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Simplified Design for Torsional Loading of Rolled Steel Members, Lin, P.H., Engineering Journal, AISC, 1977

2010 Specification for Structural Steel Buildings (ANSI/AISC 360-10), Fourth Printing (https://www.aisc.org/content.aspx?id=2884)

$\mathrm{with}\left(\mathrm{AISCShapes}\right)\:$

$\mathrm{with}\left(\mathrm{Units}\left[\mathrm{Standard}\right]\right)\:$

$$\begin{gathered} \mathrm{Cw}≔\mathrm{Property}\left(''W10X54''\,''Cw''\right)\; \\ \mathrm{Property}\left(''W10X54''\,''Cw''\,''metadata''\right) \end{gathered}$$

$\mathrm{Cw}≔2320.0⟦{\mathrm{in}}^6⟧$

$''Warping\; constant''$

$$\begin{gathered} \mathrm{JT}≔\mathrm{Property}\left(''W10X54''\,''J''\right)\; \\ \mathrm{Property}\left(''W10X54''\,''J''\,''metadata''\right) \end{gathered}$$

$\mathrm{JT}≔1.82⟦{\mathrm{in}}^4⟧$

$''Torsional\; moment\; of\; inertia''$

$$\begin{gathered} d≔\mathrm{Property}\left(''W10X54''\,''d''\right)\; \\ \mathrm{Property}\left(''W10X54''\,''d''\,''metadata''\right) \end{gathered}$$

$d≔10.1⟦\mathrm{in}⟧$

$''Overall\; depth\; of\; member,\; or\; width\; of\; shorter\; leg\; for\; angles,\; or\; width\; of\; the\; outstanding\; legs\; of\; long\; legs\; back-to-back\; double\; angles,\; or\; the\; width\; of\; the\; back-to-back\; legs\; of\; short\; legs\; back-to-back\; double\; angles''$

$$\begin{gathered} \mathrm{Sx}≔\mathrm{Property}\left(''W10X54''\,''Sx''\right)\; \\ \mathrm{Property}\left(''W10X54''\,''Sx''\,''metadata''\right) \end{gathered}$$

$\mathrm{Sx}≔60.0⟦{\mathrm{in}}^3⟧$

$''Elastic\; section\; modulus\; about\; the\; x-axis''$

$$\begin{gathered} \mathrm{Sy}≔\mathrm{Property}\left(''W10X54''\,''Sy''\right)\; \\ \mathrm{Property}\left(''W10X54''\,''Sy''\,''metadata''\right) \end{gathered}$$

$\mathrm{Sy}≔20.6⟦{\mathrm{in}}^3⟧$

$''Elastic\; section\; modulus\; about\; the\; y-axis''$

$$\begin{gathered} \mathrm{rx}≔\mathrm{Property}\left(''W10X54''\,''rx''\right)\; \\ \mathrm{Property}\left(''W10X54''\,''rx''\,''metadata''\right) \end{gathered}$$

$\mathrm{rx}≔4.37⟦\mathrm{in}⟧$

$''Radius\; of\; gyration\; about\; the\; x-axis\; =\; sqrt(Ix/A)''$

$$\begin{gathered} A≔\mathrm{Property}\left(''W10X54''\,''A''\right)\; \\ \mathrm{Property}\left(''W10X54''\,''A''\,''metadata''\right) \end{gathered}$$

$A≔15.8⟦{\mathrm{in}}^2⟧$

$''Cross-sectional\; area\; of\; member''$

$$\begin{gathered} \mathrm{Zx}≔\mathrm{Property}\left(''W10X54''\,''Zx''\right)\; \\ \mathrm{Property}\left(''W10X54''\,''Zx''\,''metadata''\right) \end{gathered}$$

$\mathrm{Zx}≔66.6⟦{\mathrm{in}}^3⟧$

$''Plastic\; section\; modulus\; about\; the\; x-axis''$

$$\begin{gathered} \mathrm{Ix}≔\mathrm{Property}\left(''W10X54''\,''Ix''\right)\; \\ \mathrm{Property}\left(''W10X54''\,''Ix''\,''metadata''\right) \end{gathered}$$

$\mathrm{Ix}≔303.0⟦{\mathrm{in}}^4⟧$

$''Moment\; of\; inertia\; about\; the\; x-axis''$

$$\begin{gathered} \mathrm{Iy}≔\mathrm{Property}\left(''W10X54''\,''Iy''\right)\; \\ \mathrm{Property}\left(''W10X54''\,''Iy''\,''metadata''\right) \end{gathered}$$

$\mathrm{Iy}≔103.0⟦{\mathrm{in}}^4⟧$

$''Moment\; of\; inertia\; about\; the\; y-axis''$

$w≔1.15⟦\frac{\mathrm{kipf}}{\mathrm{ft}}⟧\:$

$F≔5⟦\mathrm{kipf}⟧\:$

$T≔5.1⟦\mathrm{ft}\mathrm{kipf}⟧\:$

$P≔96⟦\mathrm{kipf}⟧\:$

$L≔15⟦\mathrm{ft}⟧\:$

$\mathrm{Fy}≔50⟦\mathrm{ksi}⟧\:$

$\mathrm{Lb}≔15⟦\mathrm{ft}⟧\:$

$\mathrm{Lx}≔15⟦\mathrm{ft}⟧\:$

$\mathrm{Ly}≔7.5⟦\mathrm{ft}⟧\:$

$E≔29000⟦\mathrm{ksi}⟧\:$

$G≔11200⟦\mathrm{ksi}⟧\:$

$\mathrm{\λ}≔\sqrt{\frac{G\cdot \mathrm{JT}}{E\cdot \mathrm{Cw}}}$

$0.01740610961⟦\frac{1}{\mathrm{in}}⟧$

$\mathrm{Mx}≔\frac{w{L}^2}{8}$

$32.34⟦\mathrm{foot}\mathrm{kipf}⟧$

$\mathrm{My}≔\frac{FL}{4.0}$

$18.75⟦\mathrm{foot}\mathrm{kipf}⟧$

$\mathrm{M0}≔\frac{TL}{4d}$

$22.72⟦\mathrm{foot}\mathrm{kipf}⟧$

$\mathrm{\β}≔\frac{4{\sinh \left(\frac{\mathrm{\λ}L}{2}\right)}^2}{\mathrm{\λ}L\sinh \left(\mathrm{\λ}L\right)}$

$\mathrm{\beta }≔0.5850278056$

$\mathrm{MT}≔\mathrm{\β}\mathrm{M0}$

$13.29⟦\mathrm{foot}\mathrm{kipf}⟧$

$\mathrm{fbx}≔\frac{\mathrm{Mx}}{\mathrm{Sx}}\+\frac{2\mathrm{MT}}{\mathrm{Sy}}$

$21.96⟦\frac{\mathrm{kipf}}{{\mathrm{inch}}^2}⟧$

$\mathrm{\Ω}≔1.67\:$

$\mathrm{Fnx}≔\frac{\mathrm{Fy}}{\mathrm{\Ω}}$

$29.94⟦\mathrm{ksi}⟧$

$\frac{\mathrm{fbx}}{\mathrm{Fnx}}$

$0.7333393767$

$\mathrm{Mrx}≔\left(\frac{\mathrm{Mx}}{\mathrm{Sx}}\+\frac{2\mathrm{MT}}{\mathrm{Sy}}\right)\mathrm{Sx}$

$109.78⟦\mathrm{foot}\mathrm{kipf}⟧$

$K≔0.85\:$

$\mathrm{Fe}≔\frac{{\mathrm{\π}}^{2}E}{{\left(\frac{KL}{\mathrm{rx}}\right)}^2}$

$233.50⟦\mathrm{ksi}⟧$

$\mathrm{Fcr}≔{0.658}^{\frac{\mathrm{Fy}}{\mathrm{Fe}}}\mathrm{Fy}$

$45.71⟦\mathrm{ksi}⟧$

$\mathrm{Pn}≔\mathrm{Fcr}A$

$722.27⟦\mathrm{kipf}⟧$

$\mathrm{Pc}≔\frac{\mathrm{Pn}}{\mathrm{\Ω}}$

$432.50⟦\mathrm{kipf}⟧$

$\mathrm{Mn}≔\min \left(\mathrm{Fy}\mathrm{Zx}\,\mathrm{Fy}\mathrm{Sx}\right)$

$250.00⟦\mathrm{foot}\mathrm{kipf}⟧$

$\mathrm{Mcx}≔\frac{\mathrm{Mn}}{\mathrm{\Ω}}$

$149.70⟦\mathrm{foot}\mathrm{kipf}⟧$

$\mathrm{Mcy}≔\frac{\mathrm{Mn}}{\mathrm{\Ω}}$

$149.70⟦\mathrm{foot}\mathrm{kipf}⟧$

$\frac{P}{\mathrm{Pc}}\+\frac{8}{9}\cdot \left(\frac{\mathrm{Mrx}}{\mathrm{Mcx}}\+\frac{\mathrm{My}}{\mathrm{Mcy}}\right)$

$.99$

$\mathrm{\φ}≔\frac{T}{2G\mathrm{JT}\mathrm{\λ}}\cdot \left(\frac{\mathrm{\λ}\cdot L}{2}-\frac{2\cdot \sinh \left(\frac{\mathrm{\λ}\cdot L}{2}\right)}{\sinh \left(\mathrm{\λ}\cdot L\right)}\right)\cdot \sinh \left(\frac{\mathrm{\λ}\cdot L}{2}\right)$

$\mathrm{\phi }≔0.2304416908$

$\mathrm{I3}≔\mathrm{Ix}{\sin \left(\frac{\left(90-\mathrm{\φ}\right)\mathrm{\π}}{180}\right)}^2\+\mathrm{Iy}{\cos \left(\frac{\left(90-\mathrm{\φ}\right)\mathrm{\π}}{180}\right)}^2$

$303.00⟦{\mathrm{in}}^4⟧$

$\mathrm{I4}≔\mathrm{Ix}{\cos \left(\frac{\left(90-\mathrm{\φ}\right)\mathrm{\π}}{180}\right)}^2\+\mathrm{Iy}{\sin \left(\frac{\left(90-\mathrm{\φ}\right)\mathrm{\π}}{180}\right)}^2$

$103.00⟦{\mathrm{in}}^4⟧$

$\mathrm{\Δvert}≔\frac{5w{L}^4}{384E\mathrm{I3}}$

$.15⟦\mathrm{in}⟧$

$\mathrm{\Δhoriz}≔\frac{F{L}^3}{48E\mathrm{I4}}$

$.20⟦\mathrm{in}⟧$