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SteelBeam-column

Beam-column

CSA S16-19Verified

Sample report sheets

Axial compression and biaxial bending of a W shape, HSS or channel to CSA S16 Cl. 13.8. Check a W, WWF, S or HP shape, a rectangular HSS or a channel under axial compression and bending about one or both axes. The section is classified by S16 Table 2 for beam-columns, and its class picks the interaction: cross-sectional strength, overall member strength with the amplification for a braced or unbraced frame about each axis, and lateral-torsional buckling strength with Cl. 13.6's moment resistance.

Derivation, step by step

SIImperial

CA$4.00 for this report, or Pro

PASS Governing: Lateral-torsional buckling strength, D/C 0.21, S16 Cl. 13.8.2 (c)10 of 10 checks below pass

Given

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

Loads
CfC_{f} kN\mathrm{kN} 0-50,000
MfxM_{\mathrm{fx}} kN⋅m\mathrm{kN} \cdot \mathrm{m} 0-10,000
MfyM_{\mathrm{fy}} kN⋅m\mathrm{kN} \cdot \mathrm{m} 0-5,000
Section
Framing
ω1x\omega_{1x} 0.4-1
ω1y\omega_{1y} 0.4-1
ω2\omega_{2} 1-2.5
Bracing
LxL_{x} mm\mathrm{mm} 100-30,000
kxk_{x} 0.5-2.5
LyL_{y} mm\mathrm{mm} 100-30,000
kyk_{y} 0.5-2.5
d = 247 mm b = 202 mm t = 11.0 mm w = 7.4 mm x y
The cross-section to scale.

Title block

Optional. Printed in the title block of the PDF.

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

Interaction

Cross-sectional strengthS16 Cl. 13.8.2 (a)Ia≤1⇒0.1998≤1\htmlClass{sym-I_a}{I_{a}} \leq 1 \quad \Rightarrow \quad \htmlClass{sym-I_a}{0.1998} \leq 1 0.20 PASS
Overall member strengthS16 Cl. 13.8.2 (b)Ib≤1⇒0.2015≤1\htmlClass{sym-I_b}{I_{b}} \leq 1 \quad \Rightarrow \quad \htmlClass{sym-I_b}{0.2015} \leq 1 0.20 PASS
Lateral-torsional buckling strengthS16 Cl. 13.8.2 (c)Ic≤1⇒0.2075≤1\htmlClass{sym-I_c}{I_{c}} \leq 1 \quad \Rightarrow \quad \htmlClass{sym-I_c}{0.2075} \leq 1 0.21 PASS

Moment only

Moment-only capacityS16 Cl. 13.8.4 (d), class 3 or not an I-shapeId≤1⇒0.1689≤1\htmlClass{sym-I_d}{I_{d}} \leq 1 \quad \Rightarrow \quad \htmlClass{sym-I_d}{0.1689} \leq 1 0.17 PASS
Moment-only capacity, lateral-torsionalS16 Cl. 13.8.4 (d), class 3 or not an I-shapeIdl≤1⇒0.1689≤1\htmlClass{sym-I_dl}{I_{\mathrm{dl}}} \leq 1 \quad \Rightarrow \quad \htmlClass{sym-I_dl}{0.1689} \leq 1 0.17 PASS

Compression

Web class under the axial loadS16 Table 2, Note 1Cf≤Cf,4⇒60 kN≤2098 kN\htmlClass{sym-C_f}{C_{f}} \leq \htmlClass{sym-C_f_4}{C_{f,4}} \quad \Rightarrow \quad \htmlClass{sym-C_f}{60\ \mathrm{kN}} \leq \htmlClass{sym-C_f_4}{2098\ \mathrm{kN}} 0.03 PASS
Compressive resistance, no momentS16 Cl. 13.3.1.1, C_f <= C_r at the entered K, with C_r on A_e (Cl. 13.3.4 a))A moment about x, so lateral-torsional buckling covers out-of-plane buckling and the member check takes K = 1 (S16 Cl. 13.3.1.1 at the entered K; kept on where Cl. 13.8.2 (c) does not run, so out-of-plane buckling is checked) N/A
Slenderness, x axisS16 Cl. 10.4.2.1kx⋅Lxrx≤200⇒1⋅3000 mm106 mm≤200\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}{106\ \mathrm{mm}}} \leq 200 0.14 PASS
Slenderness, y axisS16 Cl. 10.4.2.1ky⋅Lyry≤200⇒1⋅1500 mm49.2 mm≤200\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}{49.2\ \mathrm{mm}}} \leq 200 0.15 PASS

Stability

Euler buckling load, x axisS16 Cl. 13.8.5, C_f must be under C_e for U_1 to applyCf<Cex⇒60 kN<15484 kN\htmlClass{sym-C_f}{C_{f}} < \htmlClass{sym-C_ex}{C_{\mathrm{ex}}} \quad \Rightarrow \quad \htmlClass{sym-C_f}{60\ \mathrm{kN}} < \htmlClass{sym-C_ex}{15484\ \mathrm{kN}} 0.00 PASS
Euler buckling load, y axisS16 Cl. 13.8.5, C_f must be under C_e for U_1 to applyCf<Cey⇒60 kN<13247 kN\htmlClass{sym-C_f}{C_{f}} < \htmlClass{sym-C_ey}{C_{\mathrm{ey}}} \quad \Rightarrow \quad \htmlClass{sym-C_f}{60\ \mathrm{kN}} < \htmlClass{sym-C_ey}{13247\ \mathrm{kN}} 0.00 PASS

Design conditions

A moment about x, so lateral-torsional buckling covers out-of-plane buckling and the member check takes K = 1S16 Cl. 13.3.1.1 at the entered K; kept on where Cl. 13.8.2 (c) does not run, so out-of-plane buckling is checkedMfx>0 kN⋅m⇒30 kN⋅m>0 kN⋅m\htmlClass{sym-M_fx}{M_{\mathrm{fx}}} > 0\ \mathrm{kN} \cdot \mathrm{m} \quad \Rightarrow \quad \htmlClass{sym-M_fx}{30\ \mathrm{kN} \cdot \mathrm{m}} > 0\ \mathrm{kN} \cdot \mathrm{m}
Braced about at least one axisS16 Cl. 13.8.2 (a) and 13.8.1bracedaxes≥1⇒1≥1\htmlClass{sym-braced_axes}{\mathrm{braced}_{\mathrm{axes}}} \geq 1 \quad \Rightarrow \quad \htmlClass{sym-braced_axes}{1} \geq 1
A moment about x, so lateral-torsional buckling appliesS16 Cl. 13.8.2 (c), when applicableMfx>0 kN⋅m⇒30 kN⋅m>0 kN⋅m\htmlClass{sym-M_fx}{M_{\mathrm{fx}}} > 0\ \mathrm{kN} \cdot \mathrm{m} \quad \Rightarrow \quad \htmlClass{sym-M_fx}{30\ \mathrm{kN} \cdot \mathrm{m}} > 0\ \mathrm{kN} \cdot \mathrm{m}
Braced about at least one axis, so Cl. 13.8.4 (d) appliesS16 Cl. 13.8.4 (d)bracedaxes≥1⇒1≥1\htmlClass{sym-braced_axes}{\mathrm{braced}_{\mathrm{axes}}} \geq 1 \quad \Rightarrow \quad \htmlClass{sym-braced_axes}{1} \geq 1

Results

Compression

Cr\htmlClass{sym-C_r}{C_{r}} Factored compressive resistance as a column, at the entered K, Cl. 13.3.1 1554 kN\mathrm{kN}
Crb\htmlClass{sym-C_rb}{C_{\mathrm{rb}}} Factored compressive resistance, overall member check, K = 1, Cl. 13.8.2 (b) 1843 kN\mathrm{kN}
Cry\htmlClass{sym-C_ry}{C_{\mathrm{ry}}} Factored compressive resistance, weak-axis or torsional buckling, Cl. 13.8.2 (c) 1554 kN\mathrm{kN}

Flexure

Mrx\htmlClass{sym-M_rx}{M_{\mathrm{rx}}} Factored moment resistance about x, Cl. 13.5 177.6 kN⋅m\mathrm{kN} \cdot \mathrm{m}
Mrxl\htmlClass{sym-M_rxl}{M_{\mathrm{rxl}}} Factored moment resistance about x, unbraced over L_y, Cl. 13.6 177.6 kN⋅m\mathrm{kN} \cdot \mathrm{m}
Mry\htmlClass{sym-M_ry}{M_{\mathrm{ry}}} Factored moment resistance about y, Cl. 13.5 46.58 kN⋅m\mathrm{kN} \cdot \mathrm{m}
Also worth checking
  • Class 4 sections: a class 4 flange is refused and a class 4 web fails; a class 4 flange takes the effective flange of Cl. 13.5 c) iii) and a class 4 web takes Cl. 14.3.4
  • Circular HSS, singly symmetric I-sections, tees and angles, which the section list does not offer
  • 350W Class H HSS, whose n = 2.24 in Cl. 13.3.1.1 the calc does not take
  • A class 4 channel's web under M_fy: it is refused, and Cl. 13.5 c) iii)'s effective section is not computed
  • Loads applied above the shear centre, Cl. 13.6.1 a), and cantilevered segments, Cl. 13.6.1 d): M_u is for a segment between effective brace points loaded at the shear centre
  • Shear, Cl. 13.4, tension, deflection and the connections

Derivation

S.1 Resistance factor

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

ϕ=0.9(Resistance factor | S16 Cl. 13.1)\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

E=200000 MPa\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

G=77000 MPa\begin{aligned} \htmlClass{sym-G}{G} &= 77000\ \mathrm{MPa} \end{aligned}
S.4

Accepted by the author: the specified minimum yield of the grade, from the cisc_sections grade table; the standard grade by family, A992 for W shapes (CISC FAQ), 350W Class C for G40.21 HSS, A500 Grade C for A500 HSS, and 350W for channels, angles and every other family (CISC 11th edition Handbook), confirmed by the owner

Fy=345 MPa(Yield strength, A992 (ASTM A992/A992M, Structural Steel Shapes))\begin{aligned} \htmlClass{sym-F_y}{F_{y}} &= 345\ \mathrm{MPa} \quad \left(\text{Yield strength, A992 (ASTM A992/A992M, Structural Steel Shapes)}\right) \end{aligned}
S.5 Root of the yield strength

Accepted by the author: definition, the root of F_y in MPa that every CSA S16-19 Table 2 limit divides by

rootF,y=Fy1 MPa=345 MPa1 MPa=18.57\begin{aligned} \htmlClass{sym-root_F_y}{\mathrm{root}_{F,y}} &= \sqrt{\frac{\htmlClass{sym-F_y}{F_{y}}}{1\ \mathrm{MPa}}} \\ &= \sqrt{\frac{\htmlClass{sym-F_y}{345\ \mathrm{MPa}}}{1\ \mathrm{MPa}}} \\ &= 18.57 \end{aligned}

Branch: Mfx+Mfy>0 kN⋅mM_{\mathrm{fx}} + M_{\mathrm{fy}} > 0\ \mathrm{kN} \cdot \mathrm{m} held

S.6 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

A0=6250 mm2(Cross-sectional area)\begin{aligned} \htmlClass{sym-A_0}{A_{0}} &= 6250\ \mathrm{mm}^{2} \quad \left(\text{Cross-sectional area}\right) \end{aligned}
S.7 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

b=202 mm(Width of flange or horizontal leg)\begin{aligned} \htmlClass{sym-b}{b} &= 202\ \mathrm{mm} \quad \left(\text{Width of flange or horizontal leg}\right) \end{aligned}
S.8 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

d=247 mm(Depth of section or height of vertical leg)\begin{aligned} \htmlClass{sym-d}{d} &= 247\ \mathrm{mm} \quad \left(\text{Depth of section or height of vertical leg}\right) \end{aligned}
S.9 Elastic modulus 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

Sx=572000 mm3(Elastic section modulus about axis XX)\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.10 Elastic modulus 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

Sy=150000 mm3(Elastic section modulus about axis YY)\begin{aligned} \htmlClass{sym-S_y}{S_{y}} &= 150000\ \mathrm{mm}^{3} \quad \left(\text{Elastic section modulus about axis YY}\right) \end{aligned}
S.11 Axial yield load

Accepted by the author: definition, CSA S16-19 Cl. 1 symbols, C_y = axial compressive load at yield stress, A F_y

Cy=A0⋅Fy=6250 mm2⋅345 MPa=2156 kN\begin{aligned} \htmlClass{sym-C_y}{C_{y}} &= \htmlClass{sym-A_0}{A_{0}} \cdot \htmlClass{sym-F_y}{F_{y}} \\ &= \htmlClass{sym-A_0}{6250\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-F_y}{345\ \mathrm{MPa}} \\ &= 2156\ \mathrm{kN} \end{aligned}
S.12 Axial load ratio

Compared with CSA S16-19 Table 2, the ratio C_f/(phi C_y) in the web limits

ρ=Cfϕ⋅Cy=60 kN0.9⋅2156 kN=0.03092\begin{aligned} \htmlClass{sym-rho}{\rho} &= \frac{\htmlClass{sym-C_f}{C_{f}}}{\htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-C_y}{C_{y}}} \\ &= \frac{\htmlClass{sym-C_f}{60\ \mathrm{kN}}}{\htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-C_y}{2156\ \mathrm{kN}}} \\ &= 0.03092 \end{aligned}
S.13 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

t=11 mm(Thickness of flange)\begin{aligned} \htmlClass{sym-t}{t} &= 11\ \mathrm{mm} \quad \left(\text{Thickness of flange}\right) \end{aligned}
S.14 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

w=7.4 mm(Thickness of web)\begin{aligned} \htmlClass{sym-w}{w} &= 7.4\ \mathrm{mm} \quad \left(\text{Thickness of web}\right) \end{aligned}
S.15 Flange width-to-thickness ratio

Compared with CSA S16-19 Cl. 11.3.1 c), b_el of a beam flange is one-half the nominal width; a channel flange takes the full width, Cl. 11.3.1 b); an HSS flange the outside less 4t, Cl. 11.3.2 b)

λf=b2⋅t=202 mm2⋅11 mm=9.182\begin{aligned} \htmlClass{sym-lambda_f}{\lambda_{f}} &= \frac{\htmlClass{sym-b}{b}}{2 \cdot \htmlClass{sym-t}{t}} \\ &= \frac{\htmlClass{sym-b}{202\ \mathrm{mm}}}{2 \cdot \htmlClass{sym-t}{11\ \mathrm{mm}}} \\ &= 9.182 \end{aligned}
S.16 Class 1 limit, flange

Compared with CSA S16-19 Table 2 (flange limit 145/sqrt(F_y), Class 1; Table 2 OCR is scrambled, classes read in ascending order)

S16 Table 2, flanges

λf1=145rootF,y=14518.57=7.807\begin{aligned} \htmlClass{sym-lambda_f1}{\lambda_{f1}} &= \frac{145}{\htmlClass{sym-root_F_y}{\mathrm{root}_{F,y}}} \\ &= \frac{145}{\htmlClass{sym-root_F_y}{18.57}} \\ &= 7.807 \end{aligned}
S.17 Class 2 limit, flange

Compared with CSA S16-19 Table 2 (flange limit 170/sqrt(F_y), Class 2)

S16 Table 2, flanges

λf2=170rootF,y=17018.57=9.152\begin{aligned} \htmlClass{sym-lambda_f2}{\lambda_{f2}} &= \frac{170}{\htmlClass{sym-root_F_y}{\mathrm{root}_{F,y}}} \\ &= \frac{170}{\htmlClass{sym-root_F_y}{18.57}} \\ &= 9.152 \end{aligned}
S.18 Class 3 limit, flange

Compared with CSA S16-19 Table 2 (flange limit 200/sqrt(F_y), Class 3)

S16 Table 2, flanges

λf3=200rootF,y=20018.57=10.77\begin{aligned} \htmlClass{sym-lambda_f3}{\lambda_{f3}} &= \frac{200}{\htmlClass{sym-root_F_y}{\mathrm{root}_{F,y}}} \\ &= \frac{200}{\htmlClass{sym-root_F_y}{18.57}} \\ &= 10.77 \end{aligned}
S.19 Web depth-to-thickness ratio

Compared with CSA S16-19 Cl. 11.3.2 d), h of a hot-rolled web is the clear distance between flanges; an HSS web the outside less 4t, Cl. 11.3.2 b)

λw=d−2⋅tw=247 mm−2⋅11 mm7.4 mm=30.41\begin{aligned} \htmlClass{sym-lambda_w}{\lambda_{w}} &= \frac{\htmlClass{sym-d}{d} - 2 \cdot \htmlClass{sym-t}{t}}{\htmlClass{sym-w}{w}} \\ &= \frac{\htmlClass{sym-d}{247\ \mathrm{mm}} - 2 \cdot \htmlClass{sym-t}{11\ \mathrm{mm}}}{\htmlClass{sym-w}{7.4\ \mathrm{mm}}} \\ &= 30.41 \end{aligned}

Branch: Mfy>0 kN⋅mM_{\mathrm{fy}} > 0\ \mathrm{kN} \cdot \mathrm{m} did not hold

Branch: MfySy>0.9⋅MfxSx\frac{M_{\mathrm{fy}}}{S_{y}} > \frac{0.9 \cdot M_{\mathrm{fx}}}{S_{x}} did not hold

S.20 Class 1 limit, web

Compared with CSA S16-19 Table 2 (web with axial compression and major-axis bending, Class 1: 1100/sqrt(F_y) (1 - 0.39 C_f/(phi C_y)))

S16 Table 2, webs in axial compression and bending

λw1=1100rootF,y⋅(1−0.39⋅ρ)=110018.57⋅(1−0.39⋅0.03092)=58.51\begin{aligned} \htmlClass{sym-lambda_w1}{\lambda_{w1}} &= \frac{1100}{\htmlClass{sym-root_F_y}{\mathrm{root}_{F,y}}} \cdot \left(1 - 0.39 \cdot \htmlClass{sym-rho}{\rho}\right) \\ &= \frac{1100}{\htmlClass{sym-root_F_y}{18.57}} \cdot \left(1 - 0.39 \cdot \htmlClass{sym-rho}{0.03092}\right) \\ &= 58.51 \end{aligned}
S.21 Class 2 limit, web

Compared with CSA S16-19 Table 2 (web with axial compression and major-axis bending, Class 2: 1700/sqrt(F_y) (1 - 0.61 C_f/(phi C_y)))

S16 Table 2, webs in axial compression and bending

λw2=1700rootF,y⋅(1−0.61⋅ρ)=170018.57⋅(1−0.61⋅0.03092)=89.8\begin{aligned} \htmlClass{sym-lambda_w2}{\lambda_{w2}} &= \frac{1700}{\htmlClass{sym-root_F_y}{\mathrm{root}_{F,y}}} \cdot \left(1 - 0.61 \cdot \htmlClass{sym-rho}{\rho}\right) \\ &= \frac{1700}{\htmlClass{sym-root_F_y}{18.57}} \cdot \left(1 - 0.61 \cdot \htmlClass{sym-rho}{0.03092}\right) \\ &= 89.8 \end{aligned}
S.22 Class 3 limit, web

Compared with CSA S16-19 Table 2 (web with axial compression, Class 3: 1900/sqrt(F_y) (1 - 0.65 C_f/(phi C_y)))

S16 Table 2, webs in axial compression and bending

λw3=1900rootF,y⋅(1−0.65⋅ρ)=190018.57⋅(1−0.65⋅0.03092)=100.2\begin{aligned} \htmlClass{sym-lambda_w3}{\lambda_{w3}} &= \frac{1900}{\htmlClass{sym-root_F_y}{\mathrm{root}_{F,y}}} \cdot \left(1 - 0.65 \cdot \htmlClass{sym-rho}{\rho}\right) \\ &= \frac{1900}{\htmlClass{sym-root_F_y}{18.57}} \cdot \left(1 - 0.65 \cdot \htmlClass{sym-rho}{0.03092}\right) \\ &= 100.2 \end{aligned}

Branch: λf≤λf1\lambda_{f} \leq \lambda_{f1} did not hold

Branch: λf≤λf2\lambda_{f} \leq \lambda_{f2} did not hold

Branch: λf≤λf3\lambda_{f} \leq \lambda_{f3} held

S.23 Flange class

Compared with CSA S16-19 Table 2, the class is the first limit the ratio meets, past Class 3 is Class 4 by Note 1

classf=3(Flange class 3)\begin{aligned} \htmlClass{sym-class_f}{\mathrm{class}_{f}} &= 3 \quad \left(\text{Flange class 3}\right) \end{aligned}

Branch: λw≤λw1\lambda_{w} \leq \lambda_{w1} held

S.24 Web class

Compared with CSA S16-19 Table 2, the class is the first limit the ratio meets, past Class 3 is Class 4 by Note 1

classw=1(Web class 1)\begin{aligned} \htmlClass{sym-class_w}{\mathrm{class}_{w}} &= 1 \quad \left(\text{Web class 1}\right) \end{aligned}

Branch: classf>3\mathrm{class}_{f} > 3 did not hold

S.25 Section class

Accepted by the author: definition, Table 2 classifies the flange and the web separately and the section takes the worse of the two

sectionclass=max⁡(classf,classw)=max⁡(3,1)=3\begin{aligned} \htmlClass{sym-section_class}{\mathrm{section}_{\mathrm{class}}} &= \max\left(\htmlClass{sym-class_f}{\mathrm{class}_{f}}, \htmlClass{sym-class_w}{\mathrm{class}_{w}}\right) \\ &= \max\left(\htmlClass{sym-class_f}{3}, \htmlClass{sym-class_w}{1}\right) \\ &= 3 \end{aligned}
S.26 Axial load at which the web turns class 4

Compared with CSA S16-19 Table 2, the Class 3 web limit h/w <= 1900/sqrt(F_y) (1 - 0.65 C_f/(phi C_y)) solved for C_f; that limit is the same in every web row

S16 Table 2, webs, h/w at the class 3 limit

Cf,4=ϕ⋅Cy⋅(1−λw⋅rootF,y1900)0.65=0.9⋅2156 kN⋅(1−30.41⋅18.571900)0.65=2098 kN\begin{aligned} \htmlClass{sym-C_f_4}{C_{f,4}} &= \frac{\htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-C_y}{C_{y}} \cdot \left(1 - \frac{\htmlClass{sym-lambda_w}{\lambda_{w}} \cdot \htmlClass{sym-root_F_y}{\mathrm{root}_{F,y}}}{1900}\right)}{0.65} \\ &= \frac{\htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-C_y}{2156\ \mathrm{kN}} \cdot \left(1 - \frac{\htmlClass{sym-lambda_w}{30.41} \cdot \htmlClass{sym-root_F_y}{18.57}}{1900}\right)}{0.65} \\ &= 2098\ \mathrm{kN} \end{aligned}

Branch: Cf,4<0 kNC_{f,4} < 0\ \mathrm{kN} did not hold

S.27 Table 1 limit, webs and walls

Compared with CSA S16-19 Table 1 (compression elements, webs supported on both edges: width-to-thickness limit 670 / sqrt(F_y))

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

λ3w=670rootF,y=67018.57=36.07\begin{aligned} \htmlClass{sym-lambda_3w}{\lambda_{3w}} &= \frac{670}{\htmlClass{sym-root_F_y}{\mathrm{root}_{F,y}}} \\ &= \frac{670}{\htmlClass{sym-root_F_y}{18.57}} \\ &= 36.07 \end{aligned}

Branch: λw>λ3w\lambda_{w} > \lambda_{3w} did not hold

S.28 Class 4 excess, web or walls across d

Accepted by the author: a selection, zero excess because the web is within the Table 1 limit in compression, so no width is removed

Ad=0 mm2(Web or walls across d not class 4 in compression)\begin{aligned} \htmlClass{sym-A_d}{A_{d}} &= 0\ \mathrm{mm}^{2} \quad \left(\text{Web or walls across d not class 4 in compression}\right) \end{aligned}
S.29 Effective area

Compared with CSA S16-19 Cl. 13.3.4 a) (effective area from widths reduced to the Table 1 limit, the gross area less the excess)

S16 Cl. 13.3.4 a), A_e from reduced widths meeting Table 1, F_e on the gross section

Ae=A0−Ad=6250 mm2−0 mm2=6250 mm2\begin{aligned} \htmlClass{sym-A_e}{A_{e}} &= \htmlClass{sym-A_0}{A_{0}} - \htmlClass{sym-A_d}{A_{d}} \\ &= \htmlClass{sym-A_0}{6250\ \mathrm{mm}^{2}} - \htmlClass{sym-A_d}{0\ \mathrm{mm}^{2}} \\ &= 6250\ \mathrm{mm}^{2} \end{aligned}
S.30 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

rx=106 mm(Radius of gyration about axis XX)\begin{aligned} \htmlClass{sym-r_x}{r_{x}} &= 106\ \mathrm{mm} \quad \left(\text{Radius of gyration about axis XX}\right) \end{aligned}
S.31 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

ry=49.2 mm(Radius of gyration about axis YY)\begin{aligned} \htmlClass{sym-r_y}{r_{y}} &= 49.2\ \mathrm{mm} \quad \left(\text{Radius of gyration about axis YY}\right) \end{aligned}
S.32 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

J=241000 mm4(St-Venant torsional constant)\begin{aligned} \htmlClass{sym-J}{J} &= 241000\ \mathrm{mm}^{4} \quad \left(\text{St-Venant torsional constant}\right) \end{aligned}
S.33 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)

Fex=1.974×106 MPa(kx⋅Lxrx)2=1.974×106 MPa(1⋅3000 mm106 mm)2=2464 MPa\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}{106\ \mathrm{mm}}}\right)^{2}} \\ &= 2464\ \mathrm{MPa} \end{aligned}
S.34 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 (pi^2 E = 1.974 TPa)

Fey=1.974×106 MPa(ky⋅Lyry)2=1.974×106 MPa(1⋅1500 mm49.2 mm)2=2124 MPa\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}{49.2\ \mathrm{mm}}}\right)^{2}} \\ &= 2124\ \mathrm{MPa} \end{aligned}
S.35 Euler stress about x, K = 1

Compared with CSA S16-19 Cl. 13.8.2 b) i), C_r per Cl. 13.3 with K = 1

Fex1=1.974×106 MPa(Lxrx)2=1.974×106 MPa(3000 mm106 mm)2=2464 MPa\begin{aligned} \htmlClass{sym-F_ex1}{F_{\mathrm{ex}1}} &= \frac{1.974 \times 10^{6}\ \mathrm{MPa}}{\left(\frac{\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-L_x}{3000\ \mathrm{mm}}}{\htmlClass{sym-r_x}{106\ \mathrm{mm}}}\right)^{2}} \\ &= 2464\ \mathrm{MPa} \end{aligned}
S.36 Euler stress about y, K = 1

Compared with CSA S16-19 Cl. 13.8.2 b) i), C_r per Cl. 13.3 with K = 1

Fey1=1.974×106 MPa(Lyry)2=1.974×106 MPa(1500 mm49.2 mm)2=2124 MPa\begin{aligned} \htmlClass{sym-F_ey1}{F_{\mathrm{ey}1}} &= \frac{1.974 \times 10^{6}\ \mathrm{MPa}}{\left(\frac{\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-L_y}{1500\ \mathrm{mm}}}{\htmlClass{sym-r_y}{49.2\ \mathrm{mm}}}\right)^{2}} \\ &= 2124\ \mathrm{MPa} \end{aligned}
S.37 Shear centre offset

Accepted by the author: CISC SST 12.1 Xo for a channel (sections-data/cisc_12.1/), and 0 for a doubly symmetric shape, whose shear centre is on the centroid

x0=0 mm(Shear centre on the centroid, not a channel)\begin{aligned} \htmlClass{sym-x_0}{x_{0}} &= 0\ \mathrm{mm} \quad \left(\text{Shear centre on the centroid, not a channel}\right) \end{aligned}
S.38 Polar radius about the shear centre

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 with y_o = 0

r0=x02+rx2+ry2=(0 mm)2+(106 mm)2+(49.2 mm)2=116.9 mm\begin{aligned} \htmlClass{sym-r_0}{r_{0}} &= \sqrt{\htmlClass{sym-x_0}{x_{0}}^{2} + \htmlClass{sym-r_x}{r_{x}}^{2} + \htmlClass{sym-r_y}{r_{y}}^{2}} \\ &= \sqrt{\left(\htmlClass{sym-x_0}{0\ \mathrm{mm}}\right)^{2} + \left(\htmlClass{sym-r_x}{106\ \mathrm{mm}}\right)^{2} + \left(\htmlClass{sym-r_y}{49.2\ \mathrm{mm}}\right)^{2}} \\ &= 116.9\ \mathrm{mm} \end{aligned}
S.39 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

Cw=2.11×1011 mm6(Warping constant)\begin{aligned} \htmlClass{sym-C_w}{C_{w}} &= 2.11 \times 10^{11}\ \mathrm{mm}^{6} \quad \left(\text{Warping constant}\right) \end{aligned}
S.40 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 = x_o^2 + y_o^2 + r_x^2 + r_y^2 with y_o = 0; C_w = 0 for an HSS)

S16 Cl. 13.3.1.2, L_z = L_x

Fez=1.974×106 MPa⋅CwLx2+G⋅JA0⋅r02=1.974×106 MPa⋅2.11×1011 mm6(3000 mm)2+77000 MPa⋅241000 mm46250 mm2⋅(116.9 mm)2=759.6 MPa\begin{aligned} \htmlClass{sym-F_ez}{F_{\mathrm{ez}}} &= \frac{\frac{1.974 \times 10^{6}\ \mathrm{MPa} \cdot \htmlClass{sym-C_w}{C_{w}}}{\htmlClass{sym-L_x}{L_{x}}^{2}} + \htmlClass{sym-G}{G} \cdot \htmlClass{sym-J}{J}}{\htmlClass{sym-A_0}{A_{0}} \cdot \htmlClass{sym-r_0}{r_{0}}^{2}} \\ &= \frac{\frac{1.974 \times 10^{6}\ \mathrm{MPa} \cdot \htmlClass{sym-C_w}{2.11 \times 10^{11}\ \mathrm{mm}^{6}}}{\left(\htmlClass{sym-L_x}{3000\ \mathrm{mm}}\right)^{2}} + \htmlClass{sym-G}{77000\ \mathrm{MPa}} \cdot \htmlClass{sym-J}{241000\ \mathrm{mm}^{4}}}{\htmlClass{sym-A_0}{6250\ \mathrm{mm}^{2}} \cdot \left(\htmlClass{sym-r_0}{116.9\ \mathrm{mm}}\right)^{2}} \\ &= 759.6\ \mathrm{MPa} \end{aligned}
S.41 Elastic buckling stress

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

S16 Cl. 13.3.1.1 (a), the least of F_ex, F_ey and F_ez

Fe=min⁡(Fex,Fey,Fez)=min⁡(2464 MPa,2124 MPa,759.6 MPa)=759.6 MPa\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}{2464\ \mathrm{MPa}}, \htmlClass{sym-F_ey}{2124\ \mathrm{MPa}}, \htmlClass{sym-F_ez}{759.6\ \mathrm{MPa}}\right) \\ &= 759.6\ \mathrm{MPa} \end{aligned}
S.42 Elastic buckling stress, weak axis or torsional

Compared with CSA S16-19 Cl. 13.8.2 c) i), C_r based on weak-axis or torsional-flexural buckling

S16 Cl. 13.8.2 (c) (i), weak-axis or torsional buckling

Fecy=min⁡(Fey,Fez)=min⁡(2124 MPa,759.6 MPa)=759.6 MPa\begin{aligned} \htmlClass{sym-F_ecy}{F_{\mathrm{ecy}}} &= \min\left(\htmlClass{sym-F_ey}{F_{\mathrm{ey}}}, \htmlClass{sym-F_ez}{F_{\mathrm{ez}}}\right) \\ &= \min\left(\htmlClass{sym-F_ey}{2124\ \mathrm{MPa}}, \htmlClass{sym-F_ez}{759.6\ \mathrm{MPa}}\right) \\ &= 759.6\ \mathrm{MPa} \end{aligned}
S.43 Slenderness parameter, weak axis or torsional

Compared with CSA S16-19 Cl. 13.3.1.1, lambda = sqrt(F_y/F_e), on the F_e of Cl. 13.8.2 c) i)

λy=FyFecy=345 MPa759.6 MPa=0.6739\begin{aligned} \htmlClass{sym-lambda_y}{\lambda_{y}} &= \sqrt{\frac{\htmlClass{sym-F_y}{F_{y}}}{\htmlClass{sym-F_ecy}{F_{\mathrm{ecy}}}}} \\ &= \sqrt{\frac{\htmlClass{sym-F_y}{345\ \mathrm{MPa}}}{\htmlClass{sym-F_ecy}{759.6\ \mathrm{MPa}}}} \\ &= 0.6739 \end{aligned}
S.44 Slenderness parameter

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

λc=FyFe=345 MPa759.6 MPa=0.6739\begin{aligned} \htmlClass{sym-lambda_c}{\lambda_{c}} &= \sqrt{\frac{\htmlClass{sym-F_y}{F_{y}}}{\htmlClass{sym-F_e}{F_{e}}}} \\ &= \sqrt{\frac{\htmlClass{sym-F_y}{345\ \mathrm{MPa}}}{\htmlClass{sym-F_e}{759.6\ \mathrm{MPa}}}} \\ &= 0.6739 \end{aligned}
S.45 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 for hot-rolled sections and Class C or A500 HSS, written as F_y over the bracket

S16 Cl. 13.3.1, n = 1.34

fs=Fy⋅(1+λc2⋅1.34)−11.34=345 MPa⋅(1+0.67392⋅1.34)−11.34=276.2 MPa\begin{aligned} \htmlClass{sym-f_s}{f_{s}} &= \htmlClass{sym-F_y}{F_{y}} \cdot \left(1 + \htmlClass{sym-lambda_c}{\lambda_{c}}^{2 \cdot 1.34}\right)^{\frac{-1}{1.34}} \\ &= \htmlClass{sym-F_y}{345\ \mathrm{MPa}} \cdot \left(1 + \htmlClass{sym-lambda_c}{0.6739}^{2 \cdot 1.34}\right)^{\frac{-1}{1.34}} \\ &= 276.2\ \mathrm{MPa} \end{aligned}
S.46 Compressive resistance

Compared with CSA S16-19 Cl. 13.3.4 a) (C_r = phi A_e F_y / (1 + lambda^(2n))^(1/n) with n = 1.34, on the effective area)

S16 Cl. 13.3.4 a), phi A_e f_s

Cr=ϕ⋅Ae⋅fs=0.9⋅6250 mm2⋅276.2 MPa=1554 kN\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}{6250\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-f_s}{276.2\ \mathrm{MPa}} \\ &= 1554\ \mathrm{kN} \end{aligned}
S.47 Compressive stress, weak axis or torsional

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

fsy=Fy⋅(1+λy2⋅1.34)−11.34=345 MPa⋅(1+0.67392⋅1.34)−11.34=276.2 MPa\begin{aligned} \htmlClass{sym-f_sy}{f_{\mathrm{sy}}} &= \htmlClass{sym-F_y}{F_{y}} \cdot \left(1 + \htmlClass{sym-lambda_y}{\lambda_{y}}^{2 \cdot 1.34}\right)^{\frac{-1}{1.34}} \\ &= \htmlClass{sym-F_y}{345\ \mathrm{MPa}} \cdot \left(1 + \htmlClass{sym-lambda_y}{0.6739}^{2 \cdot 1.34}\right)^{\frac{-1}{1.34}} \\ &= 276.2\ \mathrm{MPa} \end{aligned}
S.48 Compressive resistance, weak axis or torsional

Accepted by the author: CSA S16-19 Cl. 13.8.2 c) i) and Cl. 13.3.1.2 Note 2; a closed section's torsional stress G J / (A r_0^2) is far above F_ey, so weak-axis flexure governs an HSS

Cry=ϕ⋅Ae⋅fsy=0.9⋅6250 mm2⋅276.2 MPa=1554 kN\begin{aligned} \htmlClass{sym-C_ry}{C_{\mathrm{ry}}} &= \htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-A_e}{A_{e}} \cdot \htmlClass{sym-f_sy}{f_{\mathrm{sy}}} \\ &= \htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-A_e}{6250\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-f_sy}{276.2\ \mathrm{MPa}} \\ &= 1554\ \mathrm{kN} \end{aligned}
S.49 Compressive resistance, zero length

Compared with CSA S16-19 Cl. 13.8.2 a) i) (C_r with lambda = 0, that is phi A_e F_y, on the effective area)

S16 Cl. 13.8.2 (a) (i), lambda = 0, on the effective area

Cr0=ϕ⋅Ae⋅Fy=0.9⋅6250 mm2⋅345 MPa=1941 kN\begin{aligned} \htmlClass{sym-C_r0}{C_{r0}} &= \htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-A_e}{A_{e}} \cdot \htmlClass{sym-F_y}{F_{y}} \\ &= \htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-A_e}{6250\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-F_y}{345\ \mathrm{MPa}} \\ &= 1941\ \mathrm{kN} \end{aligned}

Branch: Mfy≤0 kN⋅m and Mfx>0 kN⋅mM_{\mathrm{fy}} \leq 0\ \mathrm{kN} \cdot \mathrm{m}\ \text{and}\ M_{\mathrm{fx}} > 0\ \mathrm{kN} \cdot \mathrm{m} held

S.50 Slenderness parameter, K = 1

Compared with CSA S16-19 Cl. 13.8.2 b) i) (for uniaxial bending C_r is based on the axis of bending with K = 1, so F_ex with K = 1 about x)

S16 Cl. 13.8.2 (b) (i), uniaxial bending about x

λb=FyFex1=345 MPa2464 MPa=0.3742\begin{aligned} \htmlClass{sym-lambda_b}{\lambda_{b}} &= \sqrt{\frac{\htmlClass{sym-F_y}{F_{y}}}{\htmlClass{sym-F_ex1}{F_{\mathrm{ex}1}}}} \\ &= \sqrt{\frac{\htmlClass{sym-F_y}{345\ \mathrm{MPa}}}{\htmlClass{sym-F_ex1}{2464\ \mathrm{MPa}}}} \\ &= 0.3742 \end{aligned}
S.51 Compressive stress, K = 1

Compared with CSA S16-19 Cl. 13.3.1.1, C_r = phi A F_y/(1 + lambda^2n)^(1/n), n = 1.34, written as F_y over the bracket

S16 Cl. 13.3.1, n = 1.34

fsb=Fy⋅(1+λb2⋅1.34)−11.34=345 MPa⋅(1+0.37422⋅1.34)−11.34=327.6 MPa\begin{aligned} \htmlClass{sym-f_sb}{f_{\mathrm{sb}}} &= \htmlClass{sym-F_y}{F_{y}} \cdot \left(1 + \htmlClass{sym-lambda_b}{\lambda_{b}}^{2 \cdot 1.34}\right)^{\frac{-1}{1.34}} \\ &= \htmlClass{sym-F_y}{345\ \mathrm{MPa}} \cdot \left(1 + \htmlClass{sym-lambda_b}{0.3742}^{2 \cdot 1.34}\right)^{\frac{-1}{1.34}} \\ &= 327.6\ \mathrm{MPa} \end{aligned}
S.52 Compressive resistance, member check

Compared with CSA S16-19 Cl. 13.8.2 b) i) with Cl. 13.3.1.1 (C_r at K = 1 on the effective area, slenderness from the axis of bending)

Crb=ϕ⋅Ae⋅fsb=0.9⋅6250 mm2⋅327.6 MPa=1843 kN\begin{aligned} \htmlClass{sym-C_rb}{C_{\mathrm{rb}}} &= \htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-A_e}{A_{e}} \cdot \htmlClass{sym-f_sb}{f_{\mathrm{sb}}} \\ &= \htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-A_e}{6250\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-f_sb}{327.6\ \mathrm{MPa}} \\ &= 1843\ \mathrm{kN} \end{aligned}

Branch: sectionclass≤2\mathrm{section}_{\mathrm{class}} \leq 2 did not hold

S.53 Section modulus used, about x

Accepted by the author: tabulated section modulus S_x of the designated section from the CISC tables, the class 3 modulus that Cl. 13.5 b) takes

S16 Cl. 13.5 (b), S for class 3

Zex=572000 mm3(Elastic section modulus about axis XX)\begin{aligned} \htmlClass{sym-Z_ex}{Z_{\mathrm{ex}}} &= 572000\ \mathrm{mm}^{3} \quad \left(\text{Elastic section modulus about axis XX}\right) \end{aligned}
S.54 Section modulus used, about y

Accepted by the author: tabulated section modulus S_y of the designated section from the CISC tables, the class 3 modulus that Cl. 13.5 b) takes about y

S16 Cl. 13.5 (b), S for class 3

Zey=150000 mm3(Elastic section modulus about axis YY)\begin{aligned} \htmlClass{sym-Z_ey}{Z_{\mathrm{ey}}} &= 150000\ \mathrm{mm}^{3} \quad \left(\text{Elastic section modulus about axis YY}\right) \end{aligned}
S.55 Moment resistance about x

Compared with CSA S16-19 Cl. 13.5 a) and b) (M_r = phi Z F_y Class 1 and 2, phi S F_y Class 3)

S16 Cl. 13.5

Mrx=ϕ⋅Zex⋅Fy=0.9⋅572000 mm3⋅345 MPa=177.6 kN⋅m\begin{aligned} \htmlClass{sym-M_rx}{M_{\mathrm{rx}}} &= \htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-Z_ex}{Z_{\mathrm{ex}}} \cdot \htmlClass{sym-F_y}{F_{y}} \\ &= \htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-Z_ex}{572000\ \mathrm{mm}^{3}} \cdot \htmlClass{sym-F_y}{345\ \mathrm{MPa}} \\ &= 177.6\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}
S.56 Moment resistance about y

Compared with CSA S16-19 Cl. 13.5 a) and b) (M_r = phi Z F_y Class 1 and 2, phi S F_y Class 3)

S16 Cl. 13.5

Mry=ϕ⋅Zey⋅Fy=0.9⋅150000 mm3⋅345 MPa=46.58 kN⋅m\begin{aligned} \htmlClass{sym-M_ry}{M_{\mathrm{ry}}} &= \htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-Z_ey}{Z_{\mathrm{ey}}} \cdot \htmlClass{sym-F_y}{F_{y}} \\ &= \htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-Z_ey}{150000\ \mathrm{mm}^{3}} \cdot \htmlClass{sym-F_y}{345\ \mathrm{MPa}} \\ &= 46.58\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}
S.57 Moment of inertia 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

Iy=1.51×107 mm4(Moment of inertia about axis YY)\begin{aligned} \htmlClass{sym-I_y}{I_{y}} &= 1.51 \times 10^{7}\ \mathrm{mm}^{4} \quad \left(\text{Moment of inertia about axis YY}\right) \end{aligned}
S.58 Critical elastic moment

Compared with CSA S16-19 Cl. 13.6.1 a) (M_u = omega_2 pi/L sqrt(E I_y G J + (pi E/L)^2 I_y C_w))

S16 Cl. 13.6

Mu=ω2⋅πLy⋅E⋅Iy⋅G⋅J+(628319 MPaLy)2⋅Iy⋅Cw=1.2⋅π1500 mm⋅200000 MPa⋅1.51×107 mm4⋅77000 MPa⋅241000 mm4+(628319 MPa1500 mm)2⋅1.51×107 mm4⋅2.11×1011 mm6=1971 kN⋅m\begin{aligned} \htmlClass{sym-M_u}{M_{u}} &= \frac{\htmlClass{sym-omega_2}{\omega_{2}} \cdot \pi}{\htmlClass{sym-L_y}{L_{y}}} \cdot \sqrt{\htmlClass{sym-E}{E} \cdot \htmlClass{sym-I_y}{I_{y}} \cdot \htmlClass{sym-G}{G} \cdot \htmlClass{sym-J}{J} + \left(\frac{628319\ \mathrm{MPa}}{\htmlClass{sym-L_y}{L_{y}}}\right)^{2} \cdot \htmlClass{sym-I_y}{I_{y}} \cdot \htmlClass{sym-C_w}{C_{w}}} \\ &= \frac{\htmlClass{sym-omega_2}{1.2} \cdot \pi}{\htmlClass{sym-L_y}{1500\ \mathrm{mm}}} \cdot \sqrt{\htmlClass{sym-E}{200000\ \mathrm{MPa}} \cdot \htmlClass{sym-I_y}{1.51 \times 10^{7}\ \mathrm{mm}^{4}} \cdot \htmlClass{sym-G}{77000\ \mathrm{MPa}} \cdot \htmlClass{sym-J}{241000\ \mathrm{mm}^{4}} + \left(\frac{628319\ \mathrm{MPa}}{\htmlClass{sym-L_y}{1500\ \mathrm{mm}}}\right)^{2} \cdot \htmlClass{sym-I_y}{1.51 \times 10^{7}\ \mathrm{mm}^{4}} \cdot \htmlClass{sym-C_w}{2.11 \times 10^{11}\ \mathrm{mm}^{6}}} \\ &= 1971\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}
S.59 Plastic or yield moment

Compared with CSA S16-19 Cl. 13.5 a), M_p = Z F_y (Z_ex is S_x for Class 3, so M_y as Cl. 13.6.1 b) uses)

Mp=Zex⋅Fy=572000 mm3⋅345 MPa=197.3 kN⋅m\begin{aligned} \htmlClass{sym-M_p}{M_{p}} &= \htmlClass{sym-Z_ex}{Z_{\mathrm{ex}}} \cdot \htmlClass{sym-F_y}{F_{y}} \\ &= \htmlClass{sym-Z_ex}{572000\ \mathrm{mm}^{3}} \cdot \htmlClass{sym-F_y}{345\ \mathrm{MPa}} \\ &= 197.3\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}

Branch: Mu>0.67⋅MpM_{u} > 0.67 \cdot M_{p} held

S.60 Lateral-torsional buckling resistance

Compared with CSA S16-19 Cl. 13.6.1 a) i) and b) i) (M_r = 1.15 phi M_p (1 - 0.28 M_p / M_u), not above phi M_p; M_p = Z F_y, or S F_y for a Class 3 section)

S16 Cl. 13.6.1

Mrxl=min⁡(1.15⋅ϕ⋅Mp⋅(1−0.28⋅MpMu),ϕ⋅Mp)=min⁡(1.15⋅0.9⋅197.3 kN⋅m⋅(1−0.28⋅197.3 kN⋅m1971 kN⋅m),0.9⋅197.3 kN⋅m)=177.6 kN⋅m\begin{aligned} \htmlClass{sym-M_rxl}{M_{\mathrm{rxl}}} &= \min\left(1.15 \cdot \htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-M_p}{M_{p}} \cdot \left(1 - \frac{0.28 \cdot \htmlClass{sym-M_p}{M_{p}}}{\htmlClass{sym-M_u}{M_{u}}}\right), \htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-M_p}{M_{p}}\right) \\ &= \min\left(1.15 \cdot \htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-M_p}{197.3\ \mathrm{kN} \cdot \mathrm{m}} \cdot \left(1 - \frac{0.28 \cdot \htmlClass{sym-M_p}{197.3\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_u}{1971\ \mathrm{kN} \cdot \mathrm{m}}}\right), \htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-M_p}{197.3\ \mathrm{kN} \cdot \mathrm{m}}\right) \\ &= 177.6\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}

Branch: sectionclass≤2\mathrm{section}_{\mathrm{class}} \leq 2 did not hold

S.61 Coefficient on M_fx

Compared with CSA S16-19 Cl. 13.8.4, coefficient 1.0 on M_fx for the Class 3 default (0.85 for Class 1/2 I-shapes per Cl. 13.8.2)

U0x=1(Class 3 | S16 Cl. 13.8.4)\begin{aligned} \htmlClass{sym-U_0x}{U_{0x}} &= 1 \quad \left(\text{Class 3 | S16 Cl. 13.8.4}\right) \end{aligned}
S.62 Coefficient on M_fy, cross-section

Compared with CSA S16-19 Cl. 13.8.4, coefficient 1.0 on M_fy for the Class 3 default (beta = 0.6 for Class 1/2 I-shapes per Cl. 13.8.2 a))

βa=1(Class 3 | S16 Cl. 13.8.4)\begin{aligned} \htmlClass{sym-beta_a}{\beta_{a}} &= 1 \quad \left(\text{Class 3 | S16 Cl. 13.8.4}\right) \end{aligned}
S.63 Coefficient on M_fy, member

Compared with CSA S16-19 Cl. 13.8.4, coefficient 1.0 on M_fy for the Class 3 default (beta = 0.6 + 0.4 lambda_y <= 0.85 for Class 1/2 I-shapes per Cl. 13.8.2)

βb=1(Class 3 | S16 Cl. 13.8.4)\begin{aligned} \htmlClass{sym-beta_b}{\beta_{b}} &= 1 \quad \left(\text{Class 3 | S16 Cl. 13.8.4}\right) \end{aligned}
S.64 Moment of inertia 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

Ix=7.06×107 mm4(Moment of inertia about axis XX)\begin{aligned} \htmlClass{sym-I_x}{I_{x}} &= 7.06 \times 10^{7}\ \mathrm{mm}^{4} \quad \left(\text{Moment of inertia about axis XX}\right) \end{aligned}
S.65 Euler load about x

Compared with CSA S16-19 Cl. 13.8.5, C_e = pi^2 E I/L^2

Cex=1.974×106 MPa⋅IxLx2=1.974×106 MPa⋅7.06×107 mm4(3000 mm)2=15484 kN\begin{aligned} \htmlClass{sym-C_ex}{C_{\mathrm{ex}}} &= \frac{1.974 \times 10^{6}\ \mathrm{MPa} \cdot \htmlClass{sym-I_x}{I_{x}}}{\htmlClass{sym-L_x}{L_{x}}^{2}} \\ &= \frac{1.974 \times 10^{6}\ \mathrm{MPa} \cdot \htmlClass{sym-I_x}{7.06 \times 10^{7}\ \mathrm{mm}^{4}}}{\left(\htmlClass{sym-L_x}{3000\ \mathrm{mm}}\right)^{2}} \\ &= 15484\ \mathrm{kN} \end{aligned}
S.66 Euler load about y

Compared with CSA S16-19 Cl. 13.8.5, C_e = pi^2 E I/L^2

Cey=1.974×106 MPa⋅IyLy2=1.974×106 MPa⋅1.51×107 mm4(1500 mm)2=13247 kN\begin{aligned} \htmlClass{sym-C_ey}{C_{\mathrm{ey}}} &= \frac{1.974 \times 10^{6}\ \mathrm{MPa} \cdot \htmlClass{sym-I_y}{I_{y}}}{\htmlClass{sym-L_y}{L_{y}}^{2}} \\ &= \frac{1.974 \times 10^{6}\ \mathrm{MPa} \cdot \htmlClass{sym-I_y}{1.51 \times 10^{7}\ \mathrm{mm}^{4}}}{\left(\htmlClass{sym-L_y}{1500\ \mathrm{mm}}\right)^{2}} \\ &= 13247\ \mathrm{kN} \end{aligned}
S.67 Axes in a braced frame

Accepted by the author: definition, a count of the axes the member is braced about, the input to the braced-frame conditions of CSA S16-19 Cl. 13.8.1 and 13.8.2 a)

bracedaxes=1(Axes in a braced frame, of x and y)\begin{aligned} \htmlClass{sym-braced_axes}{\mathrm{braced}_{\mathrm{axes}}} &= 1 \quad \left(\text{Axes in a braced frame, of x and y}\right) \end{aligned}

Branch: Cf<CexC_{f} < C_{\mathrm{ex}} held

Branch: Cf<CeyC_{f} < C_{\mathrm{ey}} held

S.68 Amplification about x

Compared with CSA S16-19 Cl. 13.8.5, U_1 = omega_1/(1 - C_f/C_e), omega_1 per Cl. 13.8.6

S16 Cl. 13.8.5

U1x=ω1x1−CfCex=0.851−60 kN15484 kN=0.8533\begin{aligned} \htmlClass{sym-U_1x}{U_{1x}} &= \frac{\htmlClass{sym-omega_1x}{\omega_{1x}}}{1 - \frac{\htmlClass{sym-C_f}{C_{f}}}{\htmlClass{sym-C_ex}{C_{\mathrm{ex}}}}} \\ &= \frac{\htmlClass{sym-omega_1x}{0.85}}{1 - \frac{\htmlClass{sym-C_f}{60\ \mathrm{kN}}}{\htmlClass{sym-C_ex}{15484\ \mathrm{kN}}}} \\ &= 0.8533 \end{aligned}
S.69 Amplification about y

Compared with CSA S16-19 Cl. 13.8.5, U_1 = omega_1/(1 - C_f/C_e), omega_1 per Cl. 13.8.6

S16 Cl. 13.8.5

U1y=ω1y1−CfCey=11−60 kN13247 kN=1.005\begin{aligned} \htmlClass{sym-U_1y}{U_{1y}} &= \frac{\htmlClass{sym-omega_1y}{\omega_{1y}}}{1 - \frac{\htmlClass{sym-C_f}{C_{f}}}{\htmlClass{sym-C_ey}{C_{\mathrm{ey}}}}} \\ &= \frac{\htmlClass{sym-omega_1y}{1}}{1 - \frac{\htmlClass{sym-C_f}{60\ \mathrm{kN}}}{\htmlClass{sym-C_ey}{13247\ \mathrm{kN}}}} \\ &= 1.005 \end{aligned}
S.70 Cross-section strength

Compared with CSA S16-19 Cl. 13.8.2 a), C_f/C_r + 0.85 U_1x M_fx/M_rx + beta U_1y M_fy/M_ry with lambda = 0, beta = 0.6 and U_1 not less than 1.0; 13.8.4 a) with coefficients 1.0

S16 Cl. 13.8.2 (a)

Ia=CfCr0+U0x⋅max⁡(U1x,1)⋅MfxMrx+βa⋅max⁡(U1y,1)⋅MfyMry=60 kN1941 kN+1⋅max⁡(0.8533,1)⋅30 kN⋅m177.6 kN⋅m+1⋅max⁡(1.005,1)⋅0 kN⋅m46.58 kN⋅m=0.1998\begin{aligned} \htmlClass{sym-I_a}{I_{a}} &= \frac{\htmlClass{sym-C_f}{C_{f}}}{\htmlClass{sym-C_r0}{C_{r0}}} + \frac{\htmlClass{sym-U_0x}{U_{0x}} \cdot \max\left(\htmlClass{sym-U_1x}{U_{1x}}, 1\right) \cdot \htmlClass{sym-M_fx}{M_{\mathrm{fx}}}}{\htmlClass{sym-M_rx}{M_{\mathrm{rx}}}} + \frac{\htmlClass{sym-beta_a}{\beta_{a}} \cdot \max\left(\htmlClass{sym-U_1y}{U_{1y}}, 1\right) \cdot \htmlClass{sym-M_fy}{M_{\mathrm{fy}}}}{\htmlClass{sym-M_ry}{M_{\mathrm{ry}}}} \\ &= \frac{\htmlClass{sym-C_f}{60\ \mathrm{kN}}}{\htmlClass{sym-C_r0}{1941\ \mathrm{kN}}} + \frac{\htmlClass{sym-U_0x}{1} \cdot \max\left(\htmlClass{sym-U_1x}{0.8533}, 1\right) \cdot \htmlClass{sym-M_fx}{30\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_rx}{177.6\ \mathrm{kN} \cdot \mathrm{m}}} + \frac{\htmlClass{sym-beta_a}{1} \cdot \max\left(\htmlClass{sym-U_1y}{1.005}, 1\right) \cdot \htmlClass{sym-M_fy}{0\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_ry}{46.58\ \mathrm{kN} \cdot \mathrm{m}}} \\ &= 0.1998 \end{aligned}
S.71 Amplification about x, member check

Compared with CSA S16-19 Cl. 13.8.2 b) iii), U_1 = 1.0 in an unbraced frame

U1xb=1(Unbraced frame about x | S16 Cl. 13.8.2 (b))\begin{aligned} \htmlClass{sym-U_1xb}{U_{1\mathrm{xb}}} &= 1 \quad \left(\text{Unbraced frame about x | S16 Cl. 13.8.2 (b)}\right) \end{aligned}
S.72 Amplification about y, member check

Compared with CSA S16-19 Cl. 13.8.5 (U_1 = omega_1 / (1 - C_f / C_e), with C_e = pi^2 E I / L^2)

S16 Cl. 13.8.5

U1yb=ω1y1−CfCey=11−60 kN13247 kN=1.005\begin{aligned} \htmlClass{sym-U_1yb}{U_{1\mathrm{yb}}} &= \frac{\htmlClass{sym-omega_1y}{\omega_{1y}}}{1 - \frac{\htmlClass{sym-C_f}{C_{f}}}{\htmlClass{sym-C_ey}{C_{\mathrm{ey}}}}} \\ &= \frac{\htmlClass{sym-omega_1y}{1}}{1 - \frac{\htmlClass{sym-C_f}{60\ \mathrm{kN}}}{\htmlClass{sym-C_ey}{13247\ \mathrm{kN}}}} \\ &= 1.005 \end{aligned}
S.73 Overall member strength

Compared with CSA S16-19 Cl. 13.8.2 b) (overall member strength: U_1 = 1.0 in an unbraced frame, Cl. 13.8.5 in a braced frame; beta = 0.6 + 0.4 lambda_y <= 0.85)

S16 Cl. 13.8.2 (b)

Ib=CfCrb+U0x⋅U1xb⋅MfxMrx+βb⋅U1yb⋅MfyMry=60 kN1843 kN+1⋅1⋅30 kN⋅m177.6 kN⋅m+1⋅1.005⋅0 kN⋅m46.58 kN⋅m=0.2015\begin{aligned} \htmlClass{sym-I_b}{I_{b}} &= \frac{\htmlClass{sym-C_f}{C_{f}}}{\htmlClass{sym-C_rb}{C_{\mathrm{rb}}}} + \frac{\htmlClass{sym-U_0x}{U_{0x}} \cdot \htmlClass{sym-U_1xb}{U_{1\mathrm{xb}}} \cdot \htmlClass{sym-M_fx}{M_{\mathrm{fx}}}}{\htmlClass{sym-M_rx}{M_{\mathrm{rx}}}} + \frac{\htmlClass{sym-beta_b}{\beta_{b}} \cdot \htmlClass{sym-U_1yb}{U_{1\mathrm{yb}}} \cdot \htmlClass{sym-M_fy}{M_{\mathrm{fy}}}}{\htmlClass{sym-M_ry}{M_{\mathrm{ry}}}} \\ &= \frac{\htmlClass{sym-C_f}{60\ \mathrm{kN}}}{\htmlClass{sym-C_rb}{1843\ \mathrm{kN}}} + \frac{\htmlClass{sym-U_0x}{1} \cdot \htmlClass{sym-U_1xb}{1} \cdot \htmlClass{sym-M_fx}{30\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_rx}{177.6\ \mathrm{kN} \cdot \mathrm{m}}} + \frac{\htmlClass{sym-beta_b}{1} \cdot \htmlClass{sym-U_1yb}{1.005} \cdot \htmlClass{sym-M_fy}{0\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_ry}{46.58\ \mathrm{kN} \cdot \mathrm{m}}} \\ &= 0.2015 \end{aligned}
S.74 Lateral-torsional buckling strength

Compared with CSA S16-19 Cl. 13.8.2 c) (lateral-torsional buckling: weak-axis C_r, M_rx per Cl. 13.6.1, M_ry per Cl. 13.5, U_1x not below 1.0 in a braced frame)

S16 Cl. 13.8.2 (c)

Ic=CfCry+U0x⋅max⁡(U1xb,1)⋅MfxMrxl+βb⋅U1yb⋅MfyMry=60 kN1554 kN+1⋅max⁡(1,1)⋅30 kN⋅m177.6 kN⋅m+1⋅1.005⋅0 kN⋅m46.58 kN⋅m=0.2075\begin{aligned} \htmlClass{sym-I_c}{I_{c}} &= \frac{\htmlClass{sym-C_f}{C_{f}}}{\htmlClass{sym-C_ry}{C_{\mathrm{ry}}}} + \frac{\htmlClass{sym-U_0x}{U_{0x}} \cdot \max\left(\htmlClass{sym-U_1xb}{U_{1\mathrm{xb}}}, 1\right) \cdot \htmlClass{sym-M_fx}{M_{\mathrm{fx}}}}{\htmlClass{sym-M_rxl}{M_{\mathrm{rxl}}}} + \frac{\htmlClass{sym-beta_b}{\beta_{b}} \cdot \htmlClass{sym-U_1yb}{U_{1\mathrm{yb}}} \cdot \htmlClass{sym-M_fy}{M_{\mathrm{fy}}}}{\htmlClass{sym-M_ry}{M_{\mathrm{ry}}}} \\ &= \frac{\htmlClass{sym-C_f}{60\ \mathrm{kN}}}{\htmlClass{sym-C_ry}{1554\ \mathrm{kN}}} + \frac{\htmlClass{sym-U_0x}{1} \cdot \max\left(\htmlClass{sym-U_1xb}{1}, 1\right) \cdot \htmlClass{sym-M_fx}{30\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_rxl}{177.6\ \mathrm{kN} \cdot \mathrm{m}}} + \frac{\htmlClass{sym-beta_b}{1} \cdot \htmlClass{sym-U_1yb}{1.005} \cdot \htmlClass{sym-M_fy}{0\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_ry}{46.58\ \mathrm{kN} \cdot \mathrm{m}}} \\ &= 0.2075 \end{aligned}

Branch: sectionclass≤2\mathrm{section}_{\mathrm{class}} \leq 2 did not hold

S.75 Moment-only capacity

Compared with CSA S16-19 Cl. 13.8.4 d), M_fx/M_rx + M_fy/M_ry <= 1.0 with M_r of Cl. 13.5

S16 Cl. 13.8.4 (d), class 3 or not an I-shape

Id=MfxMrx+MfyMry=30 kN⋅m177.6 kN⋅m+0 kN⋅m46.58 kN⋅m=0.1689\begin{aligned} \htmlClass{sym-I_d}{I_{d}} &= \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-M_fx}{30\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_rx}{177.6\ \mathrm{kN} \cdot \mathrm{m}}} + \frac{\htmlClass{sym-M_fy}{0\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_ry}{46.58\ \mathrm{kN} \cdot \mathrm{m}}} \\ &= 0.1689 \end{aligned}
S.76 Moment-only capacity with lateral-torsional buckling

Compared with CSA S16-19 Cl. 13.8.4 d), M_fx/M_rx + M_fy/M_ry <= 1.0 with M_rx of Cl. 13.6.1 where the member is laterally unsupported

S16 Cl. 13.8.4 (d), class 3 or not an I-shape

Idl=MfxMrxl+MfyMry=30 kN⋅m177.6 kN⋅m+0 kN⋅m46.58 kN⋅m=0.1689\begin{aligned} \htmlClass{sym-I_dl}{I_{\mathrm{dl}}} &= \frac{\htmlClass{sym-M_fx}{M_{\mathrm{fx}}}}{\htmlClass{sym-M_rxl}{M_{\mathrm{rxl}}}} + \frac{\htmlClass{sym-M_fy}{M_{\mathrm{fy}}}}{\htmlClass{sym-M_ry}{M_{\mathrm{ry}}}} \\ &= \frac{\htmlClass{sym-M_fx}{30\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_rxl}{177.6\ \mathrm{kN} \cdot \mathrm{m}}} + \frac{\htmlClass{sym-M_fy}{0\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_ry}{46.58\ \mathrm{kN} \cdot \mathrm{m}}} \\ &= 0.1689 \end{aligned}

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Questions

What do omega_1x, omega_1y and omega_2 mean in CSA S16 Cl. 13.8?

omega_1 about each axis is S16's equivalent uniform moment factor for the amplification, 0.6 - 0.4 kappa and not below 0.4; omega_2 is the lateral-torsional buckling moment gradient factor of Cl. 13.6.

How is an HSS classified under S16 Table 2?

By S16 Table 2, with the walls taken on (b - 4t)/t.

Revision history

After an independent checker's review: a class 4 section now takes the effective area in compression (Cl. 13.3.4 a), a channel is capped at class 3 with the elastic modulus, K below 1.0 is refused in an unbraced frame, an HSS drops torsional buckling, and weak-axis column checking holds when no moment acts
ValueWasNowUnit
lambda_3w36.0716
A_d0mm2\mathrm{mm}^{2}
A_e6250mm2\mathrm{mm}^{2}
F_eb2464.33MPa\mathrm{MPa}
M_ref197.34kN⋅m\mathrm{kN} \cdot \mathrm{m}
fix(units,examples): zero keeps its unit, zero cope, shear tab rules
ValueWasNowUnit
x_00 \mathrm{m}0mm\mathrm{mm}

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