# Biaxial flexure of steel I-beams

If an I-shaped section is loaded in two directions (in both weak and strong axes), how can I find the maximum load-carrying capacity of them beam?

There is an interaction equation in AISC 360-10, but it's for beams which are also loaded axially. Can that interaction equation be used for only flexure in two directions, taking the axial force to be 0?

• My answer is too simple for me to dare post as an actual answer, but yes. You can use the interaction equation with N = 0. – Wasabi Jun 3 '15 at 20:36
• Can you clarify how this biaxial flexure is being produced? – CableStay Aug 26 '15 at 20:31

The simple answer is: "yes, you can use the given equations with Pr=0."

There are three reasons for my answer:

1. The code section (AISC 360, Chapter H) doesn't specifically state that Pr cannot be equal to zero.
2. The equations in code section H1 are referenced by other sections that apply to both compression and tension (e.g. H2), so there must be some continuity to the equation when going from tension to compression.
3. The graphs in the commentary on Chapter H show graphs that go from zero axial force with moment to axial force with zero moment.
• Can you share the link of that graph I couldn't find it. – Murad Nazari Jun 3 '15 at 20:55
• @MuradNazari I don't think that AISC 360-10 is readily available on the internet, but if you are looking in the book, the commentary for Chapter H starts on page 16.1-331. – hazzey Jun 3 '15 at 21:08
• @MuradNazari, for the record, a Google search for "AISC 360-10 pdf" gave me the code on the very first result. The commentary hazzey is talking about is as of page 331 (in the document. its actually page 387 of the file). – Wasabi Jun 4 '15 at 3:07
• To follow up on Wasabi's comment -- the current printing of AISC 360-10 is available as a PDF from the AISC website: ASIC – CableStay Aug 26 '15 at 16:19

This is a very interesting question. It might even be worth submitting to Modern Steel Construction's Steel Interchange. Per the AISC 360-10 user note at the beginning of Chapter F, you are allowed to use the interaction equations in Sections H1 for beams in biaxial flexure without axial load.

Interestingly, Salmon & Johnson's Steel Structures Design and Behavior 5th Edition, Section 7.11 suggests that the beam-column interaction equation is unconservative without axial load. The author suggests not relying on any plastic capacity. That is, for the yielding limit state, Mnx = FySx and Mny = FySy.

Based on my Google searches, it appears that Australian standard AS4100 Section 8.3.4 uses this approach.

• Does the text say why it's unconservative? I'm having a hard time seeing why. I would agree on not using plastic capacity unless the column is properly braced to preclude buckling. Also, not sure if this is a regional thing, but plastic section modulus in the States is $Z$, not $S$. – grfrazee Aug 26 '15 at 23:36
• Unfortunately, the author doesn't site a source. I haven't seen any experimental research on pure biaxial bending without axial load, so I've no idea how he's drawing this conclusion. And not incidentally, the Segui text actually disagrees -- he argues that the interaction equation is conservative for zero axial load. I've looked around a bit more and at least for small axial demand/axial capacity, it appears conservative. I intended elastic section modulus, to limit extreme fiber stress to yield and not rely on plastic capacity. – CableStay Aug 26 '15 at 23:45