Momento Curvatura

DIAGRAMA MOMENTO CURVATURA SECCIONES SIMPLEMENTE REFORZADAS REFORZADAS SECCION: Peralte F'c= Fy= Es= b=

210.00 4200.00 2000 200000 000 0.0 .00 0 30.00

K3=

0.967972

c 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010 0.0011 0.0012 0.0013 0.0014 0.0015 0.0016 0.0017 0.0018 0.0019 0.0020 0.0021 0.0022 0.0023 0.0024 0.0025 0.0026 0.0027 0.0028 0.0029 0.0030 0.0031 0.0032 0.0033 0.0034 0.0035 0.0036 0.0037 0.0038 0.0039 0.0040

ε

K1 0.049167 0.096670 0.142500 0.186667 0. 0.229200 0. 0.270000 0. 0.309200 0. 0.346667 0. 0.382500 0. 0.416667 0. 0.449200 0. 0.480000 0. 0.509167 0. 0.536667 0. 0.562500 0. 0.586667 0. 0.609967 0. 0.630000 0. 0.649167 0. 0.666667 0. 0.682340 0. 0.696212 0. 0.708500 0. 0.719444 0. 0.729200 0. 0.737782 0. 0.745600 0. 0.752381 0. 0.758700 0. 0.763889 0. 0.768500 0. 0.772917 0. 0.776600 0. 0.779902 0. 0.782600 0. 0.785185 0. 0.787400 0. 0.789035 0. 0.790900 0. 0.791667 0.

kg/cm2 kg/cm2 kg/c kg/cm2 m2 cm

d1=

K2 0.334745 0.336207 0.337700 0.339286 0. 0.340900 0. 0.342593 0. 0.344300 0. 0.346154 0. 0.348000 0. 0.350000 0. 0.352000 0. 0.354167 0. 0.356383 0. 0.358696 0. 0.361110 0. 0.363636 0. 0.366280 0. 0.369048 0. 0.371951 0. 0.375000 0. 0.378290 0. 0.381805 0. 0.385400 0. 0.389103 0. 0.392800 0. 0.396378 0. 0.400000 0. 0.403406 0. 0.406700 0. 0.410100 0. 0.413300 0. 0.416442 0. 0.419500 0. 0.422437 0. 0.425300 0. 0.428110 0. 0.430800 0. 0.433486 0. 0.436100 0. 0.438596 0.

55.00 cm

C 20.45 20.59 20.72 20.87 20 21.01 21 21.16 21 21.31 21 21.46 21 21.62 21 21.78 21 21.95 21 22.12 22 22.29 22 22.47 22 22.66 22 22.85 22 23.03 23 23.24 23 23.45 23 23.66 23 23.88 23 24.11 24 24.34 24 24.58 24 24.82 24 25.06 25 25.30 25 25.54 25 25.78 25 26.03 26 26.27 26 26.50 26 26.74 26 26.97 26 27.20 27 27.43 27 27.66 27 27.88 27 28.10 28 28.32 28

Fs1 338 669 992 1309 1618 1919 2214 2500 2779 3050 3313 3567 3814 4052 4200 4200 4200 4200 4200 4200 4200 4200 4200 4200 4200 4200 4200 4200 4200 4200 4200 4200 4200 4200 4200 4200 4200 4200 4200 4200

As1=

C 20.45 20.59 20.72 20.87 21.01 21.16 21.31 21.46 21.62 21.78 21.95 22.12 22.29 22.47 22.22 21.31 20.49 19.84 19.26 18.75 18.32 17.95 17.64 17.38 17.14 16.94 16.77 16.61 16.48 16.36 16.27 16.17 16.10 16.03 15.97 15.92 15.88 15.84 15.81 15.79

Area acero 18.15 cm2

M (ton-m) 2.95 5.83 8.64 11.38 14.05 16.64 19.15 21.58 23.94 26.22 28.42 30.54 32.57 34.52 35.81 36.02 36.20 36.34 36.47 36.57 36.64 36.70 36.74 36.77 36.79 36.81 36.81 36.82 36.82 36.81 36.80 36.79 36.78 36.77 36.75 36.73 36.71 36.69 36.67 36.65

DIAGRAMA MOMENTO-CURVATURA

40.00 35.00 30.00 25.00

   O    T    N    E 20.00    M    O    M 15.00

10.00 5.00 0.00 0.0

5.0

10.0

15.0

CURVATURAx10-5

20.0

25.0

30.0

fx10-5

0.5 1.0 1.4 1.9 2.4 2.8 3.3 3.7 4.2 4.6 5.0 5.4 5.8 6.2 6.7 7.5 8.3 9.1 9.9 10.7 11.5 12.3 13.0 13.8 14.6 15.3 16.1 16.9 17.6 18.3 19.1 19.8 20.5 21.2 21.9 22.6 23.3 24.0 24.7 25.3

DIAGRAMA MOMENTO CURVATURA EJECUTA: CTRL-a DATOS: P= 0.00 F'c= 210.00 b= 30.00 Fy= 4200 Es= 2,000,000 H= 50.00 Gt= 26.71 K3= 0.967972 c 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010 0.0011 0.0012 0.0013 0.0014 0.0015 0.0016 0.0017 0.0018 0.0019 0.0020 0.0021 0.0022 0.0023 0.0024 0.0025 0.0026 0.0027 0.0028 0.0029 0.0030 0.0031 0.0032 0.0033 0.0034 0.0035 0.0036 0.0037 0.0038 0.0039 0.0040

ε

K1 0.049167 0.096670 0.142500 0.186667 0.229200 0.270000 0.309200 0.346667 0.382500 0.416667 0.449200 0.480000 0.509167 0.536667 0.562500 0.586667 0.609967 0.630000 0.649167 0.666667 0.682340 0.696212 0.708500 0.719444 0.729200 0.737782 0.745600 0.752381 0.758700 0.763889 0.768500 0.772917 0.776600 0.779902 0.782600 0.785185 0.787400 0.789035 0.790900 0.791667

d1 kg kg/cm2 cm kg/cm2 kg/cm2 cm cm

K2 0.334745 0.336207 0.337700 0.339286 0.340900 0.342593 0.344300 0.346154 0.348000 0.350000 0.352000 0.354167 0.356383 0.358696 0.361110 0.363636 0.366280 0.369048 0.371951 0.375000 0.378290 0.381805 0.385400 0.389103 0.392800 0.396378 0.400000 0.403406 0.406700 0.410100 0.413300 0.416442 0.419500 0.422437 0.425300 0.428110 0.430800 0.433486 0.436100 0.438596

d1= d2= Po= Pb= Mb= To= C 15.31 15.40 15.49 15.58 15.68 15.77 15.87 15.97 16.08 16.18 16.29 16.40 15.34 14.40 13.59 12.91 12.31 11.81 11.37 10.99 10.65 10.36 10.10 9.88 9.67 9.49 9.32 9.17 9.03 8.91 8.79 8.68 8.58 8.49 8.40 8.32 8.24 8.17 8.10 8.03

Peralte Area acero 5.00 cm As1= 6.97 cm2 45.00 cm As2= 14.25 cm2 21.22 353.09 ton 89.91 ton 35.93 ton-m AM= -89.12 ton AP= Fs1 135 270 406 543 681 820 959 1099 1240 1382 1525 1668 1753 1827 1897 1961 2019 2076 2129 2179 2228 2276 2324 2370 2415 2460 2504 2548 2590 2633 2675 2715 2755 2795 2834 2872 2910 2948 2984 3022

Fs2 EQUILIBRIO -388 0.000 -769 0.000 -1143 0.000 -1510 0.000 -1871 0.000 -2223 0.000 -2569 0.000 -2907 0.000 -3238 0.000 -3562 0.000 -3877 0.000 -4185 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000 -4200 0.000

