Eurocode 4 [PDF]

Zitiervorschau

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Eurocodes Background and Applications Dissemination of information for training 18-20 February 2008, Brussels

Eurocode 4 Serviceability limit states of composite beams Univ. - Prof. Dr.-Ing. Gerhard Hanswille Institute for Steel and Composite Structures University of Wuppertal Germany 1

Contents

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Part 1:

Introduction

Part 2:

Global analysis for serviceability limit states

Part 3:

Crack width control

Part 4:

Deformations

Part 5:

Limitation of stresses

Part 6:

Vibrations

2

Serviceability limit states

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Serviceability limit states

Limitation of stresses Limitation of deflections

crack width control

vibrations web breathing 3

Serviceability limit states

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

{

}

characteristic combination:

Ed = E ∑ Gk, j + Pk + Qk,1 + ∑ ψ 0,i Qk,i

frequent combination:

Ed = E ∑ Gk, j + Pk + ψ1,1 Qk,1 + ∑ ψ 2,i Qk,i

quasi-permanent combination:

Ed = E ∑ Gk, j + Pk + ∑ ψ 2,i Qk,i

{ {

}

}

serviceability limit states Ed ≤ Cd: - deformation - crack width - excessive compressive stresses in concrete

Cd=

- excessive slip in the interface between steel and concrete - excessive creep deformation - web breathing - vibrations

4

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Part 2: Global analysis for serviceability limit states

5

Global analysis - General

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Calculation of internal forces, deformations and stresses at serviceability limit state shall take into account the following effects:

ƒ ƒ ƒ ƒ ƒ

shear lag;

ƒ ƒ

inelastic behaviour of steel and reinforcement, if any;

creep and shrinkage of concrete; cracking of concrete and tension stiffening of concrete; sequence of construction; increased flexibility resulting from significant incomplete interaction due to slip of shear connection;

torsional and distorsional warping, if any.

6

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Shear lag- effective width shear lag real stress distribution

bei < 0,2 bi

bei

σ max

σ(y)

σ( y ) = σmax

σmax

σ max

⎡ y⎤ ⎢1− b ⎥ ⎣ i⎦

4

b 5 bei

be

stresses taking into account the effective width

The flexibility of steel or concrete flanges affected by shear in their plane (shear lag) shall be used either by rigorous analysis, or by using an effective width be

σmax

y bi

bei σR

σ(y)

bei ≥ 0,2 bi ⎡b ⎤ σR = 1,25 ⎢ ei − 0,2⎥ σmax ⎣ bi ⎦

y bi

⎡ y⎤ σ( y ) = σR + [σmax − σR ] ⎢1− ⎥ ⎣ bi ⎦

4

7

Effective width of concrete flanges Le=0,25 (L1 + L2) for beff,2 Le=0,85 L1 for beff,1

Le=2L3 for beff,2

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

be,1 bo be,2

Le=0,70 L2 for beff,1

b1

L1

L1/4

L1/2

L2/4

L2/2

L2/4

beff,0

beff,2 beff,1

end supports:

beff,2

b2

L3

L2

L1/4

bo

beff,1

midspan regions and internal supports: beff = b0 + be,1+be,2

beff = b0 + β1 be,1+β2 be,2

be,i= Le/8

βi = (0,55+0,025 Le/bi) ≤ 1,0

Le – equivalent length 8

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Effects of creep of concrete

redistribution of the sectional forces due to creep

Initial sectional forces

primary effects

Mc,o

-zi,c zi,st

-Nc,o

-Mc,r Nc,r

ast

ML Mst,o

Mst,r

Nst,o

-Nst,r

The effects of shrinkage and creep of concrete and non-uniform changes of temperature result in internal forces in cross sections, and curvatures and longitudinal strains in members; the effects that occur in statically determinate structures, and in statically indeterminate structures when compatibility of the deformations is not considered, shall be classified as primary effects. 9

Effects of creep and shrinkage of concrete

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Types of loading and action effects: In the following the different types of loading and action effects are distinguished by a subscript L : L=P for permanent action effects not changing with time L=PT

time-dependent action effects developing affine to the creep coefficient

L=S

action effects caused by shrinkage of concrete

L=D

action effects due to prestressing by imposed deformations (e.g. jacking of supports)

MPT(t)

time dependent action effects ML=MPT:

MPT (t=∞)

action effects caused by prestressing due to imposed deformation ML=MD: MD

MPT(ti)

ϕ(ti,to) ϕ(t∞,to)

δ

ϕ(t,to)

ML=MD

+ 10

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Modular ratios taking into account effects of creep

centroidal axis of the concrete section

zis,L

ast

-zic,L

zc

zist,L

zi,L zst

Modular ratios:

centroidal axis of the transformed composite section centroidal axis of the steel section (structural steel and reinforcement)

nL = no [ 1+ ψL ϕ( t, t o ) ]

action short term loading

no =

Ea Ecm creep multiplier Ψ=0

permanent action not changing in time

ΨP=1,10

shrinkage

ΨS=0,55

prestressing by controlled imposed deformations

ΨD=1,50

time-dependent action effects

ΨPT=0,55 11

Elastic cross-section properties of the composite section taking into account creep effects

-zis,L

-zic,L ast

zist,L

zc zi,L zst

centroidal axis of the concrete section centroidal axis of the composite section centroidal axis of the steel section

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Modular ratio taking into account creep effect:

nL =n0 (1+ ψL ϕ( t, t 0 ) ) no =

Est Ecm ( t o )

Transformed cross-section properties of Distance between the centroidal axes of the concrete section: the concrete and the composite section: A c,L = A c / nL

Jc,L = Jc / nL

Transformed cross-section area of the composite section: A i,L =A St +A c,L

zic,L =− A st ast /A i,L

Second moment of area of the composite section: J i,L = Jst + Jc,L + A st A c,L a2st / A i,L 12

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Effects of cracking of concrete and tension stiffening of concrete between cracks

Ns

mean strain εsm=εs,2- βΔεs,r

σc(x) Δε s = β

fct,eff ρs E s

Δε s = β Δε s,r

ρs = A s / A c

fully cracked section

Δε s,r

εsr,1 A

B

stage A: stage B: stage C:

εsr,2

εsm,y

εsy

C

uncracked section initial crack formation stabilised crack formation

σs,2

σc(x)

Nsm Ns,cr

τv

σs(x)

β = 0,4

Nsy

σc(x)

σs(x) Ns

Ns ε s,2 =

ε

εsm fct Ec

σ s2 Es

β Δε s,r

εs(x) εc(x)

x

13

Influence of tension stiffening of concrete on stresses in reinforcement εsm

Ns zs

Ms≈0

a Ma za

Na

mean strain in the concrete slab:

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

equilibrium:

M εa

Ma = M − Ns a

-κ

Na = − Ns compatibility: εsm = εa + κ a Ns a2 M a Ns εsm + + = Ea A a Ea A a Ea Ja

εs,2 εs,m

Δεs=β Δεs,r

mean strain in the concrete slab:

εs εc

ε sm = ε s2 − β Δε sr =

fct,eff Ns −β Es A s ρs Es 14

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Redistribution of sectional forces due to tension stiffening

tension stiffening

fully cracked section

ΔNts

Ns,2 -zst,s

-Ms,2

a

zst,a

z2=zst

+

-Ma,2 -Na,2

Ns

A J α st = st st A a Ja

f A ΔNts = β ct,eff s ρs α st

Ns,2

Nsε MEd

-ΔNts

Js Jst

Na = Na2 − ΔNts = MEd

M

-MEd

-Ma -Na

Jst = J2

Sectional forces:

Ms = MEd

-Ms

=

ΔNts a

Ns = Ns2 + ΔNts = MEd

ΔNts Ns

Ns

A s z st,s Jst

A a z st,a Jst

Ma = Ma2 + ΔNts a = MEd

+ ΔNts

− ΔNts

Ja + ΔNts a Jst

15

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Stresses taking into account tension stiffening of concrete fully cracked

tension stiffening

Ns,2

a

zst

reinforcement:

-Ms

+

zst,a

za

ΔNts

-Ms,2

-zst,s

-Ma,2

Ns

ΔNts a

-Na,2

-MEd

= -Ma

-ΔNts

-Na

structural steel:

f σ s = σ s,2 + β ctm ρs α st

σa = σa,2 −

M f σ s = Ed z st,s + β ctm Jst ρs α st

M ΔNts ΔNts a za σa = Ed z st − + Jst Aa Ja

ΔNts ΔNts a + za Aa Ja

α st =

A st Jst A a Ja

ΔNts = β

fctm A s ρs α st 16

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Influence of tension stiffening on flexural stiffness

εsm

Ns -Ms a

zst

-Ma -Na

κ=

-M κ εa

Est J1 Est J2,ts

EstJ2

EstJ2

EstJ2,ts EstJ1

M Ma M − Ns a = = Est I2,ts Est Ja Est Ja

Effective flexural stiffness: Est J2,ts =

EJ

M

Curvature:

M

κ MR

MRn

Ea Ja (N − Ns,ε ) a 1− s M

EstJ1 uncracked section EstJ2 fully cracked section EstJ2,ts effective flexural stiffness taking into account tension stiffening of concrete 17

G. Hanswille

Univ.-Prof. Dr.-Ing. Effects of cracking of concrete - General Institute for Steel and Composite Structures method according to EN 1994-1-1 University of Wuppertal-Germany

EaJ1 EaJ2

• Determination of internal forces by uncracked analysis for the characteristic combination.

– un-cracked flexural stiffness – cracked flexural stiffness

L1

EaJ1

L1,cr

L2,cr

EaJ2

L2

EaJ1

ΔM

• Determination of the cracked regions with the extreme fibre concrete tensile stress σc,max= 2,0 fct,m. • Reduction of flexural stiffness to EaJ2 in the cracked regions. • New structural analysis for the new distribution of flexural stiffness.

