Solution

{{outputs.dimensions}} Properties

{% for i in outputs.listnum %}

{{ outputs.listprop2[i] | katex}}: {{outputs.beam[outputs.listprop[i]] | katex}} {{outputs.units[outputs.listprop[i]] | katex}}

{%endfor%}

Section Class

{%if outputs.beamtype == "W" or outputs.beamtype == "C"%}

Class 1:

$\frac{b_{f} /2}{t_f} \leq \frac{145}{\sqrt{F_y}}$ for the flange and $\frac{d – 2t_f}{t_w} \leq \frac{1100}{\sqrt{F_y}}$ for the web.

Class 2:

$\frac{b_{f} /2}{t_f} \leq \frac{170}{\sqrt{F_y}}$ for the flange and $\frac{d – 2t_f}{t_w} \leq \frac{1700}{\sqrt{F_y}}$ for the web.

Class 3:

$\frac{b_{f} /2}{t_f} \leq \frac{200}{\sqrt{F_y}}$ for the flange and $\frac{d – 2t_f}{t_w} \leq \frac{1900}{\sqrt{F_y}}$ for the web.

Otherwise, Class 4.

{% set flange_eq = "\\frac{b_f / 2}{t_f} = \\frac{" + outputs.beam['bf'] + "}{2\\times" + outputs.beam['tf'] + "}=" + outputs.flange %}

{% if outputs.CT == 'Class 4' %}

{% set flange_eq = flange_eq + "\\geq" + outputs.flangetest %}

{%else%}

{% set flange_eq = flange_eq + "\\leq" + outputs.flangetest %}

{%endif%}

Flange:

{{ flange_eq | katex }}

{% set web_eq = "\\frac{d - 2 t_f}{t_w} = \\frac{" + outputs.beam['d'] + "-" + outputs.beam['tf'] + "\\times 2 }{" + outputs.beam['tw'] + "}=" + outputs.web %}

{% if CT == 'Class 4' %}

{% set web_eq = web_eq + "\\geq" + outputs.webtest %}

{%else%}

{% set web_eq = web_eq + "\\leq" + outputs.webtest %}

{%endif%}

Web:

{{ web_eq | katex }}

{%elif outputs.beamtype == "WT" %}

For Class 1 classification, the following conditions must be met:

$\frac{b_{el}}{t} \leq \frac{145}{\sqrt{F_y}}$ for flange

For Class 2 classification, the following conditions must be met:

$\frac{b_{el}}{t} \leq \frac{170}{\sqrt{F_y}}$ for flange

For Class 3 classification, the following conditions must be met:

$\frac{b_{el}}{t} \leq \frac{200}{\sqrt{F_y}}$ for flange

{% set flange_eq = "\\frac{b_{el}}{t} = \\frac{" + outputs.beam['bf'] + "} {" + outputs.beam['tf'] + "}=" + outputs.flange %}

{% if CT == 'Class 4' %}

{% set flange_eq = flange_eq + "\\geq" + outputs.flangetest %}

{%else%}

{% set flange_eq = flange_eq + "\\leq" + outputs.flangetest %}

{%endif%}

For the flange:

{{ flange_eq | katex }}

{%endif%}

The section is {{outputs.CT}}.

{%if outputs.CT == "Class 1" or outputs.CT == "Class 2" or outputs.CT == "Class 3"%}

{%if outputs.CT == "Class 1" or outputs.CT == "Class 2"%}

Plastic Flexural Moment

Plastic Flexural Moment: $M_p = Z_xF_y$

$M_p=$ {{outputs.M}} $kN \cdot m$

{%else%}

Yield Flexural Moment

Yield Flexural Moment: $M_y = S_xF_y$

$M_y =$ {{outputs.M}} $kN \cdot m$

{%endif%}

{% if outputs.type == '1' %}

Equivalent Moment Factor

Equivalent Moment Factor: $\omega_2 = \frac{4M_{max}}{\sqrt{M^{2}_{max} + 4M^2_a + 7M^2_b + 4M^2_c}}$

$M_a, M_b,$ and $M_c$ are the absolute values of the first-quarter, mid-point, and third-quarter in the unbraced segment. $M_{max}$ is the maximum moment magnitude in the segment.

For a beam under uniformly distributed load, $\omega_2 =$ {{outputs.w2}}

Critical Elastic Flexural Moment

Critical Elastic Flexural Moment: $M_u = \frac{\omega_2\pi}{L}\sqrt{EI_yGJ + (\frac{\pi E}{L})^2I_yC_w}$

$G =$ 77000 $MPa$ and $E =$ 200000 $MPa$

$M_u =$ {{outputs.Mu}} $kN \cdot m$

Factored Flexural Moment Resistance

{%if (outputs.CT == "Class 1") or (outputs.CT == "Class 2")%}

{%if (outputs.MuM >= 2.15)%}

$\frac{M_u}{M_p}$ $=$ {{outputs.MuM}} $\geq 2.15$ therefore plastic capacity governs.

Factored Flexural Moment Resistance: $M_{rx} = \phi M_{p}$

$M_{rx} =$ {{outputs.Mrx}} $kN \cdot m$

{%elif (outputs.MuM < 2.15) and (outputs.MuM > 0.67)%}

$0.67 < \frac{M_u}{M_p} =$ {{outputs.MuM}} $< 2.15$ therefore inelastic lateral torsional buckling governs.

Factored Flexural Moment Resistance: $M_{rx} = \phi 1.15M_{p}(1 - 0.28\frac{M_u}{M_p})$

$M_{rx} =$ {{outputs.Mrx}} $kN \cdot m$

{%else%}

$\frac{M_u}{M_p} =$ {{outputs.MuM}} $\leq 0.67$ therefore elastic lateral torsional buckling governs.

Factored Flexural Moment Resistance: $M_{rx} = \phi M_{u}$

$M_{rx} =$ {{outputs.Mrx}} $kN \cdot m$

{%endif%}

{%else%}

{%if outputs.MuM >= 2.15%}

$\frac{M_u}{M_y} =$ {{outputs.MuM}} $\geq$ 2.15 therefore yield capacity governs.

Factored Flexural Moment Resistance: $M_{rx} = \phi M_{y}$

$M_{rx} =$ {{outputs.Mrx}} $kN \cdot m$

{% elif (outputs.MuM<2.15) and (outputs.MuM> 0.67)%}

$0.67 < \frac{M_u}{M_y} =$ {{outputs.MuM}} $< 2.15$ therefore inelastic lateral torsional buckling governs.

Factored Flexural Moment Resistance: $M_{rx} = \phi 1.15M_{y}(1 - 0.28\frac{M_u}{M_y})$

$M_{rx} =$ {{outputs.Mrx}} $kN \cdot m$

{%else%}

$\frac{M_u}{M_y} =$ {{outputs.MuM}} $\leq 0.67$ therefore elastic lateral torsional buckling governs.

Factored Flexural Moment Resistance: $M_{rx} = \phi M_{u}$

$M_{rx} =$ {{outputs.Mrx}} $kN \cdot m$

{%endif%}

{%endif%}

{%if outputs.SR != None %}

Factored Shear Resistance

Factored Shear resistance, $V_r = \phi A_wF_s$ where:

$A_w = d * w$

$F_s = 0.66F_y$ when $\frac{d-2 t_f}{t_w} \leq \frac{1014}{\sqrt{F_y}}$

$F_s = \frac{670\sqrt{F_y}}{\frac{h}{w}}$ when $\frac{1014}{\sqrt{F_y}} < \frac{d-2 t_f}{t_w} \leq \frac{1435}{\sqrt{F_y}}$

$F_s = \frac{961200}{(\frac{h}{w})^2}$ when $\frac{d-2 t_f}{t_w} > \frac{1435}{\sqrt{F_y}}$

$V_r =$ {{outputs.SR}} $kN$

{%endif%}

{% elif outputs.type == '2' %}

Factored Flexural Moment Resistance

{%if outputs.CT == "Class 1" or outputs.CT == "Class 2"%}

Factored Flexural Moment Resistance: $M_{rx} = \phi Z_x F_y$

{%else%}

Factored Flexural Moment Resistance: $M_{rx} = \phi S_xF_y$

{%endif%}

$M_{rx} =$ {{outputs.Mrx}} $kN \cdot m$

{% if outputs.SR != None %}

Factored Shear Resistance

Factored Shear resistance, $V_r = \phi A_wF_s$ where:

$A_w = d * w$

$F_s = 0.66F_y$ when $\frac{d-2 t_f}{t_w} \leq \frac{1014}{\sqrt{F_y}}$

$F_s = \frac{670\sqrt{F_y}}{\frac{h}{w}}$ when $\frac{1014}{\sqrt{F_y}} < \frac{d-2 t_f}{t_w} \leq \frac{1435}{\sqrt{F_y}}$

$F_s = \frac{961200}{(\frac{h}{w})^2}$ when $\frac{d-2 t_f}{t_w} > \frac{1435}{\sqrt{F_y}}$

$V_r =$ {{outputs.SR}} $kN$

{%endif%}

{%endif %}

{%else%}

This application computes calculations for Class 1, 2 and 3 sections only.

{%endif%}

Design of Flexural Members with W-Section (CSA S16-19)