МЕТОДИЧЕСКО РЪКОВОДСТВО ЗА РАЗРАБОТВАНЕ НА КУРСОВ...

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q, [kN/m]

5 N8/ m Mmax= 0,125.q.l2, [kN.m]

6 N14 N8/ sw=20 cm

max/min=Nf,k/Af Mf,k/Wf

2012

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II. , .

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1.

1.1

EN 1990 .

1.2

ISO 3898:1987.

A (Accidental action)

A (Cross sectional area)

Ac (Cross sectional area of concrete)

Ap

(Area of a prestressing tendon or tendons)

As (Cross sectional area of reinforcement)

As,min (inimum cross sectional area of reinforcement)

Asw

(Cross sectional area of shear reinforcement

D (Diameter of mandrel)

DEd (Fatigue damage factor)

E (Effect of action)

Ec, Ec(28) ()

ssssc= 0 28-(Tangent modulus of elasticity of normal

weight concrete at a stress of ssssc= 0 and at 28 days)

Ec,eff (Effective modulus of concrete)

Ecd

(Design value of modulus of elasticity of concrete)

Ecm ()

(Secant modulus of elasticity of concrete)

Ec(t) ()

ssssc= 0

(Tangent modulus of elasticity of normal weight concrete at a stress of ssssc= 0

and at time t)

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14

Ep

(Design value of modulus of elasticity of prestressing)

Es

(Design value of modulus of elasticity of reinforcing steel)

EI (Bending stiffness)

EQU (Static equilibrium)

F (Action)

Fd (Design value of an action)

Fk

(Characteristic value of an action)

Gk

(Characteristic permanent action)

I

(Second moment of area of concrete section)

L (Length)M (Bending moment)

MEd

(Design value of the applied internal bending moment)

N (Axial force)

NEd ()

(Design value of the applied axial force (tension or compression))

P (Prestressing force)

P0

(Initial force at the active end of the tendon immediately after stressing)

Qk

(Characteristic variable action)

Qfat (Characteristic fatigue load)

R (Resistance)

S (Internal forces and moments)

S (First moment of area)

SLS (Serviceability limit state)

T (Torsional moment)

TEd

(Design value of the applied torsional moment)ULS (Ultimate limit state)

V (Shear force)

VEd

(Design value of the applied force)

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15

a (Distance)

a (Geometrical data)

DDDDa (Deviation for geometrical data)

b

(Overall width of a cross-section, or actual flange width in a T or L beam)

bw T, I L

(Width of the web on T, L or L beams)

d , (Diameter; Depth)

d

(Effective depth of a cross-section)

dg

(Largest nominal maximum aggregate size)

e (Eccentricity)

fc (Compressive strength of concrete)

fcd (Design value of concrete compressive strength)

fck 28-

(Characteristic compressive cylinder strength of concrete at 28 days)

fcm

(Mean value of concrete cylinder compressive strength)

fctk (Characteristic axial tensile strength of concrete)

fctm

(Mean value of axial tensile strength of concrete)

fp

(Tensile strength of prestressing steel)

fpk

(Characteristic tensile strength of prestressing steel)

fp0,1

0,1%

(0,1% proof-stress of pressing steel)fp0,1k

0,1%

(Characteristic 0,1% proof-stress of pressing steel)

f0,2k

0,2%

(Characteristic 0,2% proof-stress or reinforcement)

ft (Tensile strength of reinforcement)

ftk

(Characteristic tensile strength of reinforcement)

fy (Yield strength of reinforcement)

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16

fyd

(Design yield strength of reinforcement)fyk

(Characteristic yield strength of reinforcement)

fywd

(Design yield of shear reinforcement)

h (Height)

h (Overall depth of a cross-section)

i (Radius of gyration)

k (Coefficient ; Factor)

l (l L) , ((or l or L ) Length ; Span)

m (Mass)

r (Radius)

1/r (Curvature at a particular section)

t (Thickness)

t (Time being considered)

t0 (The age of concrete at the time of loading)

u

(Perimeter of concrete cross-section , having area Ac)

u,v,w (Components of the displacement of a point)

x (Neutral axis depth)

x,y,z (Coordinates)

z (Level arm of internal forces)

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17

aaaa , (Angle ; ratio)

bbbb , , (ratio ; coefficient)

gggg (Partial factor)

ggggA

(Partial factor for accidental actions A)

ggggC (Partial factor for concrete)

ggggF F (Partial factor for actions , F)

ggggF,fat ,

(Partial factor for fatigue actions)

ggggC,fat

(Partial factor for fatigue of concrete)

ggggG , G

(Partial factor for permanent actions , G)

ggggM (

,

)

(Partial factor for a material property, taking account of

uncertainties in the material property itself, in geometric

deviation and in the design model used)

ggggP ,

(Partial factor for actions with prestressing, P)

ggggQ Q

(Partial factor for variable actions, Q)

ggggS ()

(Partial factor for reinforcing or prestressing steel)

ggggS,fat ()

,

(Partial factor for reinforcing or prestressing steel underfatigue loading)

ggggf

(Partial factor for actions without taking account ofmodel uncertainties)

ggggg

(Partial factor for permanent actions without taking

account of model uncertainties)

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18

ggggm ,

(Partial factor for a material property, taking accountonly of uncertainties in the material property)

dddd ;

(Increment/redistribution ratio)

zzzz /

(Reduction factor/distribution coefficient)eeeec

(Compressive strain in the concrete)

eeeec1 fc

(Compressive strain in the concrete at the peak stress fc)

eeeecu (Ultimate compressive strain in the concrete)

eeeeu

(Strain of reinforcement or prestressing steel at maximum

load)

eeeeuk

(Characteristic strain of reinforcement or prestressing

steel at maximum load)

qqqq (Angle)

llll (Slenderness ratio)

mmmm

(Coefficient of friction between the tendors and their ducts)

nnnn o(Poissons ratio)

nnnn

(Strength reduction factor for concrete cracked in shear)

xxxx

(Ratio of bond strength of prestressing and reinforcing steel)rrrr kg/m3

(Oven-dry density of concrete in kg/m3)

rrrr1000 (%) 1000

20

(Value of relaxation loss (in %), at 1000 hours after

tensioning and at a mean temperature of 20oC)

rrrrl

(Reinforcement ratio for longitudinal reinforcement)

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19

rrrrw

(Reinforcement ratio for shear reinforcement)

ssssc (Compressive strength in the concrete)

sssscp

(Compressive strength in the concrete from axial load or prestressing)

sssscu eeeecu

(Compressive strength in the concrete at the ultimate

compressive strain eeeecu)

tttt (Torsional shear stress)

FFFF

(Diameter of a reinforcing bar or of a prestressing duct)

FFFFn

(Equivalent diameter of a bundle of reinforcing bars)

ffff(t,t0) (

t0t 28-

)

(Creep coefficient, defining creep between times t and t0,

related to elastic deformation at 28 days)

ffff( ,t0)

(Final value of creep coefficient)

yyyy ,

(Factor defining representative values of variable actions)

yyyy0

(for combination values)

yyyy1

(for frequent values)

yyyy2

(for quasi-permanent values)

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1. 3.

1.4

1.4.1 :

1 N = 0.1 kgf ;( 1N = 0.09807 kgf );1 kgf = 10 N ; [kgf] x 10 = [N] ; ( [kgf] x 9.807 = [N] );

1 daN = 10 N = 1 kgf ;

1 kN = 1000 N = 103N = 100 kgf = 0.1 tf ; 1 tf = 10 kN ;

1 kN = 224.82 lb ;

1 MN = 1000000 N = 106N = 1000 kN = 10

3kN = 100 tf ;

1 kip = 1000 pound force =1000 lb ;

[kip] x 4.448 = [kN] ; [kip] x 0.4448 (0.4536) = [tf] ;

[kip] x 4448 = [N] ; [kip] x 444.8 (453.6) = [kgf] ;

1 (pound-force)(lb) x 4.448 (4.536) = [N] ;

1 (pound-force)(lb) x 0.4448 (0.4536) = [kgf] ;

1 lb-force = 0.4448 (0.4536) kgf ;1 (pound-force)(lb) x 0.004448 (0.004536) = [kN]

1 kN/m = 1000 N/m = 103N/m = 100 kgf/m = 0.1 tf/m ;

[kip/inch] x 175.118 = [kN/m];

[kip/foot] x 14.596 = [k/f] x 14.596 = [kN/m]

a A x X b B o O

g G p P

d D r R

e s S

z t T

h H u U

q Q f F

i I c C

k K y Y

l L w W

m M

n N

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1.4.2 M :

1 kN.cm = 1 kgf.m = 0.001 tf.m ; 1 kgf.m = 0.01 kN.m ;

1 kN.m = 0.1 tf.m = 100 kgf.m = 10000 kgf.cm ;

1 tf.m = 1000 kN.cm = 10 kN.m ;

1 MN.cm = 1 tf.m = 100000 kgf.cm = 1000 kN.cm = 107N.mm ;

[in kips] x 11.521 = [kgf.m];

1 inch pound force x 0.01152 =[kgf.m]; x 0.113 = [N.m] ;

1 foot pound force x 0.1383 = [kgf.m] ; x1.356 = [N.m] ;

1 meter kg force x 9.807 = [N.m]

1.4.3 :

1 GPa = 109Pa = 10

3MPa = 10

3N/mm

2= 10

4kg/cm

2;

[GPa] x 1000 = [N/mm2] = [MPa]; [GPa] x 10000 = [kg/cm2] ;

1 Mpa = 10 kgf/cm2= 0.1 kN/cm

2= 1 N/mm

2=1 MN/m

2= 10

6Pa =

= 1000 kN/m2= 1000 kPa =10

3kPa = 100 tf/m

2;

[MPa]/10 = [kN/cm2] ;

1 kPa = 103Pa = 0.001 MPa = 0.01 kgf/cm

2;

1 Pa = 1 N/m2= 0.1(0.09807) kgf/m

2=

= 0.00001(0.000009807) kgf/cm2;

1 kN/cm2

= 100 kgf/cm2= 10 Mpa ; [kN/mm

2] x 10

3= [MPa] ;

1 kN/m2= 1000 N/m

2= 0.01 kgf/cm

2= 0.001 MPa = 1000 Pa ;

1 kN/m2= 20.885 lb/ft

2= 100 kgf/m

2= 0.1 tf/m

2;

1 tf/m2= 0.1 kgf/cm

2= 0.01 MPa ;

1 kgf/cm2= 100 kN/m2= 0.1 Mpa = 0.1 N/mm2 = 0.01 kN/cm2;

1 ksi (kip/square inch) x 70.31 = [kgf/cm2] ; 1 ksi x 6.895 = [MPa] ;

1 ksi x 6895 = [ kN/m2

] ;

[psi (pound force / square inch)] x 0.0703 = [kgf/cm2];

[psi] x 6.895 = [Pa] ; [psi] x 0.006895 = [MPa];

1 psf (pound-force (lb) / square foot) x 4.882 = [kgf/m2] ;

1 psf x 47.879 = [N/m2] = [Pa] ;

1 psf = 1 lb/ft2= 0.004882 tf/m2= 0.04882 kN/m2 ;

1 psf x0.047879.44 = [kN/m2 ];

[psf] x 0.047879 = [lb/ft2] x 0.047879 = [kN/m

2];

1 psi (pound-force / square inch) x 0.07031 = ]kgf/cm2] ;

1 psi x 6895 = [N/m2] ; 1 psi x 6895 = [N/m2]=[Pa] ;1 psi x 6.895 = [kPa] ;1 psi x 0.006895 = [MPa];1 MPa = 145.03 psi;

1 psi x 6895055 = [kN/m2

]

1.4.4 :

1 kN/m2

= 100 kgf/m2= 0.1 tf/m

2;

1 kgf/m2= 0.01 kN/m

2= 0.001 tf/m

2;

1 tf/m2 = 1000 kgf/m

2= 10 kN/m

2

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1.4.5 :

1 kgf/m3= 0.01 kN/m

3;

1 tf/m3 = 10 kN/m3;

1 kN/m3

= 100 kgf/m3= 0.1 tf/m

3= 6.351 lb/ft

3;

1 pcf = 1 p/cf = 1 pound / cubic foot ;

1 pcf x 0.1570796 = [kN/m3] ;

150 pcf = 23.5619 kN/m3

2.

2.1

EN 1990.

EN 1990 2

, :

- -

EN 1990;

- EN 1991;

- EN 1990

- ,

EN 1992-1-1.

2.2

EN 1991, :

EN 1991-1.1 ,

EN 1991-1.2

EN 1991-1.3

EN 1991-1.4 EN 1991-1.5

EN 1991-1.6

EN 1991-1.7

EN 1997.

2.3 /

/

Gset ,

/

.

(SLS).

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2

23

2.4

(,

, ).

.

.

.

2.5

(SLS).

(ULS)

. ( ) -

, , ..

, .

2.6

(- ULS),

c

s . c s .

. 2.6.0.

2.6.0

(ULS)

c s

s

1,5 1,15 1,15

1,2 1,0 1,0

(,

,

(Serviceability Limit State) - SLS ,

c s .

1.

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24

2.6.1

:

Fd= F yFk (2.1)

:

F ;

y- . 2.6.1.0;

y0

,

y1

,

y2

Fk

2.6.1.0 -

Y0 Y1 Y2

:

A: ,

B: , C: , D:

E:

0,7

0,70,70,7

1,0

0,5

0,50,70,7

0,9

0,3

0,30,60,6

0,8

:

F: 30 kN

G: 30 kN, 160 kN

H:

0,7

0,70

0,7

0,50

0,6

0,30

1000 m

1000 m

0,7

0,5

0,5

0,2

0,2

0 0,6 0,2 0

T() 0,6 0,5 0

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25

2.6.1.1

Gd= G Gk - (2.2)

Gk

Gk,supGk,inf,

:

Gd,sup= G Gk,sup , (2.3)

Gd,inf= G Gk,inf. (2.4)

Gk,sup , Gk,inf ,

G,sup G,inf ,

:

Gd,sup= G,sup Gk,sup , (2.5)

Gd,inf= G,inf Gk,inf. (2.6)

2.6.1.2

Qd= QyGk, (2.7)

Gk-

2.6.1.3

Ad= AAk, (2.8)

Ak-

2.6.1.4

Pd= pPk, (2.9)

Pk-

2.6.1.5

AEd= AEk, (2.10)

AEk-

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2

26

2.7

:

-

;

- ,

,

;

-

,

.

ULS , :

- :

Gj kj p k Q1 k1 Qi 0i ki

j 1 i 1

G P Q Q >

+ + + ; (2.11)

-:

Gj kj p k Q1 01 k1 Qi 0i ki

j 1 i 1

G P Q Q >

+ + + ; (2.12)

j Gj kj p k Q1 k1 Qi 0i ki

j 1 i 1

G P Q Q >

+ + + ; (2.13)

x= 0,85 1,0 , x Gj;- :

GAj kj pA k d 11 k1 2i ki

j 1 i 1

G P Q Q

+ + + + (2.14)

- :

kj k 1 Ed 11 k1 2i ki

j 1 i 1

G P A Q Q

+ + + + (2.15)

. 2.7.1

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2

27

2.7.1

1

-

-

-

A:

;

,

,

:

-

-

-

G,sup

G,inf

QA

1,10

0,90

1,50

1,00

1,00

1,00

1,00

B:

,

,

,

,

(-)

-

-

-

G,supG,inf

Q

A

1,35

1,00

1,50

-

1,00

1,00

1,00

1,00

C:

(-)

-

-

-

G,supG,inf

Q

A

1,00

1,00

1,30

-

1,00

1,00

1,00

1,00

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2

28

3

3.1

3.1.1

,

(2000 2800 kg/m3) .

/

5 -

().

3.1.2

: ( 5% ) fck ,

fck,cube, EN 206-1.150 mm 300

mm. 150mm.

, 20oC /

28 .

fck

. 3.1.2.2

sc c0,4f ,

(). sc = fc.

sc c0,4f> ,

. sc c0,8f

,

0,2 .

(.3.1.2.1)

:

fcm= fck+ 8 (Mpa) (3.1)

()

28

, , ,

EN 12390.

t > 28 ,

t kt. kt

. 0,85.

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29

fck(t) t

( , ):

fck(t) = fcm(t) 8 (MPa) 3

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30

t ,

. 20oC

EN 12390 fcm(t)

(3.1) (3.2)

fcm(t) = bcc(t) fcm (3.2)

bcc(t) = exp1/2

28s 1

t

(3.3)

:

fcm(t) t ;fcm 28 ,

. 3.1;t ;

s , ;

= 0,20 CEM 42,5 R, CEM 52,5 N

CEM 52,5 R(R)

= 0,38 CEM 32,5 N (S)

28 , (3.2) (3.3) .

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Ta3.1.2.2 -

12/15 90/105 A

fck(MPa) 12 16 20 25 30 35 40 45 50 55 60 70 80 90fck,cube

(MPa)

15 20 25 30 37 45 50 55 60 67 75 85 95 105

fcm(MPa) 20 24 28 33 38 43 48 53 58 63 68 78 88 98 f

fctm

(MPa)

1,6 1,9 2,2 2,6 2,9 3,2 3,5 3,8 4,1 4,2 4,4 4,6 4,8 5,0 fctm=0,3fctm=2,1

fctk,0,05

(MPa)

1,1 1,3 1,5 1,8 2,0 2,2 2,5 2,7 2,9 3,0 3,1 3,2 3,4 3,5 fctk;0,05

fctk,0,95

(MPa)

2,0 2,5 2,9 3,3 3,8 4,2 4,6 4,9 5,3 5,5 5,7 6,0 6,3 6,6 fctk;0,95

Ecm

(GPa)

27 29 30 31 33 34 35 36 37 38 39 41 42 44 Ecm=2

(fcm eeeec1(

o/oo) 1,8 1,9 2,0 2,1 2,2 2,25 2,3 2,4 2,45 2,5 2,6 2,7 2,8 2,8

eeeecu3(o/o

eeeecu1(o/oo) 3,5 3,2 3,0 2,8 2,8 2,8

eeeecu3(o/o

eeeec2(o/oo) 2,0 2,2 2,3 2,4 2,5 2,6

eeeecu3(o/o

eeeecu2(o/oo) 3,5 3,1 2,9 2,7 2,6 2,6

eeeecu3(o/o

n 2,0 1,7

5

1,6 1,45 1,4 1,4eeeecu3(

o/o

eeeec3(o

/oo) 1,75 1,8 1,9 2,0 2,2 2,3 eeeecu3(o/o

eeeecu3(o/oo) 3,5 3,1 2,9 2,7 2,6 2,6

eeeecu3(o/o

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3.1.2.3 -

r

8

flck(MPa) 12 16 20 25 30 35 40 45 50 55 60 70 80

flck,cube

(MPa)

13 18 22 28 33 38 44 50 55 60 66 77 88

flcm(MPa)

17 22 28 33 38 43 48 53 58 63 68 78 88

flctm

(MPa)flctm= fctmhhhh1

flctk,0,05

(MPa)flctk,0,05= fctk,0,05hhhh1

flctk,0,95

(MPa)flctk,0,95= fctk,0,95hhhh1

Elcm

(GPa)Elcm= EcmhhhhE

eeeelc1(o/oo)

k = 1,1kflcm/ Elcm

k = 1,0

eeeelcu1(o/oo) eeeelc1

eeeelc2(o/oo) 2,0 2,2 2,3 2,4 2,5

eeeelcu2(o/oo) 3,5 3,1hhhh1 2,9hhhh1 2,7hhhh1 2,6hhhh1

n 2,0 1,75 1,6 1,45 1,4

eeeelc3(o/oo) 1,75 1,8 1,9 2,0 2,2

eeeelcu3(o/oo) 3,5hhhh1 3,1hhhh1 2,9hhhh1 2,7hhhh1 2,6hhhh1

5/19/2018 2 ...

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2

33

- ,

. 3.1.8(1).

fct,sp , fct

:

fct= 0,9 fct,sp (3.4)

, .

fctm(t) :

fctm(t) = (bcc(t))a. fctm (3.5)

bcc(t) (3.3)

a= 1 t

5/19/2018 2 ...

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2

34

3.1.4 ,

.

.

.

,

-.

. 3.1.4.1

(t t0te).

3.1.4.1

t : cs(t,ts)

; ci (t0)

cc(t,t0), .

t :

c(t) = cs(t,ts) + ci(t0) + cc(t,t0) (3.7)

,

cs .

5/19/2018 2 ...

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2

35

,

.

.

: -, ,

; ; -

.

f(t,t0)

Ec, 1,05Ecm.

, . 3.1.4.2

, , -

0,45fck(t0) t0 .

5/19/2018 2 ...

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2

36

) RH=50%

:

- 4 5

1

- 0t 100> e

0t 100= (

)

b) - RH=80%

3.1.4.2 - ffff( 0, t )

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2

37

ecc( 0, t ) t=

sc t0,

:

ecc( 0, t ) = f( 0, t ) .(sc/Ec) (3.8)

t0 0,45fck(t0), . 3.1.4.3.

,

.

, :

fk( 0, t ) = f( 0, t ) exp(1,5(ks 0,45)) (3.9)

:

fk( 0, t ) ,

f( 0, t ) ;

ks -sc/fcm(t0), sc

;

fcm(t0) .

3.1.4.3

5/19/2018 2 ...

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2

38

, . 3.1.4.2

40oC 40

oC RH = 40%

RH = 100%. :

F( 0, t ) ;

t0 , , ;h0 , 2Ac/u, Ac

;

u , ;S S, 3.1.2;

N N, 3.1.2;

R R, 3.1.2;

. ,

.

:

.

. -

, .

ecs:

ecs= ecd+ eca (3.10)

:ecs ;

ecd ;

eca .

, cd,

kh.ecd,0. ecd,0Ta. 3.1.4.4 (

30%).

5/19/2018 2 ...

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2

39

3.1.4.4 - eeeecd,0(o/oo)

CEM N

fck/fck,cube

(MPa) (%)

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

:

ecd(t) = bds(t,ts).kh.ecd,0 (3.11)

,

kh, h0, Ta. 3.1.4.5:

T3.1.4.5 - kh(3.11)

h0 kh

100 1,0

200 0,85

300 0,75

500 0,70

bds

(t,ts) = s

3s 0

(t t )

(t t ) 0,04 h

+

(3.12)

:

t , ;

ts() (

).

.

h0 (mm) = 2Ac/u

:

Ac ;

u , .

5/19/2018 2 ...

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2

40

:

eca(t) = bas(t)eca( ) (3.13)

bas(t) = 1 exp(-0,2t0,5) (3.14)

t .

3.1.5 -

scec . 3.1.5.1 (

, )

, (3.15):2

c

cm

k

f 1 (k 2)

=

+ (3.15)

: c c1/ = ;

ec1 . 3.1.2.2;

cm c1

cm

1,1Ek

f

= (fcmTa. 3.1.2.2).

(3.15) 0< |ec| < |ecu1| ecu1

.

-,

.

5/19/2018 2 ...

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2

41

3.1.5.1 -

( 0,4fcm

Ecm)

a0 = arctg Ecm0 ; Ecm0

28 ;

a= arctg Ecm; Ecm -

sc=0,4fcm;ac1= arctg Ec1; Ec1-

sc=fcm;

ec1= 0,7fcm0,31

| -2,8|o/oo ; ecu1= | -3,5|

o/oo fck C50/60.

3.1.6

:

fcd= accfck/ gc (3.16)

: gc , 2.6 . 2.6.0;

acc ,

,

.

acc0,8

1,0 .

1.

5/19/2018 2 ...

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2

42

:

flcd= alccflck/ gc (3.17)

alcc 1.

fctd :

fctd= actfctk,0,05/ gc (3.18)

: gc , 2.6 . 2.6.0;

act ,

, .

act0,8

1,0 .

1.

fctd

:

flctd= alctflctk/ gc (3.19)

alcc 1.

3.1.7 -

, . 3.1.7.1 (

):

n

cc cd

c2

f 1 1

=

0 c c2 (3.20)

c= fcd c2 c cu2 (3.21)

:

n , . 3.1.2.2;

c2 ,

. 3.1.2.2;cu2 , . 3.1.2.2.

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2

43

3.1.7.1 - -,

-

, (3.20),

, . 3.1.7.2 (

) c3 cu3

. 3.1.2.2.

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2

44

3.1.7.2 - -

(. 3.1.7.3). ,

, , ,

:

l= 0,8 fck 50MPa (3.22)

l= 0,8 (fck-50)/400 50 < fck 90MPa (3.23)

:

h= 1,0 fck 50MPa (3.24)

h= 1,0 (fck-50)/200 50 < fck 90MPa (3.25)

5/19/2018 2 ...

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2

45

3.1.7.3 -

3.1.8

.

:

fctm,fl= max{(1,6 h/1000)fctm; fctm} (3.26)

:h mm;

fctm . 3.1.2.2

, (3.26),

.

3.1.9

-: -

- . .

-. 3.1.9.1 -

- (

),

:

fck,c= fck(1,000 + 5,0 s2/fck) s2 ck0,05f (3.27)

fck,c= fck(1,125 + 2,5 s2/fck) s2 ck0,05f> (3.28)

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2

46

ec2,c= ec2(fck,c/fck)2 (3.29)

ecu2,c= ecu2+ 0,2s2/fck (3.30)

s2 (=s3)

,

, ec2ecu2. 3.1.2.2.

,

.

A -

3.1.9.1 - -

5/19/2018 2 ...

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2

47

3.2.

3.2.1

, ,

.

EN 10080. ,

fyk= 400 600 MPa.

, EN 10080,

3.2.2 3.2.6

C.

, EN 10080.

3.2.2

: (fyk f0,2k);

(fy,max); (ft); (euk ft/fyk);

; (fR C); ; ; ;

.

-40oC 100oC , .

3.2.3

fyk ( , 0,2%

f0,2k) ftk,

.

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2

48

3.2.4

,

(ft/fy)k

euk - . 3.2.4.1 -

.

(ft/fy)keuk. 3.2.4.2.

a) b)

3.2.4.1 - -

(

)

5/19/2018 2 ...

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2

49

3.2.4.2

,

(-

)

(%)

A C A B C -

fykfo,2k

(MPa) 400 o 600 5,0

k=(ft/fy)k

1,05 1,08 1,15

12

0,035

0,040

0,056

5,0

5/19/2018 2 ...

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2

50

3.2.5

. 3.2.5.1, EN 10080.

EN ISO 17760.

.

, , -

25% ,

,

. -,

.

3.2.5.1 -

1

20 mm ,

,

3,

2,

,

-

14 mm

,

4

2,4

-

14 mm

2-

14 mm

4

2,4

:1

2 0,57 .

3 16 mm .

4 28 mm .

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2

51

3.2.6

,

EN 10080.

3.2.7 -

, 3.2.2

.

(. 3.2.7.1):

a)

eud

kfyk/gs euk, k= (ft/fk)k;

b)

.

7850 kg/m3. Es

200 GPa.

3.2.7.1 - -

()

5/19/2018 2 ...

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2

52

BSt500 -

RSt37-2 DIN 17100 ( )

N

mm

cm N

mm

kg/m1 2 3 4 5 6 7 8 9 10

8 0,503 1,005 1,508 2,011 2,513 3,016 3,519 4,021 4,524 5,027 8 0,395

10 0,785 1,571 2,356 3,142 3,927 4,712 5,498 6,283 7,069 7,854 10 0,617

12 1,131 2,262 3,393 4,524 5,655 6,786 7,917 9,048 10,176 11,310 12 0,888

14 1,539 3,079 4,618 6,158 7,697 9,236 10,776 12,315 13,854 15,394 14 1,208

16 2,011 4,021 6,032 8,042 10,053 12,064 14,074 16,085 18,096 20,106 16 1,578

18 2,545 5,089 7,634 10,179 12,723 15,286 17,813 20,358 22,902 25,447 18 1,998

20 3,142 6,283 9,425 12,566 15,708 18,850 21,991 25,133 28,274 31,416 20 2,466

22 3,801 7,603 11,404 15,205 19,007 22,808 26,609 30,411 34,212 38,013 22 2,984

25 4,910 9,821 14,732 19,642 24,553 29,464 34,375 39,285 44,196 49,107 25 3,85528 6,158 12,315 18,473 24,630 30,788 36,945 43,103 49,260 55,418 61,575 28 4,834

32 8,042 16,085 24,127 32,170 40,212 48,255 56,297 64,340 72,382 80,425 32 6,313

40 12,571 25,142 37,714 50,285 62,857 75,428 88,000 100,571 113,142 125,714 40 9,868

mm

cm .

mm

kg/m1 2 3 4 5 6 7 8 9 10

5,5 0,238 0,476 0,714 0,952 1,190 1,428 1,666 1,904 2,142 2,380 5,5 0,187

6 0,283 0,565 0,848 1,131 1,414 1,696 1,979 2,262 2,545 2,827 6 0,222

6,5 0,332 0,664 0,996 1,328 1,660 1,992 2,324 2,656 2,988 3,320 6,5 0,260

7 0,385 0,770 1,155 1,540 1,925 2,310 2,695 3,080 3,465 3,850 7 0,302

8 0,503 1,005 1,508 2,011 2,513 3,016 3,519 4,021 4,524 5,027 8 0,3959 0,636 1,272 1,908 2,544 3,180 3,816 4,452 5,088 5,724 6,360 9 0,499

10 0,785 1,571 2,356 3,142 3,927 4,712 5,498 6,283 7,069 7,854 10 0,617

11 0,950 1,900 2,850 3,800 4,750 5,700 6,650 7,600 8,550 9,500 11 0,746

12 1,131 2,262 3,393 4,524 5,655 6,786 7,917 9,048 10,176 11,310 12 0,888

14 1,539 3,079 4,618 6,158 7,697 9,236 10,776 12,315 13,854 15,394 14 1,208

16 2,011 4,021 6,032 8,042 10,053 12,064 14,074 16,085 18,096 20,106 16 1,578

5/19/2018 2 ...

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2

53

3.3

3.3.1

EN 10138 .

(, ) :

, 0,1%

(fp0,1k),

(fpk/fp0,1k) and (euk)

; ; .

( ..

), (EN 1992-1-2).

.

( )

.

.

EC-2 :

1 : ;

2: ; 3:

.

1 EN 10138.

3.3.2

0,1%

(fp0,1k) (fpk)

,

0,1%

, . 3.3.2.1.

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2

54

3.3.2.1 - -(

)

3.3.3

EN 10138.

,

, EN 10138.

,

EN ISO 15630.

, , -

,

,

fpk/fp0,1k k . k .

1,1.

3.3.4

.

EN 10138 .

5/19/2018 2 ...

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2

55

3.3.5

fp0,1k, fpkeuk.

Ep 205 GPa. 195 o 210 GPa, .

, , .

Ep

195 GPa. 185 o 205 GPa,

. ,

, .

7850 kg/m

3

.-

-40oC +100

oC.

fpd

fp0,1k/ gs (. 3.3.5.1).

-(.3.3.5.1):

a) eud.

/,

,

, . 3.3.5.1,

b)

eud.

0,9euk . - ,

eud= 0,02 fp0,1k/fpk= 0,9.

5/19/2018 2 ...

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2

56

3.3.5.1 - -

()

4

4.1

,

(N 1990).

, ( EN 1990),

.

,

.

,

.

( EN 206-1)

. , ,

.

.

5/19/2018 2 ...

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2

57

4.2

,

.

4.2.1,

EN 206-1.

. 4.2.1,

, , :

(

) ; (EN 206-1, ISO 9690);

(EN 206-1);

(EN 206-1, );

, : ; ;

(EN 206-1).

4.2.1 EN 206-1

,

1.

X0

:

/,

:

2. , XC1

,

XC2 , ,

XC3

,

XC4 ,

,

XC2

3. , , XD1 ,

XD2 , ,

,

5/19/2018 2 ...

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2

58

XD3 ,

,

.

.

4. , XS1

XS2

XS3 ,

5 . / XF1

,

XF2

,

XF3

,

XF4

,

.

,

,

.

,

6.

XA1

2 EN 206-1

XA2

2 EN 206-1

` XA3

2 EN 206-1

4.3

.

: , , ,

, , , ,

(, , ).

5/19/2018 2 ...

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2

59

4.4 M

4.4.1

4.4.1.1

-

( ,

) -.

.

cmin(4.4.1.2)

devc (4.4.1.3):

cnom= cmin+ devc (4.1)

4

4.4.1.2 M, cmin

, cmin, , :

, (), (EN 1992-1-2).

-cmin

, .

cmin= max {cmin,b; cmin,dur+ Dcdur,g- Dcdur,st- Dcdur,add; 10 mm} (4.2)

: cmin,b ;

cmin,dur - ; m

Dcdur,g - ;

Dcdur,st -

;

Dcdur,add

.

,

-cmin,b, . 4.4.1.2.1

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2

60

4.4.1.2.1- , cmin,b,

cmin,b*

(n)

n= bn 55 mm , nb

* -32 mm,

cmin,b5 mm.

: cmin,b,

,

. :

- : ;

- : -

-

80 mm.

:

- 1,5 x;- 2,5 x.

,

, cmin,dur .: cmin,dur

. (

50 ) S4 ,

. 4.4.1.2.2,

. 4.4.1.2.3. S1.

cmin,dur . 4.4.1.2.4

() . 4.4.1.2.5 ().

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2

61

4.4.1.2.2

4.4.1.2.1

,

,

,

XC1 XC2 XC3 XC4 XD1 XD2 XD3 XS1 XS2 XS3

C20/25 C25/30 C30/37 C30/37 C35/45 C30/37 C35/45

/

X0 XF1 XF2 XF3 XA1 XA2 XA3

C12/15 C30/37 C25/30 C30/37 C30/37 C35/45

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2

62

4.4.1.2.3 -

4.4.1.2.1

X0 XC1 XC2/XC3 XC4 XD1 XD2/XS1 XD3/XS2/

XS3

-

100

2

2

2

2

2

2

2

1) 2)

C30/37

1

C30/37

1

C35/45

1

C40/50

1

C40/50

1

C40/50

1

C45/55

1

(-

-

-

)

1

1

1

1

1

1

1

-

--

1

1

1

1

1

1

1

4.4.1.2.3 :

1)

.

(, , ).2),

4% .

5/19/2018 2 ...

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63

4.4.1.2.4 - cmin,dur,

EN 10080

cmin,dur(mm) -

4.4.1.2.1

X0 XC1 XC2/XC3 XC4 XD1/XS1 XD2/XS2 XD3/XS3

S1 10 10 10 15 20 25 30S2 10 10 15 20 25 30 35S3 10 10 20 25 30 35 40S4 10 15 25 30 35 40 45S5 15 20 30 35 40 45 50S6 20 25 35 40 45 50 55

4.4.1.2.5 cmin,dur,

e

cmin,dur(mm)-

4.4.1.2.1

X0 XC1 XC2/XC3 XC4 XD1/XS1 XD2/XS2 XD3/XS3

S1 10 15 20 25 30 35 40S2 10 15 25 30 35 40 45S3 10 20 30 35 40 45 50S4 10 25 35 40 45 50 55S5 15 30 40 45 50 55 60S6 20 35 45 50 55 60 65

Dcdur,g.

Dcdur,g .

0 mm.

, Dcdur,st .

, . Dcdur,st

. 0 mm.

, , ,

Dcdur,add.

Dcdur,add . 0 mm.

5/19/2018 2 ...

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2

64

, ,

,

, : - C25/30; ,

, (-28 ); .

.

( )

5 mm.

/

( XF XA),

(EN 206-1, 6).

, 4.4, .

, ,

EN 206-1.

( , ).

cmin k1

1, k2XM2 k3XM3.

: XM1 , ,

.

XM2 ,

, . XM3

, ,

. k1, k2 k3

. 5 mm,

10 mm 15 mm.

4.4.1.3

cnom,

(Dcdev).

.

Dcdev .

10 mm.

ENV 13670-1.

.

.

5/19/2018 2 ...

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2

65

, (

).

Dcdev .

: Dcdev . :

- ,

,

Dcdev:

10 mm Dcdev 5 mm;

-

(

), Dcdev

:

10 mm Dcdev 0 mm;

, , -

. ,

-k1mm ,

( ) k2 mm

, . ,

, , .

k1 k2 .

40 mm 75 mm.

4.1

.

:

S4 50 ,

: XC1

; XC3 ()

, , ;25/30; BSt500 , N14 .

: . 4.4.1.2.3

.

. 4.4.1.2.2 cmin=14mm (),

. 4.4.1.2.4 cmin,dur= 15mm XC1; cmin,dur= 25 mm XC3.

(4.2) .

:

cmin = max [14,15] = 15mm XC1 ; cmin = max[14,25] = 25mm XC3.

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66

Dcdev=10mm. (4.1)

:

cnom= cmin+ Dcdev= 15 + 10 = 25mm XC1; cnom= 25 +10 = 35mm XC3.

4.2

.:

S4 50 ,

: XC1

; XC3 ()

, , ;

25/30; BSt500 , N8 .

: . 4.4.1.2.3

, S3.

. 4.4.1.2.2 cmin=8 mm (), . 4.4.1.2.4:

cmin,dur= 10mm 1; cmin,dur= 20 mm XC3.

(4.2) .

:

cmin = max [8,10] = 10mm 1; cmin = max[8,20] = 20mm XC3.

:

Dcdev=10mm. (4.1) :

cnom= cmin+ Dcdev= 10 + 10 = 20mm 1; cnom= 10 +20 = 30mm XC3.

5

5.1

, ,

. ,

.

,

, :

; ;

; ;

,

.

.

,

, .

5/19/2018 2 ...

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2

67

5.2

, , , , , ,

.

, .

, -3

. .

, --5

.

, ,

, :

- () ,- , 4

, -2.

, . , :

- -1500 mm;

- -4 ;

- --1/10

50 mm;- , 10

.

,

50 mm 40 mm., 4

-3

. .

5.3

.

.

.

5/19/2018 2 ...

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2

68

i ,

:

i 0 h m. . = , (5.1)

:

0 ;

h : h h2 / l; 2 / 3 1 = ;

m : m 0,5(1 1 /m) = +

l [m],m ,

.

0 .

1/200.

(5.1) l m ,

(. 5.3.1):

- : l= , m=1;

- : l= , m=

,

.

- ,:

l= ; m= (),

.

, a) b):

a) ei, b)

ei= i 0l / 2 (5.2)

l0.

ei=l0/400 , h 1. =

c)

Hi, :(5.3.1-a1):

Hi= iN (5.3)

N .

5/19/2018 2 ...

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2

69

: , ,. Hi

i

,

.

(. 5.3.1-b):

i i b aH (N N )= (5.4)

(. 5.3.1-c1):

i i b aH (N N )= /2 (5.5)

E(. 5.3.1-c2):

i i aH .N= (5.6)

NaNb , Hi.

ei = l0/400 ,

.

a1) a2)

a) /

5.3.1 -

5/19/2018 2 ...

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2

70

b) c1) c2)

5.3.1 -

5.4

, ,

(ULS),

(SLS).

:

- ;

- ;- .

,

(ULS), ,

. , .

(SLS) .

5/19/2018 2 ...

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2

71

5.5

(ULS).

(ULS) ,

.

:

a)

b) 0,5 o 2,

, :

1 2 uk k x / d + f

ck

50 MPa (5.7a)

3 4 uk k x / d + fck> 50 MPa (5.7b)

5k B C

6k A

: ;

xu ;d .

k1, k2, k3, k4,k5 k6.

k10,44, k21,25(0,6 + 0,0014/ecu2), k3= 0,54,

k4= 1,25(0.6+0,0014/ecu2), k5= 0,7 k6= 0,8.ecu2 . 3.1.2.2 3.1.2.3.

,

(

).

, .

5.6

5.6.1

, ,

(ULS).

,

. ()

().

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5.6.2 ,

,

, :

- , :

xu/d 0,25 C50/60;

xu/d 0,15 C55/67;

- B C;

-

0,5 2.

,

.

.

5.6.3

1,2 . ,

(

) .

,

, s

-(. 5.6.3.1).

5.6.3.1- s

xu/d 0,45

, -C50/60, 0,35 , -

C55/67.

s

.

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pl,d ,

kl, . pl,d .

()

, -C50/60 C90/105

. 5.6.3.2.

C55/67 C90/105

.

l=3,0. pl,d

kl:

kl= / 3 (5.8)

l

d.

l

:

l= MSd/ (VSd. d) (5.9)

5.6.3.2 , pl,d ,

. = 3,0

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74

5.6.4

( )

.

h ().

,

, .

,

.

: ,

; ,

; .

5.7

ULS, SLS,

.

.

5.8 ,

5.8.1

: , ,

.

: ,.

5.8.2

, -10%

.

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75

5.8.3

5.8.3.1

, l-

llim . llim

. :

llim= 20.A.B.C/ n (5.10)

: A = 1/(1+ 0,2fef) ( fef , A= 0,7);

B = 1 2+ (w , B= 1,1);

C = 1,7 rm (rm , C= 0,7);fef - ;

w =As fyd/ (Acfcd) ;

As ;

n = NEd/ (Acfcd) ;

rm= M01/ M02 ;M01, M02 , |M02| |M01|.

M01M02

, rm (C 1,7).

(.e. C > 1,7).

rm1,0 (.e. C=0,7);

-

,

;

- .

.

() , (b)

(c)

DIN 1045-1 l:

l llim= 25, if n 0,4 (5.11a)l llim= 16/ n , n>0, (5.11b)

l llim 25(2 e01/e02), (5.12)

|e02| |e01| ; NRd= NEdand MRd NEd h/20, h

.

, :

; :

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2

76

cm c

Ed

E I1 1

H F 0,2 0,1m

+

m 3 , (5.13a)

cm c

Ed

E I1 1

H F 6

m 4 (5.13b)

: m ;

H [m] ;

EdF - ( F= 1,0);cm cE I - ,

.

5.8.3.2

:

= l0/i (5.14)

: l0 ;

i .

. 5.8.3.2.1

l0:

(. 5.8.3.2.1(f)):

1 20

1 2

k kl 0,5l. 1 . 1

0, 45 k 0,45 k

= + +

+ + (5.15)

(. 5.8.3.2.1(g)):

1 2 1 20

1 2 1 2

k .k k k l l.max 1 10 ; 1 . 1

k k 1 k 1 k

= + + +

+ + + (5.16)

: k1, k2,

1 2:

k = ( / M).(EI / l) , (5.17)

M, . 5.8.3.2.1 (f) (g);

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77

EI ;

l .k=0 k=

. k1k2 0,1.

a) b) c) d) e) f) g)

5.8.3.2.1

(5.17) :

k1(k2) =cm col col

cm b b

E I / l,

E I / l

(5.18)

: Icol Ib

;

lcollb.

a ,

( ),

- a=1,0 ; - a=0,5; - a=0. / ,

,

():

l0= bE I / N (5.19)

: EI ;

NB , EI

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78

5.8.4

, fef, ,

, (), :

ef ( ,t0) 0Eqp 0Ed.M / M

= (5.20)

:

f( ,t0)

M0Eqp

- (SLS);

M0Ed

- (ULS).

M0Eqp/M0Ed ,

.

, .. fef = 0 ,

: ( ,t0) 2 ; 75 ; 0Ed EdM / N h. M0Ed

, h

.

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5.8.5,

5.8.5.1

-

l0. , ,

.

5.8.5.2

:

MEd= M0Ed+ M2 (5.21)

:

M0Ed ,

;M2 .

MEd M0Ed M2.

.

M0Ed

, M2

, 5.8.3.2.

M01M02

M0e:

M0e = 0,6 M02+ 0,4 M01 0,4 M02 (5.22)

M01M02,

. . |M02| |M01|.

M2(5.21) :

M2= NEd e2, (5.23)

:

NEd ;

e2 = (1/r)2

0l / c ;

1/r , 5.8.5.3;l0 , 5.8.3.2;

c , ;

= 10 2( ) .

, - (

8 ).

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2

80

5.8.5.3

(

) :

1/r = Kr.Kf.1/r0 (5.24):

Kr , ,

(5.26);

Kf , , (5.27);

1/r0= eyd/ (0,45d) ;

eyd= fyd/ Es ;

d , (5.25);

,

, :

d= (h/2) + is (5.25)

: is.

Kr(5.24) :

r u u balK (n n) / (n n ) 1= (5.26)

:

n = NEd/ (Acfcd), ;

NEd ;

nu = 1 +

nbal n .0,4;

= Asfyd/ (Ac fcd)As ;

Ac .

:

Kf= 1 + bfef 1 (5.27)

:

fef , 5.8.4;

= 0,35 + fcd/200 /150 , 5.8.3.1.

Ma, M0Ed

(5.21), :

Ma= NEdea (5.28)

:

ea= l 0l / 2 - ;

l0- ;

l

1

100 l

= , 0,672

1,0

l

, l [m]

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5.8.6

, .

, -.

, :

y z/ 2 z y/ 2 (5.29a)

ey/h ez/b (. 5.10)

:

y eq

z eq

e / h0,2

e / b

z eq

y eq

e / b0,2

e / h (5.29b)

:b,h ;

beq= iy. 12 eq zh i 12= ;

ly, lz l0/i y- z-;

iy, iz y- z-;

ez = MEdy/ NEd z;

ey = MEdz/ NEd y;

MEdy y,

;

MEdz z,

;NEd e

.

(5.29) ,

, (

5.8.2 5.8.3).

, :

aa

EdyEdz

Rdz Rdy

MM1,0

M M

+

(5.30)

:

MEdz/y ,

;

MRdz/y ;

a ;

: a=2

: :

NEd/ NRd 0,1 0,7 1,0

a = 1,0 1,5 2,0

.

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NEd ;

NRd = Acfcd+ Asfyd,

,

:

Ac ;

As

5.8.6.1 eyez

5.9

, ,

,

.

l/300

. l .

, , ,

:

- :( )

0t

1/3

l 50

b h / b h / b 2,5 (5.31a)

- :( )

0t

1/3

l 70

b h / b h / b 3,5 (5.31b)

: l0t ,

;

h lot;

b .

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83

5.1

1 2(. 5.1.1).

40/40 cm, 25/50 cm.

-

b - b

5.1.1

: , S.

- C25/30: Ecm= 31 GPa = 3100 kN/cm2 (. 3.1.2.2),

fcd= 2c ckb

.f 0,85.2514,17MPa 1,1417kN / cm1,5

= = =

- BSt500 : Es= 200 GPa = 20000 kN/cm2;

fyd=yk 2

s

f 5043, 478kN / cm

1,15= = ;

As1=As2= 10,18 cm2

(4N18) ; As0= 5,09 cm2 (2N18) ;

ULS:

NEd=1800 kN , NEd=1400 kN .

MoM02=90,0 kNm , M01= 42,0 kNm.

: SLS ULS : M0Eqp/ M0Ed= 0,72.: Ac= 400.400=160000 mm

2= 1600 cm

2, u = 400+400=800 mm,

h0= 2Ac/u = 2.160000/800= 400 mm . 3.1.4.2 t0= 30

0( ,t )

=2,3. (5.20) :

0ef ( , t ) 0Eqp 0Ed.M / M 2,3.0,72 1,656 ( 2)

= = = > .

A= 1/(1+ 0,2 fef) = 1/(1+0.2.1,656) = 0,751 ; w= fydAs/ (fcdAc)=43,478.25,45

0,6061,1417

= ;

As= As1+As2+As0= 2.10,18 +5,09 = 25,45 cm2; B= 1 2. 1 2.0,606 1,487+ = + = ;

n = NEd/(fcdAc) =1800

0,9808

1,147.1600

= ; rm=01

02

M 420,467

M 90

= = ;

C=1,7- rm=1,7-(-0,467)=2,167 ;

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lim 20.A.B.C / n 20.0,751.1,487.2,167 / 0,9808 48,8708 = = =

(5.13.b) m=4 ().

:

53

9 2

cm c

40.40E I 4 3100. 2,645.10 kNcm 2,645.10 kNm

12

= = =

2.

SLS: EdF 0, 72.(2.1400 2.1800) 4608kN= + = .:

5cm c

Ed

E I1 1 2,645.100,451 1,65

H F 16,8 4608= =

llim= 48,8708 .

: eyd= fyd/ Es= 43,478/20000= 0,00217;

:

yd 1

0

1 0,00217

0,000134cmr 0,45.d 0,45.36

= = =;

: n=0,9808; nu= 1+w= 1 + 0,606= 1,606; nbal = 0,4;

Kr =u

u bal

n n 1,606 0,98080,5184 1

n n 1,606 0,4

= = 1;

: 1/r = KrKf/ r0= 0,5184.1,1088.0,000134 = 0,000077 cm-1

.

e2= (1/r)2

0l / c = 0,000077.708,962/10 = 3,87 cm.

M2= NEd.e2= 1800. 0,0387 = 69,66 kNm.

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85

:

l

1 2 1 20,00488

200 200l 4,2 = = = ;

ea= l 0.l / 2 = 0,00488..708,96/2 = 1,73 cm;

Ma= NEd. ea= 1800.0,0173 = 31,14 kNm 2: M = M02+ Ma+ M2= 90 + 31,14 + 69,66 = 190,8 kNm.

5.2

5.1

().

.

: :

1 20

1 2

k kl 0,5l. 1 . 1

0,45 k 0,45 k

= + + + +

=

=1, 2092 0,1

0,5.420. 1 . 10,45 1,2092 0,45 0,1

+ +

+ + = 285,18cm

l= l0/ i = 285,18/11,547 = 24,69 < llim= 48,8708

.

6 (ULS)

6.1

6.1.1

EC-2

.

1.

.

2. (

).

3.

( . 3.1.7.1 3.2.7.1).

4.

.

5. .

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6.1.2

A,B C

. 6.1.2.1.

() euk= 3,5

o/oo(. 3.1.7.1).

( )

e= 2o/oo (.3.1.7.1).

euk = 25o/oo (.3.1.7.2

-).

1: . -.

.

, () es1 eyd.

. ,( ).

2:. eud,

|ec| |ecu|. .

.

6.1.2.1 -

3:( ). ecu.

eud>ecl eyd.

. - .

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4:( ). es1< eyd .

.

-

.

4a:, -. ecu.

. ,

.

5: . - . - ,

cu , , -

|es2| (|es3|).

C, :

xc= h(1-ec2/ecu2) xc= h(1-ec3/ecu3).

. ,

, .

6.1.3

6.1.3.1

lim

x0,45

d = = - : C12/15 o C50/60 (6.1)

6.1.3.2

cu3lim

cu3 yd

| |x0,617

d | |

= = =

+- : C/12 o C50/60 - BSt500-

. 6.1.3.2.1 (6.2)

6.1.3.2.1

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6.1.4 ,

(. 6.1.4.1):

6.1.4.1NEd= Fcd+ Fsld, (6.3)

MEds= - Fcd(d zs1 a) + Fsldzs1 Fcd(d-a) . (6.4)

:

Fcd= avbxfcd (6.5)

: a = kax (6.6)

:

MEds= -avbxfcd(d-kax) , (6.7)

() :

mEds= avx(1-kax) =Eds

2cd

M

bd f (6.8)

:

w= mEds/h (6.9)

:

Asl= cd Edsl

1( bdf N )

+ (6.10)

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NEd= 0 : w=sl sll

sl

cd cd

A

bd f f

= (6.11)

rsl :

rsl= Asl/bd (6.12)

C12/15 C50/60 . 6.1.4.2

.

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6.1.4.2 () ,

C12/15 - C50/60 (ggggc= 1,50) BSt500 (ggggs= 1,15)

mEds;m w x z=hw x z=hw x z=hw x z=h ka aaaav eeeec2[o

/oo] eeees1[o

/oo] sssss1[kN/cm2

]0,01 0,0101 0,030 0,990 0,346 0,337 -0,77 25,00 43,478

0,02 0,0203 0,044 0,985 0,360 0,463 -1,15 25,00 43,478

0,03 0,0306 0,055 0,980 0,360 0,553 -1,46 25,00 43,478

0,04 0,0410 0,066 0,976 0,368 0,622 -1,76 25,00 43,478

0,05 0,0515 0,076 0,971 0,377 0,676 -2,06 25,00 43,478

0,06 0,0621 0,086 0,967 0,387 0,718 -2,37 25,00 43,478

0,07 0,0728 0,097 0,962 0,396 0,751 -2,68 25,00 43,478

0,08 0,0836 0,107 0,956 0,405 0,778 -3,01 25,00 43,478

0,09 0,0946 0,118 0,951 0,413 0,801 -3,35 25,00 43,478

0,10 0,1057 0,131 0,946 0,416 0,810 -3,50 23,30 43,478

0,11 0,1170 0,145 0,940 0,416 0,810 -3,50 20,71 43,478

0,12 0,1285 0,159 0,934 0,416 0,810 -3,50 18,55 43,478

0,13 0,1401 0,173 0,928 0,416 0,810 -3,50 16,73 43,4780,14 0,1518 0,188 0,922 0,416 0,810 -3,50 15,16 43,478

0,15 0,1638 0,202 0,916 0,416 0,810 -3,50 13,80 43,478

0,16 0,1759 0,217 0,910 0,416 0,810 -3,50 12,61 43,478

0,17 0,1882 0,232 0,903 0,416 0,810 -3,50 11,56 43,478

0,18 0,2007 0,248 0,897 0,416 0,810 -3,50 10,62 43,478

0,19 0,2134 0,264 0,890 0,416 0,810 -3,50 9,78 43,478

0,20 0,2263 0,280 0,884 0,416 0,810 -3,50 9,02 43,478

0,21 0,2395 0,296 0,877 0,416 0,810 -3,50 8,33 43,478

0,22 0,2528 0,312 0,870 0,416 0,810 -3,50 7,71 43,478

0,23 0,2665 0,329 0,863 0,416 0,810 -3,50 7,13 43,478

0,24 0,2804 0,346 0,856 0,416 0,810 -3,50 6,61 43,478

0,25 0,2946 0,364 0,849 0,416 0,810 -3,50 6,12 43,478

0,26 0,3091 0,382 0,841 0,416 0,810 -3,50 5,67 43,4780,27 0,3239 0,400 0,834 0,416 0,810 -3,50 5,25 43,478

0,28 0,3391 0,419 0,826 0,416 0,810 -3,50 4,86 43,478

0,29 0,3546 0,438 0,818 0,416 0,810 -3,50 4,49 43,478

0,2961 0,3643 0,450 0,813 0,416 0,810 -3,50 4,28 43478

0,30 0,3705 0,458 0,810 0,416 0,810 -3,50 4,15 43,478

0,31 0,3869 0,478 0,801 0,416 0,810 -3,50 3,82 43,478

0,32 0,4038 0,499 0,793 0,416 0,810 -3,50 3,52 43,478

0,33 0,4211 0,520 0,784 0,416 0,810 -3,50 3,23 43,478

0,34 0,4390 0,542 0,774 0,416 0,810 -3,50 2,95 43,478

0,35 0,4575 0,565 0,765 0,416 0,810 -3,50 2,69 43,478

0,36 0,4768 0,589 0,755 0,416 0,810 -3,50 2,44 43,478

0,37 0,4968 0,614 0,745 0,416 0,810 -3,50 2,20 43,478

0,371 0,4994 0,617 0,743 0,416 0,810 -3,50 2,17 43,478

0,38 0,5177 0,640 0,734 0,416 0,810 -3,50 1,97 39,782

0,39 0,5394 0,666 0,723 0,416 0,810 -3,50 1,75 35,545

0,40 0,5625 0,695 0,711 0,416 0,810 -3,50 1,54 31,305

0,41 0,5873 0,725 0,698 0,416 0,810 -3,50 1,33 27,319

0,42 0,6131 0,758 0,685 0,416 0,810 -3,50 1,12 23,333

0.43 0,6417 0,792 0,670 0,416 0,810 -3,50 0,92 19,507

0,44 0,6717 0,830 0,655 0,416 0,810 -3,50 0,72 15,681

0,45 0,7064 0,873 0,637 0,416 0,810 -3,50 0,51 11,884

0,46 0,7455 0,921 0,617 0,416 0,810 -3,50 0,30 8,086

0,47 0,7939 0,980 0,592 0,416 0,810 -3,50 0,07 4,043

0,473 0,8094 1,000 0,584 0,416 0,810 -3,50 0,00 0,000

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6.1.1

; - C16/20 ,

fck = 1,6 kN/cm2, fcd= fck/gc= 1,6/1,5 = 1,067 kN/cm

2; - BSt500 .

N 8mm, fyk = 50 kN/cm2, fyd = fsk/gs = 50/1,15 = 43,478 kN/cm

2 ; h = 10 cm;

cnom = 20mm ( 4.2);

d1= cnom+ N/2 = 20 + 8/2= 24 mm=2,4cm;

d= h d1 = 10 2,4 = 7,6 cm; MEds = 14,5 kMm = 1450 kNcm;

As1= ?

: :

mEds =Eds

2 2

cd

M 14500,235

bd f 100.7, 6 .1, 067= = . 6.1.4.2

:

h= 0,8595 , x= 0,3375< xlim= 0,45 (0,617) .

:

Asl=2Eds

yd

M 14505,106cm / m'

f d 43, 478.0,8595.7, 6= = ;

s1NA = 0,503 cm2(N8) :

n = sl

s1N

A 5,10610,15 N8 / m ' 14N / m '

A 0,503= = < .

6.1.5 () (),

, x> xlim= 0,45 (0,617) , MEds> MEds,lim . 6.1.5.1.

:

MEds,lim= mEds,limbd2fcd (6.13)

. 6.1.4.2 mEds,lim= 0,296 (0,371)

lim= 0,813 (743).

MEds,lim:

Asl,lim= Eds,lim

yd lim

Mf d

(6.14)

:

MEds= MEds MEds,lim .

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6.1.5.1

MEds ,

() (, ) .

:

As2= As1=Eds

2 yd

M 1

d d f

(6.15)

: As1= Asl,lim+ As1

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6.1.2

o 25/45cm;

- C16/20 , fck = 1,6 kN/cm2 , fcd=acc fck/gc = 0,85.1,6/1,5 = 0,907 kN/cm

2;

- BSt500 ; fyk= 50 kN/cm2, fyd= fyk/gs= 50/1,15 = 43,478 kN/cm

2; h = 45 cm;

d1= 4cm; d2= 4cm.

d= h d1= 45 4 = 41 cm; MEds= 156,28 kMm = 15628 kNcm.

: mEds =Eds

2 2

cd

M 156280,41

bd f 25.41 .0,907= = . 6.1.4.2

:

x= 0,725 > xlim= 0,45 (0,617) : As1, As2.

xlim= 0,45 (0,617) mEds,lim= 0,296 (0,371) ; lim= 0,813 (0,743).

:

MEds,lim= mEds,limbd

2

fcd= 0,296.25.41

2

.0,907 = 11282,54 kNcm (14141,29 kNcm)MEds,lim:

Asl,lim=Eds,lim

yd lim

M

f d= 2 2

11282,547,785cm (10,677cm )

43,478.0,813.41=

:

MEds= MEds MEds,lim = 15628 11282,54 = 4345,46 kNcm ( 1486,71 kNcm)

:

As2= As1=Eds

2 yd

M 1

d d f

= 2 2

4345, 46 12,701cm (0,924cm )

41 4 43, 478=

(

6.1.3.1) :As1= Asl,lim+ As1=7,785 + 2,701= 10,486cm

2 (11,601cm

2)

: As2= 2,701cm2 (0,924cm

2)

(

6.1.3.2) :

As1+ As2= 10,486 + 2,701 = 13,187cm2 (12,525cm

2- 5,285%

).

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6.1.3

(

6.1.2) As1= 12,566cm2(4N20), As2= 3,079cm

2(2N14), xlim= 0,45 (0,617) .

MEds= 160 kNm.

: mEds=Eds

2 2

cd

M 160000,4197

bd f 25.41 .0,907= = > mEds,lim= 0,296 (6.1.2)

MEds,lim= mEds,limbd2fcd= 0,296.25.41

2.0,907 = 11282,536 kNcm

MEds= MEds MEds,lim = 16000 11282,536 = 4717,464 kNcm

As2=Eds

2 yd

M 1

d d f

=

4717,464 1

41 4 43,478= 2,932cm

2< 3,079cm

2(2N14);

As1=Eds,lim

yd lim

M

f d+ 2,932 = 2

11282,5362,932 10,717cm

43,478.0,813.41+ = < 12,566cm

2(4N20)

MEds= 160 kNm.

6.1.6 () (T L - ),

T L - ,

.

beff.

