Eurocode 4
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G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Eurocodes Background and Applications Dissemination of information for training 18-20 February 2008, Brussels
Eurocode 4 Serviceability limit states of composite beams Univ. - Prof. Dr.-Ing. Gerhard Hanswille Institute for Steel and Composite Structures University of Wuppertal Germany 1
Contents
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Part 1:
Introduction
Part 2:
Global analysis for serviceability limit states
Part 3:
Crack width control
Part 4:
Deformations
Part 5:
Limitation of stresses
Part 6:
Vibrations
2
Serviceability limit states
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Serviceability limit states
Limitation of stresses Limitation of deflections
crack width control
vibrations web breathing 3
Serviceability limit states
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
{
}
characteristic combination:
Ed = E ∑ Gk, j + Pk + Qk,1 + ∑ ψ 0,i Qk,i
frequent combination:
Ed = E ∑ Gk, j + Pk + ψ1,1 Qk,1 + ∑ ψ 2,i Qk,i
quasi-permanent combination:
Ed = E ∑ Gk, j + Pk + ∑ ψ 2,i Qk,i
{ {
}
}
serviceability limit states Ed ≤ Cd: - deformation - crack width - excessive compressive stresses in concrete
Cd=
- excessive slip in the interface between steel and concrete - excessive creep deformation - web breathing - vibrations
4
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Part 2: Global analysis for serviceability limit states
5
Global analysis - General
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Calculation of internal forces, deformations and stresses at serviceability limit state shall take into account the following effects:
shear lag;
inelastic behaviour of steel and reinforcement, if any;
creep and shrinkage of concrete; cracking of concrete and tension stiffening of concrete; sequence of construction; increased flexibility resulting from significant incomplete interaction due to slip of shear connection;
torsional and distorsional warping, if any.
6
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Shear lag- effective width shear lag real stress distribution
bei < 0,2 bi
bei
σ max
σ(y)
σ( y ) = σmax
σmax
σ max
⎡ y⎤ ⎢1− b ⎥ ⎣ i⎦
4
b 5 bei
be
stresses taking into account the effective width
The flexibility of steel or concrete flanges affected by shear in their plane (shear lag) shall be used either by rigorous analysis, or by using an effective width be
σmax
y bi
bei σR
σ(y)
bei ≥ 0,2 bi ⎡b ⎤ σR = 1,25 ⎢ ei − 0,2⎥ σmax ⎣ bi ⎦
y bi
⎡ y⎤ σ( y ) = σR + [σmax − σR ] ⎢1− ⎥ ⎣ bi ⎦
4
7
Effective width of concrete flanges Le=0,25 (L1 + L2) for beff,2 Le=0,85 L1 for beff,1
Le=2L3 for beff,2
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
be,1 bo be,2
Le=0,70 L2 for beff,1
b1
L1
L1/4
L1/2
L2/4
L2/2
L2/4
beff,0
beff,2 beff,1
end supports:
beff,2
b2
L3
L2
L1/4
bo
beff,1
midspan regions and internal supports: beff = b0 + be,1+be,2
beff = b0 + β1 be,1+β2 be,2
be,i= Le/8
βi = (0,55+0,025 Le/bi) ≤ 1,0
Le – equivalent length 8
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Effects of creep of concrete
redistribution of the sectional forces due to creep
Initial sectional forces
primary effects
Mc,o
-zi,c zi,st
-Nc,o
-Mc,r Nc,r
ast
ML Mst,o
Mst,r
Nst,o
-Nst,r
The effects of shrinkage and creep of concrete and non-uniform changes of temperature result in internal forces in cross sections, and curvatures and longitudinal strains in members; the effects that occur in statically determinate structures, and in statically indeterminate structures when compatibility of the deformations is not considered, shall be classified as primary effects. 9
Effects of creep and shrinkage of concrete
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Types of loading and action effects: In the following the different types of loading and action effects are distinguished by a subscript L : L=P for permanent action effects not changing with time L=PT
time-dependent action effects developing affine to the creep coefficient
L=S
action effects caused by shrinkage of concrete
L=D
action effects due to prestressing by imposed deformations (e.g. jacking of supports)
MPT(t)
time dependent action effects ML=MPT:
MPT (t=∞)
action effects caused by prestressing due to imposed deformation ML=MD: MD
MPT(ti)
ϕ(ti,to) ϕ(t∞,to)
δ
ϕ(t,to)
