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AISC Steel Construction Manual, 14th ed., uniform force method

Connection transfer force

Verified against AISC Steel Construction Manual, 14th ed., 2026-09-30

Axial force and shear on a braced beam-to-column connection, by the uniform force method. Resolve the forces meeting at a braced beam-to-column joint into the axial force and shear on the connection to the column. Each vertical brace is split by the uniform force method into the share its gusset delivers along the beam and across it, and the horizontal braces add their component along the beam. The beam's own axial force and shear complete the sums.

Given

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\(e_{c}\) \(\mathrm{mm}\) 2-600
\(d_{b}\) \(\mathrm{mm}\) 100-1,200
\(L_{c1}\) \(\mathrm{mm}\) 50-2,000
\(L_{c2}\) \(\mathrm{mm}\) 50-2,000
\(P_{f1}\) \(\mathrm{kN}\) -5,000-5,000
\(\phi_{1}\) 5-85
\(P_{f2}\) \(\mathrm{kN}\) -5,000-5,000
\(\phi_{2}\) 5-85
\(P_{f3}\) \(\mathrm{kN}\) -5,000-5,000
\(\phi_{3}\) 0-90
\(P_{f4}\) \(\mathrm{kN}\) -5,000-5,000
\(\phi_{4}\) 0-90
\(H_{f}\) \(\mathrm{kN}\) -5,000-5,000
\(V_{f}\) \(\mathrm{kN}\) -5,000-5,000
ec = 7 mm db = 310 mm Pf1 = 100 kN Lc1 = 245 mm β1 = 122 mm α1 = 270 mm Pf2 = -123 kN Lc2 = 165 mm β2 = 82 mm α2 = 257 mm w.p. Af = 12 kN Sf = 203 kN
Elevation of the joint, to scale: V-1 below the beam, V-2 above, the work point (w.p.), the distances beta up the column face and alpha along the beam to the control points, and the transfer forces at the column face. The horizontal braces are in plan and not drawn.

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This printed copy omits the derivation. The full report, with every step, is the PDF at https://calc.struct.work/calc/connection-transfer-force.pdf.

Results

Quantity Description Value Unit
\(\htmlClass{sym-A_f}{A_{f}}\) Connection axial force on the column 12.42 \(\mathrm{kN}\)
\(\htmlClass{sym-S_f}{S_{f}}\) Connection shear on the column 203.2 \(\mathrm{kN}\)

Derivation

S.1 \[\begin{aligned} \htmlClass{sym-e_b}{e_{b}} &= \frac{\htmlClass{sym-d_b}{d_{b}}}{2} \\ &= \frac{\htmlClass{sym-d_b}{310\ \mathrm{mm}}}{2} \\ &= 155\ \mathrm{mm} \end{aligned}\]
S.2 \[\begin{aligned} \htmlClass{sym-beta_1}{\beta_{1}} &= \frac{\htmlClass{sym-L_c1}{L_{c1}}}{2} \\ &= \frac{\htmlClass{sym-L_c1}{245\ \mathrm{mm}}}{2} \\ &= 122.5\ \mathrm{mm} \end{aligned}\]
S.3 \[\begin{aligned} \htmlClass{sym-alpha_1}{\alpha_{1}} &= \htmlClass{sym-beta_1}{\beta_{1}} \cdot \tan\left(\operatorname{radians}\left(\htmlClass{sym-phi_1}{\phi_{1}}\right)\right) + \htmlClass{sym-e_b}{e_{b}} \cdot \tan\left(\operatorname{radians}\left(\htmlClass{sym-phi_1}{\phi_{1}}\right)\right) - \htmlClass{sym-e_c}{e_{c}} \\ &= \htmlClass{sym-beta_1}{122.5\ \mathrm{mm}} \cdot \tan\left(\operatorname{radians}\left(\htmlClass{sym-phi_1}{45}\right)\right) + \htmlClass{sym-e_b}{155\ \mathrm{mm}} \cdot \tan\left(\operatorname{radians}\left(\htmlClass{sym-phi_1}{45}\right)\right) - \htmlClass{sym-e_c}{7\ \mathrm{mm}} \\ &= 270.5\ \mathrm{mm} \end{aligned}\]
S.4 \[\begin{aligned} \htmlClass{sym-r_1}{r_{1}} &= \sqrt{\left(\htmlClass{sym-alpha_1}{\alpha_{1}} + \htmlClass{sym-e_c}{e_{c}}\right)^{2} + \left(\htmlClass{sym-beta_1}{\beta_{1}} + \htmlClass{sym-e_b}{e_{b}}\right)^{2}} \\ &= \sqrt{\left(\htmlClass{sym-alpha_1}{270.5\ \mathrm{mm}} + \htmlClass{sym-e_c}{7\ \mathrm{mm}}\right)^{2} + \left(\htmlClass{sym-beta_1}{122.5\ \mathrm{mm}} + \htmlClass{sym-e_b}{155\ \mathrm{mm}}\right)^{2}} \\ &= 392.4\ \mathrm{mm} \end{aligned}\]
S.5 \[\begin{aligned} \htmlClass{sym-F_v1}{F_{v1}} &= \frac{\htmlClass{sym-e_b}{e_{b}}}{\htmlClass{sym-r_1}{r_{1}}} \\ &= \frac{\htmlClass{sym-e_b}{155\ \mathrm{mm}}}{\htmlClass{sym-r_1}{392.4\ \mathrm{mm}}} \\ &= 0.395 \end{aligned}\]
S.6 \[\begin{aligned} \htmlClass{sym-F_h1}{F_{h1}} &= \frac{\htmlClass{sym-alpha_1}{\alpha_{1}}}{\htmlClass{sym-r_1}{r_{1}}} \\ &= \frac{\htmlClass{sym-alpha_1}{270.5\ \mathrm{mm}}}{\htmlClass{sym-r_1}{392.4\ \mathrm{mm}}} \\ &= 0.6893 \end{aligned}\]
S.7 \[\begin{aligned} \htmlClass{sym-V_c1}{V_{c1}} &= \frac{\htmlClass{sym-P_f1}{P_{f1}} \cdot \htmlClass{sym-beta_1}{\beta_{1}}}{\htmlClass{sym-r_1}{r_{1}}} \\ &= \frac{\htmlClass{sym-P_f1}{100\ \mathrm{kN}} \cdot \htmlClass{sym-beta_1}{122.5\ \mathrm{mm}}}{\htmlClass{sym-r_1}{392.4\ \mathrm{mm}}} \\ &= 31.21\ \mathrm{kN} \end{aligned}\]
S.8 \[\begin{aligned} \htmlClass{sym-H_c1}{H_{c1}} &= \frac{\htmlClass{sym-P_f1}{P_{f1}} \cdot \htmlClass{sym-e_c}{e_{c}}}{\htmlClass{sym-r_1}{r_{1}}} \\ &= \frac{\htmlClass{sym-P_f1}{100\ \mathrm{kN}} \cdot \htmlClass{sym-e_c}{7\ \mathrm{mm}}}{\htmlClass{sym-r_1}{392.4\ \mathrm{mm}}} \\ &= 1.784\ \mathrm{kN} \end{aligned}\]
S.9 \[\begin{aligned} \htmlClass{sym-beta_2}{\beta_{2}} &= \frac{\htmlClass{sym-L_c2}{L_{c2}}}{2} \\ &= \frac{\htmlClass{sym-L_c2}{165\ \mathrm{mm}}}{2} \\ &= 82.5\ \mathrm{mm} \end{aligned}\]
S.10 \[\begin{aligned} \htmlClass{sym-alpha_2}{\alpha_{2}} &= \htmlClass{sym-beta_2}{\beta_{2}} \cdot \tan\left(\operatorname{radians}\left(\htmlClass{sym-phi_2}{\phi_{2}}\right)\right) + \htmlClass{sym-e_b}{e_{b}} \cdot \tan\left(\operatorname{radians}\left(\htmlClass{sym-phi_2}{\phi_{2}}\right)\right) - \htmlClass{sym-e_c}{e_{c}} \\ &= \htmlClass{sym-beta_2}{82.5\ \mathrm{mm}} \cdot \tan\left(\operatorname{radians}\left(\htmlClass{sym-phi_2}{48}\right)\right) + \htmlClass{sym-e_b}{155\ \mathrm{mm}} \cdot \tan\left(\operatorname{radians}\left(\htmlClass{sym-phi_2}{48}\right)\right) - \htmlClass{sym-e_c}{7\ \mathrm{mm}} \\ &= 256.8\ \mathrm{mm} \end{aligned}\]
S.11 \[\begin{aligned} \htmlClass{sym-r_2}{r_{2}} &= \sqrt{\left(\htmlClass{sym-alpha_2}{\alpha_{2}} + \htmlClass{sym-e_c}{e_{c}}\right)^{2} + \left(\htmlClass{sym-beta_2}{\beta_{2}} + \htmlClass{sym-e_b}{e_{b}}\right)^{2}} \\ &= \sqrt{\left(\htmlClass{sym-alpha_2}{256.8\ \mathrm{mm}} + \htmlClass{sym-e_c}{7\ \mathrm{mm}}\right)^{2} + \left(\htmlClass{sym-beta_2}{82.5\ \mathrm{mm}} + \htmlClass{sym-e_b}{155\ \mathrm{mm}}\right)^{2}} \\ &= 354.9\ \mathrm{mm} \end{aligned}\]
S.12 \[\begin{aligned} \htmlClass{sym-F_v2}{F_{v2}} &= \frac{\htmlClass{sym-e_b}{e_{b}}}{\htmlClass{sym-r_2}{r_{2}}} \\ &= \frac{\htmlClass{sym-e_b}{155\ \mathrm{mm}}}{\htmlClass{sym-r_2}{354.9\ \mathrm{mm}}} \\ &= 0.4367 \end{aligned}\]
S.13 \[\begin{aligned} \htmlClass{sym-F_h2}{F_{h2}} &= \frac{\htmlClass{sym-alpha_2}{\alpha_{2}}}{\htmlClass{sym-r_2}{r_{2}}} \\ &= \frac{\htmlClass{sym-alpha_2}{256.8\ \mathrm{mm}}}{\htmlClass{sym-r_2}{354.9\ \mathrm{mm}}} \\ &= 0.7234 \end{aligned}\]
S.14 \[\begin{aligned} \htmlClass{sym-V_c2}{V_{c2}} &= \frac{\htmlClass{sym-P_f2}{P_{f2}} \cdot \htmlClass{sym-beta_2}{\beta_{2}}}{\htmlClass{sym-r_2}{r_{2}}} \\ &= \frac{\left(\htmlClass{sym-P_f2}{-123\ \mathrm{kN}}\right) \cdot \htmlClass{sym-beta_2}{82.5\ \mathrm{mm}}}{\htmlClass{sym-r_2}{354.9\ \mathrm{mm}}} \\ &= -28.59\ \mathrm{kN} \end{aligned}\]
S.15 \[\begin{aligned} \htmlClass{sym-H_c2}{H_{c2}} &= \frac{\htmlClass{sym-P_f2}{P_{f2}} \cdot \htmlClass{sym-e_c}{e_{c}}}{\htmlClass{sym-r_2}{r_{2}}} \\ &= \frac{\left(\htmlClass{sym-P_f2}{-123\ \mathrm{kN}}\right) \cdot \htmlClass{sym-e_c}{7\ \mathrm{mm}}}{\htmlClass{sym-r_2}{354.9\ \mathrm{mm}}} \\ &= -2.426\ \mathrm{kN} \end{aligned}\]
S.16

AISC 14th ed. Part 13, Uniform Force Method, Fig. 13-2

\[\begin{aligned} \htmlClass{sym-A_f}{A_{f}} &= \htmlClass{sym-H_f}{H_{f}} + \htmlClass{sym-P_f1}{P_{f1}} \cdot \htmlClass{sym-F_h1}{F_{h1}} + \htmlClass{sym-P_f2}{P_{f2}} \cdot \htmlClass{sym-F_h2}{F_{h2}} + \htmlClass{sym-P_f3}{P_{f3}} \cdot \cos\left(\operatorname{radians}\left(\htmlClass{sym-phi_3}{\phi_{3}}\right)\right) + \htmlClass{sym-P_f4}{P_{f4}} \cdot \cos\left(\operatorname{radians}\left(\htmlClass{sym-phi_4}{\phi_{4}}\right)\right) \\ &= \left(\htmlClass{sym-H_f}{-50\ \mathrm{kN}}\right) + \htmlClass{sym-P_f1}{100\ \mathrm{kN}} \cdot \htmlClass{sym-F_h1}{0.6893} + \left(\htmlClass{sym-P_f2}{-123\ \mathrm{kN}}\right) \cdot \htmlClass{sym-F_h2}{0.7234} + \htmlClass{sym-P_f3}{50\ \mathrm{kN}} \cdot \cos\left(\operatorname{radians}\left(\htmlClass{sym-phi_3}{46}\right)\right) + \htmlClass{sym-P_f4}{70\ \mathrm{kN}} \cdot \cos\left(\operatorname{radians}\left(\htmlClass{sym-phi_4}{47}\right)\right) \\ &= 12.42\ \mathrm{kN} \end{aligned}\]
S.17

AISC 14th ed. Part 13, Uniform Force Method, Fig. 13-2

\[\begin{aligned} \htmlClass{sym-S_f}{S_{f}} &= \htmlClass{sym-V_f}{V_{f}} + \htmlClass{sym-P_f1}{P_{f1}} \cdot \htmlClass{sym-F_v1}{F_{v1}} - \htmlClass{sym-P_f2}{P_{f2}} \cdot \htmlClass{sym-F_v2}{F_{v2}} \\ &= \htmlClass{sym-V_f}{110\ \mathrm{kN}} + \htmlClass{sym-P_f1}{100\ \mathrm{kN}} \cdot \htmlClass{sym-F_v1}{0.395} - \left(\htmlClass{sym-P_f2}{-123\ \mathrm{kN}}\right) \cdot \htmlClass{sym-F_v2}{0.4367} \\ &= 203.2\ \mathrm{kN} \end{aligned}\]

Questions

What is e_c?

The distance from the column's centreline to the face the gusset bolts to: half the column depth when the beam frames into the flange, half the web thickness when it frames into the web.

Which brace carries P_f1, and what are phi_1 and phi_3 measured from?

V-1 is the vertical brace below the beam and V-2 the one above it, with their angles measured from the column. H-1 and H-2 are horizontal braces in plan, measured from the beam.

What sign do A_f, S_f and the brace forces take?

Tension is positive in every brace and in the beam's axial force; the beam's shear is positive downward. A positive A_f pulls the connection off the column.

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