Single angle in tension

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 \[\begin{aligned} \htmlClass{sym-b}{b} &= 76.2\ \mathrm{mm} \quad \left(\text{Depth of section or height of vertical leg}\right) \end{aligned}\]

S.4 \[\begin{aligned} \htmlClass{sym-t}{t} &= 6.35\ \mathrm{mm} \quad \left(\text{Thickness of flange}\right) \end{aligned}\]

S.5 \[\begin{aligned} \htmlClass{sym-A_0}{A_{0}} &= 927\ \mathrm{mm}^{2} \quad \left(\text{Cross-sectional area}\right) \end{aligned}\]

S.6 \[\begin{aligned} \htmlClass{sym-r_x0}{r_{x0}} &= 29.8\ \mathrm{mm} \quad \left(\text{Maximum radius of gyration about major principal axis(single angle)}\right) \end{aligned}\]

S.7 \[\begin{aligned} \htmlClass{sym-r_y0}{r_{y0}} &= 15\ \mathrm{mm} \quad \left(\text{Minimum radius of gyration about minor principal axis(single angle)}\right) \end{aligned}\]

S.8 \[\begin{aligned} \htmlClass{sym-x_0}{x_{0}} &= 25.8\ \mathrm{mm} \quad \left(\text{Distance between centroid and shear centre along major principalaxis}\right) \end{aligned}\]

S.9 \[\begin{aligned} \htmlClass{sym-I_x1}{I_{x1}} &= \htmlClass{sym-A_0}{A_{0}} \cdot \htmlClass{sym-r_x0}{r_{x0}}^{2} \\ &= \htmlClass{sym-A_0}{927\ \mathrm{mm}^{2}} \cdot \left(\htmlClass{sym-r_x0}{29.8\ \mathrm{mm}}\right)^{2} \\ &= 823213\ \mathrm{mm}^{4} \end{aligned}\]

S.10 \[\begin{aligned} \htmlClass{sym-I_y1}{I_{y1}} &= \htmlClass{sym-A_0}{A_{0}} \cdot \htmlClass{sym-r_y0}{r_{y0}}^{2} \\ &= \htmlClass{sym-A_0}{927\ \mathrm{mm}^{2}} \cdot \left(\htmlClass{sym-r_y0}{15\ \mathrm{mm}}\right)^{2} \\ &= 208575\ \mathrm{mm}^{4} \end{aligned}\]

S.11 \[\begin{aligned} \htmlClass{sym-M_fx}{M_{\mathrm{fx}}} &= 0.707 \cdot \left(\htmlClass{sym-e}{e} + \frac{\htmlClass{sym-t_p}{t_{p}}}{2}\right) \cdot \htmlClass{sym-T_f}{T_{f}} \\ &= 0.707 \cdot \left(\htmlClass{sym-e}{45\ \mathrm{mm}} + \frac{\htmlClass{sym-t_p}{10\ \mathrm{mm}}}{2}\right) \cdot \htmlClass{sym-T_f}{40\ \mathrm{kN}} \\ &= 1.414\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.12 \[\begin{aligned} \htmlClass{sym-M_fy}{M_{\mathrm{fy}}} &= \left| 0.707 \cdot \left(\htmlClass{sym-e}{e} - \frac{\htmlClass{sym-t_p}{t_{p}}}{2} - \htmlClass{sym-t}{t}\right) - \htmlClass{sym-x_0}{x_{0}} \right| \cdot \htmlClass{sym-T_f}{T_{f}} \\ &= \left| 0.707 \cdot \left(\htmlClass{sym-e}{45\ \mathrm{mm}} - \frac{\htmlClass{sym-t_p}{10\ \mathrm{mm}}}{2} - \htmlClass{sym-t}{6.35\ \mathrm{mm}}\right) - \htmlClass{sym-x_0}{25.8\ \mathrm{mm}} \right| \cdot \htmlClass{sym-T_f}{40\ \mathrm{kN}} \\ &= 80.38\ \mathrm{N} \cdot \mathrm{m} \end{aligned}\]

S.13

S16 Cl. 10.4.1, a tension member's L/r, no K

\[\begin{aligned} \htmlClass{sym-lambda_s}{\lambda_{s}} &= \frac{\htmlClass{sym-L}{L}}{\htmlClass{sym-r_y0}{r_{y0}}} \\ &= \frac{\htmlClass{sym-L}{1500\ \mathrm{mm}}}{\htmlClass{sym-r_y0}{15\ \mathrm{mm}}} \\ &= 100 \end{aligned}\]

