Concentrated load on a beam web

S.1 \[\begin{aligned} \htmlClass{sym-E}{E} &= 200\ \mathrm{GPa} \end{aligned}\]

S.2 \[\begin{aligned} \htmlClass{sym-d}{d} &= 247\ \mathrm{mm} \quad \left(\text{Depth of section or height of vertical leg}\right) \end{aligned}\]

S.3 \[\begin{aligned} \htmlClass{sym-b}{b} &= 202\ \mathrm{mm} \quad \left(\text{Width of flange or horizontal leg}\right) \end{aligned}\]

S.4 \[\begin{aligned} \htmlClass{sym-t}{t} &= 11\ \mathrm{mm} \quad \left(\text{Thickness of flange}\right) \end{aligned}\]

S.5 \[\begin{aligned} \htmlClass{sym-w}{w} &= 7.4\ \mathrm{mm} \quad \left(\text{Thickness of web}\right) \end{aligned}\]

S.6 \[\begin{aligned} \htmlClass{sym-k}{k} &= 32\ \mathrm{mm} \quad \left(\text{Distance between outer face of flange and web-toe of fillet}\right) \end{aligned}\]

S.7 \[\begin{aligned} \htmlClass{sym-S_x}{S_{x}} &= 572000\ \mathrm{mm}^{3} \quad \left(\text{Elastic section modulus about axis XX}\right) \end{aligned}\]

S.8 \[\begin{aligned} \htmlClass{sym-phi_be}{\phi_{\mathrm{be}}} &= 0.75 \quad \left(\text{End reaction | S16 Cl. 13.1 f), 14.3.2}\right) \end{aligned}\]

S.9

S16 Cl. 14.3.2

\[\begin{aligned} \htmlClass{sym-B_r1}{B_{r1}} &= \htmlClass{sym-phi_be}{\phi_{\mathrm{be}}} \cdot \htmlClass{sym-w}{w} \cdot \left(\htmlClass{sym-N}{N} + 4 \cdot \htmlClass{sym-t}{t}\right) \cdot \htmlClass{sym-F_y}{F_{y}} \\ &= \htmlClass{sym-phi_be}{0.75} \cdot \htmlClass{sym-w}{7.4\ \mathrm{mm}} \cdot \left(\htmlClass{sym-N}{250\ \mathrm{mm}} + 4 \cdot \htmlClass{sym-t}{11\ \mathrm{mm}}\right) \cdot \htmlClass{sym-F_y}{350\ \mathrm{MPa}} \\ &= 571.1\ \mathrm{kN} \end{aligned}\]

S.10

S16 Cl. 14.3.2

\[\begin{aligned} \htmlClass{sym-B_r2}{B_{r2}} &= 0.6 \cdot \htmlClass{sym-phi_be}{\phi_{\mathrm{be}}} \cdot \htmlClass{sym-w}{w}^{2} \cdot \sqrt{\htmlClass{sym-F_y}{F_{y}} \cdot \htmlClass{sym-E}{E}} \\ &= 0.6 \cdot \htmlClass{sym-phi_be}{0.75} \cdot \left(\htmlClass{sym-w}{7.4\ \mathrm{mm}}\right)^{2} \cdot \sqrt{\htmlClass{sym-F_y}{350\ \mathrm{MPa}} \cdot \htmlClass{sym-E}{200\ \mathrm{GPa}}} \\ &= 206.2\ \mathrm{kN} \end{aligned}\]

S.11 \[\begin{aligned} \htmlClass{sym-h_c}{h_{c}} &= \htmlClass{sym-d}{d} - 2 \cdot \htmlClass{sym-k}{k} \\ &= \htmlClass{sym-d}{247\ \mathrm{mm}} - 2 \cdot \htmlClass{sym-k}{32\ \mathrm{mm}} \\ &= 183\ \mathrm{mm} \end{aligned}\]

S.12 \[\begin{aligned} \htmlClass{sym-alpha}{\alpha} &= \frac{\frac{\htmlClass{sym-h_c}{h_{c}}}{\htmlClass{sym-w}{w}}}{\frac{\htmlClass{sym-L}{L}}{\htmlClass{sym-b}{b}}} \\ &= \frac{\frac{\htmlClass{sym-h_c}{183\ \mathrm{mm}}}{\htmlClass{sym-w}{7.4\ \mathrm{mm}}}}{\frac{\htmlClass{sym-L}{6000\ \mathrm{mm}}}{\htmlClass{sym-b}{202\ \mathrm{mm}}}} \\ &= 0.8326 \end{aligned}\]

S.13 \[\begin{aligned} \htmlClass{sym-M_y}{M_{y}} &= \htmlClass{sym-S_x}{S_{x}} \cdot \htmlClass{sym-F_y}{F_{y}} \\ &= \htmlClass{sym-S_x}{572000\ \mathrm{mm}^{3}} \cdot \htmlClass{sym-F_y}{350\ \mathrm{MPa}} \\ &= 200.2\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

S.14 \[\begin{aligned} \htmlClass{sym-C_rs}{C_{\mathrm{rs}}} &= 6.62\ \mathrm{TPa} \quad \left(\text{M\_f < M\_y | AISC J10.4}\right) \end{aligned}\]

S.15

AISC J10.4 (a)

\[\begin{aligned} \htmlClass{sym-B_r3}{B_{r3}} &= \frac{0.85 \cdot \htmlClass{sym-C_rs}{C_{\mathrm{rs}}} \cdot \htmlClass{sym-w}{w}^{3} \cdot \htmlClass{sym-t}{t}}{\htmlClass{sym-h_c}{h_{c}}^{2}} \cdot \left(1 + 0.4 \cdot \htmlClass{sym-alpha}{\alpha}^{3}\right) \\ &= \frac{0.85 \cdot \htmlClass{sym-C_rs}{6.62\ \mathrm{TPa}} \cdot \left(\htmlClass{sym-w}{7.4\ \mathrm{mm}}\right)^{3} \cdot \htmlClass{sym-t}{11\ \mathrm{mm}}}{\left(\htmlClass{sym-h_c}{183\ \mathrm{mm}}\right)^{2}} \cdot \left(1 + 0.4 \cdot \htmlClass{sym-alpha}{0.8326}^{3}\right) \\ &= 921.9\ \mathrm{kN} \end{aligned}\]

S.16 \[\begin{aligned} \htmlClass{sym-B_r}{B_{r}} &= \min\left(\htmlClass{sym-B_r1}{B_{r1}}, \htmlClass{sym-B_r2}{B_{r2}}, \htmlClass{sym-B_r3}{B_{r3}}\right) \\ &= \min\left(\htmlClass{sym-B_r1}{571.1\ \mathrm{kN}}, \htmlClass{sym-B_r2}{206.2\ \mathrm{kN}}, \htmlClass{sym-B_r3}{921.9\ \mathrm{kN}}\right) \\ &= 206.2\ \mathrm{kN} \end{aligned}\]

S.17

S16 Cl. 21.3 (b), halved at the end by AISC J10.1

\[\begin{aligned} \htmlClass{sym-B_r4}{B_{r4}} &= \frac{0.9 \cdot 7 \cdot \htmlClass{sym-t}{t}^{2} \cdot \htmlClass{sym-F_y}{F_{y}}}{2} \\ &= \frac{0.9 \cdot 7 \cdot \left(\htmlClass{sym-t}{11\ \mathrm{mm}}\right)^{2} \cdot \htmlClass{sym-F_y}{350\ \mathrm{MPa}}}{2} \\ &= 133.4\ \mathrm{kN} \end{aligned}\]

S.18

AISC J10.5, halved at the end

\[\begin{aligned} \htmlClass{sym-B_r5}{B_{r5}} &= \frac{0.9 \cdot 12 \cdot \htmlClass{sym-w}{w}^{3} \cdot \sqrt{\htmlClass{sym-E}{E} \cdot \htmlClass{sym-F_y}{F_{y}}}}{\htmlClass{sym-h_c}{h_{c}}} \\ &= \frac{0.9 \cdot 12 \cdot \left(\htmlClass{sym-w}{7.4\ \mathrm{mm}}\right)^{3} \cdot \sqrt{\htmlClass{sym-E}{200\ \mathrm{GPa}} \cdot \htmlClass{sym-F_y}{350\ \mathrm{MPa}}}}{\htmlClass{sym-h_c}{183\ \mathrm{mm}}} \\ &= 200.1\ \mathrm{kN} \end{aligned}\]

