• Author / Uploaded
• Antonio Martinez

Citation preview

EJERCICIO # 3 Diseñar la viga presforzada optimizando la carga de presfuerzo y la ubicación de la excentricidad (Constante y Variable) 60 cm 5

15

5

15

𝑞𝑣𝑖𝑣𝑎 = 500 𝐾𝑔/𝑚2

70 cm

40

L = 12m 15

22,5

15

22,5

1 PASO Determinamos el centro de gravedad de la seccion transversal AREA

60 15 𝐴𝑡 = 60 ∗ 15 + 15 ∗ 40 + 25 ∗ 15 At= 1875 cm^2

15

𝑌𝑔 =

𝐴1 ∗ 𝑦1 + 𝐴2 ∗ 𝑦2 + 𝐴3 ∗ 𝑦3 𝐴𝑡

𝑌𝑔 =

900 ∗ 7,5 + 600 ∗ 35 + 375 ∗ 62,5 1875 Yg=

40

25 A1= A2= A3= Y1= Y2= Y3= X1= X2= X3=

900 600 375 7.5 35 62.5 30 30 30

27.300 cm 0.1875

𝑋𝑔 =

𝐴1 ∗ 𝑥1 + 𝐴2 ∗ 𝑥2 + 𝐴3 ∗ 𝑥3 𝐴𝑡

𝑋𝑔 =

900 ∗ 30 + 600 ∗ 30 + 375 ∗ 30 1875

Xg=

30 cm

Ys=

42.700 cm

Yi=

18.300 cm

70

2 PASO Determinacion de la Inercia Centroidal 𝐼𝑥𝑥 = 𝐼𝑥 + 𝐴 ∗ 𝑦 2 Ix1= Ix2= Ix3=

16875 cm^4 80000 cm^4 7031.25 cm^4

35,2 7,7

𝐼𝑥𝑥 = 16875 + 900 ∗ 19,8

2

+ 80000 + 600 ∗ 7,7

2

+ 7031,25 + 375 ∗ 35,2

2

19,8

956956.25

Ixx=

cm^4

b1 h1 b2 h2 b3 h3 369711 115574 471671.25

60 15 15 40 25 15 19.8 7.7 35.2

3 PASO Determinacion del Modulo Estatico

𝑊𝑠 =

𝑊𝑠 = Ws=

956956,25𝑐𝑚4 42,7𝑐𝑚 22411.1534 cm^3 𝑊𝑖 =

𝑊𝑖 =

𝐼𝑥𝑥 𝑌𝑠

𝐼𝑥𝑥 𝑌𝑖

956956,25𝑐𝑚4 18,3 𝑐𝑚

Wi= 52292.69126 cm^3 4 PASO Determinacion de los Momentos Flectores 𝑞𝑣𝑖𝑔𝑎 = Υº𝐻º ∗ 𝐴𝑟𝑒𝑎 qviga= qviva=

468.75 Kg/m 125 Kg/m

ΥºHº=2500 Kg/m^3

4.6875 1.25 1200

2500

x (m) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12

Rviga (kg) R viva (kg) M viga( kg*cm) M viva (kg*cm) 2812.5 750 0.00 0.00 2812.5 750 134765.63 35937.50 2812.5 750 257812.50 68750.00 2812.5 750 369140.63 98437.50 2812.5 750 468750.00 125000.00 2812.5 750 556640.63 148437.50 2812.5 750 632812.50 168750.00 2812.5 750 697265.63 185937.50 2812.5 750 750000.00 200000.00 2812.5 750 791015.63 210937.50 2812.5 750 820312.50 218750.00 2812.5 750 837890.63 223437.50 2812.5 750 843750.00 225000.00 2812.5 750 837890.63 223437.50 2812.5 750 820312.50 218750.00 2812.5 750 791015.63 210937.50 2812.5 750 750000.00 200000.00 2812.5 750 697265.63 185937.50 2812.5 750 632812.50 168750.00 2812.5 750 556640.63 148437.50 2812.5 750 468750.00 125000.00 2812.5 750 369140.63 98437.50 2812.5 750 257812.50 68750.00 2812.5 750 134765.63 35937.50 2812.5 750 0.00 0.00

5 PASO Determinacion de la Excentricidad Constante

𝑑 = 5 cm +

𝜙 = 8𝑐𝑚

8 𝑐𝑚 = 9 𝑐𝑚 2

𝑒𝑚𝑎𝑥 = 𝑌𝑖 − 𝑑 𝑒𝑚𝑎𝑥 = 18,3 𝑐𝑚 − 9 𝑐𝑚

𝑑

𝑟 = 5 𝑐𝑚

emax=

9.300 cm

9

6 PASO Determinacion de la carga Balanceada 𝑃 ∗ 𝑒𝑚𝑎𝑥 =

𝑃=

𝑃=

𝑞 ∗ 𝐿2 8

q= L=

5.9375 kg/cm 1200 cm

𝑞 ∗ 𝐿2 8 ∗ 𝑒𝑚𝑎𝑥

5,9375𝑘𝑔/𝑐𝑚 ∗ 1200𝑐𝑚 8 ∗ 9,3 𝑐𝑚

2

P= 114919.3548 Kg P= 114.9193548 Tn 7 PASO Verificacion de Tensiones Fibra Superior 𝜎𝑠 = −

𝑃𝑒 𝑃𝑒 ∗ 𝑒 𝑀𝑣𝑖𝑔𝑎 𝑀𝑣𝑖𝑣𝑎 + − − 𝐴 𝑊𝑠 𝑊𝑠 𝑊𝑠

Fibra Inferior 𝜎𝑖 = −

𝑃𝑒 𝑃𝑒 ∗ 𝑒 𝑀𝑣𝑖𝑔𝑎 𝑀𝑣𝑖𝑣𝑎 − + + 𝐴 𝑊𝑖 𝑊𝑖 𝑊𝑖

x (m)

M viga

Mviva

σs

σi

0 1 2 3 4 5 6 7 8 9 10 11 12

0.00 257812.50 468750.00 632812.50 750000.00 820312.50 843750.00 820312.50 750000.00 632812.50 468750.00 257812.50 0.00

0.00 68750.00 125000.00 168750.00 200000.00 218750.00 225000.00 218750.00 200000.00 168750.00 125000.00 68750.00 0.00

-13.602 -28.173 -40.096 -49.368 -55.992 -59.966 -61.290 -59.966 -55.992 -49.368 -40.096 -28.173 -13.602

-81.728 -75.483 -70.374 -66.400 -63.561 -61.858 -61.290 -61.858 -63.561 -66.400 -70.374 -75.483 -81.728

Al Realizar la viga con excentricidad constante solo garantizamos que las tenciones en L/2 sean uniformes

8 PASO Determinacion de la Excentricidad Variable

𝑑 = 5 cm +

8 𝑐𝑚 = 9 𝑐𝑚 2

9

𝑒𝑚𝑎𝑥 = 𝑌𝑖 − 𝑑

𝜙 = 8𝑐𝑚

𝑒𝑚𝑎𝑥 = 18,3𝑐𝑚 − 9 𝑐𝑚 𝑑

𝑟 = 5 𝑐𝑚

𝑦=

emax=

x (m)

Y (cm) 0 1 2 3 4 5 6

4 ∗ 𝐶 ∗ 𝑥2 − 𝑒𝑚𝑎𝑥 𝐿2

9.300 cm

e (cm)

-9.30 -9.04 -8.27 -6.98 -5.17 -2.84 0.00

9.30 9.04 8.27 6.98 5.17 2.84 0.00

9 PASO Determinacion de la carga Balanceada 𝑃 ∗ 𝑒𝑚𝑎𝑥 =

𝑃=

𝑃=

𝑞 ∗ 𝐿2 8

q= L=

𝑞 ∗ 𝐿2 8 ∗ 𝑒𝑚𝑎𝑥

5,937 𝑘𝑔/𝑐𝑚 ∗ 1200𝑐𝑚 8 ∗ 9,3 𝑐𝑚 P= 114919.3548 Kg P= 114.9193548 Tn

2

5.9375 kg/cm 1200 cm

0.00 2.84 5.17 6.98 8.27 9.04 9.30

10 PASO Verificacion de Tensiones Fibra Superior 𝜎𝑠 = −

𝑃𝑒 𝑃𝑒 ∗ 𝑒 𝑀𝑣𝑖𝑔𝑎 𝑀𝑣𝑖𝑣𝑎 + − − 𝐴 𝑊𝑠 𝑊𝑠 𝑊𝑠

Fibra Inferior 𝜎𝑖 = −

𝑃𝑒 𝑃𝑒 ∗ 𝑒 𝑀𝑣𝑖𝑔𝑎 𝑀𝑣𝑖𝑣𝑎 − + + 𝐴 𝑊𝑖 𝑊𝑖 𝑊𝑖

x (m)

M viga

Mviva

σs

σi

0 1 2 3 4 5 6

0.00 257812.50 468750.00 632812.50 750000.00 820312.50 843750.00

0.00 68750.00 125000.00 168750.00 200000.00 218750.00 225000.00

-61.290 -61.290 -61.290 -61.290 -61.290 -61.290 -61.290

-61.290 -61.290 -61.290 -61.290 -61.290 -61.290 -61.290

Todas las tensiones son Uniformes en toda la viga 11 PASO Deflecciones

e= 9.3 q= 5.9375 L= 1200 E= 200000 Ixx= 956956.25 P= 114919.3548 ∆= ∆=

cm kg/cm cm cm^4 Kg

5 𝑞 ∗ 𝐿4 ∗ 384 𝐸 ∗ 𝐼

5 5,937𝑘𝑔/𝑐𝑚 ∗ 1200𝑐𝑚4 ∗ 384 200000 ∗ 956956,25 𝑐𝑚4 Δ=

∆=

∆=

0.838 cm

5 𝑃 ∗ 𝑒 ∗ 𝐿2 ∗ 48 𝐸∗𝐼

Iguales

5 114919,355 ∗ 9,3 ∗ 1200𝑐𝑚2 ∗ 48 200000 ∗ 956956,25 𝑐𝑚4 Δ=

0.838 cm

p

A 110436.62

E 3500

-31.553

WS 43.86

MVIGA MVIVA 69568.73 1093750 3750000