Program_OBK.pdf

7/29/2019 Program_OBK.pdf

http://slidepdf.com/reader/full/programobkpdf 1/48

SVEUČILIŠTE U SPLITU

GRA ĐEVINSKO-ARHITEKTONSKI FAKULTET

Alen Harapin, Domagoj Matešan, Danijela Brzović,

Marija Smilović, Nikola Grgić

RADNI MATERIJALI ZA POMOĆ PRI IZRADIPROGRAMSKOG ZADATKA IZ PREDMETA:

OSNOVE BETONSKIH KONSTRUKCIJA

Split, 2008.



Sveu č ilište u Splitu, Gra đ evinsko-arhitektonski fakultet Katedra za betonske konstrukcije i mostove

Osnove betonskih konstrukcija – Interna skripta str. 2

NAPOMENA

Ovi radni materijali izrađeni su kao vodilja studentima pri rješavanju programa iz predmeta „Osnove betonskihkonstrukcija“ na Sveučilišnom studiju Građevinsko-arhitektonskog fakulteta u Splitu.

Programski zadatak je dimenzionirati jednu relativno jednostavnu armiranobetonsku građevinu. Ovi radnimaterijali rađeni su s namjerom da budu vodič u izradi programa, ali i uzorni primjerak kako konačni projekt treba

izgledati.

Autori





SADRŽAJ

1. Tehnički opis....................................................................................................................................................6 2. ANALIZA OPTEREĆENJA ......................................................... ................................................................. ....7

2.7 Pozicije 100 – Etaže...............................................................................................................................7 2.8 Pozicije 200 – Krov .............................................................. ......................................................... .........8 2.9 Stubište..................................................................................................................................................9

3. PRORAČUN PLOČA pozicija 200 ....................................................... .......................................................... 10 3.1 Proračun pozicije 201...........................................................................................................................11 3.2 Dimenzioniranje pozicije 201................................................................................................................14 3.3 Kontrola pukotina ploče pozicije 201....................................................................................................15

4. PRORAČUN PLOČA pozicija 100 ....................................................... .......................................................... 18 4.7 Proračun pozicije 101...........................................................................................................................19 4.8 Proračun pozicije 102...........................................................................................................................20 4.9 Proračun pozicije 103...........................................................................................................................21 4.10 Proračun pozicije 104...........................................................................................................................22 4.11 Proračun pozicije 106 – Stubište..........................................................................................................23 4.12 Proračun pozicije 105 – Ploča uz stubište............................................................................................23 4.13 Prikaz dobivenih rezultata na pločama pozicija 100.............................................................................24 4.14 Dimenzioniranje ploča pozicija 100 ............................................................. .........................................25

5. PRORAČUN KONTINUIRANOG NOSAČA pozicija 200...............................................................................29 5.1 Skica sustava.......................................................................................................................................29 5.2 Analiza opterećenja..............................................................................................................................30 5.3 Proračun nosača..................................................................................................................................31 5.4 Dimenzioniranje nosača na moment savijanja ................................................................ .....................32 5.5 Dimenzioniranje nosača na poprečnu silu............................................................................................34

6. PRORAČUN KONTINUIRANOG NOSAČA pozicija 100...............................................................................36 6.1 Skica sustava.......................................................................................................................................36 6.2 Analiza opterećenja..............................................................................................................................37 6.3 Proračun nosača..................................................................................................................................38 6.4 Dimenzioniranje nosača na moment savijanja ................................................................ .....................41 6.5 Dimenzioniranje nosača na poprečnu silu............................................................................................43 6.6 Kontrola pukotina u 1. polju..................................................................................................................45 6.7 Kontrola progiba za 1. polje..................................................................................................................47





ZADATAK Na priloženim skicama dana je shema nosivih armirano betonskih konstrukcija jednog objekta. Potrebno je

izraditi planove pozicija, proračun i planove armature nosivih armirano betonskih konstrukcija.

U tablici su dane zadane veličine.

Oznaka Veličina Jedinica Opis

L1 7.0 (m) 'raster' u uzdužnom smijeru

L2 6.4 (m) 'raster' u poprečnom smijeru

H 3.6 (m) visina etaža

p 4.0 (kN/m2) pokretno opterećenje

σtla,dop 0.50 (MN/m2) dopušteno naprezanje u tlu

Zv 3 zona vjetra

Zp 8 zona potresa

S B 500B armatura

C C 30/37 klasa betona

(i) Presjek

H

H

x1 x2 x3 x4

L L L3 x L

1 1 1

1





(ii) Međuetaže

L L L3 x L

L

L

L

3 x L

y4

y3

y2

y1

x1 x2 x3 x4

11 x 3312 x 15

11 x 3312 x 15

1 1 1

1

2

2

2

2

(iii) Krovna konstrukcija

y4

y3

y2

y1

x1 x2 x3 x4

L L L3 x L

L

L

L

3 x L

1 1 1

1

2

2

2

2





1. TEHNIČKI OPIS

Predmet ovog projekta su proračun međukatnih i krovnih konstrukcija jedne armiranobetonske građevine.

Predmetna građevina sastoji se od prizemlja i kata, a namjena građevine je poslovna. Završna ploča kata jeujedno i ravni krov građevine.

Sve međ

ukatne konstrukcije su AB ploče debljine d=16.0 cm.

Za vertikalnu komunikaciju između katova predviđene je armiranobetonsko stepenište debljine nosive pločed=16.0 cm.

Za predmetnu građevinu su izvršeni geotehnički istražni radovi. Na osnovi tih istraživanja određeno je dozvoljenonaprezanje u tlu, koje na dubini temeljenja se iznosi σdop = 0.44 MPa.

Prema važećim propisima za proračun utjecaja vjetra na građevinu, građevina se gradi se u III vjetrovnoj zoni.

Građevina se gradi na lokaciji koja prema važećim propisima spada u VIII seizmičku zonu, tj. u kojoj se zapovratni period od 500 god, uz vjerojatnost pojave 66.6% očekuje pojava potresa VIII° MCS skale.





2. ANALIZA OPTEREĆENJA

2.7 Pozicije 100 – Etaže

a) stalno opterećenje

AB ploca

Toplinska izolacija

AB estrih

Završna obrada podaParket u ljepilu

Parna brana

16.0 cm

4.0 cm

5.0 cm

2.0 cm

0.5 cm

PE folija --- cm

d (m) γ (kN/m3) d × γ

Pregrade 1.00

Završna obrada poda – parket 0.02 12.0 0.24

AB estrih 0.05 25.0 1.25

Toplinska izolacija 0.04 5.0 0.20

Hidroizolacija 0.005 20.0 0.10

AB. ploča 0.16 25.0 4.00

Ukupno stalno opterećenje: g100 = 6.80 (kN/m2)

b) pokretno opterećenje

Pokretno opterećenje se uzima prema pravilniku: HRN EN 1991-2-1.

U našem slučaju, zadano je zadatkom.

q100 = 4.0 kN/m2





2.8 Pozicije 200 – Krov


AB ploca

Toplinska izolacijaVišeslojna hidro izolacija

Betonske ploce naplasticnim podlošcima

Parna branaBeton za pad

16.0 cm

8.0 cm0.5 cm

5.0 cm

0.5 cm

8.0 cm

d (m) γ (kN/m3) d × γ

Betonske ploče na plastičnim podlošcima 0.05 25.0 1.25

Hidroizolacija + parna brana 0.01 20.0 0.20Toplinska izolacija 0.08 5.0 0.40

Beton za pad 0.08 24.0 1.92

AB. ploča 0.16 25.0 4.00

Ukupno stalno opterećenje: g200 = 7.80 (kN/m2)


Za pokretno opterećenje uzima se opterećenje snijegom i vjetrom. Opterećenje snijegom za ravnekrovove, u područ jima gdje je snijeg rijedak (prema pravilniku) iznosi 0.50 kN/m2, pa se za pokretnoopterećenje neprohodnih ravnih krovova može uzeti zamjenjujuća vrijednost:

q200 = s + w ≈ 1.0 kN/m2





2.9 Stubište


1 6

33

1 5

Gazište ... 2 cmCem. mort ... 1 cmStepenikBet. ploca ... 16 cm

α

1 7 . 6

cm6.174.24cos

16cos

hh

4.24;455.033

15

š

vtg

st

st

==α

=′

°=α===α

d (m) γ (kN/m3) d × γ

Završna obrada gazišta – kamena ploča 0.02 28.0 0.56

Cementni namaz (max. 1.0 cm) 0.01 20.0 0.20

Stuba 0.075 24.0 1.80

AB. ploča (h'=17.6 cm) 0.176 25.0 4.40

Ukupno stalno opterećenje: gst = 6.96 (kN/m2)


Pokretno opterećenje se uzima prema pravilniku: HRN EN 1991-2-1.

