1992 ( 2)

. .,

1/3

EN1992 ( 2):

11 1: 2: 3: 4: 5: 6: 7: 8:

9:

10:

11: 12:

:

:

C : ( )

D :

:

F :

G :

:

I :

J :

2.3.3 (3)2.4.2.1 (1)2.4.2.2 (1)2.4.2.2 (2)2.4.2.2 (3)2.4.2.3 (1)2.4.2.4 (1)2.4.2.4 (2)2.4.2.5 (2)3.1.2 (2)P3.1.2 (4)3.1.6 (1)P3.1.6 (2)P3.2.2 (3)P3.2.7 (2)3.3.4 (5)3.3.6 (7)4.4.1.2 (3)4.4.1.2 (5)4.4.1.2 (6)4.4.1.2 (7)4.4.1.2 (8)4.4.1.2 (13)

4.4.1.3 (2)4.4.1.3 (3)4.4.1.3 (4)5.1.2 (1)P5.2 (5)5.5 (4)5.6.3 (4)5.8.3.1 (1)5.8.3.3 (1)5.8.3.3 (2)5.8.5 (1)5.8.6 (3)5.10.1 (6)5.10.2.1 (1)P5.10.2.1 (2)5.10.2.2 (4)5.10.2.2 (5)5.10.3 (2)5.10.8 (2)5.10.8 (3)5.10.9 (1)P6.2.2 (1)6.2.2 (6)6.2.3 (2)6.2.3 (3)

6.2.4 (4)6.2.4 (6)6.4.3 (6)6.4.4 (1)6.5.2 (2)6.5.4 (4)6.5.4 (6)6.8.4 (1)6.8.4 (5)6.8.6 (1)6.8.6 (2)6.8.7 (1)7.2 (2)7.2 (3)7.2 (5)7.3.1 (5)7.3.2 (4)7.4.2 (2)8.2 (2)8.3 (2)8.6 (2)8.8 (1)9.2.1.1 (1)9.2.1.1 (3)9.2.1.2 (1)

9.2.1.4 (1)9.2.2 (4)9.2.2 (5)9.2.2 (6)9.2.2 (7)9.2.2 (8)9.3.1.1(3)9.4.3(1)9.5.2 (1)9.5.2 (2)9.5.2 (3)9.5.3 (3)9.6.2 (1)9.6.3 (1)9.7 (1)9.8.1 (3)9.8.2.1 (1)9.8.3 (1).8.3 (2)9.8.4 (1)9.8.5 (3)9.8.5 (4)9.10.2.2 (2)9.10.2.3 (3)9.10.2.3 (4)9.10.2.4 (2)

11.3.5 (1)P11.3.5 (2)P11.3.7 (1)11.6.1 (1)11.6.1 (2)11.6.2 (1)11.6.4.1 (1)12.3.1 (1)12.6.3 (2)A.2.1 (1)A.2.1 (2)A.2.2 (1)A.2.2 (2)A.2.3 (1)C.1 (1)C.1 (3)E.1 (2)J.1 (3)J.2.2 (2)J.3 (2)J.3 (3)

library.tee.gr/digital/m2464/m2464_contents.htm

The Belgian Building Research Institute : www.bbri.be/antenne_norm/eurocodes/pdf/worked-examples-ec2-

def080723.pdf

www.bbri.be/antenne_norm/eurocodes/fr/pub_bbri.html#2

European Concrete Platform http://www.europeanconcrete.eu/publications/eurocodes/112-

worked-examples-for-eurocode-2 http://www.europeanconcrete.eu/publications/eurocodes/114-

commentarytoeurocode2

: EN1992 1992 :

1998

/

1

1.1 &

1-1 1: 2:

3: 4: 5: 6: 7: 8: 9: 10:

11: 12:

3 -

: , , , : -,

:

fcm = fck + 8 (MPa)fcm(t) = cc(t) . fcm , =20C

s=0.2 CEM 42,5R, CEM53,5N, CEM53,5R s=0.25 CEM 32,5R, CEM42,5N s=0.3 CEM 32,5N

fcd = cc fck / c cc =1 , 0.85 (t>28.)c =1.5

=

5,0281exp)(t

stcc

: /

:

fct = 0,9 fct,sp ()fctm(t) = (cc(t)) fctm

= 1 t < 28 = 2/3 t 28

fctm,fl=max{(1,6-h/1000)fctm; fctm}Ecm = 22 (fcm/10)0,3, fcm (MPa)Ecm(t) = (fcm(t) / fcm)0,3 Ecm

fct = 0,50fct,fl 0,90fct,spfctm = 0,3 fck2/3

Ecm = 9500(fc+8)1/3

: -

-

: (, , . , )

c2c1 c3

)2(1

2

+

=k

kfcm

c

:

/

2=0.5 n s wd fcd

c2

cc (,t0) = (, t0)(c/Ec0) , Ec0=1.05 Ecm (,t0): /, , , , k=c/fcm(to),

c>0.45fc(to).

< 0.45 fck(to) : :

:

EC 2 cs = cd + ca

( )cd(t) = ds(t, ts) cd, =

ds(t, ts) kh cd,0

h0= 2Ac/u (

) : cd,a =as(t)ca() ,

as(t) =1exp(0,2 t 0,5 ) ca()=2,5 (fck10) 10-6

fck/fck,cube (MPa)

( 0/0)

20 40 60 80 90 100 20/25 0.62 0.58 0.49 0.30 0.17 0.00 40/50 0.48 0.46 0.38 0.24 0.13 0.00 60/75 0.38 0.36 0.30 0.19 0.10 0.00 80/95 0.30 0.28 0.24 0.15 0.08 0.00 90/105 0.27 0.25 0.21 0.13 0.07 0.00

h0 kh 100 1.0 200 0.85 300 0.75 500 0.70

-

(,,C EN10080)

1: 2: 3:

1-1 1: 2: 3:

4:

5: 6: 7: 8: 9: 10:

11: 12:

4:

: , , : /

( 206-1)

:XO XC (14)XD (13)XS (13)XF (14)XA (13)

() : cnom = cmin + cdev

&

1992, 2cmin,b (= )

cmin = max 10mmcmin,dur + cdur, - cdur,st - cdur,add

cdev: 10mm 5mm c > 40mm c > 75mm

cmin,dur

cmin,dur S4

50

. E . .1

4.3

4.4

cmin,dur

25/50, XS1, C30/37, 314, 8/150mm, dg=20mm, 50 .

XS1 : .1 C30/37

50 S4 cmin,b = 14mm ( ) : 4.4 XS1 cmin,dur = 35mm cdur, = cdur,st = cdur,add = 0 Cmin = max (cmin,b , 10mm, cmin,dur ) = 35mm cdev= 10mm ():

cmin=35+15=50mm

1-1 1: 2: 3: 4:

5: 6: 7: 8: 9: 10:

11: 12:

5

/ /

: / (& -)

-

:

20%

/ 0.5 < li/li+1 < 2: C : 0.44 + 1,25 (0,6 + 0,0014/cu2) (xu/d) > 0.7

fck 50MPa() 0.44 + 1,25 (xu/d), cu2=0.0035

0.54 + 1,25 (0,6 + 0,0014/cu2) (xu/d) > 0.8

fck > 50MPa() 0.56 + 1,25 (xu/d)

allow>sallow= pl,d k: k=(/3)1/2, =MSd/ (VSd d)

allow k = 3,0

!!

s=fyk

fyk

8, 10 8m, , ( ) 50kN/m 35kN/m.

qsd=1.34 G + 1.5 Q = 120 kN/mLC_i

LC_ii

LC_iii

LC_iv

LC_v

:Concrete Design to EN1992

qsd

qsd gg

qsd g

qsd g

qsd g qsd

gg g g g1062 kNm

502 kNm

743 kNm

624 kNm

616 kNm

A(kNm) B(kNm) C(kNm) D(kNm)

(i) 0 357 -987 513 -987 357 0(ii) 0 90 -842 658 -842 90 0(iii) 0 502 -1062 616 -717 122 0(iv) 0 650 -681 128 -336 300 0(v) 0 681 -606 19 -606 681 0

30% [ (iii)](iii) 0 624 -743 770 -717 122 0

(iii) , (ii) 12%

(ii) 0 115 -743 757 -743 115 0

-

( D)

.. ( ) ...

(. )

/ (): :

Rd,max=fcd .:

Rd,max=0.6 fcd , =1-fck/250

C-T . :2Rd,max= 0.85 fcd

C-T > . :3Rd,max= 0.75 fcd

Rd,max

:

CCC

C

CC

3c16 a 0.25

3c16 a 0.25

832832

6c16 a 0.20

832

832

3c16 a 0.25

3c16 a 0.25

832

:fib Bul 61 : Design examples for strut-and-tie models

:fib Bul 61 : Design examples for strut-and-tie models

:fib Bul 61 : Design examples for strut-and-tie models

:fib Bul 61 : Design examples for strut-and-tie models

:fib Bul 61 : Design examples for strut-and-tie models

( B - Bernoulii) - (

D Discontinuity)

: D / /

: fyd : Rd,max=fcd :

Rd,max=0.6fcd, , =1-fck/250 :

Rd,max= fcd, -

Rd,max=0.85fcd -

Rd,max=0.85fcd

5.40 x 3.0m, b = 250 mm), / 400 x 250 mm, 620.

C25/30, B500c

150 kN/m .

(..FEM), L/h=2, C2 0.67h2m .

R = (150+150) 5.40/2 = 810kN, =arctan(2000/1300)=56.98 :

1: C1= q L/2 = 405 kN,

3: C3= R/sin= 966 kN

2: T1=C3 cos= 526 kN

4: 2 = qL/2= 405 kN 1:Asl 526kN/500MPa/1.15= 1210mm2,, 518 2:Asl 405kN/500MPa/1.15= 931mm2,, 320 :

3 C-C-T:2Rd,max=k2 fcd (1-fck/250)=12.75MPa (k2=0.85)c1=810kN/0.4m/0.25m=8.1MPa < Rd2,max

/ (/, ).

: - & ,

: S-N

2

2

1-1 1: 2: 3: 4: 5:

6: 7: 8: 9: 10:

11: 12:

6 :

. . (, ) = 0,8 fck50 MPa =0.80 =0.8[(fck50)/400 ] 50

(.. )

( Bresler xa+ya=k)

: MRd (?)

CEB (Bul 141) BS8110 : Mx/dx > My/dy :

Mx=Mx + My (dx/dy) Mx/dx < My/dy :

My=My + Mx (dy/dx)

NEd/NRd 0.1 0.7 1.0

a 1 1.5 2

VRd,c = [CRd,c k (100 l fck)1/3 + k1 cp] bwd ,

CRd,c = 0,18/c, k=1+(200/d)0.5, k1 =0.15

: VRd1=[Rd k (1.2+40l)+0.15 cp] bwd, Rd =0.052 fck2/3 / c

VRd,cmin = (vmin + k1 cp) bwd , vmin=0,035 k3/2 fck1/2

V < VRd,c ( min , 0.08fck /fyk): 30/55, C20/25,

: min=0.0022 VRd1 = 55.80kNEC2: min=0.0011 VRd,c = 41.75kN

min=0.0022 VRd,c = 51.91kN

1992

VRd,c VRd1VRd,s VwdVRd,max VRd2

vRd,c (N/mm2) fck=25MPa fck = 20MPa x(0.87), 30MPa x(1.13), 35MPa x(1.23), 40MPa x(1.36)

: 30/55, C20/25,

: min=0.0022 VRd1 = 55.80kNEC2: min=0.0011 VRd,c = 41.75kN

min=0.0022 VRd,c = 51.91kN

100As/ bwd, mm

150 175 200 225 250 300 400 500 >600

0.15 0.55 0.54 0.53 0.52 0.51 0.49 0.45 0.42 0.38

0.25 0.57 0.56 0.55 0.54 0.53 0.51 0.47 0.43 0.39

0.50 0.61 0.60 0.59 0.58 0.57 0.55 0.50 0.46 0.42

0.75 0.65 0.64 0.63 0.62 0.61 0.59 0.54 0.50 0.45

1.00 0.70 0.68 0.67 0.66 0.65 0.62 0.58 0.53 0.48

1.50 0.78 0.77 0.76 0.74 0.73 0.70 0.65 0.59 0.54

2.00 0.87 0.86 0.84 0.83 0.81 0.78 0.72 0.66 0.60

3.00 1.04 1.03 1.01 0.99 0.97 0.94 0.86 0.79 0.72

V > VRd,c , . . :

