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• Enrique Segura Saavedra

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DIAGRAMA DE INTERACCION COLUMNAS RECTANGULARES

40 28

6

6

40

0 0 3 0

Ø 1/2" Ø 5/8" Ø 3/4" Ø 1" P1

Es = Fy = f'c = εc =

0 0 2 0

Ø 1/2" Ø 5/8" Ø 3/4" Ø 1" P2

0 0 3 0

Ø 1/2" Ø 5/8" Ø 3/4" Ø 1" P3

2039400 Kg/cm2 4200 Kg/cm2 210 Kg/cm2 0.003

DEFORMACION UNITARIA DE FLUENCIA EN EL ACERO Eje neutro para la condicion de falla balanceada εy = Fy / Es =

4200 Kg/cm2 2039400 Kg/cm2

=

0.00205943

PUNTO 1 CALCULO DE LA FUERZA DE COMPRESION EN EL CONCRETO Cc =

,85 f'c b d =

0.85

210

40

40 =

CALCULO DE LA FUERZA DE COMPRESION DEL ACERO P1 = As1 fs1 =

7.8

4200 =

32760 Kg

285600

P2 = As2 fs2 = P3 = As3 fs3 =

5.2 7.8

4200 = 4200 =

21840 Kg 32760 Kg

CALCULO CARGA AXIAL NOMINAL Pn =

Cc + P1 + P2 + P3 =

372960 Kg =

372.96 Ton

CALCULO DEL MOMENTO FLECTOR NOMINAL Mn =

285600 32760 21840 32760

0 14 0 14

+ + =

0 Kg-cm 0 T-cm 0 T-m

PUNTO 2 ε1 εc = εc = 0.003

ε2 ε3

34 6

28 40

6

CALCULO DE DEFORMACIONES UNITARIAS ε1 =

0.003

34 = 40

0.00255

0.00205943

ε2 =

0.003

20 = 40

0.0015

0.00205943

ε3 =

0.003

6= 40

0.00045

0.00205943

CALCULO DE ESFUERZOS EN EL ACERO fs1 =

4200 Kg/cm2

fs2 =

3059.1 Kg/cm2

fs1 =

917.73 Kg/cm2

CALCULO DE LA FUERZA DE COMPRESION EN EL HORMIGON Cc =

,85 f'c b d =

0.85

210

34

40 =

242760

CALCULO DE LA FUERZA DE COMPRESION DEL ACERO P1 = As1 fs1 = P2 = As2 fs2 = P3 = As3 fs3 =

7.8 5.2 7.8

4200 = 3059.1 = 917.73 =

32760 Kg 15907.32 Kg 7158.294 Kg

298585.614 Kg =

298.585614 Ton

CALCULO CARGA AXIAL NOMINAL Pn =

Cc + P1 + P2 + P3 =

CALCULO DEL MOMENTO FLECTOR NOMINAL Mn =

242760 32760 15907.32 7158.294

e=

3 14 0 14

+ + =

1086703.88 Kg-cm 1086.70388 T-cm 10.8670388 T-m

3.63950516

PUNTO e = 10% H =

4 cm

ENTRE

40 y

39.15467

ε1 εc = εc = 0.003

ε2 ε3

33.2814695 6

28 39.15467

6

CALCULO DE DEFORMACIONES UNITARIAS ε1 =

0.003

33.15467 = 39.15467

0.00254028

0.00205943

ε2 =

0.003

19.15467 = 39.15467

0.00146762

0.00205943

ε3 =

0.003

5.15467 = 39.15467

0.00039495

0.00205943

CALCULO DE ESFUERZOS EN EL ACERO fs1 =

4200 Kg/cm2

fs2 =

2993.05554 Kg/cm2

fs1 =

805.454419 Kg/cm2

CALCULO DE LA FUERZA DE COMPRESION EN EL HORMIGON Cc =

,85 f'c b d =

0.85

210 33.2814695

40 =

237629.6922

CALCULO DE LA FUERZA DE COMPRESION DEL ACERO P1 = As1 fs1 = P2 = As2 fs2 = P3 = As3 fs3 =

7.8 4200 = 5.2 2993.05554 = 7.8 805.454419 =

32760 Kg 15563.8888 Kg 6282.54447 Kg

CALCULO CARGA AXIAL NOMINAL Pn =

Cc + P1 + P2 + P3 =

292236.126 Kg =

292.236126 Ton

CALCULO DEL MOMENTO FLECTOR NOMINAL Mn =

237629.692 3.35926525 + 32760 14 + 15563.8888 06282.54447 14 =

e=

1168945.54 Kg-cm 1168.94554 T-cm 11.6894554 T-m

4.00000357

PUNTO e = 10% H =

4 cm

ENTRE

34 y

ε1 εc = εc = 0.003

ε2 ε3

28.9 6

28

6

20

34 CALCULO DE DEFORMACIONES UNITARIAS ε1 =

0.003

28 = 34

0.00247059

0.00205943

ε2 =

0.003

14 = 34

0.00123529

0.00205943

ε3 =

0.003

0= 34

0

0.00205943

CALCULO DE ESFUERZOS EN EL ACERO fs1 =

4200 Kg/cm2

fs2 =

2519.25882 Kg/cm2

fs1 =

0 Kg/cm2

CALCULO DE LA FUERZA DE COMPRESION EN EL HORMIGON Cc =

,85 f'c b d =

0.85

210

28.9

40 =

CALCULO DE LA FUERZA DE COMPRESION DEL ACERO P1 = As1 fs1 = P2 = As2 fs2 = P3 = As3 fs3 =

