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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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The nominal flexural strength, Mn, should be the lower value obtained according to the limit states of yielding (plastic moment) and lateral-torsional buckling.
Yielding Limit State
It is a situation where the beam's cross-section is dimensioned to remain stable until it becomes plastic above the effect of bending if the local buckling of the compression flange and the lateral torsional buckling are prevented. In this case, the characteristic moment strength can be considered equal to the plastic moment strength. If these elements are compact, the nominal flexural strength, Mn, equals to plastic bending moment strength.
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| body | --uriencoded--$$ \normalsize M_n = M_p= F_%7By%7DZ_x \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; \; \; \; \; \;\;\;\;\; (F2-1) $$ |
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Lateral Torsional Buckling Limit State
The part of the beam in the compression flange area acts like an axially loaded compression frame. The beam section cannot reach its full bending capacity if the compression flange's stability is insufficient. Similar to the compression elements, global buckling or loss of stability in cross-sectional parts that may occur with local buckling determines the collapse limit state. This type of buckling of the compression flange is called lateral torsional buckling.
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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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| 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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