S.19 \[\begin{aligned} \htmlClass{sym-h}{h} &= \htmlClass{sym-d}{d} - 2 \cdot \htmlClass{sym-t}{t} \\ &= \htmlClass{sym-d}{247\ \mathrm{mm}} - 2 \cdot \htmlClass{sym-t}{11\ \mathrm{mm}} \\ &= 225\ \mathrm{mm} \end{aligned}\]

S.20 \[\begin{aligned} \htmlClass{sym-F_ye}{F_{\mathrm{ye}}} &= \min\left(\htmlClass{sym-F_y}{F_{y}}, \htmlClass{sym-F_y1}{F_{y1}}\right) \\ &= \min\left(\htmlClass{sym-F_y}{350\ \mathrm{MPa}}, \htmlClass{sym-F_y1}{300\ \mathrm{MPa}}\right) \\ &= 300\ \mathrm{MPa} \end{aligned}\]

S.21

S16 Cl. 14.4.2, 12 w of web

\[\begin{aligned} \htmlClass{sym-A_0}{A_{0}} &= 12 \cdot \htmlClass{sym-w}{w}^{2} + 2 \cdot \htmlClass{sym-t_1}{t_{1}} \cdot \htmlClass{sym-b_1}{b_{1}} \\ &= 12 \cdot \left(\htmlClass{sym-w}{7.4\ \mathrm{mm}}\right)^{2} + 2 \cdot \htmlClass{sym-t_1}{10\ \mathrm{mm}} \cdot \htmlClass{sym-b_1}{80\ \mathrm{mm}} \\ &= 2257\ \mathrm{mm}^{2} \end{aligned}\]

S.22 \[\begin{aligned} \htmlClass{sym-I_s}{I_{s}} &= \frac{\htmlClass{sym-t_1}{t_{1}} \cdot \left(2 \cdot \htmlClass{sym-b_1}{b_{1}} + \htmlClass{sym-w}{w}\right)^{3}}{12} \\ &= \frac{\htmlClass{sym-t_1}{10\ \mathrm{mm}} \cdot \left(2 \cdot \htmlClass{sym-b_1}{80\ \mathrm{mm}} + \htmlClass{sym-w}{7.4\ \mathrm{mm}}\right)^{3}}{12} \\ &= 3.909 \times 10^{6}\ \mathrm{mm}^{4} \end{aligned}\]

S.23 \[\begin{aligned} \htmlClass{sym-r_s}{r_{s}} &= \sqrt{\frac{\htmlClass{sym-I_s}{I_{s}}}{\htmlClass{sym-A_0}{A_{0}}}} \\ &= \sqrt{\frac{\htmlClass{sym-I_s}{3.909 \times 10^{6}\ \mathrm{mm}^{4}}}{\htmlClass{sym-A_0}{2257\ \mathrm{mm}^{2}}}} \\ &= 41.62\ \mathrm{mm} \end{aligned}\]

S.24 \[\begin{aligned} \htmlClass{sym-lambda_s}{\lambda_{s}} &= \frac{0.75 \cdot \htmlClass{sym-h}{h}}{\htmlClass{sym-r_s}{r_{s}}} \cdot \sqrt{\frac{\htmlClass{sym-F_ye}{F_{\mathrm{ye}}}}{1.974\ \mathrm{TPa}}} \\ &= \frac{0.75 \cdot \htmlClass{sym-h}{225\ \mathrm{mm}}}{\htmlClass{sym-r_s}{41.62\ \mathrm{mm}}} \cdot \sqrt{\frac{\htmlClass{sym-F_ye}{300\ \mathrm{MPa}}}{1.974\ \mathrm{TPa}}} \\ &= 0.04999 \end{aligned}\]

S.25

S16 Cl. 13.3.1, n = 1.34

\[\begin{aligned} \htmlClass{sym-C_r1}{C_{r1}} &= 0.9 \cdot \htmlClass{sym-A_0}{A_{0}} \cdot \htmlClass{sym-F_ye}{F_{\mathrm{ye}}} \cdot \left(1 + \htmlClass{sym-lambda_s}{\lambda_{s}}^{2 \cdot 1.34}\right)^{\frac{-1}{1.34}} \\ &= 0.9 \cdot \htmlClass{sym-A_0}{2257\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-F_ye}{300\ \mathrm{MPa}} \cdot \left(1 + \htmlClass{sym-lambda_s}{0.04999}^{2 \cdot 1.34}\right)^{\frac{-1}{1.34}} \\ &= 609.3\ \mathrm{kN} \end{aligned}\]

S.26 \[\begin{aligned} \htmlClass{sym-phi_w}{\phi_{w}} &= 0.67 \quad \left(\text{Weld resistance factor | S16 Cl. 13.1 h)}\right) \end{aligned}\]

S.27

S16 Cl. 13.13.2.2, weld metal

\[\begin{aligned} \htmlClass{sym-V_r1}{V_{r1}} &= 4 \cdot 0.67 \cdot \htmlClass{sym-phi_w}{\phi_{w}} \cdot 0.707 \cdot \htmlClass{sym-D}{D} \cdot \htmlClass{sym-h_c}{h_{c}} \cdot \htmlClass{sym-X_u}{X_{u}} \\ &= 4 \cdot 0.67 \cdot \htmlClass{sym-phi_w}{0.67} \cdot 0.707 \cdot \htmlClass{sym-D}{6\ \mathrm{mm}} \cdot \htmlClass{sym-h_c}{183\ \mathrm{mm}} \cdot \htmlClass{sym-X_u}{490\ \mathrm{MPa}} \\ &= 683\ \mathrm{kN} \end{aligned}\]

S.28

S16 Cl. 13.10 (a), clear of the fillet

\[\begin{aligned} \htmlClass{sym-V_r3}{V_{r3}} &= 0.9 \cdot 1.5 \cdot \htmlClass{sym-F_ye}{F_{\mathrm{ye}}} \cdot 2 \cdot \left(\htmlClass{sym-b_1}{b_{1}} - \left(\htmlClass{sym-k}{k} - \htmlClass{sym-t}{t}\right)\right) \cdot \htmlClass{sym-t_1}{t_{1}} \\ &= 0.9 \cdot 1.5 \cdot \htmlClass{sym-F_ye}{300\ \mathrm{MPa}} \cdot 2 \cdot \left(\htmlClass{sym-b_1}{80\ \mathrm{mm}} - \left(\htmlClass{sym-k}{32\ \mathrm{mm}} - \htmlClass{sym-t}{11\ \mathrm{mm}}\right)\right) \cdot \htmlClass{sym-t_1}{10\ \mathrm{mm}} \\ &= 477.9\ \mathrm{kN} \end{aligned}\]

S.29 \[\begin{aligned} \htmlClass{sym-B_rs}{B_{\mathrm{rs}}} &= \min\left(\htmlClass{sym-V_r1}{V_{r1}}, \htmlClass{sym-V_r3}{V_{r3}}\right) \\ &= \min\left(\htmlClass{sym-V_r1}{683\ \mathrm{kN}}, \htmlClass{sym-V_r3}{477.9\ \mathrm{kN}}\right) \\ &= 477.9\ \mathrm{kN} \end{aligned}\]

S.30 \[\begin{aligned} \htmlClass{sym-C_r2}{C_{r2}} &= \htmlClass{sym-B_rs}{B_{\mathrm{rs}}} + \htmlClass{sym-B_r}{B_{r}} \\ &= \htmlClass{sym-B_rs}{477.9\ \mathrm{kN}} + \htmlClass{sym-B_r}{206.2\ \mathrm{kN}} \\ &= 684.1\ \mathrm{kN} \end{aligned}\]

S.31 \[\begin{aligned} \htmlClass{sym-C_r}{C_{r}} &= \min\left(\htmlClass{sym-C_r1}{C_{r1}}, \htmlClass{sym-C_r2}{C_{r2}}\right) \\ &= \min\left(\htmlClass{sym-C_r1}{609.3\ \mathrm{kN}}, \htmlClass{sym-C_r2}{684.1\ \mathrm{kN}}\right) \\ &= 609.3\ \mathrm{kN} \end{aligned}\]