Small P-delta on only compressed members.

[ishvaaag]

I have read at some RISA document (Practical Analysis with the AISC 13th Edition) that small Pdelta can be disregarded for members mainly in axial compression, because their flexural stiffness doesn't contribute to the lateral stability of the structure. But the flexural stiffness not only contributes but warrants the limit axial capacity in such members; and in fact EI (the flexural stiffness) is seen to be a component of the Euler formula from where every consideration on buckling has been derived.

So in my view and except that the intent of AISC is shown to be otherwise, the stated view is incorrect, and P-small delta is a concern to be cared for in such members with mainly axial forces; other thing is that we may devise procedures that limit P-small delta influence to negligible effect through sound engineering judgement for every kind of member.

 

[JoshPlumSE]

Ishvaag -

I think you misread that portion of the paper.... Or, the paper wasn't clear enough. Which is a distinct possibility given the author.

It's not that P-Little Delta can be ignored for member mainly in axial compression. But, rather that the P-Little Delta effect is trivial for most real moment frame structures. AISC specification essentially admits this when they say you can omit Little Delta from the analysis for members which have an axial force less than 15% of the Euler buckling load.

I'll give an example of this in a follow up posting....

 

[JoshPlumSE]

Example:

A 15 foot tall W12x65 column:

P_euler = 4708 kips (for strong axis)
P_euler = 1537 kips (for weak axis)

P_yield = 955 kips

phi*Pn = 662 kips = 14% of P_euler

What's important to note about this is that the code comparison to the Euler buckling limit is based on the direction of buckling which corresponds to the moment frame..... NOT the weak direction of the column.

The amount of axial load that we can allow in this column is severly limited by the weak axis buckling of the column. However, because this column only bends about it's strong axis the P-Little Delta effect will be ingsignificant.

Now, you do have to be careful for cases where your strong and weak axis have approximate the same moment of inertia (tubes and pipes), or cases where you bend about the weak axis (weak axis moment frames or cantilever columns). However, I would consider those to be specialty cases.

 

[JoshPlumSE]

One thing to mention is that if you are EVER unsure whether P-little delta will be signicant or not you can easily account for it in RISA just by adding three joints along the height of the column between stories. You can test this out using a comparison between the RISA results and the AISC benchmark problems (as shown in the attached pdf file).

P.S. Sorry to post so many times about this subject.... But, I wanted each post to be small and easy to read.

 

[WillisV]

Josh - why don't you just use finite beam elements that take into account little pdelta effects in their base formulation?

 

[ishvaaag]

Thank, Josh ... I already understand that everything must be understood as you here state. But certainly feel that some of the remarks at the document then should be reworded, to avoid the feeling of that, say, P-small delta can be forfeited in braces "because their flexural action doesn't contribute to lateral stability", because if , for example, someone is caught with a slender brace loaded enough to above the limit relative to the Euler critical buckling load (say on weak axis) they can be missing some important consideration.

On the other hand and once understood how RISA was operating respect P-Delta I was soon using it with generous divisions of the columns to capture the P-small delta effect; the intent was always to get some "advanced steel design"-like setup where (disregarding that RISA is always doing the code check for the steel member) you may enter easily manually to check yourself in nearly a strength of material approach, or if you want, in almost mere sectional analysis -since with the subdivisions and P-Delta it was admitted (I was reading in the mid nineties') a sure K=1 was admitted- and so manual checks were still an interesting possibility.

I am still intrigued with the cases of curved members ... in what I am aligning with the issues of the Achilles turtle or the worries about the early era of differential calculus, for a polygon is never a curve, and, say, for a circumference, rotation relative to developed length is always the same -so P-small delta shows there to be a stubborn companyon at the orders of magnitude bigger curvatures of curved members, even if reducible by subdivision; it is my understanding that it is mathematically sound and proved at the basic theory for Finite Element Methods that our finite and discrete representation in the mathematical model when well stated with the lumped properties and basic relationships and in fine mesh enough converge to the actual solution, and furthermore that this can be "verified" for cases where a symbolical solution exists (in what maybe somewhat adventurously they would be admitting the proof for the part as the proof for the whole ... if not ... why to resource to that?, I mean if undoubtedly proven, why to resource to such kind of example proof?), and furthermore, lots of the practicalities of numerical analysis (everything not basically symbolical, rare in a matter dealing on numbers) is based precisely on the developments of differential and integral calculus ... based precisely in discrete representations of continuity, a fish eating its tail situation. See, I would love to know more about mathematics.

