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Annex G[informative]

Balanced summation model for the calculation of the fire resistanceof composite columns with partially encased steel sections, for

bending around the weak axis, exposed to fire all around the columnaccording to the standard temperature-time curve .

Z

Yh

bc,fi

e f

h w,fi

bc,fi

b

e wu2

u1

Figure G.1: Reduced cross-section for structural fire design

G.1 Introduction

(1) This calculation model is based on the principles and rules given in 4.3.5.1, but has been developedonly for bending around the axis Z such as:

Rd,pl,fizz,Rd,fi NN = (G.1)

(2) For the calculation of the design value of the plastic resistance to axial compression Rd,pl,fiN and ofthe effective flexural stiffness z,eff,fi)EI( in the fire situation, the cross-section is divided into fourcomponents:

- the flanges of the steel profile;

- the web of the steel profile;

- the concrete contained by the steel profile and

- the reinforcing bars.

(3) Each component may be evaluated on the basis of a reduced characteristic strength, a reducedmodulus of elasticity and a reduced cross-section in function of the standard fire resistance R30, R60,R90 or R120.

(4) The design value of the plastic resistance to axial compression and the effective flexural stiffness ofthe cross-section may be obtained, according to (4) and (5) of 4.3.5.1, by a balanced summation of thecorresponding values of the four components.

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(5) Strength and deformation properties of steel and concrete at elevated temperatures complies with thecorresponding principles and rules of 3.1 and 3.2.

G.2 Flanges of the steel profile

(1) The average flange temperature may be determined from:

( )VAk mtt,ot,f += (G.2)where:

t is the duration in minutes of the fire exposure

VAm is the section factor in m-1

, with Am = 2 (h + b) in [m] and V = h b in [m]

t,o is a temperature in C given in Table G.1

tk is an empirical coefficient given in Table G.1.

Table G.1

Standard Fire Resistancet,o

[C]tk

[mC]R30 550 9,65R60 680 9,55R90 805 6,15

R120 900 4,65

(2) On behalf of the temperature t,f = the corresponding maximum stress level and the modulus ofelasticity are determined from:

,yf,ayt,f,ay kff = and (G.3)

,Ef,at,f,a kEE = with ,yk and ,Ek following Table 3.2 of 3.2.1 (G.4)(3) The design value of the plastic resistance to axial compression and the flexural stiffness of the twoflanges of the steel profile in the fire situation are determined from:

( )a,fi,Mt,f,ayff,Rd,pl,fi feb2N = and (G.5)

( ) 6beE)EI( 3ft,f,az,f,fi = (G.6)G.3 Web of the steel profile

(1) The part of the web with the height fi,wh and starting at the inner edge of the flange may be neglected(see Figure G.1). This part is determined from:

( ) ( )( )hH0,16-11e2h5,0h tffi,w = where tH is given in Table G.2. (G.7)

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Table G.2Standard Fire Resistance

tH

[mm] R 30 350 R 60 770 R 90 1100 R 120 1250

(2) The maximum stress level is obtained from:( )h0,16H-1ff tw,ayt,w,ay = (G.8)

(3) The design value of the plastic resistance to axial compression and the flexural stiffness of the web ofthe steel profile in the fire situation are determined from:

( )[ ] a,fi,Mt,w,ayfi,wfww,Rd,pl,fi fh2e2heN = (G.9)( )[ ] 12eh2e2hE)EI( 3wfi,wfw,az,w,fi = (G.10)

G.4 Concrete

(1) An exterior layer of concrete with a thickness fi,cb may be neglected in the calculation (see FigureG.1). The thickness fi,cb is given in Table G.3, with VAm , the section factor in m-1 of the entirecomposite cross-section.

Table G.3

Standard Fire Resistance fi,cb [mm]

R30R60

R90

R120

4,015,0

0,5 ( VAm ) + 22,52,0 ( VAm ) + 24,0

(2) The average temperature in concrete t,c is given in Table G.4 in function of the section factorVAm of the entire composite cross-section and for the standard fire resistance classes.

Table G.4R30 R60 R90 R120

VAm[m-1]

t,c[C]

VAm[m-1]

t,c[C]

VAm[m-1]

t,c[C]

VAm[m-1]

t,c[C]

42346-

-

-

-

136300400

-

-

-

-

49

2150-

-

-

214300400600

-

-

-

46

133354-

-

256300400600800

-

-

459

23384143

265300400600800900

1000

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(3) On behalf of the temperature t,c = the secant modulus of concrete is obtained from:

,cu,cc,cu,csec,,c kffE == with ,ck and ,cu following Table 3.3 of 3.2.2 (G.11)

(4) The design value of the plastic resistance to axial compression and the flexural stiffness of theconcrete in the fire situation are determined from:

( ) ( )( ){ }c,fi,M,csfi,cwfi,cfc,Rd,pl,fi fAb2ebb2e2h86,0N = (G.12)

where As

is the cross-section of the reinforcing bars, and 0,86 is a calibration factor.

( ) ( ) ( )( ){ }[ ]z,s

3w

3

fi,cfi,cfsec,,cz,c,fi I12eb2bb2e2hEEI = (G.13)where z,sI

is the second moment of area of the reinforcing bars related to the central axis Z of thecomposite cross-section.

G.5 Reinforcing bars

(1) The reduction factor t,yk of the yield point and the reduction factor t,Ek of the modulus of elasticity

of the reinforcing bars, are defined in function of the standard fire resistance and the geometrical averageu of the axis distances of the reinforcement to the outer borders of the concrete (see Tables G.5 andG.6).

Table G.5: Reduction factor ky,t for the yield point fsy of the reinforcing bars

u[mm]StandardFire Resistance

40 45 50 55 60

R30 1 1 1 1 1R60 0,789 0,883 0,976 1 1R90 0,314 0,434 0,572 0,696 0,822

R120 0,170 0,223 0,288 0,367 0,436

Table G.6: Reduction factor kE,t for the modulus of elasticity Es of the reinforcing bars

u[mm]StandardFire Resistance

40 45 50 55 60

R30 0,830 0,865 0,888 0,914 0,935R60 0,604 0,647 0,689 0,729 0,763R90 0,193 0,283 0,406 0,522 0,619

R120 0,110 0,128 0,173 0,233 0,285

(2) The geometrical average u of the axis distances 1u and 2u is obtained from:

21 uuu = (G.14)where:

1u is the axis distance from the outer reinforcing bar to the inner flange edge [mm]2u is the axis distance from the outer reinforcing bar to the concrete surface [mm]

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Note: If ( )21 u-u > 10 mm, then ( )10+uuu 22= , or ( )12 u-u > 10 mm, then ( )10+uuu 11= .

(3) The design value of the plastic resistance to axial compression and the flexural stiffness of thereinforcing bars in the fire situation are obtained from:

s,fi,Msyt,yss,Rd,pl,fi fkAN = (G.15)( ) z,sst,Ez,s,fi IEkEI = (G.16)G.6 Calculation of the axial buckling load at elevated temperatures

(1) According to (4) of G.1, the design value of the plastic resistance to axial compression and theeffective flexural stiffness of the cross-section in the fire situation are determined from:

s,Rd,pl,fic,Rd,pl,fiw,Rd,pl,fif,Rd,pl,fiRd,pl,fi NNNNN +++= (G.17)

( ) ( ) ( ) ( ) ( ) z,s,fi,sz,c,fi,cz,w,fi,wz,f,fi,fz,eff,fi EIEIEIEIEI +++= (G.18)where i, is a reduction coefficient depending on the effect of thermal stresses. The values of i, aregiven in Table G.7.

Table G.7

Standard Fire Resistance f, w, c, s,

R30 1,0 1,0 0,8 1,0R60 0,9 1,0 0,8 0,9R90 0,8 1,0 0,8 0,8

R120 1,0 1,0 0,8 1,0

(2) The Euler buckling load or elastic critical load follows by:

( ) 2z,eff,fiz,cr,fi EIN pi l= (G.19)where:

l is the buckling length of the column in the fire situation.

(3) The non-dimensional slenderness ratio is obtained from:

cr,zfi,Rpl,fi, NN= (G.20)

where:

R,pl,fiN is the value of Rd,pl,fiN according to (1) when the factors c,fi,Ma,fi,M , and s,fi,M are takenas 1,0.

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(4) Using and the buckling curve c of EN 1993-1-1, the reduction coefficient z may be calculated andthe design axial buckling load in the fire situation is obtained from:

Rd,pl,fizz,Rd,fi NN = (G.21)

(5) Limitations of the method of this Annex are given as follows for the different standard fire resistanceclasses:

R30: b and h 230 mm l 13,5b

R60: for 230 mm b < 300 mm or h/b > 3 l 10b

for b 300 mm and h/b 3 l 13,5b

R90andR120

b 300 mm and h 300 mm

for h / b > 3 l 10b

for h / b 3 l 13,5b

(6) The design values of the resistance of members in axial compression or the design axial bucklingloads z,Rd,fiN are shown in Figures G.2 and G.3 in function of the buckling length l for the profile seriesHEA and the material grades S355 of the steel profile, C40/50 of the concrete, S500 of the reinforcingbars and for the different standard fire resistance classes R60, R90 and R120.

These design graphs are based on the partial material safety factors 0,1c,fi,Ms,fi,Ma,fi,M === .

G.7 Eccentricity of loading

(1) For a column submitted to a load with an eccentricity , the design buckling load ,Rd,fiN may beobtained from:

( )Rd,RdRd,fi,Rd,fi NNNN = (G.22)where:

RdN and ,RdN represent the axial buckling load and the buckling load in case of an eccentric loadcalculated according to EN 1994-1-1, for normal temperature design.

(2) The application point of the eccentric load remains inside the composite cross-section of the column.

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