Section Properties

• height of web, $h_w=d-t_{\mathrm{tf}}-t_{\mathrm{bf}}=\mathrm{1,510}-50-50=\mathrm{1,410}\text{ mm}$
• area of the web, $A_w=t_w\times h_w=32\left(\mathrm{1,410}\right)=\mathrm{45,120}{\text{ mm}}^2$
• area of the flanges, $A_f=t_f\times b_f=50\left(450\right)=\mathrm{22,500}{\text{ mm}}^2$
• gross area = net area, $A_g=A_n=A_w+A_{\mathrm{tf}}+A_{\mathrm{bf}}=\mathrm{45,120}+\mathrm{22,500}+\mathrm{22,500}=\mathrm{90,120}{\text{ mm}}^2$
• centroid of section (wrt bottom):
  $y_c=\frac{D}{2}=\frac{\mathrm{1,510}}{2}=755\text{ mm}$
• Moment of inertia, major axis:
  $I_z=2\left[A_f{\left(y_c-\frac{t_f}{2}\right)}^2+\frac{b_f{t_f}^3}{12}\right]+\frac{t_w{h_w}^3}{12}$
  $=2\left[\mathrm{22,500}{\left(755-\frac{50}{2}\right)}^2+\frac{450{\left(50\right)}^3}{12}\right]+\frac{32{\left(\mathrm{1,410}\right)}^3}{12}$
  $=\left[2\left(11.99+0.05\right)+7.48\right]\times {\left(10\right)}^9=31.47{\left(10\right)}^9{\text{ mm}}^4$
• Moment of inertia, minor axis: $I_y=2\left(\frac{t_f{b_f}^3}{12}\right)+\frac{h_w{t_w}^3}{12}$
• $=2\left(\frac{50{\left(450\right)}^3}{12}\right)+\frac{\mathrm{1,410}{\left(32\right)}^3}{12}=\left[2\left(379.7\right)+3.85\right]{\left(10\right)}^6=763.2{\left(10\right)}^6{\text{ mm}}^4$
• Elastic section modulus, major axis: $Z_z=\frac{I_z}{y_c}=\frac{31.47{\left(10\right)}^9}{755}=41.68{\left(10\right)}^6{\text{ mm}}^3$
• Elastic section modulus, minor axis: $Z_y=\frac{I_y}{\raisebox{1ex}{$b_f$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=\frac{763.2{\left(10\right)}^6}{\raisebox{1ex}{$450$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=3.392{\left(10\right)}^6{\text{ mm}}^3$
• Plastic neutral axis (wrt bottom): $y_{\text{pna}}=y_c=755\text{ mm}$
• Plastic section modulus, major axis: S_z = 488.75 (10)⁶ mm³
• Plastic section modulus, major axis: S_y = 5.423 (10)⁶ mm³
• Radius of gyration about the major axis: $r_z=\sqrt{\frac{I_z}{A_g}}=\sqrt{\frac{31.47{\left(10\right)}^9}{\mathrm{90,120}}}=590.9\text{ mm}$
• Radius of gyration about the minor axis: $r_y=\sqrt{\frac{I_y}{A_g}}=\sqrt{\frac{763.2{\left(10\right)}^6}{\mathrm{90,120}}}=92.0\text{ mm}$
• Torsional constant: $J=\frac{1}{3}\left(b_{\mathrm{bf}}{t_{\mathrm{bf}}}^3+b_{\mathrm{tf}}{t_{\mathrm{tf}}}^3+h_w{t_w}^3\right)=\frac{1}{3}\left[450{\left(50\right)}^3+450{\left(50\right)}^3+\mathrm{1,400}{\left(32\right)}^3\right]=52.9{\left(10\right)}^6{\text{ mm}}^4$
• Distance between flange centroids: $d_f=d-\frac{t_{\mathrm{bf}}+t_{\mathrm{tf}}}{2}=\mathrm{1,510}-\frac{50+50}{2}=\mathrm{1,460}\text{ mm}$
• Moment of inertia about y of the compression flange: $I_{\mathrm{cy}}=\frac{b_f{b_f}^3}{12}=\frac{50{\left(450\right)}^3}{12}=379.7{\left(10\right)}^6{\text{ mm}}^4$
• Warping constant: $I_w=I_{\mathrm{cy}}{d_f}^2\left(1-\frac{I_{\mathrm{cy}}}{I_y}\right)=379.7{\left(10\right)}^6{\left(\mathrm{1,460}\right)}^2\left(1-\frac{379.7{\left(10\right)}^6}{763.2{\left(10\right)}^6}\right)=406.7{\left(10\right)}^{12}{\text{ mm}}^6$

