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 k_y,θ and k_E,θ. k_y,θ is the reduction factor for the yield strength and k_E,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:	A_V,z, *A_V,y*	The shear areas for directions z and y
	*k_y,θ*	The reduction factor for the yield strength of steel
	*f_y*	The yield strength
	*γ_M,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
	*t_w*	The thickness of plate
	*τ_ba*	The simple post-critical shear resistance
	*γ_M,fi*	The partial safety factor for fire design situation

The design shear resistance V_fi,θ,Rd,z is calculated as the minimum of V_fi,θ,Rd,z and V_{fi,θ,ba,Rd,z}, the design shear resistance V_fi,θ,Rd,y is calculated as the minimum of V_fi,θ,Rd,y and V_{fi,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 τ_t is calculated for torsional moment T_t. Following expression is used for open cross-sections:

Where is:	*T_t*	The torsional moment
	*I_t*	The rigidity moment in St.Venant torsion
	t	The thickness of the plate in the considered point

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

Where is:	*T_t*	The torsional moment
	*Ω_t*	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 τ_ω is calculated for torsional moment T_ω using following formula:

Where is:	*T_ω*	The torsional moment
	*S_ω*	The warping constant in the considered point
	*I_ω*	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:	*k_y,θ*	The reduction factor for the yield strength of steel
	*f_y*	The yield strength
	*γ_M,fi*	The partial safety factor for fire design situation

The values of shear resistance V_Rd,y and V_Rd,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
	*k_y,θ*	The reduction factor for the yield strength of steel
	*f_y*	The yield strength
	*γ_M,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
	*k_y,θ*	The reduction factor for the yield strength of steel
	*f_y*	The yield strength
	*γ_M,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:	*χ_fi*	The reduction factor for flexural buckling in the fire design situation
	A	The cross-sectional area
	*k_y,θ*	The reduction factor for the yield strength of steel
	*f_y*	The yield strength
	*γ_M,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 χ_fi corresponds to the relative slenderness and is given by the expression

where

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

Where is:		The relative slenderness for temperature 20°C
	*k_y,θ*	The reduction factor for the yield strength of steel
	*k_E,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 N_b,fi,θ,Rd.

Calculation of bending resistance

The flexural resistance is calculated using the expression

Where is:	W	The plastic cross-sectional modulus
	*k_y,θ*	The reduction factor for the yield strength of steel
	*f_y*	The yield strength
	*γ_M,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:	*χ_LT,fi*	The reduction factor for lateral-torsional buckling for the fire design situation
	*M_c,fi,θ,Rd*	The bending resistance calculated in accordance with previous chapter

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

where

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

Where is:		The relative slenderness for temperature 20°C
	*k_y,θ*	The reduction factor for the yield strength of steel
	*k_E,θ*	The reduction factor for the slope of the linear elastic range