• Author / Uploaded
• Alfredo Zertuche

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ARMADURAS HIPERESTÁTICAS INTERNA Y EXTERNAMENTE

DE T E RMI NE LA S RE AC C I ONES DE LOS E LE M E NTOS M E DI ANTE E L M É TODO GE N E RAL DE LA SI G UI E NTE A RM A DURA:

RAx

RAy

REy

RHy

PA SO# 1 : P LA N TEAMI ENTO DE LA A RM A DURA I SOST ÁTI CA F UN DAMENTAL( ‘)

50 T

40.625 T

59.375 T

PA SO# 2 : P LA N TEAMI ENTO DE LA A RM A DURA I SOST ÁTI CA C ON UN A CA RG A UN I TARI A E N E L P UN TO E ( ‘’)

0.5

0.5

PA SO# 3 : P LA N TEAMI ENTO DE LA A RM A DURA I SOST ÁTI CA C ON UN A CA RG A UN I TARI A E N E L E LE M E NTO DG ( ‘‘’)

0

0

PASO#4: CÁLCULO DE LOS ELEMENTOS DE LA ARMADURA ISOSTÁTICA FUNDAMENTAL (‘) Ʃ𝐹𝑦 = 0

NODO A FAB’ 5

50 Ton.

3 40.625 𝑇𝑜𝑛. −𝐹𝐴𝐵 =0 5 𝐹𝐴𝐵′ = 67.708 𝑇𝑜𝑛. © ′

3 4

FAC’

Ʃ𝐹𝑥 = 0 4 − 50 = 0 5 𝐹𝐴𝐶 ′ = 104.167 𝑇𝑜𝑛. 𝕋

𝐹𝐴𝐶 ′ − 67.708

40.626 Ton.

NODO C Ʃ𝐹𝑦 = 0

FCB’

𝐹𝐶𝐵′ − 50 𝑇𝑜𝑛. = 0 𝐹𝐶𝐵′ = 50 𝑇𝑜𝑛. 𝕋 104.167 Ton.

FCE’

Ʃ𝐹𝑥 = 0 𝐹𝐶𝐸 ′ − 104.167 𝑇𝑜𝑛. = 0 𝐹𝐶𝐸 ′ = 104.167 𝑇𝑜𝑛. 𝕋

50 Ton.

Ʃ𝐹𝑦 = 0

NODO B

3 3 𝐹𝐵𝐸′ + 67.708 − 50 𝑇𝑜𝑛. = 0 5 5 50 T

FBD’

𝐹𝐵𝐸 ′ = 15.625 𝑇𝑜𝑛. © Ʃ𝐹𝑥 = 0

5

3 4

67.708 T

5

3

50 𝑇𝑜𝑛. +67.708

4

50 T

FBE’

4 4 − 15.625 − 𝐹𝐵𝐷′ = 0 5 5

𝐹𝐵𝐷′ = 91.667 𝑇𝑜𝑛. ©

NODO D

Ʃ𝐹𝑥 = 0 91.667 𝑇𝑜𝑛. − 𝐹𝐷𝐹′ = 0 𝐹𝐷𝐹 ′ = 91.667 𝑇𝑜𝑛. © 91.667 T

FDF’

Ʃ𝐹𝑦 = 0 𝐹𝐷𝐸 ′ = 0 FDE’

Ʃ𝐹𝑦 = 0 NODO E 15.626 T 5

3 4

104.167 T

FEF’ 5

3

4

3 3 𝐹𝐸𝐹′ − 15.625 =0 5 5 𝐹𝐸𝐹 ′ = 15.625 𝑇𝑜𝑛. 𝕋

FEG’ Ʃ𝐹𝑥 = 0

𝐹𝐸𝐺 ′ + 15.625

4 4 + 15.625 − 104.167 𝑇𝑜𝑛. = 0 5 5

𝐹𝐸𝐺 ′ = 79.167 𝑇𝑜𝑛. 𝕋

Ʃ𝐹𝑦 = 0

NODO H

FHF’

3 59.375 𝑇𝑜𝑛. −𝐹𝐻𝐹′ =0 5

5

3 4

𝐹𝐻𝐹 ′ = 98.958 𝑇𝑜𝑛. © FHG’ Ʃ𝐹𝑥 = 0 4 98.958 − 𝐹𝐻𝐺′ = 0 5 𝐹𝐻𝐺 ′ = 79.167 𝑇𝑜𝑛. 𝕋

59.375 T

NODO G

Ʃ𝐹𝑦 = 0

FGF’

79.167 T

79.167 T

F𝐺𝐹 ′ − 50 𝑇𝑜𝑛. = 0 𝐹𝐺𝐹 ′ = 50 𝑇𝑜𝑛. 𝕋

PA SO # 5 : C Á LC ULO DE LOS E LE M E NTOS DE LA A RM ADURA I SOST ÁTICA ( ‘) C ON CA RG A UN I TARI A E N E L P UN TO E

Ʃ𝐹𝑦 = 0

NODO A UAB’ 5

3 0.5 − 𝑈𝐴𝐵 =0 5 𝑈𝐴𝐵′ = 0.833 © ′

3 4

UAC’

Ʃ𝐹𝑥 = 0 4 =0 5 𝑈𝐴𝐶 ′ = 0.667 𝕋

𝑈𝐴𝐶 ′ − 0.833

0.5

NODO C Ʃ𝐹𝑦 = 0

UCB’

𝑈𝐶𝐵′ − 0 = 0 𝑈𝐶𝐵′ = 0 0.667

Ʃ𝐹𝑥 = 0 𝑈𝐶𝐸 ′ − 0.667 = 0 𝑈𝐶𝐸 ′ = 0.667 𝕋

UCE’

Ʃ𝐹𝑦 = 0

NODO B

3 3 0.833 − 𝑈𝐵𝐸′ =0 5 5 𝑈𝐵𝐸 ′ = 0.833 𝕋

UBD’

Ʃ𝐹𝑥 = 0 5

3 4

0.833

5

3

0.833

4

UBE’

4 4 + 0.833 − 𝑈𝐵𝐷′ = 0 5 5 𝑈𝐵𝐷′ = 0.667 ©

NODO D

Ʃ𝐹𝑥 = 0 0.667 − 𝑈𝐷𝐹′ = 0 𝑈𝐷𝐹 ′ = 0.667 © 0.667

UDF’

Ʃ𝐹𝑦 = 0 𝑈𝐷𝐸 ′ = 0 UDE’

Ʃ𝐹𝑦 = 0 NODO E 0.833

UEF’ 5

5

3

3

4

4

0.667

3 3 𝑈𝐸𝐹 − 1 + 0.833 =0 5 5 ′

𝑈𝐸𝐹 ′ = 0.833 𝕋

UEG’ Ʃ𝐹𝑥 = 0 1

𝑈𝐸𝐺 ′ + 0.833

4 4 − 0.833 − 0.667 = 0 5 5

𝑈𝐸𝐺 ′ = 0.667 𝕋

Ʃ𝐹𝑦 = 0

NODO H

UHF’

3 0.5 − 𝑈𝐻𝐹′ =0 5

5

3 4

𝑈𝐻𝐹 ′ = 0.833 © UHG’ Ʃ𝐹𝑥 = 0 4 0.833 − 𝑈𝐻𝐺′ = 0 5 𝑈𝐻𝐺 ′ = 0.667 𝕋

0.5

NODO G

Ʃ𝐹𝑦 = 0

UGF’

0.667

0.667

𝑈𝐺𝐹 ′ = 0

PA SO# 6 : C Á LC ULO DE LOS E LE M E NTOS DE LA A RM ADURA C ON CA RG A UN I TARIA E N E L E LE M E NTO DG

Ʃ𝐹𝑦 = 0 NODO D 0.833

3 𝑢𝐷𝐸′ − 1 =0 5 𝑢𝐷𝐸′ = 0.6 ©

uDB’=0

5

3

uEG’

4 1 − 𝑢𝐷𝐹′ = 0 5

4

uDE’

