# 1.10 Flexural-torsional buckling

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#### Definitions

There are three ways a compression member can buckle, or become unstable. These are flexural buckling , torsional buckling , and flexural-torsional buckling .

#### Where to find information for flexural-torsional information

The Manual provides specifications for flexural-torsional buckling in the Specification section,Section E3 (p. 16.1-28), and Appendix E3 (p. 16.1-94. Section E3 is specifically for double-angles and tee-shapedcompression member whose elements have width-thickness ratios less than ${\lambda }_{r}$ .

Torsional variables can be found in the Dimensions and Properties section of the Manual in the first section. Torsional properties start on page 1-89 andFlexural-torsional properties on page 1-96.

#### Center of gravity and shear center

The shear center is always located on the axis of symmetry, therefore, if a member has two axes of symmetry, the shearcenter will be the intersection of the two axes. Channels have a shear center that is not located on the member; thevalue, ${e}_{0}$ , tabulated in the Manual is the distance from the channel to the shear center.

For members like an I-shaped member, the center of gravity and the shear center are the exact same point where the twoaxes of symmetry intersect. for channels, the shear center and the center of gravity are different, which creates acouple and makes the twisting that causes torsional buckling.

#### Design strength for double-angle and tee-shaped compression members

Double-angles and tee-shaped members with a width-thickness ratio less than ${\lambda }_{r}$ should use the formula:

${\phi }_{c}=0.85$
${P}_{n}={A}_{g}{F}_{crft}$

where the "ft" of ${F}_{crft}$ stands for "flexural-torsional," and is expressed as:

${F}_{crft}=\frac{{F}_{cry}+{F}_{crz}}{2H}(1-\sqrt{1-\frac{4{F}_{cry}{F}_{crz}H}{({F}_{cry}+{F}_{crz})^{2}}})$

Here, ${F}_{crz}$ is expressed as:

${F}_{crz}=\frac{GJ}{A\overline{{r}_{0}}^{2}}$

where

• $\overline{{r}_{0}}$ = the polar radius of gyration about the shear center, in.
• $G=\frac{E}{2(1+\vartheta )}$
• $J$ = torsional stiffness
• $H=1-\frac{{y}_{0}^{2}}{\overline{{r}_{0}}^{2}}$
• ${y}_{0}$ = distance between shear center and centroid, in.
• ${F}_{cry}$ = equation given in Section E2 for flexural buckling about the y-axis of symmetry.

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