Inverted Tripod as a cantilevered column
Inverted Tripod as a cantilevered column
(OP)
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?
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?






RE: Inverted Tripod as a cantilevered column
In addition to looking at the column as a whole, you will need to make sure that none of the three legs buckle before the global section buckles.
Also, if there are shears and moments, what provisions are being made to transfer shear between the three to make it act compositely? Are you even attempting to make this happen?
RE: Inverted Tripod as a cantilevered column
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The legs go up about 40'. Yes, the idea is to make a diaphragm on the top connecting the three legs together. This helps a better load distribution (almost keeps the load on the centroid of the tripod).
RE: Inverted Tripod as a cantilevered column
Other than that, I think the design can be carried out by any of the conventional methods (computer modelling/analysis is more practical though).
Watch out for OT of the footing.
RE: Inverted Tripod as a cantilevered column
RE: Inverted Tripod as a cantilevered column
I evision a big old (non-composite) bold tree, it has a huge trunk with 3 diverging braches that supporting some structures above, which is/are linked to the branches by eith moment connections or else. Analytically, I would first assume the branches are fixed at the top of the trunk, then each branch will get its own share of loads, derived from typical frame analysis, or static method. Now the loads will pass down to the top of the trunk, calculate the resultant forces as usual using static method, here you go. (Provide aboundant/closely spaced shear reinforcement at the root of the branches, and the top of the trunk, for which stress induced spliting is very likely to occur)
Still, watch out for global stability, especially in areas with high wind, seismic events. Settlement could be a big problem too.
RE: Inverted Tripod as a cantilevered column
Sorry for my English mistakes.
RE: Inverted Tripod as a cantilevered column
Following the description of the structure you made, I understand that two of the tripod bars belong to the same plan and are linked directly to the upper beam, while the third is a cantilever and it make part of another plan. With this description, I understand that only two legs of the tripod are receiving the load of the beam and therefore should be designed to flexo compression. In that case you will design a column in "Y" form. If you put more details, perhaps I can help better.
4ntunes
RE: Inverted Tripod as a cantilevered column
RE: Inverted Tripod as a cantilevered column
Sorry, but I dont understand what you mean with the word "grab". Could you explain? Thanks.
RE: Inverted Tripod as a cantilevered column
RE: Inverted Tripod as a cantilevered column
RE: Inverted Tripod as a cantilevered column
RE: Inverted Tripod as a cantilevered column
The box girder provides considerable rotational restraint to the tripod column at the top and I think a fixed-free case may be conservative.
In the direction parallel to the span, the system acts like a moment frame. I am looking for some guidance regarding the computation of the "k" value for use in concrete frame design.
RE: Inverted Tripod as a cantilevered column
In the other direction, it is a frame which can be resolved by ordinary methods of structural analysis.
BA
RE: Inverted Tripod as a cantilevered column
In short, if you use P-Delta for your analysis, you may check with the length of the dividing segments and K=1, or if wanting more precision, K from non-sway chart. I wouldn't use less than 4 segments or so per straight part except if short. The method is also nice in that the length for the buckling checks entirely coincides with that of the segments used in the model.
RE: Inverted Tripod as a cantilevered column
RE: Inverted Tripod as a cantilevered column
Are you suggesting that each piece of the branch can be analyzed as a pin-ended member with K=1.0? What happens for the trunk?
I am having a hard time visualizing the buckling in this column. I must mention though that this column is huge (it has an area of about 40000 in2 at the base and about 22000 in2 at the point where it separates into 3 legs).
RE: Inverted Tripod as a cantilevered column
RE: Inverted Tripod as a cantilevered column
RE: Inverted Tripod as a cantilevered column
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.
RE: Inverted Tripod as a cantilevered column
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?
RE: Inverted Tripod as a cantilevered column
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
RE: Inverted Tripod as a cantilevered column
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.
RE: Inverted Tripod as a cantilevered column
It is not deflecting as a cantilever with a free end.
RE: Inverted Tripod as a cantilevered column
RE: Inverted Tripod as a cantilevered column
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.
RE: Inverted Tripod as a cantilevered column
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.
RE: Inverted Tripod as a cantilevered column
RE: Inverted Tripod as a cantilevered column
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
RE: Inverted Tripod as a cantilevered column
RE: Inverted Tripod as a cantilevered column
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.
RE: Inverted Tripod as a cantilevered column
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.
RE: Inverted Tripod as a cantilevered column
"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"
RE: Inverted Tripod as a cantilevered column
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
RE: Inverted Tripod as a cantilevered column
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.
RE: Inverted Tripod as a cantilevered column
RE: Inverted Tripod as a cantilevered column
BA