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

Class 1:

$\frac{b_{f} /2}{t_f} \leq \frac{145}{\sqrt{F_y}}$ for the flange and $\frac{d – 2 t_f}{t_w} \leq \frac{1100}{\sqrt{F_y}}(1-0.39 \frac{C_f}{\phi C_y})$ for the web.

Class 2:

$\frac{b_{f} /2}{t_f} \leq \frac{170}{\sqrt{F_y}}$ for the flange and $\frac{d – 2 t_f}{t_w} \leq \frac{1700}{\sqrt{F_y}}(1-0.61 \frac{C_f}{\phi C_y})$ for the web.

Class 3:

$\frac{b_{f} /2}{t_f} \leq \frac{200}{\sqrt{F_y}}$ for the flange and $\frac{d – 2 t_f}{t_w} \leq \frac{1900}{\sqrt{F_y}}(1-0.65 \frac{C_f}{\phi C_y})$ for the web.

Otherwise, Class 4.

{%if outputs.flag == 'section is atleast class 2'%}

Flange:

$\frac{b_f/2}{t_f} =$ {{outputs.flange | katex}} $\leq \frac{170}{\sqrt{F_y}} =$ {{outputs.flangetest1 | katex}}

web:

$\frac{d – 2t_f}{t_w} =$ {{outputs.web | katex}} $\leq \frac{1700}{\sqrt{F_y}}(1-0.61 \frac{C_f}{\phi C_y})=$ {{outputs.webtest1 | katex}}

Hence, {{outputs.flag}}.

{%elif outputs.flag == 'section is class 3'%}

Flange:

$\frac{b_f/2}{t_f} =$ {{outputs.flange | katex}} $\leq \frac{200}{\sqrt{F_y}} =$ {{outputs.flangetest2 | katex}}

web:

$\frac{d – 2t_f}{t_w} =$ {{outputs.web | katex}} $\leq \frac{1900}{\sqrt{F_y}}(1-0.65 \frac{C_f}{\phi C_y})=$ {{outputs.webtest2 | katex}}

Hence {{outputs.flag}}.

{%elif outputs.flag == 'section is class 4'%}

Flange:

$\frac{b_f/2}{t_f} =$ {{outputs.flange | katex}} $> \frac{200}{\sqrt{F_y}} =$ {{outputs.flangetest2 | katex}}

Web:

$\frac{d – 2t_f}{t_w} =$ {{outputs.web | katex}} $> \frac{1900}{\sqrt{F_y}}(1-0.65 \frac{C_f}{\phi C_y})=$ {{outputs.webtest2 | katex}}

Hence {{outputs.flag}}.

{%endif%}

{%if (outputs.flag == 'section is atleast class 2') or (outputs.flag == 'section is class 3')%}

Design Interaction Equation

{%if outputs.flag == 'section is atleast class 2'%}

$\frac{C_f}{C_r} + \frac{0.85 U_{1x} M_{fx}}{M_{rx}} \leq 1.0$ and $\frac{M_{fx}}{M_{rx}} \leq 1.0$

{%else%}

$\frac{C_f}{C_r} + \frac{U_{1x} M_{fx}}{M_{rx}} \leq 1.0$ and $\frac{M_{fx}}{M_{rx}} \leq 1.0$

{%endif%}

Limit State 1: Cross-Sectional Strength

{%if outputs.frametype == 'Sway (unbraced)'%}

This limit state only needs to be checked for members in braced frames.

{%elif outputs.frametype == 'Non-sway (braced)'%}

Factored Compression Resistance: $C_r = \phi A F_y =$ {{outputs.Cr1}} $kN$

{%if outputs.flag == 'section is atleast class 2'%}

Factored Flexural Moment Resistance about the Major Axis: $M_{rx}=\phi Z_x F_y=$ {{outputs.Mrx1}} $kN \cdot m$

{%else%}

Factored Flexural Moment Resistance about the Major Axis: $M_{rx}=\phi S_x F_y$ ={{outputs.Mrx1}} $kN \cdot m$

{%endif%}

Euler Buckling Load about the Major Axis: $C_{ex}=\frac{\pi E I_x}{L^2}=$ {{outputs.Cex1}} $kN$

Ratio of Smaller to Larger Factored Moment about the Major Axis: $\kappa = \frac{M_{f,min}}{M_{f,max}}$ ={{outputs.kappa1}}

Equivalent Moment Factor about the Major Axis: $\omega_{1x} = 0.6-0.4 \kappa \geq 0.4 =$ {{outputs.w1x1}}

Factor accounting for Moment Gradient and P-delta Effects about the Major Axis:

$U_{1x} = \frac{\omega_{1x}}{1-\frac{C_f}{C_{ex}}} \geq 1.0 =$ {{outputs.U1x1}}

{%if outputs.check1 <= 1%}

$\frac{C_f}{C_r} + \frac{0.85 U_{1x} M_{fx}}{M_{rx}} =$ {{outputs.check1}}  $\leq 1.0$ (Passed)

{%elif outputs.check1 >1%}

$\frac{C_f}{C_r} + \frac{0.85 U_{1x} M_{fx}}{M_{rx}} =$ {{outputs.check1}} $> 1.0$ (Failed)

{%endif%}

{%endif%}

{%if outputs.frametype == 'Sway (unbraced)'%}

Limit State 2: Overall Member Strength

Effective Length Factor: $K =$ 1.0

Slenderness Parameter about the Major Axis: $\lambda_x = \frac{KL}{r_x}\sqrt{\frac{F_y}{\pi^2 E}} =$ {{outputs.lamx}}

Residual Stress Parameter: $n =$ 1.34

Factored Compression Resistance about the Major Axis: $C_{rx} = \phi A F_y (1+\lambda_x^{2n})^{-1/n} =$ {{outputs.Crx2}} $kN$

{%if outputs.flag == 'section is atleast class 2'%}

Factored Flexural Moment Resistance about the Major Axis: $M_{rx}=\phi Z_x F_y=$ {{outputs.Mrx2}} $kN \cdot m$

{%else%}

Factored Flexural Moment Resistance about the Major Axis: $M_{rx}=\phi S_x F_y$ ={{outputs.Mrx2}} $kN \cdot m$

{%endif%}

{%if outputs.frametype == 'Non-sway (braced)'%}

Euler Buckling Load about the Major Axis: $C_{ex}=\frac{\pi E I_x}{L^2}=$ {{outputs.Cex2}} $kN$

Ratio of Smaller to Larger Factored Moment about the Major Axis: $\kappa = \frac{M_{f,min}}{M_{f,max}}$ = {{outputs.kappa2}}

Equivalent Moment Factor about the Major Axis: $\omega_{1x} = 0.6-0.4 \kappa \geq 0.4 =$ {{outputs.w1x2}}

Factor accounting for Moment Gradient and P-delta Effects about the Major Axis: $U_{1x} = \frac{\omega_{1x}}{1-\frac{C_f}{C_{ex}}} =$ {{outputs.U1x2}}

{%elif outputs.frametype == 'Sway (unbraced)'%}

Factor accounting for Moment Gradient and P-delta Effects about the Major Axis: $U_{1x} =$ {{outputs.U1x2}}

{%endif%}

{%if outputs.check2 <= 1%}

$\frac{C_f}{C_r} + \frac{0.85 U_{1x} M_{fx}}{M_{rx}} =$ {{outputs.check2}}  $\leq 1.0$ (Passed)

{%elif outputs.check2 >1%}

$\frac{C_f}{C_r} + \frac{0.85 U_{1x} M_{fx}}{M_{rx}} =$ {{outputs.check2}} $> 1.0$ (Failed)

{%endif%}

{%elif outputs.frametype == 'Non-sway (braced)'%}

Limit State 2: Overall Member Strength

$K =$ 1.0

Slenderness Parameter about the Major Axis: $\lambda_x = \frac{KL}{r_x}\sqrt{\frac{F_y}{\pi^2 E}} =$ {{outputs.lamx}}

Residual Stress Parameter: $n =$ 1.34

Factored Compression Resistance about the Major Axis: $C_{rx} = \phi A F_y (1+\lambda_x^{2n})^{-1/n} =$ {{outputs.Crx2}} $kN$

{%if outputs.flag == 'section is atleast class 2'%}

$M_{rx}=\phi Z_x F_y=$ {{outputs.Mrx2}} $kN \cdot m$

{%else%}

$M_{rx}=\phi S_x F_y$ ={{outputs.Mrx2}} $kN \cdot m$

{%endif%}

{%if outputs.frametype == 'Non-sway (braced)'%}

$C_{ex}=\frac{\pi E I_x}{L^2}=$ {{outputs.Cex2}} $kN$

$\kappa = \frac{M_{f,min}}{M_{f,max}}$ = {{outputs.kappa2}}

$\omega_{1x} = 0.6-0.4 \kappa \geq 0.4 =$ {{outputs.w1x2}}

$U_{1x} = \frac{\omega_{1x}}{1-\frac{C_f}{C_{ex}}} =$ {{outputs.U1x2}}

{%elif outputs.frametype == 'Sway (unbraced)'%}

$U_{1x} =$ {{outputs.U1x2}}

{%endif%}

{%if outputs.check2 <= 1%}

$\frac{C_f}{C_r} + \frac{0.85 U_{1x} M_{fx}}{M_{rx}} =$ {{outputs.check2}}  $\leq 1.0$ (Passed)

{%elif outputs.check2 >1%}

$\frac{C_f}{C_r} + \frac{0.85 U_{1x} M_{fx}}{M_{rx}} =$ {{outputs.check2}} $> 1.0$ (Failed)

{%endif%}

{%endif%}

Limit State 3: Lateral-Torsional Buckling Strength

$K=$ 1.0

$\lambda_y = \frac{KL}{r_y} \sqrt{\frac{F_y}{\pi^2 E}} =$ {{outputs.lamy}}

$n =$ 1.34

$C_{ry} = \phi A F_y (1+\lambda_y^{2n})^{-1/n} =$ {{outputs.Cry3}} $kN$

$\kappa = \frac{M_{f,min}}{M_{f,max}}$ = {{outputs.kappa3}}

$\omega_{2x} = 1.75+1.05 \kappa + 0.3 \kappa^2 =$ {{outputs.w2x3}}

Critical Elastic Flexural Moment: $M_u = \frac{\omega_2\pi}{L}\sqrt{EI_yGJ + (\frac{\pi E}{L})^2I_yC_w}$ = {{outputs.Mux3}} $kN \cdot m$

{%if outputs.flag == 'section is atleast class 2'%}

Plastic Flexural Moment: $M_p = Z_x F_y$ = {{outputs.Mpyx3}} $kN \cdot m$

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

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

$M_{rx} = \phi M_{p}$ $=$ {{outputs.Mrx3}} $kN \cdot m$

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

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

$M_{rx} = \phi 1.15M_{p}(1 - 0.28\frac{M_u}{M_p})$ $=$ {{outputs.Mrx3}} $kN \cdot m$

{%else%}

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

$M_{rx} = \phi M_{u}$ $=$ {{outputs.Mrx3}} $kN \cdot m$

{%endif%}

{%else%}

Yield Flexural Moment: $M_y = S_x F_y$ = {{outputs.Mpyx3}} $kN \cdot m$

{%if outputs.ratio >= 2.15%}

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

$M_{rx} = \phi M_{y}$ $=$ {{outputs.Mrx3}} $kN \cdot m$

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

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

$M_{rx} = \phi 1.15M_{y}(1 - 0.28\frac{M_u}{M_y})$ $=$ {{outputs.Mrx3}} $kN \cdot m$

{%else%}

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

$M_{rx} = \phi M_{u}$ $=$ {{outputs.Mrx3}} $kN \cdot m$

{%endif%}

{%endif%}

{%if outputs.frametype == 'Non-sway (braced)'%}

$C_{ex}=\frac{\pi E I_x}{L^2}=$ {{outputs.Cex3}} $kN$

$\omega_{1x} = 0.6-0.4 \kappa \geq 0.4 =$ {{outputs.w1x3}}

$U_{1x} = \frac{\omega_{1x}}{1-\frac{C_f}{C_{ex}}} \geq 1.0 =$ {{outputs.U1x3}}

{%elif outputs.frametype == 'Sway (unbraced)'%}

$U_{1x} =$ {{outputs.U1x3}}

{%endif%}

{%if outputs.check3 <= 1%}

$\frac{C_f}{C_r} + \frac{0.85 U_{1x} M_{fx}}{M_{rx}} =$ {{outputs.check3}}  $\leq 1.0$ (Passed)

{%elif outputs.check3 >1%}

$\frac{C_f}{C_r} + \frac{0.85 U_{1x} M_{fx}}{M_{rx}} =$ {{outputs.check3}} $> 1.0$ (Failed)

{%endif%}

Limit State 4: Moment Check

{%if outputs.check4 <= 1%}

$\frac{M_{fx}}{M_{rx}}=$ {{outputs.check4}}  $\leq 1.0$ (Passed)

{%elif outputs.check4 >1%}

$\frac{M_{fx}}{M_{rx}}=$ {{outputs.check4}} $> 1.0$ (Failed)

{%endif%}

{%else%}

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

{%endif%}

W-Section Beam-Columns under Major Axis Bending (CSA S16-19)