U našem slučaju, uzet ćemo ga jednako kao na pločama.

qst = 4.0 kN/m2





3. PRORAČUN PLOČA POZICIJA 200

y4

y3

y2

y1

x1 x2 x3 x4

L L L3 x L

L

L

L

3 x L

1 1 1

1

2

2

2

2

201





3.1 Proračun pozicije 201

L = 6.40 m

g, q

Mp1

L = 6.40 m L = 6.40 m

Ml Ml

Mp2Mp1

2 2 2

Polje 1

mkNm5.393.350.16.2535.1MMM

mkNm3.34.600.1080.0LqkM

mkNm6.254.680.7080.0LgkM

qqggsd

222q

222g

=⋅+⋅=⋅γ+⋅γ=

=⋅⋅=⋅⋅=

=⋅⋅=⋅⋅=

Polje 2

mkNm3.120.150.10.835.1MMM

mkNm0.14.600.1025.0LqkM

mkNm0.84.680.7025.0LgkM

qqggsd

222q

222g

=⋅+⋅=⋅γ+⋅γ=

=⋅⋅=⋅⋅=

=⋅⋅=⋅⋅=

Ležaj

mkNm2.491.450.19.3135.1MMM

mkNm1.44.600.1100.0LqkM

mkNm9.314.680.7100.0LgkM

qqggsd

222q

222g

=⋅+⋅=⋅γ+⋅γ=

=⋅⋅−=⋅⋅=

=⋅⋅−=⋅⋅=

U nastavku je, samo kao orijentacija, prikazan proračun numeričkim modelom. Napravljena su dva modela:

(i) Model sa slobodnim oslanjanjem na sve strane

(ii) Model sa slobodnim oslanjanjem samo u smjeru y osi.

Vidljivo je dobro slaganje „ručno“ i „računalno“ proračunatih veličina. Također je vidljivo da momenti u smjeru x (Mx),koje smo zanemarili ipak postoje.





Prikaz rezultata na numeričkom modelu (i): Model sa slobodnim oslanjanjem na sve strane

Mx (kNm/m)

My (kNm/m)

6.38

6.38

6.45

5.95

6.27

6.05

5.99

6.05

6.27

5.95

6.45

6.38

6.38

6.25

5.98

4.260.64

-3.28-6.19

-7.14

-4.96

-0.03

3.75-0.58

-5.23-8.48

-9.32

-6.72

-1.28

5.62

3.48

1.58

1.583.48

5.62

5.99

3.75

-0.58

-5.23

-8.48

-9.32-6.72

-1.28

5.95 -1.49-6.04

-8.22

-8.22

-6.04

-1.49

-2.94-8.85

-12.21

-13.11

-11.52

-7.39

-1.51

-2.94

-8.85

-12.21

-13.11-11.52

-7.39-1.51

5.95

-8.22

6.38

6.38

6.25

5.98

5.75

6.25

5.98

5.75

-0.30

2.74

4.88

4.53

1.43

-3.73

-8.22-4.36

-0.10

-0.10

-4.36

-3.731.43

4.53

4.88

2.74

-0.30

5.98

6.25

6.386.38

6.45

6.05

6.27

6.27

6.05

6.45

5.75

-13.11

-12.21

-8.85

-2.94

-11.52

-1.51-7.39

-1.49-6.04

-6.04

-1.49

-2.94-8.85

-12.21

-13.11

-11.52

-7.39

-1.51

5.75

X Y

Z

48.06

48.06

48.06

48.06

35.86

16.26

7.59

7.59

35.86

7.59

7.59

-35.44

-37.21

-19.69

7.93

-25.09-6.82

15.80

-3.50

-10.52

-3.5015.80

7.93-19.69

-35.44

-37.21

-25.09

-6.82

11.31

-11.23

-18.33

-11.2311.31

4.71-24.75

-39.03

-39.89

-27.52

-6.11

4.71

-24.75

-39.03

-39.89

-27.52-6.11

-39.03

-24.75

4.71

11.31-11.23

-18.33

-11.23

11.31

4.71-24.75

-39.03

-39.89

-27.52

-6.11

-39.89

-27.52-6.11

-37.75

-33.70

-28.82

-23.51

-17.72

-11.61

-39.79

-38.35

-37.21

-38.35

-39.79

-37.75

-33.70

-28.82

-23.51

-17.72

-11.61

-7.25

-17.04-13.46

-9.54

-6.44

-3.99

-1.93

-1.07

-16.65

-13.18

-10.52

-13.18

-16.65

-17.04

-13.46

-9.54

-6.44

-3.99

-1.93

-1.07

X Y

Z





Prikaz rezultata na numeričkom modelu (ii): Model sa slobodnim oslanjanjem samo u smjeru y osi

Mx (kNm/m)

My (kNm/m)

4.87

2.88

-0.89

-5.20-2.00

0.67

1.57

0.67

-2.00

-5.20-0.892.884.87

5.04

3.40

4.87

4.81

4.63

4.223.42

1.940.05

1.61

1.66

1.68

1.63

1.44

0.86-0.20

1.54

1.52

1.51

1.51

1.52

1.54

1.57

1.61

1.66

1.68

1.63

1.44

0.86

-0.20

-5.20

-5.19

-5.19

-5.19

-5.19

-5.19

-5.19

-5.21

-5.23

-5.26

-5.32

-5.25

-5.53

-2.19

-5.21

-5.23

-5.26

-5.32

-5.25-5.53-2.19

-5.20

-5.19

2.88

4.87

5.043.40

0.87

5.043.40

0.87

0.87

3.374.99

4.82

2.86

-0.90

-5.19-2.02

0.63

0.63

-2.02

-0.902.864.82

4.99

3.37

0.87

3.40

5.04

4.872.88-0.89

0.67

-2.00

-2.000.67

-0.89

0.87

-5.32

-5.26

-5.23

-5.21

-5.25

-2.19-5.53

-5.19

-5.19

-5.19

-5.19

-5.21

-5.23

-5.26

-5.32-5.25

-5.53

-2.19

0.87

X Y

Z

40.11

40.11

40.11

40.11

40.02

39.58

37.52

37.52

40.02

37.52

37.52

-36.75

-38.11

-21.68

7.10

-25.75-6.65

15.41

-5.08

-11.91

-5.0815.41

7.10-21.68-36.75

-38.11

-25.75

-6.65

15.48

-5.06

-11.91

-5.0615.48

7.10-21.78-36.92

-38.27

-25.86

-6.67

7.10

-21.78

-36.92

-38.27

-25.86-6.67

-36.92

-21.78

7.10

15.48-5.06

-11.91

-5.06

15.48

7.10-21.78-36.92

-38.27

-25.86

-6.67

-38.27-25.86

-6.67-38.39

-38.53

-38.63-38.59

-38.15

-36.75

-38.19

-38.13

-38.11

-38.13

-38.19

-38.39

-38.53

-38.63

-38.59

-38.15

-36.75

-34.52

-11.94

-12.01

-12.10

-12.20

-12.25

-11.93

-10.85

-11.90

-11.91

-11.91

-11.91

-11.90

-11.94

-12.01

-12.10

-12.20-12.25

-11.93

-10.85





3.2 Dimenzioniranje pozicije 201

Beton: C 30/37; f ck=30.0 MPa MPa0.205.1

0.30f f

c

ckcd ==

γ=

Armatura: B 500B; f yk=500.0 MPa MPa8.43415.1

0.500f f

s

ykyd ==

γ=

Polje 1:

1 6

100

1 3

3

As1

117.00.213100

3950

f db

M

kNm5.39M

2

cd

2

eff

sdsd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.9200.203‰5.2‰0.10 2c1s =ζ=ξ=ε=ε

2

yd

sd1s cm59.7

5.4313920.0

3950

f d

MA =

⋅⋅=

⋅⋅ζ=

Polje 2:

1 6

100

1 3

3As1

036.00.213100

1230

f db

M

kNm3.12M

2cd

2eff

sdsd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.9650.099‰1.1‰0.10 2c1s =ζ=ξ=ε=ε

2

yd

sd1s cm25.2

5.4313965.0

1230

f d

MA =

⋅⋅=

⋅⋅ζ=





Ležaj:

1 6

100

1 3

3

As1

146.00.213100

4920

f db

M

kNm2.49M

2cd

2eff

sdsd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.9020.241‰2.3‰0.10 2c1s =ζ=ξ=ε=ε

2

yd

sd1s cm65.9

5.4313902.0

4920

f d

MA =

⋅⋅=

⋅⋅ζ=

3.3 Kontrola pukotina ploče pozicije 201

Mjerodavni momenti u ploči računaju se za radnu kombinaciju opterećenja ( p0.1g0.1q ⋅+⋅= ).

Polje 1

mkNm9.283.30.16.250.1MMM

mkNm3.34.600.1080.0LqkM

mkNm6.254.680.7080.0LgkM

qqggsd

222q

222g

=⋅+⋅=⋅γ+⋅γ=

=⋅⋅=⋅⋅=

=⋅⋅=⋅⋅=

Granična vrijednost širine pukotine: mm3.0wg = (EC-2 – uobičajena sredina)

Proračunska vrijednost širine pukotine: (EC-2):

smrmk sw ε⋅⋅β= Armatura ploče donja zona: R-785 (As1=7.85 cm2/m).Skica armature u polju:

1 6

100

1 3

3

As1=R-785

Prognozna širina pukotine:

smrmk sw ε⋅⋅β= β=1.7 - odnos računske i srednje širine pukotina

Proračun srednje deformacije armature:

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

σσ

⋅β⋅β−⋅σ

=ζ⋅σ

=ε2

s

sr 21

s

s

s

ssm 1

EE

As1 = R-785 = 7.85 cm2/m’

MPa0.00032GPa0.32Ecm

== - modul elastičnosti betona

MPa0.000200GPa0.200Es == - modul elastičnosti armature





25.632

200

E

E

cm

seI ===α

cm11.385.725.6

13100211

100

85.725.6

A

db211

b

Ax

1SeI

1SeI =⎟⎟ ⎠

⎞⎜⎜⎝

⎛

⋅⋅⋅

++−⋅⋅

=⎟⎟ ⎠

⎞⎜⎜⎝

⎛

⋅α⋅⋅

++−⋅⋅α

=

MPa7.307cm

kN

77.3085.7311.313

2890

A3xd

M

Az

M2

s

sd

s

sd

s ==⋅⎟ ⎠ ⎞⎜

⎝ ⎛ −=⋅⎟

⎠ ⎞⎜

⎝ ⎛ −≈⋅=σ

( )

( ) ( )

MPa8.131cm

kN18.13

85.7311.3

13

1238

A3x

d

M

Az

M

kNm38.12kNcm7.12376

1610029.0M

MPa9.20.303.0f 3.0f

MPa0.30f ;f 3.0f ;6

hbf M;

Az

M

2

s

cr

s

cr sr

2

cr

3232ckm,ct

ck32

ckm,ct

2

m,ctcr s

cr sr

==

⋅⎟ ⎠

⎞⎜⎝

⎛ −

=

⋅⎟ ⎠

⎞⎜⎝

⎛ −

≈⋅

=σ

==⋅

⋅=

=⋅=⋅=

=⋅≈⋅

⋅=⋅

=σ

0.11 =β - Rebrasta armatura

5.02 =β - Dugotrajno opterećenje

322

s

sr 21

s

ssm 1039.1908.0

0.200000

7.307

7.307

8.1315.00.11

0.200000

7.3071

E−⋅=⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟ ⎠

⎞⎜⎝

⎛ ⋅⋅−⋅=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

σσ

⋅β⋅β−⋅σ

=ε

Proračun srednjeg razmaka pukotina:

[ ]mmkk25.050sr

21rm ρφ

⋅⋅⋅+=

mm10=φ - Promjer najdeblje šipke (R-785 Ö Ø10/10)

8.0k1 = - Rebrasta armatura

5.0k2 = - Savijanje

( )0105.0

35.2100

85.7

A

A

eff ,c

sr =

⋅⋅==ρ - Djelotvorni koeficijent armiranja glavnom vlačnom armaturom

1 6

100

1 3

3 7 . 5

mm2.1450105.0

105.08.025.050kk25.050s

r 21rm =⋅⋅⋅+=

ρφ

⋅⋅⋅+=


mm3.0wm343.02.14500139.07.1sw gsmrmk =>=⋅⋅=ε⋅⋅β=





Pukotine ne zadovoljavaju, potrebno je povećati armaturu!Armatura ploče donja zona: Q-785, s preklopom ≥ 50 cm

2

m

mm1s1s cm63.9

220

5022085.7

š

pšAA =

+⋅=

+⋅=′

cm40.363.925.6

13100211

100

63.925.6

A

db211

b

Ax

1SeI

1SeI =⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⋅

⋅⋅++−⋅

⋅=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⋅α

⋅⋅++−⋅

⋅α=

MPa0.253cm

kN30.25

63.9340.3

13

2890

A3x

d

M

Az

M2

s

sd

s

sds ==

⋅⎟ ⎠

⎞⎜⎝

⎛ −=

⋅⎟ ⎠

⎞⎜⎝

⎛ −≈

⋅=σ

322

s

sr 21

s

ssm 1009.1864.0

0.200000

0.253

0.253

8.1315.00.11

0.200000

0.2531

E−⋅=⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟ ⎠

⎞⎜⎝

⎛ ⋅⋅−⋅=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

σσ

⋅β⋅β−⋅σ

=ε


[ ]mmkk25.050sr

21rm ρφ

⋅⋅⋅+=

mm10=φ - Promjer najdeblje šipke (R-785 Ö Ø10/10)


5.0k2 = - Savijanje

( )0128.0

35.2100

63.9

A

A

eff ,c

sr =


1 6

100

1 3

3 7 . 5

mm1.1280128.0

105.08.025.050kk25.050s

r 21rm =⋅⋅⋅+=

ρφ

⋅⋅⋅+=


mm3.0wm237.01.12800109.07.1sw gsmrmk =<=⋅⋅=ε⋅⋅β=

Pukotine zadovoljavaju!





4. PRORAČUN PLOČA POZICIJA 100

30 670 30 670 30 670 30700 700 700

2100

3 0

6 1 0

3 0

6 1 0

3 0

6 1 0

3 0

6 4 0

6 4 0

6 4 0

1 9 2 0

101 101

101 101

103103

102

102

105

106

106

169 363 168

y4

y3

y2

y1

11 x 3312 x 15

11 x 3312 x 15






M

y

L = 7.0

g, q

M

2

1

L

= 6 . 4

x

101-104M

101-102M

M

y,4

g, q/2

M

x,4

101-104M

101-102M=M

y,1

q/2

M

x,1+ _

Shema 1Shema 4

⎪⎩

⎪⎨⎧

=

==

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−=

−=

=

=

=⇒≈==0507.0k

0344.0k1Shema

0798.0k

0598.0k

0324.0k

0217.0k

4Shema90.091.00.7

4.6

l

l1y

1x

by

ax

4y

4x

x

y

( ) ( )

( ) ( ) mkNm8.74.31.250.12.735.1MMMM

mkNm0.184.31.250.12.735.1MMMM

mkNm4.30.7200.40344.0L2qkM

mkNm1.20.7200.40217.0L2qkM

mkNm2.70.780.60217.0LgkM

1,x2q

4,x2qq

4,xgg

xmin,sd

1,x2q

4,x2qq

4,xgg

xmax,sd

221

1x

1,x2q

221

4x

4,x2q

221

4x

4,xg

=−⋅+⋅=−⋅γ+⋅γ=

=+⋅+⋅=+⋅γ+⋅γ=

=⋅⋅=⋅⋅=

=⋅⋅=⋅⋅=

=⋅⋅=⋅⋅=

( ) ( )

y,4 4 2 2g y 2

y,4 4 2 2q 2 y 2

y,1 1 2 2q 2 x 2

y y,4 y,4 y,1sd,max g g q q 2 q 2

ysd,min

M k g L 0.0324 6.80 6.4 9.0 kNm m

M k q 2 L 0.0324 4.00 2 6.4 2.7 kNm m

M k q 2 L 0.0507 4.00 2 6.4 4.2 kNm m

M M M M 1.35 9.0 1.50 2.7 4.2 22.5 kNm m

M

= ⋅ ⋅ = ⋅ ⋅ =

= ⋅ ⋅ = ⋅ ⋅ =

= ⋅ ⋅ = ⋅ ⋅ =

= γ ⋅ + γ ⋅ + = ⋅ + ⋅ + =

= ( ) ( )y,4 y,4 y,1g g q q 2 q 2M M M 1.35 9.0 1.50 2.7 4.2 9.9 kNmmγ ⋅ + γ ⋅ − = ⋅ + ⋅ − =

( ) ( ) mkNm4.447.1150.19.1935.1MMM

mkNm7.110.700.40598.0LqkM

mkNm9.190.780.60598.0LgkM

102101qq

102101gg

102101sd

221

ax

102101q

221

ax

102101g

−=−⋅+−⋅=⋅γ+⋅γ=

−=⋅⋅−=⋅⋅=

−=⋅⋅−=⋅⋅=

−−−

−

−

( ) ( ) mkNm6.491.1350.12.2235.1MMM

mkNm1.134.600.40798.0LqkM

mkNm2.224.680.60798.0LgkM

104101qq

104101gg

104101sd

222

by

104101q

222

by

104101g

−=−⋅+−⋅=⋅γ+⋅γ=

−=⋅⋅−=⋅⋅=

−=⋅⋅−=⋅⋅=

−−−

−

−






M

y

L = 7.0

g, q

M

2

1

L

= 6 . 4

x

102-101M

M

y,3

g, q/2

M

x,3

101-102M

=M

y,1

q/2

M

x,1

+ _

102-101M102-101M

Shema 3 Shema 1

⎪⎩

⎪⎨⎧

=

==

⎪⎪⎩

⎪⎪⎨

⎧

−=

=

=

=⇒≈==0507.0k

0344.0k1Shema

0644.0k

0270.0k

0274.0k

3Shema90.091.00.74.6

l

l1y

1x

ax

3y

3x

x

y

( ) ( )

( ) ( ) mkNm2.114.37.250.11.935.1MMMM

mkNm4.214.37.250.11.935.1MMMM

mkNm4.30.7200.40344.0L2qkMmkNm7.20.7200.40274.0L2qkM

mkNm1.90.780.60274.0LgkM

1,x2q

3,x2qq

3,xgg

xmin,sd

1,x2q

3,x2qq

3,xgg

xmax,sd

221

1x

1,x2q

2213x3,x 2q

221

3x

3,xg

=−⋅+⋅=−⋅γ+⋅γ=

=+⋅+⋅=+⋅γ+⋅γ=

=⋅⋅=⋅⋅==⋅⋅=⋅⋅=

=⋅⋅=⋅⋅=

( ) ( )

y,3 3 2 2g y 2

y,3 3 2 2q 2 y 2

y,1 1 2 2q 2 x 2


ysd,min

M k g L 0.0270 6.80 6.4 7.5 kNm m

M k q 2 L 0.0270 4.00 2 6.4 2.2 kNm m

M k q 2 L 0.0507 4.00 2 6.4 4.2 kNm m

M M M M 1.35 7.5 1.50 2.2 4.2 19.7 kNm m

M

= ⋅ ⋅ = ⋅ ⋅ =

= ⋅ ⋅ = ⋅ ⋅ =

= ⋅ ⋅ = ⋅ ⋅ =

= γ ⋅ + γ ⋅ + = ⋅ + ⋅ + == ( ) ( )y,3 y,3 y,1

g g q q 2 q 2M M M 1.35 7.5 1.50 2.2 4.2 7.1 kNmmγ ⋅ + γ ⋅ − = ⋅ + ⋅ − =

( ) ( ) mkNm9.476.1250.15.2135.1MMM

mkNm6.120.700.40644.0LqkM

mkNm5.210.780.60644.0LgkM

101102qq

101102gg

101102sd

221

ax

101102q

221

ax

101102g

−=−⋅+−⋅=⋅γ+⋅γ=

−=⋅⋅−=⋅⋅=

−=⋅⋅−=⋅⋅=

−−−

−

−






M

y

L = 7.0

g, q

M

2

1

L

= 6 . 4

x

103-101M

M

y,3

g, q/2

M

x,3

=M

y,1

q/2

M

x,1

+ _

103-101M

103-101M

103-101M

k y,3

x,3k

k y,1

x,1k

Shema 3 Shema 1

⎪⎩

⎪⎨⎧

=

==

⎪⎪⎩

⎪⎪⎨

⎧

−=

=

=

=⇒≈==0353.0k

0500.0k1Shema

0741.0k

0151.0k

0335.0k

3Shema10.10941.14.60.7

l

l1y

1x

ax

3y

3x

x

y

( ) ( )

( ) ( ) mkNm8.35.35.150.10.535.1MMMM

mkNm3.145.35.150.10.535.1MMMM

mkNm5.30.7200.40353.0L2qkM

mkNm5.10.7200.40151.0L2qkMmkNm0.50.780.60151.0LgkM

1,x2q

3,x2qq

3,xgg

xmin,sd

1,x2q

3,x2qq

3,xgg

xmax,sd

221

1y

1,x2q

221

3y

3,x2q

221

3y

3,xg

=−⋅+⋅=−⋅γ+⋅γ=

=+⋅+⋅=+⋅γ+⋅γ=

=⋅⋅=⋅⋅=

=⋅⋅=⋅⋅==⋅⋅=⋅⋅=

( ) ( )

y,3 3 2 2g x 2

y,3 3 2 2q 2 x 2

y,1 1 2 2

q 2 x 2


y

sd,min

M k g L 0.0335 6.80 6.4 9.3 kNm m

M k q 2 L 0.0335 4.00 2 6.4 2.7 kNm m

M k q 2 L 0.0500 4.00 2 6.4 4.2 kNm m

M M M M 1.35 9.3 1.50 2.7 4.1 22.8 kNm m

M

= ⋅ ⋅ = ⋅ ⋅ =

= ⋅ ⋅ = ⋅ ⋅ =

= ⋅ ⋅ = ⋅ ⋅ == γ ⋅ + γ ⋅ + = ⋅ + ⋅ + =

= ( ) ( )y,3 y,3 y,1

g g q q 2 q 2M M M 1.35 9.3 1.50 2.7 4.1 10.5 kNmmγ ⋅ + γ ⋅ − = ⋅ + ⋅ − =

( ) ( ) mkNm0.461.1250.16.2035.1MMM

mkNm1.124.600.40741.0LqkM

mkNm6.204.680.60741.0LgkM

101102qq

101102gg

101102sd

222

ax

101102q

222

ax

101102g

−=−⋅+−⋅=⋅γ+⋅γ=

−=⋅⋅−=⋅⋅=

−=⋅⋅−=⋅⋅=

−−−

−

−






M

y

L = 7.0

g, q

M

2

1

L

= 6 . 4

x

104-101M

M

y,3

g, q/2

M

x,3

=M

y,1

q/2

M

x,1

+ _

104-101M

104-101M

104-101M

k y,3

x,3k

k y,1

x,1k104-105M104-105M

Shema 5 Shema 1

⎪⎩

⎪⎨⎧

=

==

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−=

−=

=

=

=⇒≈==0353.0k

0500.0k1Shema

0467.0k

0676.0k

0158.0k

0287.0k

5Shema10.10941.14.60.7

l

l1y

1x

by

ax

3y

3x

x

y

( ) ( )

( ) ( ) mkNm2.45.35.150.13.535.1MMMM

mkNm7.145.35.150.13.535.1MMMM

mkNm5.30.7200.40353.0L2qkM

mkNm5.10.7200.40158.0L2qkM

mkNm3.50.780.60158.0LgkM

1,x2q

3,x2qq

3,xgg

xmin,sd

1,x2q

3,x2qq

3,xgg

xmax,sd

221

1y

1,x2q

221

3y

3,x2q

221

3y

3,xg

=−⋅+⋅=−⋅γ+⋅γ=

=+⋅+⋅=+⋅γ+⋅γ=

=⋅⋅=⋅⋅=

=⋅⋅=⋅⋅=

=⋅⋅=⋅⋅=

( ) ( )

y,3 3 2 2

g x 2

y,3 3 2 2

q 2 x 2

y,1 1 2 2q 2 x 2


ysd,min

M k g L 0.0287 6.80 6.4 8.0 kNm m

M k q 2 L 0.0287 4.00 2 6.4 2.4 kNm m

M k q 2 L 0.0500 4.00 2 6.4 4.2 kNm m

M M M M 1.35 8.0 1.50 2.4 4.1 20.6 kNm m

M

= ⋅ ⋅ = ⋅ ⋅ =

= ⋅ ⋅ = ⋅ ⋅ == ⋅ ⋅ = ⋅ ⋅ =

= γ ⋅ + γ ⋅ + = ⋅ + ⋅ + =

= ( ) ( )y,3 y,3 y,1g g q q 2 q 2M M M 1.35 8.0 1.50 2.4 4.1 8.3 kNmmγ ⋅ + γ ⋅ − = ⋅ + ⋅ − =

( ) ( ) mkNm0.421.1150.18.1835.1MMM

mkNm1.114.600.40598.0LqkM

mkNm8.184.680.60676.0LgkM

101104qq

101104gg

101104sd

222

ax

101104q

222

ax

101104g

−=−⋅+−⋅=⋅γ+⋅γ=

−=⋅⋅−=⋅⋅=

−=⋅⋅−=⋅⋅=

−−−

−

−

( ) ( ) mkNm9.342.950.16.1535.1MMM

mkNm2.90.700.40467.0LqkM

mkNm6.150.780.60467.0LgkM

105104qq

105104gg

104101sd

221

by

105104q

221

by

105104g

−=−⋅+−⋅=⋅γ+⋅γ=

−=⋅⋅−=⋅⋅=

−=⋅⋅−=⋅⋅=

−−−

−

−





4.11 Proračun pozicije 106 – Stubište

30 670 30

700

169 363 168

531

Ls t =5.31 m

g, q

M l

M p

mkN00.4q

mkN96.6g

=

=

mkNm1.311.850.10.1435.1MMM

mkNm1.8

14

31.50.4

14

LqM

mkNm0.1414

31.596.6

14

LgM

p,qqp,ggp,sd

22st

p,q

22st

p,g

=⋅+⋅=⋅γ+⋅γ=

=⋅

=⋅

=

=⋅

=⋅

=

mkNm2.541.1450.15.2435.1MMM

mkNm1.148

31.50.4

8

LqM

mkNm5.248

31.596.68

LgM

l,qql,ggl,sd

22st

l,q

22st

l,g

=⋅+⋅=⋅γ+⋅γ=

=⋅

=⋅

=

=⋅

=⋅

=

4.12 Proračun pozicije 105 – Ploča uz stubište

Ploča je vrlo kratka. Na njoj se realizira moment upetosti s obje strane. Prora čun nije potreban. Armirati ukontinuitetu s pločama 104 i stubištem 106.





4.13 Prikaz dobivenih rezultata na pločama pozicija 100

Konačne momente na ležajevima dobivamo na način da uprosječimo momente s lijeve i desne strane.

( )mkNm2.46

2

9.474.44=

+… itd.

