Dic Rectangular Jm

CURVA DE CAPACIDAD:DIAGRAMA DE INTERACCIÓN DE COLUMNA RECTANGULAR CON REFUERZO SIMÉTRICO Se está tomando la fibra superior como comprimida y la fibra inferior como traccionada con esto calcularemos el diagrama positivo, para encontrar el diagrama negativo, se debe tomar la fibrainferior comprimida y la superior traccionada

DIAGRAMA PARA MOMENTO M3 SECCION DE COLUMNA b = 40.00 cm h = 50.00 cm

6.153 cm

MATERIALES fy = Es = f'c = εc =

9.424 cm

2ø

1

'' +

1ø

4200 kg/cm2 2100000 kg/cm2 210 kg/cm2 0.003

2ø

3/4'' +

0ø

0

'' = 5.70 cm2

= As2

2ø

3/4'' +

0ø

0

'' = 5.70 cm2

= As3

2ø

3/4'' +

0ø

0

'' = 5.70 cm2

= As4

'' +

1ø

3/4'' = 12.99 cm2 Ast = 43.08 cm2

9.424 cm 1

OTRAS CONSIDERACIONES 6.153 cm rec. = 4.00 cm estribo = 3/8'' β = 0.85 (β = 0.85 para f'c ≤ 280 kg/cm2, disminuye 0.05 cada 70 kg/cm2, y no debe ser menor que 0.65)

CP

=

= As1

9.424 cm

2ø

CENTROIDE PLÁSTICO DE LA SECCIÓN

3/4'' = 12.99 cm2

9.424 cm

:

= As5

Se tomará con respecto a la parte inferior (fibra mas traccionada)

25.000 cm

a) PUNTO DE CARGA MAXIMA: COMPRESION PURA

: CONTROLA LA COMPRESION

𝑃. 𝑚𝑎𝑥 = 0.80 ∗ (0.85 ∗ 𝑓´𝑐 ∗ 𝐴𝑔 − 𝐴𝑠 + As ∗ fy) P.max

=

424.197 Tn

M.max =

b) PUNTO DE CARGA MAXIMA: TRACCION PURA

𝑇. 𝑚𝑎𝑥 = 𝐴𝑠 ∗ 𝑓𝑦

T.max

=

180.936 Tn

M.max =

0.000 Tn.m (no hay excentricidad por ser un refuerzo simétrico, el centroide plástico coincide con la resultante de compresiones) : CONTROLA LA COMPRESION

0.000 Tn.m (no hay excentricidad por ser un refuerzo simétrico, el centroide plástico coincide con la resultante de tracciones)

JUAN MANUEL CHERO DAMIAN

c) PUNTO DE CARGA BALANCEADA d = ab = Cb =

43.847 cm 22.362 cm 26.308 cm

: EL CONCRETO Y EL ACERO LLEGAN A SU MAXIMA CAPACIDAD AL MISMO TIEMPO

𝑎𝑏 =

𝜀𝑐 𝑓𝑦 𝜀𝑐 + 𝐸

∗𝛽∗𝑑

𝐶𝑏 =

𝑎𝑏 𝛽

𝑠

εc

0.85*f'c C's1 Cc =0.85*f'c*a*b

εs1 𝑪𝒃

𝒂𝒃 εs2

d

𝑯 − 𝑪𝑷

C's2

εs3

C's3

=

CENTROIDE PLÁSTICO EJE NEUTRO

T4

εs4

T5

εs5 SECCION

DIAG. DEFORMACIONES

capa As dist. E.N. 1 12.99 cm2 20.155 cm 2 5.70 cm2 10.732 cm 3 5.70 cm2 1.308 cm 4 5.70 cm2 -8.115 cm 5 12.99 cm2 -17.539 cm CONCRETO Cc = 0.85*f'c*a*b

εs 0.002298376 0.001223782 0.000149188 -0.00092541 -0.002

𝑪𝑷=

DIAG. FUERZAS

fs 4826.590 kg/cm2 2569.943 kg/cm2 313.295 kg/cm2 -1943.352 kg/cm2 -4200.000 kg/cm2 Pb

F 54.558 Tn 14.649 Tn 1.786 Tn -11.077 Tn -54.558 Tn 159.665 Tn 165.022 Tn

dist. C.P. 18.847 cm 9.424 cm 0.000 cm -9.424 cm -18.847 cm 13.819 cm Mb

JUAN MANUEL CHERO DAMIAN

M 10.283 Tn.m 1.380 Tn.m 0.000 Tn.m 1.044 Tn.m 10.283 Tn.m 22.064 Tn.m 45.054 Tn.m

25.000 cm

25.000 cm

d) CONSIDERANDO LA PROFUNDIDAD DEL EJE NEUTRO IGUAL A UN PORCENTAJE DEL PERALTE EFECTIVO d = a = C =

C =

0.10d

43.847 cm 3.727 cm 4.385 cm

εc

0.85*f'c

𝑪

𝒂 εs1

T1

T3

εs3

DIAG. DEFORMACIONES εs -0.00120974 -0.00765731 -0.01410487 -0.02055244 -0.027

𝑪𝑷=

DIAG. FUERZAS

fs -2540.458 kg/cm2 -16080.343 kg/cm2 -29620.229 kg/cm2 -43160.114 kg/cm2 -56700.000 kg/cm2 P

F -33.001 Tn -23.940 Tn -23.940 Tn -23.940 Tn -54.558 Tn 26.611 Tn -132.768 Tn

=

CENTROIDE PLÁSTICO

T5

εs5

capa As dist. E.N. 1 12.99 cm2 -1.768 cm 2 5.70 cm2 -11.192 cm 3 5.70 cm2 -20.615 cm 4 5.70 cm2 -30.039 cm 5 12.99 cm2 -39.462 cm CONCRETO Cc = 0.85*f'c*a*b

𝑯 − 𝑪𝑷

T4

εs4

SECCION

EJE NEUTRO

T2

εs2 d

Cc =0.85*f'c*a*b

dist. C.P. 18.847 cm 9.424 cm 0.000 cm -9.424 cm -18.847 cm 23.136 cm M

JUAN MANUEL CHERO DAMIAN

M -6.220 Tn.m -2.256 Tn.m 0.000 Tn.m 2.256 Tn.m 10.283 Tn.m 6.157 Tn.m 10.220 Tn.m

25.000 cm

25.000 cm

e) CONSIDERANDO LA PROFUNDIDAD DEL EJE NEUTRO IGUAL A UN PORCENTAJE DEL PERALTE EFECTIVO d = a = C =

C =

0.20d

43.847 cm 7.454 cm 8.769 cm

εc 𝑪

0.85*f'c 𝒂

εs1

T3

εs3

DIAG. DEFORMACIONES

capa As dist. E.N. 1 12.99 cm2 2.617 cm 2 5.70 cm2 -6.807 cm 3 5.70 cm2 -16.231 cm 4 5.70 cm2 -25.654 cm 5 12.99 cm2 -35.078 cm CONCRETO Cc = 0.85*f'c*a*b

εs 0.000895129 -0.00232865 -0.00555244 -0.00877622 -0.012

𝑪𝑷=

T5

εs5

DIAG. FUERZAS

fs 1879.771 kg/cm2 -4890.172 kg/cm2 -11660.114 kg/cm2 -18430.057 kg/cm2 -25200.000 kg/cm2 P

F 24.418 Tn -23.940 Tn -23.940 Tn -23.940 Tn -54.558 Tn 53.222 Tn -48.738 Tn

=

CENTROIDE PLÁSTICO

T4

εs4

SECCION

𝑯 − 𝑪𝑷

T2

εs2 d

Cc =0.85*f'c*a*b C's1 EJE NEUTRO

dist. C.P. 18.847 cm 9.424 cm 0.000 cm -9.424 cm -18.847 cm 21.273 cm M

JUAN MANUEL CHERO DAMIAN

M 4.602 Tn.m -2.256 Tn.m 0.000 Tn.m 2.256 Tn.m 10.283 Tn.m 11.322 Tn.m 26.207 Tn.m

25.000 cm

25.000 cm

f) CONSIDERANDO LA PROFUNDIDAD DEL EJE NEUTRO IGUAL A UN PORCENTAJE DEL PERALTE EFECTIVO d = a = C =

C =

0.30d

43.847 cm 11.181 cm 13.154 cm

εc 𝑪

0.85*f'c

T3

εs3

DIAG. DEFORMACIONES εs 0.001596753 -0.00055244 -0.00270162 -0.00485081 -0.007

𝑪𝑷=

DIAG. FUERZAS

fs 3353.181 kg/cm2 -1160.114 kg/cm2 -5673.410 kg/cm2 -10186.705 kg/cm2 -14700.000 kg/cm2 P

F 43.558 Tn -6.613 Tn -23.940 Tn -23.940 Tn -54.558 Tn 79.833 Tn 14.340 Tn

=

CENTROIDE PLÁSTICO

T5

εs5

capa As dist. E.N. 1 12.99 cm2 7.001 cm 2 5.70 cm2 -2.422 cm 3 5.70 cm2 -11.846 cm 4 5.70 cm2 -21.269 cm 5 12.99 cm2 -30.693 cm CONCRETO Cc = 0.85*f'c*a*b

𝑯 − 𝑪𝑷 EJE NEUTRO

T4

εs4

SECCION

C's1 T2

εs2 d

Cc =0.85*f'c*a*b

𝒂

εs1

dist. C.P. 18.847 cm 9.424 cm 0.000 cm -9.424 cm -18.847 cm 19.409 cm M

JUAN MANUEL CHERO DAMIAN

M 8.209 Tn.m -0.623 Tn.m 0.000 Tn.m 2.256 Tn.m 10.283 Tn.m 15.495 Tn.m 35.620 Tn.m

25.000 cm

25.000 cm

g) CONSIDERANDO LA PROFUNDIDAD DEL EJE NEUTRO IGUAL A UN PORCENTAJE DEL PERALTE EFECTIVO d = a = C =

C =

0.40d

43.847 cm 14.908 cm 17.539 cm

εc

0.85*f'c Cc =0.85*f'c*a*b

𝑪

εs1

C's1

𝒂

εs2 d

T3

εs3

T4

εs4

DIAG. DEFORMACIONES

capa As dist. E.N. 1 12.99 cm2 11.386 cm 2 5.70 cm2 1.962 cm 3 5.70 cm2 -7.461 cm 4 5.70 cm2 -16.885 cm 5 12.99 cm2 -26.308 cm CONCRETO Cc = 0.85*f'c*a*b

εs 0.001947565 0.000335673 -0.00127622 -0.00288811 -0.0045

𝑪𝑷=

T5

εs5 SECCION

C's2 EJE NEUTRO CENTROIDE PLÁSTICO

DIAG. FUERZAS

fs 4089.886 kg/cm2 704.914 kg/cm2 -2680.057 kg/cm2 -6065.029 kg/cm2 -9450.000 kg/cm2 P

F 53.128 Tn 4.018 Tn -15.276 Tn -23.940 Tn -54.558 Tn 106.443 Tn 69.815 Tn

dist. C.P. 18.847 cm 9.424 cm 0.000 cm -9.424 cm -18.847 cm 17.546 cm M

JUAN MANUEL CHERO DAMIAN

M 10.013 Tn.m 0.379 Tn.m 0.000 Tn.m 2.256 Tn.m 10.283 Tn.m 18.677 Tn.m 41.607 Tn.m

𝑯 − 𝑪𝑷

=

25.000 cm

25.000 cm

h) CONSIDERANDO LA PROFUNDIDAD DEL EJE NEUTRO IGUAL A UN PORCENTAJE DEL PERALTE EFECTIVO d = a = C =

C =

0.50d

43.847 cm 18.635 cm 21.924 cm

εc

0.85*f'c Cc =0.85*f'c*a*b

𝑪

εs1

C's1

𝒂

d

T3

εs3

𝑪𝑷=

T5

DIAG. DEFORMACIONES εs 0.002158052 0.000868539 -0.00042097 -0.00171049 -0.003

EJE NEUTRO

T4

εs5

capa As dist. E.N. 1 12.99 cm2 15.771 cm 2 5.70 cm2 6.347 cm 3 5.70 cm2 -3.076 cm 4 5.70 cm2 -12.500 cm 5 12.99 cm2 -21.924 cm CONCRETO Cc = 0.85*f'c*a*b

=

CENTROIDE PLÁSTICO

εs4

SECCION

𝑯 − 𝑪𝑷

C's2

εs2

DIAG. FUERZAS

fs 4531.908 kg/cm2 1823.931 kg/cm2 -884.046 kg/cm2 -3592.023 kg/cm2 -6300.000 kg/cm2 P

F 54.558 Tn 10.396 Tn -5.039 Tn -20.475 Tn -54.558 Tn 133.054 Tn 117.937 Tn

dist. C.P. 18.847 cm 9.424 cm 0.000 cm -9.424 cm -18.847 cm 15.682 cm M

JUAN MANUEL CHERO DAMIAN

M 10.283 Tn.m 0.980 Tn.m 0.000 Tn.m 1.929 Tn.m 10.283 Tn.m 20.866 Tn.m 44.341 Tn.m

25.000 cm

25.000 cm

i) CONSIDERANDO LA PROFUNDIDAD DEL EJE NEUTRO IGUAL A UN PORCENTAJE DEL PERALTE EFECTIVO d = a = C =

C =

0.60d

43.847 cm 22.362 cm 26.308 cm

εc

0.85*f'c C's1 Cc =0.85*f'c*a*b

εs1 𝑪

𝒂 εs2

d

𝑯 − 𝑪𝑷

C's2

εs3

C's3

=

CENTROIDE PLÁSTICO EJE NEUTRO

T4

εs4

T5

εs5 SECCION

DIAG. DEFORMACIONES

capa As dist. E.N. 1 12.99 cm2 20.155 cm 2 5.70 cm2 10.732 cm 3 5.70 cm2 1.308 cm 4 5.70 cm2 -8.115 cm 5 12.99 cm2 -17.539 cm CONCRETO Cc = 0.85*f'c*a*b

εs 0.002298376 0.001223782 0.000149188 -0.00092541 -0.002

𝑪𝑷=

DIAG. FUERZAS

fs 4826.590 kg/cm2 2569.943 kg/cm2 313.295 kg/cm2 -1943.352 kg/cm2 -4200.000 kg/cm2 P

F 54.558 Tn 14.649 Tn 1.786 Tn -11.077 Tn -54.558 Tn 159.665 Tn 165.022 Tn

dist. C.P. 18.847 cm 9.424 cm 0.000 cm -9.424 cm -18.847 cm 13.819 cm M

JUAN MANUEL CHERO DAMIAN

M 10.283 Tn.m 1.380 Tn.m 0.000 Tn.m 1.044 Tn.m 10.283 Tn.m 22.064 Tn.m 45.054 Tn.m

25.000 cm

25.000 cm

j) CONSIDERANDO LA PROFUNDIDAD DEL EJE NEUTRO IGUAL A UN PORCENTAJE DEL PERALTE EFECTIVO d = a = C =

C =

0.70d

43.847 cm 26.089 cm 30.693 cm

εc

0.85*f'c C's1 Cc =0.85*f'c*a*b

εs1 𝒂

𝑪 εs2 d

εs3

C's3 T4

εs4

DIAG. DEFORMACIONES

capa As dist. E.N. 1 12.99 cm2 24.540 cm 2 5.70 cm2 15.117 cm 3 5.70 cm2 5.693 cm 4 5.70 cm2 -3.731 cm 5 12.99 cm2 -13.154 cm CONCRETO Cc = 0.85*f'c*a*b

εs 0.002398608 0.001477528 0.000556447 -0.00036463 -0.00128571

EJE NEUTRO

𝑪𝑷=

DIAG. FUERZAS

fs 5037.077 kg/cm2 3102.808 kg/cm2 1168.539 kg/cm2 -765.731 kg/cm2 -2700.000 kg/cm2 P

F 54.558 Tn 17.686 Tn 6.661 Tn -4.365 Tn -35.073 Tn 186.276 Tn 225.743 Tn

=

CENTROIDE PLÁSTICO

T5

εs5 SECCION

𝑯 − 𝑪𝑷

C's2

dist. C.P. 18.847 cm 9.424 cm 0.000 cm -9.424 cm -18.847 cm 11.955 cm M

JUAN MANUEL CHERO DAMIAN

M 10.283 Tn.m 1.667 Tn.m 0.000 Tn.m 0.411 Tn.m 6.610 Tn.m 22.270 Tn.m 41.241 Tn.m

