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PRORAČUN PREDNAPREGNUTOG NOSAČA

Poprečni presek nosača: Geometrijske karakteristike nosača:

F= 0.32925

I= 0.0437567

0.0601

0.0927

d= 1.2 m

Yt= 0.7277 m

0.1826 m

0.2814 m

i = 0.3646 m

zaštitni sloj betona a= 2 cm0.563436 m

KARAKTERISTIKE BETONA:faza utezanja MB 60 Eb= 38 GPa

dop. napon zatezanja= -4.00 MPa

dop. napon pritiska= 26.00 MPa

faza utezanja MB 60 Eb= 38 GPa

dop. napon zatezanja= -2.40 MPa

dop. napon pritiska= 20.50 MPa

fb= 33.00 MPa

KABLOVI:

1860 MPa Eb= 195 GPa

8.5 % Ø= 15.2 mm

red

1 3 5 1.39 1300 542.1 -367.3931

2 3 10 1.39 1300 542.1 -340.2881

3 3 15 1.39 1300 542.1 -313.18314 3 20 1.39 1300 542.1 -286.0781

5 2 40 1.39 1300 361.4 -118.438736 0 -0

7 0 -08 0 -0

9 0 -0

16.429 cm

2529.8 kN dužina sidrenja 50Ø= 0.76 m

180.7 kN16.68

nk= 14 kom

GLAVNI NOSAČ GN 1.1 T120/55

m2

m4

W1= m3

W2= m3

K1=

K2=

težište rez. kabla ebk=

pk=

εk=

brojužadi ak [cm] A [cm2]

napon

utezanja f k°[MPa] Nk°[kN] Mk°[kNm]

ak=

Nk°=

Nk1=Fk= cm2

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GEOMETRIJA NOSAČA:   REAKCIJE OSLONCA:

L= 21.77 m 90.35 90.35 kN

Lp= 0.4000 m 67.70 71.20 kN

0.00 0.00 kN

SPOLJAŠNJI UTICAJI: 47.08 49.52 kN

g= 8.30 kN/m' -15.50 -16.30 kN

A1 [kN] A2 [kN] A3 [kN] A4 [kN] A5 [kN] A6 [kN] A7 [kN] A8 [kN]ei [m] 5.8 5.36 5.36Δg1=   46.3 46.3 46.3

Δg2=   0 0 0

s= 32.2 32.2 32.2w= -10.6 -10.6 -10.6

0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.20 17.90 13.54 0.00 9.42 -3.10 -0.170.40 35.47 27.08 0.00 18.83 -6.20 -0.66

1.60 133.93 108.31 0.00 75.33 -24.80 97.79

3.20 246.61 216.63 0.00 150.66 -49.59 210.47

5.80 384.40 392.63 0.00 273.06 -89.89 348.26

6.47 410.81 406.97 0.00 283.03 -93.17 374.67

7.14 433.50 421.30 0.00 293.00 -96.45 397.36

7.81 452.46 435.64 0.00 302.97 -99.74 416.33

8.48 467.70 449.97 0.00 312.94 -103.02 431.56

9.68 485.70 475.70 0.00 330.83 -108.91 449.57

10.28 490.21 488.57 0.00 339.78 -111.85 454.0710.89 491.71 514.16 0.00 357.58 -114.80 455.57

11.16 491.39 507.31 0.00 352.82 -116.15 455.2513.26 468.21 454.90 0.00 316.37 -104.15 432.07

14.45 438.84 425.28 0.00 295.76 -97.36 402.7015.64 397.73 436.22 0.00 303.37 -99.87 361.59

17.08 332.68 334.28 0.00 232.48 -76.53 296.54

18.51 250.62 232.35 0.00 161.59 -53.19 214.48

19.94 151.55 130.42 0.00 90.70 -29.86 115.41

21.37 35.47 28.48 0.00 19.81 -6.52 -0.66

21.57 17.90 14.24 0.00 9.90 -3.26 -0.1721.77 0.00 0.00 0.00 0.00 0.00 0.00

Rg=

RΔg1=

RΔg2=

Rs=

Rw=

x [m]   Mg [kNm]MΔg1[kNm]

MΔg2[kNm]   Ms [kNm]   Mw [kNm]

podizanje nosačaMg [kNm]

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GUBICI SILE PRETHODNOG NAPREZANJA:

t=0 t=∞

2529.8 kN 2403.31 kN

n= 5.13 0.04 fk°

0.00045

2.6

k= 0.95 χ=   0.8

2403.31 kN

171.67 kN

k= 0.8

1922.65 kN

137.33 kN

UTICAJI OD PRETHODNOG NAPREZANJA:

faza prednaprezanja faza eksploatacije broj težištenk

0.00 0.00 0.00 0.00 0.00 0 0.000

0.20 0.00 0.00 0.00 0.00 0 0.0000.40 -349.02 515.00 -279.22 412.00 3 0.678

1.60 -349.02 515.00 -279.22 412.00 3 0.678

3.20 -969.82 1544.99 -775.86 1235.99 9 0.628

5.80 -1241.60 2059.98 -993.28 1647.98 12 0.603

6.47 -1354.11 2403.31 -1083.29 1922.65 14 0.563

7.14 -1354.11 2403.31 -1083.29 1922.65 14 0.5637.81 -1354.11 2403.31 -1083.29 1922.65 14 0.563

8.48 -1354.11 2403.31 -1083.29 1922.65 14 0.563

9.68 -1354.11 2403.31 -1083.29 1922.65 14 0.563

10.28 -1354.11 2403.31 -1083.29 1922.65 14 0.563

10.89 -1354.11 2403.31 -1083.29 1922.65 14 0.563

11.16 -1354.11 2403.31 -1083.29 1922.65 14 0.563

13.26 -1354.11 2403.31 -1083.29 1922.65 14 0.563

14.45 -1354.11 2403.31 -1083.29 1922.65 14 0.563

15.64 -1354.11 2403.31 -1083.29 1922.65 14 0.563

17.08 -969.82 1544.99 -775.86 1235.99 9 0.62818.51 -969.82 1544.99 -775.86 1235.99 9 0.628

19.94 -349.02 515.00 -279.22 412.00 3 0.67821.37 -349.02 515.00 -279.22 412.00 3 0.678

21.57 0.00 0.00 0.00 0.00 0 0.00021.77 0.00 0.00 0.00 0.00 0 0.000

Nk°= Nk0=

Δσrel(%)=

pretpostavljaju se poćetni gubici sileprethodnog naprezanja od 5%

εsk,t=

ϕt=

Nk0= pretpostavljaju se poćetni gubici sileprethodnog naprezanja od 20%Nk01=

Nkt=

Nk01=

x [m]   Mk0 [kNm]   Nk0 [kNm]   Mkt [kNm]   Nkt [kNm]   ebk [m]

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UKUPNI UTICAJI U PRESEKU:

x [m]

faza prednaprezanja podzanje nosača   faza eksploatacije 1 faza eksploatacije 2

g+pn g+pn g+pn+Δg1+Δg2+s g+pn+Δg1+w

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.20 17.90 0.00 -0.17 0.00 40.86 0.00 28.34 0.00

0.40 -313.55 515.00 -349.69 515.00 -197.83 412.00 -222.87 412.00

1.60 -215.09 515.00 -251.23 515.00 38.35 412.00 -61.77 412.00

3.20 -102.41 515.00 -138.55 515.00 -161.97 412.00 134.42 412.00

5.80 -857.20 2059.98 -893.34 2059.98 56.82 1647.98 -306.14 1647.98

6.47 -943.30 2403.31 -979.44 2403.31 17.52 1922.65 -358.68 1922.65

7.14 -920.61 2403.31 -956.75 2403.31 64.52 1922.65 -324.94 1922.65

7.81 -901.65 2403.31 -937.79 2403.31 107.78 1922.65 -294.92 1922.65

8.48 -886.41 2403.31 -922.55 2403.31 147.33 1922.65 -268.63 1922.65

9.68 -868.41 2403.31 -904.55 2403.31 208.95 1922.65 -230.79 1922.6510.28 -863.91 2403.31 -900.05 2403.31 235.26 1922.65 -216.37 1922.65

10.89 -862.41 2403.31 -898.54 2403.31 280.16 1922.65 -192.22 1922.6511.16 -862.72 2403.31 -898.86 2403.31 268.23 1922.65 -200.73 1922.6513.26 -885.90 2403.31 -922.04 2403.31 156.19 1922.65 -264.32 1922.65

14.45 -915.27 2403.31 -951.41 2403.31 76.59 1922.65 -316.54 1922.65

15.64 -956.39 2403.31 -992.52 2403.31 54.03 1922.65 -349.22 1922.6517.08 -637.14 1544.99 -673.28 1544.99 123.59 1235.99 -185.43 1235.99

18.51 -719.20 1544.99 -755.34 1544.99 -131.30 1235.99 -346.08 1235.99

19.94 -197.47 515.00 -233.61 515.00 93.45 412.00 -27.11 412.00

21.37 -313.55 515.00 -349.69 515.00 -195.45 412.00 -221.78 412.00

21.57 17.90 0.00 -0.17 0.00 42.05 0.00 28.88 0.00

21.77 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Tmax= 90.35 kN 90.35 kN 205.12 kN 158.04 kN

TABLICA SA DOP. NAPONIMA U PRETHODNONAPREGNUTOM BETONU PREMA PNB 71

Mk0 [kNm]   Nk0 [kN]   Mk0 [kNm]   Nk0 [kN]   Mk0 [kNm]   Nkt [kN]   Mk0 [kNm]   Nkt [kN]

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KONTROLA IVIČNIH U PRESEKU:

x [m]

faza prednaprezanja podzanje nosača   faza eksploatacije 1 faza eksploatacije 2

g+pn g+pn g+pn+Δg1+Δg2+s g+pn+Δg1+w

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.20 -0.30 0.19 0.00 0.00 -0.68 0.44 -0.47 0.31

0.40 6.78 -1.82 7.38 -2.21 4.54 -0.88 4.96 -1.15

1.60 5.14 -0.76 5.74 -1.15 0.61 1.67 2.28 0.58

3.20 3.27 0.46 3.87 0.07 3.95 -0.50 -0.98 2.70

5.80 20.51 -3.00 21.11 -3.39 4.06 5.62 10.10 1.70

6.47 22.99 -2.88 23.59 -3.27 5.55 6.03 11.80 1.97

7.14 22.61 -2.64 23.21 -3.03 4.77 6.54 11.24 2.33

7.81 22.29 -2.43 22.90 -2.82 4.05 7.00 10.74 2.66

8.48 22.04 -2.27 22.64 -2.66 3.39 7.43 10.31 2.94

9.68 21.74 -2.07 22.34 -2.46 2.36 8.09 9.68 3.3510.28 21.67 -2.03 22.27 -2.42 1.93 8.38 9.44 3.50

10.89 21.64 -2.01 22.24 -2.40 1.18 8.86 9.04 3.7611.16 21.65 -2.01 22.25 -2.40 1.38 8.73 9.18 3.6713.26 22.03 -2.26 22.63 -2.65 3.24 7.53 10.24 2.99

14.45 22.52 -2.58 23.12 -2.97 4.57 6.67 11.10 2.42

15.64 23.21 -3.02 23.81 -3.41 4.94 6.42 11.65 2.0717.08 15.29 -2.18 15.89 -2.57 1.70 5.09 6.84 1.75

18.51 16.65 -3.07 17.25 -3.46 5.94 2.34 9.51 0.02

19.94 4.85 -0.57 5.45 -0.96 -0.30 2.26 1.70 0.96

21.37 6.78 -1.82 7.38 -2.21 4.50 -0.86 4.94 -1.14

21.57 -0.30 0.19 0.00 0.00 -0.70 0.45 -0.48 0.31

21.77 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

faza prednaprezanja

23.21 < 26.00 MPa OK

-3.07 < -4.00 MPa OK

podzanje nosača23.81 < 26.00 MPa OK

-3.46 < -4.00 MPa OK

faza eksploatacije11.80 < 20.50 MPa OK

-1.15 < -2.40 MPa OK

σb1 [MPa] σb2 [MPa] σb1 [MPa] σb2 [MPa] σb1 [MPa] σb2 [MPa] σb1 [MPa] σb2 [MPa]

maxσo=maxσo=

maxσo=maxσo=

maxσo=maxσo=

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POTREBNA ARMATURA U PRESEKU:

x [m]

aza prednaprezanj podzanje nosača   aza eksploatacije 1 aza eksploatacije 2 maksimalna

t=0 t=0 t=∞ t=∞   armatura

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.75 0.00

0.20 2.75 0.00 0.00 0.01 6.28 0.00 4.36 0.00 6.28 0.01

0.40 0.00 6.19 0.00 7.80 0.00 2.60 0.00 3.71 0.00 7.80

1.60 0.00 1.84 0.00 3.42 0.00 0.00 0.00 0.00 0.00 3.42

3.20 0.00 0.00 0.00 0.00 0.00 1.03 2.15 0.00 2.15 1.03

5.80 0.00 7.22 0.00 8.96 0.00 0.00 0.00 0.00 0.00 8.96

6.47 0.00 5.96 0.00 7.49 0.00 0.00 0.00 0.00 0.00 7.49

7.14 0.00 5.07 0.00 6.51 0.00 0.00 0.00 0.00 0.00 6.51

7.81 0.00 4.38 0.00 5.74 0.00 0.00 0.00 0.00 0.00 5.74

8.48 0.00 3.85 0.00 5.15 0.00 0.00 0.00 0.00 0.00 5.15

9.68 0.00 3.26 0.00 4.48 0.00 0.00 0.00 0.00 0.00 4.4810.28 0.00 3.12 0.00 4.32 0.00 0.00 0.00 0.00 0.00 4.32

10.89 0.00 3.08 0.00 4.27 0.00 0.00 0.00 0.00 0.00 4.2711.16 0.00 3.09 0.00 4.28 0.00 0.00 0.00 0.00 0.00 4.2813.26 0.00 3.83 0.00 5.13 0.00 0.00 0.00 0.00 0.00 5.13

14.45 0.00 4.87 0.00 6.29 0.00 0.00 0.00 0.00 0.00 6.29

15.64 0.00 6.50 0.00 8.07 0.00 0.00 0.00 0.00 0.00 8.0717.08 0.00 5.15 0.00 6.88 0.00 0.00 0.00 0.00 0.00 6.88

18.51 0.00 8.73 0.00 10.34 0.00 0.00 0.00 0.00 0.00 10.34

19.94 0.00 1.09 0.00 2.64 0.24 0.00 0.00 0.00 0.24 2.64

21.37 0.00 6.19 0.00 7.80 0.00 2.49 0.00 3.66 0.00 7.80

21.57 2.75 0.00 0.00 0.01 6.47 0.00 4.44 0.00 6.47 0.01

21.77 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

 ARMATURA B500B

20.00

20.00 cm

55.00 cm

115.00 cm

115.00 cm

Fa1 [cm2]   Fa2 [cm2]   Fa1 [cm2]   Fa2 [cm2]   Fa1 [cm2]   Fa2 [cm2]   Fa1 [cm2]   Fa2 [cm2]   Fa1 [cm2]   Fa2 [cm2]

σdop= kN/cm2

b1=

b2=

h1=

h2=

Fa=Za/σdop

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PRESEK PORED OSLONCA x= 0.4 m

F= 0.32925I= 0.0437567

S= 0.05296

b= 0.20 m

412.00 kN

205.12 kN

1.2513 MPa

1.2413 MPa

-0.7644 MPa

OK < 1.15 bez osiguranja

OK < 2.75 sa osiguranjem

osiguranje uzengijama Ø10

0.785200.00 MPam= 2.00

20.54 cm

USVOJENE UZENGIJE Ø10/10cm

PRESEK PORED OSLONCA x= 0.2 m

F= 0.32925

I= 0.0437567S= 0.05296

b= 0.20 m

0.00 kN

205.12 kN

0.0000 MPa

1.2413 MPa

-1.2413 MPaPREKORAČEN NAPON   > 1.15 bez osiguranja

OK < 2.75 sa osiguranjem

osiguranje uzengijama Ø100.785

200.00 MPa

m= 2.00

12.65 cm

USVOJENE UZENGIJE Ø10/10cm

m2

m4

m3

Nkt=

Tt,max=

σb=Nkt/F=

τ=T·S/(I·b)=

σ2=σb/2-√(σb2/4+τ2)=

Fau= cm2

y, dop=

epot=

m2

m4m3

Nkt=

Tt,max=

σb=Nkt/F=

τ=T·S/(I·b)=

σ2=σb/2-√(σb2/4+τ2)=

Fau= cm2

y, dop=

epot=

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PRORAČUN UGIBA

relativna vlažnost sredine 70% i dimenziji nosača dm=   20.00 cm

3.00

2.60

TRAJNI UGIB OD PREDNAPREZANJA

-15.6676 cm

δko=   -4.4764 cm

δkt=   -3.5812 cm

-3.0165 cm

δgo=   1.4599 cm

-3.654 cm

-4.982 cm

1.4374 cm

0.0000 cm

0.9997 cm

-0.3291 cm

koeficijent tečenja pri prednaprezanjuϕg=

koeficijent tečenja pri nanošenju Δg t=28 dana ϕΔg=  

Δkt = δkt + 0.5*(δko + δkt)*ϕg=

elasticni ugib od prednaprezanja u t0

elasticni ugib od prednaprezanja u t∞

UGIB U t0

Δ0 = δko + δgo=

elasticni ugib od sopstvene težine u t0

UKUPNI TRAJNI UGIB U t∞  t1=Δkt + δg(1+ϕg ) + δΔg1(1+ϕΔg) + δso=

  t2=Δkt + δg(1+ϕg ) +δΔg1(1+ϕΔg+ δwo=

δΔg1= elasticni ugib od Δg1 u t0

δΔg2= elasticni ugib od Δg2 u t0

δso= elasticni ugib od snega u t0

δwo= elasticni ugib od vetra u t0

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PRORAČUN KOEFICIJENTA SIGURNOSTI PROTIV POJAVE PRSLINA

primer za presek u x=10.89F= 3292.5

60128.324

56.344 cm

1922.65 kN

3.40 MPa

2.34 MPa

23.856 MPa napon u betonu od krajnje sile Nkt

1574.96 kNm momenat savijanja pri pojavi prslina

x [m] Mpr [kNm]0.00 0.00 140.55   0.000

0.20 40.86 140.55   3.440

0.40 81.38 495.01   6.082

1.60 317.57 495.01   1.559

3.20 613.89 1142.13   1.860

5.80 1050.09 1434.79   1.366

6.47 1100.81 1574.96   1.431

7.14 1147.81 1574.96   1.372

7.81 1191.07 1574.96   1.3228.48 1230.61 1574.96   1.280

9.68 1292.24 1574.96   1.219

10.28 1318.55 1574.96   1.194

10.89 1363.45 1574.96   1.155

11.16 1351.52 1574.96   1.165

13.26 1239.48 1574.96   1.271

14.45 1159.88 1574.96   1.358

15.64 1137.32 1574.96   1.385

17.08 899.44 1142.13   1.270

18.51 644.56 1142.13   1.77219.94 372.67 495.01   1.328

21.37 83.76 495.01   5.910

21.57 42.05 140.55   3.343

21.77 0.00 140.55   0.000

cm2

W1= cm3

težište rez. kabla ebk=

Nkt=

bzm=

bz=

σbK,∞=

Mpr=

Mqmax [kNm] γp=Mpr/Mqmax >1.15

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DIJAGRAMI KABLOVA POPRECNI PRESEK 0   cm²

1559   Mpa   21   cm   0   cm²

1763   Mpa   h=   103.57143 cm   4.5   cm

7.99 ‰ D= 120   cm   4.5   cm

45 ‰ B= 55   cm   Ap= 16.68   cm²

a=   5.5120   b= 20   cm   16.429   cm

b=  1514.9589 δ=   0.2028   5.08 ‰

Δεp=   10.0000   ‰

s= 0.1956293 Σdilatacije ‰   2.432 -0.089 1.892 11.432 15.080

αb/σa,p   0.7259 -0.0450 37.838537 40 159.8080

Dbu Dbu2 Dau Zau Zp

sile kN   2669 4 0 0 2666   0

ζ=   0.3888212 0.3321203

Dbu Dbu2 Dau Zau Zp ΣSila   kN 2669 4 0 0 2666

krak   cm 95.7 82.8 99.1 11.9 0.0

moment kNm   2554 -3 0 0 0   2551

2551 kNm

x [m]

faza prednaprezanja podzanje nosača   faza eksploatacije 1 faza eksploatacije 2

g+pn g+pn g+pn+Δg1+Δg2+s g+pn+Δg1+w

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.20 17.90 142.51 -0.17 15369.23 40.86 62.44 28.34 90.02

0.40 -313.55 8.14 -349.69 7.30 -197.83 12.90 -222.87 11.45

1.60 -215.09 11.86 -251.23 10.16 38.35 66.53 -61.77 41.30

3.20 -102.41 24.91 -138.55 18.41 -161.97 15.75 134.42 18.98

5.80 -857.20 2.98 -893.34 2.86 56.82 44.90 -306.14 8.33

6.47 -943.30 2.70 -979.44 2.60 17.52 145.59 -358.68 7.11

7.14 -920.61 2.77 -956.75 2.67 64.52 39.54 -324.94 7.85

7.81 -901.65 2.83 -937.79 2.72 107.78 23.67 -294.92 8.65

8.48 -886.41 2.88 -922.55 2.77 147.33 17.32 -268.63 9.509.68 -868.41 2.94 -904.55 2.82 208.95 12.21 -230.79 11.05

10.28 -863.91 2.95 -900.05 2.83 235.26 10.84 -216.37 11.7910.89 -862.41 2.96 -898.54 2.84 280.16 9.11 -192.22 13.27

11.16 -862.72 2.96 -898.86 2.84 268.23 9.51 -200.73 12.7113.26 -885.90 2.88 -922.04 2.77 156.19 16.33 -264.32 9.65

14.45 -915.27 2.79 -951.41 2.68 76.59 33.31 -316.54 8.06

15.64 -956.39 2.67 -992.52 2.57 54.03 47.22 -349.22 7.31

17.08 -637.14 4.00 -673.28 3.79 123.59 20.64 -185.43 13.76

18.51 -719.20 3.55 -755.34 3.38 -131.30 19.43 -346.08 7.37

19.94 -197.47 12.92 -233.61 10.92 93.45 27.30 -27.11 94.1121.37 -313.55 8.14 -349.69 7.30 -195.45 13.05 -221.78 11.50

21.57 17.90 142.51 -0.17 15369.23 42.05 60.68 28.88 88.33

21.77 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

PRORAČUN KAPACITETA NOSIVOSTI PRESEKA I GLOBALNI KOEFICIJENT SIGURNOSTI OD LOMANOSAČA SAVIJANJEM

Aa1=

σp1=   dpl= Aa2=σp2=   a1=εp1=   a2=εp1=

ap=

ε,dec=

εb εbd εa2 εa1 εp

Mnos=   s=M/Mnos >1.8

M [kNm]   s σb1 [MPa] σb2 [MPa] σb1 [MPa] σb2 [MPa] σb1 [MPa] σb2 [MPa]