Section Properties

• gross area = net area, $A_g=A_n=(d-t)\times t+b\times t=\mathrm{9,724}{\text{ mm}}^2$
• centroid of section: $y_{\mathrm{c1}}-\frac{\sqrt{2}\left(d^2+dt-t^2\right)}{2d-t}=84.2\text{ mm}$
• centroid of section: $x_c=\sqrt{2}\times d=141.2\text{ mm}$
• Moment of inertia, major axis: $I_z=\frac{d^4-{\left(b-t\right)}^4}{12}-\frac{0.5t\times d^2{\left(b-t\right)}^4}{d+b-t}=14.85{\left(10\right)}^6{\text{ mm}}^4$
• Moment of inertia, minor axis: $I_y=\frac{d^4-{\left(b-t\right)}^4}{12}=56.95{\left(10\right)}^6{\text{ mm}}^4$
• Elastic section modulus, major axis: $Z_z=\frac{I_z}{y_{\mathrm{c1}}}=\frac{14.85{\left(10\right)}^6}{84.2}=176.4{\left(10\right)}^3{\text{ mm}}^3$
• Plastic section modulus, major axis: S_z = 326.6 (10)³ mm³
• Elastic section modulus, minor axis: $Z_y=\frac{I_y}{x_c}=\frac{56.95{\left(10\right)}^6}{141.4}=402.7{\left(10\right)}^3{\text{ mm}}^3$
• Plastic section modulus, major axis: S_y = 643.9 (10)³ mm³
• Radius of gyration about the major axis: $r_z=\sqrt{\frac{I_z}{A_g}}=\sqrt{\frac{14.85{\left(10\right)}^6}{9.724{\left(10\right)}^3}}=39.1\text{ mm}$
• Radius of gyration about the minor axis: $r_y=\sqrt{\frac{I_y}{A_g}}=\sqrt{\frac{56.95{\left(10\right)}^6}{9.724{\left(10\right)}^3}}=76.5\text{ mm}$
• Torsional constant: $J=\frac{2\left(\mathrm{b}-\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)t^3}{3}=2.191{\left(10\right)}^6{\text{ mm}}^4$
• Warping constant: $I_w=0$

Section Classification

Section slenderness

Flange slenderness: ${\lambda }_{\mathrm{ef}}=\frac{b-t}{t}\sqrt{\frac{f_y}{250}}=7.082$

Web slenderness: ${\lambda }_{\mathrm{ew}}=\frac{d-t}{t}\sqrt{\frac{f_y}{250}}=7.082$

Bending about the Z axis puts the bottom flange in uniform compression (table 5.2 of AS 4100). Thus, λ_ef < λ_{ep_f} = 8 < λ_{ey_f} = 15. The ratio $\raisebox{1ex}{${\lambda }_{\mathrm{ef}}$}\!\left/ \!\raisebox{-1ex}{${\lambda }_{\mathrm{ey\_f}}$}\right.=\frac{7.082}{15}=0.472$

Bending about the Z axis puts one end of the web in tension and the other in compression (table 5.2 of AS 4100). Thus, λ_{ep_w} 8 < λ_ew < λ_{ey_w} = 22. The ratio $\raisebox{1ex}{${\lambda }_{\mathrm{ew}}$}\!\left/ \!\raisebox{-1ex}{${\lambda }_{\mathrm{ey\_w}}$}\right.=\frac{7.082}{22}=0.322$

As the ratio for the flange is higher, the flange is the critical element per cl. 5.2.2. The section is considered "compact" (λ_ef < λ_{ep_f}.

Similarly, the section is "compact" in bending about the Y axis.

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 Y 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 = 1.6 m.

