Steel column buckling, braced on a diagonal

[HagarTopeka]

I'm trying to optimize positioning of three masses to achieve the minimum kL/r about any axis.

I initially maximized moment of inertia, within my geometric constraints, about the structure's xx and yy axes, but got the correct and flawed answer, Drawing 1 in the attached file. I've realized that I need to step up to a vector system considering any axis, and that I need to minimize kL/r, since my unbraced lengths are different in different axes.

However, I'm having trouble determining L when a brace is not perpendicular to the axis being considered. The situation is shown in Drawing 2. The brace is more flexible than the column, so I am considering k to be constant for buckling about all axes.

How you determine the unbraced length of a column when it is braced in a direction that is not perpendicular to the axis of bending? I need to go deeper than the conservative answer of considering the column to be braced only in the perpendicular direction.

Thank you for your help!

 

[BAretired]

I don't understand what you are attempting to accomplish with the three masses. Drawing 1 is unclear.

Drawing 2 shows a column with a horizontal brace placed at some arbitrary angle to the two principal axes. The major axis is unbraced unless the brace is parallel to the minor axis in which case, the minor axis is unbraced.

The minor axis may be considered braced by a combination of the bracing member and the major axis provided each are capable of resisting a force determined by resolving the force required to resist minor axis buckling into two components, one parallel to the brace, the other parallel to the minor axis of the column.

When considering major axis buckling, the total axial load should be considered acting simultaneously with the horizontal component calculated above. Deflection in both principal directions must be within code limits. If the angle between the brace and minor axis is less than, say thirty degrees, it may be prudent to assume both axes are unbraced (engineering judgment).

BA

 

[HagarTopeka]

I'm sorry, I should have been more clear with Drawing 1.

 

[HagarTopeka]

I am designing this compression member for a hobby-scale project which will be treated as if it is about to collapse whenever it is loaded, and therefore am on the high-risk side of engineering judgement. I want to optimize the x and y positions of the three tubes, within x and y constraints, to maximize the Euler critical buckling force for a given weight. I have been treating the trussing between the tubes as negligible, but if there's an easy way to incorporate them into an analysis that would be best.

 

[BAretired]

Putting the tubes in an equilateral triangle seems to be the way to maximize overall moment of inertia. The value of 'I' can be approximated by neglecting the web members and considering only the tubes.

To take into account the web members, calculate deflection under a unit load at midspan, placed normal to the axial direction and normal to the line joining any two tubes. From that, calculate the equivalent I. The true I is likely about 10% less than the approximate value found by neglecting the web members.

BA

 

[KootK]

You've got two KL/r to address here.

First, look at buckling of the entire assembly. Here, the radius of gyration will be calculated using all three tubes and a moment of inertia that accounts for the shear flexibility of the webbing. BA's method above can be used for this.

Next, look at the buckling of the individual tubes between braced panel points. Your setup will successfully brace the tubes about both axes and against torsional buckling too. Your strategy should be to ensure that the single tube buckling load exceeds the whole assembly buckling load. Based on what I see, it shouldn't be an issue.

The composite assembly could also buckle torsionally. That's complicated and a pretty remote possibility.

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