1 cot 2.5 (45 21.8) : VRd = min {VRd,max ,VRd,s }

VRd,max= bw z fcd/(cot+tan)

= 0,6 [1- fck /250]

45 21.8

VRd,s = (Asw/s) z fywd cot, Vcd =0

VRd =min {VRd3 ,VRd2}

VRd,2= fcdbwz

= 0,7 [1- fck /200]

=45, =0.9d

Vwd= (Asw/s) (0.9d) fywd

VRd,c < VEd

=21.8

VRd,max

: VEd < VRd,max

oVs=Ved=(Asw/s) z fywd cot

21.8

cot = 2.5, =21.8 ( Asw )

Vd < VRd,max :

- - VEd= VRd,max

20/80, C30/35, B500c, VEd=600kN

d=800-50=750mm, z=0.9d=675mm, =21.8

: VRd,c=60.72kN < VEd=600kN VRd,s = Ved = 600kN = (Asw/s) z fywd cot (=21.8 ) Asw/s = 817mm2/m : VRd,max=491.6kN < VEd VRd,max=VEd

= 0.5arcsin(VEd/ fcd bw z) = 29 (cot=1.8)

Asw/s=1135mm2/m

12 (): 2*113.1mm2/s=1135mm2/ms=0.19m w=Asw/bws = 0.03 > w,min=0.08(fck/fyk)=0.000876 sl,max=0.75d=0.56m , 12/190

Ftd=0.5VEd cot= 540kN (

)

( ):

vEdi = VEd / (z bi) vRdivRdi = c fctd + n + fyd ( sin + cos) 0,5 fcd

vEdi f (sin + cos )yd

cf + ctd n

bi

bi

c

0,25 0,5

0,35 0,6

0,45 0,7

0,50 0,8

: V+T:

(TEd /TRd,max) + (VEd /VRd,max) 1,0

(Tsd /TRd1)2 + (Vsd/VRd2)2 1,0 .

(6.4, 9.4.3) : : 2d ( 1.5d - ) -

: : vEd < vRd,max=0.5fcd

u1 (2d)

vEd < VRd,c/bwd = vRd,c

: rcont = 2d + lH + 0,5c : rcont = min [2d + 0.56(L1L2)1/2, 2d + 0.69 L1]

1992 & : 45

,

W1 - u1

: =1.5, 1.40, 1.15 , ( ).

duVv

i

EdEd =

)/(,1Wu

VMk1 211 ccfk

1Ed

Ed =>+=

12

22121 dc2d16dc4cc2/cW ++++=1

:vEd < vRd,c= CRd,c k (100 l fck)1/3 + 0.10cp

vRd,cmin = vmin + 0.10cp , vmin=0,035 k3/2 fck1/2

vRd,c < vEd < 1.6 vRd,c (=33.7):

( ), sw,min (1,5sin+cos)/(srst) 0,08(fc)/fyksr

: uout,ef = VEd / (vRd,c d)

kd uout uout,ef (.: k=1.5)

2d

d

d

> 2d

kd

A B

kd

1

2

1

2d

300x400mm 6m 225mm, 600kN, 45kNm.

20mm 12, d=225-20-12=193mm

2d: u1=2(300+400)+4193=3825mm {V+M},

:

W1=1512,6103 mm2, c1/c2=300/400=1.33 , k=0.63 ( 6.5).=1+063 (45kNm/600kN) (3825mm/1512,6 103 mm2)=1,12

duVv

i

EdEd = )/(,1W

uVMk1 211 ccfk

1Ed

Ed =>+=

12

22121 dc2d16dc4cc2/cW ++++=1

:

- 2d:

A . :

-: (5090-2(300+400))/2 =587mm ~ 3,04d.

(2d), st=1.5d:

d vVu

C,Rd

Edouter

=

0.5d 2dd