área del acero
As = 𝑀𝑢
∅𝐹ʹ𝑦(𝑑ʹ−𝑎2) = As=
(𝑎)(0.85)(𝐹ʹ𝑐)(𝑆)𝐹ʹ𝑦
21.05∗105
0.9(0.85)(210)(1.90) = 48.41 a – 0.5 a2
0.5𝑎2 − 48.41𝑎 + 68.963 = 0
𝑎 =48.41 ± �48.412 − 4(0.5)(68.963)
1 = 48.41 ± 46.964
𝒂 = 𝟏.𝟒𝟒𝟔 𝒄𝒎
As1= 21.05∗105
0.9(4200)(48.41−(0.5∗1.446))= 11.678 𝑐𝑚2
As2= 1.446(0.85)(210)(190)4200
= 11.676 𝑐𝑚2
As=11.678 𝑐𝑚2
Verificación de acero mínimo
As> As𝑚𝑖𝑛𝑖𝑚𝑜
11.678>0.0018 (48.41) (190)
11.678> 16.556
As =16.556 𝒄𝒎𝟐
22
Nv⅝ = 16.556
2= 8.278 ≅ 9
S = 190−2(7.5)−1.588
8= 21.67 ≅ 22 𝑐𝑚 ≅ 0.22 𝑚
Lv =190 – 2(7.5) = 175 cm = 1.75 𝑚
9 ∅ ⅝ @ 0.22
RESUMEN DE DATOS:
DATOS Z1 Z2 Z3 Z4
hs 0.6m 0.6m 0.6m 0.6m
hz 0.6m 0.6m 0.6m 0.6m
Lv1=Lv2 1.575m 1.6m 1.55m 1.5m
dʹ 0.4841m 0.4841m 0.4841m 0.4841m
Az 13.63m2 7.22m2 6.817m2 3.61m2
S 3.7m 2.05m 1.949m 1.9m
T 3.7m 3.6m 3.498m 1.9m
m 1.034m 0.934m 0.884m 0,884
n 1.034m 0.934m 0.884m 0,884
ф 5/8" 5/8" 5/8" 5/8"
N°v. Long. 17 16 16 9
Sepr. Long. 0,22 0,24 0,23 0,22
N°v. Transv. 17 9 9 9
Sepr. Transv. 0,22 0,24 0,23 0,22
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