Inverted Tripod as a cantilevered column

[slickdeals]

Folks,
I am analyzing an inverted tripod as a column (you gotta love the architects). The column is significantly large.

However, this column is also subjected to moments in both directions. In one direction, this column is a cantilever, while in the other direction, it is part of a moment frame.

How would one design the three legs of the tripod? How does one calculate the effective length factors?

It is a cantilever in one direction, so K=2.0? Is that a conservative way of estimating? Are there accurate ways to determine the K factor for analysis?

 

[ishvaaag]

The difficulty in visualizing proceeds of that this is not properly a column, but a system of members or a substructure. Each branch is a member, and is asking be considered as such for a start to treat the substructure with the artful tricks of the trade. Sloped braces in trusses do not cause much concern, say 0.8L in plane, 1.0L out of plane ... not because couldn't be discussed, that can, but because experience has proven the provided values for the cases be valid enough to not cause problems with the ordinary assumptions.

You have a member below and 3 above.

Now consider the following model for buckling analysis

for the bottom member
use its length
Full fixity at the foundation (assume footing without rotation)
Free end atop

for every top branch
use its length
for the K factor SWAY chart
amount of fixity at bottom provided by only the bottom colum (conservatively)
Free end atop

What I have described is a cantilever supporting 3 cantilevers with relative fixity where the branches derive from the trunk. This may be a conservative model but still practical if no other was available.

However this model in 2 levels may be perfected for a correct statement of the K factors. Anyway is useful to get a start on how to enter the K factor evaluation problem.

You need to make an structural model and analyze it for X and Y lateral forces. The amount of displacement between levels for the case selects the chart you need to use. Say under 1/700 deformation sway displacement makes a non sway level and required chart, and over, sway level and required chart.

Now you only need to get a K factor for the standing conditions, so ratios of rigidities at the joints and restraints at far ends of beams (the other branches as well?) enter the equation. See, you have variable sections, as well. Jackson and Moreland charts constant I are not for such sections. Furthermore, were developed for loads at the joints, and were not for the endorsement in codes many wouldn't use them. So you need to delve for information on buckling at manuals, codes, etc and the information may not be readily available.

So it is not as much the difficulty of conceiving a correct model for buckling of this substructure but the practicalities of making it well that makes preferable the FEM method I have in the previous post outlined. By using a reasonable number of segments, you model approximately as well the variation of section in the members, and the program does it all: the P-Delta analysis, and the sizing in this case of reinforcement. You only need to make the assumption of K=1, or let or mandate alternatively the use of the nonsway chart or enter its values, and establish the correct restraints and reduced stiffnesses if any. For a model with few members, not much time.

 

[slickdeals]

@Ishwaag,
The member has been modeled with small segments each of length approximately 5 feet (this includes both the branches and the trunk). It seems like if I have done this, you are suggesting to use a K=1?

 

[BAretired]

If the three members of the tripod were pinned at the top and fully fixed at the bottom of the tripod, they would each have an effective length of 2.0 (2.1 recommended).

Actually, the bottom of the tripod is supported by a trunk which is presumed fixed at the bottom and rotates at the top. The effective length of each branch of the tripod is greater than 2.0 because of the support condition at the intersection point.

The effective length of the structure as a whole (trunk plus tripod) is, in my view 2.0. It is a column of varying cross section and can be solved by hand methods using Newmark's Numerical Procedures or similar technique.

If the trunk is supported by a foundation which permits rotation at the base, the K factor will depend on the magnitude of that rotation.


BA

 

[ishvaaag]

If the analysis has the effect of a P-Delta analysis integrated, and you have taken what should be a degraded stiffness for the column, say 0.7 Igross (if moments do not cause even further degradation), yes, and you can take the 5 feet to check each segment member with K=1 or less.

The only problem I find with this setup is the (for the proportions of your columns) quite short segments being used, for being quite short, shear effects might be overemphasized respect bending action on such elements.

So it would be better to use the same scheme and use longer segments. If you can make with the program use of tapered elements (I think RISA 3D can), even better, you even ameliorate the accuracy.

