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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
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
3
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
4
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
5
G. HanswilleUniv.-Prof. Dr.-Ing.
Institute for Steel and Composite Structures
University of Wuppertal-Germany
Part 2:
Global analysis for serviceability limit states
6
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.
7
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
8
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
9
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.
10
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)
11
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
12
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
13
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
fρ
β=εΔ
css A/A=ρ
4,0=β
14
G. HanswilleUniv.-Prof. Dr.-Ing.
Institute for Steel and Composite Structures
University of Wuppertal-Germany
-κ
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
Nρ
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
15
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 =
16
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
17
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:
18
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
19
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.
20
G. HanswilleUniv.-Prof. Dr.-Ing.
Institute for Steel and Composite Structures
University of Wuppertal-Germany
Part 3:
Limitation of crack width
21
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.
22
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.
23
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
24
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+=
25
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=
26
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
27
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
28
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
29
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
30
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
=ε
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
31
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
Eσ
≥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
32
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
33
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
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
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
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
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
Eσ
≥ρ+ρ
β−σ
=ε−ε
ss
max,rd34,0c4,3sρ
+=
crack width for high bond bars:
c - concrete cover of reinforcement
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:
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
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:
41
G. HanswilleUniv.-Prof. Dr.-Ing.
Institute for Steel and Composite Structures
University of Wuppertal-Germany
Part 4:
Deformations
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
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
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.
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
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.
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 ′′′−=
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 −+
=α
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
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
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
=ξ
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
=β
eff,o
ceff,c n
AA =
effective modular ratio for the concrete slab
εa
εc
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
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
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
54
G. HanswilleUniv.-Prof. Dr.-Ing.
Institute for Steel and Composite Structures
University of Wuppertal-Germany
Part 5:
Limitation of stresses
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.
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
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
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
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
60
G. HanswilleUniv.-Prof. Dr.-Ing.
Institute for Steel and Composite Structures
University of Wuppertal-Germany
Part 6:
Vibrations
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
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.
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.
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:
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)
cδ
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 =
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
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
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
69
G. HanswilleUniv.-Prof. Dr.-Ing.
Institute for Steel and Composite Structures
University of Wuppertal-Germany
Thank you very much for your kind attention
70
G. HanswilleUniv.-Prof. Dr.-Ing.
Institute for Steel and Composite Structures
University of Wuppertal-Germany
Thank you very much for your kind attention