Axial compression of a W section or HSS

Class 4 effective area and the least of flexural and torsional buckling, W shape or HSS. Check a W, WWF, S, HP or M shape or a rectangular or square HSS in concentric compression. Each element is classified and a class 4 element's excess width comes off the area. The elastic buckling stress is the least of flexural buckling about either axis and torsional buckling, and sets the compressive resistance.

Derivation, step by step

SI Imperial

First report free, then a free sign-in

PASS Governing: Compressive resistance, D/C 0.78, S16 Cl. 13.1, C_r >= C_f3 of 3 checks below pass

Given

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Type any value, everything below recomputes. Hover or tap a symbol to trace it.

Loads

C_{f}

Factored axial compression

\mathrm{kN}

0-50,000

Section

F_{y}

Yield strength

\mathrm{MPa}

200-700

Buckling about x

L_{x}

Unbraced length about x, and for twisting

\mathrm{mm}

100-20,000

k_{x}

Effective length factor, about x 0.5-2.5

Buckling about y

L_{y}

Unbraced length, about y

\mathrm{mm}

100-20,000

k_{y}

Effective length factor, about y 0.5-2.5

The cross-section to scale.

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

Checks

! marks a check that passes with D/C at or above 0.95: close to its limit. The band is a site convention, not a code limit.

Check	D/C	Result
Slenderness, x axisS16 Cl. 10.4.2.1 $\frac{\htmlClass{sym-k_x}{k_{x}} \cdot \htmlClass{sym-L_x}{L_{x}}}{\htmlClass{sym-r_x}{r_{x}}} \leq 200 \quad \Rightarrow \quad \frac{\htmlClass{sym-k_x}{1} \cdot \htmlClass{sym-L_x}{3000\ \mathrm{mm}}}{\htmlClass{sym-r_x}{108\ \mathrm{mm}}} \leq 200$	0.14	PASS
Slenderness, y axisS16 Cl. 10.4.2.1 $\frac{\htmlClass{sym-k_y}{k_{y}} \cdot \htmlClass{sym-L_y}{L_{y}}}{\htmlClass{sym-r_y}{r_{y}}} \leq 200 \quad \Rightarrow \quad \frac{\htmlClass{sym-k_y}{1} \cdot \htmlClass{sym-L_y}{1500\ \mathrm{mm}}}{\htmlClass{sym-r_y}{50.4\ \mathrm{mm}}} \leq 200$	0.15	PASS
Compressive resistanceS16 Cl. 13.1, C_r >= C_f $\htmlClass{sym-C_f}{C_{f}} \leq \htmlClass{sym-C_r}{C_{r}} \quad \Rightarrow \quad \htmlClass{sym-C_f}{1500\ \mathrm{kN}} \leq \htmlClass{sym-C_r}{1920\ \mathrm{kN}}$	0.78	PASS

Results

Quantity	Description	Value	Unit
$\htmlClass{sym-C_r}{C_{r}}$	Factored compressive resistance	1920	$\mathrm{kN}$

Also worth checking

Combined axial load and bending, Cl. 13.8 and 13.9: this is concentric compression only
Tension, shear and deflection
Twisting at the ends: the torsional length K_z L_z is taken as L_x with K_z = 1.0, which Cl. 13.3.1.2 allows only where both ends are restrained from twisting (Cl. 9.4); a member free to twist at an end, such as a cantilever column, takes a larger K_z L_z
The braces and supports that set L_x and L_y: their strength (0.02 times C_f at each brace point) and stiffness, Cl. 9.5 and 9.6, and twist restraint at the supports, Cl. 9.4
A 350W Class H HSS or a flame-cut welded I-shape, whose n = 2.24 in Cl. 13.3.1.1 the calc does not take; it takes n = 1.34 throughout
Local effects at the load points, such as web crippling and bearing, and the connections
Built-up and non-table sections, and any member out of straightness beyond what S16's column curve assumes

Derivation

S.1 Resistance factor

Compared with CSA S16-19 Cl. 13.1 a) (phi = 0.90, structural steel)

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

S.2 Steel modulus

Compared with CSA S16-19 Cl. 3, symbols (E = 200 000 MPa assumed)

\begin{aligned} \htmlClass{sym-E}{E} &= 200000\ \mathrm{MPa} \end{aligned}

S.3 Shear modulus

Compared with CSA S16-19 Cl. 3, symbols (G = 77 000 MPa assumed)

\begin{aligned} \htmlClass{sym-G}{G} &= 77000\ \mathrm{MPa} \end{aligned}

S.4 Table 1 limit, webs and walls

Compared with CSA S16-19 Cl. 11.2 Table 1 (webs of I-shapes and sidewalls of rectangular HSS: 670/sqrt(F_y))

S16 Cl. 11, Table 1, webs and HSS walls in compression

\begin{aligned} \htmlClass{sym-lambda_3w}{\lambda_{3w}} &= \frac{670}{\sqrt{\frac{\htmlClass{sym-F_y}{F_{y}}}{1\ \mathrm{MPa}}}} \\ &= \frac{670}{\sqrt{\frac{\htmlClass{sym-F_y}{350\ \mathrm{MPa}}}{1\ \mathrm{MPa}}}} \\ &= 35.81 \end{aligned}

S.5 Width

Accepted by the author: CISC SST 12.1 section table value (sections-data/cisc_12.1/), a given and not a formula; an HSS wall is the table's design thickness Tdes; W250x58 B

\begin{aligned} \htmlClass{sym-b}{b} &= 203\ \mathrm{mm} \quad \left(\text{Width of flange or horizontal leg}\right) \end{aligned}

S.6 Flange or wall thickness

Accepted by the author: CISC SST 12.1 section table value (sections-data/cisc_12.1/), a given and not a formula; an HSS wall is the table's design thickness Tdes; W250x58 T

\begin{aligned} \htmlClass{sym-t}{t} &= 13.5\ \mathrm{mm} \quad \left(\text{Thickness of flange}\right) \end{aligned}

S.7 Depth

Accepted by the author: CISC SST 12.1 section table value (sections-data/cisc_12.1/), a given and not a formula; an HSS wall is the table's design thickness Tdes; W250x58 D

\begin{aligned} \htmlClass{sym-d}{d} &= 252\ \mathrm{mm} \quad \left(\text{Depth of section or height of vertical leg}\right) \end{aligned}

S.8 Web or wall thickness

Accepted by the author: CISC SST 12.1 section table value (sections-data/cisc_12.1/), a given and not a formula; an HSS wall is the table's design thickness Tdes; W250x58 W

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

S.9 Flange width-to-thickness ratio

Compared with CSA S16-19 Cl. 11.3.1 c) (flange of beam or T-section: b_el = half the full width)

\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}{203\ \mathrm{mm}}}{\htmlClass{sym-t}{13.5\ \mathrm{mm}}} \\ &= 7.519 \end{aligned}

S.10 Table 1 limit, flanges

Compared with CSA S16-19 Cl. 11.2 Table 1 (flanges of I-sections: 200/sqrt(F_y))

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}

Branch: $\lambda_{f} > \lambda_{3}$ did not hold

S.11 Class 4 excess, flanges or walls across b

Compared with CSA S16-19 Cl. 13.3.4 a) and Table 1 (area is reduced to A_e only where flange b_el/t exceeds 200/sqrt(F_y), b_el = b/2 per Cl. 11.3.1 c), so A_b = 0 for W250x58 at 7.5 against 10.7)

\begin{aligned} \htmlClass{sym-A_b}{A_{b}} &= 0\ \mathrm{mm}^{2} \quad \left(\text{Flanges not class 4}\right) \end{aligned}

S.12 Clear web depth

Compared with CSA S16-19 Cl. 11.3.2 d) (webs of hot-rolled sections: h = clear distance between flanges)

\begin{aligned} \htmlClass{sym-h}{h} &= \htmlClass{sym-d}{d} - 2 \cdot \htmlClass{sym-t}{t} \\ &= \htmlClass{sym-d}{252\ \mathrm{mm}} - 2 \cdot \htmlClass{sym-t}{13.5\ \mathrm{mm}} \\ &= 225\ \mathrm{mm} \end{aligned}

S.13 Web depth-to-thickness ratio

Compared with CSA S16-19 Cl. 11.2 Table 1 (web ratio h/w compared with 670/sqrt(F_y))

\begin{aligned} \htmlClass{sym-lambda_w}{\lambda_{w}} &= \frac{\htmlClass{sym-h}{h}}{\htmlClass{sym-w}{w}} \\ &= \frac{\htmlClass{sym-h}{225\ \mathrm{mm}}}{\htmlClass{sym-w}{8\ \mathrm{mm}}} \\ &= 28.12 \end{aligned}

Branch: $\lambda_{w} > \lambda_{3w}$ did not hold

S.14 Class 4 excess, web or walls across d

Compared with CSA S16-19 Cl. 13.3.4 a) and Table 1 (area is reduced to A_e only where web h/w exceeds 670/sqrt(F_y), h = clear distance between flanges per Cl. 11.3.2 d), so A_d = 0 for W250x58 at 28.1 against 35.8)

\begin{aligned} \htmlClass{sym-A_d}{A_{d}} &= 0\ \mathrm{mm}^{2} \quad \left(\text{Web not class 4}\right) \end{aligned}

S.15 Gross area

Accepted by the author: CISC SST 12.1 section table value (sections-data/cisc_12.1/), a given and not a formula; an HSS wall is the table's design thickness Tdes; W250x58 A_Th

\begin{aligned} \htmlClass{sym-A_0}{A_{0}} &= 7420\ \mathrm{mm}^{2} \quad \left(\text{Cross-sectional area}\right) \end{aligned}

S.16 Effective area

Compared with CSA S16-19 Cl. 13.3.4 a) (effective area from reduced element widths; A_b and A_d deductions not typeset, only 0 at defaults)

S16 Cl. 13.3.4 a), effective area from reduced widths meeting Table 1

\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}{7420\ \mathrm{mm}^{2}} - \htmlClass{sym-A_b}{0\ \mathrm{mm}^{2}} - \htmlClass{sym-A_d}{0\ \mathrm{mm}^{2}} \\ &= 7420\ \mathrm{mm}^{2} \end{aligned}

S.17 Radius of gyration about x

Accepted by the author: CISC SST 12.1 section table value (sections-data/cisc_12.1/), a given and not a formula; an HSS wall is the table's design thickness Tdes; W250x58 Rx

\begin{aligned} \htmlClass{sym-r_x}{r_{x}} &= 108\ \mathrm{mm} \quad \left(\text{Radius of gyration about axis XX}\right) \end{aligned}

S.18 Radius of gyration about y

Accepted by the author: CISC SST 12.1 section table value (sections-data/cisc_12.1/), a given and not a formula; an HSS wall is the table's design thickness Tdes; W250x58 Ry

\begin{aligned} \htmlClass{sym-r_y}{r_{y}} &= 50.4\ \mathrm{mm} \quad \left(\text{Radius of gyration about axis YY}\right) \end{aligned}

S.19 Euler stress about x

Compared with CSA S16-19 Cl. 13.3.1.2 (F_ex = pi^2 E/(K_x L_x/r_x)^2, pi^2 E = 1.974 TPa)

\begin{aligned} \htmlClass{sym-F_ex}{F_{\mathrm{ex}}} &= \frac{1.974 \times 10^{6}\ \mathrm{MPa}}{\left(\frac{\htmlClass{sym-k_x}{k_{x}} \cdot \htmlClass{sym-L_x}{L_{x}}}{\htmlClass{sym-r_x}{r_{x}}}\right)^{2}} \\ &= \frac{1.974 \times 10^{6}\ \mathrm{MPa}}{\left(\frac{\htmlClass{sym-k_x}{1} \cdot \htmlClass{sym-L_x}{3000\ \mathrm{mm}}}{\htmlClass{sym-r_x}{108\ \mathrm{mm}}}\right)^{2}} \\ &= 2558\ \mathrm{MPa} \end{aligned}

S.20 Euler stress about y

Compared with CSA S16-19 Cl. 13.3.1.2 (F_ey = pi^2 E/(K_y L_y/r_y)^2)

\begin{aligned} \htmlClass{sym-F_ey}{F_{\mathrm{ey}}} &= \frac{1.974 \times 10^{6}\ \mathrm{MPa}}{\left(\frac{\htmlClass{sym-k_y}{k_{y}} \cdot \htmlClass{sym-L_y}{L_{y}}}{\htmlClass{sym-r_y}{r_{y}}}\right)^{2}} \\ &= \frac{1.974 \times 10^{6}\ \mathrm{MPa}}{\left(\frac{\htmlClass{sym-k_y}{1} \cdot \htmlClass{sym-L_y}{1500\ \mathrm{mm}}}{\htmlClass{sym-r_y}{50.4\ \mathrm{mm}}}\right)^{2}} \\ &= 2228\ \mathrm{MPa} \end{aligned}

S.21 Warping constant

Accepted by the author: CISC SST 12.1 section table value (sections-data/cisc_12.1/), a given and not a formula; an HSS wall is the table's design thickness Tdes; W250x58 Cw

\begin{aligned} \htmlClass{sym-C_w}{C_{w}} &= 2.68 \times 10^{11}\ \mathrm{mm}^{6} \quad \left(\text{Warping constant}\right) \end{aligned}

S.22 Warping term

Compared with CSA S16-19 Cl. 13.3.1.2 (pi^2 E C_w/(K_z L_z)^2 term, K_z = 1.0 and L_z = L_x assumed; C_w = 0 for HSS)

\begin{aligned} \htmlClass{sym-F_ez0}{F_{\mathrm{ez}0}} &= \frac{1.974 \times 10^{6}\ \mathrm{MPa} \cdot \htmlClass{sym-C_w}{C_{w}}}{\htmlClass{sym-L_x}{L_{x}}^{2}} \\ &= \frac{1.974 \times 10^{6}\ \mathrm{MPa} \cdot \htmlClass{sym-C_w}{2.68 \times 10^{11}\ \mathrm{mm}^{6}}}{\left(\htmlClass{sym-L_x}{3000\ \mathrm{mm}}\right)^{2}} \\ &= 58.78\ \mathrm{kN} \cdot \mathrm{m²} \end{aligned}

S.23 Torsional constant

Accepted by the author: CISC SST 12.1 section table value (sections-data/cisc_12.1/), a given and not a formula; an HSS wall is the table's design thickness Tdes; W250x58 J

\begin{aligned} \htmlClass{sym-J}{J} &= 409000\ \mathrm{mm}^{4} \quad \left(\text{St-Venant torsional constant}\right) \end{aligned}

S.24 Polar radius of gyration

Compared with CSA S16-19 Cl. 13.3.1.2 (r_o^2 = x_o^2 + y_o^2 + r_x^2 + r_y^2, x_o = y_o = 0 doubly symmetric)

\begin{aligned} \htmlClass{sym-r_0}{r_{0}} &= \sqrt{\htmlClass{sym-r_x}{r_{x}}^{2} + \htmlClass{sym-r_y}{r_{y}}^{2}} \\ &= \sqrt{\left(\htmlClass{sym-r_x}{108\ \mathrm{mm}}\right)^{2} + \left(\htmlClass{sym-r_y}{50.4\ \mathrm{mm}}\right)^{2}} \\ &= 119.2\ \mathrm{mm} \end{aligned}

S.25 Torsional buckling stress

Compared with CSA S16-19 Cl. 13.3.1.2 (F_ez = (pi^2 E C_w/(K_z L_z)^2 + G J)/(A r_o^2), K_z L_z taken as L_x, r_o^2 = r_x^2 + r_y^2 for a doubly symmetric section; C_w = 0 for an HSS, where Note 2 says torsional buckling is not a limit state, so including it is conservative)

S16 Cl. 13.3.1.2, L_z = L_x

\begin{aligned} \htmlClass{sym-F_ez}{F_{\mathrm{ez}}} &= \frac{\htmlClass{sym-F_ez0}{F_{\mathrm{ez}0}} + \htmlClass{sym-G}{G} \cdot \htmlClass{sym-J}{J}}{\htmlClass{sym-A_0}{A_{0}} \cdot \htmlClass{sym-r_0}{r_{0}}^{2}} \\ &= \frac{\htmlClass{sym-F_ez0}{58.78\ \mathrm{kN} \cdot \mathrm{m²}} + \htmlClass{sym-G}{77000\ \mathrm{MPa}} \cdot \htmlClass{sym-J}{409000\ \mathrm{mm}^{4}}}{\htmlClass{sym-A_0}{7420\ \mathrm{mm}^{2}} \cdot \left(\htmlClass{sym-r_0}{119.2\ \mathrm{mm}}\right)^{2}} \\ &= 856.5\ \mathrm{MPa} \end{aligned}

S.26 Elastic buckling stress

Compared with CSA S16-19 Cl. 13.3.1.1 a) (F_e the least of F_ex, F_ey, F_ez for doubly symmetric sections)

S16 Cl. 13.3.1.1 a)

\begin{aligned} \htmlClass{sym-F_e}{F_{e}} &= \min\left(\htmlClass{sym-F_ex}{F_{\mathrm{ex}}}, \htmlClass{sym-F_ey}{F_{\mathrm{ey}}}, \htmlClass{sym-F_ez}{F_{\mathrm{ez}}}\right) \\ &= \min\left(\htmlClass{sym-F_ex}{2558\ \mathrm{MPa}}, \htmlClass{sym-F_ey}{2228\ \mathrm{MPa}}, \htmlClass{sym-F_ez}{856.5\ \mathrm{MPa}}\right) \\ &= 856.5\ \mathrm{MPa} \end{aligned}

S.27 Slenderness parameter

Compared with CSA S16-19 Cl. 13.3.1.1 (lambda = sqrt(F_y/F_e))

\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}{856.5\ \mathrm{MPa}}}} \\ &= 0.6392 \end{aligned}

S.28 Compressive stress

Compared with CSA S16-19 Cl. 13.3.1.1 (C_r = phi A F_y / (1 + lambda^2n)^(1/n), n = 1.34 hot-rolled, lambda = sqrt(F_y/F_e))

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}{0.6392}^{2 \cdot 1.34}\right)^{\frac{-1}{1.34}} \\ &= 287.5\ \mathrm{MPa} \end{aligned}

S.29 Compressive resistance

Compared with CSA S16-19 Cl. 13.3.4 a) and 13.3.1.1 (C_r = phi A_e F_y/(1+lambda^2n)^(1/n))

S16 Cl. 13.3.4 a), phi A_e f_s

\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}{7420\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-f_s}{287.5\ \mathrm{MPa}} \\ &= 1920\ \mathrm{kN} \end{aligned}

Report

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Questions

Which n does an HSS take in S16 Cl. 13.3.1.1?

1.34, a class C cold-formed section. A stress-relieved class H HSS would take 2.24 and a higher resistance.

Where do A_0, r_x, r_y, J and C_w come from?

The CISC Structural Section Tables, SST 12.1: area, radii, J and C_w, and the element dimensions.

Revision history

2026-11-05 After an independent checker's review: added slenderness checks of KL/r against 200 on each axis (Cl. 10.4.2.1), an HSS no longer takes torsional buckling (Cl. 13.3.1.2 Note 2), the class 4 reduction cites Cl. 13.3.4 a), and the twist-restraint assumption is listed as not checked

Value	Was	Now	Unit
`slender_x_ok`
`slender_y_ok`

2026-10-26 fix(units,examples): zero keeps its unit, zero cope, shear tab rules

Value	Was	Now	Unit
`A_b`	0 \mathrm{m}^{2}	0	$\mathrm{mm}^{2}$
`A_d`	0 \mathrm{m}^{2}	0	$\mathrm{mm}^{2}$

Axial compression of a W section or HSS

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