Serviceability limit state - Eurocodes

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 





Part 1: 

Part 2: 

Part 3: 

Part 4: 

Part 5: 

Part 6: 

Introduction 

Global analysis for serviceability limit states 

Crack width control 

Deformations 

Limitation of stresses 

Vibrations 

2

Serviceability limit states 

G. Hanswille 







Limitation of deflections 

crack width control 

vibrations 

web breathing 

3


G. Hanswille 





characteristic combination: 

E 

{ ∑ G + P + Q + ∑ ψ Q } 

d = E k,j k k,1 0,i k, i 

frequent combination: Ed 

= E { ∑ Gk,j 

+ Pk 

+ ψ1,1 

Qk,1 

+ ∑ ψ2,i 

Qk, 

i } 

quasi-permanent combination: Ed 

= E { ∑ Gk,j 

+ Pk 

+ ∑ ψ2,i 

Qk, 

i } 

serviceability limit states 

E d 

≤ C d 

: 

C d = 

- deformation 

- crack width 

- excessive compressive stresses in concrete 

- excessive slip in the interface between steel 

and concrete 

- excessive creep deformation 

- web breathing 

- vibrations 

4

G. Hanswille 





Part 2: 

Global analysis for serviceability limit states 

5

Global analysis - General 

G. Hanswille 





Calculation of internal forces, deformations and stresses at 

serviceability limit state shall take into account the following 

effects: 

• shear lag; 

• creep and shrinkage of concrete; 

• cracking of concrete and tension stiffening of concrete; 

• sequence of construction; 

• increased flexibility resulting from significant incomplete 

interaction due to slip of shear connection; 

• inelastic behaviour of steel and reinforcement, if any; 

• torsional and distorsional warping, if any. 

6

Shear lag- effective width 

G. Hanswille 





real stress distribution 

σ max 

σ max 

b 

shear lag 

σ(y) 

b 

b 

ei < 

i 

0,2 

⎡ y ⎤ 

σ ( y) =σ max ⎢1 

− 

b 

⎥ 

⎣ i ⎦ 

4 

stresses taking into 

account the effective 

width 

b ei 

b ei 

σ max 

5 b ei 

y 

b i 

b e 

The flexibility of steel or 

0, 2 

σ max 

bi 

concrete flanges affected by 

σ(y) 

shear in their plane (shear 

⎡b 

σ 

ei 

R = 1,25 ⎢ 

lag) shall be used either by 

⎣ bi 

rigorous analysis, or by using 

an effective width b e 

y 

σ (y) =σ R + σ 

b i 

b 

ei ≥ 

⎤ 

− 0,2⎥ 

σ 

⎦ 

[ −σ ] 

σ R 

max 

R 

max 

⎡ 

⎢1 

− 

⎣ 

y 

b 

i 

⎤ 

⎥ 

⎦ 

4 

7

Effective width of concrete flanges 

G. Hanswille 





L e =0,25 (L 1 + L 2 ) for b eff,2 b b 

L e =2L 3 for b e,1 o 

b e,2 

eff,2 

L e =0,85 L 1 for b eff,1 

L e =0,70 L 2 for b eff,1 

b 1 

b o 

b 2 

L 1 L 2 

L 3 

L 1 /4 L 1 /2 L 1 /4 L 2 /4 L 2 /2 L 2 /4 

b eff,0 

b eff,1 

b eff,1 

b eff,2 

end supports: b eff 

= b 0 

+ β 1 

b e,1 

+β 2 

b e,2 

β i 

= (0,55+0,025 L e 

/b i 

) ≤ 1,0 

b eff,2 

midspan regions and 

internal supports: 

b eff 

= b 0 

+ b e,1 

+b e,2 

b e,i 

= L e 

/8 

L e 

– equivalent length 

8

Effects of creep of concrete 

G. Hanswille 





primary effects 

Initial sectional 

forces 

redistribution of the sectional 

forces due to creep 

M c,o 

-M c,r 

-z i,c 

a st 

M L 

-N c,o 

N c,r 

z i,st 

M st,o 

M st,r 

N st,o 

-N st,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 





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 

L=S 

L=D 

time-dependent action effects developing affine to the creep coefficient 

action effects caused by shrinkage of concrete 

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

supports) 

M PT (t=∞) 

M PT (t) 

time dependent action 

effects M L =M PT : 

action effects caused by 

prestressing due to imposed 

deformation M L =M D : 

M D 

ϕ(t i ,t o ) 

M PT 

(t i 

) 

ϕ(t ∞ ,t o ) 

ϕ(t,t o ) 

M L =M D 

+ 

δ 

10

Modular ratios taking into account 

effects of creep 

G. Hanswille 





centroidal axis of the concrete section 

z is,L 

a st 

z st 

-z ic,L 

z ist,L 

z c 

z i,L 

centroidal axis of the transformed 

composite section 

centroidal axis of the steel section 

(structural steel and reinforcement) 

Modular ratios: 

E 

n L = no[ 1+ ψL 

ϕ(t,t 

o ) ] no 

= 

E 

a 

cm 

action 

creep multiplier 

short term loading Ψ=0 

permanent action not changing in time Ψ P 

=1,10 

shrinkage Ψ S 

=0,55 

prestressing by controlled imposed deformations Ψ D 

=1,50 

time-dependent action effects Ψ PT 

=0,55 

11

Elastic cross-section properties of the 

composite section taking into account creep 

effects 

G. Hanswille 





-z ic,L 

z ist,L 

z c 

z i,L 

z st 

centroidal axis of the 

concrete section 




steel section 

Modular ratio taking into 

account creep effect: 

nL= 

n0 

(1+ ψL 

ϕ(t,t 

0)) 

n 

o = 

Est 

E (t 

cm 

o 

) 

Transformed cross-section properties of 

the concrete section: 

Distance between the centroidal axes of 

the concrete and the composite section: 

A = A / n J = J 

c,L 

c 

L 

c,L 

c 

/ n 

L 

z = − A 

ic,L 

st 

a 

st 

/ A 

i,L 

Transformed cross-section area of the 

composite section: 

Second moment of area of the 

composite section: 

A = A + A 

i,L 

St 

c,L 

J = J + J + 

i,L 

st 

c,L 

A 

st 

A 

c,L 

a 

2 

st 

/ 

A 

i,L 

12

Effects of cracking of concrete and tension 

stiffening of concrete between cracks 

G. Hanswille 





N s 

N sy 

mean strain ε sm 

=ε s,2 

- βΔε s,r 

Δε 

s 

f 

= β 

ρ 

β =0,4 

ct,eff 

s 

E 

s 

σ c 

(x) 

σ s 

(x) 

σ c 

(x) 

τ v 

σ s,2 

Δεs = βΔε s,r 

ρ 

s 

= A 

s 

/ A 

c 

σ c 

(x) 

N sm 

Δε s,r 

fully 

cracked 

section 

ε sr,1 

ε sr,2 

ε sm,y 

ε sy 

N s 

ε 

ε 

s,2 

σ s 

(x) 

σ 

= 

E 

s2 

s 

N s 

N s,cr 

B C 

A 

stage A: 

stage B: 

stage C: 

uncracked section 

initial crack formation 

stabilised crack formation 

ε sm 

f 

E 

ct 

c 

ε s(x) 

ε c(x) 

βΔε s,r 

x 

13

Influence of tension stiffening of concrete on 

stresses in reinforcement 

G. Hanswille 





ε sm 

N s 

equilibrium: 

z s 

M s 

≈0 

M 

Ma 

= M−Ns 

a 

a 

M a ε a 

-κ 

N 

= − 

a N s 

z a 

N a 

compatibility: 

mean strain in the concrete slab: 

ε 

sm 

= ε 

a 

+ κa 

ε 

sm 

N 

+ 

s 

E A 

a 

a 

N 

+ 

E 

s 

a 

a 

A 

2 

a 

= 

M a 

E 

a 

J 

a 

ε s,2 

ε s,m 

Δε s 

=βΔε s,r 

mean strain in the concrete 

slab: 

ε c 

ε s 

ε 

sm 

= ε 

s2 

−βΔε 

sr 

N 

= 

s 

E A 

s 

s 

f 

−β 

ρ 

ct,eff 

s 

E 

s 

14

Redistribution of sectional forces due to tension 

stiffening 

G. Hanswille 





fully cracked section 

tension stiffening 

z 2 =z st 

-z st,s 

z st,a 

a 

N s,2 

-M s,2 

ΔN ts 

-M a,2 

+ = 

ΔN ts 

a 

-M s 

N s 

-M a 

-M Ed 

-N a,2 -ΔN ts 

-N a 

N s 

α 

N sε 

st 

A 

= 

A 

N s 

st 

a 

J 

J 

st 

a 

ΔN 

ts 

= β 

ΔN ts 

N s,2 

M Ed 

f 

ct,eff 

ρ 

M 

s 

α 

A 

st 

s 

N 

Sectional forces: 

N 

a 

s 

= N 

M = M 

M 

s 

= N 

a 

a2 

s2 

= M 

Ed 

+ ΔN 

− ΔN 

a2 

J 

J 

s 

st 

ts 

ts 

+ ΔN 

= M 

= M 

ts 

a 

Ed 

Ed 

A 

A 

a 

= M 

s 

J 

J 

z 

Ed 

st 

z 

st 

st,s 

st,a 

J st = J 2 

J 

J 

a 

st 

+ ΔN 

− ΔN 

ts 

+ ΔN 

ts 

ts 

a 

15

G. Hanswille 

Stresses taking into account tension stiffening of 



concrete 



fully cracked 

tension stiffening 

N s 

N s,2 

-M s,2 

ΔN ts 

-M s 

z st 

z st,a 

a 

-M a,2 

+ = 

ΔN ts 

a 

-M a 

-M Ed 

-z st,s 

st 

z a 


-N a 

σ 

reinforcement: 

s 

= 

σ 

s,2 

+ β 

f 

ρ 

ctm 

s 

α 

st 

structural steel: 

σ 

a 

= 

σ 

a,2 

ΔN 

− 

A 

ts 

a 

+ 

ΔN 

J 

ts 

a 

a 

z 

a 

α 

st 

= 

A 

A 

st 

a 

J 

J 

st 

a 

σ 

s 

= 

M 

J 

Ed 

st 

z 

st,s 

+ 

β 

f 

ρ 

ctm 

s 

α 

st 

σ 

a 

= 

M 

J 

Ed 

st 

z 

st 

− 

ΔN 

A 

ts 

a 

+ 

ΔN 

J 

ts 

a 

a 

z 

a 

ΔN 

ts 

= 

β 

f 

ρ 

ctm 

s 

A 

α 

s 

16

G. Hanswille 


Influence of tension stiffening on flexural stiffness 




N s 

ε sm 

Curvature: 

z st 

a 

-M s 

-M a 

-M 

κ 

-N a 

ε a 

κ = 

E 

st 

M 

I 

2,ts 

M 

= 

a 

E J 

st 

a 

Effective flexural 

stiffness: 

M− 

N 

= 

E J 

st 

s 

a 

a 

M 

EJ 

E 

st 

J 

2,ts 

= 

Ea 

Ja 

(Ns 

− N 

1− 

M 

s, ε 

) 

a 

E st 

J 2 

E st 

J 2,ts 

E st 

J 1 

κ 

E st J 1 

E st J 2,ts 

E st J 2 

M R 

M Rn 

M 

E st J 1 uncracked section 

E st J 2 fully cracked section 

E st J 2,ts effective flexural 

stiffness taking into 

account tension 

stiffening of concrete 

17

Effects of cracking of concrete - General 

method according to EN 1994-1-1 

G. Hanswille 





E a 

J 1 

E a 

J 2 

– un-cracked flexural stiffness 

– cracked flexural stiffness 

L 1 L 2 

L1,cr L 2,cr 

E a J 1 E a J E a J 1 

2 

ΔM 

• Determination of internal forces by uncracked 

analysis for the characteristic 

combination. 

• Determination of the cracked regions 

with the extreme fibre concrete tensile 

stress σ c,max = 2,0 f ct,m . 

• Reduction of flexural stiffness to E a J 2 in 

the cracked regions. 

• New structural analysis for the new 

distribution of flexural stiffness. 

cracked analysis 

un-cracked analysis 

ΔM Redistribution of 

bending moments due to 

cracking of concrete 

18

Effects of cracking of concrete – 

simplified method 

G. Hanswille 





L 1 

L 2 

0,15 L 1 0,15 L 2 

E a 

J 1 

L 

min 

E a 

J 2 

/L ≥0, 

6 

max 

ΔM II 

For continuous composite beams with 

the concrete flanges above the steel 

section and not pre-stressed, including 

beams in frames that resist horizontal 

forces by bracing, a simplified method 

may be used. Where all the ratios of 

the length of adjacent continuous 

spans (shorter/longer) between 

supports are at least 0,6, the effect of 

cracking may be taken into account by 

using the flexural stiffness E a J 2 over 

15% of the span on each side of each 

internal support, and as the uncracked 

values E a J 1 elsewhere. 

19

G. Hanswille 





Part 3: 

Limitation of crack width 

20

Control of cracking 

G. Hanswille 





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 σ s 

in the reinforcement and the design 

crack width. 

21

Recommended values for w max 

G. Hanswille 





Exposure 

class 

reinforced members, prestressed 

members with unbonded tendons 

and members prestressed by 

controlled imposed deformations 

quasi - permanent 

load combination 

prestressed members with 

bonded tendons 

frequent load combination 

XO, XC1 0,4 mm (1) 0,2 mm 

XC2, XC3,XC4 0,2 mm (2) 

0,3 mm 

XD1,XD2,XS1, 

XS2,XS3 

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) 

G. Hanswille 





Class Description of environment Examples 

XO 

for concrete without reinforcement, for 

concrete with reinforcement : very dry 

no risk of corrosion or attack 

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

N s 

ε 

σ 

σ c,1 

Cracking of concrete (initial crack formation) 

L es 

σ c,1 

w 

ε s 

ε c 

L es 

σ s 

σ s,1 

L es 

L es 

σ s,1 

σ s,2 

L es 

ρ 

Δσ s 

s 

A 

= 

A 

s 

c 

N s 

Equilibrium in longitudinal direction: 

σ 

s 

A 

A s 

ρ s 

f ctm 

s 

= 

σ 

s,1 

A 

s 

+ σ 

c,1 

A 

c 

G. Hanswille 





Compatibility at the end of the introduction 

length: 

σs,1 

σc,1 

εs,1 

=εc,1 

⇒ = 

E E 

σs,1 

= 

σs 

⎡ 

⎢ 

⎣ 

s 

c 

ρs 

no 

⎤ 

1+ ρ 

⎥ 

s no 

⎦ 

s 

o 

E 

n o = 

E 

Change of stresses in reinforcement 

due to cracking: 

σ 

Δσ 

s 

s = σs 

− σs,1 

= 

1+ 

ρ n 

N 

s,r 

= f 

ctm 

A 

c 

( 1+ 

ρ n ) 

s 

o 

cross-section area of reinforcement 

reinforcement ratio 

mean value of tensile strength of concrete 

s 

c 

24

N s 

ε 

Cracking of concrete – introduction length 

w 

ε s 

N s 

Change of stresses in reinforcement 

due to cracking: 

σ 

Δσ 

s 

s = σs 

− σs,1 

= 

1+ 

ρs 

no 

Equilibrium in longitudinal direction 

G. Hanswille 





σ 

L es 

ε c 

L es 

Δσ s 

L 

L 

es 

es 

U 

π 

s 

d 

τ 

s 

sm 

τ 

= Δσs 

As 

πd 

= Δσs 

4 

sm 

2 

s 

introduction length L Es 

ρ 

s 

A 

= 

A 

s 

c 

σ c,1 

σ s 

σ s,1 

L es 

L es 

L 

es 

= 

σs 

d 

4 τ 

s 

sm 

1 

1+ 

n ρ 

o 

s 

E 

n o = 

E 

s 

c 

σ c,1 

σ s,1 

σ s,2 

L es 

τ sm 

U s 

A s 

ρ s 

τ sm 

-perimeter of the bar 

-cross-section area 

-reinforcement ratio 

-mean bond strength 

crack width 

w = 2L 

es 

( ε 

sm 

− 

ε 

cm 

) 

25

N s 

ε 

ε s,m 

ε c,m 

Determination of the mean strains of 

reinforcement and concrete in the stage of initial 

crack formation 

w 

ε s 

(x) 

ε c 

(x) 

Δε s,cr 

N s 

ε cr 

ε s 

Mean bond strength: 

τs,m 

= 

Les 

σ 

s,m 

= 

L 

∫ 

o 

1 Es 

σ 

s 

−β 

τs 

(x)dx 

Δσ 

s 

⇒ β 

G. Hanswille 





≈ 1,8 fctm 

Mean stress in the reinforcement: 

= 

σ 

s 

− Δσ 

Δσ 

s 

sm 

σ 

σ s,m 

L es 

σ s (x) 

L es 

βΔσ s 

Δσ s 

σ s σ s,1 

σ c,1 

L es 

L es 

1 

Δσsm= 

Les 

L es 

∫ 

0 

Δσs(x) 

dx 

4 x 

Δσs(x) 

= ∫ τ 

s 0 s (x) dx 

U 

Mean strains in reinforcement and concrete: 

ε 

s,m 

εc,m 

= 

= 

ε 

s,2 

β εcr 

− 

β 

Δε 

s,cr 

x 

26

N s 

Determination of initial crack width 

w 

N s 

crack width 

G. Hanswille 





ε 

ε s,m 

ε s 

(x) 

ε c 

(x) 

ε s,2 

Δε s,cr 

w = 2L 

es 

( ε 

sm 

− ε 

cm 

εs ,m − εcm 

= ( 1− β) 

εs,2 

) 

ε c,m 

ε cr 

L es 

L es 

L 

es 

= 

σs 

d 

4 τ 

s 

sm 

1 

1+ 

n ρ 

o 

s 

σ s 

τsm 

≈ 1,8 f ctm 

σ s,m 

βΔσ s 

Δσ s 

L es 

L es 

σ s 

σ s,1 

σ c,1 

x 

2 

(1− β) 

σs 

d 

w = 

s 

2 τsm 

Es 

1 

1+ 

no 

ρs 

with β= 0,6 for short term loading und 

β= 0,4 for long term loading 

27

Maximum bar diameters acc. to EC4 

G. Hanswille 





σ s 

[N/mm 2 ] 

maximum bar diameter 

d ∗ s 

for 

w k = 0,4 w k = 0,3 w k = 0,2 

160 40 32 25 

200 32 25 16 

240 20 16 12 

280 16 12 8 

320 12 10 6 

360 10 8 5 

400 8 6 4 

Crack width w: 

sm 

2 

s 

(1 − β) 

σ d 

w = 

2 τ E 

s 

s 

1 

1+ 

n ρ 

Maximum bar diameter for a 

required crack width w: 

d 

s 

= 

w 

2 τ 

sm 

σ 

E 

s 

2 

s 

o 

s 

(1+ 

n 

(1− β) 

o 

≈ 

ρ 

s 

6 

) 

f 

σ 

2 

s 

ct,m 

With τ sm 

= 1,8 f ct,mo 

and the reference 

value for the mean tensile strength of 

concrete f ctm,o 

= 2,9 N/mm 2 follows: 

d 

s 

E 

s 

450 6 5 - 

β= 0,4 for long term loading and 

repeated loading 

d 

d 

* 

s 

* 

s 

= 

w 

k 

w 

≈ 6 

k 

3,6 

f 

f 

ctm,o 

2 

σs 

ctm,o 

2 

σs 

E 

s 

E 

s 

(1+ 

n 

(1− β) 

o 

ρ 

s 

) 

28

Crack width for stabilised crack formation 

G. Hanswille 





f 

E 

N s 

ct 

c 

ε 

ε 

s,2 

ε s(x) 

ε c(x) 

σ 

= 

E 

s2 

s 

s r,max 

= 2 L es 

w 

β= 0,6 for short term loading 



Crack width for high bond bars 

w = sr,max 

( εsm 

− εcm) 

Mean strain of reinforcement and 

concrete: 

εs,m 

= εs,2 

− β Δ εs 

A 

ε s(x) 

- ε 

c fctm 

f 

c(x) εs,m 

= εs,2 

− β = εs,2 

− β 

Es 

As 

Es 

ρ 

f 

ε 

ctm 

cm =β 

Ec 

s r,min 

= L es 

ε 

sm 

− ε 

cm 

= 

σ 

E 

s 

s 

f 

− β 

E 

ctm 

s 

ρ 

s 

(1+ 

n 

o 

ρ 

s 

ctm 

) 

s 

29

Crack width for stabilised crack formation 

G. Hanswille 





N s 

w 

The maximum crack spacing s r,max 

in the 

stage of stabilised crack formation is twice 

the introduction length L es 

. 

ε 

ε 

s,2 

= 

σ 

E 

s 

s 

w = s 

r,max 

( ε 

sm 

− 

ε 

cm 

) 

ε s(x) 

ε c(x) 

f 

E 

ct 

c 

ε s(x) 

- ε c(x) 

L 

es 

= 

f 

U 

ctm 

s 

A 

τ 

c 

sm 

f 

= 

ρ 

ctm 

s 

4 

τ 

d 

s 

sm 

maximum crack width for s r 

= s r,max 

s r,max 

= 2 L es 

s r,min 

= L es 




= f d ⎛ σ f 

⎞ 

w 

ctm s 

⎜ s 

− β 

ctm 

(1+ 

no 

ρs 

⎟ 

2 τ ρ ⎝ E ρ E 

) 

sm s s s s ⎠ 

30

Crack width and crack spacing according 

Eurocode 2 

G. Hanswille 





Crack width 

w = s 

r,max 

( ε 

sm 

− 

ε 

cm 

) 

ε 

sm 

− ε 

cm 

= 

σ 

E 

s 

s 

− 

β 

f 

E 

ctm 

s 

ρ 

s 

(1+ 

n 

o 

ρ 

s 

) 

≥ 

0,6 

σ 

E 

s 

s 


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

Crack spacing 

In Eurocode 2 for the maximum crack spacing a semiempirical 

equation based on test results is given 

s 

s 

r,max = 3,4c + k1⋅ 

k2 

⋅0,425 

d s 

-diameter of the bar 

ρs 

d 

c- concrete cover 

k 1 

k 2 

coefficient taking into account bond properties of 

the reinforcement with k 1 

=o,8 for high bond bars 

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 M cr and 

the normal force of the concrete slab in the 

stage of initial cracking 

G. Hanswille 





h c 

cracking moment M cr 

z o 

z i,st 

N c+s 

z io 

a st 

M 

M 

cr 

cr 


: 

σc 

+ σc, 

ε = ct,eff = 

M c+s 

σ c 

M cr 

= 

= 

[ f − σ ] 

ct,eff 

[ f − σ ] 

ct,eff 

f 

c, ε 

c, ε 

k 

z 

z 

1 

n 

o 

f 

ctm 

o 

J 

+ h 

ic,o 

c 

n 

io 

/ 2 

o 

(1+ 

h 

J 

c 

io 

/(2 

z 

o 

) 

primary effects due to shrinkage 

M c,ε 

N c,ε 

σ cε 

sectional normal force of the concrete 

slab: 

N 

N 

cr 

cr 

= M 

= 

A 

cr 

c 

(f 

A 

co 

ct,eff 

zo 

+ A 

J 

1+ 

h 

− σ 

c 

io 

s 

c, ε 

/(2 

) 

z 

is 

+ N 

c+ 

s, ε 

( 1+ ρ n ) 

z 

o 

) 

s 

o 

+ N 

c+ 

s, ε 

N 

cr 

= A 

c 

f 

ct,eff 

(1+ ρ 

s 

n 

0 

) 

⎡ 

⎢ 

⎢ 

⎢ 

⎢ 

⎣ 

1 + 

h 

c 

k c 

k c,ε 

≈ 0,3 

1 

/(2 

z 

o 

) 

+ 

N 

c+ 

s, ε 

A 

c 

− 

f 

A 

c 

ct,eff 

σ 

c, ε 

( 1+ ρ n ) 

1+ 

hc 

/(2 zo 

) 

(1 + ρ n ) 

s 

s 

0 

o 

⎤ 

⎥ 

⎥ 

⎥ 

⎥ 

⎦ 

32

Simplified solution for the cracking moment 

and the normal force in the concrete slab 

G. Hanswille 






M c+s 

σ c 

simplified solution for the normal 

force in the concrete slab: 

h c 

z o 

z i,st 

N c+s 

M cr 

N 

cr 

≈ 

A 

c 

f 

ctm 

k 

s 

⋅k 

⋅k 

c 

primary effects due to shrinkage 

k = 0,8 

k s = 0,9 

coefficient taking into account the effect of 

non-uniform self-equilibrating stresses 

coefficient taking into account the slip 

effects of shear connection 

M c+s,ε 

N c+s,ε 

σ cε 

k 

c 

= 

1+ 

1 

hc 

2 z 

o 

+ 

0,3 

≤1,0 

cracking moment 

shrinkage 

33

Determination of minimum reinforcement 

G. Hanswille 





M c 

M c,ε 

A 

s 

h c 

z o 

o 

N c 

M cr 

N c,ε 

z i,o 

cracking 

moment 

shrinkage 

M a,ε 

N a,ε 

Ac 

fct,eff 

1 

≥ k ks 

kc 

kc 

= + 0,3 ≤ 1, 0 

σ 

1+ 

h z 

s 

c 

o 

d 

s 

= d 

∗ 

s 

f 

ct,eff 

f 

ct, 

f cto = 2,9 N/mm 2 

k = 0,8 

k s = 0,9 

k c 

d ∗ s 

d s 

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 


modified bar diameter for other concrete strength classes 

σ s stress in reinforcement acc. to Table 1 

effective concrete tensile strength 

f ct,eff 

34

Control of cracking due to direct loading – 

Verification by limiting bar spacing or bar diameter 

G. Hanswille 





A c 

fully cracked tension stiffening 

N s,2 

-z st,s 

-M s,2 -M s 

ΔN ts 

N s 

z st 

z st,a 

a 

-M a,2 

+ = 

ΔN ts 

a 

-M a 

-M Ed 

A s 

A a 

z a 


-N a 

The calculation of stresses is 

based on the mean strain in the 

concrete slab. The factor β 

results from the mean value of 

crack spacing. With s rm ≈ 2/3 

s r,max results β ≈2/3 ·0,6 = 0,4 

stresses in reinforcement 

taking into account tension 

stiffening for the bending 

moment M Ed 

of the quasi 

permanent combination: 

A 

ρ 

s 

s = β =0,4 

A 

c 

σ 

σ 

s 

s 

= σ 

= 

M 

α 

s,2 

J 

st 

Ed 

2 

+ 

z 

A 

= 

A 

2 

a 

Δσ 

st,s 

J 

J 

2 

a 

ts 

+ 

β 

f 

ρ 

ct,eff 

s 

α 

st 

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 

G. Hanswille 





Table 1: Maximum bar diameter 

Table 2: Maximum bar spacing 

σ s 

[N/mm 2 ] 


d ∗ s 

for 

w k = 0,4 w k = 0,3 w k = 0,2 

160 40 32 25 

200 32 25 16 

240 20 16 12 

280 16 12 8 

320 12 10 6 

360 10 8 5 

400 8 6 4 

450 6 5 - 

σ s 

[N/mm 2 ] 

160 

200 

240 

280 

320 

360 

maximum bar spacing in [mm] 

for 

w k = 0,4 

300 

300 

250 

200 

150 

100 

w k = 0,3 

300 

250 

200 

150 

100 

50 

w k = 0,2 

200 

150 

100 

50 

- 

- 

36

Direct calculation of crack width w for 

composite sections based on EN 1992-2 

G. Hanswille 





A s 

-M s 

N s 

-M a 

σ s, 

M Ed 

-z st,s 

z st 

crack width for high bond bars: 

σ 

α 

s 

st 

= 

M 

J 

A 

= 

A 

Ed 

st 

st 

a 

J 

J 

a 

z 

st 

st,s 

+ β 

f 

ρ 

ct,eff 

s 

c 

α 

st 

A 

ρ 

s 

s = β = 0, 4 

A 

w = s 

ε 

s 

sm 

r,max 

− ε 

r,max 

cm 

-N a 

( εsm 

− εcm) 

σs 

f 

− β 

ctm 

E E ρ 

s 

= 

s 

= 3,4c + 0,34 

s 

d 

ρ 

s 

s 

s 

(1+ 

n 

o 

ρ 

s 

) 

≥ 

0,6 

σ 

E 

s 

c - concrete cover of reinforcement 

37

Stresses in reinforcement in case of bonded 

tendons – initial crack formation 

G. Hanswille 





σ s,1 

Δσ p1 

A p , d p 

A s , d s 

Equilibrium at the crack: 

σs 

As 

+ Δσp 

Ap 

= N= 

σ 

σ s =σ s1 +Δσ s As 

= π ds 

τsm 

L 

s 

σ p 

Δσp 

Ap 

= π dp 

τpm 

L 

N 

σs 

− σs1 

δs 

= δp 

⇒ Les 

= 

Es 

Δσ s 

Δσ p 

Stresses: 

N 

L es 

σs 

= 

Δσp 

As 

+ ξ1 

Ap 

L ep 

τ pm τ sm 

τ 

σ p =σ po +σ p1 +Δσ p 

f 

ct,eff 


e,s 

ep 

Compatibility at the crack: 

ξ 

1 

= 

τ 

pm 

sm 

d 

d 

s 

v 

Δσ 

p 

= 

A 

E 

p 

A 

c 

− Δσ 

s 

ξ 

1 

+ ξ 

(1+ 

n 

p1 

With E s ≈E p and σ s1 =Δσ p1 =0 results: 

N 

1 

L 

A 

ep 

p 

o 

ρ 

tot 

) 

38

Stresses in reinforcement for final crack 

formation 

G. Hanswille 





Δσ p2 

σ s2 

Δσ p2 

Δσ p1 

σ s1 

σ s 

A s , d s 

A p , d p 

σ c 

σ c =f ct,eff 

x 

Equilibrium at the crack: 

N − P 

Maximum crack spacing: 

f 

s 

ct 

δ 

s 

A 

r,max 

σ 

c 

s2 

o 

= δ 

s 

= 

d 

= 

2τ 

− σ 

=σ 

p 

r,max 

= 

2 

s1 

s2 

sm 

σ 

s 

s 

= 

s2 

A 

[ τ n d π + τ n d π ] 

f 

(A 

sm 

ct,eff 

s 

r,max 

s 

A 

+ ξ 

c 

2 

U 

A 

− β( σs2 

− σs1) 