ML=MD
+ 10
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Modular ratios taking into account effects of creep
centroidal axis of the concrete section
zis,L
ast
-zic,L
zc
zist,L
zi,L zst
Modular ratios:
centroidal axis of the transformed composite section centroidal axis of the steel section (structural steel and reinforcement)
nL = no [ 1+ ψL ϕ( t, t o ) ]
action short term loading
no =
Ea Ecm creep multiplier Ψ=0
permanent action not changing in time
ΨP=1,10
shrinkage
ΨS=0,55
prestressing by controlled imposed deformations
ΨD=1,50
time-dependent action effects
ΨPT=0,55 11
Elastic cross-section properties of the composite section taking into account creep effects
-zis,L
-zic,L ast
zist,L
zc zi,L zst
centroidal axis of the concrete section centroidal axis of the composite section centroidal axis of the steel section
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Modular ratio taking into account creep effect:
nL =n0 (1+ ψL ϕ( t, t 0 ) ) no =
Est Ecm ( t o )
Transformed cross-section properties of Distance between the centroidal axes of the concrete section: the concrete and the composite section: A c,L = A c / nL
Jc,L = Jc / nL
Transformed cross-section area of the composite section: A i,L =A St +A c,L
zic,L =− A st ast /A i,L
Second moment of area of the composite section: J i,L = Jst + Jc,L + A st A c,L a2st / A i,L 12
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Effects of cracking of concrete and tension stiffening of concrete between cracks
Ns
mean strain εsm=εs,2- βΔεs,r
σc(x) Δε s = β
fct,eff ρs E s
Δε s = β Δε s,r
ρs = A s / A c
fully cracked section
Δε s,r
εsr,1 A
B
stage A: stage B: stage C:
εsr,2
εsm,y
εsy
C
uncracked section initial crack formation stabilised crack formation
σs,2
σc(x)
Nsm Ns,cr
τv
σs(x)
β = 0,4
Nsy
σc(x)
σs(x) Ns
Ns ε s,2 =
ε
εsm fct Ec
σ s2 Es
β Δε s,r
εs(x) εc(x)
x
13
Influence of tension stiffening of concrete on stresses in reinforcement εsm
Ns zs
Ms≈0
a Ma za
Na
mean strain in the concrete slab:
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
equilibrium:
M εa
Ma = M − Ns a
-κ
Na = − Ns compatibility: εsm = εa + κ a Ns a2 M a Ns εsm + + = Ea A a Ea A a Ea Ja
εs,2 εs,m
Δεs=β Δεs,r
mean strain in the concrete slab:
εs εc
ε sm = ε s2 − β Δε sr =
fct,eff Ns −β Es A s ρs Es 14
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Redistribution of sectional forces due to tension stiffening
tension stiffening
fully cracked section
ΔNts
Ns,2 -zst,s
-Ms,2
a
zst,a
z2=zst
+
-Ma,2 -Na,2
Ns
A J α st = st st A a Ja
f A ΔNts = β ct,eff s ρs α st
Ns,2
Nsε MEd
-ΔNts
Js Jst
Na = Na2 − ΔNts = MEd
M
-MEd
-Ma -Na
Jst = J2
Sectional forces:
Ms = MEd
-Ms
=
ΔNts a
Ns = Ns2 + ΔNts = MEd
ΔNts Ns
Ns
A s z st,s Jst
A a z st,a Jst
Ma = Ma2 + ΔNts a = MEd
+ ΔNts
− ΔNts
Ja + ΔNts a Jst
15
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Stresses taking into account tension stiffening of concrete fully cracked
tension stiffening
Ns,2
a
zst
reinforcement:
-Ms
+
zst,a
za
ΔNts
-Ms,2
-zst,s
-Ma,2
Ns
ΔNts a
-Na,2
-MEd
= -Ma
-ΔNts
-Na
structural steel:
f σ s = σ s,2 + β ctm ρs α st
σa = σa,2 −
M f σ s = Ed z st,s + β ctm Jst ρs α st
M ΔNts ΔNts a za σa = Ed z st − + Jst Aa Ja
ΔNts ΔNts a + za Aa Ja
α st =
A st Jst A a Ja
ΔNts = β
fctm A s ρs α st 16
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Influence of tension stiffening on flexural stiffness
εsm
Ns -Ms a
zst
-Ma -Na
κ=
-M κ εa
Est J1 Est J2,ts
EstJ2
EstJ2
EstJ2,ts EstJ1
M Ma M − Ns a = = Est I2,ts Est Ja Est Ja
Effective flexural stiffness: Est J2,ts =
EJ
M
Curvature:
M
κ MR
MRn
Ea Ja (N − Ns,ε ) a 1− s M
EstJ1 uncracked section EstJ2 fully cracked section EstJ2,ts effective flexural stiffness taking into account tension stiffening of concrete 17
G. Hanswille
Univ.-Prof. Dr.-Ing. Effects of cracking of concrete - General Institute for Steel and Composite Structures method according to EN 1994-1-1 University of Wuppertal-Germany
EaJ1 EaJ2
• Determination of internal forces by uncracked analysis for the characteristic combination.
– un-cracked flexural stiffness – cracked flexural stiffness
L1
EaJ1
L1,cr
L2,cr
EaJ2
L2
EaJ1
ΔM
• Determination of the cracked regions with the extreme fibre concrete tensile stress σc,max= 2,0 fct,m. • Reduction of flexural stiffness to EaJ2 in the cracked regions. • New structural analysis for the new distribution of flexural stiffness.
