Built-up column with battens, 4 m, 1000 kN

Given/assumptions: factored load $P_u=1000$ kN; unsupported length 4 m; ends held in position but not restrained against rotation → $K=1.0$, $KL=4$ m. Two channels toe to toe, battens, bolted connections (M20, 4.6); E250 ($f_y=250$, $f_u=410$).

1. Section: 2 ISMC 225 toe to toe (flanges facing)

$A=2\times3301=6602$ mm², $I_{xx}=2\times2694.6\times10^4$ mm⁴, $r_{xx}=90.3$ mm, $I_{yy,ch}=187.2\times10^4$ mm⁴, $C_{yy}=23.0$ mm, $b_f=80$ mm, $r_{yy,ch}=23.8$ mm. Built-up members are buckling class c.

• $\lambda_{xx}=\dfrac{KL}{r_{xx}}=\dfrac{4000}{90.3}=44.3$
• Battened column: $\lambda_e=1.1\lambda=48.7$ (cl. 7.6.1.5)
• $f_{cd}=185.4$ N/mm² (curve c, $f_y=250$, $\gamma_{m0}=1.10$) → $P_d=A f_{cd}=1224$ kN $\ge1000$ kN ✓

2. Clear gap (so that $I_{yy}\ge I_{xx}$)

The centroid of each section is at $x_c=57.0+s/2$ from the column axis:

$$x_c\ge\sqrt{\frac{I_{xx}-I_{yy,ch}}{A_{ch}}}=87.2\ \text{mm}\Rightarrow s\ge60.4\ \text{mm}$$

Provide $s=70$ mm. $I_{yy}=5962\times10^4$ mm⁴, $r_{yy}=95.0$ mm, $1.1\lambda_{yy}=46.3$, $P_{dy}=1247$ kN $\ge P_u$ ✓

3. Spacing of battens (cl. 7.7.1.4)

The slenderness of one section between battens must be $\le\min(50,\ 0.7\lambda_e)=34.1$:

$$C\le 34.1\times23.8=811\ \text{mm}$$

For the length of 4000 mm provide 5 bays, i.e. 6 battens at $C=800$ mm c/c.

4. Size of battens (cl. 7.7.2)

• Distance between the connection lines (middle of the flanges) $S=s+b_f=150$ mm; distance between the centroids of the channels $=s+2(b_f-C_{yy})=184$ mm
• End battens: depth $\ge184$ mm → 260 mm (to fit 4 bolts); intermediate battens: depth $\ge\max(0.75\times184,\ 2b_f)=160$ mm → 260 mm
• Thickness $\ge S/50=3$ mm → 8 mm; length of batten $=S+80=230$ mm.

5. Design forces on a batten

Transverse shear $V_t=2.5\%P_u=25.0$ kN, number of batten planes $N=2$:

$$V_b=\frac{V_tC}{NS}=\frac{25.0\times800}{2\times150}=66.67\ \text{kN},\qquad M=\frac{V_tC}{2N}=5.000\ \text{kN·m}$$

• Plate in bending: $Z=\dfrac{t d^2}{6}=90133$ mm³; $\sigma=\dfrac{M}{Z}=55.5$ N/mm² $<f_y/\gamma_{m0}=227$ N/mm² ✓ ($M_d=20.48$ kN·m)
• Shear: $\tau=\dfrac{V_b}{dt}=32.1$ N/mm² $<\dfrac{f_y}{\sqrt3\gamma_{m0}}=131$ N/mm² ✓

6. Bolts connecting battens to the sections (M20, 4.6)

4 bolts in a vertical line at each end (pitch 60 mm, end 40 mm): $V_{dsb}=45.3$ kN, bearing on 8 mm $=79.5$ kN → bolt value $45.3$ kN.

• Direct shear per bolt $=V_b/n=16.67$ kN; moment effect on the extreme bolt $=\dfrac{M y_{max}}{\Sigma y^2}=\dfrac{5.0\times10^6\times90}{18000}=25.00$ kN
• Resultant $R=\sqrt{16.67^2+25.00^2}=30.05$ kN $<45.3$ kN ✓

Answer: 2 ISMC 225 toe to toe (flanges facing), clear gap 70 mm; battens 260 × 8 mm (end and intermediate), 230 mm long, at 800 mm c/c, with 4 M20 bolts at each end.