WT in compression and bending

S.1 \[\begin{aligned} \htmlClass{sym-phi}{\phi} &= 0.9 \quad \left(\text{Resistance factor | S16 Cl. 13.1}\right) \end{aligned}\]

S.2

S16 Cl. 3.2, E = 200 000 MPa

\[\begin{aligned} \htmlClass{sym-E}{E} &= 200\ \mathrm{GPa} \end{aligned}\]

S.3

S16 Cl. 3.2, G = 77 000 MPa

\[\begin{aligned} \htmlClass{sym-G}{G} &= 77\ \mathrm{GPa} \end{aligned}\]

S.4 \[\begin{aligned} \htmlClass{sym-A_0}{A_{0}} &= 14300\ \mathrm{mm}^{2} \quad \left(\text{Cross-sectional area}\right) \end{aligned}\]

S.5 \[\begin{aligned} \htmlClass{sym-b}{b} &= 304\ \mathrm{mm} \quad \left(\text{Width of flange or horizontal leg}\right) \end{aligned}\]

S.6 \[\begin{aligned} \htmlClass{sym-t}{t} &= 23.9\ \mathrm{mm} \quad \left(\text{Thickness of flange}\right) \end{aligned}\]

S.7 \[\begin{aligned} \htmlClass{sym-d}{d} &= 456\ \mathrm{mm} \quad \left(\text{Depth of section or height of vertical leg}\right) \end{aligned}\]

S.8 \[\begin{aligned} \htmlClass{sym-w}{w} &= 15.9\ \mathrm{mm} \quad \left(\text{Thickness of web}\right) \end{aligned}\]

S.9 \[\begin{aligned} \htmlClass{sym-y}{y} &= 122\ \mathrm{mm} \quad \left(\text{Vertical distance between centroid and outside face of flange orhorizontal leg}\right) \end{aligned}\]

S.10 \[\begin{aligned} \htmlClass{sym-r_x}{r_{x}} &= 143\ \mathrm{mm} \quad \left(\text{Radius of gyration about axis XX}\right) \end{aligned}\]

S.11 \[\begin{aligned} \htmlClass{sym-r_y}{r_{y}} &= 62.7\ \mathrm{mm} \quad \left(\text{Radius of gyration about axis YY}\right) \end{aligned}\]

S.12 \[\begin{aligned} \htmlClass{sym-y_0}{y_{0}} &= 110\ \mathrm{mm} \quad \left(\text{Distance between centroid and shear centre along geometric axis YY}\right) \end{aligned}\]

S.13 \[\begin{aligned} \htmlClass{sym-J}{J} &= 2.08 \times 10^{6}\ \mathrm{mm}^{4} \quad \left(\text{St-Venant torsional constant}\right) \end{aligned}\]

S.14 \[\begin{aligned} \htmlClass{sym-C_w}{C_{w}} &= 1.24 \times 10^{10}\ \mathrm{mm}^{6} \quad \left(\text{Warping constant}\right) \end{aligned}\]

S.15

S16 Cl. 11, Table 1, flanges

\[\begin{aligned} \htmlClass{sym-lambda_3}{\lambda_{3}} &= \frac{200}{\sqrt{\frac{\htmlClass{sym-F_y}{F_{y}}}{1\ \mathrm{MPa}}}} \\ &= \frac{200}{\sqrt{\frac{\htmlClass{sym-F_y}{350\ \mathrm{MPa}}}{1\ \mathrm{MPa}}}} \\ &= 10.69 \end{aligned}\]

S.16 \[\begin{aligned} \htmlClass{sym-lambda_f}{\lambda_{f}} &= \frac{0.5 \cdot \htmlClass{sym-b}{b}}{\htmlClass{sym-t}{t}} \\ &= \frac{0.5 \cdot \htmlClass{sym-b}{304\ \mathrm{mm}}}{\htmlClass{sym-t}{23.9\ \mathrm{mm}}} \\ &= 6.36 \end{aligned}\]

S.17 \[\begin{aligned} \htmlClass{sym-A_b}{A_{b}} &= 0\ \mathrm{m}^{2} \quad \left(\text{Flange not class 4}\right) \end{aligned}\]

S.18 \[\begin{aligned} \htmlClass{sym-lambda_s}{\lambda_{s}} &= \frac{\htmlClass{sym-d}{d}}{\htmlClass{sym-w}{w}} \\ &= \frac{\htmlClass{sym-d}{456\ \mathrm{mm}}}{\htmlClass{sym-w}{15.9\ \mathrm{mm}}} \\ &= 28.68 \end{aligned}\]

S.19

S16 Cl. 11, Table 1, stem of a tee

\[\begin{aligned} \htmlClass{sym-lambda_3s}{\lambda_{3s}} &= \frac{340}{\sqrt{\frac{\htmlClass{sym-F_y}{F_{y}}}{1\ \mathrm{MPa}}}} \\ &= \frac{340}{\sqrt{\frac{\htmlClass{sym-F_y}{350\ \mathrm{MPa}}}{1\ \mathrm{MPa}}}} \\ &= 18.17 \end{aligned}\]

S.20

S16 Cl. 13.3.4 a), stem past Table 1

\[\begin{aligned} \htmlClass{sym-A_d}{A_{d}} &= \left(\htmlClass{sym-lambda_s}{\lambda_{s}} - \htmlClass{sym-lambda_3s}{\lambda_{3s}}\right) \cdot \htmlClass{sym-w}{w}^{2} \\ &= \left(\htmlClass{sym-lambda_s}{28.68} - \htmlClass{sym-lambda_3s}{18.17}\right) \cdot \left(\htmlClass{sym-w}{15.9\ \mathrm{mm}}\right)^{2} \\ &= 2656\ \mathrm{mm}^{2} \end{aligned}\]

S.21 \[\begin{aligned} \htmlClass{sym-A_e}{A_{e}} &= \htmlClass{sym-A_0}{A_{0}} - \htmlClass{sym-A_b}{A_{b}} - \htmlClass{sym-A_d}{A_{d}} \\ &= \htmlClass{sym-A_0}{14300\ \mathrm{mm}^{2}} - \htmlClass{sym-A_b}{0\ \mathrm{m}^{2}} - \htmlClass{sym-A_d}{2656\ \mathrm{mm}^{2}} \\ &= 11644\ \mathrm{mm}^{2} \end{aligned}\]

S.22

S16 Cl. 13.3.1.2, x_0 = 0

\[\begin{aligned} \htmlClass{sym-r_0}{r_{0}} &= \sqrt{\htmlClass{sym-y_0}{y_{0}}^{2} + \htmlClass{sym-r_x}{r_{x}}^{2} + \htmlClass{sym-r_y}{r_{y}}^{2}} \\ &= \sqrt{\left(\htmlClass{sym-y_0}{110\ \mathrm{mm}}\right)^{2} + \left(\htmlClass{sym-r_x}{143\ \mathrm{mm}}\right)^{2} + \left(\htmlClass{sym-r_y}{62.7\ \mathrm{mm}}\right)^{2}} \\ &= 191\ \mathrm{mm} \end{aligned}\]

S.23

S16 Cl. 13.3.1.2

\[\begin{aligned} \htmlClass{sym-F_ex}{F_{\mathrm{ex}}} &= \pi^{2} \cdot \frac{\htmlClass{sym-E}{E}}{\left(\frac{\htmlClass{sym-k_x}{k_{x}} \cdot \htmlClass{sym-L_x}{L_{x}}}{\htmlClass{sym-r_x}{r_{x}}}\right)^{2}} \\ &= \pi^{2} \cdot \frac{\htmlClass{sym-E}{200\ \mathrm{GPa}}}{\left(\frac{\htmlClass{sym-k_x}{1} \cdot \htmlClass{sym-L_x}{3000\ \mathrm{mm}}}{\htmlClass{sym-r_x}{143\ \mathrm{mm}}}\right)^{2}} \\ &= 4.485\ \mathrm{GPa} \end{aligned}\]

S.24

S16 Cl. 13.3.1.2

\[\begin{aligned} \htmlClass{sym-F_ey}{F_{\mathrm{ey}}} &= \pi^{2} \cdot \frac{\htmlClass{sym-E}{E}}{\left(\frac{\htmlClass{sym-k_y}{k_{y}} \cdot \htmlClass{sym-L_y}{L_{y}}}{\htmlClass{sym-r_y}{r_{y}}}\right)^{2}} \\ &= \pi^{2} \cdot \frac{\htmlClass{sym-E}{200\ \mathrm{GPa}}}{\left(\frac{\htmlClass{sym-k_y}{1} \cdot \htmlClass{sym-L_y}{3000\ \mathrm{mm}}}{\htmlClass{sym-r_y}{62.7\ \mathrm{mm}}}\right)^{2}} \\ &= 862.2\ \mathrm{MPa} \end{aligned}\]

S.25

S16 Cl. 13.3.1.2, K_z = 1 and L_z = L_x

\[\begin{aligned} \htmlClass{sym-F_ez}{F_{\mathrm{ez}}} &= \frac{\frac{\pi^{2} \cdot \htmlClass{sym-E}{E} \cdot \htmlClass{sym-C_w}{C_{w}}}{\htmlClass{sym-L_x}{L_{x}}^{2}} + \htmlClass{sym-G}{G} \cdot \htmlClass{sym-J}{J}}{\htmlClass{sym-A_0}{A_{0}} \cdot \htmlClass{sym-r_0}{r_{0}}^{2}} \\ &= \frac{\frac{\pi^{2} \cdot \htmlClass{sym-E}{200\ \mathrm{GPa}} \cdot \htmlClass{sym-C_w}{1.24 \times 10^{10}\ \mathrm{mm}^{6}}}{\left(\htmlClass{sym-L_x}{3000\ \mathrm{mm}}\right)^{2}} + \htmlClass{sym-G}{77\ \mathrm{GPa}} \cdot \htmlClass{sym-J}{2.08 \times 10^{6}\ \mathrm{mm}^{4}}}{\htmlClass{sym-A_0}{14300\ \mathrm{mm}^{2}} \cdot \left(\htmlClass{sym-r_0}{191\ \mathrm{mm}}\right)^{2}} \\ &= 312.2\ \mathrm{MPa} \end{aligned}\]

S.26

S16 Cl. 13.3.1.1 b) and c)

\[\begin{aligned} \htmlClass{sym-F_e}{F_{e}} &= \operatorname{root}_{\min F > 0}\left((F - \htmlClass{sym-F_ex}{F_{\mathrm{ex}}})(F - \htmlClass{sym-F_ey}{F_{\mathrm{ey}}})(F - \htmlClass{sym-F_ez}{F_{\mathrm{ez}}}) - F^{2}(F - \htmlClass{sym-F_ey}{F_{\mathrm{ey}}})\left(\frac{0\ \mathrm{m}}{\htmlClass{sym-r_0}{r_{0}}}\right)^{2} - F^{2}(F - \htmlClass{sym-F_ex}{F_{\mathrm{ex}}})\left(\frac{\htmlClass{sym-y_0}{y_{0}}}{\htmlClass{sym-r_0}{r_{0}}}\right)^{2}\right) \\ &= \operatorname{root}_{\min F > 0}\left((F - \htmlClass{sym-F_ex}{4.485\ \mathrm{GPa}})(F - \htmlClass{sym-F_ey}{862.2\ \mathrm{MPa}})(F - \htmlClass{sym-F_ez}{312.2\ \mathrm{MPa}}) - F^{2}(F - \htmlClass{sym-F_ey}{862.2\ \mathrm{MPa}})\left(\frac{0\ \mathrm{m}}{\htmlClass{sym-r_0}{191\ \mathrm{mm}}}\right)^{2} - F^{2}(F - \htmlClass{sym-F_ex}{4.485\ \mathrm{GPa}})\left(\frac{\htmlClass{sym-y_0}{110\ \mathrm{mm}}}{\htmlClass{sym-r_0}{191\ \mathrm{mm}}}\right)^{2}\right) \\ &= 271\ \mathrm{MPa} \end{aligned}\]

S.27 \[\begin{aligned} \htmlClass{sym-lambda_e}{\lambda_{e}} &= \sqrt{\frac{\htmlClass{sym-F_y}{F_{y}}}{\htmlClass{sym-F_e}{F_{e}}}} \\ &= \sqrt{\frac{\htmlClass{sym-F_y}{350\ \mathrm{MPa}}}{\htmlClass{sym-F_e}{271\ \mathrm{MPa}}}} \\ &= 1.136 \end{aligned}\]

S.28

S16 Cl. 13.3.1, n = 1.34

\[\begin{aligned} \htmlClass{sym-f_s}{f_{s}} &= \htmlClass{sym-F_y}{F_{y}} \cdot \left(1 + \htmlClass{sym-lambda_e}{\lambda_{e}}^{2 \cdot 1.34}\right)^{\frac{-1}{1.34}} \\ &= \htmlClass{sym-F_y}{350\ \mathrm{MPa}} \cdot \left(1 + \htmlClass{sym-lambda_e}{1.136}^{2 \cdot 1.34}\right)^{\frac{-1}{1.34}} \\ &= 181.6\ \mathrm{MPa} \end{aligned}\]

S.29

S16 Cl. 13.3.4 a)

\[\begin{aligned} \htmlClass{sym-C_r}{C_{r}} &= \htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-A_e}{A_{e}} \cdot \htmlClass{sym-f_s}{f_{s}} \\ &= \htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-A_e}{11644\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-f_s}{181.6\ \mathrm{MPa}} \\ &= 1.903\ \mathrm{MN} \end{aligned}\]

S.30 \[\begin{aligned} \htmlClass{sym-I_x}{I_{x}} &= 2.92 \times 10^{8}\ \mathrm{mm}^{4} \quad \left(\text{Moment of inertia about axis XX}\right) \end{aligned}\]

S.31 \[\begin{aligned} \htmlClass{sym-I_y}{I_{y}} &= 5.61 \times 10^{7}\ \mathrm{mm}^{4} \quad \left(\text{Moment of inertia about axis YY}\right) \end{aligned}\]

S.32 \[\begin{aligned} \htmlClass{sym-S_xt}{S_{\mathrm{xt}}} &= 874000\ \mathrm{mm}^{3} \quad \left(\text{Elastic section modulus about axis XX}\right) \end{aligned}\]

S.33 \[\begin{aligned} \htmlClass{sym-S_xc}{S_{\mathrm{xc}}} &= \frac{\htmlClass{sym-I_x}{I_{x}}}{\htmlClass{sym-y}{y}} \\ &= \frac{\htmlClass{sym-I_x}{2.92 \times 10^{8}\ \mathrm{mm}^{4}}}{\htmlClass{sym-y}{122\ \mathrm{mm}}} \\ &= 2.393 \times 10^{6}\ \mathrm{mm}^{3} \end{aligned}\]

S.34

AISC 360-22 F9-3

\[\begin{aligned} \htmlClass{sym-M_y}{M_{y}} &= \htmlClass{sym-S_xt}{S_{\mathrm{xt}}} \cdot \htmlClass{sym-F_y}{F_{y}} \\ &= \htmlClass{sym-S_xt}{874000\ \mathrm{mm}^{3}} \cdot \htmlClass{sym-F_y}{350\ \mathrm{MPa}} \\ &= 305.9\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.35

AISC 360-22 F9-1 and F9-2, M_p = 1.6 M_y, under F_y Z_x for every tabulated WT

\[\begin{aligned} \htmlClass{sym-M_nx1}{M_{\mathrm{nx}1}} &= 1.6 \cdot \htmlClass{sym-M_y}{M_{y}} \\ &= 1.6 \cdot \htmlClass{sym-M_y}{305.9\ \mathrm{kN} \cdot \mathrm{m}} \\ &= 489.4\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.36

AISC 360-22 F9-8

\[\begin{aligned} \htmlClass{sym-L_p}{L_{p}} &= 1.76 \cdot \htmlClass{sym-r_y}{r_{y}} \cdot \sqrt{\frac{\htmlClass{sym-E}{E}}{\htmlClass{sym-F_y}{F_{y}}}} \\ &= 1.76 \cdot \htmlClass{sym-r_y}{62.7\ \mathrm{mm}} \cdot \sqrt{\frac{\htmlClass{sym-E}{200\ \mathrm{GPa}}}{\htmlClass{sym-F_y}{350\ \mathrm{MPa}}}} \\ &= 2.638\ \mathrm{m} \end{aligned}\]

S.37

AISC 360-22 F9-9

\[\begin{aligned} \htmlClass{sym-L_r}{L_{r}} &= \frac{1.95 \cdot \frac{\htmlClass{sym-E}{E}}{\htmlClass{sym-F_y}{F_{y}}} \cdot \sqrt{\htmlClass{sym-I_y}{I_{y}} \cdot \htmlClass{sym-J}{J}}}{\htmlClass{sym-S_xt}{S_{\mathrm{xt}}}} \cdot \sqrt{\frac{2.36 \cdot \frac{\htmlClass{sym-F_y}{F_{y}}}{\htmlClass{sym-E}{E}} \cdot \htmlClass{sym-d}{d} \cdot \htmlClass{sym-S_xt}{S_{\mathrm{xt}}}}{\htmlClass{sym-J}{J}} + 1} \\ &= \frac{1.95 \cdot \frac{\htmlClass{sym-E}{200\ \mathrm{GPa}}}{\htmlClass{sym-F_y}{350\ \mathrm{MPa}}} \cdot \sqrt{\htmlClass{sym-I_y}{5.61 \times 10^{7}\ \mathrm{mm}^{4}} \cdot \htmlClass{sym-J}{2.08 \times 10^{6}\ \mathrm{mm}^{4}}}}{\htmlClass{sym-S_xt}{874000\ \mathrm{mm}^{3}}} \cdot \sqrt{\frac{2.36 \cdot \frac{\htmlClass{sym-F_y}{350\ \mathrm{MPa}}}{\htmlClass{sym-E}{200\ \mathrm{GPa}}} \cdot \htmlClass{sym-d}{456\ \mathrm{mm}} \cdot \htmlClass{sym-S_xt}{874000\ \mathrm{mm}^{3}}}{\htmlClass{sym-J}{2.08 \times 10^{6}\ \mathrm{mm}^{4}}} + 1} \\ &= 18.43\ \mathrm{m} \end{aligned}\]

S.38

AISC 360-22 F9-11

\[\begin{aligned} \htmlClass{sym-B}{B} &= 2.3 \cdot \frac{\htmlClass{sym-d}{d}}{\htmlClass{sym-L_y}{L_{y}}} \cdot \sqrt{\frac{\htmlClass{sym-I_y}{I_{y}}}{\htmlClass{sym-J}{J}}} \\ &= 2.3 \cdot \frac{\htmlClass{sym-d}{456\ \mathrm{mm}}}{\htmlClass{sym-L_y}{3000\ \mathrm{mm}}} \cdot \sqrt{\frac{\htmlClass{sym-I_y}{5.61 \times 10^{7}\ \mathrm{mm}^{4}}}{\htmlClass{sym-J}{2.08 \times 10^{6}\ \mathrm{mm}^{4}}}} \\ &= 1.816 \end{aligned}\]

S.39

AISC 360-22 F9-10

\[\begin{aligned} \htmlClass{sym-M_cr}{M_{\mathrm{cr}}} &= 1.95 \cdot \frac{\htmlClass{sym-E}{E}}{\htmlClass{sym-L_y}{L_{y}}} \cdot \sqrt{\htmlClass{sym-I_y}{I_{y}} \cdot \htmlClass{sym-J}{J}} \cdot \left(\htmlClass{sym-B}{B} + \sqrt{1 + \htmlClass{sym-B}{B}^{2}}\right) \\ &= 1.95 \cdot \frac{\htmlClass{sym-E}{200\ \mathrm{GPa}}}{\htmlClass{sym-L_y}{3000\ \mathrm{mm}}} \cdot \sqrt{\htmlClass{sym-I_y}{5.61 \times 10^{7}\ \mathrm{mm}^{4}} \cdot \htmlClass{sym-J}{2.08 \times 10^{6}\ \mathrm{mm}^{4}}} \cdot \left(\htmlClass{sym-B}{1.816} + \sqrt{1 + \htmlClass{sym-B}{1.816}^{2}}\right) \\ &= 5.46\ \mathrm{MN} \cdot \mathrm{m} \end{aligned}\]

S.40

AISC 360-22 F9-6

\[\begin{aligned} \htmlClass{sym-M_nx2}{M_{\mathrm{nx}2}} &= \htmlClass{sym-M_nx1}{M_{\mathrm{nx}1}} - \frac{\left(\htmlClass{sym-M_nx1}{M_{\mathrm{nx}1}} - \htmlClass{sym-M_y}{M_{y}}\right) \cdot \left(\htmlClass{sym-L_y}{L_{y}} - \htmlClass{sym-L_p}{L_{p}}\right)}{\htmlClass{sym-L_r}{L_{r}} - \htmlClass{sym-L_p}{L_{p}}} \\ &= \htmlClass{sym-M_nx1}{489.4\ \mathrm{kN} \cdot \mathrm{m}} - \frac{\left(\htmlClass{sym-M_nx1}{489.4\ \mathrm{kN} \cdot \mathrm{m}} - \htmlClass{sym-M_y}{305.9\ \mathrm{kN} \cdot \mathrm{m}}\right) \cdot \left(\htmlClass{sym-L_y}{3000\ \mathrm{mm}} - \htmlClass{sym-L_p}{2.638\ \mathrm{m}}\right)}{\htmlClass{sym-L_r}{18.43\ \mathrm{m}} - \htmlClass{sym-L_p}{2.638\ \mathrm{m}}} \\ &= 485.2\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.41

AISC 360-22 Table B4.1b case 10, flanges of tees

\[\begin{aligned} \htmlClass{sym-lambda_pf}{\lambda_{\mathrm{pf}}} &= 0.38 \cdot \sqrt{\frac{\htmlClass{sym-E}{E}}{\htmlClass{sym-F_y}{F_{y}}}} \\ &= 0.38 \cdot \sqrt{\frac{\htmlClass{sym-E}{200\ \mathrm{GPa}}}{\htmlClass{sym-F_y}{350\ \mathrm{MPa}}}} \\ &= 9.084 \end{aligned}\]

S.42

AISC 360-22 Table B4.1b case 10, flanges of tees

\[\begin{aligned} \htmlClass{sym-lambda_rf}{\lambda_{\mathrm{rf}}} &= 1 \cdot \sqrt{\frac{\htmlClass{sym-E}{E}}{\htmlClass{sym-F_y}{F_{y}}}} \\ &= 1 \cdot \sqrt{\frac{\htmlClass{sym-E}{200\ \mathrm{GPa}}}{\htmlClass{sym-F_y}{350\ \mathrm{MPa}}}} \\ &= 23.9 \end{aligned}\]

S.43

AISC 360-22 F9.3(a)(1), compact flange: does not apply

\[\begin{aligned} \htmlClass{sym-M_nx3}{M_{\mathrm{nx}3}} &= 1 \cdot \htmlClass{sym-M_nx1}{M_{\mathrm{nx}1}} \\ &= 1 \cdot \htmlClass{sym-M_nx1}{489.4\ \mathrm{kN} \cdot \mathrm{m}} \\ &= 489.4\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.44

AISC 360-22 F9 but lateral-torsional buckling, S16 Cl. 13.1 a) phi

\[\begin{aligned} \htmlClass{sym-M_rx}{M_{\mathrm{rx}}} &= \htmlClass{sym-phi}{\phi} \cdot \min\left(\htmlClass{sym-M_nx1}{M_{\mathrm{nx}1}}, \htmlClass{sym-M_nx3}{M_{\mathrm{nx}3}}\right) \\ &= \htmlClass{sym-phi}{0.9} \cdot \min\left(\htmlClass{sym-M_nx1}{489.4\ \mathrm{kN} \cdot \mathrm{m}}, \htmlClass{sym-M_nx3}{489.4\ \mathrm{kN} \cdot \mathrm{m}}\right) \\ &= 440.5\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.45

AISC 360-22 F9, the lowest, S16 Cl. 13.1 a) phi

\[\begin{aligned} \htmlClass{sym-M_rxl}{M_{\mathrm{rxl}}} &= \htmlClass{sym-phi}{\phi} \cdot \min\left(\htmlClass{sym-M_nx1}{M_{\mathrm{nx}1}}, \htmlClass{sym-M_nx2}{M_{\mathrm{nx}2}}, \htmlClass{sym-M_nx3}{M_{\mathrm{nx}3}}\right) \\ &= \htmlClass{sym-phi}{0.9} \cdot \min\left(\htmlClass{sym-M_nx1}{489.4\ \mathrm{kN} \cdot \mathrm{m}}, \htmlClass{sym-M_nx2}{485.2\ \mathrm{kN} \cdot \mathrm{m}}, \htmlClass{sym-M_nx3}{489.4\ \mathrm{kN} \cdot \mathrm{m}}\right) \\ &= 436.7\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.46

S16 Cl. 13.8.5, L the unbraced length (Cl. 10.3.2, K = 1)

\[\begin{aligned} \htmlClass{sym-C_ex}{C_{\mathrm{ex}}} &= \frac{\pi^{2} \cdot \htmlClass{sym-E}{E} \cdot \htmlClass{sym-I_x}{I_{x}}}{\htmlClass{sym-L_x}{L_{x}}^{2}} \\ &= \frac{\pi^{2} \cdot \htmlClass{sym-E}{200\ \mathrm{GPa}} \cdot \htmlClass{sym-I_x}{2.92 \times 10^{8}\ \mathrm{mm}^{4}}}{\left(\htmlClass{sym-L_x}{3000\ \mathrm{mm}}\right)^{2}} \\ &= 64.04\ \mathrm{MN} \end{aligned}\]

S.47

S16 Cl. 13.8.5

\[\begin{aligned} \htmlClass{sym-U_1x}{U_{1x}} &= \frac{\htmlClass{sym-omega_1}{\omega_{1}}}{1 - \frac{\htmlClass{sym-C_f}{C_{f}}}{\htmlClass{sym-C_ex}{C_{\mathrm{ex}}}}} \\ &= \frac{\htmlClass{sym-omega_1}{1}}{1 - \frac{\htmlClass{sym-C_f}{130\ \mathrm{kN}}}{\htmlClass{sym-C_ex}{64.04\ \mathrm{MN}}}} \\ &= 1.002 \end{aligned}\]

S.48

S16 Cl. 13.8.4 b)

\[\begin{aligned} \htmlClass{sym-I_m}{I_{m}} &= \frac{\htmlClass{sym-C_f}{C_{f}}}{\htmlClass{sym-C_r}{C_{r}}} + \frac{\htmlClass{sym-U_1x}{U_{1x}} \cdot \htmlClass{sym-M_fx}{M_{\mathrm{fx}}}}{\htmlClass{sym-M_rx}{M_{\mathrm{rx}}}} \\ &= \frac{\htmlClass{sym-C_f}{130\ \mathrm{kN}}}{\htmlClass{sym-C_r}{1.903\ \mathrm{MN}}} + \frac{\htmlClass{sym-U_1x}{1.002} \cdot \htmlClass{sym-M_fx}{3.55\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_rx}{440.5\ \mathrm{kN} \cdot \mathrm{m}}} \\ &= 0.07638 \end{aligned}\]

S.49

S16 Cl. 13.8.2 c) v), not less than 1.0

\[\begin{aligned} \htmlClass{sym-U_1xl}{U_{1\mathrm{xl}}} &= \max\left(\htmlClass{sym-U_1x}{U_{1x}}, 1\right) \\ &= \max\left(\htmlClass{sym-U_1x}{1.002}, 1\right) \\ &= 1.002 \end{aligned}\]

S.50

S16 Cl. 13.8.4 c)

\[\begin{aligned} \htmlClass{sym-I_l}{I_{l}} &= \frac{\htmlClass{sym-C_f}{C_{f}}}{\htmlClass{sym-C_r}{C_{r}}} + \frac{\htmlClass{sym-U_1xl}{U_{1\mathrm{xl}}} \cdot \htmlClass{sym-M_fx}{M_{\mathrm{fx}}}}{\htmlClass{sym-M_rxl}{M_{\mathrm{rxl}}}} \\ &= \frac{\htmlClass{sym-C_f}{130\ \mathrm{kN}}}{\htmlClass{sym-C_r}{1.903\ \mathrm{MN}}} + \frac{\htmlClass{sym-U_1xl}{1.002} \cdot \htmlClass{sym-M_fx}{3.55\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_rxl}{436.7\ \mathrm{kN} \cdot \mathrm{m}}} \\ &= 0.07645 \end{aligned}\]