7.8 4200 = 5.2 2519.25882 = 7.8 0=

32760 13100.1459 0

CALCULO CARGA AXIAL NOMINAL Pn =

Cc + P1 + P2 - P3 =

252206.146 Kg =

252.206146 Ton

CALCULO DEL MOMENTO FLECTOR NOMINAL Mn =

e= PUNTO 3

206346 32760 13100.1459 0

6.35932282 cm

5.55 14 0 14

+ + =

1603860.3 Kg-cm 1603.8603 T-cm 16.038603 T-m

206346

ε1 εc = εc = 0.003

ε2 ε3

25.5 6

28 30

6

CALCULO DE DEFORMACIONES UNITARIAS ε1 =

0.003

24 = 30

0.0024

0.00205943

ε2 =

0.003

10 = 30

0.001

0.00205943

ε3 =

0.003

4= 30

0.0004

0.00205943

CALCULO DE ESFUERZOS EN EL ACERO fs1 =

4200 Kg/cm2

fs2 =

2039.4 Kg/cm2

fs1 =

815.76 Kg/cm2

CALCULO DE LA FUERZA DE COMPRESION EN EL HORMIGON Cc =

,85 f'c b d =

0.85

210

25.5

40 =

CALCULO DE LA FUERZA DE COMPRESION DEL ACERO P1 = As1 fs1 = P2 = As2 fs2 = P3 = As3 fs3 =

7.8 5.2 7.8

4200 = 2039.4 = 815.76 =

32760 10604.88 6362.928

CALCULO CARGA AXIAL NOMINAL Pn =

Cc + P1 + P2 - P3 =

219071.952 Kg =

CALCULO DEL MOMENTO FLECTOR NOMINAL

219.071952 Ton

182070

Mn =

182070 32760 10604.88 6362.928

e=

7.25 14 0 14

+ + =

1867728.49 Kg-cm 1867.72849 T-cm 18.6772849 T-m

8.52563952 cm

PUNTO 4 ε1 εc = εc = 0.003

ε2 ε3

17 6

28

6

20 CALCULO DE DEFORMACIONES UNITARIAS ε1 =

0.003

14 = 20

0.0021

0.00205943

ε2 =

0.003

0= 20

0

0.00205943

ε3 =

0.003

14 = 20

0.0021

0.00205943

CALCULO DE ESFUERZOS EN EL ACERO fs1 =

4200 Kg/cm2

fs2 =

0 Kg/cm2

fs1 =

4200 Kg/cm2

CALCULO DE LA FUERZA DE COMPRESION EN EL HORMIGON Cc =

,85 f'c b d =

0.85

210

CALCULO DE LA FUERZA DE COMPRESION DEL ACERO

17

40 =

121380

P1 = As1 fs1 = P2 = As2 fs2 = P3 = As3 fs3 =

7.8 5.2 7.8

4200 = 0= 4200 =

32760 0 32760

CALCULO CARGA AXIAL NOMINAL Pn =

Cc + P1 + P2 - P3 =

121380 Kg =

121.38 Ton

CALCULO DEL MOMENTO FLECTOR NOMINAL Mn =

121380 32760 0 32760

11.5 14 0 14

+ + =

2313150 Kg-cm 2313.15 T-cm 23.1315 T-m

PUNTO 5 ε1 εc = εc = 0.003

ε2 ε3

6.34015 6

28

6

7.459 CALCULO DE DEFORMACIONES UNITARIAS ε1 =

0.003

1.459 = 7.459

0.00058681

0.00205943

ε2 =

0.003

12.541 = 7.459

0.00504397

0.00205943

ε3 =

0.003

26.541 = 7.459

0.01067476

0.00205943

CALCULO DE ESFUERZOS EN EL ACERO fs1 =

1196.736 Kg/cm2

fs2 =

4200 Kg/cm2

fs1 =

4200 Kg/cm2

CALCULO DE LA FUERZA DE COMPRESION EN EL HORMIGON Cc =

,85 f'c b d =

0.85

210

6.34015

40 =

45268.671

CALCULO DE LA FUERZA DE COMPRESION DEL ACERO P1 = As1 fs1 = P2 = As2 fs2 = P3 = As3 fs3 =

7.8 5.2 7.8

1196.736 = 4200 = 4200 =

9334.54077 21840 32760

CALCULO CARGA AXIAL NOMINAL Pn =

Cc + P1 + P2 - P3 =

3.2117749 Kg =

0.00321177 Ton

CALCULO DEL MOMENTO FLECTOR NOMINAL Mn =

e=

45268.671 9334.54077 21840 32760

16.829925 14 0 14

+ =

1351191.91 Kg-cm 1351.19191 T-cm 13.5119191 T-m

420699.442 Mu

punt 1 punt 2 punt 3 punt 4 punt 5

Pu

.9 Mu

.9 Pu

.7 Mu

.7 Pu

0 372.96 0 335.664 0 261.072 1086.70388 298.585614 978.033496 268.727053 760.692719 209.00993 1867.72849 219.071952 1680.95564 197.164757 1307.40994 153.350366 2313.15 121.38 2081.835 109.242 1619.205 84.966 1351.19191 0.00321177 1216.07272 0.0028906 945.834336 0.00224824

e = Mu / Pu = 0 596.736 895.104 1193.472 1491.84

La carga axial que define una variación en el factor de reducción de capacidad es: .1 f'c Ag =