ΔM Redistribution of bending moments due to cracking of concrete un-cracked analysis cracked analysis 18
Effects of cracking of concrete – simplified method
L1
EaJ1
L2
0,15 L1
0,15 L2
EaJ2 ΔMII
Lmin / Lmax ≥ 0,6
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
For continuous composite beams with the concrete flanges above the steel section and not pre-stressed, including beams in frames that resist horizontal forces by bracing, a simplified method may be used. Where all the ratios of the length of adjacent continuous spans (shorter/longer) between supports are at least 0,6, the effect of cracking may be taken into account by using the flexural stiffness Ea J2 over 15% of the span on each side of each internal support, and as the uncracked values Ea J1 elsewhere. 19
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Part 3: Limitation of crack width
20
Control of cracking
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
General considerations minimum reinforcement If crack width control is required, a minimum amount of bonded reinforcement is required to control cracking in areas where tension due to restraint and or direct loading is expected. The amount may be estimated from equilibrium between the tensile force in concrete just before cracking and the tensile force in the reinforcement at yielding or at a lower stress if necessary to limit the crack width. According to Eurocode 4-1-1 the minimum reinforcement should be placed, where under the characteristic combination of actions, stresses in concrete are tensile.
control of cracking due to direct loading Where at least the minimum reinforcement is provided, the limitation of crack width for direct loading may generally be achieved by limiting bar spacing or bar diameters. Maximum bar spacing and maximum bar diameter depend on the stress σs in the reinforcement and the design crack width.
21
Recommended values for wmax
Exposure class
XO, XC1
reinforced members, prestressed members with unbonded tendons and members prestressed by controlled imposed deformations
prestressed members with bonded tendons
quasi - permanent load combination
frequent load combination
0,4 mm (1)
0,2 mm
XC2, XC3,XC4 XD1,XD2,XS1, XS2,XS3
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
0,2 mm (2) 0,3 mm decompression
(1) For XO and XC1 exposure classes, crack width has no influence on durability and this limit is set to guarantee acceptable appearance. In absence of appearance conditions this limit may be relaxed. (2) For these exposure classes, in addition, decompression should be checked under the quasi-permanent combination of loads. 22
Exposure classes according to EN 1992-1-1 (risk of corrosion of reinforcement) Class
Description of environment
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Examples
no risk of corrosion or attack
XO
for concrete without reinforcement, for concrete with reinforcement : very dry
concrete inside buildings with very low air humidity
Corrosion induced by carbonation
XC1
dry or permanently wet
concrete inside buildings with low air humidity
XC2
wet, rarely dry
concrete surfaces subjected to long term water contact, foundations
XC3
moderate humidity
external concrete sheltered from rain
XC4
cyclic wet and dry
concrete surfaces subject to water contact not within class XC2
Corrosion induced by chlorides
XD1
moderate humidity
concrete surfaces exposed to airborne chlorides
XD2
wet, rarely dry
swimming pools, members exposed to industrial waters containing chlorides
XD3
cyclic wet and dry
car park slabs, pavements, parts of bridges exposed to spray containing
Corrosion induced by chlorides from sea water
XS1
exposed to airborne salt
structures near to or on the coast
XS2
permanently submerged
parts of marine structures
XS3
tidal, splash and spray zones
parts of marine structures 23
Cracking of concrete (initial crack formation)
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
w Ns
Ns ε
εs εc Les
σ
Δσs
σs
σc,1
σc,1
Les
Les
σs,1
Les
ρs =
Les
σs A s = σs,1 A s + σc,1 A c Compatibility at the end of the introduction length: σ σ ε s,1 = εc,1 ⇒ s,1 = c,1 Es Ec Es ⎡ ρs no ⎤ n = o σs,1 = σs ⎢ ⎥ Ec ⎣ 1+ ρs no ⎦ Change of stresses in reinforcement due to cracking: σs Δσ s = σ s − σ s,1 = 1+ ρ s no
Ns,r = fctm A c (1+ ρs no )
σs,2
σs,1
Equilibrium in longitudinal direction:
As Ac
As cross-section area of reinforcement ρs reinforcement ratio fctm mean value of tensile strength of concrete 24
Cracking of concrete – introduction length
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
w Ns
Ns ε
Δσs = σs − σs,1 =
εs εc Les
σ
σc,1
Change of stresses in reinforcement due to cracking:
Equilibrium in longitudinal direction L es Us τsm = Δσs A s π d2s L es π ds τsm = Δσs 4
Les Δσs
σs
introduction length LEs
σs,1
L es Les σc,1 σs,1
Les
σs 1+ ρs no
Les
τsm σs,2 Us As ρs τsm
-perimeter of the bar -cross-section area -reinforcement ratio -mean bond strength
σ d 1 = s s 4 τsm 1+ no ρs
ρs =
As Ac
no =
Es Ec
crack width
w = 2 L es (εsm − εcm ) 25
Determination of the mean strains of G. Hanswille Univ.-Prof. Dr.-Ing. reinforcement and concrete in the stage of initial Institute for Steel and Composite Structures crack formation University of Wuppertal-Germany w
Mean bond strength:
Ns
Ns τs,m =
ε
εs,m
εs(x)
Δεs,cr
εcr
εc(x)
εc,m
εs
Les
Les
σs(x)
Mean stress in the reinforcement: σ s,m = σ s − β Δσ s ⇒ β =
βΔσs σs
x
4 x Δσs ( x ) = ∫ τs ( x ) dx Us 0
Δσs σs,1
Les
σ s − Δσ sm Δσ s
Mean strains in reinforcement and concrete:
σc,1
Les
∫ τs ( x ) dx ≈ 1,8 fctm
Les o
1 Les Δσsm = ∫ Δσs( x) dx Les 0
σ σs,m
1 LEs
ε s,m = ε s,2 − β Δε s,cr εc,m = β εcr 26
Determination of initial crack width
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
w Ns
crack width
Ns
ε
εs,m εs(x)
w = 2 L es (εsm − εcm )
εs,2 Δεs,cr
εc(x)
εc,m
εs,m − εcm = (1 − β) εs,2
εcr L es =
Les
Les σs
σs ds 1 4 τsm 1+ no ρs
τsm ≈ 1,8 fctm βΔσs Δσs
σs,m
σs,1
σs
σc,1
Les
Les x
(1 − β) σ2s ds 1 w= 2 τsm Es 1+ no ρs
with β= 0,6 for short term loading und β= 0,4 for long term loading 27
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Maximum bar diameters acc. to EC4
σs [N/mm2]
∗ maximum bar diameterds for
wk= 0,4
wk= 0,3
wk= 0,2
160
40
32
25
200
32
25
16
240
20
16
12
280
16
12
8
320
12
10
6
360
10
8
5
400
8
6
4
450
6
5
-
β= 0,4
for long term loading and repeated loading
Crack width w: (1 − β) σ 2s ds 1 w= 2 τsm E s 1+ no ρs
σ 2s ds ≈ 6 fct,m E s
Maximum bar diameter for a required crack width w:
ds = w
2 τsm Es ( 1+ no ρs ) σ2s (1 − β)
With τsm= 1,8 fct,mo and the reference value for the mean tensile strength of concrete fctm,o= 2,9 N/mm2 follows: d*s = w k d*s ≈ 6
3,6 fctm,o E s ( 1+ no ρs ) σ 2s (1 − β)
w k fctm,o E s σ 2s 28
Crack width for stabilised crack formation
Crack width for high bond bars
w
w = sr,max (ε sm − ε cm )
Ns ε
ε s,2 =
Mean strain of reinforcement and concrete:
σ s2 Es
εs(x) fct Ec
εs(x)- εc(x)
εc(x) sr,max= 2 Les β= 0,6
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
εs,m = εs,2 − β Δ εs A f f εs,m = εs,2 − β c ctm = εs,2 − β ctm Es A s E s ρs f εcm = β ctm Ec
sr,min= Les
for short term loading
εsm − εcm =
σs f − β ctm (1 + no ρs ) Es Es ρs
β= 0,4 for long term loading and repeated loading 29
Crack width for stabilised crack formation
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
w The maximum crack spacing sr,max in the stage of stabilised crack formation is twice the introduction length Les.
Ns ε
σ εs,2 = s Es εs(x)
fct Ec
w = sr,max (ε sm − εcm ) εs(x)- εc(x)
εc(x)
fctm ds fctm A c Les = = Us τsm ρs 4 τsm maximum crack width for sr= sr,max
sr,max= 2 Les
sr,min= Les
β= 0,6
for short term loading
β= 0,4
for long term loading and repeated loading
fctm ds ⎛ σs ⎜⎜ w= −β 2 τsm ρs ⎝ Es
⎞ fctm (1 + no ρs ) ⎟⎟ ρs E s ⎠
30
Crack width and crack spacing according Eurocode 2
Crack width
w = sr,max (ε sm − εcm ) σ εsm − εcm = s − β Es β= 0,6 β= 0,4
Crack spacing
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
fctm (1 + no ρs ) ≥ 0,6 Es ρs
σs Es
for short term loading for long term loading and repeated loading
In Eurocode 2 for the maximum crack spacing a semiempirical equation based on test results is given
sr,max = 3,4 c + k1 ⋅ k 2 ⋅ 0,425
ds ρs
ds-diameter of the bar c- concrete cover
k1
coefficient taking into account bond properties of the reinforcement with k1=o,8 for high bond bars
k2
coefficient which takes into account the distribution of strains (1,0 for pur tension and 0,5 for bending) 31
Determination of the cracking moment Mcr and the normal force of the concrete slab in the stage of initial cracking
cracking moment Mcr:
cracking moment Mcr Mc+s
hc zio zi,st
σc + σc,ε = fct,eff = k1 fctm
σc
Mcr
Nc+s
zo
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
[
Mcr = fct,eff − σc,ε
[
ast
Mcr = fct,eff − σc,ε
Nc,ε
]z
no Jio o
+ hc / 2 no Jio
ic,o (1+ hc
/( 2 z o )
sectional normal force of the concrete slab:
primary effects due to shrinkage Mc,ε
]z
σcε
Ncr = Mcr Ncr =
A co z o + A s zis + Nc + s,ε Jio
A c ( fct,eff − σc,ε ) (1+ ρs no ) + Nc + s,ε 1+ hc /( 2 z o )
kc,ε≈ 0,3 kc ⎡ ⎢ 1 Ncr = A c fct,eff (1 + ρ s n0 ) ⎢ ⎢ 1 + hc /(2 z o ) ⎢ ⎣
+
A c σc,ε (1+ ρs no ) 1+ hc /( 2 z o ) A c fct,eff (1 + ρs n0 )
Nc + s,ε −
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
32
Simplified solution for the cracking moment and the normal force in the concrete slab cracking moment Mcr Mc+s
hc
zo
simplified solution for the normal force in the concrete slab:
σc
Mcr
Nc+s
Ncr ≈ A c fctm k s ⋅ k ⋅ k c
zi,st
k = 0,8
primary effects due to shrinkage Mc+s,ε
Nc+s,ε
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
ks= 0,9
σcε
coefficient taking into account the effect of non-uniform self-equilibrating stresses coefficient taking into account the slip effects of shear connection
kc =
1 h 1+ c 2 zo
cracking moment
+ 0,3 ≤ 1,0
shrinkage 33
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Determination of minimum reinforcement
Mc hc
zo zi,o
Nc
Mc,ε
Mcr
cracking moment
Nc,ε shrinkage
Ma,ε Na,ε
A c fct,eff As ≥ k k s kc σs k = 0,8 ks = 0,9 kc
d∗s ds σs fct,eff
kc =
f 1 + 0,3 ≤ 1,0 ds = d∗s ct,eff 1+ hc z o fct,o
fcto= 2,9 N/mm2
Influence of non linear residual stresses due to shrinkage and temperature effects flexibility of shear connection Influence of distribution of tensile stresses in concrete immediately prior to cracking maximum bar diameter modified bar diameter for other concrete strength classes stress in reinforcement acc. to Table 1 effective concrete tensile strength 34
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Control of cracking due to direct loading – Verification by limiting bar spacing or bar diameter fully cracked
Ac
tension stiffening
Ns,2
As
-Ms,2
-zst,s
a
za
The calculation of stresses is based on the mean strain in the concrete slab. The factor β results from the mean value of crack spacing. With srm≈ 2/3 sr,max results β ≈ 2/3 ·0,6 = 0,4
-Ms
+
zst zst,a Aa
ΔNts
-Ma,2 -Na,2
Ns
ΔNts a
-MEd
=
-ΔNts
-Ma -Na
stresses in reinforcement σ = σ s,2 + Δσ ts taking into account tension s fct,eff MEd stiffening for the bending σs = z st,s + β moment MEd of the quasi J2 ρs α st permanent combination: A 2 J2 As α = st ρs = β = 0,4 A a Ja Ac
The bar diameter or the bar spacing has to be limited 35
Maximum bar diameters and maximum bar spacing for high bond bars acc. to EC4
Table 1: Maximum bar diameter σs [N/mm2]
∗
maximum bar diameterds for
wk= 0,4
wk= 0,3
wk= 0,2
160
40
32
25
200
32
25
16
240
20
16
12
280
16
12
8
320
12
10
6
360
10
8
5
400
8
6
4
450
6
5
-
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Table 2: Maximum bar spacing σs [N/mm2]
maximum bar spacing in [mm] for
wk= 0,4
wk= 0,3
wk= 0,2
160
300
300
200
200
300
250
150
240
250
200
100
280
200
150
50
320
150
100
-
360
100
50
-
36
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Direct calculation of crack width w for composite sections based on EN 1992-2 As
-zst,s
-Ms
σs,
Ns
-Ma
MEd
zst
-Na crack width for high bond bars:
w = sr,max (ε sm − εcm )
fct,eff MEd σs = zst,s + β Jst ρs α st α st =
A st Jst A a Ja
ρs =
As Ac
β = 0,4
σ εsm − εcm = s − β Es
fctm (1 + no ρs ) ≥ 0,6 Es ρs
σs Es
d sr,max = 3,4 c + 0,34 s ρs c - concrete cover of reinforcement 37
Stresses in reinforcement in case of bonded tendons – initial crack formation Ap, dp
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Equilibrium at the crack:
As, ds
σs A s + Δσp A p = N = fct,eff A c ( 1 + no ρ tot ) Equilibrium in longitudinal direction: σs=σs1+Δσs
σ s A s = π ds τsm L e,s
σp
Δσp A p = π dp τpm L ep N
σp=σpo+σp1+Δσp
δ s = δp ⇒
Δσs Δσp
σs,1
Compatibility at the crack: Δσp − Δσp1 σ s − σ s1 L es = L ep Es Ep
With Es≈Ep and σs1=Δσp1=0 results: Stresses:
Δσp1
σs =
Les Lep τpm
τsm
ξ1 =
N A s + ξ1 A p τpm ds τsm dv
Δσp =
ξ1 N A s + ξ1 A p
38
Stresses in reinforcement for final crack formation Ap, dp
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Equilibrium at the crack:
N − Po = σs2 A s + Δ σp2 A p
As, ds
Maximum crack spacing: σs
fct A c =
Δσp2
sr,max =
σs2
[
sr,max τsm ns ds π + τpm np dp π 2 ds fct,eff A c 2 τsm ( A s + ξ2 A p )
Equilibrium in longitudinal direction:
Δσp2
σs2 − σs1 =
Δσp1
sr,max Us τsm 2 As
σp2 − σp1 =
sr,max Up τpm 2 Ap
Compatibility at the crack:
σs1 σc
]
σc=fct,eff
sr,max
x
δ s = δp =
σ s2 − β (σs2 − σs1) Δσp,2 − β( Δσp2 − Δσp1) = Es Ep
mean crack spacing: sr,m≈2/3 sr,max 39
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Determination of stresses in composite sections with bonded tendons
Ap
As
σs,σp
-zst,s
MEd zst
Stresses σ*s in reinforcement at the crack location neglecting different bond behaviour of reinforcement and tendons: σ*s =
MEd f z st,s + β ctm Jst ρ tot α st
α st =
Stresses in reinforcement taking into account the different bond behaviour: ⎡ Ac Ac σ s = σ *s + 0,4 fct,eff ⎢ − ⎢ A s + ξ12 A p A s + A p ⎣ Δσp = σ *s
A st Jst A a Ja
⎤ ⎡ 1 1 ⎤ ⎥ = σ *s + 0,4 fct,eff ⎢ − ⎥ ρeff ρ tot ⎦ ⎥ ⎣ ⎦
⎡ A ξ12 A c c ⎢ − 0,4 fct,eff − ⎢ A s + A p A s + ξ12 A p ⎣
⎤ ⎡ 1 ξ12 ⎤ ⎥ = σ *s − 0,4 fct,eff ⎢ − ⎥ ρ ρ ⎥ ⎥⎦ ⎢ tot eff ⎣ ⎦
β = 0,4
ρ tot = ρeff =
A s + Ap Ac A s + ξ12 A p Ac 40
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Part 4: Deformations
41
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Deflections
Effects of cracking of concrete L1
L2
0,15 L1
0,15 L2
EaJ2 EaJ1 ΔM
Deflections due to loading applied to the composite member should be calculated using elastic analysis taking into account effects from - cracking of concrete, - creep and shrinkage, - sequence of construction,
Sequence of construction
gc
- influence of local yielding of structural steel at internal supports, - influence of incomplete interaction.
F F
steel member composite member 42
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Deformations and pre-cambering
combination
limitation
general
quasi permanent
risk of damage of adjacent parts of the structure (e.g. finish or service work)
quasi – permanent (better frequent)
δ1 δ2 δ3 δ4
δmax ≤ L / 250
δ1
δ w ≤ L / 500
Pre-cambering of the steel girder:
δmax maximum deflection δw
δmax
δc deflection of the composite girder
δp = δ1+ δ2+ δ3 +ψ2 δ4
δp
δc
δ1 deflection of the steel girder
δw
effective deflection for finish and service work
δ1 – self weight of the structure δ2 – loads from finish and service work δ3 – creep and shrinkage δ4 – variable loads and temperature effects 43
Effects of local yielding on deflections
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
For the calculation of deflection of un-propped beams, account may be taken of the influence of local yielding of structural steel over a support. For beams with critical sections in Classes 1 and 2 the effect may be taken into account by multiplying the bending moment at the support with an additional reduction factor f2 and corresponding increases are made to the bending moments in adjacent spans. f2 = 0,5 if fy is reached before the concrete slab has hardened; f2 = 0,7 if fy is reached after concrete has hardened. This applies for the determination of the maximum deflection but not for pre-camber.
44
More accurate method for the determination of the effects of local yielding on deflections
L1
lcr
lcr
L2
+ z2
EaJ1
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
-
Mel,Rk
σa=fyk
fyk
Mpl,Rk
EaJ2 (EJ)eff EaJ2
EaJeff ΔM
EaJeff
Mel,Rk
MEd
Mpl,Rk 45
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Effects of incomplete interaction on deformations
The effects of incomplete interaction may be ignored provided that:
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.
P PRd P
P
cD s
s
su 46
Differential equations in case of incomplete interaction
Mc
Ec, Ac, Jc
vL
ac N c a
zc Ea, Aa, Ja
vL
Ma Na
za (w)
Slip:
aa
Vc+dVc
Vc
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Mc+ dMc Nc+dNc Ma+ dMa Na+dNa
Va
uc , u′c = ε c
ua , u′a = ε a
Va+dVa dx
sv = ua − uc + w ′ a
u′c′ + c s (ua − uc + w ′ a ) = 0 u′c′ − c s (ua − uc + w ′ a) = 0 (Ec Jc + Ea Ja ) w ′′′′ − c s a (u′a − u′c + w ′′ a) = q
Ec A c Ea A a
x
Nc = E c A c u′c
Mc = − Ec Jc w′′
Na = E a A a u′a
Ma = − Ea Ja w ′′ Va = − Ea Ja w ′′′
Vc = − E c Jc w ′′′ ≈ 0
47
Deflection in case of incomplete interaction for single span beams
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
concrete section composite section
F
Aco=Ac/no, Jco= Jc/no w L
steel section
⎡ 2 ( λ )⎤ sinh ⎢ ⎥ 12 48 FL 2 − 1+ w= 48 E a Ii,o ⎢⎢ α λ2 α λ3 sinh( λ ) ⎥⎥ ⎣ ⎦
Aa, Ja
Aio, Jio
no=Ea/Ec
3
λ2 =
q L
β=
λ ⎡ ⎤ cosh( ) − 1 ⎥ 5 q L ⎢ 48 1 384 1 2 + − w= 1 ⎢ ⎥ 384 E a Ji,o ⎢ 5 α λ2 5 α λ4 cosh( λ ) ⎥ 2 ⎦ ⎣ 4
1+ α αβ
E a A c,o A a A i,o c s L ²
α=
1 Ji,o Ja + Jc,o
−1 48
Mean values of stiffness of headed studs
spring constant per stud:
cs
spring constant of the shear connection: type of shear connection
P PRd P
P
cD
s
s
su
eL
nt=2
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
CD =
su PRd
C n cs = D t eL
CD [kN / cm]
headed stud ∅ 19mm in solid slabs
2500
headed stud ∅ 22mm in solid slabs
3000
headed studs ∅ 25mm in solid slab
3500
headed stud ∅ 19mm with Holorib-sheeting and one stud per rib
1250
headed stud ∅ 22mm with Holorib-sheeting and one stud per rib
1500
49
Simplified solution for the calculation of deflections in case of incomplete interaction q(ξ ) = q sin πξ
The influence of the flexibility of the shear connection is taken into account by a reduced value for the modular ratio.
L
wo = q
x ξ= L
Ec, Ac, Jc zc
εc
L4 π4
a
Ea, Aa, Ja
εa
EcmJc + Ea Ja +
Mc Nc Ma
za
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Na
1 βo Ecm A c Ea A a Ea A a + βo Ecm A c
=q a2
L4
1 π 4 Ea Jio,eff
A c,eff A a 2 Jio,eff = Jc,o + Ja + a A c,eff + A a A A c,eff = c no,eff effective modular ratio for the concrete slab
no,eff = no ( 1 + βs )
π2 Ecm A c βs = L2 c s 50
Comparison of the exact method with the simplified method q
1,5
w/wc
cD = 1000 KN/cm
1,4 L
1,2
beff
1,1
Ecm = 3350 KN/cm² 99 51
η=0,4
1,3
w
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
η=0,8 L [m]
1,0 5,0
10,0
20,0
exact solution simplified solution with no,eff
w/wc
450 mm
15,0
1,25
cD = 2000 KN/cm
1,2
wo- deflection in case of neglecting effects from slip of shear connection
η
degree of shear connection
1,15
η=0,4
1,1 1,05 1,0 5,0
η=0,8 L [m] 10,0
15,0
20,0 51
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Deflection in case of incomplete interactioncomparison with test results
1875
1875
F
load case 1
F F/2
load case 2
F/2
Deflection at midspan
F [kN] 200
1875
3750
1875 150
7500
load case 2 100
1500 50
load case 1
445
IPE 270
270
175
50
0
20
40
60
δ [mm] 52
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Deflection in case of incomplete interactionComparison with test results
F
F 160 120 780
s
push-out test
80 40
s[mm] 10
125 50
20
30
40
second moment of area cm4
Load case 1 F= 60 kN
Load case 2 F=145 kN
Test
-
11,0 (100%)
20,0 (100 %)
Theoretical value, neglecting flexibility of shear connection
Jio= 32.387,0
7,8 (71%)
12,9 (65%)
Theoretical value, taking into account flexibility of shear connection
Jio,eff= 21.486,0
11,7 (106%)
19,4 (97%)
Deflection at midspan in mm
53
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Part 5: Limitation of stresses
54
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Limitation of Stresses
σc
MEd
- +
σa +
σa
σs MEd
Stress limitation is not required for beams if in the ultimate limit state, -
no+ verification of fatigue is required and
-
no prestressing by tendons and /or
- - no prestressing by controlled imposed deformations is provided.
combination
stress limit
recommended values ki
structural steel
characteristic
σEd ≤ ka fyk
ka = 1,00
reinforcement
characteristic
σEd ≤ ks fsk
ks = 0,80
concrete
characteristic
σEd ≤ kc fck
kc= 0,60
headed studs
characteristic
PEd ≤ ks PRd
ks = 0,75 55
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Local effects of concentrated longitudinal shear forces composite section
steel section
bc
gc
beff zio
y
x
z MEd
MEd(x)
Ac,eff
+
Concentrated longitudinal shear force at sudden change of cross-section
VEd(x)
VEd
-
Lv=beff
+
longitudinal shear forces
Nc
v L,Ed,max =
2 MEd A c,eff zio Ea / Ec Jio b eff
vL,Ed,max
+ 56
Local effects of concentrated longitudinal shear forces
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
gc,d system
L = 40 m FE-Model
cross-section
bc=10 m 300 500x20
y
P
shear connectors
14x2000 800x60
z
CD = 3000 kN/cm per stud δ 57
Ultimate limit state - longitudinal shear forces
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
EN 1994-2
ULS
x [cm]
P
FE-Model: cD
s
FE-Model P L = 40 m
cD
s
x 58
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Serviceability limit state - longitudinal shear forces vL,Ed[kN/m]
SLS
500
EN 1994-2 x [cm]
0 200
400
600
800
1000 1200 1400 1600 1800 2000
-500
P
-1000
FE-Model: cD
-1500
s
-2000 -2500
FE-Model P
-3000
L = 40 m
-3500
cD
s
x -4000
59
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Part 6: Vibrations
60
Vibration- General
EN 1994-1-1:
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
The dynamic properties of floor beams should satisfy the criteria in EN 1990,A.1.4.4
EN 1990, A1.4.4: To achieve satisfactory vibration behaviour of buildings and their structural members under serviceability conditions, the following aspects, among others, should be considered:
the comfort of the user the functioning of the structure or its structural members Other aspects should be considered for each project and agreed with the client
61
Vibration - General
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
EN 1990-A1.4.4:
For serviceability limit state of a structure or a structural member not to be exceeded when subjected to vibrations, the natural frequency of vibrations of the structure or structural member should be kept above appropriate values which depend upon the function of the building and the source of the vibration, and agreed with the client and/or the relevant authority. Possible sources of vibration that should be considered include walking, synchronised movements of people, machinery, ground borne vibrations from traffic and wind actions. These, and other sources, should be specified for each project and agreed with the client. Note in EN 1990-A.1.4.4: Further information is given in ISO 10137.
62
Vibration – Example vertical vibration due to walking persons
Span
The pacing rate fs dominates the dynamic effects and the resulting dynamic loads. The speed of pedestrian propagation vs is a function of the pacing rate fs and the stride length ls.
lengt h
F(x,t)
pacing rate fs [Hz]
forward speed vs = fs ls [m/s]
stride length ls [m]
slow walk
∼1,7
1,1
0,6
normal walk
∼2,0
1,5
0,75
fast walk
∼2,3
2,2
1,00
slow running (jog)
∼2,5
3,3
1,30
fast running (sprint)
> 3,2
5,5
1,75
ls time t
xk
F(x,t) tk ts ls
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
xk x
63
Vibration –vertical vibrations due to walking of one person
Fi(t)
During walking, one of the feet is always in contact with the ground. The load-time function can be described by a Fourier series taking into account the 1st, 2nd and 3rd harmonic.
right foot left foot
3 ⎡ ⎤ F( t ) = Go ⎢1 + αn sin (2 n π fs t − Φ n )⎥ ⎢⎣ n =1 ⎥⎦
∑
time t F(t) 1. step
2. step 3. step
both feet
ts=1/fs
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Go αn n fs Φn
weight of the person (800 N) coefficient for the load component of n-th harmonic number of the n-th harmonic pacing rate phase angle oh the n-th harmonic Fouriercoefficients and phase angles:
α1=0,4-0,5 Φ1=0 α2=0,1-0,25 Φ2=π/2 α3=0,1-0,15 Φ3=π/2 64
Vibration – vertical vibrations due to walking of persons acceleration
F(t)
Fn π sin (2 π fE t ) 1 − e − δ fE t Mgen δ
( t ) = k a w
Mgen w(t)
c
δ
a
Fn(t) m w(xk,t)
L/2 ka Fn(t) w(t) L
)
t=
L vs
maximum acceleration a, vertical deflection w and maximum velocity v a w max = Fn π (2 π fE )2 a =k 1 − e − fE δ L / v s max
xk
(
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
fE Fn δ vs ka Mgen
Mgen
( δ
)
v max =
a 2 π fE
natural frequency load component of n-th harmonic logarithmic damping decrement forward speed of the person factor taking into account the different positions xk during walking along the beam generated mass of the system (single span beam: Mgen=0,5 m L) 65
Logarithmic damping decrement
results of measurements in buildings
Damping ratioξ [%]
with finishes without finishes
6 5 4
2
δ = 2π ξ 3
For the determination of the maximum acceleration the damping coefficient ζ or the logarithmic damping decrement δ must be determined. Values for composite beams are given in the literature. The logarithmic damping decrement is a function of the used materials, the damping of joints and bearings or support conditions and the natural frequency. For typical composite floor beams in buildings with natural frequencies between 3 and 6 Hz the following values for the logarithmic damping decrement can be assumed:
3
1
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
δ=0,10 floor beams without not loadbearing inner walls
6
9 fE [Hz]
12
δ=0,15 floor beams with not loadbearing inner walls 66
Vibration –vertical vibrations due to walking of persons
F( t ) = Go +
3
∑ Fn
sin (2 n π fs t − Φ n )
n =1
F(t)/Go fs=1,5-2,5 Hz 0,4
2fs=3,0-5,0 Hz
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
People in office buildings sitting or standing many hours are very sensitive to building vibrations. Therefore the effects of the second and third harmonic of dynamic load-time function should be considered, especially for structure with small mass and damping. In case of walking the pacing rate is in the rage of 1.7 to 2.4 Hz. The verification can be performed by frequency tuning or by limiting the maximum acceleration.
In case of frequency tuning for composite structures in office buildings the natural frequency 3fs=4,5-7,5 Hz normally should exceed 7,5 Hz if the first, second and third harmonic of the dynamic load-time function can cause significant acceleration.
0,2 0,1 2,0
4,0
6,0
8,0
Otherwise the maximum acceleration or velocity should be determined and limited to acceptable values in accordance with ISO 10137 67
Limitation of acceleration-recommended values acc. to ISO 10137
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
acceleration [m/s2] natural frequency of typical composite beams
0,1 0,05
Multiplying factors Ka for the basic curve Residential (flats, hospitals) Quiet office General office (e. g. schools)
0,01 0,005
basic curve ao
Ka=1,0 Ka=2-4 Ka=4
a ≤ ao K a 1
5
10
50 100
frequency [Hz] 68
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Thank you very much for your kind attention 69
G. Hanswille Univ.-Prof. Dr.-Ing. Institute for Steel and Composite Structures University of Wuppertal-Germany
Thank you very much for your kind attention
70
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