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Calculation of resistance

The resistance of structures in fire situations is affected by the reduction of strength and deformation properties caused by increased temperature. These changes are described by reduction factors ky,θ and kE,θ. ky,θ is the reduction factor for the yield strength and kE,θ is the reduction factor for the slope of the linear elastic range. Values are given in EN 1993-1-2, interval is between 1.0 for 20°C and 0.0 for 1200°C. Thus, the verification of steel elements is based on reduced values of yield strength and modulus of elasticity.

Calculation of shear resistance

The shear resistance for directions z and y is calculated using expressions

Where is:

AV,z, AV,y

  • The shear areas for directions z and y

ky,θ

  • The reduction factor for the yield strength of steel

fy

  • The yield strength

γM,fi

  • The partial safety factor for fire design situation

If the corresponding plate of the cross-section are fixed on both ends in the perpendicular directions, the effect of local buckling is taken into consideration. The effect of the local buckling may be reduced with the help of web stiffeners. The simple post-critical method is used for the buckling of walls:

Where is:

d

  • The height of plate

tw

  • The thickness of plate

τba

  • The simple post-critical shear resistance

γM,fi

  • The partial safety factor for fire design situation

The design shear resistance Vfi,θ,Rd,z is calculated as the minimum of Vfi,θ,Rd,z and Vfi,θ,ba,Rd,z, the design shear resistance Vfi,θ,Rd,y is calculated as the minimum of Vfi,θ,Rd,y and Vfi,θ,ba,Rd,y. The buckling caused by shear isn't considered for plates supported only on one end.

Shear due to torsion

The St. Venant and warping torsion is considered during the analysis. For St. Venant torsion, the shear stress τt is calculated for torsional moment Tt. Following expression is used for open cross-sections:

Where is:

Tt

  • The torsional moment

It

  • The rigidity moment in St.Venant torsion

t

  • The thickness of the plate in the considered point

The stress τt for closed cross-sections is given by the expression

Where is:

Tt

  • The torsional moment

Ωt

  • The area bounded by the centre lines of plates increased by a factor of two

t

  • The thickness of the plate in the considered point

For warping torsion, the shear stress τω is calculated for torsional moment Tω using following formula:

Where is:

Tω

  • The torsional moment

Sω

  • The warping constant in the considered point

Iω

  • The sectional moment of inertia

t

  • The thickness of the plate in the considered point

The shear stress is verified according to the following expression:

Where is:

ky,θ

  • The reduction factor for the yield strength of steel

fy

  • The yield strength

γM,fi

  • The partial safety factor for fire design situation

The values of shear resistance VRd,y and VRd,z are reduced due to torsion for combination of shear forces and torsional moments. The reduction is based on following expression

for I- and U-profiles and

for other cross-sections.

Calculation of tensile resistance

The tensile resistance is calculated using the expression

Where is:

A

  • The cross-sectional area

ky,θ

  • The reduction factor for the yield strength of steel

fy

  • The yield strength

γM,fi

  • The partial safety factor for fire design situation

Calculation of compressive resistance

The compressive resistance is calculated using the expression

Where is:

A

  • The cross-sectional area

ky,θ

  • The reduction factor for the yield strength of steel

fy

  • The yield strength

γM,fi

  • The partial safety factor for fire design situation

Calculation of compressive resistance including buckling consideration

The compressive resistance is calculated using the expression

Where is:

χfi

  • The reduction factor for flexural buckling in the fire design situation

A

  • The cross-sectional area

ky,θ

  • The reduction factor for the yield strength of steel

fy

  • The yield strength

γM,fi

  • The partial safety factor for fire design situation

The resistance including buckling consideration is calculated in directions y and z or in directions of main axes η and ζ. The reduction factor for flexural buckling χfi corresponds to the relative slenderness and is given by the expression

where

where

The relative slenderness for the temperature θ is given by the expression

Where is:

  • The relative slenderness for temperature 20°C

ky,θ

  • The reduction factor for the yield strength of steel

kE,θ

  • The reduction factor for the slope of the linear elastic range

The minimum of values for both directions is considered as the compressive resistance Nb,fi,θ,Rd.

Calculation of bending resistance

The flexural resistance is calculated using the expression

Where is:

W

  • The plastic cross-sectional modulus

ky,θ

  • The reduction factor for the yield strength of steel

fy

  • The yield strength

γM,fi

  • The partial safety factor for fire design situation

κ1

  • The adaptation factor for non-uniform temperature across the cross-section

κ2

  • The adaptation factor for non-uniform temperature along the beam

The bending resistance is calculated in four points (corners of the cross-sections) for cross-sections in classes 3 and 4.

Effect of lateral-torsional buckling

The flexural resistance including an effect of lateral-torsional buckling is calculated using the expression

Where is:

χLT,fi

  • The reduction factor for lateral-torsional buckling for the fire design situation

Mc,fi,θ,Rd

  • The bending resistance calculated in accordance with previous chapter

The reduction factor for lateral-torsional buckling χLT,fi corresponds to the relative slenderness and is given by the expression

where

where

The relative slenderness for the temperature θ is given by the expression

Where is:

  • The relative slenderness for temperature 20°C

ky,θ

  • The reduction factor for the yield strength of steel

kE,θ

  • The reduction factor for the slope of the linear elastic range

Calculation of resistance for bimoment

The stress caused by the bimoment is calculated with the help of elastic theory. Therefore, these cross-sections are automatically designed as classes 3 or 4. The resistance is calculated in four points (corners of the cross-sections), where are the worst results expected. The resistance is calculated using following expression:

Where is:

Iω

  • The sectional moment of inertia

ω

  • The warping coordinate

ky,θ

  • The reduction factor for the yield strength of steel

fy

  • The yield strength

γM,fi

  • The partial safety factor for fire design situation

κ1

  • The adaptation factor for non-uniform temperature across the cross-section

κ2

  • The adaptation factor for non-uniform temperature along the beam

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