Eurocode 4

(1)

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

Serviceability limit states of

composite beams

Eurocode 4

Eurocodes

Background and Applications

Dissemination of information for training 18-20 February 2008, Brussels

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

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Part 1:

Introduction

Part 2:

Global analysis for serviceability limit states

Part 3:

Crack width control

Part 4:

Deformations

Part 5:

Limitation of stresses

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Serviceability limit states

Limitation of stresses

Limitation of deflections

crack width control

vibrations

(4)

Serviceability limit states

{

∑ + + +∑ ψ

}

= _k_,_j _k _k_,₁ ₀_i, _k_i, d E G P Q Q E characteristic combination: frequent combination: Ed=E

{

∑ Gk,j + Pk + ψ1,1Qk,1 +∑ ψ2i, Qk i,

}

quasi-permanent combination: Ed=E

{

∑ Gk,j + Pk + ∑ ψ2i, Qki,

}

serviceability limit states E_d ≤ C_d:

- deformation - crack width

- excessive compressive stresses in concrete

C

_d

=

- excessive slip in the interface between steel

and concrete

- excessive creep deformation - web breathing

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Part 2:

Global analysis for serviceability limit states

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Global analysis - General

Calculation of internal forces, deformations and stresses at

serviceability limit state shall take into account the following

effects:

shear lag;

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;

inelastic behaviour of steel and reinforcement, if any;

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Shear lag- effective width

σmax

b

_e

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 b_e 2 , 0 b b i ei_≥ σ_max b_ei b_i 5 b_ei y b_i y σ_max b_ei σ(y) σ(y) 2 , 0 b b i ei_< σ_R [ ] 4 i R max R max i ei R b y 1 ) y ( 2 , 0 b b 25 , 1 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − σ − σ + σ = σ σ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = σ 4 i max b y 1 ) y ( _⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − σ = σ shear lag

real stress distribution

stresses taking into account the effective width

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midspan regions and internal supports: b_eff= b₀ + b_e,1+b_e,2 b_e,i= L_e/8

L – equivalent length end supports: b_eff= b₀ + β₁b_e,1+β₂b_e,2

β ) ≤ 1,0

Effective width of concrete flanges

L_e=0,85 L₁ for b_eff,1 _L e=0,70 L2 for beff,1 L_e=2L₃ for b_eff,2 L₁ L₂ L3 b_eff,0 b_eff,1 _b beff,1 eff,2 b_eff,2 L₁/4 L₁/2 L₁/4 L₂/4 L₂/2 L₂/4 L_e=0,25 (L₁ + L₂) for b_eff,2 _b o b_e,1 b_e,2 b_o b₁ b₂

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Initial sectional forces

redistribution of the sectional forces due to creep

-N_c,o M_c,o M_st,o N_st,o N_c,r -M_c,r M_st,r -N_st,r

z

_i,st

-z

_i,c M_L

a

_st

Effects of creep of concrete

primary effects

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.

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Effects of creep and shrinkage of concrete

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)

M

_PT

(t)

M_PT(t=∞)

ϕ(t,t

)

time dependent action effects M_L=M_PT:

action effects caused by prestressing due to imposed

deformation M_L=M_D:

δ M =M

M_D

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Modular ratios taking into account

effects of creep

[

]

cm a o o L o L

E

n

)

t

,

t

(

1 n

n

=

+

ψ

ϕ

=

Modular ratios:

centroidal axis of the concrete section centroidal axis of the transformed composite section

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

-z_ic,L z_ist,L zi,L z_c z_is,L a_st z_st

action creep multiplier

short term loading Ψ=0

permanent action not changing in time Ψ_P=1,10

shrinkage Ψ_S=0,55

prestressing by controlled imposed deformations Ψ_D=1,50 time-dependent action effects Ψ_PT=0,55

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G. Hanswille

Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany ) t ( E E n o cm st o =

Modular ratio taking into account creep effect: centroidal axis of the

concrete section L ,i 2 st L , c st L , c st L ,i J J A A a / A J = + + L , c St L ,i A A A = + L ,i st st L , ic A a /A z =− L c L , c L c L , c A /n J J /n A = = -z_ic,L z_ist,L zi,L z_c -z_is,L a_st z_st

Transformed cross-section properties of the concrete section:

Transformed cross-section area of the

composite section: Second moment of area of the _{composite section:}

Distance between the centroidal axes of the concrete and the composite section:

) ) t , t ( 1 ( n n_L= ₀ +ψ_Lϕ ₀

Elastic cross-section properties of the

composite section taking into account creep

effects

centroidal axis of the composite section centroidal axis of the

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Effects of cracking of concrete and tension

stiffening of concrete between cracks

ε ε_s(x) ε_c(x) N_s N_s c ct E f s 2 s 2 , s E σ = ε r , s ε Δ β ε_sr,1 ε_sr,2 ε_sm,y ε_sy

N

_s N_sy N_sm N_s,cr B C σ_s,2 σ_s(x) σ_c(x)

τ

_v x σ_c(x)

stage A: uncracked section stage B: initial crack formation

stage C: stabilised crack formation

fully cracked section A σ_c(x) σ_s(x) mean strain ε_sm=ε_s,2- βΔε_s,r r , s ε Δ

ε

_sm r , s s=βΔε ε Δ s s eff , ct s E f ρ β = ε Δ c s s=A /A ρ 4 , 0 = β

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

M

M_s≈0 M_a N_a ε_a a z_s N_s equilibrium: a N M M_a = − _s s a N N =− ε_s,m ε_s,2 Δεs=β Δεs,r ε_c ε_s compatibility: a a sm =ε +κ ε a a a a 2 s a a s sm J E a M A E a N A E N ₊ ₌ + ε s s eff , ct s s s sr 2 s sm E f A E N ρ β − = ε Δ β − ε = ε

mean strain in the concrete slab:

z_a

Influence of tension stiffening of concrete on

stresses in reinforcement

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Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany N_s,2 -M_s,2 ΔN_ts -M_a,2 -N_a,2 _-ΔN ts ΔN_ts a a N_s,2

-M

_Ed M_Ed N_s N_sε z_st,a -z_st,s N_s M ts st s , st s Ed ts 2 s s N J z A M N N N = +Δ = + Δ st s s eff , ct ts A f N α ρ β = Δ ts st a , st a Ed ts 2 a a N J z A M N N N = −Δ = − Δ a N J J M a N M M _ts st a Ed ts 2 a a = +Δ = + Δ

Sectional forces:

st s Ed s J J M M = a a st st st J A J A = α N_s -M_s -M_a -N_a

fully cracked section tension stiffening

+

=

z

₂

=z

_st

ΔN_ts

Redistribution of sectional forces due to tension

stiffening

2 st

J

=

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Stresses taking into account tension stiffening of

concrete

N_s,2 -M_s,2 ΔN_ts -M_a,2 -N_a,2 -ΔN_ts ΔN_ts a z_st,a -z_st,s st s s ctm ts A f N α ρ β = Δ a a st st st J A J A = α N_s -M_s -M_a -N_a

fully cracked tension stiffening

+

=

z

_st -MEd st s ctm s , st st Ed s st s ctm 2 , s s f z J M f α ρ β + = σ α ρ β + σ = σ a a ts a ts st st Ed a a a ts a ts 2 , a a z J a N A N z J M z J a N A N Δ + Δ − = σ Δ + Δ − σ = σ

reinforcement: structural steel: z_a

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Influence of tension stiffening on flexural stiffness

E_stJ₁ uncracked section E_stJ₂fully cracked section E_stJ_2,ts effective flexural

stiffness taking into account tension stiffening of concrete E_stJ₁ E_stJ₂ E_stJ_2,ts

M

κ κ ε_sm

-M

N_s -M_s -M_a -N_a ε_a a z_st a st s a st a ts , 2 st E J a N M J E M I E M ₌ ₌ − = κ EJ M_R M_Rn E_st J₁ E_st J_2,ts E_stJ₂ M a ) N N ( 1 J E J E , s s a a ts , 2 st ε − − =

M

Curvature: Effective flexural stiffness:

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• Determination of internal forces by un-cracked analysis for the characteristic combination.

• Determination of the cracked regions with the extreme fibre concrete tensile stress σ_c,max= 2,0 f_ct,m.

• Reduction of flexural stiffness to E_aJ₂ in the cracked regions.

• New structural analysis for the new distribution of flexural stiffness.

L₁ _L L₂ 1,cr L2,cr E_aJ₂ E_aJ₁ E_aJ₁ ΔM un-cracked analysis cracked analysis ΔM Redistribution of bending moments due to cracking of concrete

E_aJ₁ – un-cracked flexural stiffness E_aJ₂ – cracked flexural stiffness

Effects of cracking of concrete - General

method according to EN 1994-1-1

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Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany L₁ L₂ ΔM_II E_aJ₁ 0,15 L₁ _{0,15 L} 2 E_aJ₂ 6 , 0 L / L_min _max ≥

Effects of cracking of concrete –

simplified method

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 E

_a

J

₂

over

15% of the span on each side of each

internal support, and as the

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Part 3:

Limitation of crack width

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Control of cracking

General considerations

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.

minimum reinforcement

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 σ_s in the reinforcement and the design crack width.