S.14

S16 Cl. 13.2, gross section yield

\[\begin{aligned} \htmlClass{sym-T_r}{T_{r}} &= \htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-A_0}{A_{0}} \cdot \htmlClass{sym-F_y}{F_{y}} \\ &= \htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-A_0}{927\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-F_y}{300\ \mathrm{MPa}} \\ &= 250.3\ \mathrm{kN} \end{aligned}\]

S.15 \[\begin{aligned} \htmlClass{sym-S_x1}{S_{x1}} &= \frac{\htmlClass{sym-I_x1}{I_{x1}}}{0.707 \cdot \htmlClass{sym-b}{b}} \\ &= \frac{\htmlClass{sym-I_x1}{823213\ \mathrm{mm}^{4}}}{0.707 \cdot \htmlClass{sym-b}{76.2\ \mathrm{mm}}} \\ &= 15281\ \mathrm{mm}^{3} \end{aligned}\]

S.16

AISC 360-22 F10.1, the yield moment, at the tips

\[\begin{aligned} \htmlClass{sym-M_y}{M_{y}} &= \htmlClass{sym-F_y}{F_{y}} \cdot \htmlClass{sym-S_x1}{S_{x1}} \\ &= \htmlClass{sym-F_y}{300\ \mathrm{MPa}} \cdot \htmlClass{sym-S_x1}{15281\ \mathrm{mm}^{3}} \\ &= 4.584\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.17

AISC 360-22 F10-1

\[\begin{aligned} \htmlClass{sym-M_nx1}{M_{\mathrm{nx}1}} &= 1.5 \cdot \htmlClass{sym-M_y}{M_{y}} \\ &= 1.5 \cdot \htmlClass{sym-M_y}{4.584\ \mathrm{kN} \cdot \mathrm{m}} \\ &= 6.876\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.18

AISC 360-22 F10-4, C_b = 1, beta_w = 0 for equal legs

\[\begin{aligned} \htmlClass{sym-M_cr}{M_{\mathrm{cr}}} &= \frac{9 \cdot \htmlClass{sym-E}{E} \cdot \htmlClass{sym-A_0}{A_{0}} \cdot \htmlClass{sym-r_y0}{r_{y0}} \cdot \htmlClass{sym-t}{t}}{8 \cdot \htmlClass{sym-L}{L}} \\ &= \frac{9 \cdot \htmlClass{sym-E}{200\ \mathrm{GPa}} \cdot \htmlClass{sym-A_0}{927\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-r_y0}{15\ \mathrm{mm}} \cdot \htmlClass{sym-t}{6.35\ \mathrm{mm}}}{8 \cdot \htmlClass{sym-L}{1500\ \mathrm{mm}}} \\ &= 13.24\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.19

AISC 360-22 F10-2

\[\begin{aligned} \htmlClass{sym-M_nx2}{M_{\mathrm{nx}2}} &= \min\left(\left(1.92 - 1.17 \cdot \sqrt{\frac{\htmlClass{sym-M_y}{M_{y}}}{\htmlClass{sym-M_cr}{M_{\mathrm{cr}}}}}\right) \cdot \htmlClass{sym-M_y}{M_{y}}, 1.5 \cdot \htmlClass{sym-M_y}{M_{y}}\right) \\ &= \min\left(\left(1.92 - 1.17 \cdot \sqrt{\frac{\htmlClass{sym-M_y}{4.584\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_cr}{13.24\ \mathrm{kN} \cdot \mathrm{m}}}}\right) \cdot \htmlClass{sym-M_y}{4.584\ \mathrm{kN} \cdot \mathrm{m}}, 1.5 \cdot \htmlClass{sym-M_y}{4.584\ \mathrm{kN} \cdot \mathrm{m}}\right) \\ &= 5.646\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.20 \[\begin{aligned} \htmlClass{sym-lambda_l}{\lambda_{l}} &= \frac{\htmlClass{sym-b}{b}}{\htmlClass{sym-t}{t}} \\ &= \frac{\htmlClass{sym-b}{76.2\ \mathrm{mm}}}{\htmlClass{sym-t}{6.35\ \mathrm{mm}}} \\ &= 12 \end{aligned}\]

S.21

AISC 360-22 Table B4.1b case 12, legs of single angles

\[\begin{aligned} \htmlClass{sym-lambda_p}{\lambda_{p}} &= 0.54 \cdot \sqrt{\frac{\htmlClass{sym-E}{E}}{\htmlClass{sym-F_y}{F_{y}}}} \\ &= 0.54 \cdot \sqrt{\frac{\htmlClass{sym-E}{200\ \mathrm{GPa}}}{\htmlClass{sym-F_y}{300\ \mathrm{MPa}}}} \\ &= 13.94 \end{aligned}\]

