# What is the unbraced length of the compression portion of an X brace?

In the diagram below, the compression member is $AC$. What is its unbraced length for determining buckling capacity? This also matters because certain codes put limits on the maximum value of $\frac{KL}{r}$ for bracing members.

I know that the conservative answer is to consider the entire length, AC. This can be too conservative when lots of braces are used (total weight of steel increases). It can also be conservative if the actual compressive force is low enough that a member size is increased solely to meet the $\frac{KL}{r}$ requirements.

I can justify to myself that the tension member braces the compression member in one direction (in the plane of the page), but can the tension member be considered to provide bracing in the orthogonal direction (in and out of the page)?

There are two reasons why I think this might be possible:

1. The tension brace provides resistance solely from being present. (i.e. something is better than nothing)
2. The other brace is in tension so it will provide additional restraint similar to a bow string.

Note: Some codes have requirements that a brace be able to withstand 5% of the axial force in the member being braced. I would think that this would be the lower limit for restraint coming from the tension member.

What is its unbraced length for determining buckling capacity?

AISC defines $L$ as the laterally unbraced length of the member (emphasis mine) (AISC 14th Ed. Steel Construction Manual, Part 16, Section E2).

I can justify to myself that the tension member braces the compression member in one direction (in the plane of the page)

I would agree.

but can the tension member be considered to provide bracing in the orthogonal direction (in and out of the page)?

Note: Some codes have requirements that a brace be able to withstand 5% of the axial force in the member being braced. I would think that this would be the lower limit for restraint coming from the tension member.

Simply put, I don't think you'll be able to argue that the tension member provides enough out-of-plane resistance to bending of the compression member since the tension member would resist the out-of-plane load via bending, which is a relatively "soft" support condition.

This also matters because certain codes put limits on the maximum value of $\frac{KL}{r}$ for bracing members.

Further discussion from the Commentary Section E2 says:

It's generally a good idea to stay under 200 for your slenderness ratio, though the Commentary discussion appears to give you an "out" if you think you need it.

• I don't know, the AISC Seismic Design manual speaks specifically to this issue and states that the effective length factor for out-of-plane bending of a diagonal brace in a concentrically braced frame is 0.5. – William S. Godfrey- S.E. Feb 19 '16 at 15:36

The second edition of the AISC Sesimic Design Manual (SDM) contains many wonderfully worked out example problems. I highly recommend that you purchase this resource if you are designing braced frames in a professional setting. It can be purchased from the AISC, and if you are ever planning to sit and take the SE exam, you'll need to buy it anyway.

Example 5.2.1 in the SDM is the complete design of the diagonal brace in an ordinary concentric moment frame. In this example, while discussing the effective length of the brace, it states:

By inspection the laterally braced length of the diagonal brace in the in-plane direction is half the overall length. For buckling out-of-plane, if both of the diagonals are continuous for their full length and are connected at the intersection point, then the effective length factor, K, is 0.5 (El-Tayem and Goel, 1986; Picard and Beaulieu, 1987).

Emphasis mine.

This results in an effective length for both the in-plane and out-of-plane bending of the brace to be 0.5 * L.

• What do they take as $L$ though? Is it the full length of the brace, or the distance from the end support to the intersection? (I assume the former) – grfrazee Feb 19 '16 at 16:07
• I have run across similar thoughts, but there are a couple of things that make me wonder about the text that you propose. First is that it sets K=0.5, but without consideration of the K value that might come from the connections of the ends. This may be an artifact of being an OMF. The second is that the whole seismic code needs to be taken together as a whole, so this section's K=0.5 needs to be considered along with the R used in the entire design (and seismic vs secondary lateral loads). – hazzey Feb 19 '16 at 16:12
• grfrazee, sorry, my answer was a bit vague. The length of the diagonal brace used by AISC in the example, as you guessed, is the distance from work point to work point at each end of the brace. Perhaps I misunderstood the original question, but it was my understanding that the OP was trying to figure out what values to use for $K$ and $L$ in order to calculate $\frac{KL}{r}$ – William S. Godfrey- S.E. Feb 19 '16 at 16:14
• hazzey, in my professional experience, braced frames are almost always design assuming pinned connections. So I'd say that K = 0.5 is not set without consideration of the end conditions. K is set with the explicit understanding that the ends of the brace are considered to be pinned. This is obviously a simplification, but it's a methodology that has been nearly universally accepted by the industry. – William S. Godfrey- S.E. Feb 19 '16 at 16:24
• I'll admit though that I don't entirely understand your comment relating the R value and secondary lateral loads. In this particular case, I would use the methods outlined in Appendix A of the AISC Spec. to determine the second-order effects on the braces. This is entirely compatible with both of the the above discussed determinations of the effective length of the braces. – William S. Godfrey- S.E. Feb 19 '16 at 16:25