DIAGRAMA MOMENTO-CURVATURA 30.00 25.00 20.00    O    T    N    E 15.00    M    O    M

10.00 5.00 0.00 0.00

10.00

20.00

30.00

40.00

CURVATURA X10-5

50.00

60.00

As1

Ec= I= fcr= Mcr= f crx10-5=

d2 As2

1.52 0.25

M (ton-m) 2.21 4.37 6.49 8.56 10.59 12.57 14.51 16.39 18.23 20.03 21.77 23.47 23.72 23.86 23.98 24.08 24.17 24.23 24.29 24.33 24.37 24.40 24.42 24.44 24.46 24.47 24.48 24.49 24.49 24.50 24.50 24.51 24.51 24.51 24.51 24.51 24.51 24.51 24.51 24.51

My= Mu= u= f x10-5 0.65 1.30 1.94 2.57 3.19 3.80 4.41 5.01 5.60 6.18 6.75 7.32 8.47 9.73 11.03 12.39 13.81 15.24 16.71 18.21 19.72 21.24 22.76 24.30 25.85 27.40 28.96 30.52 32.10 33.67 35.25 36.85 38.45 40.05 41.66 43.28 44.90 46.52 48.16 49.78

23.72 24.51 5.87

220000 kg/cm2 312500 cm4 23.19 kg/cm2 2.90 ton-m 0.42

DIAGRAMA MOMENTO CURVATURA EJECUTA: CTRL-b DATOS: P= 0.00 kg Peralte Area acero F'c= 210.00 kg/cm2 d1= 5.00 cm As1= 11.40 b= 45.00 cm d2= 30.00 cm As2= 3.96 Fy= 4200.00 kg/cm2 d3= 55.00 cm As3= 11.40 Es= 2,000,000 kg/cm2 Po= 589.57 ton 26.76 H= 60.00 cm Pb= 222.62 ton 0.99% Gt= 30.00 cm Mb= 59.84 ton-m AM= K3= 0.967972 To= -112.39 ton AP= c 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010 0.0011 0.0012 0.0013 0.0014 0.0015 0.0016 0.0017 0.0018 0.0019 0.0020 0.0021 0.0022 0.0023 0.0024 0.0025 0.0026 0.0027 0.0028 0.0029 0.0030 0.0031 0.0032 0.0033 0.0034 0.0035 0.0036 0.0037 0.0038 0.0039 0.0040

ε

K1 0.049167 0.096670 0.142500 0.186667 0.229200 0.270000 0.309200 0.346667 0.382500 0.416667 0.449200 0.480000 0.509167 0.536667 0.562500 0.586667 0.609967 0.630000 0.649167 0.666667 0.682340 0.696212 0.708500 0.719444 0.729200 0.737782 0.745600 0.752381 0.758700 0.763889 0.768500 0.772917 0.776600 0.779902 0.782600 0.785185 0.787400 0.789035 0.790900 0.791667

K2 0.334745 0.336207 0.337700 0.339286 0.340900 0.342593 0.344300 0.346154 0.348000 0.350000 0.352000 0.354167 0.356383 0.358696 0.361110 0.363636 0.366280 0.369048 0.371951 0.375000 0.378290 0.381805 0.385400 0.389103 0.392800 0.396378 0.400000 0.403406 0.406700 0.410100 0.413300 0.416442 0.419500 0.422437 0.425300 0.428110 0.430800 0.433486 0.436100 0.438596

C 13.86 13.94 14.03 14.11 14.20 14.29 14.38 13.78 12.83 12.12 11.57 11.14 10.51 9.92 9.42 9.00 8.64 8.34 8.08 7.86 7.67 7.50 7.36 7.23 7.12 7.03 6.94 6.86 6.79 6.72 6.67 6.61 6.56 6.52 6.47 6.43 6.40 6.36 6.33 6.30

Fs1 128 257 386 517 648 780 913 1020 1099 1175 1249 1323 1364 1388 1408 1422 1432 1442 1449 1455 1461 1468 1475 1483 1491 1500 1508 1518 1527 1538 1549 1559 1570 1582 1594 1605 1617 1629 1641 1655

Fs2 -233 -461 -683 -901 -1113 -1319 -1520 -1882 -2409 -2952 -3506 -4065 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200

d1 cm2 cm2 cm2 cm2

Fs3 -594 -1178 -1753 -2318 -2873 -3418 -3954 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200 -4200

DIAGRAMA MOMENTO-CURVATURA 35.00 30.00 25.00    O    T 20.00    N    E    M 15.00    O    M

10.00 5.00 0.00 0.00

10.00

20.00

30.00

40.00

CURVATURA X10-5

50.00

60.00

70.00

As1

d3

As2 As3

2.31 0.38 EQUILIBRIO 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

My= Mu= u= M (ton-m) 3.64 7.21 10.71 14.15 17.53 20.83 24.06 25.90 26.56 27.21 27.85 28.48 28.71 28.80 28.87 28.93 28.98 29.02 29.05 29.08 29.10 29.12 29.13 29.14 29.14 29.15 29.15 29.15 29.15 29.15 29.15 29.15 29.15 29.14 29.14 29.14 29.13 29.13 29.12 29.12

25.90 29.12 10.93 f x10-5 0.72 1.43 2.14 2.83 3.52 4.20 4.87 5.80 7.01 8.25 9.51 10.77 12.36 14.12 15.92 17.78 19.68 21.58 23.51 25.45 27.39 29.32 31.25 33.17 35.09 37.00 38.92 40.82 42.73 44.62 46.51 48.41 50.30 52.18 54.06 55.95 57.83 59.71 61.59 63.45

CURVA ESFUERZO - DEFORMACION DEL CONCRETO Y DEL ACERO SECCION: f'c= K3= f"c= eo= Z=

210.00 kg/cm2 0.967972 203.27 kg/cm2 0.0020 0.0018 75.00

ec 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010 0.0011 0.0012 0.0013 0.0014 0.0015 0.0016 0.0017 0.0018 0.0019 0.0020 0.0021 0.0022 0.0023 0.0024 0.0025 0.0026 0.0027 0.0028 0.0029 0.0030 0.0031 0.0032 0.0033 0.0034 0.0035 0.0036 0.0037 0.0038 0.0039 0.0040

K1 0.000000 0.049167 0.096670 0.142500 0.186667 0.229200 0.270000 0.309200 0.346667 0.382500 0.416667 0.449200 0.480000 0.509167 0.536667 0.562500 0.586667 0.609967 0.630000 0.649167 0.666667 0.682340 0.696212 0.708500 0.719444 0.729200 0.737782 0.745600 0.752381 0.758700 0.763889 0.768500 0.772917 0.776600 0.779902 0.782600 0.785185 0.787400 0.789035 0.790900 0.791667

  f  C 

Fy= Es= ey=

K2

K1 0.000000 0.049167 0.096667 0.142500 0.186667 0.229167 0.270000 0.309167 0.346667 0.382500 0.416667 0.449167 0.480000 0.509167 0.536667 0.562500 0.586667 0.609167 0.630000 0.649167 0.666667

0.334745 0.336207 0.337700 0.339286 0.340900 0.342593 0.344300 0.346154 0.348000 0.350000 0.352000 0.354167 0.356383 0.358696 0.361110 0.363636 0.366280 0.369048 0.371951 0.375000 0.378290 0.381805 0.385400 0.389103 0.392800 0.396378 0.400000 0.403406 0.406700 0.410100 0.413300 0.416442 0.419500 0.422437 0.425300 0.428110 0.430800 0.433486 0.436100 0.438596

é 2e   æ e    ö = f  C  ê C  - çç C  ÷÷ êë e  O è e  O  ø "

2

4200.00 kg/cm2 2000000.00 kg/cm2 0.0021

K2 0.333333 0.334746 0.336207 0.337719 0.339286 0.340909 0.342593 0.344340 0.346154 0.348039 0.350000 0.352041 0.354167 0.356383 0.358696 0.361111 0.363636 0.366279 0.369048 0.371951 0.375000

ù ú úû

fc 0.00 19.82 38.62 56.41 73.18 88.93 103.67 117.39 130.10 141.78 152.46 162.11 170.75 178.37 184.98 190.57 195.14 198.70 201.24 202.77 203.27 201.24 199.21 197.18 195.14 193.11 191.08 189.04 187.01 184.98 182.95 180.91 178.88 176.85 174.82 172.78 170.75 168.72 166.68 164.65 162.62

es 0.0000 0.0011 0.0021 0.0032 0.0042 0.0053 0.0063 0.0074 0.0084 0.0095 0.0105 0.0116 0.0126 0.0137 0.0147 0.0158 0.0168 0.0179 0.0189 0.0200 0.0210 0.0221 0.0231 0.0242 0.0252 0.0263 0.0273 0.0284 0.0294 0.0305 0.0315 0.0326 0.0336 0.0347 0.0357 0.0368 0.0378 0.0389 0.0399 0.0410 0.0420

fs 0.0 2100.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0 4200.0

fyC

  f  C 

=

f  C  [1 - 100(e  C  - e  O )] "

B

  f  C "

0.00175

  c    f

   O    Z    R    E    U    F    S    E

C

  s    f    O    Z    R    E    U    F    S    E

esy

 A 0.000

= 0.002

0.004

εo

DEFORMACION

C

0.0018 Laboratorio fy esy Es

4798.0 0.020 239,899

0.0021

DEFORMACION

S

BARRA DE CONSTRUCCION DIMENSIONES Y PESOS ACEROS AREQUIPA DIAMETRO AREA PESO cm2 kg/m

No

LISO LISO

Ø 6 mm Ø 1/4" Ø 8 mm Ø 3/8" Ø 12 mm Ø 1/2" Ø 5/8" Ø 3/4" Ø 1" Ø 1 3/8"

3 4 5 6 8 11

0.28 0.32 0.50 0.71 1.13 1.29 2.00 2.84 5.10 10.06

0.22 0.25 0.40 0.56 0.89 0.99 1.55 2.24 3.97 7.91

SIDERPERU AREA PESO cm2 kg/m 0.28 0.32 0.50 0.71 1.13 1.27 1.98 2.85 5.07 9.58

EQUIVALENCIAS ENTRE UNIDADES 1 Lb/plg2= 0.00689 N/mm2 1 Lb/plg2= 0.07031 kg/cm2 1 N/mm2= 10.2041 kg/cm2 1 MPa= 1 N/mm2 MATERIAL MKS S.I. INGLES kg/cm2 N/mm2 Kips/plg2 f'c= 210 20.6 3 Ec= 200,000 19,600 2,845 fy= 4,200 412 60 Es= 2,000,000 196,000 28,445

f'c= u=

210 kg/cm2 0.20

Ec

= 15000

f  ' c

=

tomar: Gc

Peso = Masa =

=

Ec

(

2 1+

m)

=

217,371 kg/cm2 220,000 kg/cm2 91,667 kg/cm2

2.400 ton/m3 0.24465 ton-seg2/m

3,000 30,000

0.22 0.25 0.39 0.56 0.89 0.99 1.55 2.24 3.98 7.91

AREA pulg2

0.05 0.11 0.20 0.31 0.44 0.79 1.48

SELECCI N DEL REFUERZO  Acero Nega= 7.92 1.56 1" 3/4" 2 5/8" 1 1/2"  Acero Posit=

3.72 1" 3/4" 5/8" 1/2"

0.00 5.70 1.98 0.00 7.68

0.73

2

0.00 0.00 3.96 0.00 3.96

11.64