Δσp,2 

− β( 

Δσp2 

− Δσ 

= 

E 

E 

s 

s 

A 

Compatibility at the crack: 

s 

2 

+ Δ σ 

s 

s 

p2 

p 

τ 

) 

sm 

pm 


A 

p 

σ 

p2 

p 

p 

− σ 

p1 

s 

= 

r,max 

2 

p 

U 

A 

p 

p 

τ 

pm 

p1 

) 

s r,max 

mean crack spacing: s r,m ≈2/3 s r,max 

39

Determination of stresses in composite 

sections with bonded tendons 

G. Hanswille 





A p 

A s 

-z st,s 

σ s, 

σ p 

M Ed 

Stresses σ * s in reinforcement 

at the crack location 

neglecting different bond 

behaviour of reinforcement 

and tendons: 

z st 

Stresses in reinforcement taking into account the 

different bond behaviour: 

σ 

* 

s 

M 

= 

J 

Ed 

st 

z 

st,s 

f 

+ β 

ρ 

ctm 

tot 

α 

st 

Ast 

Jst 

αst 

= 

β =0, 4 

A J 

a 

a 

σ 

s 

Δσ 

= σ 

p 

* 

s 

= σ 

+ 0,4 f 

* 

s 

ct,eff 

− 0,4 f 

⎡ 

⎢ 

⎢ 

⎣ 

A 

ct,eff 

s 

A 

+ ξ 

c 

2 

1 

A 

p 

⎡ 

A 

⎢ c 

⎢ A + 

⎣ 

s A 

p 

− 

Ac 

A + A 

− 

A 

s 

s 

ξ 

2 

1 

A 

+ ξ 

p 

c 

2 

1 

A 

⎤ 

⎥ 

⎥ 

⎦ 

p 

= σ 

* 

s 

⎤ 

⎥ = σ 

⎥ 

⎦ 

+ 0,4 f 

* 

s 

− 0,4 f 

ct,eff 

ct,eff 

⎡ 1 

⎢ 

⎣ρ 

eff 

⎡ 1 

⎢ 

⎢⎣ 

ρ 

tot 

1 

− 

ρ 

tot 

ξ 

− 

ρ 

2 

1 

eff 

⎤ 

⎥ 

⎦ 

⎤ 

⎥ 

⎥⎦ 

ρ 

ρ 

tot 

eff 

= 

= 

A 

A 

s 

A 

s 

+ A 

c 

+ ξ 

A 

2 

1 

c 

p 

A 

p 

40

G. Hanswille 





Part 4: 

Deformations 

41

Deflections 

G. Hanswille 





Effects of cracking of concrete 

L 1 

L 2 

0,15 L 1 0,15 L 2 

E a 

J 2 

E a 

J 1 

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 

g c 

F steel member 

- influence of local yielding of 

structural steel at internal supports, 

- influence of incomplete interaction. 

F 

composite member 

42

Deformations and pre-cambering 

G. Hanswille 





δ 1 

δ 2 

δ 3 

δ 4 

combination 

general quasi - 

permanent 

risk of damage of adjacent 

parts of the structure (e.g. 

finish or service work) 

quasi – 

permanent 

(better frequent) 

limitation 

δ max ≤ L / 

δ 1 deflection of the steel girder 

250 

δ c deflection of the composite 

girder 

δ w ≤ L / 500 Pre-cambering of the steel 

girder: 

δ δ 

1 

p = δ 1 + δ 2 + δ 3 +ψ 2 δ 4 

δ p 

δ max maximum deflection 

δ c δ w 

δ w 

δ max 

effective deflection for finish 

and service work 

δ 1 – self weight of the structure 

δ 2 – loads from finish and service work 

δ 3 – creep and shrinkage 

δ 4 – variable loads and temperature effects 

43

Effects of local yielding on deflections 

G. Hanswille 





For the calculation of deflection of un-propped beams, account may 

be taken of the influence of local yielding of structural steel over a 

support. 

For beams with critical sections in Classes 1 and 2 the effect may be 

taken into account by multiplying the bending moment at the support 

with an additional reduction factor f 2 and corresponding increases are 

made to the bending moments in adjacent spans. 

f 2 = 0,5 if f y is reached before the concrete slab has 

hardened; 

f 2 = 0,7 if f y is reached after concrete has hardened. 

This applies for the determination of the maximum deflection but not 

for pre-camber. 

44

More accurate method for the determination of 

the effects of local yielding on deflections 

G. Hanswille 





L 1 L 2 

l cr l cr 

+ 

E a J 1 

z 2 - 

- M el,Rk 

σ a =f yk f yk 

E a J 2 

(EJ) eff 

E a J eff 

ΔM 

E a J eff 

M pl,Rk 

E a J 2 

M el,Rk 

M pl,Rk 

M Ed 

45

G. Hanswille 


Effects of incomplete interaction on deformations Institute for Steel and 



P Rd 

P 

The effects of incomplete interaction may be ignored 

provided that: 

• The design of the shear connection is in accordance 

with clause 6.6 of Eurocode 4, 

• either not less shear connectors are used than half 

the number for full shear connection, or the forces 

resulting from an elastic behaviour and which act on 

the shear connectors in the serviceability limit state 

do not exceed P Rd 

and 

• in case of a ribbed slab with ribs transverse to the 

beam, the height of the ribs does not exceed 80 mm. 

c D 

P 

s 

P 

s 

s u 

46

Differential equations in case of incomplete 

interaction 

G. Hanswille 





a 

z a 

(w) 

E c 

, A c 

, J c 

v L 

a c N c 

z c 

v L 

a a 

E a 

, A a 

, J a 

M c 

M a 

N a 

V a 

V c 

V c 

+dV c 

M c 

+ dM c 

N c 

+dN c 

M a 

+ dM a 

N a 

+dN a 

u 

u 

, 

′ 

c u c 

, 

′ 

a u a 

= ε 

= ε 

a 

c 

Slip: 

sv = ua 

−uc 

+ w′ 

a 

x 

Ec 

Ac 

Ea 

Aa 

(Ec 

Jc 

c 

V a 

+dV a 

dx 

u′′ 

c + cs 

(ua 

−uc 

+ w′ 

a) = 0 

u′′ 

c − cs 

(ua 

−uc 

+ w′ 

a) = 0 

+ Ea 

Ja 

) w′′′′ 

− cs 

a(u′ 

a − u′ 

c + w′′ 

a) = q 

N = E A u′ 

a 

c 

a 

c 

N = E A u′ 

a 

a 

c 

Mc = − Ec 

Jc 

w ′′ 

Ma = −Ea 

Ja 

w ′′ 

V 

c 

= −E 

c 

J 

c 

Va = −EaJa 

w′′ 

′ 

w′′′ 

≈ 0 

47

Deflection in case of incomplete interaction for 

single span beams 

G. Hanswille 





F 

concrete section 


A co 

=A c 

/n o 

, J co 

= J c 

/n o 

n o 

=E a 

/E c 

w 

L 

w 

= 

3 

F L 

48 Ea 

Ii,o 

⎡ 

⎢ 

⎢1 

+ 

⎢ 

⎣ 

12 

α λ 

2 

− 

48 

α λ 

3 

q 

λ ⎤ 

sinh 

2 

( ) 

