Tension and bending

A W shape, HSS or channel under axial tension and strong-axis bending, CSA S16 Cl. 13.9. Check a W, WWF, S or HP shape, a rectangular HSS or a channel under a factored tension and a moment about its strong axis. The section is classified by S16 Table 2; the tension and moment are combined by Cl. 13.9 with the laterally supported moment resistance, and again, less the tension's relief, with the lateral-torsional buckling resistance over the unbraced length.

Derivation, step by step

SI Imperial

First report free, then a free sign-in

PASS Governing: Web class, D/C 0.30, S16 Cl. 11.2 Table 2, class 3 web limit5 of 5 checks below pass

Given

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

Loads

T_{f}

Factored tension

\mathrm{kN}

0-50,000

M_{f}

Factored moment about x

\mathrm{kN} \cdot \mathrm{m}

0-10,000

\omega_{2}

\omega_{2}

, moment gradient 1-2.5

Section

F_{y}

Yield strength

\mathrm{MPa}

200-700

Bracing

L_{x}

Unbraced length about x

\mathrm{mm}

100-30,000

L_{y}

Unbraced length about y, and laterally

\mathrm{mm}

100-30,000

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/tension-bending.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.

Slenderness

Check	D/C	Utilisation	Result
Slenderness, x axisS16 Cl. 10.4.2.2 $\htmlClass{sym-lambda_x}{\lambda_{x}} \leq 300 \quad \Rightarrow \quad \htmlClass{sym-lambda_x}{28.3} \leq 300$	0.09		PASS
Slenderness, y axisS16 Cl. 10.4.2.2 $\htmlClass{sym-lambda_y}{\lambda_{y}} \leq 300 \quad \Rightarrow \quad \htmlClass{sym-lambda_y}{60.98} \leq 300$	0.20		PASS

Section class

Check	D/C	Utilisation	Result
Web classS16 Cl. 11.2 Table 2, class 3 web limit $\htmlClass{sym-lambda_w}{\lambda_{w}} \leq \htmlClass{sym-lambda_w3}{\lambda_{w3}} \quad \Rightarrow \quad \htmlClass{sym-lambda_w}{30.41} \leq \htmlClass{sym-lambda_w3}{101.6}$	0.30		PASS

Strength

Check	D/C	Utilisation	Result
Combined tension and bendingS16 Cl. 13.9.1 $\htmlClass{sym-I_a}{I_{a}} \leq 1 \quad \Rightarrow \quad \htmlClass{sym-I_a}{0.197} \leq 1$	0.20		PASS
Bending with tension reliefS16 Cl. 13.9.3 $\htmlClass{sym-I_b}{I_{b}} \leq 1 \quad \Rightarrow \quad \htmlClass{sym-I_b}{0.136} \leq 1$	0.14		PASS

Results

Quantity	Description	Value	Unit
$\htmlClass{sym-T_r}{T_{r}}$	Tensile resistance, gross section	1969	$\mathrm{kN}$
$\htmlClass{sym-M_rx}{M_{\mathrm{rx}}}$	Moment resistance, laterally supported	180.2	$\mathrm{kN} \cdot \mathrm{m}$
$\htmlClass{sym-M_r}{M_{r}}$	Moment resistance over L_y	180.2	$\mathrm{kN} \cdot \mathrm{m}$

Also worth checking

Bending about the minor axis: Cl. 13.9.1 and 13.9.3 are taken with M_fy = 0
Cl. 13.9.2, the 0.85 M_fx form for class 1 and 2 I-shapes, which also requires M_fx/M_r <= 1 with M_r from Cl. 13.6 and no tension relief; the calc takes Cl. 13.9.1 and 13.9.3 as the governing reading, with 13.9.2 an alternative to them
T_r is the gross section's yield, Cl. 13.2 a) i), only: net section fracture, Cl. 13.2 a) iii) and Cl. 12, and block shear, Cl. 13.2 a) ii) and Cl. 13.11, are not checked
Load applied above the shear centre, Cl. 13.6.1 a): M_u is for load at the shear centre, with omega_2 as entered
Cantilevers, Cl. 13.6.1 d): L_y is taken as the length of a segment between effective brace points
The strength and stiffness of the lateral bracing and the twist restraint at the supports, Cl. 9.3 and 9.4
Torsion of a channel loaded off its shear centre, Cl. 9.9
A load that reverses to compression, which needs Cl. 13.8
Shear, deflection, the connections and local effects at the load points
Class 4 webs, which fail the web class check, and class 4 channel flanges, which fail the flange class check

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. 3.2 (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. 3.2 (G = 77 000 MPa assumed)

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

S.4 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

\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}{350\ \mathrm{MPa}}}{1\ \mathrm{MPa}}} \\ &= 18.71 \end{aligned}

S.5 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; W250x49 A_Th

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

S.6 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; W250x49 Rx

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

S.7 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; W250x49 Ry

\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.8 Slenderness ratio about x

Compared with CSA S16-19 Cl. 10.4.1 (the slenderness ratio of a member in tension is the unbraced length L over the radius of gyration, no K)

S16 Cl. 10.4.1, a tension member takes L/r with no K

\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}{106\ \mathrm{mm}}} \\ &= 28.3 \end{aligned}

S.9 Slenderness ratio about y

Compared with CSA S16-19 Cl. 10.4.1 (the slenderness ratio of a member in tension is the unbraced length L over the radius of gyration, no K)

S16 Cl. 10.4.1, a tension member takes L/r with no K

\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}{49.2\ \mathrm{mm}}} \\ &= 60.98 \end{aligned}

S.10 Tensile resistance

Compared with CSA S16-19 Cl. 13.2 a) i) (T_r = phi A_g F_y, phi = 0.90 per Cl. 13.1 a); A_0 is the gross area)

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}{6250\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-F_y}{350\ \mathrm{MPa}} \\ &= 1969\ \mathrm{kN} \end{aligned}

S.11 Flange 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; W250x49 B

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

S.12 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; W250x49 D

\begin{aligned} \htmlClass{sym-d}{d} &= 247\ \mathrm{mm} \quad \left(\text{Depth of section or height of vertical leg}\right) \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; W250x49 T

\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; W250x49 W

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

S.15 Flange outstand

Compared with CSA S16-19 Cl. 11.3.1 c) (outstand of a beam flange is one-half of the full nominal flange width, b_out = b/2)

\begin{aligned} \htmlClass{sym-b_out}{b_{\mathrm{out}}} &= \frac{\htmlClass{sym-b}{b}}{2} \\ &= \frac{\htmlClass{sym-b}{202\ \mathrm{mm}}}{2} \\ &= 101\ \mathrm{mm} \end{aligned}

S.16 Flange width-to-thickness ratio

Accepted by the author: definition, the flange outstand over its thickness b_out/t that Table 2 limits (b_out is the S16 Cl. 11.3.1 c) half flange, its own row)

\begin{aligned} \htmlClass{sym-lambda_f}{\lambda_{f}} &= \frac{\htmlClass{sym-b_out}{b_{\mathrm{out}}}}{\htmlClass{sym-t}{t}} \\ &= \frac{\htmlClass{sym-b_out}{101\ \mathrm{mm}}}{\htmlClass{sym-t}{11\ \mathrm{mm}}} \\ &= 9.182 \end{aligned}

S.17 Class 1 limit, flange

Compared with CSA S16-19 Table 2 (flanges of I-sections in flexure: b_el/t not over 145/sqrt(F_y) for Class 1; table extraction is garbled but the 145, 170, 200 limits are present)

S16 Table 2, flanges

\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}{18.71}} \\ &= 7.751 \end{aligned}

S.18 Class 2 limit, flange

Compared with CSA S16-19 Table 2 (flanges of I-sections in flexure: b_el/t not over 170/sqrt(F_y) for Class 2; table extraction is garbled but the 145, 170, 200 limits are present)

S16 Table 2, flanges

\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}{18.71}} \\ &= 9.087 \end{aligned}

S.19 Class 3 limit, flange

Compared with CSA S16-19 Table 2 (flanges of I-sections in flexure: b_el/t not over 200/sqrt(F_y) for Class 3; table extraction is garbled but the 145, 170, 200 limits are present)

S16 Table 2, flanges

\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}{18.71}} \\ &= 10.69 \end{aligned}

S.20 Web depth-to-thickness ratio

Compared with CSA S16-19 Cl. 11.3.2 d) (web of a hot-rolled section, h = clear distance between flanges, so h/w = (d - 2t)/w)

\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}

S.21 Class 1 limit, web

Compared with CSA S16-19 Table 2 (webs of I-sections in flexure: h/w not over 1100/sqrt(F_y) for Class 1)

S16 Table 2, webs in flexure

\begin{aligned} \htmlClass{sym-lambda_w1}{\lambda_{w1}} &= \frac{1100}{\htmlClass{sym-root_F_y}{\mathrm{root}_{F,y}}} \\ &= \frac{1100}{\htmlClass{sym-root_F_y}{18.71}} \\ &= 58.8 \end{aligned}

S.22 Class 2 limit, web

Compared with CSA S16-19 Table 2 (webs of I-sections in flexure: h/w not over 1700/sqrt(F_y) for Class 2)

S16 Table 2, webs in flexure

\begin{aligned} \htmlClass{sym-lambda_w2}{\lambda_{w2}} &= \frac{1700}{\htmlClass{sym-root_F_y}{\mathrm{root}_{F,y}}} \\ &= \frac{1700}{\htmlClass{sym-root_F_y}{18.71}} \\ &= 90.87 \end{aligned}

S.23 Class 3 limit, web

Compared with CSA S16-19 Table 2 (webs of I-sections in flexure: h/w not over 1900/sqrt(F_y) for Class 3)

S16 Table 2, webs in flexure

\begin{aligned} \htmlClass{sym-lambda_w3}{\lambda_{w3}} &= \frac{1900}{\htmlClass{sym-root_F_y}{\mathrm{root}_{F,y}}} \\ &= \frac{1900}{\htmlClass{sym-root_F_y}{18.71}} \\ &= 101.6 \end{aligned}

Branch: $\lambda_{f} \leq \lambda_{f1}$ did not hold

Branch: $\lambda_{f} \leq \lambda_{f2}$ did not hold

Branch: $\lambda_{f} \leq \lambda_{f3}$ held

S.24 Flange class

Compared with CSA S16-19 Cl. 11.2 Table 2 (flange of an I-section in major-axis bending is Class 3 where 170/sqrt(F_y) < b_el/t <= 200/sqrt(F_y); W250x49 at 350 MPa 9.18 against 9.09 and 10.69)

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

Branch: $\lambda_{w} \leq \lambda_{w1}$ held

S.25 Web class

Compared with CSA S16-19 Cl. 11.2 Table 2 (I-section web with C_f = 0 is Class 1 where h/w <= 1100/sqrt(F_y); W250x49 at 350 MPa 30.4 against 58.8)

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

S.26 Section class

Accepted by the author: definition, the section takes the less favourable (higher numbered) of the flange and web classes, max(class_f, class_w)

\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}

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

Branch: $\mathrm{class}_{f} \leq 3$ held

S.27 Section modulus used

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

\begin{aligned} \htmlClass{sym-Z_e}{Z_{e}} &= 572000\ \mathrm{mm}^{3} \quad \left(\text{Elastic section modulus about axis XX}\right) \end{aligned}

S.28 Moment at the modulus used, Z F_y, S F_y or S_e F_y

Accepted by the author: Statics: M = Z_e F_y, the fully plastic moment for Z or the first-yield moment for S (stress times modulus)

\begin{aligned} \htmlClass{sym-M_p}{M_{p}} &= \htmlClass{sym-Z_e}{Z_{e}} \cdot \htmlClass{sym-F_y}{F_{y}} \\ &= \htmlClass{sym-Z_e}{572000\ \mathrm{mm}^{3}} \cdot \htmlClass{sym-F_y}{350\ \mathrm{MPa}} \\ &= 200.2\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}

S.29 Moment resistance, laterally supported

Compared with CSA S16-19 Cl. 13.5 a) and b) (M_r = phi Z F_y for Class 1 and 2, phi S F_y for Class 3; Z_e carries the class's modulus)

S16 Cl. 13.5

\begin{aligned} \htmlClass{sym-M_rx}{M_{\mathrm{rx}}} &= \htmlClass{sym-phi}{\phi} \cdot \htmlClass{sym-M_p}{M_{p}} \\ &= \htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-M_p}{200.2\ \mathrm{kN} \cdot \mathrm{m}} \\ &= 180.2\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}

S.30 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; W250x49 Iy

\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.31 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; W250x49 J

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

S.32 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; W250x49 Cw

\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.33 Critical elastic moment

Compared with CSA S16-19 Cl. 13.6.1 a) ii) (M_u = omega_2 pi/L sqrt(E I_y G J + (pi E/L)^2 I_y C_w), L = L_y; pi E = 628.3 GPa)

S16 Cl. 13.6

\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} \cdot \pi}{\htmlClass{sym-L_y}{3000\ \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}{3000\ \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}}} \\ &= 463.4\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}

Branch: $M_{u} > 0.67 \cdot M_{p}$ held

S.34 Moment resistance, laterally unsupported

Compared with CSA S16-19 Cl. 13.6.1 a) i) and b) i) (1.15 phi M (1 - 0.28 M/M_u) not over phi M, M = Z_e F_y: M_p for Class 1 and 2, M_y for Class 3; Class 4 basis differs, see report)

S16 Cl. 13.6.1 a) or b)

\begin{aligned} \htmlClass{sym-M_r}{M_{r}} &= \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}{200.2\ \mathrm{kN} \cdot \mathrm{m}} \cdot \left(1 - \frac{0.28 \cdot \htmlClass{sym-M_p}{200.2\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_u}{463.4\ \mathrm{kN} \cdot \mathrm{m}}}\right), \htmlClass{sym-phi}{0.9} \cdot \htmlClass{sym-M_p}{200.2\ \mathrm{kN} \cdot \mathrm{m}}\right) \\ &= 180.2\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}

S.35 Tension and bending interaction

Compared with CSA S16-19 Cl. 13.9.1 (T_f/T_r + M_fx/M_rx + M_fy/M_ry not over 1.0, with M_fy = 0; M_rx per Cl. 13.5)

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_f}{M_{f}}}{\htmlClass{sym-M_rx}{M_{\mathrm{rx}}}} \\ &= \frac{\htmlClass{sym-T_f}{60\ \mathrm{kN}}}{\htmlClass{sym-T_r}{1969\ \mathrm{kN}}} + \frac{\htmlClass{sym-M_f}{30\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_rx}{180.2\ \mathrm{kN} \cdot \mathrm{m}}} \\ &= 0.197 \end{aligned}

S.36 Bending less tension interaction

Compared with CSA S16-19 Cl. 13.9.3 a) and b) (M_fx/M_rx - T_f Z_x/(M_r A) for Class 1 and 2, S_x for Class 3; M_fy = 0, M_r per Cl. 13.6.1; a Class 4 I-shape or HSS takes the effective S_e where b) says S_x, conservative, see report)

S16 Cl. 13.9.3 a) and b)

\begin{aligned} \htmlClass{sym-I_b}{I_{b}} &= \frac{\htmlClass{sym-M_f}{M_{f}}}{\htmlClass{sym-M_r}{M_{r}}} - \frac{\htmlClass{sym-T_f}{T_{f}} \cdot \htmlClass{sym-Z_e}{Z_{e}}}{\htmlClass{sym-M_r}{M_{r}} \cdot \htmlClass{sym-A_0}{A_{0}}} \\ &= \frac{\htmlClass{sym-M_f}{30\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-M_r}{180.2\ \mathrm{kN} \cdot \mathrm{m}}} - \frac{\htmlClass{sym-T_f}{60\ \mathrm{kN}} \cdot \htmlClass{sym-Z_e}{572000\ \mathrm{mm}^{3}}}{\htmlClass{sym-M_r}{180.2\ \mathrm{kN} \cdot \mathrm{m}} \cdot \htmlClass{sym-A_0}{6250\ \mathrm{mm}^{2}}} \\ &= 0.136 \end{aligned}

Report

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Questions

Why does S16 Cl. 13.9 have two interactions?

The first limits the section's strength under both. The second is lateral-torsional buckling, which the tension resists: it takes the unbraced moment resistance and subtracts the tension's share of the compression flange's stress.

Is the net section checked?

No: T_r is the gross section's yield; net section fracture and block shear are not checked.

Revision history

2026-11-05 After an independent checker's review: a channel's moment resistance now rests on S_x F_y (Cl. 13.6.1 b) and the channel is capped at class 3, a square HSS takes M_r from Cl. 13.5, a class 4 flange starts from the gross M_y, and the net-section item cites Cl. 13.2 a) iii) and block shear

Value	Was	Now	Unit
`S_x`	572000		$\mathrm{mm}^{3}$

Tension and bending

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Derivation

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Slenderness

Section class

Strength

Results

Derivation

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