Eurocode 4

70
1 G. Hanswille 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

Transcript of Eurocode 4

Page 1: Eurocode 4

1

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Serviceability limit states of composite beams

Eurocode 4

EurocodesBackground and Applications

Dissemination of information for training18-20 February 2008, Brussels

Institute for Steel and Composite StructuresUniversity of Wuppertal

Germany

Univ. - Prof. Dr.-Ing. Gerhard Hanswille

Page 2: Eurocode 4

2

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

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

Part 6: Vibrations

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Serviceability limit states

Serviceability limit states

Limitation of stresses

Limitation of deflections

crack width control

vibrations

web breathing

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Serviceability limit states

{ }∑ ∑ ψ+++= i,ki,01,kkj,kd QQPGEEcharacteristic combination:

frequent combination: { }∑ ∑ ψ+ψ++= i,ki,21,k1,1kj,kd QQPGEE

quasi-permanent combination: { }∑ ∑ ψ++= i,ki,2kj,kd QPGEE

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

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Part 2:

Global analysis for serviceability limit states

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Global analysis - General

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

shear lag;

creep and shrinkage of concrete;

cracking of concrete and tension stiffening of concrete;

sequence of construction;

increased flexibility resulting from significant incompleteinteraction due to slip of shear connection;

inelastic behaviour of steel and reinforcement, if any;

torsional and distorsional warping, if any.

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Shear lag- effective width

σmax

σmax

b

be

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

2,0bb

i

ei ≥

σmax

bei

bi

5 bei

y

bi

y

σmax

bei

σ(y)

σ(y)

2,0bb

i

ei <

σR

[ ]4

iRmaxR

maxi

eiR

by1)y(

2,0bb25,1

⎥⎦

⎤⎢⎣

⎡−σ−σ+σ=σ

σ⎥⎦

⎤⎢⎣

⎡−=σ

4

imax b

y1)y( ⎥⎦

⎤⎢⎣

⎡−σ=σ

shear lag

real stress distribution

stresses taking into account the effective width

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

midspan regions and internal supports:

beff = b0 + be,1+be,2

be,i= Le/8

Le – equivalent lengthend supports: beff = b0 + β1 be,1+β2 be,2

βi = (0,55+0,025 Le/bi) ≤ 1,0

Effective width of concrete flanges

Le=0,85 L1 for beff,1 Le=0,70 L2 for beff,1

Le=2L3 for beff,2

L1 L2L3

beff,0

beff,1beff,1beff,2

beff,2

L1/4 L1/2 L1/4 L2/2L2/4 L2/4

Le=0,25 (L1 + L2) for beff,2 bobe,1 be,2

bob1 b2

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Initial sectional forces

redistribution of the sectional forces due to creep

-Nc,o

Mc,o

Mst,o

Nst,o

Nc,r

-Mc,r

Mst,r

-Nst,r

zi,st

-zi,cML

ast

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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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

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 timeL=PT time-dependent action effects developing affine to the creep coefficientL=S action effects caused by shrinkage of concreteL=D action effects due to prestressing by imposed deformations (e.g. jacking of

supports)

MPT(t)MPT (t=∞)

ϕ(t∞,to)ϕ(ti,to)ϕ(t,to)

time dependent action effects ML=MPT:

action effects caused by prestressing due to imposed

deformation ML=MD:

δ

+ML=MD

MD

MPT(ti)

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Modular ratios taking into account effects of creep

[ ]cm

aooLoL E

En)t,t(1nn =ϕψ+=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)

-zic,L

zist,Lzi,L

zczis,Last

zst

action creep multipliershort term loading Ψ=0

permanent action not changing in time ΨP=1,10shrinkage ΨS=0,55prestressing by controlled imposed deformations ΨD=1,50time-dependent action effects ΨPT=0,55

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

)t(EEn

ocm

sto =

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

concrete section

L,i2stL,cstL,cstL,i A/aAAJJJ ++=L,cStL,i AAA +=

L,iststL,ic A/aAz −=LcL,cLcL,c n/JJn/AA ==

-zic,L

zist,Lzi,L

zc-zis,L

ast

zst

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(nn 0L0L ϕψ+=

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

centroidal axis of the composite section

centroidal axis of the steel section

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G. HanswilleUniv.-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

ε

εs(x)

εc(x)

Ns Ns

c

ctEf

s

2s2,s E

σ=ε

r,sεΔβ

εsr,1 εsr,2 εsm,y εsy

Ns

Nsy

Nsm

Ns,cr

B C

σs,2σs(x)

σc(x)

τv

x

σc(x)

stage A: uncracked sectionstage B: initial crack formationstage C: stabilised crack formation

fully cracked section

A

σc(x)

σs(x)

mean strain εsm=εs,2- βΔεs,r

r,sεΔ

εsm

r,ss εΔβ=εΔ

ss

eff,cts E

β=εΔ

css A/A=ρ

4,0=β

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

εsm

MMs≈0

Ma

Na

εa

a

zs

Ns equilibrium:

aNMM sa −=

sa NN −=

εs,m

εs,2Δεs=β Δεs,r

εc

εs

compatibility:

aasm κ+ε=ε

aaaa

2s

aa

ssm JE

aMAEaN

AEN

=++ε

ss

eff,ct

ss

ssr2ssm E

fAE

β−=εΔβ−ε=ε

mean strain in the concrete slab:

mean strain in the concrete slab:

za

Influence of tension stiffening of concrete on stresses in reinforcement

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Ns,2

-Ms,2

ΔNts

-Ma,2

-Na,2 -ΔNts

ΔNts aa

Ns,2

-MEd

MEd

Ns

Nsε

zst,a

-zst,s

Ns

M

tsst

s,stsEdts2ss N

JzA

MNNN Δ+=Δ+=sts

seff,ctts

AfN

αρβ=Δ

tsst

a,staEdts2aa N

JzA

MNNN Δ−=Δ−=

aNJJMaNMM ts

st

aEdts2aa Δ+=Δ+=

Sectional forces:

st

sEds J

JMM =

aa

ststst JA

JA=α

Ns

-Ms

-Ma

-Na

fully cracked section tension stiffening

+ =z2=zst

ΔNts

Redistribution of sectional forces due to tension stiffening

2st JJ =

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Stresses taking into account tension stiffening of concrete

Ns,2

-Ms,2

ΔNts

-Ma,2

-Na,2 -ΔNts

ΔNts azst,a

-zst,s

sts

sctmts

AfNαρ

β=Δ

aa

ststst JA

JA=α

Ns

-Ms

-Ma

-Na

fully cracked tension stiffening

+ =zst-MEd

sts

ctms,st

st

Eds

sts

ctm2,ss

fzJ

M

f

αρβ+=σ

αρβ+σ=σ

aa

ts

a

tsst

st

Eda

aa

ts

a

ts2,aa

zJ

aNANz

JM

zJ

aNAN

Δ+

Δ−=σ

Δ+

Δ−σ=σ

reinforcement: structural steel:

za

a

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Influence of tension stiffening on flexural stiffness

EstJ1 uncracked sectionEstJ2 fully cracked sectionEstJ2,ts effective flexural

stiffness taking into account tension stiffening of concrete

EstJ1

EstJ2EstJ2,ts

M

κ

κ

εsm

-M

Ns

-Ms

-Ma

-Na

εa

azst

ast

s

ast

a

ts,2st JEaNM

JEM

IEM −

===κ

EJ

MR MRn

Est J1

Est J2,ts

EstJ2

Ma)NN(

1

JEJE,ss

aats,2st

ε−−

=

M

Curvature:

Effective flexural stiffness:

Page 18: Eurocode 4

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

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

• 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.

L1 L2L1,cr L2,cr

EaJ2EaJ1 EaJ1

ΔM

un-cracked analysiscracked analysis

ΔM Redistribution of bending moments due to cracking of concrete

EaJ1 – un-cracked flexural stiffness

EaJ2 – cracked flexural stiffness

Effects of cracking of concrete - General method according to EN 1994-1-1

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

L1 L2

ΔMII

EaJ1

0,15 L1 0,15 L2

EaJ2

6,0L/L maxmin ≥

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 Ea J2 over 15% of the span on each side of each internal support, and as the un-cracked values Ea J1 elsewhere.

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Part 3:

Limitation of crack width

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

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 σs in the reinforcement and the design crack width.

Page 22: Eurocode 4

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Recommended values for wmax

reinforced members, prestressedmembers with unbonded tendons

and members prestressed by controlled imposed deformations

prestressed members with bonded tendons

quasi - permanentload 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 ondurability 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 bechecked under the quasi-permanent combination of loads.

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

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

Class Description of environment Examplesno 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

XS3 tidal, splash and spray zones parts of marine structures

Page 24: Eurocode 4

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Cracking of concrete (initial crack formation)

ε εs

εc

LesLes

NsNs

w

c

ss A

A=ρ

As cross-section area of reinforcementρs reinforcement ratiofctm mean value of tensile strength of concrete

c

so E

En =

c1,cs1,sss AAA σ+σ=σ

Compatibility at the end of the introduction length:

Equilibrium in longitudinal direction:

c

1,c

s

1,s1,c1,s EE

σ=

σ⇒ε=ε

⎥⎦

⎤⎢⎣

⎡ρ+

ρσ=σ

osos

s1,s n1n

os

s1,sss n1 ρ+

σ=σ−σ=σΔ

Change of stresses in reinforcement due to cracking:

LesLes

σsσs,1

σc,1

Δσs

σc,1

σs,1

σs,2

Les

σ

( )oscctmr,s n1AfN ρ+=

Page 25: Eurocode 4

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Cracking of concrete – introduction length

ε εs

εc

LesLes

NsNs

w

c

ss A

A=ρ

Us -perimeter of the barAs -cross-section areaρs -reinforcement ratioτsm -mean bond strength

c

so E

En =

4ddL

AUL2s

ssmses

sssmsesπ

σΔ=τπ

σΔ=τ

oss

1,sss n1 ρ+σ

=σ−σ=σΔ

Change of stresses in reinforcement due to cracking:

Equilibrium in longitudinal direction

LesLes

σsσs,1

σc,1

Δσs

σc,1

σs,1

σs,2

Les

τsm

σ

sosm

sses n1

14

dLρ+τ

σ=

introduction length LEs

crack width

)(L2w cmsmes ε−ε=

Page 26: Eurocode 4

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Determination of the mean strains of reinforcement and concrete in the stage of initial crack formation

ctmL

os

esm,s f8,1dx)x(

L1 Es

≈τ=τ ∫

Mean bond strength:

s

smsssm,s σΔ

σΔ−σ=β⇒σΔβ−σ=σ

∫ τ=σΔx

0s

ss dx)x(

U4)x(∫ σΔ=σΔ

esL

0s

essm dx)x(

L1

εεs

εc(x)

LesLes

NsNs

w

LesLes

σs σs,1

σc,1

Δσs

σ

εcr

Δεs,cr

Mean strains in reinforcement and concrete:

crm,c εβ=ε

Mean stress in the reinforcement:εs,m

εc,m

βΔσs

x

σs,m

σs(x)

εs(x)

cr,s2,sm,s εΔβ−ε=ε

Page 27: Eurocode 4

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Determination of initial crack width

ε εs(x)

εc(x)

LesLes

Ns

w

LesLes

σsσs,1

σc,1

Δσs

σs

εcr

Δεs,crεs,m

εc,m

βΔσs

x

σs,m

Ns crack width

)(L2w cmsmes ε−ε=

sosm

sses n1

14

dLρ+τ

σ=

2,scmm,s )1( εβ−=ε−ε

εs,2

sossms

2s

n11

E2d)1(w

ρ+τσβ−

=

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

ctmsm f8,1≈τ

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Maximum bar diameters acc. to EC4

maximum bar diameter forσs

[N/mm2] 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 -

∗sd

sm,ct

s2s

sossm

s2s

Ef6d

n11

E2d)1(w σ

≈ρ+τ

σβ−=

Crack width w:

Maximum bar diameter for a required crack width w:

)1()n1(E2wd 2

s

sossms

β−σ

ρ+τ=

2s

so,ctmk*s

2s

soso,ctmk

*s

Efw6d

)1(

)n1(Ef6,3wd

σ≈

β−σ

ρ+=

With τsm= 1,8 fct,mo and the reference value for the mean tensile strength of concrete fctm,o= 2,9 N/mm2 follows:

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

Page 29: Eurocode 4

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Crack width for stabilised crack formation

ε

εs(x)

εc(x)

Ns

w

sr,max= 2 Les

c

ctEf

s

2s2,s E

σ=ε

εs(x)- εc(x)

sr,min= Les

)(sw cmsmmax,r ε−ε=

Crack width for high bond bars

cctm

cm

ssctm

2,sss

ctmc2,sm,s

s2,sm,s

Ef

Ef

AEfA

β=ε

ρβ−ε=β−ε=ε

εΔβ−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

)n1(Ef

E soss

ctm

s

scmsm ρ+

ρβ−

σ=ε−ε

Page 30: Eurocode 4

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Crack width for stabilised crack formation

ε

εs(x)

εc(x)

Ns

w

sr,max= 2 Les

c

ctEf

ss

2,s Eσ

εs(x)- εc(x)

sr,min= Les

sms

sctm

smscctm

es 4df

UAfL

τρ=

τ=

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

)(sw cmsmmax,r ε−ε=

⎟⎟⎠

⎞⎜⎜⎝

⎛ρ+

ρβ−

σρτ

= )n1(E

fE2

dfw soss

ctm

s

s

ssm

sctm

maximum crack width for sr= sr,max

β= 0,6 for short term loading

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

Page 31: Eurocode 4

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Crack width and crack spacing according Eurocode 2

)(sw cmsmmax,r ε−ε=Crack width

s

s21max,r

d425,0kkc4,3sρ

⋅⋅+= ds-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)

ss

soss

ctmss

cmsm E6,0)n1(

Ef

≥ρ+ρ

β−σ

=ε−ε

Crack spacing

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

Page 32: Eurocode 4

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

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

cracking moment Mcr:

[ ]

[ ])z2/(h1(z

JnfM

2/hzJn

fM

oco,ic

ioo,ceff,ctcr

co

ioo,ceff,ctcr

+σ−=

+σ−=

ε

ε

( )ε+

ε

ε+

++

ρ+σ−=

++

=

,scoc

os,ceff,ctccr

,scio

issococrcr

N)z2/(h1

n1)f(AN

NJ

zAzAMNprimary effects due to shrinkage

cracking moment Mcr

hc

zio

zo

zi,st

Nc+s

Mc,ε

Mcr

Nc,ε

σcε

ast

ctm1eff,ct,cc fkfMc+s σc

==σ+σ ε

sectional normal force of the concrete slab:

( )

⎥⎥⎥⎥

⎢⎢⎢⎢

ρ++

ρ+σ−

++

ρ+=

εε+

)n1(fA)z2/(h1n1A

N

)z2/(h11)n1(fAN

0seff,ctc

oc

os,cc,sc

oc0seff,ctccr

kc

kc,ε≈ 0,3

Page 33: Eurocode 4

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G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

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 Mcr

hc zo

zi,st

Nc+sMcr

Mc+s

Mc+s,ε

Nc+s,ε

σcε

σc

csctmccr kkkfAN ⋅⋅≈

0,13,0

z2h1

1k

o

cc ≤+

+=

shrinkage

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

ks= 0,9 coefficient taking into account the slip effects of shear connection

cracking moment

Page 34: Eurocode 4

34

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

k = 0,8 Influence of non linear residual stresses due to shrinkage and temperature effects ks = 0,9 flexibility of shear connectionkc Influence of distribution of tensile stresses in concrete immediately prior to

crackingmaximum bar diameter

ds modified bar diameter for other concrete strength classes σs stress in reinforcement acc. to Table 1fct,eff effective concrete tensile strength

css

eff,ctcs kkk

fAA

σ≥ 0,13,0

zh11k

occ ≤+

+=

Mcr

Mc

NcNc,ε

Mc,ε

cracking moment

Na,ε

Ma,ε

shrinkage

hc zo

o,ct

eff,ctss f

fdd ∗=

∗sd

fcto= 2,9 N/mm2

zi,o

Determination of minimum reinforcement

Page 35: Eurocode 4

35

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

stresses in reinforcement taking into account tension stiffening for the bending moment MEd of the quasi permanent combination:

c

ss A

A=ρ

sts

eff,cts,st

2

Eds

ts2,ss

fz

JM

αρβ+=σ

σΔ+σ=σ

aa

22st JA

JA=α4,0=β

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

Ns,2

-Ms,2

ΔNts

-Ma,2

-Na,2 -ΔNts

ΔNts azst,a

-zst,s

Ns

-Ms

-Ma

-Na

fully cracked tension stiffening

+ =zst

-MEd

za

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 βresults from the mean value of crack spacing. With srm≈ 2/3 sr,max results β ≈ 2/3 ·0,6 = 0,4

Ac

As

Aa

Page 36: Eurocode 4

36

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

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

maximum bar diameter forσs

[N/mm2] 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 -

-50100360

-100150320

50150200280

100200250240

150250300200

200300300160

wk= 0,2wk= 0,3wk= 0,4

maximum bar spacing in [mm] for

σs

[N/mm2]

∗sd

Table 1: Maximum bar diameter Table 2: Maximum bar spacing

Page 37: Eurocode 4

37

G. HanswilleUniv.-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

zst

-zst,s

As σs,

MEd

sts

eff,cts,st

stEd

sf

zJ

Mαρ

β+=σ

aa

ststst JA

JA=α

c

ss A

A=ρ 4,0=β

)(sw cmsmmax,r ε−ε=

Ns

-Ms

-Ma

-Na

ss

soss

ctmss

cmsm E6,0)n1(

Ef

≥ρ+ρ

β−σ

=ε−ε

ss

max,rd34,0c4,3sρ

+=

crack width for high bond bars:

c - concrete cover of reinforcement

Page 38: Eurocode 4

38

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Stresses in reinforcement in case of bonded tendons – initial crack formation

As, ds

Ap, dp

Les

Lep

τsmτpm

Δσp

σs=σs1+Δσs

Equilibrium at the crack:

)n1(AfNAA totoceff,ctppss ρ+==σΔ+σEquilibrium in longitudinal direction:

s,esmsss LdA τπ=σ

eppmppp LdA τπ=σΔ

Compatibility at the crack:

epp

1ppes

s

1ssps L

EL

EσΔ−σΔ

=σ−σ

⇒δ=δ

v

s

sm

pm1

p1s

1p

p1ss

dd

AAN

AAN

τ

τ=ξ

ξ+

ξ=σΔ

ξ+=σ

N

σs,1

Δσp1

Δσs

σp

σp=σpo+σp1+Δσp

Stresses:

With Es≈Ep and σs1=Δσp1=0 results:

Page 39: Eurocode 4

39

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Stresses in reinforcement for final crack formation

Maximum crack spacing:

p

1p2p2,p

s

1s2s2sps E

)(E

)( σΔ−σΔβ−σΔ=

σ−σβ−σ=δ=δ

[ ]

)AA(2Afd

s

dndn2

sAf

p2

ssm

ceff,ctsmax,r

pppmsssmmax,r

cct

ξ+τ=

πτ+πτ=

Compatibility at the crack:

pmp

pmax,r1p2psm

s

smax,r1s2s A

U2

sAU

2s

τ=σ−στ=σ−σ

Equilibrium in longitudinal direction:

Equilibrium at the crack:

p2ps2so AAPN σΔ+σ=−

σs

As, ds

Ap, dp

σs2

Δσp2

Δσp2

Δσp1

σs1

sr,max

σcσc=fct,eff

x

mean crack spacing: sr,m≈2/3 sr,max

Page 40: Eurocode 4

40

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Determination of stresses in composite sections with bonded tendons

⎥⎥⎦

⎢⎢⎣

ρξ

−ρ

−σ=⎥⎥⎦

⎢⎢⎣

ξ+

ξ−

+−σ=σΔ

⎥⎦

⎤⎢⎣

⎡ρ

−ρ

+σ=⎥⎥⎦

⎢⎢⎣

+−

ξ++σ=σ

eff

21

toteff,ct

*s

p21s

c21

ps

ceff,ct

*sp

toteffeff,ct

*s

ps

c

p21s

ceff,ct

*ss

1f4,0AA

AAA

Af4,0

11f4,0AA

AAA

Af4,0

c

p21s

eff

c

pstot

AAA

AAA

ξ+=ρ

+=ρ

Stresses σ*s in reinforcement

at the crack location neglecting different bond behaviour of reinforcement and tendons:

sttot

ctms,st

st

Ed*s

fzJ

Mαρ

β+=σ

aa

ststst JA

JA=α 4,0=β

zst

-zst,s

AsAp σs,σp

MEd

Stresses in reinforcement taking into account the different bond behaviour:

Page 41: Eurocode 4

41

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Part 4:

Deformations

Page 42: Eurocode 4

42

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

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

ΔMEaJ1

0,15 L1 0,15 L2

EaJ2

Sequence of construction

Effects of cracking of concrete

F

F

steel member

composite member

gc

Page 43: Eurocode 4

43

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Deformations and pre-cambering

δ1δ2

δ3

δ4

δ1 – self weight of the structureδ2 – loads from finish and service workδ3 – creep and shrinkageδ4 – variable loads and temperature effects

δp

δmax

δw

combination limitationgeneral quasi -

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

quasi –permanent(better frequent)

250/Lmax ≤δ

500/Lw ≤δ

δ1

δc

δ1 deflection of the steel girder

δc deflection of the composite girder

Pre-cambering of the steel girder:

δp = δ1+ δ2+ δ3 +ψ2 δ4

δmax maximum deflection

δw effective deflection for finish and service work

Page 44: Eurocode 4

44

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

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 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.

Page 45: Eurocode 4

45

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

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

EaJ1

ΔM

EaJ2

L1 L2lcr lcr

EaJeff

z2 Mel,Rk

σa=fyk fyk

--

+

Mpl,Rk

(EJ)eff

EaJ2

Mel,Rk Mpl,RkMEd

EaJeff

Page 46: Eurocode 4

46

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Effects of incomplete interaction on deformations

P

ss

PP

cD

su

PRd

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 PRd and

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

Page 47: Eurocode 4

47

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Differential equations in case of incomplete interaction

Nc

Na

Ma

Mc

Va

Nc+dNc

Na+dNa

Ma+ dMa

Mc+ dMc

Va+dVadx

a

za (w)

zc

x

Ec, Ac, Jc

Ea, Aa, Ja

vL

vL

awuus cav ′+−=

VcVc+dVc

aa

ac

Slip:

ccc u,u ε=′

aaa u,u ε=′

q)awuu(acw)JEJE(0)awuu(cuAE0)awuu(cuAE

casaacc

cascaa

casccc

=′′+′−′−′′′′+

=′+−−′′=′+−+′′

cccc uAEN ′=

aaaa uAEN ′=

wJEM ccc ′′−=

wJEM aaa ′′−=

0wJEV ccc ≈′′′−=

wJEV aaa ′′′−=

Page 48: Eurocode 4

48

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

⎥⎥⎥

⎢⎢⎢

λ

−λ

λα−

λα+=

)2

cosh(

1)2

cosh(15

38415

481JELq

3845w 42o,ia

4

q

1JJ

J1

o,ca

o,i −+

²LcAAAE

so,i

ao,ca=β

βαα+

=λ12

F

L

⎥⎥⎥

⎢⎢⎢

λ

λ

λα−

λα+=

)sinh(

)2

(sinh48121IE48

LFw2

32o,ia

3

L

Deflection in case of incomplete interaction for single span beams

Aio, Jio

composite sectionconcrete section

steel section Aa, Ja

Aco=Ac/no, Jco= Jc/no

no=Ea/Ec

w

Page 49: Eurocode 4

49

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Mean values of stiffness of headed studs

P

ss

PP

cD

su

PRd

eLnt=2

Rd

uD P

sC =spring constant per stud:

spring constant of the shearconnection: L

tDs e

nCc =

type of shear connection

headed stud ∅ 19mmin solid slabs

2500

headed stud ∅ 22mmin solid slabs

3000

headed studs ∅ 25mmin solid slab

3500

headed stud ∅ 19mmwith Holorib-sheeting and one stud per rib

1250

headed stud ∅ 22mmwith Holorib-sheeting and one stud per rib

1500

]cm/kN[CD

cs

Page 50: Eurocode 4

50

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Simplified solution for the calculation of deflections in case of incomplete interaction

( ) πξ=ξ sinqq

L

Lx

za

zc

Ec, Ac, Jc

Ea, Aa, Ja

a

Na

Ma

McNc

2

aeff,c

aeff,cao,ceff,io a

AAAA

JJJ+

++=

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

eff,ioa4

4

2

ccmoaa

aaccmoaaccm

4

4

o JE1Lq

aAEAE

AEAEJEJE

1Lqwπ

=

β+

β++π

=

)1(nn soeff,o β+=s

2ccm

2

s cLAEπ

eff,o

ceff,c n

AA =

effective modular ratio for the concrete slab

εa

εc

Page 51: Eurocode 4

51

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Comparison of the exact method with the simplified method

w/wc

L [m]1,1

1,0

1,2

1,3

1,4

1,5

5,0 10,0 15,0 20,0

w/wc

L [m]

1,1

1,0

1,15

1,2

1,25

5,0 10,0 15,0 20,0

cD = 2000 KN/cm

q

L

w

beff

450 mm

Ecm = 3350 KN/cm²

5199

exact solutionsimplified solution with no,eff

1,05 η=0,8

η=0,4

η=0,8

η=0,4

cD = 1000 KN/cm

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

η degree of shear connection

Page 52: Eurocode 4

52

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

1875 1875

1875 18753750

7500

F

F

F/2 F/2

load case 2

load case 1

δ[mm]

F [kN]

20 40 60

50

100

150

200

0

Deflection at midspan

150044

5

270

175

50

IPE 270

load case 2

load case 1

Deflection in case of incomplete interaction-comparison with test results

Page 53: Eurocode 4

53

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

780

50125Load case 1

F= 60 kNLoad case 2F=145 kN

second moment of area

cm4Deflection at midspan in mm

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%)

F 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

Page 54: Eurocode 4

54

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Part 5:

Limitation of stresses

Page 55: Eurocode 4

55

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Limitation of Stresses

σc MEd

σa

+

-

MEd

σa

σs

+

-

+

-

combination stress limit recommended values ki

structural steel characteristic σEd ≤ ka fyk

σEd ≤ ks fsk

σEd ≤ kc fck

PEd ≤ ks PRd

ka = 1,00

reinforcement characteristic ks = 0,80

concrete characteristic kc= 0,60

headed studs characteristic ks = 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 deformations is provided.