2 ⎥ 

sinh( λ) 

⎥ 

⎥ 

⎦ 

steel section 

Aa , J a 

A io 

, J io 

2 

λ 

= 

1+ α 

α β 

L 

β 

= 

Ea 

Ac,o 

Ai,o 

c 

s 

Aa 

L² 

w 

= 

5 

384 

q 

E 

a 

L 

J 

4 

i,o 

⎡ 

⎢ 

⎢1 

+ 

⎢ 

⎣ 

48 

5 

α 

1 

λ 

2 

− 

384 

5 

α 

1 

λ 

4 

λ ⎤ 

cosh( ) − 1 

2 ⎥ 

λ 

⎥ 

cosh( ) ⎥ 

2 ⎦ 

α 

= 

Ja 

1 

Ji,o 

+ Jc,o 

− 1 

48

Mean values of stiffness of headed studs 

G. Hanswille 





c s 

spring constant per stud: 

spring constant of the shear 

connection: 

s 

C u 

D = P 

Rd 

CD 

n 

c s = 

e 

L 

t 

P 

type of shear connection 

C D 

[kN/ cm] 

P Rd 

e L 

n t 

=2 

P 

P 

headed stud ∅ 19mm 

in solid slabs 


in solid slabs 

2500 

3000 

s u 

c D 

s 

s 

headed studs ∅ 25mm 

in solid slab 


with Holorib-sheeting and 

one stud per rib 

3500 

1250 


with Holorib-sheeting and 

one stud per rib 

1500 

49

Simplified solution for the calculation of 

deflections in case of incomplete interaction 

G. Hanswille 





q 

( ξ) = q sinπξ 

The influence of the flexibility of the shear connection is 

taken into account by a reduced value for the modular 

ratio. 

x 

ξ = 

L 

L 

w 

o 

L 

= q 

π 

4 

4 

E 

cm 

J 

c 

+ E 

a 

J 

a 

+ 

1 

β E 

E 

o 

a 

A 

cm 

a 

A 

+β 

c 

o 

E 

E 

a 

A 

cm 

a 

A 

c 

a 

2 

L 

= q 

π 

4 

4 

E 

a 

1 

J 

io,eff 

z c 

E c 

, A c 

, J c 

E a 

, A a 

, J a 

a 

z a 

ε c 

ε a 

M c 

Ac,eff 

Aa 

2 

Jio,eff 

= Jc,o 

+ Ja 

+ 

a 

N c Ac,eff 

+ Aa 

Ac 

A c,eff = 

M a 

no,eff 

N a 

effective modular ratio for the 

concrete slab 

n 

o,eff 

= n 

o 

(1+ 

β 

s 

) 

β 

s 

π 

= 

2 

E 

L 

cm 

2 

cs 

A 

c 

50

Comparison of the exact method with the 

simplified method 

G. Hanswille 





L 

q 

1,5 

1,4 

1,3 

w/w c 

η=0,4 

c D = 1000 KN/cm 

99 

51 

450 mm 

w o - 

η 

w 

b eff 

E cm 

= 3350 KN/cm² 

deflection in case of 

neglecting effects from slip 

of shear connection 

degree of shear connection 

1,2 

1,1 

L [m] 

1,0 

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

1,25 

1,2 

1,15 

1,1 

w/w c 

η=0,8 

η=0,4 

1,05 η=0,8 

exact solution 

simplified solution with n o,eff 

c D = 2000 KN/cm 

L [m] 

1,0 

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

51

Deflection in case of incomplete interactioncomparison 

with test results 

G. Hanswille 





1875 1875 

F 

load case 1 

load case 2 

F 

F/2 F/2 

200 

F [kN] 

Deflection at 

midspan 

1875 3750 

1875 

7500 

150 

load case 2 

1500 

50 

100 

IPE 270 

270 175 

445 

50 

0 

load case 1 

20 40 60 

δ 

[mm] 

52

Deflection in case of incomplete interaction- 

Comparison with test results 

G. Hanswille 





F 

F 

160 push-out test 

s 

780 

120 

80 

125 

50 

second moment 

of area 

cm 4 

40 

s[mm] 

10 20 30 40 

Load case 1 

F= 60 kN 

Load case 2 

F=145 kN 

Deflection at midspan in mm 

Test - 11,0 (100%) 20,0 (100 %) 

Theoretical value, neglecting flexibility 

of shear connection 

Theoretical value, taking into account 

flexibility of shear connection 

J io = 32.387,0 7,8 (71%) 12,9 (65%) 

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

53

G. Hanswille 





Part 5: 


54

Limitation of Stresses 

G. Hanswille 





σ c 

- + 

σ a 

+ 

σ s 

M Ed 

M Ed 

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. 

