Solution

{%if (outputs.M_fxmax < outputs.M_fxmin) or (outputs.M_fymax < outputs.M_fymin)%}

Please ensure the maximum and minimum moments are inputted accurately.

{%elif M_fxmax == 0%}

This application is efficient for biaxial bending or major axis bending only.

{%else%}

{{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.

If $\frac{M_{fy}}{S_y} \leq \frac{0.9 M_{fx}}{S_x}$ then $\frac{d – 2 t_f}{t_w} \leq \frac{1100}{\sqrt{F_y}}(1-0.39 \frac{C_f}{\phi C_y})$ for the web.

If $\frac{M_{fy}}{S_y} > \frac{0.9 M_{fx}}{S_x}$ then $\frac{d – 2t_f}{t_w} \leq \frac{525}{\sqrt{F_y}}$ for the web

Class 2:

$\frac{b_{f} /2}{t_f} \leq \frac{170}{\sqrt{F_y}}$ for the flange.

If $\frac{M_{fy}}{S_y} \leq \frac{0.9 M_{fx}}{S_x}$ then $\frac{d – 2 t_f}{t_w} \leq \frac{1700}{\sqrt{F_y}}(1-0.61 \frac{C_f}{\phi C_y})$ for the web.

If $\frac{M_{fy}}{S_y} > \frac{0.9 M_{fx}}{S_x}$ then $\frac{d – 2t_f}{t_w} \leq \frac{525}{\sqrt{F_y}}$ for the web.

Class 3:

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

{%if outputs.webflag == 'greater'%}

$\frac{M_{fy}}{S_y}=$ {{outputs.webcheck_y | katex}} $> \frac{0.9 M_{fx}}{S_x}=$ {{outputs.webcheck_x | katex}}

$\frac{d – 2t_f}{t_w} =$ {{outputs.web | katex}} $\leq \frac{525}{\sqrt{F_y}}$ $=$ {{outputs.webtest1 | katex}}

{%elif outputs.webflag == 'less or equal'%}

$\frac{M_{fy}}{S_y}=$ {{outputs.webcheck_y | katex}} $\leq \frac{0.9 M_{fx}}{S_x}=$ {{outputs.webcheck_x | katex}}

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

{%endif%}

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'%}

$\frac{C_f}{C_{r}} + \frac{0.85 U_{1x} M_{fx}}{M_{rx}} + \frac{\beta U_{1y} M_{fy}}{M_{ry}} \leq 1.0$ and $\frac{M_{fx}}{M_{rx}} + \frac{M_{fy}}{M_{ry}} \leq 1.0$

{%else%}

$\frac{C_f}{C_{r}} + \frac{U_{1x} M_{fx}}{M_{rx}} + \frac{U_{1y} M_{fy}}{M_{ry}} \leq 1.0$ and $\frac{M_{fx}}{M_{rx}} + \frac{M_{fy}}{M_{ry}} \leq 1.0$

{%endif%}

Limit State 1: Cross-Sectional Strength

{%if (outputs.frametype_x == 'Sway (unbraced)') and (outputs.frametype_y == 'Sway (unbraced)')%}

This limit state only needs to be checked for members in braced frames simply because for unbraced frames it can never govern.

{%else%}

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

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

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

Factored Moment Resistance about the Minor Axis: $M_{ry} = \phi Z_y F_y$ $=$ {{outputs.Mry1}} $kN \cdot m$

{%else%}

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

Factored Moment Resistance about the Minor Axis: $M_{ry} = \phi S_y F_y$ $=$ {{outputs.Mry1}} $kN \cdot m$

{%endif%}

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

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

Ratio of Smaller to Larger Factored Moment about the Major Axis: $\kappa_x = \frac{M_{f,min}}{M_{fmax}}=$ {{outputs.kappax1}}

Equivalent Moment Factor for the Major Axis: $\omega_{1x} = 0.6-0.4 \kappa_x \geq 0.4 =$ {{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}}

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

Factor accounting for Moment Gradient and P-delta Effects about the Major Axis: $U_{1x} =$ {{outputs.U1x1}}

{%endif%}

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

Euler Buckling Load about the Minor Axis: $C_{ey} = \frac{\pi^2 E I_y}{L^2} =$ {{outputs.Cey1}} $kN$

Ratio of Smaller to Larger Factored Moment about the Minor Axis: $\kappa_y = \frac{M_{f.min}}{M_{f,max}}=$ {{outputs.kappay1}}

Equivalent Moment Factor for the Minor Axis: $\omega_{1y} = 0.6-0.4 \kappa_y \geq 0.4 =$ {{outputs.w1y1}}

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

$U_{1y} = \frac{\omega_{1y}}{1-\frac{C_f}{C_{ey}}} \geq 1.0 =$ {{outputs.U1y1}}

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

Factor accounting for Moment Gradient and P-delta Effects about the Minor Axis: $U_{1y}=$ {{outputs.U1y1}}

{%endif%}

Effective Length Factor: $K =$ 1.00

Slenderness Parameter about the Minor Axis: $\lambda_y = \frac{KL}{r_y} \sqrt{\frac{F_y}{\pi^2 E}} =$ 0

$\beta =$ $0.6 + 0.4 \lambda_y =$ 0.6 $\leq 0.85$

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

{%if outputs.check1 <= 1%}

$\frac{C_f}{C_{r}} + \frac{0.85 U_{1x} M_{fx}}{M_{rx}} + \frac{\beta U_{1y} M_{fy}}{M_{ry}} =$ {{outputs.check1}}  $\leq 1.0$ (Passed)

{%elif outputs.check1 >1%}

$\frac{C_f}{C_{r}} + \frac{0.85 U_{1x} M_{fx}}{M_{rx}} + \frac{\beta U_{1y} M_{fy}}{M_{ry}} =$ {{outputs.check1}} $> 1.0$ (Failed)

{%endif%}

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

{%if outputs.check1 <= 1%}

$\frac{C_f}{C_{r}} + \frac{U_{1x} M_{fx}}{M_{rx}} + \frac{U_{1y} M_{fy}}{M_{ry}}=$ {{outputs.check1}}  $\leq 1.0$ (Passed)

{%elif outputs.check1 >1%}

$\frac{C_f}{C_{r}} + \frac{U_{1x} M_{fx}}{M_{rx}} + \frac{U_{1y} M_{fy}}{M_{ry}}=$ {{outputs.check1}} $> 1.0$ (Failed)

{%endif%}

{%if (outputs.frametype_x == 'Sway (unbraced)') and (outputs.frametype_y == 'Sway (unbraced)') %}

Limit State 2: Overall Member Strength

Effective Length Factor: $K =$ 1.00

{%if (outputs.M_fymax == 0) and (outputs.M_fymin == 0)%}

Slenderness Parameter about the Major Axis: $\lambda_x = \frac{KL}{r_x}\sqrt{\frac{F_y}{\pi^2 E}} =$ {{outputs.lam2}}

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.Cr2}} $kN$

{%else%}

Slenderness Parameter about the Minor Axis: $\lambda_y = \frac{KL}{r_y} \sqrt{\frac{F_y}{\pi^2 E}} =$ {{outputs.lam2}}

Residual Stress Parameter: $n =$ 1.34

Factored Compression Resistance about the Minor Axis: $C_{ry} = \phi A F_y (1+\lambda_y^{2n})^{-1/n} =$ {{outputs.Cr2}} $kN$

{%endif%}

{%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}} $kN \cdot m$

Factored Flexural Moment Resistance about the Minor Axis: $M_{ry} = \phi Z_y F_y =$ {{outputs.Mry2}} $kN \cdot m$

{%else%}

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

Factored Flexural Moment Resistance about the Minor Axis: $M_{ry} = \phi S_y F_y$ $=$ {{outputs.Mry2}} $kN \cdot m$

{%endif%}

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

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

Ratio of Smaller to Larger Factored Moment about the Major Axis: $\kappa_x = \frac{M_{f,min}}{M_{fmax}}=$ {{outputs.kappax2}}

Equivalent Moment Factor for the Major Axis: $\omega_{1x} = 0.6-0.4 \kappa_x \geq 0.4 =$ {{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_x == 'Sway (unbraced)')%}

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

{%endif%}

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

Euler Buckling Load about the Minor Axis: $C_{ey} = \frac{\pi^2 E I_y}{L^2} =$ {{outputs.Cey2}} $kN$

Ratio of Smaller to Larger Factored Moment about the Minor Axis: $\kappa_y = \frac{M_{f.min}}{M_{f,max}}=$ {{outputs.kappay2}}

Equivalent Moment Factor for the Minor Axis: $\omega_{1y} = 0.6-0.4 \kappa_y \geq 0.4 =$ {{outputs.w1y2}}

Factor accounting for Moment Gradient and P-delta Effects about the Minor Axis: $U_{1y} = \frac{\omega_{1y}}{1-\frac{C_f}{C_{ey}}} =$ {{outputs.U1y2}}

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

Factor accounting for Moment Gradient and P-delta Effects about the Minor Axis: $U_{1y}=$ {{outputs.U1y2}}

{%endif%}

$\beta = 0.6 + 0.4 \lambda_y \leq 0.85 =$ {{outputs.beta}}

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

{%if outputs.check2 <= 1%}

$\frac{C_f}{C_{r}} + \frac{0.85 U_{1x} M_{fx}}{M_{rx}} + \frac{\beta U_{1y} M_{fy}}{M_{ry}} =$ {{outputs.check2}}  $\leq 1.0$ (Passed)

{%elif outputs.check2 >1%}

$\frac{C_f}{C_{r}} + \frac{0.85 U_{1x} M_{fx}}{M_{rx}} + \frac{\beta U_{1y} M_{fy}}{M_{ry}} =$ {{outputs.check2}} $> 1.0$ (Failed)

{%endif%}

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

{%if outputs.check2 <= 1%}

$\frac{C_f}{C_{r}} + \frac{U_{1x} M_{fx}}{M_{rx}} + \frac{U_{1y} M_{fy}}{M_{ry}}=$ {{outputs.check2}}  $\leq 1.0$ (Passed)

{%elif outputs.check2 >1%}

$\frac{C_f}{C_{r}} + \frac{U_{1x} M_{fx}}{M_{rx}} + \frac{U_{1y} M_{fy}}{M_{ry}}=$ {{outputs.check2}} $> 1.0$ (Failed)

{%endif%}

{%else%}

Limit State 2: Overall Member Strength

$K =$ 1.00

{%if (outputs.M_fymax == 0) and (outputs.M_fymin == 0)%}

Slenderness Parameter about the Major Axis: $\lambda_x = \frac{KL}{r_x}\sqrt{\frac{F_y}{\pi^2 E}} =$ {{outputs.lam2}}

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.Cr2}} $kN$

{%else%}

$\lambda_y = \frac{KL}{r_y} \sqrt{\frac{F_y}{\pi^2 E}} =$ {{outputs.lam2}}

Residual Stress Parameter: $n =$ 1.34

Factored Compression Resistance about the Minor Axis: $C_{ry} = \phi A F_y (1+\lambda_y^{2n})^{-1/n} =$ {{outputs.Cr2}} $kN$

{%endif%}

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

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

$M_{ry} = \phi Z_y F_y =$ {{outputs.Mry2}} $kN \cdot m$

{%else%}

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

$M_{ry} = \phi S_y F_y$ $=$ {{outputs.Mry2}} $kN \cdot m$

{%endif%}

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

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

$\kappa_x = \frac{M_{f,min}}{M_{fmax}}=$ {{outputs.kappax2}}

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

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

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

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

{%endif%}

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

$C_{ey} = \frac{\pi^2 E I_y}{L^2} =$ {{outputs.Cey2}} $kN$

$\kappa_y = \frac{M_{f.min}}{M_{f,max}}=$ {{outputs.kappay2}}

$\omega_{1y} = 0.6-0.4 \kappa_y \geq 0.4 =$ {{outputs.w1y2}}

$U_{1y} = \frac{\omega_{1y}}{1-\frac{C_f}{C_{ey}}} =$ {{outputs.U1y2}}

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

$U_{1y}=$ {{outputs.U1y2}}

{%endif%}

$\beta = 0.6 + 0.4 \lambda_y \leq 0.85 =$ {{outputs.beta}}

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

{%if outputs.check2 <= 1%}

$\frac{C_f}{C_{r}} + \frac{0.85 U_{1x} M_{fx}}{M_{rx}} + \frac{\beta U_{1y} M_{fy}}{M_{ry}} =$ {{outputs.check2}}  $\leq 1.0$ (Passed)

{%elif outputs.check2 >1%}

$\frac{C_f}{C_{r}} + \frac{0.85 U_{1x} M_{fx}}{M_{rx}} + \frac{\beta U_{1y} M_{fy}}{M_{ry}} =$ {{outputs.check2}} $> 1.0$ (Failed)

{%endif%}

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

{%if outputs.check2 <= 1%}

$\frac{C_f}{C_{r}} + \frac{U_{1x} M_{fx}}{M_{rx}} + \frac{U_{1y} M_{fy}}{M_{ry}}=$ {{outputs.check2}}  $\leq 1.0$ (Passed)

{%elif outputs.check2 >1%}

$\frac{C_f}{C_{r}} + \frac{U_{1x} M_{fx}}{M_{rx}} + \frac{U_{1y} M_{fy}}{M_{ry}}=$ {{outputs.check2}} $> 1.0$ (Failed)

{%endif%}

Limit State 3: Lateral-Torsional Buckling Strength

$K=$ 1.00

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

$n =$ 1.34

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

$\kappa_x = \frac{M_{f,min}}{M_{fmax}}=$ {{outputs.kappax3}}

$\omega_{2x} = 1.75 + 1.05 \kappa_x + 0.3 \kappa_x^2 \leq 2.5 =$ {{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 about the major axis.

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

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

$0.67 < \frac{M_u}{M_p} =$

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