Page 56: Eurocode 4

56

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

composite section steel sectiongc

MEd(x)

VEd

MEd

VEd(x)

+

-+

bc

beff

y

z

ziox

Nc

vL,Ed,max

Lv=beff

Concentrated longitudinal shear force at sudden change of cross-section

effioca

ioeff,cEdmax,Ed,L bJE/E

zAM2v =

Ac,eff

longitudinal shear forces

+-

Local effects of concentrated longitudinal shear forces

Page 57: Eurocode 4

57

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

z

y

bc=10 m

300

500x2014x2000

800x60

δ

P

CD = 3000 kN/cmper stud

L = 40 m

gc,d

cross-section

FE-Model

system

shear connectors

Local effects of concentrated longitudinal shear forces

Page 58: Eurocode 4

58

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Ultimate limit state - longitudinal shear forces

FE-Model:

FE-Model

L = 40 m

x

s

P

cD

s

P

cD

EN 1994-2

x [cm]

ULS

Page 59: Eurocode 4

59

G. HanswilleUniv.-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

vL,Ed[kN/m]

x [cm]

x

s

P

cD

s

P

cD

SLS

Page 60: Eurocode 4

60

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Part 6:

Vibrations

Page 61: Eurocode 4

61

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

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

Page 62: Eurocode 4

62

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

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.

Note in EN 1990-A.1.4.4: Further information is given in ISO 10137.

Page 63: Eurocode 4

63

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

xk

ls

Span length

xls

time tF(x,t)

F(x,t)

tsxk

tk

Vibration – Example vertical vibration due to walking persons

pacing rate

fs [Hz]

forward speedvs = fs ls[m/s]

stride length

ls[m]

slow walk ∼1,7 1,1 0,6normal walk ∼2,0 1,5 0,75fast walk ∼2,3 2,2 1,00slow running (jog)

∼2,5 3,3 1,30

fast running (sprint)

> 3,2 5,5 1,75

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.

Page 64: Eurocode 4

64

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Vibration –vertical vibrations due to walking of one person

time t

Fi(t)left foot

right foot

F(t)

both feet

1. step 2. step 3. step

ts=1/fs

( )⎥⎥⎦

⎢⎢⎣

⎡Φ−πα+= ∑

=

3

1nnsno tfn2sin1G)t(F

Go weight of the person (800 N)αn coefficient for the load component of n-th harmonicn number of the n-th harmonicfs pacing rateΦn 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

α3=0,1-0,15 Φ3=π/2

Fourier-coefficients and phase angles:

Page 65: Eurocode 4

65

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Vibration – vertical vibrations due to walking of persons

( )sE v/Lf

gen

namax e1

MFka δ−−

δπ

=

Mgen

F(t)

w(t)maximum acceleration a, vertical deflection w and maximum velocity v

acceleration

Emax

2E

max

f2av

)f2(aw

π=

π=

fE natural frequencyFn load component of n-th harmonicδ logarithmic damping decrementvs forward speed of the personka factor taking into account the different

positions xk during walking along the beamMgen generated mass of the system

(single span beam: Mgen=0,5 m L)

m

L

Fn(t)xk

w(xk,t)

L/2ka Fn(t)

w(t)

( )tfE

gen

na Ee1)tf2(sin

MFk)t(w δ−−π

δπ

=sv

Lt =

Page 66: Eurocode 4

66

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Logarithmic damping decrement

For the determination of the maximum acceleration the damping coefficient ζ or the logarithmic damping decrement δmust 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-bearing inner walls

1

2

3

4

5

6

Dampingratioξ [%]

3 6 9 12fE [Hz]

ξπ=δ 2

results of measurements in buildings

with finishes

without finishes

Page 67: Eurocode 4

67

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

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)/Go

( )∑=

Φ−π+=3

1nnsno tfn2sinFG)t(F

0,4

0,2

2,0 4,0 6,0 8,0

0,1

fs=1,5-2,5 Hz

2fs=3,0-5,0 Hz

3fs=4,5-7,5 Hz

Page 68: Eurocode 4

68

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

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]

frequency [Hz]

basic curve ao

Multiplying factors Ka for the basic curve

Residential (flats, hospitals) Ka=1,0Quiet office Ka=2-4General office (e. g. schools) Ka=4

ao Kaa ≤

natural frequency of typical compositebeams

Page 69: Eurocode 4

69

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Thank you very much for your kind attention

Page 70: Eurocode 4

70

G. HanswilleUniv.-Prof. Dr.-Ing.

Institute for Steel and Composite Structures

University of Wuppertal-Germany

Thank you very much for your kind attention