Many times it is advantageous to built-up (to make) a member (column) by joining two or more members. The joining the members together is provided at discrete points at distances of $a$ (it is not realized continuously along the length).

The common built-up members used for columns are built-ups with lacings and built-ups with battenings:

Buckling of all types of members involved has to be taken into account.

The asssessment is made as if the profile is a uniform beam. That means $I_y = 2 I_{y,1}$

- Web is absent, so shear stiffness is lowered.
- It might happen that buckling appears
- in the scope of the whole column,
- in the scope of particular chord between stiffening members (located at distances $a$).

the assessment is made in the middle of the length. The verification is

$$ \begin{equation} \frac{N_{ch,Ed}}{N_{b,Rd}} \geq 1.0, \end{equation} $$where $N_{ch,Ed}$ is design compression force in the chord at mid-length, $N_{b,Rd}$ is design value of buckling resistance.

$$ \begin{equation} N_{ch,Ed} \simeq \frac{N_{Ed}}{2} + \frac{M_{Ed}}{h_0} \end{equation} $$In the code is more accurate

$$ \begin{equation} N_{ch,Ed} = \frac{1}{2} N_{Ed} + \frac{M_{Ed}}{W_{eff}}\cdot A_{ch}. \end{equation} $$Term $N_{Ed}/2$ is external load divided into two chords and $M_{Ed}$ is taken as

$$ \begin{equation} M_{Ed} = \frac{N_{Ed}\cdot e_0 + M^1_{Ed}}{1-\frac{N_{Ed}}{N_{cr}} - \frac{N_{Ed}}{S_{V}}} \end{equation} $$Moment $M^1_{Ed}$ only if an external moment is applied, term $N_{Ed}/N_{cr}$ involves deflections (2nd order effects) and the fraction with $S_V$ represents deflections from shear; $S_V$ is shear stifness of the lacings (for details see table in code), $N_{cr} = {\pi^2 EI_{eff}}/{L^2_{cr}}$.

also assessment at the support has to be made. At the support the shear force produces moment on chord $(V_{Ed}/2)\cdot(a/2)$ and on battening the moment equals to double that moment (at a node sum of moments has to be zero).

That means assessment of the chord is made for combination of normal force $N_{ch,Ed}$ together with moment $M_{ch,Ed} = V_{Ed}\cdot a / 4$. Assessment for battening member for moment $M_{bat,Ed} = V_{Ed}\cdot a/2$.

Also diagonals are considered for buckling. Axial forces in diagonals are derived from the shear force $V_{Ed}$ at the support.

In order to determine forces within diagonals we have to evaluate the shear force at the support firstly. The shear force at the support is then

$$ \begin{eqnarray} w(x) &=& f(x) = e_0 \sin \frac{\pi x }{L} \\ M(x) &=& \frac{d^2 w}{dx^2} = M_{Ed} \sin \frac{\pi x }{L} \\ V(x) &=& \frac{d M}{dx} = \frac{\pi M_{Ed}}{L} \cos\frac{\pi x}{L}\\ V_{Ed} &=& \frac{\pi M_{Ed}}{L} \end{eqnarray} $$$M_{Ed}$ is known and $V_{Ed}$ is the largest possible from $V(x)$.