Ʃ𝐹𝑥 = 0

1

𝑢𝐷𝐹′ = 0.8 ©

Ʃ𝐹𝑦 = 0 NODO G 1

3 1 − 𝑢𝐺𝐹′ = 0 5

uGF' 5

3

𝑢𝐺𝐹 ′ = 0.6 ©

4

uGE’

uGH’=0 Ʃ𝐹𝑥 = 0 1

𝑢𝐺𝐸 ′ − 1

4 =0 5

𝑢𝐺𝐸 ′ = 0.8 ©

NODO E 0.6

uEF’ Ʃ𝐹𝑦 = 0

5

3

4

𝑢𝐸𝐹 ′ 0.8

1

3 − 0.6 = 0 5

𝑢𝐸𝐹 ′ = 1 𝕋

PASO #7: CÁLCULO DE LAS DEFORMACIONES

∆E’=

𝐅′𝐔′𝐋 Ʃ 𝐀𝐄

∆DG’=

δEE’’=

𝐔′ 𝐔′ 𝐋 Ʃ 𝐀𝐄

δE-DG’’’=

𝐅 ′ 𝐮′𝐋 Ʃ 𝐀𝐄

δDG-E’’=

𝒖′ 𝐔 ′ 𝐋 Ʃ 𝐀𝐄

𝐔 ′ 𝒖′ 𝐋 Ʃ 𝐀𝐄

δDG-DG’’’=

𝒖′ 𝒖′ 𝐋 Ʃ 𝐀𝐄

𝑈 ′ 𝑈′𝐿 𝐴𝐸

𝑢′𝑢′𝐿 𝐴𝐸

𝑈′𝑢′𝐿 𝐴𝐸

(TON) F=F'+U'REy+u'XDG

0.000

3.469

0.000

0.000

-5.95

277917.56

0.000

1.780

0.000

0.000

54.72

0

0.000

0.000

0.000

0.000

0.000

50.00

-0.667

0

244567.56

0.000

1.780

0.000

0.000

-42.22

-15625

0.833

0

-108463.54

0.000

5.782

0.000

0.000

-77.38

100

54167

0.667

0

144517.56

0.000

1.780

0.000

0.000

4.72

300

60

0

0

-0.6

0.000

0.000

0.000

1.800

0.000

-11.69

D-F

400

100

-91667

-0.667

-0.8

244567.56 293334.4 1.780

2.560

2.134

-57.80

D-G

500

60

-----------

-----------

1

-----------

8.333

-----------

19.48

E-F

500

60

15625

0.833

1

108463.54 130208.3 5.782

8.333

6.942

-26.65

E-G

400

100

79167

0.667

-0.8

211217.56 -253334.4 1.780

2.560

-2.134

14.14

F-G

300

60

50000

0

-0.6

0.000

1.800

0.000

38.31

F-H

500

100

-98958

-0.833

0

412160.07

0.000

3.469

0.000

0.000

-37.20

G-H

400

100

79167

0.667

0

211217.56

0.000

1.780

0.000

0.000

29.72

2028169.26 20208.3 29.181 25.387 E E E E

6.942 E

𝐹′𝑈′ 𝐿 𝐴𝐸

ELEMENTO

L (cm)

A (cm2)

F' (Kg)

U'=U''=U'''

u'=u''=u'''

A-B

500

100

-67708

-0.833

0

282003.82

A-C

400

100

104167

0.667

0

B-C

300

60

50000

0

B-D

400

100

-91667

B-E

500

60

C-E

400

D-E

SUMA

-----------

𝐹 ′ 𝑢′𝐿 𝐴𝐸

-----------

-150000 0.000

PASO #8: CÁLCULO DE LAS RESTRICCIONES

∆E= ∆E’+δEE‘’(REy)+δE-DG(XDG)=0 ∆DG= ∆DG’+δDG-E‘’(REy)+δDG-DG(XDG)=0 29.181(REy)+6.942(XDG)= -2028169.226 6.942(REy)+25.387(XDG)= -20208.33333 REy= -74.14 Ton. XDG= 19.48 Ton. 𝕋

PASO #9: CÁLCULO DE LAS REACCIONES RESTANTES

Ʃ𝐹𝑥 = 0 −𝑅𝐴𝑥 + 50 = 0 𝑅𝐴𝑥 = 50 𝑇𝑜𝑛. Ʃ𝑀𝑎 = 0 50 4 − 74.14 8 + 50 12 − 𝑅𝐻𝑦 16 + 50(3) = 0

𝑅𝐻𝑦 = 22.305 𝑇𝑜𝑛. Ʃ𝐹𝑦 = 0 𝑅𝐴𝑦 + 22.305 − 50 − 50 + 74.14 = 0 𝑅𝐴𝑦 = 3.555 𝑇𝑜𝑛.

ARMADURA HIPERESTÁTICA ORIGINAL

50 T

3.55 T

74.14 T

22.31 T