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For design members for flexure, sections are automatically classified as compact, non-compact, or slender-element sections, according to AISC 360-16.
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Symbols
Cb : The lateral-torsional buckling modification factor
E : Modulus of elasticity of steel = 29,000 ksi (200 000 MPa)
Fcr : Lateral-torsional buckling stress for the section as determined by analysis, ksi (MPa
Fy : Specified minimum yield stress of the type of steel being used, ksi
Its : Effective radius of inertia
J : Torsional constant
ho : Distance between the flange centroids, in. (mm)
Lb : Length between points that are either braced against lateral displacement of the compression flange or braced against twist of the cross-section, in. (mm)
Lp : The limiting laterally unbraced length for the limit state of yielding, in. (mm)
Lr: The limiting unbraced length for the limit state of inelastic lateral-torsional buckling, in. (mm),
Mmax : absolute value of maximum moment in the unbraced segment, kip-in. (N-mm)
MA : absolute value of moment at quarter point of the unbraced segment, kip-in. (N-mm)
MB : absolute value of moment at centerline of the unbraced segment, kip-in. (N-mm)
MC : absolute value of moment at three-quarter point of the unbraced segment, kip-in. (N-mm)
Mn : The nominal flexural strength
Mp : Plastic bending moment
ry : Radius of gyration about the y-axis
Sx : Elastic section modulus taken about the x-axis, in.3 (mm3)
Zx = Plastic section modulus about the x-axis, in.3 (mm3)
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| body | --uriencoded--$$ \normalsize M_n = M_p= F_%7By%7DZ_x \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; (F2-1) $$ |
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| body | --uriencoded--$$ \normalsize M_n = C_b \Bigg [ M_p-(M_p-0.7F_yS_x) \bigg( \dfrac%7BL_b-L_p%7D%7BL_r-L_p%7D \bigg) \Bigg ] \le M_p \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; (F2-2) $$ |
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| Mathinline |
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| body | --uriencoded--$$ \normalsize M_n = F_%7Bcr%7DS_x \le M_p \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; (F2-3) $$ |
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| body | --uriencoded--$$ \normalsize F_%7Bcr%7D = \dfrac %7BC_b \pi%5e2E%7D%7B \Big( \dfrac%7BL_b%7D%7Br_%7Bts%7D%7D\Big)%5e2 %7D \sqrt%7B1+0.078 \dfrac%7BJc%7D%7BS_xh_o%7D \Big(\dfrac%7BL_b%7D%7Br_%7Bts%7D%7D\Big)%5e2%7D \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; (F2-4) $$ |
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The lateral-torsional buckling modification factor, Cb, is calculated below.
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| body | --uriencoded--$$ \normalsize C_%7Bb%7D = \dfrac %7B12.5M_%7Bmax%7D%7D%7B 2.5M_%7Bmax%7D+3M_A+4M_B+3M_C%7D \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; (F1-1) $$ |
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Limiting laterally unbraced length for the limit state of yielding, Lp is calculated below.
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| body | --uriencoded--$$ \normalsize L_p =1.76r_y \sqrt%7B \dfrac%7BE%7D%7BF_y%7D %7D \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; (F2-5) $$ |
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Limiting unbraced length for the limit state of inelastic lateral-torsional buckling, Lr, is calculated below.
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| body | --uriencoded--$$ \normalsize L_r =1.95r_%7Bts%7D \dfrac%7BE%7D%7B0.7F_y%7D \sqrt%7B \dfrac%7BJc%7D%7BS_xh_o%7D + \sqrt%7B \bigg( \dfrac%7BJc%7D%7BS_xh_o%7D \bigg)%5e2 + 6.76 \bigg( \dfrac%7B0.7F_y%7D%7BE%7D \bigg)%5e2 %7D%7D \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; (F2-6) $$ |
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| body | --uriencoded--$$ \normalsize r_%7Bts%7D%5e2 = \dfrac%7B\sqrt%7BI_yC_w%7D%7D%7BS_x%7D \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; (F2-7) $$ |
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For doubly symmetric I-shapes,
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| body | $$ \normalsize c=1 \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; (F2-8a) $$ |
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For channels
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| body | --uriencoded--$$ \normalsize c= \dfrac%7Bh-o%7D%7B2%7D \sqrt%7B \dfrac%7BI_y%7D%7BC_w%7D %7D \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; (F2-8b) $$ |
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For doubly symmetric I-shapes with rectangular flanges, Cw = Iyho2 / 4 and rts is calculated below.
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| body | --uriencoded--$$ \normalsize r_%7Bts%7D%5e2 = \dfrac%7BI_yh_o%7D%7B2S_x%7D $$ |
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