Ejemplo 2. Disear una losa de cimentacin para la planta de columnas mostrada en la siguiente figura. Todas las columnas son de 40 x 40. Use fc = 210 kg/cm2 para todo el concreto y fy = 4200 kg/cm2 para todo el acero. Para e l suelo use qa = 8.5 ton/m2, s = 1.75 ton/m3, Df = 1.80 m y Ks = 3600 ton/m3.

Solucin:1.- Clculos preliminares.P = (2 x37) + (2 x 48) + 40 + 42 + (4 x 145) + (2 x 109) = 1050 ton Pu = (2 x 52) + (2 x 68) +56 +60 +(4 x 204) + (2 x 154) = 1480 ton.Lx = 12.60 mLy = 22.30Mx = (Pi * Yi) = (145 + 145 + 109) (3.65) + (37 + 48 +42) (10.95) + (145 + 145 109) (-3.65) + (37 + 48 + 40) (-10.95) = 21.90 ton-m.My = (Pi * Xi) = (40 + 109 + 109 + 42)(6.1) + (37 + 145 + 145 +37)(-6.1) = -390.4 ton-m.X0 = Sxi = 6.10 + 6.10 = 12.20 m.Y0 = Syi = 7.30 + 7.30 + 7.30 = 21.90 m2.- Dimensiones.Lx = X0 + (Cx1 + Cx2)/2 = 12.20 + (0.40 + 0.40)/2 = 12.60 mLy = Y0 + (Cy1 + Cy2)/2 = 21.90 + (0.40 + 0.40)/2 = 22.30 mAL = Lx *Ly = 12.60 * 22.30 = 280.98 m2 Pq= q * AL = 3.735 * 280.98 = 1049.46 tonPT = P + Pq = 1050 + 1049.46 = 2099.46 tonex = [My]/PT = 390.4/2099.46 = 0.186 mey = [Mx]/PT = 21.9/2099.46 = 0.010 mLx = Lx 2*ex = 12.60 2(0.186) = 12.23 mLy = Ly 2*ey = 22.30 2(0.010) = 22.28 mAZ= Lx* Ly= 12.23 * 22.28 = 272.48 m2qmax = pT /AZ = 2099.46/272.48 = 7.7 ton/m2 < qa = 8.5 ton/m2

3.-Peralte.- Corte en dos direcciones.qs = PuT/AZ = 1480 ton/ 272.48 m = 5. 43 ton/m2Proponer peralte de 80 cmPor lo tanto: d=h-10 d=80 m-10 cm d=70 cm

Calculo de cortante Vucolumnas centradas Pu = 204 ton Ac= (40+d) (40+d ) b0 = 2(40+d) +2(40+d) Ac= 1.10 *1.10 b0 = 4 x 110 cm Ac=1.21 m b0 = 440 cm

Vu2 = Pu qs*Ac Vc2 = 2[(0.53) (0.75) () b0*d] Vu2 = 204 5.43 (1.21) Vc2 = 2[(0.53) (0.75) () (440) (70)] Vu2 = 197.43 ton Vc2 = 354836.85 kg /1000 Vc2 = 354.84 ton.

Vu2 > Vu. 354.84 ton >197.43 ton Ok. columnas de bordePu=68ton Ac= (40+d/2) (40+d ) b0 = 2(40+d/2) +(40+d) Ac= (75 *110)*(1/100) b0 = 150+110 cm Ac=0.825 m b0 = 260 cm

Vu2 = Pu qs*Ac Vc2 = 2[(0.53) (0.75) () b0*d] Vu2 = 68 5.43 (0.825) Vc2 = 2[(0.53) (0.75) () (260) (70)] Vu2 = 63.52 ton Vc2 = 209675.73 kg /1000 Vc2 = 209.68 ton.

Vu2 > Vu. 209.68 ton >63.52 ton Ok.

Pu=154ton Ac= (40+d/2) (40+d ) b0 = 2(40+d/2) +2(40+d) Ac= (75 *110)*(1/100) b0 = 150+220 cm Ac=0.825 m b0 = 370 cm

Vu2 = Pu qs*Ac Vc2 = 2[(0.53) (0.75) () b0*d] Vu2 = 154 5.43(0.825) Vc2 = 2[(0.53) (0.75) () (370) (70)] Vu2 = 149.52 ton Vc2 = 209675.73 kg /1000 Vc2 = 209.68 ton.

Vu2 > Vu. 209.68 ton >149.52 ton Ok.

Pu=204ton Ac= (40+d/2) (40+d ) b0 = 2(40+d/2) +2(40+d) Ac= (75 *110)*(1/100) b0 = 150+220 cm Ac=0.825 m b0 = 370 cm

Vu2 = Pu qs*Ac Vc2 = 2[(0.53) (0.75) () b0*d] Vu2 = 204 5.43(0.825) Vc2 = 2[(0.53) (0.75) () (370) (70)] Vu2 = 199.52 ton Vc2 = 209675.73 kg /1000 Vc2 = 209.68 ton.

Vu2 > Vu. 209.68 ton >199.52 ton Ok. Elegimos el borde mas cargada con Pu=204 ton. Vu2 > Vu. 209.68 ton >199.52 ton Ok. columnas de esquina Pu=52 ton Ac= (40+d/2) (40+d/2 ) b0 = (40+d/2) +(40+d/2) Ac= (75 *75)*(1/100) b0 = 75+75 cm Ac= 0.5625 m b0 = 150 cm

Vu2 = Pu qs*Ac Vc2 = 2[(0.53) (0.75) () b0*d] Vu2 = 52 5.43(0.5625) Vc2 = 2[(0.53) (0.75) () (150) (70)] Vu2 = 48.94 ton Vc2 = 120966.76 kg /1000 Vc2 = 120.96 ton.

Vu2 > Vu. 120.96 ton >48.94 ton Ok.

columnas de esquina Pu=60 ton Ac= (40+d/2) (40+d/2 ) b0 = (40+d/2) +(40+d/2) Ac= (75 *75)*(1/100) b0 = 75+75 cm Ac= 0.5625 m b0 = 150 cm

Vu2 = Pu qs*Ac Vc2 = 2[(0.53) (0.75) () b0*d] Vu2 = 52 5.43(0.5625) Vc2 = 2[(0.53) (0.75) () (150) (70)] Vu2 = 56.94 ton Vc2 = 120966.76 kg /1000 Vc2 = 120.96 ton.

Vu2 > Vu. 120.96 ton >56.94ton Ok. columnas de esquina Pu=52 ton Ac= (40+d/2) (40+d/2 ) b0 = (40+d/2) +(40+d/2) Ac= (75 *75)*(1/100) b0 = 75+75 cm Ac= 0.5625 m b0 = 150 cm

Vu2 = Pu qs*Ac Vc2 = 2[(0.53) (0.75) () b0*d] Vu2 = 52 5.43(0.5625) Vc2 = 2[(0.53) (0.75) () (150) (70)] Vu2 = 52.94 ton Vc2 = 120966.76 kg /1000 Vc2 = 120.96 ton.

Vu2 > Vu. 120.96 ton >52.94 ton Ok. Elegimos el borde mas cargada con Pu=60 ton. Vu2 > Vu. 120.96 ton >56.94ton Ok. Usar h = 80 cm, d= 70 cm.

Corte en una direccinFranja X1Bf=3.85 m Lf=12.60 m Af= Bf*Lf=48.51m Pui=52+68+56=176 ton pui+Xi =52(.20)+68(6.3)+56(12.4)=113.20ton-m xp = 6.44m e=xp-; e=6.44-=0.14> 0 reaccin del suelo hacia la derechaLe=2|e|; Le=2(0.14)=0.28 m Lf'=Lf-Le; Lf=12.60-0.28=12.32mWs=pui/Lf'; Ws==14.29 ton/m

Franja X2=X3Bf=7.3 m Lf=12.60 m Af= Bf*Lf=91.98 m Pui=204+204+154=562 ton pui+Xi =204(.20)+204(6.3)+154(12.4)= 3235.6 ton-m xp = 5.76 m e=xp-; e=5.76-=-0.54 > 0 reaccin del suelo hacia la derechaLe=2|e|; Le=2(0.54)= 1.08 m Lf'=Lf-Le; Lf=12.60-1.08=11.52 mWs=pui/Lf'; Ws==48.78 ton/m

Franja X4Bf=3.85 m Lf=12.60 m Af= Bf*Lf=48.51m Pui=52+68+60=180 ton pui+Xi =52(.20)+68(6.3)+60(12.4)= 1182.8 ton-m xp = 6.57 m e=xp-; e=6.57-=0.27 > 0 reaccin del suelo hacia la derechaLe=2|e|; Le=2(0.27)=0.54 m Lf'=Lf-Le; Lf=12.60-0.54=12.06 mWs=pui/Lf'; Ws==14.93 ton/m

franja Y1Bf=3.25 m Lf=22.3 m Af= Bf*Lf=72.475m Pui=52+204+204+52=512 ton pui+Xi =(0.2)52+(7.5)204+(14.8)204+(22.1)52=5708.8 ton-m xp = 11.15 m e=xp-; e=11.15-=0 > 0 reaccin del suelo hacia la derechaLe=2|e|; Le=2(0)=0 m Lf'=Lf-Le; Lf=22.3-0=22.3 mWs=pui/Lf'; Ws==22.97 ton/m

franja Y2=Y3Bf=6.1 m Lf=22.3 m Af= Bf*Lf=136.03 m Pui=68+204+204+68=544 ton pui+Xi =(0.2)68+(7.5)204+(14.8)204+(22.1)68=6065.6 ton-m xp = 11.15 m e=xp-; e=11.15-=0 > 0 reaccin del suelo hacia la derechaLe=2|e|; Le=2(0)=0 m Lf'=Lf-Le; Lf=22.3-0=22.3 mWs=pui/Lf'; Ws==24.39 ton/m

franja Y4Bf=3.25 m Lf=22.3 m Af= Bf*Lf=72.475m Pui=56+154+154+60=424 ton pui+Xi =(0.2)56+(7.5)154+(14.8)154+(22.1)60=4771.4 ton-m xp = 11.25 m e=xp-; e=11.25-=0.10 > 0 reaccin del suelo hacia la derechaLe=2|e|; Le=2(.10)= 0.21 m Lf'=Lf-Le; Lf=22.3-0.21=22.09 mWs=pui/Lf'; Ws==19.20 ton/m

Peralte de la losa d=h+10 d= 70+10 d=80 cm

Refuerzo a flexin

b=1.0 m ; b=100 cm

Asmin = min*b*h Asmax =max*b*hAsmin = 0.002(100)(80)=16 cm Asmax =0.01605(100)(80)=128.4 cm

Mumin= 625*hMumin= 625*80Mumin= 625.8 (80)= 4 ton-m