S.22

AISC 360-22 Table B4.1b case 12, legs of single angles

\[\begin{aligned} \htmlClass{sym-lambda_r}{\lambda_{r}} &= 0.91 \cdot \sqrt{\frac{\htmlClass{sym-E}{E}}{\htmlClass{sym-F_y}{F_{y}}}} \\ &= 0.91 \cdot \sqrt{\frac{\htmlClass{sym-E}{200\ \mathrm{GPa}}}{\htmlClass{sym-F_y}{300\ \mathrm{MPa}}}} \\ &= 23.5 \end{aligned}\]

S.23 \[\begin{aligned} \htmlClass{sym-leg}{\mathrm{leg}} &= 1 \quad \left(\text{Compact leg}\right) \end{aligned}\]

S.24

AISC 360-22 F10.3(a), compact: does not apply, F10-1 stands

\[\begin{aligned} \htmlClass{sym-M_nx3}{M_{\mathrm{nx}3}} &= 1.5 \cdot \htmlClass{sym-M_y}{M_{y}} \\ &= 1.5 \cdot \htmlClass{sym-M_y}{4.584\ \mathrm{kN} \cdot \mathrm{m}} \\ &= 6.876\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.25

AISC 360-22 F10, the lowest of the limit states

\[\begin{aligned} \htmlClass{sym-M_nx}{M_{\mathrm{nx}}} &= \min\left(\htmlClass{sym-M_nx1}{M_{\mathrm{nx}1}}, \htmlClass{sym-M_nx2}{M_{\mathrm{nx}2}}, \htmlClass{sym-M_nx3}{M_{\mathrm{nx}3}}\right) \\ &= \min\left(\htmlClass{sym-M_nx1}{6.876\ \mathrm{kN} \cdot \mathrm{m}}, \htmlClass{sym-M_nx2}{5.646\ \mathrm{kN} \cdot \mathrm{m}}, \htmlClass{sym-M_nx3}{6.876\ \mathrm{kN} \cdot \mathrm{m}}\right) \\ &= 5.646\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.26

S16 Cl. 13.1 a), phi on the nominal strength

\[\begin{aligned} \htmlClass{sym-M_rx}{M_{\mathrm{rx}}} &= \htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-M_nx}{M_{\mathrm{nx}}} \\ &= \htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-M_nx}{5.646\ \mathrm{kN} \cdot \mathrm{m}} \\ &= 5.082\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.27 \[\begin{aligned} \htmlClass{sym-x_c}{x_{c}} &= \frac{1.414 \cdot \htmlClass{sym-t}{t}}{2} + \htmlClass{sym-x_0}{x_{0}} \\ &= \frac{1.414 \cdot \htmlClass{sym-t}{6.35\ \mathrm{mm}}}{2} + \htmlClass{sym-x_0}{25.8\ \mathrm{mm}} \\ &= 30.29\ \mathrm{mm} \end{aligned}\]

S.28

AISC 360-22 F10.3, S_c to the toe, its inside corner

\[\begin{aligned} \htmlClass{sym-x_t}{x_{t}} &= 0.707 \cdot \left(\htmlClass{sym-b}{b} + \htmlClass{sym-t}{t}\right) - \htmlClass{sym-x_c}{x_{c}} \\ &= 0.707 \cdot \left(\htmlClass{sym-b}{76.2\ \mathrm{mm}} + \htmlClass{sym-t}{6.35\ \mathrm{mm}}\right) - \htmlClass{sym-x_c}{30.29\ \mathrm{mm}} \\ &= 28.07\ \mathrm{mm} \end{aligned}\]

S.29 \[\begin{aligned} \htmlClass{sym-S_y1}{S_{y1}} &= \frac{\htmlClass{sym-I_y1}{I_{y1}}}{\htmlClass{sym-x_t}{x_{t}}} \\ &= \frac{\htmlClass{sym-I_y1}{208575\ \mathrm{mm}^{4}}}{\htmlClass{sym-x_t}{28.07\ \mathrm{mm}}} \\ &= 7430\ \mathrm{mm}^{3} \end{aligned}\]

S.30 \[\begin{aligned} \htmlClass{sym-S_yh}{S_{\mathrm{yh}}} &= \frac{\htmlClass{sym-I_y1}{I_{y1}}}{\htmlClass{sym-x_c}{x_{c}}} \\ &= \frac{\htmlClass{sym-I_y1}{208575\ \mathrm{mm}^{4}}}{\htmlClass{sym-x_c}{30.29\ \mathrm{mm}}} \\ &= 6886\ \mathrm{mm}^{3} \end{aligned}\]

S.31

AISC 360-22 F10-1, M_y at first yield, the heel

\[\begin{aligned} \htmlClass{sym-M_ny1}{M_{\mathrm{ny}1}} &= 1.5 \cdot \htmlClass{sym-F_y}{F_{y}} \cdot \htmlClass{sym-S_yh}{S_{\mathrm{yh}}} \\ &= 1.5 \cdot \htmlClass{sym-F_y}{300\ \mathrm{MPa}} \cdot \htmlClass{sym-S_yh}{6886\ \mathrm{mm}^{3}} \\ &= 3.099\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.32

AISC 360-22 F10.3(a), compact: does not apply

\[\begin{aligned} \htmlClass{sym-M_ny2}{M_{\mathrm{ny}2}} &= 1 \cdot \htmlClass{sym-M_ny1}{M_{\mathrm{ny}1}} \\ &= 1 \cdot \htmlClass{sym-M_ny1}{3.099\ \mathrm{kN} \cdot \mathrm{m}} \\ &= 3.099\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.33

AISC 360-22 F10, the lowest, S16 Cl. 13.1 a) phi

\[\begin{aligned} \htmlClass{sym-M_ry}{M_{\mathrm{ry}}} &= \htmlClass{sym-phi}{\phi} \cdot \min\left(\htmlClass{sym-M_ny1}{M_{\mathrm{ny}1}}, \htmlClass{sym-M_ny2}{M_{\mathrm{ny}2}}\right) \\ &= \htmlClass{sym-phi}{0.9} \cdot \min\left(\htmlClass{sym-M_ny1}{3.099\ \mathrm{kN} \cdot \mathrm{m}}, \htmlClass{sym-M_ny2}{3.099\ \mathrm{kN} \cdot \mathrm{m}}\right) \\ &= 2.789\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.34

S16 Cl. 13.9.1

\[\begin{aligned} \htmlClass{sym-I_a}{I_{a}} &= \frac{\htmlClass{sym-T_f}{T_{f}}}{\htmlClass{sym-T_r}{T_{r}}} + \frac{\htmlClass{sym-M_fx}{M_{\mathrm{fx}}}}{\htmlClass{sym-M_rx}{M_{\mathrm{rx}}}} + \frac{\htmlClass{sym-M_fy}{M_{\mathrm{fy}}}}{\htmlClass{sym-M_ry}{M_{\mathrm{ry}}}} \\ &= \frac{\htmlClass{sym-T_f}{40\ \mathrm{kN}}}{\htmlClass{sym-T_r}{250.3\ \mathrm{kN}}} + \frac{\htmlClass{sym-M_fx}{1.414\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_rx}{5.082\ \mathrm{kN} \cdot \mathrm{m}}} + \frac{\htmlClass{sym-M_fy}{80.38\ \mathrm{N} \cdot \mathrm{m}}}{\htmlClass{sym-M_ry}{2.789\ \mathrm{kN} \cdot \mathrm{m}}} \\ &= 0.4669 \end{aligned}\]

S.35

S16 Cl. 13.9.3 b)

\[\begin{aligned} \htmlClass{sym-I_b}{I_{b}} &= \frac{\htmlClass{sym-M_fx}{M_{\mathrm{fx}}}}{\htmlClass{sym-M_rx}{M_{\mathrm{rx}}}} + \frac{\htmlClass{sym-M_fy}{M_{\mathrm{fy}}}}{\htmlClass{sym-M_ry}{M_{\mathrm{ry}}}} - \frac{\htmlClass{sym-T_f}{T_{f}} \cdot \htmlClass{sym-S_x1}{S_{x1}}}{\htmlClass{sym-M_rx}{M_{\mathrm{rx}}} \cdot \htmlClass{sym-A_0}{A_{0}}} \\ &= \frac{\htmlClass{sym-M_fx}{1.414\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_rx}{5.082\ \mathrm{kN} \cdot \mathrm{m}}} + \frac{\htmlClass{sym-M_fy}{80.38\ \mathrm{N} \cdot \mathrm{m}}}{\htmlClass{sym-M_ry}{2.789\ \mathrm{kN} \cdot \mathrm{m}}} - \frac{\htmlClass{sym-T_f}{40\ \mathrm{kN}} \cdot \htmlClass{sym-S_x1}{15281\ \mathrm{mm}^{3}}}{\htmlClass{sym-M_rx}{5.082\ \mathrm{kN} \cdot \mathrm{m}} \cdot \htmlClass{sym-A_0}{927\ \mathrm{mm}^{2}}} \\ &= 0.1773 \end{aligned}\]