18.0

22.5

44.4

49.6

21.4

19.7

47.9 44.447.9

18.0

22.5

49.6

14.3

22.8

42.0 46.0

18.0

22.5

44.4

49.6

21.4

19.7

47.9 44.447.9

18.0

22.5

49.6

42.0 46.0

14.7

20.6 34.9 54.2

34.9

31.1

31.1

46.2 46.2

47.845.8

47.845.8

46.2 46.2





4.14 Dimenzioniranje ploča pozicija 100


0.30f f

c

ckcd ==

γ=


0.500f f

s

ykyd ==

γ=

Ploč a 101 – polje

1 6

100

1 3

3

As1

067.00.213100

2250

f db

M

mkNm5.22M

2cd

2sd

sd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.949‰6.1‰0.10 2c1s =ζ=ε=ε

mcm19.45.4313949.0

2250

f d

MA 2

yd

sd1s =

⋅⋅=

⋅⋅ζ=


1 6

100

1 3

3

As1

063.00.213100

2140

f db

M

mkNm40.21M

2cd

2sd

sd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.949‰6.1‰0.10 2c1s =ζ=ε=ε

mcm98.35.4313949.0

2140

f d

MA 2

yd

sd1s =

⋅⋅=

⋅⋅ζ=






1 6

100

1 3

3

As1

067.00.213100

2280

f db

M

mkNm80.22M

2cd

2sd

sd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.950‰6.1‰0.10 2c1s =ζ=ε=ε

mcm24.45.4313950.0

2280

f d

MA 2

yd

sd1s =

⋅⋅=

⋅⋅ζ=


1 6

100

1 3

3

As1

061.00.213100

2060

f db

M

mkNm60.20M

2cd

2sd

sd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.953‰5.1‰0.10 2c1s =ζ=ε=ε

mcm82.35.4313953.0

2060

f d

MA 2

yd

sd1s =

⋅⋅=

⋅⋅ζ=

Ležaj 101 – 102

1 6

100

1 3

3

As1

136.00.213100

4620

f db

M

mkNm20.46M

2cd

2sd

sd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.907‰0.3‰0.10 2c1s =ζ=ε=ε

mcm01.95.4313907.0

4620

f d

MA 2

yd

sd1s =

⋅⋅=

⋅⋅ζ=





Ležaj 101 – 103

1 6

100

1 3

3

As1

141.00.213100

4780

f db

M

mkNm80.47M

2cd

2sd

sd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.904‰1.3‰0.10 2c1s =ζ=ε=ε

mcm35.95.4313904.0

4780

f d

MA 2

yd

sd1s =

⋅⋅=

⋅⋅ζ=

Ležaj 101 – 104

1 6

100

1 3

3

As1

136.00.213100

4580

f db

M

mkNm80.45M

2cd

2sd

sd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.910‰9.2‰0.10 2c1s =ζ=ε=ε

mcm90.85.4313910.0

4580

f d

MA 2

yd

sd1s =

⋅⋅=

⋅⋅ζ=

4.15 Dimenzioniranje stubišta

Polje

1 6

100

1 3

3As1

092.00.213100

3110

f db

M

mkNm1.31M

2cd

2sd

sd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.934‰1.2‰0.10 2c1s =ζ=ε=ε

mcm88.55.4313934.0

3110

f d

MA 2

yd

sd1s =

⋅⋅=

⋅⋅ζ=





Ležaj stubište – 105

1 6

100

1 3

3

As1

160.00.213100

5420

f db

M

mkNm20.54M

2cd

2sd

sd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.892‰5.3‰0.10 2c1s =ζ=ε=ε

mcm75.105.4313892.0

5420

f d

MA 2

yd

sd1s =

⋅⋅=

⋅⋅ζ=

Iskaz potrebne armature po pozicijama:

4.19 cm2/m

49.6

4.19 cm2/m

4.19 cm2/m 4.19 cm2/m

3.98 cm2/m

3.98 cm2/m

4.24 cm2/m3.82 cm2/m

9.01 cm2/m 9.01 cm2/m

9.01 cm2/m9.01 cm2/m

9.35 cm2/m8.90 cm2/m

9.35 cm2/m8.90 cm2/m

5.88 cm2/m

5.88 cm2/m

10.75 cm2/m

Tablica stvarne površine mreže R/Q-785 za različite širine preklopa:

Površina As

[cm2/m']

R/Q- 785 30 8.92R/Q- 785 40 9.28R/Q- 785 50 9.63

R/Q- 785 60 9.99R/Q- 785 70 10.35

MrežaPreklop

(cm)





5. PRORAČUN KONTINUIRANOG NOSAČA POZICIJA 200

5.1 Skica sustava

30 670 30 670 30 670 30700 700 700

2100

3 0

6 1 0

3 0

6 1 0

3 0

6 1 0

3 0

6 4 0

6 4 0

6 4 0

1 9 2 0

y4

y3

y2

y1

x1 x2 x3 x4

L = 7.00 m

g , q

M

L = 7.00 m L = 7.00 m1

Ø

×

Õ Ö

Ø

×

Õ Ö

Ø

×

Õ Ö

n,200 n,200

1 1

0 1 1 0

p1 M p1Mp2

M l1 M l1

V 0-d

V1-l

V1-d

V1-d

V1-l

V 0-l





5.2 Analiza opterećenja

Stalno optereć enje:

Opterećenje od ploče: pl,200 200 2g g L 7.80 6.40 49.92 kN m'= ⋅ = ⋅ =

Opterećenje od grede: gr,200 gr sv abg b h 0.30 0.44 25.0 3.30 kN m'= ⋅ ⋅ γ = ⋅ ⋅ =

30

4 4

1 6

6 0

Ukupno opterećenje na nosaču: n,200 pl,200 gr,200g g g 49.92 3.30 53.22 kN m'= + = + =

Korisno optereć enje:

Opterećenje od ploče: pl,200 200 2q q L 1.00 6.40 6.40 kN m'= ⋅ = ⋅ =

Ukupno opterećenje na nosaču: n,200 pl,200q q 6.4 kN m'= =





5.3 Proračun nosača

Momenti:

( ) ( ) kNm2.3994.3150.18.26035.1MMM

kNm4.310.740.6100.0LqkM

kNm8.2600.722.53100.0LgkM

kNm0.1009.750.12.6535.1MMM

kNm9.70.740.6025.0LqkM

kNm2.650.722.53025.0LgkM

kNm3.3191.2550.16.20835.1MMM

kNm1.250.740.6080.0LqkM

kNm6.2080.722.53080.0LgkM

l,qql,ggl,sd

221ll,q

221ll,g

2p,qq2p,gg1p,sd

2212p2p,q

2212p2p,g

1p,qq1p,gg1p,sd

2211p1p,q

2211p1p,g

=−⋅+−⋅=⋅γ+⋅γ=

−=⋅⋅−=⋅⋅=

−=⋅⋅−=⋅⋅=

=⋅+⋅=⋅γ+⋅γ=

=⋅⋅=⋅⋅=

=⋅⋅=⋅⋅=

=⋅+⋅=⋅γ+⋅γ=

=⋅⋅=⋅⋅=

=⋅⋅=⋅⋅=

Poprečne sile:

g,0 d 0 d 1

q,0 d 0 d 1

sd,0 d g g,0 d q q,0 d

g,1 l 1 l 1

q,1 l 1 l 1

V k g L 0.400 53.22 7.0 149.0 kN

V k q L 0.400 6.40 7.0 17.9 kN

V V V 1.35 149.0 1.50 17.9 228.0 kN

V k g L 0.600 53.22 7.0 223.5 kN

V k q L 0.600

− −

− −

− − −

− −

− −

= ⋅ ⋅ = ⋅ ⋅ =

= ⋅ ⋅ = ⋅ ⋅ =

= γ ⋅ + γ ⋅ = ⋅ + ⋅ =

= ⋅ ⋅ = − ⋅ ⋅ = −

= ⋅ ⋅ = − ⋅

( ) ( )sd,1 l g g,1 l q q,1 l

g,1 d 1 d 1

q,1 d 1 d 1

sd,1 d g g,1 d q q,1 d

6.40 7.0 26.9 kN

V V V 1.35 223.5 1.50 26.9 342.1 kN

V k g L 0.500 53.22 7.0 186.3 kN

V k q L 0.500 6.40 7.0 22.4 kN

V V V 1.35 186.3 1.50 22.

− − −

− −

− −

− − −

⋅ = −

= γ ⋅ + γ ⋅ = ⋅ − + ⋅ − = −

= ⋅ ⋅ = ⋅ ⋅ =

= ⋅ ⋅ = ⋅ ⋅ =

= γ ⋅ + γ ⋅ = ⋅ + ⋅ 4 285.1 kN=





5.4 Dimenzioniranje nosača na moment savijanja


0.30f f

c

ckcd ==

γ=


0.500f f

s

ykyd ==

γ=

Polje p1:

Utjecajna širina: cm640cm1495

70085.030be

5

lbb eff

00eff <=

⋅+=⇒≤+=

4 4

1 6

6 0

30

149

035.00.255149

31930

f db

M

kNm3.319M

2cd

2eff

sdsd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.9650.099‰1.1‰0.10 2c1s =ζ=ξ=ε=ε

2

yd

sd1s

pl

cm83.135.4355965.0

31930f d

MA

hcm4.555099.0dx

=⋅⋅

=⋅⋅ζ

=

<=⋅=⋅ξ=

Odabrano 4Ø22 (As=15.21 cm2)

Polje p2:


70070.030be

5

lbb eff

00eff <=

⋅+=⇒≤+=

4 4

1 6

6 0

30

149

013.00.255128

10000

f db

M

kNm0.100M

2cd

2eff

sdsd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.9810.057‰6.0‰0.10 2c1s =ζ=ξ=ε=ε

2

yd

sd1s

pl

cm26.45.4355981.0

10000f d

MA

hcm1.355057.0dx

=⋅⋅

=⋅⋅ζ

=

<=⋅=⋅ξ=






Ležaj 1:

4 4

1 6

6 0

30

220.00.25530

39920

f db

M

kNm2.399M

2cd

2w

sdsd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.1590.892‰5.3‰0.10 lim,sdlim2c1s =µ=ζ=ε=ε

kNm6.2880.25530159.0f dbM 2cd2wlim,sdlim,Rd =⋅⋅⋅=⋅⋅⋅µ=

sdlim,Rd MM < - dvostruko armiranje

( )( )

( )( )

2

yd2

lim,Rdsd

ydlim

lim,Rd1s cm61.1809.552.13

5.43555

2886039920

5.4355892.0

28860

f dd

MM

f d

MA =+=

⋅−−

+⋅⋅

=⋅−

−+

⋅⋅ζ=


( )( )

( )( )

2

yd2

lim,Rdsd2s cm09.5

5.435552886039920

f dd

MMA =

⋅−−

=⋅−

−=

Odabrano 2Ø22 (As=7.60 cm2

)





5.5 Dimenzioniranje nosača na poprečnu silu

Ležaj 0

C 30/37τRd=0.34 MPa

d1=5 cm

Vsd =228.0 kN

Nsd=0.0 kN

4 4

1 6

6 0

30

( )

( )

Rd1 Rd l cp w

w

sl

c

sdcp

c

Rd1

Rd1 sd

V k 1.2 40 0.15 b d

b 30 cm; d 55 cm

k 1.6 d 1.6 0.55 1.05

A 4 22 2 22 22.810.013 1.3%

A 30 60 1800

N 0.0A

V 0.034 1.05 1.2 40 0.013 0.15 0.0 30 55

V 101.32 kN V

0.7

⎡ ⎤= τ ⋅ ⋅ + ⋅ ρ + ⋅ σ ⋅ ⋅ =⎣ ⎦= =

= − = − =

φ + φρ = = = = =

⋅

σ = =

⎡ ⎤= ⋅ ⋅ + ⋅ + ⋅ ⋅ ⋅⎣ ⎦= <

ν = − ck

Rd2 cd w sd

f 300.7 0.55 0.5

200 200

V 0.5 f b z 0.5 0.55 2.0 30 0.9 55 816.8 kN V

= − = ≥

= ⋅ ν ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ = >

Potrebna računska poprečna armatura!

2wwminmin,sw

wmax,w

2Rd

sd

cm495.02

30300011.0

m

bsA

cm30s;cm30

cm33556.0d6.0s

42.0

8.816

1.342

V

V

=⋅⋅

=⋅⋅ρ

=

=⎩⎨⎧ =⋅=⋅

≤

=≈

Odabrane minimalne spone: Ø10/30 (Asw=0.79 cm2)

( )

kN7.2144.1133.101VVV

kN4.11330

559.05.4379.02

s

zf AmV

1RdwdRd

w

d,ywswwd

=+=+=

=⋅⋅⋅⋅

=⋅⋅⋅

=

Na mjestu maksimalne poprečne sile:

( )( )

( )cm85.26

3.1010.228559.05.4379.02

VV

zf Ams

1Rdwd

d,ywsww =

−⋅⋅⋅⋅

=−

⋅⋅⋅≤

Postaviti spone Ø10/25 (Asw=0.79 cm2)





Ležaj 1

C 30/37τRd=0.34 MPa

Vsd =342.1 kN

Nsd=0.0 kN 4 4

1 6

6 0

30

( )

( )

Rd1 Rd l cp w

w

sl

c

sdcp

c

Rd1

Rd1 sd

V k 1.2 40 0.15 b d

b 30 cm; d 55 cm

k 1.6 d 1.6 0.55 1.05

A 2 22 5 22 26.610.015 1.5%

A 30 60 1800

N0.0

A

V 0.034 1.05 1.2 40 0.015 0.15 0.0 30 55

V 106.03 kN V

0.7

⎡ ⎤= τ ⋅ ⋅ + ⋅ ρ + ⋅ σ ⋅ ⋅ =⎣ ⎦= =

= − = − =

φ + φρ = = = = =

⋅

σ = =

⎡ ⎤= ⋅ ⋅ + ⋅ + ⋅ ⋅ ⋅⎣ ⎦= <

ν = − ck

Rd2 cd w sd

f 300.7 0.55 0.5

200 200

V 0.5 f b z 0.5 0.55 2.0 30 0.9 55 816.8 kN V

= − = ≥

= ⋅ ν ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ = >


2wwminmin,sw

wmax,w

2Rd

sd

cm495.02

30300011.0

m

bsA

cm30s;cm30

cm33556.0d6.0

s

42.08.816

1.342

V

V

=⋅⋅

=⋅⋅ρ

=

=⎩⎨⎧ =⋅=⋅

≤

=≈

Odabrane minimalne spone: Ø10/30 (Asw=0.79 cm2),

kN4.2194.1130.106VVV 1RdwdRd =+=+=


( )( )

( )cm41.14

03.1061.342

559.05.4379.02

VV

zf Ams

1Rdwd

d,ywsww =

−⋅⋅⋅⋅

=−

⋅⋅⋅≤

Postaviti spone Ø10/12.5 (Asw=0.79 cm2)

Tablica nosivosti na poprečne sile za različite razmake postavljanja spona:

Površina AswNosivost betona

VRd1

Nosivost sponaVwd

Ukupna nosivostVRd3 = VRd1 + Vwd

[cm2] [kN] [kN] [kN]

Ø 10 / 30 0.79 101.30 112.74 214.04Ø 10 / 20 0.79 101.30 169.12 270.42Ø 10 / 15 0.79 101.30 225.49 326.79

Ø 10 / 12.5 0.79 101.30 270.59 371.89

Spone





6. PRORAČUN KONTINUIRANOG NOSAČA POZICIJA 100

6.1 Skica sustava

L = 7.00 m

g , q

M

L = 7.00 m L = 7.00 m1

Õ Ö

Õ Ö

Ø

×

Õ Ö

n,100 n,100

1 1

0 1 1 0

p1 M p1Mp2

M l1 M l1

V 0-d

V1-l

V1-d

V1-d

V1-l

V 0-l

30 670 30 670 30 670 30700 700 700

2100

3 0

6 1 0

3 0

6 1 0

3 0

6 1 0

3 0

6 4 0

6 4 0

6 4 0

1 9 2 0

y4

y3

y2

y1

x1 x2 x3 x4

169 363 168

ÕÕ ÖÖ

Ø

×

Ø

×

Õ

Ø

×

Ö

Ø

×

Ö

Ø

×

Ö

Ø

×

ÕÕ

Ø

×

Õ





6.2 Analiza opterećenja

Stalno optereć enje:

Opterećenje od ploče: 2pl,100,max 100

L 6.40g g 6.80 21.76 kN m'

2 2= ⋅ = ⋅ =

Zamjenjujuće opterećenje od ploče:( ) ( )2 3 2 3

pl,100,zam pl,100,max

3200.46; k 1 2 1 2 0.46 0.46 0.674

700g k g 0.674 21.76 14.67 kN m'

α = = = − ⋅ α + α = − ⋅ + =

= ⋅ = ⋅ =

Opterećenje od zida prema stubišta: zid,100 z z zidg d h 0.20 3.6 18.0 12.96 kN m'= ⋅ ⋅ γ ≈ ⋅ ⋅ =

y3

×

Ø g p l , 2 0 0 , m

a x

g p l , 2 0 0 , z

a m

320

700

Opterećenje od grede: 'mkN30.30.2544.030.0hbg absvgr 100,gr =⋅⋅=γ⋅⋅=

30

4 4

1 6

6 0

Ukupno opterećenje na nosaču 1. polje: n,100,1 pl,100,zam gr,100g 2 g g 2 14.67 3.30 32.64 kN m'= ⋅ + = ⋅ + =

Ukupno opterećenje na nosaču 2. polje:

n,100,2 pl,100,zam zid,100 gr,100g 1 g g g 1 14.67 12.96 3.30 30.93 kN m'= ⋅ + + = ⋅ + + =

Korisno optereć enje:

Opterećenje od ploče: 'mkN80.12240.6

00.42

Lqq 2

100max,100,pl =⋅=⋅=

Zamjenjujuće opterećenje od ploče:'mkN63.880.12674.0qkq

674.0k

max,100,plzam,100,pl =⋅=⋅=

=

Ukupno opterećenje na nosaču 1. polje: 'mkN26.1763.82q2q zam,100,pl1,100,n =⋅=⋅=

Ukupno opterećenje na nosaču 2. polje: 'mkN63.863.81q1q zam,100,pl2,100,n =⋅=⋅=

NAPOMENA: Vidljivo je da je optereć enje na nosač u srednjeg polja nešto manje. Da bi pojednostavnili

prorač un, usvojit ć emo isto optereć enje u svim poljima.





6.3 Proračun nosača

L = 7.00 m

g

M

L = 7.00 m L = 7.00 m1

n,100

1 1

0 1 1 0

g,p1 Mg,p1Mg,p2

M g,l1 M g,l1

V g,0-d

V g,1-l

V g,1-d

V g,1-d

V g,1-l

V g,0-l

q

M

n,100

0 1 1 0

q,p1,max M q,p1,maxV q,0-d,max

V q,1-l,max

qn,100

0 1 1 0

M q,p2,max

qn,100

0 1 1 0

V q,1-d,max

V q,1-d,max

qn,100

M q,l1,min

M q,p2,min

M q,p1,min M q,p1,min





Momenti:

( )

( )

( )

( ) ( ) kNm4.3640.9950.19.15935.1MMM

kNm3.1944.1450.19.15935.1MMM

kNm0.990.726.17117.0LqkM

kNm4.140.726.17017.0LqkM

kNm9.1590.764.32100.0LgkM

kNm5.93.4250.10.4035.1MMM

kNm1.1494.6350.10.4035.1MMM

kNm3.420.726.17050.0LqkM

kNm4.630.726.17075.0LqkM

kNm0.400.764.32025.0LgkM

kNm0.1411.2150.19.12735.1MMM

kNm8.3004.8550.19.12735.1MMM

kNm1.210.726.17025.0LqkM

kNm4.850.726.17101.0LqkM

kNm9.1270.764.32080.0LgkM

min,l,qql,ggmin,l,sd

max,l,qql,ggmax,l,sd

221lmin,l,q

221lmax,l,q

221ll,g

min,2p,qq2p,ggmin,2p,sd

max,2p,qq2p,ggmax,2p,sd

221max,2pmin,2p,q

221max,2pmax,2p,q

2212p2p,g

min,1p,qq1p,ggmin,1p,sd


221min,1pmin,1p,q

221max,1pmax,1p,q

2211p1p,g

−=−⋅+−⋅=⋅γ+⋅γ=

−=⋅+−⋅=⋅γ+⋅γ=

−=⋅⋅−=⋅⋅=

=⋅⋅=⋅⋅=

−=⋅⋅−=⋅⋅=

−=−⋅+⋅=⋅γ+⋅γ=

=⋅+⋅=⋅γ+⋅γ=

−=⋅⋅−=⋅⋅=

=⋅⋅=⋅⋅=

=⋅⋅=⋅⋅=

=−⋅+⋅=⋅γ+⋅γ=

=⋅+⋅=⋅γ+⋅γ=

−=⋅⋅−=⋅⋅=

=⋅⋅=⋅⋅=

=⋅⋅=⋅⋅=

Poprečne sile:

( )

( )

( ) ( )

( ) kN2.1390.1050.12.11435.1VVV

kN8.2594.7050.12.11435.1VVV

kN0.100.726.17083.0LqkVkN4.700.726.17583.0LqkV

kN2.1140.764.32500.0LgkV

kN8.2965.7450.11.13735.1VVV

kN9.1811.250.11.13735.1VVV

kN5.740.726.17617.0LqkV

kN1.20.726.17017.0LqkV

kN1.1370.764.32600.0LgkV

kN4.1140.650.14.9135.1VVVkN0.2054.5450.14.9135.1VVV

kN0.60.726.17050.0LqkV

kN4.540.726.17450.0LqkV

kN4.910.764.32400.0LgkV

min,d1,qqd1,ggmin,d1,sd

max,d1,qqd1,ggmax,d1,sd

1min,d1min,d1,q

1max,d1max,d1,q

1d1d1,g

l1,qql1,ggmin,l1,sd

l1,qql1,ggmax,l1,sd

1min,l1min,l1,q

1max,l1max,l1,q

1l1l1,g

min,d0,qqd0,ggmin,d0,sd

max,d0,qqd0,ggmax,d0,sd

1min,d0min,d0,q

1max,d0max,d0,q

1d0d0,g

=−⋅+⋅=⋅γ+⋅γ=

=⋅+⋅=⋅γ+⋅γ=

−=⋅⋅−=⋅⋅==⋅⋅=⋅⋅=

=⋅⋅=⋅⋅=

−=−⋅+−⋅=⋅γ+⋅γ=

−=⋅+−⋅=⋅γ+⋅γ=

−=⋅⋅−=⋅⋅=

=⋅⋅=⋅⋅=

−=⋅⋅−=⋅⋅=

=−⋅+⋅=⋅γ+⋅γ==⋅+⋅=⋅γ+⋅γ=

−=⋅⋅−=⋅⋅=

=⋅⋅=⋅⋅=

=⋅⋅=⋅⋅=

−−−

−−−

−−

−−

−−

−−−

−−−

−−

−−

−−

−−−

−−−

−−

−−

−−





Anvelopa momenata i poprečnih sila na nosaču:

L = 7.00 m

g q

L = 7.00 m L = 7.00 m1

n,100

1 1

0 1 1 0

300.8

M [kNm]sd

141.0 149.1

-9.5

-194.3

-363.4

300.8

141.0

-194.3

-363.4

205.0

114.4

-181.9

-296.8

259.8

139.2

-259.8

-139.2

181.9

296.8

-205.0

-114.4

V [kN]sd

n,100

x = 0.45 L1 x = 0.45 L1





6.4 Dimenzioniranje nosača na moment savijanja


0.30f f

c

ckcd ==

γ=


0.500f f

s

ykyd ==

γ=

Polje p1:


70085.030be

5

lbb eff

00eff <=

⋅+=⇒≤+=

4 4

1 6

6 0

30

149

033.00.255149

30080

f db

M

kNm8.300M

2cd

2eff

sdsd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.9680.091‰0.1‰0.10 2c1s =ζ=ξ=ε=ε

2

yd

sd1s

pl

cm99.125.4355968.0

30080

f d

MA

hcm0.555091.0dx

=⋅⋅

=⋅⋅ζ

=

<=⋅=⋅ξ=

Odabrano 4Ø22 (As=15.21 cm2

)

Polje p2 – pozitivni moment:


70070.030be

5l

bb eff 0

0eff <=⋅

+=⇒≤+=

4 4

1 6

6 0

30

149

019.00.255128

14910

f db

M

kNm1.149M

2cd

2eff

sdsd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.9770.065‰7.0‰0.10 2c1s =ζ=ξ=ε=ε

2

yd

sd1s

pl

cm38.65.4355977.0

14910

f d

MA

hcm6.355065.0dx

=⋅⋅

=⋅⋅ζ

=

<=⋅=⋅ξ=






Polje p2 – negativni moment:

4 4

1 6

6 0

30

005.00.25530

950

f db

M

kNm5.9M

2cd

2w

sdsd

sd

=⋅⋅

=⋅⋅

=µ

−=

Očitano: 0.988‰4.0‰0.10 2c1s =ζ=ε=ε

2

yd

sd1s cm40.0

5.4355988.0

950

f d

MA =

⋅⋅

=

⋅⋅ζ

=

2cmin,1s cm80.16030

100

1.0A%1.0A =⋅⋅=⋅=


Ležaj 1:

220.00.25530

36340

f db

M

kNm4.363M

2

cd

2

w

sdsd

sd

=⋅⋅

=⋅⋅

=µ

=

Očitano: 0.1590.892‰5.3‰0.10 lim,sdlim2c1s =µ=ζ=ε=ε

kNm6.2880.25530159.0f dbM 2cd

2wlim,sdlim,Rd =⋅⋅⋅=⋅⋅⋅µ=

sdlim,Rd MM < - dvostruko armiranje

( )( )

( )( )

2

yd2

lim,Rdsd

ydlim

lim,Rd1s cm96.1644.352.13

5.43555

2886036340

5.4355892.0

28860

f dd

MM

f d

MA =+=

⋅−−

+⋅⋅

=⋅−

−+

⋅⋅ζ=


( )( ) ( )( )2

yd2

lim,Rdsd2s cm44.3

5.435552886036340

f ddMMA =⋅− −=⋅−−=






6.5 Dimenzioniranje nosača na poprečnu silu

Ležaj 0

C 30/37τRd=0.34 MPa

Vsd =205.0 kN

Nsd=0.0 kN 4 4

1 6

6 0

30

( )

( )

Rd1 Rd l cp w

w

sl

c

sdcp

c

Rd1

Rd1 sd

V k 1.2 40 0.15 b d

b 30 cm; d 55 cm

k 1.6 d 1.6 0.55 1.05

A 4 22 2 22 22.810.013 1.3%

A 30 60 1800

N 0.0A

V 0.034 1.05 1.2 40 0.013 0.15 0.0 30 55

V 101.3 kN V

f 0.7

⎡ ⎤= τ ⋅ ⋅ + ⋅ ρ + ⋅ σ ⋅ ⋅ =⎣ ⎦= =

= − = − =

φ + φρ = = = = =

⋅

σ = =

⎡ ⎤= ⋅ ⋅ + ⋅ + ⋅ ⋅ ⋅⎣ ⎦= <

ν = − ck

Rd2 cd w sd

300.7 0.55 0.5

200 200

V 0.5 f b z 0.5 0.55 2.0 30 0.9 55 816.8 kN V

= − = ≥

= ⋅ ν ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ = >


2wwminmin,sw

wmax,w

2Rd

sd

cm495.02

30300011.0

m

bsA

cm30s;cm30

cm36606.0d6.0s

28.0

8.816

0.228

V

V

=⋅⋅

=⋅⋅ρ

=

=⎩⎨⎧ =⋅=⋅

≤

=≈

Odabrane minimalne spone: Ø10/30 (Asw=0.79 cm2)

( )

sdRd

1RdwdRd

w

d,ywswwd

VV

kN7.2143.1014.113VVV

kN4.11330

559.05.4379.02

s

zf AmV

≥

=+=+=

=⋅⋅⋅⋅

=⋅⋅⋅

=





Ležaj 1

C 30/37τRd=0.34 MPa

Vsd =296.8 kN

Nsd=0.0 kN 4 4

1 6

6 0

30

( )

( )

Rd1 Rd l cp w

w

sl

c

sdcp

c

Rd1

Rd1 sd

V k 1.2 40 0.15 b d

b 30 cm; d 55 cm

k 1.6 d 1.6 0.55 1.05

A 2 22 5 22 26.610.0015 0.15%

A 30 60 1800

N0.0

A

V 0.034 1.05 1.2 40 0.015 0.15 0.0 30 55

V 106.03 kN V

0.

⎡ ⎤= τ ⋅ ⋅ + ⋅ ρ + ⋅ σ ⋅ ⋅ =⎣ ⎦= =

= − = − =

φ + φρ = = = = =

⋅

σ = =

⎡ ⎤= ⋅ ⋅ + ⋅ + ⋅ ⋅ ⋅⎣ ⎦= <

ν = ck

Rd2 cd w sd

f 307 0.7 0.55 0.5

200 200

V 0.5 f b z 0.5 0.55 2.0 30 0.9 55 816.8 kN V

− = − = ≥

= ⋅ ν ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ = >


2wwminmin,sw

wmax,w

2Rd

sd

cm495.02

30300011.0

m

bsA

cm30s;cm30

cm36606.0d6.0

s

42.08.8161.342

VV

=⋅⋅

=⋅⋅ρ

=

=⎩⎨⎧ =⋅=⋅

≤

=≈

Odabrane minimalne spone: Ø10/30 (Asw=0.79 cm2), kN07.219VRd =


( )( )

( )cm83.17

03.1068.296

559.05.4379.02

VV

zf Ams

1Rdwd

d,ywsww =

−⋅⋅⋅⋅

=−

⋅⋅⋅≤

Postaviti spone Ø10/15 (Asw=0.79 cm2)

Tablica nosivosti na poprečne sile za različite razmake postavljanja spona:

Površina AswNosivost betona

VRd1

Nosivost sponaVwd

Ukupna nosivostVRd3 = VRd1 + Vwd

[cm2] [kN] [kN] [kN]

Ø 10 / 30 0.79 101.30 112.74 214.04Ø 10 / 25 0.79 101.30 135.29 236.59Ø 10 / 20 0.79 101.30 169.12 270.42Ø 10 / 15 0.79 101.30 225.49 326.79

Spone





6.6 Kontrola pukotina u 1. polju

Širinu pukotina kontroliramo za nefaktorizirano opterećenje i bez utjecaja puzanja.

L = 7.00 m L = 7.00 m L = 7.00 m11 1

300.8

M [kNm]sd

141.0 149.1

-9.5

-194.3

-363.4

300.8

141.0

-194.3

-363.4

Ø22

Ø22

Ø22Ø22

A B C

Ø22

Ø22

Ø22

Ø22

Ø22

4 4

1 6

6 0

30

Presjek A-A

4Ø22

2Ø14

2Ø22

Ø10 4 4

1 6

6 0

30

Presjek B-B

2Ø22

2Ø14

5Ø22

Ø10 4 4

1 6

6 0

30

Presjek C-C

2Ø22

2Ø14

2Ø22

Ø10

kNm3.2134.850.19.1270.1MMM

kNm4.850.726.17101.0LqkM

kNm9.1270.764.32080.0LgkM


221max,1pmax,1p,q

2211p1p,g

=⋅+⋅=⋅γ+⋅γ=

=⋅⋅=⋅⋅=

=⋅⋅=⋅⋅=


smrmk sw ε⋅⋅β=

β=1.7 - odnos računske i srednje širine pukotina

Proračun srednje deformacije armature:

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

σσ

⋅β⋅β−⋅σ

=ζ⋅σ

=ε2

s

sr 21

s

s

s

ssm 1

EE

As1 = 4Ø22 = 15.21 cm2

MPa0.00032GPa0.32Ecm == - modul elastičnosti betona

MPa0.000200GPa0.200Es == - modul elastičnosti armature

25.632

200

E

E

cmseI ===α





cm8.1521.1525.6

5530211

30

21.1525.6

A

db211

b

Ax

1SeI

1SeI =⎟⎟ ⎠

⎞⎜⎜⎝

⎛

⋅⋅⋅

++−⋅⋅

=⎟⎟ ⎠

⎞⎜⎜⎝

⎛

⋅α⋅⋅

++−⋅⋅α

=

MPa0.282cm

kN20.28

21.153

8.1555

21330

A3x

d

M

Az

M2

s

sd

s

sds ==

⋅⎟ ⎠

⎞⎜⎝

⎛ −=

⋅⎟ ⎠

⎞⎜⎝

⎛ −≈

⋅=σ

( )

( ) ( )

MPa0.69cm

kN90.6

21.153

8.1555

5220

A3x

d

M

Az

M

kNm20.52kNcm0.52206

603029.0M

MPa9.20.303.0f 3.0f

MPa0.30f ;f 3.0f ;6hbf M;

AzM

2

s

cr

s

cr sr

2

cr

3232ckm,ct

ck32

ckm,ct

2

m,ctcr s

cr sr

==⋅⎟

⎠

⎞⎜⎝

⎛ −=

⋅⎟ ⎠

⎞⎜⎝

⎛ −≈

⋅=σ

==⋅

⋅=

=⋅=⋅=

=⋅≈⋅⋅=⋅

=σ



00137.00.282

0.695.00.11

0.200000

0.2821

E

22

s

sr 21

s

ssm =

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟ ⎠

⎞⎜⎝

⎛ ⋅⋅−⋅=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

σσ

⋅β⋅β−⋅σ

=ε


[ ]mmkk25.050sr

21rm ρφ

⋅⋅⋅+=

mm22=φ - Promjer najdeblje šipke


5.0k2 = - Savijanje

( )0406.0

55.230

21.15

A

A

eff ,c

sr =


4 4

1 6

6 0

30

4Ø22

2Ø14

2Ø22

5

mm2.1040406.0

225.08.025.050kk25.050s

r 21rm =⋅⋅⋅+=

ρφ

⋅⋅⋅+=


mm3.0wmm242.02.10400137.07.1sw gsmrmk =<=⋅⋅=ε⋅⋅β=





6.7 Kontrola progiba za 1. polje

Progib kontroliramo za nefaktorizirano opterećenje i bez utjecaja puzanja.

( ) ( ) kNm9.2580.990.19.1590.1MMM

kNm0.990.726.17117.0LqkM

kNm9.1590.764.32100.0LgkM

kNm3.2134.850.19.1270.1MMM

kNm4.850.726.17101.0LqkM

kNm9.1270.764.32080.0LgkM

min,l,qql,ggmin,l,sd

221lmin,l,q

221ll,g


221max,1pmax,1p,q

2211p1p,g

−=−⋅+−⋅=⋅γ+⋅γ=

−=⋅⋅−=⋅⋅=

−=⋅⋅−=⋅⋅=

=⋅+⋅=⋅γ+⋅γ=

=⋅⋅=⋅⋅=

=⋅⋅=⋅⋅=

Progib homogenog presjeka:

4 4

1 6

30

4Ø22

2Ø14

2Ø22

5

As1 = 4∅22 = 15.21 cm2

As2 = 2∅22 = 7.60 cm2

4

223

2

12s

2

21seI

3

I

cm0.1006290.100890.000540

52

6060.7526021.1525.6

126030

d2

hAd

2

hA

12

bhI

=+=

=⎥⎥⎦⎤

⎢⎢⎣⎡ ⎟

⎠ ⎞⎜

⎝ ⎛ −⋅+⎟

⎠ ⎞⎜

⎝ ⎛ −⋅⋅+⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟ ⎠

⎞⎜⎝

⎛ −⋅+⎟

⎠

⎞⎜⎝

⎛ −⋅⋅α+=

cm

10000106.0

1006293200

21330

IE

M

r

1

cmkN0.3200mGN0.32EE

Ieff ,c

Sd

I

22cmeff ,c

=⋅

=⋅

=

===

L = 7.00 m L = 7.00 m11

0 1

213.3

M [kNm]sd

-258.9x = 0.45 L1

Granični progib:

cm80.2250

700

250

Llim ===ν

Beton: C 30/37; f ck=30.0 MPa

( ) ( ) MPa9.20.303.0f 3.0f

MPa00032E

3232ckm,ct

cm

=⋅=⋅=

=

Čelik: B500B; Es=200.0 GPa

25.60.32

0.200

E

E

cm

seI ===α

tot

2tot r

1Lk ⋅⋅=ν

( ) ( ) 091.021.11.01104.01.0148

5k

21.13.2139.2580.0MMM FBA

=⋅−⋅=β⋅−⋅=

=+=+=β




Progib potpuno raspucanog presjeka:

cm8.15x =

( ) ( )[ ]

( ) ( ) ( )[ ]4

2223

222s

21seI

23

II

cm0.0001910.6001510.40039

580.1560.780.155521.1525.62

80.1580.1530

12

80.1530

dxAxdA2

xbx

12

bxI

=+=

−⋅+−⋅⋅+⎟ ⎠

⎞

⎜⎝

⎛ ⋅⋅+⋅

=

−⋅+−⋅⋅α+⎟ ⎠

⎞⎜⎝

⎛ ⋅+=

cm

10000349.0

1910003200

21330

IE

M

r

1

IIeff ,c

Sd

II

=⋅

=⋅

=

Ukupni progib

MPa0.282s =σ

MPa0.69sr

=σ



97.00.282

0.695.00.111

22

s

sr 21 =⎟

⎠

⎞⎜⎝

⎛ ⋅⋅−=⎟⎟ ⎠

⎞⎜⎜⎝

⎛

σ

σ⋅β⋅β−=ζ

( ) ( )cm

10000342.00000349.097.00000106.097.01

r

1

r

11

r

1cm

10000349.0

r

1

cm

10000106.0

r

1

IIIm

II

I

=⋅+⋅−=⋅ζ+⋅ζ−=

=

=

cm80.2cm52.10000342.00.700091.0r

1Lk

cm0.700L

091.0k

lim2

tot

20t,tot =ν<=⋅⋅=⋅⋅=ν

=

=

=

Documents

Program_OBK.pdf