σ a 

- 

combination stress limit recommended 

values k i 

structural steel characteristic σ Ed 

≤ k a 

f yk 

k a 

=1,00 

headed studs characteristic P Ed 

≤ k s 

P Rd 

k s 

=0,75 

reinforcement characteristic σ Ed 

≤ k s 

f sk 

k s 

=0,80 

concrete characteristic σ Ed 

≤ k c 

f ck 

k c 

= 0,60 

55

Local effects of concentrated longitudinal 

shear forces 

G. Hanswille 






steel section 

g c 

b c 

b eff 

x 

y 

z io 

z 

A c,eff 

M Ed 

(x) 

V Ed 

(x) 

+ 

M Ed 

V Ed 

+ 

- 

Concentrated longitudinal shear force at 

sudden change of cross-section 

L v =b eff 

2 MEd 

A 

v L,Ed,max = 

E / E J 

a 

c 

c,eff 

io 

b 

z 

eff 

io 

longitudinal shear forces 

N c 

v L,Ed,max 

+ 

- 

56

Local effects of concentrated longitudinal 

shear forces 

G. Hanswille 





system 

g c,d 

FE-Model 

L = 40 m 

cross-section 

b c 

=10 m 

14x2000 

z 

500x20 

800x60 

300 

y 

P 

shear connectors 

C D 

= 3000 kN/cm 

per stud 

δ 

57

Ultimate limit state - longitudinal shear forces 

G. Hanswille 





x 

L = 40 m 

ULS 

P 

FE-Model: 

c D 

s 

FE-Model 

P 

s 

c D 

EN 1994-2 

x [cm] 

58

Serviceability limit state - longitudinal 

shear forces 

G. Hanswille 





v L,Ed 

[kN/m] 

500 

SLS 

EN 1994-2 

0 

x [cm] 

200 400 600 800 1000 1200 1400 1600 1800 2000 

-500 

-1000 

P FE-Model: 

-1500 

c D 

s 

-2000 

-2500 

FE-Model 

-3000 

-3500 

-4000 

x 

L = 40 m 

P 

c D 

s 

59

G. Hanswille 





Part 6: 

Vibrations 

60

Vibration- General 

G. Hanswille 





EN 1994-1-1: 

EN 1990, A1.4.4: 

The dynamic properties of floor beams should satisfy 

the criteria in EN 1990,A.1.4.4 

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 





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 

G. Hanswille 





Span length 

F(x,t) 

The pacing rate f s 

dominates the dynamic 

effects and the resulting dynamic loads. The 

speed of pedestrian propagation v s 

is a 

function of the pacing rate f s 

and the stride 

length l s 

. 

F(x,t) 

time t 

l s 

x k 

pacing 

rate 

f s 

[Hz] 

forward 

speed 

v s 

= f s 

l s 

[m/s] 

stride 

length 

l s 

[m] 

slow walk ∼1,7 1,1 0,6 

t k 

normal walk ∼2,0 1,5 0,75 

fast walk ∼2,3 2,2 1,00 

l s 

x k 

t s 

x 

slow running 

(jog) 

fast running 

(sprint) 

∼2,5 3,3 1,30 

> 3,2 5,5 1,75 

63

Vibration –vertical vibrations due to 

walking of one person 

G. Hanswille 





F i 

(t) 

right foot 

left foot 

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. 

F(t) 

time t 

⎡ 3 

⎤ 

F(t) = Go 

⎢1 

+ ∑ αn 

sin ( 2 n π fs 

t − Φn 

) ⎥ 

⎢⎣ 

n= 

1 

⎥ ⎦ 

both feet 

Fouriercoefficients 

and α 2 

α 1 

=0,4-0,5 Φ 1 

=0 

=0,1-0,25 Φ 2 

=π/2 

phase angles: 

α 3 

=0,1-0,15 Φ 3 

=π/2 

t s 

=1/f s 

G o 

weight of the person (800 N) 

1. step 2. step 3. step α n 

coefficient for the load component of n-th harmonic 

n number of the n-th harmonic 

f s 

pacing rate 

Φ n 

phase angle oh the n-th harmonic 

64

Vibration – vertical vibrations due to walking 

of persons 

G. Hanswille 





c 

F(t) 

M gen 

w(t) 

δ 

F n 

(t) 

m 

w(x k 

,t) 

L/2 

k a 

F n 

(t) 

x k 

w(t) 

L 

acceleration 

F 

w(t) = ka 

M 

maximum acceleration a, vertical deflection w and 

maximum velocity v 

a 

w 

F 

( 

f L / ) 

n π 

max = 

− 

E 

δ v 

a 

s 

max = ka 

1− 

e 

(2 π f 

Mgen 

δ 

a 

vmax 

= 

2 π f 

f E 

F n 

δ 

v s 

k a 

M gen 

n 

gen 

π 

δ 

sin (2 π f 

E 

t) 

( 

−δ f t 

) 

1− 

e 

natural frequency 

load component of n-th harmonic 

logarithmic damping decrement 

forward speed of the person 

factor taking into account the different 

positions x k 

during walking along the beam 

generated mass of the system 

(single span beam: M gen 

=0,5 m L) 

E 

t = 

L 

v s 

E 

E 

) 

2 

65

Logarithmic damping decrement 

G. Hanswille 





6 

5 

4 

3 

2 

1 

results of measurements in buildings 

Damping 

ratioξ [%] 

δ = 2 

π 

ξ 

with finishes 

without finishes 

3 6 9 12 

f E 

[Hz] 

For the determination of the maximum 

acceleration the damping coefficient ζ or 

the logarithmic damping decrement δ 

must be determined. Values for composite 

beams are given in the literature. The 

logarithmic damping decrement is a 

function of the used materials, the 

damping of joints and bearings or support 

conditions and the natural frequency. 

For typical composite floor beams in 

buildings with natural frequencies 

between 3 and 6 Hz the following values 

for the logarithmic damping decrement 

can be assumed: 

δ=0,10 

floor beams without not loadbearing 

inner walls 

δ=0,15 floor beams with not loadbearing 

inner walls 

66

Vibration –vertical vibrations due to walking of 

persons 

G. Hanswille 





3 

∑ 

F(t) = Go 

+ Fn 

sin ( 2 n π fs 

t − Φn 

) 

n= 

1 

F(t)/G o 

0,4 

0,2 

0,1 

f s 

=1,5-2,5 Hz 

2f s 

=3,0-5,0 Hz 

3f s 

=4,5-7,5 Hz 

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

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 

67

Limitation of acceleration-recommended 

values acc. to ISO 10137 

G. Hanswille 





acceleration [m/s 2 ] 

0,1 

0,05 

0,01 

0,005 

natural frequency of 

typical composite 

beams 

basic curve a o 

Multiplying factors K a 

for the basic curve 

Residential (flats, hospitals) K a 

=1,0 

Quiet office K a 

=2-4 

General office (e. g. schools) K a 

=4 

a ≤ 

a o K a 

1 5 10 50 100 

frequency [Hz] 

68

G. Hanswille 





Thank you very 

much for your kind 

attention 

69

G. Hanswille 





Thank you very 

much for your kind 

attention 

70

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