25.000 cm

25.000 cm

k) CONSIDERANDO LA PROFUNDIDAD DEL EJE NEUTRO IGUAL A UN PORCENTAJE DEL PERALTE EFECTIVO d = a = C =

C =

0.80d

43.847 cm 29.816 cm 35.078 cm

εc

0.85*f'c C's1

εs1 𝑪

Cc =0.85*f'c*a*b

𝒂 εs2

d

εs3

C's3

SECCION

T5

DIAG. DEFORMACIONES

capa As dist. E.N. 1 12.99 cm2 28.925 cm 2 5.70 cm2 19.501 cm 3 5.70 cm2 10.078 cm 4 5.70 cm2 0.654 cm 5 12.99 cm2 -8.769 cm CONCRETO Cc = 0.85*f'c*a*b

εs 0.002473782 0.001667837 0.000861891 5.59456E-05 -0.00075

𝑪𝑷=

EJE NEUTRO

εs5

DIAG. FUERZAS

fs 5194.943 kg/cm2 3502.457 kg/cm2 1809.971 kg/cm2 117.486 kg/cm2 -1575.000 kg/cm2 P

F 54.558 Tn 19.964 Tn 10.317 Tn 0.670 Tn -20.459 Tn 212.887 Tn 277.936 Tn

=

CENTROIDE PLÁSTICO

C's4

εs4

𝑯 − 𝑪𝑷

C's2

dist. C.P. 18.847 cm 9.424 cm 0.000 cm -9.424 cm -18.847 cm 10.092 cm M

JUAN MANUEL CHERO DAMIAN

M 10.283 Tn.m 1.881 Tn.m 0.000 Tn.m -0.063 Tn.m 3.856 Tn.m 21.484 Tn.m 37.441 Tn.m

25.000 cm

25.000 cm

l) CONSIDERANDO LA PROFUNDIDAD DEL EJE NEUTRO IGUAL A UN PORCENTAJE DEL PERALTE EFECTIVO d = a = C =

C =

0.90d

43.847 cm 33.543 cm 39.462 cm

εc

0.85*f'c C's1

εs1 𝑪

Cc =0.85*f'c*a*b

𝒂 εs2

d

εs3

C's3

T5

εs5 SECCION

DIAG. DEFORMACIONES

capa As dist. E.N. 1 12.99 cm2 33.310 cm 2 5.70 cm2 23.886 cm 3 5.70 cm2 14.462 cm 4 5.70 cm2 5.039 cm 5 12.99 cm2 -4.385 cm CONCRETO Cc = 0.85*f'c*a*b

εs 0.002532251 0.001815855 0.001099459 0.000383063 -0.00033333

𝑪𝑷= EJE NEUTRO

DIAG. FUERZAS

fs 5317.727 kg/cm2 3813.295 kg/cm2 2308.863 kg/cm2 804.432 kg/cm2 -700.000 kg/cm2 P

F 54.558 Tn 21.736 Tn 13.161 Tn 4.585 Tn -9.093 Tn 239.498 Tn 324.444 Tn

=

CENTROIDE PLÁSTICO

C's4

εs4

𝑯 − 𝑪𝑷

C's2

dist. C.P. 18.847 cm 9.424 cm 0.000 cm -9.424 cm -18.847 cm 8.228 cm M

JUAN MANUEL CHERO DAMIAN

M 10.283 Tn.m 2.048 Tn.m 0.000 Tn.m -0.432 Tn.m 1.714 Tn.m 19.707 Tn.m 33.320 Tn.m

25.000 cm

25.000 cm

m) CONSIDERANDO LA PROFUNDIDAD DEL EJE NEUTRO IGUAL A UN PORCENTAJE DEL PERALTE EFECTIVO d = a = C =

C =

1.00d

43.847 cm 37.270 cm 43.847 cm

εc

0.85*f'c C's1

εs1 𝑪

Cc =0.85*f'c*a*b

𝒂 εs2

d

C's4 T5

εs5 DIAG. DEFORMACIONES

capa As dist. E.N. 1 12.99 cm2 37.694 cm 2 5.70 cm2 28.271 cm 3 5.70 cm2 18.847 cm 4 5.70 cm2 9.424 cm 5 12.99 cm2 0.000 cm CONCRETO Cc = 0.85*f'c*a*b