ΔM Redistribution of bending moments due to cracking of concrete un-cracked analysis cracked analysis 18

Effects of cracking of concrete – simplified method

L1

EaJ1

L2

0,15 L1

0,15 L2

EaJ2 ΔMII

Lmin / Lmax ≥ 0,6

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

For continuous composite beams with the concrete flanges above the steel section and not pre-stressed, including beams in frames that resist horizontal forces by bracing, a simplified method may be used. Where all the ratios of the length of adjacent continuous spans (shorter/longer) between supports are at least 0,6, the effect of cracking may be taken into account by using the flexural stiffness Ea J2 over 15% of the span on each side of each internal support, and as the uncracked values Ea J1 elsewhere. 19

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Part 3: Limitation of crack width

20

Control of cracking

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

General considerations minimum reinforcement If crack width control is required, a minimum amount of bonded reinforcement is required to control cracking in areas where tension due to restraint and or direct loading is expected. The amount may be estimated from equilibrium between the tensile force in concrete just before cracking and the tensile force in the reinforcement at yielding or at a lower stress if necessary to limit the crack width. According to Eurocode 4-1-1 the minimum reinforcement should be placed, where under the characteristic combination of actions, stresses in concrete are tensile.

control of cracking due to direct loading Where at least the minimum reinforcement is provided, the limitation of crack width for direct loading may generally be achieved by limiting bar spacing or bar diameters. Maximum bar spacing and maximum bar diameter depend on the stress σs in the reinforcement and the design crack width.

21

Recommended values for wmax

Exposure class

XO, XC1

reinforced members, prestressed members with unbonded tendons and members prestressed by controlled imposed deformations

prestressed members with bonded tendons

quasi - permanent load combination

frequent load combination

0,4 mm (1)

0,2 mm

XC2, XC3,XC4 XD1,XD2,XS1, XS2,XS3

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

0,2 mm (2) 0,3 mm decompression

(1) For XO and XC1 exposure classes, crack width has no influence on durability and this limit is set to guarantee acceptable appearance. In absence of appearance conditions this limit may be relaxed. (2) For these exposure classes, in addition, decompression should be checked under the quasi-permanent combination of loads. 22

Exposure classes according to EN 1992-1-1 (risk of corrosion of reinforcement) Class

Description of environment

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Examples

no risk of corrosion or attack

XO

for concrete without reinforcement, for concrete with reinforcement : very dry

concrete inside buildings with very low air humidity

Corrosion induced by carbonation

XC1

dry or permanently wet

concrete inside buildings with low air humidity

XC2

wet, rarely dry

concrete surfaces subjected to long term water contact, foundations

XC3

moderate humidity

external concrete sheltered from rain

XC4

cyclic wet and dry

concrete surfaces subject to water contact not within class XC2

Corrosion induced by chlorides

XD1

moderate humidity

concrete surfaces exposed to airborne chlorides

XD2

wet, rarely dry

swimming pools, members exposed to industrial waters containing chlorides

XD3

cyclic wet and dry

car park slabs, pavements, parts of bridges exposed to spray containing

Corrosion induced by chlorides from sea water

XS1

exposed to airborne salt

structures near to or on the coast

XS2

permanently submerged

parts of marine structures

XS3

tidal, splash and spray zones

parts of marine structures 23

Cracking of concrete (initial crack formation)

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

w Ns

Ns ε

εs εc Les

σ

Δσs

σs

σc,1

σc,1

Les

Les

σs,1

Les

ρs =

Les

σs A s = σs,1 A s + σc,1 A c Compatibility at the end of the introduction length: σ σ ε s,1 = εc,1 ⇒ s,1 = c,1 Es Ec Es ⎡ ρs no ⎤ n = o σs,1 = σs ⎢ ⎥ Ec ⎣ 1+ ρs no ⎦ Change of stresses in reinforcement due to cracking: σs Δσ s = σ s − σ s,1 = 1+ ρ s no

Ns,r = fctm A c (1+ ρs no )

σs,2

σs,1

Equilibrium in longitudinal direction:

As Ac

As cross-section area of reinforcement ρs reinforcement ratio fctm mean value of tensile strength of concrete 24

Cracking of concrete – introduction length

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

w Ns

Ns ε

Δσs = σs − σs,1 =

εs εc Les

σ

σc,1

Change of stresses in reinforcement due to cracking:

Equilibrium in longitudinal direction L es Us τsm = Δσs A s π d2s L es π ds τsm = Δσs 4

Les Δσs

σs

introduction length LEs

σs,1

L es Les σc,1 σs,1

Les

σs 1+ ρs no

Les

τsm σs,2 Us As ρs τsm

-perimeter of the bar -cross-section area -reinforcement ratio -mean bond strength

σ d 1 = s s 4 τsm 1+ no ρs

ρs =

As Ac

no =

Es Ec

crack width

w = 2 L es (εsm − εcm ) 25

Determination of the mean strains of G. Hanswille Univ.-Prof. Dr.-Ing. reinforcement and concrete in the stage of initial Institute for Steel and Composite Structures crack formation University of Wuppertal-Germany w

Mean bond strength:

Ns

Ns τs,m =

ε

εs,m

εs(x)

Δεs,cr

εcr

εc(x)

εc,m

εs

Les

Les

σs(x)

Mean stress in the reinforcement: σ s,m = σ s − β Δσ s ⇒ β =

βΔσs σs

x

4 x Δσs ( x ) = ∫ τs ( x ) dx Us 0

Δσs σs,1

Les

σ s − Δσ sm Δσ s

Mean strains in reinforcement and concrete:

σc,1

Les

∫ τs ( x ) dx ≈ 1,8 fctm

Les o

1 Les Δσsm = ∫ Δσs( x) dx Les 0

σ σs,m

1 LEs

ε s,m = ε s,2 − β Δε s,cr εc,m = β εcr 26

Determination of initial crack width

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

w Ns

crack width

Ns

ε

εs,m εs(x)

w = 2 L es (εsm − εcm )

εs,2 Δεs,cr

εc(x)

εc,m

εs,m − εcm = (1 − β) εs,2

εcr L es =

Les

Les σs

σs ds 1 4 τsm 1+ no ρs

τsm ≈ 1,8 fctm βΔσs Δσs

σs,m

σs,1

σs

σc,1

Les

Les x

(1 − β) σ2s ds 1 w= 2 τsm Es 1+ no ρs

with β= 0,6 for short term loading und β= 0,4 for long term loading 27

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Maximum bar diameters acc. to EC4

σs [N/mm2]

∗ maximum bar diameterds for

wk= 0,4

wk= 0,3

wk= 0,2

160

40

32

25

200

32

25

16

240

20

16

12

280

16

12

8

320

12

10

6

360

10

8

5

400

8

6

4

450

6

5

-

β= 0,4

for long term loading and repeated loading

Crack width w: (1 − β) σ 2s ds 1 w= 2 τsm E s 1+ no ρs

σ 2s ds ≈ 6 fct,m E s

Maximum bar diameter for a required crack width w:

ds = w

2 τsm Es ( 1+ no ρs ) σ2s (1 − β)

With τsm= 1,8 fct,mo and the reference value for the mean tensile strength of concrete fctm,o= 2,9 N/mm2 follows: d*s = w k d*s ≈ 6

3,6 fctm,o E s ( 1+ no ρs ) σ 2s (1 − β)

w k fctm,o E s σ 2s 28

Crack width for stabilised crack formation

Crack width for high bond bars

w

w = sr,max (ε sm − ε cm )

Ns ε

ε s,2 =

Mean strain of reinforcement and concrete:

σ s2 Es

εs(x) fct Ec

εs(x)- εc(x)

εc(x) sr,max= 2 Les β= 0,6

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

εs,m = εs,2 − β Δ εs A f f εs,m = εs,2 − β c ctm = εs,2 − β ctm Es A s E s ρs f εcm = β ctm Ec

sr,min= Les

for short term loading

εsm − εcm =

σs f − β ctm (1 + no ρs ) Es Es ρs

β= 0,4 for long term loading and repeated loading 29

Crack width for stabilised crack formation

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

w The maximum crack spacing sr,max in the stage of stabilised crack formation is twice the introduction length Les.

Ns ε

σ εs,2 = s Es εs(x)

fct Ec

w = sr,max (ε sm − εcm ) εs(x)- εc(x)

εc(x)

fctm ds fctm A c Les = = Us τsm ρs 4 τsm maximum crack width for sr= sr,max

sr,max= 2 Les

sr,min= Les

β= 0,6

for short term loading

β= 0,4

for long term loading and repeated loading

fctm ds ⎛ σs ⎜⎜ w= −β 2 τsm ρs ⎝ Es

⎞ fctm (1 + no ρs ) ⎟⎟ ρs E s ⎠

30

Crack width and crack spacing according Eurocode 2

Crack width

w = sr,max (ε sm − εcm ) σ εsm − εcm = s − β Es β= 0,6 β= 0,4

Crack spacing

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

fctm (1 + no ρs ) ≥ 0,6 Es ρs

σs Es

for short term loading for long term loading and repeated loading

In Eurocode 2 for the maximum crack spacing a semiempirical equation based on test results is given

sr,max = 3,4 c + k1 ⋅ k 2 ⋅ 0,425

ds ρs

ds-diameter of the bar c- concrete cover

k1

coefficient taking into account bond properties of the reinforcement with k1=o,8 for high bond bars

k2

coefficient which takes into account the distribution of strains (1,0 for pur tension and 0,5 for bending) 31

Determination of the cracking moment Mcr and the normal force of the concrete slab in the stage of initial cracking

cracking moment Mcr:

cracking moment Mcr Mc+s

hc zio zi,st

σc + σc,ε = fct,eff = k1 fctm

σc

Mcr

Nc+s

zo

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

[

Mcr = fct,eff − σc,ε

[

ast

Mcr = fct,eff − σc,ε

Nc,ε

]z

no Jio o

+ hc / 2 no Jio

ic,o (1+ hc

/( 2 z o )

sectional normal force of the concrete slab:

primary effects due to shrinkage Mc,ε

]z

σcε

Ncr = Mcr Ncr =

A co z o + A s zis + Nc + s,ε Jio

A c ( fct,eff − σc,ε ) (1+ ρs no ) + Nc + s,ε 1+ hc /( 2 z o )

kc,ε≈ 0,3 kc ⎡ ⎢ 1 Ncr = A c fct,eff (1 + ρ s n0 ) ⎢ ⎢ 1 + hc /(2 z o ) ⎢ ⎣

+

A c σc,ε (1+ ρs no ) 1+ hc /( 2 z o ) A c fct,eff (1 + ρs n0 )

Nc + s,ε −

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

32

Simplified solution for the cracking moment and the normal force in the concrete slab cracking moment Mcr Mc+s

hc

zo

simplified solution for the normal force in the concrete slab:

σc

Mcr

Nc+s

Ncr ≈ A c fctm k s ⋅ k ⋅ k c

zi,st

k = 0,8

primary effects due to shrinkage Mc+s,ε

Nc+s,ε

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

ks= 0,9

σcε

coefficient taking into account the effect of non-uniform self-equilibrating stresses coefficient taking into account the slip effects of shear connection

kc =

1 h 1+ c 2 zo

cracking moment

+ 0,3 ≤ 1,0

shrinkage 33

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Determination of minimum reinforcement

Mc hc

zo zi,o

Nc

Mc,ε

Mcr

cracking moment

Nc,ε shrinkage

Ma,ε Na,ε

A c fct,eff As ≥ k k s kc σs k = 0,8 ks = 0,9 kc

d∗s ds σs fct,eff

kc =

f 1 + 0,3 ≤ 1,0 ds = d∗s ct,eff 1+ hc z o fct,o

fcto= 2,9 N/mm2

Influence of non linear residual stresses due to shrinkage and temperature effects flexibility of shear connection Influence of distribution of tensile stresses in concrete immediately prior to cracking maximum bar diameter modified bar diameter for other concrete strength classes stress in reinforcement acc. to Table 1 effective concrete tensile strength 34

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Control of cracking due to direct loading – Verification by limiting bar spacing or bar diameter fully cracked

Ac

tension stiffening

Ns,2

As

-Ms,2

-zst,s

a

za

The calculation of stresses is based on the mean strain in the concrete slab. The factor β results from the mean value of crack spacing. With srm≈ 2/3 sr,max results β ≈ 2/3 ·0,6 = 0,4

-Ms

+

zst zst,a Aa

ΔNts

-Ma,2 -Na,2

Ns

ΔNts a

-MEd

=

-ΔNts

-Ma -Na

stresses in reinforcement σ = σ s,2 + Δσ ts taking into account tension s fct,eff MEd stiffening for the bending σs = z st,s + β moment MEd of the quasi J2 ρs α st permanent combination: A 2 J2 As α = st ρs = β = 0,4 A a Ja Ac

The bar diameter or the bar spacing has to be limited 35

Maximum bar diameters and maximum bar spacing for high bond bars acc. to EC4

Table 1: Maximum bar diameter σs [N/mm2]

∗

maximum bar diameterds for

wk= 0,4

wk= 0,3

wk= 0,2

160

40

32

25

200

32

25

16

240

20

16

12

280

16

12

8

320

12

10

6

360

10

8

5

400

8

6

4

450

6

5

-

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Table 2: Maximum bar spacing σs [N/mm2]

maximum bar spacing in [mm] for

wk= 0,4

wk= 0,3

wk= 0,2

160

300

300

200

200

300

250

150

240

250

200

100

280

200

150

50

320

150

100

-

360

100

50

-

36

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Direct calculation of crack width w for composite sections based on EN 1992-2 As

-zst,s

-Ms

σs,

Ns

-Ma

MEd

zst

-Na crack width for high bond bars:

w = sr,max (ε sm − εcm )

fct,eff MEd σs = zst,s + β Jst ρs α st α st =

A st Jst A a Ja

ρs =

As Ac

β = 0,4

σ εsm − εcm = s − β Es

fctm (1 + no ρs ) ≥ 0,6 Es ρs

σs Es

d sr,max = 3,4 c + 0,34 s ρs c - concrete cover of reinforcement 37

Stresses in reinforcement in case of bonded tendons – initial crack formation Ap, dp

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Equilibrium at the crack:

As, ds

σs A s + Δσp A p = N = fct,eff A c ( 1 + no ρ tot ) Equilibrium in longitudinal direction: σs=σs1+Δσs

σ s A s = π ds τsm L e,s

σp

Δσp A p = π dp τpm L ep N

σp=σpo+σp1+Δσp

δ s = δp ⇒

Δσs Δσp

σs,1

Compatibility at the crack: Δσp − Δσp1 σ s − σ s1 L es = L ep Es Ep

With Es≈Ep and σs1=Δσp1=0 results: Stresses:

Δσp1

σs =

Les Lep τpm

τsm

ξ1 =

N A s + ξ1 A p τpm ds τsm dv

Δσp =

ξ1 N A s + ξ1 A p

38

Stresses in reinforcement for final crack formation Ap, dp

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Equilibrium at the crack:

N − Po = σs2 A s + Δ σp2 A p

As, ds

Maximum crack spacing: σs

fct A c =

Δσp2

sr,max =

σs2

[

sr,max τsm ns ds π + τpm np dp π 2 ds fct,eff A c 2 τsm ( A s + ξ2 A p )

Equilibrium in longitudinal direction:

Δσp2

σs2 − σs1 =

Δσp1

sr,max Us τsm 2 As

σp2 − σp1 =

sr,max Up τpm 2 Ap

Compatibility at the crack:

σs1 σc

]

σc=fct,eff

sr,max

x

δ s = δp =

σ s2 − β (σs2 − σs1) Δσp,2 − β( Δσp2 − Δσp1) = Es Ep

mean crack spacing: sr,m≈2/3 sr,max 39

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Determination of stresses in composite sections with bonded tendons

Ap

As

σs,σp

-zst,s

MEd zst

Stresses σ*s in reinforcement at the crack location neglecting different bond behaviour of reinforcement and tendons: σ*s =

MEd f z st,s + β ctm Jst ρ tot α st

α st =

Stresses in reinforcement taking into account the different bond behaviour: ⎡ Ac Ac σ s = σ *s + 0,4 fct,eff ⎢ − ⎢ A s + ξ12 A p A s + A p ⎣ Δσp = σ *s

A st Jst A a Ja

⎤ ⎡ 1 1 ⎤ ⎥ = σ *s + 0,4 fct,eff ⎢ − ⎥ ρeff ρ tot ⎦ ⎥ ⎣ ⎦

⎡ A ξ12 A c c ⎢ − 0,4 fct,eff − ⎢ A s + A p A s + ξ12 A p ⎣

⎤ ⎡ 1 ξ12 ⎤ ⎥ = σ *s − 0,4 fct,eff ⎢ − ⎥ ρ ρ ⎥ ⎥⎦ ⎢ tot eff ⎣ ⎦

β = 0,4

ρ tot = ρeff =

A s + Ap Ac A s + ξ12 A p Ac 40

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Part 4: Deformations

41

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Deflections

Effects of cracking of concrete L1

L2

0,15 L1

0,15 L2

EaJ2 EaJ1 ΔM

Deflections due to loading applied to the composite member should be calculated using elastic analysis taking into account effects from - cracking of concrete, - creep and shrinkage, - sequence of construction,

Sequence of construction

gc

- influence of local yielding of structural steel at internal supports, - influence of incomplete interaction.

F F

steel member composite member 42

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Deformations and pre-cambering

combination

limitation

general

quasi permanent

risk of damage of adjacent parts of the structure (e.g. finish or service work)

quasi – permanent (better frequent)

δ1 δ2 δ3 δ4

δmax ≤ L / 250

δ1

δ w ≤ L / 500

Pre-cambering of the steel girder:

δmax maximum deflection δw

δmax

δc deflection of the composite girder

δp = δ1+ δ2+ δ3 +ψ2 δ4

δp

δc

δ1 deflection of the steel girder

δw

effective deflection for finish and service work

δ1 – self weight of the structure δ2 – loads from finish and service work δ3 – creep and shrinkage δ4 – variable loads and temperature effects 43

Effects of local yielding on deflections

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

For the calculation of deflection of un-propped beams, account may be taken of the influence of local yielding of structural steel over a support. For beams with critical sections in Classes 1 and 2 the effect may be taken into account by multiplying the bending moment at the support with an additional reduction factor f2 and corresponding increases are made to the bending moments in adjacent spans. f2 = 0,5 if fy is reached before the concrete slab has hardened; f2 = 0,7 if fy is reached after concrete has hardened. This applies for the determination of the maximum deflection but not for pre-camber.

44

More accurate method for the determination of the effects of local yielding on deflections

L1

lcr

lcr

L2

+ z2

EaJ1

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

-

Mel,Rk

σa=fyk

fyk

Mpl,Rk

EaJ2 (EJ)eff EaJ2

EaJeff ΔM

EaJeff

Mel,Rk

MEd

Mpl,Rk 45

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Effects of incomplete interaction on deformations

The effects of incomplete interaction may be ignored provided that:

ƒ

The design of the shear connection is in accordance with clause 6.6 of Eurocode 4,

ƒ

either not less shear connectors are used than half the number for full shear connection, or the forces resulting from an elastic behaviour and which act on the shear connectors in the serviceability limit state do not exceed PRd and

ƒ

in case of a ribbed slab with ribs transverse to the beam, the height of the ribs does not exceed 80 mm.

P PRd P

P

cD s

s

su 46

Differential equations in case of incomplete interaction

Mc

Ec, Ac, Jc

vL

ac N c a

zc Ea, Aa, Ja

vL

Ma Na

za (w)

Slip:

aa

Vc+dVc

Vc

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Mc+ dMc Nc+dNc Ma+ dMa Na+dNa

Va

uc , u′c = ε c

ua , u′a = ε a

Va+dVa dx

sv = ua − uc + w ′ a

u′c′ + c s (ua − uc + w ′ a ) = 0 u′c′ − c s (ua − uc + w ′ a) = 0 (Ec Jc + Ea Ja ) w ′′′′ − c s a (u′a − u′c + w ′′ a) = q

Ec A c Ea A a

x

Nc = E c A c u′c

Mc = − Ec Jc w′′

Na = E a A a u′a

Ma = − Ea Ja w ′′ Va = − Ea Ja w ′′′

Vc = − E c Jc w ′′′ ≈ 0

47

Deflection in case of incomplete interaction for single span beams

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

concrete section composite section

F

Aco=Ac/no, Jco= Jc/no w L

steel section

⎡ 2 ( λ )⎤ sinh ⎢ ⎥ 12 48 FL 2 − 1+ w= 48 E a Ii,o ⎢⎢ α λ2 α λ3 sinh( λ ) ⎥⎥ ⎣ ⎦

Aa, Ja

Aio, Jio

no=Ea/Ec

3

λ2 =

q L

β=

λ ⎡ ⎤ cosh( ) − 1 ⎥ 5 q L ⎢ 48 1 384 1 2 + − w= 1 ⎢ ⎥ 384 E a Ji,o ⎢ 5 α λ2 5 α λ4 cosh( λ ) ⎥ 2 ⎦ ⎣ 4

1+ α αβ

E a A c,o A a A i,o c s L ²

α=

1 Ji,o Ja + Jc,o

−1 48

Mean values of stiffness of headed studs

spring constant per stud:

cs

spring constant of the shear connection: type of shear connection

P PRd P

P

cD

s

s

su

eL

nt=2

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

CD =

su PRd

C n cs = D t eL

CD [kN / cm]

headed stud ∅ 19mm in solid slabs

2500

headed stud ∅ 22mm in solid slabs

3000

headed studs ∅ 25mm in solid slab

3500

headed stud ∅ 19mm with Holorib-sheeting and one stud per rib

1250

headed stud ∅ 22mm with Holorib-sheeting and one stud per rib

1500

49

Simplified solution for the calculation of deflections in case of incomplete interaction q(ξ ) = q sin πξ

The influence of the flexibility of the shear connection is taken into account by a reduced value for the modular ratio.

L

wo = q

x ξ= L

Ec, Ac, Jc zc

εc

L4 π4

a

Ea, Aa, Ja

εa

EcmJc + Ea Ja +

Mc Nc Ma

za

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Na

1 βo Ecm A c Ea A a Ea A a + βo Ecm A c

=q a2

L4

1 π 4 Ea Jio,eff

A c,eff A a 2 Jio,eff = Jc,o + Ja + a A c,eff + A a A A c,eff = c no,eff effective modular ratio for the concrete slab

no,eff = no ( 1 + βs )

π2 Ecm A c βs = L2 c s 50

Comparison of the exact method with the simplified method q

1,5

w/wc

cD = 1000 KN/cm

1,4 L

1,2

beff

1,1

Ecm = 3350 KN/cm² 99 51

η=0,4

1,3

w

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

η=0,8 L [m]

1,0 5,0

10,0

20,0

exact solution simplified solution with no,eff

w/wc

450 mm

15,0

1,25

cD = 2000 KN/cm

1,2

wo- deflection in case of neglecting effects from slip of shear connection

η

degree of shear connection

1,15

η=0,4

1,1 1,05 1,0 5,0

η=0,8 L [m] 10,0

15,0

20,0 51

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Deflection in case of incomplete interactioncomparison with test results

1875

1875

F

load case 1

F F/2

load case 2

F/2

Deflection at midspan

F [kN] 200

1875

3750

1875 150

7500

load case 2 100

1500 50

load case 1

445

IPE 270

270

175

50

0

20

40

60

δ [mm] 52

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Deflection in case of incomplete interactionComparison with test results

F

F 160 120 780

s

push-out test

80 40

s[mm] 10

125 50

20

30

40

second moment of area cm4

Load case 1 F= 60 kN

Load case 2 F=145 kN

Test

-

11,0 (100%)

20,0 (100 %)

Theoretical value, neglecting flexibility of shear connection

Jio= 32.387,0

7,8 (71%)

12,9 (65%)

Theoretical value, taking into account flexibility of shear connection

Jio,eff= 21.486,0

11,7 (106%)

19,4 (97%)

Deflection at midspan in mm

53

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Part 5: Limitation of stresses

54

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Limitation of Stresses

σc

MEd

- +

σa +

σa

σs MEd

Stress limitation is not required for beams if in the ultimate limit state, -

no+ verification of fatigue is required and

-

no prestressing by tendons and /or

- - no prestressing by controlled imposed deformations is provided.

combination

stress limit

recommended values ki

structural steel

characteristic

σEd ≤ ka fyk

ka = 1,00

reinforcement

characteristic

σEd ≤ ks fsk

ks = 0,80

concrete

characteristic

σEd ≤ kc fck

kc= 0,60

headed studs

characteristic

PEd ≤ ks PRd

ks = 0,75 55

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Local effects of concentrated longitudinal shear forces composite section

steel section

bc

gc

beff zio

y

x

z MEd

MEd(x)

Ac,eff

+

Concentrated longitudinal shear force at sudden change of cross-section

VEd(x)

VEd

-

Lv=beff

+

longitudinal shear forces

Nc

v L,Ed,max =

2 MEd A c,eff zio Ea / Ec Jio b eff

vL,Ed,max

+ 56

Local effects of concentrated longitudinal shear forces

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

gc,d system

L = 40 m FE-Model

cross-section

bc=10 m 300 500x20

y

P

shear connectors

14x2000 800x60

z

CD = 3000 kN/cm per stud δ 57

Ultimate limit state - longitudinal shear forces

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

EN 1994-2

ULS

x [cm]

P

FE-Model: cD

s

FE-Model P L = 40 m

cD

s

x 58

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Serviceability limit state - longitudinal shear forces vL,Ed[kN/m]

SLS

500

EN 1994-2 x [cm]

0 200

400

600

800

1000 1200 1400 1600 1800 2000

-500

P

-1000

FE-Model: cD

-1500

s

-2000 -2500

FE-Model P

-3000

L = 40 m

-3500

cD

s

x -4000

59

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Part 6: Vibrations

60

Vibration- General

EN 1994-1-1:

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

The dynamic properties of floor beams should satisfy the criteria in EN 1990,A.1.4.4

EN 1990, A1.4.4: To achieve satisfactory vibration behaviour of buildings and their structural members under serviceability conditions, the following aspects, among others, should be considered:

the comfort of the user the functioning of the structure or its structural members Other aspects should be considered for each project and agreed with the client

61

Vibration - General

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

EN 1990-A1.4.4:

For serviceability limit state of a structure or a structural member not to be exceeded when subjected to vibrations, the natural frequency of vibrations of the structure or structural member should be kept above appropriate values which depend upon the function of the building and the source of the vibration, and agreed with the client and/or the relevant authority. Possible sources of vibration that should be considered include walking, synchronised movements of people, machinery, ground borne vibrations from traffic and wind actions. These, and other sources, should be specified for each project and agreed with the client. Note in EN 1990-A.1.4.4: Further information is given in ISO 10137.

62

Vibration – Example vertical vibration due to walking persons

Span

The pacing rate fs dominates the dynamic effects and the resulting dynamic loads. The speed of pedestrian propagation vs is a function of the pacing rate fs and the stride length ls.

lengt h

F(x,t)

pacing rate fs [Hz]

forward speed vs = fs ls [m/s]

stride length ls [m]

slow walk

∼1,7

1,1

0,6

normal walk

∼2,0

1,5

0,75

fast walk

∼2,3

2,2

1,00

slow running (jog)

∼2,5

3,3

1,30

fast running (sprint)

> 3,2

5,5

1,75

ls time t

xk

F(x,t) tk ts ls

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

xk x

63

Vibration –vertical vibrations due to walking of one person

Fi(t)

During walking, one of the feet is always in contact with the ground. The load-time function can be described by a Fourier series taking into account the 1st, 2nd and 3rd harmonic.

right foot left foot

3 ⎡ ⎤ F( t ) = Go ⎢1 + αn sin (2 n π fs t − Φ n )⎥ ⎢⎣ n =1 ⎥⎦

∑

time t F(t) 1. step

2. step 3. step

both feet

ts=1/fs

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Go αn n fs Φn

weight of the person (800 N) coefficient for the load component of n-th harmonic number of the n-th harmonic pacing rate phase angle oh the n-th harmonic Fouriercoefficients and phase angles:

α1=0,4-0,5 Φ1=0 α2=0,1-0,25 Φ2=π/2 α3=0,1-0,15 Φ3=π/2 64

Vibration – vertical vibrations due to walking of persons acceleration

F(t)

Fn π sin (2 π fE t ) 1 − e − δ fE t Mgen δ

 ( t ) = k a w

Mgen w(t)

c

δ

a

Fn(t) m w(xk,t)

L/2 ka Fn(t) w(t) L

)

t=

L vs

maximum acceleration a, vertical deflection w and maximum velocity v a w max = Fn π (2 π fE )2 a =k 1 − e − fE δ L / v s max

xk

(

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

fE Fn δ vs ka Mgen

Mgen

( δ

)

v max =

a 2 π fE

natural frequency load component of n-th harmonic logarithmic damping decrement forward speed of the person factor taking into account the different positions xk during walking along the beam generated mass of the system (single span beam: Mgen=0,5 m L) 65

Logarithmic damping decrement

results of measurements in buildings

Damping ratioξ [%]

with finishes without finishes

6 5 4

2

δ = 2π ξ 3

For the determination of the maximum acceleration the damping coefficient ζ or the logarithmic damping decrement δ must be determined. Values for composite beams are given in the literature. The logarithmic damping decrement is a function of the used materials, the damping of joints and bearings or support conditions and the natural frequency. For typical composite floor beams in buildings with natural frequencies between 3 and 6 Hz the following values for the logarithmic damping decrement can be assumed:

3

1

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

δ=0,10 floor beams without not loadbearing inner walls

6

9 fE [Hz]

12

δ=0,15 floor beams with not loadbearing inner walls 66

Vibration –vertical vibrations due to walking of persons

F( t ) = Go +

3

∑ Fn

sin (2 n π fs t − Φ n )

n =1

F(t)/Go fs=1,5-2,5 Hz 0,4

2fs=3,0-5,0 Hz

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

People in office buildings sitting or standing many hours are very sensitive to building vibrations. Therefore the effects of the second and third harmonic of dynamic load-time function should be considered, especially for structure with small mass and damping. In case of walking the pacing rate is in the rage of 1.7 to 2.4 Hz. The verification can be performed by frequency tuning or by limiting the maximum acceleration.

In case of frequency tuning for composite structures in office buildings the natural frequency 3fs=4,5-7,5 Hz normally should exceed 7,5 Hz if the first, second and third harmonic of the dynamic load-time function can cause significant acceleration.

0,2 0,1 2,0

4,0

6,0

8,0

Otherwise the maximum acceleration or velocity should be determined and limited to acceptable values in accordance with ISO 10137 67

Limitation of acceleration-recommended values acc. to ISO 10137

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

acceleration [m/s2] natural frequency of typical composite beams

0,1 0,05

Multiplying factors Ka for the basic curve Residential (flats, hospitals) Quiet office General office (e. g. schools)

0,01 0,005

basic curve ao

Ka=1,0 Ka=2-4 Ka=4

a ≤ ao K a 1

5

10

50 100

frequency [Hz] 68

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Thank you very much for your kind attention 69

G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany

Thank you very much for your kind attention

70