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Recommended values for w

_max

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

XO, XC1 0,4 mm (1) 0,2 mm XC2, XC3,XC4 0,2 mm (2) XD1,XD2,XS1, XS2,XS3 decompression 0,3 mm Exposure class

(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.

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Exposure classes according to EN 1992-1-1

(risk of corrosion of reinforcement)

Class Description of environment 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

XS2 permanently submerged parts of marine structures

Corrosion induced by chlorides from sea water

XS1 exposed to airborne salt structures near to or on the coast

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Cracking of concrete (initial crack formation)

ε _ε s ε_c L_es L_es N_s N_s w s s A A = ρ

A_s cross-section area of reinforcement ρ_s reinforcement ratio

f mean value of tensile strength of concrete

c s o E E n = c 1 , c s 1 , s s s A = σ A +σ A σ

Compatibility at the end of the introduction length:

Equilibrium in longitudinal direction:

c 1 , c s 1 , s 1 , c 1 , s E E σ = σ ⇒ ε = ε ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ρ + ρ σ = σ o s o s s 1 , s n 1 n o s s 1 , s s s n 1+ρ σ = σ − σ = σ Δ

Change of stresses in reinforcement due to cracking: L_es L_es σ_s σ_s,1 σ_c,1 Δσ_s σ_c,1 σ_s,1 σ_s,2 σ

(

_s _o

)

c ctm r , s f A 1 n N = + ρ

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Cracking of concrete – introduction length

ε _ε s ε_c L_es L_es N_s N_s w c s s A A = ρ

U_s -perimeter of the bar A_s -cross-section area ρ_s -reinforcement ratio τ_sm -mean bond strength

c s o E E n = 4 d d L A U L 2 s s sm s es s s sm s es π σ Δ = τ π σ Δ = τ o s s 1 , s s s n 1+ρ σ = σ − σ = σ Δ

Change of stresses in reinforcement due to cracking:

Equilibrium in longitudinal direction

L_es L_es σ_s σ_s,1 σ_c,1 Δσ_s σ_c,1 σ_s,1 σ_s,2 L_es τ_sm σ s o sm s s es n 1 1 4 d L ρ + τ σ = introduction length L_Es crack width

)

(

L

2 w

=

_es

ε

_sm

−

ε

_cm

(26)

G. Hanswille

Determination of the mean strains of

reinforcement and concrete in the stage of initial

crack formation

ctm L o s es m , s (x)dx 1,8 f L 1 Es ≈ τ = τ ∫

Mean bond strength:

s sm s s s m , s _Δ_σ σ Δ − σ = β ⇒ σ Δ β − σ = σ ∫τ = σ Δ x 0 s s s (x)dx U 4 ) x ( ∫ Δσ = σ Δ Les 0 s es sm (x)dx L 1 ε ε_s ε_c(x) L_es L_es N_s N_s w L_es L_es σ_s σ_s,1 σ_c,1 Δσ_s σ

ε

_cr Δε_s,cr

Mean strains in reinforcement and concrete:

cr m

,

c = β ε

ε

Mean stress in the reinforcement:

ε_s,m ε_c,m βΔσ_s σ_s,m σ_s(x) ε_s(x) cr , s 2 , s m , s

=

ε

−

β

Δ

ε

(27)

Determination of initial crack width

ε _ε s(x) ε_c(x) L_es L_es N_s w L_es L_es σ_s σs,1 σ_c,1 Δσ_s σ_s

ε

_cr

Δε

_s,cr ε_s,m ε_c,m βΔσ_s x σ_s,m N_s _{crack width}

)

(

L

2 w

=

_es

ε

_sm

−

ε

_cm s o sm s s es n 1 1 4 d L ρ + τ σ = 2 , s cm m , s −ε = (1−β) ε ε

ε

_s,2 s o s sm s 2 s n 1 1 E 2 d ) 1 ( w ρ + τ σ β − =

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

ctm sm ≈1,8 f

(28)

Maximum bar diameters acc. to EC4

maximum bar diameter for σ_s [N/mm2_] w_k= 0,4 w_k= 0,3 w_k= 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 -∗ s d s m , ct s 2 s s o s sm s 2 s E f 6 d n 1 1 E 2 d ) 1 ( w ≈ σ ρ + τ σ β − = Crack width w:

Maximum bar diameter for a required crack width w:

) 1 ( ) n 1 ( E 2 w d 2 s s o s sm s β − σ ρ + τ = 2 s s o , ctm k * s 2 s s o s o , ctm k * s E f w 6 d ) 1 ( ) n 1 ( E f 6 , 3 w d σ ≈ β − σ ρ + =

With τ_sm= 1,8 f_ct,moand the reference value for the mean tensile strength of concrete f_ctm,o= 2,9 N/mm2 _follows:

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

(29)

Crack width for stabilised crack formation

ε ε_s(x) ε_c(x) N_s w s_r,max= 2 L_es c ct E f s 2 s 2 , s E σ = ε ε_s(x)- ε_c(x) s_r,min= L_es

)

(

s

w

=

_r_,_max

ε

_sm

−

ε

_cm

Crack width for high bond bars

c ctm cm s s ctm 2 , s s s ctm c 2 , s m , s s 2 , s m , s E f E f A E f A β = ε ρ β − ε = β − ε = ε ε Δ β − ε = ε

Mean strain of reinforcement and concrete:

β= 0,6 for short term loading β= 0,4 for long term loading and

repeated loading ) n 1 ( E f E _s _s o s ctm s s cm sm − β _ρ + ρ σ = ε − ε

(30)

Crack width for stabilised crack formation

ε ε_s(x) ε_c(x) N_s w s_r,max= 2 L_es c ct E f s s 2 , s E σ = ε ε_s(x)- ε_c(x) s_r,min= L_es sm s s ctm sm s c ctm es 4 d f U A f L τ ρ = τ =

The maximum crack spacing s_r,max in the stage of stabilised crack formation is twice the introduction length L_es.

)

(

s

w

=

_r_,_max

ε

_sm

−

ε

_cm ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ρ + ρ β − σ ρ τ = (1 n ) E f E 2 d f w _o _s s s ctm s s s sm s ctm

maximum crack width for s_r= s_r,max

β= 0,6 for short term loading

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

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Crack width and crack spacing according

Eurocode 2

)

(

s

w

=

_r_,_max

ε

_sm

−

ε

_cm Crack width s s 2 1 max , r d 425 , 0 k k c 4 , 3 s ρ ⋅ ⋅ +

= d_s-diameter of the bar c- concrete cover

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

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)

s s s o s s ctm s s cm sm E 6 , 0 ) n 1 ( E f E σ ≥ ρ + ρ β − σ = ε − ε Crack spacing

β= 0,6 for short term loading

(32)

G. Hanswille

Determination of the cracking moment M

_cr

and

the normal force of the concrete slab in the

stage of initial cracking

cracking moment M_cr:

[

]

[

]

) z 2 /( h 1 ( z J n f M 2 / h z J n f M o c o , ic io o , c eff , ct cr c o io o , c eff , ct cr + σ − = + σ − = ε ε

(

)

ε + ε ε + + + ρ + σ − = + + = , s c o c o s , c eff , ct c cr , s c io is s o co cr cr N ) z 2 /( h 1 n 1 ) f ( A N N J z A z A M N

primary effects due to shrinkage cracking moment M_cr h_c z_io z_o z_i,st N_c+s M_c,ε M_cr N_c,ε σ_cε a_st ctm 1 eff , ct , c c f k f M_c+s σ_c = = σ + σ _ε

sectional normal force of the concrete slab:

(

)

⎥ ⎥ ⎥ ⎥ ⎤ ⎢ ⎢ ⎢ ⎢ ⎡ ρ + + ρ + σ − + + ρ + = ε ε + ) n 1 ( f A ) z 2 /( h 1 n 1 A N ) z 2 /( h 1 1 ) n 1 ( f A N 0 s eff , ct c o c o s , c c , s c o c 0 s eff , ct c cr k_c k_c,ε≈ 0,3

(33)

