How to select worst case loads in non-symmetric beams

[canadiancastor]

I'm looking at setting up calculations for steel angles (according to CSA S16-19) that will take into consideration any compression, flexion or combination of the two. I don't have a problem with calculating the section strengths, but I'm a bit confused with what I should take for maximum loading. Say I have a complex beam that has positive and negative moments (strong AND weak axis) and positive and negative axial load (probably impossible in practice, but I want to code this only once).

Question #1: Would it be allowed to run calculations section by section, that is find flexural and axial loads (Mfx, Mfy, Cf) at each point along the beam and compute strength and interaction checks? What seems to be breaking my brain with this approach is that everything we calculate in the standard is for "beam" loading, and not "section" loading or stress. If this is allowed, are there any limits? Could it be imagined that I have huge flexure on one end, huge axial on the other end, and basically no "actual" interaction between both sides?

Question #2: If we are not allowed to run calculations section by section as discussed above, as I suspect, that means we need to find the maximum and minimum values of all of the loading before we start our calculation checks. Since my beam in not symmetric, this would mean getting max(Mfx), min(Mfx), max(Mfy), min(Mfy), max(Cf), min(Cf) and checking all of those against their corresponding strengths. Then for interaction, I would have to permute all of these, so for the flexural / compression interaction, that would mean that I would have 8 cases per load combination to check:
max(Mfx)/Mrx+, max(Mfy)/Mry+, max(Cf)/Tr+
max(Mfx)/Mrx+, max(Mfy)/Mry+, min(Cf)/Cr-
max(Mfx)/Mrx+, min(Mfy)/Mry-, max(Cf)/Tr+
max(Mfx)/Mrx+, min(Mfy)/Mry-, min(Cf)/Cr-
min(Mfx)/Mrx-, min(Mfy)/Mry-, max(Cf)/Tr+
min(Mfx)/Mrx-, min(Mfy)/Mry-, min(Cf)/Cr-
min(Mfx)/Mrx-, max(Mfy)/Mry+, max(Cf)/Tr+
min(Mfx)/Mrx-, max(Mfy)/Mry+, min(Cf)/Cr-
Seems like there should be a simpler (albeit probably conservative) way to find the worst case? I guess if I could discard all of the cases where a "max" is negative or a "min" is positive? Any other ideas?

 

[RenHen]

I'm not familiar with CSA S16-19, but in general, it's permissible to design members by section; the maximum moment that you calculate for the beam is the maximum moment on the beam section cut at that location. There are no limits on a section by section approach other than engineering judgement, which is more relevant if you're doing it by hand & identifying specific sections to check, but if you're coding something to check the interactions at every location along the beam, that should cover it. That's similar to how structural analysis software like RISA-3D checks members.

 

[canadiancastor]

What does not seem to work in the section by section approach is when whole member stability is mixed with interaction. I have an example below that could be expected to happen in practice. We have a column with large self weight and a moment at the top. When doing interaction checks, I don't think it's so simple as to say you want to analyse "section by section". I think it's quite reasonable to simply take the maximum of M and maximum of P for interaction and stability checks.

 

[RenHen]

It's totally reasonable to combine the maximums regardless of where they occur as well; it's conservative but also the approach I tend to take unless I need a more granular approach to justify something working.

 

[Smoulder]

Don't know CSA specifically but surely the same as other codes. You have two types of check. #1 is section checks. You're looking at local overload which is exceeding yield for steel. You combine bending, compression, torsion, shear that all occur at the same section. What's happening at the other end of the beam doesn't matter. #2 is member checks which are stability checks. Here it matters what's happening all along the beam unless you're happy to be conservative. Codes are written so that you use the maximum stress but provide adjustments (increases) when the stress varies favourably along the beam length. But you might not get that adjustment in the code for varyimg compression because uniform compression is the common case unlike bending. You're doing angles which males it harder again because they're very complex and most codes only give simplified rules.

Don't try to find critical cases by logic. That's hand calculation mindset. You're programming so brute force check every load case. It would take hundreds or thousands of projects for the computing time to break even with the programming time. Even that's not a true comparison because the extra seconds per project don't add up because you can't use those seconds meaningfully.

 

[canadiancastor]

@Smoulder What you are saying make a lot of sense. Maybe I am just not familiar with the relevant clauses, but my impression was that the Canadian Standards are always doing member checks. Splitting all checks into these two categories is a great idea: I can loop through the whole section for section checks then only use max values for the member checks.

 

[Smoulder]

I looked up CSA 2009. You're right the wording is always member checks but it's doing what I said. Look at 13.8.2 which is Member Strength and Stability Class 1 and 2 Sections. Part (a) is section checks. Lambda = zero for comoression which removes buckling so you get section capacity when you do the member capacity calculation. Bending capaacity is for laterally supported beams which is code for section capacity. Part (b) is compression buckling only. Part (c) is compression and bending buckling. Now go to 13.6 which is Bending of Laterally Unsupported Members. The omega2 factor increases capacity when bending moment isn't constant. That's why you use maximum bending moment in the design check because the capacity side has been increased to compensate. If you did section by section check in 13.8.2(c) you'd be unsafe.