1 Konstrukcija linijezatežućih sila - imksus.grf.rsimksus.grf.rs/nastava/beton/MTI HVE...

1

Konstrukcija linije zatežućih sila

−

⋅ζ⋅

σ=

σ−

σ⋅=∑ = u

au

vv

u

v

aua N

hM1N

zMA)0N(

uau

vaau Nh

MAZ −⋅ζ

=σ⋅=

−⋅+=⋅+= 1uu1auuau a

2dNMyNMM

va1aaauv1a AAZ σ⋅=σ⋅=⇒σ=σ

2

.min

uvaau h9.0

MAZ×

≈σ×=

.min.min h9.0z

9.0.consthh

×≈⇒

≈ζ==

uauu MM0N =⇒=

u.min

auvaau N

h9.0MAZ −×

≈σ×=

Konstrukcija linije zatežućih sila

čisto savijanje:

3

L = 12.0 m

Reakcije oslonaca

Bg = (30×122/2)/9 = 240 kNAg = 30×12 – 240 = 120 kN

Ap1 = Bp1 = 40×9/2 = 180 kN

Bp2 = [(60×3×(9+3/2)]/9Bp2 = 210 kNAp2 = 60×3 – 210 = -30 kN

4

Dijagrami momenata savijanjaoslonac:Mg = 30×32/2 = 135 kNmpolje:xmax = 120/30 = 4.0 mMg = 120×4 – 30×42/2Mg = 240 kNm

polje:Mp1 = 40×92/2 = 405 kNm

oslonac:Mp2 = 60×32/2 = 270 kNm

5

Anvelopa momenata savijanja

( )

( )21

2

1

21

2

1

PP8.1G0.1P8.1G0.1P8.1G0.1

G1.0

PP8.1G6.1P8.1G6.1P8.1G6.1

G6.1

+⋅+⋅

⋅+⋅

⋅+⋅⋅

+⋅+⋅

⋅+⋅⋅+⋅

⋅

6


9.0 m

Mu

= 11

09.4

kN

m Mu

= 70

2.0

kNm

4.3 m1.6×G+1.8×P1

1.6×G+1.8×P2

3.0 m

7


1.0 m8.0 m

1.0 m8.0 m

1.0×G+1.8×P1

Mu

= 11

09.4

kN

m Mu

= 70

2.0

kNm

4.3 m 1.6×G+1.8×P1

1.0×G+1.8×P2

1.6×G+1.8×P2

1.6×G+1.8×P1 1.0×G+1.8×P1

1.0×G+1.8×P2 1.6×G+1.8×P2

8


1.0×G+1.8×P2

1.6×G+1.8×P2

1.0×G+1.8×P1

1.6×G+1.8×P1

nu

lta t

acka

dijagra

ma M

g

MG i MP istog znaka

MG i MP istog znaka MG i MP razlicitog znaka

MG i MP razlicitog znaka

9


1.0 m8.0 m

1.0 m8.0 m

4.3 m

Mu

= 11

09.4

kN

m Mu

= 70

2.0

kNm

1.6×G+1.8×P1 1.0×G+1.8×P1

1.0×G+1.8×P2 1.6×G+1.8×P2

10

Dimenzionisanje - presek u polju

aI = 2.5 + 1.0 + 2.5/2 = 4.75 cmaII = 4.75 + 3.0 + 2.5 = 10.25 cm

a1 = (5×4.75 + 4×10.25)/9a1 = 7.19 cm

h = 75 - 7.19 = 67.81 cmh ≈ 68 cm = hrač.

40

21.7

5

4.75

4.75

5.5

4.75

4.75

5RØ25

4RØ25

UØ10/30

30.5

16.2

5

1560

75

2RØ25

2RØ12

22 2RØ12

11

Dimenzionisanje - oslonac

aI = 2.5 + 1.0 + 2.5/2 = 4.75 cmaII = 4.75 + 3.0 + 2.5 = 10.25 cm

a1 = (4×4.75 + 2×10.25)/6a1 = 6.58 cm

h = 75 - 6.58 = 68.42 cmh ≈ 68 cm = hrač.

40

21.7

5

4.75

4.75

5.5

4.75

4.75

4RØ25

2RØ25

UØ10/15

30.5

16.2

5

1560

75

2RØ25

22

2RØ12

2RØ12

12

Analitički izrazi za momente savijanja

2)xL(gM:lx

2xgxAM:lx

2

g

2

gg

−⋅−=≥

⋅−⋅=≤

0M:lx2

xpxAM:lx

1p

21

1p1p

=≥

⋅−⋅=≤

2)xL(pM:lx

xAM:lx2

22p

2p2p

−⋅−=≥

⋅=≤

13Vrednosti momenata savijanja - Excelx Mg Mp1 Mp2

1.6×G+1.8×P1

1.6×G+1.8×P2

1.0×G+1.8×P1

1.0×G+1.8×P2 MAX MIN

0 0 0 0 0 0 0 0 0 00.5 56.25 85 -15 243 63 209.3 29.3 243 0

1 105 160 -30 456 114 393 51 456 01.5 146.3 225 -45 639 153 551.3 65.3 639 0

2 180 280 -60 792 180 684 72 792 02.5 206.3 325 -75 915 195 791.3 71.3 915 0

3 225 360 -90 1008 198 873 63 1008 03.5 236.3 385 -105 1071 189 929.3 47.3 1071 0

4 240 400 -120 1104 168 960 24 1104 04.5 236.3 405 -135 1107 135 965.3 -6 .8 1107 -6 .8

5 225 400 -150 1080 90 945 -45 1080 -455.5 206.3 385 -165 1023 33 899.3 -90 .8 1023 -90 .8

6 180 360 -180 936 -36 828 -144 936 -1446.5 146.3 325 -195 819 -117 731.3 -204.8 819 -204.8

7 105 280 -210 672 -210 609 -273 672 -2737.5 56.25 225 -225 495 -315 461.3 -348.8 495 -348.8

8 0 160 -240 288 -432 288 -432 288 -4328.5 -63 .8 85 -255 51 -561 89.3 -522.8 89.3 -561

9 -135 0 -270 -216 -702 -135 -621 0 -7029.5 -93 .8 0 -188 -150 -487.5 -93 .8 -431.3 0 -487.5

10 -60 0 -120 -96 -312 -60 -276 0 -31210.5 -33 .8 0 -67 .5 -54 -175.5 -33 .8 -155.3 0 -175.5

11 -15 0 -30 -24 -78 -15 -69 0 -7811.5 -3 .75 0 -7 .5 -6 -19 .5 -3 .8 -17 .3 0 -19 .5

12 0 0 0 0 0 0 0 0 0

14

-800

-400

0

400

800

12000 1 2 3 4 5 6 7 8 9 10 11 12

Mu

[kN

m]

1.6×G+1.8×P11.6×G+1.8×P2G+1.8×P1G+1.8×P2

Anvelopa momenata savijanja - Excel

15

Mu

= 11

09.4

kN

m Mu

= 70

2.0

kNm

Z au

≈ 18

12.8 Zau ≈ 1109.4/(0.9×0.68) = 1812.8 kN

Z au

≈ 11

47.1

Zau ≈ 702/(0.9×0.68) = 1147.1

.min

uu

.min

auvaau h9.0

MNh9.0

MAZ×

=−×

≈σ×=

M/z linija

19

Tg

Tp1

Tp2

1,6Tg+1,8(Tp1+Tp2)

21

M/z linija

kN3.1964091.4aZ v)1(

a)1(

au =×=σ×=

Z au

≈ 18

12.8

Z au

≈ 11

47.1

Z(1)

au=1

96.3

22

Linija zatežućih silaZ a

u ≈

1812

.8

Z au

≈ 11

47.1

9×Z(1

)au

=176

7.1

23

M/z linija

ζ = 0.964Aa,potr. = 42.33 cm2 (usv. 44.18 cm2)

ζ = 0.894Aa,potr. = 28.85 cm2 (usv. 29.45 cm2)

Z au

≈ 18

12.8

Z au

≈ 11

47.1

9×Z(1

)au

=176

7.1 6×

Z(1)

au=1

178.

1

24

Linija zatežućih silaZ a

u ≈

1812

.8

Z au

≈ 11

47.1

v=0.51

v=0.51

v=0.51v=0.51

POMERANJE LINIJE ZATEŽUCIH SILA:v = 0.75×h = 0.75×68 = 51 cm

M/z linije

25

Linija pokrivanja armaturom

5RØ25

2RØ25

4RØ25

7Ø25

2Ø25

4Ø25

2Ø25

>4Ø25 >4Ø25

>7Ø25 >7Ø25

26


5RØ25

2RØ25

5Ø25

2Ø25

2Ø25

>4Ø25 >4Ø25

>7Ø25 >7Ø25

2Ø25

2Ø25

2Ø25

>5Ø >5Ø

>2Ø>2Ø

27


3RØ25

2RØ25

3Ø25

2Ø25

2Ø25

>4Ø25 >4Ø25

>7Ø25 >7Ø25

2Ø25

2Ø25

2Ø25

>5Ø >5Ø

>2Ø>2Ø

>3Ø >3Ø

2Ø25

28

Linija pokrivanja armaturomZ a

u ≈

1812

.8Z a

u ≈

1147

.1

A B

POMERENA LINIJA ZATEŽUCIH SILA LINIJA ZATEŽUCIH SILA

LINIJA POKRIVANJA ARMATUROM

v=51

v=51

v=51v=51


(M, N)(M, N, T)

29Z a

u ≈

1812

.8Z a

u ≈

1147

.1

A

B

3

1

1

POMERENA LINIJA ZATEŽUCIH SILA LINIJA ZATEŽUCIH SILA

LINIJA POKRIVANJA ARMATUROM

v=51

v=51

v=51v=51

UKUPNA DUŽINA ŠIPKE (700 cm)

STATICKI POTREBNA DUŽINA ŠIPKE (460 cm)

2

1

5

2Ø25

1Ø25

2Ø25

2Ø25

2Ø25

69

35

65

125

205

25

85

165

180280

DUŽINA SIDRENJA:

Ls1 = 79.4 ˜ 80 cm (dobra adhezija)Ls2 = 119 ˜ 120 cm (loša adhezija)


NOSIVOST JEDNE ŠIPKE ARMATURE:

Z(1)au = a(1)

a ×σv = 4.91×40 = 196.3 kN

Zau ≈ Mau / (0.9×h) - NuMau = Mu + Nu × (d 2 - a1)Nu = 0: Mau = Mu ; Zau ≈ Mu / (0.9×h)

(M, N, T) (M, N)M/Z linija

32

Sidrenje armature

Ø75.3175.18.1

4004Ø

8.14ØL

p

vs ×=

××=

τ×σ

×=

donja zona (dobra adhezija)

22Ø...cm70cm8.692.275.31L 1s ≈=×=25Ø...cm80cm4.795.275.31L 1s ≈=×=

gornja zona (loša adhezija)

22Ø...cm105cm705.1L 2s =×=

1s1s2p

1p2s1p2p L5.1LL

32

×=×ττ

=⇒τ=τ

25Ø...cm120cm805.1L 2s =×=

33

Sidrenje armature1Ø

25v=51 v=51

L.Z.S. LINIJA M z

v=51 v=51 305205

LØ25s1 = 80

STATICKI POTREBNADUŽINA ŠIPKE (390 cm)

UKUPNA DUŽINA ŠIPKE (550 cm)

LØ25s1 = 80

5 5

A B

p

vs 8.14

ØLτ×

σ×=

38

1215

6015

5

15 15

BA

1

1

1-1

3

3

3-32-2

2

2

75

39

1215

6015

5

15 15

BA

1

1

1-1

3

3

5

75

5 54x7.5=30

40

15

5 530

40

3-3

5

75

15 15

5 530

40

2-2

5

75

2

2

1 2RØ25 6 2RØ25

6 2RØ25

35

1

1 2RØ25

6

75

1 2RØ251 2RØ25

40

1215

6015

5

15 15

BA

1

1

1-1

3

3

5

75

5 54x7.5=30

40

15

5 530

40

3-3

5

75

15 15

5 530

40

2-2

5

75

2

2

1 2RØ25 6 2RØ253 2RØ25

6 2RØ25

3 2RØ25 L=925 (2)