Simplified solution for the cracking moment

and the normal force in the concrete slab

simplified solution for the normal force in the concrete slab:

primary effects due to shrinkage cracking moment M_cr h_c z_o z_i,st N_c+s Mcr M_c+s M_c+s,ε N_c+s,ε σ_c_ε σ_c c s ctm c cr A f k k k N ≈ ⋅ ⋅ 0 , 1 3 , 0 z 2 h 1 1 k o c c + ≤ + = shrinkage

k = 0,8 coefficient taking into account the effect of non-uniform self-equilibrating stresses k_s= 0,9 coefficient taking into account the slip

effects of shear connection

(34)

k = 0,8 Influence of non linear residual stresses due to shrinkage and temperature effects k_s = 0,9 flexibility of shear connection

k_c Influence of distribution of tensile stresses in concrete immediately prior to cracking

maximum bar diameter

d_s modified bar diameter for other concrete strength classes σ_s stress in reinforcement acc. to Table 1

c s s eff , ct c s k k k f A A σ ≥ 0,3 1,0 z h 1 1 k o c c = ₊ + ≤ M_cr M_c N_c Nc,ε M_c,ε cracking moment N_a,ε M_a,ε shrinkage h_c z_o o , ct eff , ct s s f f d d = ∗ ∗ s d f_cto= 2,9 N/mm2 z_i,o

(35)

stresses in reinforcement taking into account tension stiffening for the bending moment M_Ed of the quasi permanent combination: c s s A A = ρ st s eff , ct s , st 2 Ed s ts 2 , s s f z J M α ρ β + = σ σ Δ + σ = σ a a 2 2 st J A J A = α 4 , 0 = β

Control of cracking due to direct loading –

Verification by limiting bar spacing or bar diameter

N_s,2 -M_s,2 ΔN_ts -M_a,2 -N_a,2 -ΔN_ts ΔN_ts a z_st,a -z_st,s N_s -M_s -M_a -N_a

fully cracked tension stiffening

+

=

z

_st

-M_Ed

z_a

a

The bar diameter or the bar spacing has to be limited

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 s_rm≈ 2/3 s_r,maxresults β ≈ 2/3 ·0,6 = 0,4

A

_c

A

_s

(36)

Maximum bar diameters and maximum bar

spacing for high bond bars acc. to EC4

maximum bar diameter for σ_s [N/mm2_] w_k= 0,4 w_k= 0,3 w_k= 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 -50 100 360 -100 150 320 50 150 200 280 100 200 250 240 150 250 300 200 200 300 300 160 w_k= 0,2 w_k= 0,3 w_k= 0,4

maximum bar spacing in [mm] for σ_s [N/mm2_] ∗ s d

(37)

Direct calculation of crack width w for

composite sections based on EN 1992-2

z_st -z_st,s A_s σ_s, M_Ed st s eff , ct s , st st Ed s f z J M α ρ β + = σ a a st st st J A J A = α c s s A A = ρ β=0,4

)

(

s

w

=

_r_,_max

ε

_sm

−

ε

_cm N_s -M_s -M_a -N_a s s s o s s ctm s s cm sm E 6 , 0 ) n 1 ( E f E σ ≥ ρ + ρ β − σ = ε − ε s s max , r

d

34 ,

0 c

4 ,

3 s

ρ

+

=

crack width for high bond bars:

(38)

Stresses in reinforcement in case of bonded

tendons – initial crack formation

A_s, d_s A_p, d_p L_es L_ep τ_sm τ_pm Δσ_p σ_s=σ_s1+Δσ_s

Equilibrium at the crack:

) n 1 ( A f N A A_s _p _p _ct_,_eff _c _o _tot s + Δσ = = + ρ σ

Equilibrium in longitudinal direction:

s , e sm s s s A = π d τ L σ ep pm p p p A = π d τ L σ Δ

Compatibility at the crack:

ep p 1 p p es s 1 s s p s L E L E σ Δ − σ Δ = σ − σ ⇒ δ = δ s pm p 1 s 1 p p 1 s s d A A N A A N τ = ξ ξ + ξ = σ Δ ξ + = σ N σ_s,1 Δσ_p1 Δσ_s σ_p σ_p=σ_po+σ_p1+Δσ_p Stresses:

(39)

Stresses in reinforcement for final crack

formation

Maximum crack spacing:

p 1 p 2 p 2 , p s 1 s 2 s 2 s p s E ) ( E ) (σ − σ ₌ Δσ −β Δσ − Δσ β − σ = δ = δ

[

]

) A A ( 2 A f d s d n d n 2 s A f p 2 s sm c eff , ct s max , r p p pm s s sm max , r c ct ξ + τ = π τ + π τ =

Compatibility at the crack:

pm p p max , r 1 p 2 p sm s s max , r 1 s 2 s A U 2 s A U 2 s τ = σ − σ τ = σ − σ

Equilibrium in longitudinal direction: Equilibrium at the crack:

p 2 p s 2 s o A A P N− =σ +Δσ σ_s A_s, d_s A_p, d_p σ_s2 Δσ_p2 Δσ_p2 Δσ_p1 σ_s1 s_r,max σ_c σc=fct,eff x

(40)

Determination of stresses in composite

sections with bonded tendons

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ρ ξ − ρ − σ = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ξ + ξ − + − σ = σ Δ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ρ − ρ + σ = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − ξ + + σ = σ eff 2 1 tot eff , ct * s p 2 1 s c 2 1 p s c eff , ct * s p tot eff eff , ct * s p s c p 2 1 s c eff , ct * s s 1 f 4 , 0 A A A A A A f 4 , 0 1 1 f 4 , 0 A A A A A A f 4 , 0 c p 2 1 s eff c p s tot A A A A A A ξ + = ρ + = ρ Stresses σ* s in reinforcement at the crack location

neglecting different bond behaviour of reinforcement and tendons: st tot ctm s , st st Ed * s f z J M α ρ β + = σ a a st st st J A J A = α β=0,4 z_st -z_st,s A_s A_p

σ

_s,

σ

_p M_Ed

Stresses in reinforcement taking into account the different bond behaviour:

(41)

Part 4:

Deformations

(42)

Deflections

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, - influence of local yielding of

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

L₁ L₂ ΔM E_aJ₁ 0,15 L₁ _{0,15 L} 2 E_aJ₂ Sequence of construction

Effects of cracking of concrete

F

steel member

composite member

(43)

Deformations and pre-cambering

δ₁ δ₂

δ₃ δ₄

δ₁– self weight of the structure

δ₂– loads from finish and service work δ₃– creep and shrinkage

δ₄– variable loads and temperature effects δ_p δ_max δ_w combination limitation general quasi -permanent risk of damage of adjacent

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

quasi – permanent (better frequent) 250 / L max ≤ δ 500 / L w ≤ δ δ₁ δ_c

δ₁ deflection of the steel girder δ_c deflection of the composite

girder

Pre-cambering of the steel girder:

δ_p=δ₁+ δ₂+ δ₃ +ψ₂ δ₄

δ_max maximum deflection

δ_w effective deflection for finish and service work

(44)

Effects of local yielding on deflections

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 f

₂

and corresponding increases are

made to the bending moments in adjacent spans.

f

₂

= 0,5 if f

_y

is reached before the concrete slab has

hardened;

f

₂

= 0,7 if f

_y

is reached after concrete has hardened.

This applies for the determination of the maximum deflection but not

for pre-camber.

(45)

More accurate method for the determination of

the effects of local yielding on deflections

E

_a

J

₁

ΔM

E

_a

J

₂

L

₁

L

₂

l

_cr

l

_cr

E

_a

J

_eff

z

₂

M

_el,Rk

σ

_a

=f

_yk

f

_yk

-+

M

_pl,Rk

(EJ)

_eff

E

_a

J

₂

M

_el,Rk

M

_Ed

M

pl,Rk

E

_a

J

_eff

(46)

Effects of incomplete interaction on deformations

P s s P P

c

_D s_u P_Rd

The effects of incomplete interaction may be ignored provided that:

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

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 P_Rd and

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

(47)

Differential equations in case of incomplete

interaction

N_c N_a M_a M_c V_a N_c+dN_c N_a+dN_a M_a+ dM_a M_c+ dM_c V_a+dV_a dx a z_a(w) z_c x E_c, A_c, J_c E_a, A_a, J_a v_L v_L a w u u s_v = _a − _c + ′ V_c Vc+dVc a_a a_c Slip: c c c, u u ′ = ε a a a, u u ′ = ε q ) a w u u ( a c w ) J E J E ( 0 ) a w u u ( c u A E 0 ) a w u u ( c u A E c a s a a c c c a s c a a c a s c c c = ′′ + ′ − ′ − ′′′ ′ + = ′ + − − ′′ = ′ + − + ′′ c c c c

E

A

u

N

=

′

a a a a E A u N = ′ w J E M_c =− _c _c ′′ w J E M_a =− _a _a ′′ 0 w J E V_c =− _c _c ′′′ ≈ w J E V_a =− _a _a ′′′

(48)

Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ λ − λ λ α − λ α + = ) 2 cosh( 1 ) 2 cosh( 1 5 384 1 5 48 1 J E L q 384 5 w 4 2 o ,i a 4 q 1 J J J 1 o , c a o ,i ₋ + = α ² L c A A A E s o ,i a o , c a = β β α α + = λ2 1 F L ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ λ λ λ α − λ α + = ) sinh( ) 2 ( sinh 48 12 1 I E 48 L F w 2 3 2 o ,i a 3 L

Deflection in case of incomplete interaction for

single span beams

A_io, J_io composite section concrete section steel section A_a, J_a A_co=A_c/n_o, J_co= J_c/n_o n_o=E_a/E_c w

(49)

Mean values of stiffness of headed studs

P s s P P

c

_D s_u P_Rd e_L nt=2 Rd u D P s C =

spring constant per stud: spring constant of the shear

connection: _L t D s e n C c =

type of shear connection 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 ] cm / kN [ C_D

c

_s

(50)

Simplified solution for the calculation of

deflections in case of incomplete interaction

( )

ξ = q sinπξ q

L

L x = ξ z_a z_c E_c, A_c, J_c E_a, A_a, J_a a N_a M_a M_c N_c 2 a eff , c a eff , c a o , c eff , io

a

A

J

+

=

The influence of the flexibility of the shear connection is taken into account by a reduced value for the modular ratio. eff , io a 4 4 2 c cm o a a a a c cm o a a c cm 4 4 o J E 1 L q a A E A E A E A E J E J E 1 L q w π = β + β + + π =

)

1 (

n

_o_,_eff

=

_o

+

β

_s s 2 c cm 2 s

c

L

A

E

π

=

β

eff , o c eff , c

n

A

=

effective modular ratio for the concrete slab

ε

_a

ε

_c

(51)

Comparison of the exact method with the

simplified method

w/w_c L [m] 1,1 1,0 1,2 1,3 1,4 1,5 5,0 10,0 15,0 20,0 w/w_c L [m] 1,1 1,0 1,15 1,2 1,25 5,0 10,0 15,0 20,0 c_D = 2000 KN/cm q L w b_eff 450 mm E_cm = 3350 KN/cm² 51 99 exact solution

simplified solution with n_o,eff

1,05 η=0,8 η=0,4 η=0,8 η=0,4 c_D = 1000 KN/cm w_o- deflection in case of

neglecting effects from slip of shear connection

(52)

Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany 1875 1875 1875 3750 1875 7500 F F F/2 F/2 load case 2 load case 1 δ F [kN] 50 100 150 200 0 Deflection at midspan 1500 445 270 175 50 IPE 270 load case 2 load case 1

Deflection in case of incomplete

interaction-comparison with test results

(53)

Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany 780 50 125 Load case 1 F= 60 kN Load case 2 F=145 kN second moment of area cm4 Deflection at midspan in mm Test - 11,0 (100%) 20,0 (100 %)

Theoretical value, neglecting flexibility of shear connection

J_io= 32.387,0 7,8 (71%) 12,9 (65%) Theoretical value, taking into account

flexibility of shear connection

J_io,eff= 21.486,0 11,7 (106%) 19,4 (97%)

F

s

s[mm]

10 20 30 40 40 80 120 160

_{push-out test}

Deflection in case of incomplete

interaction-Comparison with test results

(54)

Part 5:

Limitation of stresses

(55)

Limitation of Stresses

σ

_c _M Ed

σ

_a + -M_Ed

σ

_a

σ

_s + -+

-combination stress limit recommended values k_i structural steel characteristic σ_Ed ≤ k_a f_yk

σ_Ed ≤ k_s f_sk σ_Ed ≤ k_c f_ck P_Ed ≤ k_s P_Rd k_a= 1,00 reinforcement characteristic k_s= 0,80 concrete characteristic k_c= 0,60

headed studs characteristic k_s= 0,75

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

(56)

composite section steel section

g_c M_Ed(x) V_Ed M_Ed V_Ed(x) + -+ b_c b_eff y z z_io x N_c vL,Ed,max L_v=b_eff

Concentrated longitudinal shear force at sudden change of cross-section

eff io c a io eff , c Ed max , Ed , L b J E / E z A M 2 v = A_c,eff

longitudinal shear forces

+

-Local effects of concentrated longitudinal

shear forces

(57)

Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany z y b_c=10 m 300 500x20 14x2000 800x60 δ P C_D = 3000 kN/cm per stud L = 40 m g_c,d cross-section FE-Model system shear connectors

Local effects of concentrated longitudinal

shear forces

(58)

Ultimate limit state - longitudinal shear forces

FE-Model: FE-Model L = 40 m x s P c_D s P c_D EN 1994-2 x [cm]

ULS

(59)

Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany 200 400 600 800 1000 1200 1400 1600 1800 2000 -4000 -3500 -3000 -2500 -2000 -1500 -1000 -500 0 500

Serviceability limit state - longitudinal

shear forces

EN 1994-2 FE-Model: FE-Model L = 40 m v_L,Ed[kN/m] x [cm] x s P c_D s P c_D

SLS

(60)

Part 6:

Vibrations

(61)

Vibration- General

EN 1994-1-1:

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

(62)

Vibration - General

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

.

