Compression members are one of the most common sorts used in building design. Load-bearing walls were the earliest structural members, though now we mostly rely on columns with non-loadbearing curtain walls. Even most beams are not limited to only flexural loads, as they end up serving in the lateral-force-resisting system as well.
Any compression member has a variety of failure modes, including compressive failure of the material and a variety of cross-section and length-dependent buckling modes. In the structural world, most “structurally efficient” designs will fail due to global buckling long before the full strength of the material has been reached, so equations to calculate the buckling strength of columns and walls were some of the first advances made in the field.
The Radius of Gyration is a Section Property used in calculating the Critical Axial Load using the Euler Column Buckling equation. It’s found by dividing the smallest Moment of Inertia of a Section by the Area, then taking the square root of the result.
Video Option: Three Worked Examples
The General Equation
The Radius of Gyration is directional, so there are different radii for different directions. Simply take the square root of the Moment of Inertia in the direction you’re looking for the Radius for over the Area of the cross-section:
Shape-Specific Equations
It can be useful to have quicker equations for the Radius of Gyration for a few specific shapes, so below are a few for quick reference.
Rectangle
The below shows the radius of LEAST gyration of the rectangle, as this is typically what’s of interest.
However, depending on your bracing pattern (i.e. in a stud wall where the least gyration direction is braced by sheathing), the radius of gyration in the strong direction may be of interest.
Solid Circle
For a circle, the radius of gyration is the same in all directions.
Hollow Circle (Pipe)
Similar to a solid circle, there is only one Radius of Gyration for a hollow circle. Given that solid round sections aren’t super common in construction (unless you’re building with logs), this is probably a more useful equation to have around.
Uses of the Radius of Gyration
The Radius of Gyration is one of the section property inputs in most buckling equations. A selection of sample equations pulled from the AISC Steel Construction Manual is given below:
Euler Column Buckling (Global Buckling)
The simplest approach to column buckling, this equation (or some variation on it) is used to help determine the elastic buckling limit for columns in most US material codes.
Lateral-Torsional Buckling (Global Buckling)
When beams are subjected to about any but their weakest axis, they are in an unstable condition. Lateral-torsional buckling occurs when some small instability or eccentricity in the loading tends to make the beam twist towards a less stiff orientation, which results in premature loss of capacity.
Hopefully, that’s enough to get you off and rolling, please don’t hesitate to contact me at eric@structed.org with any further questions you may have! I’ll be happy to add articles or videos to address any questions.
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