35

890

35

1

2

1 2RØ25, 3 2RØ25

6

1 2RØ25

75

1 2RØ25

3

2 1RØ25 L=860 (1)

35

825

2 1RØ251 2RØ25

3 2RØ25

2

3

2 1RØ25

41

1215

6015

55

15 15

BA

1

1

1-1

3

3

5.5

5

75

5 54x7.5=30

40

15

5 530

40

3-3

5

75

15 15

5 530

40

2-2

5

75

2

2

1 2RØ25

4 2RØ25

6 2RØ253 2RØ25

6 2RØ25

4 2RØ25 L=760 (2)760

3 2RØ25 L=925 (2)

35

890

35

4 2RØ25

1

2

5

5 1RØ254 2RØ25, 1 2RØ25, 3 2RØ25

6

5 1RØ255 1RØ25 1 2RØ25

75

1 2RØ25

5 1RØ25

4 2RØ25

5

3

4 4

2 1RØ25 L=860 (1)

35

825

2 1RØ251 2RØ25

3 2RØ25

2

3

2 1RØ25

42

1215

6015

55

17015 15 25 375 25 25 175 10275

BA

1

1

1-1

3

3

55.

522

521

.5

75

5 54x7.5=30

40

30 55

15

5 530

40

7.5 55 15 7.53-3

16

55.

522

521

.5

75

16

15 15

5 530

40

7.5 55 15 7.52-2

55.

522

521

.5

75

16

2

2

1 2RØ25

4 2RØ25

11 2RØ12

6 2RØ25

7 2RØ2210 2RØ22

9 2RØ22

8 2RØ22

3 2RØ25

11 2RØ12

11 2RØ12

11 2RØ12

7 2RØ22

9 2RØ22

8 2RØ22

11 2RØ12

11 2RØ12

10 2RØ22 10 2RØ227 2RØ22, 8 2RØ227 2RØ22, 9 2RØ22 7 2RØ22, 8 2RØ22

11 ±1RØ126 2RØ25

4 2RØ25 L=760 (2)760

3 2RØ25 L=925 (2)

35

890

359 2RØ22 L=395 ( 2)

395

8 2RØ22 L=500 (2)500

7 2RØ22 L=865 (2)830

35

10 2RØ22 L=480 (2)

35

445

11 ±1RØ12 L=640 (8)

20

620 11 ±1RØ12 L=640 (8)

20

620

7 2RØ22

4 2RØ25

11 ±1RØ12

11 ±1RØ12

11 ±1RØ12

75

1

2

5

5 1RØ254 2RØ25, 1 2RØ25, 3 2RØ25

6

5 1RØ25

7

8

99

8

10

8 2RØ22

5 1RØ25 1 2RØ25

75

1 2RØ25

5 1RØ25

4 2RØ25

5

3

4 4

2 1RØ25 L=860 (1)

35

825

2 1RØ251 2RØ25

3 2RØ25

2

3

2 1RØ25

43

1215

6015

55

17015 15 25 375 25 25 175 10275

BA

1

1

1-1

3

3

55.

522

521

.5

75

5 54x7.5=30

40

30 55

15

5 530

40

7.5 55 15 7.53-3

16

55.

522

521

.5

75

16

15 15

5 530

40

7.5 55 15 7.52-2

55.

522

521

.5

75

16

2

212 18RUØ10/ 10 12 16RUØ10/ 25 12 19RUØ10/ 15 12 6RUØ10/ 15 12 8RUØ10/ 25

1 2RØ25

4 2RØ25

11 2RØ12

6 2RØ25

7 2RØ2210 2RØ22

9 2RØ22

8 2RØ22

3 2RØ25

11 2RØ12

11 2RØ12

11 2RØ12

13 9RUØ10/ 15

7 2RØ22

9 2RØ22

8 2RØ22

11 2RØ12

11 2RØ12


11 ±1RØ126 2RØ25

4 2RØ25 L=760 (2)760

3 2RØ25 L=925 (2)

35

890

359 2RØ22 L=395 (2)

395

8 2RØ22 L=500 (2)500

7 2RØ22 L=865 (2)830

35

10 2RØ22 L=480 (2)

35

445

11 ±1RØ12 L=640 (8)

20

620 11 ±1RØ12 L=640 (8)

20

620

12 RUØ10/15

7 2RØ22

4 2RØ25

11 ±1RØ12

11 ±1RØ12

11 ±1RØ12

12 RUØ10/25

12 RUØ10/15

13 RUØ10/15

12 19RUØ10/ 15 L=232 ( 67)

35

70

35

70

1 11 1

75

1

2

5

5 1RØ254 2RØ25, 1 2RØ25, 3 2RØ25

6

5 1RØ25

7

8

99

8

10

8 2RØ22

5 1RØ25 1 2RØ25

75

1 2RØ25

5 1RØ25

4 2RØ25

5

3

4 4

2 1RØ25 L=860 (1)

35

825

2 1RØ251 2RØ25

3 2RØ25

2

3

2 1RØ25

44

1215

6015

55

17015 15 25 375 25 25 175 10275

BA

1

1

1-1

3

3

55.

522

521

.5

75

5 54x7.5=30

40

30 55

15

5 530

40

7.5 55 15 7.53-3

16

55.

522

521

.5

75

16

15 15

5 530

40

7.5 55 15 7.52-2

55.

522

521

.5

75

16

2

212 18RUØ10/ 10 12 16RUØ10/ 25 12 19RUØ10/ 15 12 6RUØ10/ 15 12 8RUØ10/ 25

1 2RØ25

4 2RØ25

11 2RØ12

6 2RØ25

7 2RØ2210 2RØ22

9 2RØ22

8 2RØ22

3 2RØ25

11 2RØ12

11 2RØ12

11 2RØ12

13 9RUØ10/ 15

7 2RØ22

9 2RØ22

8 2RØ22

11 2RØ12

11 2RØ12


11 ±1RØ126 2RØ25

4 2RØ25 L=760 (2)760

3 2RØ25 L=925 (2)

35

890

359 2RØ22 L=395 (2)

395

8 2RØ22 L=500 (2)500

7 2RØ22 L=865 (2)830

35

10 2RØ22 L=480 (2)

35

445

11 ±1RØ12 L=640 (8)

20

620 11 ±1RØ12 L=640 (8)

20

620

12 RUØ10/15

7 2RØ22

4 2RØ25

11 ±1RØ12

11 ±1RØ12

11 ±1RØ12

12 RUØ10/25

12 RUØ10/15

13 RUØ10/15

12 19RUØ10/ 15 L=232 ( 67)

35

70

35

70

1 11 1

75

1

2

5

5 1RØ254 2RØ25, 1 2RØ25, 3 2RØ25

6

5 1RØ25

7

8

99

8

10

BETON (MB 30):

ARMATURA (RA 400/500):

589.473.645

KOLICINE MATERIJALA:

11.04.2011.420x580R 1:25/10MB 30 RA 400/500

GODIŠNJI ZADATAK - LIST br.1

PLAN ARMATURE POS 1 (var.1)M. STOJANOVIC

3.645 m3

589.47 kg

= 161.7 kg/m3 betona

8 2RØ22

5 1RØ25 1 2RØ25

75

1 2RØ25

5 1RØ25

4 2RØ25

5

3

4 4

2 1RØ25 L=860 (1)

35

825

2 1RØ251 2RØ25

3 2RØ25

2

3

2 1RØ25

1 Konstrukcija linijezatežućih sila - imksus.grf.rsimksus.grf.rs/nastava/beton/MTI HVE...

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