:

1) : hf x ;

2) : hf< x .

6.1.6.1 T L , hf x

hf x : MEds Mf= beff hf fcd (d- hf /2) - . 6.1.6.1.1, Mf

, ,

beff.

6.1.6.1.1

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2

95

Asl:

mEds =Eds

2

eff cd

M

b d f . 6.1.4.2 h As1=

Eds

yd

M

f d

6.1.6.2 T L , hf x : MEds >Mf= beffhffcd(d- hf/2) - . 6.1.6.2.1.

(.6.1.6.2.1) :

I (beff bw) hf;

II- bwx.

I Asl,Ihf

beff- bw.

:

MEds,I= (beff bw) hffcd(d- hf/2) (6.16)

mEds =Eds,I

2

eff w f cd

M

(b b )(d h / 2) f (6.17)

mEds = . 6.1.4.2 h As1,I

Asl,I=.Eds,I

yd f

M

f (d h / 2) (6.18)

II

, . 6.6:

MEds,II= MEds MEds,I (6.19)

6.1.6.2.1

mEds =Eds,II

2

w cd

M

b d f mEds,lim . 6.1.4.2 h As1,II=

Eds,II

yd

M

f d

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2

96

T L- :

Asl = Asl,I + Asl,II As,max= 0,04 Ac ; Ac= bwd;

mEds =

Eds,II

2w cd

M

b d f > mEds,lim ,

6.1.5.

: - Asl, II, 2;

- : Asl,lim, II + Asl,,II,

Asl, II, 2= Asl,,II

T L- :

Asl, II, 2 As,max= 0,04 Ac ; Ac= bwd.

T L- :

Asl= Asl, I+ Asl,lim, II + Asl,,II As,max= 0,04 Ac ; Ac= bwd.

6.1.4

- : bw = 30cm, h = 45cm; beff = 120cm; hf = 10cm; - C16/20,

fck = 1,6 kN/cm2, fcd=acc fck/gc =0,85.1,6/1,5 = 0,907 kN/cm

2; - BSt500;

fyk = 50 kN/cm2, fyd = fyk/gs = 50/1,15 = 43,478 kN/cm

2 ; h = 45 cm; d1 = 4,0cm;

d= h d1= 45 4,0 = 41 cm;

MEds= 365,92 kMm = 36592 kNcm.

: Mf = beff hf fcd (d- hf /2) = 120.10.0,907.(41 10/2) = 39182,4 kNcm =

= 391,824 kNm

MEds = 329,328 kNm Mf= 391,824 kNm hf xI :

: mEds =Eds

2

eff cd

M

b d f= 2

32932,80,18

120.41 0,907= . 6.1.4.2 h= 0,897

As1=Eds

yd

M

f d

= 232932,8

20,595cm

43,478.0,897.41

= : 2N25 + 2N28 22,132cm2

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6.1.5

-

6.1.4 . MEds= 420 kMm = 42000 kNcm.

:

Mf= beffhffcd(d- hf/2) = 120.10.0,907.(41 10/2) = 39182,4 kNcm = 391,824 kNmMEds = 420,00 kNm Mf= 391,824 kNm hf< xII .

I (beff bw) hfAsl,I .

:

MEds,I = (beff bw) hf fcd (d- hf/2) = (120-30).10.0,907. (41-10/2) = 29386,8 kNcm =

= 293,868 kNm

mEds =Eds,I

2

eff w f cd

M

(b b )(d h / 2) f =

2

29386,8

(120 30)(41 10 / 2) 0,907 = 0,278

. 6.1.4.2 h=0,828

Asl,I=.Eds,I

yd f

M

f (d h / 2) =

29386,8

43,478.0,828.(41 10 / 5)= 22,675cm

2

II , :

MEds,II= MEds MEds,I= 420 293,868 = 126,132 kNm

: mEds =Eds,II

2

w cd

M

b d f = 2

12613,2

30.41 .0,907= 0,2757

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6.1.7 () , (MEd) -

6.1.7.1

ss1:

MEd/ NEd = eo 0,5h a (6.20)

6.1.7.1.0

6.1.7.1.1, (mEds mEds,lim)

: MEds= MEd+ NEd(h/2 d1) (6.21)

mEds=Eds

2

cd

M

bd f mEds,lim= 0,296 (0,371) - . 6.1.4.2 ,

- BSt500

mEds . 6.1.4.2 h; z = h.d.

:

Asl=Eds

Ed

sl

M1N

z

(6.22)

: ssl = fyd.

:

s2,min= 0,1.N/fyd 0,002 Ac.

:

- b>30 cm, , s2,min;

- h>30 cm, (h/2).

s2

()

-.

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6.1.7.1.2 (mEds > mEds,lim) -

: MEds= MEd+ NEd(h/2 d1)

: mEds

= Eds2

cd

M

bd f> m

Eds,lim= 0,296 (0,371) - . 6.1.4.2 ,

- BSt500

mEds . 6.1.4.2 h; z = h.d.

:

Asl=Eds

Ed

sl

M1N

z

: ssl = fyd.

6.1.7.1.3(mEds > mEds,lim) -


: mEds=Eds

2

cd

M

bd f> mEds,lim= 0,296 (0,371)

- BSt500;

mEds,lim= 0,296 (0,371) . 6.1.4.2 hlim= 0,813 (0,743)

:

MEds,lim= mEds,lim.bd2fcd,


: ss2= fyd= 43,478 kN/cm2 - BSt500.

:

As2=Eds

yd 2

M

f (d d )

(6.23)

: As2,minAs2, choose. As,2 = max { As2(6.22) ; As2,choose}.

As,2 > As,2(6.22) :

MEds= As2().fyd.(d d2)

: MEd,lim= MEds- MEds

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2

100

: s2 s22cd

.A

f .b.d

= , (6.24)

ss2= fyd

: m = mEds 22 d1d

(6.25)

m . 6.1.4.2 es1, ec, x, av, ka, h

: x = x.d

2s2 c

x d

x

= (6.26)

ss2= es,2. Es> fyd

ss1= es,1. Es> fyd :ss1= fyd

:

As1=Ed,lim Eds

Ed

s1 2

M M1N

.d d d

+

(6.27)

: iH 0=

NEd = Fc+ Fs2 Fs1 (6.28)

: Fc= av.fcd.b.x ; Fs2= As2. fyd; Fs1= As1. fyd.

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6.1.6

. 6.8 : b= 25cm, h= 45cm, d=41cm, d1= d2= 4cm.

- C16/20, fck = 1,6 kN/cm2 , fcd=acc fck/gc =0,85.1,6/1,5 = 0,907 kN/cm

2;

- BSt500 ; fyk= 50 kN/cm2, fyd= fyk/gs= 50/1,15 = 43,478 kN/cm

2.

6.1.6.1

: MEd = 70kNm , NEd = 180kN , eo = MEd /NEd = 70/180=0,389m.

As1 , As2= 0.

: MEds= MEd+ NEd(h/2 d1) = 70 + 180.(0,45/2 0,04) = 103,3kNm

:

mEds= Eds2cd

Mbd f

=2

103,3.100 0,27125.41 .0,907

= < mEds,lim= 0,296 (0,371) - . 6.1.4.2 ,

- BSt500.

mEds =0,271 . 6.1.4.2 h=0,8332 ; z = h.d = 0,8332.41=34,1612cm.

:

Asl=Eds

Ed

sl

M1N

z

= 2

1 103,3.100180 2,815cm

43,478 34,1612

=

: ssl = fyd= 43,478 kN/cm2.

As1,min= 0,1.NEd/fyd= 0,1.180/43,478 = 0,414 cm2;

As1,min= 0,002.Ac=0,002.25.41=2,05cm2;

As1 = 2,815cm2> As1,min= 2,05cm

2

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2

102

6.1.6.2

: MEd = 100kNm , NEd = 180kN , eo = MEd /NEd = 100/180=0,556m.

As1 , As2= 0.

:MEds= MEd+ NEd(h/2 d1) = 100 + 180.(0,45/2 0,04) = 133,3kNm

:

m = mEds=Eds

2

cd

M

bd f=

2

133,3.1000,35

25.41 .0,907= > mEds,lim= 0,296 - . 6.1.4.2 ,

- BSt500;

m=0,35 . 6.1.4.2 x= 0,565; h=0,765; ka = 0,416;av = 0,810;

ec= 3,50/00 ;es= 2,69

0/00.

ss1= es.Es= 0,00269.2.105= 538 MPa > fyd,

ss1= fyd= 43,478 kN/cm2.

z = h. d = 0,765.41 = 31,365 cm.

:

Asl=Eds

Ed

sl

M1N

z

= 2

1 133,3.100180 5,635cm

43,478 31,365

=

As1,min = 0,1.NEd/fyd = 0,1.180/43,478 = 0,414 cm2;

As1,min= 0,002.Ac=0,002.25.41=2,05cm2

As1 = 5,635cm2> As1,min= 2,05cm

2

: a= ka.x.d = 0,416.0,565.41 = 9,636cm (6.20): MEd / NEd = eo 0,5h a

.

eo= MEd/NEd = 100/180=0,556m > 0,5.0,45 0,09636 = 0,128m

: As2= As2,min= 2,05cm2

6.1.6.3

: MEd = 100kNm , NEd= 180kN , eo= MEd/NEd = 100/180=0,556m

As2 As1 .

As1As2.

: MEds= MEd+ NEd(h/2 d1) = 100 + 180.(0,45/2 0,04) = 133,3kNm

m = mEds=Eds

2

cd

M

bd f=

2

133,3.1000,35

25.41 .0,907= > mEds,lim= 0,296 - . 6.1.4.2 ,

- BSt500

mEds,lim= 0,296 (0,371) . 6.1.4.2 hlim= 0,813 (0,743)

:

MEds,lim= mEds,lim.bd2fcd= 0,296.25.41

2.0,907 = 11282,536 kNcm = 112,82536 kNm.

MEds= MEds MEds,lim = 133,3 112,82536 = 20,474 kNm

: ss2= fyd= 43,478 kN/cm2 - BSt500.

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103

:

As2=Eds

yd 2

M

f (d d )

= 2

20,474.1001,273cm

43,478.(41 4)=

: As2,minAs2, choose.As,2 = max { As2(6.22) ; As2,choose}.

As2,min=2Ed

yd

0,1.N 0,1.1800,414cm

f 43, 478= =

As2,min = 0,002.Ac= 0,002.b.d= 0,002.25.41= 2,05cm2

: As2 = 2,05cm2 2N12 As2,choose= 2,262cm

2

: As2,choose > As2 : MEds:

MEds= As2,choose.fyd.(d d2) = 2,262.43,478.(41-4)= 3638,847kNcm=36,3885kNm

: MEd,lim= MEds- MEds = 133,3 36,3885 = 96,9115kNm

: s2 s22cd

.A

f .b.d

= =

43,478.2,2620,1057

0,907.25.41=

: ss2= fyd=43,478 kN/cm2

: m = mEds2

2

d1

d

=

40,35 0,1057. 1 0, 255

41

=

m . 6.1.4.2 es1= 5,8950/00, ec=3,5

0/00, x=0,373,

av= 0,810, ka=0,416 , h= 0,845.

: x = x.d = 0,373.41= 15,293 cm

2s2 c

x d

x

= =

15, 293 40,0035 0,00258

15,293

=

ss2= es,2. Es= 0,00258.2.105=516,909MPa > fyd= 434,78 MPa

ss1= es,1. Es= 0,005895.Es> fyd :

ss1= fyd= 43,478kN/cm2

:

As1=Ed,lim Eds

Ed

s1 2

M M1N

.d d d

+

=

21 96, 9115.100 36, 3885.100 180 4,556cm43,478 0,845.41 41 4

+ =

: iH 0=

NEd = Fc+ Fs2 Fs1; 180 280,882 + 98,347 198,085 = 181,14 ,: Fc= av.fcd.b.x =0,810.0,907.25.15,293=280,882kN;

Fs2= As2. fyd= 2,262.43,478 = 98,347kN ; Fs1= As1. fyd= 4,556.43,478 = 198,085kN.

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6.1.7.2 -

:

MEd/ NEd = eo< 0,5h a (6.29)

-

As1 . -

. 6.1.7.2.1.a,b. ,

- - 30cm,

As0- . 6.1.7.2.1.b.

a) b) c)

6.1.7.2.1

. 6.1.7.2.1 , -

As1 . 6.1.7.2.2.. ,

As1 .

6.1.7.2.2

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2

105

. 6.1.7.2.2.b -

,

. . 6.1.7.2.3 .

6.1.7.2.3

N x q b ax = xh ; FC= bfcdh b; hc= ah;

sS2= fyd ; sS1= (0,95 q+ 0,05) fyd ;

FS2= As2.sS2; FS1= As1.sS1 ;

:

SH = 0:NEd= FS1+ FS2+ FC,(1)

S= 0

FS2(d d2) +FC(d- hc) = NEd (0,5 h d1+e), (2)

S= 0

FS1(d d2) +FC(hcd2) = NEd (0,5 h d2- e), (3)

1 1,0000 0,00 0,7500 0,3855

2 1,0263 0,05 0,7625 0,3961

3 1,0555 0,10 0,7750 0,4032

4 1,0882 0,15 0,7875 0,4101

5 1,1250 0,20 0,8000 0,4166

6 1,1666 0,25 0,8125 0,4230

7 1,2142 0.30 0,8250 0,42938 1,2692 0,35 0,8375 0,4353

9 1,3333 0,40 0,8500 0,4411

10 1,4090 0,45 0,8625 0,4468

11 1,5000 0,50 0,8750 0,4523

12 1,6111 0,55 0,8875 0,4577

13 1,7500 0,60 0,9000 0,4629

14 1,9285 0,65 0,9125 0,4680

15 2,1666 0,70 0,9250 0,4729

16 2,5000 0,75 0,9375 0,4777

17 3,0000 0,80 0,9500 0,4824

18 3,8333 0,85 0,9625 0,4870

19 5,5000 0,90 0,9750 0,4914

20 10,5000 0,95 0,9875 0,4957

21 1,00 1,0000 0,5000

6.1.8 () (MEd) (NEd)

6.1.8.1 (NEd)

.

:

As,tot= NEd/fyd (6.30)

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106

6.1.8.2

6.1.8.2.1

:

e = Ed s1Ed

Mz

N> (6.31)

: As1As2 . 6.1.8.2.1.

M 0= :

As1= Ed s2

yd s1 s2

N (z e)f (z z )

++

(6.32)

As2=Ed s1

yd s1 s2

N (z e)

f (z z )

+ (6.33)

,

As1.

6.1.8.3

6.1.8.3.1, (mEds mEds,lim)

:

e = Ed s1Ed

Mz

N (6.34)

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107

:

MEds= MEd NEd(d - 0,5h) = Fcz + Fs2(d - d2) (6.35)

NEd= Fs1 Fc Fs2 (6.36)

: mEds=Eds

2

cd

Mbd f

mEds,lim= 0,296 (0,371) - . 6.1.4.2 ,

- BSt500

mEds . 6.1.4.2 h; z = h.d.

:

Asl=Eds

Ed

sl

M1N

z

+

(6.37)

: ssl = fyd.

: s2,min 0,002 Ac.

:

- b>30 cm, ,

s2,min;

- h>30 cm,

(h/2).

6.1.8.3.2, (mEds > mEds,lim)


: mEds=Eds

2

cd

M

bd f> mEds,lim= 0,296 (0,371)

- BSt500;

mEds,lim= 0,296 (0,371) . 6.1.4.2 hlim= 0,813 (0,743)

:

MEds,lim= mEds,lim.bd2

fcd,


: ss2= fyd= 43,478 kN/cm2 - BSt500.

:

As2=Eds

yd 2

M

f (d d )

(6.38)

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108

As2,minAs2, choose. As,2 = max { As2; As2,choose}.

As,2 > As,2 : MEds= As2().fyd.(d d2)

: MEd,lim= MEds- MEds

: s2 s22cd

.A

f .b.d

= , (6.39)

: ss2= fyd

: m = mEds2

2

d1

d

(6.40)

m . 6.1.4.2 es1

, ec, x, a

v, k

a, h

: x = x.d

2s2 c

x d

x

= (6.41)



:

As1=Ed,lim Eds

Ed

s1 2

M M1

N.d d d

+ + (6.42)

: iH 0=

NEd = Fs1 Fc Fs2 (6.43)

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109

6.1.7

: b= 25cm, h= 45cm, d=41cm, d1= d2= 4cm.

- C16/20, fck = 1,6 kN/cm2 , fcd=acc fck/gc =0,85.1,6/1,5 = 0,907 kN/cm

2;

- BSt500 ; fyk= 50 kN/cm2, fyd= fyk/gs= 50/1,15 = 43,478 kN/cm2.

6.1.7.1

: MEd = 70kNm , NEd= 180kN , eo= MEd/NEd = 70/500=0,14m , zs1= zs2= 18,5cm,

As1 , As2= 0.

e = 14cm < zs1= 18,5cm

:

As1 =2Ed s2

yd s1 s2

N (z e) 500.(18,5 14)10,1014cm

f (z z ) 43,478.(18,5 18,5)

+ += =

+ +

As2=2Ed s1

yd s1 s2

N (z e) 500.(18,5 14)1,3986cm

f (z z ) 43,478.(18,5 18,5)

= =

+ +

6.1.7.2 (mEds mEds,lim)

: MEd = 100kNm , NEd= 200kN , eo= MEd/NEd = 100/200=0,5m , zs1= zs2= 18,5cm,

As1.

e = 50cm < zs1= 18,5cm

MEds= MEd NEd(d - 0,5h) = 100-200(0,41-0,5.0,45)=64 kNm

: mEds=Eds

2

cd

M

bd f=

2

63.1000,165

25.41 .0,907= mEds,lim= 0,296 (0,371)-. 6.1.4.2

,

mEds 0,165 . 6.1.4.2 h=0,9065; z = h.d = 0,9065.41= 37,166.

:

Asl=Eds

Ed

sl

M1N

z

+

= 2

1 63.100200 8, 498cm

43,478 37,166

+ =

,

: ssl = fyd.

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110

6.1.7.3 - (mEds > mEds,lim)

: MEd = 180,856kNm , NEd= 100kN , eo= MEd/NEd = 180,856/100=1,808m ,

zs1= zs2= 18,5cm, As1 As2 .

e = 180,8cm > zs1= 18,5cm MEds= MEd NEd(d - 0,5h) = 180,856 -100(0,41-0,5.0,45)=162,35679 kNm

: m= mEds=Eds

2

cd

M

bd f=

2

162,35679.1000,4259

25.41 .0,907= >mEds,lim = 0,296 (0,371)

. 6.1.4.2

mEds,lim= 0,296 (0,371) . 6.1.4.2 hlim= 0,813 (0,743)

:

MEds,lim= mEds,lim.bd2fcd= 0,296.0,907.25.41

2= 11282,5358 kNcm

MEds= MEds MEds,lim = 16235,679 11282,5358 = 4953,144 kNcm

: ss2= fyd= 43,478 kN/cm2 - BSt500.

:

As2=Eds

yd 2

M

f (d d )

= 2

4953,1443,079cm

43,478.(41 4)=

- 2N14 As2= 3,079cm

2

: s2 s22cd

.A

f .b.d

= =

43, 478.3,0790,144

43,478.(41 4)=

,

: ss2= fyd

: m = mEds2

2

d1

d

=0,4259 0,144.

41 0,3

41

=

m=0,3 . 6.1 es1= 0,00415, ec= 0,0035, x= 0,458 ,h= 0,810

:

2s2 c

d / d 0,458 4 / 410,0035 0,00275

0,458

= = =

ss2= es,2. Es= 0,00275.20000= 55,088 kN/cm2> fyd :ss2= fyd

ss1= es,1. Es= 0,00415.20000= 83 kN/cm2 > fyd :ss1= fyd

:

As1=Ed,lim Eds

Ed

s1 2

M M1N

.d d d

+ +

= 2

1 11282, 5358 4953,144100 13,193cm

43,478 0,81.41 41 4

+ + =

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2

111

6.1.9 () , (MEd)(NEd)

,

. - .6.1.9.0-.

6.1.9.0-

fck 50MPa - h= 1,0.

(

) . 6.1.9.0-b

: mEd =Ed

2

cd

M

bd f (6.44)

nEd=Ed

cd

N

bdf; (6.45)

(+)

(-)

mEdnEd wtot

As1= As2()

: As,tot= As1+ As2= totyd cd

bd

f / f (6.46)

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2

112

6.1.9.0-b

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2

113

III.

1

, , , : ( ), (), , ; .

, , , ,, . -.

, - - 5 .

:- : ;

- ;- - ;

- -2; ,; . ( -2) EN 10080;

- ;- - l/250 ( )

(SLS) -2.

,

. , .

1.1 A

1.1.1

1.2.1.

s()

( ) - s,min :

ctms,min l

yk

fA 0, 26 b d

f= , -0,0013btd

( ) s , s,max = 0,04Ac , = bt.d.

-20%

As.

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smax,slab:- 3h 400 mm , h ;- 3,5 h 450 mm .

:- 2h 250 mm ;- 3h 400 mm .

1.1.2 A

.

, , - 25% . , -0,2

, . , , 15% .

1.1.3

, .

1.1.4

() ,

1.1.4.1

1.1.4.1 -

1.1.5

, , 200 mm. VEd Rd,cV> ( ) ,

., |VEd| Rd,max1/3V , ,

.

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:

maxs 0,75d(1 cot )= + ,

.:

maxs d=

1,5d.

2

, --2. () .(, , ).

, -.

- , (,

, .)

2.1hfl

: () - (1/20 -1/25) l ; (1/30 1/33) l, l ( ). l 1,5 3,5 m. - , . 4 6 m.

l ( ) (

) -:- lclaa : l = lcl+ hf;- . l ( ) (

) :- -:

- l = lcl+ hf/2 + bsup/2; bsup ;- ;

.(

)

(lcl) .

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2.2

(.2.2.1) -2

: , ; EN 10080;

2.2.1 - 100 cm.

cnom.

cnom = cmin + devc

cmin , : ; ; .

cmin= max min,b min,dur dur, dur,st dur,add{c ;c c c c ;10mm}+

:cmin,b e 4.2 -2;

cmin,dur e .cmin,dur. (50 ) S4. cmin,dur4.3N 4.4N -2 (. 4.4.1.2.3 4.4.1.2.4)

dur,c 0 mm;

dur,stc

- 0 mm;

dur,addc

- 0 mm;

devc

- 10 mm; - 40 mm ,

() 75 mm , .

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. REI 60(-), hf = 80mm

= 20mm.

2.3 / - l/d

-2 , .. , (-l/250), -2. , / (l/d) . 2.3.1, -2 .

s

( ) 310 (fyk = 500 MPa). . 2.3.1310/ s . :

s yk s,req s,prov310 / 500 / (f A / A ) =

:s,prov;s,req

.

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2.3.1 - / (l / d)

(30 St 500)

rrrr=1,5%

rrrr=0,5%

,

, ,,

,

, ()

1,0

1,3

1,5

1,2

0,4

14

18

20

17

6

20

26

30

24

8

1: ., -.2: -. -.

3: , , -, ,1/250 . , .

s ( ) 310 ( fyk = 500 MPa). . 2.3.1 310/ s .

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:

s yk s,req s,prov310 / 500 / (f A / A ) =

:

s,prov;s,req

.

2.4

() 1m .

. 2.4.1.

l = leff; lcl= ln

- 2.4.1 l = leff

l () ()-:

- lclaa : l = lcl+ hf;- .

l () ():

- -:- l = lcl+ hf/2 + bsup/2; bsup ;- ;

.

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120

, ( ) :

2max d 1 dM ( .g .q ).l = + ;

2max 3 d 4 dQ ( .g .q ).l = + ;

2min d 2 dM ( .g .q ).l = + ;

2min 3 d 5 dQ ( .g .q ).l = + ;

1 2 3 4 5, , , , , .2.7.3.3.1 , - . 20%. ) , b) . 2.4.2.

) :

d= MEd- EdM = MEd -EdC .t

8

b) , :2

Ed,l Ed Ed,l d d

t tM M V (g q )

2 8= + ;

2

Ed,r Ed Ed,r d d

t tM M V (g q )

2 8= +

():

2

d d

t(g q )

8+

. ( )

, . = gd+ qd, gd= gk. f , f =1,35; qd= qk . f , f =1,5 .

. , , gd= gk. f , f =1,0.

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2.4.2 - :) , b)

( ) (lcl= l) = gd+ qd - . 2.4.3.

gd = gk f ,

f =1,35.

qd = qk f ,

f =1,50.

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2.4.3

2.5

b=100 cm.

58/ 14/.,

().

40 50 cm. . , , . (, ) 14/.

. , . .1.3.120% .

( )

58/ 40-50 cm. lcl/4..

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2.6 , :

.

=

, [kN/m3]

25,0 24,0 78,5

()

16,0 - 18,0

6,0 8,0 20,0 - 22,0 20,0 22,0 24,0 22,0 21,0 27,0 18,0- 0,5 6,0 8,0 12,0

- H~1,00 m () 1 kN/m .

- () , . -, 0,7- .

- : , -.

, -, .- , 30% (k=0,7).

- , .,1.

-, 1 kN/m2.

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=

: if

; ; ;

;

- , ;; .

G

f 1,35

, , . Gf

1,35

, , ,

, .

Q

f 1,5

i

f .

ULS SLS, ,

.

: !

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125

2.7

2.7.1,

, ().

, . / ,

, -., ,

-, .

2.7.1.1 -

]m/kN[m1 22 . 2.7.1.1.0

.

2.7.1.1.0

2.7.1.1.1:

- - ]m/kN[,.1.1.h 2.o.

- - ]m/kN[,.1.1.h 2o

- - ]m/kN[,.1.1.h 2..of

- - ]m/kN[,.1.1.h 2.o.

[ ] 2k m/kN,G

2.7.1.1.2 :

- - ]m/kN[,..1.1.h 2Gf.

o.

- - ]m/kN[,..1.1.h 2Gf

o

- - ]m/kN[,..1.1.h 2Gf..

of

- - ]m/kN[,..1.1.h 2Gf.

o.

[ ] 2d m/kN,G

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2.7.1.1.3:

[ ]

[ ] m/kN,.qQ

)1(m/kN,q

2Qfkd

2

k

=

2.7.1.2-

( ,

1/2 hf ,- - )-

.2.7.1.2.1

2.7.1.2.1

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2.7.1.3- -. 2.7.1.3.1

q1m- , 1m.2dd m/kN,QGq += [ ] [ ] [ ]'m/kN'm1.m/kNq 2 =

2.7.1.3.1

:

[ ]'m/m.kN8

.qM

2

0Edsmax

l= ----

[ ]'m/kN2

.qV 0Edmax

l= ----

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2.7.1.4

,

, -, .

-

.

[ ]cm26

d26"/".cl

minl==

2.7.1.4.1

[ ] [ ]mm)12(,10,8cm,2

cdh nomf =

++=

[ ]cm,2

chd nomf

= -

cm2acm5,215,1ccc

cm2a60REI

.cm1c

;cm5,1c

:4S1XC:

c

devminnom

dev

min

nom

=>=+=+=

=

=

=

2

cd

EdsmaxEds

d.b.f

Mm = ---- /

[ ]

[ ]

[ ]cm100b,b

...;5,1

85,0cm/kN,

f.f

;cm/kN,f

cc2

c

ckcccd

2

cd

=

=

=

=

Edsm

. 2.4.1.4.2. .

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2.4.1.4.2

C12/15 - C50/60

fcd= a.fck/gc;

a=0,85 ;gc = 1,5

;- 25/30 :fcd = 1,4166 kN/cm

2

w=mEd/z=mEd/h;

x=x/d;

z=h=z/d;

ec2[0/00]; es1[

0/00];

:mEd= MEd/b.d

2.fcd

:As1= MEd/fyd.h.d;

As1, [cm2]

- BSt500S

fyk =500 MPA=50kN/cm

2

fyd = fyk/ gs= 50 / 1,15;

fyd = 43,478 kN/cm2

mEds wwww xxxx z=hz=hz=hz=h ka aaaav eeeec2 eeees1

0,01 0,0101 0,030 0,990 0,346 0,337 -0,77 25,00

0,02 0,0203 0,044 0,985 0,360 0,463 -1,15 25,00

0,03 0,0306 0,055 0,980 0,360 0,553 -1,46 25,00

0,04 0,0410 0,066 0,976 0,368 0,622 -1,76 25,00

0,05 0,0515 0,076 0,971 0,377 0,676 -2,06 25,00

0,06 0.0621 0,086 0,967 0,387 0,718 -2,37 25,00

0,07 0,0728 0,097 0,962 0,396 0,751 -2,68 25,00

0,08 0,0836 0,107 0,956 0,405 0,778 -3,01 25,00

0,09 0,0946 0,118 0,951 0,413 0,801 -3,35 25,00

0,10 0,1057 0,131 0,946 0,416 0,810 -3,50 23,30

0,11 0,1170 0,145 0,940 0,416 0,810 -3,50 20,71

0,12 0,1285 0,159 0,934 0,416 0,810 -3,50 18,55

0,13 0,1401 0,173 0,928 0,416 0,810 -3,50 16,730,14 0,1518 0,188 0,922 0,416 0,810 -3,50 15,16

0,15 0,1638 0,202 0,916 0,416 0,810 -3,50 13,80

0,16 0,1759 0,217 0,910 0,416 0,810 -3,50 12,61

0,17 0,1882 0,232 0,903 0,416 0,810 -3,50 11,56

0,18 0,2007 0,248 0,897 0,416 0,810 -3,50 10,62

0,19 0,2134 0,264 0,890 0,416 0,810 -3,50 9,78

0,20 0,2263 0,280 0,884 0,416 0,810 -3,50 9,02

0,21 0,2395 0,296 0,877 0,416 0,810 -3,50 8,33

0,22 0,2528 0,312 0,870 0,416 0,810 -3,50 7,71

0,23 0,2665 0,329 0,863 0,416 0,810 -3,50 7,13

0,24 0,2804 0,346 0,856 0,416 0,810 -3,50 6,61

0,25 0,2946 0,364 0,849 0,416 0,810 -3,50 6,12

0,26 0,3091 0,382 0,841 0,416 0,810 -3,50 5,670,27 0,3239 0,400 0,834 0,416 0,810 -3,50 5,25

0,28 0,3391 0,419 0,826 0,416 0,810 -3,50 4,86

0,29 0,3546 0,438 0,818 0,416 0,810 -3,50 4,49

0,296 0,3643 0,450 0,813 0,416 0,810 -3,50 4,28

0,30 0,3705 0,458 0,810 0,416 0,810 -3,50 4,15

0,31 0,3869 0,478 0,801 0,416 0,810 -3,50 3,82

0,32 0,4038 0,499 0,793 0,416 0,810 -3,50 3,52

0,33 0,4211 0,520 0,784 0,416 0,810 -3,50 3,23

0,34 0,4390 0,542 0,774 0,416 0,810 -3,50 2,95

0,35 0,4575 0,565 0,765 0,416 0,810 -3,50 2,69

0,36 0,4768 0,589 0,755 0,416 0,810 -3,50 2,44

0,37 0,4968 0,614 0,745 0,416 0,810 -3,50 2,20

0,371 0,4994 0,617 0,743 0,416 0,810 -3,50 2,17

[ ]'m/cm,d..f

MA 2

yd

Edsmaxs

=

l-

[ ]

[ ] cm/kNf

f

cm/kN,f

s

2

s

yk

yd

2

yd

=

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:

[ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]mmh.mmb.04,0mm,A

mmd.mmb.MPaf

MPaf.26,0mm,A

f

2

max,s

yk

ctm2

min,s

=

=

l

l

max,s

smin,s AAA lll

[ ]'m/,A

An

1,s

s = l - ;

As,1- . c. 2.4.1.4.3

2.4.1.4.3

( )[ ]'m/,985n14n5 opt =

.cl

max L.nN = -

N2

N

N2

N

=

=

[ ][ ] [ ]

[ ] %,cmd.cmb

cmA.100

2

s = ll

N

[mm]cm N

[mm]

[kg/m]1 2 3 4 5 6 7 8 9 10

8 0,503 1,005 1,508 2,011 2,513 3,016 3,519 4,021 4,524 5,027 8 0,395

10 0,785 1,571 2,356 3,142 3,927 4,712 5,498 6,283 7,069 7,854 10 0,617

12 1,131 2,262 3,393 4,524 5,655 6,786 7,917 9,048 10,176 11,310 12 0,888

14 1,539 3,079 4,618 6,158 7,697 9,236 10,776 12,315 13,854 15,394 14 1,208

16 2,011 4,021 6,032 8,042 10,053 12,064 14,074 16,085 18,096 20,106 16 1,578

18 2,545 5,089 7,634 10,179 12,723 15,286 17,813 20,358 22,902 25,447 18 1,998

20 3,142 6,283 9,425 12,566 15,708 18,850 21,991 25,133 28,274 31,416 20 2,46622 3,801 7,603 11,404 15,205 19,007 22,808 26,609 30,411 34,212 38,013 22 2,984

25 4,910 9,821 14,732 19,642 24,553 29,464 34,375 39,285 44,196 49,107 25 3,855

28 6,158 12,315 18,473 24,630 30,788 36,945 43,103 49,260 55,418 61,575 28 4,834

32 8,042 16,085 24,127 32,170 40,212 48,255 56,297 64,340 72,382 80,425 32 6,313

40 12,571 25,142 37,714 50,285 62,857 75,428 88,000 100,571 113,142 125,714 40 9,868

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2.7.2 , -

2.7.2.1- . 2.7.2.1.0:

2.7.2.1.0

2.7.2.1.1:

- - ]m/kN[,.1.1.h 2.o.

- - ]m/kN[,.1.1.h 2o

- - ]m/kN[,.1.1.h 2..of

- - ]m/kN[,.1.1.h 2.o.

[ ] 2k m/kN,G

2.7.2.1.2:

- - ]m/kN[,..1.1.h 2Gf.

o.

- - ]m/kN[,..1.1.h 2Gf

o

- - ]m/kN[,..1.1.h 2Gf..

of

- - ]m/kN[,..1.1.h 2Gf.

o.

[ ]2

d m/kN,G

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2.7.2.1.3:.2.7.2.1.3.0:

. 2.7.2.1.3.0

- - ( ) [ ];'m/kN,.1.hH.b .of.

- - ( ) ]'m/kN[,.1.hH.b.2 .of..

]'m/kN[,Fk

2.7.2.1.4:.2.7.2.1.3.0:

- - ( ) [ ];'m/kN,..1.hH.b Gf.

of.

- - ( ) ]'m/kN[,..1.hH.b.2 Gf.

of..

]'m/kN[,Fd

2.7.2.1.5:

[ ][ ]2Qfkd

2

k

m/kN,.qQ

m/kN,q

=

2.7.2.2 - .. 2.7.1.2-. 2.7.2.2.1:

2.7.2.2.1

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2.7.2.3 qFd1m- , 1m.

[ ][ ]'m/kN,F

m/kN,QGq

d

2

dd+= [ ] [ ] [ ][ ] [ ]kN'm1].'m/kN[F

'm/kN'm1.m/kNq

d

2

=

=

l0,[m]

q[ kN/m']

MEds

MEdsmax

VEd

VEdmax

VEdmax

+

-

Fd[ kN]

l0/2,[m]A B

2.7.2.3.1 :

[ ]'m/m.kN,4

.F

8

.qM 0d

2

0

Edsmax

ll+= ----

[ ]'m/kN,BF.qV d0Edmax += l ----

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2.7.2.4. 2.7.1.4.1, . 2.4.1.4.2 2.4.1.4.3 :

[ ]cm,2

chdnomf

=

2

cd

EdsmaxEds

d.b.f

Mm =

[ ]'m/cm,d..f

MA 2

yd

Edsmaxs

=

l

[ ] [ ][ ]

[ ] [ ]

[ ] [ ] [ ]mmh.mmb.04,0mm,A

mmd.mmb.MPaf

MPaf.26,0mm,A

f

2

max,s

yk

ctm2

min,s

=

=

l

l

m

МЕТОДИЧЕСКО РЪКОВОДСТВО ЗА РАЗРАБОТВАНЕ НА КУРСОВ...

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Transcript of МЕТОДИЧЕСКО РЪКОВОДСТВО ЗА РАЗРАБОТВАНЕ НА КУРСОВ...