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 Vz
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 Vz is calculated in the same way as for solid cross-sections:
Where is: | AV,z |
|
ky,θ |
| |
fy |
| |
γM,fi |
|
The final verification is done according to the following expression:
Verification of shear resistance Vy
The force Vy 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: | AV,z |
|
ky,θ |
| |
fy |
| |
γM,fi |
|
The bending resistance for bending moment My 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: | Wpl,y |
|
Wy |
| |
Wy,eff |
|
The bending moment Mz 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 |
|
A |
| |
Iz |
|
The similar recalculation is used for the shear force Vy in the direction of the strong axis. The shear force will be transferred into bending moment Mz for partial cross-section. Following expression is used:
Where is: | l1 |
|
The bending resistance of partial cross-section for bending moment Mz 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: | Wpl,z |
|
Wz |
| |
Wz,eff |
|
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 |
|
dN |
| |
Mz,Sd |
|
Verification of buckling resistance
The buckling resistance perpendicular to the strong axis is given by expression
Where is: | χfi,y |
|
A |
| |
ky,θ |
| |
fy |
| |
γM,fi |
| |
βA |
|
The slenderness λy in the direction perpendicular to the strong axis y is given by formula
Where is: | Lcr,y |
|
iy |
|
The relative slenderness is given by the expression
Where is: | λy |
|
λ1 |
| |
βA |
| |
ky,θ |
| |
kE,θ |
|
The slenderness value λ1 is given by the formula
Where is: | E |
|
fy |
|
The reduction factor χfi,y corresponds to the relative slenderness and is given by the expression
where
where
The partial cross-section fails if the specified axial force is greater than the resistance Nfi,θ,b,Rd,y.
The calculation of buckling resistance perpendicular to the weak axis follows. The elastic flexural buckling force Ncr is given by the expression
Where is: | lcr,z |
|
kE,θ |
| |
E |
| |
Ieff |
|
Following formula is used for Ieff for lacing
Where is: | h0 |
|
A |
|
The second moment of area I1 is calculated for built-up cross-sections with battens using the expression
Where is: | A |
|
h0 |
| |
Iz |
|
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 Ieff is given by the expression
The partial cross-section fails if the specified axial force is greater than the resistance Ncr.
The verification of the shear stiffness SV follows. The shear stiffness is given by the follwoing formula for battens
or
However, following expression has to be fulfilled
Where is: | l1 |
|
r |
| |
Ib |
| |
h0 |
|
The axial force shouldn't exceed the shear stiffness SV. 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 MS is given by the expression
Where is: | e0 |
|
The buckling resistance is given by expression
Where is: | χy |
|
A |
| |
ky,θ |
| |
fy |
| |
γM,fi |
| |
βA |
|
where the factor χz corresponds to the slenderness λ, that is given by the expression
Where is: | l1 |
|
imin |
|
The relative slenderness is given by the formula
where
The factor χz corresponds to the relative slenderness and is calculated with the help of following expression
where
where
The shear force VS is calculated for the batten
The moment Mz,Sd for the partial cross-section is given by the formula
Where is: | l1 |
|
Vy |
|
The bending resistance of partial cross-section for bending moment My 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: | Wpl,y |
|
Wy |
| |
Wy,eff |
|
The bending resistance of partial cross-section for bending moment Mz 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: | Wpl,z |
|
Wz |
| |
Wz,eff |
|
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 |
|
dN |
| |
ky |
|
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: | Vy |
|
r |
| |
d |
| |
h0 |
|
The resistance of lacing is calculated using formula
Where is: | Ad |
|
ky,θ |
| |
fy |
| |
γM,fi |
|
Following expression has to be fulfilled
The axial force in the lacing including buckling consideration is given by the following expression
Where is: | Vy |
|
VS |
| |
d |
| |
r |
| |
h0 |
|
The slenderness of the web is estimated according to the following formula:
Where is: | d |
|
Ad |
|
The relative slenderness is given by the formula
where is
The factor χSp corresponds to the relative slenderness and is calculated using expression
where
where
The buckling resistance of the web is given by the expression
The webs are OK if the following expression is fulfilled: