calcsheetSteelWT in tension
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CSA S16-19 Cl. 10.4, 13.2 and 13.9; AISC 360-22 F9

WT in tension

Verified against CSA S16-19, AISC 360-22 and CISC SST 12.1, 2026-10-01

A WT brace under tension and a moment compressing its stem, AISC 360-22 F9 resistance, CSA S16 Cl. 13.9. Check a WT brace under a factored tension and a moment that puts its stem in compression. The slenderness and the gross section's yield are CSA S16's; the moment resistance is AISC 360-22 F9's yield, lateral-torsional buckling and stem local buckling; and the tension and moment are combined by S16 Cl. 13.9.

Given

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\(T_{f}\) \(\mathrm{kN}\) 0-50,000
\(M_{\mathrm{fx}}\) \(\mathrm{kN} \cdot \mathrm{m}\) 0-10,000
\(L_{x}\) \(\mathrm{mm}\) 100-30,000
\(L_{y}\) \(\mathrm{mm}\) 100-30,000
\(F_{y}\) \(\mathrm{MPa}\) 150-700
\(\mathrm{section}\)
d = 456 mm b = 304 mm
The WT's cross-section to scale.

Title block

Optional. Printed under the title of the PDF.

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

Checks

Check D/C Result
Slenderness x ok\(\htmlClass{sym-lambda_x}{\lambda_{x}} \leq 300 \quad \Rightarrow \quad \htmlClass{sym-lambda_x}{20.98} \leq 300\) 0.07 PASS
Slenderness y ok\(\htmlClass{sym-lambda_y}{\lambda_{y}} \leq 300 \quad \Rightarrow \quad \htmlClass{sym-lambda_y}{47.85} \leq 300\) 0.16 PASS
Tension bending ok\(\htmlClass{sym-I_a}{I_{a}} \leq 1 \quad \Rightarrow \quad \htmlClass{sym-I_a}{0.04451} \leq 1\) 0.04 PASS
Bending less tension ok\(\htmlClass{sym-I_b}{I_{b}} \leq 1 \quad \Rightarrow \quad \htmlClass{sym-I_b}{-0.01988} \leq 1\) -0.02 PASS

Results

Quantity Description Value Unit
\(\htmlClass{sym-T_r}{T_{r}}\) Tensile resistance, gross section 4.505 \(\mathrm{MN}\)
\(\htmlClass{sym-M_rx}{M_{\mathrm{rx}}}\) Moment resistance about x 223.6 \(\mathrm{kN} \cdot \mathrm{m}\)

Derivation

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 \[\begin{aligned} \htmlClass{sym-d}{d} &= 456\ \mathrm{mm} \quad \left(\text{Depth of section or height of vertical leg}\right) \end{aligned}\]
S.4 \[\begin{aligned} \htmlClass{sym-b}{b} &= 304\ \mathrm{mm} \quad \left(\text{Width of flange or horizontal leg}\right) \end{aligned}\]
S.5 \[\begin{aligned} \htmlClass{sym-t}{t} &= 23.9\ \mathrm{mm} \quad \left(\text{Thickness of flange}\right) \end{aligned}\]
S.6 \[\begin{aligned} \htmlClass{sym-w}{w} &= 15.9\ \mathrm{mm} \quad \left(\text{Thickness of web}\right) \end{aligned}\]
S.7 \[\begin{aligned} \htmlClass{sym-A_0}{A_{0}} &= 14300\ \mathrm{mm}^{2} \quad \left(\text{Cross-sectional area}\right) \end{aligned}\]
S.8 \[\begin{aligned} \htmlClass{sym-y_c}{y_{c}} &= 122\ \mathrm{mm} \quad \left(\text{Vertical distance between centroid and outside face of flange orhorizontal leg}\right) \end{aligned}\]
S.9 \[\begin{aligned} \htmlClass{sym-I_x}{I_{x}} &= 2.92 \times 10^{8}\ \mathrm{mm}^{4} \quad \left(\text{Moment of inertia about axis XX}\right) \end{aligned}\]
S.10 \[\begin{aligned} \htmlClass{sym-I_y}{I_{y}} &= 5.61 \times 10^{7}\ \mathrm{mm}^{4} \quad \left(\text{Moment of inertia about axis YY}\right) \end{aligned}\]
S.11 \[\begin{aligned} \htmlClass{sym-J}{J} &= 2.08 \times 10^{6}\ \mathrm{mm}^{4} \quad \left(\text{St-Venant torsional constant}\right) \end{aligned}\]
S.12 \[\begin{aligned} \htmlClass{sym-r_x}{r_{x}} &= 143\ \mathrm{mm} \quad \left(\text{Radius of gyration about axis XX}\right) \end{aligned}\]
S.13 \[\begin{aligned} \htmlClass{sym-r_y}{r_{y}} &= 62.7\ \mathrm{mm} \quad \left(\text{Radius of gyration about axis YY}\right) \end{aligned}\]
S.14 \[\begin{aligned} \htmlClass{sym-S_xc}{S_{\mathrm{xc}}} &= 874000\ \mathrm{mm}^{3} \quad \left(\text{Elastic section modulus about axis XX}\right) \end{aligned}\]
S.15

S16 Cl. 10.4.1, L/r for a member in tension

\[\begin{aligned} \htmlClass{sym-lambda_x}{\lambda_{x}} &= \frac{\htmlClass{sym-L_x}{L_{x}}}{\htmlClass{sym-r_x}{r_{x}}} \\ &= \frac{\htmlClass{sym-L_x}{3000\ \mathrm{mm}}}{\htmlClass{sym-r_x}{143\ \mathrm{mm}}} \\ &= 20.98 \end{aligned}\]
S.16

S16 Cl. 10.4.1, L/r for a member in tension

\[\begin{aligned} \htmlClass{sym-lambda_y}{\lambda_{y}} &= \frac{\htmlClass{sym-L_y}{L_{y}}}{\htmlClass{sym-r_y}{r_{y}}} \\ &= \frac{\htmlClass{sym-L_y}{3000\ \mathrm{mm}}}{\htmlClass{sym-r_y}{62.7\ \mathrm{mm}}} \\ &= 47.85 \end{aligned}\]
S.17

S16 Cl. 13.2, gross section yield

\[\begin{aligned} \htmlClass{sym-T_r}{T_{r}} &= \htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-A_0}{A_{0}} \cdot \htmlClass{sym-F_y}{F_{y}} \\ &= \htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-A_0}{14300\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-F_y}{350\ \mathrm{MPa}} \\ &= 4.505\ \mathrm{MN} \end{aligned}\]
S.18

AISC 360-22 F9-1, F9-3 and F9-4, M_p = M_y for a tee stem in compression

\[\begin{aligned} \htmlClass{sym-M_rx1}{M_{\mathrm{rx}1}} &= \htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-S_xc}{S_{\mathrm{xc}}} \cdot \htmlClass{sym-F_y}{F_{y}} \\ &= \htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-S_xc}{874000\ \mathrm{mm}^{3}} \cdot \htmlClass{sym-F_y}{350\ \mathrm{MPa}} \\ &= 275.3\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]
S.19

AISC 360-22 F9-12, stem in compression

\[\begin{aligned} \htmlClass{sym-B}{B} &= \left(-2.3\right) \cdot \frac{\htmlClass{sym-d}{d}}{\htmlClass{sym-L_y}{L_{y}}} \cdot \sqrt{\frac{\htmlClass{sym-I_y}{I_{y}}}{\htmlClass{sym-J}{J}}} \\ &= \left(-2.3\right) \cdot \frac{\htmlClass{sym-d}{456\ \mathrm{mm}}}{\htmlClass{sym-L_y}{3000\ \mathrm{mm}}} \cdot \sqrt{\frac{\htmlClass{sym-I_y}{5.61 \times 10^{7}\ \mathrm{mm}^{4}}}{\htmlClass{sym-J}{2.08 \times 10^{6}\ \mathrm{mm}^{4}}}} \\ &= -1.816 \end{aligned}\]
S.20

AISC 360-22 F9-10

\[\begin{aligned} \htmlClass{sym-M_cr}{M_{\mathrm{cr}}} &= 1.95 \cdot \frac{\htmlClass{sym-E}{E}}{\htmlClass{sym-L_y}{L_{y}}} \cdot \sqrt{\htmlClass{sym-I_y}{I_{y}} \cdot \htmlClass{sym-J}{J}} \cdot \left(\htmlClass{sym-B}{B} + \sqrt{1 + \htmlClass{sym-B}{B}^{2}}\right) \\ &= 1.95 \cdot \frac{\htmlClass{sym-E}{200\ \mathrm{GPa}}}{\htmlClass{sym-L_y}{3000\ \mathrm{mm}}} \cdot \sqrt{\htmlClass{sym-I_y}{5.61 \times 10^{7}\ \mathrm{mm}^{4}} \cdot \htmlClass{sym-J}{2.08 \times 10^{6}\ \mathrm{mm}^{4}}} \cdot \left(\left(\htmlClass{sym-B}{-1.816}\right) + \sqrt{1 + \left(\htmlClass{sym-B}{-1.816}\right)^{2}}\right) \\ &= 361.1\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]
S.21

AISC 360-22 F9-13, M_n = M_cr <= M_y

\[\begin{aligned} \htmlClass{sym-M_rx2}{M_{\mathrm{rx}2}} &= \htmlClass{sym-phi}{\phi} \cdot \min\left(\htmlClass{sym-M_cr}{M_{\mathrm{cr}}}, \htmlClass{sym-S_xc}{S_{\mathrm{xc}}} \cdot \htmlClass{sym-F_y}{F_{y}}\right) \\ &= \htmlClass{sym-phi}{0.9} \cdot \min\left(\htmlClass{sym-M_cr}{361.1\ \mathrm{kN} \cdot \mathrm{m}}, \htmlClass{sym-S_xc}{874000\ \mathrm{mm}^{3}} \cdot \htmlClass{sym-F_y}{350\ \mathrm{MPa}}\right) \\ &= 275.3\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]
S.22 \[\begin{aligned} \htmlClass{sym-lambda_s}{\lambda_{s}} &= \frac{\htmlClass{sym-d}{d}}{\htmlClass{sym-w}{w}} \\ &= \frac{\htmlClass{sym-d}{456\ \mathrm{mm}}}{\htmlClass{sym-w}{15.9\ \mathrm{mm}}} \\ &= 28.68 \end{aligned}\]
S.23

AISC 360-22 Table B4.1b case 14, stems of tees

\[\begin{aligned} \htmlClass{sym-lambda_p}{\lambda_{p}} &= 0.84 \cdot \sqrt{\frac{\htmlClass{sym-E}{E}}{\htmlClass{sym-F_y}{F_{y}}}} \\ &= 0.84 \cdot \sqrt{\frac{\htmlClass{sym-E}{200\ \mathrm{GPa}}}{\htmlClass{sym-F_y}{350\ \mathrm{MPa}}}} \\ &= 20.08 \end{aligned}\]
S.24

AISC 360-22 Table B4.1b case 14, stems of tees

\[\begin{aligned} \htmlClass{sym-lambda_r}{\lambda_{r}} &= 1.52 \cdot \sqrt{\frac{\htmlClass{sym-E}{E}}{\htmlClass{sym-F_y}{F_{y}}}} \\ &= 1.52 \cdot \sqrt{\frac{\htmlClass{sym-E}{200\ \mathrm{GPa}}}{\htmlClass{sym-F_y}{350\ \mathrm{MPa}}}} \\ &= 36.33 \end{aligned}\]

Branch: \(\lambda_{s} \leq \lambda_{p}\) did not hold

Branch: \(\lambda_{s} \leq \lambda_{r}\) held

S.25

AISC 360-22 F9-18

\[\begin{aligned} \htmlClass{sym-F_cr}{F_{\mathrm{cr}}} &= \left(1.43 - 0.515 \cdot \htmlClass{sym-lambda_s}{\lambda_{s}} \cdot \sqrt{\frac{\htmlClass{sym-F_y}{F_{y}}}{\htmlClass{sym-E}{E}}}\right) \cdot \htmlClass{sym-F_y}{F_{y}} \\ &= \left(1.43 - 0.515 \cdot \htmlClass{sym-lambda_s}{28.68} \cdot \sqrt{\frac{\htmlClass{sym-F_y}{350\ \mathrm{MPa}}}{\htmlClass{sym-E}{200\ \mathrm{GPa}}}}\right) \cdot \htmlClass{sym-F_y}{350\ \mathrm{MPa}} \\ &= 284.2\ \mathrm{MPa} \end{aligned}\]
S.26

AISC 360-22 F9-16

\[\begin{aligned} \htmlClass{sym-M_rx3}{M_{\mathrm{rx}3}} &= \htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-S_xc}{S_{\mathrm{xc}}} \cdot \htmlClass{sym-F_cr}{F_{\mathrm{cr}}} \\ &= \htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-S_xc}{874000\ \mathrm{mm}^{3}} \cdot \htmlClass{sym-F_cr}{284.2\ \mathrm{MPa}} \\ &= 223.6\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]
S.27

AISC 360-22 F9, the lowest of the limit states

\[\begin{aligned} \htmlClass{sym-M_rx}{M_{\mathrm{rx}}} &= \min\left(\htmlClass{sym-M_rx1}{M_{\mathrm{rx}1}}, \htmlClass{sym-M_rx2}{M_{\mathrm{rx}2}}, \htmlClass{sym-M_rx3}{M_{\mathrm{rx}3}}\right) \\ &= \min\left(\htmlClass{sym-M_rx1}{275.3\ \mathrm{kN} \cdot \mathrm{m}}, \htmlClass{sym-M_rx2}{275.3\ \mathrm{kN} \cdot \mathrm{m}}, \htmlClass{sym-M_rx3}{223.6\ \mathrm{kN} \cdot \mathrm{m}}\right) \\ &= 223.6\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]
S.28

S16 Cl. 13.9.1

\[\begin{aligned} \htmlClass{sym-I_a}{I_{a}} &= \frac{\htmlClass{sym-T_f}{T_{f}}}{\htmlClass{sym-T_r}{T_{r}}} + \frac{\htmlClass{sym-M_fx}{M_{\mathrm{fx}}}}{\htmlClass{sym-M_rx}{M_{\mathrm{rx}}}} \\ &= \frac{\htmlClass{sym-T_f}{130\ \mathrm{kN}}}{\htmlClass{sym-T_r}{4.505\ \mathrm{MN}}} + \frac{\htmlClass{sym-M_fx}{3.5\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_rx}{223.6\ \mathrm{kN} \cdot \mathrm{m}}} \\ &= 0.04451 \end{aligned}\]
S.29

S16 Cl. 13.9.3 b), S at the stem tip

\[\begin{aligned} \htmlClass{sym-I_b}{I_{b}} &= \frac{\htmlClass{sym-M_fx}{M_{\mathrm{fx}}}}{\htmlClass{sym-M_rx}{M_{\mathrm{rx}}}} - \frac{\htmlClass{sym-T_f}{T_{f}} \cdot \htmlClass{sym-S_xc}{S_{\mathrm{xc}}}}{\htmlClass{sym-M_rx}{M_{\mathrm{rx}}} \cdot \htmlClass{sym-A_0}{A_{0}}} \\ &= \frac{\htmlClass{sym-M_fx}{3.5\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_rx}{223.6\ \mathrm{kN} \cdot \mathrm{m}}} - \frac{\htmlClass{sym-T_f}{130\ \mathrm{kN}} \cdot \htmlClass{sym-S_xc}{874000\ \mathrm{mm}^{3}}}{\htmlClass{sym-M_rx}{223.6\ \mathrm{kN} \cdot \mathrm{m}} \cdot \htmlClass{sym-A_0}{14300\ \mathrm{mm}^{2}}} \\ &= -0.01988 \end{aligned}\]

Questions

Why must M_fx put the stem in compression?

The moment is taken that way only: the usual WT brace is connected by its flange, and the eccentricity then bends the stem tip into compression.

Which section modulus does the second interaction take?

The stem tip's, where the moment's compression is and the tension relieves it.

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