0.1

210

1600 =

33600 Kg 33.6 T

400 350

POR NO SER ZUNCHADA e = 10% h

300

0.1

DE LA ECUACION =

40 = e = Mu / Pu =

4 cms 4 cms

CARGA AXIAL QUE DEFINE REDUCCION DE CARGA

Pnominal

e=

250 200 150 100

0.1

210

1600 =

33600 Kg 33.6 T

50 0

0

Kg

Kg

Kg

Kg

Kg

Kg

Kg

4M 0 149.184 223.776 298.368 372.96

0 925.26 1387.89 1850.52 2313.15

Max P 204.565288 204.565288 204.565288 204.565288 204.565288

Min P 0 925.26 1387.89 1850.52 2313.15

33.6 33.6 33.6 33.6 33.6

DIAGRAMA DE INTERACCION 400 350 300 P nominal, M nominal

250

90% Pn Mn

200

70% Pn Mn

150

e = 10% h

100

MAX P

50

MIN P

0 0

500

1000

1500

2000

2500

0 0

500

1000

1500

2000

2500

Mnominal

CALCULO ε1 =

ε2 =

ε3 =

ε4 =

ε5 =

ε6 =

ε7 =

ε8 =

ε9 =

ε10 =

ε1 =

ε2 =

ε3 =

ε4 =

ε5 =

ε6 =

ε7 =

ε8 =

ε9 =

ε10 =

CALCULO fs1 = fs2 = fs3 = fs4 = fs5 = fs6 = fs7 = fs8 = fs9 = fs10 =

fs1 = fs2 = fs3 = fs4 = fs5 = fs6 = fs7 = fs8 = fs9 = fs10 = CALCULO Cc =

CALCULO

P1 = As1 f P2 = As2 f P3 = As3 f P4 = As4 f P5 = As5 f P6 = As6 f P8 = As7 f P8 = As8 f P9 = As9 f P10 = As1 Pctr = Asc P11 = As1 P12 = As1 P13 = As1 P14 = As1 P15 = As1 P16 = As1 P17 = As1 P18 = As1 P19 = As1

P20 = As2 CALCULO Pn = CALCULO Mn =

e= PUNTO 4

ε 0.003

CALCULO ε1 =

ε2 =

ε3 =

ε4 =

ε5 =

ε6 =

ε7 =

ε8 =

ε9 =

ε10 =

ε1 =

ε2 =

ε3 =

ε4 =

ε5 =

ε6 =

ε7 =

ε8 =

ε9 =

ε10 =

CALCULO fs1 = fs2 = fs3 = fs4 = fs5 = fs6 = fs7 = fs8 = fs9 = fs10 = fs1 = fs2 = fs3 = fs4 = fs5 = fs6 = fs7 = fs8 = fs9 = fs10 = CALCULO Cc =

CALCULO

P1 = As1 f P2 = As2 f P3 = As3 f P4 = As4 f P5 = As5 f P6 = As6 f P8 = As7 f P8 = As8 f P9 = As9 f P10 = As1

P11 = As1 P12 = As1 P13 = As1 P14 = As1 P15 = As1 P16 = As1 P17 = As1 P18 = As1 P19 = As1 P20 = As2 CALCULO Pn = CALCULO Mn =

e=

PUNTO 5

ε 0.003

CALCULO ε1 =

ε2 =

ε3 =

ε4 =

ε5 =

ε6 =

ε7 =

ε8 =

ε9 =

ε10 =

ε1 =

ε2 =

ε3 =

ε4 =

ε5 =

ε6 =

ε7 =

ε8 =

ε9 =

ε10 =

CALCULO fs1 = fs2 = fs3 = fs4 = fs5 = fs6 = fs7 = fs8 = fs9 = fs10 = fs1 = fs2 = fs3 = fs4 = fs5 = fs6 = fs7 = fs8 = fs9 = fs10 = CALCULO Cc =

CALCULO

P1 = As1 f P2 = As2 f P3 = As3 f P4 = As4 f

P5 = As5 f P6 = As6 f P8 = As7 f P8 = As8 f P9 = As9 f P10 = As1

P11 = As1 P12 = As1 P13 = As1 P14 = As1 P15 = As1 P16 = As1 P17 = As1 P18 = As1 P19 = As1 P20 = As2 CALCULO Pn = CALCULO Mn =

e=

punt 1 punt 2 punt 3 punt 4 punt 5

La carga a .1 f'c Ag =

POR NO SE e=

DE LA ECU

CARGA AX

DIAGRAMA DE INTERACCION COLUMNAS RECTANGULARES 120 105

7.5

7.5

40

0 0 0 16

Es = Fy = f'c = εc =

Ø 1/2" Ø 5/8" Ø 3/4" Ø 1"

Ps

2039400 Kg/cm2 4200 Kg/cm2 210 Kg/cm2 0.003

DEFORMACION UNITARIA DE FLUENCIA EN EL ACERO Eje neutro para la condicion de falla balanceada εy = Fy / Es =

Ag =

4200 Kg/cm2 2039400 Kg/cm2 PI r2 =

=

0.002059429

11309.73355 cm2

PUNTO 1 CALCULO DE LA FUERZA DE COMPRESION EN EL CONCRETO Cc =

,85 f'c b d =

0.85

210

11309.73355

CALCULO DE LA FUERZA DE COMPRESION DEL ACERO Ps = As fs =

81.6

4200 =

342720

CALCULO CARGA AXIAL NOMINAL Pn =

Cc + Ps =

2361507.44 Kg =

2361.507439

CALCULO DEL MOMENTO FLECTOR NOMINAL Mn =

2018787.44 8400 8400 8400 8400 0 0 0 0 0 0 0 0 0 0 0 0 8400 8400 8400 8400

0= 10.242 29.167 43.652 51.491 0 0 0 0 0 0 0 0 0 0 0 0 -51.491 -43.652 -29.167 -10.242

= = = = = = = = = = = = = = = = = = = =

0.000 86032.800 245002.800 366676.800 432524.400 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -432524.400 -366676.800 -245002.800 -86032.800 0.000 Kg-cm 0.000 T-cm 0.000 T-m

PUNTO 2 ε1 εc = εc =

ε2

0.003

ε3

102 7.5

105 120 120

7.5

CALCULO DE DEFORMACIONES UNITARIAS ε1 =

0.003 49.75775809 = 120

0.001243944

1

ε2 =

0.003 30.83256277 = 120

0.000770814

2

ε3 =

0.003 16.34784535 = 120

0.000408696

3

ε4 =

0.003 8.508772779 = 120

0.000212719

4

ε5 =

0.003

60 = 120

0

5

ε6 =

0.003

60 = 120

0

6

ε7 =

0.003

60 = 120

0

7

ε8 =

0.003

60 = 120

0

8

ε9 =

0.003

60 = 120

0

9

ε10 =

0.003

60 = 120

0

10

ε1 =

0.003 70.24224191 = 120

0.001756056

1

ε2 =

0.003 89.16743723 = 120

0.002229186

2

ε3 =

0.003 103.6521546 = 120

0.002591304

3

ε4 =

0.003 111.4912272 = 120

0.002787281

4

ε5 =

0.003

0

5

60 = 120

ε6 =

0.003

60 = 120

0

6

ε7 =

0.003

60 = 120

0

7

ε8 =

0.003

60 = 120

0

8

ε9 =

0.003

60 = 120

0

9

ε10 =

0.003

60 = 120

0

10

CALCULO DE ESFUERZOS EN EL ACERO fs1 = fs2 = fs3 = fs4 = fs5 = fs6 = fs7 = fs8 = fs9 = fs10 =

2536.8993 1571.99821 833.494895 433.81978 0 0 0 0 0 0

Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2

fs1 = fs2 = fs3 = fs4 = fs5 = fs6 = fs7 = fs8 = fs9 = fs10 =

3581.3007 4200 4200 4200 0 0 0 0 0 0

Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2

CALCULO DE LA FUERZA DE COMPRESION EN EL HORMIGON Cc =

,85 f'c Asc =

0.85

210

10245.93773

area =

1799.639964 8446.297764 10245.93773 cm2

102

CALCULO DE LA FUERZA DE COMPRESION DEL ACERO P1 = As1 fs1 = P2 = As2 fs2 = P3 = As3 fs3 = P4 = As4 fs4 = P5 = As5 fs5 = P6 = As6 fs6 = P8 = As7 fs7 = P8 = As8 fs8 = P9 = As9 fs9 = P10 = As10 fs10 =

10.2 2536.8993 = 10.2 1571.99821 = 10.2 833.494895 = 10.2 433.81978 = 0 0= 0 0= 0 0= 0 0= 0 0= 0 0=

25876.37282 16034.38177 8501.647933 4424.961757 0 0 0 0 0 0

P11 = As11 fs11 = P12 = As12 fs12 = P13 = As13 fs13 = P14 = As14 fs14 = P15 = As15 fs15 = P16 = As16 fs16 = P17 = As17 fs17 = P18 = As18 fs18 = P19 = As19 fs19 = P20 = As20 fs20 =

10.2 10.2 10.2 10.2 0 0 0 0 0 0

36529.26718 42840 42840 42840 0 0 0 0 0 0 219886.6315

3581.3007 4200 4200 4200 0 0 0 0 0 0

= = = = = = = = = =

CALCULO CARGA AXIAL NOMINAL Pn =

Cc + SUM Pi =

2048786.52 Kg =

CALCULO DEL MOMENTO FLECTOR NOMINAL Mn =

1828899.88 25876.3728 16034.3818 8501.64793 4424.96176 0 0 0 0

9 -10.24224191 -29.16743723 -43.65215465 -51.49122722 0 0 0 0

= = = = = = = = =

16460098.96 -265032.0701 -467681.8238 -371115.2503 -227846.7113 0 0 0 0

2048.786516

0 0 0 36529.2672 42840 42840 42840 0 0 0 0 0 0

e=

0 0 0 10.24224191 29.16743723 43.65215465 51.49122722 0 0 0 0 0 0

= = = = = = = = = = = = =

0.10166037

PUNTO e = 10% H =

12 cm

0 0 0 374141.5911 1249533.011 1870058.305 2205884.174 0 0 0 0 0 0 20828040.19 Kg-cm 208.2804019 T-cm 2.082804019 T-m

ENTRE

0

ε1 εc = εc =

ε2

0.003

ε3

71.999675 7.5

105 84.7055 120

7.5

CALCULO DE DEFORMACIONES UNITARIAS ε1 =

0.003 14.46325809 = 84.7055

0.000512243

1

ε2 =

0.003 -4.461937234 = 84.7055

-0.000158028

2

ε3 =

0.003 -18.94665465 = 84.7055

-0.00067103

3

ε4 =

0.003 -26.78572722 = 84.7055

-0.000948665

4

ε5 =

0.003

0

5

24.7055 =

84.7055 ε6 =

0.003

24.7055 = 84.7055

0

6

ε7 =

0.003

24.7055 = 84.7055

0

7

ε8 =

0.003

24.7055 = 84.7055

0

8

ε9 =

0.003

24.7055 = 84.7055

0

9

ε10 =

0.003

24.7055 = 84.7055

0

10

ε1 =

0.003 34.94774191 = 84.7055

0.001237738

1

ε2 =

0.003 53.87293723 = 84.7055

0.001908008

2

ε3 =

0.003 68.35765465 = 84.7055

0.002421011

3

ε4 =

0.003 76.19672722 = 84.7055

0.002698646

4

ε5 =

0.003

24.7055 = 84.7055

0

5

ε6 =

0.003

24.7055 = 84.7055

0

6

ε7 =

0.003

24.7055 = 84.7055

0

7

ε8 =

0.003

24.7055 = 84.7055

0

8

ε9 =

0.003

24.7055 = 84.7055

0

9

ε10 =

0.003

24.7055 = 84.7055

0

10

CALCULO DE ESFUERZOS EN EL ACERO fs1 = fs2 = fs3 = fs4 = fs5 = fs6 = fs7 = fs8 = fs9 = fs10 =

1044.66777 -322.281604 -1368.49936 -1934.70833 0 0 0 0 0 0

Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2

fs1 = fs2 = fs3 = fs4 = fs5 = fs6 = fs7 = fs8 = fs9 = fs10 =

2524.24311 3891.19248 4200 4200 0 0 0 0 0 0

Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2

CALCULO DE LA FUERZA DE COMPRESION EN EL HORMIGON Cc =

,85 f'c Asc =

0.85

210

5105.196239

area =

896.6983228 4208.497917 5105.196239 cm2

71.999675

CALCULO DE LA FUERZA DE COMPRESION DEL ACERO P1 = As1 fs1 = P2 = As2 fs2 = P3 = As3 fs3 = P4 = As4 fs4 = P5 = As5 fs5 =

10.2 10.2 10.2 10.2 0

1044.66777 -322.281604 -1368.49936 -1934.70833 0

= = = = =

10655.61124 -3287.272358 -13958.69346 -19734.02495 0

P6 = As6 fs6 = P8 = As7 fs7 = P8 = As8 fs8 = P9 = As9 fs9 = P10 = As10 fs10 =

0 0 0 0 0

P11 = As11 fs11 = P12 = As12 fs12 = P13 = As13 fs13 = P14 = As14 fs14 = P15 = As15 fs15 = P16 = As16 fs16 = P17 = As17 fs17 = P18 = As18 fs18 = P19 = As19 fs19 = P20 = As20 fs20 =

0 0 0 0 0

= = = = =

0 0 0 0 0

10.2 2524.24311 = 10.2 3891.19248 = 10.2 4200 = 10.2 4200 = 0 0= 0 0= 0 0= 0 0= 0 0= 0 0=

25747.27969 39690.16329 42840 42840 0 0 0 0 0 0 124793.0635

CALCULO CARGA AXIAL NOMINAL Pn =

Cc + SUM Pi =

1036070.59 Kg =

1036.070592

CALCULO DEL MOMENTO FLECTOR NOMINAL Mn =

911277.529 10655.6112 -3287.27236 -13958.6935 -19734.0249 0 0 0 0 0 0 0 25747.2797 39690.1633 42840 42840 0 0 0 0 0 0

24.0001625 -10.24224191 -29.16743723 -43.65215465 -51.49122722 0 0 0 0 0 0 0 10.24224191 29.16743723 43.65215465 51.49122722 0 0 0 0 0 0

= = = = = = = = = = = = = = = = = = = = = =

21870808.77 -109137.348 95881.31016 609327.0457 1016129.163 0 0 0 0 0 0 0 263709.8671 1157660.347 1870058.305 2205884.174 0 0 0 0 0 0 28980321.64 Kg-cm 289.8032164 T-cm

e=

0.27971377

2.898032164 T-m

PUNTO 3 ε1 εc = εc =

ε2

0.003

ε3

76.5 7.5

105 90 120

7.5

CALCULO DE DEFORMACIONES UNITARIAS ε1 =

0.003 19.75775809 = 90

0.000658592

1

ε2 =

0.003 0.832562766 = 90

2.77521E-05

2

ε3 =

0.003 -13.65215465 = 90

-0.000455072

3

ε4 =

0.003 -21.49122722 = 90

-0.000716374

4

ε5 =

0.003

30 = 90

0

5

ε6 =

0.003

30 = 90

0

6

ε7 =

0.003

30 = 90

0

7

ε8 =

0.003

30 = 90

0

8

ε9 =

0.003

30 = 90

0

9

ε10 =

0.003

30 =

0

10

90 ε1 =

0.003 40.24224191 = 90

0.001341408

1

ε2 =

0.003 59.16743723 = 90

0.001972248

2

ε3 =

0.003 73.65215465 = 90

0.002455072

3

ε4 =

0.003 81.49122722 = 90

0.002716374

4

ε5 =

0.003

30 = 90

0

5

ε6 =

0.003

30 = 90

0

6

ε7 =

0.003

30 = 90

0

7

ε8 =

0.003

30 = 90

0

8

ε9 =

0.003

30 = 90

0

9

ε10 =

0.003

30 = 90

0

10

CALCULO DE ESFUERZOS EN EL ACERO fs1 = fs2 = fs3 = fs4 = fs5 = fs6 = fs7 = fs8 = fs9 = fs10 =

1343.1324 56.5976169 -928.073473 -1460.97363 0 0 0 0 0 0

Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2

fs1 = fs2 = fs3 = fs4 = fs5 = fs6 = fs7 = fs8 = fs9 = fs10 =

2735.6676 4022.20238 4200 4200 0 0 0 0 0 0

Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2

CALCULO DE LA FUERZA DE COMPRESION EN EL HORMIGON Cc =

,85 f'c Asc =

0.85

210

7609.619489

area =

951.8297839 6657.789705 7609.619489 cm2

76.5

CALCULO DE LA FUERZA DE COMPRESION DEL ACERO P1 = As1 fs1 = P2 = As2 fs2 = P3 = As3 fs3 = P4 = As4 fs4 = P5 = As5 fs5 = P6 = As6 fs6 = P8 = As7 fs7 = P8 = As8 fs8 = P9 = As9 fs9 = P10 = As10 fs10 = Pctr = Asctr fsctr = P11 = As11 fs11 = P12 = As12 fs12 = P13 = As13 fs13 = P14 = As14 fs14 = P15 = As15 fs15 = P16 = As16 fs16 = P17 = As17 fs17 = P18 = As18 fs18 = P19 = As19 fs19 =

10.2 10.2 10.2 10.2 0 0 0 0 0 0 2 10.2 10.2 10.2 10.2 0 0 0 0 0

1343.1324 56.5976169 -928.073473 -1460.97363 0 0 0 0 0 0 0 2735.6676 4022.20238 4200 4200 0 0 0 0 0

= = = = = = = = = = = = = = = = = = = =

13699.95043 577.295692 -9466.349423 -14901.93099 0 0 0 0 0 0 0 27903.80957 41026.46431 42840 42840 0 0 0 0 0

P20 = As20 fs20 =

0

0=

0 144519.2396

CALCULO CARGA AXIAL NOMINAL Pn =

Cc + SUM Pi =

1502836.32 Kg =

1502.836318

CALCULO DEL MOMENTO FLECTOR NOMINAL Mn =

1358317.08 13699.9504 577.295692 -9466.34942 -14901.931 0 0 0 0 0 0 0 27903.8096 41026.4643 42840 42840 0 0 0 0 0 0

e=

21.75 -10.24224191 -29.16743723 -43.65215465 -51.49122722 0 0 0 0 0 0 0 10.24224191 29.16743723 43.65215465 51.49122722 0 0 0 0 0 0

= = = = = = = = = = = = = = = = = = = = = =

0.24037989

29543396.46 -140318.2064 -16838.23586 413226.5489 767318.7147 0 0 0 0 0 0 0 285797.5677 1196636.823 1870058.305 2205884.174 0 0 0 0 0 0 36125162.15 Kg-cm 361.2516215 T-cm 3.612516215 T-m

PUNTO 4 ε1 εc = εc =

ε2

0.003

ε3

51 7.5

105 60 120

7.5

CALCULO DE DEFORMACIONES UNITARIAS ε1 =

0.003 -10.24224191 = 60

-0.000512112

1

ε2 =

0.003 -29.16743723 = 60

-0.001458372

2

ε3 =

0.003 -43.65215465 = 60

-0.002182608

3

ε4 =

0.003 -51.49122722 = 60

-0.002574561

4

ε5 =

0.003

0= 60

0

5

ε6 =

0.003

0= 60

0

6

ε7 =

0.003

0= 60

0

7

ε8 =

0.003

0= 60

0

8

ε9 =

0.003

0= 60

0

9

ε10 =

0.003

0= 60

0

10

ε1 =

0.003 10.24224191 = 60

0.000512112

1

ε2 =

0.003 29.16743723 = 60

0.001458372

2

ε3 =

0.003 43.65215465 = 60

0.002182608

3

ε4 =

0.003 51.49122722 = 60

0.002574561

4

ε5 =

0.003

0

5

0= 60

ε6 =

0.003

0= 60

0

6

ε7 =

0.003

0= 60

0

7

ε8 =

0.003

0= 60

0

8

ε9 =

0.003

0= 60

0

9

ε10 =

0.003

0= 60

0

10

CALCULO DE ESFUERZOS EN EL ACERO fs1 = fs2 = fs3 = fs4 = fs5 = fs6 = fs7 = fs8 = fs9 = fs10 =

-1044.40141 -2974.20357 -4200 -4200 0 0 0 0 0 0

Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2

fs1 = fs2 = fs3 = fs4 = fs5 = fs6 = fs7 = fs8 = fs9 = fs10 =

1044.40141 2974.20357 4200 4200 0 0 0 0 0 0

Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2

CALCULO DE LA FUERZA DE COMPRESION EN EL HORMIGON Cc =

,85 f'c Asc =

0.85

210

4578.930556

area = 51

69

533.8904382 6196.912558 4578.930556 cm2

CALCULO DE LA FUERZA DE COMPRESION DEL ACERO P1 = As1 fs1 = P2 = As2 fs2 = P3 = As3 fs3 = P4 = As4 fs4 = P5 = As5 fs5 = P6 = As6 fs6 = P8 = As7 fs7 = P8 = As8 fs8 = P9 = As9 fs9 = P10 = As10 fs10 =

10.2 -1044.40141 = 10.2 -2974.20357 = 10.2 -4200 = 10.2 -4200 = 0 0= 0 0= 0 0= 0 0= 0 0= 0 0=

-10652.89435 -30336.87646 -42840 -42840 0 0 0 0 0 0

P11 = As11 fs11 = P12 = As12 fs12 = P13 = As13 fs13 = P14 = As14 fs14 = P15 = As15 fs15 = P16 = As16 fs16 = P17 = As17 fs17 = P18 = As18 fs18 = P19 = As19 fs19 = P20 = As20 fs20 =

10.2 1044.40141 = 10.2 2974.20357 = 10.2 4200 = 10.2 4200 = 0 0= 0 0= 0 0= 0 0= 0 0= 0 0=

10652.89435 30336.87646 42840 42840 0 0 0 0 0 0 -1.45519E-11

CALCULO CARGA AXIAL NOMINAL Pn =

Cc + SUM Pi =

817339.104 Kg =

CALCULO DEL MOMENTO FLECTOR NOMINAL Mn =

817339.104 -10652.8944 -30336.8765 -42840 -42840 0 0 0

34.5 -10.24224191 -29.16743723 -43.65215465 -51.49122722 0 0 0

= = = = = = = =

28198199.1 109109.521 884848.9401 1870058.305 2205884.174 0 0 0

817.3391043

0 0 0 0 10652.8944 30336.8765 42840 42840 0 0 0 0 0 0

e=

0 0 0 0 10.24224191 29.16743723 43.65215465 51.49122722 0 0 0 0 0 0

= = = = = = = = = = = = = =

0.46905869

0 0 0 0 109109.521 884848.9401 1870058.305 2205884.174 0 0 0 0 0 0 38338000.98 Kg-cm 383.3800098 T-cm 3.833800098 T-m

PUNTO 5 ε1 εc = εc =

ε2

0.003

ε3

12.570072 7.5

105 14.78832 120

7.5

CALCULO DE DEFORMACIONES UNITARIAS ε1 =

0.003 -55.45392191 = 14.78832

-0.011249538

1

ε2 =

0.003 -74.37911723 = 14.78832

-0.015088756

2

ε3 =

0.003 -88.86383465 = 14.78832

-0.018027166

3

ε4 =

0.003 -96.70290722 = 14.78832

-0.019617423

4

ε5 =

0.003

-45.21168 = 14.78832

0

5

ε6 =

0.003

-45.21168 = 14.78832

0

6

ε7 =

0.003

-45.21168 = 14.78832

0

7

ε8 =

0.003

-45.21168 = 14.78832

0

8

ε9 =

0.003

-45.21168 = 14.78832

0

9

ε10 =

0.003

-45.21168 = 14.78832

0

10

ε1 =

0.003 -34.96943809 = 14.78832

-0.007093998

1

ε2 =

0.003 -16.04424277 = 14.78832

-0.00325478

2

ε3 =

0.003 -1.559525354 = 14.78832

-0.00031637

3

ε4 =

0.003 6.279547221 = 14.78832

0.001273887

4

ε5 =

0.003

-45.21168 = 14.78832

0

5

ε6 =

0.003

-45.21168 = 14.78832

0

6

ε7 =

0.003

-45.21168 = 14.78832

0

7

ε8 =

0.003

-45.21168 = 14.78832

0

8

ε9 =

0.003

-45.21168 = 14.78832

0

9

ε10 =

0.003

-45.21168 = 14.78832

0

10

CALCULO DE ESFUERZOS EN EL ACERO fs1 = fs2 = fs3 = fs4 = fs5 = fs6 = fs7 = fs8 = fs9 = fs10 =