The maximum moment, M_m = M₃ for symmetric loading conditions. Further, the quarter point moments, M₂ and M₄ are both equal to M_m/2 for a single point load at mid-span. Substituting these values into the equation in cl. 5.6.1.1(a)(iii), the moment modification factor for a simply-supported beam with a concentrated load at mid-span simplifies to:

The reference buckling moment:

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

The nominal member capacity:

Shear Capacity

The shear area along the Z axis and Y axis:

The nominal shear capacity along the Z axis and Y axis:

The factored shear capacity:

The shear buckling capacity factor: $\frac{d-t}{t}=\frac{200-26}{26}=6.69<\frac{82}{\sqrt{\raisebox{1ex}{$f_y$}\!\left/ \!\raisebox{-1ex}{$250$}\right.}}=\frac{82}{\sqrt{\frac{280}{250}}}=77.5$, so V_vy = V_wy

Compression Capacity

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

• λ_ef = 7.082 < λ_{ey_f} = 16
• λ_ew = 7.082 < λ_{ey_w} = 35

Therefore, A_e = A_e (no reduction for the effective area); the form factor (cl. 6.2.2), k_f = 1

The nominal section compression capacity:

The factored section compression capacity:

The member compression capacity about the Z axis.

where

${ɑ}_b$	=	0.5 per Table 6.3.3(1) for kf = 1
$k_e$	=	1.0
${\lambda }_{\mathrm{nz}}$	=	$k_e\frac{L}{r_z}\sqrt{k_f}\sqrt{\frac{f_y}{250}}=1.0\frac{\mathrm{1,600}}{39.1}\sqrt{1.0}\sqrt{\frac{280}{250}}=43.3$
${ɑ}_a$	=	$\frac{\mathrm{2,100}\left({\lambda }_n-13.5\right)}{{{\lambda }_n}^2-15.3{\lambda }_n+\mathrm{2,050}}=\frac{\mathrm{2,100}\left(43.3-13.5\right)}{{\left(43.3\right)}^2-15.3\left(43.3\right)+\mathrm{2,050}}=19.2$
$\lambda$	=	${\lambda }_n+{ɑ}_a{ɑ}_b=43.3+19.2\times 0.5=52.9$
$\eta$	=	$0.00326\left(\lambda -13.5\right)=0.00326\left(52.9-13.5\right)=0.128\ge 0$ , ok
$\xi$	=	$\frac{{\left(\frac{\lambda }{90}\right)}^2+1+\eta }{2{\left(\frac{\lambda }{90}\right)}^2}=\frac{{\left(\frac{52.9}{90}\right)}^2+1+0.128}{2{\left(\frac{52.9}{90}\right)}^2}=2.133$
${ɑ}_{\mathrm{cz}}$	=	$\xi \left[1-\sqrt{1-{\left(\frac{90}{\xi \lambda }\right)}^2}\right]=2.133\left[1-\sqrt{1-{\left(\frac{90}{2.133\times 52.9}\right)}^2}\right]=0.846$

The factored member compression capacity about the Z axis:

The member compression capacity about the Y axis:

where

${\lambda }_{\mathrm{ny}}$	=	$k_e\frac{L}{r_y}\sqrt{k_f}\sqrt{\frac{f_y}{250}}=1.0\frac{\mathrm{1,600}}{76.5}\sqrt{1.0}\sqrt{\frac{280}{250}}=22.1$
${ɑ}_a$	=	$\frac{\mathrm{2,100}\left({\lambda }_n-13.5\right)}{{{\lambda }_n}^2-15.3{\lambda }_n+\mathrm{2,050}}=\frac{\mathrm{2,100}\left(22.1-13.5\right)}{{\left(22.1\right)}^2-15.3\left(22.1\right)+\mathrm{2,050}}=8.21$
$\lambda$	=	${\lambda }_n+{ɑ}_a{ɑ}_b=22.1+8.21\times 0.5=26.2$
$\eta$	=	$0.00326\left(\lambda -13.5\right)=0.00326\left(26.2-13.5\right)=0.041\ge 0$ , OK
$\xi$	=	$\frac{{\left(\frac{\lambda }{90}\right)}^2+1+\eta }{2{\left(\frac{\lambda }{90}\right)}^2}=\frac{{\left(\frac{26.2}{90}\right)}^2+1+0.041}{2{\left(\frac{26.2}{90}\right)}^2}=6.644$
${ɑ}_{\mathrm{cy}}$	=	$\xi \left[1-\sqrt{1-{\left(\frac{90}{\xi \lambda }\right)}^2}\right]=6.644\left[1-\sqrt{1-{\left(\frac{90}{6.644\times 26.2}\right)}^2}\right]=0.957$

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: no axial force so no reduction:

Uniaxial bending capacity about the Y axis: no axial force so no reduction:

Member combined capacity out-of-plane: no axial force so no reduction: