At normal temperatures, structural design requires the structure to support the design ultimate loads (the ultimate limit state) while also limiting deformation and vibrations under serviceability circumstances (the serviceability limit state). The fire resistance design of steel columns and beams is aimed at maintaining the structural integrity of the structure at elevated fire temperature within a stipulated time. At room temperature, the structure’s design effort is primarily focused on preventing excessive deformations.

The basic goal of fire-resistant design is to prevent collapse before the stated fire resistance duration expires. During a fire, large deformations are normal and do not need to be calculated in the fire design. For a particular fire loading, the load-bearing function of a steel member is assumed to be lost at time *t*, when:

E_{fi,d }= R_{fi,d,t} ——– (1)

where;

E_{fi,d} is the design value of the relevant effects of actions in the fire situation;

R_{fi,d,t} is the design value of the resistance of the member in the fire situation at time *t.*

R_{fi,d,t} represents the design resistance of a member in a fire situation at time *t*, which can be M_{fi,t,Rd} (design bending moment resistance in a fire situation), N_{fi,t,Rd} (design axial resistance in a fire situation), or any other force (separately or in combination), and the corresponding values of M_{fi,Ed }(design bending moment in a fire situation), N_{fi,Ed} (design axial force in the fire situation), etc. represent E_{fi,d}.

The design resistance R_{fi,d,t }at time *t* shall be determined by reducing the design resistance for normal temperature design according to EN 1993-1-1 to account for the mechanical properties of steel at elevated temperatures (assuming a consistent temperature across the cross section). The differences in the equations for cold and fire design are mostly related to the shape of the stress-strain diagram at ambient temperature and the shape of the diagram at elevated temperature. Figure 2 shows this schematically.

When employed at greater temperatures, some adaptation to the design equations established for room temperature circumstances is required. If a non-uniform temperature distribution is employed, the normal temperature design resistance to EN 1993-1-1 should be changed based on this temperature distribution.

**Fire Resistance Design of Steel Columns**

The most important design requirements of steel structures is the verification of the buckling and shear resistance of the column. For uniaxial or biaxial bending of steel columns, the interaction factors should be duly considered.

**Buckling Resistance**

The design value of the compression force in the fire situation, N_{b,fi,Ed}, at each cross section should satisfy the following condition:

*N _{b,fi,Ed}/N_{b,fi,t ,Rd }≤ 1.0* ——– (2)

where the design buckling resistance N_{b,fi,t,Rd }at time *t* of a compression member with a Class 1, Class 2 or Class 3 cross section with a uniform temperature θ_{a} should be determined from:

*N _{b,fi,t ,Rd }= χ_{fi}Ak_{y,θ}f_{y} / γ_{M,fi }* ——– (3a)

and for Class 4 cross sections

*N _{b,fi,t ,Rd }= χ_{fi}A_{eff }k_{0.2p,θ }f_{y} /γ_{M,fi }*——– (3b)

where;

k_{y,θ} is the reduction factor for the yield strength of steel at uniform temperature θ_{a} ,reached at time *t*.

k_{0.2p,θ} is the reduction factor for the 0.2% proof strength of steel at uniform temperature θ_{a}, reached at time *t,*.

A_{eff }is the effective area of the cross section when subjected only to uniform compression;

χ_{fi }is the reduction factor for flexural buckling in the fire design situation, given by Eq. (5.45).

The value of χ_{fi} should be taken as the lower of the values of χ_{y,fi} and χ_{z,fi }determined according to:

χ_{fi }= 1/[φ_{θ }+ φ_{θ}^{2} −λ_{θ}^{2}] ——– (4)

where;

φ_{θ} = 0.5[1 + αλ_{θ} + λ_{θ}^{2}] ——– (5)

and the imperfection factor, α , proposed by Franssen et al (2005) is given by;

α = 0.65 √235/f_{y} ——– (6)

The non-dimensional slenderness λ_{θ} for the temperature θ_{a}, is given, for Class 1, 2 and 3 by;

λ_{θ} = λ √(k_{y,θ}/k_{E,θ}) ——– (7a)

and for Class 4 cross sections

λ_{θ }= λ √k_{0.2 p,θ}/ k_{E,θ}) ——– (7b)

where;

λ is the non-dimensional slenderness at room temperature given by Eq. (8a) or Eq. (8b) using the buckling length in fire situation* l _{fi }*. The non-dimensional slenderness at room temperature, λ, is given by;

λ = √(Af

_{y}/N

_{cr}) ——– (8a)

for Class 1, 2 and 3 cross sections or;

λ = √(A

_{eff}f

_{y}/N

_{cr}) ——– (8b)

for Class 4 cross sections.

where N_{cr} is the elastic critical force for flexural buckling based on the gross cross sectional properties and in the buckling length in fire situation, *l _{fi} *given by;

N_{cr} = π^{2}EI/l_{fi}^{2} ——– (9)

where;

E is the Young’s modulus at room temperature;*I* is the second moment of area about y-y or x-x axis based on the gross cross sectional properties;*l _{fi} *is the buckling length in fire situation.

The buckling length* l _{fi }*of a column for the fire design situation should generally be determined as for normal temperature design. In the case of a braced frame, the buckling length

*l*of a continuous column may be determined by considering it as fixed to the fire compartments above and below, provided that the fire resistance of the building components that separate these fire compartments is not less than the fire resistance of the column.

_{fi}**Shear Resistance**

The design value of the shear force in a fire situation, V_{fi,Ed} at each cross section should satisfy;

V_{fi,Ed}/V_{fi,t ,Rd} ≤ 1.0 ——– (10)

where the design shear resistance V_{fi,t ,Rd }at time t for a Class 1, Class 2 or Class 3 cross section should be determined from:

V_{fi,t ,Rd }= k_{y,θ,web}V_{Rd }[γ_{M0}/ γ_{M,fi}] ——– (11)

where;

V_{Rd} is the shear resistance of the gross cross section for normal temperature design, according to EN 1993-1-1, and given in Eq. (12);

θ_{web} is the average temperature of the web;

k_{y,θ,web} is the reduction factor for the yield strength of steel at the web temperature θ_{web}.

It should be noted that when a uniform temperature is considered in the design, the average temperature in the web is equal to the uniform temperature in the section. Alternatively, the temperature in the web can be determined using the section factor of the web. For an I-section, the section factor can be approximated as k_{sh}A_{m} / V = k_{sh}2/t_{w} , where the correction factor for the shadow effect is taken for the full section or, in a more accurate way, as the view factor evaluated as shown in Section 4.9.

The shear resistance of the gross cross section for normal temperature design is given (according to EN 1993-1-1) by;

V_{pl,Rd }= A_{v}(f_{y}/3)/γ_{M0} ——– (12)

where A_{v }is the shear area

Substituting Eq. (11) into Eq. (12), and considering a uniform temperature distribution, gives the following expression for the design shear resistance;

V_{fi,t ,Rd} = A_{v}k_{y,θ}f_{y}/√3γ_{M,fi} ——– (13)

**Design Example** (Franssen and Real, 2015)

Consider a 3.5 m long HE 180 B column in S275 grade steel, located in an intermediate storey of a braced frame and subject to a compression load of N_{fi,Ed} = 495 kN in the fire situation. Assuming that the column doesn’t have any fire protection and that the required fire resistance is R30, verify the fire resistance in each of the following domains:

a) Temperature;

b) Time;

c) Resistance.

**Solution:**

Classification of the cross section:

The relevant geometrical characteristics of the profile for the cross section classification are;

h = 180 mm

b = 180 mm

t_{w} = 8.5 mm

t_{f} = 14 mm

r = 15 mm

c = b/2 − t_{w}/2 − r = 70.75 mm (flange)

c = h − 2t_{f }− 2r = 122 mm (web)

As the steel grade is S275

ε = 0.85 √(235/f_{y}) = 0.786

The class of the flange in compression is

c/t_{f} = 70.75/14 = 5.1 < 9ε = 7.07 ⇒ Class 1

The class of the web in compression is

d/t_{w} = 122/8.5 = 14.4 < 33ε = 25.9 ⇒ Class 1

The cross section of the HE 180 B in fire situation is Class 1. This classification could be directly obtained using the table for cross sectional classification of Annex F, Vila Real et al (2009b).

**Evaluation of the critical temperature:**

For the HE 180 B:

Area, A = 6525 mm^{2}

Second moment of area, I_{z} = 13630000 mm^{4}

The design value of the compression load in fire situation: N_{fi,Ed }= 495 kN

The buckling length for intermediate storey is: *l _{fi} *= 0.5L = 0.5 × 3.5 = 1.75 m

The Euler critical load takes the value:

N_{cr }= (π^{2}EI)/l_{fi}^{2} = 9224414 N

The non-dimensional slenderness at elevated temperature is given by Eq. (5.48)

λ_{θ} = λ ⋅√(k_{y,θ}/k_{E,θ})

This is temperature dependent and an iterative procedure is needed to calculate the critical temperature. Starting with a temperature of 20ºC at which k_{y,θ} = k_{E,θ }= 1.0, equations (5.48), and (5.49) give:

λ_{θ} = λ ⋅√(k_{y,θ}/k_{E,θ}) = λ =√(Af_{y}/N_{cr}) = √((6525 × 275)/9224414) = 0.441

The reduction factor for flexural buckling χ is evaluated using Eq. (5.45):

α = 0.65 √(235/f_{y}) = 0.65 × √(235/275) = 0.601

and

φ = 0.5 × (1 + 0.601 × 0.4361 + 0.436^{2}) = 0.730

Therefore the reduction factor for flexural buckling is:

χ_{fi }= 1/[0.730 + √(0.730^{2} − 0.441^{2})] = 0.763

The design value of the buckling resistance N_{b,fi,t,Rd} at time t = 0, is obtained from Eq. (5.44):

N_{b,fi,0,Rd} = χ_{fi}Af_{y} / γ_{M,fi }= 1368 kN

and the degree of utilisation takes the value:

μ_{0} =N_{fi,Ed}/N_{fi,0,Rd} = 495/1368 = 0.362

For this degree of utilisation Eq. (5.104) gives a critical temperature, θ_{a,cr }= 635 ºC. Using this temperature, the non-dimensional slenderness λ_{θ} can be corrected, which leads to another critical temperature. The iterative procedure should continue until convergence is reached, as illustrated in the next table:

After three iterations a critical temperature of θ_{a,cr }= 623 ºC is obtained.

The verification of the fire resistance of the column may be now made.

a) The section factor of the HE 180 B is A_{m}/V = 159 m^{−1} .

The box value for the section factor;

[A_{m}/V]_{b} = (2 × (b + h))/A = [2 × (0.18 + 0.18)]/(65.25 × 10^{−4}) = 110.3 m^{−1}

and the shadow factor k_{sh}

k_{sh} = 0.9 [A_{m}/V]_{box} / [A_{m}/V] = (0.9 × 110.3 ]/159 = 0.624

The modified section factor is: k_{sh}[A_{m}/V]_{b} = 0.624 × 159 = 99.2 m^{−1}

This value could be directly obtaining from the table of Annex E, Vila Real et al (2009a). Interpolating, from table of the Annex A.4 yields the following temperature after 30 minutes:

θ_{d} = 766 ºC

and

θ_{d} > θ_{a,cr }⇒ not satisfactory.

b) By double interpolation of table of the Annex A.4 the time needed to reach a temperature of 623 ºC is;

t_{fi,d} = 17.4 min

and

t_{fi,d }< t_{fi,requ }⇒ not satisfactory.

c) The reduction factors for the yield strength and the Young’s modulus after 30 minutes of fire exposure are, interpolating in Table 5.2 for a temperature of 766 ºC:

k_{y,θ} = 0.1508 and k_{E,θ }= 0.1036

The design value of the buckling resistance is obtained from;

N_{b,fi,t ,Rd }= χ_{fi}Ak_{y,θ}f_{y}/γ_{M,fi}

The non-dimensional slenderness at 766 ºC, is

λ_{θ} = λ ⋅√(k_{y,θ}/k_{E,θ}) = 0.441√(0.1508/0.1036) = 0.532

and using

φ_{θ} = 0.5[ 1 + αλ_{θ} +λ_{θ}^{2}]

with

α = 0.65 √(235/f_{y})

gives

φ_{θ} = 0.8014

and the reduction factor for the flexural buckling is:

χ_{fi }= 1 / [φ_{θ} + φ_{θ}^{2 }− λ_{θ}^{2}] = 0.714

The design value of the buckling resistance after 30 minutes of fire exposure, takes the value:

N_{b,fi,t ,Rd }= χ_{fi}Ak_{y,θ}f_{y} /γ_{M,fi} = (0.714 × 6525 × 0.1508 × 275 × 10^{−3})/1.0 = 193 kN

and

N_{b,fi,t ,Rd} < N_{fi,d }⇒ not satisfactory.

The column does not fulfil the required fire resistance R30.

**References**

Franssen J. and Real P. V. (2015): Fire Design of Steel Structures (2nd Edition). ECCS – European Convention for Constructional Steelwork