-4200 -4200 -4200 -4200 0 0 0 0 0 0

Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2

fs1 = fs2 = fs3 = fs4 = fs5 = fs6 = fs7 = fs8 = fs9 = fs10 =

-4200 -4200 -645.204325 2597.96419 0 0 0 0 0 0

Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2 Kg/cm2

CALCULO DE LA FUERZA DE COMPRESION EN EL HORMIGON Cc =

,85 f'c Asc =

0.85

210

630.0777938

area = 12.570072

1742.946234 8936.709525 630.0777938 cm2

107.429928

CALCULO DE LA FUERZA DE COMPRESION DEL ACERO P1 = As1 fs1 = P2 = As2 fs2 = P3 = As3 fs3 = P4 = As4 fs4 =

10.2 10.2 10.2 10.2

-4200 -4200 -4200 -4200

= = = =

-42840 -42840 -42840 -42840

P5 = As5 fs5 = P6 = As6 fs6 = P8 = As7 fs7 = P8 = As8 fs8 = P9 = As9 fs9 = P10 = As10 fs10 =

0 0 0 0 0 0

P11 = As11 fs11 = P12 = As12 fs12 = P13 = As13 fs13 = P14 = As14 fs14 = P15 = As15 fs15 = P16 = As16 fs16 = P17 = As17 fs17 = P18 = As18 fs18 = P19 = As19 fs19 = P20 = As20 fs20 =

0 0 0 0 0 0

= = = = = =

0 0 0 0 0 0

10.2 -4200 = 10.2 -4200 = 10.2 -645.204325 = 10.2 2597.96419 = 0 0= 0 0= 0 0= 0 0= 0 0= 0 0=

-42840 -42840 -6581.084114 26499.23475 0 0 0 0 0 0 -237121.8494

CALCULO CARGA AXIAL NOMINAL Pn =

Cc + SUM Pi =

-124652.963 Kg =

-124.6529632

CALCULO DEL MOMENTO FLECTOR NOMINAL Mn =

112468.886 -42840 -42840 -42840 -42840 0 0 0 0 0 0 0 -42840 -42840 -6581.08411 26499.2348 0 0 0 0 0 0

53.714964 -10.24224191 -29.16743723 -43.65215465 -51.49122722 0 0 0 0 0 0 0 10.24224191 29.16743723 43.65215465 51.49122722 0 0 0 0 0 0

= = = = = = = = = = = = = = = = = = = = = =

6041262.173 438777.6432 1249533.011 1870058.305 2205884.174 0 0 0 0 0 0 0 -438777.6432 -1249533.011 -287278.5015 1364478.118 0 0 0 0 0 0 11194404.27 Kg-cm

e=

111.9440427 T-cm 1.119440427 T-m

-0.89804558

Mu punt 1 punt 2 punt 3 punt 4 punt 5

0 208.280402 361.251622 383.38001 111.944043

Pu 2361.507439 2048.786516 1502.836318 817.3391043 -124.6529632

.9 Mu 0 187.452362 325.126459 345.042009 100.749638

.9 Pu .7 Mu 2125.356695 0 1843.907864 145.7962813 1352.552687 252.8761351 735.6051939 268.3660068 -112.1876668 78.36082988

La carga axial que define una variación en el factor de reducción de capacidad es: .1 f'c Ag =

0.1

210

4800 =

POR NO SER ZUNCHADA e = 10% h e=

0.1

DE LA ECUACION =

120 = e = Mu / Pu =

12 cms 12 cms

CARGA AXIAL QUE DEFINE REDUCCION DE CARGA 0.1

210

4800 =

100800 Kg 100.8 T

Nro de varillas = Nro varillas al lado = Nro espacios al lado = Radio de distribucion = Angulo interno = Long 1 2 3 4 5 6 7 8 9 10

=

Kg

ang acum dist acum 11.25 33.75 56.25 78.75 0 0 0 0 0 0

10.24224191 29.16743723 43.65215465 51.49122722 0 0 0 0 0 0

2018787.439 Kg

16 8 4 52.5 22.5 Nro var int 2 2 2 2 0 0 0 0 0 0

Ton

=

1828899.884 Kg

Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg

Ton

y

105

=

Kg Kg Kg Kg Kg

911277.5287 Kg

Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg

Ton

=

Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg

1358317.079 Kg

Kg Kg

Ton

=

817339.1043 Kg

Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg

Ton

=

Kg Kg Kg Kg

112468.8862 Kg

Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg Kg

Ton

.7 Pu 1653.05521 1434.15056 1051.98542 572.137373 -87.2570742

e = Mu / Pu = 0 113.3523571 170.0285356 226.7047142 283.3808927

12 0 944.6029757 1416.904464 1889.205951 2361.507439

M 0 153.352004 230.028006 306.704008 383.38001

Max P=.8 P 1322.44417 1322.44417 1322.44417 1322.44417 1322.44417

Min P 0 153.352004 230.028006 306.704008 383.38001

100.8 100.8 100.8 100.8 100.8

DIAGRAMA DE INTERACCION

100800 Kg 100.8 T

2500 2000

Pnominal

1500

P nominal, M nominal 90% Pn Mn

1000

70% Pn Mn MAX P

500

MIN P 0 0 -500

100

200

300

Mnominal

400

500

P nominal, M nominal 90% Pn Mn 70% Pn Mn MAX P MIN P