εs 0.002579026 0.001934269 0.001289513 0.000644756 0

=

C's3 CENTROIDE PLÁSTICO

εs3

εs4

SECCION

𝑯 − 𝑪𝑷

C's2

𝑪𝑷=

EJE NEUTRO

DIAG. FUERZAS

fs 5415.954 kg/cm2 4061.966 kg/cm2 2707.977 kg/cm2 1353.989 kg/cm2 0.000 kg/cm2 P

F 54.558 Tn 23.153 Tn 15.435 Tn 7.718 Tn 0.000 Tn 266.108 Tn 366.973 Tn

dist. C.P. 18.847 cm 9.424 cm 0.000 cm -9.424 cm -18.847 cm 6.365 cm M

JUAN MANUEL CHERO DAMIAN

M 10.283 Tn.m 2.182 Tn.m 0.000 Tn.m -0.727 Tn.m 0.000 Tn.m 16.938 Tn.m 28.675 Tn.m

25.000 cm

25.000 cm

RESUMEN DE LAS FUERZAS CALCULADAS PARA REFUERZO SIMETRICO (ESPEJO)

ID P.max d 0.9d 0.8d 0.7d 0.6d Pb 0.5d 0.4d 0.3d 0.2d 0.1d T.max

Mn 0.000 Tn.m 28.675 Tn.m 33.320 Tn.m 37.441 Tn.m 41.241 Tn.m 45.054 Tn.m 45.054 Tn.m 44.341 Tn.m 41.607 Tn.m 35.620 Tn.m 26.207 Tn.m 10.220 Tn.m 0.000 Tn.m

Pn 424.197 Tn 366.973 Tn 324.444 Tn 277.936 Tn 225.743 Tn 165.022 Tn 165.022 Tn 117.937 Tn 69.815 Tn 14.340 Tn -48.738 Tn -132.768 Tn -180.936 Tn

ø 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.733333 0.858333 0.9 0.9 0.9 0.9

Mu 0.000 Tn.m 18.639 Tn.m 21.658 Tn.m 24.337 Tn.m 26.807 Tn.m 29.285 Tn.m 29.285 Tn.m 32.516 Tn.m 35.713 Tn.m 32.058 Tn.m 23.586 Tn.m 9.198 Tn.m 0.000 Tn.m

Pu 275.728 Tn 238.532 Tn 210.889 Tn 180.658 Tn 146.733 Tn 107.265 Tn 107.265 Tn 86.487 Tn 59.924 Tn 12.906 Tn -43.864 Tn -119.491 Tn -162.842 Tn

JUAN MANUEL CHERO DAMIAN

Mn(-) 0.000 Tn.m -28.675 Tn.m -33.320 Tn.m -37.441 Tn.m -41.241 Tn.m -45.054 Tn.m -45.054 Tn.m -44.341 Tn.m -41.607 Tn.m -35.620 Tn.m -26.207 Tn.m -10.220 Tn.m 0.000 Tn.m

Mu(-) 0.000 Tn.m -18.639 Tn.m -21.658 Tn.m -24.337 Tn.m -26.807 Tn.m -29.285 Tn.m -29.285 Tn.m -32.516 Tn.m -35.713 Tn.m -32.058 Tn.m -23.586 Tn.m -9.198 Tn.m 0.000 Tn.m

DIAGRAMA DE INTERACCION: CALCULO MANUAL VS ETABS 500

400

300 CARGAS M3n(+) MANUAL

200

Título del eje

M3n(-) MANUAL M3u(+) MANUAL 100

M3u(-) MANUAL M3u(+) ETABS

0 -50

-40

-30

-20

-10

0

10

20

-100

-200

-300

Título del eje

JUAN MANUEL CHERO DAMIAN

30

40

50

M3u(-) ETABS M3n(+) ETABS M3n(-) ETABS