Given

w=56kN/m

factored load =56$\times1.5=8^{4}$kN/m

length of span=2m

Required =design laterally unsupported beam

1) V=$\frac{wd}{2}=\frac{84\times 2}{2}$=84kN

m$=\frac{wl^{2}}{8}=\frac{8^{4}\times 2^{2}}{8}$=42kN.m

2) Trial section

Zpreq=$\frac{M}{fbd}$

Assume $\lambda$=100 $\frac{h}{tf}$=25mm

fcrb=291.4 Md

250 $ \ \ \ \ $ 152.3

291.4 $ \ \ $ fbd

300 $ \ \ \ \ \ $163.6

fbd=161.65

Zpreq=$\frac{42\times 10^{6}}{161.65}$

Zpreq=259.82$\times10^{3}mm^{3}$

ISMB $ \ \ \ $ 200

Zp=255.36$\times10^{3}$

D=200 $ \ \ \ $ bf=100mm $\ \ \ \ $ tf=10.8mm

tw=5.7mm $ \ \ \ \ $ $\gamma $ xx=83.2

$\gamma$yy=21.5 $ \ \ \ \ $ Ixx=2235.4$\times10^{6}mm^{4}$

Zxx=224$\times10^{3}$ $ \ \ \ \ \ $ R$_{1}$=11mm

$\frac{D}{tf}=\frac{200}{10.8}$=18.52

$\frac{kL}{\gamma min}=\frac{2000}{21.5}$=93

18 $ \\ \ \ \ $18.5220

90 $ \ \ \ \ \ $ 402.2 $ \ \ \ $ x $ \ \ \ \ $ 380.4

93 $ \ \ \ \ \ $ x $ \ \ \ $ y

100 $ \ \ \ \ \ $ 34364 $ \ \ \ \ $ 325.8

18 $ \ \ \ \ \ \ \ \ \ $ 402.2

18.52 $ \ \ \ \ \ \ \ \ $ x

20 $ \ \ \ \ \ \ $ 380.4

18 $ \ \ \ \ \ \ \ $ 346.4

18.52 $ \ \ \ \ \ $ y

20 $ \ \ \ \ \ $ 325.8

x=396.53 $ \ \ \ \ \ \ $ y=341.64

90 $ \ \ \ \ \ \ $ 396.53

93 $ \ \ \ \ \ \ $ fcr

10 $ \ \ \ \ \ \ $ 341.04

fcrb=379.88

350 $ \ \ \ \ \ \ \ $ 172.7

379.88 $ \ \ \ \ \ $ fbd

400 $ \ \ \ \ \ \ \ \ \ $ 184.1

fbd=179.21mpa

md=Bd$\ast ZP \ast fbd$

=1$\ast(255.56)\times $179.89

=45.94kN $ \gt$ m(42)

Anaysis is safe