Verification of built-up cross-sections

The verification of built-up cross-sections starts with the classification. The classification is identical to the process for solid cross-sections. The verification of built-up members is done according to the following rules.

Verification of shear resistance V_z

If the axis z is perpendicular to the strong axis of the cross-section (this is fulfilled for the most of cases), the shear resistance for the force V_z is calculated in the same way as for solid cross-sections:

Where is:	*A_V,z*	The shear area for direction z
	*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 final verification is done according to the following expression:

Verification of shear resistance V_y

The force V_y is usually parallel to the strong axis of the cross-section. It means, that the shear resistance depends on the stiffness of battens.

Verification of the resistance for tension, compression and bending

The resistance of the partial member in tension or in plain compression is calculated using the expression

Where is:	*A_V,z*	The area of partial cross-section
	*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 bending resistance for bending moment M_y is calculated for the classes 1 and 2 according to the following formula:

The formula for the class 3:

The formula for the class 4:

Where is:	*W_pl,y*	The plastic section modulus of partial cross-section about the axis y
	*W_y*	The elastic section modulus of partial cross-section about the axis y
	*W_y,eff*	The effective section modulus of partial cross-section about the axis y

The bending moment M_z is recalculated into the increment of axial force in partial cross-section dN. The recalculation uses following expression

The formula for members with battens:

Where is:	h₀	The distance of points of inertia of partial cross-sections
	A	The area
	*I_z*	The second moment of inertia of partial cross-section

The similar recalculation is used for the shear force V_y in the direction of the strong axis. The shear force will be transferred into bending moment M_z for partial cross-section. Following expression is used:

Where is:

l₁

The distance of battens

The bending resistance of partial cross-section for bending moment M_z is calculated for the classes 1 and 2 according to the following formula:

The formula for the class 3:

The formula for the class 4:

Where is:	*W_pl,z*	The plastic section modulus of partial cross-section about the axis z
	*W_z*	The elastic section modulus of partial cross-section about the axis z
	*W_z,eff*	The effective section modulus of partial cross-section about the axis z

The verification of the combination of axial force and bending moments is done according to the rules similar to the verification of solid cross-sections. This expression is used:

Where is:	n	The number of partial cross-sections
	dN	The increment of axial force due to bending moment M_z
	*M_z,Sd*	The bending moment in partial cross-section due to shear force V_y

Verification of buckling resistance

The buckling resistance perpendicular to the strong axis is given by expression

Where is:	*χ_fi,y*	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
	*β_A*	The factor considering the class of cross-section, β_A=A_eff/A for class 4 and β_A=1 for other classes

The slenderness λ_y in the direction perpendicular to the strong axis y is given by formula

Where is:	*L_cr,y*	The buckling length for buckling perpendicular to the axis y
	*i_y*	The radius of gyration for axis y

The relative slenderness is given by the expression

Where is:	*λ_y*	The slenderness in the direction perpendicular to the axis y
	λ₁	The slenderness value to determine the relative slenderness
	*β_A*	The factor considering the class of cross-section, β_A=A_eff/A for class 4 and β_A=1 for other classes
	*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 slenderness value λ₁ is given by the formula

Where is:	E	The modulus of elasticity
	*f_y*	The yield strength

The reduction factor χ_fi,y corresponds to the relative slenderness and is given by the expression

where

The partial cross-section fails if the specified axial force is greater than the resistance N_fi,θ,b,Rd,y.

The calculation of buckling resistance perpendicular to the weak axis follows. The elastic flexural buckling force N_cr is given by the expression

Where is:	*l_cr,z*	The buckling length for buckling perpendicular to the axis z
	*k_E,θ*	The reduction factor for the slope of the linear elastic range
	E	The modulus of elasticity
	*I_eff*	The effective value of the moment of inertia, that depends on the type of connection of partial cross-sections

Following formula is used for I_eff for lacing

Where is:	h₀	The distance of points of inertia of partial cross-sections
	A	The cross-sectional area of partial cross-section

The second moment of area I₁ is calculated for built-up cross-sections with battens using the expression

Where is:	A	The cross-sectional area of partial cross-section
	h₀	The distance of points of inertia of partial cross-sections
	*I_z*	The second moment of area of partial cross-section

The radius of gyration i₀ is given by the expression

For the slenderness

the factor μ is selected. The effective value of the moment of inertia I_eff is given by the expression

The partial cross-section fails if the specified axial force is greater than the resistance N_cr.

The verification of the shear stiffness S_V follows. The shear stiffness is given by the follwoing formula for battens

However, following expression has to be fulfilled

Where is:	l₁	The distance of battens
	r	The number of planes of lacings
	*I_b*	The in plane second moment of area of one batten
	h₀	The distance of points of inertia of partial cross-sections

The axial force shouldn't exceed the shear stiffness S_V. Also following expression has to be fulfilled

The force in the middle of the batten is calculated using formula

The force in lacing is

Where the moment M_S is given by the expression

Where is:

e₀

The bow imperfection given by the expression l_cr,z/500

The buckling resistance is given by expression

Where is:	*χ_y*	The reduction factor for flexural buckling
	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
	*β_A*	The factor considering the class of cross-section, β_A=A_eff/A for class 4 and β_A=1 for other classes

where the factor χ_z corresponds to the slenderness λ, that is given by the expression

Where is:	l₁	The distance of battens
	*i_min*	The minimum radius of gyration for partial cross-section

The relative slenderness is given by the formula

where

The factor χ_z corresponds to the relative slenderness and is calculated with the help of following expression

where

The shear force V_S is calculated for the batten

The moment M_z,Sd for the partial cross-section is given by the formula

Where is:	l₁	The distance of battens
	*V_y*	The entered shear force

The bending resistance of partial cross-section for bending moment M_y is calculated for the classes 1 and 2 according to the following formula:

The formula for the class 3:

The formula for the class 4:

Where is:	*W_pl,y*	The plastic section modulus of partial cross-section about the axis y
	*W_y*	The elastic section modulus of partial cross-section about the axis y
	*W_y,eff*	The effective section modulus of partial cross-section about the axis y

The bending resistance of partial cross-section for bending moment M_z is calculated for the classes 1 and 2 according to the following formula:

The formula for the class 3:

The formula for the class 4:

Where is:	*W_pl,z*	The plastic section modulus of partial cross-section about the axis z
	*W_z*	The elastic section modulus of partial cross-section about the axis z
	*W_z,eff*	The effective section modulus of partial cross-section about the axis z

The verification is done for two points: the mid point of the distance between two battens and in the connection of batten.

The verification in the mid point of the distance between two battens

Where is:	n	The number of partial cross-sections
	dN	The increment of axial force due to bending moment M_z
	*k_y*	The factor calculated according to the rules for solid cross-sections

The verification in the connection of batten

Verification of lacing

The axial force in the lacing without buckling consideration is given by the following expression

Where is:	*V_y*	The entered shear force
	r	The number of planes of lacings
	d	The length of lacing
	h₀	The distance of points of inertia of partial cross-sections

The resistance of lacing is calculated using formula

Where is:	*A_d*	The cross-sectional area of lacing
	*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

Following expression has to be fulfilled

The axial force in the lacing including buckling consideration is given by the following expression

Where is:	*V_y*	The entered shear force
	*V_S*	The shear force in the point of lacing
	d	The length of lacing
	r	The number of planes of lacings
	h₀	The distance of points of inertia of partial cross-sections