Once you have an stabilized lateral model under P-Delta in which material nonlinearity is contemplated, its nodes may be looked at as with displacement prevented, and one only needs then check for the in-member P-small delta buckling behavior, that you may deal with the use of the non-sway chart that gives a factor, always less than 1 and is nice to use except where maximum economy or accuracy is required.

The extant curvature from P-Delta analysis might exceed the one sometimes implicit in the checks for the use for in-member P-small delta situations. This could be something to check but normally won't be an issue but for some cases usually in the mid to slender members range; very slender simply won't meet directly P-Delta.

 

[slickdeals]

Attached is a deflected shape under an eccentric gravity load (this is deflection perpendicular to the span).

It is not deflecting as a cantilever with a free end.

 

[slickdeals]

Attachment

 

[ishvaaag]

Just extending a bit the explanation after a walk...

With the need of checking for true strength it became apparent that geometrical and material nonlinearities needed to be cared for. Elastc analysis being the dominant tool (still is), tools were provided for both things. The Jackson and Moreland (I know maybe attribution to others might be more precise but don't remember the names) charts are instruments devised to cover the effects of geometrical nonlinearities when an elastic analysis is available.

With the advent of P-Delta analysis being included in analysis programs, and if one subdivides members in segments, most of the effects of the geometrical nonlinearities are already represented in the derived forces, even (if you segment the columns) those pertaining to the gross effect of P-small delta, for its nodes have been also brought to the amplified forces position by the procedure. From then, only P-small delta geometrical nonlinearities at the level of the segment need be covered, using the segment length. Each segment can then be taken laterally restrained and a proper K may be derived by proper use of the non-sway chart. The only concern then is that the actual (at the being verified level of forces, typically factored) extant member "imperfection" for such segment under the forces with P-Delta, measured from its deviation of an straight axis, would exceed the implicit theoretical imperfection allowed in the formulations that make use of the K extracted from a non sway chart, that for steel are L/1000 and L/1500 that I remember in USA an Europe (now I would have to check which corresponds to each country-area). If so, the check should be corrected, for such P-small delta would require more magnification.

In short, any magnification coming from a sway chart you have already extant in the forces resulting from your analysis with P-Delta. Everything else is P-small delta to the level of the segment, and material nonlinearity that in simplified way is more or less checked now as it was: by a reduction of the stiffness and other section properties used in the calculation.

 

[ishvaaag]

Now the subject of short segments, for your structure. With short segments, more energy of deformation should be going to shear deformation than actually is correct. Hence, if you have shear over-represented at each segment, it is diminished bending, and I don't know if it should have less lateral displacement, but, the tip atop should have lesser deflection towards the soil since it combs less.

It would be interesting to see this proving true in your model at two segment lengths (or other model, one always has more than enough things to do) even if modeling of the variable section segments may make appear inaccuracies of the same order of magnitude.

 

[slickdeals]

Can you comment on the deflected shape?

 

[BAretired]

The middle line appears straight for most of the length above the start of the tripod. The outer two lines, assuming they started out straight would indicate double curvature as though they were receiving some torsional resistance from the box girder.

What force are you analyzing? If you are looking at a single lateral force on the girder at one column, then the girder would provide some torsional restraint by virtue of its attachment to other columns. If you consider wind load on the entire length of girder, then there should be no torsional restraint from the girder, i.e. the girder is free to rotate to accommodate the rotation of all of the columns.

BA

 

[ishvaaag]

I don't see anything illogical in the deformed shape. A complete structure model would help more to analyze if something is going wrong judging from the restraints girder to supports and at ends. Whilst combing, the compressed and tensioned branches fight to deform something that has stiffness and this imparts reaction on them interpretable as BAretired says as a restriction to rotation. If the girder box was a mechanism unable to show stiffness to the supporting branches, the branches would be free to deform and the tree would have cantilever deformation. So even in a 2D analysis the branches can take as in the figure sway frame deformation if the connections so provide.

 

[slickdeals]

One follow up question:
Since the member is modeled using a finite number of pieces, can the delta ns factor (to account for non-sway moment magnification) be set to 1.0 since the member curvature is modeled by breaking it into small segments (whose nodes do not lie in the same line as they go up in height), meaning the node above is not directly in line with the node below in one direction.

 

[ishvaaag]

All members after P-Delta, stiffness degradation to proper value can be calculated with K=1, since it is a safe value for all. The situation is after that analysis for each segment such if each segment, whichever the position in the structure, is pinned-pinned and then applied the end forces from P-Delta analysis, having the actual deviation from straightness that such analysis gives. It is only the matter of that such deviation might at some case exceed the value implied in code checks what might make such check unsafe, due to being out of the usual assumptions. From a worksheet I made for Fcr following Eurocode says 1/1000 deviation so should be 1/1500 what assumed in USA. Anyway P-small delta buckling checks are for the buckling of members. Even when exceeding such "allowed for imperfections" (now, actually the deviation from straightness, if you have a quite short stub segment, ***buckling*** is of not concern. Your stub is prevented of movement at ends. It is short, there's no magnification, point. P-small delta is, as well, for moment magnification on what given on a prior calculation, hence with reasonable segmentation the procedure can be made entirely safe.

So we have a situation where making short segments ensures no residual P-small delta effects are of not concern (we are not to forget that the gross effects of the P-small delta, when considered over the full length of the actual member, have been already been captured by its segmentation and the P-Delta analysis), but may worsen the accuracy of the solution of the FEM method. This depends of the formulations of the beam, plate, shell elements used and shouldn't be a problem for practical analyses with ordinary safety factors and looked at with engineering common sense. If the beam element has no implicit consideration of the shear deformation in more than that of bending, the observation gets void.

When you model a curved arch with straight segments you are meeting a problem of a very similar kind; and yet it is not unusual to make such things. Everyone that uses such method knows that there are some problems in the accuracy of the solution yet no other might be available in the actual case. The analysis program allowing for it, curved elements might be used etc, or shell elements directly amenable to the representation of any curvature, but even after such calculations we would be entering the realm of the buckling of shells, etc if we don't reduce in some way the results to linear elements, etc.

All these worries are on a grade of precision of less order than those actually found in the gross numbers of divisions made for shear walls and many other cases. Only the fact of that in some cases a structure might need the more accurate analysis at hand with reasonable engineering effort make worth treating the subject.

The general accuracy of the process of designing structures following the code most surely make that any error following the procedure after significant segmentation may fall well below under 1% of the intent of the code if an accurate solution was at hand, and we all know that even designing consistently and with sound practices ordinary structures variations well over 10% should be expected (even allowing for the statistical paremeters involved establishing the characteristical values). So really no problem for practical application.

 

[ishvaaag]

In the prior post, where reads in the second paragraph

"So we have a situation where making short segments ensures no residual P-small delta effects are of not concern"

must read

"So we have a situation where making short segments ensures no residual P-small delta effects are of concern"

 

[BAretired]

I am attaching a marked up copy of your deflection diagram. I first thought that the tripod was a three dimensional tripod atop a column. I now believe that all branches of the tripod are in the same plane, so it is actually more like a trident (correct me if I am wrong).

I have labeled most of the nodes for ease in discussions. The black object at the top with yellow fill is the box girder which is probably a rigid body. I am not sure how the longer branches of the tree fasten to the girder. Are they fastened top and bottom or just top?

It appears to me that there is too much curvature in the left branch just above h', but it depends on the loading and relative stiffnesses of the branches. I assume the left and right branch are similar, but not necessarily the same as the central branch.



BA

 

[slickdeals]

@BA:
No, all three branches are not in the same plane. They are indeed a tripod. The deflection diagram I sent was a flat 2D view (from 3D).

There are 2 thick diaphragms (about 4' thick) and just as wide as the box girder to which these three legs are compositely tied into. Meaning the three legs are rigidly connected at the top.

I need one clarification however. Would P-[δ] effects be captured if a curved column is broken up into a finite number of straight segments. I would appreciate a short answer.

@Ishvaag,
I appreciate you trying to help here, but each one of your dissertations is too complicated. I would appreciate simple responses where I don't have to re-read your post a dozen times to try to make sense. No offense intended, thanks for helping.

 

[ishvaaag]

Well, don't know if too short, but, yes, P-small delta would be captured, except to the level of each segment, for which should be analyzed if not a stub.

 

[BAretired]

A finite number of straight line segments is a good representation of a curved member. Accuracy improves by increasing the number of segments.

BA