Design Forces

Design maximum moment:

• Max. moment, M^*: $M_{\text{max}}=\frac{w_{\text{u}}{l}^2}{8}=\frac{300{\left(12\right)}^2}{8}=\mathrm{5,400}\text{ kN·m}=M_4$ at mid-span
• Moment at ¼ point = Moment at ¾ point: $M_{\text{2}}=M_{\text{4}}=\frac{3w_{\text{u}}{l}^2}{32}=\frac{3\left(300\right){\left(12\right)}^2}{32}=\mathrm{4,050}\text{ kN·m}$

Design axial force:

• N^* = 1,000 kN (compression)

Section Slenderness Ratio

Flange section slenderness parameter:

Hence, the flanges are compact.

Web section slenderness parameter:

Hence, the web is compact.

Bending Capacity

The section bending capacity about the major axis (cl. 5.2 of AS 4100) is determined using the section modulus as:

The nominal section capacity about the Z axis:

The factored section capacity about the Z axis:

The section bending capacity about the minor axis (cl. 5.2 of AS 4100) is determined using the section modulus as:

The nominal section capacity about the Y axis:

The factored section capacity about the Z axis:

The section bending capacity against lateral torsional buckling is checked per cl. 5.6.1 of AS 4100. The twist restraint factor, load height factor, and lateral rotation restraint factor are all equal to unity (1.0). Thus, l_e = L×k_tk_lk_r = L = 12 m.

The moment modification factor is given as:

The reference buckling moment:

where

A	=	$\frac{{\pi }^{2}E{I}_y}{{l_e}^2}=\frac{{\pi }^2\left(200\right)\left[763.2,{\left(10\right)}^6\right]}{{\left(\mathrm{12,000}\right)}^2}=\mathrm{10,462}$
B	=	$GJ=80\times 52.9{\left(10\right)}^6=4.23{\left(10\right)}^9$
C	=	$\frac{{\pi }^{2}E{I}_w}{{l_e}^2}=\frac{{\pi }^2\left(200\right)\left[406.7,{\left(10\right)}^{12}\right]}{{\left(\mathrm{12,000}\right)}^2}=5.575{\left(10\right)}^9$

The slenderness reduction factor per AS 4100 Eq. 5.6.1.1(2):

The nominal member capacity:

The factored section capacity about the Z axis:

Shear Capacity

The shear area along the Z axis:

The nominal shear capacity along the Z axis:

The factored shear capacity:

The shear area along the Y axis:

Thus, shear buckling does not control (i.e., V_u = V_w).

The nominal shear capacity along the Z axis:

The factored shear capacity:

Compression Capacity

The plate element yield slenderness limits for heavily welded flanges = 14, webs = 35 (Table 6.2.4 of AS 4100 1998)

The calculated effective bottom flange width and effective depth (cl. 6.2.1 of AS 4100):

• λ_ef = 4.88 < λ_{ey_f} = 14
• λ_ew = 51.4 > λ_{ey_w} = 35

Therefore, the effective area is not reduced for the flanges; but it is reduced for the web:

The effective area:

The form factor (cl. 6.2.2), k_f = 75,720 / 90,120 = 0.840

The nominal section compression capacity:

The factored section compression capacity:

The member compression capacity about the Z axis.

where

${ɑ}_b$	=	1.0 per Table 6.3.3(2) for welded I sections with kf < 1 and flanges > 40 mm thick
$k_e$	=	1.0
${\lambda }_{\mathrm{nz}}$	=	$k_e\frac{L}{r_z}\sqrt{k_f}\sqrt{\frac{f_y}{250}}=1.0\frac{\mathrm{12,000}}{590.9}\sqrt{0.840}\sqrt{\frac{340}{250}}=21.7$
${ɑ}_a$	=	$\frac{\mathrm{2,100}\left({\lambda }_n-13.5\right)}{{{\lambda }_n}^2-15.3{\lambda }_n+\mathrm{2,050}}=\frac{\mathrm{2,100}\left(21.7-13.5\right)}{{\left(21.7\right)}^2-15.3\left(21.7\right)+\mathrm{2,050}}=7.88$
$\lambda$	=	${\lambda }_n+{ɑ}_a{ɑ}_b=21.7+7.88\times 1.0=29.6$
$\eta$	=	$0.00326\left(\lambda -13.5\right)=0.00326\left(29.6-13.5\right)=0.052\ge 0$ , ok
$\xi$	=	$\frac{{\left(\frac{\lambda }{90}\right)}^2+1+\eta }{2{\left(\frac{\lambda }{90}\right)}^2}=\frac{{\left(\frac{29.6}{90}\right)}^2+1+0.052}{2{\left(\frac{29.6}{90}\right)}^2}=5.370$
${ɑ}_{\mathrm{cz}}$	=	$\xi \left[1-\sqrt{1-{\left(\frac{90}{\xi \lambda }\right)}^2}\right]=5.370\left[1-\sqrt{1-{\left(\frac{90}{5.370\times 29.6}\right)}^2}\right]=0.945$

The factored member compression capacity about the Z axis:

The member compression capacity about the Y axis:

where

${\lambda }_{\mathrm{ny}}$	=	$k_e\frac{L}{r_y}\sqrt{k_f}\sqrt{\frac{f_y}{250}}=1.0\frac{\mathrm{12,000}}{92.0}\sqrt{0.840}\sqrt{\frac{340}{250}}=139.4$
${ɑ}_a$	=	$\frac{\mathrm{2,100}\left({\lambda }_n-13.5\right)}{{{\lambda }_n}^2-15.3{\lambda }_n+\mathrm{2,050}}=\frac{\mathrm{2,100}\left(139.4-13.5\right)}{{\left(139.4\right)}^2-15.3\left(139.4\right)+\mathrm{2,050}}=13.66$
$\lambda$	=	${\lambda }_n+{ɑ}_a{ɑ}_b=139.4+13.66\times 1.0=153.1$
$\eta$	=	$0.00326\left(\lambda -13.5\right)=0.00326\left(153.1-13.5\right)=0.455\ge 0$ , OK
$\xi$	=	$\frac{{\left(\frac{\lambda }{90}\right)}^2+1+\eta }{2{\left(\frac{\lambda }{90}\right)}^2}=\frac{{\left(\frac{153.1}{90}\right)}^2+1+0.455}{2{\left(\frac{153.1}{90}\right)}^2}=0.752$
${ɑ}_{\mathrm{cy}}$	=	$\xi \left[1-\sqrt{1-{\left(\frac{90}{\xi \lambda }\right)}^2}\right]=0.752\left[1-\sqrt{1-{\left(\frac{90}{0.752\times 153.1}\right)}^2}\right]=0.283$

The factored member compression capacity about the Y axis:

Tension Capacity

Assume an end connection which provides uniform force distribution (i.e., k_t = 1.0)

Check Against Combined Actions

Uniaxial bending capacity about the Z axis:

$\varphi M_{\mathrm{rz}}>\varphi M_{\mathrm{sz}}=\mathrm{14,920}\text{ kN·m}$, therefore $\varphi M_{\mathrm{rz}}=\mathrm{14,920}\text{ kN·m}$

Uniaxial bending capacity about the Y axis:

$\varphi M_{\mathrm{ry}}>\varphi M_{\mathrm{sy}}=\mathrm{1,557}\text{ kN·m}$, therefore $\varphi M_{\mathrm{ry}}=\mathrm{1,557}\text{ kN·m}$

Member bending capacity: in-plane about the Z axis:

Member bending capacity: in-plane about the Y axis:

Member combined capacity out-of-plane:

Ratio for biaxial bending: here, ϕM_cz = the minimum of ϕM_iz and ϕM_oz