(63)

Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany x_k l_s Span le_ngth x l_s time t F(x,t) F(x,t) t_s x_k t_k

Vibration – Example vertical vibration due

to walking persons

pacing rate f_s [Hz] forward speed v_s = f_s l_s [m/s] stride length l_s [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 The pacing rate f_s dominates the dynamic effects and the resulting dynamic loads. The speed of pedestrian propagation v_s is a

function of the pacing rate f_s and the stride length l_s.

(64)

Vibration –vertical vibrations due to

walking of one person

time t F_i(t) left foot right foot F(t) both feet

1. step 2. step 3. step

(

)

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ Φ − π α + =

∑

= 3 1 n n s n o 1 sin 2 n f t G ) t ( F

G_o weight of the person (800 N)

α_n coefficient for the load component of n-th harmonic n number of the n-th harmonic

f_s pacing rate

Φ_n phase angle oh the n-th harmonic

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.

α₁=0,4-0,5 Φ₁=0 α₂=0,1-0,25 Φ₂=π/2 α =0,1-0,15 Φ =π/2 Fourier-coefficients and phase angles:

(65)

Vibration – vertical vibrations due to walking

of persons

(

fE L/vs

)

gen n a max

1 e

M

F

k

a

−

− δ

δ

π

=

M_gen F(t) c δ w(t)

maximum acceleration a, vertical deflection w and maximum velocity v acceleration E max 2 E max f 2 a v ) f 2 ( a w π = π = f_E natural frequency

F_n load component of n-th harmonic δ logarithmic damping decrement v_s forward speed of the person

k_a factor taking into account the different

positions x_k during walking along the beam M_gen generated mass of the system

(single span beam: M_gen=0,5 m L) m L F_n(t) x_k w(x_k,t) L/2 k_a F_n(t) w(t)

(

f t

)

E gen n a sin (2 f t) 1 e E M F k ) t ( w π − −δ δ π = s v L t =

(66)

Logarithmic damping decrement

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:

δ=0,10 floor beams without not load-bearing inner walls

δ=0,15 floor beams with not load-1 2 3 4 5 6

Damping

ratioξ [%]

3 6 ₉ ₁₂ f [Hz]

ξ

π

=

δ 2

results of measurements in buildings with finishes

(67)

Vibration –vertical vibrations due to walking of

persons

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 normally should exceed 7,5 Hz if the first, second and third harmonic of the dynamic load-time

function can cause significant acceleration. Otherwise the maximum acceleration or

velocity should be determined and limited to acceptable values in accordance with

ISO 10137 F(t)/G_o

(

)

∑

= Φ − π + = 3 1 n n s n o F sin 2 n f t G ) t ( F 0,4 0,2 2,0 4,0 6,0 8,0 0,1 f_s=1,5-2,5 Hz 2f_s=3,0-5,0 Hz 3f_s=4,5-7,5 Hz

(68)

Limitation of acceleration-recommended

values acc. to ISO 10137

1 5 10 50 100 0,01 0,05 0,1 0,005 acceleration [m/s2_] basic curve a_o

Multiplying factors K_a for the basic curve Residential (flats, hospitals) K_a=1,0

Eurocode 4

Serviceability limit states of

composite beams

Eurocode 4

Eurocodes

Background and Applications

Contents

Part 1:

Introduction

Part 2:

Global analysis for serviceability limit states

Part 3:

Crack width control

Part 4:

Deformations

Part 5:

Limitation of stresses

Serviceability limit states

Serviceability limit states

Limitation of stresses

Limitation of deflections

crack width control

vibrations

Serviceability limit states

{

}

{

}

{

}

C

=

Part 2:

Global analysis for serviceability limit states

Global analysis - General

Calculation of internal forces, deformations and stresses at

serviceability limit state shall take into account the following

effects:



shear lag;



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;



inelastic behaviour of steel and reinforcement, if any;

Shear lag- effective width

b

b

Effective width of concrete flanges

z

-z

a

Effects of creep of concrete

primary effects

Effects of creep and shrinkage of concrete

Types of loading and action effects:

M

(t)

ϕ(t,t

)

Modular ratios taking into account

effects of creep

[

]

E

E

n

)

t

,

t

(

1

n