 

[JoshPlumSE]

Willis -

There really is no different between the base formulation of our beam elements and any of our main competitors. Though there are certainly differences in the plate element and solid element forumlations.

I believe, however, that you are referring to the process of using a set of "P" forces in your members to modify the stiffness matix to account for P-Delta. I tend to refer to this as a geometric modification to the stiffness matrix. There are some advantages (and some disadvantages) to using this method for P-Delta. That discussion is probably too much to get into for this thread. All that being said, RISA will probably add this method into the program at some point in the relatively near future.

Now, this geometric stiffness method does NOT necessarily account for the P-little delta effect. So, that topic is something of a side topic. If you take a program which is based on the geometric stiffness modifications, you will still probably need to add joints along the mid-span of the columns in order to match the AISC bench mark problems for the little delta effect.

 

[WillisV]

Josh,

I agree that most commercial software uses the same element formulation, thus requiring splitting of the member to capture little P-delta. The geometric stiffness method you mention is useful for one-shot p-deltas (e.g. RAM and as an option in ETABS) which is pretty useful (and close enough in many cases) for building design does but is not technically rigorous and can provide odd results in some cases.

What I was actually referring to is the stability method (introducing stability functions into the formulation of the stiffness matrix) as originally discussed in Timoshenko which considers the buckling of the member in the matrix formulation itself. This method generally assumes the buckled shape is a sinewave, which is reasonable for most design. Using this method you can capture little p-delta effects without splitting the member into multiple pieces.

 

[JoshPlumSE]

Ishvaag -

I understand your point. That paper was written in 2007. I would probably be a little more gentle with my language were I to write it today. Rather than titling that section "Why You Can Ignore P-Little Delta", I would probably call it "When You Can Ignore P-Little Delta".

Also, the section about being allowed to ignore this effect for braces is based on a code interpretation. I give the exact language of the code and the reason why I believe this applies to braces.... So, you are free to agree or disagree.

However, if you look at a typical pinned-pinned brace you will find that including the effect of P-Little Delta should have have ANY effect on the code check of that member because the K value is equal to 1.0 and the Buckling capacity of the member has already been accounted for in the member capacity.

Will the softening of this member as it reaches it's buckling value contribute to destabilizing the structure? Not unless it is receiveing moment during the analysis? If not, then there is no moment to be amplified by the axial force.

Then again, you will probably receive some moment from self weight of the brace. If you disregard that moment then I'd say my interpretation is correct. If you consider the self weight moment to contribute to lateral resistance, then your interpretation is correct.

 

[JoshPlumSE]

Willis -

I am aware of the method you describe. It is really just a variation on the geometric stiffness method. One that includes joint rotation (based on a sine wave shape function as you describe) in the stiffness matrix adjustment rather than just joint deflection. This variation would account for P-Little Delta for most cases.

I'm not sure what the companies that do the geometric stiffness method do not incorporate this variation. But, I have a theory. Companies that use the Geometric stiffness method will normally choose ONE single load combination to define the axial force in their members. This is usually a load combination like 1.2*DL + 1.6*LL which will be most conservative for axial forces.

The advantage of doing this is that they only have to re-adjust the stiffness matrix once... rather than once for every load combination. Like you said, this method is probably good enough for most buildings. But, can lead to strange results in other circumstances.....

With that in mind, I suspect that adding the little delta variation into the stiffness matrix adjustment makes the assumption of using a single gravity only LC a bad one. Or, at least if you make that assumption then you will again lose the little delta behavior. This would be because there is little column flexure under the gravity only LC.

Like I said, I am not sure why the other guys don't use this variation of Geometric stiffness. I am only guessing.

 

[WillisV]

Josh,

Yes - thats exactly what I am referring to, and I agree with your assessment of why it might not be included in typical one-shot geometric stiffness programs. However, you could certainly include it within your formulation of RISA's iterative analysis and thereby remove the requirement for splitting of the member.

Of course this methodology was developed before we had the computing horsepower to just split the member as many times as needed because who cares how big the stiffness matrix is anymore eh? Programs like SAP will split the member internally upon request while still leaving it a single member for modeling and design results, which is probably the quickest way to skin the cat these days.